A model-independent framework for determining finite-volume effects of spatially nonlocal operators
JJLAB-THY-21-3315
A model-independent framework for determiningfinite-volume effects of spatially nonlocal operators
Ra´ul A. Brice˜no
1, 2, ∗ and Christopher J. Monahan
1, 3, † Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606, USA Department of Physics, Old Dominion University, Norfolk, Virginia 23529, USA Department of Physics, William & Mary,P.O. Box 8795 Williamsburg, Virginia 23187, USA (Dated: February 4, 2021)We present a model-independent framework to determine finite-volume corrections of ma-trix elements of spatially-separated current-current operators. We define these matrix ele-ments in terms of Compton-like amplitudes, i.e. amplitudes coupling single-particle statesvia two current insertions. We show that the infrared behavior of these matrix elements isdominated by the single-particle pole, which is approximated by the elastic form factors ofthe lowest-lying hadron. Therefore, given lattice data on the relevant elastic form factors,the finite-volume effects can be estimated non-perturbatively and without recourse to effec-tive field theories. For illustration purposes, we investigate the implications of the proposedformalism for a class of scalar theories in two and four dimensions.
I. INTRODUCTION
In recent years, there has been significant progress in the direct determination of the structureand interactions of hadrons from quantum chromodynamics (QCD), the gauge theory of the strongnuclear force. This has been made possible through algorithmic and theoretical advances in latticeQCD (the discretization of QCD on a finite Euclidean hypercubic lattice), which is presently theonly systematic ab initio computational tool available to access hadronic properties. A key classof quantities that are at the cusp of being accessible using lattice QCD are generalized partondistributions (GPDs) [1–3]. GPDs capture the three-dimensional spatial distribution of hadronicconstituents, and they generalize the elastic form factors that characterize the interactions ofhadrons with electroweak probes.Our knowledge of GPDs, which are accessible through deeply-virtual Compton scattering anddeeply-virtual meson production, is restricted to certain kinematic regions (for reviews see, forexample, [4–6]). Until recently, lattice calculations were limited in their ability to determineGPDs, or their collinear counterparts, parton distribution functions (PDFs). The advent of newtheoretical tools, including large-momentum effective theory (LaMET) [7], factorizable matrixelements [8–10], and pseudodistributions [11–13], has offered, for the first time, the possibilityof calculations of the three-dimensional structure of hadrons from the QCD Lagrangian. Thecommon thread running through these formalisms is that structural information is obtained frommatrix elements of currents that are separated in space, but local in time. This ensures thatthe matrix elements are insensitive to the time-signature of the correlation functions [14]. Thefirst lattice calculations of GPDs, applying the LaMET approach, appeared within the last year[15, 16]. Recently, the first lattice calculations of generalized form factors, which correspond tothe leading Mellin moments of GPDs, have appeared [17, 18]. For recent reviews see, for example,[19–23].Lattice calculations of matrix elements relevant to GPDs present numerous technical chal-lenges, including difficult signal-to-noise complications associated with fast-moving hadrons andsignificant systematic uncertainties. In addition to standard discretization errors, there are sys-tematic uncertainties particular to PDFs and GPDs, including power divergences that arise from ∗ [email protected] † [email protected] a r X i v : . [ h e p - l a t ] F e b Wilson line operators on the lattice, higher twist effects, and enhanced finite volume effects. InRef. [24] we identified and a proposed a method for removing these enhanced finite volume ef-fects in operators composed of spatially-separated currents. The first non-perturbative studiesof finite volume effects have found mixed results [25–28], which may indicate that Wilson-lineoperators have reduced finite-volume effects relative to spatially-separated currents, first studiednon-perturbatively in [29].Here we present a model-independent approach to determining the finite-volume error for ma-trix elements of spatially-extended two-current operators, relevant for the analysis of factorizablematrix elements. Following Ref. [30], we define the matrix elements of spatially-separated cur-rents in terms of Compton-like amplitudes, i.e. amplitudes coupling single-particle states via twocurrent insertions of, in principle, arbitrary Lorentz structure. Furthermore, we explain that ulti-mately one only needs the lowest-lying singularity of these amplitudes, namely the single-particlepole pieces. These pieces are completely constrained by the mass of the desired particle and theelastic form factors, which are among the quantities best constrained via lattice QCD, for examplesee Refs. [31–54]. In other words, one can obtain the leading order finite-volume errors withoutmaking use of an effective field theory (EFT), which in general may have poor convergence. Thisframework is similar in spirit to that of Refs. [55, 56] for isolating the leading-order finite-volumeerror of the hadron-vacuum polarization contribution to the anomalous magnetic moment of themuon.We find that, by comparing our results to the leading order effects determined using a scalarEFT, our current approach reinforces the conclusions of Ref. [24], without relying on a specificperturbative EFT analysis. We note, however, that the finite volume effects determined at leadingorder in the scalar EFT are generally of the same order of magnitude of, but parametrically smallerthan, the effects determined from the full form factors.We start by presenting the main result and introducing the framework for spatially-separatedcurrent operators and Compton-like amplitudes in Sec. II. We apply our approach to a simplescalar model, testing several form factor parametrizations, in two dimensions in Sec. III and infour dimensions in Sec. IV. We summarize in Sec. V.
