Multi-$\pi^+$ systems in finite volume

We present a formalism to describe two-$\pi^+$ and three-$\pi^+$ dynamics in finite volume, the formalism is based on combination of a variational approach and the Faddeev method. Both pair-wise and three-body interactions are included in the presentation. Impacts of finite lattice spacing and the cubic lattice symmetry are also discussed. To illustrate application of the formalism, the pair-wise contact interaction that resembles the leading order interaction terms in chiral effective theory is used to analyze recent lattice results.

We present a formalism to describe two-π + and three-π + dynamics in finite volume, the formalism is based on combination of a variational approach and the Faddeev method. Both pair-wise and three-body interactions are included in the presentation. Impacts of finite lattice spacing and the cubic lattice symmetry are also discussed. To illustrate application of the formalism, the pair-wise contact interaction that resembles the leading order interaction terms in chiral effective theory is used to analyze recent lattice results.

I. INTRODUCTION
Understanding of few-hadron interactions is crucial in nuclear/hadron physics. Few-hadron dynamics provides a unique access to various fundamental parameters of Quantum Chromodynamics (QCD), for quarks and gluons only manifest themselves within hadrons due to color confinement. For instance, the u-and d-quark mass difference can be extracted from η → 3π decay process [1][2][3][4][5][6][7]. Remarkable few-body phenomena, such as the Efimov states [8,9] and halo nuclei [10,11], have been predicted and observed in strong-interaction physics. Few-body systems also offer an outlook for many-body effects, e.g., three-nucleon forces [12][13][14] and fractional quantum Hall effects [15].
Latest advances have now made lattice QCD (LQCD) a powerful quantitative tool to study hadron physics from the first principles. In recent years, realistic LQCD calculations of multi-hadron systems have been made possible [16][17][18][19][20][21][22][23][24][25][26][27]. However, LQCD calculations are usually performed in a periodic box in the Euclidean space-time, and only discrete energy spectra are extracted from time dependent correlation functions. That is to say, the multihadron dynamics is encoded in a set of discrete energy levels in finite volume. Therefore, mapping out infinitevolume few-body dynamics from finite-volume energy spectra is a key step toward understanding multi-hadron systems from LQCD calculations.
In the two-body sector, a pioneering approach proposed by Lüscher [28] tends to build connections between infinite volume reaction amplitudes and energy levels in a periodic cubic box, which was later on further extended to the cases of moving frames and coupled channels [29][30][31][32][33][34][35][36][37][38]. In past few years, progresses have been made on the study of few-body systems above three-body thresholds in finite volume . To certain extent, most of these developments may be regarded as extensions of the Lüscher formula. Lüscher's formula in two-body sector may demonstrate a clear advantage: the quantization * pguo@csub.edu † bingwei@scu.edu.cn condition is given in terms of infinite volume two-body scattering amplitude or phase shift, it provides a direct connection between infinite-volume reaction amplitudes and finite-volume energy spectra. Unfortunately, above three-body threshold, in addition to complication of finite volume dynamics, the infinite volume few-body reaction amplitudes usually can not be easily parameterized in an analytic form, solutions of these amplitudes are given by coupled integral equations, such as Faddeev equations [68,69]. Dealing with questions regarding infinite and finite-volume physics simultaneously presents great challenges. As illustrated in Ref. [65,66], the quantization condition of few-body systems may be presented in terms of finite-volume Green's function and effective interaction between particles embodied by a potential. Since infinite volume scattering amplitudes are not explicitly involved in variational approach, it may be more efficient for practical analysis of LQCD calculation results.
In present work, the variational approach proposed in [64][65][66][67] is applied to multi-π + systems. In 3π + system, both pair-wise interactions and three-body interactions are included in our presentation. We also take into consideration relativistic pion kinematics, the finite lattice spacing and cubic lattice symmetry effects. As our initial attempt, the quantization conditions are used to analyze LQCD data published in Ref. [27] by accounting for only pair-wise contact interaction in the center-of-mass (CM) frame. The 2π + scattering length and effective range are extracted from analysis. With a single parameter, the coupling constant of the pair-wise contact interaction, we are able to make prediction on energy spectra, albeit with relatively large uncertainties.
