Chiral Perturbation for Large Momentum Effective Field Theory

Large momentum effective field theory (LaMET) enables the extraction of parton distribution functions (PDFs) directly on a Euclidean lattice through a factorization theorem that relates the computed quasi-PDFs to PDFs. We apply chiral perturbation theory (ChPT) to LaMET to further separate soft scales, such as light quark masses and lattice size, to obtain leading model independent extrapolation formulas for extrapolations to physical quark masses and infinite volume. We find that the finite volume effect is reduced when the nucleon carries a finite momentum. For nucleon momentum greater than $1$ GeV and the lattice size $L$ and pion mass $ m_\pi $ satisfying $m_\pi L\geq 3$, the finite volume effect is less than $1\%$ 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 size effectively larger in that direction. We also find that the quark mass dependence in the infinite volume limit computed with non-zero nucleon momentum reproduces the previous result computed at zero momentum, as expected. Our approach can be generalized to other parton observables in LaMET straight forwardly.

Large momentum effective field theory (LaMET) enables the extraction of parton distribution functions (PDF's) directly on a Euclidean lattice through a factorization theorem that relates the computed quasi-PDF's to PDF's. We apply chiral perturbation theory (ChPT) to LaMET to further separate soft scales, such as light quark masses and lattice size, to obtain leading model independent extrapolation formulas for extrapolations to physical quark masses and infinite volume. We find that the finite volume effect is reduced when the nucleon carries a finite momentum. For nucleon momentum greater than 1 GeV and the lattice size L and pion mass mπ satisfying mπL ≥ 3, the finite volume effect is less than 1% 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 size effectively larger in that direction. We also find that the quark mass dependence in the infinite volume limit computed with non-zero nucleon momentum reproduces the previous result computed at zero momentum, as expected. Our approach can be generalized to other parton observables in LaMET straight forwardly.
Despite the progress made in LaMET, the LaMET factorization theorem of Eq. (1) makes no attempt to separate the 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 asq but no volume dependence. As lattice exploration of the m u,d and L parameter space requires a significant amount of computing resources, model independent formulas to guide the extrapolations to physical quark masses and infinite volume are of practical importance. An effective field theory (EFT) approach such as chiral perturbation theory (ChPT) is ideal for this purpose, as EFT only relies on the symmetries and the scale separation of the system, hence the results are model independent.
In this work, we establish the procedure to apply ChPT to LaMET. The previous success of ChPT can then be directly carried over to LaMET straight forwardly. As an example, we work out the light quark mass dependence and finite volume corrections to nucleon quasi-PDF's. Other applications such as the quenched, partially quenched, and mixed action artifacts, generalizing from SU(2) to SU(3), as well as the off-forward kinematics study of GPD's and so 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 application to other quasi-observables can follow the same procedure.
For the unpolarized nucleon quasi-PDF, the equal-time correlator computed on the lattice is where |N (P ) is a nucleon state with momentum P µ = ( M 2 + P 2 z , 0, 0, P z ) and the Wilson line is with g s is the strong coupling constant and λ µ = (0, 0, 0, −1). The Fourier transform of this correlator yields the unpolarized quasi-PDFq Using γ t instead of γ z in Eq.(3) to avoid mixing with another operator of the same mass dimension [13,27] will not affect 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 with the nucleon polarization vector s µ = (P z , 0, 0, M 2 + P 2 z )/M . And the corresponding quasi-PDF for quark helicity distribution is For the transversely polarized quasi-PDF, the equal-time correlator computed on the lattice is with the nucleon polarization vector s µ = (0, 1, 0, 0). The corresponding quasi-PDF for quark helicity distribution is Replacing γ z in Eq.(8) by γ t to avoid mixing with another operator of the same mass dimension [13,27] will not affect the chiral and finite volume corrections computed in this work. Under the operator product expansion (OPE), the quark bilinear operators become with λ µ Γ µ = γ z , γ z γ 5 , γ x ⊥ γ z γ 5 for the unpolarized, helicity and transversity cases, respectively. The λ µ λ µ1 λ µ2 . . . λ µn tensor is symmetric but not traceless. But the nucleon matrix elements of the trace parts are O(M 2 /P 2 z , Λ 2 QCD /P 2 z ) corrections whose sizes are power suppressed [1,19]. Therefore, we only need to concentrate on the symmetric traceless parts:  with d the spacetime dimension and These operators are irreducible representations of the Lorentz group and are of the leading twist (twist-2). Their nucleon matrix elements give rise to moments of nucleon PDFs.
We will use the technique developed in Refs. [144][145][146] to match the quark level twist-2 operators to hadronic level operators using ChPT. The Lagrangian of ChPT is given by [136,137] where the pion decay constant F π = 93 GeV, the pion field the quark mass matrix M = diag(m u , m d ), and η is the parameter connecting the quark mass and pion mass at the leading order. We will work in the isospin symmetric limit m u = m d . N is the SU(2) doublet nucleon field. The nucleon velocity v µ = P µ /M is the ratio of the nucleon momentum P µ and the nucleon mass where the pion vector current V µ = i 2 (u∂ µ u † + u † ∂ µ u) and u 2 = Σ. g A =1.25 is the axial-vector coupling. The nucleon The small expansion parameter in the 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 = 4πF π arising from the loop expansion [136].
