Finite-t and target mass corrections in off-forward hard reactions
aa r X i v : . [ h e p - ph ] N ov Finite- t and target mass corrections in off-forwardhard reactions V. M. Braun, A. N. Manashov, , Institut f¨ur Theoretische Physik, Universit¨at Regensburg,D-93040 Regensburg, Germany Department of Theoretical Physics, St.-Petersburg State University,199034, St.-Petersburg, RussiaNovember 8, 2018
Abstract
We describe a systematic approach [1] to the calculation of kinematic corrections ∝ t/Q , m /Q in hard exclusive processes which involve momentum transfer from theinitial to the final hadron state. As an example, the complete expression is derivedfor the time-ordered product of two electromagnetic currents that includes all kinematiccorrections due to the quark distribution to twist-four accuracy. The results are applicablee.g. to the studies of deeply-virtual Compton scattering. There is hope that hard exclusive scattering processes in Bjorken kinematics can provide onewith a three-dimensional picture of the proton in longitudinal and transverse plane [2], encodedin generalized parton distributions (GPDs) [3, 4]. One of the most important reactions in thiscontext is Compton scattering with one real and one highly-virtual photon (DVCS) which hasreceived a lot of attention. The QCD description of DVCS is based on the operator productexpansion (OPE) of the time-ordered product of two electromagnetic currents. In this languagethe GPDs appear as leading-twist operator matrix elements. In order to probe the transverseproton structure one needs to measure the dependence of the amplitude on the momentumtransfer to the target t = ( P ′ − P ) in a broad range. Since the available photon virtualities Q are limited to a few GeV range, corrections of the type ∝ t/Q (which are formally higher-twisteffects), are significant and have to be taken into account.Such corrections are usually dubbed “kinematic” since they only involve ratios of kinematicvariables and at first sight have nothing to do with nonperturbative effects (e.g. one may1onsider a theoretical limit Λ ≪ t ≪ Q ). The separation of kinematic corrections ∝ t/Q from generic twist-four corrections O (Λ /Q ) proves, however, to be surprisingly difficult.The problem is well known and its importance for phenomenology has been acknowledged bymany authors [4, 5, 6, 7, 8, 9, 10, 11, 12, 13].The challenge is that, unlike target mass corrections in inclusive reactions [15], which aredetermined solely by the contributions of leading twist operators, the ∼ t/Q corrections to off-forward processes (and for spin-1/2 targets also ∼ m /Q corrections) also receive contributionsfrom higher-twist-four operators that can be reduced to total derivatives of the twist-two ones.Indeed, let O µ ...µ n be a multiplicatively renormalizable (conformal) local twist-two operator,symmetrized and traceless over all indices. The operators O = ∂ O µ ...µ n , O = ∂ µ O µ ...µ n (1)are, on the one hand, twist-four, and on the other hand their matrix elements are related to theleading twist matrix elements times the momentum transfer squared (up to, possibly, targetmass corrections). Thus, both operators contribute to the ∝ t/Q , ∝ m /Q accuracy andmust be taken into account.Moreover, all these contributions are intertwined by electromagnetic gauge and Lorentz in-variance. Implementation of the electromagnetic gauge invariance beyond the leading twistaccuracy has been at the center of many discussions, starting from Ref. [14]. By contrast,importance of the translation invariance condition has never been emphasized, to the best ofour knowledge. In particular the distinction between the