aa r X i v : . [ nu c l - t h ] J a n Coulomb corrections to nuclear reactions in artificial traps
Peng Guo
1, 2, ∗ Department of Physics and Engineering, California State University, Bakersfield, CA 93311, USA Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA (Dated: January 28, 2021)In present work, we discuss the effect of Coulomb interaction to the dynamics of two-particlesystem bound in various traps. The strategy of including Coulomb interaction into the quantizationcondition of trapped system is discussed in a general and non-perturbative manner. In most cases,Coulomb corrections to quantization condition largely rely on numerical approach or perturbationexpansion. Only for some special cases, such as the spherical hard wall trap, a closed-form ofquantization condition with all orders of Coulomb corrections can be obtained.
PACS numbers:Keywords:
I. INTRODUCTION
Recent advances in lattice quantum Chromodynamics(LQCD), ab initio nuclear many-body theory and devel-opments in computer technology have now made it pos-sible for the high precision computation of hadron andnuclei systems from the first principle. However, mostof these computations are performed in various traps,for instance, harmonic oscillator trap in nuclear physicsand periodic cubic box in LQCD. The typical observ-ables from these ab initio computations are discrete en-ergy spectrum of trapped systems. Therefore, extractingparticle interactions from discrete energy spectrum in thetrap and building connection between trapped dynamicsand infinite volume dynamics have became an importantsubject in both LQCD and nuclear physics communi-ties in recent years. In elastic two-particle sector, sucha connection between trapped system and infinite vol-ume system can be formulated in a closed form, such asL¨uscher formula [1] in a periodic cubic box in LCQD andBERW formula [2] in a harmonic oscillator trap in nu-clear physics community. Since then, L¨uscher and BERWformula have been quickly extended into both coupled-channel and few-body sectors, see e.g. Refs. [3–31]. BothL¨uscher and BERW formula have the form ofdet [cot δ ( E ) − M ( E )] = 0 , (1)where δ ( E ) refers to the diagonal matrix of scatteringphase shifts, and the analytic matrix function M ( E )is associated to the geometry and dynamics of trap it-self. L¨uscher and BERW formula as the matter of fact isthe result of the presence of two well separated physicalscales: (1) short-range interaction between two particlesand (2) size of trap. Hence the short-range dynamics thatis described by scattering phase shift and long-range cor-relation effect due to the trap can be factorized.The aim of present work is to extend such a relation toinclude long-range Coulomb interaction between charged ∗ Electronic address: [email protected] particles. Coulomb interaction becomes dominant forcharged particles interactions at low energy [32], includ-ing Coulomb interaction may be crucial for charged sys-tem interaction in a trap, see e.g. charge hadron systemin LQCD [33]. In fact, some early works on includingCoulomb corrections in finite volume has already beenpresented in Refs. [34, 35]. The discussion in Refs. [34, 35]was primarily based on effective perturbation field theoryapproach. It has been well known fact that both incom-ing plane wave and scattered spherical