aa r X i v : . [ h e p - l a t ] S e p Modeling few-body resonances in finite volume
Peng Guo
1, 2, ∗ Department of Physics and Engineering, California State University, Bakersfield, CA 93311, USA Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA (Dated: September 10, 2020)Under the assumption of separable interactions, we illustrate how the few-body quantizationcondition may be formulated in terms of phase shifts in general, which may be useful for describingand modeling of few-body resonances in finite volume.
I. INTRODUCTION
Few-hadron dynamics plays an important role inhadron and nuclear physics. There have been many goodexamples of physics processes that can only be under-stood through few-body interactions, such as, the u - and d -quark mass difference in η → π [1–7], Efimov states[8, 9] and halo nuclei [10, 11]. The understanding of few-body interaction is also crucial in recent experimental ef-forts of exotic hadrons study, since most of exotic hadronstates are expected to appear as few-hadron resonances.On the theory side, lattice Quantum Chromodynamics(LQCD) provides an ab-initio method for the study ofexotic hadron states. However, LQCD computation isusually performed in Euclidean space with certain peri-odic boundary condition, normally only discrete energyspectrum are measured in numerical simulation. Hence,mapping out few-hadron dynamics from discrete energyspectrum is a key step for the study of exotic hadronstates in LQCD. In two-body sector, L¨uscher formula[12] and its variants [13–22] provide an elegant form ofmapping out two-body phase shift from discrete energylevels.In past few years, many progresses from different ap-proaches [23–54] have been made going beyond three-body threshold. Although few-body quantization con-ditions are formulated differently among these groups, ithas been very clear [52] that in few-body sectors, the few-body amplitudes are not directly extracted from latticeresults. Particle interactions or its associated subprocessamplitudes are in fact essential ingredients in quantiza-tion condition. The infinite volume few-body amplitudesthat are generated by particle interactions through cou-pled integral equations must be computed in a separatestep once these dynamical ingredients are determined. Inorder to make predictions or fit lattice results, dynami-cal ingredients of quantization condition, such as inter-action potentials or off-shell subprocess amplitudes mustbe modeled one way or another. In addition, numberof partial waves involved in some physical processes maybe large, which may add some extra complications ontop of the uncertainty in modeling itself. Therefore, tohave a reliable and controllable predictions, the modeling ∗ [email protected] of dynamical ingredients must be constrained or guidedby experimental data or effective theory. Nevertheless,there are two physical regions in which predictions andcalculations may be made fairly reliable: (1) near thresh-old which is the region where the physical reaction canbe described rather precisely by non-relativistic potentialtheory or relativistic effective perturbation theory; (2)near resonances region where resonance properties maybe less affected by modeling and other partial waves.In present work, we focus on near resonance region andaim to provide an approximate means for the modeling offew-body resonances in finite volume. Based on separableinteraction potential assumption, we illustrate how thefew-body quantization condition may be formulated interms of subprocess phase shifts. Hence, the resonancesmay be modeled and inserted into quantization condi-tion through phase shifts. Both two-body and three-bodysubprocess amplitudes appear L¨uscher formula-like andsolutions are given by algebra equations. The aim of thiswork is to illustrate a simple way of parameterization ofdynamics of few-body resonance in finite volume. Underthe separable short-range potential approximation, thefew-body formalism is much simplified with the trade-offthat the approximation may be only valid for the descrip-tion of sharp resonances dynamics. As a simple illustra-tion, the formalism is only presented in non-relativistickinematics, the extension to relativistic kinematics maybe possible by replacing non-relativistic few-body propa-gators with relativistic ones, see [49, 51]. The relativisticextension of formalism will be presented in future publi-cations.The paper is organized as follows. With separable in-teractions approximation, the technical details of formu-lating quantization conditions in terms of phase shifts arepresented in Sec. II. A summary is given in Sec. III. II. QUANTIZATION CONDITION UNDERSEPARABLE INTERACTIONS ASSUMPTION
