Charged Particles Interaction in Both a Finite Volume and a Uniform Magnetic Field

PHYSICAL REVIEW D 103, 094520 (2021)

Charged particles interaction in both a finite volume and a uniform magnetic field

Peng Guo 1,2,* and Vladimir Gasparian 1 1Department of Physics and Engineering, California State University, Bakersfield, California 93311, USA 2Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA

(Received 5 January 2021; accepted 10 May 2021; published 27 May 2021)

A formalism for describing charged particles interaction in both a finite volume and a uniform magnetic field is presented. In the case of short-range interaction between charged particles, we show that the factorization between short-range physics and finite volume long-range correlation effect is possible; a Lüscher formulalike quantization condition is thus obtained.

DOI: 10.1103/PhysRevD.103.094520

I. INTRODUCTION which provides a relation between energy levels in a trap and the infinite volume scattering phase shift. In finite In recent years, a great effort in the nuclear and hadron volume in LQCD computation, such a relation in the elastic physics community has been put into constructing the two-body sector is known as the Lüscher formula [1], scattering dynamics of few-particle interactions from the which shows a clear factorization of short-range dynamics discrete bound state energy spectrum that is computed in and long-range correlation effects because of the periodic various types of traps, such as the commonly used periodic boundary condition. The short-range dynamics and long- finite box in lattice QCD (LQCD) and the harmonic range correlations are described by the physical scattering oscillator trap in nuclear physics computation. The ultimate phase shift and Lüscher’s zeta function respectively. goal is of course to study and explore the nature of particle Lüscher’s formula has been proving very successful interactions that plays an essential role in many fields of in the LQCD community, and it has been quickly extended physical science, such as nuclear physics and astrophysics. into both coupled-channel and few-body sectors, However, the current state-of-art ab initio computations in see [2–41]. In nuclear physics where a harmonic oscillator nuclear and hadron physics are normally performed in a trap is commonly used, such a relation is given by the harmonic oscillator trap and in a finite volume respectively. Busch-Englert-Rzażewski-Wilkens (BERW) formula [42– Instead of computing few-body scattering amplitudes, the 53]. In addition to the periodic boundary condition and discrete bound state energy levels are usually directly harmonic trap, other types of traps or boundary conditions measured and extracted from these ab initio computations. are also commonly used in different physics fields, such as Therefore, finding a relation that converts a discrete bound the hard wall trap [54,55]. Regardless of the difference state energy spectrum into a continuum scattering state is a among various traps, the same strategy is shared: as the two key step. physical scales are clearly separated, a closed asymptotic In fact, relating the energy shift caused by particle form can be found, in which short-range dynamics is interactions to the on-shell scattering parameters such as described by a scattering phase shift and the long-range phase shift has a long history across many fields in physics. effect is given by an analytic form that describes how the In general cases, the dynamics of particles interaction in propagation of particles is affected by the trap, e.g., traps is associated with the infinite volume off-shell Lüscher’s zeta function in a periodic boundary condition. reaction amplitudes in a highly nontrivial way. In the present work, we aim to establish a similar relation Fortunately, when the separation of two physics scales, to Lüscher’s and the BERW formula for the charged the size of trap and the range of particles interaction, is particles interacting in both a uniform magnetic field and clearly established, a simple asymptotic form can be found, a periodic box. We remark that only a short-range interaction which represents nuclear force or hadron interactions *[email protected] is considered in this work; the Coulomb interaction has not been incorporated in the current framework yet. We also Published by the American Physical Society under the terms of emphasize that the Coulomb repulsion may be important the Creative Commons Attribution 4.0 International license. – Further distribution of this work must maintain attribution to near threshold [56 67], especially since the long-range the author(s) and the published article’s title, journal citation, nature of the Coulomb interaction may complicate the and DOI. Funded by SCOAP3. factorization of physics at different scales and distort the

2470-0010=2021=103(9)=094520(18) 094520-1 Published by the American Physical Society PENG GUO and VLADIMIR GASPARIAN PHYS. REV. D 103, 094520 (2021) asymptotic wave functions, the Coulomb interaction must ð∇ þ iqAðrÞÞ2 Hˆ ¼ − r ; ð Þ be included in future work. We will show that with only a r 2μ 3 short-range potential, the factorization of short-range physics and the long-range correlation effect is possible. Hence where μ and q denote the reduced mass and charge of two a relation in a compact form that relates the discrete energy charged particles respectively. spectrum to scattering phase shifts can be found. Such a relation may be useful for the study of a charged hadron A. Magnetic periodic boundary condition πþ system such as the system in LQCD computation. In In a periodic finite box, the discrete translation invari- finite volume, in order to preserve translation symmetry of ance is broken by a coordinate dependent vector potential, a system in a magnetic field, the magnetic flux through a perpendicular surface of a cubic box to a uniform magnetic Ayðr þ nLÞ¼AyðrÞþBnL · ex: field must be 2π multiplied by a rational number np=nq, where np and nq are integers and relatively prime to each In order to maintain the discrete translation symmetry in a other. Therefore, the original energy level without a uniform magnetic field, the conventional translation oper- magnetic field is split into nq sublevels due to the ator must be generalized to the magnetic translation application of the magnetic field. We also remark that operator, the ultimate goal of the current work is to set up a ˆ ið−i∇rþqAðrÞ−qB×rÞ·nL foundation for exploring the possibility of the topological TrðnLÞ¼e : ð4Þ edge type states [68–70] in lattice QCD in future work. However, by using background-field methods in lattice Hence QCD [71–74], the finite volume energy levels of particles ˆ iqBryex·nL interacting in a magnetic field background may also be TrðnLÞψ εðrÞ¼e ψ εðr þ nLÞ: ð5Þ used to determine the coefficient of the leading local four- nucleon operator contributing to the neutral- and charged- In addition, to warrant a state remain translational current breakup of the deuteron. invariant through a closed path, the magnetic flux through The paper is organized as follows. The general formal- an enlarged closed path in an x-y plane that is defined by ism of charged bosons interaction in both a finite volume magnetic unit vectors and a uniform magnetic field is presented in details in Sec. II. The S-wave contribution and regularization of nqLex × Ley × Lez; ultraviolet divergence are discussed in Sec. III. A summary is given in Sec. IV. must be quantized, see Refs. [75,76],

2 qBnqL ¼ 2πnp; ð6Þ II. FINITE VOLUME DYNAMICS OF CHARGED BOSONS IN A UNIFORM MAGNETIC FIELD where np and nq are two relatively prime integers. In this section, we briefly summarize the dynamics of Therefore, the discrete translation in a enlarged magnetic ˆ charged bosons interacting in both a finite periodic box and unit box leaves the Hamiltonian invariant, and TrðnBLÞ a uniform magnetic field. The uniform magnetic field is forms a magnetic translation group, where chosen along the z axis, B ¼ Bez, and the Landau gauge for the vector potential is adopted in this work, nB ¼ nxnqex þ nyey þ nzez;nx;y;z ∈ Z: ð7Þ

According to Bloch’s theorem, in a periodic box, AðxÞ¼Bð0;x;0Þ: ð1Þ ˆ periodicity of a system requires that TrðnBLÞψ εðrÞ can only differ from ψ εðrÞ by a phase factor, which can be The complete presentation and more rigorous discussion chosen as are given in the Appendixes A and B. The dynamics of the relative motion of two charged iPB n L e 2 · B ; identical nonrelativistic spinless particles in a uniform magnetic field is described by the Schrödinger equation, where ðHˆ þ VðrÞÞψ ðrÞ¼εψ ðrÞ; ð Þ r ε ε 2 2π nx PB ¼ ex þ nyey þ nzez ;nx;y;z ∈ Z: ð8Þ L nq where ψ εðrÞ and ε are the wave function and energy for the relative motion of a two charged boson system. The Hence the magnetic periodic boundary condition is ˆ Hamiltonian operator Hr is defined by given by

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PB i ·nBL −iqBryex·nBL L ðxÞ ψ εðr þ nBLÞ¼e 2 e ψ εðrÞ: ð9Þ where n is the Laguerre polynomial, and

0 0 0 The magnetic periodic boundary condition can also be ρ ¼ rxex þ ryey; ρ ¼ rxex þ ryey obtained by considering a separable form of a total wave function, see Appendix A2. are relative coordinates defined in the x-y plane.

