Consistency checks for two-body finite-volume matrix elements: II. Perturbative systems
JJLAB-THY-19-3113CERN-TH-2020-015
Consistency checks for two-body finite-volume matrix elements:II. Perturbative systems
Ra´ul A. Brice˜no,
1, 2, ∗ Maxwell T. Hansen, † and Andrew W. Jackura
1, 2, ‡ Thomas Jefferson National Accelerator Facility, 12000 Jefferson Avenue, Newport News, VA 23606, USA Department of Physics, Old Dominion University, Norfolk, Virginia 23529, USA Theoretical Physics Department, CERN, 1211 Geneva 23, Switzerland (Dated: February 4, 2020)Using the general formalism presented in Refs. [1, 2], we study the finite-volume effects for the + J → matrix element of an external current coupled to a two-particle state of identical scalarswith perturbative interactions. Working in a finite cubic volume with periodicity L , we derive a1 /L expansion of the matrix element through O (1 /L ) and find that it is governed by two universalcurrent-dependent parameters, the scalar charge and the threshold two-particle form factor. Weconfirm the result through a numerical study of the general formalism and additionally through anindependent perturbative calculation. We further demonstrate a consistency with the Feynman-Hellmann theorem, which can be used to relate the 1 /L expansions of the ground-state energy andmatrix element. The latter gives a simple insight into why the leading volume corrections to thematrix element have the same scaling as those in the energy, 1 /L , in contradiction to earlier work,which found a 1 /L contribution to the matrix element. We show here that such a term arises atintermediate stages in the perturbative calculation, but cancels in the final result. I. INTRODUCTION
Understanding the emergence of hadrons from the interactions of their constituent quarks and gluons has remaineda challenge, even many decades after the formulation of the fundamental theory of quantum chromodynamics (QCD).In recent years, significant progress has been made in determining the properties single-hadron ground states via nu-merical calculations using lattice QCD [3–5]. Most states, however, manifest as resonances in multi-hadron scatteringprocesses, and are rigorously defined only as poles in analytically continued scattering amplitudes. In addition, whilehadronic amplitudes allow the extraction of masses, widths and couplings, to constrain structural, information includ-ing charge radii or parton distribution functions, one must calculate and analytically continue electroweak transitionamplitudes, in which an external current is coupled to the multi-hadron scattering states.Determining scattering and transition amplitudes in lattice QCD calculations is complicated by the fact thatthe latter are necessarily performed in a finite Euclidean spacetime, where one cannot directly construct asymptoticstates. Presently, the most systematic method to overcome this issue is to derive and apply non-perturbative mappingsbetween finite-volume spectra and matrix elements (which are directly calculable) and infinite-volume scattering andtransition amplitudes. This methodology was first introduced by L¨uscher [6, 7], in the context of relating the finite-volume energies of two pions, in a cubic periodic volume of length L , to the elastic → scattering amplitude.Within this framework, on-shell intermediate states yield power-law finite-volume corrections, O (1 /L n ), while thecontribution from off-shell quantities is exponentially suppressed, scaling as e − m π L , where m π is the pion mass.For sufficiently large box sizes, the second class of corrections can be neglected, giving a systematic path towardsextracting scattering observables. In the past decades, L¨uscher’s formalism has been extended to include non-zeromomentum in the finite-volume frame as well as coupled two-particle channels and particles with spin [8–16]. LatticeQCD applications of the methodology have proven highly effective in the determination of two-hadron bound andresonant states [17–35], including those at energies where multiple channels are kinematically open [36–44]. Thissuccess in the two-hadron sector has also motivated the extension to → and → scattering [45–54], with thefirst lattice QCD computations of the 3 π + system published last year [55–58]. Extensions of these finite-volume mappings have also been derived to extract electroweak transition amplitudesfrom lattice QCD calculations. As first shown in Ref. [61] in the context of K → ππ decays, finite-volume matrixelements are related to electroweak transition amplitudes through a mapping that depends on both the box sizeand the scattering amplitude of the multi-particle final state. This has been generalized to arbitrary + J → amplitudes [9, 12, 62–65] and applied in lattice QCD studies of K → ππ decay [66–69] as well as γ (cid:63) → ππ [70, 71] ∗ e-mail: [email protected] † e-mail: [email protected] ‡ e-mail: [email protected] For recent reviews on this topic we point the reader to Refs. [59, 60]. a r X i v : . [ h e p - l a t ] J a n and πγ (cid:63) → ππ [72–74] transition amplitudes. The ideas have further been generalized to + J → + J matrixelements, with long range-contributions form multi-particle intermediate states [75–77]. Most recently, the formalismfor + J → electroweak transition amplitudes has been developed [1, 2], generalizing previous studies based infixed-order calculations in a specific effective field theory [13, 78]. As compared to the + J → methodology, therelations in Refs. [1, 2] are more complicated due to the presence of additional finite-volume effects from triangle-diagram topologies.Because the + J → finite-volume mapping is complicated, it is necessary to provide various non-trivial checks onthe formalism, and calculate limiting cases in which more straightforward predictions may be extracted. With this inmind, in a previous study [79] we provided two important checks on the general relations. First we demonstrated thatthe formalism, in conjunction with the Ward-Takahashi identity, protects the electromagnetic charge of finite-volumestates. Though obvious from the general properties of the theory, in the context of our mapping this required exactcancellations between various combinations of finite-volume functions and thus provided a clear demonstration thatall effects have been properly incorporated. We then explored the bound-state limit of matrix elements and recoveredthe expected result, that finite-volume corrections scale as e − κL for large L , where κ is the binding momentum ofthe two-particle bound state. As is already well known for finite-volume bound-state energies [80–82], in the caseof a shallow bound state, κL (cid:28) m π L , many terms in the large-volume expansion (scaling as powers of e − κL ) givesignificant contributions. As a result, we find it is crucial to consider the all-orders framework of Refs. [1, 2] to extractreliable predictions of bound-state form factors.In the present article, we continue our series of consistency checks by studying the 1 /L expansion of the finite-volume matrix element, (cid:104) E , L | J (0) | E , L (cid:105) , where | E , L (cid:105) is the ground state of a perturbative two-scalar systemand J ( x ) is a scalar current. In contrast to our previous check, here we restrict attention to finite-volume scatteringstates meaning that the energy, E ( L ), approaches the two-scalar threshold as L → ∞ . For finite L , both E ( L )and (cid:104) E , L | J (0) | E , L (cid:105) admit 1 /L expansions with coefficients depending on the geometry of the finite volume aswell as infinite-volume parameters governing the interactions. In the case of the energy, the expansion is well-knownand has been studied, through various orders in 1 /L , in Refs. [6, 83–88]. An analogous study for matrix elementswas performed in Ref. [89], in which non-relativistic quantum mechanics is used to expand an n -particle ground-statematrix element through 1 /L .In this work, we derive the 1 /L expansion of L (cid:104) E , L | J (0) | E , L (cid:105) through O (1 /L ). We compare our result withRef. [89] and find significant disagreement, including a difference in the behavior of the leading volume correction,with our result scaling as 1 /L and that of the earlier work as 1 /L . To confirm our own determination, we cross-checkboth through a numerical study of the general formalism and through an independent perturbative calculation. Inaddition we use the Feynman-Hellmann theorem to relate (cid:104) E , L | J (0) | E , L (cid:105) to a mass derivative of E ( L ) and showthat this enforces certain common features between the two expansions, e.g. that both start at O (1 /L ). Finally, inour perturbative cross-check, we identify classes of terms that, if omitted, lead to the behavior reported by Ref. [89].The remainder of this article is organized as follows: We first review the 1 /L expansion of E ( L ) in Sec. II A, basedin the L¨uscher scattering formalism. Then, in Sec. II B, we derive the corresponding expansion of (cid:104) E , L | J (0) | E , L (cid:105) using the relations of Refs. [1, 2], and also describe how the results for the energy and matrix element are relatedvia the Feynman-Hellman theorem. The main expressions are succinctly summarized in Eqs. (1) and (3) below. InSec. II C we provide a numerical check of our expansion against the all-orders formalism and in II D we provide adetailed comparison of our result with Ref. [89]. Section III then describes the perturbative confirmation of our resultsand gives additional insight into the discrepancy with Ref. [89]. We briefly conclude in Sec. IV. We also include anappendix to derive one of the technical results required for Sec. II B, concerning the imaginary part of the trianglediagram entering the infinite-volume + J → matrix element. II. THRESHOLD EXPANSION
In this section we review the 1 /L expansion of the ground-state two-particle energy, E ( L ), and then turn to themain result of this work, the corresponding expansion of the finite-volume matrix element. The expressions holdfor a generic, relativistic quantum field theory in a periodic, cubic spatial volume with side-length L , provided thelowest-lying two-particle state consists of two identical scalars with mass m . We additionally require that the center-of-momentum frame (CMF) and finite-volume frame coincide, i.e. that the particles have zero momentum, P = , inthe finite volume.For convenience we summarize the two key results here:1. In Sec. II A we review the well-known expansion of the two-particle energy [6, 83–88] E ( L ) = 2 m + 4 πamL (cid:20) − I aπL + (cid:0) I − J (cid:1) a π L + (cid:16) π ra − π m a − (cid:0) I − IJ + K (cid:1) (cid:17) a π L (cid:21) + O (1 /L ) , (1)where a and r are the scattering length and effective range respectively, defined in Eq. (7) below, and m is thephysical mass. The three geometric constants I = − .
