Two-particle multichannel systems in a finite volume with arbitrary spin
JJLAB-THY-14-1833
Two-particle multichannel systems in a finite volume with arbitrary spin
Raúl A. Briceño ∗ Jefferson Laboratory, 12000 Jefferson Avenue, Newport News, VA 23606, USA (Dated: July 16, 2018)The quantization condition for two-particle systems with arbitrary number of two-body opencoupled channels, spin, momentum, and masses in a finite volume with either periodic or twistedboundary conditions is presented. Although emphasis is placed in cubic volumes, the result holdsfor asymmetric volumes. The result is relativistic, holds for all momenta below the three- andfour-particle thresholds, and is exact up to exponential volume corrections that are governed by
L/r , where L is the spatial extent of the volume and r is the range of the interactions between theparticles. For hadronic systems the range of the interaction is set by the inverse of the pion mass, m π , and as a result the formalism presented is suitable for m π L (cid:29) . The condition presented isin agreement with all previous studies of two-body systems in a finite volume. Implications of theformalism for the studies of multichannel baryon-baryon systems are discussed. I. INTRODUCTION
There is a wealth of experimental investigation of low-energy scattering processes involving two hadrons. Yet tothis day, comparison of the available experimental datawith the underlying fundamental theory of the strong in-teraction, quantum chromodynamics (QCD), has beenlimited. This is due to the fact that at moderately lowenergies QCD is non-perturbative. Currently the onlyreliable approach for studying QCD at low energies islattice QCD (LQCD). LQCD calculations are necessarilyperformed in a finite Euclidean spacetime. Therefore, itis necessary to construct a formalism that connects the fi-nite Euclidean spacetime volume observables determinedvia LQCD to the Minkowski-spacetime infinite-volumequantities of interest. The most easily determined quan-tities via LQCD are the low-energy spectra. For suffi-ciently large volumes satisfying m π L> ∼ , where L is thespatial extent of the finite volume and m π is the pionmass, finite volume effects have minimal impact in thedetermination of the low-lying single-particle spectrumof QCD [1].Although it has been previously pointed out that theEuclidean nature of the calculations imposes challengeson the determination of few-body scattering quantitiesfor arbitrary momenta in the infinite volume limit [2],the fact that LQCD calculations are performed in a fi-nite volume (FV) allows for the extraction of scatter-ing parameters from the spectrum through the Lüscher method [3–19]. This method, which has been widely usedto extract scattering phase shifts and binding energies oftwo-hadron systems from LQCD (see for example Refs.[20–40]), has been generalized to multi-coupled channeltwo-body systems with total spin S ≤ / [41–46]. Therehas also been some progress in generalizing this formal-ism onto three-particle systems [47–50].Although LQCD calculations are commonly performedwith the periodic boundary conditions (PBCs) imposed ∗ [email protected] upon the quark fields in the spatial extents, PBCs area subset of a more general class of boundary conditionsknown as twisted boundary condition (TBCs) [51, 52].TBCs require that fields are proportional to their imagesup to an overall phase, ψ ( x + n L ) = e i θ · n ψ ( x ) , where ≤ θ i < π is the twist angle in the ith Cartesian direc-tion. As a result the free finite volume momenta satisfy p = πL n + θ L , where n is an integer triplet. PBCs arerecovered when the twist angle, θ , is set to zero. It isevident that, at least in the one-body sector, by dial-ing the twist one can in principle access a continuousset of momenta. This is advantageous when performingcalculations in a finite volume where spectra are neces-sarily discretized, and has been explored extensively inthe one-body sector [53–61] as well as the two-body sec-tor [35, 46, 52, 62–65].Here we remove all previous restrictions made inthe literature and present the most general, model-independent relativistically covariant framework for de-termining the finite volume (FV) spectrum for two-particle multichannel systems with arbitrary spin, mo-menta and twist. Although this formalism is developedwith LQCD calculations in mind, the result gives a map-ping between the finite volume spectrum and the infinitevolume scattering amplitude of the system and is inde-pendent of the details of the theory at hand. It is suitablefor studying hadronic physics as well as atomic physicsin a finite volume (see Refs. [66, 67] for examples of theLüscher method applied on atomic systems).Section II reviews basic tools for constructing two-particle states with arbitrary spin. In particular, we re-view details regarding the | lS, Jm J (cid:105) and helicity bases .Section III presents the generalization of the Lüscher for-malism for two-particle systems with arbitrary quantumnumbers, masses and momenta in a periodic finite volume All throughout this work, J will denote the total angular mo-mentum, l is the orbital angular momentum, S will be the two-particle spin, and m J is the azimuthal component of J . Capital “ L ” will solely refer to the spatial extent of the finite volume.The | lS, Jm J (cid:105) basis will be referred to as as the lS basis. a r X i v : . [ h e p - l a t ] F e b and we pay close attention to the evaluation of finite vol-ume s-channel loops. Section IV discusses how this resultis generalized for systems with arbitrary twist and asym-metry volumes. The result presented, Eq. 22, is in agree-ment with all previous studies of two-body systems in afinite volume [3–5, 10–20, 35, 41–46, 63–65]. Section V re-views the implication of this formalism for baryon-baryonsystem, and we discuss its impact on future studies ofhyperon-nucleon and hyperon-hyperon scattering param-eters from LQCD. Precise determination of such interac-tions will impact our understanding of the compositionof dense nuclear matter. Furthermore, although therehas been a great deal of activity at elucidating the poorsignal/noise problem that is inherit of performing LQCDcalculation with finite baryon density and/or chemicalpotential [68–71], nucleon-nucleon LQCD calculations re-main to be more computationally costly than those withhigher strange content. Consequently, it is expected thatLQCD calculations will have a bigger immediate impactin disentangling pertinent information of nuclear systemswith non-zero strangeness. II. CONSTRUCTION OF TWO-PARTICLESTATES WITH ARBITRARY SPIN
