Charged particles interaction in both a finite volume and a uniform magnetic field
CCharged particles interaction in both a finite volume and a uniform magnetic field
Peng Guo
1, 2, ∗ and Vladimir Gasparian 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: January 5, 2021)A formalism for describing charged particles interaction in both a finite volume and a uniformmagnetic field is presented. In the case of short-range interaction between charged particles, weshow that the factorization between short-range physics and finite volume long-range correlationeffect is possible, a L¨uscher formula-like quantization condition is thus obtained.
I. INTRODUCTION
In recent years, a great effort in nuclear and hadronphysics community has been put into constructing thescattering dynamics of few-particle interactions from dis-crete bound state energy spectrum that is computed invarious type of traps, such as commonly used periodicfinite box in lattice QCD (LQCD) and harmonic oscilla-tor trap in nuclear physics computation. The ultimategoal is of course to study and explore the nature of par-ticle interactions that plays an essential role in manyfields of physical science, such as nuclear physics andastrophysics. However, the current state-of-art ab ini-tio computations in nuclear and hadron physics are nor-mally performed in a harmonic oscillator trap and in a fi-nite volume respectively. Instead of computing few-bodyscattering amplitudes, the discrete bound state energylevels are usually directly measured and extracted fromthese ab initio computations. Therefore, finding a rela-tion that convert discrete bound state energy spectruminto continuum scattering state is a key step.In fact, relating the energy shift caused by particleinteractions to the on-shell scattering parameters suchas phase shift has the long history across many fieldsin physics. In general cases, the dynamics of particlesinteraction in a traps is associated to the infinite vol-ume off-shell reaction amplitudes in a highly non-trivialway. Fortunately, when the separation of two physicsscales, the size of trap and the range of particles inter-action, is clearly established, a simple asymptotic formcan be found, which provides a relation between energylevels in a trap and infinite volume scattering phase shift.In finite volume in LQCD computation, such a relationin elastic two-body sector is known as L¨uscher formula[1], which shows a clear factorization of short-range dy-namics and long-range correlation effects because of pe-riodic boundary condition. The short-range dynamicsand long-range correlations are described by the physicalscattering phase shift and L¨uscher’s zeta function respec-tively. L¨uscher formula has been proving very successfulin LQCD community, and it has been quickly extendedinto both coupled-channel and few-body sectors, see [2–28]. In nuclear physics where a harmonic oscillator trap ∗ [email protected] is commonly used, such a relation is given by BERW for-mula [29]. In addition to periodic boundary conditionand harmonic trap, other type of traps or boundary con-ditions are also commonly used in different physics fields,such as hard wall trap [30, 31]. Regardless differenceamong various traps, the same strategy is shared: as thetwo physical scales are clearly separated, a closed asymp-totic form can be found, in which short-range dynamics isdescribed by scattering phase shift and long-range effectis given by an analytic form that describes how the prop-agation of particles is affected by the trap, e.g. L¨uscher’szeta function in periodic boundary condition.In present work, we aim to establish a similar relationto L¨uscher and BERW formula for the charged particlesinteracting in both a uniform magnetic field and a peri-odic box. We will show that for a short-range potential,the factorization of short-range physics and long-rangecorrelation effect is possible. Hence a relation in a com-pact form that relates discrete energy spectrum to scat-tering phase shifts can be found. Such a relation may beuseful for the study of charged hadron system such as π + system in LQCD computation. In finite volume, in orderto preserve translation symmetry of system in magneticfield, the magnetic flux though perpendicular surface ofcubic box to a uniform magnetic field must be 2 π mul-tiplied by a rational number n p /n q , where n p and n q are integers and relatively prime to each other. There-fore, the original energy level without magnetic field issplit into n q sublevels due to the application of magneticfield.The paper is organized as follows. The general formal-ism of charged bosons interaction in both a finite volumeand a uniform magnetic field is presented in details inSection II. The S -wave contribution and regularizationof ultraviolet divergence are discussed in Section III. Asummary is given in Section IV. II. FINITE VOLUME DYNAMICS OFCHARGED BOSONS IN A UNIFORMMAGNETIC FIELD
In this section, we discuss the dynamics of chargedbosons interacting in both a finite periodic box and auniform magnetic field. The uniform magnetic field ischosen along z -axis, B = B e z , and Landau gauge for a r X i v : . [ h e p - l a t ] J a n vector potential is adopted in this work, A ( x ) = B (0 , x, . (1)The dynamics of two charged identical non-relativisticspinless particles in a uniform magnetic field is describedby Schr¨odinger equation, (cid:16) ˆ H B + V ( r ) (cid:17) Ψ E ( x , x ) = E Ψ E ( x , x ) , (2)where ˆ H B = − (cid:88) i =1 ( ∇ i + ie A ( x i )) m . (3) m stands for the mass of identical bosons, and x i denotesthe position of i-th particle. The short-range interactionbetween two particles is represented by V ( r ), where r = x − x is relative coordinate between two particles.The center of mass (CM) and relative motions can beseparated rather straightforwardly,ˆ H B = ˆ H R + ˆ H r , (4)where ˆ H R = − ( ∇ R + iQ A ( R )) M (5)and ˆ H r = − ( ∇ r + iq A ( r )) µ , (6)see detailed presentation of separation of CM and relativemotions in Appendix A 1. Q = 2 e and q = e M = 2 m and µ = m R = x + x (cid:16) ˆ H r + V ( r ) (cid:17) ψ ε ( r ) = εψ ε ( r ) , (7)where ψ ε ( r ) and ε are wave function and energy for rela-tive motion of two charged boson system. The solutionsof CM motion can be found in Appendix A 3. A. Magnetic periodic boundary condition
In a periodic finite box, though the short-range poten-tial V is periodic V ( r + n L ) = V ( r ) , n ∈ Z , (8)where L denotes the size of box, the ˆ H r is not discretetranslation invariant ˆ H r + n L (cid:54) = ˆ H r due to the fact that vector potential is coordinate depen-dent, and breaks discrete translation symmetry A y ( r + n L ) = A y ( r ) + B n L · e x . The momentum operator, ˆp = − i ∇ r , doesn’t commutewith ˆ H r : [ ˆp , ˆ H r ] (cid:54) = 0 . Hence canonical momentum is no longer a conservedquantity as the consequence of breaking down of discretetranslation symmetry in a uniform magnetic field. It hasbeen shown in Refs. [32–34] that a pseudo-momentumoperator ˆK r = − i ∇ r + q A ( r ) − q B × r = − i ∇ r + qB ( r y , ,