wave are distortedby long-range Coulomb interaction [36], ψ ( ∞ ) l ( r, q ) r →∞ ∼ sin( qr − π l + Zµq ln 2 qr ) qr + t l ( q ) e i ( qr − π l + Zµq ln 2 qr ) qr , (2)where Z is Coulomb interaction strength, and µ and q refers to the effective mass and incoming momen-tum of two-particle system. t l ( q ) is the partial wavescattering amplitude. Hence perturbation breaks downand Coulomb corrections must be dealt with non-perturbatively, see e.g. [32]. When it comes to formulat-ing L¨uscher and BERW formula in the presence of long-range Coulomb force, Coulomb propagator must be usedinstead of free particle propagator. In this work, we of-fer a general perspective for the formulating L¨uscher andBERW formula in presence of long-range Coulomb force.All the discussion are based on Lippmann-Schwinger (LS)equation approach, hence, the discussion can be madegeneral in non-perturbative way for various types of trap.However, except the hard-sphere wall trap, the analyticform of Green’s function in a trap is usually not avail-able, Dyson equation must be solved either numericallyor in terms of perturbation expansion.The paper is organized as follows. The derivation ofthe Coulomb corrections to the quantization condition oftrapped system is presented in Sec. II. The discussionsand summary are given in Sec. III. II. CONNECTING BOUND STATES IN A TRAPTO INFINITE VOLUME SCATTERING STATEWITH COULOMB FORCE
In this section, we present a general formalism on thetopic of bridging discrete bound state energy spectrumin a trap and infinite volume scattering dynamics in thepresence of Coulomb force. The commonly used trapsare periodic finite box in LQCD [33], harmonic poten-tial in nuclear physics [37–40], and spherical hard wallin some of the lattice implementations of chiral effectivefield theory [41, 42]. The brief discussion of formal scat-tering in presence of both a short-range interaction and along-range Coulomb interaction is given in Appendix A. a. Dynamics in a trap: the relative motion of twocharged spinless particles interacting with both Coulomband short-range interactions in a trap is described bySchr¨odinger equation h ε − ˆ H t − V C ( r ) i ψ ( t ) ε ( r ) = Z trap d r ′ V S ( r , r ′ ) ψ ( t ) ε ( r ′ ) , (3)where the Hamiltonian operator of the trap is given byˆ H t = ˆ H + V trap ( r ) , (4)with ˆ H = − ∇ r µ (5)and V trap ( r ) representing the free particle Hamiltonianoperator and trap potential respectively. µ stands for thereduced mass of two particles, and ε is energy of trappedparticles associated with relative motion. V C ( r ) = − Zr (6)and V S ( r , r ′ ) denote the Coulomb and short-range inter-actions between particles respectively. b. Dynamics in infinite volume: the dynamics oftwo charged interacting particles through the same short-range interaction V S ( r , r ′ ) in infinite volume is given by h ε ∞ − ˆ H − V C ( r ) i ψ ( ∞ ) ε ∞ ( r ) = Z ∞−∞ d r ′ V S ( r , r ′ ) ψ ( ∞ ) ε ∞ ( r ′ ) , (7)where ε ∞ stands for the relative motion energy of parti-cles in infinite volume. ε ∞ is related to ε in the trap bytotal energy conservation, ε ∞ + P M = ε + E ( t ) CM = E, (8)where P M and E ( t ) CM are the center of mass (CM) energyof system in infinite volume and in the trap respectively. c. Short-range interaction: the separable zero-range potential is assumed in follows for the derivationof quantization condition. In coordinate space, it has theform of, see Refs.[30, 31], V S ( r , r ′ ) = δ ( r ) δ ( r ′ )( rr ′ ) X lm V ( S ) l ( rr ′ ) l Y lm ( ˆr ) Y ∗ lm ( ˆr ′ ) . (9) A. Coulomb force modified dynamical equations ina trap
