Selection rules for the decay of a particle into two identical massless particles of any spin

The well-known Landau-Yang (LY) theorem on the decay of a neutral particle into two photons is generalized for analyzing the decay of a neutral or charged particle into two identical massless particles of any spin. Selection rules categorized by discrete parity invariance and Bose/Fermi symmetry are worked out in the helicity formulation. The general form of the Lorentz-covariant triple vertices are derived and the corresponding decay helicity amplitudes are explicitly calculated in the Jacob-Wick convention. After checking the consistency of all the analytic results obtained by two complementary approaches, we extract out the key aspects of the generalized LY theorem.


Introduction
In the Standard Model (SM) [1,2,3,4], the photon and gluons are spin-1 massless particles. The direct observation of gravitational waves [5,6,7] strongly indicates the presence of spin-2 massless bosons called gravitons at the quantum level. On the other hand, it is not yet clear whether the lightest neutrino is massless or not [8] and it is a hotly-debated issue whether there can exist any massless particle with its spin larger than two [9,10,11]. In this situation, any conceptual and/or practical studies of massless particles with arbitrary spin are still of great interest.
More than seventy years ago, Landau [12] and Yang [13] have shown that selection rules governing the decay of a particle into two photons, which are the prototype of spin-1 massless particles, can be derived from the general principle of invariance under rotation and inversion. Concisely speaking, in terms of the number n of possible two-photon states for the J and intrinsic parity η of the decaying particle can the selection rules be summarized collectively with the compact notation n [J] η as with a positive integer k. One consequence of the so-called Landau-Yang (LY) theorem is that no on-shell spin-1 particle can decay into two on-shell massless photons. [Therefore, both the gluon-fusion production and the two-photon decay of the resonance with mass of 125 GeV discovered and confirmed at the LHC [14,15] exclude the possibility of the spin being unity. i.e. J = 1 [16].] In this letter, we generalize the LY theorem for investigating the decay X → M M of a neutral or charged particle into two identical massless particles of any spin. We note in passing that, based on the approaches by Landau [12], the generalization of the LY theorem to the case with spin-0 and spin-1/2 massless particles has been investigated in Ref. [17]. Besides, although not studied here, we note that the LY theorem may be avoided if some of the basic assumptions for the theorem are not imposed [18,19,20,21,22,23].
Firstly, we work out the selection rules categorized by discrete parity invariance and Bose/Fermi symmetry due to two identical bosons/fermions on the decay in the helicity formulation based on the Jacob-Wick a sychoi@jbnu.ac.kr b jaehoonjeong229@gmail.com convention [24] in Section 2. Secondly, we derive the general form of the Lorentz covariant triple vertices explicitly in Section 3 and then calculate the corresponding decay helicity amplitudes in detail in Section 4. ‡ After checking the consistency of all the analytic results obtained by two complementary approaches, we summarize the key aspects of the generalized LY theorem and conclude in Section 5. All the formulae useful for explicitly deriving the decay helicity amplitudes in the main text are listed in Appendices A and B.