II. FRAMEWORK
We begin by defining the infinite-volume matrix element [24] M ∞ ( ξξξ, P f , P i ) ≡ (cid:104) P f |J (0 , ξξξ ) J (0) | P i (cid:105) , (1)where | P i (cid:105) and | P f (cid:105) are the initial and final single-particle states, and ξ µ = (0 , ξξξ ) is the separationof the currents. For simplicity, we consider the case where the initial and final states are identicalscalars of mass m , and we assume the currents, J , to be scalars. For further simplification, we set ξξξ = ξ ˆ z , where ˆ z is a unit vector in the z -direction. It is relatively straightforward to generalizethese ideas to arbitrary Lorentz structures.For clarity we quote here our main result. The difference between the infinite volume matrixelement, Eq. (1), and its finite-volume analogue is δ M L ( ξ ˆ z, P f , P i ) ≡ M L ( ξ ˆ z, P f , P i ) − M ∞ ( ξ ˆ z, P f , P i )= ie − iP f,z ξ (cid:90) q e iq z ( ξ − L ) F ( − ( P f − q ) ) F ( − ( P i − q ) ) q − m + i(cid:15) + O ( e − mL ) , (2)where we have introduced the D -dimensional measure, (cid:90) q = (cid:90) d D q (2 π ) D . Our result expresses the leading finite volume correction in terms of the scalar form fac-tor F ( − ( P f − q ) ) = (cid:104) P f |J (0) | q (cid:105) . For arbitrary currents, J A and J B , one can use + + q
Eq. (2) by replacing the scalar form factors with the corresponding elastic matrix elements, (cid:104) P f |J B (0) | q (cid:105)(cid:104) q |J A (0) | P i (cid:105) . Depending on the quantum numbers of the current, the J A (0) | P i (cid:105) may not have the quantum numbers of the incoming state. In this case, the pole appearing in theintegrand will correspond to the mass of the lowest lying particle with the appropriate quantumnumbers.In what follows, we show that this integral provides a model-independent determination of thecoefficient of the leading-order finite volume correction for matrix elements of spatially separatedcurrents. This correction is O ( e − m | L − ξ | ), as found using a scalar effective theory in Ref. [24].To arrive at Eq. (2), we follow Ref. [24] and introduce the Fourier transform of this matrixelement. In contrast to Ref. [24], however, we study this Fourier transform non-perturbativelyusing all-orders perturbation theory, rather than applying a perturbative analysis in the contextof a scalar EFT. All-orders perturbation theory leads to results that are consistent with dispersiveapproaches and unitarity, and these enable us to introduce a model-independent definition of theleading finite-volume corrections to this matrix element. A. Derivation
In general, this matrix element can be defined as the Fourier transform of the “Compton-likeamplitude” , defined in Ref. [30], M ∞ ( ξξξ, P f , P i ) = (cid:90) q e i q · ξξξ ( − i ) T ( s, Q , Q if ) , (3)where Q = − q and Q if = − ( P f + q − P i ) are the virtualities of the two currents, and s =( P f + q ) . We define the Compton-like amplitude T to have factors of i for each current insertion.In Fig. 1 we depict T diagrammatically, showing only the single-particle pole and the two particlecontribution. Given that in general M ∞ is finite, the integral on the right hand side of Eq. (3)is convergent.Working to all orders in perturbation theory, one can isolate the singularities of the amplitudesin the complex s -plane, as well as in the plane of the other kinematic variables. The closestsingularity to the region of integration is the single-particle pole. As a result, this singularitydescribes the long-distance behavior of the matrix elements, which is the focus of this work.This contribution can be written in terms of the single-current matrix elements and the elastichadronic form factors as i T pole ( s, Q , Q if ) = i (cid:104) P f |J (0) | q + P f (cid:105) is − m + i(cid:15) i (cid:104) q + P f |J (0) | P i (cid:105) = iF ( Q ) is − m + i(cid:15) iF ( Q if ) , (4) We reserve the term “Compton amplitude” for the specific case of the insertion of two vector currents. which we depict in Fig. 1.Having expressed the matrix element in terms of the Compton-like amplitude, we can nowevaluate the finite-volume corrections. There are two classes of finite-volume effects. The first arestandard errors associated with virtual particles “ wrapping around the volume ”. In Appendix Awe show that these effects can be encoded by replacing T with its finite-volume analogue T L introduced in Ref. [30]. These effects lead to the standard O ( e − mL ) errors. The second classof finite-volume artifacts arise from the spatial separation of the two currents, ξ . As explainedin Ref. [24], this class of errors scale as O ( e − m | L − ξ | ). Therefore, for mL (cid:29) e − m | L − ξ | behavior, without recourse to an effective field theory.The finite-volume analogue of Eq. (3) can be written as the finite-volume Fourier transformof T L . As a result, we can write the finite-volume correction to the matrix element as δ M L ( ξξξ, P f , P i ) = 1 L D − (cid:88) q (cid:90) d q π e i q · ξξξ ( − i ) T L ( s, Q , Q if ) − (cid:90) q e i q · ξξξ ( − i ) T ( s, Q , Q if )= 1 L D − (cid:88) q (cid:90) d q π e i q · ξξξ ( − i ) T ( s, Q , Q if ) − (cid:90) q e i q · ξξξ ( − i ) T ( s, Q , Q if ) + O ( e − mL )= (cid:88) n (cid:54) =0 (cid:90) q e i q · ( ξξξ + L n ) ( − i ) T ( s, Q , Q if ) + O ( e − mL ) (5)= (cid:88) n (cid:54) =0 M ∞ ( ξξξ + L n , P f , P i ) + O ( e − mL ) . (6)where in the second equality we made use of the arguments presented in Appendix A to replace T L with T up to O ( e − mL ) errors. We have used the Poisson summation formula in the third equality.The last equality, although formally correct, is not in general useful. This result states that thefinite-volume corrections of the matrix elements at ξξξ depend on the value of M ( ξξξ + L n , P ) where | n | (cid:54) = 0, which is a-priori unknown. Therefore, we use instead the second-to-last equality, Eq. (5),to estimate the large distance value of the matrix element and, from this, infer the finite-volumeeffects.Depending on the specific choice of ξξξ , the sum over n includes finite-volume errors that are O ( e − mL ) or smaller, which we neglect in this analysis. In the following derivation, we choose ξξξ = ξ ˆ z , which is the case most typically used in calculations. With this choice, the finite-volumecorrection reduces to δ M L ( ξ ˆ z, P f , P i ) = (cid:90) q e i q · ˆ z ( ξ − L ) ( − i ) T ( s, Q , Q if ) + O ( e − mL ) (7)= M ∞ ( ξ − L, P f , P i ) + O ( e − mL ) . (8)The generalization to other geometries is straightforward, and requires including other modes inthe finite volume sum. For example, if ξξξ = ξ √ (1 , , n = ( − , ,
