Statistical Models for Nucleon Structure Function 1527568954, 9781527568952

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Copyright © 2024. Cambridge Scholars Publishing. All rights reserved.

Statistical Models for Nucleon Structure Function

Mirez, Tarrillo, Carlos Alberto, and Luis Augusto Trevisan. Statistical Models for Nucleon Structure Function, Cambridge Scholars Publishing, 2024. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/wisc/detail.action?docID=31136544. Created from wisc on 2024-04-11 00:44:35.

Copyright © 2024. Cambridge Scholars Publishing. All rights reserved. Mirez, Tarrillo, Carlos Alberto, and Luis Augusto Trevisan. Statistical Models for Nucleon Structure Function, Cambridge Scholars Publishing, 2024. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/wisc/detail.action?docID=31136544. Created from wisc on 2024-04-11 00:44:35.

Statistical Models for Nucleon Structure Function By

Copyright © 2024. Cambridge Scholars Publishing. All rights reserved.

Carlos Alberto Mirez Tarrillo and Luis Augusto Trevisan

Mirez, Tarrillo, Carlos Alberto, and Luis Augusto Trevisan. Statistical Models for Nucleon Structure Function, Cambridge Scholars Publishing, 2024. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/wisc/detail.action?docID=31136544. Created from wisc on 2024-04-11 00:44:35.

Statistical Models for Nucleon Structure Function By Carlos Alberto Mirez Tarrillo and Luis Augusto Trevisan This book first published 2024 Cambridge Scholars Publishing Lady Stephenson Library, Newcastle upon Tyne, NE6 2PA, UK British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Copyright © 2024 by Carlos Alberto Mirez Tarrillo and Luis Augusto Trevisan All rights for this book reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

Copyright © 2024. Cambridge Scholars Publishing. All rights reserved.

ISBN (10): 1-5275-6895-4 ISBN (13): 978-1-5275-6895-2

Mirez, Tarrillo, Carlos Alberto, and Luis Augusto Trevisan. Statistical Models for Nucleon Structure Function, Cambridge Scholars Publishing, 2024. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/wisc/detail.action?docID=31136544. Created from wisc on 2024-04-11 00:44:35.

CONTENTS

Preface ...................................................................................................... vii Chapter 1 .................................................................................................... 1 Introduction I. Review on Partons Model Chapter 2 .................................................................................................... 6 Partons Model and the Nucleon’s Structure Function II. Models with Fermi-Dirac Distributions Chapter 3 .................................................................................................. 32 Angelini and Pazzi Works

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Chapter 4 .................................................................................................. 35 The Cleymans-Thews Model Chapter 5 .................................................................................................. 37 The Mac and Ugaz Work Chapter 6 .................................................................................................. 46 Thermodynamic Model for Proton Spin Chapter 7 .................................................................................................. 50 The Bickerstaff and Londergan Work Chapter 8 .................................................................................................. 58 The Devanathan–Karthiyayini-Ganesamurthy Model Chapter 9 .................................................................................................. 62 The Thermodynamical Bag Model for the Nucleon’s Spin Chapter 10 ................................................................................................ 66 The Statistical by Soffer–Bourrely–Bucella – The Polarized Case

Mirez, Tarrillo, Carlos Alberto, and Luis Augusto Trevisan. Statistical Models for Nucleon Structure Function, Cambridge Scholars Publishing, 2024. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/wisc/detail.action?docID=31136544. Created from wisc on 2024-04-11 00:44:35.

vi

Contents

Chapter 11 ................................................................................................ 69 The Bhalerao Statistical Model Chapter 12 ................................................................................................ 72 Statistical Quark Model with Linear Confining Potential Chapter 13 ................................................................................................ 86 A Very Simple Statistical Model to Quark’s Asymmetries Chapter 14 ................................................................................................ 92 The Statistical Model by Zhang, Shao and Ma Chapter 15 ................................................................................................ 96 Statistical Model with Q2 Dependence III. Models with Non-Extensivity Chapter 16 .............................................................................................. 100 Thermodynamics with Fractal Structure, Tsallis Statistics and Hadrons Chapter 17 .............................................................................................. 118 Fractal Structure and Non-Extensive Statistics

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Chapter 18 .............................................................................................. 140 The Nonextensive Thermodynamic Model – Tsallis Temperature Chapter 19 .............................................................................................. 144 The Polarized Nonextensive Statistical Model Chapter 20 .............................................................................................. 155 On the Difference between the Radii of Gluons and Quarks Chapter 21 .............................................................................................. 171 Nuclear EMC Effect in Non-Extensive Statistical Model Chapter 22 .............................................................................................. 174 Overall Comparisons and Conclusion References .............................................................................................. 180

Mirez, Tarrillo, Carlos Alberto, and Luis Augusto Trevisan. Statistical Models for Nucleon Structure Function, Cambridge Scholars Publishing, 2024. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/wisc/detail.action?docID=31136544. Created from wisc on 2024-04-11 00:44:35.

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PREFACE

During the eighties and nineties, many statistical/thermodynamical models emerged to describe the nucleon structure functions and the energy distribution of quarks. Most of these models describe the compound quarks and gluons inside the nucleon as a Fermi-Dirac or Bose-Einstein gas, confined in an MIT bag with continuous energy levels. These models obtained some relevant features of the nucleons, like the asymmetries between and , the spin-dependent structure functions, and the ratio , for instance. In this work, we reviewed the hadronic models. They use statistical/thermodynamic features to describe the hadrons’ structure functions, polarized and unpolarized. The revised works were described, as far as possible, in chronological order. We believe this book is convenient for researchers and students because it put together several studies about the thermodynamic features of nucleons and their structure function. Because this subject is yet to undergo research, there is no unique approach or model that is in complete accord. I wish to thank Professor Lauro Tomio for his criticism; Professor Tobias Frederico for discussions, and mainly Professor Airton Deppman for the discussion, and for allowing me to use some files in this manuscript.

Mirez, Tarrillo, Carlos Alberto, and Luis Augusto Trevisan. Statistical Models for Nucleon Structure Function, Cambridge Scholars Publishing, 2024. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/wisc/detail.action?docID=31136544. Created from wisc on 2024-04-11 00:44:35.

Copyright © 2024. Cambridge Scholars Publishing. All rights reserved. Mirez, Tarrillo, Carlos Alberto, and Luis Augusto Trevisan. Statistical Models for Nucleon Structure Function, Cambridge Scholars Publishing, 2024. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/wisc/detail.action?docID=31136544. Created from wisc on 2024-04-11 00:44:35.

CHAPTER 1

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INTRODUCTION

QCD (Quantum Chromodynamics) is the theory of strong interaction in the standard model. This theory describes the short-range interactions among the subnuclear components, the quarks. The gluons mediate the interactions and the main difference with the QED (Quantum Electrodynamics) is the use of a Lagrangian, where gluons may interact among them (in opposition to the photons from QED, that don’t interact). Although this theory describes many relevant aspects, some open questions have been studied with effective alternative models, respecting the basic principles of QCD. One of the questions that concerns the structure-function is the distribution of energy of quarks inside the nucleon. We show some reasons to use effective models in the following: The perturbative theories (short distances) describe the interactions inside the nucleon (strong interactions between quarks and gluons). However, including more diagrams and details makes this method too complex. We need to include many loopings and renormalizations, which is almost impracticable. In principle, this is already a 3-body problem, without an analytical solution, even in classical cases. On the other hand, the lattice quantum field theory also demands great computational efforts. Some effective models do not include Fermi-Dirac and Bose-Einstein statistics as relevant physical effect, such as, the valon model and perturbative chiral. The parameters fit according to the experimental data available. Field and Feynman6 pointed out that including the Fermi-Dirac , statistics is relevant to describing the sea asymmetry in the nucleon which explains the violation of the Gottfried sum rule7. Another very important characteristic is the confining and the asymptotic freedom, predicted by QCD, considered in the models through the effective potential. The effective temperature for each model is also relevant, and it is interesting to compare the obtained temperature of the different models. The Deep Inelastic scattering (DIS) process between leptons and nucleons has been an indispensable tool in describing the hadron structure. Interest in describing the partonic distribution and the different phenomena

Mirez, Tarrillo, Carlos Alberto, and Luis Augusto Trevisan. Statistical Models for Nucleon Structure Function, Cambridge Scholars Publishing, 2024. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/wisc/detail.action?docID=31136544. Created from wisc on 2024-04-11 00:44:35.

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2

Chapter 1

involved, such as the sea asymmetry, strangeness in the nucleon, EMC has generated (Euro Muon Collaboration) effect, and the ratio many theoretical models in recent decades. These models have the appeal of simplicity and are physically wellfounded. The phenomenology to explain such models is in the following way: even though valence quarks must lie in discrete energy levels, they can emit gluons that may split in a quark-antiquark pair, with continuous energy. In the framework of the MIT bag model5, an estimate for the structurefunction was presented by Jaffe8. As it can be speculated, with partons bound in the wee volume of the nucleon, we have not only the dynamic but also the statistical properties; for example, the Pauli exclusion principle should have a relevant effect on PDF (Particle Distribution Function). Most statistical/thermodynamic models proposed in the eighties and nineties consider the confinement given by the MIT bag model, and treat the quark/gluons inside de nucleon as a Fermi-Dirac and BoseEinstein gas of free particles with a continuous spectrum). In this book, we may check the main features of the statistical model. In the next chapters, we describe the statistical models, clarifying focus, motivation, and results. The following papers are studied in chronological order, as follows: 1. Angelini and Pazzi’s works (1982-1983)9 used a statistical model with a Boltzmann distribution and scaling violation. 2. The Cleymans-Thews’ model (1988)10 started with the transition rate of scattering in the framework of the temperature-dependent field theory and explored a statistical way to generate compatible pdfs. 3. The Mac and Ugaz’s work (1989)11 incorporated first-order QCD corrections, introduced by Altarelli and Parisi12. 4. Bickerstaff and Londergan (1990)15 interpreted the finitetemperature property to mimic some volume-dependent effect due to confining. They also discussed the theoretical validity of the ideal gas assumption in detail. 5. The Ganesamurthy, Devanathan, Rajasekaran and Karthiyaini (1994,1996)16,17,18 proposed a thermodynamical bag model, which evolves as a function of . The structure-function they got is practicable for and has the correct asymptotic behavior for ; in addition, they parametrized on and exhibited the scaling behavior.

Mirez, Tarrillo, Carlos Alberto, and Luis Augusto Trevisan. Statistical Models for Nucleon Structure Function, Cambridge Scholars Publishing, 2024. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/wisc/detail.action?docID=31136544. Created from wisc on 2024-04-11 00:44:35.

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Introduction

3

6. Soffer-Bourrely-Bucella have been working with parametrization based on the Fermi-Dirac and Bose-Einstein distributions (1995today)19,20,22,23,21. We reviewed the polarized case (1995-today). 7. Bhalerao and Bhalerao et al. (1996, 2000)25,26 introduced finite size corrections to the statistical model and got more accurate results for unpolarized and polarized structure functions. 8. Trevisan, Tomio, Mirez, Frederico (1999 and 2008)27,28 presented a statistical model based on the Dirac equation with a linear confining potential. They also obtained the strangeness in the nucleon. 9. Trevisan and Mirez presented “A very simple statistical model to quarks asymmetry”267 that considers the meson-hadron fluctuations as energy states with some probability varying according to the temperature. 10. Zhang, Shao and Ma (2009)29 intended to present a statistical model using few parameters to fit the data. They also studied the EMC79 effect with the statistical model. 11. Deppman33, and Deppman et al. 34, presented a relation between the fractal structure of the nucleons and Tsallis statistic (nonextensivity). 12. Trevisan and Mirez35 presented a statistical model that considered the nonextensivity introduced by Tsallis112,113. 13. Trevisan, Mirez, and Silva presented a model with different sizes for quarks and gluons to fix the low moment carried by gluons in the previous statistical model.37 14. Trevisan and Mirez gave a version of the nonextensive statistical model applied to the EMC effect. 15. On the framework of the valon model38,30,39, Mirjalili and collaborators studied the statistical approach40,41. Interestingly, despite the fact that the models basically start from the same physical description, there are some remarkable variations such as scale variance, the dependence of the temperature with the Bjorken variable , and different ways of taking into account the polarization.

Mirez, Tarrillo, Carlos Alberto, and Luis Augusto Trevisan. Statistical Models for Nucleon Structure Function, Cambridge Scholars Publishing, 2024. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/wisc/detail.action?docID=31136544. Created from wisc on 2024-04-11 00:44:35.

Copyright © 2024. Cambridge Scholars Publishing. All rights reserved. Mirez, Tarrillo, Carlos Alberto, and Luis Augusto Trevisan. Statistical Models for Nucleon Structure Function, Cambridge Scholars Publishing, 2024. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/wisc/detail.action?docID=31136544. Created from wisc on 2024-04-11 00:44:35.

PART I

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REVIEW ON PARTONS MODEL

Mirez, Tarrillo, Carlos Alberto, and Luis Augusto Trevisan. Statistical Models for Nucleon Structure Function, Cambridge Scholars Publishing, 2024. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/wisc/detail.action?docID=31136544. Created from wisc on 2024-04-11 00:44:35.

CHAPTER 2 PARTONS MODEL AND THE NUCLEON’S STRUCTURE FUNCTION

2.1 Introduction This chapter is a brief introduction to the main concepts of QCD quark models31,32, which we will use in the following chapters. For the process , we initially describe the kinematics for the proton model. After, we do the elastic and inelastic electron-proton scattering process. Finally, we study the momentum distribution of the Bjorken scale.

2.2 Process

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The Feynman’s diagram for scattering is illustrated in Fig. (2.1). Applying Feynman’s rules, we calculate the invariant amplitude (2.1) From Fig. (2.1), we have the quad-moment . We can calculate the non-polarized shock section by simply squaring the amplitude and summing the spins (we sum the spins separately for each electron and muon), in the form

(2.2) where the tensor for the electron vertex is

Mirez, Tarrillo, Carlos Alberto, and Luis Augusto Trevisan. Statistical Models for Nucleon Structure Function, Cambridge Scholars Publishing, 2024. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/wisc/detail.action?docID=31136544. Created from wisc on 2024-04-11 00:44:35.

Partons Model and the Nucleon’s Structure Function

7

(2.3) The same is true for the tensor

.

Figure 2.1: Feynman diagram for electron-muon scattering.

Applying properties of the traces, we have from Fig. (2.1)

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(2.4) where is the electron mass. For muon, we have the same procedure to do

(2.5) where is the muon mass. Now by multiplying both terms

, we get

(2.6) At the “relativistic limit” we assume that amplitude will be reduced to the following expression

, so squared

Mirez, Tarrillo, Carlos Alberto, and Luis Augusto Trevisan. Statistical Models for Nucleon Structure Function, Cambridge Scholars Publishing, 2024. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/wisc/detail.action?docID=31136544. Created from wisc on 2024-04-11 00:44:35.

Chapter 2

8

(2.7) The Mandelstam variables at the relativistic limit are (2.8) (2.9) (2.10) therefore the spreading amplitude takes the final form

(2.11)

2.2.1 Process e

e

in the lab frame

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For the scattering form

considering the electron mass amplitude

and the muon mass

, we have the

(2.12) The scattering process in the laboratory reference frame is illustrated in Fig. (2.2)

Mirez, Tarrillo, Carlos Alberto, and Luis Augusto Trevisan. Statistical Models for Nucleon Structure Function, Cambridge Scholars Publishing, 2024. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/wisc/detail.action?docID=31136544. Created from wisc on 2024-04-11 00:44:35.

Partons Model and the Nucleon’s Structure Function

Figure 2.2: Process

9

, in the lab frame.

The scattering process in the laboratory reference frame is illustrated in , , and Fig. (2.2). So we have , if we replace in Eq. (2.12), we get

(2.13) , then

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If we consider that the muon is initially at rest, that is we have

(2.14) and since the kinematic relations

(2.15) then, for the amplitude we have

(2.16) and we squared

, where

,

so

Mirez, Tarrillo, Carlos Alberto, and Luis Augusto Trevisan. Statistical Models for Nucleon Structure Function, Cambridge Scholars Publishing, 2024. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/wisc/detail.action?docID=31136544. Created from wisc on 2024-04-11 00:44:35.

Chapter 2

10

that results

Having these relations, and using the formula that relates amplitude and shock sections (see Eq. (4.27) from Ref. 31), we obtain

(2.17) By performing the integration into the Dirac delta functions, we have

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(2.18) Inserting Eq. (2.16) into Eq. (2.17) and using the delta integration of Eq. (2.18), we get

(2.19) Performing

integration over and replacing with , we get the differential shock section in the lab frame

(2.20) where

.

Mirez, Tarrillo, Carlos Alberto, and Luis Augusto Trevisan. Statistical Models for Nucleon Structure Function, Cambridge Scholars Publishing, 2024. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/wisc/detail.action?docID=31136544. Created from wisc on 2024-04-11 00:44:35.

Partons Model and the Nucleon’s Structure Function

11

2.3 Electron-Proton scattering 2.3.1 Elastic Scattering

Figure 2.3: Elastic scattering process electron-proton:

The scattering amplitude for the Fig. (2.3) is given by the following expression:

(2.21)

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where the transition currents of the electron (the electron mates with the photon as a Dirac particle) and the proton (not a Dirac particle because it has an internal structure) are respectively (2.22) (2.23) As we do not know the structure of the proton, we will use the most general combination of Dirac arrays between square brackets. Matrix are discarded due to parity conservation because of the terms like object (parameterized the coupling of anti-commutation matrix. The the proton with the photon) has the general form expressed as (2.24) where and are two independent form factors and is the anomalous magnetic moment. Using Eq. (2.24) to calculate the differential shock

Mirez, Tarrillo, Carlos Alberto, and Luis Augusto Trevisan. Statistical Models for Nucleon Structure Function, Cambridge Scholars Publishing, 2024. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/wisc/detail.action?docID=31136544. Created from wisc on 2024-04-11 00:44:35.

12

Chapter 2

section of the electron-proton elastic scattering we obtain the Rosembluth31 formula.

(2.25)

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The form factors and (anomalous magnetic moment coupling) represent the fact that the proton is not an elementary particle and these as a function factors are determined experimentally by measuring and . These form factors are energy-dependent and of . In Eq. (2.25) the value of “ " is the anomalous magnetic moment of the proton. If we increase the energy and the proton is broken, then we can study its internal structure. If the proton were a point-like particle (without structure) like the electron (or muon), having a charge “ ", and magnetic moment of Dirac ", the result for the scattering would also be valid for the “ proton case, exchanging the mass of muon for the proton. So in Eq. (2.25), and for every . So we get Eq. (2.20) we have

(2.26) where the factor

(2.27) originates from the retreat of the target.

2.3.2 Inelastic scattering For high energy scattering where , the proton becomes a complicated multi-particle system, illustrated by Fig. (2.4)

Mirez, Tarrillo, Carlos Alberto, and Luis Augusto Trevisan. Statistical Models for Nucleon Structure Function, Cambridge Scholars Publishing, 2024. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/wisc/detail.action?docID=31136544. Created from wisc on 2024-04-11 00:44:35.

Partons Model and the Nucleon’s Structure Function

13

Figure 2.4: Diagram for the scattering

The differential shock section is of the form (2.28) is described more generally

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(2.29) represents the leptonic tensor. The most general form of the where should be constructed with and the independent hadronic tensor . moments and So we have for the hadronic tensor (2.30) we disregard the antisymmetric contributions to because they tensor is disappear after we insert them into Eq. (2.29) because the symmetrical. is reserved for a parity-violating structure when a beam of neutrinos interacts rather than electrons so that the virtual photon is replaced by a weak boson. (ou in the space of The conservation of current the moments) implies that (2.31)

Mirez, Tarrillo, Carlos Alberto, and Luis Augusto Trevisan. Statistical Models for Nucleon Structure Function, Cambridge Scholars Publishing, 2024. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/wisc/detail.action?docID=31136544. Created from wisc on 2024-04-11 00:44:35.

