MMulti- π + systems in finite volume Peng Guo
1, 2, 3, ∗ and Bingwei Long † College of Physic, Sichuan University, Chengdu, Sichuan 610065, China 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: May 6, 2020)We present a formalism to describe two- π + and three- π + dynamics in finite volume, the formalismis based on combination of a variational approach and the Faddeev method. Both pair-wise andthree-body interactions are included in the presentation. Impacts of finite lattice spacing and thecubic lattice symmetry are also discussed. To illustrate application of the formalism, the pair-wisecontact interaction that resembles the leading order interaction terms in chiral effective theory isused to analyze recent lattice results. I. INTRODUCTION
Understanding of few-hadron interactions is crucial innuclear/hadron physics. Few-hadron dynamics providesa unique access to various fundamental parameters ofQuantum Chromodynamics (QCD), for quarks and glu-ons only manifest themselves within hadrons due to colorconfinement. For instance, the u - and d -quark mass dif-ference can be extracted from η → π decay process [1–7]. Remarkable few-body phenomena, such as the Efimovstates [8, 9] and halo nuclei [10, 11], have been predictedand observed in strong-interaction physics. Few-bodysystems also offer an outlook for many-body effects, e.g.,three-nucleon forces [12–14] and fractional quantum Halleffects [15].Latest advances have now made lattice QCD (LQCD)a powerful quantitative tool to study hadron physics fromthe first principles. In recent years, realistic LQCD calcu-lations of multi-hadron systems have been made possible[16–27]. However, LQCD calculations are usually per-formed in a periodic box in the Euclidean space-time, andonly discrete energy spectra are extracted from time de-pendent correlation functions. That is to say, the multi-hadron dynamics is encoded in a set of discrete energylevels in finite volume. Therefore, mapping out infinite-volume few-body dynamics from finite-volume energyspectra is a key step toward understanding multi-hadronsystems from LQCD calculations.In the two-body sector, a pioneering approach pro-posed by L¨uscher [28] tends to build connections betweeninfinite volume reaction amplitudes and energy levels ina periodic cubic box, which was later on further extendedto the cases of moving frames and coupled channels [29–38]. In past few years, progresses have been made onthe study of few-body systems above three-body thresh-olds in finite volume [39–67]. To certain extent, mostof these developments may be regarded as extensions ofthe L¨uscher formula. L¨uscher’s formula in two-body sec-tor may demonstrate a clear advantage: the quantization ∗ [email protected] † [email protected] condition is given in terms of infinite volume two-bodyscattering amplitude or phase shift, it provides a directconnection between infinite-volume reaction amplitudesand finite-volume energy spectra. Unfortunately, abovethree-body threshold, in addition to complication of finitevolume dynamics, the infinite volume few-body reactionamplitudes usually can not be easily parameterized in ananalytic form, solutions of these amplitudes are given bycoupled integral equations, such as Faddeev equations[68, 69]. Dealing with questions regarding infinite andfinite-volume physics simultaneously presents great chal-lenges. As illustrated in Ref. [65, 66], the quantizationcondition of few-body systems may be presented in termsof finite-volume Green’s function and effective interactionbetween particles embodied by a potential. Since infinitevolume scattering amplitudes are not explicitly involvedin variational approach, it may be more efficient for prac-tical analysis of LQCD calculation results.In present work, the variational approach proposed inRefs. [64–67] is applied to multi- π + systems. In 3 π + system, both pair-wise interactions and three-body in-teractions are included in our presentation. We also takeinto consideration relativistic pion kinematics, the finitelattice spacing and cubic lattice symmetry effects. As ourinitial attempt, the quantization conditions are used toanalyze LQCD data published in Ref. [27] by accountingfor only pair-wise contact interaction in the center-of-mass (CM) frame. The 2 π + scattering length and effec-tive range are extracted from analysis. With a single pa-rameter, the coupling constant of the pair-wise contactinteraction, we are able to make prediction on energyspectra, albeit with relatively large uncertainties.The paper is organized as follows. The formalism offinite volume multi- π + systems at the continuum limit,where the lattice spacing vanishes, is presented in detailin Section II. The finite lattice spacing effect and cubiclattice symmetry are discussed in Section III. Numericalresults are given in Section IV, followed by a summaryin Section V. a r X i v : . [ h e p - l a t ] M a y II. RELATIVISTIC MULTI- π + SYSTEMS INFINITE VOLUME AT CONTINUUM LIMIT
For the future reference, we give in this section a com-plete presentation of our formalism that describes finite-volume dynamics of multi- π + systems. The formalismadopted in this work is based on the variational ap-proach combined with the Faddeev method which waspreviously discussed in Refs. [64–67]. The relativistic,finite-volume multi- π + dynamics is for the moment for-mulated at the continuum limit with vanishing latticespacing ( a = 0), in the sense that ultraviolet divergenceis to be removed through proper regularization proce-dure. The finite lattice spacing effect will be installedrather straight-forwardly later on in Section III. In whatfollows, we will go through dynamical equations of thetwo-pion and three-pion systems, and close the sectionby elaborating how ultraviolet divergences are treatedand how renormalization is carried out. A. Dynamical equations of π + system The relativistic 2 π + system in finite volume is governedby the homogeneous Lippmann-Schwinger (LS) equation[65–67]: φ π ( r ) = (cid:90) L d r (cid:48) G ( P )2 π ( r − r (cid:48) ; E ) V ( r (cid:48) ) φ π ( r (cid:48) ) , (1)where r and P are the relative coordinate and total mo-mentum of the pions, respectively. With the assumptionof zero lattice spacing ( a = 0), r is continuously dis-tributed in finite volume, and (cid:82) L d r (cid:48) stands for contin-uous integration bound by the edges of a periodic cubicbox. The spherical two-body potential is represented by V ( r ) where r = | r | . The wave function for relative mo-tion of the two pions satisfies periodic boundary condi-tion [64–66], φ π ( r + n L ) = e − i P · n L φ π ( r ) , n ∈ Z , (2)where L is the size of the cubic lattice. Imposing the theperiodic boundary condition on the plane wave of CMmotion yields P = 2 πL d , d ∈ Z . (3)We remark that throughout the entire paper, the explicitenergy dependence of both the wave function and thescattering amplitude is dropped for the convenience ofpresentation. The two-pion Green’s function is given by G ( P )2 π ( r ; E ) = (cid:88) p e i ( p − P ) · r (cid:101) G ( P )2 π ( p ; E ) , (cid:101) G ( P )2 π ( p ; E ) = 1 L E p + E P − p )2 E p E P − p E − ( E p + E P − p ) , (4) where E p = (cid:112) m π + p and p = π n L , n ∈ Z .Defining finite-volume two-pion amplitude, t ( P )2 π ( k ) = − (cid:90) L d r e − i ( k − P ) · r V ( r ) φ π ( r ) , (5)where k = π n L , n ∈ Z , one can transform Eq. (1) intothe homogeneous momentum-space LS equation: t ( P )2 π ( k ) = (cid:88) p (cid:101) V ( | k − p | ) (cid:101) G ( P )2 π ( p ; E ) t ( P )2 π ( p ) , (6)where the momentum-space potential (cid:101) V ( k ) = (cid:90) L d r e − i