0) and n = (0 , − , ξξξ , i.e. ξ < L/
2, these two modes would providethe leading order contributions to the finite-volume errors.Having established the relationship between the desired matrix elements and the Compton-likeamplitude, we can identify the relation between the long-range contributions to the matrix ele-ments and the low-energy contributions to the amplitude. In particular, one expects the integralto be saturated by the small | q | region. Thus, although the pole contribution is not, in general,a good description of M ∞ for small values of ξ , this contribution does provide a reasonable ap-proximation of the finite-volume corrections to M ∞ . In other words, we approximate Eq. (7)using Eq. (4) for the pole contribution, δ M L ( ξ ˆ z, P f , P i ) = (cid:90) q e i q · ˆ z ( ξ − L ) ( − i ) F ( Q ) F ( Q if ) − ( q + P f ) + m − i(cid:15) + O ( e − mL ) , (9)which is our main result, Eq. (2).This approximation can be further justified as follows. The expected form of this integral,Eq. (9), is O ( e − m | L − ξ | ), as was shown in Ref. [24]. The neglected terms in T will have singularitiesassociated with a higher energy scale, M (cid:29) m . In general this energy scale could correspond toexcited states or thresholds. Performing a spectral decomposition for these singularities, one canimmediately conclude that these contributions will lead to finite-volume corrections of the form O ( e − M | L − ξ | ). For moderately small values of ξ , these errors can be safely neglected.Expressing the finite volume corrections in this form has several advantages over the per-turbative EFT expansion applied in Ref. [24]. First and foremost, this representation is non-perturbative in the dynamics. We have made no assumption about the power counting of anyunderlying EFT. Instead, we have made the purely kinematic assumption that the single-particlepole is the dominant singularity. Moreover, this can be systematically improved by includingfurther singularities, starting with the two-particle cut, the form of which was recently derived inRef. [30]. Finally, this representation extends straightforwardly to finite-volume effects for ma-trix elements involving initial and final states that have different momenta, which is particularlyrelevant for calculations of GPDs.In practice, the main challenge to evaluating finite-volume corrections using this representationis that, formally, knowledge of the form factors over a large kinematic region is required. Infact, each integral in Eq. (9) ranges from negative to positive infinity. State-of-the-art latticecalculations generally span a small region at low momentum transfer, but calculations up to Q (cid:39) . have been carried out [36, 42, 43]. As a result, one might worry that thisframework could be unrealistic to implement. But, as has been mentioned before, these integralsare dominated by the small q region. Thus, we envision using parametrizations of the form factorsthat describe the lattice QCD form factors accurately and vanish rapidly enough as | q | → ∞ . Inthe following sections we test these ideas for a simple dipole parametrization of the form factors.Having a covariant parametrization for the form factor, one may proceed to evaluate analyt-ically the integral shown in Eq. (9) using the techniques used in Ref. [24]. Alternatively, onecan further approximate Eq. (9) by evaluating the q integral and only keeping the pion polecontribution, to give δ M L, pole ( ξ ˆ z, P f , P i ) ≡ (cid:90) d D − q (2 π ) D − e i q · ˆ z ( ξ − L ) F ( Q ) F ( Q if )2 (cid:112) ( q + P f ) + m (cid:12)(cid:12)(cid:12)(cid:12) q = − E + √ ( q + P f ) + m , (10)where the subscript “pole” indicates that only the pion pole contribution has been retained. B. Comparison with scalar EFT
Before we apply this formalism to a scalar model, we note that we can directly compare theform-factor representation of Eq. (9) to the corresponding expression obtained from a scalar EFTat leading order studied in Ref. [24]. To compare these results, we replace each form factor inthe Compton-like amplitude, Eq. (4), by its value evaluated at Q = 0, i.e. with the charge ofthe particle iF (0) = g , and set the initial and final momenta equal, P = P i = P f . With thesereplacements, the Compton-like amplitude reduces to T LO ( s, Q , Q ) = − g s − m + i(cid:15) g. (11)Inserting this into Eq. (9), we find δ M L ;LO ( ξ ˆ z, P ) = (cid:90) q e i q · ˆ z ( ξ − L ) ( − i ) g − ( q + P ) + m − i(cid:15) , = (cid:90) q E e i q · ˆ z ( ξ − L ) g ( q E + P E ) + m , (12)where we have replaced q = iq E , P = iP E . This is the same expression presented in Eq. (15) ofRef. [24] for the leading order contribution.In the following sections, we refer to this prescription as the “charge prescription” and comparethe results for the corresponding finite volume effects with those estimated using parametrizationsof the form factors. III. MODEL CALCULATIONS IN 2D