Few-body quantization condition in finite volume canbe formulated from homogeneous Faddeev type equa-tions, see [50–53]. As a simple example, we considerthree non-relativistic identical bosons of mass m interact-ing with both pair-wise interaction and three-body forcein follows. Due to exchange symmetry, only two indepen-dent Faddeev amplitudes are required: T (2 b ) and T (3 b ) that are associated with pair-wise two-body interaction, V (2 b ) , and three-body interaction, V (3 b ) , by T (2 b, b ) ( k , k ) = −h k k | mV (2 b, b ) | Ψ i , (1)where Ψ stands for the three-body total wave function.The ( k , k ) ∈ π n L , n ∈ Z refer to particle-1 and -2momenta respectively, and third particle momentum isconstrained by total momentum conservation, k = − k − k . In follows, we also use symbols ( k , k (13)2 ) to describetwo independent relative momenta of three particles,where k = k − k k + k , k (13)2 = r (cid:18) k + k − k (cid:19) = − r k . (2)The stationary states of three-body dynamics in finitevolume is described by homogeneous Faddeev type equa-tions, see [50–53], T (2 b ) ( k , k ) = − L X p τ (2 b ) ( k ; p + k ) mE − p + k +( p + k ) × (cid:20) T (2 b ) ( k , p ) + T (3 b ) ( p , k ) (cid:21) , (3)and T (3 b ) ( k , k )= − L X p , p τ (3 b ) ( K ; P ) mE − p + p +( p + p ) T (2 b ) ( p , p ) , (4)where ( p , p ) ∈ π n L , n ∈ Z . The symbol ( K , P ) stand for 6-dimensional vectors, theyare related to relative momenta ( k , k (13)2 ) by K = { k , k (13)2 } = { k + k , − r k } , P = { p , p (13)2 } = { p + p , − r p } . (5)The length of 6 D vectors are given by K = q k + k = vuut X i =1 k i ,P = q p + p = vuut X i =1 p i . (6) Symbols τ (2 b ) and τ (3 b ) that are associated with two-body interaction V (2 b ) and three-body interaction V (3 b ) respectively are used to describe off-shell subprocesstransition amplitudes between initial and final momentastates. For example, τ (2 b ) in (13) isobar channel withparticle-2 carrying a momentum k satisfies two-bodyinhomogeneous Lippmann-Schwinger equations, τ (2 b ) ( k ; k ′ ) = − m e V (2 b ) ( | k − k ′ | )+ 1 L X p m e V (2 b ) ( | k − p − k | ) mE − p + k +( p + k ) τ (2 b ) ( p + k k ′ ) , (7)and similarly τ (3 b ) satisfies a three-body equation, τ (3 b ) ( K ; K ′ ) = − m e V (3 b ) ( | K − K ′ | )+ 1 L X p , p m e V (3 b ) ( | K − P | ) mE − p + p +( p + p ) τ (3 b ) ( P ; K ′ ) . (8) τ (2 b ) and τ (3 b ) are dynamical input of finite volume Fad-deev equations in Eq.(3) and Eq.(4), and must be solvedfirst.The quantization condition without cubic irreduciblerepresentation projection is given by0 = det (cid:20) L δ k , p δ k , p + L δ k , p τ (2 b ) ( k ; p + k ) mE − k − ( p + k ) − L X p τ (2 b ) ( k ; p + k ) τ (3 b ) ( { p + k , − q k } ; P ) h mE − k − ( p + k ) i ( mE − P ) , (9)where τ (2 b ) and τ (3 b ) in principle are given by the solu-tions of Eq.(7) and Eq.(8) respectively. In Sec.II A, wewill show that with separable interaction approximationEq.(7) and Eq.(8) may be converted into algebra equa-tions. Hence, the solutions of τ (2 b ) and τ (3 b ) are L¨uscherformula-like, and can be formulated in terms of conven-tional two-body phase shifts in 3 D and unconventionalbut mathematically convenient three-body phase shiftsin 6 D . We remark that the momentum sum in quantiza-tion condition must be regulated in numerical evaluationof discrete energy levels, which are either associate to ul-traviolet divergence or normalization of determinant con-dition. In current work, since the technical regularizationis not our focus of presentation, the specific procedure ofregularization has been left-out, we refer interesting read-ers to Refs. [28, 29]. A. Separable interactions and algebra solutions of τ (2 b ) and τ (3 b ) Under the assumption