Consistency checks for two-body finite-volume matrix elements: I. Conserved currents and bound states
JJLAB-THY-19-3040CERN-TH-2019-149
Consistency checks for two-body finite-volume matrix elements:I. Conserved currents and bound states
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: September 24, 2019)Recently, a framework has been developed to study form factors of two-hadron states probedby an external current. The method is based on relating finite-volume matrix elements, computedusing numerical lattice QCD, to the corresponding infinite-volume observables. As the formalismis complicated, it is important to provide non-trivial checks on the final results and also to explorelimiting cases in which more straightforward predications may be extracted. In this work we provideexamples on both fronts. First, we show that, in the case of a conserved vector current, the formalismensures that the finite-volume matrix element of the conserved charge is volume-independent andequal to the total charge of the two-particle state. Second, we study the implications for a two-particle bound state. We demonstrate that the infinite-volume limit reproduces the expected matrixelement and derive the leading finite-volume corrections to this result for a scalar current. Finally, weprovide numerical estimates for the expected size of volume effects in future lattice QCD calculationsof the deuteron’s scalar charge. We find that these effects completely dominate the infinite-volumeresult for realistic lattice volumes and that applying the present formalism, to analytically remove aninfinite-series of leading volume corrections, is crucial to reliably extract the infinite-volume chargeof the state.
I. INTRODUCTION
One of the overarching goals of modern-day nuclear physics is the characterization and fundamental understandingof the low-lying strongly-interacting spectrum. There is, by now, overwhelming evidence that the detailed propertiesof all low-lying states are governed by the dynamics of quark and gluon fields in the mathematical framework ofquantum chromodynamics (QCD). But still, it remains a significant challenge to extract low-energy predictions fromthe underlying theory.The vast majority of QCD states emerge as either bound states or resonances of multi-hadron configurations. Anexample is the deuteron, a shallow bound state of the isoscalar proton-neutron channel with a binding energy of m n + m p − M d ≈ . f (980) resonance couplesstrongly to ππ and KK states, and has been postulated to be both a tetraquark [3] and a KK molecule [4]. The challenge of resolving the inner structure of composite hadrons is twofold: First, QCD is non-perturbative, sothat systematic low-energy calculations are challenging. This has been addressed with substantial success using low-energy effective theories, methods based in amplitude analysis and numerical calculations using lattice QCD (LQCD).In contrast to the first two methods, LQCD has the unique advantage of relating the fundamental QCD lagrangianto low-energy predictions. Second, composite states generally manifest as dynamical enhancements of multi-hadronscattering rates, meaning that the detailed observation depends on the production mechanism and decay channel of theresonance in question. This ambiguity is resolved, at least in principle, by recognizing that across all production anddecay channels, a given resonance always leads to the same pole in an analytic continuation of scattering amplitudesto complex energies.These two points have motivated the community to develop a systematic framework for extracting hadronic scat-tering amplitudes via LQCD. From the energy dependence of such amplitudes one can then quantitatively describethe bound and resonant states of the theory. In addition, by extracting transition amplitudes involving externalcurrents, one can in principle access structural information of these states. In this work, we focus on an example inthe latter class of the amplitudes, namely + J → transition amplitudes. We consider a method, first introducedin Refs. [7, 8], that allows one to determine such quantities from numerical LQCD. ∗ e-mail: [email protected] † e-mail: [email protected] ‡ e-mail: [email protected] Similar outstanding puzzles are present in the heavy quark sector; see Refs. [5] and [6] for recent reviews. a r X i v : . [ h e p - l a t ] S e p The primary formal challenge arises from the fact that LQCD calculations are necessarily performed in a finiteEuclidean spacetime, where the definition of asymptotic states is obscured. One of the leading methods to overcomethis 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. This approachwas first introduced by L¨uscher [13, 14], in seminal work relating the spectrum of two-particle states in a cubicvolume with periodicty L , to the corresponding infinite-volume amplitudes. The idea has since been extended forarbitrary two-particle scattering [15–23] and more recently to three particles [24–33], with the latter currently limitedto identical scalars (or pseudoscalars). The two-particle relations have made possible the determination of hadronicscattering amplitudes for a wide range of particle species [34–52], including energies where multiple channels arekinematically open [53–61]. Most recently, the first LQCD calculations to constrain three-particle interactions usingexcited states were performed in Refs. [62–64].Electroweak interactions involving scattering states can also be accessed using LQCD, via a generalization of themethods described above. The seminal example in this sector is the work of Ref. [65], providing a formal method fordetermining the electroweak decay, K → ππ . More generally, in processes for which the effects of the electroweak sectorcan be treated perturbatively, the relevant amplitudes are given via the evaluation of QCD matrix elements, built fromthe appropriate currents together with multi-particle external states. These ideas have been successfully developed forthe case that either the initial or the final state couples strongly to two-particle scattering states [16, 19, 65–69] andimplemented in a number lattice QCD studies, most prominently to determine the K → ππ decay amplitudes [70–73] as well as the electromagnetic process πγ (cid:63) → ππ [74–78]. This progress motivates the consideration of morecomplicated electroweak transitions, in particular those with two hadrons in both the initial and final state.As we discuss in detail in Sec. II, + J → transition amplitudes allow one to extract elastic form factors of boundstates and resonances, thereby providing direct information on the structure of these states and possibly resolvingwhich models are most descriptive [79–81]. As compared to the transitions described in the preceding paragraph, thenecessary formalism for these quantities is significantly more complicated [7, 8, 20, 82]. Building on previous work,in Ref. [7] two of us derived a model-independent relation between the corresponding finite-volume matrix elements,schematically denoted (cid:104) |J | (cid:105) L (where L indicates the side-length of the periodic cubic volume), and the + J → transition amplitude, W . In Ref. [8] we improved the method by simplifying technical details relating to the on-shell projection of the single-particle form factor and by using Lorentz covariant poles in the various finite-volumekinematic functions that arise. We stress that the two approaches are equivalent and only differ in the exact definitionsof unphysical, intermediate quantities. The results are derived to all orders in the perturbative expansion of a genericrelativistic field theory, for any type of two-scalar channels, with generalizations to spin and coupled channels left tofuture work. Details of this formalism are reviewed in Sec. III.The purpose of this work is to provide two non-trivial checks on the general relations of Refs. [7, 8], and alsoto demonstrate their predictive power even in simplified special cases. As a first check, in Sec. IV we demonstratethat the method is consistent with the consequences of the conserved vector current. In particular, the formalismpredicts that the charge of a two-hadron finite-volume state is exactly equal to the sum of the constituent chargesand independent of L . This relies on non-trivial relations between various L -dependent geometric functions, and arelation between the → and + J → amplitudes that follows from the Ward-Takahashi identity. The secondcheck, presented in Sec. V B, considers the analytic continuation of the formalism below two-particle threshold, fortheories with an S -wave bound state. We show that the finite- and infinite-volume matrix elements coincide (oncenormalization factors are accounted for) up to term scaling as e − κ B L , where κ = m − M / m , binding to a mass of M B .Presently, LQCD calculations of light nuclei properties are being performed at unphysically heavy quark masses, forwhich the binding momenta exceed their real-world values [41, 47, 84–86]. In addition the properties of states can beshifted, e.g. the dineutron, in nature a virtual bound state, is found to be a standard bound state for m π (cid:38)