The form factor representation of the finite volume corrections, Eq. (9), is model independentand depends only on the kinematic approximation that the integral of the Compton-like amplitudeis dominated by the closest pole singularity. In this section we test the implications of thisrepresentation in a scalar theory in two dimensions. For simplicity, we set the initial and finalmomenta to be equal, and use the notation P = P i = P f = P z ˆ z , which is the case relevant tocalculations of collinear hadron structure, and label matrix elements by P z . The extension to theoff-forward case, relevant to calculations of GPDs, is straightforward.Based on the results of Ref. [24], this formalism is likely to be most immediately useful forthe estimation of the finite-volume effects for matrix elements of the pion. With this in mind,although the following discussion is quite general and assumes only that the hadronic state haszero spin, we will refer to this state as the pion.Assuming the single-particle pole dominates the integral, in two dimensions the infinite-volumematrix element in Eq. (3) can be approximated as M D ∞ ( ξ ˆ z, P z ˆ z ) ≈ i (cid:90) d q (2 π ) e iq z ξ (cid:0) F ( Q ) (cid:1) ( q + P ) − m + i(cid:15) . (13)Similarly, the finite-volume matrix element, given in Eq. (9), reduces to a two dimensional integral, δ M L ( ξ ˆ z, P z ˆ z ) ≈ (cid:90) d q (2 π ) e iq z ( ξ − L ) (cid:0) F ( Q ) (cid:1) ( q + P ) − m + i(cid:15) . (14)In order to explore the implications for this formalism, we evaluate the integral of Eq. (14)using three different approximations. First, we parametrize the form factor using a dipole formand evaluate these integrals exactly. Second, using the dipole form, we approximate the integraleven further by evaluating only the pion pole contribution. Finally, we compare our results tothe “charge prescription”. A. Dipole form factor
One standard parametrization for form factors is the dipole form, iF ( Q ) = − ig m F q − m F + i(cid:15) , (15)where m F is typically attributed to a resonance with the quantum numbers of the current. Sucha pole, in general, violates unitarity. Instead, unless the form factor has the quantum numbers ofa single particle states [e.g. the axial vector in QCD], the closest singularity of the form factor willbe the first branch-cut associated with multi-particle production in the time-like region. Despitethis, the dipole approximation provides a reasonable prescription of form factors in the spacelikeregion.With this expression for the form factor, the integral of Eq. (14) can be expressed as M D ∞ , dip . ( ξ ˆ z, P z ˆ z ) = ig m F (cid:90) d q (2 π ) e iq z ξ ( q − m F + i(cid:15) ) (( q + P ) − m + i(cid:15) ) . (16)This integral can be evaluated using standard Schwinger tricks, as carried out in, for example,Ref. [24]. Instead, we choose to first evaluate the q integral using Cauchy’s residue theorem.The integrand has four poles, q = ± ( ω qF − i(cid:15) ) , q = − E ± ( ω qP − i(cid:15) ) , (17)where we have introduced E = (cid:112) P z + m as the energy of the pion and defined ω qF = (cid:113) q z + m F , ω qP = (cid:112) ( q z + P z ) + m . (18)The first of these poles arises from the form factor, and the second from the pion propagator.Closing the contour in the upper half plane, we obtain M D ∞ , dip . ( ξ ˆ z, P z ˆ z ) = M D ∞ , pole ( ξ ˆ z, P z ˆ z ) + M D ∞ , FF ( ξ ˆ z, P z ˆ z ) (19)where M D ∞ , pole and M D ∞ , FF are the pion and form factor pole contributions, respectively, and areexplicitly defined as M D ∞ , pole ( ξ ˆ z, P z ˆ z ) = ( gm F ) π (cid:90) ∞−∞ d q z e iq z ξ ω qP (cid:0) ( E − ω qP ) − ω qF (cid:1) , (20) M D ∞ , FF ( ξ ˆ z, P z ˆ z ) = ( gm F ) π (cid:90) ∞−∞ d q z e iq z ξ ω qF (cid:34) ω qP − ( E + ω qF ) ( E + 3 ω qF ) (cid:0) ( E + ω qF ) − ω qP (cid:1) (cid:35) . (21)Note that Eq. (20) is just the two-dimensional and model-dependent version of Eq. (10). Weseparate the pole contributions to the matrix element so that we can see the role of each term,and in particularly showcase the dominance of the pion pole contribution. One can then evaluatethe remaining integrals numerically.In a finite volume the momentum integral is replaced a sum of lattice modes. A similar analysisto the infinite-volume case leads to an expression for the finite volume corrections, δ M DL, dip . ( ξ ˆ z, P z ˆ z ) = δ M DL, pole ( ξ ˆ z, P z ˆ z ) + δ M DL, FF ( ξ ˆ z, P z ˆ z ) (22)where, δ M DL, pole ( ξ ˆ z, P z ˆ z ) = ( gm F ) π (cid:90) ∞−∞ d q z e iq z ( ξ − L ) ω qP (cid:0) ( E − ω qP ) − ω qF (cid:1) , (23) δ M DL, FF ( ξ ˆ z, P z ˆ z ) = ( gm F ) π (cid:90) ∞−∞ d q z e iq z ( ξ − L ) ω qF (cid:34) ω qP − ( E + ω qF ) ( E + 3 ω qF ) (cid:0) ( E + ω qF ) − ω qP (cid:1) (cid:35) . (24)Note that, consistent with our main results, Eq. (2), we have only kept the finite-volume modethat leads to the leading finite-volume error. All other modes give errors of O ( e − mL ) or smallerfor moderately small values of ξ , and are therefore neglected. B. Charge prescription
As a comparison, we also evaluate the charge prescription, considered in Ref. [24]. FromEq. (12), we can write this in two dimensions as δ M DL, LO ( ξ ˆ z, P z ˆ z ) = g (cid:90) d q E (2 π ) e iq z ( ξ − L ) q E + P E ) + m , (25)Introducing a Schwinger representation of the propagator, and carrying out the momentum andSchwinger integrals, leads to δ M DL, LO ( ξ ˆ z, P z ˆ z ) = g π e − iP z ( ξ − L ) K ( m | ξ − L | ) , (26)where K n ( z ) is the modified Bessel function of the second kind. FIG. 2: The two-dimensional finite-volume matrix elements versus the infinite-volume matrix element as afunction of the separation of the two currents for a range of parameters, with initial and final momentumfixed to zero. For each set of parameters, the bottom half of the panels show the percentage finite-volumeerrors as defined in Eq. (27).
