Hamiltonian effective field theory in elongated or moving finite volume

We extend previous work concerning rest-frame partial-wave mixing in Hamiltonian effective field theory to both elongated and moving systems, where two particles are in a periodic elongated cube or have nonzero total momentum, respectively. We also consider the combination of the two systems when directions of the elongation and the moving momentum are aligned. This extension should also be applicable in any Hamiltonian formalism. As a demonstration, we analyze lattice QCD results for the spectrum of an isospin-2 $\pi\pi$ scattering system and determine the $s$, $d$, and $g$ partial-wave scattering information. The inclusion of lattice simulation results from moving frames significantly improves the uncertainty in the scattering information.


I. INTRODUCTION
Lattice simulations of relativistic quantum-field theories are performed in a Euclidean four-dimensional finite volume. Scattering states are contained in the finite box with discretized energy levels. Understanding the relationship between these finite-volume energy levels and experimental scattering observables such as the phase shift and inelasticity is of significance. For the case of elastic two-body scattering in the rest frame, Lüscher [1][2][3] found a model-independent formula which is now known as Lüscher's formula.
An equivalent approach is provided in Hamiltonian effective field theory (HEFT) [4][5][6][7][8][9][10][11][12], a Hamiltonian extension of chiral effective field theory. In the standard approach, a Hamiltonian which respects the constraints of chiral effective field theory is fit to the finitevolume energy spectrum of lattice field theory and the infinite-volume scattering observables are obtained from the constrained Hamiltonian. The approach bridges finite-volume lattice field theory and experimental observables while providing insight into the composition of the scattering states in terms of noninteracting multiparticle basis states.
Different partial waves are mixed in the finite volume as a result of broken spherical symmetry. This mixing complicates the construction of the Hamiltonian matrix. For example, its incorporation significantly increases the dimension of the matrix. A recent work [12] established a formalism for disentangling partial-wave mixing and maximally reducing the dimension of the Hamiltonian matrix in the finite volume via an optimal set of rest-frame basis states. In this work, we will generalize this formalism to both elongated and moving systems with nontrivial total momentum. We will also consider the combination of these two systems when the direction of elongation and that of the moving momentum are aligned.
The Lüscher formula has already been extended and applied to the case of rectangular cuboid boxes [13][14][15][16][17][18][19][20][21][22][23]. This work will first consider a more general case where the box is allowed to be a general parallelepiped, as illustrated in Fig. 1. We then focus on a special class of the parallelepiped termed an elongated cube.
The Lüscher formula has also been extended to moving systems [24][25][26][27][28][29]. To realize the extension in a Hamiltonian formalism, one needs a Hamiltonian making contact with both the infinite-volume scattering observables parametrized in the rest frame and the finitevolume spectrum in the moving frame. This can be achieved within the formalism proposed in Refs. [30,31]. In that formalism, different forms of the moving-frame Lüscher formula are unified as different momentum transformations. Furthermore it leads to a new momentum transformation which is not only useful in the Hamiltonian formalism, but can also be used in the finite-volume three-particle quantization condition [32].
The symmetry in a moving frame is quite compatible with a cube elongated in the same direction as the nonzero total momentum. This case will be termed the elongated moving system, and disentangling partial-wave mixing in the elongated moving system is the main concern of this work. We will also demonstrate how the formalism works by analyzing lattice QCD results from Ref. [33] for the spectrum of an isospin-2 ππ scattering system. As also noted in Ref. [12], the discussion in this work should apply not only in HEFT, but also in any Hamiltonian formalism, e.g., the harmonic oscillator basis effective theory [34][35][36].