In order to understand the claim that the results pre-sented in Secs. III & IV are covariant and applicable forsystems with arbitrary spin it is important to first re-view the basics of the construction of two-particle statesin the lS basis as well as the helicity basis. In order toconstruct single particle states with arbitrary helicity λ and momentum p = p (sin θ cos φ, sin θ sin φ, cos θ ) , it isconvenient to first define a state with zero total momen-tum, definite spin (s) and azimuthal component of spin( λ ), | , s, λ (cid:105) . By acting on this state with a boost alongthe z-axis, ˆ L z ( p ) , followed by a rotation, ˆ R φ,θ, − φ , thattakes the momentum from the z-axis to the desired di-rection of the momentum, one obtains the desired statewith definitive helicity [72–75] | p , sλ (cid:105) = ˆ R φ,θ, − φ ˆ L z ( p ) | , sλ (cid:105) . (1)Two-particles states can be built out of direct product ofthese, | p , s λ ; p , s λ (cid:105) = | p , s λ (cid:105) ⊗ | p , s λ (cid:105) . In general a rotation can be defined by a unitary operator withthree angles as arguments ˆ R α,β,γ = e − iα ˆ J z e − iβ ˆ J y e − iγ ˆ J z , where ˆ J k is the angular momentum operator in the kth cartesian axis.To define a three-dimensional vector only two angles are needed.As a result there is some freedom when defining the rotationoperator. In this work the operators is chosen to be ˆ R φ,θ, − φ , suchthat when θ = 0 and it acts on a state with angular momentumquantized along the z-axis the overall phase vanishes. When restricting oneself to the center of mass (c.m.)frame this simplifies to | q ∗ , λ λ (cid:105) = | q ∗ , s λ (cid:105) ⊗ | − q ∗ , s λ (cid:105) = ˆ R φ,θ, − φ ˆ K z ( q ∗ ) | , s λ (cid:105)⊗ ˆ R φ,θ, − φ ˆ K − z ( q ∗ ) | , s λ (cid:105) , (2)where q ∗ is the relative momenta between the two-particles in the c.m. frame . Note that the explicit s and s labels have been suppressed. From these statesone can readily construct states with definite total angu-lar momentum ( J, m J ) [72–75] | Jm J , λ λ (cid:105) = N J ˆ d Ω D J ∗ m J ,λ ( φ, θ, − φ ) | q ∗ , λ λ (cid:105) , (3)where d Ω = sin θ dθ dφ , λ = λ − λ , D J ∗ m J ,λ ( φ, θ, − φ ) isthe complex conjugate of the Wigner- D matrix defined as D Jm J ,λ ( φ, θ, − φ ) = (cid:104) Jm J | ˆ R φ,θ, − φ | Jλ (cid:105) , and N J = (cid:113) J +14 π as to ensure that the states are properly normalized. Itis straightforward to show that in fact these states trans-form as states with definite angular momentum underrotations ˆ R α,β,γ | Jm J , λ λ (cid:105) = (cid:88) m (cid:48) D Jm J (cid:48) ,m J ( α, β, γ ) | Jm J (cid:48) , λ λ (cid:105) . Although it is customary to use the helicity basis whenconsidering relativistic systems, one can always performcalculation in the lS basis, | lS, Jm J (cid:105) . In order to prop-erly define these states, one may first define the angularmomentum operator ˆ J as a sum of the spin and orbitalangular momentum operators, ˆ J = ˆ S + ˆ l [73, 74]. Ithas been shown that the spin operator can be writtenas a combination of all the generators of the Poincarégroup, i.e., ˆ J (rotations), ˆ K (boosts) and ( ˆ P , ˆ P ) (trans-lations) [73, 74] ˆ S = 1 M (cid:18) ˆ P ˆ J − ˆ P × ˆ K − P + M ˆ P ( ˆ P · ˆ J ) (cid:19) , (4)where M is the mass of the particle of interest. Theorbital angular momentum operator can then definedas ˆ l = ˆ J − ˆ S . Having defined these two opera-tors, one can perform two distinct classes rotations,the first where ˆ S is used as the generator of the rota-tion ˆ R Sα,β,γ = e − iα ˆ S z e − iβ ˆ S y e − iγ ˆ S z and the second for ˆ l ˆ R lα,β,γ = e − iα ˆ l z e − iβ ˆ l y e − iγ ˆ l z . As one would expect, theseoperators can be interpreted as acting on the spin and In general, this describes the component of the wavefunction thatonly depends on the relative coordinates. Center of mass frame coordinates and functions will be givena superscript “ ∗ ” to distinguish them from the lattice framecoordinates. spatial components of the single-particle states respec-tively [74], ˆ R Sφ,θ, − φ | p , sm s (cid:105) = (cid:88) m s (cid:48) D sm s (cid:48) ,m s ( φ, θ, − φ ) | p , sm s (cid:48) (cid:105) , ˆ R lφ,θ, − φ | p , sm s (cid:105) = | p (cid:48) = ˆ R p , sm s (cid:105) , (5)where ˆ R is the three-dimensional representation of therotation acting on p . Effectively one can conclude thatthe ˆ R Sφ,θ, − φ operator acts on a relativistic state as if itwere at rest, while the ˆ R lφ,θ, − φ operator acts on it as if ithad zero spin.For two-particle systems this can be generalized bydefining the two body spin and orbital angular momen-tum operators ˆ S = ˆ S + ˆ S , ˆ l = ˆ l + ˆ l , (6)where ˆ S ( ˆ S ) and ˆ l (ˆ l ) are, respectively, the spin andorbital angular momentum operators that act on the “1”(“2”) particle state. By restricting oneself to the c.m.frame, the two-particle state with total spin S can bedefined in terms of single particle states in the standardway, | q ∗ , Sm S (cid:105) = (cid:88) m s ,m s | q ∗ , s m s (cid:105) ⊗ | − q ∗ , s m s (cid:105)×(cid:104) s m s , s m s | s s , Sm S (cid:105) , where (cid:104) s m s , s m s | s s , Sm S (cid:105) is the Clebsch-Gordancoefficient. Similarly to the one-particle system, one canshow that under R S and R l these states transform as ˆ R Sφ,θ, − φ | q ∗ , Sm S (cid:105) = (cid:88) m S (cid:48) D Sm S (cid:48) ,m S ( φ, θ, − φ ) | q ∗ , Sm S (cid:48) (cid:105) , ˆ R lφ,θ, − φ | q ∗ , sm S (cid:105) = | q ∗(cid:48) = ˆ R q ∗ , Sm S (cid:105) . (7)States with definite orbital angular momentum can beconstructed by integration over all angles of the relativemomentum with the appropriate spherical harmonic | lm l , Sm S (cid:105) = ˆ d Ω Y lm l ( ˆ q ∗ ) | q ∗ , Sm S (cid:105) , (8) ⇒ ˆ R lφ,θ, − φ | lm l , Sm S (cid:105) = (cid:88) m l (cid:48) D lm l (cid:48) ,m l ( φ, θ, − φ ) | lm l (cid:48) , Sm S (cid:105) , (9)where m l is the azimuthal component of the orbital angu-lar momentum. Using Clebsch-Gordan coefficients thesestates can be added appropriately to give a state withtotal angular momentum | lS, Jm J (cid:105) = (cid:88) m l ,m S | lm l , Sm S (cid:105)(cid:104) lm l , Sm S | lS, Jm J (cid:105) . (10)One can show that these states can be written as alinear combination of the states with definite helicity with an overlap factor equal to [72] (cid:104) Jm J , λ λ | lS, Jm