Target Mass Effects in Parton Quasi-Distributions
TTarget Mass E ff ects in Parton Quasi-Distributions A. V. Radyushkin
Physics Department, Old Dominion University, Norfolk, VA 23529, USAThomas Je ff erson National Accelerator Facility, Newport News, VA 23606, USA Abstract
We study the impact of non-zero (and apparently large) value of the nucleon mass M on the shape of parton quasi-distributions Q ( y , p ), in particular on its change with the change of the nucleon momentum p . We observe that the usual target-mass correctionsinduced by the M -dependence of the twist-2 operators are rather small. Moreover, we show that within the framework based onparametrizations by transverse momentum dependent distribution functions (TMDs) these corrections are canceled by higher-twistcontributions. We identify a novel source of kinematic target-mass dependence of TMDs and build models corrected for suchdependence. We find that resulting changes may be safely neglected for p (cid:38) M .
1. Introduction
The parton quasi-distributions (PQDs) Q ( y , p ) recently pro-posed by X. Ji [1] convert into usual twist-2 parton distributionfunctions (PDFs) f ( y ) when the hadron momentum p tends toinfinity. Unlike PDFs that are defined through a correlator ofquark fields separated by a light-like interval z , the definition ofQPDs refers to the interval z that has only a space z compo-nent. This opens a possibility to extract PQDs from Euclideanlattice gauge calculations.It is expected that, for a finite p ≡ P , the di ff erence between Q ( y , P ) and f ( y ) is explained by the higher-twist and target-mass corrections in powers of Λ / P and M / P , respectively.The target-mass dependence of the twist-2 matrix elements iswell-known since mid 70’s [2, 3]. In Ref. [4] (see also [5, 6, 7]),this information was used to connect x n moments of PDFs f ( x )and y n moments of the twist-2 part of QPDs Q ( y , P ). In Ref. [8],this connection was converted into a direct relation between thefunctional forms of Q twist − ( y , P ) and f ( y ). In the present paper,we give our derivation of this relation and emphasize that for y > y < f ( y ), it reduces to a simplerescaling by factors depending on the ratio M / P .We also observe that the P -evolution pattern exhibited by thecorresponding components of Q twist − ( y , P ) is rather di ff erentfrom the nonperturbative evolution of PQDs Q ( y , P ) in the mod-els considered in our recent paper [9]. Furthermore, the com-parison of the two cases indicates that the M / P target-masscorrections in Q twist − ( y , P ) are much smaller than the Λ / P higher-twist corrections in our model PQDs Q ( y , P ).According to Ref. [9], the PQDs are completely determinedby the transverse momentum dependent distributions F ( x , k ⊥ ).Thus, our next goal is to find the twist-2 part F twist − ( x , k ⊥ )of the total TMD. Using the formalism of virtuality distribu-tion functions (VDFs) [10, 11] we find the explicit form ofsuch a TMD [it coincides with the results of earlier studies[12, 13] based on a particular on-mass-shell Ansatz for the parton-hadron blob χ ( k , p )]. The form of F twist − ( x , k ⊥ ) is fullyspecified by the PDF f ( x ), and its k ⊥ -support is limited by k ⊥ ≤ x (1 − x ) M . As a consequence, the average transversemomentum induced by such a TMD is rather small. In particu-lar, for a toy PDF f ( x ) = (1 − x ) , it is given by (cid:104) k ⊥ (cid:105) = M / in case of the nucleon, that is con-siderably smaller than a folklore value of (300 MeV) .Our further study shows that the twist-2 part is not the onlysource of kinematic target-mass corrections: they also comefrom the higher-twist contributions. After incorporating theanalysis of target-mass dependence for Feynman diagrams inthe α -representation and studying equations of motion for thefull TMD F ( x , k ⊥ ), we conclude that F ( x , k ⊥ ) should dependon k ⊥ through the combination k ⊥ + x M , and that this isthe only “kinematically required” target-mass e ff ect for the fullTMD. Making this modification in the models used in Ref. [9],we observe that these M / P -corrections may be neglected wellbefore the PQDs closely approach the limiting PDF form.The paper is organized as follows. In Section 2, we startwith recalling the definition of PQDs and their relation to VDFsestablished in Ref. [9]. Then, using the α -representation, weanalyze the target-mass dependence of Feynman diagrams. InSection 3, we investigate the M -dependence of the twist-2 partof the PQD Q ( y , P ). Using the VDF formalism, we also findthe twist-2 parts of the relevant VDF and TMD. In section 4,we study the M -dependence of higher-twist contributions andequations of motion for TMDs. Since the basic relations be-tween various types of parton distributions are rather insensi-tive to complications brought in by spin, in Sections 2 – 4 werefer to a simple scalar model. In Section 5, we discuss mod-ifications related to quark spin and gauge nature of gluons inquantum chromodynamics (QCD). In Section 6, we discuss the k ⊥ → k ⊥ + x M modification of models for soft TMDs usedin Ref. [9], and present numerical results for nonperturbativeevolution of PQDs obtained in this modeling. Summary of thepaper and our conclusions are given in Section 7. Preprint submitted to Physics Letters B September 19, 2018 a r X i v : . [ h e p - ph ] M a y . Quasi-Distributions The parton quasi-distributions originate from equal-timebilocal operator formed from two fields φ (0) φ ( z ) separated inspace only [1], which corresponds to z = (0 , , , z ) [or, forbrevity, z = z ]. Then the PQDs are defined by (cid:104) p | φ (0) φ ( z ) | p (cid:105) = (cid:90) ∞−∞ dy Q ( y , p ) e iyp z . (2.1)In our paper [9] we have analyzed the PQDs in the context ofa general VDF representation [10, 11] (cid:104) p | φ (0) φ ( z ) | p (cid:105) ≡ B ( z , p ) = (cid:90) ∞ d σ (cid:90) − dx Φ ( x , σ ; M ) × e − ix ( pz ) − i σ ( z − i (cid:15) ) / (2.2)(where M = p ) that basically reflects the fact that the matrixelement (cid:104) p | φ (0) φ ( z ) | p (cid:105) depends on z through ( pz ) and z , andmay be treated as a double Fourier representation with respectto these variables.The VDF representation holds for any p and z , but it is con-venient to take the frame in which p = { E , ⊥ , p = P } . When z has only the minus component z − , the matrix element (cid:104) p | φ (0) φ ( z − ) | p (cid:105) = (cid:90) − dx f ( x ) e − ixp + z − (2.3)is parameterized by the parton distribution function (PDF) f ( x )that depends on the fraction x of the target momentum compo-nent p + carried by the parton. The relation between the VDF Φ ( x , σ ) and the collinear twist-2 PDF f ( x ) is formally given by f ( x ) = (cid:90) ∞ Φ ( x , σ ) d σ . (2.4)The σ -integral diverges when Φ ( x , σ ) has a ∼ /σ hard partgenerating perturbative evolution of PDFs. Our primary con-cern is nonperturbative evolution, so we will always imply thesoft part of Φ ( x , σ ) for which the σ -integral converges.If we take z having just the third component, z = z , we have (cid:104) p | φ (0) φ ( z ) | p (cid:105) = (cid:90) ∞ d σ (cid:90) − dx Φ ( x , σ ) e ixp z + i σ z / . (2.5)This gives a relation between PQDs and VDFs, Q ( y , P ) = (cid:90) ∞ d σ (cid:114) i P πσ (cid:90) − dx Φ ( x , σ ) e − i ( x − y ) P /σ . (2.6)For large P , we have (cid:114) i P πσ e − i ( x − y ) P /σ = δ ( x − y ) + σ P δ (cid:48)(cid:48) ( x − y ) + . . . (2.7)and Q ( y , P → ∞ ) tends to the integral (2.4) producing f ( y ).The deviation of Q ( y , P ) from f ( y ) for large P may be de-scribed by higher-twist corrections in powers of Λ / P (where Λ is a scale like average primordial transverse momentum) andtarget mass corrections in powers of M / P .As shown in our paper [9] , PQDs are completely determinedby TMDs, so building models for TMDs we generate evolu-tion patterns showing how Q ( y , P ) may depend on P due to thetransverse-momentum e ff ects. χ ( k, p ) k k p p z . . Figure 1: Structure of parton-hadron matrix element.
To discuss the origin of the target-mass dependence of VDFsit is convenient to switch to the momentum space description ofthe bilocal matrix element (cid:104) p | φ (0) φ ( z ) | p (cid:105) = π (cid:90) d k e − ikz χ ( k , p ) (2.8)in terms of the function χ ( k , p ) (see Fig. 1) which is an analogof the Bethe-Salpeter amplitude [14].A crucial observation is that the contribution of any (uncut)diagram to χ ( k , p ) may be written as i χ d i ( k , p ) = i l P (c . c . )(4 π i ) L (cid:90) ∞ l (cid:89) j = d α j [ D ( α )] − × exp (cid:40) ik A ( α ) D ( α ) + i ( p − k ) B s ( α ) + ( p + k ) B u ( α ) D ( α ) (cid:41) × exp ip B p ( α ) D ( α ) − i (cid:88) j α j ( m j − i (cid:15) ) (2.9)(see, e.g., [15]) , where P (c . c . ) is the relevant product ofcoupling constants, L is the number of loops of the di-agram, and l is the number of its lines. The functions A ( α ) , B s ( α ) , B u ( α ) , C ( α ) , D ( α ) are sums of products of the non-negative α j -parameters. Using Eq. (2.11) we get the represen-tation i χ ( k , p ) = (cid:90) ∞ d λ (cid:90) − dx e i λ [ k − x ( kp ) + i (cid:15) ] F ( x , λ ; p ) (2.10)with a function F ( x , λ ; p ) specific for each diagram F d i ( x , λ ; p ) = i l P (c . c . )(4 π i ) L (cid:90) ∞ l (cid:89) j = d α j [ D ( α )] − × δ (cid:32) λ − A ( α ) + B s ( α ) + B u ( α ) D ( α ) (cid:33) δ (cid:32) x − B s ( α ) − B u ( α ) A ( α ) + B s ( α ) + B u ( α ) (cid:33) × exp ip B s ( α ) + B u ( α ) + B p ( α ) D ( α ) − i (cid:88) j α j ( m j − i (cid:15) ) . (2.11)2ransforming Eq. (2.10) to the coordinate representation andchanging λ = /σ gives (cid:104) p | φ (0) φ ( z ) | p (cid:105) = (cid:90) ∞ d σ (cid:90) − dx e − ix ( pz ) − i σ ( z − i (cid:15) ) / × e − ix M /σ F ( x , /σ ; M ) . (2.12)Note that the quadratic dependence on x in the exponentialwas produced by the k → z Fourier transformation: origi-nally all terms in the exponential of Eq. (2.10) have linear de-pendence on x . Basically, one gets − x M after manipulating k − x ( kp ) into ( k − xp ) − x M .Absorbing the factor exp[ − ix M /σ ] into F ( x , /σ ) anddefining the Virtuality Distribution Function Φ ( x , σ ; M ) = exp[ − ix M /σ ] F ( x , /σ ; M ) (2.13)gives the VDF representation (2.2).Taking z that has z − and z ⊥ components only, i.e., projectingon the light front z + =
0, we define the
Transverse MomentumDependent Distribution in the usual way as a Fourier transformwith respect to remaining coordinates z − and z ⊥ . The TMD maybe written in terms of VDF as F ( x , k ⊥ ) = i π (cid:90) ∞ d σσ Φ ( x , σ ; M ) e − i ( k ⊥ − i (cid:15) ) /σ . (2.14)Since Φ ( x , σ ; M ) must have the exp[ − ix M /σ ] factor, theTMD F ( x , k ⊥ ) must depend on k ⊥ through the k ⊥ + x M com-bination. Thus, this part of the M -dependence is kinemati-cal, and hence predictable if we know the k ⊥ dependence of F ( x , k ⊥ ).In addition, Φ ( x , σ ; M ), and hence also F ( x , k ⊥ ) have a“dynamical” or “kinematically unpredictable” M -dependencecontained in F ( x , /σ ; M ) that comes from the last line in the α -representation (2.11).