of separable short-range poten-tials for both V (2 b ) and V (3 b ) , the partial wave expansionof potentials thus have the forms of e V (2 b ) ( | k − k ′ | )= X LM Y LM ( ˆk ) g (2 b ) L ( k ) V (2 b ) L g (2 b ) L ( k ′ ) Y ∗ LM ( ˆk ′ ) , (10)and e V (3 b ) ( | K − K | )= X [ J ] Y [ J ] ( ˆK ) g (3 b )[ J ] ( K ) V (3 b ) J g (3 b )[ J ] ( K ′ ) Y ∗ [ J ] ( ˆK ′ ) , (11)where Y LM ( ˆk ) is 3 D spherical harmonic function withquantum numbers | LM i representing orbital angular mo-mentum configurations between particle-1 and -3, whileparticle-2 acts as a spectator and is not involved in inter-action. Y [ J ] ( ˆK ) stands for the 6 D hyperspherical har-monic basis function, see Refs. [55, 56] and also Ap-pendix A, the quantum numbers [ J ] represent a spe-cific angular momentum configuration of three particleswith a total angular momentum- J . Y [ J ] ( ˆK ) may be con-structed through two 3 D spherical harmonic functions.For example, considering a configuration with angularmomentum state | L M i between particle-1 and -3coupled with particle-2 in relative angular momentumstate | L (13)2 M (13)2 i into total angular momentum state | [ J ] i = | JM L L (13)2 i , thus Y [ J ] ( ˆK ) is given by Y [ J ] ( ˆK ) = X M ,M (13)2 h L M , L (13)2 M (13)2 | JM i× Y L M ( ˆk ) Y L (13)2 M (13)2 ( ˆk (13)2 ) P JL L (13)2 ( φ ) , (12)where φ = tan − k k (13)2 . The function P JL L (13)2 ( φ ) is related to Jacobi polyno-mial by, also see [55, 56], P JL L (13)2 ( φ ) = N JL L (13)2 (sin φ ) L (cos φ ) L (13)2 × P ( L + ,L (13)2 + ) J − L − L (13)22 (cos 2 φ ) , (13)the normalization factor N JL L (13)2 is determined by or-thonormal relation, Z π dφ sin φ cos φ P JL L (13)2 ( φ ) P J ′ L L (13)2 ( φ ) = δ J,J ′ . (14)The form factors, g (2 b ) L and g (3 b )[ J ] , and potential strengths, V (2 b ) L and V (3 b ) J , may be considered as model parameters.Usually, the form factors, such as g (2 b ) L , must show thecorrect threshold behavior, g (2 b ) L ( k → ∼ k L . (15) The potential strengths V (2 b ) L and V (3 b ) J may be used tomodel two-body and three-body resonances, for example,the two-particle resonance of mass m (2 b ) R in (13) isobarpair channel with particle-2 carrying momentum k maybe given by V (2 b ) L ∝ E − k m ) − m (2 b ) R . (16)A three-particle resonance of mass m (3 b ) R thus may bemodeled similarly by V (3 b ) L ∝ E − m (3 b ) R . (17)Separable interactions suggest that τ (2 b ) in Eq.(7) and τ (3 b ) in Eq.(8) may be given by L¨uscher formula-like al-gebra equations, see detailed discussion in Appendix A, q mE − k π τ (2 b ) ( k ; k ′ )= X LM,L ′ M ′ Y LM ( ˆk ) g (2 b ) L ( k ) g (2 b ) L ′ ( k ′ ) Y ∗ L ′ M ′ ( ˆk ′ ) g (2 b ) L ( q mE − k ) g (2 b ) L ′ ( q mE − k ) × i L − L ′ (cid:20) δ LM,L ′ M ′ cot δ (2 b ) L ( r mE − k ) − M (2 b, k ) LM,L ′ M ′ ( r mE − k ) (cid:21) − , (18)and( mE ) π τ (3 b ) ( K ; K ′ )= X [ J ] , [ J ′ ] Y [ J ] ( ˆK ) g (3 b )[ J ] ( K ) g (3 b )[ J ′ ] ( K ′ ) Y ∗ [ J ′ ] ( ˆK ′ ) g (3 b )[ J ] ( √ mE ) g (3 b )[ J ′ ] ( √ mE ) × i J − J ′ h δ [ J ] , [ J ′ ] cot δ (3 b ) J ( √ mE ) − M (3 b )[ J ] , [ J ′ ] ( √ mE ) i − . (19)Generalized L¨uscher zeta functions, M (2 b, k ) in 3 D and M (3 b ) in 6 D , are given respectively by k π M (2 b, k ) LM,L ′ M ′ ( k ) = δ LM,L ′ M ′ ik π + δ LM,L ′ M ′ Z p dp (2 π ) g (2 b ) L ( p ) g (2 b ) L ( k ) ! k − p − L X p = π n L + k , n ∈ Z g (2 b ) L ( p ) g (2 b ) L ′ ( p ) g (2 b ) L ( k ) g (2 b ) L ′ ( k ) Y ∗ LM ( ˆp ) Y L ′ M ′ ( ˆp ) k − p , (20)and( mE ) π M (3 b )[ J ] , [ J ′ ] ( √ mE ) = δ [ J ] , [ J ′ ] i ( mE ) π + δ [ J ] , [ J ′ ] Z P dP (2 π ) g (3 b ) J ( P ) g (3 b ) J ( √ mE ) ! mE − P − L X p , p g (3 b ) J ( P ) g (3 b ) J ′ ( P ) g (3 b ) J ( √ mE ) g (3 b ) J ′ ( √ mE ) Y ∗ [ J ] ( ˆP ) Y [ J ′ ] ( ˆP ) mE − P . (21)The two-body phase shift δ (2 b ) L is defined in a conven-tional way, which may be modeled and constrained by ex-perimental data. The unconventional three-body phaseshift δ (3 b ) J may be interpreted as scattering of one parti- cle off a short-range potential in 6 D . It may only serveas a mathematically convenient tool for the modeling ofthree-body resonance of total spin- J . B. Quantization condition with separableinteractions approximation
