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Center for Theoretical Physics

arXiv:hep-ph/9207259v1 24 Jul 1992SPIN STRUCTURE FUNCTIONS*R. L. JaffeCenter for Theoretical PhysicsLaboratory for Nuclear Scienceand Department of PhysicsMassachusetts Institute of TechnologyCambridge, Massachusetts 02139U.S.A.Invited talk presented at:Baryon ’92Yale UniversityNew Haven, ConnecticutJune 2 – 4, 1992Typeset in TEX by Roger L. GilsonCTP#2114July 1992* This work is supported in part by funds provided by the U. S. Department of Energy(D.O.E.) under contract #DE-AC02-76ER03069.0

I.INTRODUCTIONThe study of spin-dependent effects in deep inelastic scattering originated with the workof the SLAC–Yale Collaboration headed by Vernon Hughes in the early 1970’s.1, 2 This subjecthas been reinvigorated by the surprising results published in 1987 by the European MuonCollaboration3 which also had a large representation from Yale. It is highly appropriate,therefore, to dedicate this talk here at Yale to the fascinating subject of the spin structure ofthe nucleon as probed at large momentum transfer in lepton scattering and related processes.Deep inelastic processes have long been recognized for their dual roles, on the one hand asprecise quantitative tests of QCD — largely through the logarithmic Q2-variation generatedby radiative corrections — and on the other hand as precise and well-understood probes ofhadron structure — an approach summarized at the most elementary level by the well-knownparton model interpretation of deep inelastic structure functions.

Often, when parton modelphenomena have clear interpretations, they have led to surprises and major revisions of ourconcept of hadrons as bound states of quarks and gluons. Three examples come to mind:1.

The “Momentum Sum Rule,” recognized in the early 1970’s4 and checked experimentallysoon after, showed that about 50% of the nucleon’s “momentum” is carried by gluons.A surprise to quark modelists, this result is actually predicted by QCD at asymptoticln Q2 (ln Q2/Λ2 >> 1).5 The fact that it seems to work even at moderate Q2 suggestseither that asymptopia is precocious or that glue is an intrinsic component of the nucleonbound state.2. The “Gottfried Sum Rule,”6 which is not a sum rule at all but instead a measurementof the isospin asymmetry of the antiquarks in the proton:Rdx¯up(x) −¯dp(x).

The1

recent NMC precision measurement of this quantity7 firmly established the existence ofan asymmetry, and requires quark modelists to entertain effects which deform the sea onaccount of the valence quarks in a hadron.3. Finally, and perhaps most significant, the EMC measurement of gep1 (x) can beinterpreted8, 9 as a direct measure of the fraction of the nucleon’s spin to be found onthe spin of the quarks.

The result: 12.0 ± 9.4 ± 13.8%, caught everyone by surprise. Areasonably sophisticated quark model estimate was 60 ± 12%.8, 10Naive quark models are apparently too naive.

Even rather sophisticated models with rela-tivistic quarks, ambient gluon fields and distant meson clouds do not easily accommodatethese unusual results.The EMC result, which originates in an unexpectedly small spin asymmetry at low-x hasstimulated a reconsideration of spin effects in deep inelastic processes. Much to the surpriseof those of us involved in this undertaking, the reconsideration has led to a much richer andmore rational picture of the role of spin in deep inelastic processes than existed before.

Theinterpretation of transverse spin and of chirality in hard processes* has had to be rewritten.Along the way, the importance of polarization in deep inelastic processes with hadronic initialstates (e.g. polarized Drell-Yan) has come to the fore.

The principal object of my talk is tooutline a unified description of spin-dependent structure functions which has emerged in theprevious few years. Much of my work on this subject was performed in collaboration withXiangdong Ji and Aneesh Manohar and my debt to them is substantial.

* I use the term hard processes in the same sense as the fine book of the same title byB. L. Ioffe, V. A. Khoze and L. N. Lipatov.11 The reader should bear in mind that “hard”is to be construed as the opposite of “soft,” not “easy.” Indeed, “hard processes” are amongthe few easy aspects of QCD!2

My talk will be organized as follows:I. IntroductionII. How to count: enumerating and interpreting the independent structure functions ofhadrons.III.

The g1 saga: some remarks on the longitudinal spin asymmetry measurement and inter-pretation.IV. h1(x, Q2): a new structure function and its relation to chirality, transversity and polarizedDrell–Yan.V.

Twist-3: measuring quark-gluon correlations with gT (x, Q2) and hL(x, Q2).II.HOW TO COUNT:Enumerating and Interpreting the Structure Functions of HadronsMerely counting the independent “structure functions” of hadrons turns out to be anon-trivial problem. Its complexity increases with the degree of departure from scaling atlarge-Q2.

Effects which scale (modulo logarithms of Q2 from radiative corrections) at large-Q2 are “twist-2.” Those which vanish as 1/Q (again, mod logarithms) are “twist-3,” and soon. The discussion here will be complete through twist-3.

