Chiral Perturbation for Large Momentum Effective Field Theory
CChiral Perturbation for Large Momentum Effective Field Theory
Wei-Yang Liu ∗ and Jiunn-Wei Chen † Department of Physics, Center for Theoretical Physics,and Leung Center for Cosmology and Particle Astrophysics,National Taiwan University, Taipei, Taiwan 106
Large momentum effective field theory (LaMET) enables the extraction of parton distributionfunctions (PDF’s) directly on a Euclidean lattice through a factorization theorem that relates thecomputed quasi-PDF’s to PDF’s. We apply chiral perturbation theory (ChPT) to LaMET to furtherseparate soft scales, such as light quark masses and lattice size, to obtain leading model independentextrapolation formulas for extrapolations to physical quark masses and infinite volume. We findthat the finite volume effect is reduced when the nucleon carries a finite momentum. For nucleonmomentum greater than 1 GeV and the lattice size L and pion mass m π satisfying m π L ≥ , thefinite volume effect is less than and is negligible for the current precision of lattice computations.This can be interpreted as a Lorentz contraction of the nucleon size in the z-direction which makesthe lattice size effectively larger in that direction. We also find that the quark mass dependencein the infinite volume limit computed with non-zero nucleon momentum reproduces the previousresult computed at zero momentum, as expected. Our approach can be generalized to other partonobservables in LaMET straight forwardly. ∗ [email protected] † [email protected] a r X i v : . [ h e p - l a t ] N ov I. INTRODUCTION
Large momentum effective theory (LaMET) enables computations of parton distribution functions (PDF’s) ofhadrons on a Euclidean lattice. LaMET relates equal-time spatial correlators (whose Fourier transforms are calledquasi-PDF’s) to PDF’s in the infinite hadron momentum limit [1, 2]. For large but finite momenta accessible ona realistic lattice, LaMET relates quasi-PDF’s to physical ones through a factorization theorem, which involves amatching coefficient and power corrections that are suppressed by the hadron momentum: ˜ q ( x, P z , a ) = (cid:90) − dy | y | Z ( xy , yP z , a, µ ) q ( y, µ ) + O ( M P z , Λ QCD P z ) , (1)where x and y are momentum fractions of the parton with respect to the hadron, P z is the hadron momentum, ˜ q isthe quasi-PDF defined computed on a lattice with lattice spacing a while ˜ q is the PDF defined at the scale µ . M isthe hadron mass and Λ QCD is the strong interaction scale. The hierarchy of scales follows πa (cid:29) P z (cid:29) M, Λ QCD (cid:29) πL , (2)with L is the size of the lattice, such that the power correction is small. ˜ q and q have the same infrared physics. Theirdifference in the ultraviolet is compensated by the matching kernel Z . The proof of factorization was developed inRefs. [3–5].Since LaMET was proposed, a lot of progress has been made in the theoretical understanding of the formalism [4, 6–62]. The method has been applied in lattice calculations of PDF’s for the nucleon [19, 25, 26, 28, 63–77], π [78–80] and K [81] mesons. Despite limited volumes and relatively coarse lattice spacings, the state-of-the-art nucleon isovectorquark PDF’s, determined from lattice data at the physical point, have shown reasonable agreement [66, 67] withphenomenological results extracted from the experimental data. Encouraged by this success, LaMET has also beenapplied to ∆ + [82] and twist-three PDF’s [83–85], as well as gluon [86], strange and charm distributions [87]. It wasalso applied to meson distribution amplitudes [20, 88–90] and generalized parton distributions (GPD’s) [91–94]. Morerecently, attempts have been made to generalize LaMET to transverse momentum dependent (TMD) PDF’s [95–102]to calculate the nonperturbative Collins-Soper evolution kernel [97, 103, 104] and soft functions [105] on the lattice.LaMET also brought renewed interests in earlier approaches [106–112] and inspired new ones [113–128]. For recentreviews, see, e.g., Refs. [129–133]. The renormalon ambiguity in the matching kernel Z was first studied in Ref. [48]which implies the power correction due to higher twist effect should be O (Λ QCD /x P z ) to cancel the renormalonambiguity. However, the study of bubble chain diagrams of Ref. [134] did not find the slow convergence of the kernelat three loop order, indicating that the renormalon effect could be mild to this order in quasi-PDFs.Despite the progress made in LaMET, the LaMET factorization theorem of Eq. (1) makes no attempt to separatethe light quark mass scales m u,d and L from scales such as Λ QCD or Λ χ (the scale of chiral symmetry breaking). Thus ˜ q is a function of all these scales, while q is expected to have the same quark mass dependence as ˜ q but no volumedependence. As lattice exploration of the m u,d and L parameter space requires a significant amount of computingresources, model independent formulas to guide the extrapolations to physical quark masses and infinite volume areof practical importance. An effective field theory (EFT) approach such as chiral perturbation theory (ChPT) is idealfor this purpose, as EFT only relies on the symmetries and the scale separation of the system, hence the results aremodel independent.ChPT has been successfully applied to many aspects of meson [135], single- [136, 137], and multi-nucleon systems (see[138–143] for reviews). In particular, ChPT has been applied to PDF’s in the meson and single nucleon systems, firstin Refs. [144–146] then in [147–152] with more applications in PDF’s as well as other light-cone dominated observablessuch as generalized parton distributions (GPD’s) [153–158]. ChPT has also been applied to multi-nucleon sectors tostudy the EMC effect [159, 160] and the connection between the EMC effect and short range correlations [161, 162].In this work, we establish the procedure to apply ChPT to LaMET. The previous success of ChPT can then bedirectly carried over to LaMET straight forwardly. As an example, we work out the light quark mass dependence andfinite volume corrections to nucleon quasi-PDF’s. Other applications such as the quenched, partially quenched, andmixed action artifacts, generalizing from SU(2) to SU(3), as well as the off-forward kinematics study of GPD’s andso on, can all be studied within this framework. II. APPLYING CHPT TO QUASI-PDFS