B. Finite volume Lippmann-Schwinger equation and 2. Quantization condition with short-range interaction quantization condition The discrete bound state energy spectrum can be found as the eigen-energy solutions of the homogeneous LS 1. Finite volume Lippmann-Schwinger equation equation in Eq. (10). The partial wave expansion in an The Schrödinger equation and magnetic periodic boun- angular momentum basis is commonly used in describing dary condition in Eq. (2) and Eq. (9) together can be an infinite volume scattering state. However in a magnetic replaced by the finite volume homogeneous Lippmann- field, due to the asymmetry of the magnetic Hamiltonian in Schwinger (LS) equation, x-y plane and along the z axis, the angular momentum basis Z in spherical coordinates is in fact not the most convenient 0 ðLÞ 0 0 0 basis in describing dynamics of charged particles in a ψ εðrÞ¼ dr GB ðr; r ; εÞVðr Þψ εðr Þ; ð10Þ 3 uniform magnetic field. Nevertheless, it can be done in LB principle. For the sake of the consistency of presentation in where the volume integration over the magnetic unit cell is both finite volume and infinite volume dynamics, let us defined by consider the partial wave expansion of Eq. (10), which yields Z Z nqL Z L Z L 2 2 2 Z dr0 ¼ dr0 dr0 dr0 : ð Þ X n L x y z 11 ðLÞ 02 0 ðB;LÞ 0 0 ðLÞ 0 L3 − q −L −L ψ ðrÞ¼ r dr G ðr; r ; εÞV 0 ðr Þψ ðr Þ: B 2 2 2 lm lm;l0m0 l l0m0 L3 l0m0 B ’ GðLÞ The finite volume magnetic Green s function B satisfies ð15Þ the dynamical equation The purpose of this work is to find a Lüscher formulalike ˆ ðLÞ 0 ðε − HrÞGB ðr; r ; εÞ simple relation that connects short-range physics associated X PB VðrÞ −i ·nBL iqBryex·nBL 0 with particles interaction and the long-range effect ¼ e 2 e δðr − r þ nBLÞ: ð12Þ n generated by the finite volume and magnetic field. Also B consider the fact that such a relation is the result of clear The solution of a finite volume magnetic Green’s separation of two physical scales: (1) the range of potential ðLÞ VðrÞ and (2) the size of a trap or finite volume. When the function GB can be constructed from its infinite volume Gð∞Þ two scales are clearly separated, the short- and long-range counterpart B by physics can be factorized, and a compact relation as the

ðLÞ 0 leading order contribution can be found by studying the GB ðr; r ; εÞ X asymptotic behavior of the wave function [8,24]. Therefore, ð∞Þ PB 0 0 i ·nBL −iqBryex·nBL for our purpose, it is sufficient to consider the zero-range ¼ GB ðr; r þ nBL; εÞe 2 e n potential, XB −iPB n L iqBr e n L ð∞Þ 0 2lþ1 2 3 ¼ e 2 · B e y x· B G ðr þ n L; r ; εÞ: ð Þ δðrÞ 2 Γ ðl þ Þ B B 13 V ðrÞ → V 2 ; ð Þ l l 2 3 2l 16 nB r ð2πÞ r

Details of the construction can be found in Appendix A4. see Appendix B for a more rigorous discussion. The analytic expression of an infinite volume magnetic Equation (15) is thus turned into an algebraic equation, ð∞Þ Green’s function G is given by B ðLÞ X 2l0þ1 2 0 3 ψ lm ðrÞ 2 Γ ðl þ 2Þ ¼ Vl0 ð∞Þ 2μqB iqB 0 0 qB 0 2 l 3 0 − ðrxþrxÞðry−ryÞ − jρ−ρ j r 0 0 ð2πÞ GB ðr;r ;εÞ¼− e 2 e 4 l m 2π pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi GðB;LÞ ðr; r0; εÞ ðLÞ 0 ∞ qB 0 2 i 2με−2qBðnþ1Þjr −r0 j lm;l0m0 ψ 0 0 ðr Þ XiL ð jρ − ρ j Þe 2 z z l m : ð Þ n 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi × l 0l0 0l0 17 × ; r r r r0→0 n¼0 1 2 2με − 2qBðn þ 2Þ Hence the quantization condition of the discrete energy ð Þ 14 spectrum is given by

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2 0 3 ðB;LÞ 0 δ 0 0 Γ ðl þ ÞG 0 0 ðr;r ;εÞ III. S-WAVE CONTRIBUTION AND CONTACT lm;l m − 2 lm;l m ¼ 0: ð Þ det 2lþ1 3 l 0l0 18 INTERACTION 2 Vl ð2πÞ r r r;r0→0 As was already mentioned in the previous section, the angular momentum basis in general is not a convenient Under the same assumption of the zero-range approxi- basis for the dynamics of charged particles in a uniform V mation given in Eq. (16), the potential strength l is related magnetic field. The partial wave expansion of the finite to the infinite volume two-body scattering phase shift volume magnetic Green’s function and ultraviolet regu- δ ðk Þ l ε by larization can be tedious in general. Fortunately, if only the S -wave contribution is dominant, the formalism can be ð4πÞ2 22lþ1Γðl þ 1ÞΓðl þ 3Þ worked out nicely. In this section, only a contact interaction þ 2 2 ¼ −k2lþ1 δ ðk Þ; 2lþ1 ε cot l ε potential 2μVl πr r→0 ð Þ V 19 VðrÞ¼ 0 δðrÞðÞ 4π 23 see detailed discussions in Appendix B. The relative is used, which may be considered as the leading order k momentum ε in infinite volume is related to the relative contribution of chiral effective field theory and may be ε þ finite volume energy by suitable for the few-body system, such as π interactions in finite volume. k2 QB 1 P2 þ P2 With a contact interaction, the finite volume quantization ε ¼ ε þ △E ; △E ¼ n þ − x y ; 2μ R R M 2 2M condition is simply given by

ð20Þ 4π ðLÞ ¼ GB ð0; 0; εÞ: ð24Þ V0 where △ER is the result of quantization of center-of-mass In infinite volume, V0 is related to the S-wave scattering (CM) motion in a uniform magnetic field. amplitude by Eliminating Vl, Eqs. (18) and (19) together yield a Lüscher formulalike simple relation, 4π ð∞Þ 2μkε 1 − G ð0; kεÞ¼− ; ð25Þ V0 4π t0ðkεÞ ðB;LÞ det ½δlm;l0m0 cot δlðkεÞ − M 0 0 ðεÞ ¼ 0; ð21Þ lm;l m where 1 where t0ðkεÞ¼ ; cot δ0ðkεÞ − i

ð∞Þ 2l0þ3 2 0 3 ðB;LÞ 0 ’ G ð0; k Þ 2 Γ ðl þ Þ G 0 0 ðr; r ; εÞ and infinite volume Green s function ε is MðB;LÞ ðεÞ¼− 2 lm;l m lm;l0m0 2lþ1 l 0l0 given by 2μkε ð2πÞ r r r;r0→0 Z 2lþ1 1 3 2 Γðl þ ÞΓðl þ Þ 1 dp 1 2μkε 2μ 2 2 Gð∞Þð0;k Þ¼ ¼ −i − : ð Þ − δlm;l0m0 : ε 3 2 2 26 π 2lþ1 ð2πÞ kε p 4π 4πr ðkεrÞ r→0 2μ − 2μ r→0 ð22Þ Thus, the quantization condition is simply given by

ðB;LÞ 4π ðLÞ 1 The second term in M 0 0 plays the role of the regulator δ ðk Þ¼− G ð0; 0; εÞ − : ð Þ lm;l m cot 0 ε 2μk B k r 27 of ultraviolet (UV) divergence and will cancel out the UV ε ε r→0 divergence in the finite volume magnetic Green’s function, ðLÞ The magnetic Green’s function GB ð0; 0; εÞ is a real MðB;LÞ so that lm;l0m0 is ultimately free of UV divergence. In function of ε. The UV divergent term general, the regularization and isolation of UV divergence ’ 1 in higher partial waves of a finite volume magnetic Green s − function is a highly nontrivial task. Fortunately, it can be kεr r→0 accomplished rather neatly for S-wave, hence, only the S-wave contribution will be considered in Sec. III. The plays the role of the UV counterterm that cancels out the ðLÞ regularization of UV divergence will be worked out UV divergent term in GB ð0; 0; εÞ, so the ultimate result is explicitly. finite and real as a function of ε.