k · r V ( r ) . (7)According to the variational approach [64–67], the quan-tization condition for the two-pion system is given bydet (cid:104) δ k , p − (cid:101) V ( | k − p | ) (cid:101) G ( P )2 π ( p ; E ) (cid:105) = 0 , ( k , p ) ∈ π n L , n ∈ Z . (8)Solving for E yields the discrete energy spectrum of twointeracting pions in finite volume at the continuum limitwhere the lattice space approaches zero. B. Dynamical equations of π + system After factoring out the CM motion (see Appendix Dfor the example of removing CM motion for a nonrela-tivistic three-particle system), the dynamics of relativis-tic finite-volume three-pion system is described by a LStype integral equation in coordinate space, φ π ( r , r ) = (cid:88) k =1 (cid:90) L d r (cid:48) d r (cid:48) × G ( P )( k ) ( r − r (cid:48) , r − r (cid:48) ; E ) V ( k ) ( r (cid:48) , r (cid:48) ) φ π ( r (cid:48) , r (cid:48) ) , (9)where r ij = x i − x j is relative coordinate between the i-th and j-th pions. r and r are chosen to describe therelative motion of 3 π + system. The pair-wise interactionsbetween i-th and j-th identical pions are represented by V ( k ) ( r , r ) = V ( r ij ) , (10)with k = 1 , , k (cid:54) = i (cid:54) = j and V ( r ) is the previouslydefined two-body potential. V (4) ( r , r ) with k = 4denotes the three-body force acting on all the pions. Thethree-pion Green’s functions are defined as follows: G ( P )( k ) ( r , r ; E )= (cid:88) p , p e i ( p − P ) · r e i ( p − P ) · r (cid:101) G ( P )( k ) ( p , p ; E ) , (11)and (cid:101) G ( P )( k ) ( p , p ; E ) = 2 E p k (cid:101) G ( P )3 π ( p , p ; E ) , k = 1 , , , (cid:101) G ( P )(4) ( p , p ; E ) = (cid:101) G ( P )3 π ( p , p ; E ) , (12)where (cid:101) G ( P )3 π ( p , p ; E ) = 1 L (cid:80) i =1 E p i E p E p E p E − ( (cid:80) i =1 E p i ) . (13)The total momentum P = p + p + p = 2 πL d , d ∈ Z , (14)where p i = 2 π n i L , n i ∈ Z , (15)is the momentum of the i -th particle. The relativistickinematic factors, 2 E p k for k = 1 , ,
3, in Eq.(12) areassociated with the relativistic normalization of the freepropagating spectator particle in presence of the pair-wise interaction between the i-th and j-th particles. Therelativistic LS equation may be derived from the Bethe-Salpeter equation, see Appendix A. The three-body wavefunction must satisfy the periodic boundary condition[66]: φ π ( r + n L, r + n L ) = e − i P · ( n L + n L ) φ π ( r , r ) , (16)where n , ∈ Z .As suggested in Refs. [64–67], the Faddeev amplitudesmay be introduced by t ( P )( k ) ( k , k ) = − (cid:90) L d r d r e − i ( k − P ) · r e − i ( k − P ) · r × V ( r ij ) φ π ( r , r ) , k (cid:54) = i (cid:54) = j,t ( P )(4) ( k , k ) = − (cid:90) L d r d r e − i ( k − P ) · r e − i ( k − P ) · r × V (4) ( r , r ) φ π ( r , r ) . (17)Equation (9) is thus turned into coupled equations: t ( P )( k ) ( k , k ) = (cid:88) p , p (cid:101) V ( k ) ( k − p , k − p ) × (cid:101) G ( P )3 π ( p , p ; E ) (cid:34) (cid:88) k (cid:48) =1 E p k (cid:48) t ( P )( k (cid:48) ) ( p , p ) + t ( P )(4) ( p , p ) (cid:35) , (18)where (cid:101) V ( k ) ’s are the Fourier transform of pair-wise andthree-body interaction potentials, (cid:101) V ( k ) ( k , k ) = (cid:90) L d r d r e − i k · r e − i k · r V ( r ij ) | k (cid:54) = i (cid:54) = jk =1 , , , (cid:101) V (4) ( k , k ) = (cid:90) L d r d r e − i k · r e − i k · r V (4) ( r , r ) . (19) = (13)2 1 + (13)2 (13)2(23) + (13)3(12) 2 + =
132 132(13)2 + + + t (2) t (2) t (1) t (3) t (4) t (4) t (4) t (2) t (1) t (3) FIG. 1: Diagrammatic representation of Eq.(25) andEq.(26), pair-wise and three-body interactions arerepresented by black solid circle and blue solid square.Two-body interactions (cid:101) V (1 , , are related to (cid:101) V ( k ), as de-fined in Eq. (7), through (cid:101) V (2) ( k , k ) = (cid:101) V ( k ) L δ k , , (cid:101) V (1) ( k , k ) = (cid:101) V ( k ) L δ k , , (cid:101) V (3) ( k , k ) = (cid:101) V ( | k − k | L δ k , − k , (20)where δ k , p denotes the 3D Kronecker delta function.
1. Exchange symmetry of π + system As is the case for any identical bosons, the wave func-tion of 3 π + system must be invariant under exchange ofany pair of pions, for example, φ π ( r , r ) ↔ = φ π ( r , r ) ↔ = φ π ( r , r ) , (21)and φ π ( r ik , r jk ) i ↔ j = φ π ( r jk , r ik ). The exchange sym-metry of three-body wave function suggests that t ( P )(1 , , are related: t ( P )(1) ( k , k ) = t ( P )(2) ( k , k ) ,t ( P )(3) ( k , k ) = t ( P )(2) ( k , k ) = t ( P )(2) ( k , k ) , (22)where P = k + k + k .Using the definition of t ( P )(2) in Eq.(17) and symmetryrelations of wave function in Eq.(21), we find useful sym-metry properties of t ( P )(2) : t ( P )(2) ( k , k ) = t ( P )(2) ( k , k ) , t ( P )(2) ( k , k ) = t ( P )(2) ( k , k ) . (23)In addition to that of the wave function, exchange sym-metry of three-body potential V (4) also constrains theamplitude t (4) : t ( P )(4) ( k , k ) = t ( P )(4) ( k , k ) = t ( P )(4) ( k , k ) = · · · . (24)
2. Further reduction of π + LS equations
Using the aforementioned symmetry relations of Fad-deev amplitudes, Eq.(18) are further reduced into thefollowing equations: t ( P )(2) ( k , k ) = (cid:88) p (cid:101) V ( | k − p | ) L (cid:101) G ( P )3 π ( p , k ; E ) × (cid:20) E k t ( P )(2) ( p , k ) + 2 E p t ( P )(2) ( k , p )+ 2 E P − p − k t ( P )(2) ( p , P − p − k ) + t ( P )(4) ( p , k ) (cid:21) , (25)and t ( P )(4) ( k , k ) = (cid:88) p , p (cid:101) V (4) ( k − p , k − p ) (cid:101) G ( P )3 π ( p , p ; E ) × (cid:20) E p t ( P )(2) ( p , p ) + 2 E p t ( P )(2) ( p , p )+ 2 E P − p − p t ( P )(2) ( p , P − p − p ) + t ( P )(4) ( p , p ) (cid:21) . (26)The diagrammatic representations of Eq.(25) andEq.(26) are shown in Fig. 1. It may be convenient toconsolidate Eqs.(25) and (26) in the matrix form: (cid:34) t ( P )(2) ( k , k ) t ( P )(4) ( k , k ) (cid:35) = (cid:88) p , p (cid:101) G ( P )3 π ( p , p ; E ) × K ( k , k ; p , p ) (cid:34) t ( P )(2) ( p , p ) t ( P )(4) ( p , p ) (cid:35) , (27)where ( k i , p i ) ∈ π n L , n ∈ Z . Elements of the matrixfunction K ( k , k ; p , p ) are given by K ( k , k ; p , p ) = 2 E p L (cid:20) δ p , k (cid:101) V ( | k − p | )+ δ p , k (cid:16) (cid:101) V ( | k − p | ) + (cid:101) V ( | k − p | ) (cid:17) (cid:21) , K ( k , k ; p , p ) = L δ p , k (cid:101) V ( | k − p | ) , K ( k , k ; p , p ) = 2 E p (cid:20) (cid:101) V (4) ( k − p , k − p )+ (cid:101) V (4) ( k − p , k − p ) + (cid:101) V (4) ( k − p , k − p ) (cid:21) , K ( k , k ; p , p ) = (cid:101) V (4) ( k − p , k − p ) . (28)The coupled equations, Eqs.(25) and (26), or the ma-trix form of them Eq.(27), will serve as basic dynamicalequations for 3 π + system. The entire 3 π + energy spec-trum in a cubic box can be produced by the quantizationcondition from Eq.(27):det (cid:104) I − (cid:101) G ( P ) K (cid:105) = 0 . (29)More technical aspects need to be spelled out in orderto apply the above quantization condition: renormaliza-tion procedure, projection of the spectrum according to irreducible representations of cubic symmetry group, andfinite lattice spacing effect. All these mentioned factorshence ultimately will add extra layer of technical compli-cation on top of Eq.(27). As a simple illustration of ourformalism, a specific choice about interactions is made inthis work: the two-body potential is the zero-range po-tential and the three-body interaction is turned off. Thecontact interaction resembles the leading order terms ofthe chiral Lagrangian [47, 48, 70]. With contact interac-tion, the 3 π + LS equation is simplified considerably. Wenow turn to renormalization of the multi- π + LS equa-tions.