Algebra solutions of τ (2 b ) in Eq.(18) and τ (3 b ) inEq.(19) suggest that partial expansion of T (2 b ) ( k , k )may have the form of T (2 b ) ( k , k ) = X LM Y LM ( ˆk ) g (2 b ) L ( k ) T (2 b ) LM ( k ) . (22)The separable form of T (2 b ) ( k , k ) thus allow one tofurther reduce Faddeev equations, Eq.(3) and Eq.(4), to T (2 b ) LM ( k ) = − L X p X L ′ M ′ τ (2 b ) LM ( p + k ) g (2 b ) L ′ ( | k + p | ) Y L ′ M ′ ( k + p ) mE − k − ( p + k ) T (2 b ) L ′ M ′ ( p )+ 1 L X p X L ′ M ′ (cid:20) L X k , p τ (2 b ) LM ( k ) mE − K τ (3 b ) ( K ; P ) g (2 b ) L ′ ( p ) Y L ′ M ′ ( ˆp ) mE − P (cid:21) T (2 b ) L ′ M ′ ( p ) , (23)where τ (2 b ) LM ( k ′ ) is defined by relation τ (2 b ) ( k ; k ′ ) = X LM Y LM ( ˆk ) g (2 b ) L ( k ) τ (2 b ) LM ( k ′ ) . (24) Therefore, a partially expanded quantization condition isgiven bydet (cid:20) δ LM,L ′ M ′ L δ k , p + 2 τ (2 b ) LM ( p + k ) g (2 b ) L ′ ( | k + p | ) Y L ′ M ′ ( k + p ) mE − k − ( p + k ) − L X k , p τ (2 b ) LM ( k ) τ (3 b ) ( K ; P ) g (2 b ) L ′ ( p ) Y L ′ M ′ ( ˆp )( mE − K ) ( mE − P ) (cid:21) = 0 . (25)As a specific example, let’s consider a simple case withonly S -wave contributions in both two-body and three-body channels, that is to say, J = L = L (13)2 = 0.Thus τ (2 b ) and τ (3 b ) are given respectively by phase shifts δ (2 b ) L =0 and δ (3 b ) J =0 only, τ (2 b ) ( k ; k ′ ) = τ (2 b, k ) ( r mE − k ) = 4 π q mE − k × δ (2 b )0 ( q mE − k ) − M (2 b, k )00 , ( q mE − k ) , (26)and τ (3 b ) ( K ; K ′ ) = τ (3 b ) ( √ mE )= 128 π ( mE ) δ (3 b )0 ( √ mE ) − M (3 b )[0] , [0] ( √ mE ) . (27)The quantization condition in this case is given by a sim-ple form,det (cid:20) L δ k , p + 2 τ (2 b, k ) ( q mE − k ) mE − k − ( p + k ) + 1 L X k , p τ (2 b, k ) ( q mE − k ) τ (3 b ) ( √ mE )( mE − K ) ( mE − P ) (cid:21) = 0 . (28)The two-body and three-body resonances hence can beinserted through modeling of δ (2 b ) L =0 and δ (3 b ) J =0 . III. SUMMARY
In summary, with separable interaction approxima-tion, we show that the subprocess transition amplitudesare L¨uscher formula-like, and the quantization conditionmay be formulated in terms of both two-body and three-body phase shifts that may be useful for describing res-onances in few-body interactions. Two-body phase shiftmay be modeled and constrained by experimental data,and three-body phase shift may serve as a convenient toolfor inserting three-body resonances with a specific spininto quantization condition.
ACKNOWLEDGMENTS
We acknowledge support from the Department ofPhysics and Engineering, California State University,Bakersfield, CA. This research was supported in part bythe National Science Foundation under Grant No. NSFPHY-1748958.
Appendix A: L¨uscher formula in D -dimensionalspace1. Scattering in D -dimensional space Let’s start with N -body Schr¨odinger equation in centerof mass frame, mE + N − X j =1 ∇ ξ j ψ ( ξ ; K ) = mV ( ξ ) ψ ( ξ ; K ) . (A1) the relative coordinates of N -particle are given by ξ j = s jj + 1 j j X i =1 x i − x j +1 ! , q j = s j j + 1) j j X i =1 k i − k j +1 ! , j = 1 , · · · , N − , (A2)where x i and k i stand for the coordinate and momentumof i-th particle respectively. D = 3( N −
1) dimensionalvector ( ξ , K ) are defined by relative coordinates and mo-menta of particles, ξ = { ξ , ξ , · · · , ξ N − } , ξ = | ξ | = vuut N − X j =1 ξ j , K = { q , q , · · · , q N − } , K = | K | = vuut N − X j =1 q j . (A3) D -dimensional Laplace operator has a separable formbetween radial and orbital terms [55, 56], ∇ D = N − X j =1 ∇ ξ j = 1 ξ D − ∂∂ξ ξ D − ∂∂ξ + ˆ L (Ω D ) ξ , (A4)where ˆ L (Ω D ) is the grand orbital operator. The eigen-states of orbital equationˆ L (Ω D ) Y [ L ] (Ω D ) = L ( L + D − Y [ L ] (Ω D ) (A5)is given by hyperspherical harmonic Y [ L ] (Ω D ) [55, 56],where [ L ] is a set of D − L . Hyperspherical har-monic Y [ L ] (Ω D ) basis define a complete set of orthonor-mal angular function in D -dimensional space, Z d Ω D Y ∗ [ L ] (Ω D ) Y [ L ′ ] (Ω D ) = δ [ L ] , [ L ′ ] . (A6)The scattering in D -dimensional space can also be de-scribed by Lippmann-Schwinger equation ψ ( ξ ; K ) = e i K · ξ + Z d ξ ′ G D ( ξ − ξ ′ ; E ) mV ( ξ ′ ) ψ ( ξ ′ ; K ) ,G D ( ξ − ξ ′ ; E ) = Z d Q (2 π ) D e