---

HARD SCATTERING IN QCD WITH POLARIZED BEAMS

arXiv:hep-ph/9207265v1 27 Jul 1992PSU/TH/100May 19, 1992HARD SCATTERING IN QCD WITH POLARIZED BEAMSJohn C. Collins*Physics DepartmentPennsylvania State UniversityState College, PA 16802, U.S.A.ABSTRACTI show that factorization for hard processes in QCD is also valid when the de-tected particles are polarized, and that the proof of the theorem determines theoperator form for the parton densities. Particular attention is given to the case oftransversely polarized incoming hadrons.1.

INTRODUCTIONPerturbative QCD is by now a staple ingredient of most phenomenology of high energycollisions. Mostly the applications have been to the case that the incoming particles areunpolarized.

However, in recent years much more attention has been devoted to the polarizedcase. Unfortunately, there has been confusion as to the exact status of the factorizationtheorem in the polarized case.

This is the theorem that underlies almost all perturbativecalculations. The confusion has particularly extended to the question of the correct definitionof the parton distribution functions [1, 2, 3].In actuality, when one examines the proofs of the factorization theorem [4, 5], one findsthat almost no reference is made to the polarization of the measured particles.

So the purposeof this paper is to explain that the factorization theorem does indeed apply, and that itdetermines, essentially uniquely, the definition of the parton distributions. (The ambiguityin the definition is directly tied to the well-known freedom to choose the renormalization* E-mail: collins@phys.psu.edu orBITNET: collins@psuleps.1

and factorization scales, and is entirely independent of the problems of polarization.) Therehas been particular confusion about the case about the case that the incoming beams havetransverse polarization.

For example of the supposed problems, see the discussion in thebook by Ioffe et al, [3]. Better and more recent discussions of some of the issues can befound in [6, 7, 8, 9, 10, 11].The processes with which we are concerned are those with a hard scattering: deeplyinelastic lepton scattering, the Drell-Yan process, jet production in hadron collisions, etc.For the sake of definiteness, I will treat in this paper two specific cases: the structurefunctions for deeply inelastic lepton scattering, and the Drell-Yan process.All the pieces of the discussion can be found in the literature.

What has been lacking isa unified presentation. In this paper, I shall be concerned solely with the ‘twist-2’ behaviorof hard scattering cross sections.

This means that at each order of perturbation theory Iconsider the terms that scale with energy like their dimension (modified by logarithms).‘Higher twist’ terms—power suppressed terms—will be ignored; they are very interesting,but their study is much harder than that of the twist-2 terms.The terminology of twist has its origins in the operator product expansion for deeplyinelastic lepton scattering, and is somewhat inappropriate here. But the usage has stuck.2.

STATEMENT OF FACTORIZATION THEOREMIn this section, I will formulate the factorization theorems for deeply inelastic scattering andfor the Drell-Yan cross section. These are typical of the most general case of factorization.2.1 Structure Functions for Deeply Inelastic ScatteringThe structure functions for deeply inelastic scattering are defined in terms of the structuretensor Wµν(p, q) by:Wµν =−gµν + qµqν/q2W1(x, Q2) +pµ −qµp · q/q2 pν −qνp · q/q2 W2(x, Q2)M2+ iM ǫµνρσqρsσG1 + p · qM2 G2−s · qpσ 1M2G2.

(1)Here, M, pµ and sµ are the mass, momentum and spin vector of the target, and qµ is themomentum of the exchanged virtual photon. (The case of a more general exchanged boson,2

like a W or a Z gives more structure functions, a situation that is an inessential complicationfor the present purpose.) As usual, we define Q2 = −q2, ν = p · q, and x = Q2/2p · q. Wewill be interested in the Bjorken limit: Q →∞with x fixed.

Our normalization of the spinvector is such that a pure state has s2 = −1.As for the spin-dependent structure functions, we will consider in this paper only thecase that the target has spin12. Then target’s spin state (no matter whether mixed or pure)is completely determined by its spin vector sµ, and there are exactly two polarized structurefunctions, G1 and G2.

(For the case of a target of general spin, see [12].) The normalizationof sµ is that it satisfies 0 ≥s2 ≥−1 and s · p = 0, and that a pure spin state has s2 = −1.Later, when we do power counting to determine the sizes of the leading contributionsto the structure functions, it will be convenient to work in the center-of-mass frame.

Thenin the Bjorken limit, we find that components of pµ and of qµ are of order Q. But also thecomponents of sµ become large.

So, following Ralston and Soper [6], we decompose sµ interms of a helicity λ and a transversity s⊥:sµ = λ pµM −qµMp · q!1q1 + 2xM2/p · q+ s⊥µ,(2)where the helicity λ is defined byλ = q · sMq · p1q1 + 2xM2/p · q,(3)and sµ⊥is orthogonal to both pµ and qµ. Then −sµsµ = λ2 + |s⊥|2, while both |λ| and |s⊥|are less than unity.

For a pure state, λ2 + |s⊥|2 = 1.It is common to call s⊥the transverse spin. However, moving particles are not in aneigenstate of transverse spin, as Jaffe and Ji [9] explained, but may be in an eigenstate of thetransverse components of the Pauli-Lubanski vector; these transverse components are calledtransversity.

Note that our spin vector sµ is proportional to the Pauli-Lubanski vector. )Polarized particles in a high energy accelerator are typically transversely polarized and carrya definite value of the spin vector.

Thus they are in a state of definite transversity. Note thatthe concept of ‘transverse’ in this context is not Lorentz invariant, when referred to a single3

particle. It is only defined when one brings another vector into the situation to represent(say) the center-of-mass of the scattering.At various stages, we will need to take the limit of zero mass, and the decomposition (2)exhibits potential singularities at M = 0.

In this limit, we may approximate eq. (2) bysµ = λpµM + s⊥µ + power law correction,(4)The factorization properties are most easily expressed in terms of scaling structure func-tions which are defined by rewriting eq.

(1) asWµν =(−gµν + qµqν/q2)F1(x, Q2) + (pµ −qµp · q/q2)(pν −qνp · q/q2)p · qF2(x, Q2)+ iMp · qǫµνρσqρsσg1 +iM(p · q)2ǫµνρσqρ(p · qsσ −s · qpσ)g2. (5)Here, F1 ≡W1, F2 ≡p · qW2/M2, g1 ≡p · qG1/M2 and g2 ≡p · q2G2/M4.In the Bjorken limit, it is convenient to use the decomposition eq.

(2) of the spin, so thatthe spin-dependent part of Wµν is:W polµν =λiǫµνρσqρpσp · q1q1 + 2xM2/p · q g1 −2xM2p · q g2!+ iǫµνρσqρsσ⊥Mp · qg2=λiǫµνρσqρpσp · qg1 + λiǫµνρσqρsσ⊥Mp · qg2 + non-leading powers. (6)We will see that in the Bjorken limit, Q →∞with x fixed, each of F1, F2, g1, and g2scales like Q0 times logarithms.

Each of these scaling structure functions is dimensionlessand, except for g2, its definition in terms of the tensor does not involve the mass of the target.The exception for g2 is a choice that directly reflects the fact that its leading contribution isassociated with operators of twist-3 rather than twist-2: redefining it to remove the factorof M would reduce its power law by one power of Q.2.2 Factorization for Unpolarized Deeply Inelastic ScatteringThe factorization theorem [4] for deeply inelastic lepton scattering applies in the Bjorken4

limit, and for the unpolarized structure functions it gives:F1(x, Q2) =XaZ 1xdξξ fa/A(ξ, µ) H1a xξ , Qµ , αs(µ)!+ remainder,1xF2(x, Q2) =XaZ 1xdξξ fa/A(ξ, µ) ξxH2a xξ , Qµ , αs(µ)!+ remainder. (7)This theorem asserts that in the Bjorken limit, the target may be regarded as a beam ofpartons, and that the scattering really takes place on these partons.