450 MeV[39, 40, 87–89]. The increased binding suppresses finite-volume effects and this has permitted exploratory calculationsof matrix elements of these states [90–94], in which volume effects are ignored.As LQCD calculations of multi-nucleon systems move towards physical quark masses, the binding momenta of thenuclei decrease and it is well-known that finite-volume effects of the naively extracted states can become a dominantsource of systematic uncertainty [18, 95, 96]. In the case of spectroscopy, an infinite series of e − κ B L corrections can beremoved by applying the L¨uscher formalism, as was done in [41, 86] as well as in a wide variety of mesonic channelswhere bound states appear [42, 46, 53, 54, 57, 58]. The results of this work stress that it is important to pursue thesame paradigm for matrix elements of loosely bound states, using the formalism of Refs. [7, 8] to non-perturbatively We point the reader to Refs. [9] and [10] for recent reviews detailing the progress of the field. See also Refs. [11] and [12] for alternativemethods to determine rates and amplitudes, which require significantly larger volumes as well as techniques to regulate the inverseLaplace transform. Reference [83] was the first work to consider the coupling of an external current to finite-volume two-hadron states. In that publication,the authors consider the use of background fields in the context of a fixed-order expansion in a particular effective field theory, analternative to the matrix elements of currents discussed here. remove binding-momentum-enhanced finite-volume artifacts. To illustrate this point, in Sec. V B we determine theleading e − κ B L corrections and compare these to the full result, which holds up to e − mL . Finally, in Sec. V C we presenta numerical example, meant to model the deuteron at physical pion masses, and show that the full formalism is neededto reliably remove the L -dependence for box sizes in the region of mL ≈ −
7. Otherwise the e − κ B L corrections canbecome comparable in size with the infinite-volume result and thereby dominate the systematic uncertainties.Though largely addressed above, we close here with a brief summary of the remaining sections. After reviewingbasic properties of the infinite-volume → and + J → amplitudes in Sec. II, in Sec. III we described thecorresponding finite-volume formalism for each type of amplitude. Then, in a very compact Sec. IV, we demonstratethat the finite-volume + J → formalism gives the expected results for matrix elements of a conserved current.Section V is dedicated to volume effects on a two-particle bound state, including a check that the L → ∞ limit givesthe required result, a calculation of the leading O ( e − κ B L ) corrections, and a numerical exploration intended to guidefuture LQCD calculations of the deuteron’s scalar charge. We briefly conclude in Sec. VI. In addition, this articleincludes three appendices, providing proofs of various technical results used in the main text. II. INFINITE-VOLUME AMPLITUDES AND BOUND STATES
In this section we review the definitions and key properties of the infinite-volume → and + J → amplitudes,with particular attention to the expressions relevant for an S -wave bound state. For simplicity, we focus on systemscomposed of two scalar particles, with degenerate mass m , distinguished by their charge with respect to an externalcurrent J µ . One of the particles carries charge Q , while the other is neutral. Here we have in mind a scalar analogof the proton-neutron system. A. 2 → In a general Lorentz frame, the two-particle system has a total energy-momentum denoted by P = ( E, P ). Boostingto the center-of-momentum frame (CMF) we define P (cid:63) = ( E (cid:63) , ), which is related to the Mandelstam variable s anda generic P by E (cid:63) ≡ s ≡ P µ P µ = E − P . (1)Two-particle scattering is described by s , as well as the back-to-back momentum orientations of the initial and finalstates in the CMF: ˆk (cid:63)i and ˆk (cid:63)f , respectively. Using these coordinates we can introduce the scattering amplitude andits partial wave expansion M ( s, ˆk (cid:63)f , ˆk (cid:63)i ) = 4 π (cid:88) (cid:96),m (cid:96) Y (cid:96)m (cid:96) ( ˆk (cid:63)f ) M (cid:96) ( s ) Y ∗ (cid:96)m (cid:96) ( ˆk (cid:63)i ) . (2)We have used that total angular momentum, (cid:96) , is conserved, and that the partial-wave amplitude is independent ofthe projection, m (cid:96) , both consequences of rotational symmetry. In the following we are interested in the case of a scalarbound state, appearing as a sub-threshold pole in M (cid:96) =0 ( s ). We therefore restrict attention to the S -wave ( (cid:96) = 0)amplitude and do not write the angular momentum index on the partial wave amplitude for the rest of this section.The elastic → scattering amplitude can be represented in terms of the K matrix, which is an on-shell represen-tation that enforces S matrix unitarity explicitly below the inelasticity threshold [97], M ( s ) = K ( s ) 11 − iρ ( s ) K ( s ) . (3)Here, ρ ( s ) is the two-body phase space, encoding the on-shell propagation of two particles. It is defined as ρ ( s ) = q (cid:63) πE (cid:63) = 116 π (cid:114) − m s , (4)where q (cid:63) is the relative momentum of the two particles in the CMF, q (cid:63) ≡ (cid:112) s/ − m . This square root introduces abranch cut in the complex s plane, illustrated in Fig. 1. Bound states are then defined as subthreshold poles on thefirst Riemann sheet, the sheet for which Im q (cid:63) > We denote CMF quantities with a (cid:63) superscript throughout. Virtual bound states (e.g. the dineutron) and resonances (e.g. the ρ ) arise on the second sheet, for which Im q (cid:63) < Sheet I s
We now turn to the less standard + J → transition amplitude, defined via (cid:104) P f , ˆk (cid:63)f , out | J µ | P i , ˆk (cid:63)i , in (cid:105) conn. ≡ W µ ( P f , ˆk (cid:63)f ; P i , ˆk (cid:63)i ) . (9)Here the initial and final states have kinematics as in Sec. II A and the current, J µ , is a local operator evaluated inposition space at the origin. Since the current can inject energy and momentum, the initial and final states carrydifferent total four-momenta, P i and P f respectively. It is also convenient to define the squared momentum transfer, Q ≡ − ( P f − P i ) , where the overall minus is included so that Q > P f − P i .The amplitude W µ can be defined for local currents with any Lorentz structure and in Sec. V B 2 we also considerspecific results for a scalar current. Here, for concreteness we focus on a conserved vector current J µ ( x ) satisfying ∂ µ J µ ( x ) = 0 . (10)Our first aim is to connect this amplitude to the bound-state form factor, defined via (cid:104) P f , B | J µ | P i , B (cid:105) = ( P f + P i ) µ F B ( Q ) , (11)where | P i , B (cid:105) is the bound state, normalized as (cid:104) P f , B | P i , B (cid:105) = (2 π ) ω P i δ ( P f − P i ) with energy ω P = (cid:112) s B + P .Eq. (11) is related to the S -wave projection of W µ , analytically continued below threshold to the bound-state pole: W µ ( P f , P i ) = ( P i + P f ) µ F B ( Q ) i ( ig ) ( s f − s B )( s i − s B ) (cid:2) O ( s i,f − s B ) (cid:3) , (12) =
Before giving detailed expressions for the finite-volume effects on a two-body bound state, in this section we brieflyreview the general formalism describing the finite-volume energies and matrix elements of two-particle systems. Inthe following, we work in a cubic, periodic volume of length L with infinite temporal extent. The total momentumof the system in the finite-volume frame is allowed to take on any value consistent with the periodicity: P = 2 π n /L with n ∈ Z . We leave a detailed proof of this claim and an analysis of its consequences to future work [99].