3. Target mass dependence of the twist-2 part
Another (and well-known) example of the kinematical targetmass dependence is given by the M -structure of the matrixelements of the twist-2 local operators.To get the twist-2 part of the bilocal operator φ (0) φ ( z ), oneshould start with the Taylor expansion in z and then changethe product of derivatives ∂ µ . . . ∂ µ n into its traceless part { ∂ µ . . . ∂ µ n } . In a short-hand notation ( z ∂ ) n → { z ∂ } n , and( zp ) n → { zp } n , so that (cid:104) p | φ (0) φ ( z ) | p (cid:105)| twist − = (cid:90) − dx f ( x ) ∞ (cid:88) n = ( − ix ) n { zp } n n ! . (3.1)Note that for z = z − we have { zp } n = ( zp ) n , which repro-duces Eq. (2.3). To proceed in a situation with z (cid:44)
0, weuse the fact that the structure of { zp } n is related to the Gegen-bauer polynomials C n (cosh θ ) equal to Chebyshev polynomials U n (cosh θ ) = sinh(( n + θ ) / sinh θ . As a result, { zp } n = ( zp ) n [1 + r ] n + − [1 − r ] n + n + r , (3.2) where r = (cid:112) − z p / ( zp ) (see, e.g., Ref. [16]). Using p = ( E , ⊥ , P ) and taking z = z , we have ( zp ) = − z P , z = − z and p = M . Thus, we have r = (cid:112) + M / P = E / P and { zp } n = ( − n z n [ P + E ] n + − [ P − E ] n + n + E . (3.3)This gives (cid:104) p | φ (0) φ ( z ) | p (cid:105)| twist − = (cid:90) − dx f ( x ) × (cid:20) E + P E e ixz ( P + E ) / + E − P E e ixz ( P − E ) / (cid:21) , (3.4)and we get the twist-2 part of PQD in the form Q twist − ( y , P ) = + ∆ (cid:2) f ( y / (1 + ∆ ) + f ( − y / ∆ ) (cid:3) , (3.5)where ∆ = E − P P = M P + . . . . This result was originally obtained (in somewhat di ff erent wayand notations) in Ref. [8]. As noticed there, the integral over y is preserved (cid:90) ∞−∞ dy Q twist − ( y , P ) = (cid:90) − dx f ( x ) . (3.6)One can check that the momentum sum rule also holds, (cid:90) ∞−∞ dy y Q twist − ( y , P ) = (cid:90) − dx x f ( x ) . (3.7)Since the PQD Q ( y , P ) for negative y may come both fromthe y > y < f ( y ), it makes sense to split f ( y ) in these two parts and analyze PQDs coming from each ofthem separately. For illustration, we take the same model as inRef. [9], namely, the function f ( x ) = (1 − x ) θ (0 < x < y is obtainedfrom the original f ( y ) by stretching it by factor (1 + ∆ ) in thehorizontal direction and squeezing by factor (1 + ∆ ) in the ver-tical one (see Fig. 2). For negative y , one should take f ( − y ) andcontract it by factor ∆ in the horizontal direction, with the samesqueeze by (1 + ∆ ) in the vertical one.Thus, if the twist-2 target mass corrections were the onlyones here, it would be very easy to reconstruct such a PDF fromthe PQD at positive y : one should just perform the (1 + ∆ ) and(1 + ∆ ) rescaling mentioned above.For comparison, we show in Fig. 3 the P -dependence of PQDdue to the nonperturbative evolution in the Gaussian model ofRef. [9]. Notice that the curve for P = Λ is close in heightto the P = M curve of Fig. 2. We expect that the scale Λ isabout 300 to 500 MeV, or from 1 / / Λ corresponds to about 3 – 5 M. One can see thatalready the P = M curve from Fig. 2 is very close to thelimiting curve (in this case ∆ = . Λ (despite thefact that Λ was taken to be 2 – 3 times smaller than M ).3 yQ twist ( y, P ) Figure 2: Twist-2 part of Q ( y , P ) for P / M = . , , y = .