The remainder termsare a power of Q smaller than the leading terms.The quantities fa/A(ξ) are parton distribution functions (or parton densities).Theiroperator definition, which will be given below, can be interpreted in light front quantizationas the number density of partons as a function of the light-cone fraction of the momentumof the parent hadron.The functions H1a and H2a should be regarded as the short-distance part of the structurefunctions for a parton target of flavor a (gluon, quark or antiquark). In lowest order in αs,each Hia is a delta function at ξ = x times the charge squared e2a of the parton:H1a(x/ξ) = 12e2aδ(x/ξ −1) + O(αs),H2a(x/ξ) = e2aδ(x/ξ −1) + O(αs).

(8)The first term in these perturbation expansions gives the parton model approximation toQCD, with F1 = 12Pa e2afa/p(x) and F2 =Pa e2axfa/p(x). In higher order, the Hia are thestructure functions at the parton level, but with subtractions made according to a standardprescription, to remove the non-ultraviolet contributions.The appearance of the Dirac delta function in eq.

(8) (and of more complication gener-alized functions in higher orders) implies that the asymptotic behavior given by eq. (7) is tobe interpreted in the sense of distribution theory– cf.

[13].The factors of 1/x and of ξ/x in the equation for F2 arise from the dependence on thetarget momentum of the definition of the structure function F2.2.3 Structure Functions v. Parton DistributionsI make a clear distinction between the concepts of a ‘structure function’ and a ‘partondistribution function’. A structure function is a term in a decomposition of the deep inelastic5

cross section, as in eq. (5); it is experimentally measurable.

A parton distribution function(or parton density) is a number density of quarks (or gluons) in a fractional momentumvariable. Mathematically it is a hadron expectation value of a certain operator; as such, itis a theoretical construct.However, the parton model gives the structure functions in terms of certain simple linearcombinations of quark and antiquark densities.

[For our purposes, the parton model isthe (useful) approximation in which one neglects the perturbative corrections to the hardscattering coefficients in the factorization formulae.] It has therefore become common in theliterature to identify the concepts of structure function and parton distribution.This identification has particularly disastrous consequences for the discussion of polarizedscattering with transversely polarized hadrons: The transverse spin contribution to deepinelastic scattering at the level of approximation of the leading term in eq.

(7) is exactlyzero (to all orders of perturbation theory), as we will review below. Attempts to identify thestructure function g2 with some kind of transverse spin distribution result in inconsistencies.2.4 Factorization for Drell-YanThe factorization theorem for the Drell-Yan process is typical of factorization theorems formore general hard scattering processes, and it is formulated as follows.The process is the inclusive production of a lepton pair of high invariant mass via anelectroweak particle.The classical case is with a high-mass virtual photon: H + H →γ∗+ anything, with γ∗→e+e−or γ∗→µ+µ−.

The cases of W and Z production can betreated in an essentially identical fashion.We let s be the square of the total center-of-mass energy and qµ be the momentum ofthe γ∗. The kinematic region to which the theorem applies is where √s and Q get large ina fixed ratio.

(Q isqq2.) The transverse momentum q⊥of the γ∗is either of order Q or isintegrated over.In the case that q⊥is integrated over, the factorization theorem for the unpolarized6

Drell-Yan cross section reads:dσdQ2dydΩ=Xa,bZ 1xAdξAZ 1xBdξB fa/A(ξA, µ) Hab xAξA, xBξB, θ, φ, Q; µQ, αs(µ)!fb/B(ξB, µ)+ remainder,(9)where y is the rapidity of the virtual photon and dΩis the element of solid angle for thelepton pair: the polar angles for this decay are θ and φ relative to some chosen axes. Thesums over a and b are over parton species, and we writexA = eysQ2s ,xB = e−ysQ2s .

(10)The function Hab is the ultraviolet-dominated hard scattering cross section, computablein perturbation theory. It plays the role of a parton level cross section and is often writtenasHab =dˆσdQ2dydΩ,(11)where the hat over the σ indicates a hard scattering cross section at the parton level.

Theparton distribution functions, f, are the same as in deeply inelastic scattering. Fig.

1 illus-trates the factorization theorem.Fig. 1.

Factorization theorem for Drell-Yan cross section.7

2.5 TwistIn both of the above factorization theorems, the dependence of the hard scattering coefficients(H1a etc) on the large momentum Q is of the form Qp times logarithms of Q, where p isthe dimension of the hard scattering coefficient.This is true for each separate order ofperturbation theory in αs. For the coefficient Hab for Drell-Yan we have p = −4, and for H1and H2 for deeply inelastic scattering we have p = 0.

The quantities multiplying the hardscattering coefficients are dimensionless parton densities. Immediate consequences are thestandard scaling laws that F1 and F2 behave like Q0 and that the Drell-Yan cross sectiondσ/dQ2dy behaves like Q−4 in the scaling limit, apart from the usual logarithmic scalingviolations.Terms of this kind, we will label ‘twist-2’ as a generalization of the usage in the operator-product expansion for deep inelastic scattering.

(Twist is the dimension minus the spin ofthe operators.) The remainder terms in the factorization theorems are therefore called highertwist.When we consider scattering with transversely polarized hadrons, there are some pro-cesses for which the twist-2 term is exactly zero.

A notorious example is the single transversespin asymmetry of high p⊥particle production in hadron-hadron scattering. Another caseis the structure function g2 for deeply inelastic scattering.

For these processes the leadingtwist is twist-3, and the corresponding asymmetries are proportional to some hadronic massscale divided by Q at large Q. The choice of the dimensional factor multiplying g2 in eq.

(5)to be M/p·q2 rather than 1/p·q3/2 is an expression of this fact.2.6 Longitudinal PolarizationThe factorization theorems stated above are for unpolarized incoming hadrons, and theyinvolve an incoherent sum over parton types. In the case that the incoming hadrons arepolarized, the theorems need generalization.For the case of longitudinal polarization, the factorization statements can be readilyformulated simply by simply extending the sum over parton types a (and b) to include asum over parton helicities.

The unpolarized parton densities will be a sum over the helicity8

densities. (The asserted theorem is still in need of the proof which will be summarized later.

)For Drell-Yan, the hard-scattering coefficient in eq. (9) should be treated as a helicity-dependent cross section at the parton level.For deeply inelastic scattering, the formulae for F1 and F2 will remain unchanged, sincethese structure functions and the corresponding parton level structure functions are, by defi-nition, spin independent.

But there will be a factorization formula for the helicity dependentstructure function g1, and this will involve the helicity asymmetry of the parton densities.Extra polarized structure functions beyond g1 will be needed for the case of a target of spingreater than12. [12]2.7 General Polarization, Including TransverseNow, a characteristic of the quantum mechanical theory of spin is that interference andcoherent phenomena occur even in circumstances where the physics is otherwise classical.Such is the case for the factorization of hard processes when the detected hadrons have ageneral polarization.

The problem is one of interference between scattering of partons ofdifferent quantum numbers.For the flavor quantum numbers of partons there is no such interference. For example,we have no contribution to F1 from an interference between scattering on an up quark anda down quark:⟨u + γ∗|final state⟩⟨final state|d + γ∗⟩.

(12)The reason is that an examination of the flavor of the final state of the hard scattering issufficient to determine which kind of parton initiated the hard scattering. Alternatively,one can examine the term in an operator definition of a parton distribution that would beappropriate to an interference term:⟨p|¯u · · · d|p⟩.

(13)Quark number conservation forces this to be zero. The dots indicate factors that are irrele-vant to the flavor structure.But for spin, there are no such constraints: The final states that can be produced fromthe scattering of left-handed quarks can be the same as the final states from right-handed9

quarks, and therefore the amplitudes can interfere. To take account of this, we must equipthe partons entering the hard scattering with a spin density matrix.Therefore the fullspecification of a parton distribution is given by the number density of partons togetherwith the parton’s density matrix.

The number density times the density matrix has lineardependence on the spin density matrix of the initial hadron. Longitudinal polarization givesthe special case that the density matrices of both the parton and the hadron are diagonal ina helicity basis.