kinematic corrections of Nachtmann’stype, i.e. due to contributions of leading-twist [6, 9, 7, 10, 11, 12, 13], and of higher-twist opera-tors in Eq. (1) is not invariant under translations along the line connecting the electromagneticcurrents in the T -product. Hence this distinction has no physical meaning; the existing esti-mates of kinematic effects, e.g. in DVCS, by the contributions of leading twist operators alonecan be misleading.On a more technical level, the problem arises because O has rather peculiar properties:the divergence of a conformal operator vanishes in the free theory (the Ferrara-Grillo-Parisi-Gatto theorem [16]). A related feature is that using QCD equations of motion (EOM) O canbe expressed in terms of quark-antiquark-gluon operators. The simplest example of such arelation is known for many years [17, 18, 19]: ∂ µ O µν = 2¯ qigG νµ γ µ q , (2)where O µν = (1 / qγ µ ↔ D ν q + ( µ ↔ ν )] is the quark part of the energy-momentum tensor. Theoperator on the r.h.s. of Eq. (2) involves the gluon field strength and, naively, its hadronicmatrix elements are of the order of Λ , which is in fact not the case. More complicatedexamples can be found in [20, 21].The general structure of such relations is, schematically( ∂ O ) N = X k a ( N ) k G Nk , (3)where G Nk are twist-four quark-antiquark-gluon operators and a ( N ) k are the numerical coeffi-cients. The subscript N stands for the number of derivatives in O N and the summation goes2ver all contributing operators which may include total derivatives (so that in practice k is acertain multi-index). The same operators, G Nk , also appear in the OPE for the product ofcurrents of interest at the twist-four level: T { j ( x ) j (0) } t =4 = X N,k c N,k ( x ) G Nk . (4)A separation of “kinematic” and “dynamical” contributions to the OPE implies that one at-tempts to reassemble this expansion in such a way that the contribution of a particular com-bination appearing in (3) is separated from the remaining twist-four contributions. The “kine-matic” power correction would correspond to taking into account this term only, and discardingcontributions of “genuine” quark-gluon operators.The guiding principle is that the separation of kinematic and dynamical effects is onlyphysically meaningful (e.g. they are separately gauge- and Lorentz-invariant) if they haveautonomous scale dependence. Different twist-four operators of the same dimension mix witheach other and satisfy a certain renormalization group (RG) equation which can be solved, atleast in principle. Let G N,k be the set of multiplicatively renormalizable twist-four operators sothat G N,k = X k ′ ψ ( N ) k,k ′ G N,k ′ . (5)Eq. (3) tells us that one of the solutions of the RG equation is known without calculation .Indeed, it provides one with an explicit expression for a twist-four operator with the anomalousdimension equal to the anomalous dimension of the leading twist operator. (For simplicity weignore the contributions of ∂ O N in this discussion; they do not pose a problem and can betaken into account using conventional methods.)Let us assume that this special solution corresponds to k = 0, i.e. G N,k =0 ≡ ( ∂ O ) N and ψ ( N ) k =0 ,k ′ = a k ′ . Inverting the matrix of coefficients, ψ ( N ) k,k ′ , and separating the term with k = 0we can write the expansion of an arbitrary twist-four operator in terms of the multiplicativelyrenormalizable ones G N,k = φ ( N ) k, ( ∂ O ) N + X k ′ =0 φ ( N ) k,k ′ G N,k ′ . (6)Inserting this expansion into Eq. (4) one obtains T { j ( x ) j (0) } tw − = X N,k c N,k ( x ) φ ( N ) k, ( ∂ O ) N + . . . , (7)where the ellipses stand for the “genuine” twist-four quark