913 633 , J = 16 .
532 316 , K = 8 .
401 924 , (2)are defined and evaluated to high precision in Refs. [6, 84].2. In Sec. II B we show that the ground-state matrix element of a scalar current at zero momentum transfer admitsan analogous expansion L (cid:104) E , L | J (0) | E , L (cid:105) = gm × (cid:20) − π m a (cid:18)(cid:0) − ma T (cid:1) a π L − (cid:0) − ma T (cid:1) I a π L + (cid:0) − ma T (cid:1)(cid:0) I − J (cid:1) a π L (cid:19)(cid:21) + O (1 /L ) , (3)where (cid:104) E , L | E , L (cid:105) = 1, g is the scalar charge of a single particle under the scalar current, J ( x ), and T ≡ mr + 64 πm g F . (4)Here F is the threshold form factor, defined in Eqs. (24) and (27) below via a straightforward relation to theinfinite-volume + J → transition amplitude. A. Finite-volume energies
For a range of CMF energies from 2 m up to the first inelastic threshold, the finite-volume spectrum is describedby the L¨uscher quantization condition [6], which is exact up to exponentially suppressed L dependence of the form e − mL . The result of Ref. [6] relates the discrete energies, E n ( L ), to the physical scattering amplitude, by expressingthe former as roots of a determinant in the space of two-particle angular momenta. The effects of higher angularmomenta first appear in powers of 1 /L well-beyond the orders that we control, so that for our purposes it is sufficientto consider the truncated quantization condition M − ( E n ) = − F ( E n , L ) , (5)where F is a known finite-volume function, and M is the S -wave scattering amplitude, related to the S -wave scatteringphase shift, δ , via M ( E ) ≡ πEq cot δ ( q ) − iq . (6)Here q is the relative momentum of the two particles in the CMF, defined via E ≡ (cid:112) m + q . We recall also that q cot δ ( q ) admits a convergent expansion about two-particle threshold, referred to as the effective range expansion: q cot δ = − a + 12 rq + O ( q ) , (7)where a is the scattering length and r the effective range.The finite-volume function, F , can be expressed in many forms, all equivalent up to exponentially suppressedcorrections (see, e.g., Refs. [1, 8, 9, 11, 15]). We begin with the following definition, F ( E, L ) = 12 lim Λ →∞ (cid:20) Λ (cid:88) ˆ k (cid:21) ω k E ( E − ω k + i(cid:15) ) , (8) ≡ i q πE + F pv ( E, L ) , (9) The leading corrections from non-trivial angular momenta enter via a finite-volume function denoted by F , ( E, L ) and defined, forexample, in Ref. [9]. This quantity encodes the mixing of the S -wave ( (cid:96) = 0) and the G -wave ( (cid:96) = 4) due to the reduced rotationalsymmetry of the cubic volume. The F , -correction enters as an additive term in Eq. (5), scaling as F , ( E, L ) = O (1 /L ). Thecorresponding G -wave correction to the ground-state energy, E ( L ), then scales as 1 /L and is therefore five orders beyond the 1 /L contributions that we keep. where (cid:20) Λ (cid:88) ˆ k (cid:21) ≡ L | k | < Λ (cid:88) k ∈ (2 π/L ) Z − ˆ d k (2 π ) Θ(Λ − | k | ) , (10)and Θ is the usual Heaviside step function, included here to implement the hard cutoff. In Eq. (9) we have separated F into its real and imaginary parts, denoting the former by F pv . The subscript “pv” stands for principal value,indicating that the real part of F is equivalently given by taking the original definition and replacing the i(cid:15) poleprescription in the integral with a principal value. Separating out the imaginary part is useful as it exactly cancelsthe imaginary part of the inverse scattering amplitude [see Eq. (6)]. It follows that Eq. (5) is exactly equivalent tothe real equation q n cot δ ( q n ) = − πE n F pv ( E n , L ) , (11)where q n ≡ E n / − m .From these relations, it is straightforward to determine the 1 /L expansion of the lowest lying two-particle energy,denoted E ( L ) and defined as smallest hamiltonian eigenvalue satisfying lim L →∞ E ( L ) = 2 m . The infinite-volumevalue motivates the definitions ∆ E ( L ) ≡ E ( L ) − m ≡ m (cid:20)(cid:114) q ( L ) m − (cid:21) , (12)= 4 πamL ∞ (cid:88) j = − γ j (cid:16) aπL (cid:17) j . (13)Here the first line serves to define ∆ E (the distance from the finite-volume state to the infinite-volume threshold)and its relation to q . Equation (13) introduces notation for a generic power series in 1 /L , and the 1 /L prefactor,as well as the factors of scattering length, simplify the form of γ j in the final result. The aim of this subsection is toreview the determination of the coefficients γ j , defining the large-volume expansion of E ( L ).The final non-trivial ingredient is the threshold expansion of F pv , which can be written as F pv ( E, L ) = 14 Eq L (cid:20) − ∞ (cid:88) j =1 (cid:18) qL π (cid:19) j I j (cid:21) , (14)where I j are numerical constants characterizing the cubic geometry, I j = lim Λ →∞ (cid:20) n< Λ (cid:88) n (cid:54) = − π ˆ Λ0 d n n (cid:21) n , j = 1 , (cid:88) n (cid:54) = n j , j ≥ , (15)with n = | n | and with the sums running over all non-zero integer vectors, n ∈ Z / { } . The coefficients, γ j , can now be determined in a two step procedure: First, one substitutes the effective rangeexpansion, Eq. (7), and the expansion of F pv , Eq. (14), into the real version of the quantization condition, Eq. (11).In this way, both sides of the equation are expressed as polynomials in q or, via the relation q = E / − m , aspolynomials in E . Second, substituting the 1 /L expansion of ∆ E ( L ) given in Eq. (12), one reaches an equalityinvolving two series of 1 /L . The result can only be satisfied for all L by tuning the values of γ j to enforce the equalityof all coefficients. One finds γ − = γ − = 0, meaning that ∆ E ( L ) scales as 1 /L . The first few non-trivial coefficientsare then given by [6, 83–88] γ = 1 , γ = −I , γ = I − J ,γ = − (cid:0) I − IJ + K (cid:1) + 2 π ra − π m a , (16)where we have adopted the notation of Ref. [84]: I = I , I = J , I = K . This result is summarized in Eq. (1). A convenient method to evaluate these is given in Ref. [87], in which an exponential damping function is used to accelerate convergence. High-precision numerical determinations of these constants can also be found in that reference.
