Lattice operators for scattering of particles with spin

We construct operators for simulating the scattering of two hadrons with spin on the lattice. Three methods are shown to give the consistent operators for P N , P V , V N and N N scattering, where P , V and N denote pseudoscalar, vector and nucleon. Explicit expressions for operators are given for all irreducible representations at lowest two relative momenta. Each hadron has a good helicity in the first method. The hadrons are in a certain partial wave L with total spin S in the second method. These enable the physics interpretations of the operators obtained from the general projection method. The correct transformation properties of the operators in all three methods are proven. The total momentum of two hadrons is restricted to zero since parity is a good quantum number in this case.


Introduction
The study of two-hadron interactions with lattice QCD requires operators that create and annihilate the two-hadron system with the desired quantum numbers. Systems of two hadrons with zero spin have been extensively studied and the corresponding scattering matrices for interactions of two pseudoscalars (P ) were extracted, for example ππ, Kπ, . . . . The simulations of two-hadron systems where one or both hadrons carry non-zero spin are scarce, and those mostly focused on the partial-wave L = 0. An example of a study that aimed also at higher partial waves considered nucleon-nucleon scattering [1]. There is a great need for lattice results on further channels or higher partial waves where one or both hadrons carry spin. We have in mind channels that involve the nucleon N or/and vectors V = J/ψ, Υ b , D * , B * , . . . which are (almost) stable under strong interactions. The P N or V N scattering is crucial for ab-initio study of baryon resonances and pentaquark candidates, P V is essential for mesonic resonances and tetraquark candidates, while N N to grasp two-nucleon interaction and deuterium. 1 We consider the most common simulations that are done in a box of size L with periodic boundary conditions in space. The momenta p of non-interacting single hadrons are multiples of 2π/L in this case.
We focus on the system with total momentum zero where parity is a good quantum number. This has a practical advantage since fixing the parity gives a strong handle on which channels and partial waves can contribute as eigenstates in a given study. On the lattice the continuum symmetry is reduced to a discrete symmetry and one extracts the eigenstates that transform according to the chosen irreducible representation (irrep) Γ of the discrete group O (2) h . Only the states with desired parity contribute to a given irrep when the total momentum is zero, while both even and odd partial waves contribute to a given irrep when the total momentum is not zero.
In this paper we present the general expressions to construct operators for scattering of particles with spin using three methods. The corresponding proofs of the correct transformation properties are given. Then the explicit expressions H (1) (p)H (2) (−p) for P V , P N , V N and N N are presented for lowest two relative momenta |p| = 0, 2π/L.
The reason to introduce two-hadron operators in a lattice simulation is to extract the eigenstates and their energies in a desired scattering channel. These energies render the scattering phase shift via the well-known Lüscher method [2], which was originally derived for the two spin-less particles. The relation has been generalized to the scattering of two JHEP01(2017)129 particles with arbitrary spin [3][4][5][6][7] and the most general case is considered in [3]. This can be directly applied to extract the scattering matrix from the energies obtained using the operators presented in this work.
The three methods to construct operators complement each other and it is verified that all three methods lead to consistent results. The projection method is a general mathematical tool which leads to one or several operators O Γ,r,n that transform according to given irrep Γ and row r. It is a "bottom-up" method, working entirely with lattice operators. But it gives little guidance on which continuum quantum numbers of total J, spin S, orbital momentum L or single-particle helicities λ 1,2 will be related with a given operator. This is remedied with the helicity and partial-wave "top-down" methods where first the operators with good continuum quantum numbers (J, P, λ 1,2 ) or (J, L, S) are constructed and then subduced to the irreps Γ of the discrete group. The hope is that the results indicate which linear combinations O Γ,r,n of various n enhance couplings to the states with desired continuum quantum numbers. An analogous strategy for single-hadron states was proposed by the Hadron Spectrum Collaboration in [8]. We do not prove whether two particle operators also retain a "memory" of their construction origin. Future simulations analogous to the single-hadron case [8] will be necessary to explore how well that may be realized in practice.
Certain aspects of constructing the lattice operators for scattering of particles with spin were already presented before: partial-wave method in [1,9], projection method for example in [5], helicity states for single-hadrons in [10] and some considerations for two hadrons in [9,11]. The study [12] combines single particle irreps to two-particle operators. For this one needs the values of all corresponding Clebsch-Gordan coefficients to embed the product irreps of the single particle operators in the irreps of the two-particle frame (see section 4.5). Despite all previous work, various practical aspects and proofs are lacking to build a two-hadron operator related to desired continuum quantum numbers in a given irrep. Here we devise two-hadron helicity operators, verify their consistency with other two methods and provide related proofs. We also argue how simple non-canonical single-hadron operators can be used as building blocks for that purpose.
The paper is organized as follows. The single-hadron operators that are employed to build two-hadron operators are considered in section 2. The required transformations properties of two-hadron operators are given in section 3. The general expressions for two-hadron operators with three methods are presented in section 4, while the proofs of correct transformation properties are given in appendix A. The explicit operators for lowest two relative momenta are collected in sections 5 and C. The necessary technicalities are delegated to appendix B.

