PRDPRVDAQPhysical Review DPhys. Rev. D2470-00102470-0029American Physical Society10.1103/527n-tjbrARTICLESLattice field theories, lattice QCDHigher order quantization conditions for two-body scattering with spinHIGHER ORDER QUANTIZATION CONDITIONS FOR TWO-BODY …CHANDLER, LEE, AND ALEXANDRUhttps://orcid.org/0009-0008-8786-5870ChandlerLucas*https://orcid.org/0000-0001-8169-3440LeeFrank X.†https://orcid.org/0000-0003-4547-1554AlexandruAndrei‡Physics Department, The George Washington Universityhttps://ror.org/00y4zzh67, Washington, DC 20052, USA
Contact author: lchandler27@gwu.edu
Contact author: fxlee@gwu.edu
Contact author: aalexan@gwu.edu
18June20261June2026113111145196March202619May2026Published by the American Physical Society2026authorsPublished by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.
We examine the Lüscher quantization condition to high order for the scattering of a spinless particle and a spin-1/2 particle in a periodic box. First, we derive the quantization conditions in a nonrelativistic framework up to total angular momentum J=11/2 in both cubic and elongated geometries, and for both rest and moving frames. Then, we introduce a method to transparently cross-check their convergence, using both quantized energy levels in the box and infinite-volume phase shifts for the same potential. We clarify how to incorporate spin-orbit coupling into the formalism and show in detail how the quantization conditions converge order by order in the various irreducible representations. In all, we validated 19 quantization conditions (12 in cubic box, 7 in elongated box). This is a necessary step in applying the method in precision studies of systems in finite volume with half-integer spin, such as meson-baryon scattering.
U.S. Department of Energy10.13039/100000015DE-FG02-95ER40907INTRODUCTION
Scattering is an indispensable tool in our understanding of nature at deeper and deeper levels, from the original Rutherford experiment on the structure of the atom, to modern nuclear and particle physics experiments on the structure and interaction of hadrons. Such physics is governed by quantum chromodynamics (QCD), the fundamental theory of the strong interaction. Theoretically, determination of hadronic properties at low to medium energies from the first principles of QCD remains fundamental but challenging. Because the interaction is strong, it requires nonperturbative methods.
In lattice QCD, numerical simulations based on Monte Carlo evaluation of the path integral of QCD are performed by placing the system on a discrete lattice with periodic boundary conditions, which leads to quantized energy spectrum in the system. Continuum but finite-volume results can be obtained by increasing the number of grid points and decreasing the lattice spacing at the same time. The energy spectrum in the finite volume is then connected to the infinite volume scattering phase shifts through quantization conditions (QC). A number of finite-volume approaches have been proposed for two-body scattering. The most well-known is the Lüscher method [1] which establishes a direct connection between the energy levels and phase shifts. Alternative approaches based on the finite volume include the HALQCD method [2–4] which extracts the interaction potential from Bethe-Salpeter amplitudes, and the integrated correlation function method (ICF) [5–7] which works directly with correlation functions and bypasses the energy spectrum determination.
The original Lüscher method has been extended to a variety of scenarios (such as moving frames, asymmetric lattices, multi- and coupled-channels, effective field theories, etc) to extract information about hadron-hadron interactions [8–26]. In the meson sector, such methods have been successfully applied in lattice QCD simulations [27–38]. With increasing computing power and algorithm improvements, such approaches are starting to bear fruit in the baryon sector [39–45] (see Ref. [46] for a recent review). The main challenge is to overcome the deterioration of signal-to-noise ratio when baryon are involved. Remarkable progress has also been made in finite volume approaches in the three-body sector [47–60] (see Ref. [61] for a recent review).
In the present work, we focus on the scattering of a spinless particle and a spin-1/2 particle in the Lüscher method. The study is motivated by the desire to apply the method in the baryon sector, especially meson-baryon scattering. We aim to conduct a comprehensive examination of the QCs in this sector, complete with cubic and elongated boxes, rest and moving frames, unequal masses, and higher partial waves (up to ℓ=6 or J=11/2). Furthermore, after deriving the QCs, we validate each one numerically to high precision. This is accomplished by separately computing the two main ingredients of the QCs for a toy potential: energy spectrum in the periodic box and infinite-volume phase shifts. It follows a similar study for systems with no spin [62].
The work is presented in four steps. In Sec. II we outline the derivation of the QCs and their projection to the irreducible representations (irreps) of various symmetry groups. In Sec. III we detail how to solve the Schrödinger equation in the box to obtain the energy spectrum for a test potential that includes a spin-orbit interaction. In Sec. IV we compute infinite volume phase shifts, discuss how to check the convergence of the QCs, and show a few examples. A brief summary and outlook is given in Sec. V. The matrix elements for the derived QCs are partially given in Appendix A. A supplement is supplied from which the full QCs can be retrieved [63]. Details of the group theory used for the derivation are provided in Appendix B.
QUANTIZATION CONDITION
There are different avenues to arrive at the same QCs, up to exponentially suppressed corrections. In this work, we work with the simplest approach, namely, nonrelativistic quantum mechanics. Despite its simplicity, the approach captures all the essential ideas and ingredients in the construction of QCs. The derivation is fairly involved mathematically and requires special functions (zeta functions) and extensive use of group theory. The introduction of spin-1/2 degrees of freedom adds even more complexity than the zero spin case and requires double-cover group theory. There exist previous studies on systems with half-integer spin [12,13,16,19] in limited scopes. A comprehensive reexamination aims not only to serve as an independent check of existing QCs in the literature but also as a foundation for new QCs at higher partial waves and in elongated boxes. In the following, we outline the essential steps in a reasonably self-contained derivation.
The starting point is two-particle scattering in the center of mass (CM) frame described by the Schrödinger equation, [-ℏ22μ∇2+V(r)]ψ(r)=Eψ(r),where μ=m1m2m1+m2 is the reduced mass of the system. We omit the spin degree of freedom in the discussion until near the end when a change of basis can be used to transform the matrix elements from a spin-0 system to a spin-1/2 one. In infinite volume, the wave function in the asymptotic region has the form of an incoming plane wave plus an outgoing spherical wave, ψ(r)=eikz+f(θ,k)eikrr,where k is the relative CM momentum and is related to the CM energy by E=ℏ2k22μ. The scattering amplitude can be expressed in partial-wave expansion, f(θ,k)=∑l=0∞(2l+1)fl(k)Pl(cosθ).Phase shifts enter via the partial-wave amplitudes, fl(k)≡e2iδl-12ik=Sl-12ik=Tlk=Klk(1-iKl),where alternative definitions via S matrix, T matrix and K matrix are also indicated. Phase shift δl is a real valued function of the interaction energy and encodes information about the nature of the interaction between the two particles, such as whether the force is repulsive (δ<0) or attractive (δ>0), whether a resonance is formed in the scattering process (a sharp change across π/2), and so on.
The standard procedure to determine the phase shift is by matching the asymptotic solution (where the interaction vanishes) in terms of spherical Bessel functions ψ∝[αljl(kr)+βlnl(k)] (or exterior solution), and the interior solution to arrive at the expression, e2iδl(k)=αl(k)+iβl(k)αl(k)-iβl(k).
In finite volume, a similar procedure can be followed. The main difference is the need to accommodate periodic boundary conditions imposed by the box. The details were first worked out by Lüscher [1]. The setup assumes that the box, of length L, is large enough such that the interaction range, R, cannot wrap around the periodic boundary of the box R<L/2. The resulting quantization condition can be expressed in a single matrix equation that includes all partial waves, det[diag{e2iδl(k)}-M(k,L)+iM(k,L)-i]=0.Here, diag{e2iδl(k)} is a shorthand for a diagonal matrix of all partial waves. The M(k,L) is a purely mathematical, dimensionless, Hermitian matrix function of CM momentum and box size. Its derivation for half-integer angular momentum systems is the focus of this work. The condition establishes a connection between quantized energy levels in a periodic box and phase shifts in infinite volume. In fact, the quantization condition is fairly general: It does not matter how the energy spectrum is generated in the box, be it quantum mechanics, effective field theories, or lattice QCD. The quantization condition will remain the same up to exponentially suppressed finite-volume corrections.
To obtain an analytic expression for the M(k,L) matrix, one starts by identifying a solution to the wave function in the exterior where the Schrödinger equation has the form of a Helmholtz equation, (∇2+k2)ψ(r)=0.A solution to this equation that satisfies the periodic boundary conditions is given by the Green’s function, G(r,k2)=1L3∑peip·rp2-k2,where the sum runs over the quantized box momenta. A complete basis can be formed by taking its derivatives (see Ref. [1]), Glm(r,k2)=Yl,m(∇)G(r,k2),where Ylm(r)=rlYlm(θ,ϕ) are the homogeneous harmonic polynomials. In order to match the exterior solution with the interior, one needs to expand Glm in terms of nl, jl, and Ylm, Glm(r,k2)=(-1)lkl+14π[nl(kr)Ylm(θ,ϕ)+∑l′=0∞∑m′=-l′l′Mlm,l′m′jl′(kr)Yl′m′(θ,ϕ)],where the M matrix is introduced as a way to connect to phase shifts in similar fashion to that in infinite volume. Performing the interior-exterior matching procedure with the wave function expanded in this basis leads to, ψ=∑lmdlmGlm(r,k2)=∑lmclm[aljl+blnl]Ylm.By equating the coefficients of nl and jl, the following condition holds, ∑l′m′cl′m′[bl′Ml′m′,lm-al′δll′δmm′]=0.Considering only nontrivial solutions results in the determinant equation, det[BM-A]=0,where A and B are diagonal matrices from al and bl, respectively. The matrix version of Eq. (5) gives the connection to the phase shifts, e2iδ=A+iBA-iB,and yields the QC as given in Eq. (6).
The matrix M emerging from the matching procedure takes the explicit form, Mlm,l′m′=(-1)lηπ3/2∑j=|l-l′|l+l′∑s=-jjijqj+1Zjs(q2,η)Clm,js,l′m′,where η is the z-elongation factor for noncubic box geometry, and the tensor coefficient are the Wigner 3j symbols, Clm,js,l′m′=(-1)m′il-j+l′(2l+1)(2j+1)(2l′+1)×(ljl′000)(ljl′ms-m′).It is expressed in terms of zeta functions defined by, Zlm(q2,η)=∑n˜Ylm(n˜)n˜2-q2,where the summation index n˜ and q are given by, n˜=(nx,ny,nz/η),q=kL2π.
The projection to half-integer angular momentum is achieved by a straightforward change of basis by coupling to spin-1/2 by J=ℓ+12, MJlM,J′l′M′=∑m,m′∑σ,σ′⟨lm,12σ|JM⟩⟨l′m′,12σ′|J′M′⟩Mlm,l′m′.where |JlM⟩ is the coupled basis of total angular momentum J, orbital angular momentum ℓ, and J substates M. Both the zeta function and the matrix M are mathematical functions dependent on dimensionless variables. One can note that the poles of the zeta function n˜=q2 correspond to noninteracting energies in the box. Interactions cause deviations from the poles and it is these deviations that are responsible for the phase shifts in the QCs.
We employ a commonly used shorthand notation in the presentation of the QCs, wlm(q2,η)≡Zlm(q2,η)ηπ3/2ql+1.This allows us to express M as a linear combination of wlm with purely numerical coefficients.
Symmetry-adapted quantization conditions
In finite volume, the continuous rotational symmetry is reduced to the finite symmetry group of the box so the quantization condition in Eq. (6) must take on the same symmetry properties. Multiplying both sides of Eq. (6) by det[M-i] gives, det[e2iδ(M-i)-(M+i)]=0.After distributing and refactoring it becomes, det[(e2iδ-1)(M-i(e2iδ+1)(e2iδ-1))]=0.Dropping the constant factor, the resulting matrix quantization condition can be written in standard form in terms of matrix elements, det[MJlM,J′l′M′-δJJ′δll′δMM′cotδJl]=0,where the delta functions should not be confused with phase shifts δJl.
The QC in Eq. (23) can be projected to the space of irreducible representations (irreps) of the box symmetry. This projection block-diagonalizes the QC matrix. The determinant then factors into smaller, independent ones, and partial-wave mixing is confined to what the symmetry actually permits. If the box energy levels are irrep-projected in matching fashion, it allows one to study the scattering system on an irrep by irrep basis.