(Note that the use of a density matrix allows the state of a deeply inelasticscattering parton to be either pure or mixed: In an inclusive cross section, where we sumover unobserved parts of the final state, a parton that enters the hard scattering can be ina mixed state even when its parent hadron is in a pure state. )The most general form for the factorization theorem for Drell-Yan isdσdQ2dydΩ=Xa,bZ 1xAdξAZ 1xBdξB ρa(αα′)/Afa/A(ξA, µ)Ha(αα′)b(ββ′) xAξA, xBξB, θ, φ, Q; µQ, αs(µ)!ρb(ββ′)/Bfb/B(ξB, µ)+ remainder.

(14)Here ρi(αα′)/H is the density matrix of partons of flavor i in hadron H, with α and α′ beingthe helicity indices of the matrix. The density matrix is of course a function of the samevariables ξ and µ as the number density fi/h(ξ, µ).

The factorization (14) differs from theunpolarized case by the presence of spin density matrices ρ for the partons and by thedependence of the hard scattering coefficient on the spin indices, α, α′, β, and β′. Thedensity matrix of a parton is necessarily a linear function of the spin vector of its parenthadron.Calculations of amplitudes in perturbation theory are often made in a helicity basis [14].In that case, it is convenient to work directly with density matrices in a helicity basis.

Theindices α, α′ etc will take on the values + and −(or L and R for left- and right-handedpolarization).Another method is to work with cut Feynman graphs, for the cross section. In thatcase, one uses a polarization sum for initial state quarks written in terms of the quark’s spin10

vector:12p/ (1 −λγ5 + γ5s/⊥) . (15)(Our conventions here are those of Bjorken and Drell [15].) The density matrix of the quarkis then12 1 + λsx + isysx −isy1 −λ,(16)where we have chosen the z-axis to be along the 3-momentum of pµ.

For the case of a spin-12hadron, the helicity λa of a quark is proportional to the helicity λA of it parent hadron, andsimilarly for the transversity:λa = ∆L,a/AλA,(17)s⊥a = ∆T,a/As⊥A. (18)Here ∆L and ∆T are the longitudinal and transverse spin asymmetries are a quark in a fullypolarized hadron.

They are functions of the variables ξ and µ, of course, and these asym-metries give the quark densities defined by Jaffe and Ji [9] by weighting by the unpolarizeddensities:fLa/A = ∆La/Afa/A,(19)fT a/A = ∆T a/Afa/A,(20)Jaffe and Ji use the notation g1a/A instead of fLa/A, but this identification invites confusionwith the g1 structure function, to which it is directly related in the parton model. I havealso changed Jaffe and Ji’s notation h for the transversity distribution to fT to correspondwith fL.We will give the operator definitions of the parton densities below, exactly as stated byJaffe and Ji, and we will justify that these are the correct definitions to use in the factorizationtheorems.

These densities satisfy linear evolution equations (Gribov-Lipatov-Altarelli-Parisi)of a similar structure to the ones in the unpolarized case.Exactly corresponding considerations apply to gluons. They also have two physical po-larization states.

(As we will review below, it is the states of a massless parton that are11

relevant for the factorization theorem.) However, the gluon has spin one, so that the offdi-agonal terms in its density matrix correspond to linear rather than transverse polarization.Moreover, an operator measuring these offdiagonal elements has helicity two, unlike the op-erator for quarks, which has helicity one, as can be seen from the operator definitions below.Thus it is a consequence of angular momentum conservation, as explained by Artru andMekhfi[7], that there are no linearly polarized gluon partons in a spin-12 hadron.

Further-more, the evolution equation for transversely polarized quarks has no mixing with gluons,and vice versa.This result depends on the azimuthal symmetry of the operators measuring the partondensities. One way of evading the result is to use a process sensitive to the intrinsic transversemomentum of the partons [16].2.8 Consequences of Chiral SymmetryIn QCD there are chiral symmetries that if unbroken would actually prohibit some of theinterference terms that would otherwise occur with transverse polarization.

At the level ofthe hard scattering, as in (12), one is working in perturbation theory with quark massesneglected. Thus the chiral symmetries are exact for the hard scattering coefficients at thetwist 2 level.An important manifestation of this is conservation of quark helicity in massless pertur-bation theory.

As we will see below, a particular consequence of this is that the twist 2contribution to the structure function g2 is exactly zero. That is, the transverse spin asym-metry in ordinary deeply inelastic scattering is of the order of a hadron mass divided by Qfor large Q.But one can easily conceive of hard scattering with two partons in the initial state, andthen there is no reason for the interference terms to vanish [6].At first glance, the same reasoning might appear to apply to the parton densities, asdefined by (13).

But there we are dealing with nonperturbative quantities. Hence the offdiagonal terms in the density matrix are explicitly allowed for two reasons.

First the chiralsymmetry is definitely broken in the nonperturbative part of QCD. Secondly, in the helicity12

basis, which is natural to use, the density matrix for the hadron is itself offdiagonal, so theycan be no constraint prohibiting offdiagonal terms in the quark density matrix.3. SUMMARY OF PROOFThe proof of a factorization theorem is made by considering a cross section as the sum overall cut Feynman graphs for the process in question.

Ideally, one would like to extend theproof to handle nonperturbative contributions. But that extension has yet not been made.The formulation of the theorem is in fact general enough to allow such an extension,and the physical picture it gives is completely reasonable.

In particular, the definitions ofthe parts of the factorization formulae that are to be used nonperturbatively, viz, the partondensities, are valid beyond perturbation theory: The parton densities have a gauge invariantdefinition in terms of hadron matrix elements of certain operators.Furthermore, in the whole of our discussion, there is no restriction on which kinds ofparticle compose the initial state, except that they should be gauge invariant and physical. Inperturbative calculations we will typically use on-shell, physically polarized gluon and quarkstates, but in principal we could also use hadron states with a bound-state wave function.For the general theory, it will make no difference.

The hard scattering functions are the onlyquantities for which a purely perturbative calculation makes sense, and for them the initialstates are always on shell, massless partons.The steps in the proof [4] are:1. Power counting.

Apply the method of Libby and Sterman [17] to determine those regionsof integration momenta of the cut graphs that give leading contributions.2. Cancellation of superleading regions.

There are contributions in which all the partonscoupling to the hard scattering are gluons with an unphysical scalar polarization. Un-fortunately, these give a power law larger than the final result.

Ward identity methodsare used to show that these contributions exactly cancel among themselves.3. All remaining contributions have a form like that of the parton model, with two gener-alizations.

First, the hard scattering is not restricted to be in the Born approximation.13

Secondly, there may be soft gluons connecting the lines associated with initial hadronsamong themselves and with initial and final state lines in the hard scattering, and theremay be extra collinear gluons with scalar polarization connecting to the hard scattering.4. Cancellation of final-state interactions.

All interactions that are too late to affect theinclusive cross section must cancel. For example, hadronization of final-state jets doesnot affect the totally inclusive structure functions.

Hence the partons initiating thesejets effectively have virtuality of order Q25. Taylor expansion.

The hard scattering is expanded in powers of the small components ofits external momenta, and in the mass parameters of its internal lines. The subgraphs ofcollinear lines (‘jet subgraphs’) are expanded in powers of the relatively small componentsof the soft lines that connect them to other jet subgraphs.6.

Cancellation of soft gluons.A Ward identity argument is used to factorize the softgluons, after which a unitarity cancellation applies.7. Factorization of collinear scalar gluons.

This again goes by a Ward identity argument.8. At this point we have jet factors and hard scattering factors.

One now applies com-binatoric arguments in the same way as in Wilson’s expansion to get the factorizationtheorem [18, 19, 20].9. The hard scattering can now be identified as a cross section for scattering of on-shellpartons with subtractions to remove the non-ultraviolet contributions.10.

Operator definition of parton densities. The jet factors can now be shown to be exactlyhadronic expectation values of certain operators.

The precise form of these operatorsis completely determined by the Taylor expansion at step 5. The operators are bilocaloperators that have simple interpretations in light front quantization.