0) (9)in fact commute with ˆ H r :[ ˆK r , ˆ H r ] = 0 . Therefore, ˆK r can be used as generator of a magnetictranslation operator,ˆ T r ( n L ) = e i ˆK r · n L , (10)and [ ˆ T r ( n L ) , ˆ H r ] = 0 , n ∈ Z . (11)However, the magnetic translation operator in gen-eral doesn’t commute with each other, for instance, for aclosed path in a single box,ˆ T r ( − L e x ) ˆ T r ( − L e y ) ˆ T r ( L e x ) ˆ T r ( L e y ) = e − iqBL (cid:54) = 1 . (12)To warrant a state that is translated through a closedpath remain same, the magnetic flux qBL through thesurface of path must be quantized, qBL = 2 πn, n ∈ Z . In fact, this conclusion can be made in a more general wayby considering a enlarged closed path in x − y plane withthe size of n q L e x × L e y where n q ∈ Z , see Refs. [32, 33].Hence, the generalized magnetic quantization conditionis given by qBn q L = 2 πn p , (13)where n p and n q are two relatively prime integers. Ina enlarged magnetic unit box defined by magnetic unitvectors: n q L e x × L e y × L e z , the magnetic translation operators now commute witheach other [ ˆ T r ( n q L e x ) , ˆ T r ( L e y )] = 1 . (14)Therefore, the discrete translation in a enlarged magneticunit box leaves Hamiltonian invariant, and ˆ T r ( n B L ) forma magnetic translation group, where n B = n x n q e x + n y e y + n z e z , n x,y,z ∈ Z . (15)The application of the magnetic field also results in thesplitting of each energy level into n q sub-energy levels.Under magnetic translation operation, the wave func-tion behaves asˆ T r ( n B L ) ψ ε ( r ) = e iqBn x n q Lr y ψ ε ( r + n B L ) . (16)According to Bloch theorem, in a periodic box, periodic-ity of system requires that ˆ T r ( n B L ) ψ ε ( r ) can only differfrom ψ ε ( r ) by a phase factor, which can be chosen as e i P B · n B L , where P B = 2 πL (cid:18) n x n q e x + n y e y + n z e z (cid:19) , n x,y,z ∈ Z . (17)Hence the magnetic periodic boundary condition is givenby ψ ε ( r + n B L ) = e i P B · n B L e − iqBn x n q Lr y ψ ε ( r ) . (18)The magnetic periodic boundary condition can also beobtained by considering separable form of total wavefunction, see Appendix A 2. B. Finite volume Lippmann-Schwinger equationand quantization condition
1. Finite volume Lippmann-Schwinger equation
The Schr¨odinger equation and magnetic periodicboundary condition in Eq.(7) and Eq.(18) together canbe replaced by finite volume homogeneous Lippmann-Schwinger (LS) equation, ψ ε ( r ) = (cid:90) L B d r (cid:48) G ( L ) B ( r , r (cid:48) ; ε ) V ( r (cid:48) ) ψ ε ( r (cid:48) ) , (19)where the volume integration over the magnetic unit cellis defined by (cid:90) L B d r (cid:48) = (cid:90) nqL − nqL dr (cid:48) x (cid:90) L − L dr (cid:48) y (cid:90) L − L dr (cid:48) z . (20) The finite volume magnetic Green’s function G ( L ) B alsomust satisfies the magnetic periodic boundary condition, G ( L ) B ( r , r (cid:48) ; ε )= e − i P B · n B L e iqBn x n q Lr y G ( L ) B ( r + n B L, r (cid:48) ; ε )= e i P B · n B L e − iqBn x n q Lr (cid:48) y G ( L ) B ( r , r (cid:48) + n B L ; ε ) , (21)hence dynamical equation for G ( L ) B is given by (cid:16) ε − ˆ H r (cid:17) G ( L ) B ( r , r (cid:48) ; ε )= (cid:88) n B e − i P B · n B L e iqBr y e x · n B L δ ( r − r (cid:48) + n B L ) . (22)The solution of finite vollume magnetic Green’s func-tion G ( L ) B can be constructed from its infinite volumecounterpart G ( ∞ ) B by, G ( L ) B ( r , r (cid:48) ; ε )= (cid:88) n B G ( ∞ ) B ( r , r (cid:48) + n B L ; ε ) e i P B · n B L e − iqBr (cid:48) y e x · n B L = (cid:88) n B e − i P B · n B L e iqBr y e x · n B L G ( ∞ ) B ( r + n B L, r (cid:48) ; ε ) , (23)details of construction can be found in Appendix A 4.The infinite volume magnetic Green’s function G ( ∞ ) B sat-isfies equation, (cid:16) ε − ˆ H r (cid:17) G ( ∞ ) B ( r , r (cid:48) ; ε ) = δ ( r − r (cid:48) ) , (24)and the analytic expression of G ( ∞ ) B is given by G ( ∞ ) B ( r , r (cid:48) ; ε ) = − µqB π e − iqB ( r x + r (cid:48) x )( r y − r (cid:48) y ) e − qB | ρ − ρ (cid:48) | × ∞ (cid:88) n =0 iL n ( qB | ρ − ρ (cid:48) | ) e i √ µε − qB ( n + ) | r z − r (cid:48) z | (cid:113) µε − qB ( n + ) , (25)where L n ( x ) is Laguerre polynomial, and ρ = r x e x + r y e y , ρ (cid:48) = r (cid:48) x e x + r (cid:48) y e y are relative coordinates defined in x − y plane.
2. Quantization condition with short-range interaction
The discrete bound state energy spectrum can befound as the eigen-energy solutions of homogeneous LSequation in Eq.(19). The partial wave expansion in angu-lar momentum basis is commonly used in describing infi-nite volume scattering state. However in magnetic field,due to asymmetry of magnetic Hamiltonian in x − y planeand along z -axis, angular momentum basis in sphericalcoordinates is in fact not most convenient basis in de-scribing dynamics of charged particles in uniform mag-netic field. Nevertheless, it can be done in principle. Forthe sake of the consistency of presentation in both finitevolume and infinite volume dynamics. Let’s consider thepartial wave expansion of Eq.(19). Using ψ ε ( r ) = (cid:88) lm ψ ( L ) lm ( r ) Y lm ( ˆr ) , (26)and G ( L ) B ( r , r (cid:48) ; ε ) = (cid:88) lm,l (cid:48) m (cid:48) Y lm ( ˆr ) G ( B,L ) lm,l (cid:48) m (cid:48) ( r, r (cid:48) ; ε ) Y ∗ l (cid:48) m (cid:48) ( ˆr (cid:48) ) , (27)we find ψ ( L ) lm ( r ) = (cid:88) l (cid:48) m (cid:48) (cid:90) L B r (cid:48) dr (cid:48) G ( B,L ) lm,l (cid:48) m (cid:48) ( r, r (cid:48) ; ε ) V l (cid:48) ( r (cid:48) ) ψ ( L ) l (cid:48) m (cid:48) ( r (cid:48) ) . (28)Since the purpose of this work is to find a L¨uscherformula-like simple relation that connects short-rangephysics associated to particles interaction V ( r ) and thelong-range effect generated by the finite volume and mag-netic field. Also considering the fact that such a relationis result of clear separation of two physical scales: (1)the range of potential V ( r ) and (2) the size of a trap orfinite volume. When the two scales are clearly separated,the short- and long-range physics can be factorized, anda compact relation as the leading order contribution canbe found by studying the asymptotic behavior of wavefunction [8, 22]. Therefore, for our purpose, it is suffi-cient to consider zero-range potential, V l ( r ) → V l δ ( r ) r l +1 Γ ( l + )(2 π ) r l , (29)see Appendix B for the more rigorous discussion. TheEq.(28) is thus turned into an algebra equation, ψ ( L ) lm ( r ) r l = (cid:88) l (cid:48) m (cid:48) V l (cid:48) l (cid:48) +1 Γ ( l (cid:48) + )(2 π ) × G ( B,L ) lm,l (cid:48) m (cid:48) ( r, r (cid:48) ; ε ) r l r (cid:48) l (cid:48) ψ ( L ) l (cid:48) m (cid:48) ( r (cid:48) ) r (cid:48) l (cid:48) | r (cid:48) → . (30)Hence the quantization condition of discrete energy spec-trum is given bydet (cid:34) δ lm,l (cid:48) m (cid:48) l +1 V l − Γ ( l (cid:48) + )(2 π ) G ( B,L ) lm,l (cid:48) m (cid:48) ( r, r (cid:48) ; ε ) r l r (cid:48) l (cid:48) | r,r (cid:48) → (cid:35) = 0 . (31)Under the same assumption of zero-range approxima-tion given in Eq.(29), the potential strength V l is relatedto the infinite volume two-body scattering phase shift δ l ( k ε ) by(4 π ) µV l + 2 l +1 Γ( l + )Γ( l + ) πr l +1 | r → = − k l +1 ε cot δ l ( k ε ) , (32) see detailed discussions in Appendix B. The relative mo-mentum k ε in infinite volume is related to the relativefinite volume energy ε by k ε µ = ε + (cid:52) E R , (cid:52) E R = QBM ( n + 12 ) − P x + P y M , (33)where (cid:52) E R is the result of quantization of CM motionin uniform magnetic field.Eliminating V l , the Eq.(31) and Eq.(32) together yielda L¨uscher formula-like simple relation,det (cid:104) δ lm,l (cid:48) m (cid:48) cot δ l ( k ε ) − M ( B,L ) lm,l (cid:48) m (cid:48) ( ε ) (cid:105) = 0 , (34)where M ( B,L ) lm,l (cid:48) m (cid:48) ( ε ) = − l (cid:48) +3 Γ ( l (cid:48) + )2 µk l +1 ε (2 π ) G ( B,L ) lm,l (cid:48) m (cid:48) ( r, r (cid:48) ; ε ) r l r (cid:48) l (cid:48) | r,r (cid:48) → − δ lm,l (cid:48) m (cid:48) l +1 Γ( l + )Γ( l + ) π k ε r ) l +1 | r → . (35)The second term in M ( B,L ) lm,l (cid:48) m (cid:48) plays the role of the regula-tor of ultraviolet (UV) divergence and will cancel out theUV divergence in finite volume magnetic Green’s func-tion, so that M ( B,L ) lm,l (cid:48) m (cid:48) is ultimately free of UV divergence.In general, the regularization and isolation of UV diver-gence in higher partial waves of finite volume magneticGreen’s function is a highly non-trivial task. Fortunately,it can be accomplished rather neatly for S -wave, hence,only S -wave contribution will be considered in SectionIII. The regularization of UV divergence will be workedout explicitly. III. S -WAVE CONTRIBUTION AND CONTACTINTERACTION As already mentioned in previous section, the angularmomentum basis in general is not convenient basis forthe dynamics of charged particles in uniform magneticfield. The partial wave expansion of finite volume mag-netic Green’s function and ultraviolet regularization canbe tedious in general. Fortunately, if only S -wave con-tribution is dominant, the formalism can be worked outnicely. In this section, only a contact interaction poten-tial V ( r ) = V π δ ( r ) (36)is used, which may be considered as the leading ordercontribution of chiral effective field theory and may besuitable for the few-body system, such as π + interactionsin finite volume.With a contact interaction, the finite volume quanti-zation condition is simply given by4 πV = G ( L ) B ( , ; ε ) . (37)In infinite volume, V is related to S -wave scattering am-plitude by 4 πV − G ( ∞ ) ( ; k ε ) = − µk ε π t ( k ε ) , (38)where t ( k ε ) = 1cot δ ( k ε ) − i , and infinite volume Green’s function G ( ∞ ) ( ; k ε ) is givenby G ( ∞ ) ( ; k ε ) = (cid:90) d p (2 π ) k ε µ − p µ = − i µk ε π − µ πr | r → . (39)Thus, the quantization condition is simply given bycot δ ( k ε ) = − π µk ε G ( L ) B ( , ; ε ) − k ε r | r → . (40)The magnetic Green’s function G ( L ) B ( , ; ε ) is a real func-tion of ε . The UV divergent term − k ε r | r → play the role of UV counter term that cancel out the UVdivergent term in G ( L ) B ( , ; ε ), so ultimate result is finiteand real as a function of ε . A. Regularization of UV divergence