J (cid:105) = (cid:18) l + 12 J + 1 (cid:19) / (cid:104) l , Sλ | Jλ (cid:105)×(cid:104) s λ , s − λ | sλ (cid:105) . (11)Since the total angular momentum is a conserved quan-tity, the → scattering amplitude, M , is diagonal in J . One may choose to evaluate its matrix elements inthe helicity basis using Eq. 3, in which case one finds (cid:104) q ∗ f , α α |M| q ∗ i , λ λ (cid:105) = (cid:88) J,m J ( N J ) [ M ] Jm J α α ,λ λ ×D J ∗ m J ,α ( φ f , θ f , − φ f ) D Jm J ,λ ( φ i , θ i , − φ i ) , (12)where [ M ] Jα α ,λ λ is the value of the scattering ampli-tude for a initial state with helicity λ , λ and final helic-ity α , α and that has been projected onto total angu-lar momentum ( J, m J ) . Alternatively, one can write thescattering amplitude in the lS basis using Eq. 10, (cid:104) q ∗ f , S (cid:48) m S (cid:48) |M| q ∗ i , Sm S (cid:105) = 4 π (cid:88) J,mJ,l,l (cid:48) m l ,m l (cid:48) Y l (cid:48) m l (cid:48) (ˆ q ∗ f ) Y ∗ lm l (ˆ q ∗ i ) ×(cid:104) lm l Sm s | lS, Jm J (cid:105)(cid:104) l (cid:48) m l (cid:48) S (cid:48) m s (cid:48) | l (cid:48) S (cid:48) , Jm J (cid:105) [ M ] Jm J l (cid:48) S (cid:48) ,lS , (13)where [ M ] Jm J l (cid:48) S (cid:48) ,lS is the value of the scattering amplitudefor an ingoing state with ( l, S ) and outgoing ( l (cid:48) , S (cid:48) ) andthat has been projected onto total angular momentum ( J, m J ) . The √ π factor for each spherical harmonic hasbeen introduced to simplify the subsequent expressionsin Sec. III A. Given that these two representations areequivalent, in the remainder of this work the lS basiswill be used. All that has been assumed in writing Eq. 13is that the scattering is diagonal in angular momentum.Therefore, Eq. 13 holds for any quantity that is diagonalin J . When considering a system with N open two-bodychannels that can couple, one can simply upgrade thescattering amplitude to also be a matrix in the number ofopen channel. For such cases, the matrix elements of M get an additional subscript associated with the incoming( “ a ” ) and outgoing ( “ b ” ) channel, [ M ] Jm J l (cid:48) S (cid:48) b,lSa . III. TWO-PARTICLE MULTICHANNELSYSTEMS WITH SPIN AND PBCS
Having reviewed the basics of relativistic two-particlestates with spin, one may proceed to determine the fi-nite volume spectra of such systems. To arrive at thequantization condition for multichannel two-particle sys-tems with arbitrary spin, masses and momenta, considera system with total energy (momentum) equal to E ( P )and c.m. energy E ∗ = √ E − P . Allow for the sys-tem to have N open channels that can mix, each com-posed of two-particles with masses m j, and m j, with m j, ≤ m j, and spin s j, and s j, . Each particle canbe either a fermion or a boson. The particles in the jth channel can go on-shell if the c.m. energy satisfies m j, + m j, < ∼ E ∗ (cid:28) E ∗ th , where E ∗ th refers to the first few-particle threshold present in the theory. For instance, fora systems such as ππ − K ¯ K with exact G-parity, we arerestricted to energies below the four-particle threshold,while for the two-nucleon systems the energy is restrictedbelow the pion production threshold. Furthermore, itwill be assumed that no single particle states can go on-shell. The c.m. relative momentum for the jth channelsatisfies k ∗ j = (cid:32) E ∗ − ( m j, + m j, )2 + ( m j, − m j, ) E ∗ (cid:33) . (14)The derivation and details of the quantization condi-tion for systems where the total spin of the open channelsis zero has been presented in Refs. [42, 43]. The remain-ing piece needed to arrive at the result with non-zerospin can be deduced from the S = 1 / and S = 1 singlechannel results [13, 17–19]. These works concluded thatthe distinguishing feature of the power law finite volumecorrections of two-particle propagators between spinlesssystems and nonzero spin systems can be attributed toClebsch-Gordan coefficients which project two-particlesstates with definite spin and orbital angular momentumto a state with total J as shown in Eq. 10. In Sec. III Athe emergence of these Clebsch-Gordan coefficients forgeneric spin systems will be shown.Arriving at the → QC can be done by introduc-tion the relativistic c.m. kernel , K ∗ rel , which is defined asthe sum of all the two-particle irreducible s-channel dia-grams. Just like the scattering amplitude, the kernel is amatrix over all the open channels and is diagonal in to-tal angular momentum. An example of a matrix elementof the kernel is illustrated in Fig. 1(b). Having defined K ∗ rel , the infinite volume scattering amplitude satisfies aself-consistent matrix, integral equation i M = − iK ∗ rel + iK ∗ rel G ∞ M , (15)where G ∞ is a diagonal matrix in the number of chan-nels, orbital angular momentum and spin. Its jth matrixelement in channel space is the infinite volume s-channelloop for the jth channel. The “ ab ” matrix element ofthe second term in the equation above, can be explicitlywritten as an integral of the form, i [ K ∗ rel G ∞ M ] ab ≡ ˆ d q (2 π ) [ K rel ( p f , q )] aj ∆ j ( q )[ M ( q, p i )] jb , (16) For bound states E ∗ < m j, + m j, , which leads to the rela-tive momentum to be imaginary k ∗ j < . Although it may besometimes desirable to approximate the finite volume effects as-sociated with the determination of a bound state energy in afinite volume [14, 15, 76, 77], the formalism presented here non-perturbatively describes such effects for bound states. where K rel denotes the functional form of the kernelin the lattice frame, the dependence on the total four-momentum P is left implicit, and summation over re-peated indices is implied. ∆ j ( q ) denotes relativistic two-particle propagator. In general, this is a matrix in spinthat mixes different azimuthal components of spin. Inthe helicity basis it diagonal and can be written as [∆ j ( q )] α ,α ; λ ,λ = z j, ( P − q ) z j, ( q ) δ α ,λ δ α ,λ [( q − P ) − m j, + i(cid:15) ][ q − m j, + i(cid:15) ] , (17)where z j,i is the residue of the ith single particle, fullydressed propagator in the jth channel.As will become evident shortly, when interested in thedetermination of the finite volume it will not be necessaryto give an explicit expression for the infinite volume func-tion G ∞ , and all that will be necessary is to determine thedifference between this object and its finite volume coun-terpart, δ G V = G V − G ∞ . In Sec. III A, δ G V is derivedfor systems with periodic boundary conditions and theresult is given in Eq. 23, and the expression