i Q · ( ξ − ξ ′ ) mE − Q , (A7)where Green’s function satisfies equation (cid:2) mE + ∇ D (cid:3) G D ( ξ − ξ ′ ; E ) = δ ( ξ − ξ ′ ) . (A8)The analytic expression of Green’s function and its par-tial wave expansion in terms of hyperspherical harmonicbasis are given respectively by G D ( ξ ; E ) = − i mE ) D − (2 π ) D − H (1) D − ( √ mEξ )( √ mEξ ) D − , (A9)and G D ( ξ − ξ ′ ; E ) ξ>ξ ′ = − i ( mE ) D − × X [ L ] Y [ L ] (Ω ξ ) H (1) L ( √ mEξ ) J L ( √ mEξ ′ ) Y ∗ [ L ] (Ω ξ ′ ) , (A10)where J L ( z ) = r π J L + D − ( z ) z D − , N L ( z ) = r π N L + D − ( z ) z D − , (A11)and H (1) L ( z ) = J L ( z ) + i N L ( z ) . (A12)Assuming potential V ( ξ ) is spherical and short-range,and also using partial wave expansion of plane wave in D -dimensional space, e i K · ξ = r π (2 π ) D X [ L ] i L Y [ L ] (Ω ξ ) Y ∗ [ L ] (Ω K ) J L ( √ mEξ ) , (A13)the asymptotic form of wave function is obtained, ψ ( ξ ; K ) Large ξ → r π (2 π ) D X [ L ] i L Y [ L ] (Ω ξ ) Y ∗ [ L ] (Ω K ) × h J L ( √ mEξ ) + if ( D ) L ( √ mE ) H (1) L ( √ mEξ ) i , (A14)where f ( D ) L is defined by r π (2 π ) D ( mE ) D − i L f ( D ) L ( √ mE ) Y ∗ [ L ] (Ω K )= − Z d ξ ′ Y ∗ [ L ] (Ω ξ ′ ) J L ( √ mEξ ′ ) mV ( ξ ′ ) ψ ( ξ ′ ; K ) . (A15)Thus f ( D ) L may be interpreted as partial wave scatteringamplitude in D -dimensional space, and it can be param-eterized in terms of D -dimensional phase shift δ ( D ) L ( k E )[55, 56] by f ( D ) L ( √ mE ) = 1cot δ ( D ) L ( √ mE ) − i . (A16)
2. Lippmann-Schwinger equation in momentumspace and separable potential approximation
The off-shell transition amplitude between initial andfinal momentum states | K i and | K ′ i may be introducedby t ( D ) ( K , K ′ ) = − Z d ξ e − i K ′ · ξ mV ( ξ ) ψ ( ξ ; K ) , (A17) thus Eq.(A7) can be converted into momentum spaceLippmann-Schwinger equation, t ( D ) ( K , K ′ ) = − m e V ( | K − K ′ | )+ Z d Q (2 π ) D m e V ( | K − Q | ) mE − Q t ( D ) ( Q , K ′ ) . (A18)The partial wave expansion of above equation yields t ( D ) L ( K, K ′ ) = − m e V L ( K, K ′ )+ Z Q D − dQ (2 π ) D m e V L ( K, Q ) mE − Q t ( D ) L ( Q, K ′ ) , (A19)where the expansion relations of potential and amplitudeare given by e V ( | K − K ′ | ) = X [ L ] Y [ L ] ( ˆK ) e V L ( K, K ′ ) Y ∗ [ L ] ( ˆK ′ ) , (A20)and t ( D ) ( K , K ′ ) = X [ L ] Y [ L ] ( ˆK ) t L ( K, K ′ ) Y ∗ [ L ] ( ˆK ′ ) . (A21)Under assumption of separable potential, e V L ( K, K ′ ) = g ( D ) L ( K ) V L g ( D ) L ( K ′ ) , (A22)where g ( D ) L and V L stand for the form factor and inter-action strength of potential, thus a closed algebra formof off-shell partial wave amplitude, t ( D ) L ( K, K ′ ), may beobtained, see [57], t ( DL ( K, K ′ ) = − g ( D ) L ( K ) g ( D ) L ( K ′ ) mV L − R Q D − dQ (2 π ) D (cid:16) g ( D ) L ( Q ) (cid:17) mE − Q . (A23)Compared with on-shell scattering amplitude f ( D ) L ( √ mE ) in Eq.(A15), we find t ( DL ( K, K ′ ) = g ( D ) L ( K ) g ( D ) L ( K ′ ) (cid:16) g ( D ) L ( √ mE ) (cid:17) × π (2 π ) D ( mE ) D − δ ( D ) L ( √ mE ) − i , (A24)and also a useful relation1 mV L = Z Q D − dQ (2 π ) D (cid:16) g ( D ) L ( Q ) (cid:17) mE − Q + (cid:16) g ( D ) L ( √ mE ) (cid:17) π mE ) D − (2 π ) D h i − cot δ ( D ) L ( √ mE ) i . (A25)Therefore off-shell partial wave amplitude, t ( D ) L ( K, K ′ )may be modeled in terms of on-shell physical quantity:phase shifts δ ( D ) L ( √ mE ).We remark that the separable potential approxima-tion is in fact based on the assumption of hypersphericalshort-range interaction. The hyperspherical partial waveexpansion of momentum space potential is given by e V ( | K − K ′ | ) = Z d ξ e − i ( K − K ′ ) · ξ V ( ξ ) ∝ X [ L ] Y [ L ] ( ˆK ) Z ξ D − dξ J L ( Kξ ) V ( ξ ) J L ( K ′ ξ ) Y ∗ [ L ] ( ˆK ′ ) . (A26)For short-range potential, asymptotically one obtains Z ξ D − dξ J L ( Kξ ) V ( ξ ) J L ( K ′ ξ ) ∼ K L V L K ′ L , (A27)which thus yield the expression in Eq.(A22). The separa-ble potential approximation may be useful for the model-ing of sharp few-body resonances that are predominantlygenerated by quark and gluon dynamics. Hence, thehadron-hadron interactions may be well approximatedby a short-range energy dependent interaction, the Breit-Wigner formula is a good example of such an approxima-tion.