A. Finite-volume energies
In the window of energies for which only two particles can propagate, the finite-volume spectrum is related to theinfinite-volume partial-wave amplitudes, defined in Eq. (3), via the L¨uscher quantization condition [14–16]. Generally,the quantization condition is a determinant over angular momentum space. If we neglect waves higher than (cid:96) = 0,however, it reduces to a simple algebraic relation M − ( s n ) = − F ( P n , L ) + O ( e − mL ) , (21)where s n = P n = E n ( L ) − P corresponds to the eigenenergy of the n th finite-volume two-particle state. Here F ( P, L ) is a known finite-volume function, F ( P, L ) = (cid:20) L (cid:88)(cid:90) k (cid:21) ω k ω Pk ( E − ω k − ω Pk + i(cid:15) ) , (22)where ω k = √ m + k and ω Pk = (cid:112) m + ( P − k ) are the on-shell energies of the two particles, and (cid:20) L (cid:88)(cid:90) k (cid:21) ≡ L (cid:88) k ∈ (2 π/L ) Z − (cid:90) d k (2 π ) . Equation (21) holds up to corrections associated with higher partial waves and only for s n below the first inelasticthreshold. B. Finite-volume matrix elements
Similarly, one can relate finite-volume matrix elements of two-particle systems to infinite-volume + J → transition amplitudes. Here, the relevant formalism was first derived in Ref. [7] using an all-orders perturbativeexpansion based in a generic relativistic effective field theory. Recently, in Ref. [8], the formal approach was improved intwo ways: First, by rearranging the separation of finite-volume effects, we were able to show that the extracted infinite-volume transition amplitudes are manifestly Lorentz covariant. Second, we re-organized the analysis so that single-particle matrix elements enter via standard form factors (rather than a non-standard spherical harmonic decompositionused in the first publication). While the two representations are formally equivalent, the work of Ref. [8] is expectedto be significantly more convenient in numerical applications going forward. Of course, all expressions used here aretaken from the improved approach.Again, assuming all but the (cid:96) = 0 partial waves are negligible, the matrix elements of the vector current fortwo-particle states can be related to W µ df , defined in Eq. (18), as follows L (cid:104) P n,f , L | J µ | P n,i , L (cid:105) = W µL, df ( P n,f , P n,i , L ) (cid:113) R ( P n,f , L ) R ( P n,i , L ) , (23)where P n,i = ( E n,i , P i ) and P n,f = ( E n,f , P f ). Here W µL, df is an L -dependent function related to W µ df in a mannerdetailed in the following paragraph. In addition, R is a generalization of the Lellouch-L¨uscher factor [65], firstintroduced in Ref. [67] R ( P n , L ) = (cid:20) ∂∂E (cid:0) F − ( P, L ) + M ( s ) (cid:1)(cid:21) − E = E n , (24)= −M − ( s n ) (cid:20) ∂∂E (cid:0) F ( P, L ) + M − ( s ) (cid:1)(cid:21) − E = E n , (25)where we have given a second form that will be particularly useful for this work. In general, R is a matrix over alltwo-particle degrees of freedom, but in the case considered it reduces to a simple derivative of the functions shown.Before defining W µL, df ( P f , P i , L ), we need to introduce a second L -dependent kinematic function, G µ ··· µ n , firstintroduced in Refs. [7, 8]. For the (cid:96) = 0 truncation it takes the form G µ ··· µ n ( P f , P i , L ) = (cid:20) L (cid:88)(cid:90) k (cid:21) k µ · · · k µ n ω k (( P f − k ) − m + i(cid:15) )(( P i − k ) − m + i(cid:15) ) (cid:12)(cid:12)(cid:12) k = ω k . (26)In this work we will specifically need the scalar and vector G -functions, denoted G and G µ respectively. These aredefined by keeping zero or one factor, respectively, of k µ in the numerator of the integrand. With these in hand, W µL, df can be defined via its relation to W µ df as follows: W µL, df ( P f , P i , L ) = W µ df ( P f , P i ) + f ( Q ) M ( s f ) (cid:104) ( P f + P i ) µ G ( P f , P i , L ) − G µ ( P f , P i , L ) (cid:105) M ( s i ) . (27)Here f ( Q ) is the form factor of the charged particle while the form factor of the neutral particle, which vanishesidentically at Q = 0, is assumed negligible for all values of momentum transfer. IV. MATRIX ELEMENTS OF THE CONSERVED VECTOR CURRENT
Having introduced the general formalism, we proceed to perform the checks outlined in the introduction. The firstcheck is to show that, for any finite-volume state, the matrix element with respect to the charge operator (cid:98) Q ≡ (cid:90) d x J ( x ) , (28)is predicted by the formal mapping to be L -independent and equal to the charge of the state. To demonstrate this, wefirst introduce another expression for the Lellouch-L¨uscher factor. Evaluating the energy derivative of F in Eq. (25),one can show R ( P n ( L ) , L ) = 1 M ( s n ( L )) (cid:20) − ∂∂E M − ( s ) + 2 E G ( P, L ) − G µ =0 ( P, L ) (cid:21) − P = P n ( L ) . (29)Here we have also adopted the shorthand G ( P, L ) ≡ G ( P, P, L ), i.e. we do not repeat the total momentum argumentwhen it is the same for the incoming and outgoing states. Note that, in this subsection, we are considering not onlythe finite-volume bound state but also excited states. We do continue to restrict attention to the S -wave only.Substituting this result into Eq. (23), and also taking the relation between W µ df and F µ [Eq. (18)], we find L (cid:104) P n,f , L | J µ | P n,i , L (cid:105) = F µ ( P f , P i ) + f ( Q ) (cid:2) ( P i + P f ) µ G ( P f , P i , L ) − G µ ( P f , P i , L ) (cid:3)(cid:113)(cid:2) − ∂ E i M − ( s i ) + 2 E i G ( P i , L ) − G µ =0 ( P i , L ) (cid:3)(cid:2) − ∂ E f M − ( s f ) + 2 E f G ( P f , L ) − G µ =0 ( P f , L ) (cid:3) (cid:12)(cid:12)(cid:12)(cid:12) P i,f = P i,f ( L ) . (30)This result will prove very powerful in the following derivations. To see the consequences of this for the charge operatorwe set µ = 0 in the vector current and also set the initial and final-states to coincide. This yields (cid:104) P n , L | (cid:98) Q | P n , L (cid:105) = F ( P ) + f (0) (cid:2) EG ( P, L ) − G ( P, L ) (cid:3) − ∂ E M − ( s ) + 2 EG ( P, L ) − G ( P, L ) (cid:12)(cid:12)(cid:12)(cid:12) P = P n ( L ) , (31)where we have used the x -independence of the matrix element to replace L J (0) → (cid:98) Q and have defined F ( P ) ≡F ( P, P ) as a convenient shorthand for systems with identical initial and final momenta.This can be further simplified via the identity F ( P ) = Q M ( s ) ∂∂E M ( s ) = − Q ∂∂E M − ( s ) , (32)which immediately follows from Eqs. (18) and (20). Substituting this into the numerator of Eq. (31) and also using f (0) = Q , we recover a very satisfying cancellation of all terms to deduce (cid:104) P n , L | (cid:98) Q | P n , L (cid:105) = Q , (33)as expected. This is a highly non-trivial verification that the general + J → finite-volume formalism is consistentthe consequences of current conservation. The derivation relies on two unexpected identities: First, the fact thatthe energy-derivative of F ( P, L ) can be expressed using the G -functions, as shown in Eq. (29), and second, that theWard-Takhashi identity relates F ( P ) to the scattering amplitude, Eq. (32). V. BOUND STATE IN A FINITE VOLUME
We now turn to the implications of the general formalism for bound-state matrix elements in a finite volume.
