Transverse-Momentum-Dependent Parton Distribution Functions from Large-Momentum Effective Theory
aa r X i v : . [ h e p - ph ] N ov Transverse-Momentum-Dependent PDFs from Large-Momentum Effective Theory
Xiangdong Ji,
1, 2
Yizhuang Liu, ∗ and Yu-Sheng Liu Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai 200240, China Department of Physics, University of Maryland, College Park, MD 20742, USA (Dated: November 12, 2019)We show that the lightcone transverse-momentum-dependent parton distribution functions(TMDPDFs), important for describing high-energy scattering processes such as Drell-Yan and semi-inclusive deep-inelastic scattering with observed small transverse momentum, can be obtained fromEuclidean lattice QCD calculations in the framework of large-momentum effective theory (LaMET).We present a LaMET factorization of the Euclidean quasi distributions in terms of the physicalTMDPDFs at leading order in (1 /P z ) expansion, with the matching coefficient solved from arenormalization group equation. We demonstrate implementation strategies on lattice with finite-length gauge links and nonperturbative renormalization. The rapidity evolution for quasi-TMDPDFand the rapidity regularization independent factorization scheme are also discussed. Introduction. —High-energy hadron processes involv-ing measuring particles with small transverse momentum( k ⊥ ∼ Λ QCD ) has been of great interest for particle andnuclear physicists for many years [1, 2]. On the one hand,many productions, such as Higgs boson production atLarge Hadron Collider, peak at relatively small trans-verse momentum and cannot be explained completelywith the standard parton distribution functions (PDFs)and perturbative quantum chromodynamics (QCD). Onthe other hand, quarks and gluons confined in a hadronsuch as the proton do carry physical transverse momentawhich warrant new nonperturbative observables to de-scribe. The transverse-momentum-dependent parton dis-tribution functions (TMDPDFs) are the simplest exten-sion of the standard textbook PDFs and have inspired alarge body of theoretical work [1–7]. However, becauseTMDPDFs involves the lightcone correlations of funda-mental fields, there has been little attempt to computethem from lattice QCD (see [8] and references therein forexception).The recent development of large-momentum effectivetheory (LaMET) proposed by one of the authors [9, 10]has opened up a possibility of directly calculating TMD-PDFs on lattice. The essence of LaMET is very simple:while Euclidean lattice does not support modes travel-ling along the lightcone at operators level, it does sup-port on-shell fast moving hadrons which allow extractionof collinear physics large-momentum factorization. How-ever, early attempts on formulating a LaMET calcula-tion on TMDPDFs have not met with complete successbecause of the presence of the soft modes [11–14]. Theso-called soft function appearing in various high-energyprocesses summarizes the soft gluon radiation effects offast moving charged particle. It involves two oppositelightcone directions and presents a crucial difficulty toimplement on lattice. However, the recent progress bythe present authors has showed that it can be calcu-lated as the large-velocity-transfer form factor of a fixed- ∗ [email protected] separation color-anticolor source pair [15]. Thus the finalobstacle to formulate a LaMET calculation for TMD-PDFs has been removed.In this paper, we present the essential ingredients fora lattice calculation of TMDPDFs. We start by defin-ing the TMDPDFs using lightcone correlations in dimen-sional regularization for ultraviolet (UV) divergences andmodified minimal-subtraction (MS) scheme, just as in thestandard PDF case. We then explore Euclidean latticedistributions, quasi-TMDPDFs, which capture the samenonperturbative physics in the large-momentum ( P z )limit. We organize a LaMET factorization which