Twist-4 effects which vanish like1/Q2 require a considerable investment in formalism and are much harder to extract fromexperiment.The familiar quark and gluon distribution of the parton model are actually special casesof general light-cone correlation functions. A couple of generic examples aref(x) ≡Z dλ2π e−iλx ⟨P |φ(0)φ(λn)| P⟩(2.1)E(x, x′) ≡Z dλ dλ′4π2 e−iλx−iλ′x′ ⟨P |φ(0)φ(λn)φ(λ′n)| P⟩(2.2)3

where pµ = (p, 0, 0, p), nµ =12p, 0, 0, −12p, P µ = pµ + M22 nµ, φ(x) is a generic field, andp, which is arbitrary, labels the frame. p =M2 corresponds to the target rest frame andp →∞is the “infinite momentum frame.” Both f(x) and E(x, x′) depend non-trivially ona renormalization scale, µ2 [For a careful discussion see Ref.

[12].] In an asymptotically freetheory like QCD in which β(g) ∼g3 for small g, the µ-dependence of f(x) and E(x, x′) is atmost a power of ln µ.

These logarithms do not change the structure of our results and theycan be considered separately, hence I will often suppress the renormalization scale.Both f(x) and E(x, x′) are ground state correlation functions with the correlation takenalong a tangent to the light-cone. F(x) has a simple, probabilistic interpretation obtained byinserting a complete set of states:f(x) =XX|⟨P |φ(0)| X⟩|2 δP +X −(1 −x)P +(2.3)so f(x) is the probability to find a quantum of φ with k+ = xP + in the target state (seeFig.

1a). E(x, x′) does not have a probabilistic interpretation (see Fig.

1b). Some generalproperties of distributions such as E(x, x′) are developed in Ref.

[13].(a)(b)Fig. 14

There are relatively simple rules for enumerating those quark and gluon distributionfunctions which contribute effects at twist-2 (i.e. O(1) at large Q2) and twist-3 (i.e.

O(1/Q)at large-Q2):141. Identify independent degrees of freedom when QCD is canonically quantized on the light-cone.The solution to this problem is well-known:15 The quark field is decomposed with light-cone projection operators, P± = 12γ±γ∓, where γ± ≡1√2(γ0 ± γ3) —ψ± ≡P±ψ .

(2.4)The gluon field, Aµ, is decomposed with respect to light cone coordinates — Aµ →A+, A1, A2, A−.In the gauge A+ = 0, then ψ+ and ⃗A⊥=A1, A2are independentvariables, whereas ψ−and A−are dependent, constrained variablesA−= A−hψ+, ⃗A⊥iψ−= ψ−hψ+, ⃗A⊥i. (2.5)The independent degrees of freedom, ψ+ and ⃗A⊥, carry helicity ± 12 and ±1, respectively.

Thedependent degrees of freedom ψ−and A−carry helicity ± 12 and 0, respectively.2. To enumerate all twist-2 distributions and exhibit their target spin dependence, countindependent helicity amplitudes for forward parton-hadron scattering using only indepen-dent degrees of freedom.The generic helicity amplitude shown in Fig.

2 gives a visual depiction of the generic light-cone correlation function of Eq. (2.1).

Since there is no momentum transfer to the target, theamplitude corresponds to forward scattering and takes place along the ˆe3-axis in coordinate5

space. Therefore helicity (≡angular momentum about ˆe3) is conserved: h + H = h′ + H′and furthermore A(h, H →h′, H′) = A(−h, −H →−h′, −H′) and A(h, H →h′, H′) =A(h′, H′ →h, H) are consequences of parity and time reversal invariance, respectively.

Withthese relations, we enumerate the independent helicity amplitudes for quarks and gluonsscattering from targets with spin-1/2 and spin-1 in Table 1.Fig. 2A(hH →h′H′)A(hH →h′H′)A(hH →h′H′)Target Spin-1/2Target Spin-1QuarksA1212→1212A12−12→12−12A−1212→12−12A121→121A12−1→12−1A−121→120A120→120GluonsA112→112A1−12→1−12A11→11A1−1→1−1A−11→1−1A10→10Table 1:Twist-2 helicity amplitudes.6

Each amplitude in Table 1 corresponds to a different quark or gluon distribution function(a function of x and ln q2). There are three quark and two gluon distributions for a spin-1/2target:f1(x) ∝A (1212→1212 ) + A ( 12−12→12−12 )g1(x) ∝A (1212→1212 ) −A ( 12−12→12−12 )h1(x) ∝A ( −1212→12−12 )(2.6)G(x) ∝( 112→112 ) + A ( 1−12→1−12 )∆G(x) ∝A ( 112→112 ) −A ( 1−12→1−12 )(2.7)The spin-average and helicity difference structure functions for quarks (f1(x) and g1(x)) andgluons (G(x) and ∆G(x)) are well-known.

There is experimental information on f1, g1 andG.The helicity-flip distribution function h1(x) is little-known because it does not figureprominently in electron scattering. Its interpretation is the subject of Section IV below.

Notethat there is no helicity-flip gluon distribution at leading twist.For a spin-1 target there are several new distribution functions beyond the obvious gen-eralizations of f1, g1, h1, G and ∆G. Two are quadrupole distributions for quarks and gluons,b1(x) ∝A ( 121→121 ) + A ( 12−1→12−1 ) −2A ( 120→120 )B1(x) ∝A ( 11→11 ) + A ( 1−1→11 ) −2A ( 10→10 )(2.8)and one is a gluon helicity-flip structure function unique to targets with J ≥1:∆(x) ∝A ( −11→1−1 ).

(2.9)b1(x) and ∆(x) have novel properties and interesting implications for nuclear physics.167

3. To enumerate twist-3 distributions which enter inelastic hard processes, count indepen-dent helicity amplitudes for forward parton-hadron scattering with one-independent andone-dependent degree of freedom.