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X∞ X 2 A. Regularization of UV divergence ðLÞ 2μqB −iqBnxnynqL △GB ðεÞ¼ e 2 In this section, we show explicitly how the UV diver- 4π ðLÞ n¼0 nx;ny∈Z gence in GB ð0; 0; εÞ is regularized and isolated out −qBjn n Le þn Le j2 qB 2 explicitly. The UV divergence only appears when e 4 x q x y y L jn n Le þn Le j × n 2 x q x y y 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 r ¼ðρ; 0Þ → 0; 2με−2qBðnþ1ÞL 2qBðnþ1ÞL 6 cot 2 coth 2 7 ×4qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 5: 2με−2qBðnþ 1Þ 2qBðnþ 1Þ hence, a small r ¼ ρ is used as UV regulator. In the end, a 2 2 final expression is obtained by taking the limit of ð Þ r ¼ ρ → 0. Starting with an explicit expression of magnetic 33 Green’s function in CM frame (PB ¼ 0), X GðLÞð0; 0;0Þ ðLÞ iqB Next, the UV divergence in B can be isolated − nxnqLnyL GB ð0; 0; εÞ¼ e 2 ðLÞ out by further splitting GB ð0; 0;0Þ into nx;ny∈Z

nXz∈Z −qBjρþn n Le þn Le j2 qB 1 × e 4 x q x y y ðLÞ ðB;LÞ ðB;LÞ 2π L G ð0; 0;0Þ¼G þ G ; ð34Þ 2πnz B UV RC kz¼ L

X∞ L ðqB jρ þ n n Le þ n Le j2Þ n 2 x q x y y × : ðB;LÞ qB 1 k2 G z ρ→0 where is UV divergent and is given by n¼0 ε − μ ðn þ 2Þ − 2μ UV ð Þ 28

qB 1 nXz∈Z X∞ L ðqB jρj2Þ GðB;LÞ ¼ n 2 : ð Þ The UV divergence is associated to the term 2 35 UV 2π L qB 1 kz 2πn − ðn þ Þ − k ¼ z n¼0 μ 2 2μ z L ρ→0 n ∈Z Z 1 Xz X∞ 1 Λ¼1 dk3 1 ∝ r ∝ Λ ¼ ; 2 2 L qB 1 kz k r 2πnz n¼0 ε − ðn þ Þ − r→0 kz¼ μ 2 2μ ðB;LÞ L The GRC is a regulated constant term, and is defined by ð29Þ

n ∈Z X 2 Xz ðLÞ iqBnxnynqL qB 1 G ðB;LÞ − hence B is linearly divergent. G ¼ e 2 RC 2π L The linear divergence can be regularized simply by n ;n ≠0 2πnz x y kz¼ subtraction. Therefore, we first split the finite volume L ∞ qB 2 qB X − 4 jnxnqLexþnyLeyj 2 magnetic Green’s function into a regularized term by e Lnð 2 jnxnqLex þnyLeyj Þ × 2 : subtraction and a term that is UV divergent, qB 1 kz n¼0 − μ ðnþ 2Þ− 2μ ðLÞ ðLÞ ðLÞ ð36Þ GB ð0; 0; εÞ¼△GB ðεÞþGB ð0; 0;0Þ; ð30Þ where ðB;LÞ (1) The regulated constant term GRC can be further simplified by using the identity ðLÞ ðLÞ ðLÞ △GB ðεÞ¼GB ð0; 0; εÞ − GB ð0; 0;0Þ: ð31Þ

X∞ qB 2 ðLÞ Lnð x Þ The subtracted term △GB ðεÞ is free of UV divergence. 2 2 qB 1 kz Using the identity n¼0 − μ ðn þ 2Þ − 2μ 2 2 pffiffiffiffiffiffiffiffiffi μ 1 kz 1 kz qB 2 nXz∈Z ¼ − Γ þ U þ ; 1; x ; ð Þ 1 1 2μ 2μEL qB 2 2qB 2 2qB 2 37 2 ¼ pffiffiffiffiffiffiffiffiffi cot ; ð32Þ L kz 2 2μE 2 2πnz E − 2μ kz¼ L where Uða; b; zÞ stands for the Kummer function, hence, ðLÞ △GB ðεÞ is thus given explicitly by we find

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Z nz∈Z 2 nz∈Z nΛ 2μ 1 X 1 k qB 1 X X nΛ 1 GðB;LÞ ¼ − Γ þ z GðB;LÞ ¼ − dn RC UV qB k2 4π L 2 2qB 2π L 0 1 z 2πnz 2πnz n¼0 − μ ðn þ 2Þ − 2μ kz¼ L kz¼ L X 2 2 iqBnxnynqL qB 2 qBr − − jnxnqLexþnyLeyj n ∈Z Z L 2 ð Þ × e 2 e 4 Xz ∞ ξ 2 1 1 2qBr2 n ;n ≠0 þ ξdξ : x y 2 2 2 2 2π L ξ qBr kz r 2πnz 0 − − − r→0 2 kz¼ 2μ 2μ 2μ 1 kz qB 2 L × U þ ; 1; jnxnqLex þ nyLeyj : 2 2qB 2 ð43Þ ð38Þ Using the asymptotic form of the Laguerre polynomial, Asymptotically, the Kummer function decays exponentially, qBr2 pffiffiffiffiffi r→0 2 2 2 L ξ2 → J0ð ξ Þ; ð44Þ qB 1 k 1 k qB 2 − x2 z z 2 2qBr2 e 4 Γ þ U þ ; 1; x 2 2qB 2 2qB 2 qffiffiffiffiffiffiffiffiffi k →∞ z 2 2 and identity in Eq. (32) again, we find → 2K0ð kzx Þ; ð39Þ GðB;LÞ ¼ △GðB;LÞ UV UV GðB;LÞ qffiffiffiffiffiffiffiffiffiffi hence RC is indeed a well-defined regulated constant. 2 ξ þqBL (2) The explicit expression of the UV divergence in Z r2 2μ ∞ pffiffiffiffiffi coth GðB;LÞ − ξdξJ ð ξ2Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 ; UV can be worked out. First of all, the infinite sum of 2πr 0 2 2 n ∈ ½0; ∞ n ∈ ½0;n 0 2 ξ þ qBr integer in Eq. (35) is split into Λ and r→0 n ∈ ½nΛ; ∞, where nΛ serves as a cutoff integer and ð45Þ nΛ ≫ 1. For large nΛ, the summation is replaced by integration, hence one can rewrite Eq. (35) to where n ∈Z Xz XnΛ qB 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðB;LÞ qB 1 Lnð jρj Þ G ¼ 2 n Z 2qBðnþ1ÞL 2 XΛ n 2 UV 2π L qB 1 kz ðB;LÞ 2μqB Λ coth 2πn − ðn þ Þ − ρ→0 2 k ¼ z n¼0 μ 2 2μ △G ¼ − − dn qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : z L UV 2π n¼0 0 1 Z 2 2qBðn þ 2Þ nXz∈Z qB 2 qB 1 ∞ Lnð jρj Þ þ dn 2 : ð40Þ qB 1 k2 ð46Þ 2π L n z ρ→0 2πnz Λ − μ ðn þ 2Þ − 2μ kz¼ L 1 As r → 0 and r → ∞, Let us rewrite it further to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ξ2 Z 2 þ qBL nXz∈Z XnΛ qB 2 z→∞ −2z −4z r qB 1 nΛ L ð jρj Þ z ∼ 1 þ 2e þ 2e ;z¼ ; GðB;LÞ ¼ − dn n 2 coth 2 2 UV 2π L qB 1 kz 2πn 0 − ðn þ Þ − ρ→0 k ¼ z n¼0 μ 2 2μ z L ð47Þ n ∈Z Z qB 1 Xz ∞ L ðqB jρj2Þ þ dn n 2 : 2 and also using identity 2π L qB 1 kz 2πn 0 − ðn þ Þ − ρ→0 k ¼ z μ 2 2μ z L Z pffiffiffiffiffi ∞ 2 pffiffiffiffiffi ð41Þ J0ð ξ Þ 2π − qBr ξdξ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ e ; ð48Þ 0 2 ξ2 þ qBr2 4π The first term in Eq. (41) is finite, so ρ can be set to zero safely. In the second term in Eq. (41), let us rescale the we finally obtain a explicit expression of UV divergence, integral dummy variable dn to pffiffiffiffiffi − qBr ðB;LÞ ðB;LÞ 2μe 2 G ¼ △G − ξ UV UV qBρ2dn → ξdξ;qBρ2n → ; ð Þ 4πr r→0 2 42 pffiffiffiffiffiffi 2μ qB 2μ ¼ △GðB;LÞ þ − : ð Þ UV 4π 4πr 49 thus, the Eq. (35) is then turned into r→0

