Gluon Gravitational Form Factors at Large Momentum Transfer
aa r X i v : . [ h e p - ph ] J a n Gluon Gravitational Form Factors at Large Momentum Transfer
Xuan-Bo Tong,
1, 2, 3
Jian-Ping Ma, and Feng Yuan CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,Chinese Academy of Sciences, Beijing 100190, China School of Physical Sciences,University of Chinese Academy of Sciences,Beijing 100049,China Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
We perform a perturbative QCD analysis of the gluonic gravitational form factors (GFFs) of theproton and pion at large momentum transfer. We derive the explicit factorization formula of theGFFs in terms of the distribution amplitudes of hadrons. At the leading power, we find that A πg ( t ) = C πg ( t ) ∼ / ( − t ) for pion, A pg ( t ) ∼ / ( − t ) and C pg ( t ) ∼ ln ( − t/ Λ ) / ( − t ) for proton, respectively,where t is the momentum transfer and Λ a non-perturbative scale to regulate the endpoint singularityin C pg calculation. Our results provide a unique perspective of the momentum dependence of theGFFs and will help to improve our understanding of the internal pressure distributions of hadrons. I. INTRODUCTION
The gravitational form factors (GFFs) are the fun-damental ingredients to probe the internal structureof hadrons. As the matrix elements of the energy-momentum tensor (EMT) [1–6], they provide importantinformation on the hadron’s mass and spin [2–14], andthe mechanical property [15–19]. In experiments, theGFFs can be constrained from the generalized partondistributions (GPD) [5, 6, 20–22] which are measured inthe hard exclusive processes like deeply virtual Comptonscattering [5, 6, 23–25] and deeply virtual meson produc-tion [26–28].Recently, a glimpse of the quark GFFs and its inter-pretation as a pressure distribution inside the proton hasbeen reported in Ref. [17]. The lattice QCD has alsobeen applied to compute the GFFs for the quarks andgluons [29–38] and deep insight has been obtained fromthese studies [18]. All these developments have attractedgreat attention in the hadron physics community andit is expected that future measurements at both JLab12 GeV [39] and the Electron-Ion collider [40, 41] willprovide more important constraints on the quark/gluonGFFs of the hadrons.In this paper, we will investigate the GFFs at large mo-mentum transfer, focusing on the gluonic contributions.This will provide a unique perspective of their behaviorsand improve the parameterizations in the wide range ofkinematics. At large momentum transfer, the form fac-tors can be calculated from perturbative QCD [42–48].Previously, a power counting method [49–51] was appliedto estimate the power behaviors for the quark GFFs [52].The power behavior arguments have also played impor-tant roles in the phenomenology studies [17, 38, 53]. Thefactorization formalism for the GFFs follows that devel-oped in the literature for the hard exclusive processesat large momentum transfer and the final results de-pend on the gauge invariant distribution amplitudes ofhadrons [54–59].Meanwhile, the gluon GFFs of nucleon play impor-tant roles in the near threshold heavy quarkonium photo-productions. These processes have gained quite an inter- est in recent years, because they promise to measure theproton mass decomposition [60–69]. In the near thresh-old region, the momentum transfer