1) compared to the limiting PDF f ( y ) = (1 − y ) θ (0 < y < The P -evolution patterns in Figs. 2 and 3 are rather di ff erent.It is interesting to find a physical reason for this di ff erence. Asshown in Ref. [9], the PQDs are completely determined by theTMDs, Q ( y , P ) = (cid:90) ∞−∞ dk (cid:90) − dx P F ( x , k + ( x − y ) P ) . (3.8)The Gaussian model mentioned above corresponds to a factor-ized Ansatz F G ( x , k ⊥ ) = f ( x ) π Λ e − k ⊥ / Λ . (3.9)So, let us find out what kind of TMD corresponds to the twist-2part of the matrix element.The first step is to find the VDF corresponding to the twist-2contribution (3.1). To this end, we start with the decompositionof the traceless combinations over the usual ones, { pz } n = ( pz ) n (cid:98) n / (cid:99) (cid:88) k = ( − k ( n − k )! k !( n − k )! (cid:32) M z pz ) (cid:33) k , (3.10)that follows from the ξ k expansion of the Gegenbauer polyno-mials C n ( ξ ). This gives a double expansion in ( pz ) and z forthe sum in Eq. (3.1), ∞ (cid:88) n = ( − i ) n x n { pz } n n ! = ∞ (cid:88) k = k ! (cid:32) x M z (cid:33) k × ∞ (cid:88) N = ( − i ) N x N ( pz ) N N ! ( N + k )!( N + k )! . (3.11)Representing ( N + k )!( N + k )! = (cid:90) dt k . . . (cid:90) t dt t N + k (3.12) - yQ ( y, P )3510 Figure 3: Evolution of Q ( y , P ) in the Gaussian model for P / Λ = , ,
10 (frombottom to top at y = .
1) compared to the limiting PDF f ( y ) = (1 − y ) θ ( y ). we get ∞ (cid:88) n = ( − i ) n x n { pz } n n ! = ∞ (cid:88) k = k ! (cid:32) M z (cid:33) k × (cid:90) x du k . . . (cid:90) u du u k e − iu ( pz ) . (3.13)At this stage, it is convenient to treat x > x < f ( x ) separately. For definiteness, we take x >
0. Notice nowthat (cid:90) dx f ( x ) (cid:90) x du k . . . (cid:90) u du u k e − iu ( pz ) = (cid:90) dx x k e − ix ( pz ) f k ( x ) , (3.14)where the functions f k ( x ) are defined by the recurrence relation f k + ( x ) = (cid:90) x dy f k ( y ) , (3.15)with f ( x ) = f ( x ). As a result, (cid:104) p | φ (0) φ ( z ) | p (cid:105)| twist − = (cid:90) dx e − ix ( pz ) ∞ (cid:88) k = k ! (cid:32) x M z (cid:33) k f k ( x ) . (3.16)Comparing with the VDF representation (2.2), we find Φ twist − ( x , σ ) = ∞ (cid:88) k = k ! (cid:16) − ixM (cid:17) k δ ( k ) ( σ ) f k ( x ) . (3.17) To proceed with the formula (2.14 ) producing the TMD weuse (cid:90) ∞ d σσ δ ( n ) ( σ ) e − i ( k ⊥ − i (cid:15) ) /σ = ( − i ) n n ! δ ( n ) ( k ⊥ ) , (3.18)4hich results in the δ ( n ) ( k ⊥ ) expansion F twist − ( x , k ⊥ ) = π ∞ (cid:88) n = ( − xM ) n δ ( n ) ( k ⊥ ) f n ( x ) (3.19)that is equivalent to the following expression for the ( k ⊥ ) n mo-ments of F twist − ( x , k ⊥ ): (cid:90) ∞ dk ⊥ ( k ⊥ ) n F twist − ( x , k ⊥ ) = π ( xM ) n n ! f n ( x ) . (3.20)It is easy to check that the moment relation (3.20) is satisfiedby the function F twist − ( x , k ⊥ ) = − x π M f (cid:48) ( x + k ⊥ / xM ) . (3.21)In the M = F twist − ( x , k ⊥ ) (cid:12)(cid:12)(cid:12)(cid:12) M → = π f ( x ) δ ( k ⊥ ) . (3.22)Thus, no transverse momentum is generated in the case of amassless target. Our illustration model f ( x ) = (1 − x ) gives F modtwist − ( x , k ⊥ ) = π ( xM ) ( x ¯ xM − k ⊥ ) θ ( k ⊥ ≤ x ¯ xM ) , (3.23)where ¯ x ≡ − x . One can check that using the TMD (3.23)in the TMD → PQD conversion formula (3.8) one obtains thePQDs dictated by Eq. (3.5) and shown in Fig. 2.The interpretation of the twist-2 approximation in termsof the transverse momentum dependent function given byEq. (3.21) is known [12, 13] from the early days of the ξ -scalingapproach [3]. It was derived by imposing the k = χ ( k , p ) through the Ansatz χ A ( k , p ) = − π δ ( k ) M f (cid:48) (cid:32) pk ) M (cid:33) , (3.24)while keeping the target mass finite p = M , see, e.g.,Ref. [12]. In a similar context, Eq. (3.21) was obtained in Refs.[17, 18] (see also [19]).Our VDF-based derivation shows that the twist-2 TMD(3.21) can be obtained without additional assumptions. Note that, because the support of f ( x ) is 0 ≤ x ≤
1, thetwist-2 TMDs (3.21) vanish for k ⊥ ≥ x ¯ xM . This should becontrasted with the usual expectation (incorporated into ourTMD models in Ref. [9] ) that TMDs are smooth functionsof k ⊥ with a support extending to k ⊥ = ∞ .From a physical point of view, the twist-2 part F twist − ( x , k ⊥ )describes a situation when a free massless quark happens some-how to be bound within a system with a total mass M . This re-sults in a kinematic transverse momentum described by a ratherartificially-looking TMD of Eq. (3.23) type. Clearly, this is justa model construction mimicking a hadron by a combination offree quarks with the total invariant mass M . Comparing TMDs, it is instructive to calculate the averagetransverse momentum (cid:107) f (cid:107) (cid:104) k ⊥ (cid:105) ≡ (cid:90) dx (cid:90) d k ⊥ k ⊥ F ( x , k ⊥ ) (3.25)that they induce. Here, (cid:107) f (cid:107) ≡ (cid:90) dx f ( x ) . (3.26)For f ( x ) = (1 − x ) , we have (cid:104) k ⊥ (cid:105) twist − = M ≈ (170 MeV) . (3.27)For comparison, the Gaussian TMD (3.9) gives (cid:104) k ⊥ (cid:105) G = Λ .Thus, taking Λ = M / Λ / P correc-tions for PQD in the Gaussian model are about 3 times largerthan the M / P corrections in the twist-2 part of the PQD. Thisobservation explains the di ff erence between Figs. 2 and 3.Note that for more realistic valence PDFs f ( x ) that are sin-gular for x =
0, the value of (cid:104) k ⊥ (cid:105) twist − is even smaller. Inparticular, for f ( x ) = (1 − x ) / √ x it equals to M /
66, resultingin (cid:104) k ⊥ (cid:105) twist − ≈ (116 MeV) , i.e. factor of 8 smaller than theexpected folklore value of 0.1 GeV .A rather exotic form of the twist-2 part of the TMD con-tradicts a natural expectation that TMDs should be smoothfunctions of k ⊥ with an unlimited support. To produce sucha smooth TMD (having, moreover, a much larger (cid:104) k ⊥ (cid:105) ), thehigher-twist terms should literally wipe out the features broughtin by the twist-2 term. This is only possible if the higher-twistterms also have the M -dependence.