C. Pair-wise contact interaction andrenormalization
Since contact interactions allow particles to get arbi-trarily close to each other, it is often necessary to regu-larize the ultraviolet part of the dynamics. This is indeedthe case with our illustrative choice of multiple-pion in-teractions. We will discuss renormalization of both 2 π + and 3 π + LS equations in what follows.
1. Renormalization of π + LS equation
The zero-range potential, V ( r ) = V δ ( r ) and (cid:101) V ( k ) = V , is the simplest case of separable potentials. Morespecifically in this case, t ( P )2 π ( k ) ≡ t ( P )2 π (30)is independent of k for fixed E , and is factored out of thetwo-body homogeneous LS equation (6):1 = V (cid:88) p (cid:101) G ( P )2 π ( p ; E ) . (31)The infinite momentum sum of two-pion Green’s function(4) is divergent, and the divergence can be regularized byimposing a sharp ultraviolet momentum cutoff Λ. It isinstructive to display the divergence in infinite volumeand at the continuum limit: | p | < Λ (cid:88) p (cid:101) G ( )2 π ( p ; 0) L →∞ → − π (cid:90) Λ dp p E p Λ →∞ → − π ln Λ m π + 1 − ln 28 π . (32)For finite values of P and E , the divergence remainssame, or equivalently, adds corrections in powers of P/ Λand √ m π E/ Λ. The cutoff dependence of | p | < Λ (cid:88) p (cid:101) G ( P )2 π ( p ; E = 0) L -> ∞ L = -
50 100 150 200 250 300 350 400 - - - - - - Λ ( GeV ) G π ( ) ( , E = ) FIG. 2: Plot of cufoff dependence of (cid:80) | p | < Λ p (cid:101) G ( )2 π ( p ; 0)(black dots) vs. π (1 − ln m π ) (red curve), where L = 10GeV − and m π = 0 . (cid:80) | p | < Λ p (cid:101) G ( )2 π ( p ; 0) − π (1 − ln m π ), where Λ = π √ L ,so that black dots and red curve overlap when Λ = Λ .compared with its infinite-volume counterpart, − π (cid:90) Λ dp p E p Λ →∞ → π (1 − ln 2Λ m π ) , as Λ → ∞ , is shown in Fig. 2.Finite lattice spacing provides a natural regularizationon ultraviolet divergence. The physical observables mustnot depend on the cutoff or choice of lattice spacing, andthis is to be assured by renormalization procedure. Thebare interaction strength, V , must be redefined to absorbultraviolet divergence of the momentum sum of Green’sfunction by 1 V = 1 V R ( µ ) + | p | < Λ (cid:88) p (cid:101) G ( )2 π ( p ; µ ) , (33)where V R ( µ ) stands for the renormalized physical cou-pling strength at the renormalization scale, µ . The two-body quantization condition (31) can be rewritten withthe renormalized coupling as1 V R ( µ ) = | p | < Λ (cid:88) p (cid:101) G ( P )2 π ( p ; E ) − | p | < Λ (cid:88) p (cid:101) G ( )2 π ( p ; µ ) , (34)where the cutoff dependencies from the summations onthe right-hand side cancel out at the limit Λ → ∞ , soultraviolet divergence is now removed from the quantiza-tion condition. - - - E ( GeV ) G ( E ) - G π ( ) ( , ) FIG. 3: Plot of (cid:80) | p | < Λ p (cid:104) (cid:101) G ( )2 π ( p ; E ) − (cid:101) G ( )2 π ( p ; µ ) (cid:105) (blackcurve) vs. (cid:80) | p | < Λ p (cid:104) E L (cid:101) G ( )3 π ( p , ; E + 2 m π ) − (cid:101) G ( )2 π ( p ; µ ) (cid:105) (redcurve) with µ = 0GeV, L = 10GeV − and m π = 0 .
2. Renormalization of π + LS equation
With three-body interaction V (4) turned off, t ( P )(4) van-ishes in Eqs. (25) and (26). Due to the zero-range natureof the interaction, the amplitude t ( P )(2) ( k , k ) associatedwith two-pion interaction between the pair (13) dependsonly on momentum k . So it is appropriate to rename t ( P )(2) ( k , k ) as t ( P )3 π ( k ) = t ( P )(2) ( k , k ) . (35)The 3 π + LS equation (27) is therefore reduced to (cid:34) − V (cid:88) p E k L (cid:101) G ( P )3 π ( p , k ; E ) (cid:35) t ( P )3 π ( k )= 2 V (cid:88) p E p L (cid:101) G ( P )3 π ( p , k ; E ) t ( P )3 π ( p ) . (36)The first line of Eq.(36) describes interaction within thepair (13), with the second pion acting as the spectator.The second line of Eq.(36) represents crossed-channel in-teractions through exchanging the second particle be-tween the pairs. The direct-channel terms in Eq.(36)resemble the leading order isobar contribution in Khuri-Treiman approach [71–77]. The crossed-channel termsare associated with rescattering corrections from otherpairs into isobar pair (13). It can be illustrated quitestraightforwardly by iterations of Eq.(36) that crossed-channel contributions are not UV divergent, and that theUV divergence emerge only in direct-channel term. So, asfar as renormalization is concerned, crossed-channel in-teractions can be “turned off”. We arrive at an equationsimilar to Eq. (31):1 = V (cid:88) p E k L (cid:101) G ( P )3 π ( p , k ; E ) , (37)where k is momentum of the spectator pion. Equations(37) and (31) are not identical due to the relativistic kine-matics brought by the spectator particle on top of thepair (13):2 E k L (cid:101) G ( P )3 π ( k , k ; E )= 1 L E k + E k + E k )2 E k E k E − ( E k + E k + E k ) , (38)compared to (cid:101) G ( P )2 π ( k ; E ) = 1 L E k + E k )2 E k E k E − ( E k + E k ) , (39)where P = k + k is the total momentum of the pair(13). Nonetheless, the multi-particle Green’s function isdominated by the location of poles of particles propaga-tor, so the dominant contributions of the 2 π + and 3 π + Green’s functions behave in a similar way,2 E k L (cid:101) G ( P )3 π ( k , k ; E ) ∼ L E k E k E − ( E k + E k + E k ) , (40)and (cid:101) G ( P )2 π ( k ; E ) ∼ L E k E k E − ( E k + E k ) . (41)Therefore, the asymptotic high energy behavior of infi-nite momentum sum in Eq.(37) and Eq.(31) should beexactly same, with a momentum cutoff, | p | < Λ (cid:88) p E k L (cid:101) G ( P )3 π ( p , k ; 0) ∼ − π ln Λ m π + finite part . (42)The renormalization procedure in 3 π + LS equation thuscan be carried out in the same way as in the 2 π + sector,see Fig. 3 for the comparison of | p | < Λ (cid:88) p (cid:104) (cid:101) G ( )2 π ( p ; E ) − (cid:101) G ( )2 π ( p ; µ ) (cid:105) and | p | < Λ (cid:88) p (cid:104) E L (cid:101) G ( )3 π ( p , ; E + 2 m π ) − (cid:101) G ( )2 π ( p ; µ ) (cid:105) . Using Eq.(33) and redefining the bare coupling V , therenormalized 3 π + LS equation can be given in a compactform: t ( P )3 π ( k ) = 2 | p | < Λ (cid:88) p E p L (cid:101) G ( P )3 π ( p , k ; E ) V R ( µ ) − (cid:101) S ( P )3 π ( k ; E, µ ) t ( P )3 π ( p ) , (43) where (cid:101) S ( P )3 π ( k ; E, µ ) = | p | < Λ (cid:88) p (cid:104) E k L (cid:101) G ( P )3 π ( p , k ; E ) − (cid:101) G ( )2 π ( p ; µ ) (cid:105) . (44)The quantization condition of 3 π + can be rewritten withthe renormalized coupling asdet δ k , p − E p L (cid:101) G ( P )3 π ( p , k ; E ) V R ( µ ) − (cid:101) S ( P )3 π ( k ; E, µ ) = 0 , ( k , p ) ∈ π n L , n ∈ Z , (45)which yields entire discrete energy spectrum of three pi-ons in a finite box at continuum limit. III. MULTI- π + DYNAMICS WITH FINITELATTICE SPACING