A. Volume effects on the energies
We start by reviewing results for finite-volume effects in the energy level, E P B ( L ), defined to coincide with themoving bound state in the infinite-volume limit,lim L →∞ E P B ( L ) = E P B ≡ (cid:113) M + P . (34)Boosting these energies to the rest frame, we also define s P B ( L ) ≡ E P B ( L ) − P ≡ s B + δs P B ( L ) , (35)with s B = lim L →∞ s P B ( L ) = M . Note that the finite-volume energies depend on P , even after boosting back tothe rest frame. In the following, we give expressions for the volume-induced shift, δs P B ( L ), for two values of totalmomentum. This represents a small subset of the more general expressions derived in Ref. [18]. The quantization condition, Eq. (21), is satisfied only at the finite-volume energies, e.g. at P B ( L ) ≡ ( E P B ( L ) , P ).We are thus strictly interested in F ( P, L ) only when it is evaluated at these points. However, taking δs B as a smallparameter, we note F ( P B ( L ) , L ) = F ( P B , L ) + O ( δs B ) , (36)where P B ≡ ( E P B , P ) is the infinite-volume bound-state momentum in a moving frame.As is discussed in detail in Ref. [18] and reviewed in Appendix C, the subthreshold L -dependence of the F -functionis governed by the binding momentum: κ ≡ m − M B /
4. In particular from Eqs. (C3) and (C16) we find F ( P B , L ) = − πM B (cid:88) m (cid:54) = e iL m · P / e − κ B L | m (cid:48) | L | m (cid:48) | , (37)where m (cid:48) ≡ m + ( γ − m · P | P | P , (38)and γ = E P B /M B . This result is to be combined with the inverse scattering amplitude, also evaluated at s P B ( L ), butthen expanded in powers of δs B to yield M − ( s P B ( L )) = δs B ∂∂s M − ( s ) (cid:12)(cid:12)(cid:12) s = s B + O ( δs ) , (39)= − δs B /g + O ( δs ) , (40)where we have used M − ( s B ) = 0. Combining Eqs. (37) and (40) then yields the elegant result δs P B ( L ) = g F ( P B , L ) + O ( e − κ B L ) , (41)which shows that the leading shift to the finite-volume bound state is given directly by the F -function, evaluated atthe infinite-volume bound-state energy.To close this section we think it useful to unpack Eq. (41) for a two specific cases. First, in the case of vanishingmomentum in the finite-volume frame, the three universal orders are given by δs [000]B = − g πM B L (cid:20) e − κ B L + √ e −√ κ B L + 43 √ e −√ κ B L (cid:21) + O ( e − κ B L ) . (42) Related results for bound states can be found in Refs. [95, 96, 100, 101]. O ( e − κ B L ) higher derivatives of the inverse amplitude enter, requiring information beyond the coupling, g .For nonzero momenta, the expressions are complicated by the relation between m (cid:48) and m , and by the volumedependence entering γ through E P B = (cid:112) M + (2 π/L ) n . Useful results can be reached, however, by expanding inall L dependence. Performing such an expansion, and neglecting terms scaling as e − κ B L /L and e −√ κ B L , we find δs [00 n ]B = − g [4 + 2 cos( nπ )] e − κ B L πM B L (cid:20) − n cos( nπ )4 + 2 cos( nπ ) 4 π κ B M L (cid:21) . (43)To compare these results to those in Ref. [18] we note that, in the earlier work, the authors introduce an L -dependentbinding momentum, defined via κ B ( L ) = m − s B ( L ) /
4. Then the finite-volume shift, δκ B ( L ) ≡ κ B ( L ) − κ B satisfiesthe relation δs P B ( L ) = − κ B δκ P B ( L ) + O ( δs ) . (44)Combining this with the relation between the coupling and the scattering phase1 g = 164 πκ B M B (cid:18) − κ B dd q (cid:63) q (cid:63) cot δ ( q (cid:63) ) (cid:19) s = s B , (45)yields Eq. (9) of Ref. [18].In closing we comment that, due to the reduction of rotational symmetry, higher partial waves do induce finite-volume corrections to the scalar bound state and corresponding matrix elements. In particular, for P = 0, (cid:96) = 0 mixeswith (cid:96) = 4 , , . . . , as can be seen by the fact that the corresponding off-diagonal components of F are nonzero. Theseadditional angular-momentum contributions are, in fact, not volume-suppressed relative to the S -wave contributions,but are suppressed by powers of the binding momentum in units of the scattering-length analogs appearing in higher-partial waves. For example the (cid:96) = 4 phase shift satisfies an expansion analogous to Eq. (6) q (cid:63) cot δ (cid:96) =4 ( s ) = M q (cid:63) + O ( q (cid:63) − ) , (46)where M has units of energy. In the case of zero spatial momentum in the finite-volume frame, one can show thatthe first non S -wave contribution to s [000]B ( L ) is suppressed relative to the leading shift by a factor of κ M B / M .Having reproduced the known expansion for the binding energy [18], we now turn to the finite- L corrections of thebound-state matrix element. B. Volume effects on the matrix elements
1. Matrix elements in the L → ∞ limit We begin by confirming that, in the L → ∞ limit, the finite-volume bound-state matrix element (as describedby the general + J → formalism) coincides with its infinite-volume counterpart. Here it is important to stressthat the various quantities we consider have a well-defined L → ∞ limit, only because we are considering them atsub-threshold kinematics and thus away from a set of finite-volume poles that becomes arbitrarily dense.We begin with Eq. (27), the relation between W µL, df and W µ df . For L → ∞ , these two quantities coincide because the G function defining their difference vanishes. This is the case because the sum within G (cf. Eq. (26)) is transformedto an integral in the limit and is exactly canceled by the second, subtracted integral. Equation (23) thus becomeslim L →∞ L (cid:104) P B ,f , L | J µ | P B ,i , L (cid:105) = lim L →∞ W µ df ( P B ,f ( L ) , P B ,i ( L )) (cid:113) R ( P B ,f ( L ) , L ) R ( P B ,i ( L ) , L ) . (47)The next step is to expand R , evaluated at the finite-volume bound-state energy, about large L . Using the form givenby Eq. (25), one readily finds R ( P B ( L ) , L ) = −M − ( s P B ( L )) (cid:20) E B g + O ( e − κ B L ) (cid:21) − , (48)= − (cid:0) s P B ( L ) − s B (cid:1) E B g (cid:104) O ( e − κ B L ) (cid:105) . (49)1We are now in position to evaluate the limit. The only subtlety is that a double-zero, arising from the Lellouch-L¨uscher factors, is exactly canceled by the double pole in W µ df . Substituting Eqs. (17) and (49) into Eq. (47), wereach lim L →∞ (cid:112) E B ,i E B ,f L (cid:104) P B ,f , L | J µ | P B ,i , L (cid:105) = (cid:104) P f , B | J µ | P i , B (cid:105) = ( P B ,i + P B ,f ) µ F B ( Q ) . (50)This is exactly the desired result, with the extra factors on the left-hand side accounting for the different normalizationconventions of finite- and infinite-volume states.In this derivation we did not make reference to the Lorentz structure of the current, only to the fact that the + J → amplitude, W , must have a double pole structure associated with the initial and final bound states. As aresult, in general the formalism fulfills the expectation that for an arbitrary current J µ ...