gluon operators (e.g. with differentanomalous dimensions). This is the solution we want to have, but the problem with it is thatfinding the coefficients φ ( N ) k, in general requires knowledge of the full matrix ψ ( N ) k,k ′ , in other wordsthe explicit solution of the twist-four RG equations, which is not available.Our starting observation is that twist-four operators in QCD come in two big groups:the so-called quasipartonic [22], that only involve “plus” components of the fields, and non-quasipartonic which also include “minus” light-cone projections. Quasipartonic operators arenot relevant for the present discussion since they have an autonomous evolution (to one-loop3ccuracy). As a consequence, ( ∂ O ) N does not appear in the expansion of quasipartonic op-erators in multiplicatively renormalizable ones, Eq. (6): the corresponding coefficients φ ( N ) k, vanish. Hence the kinematic power correction ∼ ( ∂ O ) N originates entirely from contributionsof non-quasipartonic operators.Renormalization of twist-four non-quasipartonic operators was studied recently in [23, 24].The main result is that in a suitable operator basis the corresponding RG equations can bewritten in terms of several SL (2)-invariant kernels. Using SL (2)-invariance we are able to provethat the anomalous dimension matrix for non-quasipartonic operators is hermitian with respectto a certain scalar product. This implies that different eigenvectors are mutually orthogonal,i.e. X k µ ( N ) k ψ ( N ) l,k ψ ( N ) m,k ∼ δ l,m , (8)where µ ( N ) k is the corresponding (nontrivial) measure. From this orthogonality relation and theexpression (3) for the relevant eigenvector one obtains, for the non-quasipartonic operators φ ( N ) k, = a ( N ) k || a ( N ) || − , (9)where || a ( N ) || = P k µ ( N ) k ( a ( N ) k ) . Inserting this expression into (7) one ends up with the desiredseparation of kinematic effects.The actual derivation is done using the two-component spinor formalism in intermediatesteps and requires some specific techniques of the SL (2) representation theory. This talk isbased on the results presented in Ref. [1]; details of the derivation will be given in a forthcomingpaper. We have been able to find the contributions related to the leading-twist operator (11) in the T -product of two electromagnetic currents T µν = i T { j emµ ( x ) j emν (0) } to twist-four accuracy. Theresult can be brought to the form T µν = − π x n x α h S µανβ V β + iǫ µναβ A β i + x h ( x µ ∂ ν + x ν ∂ µ ) X + ( x µ ∂ ν − x ν ∂ µ ) Y io , (10)where ∂ µ = ∂/∂x µ , S µανβ = g µα g νβ + g να g µβ − g µν g αβ and a totally antisymmetric tensor isdefined such that ǫ = 1. The expansion of invariant functions V β and A β starts from twisttwo, wheareas X and Y are already twist-four. In order to write the result we first need tointroduce some notations.We define nonlocal (light-ray) vector O V and axial-vector O A operators of the leading-twist-two as the generating functions for local twist-two operators O ( z x, z x ) = (cid:2) ¯ q ( z x ) /x ( γ ) Q q ( z x ) (cid:3) l.t. . (11)Here x µ is an arbitrary four-vector (not necessarily light-like), z and z are real numbers and Q is the matrix of quark electromagnetic charges. Here and below the Wilson line between the4uark fields is implied. The leading-twist projector [ . . . ] l.t. stands for the subtraction of tracesof the local operators so that by definition (cid:2) ¯ q ( z x ) /x Q q ( z x ) (cid:3) l.t. == X N N ! x µ x µ . . . x µ N n ¯ q (0) γ µ [ z ← D µ + z → D µ ] . . . [ z ← D µ N + z → D µ N ] Q q (0) − traces o . (12)The leading-twist light-ray operators satisfy the Laplace equation ∂ x O ( z x, z