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Putting all the pieces together, we thus find At the limit of qB → 0, using the asymptotic form of ðLÞ 2 2 GB ð0; 0; εÞ 1 kz − 2με 1 kz − 2με qB pffiffiffiffiffiffi Γ þ U þ ; 1; r2 2μ qB 2μ 2 2qB 2 2qB 2 ¼ △GðLÞðεÞþGðB;LÞ þ △GðB;LÞ þ − ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B RC UV qB→0 4π 4πr r→0 ð1Þ 2 → iπH0 ð 2με − kzrÞþOðqBÞ; ð54Þ ð50Þ and also taking nq ¼ 1, thus one finds △GðLÞðεÞ GðB;LÞ △GðB;LÞ where B , RC , and UV are all free of UV divergence and given by Eq. (33), Eq. (38), and Eq. (46) nXz∈Z ðLÞ qB→0 2μ respectively. G ð0; 0; εÞ → − B L 2πnz kz¼ L qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B. Regulated S-wave quantization condition X i Hð1Þ 2με − k2jn Le þ n Le j þ OðqBÞ: With explicitly isolated UV divergence in the finite × 4 0 z x x y y volume magnetic Green’s function in Eq. (50), the UV nx;ny divergent terms in quantization condition given by Eq. (27) ð55Þ cancel out, thus we find a regulated quantization condition Next, using the identity δ ðk Þ¼MðB;LÞðεÞ; ð Þ cot 0 ε 0;0 51 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X i − Hð1Þ 2με − k2jn Le þ n Le j where 4 0 z x x y y nx;ny MðB;LÞðεÞ 0;0 1 X 1 h i pffiffiffiffiffiffi ¼ ; ð56Þ 4π ðLÞ ðB;LÞ ðB;LÞ qB L2 2με − k2 − k2 − k2 ¼ − △G ðεÞþG þ △G − : ð Þ 2πnx;y z x y B RC UV 52 kx;y¼ ;nx;y∈Z 2μkε kε L △GðLÞðεÞ GðB;LÞ △GðB;LÞ The expression of B , RC , and UV are given one thus can easily show that by Eq. (33), Eq. (38), and Eq. (46) respectively. ðLÞ qB→0 ðLÞ GB ð0; 0; εÞ → G0 ð0;kεÞþOðqBÞ; ð57Þ C. Lüscher formula at the limit of qB → 0 where Using the identity given in Eq. (37), Eq. (28) can be rewritten as 2μ X eip·r GðLÞðr;kÞ¼ : ð Þ 0 3 2 2 58 nXz∈Z 2 L k − p ðLÞ 2μ 1 1 kz −2με p¼2πn;n∈Z3 G ð0;0;εÞ¼− Γ þ L B 4πL 2 2qB k ¼2πnz ðB;LÞ z L Hence, the finite volume magnetic zeta function M0;0 ðεÞ X iqBn n n L2 qB → 0 − x y q −qBjn n Le þn Le j2 at the limit of is given by × e 2 e 4 x q x y y n ;n x y qB→0 MðB;LÞðεÞ → MðLÞðk ÞþOðqBÞ; ð Þ 1 k2 −2με qB 0;0 0;0 ε 59 U þ z ;1; jn n Le þn Le j2 : × 2 2qB 2 x q x y y which is consistent with the perturbation result given in ð53Þ ðLÞ Ref. [71]. The M0;0 ðkεÞ denotes the regular finite volume zeta function, see [1,2,8], and is defined by

2 2 kε −p Z 2 n≠0 2 2 ∞ kε n 4π X e Λ2 Λ X ∞ ðnLΛξÞ k X ð Þ ðLÞ − þ 2 Λ2 M ðk Þ¼− þ pffiffiffi dξe 4 ðΛξÞ þ ; ð Þ 0;0 ε k L3 k2 − p2 k π n!ð2n − 1Þ 60 ε 2π 3 ε ε 3 1 n¼0 p¼ L n;n∈Z n∈Z where Λ is an arbitrary UV regulator. ðB;LÞ is illustrated in the comparison of the curves of M0;0 ðεÞ The comparison of the finite volume magnetic zeta n ¼ 1 n ¼ 2 ðB;LÞ in the upper and lower panels with q and q function M0;0 ðεÞ and regular finite volume zeta function ðB;LÞ respectively; the number of M0;0 ðεÞ curves double as the MðLÞðkÞ 0;0 are shown in Fig. 1. The splitting of energy levels value of nq is doubled.

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Bakersfield, CA. This research (P. G.) was also supported in part by the U.S. National Science Foundation under Grant No. NSF PHY-1748958.

APPENDIX A: TWO CHARGED BOSONS IN A UNIFORM MAGNETIC FIELD The dynamics of two charged nonrelativistic identical bosons in a uniform magnetic field is described by THE Schrödinger equation, X2 ð∇ þ ieAðx ÞÞ2 E þ i i − Vðx − x Þ Ψ ðx ; x Þ¼0; 2m 1 2 E 1 2 i¼1 ðA1Þ

where m is the mass of identical bosons. xi denotes the position of the ith particle, and the short-range interaction between two particles is represented by Vðx1 − x2Þ. AðxiÞ stands for the vector potential of a uniform magnetic field. Throughout the entire work, the uniform magnetic field is assumed along the z axis, B ¼ Bez, and the vector potential in Landau gauge is used,

AðxiÞ¼Bð0;xi; 0Þ: ðA2Þ

The solutions of the Schrödinger equation in other gauges FIG. 1. Comparison of the finite volume magnetic zeta function are obtained by a gauge transformation through a scalar k MðB;LÞðεÞ ε 0;0 (black solid) and regular finite volume zeta function field, χðxiÞ, ðLÞ kM0;0 ðkÞ (red solid) in Eqs. (52) and (60) respectively. The −1 parameters are chosen as μ ¼ 0.2 GeV, L ¼ 10 GeV , np ¼ 1, AðxiÞ → AðxiÞ − ∇χðxiÞ; ðA3Þ and nq ¼ 1, 2 in the upper and lower panels respectively. and P 2 IV. SUMMARY ie χðxiÞ ΨEðx1; x2Þ → e i¼1 ΨEðx1; x2Þ: ðA4Þ A formalism for describing charged spinless bosons interaction in both a finite volume and a magnetic field is presented in this work. We show that for a short-range potential, a Lüscher formulalike relation that relates dis- 1. Separation of center of mass and relative motions crete energy spectrum to scattering phase shifts can be obtained. The regularization of UV divergence is worked The center-of-mass motion (CM) and relative motion of out explicitly for S-wave contribution; the regulated S-wave two particles can be separated by introducing CM and quantization condition may be useful for the LQCD study relative coordinates respectively of charged boson system, such as a πþ or Kþ system. In x1 þ x2 finite volume and in a magnetic field, translation symmetry R ¼ ; r ¼ x1 − x2: ðA5Þ of a system is only preserved when the magnetic flux, 2 2 ΦB ¼ qBL , is given by 2π multiplied by a rational number Therefore, the Hamiltonian has a separable form and the np=nq where np and nq are relatively prime integers. The total two particles wave function is given by the product of presence of a magnetic field thus results in the splitting of CM and relative wave functions, the energy level into nq subenergy levels.

ΨEðx1; x2Þ¼ΦE−εðRÞψ εðrÞ; ðA6Þ ACKNOWLEDGMENTS

P. G. acknowledges support from the Department of where the CM wave function, ΦE−εðRÞ, and relative wave Physics and Engineering, California State University, function, ψ εðrÞ, satisfy Schrödinger equations respectively,

094520-8 CHARGED PARTICLES INTERACTION IN BOTH A FINITE … PHYS. REV. D 103, 094520 (2021) ð∇ þ iQAðRÞÞ2 Kˆ ¼ −i∇ þ qAðrÞ − qB r ¼ −i∇ þ qBðr ; 0; 0Þ ðE − εÞþ R Φ ðRÞ¼0; ð Þ r r × r y 2M E−ε A7 ðA11Þ and ˆ in fact commutes with Hr: ð∇ þ iqAðrÞÞ2 ε þ r − VðrÞ ψ ðrÞ¼0: ð Þ ½Kˆ ; Hˆ ¼0: 2μ ε A8 r r ˆ The total and reduced mass of two particles are Therefore, Kr can be used as generator of a magnetic translation operator, m M ¼ 2m and μ ¼ ˆ ˆ iKr·nL ið−i∇rþqAðrÞ−qB×rÞ·nL 2 TrðnLÞ¼e ¼ e ; ðA12Þ respectively, and similarly where eið−i∇rÞ·nL is the pure translation operator, and e Q ¼ 2e q ¼ and ið−i∇rÞ·nL 2 e ψ εðrÞ¼ψ εðr þ nLÞ: ðA13Þ are total and reduced charges respectively. So that

ˆ iqðAðrÞ−B×rÞ·nL 2. Magnetic translation group and magnetic periodic TrðnLÞψ εðrÞ¼e ψ εðr þ nLÞ; ðA14Þ boundary condition Tˆ Now, let us consider putting charged particles in a and r commutes with the Hamiltonian, periodic cubic box with size L, and interaction between ˆ ˆ two particles is also periodic, ½Tr; H¼0;

Vðr þ nLÞ¼VðrÞ; n ∈ Z3: ðA9Þ which leaves the Hamiltonian invariant. However, the magnetic translation operators do not commute with each Without a magnetic field, the discrete translation symmetry other in general, of a system in finite volume yields the conserved total ˆ ˆ −iqBL2n n ˆ ˆ momentum of a system with discrete values: TrðnxLexÞTrðnyLeyÞ¼e x y TrðnyLeyÞTrðnxLexÞ; ð Þ 2πn A15 P ¼ ; n ∈ Z3: L where ðnx;nyÞ ∈ Z. In a magnetic field, though the potential V is periodic, As shown in Refs. [75,76], when the values of qB are ˆ taken as the Hr is not discrete translation invariant 1 2π n Hˆ ≠ Hˆ qB ¼ eB ¼ p ; ð Þ rþnL r 2 A16 2 L nq due to the fact that the vector potential is coordinate dependent and breaks discrete translation symmetry where np and nq are integers that are relatively prime. The magnetic translation operators with an enlarged unit cell n L e Aðr þ nLexÞ¼AðrÞþBnLey;n∈ Z: ðA10Þ formed by the increased size of q in the x direction thus commute with each other, The momentum operator, pˆ ¼ −i∇r, does not commute ˆ ˆ ˆ with Hr: ½TrðnxðnqLÞexÞ; TrðnyLeyÞ ¼ 0: ðA17Þ ˆ ½pˆ ; Hr ≠ 0: Therefore magnetic translation operators with an enlarged magnetic unit box that is defined by Hence canonical momentum is no longer a conserved quantity as the consequence of breaking down of discrete nqLex × Ley × Lez translation symmetry in a uniform magnetic field. It has been shown in Refs. [75–77] that a pseudomomentum form a discrete group that are commonly referred to as a operator magnetic translation group.