The terms thatwere obtained in step 7 from the collinear scalar gluons turn these into gauge invariantoperators.3.1 Use of Physical GaugeIt is possible to use a physical gauge in trying to prove factorization. Several of the unpleasantsteps involving gluons with scalar polarization can then be omitted, and the result appears14

to be a much simpler proof.However, there are unphysical singularities that prevent the unitarity arguments in step4 from being applied in as strong a manner as is needed. Furthermore, the Ward identityarguments to cancel soft gluons rely on contour deformations that are obstructed by thesesame singularities.

Thus it seems best to work in an ordinary covariant gauge and to acceptthe added complications [4, 5].4. POWER COUNTINGLibby and Sterman [17] showed that to classify the important regions of momentum spacein a high energy limit it is useful to measure momenta and masses in units of the largemomentum scale Q.

Thus one writes a generic momentum and mass in the formkµ = Q˜kµ,m = Q ˜m. (21)By simple dimensional analysis, the large Q limit is equivalent to the limit of zero mass andon-shell external momenta; for example a cross section might be writtenσ(Q2; m, k) = Q−2σ(1; ˜m, ˜k),(22)with ˜m →0 and ˜k2 →0 when Q →∞Thus the complications of a high-energy limit can beinvestigated by examining the singularities in zero-mass limit.

The method of Coleman andNorton [21] shows in a physically appealing fashion how to determine the configuration ofloop momenta that give these singularities. Tkachov and collaborators [13, 19] have shownhow systematic exploitation of this idea can considerably streamline proofs of the operatorproduct expansion and of other results on the asymptotics of Euclidean Green functions.In our case, the significant configurations involve internal lines of three kinds: (a) Linesthat carry momenta collinear to momenta of external particles.

(b) Lines that carry softmomenta. (c) Lines that carry large ultraviolet momenta.

(If only large momentum lines areimportant, then the cross section under discussion is infra-red safe, and the problem may betreated by classical renormalization-group methods. )15

Then the importance of each configuration is determined by expanding in powers of asuitable small variable λ about the singular point in the massless limit.Let us now examine these arguments as applied to deeply inelastic lepton scattering. Wewill present them in a sufficiently general manner, that the extension to other processes willbe simple.4.1 Deeply Inelastic Scattering at Twist-2 LevelTo explain the power counting, it will be convenient to use light-front coordinates in whichp+ = Q/x√2,p−= M2/Q√2,p⊥= 0,(23)and−q+ = q−= Q/√2,q⊥= 0.

(24)(Our metric is such that V 2 = 2V +V −−V 2⊥for any vector V µ. )Let us also assume that we sum over cuts of graphs for the structure tensor beforeapplying the Libby-Sterman argument to deeply inelastic scattering.

The sum over cutsmeans that no final-state interactions need enter our argument.At the leading power of Q, contributions to the structure tensor come from regionssymbolized in fig. 2.

In the upper part of this diagram, which we will call the hard subgraphH, all the internal lines have large momenta, that is the scaled momenta have virtuality oforder unity. In the lower part, which we will call the collinear or jet subgraph, J, all thelines have momenta collinear with pµ.

That is, the corresponding scaled momenta are closeto a light-like vector with only a nonzero + component. All but two of the lines joining thesubgraphs are gluons with scalar polarization.It is a relatively simple generalization [17] of the arguments below that shows thatgraphs with extra quarks and/or transversely polarized gluons joining the two subgraphs aresuppressed by a power Q for each extra line.4.2 Simple Quark ConnectionFirst we consider the case that there is just a single quark line connecting the subgraphs Hand J on each side of the final-state cut.

As we now show, it is fairly easy to massage the16

Fig. 2.

Regions for twist-2 contributions to for deep inelastic scattering.contribution of fig. 2 into a parton-model-like form that implies the scaling properties of thestructure functions that were stated earlier.We must first find the leading part of the trace over Dirac matrices.

For this purpose, wedecompose the top part — the hard subgraph H — and the bottom part — the jet subgraphJ — according to the Dirac structure on the fermion line connecting the two subgraphs:J = JS1 + JVκ γκ + JTκλσκλ + JPVκγκγ5 + JPSγ5,H = HS1 + HVκ γκ + HTκλσκλ + HPVκγκγ5 + HPSγ5. (25)In the last line we have suppressed the indices µν of the structure tensor.

When we performthe trace over the Dirac matrices and the integral over the explicit loop momentum kµ, wefind that:W =Zd4k(2π)4 tr(JH)=Zd4k(2π)4 4JSHS + JV · HV −2JT · HT −JPV · HPV + JPSHPS. (26)We suppose that we are looking only at the region of momentum appropriate to fig.

2, sothat in particular k+ = O(Q), and |k−|, |k⊥| ≪Q. The jet part J depends on the momentapµ and kµ, and on the hadron’s spin state defined by λ and s⊥.

In the rest frame of thehard scattering, all the vectors involved in H have components of order Q, so that we mayregard all components of the decomposition of H as being of order QD, where D is the massdimension of H.17

As we increase Q while keeping the longitudinal momentum fraction, virtuality andtransverse components of kµ fixed, we may consider J as being obtained by a boost fromthe rest frame of pµ. Then the terms in the decomposition of J scale with Q as follows:J+V , J+A, J+iT∝Q1,JiV , JiA, J+−T, JijT , JS, JPS ∝Q0,J−V , J−A, J−iT∝Q−1,(27)the proof following from the effect of boost transformations.

The indices i and j refer topurely transverse components.It follows that the leading terms in the trace in eq. (26) are given byW =Zd4k(2π)4 4JV +HV −−2JT +iHT −i −JPV +HPV −(1 + O(mass/Q × logarithms)) .

(28)4.3 Relation to Quark DistributionIn fig. 2 we aim to associate the subgraph H with a contribution to the hard scattering coef-ficient and the subgraph J with contribution to a parton distribution.

Since all componentsof momentum inside H are of order Q, we may, within H, neglect the transverse momentumand virtuality of kµ and writeW =Zdξ tr H(q, ξ)"p+Z dk−d2k⊥(2π)4J(k, p, s)#+ nonleading power,(29)where ξ ≡k+/p+, while H has been approximated by something with an incoming onshellquark that has zero transverse momentum. We also set the quark masses in H to zero.It remains to discuss the polarization structure.In the frame we have chosen, it is manifest that the leading power for tr JH comes fromthe terms in eq.

(28). To relate this to a standard spin projection for Dirac particles, recallthe conventional projection for a spinor wave function:(p/ + m) (1 + γ5s/).

(30)18

This is singular in the zero mass limit. After application of the decomposition (2) in termsof helicity and transverse spin, an expression is obtained that has a well-behaved zero-masslimit:p/ (1 + γ5s/⊥−λγ5) .

(31)After some reorganization of the γ matrices implicit in eq. (28), we get a contribution ofthe formWµν =Z 1xdξξ12 tr Hµνˆk/[1 + γ5(λq + s/q⊥)]f(ξ) + twist higher than 2.

(32)where H means the massless on-shell limit of H, and ˆkµ ≡(ξp+, 0, 0⊥). We have restoredthe µν indices that correspond to the external photon.The quantity f(ξ) in this equation represents the contribution of the lower part of fig.

2to the quark density. We need to sum over all possible graphs and all possible regions, andto apply the same combinatoric arguments as for the operator product expansion.

Then weshould expect to get the following definition of the quark density:f(ξ) =Z dk−d2k⊥(2π)4tr J(k, p, s)γ+2 ,(33)while the quark helicity, λq, and quark transversity, sq⊥, are defined byλqf(ξ) =Z dk−d2k⊥(2π)4tr J(k, p, s)γ5γ+2,(34)andsq⊥f(ξ) =Z dk−d2k⊥(2π)4tr J(k, p, s)γ+2 γ5γ⊥. (35)The reasoning given above for these to be appropriate definitions can be found in [6], wherereferences to earlier work on unpolarized parton densities in light front quantization can befound.Equation (32) is clearly of the form of the desired factorization theorem.