The lattice QCD simulations are usually performed ina cubic box with finite lattice spacing. We discuss inthis section impacts of finite lattice spacing and cubiclattice symmetry on multi- π + energy spectra. We willsubject the pion momenta to restriction imposed by thelattice spacing, but will not keep track of any other par-ticular lattice artifacts, such as lattice action . The resultwill serve as a phenomenology motivated estimation of fi-nite spacing effects on observables, especially the excitedstates, as we will see. A. Finite lattice spacing effect
With a finite lattice spacing a , the coordinate of parti-cles become discrete, a continuous integration over coor-dinates must be replaced by a discrete sum over latticesites: (cid:90) L d r → a (cid:88) n , (46)where the sum of n is finite and is bound by the latticespacing a :( n x , n y , n z ) ∈ [ − N, N ] and N = L a − . The infinite momentum sum (cid:80) p where p = πL n and n ∈ Z at continuum limit is replaced by a finitesum with momenta restricted to the first Brillouin zone:( n x , n y , n z ) ∈ [ − N, N ].In addition, the continuous relativistic energy momen-tum dispersion relation, E p = (cid:112) m π + p , is replaced bylattice dispersion relation with a explicit dependence onthe finite lattice spacing a ,2 sinh aE p (cid:115) am π + 4 − (cid:88) i = x,y,z cos ap i , (47) - - - E ( GeV ) G π a , ( , E ) - G π a , ( , ) FIG. 4: Plot of (cid:80) p (cid:104) (cid:101) G ( a, )2 π ( p ; E ) − (cid:101) G ( a, )2 π ( p ; µ ) (cid:105) forvarious finite lattice spacing a ’s with µ = 0GeV L = 10GeV − and m π = 0 . a = 0 . − (Red), 0 . − (Orange), 0 . − (Blue), and0GeV − (Black).where p i = πL n i , i = x, y, z and n i ∈ [ − N, N ]. There-fore, to convert all the continuum limit multi- π + dynam-ical equations presented in Section II to equations withexplicit finite lattice spacing effect build in, the followingrelations between the physical quantities at continuumlimit and their finite lattice spacing counterparts mustbe considered: E ↔ a sinh aE , m π ↔ a sinh am π , p i ↔ a sin ap i . (48)For examples, the momentum space Green’s function de-fined in Eq.(4) and Eq.(13) are now replaced by finitelattice spacing correspondences, (cid:101) G ( a, P )2 π ( p ; E ) = a L aE p + 2 sinh aE P − p )4 sinh aE p aE P − p × aE ) − (2 sinh aE p + 2 sinh aE P − p ) , (49)for 2 π + , and (cid:101) G ( a, P )3 π ( p , p ; E ) = a L (cid:80) i =1 aE p i )4 sinh aE p aE p aE p × aE ) − ( (cid:80) i =1 aE p i ) , (50)for 3 π + , where p = P − p − p . The symbol a insuperscript is used to label the functions defined witha finite lattice spacing. The examples of finite latticespacing effects are illustrated in Fig. 4 and Fig. 5. π + LS equation with finite lattice spacing
With all the ingredients mentioned previously, the 2 π + LS equation for a general potential at a finite lattice spac- - - - - - - - - r =( r x ,0,0 ) R e [ G π a , ( r , E ) ] FIG. 5: Plot of real part of finite spacing version of G ( a, )2 π ( r ; E ) = (cid:80) p e i p · r (cid:101) G ( a, )2 π ( p ; E ) (red dots) comparedwith its infinite volume counterpart (black curve) withchosen parameters: r = ( r x , , E = 1GeV L = 10GeV − , m π = 0 . a = 0 . − .ing is obtained by replacing Eq.(6) by t ( a, P )2 π ( k ) = (cid:88) p (cid:101) V ( | k − p | ) (cid:101) G ( a, P )2 π ( p ; E ) t ( a, P )2 π ( p ) , ( k , p ) ∈ π n L , ( n x , n y , n z ) ∈ [ − N, N ] . (51)For contact interaction, a finite lattice spacing may playthe role of natural ultraviolet cutoff, Eq.(34) is thus re-placed by1 V R ( µ ) = (cid:88) p (cid:101) G ( a, P )2 π ( p ; E ) − (cid:88) p (cid:101) G ( a, )2 π ( p ; µ ) , (52)where sum of p = π n L is restricted in first Brillouin zonewith ( n x , n y , n z ) ∈ [ − N, N ] and N = L a − π + LS equation with finite lattice spacing
With same strategy, the finite lattice spacing versionof 3 π + LS equation is obtained by replacing Eq.(43) by t ( a, P )3 π ( k ) = 2 (cid:88) p aE p L a (cid:101) G ( a, P )3 π ( p , k ; E ) V R ( µ ) − (cid:101) S ( a, P )3 π ( k ; E, µ ) t ( a, P )3 π ( p ) , ( k , p ) ∈ π n L , ( n x , n y , n z ) ∈ [ − N, N ] , (53)where (cid:101) S ( a, P )3 π ( k ; E, µ )= (cid:88) p (cid:20) aE k L a (cid:101) G ( a, P )3 π ( p , k ; E ) − (cid:101) G ( a, )2 π ( p ; µ ) (cid:21) , (54)and p = π n L , ( n x , n y , n z ) ∈ [ − N, N ]. B. Cubic lattice symmetry group and itsirreducible representations