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iQðAðRÞ−B RÞ n1Bþn2BL n þn The translation operator for two charged particles can be × · 2 1B 2B e ΦE−εðR þ 2 LÞ introduced by ΦE−εðRÞ ψ ðrÞ ˆ ˆ ˆ ε Tx ;x ðn1BL; n2BLÞ¼Tx ðn1BLÞTx ðn2BLÞ; ð Þ ¼ : ðA25Þ 1 2 1 2 A18 iqðAðrÞ−B×rÞ·ðn1B−n2BÞL e ψ εðr þðn1B − n2BÞLÞ where The separable form of the CM motion and relative motion in Eq. (A25) suggests that both sides must be ˆ ið−i∇x þeAðxiÞ−eB×xiÞ·nL Tx ðnLÞ¼e i : ðA19Þ equal to a phase factor that is independent of both CM and i relative coordinates. It allows us to introduce an arbitrary parameter PB that is associated with a pure translation n L n L Both 1B and 2B are defined in an enlarged magnetic operator; the phase factor may be chosen having the form of unit box, iP n1Bþn2BL e B· 2 : niBL ¼ nixðnqLÞex þ niyLey þ nizLez;nix;iy;iz ∈ Z: ðA20Þ Hence the CM wave function satisfies Bloch type magnetic periodic boundary condition, We may rewrite two particles translation operator in terms nBL −iðP þQBR e Þ nBL Φ R þ ¼ e B y x · 2 Φ ðRÞ: ð Þ of CM and relative motion quantities, E−ε 2 E−ε A26 ˆ ˆ ˆ n1B þ n2B The boundary condition for a relative wave function is Tx ;x ðn1BL;n2BLÞ¼Trðn1BL − n2BLÞTR L ; 1 2 2 given by ðA21Þ PB i ·nBL −iqBryex·nBL ψ εðr þ nBLÞ¼e 2 e ψ εðrÞ; ðA27Þ ˆ where Tr is defined in Eq. (A12), and where we have also assumed iP n L nBL ið−i∇ þQAðRÞ−QB RÞ nBL e B· B ¼ 1; ð Þ Tˆ ¼ e R × · 2 : ð Þ A28 R 2 A22 thus Note that nx ny nz PB ¼ 2π ex þ ey þ ez ;nx;y;z ∈ Z: 2π n nqL L L QB ¼ 2eB ¼ p ; L 2 n ð2Þ q Although PB resembles the total momentum of a system in absence of a magnetic field, PB is not a conserved quantity and the translation operation of the CM motion may be in the magnetic field. In fact, the conserved quantity can be considered as a motion of a composite charge particle with identified as a pseudomomentum, see e.g., Ref. [77], total charges of Q ¼ 2e in a periodic box with size L=2. The magnetic translation invariance of a system yields KR ¼ PB þ QðAðRÞ − B × RÞ; ðA29Þ

Tˆ ðn L; n LÞΨ ðx ; x Þ¼Ψ ðx ; x Þ: ð Þ x1;x2 1B 2B E 1 2 E 1 2 A23 which is associated with the generator of a magnetic translation operator for CM motion, Using Eq. (A19), the boundary conditions for two particles nBL ˆ nBL ˆ iKR· ˆ ˆ in both finite volume and a uniform magnetic field is TR ¼ e 2 ; ½KR; H¼0: ðA30Þ given by 2

Ψ ðx þ n L; x þ n LÞ E 1 P1B 2 2B 2 −ie ðAðxiÞ−B×xiÞ·niBL ¼ e i¼1 ΨEðx1; x2Þ: ðA24Þ 3. CM motion solutions The CM motion of two charged bosons in a uniform In terms of CM and relative wave functions, we have magnetic field is described by

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ˆ PB H Φ ðRÞ¼E Φ ðRÞ; ð Þ i 2 ·nBL −iqBryex·nBL R ER R ER A31 ψ εðr þ nBLÞ¼e e ψ εðrÞ: ðA40Þ where The integral representation of Schrödinger Eq. (A38) and magnetic periodic boundary condition in Eq. (A40) 1 ˆ 2 2 2 together is given by the finite volume Lippmann-Schwinger HR ¼ − ½∂R þð∂R þ iQBRxÞ þ ∂R ; ðA32Þ 2M x y z equation, Z and ΦE must satisfy the boundary condition R 0 ðLÞ 0 0 0 ψ εðrÞ¼ dr GB ðr; r ; εÞVðr Þψ εðr Þ; ðA41Þ 3 LB nBL −iðP þQBR e Þ nBL Φ R þ ¼ e B y x · 2 Φ ðRÞ: ð Þ ER ER A33 2 3 where LB stands for the volume of the magnetic unit box defined by unit vectors The solution that satisfies the magnetic periodic boundary condition can be found in [75], nqLex × Ley × Lez; ny∈Z X k P 2 y ik ðR þ BxÞ −iP R Φ ðRÞ¼ ϕ R þ e y y QB e Bz z ; and ER;n L n x QB 4πnynp Z Z Z Z ky¼ L −PBy nqL L L 0 2 0 2 0 2 0 ð Þ dr ¼ drx dry drz: ðA42Þ A34 3 nqL L L LB − 2 −2 −2 ϕ where n is the eigen-solution of a 1D harmonic oscillator ’ GðLÞ potential, The finite volume magnetic Green s function B also must satisfy the magnetic periodic boundary condition, 1 QB 1 2 2 2 ðLÞ − ½∂ − ðQBÞ RxϕnðRxÞ¼ n þ ϕnðRxÞ: 0 2M Rx M 2 GB ðr; r ; εÞ PB ðLÞ −i ·nBL iqBryex·nBL 0 ðA35Þ ¼ e 2 e GB ðr þ nBL; r ; εÞ iPB n L −iqBr0 e n L ðLÞ 0 ¼ e 2 · B e y x· B G ðr; r þ n L; εÞ: ð Þ The eigen-energy of CM motion is given by B B A43 QB 1 P2 The magnetic periodic boundary conditions and Eq. (A41) Bz ðLÞ ER;n ¼ n þ þ ;n¼ 0; 1; 2; …; ðA36Þ M 2 2M suggest that GB is the solution of the differential equation,

ˆ ðLÞ 0 and the analytic expression of ϕn is ðε − HrÞGB ðr; r ; εÞ X PB −i ·nBL iqBryex·nBL 0 1 pffiffiffiffiffiffiffi ¼ e 2 e δðr − r þ nBLÞ: ðA44Þ 1 QB 4 QB 2 − Rx n ϕnðRxÞ¼pffiffiffiffiffiffiffiffiffi e 2 Hnð QBRxÞ: ðA37Þ B 2nn! π Now, one of the key steps therefore is to find an analytic ðLÞ solution of finite volume magnetic Green’s function GB . ðLÞ The GB can be constructed from an infinite volume ð∞Þ 4. Relative motion and finite volume magnetic Green’s function G , where Lippmann-Schwinger equation B Z The relative motion of two charged particles in a uniform X∞ ∞ dk dk Gð∞Þðr;r0;εÞ¼ y z magnetic field is described by B 2π 2π n¼0 −∞ k k 0 0 ðHˆ þ VðrÞÞψ ðrÞ¼εψ ðrÞ; ð Þ y 0 y ikyðry−ryÞ ikzðrz−rzÞ r ε ε A38 ϕnðrx þ qBÞϕnðrx þ qBÞe e × 2 ; ε − qB ðn þ 1Þ − kz where μ 2 2μ ðA45Þ 1 ˆ 2 2 2 Hr ¼ − ½∂r þð∂r þ iqBrxÞ þ ∂r ; ðA39Þ 2μ x y z ð∞Þ and GB satisfies the equation, and ψ ε must satisfy the magnetic periodic boundary ðε − Hˆ ÞGð∞Þðr; r0; εÞ¼δðr − r0Þ: ð Þ condition r B A46