In this section, we show explicitly how the UV diver-gence in G ( L ) B ( , ; ε ) is regularized and isolated out ex-plicitly. The UV divergence only appear when r = ( ρ , → , hence, a small r = ρ is used as UV regulator, in theend, final expression is obtained by taking the limit of r = ρ →
0. Starting with explicit expression of magneticGreen’s function in CM frame ( P B = ), G ( L ) B ( , ; ε ) = (cid:88) n x ,n y ∈ Z e − iqB n x n q Ln y L × e − qB | ρ + n x n q L e x + n y L e y | qB π L n z ∈ Z (cid:88) k z = πnzL × ∞ (cid:88) n =0 L n ( qB | ρ + n x n q L e x + n y L e y | ) ε − qBµ ( n + ) − k z µ | ρ → . (41)The UV divergence is associated to the term1 L n z ∈ Z (cid:88) k z = πnzL ∞ (cid:88) n =0 ε − qBµ ( n + ) − k z µ ∝ (cid:90) Λ= r dk k ∝ Λ = 1 r | r → , (42) hence G ( L ) B is linearly divergent.The linear divergence can be regularized simply by sub-traction. Therefore, we firstly split finite volume mag-netic Green’s function into regularized term by subtrac-tion and a term that is UV divergent, G ( L ) B ( , ; ε ) = (cid:52) G ( L ) B ( ε ) + G ( L ) B ( , ; 0) , (43)where (cid:52) G ( L ) B ( ε ) = G ( L ) B ( , ; ε ) − G ( L ) B ( , ; 0) . (44)The subtracted term (cid:52) G ( L ) B ( ε ) is free of UV divergence.Using identity1 L n z ∈ Z (cid:88) k z = πnzL E − k z µ = 2 µ √ µE cot √ µEL , (45) (cid:52) G ( L ) B ( ε ) is thus given explicitly by (cid:52) G ( L ) B ( ε ) = 2 µqB π ∞ (cid:88) n =0 (cid:88) n x ,n y ∈ Z e − iqBnxnynqL × e − qB | n x n q L e x + n y L e y | L n ( qB | n x n q L e x + n y L e y | ) × cot √ µε − qB ( n + ) L (cid:113) µε − qB ( n + ) + coth √ qB ( n + ) L (cid:113) qB ( n + ) . (46)Next, the UV divergence in G ( L ) B ( , ; 0) can be iso-lated out by further split G ( L ) B ( , ; 0) into G ( L ) B ( , ; 0) = G ( B,L ) UV + G ( B,L ) RC , (47)where G ( B,L ) UV is UV divergent and is given by, G ( B,L ) UV = qB π L n z ∈ Z (cid:88) k z = πnzL ∞ (cid:88) n =0 L n ( qB | ρ | ) − qBµ ( n + ) − k z µ | ρ → . (48)The G ( B,L ) RC is a regulated constant term, and is definedby G ( B,L ) RC = (cid:88) n x ,n y (cid:54) =0 e − iqBnxnynqL qB π L n z ∈ Z (cid:88) k z = πnzL × ∞ (cid:88) n =0 e − qB | n x n q L e x + n y L e y | L n ( qB | n x n q L e x + n y L e y | ) − qBµ ( n + ) − k z µ . (49)(1) The regulated constant term G ( B,L ) RC can be furthersimplified by using identity ∞ (cid:88) n =0 L n ( qB x ) − qBµ ( n + ) − k z µ = − µqB Γ( 12 + k z qB ) U ( 12 + k z qB , , qB x ) , (50)where U ( a, b, z ) stands for Kummer function, hence, wefind G ( B,L ) RC = − µ π L n z ∈ Z (cid:88) k z = πnzL Γ( 12 + k z qB ) × (cid:88) n x ,n y (cid:54) =0 e − iqBnxnynqL e − qB | n x n q L e x + n y L e y | × U ( 12 + k z qB , , qB | n x n q L e x + n y L e y | ) . (51)Asymptotically, Kummer function decay exponentially, e − qB x Γ( 12 + k z qB ) U ( 12 + k z qB , , qB x ) k z →∞ → K ( (cid:112) k z x ) , (52)hence G ( B,L ) RC is indeed a well-defined regulated constant.(2) The explicit expression of UV divergence in G ( B,L ) UV can be worked out. As ρ = r →
0, let’s define qBρ = ξdξ, qBρ n = ξ , (53)thus, qBρ ∞ (cid:88) n =0 ρ = r → = (cid:90) ∞ ξdξ, (54)and the Eq.(48) is then turned into G ( B,L ) UV = 12 π L n z ∈ Z (cid:88) k z = πnzL (cid:90) ∞ ξdξ L ξ qBr ( qBr ) − ξ µ − qBr µ − k z r µ | r → . (55)Using asymptotic form of Laguerre polynomial, L ξ qBr ( qBr r → → J ( (cid:112) ξ ) , (56)and identity in Eq.(45) again, we find G ( B,L ) UV = − µ πr (cid:90) ∞ ξdξJ ( (cid:112) ξ ) coth (cid:113) ξ r + qBL (cid:112) ξ + qBr | r → . (57)As r → r → ∞ ,coth z z →∞ ∼ e − z + 2 e − z · · · , z = (cid:113) ξ r + qBL , (58)and also using identity (cid:90) ∞ ξdξ J ( (cid:112) ξ )2 (cid:112) ξ + qBr = 2 π π e −√ qBr , (59) we finally obtain a explicit expression of UV divergence, G ( B,L ) UV = − µe −√ qBr πr | r → = 2 µ √ qB π − µ πr | r → . (60)Putting all pieces together, we thus find G ( L ) B ( , ; ε ) = (cid:52) G ( L ) B ( ε ) + G ( B,L ) RC + 2 µ √ qB π − µ πr | r → , (61)where (cid:52) G ( L ) B ( ε ) and G ( B,L ) RC are both free of UV diver-gence and given by Eq.(46) and Eq.(51) respectively. B. Regulated S -wave quantization condition With explicitly isolated UV divergence in finite volumemagnetic Green’s function in Eq.(61), the UV divergentterms in quantization condition given by Eq.(40) cancelout, thus we find a regulated quantization conditioncot δ ( k ε ) = M ( B,L )0 , ( ε ) , (62)where M ( B,L )0 , ( ε ) = − π µk ε (cid:52) G ( L ) B ( ε ) − π µk ε G ( B,L ) RC − √ qBk ε . (63)The expression of (cid:52) G ( L ) B ( ε ) and G ( B,L ) RC are given byEq.(46) and Eq.(51) respectively. C. L¨uscher formula at the limit of qB → At the limit of qB →
0, the finite volume magneticGreen’s function thus approach regular finite volumeGreen’s function, G ( L ) B ( r , r (cid:48) ; ε ) eB → → G ( L )0 ( r − r (cid:48) , k ε ) , (64)where G ( L )0 ( r , k ) = 2 µL (cid:88) p = πL n , n ∈ Z e i p · r k − p . (65)Hence, finite volume magnetic zeta function M ( B,L )0 , ( ε )at the limit of qB → M ( L )0 , ( k ε ) = − πk ε L (cid:88) p = πL n , n ∈ Z e k ε − p k ε − p + Λ k ε √ π (cid:34) n (cid:54) =0 (cid:88) n ∈ Z (cid:90) ∞ dξe − ( n L Λ ξ )24 + k ξ )2 + ∞ (cid:88) n =0 ( k ε Λ ) n n !(2 n − (cid:35) , (66)where Λ is an arbitrary UV regulator. - - ϵ ( GeV ) M ( ϵ ) n p = n q = - - ϵ ( GeV ) M ( ϵ ) n p = n q = FIG. 1: The comparison of the finite volume magneticzeta function M ( B,L )0 , ( ε ) (black solid) and regular finitevolume zeta function M ( L )0 , ( k ) (red solid) in Eq.(63)and Eq.(66) respectively. The parameters are chosen as: µ = 0 . L = 10GeV − , n p = 1 and n q = 1 , M ( B,L )0 , ( ε ) and regular finite volume zeta func-tion M ( L )0 , ( k ) are shown in Fig. 1. The splitting of en-ergy levels are illustrated in the comparison of the curvesof M ( B,L )0 , ( ε ) in upper and lower panels with n q = 1and n q = 2 respectively, the number of M ( B,L )0 , ( ε ) curvesdouble as the value of n q is doubled. IV. SUMMARY
A formalism for describing charged spinless bosons in-teraction in both a finite volume and a magnetic field ispresented in this work. We show that for a short-rangepotential, a L¨uscher formula-like relation that relates dis-crete energy spectrum to scattering phase shifts can beobtained. The regularization of UV divergence is workedout explicitly for S -wave contribution, the regulated S -wave quantization condition may be useful for the LQCDstudy of charged boson system, such as π + or K + sys- tem. In finite volume and in magnetic field, translationsymmetry of system is only preserved when the magneticflux, Φ B = qBL , is given by 2 π multiplied by a rationalnumber n p /n q where n p and n q are relatively prime in-tegers. The presence of magnetic field thus result in thesplitting of energy level into n q sub-energy levels. ACKNOWLEDGMENTS
P.G. acknowledges support from the Department ofPhysics and Engineering, California State University,Bakersfield, CA. This research (PG) was also supportedin part by the National Science Foundation (US) underGrant No. NSF PHY-1748958.
