Fortschritte der Physik / Progress of Physics: Band 29, Heft 11/12 [Reprint 2022 ed.] 9783112656044, 9783112656037

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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 R E P U B L I K VON F. KASCHLUHN, A. LÖSCHE, R. RITSCHL UND R. ROMPE

H E F T 11/12 . 1981 • B A N D 29

A K A D E M I E

31728

-

V E R L A G

EVP 2 0 , - M

•

B E R L I N

ISSN 0 0 1 5 - 8 2 0 8

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Zeitschrift „Fortschritte der Physik" Herauageber: Prof. Dr. Frank Kaachlulm, Prof. Dr. Artur LSsche, Prof. Dr. Rudolf Ritsehl, Prof. Dr. Robert Rompe, im Auftrag der Physikalischen Gesellschaft der Deutschen Demokratischen Republik. Verlag: Akademie-Verlag, DDR - 1086 Berlin, Leipziger Straße 3 - 4 ; Fernruf: 2236221 und 2236229; Telex-Nr.: 114420; 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 StraDe 2. Veröffentlicht unter der Lizenznummer 1324 des Presseamtes heim Vorsitzenden des Ministerrates der Deutschen Demokratischen Republik. Gesamtherateilung: VEB Druckhaus „Maxim Gorki", DDR - 7400 Altenburg, Carl-von-Oasietzky-StraDe 30/31. Erscheinungsweise: Die Zeitschrift „Fortschritte der Physik" erscheint monatlich. Die 12 Hefte eines Jahres bilden einen Band. Bezugspreis je Band 180,— M zuzüglich Versandspesen (Preis für die DDR: 120,— M). Preis je Heft 15,— M (Preis für die D D R : 10,—M). Bestellnummer dieses Heftes: 1027/29/11/12. $ 1981 by Akademie-Verlag Berlin. Printed in the German Democratie Republio. AN (EDV) 57618

ISSN 0015-8208 Fortschritte der Physik 30, 5 0 5 - 5 5 0 (1981)

A Survey of Polarization Asymmetries Predicted by QCD F . BALDRACCHINI

Instituto di Fisica Teorica dell'Università di Trieste, Italy N . S . CRAIGIE

International Centre for Theoretical Physics, Trieste, Italy and Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, Italy V . ROBERTO

Istituto di Fisica Teorica dell'Università di Trieste, Italy and M.SOCOLOVSKY

International Centre for Theoretical Physics, Trieste, Italy and Centro de Investigación y Estudios Avanzados, México, D. F. Abstract We give a comprehensive review of the polarization asymmetries predicted by QCD and the parton model for hard processes. The article discusses the interrelation between these processes which include deep inelastic inclusive, massive lepton pairs, reflected and transmitted asymmetries in hadroii production at large pT, prompt photon production, etc. We also discuss the validity of the formulae derived and the potential effects of non-leading logarithmic and power orders.

I. Introduction Recently there has been a considerable interest in high energy and deep inelastic experiments with polarization beams and targets, since this provides a valuable means of testing the parton model and QCD. In particular, these experiments will provide a very valuable way of studying factorization beyond the leading logarithmic and power orders. In this article we survey the large number of interrelate processes, which admit a parton model interpretation and for which cross-sections and polarization asymmetries are calculable in the leading orders of QCD. We also provide estimates of the expected sizers of the various asymmetries in the leading order in QCD, however one must stress that 1

Zeitschrift „Fortschritte der Physik", Bd. 29, Heft 11/12

506

F . BAIDKACCHIKI, N . S . CRAIGIE, V . KOBERTO, a n d M . SOCOLOVSKY

much of the basic input needed in such estimates is not yet available, because this class of polarization experiments has barely begun [1]. This article complements and extends the work of Sivers and collaborators [2], who discussed mainly what we shall call reflected (beam-target) asymmetries in large ^inclusive experiments. We shall discuss how transmitted asymmetries may also be mea'sured with a polarized beam or target facility. The other processes analysed in this article include deep inelastic electron or muon and neutrino scattering with polarized targets and the various asymmetries in massive muon pair production and prompt photons at large pT. The article is divided up as follows. Sec. I I is devoted to the notion of factorization and how the basic asymmetries are derived in any short distance (i.e. hard) process, which admits a parton model interpretation and by the same token, perturbative QCD thought to be applicable. In Sec. I l l we give the basic formulae for the asymmetries for the various processes listed in the following list of contents: 3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7.

Deep inelastic electron proton scattering ep e' + X ; Deep inelastic neutrino (antineutrino) nucleón scattering; Semi-inclusive deep inelastic electroproduction; Asymmetries in mass lepton pair production; Reflected and transmitted double asymmetries in hadron production at large pT\ Double asymmetries in prompt photon production at large pT; Single spin asymmetry-pout correlation in PP -»• TC -f- jet + X .