The projection is achieved by a decomposition into the basis vectors of the symmetry group, expressed as, |ΓαJln⟩=∑MCJlMΓαn|JlM⟩,where Γ stands for a given irrep, α runs from 1 to the dimension of the irrep, n runs from 1 to n(Γ,J), the multiplicity of J in irrep Γ. Using this decomposition, the matrix is projected onto, ⟨ΓαJln|M|Γ′α′J′l′n′⟩=∑MM′(CJlMΓαn)*CJ′l′M′Γ′α′n′MJlM,J′l′M′=δΓΓ′δαα′MJln,J′l′n′Γ,where Shur’s lemma in group representation theory is used to achieve irrep orthogonality. Since the components α are not observables, one can average over them for multidimensional irreps, leading to the final form, ∏Γdet[MJln,J′l′n′Γ-δJJ′δll′δnn′cotδJl]=0.We see that the QC is completely factorized by the irreps of symmetry groups in the box.
The construction of the matrix elements MJln,J′l′n′Γ is technically involved, relying on standard but complicated group-theory steps. We have derived the QCs up to J=11/2. The QCs are rather lengthy, whose presentation is relegated to Appendix A. Even so, only partial results up to J=5/2 are printed in the tables. To facilitate the use of the QCs, we supply a supplemental file [63] from which all of the QCs derived in this work can be recovered in an easy-to-read and easy-to-use format that also eliminates typos if the QCs are copied. For completeness, all group theory details used to derive the QCs are placed in Appendix B.
In the following, we discuss a few aspects of the QCs.
Kinematics
We define P as the total momentum of the two-particle system in the lab frame, P=p1+p2.The momenta in the periodic box are quantized, P=2πLd˜=2πL(dx,dy,dz/η),p1=2πLn˜1=2πL(n1x,n1y,n1z/η),p2=2πLn˜2=2πL(n2x,n2y,n2z/η).The tilde indicates the possibility of an elongation of the box in the z direction via the factor η. Sometimes, the momenta can be referenced by the triplet of integers, with the above definition in mind. Rest frame refers to P=0 (the two particles have back-to-back momentum in the lab frame). Moving frames refer to nonzero P. In the literature, they are sometimes referenced simply by the boost vector d˜, or simply d (suppressing the tilde).
In the lab frame the total energy is, Elab=p122m1+p222m2.In the CM frame, the total energy is given by Ecm=k22m˜,where m˜ is the reduced mass and k the relative CM momentum (the two particles are back-to-back in the CM frame by definition).
For moving frames the CM energy is always lower than the lab energy, Ecm=Elab-P22(m1+m2),rendering moving frames a good option to reach smaller CM energies in the QCs.
We can find a relation for k in terms of the lab momenta through a Galilean transformation. Assuming that particle 1 has the same sign as P we have, p1=k+m1m1+m2P,p2=-k+m2m1+m2P.Solving for k gives, k=12(p1-p2)+m2-m12(m1+m2)P,=p1+12(m2-m1m1+m2-1)P.In terms of the box momenta, it reads k=2πL(n˜-12Ad˜),where p1 is represented by n˜=(nx,ny,nz/η), and A≡1+m2-m1m2+m1.If instead particle 2 has the same sign as P then this amounts to a swapping of m1 and m2 in A. We will show below that this has no practical effect on the quantization condition.
In the moving frame, the CM wave function picks up a complex phase under periodic boundary condition [8,10–12], ψ(r+nL)=eiπAn˜·d˜ψ(r).Because the phase factor depends on the particle masses and the boost d it can be difficult to implement in practice. We will show by working in the lab frame one can avoid this condition and make connection to CM frame through kinematics.
In moving frames, the summation grid for the zeta functions in Eq. (17) needs to be modified to incorporate the boost, Zlm(q2,d˜,η)=∑n˜∈Pd˜(η)Ylm(n˜)n˜2-q2.The associated shorthand retains the same form, wlm(q2,d˜,η)≡Zlm(q2,d˜,η)ηπ3/2ql+1.The new summation grid is, Pd˜(η)={n˜∈R3|n˜=η^-1(m-12Ad),m∈Z3}.The z-elongation projector η^-1 is defined as η^-1b=(bx,by,bz/η) for any b vector. The evaluation of these zeta functions is nontrivial but has been discussed in Refs. [11,12,18,31].
We can now see how the QC is affected by unequal masses. The mass dependence is contained in the zeta function. Should we swap the labels of m1 and m2 this will only change the A factor in the summation grid, producing an overall minus sign in the summation index. It leads to an overall sign change in the zeta function which does not change the form of the QC. However, the order does matter for the total energy of the system, see Eq. (29).
Group symmetry of rest and moving frames
The symmetry group of the cubic box for rest frames is the octahedral group, O with 24 group elements. Including parity (improper rotations) the group is doubled to Oh with 48 elements. With spin-1/2, it is further doubled to O2h with 96 elements. In the z-elongated box, the corresponding groups are D4 with 8 elements, D4h with 16 elements, and D24h with 32 elements. If moving frames are considered, these symmetry groups are further reduced to the so-called little groups. The little group corresponding to a moving frame is the subgroup that preserves the direction of motion. That is, if R is a rotation in the group, then Rd=d. Therefore, the little group depends on the direction of motion of the two-particle system and the box geometry. For spin-1/2 systems, the double cover little groups are named C2nv where n denotes that the box has an n-fold rotational symmetry about the direction of motion. For example, the little group for a moving frame in the z direction is C24v, whereas the little group for a moving frame in the xy plane is C22v. Table I summarizes the little groups in cubic and elongated boxes. We will consider three distinct little groups in this work: C22v,C23v, and C24v. Note that there is no C23v but the possibility of d=(1,0,0) little group C22v in the elongated box. Details of all the double cover groups considered in this work can be found in Appendix B.
I10.1103/527n-tjbr.t1
Double cover little groups for moving frames in cubic and elongated boxes. The lowest distinct patterns for the boost vector d=(nx,ny,nz) are shown, but integer multiples of d belong to the same little groups. Furthermore, all momenta related via a lattice symmetry with the ones below have the same little groups (this means all permutations for cubic, and nx and ny permutations for z elongated).
When moving from the infinite rotational symmetries of the infinite volume to the reduced symmetries of the box one ends up with a many-to-one matching from J to the box-irreps. This means that each QC only couples to a subset of J values in an infinite tower. This is in full display in the summary Table VII. The lowest partial wave phase shift can be predicted using the energy levels in the box if the higher partial waves can be neglected. This is known as the Lüscher formula. As the symmetry is reduced the mixing of partial waves increases. For example, in the rest frame the gap between the lowest two J in the G1g irrep in the cubic box is 3 (J=12 and J=72), whereas in the elongated box it is 1 (J=12 and J=32). For moving frames, the partial waves mix (for fixed J, the irrep couples to J=l+sandJ=l-s), causing this gap to always be 1.
Simplification of QCs
There are symmetries in the wlm functions that can be used to simplify the matrix elements. They arise from how the spherical harmonics transform under the group operations.
The standard property Yl-m=(-1)mYlm* translates directly to wl-m=(-1)mwlm*.
When a system is invariant under a mirror reflection about the xy plane, it leads to Ylm(θ,ϕ)=Ylm(π-θ,ϕ)=(-1)(l-m)Ylm(θ,ϕ). Which means wlm=0forl-m=odd. In particular wl0=0forl=1,3,5,…. This is valid for all systems with parity symmetry, which leads to a separation into sectors by parity. Note that in moving frames, parity is no longer a symmetry.
Due to the eimϕ dependence in Ylm, a system that is invariant under a π/2 rotation about the z axis has the constraint that eimπ/2=1. This means that wlm=0 for m≠0,4,8,…, regardless of l.
When the system is invariant under a mirror reflection about the xz plane then Ylm(θ,ϕ)=Ylm(θ,2π-ϕ)=Ylm*(θ,ϕ). This means all the wlm functions are real valued. However, the matrix elements M can have complex valued coefficients depending on basis vectors.
Invariance under a swapping of x↔y gives the constraint that wlm=eiπ2mwlm*. This constrains the relations between the real and imaginary parts of wlm depending on mmod4.
Similar to constraint (v), invariance under a mapping of x→-x leads to the constraint that wlm=eiπmwlm*. This causes the wlm to either be real for even m or purely imaginary for odd m.
These properties greatly simplify the presentation of the matrix elements. See Table II for a summary of all the constraints in each frame. Moreover, the M matrix is Hermitian, so we only need to list the lower triangular part of the matrix.
II10.1103/527n-tjbr.t2
Simplifications of the wlm in various frames for both cubic and z-elongated boxes due to symmetries of the spherical harmonics. We use the notation wlm≡wrlm+Iwilm. In the rightmost column we indicate which listed rule is responsible for the simplification. Rule (i) applies in all cases, as such, we only look at these rules in relation to m≥0.
dwlm SimplificationsRule{0,0,0}wlm=wrlm(iv)wlm≠0 for l even and m≡0mod4(ii) and (iii){0,0,1}wlm=wrlm(iv)wlm≠0 for m≡0mod4(iii){1,1,0}wrlm=0 for l-m odd(ii)wilm=0 for m≡0mod4(v)wilm=wrlm for m≡1mod4(v)wrlm=0 for m≡2mod4(v)wilm=-wrlm for and m≡3mod4(v){1,1,1}wilm=0 for m≡0mod4(v)wilm=wrlm for m≡1mod4(v)wrlm=0 for m≡2mod4(v)wilm=-wrlm for and m≡3mod4(v)
An important consideration in the use of these matrix elements is to note the conventions and notations used. There are multiple conventions in use in the literature for basis vectors, the definition of the w functions, etc. But since the QCs are invariant under a change of basis, the physical content is the same. Numerically, they all produce the same determinant.
In moving frames, condition (ii) no longer applies. For any moving frame other than d=(0,0,1), condition (iii) fails; and for moving frames with any momentum in the y direction, condition (iv) fails as well. This greatly reduces the simplification of the matrix elements. However, we can find a few more simplifications through the closure relation of the zeta functions [1], ∑m′=llDm′m(l)(α,β,γ)Zlm′=Zlm,where D is the Wigner rotation matrix for a transformation in the little group. Because this relation stems from the spherical harmonics we can apply it to all box geometries and boost directions. Using this relation for each group element gives a constraint though each constraint is not independent. A table detailing these constraints for various boost directions is given in Table III. The closure relation only provides new relations for d=(0,0,0) and d=(1,1,1) in the cubic box. All other simplifications from the closure relation are equivalent to previous simplifications.
III10.1103/527n-tjbr.t3
Additional relations derived from the closure condition in Eq. (40) for rest frame d={0,0,0} and moving frame d={1,1,1} in cubic box. They are used to further simply the matrix elements in the two cases.
d=0,0,0, group O2hw44→514w40w64→-72w60w84→131411w80w88→136522w80w104→-6665w100w108→-187130w100d={1,1,1}, group C3v2wr11→-w102wi22→-wr21wr31→3w304wr33→-145w30wi42→22wr41wr43→7wr41w44→514w40wr53→57w50+3314wr51w54→-514w50-8wr5121wr55→542wr51-w507wi62→14(-10wr61-6wr63)wi66→144(255wr63-922wr61)w64→-72w60wr65→611wr61+1511wr63wi76→363wi72-32i(21w70-6wr71+92wr73)33143w74→-2542w70-2483wr71+120wr736611wr75→1442w70-943wr71+21wr733311wr77→-1111311(26w70+21wr71+27wr73)wi82→12(66wr83-70wr81)wi86→33326wr81-3526wr83w84→131411w80wr85→31513wr83-27713wr81wr87→22113wr83-5513wr81w88→136522w80From nonrelativistic to relativistic QCs
Although the QCs are derived in a nonrelativistic framework, they can be transformed into their relativistic versions with only a few small modifications.
First, relativistic kinematics must replace the nonrelativistic one. Elab=p12+m12+p22+m22,Ecm=k2+m12+k2+m22.The two are related by, Elab=Ecm2+P2.Boost velocity and relativistic factor are given by, v=PElab,γ=11-v2=ElabEcm.The kinematical sequence in a calculation typically goes like this: for a given boost P, first Elab is determined in the box (Schrödinger equation, lattice QCD, etc), then γ factor, then Ecm, then CM k, then zeta function via dimensionless q.
Second, the summation grid in Eq. (39) of the zeta function is modified by, Pd(η)={n˜∈R3|n˜=γ^-1η^-1(m-12Ad),m∈Z3}.where the γ^-1 projection is applied in the direction of the boost d and the factor in Eq. (35) is replaced by, A=1+m12-m22Ecm2.Third, the wlm in Eq. (38) picks up a γ factor (see, e.g., Ref [12]), wlm(q2,d,η)≡Zlm(q2,d,η)γηπ3/2ql+1.