B. Finite-volume matrix elements
We now turn to the 1 /L expansion of the finite-volume + J → matrix element, where J is a generic scalarcurrent density. As above, we assume that the total momentum vanishes in the finite-volume frame, and we truncateall infinite-volume amplitudes to the S wave. Then the formalism presented in Refs. [1, 2] simplifies to L (cid:104) E (cid:48) n , L | J (0) | E n , L (cid:105) = W L, df ( E (cid:48) n , E n , L ) (cid:112) R ( E (cid:48) n , L ) R ( E n , L ) , (17)where | E n , L (cid:105) is the n th finite-volume excited state, normalized to unity. As with the L¨uscher quantization condition,this relation holds up to the first inelastic threshold and is exact up to exponentially suppressed corrections of theform e − mL .The right-hand side is composed of the Lellouch-L¨uscher factor, R , defined via R ( E n , L ) ≡ lim E → E n E − E n F − ( E, L ) + M ( E ) , (18)= −M − ( E n ) lim E → E n (cid:20) ∂∂E (cid:16) F pv ( E, L ) + q πE cot δ (cid:17)(cid:21) − , (19)and W L, df , a finite-volume quantity that contains the infinite-volume + J → transition amplitude W L, df ( E (cid:48) , E, L ) = W df ( E (cid:48) , E ) + f ( Q ) M ( E (cid:48) ) G ( E (cid:48) , E, L ) M ( E ) . (20)Here f ( Q ) is the single-particle form factor with momentum transfer Q ≡ − ( E (cid:48) − E ) . In the forward limit thisbecomes the scalar charge, denoted by g ≡ f (0). In the second term in Eq. (20) we have also introduced G , adouble-pole finite-volume function given explicitly by G ( E (cid:48) , E, L ) = lim Λ →∞ (cid:20) Λ (cid:88) ˆ k (cid:21) ω k E (cid:48) ( E (cid:48) − ω k + i(cid:15) ) 1 E ( E − ω k + i(cid:15) ) . (21)The final ingredient in the definition of W L, df is the first term on the right-hand side of Eq. (20), the infinite-volumedivergence-free transition amplitude, W df . Here ‘divergence free’ refers to the subtraction of diagrams where thecurrent probes one of the external legs. (See Refs. [1, 2] for a detailed discussion of the relation between W df andinfinite-volume matrix elements.) Though the long-distance poles have been removed, W df does still contain twoother types of kinematic singularities: ( i ) threshold singularities arising from the two-particle initial and final stateinteractions, analogous to those in the standard → scattering amplitude, and ( ii ) anomalous triangle singularities,which occur at the boundaries of the kinematic region where all intermediate states of the triangle topology can goon shell.For the remainder of this article, we focus on the special case where E (cid:48) = E , i.e. we evaluate the matrix element atzero momentum transfer. One of the many simplifying features of this limit is that the anomalous triangle singularities,type ( ii ) above, then only arise at threshold and are completely given by the imaginary part of the integral defining G ( E, E, L ). Since the sum in G is pure real, this is equal (up to a minus) toIm G ( E, E, L ) = Im lim Λ →∞ (cid:20) Λ (cid:88) ˆ k (cid:21) ω k E ( E − ω k + i(cid:15) ) = − πEq , (22)where the final equality is proven in Appendix A. It will prove convenient in the following to also introduce notationfor the real part of G . We define G pv ( E, L ) ≡ Re G ( E, E, L ) ≡ lim Λ →∞ (cid:20) Λ (cid:88) ˆ k (cid:21) ω k E ( E − ω k + i(cid:15) ) + i πEq . (23)One can remove both the usual threshold singularities ( ∝ q ) and the threshold triangle singularities ( ∝ /q ) byintroducing a zero-momentum-transfer two-hadron form factor, F ( E ), related to W df ( E, E ) via W df ( E, E ) = M ( E ) (cid:20) F ( E ) + i g πEq (cid:21) M ( E ) . (24)Here the S -wave scattering amplitude, M , removes the initial- and final-state two-particle interactions so that F doesnot contain the threshold cusp appearing in M and W df . The second term, taken directly from Eq. (22), then removesthe remaining singular behavior.This completes our general discussion of the building blocks entering Eq. (17). Substituting the finite- and infinite-volume functions at zero momentum transfer into the general result, we deduce an all-orders expression for thefinite-volume matrix element in the S -wave only approximation L (cid:104) E n , L | J | E n , L (cid:105) = F ( E n ) + g G pv ( E n , L ) − ∂∂E (cid:16) F pv ( E, L ) + 116 πE q cot δ (cid:17)(cid:12)(cid:12)(cid:12) E = E n . (25)Given finite-volume energies and matrix elements, e.g. computed from lattice QCD, Eq. (25) can be used to solve forthe unknown F . Together with the scattering amplitude, M , and the single-particle charge, g , this yields a predictionfor the full + J → transition amplitude in the kinematic region around the zero-momentum-transfer point.In the present article, however, our aim is to analytically study the L dependence of the threshold matrix el-ement, (cid:104) E , L | J | E , L (cid:105) , by expanding the right-hand side of Eq. (25) in powers of 1 /L . Specifically, we expand L (cid:104) E , L | J | E , L (cid:105) through O ( L − ), corresponding to four non-trivial orders in the matrix element’s large volumebehavior. To set up the calculation we introduce an expression analogous to Eq. (13) above L (cid:104) E , L | J | E , L (cid:105) = gm ∞ (cid:88) j =0 β j (cid:16) aπL (cid:17) j , (26)where, as before, we have removed various factors to simplify the expressions of β j that arise in our final result.We next expand all quantities entering Eq. (25) about E = 4 m , equivalently about q = 0, beginning with F ( E ) ≡ F + O ( q ) . (27)We will see below that F first contributes to L (cid:104) E , L | J | E , L (cid:105) at O (1 /L ), i.e. to β , implying that the O ( q )corrections first enter at O (1 /L ) ( β ) and are beyond the order we work. Next, the finite-volume G function hasa similar expansion to that given in Eq. (14), but with the leading L scaling enhanced by the ( E − m ) ∼ q polein the summand. Using Eq. (A2) in the appendix, one can readily recover the full expansion through a derivativerelation to F pv : G pv ( E, L ) = − E ∂∂q (cid:2) EF pv ( E, L ) (cid:3) , (28)= 14 Eq L (cid:20) ∞ (cid:88) j =1 ( j − (cid:18) qL π (cid:19) j I j (cid:21) . (29)Note that, when evaluated at the finite-volume ground state energy, q = O (1 /L ) implying G pv = O ( L ). In Eq. (25)this leads to an O ( L ) scaling of the numerator, which is, however, canceled by the same scaling in the denominatorso that L (cid:104) E , L | J | E , L (cid:105) is finite as L → ∞ .To conclude the exercise we rewrite the denominator of Eq. (25) as a derivative with respect to q and expand theremaining functions to reach L (cid:104) E , L | J | E , L (cid:105) = F + g Eq L (cid:20) ∞ (cid:88) j =1 ( j − (cid:18) qL π (cid:19) j I j (cid:21) − π ∂∂q (cid:18) πq L (cid:20) − ∞ (cid:88) j =1 (cid:18) qL π (cid:19) j I j (cid:21) + 12 rq (cid:19) + O (1 /L ) , (30)where it is understood that q is set to q ≡ E ( L ) / − m everywhere on the right-hand side. In the denominator wehave also substituted the threshold expansion of q cot δ , through the order we require, and used the fact that the q derivative annihilates the constant term. Expanding this expression and matching to Eq. (26) yields the main resultof this work: β = 1, β = β = 0, β = − π m a + 2 π ma T , (31) β = (cid:20) π m a − π ma T (cid:21) I , (32) β = (cid:20) − π m a + 6 π ma T (cid:21) ( I − J ) , (33) -0.2-0.10.0 51050-0.66-0.64-0.62 10 100 -0.2-0.10.0 51050-0.64-0.62-0.60 10 100 0.00 . mL (cid:16) aπL (cid:17) . . mL (cid:16) aπL (cid:17) ( m L ) × M J ( m L ) × M J (a) mr = 0, m F = 0 (b) mr = 0 . m F = 0 (c) mr = 0 . m F = 0 . mL ) × M J ( L ) (top row) and ( mL ) × M J ( L ) (bottom row) vs mL , with M J ( L ) as defined in Eqs. (25)and (36). In each panel the solid line shows ( mL ) n × M J ( L ) and the horizontal dashed line shows the expected asymptote,predicted by the analytic 1 /L expansion. All plots are evaluated at fixed g/m = 1 . ma = 0 .