3. L¨uscher formula in D -dimensional space andseparable potential approximation Scattering solution in finite volume may be describedby inhomogeneous Lippmann-Schwinger equation, τ ( D ) ( K , K ′ ) = − m e V ( | K − K ′ | )+ 1 L D X p , ··· , p N − m e V ( | K − Q | ) mE − Q τ ( D ) ( Q , K ′ ) , (A28)where p i ∈ π n L , n ∈ Z and Q = P Ni =1 p i . Consider-ing partial wave expansion again, τ ( D ) ( K , K ′ ) = X [ L ] , [ L ′ ] Y [ L ] ( ˆK ) τ ( D )[ L ] , [ L ′ ] ( K, K ′ ) Y ∗ [ L ′ ] ( ˆK ′ ) , (A29)one finds τ ( D )[ L ] , [ L ′ ] ( K, K ′ ) = − δ [ L ] , [ L ′ ] m e V L ( K, K ′ )+ X [ l ] L D X p , ··· , p N − m e V L ( K, Q ) Y ∗ [ L ] ( ˆQ ) Y [ l ] ( ˆQ ) mE − Q × τ ( D )[ l ] , [ L ′ ] ( Q, K ′ ) . (A30)Again, the separable potential given in Eq.(A22) suggeststhat τ ( D )[ L ] , [ L ′ ] may have the separable form of τ ( D )[ L ] , [ L ′ ] ( K, K ′ ) = g L ( K ) C [ L ] , [ L ′ ] ( E ) g L ′ ( K ′ ) , (A31) where C [ L ] , [ L ′ ] ( E ) satisfies a matrix equation, C [ L ] , [ L ′ ] ( E ) = − δ [ L ] , [ L ′ ] mV L + X [ l ] L D X p , ··· , p N − × g ( D ) L ( Q ) g ( D ) l ( Q ) Y ∗ [ L ] ( ˆQ ) Y [ l ] ( ˆQ ) mE − Q C [ l ] , [ L ′ ] ( E ) . (A32)Hence, a closed algebra form of solution of off-shell solu-tion of finite volume amplitude, τ ( D )[ L ] , [ L ′ ] , is obtained, τ ( D )[ L ] , [ L ′ ] ( K, K ′ ) = g L ( K ) g L ′ ( K ′ ) g L ( √ mE ) g L ′ ( √ mE ) h D ( √ mE ) i − L ] , [ L ′ ] , (A33)where D [ L ] , [ L ′ ] ( √ mE ) = − δ [ L ] , [ L ′ ] g L ( √ mE ) mV L g L ′ ( √ mE )+ 1 L D X p , ··· , p N − g L ( Q ) g L ′ ( Q ) g L ( √ mE ) g L ′ ( √ mE ) Y ∗ [ L ] ( ˆQ ) Y [ L ′ ] ( ˆQ ) mE − Q . (A34)Using relation given in Eq.(A25), thus D [ L ] , [ L ′ ] is linkedto generalized L¨uscher formula in D -dimensional space,2 π (2 π ) D ( mE ) D − i L − L ′ D [ L ] , [ L ′ ] ( √ mE )= δ [ L ] , [ L ′ ] cot δ ( D ) L ( √ mE ) − M [ L ] , [ L ′ ] ( √ mE ) , (A35)where M [ L ] , [ L ′ ] is generalized L¨uscher’s zeta function in D -dimensional space, π mE ) D − (2 π ) D M [ L ] , [ L ′ ] ( √ mE ) = iδ [ L ] , [ L ′ ] π mE ) D − (2 π ) D − L D X p , ··· , p N − i L − L ′ g L ( Q ) g L ′ ( Q ) g L ( √ mE ) g L ′ ( √ mE ) Y ∗ [ L ] ( ˆQ ) Y [ L ′ ] ( ˆQ ) mE − Q + δ [ L ] , [ L ′ ] Z Q D − dQ (2 π ) D g ( D ) L ( Q ) g ( D ) L ( √ mE ) ! mE − Q . (A36)Therefore, the inverse of τ ( D )[ L ] , [ L ′ ] is explicitly related toL¨uscher formula by2 π (2 π ) D ( mE ) D − " g L ( √ mE ) g L ′ ( √ mE ) g L ( K ) g L ′ ( K ′ ) τ ( D ) ( K, K ′ ) − L ] , [ L ′ ] = i L ′ − L h δ [ L ] , [ L ′ ] cot δ ( D ) L ( √ mE ) − M [ L ] , [ L ′ ] ( √ mE ) i . (A37)Generalized L¨uscher zeta function can also be derivedby considering hyperspherical harmonic basis functionexpansion of Green’s function. In infinite volume, thehyperspherical harmonic basis expansion of Green’s func-tion is given by, Z d Q (2 π ) D e i Q · ( ξ − ξ ′ ) mE − Q ξ>ξ ′ = − i ( mE ) D − × X [ L ] Y [ L ] (Ω ξ ) H (1) L ( √ mEξ ) J L ( √ mEξ ′ ) Y ∗ [ L ] (Ω ξ ′ ) . (A38)Similarly to expansion of infinite volume Green’s func-tion, the expansion of finite volume Green’s function may be written as,1 L D X p , ··· , p N − e i Q · ( ξ − ξ ′ ) mE − Q ξ>ξ ′ = ( mE ) D − X [ L ] , [ L ′ ] Y [ L ] (Ω ξ ) × h δ [ L ] , [ L ′ ] N L ( √ mEξ ) − M [ L ] , [ L ′ ] ( √ mE ) J L ( √ mEξ ) i × J L ′ ( √ mEξ ′ ) Y ∗ [ L ′ ] (Ω ξ ′ ) . (A39)Combining Eq.