Sec. IV will be devoted to estimates of the asymmetries given in Sec. I l l , while finally in Sec. V we shall discuss the role of the next to leading orders in QCD (both logarithmic and powers). Here we try to emphasise the importance of polarization experiments in giving us an insight into many aspects of QCD not yet understood properly. The totality of the experiments discussed here will be a powerful means of testing the basic premises of the parton model and the application of perturbative QCD.. We however cannot go into the feasibility of each of the experiments discussed, except we might add as a theoretician one never ceases to be amazed at the ability of our experimental colleagues to measure even the most unlikely of processes, thus often providing a crucial new valuable insight. II. General Parton Spin Formalism and Factorization in the Leading Order in QCD In this section we shall set up our procedure with which spin asymmetries associated with processes admitting a parton model (PM) interpretation are defined in the framework of quantum chromodynamics (QCD)inthe leading logarithmic regime (LLR). Beforedoing that, however, we shall cjiscuss briefly the important concept of factorization which is at the base of all our next developments. In the naive PM [3] when considering processes involving hadrons A, B, ... and leptons I, I' in the kinematical region in which all the relevant observable variables (energies, momentum transfers) are large, enough, one writes the physical cross-section as a convolution of distribution and fragmentation functions (which measure probabilities of finding partons in hadrons and hadrons as products of parton decays with appropriate fractions of longitudinal momenta, respectively) with cross-sections involving partons and the gauge particles assiociated with the electroweak interactions. That is, one obtains a complete factorized probabilistic picture. As is well known the previous scheme still goes through in the LLR of QCD and this is shown for example in Refs. [4] and [5], where non-covariant gauge techniques are used. We are led to a picture in which physical cross-sections are calculable again as convolutions, but now in terms of parton densities which depend on the frequency scale at which

Polarization Asymmetries

507

the hard process occurs and; with the parton cross-sections evaluated to lowest order of QCD perturbation theory. Indeed* as a brief sketch of the arguments involved in the factorization proof, consider a process involving at least the two quark-hadron channels (a, A) and (b, B) in which Q2 = pA • pB defines the large mass scale (Fig. 1 (a)). Diagrams involving gluons which are exchanged between different parton-hadron channels (see Fig. 1 (b)) will in general not lead to factorization of parton probabilities and fragmentation functions because such interference terms correspond to the kind of coherent effects one normally excludes in the naive parton model description. In a covariant gauge one would expect such interference terms to be overwhelmingly important, however factorization still emerges because one can choose a non-covariant gauge (rj • A = 0 , where the vector

A

A

A

) r

•l

d)

A

)

Fig. 1. (a) Block-wise factorization of Bethe-Salpeter amplitudes in the parton hadron channels (a, A) and (b, B), respectively (b) Crossed graph which could mix (a, A) and (6, B) channels and destroy factorization (c) Gluon exchange in LLB, in the rj gauge (d) Real and virtual gluons in a given channel 1*

508

F . BALDEACCHXOT, N . S. CRAIGIE, V . ROBERTO, a n d M. SOCOLOVSKY

Y\ = pA + pB), for which such cross diagrams give zero contribution in the leading order in QGD. To see how this works/one makes a Sudakov decomposition of the momentum in each channel according to ka = xapA + yar\ + kaT and kb = xbpB + JW? + kbT and similarly for the primed quantities. In the region of integration responsible for the leading logarithmic behaviour (LLR) (xa, xb, xa', xb ~ 1, ya, yb, ya', yb 1 and kT2 2pA • pB), the vertex MaV^J^a c a n he replaced by 2pAfJ6a and the gluon insertion leads to the factor [pA"pB>D^(k, V) + 0(y, where DMr(k, V) =

+

+ k^/k

• n - rfkpkj(k

• V)*]/k*

(2.1)

is the gluon propagator in the r) gauge. This type of contribution vanishes in the leading order (LLR) since the second term in the gluon propagator cancels the contribution from the (—gr'") term and the third term is of 0(y2) and contributes only in non-leading orders. Thus in this gauge the leading order in QCD corresponds to the factorized insertions shown in Fig. 1 (c), plus the corresponding virtual insertions on, respectively, the pA and pB lines. The infra-red divergence corresponding to soft gluons cancel between real and virtual gluons in each channel (Fig. 1 (d)) and the coherence effects in leading order lay hidden in the anomalous dimensions, which govern the scaling violations of the parton densities. In the next to leading order important coherent effects will be associated with the gauge fixing vector rj. Apart from these distinctions, we have block-wise factorization of the various Green functions describing the process in the LLR of QCD as in the covariant parton model description, leading to a formula of the type OAB- (PA, PB, • • •) = E / dxaD/(xa, a.b

Q») f dxbDE\xb, Q*) - o~(xapA, xbpB,...),

. (2.2)

where Q2) = ^

J ^

d^((ka + Qf) AA"(ka) Z£(ka*).

(2.3)

In deriving (2.2) one makes use of the fact that in the LLR the relevant part of the parton-hadron Bethe-Salpeter ((B-S) Green's functions is given by TAa = Pij(ka) AA°(ka), where P?- is the spin projection operator for parton a in the limit ka2 = 0. Z^ is the wave function renormalization factor associated with the ath parton and

°>s»)

w

(7 k2> «») (2-12) ^

or since the non-singlet quark distribution functions do not mix with gluon distributions, we have in that case the simpler form 1

Q* = S2D(^Qo2)+f

^ ^ P -

J j S

2

P ( z ) S

2

(2.13)

D ^ , k j ,

where d2P(z)

— l i m cF

2z 1 - Z + Ô

0

and we denote the corresponding anomalous dimensions l ynâ"" = J

2)

dzz*-n2P{z)

=

j^l + 4 f i ] .