relatesEuclidean quasi-TMDPDFs to Minkowski TMDPDFs atlarge but finite P z , valid to all orders in QCD perturba-tion theory. We discuss subtleties related to the latticeimplementation of the quasi-TMDPDFs: finite-lengthgauge links, self-energy subtraction, and UV renormaliza-tion. The universality feature of LaMET indicates thatthere are infinite many possible quasi-TMDPDFs whichall produce the same result in the large-momentum limit. Factorization of quasi-TMDPDFs . —Defining a properTMDPDF that becomes a common standard for bothphenomenological data fitting and easy perturbativeQCD calculations turns out be more challenging than forthe standard PDFs [16]. The lightcone divergences fromsoft-gluon contributions cannot be simply regulated indimensional regularization: an extra rapidity regulatorhas to be introduced. Many proposals have been madein the literature which result in different ways to iso-late the soft physics (see [14] for a nice summary). Aphysics-motivated definition by Collins [6, 7, 14], calledphysical TMDPDF, turns out to be independent of thisextra complication. This definition involves a combina-tion of the na¨ıve or unsubtracted lightcone parton den-sity f ( x, τ, b ⊥ , µ ) with “one-half” of the soft contribution S ( τ, τ ′ , b ⊥ , µ ) subtracted, f TMD ( x, ζ, b ⊥ , µ ) = f ( x, τ, b ⊥ , µ ) p S ( τ, τ ′ , b ⊥ , µ ) (1)where x is the longitudinal momentum fraction of par-tons; b ⊥ is the transverse separation conjugate to trans-verse momentum k ⊥ ; τ and τ ′ are generic rapidity reg-ulators which in the end give rise to ζ dependence inthe ratio. In this work we consider the quark-TMDPDFin the non-singlet channels with unpolarized hadron. Itis straightforward to generalize the discussions to otherchannel including spin dependence and gluons. The un-subtracted numerator is defined as: f ( x, τ, b ⊥ , µ ) (2)= 12 Z dξ − π e − ixξ − P + h P S | Ψ n ( ξ − , ,~b ⊥ ) γ + Ψ n (0) | P S i (cid:12)(cid:12)(cid:12) τ where | P S i is the hadron state labeled by momen-tum P and spin S ; the “gauge-invariant” quark fieldis Ψ n ( ξ ) = U n ( ±∞ , ξ ) ψ ( ξ ) with the quark color indicesare implicit; ψ is the quark field in lightcone coordinate( ξ − , ξ + , ~ ⊥ ); n µ is the lightlike vector in the minus light-cone direction n = √ (ˆ e t − ˆ e z ) = (1 , ,~ ⊥ ); U n ( ±∞ , ξ ) = P exp h − ig R ±∞ dsn · A ( ns + ξ ) i is a gauge link along n direction pointing to positive or negative infinity; τ isa generic rapidity regulator which only acting on gaugelinks. The sign choice in the gauge link U is pro-cess dependent: For the Drell-Yan process one shouldchoose negative sign while for the semi-inclusive deepinelastic scattering (SIDIS) one should choose positivesign [16, 17]. For unpolarized TMDPDF the two choicesare equivalent. The lightcone soft function is defined bytwo conjugate lightlike Wilson lines, S ( τ, τ ′ , b ⊥ ,µ ) =1 N c tr h | ¯ T [ U † n ′ ( −∞ ,~b ⊥ ) τ ′ U † n ( ±∞ ,~b ⊥ ) τ ] T [ U n ( ±∞ , ⊥ ) τ U n ′ ( −∞ , ⊥ ) τ ′ ] | i (3)with n ′ = √ (ˆ e t + ˆ e z ) is in the plus lightcone direction.Again ± corresponds to the process dependency men-tioned before.To calculate TMDPDFs in Euclidean space, one mustdefine Euclidean correlators corresponding to both un-subtracted TMDPDF f and the soft function S . Thequasi distribution for the unsubtracted parton density isstraightforward, one can define [9–14] e f ( x, ζ z , b ⊥ , µ ) (4)= Z dz π e ixzP z h P S | Ψ n z (0 , z ,~b ⊥ )Γ Ψ n z ( − z | P S i where ζ z = (2 xP z ) ; Ψ n z ( ξ ) = U n z (+ ∞ , ξ ) ψ ( ξ ) in thecoordinate ( t, z, ~ ⊥ ) and U n z is a gauge link extended toinfinity along z -direction, which naturally corresponds tothe choice for the Drell-Yan process; Γ can be chosen as γ z , γ t , or any combination of the two. It is understoodthat the self-interactions of gauge links are subtractedin order to remove the pinch-pole singularity [15, 18],which will be