The energy spectrum of a quantum system is normallyorganized and labeled according to irreducible represen-tations (irreps) of the symmetry groups of the system.These irreps carry so-called “good” quantum numbersthat help in practice identify states of the system. For ex-ample, in infinite volume, hadronic bound states are typ-ically labeled in terms of total angular momentum andparity based on the system’s behavior under rotationsand space inversion. To decouple states with differentquantum numbers, one can project the dynamic equa-tions of the system onto each irrep of symmetry groups.The end result is that each eigen-energy belongs to cer-tain irrep, or irreps in the case of degeneracy.Similar operations can be carried out for finite-volumesystems as well. The energy spectra of finite-volumemulti- π + system are expected to be labeled by irreps ofthe cubic lattice symmetry group. For instance, the cu-bic symmetry group for a system with vanishing totalmomentum, P = 0, is the octahedral group O h , whichconsists of 48 symmetry operations, including 24 discretespace rotations and inversions of all axes. The irreps ofoctahedral group O h include one-dimensional representa-tions, A ± and A ± , two-dimensional representation, E ± ,and three-dimensional representations, T ± and T ± . Thesuperscripts ± are used to label even or odd parity stateof the system. A brief introduction on the subject ofirrep projection of dynamical equations is given in thissection, with the 2 π + LS equation used as a specific ex-ample. More elaborate explanations on the subject canbe found in, e.g., Refs. [54, 78]. The irrep-projected 2 π + and 3 π + LS equations in the CM frame will be presentedin the following subsections.We start with the projection operator for the cubicgroup [78], P ( λ ) α,α = d λ (cid:88) g ∈G Γ ( λ ) ∗ α,α ( g ) O ( g ) , (55)where G and g stand for the the cubic symmetry groupand its elements. Here λ is used to label a specificirrep, and d λ denotes the dimension of irrep λ , e.g. , d λ = 1 , , , , λ = A ± , A ± , E ± , T ± , T ± , respec-tively. With g running over all elements of G , Γ ( λ ) ( g ) area set of d λ -by- d λ matrices that furnish irrep λ . O ( g ) rep-resents the symmetry operation implemented on quan-tum states. These operations are perhaps most easilyexplained by their action on momenta or coordinates ofthe particles. For instance, p (cid:48) = O ( g ) p = g p , where | p (cid:48) | = | p | for all g ∈ G , because operations in O h are either rotation or inversion. The complete listof symmetry operations of the octahedral group O h aregiven in Appendix C. Momenta of a particle can be grouped into various sets,and momenta in the same set are connected by symmetryoperations. Any set of such momenta can be representedby a single reference vector p , and the rest of membersof this particular set can be reached by p = g p ( g ∈ G ) . The reference vector used in this work is the same conceptas used in Ref. [54], while in Ref. [78] the set of momentarepresented by p is called the “star” of p . With a cutoffon the lattice momenta, p = 2 π n L , ( n x , n y , n z ) ∈ [ − N, N ] , the reference vectors, p ’s, may be chosen as p ∈ { πL n i,j,k } , where n i,j,k = ( k, j, i ) , and ( i, j, k ) ∈ [0 , N ] satisfies k ≤ j ≤ i . The total numberof reference vectors p for a fixed N is thus given by N (cid:88) i =0 i (cid:88) j =0 j (cid:88) k =0 = 16 ( N + 1)( N + 2)( N + 3) . In terms of reference vectors and symmetry operations,the sum of momenta over an arbitrary function can bereorganized as follows: (cid:88) p f ( p ) = (cid:88) p ϑ ( p )48 (cid:88) g ∈G f ( g p ) , (56)where ϑ ( p ) is the multiplicity of distinct momentawithin the set represented by p . For instance, for p = 2 πL (0 , , ,ϑ ( p ) = 6 and the six distinct momenta are p = g p ∈ πL { (0 , , ± , (0 , ± , , ( ± , , } . The multiplicity function ϑ ( p ) used in this work has thesame meaning as the multiplicity of a given shell definedin Ref. [54]. Therefore, the projector defined in Eq.(55)acting on amplitudes can be interpreted as weighted av-erage within the momentum set represented by p , andthe projected amplitudes can be labeled by p ’s. C. Irrep projection of π + LS equation
As an example, we discuss projection of two-pion am-plitudes for P = onto irrep λ that is given by t ( λ )2 π ( k ) = P ( λ ) α,α t ( )2 π ( k ) = d λ (cid:88) g ∈G Γ ( λ ) ∗ α,α ( g ) t ( )2 π ( g k ) . (57)The projected amplitude, t ( λ )2 π ( k ), actually does not de-pend on quantum number α due to the cubic symmetryof the system. Therefore, it may be convenient to ex-press the projection operator in terms of the character ofirreps: P ( λ ) = 1 d λ (cid:88) α P ( λ ) α,α = 148 (cid:88) g ∈G χ ( λ ) ∗ ( g ) O ( g ) , (58)where χ ( λ ) ( g ) = (cid:88) α Γ ( λ ) α,α ( g )is the character of irrep λ , and it satisfies orthogonalityrelation: 148 (cid:88) g ∈G χ ( λ ) ∗ ( g ) χ ( λ (cid:48) ) ( g ) = δ λ,λ (cid:48) . (59)With the help of P ( λ ) , one can rewrite the projection oftwo-pion amplitudes in a more compact form: t ( λ )2 π ( k ) = 148 (cid:88) g ∈G χ ( λ ) ∗ ( g ) t ( )2 π ( g k ) . (60)The above equation can be inverted using the orthogo-nality relation: t ( )2 π ( g k ) = (cid:88) λ χ ( λ ) ( g ) t ( λ )2 π ( k ) . (61)Projection of dynamic equations established in previ-ous sections is obtained by applying projection formulaof amplitudes. Applying Eqs. (60), (61), and (56) toEq. (6), we arrive at t ( λ )2 π ( k ) = (cid:88) p ϑ ( p ) (cid:101) V ( λ ) ( k , p ) (cid:101) G ( )2 π ( p ; E ) t ( λ )2 π ( p ) , (62)where (cid:101) V ( λ ) ( k , p ) is the projected potential: δ λ,λ (cid:48) (cid:101) V ( λ ) ( k , p )= 148 (cid:88) g p ,g k ∈G χ ( λ ) ∗ ( g k ) (cid:101) V ( | g k k − g p p | ) χ ( λ (cid:48) ) ( g p )= δ λ,λ (cid:48) (cid:88) g ∈G χ ( λ ) ∗ ( g ) (cid:101) V ( | g k − p | ) . (63)The energy spectrum stemming from the irrep-projecteddynamical equation is of course labeled by the same irrep.With the contact interaction, (cid:101) V ( k ) = V , only A +1 irrepsurvives after projection of the potential: (cid:101) V ( λ ) ( k , p ) = δ λ,A +1 V . Therefore, projection of Eq.(52) yields non-trivial solu-tions only in A +1 : δ λ,A +1 V R ( µ ) = (cid:88) p ϑ ( p ) (cid:101) G ( a, )2 π ( p ; E ) − (cid:88) p ϑ ( p ) (cid:101) G ( a, )2 π ( p ; µ ) . (64) D. Irrep projection of π + LS equation
Since the three-pion amplitudes depend on two mo-mentum variables, the irrep projection can be done byfirst projecting out each momentum dependence sepa-rately onto the corresponding irrep, and then couplingtwo individual projections to an irrep of the 3 π + 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 3 π + amplitude t ( )( k ) ( k , k ) is accomplished by the following operator: P ( λ ) α (1 ,