In this section, we apply ChPT to both unpolarized and polarized isovector nucleon quasi-PDF’s. The applicationto other quasi-observables can follow the same procedure.For the unpolarized nucleon quasi-PDF, the equal-time correlator computed on the lattice is h ( z, P z ) = 12 P z (cid:104) N ( P ) | ¯ ψ ( z ) γ z W ( z, ψ (0) | N ( P ) (cid:105) , (3)where | N ( P ) (cid:105) is a nucleon state with momentum P µ = ( (cid:112) M + P z , , , P z ) and the Wilson line is W ( z,
0) = exp (cid:20) ig s λ µ (cid:90) z dz (cid:48) A µ ( z (cid:48) λ ν ) (cid:21) , (4)with g s is the strong coupling constant and λ µ = (0 , , , − . The Fourier transform of this correlator yields theunpolarized quasi-PDF ˜ q ( x, P z ) = P z (cid:90) ∞−∞ dz π e ixP z z h ( z, P z ) . (5)Using γ t instead of γ z in Eq.(3) to avoid mixing with another operator of the same mass dimension [13, 27] will notaffect the chiral and finite volume corrections computed in this work.For the longitudinally polarized quasi-PDF, the equal-time correlator computed on the lattice is ∆ h ( z, P z ) = 12 M s z (cid:104) N ( P, s ) | ¯ ψ ( z ) γ z γ W ( z, ψ (0) | N ( P, s ) (cid:105) , (6)with the nucleon polarization vector s µ = ( P z , , , (cid:112) M + P z ) /M . And the corresponding quasi-PDF for quarkhelicity distribution is ∆˜ q ( x, P z ) = P z (cid:90) ∞−∞ dz π e ixP z z ∆ h ( z, P z ) . (7)For the transversely polarized quasi-PDF, the equal-time correlator computed on the lattice is δh ( z, P z ) = 12 P z s x (cid:104) N ( P, s ) | ¯ ψ ( z ) γ x γ z γ W ( z, ψ (0) | N ( P, s ) (cid:105) , (8)with the nucleon polarization vector s µ = (0 , , , . The corresponding quasi-PDF for quark helicity distribution is ∆˜ q ( x, P z ) = P z (cid:90) ∞−∞ dz π e ixP z z ∆ h ( z, P z ) . (9)Replacing γ z in Eq.(8) by γ t to avoid mixing with another operator of the same mass dimension [13, 27] will notaffect the chiral and finite volume corrections computed in this work.Under the operator product expansion (OPE), the quark bilinear operators become λ µ ¯ ψ ( z )Γ µ W ( z, ψ (0) (cid:39) ∞ (cid:88) n =0 ( iz ) n n ! λ µ λ µ λ µ . . . λ µ n ¯ ψ Γ µ iD µ iD µ . . . iD µ n ψ, (10)with λ µ Γ µ = γ z , γ z γ , γ x ⊥ γ z γ for the unpolarized, helicity and transversity cases, respectively. The λ µ λ µ λ µ . . . λ µ n tensor is symmetric but not traceless. But the nucleon matrix elements of the trace parts are O ( M /P z , Λ QCD /P z ) corrections whose sizes are power suppressed [1, 19]. Therefore, we only need to concentrate on the symmetric tracelessparts: O µµ µ ...µ n q = ¯ ψγ ( µ iD µ iD µ . . . iD µ n ) ψ, (11) ∆ O µµ µ ...µ n q = ¯ ψγ ( µ γ iD µ iD µ . . . iD µ n ) ψ, (12) δ O xµµ µ ...µ n q = ¯ ψγ [ x γ ( µ ] γ iD µ iD µ . . . iD µ n ) ψ, (13)where ( . . . ) means symmetrization of the enclosed Lorentz indices with the trace parts subtracted and [ . . . ] meansthe enclosed indices are antisymmetrized. These operators are irreducible representations of the Lorentz group andare of the leading twist (twist-2). Their nucleon matrix elements give rise to moments of nucleon PDFs. (a) (b) (c)(d) (e) (f) FIG. 1: Leading Feynman diagrams in ChPT for nucleon twist-2 matrix elements. The dashed lines are pions, solidlines are nucleons, and the filled squares are the operators. Diagram (a) is the tree level diagram and diagram (d)denotes the wavefunction renormalization. Diagrams (e) and (f) only contribute to the unpolarized n = 0 case topreserve the fermion number conservation. They are higher order in the power counting when n > .We will use the technique developed in Refs.[144–146] to match the quark level twist-2 operators to hadronic leveloperators using ChPT. The Lagrangian of ChPT is given by [136, 137] L = F π ∂ µ Σ ∂ µ Σ † ) + η tr( M Σ † + M † Σ) +
N iv · DN + 2 g A N S · AN + . . . , (14)where the pion decay constant F π = 93 GeV, the pion field Σ = e iFπ Π , Π = (cid:18) π √ π + √ π − − π (cid:19) ,the quark mass matrix M = diag ( m u , m d ) , and η is the parameter connecting the quark mass and pion mass at theleading order. We will work in the isospin symmetric limit m u = m d . N is the SU(2) doublet nucleon field. Thenucleon velocity v µ = P µ /M is the ratio of the nucleon momentum P µ and the nucleon mass M . D µ = ∂ µ − iV µ ,where the pion vector current V µ = i ( u∂ µ u † + u † ∂ µ u ) and u = Σ . g A =1.25 is the axial-vector coupling. The nucleonspin vector S µ = i σ µν v ν γ . The axial vector current A µ = i ( u∂ µ u † − u † ∂ µ u ) . The small expansion parameter (cid:15) inthe perturbation theory is the ratio of the light to heavy scales in the problem. The light scales are the pion mass m π and the typical momentum transfer q , while the heavy scales are the nucleon mass M and the induced scales Λ χ = 4 πF π arising from the loop expansion [136].Although the /M expansion of ChPT looks like a non-relativistic expansion, the ChPT formulation is actually fullyrelativistic. The expansion requires the nucleon momentum M v + k has small off-shellness which means v · k + k /M (cid:28) M , but there is no restriction on v . Therefore ChPT can still be applied in our analysis where the nucleon is relativistic.Now, each operator in Eqs.(11)-(13) can be rewritten as a sum of an infinite number of hadronic operators of thesame symmetries. The hadronic operators can then be organized by their mass dimensions d O with each operatorcounted as order (cid:15) d O . Therefore, operators of smaller mass dimension are more important in the power counting. Theisovector combinations of the lowest dimensional operators in the matching are: O µµ µ ...µ n u − d = c ( n )1 ¯ N v ( µ v µ . . . v µ n ) ( uτ u † + u † τ u ) N + ˜ c ( n )1 ¯ N S ( µ v µ . . . v µ n ) ( uτ u † − u † τ u ) N + · · · , (15) ∆ O µµ µ ...µ n u − d = c ( n )2 ¯ N S ( µ v µ . . . v µ n ) ( uτ u † + u † τ u ) N + ˜ c ( n )2 ¯ N v ( µ v µ . . . v µ n ) ( uτ u † − u † τ u ) N + · · · , (16) δ O xµµ µ ...µ n u − d = c ( n )3 ¯ N S [ x v ( µ ] v µ . . . v µ n ) ( u † τ u † + uτ u ) N + ˜ c ( n )3 ¯ N S [ x S ( µ ] v µ . . . v µ n ) ( u † τ u † − uτ u ) N + · · · . (17)These nucleon operators are shown as the filled squares in diagrams (a)-(d) of Fig. 1. For the unpolarized case, the n = 0 pionic operator O µu − d,π (cid:39) a (0) F π tr (cid:0) Σ † τ i∂ µ Σ + Σ τ i∂ µ Σ † (cid:1) (18)can also appear in diagrams (e) and (f). They are of the same order in the power counting as diagrams (a)-(d). For n > , (e) and (f) become higher order diagrams. III. RESULTS