Moreover, thetrace with H in eq. (26) is such that the result has the normalization of the structure tensorfor deep inelastic scattering offa quark target with momentum ξp, helicity λq and transversespin sq⊥.

Furthermore fi(ξ) must be interpreted as the number density of partons. (The19

factor 1/ξ in eq. (32) is needed to interpret f(ξ) as a number density because of the relativisticnormalization of the states.) There will be technicalities to generalize these results to realQCD, but the power counting arguments will remain unchanged.4.4 Transverse PolarizationA twist-2 contribution to the structure functions (M/√p · q)g2 and g1 is one that is oforder Q0 (modulo the usual logarithms), since they are dimensionless: all the factors in thetensors multiplying them in eq.

(7) are dimensionless ratios of momenta that are of orderQ. The power counting argument just presented shows that our basic expectation is that(M/√p · q)g2 scales like Q0.

(In doing the power counting, we consider in the first instance, the structure tensor Wµν,which is dimensionless. Then we derive results for the structure functions by considering thepossible tensors in eq.

(5), but treating the tensors in combinations that scale as Q0 in theBjorken limit. The coefficients of these tensors then also scale as Q0 (times logarithms).

Theunpolarized structure functions F1 and F2 are such coefficients. Since the spin vector for alongitudinally polarized proton can be taken as λpµ/M, the coefficient of g1 is iλǫµνρσqρsσ/p·q, so that g1 scales as Q0.

But the part of the spin vector that goes with g2 is the transversepart sµ⊥, which is invariant when one goes to the Bjorken limit; thus it is the combination(M/√p · q)g2 that must be considered in our scaling argument. )Now, when one actually performs the calculation of Feynman graphs to the leading powerof the hard subgraphs H, one gets zero for the part corresponding to g2.

The most basicway of seeing this is to observe that the leading power of Q is given by inserting masslesspropagators everywhere in the hard part. Then H must contain an odd number of Diracmatrices, and this gives zero in the trace with k/γ5s/p⊥in eq.

(32). (Similar reasoning showsthat the other two terms in eq.

(32) are generally nonzero.) This argument works to allorders of perturbation theory, and demonstrates that (M/√p · q)g2 is suppressed by at leastone power of Q.

Since the first nonleading power term is presumably nonzero, it is in factg2, with the conventional definition, that scales.A fancier way of saying the same thing is to observe that in the zero mass limit, both20

QCD and the electromagnetic vertices are chirally invariant.Chiral invariance prohibitshelicity flip for the quarks, and g2 corresponds to an offdiagonal term in the density matrix,so that it necessarily involves helicity flip.If one calculates Feynman graphs for H, using an on-shell projection, but leaving thequark masses nonzero, then the result is of course proportional to the quark mass. However,to treat this as the dominant contribution to g2 is an entirely incorrect application both ofparton model ideas and of QCD.

In the first place, as the above argument makes clear, thereare other terms in the projection over Dirac matrices besides the ones that give the twist-2terms. These terms cannot be interpreted as the product of the scattering of on-shell quarkstimes a quark number density.

Rather they correspond to coupling to the matrix elementsof the twist-3 operators that were listed by Jaffe and Ji [9]. Furthermore, there are regionsother than those of fig.

2 that contribute at the twist 3 level; the corresponding operatorsinvolve, for example, the correlation of a gluon with quarks [10, 11].When one attempts to force a connection in the standard fashion between a transversespin dependence of the quark densities and the transverse structure functions, contradictionsarise [3, 2]. These have given an undeserved idea in the folklore that transverse spin cannotbe treated within the parton model and its QCD realization.It should also be clear that if one represents the size of the twist 3 contributions as beingof order M/Q relative to a typical twist 2 term, then M should be some kind of hadronicmass scale, hundreds of MeV, at the least.

The effect of putting a current quark mass intothe calculation of the hard subgraph is a rather small effect, and hardly can be expected tobe the dominant nonleading contribution.4.5 Most General CaseTo the extent that the gauge properties of QCD are irrelevant, the argument given abovecan be readily turned into a full proof. (One has a minor generalization that it is necessaryalso to consider the possibility of gluon lines joining the jet and hard subgraphs.

)Moreover, the argument can be further generalized, for example to processes like Drell-Yan with two hadrons in the initial state. If both partons are both transversely polarized,21

then the helicity conservation argument no longer prohibits a transverse spin dependence ofthe cross section for such a process, rather the contrary.Our argument shows clearly that there is no problem in defining the concept of a trans-versely polarized quark. Such an argument was (to my knowledge) first constructed, in thecontext of the Drell-Yan process, by Ralston and Soper [6].

In that process, transverselypolarized quarks do indeed give contributions to the cross section, at the level of twist-2terms. Ralston and Soper were working at a time before the full proof of the factorizationtheorem had become worked out.However, as far as proofs of factorization in a gauge theory are concerned, there are threeessential complications.

First, if there is a quark connection, then it is possible to have extragluons connecting H and J. Second, it is possible to have Faddeev-Popov ghost connections,for which there is no physical parton distribution.

Third, if there are only gluons connectingthe two subgraphs, then the leading power is Q2 times the canonical power. We will treatthese complications in the next section.

The important point is to see that these issues arethe same as in unpolarized scattering, so that we need only quote previous results.5. SUPER-LEADING TERMSConsider now the case of a gluon connecting the two subgraphs in fig.

2. We may representthis byJα...(−gαβ)Hβ...,(36)where the dots (...) represent the indices for the other lines and for the virtual photonsattached to H. By exactly the same argument as in the previous subsection, the leadingpower in (36) comes from the term−J+p+ p+H−.

(37)We have multiplied and divided by p+ to exhibit a jet factor that is boost invariant. It iseasy to check if all the lines joining the two subgraphs are gluons, then we get a contributionto the structure functions that is a factor of Q2 larger than the twist-2 contribution that wegot in the quark case.

We call this contribution ‘super-leading’.22

Moreover, if we start with a contribution with just a pair of quark lines, and if we addan extra gluon line joining the top and bottom, then the term (37) in the gluon polarizationgives a contribution that has the same power law as before the gluon was added. This isspecific to the case of a vector field.

If we add an extra fermion or an extra scalar line (inthe case that we have a model with elementary spin zero fields), then the numerator factorsare insufficient to compensate the extra large denominator in H, and we lose a power of Q.The factor (37) results in a factor Q greater than ‘normal’.To handle these contributions, we make the following decomposition of the numerator−gαβ of the gluon propagator in (36):−gαβ = nαkβn · k + −gαβ + nαkβn · k!= nαkβn · k + hαβ. (38)Here, kα is the momentum of the gluon, supposed collinear to pµ, and nα is a vector withjust a −component: nα = δα−.

(This vector gives a covariant definition of the fractionalmomentum carried by k: ξ = n · k/n · p.) The leading term (37) is entirely contained in thefirst term in eq. (38), while the contribution given by hαβ is exactly one power of Q smaller.When we expand each of the gluons between H and J by using eq.

(38), we will say thatthe gluons with the nαkβ/n · k term have scalar polarization, while those with the hαβ termhave transverse polarization.The largest superleading term is obtained when all the connections between H and Jare gluons with the nαkβ/n · k term. Since every gluon attached to H is given a factorkβ, there is a cancellation [22] by a Ward identity.

This cancellation is exact in an abeliantheory. But since we must exclude graphs that are one-particle-reducible in each group ofcollinear external lines, we obtain extra terms in a nonabelian theory.

For example, thereare commutators between the different scalar gluons. The same argument must be appliedrecursively to the commutators.

The details of such an argument have never been workedout in detail, even in the unpolarized case, to the best of my knowledge. In any event, theargument has nothing to do with the polarization of any of the external particles involved.When there is only one transverse gluon, with the other connecting lines being scalar23

gluons, the same argument applies. The transverse gluon gives some nonzero terms in theWard identity, involving the transverse gluon.

But there is no connection on other side of thefinal state, which means the process cannot happen. In a nonabelian theory, there shouldalso be terms in the Ward identity that bring in ghosts and that cancel the contributionswhen Faddeev-Popov ghosts connect the hard and jet subgraphs.