x ) = 0 . Theexplicit form of the projector [ . . . ] l.t. is irrelevant for what follows. Useful representations canbe found e.g. in [9, 25].Thanks to crossing symmetry the vector and axial-vector operators always appear to beantisymmetrized and symmetrized over the quark and antiquark positions, respectively, so wedefine the corresponding combinations: O ( − ) V ( z , z ) = (cid:2) ¯ q ( z x ) /x Q q ( z x ) (cid:3) l.t. − ( z ↔ z ) , (13) O (+) A ( z , z ) = (cid:2) ¯ q ( z x ) /x γ Q q ( z x ) (cid:3) l.t. + ( z ↔ z ) . The leading-twist expressions are well known and can be written as (cf. [25]) V t =2 µ = 12 ∂ µ Z du O ( − ) V ( u, , A t =2 µ = 12 ∂ µ Z du O (+) A ( u, . (14)Note that the separation of the leading-twist terms [ . . . ] l.t. from the nonlocal operators producesa series of kinematic power corrections to the amplitudes, which are similar to Nachtmann targetmass corrections in deep-inelastic lepton-nucleon scattering [15]. Such corrections are discussedin detail in [8, 9, 7, 10, 11, 12, 13].For the twist-three functions we obtain V t =3 µ = h i P ν , Z du n iǫ µαβν x α ∂ β e O (+) A ( u ) + (cid:16) S µανβ x α ∂ β + ln u ∂ µ x ∂ ν (cid:17) e O ( − ) V ( u ) oi , A t =3 µ = h i P ν , Z du n iǫ µαβν x α ∂ β e O ( − ) V ( u ) + (cid:16) S µανβ x α ∂ β + ln u ∂ µ x ∂ ν (cid:17) e O (+) A ( u ) oi . (15)Here P ν is the momentum operator [ i P ν , q ( y )] = ∂∂y ν q ( y ) , and we used the notation e O ( ± ) a ( z ) = 14 Z z dw O ( ± ) a ( z, w ) . (16)One can easily verify that x µ V t =3 µ = ∂ µ V t =3 µ = 0 and similarly x µ A t =3 µ = ∂ µ A t =3 µ = 0. Note thatthe terms in ln u in Eqs. (15) are themselves twist-four and can be omitted if the calculation isdone to twist-three accuracy. The resulting simplified expression is in agreement with Refs. [6,7]. These terms must be included, however, in order to ensure correct separation of twist-threeand twist-four contributions.The flavor-nonsinglet twist-four contributions to Eq. (10) present our main result. In thiscase we prefer to write the answer in terms of integrals over the position of the local conformal5perators, cf. Eq. (17). This form is usually referred to as the conformal OPE [26]. Forexample, a light-ray operator can be written as the conformal expansion O ( z x, z x ) = X N κ N z N Z du ( u ¯ u ) N +1 (cid:2) O N ( z u x ) (cid:3) l.t. , (17)where κ N = 2(2 N + 3) / ( N + 1)!and we use the shorthand notation ¯ u = 1 − u , z = z − z , z u = ¯ uz + uz . The conformaloperator O N is defined as O N ( y ) =( ∂ z + ∂ z ) N C / N (cid:18) ∂ z − ∂ z ∂ z + ∂ z (cid:19) O ( z x + y, z x + y ) (cid:12)(cid:12)(cid:12) z i =0 , (18)where C / N ( x ) is the Gegenbauer polynomial.The leading-twist contribution to the OPE of two electromagnetic currents can be writtenin the same form, for comparison: V t =2 µ = ∂ µ X N, odd κ N N + 2 Z du u N ¯ u N +2 [ O VN ( ux )] l.t. . (19)Here O VN ( ux ) is the conformal operator (18) at the space-time position ux .We obtain V t =4 µ = 12 X N, odd κ N ( N + 2) Z du (cid:26) ( u ¯ u ) N +1 x µ [ b O VN ( ux )] l.t. + N u N − ¯ u N +2 h u + 1 N + 2 i x ∂ µ [( c O ) VN ( ux )] l.t. (cid:27) , A t =4 µ = 14 X N, even κ N N ( N + 2) Z du u N − ¯ u N +2 h u + 1 N + 2 i x ∂ µ [ b O AN ( ux )] l.t. , X t =4 = 14 X N, odd κ N ( N + 2) Z du u N − ¯ u N +1 h − N + 1 N + 2 ¯ u i [ b O VN ( ux )] l.t. , Y t =4 = − X N, odd κ N ( N + 2) Z du u N − ¯ u N +1 h − N + 1 N + 2 ¯ u + 2 N + 1 N + 3 ¯ u i [ b O VN ( ux )] l.t. . (20)Here b O N is defined as the divergence of the leading-twist conformal operator, cf. O in Eq. (1): b O N ( y ) = 1 N + 1 ∂∂x µ (cid:2) i P µ , O N ( y ) (cid:3) = (cid:2) i P µ , O µµ ...