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ðLÞ The LS equation (A41) is equivalently given in terms of Therefore, GB is also given by ð∞Þ GB by ðLÞ 0 Z GB ðr;r ;εÞ ∞ 0 ð∞Þ 0 0 0 X P ψ ðrÞ¼ dr G ðr; r ; εÞVðr Þψ ðr Þ: ð Þ −iðPBx−qBr Þn n L −i Byn L −iqBðr þr0 þn n LÞðr −r0 þn LÞ ε B ε A47 ¼ e 2 y x q e 2 y e 2 x x x q y y y −∞ nx;ny∈Z

nXz∈Z The integration over infinite volume in Eq. (A47) −qBjρ−ρ0þn n Le þn Le j2 qB 1 e 4 x q x y y can be folded up to an infinite sum of integration in a × 2π L k ¼2πnzþPBz magnetic cell, z L 2 ∞ qB 0 Z XL ð jρ−ρ0 þn n Le þn Le j2Þeikzðrz−rzÞ X n 2 x q x y y : ð∞Þ × 2 ψ ðrÞ¼ dr0G ðr; r0 þ n L; εÞ qB 1 kz ε B B n¼0 ε− μ ðnþ 2Þ− 2μ L3 nB B 0 0 ðA52Þ × Vðr þ nBLÞψ εðr þ nBLÞ: ðA48Þ

Using the magnetic periodic boundary condition given in ðLÞ APPENDIX B: CONNECTING BOUND STATES Eq. (A40), GB is thus identified as IN A TRAP TO INFINITE VOLUME X SCATTERING STATE ðLÞ ð∞Þ PB 0 0 0 i ·nBL −iqBryex·nBL GB ðr; r ; εÞ¼ GB ðr; r þ nBL; εÞe 2 e In this Appendix, we present a general formalism and n XB discussion on the topic of building connections between a PB ð∞Þ −i ·nBL iqBryex·nBL 0 discrete energy spectrum of bound state in a trap and ¼ e 2 e GB ðr þ nBL; r ; εÞ: nB infinite volume scattering dynamics. The type of trap is not specified in what follows; the typical and commonly used ð Þ A49 traps are periodic finite box in LQCD, harmonic potential in nuclear physics, etc. Hence, explicitly we find The relative motion of two interacting particles in a trap is described by the Schrödinger equation X ðLÞ PBx 0 −ið −qBryÞnxnqL GB ðr; r ; εÞ¼ e 2 Z n ∈Z ˆ ðtrapÞ 0 0 ðtrapÞ 0 ðtrapÞ x Htrapψ ε ðrÞþ dr Vðr; r Þψ ε ðr Þ¼εψ ε ðrÞ; trap nXy;z∈Z 1 0 0 eikyðry−ryÞeikzðrz−rzÞ ð Þ × L2 B1 2πny;z PBy;Bz ky;z¼ L þ 2 Hˆ ∞ ky 0 ky where trap stands for the trap Hamiltonian operator, and X ϕnðrx þ nxnqL þ Þϕnðrx þ Þ qB qB : the interaction between particles is described by a nonlocal × 2 0 qB 1 kz Vðr; r Þ n¼0 ε − μ ðn þ 2Þ − 2μ short-range interaction in general. The effect of a trap is usually reflected by both the trap Hamiltonian and ð Þ A50 boundary condition of the wave function in a trap. In the case of charged particles trapped in both a periodic box and ð∞Þ ˆ The other representation of GB is given in a uniform magnetic field, Htrap and the boundary condition ˆ Refs. [78,79] by are thus given by Hr in Eq. (A39) and magnetic periodic boundary condition in Eq. (A40) respectively. The energy ð∞Þ iqB 0 0 qB 0 2 0 − 2 ðrxþrxÞðry−ryÞ − 4 jρ−ρ j spectrum hence becomes discrete. GB ðr;r ; εÞ¼e e Z ∞ qB 0 In infinite volume, the dynamics of two interacting qB ∞ dk X L ð jρ − ρ0j2Þeikzðrz−rzÞ 0 z n 2 ; particles through the same short-range interaction Vðr; r Þ × qB k2 2π −∞ 2π 1 z is given by n¼0 ε − μ ðn þ 2Þ − 2μ Z ðA51Þ ∞ Hˆ ψ ð∞ÞðrÞþ dr0Vðr; r0Þψ ð∞Þðr0Þ¼ε ψ ð∞ÞðrÞ; 0 ε∞ ε∞ ∞ ε∞ −∞ where ðB2Þ ρ ¼ r e þ r e ; ρ0 ¼ r0 e þ r0 e : x x y y x x y y where

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∇2 range physics due to the presence of a trap can be Hˆ ¼ − r : 0 2μ factorized. The derivation of tje Lüscher formula or BERW formula The energy spectrum of the scattering solution in infinite can be illustrated by considering momentum space volume is continuous. With an incoming plane wave, Lippmann-Schwinger equation under the assumption of pffiffiffiffiffiffiffiffiffiffiffi separable potential, see e.g., [28–30], and an example of iq·r e where q ¼ 2με∞; derivation of BERW formula in momentum space is given in Appendix C. Here the result is only summarized briefly the asymptotic wave function of scattering states is thus symbolically; the reaction amplitudes in both trap and described by on-shell scattering amplitudes, infinite volume may be introduced respectively by X ð∞Þ r→∞ ðþÞ ˆ ˆ ψ ðrÞ → ð2l þ 1ÞP ðqˆ rˆÞil½j ðqrÞþit ðqÞh ðqrÞ; tˆtrap ¼ −V ψˆ and tˆ∞ ¼ −Vψˆ ∞; ε∞ l · l l l l They satisfy integral LS equations, ðB3Þ ˆ ˆ tˆtrapðεÞ¼VGtrapðεÞtˆtrapðεÞ; ðB7Þ where tlðqÞ denotes the elastic on-shell partial wave scattering amplitude and can be parametrized by a phase and shift function δlðqÞ, ˆ ˆ ˆ 1 tˆ∞ðqÞ¼−V þ VG∞ðqÞtˆ∞ðqÞ; ðB8Þ tlðqÞ¼ : ðB4Þ cot δlðqÞ − i where We also remark that in a general case, depending on the 1 ε Gˆ ðεÞ¼ ð Þ trap, the infinite volume relative energy ∞ is related to trap ˆ B9 finite volume relative energy ε by the shared total energy. ε − Htrap For instance, in the case of charged particles trapped in both a periodic box and a uniform magnetic field, and

P2 ˆ 1 G∞ðqÞ¼ ðB10Þ ε∞ þ ¼ ε þ ER;n ¼ E; ðB5Þ q2 ˆ 2M 2μ − H0 E where CM energy R;n is given by Eq. (A36). are Green’s function in a trap and in infinite volume The dynamics of particles in a trap and in infinite volume respectively. Under the assumption of a separable potential are associated by the short-range interaction potential that is equivalent to the zero-range interaction, between two particles. As far as the range of potential is X far smaller than the size of the trap, a compact expression ˜ 0 0 l ˆ ˆ 0 Vðk; k Þ¼ ðkk Þ VlYlmðkÞYlmðk Þ; ðB11Þ between phase shift of scattering states and a function, lm Mlm;l0m0 ðεÞ, that reflect geometric and dynamical proper- ties of the trap can be found, Eq. (B7) and Eq. (B8) are turned into algebraic equations, and can be solved analytically [30]. Eliminating Vˆ from det½δlm;l0m0 cot δlðqÞ − Mlm;l0m0 ðεÞ ¼ 0: ðB6Þ two equations, the quantization condition is thus obtained In the case of finite volume in LQCD, this relation is a well- 1 ˆ ˆ det − G∞ðqÞþGtrapðεÞ ¼ 0; ðB12Þ known Lüscher formula [1], the matrix function tˆ∞ðqÞ Mlm;l0m0 ðεÞ is thus a zeta function. In finite volume, the angular momentum is no longer a good quantum number which is equivalent to Eq. (B6). due to the breaking rotation symmetry in finite volume. In Though the plane wave basis in momentum space may the case of harmonic trap in nuclear physics, the relation is be a very convenient basis in finite volume and other types known as the BERW formula [42–51], where the function of traps, for the charged particles in a uniform magnetic Mlm;l0m0 becomes diagonal in angular momentum basis. field, the momentum is no longer the conserved quantity The simple form of the quantization condition in Eq. (B6) is due to the breaking translation symmetry by magnetic field. the result of the presence of two distinguishable scales: Introducing a reaction amplitude in momentum space (1) short-range interaction between two particles and becomes a tricky business. Therefore, in what follows, (2) size of trap. Hence the short-range dynamics that is instead of working in momentum space, we will present the described by phase shift or scattering amplitude and long- general discussion of derivation of a quantization condition