Appendix A: Two charged bosons in a uniformmagnetic field
The dynamics of two charged non-relativistic identi-cal bosons in a uniform magnetic field is described bySchr¨odinger equation, (cid:34) E + (cid:88) i =1 ( ∇ i + ie A ( x i )) m − V ( x − x ) (cid:35) Ψ E ( x , x ) = 0 , (A1)where m is the mass of identical bosons. x i denotes theposition of i-th particle, and the short-range interactionbetween two particles is represented by V ( x − x ). A ( x i )stands for the vector potential of uniform magnetic field.Throughout the entire work, the uniform magneticfield is assumed along the z -axis, B = B e z , the vectorpotential in Landau gauge is used, A ( x i ) = B (0 , x i , . (A2)The solutions of Schr¨odinger equation in other gaugesare obtained by a gauge transformation through a scalarfield, χ ( x i ), A ( x i ) → A ( x i ) − ∇ χ ( x i ) , (A3)and Ψ E ( x , x ) → e ie (cid:80) i =1 χ ( x i ) Ψ E ( x , x ) . (A4)
1. Separation of center of mass and relativemotions
The center of mass motion (CM) and relative motionof two particles can be separated by introducing CM andrelative coordinates respectively R = x + x , r = x − x . (A5)Therefore, the Hamiltonian has a separable form and thetotal two particles wave function is given by the productof CM and relative wave functions,Ψ E ( x , x ) = Φ E − ε ( R ) ψ ε ( r ) , (A6)where CM wave function, Φ E − ε ( R ), and relative wavefunction, ψ ε ( r ), satisfy Schr¨odinger equations respec-tively, (cid:34) ( E − ε ) + ( ∇ R + iQ A ( R )) M (cid:35) Φ E − ε ( R ) = 0 , (A7)and (cid:34) ε + ( ∇ r + iq A ( r )) µ − V ( r ) (cid:35) ψ ε ( r ) = 0 . (A8)The total and reduced mass of two particles are M = 2 m and µ = m Q = 2 e and q = e
2. Magnetic translation group and magneticperiodic boundary condition
Now, let’s consider putting charged particles in a peri-odic cubic box with size L , and interaction between twoparticles is also periodic, V ( r + n L ) = V ( r ) , n ∈ Z . (A9)Without magnetic field, the discrete translation symme-try of system in finite volume yields the conserved totalmomentum of system with discrete values: P = 2 π n L , n ∈ Z . In magnetic field, translation symmetry is explicitly bro-ken by position dependent vector potential A ( r ), A ( r + nL e x ) = A ( r ) + BnL e y , n ∈ Z , (A10)hence, the Hamiltonian is no longer invariant un-der translation operation in general. Fortunately, themagnetic translation operators can be introduced, seeRefs. [32–34]. For instance, the magnetic translation op-erator for relative motion can be defined byˆ T r ( n L ) = e i ( − i ∇ r + q A ( r ) − q B × r ) · n L , (A11)where e i ( − i ∇ r ) · n L is pure translation operator, and e i ( − i ∇ r ) · n L ψ ε ( r ) = ψ ε ( r + n L ) . (A12) So thatˆ T r ( n L ) ψ ε ( r ) = e iq ( A ( r ) − B × r ) · n L ψ ε ( r + n L ) , (A13)and ˆ T r commutes with Hamiltonian,[ ˆ T r , ˆ H ] = 0 , which leaves Hamiltonian invariant. However, the mag-netic translation operators do not commute with eachother in general,ˆ T r ( n x L e x ) ˆ T r ( n y L e y ) = e − iqBL n x n y ˆ T r ( n y L e y ) ˆ T r ( n x L e x ) , (A14)where ( n x , n y ) ∈ Z .As shown in Refs. [32, 33], when the values of qB aretaken as qB = 12 eB = 2 πL n p n q , (A15)where n p and n q are integers that are relatively prime.The magnetic translation operators with enlarged unitcell formed by increased size of n q L in e x direction thuscommute with each other, (cid:104) ˆ T r ( n x ( n q L ) e x ) , ˆ T r ( n y L e y ) (cid:105) = 0 . (A16)Therefore magnetic translation operators with enlargedmagnetic unit box that is defined by n q L e x × L e y × L e z form a discrete group that are commonly referred as mag-netic translation group.The translation operator for two charged particles canbe introduced byˆ T x , x ( n B L, n B L ) = ˆ T x ( n B L ) ˆ T x ( n B L ) , (A17)where ˆ T x i ( n L ) = e i ( − i ∇ x i + e A ( x i ) − e B × x i ) · n L . (A18)Both n B L and n B L are defined in enlarged magneticunit box, n iB L = n ix ( n q L ) e x + n iy L e y + n iz L e z , n ix,iy,iz ∈ Z . (A19)We may rewrite two particles translation operator interms of CM and relative motion quantities,ˆ T x , x ( n B L, n B L ) = ˆ T r ( n B L − n B L ) ˆ T R ( n B + n B L ) , (A20)where ˆ T r is defined in Eq.(A11), andˆ T R ( n B L e i ( − i ∇ R + Q A ( R ) − Q B × R ) · n BL . (A21)Note that QB = 2 eB = 2 π ( L ) n p n q , the translation operation of CM motion may be consid-ered as motion of composite charge particle with totalcharges of Q = 2 e in a periodic box with size of L/ T x , x ( n B L, n B L )Ψ E ( x , x ) = Ψ E ( x , x ) . (A22)Using Eq.(A18), the boundary conditions for two parti-cles in both finite volume and a uniform magnetic fieldis given by,Ψ E ( x + n B L, x + n B L )= e − ie (cid:80) i =1 ( A ( x i ) − B × x i ) · n iB L Ψ E ( x , x ) . (A23)In terms of CM and relative wave functions, we have e iQ ( A ( R ) − B × R ) · n B + n B L Φ E − ε ( R + n B + n B L )Φ E − ε ( R )= ψ ε ( r ) e iq ( A ( r ) − B × r ) · ( n B − n B ) L ψ ε ( r + ( n B − n B ) L ) . (A24)The separable form of CM motion and relative motionin Eq.(A24) suggests that both sides must be equal to aphase factor that is independent of both CM and rela-tive coordinates. It allows us to introduce an arbitraryparameter P B that is associated to pure translation op-erator, the phase factor may be chosen having form of e i P B · n B + n B L . Hence the CM wave function satisfies Bloch type mag-netic periodic boundary condition,Φ E − ε ( R + n B L e − i ( P B + QBR y e x ) · n BL Φ E − ε ( R ) . (A25)The boundary condition for relative wave function isgiven by ψ ε ( r + n B L ) = e i P B · n B L e − iqBr y e x · n B L ψ ε ( r ) , (A26)where we have also assumed e i P B · n B L = 1 , (A27)thus P B = 2 π (cid:18) n x n q L e x + n y L e y + n z L e z (cid:19) , n x,y,z ∈ Z . Although P B resemble the total momentum of system inabsence of magnetic field, P B is not a conserved quantityin magnetic field. In fact, the conserved quantity can beidentified as pseudo-momentum, see e.g. Ref. [34], K R = P B + Q ( A ( R ) − B × R ) , (A28)which is associated to the generator of magnetic transla-tion operator for CM motion,ˆ T R ( n B L e i ˆK R · n BL , [ ˆK R , ˆ H ] = 0 . (A29)