Selection rules in the helicity formalism
The helicity formalism [24,26] is one of the most efficient tools for discussing the two-body decay of a spin-J particle X into two of a massless particle M of spin s, treating any massive and massless particles on an equal footing. For the sake of a transparent and straightforward analytic analysis, we describe the two-body decay X → M M in the X rest frame (XRF) in terms of the momenta, {p, k 1 , k 2 }, and helicities, {σ, λ 1 , λ 2 }, of the particles, as depicted in Figure 1. : Kinematic configuration for the helicity amplitude of the two-body decay X → M M of X into two identical massless particles M M in the X rest frame (XRF). The notations, {p, k 1,2 } and {σ, λ 1,2 }, are the momenta and helicities of the decaying particle X and two massless particles M M , respectively. The polar and azimuthal angles, θ and φ, are defined with respect to an appropriately chosen coordinate system.
Before going into a detailed description of the general Lorentz-covariant form XM M vertex in Section 3, we study general restrictions on the decay helicity amplitude due to space inversion and Bose/Fermi symmetry for two identical massless bosons/fermions in the final state.
The helicity amplitude of the decay X → M M is decomposed in terms of the polar and azimuthal angles, θ and φ, defining the direction of one massless particle in a fixed coordinate system as with the constraint |λ 1 − λ 2 | ≤ J in the Jacob-Wick convention [24,26] (see Figure 1 for the kinematic configuration). Here, the helicity σ of the spin-J massive particle X takes one of 2J + 1 values between −J and J. In contrast, the helicities, λ 1,2 , can take only two values ±s because only the maximal-magnitude helicity values are allowed for any massless particle. Therefore, there exist at most four independent (λ 1 , λ 2 ) combinations (±s, ±s) and (±s, ∓s), to be denoted by the shortened notations (±, ±) and (±, ∓) in the following. We note that the reduced helicity amplitudes C λ 1 ,λ 2 do not depend on the X helicity σ due to ‡ Another convenient procedure for describing the XM M triple vertex is to use a spinor formalism developed for handling massive as well as massless particles in Ref. [25].
rotational invariance and the polar-angle dependence is fully encoded in the Wigner d function d J σ,λ 1 −λ 1 (θ) given in the convention of Rose [27]. Figure 2: Feynman rules for the general XM M triple vertex of a spin-J particle X and two identical spin-s massless particles M M . The indices, µ, α and β, stand for the sequences of µ = µ 1 · · · µ J , α 1 · · · α s and β 1 · · · β s , collectively. The symmetric and anti-symmetric momentum combinations, p = k 1 + k 2 and k = k 1 − k 2 , are introduced for systematic classifications of the triple vertex tensor.
Bose or Fermi symmetry for the two identical integer or half-integer spin particles in the final state leads to the relation for the reduced decay helicity amplitudes: derived by the (anti)-symmetrization of the final state of two identical particles. If the decay process conserves parity, the reduced helicity amplitudes satisfy the space-inversion or parity relation with η being the intrinsic X parity. Once more, we emphasize that there are only four helicity combinations (±, ±) and (±, ∓) for the final-state system of two massless particles.
Remarkably, the identical-particle (ID) condition (4) and parity (PA) relation (5) enable us to straightforwardly derive the selection/exclusion rules for the two-body decay X → M M classified according to the spin J, the spin s, the X intrinsic parity η and also whether J < 2s or not. The opposite-helicity amplitudes, C ±,∓ , vanish for J < 2s and the same-helicity amplitudes, C ±,± , vanish for any odd integer J. Consequently, one important exclusion rule is that any decay with odd J less than 2s is forbidden irrespective of the X parity. Specifically, the well-known rule that any spin-1 particle cannot decay into two identical massless spin-1 particles such as photons and color-neutral gluons can be guaranteed because J = 1 is odd and J = 1 < 2s = 2 for s = 1. Moreover, for J ≥ 2s, the same-helicity amplitudes still vanish for any odd J. According to the ID condition (4) and the PA relation (5), the opposite-helicity amplitudes survive only when η (−1) 2s = +1 with the relation C +,− = (−1) J C −,+ . Based on the above observations, a few interesting selection rules can be extracted out as follows: • No odd-J particle can decay into two identical spin-0 massless scalar bosons (s = 0).
• The allowed decay of a spin-1 massive particle into two identical massless particles is only into two identical spin-1/2 massless fermions, when the intrinsic parity η of the decaying particle is odd.
• Any even-parity and odd-spin particle cannot decay into two identical massless fermions.
• Any odd-parity and odd-spin particle cannot decay into two identical massless bosons. Table 1 summarizes all the selection/exclusion rules on the special decay X → M M . We note that the case with s = 0 is more restricted than the case with non-zero spin s, as marked with the comment [forbidden].
Interestingly, as pointed out in Ref. [25], the general selection rules enable us to re-interpret the Weinberg-Witten (WW) theorem [9]. Since conserved currents and stress tensors measure the charge and momentum on single particle states respectively, let us consider the decay of a massive neutral particle into two oppositehelicity massless particles, i.e. λ 1 = −λ 2 = ±, because two momenta k 1 and k 2 are out going and so λ 1 = −λ 2 in order for k 1 and k 2 to represent the same particle. In this case, as can be checked with Table 1, the inequality s ≤ J/2 must be satisfied, i.e. s ≤ 1/2 for J = 1 and s ≤ 1 for J = 2. That is to say, massless particles with spin s > 1/2 cannot couple to a Lorentz covariant conserved current corresponding to J = 1 and those with spin s > 1 cannot couple to a conserved energy-momentum stress tensor corresponding to J = 2. Table 1: Selection rules for the decay X → M M . The values, J and s, are the spins of the massive X and massless M particles, respectively, and the signature ± denotes the sign of the product ζ ≡ η(−1) J−2s of the X intrinsic parity η and the factor (−1) J−2s . The mark [f obidden] means that the corresponding term is absent when s = 0.
In the next section, we show how the selection rules derived in the helicity formalism are reflected and encoded in the Lorentz-covariant form of the triple XM M vertex in the integer and half-integer s cases separately.