(2.14)

In fact, since zero mass quarks have no spin asymmetry, it turns out for the qq term, dAaA" only contributions in order a s 2 beyond LLB.

has

511

Polarization Asymmetries

We shall return to a more detailed discussion of these evolution equations in Sec. IV, where all the relevant cases will be discussed. Finally we note that if we form longitudinal polarization or helicity asymmetries, the same considerations go through and one obtains corresponding formulae with the spin differential replaced by helicity differences, which we denote by A F ( X ) = 1/2 [JP(+) — F(—)] in order to distinguish the latter from the former. If we include effects of longitudinal gluons in next to leading order, then the above formulae must be modified to include the additional terms. a

a

III. Survey of the Main Asymmetries Predicted in the Leading Order in QCD In this section we shall apply the techniques developed in Sec. I I to a number of hard processes, which admit a parton model description and derive formulae describing in the leading order of QCD spin and/or helicity asymmetries associated with them. In Sec. I Y we shall provide detailed estimates for a number of the cases below, while for the others, rough estimates and a review of present-day literature will be presented. 3.1. Total inclusive deep inelastic electroproduction Consider the process e(A) + N(yl) (e') + X , where X refers to the initial lepton helicity, A is the longitudinal polarization of the nucléon (along the beam direction) and (e') denotes the unpolarized final lepton. The double differential cross-section is given by

\at

«W/eWïrM)-Ke'>X

= tS tt

—T— S 8 ->r 11 i

xe'2 £

y

K«2 +

u*)

+

-

D

^ x > G 2 ; x>

(3.1) sin2 x 6 / 2 , v = E — E ' , E and E' are the beam and final lepton energies, respectively, and 0 the scattering angle in the laboratory frame, J J refers to the sum over the helicities of the

where

x

=

—t/{s

+

u)

=

Q

2

/ 2 M

N

v , s =

2 M

S

E , u

- 2 M

=

N

E ' , t =

- Q

2

=

—4EE'

strucked parton i (quarks and antiquarks) and e^is its squared charge. D ( A , A') measures the probability of finding a quark -(antiquark) i with helicity A' in a nucléon with longitudinal polarization A. From Eq. (3.1) it is easy to obtain for the longitudinal N

asymmetry 3 ) ALL =

(

^"t"'—r the expression

+ ) + —)

i

with

y

—

v / E

and

J J i

=

In deriving Eq. (3.2) we have used the relations

DN \-f-,

+)

q,q

= DN\—, — ) and Z V ( + , —) = DNL(—, + ) which are a consequence of parity invariance of the strong interactions. The spin structure of the functions A^D^1 is defined in 3)

Since in the CM frame the nucléon is in a definite helicity state we may also refer to A n as a helicity asymmetry.

512

F. Baldbacchiot, N. S. Craigie, V. Roberto, and M. Socolovsky

Eq. (2.11). We note in Eq. (3.2) the presence of a ^/-dependent kinematical factor in front of what may be called "parton asymmetry". A rough estimate of the latter can be given in the framework of the SU(6) model, which is reliable near x — 1 and which gives 5/9 and zero for proton and neutron targets, respectively. One can also analyse the spin dependence of deep inelastic scattering in terms of the four structure functions Wlt W2, @i and 02 defined by the decomposition [3] W>"

=

Z ( j ? , s \ J " X

=

i - g " '

+

|a;) (a;| J

- )

w !

+

f

\p,

s ) d(Mx*

( p t f J M

+ il

+

-

+

( p +

...)

- M

W

g0J

q)*)

2

(3.3)

G,j.

The structure functions Gx and G2 are simply related to the spin and helicity differentials W

a n d

A2Dn*

4

by

) 2 G ^ x , Q2)

=

£

e

i

2

A

2

D

N

i

( x ,

Q*)

. 2[G1

+

G2]

=

Z i

^

d

M

x

,

Q

2

(3-4)

).

The G structure functions satisfy the Bjorken sum rules [7] (see Ref. [3] for details) in the form i 2 j d x

[ G ^ x )

-

G ? { x ) }

=

j

^

,

(3-5)

o

where

GA/GV

~ 1.23 is the ratio of axial vector and vector coupling constants. i /

I dxG2P(x)

-

0

J

dxG

2

"(x)

-

0 .

(3.6)

0

The latter corresponds to angular momentum conservation. This implies, respectively, for the spin and helicity parton densities the identical sum rules

J

i dx

[• ¡a + X provide a novel polarization asymmetry experiment [([a~) + X(((jl) : outgoing unpolarized muon) in the leading order of QCD are ^

(vN(4)

Q*; A, - ) + (1 - y)2 Z D^(x, Q«; A,