Applications of perturbative QCD to hard exclusive electroproduction
processes in the Bjorken limit at small invariant momentum transfer t bring in
a new type of parton distributions which have hybrid properties, resembling
both the parton distribution functions and the distribution amplitudes. Their
t-dependence is analogous to that of hadronic form factors. We discuss general
properties of these new parton distributions, their relation to usual parton
densities and the evolution equations which they satisfy.Comment: 4 pages, to be published in the Proceedings of the International
  Conference on Deep Inelastic Scattering, Chicago, April 1997, AI

Radyushkin, Anatoly V.

English

arXiv

Applications of perturbative QCD to hard exclusive electroproduction processes in the Bjorken limit at small invariant momentum transfer t bring in a new type of parton distributions which have hybrid properties, resembling both the parton distribution functions and the distribution amplitudes. Their t-dependence is analogous to that of hadronic form factors. The authors discuss general properties of these new parton distributions, their relation to usual parton densities and the evolution equations which they satisfy

Radyushkin, A.V.

UNT (University of North Texas) Digital Library

arXiv:hep-ph/9706492   24 Jun 97e-print JLAB-THY-97-25June 1997Asymmetric Parton DistributionsAnatoly V. RadyushkinPhysics Department, Old Dominion University, Norfolk, VA 23529, USAJeerson Lab, Newport News,VA 23606, USAApplications of perturbative QCD to hard exclusive electroproduction processes in the Bjorkenlimit at small invariant momentum transfer t bring in a new type of parton distributions whichhave hybrid properties, resembling both the parton distribution functionsfa(x) and the distributionamplitudes. Their t-dependence is analogous to that of hadronic form factors. We discuss generalproperties of these new parton distributions, their relation to usual parton densities and the evolutionequations which they satisfy.INTRODUCTIONThe use of phenomenological functions accumulating information about nonperturbative long-distance dynamics isthe standard feature of applications of pQCD to hard processes. The well-known examples are the parton distributionfunctions fp=H (x) [1] used in pQCD approaches to hard inclusive processes and distribution amplitudes '(x) [2]etc. which naturally emerge in the asymptotic QCD analyses of hard exclusive processes. Recently, it was arguedthat the same gluon density fg(x) used for description of hard inclusive processes also determines the amplitudes ofhard exclusive J= [3] and -meson [4] electroproduction. It was also proposed [5] to use another exclusive processof deeply virtual Compton scattering p ! p0 (DVCS) for measuring parton distribution functions inaccessiblein inclusive measurements. An important feature (noticed long ago [6{8]) is that kinematics of DVCS and hardelastic electroproduction processes, in a frame where the hadron moves fast, requires the presence of the longitudinalcomponent in the momentum transfer r  p   p0 from the initial hadron to the nal: rk = p. This means thatthe matrix element hp0j : : : jpi is essentially asymmetric (\skewed"). For DVCS and -electroproduction in the regionQ2 >> jtj;m2H , the \skewedness" parameter  coincides with the Bjorken variable xBj = Q2=2(pq). For diractiveprocesses, both the skewedness  of the nonperturbative matrix element hp0j : : : jpi and the absolute value of themomentum transfer t  (p0   p)2 are small. Studying the DVCS process, one should be able to consider the wholeregion 0    1 and t  1GeV 2 [9]. In this situation, one deals with essentially nonforward (or o-forward interminology of ref. [5]) matrix element hp0j : : : jpi. The pQCD approaches incorporating asymmetric/o-forwardparton distributions were formulated in refs. [5,9{12]. A detailed analysis of pQCD factorization for hard mesonelectroproduction processes was given in ref. [13]. The basic idea of our papers [10,11] is that one should treat theinitial momentum p and the longitudinal part of the momentum transfer r on equal footing by introducing doubledistributions F (x; y), which specify the fractions of p and rk carried by the active parton: kk = xp + yrk. Thesedistributions look like distribution functions with respect to x and like distribution amplitudes with respect