from the nucleon tar-get is relatively large, ( − t ∼ and 10GeV for J/ψ and Υ, respectively). Therefore, our results for the gluonGFFs at large momentum transfer shall make a valuablecontribution to understanding the t -dependence in theseprocesses.The gravitational form factors of the hadrons are thetransition matrix elements of the energy momentum ten-sor. The gluon sector reads, T µνg = G aµα G aνα + 14 g µν G aαβ G aαβ , (1)where G aµν = ∂ µ A aν − ∂ ν A aµ − g s f abc A bµ A cν is the strengthtensor of the gluon field A aµ . For the proton, the GFFsare parametrized as [5, 6], h P ′ , s ′ | T µνg | P, s i = ¯ U s ′ ( P ′ ) h A g ( t ) γ { µ ¯ P ν } + C g ( t ) ∆ µ ∆ ν − g µν ∆ M + · · · (cid:21) U s ( P ) , (2)where P and P ′ are the initial and final state hadronmomentum, respectively, ∆ = P ′ − P is the momentumtransfer and t = ∆ , ¯ P = ( P + P ′ ) / a { µ b ν } = ( a µ b ν + a ν b µ ) / U s ( P ) is the spinor of thenucleon with the spin s and mass M , which is nomarlizedas ¯ U s ( P ) U s ( P ) = 2 M . Here, we follow the notations inRefs. [5, 6], where C form factor has also been referred as D or d form factor in Refs. [15–19] with different normal-ization: D ( t ) = 4 / d ( t ) = 4 C ( t ). In addition, we onlykeep the A and C form factors in the above equation forsimplicity.All the gluon form factors depend on the renormaliza-tion scale, since the gluon piece of EMT is not conservedindividually and only the total GFFs are renormaliza-tion independent. Generally, the A -form factors describethe distributions of the quark or gluon momentum insidethe hadron, whereas the C -form factors characterize themechanical properties.In the following, we first show the derivations of thegluon GFFs of pion, where we compute both A and C form factors. Different from previous analysis, we findthat both form factors scale as 1 / ( − t ) at large momen-tum transfer. Then, we derive the gluon GFFs of nucleon.Different from the pion case, the nucleon’s C form factoris power suppressed respect to the A form factor. Themethod developed in these calculations can be extendedto all other form factors. II. GRAVITATIONAL FORM FACTOR FORPION
We start our analysis with the pion GFFs [15, 16, 18], h P ′ | T µνg | P i = 2 ¯ P µ ¯ P ν A πg ( t )+ 12 (∆ µ ∆ ν − g µν ∆ ) C πg ( t ) + · · · , (3)where we have neglected C form factor in the above pa-rameterization. As shown in Fig. 1, there is one diagramthat contributes at the leading order of perturbation the-ory. The circle cross in the diagram denotes the localoperator of the gluon EMT in Eq. (3).Considering the leading asymptotic behaviour of large − t , the light-cone Fock state expansion of the pion havebeen performed with only minimal numbers of parton.The gluon EMT operator transport the two hard gluonexchanges between the quark line and generate the hardpart of the GFFs. Compared to the hard scale t , onecan neglect the transverse momenta of partons in thehard part, since they are expected to be on the or-der of Λ QCD . Integrating out the k ⊥ in the pion wavefunction, we obtain the disribution amplitude φ ( x ) = R d k ⊥ (2 π ) ψ ( x, k ⊥ ) [54]. This finally leads to a factorizationformula for the GFFs of the pion at large t : A πg ( t, µ ) = Z dx dy φ ∗ ( y , µ ) φ ( x , µ ) A πg ( x , y , t, µ ) ,C πg ( t, µ ) = Z dx dy φ ∗ ( y , µ ) φ ( x , µ ) C πg ( x , y , t, µ ) , (4)where A πg or C πg is the perturbative calculable hard partof the GFFs. Before we present a detailed result forthe hard part, a power counting analysis can be de-rived [49, 50]. The diagram of Fig. 1 is very similarto that for the electromagnetic form factor calculationat large momentum [43]. Therefore, we can apply thesame power counting and deduce that they should scaleas 1 / ( − t ) at large − t . Of course, we have to make surethat they do contribute to nonzero A g and C g .Carrying out the calculations of Fig. 1, it is interestingto find out that the A g and C g form factors have thesame hard coeffcient, A πg ( x , y , t ) = C πg ( x , y , t ) = 8 πα s C F − t (cid:18) x ¯ x + 1 y ¯ y (cid:19) , (5) (a) (b) Figure 1: Representative diagrams of two classes that contribute to the proton GFFs at the large − t limit.1 FIG. 1. Leading order diagram contribution to the gluonicgravitational form factor of pion at large momentum trans-fer, where the incoming and out going hadron states havemomenta P and P ′ . The cross symbol in the middle of thediagram represents the operator of the gluonic component inthe energy-momentum tensor of Eq. (1). where C F = 4 / x = 1 − x is used.A number of interesting features can be found from theabove result. First, A πg and C πg GFFs of the pion havethe same power counting of t . This is different from thenucleon case below, where C pg is power suppressed com-pared to A pg . Second, they share exactly the same large- t behavior. This is a surprising result. It will be interestedto check higher order corrections. In general, we expectthis will change.The hadron GFFs can be derived from the sum rulesof the GPDs [5]. The quark GPDs at large momentumtransfer have been calculated in Ref. [57]. We can followthe same procedure to compute the gluon GPD of pion atlarge momentum, and we find that it leads to the sameresult for the gluon GFFs as above. This provides animportant cross check for our derivations.In addition, we can derive the quark GFFs for pionfrom the quark GPD results from Ref. [57]. In termsof the same factorization formula, we obtain the hardcoefficients for A q and C q as, A πq ( x , y , t ) = 8 πα s C F − t x + y + 1¯ x ¯ y , C πq ( x , y , t ) = 8 πα s C F − t x + y − x ¯ y . (6)It is interesting to note that, different from the gluoncase, A and C form factors are not the same for thequark. However, they have the same power behavior.This is different from the power counting analysis de-rived in Ref. [52]. The physics implication of this resultdeserves further investigations.Physically, the C π ( t ) characterizes the mechanicalproperties such as pressure distribution and shear forcesinside the pion system [16]. It also determines the me-chanical radius of the hadron [16, 70]. The above resultsprovide important perspectives on these interpretations.Another important point from our results is that C πg ( t )is positive at large ( − t ), whereas there is a strong argu-ment that C πg is negative at low ( − t ) [16] and a recentlattice calculation also confirms that [38]. That meansthat C g ( t ) will change sign at higher ( − t ). We hope fu-ture lattice simulation can extend to higher momentumtransfer to test this prediction. (a) (b) Figure 1: Representative diagrams of two classes that contribute to the proton GFFs at the large − t limit.1 (a) (b) Figure 1: Representative diagrams of two classes that contribute to the proton GFFs at the large − t limit.1 FIG. 2. Representative diagrams of two classes that con-tribute to the gluon GFFs of the proton at the large − t limit.The cross symbol in the middle of diagrams represents thegluonic energy-momentum tensor operator. The three quarklines denote the leading light-cone wave function configura-tion for the proton state. III. GRAVITATIONAL FORM FACTOR FORNUCLEON