2) = (cid:88) α ,α (cid:18) λ λ α α (cid:12)(cid:12)(cid:12)(cid:12) λα (cid:19) P ( λ ) α ,α (1) P ( λ ) α ,α (2) , (65)where P ( λ ) α ,α (1) and P ( λ ) α ,α (2) are the uncoupled irrepprojection operators on momenta k and k , respectively,and (cid:18) λ λ α α (cid:12)(cid:12)(cid:12)(cid:12) λα (cid:19) is the Clebsch-Gordan coefficient that couple irreps λ and λ to irrep λ on the α -th row. Again, due to the cu-bic symmetry of the system, projected three-body ampli-tudes do not depend on α , so the projection of t ( )( k ) ( k , k )can be written as t ( λ )( k ) ( k , k ) = 1 d λ (cid:88) α P ( λ ) α (1 , t ( )( k ) ( k , k )= d λ d λ d λ (cid:88) α ,α ,α (cid:18) λ λ α α (cid:12)(cid:12)(cid:12)(cid:12) λα (cid:19) × (cid:88) g ,g ∈G Γ ( λ ) ∗ α ,α ( g )Γ ( λ ) ∗ α ,α ( g ) t ( )( k ) ( g k , g k ) . (66)The general projection of three-particle amplitudescan be cumbersome. However for the interaction un-der consideration in the present paper, no three-bodyforce and only pair-wise contact interaction, the proce-dure is greatly simplified. This is because, as shown inSec. II C, the three-pion amplitude depends only on asingle momentum: t ( P )3 π ( k ). With no dependence on k ,only the trivial irrep A +1 survives after projection of k dependence: λ = A +1 and Γ ( λ ) α ,α = 1. The coupledirrep of the 3 π + system is thus determined by k depen-dence alone: λ = λ and α = α . Therefore, one canapply straightforwardly most of the results for the two-pion case in Sec. III C to the irrep projection of the 3 π + LS equation.With pair-wise contact interaction, the irrep projectionof t ( )3 π ( k ) in the CM frame is given by t ( λ )3 π ( k ) = 148 (cid:88) g ∈G χ ( λ ) ∗ ( g ) t ( a, )3 π ( g k ) . (67)0Similar to Eq.(61), we also have t ( a, )3 π ( g k ) = (cid:88) λ χ ( λ ) ( g ) t ( λ )3 π ( k ) . (68)Noticing that (cid:101) S ( a, )3 π ( k ; E, µ ) remains invariant undersymmetry operations, (cid:101) S ( a, )3 π ( g k ; E, µ ) = (cid:101) S ( a, )3 π ( k ; E, µ ) , (69)and that the projection on the right-hand side of Eq.(53)yields148 (cid:88) g ,g ∈G χ ( λ ) ∗ ( g ) χ ( λ (cid:48) ) ( g ) × (cid:20) aE g p L a (cid:101) G ( a, )3 π ( g p , g k ; E ) (cid:21) = δ λ,λ (cid:48) (cid:88) g ∈G χ ( λ ) ∗ ( g )4 sinh aE p L a (cid:101) G ( a, )3 π ( p , g k ; E ) , (70)one can show that irrep projection of the 3 π + LS equation(53) ultimately leads to the following: t ( λ )3 π ( k ) = 2 (cid:88) p ϑ ( p ) (cid:101) C ( λ )3 π ( k , p ; E ) V R ( µ ) − (cid:101) S ( a, )3 π ( k ; E, µ ) t ( λ )3 π ( p ) , (71)where (cid:101) C ( λ )3 π ( k , p ; E )= 148 (cid:88) g ∈G χ ( λ ) ∗ ( g )4 sinh aE p L a (cid:101) G ( a, )3 π ( p , g k ; E ) . (72)The O h -irrep projected 3 π + quantization condition, un-der our assumptions, has a simple form:det δ k , p − ϑ ( p ) (cid:101) C ( λ )3 π ( k , p ; E ) V R ( µ ) − (cid:101) S ( a, )3 π ( k ; E, µ ) = 0 . (73)The above equation suggests that except for A − irrep,only trivial solutions — free particle states— can befound near the ground state with all three pions at rest,because the only irrep in which (cid:101) C ( λ )3 π ( , ; E ) does notvanish is λ = A − . IV. NUMERICAL RESULTS
In this section, we use the quantization conditions (64)and (73) to produce 2 π + and 3 π + energy spectra for var-ious lattice sizes. A single parameter, the renormalizedcoupling of the two-body contact interaction V R ( µ ), isfitted to the lattice results of Ref. [27]. The two-pion scattering parameters, the scatteringlength a and effective range r , can immediately be pro-duced once V R ( µ ) is determined from the fit. a and r are usually defined by the effective range expansion atlow energies: p cot δ ( E ) = − a + r p + O ( p ) , (74)where p = 12 (cid:112) E − m π is relative momentum of the pions in the CM frame. Onthe other hand, as detailed in Appendix B, the isospin-2 S -wave phase shift of π + π + scattering through a pair-wise contact potential is given bycot δ (2)0 ( E ) = − π (cid:113) − m π E (cid:20) V R ( µ ) − ReG ( E ) + G ( µ ) (cid:21) , (75)where G ( E ) = (cid:113) − m π E π ln (cid:113) − m π E + 1 (cid:113) − m π E − . Comparing Eqs. (74) and (75), one finds a and r interms of V R ( µ ):1 a m π = 16 π (cid:20) V R ( µ ) + G ( µ ) (cid:21) ,r m π = 1 a m π + 4 π . (76)The values of π + π + scattering length a and effectiverange r extracted in this study are compared in Table Iwith 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 producethe CM-frame 2 π + and 3 π + energy spectra in A and E irreps with Eqs. (64) and (73), at lattice spacing a = 0 . − and pion mass m π = 0 . µ = 0GeV, so that G ( µ ) = π is real.The value of coupling constant of contact interaction isextracted, V R (0) ∼ . ± . . (77)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). Thissuggests that more sophisticated pair-wise interactionsand/or three-body interactions may be needed.1
16 18 20 22 24 26 28 300.40.60.81.01.2 L ( GeV - ) E π in A + irrep ( GeV ) P [ ] p [ ] , p [ ] p [ ] , p [ ] p [ ] , p [ ] (a) 2 π + energy spectrum in A +1 irrep.
16 18 20 22 24 26 28 300.60.81.01.21.4 L ( GeV - ) E π in A - irrep ( GeV ) P [ ] p [ ] , p [ ] p [ ] , p [ ] p [ ] , p [ ] (b) 3 π + energy spectrum in A − irrep.
16 18 20 22 24 26 28 300.60.81.01.21.4 L ( GeV - ) E π in E - irrep ( GeV ) P [ ] p [ ] , p [ ] p [ ] , p [ ] p [ ] , p [ ] (c) 3 π + energy spectrum in E − irrep. FIG. 6: 2 π + and 3 π + energy spectrum in irreps A and E and CM frame: red bands are spectrum produced byusing Eq.(64) and Eq.(73) for 2 π + and 3 π + respectively, the lattice results (black circles) are taken from [27]. Thefree multi- π + energy spectrum (blue dashed curves) by using E = a sinh − ( (cid:80) i sinh aE p i ) and (cid:80) i p i = P are alsoplotted as reference. The lattice spacing used in this work and in [27] is a = 0 . − , m π = 0 . π + π + scattering length a and effectiverange r . a m π r m π m π (GeV)This work 0 . ± .
17 4 . ± . . . ± . . ± . . . ± . − . ± .
13 0 . . ± .
006 28 . ± .
