Fortschritte der Physik / Progress of Physics: Band 28, Heft 10 1980 [Reprint 2021 ed.] 9783112522967, 9783112522950

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FORTSCHRITTE DER PHYSIK H E R A U S G E G E B E N IM AUFTRAGE D E R P H Y S I K A L I S C H E N

GESELLSCHAFT

DER DEUTSCHEN DEMOKRATISCHEN VON F. K A S C H L U H N ,

REPUBLIK

LÖSCHE. R. R I T S C H L U N D R. R O M P E

H E F T 10 • 1 9 8 0 • B A N D 28

A K A D E M I E

- V E R L A G

EVP 1 0 , - M

ISSN 0015 - 8208

31728

.

B E R L I N

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Zeitschrift „Fortschritte der Physik" Herausgeber: Prof. Dr. Frank Kaschluhn, Prof. Dr. Artur Ldsohe, Prof. Dr. Rudolf Ritsehl, Prof. Dr. Robert Rompe, im Auftrag der Physikalischen Gesellschaft der Deutschen Demokratischen Republik. Verlag: Akademie-Verlag, DDR - 1080 Berlin, Leipziger Straße 3 - 4 ; Fernruf: 22 36 221 und 22 3 6 2 2 9 ; Telex-Nr. 1 1 4 4 2 0 ; Bank: Staatsbank der DDR, Berlin, Konto-Nr. 6836-26-20712. Chefredakteur: Dr. Lutz Rothkirch. Anschrift der Redaktion: Sektion Physik der Humboldt-Universität zu Berlin, DDR - 1040 Berlin, Hessische StraBe 2. Veröffentlicht unter der Lizenznummer 1324 des Presseamtes beim Vorsitzenden des Ministerrates der Deutschen Demokratischen Republik. Gesamtherstellung: VE13 Druckhaus „Maxim Gorki", DDR - 7400 Altenburg, Carl-von-Ossietzky-Strafie 30/31. Erscheinungsweise: Die Zeitschrift „Fortschritte der Physik" erscheint monatlioh. Die 12 Hefte eines Jahres bilden einen Band. Bezugspreis je Band: 180,— M zuzüglich Versandspesen (Preis für die D D R : 120,— U). Preis je Heft 15,— M (Preis fttr die D D R : 1 0 , - M). Bestellnummer dieses Heftes: 1027/28/10. /( X980 by Akademie-Verlag Berlin. Printed in the German Democratio Republio. AN (EDV) 5 7 6 1 8

Fortschritte der Physik 28, 5 0 1 - 5 2 5 (1980)

Problems of Deep Inelastic Lepton-Nucleus Interaction Y A . Z . DABBAIDZE, S . M . ESAKIA, V . R . GABSEVANISHVILI, Z . R . MENTESHASHVILI

High Energy Nuclear Physics Laboratory, Tbilisi, State University, Tbilisi,

USSR

Abstract The deep inelastic scattering of electrons (muons) and neutrinos on the nuclei and the process of lepton pair production in hadron-nucleus collisions are considered. The expressions for the corresponding differential cross-sections are derived provided the final lepton and the spectator fragment of the initial nucleus are detected in coincidence. The structure functions of the processes under consideration are analysed by the principle of automodelity. Some scale-invariant regularities are established for them. In the framework of "light front" formalism for many-body systems the relations of these structure functions to the usual structure functions of the deep inelastic leptonhadron interaction are obtained. A hypothesis is put forward on the scale invariance of structure functions with respect to the ^-variable which is some complicated dimensionless combination of kinematic invariants. Contents 1. Introduction 501 2. Deep inelastic scattering of charged leptons on the nucleus. The process IA —> I'(A — 1) X. 503 3. Deep inelastic scattering of neutrino (antineutrino) on the nucleus. The process vA l(A - 1) X 506 4. Lepton pair production in hadron-nucleus collisions. The process a A —> l+l~(A — 1) X . 509 5. Dimensional analysis and the principle of automodelity 512 6. Relations between the nucleus and the nucléon structure functions 516 7. Discussion and conclusions 519 Appendix A Appendix B