3. CM motion solutions
The CM motion of two charged bosons in a uniformmagnetic field is described byˆ H R Φ E R ( R ) = E R Φ E R ( R ) , (A30)whereˆ H R = − M (cid:2) ∂ R x + ( ∂ R y + iQBR x ) + ∂ R z (cid:3) , (A31)and Φ E R must satisfies boundary conditionΦ E R ( R + n B L e − i ( P B + QBR y e x ) · n BL Φ E R ( R ) . (A32)The solution that satisfies magnetic periodic boundarycondition can be found in [32],Φ E R,n ( R )= 2 L n y ∈ Z (cid:88) k y = πnynpL − P By φ n ( R x + k y QB ) e ik y ( R y + PBxQB ) e − iP Bz R z , (A33)where φ n is eigen-solution of 1D harmonic oscillator po-tential, − M (cid:2) ∂ R x − ( QB ) R x (cid:3) φ n ( R x ) = QBM ( n + 12 ) φ n ( R x ) . (A34)The eigen-energy of CM motion is given by E R,n = QBM ( n + 12 ) + P Bz M , n = 0 , , , · · · , (A35)and analytic expression of φ n is φ n ( R x ) = 1 √ n n ! (cid:18) QBπ (cid:19) e − QB R x H n ( (cid:112) QBR x ) . (A36)
4. Relative motion and finite volumeLippmann-Schwinger equation
The relative motion of two charged particles in a uni-form magnetic field is described by (cid:16) ˆ H r + V ( r ) (cid:17) ψ ε ( r ) = εψ ε ( r ) , (A37)whereˆ H r = − µ (cid:2) ∂ r x + ( ∂ r y + iqBr x ) + ∂ r z (cid:3) , (A38)and ψ ε must satisfies magnetic periodic boundary condi-tion ψ ε ( r + n B L ) = e i P B · n B L e − iqBr y e x · n B L ψ ε ( r ) . (A39)0The integral representation of Schr¨odinger equation(A37) and magnetic periodic boundary condition inEq.(A39) together is given by finite volume Lippmann-Schwinger equation, ψ ε ( r ) = (cid:90) L B d r (cid:48) G ( L ) B ( r , r (cid:48) ; ε ) V ( r (cid:48) ) ψ ε ( r (cid:48) ) , (A40)where L B stands for the volume of magnetic unit boxdefined by unit vectors n q L e x × L e y × L e z , and (cid:90) L B d r (cid:48) = (cid:90) nqL − nqL dr (cid:48) x (cid:90) L − L dr (cid:48) y (cid:90) L − L dr (cid:48) z . (A41)The finite volume magnetic Green’s function G ( L ) B alsomust satisfy the magnetic periodic boundary condition, G ( L ) B ( r , r (cid:48) ; ε )= e − i P B · n B L e iqBr y e x · n B L G ( L ) B ( r + n B L, r (cid:48) ; ε )= e i P B · n B L e − iqBr (cid:48) y e x · n B L G ( L ) B ( r , r (cid:48) + n B L ; ε ) . (A42)The magnetic periodic boundary conditions andEq.(A40) suggest that G ( L ) B is the solution of differentialequation, (cid:16) ε − ˆ H r (cid:17) G ( L ) B ( r , r (cid:48) ; ε )= (cid:88) n B e − i P B · n B L e iqBr y e x · n B L δ ( r − r (cid:48) + n B L ) . (A43)Now, one of the key steps therefore is to find an ana-lytic solution of finite volume magnetic Green’s function G ( L ) B . The G ( L ) B can be constructed from infinite volumemagnetic Green’s function G ( ∞ ) B , where G ( ∞ ) B ( r , r (cid:48) ; ε ) = ∞ (cid:88) n =0 (cid:90) ∞−∞ dk y π dk z π × φ n ( r x + k y qB ) φ ∗ n ( r (cid:48) x + k y qB ) e ik y ( r y − r (cid:48) y ) e ik z ( r z − r (cid:48) z ) ε − qBµ ( n + ) − k z µ , (A44)and G ( ∞ ) B satisfies equation, (cid:16) ε − ˆ H r (cid:17) G ( ∞ ) B ( r , r (cid:48) ; ε ) = δ ( r − r (cid:48) ) . (A45)The LS equation (A40) is equivalently given in terms of G ( ∞ ) B by ψ ε ( r ) = (cid:90) ∞−∞ d r (cid:48) G ( ∞ ) B ( r , r (cid:48) ; ε ) V ( r (cid:48) ) ψ ε ( r (cid:48) ) . (A46) The integration over infinite volume in Eq.(A46) can befolded up to infinite sum of integration in magnetic cell, ψ ε ( r ) = (cid:88) n B (cid:90) L B d r (cid:48) G ( ∞ ) B ( r , r (cid:48) + n B L ; ε ) × V ( r (cid:48) + n B L ) ψ ε ( r (cid:48) + n B L ) . (A47)Using magnetic periodic boundary condition given inEq.(A39), G ( L ) B is thus identified as G ( L ) B ( r , r (cid:48) ; ε )= (cid:88) n B G ( ∞ ) B ( r , r (cid:48) + n B L ; ε ) e i P B · n B L e − iqBr (cid:48) y e x · n B L = (cid:88) n B e − i P B · n B L e iqBr y e x · n B L G ( ∞ ) B ( r + n B L, r (cid:48) ; ε ) . (A48)Hence, explicitly we find G ( L ) B ( r , r (cid:48) ; ε ) = (cid:88) n x ∈ Z e − i ( PBx − qBr y ) n x n q L × L n y,z ∈ Z (cid:88) k y,z = πny,zL + PBy,Bz e ik y ( r y − r (cid:48) y ) e ik z ( r z − r (cid:48) z ) × ∞ (cid:88) n =0 φ n ( r x + n x n q L + k y qB ) φ ∗ n ( r (cid:48) x + k y qB ) ε − qBµ ( n + ) − k z µ . (A49)The other representation of G ( ∞ ) B are given in Refs. [35,36] by G ( ∞ ) B ( r , r (cid:48) ; ε ) = e − iqB ( r x + r (cid:48) x )( r y − r (cid:48) y ) e − qB | ρ − ρ (cid:48) | × qB π (cid:90) ∞−∞ dk z π ∞ (cid:88) n =0 L n ( qB | ρ − ρ (cid:48) | ) e ik z ( r z − r (cid:48) z ) ε − qBµ ( n + ) − k z µ , (A50)where ρ = r x e x + r y e y , ρ (cid:48) = r (cid:48) x e x + r (cid:48) y e y . Therefore, G ( L ) B is also given by G ( L ) B ( r , r (cid:48) ; ε ) = (cid:88) n x ,n y ∈ Z e − i ( PBx − qBr y ) n x n q L e − i PBy n y L × e − iqB ( r x + r (cid:48) x + n x n q L )( r y − r (cid:48) y + n y L ) × e − qB | ρ − ρ (cid:48) + n x n q L e x + n y L e y | qB π L n z ∈ Z (cid:88) k z = πnzL + PBz × ∞ (cid:88) n =0 L n ( qB | ρ − ρ (cid:48) + n x n q L e x + n y L e y | ) e ik z ( r z − r (cid:48) z ) ε − qBµ ( n + ) − k z µ . (A51)1 Appendix B: Connecting bound states in a trap toinfinite volume scattering state