for systemswith arbitrary twist and asymmetry volumes is given inSec. IV.In order to define the relation between the scatteringamplitude and the S -matrix, it is convenient to intro-duce a matrix that is diagonal over the N open channels P = diag ( (cid:112) n q ∗ , (cid:112) n q ∗ , . . . , (cid:112) n N q ∗ N ) / √ πE ∗ , where n j is the symmetry factor for the jth channel and is equalto / if the two-particles are identical and 1 otherwise.The S -matrix is diagonal in the total angular momentumbasis. For a system with total angular momentum J , thescattering amplitude M J is related to the S -matrix forthat channel via, [42] i M J = P − ( S J − I ) P − . (18)For spinless systems the orbital angular momentum isequal to the total angular momentum. For systems withnonzero spin, this will in general not be true. For instancein the spin-triplet positive parity two-nucleon channel,considered in Ref [19], S would be a × matrix thatcouples the S and D , S = (cid:18) S S S SD S DS S D (cid:19) . (19)Each of the four matrix elements are × matrices pro-portional to the identity.In general, one can have an S -matrix that not onlycouples orbital angular momentum states but also flavorstates and/or spin state. For example, consider a spin-singlet ΛΛ system in a S-wave. The ground states ofthis channel, the H-dibaryon, has been observed to be abound state for unphysical values of m π [23, 24, 78, 79].In flavor space ΛΛ also mixes with the I = 0 Ξ N and ΣΣ channels. For low-energies, the positive parity J =0 S -matrix can be approximated as a × matrix in fla-vor space. The ΛΛ ground state must be an spin-singletstate due to the Pauli-exclusion principle, but the spin of = ++ + ... + ... ++= ++ i G V i M V } + ) ( = ) ( + ) ( ) ( V V ) ( } } = + + ... iK ⇤ rel (a) = + + + ... + ... ++= + + = )( V V )()( + )( } } i G V )( i M V } + − i K rel (b) = ++ + ... + ... ++= ++ i G V i M V } + − i K rel ) ( = ) ( + ) ( ) ( V V ) ( } } = + + ... (c) FIG. 1. a) Shown is the self consistent definition of M V , which is defined as the sum of all → finite volume diagrams,Eq. 20. The solid lines denote two-particles in the “1” channel, dashed lines denote particle in the “2” channel. M V is writtenin terms of the c.m. kernel, K ∗ rel , and the fully dressed single particle propagators. b) Shown is K ∗ rel for the first channel, whichis the sum of all two-particle irreducible s-channel diagrams. Explicitly shown are examples of diagrams that are included in thekernel: contact interactions, t- and u-channel diagrams and possible meson exchange diagrams, if allowed by the symmetriesof the system of interest. If the two initial and final states of the kernel are baryons these exchange diagrams are presented,otherwise they are not allowed by G-parity. In general, all diagrams allowed by the underlying theory where the intermediateparticles cannot all simultaneously go on-shell are absorbed into the kernel. As described in the text, in this study we arerestricted to energies where only two-particle states are allowed to go on-shell. c) Shown is the definition of the fully dressedone particle propagator in terms of the the one particle irreducible (1PI) diagrams. the Ξ N is not constrained by symmetry considerations.Therefore, the P and P Ξ N states mix and the cor-responding J = 1 S -matrix, which can be approximatedto be a × matrix, has nonzero elements coupling thesetwo channels. For sufficiently high energies, the P - F mixing of the Ξ N state may in general not be neglected.Furthermore, although the S -matrix does not couple S and P Ξ N states, these may in general mix in a finitevolume [80].Even though the scattering amplitude may in generalnot be diagonal in spin, spin is conserved in the infinitevolume loops, G ∞ . This is a consequence of the factthat the single particle propagators are diagonal in he-licity. This also explains why spin is conserved in thefinite volume loops, G V . The only difference between G V and G ∞ is that the momenta of the intermediate parti-cles is discretized for the former but continuous for thelatter. This results in partial wave mixing in a finite vol-ume. This is in agreement with what has previously beenfound for systems with nonzero spin [13, 17–19] and willbe reviewed in Sec. III A.The finite volume spectrum can be obtained from thepoles of the sum of all amputated → finite volumediagrams, M V , which is represented in Fig. 1. Thisis the analogous finite volume object to the infinite vol-ume scattering amplitude, and it asymptotes to M as The poles of this object satisfy the same quantization conditionas those of the finite volume correlation function [12, 42–44]. the volume is taken to infinity. This object satisfies thefollowing matrix, summation equation i M V = − iK ∗ rel + iK ∗ rel G V M V , (20)where G V is the finite volume s-channel loop, and in par-ticular the matrix elements of the second term in theequation above can be written as i [ K ∗ rel G V M ] ab ≡ L (cid:88) q ˆ dq π [ K rel ( p f , q )] aj ∆ j ( q )[ M ( q, p i )] jb . (21)As thoroughly discussed in Ref. [12] for the spin-singlet, single channel scenario, the only power law finitevolume corrections of G V arise from the pole structureof the intermediate two-particle propagator. Therefore,the difference between this loop and the infinite volumecounterpart, δ G V ≡ G V − G ∞ , depends on the on-shellmomentum. The on-shell condition fixes the magnitudeof the momentum running through the kernels but notits direction. Therefore it is convenient to decompose theproduct of the kernels and δ G V into spherical harmonics.These depend not only on the directionality of the in-termediate momentum but also on those of the incomingand outgoing momenta. In Refs. [42–44] it was demon-strated that this persists to be true for coupled channelsystems with S = 0 or S = 1 . In fact, this observation isindependent of the spin structure of the system of inter-est and the number of channels. For arbitrary number ofchannels one may simply upgrade δ G V to be not just amatrix in the spherical harmonic space but also a matrixin the open channels. If the system has non-zero spin,then it is convenient to represent the kernel and δ G V notas matrices in orbital angular momentum but rather totalangular momentum. Just like the scattering amplitude,the kernel is diagonal in the total angular momentum. Asa result, its matrix elements can be written in the sameform as the scattering amplitude, Eq. 13. A derivationof