We are interested in the finite volume effect for the nucleon quasi-PDF evaluated at nucleon momentum P z on aEuclidean lattice. We will work with a lattice with length L in the three spatial directions but the size of the timedirection is infinite. Assuming the nucleon and pion fields both satisfy periodic boundary conditions in the spatialdirections, such that their momenta are quantized as (cid:126)p n = 2 π(cid:126)n/L in the reciprocal lattice space, with (cid:126)n = ( n x , n y , n z ) and n i are integers. Poisson’s formula provides a nice way to separate a discrete momentum sum into a momentumintegration in the infinite volume limit and corrections caused by finite volume effect: L (cid:88) (cid:126)n f ( (cid:126)p n = 2 π(cid:126)nL ) = (cid:90) d (cid:126)p (2 π ) f ( (cid:126)p ) + (cid:88) (cid:126)n (cid:54) =0 (cid:90) d (cid:126)p (2 π ) e i(cid:126)n · (cid:126)pL f ( (cid:126)p ) . (19)Our results for the nucleon twist-2 matrix elements at one loop order are (cid:104)O µµ ··· µ m u − d (cid:105) = (cid:104)O µµ ··· µ m u − d (cid:105) − (1 − δ m ) g A + 116 π F π m π log m π Λ n + m π π F π (cid:88) (cid:126)n (cid:54) =0 (cid:18) K ( nm π L ) nm π L + 3 g A J (cid:18) nm π L, (cid:126)n · (cid:126)vn (cid:19)(cid:19) , (cid:104) ∆ O µµ ··· µ m u − d (cid:105) = (cid:104) ∆ O µµ ··· µ m u − d (cid:105) − g A + 116 π F π m π log m π (∆Λ n ) − m π π F π (cid:88) (cid:126)n (cid:54) =0 (cid:18) K ( nm π L ) nm π L + 2 g A J (cid:18) nm π L, (cid:126)n · (cid:126)vn (cid:19)(cid:19) , (cid:104) δ O µµ ··· µ m u − d (cid:105) = (cid:104) δ O µµ ··· µ m u − d (cid:105) − g A + 132 π F π m π log m π ( δ Λ n ) − m π π F π m π (cid:88) (cid:126)n (cid:54) =0 (cid:18) K ( nm π L ) nm π L + 4 g A J (cid:18) nm π L, (cid:126)n · (cid:126)vn (cid:19)(cid:19) , (20)where J (cid:18) nm π L, (cid:126)n · (cid:126)vn (cid:19) = − K ( nm π L ) nm π L + 13 (cid:18) (cid:126)n · (cid:126)v ) n (cid:19) K ( nm π L )+ (cid:90) ∞ dαα (1 − cos( α(cid:126)n · (cid:126)vm π L )) (cid:20) K ( nm π L (cid:112) α ) − (cid:18) (cid:126)n · (cid:126)v ) n (cid:19) nm π L (cid:112) α K ( nm π L (cid:112) α ) (cid:21) , (21)and where the subscript indicates that the corresponding matrix elements are evaluate first in the infinite volumelimit, then in the chiral limit such that m π L → ∞ . n = (cid:113) n x + n y + n z and (cid:126)n · (cid:126)v = n z P z /M . n i in Eq. (20) plays therole to label the number of times the pion crosses the boundary of lattice in the i -direction. These matrix elementsdetermine the m -th moment of the PDF defined as (cid:82) dxx m q ( x ) . The δ m in the unpolarized case yields the required ( u − d ) quark number conservation in the proton. This implies that there is a δ ( x ) contribution in q ( x ) because itcontributes to the zero-th moment but not any other moment.In Eq. (20), the n independent part is the infinite volume result whose leading quark-mass dependence reproducesthe previous results of Refs. [164, 165]. The scales Λ n , ∆Λ n , δ Λ n are associated with counterterms at the m π orderthat need to be fit to data. Converting from the moments to distributions in the momentum fraction x , both the (a)(b) FIG. 2: Finite volume effect for the unpolarized twist-2 matrix element of Eq.(20) for m (cid:54) = 0 shown as a function of(a) P z and (b) m π L . The absolute value of the finite volume effect for a moving proton ( P z (cid:54) = 0 ) is always smallerthan a rest one ( P z = 0 ) for any value of m π L . For P z /M ≥ and m π L ≥ , the finite volume effect is less than .PDF and quasi-PDF has the leading quark mass dependence q u − d ( x ) = q u − d, ( x ) (cid:18) − g A + 116 π F π m π log m π Λ χ (cid:19) + c u − d, ( x ) m π + δ ( x ) (cid:18) g A + 116 π F π m π log m π Λ χ − m π (cid:90) − dxc u − d, ( x ) (cid:19) , ∆ q u − d ( x ) =∆ q u − d, ( x ) (cid:18) − g A + 116 π F π m π log m π Λ χ (cid:19) + ∆ c u − d, ( x ) m π ,δq u − d ( x ) = δq u − d, ( x ) (cid:18) − g A + 132 π F π m π log m π Λ χ (cid:19) + δc u − d, ( x ) m π , (22)where the functions q, c, ∆ q, ∆ c, δq, δc are m π independent. Quark number conservation (cid:82) − dxq u − d ( x ) = (cid:82) − dxq u − d, ( x ) =1 is preserved. The delta function in q u − d ( x ) appears because we truncate the chiral expansion at one loop order.Should we go to higher loop orders, the delta function will be smeared into a more smooth function.In Fig.2, we show the finite volume effect of the unpolarized twist-2 matrix elements of Eq.(20) for m (cid:54) = 0 by takingthe ratio of the matrix elements at finite and infinite volumes. We see that the finite volume effect is not monotonicin P z nor in m π L , due to partial cancelations of several different contributions. However, the absolute value of thefinite volume effect for a moving proton ( P z (cid:54) = 0 ) is always smaller than a rest one ( P z = 0 ) for any value of m π L . For P z /M ≥ and m π L ≥ , the finite volume effect is less than and is negligible for the current precision of latticecomputations. This can be interpreted as an effect due to the Lorentz contraction in the z-direction which makes thebox size effectively bigger in that direction. (a) (b)(c) (d) FIG. 3: The (a) real and (b) imaginary parts of the equal time correlator for an unpolarized proton in the infinitevolume limit h ∞ u − d ( z ) . This quantity is constructed using the proton isovector PDF extracted by the CTEQ-JLabcollaboration (CJ12) [163] then matched to a quasi-PDF with P z =1.3 GeV. The finite volume contribution is shownin (c) and (d), which is much smaller than other errors in a typical lattice QCD computation. (a) (b) FIG. 4: Same as Fig. 3 but with different P z ’s.The finite volume effect for equal time correlators of Eqs. (3), (6), and (8) can be derived from Eq.