General Lorentz-covariant triple vertex tensors
Generically, the decay amplitude of one on-shell particle X of mass m and integer spin J into two of a massless particle M of spin s can be written in terms of the triple vertex tensor Γ as with λ 1,2 = ±s = ± for the two-body decay X → M M , where p and σ are the momentum and helicity of the particle X, and k 1,2 and λ 1,2 are the momenta and helicities of two massless particles, respectively. Here, p = k 1 + k 2 and k = k 1 − k 2 , are symmetric and anti-symmetric under the interchange of two momenta, k 1 and k 2 . If s is an integer, then the wave tensorsū α 1 ···α 2 1 (k 1 , λ 1 ) and v β 1 ···βs 2 (k 2 , λ 2 ) are given bȳ v β 1 ···βs and if s = n + 1/2 is a half-integer, the wave tensors are given bȳ whereū 1 (k 1 , ±) = u † 1 (k 1 , ±)γ 0 and the u 1 (k 1 , ±) and v 2 (k 2 , ±) are the spin-1/2 particle u 1 and anti-particle v 2 spinors. (See Appendix A for their detailed expressions.) An on-shell boson of integer spin J, non-zero mass m, momentum p and helicity σ is defined by a rank-s tensor ǫ µ 1 ···µ J (p, σ) [28,29,30] that is completely symmetric, traceless and divergence-free and it satisfies the on-shell wave equation (p 2 − m 2 ) ǫ µ 1 ···µ J (p, σ) = 0 for any helicity value of σ taking an integer between −J and J. The wave tensor can be expressed explicitly by a linear combination of s products of spin-1 wave vectors with appropriate Clebsch-Gordon coefficients.

Massless particles of integer spin
The wave tensor of a massless particle of integer spin s is given simply by a product of s spin-1 wave vectors, each of which carries the same helicity of ±1, as where, for notational convenience, the notation ± is used for ±1. The wave tensor (14) also is completely symmetric, traceless and divergence-free and it satisfies the wave equation k 2 ǫ α 1 ···αs (k, λ) = 0 with k 2 = 0 automatically.
To begin with, let us consider as a special case the decay X → γγ of a massive integer-spin boson X into two massless spin-1 photons, originally investigated by Landau [12] and Yang [13]. Imposing the on-shell conditions valid for two spin-1 massless photons § and performing the Bose symmetrization of two identical γ states in the final state, we can write the general Xγγ vertex in a greatly-simplified form [31] as satisfying the orthogonality conditions, k α 1 Γ X → γγ µ; α,β (p, k) = k β 2 Γ X → γγ µ; α,β (p, k) = 0, with µ denoting collectively µ 1 · · · µ J , the projection factors, η ± = [1 ± (−1) J ]/2, and two momentum combinations, p = k 1 + k 2 and k = k 1 − k 2 , which are symmetric and antisymmetric under the interchange of two massless particles, i.e. k 1 ↔ k 2 and α ↔ β, respectively. The antisymmetric tensor αβpk = ε αβρσ p ρ k σ in terms of the totally antisymmetric Levi-Civita tensor with the sign convention ε 0123 = +1. For the sake of notation, the following orthogonal tensors are introduced, with i = 1, · · · , J. The totally symmetric wave tensor ǫ µ 1 ···µ J (p, σ) to be coupled to the triple vertex (16) guarantees the automatic symmetrization of all the x ± b and y ± b terms under any µ-index permutations. It is straightforward to derive the following selection rules from the expression of the Xγγ vertex in Eq. (16), § While the on-shell conditions (15) are maintained, the wave vector ǫ µ i of a spin-1 massless particle can be adjusted by adding any term proportional to k µ i , guaranteeing that its spatial part orthogonal to ki remains as two physical degrees of freedom.
• The Y − 1 term survives for positive odd integers, J = 3, 5, and so on, satisfying J > 2. One immediate consequence is that any massive on-shell spin-1 particle with J = 1 cannot decay into two on-shell identical massless spin-1 particles such as photons, because the three x ± b and y + b terms vanish with η + = 0 for J = 1 and the y − b term surviving only for odd J contribute to the vertex only for J ≥ 3. For a massless boson M = b of an arbitrary integer spin s, the general Lorentz-covariant form of the Xbb vertex can be written compactly by introducing two scalar-type operators and two tensor-type operators as with i = 1, · · · , s. The general form of the vertex tensor is then cast into a compact form as with η ± = [1 ± (−1) J ]/2 and the step function θ(J − 2s) = 1 and 0 for J ≥ 2s and J < 2s. Here, the indices, µ, α and β, stand collectively for µ = µ 1 · · · µ J , α = α 1 · · · α s and β = β 1 · · · β s . One immediate consequence is that the case with odd J less than 2s is forbidden irrespective of the X intrinsic parity η, which is consistent with the corresponding selection rule listed in Table 1. Specifically, any spin-1 particle cannot decay into two identical spin-1 massless particles with s = 1.
Interchanging two identical massless fermions, i.e taking the opposite fermion flow line [32,33], we can rewrite the helicity amplitude of the decay X → f f with a massless fermion M = f as with the superscript T denoting the transpose of the matrix. Introducing the charge-conjugation operator C satisfying C † = C −1 and C T = −C relating the v spinor to the u spinor as withū = u † γ 0 , we can rewrite the amplitude as Since Fermi statistics requiresM = −M, the triple vertex tensor must satisfy the relation which enables us to classify all the allowed terms systematically [34,35,36].
The basic relation for the charge-conjugation invariance of the Dirac equation is C γ T µ C −1 = −γ µ with a unitary matrix C. Repeatedly using the basic relation, we can derive There are no further independent terms as any other operator can be replaced by a linear combination of 1, γ 5 , γ µ , and γ µ γ 5 by use of the so-called Gordon identities, when coupled to the u and v spinors.
Because of the conditions (25) and (26), the vector structure γ µ can be contracted only with the wave vector ǫ µ (p, σ) of the decaying particle X with the helicity σ = ±1, 0 = ±, 0. Effectively we can make the following replacements because two corresponding terms in each equation give rise to the same helicity amplitudes as shown explicitly in Appendix B. Then, for a massless fermion M = f , the general form of the Xf f triple vertex tensor can be written as in terms of four independent parameters, x ± f and y ± f , with n = s − 1/2. The scalar part proportional to g ⊥ α i β i k µ 2i k µ 2i+1 of T + α i β i ,µ 2i µ 2i+1 in Eq. (20) can be discarded because its contribution vanishes in the opposite helicity case with λ 1 = −λ 2 enforced by the helicity-preserving (axial-)vector currents. It is straightforward to check that the expression (34) satisfies the Fermi symmetry condition (30) for two identical fermions. One non-trivial observation is that a spin-1 particle can decay only into two identical spin-1/2 particles [36] through an axial-vector γ µ γ 5 current as can be checked with the last expression in Eq. (34).