C. Results in two dimensions
We have presented three schemes for estimating the finite-volume error. These correspond tousing the covariant dipole form factor, keeping only the pion pole, and the charge prescriptions,defined by Eqs. (22), (23), and (26), respectively. We compare our results for the three schemes inFigures 2, and 3. In Figure 2 we compare the finite-volume and infinite-volume matrix elementsas functions of the separation of the two currents, with initial and final momentum fixed to zero.We use six sets of parameters { m F , L, z } , expressed each combination in terms of m . We stateour choices for the combination of parameters in the upper right corner of each panel.For each choice of parameters we calculate δ M DL, dip . , δ M DL , and δ M DL . For the finite-volumematrix element we use δ M DL, dip . . Both M D ∞ , dip . and M DL, dip . are shown in units of M D ∞ , dip . evaluated at ξ = 0. We highlight that at small current separations, ξ ∼
0, the leading finitevolume correction is e −| L − ξ | m ∼ e − Lm . In other words, for small ξ the dominant error is of theorder of the suppressed errors. For clarity of presentation, however, we only keep the error thatscales e −| L − ξ | m in the plots, even when ξ = 0.We determine the percentage finite-volume error∆ M DL ≡ (cid:12)(cid:12)(cid:12)(cid:12) δ M DL M D ∞ (cid:12)(cid:12)(cid:12)(cid:12) × , (27) FIG. 3: Same as in Figure 2, but the parameters m F and L are fixed to be 1 . m and 4 /m respectively.Instead, the momenta are varied from 0 to 1 . m . for each choice of parameters and framework for evaluating the finite-volume corrections. Weplot this percentage as a function of ξ in the bottom half of each one of the panels.From Figure 2 we observe two important features. First, regardless of the choice of the formfactor, the charge prescription consistently underestimates the finite-volume corrections. Second,as expected, the pion pole prescription dominates the finite-volume corrections. This is because,as explained in Sec. (II A), the pion pole is the closest singularity of the Compton amplitude tothe region of integration. As noted in Ref. [24], for the choice of mL ∼
4, finite-volume effectscan be sizable. Here we find that for a form factor with a small pole mass, i.e. m F < . m , theseeffects can be of the order of 10%-20% even for small values of ξ , ξm (cid:39) .
5. These effects arereduced to the percent level for a slightly larger volume of mL ∼ m F can be understood in terms of thesingularities of the integrand. The smaller the value of m F , the closer the pole of the form factoris to the kinematic region of integration in Eq. (20).In Figure 3 we explore the momentum dependence of matrix elements with m F = 1 . m and mL = 4, which suffer the largest finite-volume effects. Once again, the plots illustrate the chargeprescription consistently underestimates of the leading finite volume effects, relative to the formfactor models. Furthermore, one observes that finite-volume errors are further suppressed atincreasingly large momenta. IV. MODEL CALCULATIONS IN FOUR DIMENSIONS
Here we follow similar steps to those presented in Sec. III for the four-dimensional case. Weagain set the initial and final momenta to be equal and align it along the ˆ z axis, P = P i = P f = P z ˆ z . As before, we also align the current displace along the ˆ z axis, ξξξ = ξ ˆ z . Applying the sameapproximations discussed before, the four-dimensional infinite-volume matrix element in Eq. (3)can be written as M ∞ ( ξ ˆ z, P z ˆ z ) ≈ i (cid:90) d q (2 π ) e iq z ξ (cid:0) F ( Q ) (cid:1) ( q + P ) − m + i(cid:15) . (28)0Similarly, the finite-volume counterpart, Eq. (9), is equal to δ M L ( ξ ˆ z, P z ˆ z ) ≈ i (cid:90) d q (2 π ) e iq z ( ξ − L ) (cid:0) F ( Q ) (cid:1) ( q + P ) − m + i(cid:15) . (29)Using the dipole form factor defined in Eq. (15), the infinite-volume matrix element can bewritten as M ∞ , dip . ( ξ ˆ z, P z ˆ z ) = ig m F (cid:90) d q (2 π ) e iq z ξ ( q − m F + i(cid:15) ) (( q + P ) − m + i(cid:15) ) . (30)In principle, one could evaluate the q integral and obtain two contributions, which would be thethree-dimensional analogues of Eqs. (23) and (22). It is more convenient, however, to retain thecovariant four-dimensional integral, introduce a Feynman parameter followed by the Schwingerparameterization, as in Ref. [24]. This allows one to reduce the four-dimensional integral to asingle integral over the Feynman parameter. Using Eqs. (16), (A9), and (A11) in Ref. [24], onecan show that this is equal to M ∞ , dip . ( ξ ˆ z, P z ˆ z ) = ξ ( gm F ) (4 π ) (cid:90) dx x e i ( x − P z ξ (cid:113) xm F + m (1 − x ) K (cid:18) ξ (cid:113) xm F + m (1 − x ) (cid:19) . (31)Then, from the last line of Eq. (5), one can write the finite-volume correction in terms of the sumover the