(20): h u − d ( z, P z ) = h ∞ u − d ( z, P z ) − m π π F π (cid:88) (cid:126)n (cid:54) =0 (cid:18) K ( nm π L ) nm π L + 3 g A J (cid:18) nm π L, (cid:126)n · (cid:126)vn (cid:19)(cid:19) + m π π F π (cid:88) (cid:126)n (cid:54) =0 (cid:18) K ( nm π L ) nm π L + 3 g A J (cid:18) nm π L, (cid:126)n · (cid:126)vn (cid:19)(cid:19) , ∆ h u − d ( z, P z ) = ∆ h ∞ u − d ( z, P z ) − m π π F π (cid:88) (cid:126)n (cid:54) =0 (cid:18) K ( nm π L ) nm π L + 2 g A J (cid:18) nm π L, (cid:126)n · (cid:126)vn (cid:19)(cid:19) ,δh u − d ( z, P z ) = δh ∞ u − d ( z, P z ) − m π π F π m π (cid:88) (cid:126)n (cid:54) =0 (cid:18) K ( nm π L ) nm π L + 4 g A J (cid:18) nm π L, (cid:126)n · (cid:126)vn (cid:19)(cid:19) . (23)To mimic the finite volume effect on an equal time correlator computed in lattice QCD, we generate the infinitevolume result by using the unpolarized isovector PDF extracted by the CTEQ-JLab collaboration (CJ12) [163]. Theprocedure is to perform the matching from PDF defined in the MS scheme to the quasi-PDF defined in the RI/MOMscheme, then we Fourier transform the quasi-PDF to produce the infinite volume equal time correlator h ∞ u − d ( z, P z ) .In Fig.3 we use Eq.(23) to show the finite volume effect in h u − d . We have used α s =0.283, p zR =1.2 GeV, µ =3.1GeV, µ R =2.4 GeV, and m π =0.220 GeV in the matching. In Fig. 4, dependence on different P z ’s is shown. Again,these figures show that finite volume effect is negligible for current lattice computations of quasi-PDFs. We have usedsimilar parameters as the lattice calculation of Ref.[166], where the size of finite volume effect is found to be smallerthan the error of the calculation and is consistent with our result within errors.In other versions of heavy baryon chiral perturbation, one could include ∆ resonances or generalize the formalismfrom SU(2) to SU(3). Some of the quark mass dependence of PDF’s and GPD’s is already computed in these theories.However, we do not expect the finite volume effect changes a lot by adding those heavier degrees of freedom. Therefore,our conclusion on the smallness of the finite volume effects of quasi-PDF’s in these theories will stay the same. IV. CONCLUSION
LaMET enables the extraction of PDFs directly on a Euclidean lattice through a factorization theorem that relatesthe computed quasi-PDF’s to PDF’s. We have applied ChPT to LaMET to further separate soft scales, such aslight quark masses and lattice size, to obtain leading model independent extrapolation formulas for extrapolations tophysical quark masses and infinite volume.We find that the finite volume effect is reduced when the nucleon carries a finite momentum. For P z /M >1 GeV and m π L ≥ , the finite volume effect is less than and is negligible for the current precision of lattice computations.This can be interpreted as a Lorentz contraction of the nucleon size in the z-direction which makes the lattice sizeeffectively larger in that direction. We also find that the quark mass dependence in the infinite volume limit computedwith non-zero nucleon momentum reproduces the previous result computed at zero momentum, as expected.In this work, we establish the procedure to apply ChPT to LaMET. The previous success of ChPT can then bedirectly carried over to LaMET straight forwardly. Other applications such as the quenched, partially quenched, andmixed action artifacts, generalizing from SU(2) to SU(3), as well as the off-forward kinematics study of GPD’s andso on, can all be studied within this framework. ACKNOWLEDGMENTS
This work is partly supported by the Ministry of Science and Technology, Taiwan, under Grant No. 108-2112-M-002-003-MY3 and the Kenda Foundation.
Appendix A: Integrals of the finite volume corrections
In this appendix, we show how the integrals involved in the diagrams of Fig. 1 are computed. First, diagrams (c)depends on the integral I = (cid:88) (cid:126)n (cid:54) =0 (cid:90) d k (2 π ) e i(cid:126)n · (cid:126)kL k − m = (cid:88) (cid:126)n (cid:54) =0 n =0 (cid:90) d k E (2 π ) e in · k E L − ik E + m = − im π (cid:88) (cid:126)n (cid:54) =0 n =0 nmL K ( nmL ) , (A1)where a Wick rotation k → ik to Euclidean space is performed after the first equal sign with n µ = ( (cid:126)n, n ) and n = (cid:113)(cid:80) i =1 n i . We have also used the d-dimensional integral [167, 168], (cid:88) n (cid:54) =0 (cid:90) d d k E (2 π ) d e in · k E L k E + ∆ ) a = 2(4 π ) d/ Γ( a ) (cid:18) nL (cid:19) a − d/ K a − d/ ( n ∆ L ) . (A2)Analogously, diagrams (b), (d), and (e) depend on the integral I = (cid:88) (cid:126)n (cid:54) =0 (cid:90) d k (2 π ) e i(cid:126)n · (cid:126)kL ( S · k ) ( k − m )( v · k ) = − im (cid:88) (cid:126)n (cid:54) =0 n =0 (cid:90) ∞ dααe − iαmn · vL (cid:90) d k (2 π ) e in · kL ( S · k ) [ k + (1 + α ) m ] = 4 im { S µ , S ν } (cid:88) (cid:126)n (cid:54) =0 n =0 (cid:90) ∞ dα αL e − iαmn · vL ∂ ∂n µ ∂n ν (cid:90) d k (2 π ) e in · kL k + (1 + α ) m ] = im π ( v µ v ν + g µν ) (cid:88) (cid:126)n (cid:54) =0 n =0 (cid:90) ∞ dααe − iαmn · vL ∂ ∂n µ ∂n ν (cid:18) n √ α mL (cid:19) K ( n (cid:112) α mL ) , (A3)where Wick rotation is performed and all 4-vectors are defined in Euclidean space after the first equal sign and wehave used the anticommutation relation in Euclidean space: { S µ , S ν } = 12 ( v µ v ν + g µν ) (A4)The derivative on K yields ∂ ∂n µ ∂n ν (cid:18) n √ α mL (cid:19) K ( n (cid:112) α mL ) = − g µν K ( n (cid:112) α mL )+ n µ n ν n (cid:112) α mL K ( n (cid:112) α mL ) . (A5)Therefore, the final result of I is I = im π (cid:88) (cid:126)n (cid:54) =0 n =0 (cid:90) ∞ dααe − iαmn · vL (cid:20) − K ( n (cid:112) α mL ) + (cid:18) n · v ) n (cid:19) (cid:112) α nmL K ( n (cid:112) α mL ) (cid:21) = im π (cid:88) (cid:126)n (cid:54) =0 (cid:90) ∞ dαα cos( αm(cid:126)n · (cid:126)vL ) (cid:20)(cid:18) (cid:126)n · (cid:126)v ) n (cid:19) (cid:112) α nmL K ( (cid:112) α nmL ) − K ( (cid:112) α nmL ) (cid:21) . (A6)In the final step, we drop the sine function in e − iαm(cid:126)n · (cid:126)vL since the summation over (cid:126)n cancels all the terms odd in (cid:126)n . [1] X. Ji, Phys. Rev. Lett. , 262002 (2013), arXiv:1305.1539 [hep-ph]. [2] X. Ji, Sci. China Phys. Mech. Astron. , 1407 (2014), arXiv:1404.6680 [hep-ph].[3] Y.-Q. Ma and J.-W. Qiu, Phys. Rev. Lett. , 022003 (2018), arXiv:1709.03018 [hep-ph].[4] T. Izubuchi, X. Ji, L. Jin, I. W. Stewart, and Y. Zhao, Phys. Rev. D98 , 056004 (2018), arXiv:1801.03917 [hep-ph].[5] Y.-S. Liu, W. Wang, J. Xu, Q.-A. Zhang, J.-H. Zhang, S. Zhao, and Y. Zhao, Phys. Rev. D , 034006 (2019),arXiv:1902.00307 [hep-ph].[6] X. Xiong, X. Ji, J.-H. Zhang, and Y. Zhao, Phys. Rev.