Chapter 2

14

by grouping terms, we observe

So only two of the four inelastic structure functions are independent, so

(2.32) so in inelastic scattering we have 02 (two) important variables (2.33)

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The invariant mass per

of the final hadronic system is related to

and

and we have the dimensionless variables

(2.34) and so the kinematic region is the resting frame of the target proton, we have

. Also consider that in

where and are the start and end energy of the electron, respectively. is similar to the , The shock section for with , so using the expression replacing

Mirez, Tarrillo, Carlos Alberto, and Luis Augusto Trevisan. Statistical Models for Nucleon Structure Function, Cambridge Scholars Publishing, 2024. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/wisc/detail.action?docID=31136544. Created from wisc on 2024-04-11 00:44:35.

Partons Model and the Nucleon’s Structure Function

15

(2.35) and considering that

(current conservation), we have (2.36)

Using the kinematic relations Eq. (2.15), we obtain in the laboratory reference

(2.37) including the flow factor and the phase space factor, we have the differential shock section even for the electron-proton inelastic scattering

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(2.38) The extra factor arises by normalization of (2.37) into Eq. (2.38), we finally get

. Inserting Eq.

(2.39) where the mass of the electron is neglected. The test for the proton to be composed of point particles is the behavior of the differential shock section, within the process the differential shock section is given as follows

(2.40)

Mirez, Tarrillo, Carlos Alberto, and Luis Augusto Trevisan. Statistical Models for Nucleon Structure Function, Cambridge Scholars Publishing, 2024. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/wisc/detail.action?docID=31136544. Created from wisc on 2024-04-11 00:44:35.

16

Chapter 2

If we compare the result with the elastic shock section with a point , we can proton, in the scattering of high-energy virtual photons write

(2.41) So the proton structure function is

(2.42)

(2.43)

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Figure 2.5: Illustration of the Eq. (2.43)

2.4 Bjorken scale and the Partons model When entering a positive variable , where is the mass of in DIS, electron-proton scattering is an elastic the quark, then at scattering of an electron by a free quark within the proton. in Eq. (2.43) we were Using the delta identity of Dirac able to rearrange the terms to introduce the dimensionless structure

(2.44)

(2.45)

Mirez, Tarrillo, Carlos Alberto, and Luis Augusto Trevisan. Statistical Models for Nucleon Structure Function, Cambridge Scholars Publishing, 2024. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/wisc/detail.action?docID=31136544. Created from wisc on 2024-04-11 00:44:35.

Partons Model and the Nucleon’s Structure Function

17

These functions are just functions expressed in terms of and and , so if the virtual photons in solve the point not constituents within the proton, we can get the following expressions for point particles (2.46) (2.47) where

(2.48)

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Note that in Eq. (2.48) there is no scaling. The proton mass is used, instead of the quark mass , to define the dimensionless variable . The presence of free quarks is signaled by the fact that the inelastic structure function is independent of to a fixed value in Eq. (2.48). In the parton model, a connection is created between fundamental particles (quarks) and hadrons. Basically, in the parton model, we have • • • •

The proton is made up of a group of partons. In deep inelastic scattering, the photon interacts with a parton. Partons are elementary particles whose interactions we can calculate. These partons are identified as quarks and gluons. The patron has negligible or null transverse momentum.

So the proton is made up of other particles, the partons; these are elemental. Several types of point protons make up the proton. They can carry different fractions of the energy and total momentum of the proton. We illustrate the momentum distribution of the pattern in Figure (2.7).

Mirez, Tarrillo, Carlos Alberto, and Luis Augusto Trevisan. Statistical Models for Nucleon Structure Function, Cambridge Scholars Publishing, 2024. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/wisc/detail.action?docID=31136544. Created from wisc on 2024-04-11 00:44:35.

Chapter 2

18

Figure 2.6: The proton consists of point quarks.

The distribution describes the probability of a constituent carrying a fraction of the momentum of the proton . So the sum of the fractions is equal to 1

(2.49)

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where the sum over gluons included).

denotes summation over all partons (quarks and

Figure 2.7: Distribution of Momentum of Partons.

Both the proton and proton move along the z axis, (i.e. transverse moment ) with longitudinal momentum and . As for dottype protons, we have (2.50)

Mirez, Tarrillo, Carlos Alberto, and Luis Augusto Trevisan. Statistical Models for Nucleon Structure Function, Cambridge Scholars Publishing, 2024. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/wisc/detail.action?docID=31136544. Created from wisc on 2024-04-11 00:44:35.

Partons Model and the Nucleon’s Structure Function

19

so the momentum of a patron is: . So for an electron colliding with a fraction with momentum and unit charge , from Eq. (2.45) and Eq. (2.47) we have the functions of dimensionless structure

(2.51)

(2.52) where we use the kinematics of Eq. (2.51) and is the dimensionless structure function is for a proton in variable defined in Eq. (2.48). The Eq. (2.51), now we add over all the proton constituent protons, Fig. (2.6) and Fig. (2.7) so we get

(2.53)

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(2.54) as and express the result It is conventional to redefine in terms of . When compared to Eq. (2.47), Eq. (2.53) takes the same . So to sum all the partons, expression when

(2.55) (2.56) being (2.57) The moment fraction is identical to the kinematic variable of the virtual photon, so the virtual photon must have exactly the value of the variable to be absorbed by a parton with a moment fraction . Due to the

Mirez, Tarrillo, Carlos Alberto, and Luis Augusto Trevisan. Statistical Models for Nucleon Structure Function, Cambridge Scholars Publishing, 2024. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/wisc/detail.action?docID=31136544. Created from wisc on 2024-04-11 00:44:35.

20

Chapter 2

“delta function” in Eq. (2.53) we can equate these two distinct physical quantities. Thus, the structure function for a parton with momentum

(2.58)

(2.59) is used. where the approach We can add the result of a proton over all the protons, in this case for the proton

(2.60)

(2.61)

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These structure functions for Callan - Gross Relation,1 where

spinning partons are related by the

(2.62) from Eq. (2.55) and Eq. (2.56) are The inelastic structure functions to a fixed functions of the variable only. They are independent from . So, it is said that they satisfy the Bjorken scale. In short we have In deep elastic scattering, we have • The proton is characterized by form factors that are independent of the energy scale. .

1

This relationship is recurring from the details of the disturbing shock section, confirming between the patron the existence of Dirac particles, spin

Mirez, Tarrillo, Carlos Alberto, and Luis Augusto Trevisan. Statistical Models for Nucleon Structure Function, Cambridge Scholars Publishing, 2024. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/wisc/detail.action?docID=31136544. Created from wisc on 2024-04-11 00:44:35.

Partons Model and the Nucleon’s Structure Function

21

The parton model • Reproduces Bjorken’s scaling behavior. • Is a model for the proton structure with structural factors (which are measured experimentally) interpreted as distribution functions of quarks and gluons within the proton, which are independent of the energy scale. • Predicts the Callan-Gross relationship: • Considers the partons as free particles inside the hadron (inelastic structure function is independent of for a value of ) In QCD, some important features • Asymptotic freedom (quarks are treated as free particles inside high energy hadrons). • Confinement, not observing free quarks and gluons, which guarantees the existence of colorless hadrons. • Gauge theory (predicts a massless particle which carries the strong interaction: the gluon). Therefore, QCD fits perfectly into the standard model.

2.4.1 Distribution function with partons

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Measurements of the large inelastic structure functions for reveal the hadron structure to quarks. The sum in Eq. (2.55) is overall protons in the proton, so

(2.63) e are the probability distributions of the “ ” quarks and where antiquarks in the proton and neglect the presence of “ " quark charm and heavier quarks. The inelastic structure function for neutrons is performed experimentally by scattering electrons by a deuterium target.

(2.64)

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22

Chapter 2

and since proton and neutron are members of an isospin doublet, their quarks are related by (2.65) (2.66) (2.67) The proton consists of valence quarks with the combination , accompanied by quark-antiquark pairs known as “sea quarks”. Fig. (2.8) illustrates this process in detail. If we assume that the sea is symmetrical in the flavors of the quark we have a valence quark and a sea quarks, then for each antiquark listed as follows. (2.68) therefore, we have (2.69)

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(2.70) where is the distribution of the sea common to all flavors, if we has 06 antiquarks from assume that sea is symmetrical. So we have the sea.

Figure 2.8: Proton with valence quarks

, gluons and sea quark-antiquark pairs

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Partons Model and the Nucleon’s Structure Function

23

Figure 2.9: Distribution of unpolarized brackets, times the distribution , , using parametrization of the following order to the main order (NNLO) in the MRST2006 parametrization269 on the scale: =20 and = .

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24

Chapter 2

The Eq. (2.68) is an approximate expression. In this thesis, possible differences in the distribution of sea quarks are the main study targets. If we add all the contributions of the protons, we get the quantum numbers of the proton: “charge 1, baryonic number 1, strangeness 0.”

(2.71)

(2.72)

(2.73) We can then combine Eq. (2.68) with Eq. (2.63) and Eq. (2.64) and we get (2.74)

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(2.75) is the sum of the squares of the load: over the six different where sea quark distributions. Since gluons create the pairs of the sea, we to have spectrum from “bremsstrahlung" to small, so that expect the quarters of “sea grow logarithmically" when . According with Eq. (2.75), we have

(2.76) and

(2.77)

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Partons Model and the Nucleon’s Structure Function

25

As for the proton, there is evidence that for large, so by applying the limit for Eq. (2.77) to then we have 270;271. Calculating the difference between Eq. (2.75) and Eq. (2.74), we get (2.78) which are valence quarks without sea quarks. Performing integration from experimental data for , we have

,

in

(2.79) (2.80) where

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(2.81) and we do the same is the fraction of the moment loaded by quarks and gluons , . In Eq. (2.81) we neglect strange for quarks quarks. We know the total momentum is 1, so (2.82) (2.83) then we may solve the Eq. (2.81) and obtain (2.84) Quarks charge about % from proton moment, in fact % for valence quarks and % for sea quarks. The remainder is carried by gluons (approximately % of proton moment) although gluons are not directly measured in deep inelastic scattering, their presence is in the production of high-energy collision jets.

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26

Chapter 2

2.5 Gottfried’s Sum Rule (GSR) From Eq. (2.71), Eq. (2.72) and Eq. (2.73) where each term represents an integral over partons, we can then perform combinations. The Gottfried7 sum rule provides information on the distribution of sea quarks. It is defined as follows

(2.85)

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, where From the general expression for , , , , and using proton-neutron symmetry, we have that (2.86) (2.87) We apply equations Eq. (2.86), Eq. (2.87) to Eq. (2.85), and we obtain

(2.88)

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Partons Model and the Nucleon’s Structure Function

27

If the sea is: “symmetrical", we apply Eq. (2.68), in Eq. (2.69) and Eq. (2.70), that is and . If

(2.89) and using Eq. (2.89), we obtain from Eq. (2.88)

(2.90) The first experimental evidence for the value of comes from the NMC246 experiment in CERN on the value of the Gottfried sum rule, showing that the value of experimentally is (2.91)

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Other recent experiments have been performed and illustrated in the following Table (2.1).

Table 2.1: Gottfried Integral Experimental values for . The experiments are NMC,246 HERMES,240 Rstowell-NuSea.229

is evidence of a breakdown of flavour The value of symmetry in the nucleon sea or a flavour asymmetry of sea quarks . and So if we consider the equations Eq. (2.71), Eq. (2.72), we can rewrite the Gottfried integral as follows

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28

Chapter 2

(2.92) where we define the following integral

(2.93) then 246

(2.94)

240

(2.95)

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229

(2.96)

The pattern distribution function (PDF) can be determined from deep inelastic lepton-nucleon scattering data and nucleon-initiated hard scattering processes. The following Table (2.2) is based on Ref. 268 where some of the main processes for determining the distribution of partons are highlighted.

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Partons Model and the Nucleon’s Structure Function

29

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Table 2.2: Lepton-nucleon processes and hard scattering processes, they show the sensitivity for the pattern distribution function (FDP) that are tested Ref. 268.

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PART II

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MODELS WITH FERMI-DIRAC DISTRIBUTIONS

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CHAPTER 3 ANGELINI AND PAZZI WORKS

In a paper published by Angelini and Pazzi,9 there was an attempt to obtain thermodynamical information on the quark matter using the experimental data on the nucleon valence quark distribution measured in DIS. The hypothesis is the following: the valence quarks are non-interacting and look like a relativistic Boltzmann gas without any kinematical limits. The authors observed the following problems with these assumptions: • •

The quark interactions, as well as any dynamical effects, should not be very sensible in the IMF except for the region. The absence of kinematical limits does not allow the quark distribution to be zero at x=1.

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The proposed valence quark distribution has the form

(3.1) where , and is the quark mass, is the temperature and is the momentum. In Lorentz frame in which the nucleon is moving with a given , in the equation Eq. (3.1) integrated over becomes:

(3.2)

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Angelini and Pazzi Works

33

where and are the components where p is perpendicular to and limit (IMF), if the variable, parallel to , respectively. In the , where is the nucleon mass, we have defined as

(3.3) where (3.4) in the case of

the expression obtained is

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(3.5) and fixes the normalization. where Angelini and Pazzi used the ABCLOS13 and CDHS14 collaborations in interactions to obtain a comparison with the experimental data. , and The results were obtained in two ways: First, it approached . The obtained the fits averaged over . The results were good for temperature was about MeV, while the value for the constant stayed between and . See the details in the table below:

Table 3.1: Results of fits using Eq. (3.5) to ABCLOS and CDHS data.

In the second way, the relation was considered as

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34

Chapter 3

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and then it is possible to establish a relation between and . The and tends to decrease when temperature seems to depend slowly on is increased. The temperature stayed around 50 MeV, and the values for were between 7.7 and 12.2. See Table (3.2) to more details.

Table 3.2: Results of fits to ABCLOS and CDHS data on using expression Eq. (3.5) in the text. When the cut is not specified, no data are . available for

We observe that this work presented an interesting approach, but is oversimplified. The next step was to consider the spin effects in distributions that lead to Fermi-Dirac and Bose-Einstein gasses. We describe that in the following chapters.

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CHAPTER 4 THE CLEYMANS-THEWS MODEL

J. Cleymans and R.L. Thews10 presented the confined quark/gluons gas with continuous energy levels. The inelastic structure function emerges by using a thermal model. This means value of the hadronic tensor multiplies the Fermi-Dirac thermal distribution for quarks by the mean energy and density, which depends on characteristic temperature and chemical potential. This leads to:

(4.1)

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where (4.2) The model describes the nucleon composed of quarks and antiquarks at a temperature around MeV, and the chemical potential is MeV. The parameters and adjust the exponential fall of the inelastic structure function ( for proton) for high and the picks. For , the structure-function is not well described, because in this model . The structure functions for deep inelastic scattering calculated in this model are applied in the Drell-Yan process to study how they determine considering the production rate of massive leptons , then the rate is: at the beginning the chemical potential The structure functions for deep inelastic scattering calculated in this model are applied in the Drell-Yan process to study how they determine considering the production rate of massive leptons , then the rate is: at the beginning the chemical potential

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36

Chapter 4

(4.3) This production rate is related to the quark density in the case of . Using , we obtain the quark distribution on the form:

(4.4)

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We remark that his model does not consider the gluons yet. In the next chapter, we will review a work that considers some gluons effects.

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CHAPTER 5 THE MAC AND UGAZ WORK

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The Mac and Ugaz model11 is the relativistic thermal model of a free particle gas, with same results of the Cleymans and Thews’s work.10 This model, however, takes into account the gluon distribution and considers the confining and also the perturbative QCD corrections, by using the Altarelli-Parisi formulas.12 The nucleon, in this model, is a relativistic gas where “free" quarks and gluons are inside a volume of a bag. This model is similar to the MIT bag model. The volume is related to the nucleon size. With a defined temperature , we may apply the Fermi-Dirac statistics for quarks, and Bose-Einstein for gluons. as the mean number of In the proton rest frame, we define quarks, with both polarizations and momentum , being the distribution of “free" confined quarks defined as

(5.1) where “ " is the energy, “ " the chemical potential. For the antiquarks, in a free we replace ‘ ” by “- ” in Eq. (5.1). The gluon distribution gas is given by

(5.2) where there is no chemical potential for gluons. The temperature in the model is defined in the nucleon rest frame. This temperature can only be defined in the rest frame. In the frame in which the proton moves on the axis, the distribution in Eq. (5.1) becomes

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Chapter 5

38

(5.3) here means the energy of a constituent parton and , and the volume, temperature, and chemical potential in the moving frame, respectively. For the case with massless quarks and gluons, the following relation is satisfied:

(5.4) where

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(5.5) is the usual Lorentz transformation for energies measured in the rest frame of a proton with mass to the frame in which the proton with mass is moving with momentum (in the z-axis) and energy . Eq. (5.1) and Eq. (5.3) give the probability of finding a parton with a certain momentum. Replacing Eq. (5.4) in Eq. (5.3) means physically that both observers must have some measure of probability per unity of volume to find the system in a given state. of the distribution By integrating the transverse momentum we obtain the longitudinal distribution of the quarks. In the moving frame, using Eq. (5.3) and Eq. (5.4), it reads

(5.6) Doing

and defining the variable changes in the equation:

(5.7) replacing Eq. (5.7) in Eq. (5.6) we have the integral

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The Mac and Ugaz Work

39

(5.8) . Taking the infinity moment limit the following approach holds

,

e

. In this limit

(5.9) then

that eliminates

. Finally the distribution function is obtained:

(5.10)

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where

is the volume in the rest frame, related to the moving frame by with the Lorentz contraction in the z-axis. This is the same result obtained by Cleymans and Thews [see Eq. (14) and Ref. 10]. The longitudinal momentum distribution for gluons and antiquarks are calculated and defined as:

(5.11) (5.12) The sum rules must be obeyed by the equations Eq. (5.10) and Eq. (5.12) (5.13) and the normalization is chosen in a way that

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40

Chapter 5

(5.14) (5.15) which leads to (5.16) Notice that in Eq. (5.13) is the number of quarks, in any frame, in this way, it may be written (5.17) Integrating in spherical coordinates in the rest frame and using Eq. (5.1)

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(5.18) and substituting the series expansion

in Eq. (5.18), we get

(5.19) with integration, we get

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The Mac and Ugaz Work

41

(5.20) Lets consider now ; using the formula 24.16 from Schaum,42 we check the relation of Eq. (5.20) with the Fourier and get the expression representation of (5.21)

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By inserting the chemical potential obtained in Eq. (5.21) into Eq. (5.10) and Eq. (5.12) we get the momentum distribution of the model. Mac and Ugaz also calculated the perturbative corrections due to QCD effects using the Altarelli-Parisi equations.12 The obtained results by Mac and Ugaz were the following:

where is the proton radius. The too big radius is because it used continuous energy levels instead of discrete ones. With the discrete energy levels, the radius would be smaller. Let us show how the model of gas with continuous energy levels and discrete energy is equivalent in the limit case. In a model with discrete levels, the probability density for a fermionic system with energy levels and temperature is

(5.22) where is the degenerescence of each level, the chemical potential, which normalizes the sum. For the antiparticles, we have

To calculate the chemical potential of the quarks, the conditions are

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42

Chapter 5

(5.23) To obtain probability density, in terms of the Bjorken scale , the Fourier transform is used. In the momentum space:

(5.24) where

(5.25) To return to the relativistic model of free gas by Mac and Ugaz11 one should use the free wave solutions of the Dirac equation. Normalizing in the volume , for positive energy: (5.26)

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and for negative energy (antiparticles) (5.27) where, in the quarks rest frame:

(5.28) Integrating the plane wave solutions solutions in the volume, we have the degenerescence factor (5.29)

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The Mac and Ugaz Work

being is

and

(identity:

43

), where the integrand in

(5.30) In this way, the total number of particles from Eq. (5.22) is given by

(5.31) we obtained the integration in the momentum space.