*The dependent gluon mode is A−with helicity 0. The dependent quark mode is ψ−withhelicity ±1/2.

To distinguish ψ−from ψ+ in helicity amplitudes we denote it 12 in helicityamplitudes. Table 2 summarizes the independent twist-3 helicity amplitudes for spin-1/2 andspin-1 targets.A(hH →h′H′)A(hH →h′H′)A(hH →h′H′)Target Spin-1/2Target Spin-1QuarksA1212→1212A12−12→12−12A−1212→12−12A121→121A12−1→12−1A−121→120A−121→120A120→120GluonsA012→1−12A01→10A00→1−1Table 2:Twist-3 Helicity AmplitudesFrom Table 2 we see that a description of the nucleon requires three twist-3 quark dis-tributions, one spin averaged,e(x) ∝A 1212→1212+ A 12−12→12−12,(2.10)* This rule does not give rise to all twist-3 distributions.

The complete description of twist-3 requires the introduction of distributions which depend on two variables like Eq. (2.2).However, the only ones which appear to arise at tree level (i.e.ignoring QCD radiativecorrections) in hard processes through order 1/Q are the ones captured by this rule.8

one helicity difference,hL(x) ∝A 1212→1212−A 12−12→12−12,(2.11)and one helicity flip,gT (x) ∝A −1212→12−12. (2.12)There is only one twist-3 gluon distribution for the nucleon and it involves helicity flipGT (x) ∝A ( 012→1−12 ).

(2.13)Some comments on this analysis:• I have suppressed flavor labels. For each twist and helicity structure there are independentdistributions for each flavor of quark and antiquark.

The flavor singlet quark distributionsmix under q2-evolution with gluon distributions of the same twist and helicity structure.Specifically f1 and G, g1 and ∆G, and gT and GT mix. e, h1 and hL do not mix withgluon distributions.• Distributions associated with target helicity flip are in actuality measured by orientingthe target spin transverse to the scattering axis-ˆe3.The transversely polarized stateis a linear superposition of helicity eigenstates (e.g.|⊥⟩=1√2 (|1/2⟩+ | −1/2⟩) and|⊤⟩=1√2 (|1/2⟩−| −1/2⟩), where |⊥⟩and |⊤⟩are transverse spin eigenstates).

Thenthe asymmetry obtained by reversing the target’s transverse spin isolates the helicity flipamplitude. Thus h1 and gT (and GT and ∆as well) are important in processes involvingtransversely polarized targets.• If we decompose the independent quark field ψ+ into chirality eigenstates, we find that(of the nucleon distribution functions) h1, hL and e are chirally odd (i.e.

the quark fields9

in the correlation function of Eq. (2.1) have opposite chirality) whereas f1, q1 and gT arechirally even (the quark fields in Eq.

(2.1) have the same chirality). This explains whyh1, hL and e do not mix with gluon distributions (all of which are chirally even).

Furtherit explains why f1, g1 and gT are well-known and much studied whereas h1, hL and e arepoorly-known and neglected: QED and perturbative QCD both conserve chirality (exceptfor small quark mass terms) so h1, hL and e are suppressed in deep inelastic electron(muon and neutrino) scattering. As we shall see in Section IV, they are not suppressed inhard processes with hadronic initial states such as Drell–Yan production of lepton pairs.The results of this section (for a spin-1/2 target like the nucleon) are summarized inTable 3, where the twist-2 and twist-3 quark and gluon distributions are classified accordingto target spin dependence and chirality.Twist-2 O(1)Twist-3 O(1/Q)SpinDependenceQuarkChiral EvenQuarkChiral OddGluonQuarkChiral EvenQuarkChiral OddGluonSpin average(unpolarized target)f1(x)—G(x)—e(x)—Helicity asymmetry(longitudinal polarization)g1(x)—∆G(x)—hL(x)—Helicity flip(transverse polarization)—h1(x)—gT (x)—GT (x)Table 3:Summary of Nucleon Quark and Gluon Distributions through Twist-3, O(1/Q).10

III.THE g1g1g1 SAGA:Brief Remarks on the Longitudinal Spin Asymmetry Measurement andInterpretationThe renewed interest in deep inelastic spin phenomena dates from a single number ap-pearing in a 1987 paper by the European Muon Collaboration.3 In Ref. [3] the EMC reporteda value for the area under g1(x, Q2) for a proton target at a nominal value of Q2 = 10 GeV2.In QCD this integral is related to axial charges measured in β-decay and to the flavor singletaxial charge which is otherwise unmeasured:8, 17Z 10dx gep1 (x, Q2) = 118(3F + D)1 −αs(Q2)/π+ 2Σ(Q2)1 +33 −8f32 −2f αs(Q2)π.

(3.1)f is the number of flavors, F and D are SU(3) invariant matrix elements of hyperon β-decayand Σ(Q2) is the renormalization scale-dependent18 flavor singlet axial charge:sµΣ(Q2) =PsXu,d,s¯qγµγ5qPsQ2. (3.2)Σ(Q2) is also the fraction of the proton’s spin to be found on the spins of its quarks.

Manyissues associated with this sum rule and its interpretation are reviewed in Ref. [10].In the most naive, non-relativistic quark model, ΣNQM = 1, because only the quarkscarry spin in the nucleon.