discussed in more detail later. According tothe recent work [15], the soft function with off-lightcone regulator [1, 4, 6, 17] can be obtained as the form factorof a heavy quark-antiquark system: e S ( ρ, b ⊥ , µ ) = v ′ h Q ¯ Q | J ( v, v ′ ,~b ⊥ ) | Q ¯ Q i v (5)where v and v ′ are four-velocities slightly off lightcone,and ρ = q v + v ′− v − v ′ + = e Y + Y ′ is parametrically large. Y isthe rapidity corresponding to v .With the elements above, we define the quasi-TMDPDF as e f TMD ( x, ζ, ζ z , b ⊥ , µ ) = e f ( x, ζ z , b ⊥ , µ ) q e S ( ρ = ζ z /ζ, b ⊥ , µ ) . (6)To ensure a perturbative matching of this to the TMD-PDF, the rapidity scale ρ of the soft function can be fixedto ρ = ζ z /ζ which compensates the rapidity gap betweenthe unsubtracted quasi-TMDPDF at ζ z and the TMD-PDF at ζ . Alternatively, one can write e S as:1 e S ( ρ, b ⊥ , µ ) = e −D ( b ⊥ ,µ ) ln ρ S I ( b ⊥ , µ ) (7)where D ( b ⊥ , µ ) is the Colins-Soper evolution kernel,which can be extracted from the P z evolution [13]. S I ( b ⊥ , µ ) is rapidity-independent and can be extractedfrom a light-meson form factor and quasi-TMD wavefunction [15]. The above discussion allows the calculationof S at any ρ and then evolve to the desired ρ = ζ z /ζ .With the definitions above, the quasi-TMDPDF admitsthe following LaMET factorization at large P z e f TMD ( x, ζ, ζ z , b ⊥ , µ ) = C (cid:0) ζ z /µ (cid:1) f TMD ( x, ζ, b ⊥ , µ )+ O Λ ζ z , M ζ z ! (8)where C can be calculated perturbatively, and M is thehadron mass.The justification for the large P z factorization of thequasi distribution is as follows. At large P z and smalltransverse momentum k ⊥ , all the hard modes are con-fined to two hard-cores in the vicinity of the two ver-tices at 0 and (0 , z, b ⊥ ). Therefore, the hard modes can-not change the momentum fraction of e f . Only collinearmodes will be able to change the momentum fractions x , thus the momentum fraction defined by k z /P z isequivalent to the momentum fraction k + /P + . As aconsequence, e f is equivalent to the unsubtracted TMD-PDF f defined in off-lightcone regularization scheme us-ing nearly-lightlike spacelike vectors, with substitution ζ → ζ z due to boost invariance. Then the factorizationtheorem in the off-lightcone scheme [1, 4] applies and in-dicates that the combination e f e f / e S contains all the non-perturbative information for the Drell-Yan cross section.Comparing with the factorization using TMDPDF, wefind that e f / p e S and f / √ S contain the same nonpertur-bative contribution up to rapidity evolution. This leadsto Eq. (8) with the explicit choice for ρ served as compen-sating any rapidity mismatch between e f TMD and f TMD . Matching kernel and its evolution. —The perturbativestructure of the matching coefficient can be derivedstraightforwardly. It is known that e f satisfies a simplerenormalization group equation [4], µ ddµ ln e f ( x, ζ z , b ⊥ , µ ) = 2 γ F ( α s ) (9)where γ F is the anomalous dimension of quark field inaxial gauge, or the anomalous dimension for the heavy-light quark current [19]. For the soft function we have: µ ddµ ln e S ( ρ, b ⊥ , µ ) = − cusp ( α s ) ln ρ + 2Γ I ( α s ) , (10)where Γ cusp is the universal lightlike cusp anomalous di-mension while Γ I denote the constant term for the cuspanomalous dimension at large rapidity. On the otherhand, for TMDPDF, one has [14, 20], µ ddµ ln f TMD ( x, ζ, b ⊥ , µ ) = Γ cusp ( α s ) ln µ ζ − γ H ( α s ) , (11)where γ H is the hard anomalous dimension.Collecting all of the results above, the matching ker-nel C ( ζ z /µ ) satisfies the following renormalization groupequation: µ ddµ ln C (cid:18) ζ z µ (cid:19) = Γ cusp ln ζ z µ + γ C (12)where γ C = 2 γ F − Γ I + 2 γ H . The general solution to therenormalization group equation reads C (cid:18) α s ( µ ) , ζ z µ (cid:19) = C (cid:16) α s ( p ζ z ) , (cid:17) (13) × exp (cid:26)Z µ √ ζ z dµ ′ µ ′ (cid:20) Γ cusp ( α s ( µ ′ )) ln ζ z µ ′ + γ C (cid:0) α s ( µ ′ ) (cid:1)(cid:21)(cid:27) . This equation allows the determination of the logarith-mic structure for C to all orders in perturbation theory,up to unknown constants related to the initial condition C ( α s , cusp , γ F , γ H all have been calculated tothree-loops [20–22], while Γ I have been worked out totwo-loop level [22].At one-loop level, the unsubtracted quasi-TMDPDFand lightcone TMDPDF can be found in Refs. [12, 14]and Refs. [14]; the soft function is [4, 14] e S ( ρ, b ⊥ , µ ) = 1 + α s C F π (2 − ln ρ ) ln µ b ⊥ e − γ E . (14)Thus the one-loop matching coefficient readsln C (1) (cid:18) α s , ζ z µ (cid:19) = α s (cid:20) c + C F π (cid:18) ℓ −
12 ln ζ z µ (cid:19)(cid:21) (15) where c = C F π ( − π ) is determined by perturbationtheory at one-loop level. We also anticipate that the two-loop matching kernel is of the formln C (2) (cid:18) α s , ζ z µ (cid:19) = α s ( c − (cid:16) γ (2) C − β c (cid:17) ln ζ z µ − (cid:18) Γ (2)cusp − β C F π (cid:19) ln ζ z µ − β C F π ln ζ z µ ) (16)where β = − π (cid:0) C A − N f T F (cid:1) is the coefficient ofone-loop β -function; c is a constant to be determined inperturbation theory at two-loop level. Self-interaction subtraction and renormalization. —Toimplement a calculation of the quasi distribution on lat-tice, a few new elements are needed.First, the Euclidean unsubtracted quasi-TMD e f ( x, ζ z , b ⊥ , µ ) shall now be calculated with a finitelength L of the staple shaped gauge link and closed witha transverse gauge link with width b ⊥ . In large L limit,there are pinch-pole singularities which is responsiblefor the heavy-quark potential term e − LV ( b ⊥ ) . The gaugelink self-energy contains linear divergence which canbe renormalized multiplicatively shown in Ref. [23, 24].Furthermore, there are additional cusp divergencesat junctions of the z -direction and transverse gaugelinks. These extra complications can be eliminated byperforming a subtraction using p Z (2 L, b ⊥ ) defined bya square root of the vacuum expectation of a spacialrectangular Wilson loop with length 2 L and width b ⊥ . The same subtraction applies to the soft functionas well. The na¨ıve soft function in the off-lightconescheme, defined similar to Eq. (3), also contains twostaples along v and v ′ directions. Each of the staplealso contributes to additional cusp divergences and adivergent time evolution factor associated to the heavyquark-antiquark state. As a result, two rectangularWilson loops along these directions are required toremove those singularities. After the subtraction, thesoft function is equal to the form factor in Eq. (5). Seeour recent work [15] for more detail.Second, the lattice version for e f contains UV diver-gences associated to two quark-link vertices and needsrenormalization. To match to the MS scheme in con-tinuum theory, one should adopt regularization indepen-dent renormalization scheme similar to those proposed inRefs. [25–27]. A possible candidate for the renormaliza-tion factor is the nonlocal operator with small ~b R separa-tion in transverse direction sandwiched between off-shellamputated quark with large Euclidean momentum: R ( b R , p E = µ R , p zE = p zR , a ) (17)= h | ψ ( p E ) Ψ n ⊥ (0 ,~b R )Γ Ψ n ⊥ (0) ψ ( p E ) | i (cid:12)(cid:12)(cid:12) amputated where a is lattice spacing. The transverse separation b R ,which has to be perturbatively small, also introduces ex-tra linear divergence from gauge link self-energy. We canmodify the procedure to subtract self-interaction by using p Z (2 L, b ⊥ − b R ) to compensate the extra contribution.Finally, R can be used to renormalize the correlator of e f in position space. The resulting renormalized correlatorhas a well-defined continuum limit ( a →
0) and can bematched to distributions in MS scheme. The matchingkernel in this nonperturbative scheme will be given infuture work.