(5.32) where the momentum distribution used by Mac and Ugaz is the integrand, therefore

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(5.33) The first order QCD corrections were considered on this work, using the distributions previously found. See Fig. (5.1).

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44

Chapter 5

Figure 5.1: first order QCD corrections

The quark-gluon emissions lead to the corrections:

(5.34) and the corrections to antiquarks are made simply by:

The Gluon splitting process leads to the corrections:

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The Mac and Ugaz Work

45

where

Between 10 and 20 GeV2/c2, between the distributions is:

is set to 0.1. The relation, in this model,

(5.35)

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The gluon corrections are important for . Also, the authors speculate that, due to the big obtained, this kind of model may be appropriate for larger systems such as heavy nuclei, heavy-ion collisions, and even in the early universe.

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CHAPTER 6 THERMODYNAMIC MODEL FOR PROTON SPIN

Before introducing the Thermodynamical/Statistical models for polarized structure functions, we recall some basic definitions and rules, which are the starting points for the models. For a more detailed review on this subject, see Greiner.43 See the expression below

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(6.1) indicates the probability of finding a quark with a Here momentum fraction polarized in the same direction as the whole proton, is the quark’s electric charge. The notation and and is also common. One of the most relevant properties of unpolarized functions is that the momentum fraction carried by quarks accounts for only half of the total momentum. This is striking proof of the existence of gluons. Analogously, from the polarized structure function, we ask how much of the proton spin is carried by quarks, how much by gluons, and how much is present in angular momentum. The models for proton and neutron spin must consider the Bjorken sum rule44,45

(6.2) . More perturbative corrections are shown on which is exact for Greiner’s book (see formula 5.247). Also, it’s usual to define: (6.3)

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Thermodynamic Model for Proton Spin

47

and (6.4) If only and quarks are being taken into account, the proton and neutron spin structure function read explicitly:

(6.5)

(6.6)

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and . where it’s supposed It’s important to remark that some works present do not consider the , recently measured.46,47 contribution Ganesamurthy, Devanathan, and Rajasekaran presented a work named “Thermodynamical model for proton spin”.17 In this work, they have followed the same initial considerations of Mac and Ugaz but have used different chemical potentials for up and down quarks as a function of temperature. In the infinite momentum frame of the nucleon, the quark distribution is in function of chemical potential and the temperature, as follows11

(6.7) The antiquark distribution is obtained by putting valence quark obeys the usual sum rules:

and the

(6.8)

(6.9) the obtained temperature is about MeV. To estimate the valence quark contribution to the proton spin, the authors assumed that there is no contribution from the sea quark and gluons.

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48

Chapter 6

The valence quark distribution function may be written in terms of the 48 and unpolarized valence distribution function

(6.10)

(6.11) is the spin dilution factor. where The phenomenological parametrization of the spin dilution function is taken as (6.12) where is the only free parameter used to satisfy the Bjorken sum rule. To include the QCD corrections, the model needs the gluon distribution, given by:11

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(6.13) The spin carried by the gluons dilution function for the gluons48

is obtained by using

for the spin (6.14)

Introducing the first order QCD correction to the and valence quark distributions49

(6.15)

(6.16) is the strong coupling constant and is the gluonic where and the values contribution to the proton spin. On the definitions of and must be replaced by their corrected values. of

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Thermodynamic Model for Proton Spin

49

Besides the Bjorken sum rule, the proton spin sum rule must be satisfied:

where is the orbital angular momentum. The model has only one free parameter , which is used to fit the . There is no dependence with . value of A good agreement with experimental data is obtained in this model using MeV. We have

and

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is in agreement with the naive quark model expectation.

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CHAPTER 7 THE BICKERSTAFF AND LONDERGAN WORK

Similarly to Cleymans and Thews’s, or the Mac and Ugaz’s work, Bickerstaff and Londergan’s15 paper also used a confined Fermi gas at a finite temperature. The authors have shown how the different confining models affect the energy-momentum tensor. The models are the MIT,5 the Freidberg-Lee soliton model63,64,59 and the color-dielectric.50-52 The authors have also shown that the same equation of state (EoS) holds in the three cases. The energy-momentum tensor is defined as: (7.1)

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where is the energy density, is the pressure and (velocity), while (1,0) in the rest frame. Therefore:

is the speed (7.2)

and (7.3) the term

for different models will be studied in the next sections.

7.1 MIT bag model In 1974, Chodos et al.5 presented the famous MIT bag model. The authors considered the confining and the asymptotic freedom, the cornerstones of QCD. These two features were introduced by employing an effective spherical and central potential, which is zero inside the bag, and infinity on the wall (the border). In simple words, the nucleons are balls within the quarks inside.

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The Bickerstaff and Londergan Work

51

The Lagrangian for this model is: (7.4) inside the bag, and 0 outside. is a function to account where for the surface effects. In the inner region, the density is uniform, with . This leads to a Dirac equation of free particles, in which the energymomentum tensor is given by: (7.5) where (7.6) is the quarks contribution. Therefore: (7.7)

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and (7.8)

7.2 Soliton model, by Freidberg-Lee The static bag models, discussed in the preceding sections, are necessarily limited in scope because of their noncovariant, nonHamiltonian formulation. These models cannot be expected to yield an annihilation, and many adequate description of hadronic collisions, other dynamical phenomena for which QCD analyses are usually limited to the lowest order in gluon exchange. Thus, it is desirable to develop alternative bag formulations which can be described by a complete Hamiltonian and are manifestly covariant.53-55 Such formalism is the soliton bag model proposed by Friedberg and Lee.56,61 The heart of the Friedberg-Lee model is the nontopological soliton, or field.56-58,61 A soliton is a solitary wave that asymptotically preserves its

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52

Chapter 7

shape and velocity in nonlinear interactions, with other solitary waves or other localized disturbances. Topology studies geometrical properties that are not changed under deformations. In this case, nontopological means properties that do not change with the interactions. This is a phenomenological representation of the quantum excitations of the selfinteracting gluon field. It is a scalar field. The energy of a uniform system as a function of the field strength has two minima, one at zero and a second, deeper minimum, at a large finite value identified as the vacuum value. In the absence of quarks, the normal state of the field is at the vacuum value. In the presence of quarks, the field finds a minimum in the vicinity of zero; the quarks dig a hole in the vacuum. This is the origin of confinement in the model. Another way to understand the soliton is to visualize it as a gas bubble immersed in a liquid medium. The parameter p can then be interpreted as the gas pressure. The situation in which the bubble is filled uniformly with the quark gas is representative of the MIT bag. This visualization gives the thermodynamic properties of the quark gas in the bubble (i.e., the nucleon). This view is quite similar to the QCD dielectric model. The Lagrangian density, without the gluons terms, is given by63,64,59

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(7.9) has a local minimum value in , and a global The potential minimum in the minimum value indicates the stability of the equation of motion. For the inner of the bag to be a region with uniform density, like a gas, the Lagrangian has the form: (7.10) where is a constant. In this approach, the quarks obey the Dirac equation for free particles with effective mass: (7.11) and (7.12)

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The Bickerstaff and Londergan Work

and

Note that the EoS is the same of the MIT, just replacing by .

53

by

.

7.3 QCD Dielectric Model The basic idea of the QCD dielectric model is to compare the vacuum to a dielectric material. The nucleon is a hole inside this material. No QCD particles (quarks, gluons) can go outside the hole, in the same way, electric particles can’t move in a dielectric material. , and the Lagrangian The quark field is re-scaled by density becomes:

(7.13) potential has a local minimum in and global in . The Inside the bag, is a little greater than 1. This means a perturbative potential. The uniform density in this model is obtained with the term:

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In this case, the resulting Dirac equation for the free quarks has an effective mass: (7.14) and the energy-momentum tensor results: (7.15) that is the same presented in the bag and Freidberg-Lee model with the replacement

e

.

7.4 The energy density in the system The Bieckerstaff-Londergan model proposed a spherical nucleon, similar to MIT, where the quarks have a uniform density, like a Fermi gas,

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54

Chapter 7

therefore with a Fermi-Dirac distribution for energy, with continuous levels. This results in the energy density and pressure as follows:

(7.16)

(7.17) and density

(7.18) where

is the degeneration of the quarks with flavour

and and quarks and antiquarks given by:

,

are the Fermi-Dirac distribution for

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(7.19)

(7.20) is a typical Fermi energy for the distribution. If there is no where vector potential, equivalent to chemical potential, this energy is defined by: (7.21) Eq. (7.16), Eq. (7.17), and Eq. (7.18) are solved self-consistently, noting the balance condition . Also, if it’s considered there is no surface contribution (like in the MIT model), the mass will be the energy density multiplies by volume (7.22)

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The Bickerstaff and Londergan Work

55

where is the volume of spherical bag. and . The routine to Thus, the model’s parameters are search for the consistent value of the model starts with a test temperature , with the corresponding energies and , from what: (7.23) where and . From these values, the is calculated from Eq. (7.22) and compared with the mass given as the initial parameter. After, the temperature is changed until the results are consistent. In the special case of interest it can be proved: (7.24) where the mass is the sum of each quark energy. Considering the virial theorem, the result is (7.25)

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where, for each quark flavor, we define:

(7.26) For the massless quarks, there is an analytical solution: (it is obtained in a similar way of the Mac and Ugaz, considering the series expansion, and the Fourier series)

(7.27) e (7.28) The model may present some coherent sets of results:

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Chapter 7

56

Table 7.1: The balance values for some protons radius and quarks mass.

7.5 Structure Functions The quark and antiquarks distributions depend, besides the Bjorken scale , on the temperature. In other words, the temperature gives the probability of a current quark having certain energy and therefore carries out some fraction of the total nucleon momentum. The model proposed by Bieckerstaff and Londergan results in the formula: (7.29)

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and (7.30) with

(7.31) and (the index 3 in

indicates the z component) (7.32)

and to

the positive root is considered.

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The Bickerstaff and Londergan Work

57

(7.33) The turn the integration easier and gives a relation between

and ,

(7.34) where,

(7.35) replacing the functions : for

and

, and integrating, we get the form

(7.36) .

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used to calculate

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CHAPTER 8 THE DEVANATHAN–KARTHIYAYINI– GANESAMURTHY MODEL

In 1994,16 the three authors above presented a model named “Thermodynamical Bag Model for nucleon structure functions”, also based on the approximation of a Fermi-Dirac and Bose-Einstein gas for quarks and gluons, respectively. It is relevant to show the differences between this model and the other. The authors observed that the models by Mac and Ugaz, Cleymans and Thews, and Bickerstaff–Londergan do not exhibit the correct asymptotic behavior as and . extends beyond into the unphysical region and vanishes as . The work starts with a review of lepton hadron scattering:

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(8.1) where denotes all possible hadronic states of the invariant mass depends on both and .

which

(8.2) In the parton model, the nucleon structure function is expressed in terms of the parton distribution functions when both are functions of only (scaling approximation). If one attempts to deduce the quark distribution function from the Fermi distribution function, the excitation of the target nucleon to the final hadronic state and its subsequent deexcitation should be incorporated.

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The Devanathan–Karthiyayini-Ganesamurthy Model

59

That is what the work revised in this chapter proposed by identifying with the mass of excited nucleon and using a parameter in normalization. After having considered the nucleon as an MIT bag, consisting of quarks and gluons and making the usual association with Fermi and bosons gases, the energy density is given as a function of the temperature and the chemical potentials and . The obtained results are similar to those explained in the Mac–Ugaz and Bickerstaff–Londergan chapters.

(8.3) (8.4) (8.5) The energy density is given after considering 6 degrees of freedom for quarks and 16 for gluons. (8.6)

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The number densities (valence quark densities), for the proton case, are: (8.7) (8.8) which are basically the same result obtained by Mac–Ugaz, but using two chemical potentials, instead one. The invariant mass of the final hadron is identified with that of the excited nucleon state, and it is obtained using the thermodynamical bag model. Assuming that the energy transfer to the nucleon results in heating the constituent quark-gluon gas. where

(8.9) is the energy density of the quark-gluon gas at temperature

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.

Chapter 8

60

The pressure balance condition once again yields the equation (8.10) or equivalently, (8.11) The bag constant is known to decrease as the temperature increases according to the formula125 ࡳ 128

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(8.12) where is a bag constant corresponding to and is the critical temperature. Above that there is the phase transition from hadronic to MeV and quark-gluon plasma. The values are MeV. We note we need two initial pieces of information on this model: . Then we have to solve Eq. (8.7) and Eq. (8.8) selfand consistently. Given the bag constant and the invariant mass as a function of and , one can obtain the variables , , and . There is a relation between and . The smaller the value of , the greater the temperature . The volume of the bag is not constant and increases with the temperature. See Table (8.1) to observe these interesting features of the model.

Table 8.1: Table showing the dependence of temperature T, bag radius R and chemical potentials and on the Bjorken variable along with the quark distribution with the quark distribution functions in DIS ( =4 GeV2)

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The Devanathan–Karthiyayini-Ganesamurthy Model

61

To obtain the structure functions, the same formulae of Mac and Ugaz is used. and Cleymans and Thews are used, but instead of , ranging from 2 to Calculations have been performed for values of , is found to depend 100 GeV2. To get perfect scaling at weakly on

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(8.13)

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CHAPTER 9 THE THERMODYNAMICAL BAG MODEL FOR THE NUCLEON’S SPIN

In 1996, Devanathan and McCarthy presented a work,18 following the work by Devanathan, Karthiyayini, Ganesamurthy16 to obtain the polarized structure functions. The main difference from the early model is the introduction of chemical potential to the spin parameter, resulting in the ). following set of number densities for the and quarks (

(9.1)

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(9.2)

(9.3)

(9.4) Similar equations hold for quarks for which the chemical potential is and the spin parameter is . The energy densities for all kinds of quarks can also be written supposing there is no rest mass. The temperature and the chemical potentials and obey the following set of equations: (9.5) (9.6)

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The Thermodynamical Bag Model for the Nucleon’s Spin

63

(9.7) (9.8) (9.9) (9.10) The multiplicative factor denotes the color degeneracy that is three, and is the volume of the bag. is the bag constant and the mass or of the quark-gluon energy of the system by . The energy density gas is an explicit function of the energy densities of the constituents.

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(9.11) The values for and that results in good agreement with the experimental data65 are and . In equations Eq. (9.5) to Eq. (9.10) the equations Eqs. (9.1–9.4) are used, with the Fourier series expansions and the relations between trigonometric and hyperbolic functions. Then the following analytical forms result for the energy density:

(9.12) and for the number density:

(9.13)

(9.14)

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64

Chapter 9

(9.15)

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(9.16) By using these analytical expressions, the equations of the state of the bag have to be solved to yield the bag variables. The relevant point is that there is no free parameter. The invariant mass of the final hadronic state in deep inelastic as the mass of the scattering is a function of and . By identifying excited nucleon, the equations of state are solved self consistently to yield and . In this model, there is a relation between the Bjorken variable and the temperature of the thermodynamical bag. The smaller the value of , the greater the temperature; thus explaining the huge production of sea quarks and gluons at smaller values of . The volume of the bag is not constant but increases with the temperature. The chemical potentials and and the spin parameters and decrease with the temperature. In the limiting case of (elastic scattering), , and the bag radius fm. The quark spin distribution functions are obtained by transforming the Fermi distribution functions equations Eqs. (9.1–9.4) to the infinite momentum frame (IMF) and integrating over the transverse momentum following the works of Mac and Ugaz,11 Cleymans and Thews,10 and Devanathan et al.16

(9.17)

(9.18)

(9.19)

(9.20) with

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The Thermodynamical Bag Model for the Nucleon’s Spin

65

(9.21)

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a similar set of equations is obtained for quarks. Devanathan and McCarthy gave a new physical interpretation for the Bjorken variable. The Bjorken variable was introduced to define a specific kinematic configuration of the reaction. They are considered the inelastic parameter with values ranging from 0 to 1. The upper limit, , is the elastic case, and the lower limit corresponds to the extreme inelastic event when the energy of the incident lepton is transferred to the target nucleon. Alternatively, in the IMF, the Bjorken variable corresponds to the fraction of the momentum carried by the quark. The main results obtained by Devanathan and McCarthy are shown in the Table (9.1).

Table 9.1: Comparison of the calculation by Devanathan and McCarthy ( GeV2) with the experimental data 65-67 on spin observables.

=3

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CHAPTER 10 THE STATISTICAL BY SOFFER–BOURRELY–BUCELLA – THE POLARIZED CASE

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The group composed of J. Soffer, C. Bourrely, and C. Bucella have presented a sequence of works with “parameterized statistical formulas”,1924 named “statistical approach of parton distributions”. In this model, the nucleon is a massless partons gas (quarks, antiquarks, gluons), balanced at some temperature, confined in a volume . The light quarks, with helicity , with the energy scale =4 GeV2, are given by the sum of two terms, similar to the Fermi-Dirac distributions (the authors named it “almost Fermi-Dirac distribution”), with parameters that fit the experimental data. For polarized functions, the formula is:

(10.1) is a constant, equivalent to the thermodynamical potential (or Here chemical potential), is the temperature, the same for all partons. Observe that the second term in the above formula represents the quarks generated by the gluon splitting process. For an unpolarized function we have , and for the polarized . The following properties hold: •

for a positively polarized quark has the The chemical potential opposite signal of the antiquark with opposite polarization:

•

The chemical potential for the gluons is zero:

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The Statistical by Soffer–Bourrely–Bucella – The Polarized Case

67

For the antiquarks, the distributions have the form:

(10.2) To the gluon distribution, the model has an almost Bose-Einstein distribution:

(10.3)

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for an small , a similar behaviour between quark and gluon functions is expected, in order to obtain it, the authors did

The free parameters in the model are listed: , , , , , , , . , , And the parameters fitted by the normalizations ( , , sum of moments equal 1,) are . To include the experimental results describing the strangeness, Soffer, Bourrely and Buccella introduced the following formulas:

(10.4) and

(10.5) , where the new parameters values are determined by the here, following sum rules:

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Chapter 10

68

•

There is no net strangeness in the nucleon

•

The second Bjorken sum rule

where

and

are the hyperon-beta decay constant,

And the additional constraint is considered

,

,

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The obtained values are .

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,

CHAPTER 11 THE BHALERAO STATISTICAL MODEL

11.1 Unpolarized The description for the nucleon proposed by Bhalerao25 is quite similar to the models already cited, in other words, “a parton gas with massless particles (quarks, antiquarks, gluons) balanced at a temperature in a spherical volume (and consequently, radius )”. Bhalerao considers the proton rest frame and the infinity momentum frame (IMF), moving with . the velocity and the index are The Lorentz factor quantities in the IMF. The particle number density in the phase space is given by:

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(11.1) where is the degenerescence, 16 for the gluons, 6 for the quarks is the four-moment and (antiquarks) of some flavor,

with leads to:

.The Bjorken scale variable is given by

, that

The kinetics limits are , valid when and . Replacing the limits and integrating over the “volumes” in the phase space results:

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70

Chapter 11

(11.2) Note that the quantity represents and has the important feature, in this model, of going to zero when goes to 0 or 1. The differential in this work is the introduction of the finite size corrections (FSC), considered in the work.69 This results in the modification:

(11.3) Considering the usual constraints in the particle number and the sum of the moments, we get the equations:

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(11.4)

(11.5) The only free parameters in the model are and . The joint solution of the system above gives the values of and the chemical potentials. The , , MeV, MeV, results are MeV. The model predicts, for Q2= 4 GeV2, and the proton radio fm.