The paper is organized as follows. The formalism of finite volume multi-π + systems at the continuum limit, where the lattice spacing vanishes, is presented in detail in Section II. The finite lattice spacing effect and cubic lattice symmetry are discussed in Section III. Numerical results are given in Section IV, followed by a summary in Section V.
II. RELATIVISTIC MULTI-π + SYSTEMS IN FINITE VOLUME AT CONTINUUM LIMIT For the future reference, we give in this section a complete presentation of our formalism that describes finitevolume dynamics of multi-π + systems. The formalism adopted in this work is based on the variational approach combined with the Faddeev method which was previously discussed in Refs. [64][65][66][67]. The relativistic, finite-volume multi-π + dynamics is for the moment formulated at the continuum limit with vanishing lattice spacing (a = 0), in the sense that ultraviolet divergence is to be removed through proper regularization procedure. The finite lattice spacing effect will be installed rather straight-forwardly later on in Section III. In what follows, we will go through dynamical equations of the two-pion and three-pion systems, and close the section by elaborating how ultraviolet divergences are treated and how renormalization is carried out.
A. Dynamical equations of 2π + system The relativistic 2π + system in finite volume is governed by the homogeneous Lippmann-Schwinger (LS) equation [65][66][67]: (1) where r and P are the relative coordinate and total momentum of the pions, respectively. With the assumption of zero lattice spacing (a = 0), r is continuously distributed in finite volume, and L 3 dr stands for continuous integration bound by the edges of a periodic cubic box. The spherical two-body potential is represented by V (r) where r = |r|. The wave function for relative motion of the two pions satisfies periodic boundary condition [64][65][66], where L is the size of the cubic lattice. Imposing the the periodic boundary condition on the plane wave of CM motion yields We remark that throughout the entire paper, the explicit energy dependence of both the wave function and the scattering amplitude is dropped for the convenience of presentation. The two-pion Green's function is given by where E p = m 2 π + p 2 and p = 2πn L , n ∈ Z 3 . Defining finite-volume two-pion amplitude, where k = 2πn L , n ∈ Z 3 , one can transform Eq. (1) into the homogeneous momentum-space LS equation: where the momentum-space potential According to the variational approach [64][65][66][67], the quantization condition for the two-pion system is given by Solving for E yields the discrete energy spectrum of two interacting pions in finite volume at the continuum limit where the lattice space approaches zero.
B. Dynamical equations of 3π + system After factoring out the CM motion (see Appendix D for the example of removing CM motion for a nonrelativistic three-particle system), the dynamics of relativistic finite-volume three-pion system is described by a LS type integral equation in coordinate space, where r ij = x i − x j is relative coordinate between the ith and j-th pions. r 13 and r 23 are chosen to describe the relative motion of 3π + system. The pair-wise interactions between i-th and j-th identical pions are represented by with k = 1, 2, 3 and k = i = j and V (r) is the previously defined two-body potential. V (4) (r 13 , r 23 ) with k = 4 denotes the three-body force acting on all the pions. The three-pion Green's functions are defined as follows: and where G (P) The total momentum where is the momentum of the i-th particle. The relativistic kinematic factors, 2E p k for k = 1, 2, 3, in Eq.(12) are associated with the relativistic normalization of the free propagating spectator particle in presence of the pairwise interaction between the i-th and j-th particles. The relativistic LS equation may be derived from the Bethe-Salpeter equation, see Appendix A. The three-body wave function must satisfy the periodic boundary condition [66]: where n 1,2 ∈ Z 3 .
As suggested in Refs. [64][65][66][67], the Faddeev amplitudes may be introduced by Equation (9) is thus turned into coupled equations: where V (k) 's are the Fourier transform of pair-wise and three-body interaction potentials,    Two-body interactions V (1,2,3) are related to V (k), as defined in Eq. (7), through where δ k,p denotes the 3D Kronecker delta function.

