Consistency checks for two-body finite-volume matrix elements: Conserved currents and bound states

Abstract

Recently, a framework has been developed to study form factors of two-hadron states probed by an external current. The method is based on relating finite-volume matrix elements, computed using numerical lattice QCD, to the corresponding infinite-volume observables. As the formalism is complicated, it is important to provide non-trivial checks on the final results and also to explore limiting cases in which more straightforward predications may be extracted. In this work we provide examples on both fronts. First, we show that, in the case of a conserved vector current, the formalism ensures that the finite-volume matrix element of the conserved charge is volume-independent and equal to the total charge of the two-particle state. Second, we study the implications for a twoparticle bound state. We demonstrate that the infinite-volume limit reproduces the expected matrix element and derive the leading finite-volume corrections to this result for a scalar current. Finally, we provide numerical estimates for the expected size of volume effects in future lattice QCD calculations of the deuteron's scalar charge. We find that these effects completely dominate the infinite-volume result for realistic lattice volumes and that applying the present formalism, to analytically remove an infinite-series of leading volume corrections, is crucial to reliably extract the infinite-volume charge of the state. * 1 Similar outstanding puzzles are present in the heavy quark sector; see Refs.[5] and [6] for recent reviews. arXiv:1909.10357v1 [hep-lat] 23 Sep 2019The primary formal challenge arises from the fact that LQCD calculations are necessarily performed in a finite Euclidean spacetime, where the definition of asymptotic states is obscured. One of the leading methods to overcome this issue is to derive and apply non-perturbative mappings between finite-volume energies and matrix elements (directly calculable via numerical LQCD) and infinite-volume scattering and transition amplitudes. 2 This approach was first introduced by Lüscher [13,14], in seminal work relating the spectrum of two-particle states in a cubic volume with periodicty L, to the corresponding infinite-volume amplitudes. The idea has since been extended for arbitrary two-particle scattering [15][16][17][18][19][20][21][22][23] and more recently to three particles [24][25][26][27][28][29][30][31][32][33], with the latter currently limited to identical scalars (or pseudoscalars). The two-particle relations have made possible the determination of hadronic scattering amplitudes for a wide range of particle species [34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52], including energies where multiple channels are kinematically open [53][54][55][56][57][58][59][60][61]. Most recently, the first LQCD calculations to constrain three-particle interactions using excited states were performed in Refs. [62][63][64].Electroweak interactions involving scattering states can also be accessed using LQCD, via a generalization of the methods described above. The ...

Consistency checks for two-body finite-volume matrix elements. II. Perturbative systems

“…To gain confidence in this formalism, we have performed a series of consistency checks, presented in Ref. [79] together with the present article. While Ref.…”

“…While Ref. [79] is concerned with the volume-independence of the charge and the finite-volume effects on bound-state matrix elements, this work is dedicated to the 1=L expansion of the lowest-lying two-hadron scattering state.…”

“…Because the 2 þ J → 2 finite-volume mapping is complicated, it is necessary to provide various nontrivial checks on the formalism, and calculate limiting cases in which more straightforward predictions may be extracted. With this in mind, in a previous study [79] we provided two important checks on the general relations. First we demonstrated that the formalism, in conjunction with the Ward-Takahashi identity, protects the electromagnetic charge of finite-volume states.…”

Matrix elements of bound states in a finite volume

“…We refer the reader to Ref. [11] for detailed discussions of this study, where these results where first presented…”

Decay amplitudes to three hadrons from finite-volume matrix elements

“…The corresponding problem for two-particle K → ππ decays was solved in a seminal paper by Lellouch and Lüscher (LL) [32], where it was shown, for the case of a kaon at rest in the finite-volume frame, that the relation between the squared finite-volume matrix element and the magnitude squared of the infinite-volume decay amplitude is an overall multiplicative factor, the LL factor. This result was subsequently generalized in many ways [33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49], with the most important extension for our purposes being the work of refs. [42,44], in which an alternative and more general formalism was developed for calculating the LL factors for arbitrary 1 → 2 processes mediated by an external operator.…”