II. HAMILTONIAN IN ELONGATED MOVING FINITE VOLUME
A. Parallelepiped and elongated cube Normally, the system under consideration in lattice field theory simulations is a periodic cube. However, there are good reasons to also consider asymmetric boxes, where longer dimensions provide access to smaller nontrivial momenta [72]. For example, Ref. [13] studied a rectangular cuboid (including the square cuboid as a special case). In general, the box can be a parallelepiped as shown in Fig. 1.
If we choose one of the vertices of the parallelepiped as the origin, the parallelepiped can be specified by the three vectors a, b, c corresponding to the edges connected to the origin. The three vectors specify a matrix which sends the three unit vectors of the coordinate axes to a, b, and c respectively, where the subscripts denote the coordinate components of the vectors. To restrict the range of x * within the parallelepiped, one can define x * via x * i = j M ij x j and constrain x within the unit cube. Correspondingly, when imposing the periodic boundary condition, the momentum k * should be discretized as where we replace the unit cube with a cube of edge length L, as is standard in lattice field theory.
Here, we do not consider all the cases in Fig. 1 since in most cases the symmetries are broken too much. When we study moving frames in the following sections, we will find that the symmetry of a cube elongated in the same direction as the moving momentum is quite compatible with the moving effects. So in this paper, we define the elongated cube as a cube elongated in a specific directiond η with a magnitude η, and we consider three d η as follows:  , corresponding to the trigonal trapezohedron, labelled as (c) in Fig. 1.
We note that the overall factors of d η are not important, and are taken as presented for further convenience. In the elongated cube, the momentum is discretized as where the ⊥ and components of a vector are defined through To be more concrete, for an infinite-volume Hamiltonian where h and V denote the kinetic and potential energy respectively, and the state |k * is normalized as to put it in an elongated cube, we need the discretization and so that the basis |n is orthonormal n |n = δ n n .
Finally, the Hamiltonian in a finite elongated cube is We note the Hamiltonian Eq. (10) applies to any elongated cubes, including those of the three scenarios (a), (b) and (c) introduced above Eq. (3). We also note, that the box studied in Ref. [13] corresponds to where an overall factor can be absorbed into L. This box corresponds to the rectangular cuboid, which is not completely equivalent to the elongated cube, since it needs elongation in more than one direction. However in some cases, η 1 = η 2 = 1 for instance, the box is an elongated cube with d η = (0, 0, 1).

B. Moving system
Since the infinite-volume potential and scattering observables are most easily parametrized in the rest frame, we need a Hamiltonian that can produce the moving-frame spectrum while still written in terms of the rest-frame potential.
As suggested in Refs. [30,31], one can introduce a momentum transformation k * → k to the infinite-volume Hamiltonian where J denotes the Jacobian of the transformation. Then one can define such that and the infinite-volume Hamiltonian will be Now if one discretizes k instead of k * , one gets a different finite-volume Hamiltonian Ref. [30] proposed a number of general momentum transformations for the moving system, and proved that those transformations can keep the relationship between the infinite-volume phase shifts and finite-volume spectrum up to exponentially suppressed corrections. That paper also studied three typical transformations. While two of the three have been used in many previous works [24][25][26], they introduce additional energy dependence. The third one (labelled as scheme C in Ref. [30]) does not have such problems and hence is suitable here. It reads where ω i (k) = k 2 + m 2 i and P is the total momentum of the moving system, and the corresponding Jacobian is In the finite cube, the total momentum can only take discrete values as P = 2π L d γ with d γ an integer vector.

C. Elongated moving system
Both the elongated and moving systems have smaller finite-volume symmetry groups than the rest-frame cube, and combining them will normally give a much smaller one. However, if the elongated direction and the moving direction are the same, their combination will not see a large reduction in symmetry. We will call this combination the elongated moving system. The corresponding finite-volume Hamiltonian is obtained by combining Eqs. (10) and (16), which reads where the momentum transformation k * → k and the corresponding Jacobian is the same as Eqs. (17) and (18) for the moving system, except that the total momentum P should now be noting either d η = d γ or η = 1 for P = 0 in this "elongated moving system". Now Eqs. (17)(18)(19)(20) are all the ingredients needed to write down the elongated moving Hamiltonian. The eigenvalues of this Hamiltonian are rest-frame energies E * n related to moving-frame energies E n via Since the kinetic energy h(k * ) depends only on the length of k * , one can separate the kinetic term as so that k * depends only on e n (and ên is simply defined to sum over all the n with the same e n ). In other words, e n denotes a degenerate shell of the basis-state Hamiltonian, H 0 , describing the energies of the noninteracting states. Note, there can be several values for e n related to vectors n, d η and d γ all providing the same degenerate value for h(k * ). It is interesting to consider how the elongated moving system reduces to more simple cases for certain values of d η , η and d γ . When the masses of the two particles are the same, we also have Thus our discussion for the degenerate shells splits into four cases as listed in Table I. There we introduce an elongated moving vector d = 0, since either we can set d η = d γ or choose one of the two vectors to vanish. In Table I, case A refers to the standard unelongated and rest-frame system. e n can be simply chosen as n 2 . Case B refers to the elongated or unelongated moving-frame system with two particles of different mass. As suggested by Eq. (17), h(k * ) now depends on n 2 , (d − n) 2 , n 2 , and n 2 ⊥ . Because both n 2 and n 2 ⊥ can be reexpressed in terms of n 2 and (d − n) 2 , e n can be chosen as (n 2 , (d − n) 2 ), or (n 2 , n · d) equivalently. Case C1 refers to the elongated rest-frame system. h(k * ) now depends on n 2 , n 2 ∝ |n · d| 2 , and n 2 ⊥ = n 2 − n 2 . Thus e n can be chosen as (n 2 , |n · d|). Case C2 refers to the elongated or unelongated moving-frame system with two particles of the same mass. In contrast to case B, n and d − n are on the same shell as indicated by Eq. (23). As (n 2 , (d − n) 2 ) and ((d − n) 2 , n 2 ) denote the same shell, e n can be chosen as an unordered pair {n 2 , (d − n) 2 }.