Discussion and conclusion. —Another important topicthat need to be emphasized is the rapidity evolution ofthe quasi-TMDPDF and the TMDPDF. In the lightconeregularization scheme, the introduction of the lightlikegauge links results in the rapidity divergences which mustbe regularized. The regularization effectively introducesrapidity cutoff for small k + or large k − modes, and thecutoff dependencies transmute to the rapidity evolutionscale ζ for TMDPDFs. However, for a physical process,there is no rapidity divergences and the large k − modesare naturally bounded by the physical hadron momen-tum. Thus, one would expect that the evolution withrespect to the hadron momentum is also equivalent tothe rapidity evolution extracted from the cutoff depen-dence. This is exactly the case for the unsubtractedquasi-TMDPDF. In fact, it can be shown that the ζ z evo-lution of e f ( x, ζ z , b ⊥ , µ ) is the Collins-Soper evolution [1]2 ζ z ddζ z ln e f ( x, ζ z , b ⊥ , µ ) = D ( b ⊥ , µ ) + G ( ζ z , µ ) (18)where D ( b ⊥ , µ ) is the nonperturbative Collins-Soper ker-nel and the G is a perturbative kernel. At one-loop level,one has: D (1) ( b ⊥ , µ ) = − α s C F π ln µ b ⊥ e − γ E , (19) G (1) ( ζ z , µ ) = α s C F π (cid:18) − ln ζ z µ (cid:19) . (20)Notice that the combination D + G is renormalizationevolution free, which was proved in Ref.[1].It is worth to mention that the definition of quasi-TMDPDF is not unique. Alternatively, we can define e f TMD ( x, ζ z , b ⊥ , µ ) = e f ( x, ζ z , b ⊥ , µ ) p S I ( b ⊥ , µ ) such thatthe factorization theorem becomes e f TMD ( x, ζ z , b ⊥ , µ ) (21)= C (cid:18) ζ z µ (cid:19) exp (cid:20) D ( b ⊥ , µ ) ln ζ z ζ (cid:21) f TMD ( x, ζ, b ⊥ , µ ) . This formula is similar to what was proposed in Ref. [14],except the definition of the soft function. Although, thebent soft functions in Ref. [11, 14] give the correct cuspanomalous dimension at one-loop level, in general it dif-fers from S I beyond one-loop correction [28]. In fact,the TMD soft function in general is controlled by cuspanomalous dimension at large hyperbolic angle ratherthan a circular angle.Our formalism allows the factorization of the DY crosssection in terms of the unsubtracted quasi-TMDPDF andsoft function. In fact, one can show that [15] dσ DY d ~b ⊥ dxdx ′ =ˆ σ ( x, x ′ , ζ z , ζ ′ z , µ ) (22) e f ( x, ζ z , b ⊥ , µ ) e f ( x ′ , ζ ′ z , b ⊥ , µ ) S I ( b ⊥ , µ )where ˆ σ is the hard kernel. This factorization scheme ismanifestly rapidity divergence free.In conclusion, we present a complete LaMET formula-tion for calculation of nonperturbative TMDPDFs. Wehave provided with the definitions of lattice calculablequasi-TMDPDFs and soft functions. We demonstratethat the quasi-TMDPDF, defined as a combination ofthe unsubtracted quasi-TMDPDF and the soft function,relates to the lightcone distribution through factoriza-tion. We also show that the matching kernel is controlledby a renormalization group equation with known anoma-lous dimensions. We also argue that the DY processcan be factorized into quasi TMDPDFs directly, withthe P z evolution plays the role of the rapidity evolu-tion. Through the factorization theorem in Eq. (8), thephysical TMDPDFs in Eq. (1) can be calculated on lat-tice. The final obstacle to formulate a quasi-TMDPDFin LaMET has been overcome. This allows first-principlepredictions for TMDPDFs and cross sections in high en-ergy experiments. Acknowledgment. —We thank Andreas Sch¨afer, FengYuan, and Yong Zhao for valuable discussions. Thiswork is supported partially by Science and Tech-nology Commission of Shanghai Municipality (GrantNo.16DZ2260200), National Natural Science Foundationof China (Grant No.11655002), and the US DOE grantDE-FG02-93ER-40762. [1] J. C. Collins and D. E. Soper,Nucl. Phys.
B193 , 381 (1981), [Erratum: Nucl.Phys.B213,545(1983)].[2] J. C. Collins, D. E. Soper, and G. F. Sterman,Nucl. Phys.
B250 , 199 (1985).[3] J. C. Collins, D. E. Soper, and G. F. Sterman, Nucl. Phys.