Now we turn to investigate the proton cases. Dueto its spin, the calculations are more involved. To ex-tract the GFFs, one needs to evaluate the EMT ma-trix elements for different nucleon helicity configurations.The A g ( t ) form factor can be obtained with the helicity-conserved matrix element, whereas C g ( t ) requires thehelicity-flipped matrix element. Again, we can followa power counting analysis [49, 50, 56] to determine thepower behaviors at large ( − t ). For example, similar to F form factor, the A g form factor scales as 1 / ( − t ) . Onthe other hand, because of helicity-flip, C g form factorwill scale as 1 / ( − t ) . The detailed calculations belowwill confirm these power counting analysis.First, we deal with the A g ( t ) GFF for the proton. Sinceit is associated with the proton heilicity-conserved ma-trix, the procedure toward the factorization will be thesame as that for the pion case, A g ( t ) = Z [ dx ][ dy ]Φ ∗ ( y , y , y )Φ ( x , x , x ) × A g ( { x } , { y } ) , (7)where { x } = ( x , x , x ), [ dx ] = dx dx dx δ (1 − x − x − x ), and Φ ( x i ) is the twist-three light-cone amplitude ofthe proton [58].In the calculations, we need to contract the gluonicEMT operator to the three quark light-cone wave func-tion configurations for the initial and final state nucleons.Because of three-gluon vertex in QCD, we have two dif-ferent classes of diagrams that contribute to the hardpart, which are shown in Fig. 2. However, at this or-der, because of anti-symmetric color structure associatedwith leading-twist distribution amplitudes in the nucleonstates, the diagram in the right panel vanishes. There-fore, we only need to consider the left panel diagram inthe perturbative calculations. In this class of diagrams,the local EMT operator is attached to a quark lines bytwo gluons and another gluon is exchanged separatelybetween two quarks lines. In total, we have 12 diagrams,which are shown in Fig. 3.For the A g form factor, it follows that for the F formfactor and the contributions from from Fig. 3 can be writ- ten as, A g ( { x } , { y } ) = 2 A + A ′ . (8)where A ′ is obtained from A by interchanging y and y .The expression of A can be summarized in the followingcompact form, A = 8 π α s C B t (cid:16) I + I + I + I (cid:17) , (9)where C B = 2 / I ij isdefined by I ij = x i + y i ¯ x i ¯ y i x i x j y i y j . (10)It has been suggested that the power behavior of theelectromagnetic form factors at large ( − t ) can be relatedto the power behavior of parton distributions at large x [71, 72]. However, this relation seems break down forthe gluonic GFF of nucleon. We know that gluon distri-bution is (1 − x ) suppressed respect to the quark distri-bution [73]. However, their GFFs have the same powerbehavior at large ( − t ), where the quark GFF can be ob-tained from the GPD calculations in Ref. [57] (see alsothe power counting analysis