III. PARTIAL-WAVE MIXING IN AN ELONGATED MOVING SYSTEM
Spherical symmetry allows the following partial-wave expansion where Y lm (k * ) are the usual spherical harmonics, as shown in Eq. (B1), and its variables are the direction angles (θ * , φ * ) of the vector k * . Different partial waves are decoupled under this potential in the infinite volume. In the finite volume, partial wave numbers (l, m) are no longer good quantum numbers, and the partial wave potentials v l with different l are coupled together in the determination of finite-volume spectra. This phenomena, called partial-wave mixing, complicates the structure of the Hamiltonian Eq. (19). In the standard case (case A in Table I), Ref. [12] proposed a method that provides an optimal set of basis states maximally reducing the dimension of the Hamiltonian. In this section, we will generalize that method to more general cases. The spherical symmetry group SO (3) is broken into one of its subgroups G in the finite volume. In the standard case A, G is the octahedral group O. In other cases in Table I, G is smaller, and turns out to be also a subgroup of a two-dimensional rotation group O(2) (or O(2)×C 2 in case C), where the rotation axis of this O(2) should be the same as the elongated moving vector d. This prefers the partial wave expansion Eq. (24) expanded in a coordinate system different from that of the discretized momentum. Appendix A discusses how the coordinate system is chosen (results are summarized in Table XIV).
In general cases, we expect the finite-volume potential in Eq. (19) can be put in a similar form as Eq. (24) as follows: where α ∞ denotes the index of the vector of the irreducible representation Γ ∞ of the group G ∞ , and the definitions for u Γ∞,α∞ andṽ Γ∞ are summarized in Table II. In the table, we also have and S P (m, θ * ) is defined via which gives In the standard case, Eq. (25) becomes the same as Eq. (24). In case B, we now take O(2) as G ∞ , and the α ∞ -independence ofṽ Γ∞ comes from the invariance of (l − m)! (l + m)! P lm (cos θ * ) P lm (cos θ * ) under m → −m. In case C, G ∞ is now O(2)×C 2 , where the C 2 symmetry comes from k → P − k, i.e., exchanging the momenta of the two equal-mass particles, and is different from the parity symmetry in the usual sense. In the latter case, one is concerned with k → −k and P → −P. It is now | cos θ * | instead of cos θ * independent ofê n .
In what follows, we will show how to construct the optimal set of basis states maximally reducing the dimension of the finite-volume Hamiltonian in general cases. The formalism is basically the same as that in Ref. [12], except with a different language introduced above. Now one can introduce to write V L as and construct the states |e n , Γ ∞ ; Γ, f, α via linear combinations of |e n ; Γ ∞ , α ∞ as follows: where Γ, f and α denote the α-th vector of the f -th occurrence of the irreducible representation Γ reduced from the Γ ∞ , and the coefficients derived from group theory can be found in Appendix A. One can then define the inner product matrices for these states as and Using these inner product matrices, one can orthonormalize |e n , Γ ∞ ; Γ, f, α to our final basis |e n ; Γ, F, α . The Wigner-Eckart theorem only permits the following general form for the V L : There are different methods to orthonormalize |e n , Γ ∞ ; Γ, f, α . We present the result of the eigenmode-based method discussed in Ref. [12] as follows: where λ F and X F Γ∞,f are the F -th eigenvalue and the (Γ ∞ , f )-component of the F -th eigenvector of the matrix P en;Γ,α respectively.