With these changes, all the QCs in this work can be used in relativistic studies.
TWO-PARTICLE ENERGIES IN THE BOX
Lüscher’s method establishes a relationship between the phase shifts in infinite volume and the energy spectrum of two-particle states in finite volume with periodic boundary conditions. Consequently, a comparison of the energy spectrum predicted by the quantization conditions with the spectrum obtained from an independent lattice Hamiltonian calculation provides a robust check of the newly derived QCs. In this section, we detail how to obtain the energy spectrum from solving the Schrödinger equation for a test potential.
Reduction of the lattice Hamiltonian
We want to obtain the energy spectrum in the continuum but in finite volume. Consider the general case of a box with dimensions L×L×ηL where η is the elongation factor in the z-direction. The Schrödinger equation HΨ=EΨ in the box frame (lab frame) with periodic boundary conditions has the Hamiltonian, H=-ℏ22m1∇12-ℏ22m2∇22+VL(|r1-r2|).Here VL is periodic version of the infinite-volume potential V, VL(|r1-r2|)=∑n1,n2V(|r1+n1L-r2-n2L|).Visually, the continuous space gets tiled into an infinite number of ηL3 boxes in which the potential is replicated. We solve the problem in a single box with periodic boundary conditions. Under this scenario, the potential is no longer rotationally symmetric. Instead, it takes on the symmetry of the box. For a system with spin, the potential has a spin-orbit interaction, in addition to the central part, V(r,p)=Vc(r)+Vso(r)l·s,where s=σℏ2 is the spin-1/2 operator in terms of Pauli matrices. This potential is implicitly dependent on the vectors r and p via the l operator. In spin space and Cartesian components it reads, V(r,p)=Vc(r)(1001)+Vso(r)(lzlx-ilylx+ily-lz)ℏ2withlx=ypz-zpy,ly=zpx-xpz,lz=xpy-ypx.Note that the potential is no longer local due to the presence of spin-orbit coupling.
The orbital angular momentum relative to CM is defined by, l=r1×p1+r2×p2-R×P,where R=(m1r1+m2r2)/(m1+m2) is the CM position and P=p1+p2 the total momentum of the system. l can be expressed in relative variables as, l=(r1-r2)×m2p1-m1p2m1+m2≡r×p,where r is the relative position and p=μ(v1-v2) the reduced momentum between the two particles. The momentum operator is p=-iℏ∇.
Implementing periodic boundary conditions requires the wave functions to satisfy, Ψ(r1+n1L,r2+n2L,ms)=Ψ(r1,r2,ms),where n1=(n1x,n1y,n1zη) and n2=(n2x,n2y,n2zη). Note that the n index appearing in periodic boundary conditions has η multiplied in the z-component, while it is divided in the context of discrete momenta, such as Eq. (28). The wave function carries an additional quantum number ms=±1/2 for the spin-1/2 particle.
We solve Eq. (48) numerically by discretizing the box into a lattice of Nx×Ny×Nz grid points with isotropic lattice spacing a. Both cubic (Nx=Ny=Nz) and z-elongated lattices (Nx=Ny=N and Nz=ηN) are considered. However, this brute force numerical approach presents a significant challenge: The Hilbert space for the two-particle state grows as N6, making the problem computationally intractable for all but the smallest lattices. This rapid growth arises because the Hamiltonian, as formulated, includes all possible momentum sectors simultaneously (or all possible rest and moving frames). Since our interest lies in probing only a handful of specific momentum sectors, this comprehensive formulation becomes an unnecessary computational hurdle. To address this challenge and to focus on individual momentum sectors more efficiently, we switch to a basis consisting of total momentum P and relative coordinates r in the lab frame, |P,r,ms⟩=∑meiP·m|m,m+r,ms⟩,where the |n1,n2,ms⟩ notation is the ket in the position representation for two particles. The change of basis is realized by projecting to P by summing over all positions separated by r. In this decomposition, the 6D basis |n1,n2,ms⟩ of all momenta is reduced to a 3D basis |P,r,ms⟩ of fixed momentum. The Hilbert space grows with N3 In the new basis, giving a significant improvement in scalability and making studies on larger lattices more feasible.
To discretize the lattice Hamiltonian we approximate the gradient operator by a two-point symmetric stencil, ∇ψ(rμ)=∑k=13ψ(r+ak^)-ψ(r-ak^)2a+O(a2).Similarly, using a centered three-point stencil, the Laplacian operator is, ∇2ψ(r)=∑k=13ψ(r+ak^)-2ψ(r)+ψ(r-ak^)a2+O(a2).The change of basis leads to the reduced problem Hψ(P,r,ms)=Eψ(P,r,ms) where the lattice Hamiltonian is given by, H|P,r,ms⟩=-ℏ22∑k=131a2[(eiPkam1+1m2)|P,r+ak^,ms⟩+(e-iPkam1+1m2)|P,r-ak^,ms⟩-2(1m1+1m2)|P,r,ms⟩]+Vc|P,r,ms⟩+Vso-iℏ4a(m1+m2)εijkσirj[(m2e-iaPk+m1)|P,r-ak^⟩-(m2eiaPk+m1)|P,r+ak^,ms⟩].The discretization error of this Hamiltonian comes in at O(a2). For a system at rest (P=0), it coincides with the familiar form in the CM frame for relative motion, H|0,r,ms⟩=-ℏ22μ∑k=1312a2(|0,r+ak^,ms⟩+|0,r-a,ms^⟩-2|0,r,ms⟩)+V(r,p)|0,r,ms⟩,where μ=m1m2/(m1+m2) is the reduced mass and V is the potential in Eq. (50). We emphasize that the Hamiltonian in Eq. (58) is more general than Eq. (59): it stays in the lab frame so the standard periodic boundary condition |P,r+nL,ms⟩=|P,r,ms⟩ applies in the reduced basis, and moving frames are automatically incorporated with P as an input in the Hamiltonian.
The low-lying spectrum of the reduced Hamiltonian can be obtained quickly for lattices up to 243 on a laptop computer. To accelerate the convergence to the continuum we use an improved version with a centered six-point stencil for the gradient and a centered seven-point stencil for the Laplacian, H|P,r,ms⟩=-ℏ22∑k=13-1180a2[-2(e3iPkam1+1m2)|P,r+3ak^,ms⟩+27(e2iPkam1+1m2)|P,r+2ak^,ms⟩-270(eiPkam1+1m2)|P,r+ak^,ms⟩-2(e-3iPkam1+1m2)|P,r-3ak^,ms⟩+27(e-2iPkam1+1m2)|P,r-2ak^,ms⟩-270(e-iPkam1+1m2)|P,r-ak^,ms⟩+490(1m1+1m2)|P,r,ms⟩]+Vc|P,r,ms⟩+Vso-iℏ60aεijkσirj[(m2e-3iPka+m1)|P,r-3ak^,ms⟩-9(m2e-2iPka+m1)|P,r-2ak^,ms⟩+45(m2e-iPka+m1)|P,r-ak^,ms⟩-(m2e3iPka+m1)|P,r+3ak^,ms⟩+9(m2e2iPka+m1)|P,r+2ak^,ms⟩-45(m2eiPka+m1)|P,r+ak^,ms⟩].The discretization error of this Hamiltonian comes in at O(a6).
Having obtained the lattice Hamiltonian in the reduced |P,r,ms⟩ basis, we need to take the continuum limit to obtain box levels from lattice levels. This is done by increasing the number of grid points and decreasing the lattice spacing simultaneously while keeping the box size fixed, lima→0N→∞Na=L.Since the discretization error is known to behave as O(a6), we perform a linear extrapolation in a6 using three lattices.
We will employ two potentials in this work. One is to check our energy determination method and another to check the quantization conditions later. The first is V(r1,r2)=(-V0+V1|r1-r2|4+cℓsℓ·s)e-β|r1-r2|2,where applies to both 6D and 3D cases if we recall r=r1-r2. The parameters are set to V0=4GeV, V1=116fm-4, cℓs=1GeV-1fm-2, β=18fm-2, m1=m2=2GeV. This potential is chosen because it is studied in detail in a previous work [64] using an independent, Fourier basis method. The potential admits a wide variety of bound and scattering states across partial waves and energies suitable for testing purposes. In a box with periodic boundary conditions, the relative coordinate r must be replaced by the displacement corresponding to the shortest distance between the periodic images of the two particles. The adjustment arises from the situation where one particle is closer to an image rather than the other particle. This results in the shortest path between them corresponding to going through the boundary rather than across the box. This can be accounted for with the definition, ri={r1i-r2i-sign(r1i-r2i)Lif|r1i-r2i|≥L2r1i-r2iif|r1i-r2i|<L2.
We perform two checks to verify the correctness of our implementation of the |P,r,ms⟩ basis. The first test ensures that this reduced basis accurately reproduces the results in the original basis as defined in Eq. (48). To achieve this, we solve a cutoff version of the 6D Hamiltonian on a 43 lattice. The modification involves imposing a hard cutoff on the potential for |r1-r2| greater than two lattice spacings. This cutoff prevents interactions between the particles and their periodic images. With this modified 6D Hamiltonian we solve for all 8192 eigenvalues, which encompass all 64 discrete total momentum sectors of the system, P=2πa(iNx,jNy,kNz)wherei=0,…,Nx-1,j=0,…,Ny-1,k=0,…,Nz-1.Separately, we compute the 128 eigenvalues from the projected 3D Hamiltonian in Eq. (58) on the same lattice and potential, using each of the 64 P’s in Eq. (64) as an input. Perfect agreement is achieved between the 6D and 3D results, for both the three-point stencil in Eq. (58) and seven-point stencil in Eq. (60) versions of the Hamiltonian. The same check is carried out for z-elongated lattices (Nx=Ny≠Nz).
The second check is to confirm that our implementation produces the correct energy spectrum in the continuum. We solve Eq. (60) without a potential cutoff on three lattices of 323 with a=1.125fm, 403 with a=0.9fm, and 483 with a=0.75fm. After continuum extraction we compare our results with those found in [64] which uses the same potential and parameters. We find perfect agreement.
Symmetry-adapted energy levels in the box
In order to match the energy levels to that of the symmetry-adapted QCs in Eq. (26), we project the eigenvectors of the lattice Hamiltonian to the irreps of the same symmetry group. To do this we construct projectors PΓ,λ according to, PΓ,λ|P,r,ms⟩=dΓg∑k[Dλ,λΓ(Sk)]*|SkP,Skr,ms⟩,where Γ is the irrep, λ the representation row, dΓ the dimension of the irrep, g the total number of elements in the symmetry group, Sk the symmetry operation, and Dλ,λΓ(Sk) its representation matrix. We say that if the norm of a projected eigenvector, PΓ|P,r,ms⟩, is nonzero then the eigenvector and its corresponding eigenvalue belong to the λ row of the irrep Γ (we assume no accidental degeneracies). An equivalent way to classify the energy levels is to project the Hamiltonian matrix into an irrep first, then diagonalizes it to obtain its eigenvalues [62].
CONVERGENCE CHECKS
In this section, we carry out a numerical check of the QCs derived in Sec. II. A QC is a function of finite-volume energy and infinite-volume phase shift. Our strategy is to obtain the energy spectrum and phase shifts independently for the same interaction potential, then feed the phase shift into the QC to predict the energy spectrum, and compare it with the independently computed spectrum. A QC is considered converged and checked if the two spectra agree with each other. This is done by including more and more partial waves in the QC.
The standard Lüscher formula utilizes the energy levels in the box determined from a certain method, but only retains the lowest partial-wave in the QC in order to make a prediction for its phase shift. By considering higher order partial waves, one can assess how reliable the prediction is for each QC in this approach.
The energy spectrum in the box has been obtained to high precision by solving the Schrödinger equation in Sec. III. The infinite volume phase shifts are computed in the following.