A model independent analysis assuming only SU(3) symmetry inhyperon β-decay and no polarized strange quarks in the nucleon gives8, 10 ΣEJ ∼= 0.60 ± 0.12.The EMC data giveΣEMCQ2EMC= 0.120 ± 0.094 ± 0.138 ,(3.3)11

far from the expectations of any model and arguably compatible with zero. Many explanationshave been offered by theorists.

Reviews with a variety of biases can be found in Refs. [19–21].

Rather than give a (most-likely superfluous) review here I will restrict myself to a fewcomments.• Data — The EMC result relies heavily on the extrapolation to x = 0 guided by the fourlowest-x points (see Fig. 3).

Confirmation is urgently needed. Several new polarizedelectron and muon experiments are in the works:→SMC – The Spin Muon Collaboration now running at CERN.

They expect to coverroughly the same x and Q2 range as EMC with better statistics and control of sys-tematic errors. They will first measure the longitudinal asymmetry from deuterium22in order to test Bjorken’s sum rule,23 so they will not immediately check the EMCresult.

They plan to run on polarized hydrogen in the near future. If one is willingto accept the validity of the Bjorken sum rule (which is an iron-clad prediction ofQCD), then the deuteron data will check the old result (modulo problems extractingneutron data from a deuteron target).→SLAC E142 and E143 — will study 3He (E142) and deuterium and hydrogen (inNH3 and ND3 targets) (E143).

These classic End Station A experiments will havelimited sensitivity to low-x at large-Q2. On the other hand, they promise to reportdata quickly.→HERMES — A large and ambitious program to place polarized gas targets in theelectron beam at HERA, it is awaiting a full approval, contingent upon demonstratedpolarization of order 50% in the electron ring.

This facility will be able to study12

a variety of targets in various polarization configurations (both longitudinal andtransverse). They will not have data until the latter half of this decade.It seems as though we shall have wait some time for clear confirmation of the EMCdata from electron or muon scattering.

There is, however, another possible sourceof information about Σ(Q2):→LSND — A group at LAMPF is presently mounting an experiment to mea-sure very low energy elastic neutrino scattering from liquid scintillator.At verylow energies this probes the axial “charge” seen by the Z0-boson — namely¯uγµγ5u −¯dγµγ5d −¯sγµγ5s.23 (Heavy quarks can be ignored — see Ref. [24].) Sincethe first two terms are known from neutron β-decay, the experiment measures thepolarized strange quark content of the proton directly.

An earlier version of thisexperiment using higher energy neutrinos at BNL — and therefore subject to fur-ther problems of interpretation — provided weak confirmation of the EMC result.25LSND can confirm the EMC data and eventually check the SU(3) symmetry assump-tions which lurk behind Eq. (3.1).

LSND has its own subtleties, some associated withthe fact that many nucleons in scintillator are bound in carbon nuclei. An interestingvariation suggested by Garvey et al.26 may circumvent some of these difficulties.• The Gluon Controversy — The triangle anomaly is closely connected toR 10 dx gep1 (x, Q2).It generates the Q2-dependence of Σ(Q2).17, 18 In 1988 several groups suggested27 thata proper interpretation of the anomaly alters the sum rule, Eq.

(3.1).The claim isthat Σ(Q2) measured by experiment consists of two pieces: one, the “true” quark model13

piece, eΣ, is Q2-independent and of order unity; the other, −αs(Q2)2π∆G(Q2), is a gluoniccontribution proportional to the integrated gluonic spin asymmetry ∆G(Q2):ΣQ2= eΣQ2−αsQ22π∆GQ2. (3.4)The proposal is summarized in Fig.

4. The authors of Ref.

[27] suggest that ∆G maybe so large that it cancels the true quark model piece, eΣ, and yields the small resultfound by EMC. Carlitz, Collins and Mueller27 suggested ways to measure ∆G in two jetproduction by polarized electrons.Fig.

4This interesting idea generated much activity but has run into a couple of problems whichcloud the original interpretation suggested in Refs. [27].

First, several groups28 pointedout that the separation between eΣ and αs∆G/2π is ambiguous. Second, no convincingargument has emerged for interpreting eΣ as the true quark model spin contribution.

Infact, a counterexample can be found in the heavy quark limit10 of QCD. Finally, there isno direct way to measure eΣ.

If eΣ need not be of order unity, ∆G need not be large and14

the predictive value of Eq. (3.4) becomes dubious.

Whatever its significance, Eq. (3.4)has stimulated discussion of experiments to measure ∆G(x, Q2) which is interesting inits own right.• Q ¯Q-Pairs — There is a prevailing view that the most natural state for the Q ¯Q-sea in thenucleon is unpolarized.

A more careful investigation shows that this is hardly obviousand possibly a factor in explaining the small value of ΣEMC. In Ref.

[29] Lipkin andI studied the spin and angular momentum content of a nucleon containing an extraQ ¯Q-pair in a variety of constituent quark models. Our most important observation wasmerely that vacuum quantum numbers, JP C = 0++, for a Q ¯Q-pair require L = S = 1.The L = S = 0 state has JP C = 0−+ and could only make a nucleon if combined witha less symmetric three quark state, which in turn would have problems with traditionalquark model successes like magnetic moments and octet axial charges.