In the trap, the integral representation of Eq.(3) isgiven by ψ ( t ) ε ( r ) = Z trap d r ′′ G ( C,t ) ( r , r ′′ ; ε ) × Z trap d r ′ V S ( r ′′ , r ′ ) ψ ( t ) ε ( r ′ ) , (10)where G ( C,t ) ( r , r ′′ ; ε ) = h r | ε − ˆ H t − ˆ V C | r ′′ i (11)stands for the Coulomb Green’s function in a trap. TheCoulomb Green’s function G ( C,t ) satisfies Dyson equa-tion, G ( C,t ) ( r , r ′′ ; ε ) = G ( t ) ( r , r ′′ ; ε )+ Z trap d r ′ G ( t ) ( r , r ′ ; ε ) V C ( r ′ ) G ( C,t ) ( r ′ , r ′′ ; ε ) , (12)where G ( t ) ( r , r ′′ ; ε ) = h r | ε − ˆ H t | r ′′ i (13)is particle Green’s function in a trap. The partial waveexpansions ψ ( t ) ε ( r ) = X lm ψ ( t ) lm ( r ) Y lm ( ˆr ) (14)and G ( C,t ) ( r , r ′′ ; ε )= X lm,l ′′ m ′′ Y lm ( ˆr ) G ( C,t ) lm,l ′′ m ′′ ( r, r ′′ ; ε ) Y ∗ l ′′ m ′′ ( ˆr ′′ ) (15)yields ψ ( t ) lm ( r ) = X l ′ m ′ Z trap r ′′ dr ′′ G ( C,t ) lm,l ′ m ′ ( r, r ′′ ; ε ) × Z trap r ′ dr ′ V ( S ) l ′ ( r ′′ , r ′ ) ψ ( t ) l ′ m ′ ( r ′ ) . (16)With separable potential given in Eq.(9), the quantiza-tion condition that determines the discrete bound stateenergy spectrum of trapped system is thus given bydet " δ lm,l ′ m ′ V ( S ) l − G ( C,t ) lm,l ′ m ′ ( r, r ′ ; ε ) r l r ′ l ′ | r,r ′ → = 0 . (17)
1. Harmonic oscillator trap
In the harmonic oscillator (HO) trap with trap poten-tial: V trap ( r ) = 12 µω r , the rotational symmetry is still preserved, thus only di-agonal elements of partial wave Green’s function con-tribute. The partial wave Dyson equation for harmonictrap Green’s function in presence of Coulomb force isgiven by G ( C,ω ) l ( r, r ′ ; ε ) = G ( ω ) l ( r, r ′ ; ε ) − Z ∞ r ′′ dr ′′ G ( ω ) l ( r, r ′′ ; ε ) Zr ′′ G ( C,ω ) l ( r ′′ , r ′ ; ε ) , (18)where G ( ω ) l is the partial-wave HO Green’s function andis given in Refs. [31, 43] by G ( ω ) l ( r, r ′ ; ε ) = − ω ( rr ′ ) Γ( l + − ε ω )Γ( l + ) × M ε ω , l + ( µωr < ) W ε ω , l + ( µωr > ) . (19) M a,b ( z ) and W a,b ( z ) are the Whittaker functions as de-fined in Ref. [44].
2. Periodic cubic box
In finite volume, the trap potential is replaced by pe-riodic boundary condition. The rotational symmetry isbroken, and angular orbital momenta are no longer goodquantum numbers. In addition, the periodic boundarycondition is not satisfied by infinite volume Coulomb po-tential: V C ( r ) = − Zr . The infinite volume Coulomb po-tential is usually replaced by infrared singularity regular-ized periodic Coulomb potential, see Refs. [34, 35], V ( L ) C ( r ) = − L X p = π n L , n ∈ Z , n = πZ | p | e i p · r , (20)where L is size of cubic box, and V ( L ) C ( r + n L ) = V ( L ) C ( r ) , n ∈ Z . (21)In momentum space, Dyson equation in finite volume isgiven by e G ( C,L ) ( p , p ′ ; ε ) = L δ p , p ′ ε − p µ − ε − p µ L p ′′ = p X p ′′ = π n L , n ∈ Z πZ | p − p ′′ | e G ( C,L ) ( p ′′ , p ′ ; ε ) . (22)The finite volume Coulomb force modified Green’s func-tion in coordinate space is thus given by finite volumeFourier transform G ( C,L ) ( r , r ′ ; ε ) = 1 L n ∈ Z X p , p ′ ∈ π n L e i p · r e G ( C,L ) ( p , p ′ ; ε ) e − i p ′ · r ′ . (23)