Explicit form of reduced helicity amplitudes
Employing all the analytic results for the scalar, vector and tensor currents listed in Appendix B enables us to explicitly calculate all the reduced helicity amplitudes C λ 1 ,λ 2 defined in Eq. (3) both in the integer-spin and half-integer-spin cases for the massless particle.
In the case with a massless integer-spin s boson M = b, the reduced helicity amplitudes for the process X → bb with the same helicities (±, ±) read surviving only for even-integer J. If parity is preserved, the term, x + b or x − b , can survive for the even or odd X intrinsic parity η = ±, respectively. On the other hand, the reduced helicity amplitudes with the opposite helicities (±, ∓), which survive only when J ≥ 2s, read for even-integer spin J ≥ 2s and even X intrinsic parity and for odd-integer spin J ≥ 2s and odd X intrinsic parity.
In the case with a massless fermion M = f with a half-integer spin s, the reduced helicity amplitudes for the process X → f f with the same helicities (±, ±) read surviving only for even-integer J. If parity is preserved, the x + f and x − f terms can survive for the odd and even X intrinsic parity η = ∓, respectively. On the other hand, the reduced helicity amplitudes, which survive only when J ≥ 2s, read for even-integer spin J ≥ 2s and odd X intrinsic parity η = −, and for odd-integer spin J ≥ 2s and even X intrinsic parity η = +.

Conclusions
We have generalized the well-known Landau-Yang (LY) theorem on the decay of a neutral particle into two photons for analyzing the two-body decay of a neutral or charged particle into two identical massless particles of any integer or half-integer spin. After having worked out the selection rules classified by discrete parity invariance and Bose/Fermi symmetry in the helicity formulation, we have derived the general form of the Lorentz covariant triple vertices and calculated the helicity amplitudes explicitly. After checking the consistency of all the analytic results obtained from two complementary approaches, we have drawn out the key aspects of the generalized LY theorem including the re-interpretation of the WW theorem.
Introducing a compact notation n [J, s] ζ consisting of the number n of independent terms, the X and M spins, J and s, and a parity ζ = η(−1) J−2s with the X intrinsic parity η, we can summarize the selection rules on the decay of a massive particle into an identical pair of massless particles of arbitrary spin: with k = 1, 2, · · · . In the special case of s = 1, the selection rules reduce to those called the LY theorem described in Eq. (1). The massless particle for the decay of a spin-1 particle into its identical pair must have a half-integer spin s = 1/2. If parity is preserved, no odd-spin and odd-parity (even-parity) particle can decay into an identical pair of massless bosons (fermions).
As natural extension of the present work, the decays of an arbitrary integer-spin particle into two identical massive particles [36] or two distinguishable charge self-conjugate particles of any spin are presently under study and its results will be reported separately. Before closing, we emphasize that the selection rules obtained in this letter can be applied without any modifications, for example, to the decay of a doublycharged massive particle into two identical singly-charged massless particles, where the charge can be of any type as well as of a typical electric charge type.

Acknowledgments
The work was in part by the Basic Science Research Program of Ministry of Education through National Research Foundation of Korea (Grant No. NRF-2016R1D1A3B01010529) and in part by the CERN-Korea theory collaboration. A correspondence with Kentarou Mawatari concerning the covariant formulation of the Landau-Yang theorem is acknowledged.

A Wave vectors and spinors in the Jacob-Wick convention
In Appendix A, we show the explicit expressions for the wave vectors and spinors of a massive particle X and two massless particles in the XRF with the kinematic configuration as shown in Figure 1. The Jacob-Wick convention of Ref. [24] is chosen for the vectors and spinors. In terms of the polar and azimuthal angles, θ and φ, the three momenta, p = k 1 + k 2 and k 1,2 , and the combination k = k 1 − k 2 read For the sake of notation, we introduce three unit vectorŝ k = (sin θ cos φ, sin θ sin φ, cos θ) = k/m , which are mutually orthonormal, i.e.k ·θ =θ ·φ =φ ·k = 0 andk ·k =θ ·θ =φ ·φ = 1.
The wave vectors for the particle with momentum p and two massless particles with momenta k 1,2 are given by ǫ(p, 0) = (0, 0, 0, 1) , where the last relation between two wave vectors is satisfied in the Jacob-Wick convention.
The angle-dependent parts are encoded solely in the Wigner-d functions and the reduced helicity amplitudes are independent of the helicity of the decaying particle. Therefore, it is sufficient to consider the cases with the maximal X helicity σ = J and the zero and two maximal helicity differences, λ 1 − λ 2 = 0, ±2s for deriving the reduced helicity amplitudes. For them, three relevant Wigner-d functions read which are to be factored out for deriving the reduced helicity amplitudes in Section 4.