Stimulated bythis, we constructed a three-component model of an octet baryon consisting of 1) thebare three-quark state |b⟩; 2) an additional 0++ pair |b[Q ¯Q]0++⟩; 3) an additional 1++pair |b[Q ¯Q]1++⟩. The last is the only other Q ¯Q state which can combine with the usualsymmetric three-quark state to produce a nucleon with the proper spin and parity.

Wefound that it is rather easy to accommodate the EMC result (and all traditional successesof the quark model) in such a model, though components 2) and 3) dominate the nucleon’swavefunction. Of course this is hardly an explanation of the EMC data.

It is merely anexample of a rather benign sea quark distribution which accommodates the data.• Flavor tagging to measure ∆u, ∆d and ∆s — There is great interest in the contributionsof individual quark flavors to the spin asymmetry. They are defined by∆qa Q2sµ ≡Ps¯qaγµγ5qaQ2Ps.

(3.5)15

The sum rule for gep1 can be rewritten as a measurement of one linear combination of ∆u,∆d and ∆s. The separate flavor contributions cannot be distinguished in electron scatter-ing.

Neutrino scattering from polarized targets could partially unravel flavor dependence,but such experiments are out of the questions for the foreseeable future.Recently, Close and Milner have suggested using leading particle effects in fragmenta-tion to tag the flavor of the struck quark.30 The basic idea is an old one:31 a π+ observedat large-x and large-z in the current fragmentation region is most likely a fragment of au-quark since ¯d quarks are suppressed at large x. Similarly leading π−’s are correlatedwith d-quarks and leading K± with ¯s and s (though this correlation is less pronouncedsince s and ¯s have a soft x-distribution).

The idea has been studied in connection withthe HERMES proposal32 but could be implemented in any experiment with adequateparticle identification.IV.h1(x, Q2)h1(x, Q2)h1(x, Q2), CHIRALITY, TRANSVERSITY AND POLARIZED DRELL–YANThe twist-2, transverse spin structure function, h1(x, q2) has been largely ignored since itsdiscovery by Ralston and Soper in 1979.33 I’d like to rectify the situation by giving h1(x, Q2)a major place in this talk.34 One benefit of this will be a clear understanding of transversespin in the parton model, a subject which has been surrounded by confusion for years.h1(x, Q2) is a chiral-odd structure function. It is projected out of a light-cone quarkcorrelation function with the Dirac matrix σµνγ5,1212πZdλ eiλx Ps ¯ψ(0)σµνiγ5ψ(λn)Ps= h1(x) (s⊥µpν −s⊥νpµ)M + hL(x)M (pµnν −pνnµ) s · n+ h3(x)M (s⊥µnν −s⊥νnµ)(4.1)16

where pµ = (p, 0, 0, p), nµ =12p(1, 0, 0, −1), sµ ≡(s · n)pµ + (s · p)nµ + sµ⊥and hL(x) andh3(x) are twist-3 and twist-4 distribution functions, respectively. If we decompose ψ into left-and right-handed components, it is clear that h1(x) is chirally-odd, as illustrated in Fig.

5a.Deep inelastic lepton scattering in QCD proceeds via the “handbag” diagram, Fig. 5b, andvarious decorations which generate log−Q2 dependences, αs(Q2) corrections and higher twistcorrections, examples of which are shown in Figs.

5c–f. All these involve only chirally-evenquark distributions because the quark couplings to the photon and gluon preserve chirality.Only the quark mass insertion, Fig.

5f, flips chirality. So up to corrections of order mq/Q,h1(x, Q2) decouples from electron scattering.Fig.

5There is no analogous suppression of h1(x, Q2) in deep inelastic processes with hadronicinitial states such as Drell–Yan. The argument can be read from the standard parton diagram17

Fig. 6for Drell–Yan (Fig.

6). Although chirality is conserved on each quark line separately, the twoquarks’ chiralities are unrelated.

It is not surprising, then, that Ralston and Soper found thath1(x, Q2) determines the transverse-target, transverse-beam asymmetry in Drell–Yan.The parton interpretation of h1 can be made transparent by decomposing the quark fieldswhich appear in Eq. (4.1) first with respect to the light-cone projection operator P± and thenwith respect to various spin projection operators which commute with P±.

The two cases ofinterest are first, the chirality projection operators, PL and PR,P LR ≡12 (1 ∓γ5)(4.2)which satisfyhP LR , P±i= 0 ,(4.3)and second, the transversity projection operators,35 Q±,Q± ≡121 ∓γ5γ⊥,(4.4)where γ⊥is either γ1 or γ2, and Q± satisfies[Q±, P±] = 0 . (4.5)18

As explained in Section II, h1 involves only independent light-cone degrees of freedom. Thehelicity structure of h1 described in Section II is apparent in a chiral basis because helicityand chirality coincide up to irrelevant mass corrections,h1(x) = 2xℜP ˆe1L†(xP)R(xP)P ˆe1(4.6)compared with f1 and g1f1(x) = 1xPR†(xP)R(xP) + L†(xP)L(xP) P(4.7)g1(x) = 1xP ˆe3R†(xP)R(xP) −L†(xP)L(xP)P ˆe3.

(4.8)In Eqs. (4.6) – (4.8) R(xP) and L(xP) are operators which annihilate independent light conecomponents of the quark field in eigenstates of PR and RL, respectively, with momentumk+ ≡xP + and integrated over ⃗k⊥.