Infinite volume phase shifts
In infinite volume, the Schrödinger equation in Eq. (1) in CM frame with a central potential V(r) admits solutions with spherical symmetry. The solution in spherical coordinates can be expanded in partial waves ψ(r)=∑lRl(r)Ylm(θ,ϕ) where Ylm is spherical harmonics. The radial wave function Rl(r) can be found by solving the standard radial equation, [-ℏ22μd2dr2+l(l+1)ℏ22μr2+V(r)]ul(r)=Eul(r),where ul(r)=rRl(r) is the auxiliary radial wave function. Depending on the potential, the solution can be bound states with quantized energies, or scattering states with continuous energies. For scattering states the boundary conditions are limr→0ul(r)=0,limr→∞ul(r)∝eiδlsin[kr-lπ2+δl],where k=2μE/ℏ2 is the relative back-to-back CM momentum and δl is the phase shift for partial-wave l. The -lπ/2 term is to account for the repulsive centrifugal barrier term l(l+1)ℏ22μr2 and is inserted to ensure that the phase shift in this definition vanishes when the potential itself vanishes. Conventionally, to obtain the phase shift it is necessary to solve a second-order differential equation from the origin to the asymptotic region, then match the solution with an appropriate sine function (the interior-exterior match procedure discussed in Sec. II). Here we resort to the variable phase method[65] which requires solving only a first-order differential equation, dδ˜l(k,r)dr=-U(r)k[cosδ˜l(k,r)j^l(kr)-sinδ˜l(k,r)n^l(kr)]2,where U(r)=2μℏ2V(r), and j^l(kr)≡(kr)jl(kr) and n^l(kr)≡(kr)nl(kr) are modified spherical Bessel functions. The phase function δ˜l(k,r) is integrated from the origin where δ˜l(k,0)=0 to the asymptotic region where the potential is negligible; then the scattering phase shift is obtained directly as the asymptotic value δl(k)=limr→∞δ˜l(k,r).
For checking the QCs in this section, we consider a second, simpler, and repulsive potential in the form, V(r)=(V0+cℓsℓ·s)e-βr2.The wave function is the eigenstate of {J2,Jz,ℓ2,s2} which we label as |JMℓ⟩ with s=1/2 suppressed for simplicity. In spherical coordinates, it has the form ψ(r)=Rℓ(r)YJMℓ(θ,ϕ) where YJMℓ is the two-component spinor spherical harmonics. The total angular momentum is J=ℓ+s. For a given partial wave ℓ, there are two possible J values: J±=ℓ±1/2. The spin-orbit coupling in this basis splits into two branches, ℓ·s=12[J(J+1)-ℓ(ℓ+1)-34]={ℓ/2forJ=ℓ+12(allℓ)-(ℓ+1)/2forJ=ℓ-12(ℓ≠0)The s-wave (ℓ=0) resides in the J=ℓ+12 branch. The potential becomes central and diagonal in this basis, VJℓ(r)=(VJ=ℓ+12(r)00VJ=ℓ-12(r)).Thus for each partial wave l, the two states J=ℓ±12 can be treated separately in the infinite volume. In the finite volume, however, the two remain coupled as in Eq. (69) or in the lattice Hamiltonian Eqs. (58) or (60). The two decoupled central potentials are VJ=ℓ+12(r)=(V0+12ℓcℓs)e-βr2,VJ=ℓ-12(r)=(V0-12(ℓ+1)cℓs)e-βr2.We see that the effect of the spin-orbit term is to modify the V0 value. As ℓ increases, the J=ℓ+12 branch is more and more repulsive. At the same time, the depth of the J=ℓ-12 decreases with ℓ so the phase shift in this branch is expected to have smaller magnitudes at fixed energies.
For the parameters, we consider two particles mimicking the pion and nucleon: a spin-0 particle of mass m1=0.138GeV and a spin-1/2 of m2=0.94GeV, and the same V0=4GeV and β=18fm-2 as in the first potential in Eq. (62). For the strength of the spin-orbit coupling, we choose a value of csℓ=25GeV-1fm-2 which leads to sizable and well-separated phase shifts but still keeps the potential in the J=ℓ-12 branch (including the centrifugal barrier) repulsive over the range of ℓ and k values considered. Phase shifts for partial waves up to l=5 are shown in Fig. 1 for both branches. The goal is to feed these phase shifts into the QCs and confirm that the energy spectrum produced is converging to the energy spectrum derived from the lattice Hamiltonian Eq. (60).
110.1103/527n-tjbr.f1
Phase shifts as a function of CM k from the test potential for partial waves up to l=5 for spin-up branch in Eq. (72) (top) and spin-down branch in Eq. (73) (bottom) states. The s-wave (l=0) resides in the spin-up branch.
Results and discussion
The range of the test potential in this section is about 12 fm so a box of size 36 fm is sufficient to make the finite-volume effects negligible. While such large volumes are not yet feasible in current lattice QCD calculations, we employ it here in order to get a precise check. We calculate the energy spectrum for three different lattices: 403 with a=0.9fm, 483 with a=0.75fm, and 603 with a=0.6fm, then extrapolate it to the continuum limit. The extrapolation is a function of a6, the error present in the Hamiltonian from the seven-point stencil approximation in Eq. (60). Figure 2 shows an example of this continuum extrapolation alongside that from the standard three-point stencil with O(a2) error. One sees that both extrapolations reach the same result, but the seven-point function converges much faster.
210.1103/527n-tjbr.f2
Continuum extrapolation of the 4th lowest level in the G1g irrep rest frame in a cubic box of L=36fm. Three lattices are used: 403 with a=0.9fm, 483 with a=0.75fm, and 603 with a=0.6fm. The red points are from the three-point stencil in Eq. (58) and the blue points are from the seven-point stencil in Eq. (60). The fitted curves and forms are also displayed.
For the elongated box, we use the same lattice spacings and length in the x and y directions, but we elongate the z direction by a factor of η=1.5. Calculation of a large number of energy states is computationally expensive due to the Hamiltonian being an ≈105×105 matrix. For this reason, we only look at the 100 lowest eigenvalues for each QC. Each energy level in a d-dimensional irrep will be d-fold degenerate. In some QCs, this degeneracy significantly reduces the number of unique energy values studied.
Our objective is to check the validity of the derived QCs in Eq. (26) to higher partial waves. In lattice QCD simulations, energy levels are used to predict phase shifts. Since each QC is a single condition that couples to an infinite tower of partial waves, generally speaking, only the lowest phase shift in a given irrep can be isolated (Lüscher formula). One naturally wonders: what about the higher partial waves? How do they influence the prediction of the lowest partial wave? To answer this question, we turn the typical usage of a QC in reverse: we feed the infinite volume phase shifts to the QC, then numerically find its roots, then compare the resulting levels to the box levels determined separately from the Schrödinger equation. We check convergence order by order according to J (for fixed J, ℓ can take J±1/2): order 1 has only the lowest J in the QC, order 2 has the next lowest J added, and so on. Usually, we check each QC up to J=112; if this does not reach convergence, we then include J=132.
Before going to higher orders, we show the overall quality of phase shift prediction from the Lüscher formula in Fig. 3 (top) for selected irreps in the cubic box. The Lüscher formula is the QC truncated to include only the lowest partial wave. For this special case, we can solve the inverse problem: given an energy level, we can compute a phase shift. Note that this is not possible for any higher order truncations of QCs. The black dots in the figure indicate the solution of the inverse problem, using as inputs the box energy level computed by solving the Schrödinger equation. The red crosses correspond to using the Lüscher formula to compute energy levels given the lowest partial wave. The G1g irrep gives the best prediction for phase shift δ120 (s-wave), with the lowest 5 energy levels reproducing the phase shifts. Note also that the crosses are very close to the black points in this case. The 6th level is completely off because it occurs very close to a noninteracting energy level. In the absence of higher partial waves, this level would be exactly equal to the noninteracting energy, but it gets slightly shifted due to the presence of higher partial waves. So these shifts are not connected to the lowest partial wave, and using the Lüscher formula produces incorrect results (in special cases, you can use these levels to extract the next partial wave [62]). The Hu irrep gives the best prediction for phase shift δ321 (p-wave). Out of the 8 lowest energy levels, levels 1, 3, 4, 5 reproduce the phase shifts, but not levels 2, 6, 7, 8 for the same reason as in the G1g case (note that the second energy level has no corresponding cross).
310.1103/527n-tjbr.f3
Top row: phase shift reconstruction from Lüscher formula (lowest partial wave) from the box energy levels in G1g and Hu irreps in the rest frame, and G2 irrep of C24v for moving frame d=(0,0,1), all in the cubic box. Black points are predicted phase shifts from box energy levels. Red crosses show the energy levels predicted by the Lüscher formula. The red and blue curves show the two largest phase shifts active in the irrep. Vertical gray lines indicate noninteracting levels. Bottom row: same as the top row, but in the elongated box.
The G2 irrep of C24v is an example of a moving frame. The story here is a bit more complex: here we have mixing between different parities and the next strongest phase-shift is quite large. In our setup, every noninteracting energy is at least doubly degenerate due to the spin, and in the presence of interactions we end up with pairs of energy levels close by, a feature that Lüscher formula cannot capture. The spin-partners have the same ℓ and belong to neighboring J’s. In the rest frame irreps, for lower values of J, the neighboring J’s belong to different irreps. While at higher J, deeper in the spectrum, the irreps start to couple to neighboring J’s and show the spin-partners. For example, in the G1g, the first neighbors to appear are J=72 and J=92. Whereas for moving frames, the irreps couple to neighboring J’s from the beginning, in the C24v, little group the G2 irrep couples to J=32 and J=52. The crosses show that the Lüsher formula predicts an energy level bracketed by the spin-pair. However even though the pair is very close in energy, the resultant phase shifts are very different, indicating that the inverse problem is very sensitive. The dominant phase shift is δ321 (same as Hu in the rest frame). Using the Lüscher formula here is clearly wrong. As we see later using higher order QC reproduces the correct spectrum very precisely.
Figure 3 (bottom) shows the situation for the same QCs in the elongated box. Overall, we see that the quality is about the same as in the cubic box, even though the levels are more packed due to elongation. Their detailed convergence tables can be found in the supplement [63].
We now turn to the impact of high partial waves in the QCs. Since the effects depend on the QC under consideration to varying degrees (some are quite small), we need an objective criterion to judge the quality of convergence. To this end, we introduce a numerical χ2-measure, χ2=(kbox-kQC)2(kbox-klat)2,where kbox is the energy level from the continuum extraction of the three lattice levels, kQC is the level predicted from the QC by root-finding, and klat is the energy level on the lattice with the finest spacing. The idea behind this criterion is to compare the difference between the predictions of the QCs and the error from the lattice. Since the difference between the box levels and levels from the finest lattice is typically O(10-7), the χ2 measure is very sensitive to small changes in the QCs. A nonconvergent energy level blows up the χ2. Note that the χ2 introduced here is not to be confused with the χ2 measure used in curve fitting. Here, convergence means χ2 approaches zero as opposed to a value close to 1.
Let us return to the G1g irrep in Fig. 3 as the first example to showcase how convergence is checked by the χ2 measure. Table IV lists the order-by-order convergence of the lowest 6 energy levels in this QC. The first 5 levels are typical of the QC convergence; as new l values are included in the QC (every other order) the χ2 value drops significantly until it approaches ≈10-3 at which point the values in χ2 start to approach machine-precision and the precision from further orders cannot be captured here. We see the importance of using a numerical measure to check convergence. Visually, the lowest 5 levels all fall on the red curve at order 1, giving the impression that they reproduce the infinite volume phase shifts. But Table IV reveals that only level 1 has a small χ2 at order 1. The next 4 levels only converge by order 3 which means two more partial waves are added to the lowest one in the QC. Only even ℓ=0, 4, 6 partial waves contribute because G1g is an even-parity irrep by (-1)ℓ. There is no multiplicity to order 4 in this QC.
IV10.1103/527n-tjbr.t4
Convergence of G1g by adding higher partial waves in the rest frame of cubic box, as measured by χ2 in Eq. (74). The 4 orders are labeled by Jln=1201,7241,9241,11261 (J total angular momentum, ℓ orbital angular momentum, n multiplicity). All energy levels are represented by CM momentum k in MeV. The value marked with a red star (*) is a noninteracting level in the box replacing the pinched energy level not captured by the QC at this order. The last line is again the pinched 6th level but with the convergence measured by χ22 in Eq. (75), see discussion in the text.
LevelBox LLatticeOrder 1Order 2Order 3Order 4χ2 order 1χ2 order 2χ2 order 3χ2 order 419.357389.357379.357389.357389.357389.357380.001196650.0007161570.06898080.0689826239.092639.092639.092339.092439.092639.092613999.48759.090.002706110.00295507355.360255.360255.3655.3655.360255.36024237.632659.142.72282.27047465.242265.242265.22865.231165.242265.24222.45×1061.51×1060.003396820.00415965571.655471.655471.652971.653471.655471.655474016.445589.814.083711.9186676.911376.911376.8827*76.891576.911276.91121.72×107197.783160.295676.911376.911376.8827*76.891576.911276.91120.4815.50×10-64.45×10-6
Level 6 deserves special attention. It has no solution even at order 1. The reason is that this level is extremely close to a noninteracting level which is a singularity (pole) of the zeta functions, see Eq. (17). We say the level is “pinched” and use a red star and the noninteracting level instead to label it in the convergence table. The resolution is in the inclusion of higher partial waves which can pull the level away from the pole. These pinched points are a major indicator in the convergence analysis. A single pinched point blows up the total χ2 value for a QC.