In this section, we present a general formalism anddiscussion on the topic of building connections betweendiscrete energy spectrum of bound state in a trap andinfinite volume scattering dynamics. The type of trap isnot specified in follows, the typical and commonly usedtraps are periodic finite box in LQCD, harmonic poten-tial in nuclear physics, etc.The relative motion of two interacting particles in atrap is described by Schr¨odinger equationˆ H trap ψ ( trap ) ε ( r )+ (cid:90) trap d r (cid:48) V ( r , r (cid:48) ) ψ ( trap ) ε ( r (cid:48) ) = εψ ( trap ) ε ( r ) , (B1)where ˆ H trap stands for the trap Hamiltonian operator,the interaction between particles is described by a non-local short-range interaction V ( r , r (cid:48) ) in general. The ef-fect of a trap is usually reflected by both trap Hamilto-nian and boundary condition of wave function in a trap.In the case of charged particles trapped in both a periodicbox and a uniform magnetic field, ˆ H trap and boundarycondition are thus given by ˆ H r in Eq.(A38) and mag-netic periodic boundary condition in Eq.(A39) respec-tively. The energy spectrum hence becomes discrete.In infinite volume, the dynamics of two interacting par-ticles through the same short-range interaction V ( r , r (cid:48) )is given byˆ H ψ ( ∞ ) ε ∞ ( r ) + (cid:90) ∞−∞ d r (cid:48) V ( r , r (cid:48) ) ψ ( ∞ ) ε ∞ ( r (cid:48) ) = εψ ( ∞ ) ε ∞ ( r ) , (B2)where ˆ H = − ∇ r µ . The energy spectrum of scattering solution in infinite vol-ume is continuous. With a incoming plane wave, e i q · r where q = (cid:112) µε ∞ , the asymptotic wave function of scattering states is thusdescribed by on-shell scattering amplitudes, ψ ( ∞ ) ε ∞ ( r ) r →∞ → (cid:88) l (2 l +1) P l ( ˆq · ˆr ) i l (cid:104) j l ( qr ) + it l ( q ) h (+) l ( qr ) (cid:105) , (B3)where t l ( q ) denotes the elastic on-shell partial wave scat-tering amplitude and can be parametrized by a phaseshift function δ l ( q ), t l ( q ) = 1cot δ l ( q ) − i . (B4)We also remark that in general case, depending on thetrap, the infinite volume relative energy ε ∞ is related tofinite volume relative energy ε by the shared total energy. For instance, in the case of charged particles trapped inboth a periodic box and a uniform magnetic field, ε ∞ + P M = ε + E R,n = E, (B5)where CM energy E R,n is given by Eq.(A35).The dynamics of particles in a trap and in infinite vol-ume are associated by the short-range interaction poten-tial between two particles. As far as the range of poten-tial is far smaller than the size of the trap, a compactexpression between phase shift of scattering states and afunction, M lm,l (cid:48) m (cid:48) ( ε ), that reflect geometric and dynam-ical properties of the trap can be found,det[ δ lm,l (cid:48) m (cid:48) cot δ l ( q ) − M lm,l (cid:48) m (cid:48) ( ε )] = 0 . (B6)In the case of finite volume in LQCD, this relationis well-known L¨uscher formula [1], the matrix function M lm,l (cid:48) m (cid:48) ( ε ) is thus zeta function. In finite volume, theangular momentum is no longer a good quantum numberdue to the breaking rotation symmetry in finite volume.In the case of harmonic trap in nuclear physics, the re-lation is known as BERW formula [29], where function M lm,l (cid:48) m (cid:48) becomes diagonal in angular momentum basis.The simple form of quantization condition in Eq.(B6) isthe result of the presence of two distinguishable scales:(1) short-range interaction between two particles and (2)size of trap. Hence the short-range dynamics that is de-scribed by phase shift or scattering amplitude and long-range physics due to the presence of a trap can be fac-torized.The derivation of L¨uscher formula or BERW for-mula can be illustrated by considering momentum spaceLippmann-Schwinger equation under the assumption ofseparable potential, see e.g. [26–28], an example ofderivation of BERW formula in momentum space is givenin Appendix C. Here the result is only summarized brieflysymbolically, the reaction amplitudes in both trap andinfinite volume may be introduced respectively byˆ t trap = − ˆ V ˆ ψ and ˆ t ∞ = − ˆ V ˆ ψ ∞ , they satisfy integral LS equations,ˆ t trap ( ε ) = ˆ V ˆ G trap ( ε )ˆ t trap ( ε ) , (B7)and ˆ t ∞ ( q ) = − ˆ V + ˆ V ˆ G ∞ ( q )ˆ t ∞ ( q ) , (B8)where ˆ G trap ( ε ) = 1 ε − ˆ H trap (B9)and ˆ G ∞ ( q ) = 1 q µ − ˆ H (B10)2are Green’s function in a trap and in infinite volume re-spectively. Under the assumption of separable potentialthat is equivalent to the zero-range interaction, (cid:101) V ( k , k (cid:48) ) = (cid:88) lm ( kk (cid:48) ) l V l Y lm ( ˆk ) Y ∗ lm ( ˆk (cid:48) ) , (B11)Eq.(B7) and Eq.(B8) are turned into algebra equations,and can be solved analytically [28]. Eliminating ˆ V fromtwo equations, the quantization condition is thus ob-tained det (cid:20) t ∞ ( q ) − ˆ G ∞ ( q ) + ˆ G trap ( ε ) (cid:21) = 0 , (B12)which is equivalent to Eq.(B6).Though the plane wave basis in momentum space maybe a very convenient basis in finite volume and othertypes of traps, for the charged particles in uniform mag-netic field, the momentum is no longer the conservedquantity due to the breaking translation symmetry bymagnetic field. Introducing a reaction amplitude in mo-mentum space becomes a tricky business. Therefore, infollows, instead of working in momentum space, we willpresent the general discussion of derivation of quantiza-tion condition in coordinate space under assumption ofseparable short-range potential again. The Fourier trans-form of separable potential given in Eq.(B11) is V ( r , r (cid:48) ) = (cid:90) d k (2 π ) d k (cid:48) (2 π ) e − i k · r (cid:101) V ( k , k (cid:48) ) e i k (cid:48) · r (cid:48) = δ ( r ) δ ( r (cid:48) )( rr (cid:48) ) (cid:88) lm V l l +1 Γ ( l + )(2 π ) ( rr (cid:48) ) l Y lm ( ˆr ) Y ∗ lm ( ˆr (cid:48) ) . (B13)
1. Dynamical equation in a trap
In the trap, the integral representation of Eq.(B1) isgiven by the Lippmann-Schwinger equation ψ ( trap ) ε ( r ) = (cid:90) trap d r (cid:48)(cid:48) G ( trap ) ( r , r (cid:48)(cid:48) ; ε ) × (cid:90) trap d r (cid:48) V ( r (cid:48)(cid:48) , r (cid:48) ) ψ ( trap ) ε ( r (cid:48) ) , (B14)where G ( trap ) ( r , r (cid:48)(cid:48) ; ε ) = (cid:104) r | ε − ˆ H trap | r (cid:48)(cid:48) (cid:105) (B15)stands for the Green’s function in a trap. The partialwave expansions ψ ( trap ) ε ( r ) = (cid:88) lm ψ ( trap ) lm ( r ) Y lm ( ˆr ) (B16)and G ( trap ) ( r , r (cid:48)(cid:48) ; ε )= (cid:88) lm,l (cid:48)(cid:48) m (cid:48)(cid:48) Y lm ( ˆr ) G ( trap ) lm,l (cid:48)(cid:48) m (cid:48)(cid:48) ( r, r (cid:48)(cid:48) ; ε ) Y ∗ l (cid:48)(cid:48) m (cid:48)(cid:48) ( ˆr (cid:48)(cid:48) ) (B17) yields ψ ( trap ) lm ( r ) = (cid:88) l (cid:48) m (cid:48) (cid:90) trap r (cid:48)(cid:48) dr (cid:48)(cid:48) G ( trap ) lm,l (cid:48) m (cid:48) ( r, r (cid:48)(cid:48) ; ε ) × (cid:90) trap r (cid:48) dr (cid:48) V l (cid:48) ( r (cid:48)(cid:48) , r (cid:48) ) ψ ( trap ) l (cid:48) m (cid:48) ( r (cid:48) ) . (B18)Under assumption of separable potential with the formof Eq.(B13), Eq.(B18) is turned into an algebra equation, ψ ( trap ) lm ( r ) r l = (cid:88) l (cid:48) m (cid:48) V l (cid:48) l (cid:48) +1 Γ ( l (cid:48) + )(2 π ) × G ( trap ) lm,l (cid:48) m (cid:48) ( r, r (cid:48)(cid:48) ; ε ) r l r (cid:48)(cid:48) l (cid:48) ψ ( trap ) l (cid:48) m (cid:48) ( r (cid:48) ) r (cid:48) l (cid:48) | r (cid:48) ,r (cid:48)(cid:48) → . (B19)hence the discrete energy spectrum is determined bydet (cid:34) δ lm,l (cid:48) m (cid:48) l (cid:48) +1 V l − Γ ( l (cid:48) + )(2 π ) G ( trap ) lm,l (cid:48) m (cid:48) ( r, r (cid:48) ; ε ) r l r (cid:48) l (cid:48) | r,r (cid:48) → (cid:35) = 0 . (B20)