in Ref. [52]).Calculation of C g is much more complicated. This isbecause it can only be extracted from the helicity flippedmatrix element of gluon EMT h P ↑ | T µνg | P ↓ i and the fi-nal result depends on the higher-twist distribution am-plitudes of nucleon. As we mentioned before, it is thequark OAM that generate the proton helicity flip and de-termine the large momentum transfer behavior of theseGFFs. To include the content of the OAM in the analysis,we follow the strategy and technology in Ref. [48]. First,we need the three-quark light-cone Fock expansion ofthe proton state [55], where the components are denotedwith obital angular momenmtum l z , e.g. | P ↓ i l z =1 ∼ R ( k x + ik y ) ψ + ( k x + ik y ) ψ , where the factors ( k xi + ik yi )beside the light-cone wave function are the manifesta-tions of the quark OAM. Since the helicities of the upand down quarks are approximately conserved in the highenergy scattering, the quark OAM in the intial and finalstates must differ by one unit. Therefore, the leading con-tributions will come from the following two matrix ele-ments, l z =0 h P ↑ | ( T µνg ) | P ↓ i l z =1 , l z = − h P ↑ ( T µνg ) | P ↓ i l z =0 . Toevaluate this two amplitude, we work in the Breit framewhere the initial and finial proton are anti-collinear. Inthis frame, the partonic quarks have the the longitudi-nal momenmta x i P and the transervse momentum k i .They emit from the proton and participate in the hardinteraction with the gluon EMT operator. Endured withthe hard gluon exchanges, these quarks recoil and thusproduce the large momentum transfer. Finally, they ob-tain the momenta y i P ′ + k ′ i and recombine into the pro-ton. The collinearity ensures the transverse momentaof the partons is order Λ QCD . However, we can notnaively ignore the transverse momentum for the lead-ing power. Since the quark OAM act like ( k xi ± ik yi ) (a) (b) Figure 1: Representative diagrams of two classes that contribute to the proton GFFs at the large − t limit.1 FIG. 3. Perturbative diagrams that contributes the hard parts of gluon GFFs. Mirrored graphs are implied. inside the phase space integral, this content of transversemomentum in the hard part will be picked up by thesefactors. For that, we should perform the internal trans-verse mommentum expansion on the hard part in thelimit of large − t . Then only linear terms of quark trans-verse momentum in the hard part contribute. There-fore, the leading hard part must have a stucture like k i C ( x , x , x , y , y , y , t ). Ultimately, the dependenceof k i will be absorbed in the twist-four amplitude of theproton, e.g. Ψ ∼ R d k k · { k ψ + k ψ } .With the above analysis, we carry out a detailedderivation for all the diagrams of Fig. 3 and C g ( t ) canbe factorized into, C g ( t ) = Z [ dx ][ dy ] { x Φ ( x , x , x ) C Φ g ( { x } , { y } )+ x Ψ ( x , x , x ) C Ψ g ( { x } , { y } ) } Φ ( y , y , y ) , (11)where Ψ and Φ are the twist-four distribution ampli-tude of the proton [59]. C g can be written as, C g = 2 C + C ′ , (12)where C ′ is obtained from C by interchanging y and y .From the detailed calculations of the diagrams in Fig .3,we obtain C Ψ ( { x } , { y } ) = H ( { x } , { y } ) , C Φ = C Ψ (1 ↔ , (13)where H ( { x } , { y } ) = C B M − t ) (4 πα s ) × (cid:20) x K ( x ¯ x + y y − y ¯ x ) + ¯ x ˜ K ( x ¯ x + y ¯ y )+ x ( ˜ K − K ) ( x ¯ x − y ¯ y ) − K (cid:0) x + y (cid:1) + x ( K + K ) (cid:0) x − y (cid:1) + ( ˜ K + ˜ K )( x ¯ x + y ¯ y ) (cid:21) . (14) The functions K i are defined as K = 1 x x y y ¯ x ¯ y , K = 1 x x x y y ¯ x ¯ y ,K = 1 x x y y ¯ x ¯ y , K = 1 x x y y ¯ x ¯ y ,K = 1 x x x y y ¯ x ¯ y , ˜ K i = K i (1 ↔ . (15)Comparing the above to the A g results, we find two im-portant features. First, we confirm the power countinganalysis, C g form factor is suppressed by 1 / ( − t ) at largemomentum transfer. Second, because of the hard coef-ficients contain additional factor of 1 /y i y j , there will bean end-point singularity in the C g form factor. We canfollow the arguments presented in Ref. [48] for the Pauliform factor F and derive that these end-point singu-larities will lead to a logarithmic enhancement at largemomentum transfer. In the sense, the large t behavior for C g ( t ) will be ln ( − t/ Λ ) / ( − t ) where Λ represents a lowmomentum scale to regulate the end-point singularity inthe above integral. The phenomenological importance ofthese logarithms have been shown for the F form fac-tor [48] and we expect the same for the C pg form factor. IV. CONCLUSION
In summary, we have carried out a perturbative analy-sis of the gluon gravitation form factors for pion and nu-cleon. The leading order contributions predict that the C πg form factor is the same as that of A πg and they bothscale as 1 / ( − t ) at large momentum. For the nucleon, the C pg is power suppressed as compared to the A pg . Becauseof the end point singularity, the C pg form factor has an ad-ditional logarithmic contribution. These results will haveprofound implications for the phenomenological studiesof these form factors and their interpretations as pressuredistributions inside hadrons.It will be interested to extend our derivations to allother gravitational form factors, including B and C formfactors. In addition, a comprehensive analysis of thequark/gluon generalized parton distributions at largemomentum transfer should be carried out as well. Theseresults will provide important guidance for future mea-surements at the EIC [40, 41], where GPDs and GFFsare among the most important topics to reveal the pro-ton tomography and mass decomposition.Theoretically, it will be important to investigate fur-ther the end-point singularity associated with C pg formfactor when the quark lines become soft. A rigorousframework needs to be developed where one can fac-torize and resum these soft parton contributions in theexclusive processes, following, e.g., recent progresses indealing with the end-point singularity in H → γγ pro-cess [74, 75]. We will come back to this issue in a future publication. Acknowledgments:
We thank Xiangdong Ji, MaximPolyakov, Peter Schweitzer for comments and sugges-tions. This material is based upon work supportedby the U.S. Department of Energy, Office of Sci-ence, Office of Nuclear Physics, under contract num-bers DE-AC02-05CH11231. J.P. and X.B. are sup-ported by National Natural Science Foundation of P.R.China(No.12075299,11821505, 11935017) and by theStrategic Priority Research Program of Chinese Academyof Sciences, Grant No. XDB34000000. X.B acknowledgesthe scholarship provided by the University of ChineseAcademy of Sciences for the joint Ph.D. training. [1] I. Kobzarev and L. Okun, Zh. Eksp. Teor. Fiz. , 1904(1962).