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Massive photons: an infrared regularization scheme for lattice QCD+QED

Standard methods for including electromagnetic interactions in lattice quantum chromodynamics calculations result in power-law finite-volume corrections to physical quantities. Removing these by extrapolation requires costly computations at multiple volumes. We introduce a photon mass to alternatively regulate the infrared, and rely on effective field theory to remove its unphysical effects. Electromagnetic modifications to the hadron spectrum are reliably estimated with a precision and cost comparable to conventional approaches that utilize multiple larger volumes. A significant overall cost advantage emerges when accounting for ensemble generation. The proposed method may benefit lattice calculations involving multiple charged hadrons, as well as quantum many-body computations with long-range Coulomb interactions.

Introduction -Approximately 95% of the visible mass of the Universe arises from the binding of quarks into nucleons by the strong interactions of quantum chromodynamics (QCD). The relative mass difference between the proton and neutron is approximately 0.07%, and is attributed to two sources of isospin symmetry breaking in the standard model, namely, differences in the down and up quark masses and their electric charges. Although these breaking effects are minute, they play an essential role in our understanding of the Universe. For example, the primordial abundance of light nuclear elements in the early Universe is exquisitely sensitive to the excess mass of the neutron compared to the proton [1,2].
Although inclusion of electromagnetism in LQCD is theoretically straightforward [24,25], it presents practical challenges due to the long-range nature of the electromagnetic (QED) interactions. Specifically, such interactions give rise to power-law finite-volume (FV) correc-tions, and their removal via extrapolation requires computationally demanding simulations performed at multiple volumes. An analytic understanding of the power-law FV effects within such setups [8,[26][27][28] has enabled reliable FV extrapolations of the single hadron spectrum.
Despite the successful application of present techniques, there are a number of reasons for considering new methods. Control over FV modifications to light nuclear binding energies seems to require particularly large volumes [26]. There are quantities in addition to the spectrum for which a precise knowledge of the QED modifications is needed, for example, corrections to hadronic matrix elements [29] and charged particle scattering [30], both of which suffer from infrared (IR) challenges. LQCD calculations are performed with multiple ultraviolet (UV) regulators, providing valuable cross-checks on the continuum extrapolation of many important quantities [31]. Multiple IR regulators can do the same for LQCD calculations that include QED, but to date, only a few other formulations have been considered [32][33][34]. Of those, only one is constructed with a local quantum field theory [33,34]. Finally, computationally efficient means of accounting for IR effects are always desirable, not just for lattice QCD+QED, but anywhere long-range Coulomb interactions are present (see, e.g., Ref. [35]).
Motivated by these considerations, we demonstrate the viability of an alternative IR regulator for lattice QCD+QED simulations: namely, the introduction of a photon mass m γ . Although a photon mass term manifestly violates gauge invariance, it maintains locality and its effects on hadronic quantities can be reliably quantified and accounted for within an effective field theory (EFT) framework. The introduction of a new scale, m γ , implies an additional extrapolation within our approach. With the aid of analytic formulas, however, we demonstrate that for the spectrum, such extrapolations can be performed at a single volume and yield results that are consistent with conventional approaches. In the remaining sections, we present the salient features of our calculation.
Analytic considerations -In continuum Euclidean spacetime, the R ξ gauge fixed action for the massive photon is given by where F µν = ∂ µ A ν − ∂ ν A µ ; throughout this study, we work in Landau gauge, corresponding to the limit ξ → 0. An Abelian theory, such as QED, with a massive vector gauge field is still perturbatively renormalizable. This well known result follows from the fact that it is possible to find a Becchi-Rouet-Stora-Tyutin (BRST) transformation that leaves the Lagrangian invariant up to a total divergence [36]. The BRST symmetry is preserved if one uses a gauge invariant UV cutoff [37], such as a spacetime lattice; thus, the renormalizability follows from the power-counting theorems for a lattice regularization [38]. We consider three forms of corrections to correlators and hadron mass differences at leading order in the finestructure constant α = e 2 /(4π). These corrections arise from either the zero mode contribution to the partition function, the presence of a finite photon mass, or FV effects. The analytic forms of these corrections are determined from an EFT for hadrons of mass M (M = m n , m p , m K + , and m K 0 ) and charge Q; the naive expansion is in m γ /M (i.e. Λ UV = M ) [39]. The EFT is a generalization of nonrelativistic QED [40] for hadrons that includes a photon mass term, and additional operators that are not constrained by gauge invariance.
treated nonperturbatively [41]. In this regime, the twopoint function for single hadrons has the form where Z is an overlap factor; the zero-mode contribution appears as x = (4παQ 2 )/(2m 2 γ L 3 T ), and vanishes as T → ∞ at fixed L and m γ .