2. Infinite volume dynamical equation
In infinite volume, with a incoming plane wave of e i q · r ,the scattering solution of two particles interaction is de-scribed by inhomogeneous integral Lippmann-Schwingerequation, ψ ( ∞ ) ε ∞ ( r , q ) = e i q · r + (cid:90) ∞−∞ d r (cid:48)(cid:48) G ( ∞ ) ( r − r (cid:48)(cid:48) ; q ) (cid:90) ∞−∞ d r (cid:48) V ( r (cid:48)(cid:48) , r (cid:48) ) ψ ( ∞ ) ε ∞ ( r (cid:48) , q ) , (B21)where q = √ µε ∞ , and the Green’s function is given by G ( ∞ ) ( r − r (cid:48)(cid:48) ; q )= (cid:90) d p (2 π ) e i p · ( r − r (cid:48)(cid:48) ) q µ − p µ = − µ π iqh (+)0 ( q | r − r (cid:48)(cid:48) | ) . (B22)Considering partial wave expansion, ψ ( ∞ ) ε ∞ ( r , q ) = (cid:88) lm Y ∗ lm ( ˆq ) ψ ( ∞ ) l ( r, q ) Y lm ( ˆr ) , (B23)and G ( ∞ ) ( r − r (cid:48)(cid:48) ; q ) = (cid:88) lm Y lm ( ˆr ) G ( ∞ ) l ( r, r (cid:48)(cid:48) ; q ) Y ∗ lm ( ˆr (cid:48)(cid:48) ) ,G ( ∞ ) l ( r, r (cid:48)(cid:48) ; q ) = − µiqj l ( qr < ) h (+) l ( qr > ) , (B24)we thus obtain ψ ( ∞ ) l ( r, q ) = 4 πi l j l ( qr )+ (cid:90) ∞ r (cid:48)(cid:48) dr (cid:48)(cid:48) G ( ∞ ) l ( r, r (cid:48)(cid:48) ; q ) (cid:90) ∞ r (cid:48) dr (cid:48) V l ( r (cid:48)(cid:48) , r (cid:48) ) ψ ( ∞ ) l ( r (cid:48) , q ) . (B25)3The separable potential given in Eq.(B13) yields an al-gebra equation ψ ( ∞ ) l ( r, q ) r l = 4 πi l j l ( qr ) r l + V l l +1 Γ ( l + )(2 π ) G ( ∞ ) l ( r, r (cid:48)(cid:48) ; q )( rr (cid:48)(cid:48) ) l ψ ( ∞ ) l ( r (cid:48) , q ) r (cid:48) l | r (cid:48) ,r (cid:48)(cid:48) → . (B26)The wave function solution is thus given by ψ ( ∞ ) l ( r, q ) r l = 4 πi l (cid:34) j l ( qr ) r l + it l ( q ) h (+) l ( qr ) r l (cid:35) , (B27)where the partial wave two-body scattering amplitude t l ( q ) is given by t l ( q ) = − µq l +1 (4 π ) V l − l +1 Γ ( l + )(2 π ) G ( ∞ ) l ( r (cid:48) ,r (cid:48)(cid:48) ; q )( r (cid:48) r (cid:48)(cid:48) ) l | r (cid:48) ,r (cid:48)(cid:48) → . (B28)
3. Quantization condition in a trap
Combining Eq.(B20) and Eq.(B28), and eliminating V l , one thus finddet (cid:20) δ lm,l (cid:48) m (cid:48) µq l +1 t l ( q ) − δ lm,l (cid:48) m (cid:48) l +3 Γ ( l + )(2 π ) G ( ∞ ) l ( r, r (cid:48) ; q )( rr (cid:48) ) l | r,r (cid:48) → + 2 l (cid:48) +3 Γ ( l (cid:48) + )(2 π ) G ( trap ) lm,l (cid:48) m (cid:48) ( r, r (cid:48) ; ε ) r l r (cid:48) l (cid:48) | r,r (cid:48) → (cid:21) = 0 . (B29)Using asymptotic form of2 l +3 Γ ( l + )2 µ (2 π ) G ( ∞ ) l ( r, r (cid:48) ; q )( rr (cid:48) ) l | r,r (cid:48) → = − iq l +1 − l +1 Γ( l + )Γ( l + ) π r l +1 | r → , (B30)and also the parameterization of t − l ( q ) = cot δ l ( q ) − i, thus the quantization condition in a trap is indeed givenby a L¨uscher formula-like relation,det [ δ lm,l (cid:48) m (cid:48) cot δ l ( q ) − M lm,l (cid:48) m (cid:48) ( ε )] = 0 , (B31)where M lm,l (cid:48) m (cid:48) ( ε ) = − l (cid:48) +3 Γ ( l (cid:48) + )2 µq l +1 (2 π ) G ( trap ) lm,l (cid:48) m (cid:48) ( r, r (cid:48) ; ε ) r l r (cid:48) l (cid:48) | r,r (cid:48) → − δ lm,l (cid:48) m (cid:48) l +1 Γ( l + )Γ( l + ) π qr ) l +1 | r → . (B32)The second term in Eq.(B32) is an ultraviolet counterterm that would cancel out the ultraviolet divergent termin G ( trap ) lm,l (cid:48) m (cid:48) , ultimate result is finite and well-defined. Appendix C: Momentum space LS equation andparticles interaction in a harmonic trap
In this section, we present some technical details ofnon-relativistic spinless particles interaction in a har-monic trap. The dynamics of non-relativistic bosonicparticles interaction in a harmonic trap is described by (cid:16) ˆ H ( ho ) + ˆ V (cid:17) Ψ ( ho ) E ( x , x ) = E Ψ ( ho ) E ( x , x ) , (C1)where ˆ H ( ho ) = (cid:88) i =1 (cid:18) − ∇ i m + 12 mω x i (cid:19) , (C2)and x i again stand for the i-th particle’s position, theˆ V represents the interaction between two particles. ω isthe angular frequency of the oscillator. The separationof CM and relative motionsˆ H ( ho ) = ˆ H ( ho ) R + ˆ H ( ho ) r , where ˆ H ( ho ) R = − ∇ R M + 12 M ω R and ˆ H ( ho ) r = − ∇ r µ + 12 µω r yields againΨ ( ho ) E ( x , x ) = Φ ( ho ) E R,n ( R ) ψ ( ho ) ε ( r ) . The CM wave function Φ ( ho ) E R,n ( R ) is the solution of 3 D harmonic oscillator potential,ˆ H ( ho ) R Φ ( ho ) E R,n ( R ) = E R,n Φ ( ho ) E R,n ( R ) , (C3)where eigen-energy is given by E R,n = ω ( n + 32 ) , n = 0 , , , · · · . (C4)The relative wave function ψ ( ho ) ε ( r ) satisfies Lippmann-Schwinger equation, ψ ( ho ) ε ( r ) = (cid:90) d r (cid:48) G ( ho ) ( r , r (cid:48) ; ε ) (cid:90) d r (cid:48)(cid:48) V ( r (cid:48) , r (cid:48)(cid:48) ) ψ ( ho ) ε ( r (cid:48)(cid:48) ) , (C5)where Green’s function satisfies equation, (cid:16) ε − ˆ H ( ho ) r (cid:17) G ( ho ) ( r , r (cid:48) ; ε ) = δ ( r − r (cid:48) ) . (C6)The analytic expression of Green’s function in harmonictrap is given by [37] G ( ho ) ( r , r (cid:48) ; ε ) = (cid:88) lm Y lm ( ˆr ) G ( ho ) l ( r, r (cid:48) ; ε ) Y ∗ lm ( ˆr (cid:48) ) ,G ( ho ) l ( r, r (cid:48) ; ε ) = − ω ( rr (cid:48) ) Γ( l + − ε ω )Γ( l + ) × M ε ω , l + ( µωr < ) W ε ω , l + ( µωr > ) , (C7)where M a,b ( x ) and W a,b ( x ) are Whittaker functions [38].4
1. Momentum space LS equation and reactionamplitude in a harmonic oscillator trap