3. Spherical hard wall
The hard-sphere boundary condition is accomplishedby the trap potential V trap ( r ) = ( , r < R , ∞ , r > R , (24)where R is the radius of the sphere. Hence, inside ofspherical hard wall: | r | < R , Coulomb force modifiedGreen’s function satisfies h ε − ˆ H − ˆ V C i G ( C,h.s. ) ( r , r ′ ; ε ) = δ ( r − r ′ ) , (25)which is just regular differential equation for CoulombGreen’s function except boundary condition. B. Coulomb force modified infinite volumedynamical equations
In infinite volume, the scattering solution of twocharged interacting particles in presence of Coulomb in-teraction is described by inhomogeneous LS equation, ψ ( ∞ ) ε ∞ ( r , q ) = ψ ( C, ∞ ) ε ∞ ( r , q )+ Z ∞−∞ d r ′′ G ( C, ∞ ) ( r , r ′′ ; q ) Z ∞−∞ d r ′ V S ( r ′′ , r ′ ) ψ ( ∞ ) ε ∞ ( r ′ , q ) , (26)where q is on-shell incoming momentum: q = p µε ∞ .ψ ( C, ∞ ) ε ∞ and G ( C, ∞ ) are Coulomb wave function andCoulomb Green’s function respectively. The partial waveexpansion ψ ( ∞ ) ε ∞ ( r , q ) = X lm Y ∗ lm ( ˆq ) ψ ( ∞ ) l ( r, q ) Y lm ( ˆr ) ,G ( C, ∞ ) ( r , r ′′ ; q ) = X lm Y lm ( ˆr ) G ( C, ∞ ) l ( r, r ′′ ; q ) Y ∗ lm ( ˆr ′′ ) , (27)and separable potential in Eq.(9) yield an algebra equa-tion ψ ( ∞ ) l ( r, q ) r l = ψ ( C, ∞ ) l ( r, q ) r l + V ( S ) l G ( C, ∞ ) l ( r, r ′′ ; q )( rr ′′ ) l ψ ( ∞ ) l ( r ′ , q ) r ′ l | r ′ ,r ′′ → . (28)
1. Coulomb wave function and Coulomb Green’s function
The analytic expression of ψ ( C, ∞ ) ε ∞ and G ( C, ∞ ) are givenin Refs. [36, 45] respectively by ψ ( C, ∞ ) l ( r, q ) = 4 π Γ( l + 1 + iγ )(2 l + 1)! e − π γ × (2 iqr ) l e iqr M ( l + 1 + iγ, L + 2 , − iqr ) , (29)and G ( C, ∞ ) l ( r, r ′′ ; q ) = 2 µ (2 iq ) Γ( l + 1 + iγ )(2 l + 1)! × ( − iqr < ) l e iqr < M ( l + 1 + iγ, l + 2 , − iqr < ) × ( − iqr > ) l e iqr > U ( l + 1 + iγ, l + 2 , − iqr > ) , (30)where M ( a, b, z ) and U ( a, b, z ) are two linearly indepen-dent Kummer functions [44], and γ = − Zµq . (31)For the convenience, let’s introduce two real functions: j ( C ) l ( γ, qr ) = C l ( γ )( qr ) l e iqr M ( l + 1 + iγ, l + 2 , − iqr ) , (32)and n ( C ) l ( γ, qr )= i ( − qr ) l e π γ e iqr U ( l + 1 + iγ, l + 2 , − iqr ) e iδ ( C ) l − i ( − qr ) l e π γ e − iqr U ( l + 1 − iγ, l + 2 , iqr ) e − iδ ( C ) l , (33)where the Sommerfeld factor and Coulomb phase shiftare defined in [36] by C l ( γ ) = 2 l | Γ( l + 1 + iγ ) | (2 l + 1)! e − π γ , (34)and e iδ ( C ) l = Γ( l + 1 + iγ )Γ( l + 1 − iγ ) . (35)Also using identity M ( l + 1 + iγ, l + 2 , − iqr )= − ( − l (2 l + 1)!Γ( l + 1 − iγ ) e πγ U ( l + 1 + iγ, l + 2 , − iqr ) − ( − l (2 l + 1)!Γ( l + 1 + iγ ) e πγ e − iqr U ( l + 1 − iγ, l + 2 , iqr ) , (36)the partial wave Coulomb wave function and CoulombGreen’s function can thus be rewritten as ψ ( C, ∞ ) l ( r, q ) = 4 πi l j ( C ) l ( γ, qr ) e iδ ( C ) l , (37)and G ( C, ∞ ) l ( r, r ′′ ; q ) = − i µqj ( C ) l ( γ, qr < ) h ( C, +) l ( γ, qr > ) , (38)where h ( C, ± ) l ( γ, qr ) = j ( C ) l ( γ, qr ) ± in ( C ) l ( γ, qr )= − − qr ) l e π γ e ± iqr U ( l + 1 ± iγ, l + 2 , ∓ iqr ) e ± iδ ( C ) l . (39) The Coulomb Green’s function in Eq.(38) thus resemblethe free particle Green’s function, G (0 , ∞ ) l ( r, r ′′ ; q ) = − i µqj l ( qr < ) h (+) l ( qr > ) , (40)where j l and h (+) l are regular spherical Bessel and Hankelfunctions.