δ G V is presented in Sec. III A.Having upgraded these objects to infinite dimensional matrices in J and the space of open channels, it is easyto see that the poles of Eq. 20 satisfy det [ M − + δ G V ] =det oc (cid:2) det lSJm J [ M − + δ G V ] (cid:3) = 0 , (22)where the determinant det oc is over the N open channelsand the determinant det lSJm J is over the | lS, Jm J (cid:105) basis,and both M and δ G V functions are evaluated on the on-shell value of the momenta, Eq. 14. The matrix elementsof δ G V for the jth channel are defined as (cid:2) δ G Vj (cid:3) Jm J ,lS ; J (cid:48) m J (cid:48) ,l (cid:48) S (cid:48) = ik ∗ j δ SS (cid:48) πE ∗ n j δ JJ (cid:48) δ m J m J (cid:48) δ ll (cid:48) + i (cid:88) l (cid:48)(cid:48) ,m (cid:48)(cid:48) (4 π ) / k ∗ l (cid:48)(cid:48) +1 j c d l (cid:48)(cid:48) m (cid:48)(cid:48) ( k ∗ j ; L ) × (cid:88) m l ,m l (cid:48) ,m S (cid:104) lS, Jm J | lm l , Sm S (cid:105)(cid:104) l (cid:48) m l (cid:48) , Sm S | l (cid:48) S, J (cid:48) m J (cid:48) (cid:105) ˆ d Ω Y ∗ l,m l Y ∗ l (cid:48)(cid:48) ,m (cid:48)(cid:48) Y l (cid:48) ,m l (cid:48) , (23)and the function c d lm is defined as c d lm ( k ∗ j ; L ) = √ πγL (cid:18) πL (cid:19) l − Z d lm [1; ( k ∗ j L/ π ) ] , Z d lm [ s ; x ] = (cid:88) r ∈P d | r | l Y l,m ( r )( r − x ) s , (24)where γ = E/E ∗ , the sum is performed over P d = (cid:8) r ∈ R | r = ˆ γ − ( m − α j d ) (cid:9) , m is a triplet integer, d is the normalized boost vector d = P L/ π , α j = (cid:104) m j, − m j, E ∗ (cid:105) [15, 16, 80], and ˆ γ − x ≡ γ − x || + x ⊥ ,with x || ( x ⊥ ) denoting the x component that is paral-lel(perpendicular) to the total momentum, P . Detailsregarding the representation of the s-channel loops asmatrices in angular momentum are shown in Sec. III A.In deriving the result, PBCs have been assumed on thespatial extents of the lattice. The boundary conditionsof the system are encoded in the form of the Z func-tions. References [64, 81] derived these for systems withnonzero momenta, arbitrary masses and twisted bound-ary conditions. For completeness, Sec. IV includes theresult in the presence of arbitrary twist and asymmetryvolumes. As discussed above, it is evident from Eq. 23that δ G V is diagonal in spin, although the scattering am-plitude may in general not be. Due to the reduction ofrotational symmetry, δ G V mixes different orbital angularmomentum states and consequently different J states, asexpected. For example, for systems with d = { (0 , , , (0 , , n ) , ( n, n, , ( n, n, n ) , ( n, m, , ( n, n, m ) , ( n, m, p ) } ,or any cubic rotation of these, the symmetry point groupsare the double cover of the octahedral ( O D h ) and the di-cyclic groups Dic , Dic , Dic , C , C & C , respectively.Table I lists the decomposition of the irreducible repre-sentations (irreps) of these three groups onto continuum states that have overlap with both half-integer and inte-ger spin systems up to J = 4 [3, 4, 10, 17, 38, 82–86].As was mentioned in Sec. I, the master equation pre-sented here, Eq. 22, is consistent with all previous results.The most general multichannel result for scalars was pre-sented in Refs. [42, 43]. If one restricts the total spin ofall of the available channels to be exactly zero ( S = 0 inEq. 23), then Clebsch-Gordan coefficients are all replacedwith Kronecker delta functions setting orbital and totalangular momenta equal to each one and one recovers theresult of these references. If S = 1 / in Eq. 23 and fur-thermore restricts there to be only a single channel, onerecovers the result of Ref. [13, 17]. If one allows for arbi-trary numbers of channels with S = 1 / then one arrivesat the result of Ref. [44]. Allowing for a single channelwith S = 1 and restricting the energies to be relativistic,i.e., γ ≈ , one arrives at the two-nucleon result shownin Refs. [18, 19, 87, 88].Although what is presented here is the master equationdescribing the full finite volume spectrum for arbitrarytwo-body systems, in practice one needs to reduce themaster equation onto the quantization condition of theirreps of the system of interest. For systems with PBCs,there has been a great deal of effort in reduction of thesemaster equation for a wide variety of scenarios [3–5,10, 13, 16, 17, 38, 83, 86, 89]. References [38, 83, 86]demonstrate how to decompose the master equation forinteger-spin systems for the irreps of the symmetry pointgroups corresponding to d = { (0 , , , (0 , , n ) , (0 , n, n ) , ( n, n, n ) , ( n, m, , ( n, n, m ) } . References [13, 17] containthe relations of the non-vanishing c lm functions as well asthe basis vectors for S = 1 / systems with d = { (0 , , , (0 , , n ) , (0 , n, n ) , ( n, n, n ) } . (a) J P O Dh ± A ± ± G ± ± T ± ± H ± ± E ± ⊕ T ± ± G ± ⊕ H ± ± A ± ⊕ T ± ⊕ T ± ± G ± ⊕ G ± ⊕ H ± ± A ± ⊕ E ± ⊕ T ± ⊕ T ± (b) | λ | ˜ η Dic Dic Dic C C + A A A A A − A A A B A E E E E B E B ⊕ B E A ⊕ B A E E B ⊕ B E B B ⊕ B A ⊕ A E A ⊕ B A E E E E B E B ⊕ B A ⊕ A A ⊕ B A E E E E B A ⊕ A A ⊕ A E A ⊕ B A TABLE I. (a) The decomposition of the irreps of the SO(3)group up to J = 4 in terms of the irreps of the O Dh [3, 4, 90–92]. (b) The decomposition of the helicity states to the irrepsof five of the little groups of O Dh : Dic , Dic , Dic , C & C [38, 83–86]. λ labels the helicity of the state and ˜ η = P ( − J , where P is the parity of the state. A. Relativistic finite volume loop with spin
In the absence of weak interactions, the free two-particle propagators are diagonal in the open channels.That is to say, in the absence of two-body interactionsthe different channels would not mix. This is depicted inFig. 