IV. EXAMPLE OF ISOSPIN-2 ππ SCATTERING
In this section, following a similar discussion in Sec. IV of Ref. [12], we apply the formalism developed herein to analyze lattice QCD results for the isospin-2 ππ scattering system. This time, the moving-frame data is included in the analysis.
As in Ref. [12], the lattice QCD results are from Ref. [33] where an anisotropic action is used. They quote the anisotropy ξ = a s /a t = 3.444(6) and the pion mass in lattice units a t m π = 0.06906 (13). The ππ-channel is also studied in their other recent works [74][75][76]. Drawing on the scale setting provided in Ref. [76], a −1 t = 5.662 GeV, m π is approximately 391 MeV.
In the analysis performed in Ref. [33], lattice results above the 4π threshold were not included. Since our formalism does not include the four-body contributions, the same cut is employed.

A. The procedures
As in Ref. [12], we work with dimensionless lattice units. The kinetic energy h is taken as and when going to the finite-volume system, we have where N = n 2 . Because the isospin is two, only s, d, and g waves need to be taken into account, as in Ref. [33]. With the partial-wave expansion of Eq. (24), the partial-wave potentials are taken to be of a simple separable form with with parameters G l and d l dimensionless. The parameters in these potentials were fit to minimize the χ 2 defined by where E Sep − E Lattice denotes the vector of the differences between the spectrum obtained in the separable potential model and the lattice simulation. The covariance matrix C denotes the covariances in the lattice spectrum of Ref. [33]. The spectrum was calculated using the method discussed in Sec. III. While a momentum cutoff N cut = 600 was used in Ref. [12], we found that N cut = 100 is already enough for the analysis. Consider a specific level (we choose the highest one of A + 1 in Fig. 2) for example, while the lattice level is 0.263773(424), the level solved from the Hamiltonian (using the parameters taken from the rest-frame fit in Ref. [12]) only shifts around 0.000001 when N cut reduces from 600 to 100. Actually, for the lattice size L ∼ 3 fm used here, 2π L √ N cut are roughly 10 and 4 GeV when N cut = 600 and 100 respectively. So N cut = 100 is totally enough here. For studies with larger L, however, one needs larger N cut . What's more, we found |χ 2 Ncut=100 − χ 2 Ncut=600 | < 0.1 in the range of parameters of interest. On the other hand, the values a t m π = 0.06906 (13) and ξ = a s /a t = 3.444(6) may bring appreciable uncertainties to our analysis. Here we do not consider them, because the analysis based on the Lüscher method implemented in Ref. [33] suggests that they only have a small effect. The dimensions of the finite-volume Hamiltonian matrices for N cut = 100 and 600 are listed in Table III for each of the irreducible representations considered. Case B is not included, as we have m 1 = m 2 in the current ππ system. It is notable that the analysis of the moving-frame lattice data corresponds to the C2 case with η = 1.  (42) with rest-frame data only and both rest-and movingframe data. Both data are from Ref. [33] and the fitting results of rest-frame data are taken from Ref. [12]. Covariances for parameters are described by the Hessians listed in Eqs. (44) and (45).
Data used

B. The results
As in Ref. [12] we set d 2 = d 4 = d B = 4.78 in the fitting. The results of the fit are shown in Table IV, where results of Ref. [12] are also included for comparison. Using those parameters, we predict the L-dependent spectrum for both rest and moving frames in Figs. 2 and 3.
Our covariance for parameters λ i is defined as [H/2] −1 , where H is the Hessian of χ 2 , the matrix of second-order partial derivatives over parameters For comparison, we also list the covariance obtained in Ref. [12] for the fitting of rest-frame data: As the different values of l are decoupling in solving for the phase shifts, the values underlined in Eqs. (44) and (45) are used in calculating the errors in the phase shifts.
In Table IV, one may be concerned with the increase of χ 2 /N dof after including the moving frame data. However, the fitting based on the Lüscher method implemented in Ref. [33] gives χ 2 /N dof = 116/(49 − 3) (they did not include the g-wave), quite close to ours. As shown in Fig. 4, the inclusion of moving frame lattice QCD results induces small variations in the phase shifts within the 1σ uncertainties of the predictions from the rest-frame data alone. However, the use of moving-frame data reduces the uncertainties in the HEFT phase shifts significantly. Moreover, the constraints provided by several lattice QCD energy levels on a small number of parameters characterising the spectrum lead to results that are relatively precise in comparison to the Lüscher method. Still the results from the two approaches are generally consistent with only one outlier arising from the Lüscher method.
We also examine the differences in the finite-volume spectra associated with the elongation Ref. [33]. Red dashed (blue solid) curves illustrate the central values of the phase shifts obtained from rest-frame data only (both rest-and moving-frame data). The coloured shading describes the associated 1σ uncertainties. The red circle (blue triangle) points from Ref. [33] illustrate the phase shifts that that can be extracted from the finite-volume spectrum of Figs. 2 and 3 using Lüscher's method.
of the lattice volume versus the nonzero total momentum of the two-particle system. Our consideration aims to understand how elongation and nonzero total momentum differ in the spectrum. One may find for the pure elongated and the pure moving systems that 1 η and γ play a similar role. Noting that γ depends on the momentum k and the total momentum P, to make a comparison we set η = 1 γ ≈ 0.868 with γ taking the value on k = 0, d = (0, 0, 1) and L/a s = 20. We predict the L-dependent spectrum for the corresponding elongated, moving and elongated moving systems in Fig. 5. These three systems provide quite different spectra. Thus the consideration of elongated, moving, and elongated-moving systems are useful for generating more data within a certain range of lattice sizes.