Exchange symmetry of 3π + system
As is the case for any identical bosons, the wave function of 3π + system must be invariant under exchange of any pair of pions, for example, (21) and φ 3π (r ik , r jk ) i↔j = φ 3π (r jk , r ik ). The exchange symmetry of three-body wave function suggests that t (P) (1,2,3) are related: where P = k 1 + k 2 + k 3 . Using the definition of t (P) (2) in Eq. (17) and symmetry relations of wave function in Eq.(21), we find useful symmetry properties of t (P) (2) : (23) In addition to that of the wave function, exchange symmetry of three-body potential V (4) also constrains the amplitude t (4) : and t (P) The diagrammatic representations of Eq.(25) and Eq.(26) are shown in Fig. 1. It may be convenient to consolidate Eqs. (25) and (26) in the matrix form: where (k i , p i ) ∈ 2πn L , n ∈ Z 3 . Elements of the matrix function K(k 1 , k 2 ; p 1 , p 2 ) are given by The coupled equations, Eqs. (25) and (26), or the matrix form of them Eq.(27), will serve as basic dynamical equations for 3π + system. The entire 3π + energy spectrum in a cubic box can be produced by the quantization condition from Eq. (27): More technical aspects need to be spelled out in order to apply the above quantization condition: renormalization procedure, projection of the spectrum according to irreducible representations of cubic symmetry group, and finite lattice spacing effect. All these mentioned factors hence ultimately will add extra layer of technical complication on top of Eq. (27). As a simply illustration of our formalism, a specific choice about interactions is made in this work: the two-body potential is the zero-range potential and the three-body interaction is turned off. The contact interaction resembles the leading order terms of the chiral Lagrangian [47,48,70]. With contact interaction, the 3π + LS equation is simplified considerably. We now turn to renormalization of the multi-π + LS equations.

C. Pair-wise contact interaction and renormalization
Since contact interactions allow particles to get arbitrarily close to each other, it is often necessary to regularize the ultraviolet part of the dynamics. This is indeed the case with our illustrative choice of multiple-pion interactions. We will discuss renormalization of both 2π + and 3π + LS equations in what follows.

Renormalization of 2π + LS equation
The zero-range potential, V (r) = V 0 δ(r) and V (k) = V 0 , is the simplest case of separable potentials. More specifically in this case, is independent of k for fixed E, and is factored out of the two-body homogeneous LS equation (6): The infinite momentum sum of two-pion Green's function (4) is divergent, and the divergence can be regularized by imposing a sharp ultraviolet momentum cutoff Λ. It is instructive to display the divergence in infinite volume and at the continuum limit: For finite values of P and E, the divergence remains same, or equivalently, adds corrections in powers of P/Λ and √ m π E/Λ. The cutoff dependence of shifted by a constant value: L , so that black dots and red curve overlap when Λ = Λ 0 . compared with its infinite-volume counterpart, as Λ → ∞, is shown in Fig. 2.
Finite lattice spacing provides a natural regularization on ultraviolet divergence. The physical observables must not depend on the cutoff or choice of lattice spacing, and this is to be assured by renormalization procedure. The bare interaction strength, V 0 , must be redefined to absorb ultraviolet divergence of the momentum sum of Green's function by where V R (µ) stands for the renormalized physical coupling strength at the renormalization scale, µ. The twobody quantization condition (31) can be rewritten with the renormalized coupling as where the cutoff dependencies from the summations on the right-hand side cancel out at the limit Λ → ∞, so ultraviolet divergence is now removed from the quantization condition.