1(a). As a consequence, it is only necessary to in-vestigate the structure of the s-channel loop appearing inone of the open channels. Therefore, to alleviate some ofthe strenuous notation that is necessary when discussingcoupled channel systems, we will momentarily drop the j subscript that explicitly reminds the reader that the jth channel of potentially infinitely many open channelsis being discussed.In order to evaluate the sum depicted in Fig. 2, it isconvenient to upgrade the kernel onto a matrix in spin.Depending on the nature of the the particles of interest,bosonic vs. fermionic, the dimensionality of the singleparticle propagator will differ. Nevertheless, the singleparticle poles will satisfy the relativistic dispersion re-lation for all particles. Alternatively, one may alwaysperform a field redefinition to assure the propagator ofbosonic and fermionic field have the same dimensions,and in doing so one can define the residues appearing inEq. 17 to be equal to one when the particles go on-shell.This allows one to write the difference between the firstterm of the finite volume loop depicted in Fig. 2 and itsinfinite volume counterpart in the following form iδG V ≡ n (cid:34) L (cid:88) q ˆ (cid:35) ˆ dq π K rel ( p f , q ) K rel ( q, p i ) z ( P − q ) z ( q )[( q − P ) − m + i(cid:15) ][ q − m + i(cid:15) ] , (25)where the kernels are being represented as matrices inhelicity and the dependence on the total four-momentum P is being suppressed, and the following notation hasbeen introduced (cid:34) L (cid:88) q ˆ (cid:35) ≡ (cid:32) L (cid:88) q − ˆ d q (2 π ) (cid:33) . (26)More explicitly, the product of the two kernels in Eq. 25should be interpreted as K rel ( p f , q ) K rel ( q, p i ) = (cid:88) λ ,λ ˆ K rel ( p f , q ) | q − P , s λ (cid:105) ⊗ | − q , s λ (cid:105)(cid:104)− q , s λ | ⊗ (cid:104) q − P , s λ | ˆ K rel ( q, p i ) . (27)This is to emphasize that the single particle propagatorsare diagonal in helicity. Because there is a complete set of states between the two kernels, one can always performa unitary transformation to represent this product in analternative basis.In general, the kernel is a function of volume, butsince the c.m. energy is restricted to satisfy m j, + m j, < ∼ E ∗ (cid:28) E ∗ th the intermediate particles appearingin the kernel, Fig. 1(b), cannot all simultaneously go on-shell. Therefore, one can show using Poisson’s resumma-tion formula, (cid:34) L (cid:88) q ˆ (cid:35) f ( q ) = (cid:88) n (cid:54) =0 ˆ d q (2 π ) f ( q ) e iL n · q . that this leads to exponentially small deviations fromthe infinite volume kernel. By neglecting these correc-tions, the result discussed here holds for volumes sat-isfying m π L (cid:29) . We will also neglect terms in δG V that are exponentially suppressed with the mass of any ofthe two-particles in the given channel since O ( e − m i L ) ≤O ( e − m π L ) . These corrections have been previously de-termine for ππ [93] and N N systems [94] in an S-wave,as well as the ππ system in a P-wave in Ref. [95, 96].The identification of the power law volume dependenceof this function is most readily done by rewriting thesummand in terms of the c.m. of coordinates. To do thisthe notation used in Ref. [12] will be used ω q,i = (cid:113) | q | + m i . (28) The Lab frame coordinates q = ( q || , q ⊥ ) and ω q,i appear-ing in the summand above can be transformed to c.m.coordinates q ∗ = ( q ∗|| , q ∗⊥ ) and ω ∗ q,i = (cid:112) q ∗ + m i usingthe standard Lorentz transformations ω ∗ q,i = γ ( ω q,i − βq || ) ,q ∗|| = γ ( q || − βω q,i ) , q ∗⊥ = q ⊥ , (29)where γ = E ∗ E , β = PE . Using these relations, writ-ing the functional form of the kernels in the c.m. frameas K ∗ rel , and neglecting exponentially suppressed correc-tions, Eq. 25 can be rewritten as [12, 97] iδG V = − in (cid:34) L (cid:88) q ˆ (cid:35) E ∗ ω ∗ q, ω q, K ∗ rel ( p ∗ f , q ∗ ) K ∗ rel ( q ∗ , p ∗ i ) z ∗ ( q ∗ ) z ∗ ( q ∗ ) k ∗ − q ∗ + i(cid:15) (cid:32) E ∗ + m − m E ∗ + 2 ω ∗ q, ω ∗ q, (cid:33) . (30)where k ∗ is the on-shell c.m. momentum and satisfiesEq. 14. In general the kernel will also depend on ω ∗ q,i , butsince this is itself a function of q ∗ the explicit dependenceon ω ∗ q,i has been suppressed.By restricting themselves to the scalar sector, Kim,Sachrajda, and Sharpe showed that this summation canbe represented as a product of infinite-dimensional matri-ces in orbital angular momentum [12]. This result can berecovered by decomposing the product of the two kernelsinto spherical components, K ∗ rel ( p ∗ f , q ∗ ) K ∗ rel ( q ∗ , p ∗ i ) = (cid:88) l,m f lm ( q ∗ ) √ πq ∗ l Y lm (ˆ q ∗ ) . (31)The function f lm is defined as to satisfy this equationand its definition in terms of the spherical decompositionof the kernels is easy to write down. Using this functionone finds that δG V can be written as [12] iδG V = n k ∗ f ( k ∗ )8 πE ∗ + i E ∗ (cid:88) l,m f ∗ lm ( k ∗ ) c d lm ( k ∗ ) , where c d lm has been defined in Eq. 24. This expressionholds for arbitrary spin systems. Section II showed thatone can decompose any object that is diagonal in angu-lar momenta, such as the scattering amplitude and thekernel, in the lS basis with matrix elements shown inEq. 13. Using these expressions along with Eqs. 27 & 31one finds iδG V = − i [ K ∗ rel ] Jm J lS ( δ G V ) Jm J ,lS ; J (cid:48) m J (cid:48) ,l (cid:48) S (cid:48) [ K ∗ rel ] J (cid:48) m (cid:48) J l (cid:48) S (cid:48) = [ − iK ∗ rel ]( iδ G V )[ − iK ∗ rel ] , (32) For further details see Ref. [97]. where we have suppressed the indices of the incoming andoutgoing state in the loop, [ K ∗ rel ] Jm J lS denote the on-shellkernels, and the matrix elements ( δ G V ) Jm J ,lS ; J (cid:48) m J (cid:48) ,l (cid:48) S (cid:48) are defined in Eq. 23. Equation 32 shows that the differ-ence between the finite volume and infinite volume loopscan be represented in a matrix representation of func-tions that only depend on the on-shell momenta. Havingshown this for a single channel allows one to quickly de-rive the relation for an arbitrary numbers of channels. Ingeneral one could have one species, “ a ” , going into theloop and another one, “ b ” , outgoing. By upgrading allthe objects appearing in Eq. 32 in the space of channels,one finds, iδG Vba = [ − iK ∗ rel ] bj ( iδ G Vj )[ − iK ∗ rel ] ja , (33)where the intermediate j -index is summed over all openchannels. By utilizing this relation along with the defini-tion of the infinite volume scattering amplitude in termsof the kernel, Eq. 15, and the definition of M V , Eq. 20,one arrives at the quantization condition, Eq. 22. IV. TWO-PARTICLE MULTICHANNELSYSTEMS WITH SPIN WITH TBCS INASYMMETRIC VOLUMESA. Cubic volumes