Although the 1/M expansion of ChPT looks like a non-relativistic expansion, the ChPT formulation is actually fully relativistic. The expansion requires the nucleon momentum M v+k has small off-shellness which means 2v·k+k 2 /M 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.
The quark level operators on the left hand sides of these equations arise from OPE's. They encode the physics below the energy scale Λ, while the Wilson coefficients encode the physics above Λ. By matching these quark level operators to the most general combinations of hadronic level operators with the same symmetries, we further introduce a scale Λ χ below Λ. Thec can also appear in diagrams (e) and (f). They are of the same order in the power counting as diagrams (a)-(d). For n > 0, (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 a Euclidean lattice. We will work with a lattice with length L in the three spatial directions but the size of the time direction is infinite. Assuming the nucleon and pion fields both satisfy periodic boundary conditions in the spatial directions, such that their momenta are quantized as p n = 2π n/L in the reciprocal lattice space, with 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 momentum integration in the infinite volume limit and corrections caused by finite volume effect: Our results for the nucleon twist-2 matrix elements at one loop order are and where the subscript 0 indicates that the corresponding matrix elements are evaluate first in the infinite volume limit, then in the chiral limit such that m π L → ∞. n = n 2 x + n 2 y + n 2 z and n · v = n z P z /M . n i in Eq. (22) plays the The absolute value of the finite volume effect for a moving proton (P z = 0) is always smaller than a rest one (P z = 0) for any value of m π L. For P z /M ≥ 1 and m π L ≥ 3, the finite volume effect is less than 1%.
role to label the number of times the pion crosses the boundary of lattice in the i-direction. These matrix elements determine the m-th moment of the PDF defined as dxx m q(x). The δ m0 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 it contributes to the zero-th moment but not any other moment. In Eq. (22), the n independent part is the infinite volume result whose leading quark-mass dependence reproduces the previous results of Refs. [164,165]. The scales Λ n , ∆Λ n , δΛ n are associated with counterterms at the m 2 π order that need to be fit to data. Converting from the moments to distributions in the momentum fraction x, both the PDF and quasi-PDF has the leading quark mass dependence where the functions q, c, ∆q, ∆c, δq, δc are m π independent. Quark number conservation 1 −1 dxq u−d (x) = 1 −1 dxq u−d,0 (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. . This quantity is constructed using the proton isovector PDF extracted by the CTEQ-JLab collaboration (CJ12) [163] then matched to a quasi-PDF with P z =1.3 GeV. The finite volume contribution is shown in (c) and (d), which is much smaller than other errors in a typical lattice QCD computation. In Fig.2, we show the finite volume effect of the unpolarized twist-2 matrix elements of Eq.(22) for m = 0 by taking the ratio of the matrix elements at finite and infinite volumes. We see that the finite volume effect is not monotonic in P z nor in m π L, due to partial cancelations of several different contributions. However, the absolute value of the finite volume effect for a moving proton (P z = 0) is always smaller than a rest one (P z = 0) for any value of m π L. For P z /M ≥ 1 and m π L ≥ 3, the finite volume effect is less than 1% and is negligible for the current precision of lattice computations. This can be interpreted as an effect due to the Lorentz contraction in the z-direction which makes the box size effectively bigger in that direction.
The finite volume effect for equal time correlators of Eqs. (3), (6), and (8) can be derived from Eq. (22): To mimic the finite volume effect on an equal time correlator computed in lattice QCD, we generate the infinite volume result by using the unpolarized isovector PDF extracted by the CTEQ-JLab collaboration (CJ12) [163]. The procedure is to perform the matching from PDF defined in the MS scheme to the quasi-PDF defined in the RI/MOM scheme, 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. (25) to show the finite volume effect in h u−d . We have used α s =0.283, p z R =1.2 GeV, µ=3.1 GeV, 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 used similar parameters as the lattice calculation of Ref. [166], where the size of finite volume effect is found to be smaller than the error of the calculation and is consistent with our result within errors.
Eq. (25) shows that the combinations of correlators ∆h u−d (z, P z )/∆h ∞ u−d (z, P z )−1 and δh u−d (z, P z )/δh ∞ u−d (z, P z )−1 are especially simple. They do not depend on the correlators ∆h u−d (z, P z ) nor δh u−d (z, P z ). Furthermore, they do not depend on z. Their dependence on m π L is very similar to Fig.2(b) and hence will not be shown again here.
In other versions of heavy baryon chiral perturbation, one could include ∆ resonances or generalize the formalism from 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 relates the computed quasi-PDF's to PDF's. We have applied ChPT to LaMET to further separate soft scales, such as light quark masses and lattice size, to obtain leading model independent extrapolation formulas for extrapolations to physical 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 ≥ 3, the finite volume effect is less than 1% 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 size effectively larger in that direction. We also find that the quark mass dependence in the infinite volume limit computed with non-zero nucleon momentum reproduces the previous result computed at zero momentum, as expected.
(A6) In the final step, we drop the sine function in e −iαm n· vL since the summation over n cancels all the terms odd in n.