(4) vanishes in Eqs. (25) and (26). Due to the zero-range nature of the interaction, the amplitude t (P) (2) (k 1 , k 2 ) associated with two-pion interaction between the pair (13) depends only on momentum k 2 . So it is appropriate to rename t (P) The 3π + LS equation (27) is therefore reduced to The first line of Eq.(36) describes interaction within the pair (13), with the second pion acting as the spectator. The second line of Eq.(36) represents crossed-channel interactions through exchanging the second particle between the pairs. The direct-channel terms in Eq.(36) resemble the leading order isobar contribution in Khuri-Treiman approach [71][72][73][74][75][76][77]. The crossed-channel terms are associated with rescattering corrections from other pairs into isobar pair (13). It can be illustrated quite straightforwardly by iterations of Eq.(36) that crossedchannel contributions are not UV divergent, and that the UV divergence emerge only in direct-channel term. So, as far as renormalization is concerned, crossed-channel interactions can be "turned off". We arrive at an equation similar to Eq. (31): where k 2 is momentum of the spectator pion. Equations (37) and (31) are not identical due to the relativistic kinematics brought by the spectator particle on top of the pair (13): compared to where P = k 1 + k 3 is the total momentum of the pair (13). Nonetheless, the multi-particle Green's function is dominated by the location of poles of particles propagator, so the dominant contributions of the 2π + and 3π + Green's functions behave in a similar way, and . (41) Therefore, the asymptotic high energy behavior of infinite momentum sum in Eq. (37) and Eq.(31) should be exactly same, with a momentum cutoff, (42) The renormalization procedure in 3π + LS equation thus can be carried out in the same way as in the 2π + sector, see Fig. 3 for the comparison of Using Eq.(33) and redefining the bare coupling V 0 , the renormalized 3π + LS equation can be given in a compact form: The quantization condition of 3π + can be rewritten with the renormalized coupling as which yields entire discrete energy spectrum of three pions in a finite box at continuum limit.
The lattice QCD simulations are usually performed in a cubic box with finite lattice spacing. We discuss in this section impacts of finite lattice spacing and cubic lattice symmetry on multi-π + energy spectra.

A. Finite lattice spacing effect
With a finite lattice spacing a, the coordinate of particles become discrete, a continuous integration over coordinates must be replaced by a discrete sum over lattice sites: where the sum of n is finite and is bound by the lattice spacing a : (n x , n y , n z ) ∈ [−N, N ] and N = L 2a − 1 .
The infinite momentum sum p where p = 2π L n and n ∈ Z 3 at continuum limit is replaced by a finite sum with momenta restricted to the first Brillouin zone: In addition, the continuous relativistic energy momentum dispersion relation, E p = m 2 π + p 2 , is replaced by lattice dispersion relation with a explicit dependence on the finite lattice spacing a, 2 sinh aE p 2 = 2 cosh am π + 4 − 2 i=x,y,z where p i = 2π L n i , i = x, y, z and n i ∈ [−N, N ]. Therefore, to convert all the continuum limit multi-π + dynamical equations presented in Section II to equations with explicit finite lattice spacing effect build in, the following relations between the physical quantities at continuum limit and their finite lattice spacing counterparts must be considered: for 2π + , and G (a,P) 3π for 3π + , where p 3 = P − p 1 − p 2 . The symbol a in superscript is used to label the functions defined with a finite lattice spacing. The examples of finite lattice spacing effects are illustrated in Fig. 4 and Fig. 5.

2π + LS equation with finite lattice spacing
With all the ingredients mentioned previously, the 2π + LS equation for a general potential at a finite lattice spac- For contact interaction, a finite lattice spacing may play the role of natural ultraviolet cutoff, Eq. (34) is thus replaced by where sum of p = 2πn L is restricted in first Brillouin zone with (n x , n y , n z ) ∈ [−N, N ] and N = L 2a − 1.

3π + LS equation with finite lattice spacing
With same strategy, the finite lattice spacing version of 3π + LS equation is obtained by replacing Eq.(43) by t (a,P) 3π where S (a,P) 3π and p = 2πn L , (n x , n y , n z ) ∈ [−N, N ].