4. Higher-twist contributions
The twist-2 contribution appears as the first term in the twistdecomposition of the original bilocal operator φ (0) φ ( z ) = ∞ (cid:88) l = (cid:32) z (cid:33) l ∞ (cid:88) N = N + l !( N + l + φ (0) { z ∂ } N ( ∂ ) l φ (0) , (4.1)(see, e.g., Ref. [16]). The operators containing powers of ∂ have higher twists, and their contribution to the light-cone ex-pansion is accompanied by powers of z . For PQDs, z wouldresult in a 1 / P suppression factor, just like for the target-masscorrections in twist-2 contribution.To analyze the interplay between the twist-2 and twist-4terms, let us take the terms bilinear in z , φ (0) φ ( z ) | bil = φ (0) { z ∂ } φ (0) + (cid:32) z (cid:33) φ (0) ∂ φ (0) . (4.2)For the matrix element, this gives2 (cid:104) p | φ (0) φ ( z ) | p (cid:105)| bil = − { zp } (cid:90) dx x f soft ( x ) + (cid:32) z (cid:33) (cid:104) p | φ (0) ∂ φ (0) | p (cid:105) . (4.3)5s we discussed, { zp } = (cid:104) ( zp ) − z M / (cid:105) contains the z M target-mass correction term.Since a VDF contains all information about the z -dependenceof the original matrix element, it should provide the VDF rep-resentation for the twist-4 matrix element (cid:104) p | φ (0) ∂ φ (0) | p (cid:105) aswell. To this end, we calculate (cid:3) z B ( z , p ) in the VDF represen-tation (2.2) involving the x > (cid:3) z B ( z , p ) = − (cid:90) ∞− d σ (cid:90) dx e − ix ( pz ) − i σ ( z − i (cid:15) ) / Φ ( x , σ ) × (cid:34) x M + x σ ( pz ) + σ z + i σ (cid:35) . (4.4)(We remind that p in the VDF representation (2.2) is the actual hadron momentum, with p = M ). Assuming a soft Φ ( x , σ )and taking z =
0, we get the twist-4 matrix element (cid:104) p | φ (0) ∂ φ (0) | p (cid:105)| soft = − (cid:90) ∞ d σ (cid:90) dx Φ soft ( x , σ ) (cid:104) x M + i σ (cid:105) = − M (cid:90) dx x f soft ( x ) + (cid:90) dx (cid:90) d k ⊥ k ⊥ F soft ( x , k ⊥ ) . (4.5)As one can see, it contains a term which a) is proportional to M and b) is completely specified by the twist-2 PDF f soft ( x ).This means that the kinematical target-mass correction terms z M are contained not only in the twist-2 part of the originalmatrix element B ( z , p ), but also in its higher-twist parts. Mostimportantly, when substituted in Eq. (4.3), this term cancels the z M term coming from the twist-2 part. As a result, we havethe expression (cid:104) p | φ (0) φ ( z ) | p (cid:105)| bil = −
12 ( zp ) (cid:90) dx x f soft ( x ) + (cid:32) z (cid:33) (cid:90) dx (cid:90) d k ⊥ k ⊥ F soft ( x , k ⊥ ) (4.6)free of the M z terms. A similar result may be easily obtained in general case ifone expands the exp[ − i σ z /
4] factor in the VDF representation(2.2) and uses the relation( − i ) l (cid:90) ∞ d σ σ l Φ ( x , σ ) = l ! (cid:90) d k ⊥ k l ⊥ F ( x , k ⊥ ) . (4.7)Then one obtains the representation of the matrix element (cid:104) p | φ (0) φ ( z ) | p (cid:105) = ∞ (cid:88) l = l !) (cid:32) z (cid:33) l (cid:90) − dx e − ix ( pz ) × (cid:90) d k ⊥ k l ⊥ F ( x , k ⊥ ) (4.8)in terms of the TMD F ( x , k ⊥ ). The sum over l gives the Besselfunction J , so we may also write (cid:104) p | φ (0) φ ( z ) | p (cid:105) = (cid:90) − dx e − ix ( pz ) × π (cid:90) ∞ dk ⊥ J (cid:32) (cid:113) − k ⊥ z (cid:33) F ( x , k ⊥ ) . (4.9) The TMD parametrizations (4.8) and (4.9) provide anotherform of the z -expansion, alternative to the twist decomposi-tion (4.1). Its advantage is that the ( pz )-dependence comesthrough the plane waves e − ix ( pz ) producing simple powers ( pz ) n rather than complicated traceless combinations { pz } n contain-ing z M target-mass dependent terms that are simply artifactsof the twist decomposition. The TMD representation (4.9) isespecially convenient in applications to PQDs. In particular, itdirectly leads to the TMD → PQD conversion formula (3.8).One may argue that, due to equations of motion,like ∂ φ = λψφ in a scalar λφ ψ theory, one may write (cid:104) p | φ (0) ∂ φ (0) | p (cid:105) as (cid:104) p | φ (0) λψ (0) φ (0) | p (cid:105) or Λ (cid:104) p | φ (0) φ (0) | p (cid:105) ,with Λ having no visible M -dependence, so that there isapparently nothing to cancel the M -dependence of { pz } inEq. (4.3). But this is exactly the disadvantage of such anapproach: the only thing it says about matrix elements of (cid:104) p | φ (0) { z ∂ } N ( ∂ ) l φ (0) | p (cid:105) type is that, compared to the twist-2case, they have extra ( Λ ) N factors of unspecified size and prop-erties.Still, it is an interesting question of how to incorporate equa-tions of motions in the VDF / TMD parametrizations of the bilo-cal matrix element. ff -shell quarks Since quarks in the nucleon are virtual, the matrix ele-ment B ( z , p ) does not satisfy the free-quark equation of motion (cid:3) z B ( z , p ) =
0. Keeping nonzero z and integrating by parts inEq. (4.4), we obtain − (cid:3) z B ( z , p ) = (cid:90) ∞ d σ (cid:90) dx e − ix ( pz ) − i σ ( z − i (cid:15) ) / × (cid:32) x M − ix σ ∂∂ x − i σ ∂∂σ − i σ (cid:33) Φ ( x , σ ) . (4.10)By equations of motion, this should be equal to the 3-bodyquark-quark-gluon contribution. For example, in a λφ ψ scalarmodel, this should be equal to (cid:104) p | φ (0) λψ ( z ) φ ( z ) | p (cid:105) . Thus,building the VDF parametrization for the matrix element of the3-body φψφ operator in a situation when ψ and one of the φ ’sare at the same point (and may be treated as one field) we shouldimpose the condition Φ φ ( ψφ ) ( x , σ ) = (cid:32) x M − ix σ ∂∂ x − i σ ∂∂σ − i σ (cid:33) Φ ( x , σ )(4.11)reflecting equations of motion. For the TMDs constructed from Φ ’s using Eq. (2.14) (with k ⊥ substituted by κ to avoid tooclumsy notations below) this gives F φ ( ψφ ) ( x , κ ) = (cid:16) x M − κ (cid:17) F ( x , κ ) + x ∂∂ x (cid:90) ∞ κ d κ F ( x , κ ) , (4.12)or, di ff erentiating with respect to κ , ∂∂κ F φ ( ψφ ) ( x , κ ) = (cid:34)(cid:32) xM − κ x (cid:33) ∂∂κ − ∂∂ x (cid:35) x F ( x , κ ) . (4.13)6or the twist-2 part, when the l.h.s. of Eq. (4.13) vanishes,we have seen in Eq. (3.21) that the function x F ( x , κ ) dependson x and κ through the combination η ≡ x + κ / xM . (4.14)Noticing that ∂η∂ x = − κ x M , ∂η∂κ = xM , (4.15)we can rewrite Eq. (4.13) in terms of x and η variables, ∂∂κ F φ ( ψφ ) ( x , κ ) = (cid:34) ∂η/∂ x ∂η/∂κ ∂∂κ − ∂∂ x (cid:35) x F ( x , κ ) . (4.16)Now, treating x F ( x , κ ) as a function G ( x , η ) of x and η , andintroducing G ( x , η ) ≡ ∂ F φ ( ψφ ) ( x , κ ) /∂κ we have G ( x , η ) = (cid:34) ∂η∂ x ∂∂η − ∂∂ x − ∂η∂ x ∂∂η (cid:35) G ( x , η ) , (4.17)and finally G ( x , η ) = − ∂∂ x G ( x , η ) . (4.18)If G ( x , η ) vanishes, then we conclude that G ( x , η ) must be afunction of η , in agreement with Eq. (3.21). If G ( x , η ) does notvanish, the only restriction imposed by the equation of motionis Eq. (4.18). Thus, we may take any reasonable model for thetwo-body function G ( x , η ) and then just incorporate Eq. (4.18)[or original Eq. (4.11)] as a restriction that should be satisfiedby the three-body function G ( x , η ), when the qGq contributionis included, say, in a DIS calculation.Of course, choosing a model for G ( x , η ) one should take carethat the resulting G ( x , η ) is also reasonable. In other words,if one has some information / expectations about the form of G ( x , η ), one should make an e ff ort to find a form of G ( x , η )that would lead to the desired (or close) form of G ( x , η ).An important lesson is that, in the context of equations ofmotion, it is natural to build models of TMDs F ( x , k ⊥ ) in theform of functions of x and k ⊥ + x M . This observation is infull accord with the general conclusion made at the end of theSection 2 that TMDs F ( x , k ⊥ ) must depend on k ⊥ through the k ⊥ + x M combination.