11.2 Polarized Continuing this work, Bhalerao et al.26 published an adaptation of the model for polarized structure function, using data from different collaborations.70,65-67;71,72;68,74 Using the notation and polarization for the particle number of flavor , the following equations must be obeyed:

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The Bhalerao Statistical Model

71

(11.6) (11.7) (11.8) (11.9) (11.10) (11.11)

(11.12) are the proton data. For the neutrons, we have , , . There is a relation between the chemical potential: ; . Thus, there are seven equations and seven unknowns; the chemical potentials for polarized particles and antiparticles, 03 kind and 02 polarization, and .

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where the values of

Table 11.1: The values of parameters and . The temperature potential are in MeV.

and the chemical

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CHAPTER 12 STATISTICAL QUARK MODEL WITH LINEAR CONFINING POTENTIAL

The statistical quark model presented by Trevisan et al.28 considers all individual quarks of the system, valence, and sea quarks, confined by a central effective interaction, with strength and equal expressions for the scalar and vector components: (12.1) This way, the energy levels of the confined quarks are obtained from the stationary Dirac equation,

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(12.2) Dirac’s In Eq. (12.1) and Eq. (12.2), and are the usual matrices, which can be written in terms of the Pauli’s matrices. With given by 114;115

(12.3) the final coupled equations will be reduced to a single second-order differential equation, (12.4) This equation is solved numerically for the radial part, after partial ), where , the radial part of wave expansion. For wave ( is related by the Airy function (Ai):

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Statistical Quark Model with Linear Confining Potential

73

(12.5) where is the corresponding th root of Ai , the current quark mass, and the energy levels,

,

is

(12.6) For the and quarks with

, the energies are given by (12.7)

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We use equal strength to and quarks. The sum of the first energy level of these quarks results in the mass. This will be corrected to the nucleon mass by introducing instanton interactions, which reduce the amount of energy. Next we consider the Fermi-Dirac distributions. The probability density for a system, with energy levels and temperature , is given by

(12.8) With the appropriated chemical potential ( ) and with the normalized wave function, we can fit the quark flavor number ) in the baryon and the violation of , gives the ( the density probability for each state, level degeneracy and normalized to one, (12.9) In the present work we considered only the light quarks, and , having the corresponding current quark masses given by . The energies were taken to be equal for the and quarks, obtained by using a confining potential model in the Dirac equation. With the above, we obtained the following normalization for the proton (neutron):

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74

Chapter 12

(12.10)

The units are such that the Boltzmann Constant , , and are all set to 1. The thermal masses for the proton ( ) and neutron ( ) were given in terms of the corresponding energy levels, as follows

(12.11)

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In order to calculate the nucleon structure function, for convenience we wrote the wave function in momentum space, taking the Fourier transform

(12.12) Using the null plane variables,

(12.13) where is the momentum fraction of the nucleon carried by the quark, is the nucleon thermal mass at a given temperature , we redefined the wave function Eq. (12.12) as (12.14) By using Eqs. (12.8–12.14), we obtained the quark structure function for each flavor

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Statistical Quark Model with Linear Confining Potential

75

(12.15) where

(12.16) describes the probability of a quark with flavour having a fraction of the total momentum of the nucleon, considering a temperature in , we the thermal model. For the corresponding antiquark distribution, have to replace by in Eq. (12.15). In the model, we consider the result given by the NMC Collaboration,76,77

(12.17) implying that the violation of the GSR is given by

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(12.18) The statistical model of Ref.27 reproduces this result with a temperature parameter adjusted to 108 MeV, with the chemical potentials 135 MeV and 78 MeV. Actually, from the analysis of the E866 experiments 85,86, the value for the violation of the GSR is . This implies a readjustment of the parameters of the same statistical model, such that 103 MeV, 147 MeV, and 88 MeV. The total quantity of the is recalculated to be . With such results, the ratio is constant: and

(12.19) Next, we implement an analysis to fit the experimental data for the . As it will be worked out in the next section, ratios / and

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76

Chapter 12

considering different effects, the thermal model can be used not only to obtain the GSR but also to obtain the cluster distribution in the nucleon.38

12.1 Additional Effects Considered

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12.1.1 Effective light-quark mass shift The difference between the interactions of and quarks in the nucleon is supposed to come from instanton contributions, which are flavour-spin dependent.78 Each quark interacts with the other, of opposite spin and different flavors. In the present approach, we use the quark distributions from the model with linear confining potential. To consider the instanton contribution effectively, we note that quarks have a more attractive channel, with lower energy than that of the quark. In the thermal model, as the initial confining potential is the same, there is no difference between the quark energies. It follows that the ratios and are constant. Once given any quark distribution , a simple mathematical way to implement the above considerations is to introduce a displacement of the distributions over the scale, in such a way that (and corresponding antiparticles) distributions have their respective and maxima at different positions. Note that working with both, particle and antiparticle distributions, one can obtain, at once, all ratios between structure functions. The physical meaning behind the displacement is that the effective potential for and quarks should be different, and also for their antiparticles. state is more energetic than a state In the Fock space, a . The displacement in the scale can be done for any probability distribution. In the following, to verify the effect of such displacement, Trevisan and collaborators chose the simplest distribution, given by Dirac’s delta function (see, for example, chapter 9 of Ref. 80):

(12.20) is a quark mass and is the total mass of nucleon. The where distribution above is valid for and quarks in case they are considered with equal masses. , the new distributions for and are When , such that can be written as shifted in relation to

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(12.21) , the rescaled function will have the maximum So, when shifted to the right of the original function, as shown in Fig. (12.1). For a reasonable fit of the observable, in the present work the authors ( ) for the proton and Md/Mu considered Mu/Md ) for the neutron. ( In Fig. (12.2), you can see the different contributions to the antiquark . The solid line shows the effect of mass rescaling. ratio and are sensitive to the mass As shown, the ratios displacement, which can be related to the effective quark confining strengths. This effect improves the overall fitting of the sea-quark ratios and neutron to proton structure function ratios. u(x)

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d(x)

Figure 12.1: The u and d quark structure functions for the proton in the thermal model, are shown as functions of the Bjorken momentum scale . The calculation for is shown by a short-dashed line. For the , we present two results: the one ) is shown by a solid line; and another given by equal masses ( (with 1.25, shown by long-dashed line) where the maximum is shifted to the right-hand-side.

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78

Chapter 12

 Figure 12.2: The u (x) distributions, without re-scaling (solid line) and after the rescaling (dashed line). Note the difference of where the functions have the maximum value and also the maximum value.

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12.1.2 Gluon Splitting Process In the actual subsection Trevisan et al. considered the perturbative QCD phenomenology: the process of gluon, emitted by quarks, being split in quark-antiquark pairs. Such gluon splitting is a well-studied process.80 or pairs from any quark. So, There are equal chances to originate we can use a generic function that describe this process: (12.22) The number of particles-antiparticles pairs in such phenomenon is given by the Gribov-Lipatov-Altarelli-Parisi equation.12 They first considered a gluon creation from an original quark distribution that, after the splitting of gluons, creates equal quantities of each flavour of light quarks.80–82 The joint probability density to obtain a quark coming from the and at some fixed low is given subsequent decays by

(12.23)

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79

where we use the index to indicate that the quarks are generated by and :80 gluon splitting processes, given by the functions

(12.24) The probability given in Eq. (12.23) is the same for the quark and . Hence, the model has two sea-quark antiquark, such that components, the thermal component, which gives the violation of , and a component that comes from QCD perturbative processes, equal for , as all quarks. This second component, when dominating, makes shown by

(12.25)

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Note that with only the considerations of the Dirac equations and will be constant. This occurs because we gluonic effects, the ratio have the same wave functions for all quarks and anti-quarks, being the difference only in the normalization. How to deal with the difference between the interactions of and quarks of the nucleon in an effective way has been shown in the previous subsection, considering a quark mass shift.

12.1.3 Quark Substructure The quark in this model has effective degrees of freedom that can have substructure. The structure function of the constituent quark/antiquark can , assuming that the be extracted from the pion structure function, asymptotic form dominates the pion light-front wave function. The asymptotic form of the wave function for a massless pion implies a constant probability for the valence quark to have a given momentum fraction (see Ref. 83). The pion structure function can be written as

(12.26) is the pion structure function for constituent quarks from the where valence wave function Ref. 83.

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80

Assuming function,

Chapter 12

for the asymptotic form of the valence wave

(12.27) and deriving in , one gets (12.28) the constituent quark/antiquark structure function. In order to analyze how this substructure of the constituent quark can affect the structure-function, the authors considered the parametrization given in Ref. 84. The structure function of a valence quark in for the pion is given by (12.29)

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where the parameters are

(12.30) structure function in the Using the such parameters, the antiquark nucleon from the constituent substructure is given by

(12.31) is the quark structure function given by the thermal model. where . As we see, such effect is relevant to obtaining a good fitting for In this way, to obtain a better fitting, we need to combine the values of (from the gluon splitting process) with the values of . Without presents a considerable considering such substructure, the difference

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Statistical Quark Model with Linear Confining Potential

81

deviation when compared to the experimental results (see Fig. (12.4)). The correction to this problem is done by considering the substructure of the antiquarks as derived from the pion structure function, using the parametrization given in Ref. 84.

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12.2 Main Results To fit the experimental data, the authors estimated three free , which gives the mass scale parameters: the thermal mass ratio shift; , for the gluon splitting; and , in case that we consider pionic processes as described in the previous section. , shifting the For the first parameter they use in Fig. (12.1), from the original to maximum value of the function . In Fig. (12.2), we show the different contributions to the ratio . Without the rescaling, with the dotted line, you can see the result of the thermal model. With the dot-dashed line, only the gluon contribution. In both cases, the ratio is constant. The ratio is always equal to one for the gluon contribution. With a dashed line, it is possible to observe two contributions: the thermal model and the gluon effect ( 1.72). The solid line presents the rescaled result for the ratio without gluon contributions. In Fig. (12.3) and Fig. (12.4), we have two important results of the model related to the antiquark distribution in the nucleon. In Fig. (12.3), and, in Fig. (12.4), the antiquark we show the antiquark ratio . difference The initial result of the thermal model, with no shift of the distributions and without quark pair contributions, is shown in Fig. (12.3) by the straight (long-dashed) constant line. The experimental data, shown in Figs. 3 and 4, are from Ref. 85, obtained with Q2 =7.35 GeV2. As shown in Fig. (12.3), the best choice for the ratio is obtained with , which implies GeV. This value enters the , calculation of gluon contributions for the quark ratio distribution shown in Fig. (12.5); and also in the neutron to proton ratio distribution, , shown in Fig. (12.6), where Q2=12 GeV2. With the solid line, the authors showed in Fig. (12.3) and Fig. (12.4), the results obtained by considering all the effects: the thermal model, mass shift, gluon splitting, and contributions of pionic formation. We note in Fig. (12.4) that both combined effects from gluon splitting and pion formation are relevant for the best fitting of the difference .

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82

Chapter 12

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, as a function of , are Figure 12.3: Our model results for the antiquark ratio compared with data results obtained from Refs. 85 and 86. Without mass-scaling displacement, we obtain the constant dashed line. The dashed-line curve shows the thermal model with . The solid line shows the model result with and .

In Fig. (12.5) and Fig. (12.6), we see that the model gives a good fit to the data, with the gluonic contributions affecting mainly the low-x region and . of the ratios results, we note that we have two As an additional remark, for different ways to extract the values from the same experimental data, as given in Ref. 87 and 88: on-shell and off-shell. The off-shell values are obtained considering nuclear effects.89 Our results are in an intermediate range between the two calculations.

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83

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Figure 12.4: Results for the difference , as functions of . Model results are compared with data from Ref. 85, scaled to fixed Q2=54 GeV2/c2. With a dashed line, the model result with is shown. The solid-line represents , and .

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84

Chapter 12

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Figure 12.4: The d/u quark-ratio distribution inside the proton is shown as function of x. The model results consider the mass-scaling displacement; without gluon splitting (dashed line) and with gluon splitting (solid line). The model was compared with data from Refs. 87 and 88 and calculations from Ref. 89. The solid circles correspond to off-shell calculations, and the crossed circles to on-shell calculations.

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Figure 12.6: The neutron to proton structure function distribution, , is shown as function of . As in Fig. (12.5), the model results consider the mass shift rescaling; without gluon splitting (dashed-line) and with gluon splitting (solidline). The model is compared to data from SLAC90 and EMC129 collaborations with Q2=12 GeV2.

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CHAPTER 13

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A VERY SIMPLE STATISTICAL MODEL TO QUARK’S ASYMMETRIES

Trevisan and Mirez267 have proposed a simple statistical model with the Fock states being the meson-hadron fluctuations. As expected, an insight into the Gottfried sum rule (GSR) violation emerged. The model also predicted the difference between the strangeness amount in proton and neutron. The violation of the Gottfried Sum Rule7 does not emerge considering only the QCD phenomenology. One popular explanation is the meson cloud around the nucleon – there are more mesons around the proton because the fluctuation is more probable in the Fock than space. 111,40 depends on , but there are The observed violation of the few phenomenological works to explain such behavior. Abbate showed a till =10 GeV2 and a small increase decrease of the difference at = 30 GeV2. Sohaily et al. related the scattering energy to the variables (Temperature, Volume, Chemical potential) of the statistical model by Zhang.29

13.1 Review on the structure function and quark sea asymmetries In that work, the following notation was adopted: is the quark ( ) distribution function inside the proton and, naturally, refers to the neutron case. We also suppose the symmetries below:

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A Very Simple Statistical Model to Quark’s Asymmetries

87

where the new indices and refer to valence quark and sea quark, respectively. The structure function for the proton is given by:

(13.1) and,

(13.2) where S is the contribution of another particles of the sea, without the strange quark. The strange quarks must remain with the index, because this is one of important point of the present work. The antiquarks have the bar and do not need the index. Writing again and we have:

(13.3)

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and

(13.4) Therefore, making the difference (

), and integrating, the result

is:

(13.5) The dependence of on which obtained the Table. (13.1):

was studied by Abatte and Forte,111

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88

Chapter 13

Table 14.1: The amounts of the Gottfried sum rule versus different values of Q2.

In the Fock representation of the hadron, besides the three quarks, the fluctuation with quark-antiquark may be represented as a state, and the combination of a valence quark with an antiquark may originate a mesonhadron state. and , one may have, for Considering the creation of the pairs , the proton, the states:

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with some possible configurations,

and for the neutron

that may be

The difference in the probability of states with and originates the violation of the Gottfried sum rule. The model presented in the next section describes the behavior of such asymmetry and gives an insight about the difference of the amount of strangeness in the proton and neutron.

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A Very Simple Statistical Model to Quark’s Asymmetries

89

13.2 A Simple Statistical Model The phenomena described above were analyzed with a simple statistical model. Each Fock State has an energy level, which is the sum of the masses of the hadrons plus the meson mass, therefore

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Table 13.2: The main fluctuations of the proton that cause the violation of the Gottfried sum rule and have strangeness.

Table 13.3: The main fluctuations of the neutron that cause the violation of the Gottfried sum rule and have strangeness.

The Fermi-Dirac distribution gives the probability some temperature

for each state , at

(13.6) where is the chemical potential for the proton or neutron. As usual

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90

Chapter 13

and the thermal mass is defined as (if states are considered):

The thermal mass is a parameter for the energy system, that is, the mean energy. It’s comparison with the states energies helps us to understand the physical meaning of the model.

13.3 Results

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In Table (13.4) we show the dependence of the statistical weight and the temperature (T). The second column shows the probability of , that is the more important contribution for the and oscillation asymmetry. Also, on the same table, you can see the difference between the sum of the four states for neutron and the four states for proton that have some strange quark contribution. The strange quark contribution to proton and neutron structure functions may have, therefore, different amounts, and the difference between them increases with the energy.

Table 13.4: The dependence of states with the temperature.

Although few states were considered, the main conclusion that comes from the model will not change with further improvements.

13.4 Summary In the work “A very simple statistical model for the quarks asymmetry” some features of the structure functions for protons and neutrons were revised, as their dependence on the energy in the scattering. The asymmetry - increases, then it slightly decreases. It was observed

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A Very Simple Statistical Model to Quark’s Asymmetries

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that there must be a small difference in the amount of the strangeness between the proton and the neutron and that this difference is more evident if the energy rises. To explain these facts in a qualitative way, it was implemented a simple statistical model, with a Fermi-Dirac distribution for the Fock states. The physical explanation for the observed data, that comes from the model is that, with low scattering energy, that is, low temperature, the states that generate the and asymmetry are more frequent, while, if the energy increases, the states with strange particles (or states that do not asymmetry) gain statistical weight. contribute to the Also, due to the small difference of mass between the states with strange quarks in protons and neutrons, there is a small difference between the strange quark contributions. Therefore, the model may explain two different phenomena. As a final remark, it was noticed that the small difference in the strangeness may be multiplied in a nuclear medium, where the number of protons and neutrons may be different, and this may affect some physical properties in the system.

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CHAPTER 14 THE STATISTICAL MODEL BY ZHANG, SHAO AND MA

In 2009, Zhang, Shao, and Ma29 proposed the statistical model described below. The nucleon is composed of free particles in thermal balance. In the nucleon rest frame, the mean number of the particle is given by: (14.1) where

is given by

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(14.2) where is the degenerescence (6 for quarks …) and the chemical potential have the opposite signal for antiparticles. is necessary The on-shell condition for the energy . is the mass of a quark with flavor . It is also and defined: , , and , where is the momentum fraction, at light cone, carried by a parton, and is the nucleon mass. The four-moment is . The integral is rewritten

(14.3) Using the delta function properties, it is obtained:

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The Statistical Model by Zhang, Shao and Ma

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(14.4)

e (14.5) The number particle equation has the form: (14.6) where

(14.7)

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Considering the isotropy for

the integral is solved analytically

(14.8) where

(14.9) and

is the polilogaritmic function, given by:

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94

Chapter 14

Applying the sum rule for the valence quarks and sum of moments, we get the thermodynamical quantities of the model. The results where , with a try temperature the chemical potential and obtained for MeV, we have MeV, volume are obtained. For the case MeV, and the nucleon’s radius fm.

14.1 The EMC effect in this model Continuing this work, the same authors92 have applied the model to study the EMC Effect, the fact that the structure function for an isolated proton is different than that one is in the nuclear medium (see Ref. 92 for the experimental and review papers about this subject). For this end, it was in the deuterium was the same of an isolated proton, and supposed that also for the deuteron structure function:

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(14.10) was assigned, and for another For the deuteron, the temperature . In this model, changing the temperature, the nuclei the temperature also change, as well as the chemical potentials. The ratio values of was fitted to the experimental data, on the kinetic . region inside the This means that, instead of using a constant value to nuclei, the new constraints now are the experimental data for the ratio, the combined value of , volume, and chemical potential that fits the curve. For the deuteron, they used the same experimental value of an isolated nucleon:

according to experimental data.108,109 In the numerical simulations, Zhang and collaborators considered three situations to study: (a) The case only with u, d flavours and gluons, MeV, (b) The case with strange quarks and MeV, (c) The case with strange quarks and

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The Statistical Model by Zhang, Shao and Ma

95

the values for were chosen because they are the suggested upper and down limits in PDG2008.107 Then, the results for the Deuteron are, (a) (b) (c)

MeV, MeV, MeV.

For the nuclei, it is interesting to see the table with the temperature T, Volumes (V and V ), the nucleon radius inside the deuteron and in an atom (r and r ), the chemical potential ( and ).

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Table 14.1: Results for case (a).

Table 14.2: Results for case (b).

Table 14.3: Results for case (c).