Again, the polarizationstate of the initial hadron is entirely irrelevant.We are therefore left with contributions that involve either two transverse gluon lines ortwo quark lines, one on each side of the final-state cut, together with arbitrarily many scalargluons. The identical argument given above is used to extract the leading term for the quarklines, then a Ward identity is used to move the scalar gluons to the quarks, and therebybuild up a gauge-invariant quark operator.

The case of two transverse gluons is handledsimilarly: Only values of the index β in hαβ that are in the transverse plane give a leadingcontribution, and the combination of one of these factors on each side of the cut is exactlywhat corresponds to the spin density matrix for a gluon.6. EXTRACTION OF SOFT GLUONSDeeply inelastic scattering is special: there are no soft gluons to worry about after the sumover final-state cuts.7.

FACTORIZATIONWe now need combinatoric arguments to go from the decomposition given above, region-by-region, to the factorization eq. (7) with an operator definition of the parton densities.

Thesearguments are of exactly the same form as for Wilson’s original operator product expansionin the Euclidean case [18, 19, 20]. Once one has established the leading regions to be thosesymbolized by fig.

2, the fact of having a different kinematic definition of the regions isirrelevant.Thus we may regard fig. 2, without the extra scalar gluons, as being the factorization.The spin structure is entirely contained in the Dirac structure we elucidated above.

Thatenables us to read offthe correct definitions of the parton densities.24

8. DEFINITION OF PARTON DISTRIBUTION FUNCTIONSElementary manipulations convert the formula (33) for the quark density into the expectationvalue of a certain nonlocal operator [23]:fi(ξ) =Z dy−2π e−iξp+y−⟨p| ¯ψi(0, y−, 0⊥) γ+2 Pe−ig R y−0dy′−A+α (0,y′−,0) tα ψi(0) |p⟩.

(39)The path ordered exponential is needed to make the operator gauge invariant, and themanipulations with the Ward identities prove that it is needed. When we work in the light-cone gauge A+ = 0, this exponential vanishes.The quark helicity and transverse spin are functions of ξ.

They are defined by replacingthe γ+/2 in eq. (39) by γ5γ+/2 and γ+γ5γ⊥/2, respectively.

(See eqs. (34) and (35).) Byconservation of angular momentum and parity, the quark helicity and transverse spin areproportional [7] to the corresponding quantities for the target, so that we can writeλifi(ξ) = λfLi(ξ),sµi⊥fi(ξ) = sµ⊥fT i(ξ),(40)where the spin variables with and without the subscript i are for the parton i and the hadrontarget, respectively.

(Angular momentum conservation here refers to the invariance of boththe theory and of definitions like eq. (39) under rotations about the z-axis.The operator definitions that translate eqs.

(34) and (35) are:λfLi(ξ) =Z dy−2π e−iξp+y−⟨p| ¯ψi(0, y−, 0⊥) γ+γ52Pe−ig R y−0dy′−A+α (0,y′−,0) tα ψi(0) |p⟩. (41)andsµ⊥fT i(ξ) =Z dy−2π e−iξp+y−⟨p| ¯ψi(0, y−, 0⊥)γ+γµ⊥γ52Pe−ig R y−0dy′−A+α (0,y′−,0)tαψi(0) |p⟩.

(42)The asymmetries ∆L = fLi(ξ)/fi(ξ) and ∆T = fT i(ξ)/fi(ξ) are, in general, functions of thefractional momentum variable ξ (and of the scale µ at which the densities are defined).Feynman rules are readily written down, as in fig. 3.

We have diagrams in which thereis an incoming particle for the state |p⟩on the left, and an outgoing particle for the state ⟨p|on the right. There is a cut for the final state.

The operator is represented by the double25

line crossing the final state. Any number of gluons may attach to the double line.

Integralsover all loop momenta are performed, and in addition there is an integral over the k−andk⊥coming out of the vertex; k+ is set equal to ξp+. Finally the product of Dirac matricesfor the explicit quark line is traced with γ+/2 for the unpolarized density, with γ5γ+/2 forthe helicity part, and with γ+γ5γ⊥/2 for the transverse polarization part.

These rules areequivalent to those written down by Artru and Mekhfi[7] in a helicity basis.Fig. 3.

Feynman rules for quark densities.The definitions given above have ultra-violet divergences when k⊥→∞. These arerenormalized in the same way as the ultra-violet divergences in the ordinary twist-2 localoperators.

Modulo the anomalies associated with the γ5, which have nothing specific to do26

with the difficulties in defining transverse polarization, these renormalizations are straight-forward. The renormalization procedure introduces explicit dependence on a renormalizationscale µ.

The renormalization group equations for the parton densities are the ordinary evo-lution equations, the (Gribov-Lipatov)-Altarelli-Parisi equations.When integer moments are taken of the above quark densities, and combined with theappropriate sign (plus or minus) times the antiquark densities, matrix elements of twisttwo local operators are obtained. For the unpolarized distributions and and for the helicitydistributions, these operators are familiar from the treatment of deeply inelastic scatteringby the operator product expansion.9.

COVARIANCE OF PARTON DISTRIBUTIONSThe definitions we have given of the quark distributions depend on the choice of a frame,and might therefore appear not to be Lorentz invariant. So we must now show that this isnot so.

The choice of coordinates can be specified by a vectornµ ≡δµ−,(43)which is lightlike and future pointing. Then the momentum-space integrals in eq.

(29) andits relatives have a covariant formulation:dk−d2k⊥= d4k δ(k · n −ξp · n),(44)while the coordinate space integrals in eq. (39) etc aredy−e−iξp+y−= dλ e−iξλp·n.

(45)The coordinate of the antiquark field in eq. (39) is λnµ, and the matrix γ+ is γ · n, so thatall the definitions are invariant under boosts along the z-axis, that is, they are invariantunder scaling of nµ to Cnµ.

Finally, the path-ordered exponential in eq. (39) has the boostinvariant formPe−ig R λn0dλ′ n·Aα(λ′n)tα(46)27

All the definitions of the parton densities now covariant, and they all explicitly depend onthe fractional momentum variable ξ. They also depend on the momentum and spin vectorof the hadron p, s, and on n. The operator expectation values in the definitions dependlinearly on the hadron’s density matrix and so their dependence on s is a constant plus alinear term.Let us define the hadron’s helicity by λ = Ms · n/p · n and its transverse polarizationby sµ⊥= sµ −λ(pµ/M −nµM/p · n) = sµ −pµs · n/p · n −nµs · nM/p · n2.

The transversepolarization satisfies n · s⊥= 0.The distributions for unpolarized quarks and for the helicity dependence, eqs. (39) and(41) can now be seen to be Lorentz invariant.

Since n · n = 0 and the distributions areindependent of the scale of nµ, the only kinematic variable on which they can depend (asidefrom ξ = k · n/p · n) is the helicity λ, and that at most linearly. Parity invariance of QCDthen shows that the unpolarized distribution is independent of the hadron polarization, whilethe helicity asymmetry is linear, just as we asserted on the left of the defining equations.

(Note that parity invariance is essential to this result: if QCD did not conserve parity, then,for example, the number of left handed quarks in a unpolarized proton need not equal thenumber of right handed quarks. )As defined in eq.

(42), the transverse distribution is given as a Lorentz vector that repre-sents the polarization vector sµq of the quark. Rotation invariance forces it to be proportionalto the hadron’s spin vector, and so the right hand side of eq.

(42) must be a linear combi-nation of sµ, nµs · n/p · n2 and pµλ, with coefficients that are independent of n and of thespin of the hadron. We have written the definition in terms of the transverse components ofa gamma matrix γµ⊥, but we could have used the complete γµ to preserve manifest Lorentzcovariance.

The definition satisfies n · sq = 0, since (γ · n)2 = 0. Thus we get a linearcombination of sµ⊥and nµs · n/p · n2.