µ n lim L →∞ (cid:112) E B ,i E B ,f L (cid:104) P B ,f , L | J µ ...µ n | P B ,i , L (cid:105) = (cid:104) P f , B | J µ ...µ n | P i , B (cid:105) . (51)
2. Large L expansion of the bound-state matrix element As shown in Sec. IV, the conserved vector current leads to volume-independent matrix elements at zero momentumtransfer. Thus, to reach an interesting large- L expansion, in this section we turn to a scalar current J and define g P S, B ( L ) ≡ E B ( L ) L (cid:104) P B , L | J | P B , L (cid:105) , (52)where the subscript indicates that this matrix element defines the scalar charge of the bound state. The infinite-volume bound-state scalar charge is recovered in the L → ∞ limit, i.e. g S, B ≡ lim L →∞ g P S, B ( L ). In direct analogy toEq. (31) above, we observe g P S, B ( L ) = F ( s ) + g S G ( P, L ) − ∂ s M − ( s ) + G ( P, L ) − G ( P, L ) /E (cid:12)(cid:12)(cid:12) P = P B ( L ) . (53)Here F ( s ) ≡ M − ( s ) W df ( P, P ) with W df given by Eq. (13), in which the vector current is replaced by a scalar.Note that the numerator includes only the scalar G -function, reflecting the scalar current considered. However, thedenominator remains identical to the vector case since the Lellouch-L¨uscher factors are independent of the current.We have also introduced g S as the scalar charge of the single-particle state, g S ≡ f (0), where f is the single particleform-factor f ( Q ) ≡ (cid:104) P f , g S | J | P i , g S (cid:105) . (54)As above, we take the coupling of the current to the other constituent particle to be negligible.With these ingredients in hand it is straightforward to expand Eq. (53) about L → ∞ to reach g P S, B ( L ) g S, B = 1 + δs P B ( L ) ∂∂s (cid:20) F B ( s ) g S, B + g ∂∂s M − ( s ) (cid:21) + g ( g S − g S, B ) g S, B G ( P B , L ) + g G ( P B , L ) E B + O ( e −√ κ B L ) . (55)We note that a great deal of structural information enters the leading finite-volume correction. The δs B ( L )-dependentterm is the correction induced from the energy shift and is thus proportional to energy derivatives of both the inverseamplitude and the + J → transition amplitude (entering via F ( s )). The second term in Eq. (55) arises due to amismatch between the scalar charge of the bound-state and the summed charges of its constituents. The final termin Eq. (55) is a direct consequence of the triangle diagram, Fig. 2(c).We close this section with a final, more explicit result for the leading-volume correction in the case where the CMFand finite-volume frames coincide, i.e. P = . Substituting the leading result for δs [000]B , and results from AppendixC for the G -functions, one finds g [000] S, B ( L ) g S, B = 1 + g e − κ B L πM B κ B (cid:20) g S − g S, B g S, B + 4 κ B M L − κ B L ∂∂s (cid:18) F B ( s ) g S, B + g ∂∂s M − ( s ) (cid:19) s = s B (cid:21) + O ( e −√ κ B L ) . (56)The leading 1 in the square brackets arrises from the triangle diagram, Fig. 2(c), and will be the dominant finite-volumeeffect provided | g S − g S,B | ≤ | g S,B | / − − − − − − q ⋆ / MeV q ⋆ cot δ ( q ⋆ ) − p − q ⋆ Figure 3. Plot of q (cid:63) cot δ ( q (cid:63) ) and − (cid:112) − q (cid:63) as a function of q (cid:63) , in units of MeV , for the effective range expansion, Eq. (6),using the scattering length and effective range for the pn -system in S . The vertical dashed line indicates the deuteron, withbinding momentum κ B ∼ .
58 MeV. − − − − −
20 4 5 6 7 8 m π Lκ B L . . . E ⋆ n ( L ) − m / M e V | n | = 0 | n | = 1 (a) m π Lκ B L . . . g P S , B ( L ) / g S , B | n | = 0 | n | = 1 (b)Figure 4. (a) Finite-volume energy spectrum as a function of m π L for pn -scattering parameters as explained in the text. Thetwo colors indicate a system at rest in the finite-volume frame (red) and a system that is boosted with one unit of momentum, P = 2 π n /L and | n | = 1 (blue-green). The solid lines show the prediction of the L¨uscher quantization condition, which holds upto e − m π L . The dashed lines show a prediction based on the leading O ( e − κ B L ) term in the large- L expansion. All four curvesasymptote to the horizontal line at M B − m ∼ − .
21 MeV, the infinite-volume binding energy. (b) Ratio of the finite-volumebound-state matrix element, g P S, B ( L ) = 2 E B ( L ) L (cid:104) P B , L | J | P B , L (cid:105) , to the infinite-volume scalar charge, g S, B , as a function of m π L . The solid curves show the prediction of the full formalism and the dashed lines show the leading term in the large- L expansion, given by Eq. (56) and its moving-frame analog. All four curves asymptote to 1. C. Numerical expectations for finite-volume dependence
In this section, we use the full + J → formalism to explore the finite-volume corrections to a bound-statematrix element in an example with scattering parameters chosen to mimic the deuteron. As above we consider thesimplest case of a scalar current and an S -wave bound state and examine the finite-volume corrections given byEqs. (53) and (55). For the → scattering amplitude, we use use the phenomenological values for the pn scatteringlength ( a = 5 .
425 fm) and effective range ( r = 1 .
749 fm) to describe the scattering amplitude and spectrum. Withthe nucleon mass at m = 934 MeV , the deuteron bound-state pole lies at √ s B = 1875 .