(A38) and Eq.(A39), we obtain1 L D X p , ··· , p N − e i Q · ( ξ − ξ ′ ) mE − Q − Z d Q (2 π ) D e i Q · ( ξ − ξ ′ ) mE − Q ξ>ξ ′ = ( mE ) D − X [ L ] , [ L ′ ] Y [ L ] (Ω ξ ) J L ( √ mEξ ) × h iδ [ L ] , [ L ′ ] − M [ L ] , [ L ′ ] ( √ mE ) i J L ′ ( √ mEξ ′ ) Y ∗ [ L ′ ] (Ω ξ ′ ) . (A40)Next, using plane wave expansion formula given inEq.(A13) and also replacing g ( D ) L ( k ) by k L , we thus findagain Eq.(A36), which may also suggest that the formfactor, g ( D ) L , may be chosen as g ( D ) L ( k ) ∼ k L . [1] J. Kambor, C. Wiesendanger, andD. Wyler, Nucl. Phys. B465 , 215 (1996),arXiv:hep-ph/9509374 [hep-ph].[2] A. V. Anisovich and H. Leutwyler,Phys. Lett.
B375 , 335 (1996),arXiv:hep-ph/9601237 [hep-ph].[3] S. P. Schneider, B. Kubis, and C. Ditsche,JHEP , 028 (2011), arXiv:1010.3946 [hep-ph].[4] K. Kampf, M. Knecht, J. Novotny, andM. Zdrahal, Phys. Rev. D84 , 114015 (2011),arXiv:1103.0982 [hep-ph].[5] P. Guo, I. V. Danilkin, D. Schott,C. Fern´andez-Ram´ırez, V. Mathieu, and A. P.Szczepaniak, Phys. Rev.
D92 , 054016 (2015),arXiv:1505.01715 [hep-ph].[6] P. Guo, I. V. Danilkin, C. Fern´andez-Ram´ırez, V. Mathieu, and A. P.Szczepaniak, Phys. Lett.
B771 , 497 (2017),arXiv:1608.01447 [hep-ph].[7] G. Colangelo, S. Lanz, H. Leutwyler, andE. Passemar, Phys. Rev. Lett. , 022001 (2017),arXiv:1610.03494 [hep-ph].[8] V. Efimov, Phys. Lett. , 563 (1970).[9] E. Braaten and H. W. Ham-mer, Phys. Rept. , 259 (2006),arXiv:cond-mat/0410417 [cond-mat].[10] M. V. Zhukov, B. V. Danilin, D. V. Fedorov,J. M. Bang, I. J. Thompson, and J. S. Vaagen,Phys. Rept. , 151 (1993).[11] H. W. Hammer, C. Ji, and D. R.Phillips, J. Phys.
G44 , 103002 (2017),arXiv:1702.08605 [nucl-th].[12] M. L¨uscher, Nucl. Phys.
B354 , 531 (1991). [13] K. Rummukainen and S. A. Got-tlieb, Nucl. Phys.
B450 , 397 (1995),arXiv:hep-lat/9503028 [hep-lat].[14] N. H. Christ, C. Kim, and T. Ya-mazaki, Phys. Rev.
D72 , 114506 (2005),arXiv:hep-lat/0507009 [hep-lat].[15] V. Bernard, M. Lage, U.-G. Meißner, and A. Rusetsky,JHEP , 024 (2008), arXiv:0806.4495 [hep-lat].[16] S. He, X. Feng, and C. Liu, JHEP , 011 (2005),arXiv:hep-lat/0504019 [hep-lat].[17] M. Lage, U.-G. Meißner, andA. Rusetsky, Phys. Lett. B681 , 439 (2009),arXiv:0905.0069 [hep-lat].[18] M. D¨oring, U.-G. Meißner, E. Oset, andA. Rusetsky, Eur. Phys. J.
A47 , 139 (2011),arXiv:1107.3988 [hep-lat].[19] R. A. Brice˜no and Z. Davoudi,Phys. Rev.
D88 , 094507 (2013),arXiv:1204.1110 [hep-lat].[20] M. T. Hansen and S. R. Sharpe,Phys. Rev.
D86 , 016007 (2012),arXiv:1204.0826 [hep-lat].[21] P. Guo, J. Dudek, R. Edwards, and A. P.Szczepaniak, Phys. Rev.
D88 , 014501 (2013),arXiv:1211.0929 [hep-lat].[22] P. Guo, Phys. Rev.