2. Coulomb force modified scattering amplitudes
In presence of Coulomb force, the total scattering am-plitude now is composed of two components: (1) theshort-range interaction scattering amplitude modified byCoulomb interaction and (2) the pure Coulomb scatter-ing amplitude. a. Coulomb force modified short-range interactionscattering amplitude: the short-range interaction scat-tering amplitude can be defined by the solution ofEq.(28), ψ ( ∞ ) l ( r, q ) = 4 πi l (cid:20) j ( C ) l ( γ, qr ) e iδ ( C ) l + it ( SC ) l ( q ) h ( C, +) l ( γ, qr ) e − iδ ( C ) l (cid:21) , (41)where t ( SC ) l ( q ) is the Coulomb force modified short-rangeinteraction scattering amplitude and is given by t ( SC ) l ( q ) = − µq (cid:18) j ( C ) l ( γ,qr ) r L | r → (cid:19) V ( S ) l − G ( C, ∞ ) l ( r ′ ,r ′′ ; q )( r ′ r ′′ ) l | r ′ ,r ′′ → e iδ ( C ) l . (42)The t ( SC ) l ( q ) is typically parameterized by both Coulombphase shift δ ( C ) l and a short-range scattering phase shift δ ( S ) l , t ( SC ) l ( q ) = 1cot δ ( S ) l − i e iδ ( C ) l . (43)Using the asymptotic form: j ( C ) l ( γ, qr ) r l | r → = q l C l ( γ ) ,Im " G ( C, ∞ ) l ( r ′ , r ′′ ; q )( r ′ r ′′ ) l | r ′ ,r ′′ → = − µq l +1 C l ( γ ) , (44)and Eq.(42) and Eq.(43), one thus find1 V ( S ) l = − µq l +1 C l ( γ ) cot δ ( S ) l ( q )+ Re " G ( C, ∞ ) l ( r ′ , r ′′ ; q )( r ′ r ′′ ) l | r ′ ,r ′′ → . (45) b. Pure Coulomb scattering amplitude: the pureCoulomb scattering amplitude is defined by Coulombwave function. By introducing h ( C, ± ) l ( γ, qr ) = e ± iδ ( C ) l H ( C, ± ) l ( γ, qr ) , (46)where H ( C, ± ) l ( γ, qr ) = J ( C ) l ( γ, qr ) ± iN ( C ) l ( γ, qr )= − − qr ) l e π γ e ± iqr U ( l + 1 ± iγ, l + 2 , ∓ iqr ) , (47)we can thus rewrite the Coulomb wave function inEq.(37) to ψ ( C, ∞ ) l ( r, q ) = 4 πi l h J ( C ) l ( γ, qr ) + it ( C ) l ( q ) H ( C, +) l ( γ, qr ) i , (48)where t ( C ) l ( q ) is the pure Coulomb scattering amplitude: t ( C ) l ( q ) = e iδ ( C ) l − i . (49) c. Total scattering amplitude: the total wave func-tion in Eq.(41) is now also given by ψ ( ∞ ) l ( r, q ) = 4 πi l (cid:20) J ( C ) l ( γ, qr ) + it l ( q ) H ( C, +) l ( γ, qr ) (cid:21) , (50)where t l ( q ) = t ( C ) l ( q ) + t ( SC ) l ( q ) = e iδ ( C ) l e iδ ( S ) l − i . (51) d. Asymptotic forms of wave functions: usingasymptotic form of H ( C, ± ) l functions, H ( C, ± ) l ( γ, qr ) r →∞ → h ( ± ) l ( qr ) e ∓ iγ ln 2 qr , (52)one can easily illustrate that ψ ( C, ∞ ) l ( r, q ) r →∞ → πi l (cid:20) sin( qr − π l − γ ln 2 qr ) qr + it ( C ) l ( q ) h (+) l ( qr ) e − iγ ln 2 qr (cid:21) , (53)and ψ ( ∞ ) l ( r, q ) r →∞ → πi l (cid:20) sin( qr − π l − γ ln 2 qr ) qr + it l ( q ) h (+) l ( qr ) e − iγ ln 2 qr (cid:21) , (54)where the factor γ ln 2 qr represents the long-rangeCoulomb distortion effect to the incoming plane waveand outgoing spherical wave. C. Quantization condition in a trap in presence ofCoulomb interaction