[2] H. Pagels, Phys. Rev. , 1250 (1966).[3] X.-D. Ji, Phys. Rev. Lett. , 1071 (1995),arXiv:hep-ph/9410274.[4] X.-D. Ji, Phys. Rev. D , 271 (1995),arXiv:hep-ph/9502213.[5] X.-D. Ji, Phys. Rev. Lett. , 610 (1997),arXiv:hep-ph/9603249.[6] X.-D. Ji, Phys. Rev. D , 7114 (1997),arXiv:hep-ph/9609381.[7] R. Jaffe and A. Manohar,Nucl. Phys. B , 509 (1990).[8] B. Filippone and X.-D. Ji,Adv. Nucl. Phys. , 1 (2001), arXiv:hep-ph/0101224.[9] S. D. Bass, Rev. Mod. Phys. , 1257 (2005),arXiv:hep-ph/0411005.[10] C. A. Aidala, S. D. Bass, D. Hasch, andG. K. Mallot, Rev. Mod. Phys. , 655 (2013),arXiv:1209.2803 [hep-ph].[11] E. Leader and C. Lorc´e, Phys. Rept. , 163 (2014),arXiv:1309.4235 [hep-ph].[12] X. Ji, Natl. Sci. Rev. , 213 (2017),arXiv:1605.01114 [hep-ph].[13] A. Deur, S. J. Brodsky, and G. F. De T´eramond,Rept. Prog. Phys. (2019), 10.1088/1361-6633/ab0b8f,arXiv:1807.05250 [hep-ph].[14] X. Ji, F. Yuan, and Y. Zhao, (2020),arXiv:2009.01291 [hep-ph].[15] M. Polyakov, Phys. Lett. B , 57 (2003),arXiv:hep-ph/0210165.[16] M. V. Polyakov and P. Schweitzer,Int. J. Mod. Phys. A , 1830025 (2018),arXiv:1805.06596 [hep-ph].[17] V. Burkert, L. Elouadrhiri, and F. Girod,Nature , 396 (2018).[18] P. Shanahan and W. Detmold,Phys. Rev. Lett. , 072003 (2019),arXiv:1810.07589 [nucl-th].[19] K. Kumeriˇcki, Nature , E1 (2019).[20] D. M¨uller, D. Robaschik, B. Geyer, F.-M. Dittes,and J. Hoˇ rejˇsi, Fortsch. Phys. , 101 (1994),arXiv:hep-ph/9812448.[21] M. Diehl, Generalized parton distributions , Ph.D. thesis(2003), arXiv:hep-ph/0307382. [22] A. Belitsky and A. Radyushkin,Phys. Rept. , 1 (2005), arXiv:hep-ph/0504030.[23] A. Radyushkin, Phys. Lett. B , 417 (1996),arXiv:hep-ph/9604317.[24] N. d’Hose, S. Niccolai, and A. Rostomyan,Eur. Phys. J. A , 151 (2016).[25] K. Kumericki, S. Liuti, andH. Moutarde, Eur. Phys. J. A , 157 (2016),arXiv:1602.02763 [hep-ph].[26] J. C. Collins, L. Frankfurt, and M. Strikman,Phys. Rev. D , 2982 (1997), arXiv:hep-ph/9611433.[27] L. Mankiewicz, G. Piller, E. Stein, M. Vantti-nen, and T. Weigl, Phys. Lett. B , 186 (1998),[Erratum: Phys.Lett.B 461, 423–423 (1999)],arXiv:hep-ph/9712251.[28] L. Favart, M. Guidal, T. Horn, andP. Kroll, Eur. Phys. J. A , 158 (2016),arXiv:1511.04535 [hep-ph].[29] M. Gockeler, R. Horsley, D. Pleiter, P. E. Rakow,A. Schafer, G. Schierholz, and W. Schroers(QCDSF), Phys. Rev. Lett. , 042002 (2004),arXiv:hep-ph/0304249.[30] M. Gockeler, P. Hagler, R. Horsley, D. Pleiter, P. E.Rakow, A. Schafer, G. Schierholz, and J. Zan-otti (QCDSF, UKQCD), Phys. Lett. B , 113 (2005),arXiv:hep-lat/0507001.[31] D. Brommel, M. Diehl, M. Gockeler, P. Hagler, R. Hors-ley, D. Pleiter, P. E. Rakow, A. Schafer, G. Schier-holz, and J. Zanotti, PoS LAT2005 , 360 (2006),arXiv:hep-lat/0509133.[32] D. Brommel,
Pion Structure from the Lattice ,Ph.D. thesis, Regensburg U. (2007).[33] P. Hagler, J. W. Negele, D. B. Renner, W. Schroers,T. Lippert, and K. Schilling (LHPC, SESAM),Phys. Rev. D , 034505 (2003), arXiv:hep-lat/0304018.[34] P. Hagler et al. (LHPC),Phys. Rev. D , 094502 (2008),arXiv:0705.4295 [hep-lat].[35] P. Hagler, Phys. Rept. , 49 (2010),arXiv:0912.5483 [hep-lat].[36] M. G¨ockeler, P. H¨agler, R. Horsley, Y. Nakamura,D. Pleiter, P. L. Rakow, A. Sch¨afer, G. Schier-holz, H. St¨uben, and J. Zanotti (QCDSF,UKQCD), Phys. Rev. Lett. , 222001 (2007),arXiv:hep-lat/0612032. [37] C. Alexandrou, J. Carbonell, M. Constantinou, P. Har-raud, P. Guichon, K. Jansen, C. Kallidonis, T. Korzec,and M. Papinutto, Phys. Rev. D , 114513 (2011),arXiv:1104.1600 [hep-lat].