094520-13 PENG GUO and VLADIMIR GASPARIAN PHYS. REV. D 103, 094520 (2021) in coordinate space under the assumption of a separable ðtrapÞ X 2l0þ1 2 0 3 ψ lm ðrÞ 2 Γ ðl þ 2Þ ¼ V 0 short-range potential again. The Fourier transform of a rl l ð2πÞ3 separable potential given in Eq. (B11) is l0m0

GðtrapÞ ðr; r00; εÞ ψ ðtrapÞðr0Þ Z lm;l0m0 l0m0 0 × 0 0 : ðB19Þ dk dk 0 0 l 00l 0l Vðr; r0Þ¼ e−ik·rV˜ ðk; k0Þeik ·r r r r r0;r00→0 ð2πÞ3 ð2πÞ3 δðrÞδðr0Þ X 22lþ1Γ2ðl þ 3Þ ¼ V 2 Y ðrˆÞY ðrˆ0Þ: hence the discrete energy spectrum is determined by ðrr0Þ2 l ð2πÞ3ðrr0Þl lm lm lm 2 0 3 ðtrapÞ 0 δ 0 0 Γ ðl þ Þ G 0 0 ðr; r ; εÞ ðB13Þ lm;l m − 2 lm;l m ¼ 0: det 2l0þ1 3 l 0l0 2 Vl ð2πÞ r r r;r0→0 ðB20Þ

1. Dynamical equation in a trap In the trap, the integral representation of Eq. (B1) is given by the Lippmann-Schwinger equation 2. Infinite volume dynamical equation In infinite volume, with an incoming plane wave Z of eiq·r, the scattering solution of two particles interaction ðtrapÞ 00 ðtrapÞ 00 ψ ε ðrÞ¼ dr G ðr; r ; εÞ is described by inhomogeneous integral Lippmann- trap Z Schwinger equation, 0 00 0 ðtrapÞ 0 dr Vðr ; r Þψ ε ðr Þ; ð Þ × B14 ð∞Þ trap ψ ε ðr;qÞ ∞ Z Z ∞ ∞ ¼eiq·r þ dr00Gð∞Þðr−r00;qÞ dr0Vðr00;r0Þψ ð∞Þðr0;qÞ; where ε∞ −∞ −∞ ð Þ 1 B21 GðtrapÞðr; r00; εÞ¼hrj jr00iðÞ ˆ B15 pffiffiffiffiffiffiffiffiffiffiffi ε − Htrap where q ¼ 2με∞, and the Green’s function is given by stands for the Green’s function in a trap. The partial wave expansions Gð∞Þðr − r00; qÞ Z ip·ðr−r00Þ X dp e 2μ ðþÞ ðtrapÞ ðtrapÞ ¼ ¼ − iqh ðqjr − r00jÞ: ð Þ ψ ðrÞ¼ ψ ðrÞY ðrˆÞðÞ 3 2 2 0 B22 ε lm lm B16 ð2πÞ q − p 4π lm 2μ 2μ

Considering partial wave expansion, and X ψ ð∞Þðr; qÞ¼ Y ðqˆÞψ ð∞Þðr; qÞY ðrˆÞ; ð Þ X ε∞ lm l lm B23 ðtrapÞ 00 ˆ ðtrapÞ 00 ˆ00 G ðr; r ; εÞ¼ YlmðrÞGlm;l00m00 ðr; r ; εÞYl00m00 ðr Þ lm lm;l00m00 ðB17Þ and X ð∞Þ 00 ð∞Þ 00 00 G ðr − r ; qÞ¼ YlmðrˆÞGl ðr; r ; qÞYlmðrˆ Þ; yield lm Z ð∞Þ 00 ðþÞ X Gl ðr; r ; qÞ¼−2μiqjlðqr<Þhl ðqr>Þ; ðB24Þ ðtrapÞ 002 00 ðtrapÞ 00 ψ lm ðrÞ¼ r dr Glm;l0m0 ðr; r ; εÞ 0 0 trap l mZ we thus obtain 02 0 00 0 ðtrapÞ 0 Z r dr V 0 ðr ;rÞψ 0 0 ðr Þ: ð Þ ∞ × l l m B18 ð∞Þ ð∞Þ trap l 002 00 00 ψ l ðr;qÞ¼4πi jlðqrÞþ r dr Gl ðr;r ;qÞ Z 0 ∞ Under the assumption of a separable potential with the form 02 0 00 0 ð∞Þ 0 × r dr Vlðr ;r Þψ l ðr ;qÞ: ðB25Þ of Eq. (B13), Eq. (B18) is turned into an algebraic equation, 0

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The separable potential given in Eq. (B13) yields an where algebraic equation

2l0þ3 2 0 3 GðtrapÞ ðr; r0; εÞ 2 Γ ðl þ 2Þ lm;l0m0 ð∞Þ M 0 0 ðεÞ¼− ψ ðr; qÞ jlðqrÞ lm;l m 2lþ1 l 0l0 l ¼ 4πil 2μq ð2πÞ r r r;r0→0 rl rl 22lþ1Γðl þ 1ÞΓðl þ 3Þ 1 2lþ1 2 3 ð∞Þ 00 ð∞Þ 0 2 2 2 Γ ðl þ Þ G ðr; r ; qÞ ψ ðr ;qÞ − δlm;l0m0 : þ V 2 l l : π ðqrÞ2lþ1 l 3 00 l 0l r→0 ð2πÞ ðrr Þ r r0;r00→0 ðB32Þ ðB26Þ The second term in Eq. (B32) is an ultraviolet counterterm The wave function solution is thus given by that would cancel out the ultraviolet divergent term in ðtrapÞ G 0 0 , and ultimate result is finite and well defined. ψ ð∞Þðr; qÞ j ðqrÞ hðþÞðqrÞ lm;l m l ¼ 4πil l þ it ðqÞ l ; ð Þ rl rl l rl B27 APPENDIX C: MOMENTUM SPACE LS where the partial wave two-body scattering amplitude tlðqÞ EQUATION AND PARTICLES INTERACTION is given by IN A HARMONIC TRAP

2μq2lþ1 In this section, we present some technical details of ð4πÞ2 nonrelativistic spinless particles interaction in a harmonic tlðqÞ¼− : ðB28Þ 22lþ1Γ2ðlþ3Þ Gð∞Þðr0;r00;qÞ 1 2 l trap. The dynamics of nonrelativistic bosonic particles − 3 0 00 l jr0;r00→0 Vl ð2πÞ ðr r Þ interaction in a harmonic trap is described by

ˆ ðhoÞ ˆ ðhoÞ ðhoÞ ðH þ VÞΨE ðx1; x2Þ¼EΨE ðx1; x2Þ; ðC1Þ

3. Quantization condition in a trap where V Combining Eqs. (B20) and (B28), and eliminating l, one thus find X2 ∇2 1 Hˆ ðhoÞ ¼ − i þ mω2x2 ; ð Þ 2m 2 i C2 2μq2lþ1 i¼1 det δlm;l0m0 tlðqÞ x i ’ Vˆ and i again stands for the th particle s position, the 2lþ3 2 3 ð∞Þ 0 ω 2 Γ ðl þ 2Þ Gl ðr; r ; qÞ represents the interaction between two particles. is the − δ 0 0 lm;l m 0 l angular frequency of the oscillator. The separation of CM ð2πÞ ðrr Þ r;r0→0 and relative motions 22l0þ3Γ2ðl0 þ 3Þ GðtrapÞ ðr; r0; εÞ þ 2 lm;l0m0 ¼ 0: ð Þ l 0l0 B29 ˆ ðhoÞ ˆ ðhoÞ ˆ ðhoÞ ð2πÞ r r r;r0→0 H ¼ HR þ Hr ;

Using the asymptotic form of where

ð∞Þ 2 22lþ3Γ2ðl þ 3Þ G ðr; r0; qÞ ðhoÞ ∇ 1 2 l Hˆ ¼ − R þ Mω2R2 0 l R 2M 2 2μð2πÞ ðrr Þ r;r0→0

22lþ1Γðl þ 1ÞΓðl þ 3Þ 1 ¼ −iq2lþ1 − 2 2 ; ð Þ and 2lþ1 B30 π r r→0 ∇2 1 Hˆ ðhoÞ ¼ − r þ μω2r2 and also the parameterization of r 2μ 2