* According to Eq. (4.7) f1(x) counts quarks with k+ =xP + irrespective of helicity, while from Eq.

(4.8) g1(x) counts quarks with helicity parallelto the target helicity minus those antiparallel. h1(x) is obscure in this basis.

If, instead, wediagonalize transversity, thenf1(x) = 1xPα†(xP)α(xP) + β†(xP)β(xP) P(4.9)h1(x) = 1xP ˆe1α†(xP)α(xP) −β†(xP)β(xP) P ˆe1(4.10)g1(x) = 2xℜP ˆe3α†(xP)β(xP) P ˆe3(4.11)where α(xP) and β(xP) annihilate independent components of the quark field in eigenstatesof Q+ and Q−, respectively.From Eq. (4.10) it is clear that h1(x) counts quarks with* The Q2-dependence of f1, g1 and h1 due to QCD radiative corrections can be restored(to leading logarithmic order) by integrating over ⃗k⊥only up to ⃗k2 <∼Q2.

For a completediscussion, see Ref. [12].19

k+ = xP + signed according to whether their transversity is parallel or antiparallel to thetarget transversity. In this basis the interpretation of g1(x) is obscure.The simple structure of Eqs.

(4.6) – (4.8) and (4.9) – (4.11) shows that transverse spineffects and longitudinal spin effects are on a completely equivalent footing in perturbativeQCD. Not knowing about h1(x), many authors, beginning with Feynman,36 have attempted tointerpret g⊥(x) as the natural transverse spin distribution function.

Since g⊥(x) is twist-3 andinteraction dependent, this attempt led to the erroneous impression that transverse spin effectswere inextricably associated with off-shellness, transverse momentum and/or quark-gluoninteractions.37 The resolution contained in the present analysis is summarized in Table 4 wherethe symmetry between transverse and longitudinal spin effects is apparent. Only ignoranceof h1 and hL prevented the appreciation of this symmetry at a much earlier date.LongitudinalTransverseSpinSpinTwist-2g1(x, Q2)h1(x, Q2)Twist-3hL(x, Q2)gT (x, Q2)Table 4:The symmetry of transverse and longitudinalspin distribution functions.It is useful to summarize some of the known properties of h1(x, Q2) and compare themwith analogous properties of g1(x, Q2).• Inequalities:g1(x, Q2) < f1(x, Q2)h1(x, Q2) < f1(x, Q2)(4.12)for each flavor of quark and antiquark.20

• Physical interpretation:h1(x, Q2) measures transversity. It is chirally odd and relatedto a bilocal generalization of the tensor operator, ¯qσµνiγ5q.

g1(x, Q2) measures helicity.It is chirally even and related to a bilocal generalization of the axial charge operator,¯qγµγ5q.• Sum rules:If we define a “tensor charge”2siδqa(Q2) ≡Ps¯qσ0iiγ5λa2 qQ2Ps,(4.13)where λa is a flavor matrix and Q2 is a renormalization scale, then δqa(Q2) is related toan integral over ha1(x, Q2),δqa(Q2) =Z 10dxha1(x, Q2) −h¯a1(x, Q2)(4.14)where ha1 and h¯a1 receive contributions from quarks and antiquarks, respectively. Theanalogous expressions for g1(x, Q2) involve axial charges,2si∆qa(Q2) ≡Ps¯qγiγ5λa2 qQ2Ps(4.15)∆qa(Q2) =Z 10dxga1(x, Q2) + g¯a1(x, Q2).

(4.16)Note the contrast: h1(x, Q2) is not normalized to a piece of the angular momentum tensor,so h1, unlike g1, cannot be interpreted as the fraction of the nucleons’ spin found on thequarks’ spin. Note the sign of the antiquark contributions: δqa is charge-conjugationodd, whereas ∆qa is charge conjugation even.

All tensor charges δqa have non-vanishinganomalous dimensions,38 but none mix with gluonic operators under renormalization.In contrast, the flavor non-singlet axial charges, ∆qa, a ̸= 0, have vanishing anomalous21

Fig. 7dimension, whereas the singlet axial charge ∆q0 ∝Σ has an anomalous dimension arisingfrom the triangle anomaly.17, 18, 10• Models:h1 and g1 are identical in non-relativistic quark models, but differ in relativisticmodels like the bag model — see Fig.

7.34• Role in polarized Drell–Yan:h1, g1 and their twist-3 counterparts gT and hL can be mea-sured in lepton-pair production with appropriately polarized beams and targets (Drell–Yan process). If both target and beam are longitudinally polarized,39ALL =Pae2aga1(x)g¯a1(y)Pae2af a1 (x)f ¯a1 (y) .

(4.17)If both target and beam are transversely polarized,33AT T = sin2 θ cos 2φ1 + cos2 θPae2aha1(x)h¯a1(y)Pae2af a1 (x)f ¯a1 (y)(4.18)and if one is longitudinal and the other transverse,34ALT = 2 sin 2θ cos φ1 + cos2 θMpQ2Pae2a (ga1(x)yg¯aT(y) −xhaL(x)h¯a1(y))Pae2af a1 (x)f ¯a1 (y). (4.19)22

In each case, AAB is the appropriate spin asymmetry. [The Q2-dependence of the distri-butions has been suppressed here.] The appearance of h1(x) at leading-twist (i.e.

scaling)in Eq. (4.18) illustrates its importance in processes in which it is not suppressed by achirality selection rule.