The reaction amplitude in a harmonic trap can be de-fined by T ( ho ) ε ( k ) = − (cid:90) d r e − i k · r (cid:90) d r (cid:48)(cid:48) V ( r (cid:48) , r (cid:48)(cid:48) ) ψ ( ho ) ε ( r (cid:48)(cid:48) ) , (C8)and T ( ho ) ε ( k ) satisfies momentum space LS equation, T ( ho ) ε ( k )= (cid:90) d k (cid:48) (2 π ) d k (cid:48)(cid:48) (2 π ) (cid:101) V ( k , k (cid:48) ) (cid:101) G ( ho ) ( k (cid:48) , k (cid:48)(cid:48) ; ε ) T ( ho ) ε ( k (cid:48)(cid:48) ) , (C9)where (cid:101) V and (cid:101) G ( ho ) are the Fourier transform of interac-tion potential V and Green’s function G ( ho ) respectively.In harmonic oscillator trap, rotation symmetry is intact,hence the angular momentum is still a good quantumnumber, the partial wave expansion of T ( ho ) ε ( k ) = (cid:88) lm T ( ho ) l ( k ) Y lm ( ˆk )and (cid:101) G ( ho ) ( k , k (cid:48) ; ε ) = (cid:88) lm Y lm ( ˆk ) (cid:101) G ( ho ) l ( k, k (cid:48) ; ε ) Y ∗ lm ( ˆk (cid:48) ) (C10)thus yields T ( ho ) l ( k )= (cid:90) ∞ k (cid:48) dk (cid:48) (2 π ) k (cid:48)(cid:48) dk (cid:48)(cid:48) (2 π ) (cid:101) V l ( k, k (cid:48) ) (cid:101) G ( ho ) l ( k (cid:48) , k (cid:48)(cid:48) ; ε ) T ( ho ) l ( k (cid:48)(cid:48) ) . (C11)The separable potential (cid:101) V l ( k, k (cid:48) ) = ( kk (cid:48) ) l V l suggests that T ( ho ) l ( k ) = k l t ( ho ) l , (C12)hence the quantization condition under assumption ofseparable potential is given by1 V l = (cid:90) ∞ k (cid:48) dk (cid:48) (2 π ) k (cid:48)(cid:48) dk (cid:48)(cid:48) (2 π ) ( k (cid:48) k (cid:48)(cid:48) ) l (cid:101) G ( ho ) l ( k (cid:48) , k (cid:48)(cid:48) ; ε ) . (C13)
2. Momentum space LS equation and scatteringamplitude in infinite volume
In infinite volume, the scattering amplitude is definedby T ( ∞ ) ε ∞ ( k , q ) = − (cid:90) d r e − i k · r (cid:90) d r (cid:48) V ( r , r (cid:48) ) ψ ( ∞ ) ε ∞ ( r (cid:48) , q ) , (C14) and it satisfies the momentum space LS equation T ( ∞ ) ε ∞ ( k , q ) = − (cid:101) V ( k , q ) + (cid:90) d k (cid:48) (2 π ) (cid:101) V ( k , k (cid:48) ) k (cid:48) µ − q µ T ( ∞ ) ε ∞ ( k (cid:48) , q ) . (C15)The partial wave expansion T ( ∞ ) ε ∞ ( k , q ) = (cid:88) lm T ( ∞ ) l ( k, q ) Y lm ( ˆk ) Y ∗ lm ( ˆq )yields, T ( ∞ ) l ( k, q ) = − (cid:101) V l ( k, q ) + (cid:90) ∞ k (cid:48) dk (cid:48) (2 π ) (cid:101) V l ( k, k (cid:48) ) k (cid:48) µ − q µ T ( ∞ ) l ( k (cid:48) , q ) . (C16)The assumption of separable potential again yields ananalytic solution of scattering amplitude, T ( ∞ ) l ( q (cid:48) , q ) = − ( q (cid:48) q ) L V l − (cid:82) ∞ k dk (2 π ) k lk µ − q µ . (C17)The on-shell partial wave scattering amplitudes T ( ∞ ) l ( q, q ), where q = √ µε ∞ , are usually parameterizedby phase shift, T ( ∞ ) l ( q, q ) = (4 π ) µq δ l ( q ) − i . (C18)Therefore, a simple relation between V L and phase shiftis obtained,1 V l − (cid:90) ∞ k dk (2 π ) k lk µ − q µ = − µq l +1 (4 π ) [cot δ l ( q ) − i ] . (C19)
3. Quantization condition in a harmonic oscillatortrap
Combing Eq.(C19) and Eq.(C13), we find (cid:90) ∞ k dk (2 π ) k (cid:48) dk (cid:48) (2 π ) ( kk (cid:48) ) l (cid:101) G ( ho ) l ( k, k (cid:48) ; ε ) − (cid:90) ∞ k dk (2 π ) k lk µ − q µ = − µq l +1 (4 π ) [cot δ l ( q ) − i ] . (C20)Using asymptotic form of spherical Bessel function, j l ( kr ) r → → √ π ( kr ) l l +1 Γ( l + ) , (C21)one can easily prove that (cid:90) ∞ k dk (2 π ) k (cid:48) dk (cid:48) (2 π ) ( kk (cid:48) ) l (cid:101) G ( ho ) l ( k, k (cid:48) ; ε )= 1(4 π ) l +2 Γ ( l + ) π G ( ho ) l ( r, r (cid:48) ; ε )( rr (cid:48) ) l | r,r (cid:48) → , (C22)5and (cid:90) ∞ k dk (2 π ) k lk µ − q µ = 1(4 π ) l +2 Γ ( l + ) π G ( ∞ ) l ( r, r (cid:48) ; q )( rr (cid:48) ) l | r,r (cid:48) → , (C23)where the analytic expression of G ( ho ) l ( r, r (cid:48) ; ε ) and G ( ∞ ) l ( r, r (cid:48) ; q ) are given in Eq.(C7) and Eq.(B24) respec-tively. Also using the asymptotic form of harmonic oscil- lator trap Green’s function,2 l +2 Γ ( l + ) π G ( ho ) l ( r, r (cid:48) ; ε )( rr (cid:48) ) l | r,r (cid:48) → = − ( µω ) l + ω l +2 ( − l +1 Γ( + l − ε ω )Γ( − l − (cid:15) n ω ) − l +1 Γ( l + )Γ( l + ) π µr l +1 | r → , (C24)and asymptotic form of G ( ∞ ) l ( r, r (cid:48) ; q ) given in Eq.(B30),the UV divergence cancel out explicitly in Eq.(C20), andthe quantization condition is thus reduced to BERW for-mula,cot δ l ( q ) − ( − l +1 ( 4 µωq ) l + Γ( + l − ε ω )Γ( − l − ε ω ) = 0 , (C25)where q and ε are associated by q µ + P M = ε + ω ( n + 32 ) . (C26) [1] M. L¨uscher, Nucl. Phys. B354 , 531 (1991).[2] K. Rummukainen and S. A. Gottlieb, Nucl. Phys.
B450 ,397 (1995), arXiv:hep-lat/9503028 [hep-lat].[3] N. H. Christ, C. Kim, and T. Yamazaki, Phys. Rev.
D72 , 114506 (2005), arXiv:hep-lat/0507009 [hep-lat].[4] V. Bernard, M. Lage, U.-G. Meißner, and A. Rusetsky,JHEP , 024 (2008), arXiv:0806.4495 [hep-lat].[5] S. He, X. Feng, and C. Liu, JHEP , 011 (2005),arXiv:hep-lat/0504019 [hep-lat].[6] M. Lage, U.-G. Meißner, and A. Rusetsky, Phys. Lett. B681 , 439 (2009), arXiv:0905.0069 [hep-lat].[7] M. D¨oring, U.-G. Meißner, E. Oset, and A. Rusetsky,Eur. Phys. J.
A47 , 139 (2011), arXiv:1107.3988 [hep-lat].[8] P. Guo, J. Dudek, R. Edwards, and A. P. Szczepaniak,Phys. Rev.
D88 , 014501 (2013), arXiv:1211.0929 [hep-lat].[9] P. Guo, Phys. Rev.
D88 , 014507 (2013), arXiv:1304.7812[hep-lat].[10] S. Kreuzer and H. W. Hammer, Phys. Lett.
B673 , 260(2009), arXiv:0811.0159 [nucl-th].[11] K. Polejaeva and A. Rusetsky, Eur. Phys. J.
A48 , 67(2012), arXiv:1203.1241 [hep-lat].[12] M. T. Hansen and S. R. Sharpe, Phys. Rev.
D90 , 116003(2014), arXiv:1408.5933 [hep-lat].[13] M. Mai and M. D¨oring, Eur. Phys. J.
A53 , 240 (2017),arXiv:1709.08222 [hep-lat].[14] M. Mai and M. D¨oring, Phys. Rev. Lett. , 062503(2019), arXiv:1807.04746 [hep-lat].[15] 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].[16] P. Guo, Phys. Rev.
D95 , 054508 (2017),arXiv:1607.03184 [hep-lat].[17] P. Guo and V. Gasparian, Phys. Lett.
B774 , 441 (2017), arXiv:1701.00438 [hep-lat].[18] P. Guo and V. Gasparian, Phys. Rev.
D97 , 014504(2018), arXiv:1709.08255 [hep-lat].[19] P. Guo and T. Morris, Phys. Rev.
D99 , 014501 (2019),arXiv:1808.07397 [hep-lat].[20] M. Mai, M. D¨oring, C. Culver, and A. Alexandru, Phys.Rev. D , 054510 (2020), arXiv:1909.05749 [hep-lat].[21] P. Guo, M. D¨oring, and A. P. Szczepaniak, Phys. Rev.