In the derivation of the master equation shown inEq. 22, periodic boundary condition on the spatial ex-tent of the cubic volume have been assumed. The peri-odicity constraint is encoded in the expression for the Z functions shown in Eq. 24, and this is generally true forarbitrary boundary conditions. As discussed in Sec. I,TBCs require that fields are proportional to their im-ages up to an overall phase. Therefore, particle “1” inthe jth channel will have a free discretized momenta V V ∞ + = FIG. 2. Shown is the close-up of a generic finite volume loop appearing in the determination of the quantization condition,Eq. 22, which is determined from the poles of the M V , defined in Eq. 20 and pictorially depicted in Fig. 1. The finite volumeloop can always be set equal to its infinite volume counterpart up to finite volume correction. In Sec. III A, it is shown thatthis correction can be written as a product of infinite-dimensional matrices that solely depend on the on-shell momenta of theintermediate particles in the loop. satisfying p j, = π n L + φ j, L , where φ j, is the three-dimensional phase for that particle. Each particle ineach channel could have an overall different phase which,when thinking of LQCD calculations, is dictated by thequark content of the hadron in mind. As a consequence,the total momentum of the systems will be shifted to P = π d L + φ j, + φ j, L . Although, it may naively seem thatthe total momentum would depend on the channel con-sidered, it is easy to convince oneself that for coupledchannel systems φ j, + φ j, is a conserved quantity, sinceantiquark fields satisfy ¯ ψ ( x + n L ) = e − i θ · n ¯ ψ ( x ) . (34)Having defined the total momentum, the relationshipbetween the total energy and the c.m. energy remainunchanged, and the c.m. on-shell momenta for the jth channel still satisfies Eq. 14. The only part of the masterequation that is modified is the finite volume function,Eq. 23. One finds that the c lm and Z functions witharbitrary twist for a cubic volume is [35, 46, 63–65], c d , φ j, , φ j, lm ( k ∗ ; L ) = √ πγL (cid:18) πL (cid:19) l − ×Z d , φ j, , φ j, lm [1; ( k ∗ L/ π ) ] , (35) Z d , φ j, , φ j, lm [ s ; x ] = (cid:88) r ∈P φ , φ d | r | l Y l,m ( r )( r − x ) s , (36)where P φ , φ d = (cid:110) r ∈ R | r = ˆ γ − ( m − α j d + ∆ ( j ) π ) (cid:111) ,where m is a triplet integer, ∆ ( j ) = − ( α j − )( φ j, + φ j, )+ ( φ j, − φ j, ) . Just as before, ˆ γ − x ≡ γ − x || + x ⊥ ,with x || ( x ⊥ ) denoting the x component that is paral-lel(perpendicular) to the total momentum, P .With this, one arrives at the conclusion that the quan-tization condition for the spectrum of a two-particlemulti-channel system with TBCs can still be written asEq. 22, where the matrix elements of δ G V for the jth channel can be obtained by replacing c d l (cid:48)(cid:48) m (cid:48)(cid:48) ( k ∗ j ; L ) with c d , φ j, , φ j, l (cid:48)(cid:48) m (cid:48)(cid:48) ( k ∗ j ; L ) in Eq. 23. One important observa-tion is that if the two-particles are degenerate and they have the same twist then twisting will have no overallimpact in the c.m. spectrum. Therefore, one may notgain any additional information for systems like π + π + or pp using TBCs. Furthermore, if isospin is exact andthe twist on the up and down quarks is the same, thiswill give the same pn c.m. spectrum as if it was at restand untwisted. Reference [64] investigated the implica-tion of the determination of the deuteron binding en-ergy when using asymmetric twists on the up and downquarks, and found that by introducing an overall twist φ p = − φ n = ( π/ , π/ , π/ finite volume artifacts of thedeuteron binding energies can be reduced from ∼ e − κL /L to ∼ e − κL /L , where κ is infinite volume binding momen-tum of the deuteron.Another important remark is that when introducing anarbitrary twist, partial wave mixing can be a subtle mat-ter. This is due to the rich structure of the c lm ’s in Eq. 35.For instance, for the scenario discussed in Ref. [64] theS-wave deuteron channel not only has physical mixingwith the D , but in general will have finite volume mix-ing with the P , P , P , D and D channels, aswell as higher partial waves, even when the up and downquark masses are exactly degenerate. B. Asymmetric volumes
References [8–10] demonstrated how the
Lüscher method can be generalized for asymmetric volumes.Adopting the notation introduced in these references,let L be the spatial extent of the z-axis and η i be theasymmetric factor of the ith axis, i.e., L x = η x L and L y = η y L . In evaluating the finite volume loop in theprevious section, Eq. 25, one must make the followingreplacement, L (cid:88) q → η x η y L (cid:88) q , (37)which leads to an overall factor of ( η x η y ) − in the c lm functions, Eq. 35. Furthermore, the free particlemomenta are altered. Let χ be an arbitrary three-dimensional vector. By introducing the notation ˜ χ = ( χ x /η x , χ y /η y , χ z ) , one can readily find that the “1” inthe jth channel has a free momentum of p j, = π ˜ n L + ˜ φ j, L ,where n is a integer triplet and φ j, is the particle’s twist.With these pieces one may arrive at the most generalform of the c lm and Z functions with arbitrary twist foran asymmetric volume c d , φ j, , φ j, lm ( k ∗ ; L ; η x , η y ) = √ πη x η y γL (cid:18) πL (cid:19) l − ×Z d , φ j, , φ j, lm [1; ( k ∗ L/ π ) ; η x , η y ] , (38) Z d , φ j, , φ j, lm [ s ; x ; η x , η y ] = (cid:88) r ∈P φ , φ d ; ηx,ηy | r | l Y l,m ( r )( r − x ) s , (39)where P φ , φ d ; η x ,η y = (cid:110) r ∈ R | r = ˆ γ − ( ˜ m − α j ˜ d + ˜ ∆ ( j ) π ) (cid:111) ,where m is a triplet integer, ˜ ∆ ( j ) = − ( α j − )( ˜ φ j, +˜ φ j, ) + ( ˜ φ j, − ˜ φ j, ) . In the limit that the total momen-tum and twist of the system vanishes, this result agreeswith Refs. [8–10]. The boost vector, ˜ d , is defined to beequal to P L/ π . It is important to note that in the limitwhere the twist angles of the two particles vanish, theboost vector for asymmetric volumes in general is not aninteger triplet. V. IMPLICATION FOR BARYON-BARYONSYSTEMS
As was discussed in the previous sections, the formal-ism presented here is universal and gives a mapping be-tween the finite volume spectrum and the infinite volumescattering amplitude for arbitrary two-particle systems.A sector of physics where this formalism will have a clearand immediate impact is on the study of light nucleiand hyper nuclei from LQCD. This is a field that hasreceived a great deal of excitement in recent years [23–29, 36, 78, 79, 98]. As was alluded to in the previous sec-tions, these are systems with rather rich structure andwith potential partial wave mixing in the infinite vol-ume and/or several inelastic thresholds. For instance,the determination of hyperon-nucleon scattering phaseshifts studied in Ref. [26] was limited by the fact thatthis formalism was not known. Having the formalism inplace, future calculations of these systems will no longerbe restricted to the study of the ground state, where pre-sumably only S -wave phase shifts are prevalent, but alsoscattering parameters will be able to be determined fromexcited states. The study in Ref. [26] explicitly