The explicit factor of m/pQ2 in Eq. (4.19) confirms the twist-3assignment of both gT and hL.• Measuring h1 in electroproduction:14h1(x) does not appear at leading twist in elec-troproduction because the diagrams of Fig.

5b – 5e preserve the quark chirality. Thequark mass insertion in Fig.

5f gives rise to a contribution of order m/Q to the mea-sured g2(x, Q2) obtained from electroproduction offa transversely polarized target. uand d quarks are abundant in the nucleon but have very small (current) masses.

Heavyquarks have non-negligible masses but are rare in the nucleon. In all, h1(x, Q2) makes anegligible impact on inclusive electroproduction.

[It is interesting to note, however, thath1(x, Q2) was discovered and its anomalous dimensions calculated in Ref. [38] on accountof this small contribution to electroproduction.

]This situation can be changed by observing a particle — most easily a single pion —in the current fragmentation region: ep →HX. The spin and twist properties of thisprocess depend on the quark fragmentation function as well as the distribution function.If no spin variables are measured in the final state, then two fragmentation functionscan enter: first ˆf1(x, Q2) which is the familiar twist-two fragmentation function.

It ischirally even, and measures the probability for a quark (a) to fragment into a hadron(H) with longitudinal momentum fraction z. [I have suppressed the labels a and H onˆf1.] ˆf1(z, Q2) is analogous to the distribution function f1(x, Q2).

Continuing the anal-ogy, there is a twist-three, chiral-odd fragmentation function, ˆe(z, Q2), analogous to the23

distribution function e(x, Q2) described in Section II. Polarizing the target transverselyto the beam, the distribution functions h1(x, Q2) and gT (z, Q2) can enter.

h1(x, Q2)can only contribute if ˆe(z, Q2) provides a compensating chirality flip. gT (z, Q2) does notflip chirality, and, to leading twist, fragments via ˆf1(z, Q2).

The resulting asymmetry isthe sum of two terms — [h1 ⊗ˆe] ⊕hgT ⊗ˆf1i— each of which is suppressed by O(1/Q)because one twist-three object (gT or ˆe) is required in each case:AHT (x, z, Q2) ∝ΛpQ2Pae2ahha1(x, Q2)ˆea/H(z, Q2) + gaT (x, Q2) ˆf a/H1(z, Q2)iPae2af a1 (x, Q2) ˆf a/H1(z, Q2). (4.20)Since ˆf1(z, Q2) can be measured in e+e−→HX and gT (x, Q2) can be measured ininclusive, polarized electroproduction, it is possible, at least in principle, to extract bothh1(x, Q2) and ˆe(z, Q2) from a measurement of the x- and z-dependence of the transversespin asymmetry in ep →eHX.14V.TWIST THREE:Measuring Quark-Gluon Correlations with gT (x, Q2)gT (x, Q2)gT (x, Q2) and hL(x, Q2)hL(x, Q2)hL(x, Q2)The twist-three quark distributions gT (x, Q2) and hL(x, Q2) are unique windows intoquark gluon correlations in the nucleon.

All higher twist effects probe quark-gluon correla-tions. gT and hL are unique in that they dominate certain observables, in contrast to generichigher-twist effects which must be extracted as corrections to QCD fits to leading twist.

Forexample, consider electron scattering from a nucleon target polarized at an angle α to theincident electron beam (Fig. 8).

The spin-dependent part of the cross-section is given by40d∆σdx dy dφ =e44π2Q2(cos α1 −y2 −y24 (κ −1)g1 −y2(κ −1)g2−sin α cos φs(κ −1)1 −y −y24 (κ −1) y2g1 + g2)(5.1)24

where x = Q2/2P · q, y = P · q/ME, κ = 1 + 4M 2x2/Q2 and φ is the dihedral angle betweenthe scattering plane and the plane defined by the beam and the target spin. As promised,effects associated with g1 scale, but effects of gT (= g1 + g2) fall at least like 1/Q.Anexperimenter can measure gT by (first measuring g1, then) orienting his target at 90◦to theelectron beam.

This should be contrasted with the elaborate theoretical analysis necessary toisolate higher-twist in (say) spin averaged electron scattering. Of course, the experiment isstill non-trivial: the asymmetry is suppressed relative to the rate by O(1/Q) necessitating ahigh statistics experiment.Fig.

8The precision with which the operator product expansion and perturbative QCD allowus to analyze electron scattering makes gT and hL very useful tools. Moments of gT andhL measure the expectation values of specific, well-defined local operators.

One must firstseparate out a “contamination” of twist-2 operators from gT and hL, namely41, 34gT (x, Q2) =Z 10dyy g1(y, Q2) + ¯g2(x, Q2)(5.2)25

hL(x, Q2) = 2xZ 10dyy2 h1(y, Q2) + ¯h2(x, Q2) . (5.3)This leaves ¯g2 and ¯h2 which depend explicitly on quark gluon interactions and on the gaugecoupling g, schematically,¯g2(x) ∼gDPs¯qγµ eGαβq PsE(5.4)¯h2(x) ∼gPs¯qiσαλγ5Gβλq Ps.