D90 , 014051 (2014), arXiv:1310.7471 [hep-ph].[7] X. Ji and J.-H. Zhang, Phys. Rev.
D92 , 034006 (2015), arXiv:1505.07699 [hep-ph].[8] X. Ji, A. SchÀfer, X. Xiong, and J.-H. Zhang, Phys. Rev. D , 014039 (2015), arXiv:1506.00248 [hep-ph].[9] X. Xiong and J.-H. Zhang, Phys. Rev. D92 , 054037 (2015), arXiv:1509.08016 [hep-ph].[10] X. Ji, J.-H. Zhang, and Y. Zhao, Nucl. Phys.
B924 , 366 (2017), arXiv:1706.07416 [hep-ph].[11] C. Monahan, Phys. Rev.
D97 , 054507 (2018), arXiv:1710.04607 [hep-lat].[12] I. W. Stewart and Y. Zhao, Phys. Rev.
D97 , 054512 (2018), arXiv:1709.04933 [hep-ph].[13] M. Constantinou and H. Panagopoulos, Phys. Rev.
D96 , 054506 (2017), arXiv:1705.11193 [hep-lat].[14] J. Green, K. Jansen, and F. Steffens, Phys. Rev. Lett. , 022004 (2018), arXiv:1707.07152 [hep-lat].[15] X. Xiong, T. Luu, and U.-G. Meißner, (2017), arXiv:1705.00246 [hep-ph].[16] W. Wang, S. Zhao, and R. Zhu, Eur. Phys. J.
C78 , 147 (2018), arXiv:1708.02458 [hep-ph].[17] W. Wang and S. Zhao, JHEP , 142 (2018), arXiv:1712.09247 [hep-ph].[18] J. Xu, Q.-A. Zhang, and S. Zhao, Phys. Rev. D97 , 114026 (2018), arXiv:1804.01042 [hep-ph].[19] J.-W. Chen, S. D. Cohen, X. Ji, H.-W. Lin, and J.-H. Zhang, Nucl. Phys.
B911 , 246 (2016), arXiv:1603.06664 [hep-ph].[20] J.-H. Zhang, J.-W. Chen, X. Ji, L. Jin, and H.-W. Lin, Phys. Rev.
D95 , 094514 (2017), arXiv:1702.00008 [hep-lat].[21] T. Ishikawa, Y.-Q. Ma, J.-W. Qiu, and S. Yoshida, (2016), arXiv:1609.02018 [hep-lat].[22] J.-W. Chen, X. Ji, and J.-H. Zhang, Nucl. Phys.
B915 , 1 (2017), arXiv:1609.08102 [hep-ph].[23] X. Ji, J.-H. Zhang, and Y. Zhao, Phys. Rev. Lett. , 112001 (2018), arXiv:1706.08962 [hep-ph].[24] T. Ishikawa, Y.-Q. Ma, J.-W. Qiu, and S. Yoshida, Phys. Rev.
D96 , 094019 (2017), arXiv:1707.03107 [hep-ph].[25] J.-W. Chen, T. Ishikawa, L. Jin, H.-W. Lin, Y.-B. Yang, J.-H. Zhang, and Y. Zhao, Phys. Rev.
D97 , 014505 (2018),arXiv:1706.01295 [hep-lat].[26] C. Alexandrou, K. Cichy, M. Constantinou, K. Hadjiyiannakou, K. Jansen, H. Panagopoulos, and F. Steffens, Nucl.Phys.
B923 , 394 (2017), arXiv:1706.00265 [hep-lat].[27] J.-W. Chen, T. Ishikawa, L. Jin, H.-W. Lin, Y.-B. Yang, J.-H. Zhang, and Y. Zhao, (2017), arXiv:1710.01089 [hep-lat].[28] H.-W. Lin, J.-W. Chen, T. Ishikawa, and J.-H. Zhang (LP3), Phys. Rev.
D98 , 054504 (2018), arXiv:1708.05301 [hep-lat].[29] J.-W. Chen, T. Ishikawa, L. Jin, H.-W. Lin, A. SchÀfer, Y.-B. Yang, J.-H. Zhang, and Y. Zhao, (2017),arXiv:1711.07858 [hep-ph].[30] H.-n. Li, Phys. Rev.
D94 , 074036 (2016), arXiv:1602.07575 [hep-ph].[31] C. Monahan and K. Orginos, JHEP , 116 (2017), arXiv:1612.01584 [hep-lat].[32] A. Radyushkin, Phys. Lett. B767 , 314 (2017), arXiv:1612.05170 [hep-ph].[33] G. C. Rossi and M. Testa, Phys. Rev.
D96 , 014507 (2017), arXiv:1706.04428 [hep-lat].[34] C. E. Carlson and M. Freid, Phys. Rev.
D95 , 094504 (2017), arXiv:1702.05775 [hep-ph].[35] R. A. Briceño, J. V. Guerrero, M. T. Hansen, and C. J. Monahan, Phys. Rev. D , 014511 (2018), arXiv:1805.01034[hep-lat].[36] T. J. Hobbs, Phys. Rev. D97 , 054028 (2018), arXiv:1708.05463 [hep-ph].[37] Y. Jia, S. Liang, L. Li, and X. Xiong, JHEP , 151 (2017), arXiv:1708.09379 [hep-ph].[38] S.-S. Xu, L. Chang, C. D. Roberts, and H.-S. Zong, Phys. Rev. D97 , 094014 (2018), arXiv:1802.09552 [nucl-th].[39] Y. Jia, S. Liang, X. Xiong, and R. Yu, Phys. Rev.
D98 , 054011 (2018), arXiv:1804.04644 [hep-th].[40] G. Spanoudes and H. Panagopoulos, Phys. Rev.
D98 , 014509 (2018), arXiv:1805.01164 [hep-lat].[41] G. Rossi and M. Testa, Phys. Rev.
D98 , 054028 (2018), arXiv:1806.00808 [hep-lat].[42] Y.-S. Liu, J.-W. Chen, L. Jin, H.-W. Lin, Y.-B. Yang, J.-H. Zhang, and Y. Zhao, (2018), arXiv:1807.06566 [hep-lat].[43] X. Ji, Y. Liu, and I. Zahed, Phys. Rev.
D99 , 054008 (2019), arXiv:1807.07528 [hep-ph].[44] S. Bhattacharya, C. Cocuzza, and A. Metz, Phys. Lett.