to y. Thelongitudinal momentum transfer rk is proportional to p: rk = p and it is convenient to parametrize matrix elementshp  rj : : : jpi by nonforward F(X; t) or, in the t! 0 limit, asymmetric distribution functions F(X) specifying thetotal light-cone fractions Xp = (x+y)p, (X )p  X0p of the initial hadron momentum p carried by the \outgoing"Also at Laboratory of Theoretical Physics, JINR, Dubna, Russiaand \returning" partonsy. Double distributions F (x; y) are universal functions in the sense that they do not dependon the skewedness parameter  while the asymmetric distribution functions F(X) form a family of X-dependentfunctions changing their shape when  is changedz. The functions F(X) also have hybrid properties. When X  ,the returning parton carries a positive fraction (X   )p of the initial momentum p, and F(X) is similar to the usualparton distribution f(X). In the region 0  X   the dierence X    is negative, i:e:; the second parton shouldbe treated as propagating together with the rst one. The partons in this case share the longitudinal momentumtransfer rk = p in fractions Y  X= and 1  Y : in the region X   the function F(X) looks like a distributionamplitude.It is very important to know spectral properties of the parton distributions F (x; y) and F(X). In ref. [12], itwas shown that both arguments of a double distribution F (x; y) satisfy the natural \parton" constraints 0  x  1,0  y  1 for any Feynman diagram contributing to F (x; y). A less obvious restriction 0  x + y  1 [10{12]guarantees that the argument X = x + y of the asymmetric distribution F(X) also changes between 0 and 1.Since X = 0 can be obtained only if both x = 0 and y = 0, due to vanishing phase space for such a congurationthe asymmetric distributions F(X) vanish for X = 0. This property is very essential, because the hard subprocessamplitudes usually contain 1=X factors. When F(0) 6= 0, one faces a non-integrable singularity F(X)=X at X = 0.ASYMMETRIC DISTRIBUTIONSThere are two leading-twist quark operators  a(0)E(0; z;A) a(z) and  a(0)5E(0; z;A) a(z), where E(0; z;A)is the usual path-ordered exponential which makes the operators gauge-invariant. In the forward case, the rst onegives the spin-averaged densities fa(x) while the second one is related to the spin-dependent ones fa(x). Here, wewill discuss the  aE(0; z;A) a operators and gluonic operators with which they mix under evolution. The relevantnonforward matrix element can be written ash p0; s0 j  a(0)ẑE(0; z;A) a(z) j p; s ijz2=0 (1)= u(p0; s0)ẑu(p; s)Z 10e iX(pz)Fa (X; t)  ei(X )(pz)Fa (X; t)dX+ u(p0; s0)ẑr̂   r̂ẑ2Mu(p; s)Z 10e iX(pz)Ka (X; t)   ei(X )(pz)Ka (X; t)dX;where ẑ  z. In Eq.(1), quark and antiquark contributions are explicitly separated (cf. [15]). As emphasized byX. Ji [5], the parametrization of this nonforward matrix element must include both the nonip functions F(X; t) andthe spin-ip functions which we denote by K(X; t). Taking the O(z) term of the Taylor expansion gives the sumrules [5]Z 10Fa (X; t)  Fa (X; t)dX = F a1 (t) ; (2)Z 10Ka (X; t)  Ka (X; t)dX = F a2 (t) (3)relating the nonforward distributions Fa (X; t), Ka (X; t) to the a-avor components of the Dirac and Pauli formfactors, respectively. For gluons, the nonforward distribution can be dened through the matrix elementhp0 j zzGa(0)Eab(0; z;A)Gb(z) j pijz2=0 (4)= u(p0)ẑu(p) (z  p)Z 1012he iX(pz) + ei(X )(pz)iFg (X; t) dX + \Kg" : (5)In the formal t = 0 limit, the nonforward distributions F(X; t), K(X; t) convert into the asymmetric distri-bution functions F(X), K(X). In the  = 0 limit, F(X) reduce to the usual parton densities Fa=0(X) =fa(X) ; Fg=0(X) = Xfg(X): For small , Fa (X) diers from fa(X) by O().yAsymmetric distribution functions dened in ref. [11] are similar to, but not coinciding with the t! 0 limit of the o-forwardparton distributions introduced by X.Ji [5].zA more detailed discussion of double distributions and their relation to nonforward distributions was given in my talk aboutDVCS [14].2EVOLUTION EQUATIONSNear the light cone z2  0, the bilocal operators  (0) : : : (z) develop logarithmic singularities ln z2, and thelimit z2 ! 