mirror-image contributions of this integral, with the largest contribution coming fromthe nearest image, M L, dip . ( ξ ˆ z, P z ˆ z ) = M ∞ , dip . ( ξ − L, P z ˆ z ) . (32)We can compare this with the charge prescription in four dimensions, Eq. (12), which wasconsidered in Ref. [24]. Applying Eq. (16) of Ref. [24] one readily obtains δ M L ;LO ( ξ ˆ z, P z ˆ z ) = (cid:90) d q E (2 π ) e iq z ( ξ − L ) g ( P E + q E ) + m , = m g π e − i P · ( ξ ˆ z + L n ) K ( | ξ − L | m ) | ξ − L | . (33)This can be further simplified by using the fact that e i | P | L = e i π | m | = 1, δ M L ;LO ( ξ ˆ z, P z ˆ z ) = m g π e − iP z ξ K ( | L − ξ | m ) | L − ξ | . (34)We now have the ingredients needed to compare numerical results. Following Sec. III C, weexpect the largest finite-volume artifacts at smaller values of m F and therefore focus our attentionon m F = 2 . m and 1 . m . In Figure 4 we show results for these two values of m F and a rangeof external momenta. As in the two-dimensional case, we see that the leading-order contributionunderestimates the finite-volume artifacts. However, the results show that the leading-ordercontribution captures the approximate error much more effectively in four dimensions than itdoes in two dimensions. Furthermore, the difference between the leading-order contribution andthe full dipole form vanishes relatively quickly for increasing values of the external momenta.Most importantly, the overall magnitude of the error is about four times smaller in fourdimensions than in two dimensions. This can be seen by comparing the values of these two casesfor m F = 1 . m , ξ = 0 and P z = 0 in Figures 2 and 4. One can reconcile this observationby again considering the singularities of the integrands. The integrands in the two-dimensionaland four-dimensional integrals, Eqs. (16) and (30) respectively, are identical; the only differenceis the dimension of the integral. The two additional angular integrals in the four-dimensionalintegral soften the singularity of the integrand and consequently the magnitude of the finitevolume corrections.1 FIG. 4: We plot the four-dimensional finite-volume matrix element versus the infinite-volume one as afunction of the separation of the two currents for a range of momenta, with all other parameters fixed.
V. CONCLUSIONS
Recent theoretical developments have enabled the determination of hadron structure directlyfrom QCD. Calculations of collinear hadron structure have matured sufficiently that a detailedunderstanding of systematic uncertainties is necessary to compare these calculations to experi-mental data. In Ref. [24] we suggested that the spatially extended composite operators used inmany lattice calculations of hadron structure may induce enhanced finite volume effects. Herewe developed an EFT-independent framework for estimating those finite volume effects.By defining the matrix elements of spatially-separated currents in terms of a Compton-likeamplitude, introduced in Ref. [30], we argued that the infrared behavior of these matrix elementsis dominated by the single-particle pole. This contribution can be determined from the elasticform factors of the lowest-lying hadronic state with the appropriate quantum numbers. Thisprovides an opportunity to estimate finite volume effects without relying on an underlying EFTthat may have, in general, poor convergence.We studied this methodology in simple scalar models, in two and four dimensions, focusingon the case of two spatially-separated scalar currents. We found that, by comparing our resultsto those derived from a scalar EFT in Ref. [24], our current approach reinforces the conclusionsof Ref. [24]. We note, however, that the finite volume effects determined at leading order in thescalar EFT are generally smaller than determinations that incorporate form factors.2 + + q
The authors would like to thank M. T. Hansen and J. V. Guerrero for useful discussions.RAB and CJM are supported in part by USDOE grant No. DE-AC05-06OR23177, under whichJefferson Science Associates, LLC, manages and operates Jefferson Lab. RAB also acknowledgessupport from the USDOE Early Career award, contract DE-SC0019229.