1, with mr and m F varied,as indicated in the labels and in the main text. where T is a combination of F and the effective range, r , defined in Eq. (4) above. The leading order term, β ,represents a pure single-hadron contribution which arises from the G function, while the two-hadron form-factor F is sub-leading, along with relativistic corrections from the single-hadron term.We close the subsection with a simple argument that explains the absence of 1 /L and 1 /L terms, and also givesinsight into the pattern of geometric constants entering β , β and β . If we work in a generic scalar field theory with thefield ϕ ( x ) creating a single particle state, one possibility is to choose J ( x ) ∝ ϕ ( x ) for the scalar current. Then, by theFeynman-Hellman theorem, the finite-volume matrix element is proportional to a mass derivative of the ground-stateenergy. Given the result E ( L ) = 2 m + O (1 /L ), this immediately implies that L (cid:104) E , L |J (0) | E , L (cid:105) = g/m + O (1 /L ),i.e. the absence of 1 /L and 1 /L terms in the energy implies the same must hold for the matrix element. Here thefactor of L , multiplying the matrix element, is required because the contribution appearing in the Hamiltonian isnot directly J ( x ) but rather ´ L d x J ( x ).Indeed, the full result can be derived from the Feynman-Hellman theorem via the relation L (cid:104) E , L |J (0) | E , L (cid:105) = g d E ( L )d m . (34)The derivative corresponds to varying the physical mass by varying the bare mass in the Lagrangian, while keepingall other bare parameters fixed. As a result, all other physical quantities predicted by the Lagrangian inherit an m dependence, while L remains constant. Through the order we work one only requires an expression for the m derivative of the scattering length. Deriving this explicitly goes beyond the scope of this article. We only note thatthe result d a d m = 12 a r + 32 πma g F = a m T , (35)leads to a perfect correspondence between the 1 /L expansions of E ( L ) and L (cid:104) E , L |J (0) | E , L (cid:105) , as can be readilyseen from Eqs. (1) and (3). C. Numerical confirmation
To verify our strategy for expanding the general formalism in powers of 1 /L , here we numerically study the difference M J ( L ) ≡ mL g (cid:104) E , L | J | E , L (cid:105) − β , (36)as a function of mL , using Eq. (25) to evaluate the finite-volume matrix element. By choosing various values of ma , mr , g/m and m F , we are able to confirm numerically that our analytic 1 /L expansion is consistent with the generalformalism.In Fig. 1 we show the behavior of ( mL ) × M J ( L ) (top row) and ( mL ) × M J ( L ) (bottom row) vs mL , with ma = 0 . g/m = 1, and various choices of mr and m F . In the first column we take mr = 0 and m F = 0,in the second mr = 0 .
25 and m F = 0, and in the third mr = 0 .
25 and m F = 0 .
5. The plots of the top rowindicate that, as mL → ∞ , ( mL ) × M J ( L ) asymptotes to zero, confirming the result β = 0. This behavior isunchanged by varying the values of mr and m F , as shown. The plots of the bottom row show that ( mL ) × M J ( L )asymptotes to a non-zero value corresponding to β ( ma/π ) in the expansion. For the numerical values considered, β ( ma/π ) = − . , − . , .
7, for the first, second, and third columns, respectively. The numerical results againconfirm that there is no contribution at O ( L − ) and that the first non-trivial correction, at O ( L − ), is in agreementwith the analytic expression for the threshold expansion. We have also checked that the large L numerical result forthe O ( L − ) coincides with our expansion. D. Comparison with Ref. [89]
In this section we compare our result, summarized in Eq. (3), to that of Ref. [89], and find clear discrepancies. Theearlier work uses a non-relativistic effective field theory to calculate the 1 /L expansion for n + J → n ground statefinite-volume matrix elements, where n is any number of identical scalar particles. For n = 2 the result of Ref. [89]becomes L (cid:104) E , L | J | E , L (cid:105) (Ref. [89]) = 2 α + 2 α a π L J + α L + 4 α a π L ( K − IJ ) − α aπL I + 2 α a π L C + O (1 /L ) , (37)where α and α are couplings relating the scalar current to creation and annihilation operators and C is anothergeometric constant, related to those specifically defined in Ref. [89] via C = 3 I J − IK − J + 3 L . The discrepancyof this result with Eq. (3) is immediately clear, in particular due to the 1 /L term. As already described at the endof Sec. II B, the Feynman-Hellmann theorem implies that a 1 /L correction to the matrix element requires the samefor the finite-volume ground state energy. Since the latter is well-known to be absent, we are confident that this termcannot arise.To give a more detailed comparison, we must next relate the effective-field-theory-independent parameters of ourcalculation, g and T , to the couplings that enter the earlier work. First, note that the relation between g and α isgiven unambiguously by matching the L → ∞ results of the two calculations:2 α (Ref. [89]) = gm . (38)By contrast, the expression for α is less clear. We can derive a partial relation by matching the 1 /L coefficients, butit is unclear whether we should only match the F term within T or if we should also absorb other infinite-volumeterms, e.g. those depending on the scalar charge g and scattering parameters. We take the relation α (Ref. [89]) = g πam ( ma T − − ζ , (39)where ζ parametrizes our ignorance of the full relation and can be used to remove the − r -dependent term within T . Here we do not allow the geometric constants I , J and K to enter the relation, asthese are only defined via the cubic geometry of the finite-volume, and it must be possible to relate the scatteringparameters and the couplings with no reference to this. We deduce L (cid:104) E , L | J | E , L (cid:105) (this work) = 2 α + α + ζL − α + ζ ) aπL I − α a m L I + O (1 /L ) , (40)Comparing to Eq. (37) we first note that the 2 α and α /L terms now agree by construction. Thus, the only non-trivial agreement is in the α /L term, which exactly corresponds between the two expressions. Otherwise the resultsare inconsistent due to ( i ) the 1 /L term of Ref. [89] and ( ii ) geometric-constant-dependent discrepancies at both O (1 /L ) and O (1 /L ).In the next section we provide a final cross-check of our result by performing an explicit perturbative calculationof the finite-volume matrix element, similar in spirit to that of Ref. [89] but based here in a relativistic effective fieldtheory. The results of this excercise verify our general expression and also shed light on the source of the incorrect1 /L scaling found in Ref. [89]. III. PERTURBATIVE EXPANSION OF MATRIX ELEMENT
In this section, we provide an alternative derivation of the matrix element near threshold using perturbation theory.This requires deriving expansions of the finite-volume two- and three-point correlation functions, using the time-dependence to isolate the ground state, and then forming a ratio to identify L (cid:104) E , L |J (0) | E , L (cid:105) . We work with ageneralized effective field theory of a scalar field with mass m .As we are interested in the two-particle threshold state, it is convenient to use an interpolator defined as the productof two scalar fields, each projected to zero spatial momentum. The two-point function is thus defined as C ( t ) = (2 m ) L e imt (cid:104) ˜ ϕ ( t ) ˜ ϕ † (0) (cid:105) , (41)where ˜ ϕ p ( t ) defines our notation for a single scalar field of momentum p at time t . This is related to the position-spaceand momentum-space field operators by Fourier transforms,˜ ϕ p ( t ) = ˆ L d x e − i p · x ϕ ( t, x ) = ˆ d p π e − ip t ˜ ϕ ( p ) . (42)We restrict attention to t > | t | , for the correlation function. The normalization of Eq. (41) is chosen such that the correlator is unity fornon-interacting limit, coinciding with the conventions chosen in Ref. [86], with the difference that we use Minkowskitime here.The two-point correlator can be written using the usual spectral representation, C ( t ) = (cid:88) n Z n e − i ∆ E n t , (43)where ∆ E n = E n − m . Since we are only interested in the threshold state, we will explicitly isolate the n = 0 termwithin C ( t ), defining C , th ( t ) = Z e − i ∆ E t . (44)As discussed in Ref. [86], one can do this systematically by using the fact that excited state corrections always leadto a time dependence of the form:exp[ − i ( (cid:112) m + (2 π/L ) n − m ) t ], with n >