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CHAPTER 15 STATISTICAL MODEL WITH Q2 DEPENDENCE

15.1 Valon Model with Q2 dependence

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Sohayli et.al. 40 presented a model where the statistical approach given by Zhang29 is considered to obtain the parton distribution inside a valon.39 In the valon model, the proton is a bound state of three valence quark clusters, so the constituents bear all the momentum of the proton. At high values, the structure of the valon depends on leading order results in the perturbative QCD. From the experimental data on of protons and neutrons, the flavor-dependent valon distributions in the nucleon are determined. The distribution of valons in the proton is independent of , but the distribution of partons in the constituent quarks (the valons) is dependent. In the work by Sohaily et al., the distributions of valons inside the proton are given by

and

where GU/p, GD/p are distributions of and type valons in the proton. The parton distribution in the nucleon can be expressed as

(15.1) and Here G(y) is the valon distribution in the nucleon, are parton distributions in the valon and in the proton, respectively. In Eq. (15.1), and are the momentum fraction of the valon

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Statistical Model with Q2 Dependence

97

(parton) carried by the partons inside the valon (proton). Integrating over the delta function we get

(15.2) So the distributions of and quarks inside the proton are given by

(15.3) and

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(15.4) The distributions of gluons and sea quarks are calculated similarly. Where and are the distributions of quarks inside the and are the distribution inside the valons, proton, and that are given by the statistical model, using Eqs. (15.3, 15.4). In the valon model, the constituent quark is a valence quark surrounded by sea quarks of all kinds and gluons. Therefore, it is possible to find a quark in the sea of the quark, for instance. Then the equation that gives the normalization becomes more complex. For the quark, we have:

(15.5) Two kinds of numerical results were presented in that work: with two , based on the violation of the and three flavors and dependence on Gottfried sum rule, as shown in the Table (15.1):

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Chapter 15

98

Table 15.1: The amounts of the Gottfried sum rule versus different values of Q2

With two flavours, and with the usual sum rules for valence quarks and momentum, besides the violation of the Gottfried sum rule at Q2=4.5 , the results are: MeV, MeV, GeV2, MeV, and the radio fm. With the strange quark MeV, MeV, MeV, included, the results are and fm.

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15.2 Q2 dependence without the valon model Mirjalili et al.41 have also studied how the parameters of the model by Zhang29 changed with the energy. They proposed the following approach for each quantity: (15.6) and got the Table (15.2):

Table 15.2: The numerical values of parameters in Eq. (15.6).

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PART III

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MODELS WITH NON-EXTENSIVITY

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CHAPTER 16

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THERMODYNAMICS WITH FRACTAL STRUCTURE, TSALLIS STATISTICS AND HADRONS

Deppman published a work in Physical Review A that named this chapter. He introduced the thermofractal concept, an application of fractals in thermodynamics. The self-similarity of the thermofractals occurs in the statistical properties of the internal structure. Mandelbrot 228 created the term fractal to designate systems presenting scaling symmetry. The interesting fact is that, for such systems, their dimension, according to the definition by Haussdorf, is not necessarily an integer.181 The concept of a fractal applies to distribution functions, where self-affinity appears. In such cases, there are usually many dimensions associated to the scale symmetry,181,182 and the system is called multifractal. Deppman remembered that the main motivation for introducing a , which would lead to nonextensive statistics, nonadditive entropy, was its applicability to fractal or multifractal systems, since this entropy would naturally lead to power-law distributions, characteristic of fractals. and the Before that, other works183–186 studied the relation between fractals. The main interest of Deppman’s work was the thermodynamical aspects of high-energy collisions. Fermi187 was the first to observe such thermodynamical features. Hagedorn 188 subsequently developed it fifty years ago by supposing a self-similar structure for the hadrons. This was done by the following definition of fireballs: “Fireball is a * statistically balanced system composed by an undetermined number of fireballs, each one of them being, in its turn, a (go to *)”

This definition makes clear the self-similarity of the fireball structure, resulting in a scale invariance typical of fractals, as already mentioned in

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Thermodynamics with Fractal Structure, Tsallis Statistics and Hadrons

101

Ref. 145, 155, 189. Following this reasoning, it’s worth mentioning Deppman, Menezes, and Megias’s work.130

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“The fractal structures can be formed by systems described by the Yang-Mills field theory. The presence of fractal structures lead to a recurrence formula that allows the determination of the effective coupling even in high perturbative orders. A consequence of the fractal structure is that the proper thermodynamical theory for describing the interacting system is the non extensive Tsallis statistics, rather than the traditional Boltzmann-Gibbs statistics, where the entropic index, q, is a measure of the non additivity of the entropy. Here is obtained, for the first time, in terms of the field theory parameters. The fractal dimension is determined as a function of the entropic index. When applied to QCD in the asymptotic approximation, the theoretical valued obtained for is in good agreement with the values found in the analyses of experimental data. Some experimental features can be explained by the theoretical results derived by this proposal, in particular the behavior of particle multiplicity as a function of the collision energy, which depends on the fractal dimension. Although fractal structure could be perceived in any field theory of Yang-Mills type, in QCD its effects are more evident. In the Figure xxx, a diagrammatic view of the Yang-Mills fields is showed."

With the above definition and a self-consistent argument, Hagedorn obtained the complete thermodynamical description of fireballs. The predictions included the limiting temperature and the mass spectrum formula, which allowed comparison to experimental data. A such recursive aspect of the definition was also used by S. Frautschi190 who proposed that hadrons are made of hadrons. With this definition, he derived some of the results obtained previously by Hagedorn. An interesting historical aspect is that Hagedorn’s thermodynamical approach was proposed some years before the quark structure of hadrons became accepted, but it had far-reaching consequences. In 2000 it was shown that simply changing the exponential function in the self-consistent thermodynamics distribution by the q-exponential relation from Tsallis statistics would result in a power-law distribution for which can describe the data in the whole range145,193. In 2012 the self-consistent principle proposed by Hagedorn was generalized by the inclusion of , leading to a well-defined thermodynamical theory when Tsallis statistics replaced Boltzmann statistics.194 The experimental evidence of the applicability comes from the data from HEP. Some authors analyzed it uses the thermodynamical formula derived from Tsallis statistics146,153,195,196,148 or using the power-law formula inspired on QCD.197,198,151,156,199

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102

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Deppman introduced the thermofractals as follows: “Define thermofractal as a class of thermodynamical systems presenting a fractal structure in its thermodynamical description in the following sense”: 1. The total energy is given by (16.1) where corresponds to the kinetic energy of constituent subsystems and corresponds to the internal energy of those subsystems, which behaves as particles with an internal structure. is 2. The constituent particles are thermofractals. The ratio can vary constant for all the subsystems. However, the ratio , which is self-similar (selfaccording to a distribution, affine), that is, at different levels of the subsystem hierarchy, the distribution of the internal energy is equal (proportional) to those in the other levels. 3. At some level in the hierarchy of subsystems the phase space is so narrow that one can consider (16.2) with being independent of the energy .

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For the description of the thermodynamical properties of such a system, the starting point is the Boltzmann factor (16.3) with being the entropy and the Boltzmann constant. Supposing the variations of the volume can be disregarded one has (16.4) so the probability in Eq. (16.3) can be written in terms of the total energy as (16.5) where is a generalized differential. Due to properties 1 and 2 of thermofractals one has (16.6)

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with (16.7) where (16.8) Since is related to the kinetic energy part of the constituent particles, it is reasonable to write, based on the energy distribution, Eq. (16.9),

(16.9) (16.10) and, for the internal energy, it is possible to write (16.11)

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is the probability density for the subsystem internal energy. where Note that due to Eq. (16.8) one has (16.12) so Eq. (16.6) is now given by

(16.13) is an effective number of subsystems. Factors not where depending on or are included in the constant . The thermodynamical potential is given by

(16.14)

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which, after integration on

, results in

(16.15)

16.1 Self-affine solution Now using property 2 it can be imposed the self-affinity in the probability functions by establishing (16.16) Eq. (16.15) and Eq. (16.16) are simultaneously satisfied if

(16.17) where is the number of levels in the subsystem hierarchy according to property 3. Defining

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(16.18) and (16.19) we obtain that

(16.20) which is the well-known Tsallis distribution. Notice that this system presents several entropic indexes depending on the hierarchical level of the thermofractal. In the next section it will be shown that it is possible to obtain a thermofractal with independent of the fractal level.

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16.2 Self-similarity By slightly modifying Eq. (16.15) and writing

(16.21) where is a fractal index, it is possible to impose the identity (16.22) corresponding to a self-similar solution for the thermofractal probability distributions. The simultaneous solution for equations Eq. (16.21) and Eq. (16.22) is obtained with

(16.23) Introducing the index by

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(16.24) and the effective temperature (16.25) one finally obtains

(16.26) which is exactly the characteristic Tsallis -exponential factor of the nonextensive statistics. Eq. (16.26) shows that, instead of the Boltzmann statistical weight, the Tsallis statistical weight given by the -exponential function should be used to describe more directly the thermodynamics of thermofractals. In fact, writing

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Chapter 16

106

(16.27) it follows from Eq. (16.26) that

(16.28) , with representing a which is the Tsallis entropy with discretizated probability based on Eq. (16.26). Notice that the change is necessary due to the different definition of the q-exponential used here (see, for instance, Ref. 158). This result is in agreement with the findings in Ref. 186, where it is shown that self-similarity in fractal systems are described by Tsallis statistics. Note that from the equations Eq. (16.24) and Eq. (16.25) one has (16.29) showing that the entropic index is related to the ratio between the Tsallis temperature and the Hagedorn temperature .

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16.3 Discussion In order to make the structure of the thermofractal clear it is interesting to analyse what happens when one considers the first level after the initial one in the fractal structure. From Eq. (16.21) one has

(16.30) where (16.31) and that , one can see that Considering that the term between brackets is the internal energy distribution. Considering the internal energy is distributed statistically among the constituent

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Thermodynamics with Fractal Structure, Tsallis Statistics and Hadrons

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subsystems, and considering that they are independent of each other it is possible to write (16.32) and (16.33) and corresponding to the energy and the probability density for with the th subsystem, respectively. Due to properties 1 and 2 of thermofractals, all density distributions are identical, since here the self-consistent solution is under consideration2. Therefore Eq. (16.30) can be written as

(16.34)

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The kinetical energy can be written in terms of the individual subsystems, as described above in the case of an ideal gas, resulting in

(16.35) . with being the kinetical energy of the th subsystem, with Notice that the term between square brackets represents the internal energy distribution of one subsystem of the original thermofractal. Therefore, according to property 1, the subsystem is also a thermofractal, can be separated into two parts, and due to property 2, its energy , with being the kinetic energy of the components of the subsystem and their internal energy. Then

2

For the self-affine solution a similar reasoning can be applied.

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(16.36) The equation above shows that it is possible to factorize the probability distributions of each subsystem, and it explicitly shows that each of them has an internal energy distribution that has the same form of the original system, according to Eq. (16.21). In Eq. (16.26), is a normalizing constant, which gives (16.37) The average energy of the thermofractal is then

(16.38) resulting in

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(16.39) From Eq. (16.8) and the mean value for it results that

(16.40) Considering also Eq. (16.29) it is possible to observe that, while the temperature regulates the average energy of the system, the temperature regulates the ratio between the kinetic energy , and the internal energy . it is possible to write the ratio Defining

(16.41) and using Eq. (16.40) the following is obtained

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(16.42)

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which represents the ratio between the internal energy of one of the thermofractal constituent subsystems and the total energy of the main fractal. Tsallis statistics approach Boltzmann It is known that as statistics, so it is interesting to analyse the thermofractal in that limit. Due also , and from Eq. (16.25) one notice that to Eq. (16.24), as there are two ways to get this limit: one by letting and the other keeping constant. If the Boltzmann limit is not obtained. In fact in this case one has , as in the case of the self-affine solution, but with independent of the hierarchical level. This is possible only for corresponding to the trivial case of a thermofractal with energy . This also indicates that the self-similar solution is not a special case of the self-affine solution, but represents a different system. The Boltzmann limit is obtained if is constant, what means that remains constant as , therefore . Hence the Boltzmann limit is obtained if almost all energy of the gas appears in the form of kinetical energy of its constituents. In this case the system is insensitive to the subsystem internal energy, thus behaving as an ideal gas that can be described by Boltzmann entropy.

16.3.1 Thermofractal dimensions Haussdorf Dimension Consider a hypothetical experiment where the energy of the thermofractal is measured with resolution . This means that energy fluctuations smaller than can be neglected, defining in this way the level of the thermofractal structure where the subsystems internal degrees of freedom can be ignored, according to property 3 above. The level is such that , so

(16.43)

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The Haussdorf fractal dimension 181,182 is determined by considering that when the energy is measured in units of the total energy scales as while the energy of each subsystem scales as which gives (16.44) is the number of boxes necessary to completely cover all where subsystem energies of a thermofractal. It follows the well-known relation

(16.45) Since at the level all subsystems have distinguishable energies at the given resolution then is the number of subsystems at this level, i.e., . From here it follows

(16.46)

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Fractal spectrum There are several parameters that characterize multifractals and, in the following paragraphs some of those multifractals parameters will be investigated. Among these quantities, the Lipshitz-Hölder mass exponent and the fractal spectrum are the most used.182 In this context the for the event is related to the mass exponent by probability (16.47) where is the linear dimension of the basic box in which the phase space is partitioned. The partition function is (16.48) This partition function is also written in another form (16.49)

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Thermodynamics with Fractal Structure, Tsallis Statistics and Hadrons

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with (16.50) so that (16.51) where3 (16.52) . The function is the using, for the sake of simplicity, multifractal spectrum. Let us consider the thermofractal which presents a probability density given by Eq. (16.26). In order to avoid confusion with the symbols used , with . One has for probability we will indicate it by (16.53) so the probability to find particles in the box with dimension around is

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(16.54) with (16.55) It follows that the mass exponent,

, is

(16.56) Using Eq. (16.43) it results

(16.57) 3

The usual notation is but here it is made use of the Tsallis temperature and the entropic index.

to avoid confusion with

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The fractal spectrum is related to the number of boxes with the same index . Therefore, consider the probability (16.58) Now the number of boxes with dimension interval is given by the relation

corresponding to the (16.59)

Using this result in Eq. (16.58) and considering Eq. (12.8), it is obtained (16.60) From the equation above one can see that (16.61) with

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(16.62) Applying Eq. (16.43) one gets

(16.63) The comparison between equations Eq. (16.56) and Eq. (16.63) results in (16.64) Note that this result was already expected from the multifractal dimension theory.181,182 Also corresponds to the Haussdorf dimension given in Eq. (16.46). In the limit it results

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(16.65) The calculations performed here are valid everywhere but for the case of corresponding to the lowest range of probabilities, which is indicated by . Due to the asymptotic behaviour of the probability density one so and also the number of boxes has , hence . But since the probability does not diverge one has (16.66) . therefore The Lipshitz-Hölder exponent is given by Eq. (16.51). With the results obtained so far one has (16.67) so

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(16.68) The exponent below.

can be observed experimentally, as discussed

16.3.2 Thermofractals and Hadrons Before considering using the thermofractal to get some knowledge about the hadron structure a few comments are needed. In the construction of thermofractal formalism, antisymmetrization was not taken into account. The effects of antisymmetrization, however, are expected to be small188,190 since the phase space is sufficiently large to consider the hadronic states of interest as a dilute gas. Another aspect is that the treatment used here is semi-relativistic, with the energy of the particles calculated as (16.69)

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114

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where the internal energy is identified with the subsystem mass, . This may be a good approximation when the temperature is small so that is sufficiently larger than . The formalism derived in the last section is very general even though it has been motivated by the definitions of hadrons given by Hagedorn188 and Frautschi.190

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Hadron Fractal Dimension To calculate the fractal properties of hadrons one needs two parameters and that characterize the hadronic thermodynamics, namely, the ratio the entropic index . These values have been thoroughly investigated in analyses of distributions from high energy collisions, 195, 146, 153, 196, 153 148 in an analysis of the hadronic mass spectrum, and the comparison of the thermodynamical calculations with LQCD data.202 The values found and .202,158 are Proceeding to calculate the thermofractal properties one has, using Eq. , and using it results . From (16.29), equations Eq. (16.42) one has . Finally, using Eq. (16.46) give us that , so from Eq. (16.65) too. can be observed experimentally through the The exponent intermittency in experimental data, which has been studied in many works on high energy collisions.206,207,204,205,209,208,210,211 Intermittency allows a direct measure of that exponent and has been used as an indication of fractal aspects in multiparticle production. The value calculated here is in fair agreement with the results of analyses of experimental data in hadronhadron collisions,212–216 which range between and . At this point, we put a few words to explain what is intermittency. Let’s imagine some measure of rapidity distribution in scattering is taken, for the rapidity interval. If another with the experimental resolution , the resulting distribution will not measure is taken, with the interval reproduce the expected statistic pattern of the first and will be different because some phenomena (such as quark-gluon plasma and gluon splitting) occur and affect the distribution. This means that a better resolution shows the time evolution of the phenomena. The agreement described above needs to be discussed in more detail. The analysis of intermittency is made through a sophisticated methodology that was developed some decades ago to extract fractal parameters from experimental data 204–207 and has been applied since then to study mainly data from heavy ion collisions in emulsion.218–220;217 But aside from the technical difficulties, there is the unavoidable problem

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described in Refs. 221;222 where it is shown that when multiple fractal sources are present the measured intermittency is weaker than the real fractal dimension would imply. Experimental data where it was supposed fewer sources tend to present stronger intermittency effects when measured with the available technique. This may explain the fact that the intermittency in nucleus-nucleus collisions, which is 0.97, is much weaker than that from hadron-hadron or collisions, which is 0.4.212 The fair agreement found between the calculation and the experimental values indicates that the thermofractal proposed here can indeed give a reliable description of the fractal aspects of multiparticle production. In addition, it can show that the intermittency found in HEP data is related to the fractal structure of the hadron. It is the hadron structure that leads to the nonextensive self-consistent thermodynamics194 as the proper thermodynamical description of the hadronic systems. The study of intermittency has been used to show multifractal aspects in the cascade dynamics behind multiparticle production. The dynamical cascade is connected to complex QCD diagrams which would describe the entire particle production process.224;223;225 Here we show the connection between intermittency and Tsallis statistics. However, a direct connection to the scattering dynamics ruled by QCD is possible, as shown below. S-matrix and entropic index

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Another important result for thermofractals is that the thermodynamical potential for the self-similar solution

(16.70) can be written in the form

(16.71) where Eq. (16.7) was used and

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(16.72) is the potential function for a non-interacting gas. Writing the potential in this form allows a direct comparison with the Dashen, Ma and Bernstein226 formula connecting thermodynamics and microscopic information on the interaction among the particles composing the gas, which appears in terms of the scattering matrix, S, in

(16.73) where the index indicates that the trace is performed for the connected diagrams in the Feynman-Dyson expansion. Direct comparison of Eq. (16.71) and Eq. (16.73) gives

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(16.74) which is a relation establishing constraints in the S matrix which will allow the interacting gas to show up nonextensive features. Eq. (16.74) relates the matrix to the entropic factor, allowing one to extract information on the microscopic interaction from the non extensive behaviour of the experimental distributions.

16.4 Conclusions The main conclusions of the Deppman’s work are listed below: •

• •

The work introduces a system that has a fractal structure in its thermodynamical functions, which is called thermofractal. It is shown that its thermodynamics is more naturally described by Tsallis statistics. A relation between the fractal dimension and the entropic index, , is found. The ratio between the Tsallis temperature, and the Boltzmann temperature, , is related to the entropic index and to the number of subsystems, , in the next level of the fractal structure.