The pinched level can give the impression that the QC is not converging. Or that if it is converging, it could still far from the box level. To check this possibility, we introduce a modified measure, χ22=(kbox-kQC)2(kbox-kfree)2,where noninteracting energy level kfree replaces klat in the denominator. The idea behind this is to note that interaction does not alter the number of energy levels in the box; it merely shifts them. This χ22 measures the convergence of the QC against that energy shift, as indicated in the last line of Table IV. Using this measure, we see that the pinched level is “unpinched” when higher partial waves are added, and normal convergence is observed. In the following, we only use the regular χ2 in the convergence tables, knowing that whenever there is a pinched level, the χ22 measure can properly account for its convergence.
Next, we look at the Hu irrep in Fig. 3 which has two pinched levels (6 and 8). In fact, levels 6, 7, and 8 are degenerate in the noninteracting case and the interaction only separates them by a small amount, leaving them all very close to the free particle pole. Typically, only one of these levels will be found at order 1. The inclusion of higher J waves is required to achieve convergence for each level individually, as illustrated in Table V. One sees that the QC converges at order 5, as opposed to G1g at order 3. It means at order 5, phase shifts δ321, δ523, δ723, δ925, δ1125 are contributing in the QC. This QC is an example of how higher partial waves and high precision work together to achieve convergence, especially when near degenerate levels are involved. Only ℓ=1, 3, 5 partial waves contribute because Hu is an odd-parity irrep. There is no multiplicity to order 5 in this QC.
V10.1103/527n-tjbr.t5
Similar to Table IV, but for the Hu irrep in the rest frame of the cubic box. The 5 orders are labeled by Jln=3211,5231,7231,9251,11251. All the klat levels match the box levels to 6 decimal places so we omit the column to save space.
LevelBox LOrder 1Order 2Order 3Order 4Order 5χ2 Order 1χ2 order 2χ2 order 3χ2 order 4χ2 order 5135.36535.360535.362735.36535.36535.3652.21×1076.12×10612.2510.670.49248.666148.624948.63448.666148.666148.66611.07×10116.51×101022232.711058.91.61350.862850.861850.862350.862750.862750.86281.42×1054.13×104302.29152.493.18×10-4461.212661.171561.191461.212361.212361.21261.55×1084.11×1077386.835711.040.0083569.748469.694969.721169.747869.747969.74846.79×1071.77×1079118.627291.120.031676.884776.8827*76.8827*76.882776.883376.8847200496.96881.75.79777.017376.882777.030977.015677.015877.01733.03×1083.12×10647288.336145.90.31877.205876.8827*76.8827*77.205677.205677.20582013.141476.960.90
The G2 irrep in Fig. 3 offers a prime example for moving frames. The idea of using moving frames is to add more coverage at smaller energies than those in the rest frame. Indeed, we see about 4 times more energy levels in this QC than in Hu at the same cutoff, but they do a poor job reproducing the infinite volume phase shifts. The reason is partial-wave mixing due to loss of parity in moving frames.
In typical lattice QCD applications, one inserts the finite-volume energy levels into the quantization conditions and solves for the phase shifts. In practice, however, one can usually only isolate the lowest partial wave in a given irrep from a single QC, so the QC is commonly truncated at leading order. This truncation discards contributions from higher partial waves, i.e., it is an approximation of the full Lüscher relation Eq. (26), and it is only reliable when those higher partial waves do not contribute greatly to the energy states in question. In the rest frame this often appears to be a mild assumption, since the lowest partial waves in an irrep are typically well separated. In moving frames, however, the loss of parity ensures partial wave mixing; both even-ℓ and odd-ℓ appear and the distance between the lowest two partial waves in any irrep, l1st-l2nd, is guaranteed to be 1, so one should generally expect larger truncation effects in moving frames. Even for levels that are well separated from the noninteracting poles, the reconstructed phase shift in G2 is still quite poor. Within our study, this kind of degradation is typical across all moving frames.
Table VI shows order-by-order convergence in G2 of C24v for all 22 levels displayed, including 4 pinched levels (16, 18, 19, 21). At order 1, χ2 is very high across the board. This is due to the fact that the levels from the finest lattice klat which are used as a measuring stick in Eq. (74), are already close to the box levels (to 6 significant figures). It takes 5 orders for most of the lower levels to converge; some higher levels are still not very convergent. At order 5, ten phase shifts δ321, δ322, δ522, δ523, δ723, δ724, δ924, δ925, δ1125, δ1126 are contributing in the QC (see Fig. 1). Due to mixing, even at order 1, we have both δ321 and δ322. We have to cut δ322 to make a prediction for δ321 Fig. 3. We see both even and odd ℓ partial waves participating due to loss of parity. Another symptom in moving frames is multiplicity starts early (at J=5/2 in this case), as opposed to no multiplicity in the rest-frame QCs G1g and Hu below J=11/2. Multiplicity also leads to larger QC matrices. The matrix size is 4×4 at order 4 for G1g, and 5×5 at order 5 for Hu. For G2 of C24v here, it is 20×20 at order 5. The size can be found by adding up the multiplicities through the orders. At order 2, the size is 6×6, whose full form is written out as an example in Eq. (A1) of Appendix A. Overall, convergence is slower than in rest-frame G1g and Hu due to partial wave mixing in moving frames.
VI10.1103/527n-tjbr.t6
Similar to Table V, but for the G2 irrep of the C24v little group for moving frame d=(0,0,1) in the cubic box. The 5 orders are labeled in pairs by Jln=(3211,3221),(5222,5232),(7232,7242),(9242,9252),(11253,11263).
LevelBox LOrder 1Order 2Order 3Order 4Order 5χ2 Order 1χ2 order 2χ2 order 3χ2 order 4χ2 order 5134.822934.713934.822434.822834.822934.82291.44×1010271490.2310.950.3050.059235.629935.619435.627835.629935.629935.62991.16×1084.82×106259.2067.360.185345.738745.648745.725845.738245.738745.73871.47×10103.00×108338550.17.80.118446.268146.108746.261646.26846.268146.26814.52×10107.35×10711384.5190.20.0659549.178848.940049.177349.178349.178849.17881.13×10115.01×106498884.33.41.35649.882349.85249.869549.882149.882349.88235.52×1089.87×10731533.1576.10.0240751.935551.856351.91851.934251.935551.93552.76×1091.35×108726572.425.80.016852.747852.55952.739952.747652.747852.74786.94×1091.22×1075631.977.10.0567957.436757.207357.401257.436157.436757.43671.05×10112.52×109644423.120.711.61058.284358.072458.279358.283258.284258.28431.02×10105.68×106246925.1087.41.561162.487362.265162.441362.484562.487362.48732.39×10101.02×1093.71×106143.382.791263.237962.990163.214963.237363.237763.23796.37×1095.52×10744844.83643.914.381369.490069.111269.481069.485969.490069.49002.12×1091.20×106250210.65.425117.91469.872869.794169.846569.87269.872369.87281.46×1081.64×10714002.86350.00.381573.031573.141473.007473.025573.030873.03155.88×1082.84×1071.79×10629245.6127.41673.371872.9724*73.294973.36273.371473.37182.20×1083.58×1068575.08.361775.519275.457275.435875.518175.518975.51927.20×1071.30×10825722.92744.089.81875.708175.0176*75.685275.70375.707275.7081.40×107701237.21335.56.91977.119577.0086*77.091677.117277.117577.11951.69×107110736.89199.03.242077.530977.466477.512177.524677.530877.53085.77×1074.91×106552083.254.78211.152177.865177.0086*77.844177.851777.86577.8651.29×1075.22×106384.684.122278.288678.141478.203678.28578.287578.28855.95×1081.98×108341964.33215.660.91
Lastly, we mention an interesting feature we observed in the K1 and K2 irreps of the C23v little group for moving frame d=(1,1,1) in the cubic box: they have the same convergence values (see Table VII). This is a consequence of the Kramer’s degeneracy theorem [66]. The theorem states that for every energy eigenstate of a half-integer spin system with time reversal symmetry, one can construct another energy eigenstate of the same energy through the time reversal symmetry. As a result, every energy eigenstate must be at least doubly degenerate for half-integer spin systems with time symmetry. This is handled naturally for most cases where the irreps are of even dimension and thus the energy levels have an even degeneracy. But in the case where a frame has one-dimensional irreps—what we have called K irreps—then those irreps must form into coupled pairs with the same spectrum of energy values. The group elements of these coupled irreps are related by K1(g)=K2(g)* and the basis vectors are reversed for spin-up and spin-down between them, i.e., if we write a basis vector as vJ,lΓ for irrep Γ then, for fixed J, vJ,J±sK1=vJ,J∓sK2. We also observed the relations in Ref. [12]. The property causes the convergence of these irreps to be identical when looked at J-by-J. The energy degeneracy are observed in the lattice values as well. Note that Kramer’s theorem only applies to the 1D irreps of spin-1/2 little groups. There is no such degeneracy in the 1D irreps of spin-0 little groups (such as A1 and A2, or B1 and B2 of C2v).
We have examined all 19 QCs using the same checks. Their phase shift reconstruction at order 1 and order-by-order convergence data can be found in the supplement [63].
SUMMARY AND OUTLOOK
We derived higher-order Lüscher quantization conditions (QCs) for the scattering of two particles with total spin s=1/2. We checked the correctness of our results numerically by comparing the QC predictions with the spectrum of two-particle states in a box. This spectrum is found by solving the Schrödinger equation of a simple nonrelativistic potential. Both the phase shifts in infinite volume and energy levels in finite volume are independently generated. Table VII gives an overview of all the QCs considered in this work, along with convergence checks.
VII10.1103/527n-tjbr.t7
Summary of convergence order by order according to the average χ2 measure in Eq. (74) for all the QCs discussed in this work. The orders in each QC are labeled by total angular momentum and multiplicity as J(n) with orbital angular momentum implicit (ℓ=J±1/2). For rest frames, ℓ is even for g irreps and odd for u irreps. For moving frames, both ℓ values are allowed. The N is the number of levels inspected. Both the QC matrices and detailed convergence data can be found in the supplement [63].
We derived QCs for numerous scenarios, including rest frames as well as selected moving frames for cubic and elongated boxes. We worked out quantization conditions up to J=11/2. The QCs agree with those in the literature up to J=7/2[12,13,19]. All higher orders in cubic and elongated boxes are new.
All QCs were checked against a simple potential with phase shifts that can be readily calculated through the variable phase method. The potential and box size were chosen to satisfy two conditions: (1) the exponentially suppressed finite-volume corrections of the QCs are negligible and (2) the energy levels are sensitive to contributions from partial waves as high as J=11/2. This allowed us to test with confidence high-order QCs and we find that the results agree to more than six significant figures. We performed these checks for several energy levels in each irrep.
For the most part, we find that elongated boxes work about as well as cubic ones. This gives a cost effective way of increasing the kinematic range for a small increase in lattice volume.
Due to the smaller number of irreps that couple to systems with spin than without [62], there is a larger amount of partial wave mixing in the QCs. This makes the extraction of phase shifts in the meson-baryon sector more difficult than for meson-meson scattering. The loss of parity in moving frames exacerbates the problem, with most moving frames having only one or two irreps.
Looking to the future, the results and methodology developed here can be used for lattice QCD calculations, both to connect finite box energy-levels to the infinite volume amplitudes and also to lay the ground-work for such calculations. In general, energy levels computed from lattice QCD can be directly connected to phase-shifts only when QCs can be truncated to the lowest partial wave in the relevant irrep. The precision of the predicted phase-shift is influenced by the size of the next higher partial wave in the irrep. The size of this effect is not known beforehand, and it depends on the energy range and box geometry. Given a good approximation of the phase-shifts relevant for a particular channel we can determine which irreps and box geometries are most likely to have reduced mixing. In the same vein, when we have an interaction model available, assuming that we can use it to compute infinite-volume phase-shifts, we can use the QCs validated in this work to determine which irreps and energy levels are most effective in constraining the parameters of the model. This information can be used to efficiently allocate computational resources.