0. In this work we are only interestedin the overlap factor Z . Following Ref. [86], this can be determined via Z = C , th (0) = (2 m ) L (cid:12)(cid:12) (cid:104) | ˜ ϕ (0) | E , L (cid:105) (cid:12)(cid:12) . (45)In a similar manner, we define the 3-point correlation function C ( t (cid:48) , t ) = (2 m ) L e im ( t (cid:48) − t ) (cid:104) ˜ ϕ ( t (cid:48) ) J (0) ˜ ϕ † ( t ) (cid:105) , (46)where J is a scalar, two-field current J ( x ) = g ϕ ( x ) ϕ † ( x ) , (47)and g is the scalar charge. In defining C ( t (cid:48) , t ) we have required t (cid:48) > > t and have set the prefactor to match thatused in the 2-point correlator. The current is renormalized in the same way as the mass-term within the Lagrangian,equivalently by requiring that g is the single-hadron scalar charge to all orders.We note here that our general result holds for any scalar current, whereas in this section we restrict attention to thesingle term of Eq. (47). This is sufficient for the cross check, since the J ( x ) induces all g and F terms and therefore0allows one to check all terms in the general expansion. The generality is lost in this case only in that the perturbativeresult obscures the fact that the 1 /L expansion, when expressed in terms of g and F is universal, i.e. holds for allscalar currents. This universality is a direct consequence of the general formalism derived in Refs. [1, 2].Following the procedure for the 2-point correlator, we isolate the threshold term from the spectral decomposition,giving C , th ( t (cid:48) , t ) = (2 m ) L e − i ∆ E ( t (cid:48) − t ) (cid:104) | ˜ ϕ (0) | E , L (cid:105) (cid:104) E , L | J (0) | E , L (cid:105) (cid:104) E , L | ˜ ϕ † (0) | (cid:105) , (48)= Z e − i ∆ E ( t (cid:48) − t ) (cid:104) E , L | J (0) | E , L (cid:105) . (49)As with the 2-point correlator, one can unambiguously separate exponentials contributing to excited states, so thatthis threshold correlator is straightforward to calculate, order by order in perturbation theory. The matrix elementwe are after is then given by the ratio (cid:104) E , L | J (0) | E , L (cid:105) = 1 Z C , th (0 , . (50)In the following subsections, we calculate Z = C , th (0) and L C , th (0 ,
0) [and thus L (cid:104) E , L | J (0) | E , L (cid:105) ] through O ( a , /L ) in a generic, effective-field-theory expansion.We remark that the perturbative check of this section differs from the derivation of Refs. [1, 2], even though bothare based in the generic properties of relativistic field theory. The key distinction is that the ground state matrixelement is identified here through terms with time dependence of the form e − i mt , corresponding in momentum spaceto the lowest lying non-interacting finite-volume pole. Of course, the full correlator has a time dependence dictated bythe interacting spectrum. This corresponds to the interacting pole positions (and the cancellation of non-interactingpoles) that was identified after the all orders summation in Refs. [1, 2].The distinction leads to important technical differences in the calculation. In particular, in Refs. [1, 2] we foundthat diagrams in which the current couples to a final single-particle [see Figs. 3(b1-c2)] did not contribute to theresidue of interacting poles that defined the matrix element of interest. In the present calculation, by contrast, thesediagrams appear at the fixed-order being considered, and turn out to be necessary in recovering Eq. (3). A. Two-point correlator
The order-by-order calculation of Z [through O ( a , /L )] is one of the central ingredients in perturbative determi-nations of the ground state two-particle energy, described in detail in Refs. [86, 88]. As illustrated in detail in Ref. [88],one of the central complications in the fixed-order calculation is that numerous contribution arise that either cancelin the final result or else are absorbed in the relation between the bare coupling and the scattering length. To avoidthese complications, here we present a new method, in which Z is derived through the expansion of finite-volumecorrelator expressed via standard identities that arise in the context of finite-volume quantization conditions.We begin with C ( t ) = (2 m ) L e imt ˆ d E (cid:48) π ˆ d E π ˆ d k (cid:48) π ˆ d k π e − iE (cid:48) t G L ( E (cid:48) , E, k (cid:48) , k ) , (51)where G L ( E (cid:48) , E, k (cid:48) , k ) ≡ ˆ L d x e ik (cid:48) x ˆ L d y e i ( P (cid:48) − k (cid:48) ) y ˆ L d z e − ikz ˆ L d w e − i ( P − k ) w (cid:104) ϕ ( x ) ϕ ( y ) ϕ † ( z ) ϕ † ( w ) (cid:105) L , (52)with k µ = ( k , ), k (cid:48) µ = ( k (cid:48) , ), P µ = ( E, ), P (cid:48) µ = ( E (cid:48) , ). We stress here that the four-point function also includesthe disconnected contractions. In addition, we note that G L is proportional to an energy conserving Delta function, δ ( E − E (cid:48) ).We next note that, following the Lehmann-Symanzik-Zimmermann reduction formula, the connected part of G L will contain a quadruple pole of the form [( k − m )( k (cid:48) − m )(( P − k ) − m )(( P (cid:48) − k (cid:48) ) − m )] − and, after projectingto zero spatial momentum, this leads to poles at k , k (cid:48) = ± m ∓ i(cid:15) . Evaluating the k and k (cid:48) integrals by enciriclingthese, we find C , th ( t ) = 1 − L e imt m d E π e − iEt E i M L ( E )( E − m + i(cid:15) ) , (53)1 + · · ·
We now turn to the three-point correlator, which can be written as C ( t (cid:48) , t ) = (2 m ) L e im ( t (cid:48) − t ) ˆ d E (cid:48) π ˆ d E π ˆ d k (cid:48) π ˆ d k π e − iE (cid:48) t (cid:48) + iEt (cid:104) ˜ ϕ ( P (cid:48) − k (cid:48) ) ˜ ϕ ( k (cid:48) ) J (0) ˜ ϕ ( P − k ) ˜ ϕ ( k ) (cid:105) , (63)where all four-momenta have a vanishing spatial component. As in the preceding section, we express the momentum-space correlator on the right-hand side of Eq. (63) order by order in Feynman diagrams, and then calculate thecontribution of each to C , th (0 , × C (0)3pt , th (0 ,
0) = (2 m ) L (cid:16) × g m L (cid:17) m d E (cid:48) π m d E π iE (cid:48) ( E (cid:48) − m + i(cid:15) ) iE ( E − m + i(cid:15) ) , (64)= gmL , (65)where the upstairs factor of L comes from the momentum-conserving delta function associated with the disconnectedpropagator. Alternatively, this same result is reached using propagators in the time-momentum representation C (0)3pt , th ( t (cid:48) , t ) = (2 m ) L e im ( t (cid:48) − t ) (cid:20) (cid:18) L e − imt (cid:48) m (cid:19) gL (cid:18) L e imt m (cid:19)(cid:18) L e − im ( t (cid:48) − t ) m (cid:19)(cid:21) = gmL , (66)where in this case the current is rewritten as J (0) = [ g/L ] (cid:80) k , p ˜ ϕ (0 , k ) ˜ ϕ † (0 , p ), leading to the volume factor asshown. The complete leading-order calculation also includes a term in which the current is disconnected from bothpropagators. However this term, like every other contribution with the current fully disconnected, is cancelled bya counterterm, chosen to enforce g as the physical value of the scalar charge. For this reason we omit current-disconnected diagrams throughout.The contribution at NLO is given by the diagrams in Figs. 3 (b1) and (b2), where the current couples to a singlehadron in the final state. As mentioned above, these do not contribute to the all orders derivation of Ref. [1, 2] butmust be included in this fixed-order calculation. The two diagrams give the same contribution to the threshold matrixelement and we find C (1)3pt , th ( t (cid:48) , t ) = − ig M , th (2 m ) L e im ( t (cid:48) − t ) m ) m d E (cid:48) π m d E π ie − iE (cid:48) t (cid:48) + iEt E ( E − m + i(cid:15) ) E (cid:48) ( E (cid:48) − m + i(cid:15) ) , (67)= − ig M , th (2 m ) L e − imt m ) m d E π e iEt E ( E − m + i(cid:15) ) , (68)= 4 g M , th (2 m ) L e − imt m ) ∂∂E e iEt E (cid:12)(cid:12)(cid:12)(cid:12) E =2 m , (69)where M , th = − πma is the threshold scattering amplitude. Setting t = t (cid:48) = 0 then yields C (1)3pt , th (0 ,