(2) Photon mass: The hadron's electromagnetic mass shift can be determined as a function of photon mass, order by order in an expansion in powers of m γ /M . With the electromagnetic mass written as M (α, m γ ), we de- which is UV finite. These IR shifts are given by The leading-order (LO) expression is nonanalytic in the squared photon mass, whereas the next-to-leadingorder (N LO) expression is analytic but arises from both loops and local contributions [42]; the next-to-next-toleading-order ( . The latter two orders are accompanied by coefficients not fixed by the hadron charge. (3) Finite volume: The effects of FV can similarly be calculated using a nonrelativistive QED approach. This is a finite photon mass generalization of that pursued by Refs. [26,27]. The FV corrections to the electromagnetic mass are written as , and for charged hadrons are given up to N LO by where I n (z) = 1 and ν ∈ Z 3 . By contrast, the leading nonvanishing correction for neutral baryons [mesons] appears at N 2 LO [next-to-next-to-next-leading-order (N 3 LO)]. Because the zero mode of the temporal photon is treated exactly in Eq. 2, the FV corrections are calculated with this mode removed-a manifestation of which is the subtracted term appearing at LO. Lattice parameters and ensembles -Electroquenched numerical calculations of the hadron spectrum were performed using a modified version of the Chroma software suite [43]. Studies were performed using dynamical SU(3) flavor symmetric isotropic QCD gauge field configurations generated using a tadpole-improved Lüscher-Weisz gauge action and clover fermion action. The configurations correspond to a single lattice spacing a = 0.1453(16) fm, three spatial extents: L ∼ 3.48 fm, 4.64 fm and 6.96 fm, and temporal extents T > L. The pion (kaon) and nucleon masses in physical units arē m π =m K = 807.0(9.1) MeV andm n = 1.634(18) GeV, respectively. This choice of masses ensures that the only appreciable FV corrections to hadron masses are those arising from QED effects. The QCD ensembles used in this work comprise 956 (L/a = 24, T /a = 48), 515 (L/a = 32, T /a = 48) and 342 (L/a = 48, T /a = 64) configurations and are a subset of those described in Ref. [44]; further details regarding the ensembles, lattice action and parameters can be found there.
The uncorrelated photon field configurations A µ were generated using two different lattice actions: a conventional massless Coulomb gauge-fixed action with the zero mode removed [11,24,25] (QED T L ) [45], and a naive lattice discretized form of Eq. 1 (QED M ), where the derivatives are replaced by finite differences. Note that in Euclidean space, Landau gauge is a complete gauge-fixing condition, and therefore in the latter case, the path integration over nonzero modes is well defined in the m γ → 0 limit. The photon mass val-ues considered in this work are given by m γ /m π ∈ [1/14, 1/7, 1/4, 1/3, 5/12, 1/2, 7/12, 1]. In both cases, results were obtained by computing correlation functions on QCD+QED gauge configurations generated by postmultiplying each QCD configuration by a single e ieQqAµ , where Q u = 2/3, Q d = Q s = −1/3. In the electroquenched approximation with SU(3) flavor symmetry, isospin splittings have missing contributions that are O(α 2 ), and therefore negligible for this study.
In the electroquenched theory, the fine structure coupling does not renormalize and therefore we take it to be equal to its experimental value α −1 = 137.036 . . ., measured in the Thomson limit. The presence of electromagnetic interactions demands renormalization of the valence bare quark masses m q , however. Since our lattice regulator breaks chiral symmetry, this leads to an additive shift in the quark mass. We tune the valence quark masses so that, in the presence of electromagnetic interactions, the neutralqq meson mass m qq obtained from the connected part of theqq correlation function is sufficiently close to the pion (kaon) massm π . For our electroquenched calculation, this choice of renormalization is robust but the quark mass renormalization in the full QCD+QED does not allow for a unique separation of the QED and QCD effects [46]. All measurements were performed using valence quark masses am u = −0.25501 and am d = am s = −0.24750 (the QCD bare quark mass is am q = −0.2450); the resulting mistuning for the charge neutral mesons was ∆m qq /m π 0.1% for all values of m γ /m π ≤ 1, where ∆m qq ≡ m qq −m π .
The mistuning from strong isospin breaking can be estimated using chiral symmetry. For the kaon, one finds while the nucleon correction is given by We can estimate the parameter α d−u from the LQCD determination of the m d −m u contribution to the nucleon mass splitting [2][3][4][5][6][7][8] and find ∆m p−n /m n 0.0002. In both cases, mistuning is a potentially sizable correction to our results, which affects both the QED T L and QED M determinations. Although a precise quark mass tuning is required for practical applications, it is not needed in the present proof-of-principle study [47].