V. SUMMARY
In this work, we have extended HEFT to accommodate both an elongated finite volume and systems with nonzero total momentum. We also consider their combination when the directions of the elongation and the total momentum are aligned. To calculate the finitevolume energy levels, we first constructed the elongated-moving Hamiltonian Eqs. (17)(18)(19)(20) via the potential parametrized in the rest frame. The spectrum solved from the Hamiltonian can approximate the real spectrum of the elongated moving system up to exponentiallysuppressed corrections. The elongation was handled in the usual way, and the moving effects were realized via a momentum transformation proposed in Ref. [30].
We then applied the formalism proposed in Ref. [12] to disentangle the partial-wave mixing in the elongated moving Hamiltonian. This formalism maximally reduces the dimension of the Hamiltonian matrix. Different from the rest frame, the elongated moving system has an additional characteristic direction, which prefers the partial-wave expansion expanded in a specific coordinate system.
Next, an example of isospin-2 ππ scattering was used to demonstrate how this formalism works. The use of moving-frame data induced small corrections in the phase shifts within the 1σ bounds of the rest-frame predictions. However the consideration of moving-frame lattice results significantly reduced the uncertainty in the phase-shift predictions. The consistency between analyses from the rest-frame Hamiltonian formalism implemented in Ref. [12] and the Lüscher method implemented in Ref. [33] is maintained by the moving-frame Hamiltonian formalism implemented here.
Finally, we examined differences between the effects of elongation and nonzero total momentum. The spectra obtained from the elongated, moving and elongated moving systems are quite different, and provide additional avenues for generating more Lattice QCD results within a certain range of lattice size. On the current status of lattice simulations, L is roughly in the range 2-6 fm.
This work has largely accomplished the outlook of Ref. [12] for the generalization of the Hamiltonian formalism. More applications to two-body channels with data from elongated and moving systems are planned. Furthermore, as mentioned in Ref. [12], the movingframe formalism developed here is necessary for a three-body formalism, since two of the three particles can have a nonvanishing total momentum. In the three-body case, a direct Hamiltonian fit should be formally simpler than the three-body Lüscher formalism. Of course, one of the remaining challenges is the significant increase in the dimension of the resultant Hamiltonian matrix. In the infinite volume, with the symmetry group G ∞ and its irreducible representations (irreps) Γ ∞ , one can label the quantum state vectors as |Γ ∞ , α ∞ , where α ∞ is the index for the vectors in the irrep Γ ∞ . For example, if we consider G ∞ = SO(3), then |Γ ∞ , α ∞ will be |l, m . In the finite volume, the symmetry group G ∞ reduces into one of its subgroups G, and the vectors for an irrep Γ are now labelled as |Γ, α .
According to the restricted representation, the infinite-volume vectors |Γ ∞ , α ∞ also behave as the vectors belonging to the representations of G, hence, they can be combined to obtain the vectors belonging to the irrep Γ as follows: where f is introduced since a specific Γ can be obtained more than once from the reduction of a single Γ ∞ . C Γ∞ is the unitary coefficient matrix. The purpose of this section is to provide some frequently used C Γ∞ . The results are summarized in Tables VI to XIII where the C 2 is generated by the parity inversion. The relation can be formally written as In the elongated moving system, one also cares about the group O(2)×C 2 . It can be formally written as The group O(2) is isomorphic to the semidirect product SO(2) C 2 formally written as where the C 2 is generated by the reflection rather than the parity inversion.
Since the irreps of direct-product groups can be constructed as the tensor products of the irreps of the two original groups, we will focus on the groups without the parity inversion in the following discussions, that are, SO(3) and O (2). Although the irreps of SO(3) and O (2) are both labelled as integers and half-integers, they have different dimensions. For SO(3), α ∞ takes −Γ ∞ , − Γ ∞ + 1, · · · , Γ ∞ , so the dimension will be 2Γ ∞ + 1. For O (2), α ∞ can only take + when Γ ∞ = 0, and can take both + and − in other cases.
In the finite volume, SO(3) will reduce into the octahedral group O, and O(2) will reduce into dihedral groups whose orders will depend on the direction of the elongated moving vector d. Groups and their irreps are summarized in Table V, where A, B and K are labels for one-dimensional irreps, E and G for two-dimensional, T for three-dimensional, and H for four-dimensional. For SO (3), those C Γ∞ are provided in many papers, e.g., Table A.2 (for bosons) and Table A.4 (for fermions) of Ref. [77]. Here we cite their results in Tables VI and VII. For O(2), we take the case d = (0, 0, 1) for example. The rotations of angles 0, π 2 , π and 3π 2 in the SO(2) and the reflection in C 2 will survive, then the resulting group Dih 4 can be formally written as C 4 C 2 . So Dih 4 can be generated by the π 2 rotation element R π 2 and the reflection element R, whose representation matrices can be chosen to be where we represent |Γ ∞ = 0, ± as follows: Note when Γ ∞ = 0, since Γ ∞ and Γ ∞ +4 share the same representation matrices as indicated in Eq. (A5), one will have C Γ∞ = C Γ∞+4 . In fact, for d = (0, 1, 1), it will be C Γ∞ = C Γ∞+2 , and for d = (1, 1, 1), it will be C Γ∞ = C Γ∞+3 . Because those irreps are only one-or twodimensional, it is easy to find out those C Γ∞ . We also note that since there is still the freedom to choose the representation matrices of the finite-volume irreps, those C Γ∞ are not unique. The results for our choice are summarized in Tables VIII to XIII.