0) = − gmL π m a (cid:16) aπL (cid:17) . (70)As with the leading-order result, this can also be reproduced using time-momentum perturbation theory.It is instructive to already collect the results for C , th (0 ,
0) and Z , through O ( a ). We find (cid:104) E , L | J (0) | E , L (cid:105) = C , th (0 , Z = gmL − × π m a (cid:16) aπL (cid:17) − π m a (cid:16) aπL (cid:17) + O ( a ) . (71)Note that the factor of 2 in the numerator spoils the cancellation, so that an O ( a/L ) term does contribute to thefinal result. This may seem surprising since all contributions considered so far involve the current coming to one ofthe external legs. Thus for each term in C , th (0 ,
0) one expects a closely related contribution to Z . The key point isthat the relative combinatoric factors differ between the LO and NLO terms and this leads to an NLO term survivingin the matrix element.3We now turn to contributions scaling as a /L . As with the NLO contributions, here the contribution to C , th (0 , Z . However, an additional termwith the current on an internal leg, does cancel against the remainder so that no 1 /L behavior enters the finalmatrix element. The relevant expressions arise from evaluating the next-to-next-to-leading-order (N2LO) diagramsof Figs. 3(c1), (c2) and (d). The first two of these give C (2 ,c )3pt , th (0 ,
0) = − gmL (cid:16) aπL (cid:17) J + O (1 /L ) , (72)and thus follow the pattern of the NLO diagram, including the factor of 2 that spoils the complete cancellation.The diagram of Fig. 3(d) contributes a similar term as can be seen from rewriting the expression as C (2 , d)3pt , th (0 ,
0) = − (2 m ) L g (2 m ) ˆ d E (cid:48) π ˆ d E π M ( E (cid:48) ) G ( E (cid:48) , E, L ) M ( E ) E (cid:48) ( E (cid:48) − m + i(cid:15) ) E ( E − m + i(cid:15) ) + O (1 /L ) . (73)Here we have introduced G ( E (cid:48) , E, L ) in the perturbative expansion by rewriting the summed loop as an integral plusa sum-integral difference. The latter contributes at 1 /L and is thus dropped to illustrate the leading behavior first.We can further simplify this by dividing G ( E (cid:48) , E, L ) into real and imaginary parts and splitting the real part into thedouble pole at threshold ( ∼ / [( E − m )( E (cid:48) − m )]), together with the sum over k (cid:54) = 0, denoted by G (cid:48) ( E (cid:48) , E, L ). Aswith F (cid:48) ( E, L ), this term is regular near E = E (cid:48) = 2 m and the resulting contribution comes from encircling the polesshown explicitly in Eq. (73). Substituting G (cid:48) (2 m, m, L ) = L π m J + O (1 /L ) , (74)together with the threshold scattering amplitude then gives C (2 ,d )3pt , th (0 ,
0) = gmL (cid:16) aπL (cid:17) J + O (1 /L ) . (75)Alternatively, this same result can be derived in time-momentum perturbation theory. In this case one finds that therelevant time-dependence arises only for k (cid:54) = . This gives C (2 ,d )3pt , th (0 ,
0) = g M , th (2 m ) L L (cid:88) k (cid:54) = m ) (2 ω k ) (2 ω k − m ) , (76)which is equivalent to Eq. (75) above. Combining C (2 ,c )3pt , th (0 ,
0) and C (2 ,d )3pt , th (0 ,
0) gives the same 1 /L dependence asin Z , such that these terms perfectly cancel in the ratio. This is our final confirmation that 1 /L scaling is absentfrom the matrix element: L (cid:104) E , L | J (0) | E , L (cid:105) .To conclude the perturbative check, we have evaluated C , th (0 ,
0) to one higher order in both a and 1 /L . Thisfollows the same pattern of the calculation so far, but induces F and r dependent contributions as well as variousgeometric constants. One finds mL g C , th (0 ,
0) = 1 − (cid:16) aπL (cid:17) J − π m a (cid:0) − ma T (cid:1) (cid:16) aπL (cid:17) − K − IJ ) (cid:16) aπL (cid:17) + O ( L − ) . (77)Combining this with Eq. (62) gives an expansion through 1 /L that is completely consistent with the result of Eq. (3).Figure 4 shows the leading 1 /L scaling for each diagram contributing to the matrix element.We speculate that the disagreement with Ref. [89] arises from the earlier work omitting the 1 /L corrections to Z and also dropping contributions to C , th (0 ,
0) with the current attached to an external leg. Equivalently, the earlierwork may have been based in the assumption that the two sets of terms cancel, as they would if it were not for theleading-order discrepancy, summarized in Eq. (71). Indeed, we find that if we drop diagrams where the current probesthe external legs [(b), (c), and (e) of Fig. 3], and also drop Z , then we exactly recover the 1 /L expansion of Ref. [89].We stress however that there is no theory nor limiting case where this result holds and, in particular, the absence ofthe 1 /L is a universal result inherited from the ground-state energy. IV. SUMMARY
Understanding the structure of strongly interacting resonances and bound states requires knowledge of two-hadronelectroweak transition amplitudes. With this in mind, a framework was presented in Refs. [1, 2] to relate finite-volume4 + · · ·
We thank Will Detmold for useful discussions. R.A.B. is supported in part by USDOE grant No. DE-AC05-06OR23177, under which Jefferson Science Associates, LLC, manages and operates Jefferson Lab. R.A.B. also ac-knowledges support from the USDOE Early Career award, contract de-sc0019229.5
Appendix A: Imaginary part of triangle diagram
In this appendix we demonstrate that G ( E, E, L ) has a simple imaginary part, given by Eq. (22) of the main text.The imaginary part arises only from the integral part of G ( E, E, L ), and thus, the quantity we are after is given byIm G ( E, E, L ) = − Im ˆ Λ d k (2 π ) ω k E ( E − ω k + i(cid:15) ) . (A1)Here we have included the hard cutoff, Λ, since the real part of the integral has an ultraviolet divergence that cancelswith that of the sum in G ( E, E, L ). As we will see, the imaginary part is ultraviolet-finite and therefore also universal.Next it is convenient to expand the integrand about the douple pole at q = k , where q = E / − m and k = k ,12 ω k E ( E − ω k + i(cid:15) ) = 12 ω k ( E + 2 ω k ) (4 E ) ( q − k + i(cid:15) ) = 14 E q − k + i(cid:15) ) + O (cid:104)(cid:0) k − q (cid:1) (cid:105) . (A2)This is useful because the sub-leading terms only contribute to the real part of the integral. We reachIm G ( E, E, L ) = − π E ˆ ∞ d k k ( q − k + i(cid:15) ) , (A3)where we have also used that the singular piece gives a convergent integral so that we can send Λ → ∞ . To identifythe imaginary part, we rewrite Eq. (A3) as a contour integral, ˆ ∞ d k k ( q − k + i(cid:15) ) = 12 ˆ ∞−∞ d k k ( q − k + i(cid:15) ) = 12 ˛ d k k ( k − q − i(cid:15) ) ( k + q + i(cid:15) ) , (A4)where, for concreteness, we envision closing the contour in the upper-half plane. Evaluating the integral, we pick upthe residue at the pole, k = q + i(cid:15) , ˛ d k k ( k − q − i(cid:15) ) ( k + q + i(cid:15) ) = 2 πi dd k k ( k + q ) (cid:12)(cid:12)(cid:12)(cid:12) k = q = i π q , (A5)and thereby conclude the desired result Im G ( E, E, L ) = − πEq . (A6) [1] R. A. Brice˜no and M. T. Hansen, Phys. Rev. D94 , 013008 (2016), arXiv:1509.08507 [hep-lat].[2] A. Baroni, R. A. Brice˜no, M. T. Hansen, and F. G. Ortega-Gama, Phys. Rev.