(5.5)To be more precise, for example,R 10 dx x2¯g1(x, Q2) is directly related to the nucleon matrixelement of the operator,42, 40θ[σ,{µ1]µ2} = g8¯ψeGσµ1γµ2 + eGσµ2γµ1ψ . (5.6)Similar sum rules for other moments of ¯g2 and ¯h2 can be found in Refs.

[42] and [34], respec-tively. Admittedly, at the present time we cannot compute the right-hand side of the sumrules.

However, models and more ambitious programs like lattice QCD would be aided byexperimental information on matrix elements such as Eq. (5.6).

A bag model calculation of¯g2(x) and ¯h2(x) is shown in Fig. 9 for the sake of a rough estimate.34Fig.

926

Twist-three distribution functions evolve with Q2 in a more complicated manner thanleading twist ones.Typically, at leading twist (and leading order), distributions obey anAltarelli–Parisi equation of the form,dd ln Q2 f(x, Q2) = αs(Q2)πZ 1xdy Pxyf(y, Q2)(5.7)with a perturbative “splitting function” P(x/y). If f(x, Q2) is known at Q2 = Q20 (withQ20 >> Λ2) it can be “evolved” to another Q2 using Eq.

(5.7). At the very least, Eq.

(5.7)allows experimenters to amalgamate data at a variety of different (large) Q2-values.The evolution of a twist-3 distribution like gT (x, Q2) is more complicated.42 It does notobey a simple evolution equation.Instead gT (x, Q2) is related to a pair of more generaldistribution functions, G(x, y, Q2) and eG(x, y, Q2), defined by relations likeZ dλ2πdµ2π eiλxeiµ(y−x)Ps ¯ψ(0)iDα(µn) n/ ψ(λn)Q2Ps= 2i ǫαβµνnβsµpνG(x, y, Q2) + lower twist(5.8)and similarly for eG(x, y, Q2). G and eG are generic twist-3 parton distributions.34 gT (x, Q2)is a simple projection:gT (x, Q2) = 12xZ 10dyheG(x, y) + eG(y, z) + G(x, y) −G(y, x)i.

(5.9)So twist-3 is inherently much more complicated than gT (x, Q2) suggests. It is a happy accidentthat only this particular projection of G and eG appears in electron scattering.

The difficultywith evolving gT (x, Q2) is that it is G(x, y, Q2) and eG(x, y, Q2) which obey Altarelli–Parisi-like integro-differential evolution equations. So a measurement of gT (x, Q2) at some Q2 =Q20 >> Λ2 does not provide enough information to predict gT (x, Q2) at some other largeQ2.43, 44 This “impediment to evolution” was recently stressed in Ref.

[44].27

In a recent paper, Ali, Braun and Hiller45 have suggested a way around this problem.They study the anomalous dimension matrices for all the local operators which contribute toG and eG. In the asymptotic limit of Nc →∞(Nc ≡number of colors) and x →1 they showthat gT (x, Q2) is an eigenfunction of the matrix evolution equations.

In simpler terms: for Nclarge and x near 1, gt(x, Q2) evolves approximately according to an Altarelli–Parisi equationlike (5.1), although the splitting function is not the naive one which would have been obtainedby ignoring the complexity of the problem (as was done, for example, in Ref. [46] in the earlydays of perturbative QCD).

The authors of Ref. [45] argue that for Nc not too large and xnot too near 1, their results remain approximately valid.

If they are right then gT (x, Q2) canbe evolved with Q2 more or less like a standard distribution function, making it possible tointerpret data taken at a variety of Q2 values in a systematic way.VI.CONCLUSIONSManipulation of the spin, twist and chirality dependence of deep inelastic processes offersus a new sensitivity to the details of nucleon structure. This varied and precise informationcomes as a consequence of our confidence in the formalism provided by perturbative QCD— yet another example of the adage that yesterday’s novelty is today’s tool (and tomorrow’sbackground!).

A carefully planned sequence of experiments including deep inelastic scatteringfrom a variety of targets (proton, deuteron, 3He, nuclei) in a variety of spin states, as well ashadronic processes such as polarized Drell–Yan, can give us much more detailed informationon the internal quark and gluon structure of the nucleon (and nuclei) than we presentlypossess. There is no other program which rivals it in precision or clarity of interpretationwithin the framework of QCD.

The experiments to date only scratch the surface of this richand challenging subject.28

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FIGURE CAPTIONSFig. 1: Generalized light-cone distribution function: a) the familiar two-particle case expressedas a probability; b) a generic three-particle distribution.Fig.

2: Forward scattering of a parton (quark or gluon) of momentum k and helicity h from atarget of momentum P and helicity H.Fig. 3: EMC data on g1(x) and its integral.Fig.

4: Graphical representation of gluonic contribution to Σ.Fig. 5: Chirality in deep inelastic scattering: a) Chirally odd contributions to h1(x); b)–e) Chi-rally even contributions to deep inelastic scattering (plus L ↔R for electromagneticcurrents); f) Chirality flip by mass insertion.Fig.

6: Chirality in Drell–Yan (plus L ↔R) production of lepton pairs.Fig. 7; Bag model estimates of h1(x) and g1(x).Fig.

8: Kinematics for polarized deep inelastic lepton scattering from a spin-1/2 target polarizedat an angle α with respect to the beam axis.Fig. 9: The proton’s twist-3 spin-dependent structure functions g2 and h2 in the Bag model: a)g2 and ¯g2; b) h2 and ¯h2.33

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