0 is singular. Calculationally, these singularities manifest themselves as ultraviolet divergences for thelight-cone operators. The divergences are removed by a subtraction prescription characterized by some scale :F(X; t)! F(X; t;). In QCD, the gluonic operator Og(uz; vz) = zzGa(uz)Eab(uz; vz;A)Gb(vz) mixes underrenormalization with the avor-singlet quark operator. In the leading logarithm approximation, the easiest way toget the evolution equations for nonforward distributions is to use evolution equations [16,17] for light-ray operators:ddOa(0; z) =Z 10Z 10XbBab(u; v)Ob(uz; vz) (u + v  1) du dv ; (6)where v  1  v and a; b = g;Q. For valence distributions, there is no mixing, and their evolution is generated by theQQ-kernel alone. Inserting Eq.(6) between chosen hadronic states and parametrizing matrix elements by appropriatedistributions, one can get DGLAP and BL-type kernels and also to calculate the new kernels Rab(x; y; ; ) andW ab (X;Z). The kernels Rab(x; y; ; ) govern the evolution of double distributions:ddF a(x; y; t;) =Z 10dZ 1 0XbRab(x; y; ; )F b(; ; t;) d : (7)The form of equations is not aected by the t-dependence, and \t" will not be indicated in what follows. Sinceintegration over y converts F a(x; y) into the parton density fa(x), whose evolution is governed by the DGLAPequation, the kernel RQQ(x; y; ; ) has the propertyZ 1 x0RQQ(x; y; ; )dy =1PQQ(x=): (8)Integrating RQQ(x; y; ; ) over x one gets the BL-type kernel:Z 1 y0RQQ(x; y; ; )dx = VQQ(y; ): (9)Another set of kernels W ab (X;Z) describes the evolution of nonforward distributions and asymmetric distributionfunctions:ddFa (X; t;) =Z 10XbW ab (X;Z)Fb (Z; t;) dZ : (10)When  = 1, the initial momentum coincides with the momentum transfer and F(X) reduces to a distributionamplitude whose evolution is governed by the BL-type kernels: W ab=1(X;Z) = Vab(X;Z): The BL-type kernelsappear as a part of the asymmetric kernel W ab (X;Z) even in the general  6= 1; 0 case [12]. As explained earlier, if Xis in the region X  , then the function F(X) can be treated as a distribution amplitude 	(Y ) with Y = X=. Forthis reason, when both X and Z are smaller than , the kernels W ab (X;Z) simply reduce to the BL-type evolutionkernels V ab(X=; Z=). Furthermore, the BL-type kernels also govern the evolution in the region corresponding totransitions from a fraction Z which is larger than  to a fraction X which is smaller than . Note, that the evolutionjump through the critical fraction  is irreversible: X   if Z  . To put it in words, when the parton momentumdegrades in the evolution process to values smaller than the momentum transfer p  r, further evolution is like thatfor a distribution amplitude: the momentum can decrease or increase up to the r-value but cannot exceed this value.When X > , the initial quark momentumXp is larger than the momentum transfer r = p, and the asymmetricdistribution function Fa (X) is a generalization of the usual distribution function fa(X) for a skewed kinematics.Hence, the evolution in the region  < X  1 ,  < Z  1 is similar to that generated by the DGLAP equation. Therelevant kernels P ab (X;Z) can be obtained from the light-ray kernels, e:g:;PQQ (X;Z) =Z  XZZ0Z 10Bab ( w (1 X=Z) ; w (1 X0=Z 0)) dw ; (11)where Z0  Z   . Note, that PQQ (X;Z) is given by a function symmetric with respect to the interchange of X;Zwith X0; Z0. In the  ! 0 limit, PQQ (X;Z) converts into the DGLAP kernel: PQQ=0(X;Z) = PQQ(X=Z)=Z.3Acknowledgement. This work was supported by the US Department of Energy under contract DE-AC05-84ER40150.[1] R.P.Feynman, The Photon-Hadron Interactions, W.A.Benjamin,Inc., 1972.[2] A.V. Efremov and A.V. Radyushkin, Theor. Math. Phys. 42 (1980) 97.[3] M.G. Ryskin, Z. Phys. C37 (1993) 89.[4] S.J. Brodsky et al. Phys.Rev. D50 (1994) 3134.[5] X. Ji, Phys.Rev.Lett. 78 (1997) 610.[6] J. Bartels and M. Loewe, Z.Phys. C12 (1982) 263.[7] L.V. Gribov, E.M. Levin and M.G. Ryskin, Phys.Reports, 100 (1983) 1.[8] F.-M. Dittes et al. Fortschr.Phys. 42 (1994) 101.[9] X. Ji, Phys.Rev. D55 (1997) 7114; hep-ph/9609381.[10] A.V. Radyushkin, Phys. Lett. B380 (1996) 417.[11] A.V. Radyushkin, Phys. Lett. B385 (1996) 333.[12] A.V. Radyushkin, preprint JLAB-THY-97-10; hep-ph/9704207.[13] J.C. Collins, L. Frankfurt, and M. Strikman, preprint CERN-TH/96-314.[14] A.V. Radyushkin, these Proceedings.[15] A.V. Efremov and A.V. Radyushkin, Lett. Nuovo Cim. 19 (1977) 83.[16] Th. Braunschweig et al. Ann. Phys. (Leipzig) 44 (1987) 403.[17] I.I. Balitsky and V.M. Braun, Nucl.Phys. B311 (1988/89) 541.4

Asymmetric Parton Distributions

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