B788 , 453 (2019), arXiv:1808.01437 [hep-ph].[45] A. V. Radyushkin, Phys. Lett.
B788 , 380 (2019), arXiv:1807.07509 [hep-ph].[46] J.-H. Zhang, X. Ji, A. SchÀfer, W. Wang, and S. Zhao, Phys. Rev. Lett. , 142001 (2019), arXiv:1808.10824 [hep-ph].[47] Z.-Y. Li, Y.-Q. Ma, and J.-W. Qiu, Phys. Rev. Lett. , 062002 (2019), arXiv:1809.01836 [hep-ph].[48] V. M. Braun, A. Vladimirov, and J.-H. Zhang, Phys. Rev.
D99 , 014013 (2019), arXiv:1810.00048 [hep-ph].[49] W. Detmold, R. G. Edwards, J. J. Dudek, M. Engelhardt, H.-W. Lin, S. Meinel, K. Orginos, and P. Shanahan (USQCD),Eur. Phys. J. A , 193 (2019), arXiv:1904.09512 [hep-lat].[50] R. S. Sufian, C. Egerer, J. Karpie, R. G. Edwards, B. Joó, Y.-Q. Ma, K. Orginos, J.-W. Qiu, and D. G. Richards, (2020),arXiv:2001.04960 [hep-lat].[51] C. Shugert, X. Gao, T. Izubichi, L. Jin, C. Kallidonis, N. Karthik, S. Mukherjee, P. Petreczky, S. Syritsyn, and Y. Zhao,in (2020) arXiv:2001.11650 [hep-lat].[52] J. R. Green, K. Jansen, and F. Steffens, Phys. Rev. D , 074509 (2020), arXiv:2002.09408 [hep-lat].[53] V. Braun, K. Chetyrkin, and B. Kniehl, (2020), arXiv:2004.01043 [hep-ph].[54] H.-W. Lin, Int. J. Mod. Phys. A , 2030006 (2020).[55] M. Bhat, K. Cichy, M. Constantinou, and A. Scapellato, (2020), arXiv:2005.02102 [hep-lat].[56] L.-B. Chen, W. Wang, and R. Zhu, Phys. Rev. D , 011503 (2020), arXiv:2005.13757 [hep-ph].[57] X. Ji, (2020), arXiv:2003.04478 [hep-ph]. [58] L.-B. Chen, W. Wang, and R. Zhu, (2020), arXiv:2006.10917 [hep-ph].[59] L.-B. Chen, W. Wang, and R. Zhu, (2020), arXiv:2006.14825 [hep-ph].[60] C. Alexandrou, G. Iannelli, K. Jansen, and F. Manigrasso (Extended Twisted Mass), (2020), arXiv:2007.13800 [hep-lat].[61] Z. Fan, X. Gao, R. Li, H.-W. Lin, N. Karthik, S. Mukherjee, P. Petreczky, S. Syritsyn, Y.-B. Yang, and R. Zhang,(2020), arXiv:2005.12015 [hep-lat].[62] X. Ji, Y. Liu, A. Schäfer, W. Wang, Y.-B. Yang, J.-H. Zhang, and Y. Zhao, (2020), arXiv:2008.03886 [hep-ph].[63] H.-W. Lin, J.-W. Chen, S. D. Cohen, and X. Ji, Phys. Rev. D91 , 054510 (2015), arXiv:1402.1462 [hep-ph].[64] C. Alexandrou, K. Cichy, V. Drach, E. Garcia-Ramos, K. Hadjiyiannakou, K. Jansen, F. Steffens, and C. Wiese, Phys.Rev.
D92 , 014502 (2015), arXiv:1504.07455 [hep-lat].[65] C. Alexandrou, K. Cichy, M. Constantinou, K. Hadjiyiannakou, K. Jansen, F. Steffens, and C. Wiese, Phys. Rev.
D96 ,014513 (2017), arXiv:1610.03689 [hep-lat].[66] H.-W. Lin, J.-W. Chen, X. Ji, L. Jin, R. Li, Y.-S. Liu, Y.-B. Yang, J.-H. Zhang, and Y. Zhao, Phys. Rev. Lett. ,242003 (2018), arXiv:1807.07431 [hep-lat].[67] C. Alexandrou, K. Cichy, M. Constantinou, K. Jansen, A. Scapellato, and F. Steffens, Phys. Rev. Lett. , 112001(2018), arXiv:1803.02685 [hep-lat].[68] J.-W. Chen, L. Jin, H.-W. Lin, Y.-S. Liu, Y.-B. Yang, J.-H. Zhang, and Y. Zhao, (2018), arXiv:1803.04393 [hep-lat].[69] C. Alexandrou, K. Cichy, M. Constantinou, K. Jansen, A. Scapellato, and F. Steffens, Phys. Rev.
D98 , 091503 (2018),arXiv:1807.00232 [hep-lat].[70] H.-W. Lin, J.-W. Chen, X. Ji, L. Jin, R. Li, Y.-S. Liu, Y.-B. Yang, J.-H. Zhang, and Y. Zhao, Phys. Rev. Lett. ,242003 (2018), arXiv:1807.07431 [hep-lat].[71] Z.-Y. Fan, Y.-B. Yang, A. Anthony, H.-W. Lin, and K.-F. Liu, Phys. Rev. Lett. , 242001 (2018), arXiv:1808.02077[hep-lat].[72] Y.-S. Liu, J.-W. Chen, L. Jin, R. Li, H.-W. Lin, Y.-B. Yang, J.-H. Zhang, and Y. Zhao, (2018), arXiv:1810.05043[hep-lat].[73] W. Wang, J.-H. Zhang, S. Zhao, and R. Zhu, (2019), arXiv:1904.00978 [hep-ph].[74] H.-W. Lin and R. Zhang, Phys. Rev.
D100 , 074502 (2019).[75] K.-F. Liu, (2020), arXiv:2007.15075 [hep-ph].[76] R. Zhang, Z. Fan, R. Li, H.-W. Lin, and B. Yoon, Phys. Rev.