B. Cubic lattice symmetry group and its irreducible representations
The energy spectrum of a quantum system is normally organized and labeled according to irreducible representations (irreps) of the symmetry groups of the system. These irreps carry so-called "good" quantum numbers that help in practice identify states of the system. For example, in infinite volume, hadronic bound states are typically labeled in terms of total angular momentum and parity based on the system's behavior under rotations and space inversion. To decouple states with different quantum numbers, one can project the dynamic equations of the system onto each irrep of symmetry groups. The end result is that each eigen-energy belongs to certain irrep, or irreps in the case of degeneracy.
Similar operations can be carried out for finite-volume systems as well. The energy spectra of finite-volume multi-π + system are expected to be labeled by irreps of the cubic lattice symmetry group. For instance, the cubic symmetry group for a system with vanishing total momentum, P = 0, is the octahedral group O h , which consists of 48 symmetry operations, including 24 discrete space rotations and inversions of all axes. The irreps of octahedral group O h include one-dimensional representations, A ± 1 and A ± 2 , two-dimensional representation, E ± , and three-dimensional representations, T ± 1 and T ± 2 . The superscripts ± are used to label even or odd parity state of the system. A brief introduction on the subject of irrep projection of dynamical equations is given in this section, with the 2π + LS equation used as a specific example. More elaborate explanations on the subject can be found in, e.g., Refs. [54,78]. The irrep-projected 2π + and 3π + LS equations in the CM frame will be presented in the following subsections.
We start with the projection operator for the cubic group [78], where G and g stand for the the cubic symmetry group and its elements. Here λ is used to label a specific irrep, and d λ denotes the dimension of irrep λ, e.g., d λ = 1, 1, 2, 3, 3 for λ = A ± 1 , A ± 2 , E ± , T ± 1 , T ± 2 , respectively. With g running over all elements of G, Γ (λ) (g) are a set of d λ -by-d λ matrices that furnish irrep λ. O(g) represents the symmetry operation implemented on quantum states. These operations are perhaps most easily explained by their action on momenta or coordinates of the particles. For instance, where |p | = |p| for all g ∈ G, because operations in O h are either rotation or inversion. The complete list of symmetry operations of the octahedral group O h are given in Appendix C.
Momenta of a particle can be grouped into various sets, and momenta in the same set are connected by symmetry operations. Any set of such momenta can be represented by a single reference vector p 0 , and the rest of members of this particular set can be reached by The reference vector used in this work is the same concept as used in Ref. [54], while in Ref. [78] the set of momenta represented by p 0 is called the "star" of p 0 . With a cutoff on the lattice momenta, the reference vectors, p 0 's, may be chosen as In terms of reference vectors and symmetry operations, the sum of momenta over an arbitrary function can be reorganized as follows: where ϑ(p 0 ) is the multiplicity of distinct momenta within the set represented by p 0 . For instance, for p 0 = 2π L (0, 0, 1) , ϑ(p 0 ) = 6 and the six distinct momenta are The multiplicity function ϑ(p 0 ) used in this work has the same meaning as the multiplicity of a given shell defined in Ref. [54]. Therefore, the projector defined in Eq.(55) acting on amplitudes can be interpreted as weighted average within the momentum set represented by p 0 , and the projected amplitudes can be labeled by p 0 's.

C. Irrep projection of 2π + LS equation
As an example, we discuss projection of two-pion amplitudes for P = 0 onto irrep λ that is given by The projected amplitude, t (λ) 2π (k 0 ), actually does not depend on quantum number α due to the cubic symmetry of the system. Therefore, it may be convenient to express the projection operator in terms of the character of irreps: where is the character of irrep λ, and it satisfies orthogonality relation: With the help of P (λ) , one can rewrite the projection of two-pion amplitudes in a more compact form: The above equation can be inverted using the orthogonality relation: Projection of dynamic equations established in previous sections is obtained by applying projection formula of amplitudes. Applying Eqs. (60), (61), and (56) to Eq. (6), we arrive at where V (λ) (k 0 , p 0 ) is the projected potential: The energy spectrum stemming from the irrep-projected dynamical equation is of course labeled by the same irrep. With the contact interaction, V (k) = V 0 , only A + 1 irrep survives after projection of the potential: Therefore, projection of Eq.(52) yields non-trivial solutions only in A + 1 : D. Irrep projection of 3π + LS equation Since the three-pion amplitudes depend on two momentum variables, the irrep projection can be done by first projecting out each momentum dependence separately onto the corresponding irrep, and then coupling two individual projections to an irrep of the 3π + system. This procedure resembles addition of angular momenta, which is essentially reduction of tensor product of two SO(3) irreps.