520 522

References

524

1. Introduction A study of deep inelastic processes involving leptons plays an essential role in understanding the nature of elementary particles. Considering leptons (electrons, muons, neutrinos) as structureless point particles and assuming the interactions between them to be known, they can be used as good probes in studying strongly interacting particles (hadrons) to describe the internal structure of which the so-called structure functions (formfactors) are introduced. The scaling behaviour of these structure functions was observed in the experiments on deep inelastic lepton-hadron interactions [ i ] . This phenomenon gave rise to a number of theoretical approaches meant to explain the observed 34

Zeitschrift „Fortschritte der Physik", Heft 10

502

Y A . Z. DARBAIDZE, S . M . ESAKIA, V . R . GABSEVANISHVILI, Z. R . MENTESHASHVILI

experimental regularities. The scaling properties of structure functions are explained in the framework of the parton model [2] and of the automodelity principle for electromagnetic and weak interactions [3]. The existence of the scale invariant (automodel) asymptotics is proved [4] in the framework of the local quantum field theory on the basis of the Jost-Lehman-Dyson representation [5]. For a more detailed discussion of some theoretical approaches to the deep inelastic lepton-hadron interactions see, e.g., the review papers [6] and the references therein. Note that in the experiments with beams of high energy electrons (see, e.g., [7] and the references therein) the scale in variance property in the Bjorken variable xB = —q2/2v was verified in the range l(GeV/c) 2 I'(A — 1) X with the spectator nucleus (A — 1) in the final state will be the object of our analysis. These processes are kinematically similar to the deep inelastic lepton-hadron processes provided the final lepton axd one of the produced hadrons are detected in coincidence IN —VhX. In Section 2 the deep inelastic scattering of charged kptons (electrons and muons) on the nucleus is considered in the lowest order in the electromagnetic interaction. In Section 3 the scattering of the neutrino (antineutrino) on the nucleus is treated in the framework of the F-v4-theory of weak interactions. In Section 4 a process of lepton pair production in hadron-nucleus collisions is studied. In Section 5 the dimensional analysis of the structure functions is carried out. The scaling properties of these functions are derived on the basis of the principle of automodelity. In Section 6 the relations of the structure functions of deep inelastic lepton-nucleus

Deep Inelastic Lepton-Nucleus Interaction

503

processes to the usual structure functions of lepton-hadron interactions are obtained in the framework of "light f r o n t " formalism for composite systems. In Section 7 the discussion of the results obtained and some conclusions are given.

2. Deep inelastic scattering of charged leptons on the nucleus. The process IA -* l'(A — 1) X

Condsider the process of the deep inelastic scattering of the electron (muon) with fourmomentum k on the nucleus A with four-momentum PA when the final lepton with fourmomentum k' and the spectator fragment (A — 1) of the nucleus A with four-momentum PA\ are detected on coincidence. Define the kinematic invariants of the process IA l'{A — 1) X in the following manner: k'f, vA = PAq, vsp = PXLiq q2 = (k (2.1) * = PaJTi, su = (k + PA)». The elements of the phase space of the particles detected in the final state are expressed in terms of these invariants as dk'jEk. = 2JII~1I2(Sia, M,2, MA2) dq2 dvA (2.2) dpsrjE*^

= (vA2 -

m/q2)-!'2

dvsp dx d I'hX is considered in [20\ Let us derive now a relation between the cross-sections of the processes IA —> V {A — 1) X and yA (A — 1) X in the limit g2 —> 0. The cross-section of photoabsorption of the longitudinally polarized photons should vanish in this limit as it corresponds to the real photons which have transversal polarization only. This leads to the relation lira Wi!A{q\

i!->0

vA, vsp, x) = lim [ ( m / -

>o

= lim [(vAW) q'-tO

V / ? 2 ) W2'A(q*, vA, vsp, *)] W2lA(q\

vA, vsp, *)].