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Thermodynamics with Fractal Structure, Tsallis Statistics and Hadrons

• • •

• •

117

It is shown that while regulates the system energy, regulates the fraction of the total energy that is accumulated in as internal energy of the subsystems. The study of the self-similar thermofractal reveals that it is a fractal with dimension determined by and . The Lipshitz-Hölder exponent is calculated in terms of , and . Assuming that hadrons present a thermofractal structure, the relevant values for the calculation are obtained from the analyses of distribution and from the observed hadronic mass spectrum, was already found in a work comparing the while the ratio thermodynamic results to the LQCD data. The comparison between the calculated fractal dimension and the value obtained from the analysis of intermittency in HEP experimental data shows a fair agreement. For a system of interacting particles presenting thermofractal structure it is found a relation between the entropic index and the -matrix for the particle interaction. This result allows, on one hand, to connect the entropic index to fundamental aspects of the interaction between the constituents, and, on the other hand, it establishes constraints on the -matrix to allow the emergence of non extensivity in the corresponding system.

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Deppman and collaborators presented, later, a second work that was focused on partition function and entropy.202 The main ideas and proposals of this work are described in the next chapter.

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CHAPTER 17 FRACTAL STRUCTURE AND NON-EXTENSIVE STATISTICS

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A. Deppman and collaborators investigated the properties of the fractal thermodynamical system and proposed a diagrammatic method for calculating the relevant quantities related to such a system (see “Fractal structure and nonextensive statistics”).252 They showed that a system with the fractal structure described there presents temperature fluctuation following an Euler Gamma Function, by previous works that evidenced the connections between those fluctuations and Tsallis statistics. Finally, the scale invariance of the fractal thermodynamical system is discussed in terms of the Callan-Symanzik equation. In the review of this work, we repeated some parts of the preceding chapter (mainly formulae) to make the understanding easier.

17.1 Motivation Deppman et al.252 cited four connections between Boltzmann and Tsallis statistics that are: 1. The cosmic rays; C. Beck145 considered a generalized statistical mechanics model for the creation process of cosmic rays that takes into account local temperature fluctuations. 2. The nonlinear Fock-Planck equation for diffusion, studied by L. Borland.251 3. Grzegorz Wilk and Zbigniew Wodarczyk investigated the multiplicity fluctuations observed in high-energy nuclear collisions attributing them to intrinsic fluctuations of the temperature of the hadronizing system formed in such processes. To account for these fluctuations, they replaced the usual Boltzmann-Gibbs (BG) statistics with the nonextensive Tsallis statistics characterized by the nonextensivity parameter q, with being a direct measure of fluctuation.

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4. The thermofractal concept, introduced by Deppman.

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On the work described in this chapter, the authors have detailed the analysis of the fourth of those connections, where a system featuring fractal structure in its thermodynamic properties, named thermofractals,142 has been shown to follow Tsallis statistics. These fractals are relatively simple systems: they are objects with an internal structure that is like an ideal gas of a specific number of subsystems, which are also fractals of the same kind. The self-similarity between fractals at different levels of the internal structure follows from its definition and reveals the typical scale invariance. Thermodynamical systems with the thermofractal structure show fractional dimensions,142 another feature shared with fractals in general. The fractal dimension can be related to the fact that the system energy is proportional to the power of the number of particles, this power being different from the unit. It is interesting to note that the concepts discussed in the preceding chapter and this chapter (the relation among fractals, self-similarity, and non-extensive statistics) may be generalized and have applications in another field (a possible relation is in the stock market). The main results in the work revised in this chapter are: (a) The temperature of the fractal system addressed here fluctuates according to the Euler Gamma Function, a kind of temperature fluctuation already associated to Tsallis statistics; (b) the scale invariance of thermofractals in terms of the CallanSymanzik equation, a result that may be of importance for applications in hadron physics.

17.2 Fractals and Tsallis Statistics From a mathematical point of view, the basic difference between , which is an Boltzmann and Tsallis statistics is the probability factor, exponential function of energy in the case of Boltzmann statistics, and in the nonextensive statistics proposed by Tsallis is a function called qexponential, given by

(17.1)

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120

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where is associated with the temperature, is the Boltzmann constant, is a normalization constant, is the so-called entropic factor, which is a measure of the deviation of the system thermodynamic behavior from the one predicted by the extensive statistics. The partition function is a basic quantity used to calculate all thermodynamic properties of a system. It is defined as

(17.2) is the density of states. The probability of finding the system where at an energy between and is, accordingly, given by (17.3) For simplicity, here we will use the quantity

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(17.4) which is, obviously, identical to the unit. Define as a particular density of states characteristic of such a fractal system. The main characteristic of the fractal system142 of interest here is that , which can be written in Boltzmann statistics as

(17.5) results to be equivalent to the integration over all possible energies of the -exponential function, that is,

(17.6) It means that, for systems with a particular density of states, Tsallis statistics can substitute Boltzmann statistics while all the details of the internal system structure are ignored. In particular, this system presents a fractal structure in some thermodynamic quantities and, consequently, it

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shows an internal structure with self-similarity, i.e., the internal components are identical to the main system after rescaling. The importance of this result is two-fold: in one way, it allows us to understand the emergence of nonextensivity; in the other, the applicability of Tsallis entropy becomes evident, with the entropic index, , being given by quantities well defined in the Boltzmann statistics; on the other hand, the structure obtained resembles in many ways strongly interacting systems, where Tsallis statistics describes well the experimental distributions.153;167–169 The particular fractal structure that leads to Tsallis statistics has a density of states given by (17.7) where and are independent quantities and part of the total energy, , is such that

. The remaining

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(17.8) The exponent in Eq. (17.7) is a constant that will be related, in the to the Tsallis distribution. following, to the entropic index, and Notice that the phase space corresponding to a variation is given, in terms of the new variable, by , since the two variables are independent. Substituting Eq. (17.7) in Eq. (17.5) it follows that

(17.9) and . Observe that now we have with integrations on the independent variables and . It will be clear in the next section that the integration in is equivalent to an integration on the compound system momentum and that the integration on is related to integration over the energy of a given component of the system, namely, its subsystems. is It is straightforward to verify that reduces to Eq. (17.1) if itself a -exponential. Defining

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(17.10) substituting Eq. (17.7) into Eq. (17.5) and integrating the last equation in , it will result in Eq. (17.9) when the following substitutions are made:

(17.11) with these substitutions the density distribution results to be (17.12) Comparing Eq. (17.4) and Eq. (17.9) one can see that the energy is equal to the probability density . distribution of the system, Hence the energy distribution of the system follows the same distribution of the energy distribution of the compound system internal energy, i.e.,

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(17.13) This result shows that some properties of the main system are also found in its compound systems. This is the self-similarity property, that defines a system with a fractal structure. The system described by the density of states given by Eq. (17.7) is a fractal.142 Moreover, the distribution given by Eq. (17.12) is the well-known Tsallis distribution, hence we can conclude that using Tsallis statistics whole complexity of the fractal system is taken into account in a more simple way, since from the nonextensive entropy associated with that statistics all thermodynamic properties emerge from the usual thermodynamic relations.170;106

17.3 Fractal Structure The results obtained in the last section show that the system with the density of states given by Eq. (17.7) presents self-similarity, allowing one to interpret it as a fractal system. In this section, such structure will be analyzed, and it will be shown that such a system is a fractal in the energymomentum space. Notice that Eq. (17.9) can be written as

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(17.14) The most evident aspect of a fractal structure is its scale invariance. For the system studied here, it means not only that the self-consistency relation represented by Eq. (16.22) must be valid, but also that, for the kinetic energy , the distributions must be the same at all levels of the fractal structure. From Eq. (17.7), it follows that the distribution for is

(17.15) which represents a Maxwellian distribution of energy. Therefore, the scale invariance of thermofractals will be accomplished with the requirement that the kinetic energy distribution and the internal energy distribution are invariant under a scale transformation, so

(17.16) and

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(17.17) remains constant, hence

(17.18) was used to refer to Here and in what follows the upper index quantities of the initial level of the thermofractal structure, or main system, to refer to quantities for the -th level of the structure. and upper index , and The energy of the initial thermofractal, or main system, is . the temperature of the internal structure to this level is It is interesting to express the scaling properties in terms of the fractal dimension, which is one of the distinguishing properties of fractals, and expresses the fact that some quantities do not scale as one could naively expect from the topological dimension of the system. In the present case, as it was shown in Ref. 142, energy and particle multiplicity do not increase in the same way, different behavior from that found in an extensive ideal

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124

gas. In fact, in Ref. 142 the subsystem energies obey a geometric ratio given by:

(17.19) where

(17.20) is the fractal dimension. Here is the ratio between the internal energy of a subsystem and that of its parent system is given in terms of the parameters and by

(17.21) The internal energy distribution scales by a factor

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(17.22) defining the quantity

, and

(17.23) Therefore, fractals with different internal energies present energy distributions that are similar and scale with the internal energy of the subsystems, that is, (17.24) Remarkably, as all energies are rescaled, it also happens that be rescaled, therefore one has

has to

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Fractal Structure and Non-Extensive Statistics

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(17.25) with determined by the equations Eq. (17.11) and Eq. (17.23). Thus the argument of the -exponential function in the probability distribution does not change when we move from one level of the system to its next level. This is the essence of selfis the self-similar distribution. similarity, and Another interesting feature is that (17.26) In what follows the structure of the system and its subsystem just described will be investigated in detail. For the sake of clarity the symbols (17.27) is dimensionless. will be used. Note that Due to property 2 of thermofractals one has at the level fractal structure

of the

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(17.28) is the total kinetic energy of the compound fractals and is their total internal energy. The following normalized energies will be adopted:

where

(17.29) and

(17.30) corresponding to the level of the fractal structure. Note with that the normalized energies are dimensionless and scale invariant. Given a fractal with non extensive temperature , the subsystem , fluctuates according to the distribution energy,

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Chapter 17

126

(17.31) and generalizing Eq. (17.9) to any subsystem level (see Eq. (17.86) in the Appendix).

one can write

(17.32) represents the energy distribution of a constituent fractal at the -th subsystem level of the main system. correspond to the kinetic energy of the -th constituent fractal Let at the -th level of the fractal subsystem structure, each one having an . Eq. (17.32) can be written in internal energy determined by terms of the kinetic and internal energy of each constituent subsystem fractal, since

(17.33) Copyright © 2024. Cambridge Scholars Publishing. All rights reserved.

Note that

(17.34) therefore, also the constant

scales as (17.35)

being the constant for the main system. This result is with consistent with the temperature scale in Eq. (17.23) and with the energy scaling relation in Eq. (17.24). It results that (17.36) with the level

the normalized total internal energy of the thermofractals at . Of course,

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(17.37) The term

can be written in terms of

as

(17.38) that would sum since it is related to the number of possible states . The delta function, here, indicates that up the total energy

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is equal to the sum of the energies and interval between With these definitions one has

, which is to be found in the .

(17.39) where

(17.40) and

(17.41)

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Chapter 17

128

Observe that the integrations inside brackets are performed on the variables corresponding to the subsystem level . In Eq. (17.39) the relation Eq. (17.34) was used for writing in place of . Since

(17.42) being the mass of the -th constituent fractal. One can identify with the mass with the internal energy of the fractal subsystem, so that , following that

(17.43) where (17.44)

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Then Eq. (17.39) results in (see Appendix, equations Eqs. (17.77) (76) and Eq. (17.86))

(17.45) . In Eq. (17.45) the potential is described where entirely in terms of the characteristics of the compound thermofractals and at the -th level of the subsystem fractal structure, with related to their kinetic and internal energies, respectively. But

being

(17.46) and

are independent of , so it results in

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(17.47) The self-similar relation present in the subsystem fractal structure can be more apparent if Eq. (17.47) is written as

(17.48) , the same where it is possible to recognize, in the term expression as in Eq. (17.28), what allows the extension of calculations to include quantities of the next subsystem level in the fractal structure, i.e., level , since (17.49) and (17.50) Copyright © 2024. Cambridge Scholars Publishing. All rights reserved.

In addition, due to Eq. (17.9),

(17.51) what allows the passage to the next subsystem level by following all the steps described above. Before going into further calculations, however, a diagrammatic description will be introduced.

17.4 Diagrammatic Representation It is possible to have a diagrammatic representation of the probability densities that can facilitate calculations of and other relevant quantities. In Fig. (17.1) the basic diagram symbols are presented, adopting

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Chapter 17

for simplicity. Each of the basic diagrams corresponds to a mathematical expression, and the correspondence can be established as follows:

Figure 17.1: Basic diagrams for the fractal structure: ( ) main fractal; ( ) vertex; ( ) final fractal.

1. A line corresponds to a term

and , where with the fractal represented by the line. 2. A vertex corresponds to the term

(17.52) is the total energy of

(17.53) 3. To each final line, i.e., those lines that do not finish in a vertex, the associated term reads

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(17.54) The simplest diagram of interest is a line with a vertex where each branch is a final line. In this case the diagram scheme results in

(17.55) Delta functions can be included to fix energy and momentum of some of the fractals at any level. As an example, consider the graph shown in Fig. (17.2). Observe that there are two levels of the subsystem structure, the initial fractal has well defined momentum (it is indicated by ), and in the second level one of the subsystems has well defined energy and momentum. Such a diagram gives

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the probability to find a constituent subsystem fractal at the third level of the initial fractal . According to the diagrammatic rules one has

Figure 17.2: Example of a tree graph representing the different levels of a fractal.

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(17.56) where determines the kinetic part of the fractal indicated by at the second level. It is also possible to consider the subsystem fractal structure in the fractals with energies opposite way: given , the probability that they form varying in the range and is a single fractal with energies given by (17.57) and . This result is a direct consequence with of the fact that thermofractals are in thermal equilibrium. After integrating on one obtains (17.58) showing the consistency of the fractal description introduced in the present work.

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132

Figure 17.3: The same diagram of Fig. (17.2) represented as a linear graph.

This is possible by rearranging terms in the summation of different contributions and using the merging property of thermofractals. The process described in Eq. (17.57) corresponds to fractal subsystems merging into a single one. In the example given above and described in Fig. (17.2)., the final system generated from the lower branch at the first level can be merged into a single fractal. The tree diagram can then be reduced to a linear diagram, as shown in Fig. (17.3), resulting in a simpler expression for the probability calculated in that example. In this case the result is

(17.59)

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17.5 Temperature fluctuation in thermofractals On the right hand side of the last equality in Eq. (17.47) the distribution of the kinetic energy of the thermofractals at the th level is given by (17.60) with

(17.61) is the scaled temperature at the th level of the thermofractal. where However, at the subsystem level there are thermofractals, and each of them presents different internal energies. One could, therefore, associated to the thermofractal at the previous write the temperature level. Then Eq. (17.60) can be written as

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(17.62) for each thermofractal found inside a thermofractal at level

, with

(17.63) Suppose now that at the th level the internal energy fluctuations are already small enough to be disregarded and the internal energy is a constant . Then, according to the diagrammatic rule 3 of thermofractals, subsystem level the energy fluctuation of the th thermofractal at the is proportional to the kinetic energy fluctuation, that is,

(17.64)

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. But the product of Gamma functions above is itself a where Gamma function, as described in the Appendix, resulting

(17.65) . Since the thermofractals at the th subsystem level are with being considered as structureless particles, the subsystem at level can be considered as an ideal gas of particles with masses . The parent thermofractal at level is, therefore, formed by thermofractals, each one considered as an ideal gas of particles but at and with total energy . The probability density different temperatures to find a set with total internal energy energy is then

(17.66)

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If, at this stage, one still disregards the thermofractal subsystem structure, the kinetic energy F can only be interpreted as a parameter, while the system energy is the only quantity that keeps some physical meaning, besides the temperature that now fluctuates inside the system. When this step is performed, the equation above is interpreted as a Gamma , that is, distribution of the inverse temperature

(17.67) The distribution of temperatures, as described by Eq. (17.67), was already considered in connection to the Tsallis distribution in a different context.139,140 On the other hand, the possibility of an equilibrated system with temperature fluctuation is rather controversial.172–175 In the present work such fluctuations are well defined in association with the fractal structure of the thermodynamics functions of the system analyzed. Temperature fluctuations arising from a multi scale system were already analyzed in Ref. 167.

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17.6 Callan–Symanzik equation for thermofractals Deppman and collaborators showed some connections between thermofractals description to quantum field theory. They showed that the Callan-Symanzik equation may be applied to thermofractals. The simplest diagrammatic representation of the thermofractal evolution from one level to the next level corresponds to a vertex with an initial system , (as described by diagram characterized by energy and momentum subsystems with in Fig. 1b), at an arbitrary level generating and . Such diagram leads to such that

(17.68) Here, the passage from one level to the next subsystem represents only an alternative description of the same system. However, one can consider that the initial thermofractal can break into pieces, each one being a thermofractal. Let be a coupling constant that gives weight to a transition from one subsystem level to another one, then one can write

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(17.69) and the term

(17.70) is, then, can be considered as an effective coupling constant. understood as a vertex function that is scale-free. Vertex functions that are invariant under scale transformation can be described by the CallanSymanzik equation, which played a fundamental role in the determination of the asymptotic freedom in Yang-Mills theory. A thermofractal version of the such equation was already derived in Ref. 179, and it will be derived here in a different way. works, as seen above, as a The thermofractal temperature scale parameter that determines the fractal structure of the subsystem at a certain level, so one can write the factor in terms of the subsystem temperature by using Eq. (17.22), i.e.,

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(17.71) , for the sake of scaling it will be assumed Since , which is a good approximation for sufficiently high. It results that the vertex function is

(17.72) Notice that, when the scale transformation on energy and momentum and , the distribution remains is performed, so that is invariant, therefore it can be left out of the scale unchanged since invariance analysis of the vertex function studied here. Taking this aspect into account and introducing for the sake of simplicity, the scale invariance of the vertex function is expressed by

(17.73)

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Chapter 17

where it made use of the scaling property of thermofractals. From the above expression, it is straightforward to conclude that

(17.74) (17.75) and with these results one can write

(17.76) where is the anomalous dimension for thermofractals, a result equivalent to the one obtained in Ref. 179. The fact that thermofractals satisfy the Callan-Symanzik equation indicates that if it is possible to describe such systems through a field theoretical approach, the Yang-Mills theory is the appropriate framework for it. These results, therefore, set the grounds for a more fundamental description of thermofractals in terms of gauge field theory.

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17.7 Final Remarks Therefore, besides giving consistency to the relation between fractals and Tsallis Statistics, the work by Deppman improved the knowledge about this subject by showing how to do a diagrammatic scheme to thermofractals and relating them with the Callan-Symanzik equation.

17.8 Appendix A: Useful Formulae The energy distribution of an ideal gas is given by

(17.77) and is the normalization constant. If the momenta of where the different particles are independent, then

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(17.78) Since

(17.79) the normalization constant must be chosen as (17.80) Notice that the integration in Eq. (17.78) can be performed in terms of the total momentum dimension . Then

, by considering a hypersphere of

(17.81) where

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(17.82) is a surface factor for the -dimensional hypersphere, with Euler Gamma Function. , then But

being the

(17.83) hence

(17.84) Therefore, from Eq. (17.81) one has

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138

(17.85) Substituting the relations for

,

and for

, it results in (17.86)

17.9 Appendix B: The Callan Symanzik equation The author Jay Wacker explained the meaning of the Callan-Symanzik equation as follows (see https://www.quora.com/What-is-the-significanceof-the-Callan):

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17.9.1 N-Point Functions The most conceptually challenging thing to start with is the notion of an “N-point” Green’s function. These N-point functions are essentially how strongly N-particles interact with each other. So you can have a 3point function between an electron-positron and a photon which describes how strongly a photon interacts with an electron. The most common N-point functions to consider are the 2-point, 3point and 4-point functions. The 2-point function describes one particle coming in and one particle coming out. This turns out to be where things like the mass, the index of refraction, and decays can be read off. 3-point and 4-point functions describe basic interactions of particles with each other. Higher point functions are typically constructed from lower point functions since these higher point interactions tend to be constructed out of lower point functions.

17.9.2 Quantum N-Point Functions These N-point functions are interesting on their own for classical physics, but new and bizarre features start to happen when these N-point functions are computed in quantum mechanics (usually Quantum Field Theory). Quantum Mechanical N-point functions become dependent on how you measure it. So at shorter distances, particles may interact more strongly or weakly with each other. The Callan-Symanzik equation is the equation that describes how these N-point functions change depending on how you probe them.