Single-hadron operators and their transformations
The single-hadron operators H(p) need to have certain transformation properties under rotations R and inversion I in order to build two-hadron operators H (1) (p)H (2) For a particle at rest m s is a good quantum number of the spin-component S z . The m s is generally not a good quantum number for H ms (p = 0); in this case it denotes the eigenvalue of S z for the corresponding field H ms (0), which has good m s . The transformations of annihilation operators H ms (p) are obtained by the hermitian conjugation where the following properties of the Wigner D-matrix and R are used throughout this work We employ the conventional definition of D used in [13,14] and discussed in appendix B.1. Note that the z-component of spin m s is a good quantum number for a particle at rest. The pseudoscalar fields P (J P = 0 − ), vector fields V (J P = 1 − ) and nucleon fields N (J P = 1 2 + ) are considered, together with their mutual two-hadron scattering channels later on. One can choose the simplest annihilation operators that transform according to (2.2), for example where p and x are three-vectors and the time index is omitted. The N 1,2 are the upper two components of Dirac four-spinor N µ=1,...,4 in the Dirac basis. The relations for the vector states |V ms=1 = (−|V x −i|V y )/ √ 2 versus annihilation operators V ms=1 = (−V x +iV y )/ √ 2 are discussed in appendix B.3. The operators P , N 1/2,−1/2 and V i=x,y,z like (2.4) will be basic building blocks of our two-hadron operators.
Since all two-hadron operators will entail transformations of annihilation fields H = P, V, N , we present them explicitly with m s,s ′ = ±1/2 and (J k ) ij = −iǫ ijk rendering real T s=1 . The fields transform as basisvectors, which is contrasted to the transformation of components in appendix B.2. The

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transformations of V i (p) in (2.5) can be used to verify the transformations (2.2) of the fields V ms (p) defined in (2.4). Note that canonical hadron fields H (c) ms (p) ≡ L(p)H ms (0) also transform under rotations as given in (2.2), 2 but those would be less practical to implement. The canonical fields are obtained from the rest fields H ms (0) (which have good m s ) with the boost L(p) from 0 to , which contains also lower components and depends on the mass m. We will employ the simpler fields V and N (2.4) that have all the required transformation properties (2.2), (2.5) to prove the correct transformations of the twohadron operators in appendix A. The fields (2.4) agree with the canonical fields for p = 0.
(3.1) Such continuum-like operators will present an intermediate step below and the only relevant rotations will be those contained in the discrete group.
On the cubic lattice the continuum rotation group is reduced to the cubic group R ∈ O with 24 elements for integer J. These are given in terms of the rotation angle −π ≤ ω < π around vector n in positive direction and we use the ordering i = 1, . . . , 24 listed in table A.1 of [16]. The double-cover group O 2 has the same 24 elements, and additional 24 elements (i = 25, . . . 48) 3 with the same n and angle π ≤ ω < 3π where ω i+24 = ω i + 2π. The number of symmetry elements gets doubled toR = {R, IR} ∈ O (2) h when inversion is a group element.
The representations (3.1) with given J and m J are reducible under O (2) h , so we seek the states/operators that transform according to the corresponding irreducible representation (irrep) Γ and row r The systems with integer J transform according to irreps Γ = A ± 1,2 , E ± , T ± 1,2 , while systems with half-integer J according to Γ = G ± 1,2 , H ± as summarized in table 1. The upper index stands for parity and the number of rows r = 1, . . . ,dim Γ is equal to the dimensionality of irrep Γ. We employ the same conventions for rows in all irreps as in [16]. 4 The explicit representations T Γ r,r ′ (R) for R ∈ O (2) are given in appendix A of [16]. 5 As indicated above, here we consider only two-hadron states with total momentum zero. Both even and odd partial waves contribute to a given irrep when total momentum is not zero. Working with operators that mix parity is no problem in principle and has already been considered for, e.g., pseudoscalar-pseudoscalar scattering. It leads to more eigenstates in the given irreducible representation and one has to apply further assumptions and tools to extract the scattering phase shift. Note that in both cases, exact or mixed parity, the lattice irreps may still couple to a tower of higher partial waves. The phase shifts for ℓ ≤ 2 in the parity mixed case have been successfully extracted, for example, via the parameterization of the S-matrix by the Hadron Spectrum Collaboration in [17,18].