D100 , 034511 (2019), arXiv:1812.10504[hep-lat].[3] S. Durr et al. , Science , 1224 (2008), arXiv:0906.3599 [hep-lat].[4] Z. Fodor and C. Hoelbling, Rev. Mod. Phys. , 449 (2012), arXiv:1203.4789 [hep-lat].[5] S. Borsanyi et al. , Science , 1452 (2015), arXiv:1406.4088 [hep-lat].[6] M. Luscher, Commun. Math. Phys. , 153 (1986).[7] M. Luscher, Nucl. Phys. B354 , 531 (1991).[8] K. Rummukainen and S. A. Gottlieb, Nucl. Phys.
B450 , 397 (1995), arXiv:hep-lat/9503028 [hep-lat].[9] C. h. Kim, C. T. Sachrajda, and S. R. Sharpe, Nucl. Phys.
B727 , 218 (2005), arXiv:hep-lat/0507006 [hep-lat].[10] S. He, X. Feng, and C. Liu, JHEP , 011 (2005), arXiv:hep-lat/0504019 [hep-lat].[11] L. Leskovec and S. Prelovsek, Phys. Rev. D85 , 114507 (2012), arXiv:1202.2145 [hep-lat].[12] M. T. Hansen and S. R. Sharpe, Phys. Rev.
D86 , 016007 (2012), arXiv:1204.0826 [hep-lat].[13] R. A. Brice˜no and Z. Davoudi, Phys. Rev.
D88 , 094507 (2013), arXiv:1204.1110 [hep-lat].[14] R. A. Brice˜no, Z. Davoudi, and T. C. Luu, Phys. Rev.
D88 , 034502 (2013), arXiv:1305.4903 [hep-lat].[15] R. A. Brice˜no, Phys. Rev.
D89 , 074507 (2014), arXiv:1401.3312 [hep-lat].[16] F. Romero-Lopez, A. Rusetsky, and C. Urbach, Phys. Rev.
D98 , 014503 (2018), arXiv:1802.03458 [hep-lat].[17] J. J. Dudek, R. G. Edwards, M. J. Peardon, D. G. Richards, and C. E. Thomas, Phys. Rev.
D83 , 071504 (2011),arXiv:1011.6352 [hep-ph].[18] S. R. Beane, E. Chang, W. Detmold, H. W. Lin, T. C. Luu, K. Orginos, A. Parreno, M. J. Savage, A. Torok, andA. Walker-Loud (NPLQCD), Phys. Rev.
D85 , 034505 (2012), arXiv:1107.5023 [hep-lat]. [19] C. Pelissier and A. Alexandru, Phys. Rev. D87 , 014503 (2013), arXiv:1211.0092 [hep-lat].[20] J. J. Dudek, R. G. Edwards, and C. E. Thomas (Hadron Spectrum), Phys. Rev.
D87 , 034505 (2013), [Erratum: Phys.Rev.D90,no.9,099902(2014)], arXiv:1212.0830 [hep-ph].[21] L. Liu, K. Orginos, F.-K. Guo, C. Hanhart, and U.-G. Meissner, Phys. Rev.
D87 , 014508 (2013), arXiv:1208.4535 [hep-lat].[22] S. R. Beane et al. (NPLQCD), Phys. Rev.
C88 , 024003 (2013), arXiv:1301.5790 [hep-lat].[23] K. Orginos, A. Parreno, M. J. Savage, S. R. Beane, E. Chang, and W. Detmold, Phys. Rev.
D92 , 114512 (2015),arXiv:1508.07583 [hep-lat].[24] E. Berkowitz, T. Kurth, A. Nicholson, B. Joo, E. Rinaldi, M. Strother, P. M. Vranas, and A. Walker-Loud, Phys. Lett.
B765 , 285 (2017), arXiv:1508.00886 [hep-lat].[25] C. B. Lang, D. Mohler, S. Prelovsek, and R. M. Woloshyn, Phys. Lett.
B750 , 17 (2015), arXiv:1501.01646 [hep-lat].[26] J. Bulava, B. Fahy, B. Horz, K. J. Juge, C. Morningstar, and C. H. Wong, Nucl. Phys.
B910 , 842 (2016), arXiv:1604.05593[hep-lat].[27] B. Hu, R. Molina, M. Doring, and A. Alexandru, Phys. Rev. Lett. , 122001 (2016), arXiv:1605.04823 [hep-lat].[28] C. Alexandrou, L. Leskovec, S. Meinel, J. Negele, S. Paul, M. Petschlies, A. Pochinsky, G. Rendon, and S. Syritsyn, Phys.Rev.
D96 , 034525 (2017), arXiv:1704.05439 [hep-lat].[29] G. S. Bali, S. Collins, A. Cox, and A. Sch¨afer, Phys. Rev.
D96 , 074501 (2017), arXiv:1706.01247 [hep-lat].[30] M. L. Wagman, F. Winter, E. Chang, Z. Davoudi, W. Detmold, K. Orginos, M. J. Savage, and P. E. Shanahan, Phys.Rev.
D96 , 114510 (2017), arXiv:1706.06550 [hep-lat].[31] C. W. Andersen, J. Bulava, B. Horz, and C. Morningstar, Phys. Rev.
D97 , 014506 (2018), arXiv:1710.01557 [hep-lat].[32] R. Brett, J. Bulava, J. Fallica, A. Hanlon, B. Horz, and C. Morningstar, Nucl. Phys.
B932 , 29 (2018), arXiv:1802.03100[hep-lat].[33] M. Werner et al. , (2019), arXiv:1907.01237 [hep-lat].[34] M. Mai, C. Culver, A. Alexandru, M. Doring, and F. X. Lee, (2019), arXiv:1908.01847 [hep-lat].[35] D. J. Wilson, R. A. Briceno, J. J. Dudek, R. G. Edwards, and C. E. Thomas, Phys. Rev. Lett. , 042002 (2019),arXiv:1904.03188 [hep-lat].[36] D. J. Wilson, J. J. Dudek, R. G. Edwards, and C. E. Thomas, Phys. Rev.
D91 , 054008 (2015), arXiv:1411.2004 [hep-ph].[37] J. J. Dudek, R. G. Edwards, C. E. Thomas, and D. J. Wilson (Hadron Spectrum), Phys. Rev. Lett. , 182001 (2014),arXiv:1406.4158 [hep-ph].[38] D. J. Wilson, R. A. Brice˜no, J. J. Dudek, R. G. Edwards, and C. E. Thomas, Phys. Rev.
D92 , 094502 (2015),arXiv:1507.02599 [hep-ph].[39] J. J. Dudek, R. G. Edwards, and D. J. Wilson (Hadron Spectrum), Phys. Rev.