63 MeV with a coupling of g = 5370 . κ B = 45 .
58 MeV, with a binding energy − .
21 MeV.The effective range expansion is shown in Fig. 3 as a function of q (cid:63) and the location of the bound state is indicated.We make two assumptions to simplify the numerical exercise: First, we assume that the infinite-volume bound-stateform factor is constant, i.e. F B ( s ) = g S, B . As seen in the fourth term of the brackets in Eq. (56), this contributionis suppressed by 1 /L , thus it is reasonable that this approximation will not strongly alter the prediction. Second, weassume that difference g S − g S, B is numerically small and set g S = g S, B .Within this set-up one can numerically evaluate Eqs. (53) and (55) and compare the results. The first step is todetermine the finite-volume bound-state energy, using the effective-range description of the pn scattering amplitudein the quantization condition, Eq. (21). Figure 4(a) shows the bound-state energy as a function of L for both | d | = 0and 1. The solid lines represents the full solution obtained from Eqs. (3) and (21) with the pn -scattering parameters.The dashed lines correspond to the leading-order approximation using Eq. (41) for the same momenta. These resultsreproduce those of Ref. [18], and we see that for lattice calculations performed at m π L ∼
4, deviations between theexact and approximated forms are significant.Turning to the two-particle matrix elements, Fig. 4(b) shows the ratio of the finite-volume bound-state matrixelement to the infinite-volume scalar charge, g P S, B ( L ) / g S, B . Solid lines represent the full solution, using Eq. (53)evaluated at the finite-volume energy, and the dashed lines give the leading-order shift of Eq. (55). Again, significantdeviations arise between the full prediction, the leading-order expansion, and the infinite-volume result. This illustratesthat, to reliably extract infinite-volume matrix elements of shallow bound states like the deuteron, it is highly beneficialto use the full formalism which removes an infinite series of terms scaling as powers of e − κ B L . In the present example,only at m π L ∼ VI. CONCLUSION
In this work, we have provided strong consistency checks on, and also explored various consequences of, the formalismderived in Refs. [7, 8], which gives a relation between finite-volume matrix elements, schematically denoted (cid:104) |J | (cid:105) L ,and the corresponding infinite-volume + J → amplitudes.First, in the case of the conserved vector current, we have shown that resulting prediction for the two-particle matrixelement of the charge operator, (cid:104) | (cid:98) Q | (cid:105) L , behaves as expected. Specifically, the matrix element is L -independent andequal to the sum of the constituent charges. Though it is clear that this relation must hold, the way it arises inthe mapping is highly non-trivial, relying on an identity relating various L -dependent geometric functions [Eq. (29)]as well as a relation between the → and + J → amplitudes that follows from the Ward-Takahashi identity[Eq. (20)].Second, for a generic local current, we have demonstrated that the mapping of Refs. [7, 8], reproduces the expectedbehavior in the case of an S -wave two-particle bound state. By analytically continuing the formal relations belowthreshold to the bound-state pole, we have confirmed that the finite- and infinite-volume matrix elements are equalup to volume corrections scaling as e − κ B L , where κ B is the binding-momentum of the state. This is an expectedextension of the well-known result for the L dependence of the bound-state energy.These two checks give confidence that our admittedly complicated formalism correctly describes two-particle finite-volume states and is ready to be implemented in a LQCD calculation, with the first application likely being the( ππ ) I=1 + J µ → ( ππ ) I=1 transition amplitude, allowing one to extract the electromagnetic form factors of the ρ .As an additional example of the utility of the general approach, we have determined the full functional form of theleading, O ( e − κ B L ) correction to the bound-state matrix element of a local scalar current. The result, Eq. (56), showsthat the coefficient of the leading exponential depends on the bound state’s coupling to the two-particle asymptoticstate, the scalar charges of both the bound state and its constituents, and also on derivatives of both the → scattering amplitude and the bound-state form factor. We work here with isospin symmetry, approximating m p = m n . + J → over a range of energies, including in a neighborhoodaround the bound-state pole. Doing so removes an infinite series of terms scaling as powers of e − κ B L and, for shallowbound-states, allows one to control an otherwise dominant source of systematic uncertainty. To stress this point, asa final exercise, we have presented numerical comparisons of the leading e − κ B L correction with the full finite-volumeshift, for a toy set-up mimicking a LQCD calculation of the deuteron’s scalar charge. For physical pion masses andvolumes in the range m π L ∼ L correction will dominate the infinite-volume charge andthat removing only the leading exponential also does not give a reliable extraction. Thus, we conclude that the fullmethod must be used to gain a reliable result for the form factors of shallow bound states as well as resonances.This work makes use of identities that will be presented in a companion article that outlines, in detail, the analyticstructure of the generalized form factors considered here [99]. An additional check is underway to reproduce analyticexpressions for the 1 /L expansion presented in Ref. [102], for the threshold-state matrix element of a scalar currentin a weakly-coupled system. VII. ACKNOWLEDGEMENTS
We thank Alessandro Baroni, Felipe Ortega-Gama, and Akaki Rusetsky for useful discussions. RAB is supportedin part by USDOE grant No. DE-AC05-06OR23177, under which Jefferson Science Associates, LLC, manages andoperates Jefferson Lab. RAB also acknowledges support from the USDOE Early Career award, contract de-sc0019229.
Appendix A: Proof of Eq. (12)