ACKNOWLEDGMENTS
This research is supported by the U.S. Department of Energy under Grant No. DE-FG02-95ER40907.
DATA AVAILABILITY
There are no publicly available research data or software supporting this manuscript. Requests for further information or data should be sent to the authors.
MATRIX ELEMENTS FOR QUANTIZATION CONDITIONS
In this section, we present all the QCs derived in this work in terms of the matrix elements of MJln,J′l′n′Γ in Eq. (26). We show only partial results up to certain J value for every QC in both cubic and elongated boxes here. Only unique matrix elements are displayed. Since M is hermitian, we only show the upper triangular elements starting at the top left, and then go down each column until we hit the diagonal, and then go to the next column. The complete matrix for any QC at any order (up to J=112) can be recovered on demand from a python notebook in an easy-to-read and easy-to-use format in the supplement [63].
As an explicit example of how to use the following tables, we will build the 2nd order QC for the G2 irrep in the C24v little group of the cubic box. If one wished to start from scratch, they would begin with the group elements in Table XVIII and follow the procedure described in Appendix B 5 to derive the basis vectors (Tables XXI–XXVI). From there, the procedure detailed in Sec. II will produce the matrix elements contained in Tables IX–XV. To obtain the QC of the desired order, one must truncate the matrix in Eq. (26). The size of the matrix depends on the multiplicity and whether partial waves mix. Referencing Table VII, we find the G2 irrep has J(n) content J(n)=32(1),52(2),⋯ and parity is not a symmetry (l’s mix), the 2nd order QC is then, |M3211,3211-δ321M3211,3221M3211,5221M3211,5231M3211,5222M3211,5232M3221,3211M3221,3221-δ322M3221,5221M3221,5231M3221,5222M3221,5232M5221,3211M5221,3221M5221,5221-δ522M5221,5231M5221,5222M5221,5232M5231,3211M5231,3221M5231,5221M5231,5231-δ523M5231,5222M5231,5232M5222,3211M5222,3221M5222,5221M5222,5231M5222,5222-δ522M5222,5232M5232,3211M5232,3221M5232,5221M5232,5231M5232,5222M5232,5232-δ523|=0.One then looks at Table X for the matrix elements of the upper triangle. The lower triangle can be filled in by using the fact that these matrices are Hermitian. This would give the final form of the QC.
A reference guide with links to the tables is given here in Table VIII.
VIII10.1103/527n-tjbr.t8
Quick reference with links to all the QC tables in this appendix.
Matrix elements for rest frame in cubic box (group O2h) in each irrep. Since the matrix elements between the even and odd parity irreps can only differ by a minus sign we have combined the tables. The top sign and left l value corresponds to the even g irrep while the bottom sign and the right l in parenthesis gives the odd u irrep.
Matrix elements for moving frame d=(1,1,1) in cubic box (little-group C3v) for the three lowest partial waves in each irrep. The elements of K1 and K2 are related by an overall sign on certain elements. We condense these into one table where the top sign is for K1 and the bottom sign is for K2. Both K1 and K2 couple to the same J(l)’s.
Matrix elements for rest frame in z-elongated box (group D24h) up to the third partial wave in each irrep. Since the matrix elements between the even and odd parity irreps can only differ by a minus sign we have combined the tables. The top sign and left l value corresponds to the even g irrep while the bottom sign and the right l in parenthesis gives the odd u irrep.
Group table of D4h for rest frame in elongated box. The inversion is labeled by preceding with the letter I in the k and Oh columns. The E representations are given in terms of Pauli matrices.
Group elements of C22v in both the cubic and elongated boxes.
kC22vnωSk{α,β,γ}A1A2B1B2G11E{1,0,0}4πt1{0,0,0}1111g12C2a{1,1,0}πt19{0,π,π2}1-1-11g193E˜{1,0,0}2πt1{0,0,2π}1111−g14C˜2a{1,1,0}3πt19{0,π,5π2}1-1-11−g195IC2z{0,0,1}π−t4{0,0,π}11-1-1−g46IC2b{1,-1,0}π−t20{0,π,7π2}1-11-1−g207I˜C2z{0,0,1}3π−t4{0,0,3π}11-1-1g48I˜C2b{1,-1,0}3π−t20{0,π,3π2}1-11-1g20GROUP THEORY DETAILS
For completeness, we give the group theory details used to derive the QCs in the previous appendix. Due to spin degree of freedom, double-cover groups must be used.
Symmetry of the cubic box
The symmetry group of the cubic box is the octahedral group and is denoted by O. This group consists of the 24 orientation preserving rotations. This group is sufficient to describe integer angular momentum in the cubic box.
To account for half-integer angular momentum in the cubic box requires the double-cover of O, which we denote O2. This group is constructed by viewing a 2π rotation as different from a 4π rotation. As such, the number of group elements doubles. Each element g∈O now has a corresponding double cover element that we denote as g˜. If one describes the rotation using the rotation axis and angle, g=R(n→,ω) then g˜=R(n→,(ω+2π)mod4π). Or if described by active z-y-z Euler angles g=R(α,β,γ) then g˜=R(α,β,(γ+2π)mod4π). Both descriptions are provided in the tables below.
For nonmoving frames, parity is another symmetry of the cubic box. This is another doubling of the group elements corresponding to O2h=O2⊗{E,-E}. Where the subscript h denotes the inclusion of parity, and E is the identity element.
A useful tool when studying a symmetry group is to look at its irreducible representations (irreps). The number of irreps of a group is equal to the number of conjugacy classes. One can easily show that the group O has 5 conjugacy classes and thus 5 irreps. To determine the dimensionality of irreps we use the fact that the sum of the squares of the dimensions of the irreps equals the order of the group. |G|=∑idi2where di is the dimension of the ith irrep. This gives the Diophantine equation 24=a2+b2+c2+d2+c2. This has one solution with dimensions 1, 1, 2, 3, and 3. Using the convention set by Mulliken [67], these irreps are called A1, A2, E, T1, and T2, respectively. A1 is the trivial representation; every element is represented by 1. The A2 representation can be constructed from the A1 irrep using two facts. The first is that irreps are orthogonal in the sense that ⟨χi,χj⟩=1|G|∑g∈Gχi(g)¯χj(g)=δij.Where χi(g) is the character of the element g in the ith irrep. The second fact needed is that all elements in a given conjugacy class have the same character. Using these two facts, and noting that the size of the conjugacy classes are 1, 3, 8, 6, and 6 we see that the only way for the orthogonality relation to hold is for the two classes with six elements to be represented with -1. The real meaning of the subscripts G1/2 has to do with the symmetry properties of the irrep basis functions, but such details are not necessary to construct the irreps here. The construction of the E irrep proceeds in a similar way. One finds that the conjugacy classes must have characters 2, 2, -1, 0, and 0, respectively. One then must choose matrices with these traces that also form a homomorphism. We choose the five found in [68]. e1=(-1001),e2=(-12-3232-12),e3=(-1232-32-12),e4=(12-32-32-12),e5=(123232-12).The T1 irrep is then the standard 3×3 rotation matrices and the T2 follows the same pattern as the A1 and A2 irreps by negating the last 12 elements in the group. The T1 elements are generated via tk=e-i(n·J)ωk,k=1,…,24,where (Jk)ij=iεijk, and n and ωk are the rotation axis and angle in the order given below.
The double cover adds three more irreps called G1, G2 and H with dimensions 2, 2, and 4, respectively. These are the “spinor” irreps and are the only ones sensitive to the double-cover and antisymmetric with respect to it. That is, if DΓ(g) is an irrep of element g∈O2 then DΓ(g)=-DΓ(g˜) for Γ∈{G1,G2,H} and DΓ(g)=DΓ(g˜) for Γ∈{A1,A2,E,T1,T2}. The G elements are generated similar to the T elements but they halve the rotation angle so as to distinguish a 2π and 4π rotation. gk=e-i(n·σ)ωk/2,k=1,…,24.The four-dimensional H irreps are generated by hk=e-i(n·J)ωk,k=1,…,24.Where J are the generators of spin-3/2Jx=(03200320100103200320),Jy=(0-i3200i320-i00i0-i3200i320),Jz=(320000120000-120000-32).
Inclusion of parity doubles the number of irreps. Forming an even and an odd version of each irrep, denoted by the subscript g and u for gerade and ungerade (even and odd). The even irreps do not change with parity DΓ(g)=DΓ(-E*g), whereas the odd irreps are antisymmetric with respect to the inversion DΓ(g)=-DΓ(-E*g).
We have organized the following tables in the manner described here. By first listing the elements of the standard group without double-cover or parity, and then listing the double-cover elements, and then the parity elements. We denote the natural representation of the group elements by Sk. These are the matrices that act on the standard (x,y,z) coordinates. Sk happens to coincide with the T1u irrep. A faithful representation of the O2h group can be made with the G1 irrep by Γfaithful={DG1(g)⊗I2|g∈O2}∪{DG1(g)⊗(100-1)|g∈O2}.
Time reversal symmetry
A natural question that arises in symmetry discussions is “What happens if we include time-reversal symmetry in our group?” After all, the addition of parity and the double-cover doubled the number of group elements and provided more irreps to project onto. Unfortunately, since the time-reversal operator is antiunitary, it does not fit nicely into the standard representation theory. Representation theory with antiunitary elements is known as corepresentation theory. The theory of corepresentations was explored by Wigner [69] and Dimmock and Wheeler [70]. In short, the inclusion of antiunitary elements breaks certain structures of standard representation theory, e.g., the number of conjugacy classes is no longer equal to the number of irreducible corepresentations (ICRs). Ref. [71] lays out the resulting structure of ICRs in terms of the irreps of the unitary part of the group. Let G=G∪a0G, with a0 the antiunitary operator. Pick an irrep Γ of G.
Γ is real, Γ=Γ*. The antiunitary symmetry will not split the representation any further, providing no additional ICRs to project onto. Effectively, the ICR retains the same structure as the irrep in the unitary group.
Γ is “quaternionic,” Γ is equivalent to its conjugate but in a way that forces a quaternionic structure. The ICR will take the form Γ⊕Γ. Simply doubling the dimension of the irrep.
Γ is complex, Γ≠Γ*. The antiunitary element then swaps Γ↔Γ* and therefore the ICR will take the form Γ⊕Γ* to encompass the extra symmetry. In the case of the K irreps here this ends up merging them into one greater ICR of doubled dimension.
In all three cases, there is no further structure to exploit from the methodology used here.
Elongated box
The symmetry group of the z-elongated box is D4, It is the subgroup of O that preserves the (0, 0, 1) vector. The construction of the irreps follows that of the cubic box. One finds there are five irreps with dimensions 1, 1, 1, 1, 2 and identifies them as the A1, A2, B1, B2, and E irreps. The double cover, D42 doubles the group elements and adds two 2-D irreps, G1 and G2. Parity doubles the number of elements again and forms even and odd versions of each irrep. This brings the total number of elements to 32 with 14 irreps. We use the same notational convention as the cubic case to denote the elements in Table XVII.
Moving frames
When the two particle system has a nonzero total momentum, we say that it is in a moving frame. We denote the direction of total momentum as d=(nx,ny,nz) with ni∈Z. These moving frames single out a certain direction in space, reducing the overall symmetry of the box. These subgroups are called little groups and their elements are all elements, k from O2h (or D24h) such that Skd=d. In this work we have considered three moving frames in the cubic box and two in the elongated case: d=(0,0,1) and d=(1,1,0) in both cases and d=(1,1,1) in just the cubic box. Tables XVI through XX detail these little groups.
For permutations of these directions say, d=(1,0,0), then the irreps will stay the same but the group elements will change to new subgroups of the parent group. This will change the basis vectors and the overall structure of the QCs. However, physical results such as energy levels, phase shifts, and angular momentum from two similar boost directions must be the same. This change of boost direction is equivalent to a rotation of the zeta functions.
