A model-independent framework for determining
finite-volume effects of spatially nonlocal operators

In general, this matrix element can be defined as the Fourier transform of the “Compton-like amplitude”¹¹1We reserve the term “Compton amplitude” for the specific case of the insertion of two vector currents., defined in Ref. Briceno et al. (2020),

where $Q^{2}=-q^{2}$ and $Q^{2}_{if}=-(P_{f}+q-P_{i})^{2}$ are the virtualities of the two currents, and $s=(P_{f}+q)^{2}$. We define the Compton-like amplitude ${\mathcal{T}}$ to have factors of $i$ for each current insertion. In Fig. 1 we depict ${\mathcal{T}}$ diagrammatically, showing only the single-particle pole and the two particle contribution. Given that in general ${\mathcal{M}}_{\infty}$ 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 amplitudes in the complex $s$-plane, as well as in the plane of the other kinematic variables. The closest singularity to the region of integration is the single-particle pole. As a result, this singularity describes 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 elastic hadronic form factors as

which we depict in Fig. 1.

Having expressed the matrix element in terms of the Compton-like amplitude, we can now evaluate the finite-volume corrections. There are two classes of finite-volume effects. The first are standard errors associated with virtual particles “wrapping around the volume”. In Appendix A we show that these effects can be encoded by replacing ${\mathcal{T}}$ with its finite-volume analogue ${\mathcal{T}}_{L}$ introduced in Ref. Briceno et al. (2020). These effects lead to the standard $\mathcal{O}(e^{-mL})$ errors. The second class of finite-volume artifacts arise from the spatial separation of the two currents, $\bm{\xi}$. As explained in Ref. Briceno et al. (2018a), this class of errors scale as $\mathcal{O}(e^{-m|L-\xi|})$. Therefore, for $mL\gg 1$ the latter will be the dominant error. Here we explain how to determine the prefactor multiplying the $e^{-m|L-\xi|}$ behavior, without recourse to an effective field theory.

The finite-volume analogue of Eq. (3) can be written as the finite-volume Fourier transform of ${\mathcal{T}}_{L}$. As a result, we can write the finite-volume correction to the matrix element as

where in the second equality we made use of the arguments presented in Appendix A to replace ${\mathcal{T}}_{L}$ with ${\mathcal{T}}$ up to $\mathcal{O}(e^{-mL})$ 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 the finite-volume corrections of the matrix elements at $\boldsymbol{\xi}$ depend on the value of ${\cal M}(\boldsymbol{\xi}+L\textbf{n},\textbf{P})$ where $|\textbf{n}|\neq 0$, 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-volume effects.

Depending on the specific choice of $\boldsymbol{\xi}$, the sum over n includes finite-volume errors that are $\mathcal{O}(e^{-mL})$ or smaller, which we neglect in this analysis. In the following derivation, we choose $\boldsymbol{\xi}=\xi\hat{z}$, which is the case most typically used in calculations. With this choice, the finite-volume correction reduces to

The generalization to other geometries is straightforward, and requires including other modes in the finite volume sum. For example, if $\boldsymbol{\xi}=\frac{\xi}{\sqrt{2}}(1,1,0)$, then $\textbf{n}=(-1,0,0)$ and $\textbf{n}=(0,-1,0)$ both contribute equally. For moderate values of $\boldsymbol{\xi}$, i.e. $\xi<L/2$, these two modes would provide the leading order contributions to the finite-volume errors.

Having established the relationship between the desired matrix elements and the Compton-like amplitude, we can identify the relation between the long-range contributions to the matrix elements and the low-energy contributions to the amplitude. In particular, one expects the integral to be saturated by the small $|\textbf{q}|$ region. Thus, although the pole contribution is not, in general, a good description of ${\mathcal{M}}_{\infty}$ for small values of $\xi$, this contribution does provide a reasonable approximation of the finite-volume corrections to ${\mathcal{M}}_{\infty}$. In other words, we approximate Eq. (7) using Eq. (4) for the pole contribution,

which is our main result, Eq. (2).