5. QCD
In spinor case, one deals with the matrix element of a B α ( z , p ) ≡ (cid:104) p | ¯ ψ (0) γ α ψ ( z ) | p (cid:105) (5.1)type. It may be decomposed into p α and z α parts: B α ( z , p ) = p α B p ( z , p ) + z α B z ( z , p ). These parts are not completely inde-pendent, since there are restrictions imposed by equations ofmotion. Consider the handbag contribution for the virtual Comptonamplitude, whose imaginary part gives the deep inelastic scat-tering (DIS) cross section. It may be written as T µν ( q , p ) = − s µναβ (cid:90) d z π z β z ˜ B α ( z , p ) e − i ( qz ) , (5.2)where z β / z comes from the spinor massless propagator S c ( z ) = − / π / z / z , ˜ B α ( z , p ) = B α ( z , p ) − B α ( − z , p ), and s µναβ ≡ − g µν g αβ + g µα g νβ + g να g µβ .To check the electromagnetic gauge invariance, we calculate s µναβ ∂∂ z µ (cid:18) z β z ˜ B α ( z , p ) (cid:19) = z β z (cid:104) ˜ B ν , β − ˜ B β , ν + g νβ ˜ B α, α (cid:105) , (5.3)where ˜ B ν , β ≡ ( ∂/∂ z β ) ˜ B ν ( z , p ), etc.The antisymmetric term will be eliminated if one takes B α ( z , p ) to be a derivative B α ( z , p ) = ∂ α B ( z , p ) of some “gener-ating” scalar function B ( z , p ). After that, B α,α = (cid:3) z B ( z , p ), andthe equation of motion for B α,α = (cid:104) p | ¯ ψ (0) /∂ψ ( z ) | p (cid:105) brings us to astudy of (cid:3) z B ( z , p ), which completely parallels that performedin the previous section.As for the remaining violation of the EM gauge invariancefor the DIS handbag, it is proportional to (cid:3) z ˜ B ( z , p ), i.e., we stillhave it, as it is caused by the virtuality of the active quarks.In a Yukawa gluon model, we have /∂ψ ( z ) = ig φ ( z ) ψ ( z ), and thisviolation will be compensated when one includes terms comingfrom the 3-body ¯ ψφψ diagrams, provided that one imposes therestriction (4.18). In QCD, one should take the operator O α q ( z , z ; A ) ≡ ¯ ψ ( z ) γ α ˆ E ( z , z ; A ) ψ ( z ) (5.4)involving the gauge link ˆ E ( z , z ; A ) along the straight line con-necting z and z . The equation of motion, applied to the rela-tive coordinate z , takes the form ∂∂ z α (cid:104) p | ¯ ψ ( X − z ) γ α ψ ( X + z ) | p (cid:105) = ( ig ) (cid:104) p | ¯ ψ ( z ) γ α A α ( z , z ) ψ ( z ) | p (cid:105) , (5.5)where A α ( z ; z ) = ( z ν − z ν ) (cid:90) dt G να ( t ( z − z ) + z ) . (5.6)As a result, we have ∂ α B α ( z , p ) = ig (cid:104) p | ¯ ψ (0) z ν γ α (cid:90) dt G να ( tz ) ψ ( z ) | p (cid:105)≡ B ¯ ψ G ψ ( z , p ) . (5.7)Taking again B α ( z , p ) = ( ∂/∂ z α ) B ( z , p ) reduces the equation ofmotion to the equation for (cid:3) z B ( z , p ) involving a scalar function B ( z , p ), and we can use all the results of Section 3, since theexplicit form of B ¯ ψ G ψ ( z , p ) was not essential there.7 . Modeling target-mass dependence of PQDs -dependence of TMDs Thus, if one uses the VDF / TMD representations (2.2), (4.9)for matrix elements, there are no kinematic z M -correctionsthat are artifacts of expansion over traceless { pz } n combina-tions. Furthermore, the PQDs are given by the conversion for-mula (3.8), and the target-mass dependence of Q ( y , P ) may onlycome from that of F ( x , k ⊥ ).According to the general statement made at the end of Sec-tion 2, the TMDs F ( x , k ⊥ ) must depend on k ⊥ through the k ⊥ + x M combination. This is a “predictable” or “kinemat-ical” target-mass dependence.We also noted there that F ( x , k ⊥ ) may have a “dynami-cal” M -dependence due to the M -dependence of the under-lying function F ( x , /σ ; M ) of Eq. (2.12). This kind of M -dependence cannot be derived from kinematics, and in thissense it is “unpredictable”. In principle, there is nothing specialin the fact that F ( x , /σ ; M ) depends on the hadron mass, justlike there is no wonder that the shape of a PDF f ( x ) may bedi ff erent if the hadron mass would be di ff erent.This is to say that some part of the M -dependence of F ( x , /σ ; M ) may be absorbed into the form of the PDF f ( x ),and would not lead to M / P corrections describing the di ff er-ence between a QPD Q ( y , P ) and its PDF f ( y ). Still, some partof the unpredictable M -dependence may lead to the M / P corrections, and it is a challenge to build VDF models thatwould “realistically” reflect that part of the M -dependence.Leaving this problem for future studies, in what follows wewill investigate the consequences of the “mandatory” change k ⊥ → k ⊥ + x M in the TMD models that have been used forgenerating nonperturbative evolution of PQDs in our paper [9]. Adding the M -dependence into our Gaussian model (3.9) bythe k ⊥ → k ⊥ + x M prescription, we get F G ( x , k ⊥ ) → f ( x ) π Λ e − ( k ⊥ + x M ) / Λ = ˜ f ( x ) π Λ e − k ⊥ / Λ , (6.1)where ˜ f ( x ) = f ( x ) e − x M / Λ . Thus, we have a simple change inthe form of the PDF, f ( x ) → ˜ f ( x ), that would not be reflectedby M / P terms in the di ff erence between ˜ Q ( x , P ) and ˜ f ( x ). Another VDF model proposed in Ref. [9], Φ m ( x , σ ) = f ( x )2 im Λ K (2 m / Λ ) e i σ/ Λ − im /σ − (cid:15)σ , (6.2)intends to reproduce the large- | z | exponential ∼ e −| z | m fall-o ff ofthe perturbative propagator D c ( z . m ) of a particle with mass m ,while removing its 1 / z singularity at small z by a “confine-ment” factor e i σ/ Λ reflecting the finite size of a hadron. Thismodel corresponds to the TMD given by F m ( x , k ⊥ ) = f ( x ) K (cid:18) (cid:113) k ⊥ + m / Λ (cid:19) π m Λ K (2 m / Λ ) . (6.3) Using the prescription k ⊥ → k ⊥ + x M amounts to the change m → m + x M in this model. To avoid a two-parameter ( Λ and m ) modeling, in our paper [9] we took m =