B308 , 833 (1988).[4] X.-D. Ji, J.-P. Ma, and F. Yuan,Phys. Rev.
D71 , 034005 (2005),arXiv:hep-ph/0404183 [hep-ph].[5] X.-D. Ji, J.-P. Ma, andF. Yuan, Phys. Lett.
B597 , 299 (2004), arXiv:hep-ph/0405085 [hep-ph].[6] J. Collins,
Proceedings, QCD Evolution Work-shop on From Collinear to Non-CollinearCase: Newport News, Virginia, April 8-9,2011 , Int. J. Mod. Phys. Conf. Ser. , 85 (2011),arXiv:1107.4123 [hep-ph].[7] J. C. Collins and T. C. Rogers,Phys. Rev. D87 , 034018 (2013),arXiv:1210.2100 [hep-ph].[8] B. Yoon, M. Engelhardt, R. Gupta, T. Bhat-tacharya, J. R. Green, B. U. Musch, J. W.Negele, A. V. Pochinsky, A. Schfer, andS. N. Syritsyn, Phys. Rev.
D96 , 094508 (2017),arXiv:1706.03406 [hep-lat].[9] X. Ji, Phys. Rev. Lett. , 262002 (2013),arXiv:1305.1539 [hep-ph].[10] X. Ji, Sci. China Phys. Mech. Astron. , 1407 (2014),arXiv:1404.6680 [hep-ph].[11] X. Ji, P. Sun, X. Xiong, andF. Yuan, Phys. Rev. D91 , 074009 (2015),arXiv:1405.7640 [hep-ph].[12] X. Ji, L.-C. Jin, F. Yuan, J.-H. Zhang,and Y. Zhao, Phys. Rev.
D99 , 114006 (2019),arXiv:1801.05930 [hep-ph].[13] M. A. Ebert, I. W. Stewart, andY. Zhao, Phys. Rev.
D99 , 034505 (2019),arXiv:1811.00026 [hep-ph].[14] M. A. Ebert, I. W. Stewart, and Y. Zhao,JHEP , 037 (2019), arXiv:1901.03685 [hep-ph].[15] X. Ji, Y. Liu, and Y.-S. Liu, (2019),arXiv:1910.11415 [hep-ph]. [16] J. Collins, Camb. Monogr. Part. Phys. Nucl. Phys. Cos-mol. , 1 (2011).[17] J. C. Collins and A. Metz,Phys. Rev. Lett. , 252001 (2004),arXiv:hep-ph/0408249 [hep-ph].[18] J. Collins, Proceedings, International Workshopon Relativistic nuclear and particle physics (LightCone 2008): Mulhouse, France, July 7-11, 2008 ,PoS
LC2008 , 028 (2008), arXiv:0808.2665 [hep-ph].[19] X.-D. Ji and M. J. Musolf, Phys. Lett.
B257 , 409 (1991).[20] M.-X. Luo, X. Wang, X. Xu, L. L. Yang, T.-Z. Yang,and H. X. Zhu, (2019), arXiv:1908.03831 [hep-ph].[21] K. G. Chetyrkin and A. G.Grozin, Nucl. Phys.
B666 , 289 (2003),arXiv:hep-ph/0303113 [hep-ph].[22] A. Grozin, J. M. Henn, G. P. Korchemsky, and P. Mar-quard, JHEP , 140 (2016), arXiv:1510.07803 [hep-ph].[23] T. Ishikawa, Y.-Q. Ma, J.-W. Qiu, andS. Yoshida, Phys. Rev. D96 , 094019 (2017),arXiv:1707.03107 [hep-ph].[24] X. Ji, J.-H. Zhang, and Y. Zhao,Phys. Rev. Lett. , 112001 (2018),arXiv:1706.08962 [hep-ph].[25] I. W. Stewart and Y. Zhao,Phys. Rev.
D97 , 054512 (2018),arXiv:1709.04933 [hep-ph].[26] M. A. Ebert, I. W. Stewart, and Y. Zhao, (2019),arXiv:1910.08569 [hep-ph].[27] P. Shanahan, M. Wagman, and Y. Zhao, (2019),arXiv:1911.00800 [hep-lat].[28] G. P. Korchemsky and A. V. Radyushkin,Nucl. Phys.