Following that idea, projection of 3π + amplitude t (0) (k) (k 1 , k 2 ) is accomplished by the following operator: where P (λ1) α1,α1 (1) and P (λ2) α2,α2 (2) are the uncoupled irrep projection operators on momenta k 1 and k 2 , respectively, and is the Clebsch-Gordan coefficient that couple irreps λ 1 and λ 2 to irrep λ on the α-th row. Again, due to the cubic symmetry of the system, projected three-body amplitudes do not depend on α, so the projection of t can be written as The general projection of three-particle amplitudes can be cumbersome. However for the interaction under consideration in the present paper, no three-body force and only pair-wise contact interaction, the procedure is greatly simplified. This is because, as shown in Sec. II C, the three-pion amplitude depends only on a single momentum: t (P) 3π (k 2 ). With no dependence on k 1 , only the trivial irrep A + 1 survives after projection of k 1 dependence: λ 1 = A + 1 and Γ (λ1) α1,α1 = 1. The coupled irrep of the 3π + system is thus determined by k 2 dependence alone: λ = λ 2 and α = α 2 . Therefore, one can apply straightforwardly most of the results for the twopion case in Sec. III C to the irrep projection of the 3π + LS equation.
With pair-wise contact interaction, the irrep projection of t Similar to Eq.(61), we also have t (a,0) Noticing that S (a,0) 3π (k 0 ; E, µ) remains invariant under symmetry operations, and that the projection on the right-hand side of Eq.(53) yields one can show that irrep projection of the 3π + LS equation (53) ultimately leads to the following: where The O h -irrep projected 3π + quantization condition, under our assumptions, has a simple form: The above equation suggests that except for A + 1 irrep, only trivial solutions -free particle states-can be found near the ground state with all three pions at rest, because the only irrep in which C (λ) 3π (0, 0; E) does not vanish is λ = A + 1 .

IV. NUMERICAL RESULTS
In this section, we use the quantization conditions (64) and (73) to produce 2π + and 3π + energy spectra for various lattice sizes. A single parameter, the renormalized coupling of the two-body contact interaction V R (µ), is fitted to the lattice results of Ref. [27].
The two-pion scattering parameters, the scattering length a 0 and effective range r 0 , can immediately be produced once V R (µ) is determined from the fit. a 0 and r 0 are usually defined by the effective range expansion at low energies: where is relative momentum of the pions in the CM frame. On the other hand, as detailed in Appendix B, the isospin-2 S-wave phase shift of π + π + scattering through a pairwise contact potential is given by Comparing Eqs. (74) and (75), one finds a 0 and r 0 in terms of V R (µ): The values of π + π + scattering length a 0 and effective range r 0 extracted in this study are compared in Table I with the values given by other works. Using Eq.(75), the π + π + S-wave phase shifts are plotted in Fig. 7.
Treating V R (µ) as a free parameter, one can produce the CM-frame 2π + and 3π + energy spectra in A + 1 and E + irreps with Eqs. (64) and (73), at lattice spacing a = 0.324GeV −1 and pion mass m π = 0.2GeV. The spectra are then matched to the lattice results reported in Ref. [27], as shown in Fig. 6. The renormalization scale is chosen at µ = 0GeV, so that G(µ) = 1 8π 2 is real. The value of coupling constant of contact interaction is extracted, As illustrated in Fig. 6, with only one parameter, Eqs. (64) and (64) struggle to match the lattice results, so the fit yields a large error on the value of V R (0). This suggests that more sophisticated pair-wise interactions and/or three-body interactions may be needed. (c) 3π + energy spectrum in E + irrep.