0. Let us dothe same here. In the context of the m -model (6.2), the resultingTMD model F ( x , k ⊥ ; M ) = f ( x ) K (cid:18) (cid:113) k ⊥ + x M / Λ (cid:19) π xM Λ K (2 xM / Λ ) (6.4)corresponds to assuming that the parton mass m is a fraction xM of the nucleon mass M . This assumption does not look abso-lutely unnatural in view of the fact that the VDF representation(2.2) involves the plane wave factor e − ix ( pz ) in which p is theactual hadron momentum p satisfying p = M .For the quasi-distribution, the model (6.4) gives Q ( y , P ; M ) = P Λ (cid:90) − dx f ( x ) e − √ ( x − y ) P / Λ + x M / Λ K (2 xM / Λ ) xM / Λ . (6.5) Now we have two parameters, the nucleon mass M and thetransverse momentum scale Λ , and we need to decide what istheir ratio. To this end, we calculate the average transverse mo-mentum in the model of Eq. (6.4) with f ( x ) = (1 − x ) , andfind (cid:113) (cid:104) k ⊥ (cid:105) mod ≈ Λ (cid:18) M Λ (cid:19) . , (6.6)with 1.5% accuracy in the interval 1 . < M / Λ <
5, i.e., for Λ between 200 and 600 MeV. The factor ( M / Λ ) . changes from1.1 to 1.3 in this region. Thus, the average transverse momen-tum is predominantly determined by Λ . Assuming a folklorevalue of 300 MeV for the average k ⊥ , we take M / Λ = M terms in Eq. (6.5) on theshape of quasi-distributions, we take again f ( y ) = (1 − y ) θ ( y ),and compare curves for M / Λ = M / Λ = P / Λ = P / M = − y ) shape. Increasing P to 2 M , weget the curves that practically coincide (see Fig. 5), still beingrather far from the asymptotic P / Λ → ∞ shape.Thus, in this scenario, when one reaches the momentum P that is su ffi ciently large to stop the nonperturbative evolution ofthe PQD Q ( y , P ), there is no need to bother about target masscorrections. Given the expected accuracy of lattice gauge cal-culations, they may be safely neglected starting with P ∼ M .
7. Summary and conclusions
In this paper, we have studied the target-mass dependenceof the parton virtuality distributions. Our main result is thatif one uses the VDF / TMD representations (2.2), (4.9) for ma-trix elements, there are no kinematic z M -corrections that arean inherent feature / artifact of expansions over traceless { pz } n combinations that appear in the twist decomposition. In ourapproach, the PQDs are given by the TMD → PQD conversion8 yQ ( y, P ) M = 0 M/ Λ = 3 P = M Figure 4: Comparison of QPDs Q ( y , P ; M ) for M = Λ and M = P = M . formula (3.8). In the P → ∞ limit of the latter, the PQD Q ( y , P )tends to the twist-2 PDF f ( y ) irrespectively of the fact that theVDF / TMD representation does not involve the twist decompo-sition.We have established that TMDs F ( x , k ⊥ ; M ) must dependon k ⊥ through the k ⊥ + x M combination. Hence, the x M addition here may be considered as a kinematic target-mass correction. Furthermore, TMDs may have a dynamic M -dependence that cannot be predicted from kinematical con-siderations. Just like the form of the k ⊥ -dependence of theTMDs, this part of the M -dependence can only be modeledin our approach.We have studied the e ff ect of the k ⊥ → k ⊥ + x M modi-fication of the TMD models used in our paper [9], and foundthat the M / P corrections become negligible well before thePQD curves Q ( y , P ) become close enough to the correspondingPDF f ( y ). Thus, we see no need to correct the lattice gaugecalculations of PQDs for M -e ff ects.A similar analysis of the target-mass e ff ects can be made forthe pion quasi-distribution amplitude studied recently on thelattice in Ref. [7] and in the VDF approach in Ref. [20]. Sincethe pion mass m π is much smaller than the nucleon mass M (even when m π is taken in its lattice version m π ∼
310 MeV[7]), while the pion size scale ∼ / Λ is not very di ff erent fromthat of the nucleon, the target-mass e ff ects in that case may becompletely ignored.A possible future extension of our findings is an applicationof the VDF / TMD approach to inclusive DIS, with the goal toinvestigate if the target-mass corrections described there by theNachtmann ξ variable [2] are a genuine feature of the processor just an artifact of the twist decomposition. Acknowledgements
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