Two-hadron operators in three methods
Here we present two-hadron operators with total momentum zero that transform according to (3.2). They will be constructed by three methods, where continuum-like operators (3.1) will appear as an intermediate step in two of the methods. Their correct transformation properties (3.1), (3.2) are proven in appendix A.

Projection method
This is a well-known mathematical method, where a projector to the desired irrep Γ and row r is used on an arbitrary operator (see for example section 4.19 of [19]) The H a are arbitrary single hadron lattice operators of desired |p| and any p and m s , for example operators (2.4) or their linear combinations; the other index (1) and (2) denotes the particle type. We have taken H a with all possible combinations of direction p and polarizations of both particles m s1 and m s2 (for vectors we chose V x , V y and V z as H a ).

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For fixed |p|, Γ and r one can get one or more linearly independent operators O |p|,Γ,r,n which are indicated by n. As an illustration we present two-linearly independent P V operators while others are listed in the section 5 and appendix C.
h in (4.1) is the operator symbolising rotations R possibly combined with inversions (R is reserved in this article for rotations only) and one uses in practicẽ where the action of rotation R or inversion I on all H = P, V, N are given in (2.2, 2.5). The representation matrices T Γ (R) for all elementsR are listed for all irreps in appendix A of [16]. The proof of correct transformation is shown for completeness in appendix A.
The projection method is very general, but it does not offer physics intuition what O |p|,Γ,r,n with different n represent in terms of the continuum quantum numbers. This will be remedied with the next two methods, that indicate which linear combinations of O n correspond in the continuum to certain partial waves or helicity quantum numbers.

Helicity method
The states with good helicity in the continuum scattering have been thoroughly discussed by Jacob and Wick [13]. They have been considered for one-hadron states on the lattice by the Hadron-Spectrum collaboration [10], but they have not been widely used for two-hadron scattering on the lattice yet.
The m s is a good quantum number for particles with p = 0 or p z ∝ e z since L z = 0 and S z = J z , while it is generally not a good quantum number for arbitrary p = 0. Here L z , S z and J z denote components of the orbital, spin and total angular momentum, respectively. The advantage of the helicity h is that it is a good quantum number for arbitrary p, where S and p are three-vectors. To obtain a helicity state, one starts from a state with momentum p z ∝ e z and |p z | = |p| that has good m s . This state is rotated from p z to desired direction of p with R p 0 , The upper index h indicates that the polarisation index (λ) stands for the helicity of the particle (not m s ). Different choices of R p 0 are possible, which lead to different phases in the definition of |p, s, λ h . We do not restrict to a specific choice of R p 0 since our resulting two-particle operator will be independent of this (except for the overall phase of operator, which is irrelevant). Simple examples of H pz,ms=λ can be read-off from (2.4) and the action of R p 0 on them is given by (2.5).

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The arbitrary rotation R rotates the momentum p and the spin S in the same way, so the helicity h ∝ S· p of the state remains the same and only the direction of the momentum changes [10,13] R|p, s, where only the phase ϕ(R) depends on R.

Construction of two-hadron helicity operators
The starting point is a two-particle state composed of |p, can be chosen as arbitrary momentum in a given shell |p|, and λ 1,2 are chosen helicities. The two-particle helicity state and the annihilation operator with the correct transformation properties under rotation (3.1) are with the proof for operators given in appendix A. Note that the labels on the left-hand side should just indicate the origin of construction; they would correspond to the continuum quantum numbers if R was summed over the continuum rotation group. The main difference with respect to the continuum case [13] is the discrete sum over the elements of the discrete group R ∈ O for integer J and R ∈ O 2 for half-integer J. Appendix B.1 provides D-matrices 6 for R in O and double-cover O 2 . The helicity of the two-particle state is λ = λ 1 − λ 2 for all cases we consider. 7 Note that the operator (4.6) is a sum of states with fixed λ 1,2 since the helicity of each particle is conserved with rotation (4.5), The phase Φ(R) depends also on the choice of R 0 through the definition of the helicity state (4.4). We do not explicitly determine Φ(R), but rather directly employ expressions (4.6) and (4.8) to get explicit results for the interpolators in section 5.
The final helicity operator with desired parity P = ±1 is obtained from (4.6) by parity One has to use the same rotation in D and in the transformation of fields. The additional rotation of 2π that is present in half of the O 2 elements renders factors (−1) J from D J (R) and (−1) s 1 (−1) s 2 from the transformations of the both fields in (4.6). 7 In the continuum this follows from the continuum integration over Euler angle γ [13].