FIG. 6: 2π + and 3π + energy spectrum in irreps A + 1 and E + and CM frame: red bands are spectrum produced by using Eq.(64) and Eq.(73) for 2π + and 3π + respectively, the lattice results (black circles) are taken from [27]. The free multi-π + energy spectrum (blue dashed curves) by using E = 2 a sinh −1 ( i sinh aEp i 2 ) and i p i = P are also plotted as reference. The lattice spacing used in this work and in [27] is a = 0.324GeV −1 , m π = 0.200GeV.   (74), where a 0 m π = 0.1019 ± 0.0088 and r 0 m π = 9.0 ± 2.4 are taken from [27]. For the reference, the physical phase shift by using parameterization given in [79] is also plotted (blue band).

V. SUMMARY
Based on the variational approach combined with the Faddeev method proposed in Refs. [64][65][66][67], the relativis-tic 2π + and 3π + dynamics in finite volume are presented in the paper. The presentation of multi-π + dynamics included both pair-wise and three-body interactions, and the effects of finite lattice spacing and projections onto irreps of the cubic lattice group were also discussed. The quantization conditions were used to analyze the lattice data published in [27]. In the present work, only contact pair-wise interactions were employed in our analysis and, as a result, the renormalized coupling strength was the sole free parameter. The scattering length a 0 and effective range r 0 were obtained, however, with rather large uncertainty: a 0 m π = 0.28 ± 0.17 and r 0 m π = 4.8 ± 2.1, which might be attributed to the lack of accuracy in describing long-range interactions among the π + 's by only a contact pair-wise interaction. It may also suggest that it is worth investigating the role of three-pion forces.
Appendix A: Reduction of Bethe-Salpeter equation to relativistic 3π + Lippmann-Schwinger equation Consider the general form of Bethe-Salpeter (BS) equation [80] for three-scalar-particle bound states: where p i = (p i0 , p i ) are the four momenta of three particles, the three-body BS wave function is labeled only by two independent particle momenta, p 1 and p 2 , since the momenta of three particles are constrained by energy-momentum conservation, p 3 = P − p 1 − p 2 . Assuming "instantaneous interaction kernel", I(p 1 , p 2 ) = I(p 1 , p 2 ), and introducing Lippmann-Schwinger equation wave function, ψ(p 1 , p 2 ) = dp10 2π dp20 2π ψ BS (p 1 , p 2 ), hence, we get ψ(p 1 , p 2 ) = dp 10 2π dp 20 2π The integration over propagators can be carried out, (2π) 6 G (P) (p 1 , p 2 ; E) = dp 10 2π dp 20 2π where E pi = p 2 i + m 2 π . The interactions of 3π + consist of pair-wise interactions and three-body interaction potential, where symbols V and V (4) are used to denote pair-wise interaction and three-body force respectively in consistent with the conventions used in Section II. q ij = pi−pj 2 refers to the relative momentum between i-th and jth pions. The relativistic kinematic factors p k |p k = 2E p k (2π) 3 δ(p k − p k ) emerge when k-th particle is free propagating and not involved in the interaction. The relativistic 3π + Lippmann-Schwinger equation is thus given explicitly by Appendix B: 2π scattering amplitude in infinite volume with contact interaction With the contact interaction, the relativistic Lippmann-Schwinger equation of 2π system can be solved analytically, so the scattering wave function of 2π in infinite volume and in CM frame is given by where r and p are the relative coordinate and momentum of two pions respectively. The total energy of two pions is related to p by E = 2E p = 2 p 2 + m 2 π . The two pions Green's function in CM frame in infinite volume is given by The two pions amplitude in CM frame is thus given by, The Green's function G (0) (0; E) diverge at r = 0, a cutoff Λ on momentum integration may be introduced to regularize ultraviolet divergence, as Λ → ∞ hence we find where The imaginary part of function G(E) is non-zero only above threshold of two pions: for E > 2m π , and zero otherwise. The ultraviolet divergence may be absorbed by redefining bare coupling, V 0 , where V R (µ) stands for the renormalized coupling strength at a renormalization scale µ. µ will be chosen below two pions threshold, so that G(µ) is real. Therefore, the cutoff dependence is cancelled out completely, and renormalized scattering amplitude is now given by The phase shift is defined by Expanding phase shift near threshold p = |p| = 1 2 where the effective expansion parameters, a 0 and r 0 are given by The octahedral group O h is the direct product of proper rotational group O that rotates a cube into itself and spatial inversion I: and the inversion matrix, All 48 elements of the octahedral group O h are usually grouped into different conjugacy classes, and the mem- bers within the same conjugacy class share the same character value for a given irrep. Using the same convention as used in [78], ten classes of the octahedral group O h are named as C i where i = 1, · · · , 10, they are associated with 48 elements by and C i = IC (i−5) for i = 6, · · · , 10. The character table for all irreps of the octahedral group O h with positive parity quantum number is given in Table II.