Analysis and results -Shell-shell and shell-point correlation functions were estimated using a single measurement per configuration, with a randomly chosen spacetime source location. Following Ref. [9], we average observables over +e and −e on each configuration in order to exactly cancel off the O(e) contributions to statistical noise. Mass differences due to electromagnetic effects can be determined from the late-time dependence of the single hadron correlation functions C A (τ ) and C B (τ ), by studying the plateau region of an effective mass dif- . By exploiting the correlations between A and B, we are able extract a clear signal for the mass difference. For the nucleons, we consider a generalized effective mass formula of the form: which neglects the backward propagation of states on a lattice of finite temporal extent T . For mesons, we account for the backward propagating state by considering a generalized effective mass formula of the form: where h(τ, a) = xa(a − T + 2τ ) + log[C(τ + a)/C(τ )]. Both formulas treat the zero mode of the temporal photon field appearing in Eq. 2 nonperturbatively (for neutral hadrons x = 0 and these expressions reduce to their conventional forms). Although this contribution is negligible compared to the hadron masses, for the lattice parameters considered it can be comparable in magnitude to the mass differences we wish to extract. Fig. 1 provides an explicit example of the behavior of ∆M eff (τ ) for the kaon mass splitting, computed both with and without the zero-mode contribution accounted for.
Mass differences were determined for all volumes and photon masses via a correlated constant least-squares fit to ∆M eff in the plateau region, as demonstrated in Fig. 1. An analogous determination from exponential fits to a ratio of correlation functions yielded consistent results. Systematic uncertainties were estimated by varying the region over which fits were performed, and all uncertainties were added in quadrature. The extracted mass shifts were subsequently extrapolated to vanishing photon mass and/or the infinite-volume limit using the fit formula: where K γ and K L indicate the order of each extrapolation. In the case of mass splittings, an appropriate linear combination of mass shift formulas was used. Note that for the QED T L extrapolations, K γ = 0; the appropriate FV formulas for δ L M N k LO retain T dependence, and may be found in Ref. [8].
We carry out two independent analyses to test the viability of our proposal: (1) an infinite-volume extrapolation of QED T L induced mass differences, as is conventionally performed, and (2) an m γ → 0 extrapolation of QED M induced mass differences using data at a single FV, but after having first removed the lowest order FV contributions δ L M . Both types of extrapolation were performed using Eq. 10, noting that many of the lowestorder contributions are fixed by theory. Results for the first analysis, using all three volumes, are provided in Table I and representative fits are shown in Fig. 2. Results for the second analysis on the smallest volume are provided in Table II for comparison, and shown in Fig. 3. Analogous QED M extrapolations, performed at each of the three volumes, are summarized in Fig. 4 and are consistent not only with each other, but also the QED T L extrapolations. In all cases, we find that the numerical and theoretical mass corrections are in excellent agreement down to at least m γ L ∼ 1.
The most computationally demanding part of our calculation involves multiple inversions of the Dirac operator. Assuming, conservatively, a linear scaling with spacetime volume, the total inversion cost for L/a = 32 is 515/956 × (32/24) 3 ∼ 1.3 times greater than that of L/a = 24. By comparison, the L/a = 48 inversion cost is ∼ 3.8 times greater. The L/a = 24 extrapolations using m γ /m π ∈ [1/4, 1/2] data, provided in Table II,  consistent with those using all values of m γ . The results are also consistent with the QED T L extrapolation using three volumes, provided in Table I, but required only 4/5 the computational cost. We therefore conclude that for the same precision and accuracy, the numerical cost of our QED M calculation of the mass splittings is comparable to or less than that of QED T L .
Conclusion -This work demonstrates that it is possible to reliably estimate infinite-volume hadron mass differences induced by electromagnetism on a single lattice volume with QED M . Conservatively, the pionless EFT employed in this work is valid for m γ 2m π and m π L 4. Provided these inequalities are satisfied, the analytic expressions obtained for the mass shift are valid up to O(m 3 γ /M 3 , α 2 ), and are independent of the pion mass; from our numerics, it appears that this order is sufficient to obtain reliable extrapolations of the mass shifts in the regime m γ /m π 1 and m γ L 1.
On preexisting lattice configurations, and for equal computational cost, we obtain an equally precise uncertainty in extrapolated differences as compared to the traditional method. This cost comparison does not account for the significant overhead of generating the configurations in the first place. The results of our analysis pave the way for a more complete treatment of QED corrections using this approach. When considering more involved LQCD calculations, such as charged-particle scattering [30], our method provides a mass gap to produce a photon, thus increasing the range of energy for which the standard Lüscher method [48,49] for obtaining the scattering phase shift can be employed. It will be interesting to explore these types of calculations, and also to use our method with chiral fermions, which do not suffer from additive quark mass renormalization. Finally, it would be interesting to see if our method of screened interactions coupled with analytic extrapolation techniques is of benefit to quantum many-body calculations.
We would like to thank W. Detmold, R. Edwards,