This approximation can be further justified as follows. The expected form of this integral, Eq. (9), is $\mathcal{O}(e^{-m|L-\xi|})$, as was shown in Ref. Briceno et al. (2018a). The neglected terms in ${\mathcal{T}}$ will have singularities associated with a higher energy scale, $M\gg m$. In general this energy scale could correspond to excited states or thresholds. Performing a spectral decomposition for these singularities, one can immediately conclude that these contributions will lead to finite-volume corrections of the form $\mathcal{O}(e^{-M|L-\xi|})$. For moderately small values of $\xi$, these errors can be safely neglected.

Expressing the finite volume corrections in this form has several advantages over the perturbative EFT expansion applied in Ref. Briceno et al. (2018a). First and foremost, this representation is non-perturbative in the dynamics. We have made no assumption about the power counting of any underlying EFT. Instead, we have made the purely kinematic assumption that the single-particle pole is the dominant singularity. Moreover, this can be systematically improved by including further singularities, starting with the two-particle cut, the form of which was recently derived in Ref. Briceno et al. (2020). Finally, this representation extends straightforwardly to finite-volume effects for matrix elements involving initial and final states that have different momenta, which is particularly relevant for calculations of GPDs.

In practice, the main challenge to evaluating finite-volume corrections using this representation is that, formally, knowledge of the form factors over a large kinematic region is required. In fact, each integral in Eq. (9) ranges from negative to positive infinity. State-of-the-art lattice calculations generally span a small region at low momentum transfer, but calculations up to $Q^{2}\simeq 6.0\;\mathrm{GeV}^{2}$ have been carried out Lin et al. (2010); Chambers et al. (2017); Koponen et al. (2017). As a result, one might worry that this framework could be unrealistic to implement. But, as has been mentioned before, these integrals are dominated by the small q region. Thus, we envision using parametrizations of the form factors that describe the lattice QCD form factors accurately and vanish rapidly enough as $|\textbf{q}|\to\infty$. In the 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 analytically the integral shown in Eq. (9) using the techniques used in Ref. Briceno et al. (2018a). Alternatively, one can further approximate Eq. (9) by evaluating the $q_{0}$ integral and only keeping the pion pole contribution, to give

where the subscript “$\mathrm{pole}$” indicates that only the pion pole contribution has been retained.

Before we apply this formalism to a scalar model, we note that we can directly compare the form-factor representation of Eq. (9) to the corresponding expression obtained from a scalar EFT at leading order studied in Ref. Briceno et al. (2018a). To compare these results, we replace each form factor in the Compton-like amplitude, Eq. (4), by its value evaluated at $Q^{2}=0$, i.e. with the charge of the particle $iF(0)=g$, and set the initial and final momenta equal, $P=P_{i}=P_{f}$. With these replacements, the Compton-like amplitude reduces to

Inserting this into Eq. (9), we find

where we have replaced $q=iq_{E}$, $P=iP_{E}$. This is the same expression presented in Eq. (15) of Ref. Briceno et al. (2018a) for the leading order contribution.

In the following sections, we refer to this prescription as the “charge prescription” and compare the results for the corresponding finite volume effects with those estimated using parametrizations of the form factors.

One standard parametrization for form factors is the dipole form,

where $m_{F}$ is typically attributed to a resonance with the quantum numbers of the current. Such a pole, in general, violates unitarity. Instead, unless the form factor has the quantum numbers of a single particle states [e.g. the axial vector in QCD], the closest singularity of the form factor will be the first branch-cut associated with multi-particle production in the time-like region. Despite this, the dipole approximation provides a reasonable prescription of form factors in the spacelike region.