Appendix D: Non-relativistic three-particle dynamics in finite volume The dynamics of three non-relativistic identical bosonic particles in finite volume is described by the U (r ij ) − U (4) (r 13 , r 23 ) where m is the mass of identical bosons. x i denotes the position of i-th particle, and r ij = x i − x j is relative coordinate between i-th and j-th particles. The pair-wise interaction between i-th and j-th particles is described by U (r ij ), and U (4) (r 13 , r 23 ) represents the threebody interaction among all particles. Both pair-wise and three-body interactions in finite volume are assumed to be short-range and periodic, that is to say U (r) = U (|r + nL|), n ∈ Z 3 , U (4) (r 13 , r 23 ) = U (4) (r 13 + n 1 L, r 23 + n 2 L), n 1,2 ∈ Z 3 , where L is the size of the cubic lattice. Therefore finite volume three-particle wave function must also satisfy periodic boundary condition, Φ(x 1 , x 2 , x 3 ) = Φ(x 1 + n x1 , x 2 + n x2 , x 3 + n x3 ), (D3) where n xi ∈ Z 3 . As suggested in [65,66], it may be more convenient to consider the integral representation of Eq.(D1), p1,p2,p3 where p 1,2,3 ∈ 2πn L , n ∈ Z 3 . The center of mass motion of three-particle system can be factorized by Φ(x 1 , x 2 , x 3 ) = e iP·R φ(r 13 , r 23 ), where R = x1+x2+x3 3 = r13+r23 3 + x 3 is center of mass position of three-particle system, and P = 2π L d with d ∈ Z 3 stands for the total momentum of three-particle in a periodic cubic box. φ(r 13 , r 23 ) is the wave function that is associated with the internal motion of three particles, and it satisfies periodic boundary condition, φ(r 13 + n 1 L, r 23 + n 2 L) = e −i P 3 ·(n1L+n2L) φ(r 13 , r 23 ), (D6) where n 1,2 ∈ Z 3 . We remark that in this work we use (r 13 , r 23 ) to describe the internal motion of three particles, and x 3 is thus associated to CM motion. So that L 3 3 i=1 dx i = L 3 dr 13 dr 23 dx 3 , and it resembles a twolight and one heavy three-body atomic system. Integrating out CM motion, U (r ij ) + U (4) (r 13 , r 23 ) φ(r 13 , r 23 ). (D8) The three-particle Green's function is defined by G (P) (r 13 , r 23 ; E) = p1,p2 e i(p1− P 3 )·r13 e i(p2− P 3 )·r23 G (P) (p 1 , p 2 ; E), where p 1,2 ∈ 2πn L , n ∈ Z 3 , and p 3 = P − p 1 − p 2 . Following the same procedures as described in Section II, only two independent scattering amplitudes are required due to exchange symmetry of three-particle wave function, where p 3 = P − p 1 − p 2 , and U and U (4) are the Fourier transform of interaction potentials U and U (4) respectively. Non-relativistic three-particle dynamical equations, Eq.(D11) and Eq.(D12), resemble their relativistic counter parts, Eq.(25) and Eq. (26), excepts some relativistic kinematic factors.