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139

17.9.3 Callan-Symanzik Equation’s Significance

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In terms of significance, “the Callan-Symanzik equation is the background for the Renormalization Group Equation which is arguably one of the most important concepts from modern physics. It shows that the constants of nature are not constant and, instead, should be thought of as dynamical entities. The renormalization group equation is why the strong force becomes strong and how all the disparate (non-gravitational) forces of nature may unify into a single entity – a Grand Unified Theory”.

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CHAPTER 18 THE NONEXTENSIVE THERMODYNAMIC MODEL – TSALLIS TEMPERATURE

Trevisan and Mirez35 presented a contribution considering the adapted Fermi-Dirac and Bose-Einstein distribution to nonextensive thermodynamics, proposed by Tsallis.113,112 The theoretical motivations of these works were to suggest the nonextensive statistical mechanics as an appropriate basis to deal with physical systems with strong correlation dynamics, long-range interactions, and memory effects. They used the steps of Bhalerao,25 without considering the finite size corrections yet.

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18.1 Fundamentals of the nonextensive statistics Let’s consider two statistically independent subsystem, and and we call individual probability density

and

, with

(18.1) the joint probability density of a composite system , the nonadditive is summarize in the relation116,117 (nonextensive) character of (18.2) , the third term in the right-hand side of Eq. (18.2) in the limit vanishes and we recover the additivity (extensivity). A possible physical interpretation to consider the Tsallis statistical can be extracted from the valon model, where each valence quark, with the quark-antiquark sea and gluons around them, is named the valon.38,30

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The Nonextensive Thermodynamic Model – Tsallis Temperature

141

18.2 The model In this model, each quark (antiquark) may be considered a subsystem with some entropy. As the quarks (antiquarks) interact among them, the total entropy for the system may be given by Eq. (18.2), such that one . could explore the possibility of having It is possible to obtain the associate quantum mean occupation number of particles species in a grand canonical ensemble, for a given energy level . For a dilute gas of particles and for small deviations from the ), it can be written as 118,119 standard statistics (

(18.3) for the case

, with

(18.4)

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. In the above, , and we use for for the case for bosons. fermion and The corresponding Fermi-Dirac and Bose-Einstein distributions are . recovered with

18.3 Results of the Model In the present model, the authors combined the values of , , and and found the chemical potentials , for the quarks e , in a set constrained by the normalization equations and the sum of the moments, including gluons. The temperature is in the interval (MeV), and in the . These parameters were set because within these interval ranges reasonable values for the nucleon radius and the asymmetry are obtained in the numerical simulation. Therefore, there are many possible ways to combine the variables and and observe the physical features and interpretations. and difference In Fig. (18.1) and Fig. (18.2), the ratio are shown, respectively, for the proton and neutron, being compared with available experimental data. The Figures are generated

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142

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using the parameters MeV, line); and

Chapter 18

MeV, and

and fm (dashed fm (solid line).

Figure 18.1: Model results for of the neutron to proton structure function as a function of , compared with experimental data from Refs. NMC,120,76,121,77 E665,122 EMC,91 BCDMS123,124 and SLAC.90,129 The dashed line is for T=55 MeV, q = 0.90; the solid line is for T=35 MeV, q = 1.03.

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143

Figure 18.2: Model results for of the difference of the proton to the neutron structure function as function of x, are compared with experimental data from Refs. NMC,120,76,121,77 EMC,91 BCDMS123,124 and SLAC.90,129 The dashed line is for T=55 MeV, q = 0.90; the solid line is for T=35 MeV, q = 1.03.

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CHAPTER 19 THE POLARIZED NONEXTENSIVE STATISTICAL MODEL

In the previous chapter, a work by Trevisan and Mirez was revised. To justify such procedure, we remind that the nonextensive features occur when there are strong correlations or interactions among the parts. As a natural sequence, they considered the case of polarized structure-function. In section II of that paper, the model was applied considering the available experimental data. Section III showed the main results, and section IV the conclusion.

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19.1 The Model In the work of Trevisan, the steps of the formalism considered in Refs.25;26 were followed, but instead of the usual Fermi-Dirac or BoseEinstein distributions, the adapted version applied for the nonextensive thermodynamics was used, as done in Ref. 35. The parameters of the model are the radius ( ), the temperature ( ) and the factor -from Tsallis, besides the chemical potentials for each kind of particle. These parameters must be constrained by the experimental data. If denotes the number of quarks (antiquarks) of flavor and spin parallel (anti-parallel) to the nucleon spin, then the Parton Distribution Functions (PDF) in the proton has to satisfy the following seven constraints: (19.1) (19.2) (19.3) (19.4)

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145

(19.5) (19.6)

(19.7) and similarly for the neutron. We use , , for the proton and , , for the neutron. These values are the same used by Bhalerao et al. 25;26, based on the experimental data available from the following collaborations: E142,70 E143,67 E154,71;72 SMC,68 and HERMES.74 in the infinite-momentum frame The parton number density in the nucleon rest frame are related to each (IMF) and the density other by:

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(19.8) where the superscript refers to the IMF, is the nucleon mass and is the parton energy in the nucleon rest frame.25 This is a general relation connecting the two frames; the only assumption made is that of massless partons, which is common in deep inelastic scattering formalism. For each particle , we have:

(19.9) where is the Bjorken variable and is the spin-color degeneracy factor. is the nucleon volume and is the probability distribution, which is given by:

(19.10) for the case

and

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146

Chapter 19

(19.11) . for the case , we have In these expressions, in units of Boltzmann constant , where is the temperature; and we have for fermion and for bosons. The corresponding Fermi-Dirac and Bose-Einstein , where remains as a free distributions are recovered with parameter in this model, is the chemical potential for each kind of quark. The following relations among the chemical potentials are used to solve the system: (19.12) and (19.13)

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19.2 Results In the model discussed in this section, we have to combine the values of , and and find the chemical potentials in a set constrained by the normalization equations Eq. (19.1) to Eq. (19.6), with sum of moments Eq. (19.7). The obtained results for the asymmetries of the quarks and the sum of the momenta of the quarks depend on the values of the variables , and . The asymmetries of the antiquarks is a consequence of using different chemical potentials for up and down quarks. The interesting range for the temperature is within the interval MeV, with within the interval . Within these ranges, reasonable values for the nucleon radius and for the asymmetry in the nucleon sea are obtained in the numerical simulations. Therefore, there is a great number of possible ways to combine the variables and , which should be followed the corresponding interpretations, in view of the physical features. In order to adjust the model, the experimental results by violation were used: Towell et al.229 for the Gottfried sum rule7

(19.14) where

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147

(19.15) This implies that the numerical results near Eq. (19.14) must be investigated in more detail. On the next tables, some combinations are shown. One important feature of the present work is that there are no isospin symmetries, that is, proton and neutron may have a small difference in properties, such as strangeness content and other quark-antiquarks quantities. Therefore, within our assumptions, (19.16) (19.17) and (19.18)

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( ) is corresponding to all anti (anti ) quarks inside the where proton. However, it is important to make clear that Eq. (19.14) is an initial reference for the numerical values. In fact, without any kind of symmetries, the violation leads to: (19.19) The features for proton and neutron are analyzed at the same temperature because the experimental process to take neutron measures usually have to involve protons together. The physical meaning of this choice is the assumption that there is a thermal equilibrium between them (in the case of taking the measure using deuterium). Under such condition, we present the results obtained for the variable and for the radius. and the difference , we have Moreover, to obtain the ratio to combine the structure functions obtained at the same temperature. On Table (19.1) and Table (19.2), we have selected 2 cases with MeV for the proton, and 1 for the neutron. By considering MeV, we selected 2 cases for the proton and 3 for the neutron.

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148

Chapter 19

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Table 19.1: Table showing different combinations of the variables , and , that keep the momentum constraint and the calculated antiquark asymmetries.

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Table 19.2: Table showing different combinations of the variables , and , that keep the momentum constraint and the calculated antiquark asymmetries.

The more interesting case is happening for MeV because the radius and are also the same for the proton and neutron. The resulting pictures about this case are exposed below. The chemical potentials are given on Table (19.3).

Table 19.3: The chemical potentials for the proton and the neutron, for the case MeV, and fm.

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Chapter 19

150

By considering different values for the strange contribution to the polarizations in the proton and neutron, an estimation about the amounts of strangeness for each nucleon is obtained, with the small asymmetry on the strange quark quantities estimated:

(19.20) In a nucleus, with a very different number of protons and neutrons, this small asymmetry may have cumulative and important effect.248 Moreover, this combination of parameters gives the best approach to all relevant quantities calculated in the model, such that

(19.21)

(19.22)

(19.23)

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and

,

,

, (19.24) (19.25) (19.26)

The experimental results are in the Table (19.4).

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151

Experimental Values of the Studied Parameters Parameter SG ī1p ī1n Value 0.244 ± 0.045111 0.1530 ± 0.0061(LO)46 í0.058 ± 0.018(5 GeV) 71

Parameter a0 a3 a8 Value 0.278 ± 0.057(LO)46 1.269 ± 0.00346 0.586± 0.03146 Table 19.4: Experimental values, and the references. The includes all the is obtained possible deviations, and the energy is GeV, but the violation of with energy between GeV and GeV.111

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In the Figures below, we have the model results compared with some Fig. available experimental data, for the the difference Eq. (19.2) and (19.1) and for the polarized structure function Eq. (19.3).

Figure 19.1: The difference

. The experimental data are from Refs. 245, 244.

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152

Chapter 19

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19.3 Conclusion In the revised work here, the results of a model which considers the Tsallis distribution for the fermions, to describe the polarized structure function of the nucleons, were presented. In the adjustment of the model with available experimental data, no kind of symmetry in the quark sea is supposed a priori. The hypothesis of equal temperature for the proton and neutron is justified, with comparisons among structure functions (polarized and unpolarized) made under such a rule. After fixing the temperature, the authors tried different combinations for the parameters (from Tsallis) and (radius) in the proton and the neutron and verified the sum rules. This feature of the model allows intuitive insights into the hadron phenomenology. The best results were obtained in the case where the parameters were MeV, and fm. By comparing this work with Ref. 35 it is possible to verify a small improvement in the radius, now closer to the usually accepted radius for the nucleons (in that work, MeV, fm and ). Another important point is the little difference obtained for the amount of strangeness in proton and neutron. This difference may have cumulative and relevant effects in the nuclear media248 and deserves further investigation. Therefore, this work confirms the plausibility of nonextensity in the statistical properties of nucleons and stresses the attention to the lack of symmetry in the number of sea particles in protons and neutrons. The recent improvements in experimental data can easily be incorporated into a numerical approach.

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. The

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Figure 19.2: The polarized structure function for the proton experimental data are from Ref. 67 (E143).

153

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154

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Figure 19.3: The polarized structure function for the proton The experimental data are from Ref. 67 (E143), 71; 72 (E154), and 70 (E142).

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CHAPTER 20 ON THE DIFFERENCE BETWEEN THE RADII OF GLUONS AND QUARKS

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20.1 Introduction Trevisan, Mirez and da Silva published a work in a special issue of MPDI physics (see https://doi.org/10.3390/physics3040073), devoted to Statistical Approaches in High Energy Physics (see https://www.mdpi.com /journal/physics/special_issues/sahep). This journal is open, and we reproduce its main parts here. “An important issue in Quantum Chromodynamics (QCD) is how much momentum gluons carry in nucleons. Some authors suppose that gluons carry out almost half of the momentum in the nucleons31,231, while others say this must be about 30%.232 The theoretical predictions also have variations, between near 50% and 20%.234,235 On this subject, many models were proposed to describe the structure function of the nucleon, among them, the statistical models.” In Refs. 27 and 28, the variables fit better by a system of equations with the usual sum rules for valence quarks, and the violation of the Gottfried sum rule (GSR) for instance,85 is an additional result. The GSR violation may be used as information to fit the temperature. Another sum rule is the sum of momenta of all particles, quarks, and gluons, which must be equivalent to 1. In Refs. 27 and 28 Trevisan and Mirez considered the effects of nonextensivity by Tsallis113,106 to study the nucleon structure function. These works have three main variables: the temperature , the of Tsallis, and the radius . The model proposed in Refs. 27 and 28 was an adaptation of the model by Bhalerao25,26 to the non-extensivity. The -exponential replaces the usual exponential in the distributions. The non-extensivity describes some situations in which the sum of the entropies of two independent subsystems A and B that have individual and , respectively, implying an additional probability densities,

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156

Chapter 20

term that depends on a variable , and not a sum of the two entropies. In the additivity (extensivity) is recovered. That is, the the limit entropy of the joint system is different from the sum of the entropy of subsystems but with some interaction. Let us denote the and entropy of the A and B system, respectively; then:

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(20.1) ( , etc.) will be discussed in The exact form of the function Section 20.2. The use of Tsallis statistics is justified by considering the valon model38,30. The valon is considered to consist of the valence quark, sea quarks, and gluons around. This subsystem, with some entropy, interacts with another valon. Therefore, the sum of the entropies reflects this fact. Each cloud of quarks and gluons around the valence quarks overlaps the neighbors so that the particles are indistinguishable (in sense that one cannot tell to which valence quark the particles belong to). In addition, the radius of each valence quark with its sea is considered the radius of the nucleon. Recently, Deppman253 gave a theoretical justification to use the Tsallis distribution to study QCD, and Cardoso et al.254 studied the thermodynamical variables in the MIT bag model considering the non-extensivity. Here, a new study, in which the momenta of gluons and quarks are computed separately, obeying, of course, the sum rule. Therefore, new independent variables occur, the radii of quarks. The paper is organized as follows. In Section 20.2, the model of 27,28 is discussed and modified to consider the sum of momenta of quarks and gluons separately. Section 20.3 gives results and discusses those. Section 20.3 provides studies with fixed temperature (and variable and radius), variable temperature (fixed and radius), and variable radius ( and temperature are fixed). The conclusions and final remarks are given in Section 20.4.

20.2 Theory and Methods In the present picture, the nucleon is considered a spherical bag with a radius . Inside the bag, there is a gas of massless partons (quarks, antiquarks, gluons) in equilibrium at temperature . In the nucleon center of mass frame, each parton can charge a maximum of half of the total momentum. The other half is shared by the

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On the Difference between the Radii of Gluons and Quarks

157

other partons. Thus, the mass (energy) that a parton can bring is at most half the mass of the nucleon. Therefore, MeV is used in what follows. The main point of this study is to use a different radius for quarks and gluons. Besides this, consider that partons are point particles that form the gas inside the spherical bag with the radius of the nucleon. Following Bhalerao,25 we consider the infinite momentum frame (IMF) to obtain the particle distribution functions (PDFs) for quarks and gluons. Below, two frames are considered, namely, the proton rest frame and IMF, both moving with velocity ( 1) along the common axis. The interest of this study lies in the limit, when the Lorentz factor, . The dependence of the particle number on the Bjorken variable, , is263,1

(20.2) for each particle . Here, is the spin-color degeneracy factor —16 for the gluons (see, for instance, Ref. 255), and 6 for the quarks (antiquarks)—of is the nucleon volume, and is the energy some flavor , probability distribution, which is given by

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(20.3) for the case kind of quark, and

, where

is the chemical potential for each

(20.4) . In these expressions, (in units of for the case is the temperature, and (+1) stays for Boltzmann constant, ), where fermions and ( 1) for bosons. The corresponding Fermi–Dirac and Bose– , where remains a free Einstein distributions are recovered with parameter in this model. There are some studies about how to apply the non-extensivity to fermions and bosons2 and references therein. Parvan and Batacharya developed2 the fermions and bosons distributions of the transverse momentum using the grand-canonical ensemble. If one expands Eq. (20.3) and compares it with Equation (56) (the thermodynamical distribution

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Chapter 20

158

part) of Ref. 2, a close similarity of the present ansatz is observed with a more rigorous study. The only difference is focusing on the longitudinal momentum in the present study. From Eq. (20.2), one obtains particle distributions, , and the total number of particles, . Following Refs. 263 and 1, one gets the following set of equations and constraints: (20.5) (20.6) (20.7) (20.8) (20.9) (20.10)

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(20.11) In these equations, ( ) means the number of up quarks, with positive (negative) spin; denotes up anti-quark with positive orientation; and similarly for other (down, , strange, ) quarks. is the difference between the sum of the up quark and up anti-quark with positive polarization and the negative polarization sum; and similarly for other cases. The following relations among the chemical potentials are used to solve the system: (20.12) and (20.13) We use the proton

, , and , , for the neutron. Equations Eqs. (20.5) and (19.2) interchange for the neutron. The experimental data can be obtained for E142,70 E143,67 E154,71,72 SMC,68 and HERMES74 collaborations. for

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On the Difference between the Radii of Gluons and Quarks

159

Besides, the main proposal of the present study is to calculate the radius of gluons and quarks separately; therefore, Equation Eq. (20.11), which still holds, splits into two equations:

(20.14) and

(20.15) according to Ref. 232. Of course, the values may vary according to the reference or the model. Eqs. (20.14) and (20.15) can be written as follows:

(20.16) and

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(20.17) The results with different combinations for the variables (temperature), (nucleon radius), and (the Tsallis parameter for non-extensivity) are given in Section 20.3 just below.

20.3 Results In the present study, the quark and gluon radii are varied in order to obtain the appropriate momentum. Recent studies232 show that the gluons carry about 30% of the momentum. This value is close to one obtained in previous studies.27,28 As usual, the violation of the GSR is an additional condition to be achieved. The experimental data229,257,272 provides with the values in the interval [0.09, 0.15]. Therefore, one can consider

(20.18) Moreover, the following convention will be used:

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(20.19) and

(20.20)

20.3.1 The Model with Constant Temperature

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Here, the variation of the radius with the is studied for the case of two different temperatures. The momentum of quarks is about 72%. Once the temperature is set, the radius and are varied. For these variations, many solutions may exist. Fig. (20.1) shows the dependence of the radius on the nonextensivity factor . The radius is seen to decrease with almost linearly, and is smaller than 1. This result means that with more gluon interactions, the system called valon (a valence quark with the corresponding quark-antiquark and gluon cloud)38 interacts more with another valon, changing its internal entropy.

Figure 20.1: Variation of the radius, , with the factor of Tsallis statistics with two different temperatures, . Dashed lines are for gluons, and solid lines are for quarks. Red lines are for MeV and blue lines are for MeV.

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On the Difference between the Radii of Gluons and Quarks

161

Each point in the quark’s line of Fig. (20.1) has five chemical potentials. Only those potentials are chosen which satisfy the sum of momenta and violation of the GSR; see Table (20.1), MeV. Table 20.1: The chemical potentials for the points of the blue continuous line of Fig. (20.1). Each point is identified by the Tsallis non-extensivity parameter, , and the radius, . The sum of all quarks’ momenta is about 0.72, and the valence quarks’ momenta sum is about 0.48. The difference - is around 0.14.

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20.3.2 Variable Temperature, and Radius Fixed In this subsection, the value of is set to , and the quark radius is taken to be fm. The dependent variables are the sum of the quarks’ momenta, the momenta of the valence quarks, and the violation of the GSR. With the and chosen, the obtained values for the dependent variables stay within the experimental ranges. Many combinations of values of and are possible. The aim here is always to observe the behaviour of the model and not to establish numerical parameters. Fig. (20.2) shows the dependence of the quarks’ momenta (total and valence) with the temperature , which varies from MeV to 34 MeV. In Table (20.2), some experimental results of the quarks’ mean momenta are given. fraction and the scattering energies

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162

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Figure 20.2: The dependence of quarks’ momenta on temperature in the range – MeV. The blue line represents the sum of all quarks’ momentum. The red line represents the valence quark momentum.