The coefficient of sµ⊥we call the transversity part ofthe quark density, fT (ξ).The term proportional to nµs · n/p · n2 corresponds to one of the twist three distributionlisted by Jaffe and Ji [9], and arises only if one replaces the γµ⊥in eq. (42) or eq.

(35) by γ−.Most importantly, this term gives no contribution in a twist-2 hard scattering calculation,28

because one immediately puts the quark spin into a trace calculation involving the quantityk/(1 −λqγ5 + γ5s/q),(47)where the spin vector is supposed to satisfy sq · k = 0, with kµ now being a light-like vectorin the + direction.We have now seen that the definitions we have made of the quark densities, both theunpolarized one and the longitudinal and the transversity spin distributions, are explicitlyLorentz invariant: they are scalar quantities independent of the choice of the vector nµ.Essentially the same considerations apply to the gluon distributions to be defined in thenext section. The only non-invariance comes into the definition of the hadron helicity λ andthe hadron transverse spin vector sµ⊥.Let us now see why this last noninvariance creates no problem.

The physics is that wedefine the parton densities to be used in a conventional twist-2 hard scattering calculation.The presence of other particles in the process gives us the definition of nµ. Consider forexample a collision of two particles of momenta p1 and p2—fig.

4. Suppose particle 1, whichis moving to the right, has left-handed helicity.

By boosting so that the reference framemoves faster than the particle, we reverse its velocity and its helicity, but clearly we havealso changed the view of the collision: particle 2 now overtakes particle 1.Fig. 4.

Collision of two particles.From the point-of-view of the rest frame of particle 1, it is being probed by a an almostlight-like particle moving in a certain direction. We can create the exactly light-like vector29

nµ as a linear combination of pµ2 with a small admixture of pµ1:nµ ∝pµ2 −cpµ1,with c ≈m22/s. The only ambiguity is that it might be convenient to choose one of the othermomenta in the hard scattering instead of p2 in this formula.

For example, in deeply inelasticscattering, one often chooses to define transverse coordinates with respect to p and q, whichare the momentum vectors relevant for the hadronic part of the process, whereas the simplerdefinition for an experiment is to define transverse with respect to p and the momentumof the incoming lepton l. (This is indeed what one means by transverse polarization in anexperiment.) These two possibilities give different definitions of the vector n in the partondensities.

In the rest frame of the incoming hadron, they differ by a rotation through anangle of order M/√s, so that at its largest the difference corresponds to a twist-3 effect.10. OPERATOR DEFINITION OF POLARIZED GLUON DISTRIBUTIONSExactly analogous definitions may be made for the density of gluons and the longitudinaland linear polarization of the gluon.

(A pure state that is a linear combination of equalamounts of left and right helicity is called transversely polarized for a spin-12 particle, butlinearly polarized for a spin-1 particle).The gauge invariant definitions arefg(ξ) = −2Xj=1Zdy−2πξp+e−iξp+y−⟨p|G+j(0, y−, 0⊥) P G+j(0)|p⟩,fhelg(ξ) =2Xj,j′=1P heljj′Zdy−2πξp+e−iξp+y−⟨p|G+j(0, y−, 0⊥) P G+j′(0)|p⟩,fling(ξ) =2Xj,j′=1P lin⊥,jj′Zdy−2πξp+e−iξp+y−⟨p|G+j(0, y−, 0⊥) P G+j′(0)|p⟩,(48)where Gµν is the gluon field strength tensor and P denotes the path-ordered exponentialof the gluon field along the light-cone that makes the operators gauge-invariant, in exactanalogy to eq. (39):P = exp"Z y−0dy′−A+α(0, y′−, 0⊥)Tα#.

(49)30

Here Tα are the generating matrices for the adjoint representation of color SU(3). The jindex runs over the two transverse dimensions, and the spin projection operators are definedbyP hel11 = P hel11 = 0,P hel12 = −P hel21 = i,P linn,jj′ = njnj′ −δjj′/2.

(50)By angular momentum conservation, the linear polarization of a gluon is zero in a spin-12hadron [7]. (The reason is that the linear polarization is measured by an operator that flipshelicity by two units.

Since no helicity is absorbed by the space-time part of the definitionof the parton densities (the integrals are azimuthally symmetric), the helicity flip in theoperator must correspond to a helicity flip term in the density matrix for the hadron.Just as for the quarks, one can take integer moments of the gluon densities and getmatrix elements of local operators.11. FACTORIZATION FOR DRELL-YANThere are some complications in the proof of factorization for Drell-Yan as compared with theproof for deeply inelastic scattering.

These same complications appear in all other processeswith two hadrons in the initial state. The complications are explained in [4, 5], and theproof that they do not wreck the factorization are entirely independent of the polarizationissue.The typical leading region corresponds to fig.

1, which looks like an obvious general-ization of the case of deeply inelastic scattering, fig. 2.

There are now two jet subgraphs,corresponding to the two initial-state hadrons. There is also the possibility of extra collineargluons joining the jet subgraphs to the hard subgraph, and these are treated in exactly thesame way as in deeply inelastic scattering.

However, and much more malignantly, there areleading regions in which soft gluons are exchanged between the two jet subgraphs. (Thereare also soft gluons exchanged with jets going into the final state from the hard scattering,but these cancel after a sum over the unobserved part of the final state.) The soft gluons canbe emitted offinternal lines and in the initial state.

Note that a soft gluon is defined to be31

one that carries momentum much less than Q, as measured in the center of mass frame. Thisis a broader definition than the one that concerns the very well-known infrared divergencesin QED, and consequently the soft gluons are not restricted to being emitted from external,onshell colored particles.Such interactions were known before QCD: they were called Pomeron exchange, andphysically they generate the observed final states.

The final states corresponding to fig. 1taken in its naivest interpretation has two jets of hadrons corresponding to the remnants ofthe two incoming hadrons, together with a large gap in rapidity between them that containsno particles.

This gap is completely filled in by the Pomeron.The leading power for the soft interactions is given by a generalization of the methodthat generates (37) from (36). After that, a proof of cancellation of these soft interactionsinvolves a tricky combination of Ward identities, analyticity and unitarity [5].

None of thispart of the proof depends on the polarization. Note that a key part of the proof rests onanalyticity arguments that were first made in nonperturbative Pomeron physics [24]: theseinvolve analyticity (i.e., causality) but not polarization, and are valid to the whole leadingpower.Once one has got the soft interactions canceled, and the collinear scalar gluons factoredout of the hard part, one is left with the task of determining the leading power part of thetraces over Dirac matrices joining the jet subgraphs and the hard subgraph.

(There is thesame task to perform for gluons.) This just involves two copies of the argument given abovefor deeply inelastic scattering.

Ralston and Soper [6] gave this argument before the fullapparatus of factorization was formulated, and we now see their argument must be true infull QCD.The only difference from deeply inelastic scattering is that the suppression of transversepolarization no longer occurs. If we have both initial hadrons transversely polarized, thenthere is a twist-2 asymmetry in the hard scattering cross section that is explicitly nonzeroat the Born graph level—an asymmetry that is well known in e+e−physics.One error does occur in the Ralston-Soper paper.

They attempt to list all the rathernumerous structure functions that are permitted in the decomposition of the dilepton angular32

distribution, and they miss some (These are not structure functions in the common misusageof the term.) The error was corrected by Donoghue and Gottlieb [25], and does not at alleffect the general principles.

In any case, only one of the polarized structure functions isactually nonzero for the Born graph.12. FACTORIZATION FOR OTHER PROCESSESIn this paper, I have restricted attention to inclusive processes with a single large scale.The methods apply to many processes.

Although not explicitly treated here, the issues inhandling the single particle distribution in a jet are isomorphic to those in the distributionof partons in a hadron. So the methods apply equally to high p⊥single-particle productionin hadron-hadron collisions, to inclusive particle production in e+e−annihilation, or toinclusive particle production in deeply inelastic lepton scattering, for example.

It would beuseful to make a complete characterization of the processes for which a factorization theoremof the standard type holds. All the considerations of the present paper apply to any of theseprocesses.One interesting possibility is that of measuring the correlation between two particles ina jet.