89 0 . V. SUMMARY AND OUTLOOK
Based on the variational approach combined with theFaddeev method proposed in Refs. [64–67], the relativis-tic 2 π + and 3 π + dynamics in finite volume are presentedin the paper. The presentation of multi- π + dynamics in-cluded both pair-wise and three-body interactions, andthe effects of finite lattice spacing and projections ontoirreps of the cubic lattice group were also discussed. Thequantization conditions were used to analyze the latticedata published in [27]. In the present work, only contactpair-wise interactions were employed in our analysis and,as a result, the renormalized coupling strength was thesole free parameter. The scattering length a and effec-tive range r were obtained, however, with rather largeuncertainty: a m π = 0 . ± .
17 and r m π = 4 . ± . π + ’s by onlya contact pair-wise interaction.The idea of our framework is somewhat similar to thatof Refs. [47, 48] in the sense both used an interaction toconnect finite-volume and infinite-volume dynamics. Butthe implementations differ in many details. Relativistickinematics of the pion and more lattice effects are con-sidered here.With the framework and computational facility de-veloped here, we can improve readily the interactionsby supplanting more sophisticated presentations for the - - - - - ( pm π ) p m π C o t [ δ ( ) ] FIG. 7: The plot of pm π cot δ (2)0 ( E ) vs. ( pm π ) by usingEq.(75) (red band) compared with the phase shift givenby effective range expansion formula (black band),Eq.(74), where a m π = 0 . ± . r m π = 9 . ± . ACKNOWLEDGMENTS
We acknowledge support from the Department ofPhysics and Engineering, California State University,Bakersfield, CA. This research was supported in partby the National Science Foundation under Grant No.NSF PHY-1748958. P.G. acknowledges GPU comput-ing resources (http://complab.cs.csubak.edu) from theDepartment of Computer and Electrical Engineeringand Computer Science at California State University-Bakersfield made available for conducting the researchreported in this work. B.L. acknowledges support bythe National Science Foundation of China under GrantNos.11775148 and 11735003.
Appendix A: Reduction of Bethe-Salpeter equationto relativistic π + Lippmann-Schwinger equation
Consider the general form of Bethe-Salpeter (BS) equa-tion [80] for three-scalar-particle bound states: ψ BS ( p , p ) = ( − i ) ( p − m π )( p − m π )( p − m π ) × (cid:90) d p (cid:48) (2 π ) d p (cid:48) (2 π ) I ( p (cid:48) − p , p (cid:48) − p ) ψ BS ( p (cid:48) , p (cid:48) ) , (A1)where p i = ( p i , p i ) are the four momenta of threeparticles, the three-body BS wave function is labeledonly by two independent particle momenta, p and p ,since the momenta of three particles are constrained byenergy-momentum conservation, p = P − p − p . As-suming ”instantaneous interaction kernel”, I ( p , p ) = I ( p , p ), and introducing Lippmann-Schwinger equa-tion wave function, ψ ( p , p ) = (cid:82) dp π dp π ψ BS ( p , p ),hence, we get ψ ( p , p ) = (cid:90) dp π dp π ( − i ) ( p − m π )( p − m π )( p − m π ) × (cid:90) d p (cid:48) (2 π ) d p (cid:48) (2 π ) I ( p (cid:48) − p , p (cid:48) − p ) ψ ( p (cid:48) , p (cid:48) ) . (A2)The integration over propagators can be carried out,(2 π ) G ( P ) ( p , p ; E )= (cid:90) dp π dp π ( − i ) ( p − m π )( p − m π )( p − m π )= 12 E p E p E p E p + E p + E p ) E − ( E p + E p + E p ) , (A3)where E p i = (cid:112) p i + m π .The interactions of 3 π + consist of pair-wise interac-tions and three-body interaction potential, I ( p (cid:48) − p , p (cid:48) − p ) = (cid:101) V (4) ( p (cid:48) − p , p (cid:48) − p )+ (cid:88) k =1 E p k (2 π ) δ ( p (cid:48) k − p k ) (cid:101) V ( | q (cid:48) ij − q ij | ) | k (cid:54) = i (cid:54) = j , (A4) where symbols (cid:101) V and (cid:101) V (4) are used to denote pair-wiseinteraction and three-body force respectively in consis-tent with the conventions used in Section II. q ij = p i − p j refers to the relative momentum between i-th and j-th pions. The relativistic kinematic factors (cid:104) p k | p (cid:48) k (cid:105) =2 E p k (2 π ) δ ( p (cid:48) k − p k ) emerge when k-th particle is freepropagating and not involved in the interaction. The rel-ativistic 3 π + Lippmann-Schwinger equation is thus givenexplicitly by ψ ( p , p ) = G ( P ) ( p , p ; E ) × (cid:20) E p (2 π ) (cid:90) d p (cid:48) (cid:101) V ( | p (cid:48) − p | ) ψ ( p , p (cid:48) )+ 2 E p (2 π ) (cid:90) d p (cid:48) (cid:101) V ( | p (cid:48) − p | ) ψ ( p (cid:48) , p )+ 2 E p (2 π ) (cid:90) d p (cid:48) (cid:101) V ( | p (cid:48) − p | ) ψ ( p (cid:48) , p + p − p (cid:48) )+ (cid:90) d p (cid:48) d p (cid:48) (cid:101) V (4) ( p (cid:48) − p , p (cid:48) − p ) ψ ( p (cid:48) , p (cid:48) ) (cid:21) . (A5) Appendix B: π scattering amplitude in infinitevolume with contact interaction With the contact interaction, V ( r ) = V δ ( r ) , the relativistic Lippmann-Schwinger equation of 2 π sys-tem can be solved analytically, so the scattering wavefunction of 2 π in infinite volume and in CM frame isgiven by φ p ( r ) = e i p · r + G ( ) ( r ; E ) V φ p ( ) , (B1)where r and p are the relative coordinate and momentumof two pions respectively. The total energy of two pionsis related to p by E = 2 E p = 2 (cid:112) p + m π . The twopions Green’s function in CM frame in infinite volume isgiven by G ( ) ( r ; E ) = (cid:90) d q (2 π ) E q e i q · r E − (2 E q ) . (B2)The two pions amplitude in CM frame is thus given by, t ( E ) = − V φ p ( ) = − V − G ( ) ( ; E ) . (B3)The Green’s function G ( ) ( ; E ) diverge at r = , a cut-off Λ on momentum integration may be introduced toregularize ultraviolet divergence, as Λ → ∞ hence wefind (cid:90) Λ d q (2 π ) E q E − (2 E q ) = G ( E ) − π ln 2Λ m π , (B4)where G ( E ) = (cid:113) − m π E π ln (cid:113) − m π E + 1 (cid:113) − m π E − . (B5)3The imaginary part of function G ( E ) is non-zeroonly above threshold of two pions: ImG ( E ) = − π (cid:113) − m π E for E > m π , and zero otherwise. Theultraviolet divergence may be absorbed by redefiningbare coupling, V ,1 V = 1 V R ( µ ) + G ( µ ) − π ln 2Λ m π , (B6)where V R ( µ ) stands for the renormalized couplingstrength at a renormalization scale µ . µ will be chosenbelow two pions threshold, so that G ( µ ) is real. There-fore, the cutoff dependence is cancelled out completely,and renormalized scattering amplitude is now given by t ( E ) = − V R ( µ ) − G ( E ) + G ( µ ) , µ < m π . (B7)The phase shift is defined by t ( E ) = π (cid:114) − m πE δ ( E ) − i ,hence we obtaincot δ ( E ) = − π (cid:113) − m π E (cid:20) V R ( µ ) − ReG ( E ) + G ( µ ) (cid:21) . (B8) Expanding phase shift near threshold p = | p | = (cid:112) E − m π ∼