Appendix A: Justification replacing T L with T Here we provide a justification of replacing T L with T in Eq. (6) up to errors that scale as O ( e − mL ).The key assumption made in Eq. (6) is that the integral of the finite-volume Compton-amplitude T L is equal to its infinite-volume counterpart, T ∞ . It is certainly not true that T L can be replaced with T ∞ in general. In fact, in general, these two quantities can differ by arbi-trary amounts. Part of the point of Ref. [30] was to address this discrepancy for kinematics thatmay be accessed using lattice QCD. Here we use the findings of Ref. [30], in addition to others.We are interested in volumes for which O ( e − mL ) errors can be neglected. In this region thereare two other sources of large finite-volume effects. The first arise from new scales in the problem,such as the separation between the external currents. This, of course, is the focus of the maintext of this work. The second arises from multi-particle states going on-shell. The manifestationof power-law finite-volume effects for two or more particle states is most famously associated withL¨uscher’s work [57]. As a result, all we need to argue is that the finite-volume effects associatedwith multi-particle states in T L can be ignored in the integral.Considering only contributions from two-particle states, we depict the two classes of potentialfinite-volume effects in Figure 5 for T L . The first are the s - and u -channel two-particle loops.We illustrate the s-channel contributions explicitly in the second term on the righthand side ofthe equality, but leave the u-channel contributions implicit. The second class are those shownin the third term on the right-hand side, associated with time-like two-particle loops. The s -and u -channel finite volume corrections are related to 1 → → → M ), and a purely kinematic finite-volume function, F . This finite-volume function can be written in terms of the Riemann zeta functions. Similarly, one can writethe finite-volume effects associated with the time-like diagrams to all orders in terms of 0 → M , and the F following formalism presented in Refs. [61, 62]. We point the reader to Refs. [58, 59] for recent reviews on the implementations and extensions of L¨uscher’sformalism. δ T L ∝ F
11 + M F . (A1)The poles of this function coincide with the finite-volume two-particle states, colloquially knownas the L¨uscher poles. Assuming no UV cutoff, there is an infinite number of these poles, which wedenote to be located at M n ≥ m where n is a discrete integer enumerating the possible states.The residues of the poles depend on the finite-volume transition matrix elements, and we willcompactly denote them as c n . Therefore, the contributions to T L coming from the L¨uscher polescan be written as, δ T L ( s ) ∝ (cid:88) n c n s − M n . (A2)Here we are being schematic and assuming we are only interested in the s -dependent terms.Similar contributions can be written for u -channel diagrams and for those associated with thetime-like form factors.The finite-volume effects associated with the time-like form factor can immediately be identi-fied to be of O ( e − mL ). This is because, as identified in the main body, the largest finite-volumeeffects come from the single-particle pole of the Compton-like amplitude. Given that the externalstates are on-shell, the residue of the finite-volume amplitude evaluated at the single-particlepole must depend on space-like virtualities. For these kinematics the finite-volume effects are O ( e − mL ).This leaves us to consider the contributions from the L¨uscher poles in the s -/ u -channel dia-grams. To see the size of these, one can simply insert Eq. (A2) as a correction to the two-particlecontributions of the s -/ u -channel diagrams. This leads to contributions that are of the same kindas the ones considered in the main body of the text, with the mass of the light exchanged particlereplaced with M n . When the exchanged particle is of mass m we know this leads to finite-volumeerrors of the order O ( e − m | L − ξ | ). Since the L¨uscher poles satisfy M n ≥ m , these correctionswould scale at worst as O ( e − m | L − ξ | ). For moderate values of ξ < L/
2, these are subleading. [1] X.-D. Ji, Phys. Rev. D , 7114 (1997), hep-ph/9609381.[2] A. V. Radyushkin, Phys. Lett. B , 417 (1996), hep-ph/9604317.[3] A. V. Belitsky and A. V. Radyushkin, Phys. Rept. , 1 (2005), hep-ph/0504030.[4] L. Favart, M. Guidal, T. Horn, and P. Kroll, Eur. Phys. J. A52 , 158 (2016), 1511.04535.[5] N. d’Hose, S. Niccolai, and A. Rostomyan, Eur. Phys. J.
A52 , 151 (2016).[6] K. Kumericki, S. Liuti, and H. Moutarde, Eur. Phys. J.
A52 , 157 (2016), 1602.02763.[7] X. Ji, Phys. Rev. Lett. , 262002 (2013), 1305.1539.[8] Y.-Q. Ma and J.-W. Qiu, Phys. Rev.
D98 , 074021 (2018), 1404.6860.[9] Y.-Q. Ma and J.-W. Qiu, Int. J. Mod. Phys. Conf. Ser. , 1560041 (2015), 1412.2688.[10] Y.-Q. Ma and J.-W. Qiu, Phys. Rev. Lett. , 022003 (2018), 1709.03018.[11] A. Radyushkin, Phys. Lett. B767 , 314 (2017), 1612.05170.[12] A. V. Radyushkin, Phys. Rev.
D96 , 034025 (2017), 1705.01488.[13] K. Orginos, A. Radyushkin, J. Karpie, and S. Zafeiropoulos, Phys. Rev.
D96 , 094503 (2017),1706.05373.[14] R. A. Briceno, M. T. Hansen, and C. J. Monahan, Phys. Rev.