[38] P. Shanahan and W. Detmold,Phys. Rev. D , 014511 (2019),arXiv:1810.04626 [hep-lat].[39] J. Dudek et al. , Eur. Phys. J. A , 187 (2012),arXiv:1208.1244 [hep-ex].[40] A. Accardi et al. , Eur. Phys. J. A , 268 (2016),arXiv:1212.1701 [nucl-ex].[41] D. Boer et al. , (2011), arXiv:1108.1713 [nucl-th].[42] G. Lepage and S. J. Brodsky,Phys. Rev. Lett. , 545 (1979), [Erratum:Phys.Rev.Lett. 43, 1625–1626 (1979)].[43] S. J. Brodsky and G. Lepage,Phys. Rev. D , 2848 (1981).[44] A. Efremov and A. Radyushkin,Phys. Lett. B , 245 (1980).[45] V. Chernyak and A. Zhitnitsky, JETP Lett. , 510(1977).[46] V. Chernyak and A. Zhitnitsky, Sov. J. Nucl. Phys. ,544 (1980).[47] V. Chernyak and A. Zhitnitsky,Phys. Rept. , 173 (1984).[48] A. V. Belitsky, X.-d. Ji, andF. Yuan, Phys. Rev. Lett. , 092003 (2003),arXiv:hep-ph/0212351.[49] S. J. Brodsky and G. R. Farrar,Phys. Rev. Lett. , 1153 (1973).[50] V. Matveev, R. Muradian, and A. Tavkhelidze,Lett. Nuovo Cim. , 719 (1973).[51] X.-d. Ji, J.-P. Ma, and F. Yuan,Eur. Phys. J. C , 75 (2004), arXiv:hep-ph/0304107.[52] K. Tanaka, Phys. Rev. D , 034009 (2018),arXiv:1806.10591 [hep-ph].[53] L. Frankfurt and M. Strikman,Phys. Rev. D , 031502 (2002), arXiv:hep-ph/0205223.[54] M. Burkardt, X.-d. Ji, and F. Yuan,Phys. Lett. B , 345 (2002), arXiv:hep-ph/0205272.[55] X.-d. Ji, J.-P. Ma, and F. Yuan,Nucl. Phys. B , 383 (2003), arXiv:hep-ph/0210430.[56] X.-d. Ji, J.-P. Ma, and F. Yuan,Phys. Rev. Lett. , 241601 (2003),arXiv:hep-ph/0301141.[57] P. Hoodbhoy, X.-d. Ji, and F. Yuan,Phys. Rev. Lett. , 012003 (2004), arXiv:hep-ph/0309085.[58] V. M. Braun, S. E. Derkachov, G. Korchem-sky, and A. Manashov, Nucl. Phys. B , 355 (1999),arXiv:hep-ph/9902375.[59] V. Braun, R. Fries, N. Mahnke, and E. Stein,Nucl. Phys. B , 381 (2000), [Erratum: Nucl.Phys.B607, 433–433 (2001)], arXiv:hep-ph/0007279.[60] D. Kharzeev, H. Satz, A. Syamtomov, and G. Zinovjev,Eur. Phys. J. C , 459 (1999), arXiv:hep-ph/9901375.[61] S. Brodsky, E. Chudakov, P. Hoyer, and J. Laget,Phys. Lett. B , 23 (2001), arXiv:hep-ph/0010343.[62] O. Gryniuk and M. Vander-haeghen, Phys. Rev. D , 074001 (2016),arXiv:1608.08205 [hep-ph].[63] Y. Hatta and D.-L. Yang,Phys. Rev. D , 074003 (2018),arXiv:1808.02163 [hep-ph].[64] Y. Hatta, A. Rajan, and D.-L.Yang, Phys. Rev. D , 014032 (2019),arXiv:1906.00894 [hep-ph].[65] K. A. Mamo and I. Zahed,Phys. Rev. D , 086003 (2020),arXiv:1910.04707 [hep-ph].[66] O. Gryniuk, S. Joosten, Z.-E. Meziani, andM. Vanderhaeghen, Phys. Rev. D , 014016 (2020),arXiv:2005.09293 [hep-ph].[67] R. Wang, J. Evslin, andX. Chen, Eur. Phys. J. C , 507 (2020),arXiv:1912.12040 [hep-ph].[68] F. Zeng, X.-Y. Wang, L. Zhang, Y.-P. Xie, R. Wang,and X. Chen, Eur. Phys. J. C , 1027 (2020),arXiv:2008.13439 [hep-ph].[69] M.-L. Du, V. Baru, F.-K. Guo, C. Han-hart, U.-G. Meißner, A. Nefediev, andI. Strakovsky, Eur. Phys. J. C , 1053 (2020),arXiv:2009.08345 [hep-ph].[70] S. Kumano, Q.-T. Song, andO. Teryaev, Phys. Rev. D , 014020 (2018),arXiv:1711.08088 [hep-ph].[71] S. Drell and T.-M. Yan, Phys. Rev. Lett. , 181 (1970).[72] G. B. West, Phys. Rev. Lett. , 1206 (1970).[73] S. J. Brodsky, M. Burkardt, and I. Schmidt,Nucl. Phys. B , 197 (1995), arXiv:hep-ph/9401328.[74] Z. L. Liu and M. Neubert, JHEP04