Combining Eq.(17) and Eq.(45) by eliminating V ( S ) l ,one thus find Coulomb force modified L¨uscher formula-like relation,det h δ lm,l ′ m ′ cot δ ( S ) l ( q ) − M ( C,t ) lm,l ′ m ′ ( ε ) i = 0 , (55)where M ( C,t ) is generalized zeta function in presence ofCoulomb force, M ( C,t ) lm,l ′ m ′ ( ε ) = − µq l +1 C l ( γ ) G ( C,t ) lm,l ′ m ′ ( r, r ′ ; ε ) r l r ′ l ′ | r,r ′ → + δ lm,l ′ m ′ µq l +1 C l ( γ ) Re (cid:16) G ( C, ∞ ) l ( r, r ′ ; q ) (cid:17) ( rr ′ ) l | r,r ′ → . (56)Both G ( C,t ) lm,l ′ m ′ and G ( C, ∞ ) l are ultraviolet divergent, andafter cancellation between two terms, generalized zetafunction is finite and well-defined. At the limit of γ → C l ( γ ) γ → → C l (0) = √ π l +1 Γ( l + ) = j l ( qr )( qr ) l | r → , (57)and Re (cid:16) G ( C, ∞ ) l ( r, r ′′ ; q ) (cid:17) ( rr ′′ ) l | r,r ′′ → γ → → µq j l ( qr ) n l ( qr ) r l | r → , (58)hence, Eq.(56) is reduced to Eq.(B32) in Ref. [30]. III. DISCUSSION AND SUMMARYA. Perturbation expansion
The key element of generalized zeta function in Eq.(56)is Coulomb interaction modified Green’s function in atrap, which is given by Dyson equation, Eq.(12). SolvingDyson equation in most cases is not an easy task, greateffort must to be made to deal with both ultraviolet di-vergence (UV) and infrared divergence (IR) caused byCoulomb interaction. Therefore the perturbation expan-sion may be more practical in general, see discussion in[34, 35]. Symbolically, the Coulomb force modified zetafunction is a real function and given byˆ M C,t ∼ C ( γ ) h ˆ G C,t − Re (cid:16) ˆ G C, ∞ (cid:17)i , (59)where the solution of ˆ G C,t is given byˆ G C,t = ˆ G t − ˆ V C ˆ G t = ∞ X n =0 ˆ G t (cid:16) ˆ V C ˆ G t (cid:17) n , (60)and ˆ G t denotes Green’s function in a trap,ˆ G t ( E ) = 1 E − ˆ H t . (61)Although the analytic expression of infinite volumeCoulomb Green’s function ˆ G C, ∞ is known already, in or-der to make sure the UV and IR divergences cancelledout properly order by order, ˆ G C, ∞ can be expanded interms of perturbation theory as wellˆ G C, ∞ = ˆ G , ∞ − ˆ V C ˆ G , ∞ = ∞ X n =0 ˆ G , ∞ (cid:16) ˆ V C ˆ G , ∞ (cid:17) n , (62)where ˆ G , ∞ ( E ) = 1 E − ˆ H . (63)Hence, Coulomb corrected zeta function may be com-puted by perturbation expansion systematically, C ( γ ) ˆ M C ∼ ∞ X n =0 h ˆ G t (cid:16) ˆ V C ˆ G t (cid:17) n − Re (cid:16) ˆ G , ∞ (cid:16) ˆ V C ˆ G , ∞ (cid:17) n (cid:17)i . (64)Perturbation expansion may be well applied to both HOtrap and periodic cubic box, also see [34, 35] for the dis-cussion in finite volume from effective field theory per-spective. a. HO trap: for the harmonic oscillator trap, iter-ating Dyson equation in Eq.(18) once, the leading orderand first order perturbation result can be written downformally by, C l ( γ ) C l (0) M ( C, th ) l ( ε ) = ( − l +1 (cid:18) µωq (cid:19) l + Γ( l + − ε ω )Γ( − l − ε ω ) , (65)and C l ( γ ) C l (0) M ( C, st ) l ( ε )= − l +2 Γ ( l + )2 µq l +1 π △ G ( C, st ) l ( r, r ′ ; ε )( rr ′ ) l | r,r ′ → , (66)where △ G ( C, st ) l ( r, r ′ ; ε )= − Z ∞ r ′′ dr ′′ G ( ω ) l ( r, r ′′ ; ε ) Zr ′′ G ( ω ) l ( r ′′ , r ′ ; ε )+ Re Z ∞ r ′′ dr ′′ G (0 , ∞ ) l ( r, r ′′ ; q ) Zr ′′ G (0 , ∞ ) l ( r ′′ , r ′ ; q ) . (67) b. Periodic cubic box: similarly, in finite volume,the leading order and first order perturbation result aregiven by C l ( γ ) C l (0) M ( C, th ) lm,l ′ m ′ ( ε )= − L n ∈ Z X p ∈ π n L p l + l ′ q l +1 Y lm ( ˆp ) Y ∗ l ′ m ′ ( ˆp )2 µε − p − δ lm,l ′ m ′ l +1 Γ( l + )Γ( l + ) π qr ) l +1 | r → , (68)and C l ( γ ) C l (0) M ( C, st ) lm,l ′ m ′ ( ε )= 12 µq l +1 L n ∈ Z X p , p ′ ∈ π n L p l Y lm ( ˆp ) ε − p µ πZ | p − p ′ | p ′ l ′ Y ∗ l ′ m ′ ( ˆp ′ ) ε − p ′ µ − δ lm,l ′ m ′ µq l +1 Re Z d p d p ′ (2 π ) p l Y lm ( ˆp ) ε − p µ πZ | p − p ′ | p ′ l Y ∗ lm ( ˆp ′ ) ε − p ′ µ . (69) B. Analytic solutions in a spherical hard wall trap
The rotational symmetry inside of a hard-sphere trapis also well-preserved, so angular orbital momentum isstill a good quantum number, only diagonal elementsof Coulomb Green’s function contribute. The par-tial wave Coulomb Green’s function inside hard-spheremust be the combination of regular and irregular func-tions of Coulomb differential equation: j ( C ) l ( γ, qr ) and n ( C ) l ( γ, qr ) defined in Eq.(32) and Eq.(33) respectively.Similar to the hard-sphere trap without Coulomb in-teraction, see Eq.(54) in Ref.[31], the closed form ofCoulomb force modified Green’s function inside hard-sphere is given by G ( C,h.s. ) l ( r, r ′′ ; ε ) = − µqj ( C ) l ( γ, qr < ) j ( C ) l ( γ, qr > ) × " n ( C ) l ( γ, qR ) j ( C ) l ( γ, qR ) − n ( C ) l ( γ, qr > ) j ( C ) l ( γ, qr > ) . (70)The real part of Coulomb Green’s function in infinitevolume is given by Re (cid:16) G ( C, ∞ ) l ( r, r ′′ ; ε ) (cid:17) = 2 µqj ( C ) l ( γ, qr < ) n ( C ) l ( γ, qr > ) . (71)Hence, after UV cancellation in Eq.(56), the analytic ex-pression of Coulomb force modified generalized zeta func-tion for hard-sphere trap is obtained M ( C,h.s. ) lm,l ′ m ′ ( ε ) = δ lm,l ′ m ′ C l (0) C l ( γ ) n ( C ) l ( γ, qR ) j ( C ) l ( γ, qR ) . (72)The quantization condition in a hard-sphere trap in pres-ence of Coulomb interaction is given in a closed-form:cot δ ( S ) l ( q ) = C l (0) C l ( γ ) n ( C ) l ( γ, qR ) j ( C ) l ( γ, qR ) , (73)where j ( C ) l ( γ, qr ) and n ( C ) l ( γ, qr ) are defined in Eq.(32)and Eq.(33) respectively. C. Summary
In summary, we present a general discussion onthe topic of formulating quantization condition oftrapped systems by including long-range Coulomb inter-action. Although all the discussion are based on non-perturbative LS equation approach, in most cases, theCoulomb force modified Green’s function in a trap mustbe solved either numerically or by perturbation expan-sion. In special cases, such as the spherical hard walltrap, the closed-form of quantization condition is ob-tained and given in Eq.(73).
Acknowledgments
We thank fruitful discussion with Bingwei Long. P.G.also acknowledges support from the Department ofPhysics and Engineering, California State University,Bakersfield, CA. The work was supported in part bythe National Science Foundation under Grant No. NSFPHY-1748958.
Appendix A: Formal scattering theory withshort-range and Coulomb interactions
In this section, the formal scattering theory in pres-ence of both a short-range interaction and a long-rangeCoulomb interaction is briefly discussed, the completediscussion can be found in Refs. [46, 47]. The connectionto trapped system is also briefly discussed symbolically.