µ N ( y ) (cid:3) x µ . . . x µ N . (21)Note that the operator O in Eq. (1), which corresponds to [ i P µ [ i P µ , O N ] in our presentnotation, does not contribute to the answer for our special choice of the correlation function6 { j µ ( x ) j ν (0) } . The T-product with symmetric positions of the currents, T { j µ ( x ) j ν ( − x ) } ,includes both operators. The corresponding expression turns out to be much more cumbersome.Conservation of the electromagnetic current implies that ∂ µ T µν ( x ) = 0 and ∂ ν T µν ( x ) = i [ P ν , T µν ( x )]. We have checked that these identities are satisfied up to twist-5 terms.For completeness we give the relation for the operator [ i P µ , ∂ µ O ( z , z )] entering the twist-three functions V t − µ , A t − µ in terms of b O N :[ i P µ , ∂ µ O ( z , z )] = 12 S + Z udu [ i P µ [ i P µ , O ( uz , uz )]]+ X N κ N ( N +1) z N Z dv v N Z du ( u ¯ u ) N +1 b O N ( vz u x ) , (22)where S + = z ∂ z + z ∂ z + 2 z + 2 z . It is also possible to rewrite, v.v., all contributions oflocal operators b O N in terms of the nonlocal light-ray operator [ i P µ , ∂ µ O ( z , z )], which can beadvantageous in certain applications. Hadronic matrix elements of the twist-4 operator b O N are of course related to those of theleading twist, O N . For illustration, we present the corresponding explicit expressions for thetwo proton states with momenta p ′ / = p , which are relevant e.g. for virtual Compton scattering.The leading-twist matrix elements can be parametrized as (cf. [3, 4]) h p ′ |O N ( n ) | p i = ¯ u ( p ′ ) /nu ( p ) N X k = even F N,k ( t )∆ k + P N − k + + 1 m ¯ u ( p ′ ) u ( p ) N +1 X k = even H N,k ( t )∆ k + P N +1 − k + , (23)where F N,k ( t ) and H N,k ( t ) are generalized form factors corresponding to moments of the leading-twist GPD and we used the notations P = ( p + p ′ ) /
2, ∆ = p ′ − p , p = ( p ′ ) = m , t = ∆ ; u ( p ) is the nucleon spinor. By analogy, we define h p ′ | b O N ( n ) | p i = ¯ u ( p ′ ) /nu ( p ) N X k = even b F N,k ( t )∆ k + P N − k + + 1 m ¯ u ( p ′ ) u ( p ) N +1 X k = even b H N,k ( t )∆ k + P N +1 − k + . (24)A short calculation yields b F N,k ( t ) = t F N,k ( t ) k (2 N + 3 − k )2( N + 1) − (cid:18) m − t (cid:19) F N,k − ( N − k + 2)( N − k + 1)2( N + 1) b H N,k ( t ) = t H N,k ( t ) k (2 N + 3 − k )2( N + 1) − (cid:18) m − t (cid:19) H N,k − ( N − k + 3)( N − k + 2)2( N + 1) − m ( N − k + 2)( N + 1) F N,k − ( t ) . (25)Note that the twist-4 matrix elements involve both finite- t and target (nucleon) mass correc-tions. Concrete applications will be considered elsewhere.7 Conclusions
To summarize, we have given a complete expression for the time-ordered product of two elec-tromagnetic currents that resums all kinematic corrections related to quark GPDs to twist-fouraccuracy. The results can be applied to various two-photon processes, e.g. to the studies ofdeeply-virtual Compton scattering and γ ∗ → ( π, η, . . . ) + γ transition form factors. The twist-four terms calculated in this work give rise to both a ∝ t/Q correction and the target masscorrection ∝ m /Q for DVCS, whereas for the transition form factors these two effects areindistinguishable as there is only one scale. The main remaining question is whether QCDfactorization itself is valid in such reactions to twist-four accuracy, at least for kinematic con-tributions. Clarification of this issue goes beyond the tasks of this study. Acknowledgments
The work by A.M. was supported by the DFG, grant BR2021/5-2, and RFFI, grant 09-01-93108. V.B. thanks Profs. A. Faessler and J. Wambach for the invitation to the school andhospitality.
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