(2.15)

506

Y A . Z . DABBAIDZE, S . M . ESAKIA, V . R . GAESEVANISHVILI, Z. R .

MENTESHASHVILI

In the limit q2 0 the cross-section of the transversally polarized photons is the total photoabsorption cross-section and, in view of (2.14a), takes the form d a ( y A

- >

-

( A

1 )

4TI2£%

X )

l i m d v

d x

s p

1>A

W ^ i q

2

,

v

{ q

2

,

A

v

s P

,

(2.16)

x ) .

q>-+o

Taking into account (2.15), we have d a { y A

( A d v

-

1 )

X )

d x

s p

—4n2oi lim 92—>0

0

q'->

\

W

z

l A

,

v

A

,

v

s p

,

(2.17)

x )

To establish the relation of this cross-section to the cross-section of deep inelastic electroproduction write (2.10) in the laboratory frame (the rest frame of the nucleus A). Neglecting the lepton mass, we have: d a ( l A d q

- > l ' ( A 2

d v

d v

A

-

1 )

s p

d x

1

X )

8(2nf

m

/

(

v

X |2 sin2 y

! E

/

-

W

1

2

m / q

M (g 2 ,

v

A

) ^

,

v

2

\

,

s p

-

k

E

k

\

(

J

\

+

x )

m

4 n i x q

A

2

2

\

2

)

, c o s

2

W

u

2

( q

2

,

v

A

,

v

s p

,

.

(2.18)

Here Ek and Ek- are the energies of the initial and final leptons, respectively, 0 is the scattering angle (the angle between the directions of momenta k and k' of the initial and final leptons). For q2 0 we have sin2 0/2 — 0 and hence cos2 0/2 —> 1 and from the cross-section (2.18) we obtain d a { l A d q

2

d v

l ' { A

-

d v

s p

A

1 )

X )

r->0

d x

.

2(2v/

(2.19)

Formulas (2.17) and (2.19) yield the following relation between the cross-sections of electroproduction and photoabsorption on the nucleus: d a ( l A d q

- > 1 ' { A 2

d v

A

d v

s p

1 ) d x

X )

2

1

2(2^)5

q ^0

\

U)(

)

d v

( A

-

1 )

X )

(2.20)

d x

s p

3. Deep inelastic scattering of the neutrino (antineutrino) on the nucleus. The process vA -* I (A — 1 ) X

Consider now the charged current induced deep inelastic neutrino-nucleus interactions in the framework of the F-/1-theory of weak interactions. Define the kinematic invariants of this process similar to the case of deep inelastic electroproduction (see formulas (2.1) and (2.2)), but now let k and k' be the four-momenta of the initial neutrino (antineutrino) and the final lepton, respectively. In the lowest order in the weak interaction the matrix element of the deep inelastic interaction of the neutrino (antineutrino) with the nucleus is of the form : T ( v A

1 { A

-

1 )

X )

=

l , - ( P

s

A

U ,

X I

J ± ( 0 )

I P

A

) .

(3.1)

Deep Inelastic Lepton-Nucleus Interaction

507

G is the weak interaction constant (GhnP2 = 1.02 • 105). J^- are weak hadronic charged currents, I ^ are the corresponding weak leptonic currents: vi -> I: l„- = u{k') y„{ 1 + y5) u(k) (3.2) vt-+U

1+ =

v(k)yli{l-ya)v(k').

When the lepton I and the spectator nucleus (A — 1) are detected in the final state, the cross-section of the deep inelastic neutrino (antineutrino)-nucleus interaction looks as follows: l(A - 1) Z ) G2 : ' I __ _ dq2 dvA dvsp dx d -

^ q)f

- *) K ->) K - 7*) K - ? 4

+

i 2

-

J ^uPY-IAW^

+ {PTi./b

+ i(PsI-1.„?. - i î - i . ^ ) ^ ï î + ¿(-P.i.^i.v -

».r

+

vi

P

+ Pf-i„