Basis vectors
The irreps of the continuum rotation group with J=0,12,32,…, are defined on the (2J+1) dimensional space spanned by the basis vectors |JM⟩. These basis vectors are the spin spherical harmonics and reduce to the standard spherical harmonics for J∈Z. These representations, while irreducible in the continuum, become reducible into the 16 irreps of O2h. This is to say, that certain subspaces in the space spanned by |JM⟩, are invariant under the symmetry transformations of the box. The basis vectors for this new subspace can be constructed from the projector PΓλ,λ′|JM⟩=dΓ|G|∑g∈G[Dλ,λ′Γ(g)]*∑M′=-JJDM′MJ(g)|JM′⟩where Γ is the irrep, λ a row from the irrep, dΓ the dimension of the irrep, |G| is the order of the group, DΓ(g) the representation of group element g, and DJ(g) the Wigner D matrix evaluated at the Euler angles corresponding to group element g. Then, if we rotate a projected vector from row λ, then we get S(g)PΓλλ|JM⟩=Dλ′λΓ(g)PΓλ′λ|JM⟩.The recipe for calculating the basis vectors is then as follows. For a given J and irrep Γ, we construct the projector for row 1. Then we perform a QR decomposition, PΓ11=Q†R, with Q a unitary matrix and R being upper triangular. If R=0 then this irrep does not couple to this J. If R≠0 then the Q† and R matrices will have the dimensions (2J+1)×k and k×(2J+1), respectively. The rank, k, of these matrices is the number of linearly independent vectors for this J. This is what is meant by the multiplicity of the irrep in J.
The k columns of Q†≡|Γ1Jn⟩, with n=1,…,k are the basis vectors corresponding to row 1. For a one-dimensional irrep, this is the end of the story. For higher dimensional irreps one can use these basis vectors to build the higher dimensions, |Γλ′Jn⟩=(PΓλ′1R-1)T,λ′=2,…,dΓ,where RR-1=Ik×k.
For improper rotations, one must insert a factor of (-1)J onto the Wigner D function in Eq. (B9) to account for the parity of the spin-spherical harmonics. The basis vectors obtained from this procedure are orthonormal, ⟨Γ′λ′,J′,n′|ΓλJn⟩=δΓΓ′δλλ′δJJ′δnn′.
All the basis vectors up to J=132 are included below. Table XXI contains the vectors for O2h, Table XXII for the D24h group in the elongated box, Table XXIII for the little group C24v for both box geometries, Table XXIV for the C23v little group in the cubic box, and Tables XXV and XXVI contain the basis vectors for the C22v little group in the cubic and elongated boxes, respectively.
XXI10.1103/527n-tjbr.t21
Basis vectors of O2h in terms of spherical harmonics Y(J,M) up to J=132 for the rest frame of the cubic box. Multiplicities are labeled by n and multidimensional components are indicated by α.
Basis vectors of D24h in terms of spi spherical harmonics Y(J,M) up to J=132 for the rest frame of the elongated box. Multiplicities are labeled by n and multidimensional components are indicated by α.
Basis vectors of C24v in terms of spherical harmonics Y(J,M) up to J=132 In both cubic and z-elongated boxes. Multiplicities are labeled by n and multidimensional components are indicated by α.
Basis vectors of C23v in terms of spherical harmonics Y(J,M) up to J=132 in the cubic box. Multiplicities are labeled by n and multidimensional components are indicated by α. The K2 basis vectors are the same as the K1 basis vectors simply swapping the l for the corresponding J. For example, the basis vectors for J=3/2,l=1 in K1 are the same as the J=3/2,l=2 basis vectors in K2.
Basis vectors of C22v in terms of spherical harmonics Y(J,M) up to J=132 in the cubic box. Multiplicities are labeled by n and multidimensional components are indicated by α.
Basis vectors of C22v in terms of spherical harmonics Y(J,M) up to J=132 in the elongated box. Multiplicities are labeled by n and multidimensional components are indicated by α.
ΓJlnαBasis vectorΓJlnαBasis vectorΓJlnαBasis vectorG112011Y(12,12)G172441Y(72,52)G1112551Y(112,72)2Y(12,-12)2Y(72,-52)2iY(112,-72)G112111Y(12,-12)G192411Y(92,-72)G1112651Y(112,52)2-iY(12,12)2Y(92,72)2Y(112,-52)G132111Y(32,-12)G192511Y(92,-92)G1112561Y(112,112)2iY(32,12)2-iY(92,92)2-iY(112,-112)G132211Y(32,-32)G192421Y(92,-32)G1112661Y(112,92)2Y(32,32)2-Y(92,32)2-Y(112,-92)G132121Y(32,32)G192521Y(92,-52)G1132611Y(132,-112)2-iY(32,-32)2iY(92,52)2-Y(132,112)G132221Y(32,12)G192431Y(92,12)G1132711Y(132,-132)2-Y(32,-12)2Y(92,-12)2iY(132,132)G152211Y(52,-32)G192531Y(92,-12)G1132621Y(132,-72)2-Y(52,32)2-iY(92,12)2Y(132,72)G152311Y(52,-52)G192441Y(92,52)G1132721Y(132,-92)2iY(52,52)2-Y(92,-52)2-iY(132,92)G152221Y(52,12)G192541Y(92,32)G1132631Y(132,-32)2Y(52,-12)2iY(92,-32)2-Y(132,32)G152321Y(52,-12)G192451Y(92,92)G1132731Y(132,-52)2-iY(52,12)2Y(92,-92)2iY(132,52)G152231Y(52,32)G192551Y(92,72)G1132641Y(132,12)2iY(52,-32)2-iY(92,-72)2Y(132,-12)G152331Y(52,52)G1112511Y(112,-92)G1132741Y(132,-12)2-Y(52,-52)2iY(112,92)2-iY(132,12)G172311Y(72,-52)G1112611Y(112,-112)G1132651Y(132,52)2-iY(72,52)2Y(112,112)2-Y(132,-52)G172411Y(72,-72)G1112521Y(112,-52)G1132751Y(132,32)2-Y(72,72)2-iY(112,52)2iY(132,-32)G172321Y(72,-12)G1112621Y(112,-72)G1132661Y(132,92)2iY(72,12)2-Y(112,72)2Y(132,-92)G172421Y(72,-32)G1112531Y(112,-12)G1132761Y(132,72)2Y(72,32)2iY(112,12)2-iY(132,-72)G172331Y(72,32)G1112631Y(112,-32)G1132671Y(132,132)2-iY(72,-32)2Y(112,32)2-Y(132,-132)G172431Y(72,12)G1112541Y(112,32)G1132771Y(132,112)2-Y(72,-12)2-iY(112,-32)2iY(132,-112)G172341Y(72,72)G1112641Y(112,12)2iY(72,-72)2-Y(112,-12)1Martin Lüscher, Two particle states on a torus and their relation to the scattering matrix, Nucl. Phys.B354, 531 (1991).NUPBBO0550-321310.1016/0550-3213(91)90366-62N. Ishii, S. Aoki, and T. Hatsuda, Nuclear force from lattice QCD, Phys. Rev. Lett.99, 022001 (2007).PRLTAO0031-900710.1103/PhysRevLett.99.0220013Takumi Iritani, Sinya Aoki, Takumi Doi, Shinya Gongyo, Tetsuo Hatsuda, Yoichi Ikeda, Takashi Inoue, Noriyoshi Ishii, Hidekatsu Nemura, and Kenji Sasaki (HAL QCD Collaboration), Systematics of the HAL QCD potential at low energies in lattice QCD, Phys. Rev. D99, 014514 (2019).PRVDAQ2470-001010.1103/PhysRevD.99.0145144Noriyoshi Ishii, Sinya Aoki, Takumi Doi, Tetsuo Hatsuda, Yoichi Ikeda, Takashi Inoue, Keiko Murano, Hidekatsu Nemura, and Kenji Sasaki, Hadron–hadron interactions from imaginary-time Nambu–Bethe–Salpeter wave function on the lattice, Phys. Lett. B712, 437 (2012).PYLBAJ0370-269310.1016/j.physletb.2012.04.0765Peng Guo and Vladimir Gasparian, Toward extracting the scattering phase shift from integrated correlation functions, Phys. Rev. D108, 074504 (2023).PRVDAQ2470-001010.1103/PhysRevD.108.0745046Peng Guo, Frank X. Lee, and Andrei Alexandru, Toward extracting scattering phase shifts from integrated correlation functions. V. Complex ϕ4 field model in 3+1 dimensions, Phys. Rev. D112, 074511 (2025).PRVDAQ2470-001010.1103/v132-nsb27Peng Guo, Paul LeVan, Frank X. Lee, and Yong Zhao, Extracting scattering phase shift in quantum mechanics on quantum computers, Phys. Rev. D113, 054512 (2026).PRVDAQ2470-001010.1103/ds7x-7bkp8K. Rummukainen and Steven A. Gottlieb, Resonance scattering phase shifts on a nonrest frame lattice, Nucl. Phys.B450, 397 (1995).NUPBBO0550-321310.1016/0550-3213(95)00313-H9C. H. Kim, C. T. Sachrajda, and Stephen R. Sharpe, Finite-volume effects for two-hadron states in moving frames, Nucl. Phys.B727, 218 (2005).NUPBBO0550-321310.1016/j.nuclphysb.2005.08.02910Ziwen Fu, Rummukainen-Gottlieb’s formula on two-particle system with different mass, Phys. Rev. D85, 014506 (2012).PRVDAQ1550-799810.1103/PhysRevD.85.01450611Luka Leskovec and Sasa Prelovsek, Scattering phase shifts for two particles of different mass and nonzero total momentum in lattice QCD, Phys. Rev. D85, 114507 (2012).PRVDAQ1550-799810.1103/PhysRevD.85.11450712M. Gockeler, R. Horsley, M. Lage, U. G. Meissner, P. E. L. Rakow, A. Rusetsky, G. Schierholz, and J. M. Zanotti, Scattering phases for meson and baryon resonances on general moving-frame lattices, Phys. Rev. D86, 094513 (2012).PRVDAQ1550-799810.1103/PhysRevD.86.09451313Veronique Bernard, Michael Lage, Ulf-G. Meissner, and Akaki Rusetsky, Resonance properties from the finite-volume energy spectrum, J. High Energy Phys.08 (2008) 024.JHEPFG1029-847910.1088/1126-6708/2008/08/02414Yan Li, Jia-jun Wu, Derek B. Leinweber, and Anthony W. Thomas, Hamiltonian effective field theory in elongated or moving finite volume, Phys. Rev. D103, 094518 (2021).PRVDAQ2470-001010.1103/PhysRevD.103.09451815M. Doring, U. G. Meissner, E. Oset, and A. Rusetsky, Scalar mesons moving in a finite volume and the role of partial wave mixing, Eur. Phys. J. A48, 114 (2012).EPJAFV1434-600110.1140/epja/i2012-12114-616Ning Li and Chuan Liu, Generalized Lüscher formula in multichannel baryon-meson scattering, Phys. Rev. D87, 014502 (2013).PRVDAQ1550-799810.1103/PhysRevD.87.01450217Peng Guo, Jozef Dudek, Robert Edwards, and Adam P. Szczepaniak, Coupled-channel scattering on a torus, Phys. Rev. D88, 014501 (2013).PRVDAQ1550-799810.1103/PhysRevD.88.01450118Xu Feng, Xin Li, and Chuan Liu, Two particle states in an asymmetric box and the elastic scattering phases, Phys. Rev. D70, 014505 (2004).PRVDAQ1550-799810.1103/PhysRevD.70.01450519Frank X. Lee and Andrei Alexandru, Scattering phase-shift formulas for mesons and baryons in elongated boxes, Phys. Rev. D96, 054508 (2017).PRVDAQ2470-001010.1103/PhysRevD.96.05450820C. Liu, X. Feng, and S. He, Two particle states in a box and the s-matrix in multi-channel scattering, Int. J. Mod. Phys. A21, 847 (2006).IMPAEF0217-751X10.1142/S0217751X0603215021Raul A. Briceno and Zohreh Davoudi, Moving multichannel systems in a finite volume with application to proton-proton fusion, Phys. Rev. D88, 094507 (2013).PRVDAQ1550-799810.1103/PhysRevD.88.09450722Raul A. Briceno, Two-particle multichannel systems in a finite volume with arbitrary spin, Phys. Rev. D89, 074507 (2014).PRVDAQ1550-799810.1103/PhysRevD.89.07450723Yan Li, Jia-Jun Wu, Curtis D. Abell, Derek B. Leinweber, and Anthony W. Thomas, Partial wave mixing in Hamiltonian effective field theory, Phys. Rev. D101, 114501 (2020).PRVDAQ2470-001010.1103/PhysRevD.101.11450124M. Döring, U. G. Meißner, E. Oset, and A. Rusetsky, Unitarized chiral perturbation theory in a finite volume: Scalar meson sector, Eur. Phys. J. A47, 139 (2011).EPJAFV1434-600110.1140/epja/i2011-11139-725Maxwell T. Hansen and Stephen R. Sharpe, Multiple-channel generalization of lellouch-lüscher formula, Phys. Rev. D86, 016007 (2012).PRVDAQ1550-799810.1103/PhysRevD.86.01600726Peng Guo and Frank X. Lee, Toward extracting scattering phase shift from integrated correlation functions. III. Coupled channels, Phys. Rev. D111, 054506 (2025).PRVDAQ2470-001010.1103/PhysRevD.111.05450627S. Aoki (CP-PACS Collaboration), Lattice QCD calculation of the rho meson decay width, Phys. Rev. D76, 094506 (2007).PRVDAQ1550-799810.1103/PhysRevD.76.09450628Xu Feng, Karl Jansen, and Dru B. Renner, Resonance parameters of the rho-meson from lattice QCD, Phys. Rev. D83, 094505 (2011).PRVDAQ1550-799810.1103/PhysRevD.83.09450529aC. B. Lang, Daniel Mohler, Sasa Prelovsek, and Matija Vidmar, Coupled channel analysis of the rho meson decay in lattice QCD, Phys. Rev. D84, 054503 (2011); PRVDAQ1550-799810.1103/PhysRevD.84.05450329bC. B. Lang, Daniel Mohler, Sasa Prelovsek, and Matija VidmarPhys. Rev. D89, 059903(E) (2014).PRVDAQ1550-799810.1103/PhysRevD.89.05990330S. Aoki (CS Collaboration), ρ meson decay in 2+1 flavor lattice QCD, Phys. Rev. D84, 094505 (2011).PRVDAQ1550-799810.1103/PhysRevD.84.09450531Dehua Guo, Andrei Alexandru, Raquel Molina, and Michael Döring, Rho resonance parameters from lattice QCD, Phys. Rev. D94, 034501 (2016).PRVDAQ2470-001010.1103/PhysRevD.94.03450132Jozef J. Dudek, Robert G. Edwards, and David J. Wilson (Hadron Spectrum), An a0 resonance in strongly coupled πη, KK¯ scattering from lattice QCD, Phys. Rev. D93, 094506 (2016).PRVDAQ2470-001010.1103/PhysRevD.93.09450633Thomas Luu and Martin J. Savage, Extracting scattering phase-shifts in higher partial-waves from lattice QCD calculations, Phys. Rev. D83, 114508 (2011).PRVDAQ1550-799810.1103/PhysRevD.83.11450834John Bulava, Brendan Fahy, Ben Hörz, Keisuke J. Juge, Colin Morningstar, and Chik Him Wong, I=1 and i=2π−π scattering phase shifts from nf=2+1 lattice QCD, Nucl. Phys.B910, 842 (2016).NUPBBO0550-321310.1016/j.nuclphysb.2016.07.02435Gunnar S. Bali, Sara Collins, Antonio Cox, Gordon Donald, Meinulf Göckeler, C. B. Lang, and Andreas Schäfer (RQCD Collaboration), ρ and K* resonances on the lattice at nearly physical quark masses and Nf=2, Phys. Rev. D93, 054509 (2016).PRVDAQ2470-001010.1103/PhysRevD.93.05450936Colin Morningstar, John Bulava, Bijit Singha, Ruairí Brett, Jacob Fallica, Andrew Hanlon, and Ben Hörz, Estimating the two-particle K-matrix for multiple partial waves and decay channels from finite-volume energies, Nucl. Phys.B924, 477 (2017).NUPBBO0550-321310.1016/j.nuclphysb.2017.09.01437M. Mai, C. Culver, A. Alexandru, M. Döring, and F. X. Lee, Cross-channel study of pion scattering from lattice QCD, Phys. Rev. D100, 114514 (2019).PRVDAQ2470-001010.1103/PhysRevD.100.11451438Zhengli Wang, Derek B. Leinweber, Chuan Liu, Liuming Liu, Peng Sun, Anthony W. Thomas, Jia jun Wu, Hanyang Xing, and Kang Yu, Spectral parameters of the ρ resonance from lattice QCD, J. High Energy Phys.08 (2025) 064.JHEPFG1029-847910.1007/JHEP08(2025)06439John Bulava, Andrew D. Hanlon, Ben Hörz, Colin Morningstar, Amy Nicholson, Fernando Romero-López, Sarah Skinner, Pavlos Vranas, and André Walker-Loud, Elastic nucleon-pion scattering at mπ=200MeV from lattice QCD, Nucl. Phys.B987, 116105 (2023).NUPBBO0550-321310.1016/j.nuclphysb.2023.11610540John Bulava, Bárbara Cid-Mora, Andrew D. Hanlon, Ben Hörz, Daniel Mohler, Colin Morningstar, Joseph Moscoso, Amy Nicholson, Fernando Romero-López, Sarah Skinner, and André Walker-Loud (Baryon Scattering (BaSc) Collaboration), Lattice QCD study of πΣ-K¯n scattering and the Λ(1405) resonance, Phys. Rev. D109, 014511 (2024).PRVDAQ2470-001010.1103/PhysRevD.109.01451141Constantia Alexandrou, Simone Bacchio, Giannis Koutsou, Theodoros Leontiou, Srijit Paul, Marcus Petschlies, and Ferenc Pittler, Elastic nucleon-pion scattering amplitudes in the Δ channel at physical pion mass from lattice QCD, Phys. Rev. D109, 034509 (2024).PRVDAQ2470-001010.1103/PhysRevD.109.03450942Adrian L. Kiratidis, Waseem Kamleh, Derek B. Leinweber, and Benjamin J. Owen, Lattice baryon spectroscopy with multi-particle interpolators, Phys. Rev. D91, 094509 (2015).PRVDAQ1550-799810.1103/PhysRevD.91.09450943Colin Morningstar, Nucleon scattering from lattice QCD, Proc. Sci.CD2024 (2026) 003 [arXiv:2504.01950].1824-803910.22323/1.479.000344Luka Leskovec, Christian B. Lang, M. Padmanath, and Sasa Prelovsek, A lattice QCD study of pion–nucleon scattering in the Roper channel, Few-Body Syst.59 (2018), 10.1007/s00601-018-1419-2.FBSYEQ0177-796310.1007/s00601-018-1334-645Zhong-Lin Ma, Zhan-Wei Liu, and Jiong-Jiong Liu, Odd-parity strange baryons Σ(12-) below 1.8 GeV with a Hamiltonian effective field theory, Phys. Rev. D113, 014037 (2026).PRVDAQ2470-001010.1103/m32c-wk6346Colin Morningstar, Low-lying baryon resonances from lattice QCD, Acta Phys. Pol. B57, 2 (2026), https://www.actaphys.uj.edu.pl/fulltext?series=Reg&vol=57&aid=2-A10.47Silas R. Beane, William Detmold, Thomas C. Luu, Kostas Orginos, Martin J. Savage, and Aaron Torok, Multi-pion systems in lattice QCD and the three-pion interaction, Phys. Rev. Lett.100, 082004 (2008).PRLTAO0031-900710.1103/PhysRevLett.100.08200448Peng Guo and Michael Döring, Lattice model of heavy-light three-body system, Phys. Rev. D101, 034501 (2020).PRVDAQ2470-001010.1103/PhysRevD.101.03450149M. Mai, M. Döring, C. Culver, and A. Alexandru, Three-body unitarity versus finite-volume π+π+π+ spectrum from lattice QCD, Phys. Rev. D101, 054510 (2020).PRVDAQ2470-001010.1103/PhysRevD.101.05451050Ben Hörz and Andrew Hanlon, Two- and three-pion finite-volume spectra at maximal isospin from lattice QCD, Phys. Rev. Lett.123, 142002 (2019).PRLTAO0031-900710.1103/PhysRevLett.123.14200251Fernando Romero-López, Stephen R. Sharpe, Tyler D. Blanton, Raúl A. Briceño, and Maxwell T. Hansen, Numerical exploration of three relativistic particles in a finite volume including two-particle resonances and bound states, J. High Energy Phys.10 (2019) 007.JHEPFG1029-847910.1007/JHEP10(2019)00752Tyler D. Blanton, Fernando Romero-López, and Stephen R. Sharpe, Implementing the three-particle quantization condition including higher partial waves, J. High Energy Phys.03 (2019) 106.JHEPFG1029-847910.1007/JHEP03(2019)10653Maxim Mai, Andrei Alexandru, Ruairí Brett, Chris Culver, Michael Döring, Frank X. Lee, and Daniel Sadasivan (GWQCD Collaboration), Three-body dynamics of the a1(1260) resonance from lattice QCD, Phys. Rev. Lett.127, 222001 (2021).PRLTAO0031-900710.1103/PhysRevLett.127.22200154Andrei Alexandru, Ruairí Brett, Chris Culver, Michael Döring, Dehua Guo, Frank X. Lee, and Maxim Mai, Finite-volume energy spectrum of the K-K-K- system, Phys. Rev. D102, 114523 (2020).PRVDAQ2470-001010.1103/PhysRevD.102.11452355Maxim Mai, Michael Döring, and Akaki Rusetsky, Multi-particle systems on the lattice and chiral extrapolations: A brief review, Eur. Phys. J. Special Topics230, 1623 (2021).EPJSAC1951-640110.1140/epjs/s11734-021-00146-556Haobo Yan, Maxim Mai, Marco Garofalo, Ulf-G. Meißner, Chuan Liu, Liuming Liu, and Carsten Urbach, ω meson from lattice QCD, Phys. Rev. Lett.133, 211906 (2024).PRLTAO0031-900710.1103/PhysRevLett.133.21190657Haobo Yan, Maxim Mai, Marco Garofalo, Yuchuan Feng, Michael Döring, Chuan Liu, Liuming Liu, Ulf-G. Meißner, and Carsten Urbach, Emergence of the π(1300) Resonance from Lattice QCD, Phys. Rev. Lett.136, 141901 (2026).PRLTAO0031-900710.1103/vfr3-5lsb58Yuchuan Feng, Chris Culver, Michael Döring, Maxim Mai, Andrei Alexandru, and Frank X. Lee, Coupled-channel approach to isotensor πππ scattering from lattice QCD, arXiv:2601.16916.59Rishabh Bubna, Effective field theory methods applied to two-body and three-body systems, Ph.D. thesis, U. Bonn (main) (2026).60Herzallah Alharazin, André Baião Raposo, John Bulava, Sebastian Dawid, Jeremy R. Green, Colin Morningstar, Fernando Romero-López, Miguel Salg, Stephen R. Sharpe, and Andres Stump, Three-body study of the tcc(3875)+ from lattice QCD, arXiv:2602.17204.61Stephen R. Sharpe, Three-particle scattering amplitudes from lattice QCD, arXiv:2601.04147.62Frank X. Lee, Andrei Alexandru, and Ruairí Brett, Higher order finite volume quantization conditions for two spinless particles, Phys. Rev. D105, 054517 (2022).PRVDAQ2470-001010.1103/physrevd.105.05451762See Supplemental Material at http://link.aps.org/supplemental/10.1103/527n-tjbr. The supplement consists of two parts. A pdf file named “qc-convergence.pdf” that has the phaseshift prediction at order 1 plus order-by-order convergence data for all the QCs. A folder named “qc-matrix” that contains python code to recover the full matrix on demand for any QC at any order.64Frank X. Lee, Colin Morningstar, and Andrei Alexandru, Energy spectrum of two-particle scattering in a periodic box, Int. J. Mod. Phys. C31, 2050131 (2020).IJMPEO0129-183110.1142/S012918312050131465F. Calogero, Variable Phase Approach to Potential Scattering (Academic Press, New York, 1967).66Martin J. Klein, On a degeneracy theorem of kramers, Am. J. Phys.20, 65 (1952).AJPIAS0002-950510.1119/1.193311867Robert S. Mulliken, Report on notation for the spectra of polyatomic molecules, J. Chem. Phys.23, 1997 (1955).JCPSA60021-960610.1063/1.174065568S. L. Altmann and P. Herzig, Point-Group Theory Tables, Oxford Science Publications (Clarendon Press, New York, 1994).69Eugene P. Wigner, Normal form of antiunitary operators, J. Math. Phys. (N.Y.)1, 409 (1960).JMAPAQ0022-248810.1063/1.170367270J. O Dimmock and R. G Wheeler, Irreducible representations of magnetic groups, J. Phys. Chem. Solids23, 729 (1962).JPCSAW0022-369710.1016/0022-3697(62)90531-071J. D. Newmarch and R. M. Golding, The character table for the corepresentations of magnetic groups, J. Math. Phys. (N.Y.)23, 695 (1982).JMAPAQ0022-248810.1063/1.525423