0, we find p cot δ ( E ) = − a + r p + O ( p ) , (B9)where the effective expansion parameters, a and r aregiven by 1 a m π = 16 π (cid:20) V R ( µ ) + G ( µ ) (cid:21) ,r m π = 16 π (cid:20) V R ( µ ) + G ( µ ) (cid:21) + 4 π . (B10) Appendix C: Character Table for the octahedralgroup O h The octahedral group O h is the direct product ofproper rotational group O that rotates a cube into it-self and spatial inversion I : O h = O × I .O h contains 48 elements, including 24 elements of rota-tional group O and 24 elements of combined operationof inversion and rotations: Ig where g ∈ O . The matrixrepresentations of 24 proper rotational group O are givenby E = , C x = − − , C y = − − , C z = − − ,C x = − , C y = −
10 1 01 0 0 , C z = − , C − x = −
10 1 0 ,C − y = − , C − z = − , C a = − , C b = − − − ,C c = − , C d = − − − , C e = − , C f = − − − ,C α = − − , C β = − −
11 0 0 , C γ = − − , C δ = ,C − α = −
11 0 00 − , C − β = − − , C − γ = − − , C − δ = , (C1)and the inversion matrix, I = − − − . (C2) All 48 elements of the octahedral group O h are usuallygrouped into different conjugacy classes, and the mem-4TABLE II: Character table for irreps of the octahedralgroup O h with positive parity χ ( C , C ) χ ( C , C ) χ ( C , C ) χ ( C , C ) χ ( C , C ) A +1 A +2 E + T +1 T +2 bers within the same conjugacy class share the same char-acter value for a given irrep. Using the same conventionas used in [78], ten classes of the octahedral group O h are named as C i where i = 1 , · · · ,
10, they are associatedwith 48 elements by C = E, C = (cid:32) C α , C β , C γ , C δ ,C − α , C − β , C − γ , C − δ (cid:33) , C = ( C x , C y , C z ) , C = (cid:32) C x , C y , C z ,C − x , C − y , C − z (cid:33) , C = ( C a , C b , C c , C d , C e , C f ) , (C3)and C i = I C ( i − for i = 6 , · · · ,
10. The character tablefor all irreps of the octahedral group O h with positiveparity quantum number is given in Table II. Appendix D: Non-relativistic three-particledynamics in finite volume
The dynamics of three non-relativistic identicalbosonic particles in finite volume is described by the (cid:34) mE + (cid:88) i =1 ∇ i − (cid:88) k =1 U ( r ij ) − U (4) ( r , r ) (cid:35) × Φ( x , x , x ) = 0 , i (cid:54) = j (cid:54) = k, (D1)where m is the mass of identical bosons. x i denotesthe position of i-th particle, and r ij = x i − x j is rel-ative coordinate between i-th and j-th particles. Thepair-wise interaction between i-th and j-th particles is de-scribed by U ( r ij ), and U (4) ( r , r ) represents the three-body interaction among all particles. Both pair-wise andthree-body interactions in finite volume are assumed tobe short-range and periodic, that is to say U ( r ) = U ( | r + n L | ) , n ∈ Z ,U (4) ( r , r ) = U (4) ( r + n L, r + n L ) , n , ∈ Z , (D2)where L is the size of the cubic lattice. Therefore finitevolume three-particle wave function must also satisfy pe-riodic boundary condition,Φ( x , x , x ) = Φ( x + n x , x + n x , x + n x ) , (D3) where n x i ∈ Z . As suggested in [65, 66], it may bemore convenient to consider the integral representationof Eq.(D1),Φ( x , x , x ) = (cid:90) L (cid:89) i =1 d x (cid:48) i L (cid:88) p , p , p e i (cid:80) i =1 p i · ( x i − x (cid:48) i ) mE − (cid:80) i =1 p i × (cid:34) (cid:88) k =1 U ( r (cid:48) ij ) + U (4) ( r (cid:48) , r (cid:48) ) (cid:35) Φ( x (cid:48) , x (cid:48) , x (cid:48) ) , (D4)where p , , ∈ π n L , n ∈ Z . The center of mass motionof three-particle system can be factorized byΦ( x , x , x ) = e i P · R φ ( r , r ) , (D5)where R = x + x + x = r + r + x is center of massposition of three-particle system, and P = πL d with d ∈ Z stands for the total momentum of three-particle in aperiodic cubic box. φ ( r , r ) is the wave function thatis associated with the internal motion of three particles,and it satisfies periodic boundary condition, φ ( r + n L, r + n L ) = e − i P · ( n L + n L ) φ ( r , r ) , (D6)where n , ∈ Z . We remark that in this work we use( r , r ) to describe the internal motion of three parti-cles, and x is thus associated to CM motion. So that (cid:82) L (cid:81) i =1 d x (cid:48) i = (cid:82) L d r (cid:48) d r (cid:48) d x (cid:48) , and it resembles a two-light and one heavy three-body atomic system. Integrat-ing out CM motion, (cid:90) L d x (cid:48) e i ( P − p − p − p ) · x (cid:48) = L δ p , P − p − p , (D7)the three-particle Lippmann-Schwinger equation,Eq.(D4), now is reduced to φ ( r , r ) = (cid:90) L d r (cid:48) d r (cid:48) G ( P ) ( r − r (cid:48) , r − r (cid:48) ; E ) × (cid:34) (cid:88) k =1 U ( r (cid:48) ij ) + U (4) ( r (cid:48) , r (cid:48) ) (cid:35) φ ( r (cid:48) , r (cid:48) ) . (D8)The three-particle Green’s function is defined by G ( P ) ( r , r ; E )= (cid:88) p , p e i ( p − P ) · r e i ( p − P ) · r (cid:101) G ( P ) ( p , p ; E ) , (cid:101) G ( P ) ( p , p ; E ) = 1 L mE − (cid:80) i =1 p i , (D9)where p , ∈ π n L , n ∈ Z , and p = P − p − p .Following the same procedures as described in SectionII, only two independent scattering amplitudes are re-quired due to exchange symmetry of three-particle wave5function, T ( P )(2) ( k , k ) = − (cid:90) L d r d r e − i ( k − P ) · r e − i ( k − P ) · r × U ( r ) φ ( r , r ) ,T ( P )(4) ( k , k ) = − (cid:90) L d r d r e − i ( k − P ) · r e − i ( k − P ) · r × U (4) ( r , r ) φ ( r , r ) . (D10)Two amplitudes, T ( P )(2) and T ( P )(4) , satisfy equations, T ( P )(2) ( k , k ) = (cid:88) p (cid:101) U ( | k − p | ) L (cid:101) G ( P ) ( p , k ; E ) × (cid:20) T ( P )(2) ( p , k ) + T ( P )(2) ( k , p )+ T ( P )(2) ( p , P − p − k ) + T ( P )(4) ( p , k ) (cid:21) , (D11) and T ( P )(4) ( k , k ) = (cid:88) p , p (cid:101) U (4) ( k − p , k − p ) × (cid:101) G ( P ) ( p , p ; E ) (cid:20) T ( P )(2) ( p , p ) + T ( P )(2) ( p , p )+ T ( P )(2) ( p , p ) + T ( P )(4) ( p , p ) (cid:21) , (D12)where p = P − p − p , and (cid:101) U and (cid:101) U (4) are the Fouriertransform of interaction potentials U and U (4) respec-tively. Non-relativistic three-particle dynamical equa-tions, Eq.(D11) and Eq.(D12), resemble their relativisticcounter parts, Eq.(25) and Eq.(26), excepts some rela-tivistic kinematic factors. [1] J. Kambor, C. Wiesendanger, and D. Wyler, Nucl. Phys. B465 , 215 (1996), arXiv:hep-ph/9509374 [hep-ph].[2] A. V. Anisovich and H. Leutwyler, Phys. Lett.
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