With this expression for the form factor, the integral of Eq. (14) can be expressed as

This integral can be evaluated using standard Schwinger tricks, as carried out in, for example, Ref. Briceno et al. (2018a). Instead, we choose to first evaluate the $q^{0}$ integral using Cauchy’s residue theorem. The integrand has four poles,

where we have introduced $E=\sqrt{P_{z}^{2}+m^{2}}$ as the energy of the pion and defined

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

where ${\mathcal{M}}^{2D}_{\infty,\rm pole}$ and ${\mathcal{M}}^{2D}_{\infty,\rm FF}$ are the pion and form factor pole contributions, respectively, and are explicitly defined as

Note that Eq. (20) is just the two-dimensional and model-dependent version of Eq. (10). We separate 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 evaluate the remaining integrals numerically.

In a finite volume the momentum integral is replaced a sum of lattice modes. A similar analysis to the infinite-volume case leads to an expression for the finite volume corrections,

where,

Note that, consistent with our main results, Eq. (2), we have only kept the finite-volume mode that leads to the leading finite-volume error. All other modes give errors of $\mathcal{O}(e^{-mL})$ or smaller for moderately small values of $\xi$, and are therefore neglected.

As a comparison, we also evaluate the charge prescription, considered in Ref. Briceno et al. (2018a). From Eq. (II.2), we can write this in two dimensions as

Introducing a Schwinger representation of the propagator, and carrying out the momentum and Schwinger integrals, leads to

where $K_{n}(z)$ is the modified Bessel function of the second kind.

We have presented three schemes for estimating the finite-volume error. These correspond to using 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 in Figures 2, and 3. In Figure 2 we compare the finite-volume and infinite-volume matrix elements as 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 state our choices for the combination of parameters in the upper right corner of each panel.

For each choice of parameters we calculate $\delta{\mathcal{M}}^{2D}_{L,\rm dip.}$, $\delta{\mathcal{M}}^{2D}_{L}$, and $\delta{\mathcal{M}}^{2D}_{L}$. For the finite-volume matrix element we use $\delta{\mathcal{M}}^{2D}_{L,\rm dip.}$. Both ${\mathcal{M}}^{2D}_{\infty,\rm dip.}$ and ${\mathcal{M}}^{2D}_{L,\rm dip.}$ are shown in units of ${\mathcal{M}}^{2D}_{\infty,\rm dip.}$ evaluated at $\xi=0$. We highlight that at small current separations, $\xi\sim 0$, the leading finite volume correction is $e^{-|L-\xi|m}\sim e^{-Lm}$. In other words, for small $\xi$ the dominant error is of the order of the suppressed errors. For clarity of presentation, however, we only keep the error that scales $e^{-|L-\xi|m}$ in the plots, even when $\xi=0$.

We determine the percentage finite-volume error

for each choice of parameters and framework for evaluating the finite-volume corrections. We plot this percentage as a function of $\xi$ 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 form factor, 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.1), the pion pole is the closest singularity of the Compton amplitude to the region of integration. As noted in Ref. Briceno et al. (2018a), for the choice of $mL\sim 4$, finite-volume effects can be sizable. Here we find that for a form factor with a small pole mass, i.e. $m_{F}<2.5m$, these effects can be of the order of 10%-20% even for small values of $\xi$, $\xi m\simeq 0.5$. These effects are reduced to the percent level for a slightly larger volume of $mL\sim 6$.

These larger finite-volume effects at smaller values of $m_{F}$ can be understood in terms of the singularities of the integrand. The smaller the value of $m_{F}$, the closer the pole of the form factor is to the kinematic region of integration in Eq. (20).

In Figure 3 we explore the momentum dependence of matrix elements with $m_{F}=1.5m$ and $mL=4$, which suffer the largest finite-volume effects. Once again, the plots illustrate the charge prescription consistently underestimates of the leading finite volume effects, relative to the form factor models. Furthermore, one observes that finite-volume errors are further suppressed at increasingly large momenta.