D96 , 014502 (2017), 1703.06072.[15] C. Alexandrou, K. Cichy, M. Constantinou, K. Hadjiyiannakou, K. Jansen, A. Scapellato, and F. Stef-fens (2020), 2008.10573.[16] J.-W. Chen, H.-W. Lin, and J.-H. Zhang, Nucl. Phys. B , 114940 (2020), 1904.12376.[17] C. Alexandrou et al., Phys. Rev. D , 034519 (2020), 1908.10706.[18] G. S. Bali, S. Collins, M. G¨ockeler, R. R¨odl, A. Sch¨afer, and A. Sternbeck, Phys. Rev. D , 014507(2019), 1812.08256.[19] C. Monahan, PoS
LATTICE2018 , 018 (2018), 1811.00678. [20] Y. Zhao, Int. J. Mod. Phys. A , 1830033 (2019), 1812.07192.[21] K. Cichy and M. Constantinou, Adv. High Energy Phys. , 3036904 (2019), 1811.07248.[22] A. V. Radyushkin (2019), 1912.04244.[23] X. Ji, Y.-S. Liu, Y. Liu, J.-H. Zhang, and Y. Zhao (2020), 2004.03543.[24] R. A. Briceno, J. V. Guerrero, M. T. Hansen, and C. J. Monahan, Phys. Rev. D98 , 014511 (2018),1805.01034.[25] B. Jo´o, J. Karpie, K. Orginos, A. V. Radyushkin, D. G. Richards, R. S. Sufian, and S. Zafeiropoulos,Phys. Rev. D , 114512 (2019), 1909.08517.[26] B. Jo´o, J. Karpie, K. Orginos, A. Radyushkin, D. Richards, and S. Zafeiropoulos, JHEP , 081(2019), 1908.09771.[27] C. Alexandrou, K. Cichy, M. Constantinou, K. Hadjiyiannakou, K. Jansen, A. Scapellato, and F. Stef-fens, Phys. Rev. D , 114504 (2019), 1902.00587.[28] H.-W. Lin and R. Zhang, Phys. Rev. D , 074502 (2019).[29] R. S. Sufian, C. Egerer, J. Karpie, R. G. Edwards, B. Jo´o, Y.-Q. Ma, K. Orginos, J.-W. Qiu, andD. G. Richards (2020), 2001.04960.[30] R. A. Briceno, Z. Davoudi, M. T. Hansen, M. R. Schindler, and A. Baroni, Phys. Rev. D101 , 014509(2020), 1911.04036.[31] M. Gockeler, T. Hemmert, R. Horsley, D. Pleiter, P. E. Rakow, A. Schafer, and G. Schierholz(QCDSF), Phys. Rev. D , 034508 (2005), hep-lat/0303019.[32] C. Alexandrou, G. Koutsou, J. W. Negele, and A. Tsapalis, Phys. Rev. D , 034508 (2006), hep-lat/0605017.[33] S. Syritsyn et al., Phys. Rev. D , 034507 (2010), 0907.4194.[34] T. Yamazaki, Y. Aoki, T. Blum, H.-W. Lin, S. Ohta, S. Sasaki, R. Tweedie, and J. Zanotti, Phys.Rev. D , 114505 (2009), 0904.2039.[35] J. Bratt et al. (LHPC), Phys. Rev. D , 094502 (2010), 1001.3620.[36] H.-W. Lin, S. D. Cohen, R. G. Edwards, K. Orginos, and D. G. Richards (2010), 1005.0799.[37] S. Collins et al., Phys. Rev. D , 074507 (2011), 1106.3580.[38] T. Bhattacharya, S. D. Cohen, R. Gupta, A. Joseph, H.-W. Lin, and B. Yoon, Phys. Rev. D ,094502 (2014), 1306.5435.[39] P. Shanahan, A. Thomas, R. Young, J. Zanotti, R. Horsley, Y. Nakamura, D. Pleiter, P. Rakow,G. Schierholz, and H. St¨uben, Phys. Rev. D , 034502 (2014), 1403.1965.[40] P. Shanahan, A. Thomas, R. Young, J. Zanotti, R. Horsley, Y. Nakamura, D. Pleiter, P. Rakow,G. Schierholz, and H. St¨uben (CSSM, QCDSF/UKQCD), Phys. Rev. D , 074511 (2014), 1401.5862.[41] S. Capitani, M. Della Morte, D. Djukanovic, G. von Hippel, J. Hua, B. J¨ager, B. Knippschild,H. Meyer, T. Rae, and H. Wittig, Phys. Rev. D , 054511 (2015), 1504.04628.[42] J. Koponen, A. Zimermmane-Santos, C. Davies, G. Lepage, and A. Lytle, Phys. Rev. D , 054501(2017), 1701.04250.[43] A. Chambers et al. (QCDSF, UKQCD, CSSM), Phys. Rev. D , 114509 (2017), 1702.01513.[44] K.-I. Ishikawa, Y. Kuramashi, S. Sasaki, N. Tsukamoto, A. Ukawa, and T. Yamazaki (PACS), Phys.Rev. D , 074510 (2018), 1807.03974.[45] D. Djukanovic, K. Ottnad, J. Wilhelm, and H. Wittig, Phys. Rev. Lett. , 212001 (2019),1903.12566.[46] Y.-C. Jang, R. Gupta, H.-W. Lin, B. Yoon, and T. Bhattacharya, Phys. Rev. D , 014507 (2020),1906.07217.[47] S. Park, T. Bhattacharya, R. Gupta, Y.-C. Jang, B. Joo, H.-W. Lin, and B. Yoon, in (2020), 2002.02147.[48] C. Alexandrou, M. Brinet, J. Carbonell, M. Constantinou, P. Harraud, P. Guichon, K. Jansen, T. Ko-rzec, and M. Papinutto (ETM), Phys. Rev. D , 045010 (2011), 1012.0857.[49] R. Gupta, Y.-C. Jang, H.-W. Lin, B. Yoon, and T. Bhattacharya, Phys. Rev. D , 114503 (2017),1705.06834.[50] J. Green, N. Hasan, S. Meinel, M. Engelhardt, S. Krieg, J. Laeuchli, J. Negele, K. Orginos, A. Pochin-sky, and S. Syritsyn, Phys. Rev. D , 114502 (2017), 1703.06703.[51] C. Alexandrou, M. Constantinou, K. Hadjiyiannakou, K. Jansen, C. Kallidonis, G. Koutsou, andA. Vaquero Aviles-Casco, Phys. Rev. D , 054507 (2017), 1705.03399.[52] E. Shintani, K.-I. Ishikawa, Y. Kuramashi, S. Sasaki, and T. Yamazaki, Phys. Rev. D , 014510(2019), 1811.07292.[53] Y.-C. Jang, R. Gupta, B. Yoon, and T. Bhattacharya, Phys. Rev. Lett. , 072002 (2020),1905.06470.[54] P. E. Shanahan and W. Detmold, Phys. Rev. D99 , 014511 (2019), 1810.04626. [55] M. T. Hansen and A. Patella, Phys. Rev. Lett. , 172001 (2019), 1904.10010.[56] M. T. Hansen and A. Patella, JHEP , 029 (2020), 2004.03935.[57] M. Luscher, Nucl.Phys. B354 , 531 (1991).[58] R. A. Briceno, J. J. Dudek, and R. D. Young, Rev. Mod. Phys. , 025001 (2018), 1706.06223.[59] M. T. Hansen and S. R. Sharpe, Ann. Rev. Nucl. Part. Sci. , 65 (2019), 1901.00483.[60] R. A. Briceno, M. T. Hansen, and A. Walker-Loud (2014), 1406.5965.[61] R. A. Briceno and M. T. Hansen, Phys. Rev. D92 , 074509 (2015), 1502.04314.[62] H. B. Meyer, Phys. Rev. Lett.107