1. Coulomb force modified scattering amplitude ininfinite volume
The infinite volume scattering amplitude in the pres-ence of both Coulomb and short-range nuclear interac-tions is defined by T ∞ = −h Ψ | ( ˆ V C + ˆ V S ) | Ψ (+) i , (A1)where | Ψ i stands for plane wave. The | Ψ (+) i is definedby LS equation, | Ψ ( ± ) i = | Ψ i + ˆ G ( E ± i V C + ˆ V S ) | Ψ ( ± ) i , (A2) where ˆ G ( E ± i
0) = 1 E − ˆ H ± i . (A3)Using Eq.(A2) and also LS equation for pure Coulombinteraction, h Ψ | = h Ψ ( − ) C | − h Ψ ( − ) C | ˆ V C ˆ G ( E + i , (A4)the total infinite volume scattering amplitude T ∞ can berewritten as, also see Refs. [46, 47], T ∞ = −h Ψ ( − ) C | ˆ V C | Ψ i − h Ψ ( − ) C | ˆ V S | Ψ (+) i . (A5)The first term in Eq.(A5) is identified as pure Coulombinteraction scattering amplitude, T C, ∞ = −h Ψ ( − ) C | ˆ V C | Ψ i = −h Ψ | ˆ V C | Ψ (+) C i . (A6)The partial wave coulomb amplitude is parameterized byCoulomb phase shifts, T ( C, ∞ ) l ∝ e iδ ( C ) l − i , where the Coulomb phase shift δ ( C ) l is defined in Eq.(35).The second term in Eq.(A5) is the result of short-rangeinteraction in the presence of Coulomb interaction, usingLS equation | Ψ (+) i = | Ψ (+) C i + ˆ G C ( E + i V S | Ψ (+) i , (A7)where ˆ G C ( E ± i
0) = 1 E − ˆ H − ˆ V C ± i , (A8)it can be shown rather straight-forwardly that secondterm satisfies equation − h Ψ ( − ) C | ˆ V S | Ψ (+) i = −h Ψ ( − ) C | ˆ V S | Ψ (+) C i − h Ψ ( − ) C | ˆ V S ˆ G C ( E + i
0) ˆ V S | Ψ (+) i . (A9)Hence, it may be useful and more convenient to define aCoulomb force modified scattering operatorˆ T SC, ∞ | Ψ (+) C i = − ˆ V S | Ψ (+) i , (A10)thus, the second term in Eq.(A5) now can be written as − h Ψ ( − ) C | ˆ V S | Ψ (+) i = h Ψ ( − ) C | ˆ T SC, ∞ | Ψ (+) C i . (A11)According to Eq.(A9), ˆ T SC, ∞ satisfies operator equationˆ T SC, ∞ = − ˆ V S + ˆ V S ˆ G C ( E + i
0) ˆ T SC, ∞ . (A12)The total scattering amplitude is now given by T ∞ = T C, ∞ + h Ψ ( − ) C | ˆ T SC, ∞ | Ψ (+) C i . (A13)Given the fact that the symbolic solution of ˆ T SC, ∞ oper-ator is given byˆ T − SC, ∞ = − ˆ V − S + ˆ G C ( E + i , (A14)and h Ψ ( − ) C | Ψ (+) C i = 1 + 2 iT C, ∞ = S C, ∞ ∝ e iδ ( C ) l (A15)is the pure Coulomb interaction S -matrix, the partialwave expansion of h Ψ ( − ) C | ˆ T SC, ∞ | Ψ (+) C i is conventionallyparameterized by both Coulomb phase shift δ ( C ) l and theshort-range interaction phase shift, δ ( S ) l , see Refs. [46,47], h Ψ ( − ) C | ˆ T SC, ∞ | Ψ (+) C i ∝ e iδ ( C ) l e iδ ( S ) l − i . (A16)Therefore, the partial wave total infinite volume scatter-ing amplitude is thus defined by a total phase shift, δ l = δ ( S ) l + δ ( C ) l , and T ( ∞ ) l ∝ e iδ l − i = e iδ ( C ) l − i + e iδ ( C ) l e iδ ( S ) l − i . (A17)
2. Charged particles in a trap in presence ofCoulomb force
In the trap, the Eq.(A12) is now modified toˆ T SC,t = − ˆ V S + ˆ V S ˆ G C,t ( E + i
0) ˆ T SC,t , (A18)where ˆ G C,t ( E ± i
0) = 1 E − ˆ H t − ˆ V C ± i , (A19) is Coulomb Green’s function in the trap, andˆ H t = ˆ H + ˆ V t is trap Hamiltonian operator. The quantization condi-tion including Coulomb interaction thus is given bydet (cid:2) V − S − G C,t ( E + i (cid:3) = 0 . (A20)
3. Quantization condition including Coulombinteraction
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