D101 , 034516 (2020), arXiv:1909.10990 [hep-lat].[77] C. Alexandrou, K. Cichy, M. Constantinou, J. R. Green, K. Hadjiyiannakou, K. Jansen, F. Manigrasso, A. Scapellato,and F. Steffens, (2020), arXiv:2011.00964 [hep-lat].[78] J.-W. Chen, L. Jin, H.-W. Lin, Y.-S. Liu, A. SchÀfer, Y.-B. Yang, J.-H. Zhang, and Y. Zhao, (2018), arXiv:1804.01483[hep-lat].[79] T. Izubuchi, L. Jin, C. Kallidonis, N. Karthik, S. Mukherjee, P. Petreczky, C. Shugert, and S. Syritsyn, Phys. Rev. D , 034516 (2019), arXiv:1905.06349 [hep-lat].[80] X. Gao, L. Jin, C. Kallidonis, N. Karthik, S. Mukherjee, P. Petreczky, C. Shugert, S. Syritsyn, and Y. Zhao, (2020),arXiv:2007.06590 [hep-lat].[81] H.-W. Lin, J.-W. Chen, Z. Fan, J.-H. Zhang, and R. Zhang, (2020), arXiv:2003.14128 [hep-lat].[82] Y. Chai et al. , (2020), arXiv:2002.12044 [hep-lat].[83] S. Bhattacharya, K. Cichy, M. Constantinou, A. Metz, A. Scapellato, and F. Steffens, (2020), arXiv:2004.04130 [hep-lat].[84] S. Bhattacharya, K. Cichy, M. Constantinou, A. Metz, A. Scapellato, and F. Steffens, Phys. Rev. D , 034005 (2020),arXiv:2005.10939 [hep-ph].[85] S. Bhattacharya, K. Cichy, M. Constantinou, A. Metz, A. Scapellato, and F. Steffens, (2020), arXiv:2006.12347 [hep-ph].[86] Z. Fan, R. Zhang, and H.-W. Lin, (2020), arXiv:2007.16113 [hep-lat].[87] R. Zhang, H.-W. Lin, and B. Yoon, (2020), arXiv:2005.01124 [hep-lat].[88] J.-H. Zhang, L. Jin, H.-W. Lin, A. SchÀfer, P. Sun, Y.-B. Yang, R. Zhang, Y. Zhao, and J.-W. Chen (LP3), Nucl.Phys.
B939 , 429 (2019), arXiv:1712.10025 [hep-ph].[89] R. Zhang, C. Honkala, H.-W. Lin, and J.-W. Chen, (2020), arXiv:2005.13955 [hep-lat].[90] J. Hua, M.-H. Chu, P. Sun, W. Wang, J. Xu, Y.-B. Yang, J.-H. Zhang, and Q.-A. Zhang, (2020), arXiv:2011.09788[hep-lat].[91] J.-W. Chen, H.-W. Lin, and J.-H. Zhang, (2019), 10.1016/j.nuclphysb.2020.114940, arXiv:1904.12376 [hep-lat].[92] C. Alexandrou, K. Cichy, M. Constantinou, K. Hadjiyiannakou, K. Jansen, A. Scapellato, and F. Steffens, (2020),arXiv:2008.10573 [hep-lat].[93] H.-W. Lin, (2020), arXiv:2008.12474 [hep-ph].[94] C. Alexandrou, K. Cichy, M. Constantinou, K. Hadjiyiannakou, K. Jansen, A. Scapellato, and F. Steffens, Phys. Rev. D , 114504 (2019), arXiv:1902.00587 [hep-lat].[95] X. Ji, P. Sun, X. Xiong, and F. Yuan, Phys. Rev. D91 , 074009 (2015), arXiv:1405.7640 [hep-ph].[96] X. Ji, L.-C. Jin, F. Yuan, J.-H. Zhang, and Y. Zhao, (2018), arXiv:1801.05930 [hep-ph].[97] M. A. Ebert, I. W. Stewart, and Y. Zhao, Phys. Rev. D , 034505 (2019), arXiv:1811.00026 [hep-ph].[98] M. A. Ebert, I. W. Stewart, and Y. Zhao, JHEP , 037 (2019), arXiv:1901.03685 [hep-ph].[99] M. A. Ebert, I. W. Stewart, and Y. Zhao, JHEP , 099 (2020), arXiv:1910.08569 [hep-ph].[100] X. Ji, Y. Liu, and Y.-S. Liu, Nucl. Phys. B , 115054 (2020), arXiv:1910.11415 [hep-ph].[101] X. Ji, Y. Liu, and Y.-S. Liu, (2019), arXiv:1911.03840 [hep-ph].[102] M. A. Ebert, S. T. Schindler, I. W. Stewart, and Y. Zhao, (2020), arXiv:2004.14831 [hep-ph].[103] P. Shanahan, M. L. Wagman, and Y. Zhao, Phys. Rev. D , 074505 (2020), arXiv:1911.00800 [hep-lat]. [104] P. Shanahan, M. Wagman, and Y. Zhao, (2020), arXiv:2003.06063 [hep-lat].[105] Q.-A. Zhang et al. (Lattice Parton), (2020), arXiv:2005.14572 [hep-lat].[106] K.-F. Liu and S.-J. Dong, Phys. Rev. Lett. , 1790 (1994), arXiv:hep-ph/9306299 [hep-ph].[107] W. Detmold and C. Lin, Phys. Rev. D , 014501 (2006), arXiv:hep-lat/0507007.[108] V. Braun and D. Müller, Eur. Phys. J. C , 349 (2008), arXiv:0709.1348 [hep-ph].[109] G. S. Bali et al. , Eur. Phys. J. C , 217 (2018), arXiv:1709.04325 [hep-lat].[110] G. S. Bali, V. M. Braun, B. Gläßle, M. Göckeler, M. Gruber, F. Hutzler, P. Korcyl, A. Schäfer, P. Wein, and J.-H. Zhang,Phys. Rev. D , 094507 (2018), arXiv:1807.06671 [hep-lat].[111] W. Detmold, I. Kanamori, C. D. Lin, S. Mondal, and Y. Zhao, PoS LATTICE2018 , 106 (2018), arXiv:1810.12194[hep-lat].[112] J. Liang, T. Draper, K.-F. Liu, A. Rothkopf, and Y.-B. Yang (XQCD), Phys. Rev. D , 114503 (2020), arXiv:1906.05312[hep-ph].[113] Y.-Q. Ma and J.-W. Qiu, Phys. Rev.