In this appendix we demonstrate that, in theories with a two-particle bound state, the + J → amplitude satisfiesEq. (12), repeated here for convenience W µ ( P f , P i ) = ( P i + P f ) µ F B ( Q ) i ( ig ) ( s f − s B )( s i − s B ) (cid:2) O ( s i,f − s B ) (cid:3) . (A1)To show this, it is necessary to return to the matrix element definition of the amplitude, W µ , given in Eq. (9).Inserting a complete set of states on either side of the current, J µ , we reach (cid:90) d P (cid:48)(cid:48) (2 π ) E P (cid:48)(cid:48) B (cid:90) d P (cid:48) (2 π ) E P (cid:48) B (cid:104) P f , ˆk (cid:63)f , out | P (cid:48)(cid:48) , B (cid:105)(cid:104) P (cid:48)(cid:48) , B | J µ | P (cid:48) , B (cid:105)(cid:104) P (cid:48) , B | P i , ˆk (cid:63)i , in (cid:105) = W µ ( P f , ˆk (cid:63)f ; P i , ˆk (cid:63)i ) + · · · , (A2)where we have kept only the bound-state sector of the Fock space, as this will be sufficient to identify the pole thatwe are after.Three additional subtleties arise here: (1) To properly implement the normalization of the bound-state, (cid:104) P (cid:48) , B | P, B (cid:105) ≡ (2 π ) E P B δ ( P (cid:48) − P ) , (A3)we must integrate over all spatial momenta with the standard Lorentz-invariant factor as shown. (2) Since the spectraldecomposition can only be performed on the full matrix element we have dropped the “conn” subscript that appearsin Eq. (9). To preserve the definition we have included the ellipsis on the right-hand side, which is understood torepresent all disconnected contributions. These will, however, play no role, since they do not contain the bound-statepole. (3) The expression we are after requires the analytic continuation of P f and P i to the sub-threshold region.This is subtle at the level of Fock states, and is more easily understood by rewriting the result in terms of operatorsprojected to definite momentum. This, in turn, reveals that the time-ordering of the operators must be carefullytreated, as we explain in more detail below.The next step is to substitute (cid:104) P f , ˆk (cid:63)f , out | P (cid:48)(cid:48) , B (cid:105) ≡ (2 π ) δ ( P f − P (cid:48)(cid:48) B ) ig , (A4) (cid:104) P (cid:48) , B | P i , ˆk (cid:63)i , in (cid:105) ≡ (2 π ) δ ( P i − P (cid:48) B ) ig . (A5)Here the four-dimensional delta function arises in direct analog to the standard relation between T matrix andscattering amplitude and leads to the definition of the bound-state coupling g . Using the spatial delta functions toevaluate the integrals in Eq. (A2), we reach (cid:90) ∞−∞ d x (cid:48)(cid:48) e ix (cid:48)(cid:48) ( E f − E P (cid:48)(cid:48) B ) (cid:90) ∞−∞ d x (cid:48) e − ix (cid:48) ( E i − E P (cid:48) B ) ig E P (cid:48)(cid:48) B (cid:104) P (cid:48)(cid:48) , B | J µ | P (cid:48) , B (cid:105) ig E P (cid:48) B = W µ ( P f , ˆk (cid:63)f ; P i , ˆk (cid:63)i ) + · · · , (A6)5where it is understood that one must set P (cid:48)(cid:48) → P f and P (cid:48)(cid:48) → P i . Here we have also written the remaining temporaldelta functions as integrals over time.Introducing the integrals over x (cid:48)(cid:48) and x (cid:48) allows us to address the subtlety mentioned as point (3) above. Studyingthe correlation functions reveals that the above expression does not correctly treat all time orderings. For the presentcase, this is resolved by restricting the integral over x (cid:48)(cid:48) from 0 to ∞ and similarly that over x (cid:48) from −∞ to 0. Doingso, and also including the i(cid:15) prescription required to project the external states in the correlator to the vacuum, onecan evaluate both integrals to reach i ( ig )2 E P (cid:48)(cid:48) B ( E f − E P (cid:48)(cid:48) B ) i ( ig )2 E P (cid:48) B ( E i − E P (cid:48) B ) (cid:104) P (cid:48)(cid:48) , B | J µ | P (cid:48) , B (cid:105) = W µ ( P f , ˆk (cid:63)f ; P i , ˆk (cid:63)i ) + · · · . (A7)This is the result that we had aimed to prove. Up to the O (( s i,f − s B ) ) terms that we neglect, one can replaceeach pole with the covariant form and also drop the ellipses. Projecting both sides to the S -wave, and substitutingEq. (11), we deduce Eq. (12). Appendix B: Ward-Takahashi identity for 2 + J →
In this appendix, we demonstrate how Eq. (20) follows from the Ward-Takahashi identity. A consequence of currentconservation, the Ward-Takahashi identity relates a given n -point Green function, coupled to an external conservedcurrent, to the corresponding ( n − C µ be a 5-pointfunction coupling the conserved vector current, J µ , to two neutral and two charged mesons. The Ward-Takahashiidentity then reads q µ C µ ( p (cid:48) , k (cid:48) ; p, k ) = Q (cid:104) C ( p (cid:48) + q, k (cid:48) ; p, k ) − C ( p (cid:48) , k (cid:48) ; p − q, k ) (cid:105) , (B1)where q µ = ( p (cid:48) + k (cid:48) ) µ − ( p + k ) µ = P (cid:48) µ − P µ , with the second equality introducing notation for the total momenta ofthe outgoing and incoming two-meson states. We have also introduced C (with no index) as the four-point functionwithout the current insertion. We further define k and k (cid:48) as the initial- and final-state momenta of the neutralparticles, respectively, and p = P − k and p (cid:48) = P (cid:48) − k (cid:48) as the corresponding momenta for the particles carrying thecharge, Q .The + J → and → amplitudes, W µ and M respectively, are related to the Green functions by amputatingthe external meson propagators and placing them on the mass shell, i.e., C amp −−−−−→ on-shell M , (B2) C µ amp −−−−−→ on-shell W µ , (B3)where the amputated Green functions are defined as C amp ( p (cid:48) , k (cid:48) ; p, k ) ≡ ( p (cid:48) − m )( k (cid:48) − m )( p − m )( k − m ) C ( p (cid:48) , k (cid:48) ; p, k ) , (B4)and the same with the µ index included on both sides. Considering only the amputation at this stage and substitutingEq. (B4) into (B1), we find q µ C µ amp ( p (cid:48) , k (cid:48) ; p, k ) = Q (cid:20) C amp ( p (cid:48) , k (cid:48) ; p + q, k ) p − m ( p + q ) − m − p (cid:48) − m ( p (cid:48) − q ) − m C amp ( p (cid:48) − q, k (cid:48) ; p, k ) (cid:21) , (B5)where the ratios of amputation factors arise since the Ward-Takahashi identity changes the momenta carried by themesons on the two sides of the equation.In the limit where p (cid:48) , k (cid:48) , p and k go on shell, the numerators on the right hand side of Eq. (B8) vanish but thedenominators do not, yielding the well-known Ward identity: q µ W µ = 0. In addition, the long-range pieces thatdefine the difference between W µ and W µ df [see Fig. 2(b)] are proportional to ( P (cid:48) + P ) µ and therefore also vanishwhen contracted with q µ . (Equivalently they are proportional to the single-particle matrix element of J µ and musttherefore also satisfy the Ward identity.) It follows that W µ df itself satisfies the identity: q µ W µ df = 0.Returning to the off-shell relation, Eq. (B8), we re-express all functions in terms of P, P (cid:48) , k and k (cid:48) to write q µ C µ amp ( P (cid:48) − k (cid:48) , k (cid:48) ; P − k, k ) =Q (cid:20) C amp ( P (cid:48) − k (cid:48) , k (cid:48) ; P (cid:48) − k, k ) ( P − k ) − m ( P (cid:48) − k ) − m − ( P (cid:48) − k (cid:48) ) − m ( P − k (cid:48) ) − m C amp ( P − k (cid:48) , k (cid:48) ; P − k, k ) (cid:21) . (B6)6Applying a P (cid:48) ν derivative on the left-hand side then gives ∂∂P (cid:48) ν [LHS] = C ν amp ( P (cid:48) − k (cid:48) , k (cid:48) ; P − k, k ) + q µ ∂ C µ amp ( P (cid:48) − k (cid:48) , k (cid:48) ; P − k, k ) ∂P (cid:48) ν , (B7)and, applying the same to the right-hand side, one finds ∂∂P (cid:48) ν [RHS] = Q ∂ C amp ( P (cid:48) − k (cid:48) , k (cid:48) ; P (cid:48) − k, k ) ∂P (cid:48) ν ( P − k ) − m ( P (cid:48) − k ) − m − ( P (cid:48) − k ) ν C amp ( P − k (cid:48) , k (cid:48) ; P − k, k ) ( P − k ) − m [( P (cid:48) − k ) − m ] − ( P (cid:48) − k ) ν ( P − k (cid:48) ) − m C amp ( P − k (cid:48) , k (cid:48) ; P − k, k ) . (B8)Next, before equating the two sides, we take the zero-momentum-transfer limit ( P (cid:48) → P ) and substitute w µ ( P − k ; P − k ) = 2( P − k ) µ Q , (B9)for the + J → matrix element at zero momentum transfer. This then gives C µ amp ( P − k (cid:48) , k (cid:48) ; P, P − k ) = Q ∂∂P µ C amp ( P − k (cid:48) , k (cid:48) ; P − k, k )+ i C amp ( P − k (cid:48) , k (cid:48) ; P − k, k ) i ( P − k ) − m w µ ( P − k ; P − k )+ w µ ( P − k (cid:48) ; P − k (cid:48) ) i ( P − k (cid:48) ) − m i C amp ( P − k (cid:48) , k (cid:48) ; P − k, k ) . (B10)Setting P = P (cid:48) has greatly simplified the expressions, but care must be taken as the second and third terms on theright-hand side will diverge when we set p (cid:48) , k (cid:48) , p and k to their on shell values. Indeed these are the same divergencesthat appear in the difference between W µ and W µ df , with the only subtlety that they were first defined in on-shellamplitudes at P (cid:48) − P (cid:54) = 0. Fortunately, in the present case the distinction is unimportant because, when appliedto the divergence-free amplitude, the zero-momentum-transfer and on-shell limits commute. We can thus move thesecond and third terms to the left-hand side and take p (cid:48) , k (cid:48) , p, k on shell to conclude W µ df ( P, ˆk (cid:63)f ; P, ˆk (cid:63)i ) = Q ∂∂P µ M ( s, ˆk (cid:63)f , ˆk (cid:63)i ) . (B11)This remarkable result gives a clear interpretation to W µ df in the forward limit.Finally, since the derivative is with respect to total momenta, we can easily project both sides to definite angularmomentum. This leads to W µ df ,(cid:96) (cid:48) m (cid:48) ,(cid:96)m ( P ) = δ (cid:96) (cid:48) (cid:96) δ m (cid:48) m Q ∂∂P µ M (cid:96) ( s ) , (B12)as claimed in Eq. (20) for the special case of S -wave systems. Appendix C: Analytic continuation of finite-volume functions below threshold
In this section we give results for the analytic continuations of the F - and G -functions below threshold. Specificallywe require results for F ( P, L ), G ( P, L ) and G µ =0 ( P, L ), where we recall that a single momentum argument within G indicates that the initial- and final-state four-momenta are equal. Each of these can be written in terms of a class offunctions naturally extending those defined in Refs. [13–16]: c ( n ) JM ( P, L ) = (cid:20) L (cid:88)(cid:90) k (cid:21) ω (cid:63) k ω k √ π k (cid:63) J Y JM ( ˆk (cid:63) )( q (cid:63) − k (cid:63) + i(cid:15) ) n . (C2) The c ( n ) JM are proportional to the dimensionless functions denoted by Z ( n ) JM in Ref. [8] c ( n ) JM ( P, L ) = L n − (2 π ) n (cid:18) πL (cid:19) J Z ( n ) JM ( P, L ) . (C1)For the analytic work presented here, the dimensionful versions prove slightly more convenient. F ( P, L ) = 12 E (cid:63) c (1)00 ( P, L ) , (C3) G ( P, L ) = 14 E (cid:63) c (2)00 ( P, L ) , (C4) G µ =0 ( P, L ) = − E E (cid:63) c (1)00 ( P, L ) + E E (cid:63) c (2)00 ( P, L ) + 14 √ P z E (cid:63) c (2)10 ( P, L ) , (C5)where the last result also assumes that P is parallel to the ˆ z axis.For P < (2 m ) , the summand of c ( n ) is a smooth function of k (cid:63) with a finite region of analyticity. As a result,the sum and integral must become exponentially close to each other, with the scale in the exponential given by thegrid-spacing of the sum (set by L ) and the size of the analytic domain (set by 4 m − P ). To make this explicit, weapply the Poisson summation formula to c ( n ) JM , evaluated at a generic sub-threshold four-momentum, P κ , satisfying m − P κ / κ . We find c ( n ) JM ( P κ , L ) = ( − n (cid:88) m (cid:54) = (cid:90) d k (cid:63) (2 π ) √ πk (cid:63) J Y JM ( ˆk (cid:63) )( κ + k (cid:63) ) n e iL m · k , (C6)where we have used the fact that the integration measure is a Lorentz invariant, d k /ω k = d k (cid:63) /ω (cid:63) k .The kinematic variables in the CMF are related to the moving frame variables via standard Lorentz transformations, k (cid:63) || = γ ( k || − ω k β ) , k (cid:63) ⊥ = k ⊥ ,ω (cid:63) k = γ ( ω k − β · k ) , (C7)where k ⊥ = k − k || , k || = ( k · ˆ β ) ˆ β , β = P /E is the velocity, and γ = E/E (cid:63) the Lorentz factor. We can then writethe phase factor in terms of the CMF momenta, m · k = m (cid:48) · k (cid:63) + ω (cid:63) k E (cid:63) m · P , (C8)with m (cid:48) defined in Eq. (38).With these relations in hand we can write the integrand solely in terms of k (cid:63) , c ( n ) JM ( P κ , L ) = ( − n (2 π ) (cid:88) m (cid:54) = (cid:90) ∞ d k (cid:63) ( k (cid:63) ) J +2 ( k (cid:63) + κ ) n e iLω (cid:63) k m · P /E (cid:63) (cid:90) d ˆk (cid:63) √ πY JM ( ˆk (cid:63) ) e iL m (cid:48) · k (cid:63) . (C9)Next, we evaluate the angular piece by introducing spherical Bessel functions and making use of the standard planewave expansion, e iL m (cid:48) · k (cid:63) = 4 π ∞ (cid:88) (cid:96) =0 i (cid:96) j (cid:96) ( L | m (cid:48) | k (cid:63) ) (cid:96) (cid:88) m (cid:96) = − (cid:96) Y (cid:96)m (cid:96) ( ˆm (cid:48) ) Y ∗ (cid:96)m (cid:96) ( ˆk (cid:63) ) , (C10)where j (cid:96) ( z ) is the spherical Bessel function of the first kind. The angular integral in Eq. (C9) becomes I JM ( Lk (cid:63) | m (cid:48) | , m ) ≡ (cid:90) d ˆk (cid:63) √ πY JM ( ˆk (cid:63) ) e iL m (cid:48) · k (cid:63) , (C11)= (4 π ) / i J j J ( L | m (cid:48) | k (cid:63) ) Y JM ( ˆm (cid:48) ) . (C12)For the cases considered here we require only I ( Lk (cid:63) | m (cid:48) | , m ) = 4 π sin ( Lk (cid:63) | m (cid:48) | ) Lk (cid:63) | m (cid:48) | , (C13) I ( Lk (cid:63) | m (cid:48) | , m ) = i π √ (cid:32) sin ( Lk (cid:63) | m (cid:48) | )( Lk (cid:63) | m (cid:48) | ) − cos ( Lk (cid:63) | m (cid:48) | ) Lk (cid:63) | m (cid:48) | (cid:33) m (cid:48) · P | m (cid:48) || P | . (C14)8To evaluate the remaining integral over k (cid:63) , we express the sinusoidal functions in I JM in terms of exponentialsand then divide the function I JM into two terms, denoted I (+) JM and I ( − ) JM : I (+) JM is defined by replacing the sinusoidalfunctions with the part of their exponential representation that decays as k (cid:63) → i ∞ , e.g. sin( x ) → e ix / (2 i ), and I ( − ) JM is defined in the same way for the part that decays as k (cid:63) → − i ∞ , e.g. sin( x ) → − e − ix / (2 i ). This leads to adecomposition of c ( n ) JM into two integrals c ( n ) JM ( P κ , L ) = ( − n (2 π ) (cid:88) x = ± (cid:88) m (cid:54) = (cid:90) ∞ d k (cid:63) ( k (cid:63) ) J +2 ( k (cid:63) + κ ) n e iLω (cid:63) k m · P /E (cid:63) I ( x ) JM ( Lk (cid:63) | m (cid:48) | , m ) . (C15)Furthermore, one can show that the I ( ± ) JM factors dominate the behavior at large, imaginary k (cid:63) . It follows that the I (+) JM ( I ( − ) JM ) integral can be evaluated by closing the contour in the upper (lower) half of the complex plane.In addition to the k (cid:63) = ± iκ pole, the integrand has branch cuts starting at k (cid:63) = ± im , associated with the squareroot in ω k . These lead to exponential corrections of the order of O ( e − mL ) that are ignored throughout, i.p. alreadyin deriving the formalism considered in this work. Therefore these contributions should also dropped in the presentevaluations. 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