States in elongated moving system
In the elongated moving system, one will deal with the states where e n denotes (n 2 , n · d), and ên means summing over all the states with the same e n , and (Γ ∞ , α ∞ ) will be (|m|, sign(m)) (we define sign(0) = +), and the angle φ depends on the choice of the axes labelled as (x,ỹ,z), which can differ from the finite-volume box's  [77].      √ 4π . There are many useful properties of the P matrix listed as follows: • [P n 2 ] l ,m ;l,m is real because of the symmetry under φ → −φ.
• [P n 2 ] l ,m ;l,m = 0 when l + l is odd because of the symmetry under n → −n.
• [P n 2 ] l ,m ;l,m = 0 when |m − m| = 2 because of the symmetry under φ → φ + π 2 . The result of the summation nx,ny e −i(m −m)φ (B4) depends on (n 2 , n z ). One need not calculate it for all (n 2 , n z ) by noting the map n → ±n + (0, 0, j) with any integer j. In fact, the summation Eq. (B4) is related to the P matrix P B en with d = (0, 0, 1) discussed in Appendix B 2.
2. P matrix of the elongated moving system From Eq. (33), the P matrix for the case B in Table II where it does not matter how to redefine the ill-defined φ * when n 2 = n 2 , because either m = m = 0 then Eq. (B5) is φ * -independent, or one of m and m is nonzero then P l m or P lm in Table II and Eq. (B2) vanishes.
To solve for it, it is worth noting that the map n → n + j d with j d any integer vector will tell us P B (n 2 ,n·d) = P B (n 2 +2j n·d+j 2 d 2 ,+n·d+j d 2 ) , and the map n → −n + j d will tell us P B (n 2 ,n·d) = (−1) m −m P B (n 2 −2j n·d+j 2 d 2 ,−n·d+j d 2 ) .
With the coordinate axes (x,ỹ,z) of spherical harmonics taking the values in Table XIV, there are many useful properties of P B (n 2 ,n·d) listed as follows: • P B (n 2 ,n·d) is real because of the symmetry under φ → −φ, which holds for all the d presented in Table XIV.
For the case C (C1 or C2), the map n → d γ − n will tell us [P C en ] |m | ± ,S m ;|m| ∓ ,Sm = 0 , where we have used that cos θ * has the same sign with n·d− d 2 γ 2 in case C. We also emphasize that P C1 = P C2 since d γ = 0 in case C1 while d γ = d in case C2.