D88 , 014507 (2013),arXiv:1304.7812 [hep-lat].[23] S. Kreuzer and H. W. Ham-mer, Phys. Lett.
B673 , 260 (2009),arXiv:0811.0159 [nucl-th].[24] S. Kreuzer and H. W. Ham-mer, Eur. Phys. J.
A43 , 229 (2010),arXiv:0910.2191 [nucl-th]. [25] S. Kreuzer and H. W. Grießhammer,Eur. Phys. J.
A48 , 93 (2012), arXiv:1205.0277 [nucl-th].[26] K. Polejaeva and A. Rusetsky,Eur. Phys. J.
A48 , 67 (2012), arXiv:1203.1241 [hep-lat].[27] R. A. Brice˜no and Z. Davoudi,Phys. Rev.
D87 , 094507 (2013),arXiv:1212.3398 [hep-lat].[28] M. T. Hansen and S. R. Sharpe,Phys. Rev.
D90 , 116003 (2014),arXiv:1408.5933 [hep-lat].[29] M. T. Hansen and S. R. Sharpe,Phys. Rev.
D92 , 114509 (2015),arXiv:1504.04248 [hep-lat].[30] M. T. Hansen and S. R. Sharpe,Phys. Rev.
D93 , 096006 (2016), [Erratum: Phys.Rev.D96,no.3,039901(2017)], arXiv:1602.00324 [hep-lat].[31] R. A. Brice˜no, M. T. Hansen, and S. R.Sharpe, Phys. Rev.
D95 , 074510 (2017),arXiv:1701.07465 [hep-lat].[32] H.-W. Hammer, J.-Y. Pang, and A. Rusetsky,JHEP , 109 (2017), arXiv:1706.07700 [hep-lat].[33] H. W. Hammer, J. Y. Pang, and A. Rusetsky,JHEP , 115 (2017), arXiv:1707.02176 [hep-lat].[34] U.-G. Meißner, G. R´ıos, and A. Ruset-sky, Phys. Rev. Lett. , 091602 (2015), [Erra-tum: Phys. Rev. Lett.117,no.6,069902(2016)],arXiv:1412.4969 [hep-lat].[35] M. Mai and M. D¨oring, Eur. Phys. J. A53 , 240 (2017),arXiv:1709.08222 [hep-lat].[36] M. Mai and M. D¨oring,Phys. Rev. Lett. , 062503 (2019),arXiv:1807.04746 [hep-lat].[37] M. D¨oring, H. W. Hammer, M. Mai, J. Y. Pang, § . A.Rusetsky, and J. Wu, Phys. Rev. D97 , 114508 (2018),arXiv:1802.03362 [hep-lat].[38] F. Romero-L´opez, A. Rusetsky, andC. Urbach, Eur. Phys. J.
C78 , 846 (2018),arXiv:1806.02367 [hep-lat].[39] P. Guo, Phys. Rev.
D95 , 054508 (2017), arXiv:1607.03184 [hep-lat].[40] P. Guo and V. Gasparian, Phys. Lett.
B774 , 441 (2017),arXiv:1701.00438 [hep-lat].[41] P. Guo and V. Gasparian,Phys. Rev.
D97 , 014504 (2018),arXiv:1709.08255 [hep-lat].[42] P. Guo and T. Morris, Phys. Rev.
D99 , 014501 (2019),arXiv:1808.07397 [hep-lat].[43] T. D. Blanton, F. Romero-L´opez, and S. R. Sharpe,JHEP , 106 (2019), arXiv:1901.07095 [hep-lat].[44] F. Romero-L´opez, S. R. Sharpe, T. D. Blanton, R. A.Brice˜no, and M. T. Hansen, JHEP , 007 (2019),arXiv:1908.02411 [hep-lat].[45] T. D. Blanton, F. Romero-L´opez, and S. R.Sharpe, Phys. Rev. Lett. , 032001 (2020),arXiv:1909.02973 [hep-lat].[46] M. Mai, M. D¨oring, C. Culver, andA. Alexandru, Phys. Rev. D , 054510 (2020),arXiv:1909.05749 [hep-lat].[47] P. Guo, M. D¨oring, and A. P. Szczepa-niak, Phys. Rev. D98 , 094502 (2018),arXiv:1810.01261 [hep-lat].[48] P. Guo, Phys. Lett.
B804 , 135370 (2020),arXiv:1908.08081 [hep-lat].[49] P. Guo and M. D¨oring, Phys. Rev.
D101 , 034501 (2020),arXiv:1910.08624 [hep-lat].[50] P. Guo, Phys. Rev.
D101 , 054512 (2020),arXiv:2002.04111 [hep-lat].[51] P. Guo and B. Long, Phys. Rev. D , 094510 (2020),arXiv:2002.09266 [hep-lat].[52] P. Guo, (2020), arXiv:2007.04473 [hep-lat].[53] P. Guo and B. Long, (2020), arXiv:2007.10895 [hep-lat].[54] M. T. Hansen, F. Romero-L´opez, and S. R. Sharpe,JHEP , 047 (2020), arXiv:2003.10974 [hep-lat].[55] M. Fabre de la Ripelle,Annals of Physics , 281 (1983).[56] M. Fabre de la Ripelle, Few-Body Systems , 1 (1993).[57] C. Lovelace, Phys. Rev.135