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Table 20.2: The mean momentum, , carried by quarks ( at different scattering energies, .

) and gluons (

)

Based on Fig. (20.2), one can conclude that the sum of the quarks’ momenta increases with the temperature, and that the contribution of the sea quarks increases faster than that of the valence quarks. Moreover, with a linear extrapolation and considering Table (20.2), it is possible to relate the temperature and the scattering energy . The linear approximation in Fig. (20.2) is given by the equation:

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On the Difference between the Radii of Gluons and Quarks

163

(20.21) . with the determination coefficient Then, based on the linear approach given by Equations Eqs. (20.21), and Table (20.2), it is interesting to relate momentum, temperature, and . Table (20.3) puts together the information about the mean quark’s , and the temperature that momentum fraction, the scattering energies comes from Eq. (20.21). Table 20.3: The momentum carried by quarks and gluons with different scattering energies and the corresponding temperature, according to Eq. (20.21).

A relation between the temperature and the transferred momentum is obtained to read:

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(20.22) with the determination coefficient (20.3).

. Fig. (20.3) represents Table

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Figure 20.3: The dependence of the temperature fm.

on

for

and

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If the table from Ref. 256 is taken into account, with the same relation for temperature and total momentum for the quarks, one has the following linear relation: (20.23) . The physical meaning of such an approach is an with increase of nucleons’ internal energies when there is an inelastic scattering, which causes heating. Some results using relation Eq. (20.23) are shown in Table (20.4).

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On the Difference between the Radii of Gluons and Quarks

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Table 20.4: The momentum carried by quarks and the corresponding temperature, . This table is based on the table from Ref. 256 and using Eq. with different (20.23).

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In Fig. (20.4), the violation of the is studied as a function of –34 MeV. As expected, the difference between and temperature, increases with temperature. The fact that the difference grows with temperature is understood as more energy allows more particles to be created, and the Pauli’s principle makes the difference become larger.

Figure 20.4: The dependence of the difference on the temperature for

and

(see Equation Eq. (12.17)) fm.

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20.3.3 Variable Radius, and Temperature Fixed

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Here, the value of is fixed to , and the temperature is set to MeV. The dependent variables are the sum of momenta and the violation of the GSR. The combination of the values of and used keeps quark momenta to be in the experimentally obtained range. Fig. (20.5) shows the variation of the sum of the quarks’ momenta with the radius. The contribution of valence quarks tends to a constant, while the sea quarks component demonstrates fast increase.

Figure 20.5: The change of quarks’ momentum with the radius, given in fm. The blue line represents the sum for all quarks, and the red line the sum for the valence quarks only for q = 0.97 and T = 35 MeV.

From Fig. (20.6), one concludes that the violation of the GSR is related to a large meson cloud.

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On the Difference between the Radii of Gluons and Quarks

Figure 20.6: The dependence of

167

on the radius for q = 0.97 and T = 35 MeV.

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20.3.4 Variable , Temperature and Radius Fixed In this subsection, the dependence of the momentum of quarks with is studied. The temperature is kept at MeV, and the radius is set to fm. With these values for and , the momenta and the GSR violation lie in the experimentally verified range for a small variation of the value. In Fig. (20.7), the quark momenta fraction is shown dependent on the Tsallis variable . One can observe that with higher non-extensivity, the sum of the quarks’ momentum decreases. This indicates that the nonextensivity is related to more gluons, which are self-interacting.

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Figure 20.7: The dependence of the quarks’ momentum on for T = 35 MeV and R = 2.0 fm. The red line represents the sum of the valence quarks, and the blue line includes the sea quarks.

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is shown depending on the factor In Fig. (20.8), the difference . It increases if the non-extensivity decreases.

Figure 20.8: The difference

as a function of for T = 35 MeV and R = 2.0 fm.

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On the Difference between the Radii of Gluons and Quarks

169

Table (20.5) shows some values for the radius of gluons and quarks in the model. As predicted, gluons are “larger” than quarks. Table 20.5: The radii ( ) and momentum sums ( ) of gluons ( ) and quarks ( ) along with the temperature, , the Tsallis variable, . The corresponding differences, (see Eq. (12.17)), are also shown. The gluons momentum is considered of about 46% of the total momentum.

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20.4 Conclusions In the present paper studies concerning the variables of the nonextensive statistical model for the nucleon structure function were given. The variations of the nucleon temperature, the radius, and the parameter of Tsallis statistics were studied. The main objective is to verify that the volume occupied by quarks and gluons may be different. To this end, the momentum carried by quarks are investigated within the existing models. It was found that not only the radius can be modified but also the temperature, . The temperature was lower than that obtained in the previous model27,28 (about 40–60 MeV) to satisfy the sum of the quarks’ momenta. to be close to the experimental data. In these studies, the quarks’ momentum is about 80%. Higher temperatures mean higher energies and more quark-antiquark creation. This increases the total momentum of the quarks. It is possible to notice that the same dependence occurs with the 256 . The linear relationship between and and scattering energy and the quarks’ total momentum are obtained. The radius is obtained to be larger (in comparison with previous results27,28) to fit the violation of Gottfried’s sum rule. The large volume needed to obtain this effect is due to the meson cloud around the valence quarks. Another point to explain is the radius obtained in the present model. Thermodynamical models usually have a radius larger than 1 fm, and some corrections are used. For instance, Bhalerao’s study has the finite-size correction, and Mac–Ugaz’s264 paper uses the perturbative correction. The thermodynamic/statistical model with effective confining potential265,266 also may correct this problem. The initial hypothesis that the gluons gas must occupy a bigger volume than the quarks gas is confirmed.

Mirez, Tarrillo, Carlos Alberto, and Luis Augusto Trevisan. Statistical Models for Nucleon Structure Function, Cambridge Scholars Publishing, 2024. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/wisc/detail.action?docID=31136544. Created from wisc on 2024-04-11 00:44:35.

170

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The q parameter is below 1 in the present model. This result is different from that obtained in Ref.3 or 4. Those works take measures in the context of quark-gluon plasma, while here, the goal is to obtain the results of the deep inelastic scattering to obtain the structure function. Moreover, the present study is an improvement of an extensive model, which makes the result near 1. It is worth mentioning that there is another 141 approach that obtained in the context of a heath bath. To consider the gluon polarization is an interesting point to be studied soon. In this case, the set of Eqs. (20.5)–(20.11) needs to get one more equation to take this into account.

Mirez, Tarrillo, Carlos Alberto, and Luis Augusto Trevisan. Statistical Models for Nucleon Structure Function, Cambridge Scholars Publishing, 2024. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/wisc/detail.action?docID=31136544. Created from wisc on 2024-04-11 00:44:35.

CHAPTER 21

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NUCLEAR EMC EFFECT IN NON-EXTENSIVE STATISTICAL MODEL

Trevisan and Mirez describe the nuclear EMC effect using the proton structure functions obtained from the non-extensive statistical quark model. We remind that such a model has three fundamental variables, the temperature , the radius, and the Tsallis parameter . The combination of different small changes allows good agreement with the experimental data. Another interesting point of the model is that it allows phenomenological interpretation, for instance, with constant and changing the radius and the temperature or changing the radius and and keeping the temperature. In 1982, the European Muon Collaboration at CERN discovered that nucleons inside a nucleus have a remarkably different momentum configuration as expected, i.e., the structure-function for a bound nucleon differs from that for an isolated one significantly,79 which was named the nuclear EMC effect. In order to account for the EMC effect, there have been many efforts and insights implemented in various models, e.g., the cluster model,94 the pion excess model,95 the x-rescaling model,96 the Q2-rescaling model,97 the nucleon swelling model,98 and the deconfinement model.99 The statistical idea applied to polarized and unpolarized structurefunction also applies to the nuclear EMC effect. See, for instance, the work by Zhang et al.100 and references therein. Angelini and Pazzi101 introduced thermodynamical analysis to the EMC effect, and utilizing the ratio of valence quark distributions at different temperatures and confinement volumes, fit the data well. Afterward, Li and Peng102 discussed the EMC effect using the Fermi-Dirac distribution for fermions and the Bose-Einstein distribution for gluons. Further, Roïzynek and Wilk103 combined nuclear Fermi motion with statistical effect104 to account for the rise at large . Therefore, to study the EMC effect considering the non-extensive statistics seems to be a natural improvement, with an interesting physical appeal.

Mirez, Tarrillo, Carlos Alberto, and Luis Augusto Trevisan. Statistical Models for Nucleon Structure Function, Cambridge Scholars Publishing, 2024. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/wisc/detail.action?docID=31136544. Created from wisc on 2024-04-11 00:44:35.

Chapter 21

172

In a model that considers only two flavors for the quarks, the following system of equations holds, in the proton case: (21.1) (21.2) (21.3) where is the number of the quark “ ”. These equations constrain the chemical potentials. To calculate the structure functions , we use the definitions:

(21.4)

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Then, the proton structure function is

and in the same way to the proton structure function in a nuclear medium . There are three free parameters, the temperature , the radius , and of Tsallis statistics. If one is changed, the another usually needs to be too, in order to keep the constraints. The proposal is to check what variable is more relevant to cause the changes in the structure-function of the nuclear medium. We show the cases: (a) (b) (c)

MeV (fixed value), with with fm; (fixed value), MeV with MeV with fm; fm (fixed value), MeV with MeV with .

fm and fm and and

It is noticeable that the ratio between the structure functions is calculated considering the smaller radius to the free nucleon and the bigger one to the nucleon in a nuclear medium (cases (a) and (b)) and in case (c), MeV represents the isolated nucleon.

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By observing the picture Fig. (21.1), it is possible to conclude that small changes in the radius are the most important factor to explain the EMC effect.

Figure 21.1: , in the range of EMC Effect (0.0 to 0.8), in the nonextensive statistical model considering the cases explained above. Experimental data are from Gomez et al, with q2=5 GeV.110

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CHAPTER 22 OVERALL COMPARISONS AND CONCLUSION

22.1 Main Results from the models and comparison to the experimental data

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To make the comparison between each model’s results and the experimental data easy, these are put together on Table (22.1) and Table (22.2). You may see that some works have used the available experimental data to fit the parameters of the model, as in Bhalerao.25 On Table (22.1), the main results of the unpolarized models are disposed altogether, in a way that it is possible to compare the parameters and results of each one. On the works by Angelini and Pazzi, Cleymans and Thews, and Mac and Ugaz, there is no violation of the Gottfried sum . In the work by Devanathan et al., there is a dependence rule, so and, therefore, of . of the temperature on

Mirez, Tarrillo, Carlos Alberto, and Luis Augusto Trevisan. Statistical Models for Nucleon Structure Function, Cambridge Scholars Publishing, 2024. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/wisc/detail.action?docID=31136544. Created from wisc on 2024-04-11 00:44:35.

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Overall Comparisons and Conclusion

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Table 22.1: Table showing a comparison among the models. B-L means the results by Bickerstaff and Londergan,15 D-K-G the results by Devanathan et al.16 Notice means no antiquarks asymmetries. The NNPDF is the neural that network parametrization of all available data.111

Mirez, Tarrillo, Carlos Alberto, and Luis Augusto Trevisan. Statistical Models for Nucleon Structure Function, Cambridge Scholars Publishing, 2024. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/wisc/detail.action?docID=31136544. Created from wisc on 2024-04-11 00:44:35.

176

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On Table (22.2), the results and parameters of the polarized models are shown, so the comparison between the models and the experimental results is possible. The model presented by Devanathan and McCarthy, in the same way as that by Devanathan et al., has parameters depending on . On the table, the initialism G-D-R refers to the work by Ganesamurthy et al.,17 BSB refers to the work by Bourrely et al.,19 and D-M to Devanathan and McCarthy’s work.18

Mirez, Tarrillo, Carlos Alberto, and Luis Augusto Trevisan. Statistical Models for Nucleon Structure Function, Cambridge Scholars Publishing, 2024. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/wisc/detail.action?docID=31136544. Created from wisc on 2024-04-11 00:44:35.

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Overall Comparisons and Conclusion

177

Table 22.2: Comparison among the models with polarization and the experimental data. Bhalerao has used the different experimental available results for proton and neutron. In the present table, the data are from Refs. 70, 65, 66, 67, 71, 72, 73, 68 and 46. In column , for the experimental results, we put the value of in GeV2. Blr(p) means Bhalerao results for proton and Blr(n) for neutron.

Mirez, Tarrillo, Carlos Alberto, and Luis Augusto Trevisan. Statistical Models for Nucleon Structure Function, Cambridge Scholars Publishing, 2024. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/wisc/detail.action?docID=31136544. Created from wisc on 2024-04-11 00:44:35.

178

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22.2 Final Remarks, Conclusions and some open questions In this book, the statistical/thermodynamic models for estimating nucleon structure functions, both polarized and unpolarized, were reviewed. The reader can see that these models are based on fermion and boson distributions. The confinement is given by the MIT bag model. A common feature of the model is the temperature, usually around MeV (few models obtained different results). The proton radius obtained in such a model was about twice the expected value in most cases (an exception to the work of Bhalerao,25 with the Finite Size Correction). This failure may be a consequence of the approximation of continuous energy levels, even for valence quarks, which should receive a different treatment. The chemical potential has to be different for quarks and , to obtain normalizations, and this difference leads to antiquark asymmetries. In the polarized case, an additional chemical potential term is introduced and the Bjorken sum rule is considered. Moreover, the temperature is not easy to estimate by experimental methods, contrary to the radius. The Thermodynamical bag model for nucleon structure functions16 shows an interesting proposal that the scattering heats the nucleon, so the observed temperature is not fixed. Therefore, there is a scale dependence. The scale dependence was also proposed in the works by Angelini and Pazzi. In this case, it is interesting to construct new models with a relation and . On the Table (22.1), the main results of the unpolarized between models are disposed altogether, in a way that it is possible to compare the parameters and results of each one. On the works by Angeliniand Pazzi, Cleymans and Thews, and Mac and Ugaz, there is no violation of the . In the work by Devanathan et al., there Gottfried sum rule, so and, therefore, on . is a dependence of the temperature on In conclusion, such kind of models help us to understand important features of nucleons’ structure with the simple and reasonable basic principle, but there are some new open questions to be solved and some improvements that must be done in these models: 1. The endpoint problem. Considering the relation for the Bjorken , in the model with discrete energy scale, levels, the structure-function does not tend to zero as . Therefore, the present formalism is not appropriate to use with the model with discrete energy levels. 2. Recent experimental results showed the resonance phenomena and the quark hadron duality at intermediate values of energy scattering

Mirez, Tarrillo, Carlos Alberto, and Luis Augusto Trevisan. Statistical Models for Nucleon Structure Function, Cambridge Scholars Publishing, 2024. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/wisc/detail.action?docID=31136544. Created from wisc on 2024-04-11 00:44:35.

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. The statistical model based on the Fermi-Dirac and Bose-Einstein, with continuous energy levels, does not predict such behavior. 3. There is a growing interest in the applications of no-extensive thermodynamics to nuclear physics. The concept of thermofractal introduced by Deppman needs further development to describe the structure-function. 4. The transversal momentum distribution (TMD) has received attention in the last few years. Therefore, it is interesting to study the possible relation between the statistical features and the TMD.

Mirez, Tarrillo, Carlos Alberto, and Luis Augusto Trevisan. Statistical Models for Nucleon Structure Function, Cambridge Scholars Publishing, 2024. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/wisc/detail.action?docID=31136544. Created from wisc on 2024-04-11 00:44:35.

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Mirez, Tarrillo, Carlos Alberto, and Luis Augusto Trevisan. Statistical Models for Nucleon Structure Function, Cambridge Scholars Publishing, 2024. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/wisc/detail.action?docID=31136544. Created from wisc on 2024-04-11 00:44:35.

Statistical Models for Nucleon Structure Function

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Mirez, Tarrillo, Carlos Alberto, and Luis Augusto Trevisan. Statistical Models for Nucleon Structure Function, Cambridge Scholars Publishing, 2024. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/wisc/detail.action?docID=31136544. Created from wisc on 2024-04-11 00:44:35.

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Bourrely, C., Jacques Soffer and Franco Buccella, “Recent tests for the statistical parton distributions”, Modern Physics Letters A, 18, number 11, (2003), 771-778, doi:10.1142/S0217732303009861, http://www.worldscientific.com/doi/abs/10.1142/S0217732303009 861 Bourrely, C. and Jacques Soffer and Franco Buccella, “Strangeness asymmetry of the nucleon in the statistical parton model”, Physics Letters B, 648, number 1, (2007), 39–45, doi:10.1016/j.physletb.2007.02.063, http://www.sciencedirect.com/science/article/pii/S037026930700295 Bourrely, Claude and Soffer, Jacques, “Statistical approach for unpolarized fragmentation functions for the octet baryons”, Phys. Rev. D, 68, issue 1, (2003), page 014003, doi:10.1103/PhysRevD.68.014003, http://link.aps.org/doi/10.1103/PhysRevD.68.014003 Soffer, J., “New developments in the statistical approach of parton distributions”, ArXiv:hep-ph/0606109v1, (2006) Soffer, J., “New developments in the quantum statistical approach of parton distributions", ArXiv0912.0496v1, (2009) Bhalerao, R.S, “Statistical model for the nucleon structure functions", Physics Letters B, 387, number 4,(1996),881, doi:10.1016/0370-2693(96)01206-3, http://www.sciencedirect.com/science/article/pii/0370269396012063 Bhalerao, R.S. and N.G. Kelkar and B. Ram, “Model for polarized and unpolarized parton density functions in the nucleon”, Physics Letters B, 476, number 34, (2000), 285–290, doi:10.1016/S03702693(00)00158-1, http://www.sciencedirect.com/science/article/pii/S03702693000015 81 Trevisan, L.A. and Frederico, T. and Tomio, L., “Strangeness content and structure function of the nucleon in a statistical quark model”, The European Physical Journal C – Particles and Fields, 11, number 2, (1999), 351-357 doi:10.1007/s100529900162, http://dx.doi.org/10.1007/s100529900162 Trevisan, L.A. and Mirez, C. and Frederico, T. and Tomio, L., “Quark Sea structure functions of the nucleon in a statistical model”, The European Physical Journal C, 56, number 2, (2008), 221–229, doi: 10.1140/epjc/s10052-008-0651-1, http://dx.doi.org/10.1140/epjc/s10052-008-0651-1 Zhang, Y. and Lijing Shao and Bo-Qiang Ma, “Statistical effect in the parton distribution functions of the nucleon”, Physics Letters B,

Mirez, Tarrillo, Carlos Alberto, and Luis Augusto Trevisan. Statistical Models for Nucleon Structure Function, Cambridge Scholars Publishing, 2024. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/wisc/detail.action?docID=31136544. Created from wisc on 2024-04-11 00:44:35.

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Mirez, Tarrillo, Carlos Alberto, and Luis Augusto Trevisan. Statistical Models for Nucleon Structure Function, Cambridge Scholars Publishing, 2024. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/wisc/detail.action?docID=31136544. Created from wisc on 2024-04-11 00:44:35.

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Mirez, Tarrillo, Carlos Alberto, and Luis Augusto Trevisan. Statistical Models for Nucleon Structure Function, Cambridge Scholars Publishing, 2024. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/wisc/detail.action?docID=31136544. Created from wisc on 2024-04-11 00:44:35.

Statistical Models for Nucleon Structure Function

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Mirez, Tarrillo, Carlos Alberto, and Luis Augusto Trevisan. Statistical Models for Nucleon Structure Function, Cambridge Scholars Publishing, 2024. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/wisc/detail.action?docID=31136544. Created from wisc on 2024-04-11 00:44:35.

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Mirez, Tarrillo, Carlos Alberto, and Luis Augusto Trevisan. Statistical Models for Nucleon Structure Function, Cambridge Scholars Publishing, 2024. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/wisc/detail.action?docID=31136544. Created from wisc on 2024-04-11 00:44:35.