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where we have expressed H h (4.4) with fields H ms (p z ) (2.4) that have good m s . The actions of inversion I and the rotation R on the fields H ms are given in (2.5). One chooses particular p with given |p| and performs rotation R p 0 from p z to p. There are several possible choices of R p 0 , but they lead only to different overall phases for the whole operator (4.8), which is irrelevant. 8 The operators for |p| = 1 (in units of 2π/L) will be explicitly presented in section 5. We take the simplest choice p = p z = (0, 0, 1) and As an illustration we present two P V operators with the same J P ,

Subduction to irreducible representations
The operators (4.8) would correspond to the irreducible representations only for the continuum rotation group R ∈ SO(3) in which case J indicates the continuum quantum number. These operators (4.8) represent reducible representation under the reduced discrete group O (2) . In simulations one needs to employ operators, which transform according to irreducible representations Γ and row r of G = O (2) . Those are obtained from the continuum-like operators O J,m J by the subduction [8,20] O The subduction coefficients S are real and are given in appendices of [8] and [20] for all irreps. For the T 1,2 irreps our convention for rows is different and the corresponding coefficients S are listed in appendix B.4. This strategy, to start from operators with continuum spin O J,m J and subduce them to irreps O Γ,r , was proposed by the Hadron Spectrum Collaboration in [8]. It was then extensively employed for single hadrons by the same collaboration. 10 The subduced operators O Γ,r were found to "carry the memory" of the spin J, where they were subduced from and dominantly couple to the lattice eigenstates with this spin. Whether that expectation holds for two-particle operators combining spin and angular momentum is unclear; if the behaviour is like for the single-particle operators one may expect that the subduced operators O [J,P,λ 1 ,λ 2 ,λ] |p|,Γ,r will dominantly represent eigenstates with continuum quantum numbers J, λ 1 , λ 2 , λ.

Partial-wave method
Often one is interested in the scattering of two hadrons in a given partial wave L. The orbital angular momentum L and total spin S are not separately conserved, so several (L, S) combinations can render the same J, m J and P , which are good quantum numbers. Nevertheless, the L and S are valuable physics quantities to label continuum annihilation field operator (4.11) The operator has parity P = P 1 P 2 (−1) L and its correct transformation property under rotation (3.1) is demonstrated in appendix A. The operator was considered for nucleonnucleon scattering already in [1], where the proof was not presented (this reference uses Y Lm L where we have Y * Lm L ). The C are Clebsch-Gordan coefficients, p is an arbitrary momentum with desired |p|, Y Lm L ( Rp) is a spherical harmonic for direction (θ, φ) given by the normalized vector Rp. Simple choices of one-particle annihilation operators H are listed in (2.4). The sum is over R ∈ O for integer as well as half-integer J, since rotations for angle ω and ω + 2π have the same effect on p. Here is an example of two P V operators with the same J and S O |p|=1,J=1,m J =0,L=0,S=1 ∝ p=±ex,±ey,±ez |p|,Γ,r carry the memory of continuum J, L, S and dominantly couple to eigenstates with these quantum numbers. This depends on how close one is to the physical limit and whether there may be an actual physical transition 11 between states of different (L, S) in the given chanel J P . If this transition is small, one can hope that operators (4.12) would be valuable handles for certain lower values of J and L, which are of most practical importance. Certain higher L ′ > L inevitably lead to the same operators (or their linear combinations), and O

Relation between partial-wave and helicity operators
Operators calculated in partial-wave method (4.11) are linear combinations of operators calculated by helicity method (4.8) where the proof of the relation is presented in the appendix D. For |p| = 1, we take the reference momentum p = R p 0 p z along the z-direction, so R p 0 and D J (R p 0 ) are identity matrices.