−1 tl ðqÞ¼cot δlðqÞ − i; yield again

ðhoÞ ðhoÞ ðhoÞ thus the quantization condition in a trap is indeed given by a ΨE ðx1; x2Þ¼ΦE ðRÞψ ε ðrÞ: Lüscher formulalike relation, R;n ΦðhoÞðRÞ The CM wave function ER;n is the solution of the ½δ 0 0 δ ðqÞ − M 0 0 ðεÞ ¼ 0; ð Þ det lm;l m cot l lm;l m B31 3D harmonic oscillator potential,

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Hˆ ðhoÞΦðhoÞðRÞ¼E ΦðhoÞðRÞ; ð Þ and R ER;n R;n ER;n C3 X ˜ ðhoÞ 0 ˆ ˜ ðhoÞ 0 ˆ 0 where eigen-energy is given by G ðk; k ; εÞ¼ YlmðkÞGl ðk; k ; εÞYlmðk ÞðC10Þ lm 3 ER;n ¼ ω n þ ;n¼ 0; 1; 2; …: ðC4Þ 2 thus yields

ðhoÞ The relative wave function ψ ε ðrÞ satisfies the ðhoÞ Tl ðkÞ Lippmann-Schwinger equation, Z ∞ k02dk0 k002dk00 Z Z ¼ V˜ ðk; k0ÞG˜ ðhoÞðk0;k00; εÞTðhoÞðk00Þ: 3 3 l l l ðhoÞ 0 ðhoÞ 0 00 0 00 ðhoÞ 00 0 ð2πÞ ð2πÞ ψ ε ðrÞ¼ dr G ðr; r ; εÞ dr Vðr ; r Þψ ε ðr Þ; ðC11Þ ðC5Þ The separable potential where Green’s function satisfies the equation ˜ 0 0 l Vlðk; k Þ¼ðkk Þ Vl ˆ ðhoÞ ðhoÞ 0 0 ðε − Hr ÞG ðr; r ; εÞ¼δðr − r Þ: ðC6Þ suggests that The analytic expression of Green’s function in a harmonic trap is given by [80] ðhoÞ ðhoÞ T ðkÞ¼klt ; ðC12Þ X l l ðhoÞ 0 ðhoÞ 0 0 G ðr; r ; εÞ¼ YlmðrˆÞGl ðr; r ; εÞYlmðrˆ Þ; lm hence the quantization condition under the assumption of a 1 Γðl þ 3 − ε Þ separable potential is given by GðhoÞðr; r0; εÞ¼− 2 4 2ω l 0 3 3 ωðrr Þ2 Γðl þ Þ Z 2 1 ∞ k02dk0 k002dk00 0 00 l ˜ ðhoÞ 0 00 2 2 ¼ ðk k Þ Gl ðk ;k ; εÞ: ðC13Þ × M ε l 1ðμωr<ÞW ε l 1ðμωr>Þ; ðC7Þ V 3 3 2ω;2þ4 2ω;2þ4 l 0 ð2πÞ ð2πÞ where Ma;bðxÞ and Wa;bðxÞ are Whittaker functions [81].

1. Momentum space LS equation and reaction 2. Momentum space LS equation and scattering amplitude in a harmonic oscillator trap amplitude in infinite volume The reaction amplitude in a harmonic trap can be In infinite volume, the scattering amplitude is defined by defined by Z Z Z Z ð∞Þ −ik r 0 0 ð∞Þ 0 Tε ðk; qÞ¼− dre · dr Vðr; r Þψ ε ðr ; qÞ; ðhoÞ −ik r 00 0 00 ðhoÞ 00 ∞ ∞ Tε ðkÞ¼− dre · dr Vðr ; r Þψ ε ðr Þ; ðC8Þ ðC14Þ ðhoÞ and Tε ðkÞ satisfies the momentum space LS equation, Z and it satisfies the momentum space LS equation dk0 dk00 ðhoÞ ˜ 0 ˜ ðhoÞ 0 00 ðhoÞ 00 Tε ðkÞ¼ Vðk;k ÞG ðk ;k ;εÞTε ðk Þ; Z ð2πÞ3 ð2πÞ3 0 ˜ 0 ð∞Þ dk Vðk; k Þ ð∞Þ 0 Tε ðk; qÞ¼−V˜ ðk; qÞþ Tε ðk ; qÞ: ð Þ ∞ ð2πÞ3 k02 q2 ∞ C9 2μ − 2μ ð Þ where V˜ and G˜ ðhoÞ are the Fourier transform of the C15 interaction potential V and Green’s function GðhoÞ respectively. In a harmonic oscillator trap, rotation symmetry is The partial wave expansion intact, hence the angular momentum is still a good quantum X number; the partial wave expansion of Tð∞Þðk; qÞ¼ Tð∞Þðk; qÞY ðkˆ ÞY ðqˆÞ ε∞ l lm lm X lm ðhoÞ ðhoÞ ˆ Tε ðkÞ¼ Tl ðkÞYlmðkÞ lm yields

094520-16 CHARGED PARTICLES INTERACTION IN BOTH A FINITE … PHYS. REV. D 103, 094520 (2021) Z Z ∞ k02dk0 V˜ ðk; k0Þ ∞ k2dk k02dk0 Tð∞Þðk; qÞ¼−V˜ ðk; qÞþ l Tð∞Þðk0;qÞ: ðkk0ÞlG˜ ðhoÞðk; k0; εÞ l l 3 02 q2 l 3 3 l 0 ð2πÞ k − 0 ð2πÞ ð2πÞ 2μ 2μ 1 22lþ2Γ2ðl þ 3Þ GðhoÞðr; r0; εÞ ðC16Þ ¼ 2 l ; ð Þ 2 0 l C22 ð4πÞ π ðrr Þ r;r0→0 The assumption of the separable potential again yields an analytic solution of scattering amplitude, and

0 L Z ð∞Þ ð∞Þ 0 ðq qÞ ∞ 2 2l 2lþ2 2 3 0 T ðq ;qÞ¼− R : ð Þ k dk k 1 2 Γ ðl þ 2ÞGl ðr;r ;qÞ l 1 ∞ k2dk k2l C17 ¼ ; 2 2 − 3 2 2 3 k q 2 0 l Vl 0 ð2πÞ k q 0 ð2πÞ − ð4πÞ π ðrr Þ r;r0→0 2μ−2μ 2μ 2μ ð Þ Tð∞Þðq; qÞ C23 The on-shellp partialffiffiffiffiffiffiffiffiffiffiffi wave scattering amplitudes l , where q ¼ 2με∞, are usually parametrized by phase shift, ðhoÞ 0 where the analytic expression of Gl ðr; r ; εÞ and 2 ð∞Þ 0 ð∞Þ ð4πÞ 1 Gl ðr; r ; qÞ are given in Eqs. (C7) and (B24) respectively. Tl ðq; qÞ¼ : ðC18Þ 2μq cot δlðqÞ − i Also using the asymptotic form of the harmonic oscillator trap Green’s function, V Therefore, a simple relation between L and phase shift is 22lþ2Γ2ðl þ 3Þ GðhoÞðr; r0; εÞ obtained, 2 l 0 l Z π ðrr Þ 0 1 ∞ k2dk k2l 2μq2lþ1 r;r →0 − ¼ − ½ δ ðqÞ − i: ð Þ lþ3 3 l ε 3 2 q2 2 cot l C19 ðμωÞ 2 Γð þ − Þ Vl 0 ð2πÞ k ð4πÞ 2lþ2 lþ1 4 2 2ω 2μ − 2μ ¼ − 2 ð−1Þ ω Γð1 − l − ϵn Þ 4 2 2ω 22lþ1Γðl þ 1ÞΓðl þ 3Þ 2μ − 2 2 ; ð Þ π r2lþ1 C24 3. Quantization condition in a harmonic oscillator trap r→0 Combining Eqs. (C19) and (C13), we find ð∞Þ 0 and asymptotic form of Gl ðr; r ; qÞ given in Eq. (B30), Z Z ∞ k2dk k02dk0 ∞ k2dk k2l the UV divergence cancels out explicitly in Eq. (C20), and ðkk0ÞlG˜ ðhoÞðk; k0; εÞ − the quantization condition is thus reduced to the BERW 3 3 l 3 2 q2 0 ð2πÞ ð2πÞ 0 ð2πÞ k 2μ − 2μ formula, 2μq2lþ1 lþ1 3 l ε ¼ − ½cot δlðqÞ − i: ðC20Þ 4μω 2 Γð þ − Þ ð4πÞ2 δ ðqÞ − ð−1Þlþ1 4 2 2ω ¼ 0; ð Þ cot l q2 1 l ε C25 Γð4 − 2 − 2ωÞ Using an asymptotic form of the spherical Bessel function, where q and ε are associated by pffiffiffi πðkrÞl q2 P2 3 j ðkrÞ ⟶r→0 ; ð Þ þ ¼ ε þ ω n þ : ð Þ l lþ1 3 C21 2μ 2M 2 C26 2 Γðl þ 2Þ one can easily prove that

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