This should be correlated with the spin of the parton that initiates the jet, and mayprovide a useful handle to probe the transverse spin distribution [26].There are many situations in which there is a second scale associated with the hardscattering. The simplest is the Drell-Yan process when q⊥≪Q.

A full factorization theoremhas been stated [27] for this process, and has been proved [28] for the analogous processesof two-particle production and the energy-energy correlation in e+e−annihilation in theback-to-back region. It would be interesting, and not too hard, to extend these theoremsto the polarized case.

The polarization-specific issues are orthogonal to and decoupled fromthe complications associated with the low q⊥region.Another case is where the hard scattering has a scale Q much less than the center-of-massenergy √s. This is called the semi-hard region or the small x region.

A leading logarithmstatement of a kind of factorization has been stated by Lipatov and coworkers [29]. Theproof leaves much to be desired, and does not go beyond the leading logarithm level.

Much33

work remains here.Another area is that of exclusive processes. The state of the the factorization theoremsand their proofs is not nearly so good as for inclusive processes.

There are considerablecomplications [30] because of regions other than the simplest short-distance scattering: forexample in hadron-hadron elastic scattering at large angle, there is competition between theshort-distance scattering and the Sudakov-suppressed Landshoffprocess. Disentangling thepolarization dependence would be interesting as a piece of theory, but may not be a highpriority because of the minute cross sections at the high values of Q where perturbativemethods unambiguously apply.Perhaps the most interesting recent developments in the theory of polarized hard scat-tering have been the realization [10, 11] that a generalization of the factorization theoremappears to be provable for the first nonleading twist.

Now the first nonleading twist thatis relevant in polarized scattering, with transversely polarized beams, is twist 3: that is,the first power corrections are a single power 1/Q down from the twist-2 terms. In manycases of single spin asymmetries, the twist-2 term vanishes.

An interesting phenomenologyshould result. Qiu and Sterman [10] have explained the validity of factorization in this case;Jaffe and Ji [9] have listed the operators that are needed to define the single-body partondistributions.

Interesting physics also lies in two parton correlations that are an essentialparton of the twist-3 results. There are experiments on the single transverse spin asymme-tries of single particle production at relatively large p⊥that are greatly in need of theoreticalinterpretation.13.

CONCLUSIONSThe factorization theorems for hard scattering are as true when the incoming hadrons arepolarized as when they are unpolarized. This is also true for processes in which one measuresthe polarization of hadrons in the final state (from fragmentation of a jet).The parton densities have unambiguous definitions, which are just matrix elements ofgauge invariant generalizations of the quark and gluon number operators that are naturalin light-front (or infinite-momentum) quantization.In the case of polarized beams, the34

operators are just those that are directly related to a spin density matrix for the partons.This has particular consequences: The quark transversity distribution that is relevantfor twist-2 processes with transversely polarized hadrons is perfectly well defined, contraryto what one might conclude from a superficial reading of the literature [2, 3]. The helicityasymmetry of the gluon density is well defined.

Its first moment is in general a nonlocaloperator, unless one uses the light cone gauge A+ = 0.ACKNOWLEDGMENTSThis work was supported in part by the U.S. Department of Energy under grant DE-FG02-90ER-40577, and by the Texas National Laboratory Research Commission. I would like tothank many colleagues for discussions, notably, S. Heppelmann, R.L.

Jaffe, X.-D. Ji, A.H.Mueller, D. Soper, and G. Sterman.REFERENCE1 See for example some of the articles in [8].2 R. Feynman, “Photon-Hadron Interactions”, (Benjamin, Reading, 1972).3 B.L. Ioffe, V.A.

Khoze and L.N. Lipatov, “Hard Processes, Vol.

1” (North-Holland,Amsterdam, 1984), p. 253.4 J.C. Collins, D.E. Soper and G. Sterman, “Factorization of Hard Processes in QCD”in “Perturbative QCD” (A.H. Mueller, ed.) (World Scientific, Singapore, 1989), andreferences therein.5 J.C. Collins, D.E.

Soper and G. Sterman, Nucl. Phys.

B261, 104 (1985) and B308, 833(1988); G.T. Bodwin, Phys.

Rev. D31, 2616 (1985) and D34, 3932 (1986).6 J.P. Ralston and D.E.

Soper, Nucl. Phys.

B152, 209 (1979).7 X. Artru and M. Mekhfi, Z. Phys. C 45, 669 (1990).8 Proceedings of the Polarized Collider Workshop, (J.C. Collins, S. Heppelmann and R.Robinett, eds.) pages 1–9 (A.I.P., New York, 1991).35

9 R.L. Jaffe and X.-D. Ji, Phys.

Rev. Lett.

67, 552 (1991).10 J.-W. Qiu and G. Sterman, Nucl. Phys.

B353, 105,137 (1991) and Phys. Rev.

Lett. 67,2264 (1991).11 I.I.

Balitsky and V.M. Braun, Nucl.

Phys. B311, 541 (1989).12 P. Hoodbhoy, R.L.

Jaffe, and A.V. Manohar, Nucl.

Phys. B312, 571 (1989); R.L.

Jaffeand A.V. Manohar, Nucl.

Phys. B321, 343 (1989); R.L.

Jaffe and A.V. Manohar, Phys.Lett.

223B, 218 (1989).13 F.V. Tkachov, “Euclidean Asymptotic Expansions of Green Functions of Quantum Fields(I) Expansions of Products of Singular Functions”, Fermilab preprint FERMILAB-PUB-91/347-T.14 R. Gastmans and T.T.

Wu, “The Ubiquitous Photon”, (Oxford University Press, 1990).This reviews helicity methods and gives a very comprehensive list of helicity amplitudesthat have been calculated.15 J.D. Bjorken and S.D.

Drell, “Relativistic Quantum Fields” (McGraw-Hill, New York,1966).16 J.C. Collins, “Fragmentation of Transversely Polarized Quarks Probed in TransverseMomentum Distributions”, Penn State preprint PSU/TH/102, in preparation.17 S. Libby and G. Sterman, Phys. Rev.

D18, 3252,4737 (1978).18 O.I. Zavialov, “Renormalized Quantum Field Theory” (Kluwer, Boston, 1989).19 G.B.

Pivovarov and F.V. Tkachov, “Euclidean Asymptotic Expansions of Green Func-tions of Quantum Fields (II) Combinatorics of the As-Operation”, Fermilab preprintFERMILAB-PUB-91/345-T.20 J.C. Collins, ‘Renormalization’ (Cambridge University Press, Cambridge, 1984).21 S. Coleman and R. Norton, Nuovo Cim.

28, 438 (1965).22 J.M.F. Labastida and G. Sterman Nucl.

Phys. B254, 425 (1985).23 J.C. Collins and D.E.

Soper, Nucl. Phys.

B194, 445 (1982), and references therein.24 C. DeTar, S.D. Ellis, and P.V.

Landshoff, Nucl. Phys.

B87, 176 (1975); J. Cardy and G.36

Winbow, Phys. Lett.

52B, 95 (1974).25 J.T. Donohue and S. Gottlieb, Phys.

Rev. D23, 2577,2581 (1983).26 J.C. Collins, S. Heppelmann, Bob Jaffe, X.-D. Ji and G. Ladinsky, “Measuring Transver-sity Densities in Singly Polarized Hadron-Hadron Collisions” preprint PSU/TH/101, inpreparation.27 J.C. Collins, D.E.

Soper and G. Sterman, Nucl. Phys.

B250, 199 (1985).28 J.C. Collins and D.E. Soper,, Nucl.

Phys. B193, 381 (1981).29 L.N.

Lipatov, in “Perturbative QCD” (A.H. Mueller, ed.) (World Scientific, Singapore,1989), and references therein.30 Yu.L.

Dokshitzer, V.A. Khoze, A.H. Mueller and S.I.Troyan, “Basics of PerturbativeQCD” (Editions Fronti`eres, Gif-sur-Yvette, 1991), give a review.37

---

출처: arXiv:9207.265 • 원문 보기