Method combining single particle irreducible representations
Let us mention also a fourth method, which is not applied here. This approach [12] combines single particle irreps Γ 1,2 (for the cubic group at p = 0 or for the little groups at p = 0) to two-particle operators in irrep Γ: Γ 1 ⊗ Γ 2 → Γ. One hopes that this way one may profit from the optimized single particle operators for given Γ 1,2 , as for example in the P P study [17]. One needs the values of all corresponding Clebsch-Gordan coefficients to embed the product irreps of the single particle operators in the irreps of the two-particle frame. The method has been, for example, used in P P scattering [17] where the Clebsch-Gordan coefficients are known [11]. The reference [12] provides which Clebsch-Gordan coefficients are non-zero for scattering of particles with spin. This approach does not relate the resulting operators to continuum quantum numbers like L or S. Nevertheless, it might turn out to be particularly valuable for constructing two-hadron operators with non-zero total momenta. |p|,Γ,r will be expressed as linear combinations of known operators O |p|,Γ,r,n obtained by the projection method. This indicates which linear combinations have to be employed in a simulation to enhance a coupling to the state with continuum quantum numbers J, P, λ 1 , λ 2 or J, L, S. This coupling is expected to be favourable when the studied system is approaching the physical limit, if certain higher partial waves are suppressed and if the physical mixing of different (L, S) in given J P channel is small.
In both, helicity and partial wave, methods we consider J ≤ 2 for integer and J ≤ 5/2 for half-integer J. The helicity operator will not be given for |p| = 0 as helicity h

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since h (4.3) reverses sign if p → −p and S → S, while p → −p only multiplies the operator by ±1 depending on parity. The m J will appear as an index in helicity and partialwave operators in cases when only one m J is present in subduction to given irrep and row (4.10), (4.12). The vector fields are expressed in the V x,y,z basis (2.4), which are most straightforward for implementation. The normalization of each operator is arbitrary. For brevity, only the first row r = 1 is displayed, which should suffice for an actual simulation. Other rows and |p| can be obtained from the general expressions for O above.
We verified that all partial-wave and helicity operators are related by (4.13). Also, we confirmed that the number of linearly independent operators in each irrep agrees with the fourth method [12] in section 4.5, as elaborated for the P V case below. The number of operators based in this method can be deduced based on table I (for p = 0) and table III (for |p| = 1) of [12].
Before listing the operators, let us point out again the key expressions that have been used to derive them. The operators from projection method are obtained using (4.1) with representations T Γ (R) listed in appendix A of [16]. The helicity operator (4.9) for |p| = 1 is subduced to irreps by (4.10), while the partial-wave operator (4.11) comes from the subduction with (4.

P V operators
Operators for other irreps are equal to 0.

|p| = 1
There is only one independent operator with |p| ≤ 1 for each irrep (and given row), except for T + 1 . The result confirms, for example, that O ,n need to be employed to study L = 0 or L = 2 partial waves. Note that both partial waves contribute to the same J P = 1 + channel even in the continuum P V scattering with S = 1. Specific higher J ≥ 3 and L ≥ 4 lead to the same T + 1 operators (or their linear combinations). In the limit when partial waves L ≥ 4 are negligible, we expect that O (L = 0, S = 1) ↔ (L = 2, S = 1) in this channel is small, which is realized, for example, in the heavy-light meson observables due to the heavy quark symmetry. 12 The correct number of independent interpolators for |p| = 1 can be verified as follows. The total number can be obtained by counting different P (p)V i (−p), which span a basis with 6 · 3 = 18 terms (6 for directions of p and 3 for polarization of V ). This agrees with the number 5 · 3 + 2 + 1 = 18 of interpolators above in all rows (5 for number of independent O T 1,2 ,r=1 and 3 for number of rows in T 1,2 ). The number of P (1)V (−1) interpolators for each irrep also agrees with the fourth method [12], discussed in section 4.5. The table I in [12] indicates that P (1) is in irrep A 2 of group Dic 4 , while V (−1) can be in [12]. This agrees with our interpolators above, so T + 1 indeed appears twice.

P N , V N and N N operators
The explicit expressions for |p| = 0, 1 and all irreps are presented in appendix C. Several linearly independent operators O Γ,r,n typically arise for V N and N N in most of irreps. The number of linearly independent operators in a given irrep is the same as the number of different (L, S) combinations that can contribute to given J P in the continuum limit (this applies for J ≤ 3/2 which are contained in a single irrep according to table 1). The partial-wave and helicity operators indicate which linear combinations of O Γ,n are most relevant to study scattering with given (J, L, S) or given (J, P, λ 1 , λ 2 ).