D98 , 074021 (2018), arXiv:1404.6860 [hep-ph].[114] Y.-Q. Ma and J.-W. Qiu, Int. J. Mod. Phys. Conf. Ser. , 1560041 (2015), arXiv:1412.2688 [hep-ph].[115] A. J. Chambers, R. Horsley, Y. Nakamura, H. Perlt, P. E. L. Rakow, G. Schierholz, A. Schiller, K. Somfleth, R. D. Young,and J. M. Zanotti, Phys. Rev. Lett. , 242001 (2017), arXiv:1703.01153 [hep-lat].[116] A. V. Radyushkin, Phys. Rev. D96 , 034025 (2017), arXiv:1705.01488 [hep-ph].[117] K. Orginos, A. Radyushkin, J. Karpie, and S. Zafeiropoulos, Phys. Rev. D , 094503 (2017), arXiv:1706.05373 [hep-ph].[118] A. Radyushkin, Phys. Lett. B , 433 (2018), arXiv:1710.08813 [hep-ph].[119] A. Radyushkin, Phys. Rev. D , 014019 (2018), arXiv:1801.02427 [hep-ph].[120] J.-H. Zhang, J.-W. Chen, and C. Monahan, Phys. Rev. D , 074508 (2018), arXiv:1801.03023 [hep-ph].[121] J. Karpie, K. Orginos, and S. Zafeiropoulos, JHEP , 178 (2018), arXiv:1807.10933 [hep-lat].[122] B. Joó, J. Karpie, K. Orginos, A. Radyushkin, D. Richards, and S. Zafeiropoulos, JHEP , 081 (2019), arXiv:1908.09771[hep-lat].[123] A. V. Radyushkin, Phys. Rev. D , 116011 (2019), arXiv:1909.08474 [hep-ph].[124] B. Joó, J. Karpie, K. Orginos, A. V. Radyushkin, D. G. Richards, R. S. Sufian, and S. Zafeiropoulos, Phys. Rev. D ,114512 (2019), arXiv:1909.08517 [hep-lat].[125] I. Balitsky, W. Morris, and A. Radyushkin, Phys. Lett. B , 135621 (2020), arXiv:1910.13963 [hep-ph].[126] A. Radyushkin, Int. J. Mod. Phys. A , 2030002 (2020), arXiv:1912.04244 [hep-ph].[127] B. Joó, J. Karpie, K. Orginos, A. V. Radyushkin, D. G. Richards, and S. Zafeiropoulos, (2020), arXiv:2004.01687[hep-lat].[128] K. Can et al. , (2020), arXiv:2007.01523 [hep-lat].[129] H.-W. Lin et al. , Prog. Part. Nucl. Phys. , 107 (2018), arXiv:1711.07916 [hep-ph].[130] K. Cichy and M. Constantinou, Adv. High Energy Phys. , 3036904 (2019), arXiv:1811.07248 [hep-lat].[131] Y. Zhao, PoS LATTICE2019 , 267 (2020).[132] X. Ji, Y.-S. Liu, Y. Liu, J.-H. Zhang, and Y. Zhao, (2020), arXiv:2004.03543 [hep-ph].[133] X. Ji, (2020), arXiv:2007.06613 [hep-ph].[134] W.-Y. Liu and J.-W. Chen, (2020), arXiv:2010.06623 [hep-ph].[135] J. Gasser and H. Leutwyler, Annals Phys. , 142 (1984).[136] E. E. Jenkins and A. V. Manohar, Phys. Lett. B , 558 (1991).[137] V. Bernard, N. Kaiser, and U.-G. Meissner, Int. J. Mod. Phys. E , 193 (1995), arXiv:hep-ph/9501384.[138] S. R. Beane, P. F. Bedaque, W. C. Haxton, D. R. Phillips, and M. J. Savage, , 133 (2000), arXiv:nucl-th/0008064.[139] S. Beane, P. F. Bedaque, M. Savage, and U. van Kolck, Nucl. Phys. A , 377 (2002), arXiv:nucl-th/0104030.[140] P. F. Bedaque and U. van Kolck, Ann. Rev. Nucl. Part. Sci. , 339 (2002), arXiv:nucl-th/0203055.[141] K. Kubodera and T.-S. Park, Ann. Rev. Nucl. Part. Sci. , 19 (2004), arXiv:nucl-th/0402008.[142] U.-G. Meißner, Nucl. Phys. News. , 11 (2014), arXiv:1505.06997 [nucl-th].[143] H.-W. Hammer, A. Nogga, and A. Schwenk, Rev. Mod. Phys. , 197 (2013), arXiv:1210.4273 [nucl-th].[144] D. Arndt and M. J. Savage, Nucl. Phys. A697 , 429 (2002), arXiv:nucl-th/0105045 [nucl-th].[145] J.-W. Chen and X.-d. Ji, Phys. Lett.
B523 , 107 (2001), arXiv:hep-ph/0105197 [hep-ph].[146] J.-W. Chen and X.-d. Ji, Phys. Rev. Lett. , 152002 (2001), [Erratum: Phys. Rev. Lett.88,249901(2002)], arXiv:hep-ph/0107158 [hep-ph].[147] W. Detmold, W. Melnitchouk, J. W. Negele, D. B. Renner, and A. W. Thomas, Phys. Rev. Lett. , 172001 (2001),arXiv:hep-lat/0103006.[148] W. Detmold, W. Melnitchouk, and A. W. Thomas, Phys. Rev. D , 054501 (2002), arXiv:hep-lat/0206001.[149] W. Detmold, W. Melnitchouk, and A. W. Thomas, Phys. Rev. D , 034025 (2003), arXiv:hep-lat/0303015.[150] W. Detmold and C. Lin, Phys. Rev. D , 054510 (2005), arXiv:hep-lat/0501007.[151] P. Hagler et al. (LHPC), Phys. Rev. D , 094502 (2008), arXiv:0705.4295 [hep-lat].[152] M. Gockeler, R. Horsley, D. Pleiter, P. E. L. Rakow, A. Schafer, G. Schierholz, and W. Schroers (QCDSF), Phys. Rev.Lett. , 042002 (2004), arXiv:hep-ph/0304249 [hep-ph].[153] J.-W. Chen and X.-d. Ji, Phys. Rev. Lett. , 052003 (2002), arXiv:hep-ph/0111048 [hep-ph].[154] A. V. Belitsky and X. Ji, Phys. Lett. B , 289 (2002), arXiv:hep-ph/0203276.[155] J.-W. Chen and I. W. Stewart, Phys. Rev. Lett. , 202001 (2004), arXiv:hep-ph/0311285.[156] J.-W. Chen, W. Detmold, and B. Smigielski, Phys. Rev. D , 074003 (2007), arXiv:hep-lat/0612027.[157] S.-i. Ando, J.-W. Chen, and C.-W. Kao, Phys. Rev. D , 094013 (2006), arXiv:hep-ph/0602200. [158] M. Diehl, A. Manashov, and A. Schafer, Eur. Phys. J. A , 315 (2006), [Erratum: Eur.Phys.J.A 56, 220 (2020)],arXiv:hep-ph/0608113.[159] J.-W. Chen and W. Detmold, Phys. Lett. B , 165 (2005), arXiv:hep-ph/0412119.[160] S. R. Beane and M. J. Savage, Nucl. Phys. A , 259 (2005), arXiv:nucl-th/0412025.[161] J.-W. Chen, W. Detmold, J. E. Lynn, and A. Schwenk, Phys. Rev. Lett. , 262502 (2017), arXiv:1607.03065 [hep-ph].[162] J. Lynn, D. Lonardoni, J. Carlson, J. Chen, W. Detmold, S. Gandolfi, and A. Schwenk, J. Phys. G , 045109 (2020),arXiv:1903.12587 [nucl-th].[163] J. Owens, A. Accardi, and W. Melnitchouk, Physical Review D , 094012 (2013).[164] J. W. Chen and X. Ji, Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics , 107 (2001).[165] D. Arndt and M. J. Savage, Nuclear Physics A , 429 (2002), arXiv:0105045 [nucl-th].[166] H. W. Lin and R. Zhang, Physical Review D , 74502 (2019).[167] A. Ali Khan, T. Bakeyev, M. Göckeler, T. R. Hemmert, R. Horsley, A. C. Irving, B. Joó, D. Pleiter, P. E. Rakow,G. Schierholz, and H. Stüben, Nuclear Physics B , 175 (2004), arXiv:0312030 [hep-lat].[168] D. Bećirević and G. Villadoro, Phys. Rev. D69