D93 , 094506 (2016), arXiv:1602.05122[hep-ph].[40] R. A. Brice˜no, J. J. Dudek, R. G. Edwards, and D. J. Wilson, Phys. Rev. Lett. , 022002 (2017), arXiv:1607.05900[hep-ph].[41] G. Moir, M. Peardon, S. M. Ryan, C. E. Thomas, and D. J. Wilson, JHEP , 011 (2016), arXiv:1607.07093 [hep-lat].[42] R. A. Briceno, J. J. Dudek, R. G. Edwards, and D. J. Wilson, Phys. Rev. D97 , 054513 (2018), arXiv:1708.06667 [hep-lat].[43] A. Woss, C. E. Thomas, J. J. Dudek, R. G. Edwards, and D. J. Wilson, JHEP , 043 (2018), arXiv:1802.05580 [hep-lat].[44] A. J. Woss, C. E. Thomas, J. J. Dudek, R. G. Edwards, and D. J. Wilson, Phys. Rev. D100 , 054506 (2019),arXiv:1904.04136 [hep-lat].[45] M. T. Hansen and S. R. Sharpe, Phys. Rev.
D90 , 116003 (2014), arXiv:1408.5933 [hep-lat].[46] M. T. Hansen and S. R. Sharpe, Phys. Rev.
D92 , 114509 (2015), arXiv:1504.04248 [hep-lat].[47] M. Mai and M. Doring, Eur. Phys. J.
A53 , 240 (2017), arXiv:1709.08222 [hep-lat].[48] H. W. Hammer, J. Y. Pang, and A. Rusetsky, JHEP , 115 (2017), arXiv:1707.02176 [hep-lat].[49] R. A. Briceno, M. T. Hansen, and S. R. Sharpe, Phys. Rev. D95 , 074510 (2017), arXiv:1701.07465 [hep-lat].[50] R. A. Briceno, M. T. Hansen, and S. R. Sharpe, Phys. Rev.
D99 , 014516 (2019), arXiv:1810.01429 [hep-lat].[51] M. Mai and M. Doring, Phys. Rev. Lett. , 062503 (2019), arXiv:1807.04746 [hep-lat].[52] R. A. Brice˜no, M. T. Hansen, and S. R. Sharpe, Phys. Rev.
D98 , 014506 (2018), arXiv:1803.04169 [hep-lat].[53] D. Guo, A. Alexandru, R. Molina, M. Mai, and M. D¨oring, Phys. Rev.
D98 , 014507 (2018), arXiv:1803.02897 [hep-lat].[54] T. D. Blanton, F. Romero-L´opez, and S. R. Sharpe, JHEP , 106 (2019), arXiv:1901.07095 [hep-lat].[55] B. H¨orz and A. Hanlon, (2019), arXiv:1905.04277 [hep-lat].[56] T. D. Blanton, F. Romero-L´opez, and S. R. Sharpe, (2019), arXiv:1909.02973 [hep-lat].[57] M. Mai, M. D¨oring, C. Culver, and A. Alexandru, (2019), arXiv:1909.05749 [hep-lat].[58] C. Culver, M. Mai, R. Brett, A. Alexandru, and M. D¨oring, (2019), arXiv:1911.09047 [hep-lat].[59] R. A. Briceno, J. J. Dudek, and R. D. Young, Rev. Mod. Phys. , 025001 (2018), arXiv:1706.06223 [hep-lat].[60] M. T. Hansen and S. R. Sharpe, Annual Review of Nuclear and Particle Science , null (2019), arXiv:1901.00483 [hep-lat].[61] L. Lellouch and M. Luscher, Commun. Math. Phys. , 31 (2001), arXiv:hep-lat/0003023 [hep-lat].[62] N. H. Christ, C. Kim, and T. Yamazaki, Phys. Rev. D72 , 114506 (2005), arXiv:hep-lat/0507009 [hep-lat].[63] R. A. Brice˜no, M. T. Hansen, and A. Walker-Loud, Phys. Rev.
D91 , 034501 (2015), arXiv:1406.5965 [hep-lat].[64] R. A. Brice˜no and M. T. Hansen, Phys. Rev.
D92 , 074509 (2015), arXiv:1502.04314 [hep-lat].[65] A. Agadjanov, V. Bernard, U.-G. Meißner, and A. Rusetsky, Nucl. Phys.
B910 , 387 (2016), arXiv:1605.03386 [hep-lat].[66] T. Blum et al. , Phys. Rev. Lett. , 141601 (2012), arXiv:1111.1699 [hep-lat].[67] P. A. Boyle et al. (RBC, UKQCD), Phys. Rev. Lett. , 152001 (2013), arXiv:1212.1474 [hep-lat].[68] T. Blum et al. , Phys. Rev.
D91 , 074502 (2015), arXiv:1502.00263 [hep-lat].[69] Z. Bai et al. (RBC, UKQCD), Phys. Rev. Lett. , 212001 (2015), arXiv:1505.07863 [hep-lat]. [70] X. Feng, S. Aoki, S. Hashimoto, and T. Kaneko, Phys. Rev. D91 , 054504 (2015), arXiv:1412.6319 [hep-lat].[71] C. Andersen, J. Bulava, B. H¨orz, and C. Morningstar, Nucl. Phys.
B939 , 145 (2019), arXiv:1808.05007 [hep-lat].[72] R. A. Brice˜no, J. J. Dudek, R. G. Edwards, C. J. Shultz, C. E. Thomas, and D. J. Wilson, Phys. Rev. Lett. , 242001(2015), arXiv:1507.06622 [hep-ph].[73] R. A. Brice˜no, J. J. Dudek, R. G. Edwards, C. J. Shultz, C. E. Thomas, and D. J. Wilson, Phys. Rev.
D93 , 114508(2016), arXiv:1604.03530 [hep-ph].[74] C. Alexandrou, L. Leskovec, S. Meinel, J. Negele, S. Paul, M. Petschlies, A. Pochinsky, G. Rendon, and S. Syritsyn, Phys.Rev.
D98 , 074502 (2018), arXiv:1807.08357 [hep-lat].[75] N. H. Christ, X. Feng, G. Martinelli, and C. T. Sachrajda, Phys. Rev.
D91 , 114510 (2015), arXiv:1504.01170 [hep-lat].[76] X. Feng, L.-C. Jin, X.-Y. Tuo, and S.-C. Xia, Phys. Rev. Lett. , 022001 (2019), arXiv:1809.10511 [hep-lat].[77] R. A. Brice˜no, Z. Davoudi, M. T. Hansen, M. R. Schindler, and A. Baroni, (2019), arXiv:1911.04036 [hep-lat].[78] V. Bernard, D. Hoja, U. G. Meissner, and A. Rusetsky, JHEP , 023 (2012), arXiv:1205.4642 [hep-lat].[79] R. A. Brice˜no, M. T. Hansen, and A. W. Jackura, (2019), arXiv:1909.10357 [hep-lat].[80] S. R. Beane, P. F. Bedaque, A. Parreno, and M. J. Savage, Phys. Lett. B585 , 106 (2004), arXiv:hep-lat/0312004 [hep-lat].[81] Z. Davoudi and M. J. Savage, Phys. Rev.
D84 , 114502 (2011), arXiv:1108.5371 [hep-lat].[82] R. A. Brice˜no, Z. Davoudi, T. Luu, and M. J. Savage, Phys. Rev.
D88 , 114507 (2013), arXiv:1309.3556 [hep-lat].[83] K. Huang and C. N. Yang, Phys. Rev. , 767 (1957).[84] S. R. Beane, W. Detmold, and M. J. Savage, Phys. Rev.
D76 , 074507 (2007), arXiv:0707.1670 [hep-lat].[85] W. Detmold and M. J. Savage, Phys. Rev.
D77 , 057502 (2008), arXiv:0801.0763 [hep-lat].[86] M. T. Hansen and S. R. Sharpe, Phys. Rev.
D93 , 014506 (2016), arXiv:1509.07929 [hep-lat].[87] M. T. Hansen and S. R. Sharpe, Phys. Rev.
D93 , 096006 (2016), arXiv:1602.00324 [hep-lat].[88] S. R. Sharpe, Phys. Rev.
D96 , 054515 (2017), [Erratum: Phys. Rev.D98,no.9,099901(2018)], arXiv:1707.04279 [hep-lat].[89] W. Detmold and M. Flynn, Phys. Rev.