Conclusions
We consider three different methods to construct two-hadron interpolators where one or both hadrons carry spin. The focus is on the case with total-momentum zero where parity P is a good quantum number. The correct transformation properties are proven for all three methods. The projection method is a general mathematical tool which leads to one or several operators O Γ,r,n that transform according to given irrep Γ and row r, but it does not give much insight on the underlying continuum quantum numbers. The partial-wave and the helicity methods indicate which linear combinations O Γ,r,n of various n have to be employed in order to hopefully enhance couplings to the states with desired continuum quantum numbers. The partial-wave method renders operators O where each hadron has good helicity λ 1,2 . The quality of these enhancements depends on how close one is to the physical limit, the smallness of the disregarded higher partial waves, and on the magnitude of the physical transitions between various (L, S) for a given J P .
Explicit expressions for P V , P N , V N and N N operators with lowest two relative momenta are provided in all irreducible representations. We verified that all three methods 12 One Qq state with J P = 1 + decays only via L = 0 in the mQ → ∞ limit [21], and O [J=1,L=0,S=1] T + 1 was used to consider D1(2430) → D * π in [22]. The other J P = 1 + state decays only via L = 2 and is narrow.

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lead to consistent results, where partial-wave and helicity methods are particularly helpful for physics interpretation. The operators shall be valuable to simulate the scattering of particles with spin in desired channels using quantum field theory on the lattice.

Acknowledgments
We want to thank kindly R. Briceño

A.1 Projection method
The arbitrary operator H (1),a H (2),a in (4.1) is a linear combination of O Γ ′ ,r ′ and the projector projects out O Γ,r : where (3.2) and the orthogonality theorem [19] were used, while n G is number of group elements.

A.2 Helicity method
Let us verify that annihilation operator (4.6) transforms under rotation R a ∈ O (2) as (3.1), where indices |p| and λ are omitted for brevity where we defined R ′ = R a R and used (2.3).

A.3 Partial-wave method
Here we verify that operator (4.11) transforms under rotation R a ∈ O (2) as (3.1). First the summation R ∈ O (4.11) can be replaced by R ∈ O (2) since both have the same effect on p 13 where (2.2) was used. We intruduce introduce R ′ ≡ R a R, which is also an element of O (2) , and use where the second step follows from Y * and pairs of Clebsch-Gordan coefficients with the same subscripts are summed to get m s1 ,m s2 (A.6) Inserting all this to (A.3) one has

B.1 Wigner D-matrices
We employ the conventional definition of Wigner D-matrix, which renders (2.1) and is used for example in [13,14]. That convention differs from the definition in Wigner's book [23], which is incorporated in Wolfram's Mathematica [24]. 15 The employed D j for the rotation R that is consistent with (2.1) is WignerD denotes the Mathematica function and −π ≤ ω < 3π is the angle of rotation R around n in positive direction. The rotations for π ≤ ω < 3π are present only in the double cover group. The Euler angles of R can be obtained with Mathematica as {α, β, γ} = EulerAngles[T ] with T = exp(−i n Jω) and (J k ) ij = −iǫ ijk (2.5). They are the same for ω and ω + 2π, and the only difference between these two rotations is given by the factor F . The sign of F = ±1 for half-integer j and π ≤ ω < π can be arbitrarily chosen for rotations i = 1, . . . , 24 in table A.1 of [16]: we choose it so that D 1/2 = exp(− i 2 n σω), which gives F = −1 for i = 2, 4, 12, 16, 18, 20, 21, 23, 24 and F = 1 for remaining fifteen. The sign of next 24 elements of O 2 with π ≤ ω < 3π is fixed by F (ω + 2π) = −F (ω). The resulting operators do not depend on the choice of the sign in F , except for irrelevant multiplicative factor. 16

B.3 Two bases for vectors
The vector states The annihilation operators V (2.4) are obtained by hermitian conjugation of V † , so annihilation operators have complex-conjugated coefficients with respect to the states above.
Operators for other irreps are equal to 0.

D Relation between partial wave and helicity operators
The proof of the relation (4.13) between partial-wave operators (4.11) and helicity operators (4.8) is presented here. The Clebsch-Gordan coefficient in the partial wave operator (4.11) can be rewritten as This operator already has a good parity P = P 1 P 2 (−1) L and one can act on it by the parity projection A similar relation can be found in eq. (4) of [11]. We close by deriving the relation (D.3), which was used above. The intermediate steps are taken from [28]. Relation

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The tensor Y LS Jm J ( Rp) can be transformed to helicity basis (denoted by prime on page 197 of [28]) which in turn can be expressed using Wigner-D matrices (page 197 of [28]) where the right-hand-side can be further expressed as (pages 173 and 197 of [28]) Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.