avoidedcoupled channels systems, e.g., I = 1 / N Σ - N Λ . Al-though this remains to be a computationally challengingproblem, there is no formal restriction for determiningnot only scattering phase shifts but also mixing anglesand thereby unfolding the rich structure of these systems.The need for performing calculations with multiple to-tal momenta has been extensively advocated in the litera-ture [5, 11–13, 17, 80, 87]. When boosting a given system its c.m. energy in a finite volume is altered. This is ev-ident from the quantization condition shown in Eq. 22.The scattering amplitude only depends on c.m. coordi-nates, while the c lm functions, Eq. 24, and consequently δG V , Eq. 23, depend on both the c.m. coordinate and to-tal momenta of the system. Therefore the c.m. energieswhere Eq. 22 vanishes will in general differ for differentboosts. This is extremely advantageous when trying toconstrain the scattering amplitude from the finite volumespectrum, since it is at these energies where the scatteringamplitude is determined. For coupled channel systems,boosting is a necessity [42, 43]. For example, the J = 1 S = 1 matrix shown in Eq. 19 depends on three func-tions of energy, the S-wave and D-wave phase shifts andthe mixing angle that couples these two channels. There-fore, from a single energy one can only constrain a linearcombinations of these functions. Performing calculationswith multiple boosts aids in disentangling these functionsfrom the spectrum. In Ref. [87] it was shown just how todo this for the S - D two nucleon channel. An alterna-tive tool for coupled channel systems is to perform calcu-lations with twisted boundary conditions [35, 46, 63–65]or asymmetric volumes [8–10].Furthermore, Refs. [19, 87] went into great detail indemonstrating that the presence of partial wave mixingin the infinite volume could lead to an unexpectedly largeeffect in the boosted c.m. finite volume spectrum. Forthe deuteron channel, it was demonstrated that at thephysical point these effects can lead to a ∼ cor-rection to the binding energies for moderate volumes of L ∼ fm. This observation is expected to also hold for S = 1 hyperon-nucleon/hyperon-hyperon systems.In Ref. [19] it was assumed that isospin is exact, whichin the infinite volume, where parity and total angular mo-mentum are good quantum numbers, leads to spin con-servation. For instance, this would suggest that P and P NN channels could not mix. In nature, up and downquark masses are not degenerate, and searches for exper-imental (e.g., see Refs. [99–102]) and theoretical conse-quences (see Ref. [103] for a review on the topic) in thetwo-body sector of this reduction of symmetry are chal-lenging. By performing calculations with non-degenerateup and down quark masses, future LQCD calculationswill be able to further constrain the mixing between dif-ferent spin channels.Although the discussion is focused on the baryon-baryon sector, this formalism will also be necessary forfuture studies of meson-meson or meson-baryon processeswhere one or both particles have spin. An example ofsuch systems is the J/ Ψ - φ scattering channel, which wasrecently studied in Ref. [35] using TBCs. This benchmarkcalculation determined the S and P J/ Ψ - φ phaseshifts using configurations with a lightest pion mass of m π = 156 MeV in hopes of finding evidence for the Y (4140) resonance [104, 105]. In obtaining their resultthe authors have made two two reasonable approxima-tions. The first approximation refers to the fact thatalthough the authors of Ref. [35] accounted for the finite1volume partial wave mixing of the S and P waves,they did not include effects due to physical mixing be-tween the P , P and P waves in their analysis. Thisis expected to be a small contribution for non-relativisticsystems, but in general the quantization condition pre-sented in this work can be used to include such effects.The second approximation refers to the unstable natureof the J/ Ψ(1 S ) and/or φ (1020) . Although Ref. [47] quan-titatively demonstrated that for a resonances such as the ρ , with a decay width of 147.8(9) MeV [106], one maynot use two-body formalism presented here and used inRef. [35], this formalism is expected to accurately de-scribe the spectrum of a system including the J/ Ψ(1 S ) and/or the φ (1020) . This is because their respective de-cay widths are 92.9(2.8) keV and 4.26(4) MeV [106] andtheir hadronic decays are in general suppressed by theOZI (Okubo-Zweig-Iizuka) rule [107–110]. VI. CONCLUSION
This paper presents the most general two-body fi-nite volume formalism that gives the relationship be-tween the finite volume spectrum and the infinite vol-ume → scattering amplitude. The result holds for anarbitrary number of open two-body channels, with arbi-trary masses, spin and momenta. The only restrictions isthat the c.m. energy lies below the three-body inelasticthreshold and that the spatial extent to the volume is sig-nificantly larger than the range of the interactions. It is evident from the result, that it is consistent with all pre-vious two-body finite volume results [3–5, 8–20, 35, 41–46, 63–65].Section II reviewed the basics of the construction ofhelicity states and their relation with the lS basis [72–75]. Section III presented a derivation of the quantizationcondition for multichannel systems with arbitrary spin,Eq. 22, using generic aspects of relativistic quantum fieldtheory. Sections IV A & IV B presented the generaliza-tion of this result for systems with arbitrary TBCs ina cubic and asymmetric volume, respectively. Althoughthe result is generic and independent of the nature of theparticles of interest, Section V discussed the implicationof this formalism for two-baryon systems. A place wherethis formalism will have immediate impact in the studiesof hyperon-nucleon and hyperon-hyperon systems. Acknowledgments RB acknowledges support from the U.S. Department ofEnergy contract DE-AC05-06OR23177, under which Jef-ferson Science Associates, LLC, manages and operatesthe Jefferson Laboratory. RB would like to thank Mar-tin Savage, Zohreh Davoudi, Thomas Luu, Robert Ed-wards, Kostas Orginos, Adam Szczepaniak, Jozef Dudek,André Walker-Loud, Igor Danilkin, Maxwell Hansen,William Detmold, and Colin Morningstar for many use-ful discussions and feedback on previous versions of thismanuscript. [1] M. Luscher, Commun.Math.Phys. , 177 (1986).[2] L. Maiani and M. Testa, Phys.Lett.
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