Selection Rules for the Decay of a Particle Into Two Identical Massless

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2 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 MM in the X rest frame (XRF) → X(p,σ) M(k , λ ) + M(k , λ ) , (2) → 1 1 2 2 in terms of the momenta, p, k , k , and helicities, σ, λ , λ , of the particles, as depicted in Figure 1. { 1 2} { 1 2}

M(k1, λ1)

θ, φ MX→MM X(p, σ) = σ;λ1,λ2 (θ, φ)

M(k2, λ2)

Figure 1: Kinematic conﬁguration for the helicity amplitude of the two-body decay X MM of X into → two identical massless particles MM in the X rest frame (XRF). The notations, p, k and σ, λ , are { 1,2} { 1,2} the momenta and helicities of the decaying particle X and two massless particles MM, 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 XMM 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 MM 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

X→MM (θ, φ) = dJ (θ) ei(σ−λ1+λ2)φ with λ λ J . (3) Mσ;λ1,λ2 Cλ1,λ2 σ, λ1−λ2 | 1 − 2|≤ with the constraint λ λ J in the Jacob-Wick convention [24, 26] (see Figure 1 for the kinematic | 1 − 2| ≤ 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 , can take only two values s because only the maximal-magnitude 1,2 ± helicity values are allowed for any massless particle. Therefore, there exist at most four independent (λ1, λ2) combinations ( s, s) and ( s, s), to be denoted by the shortened notations ( , ) and ( , ) in the ± ± ± ∓ ± ± ± ∓ following. We note that the reduced helicity amplitudes do not depend on the X helicity σ due to Cλ1,λ2 ‡Another convenient procedure for describing the XMM triple vertex is to use a spinor formalism developed for handling massive as well as massless particles in Ref. [25].

2 J rotational invariance and the polar-angle dependence is fully encoded in the Wigner d function dσ,λ1−λ1 (θ) given in the convention of Rose [27].

Mα k1

µ µ X = i Γα,β(p, k) p

Mβ

Figure 2: Feynman rules for the general XMM triple vertex of a spin-J particle X and two identical spin-s massless particles MM. The indices, µ, α and β, stand for the sequences of µ = µ µ , α α 1 · · · J 1 · · · s and β β , collectively. The symmetric and anti-symmetric momentum combinations, p = k + k and 1 · · · s 1 2 k = k k , are introduced for systematic classiﬁcations of the triple vertex tensor. 1 − 2

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:

= ( 1)J with λ λ J , (4) Cλ1,λ2 − Cλ2,λ1 | 1 − 2|≤ 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

J−2s = η( 1) − − with λ λ J , (5) Cλ1,λ2 − C λ1, λ2 | 1 − 2|≤ 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 straight- forwardly derive the selection/exclusion rules for the two-body decay X MM 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, ± ∓, vanish for J < 2s and the same-helicity amplitudes, ± ±, vanish for any odd integer J. C , C , 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 2s J when η ( 1) = +1 with the relation − = ( 1) − . − C+, − 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.

3 Table 1 summarizes all the selection/exclusion rules on the special decay X MM. 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 opposite- helicity massless particles, i.e. λ = λ = , because two momenta k and k are out going and so λ = λ 1 − 2 ± 1 2 1 − 2 in order for k1 and k2 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.

J J < 2s J 2s [s = 0] ≥ ζ = ζ =+ ζ = ± − odd forbidden forbidden − = − [forbidden] C+, −C ,+

ζ = ζ =+ ζ = ± − even = − − − = − [forbidden] , +,+ = −,− = − − [forbidden] C+,+ ±C , C+, C ,+ C C C+,+ −C ,

Table 1: Selection rules for the decay X MM. 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 [fobidden] 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 XMM vertex in the integer and half-integer s cases separately.

3 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

X→MM α1···αs µ1···µJ β1···βs =u ¯ (k1, λ1) Γ (p, k) v (k2, λ2) ǫµ1···µ (p,σ) , (6) Mσ;λ1,λ2 1 α1···αs,β1···βs 2 J with λ = s = for the two-body decay X MM, where p and σ are the momentum and helicity of 1,2 ± ± → the particle X, and k1,2 and λ1,2 are the momenta and helicities of two massless particles, respectively. Here, p = k1 + k2 and k = k1 k2, are symmetric and anti-symmetric under the interchange of two momenta, k1 − α1···α2 β1···βs and k2. If s is an integer, then the wave tensorsu ¯1 (k1, λ1) and v2 (k2, λ2) are given by

u¯α1···αs (k , s) = ǫ∗α1···αs (k , s) , (7) 1 1 ± 1 1 ± vβ1···βs (k , s) = ǫ∗β1···βs (k , s) , (8) 2 2 ± 2 2 ± and if s = n + 1/2 is a half-integer, the wave tensors are given by

u¯α1···αs (k , s) = ǫ∗α1···αn (k , n)u ¯ (k , ) , (9) 1 1 ± 1 1 ± 1 1 ± vβ1···βs (k , s) = ǫ∗β1···βn (k , n) v (k , ) , (10) 2 2 ± 2 2 ± 2 2 ±

4 whereu ¯ (k , )= u† (k , )γ0 and the u (k , ) and v (k , ) are the spin-1/2 particle u and anti-particle 1 1 ± 1 1 ± 1 1 ± 2 2 ± 1 v2 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 (p,σ) [28, 29, 30] that is completely symmetric, traceless and divergence-free µ1···µJ εαβµiµj ǫ (p,σ) = 0 , (11) µ1···µi···µj ···µJ gµiµj ǫ (p,σ) = 0 , (12) µ1···µi···µj ···µJ pµi ǫ (p,σ) = 0 , (13) µ1···µi···µJ

2 2 and it satisﬁes the on-shell wave equation (p m ) ǫµ1···µ (p,σ) = 0 for any helicity value of σ taking − J 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.

3.1 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 ± ǫα1···αs (k, s) = ǫα1 (k, ) ǫαs (k, ) , (14) ± ± · · · ± 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 k2ǫα1···αs (k, λ) = 0 with k2 = 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 §

k ǫ (k , λ )=0 and k2 = 0 [i = 1, 2] , (15) i · i i i i 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

X → γγ + J Γ (p, k) = η+ x g⊥αβ kµ1 kµ /m µ; α,β b · · · J − J+2 + η+ x ı αβpk kµ1 kµ /m b h i · · · J + 2 J−2 + η+ y g⊥αµ1 g⊥βµ2 + g⊥βµ1 g⊥αµ2 g⊥αβ kµ1 kµ2 /m kµ3 kµ /m b − · · · J − J + η− y ı( g⊥αµ1 βµ2pk + g⊥βµ1 αµ2pk ) kµ3 kµ /m , (16) b h i h i · · · J α X → γγ β X → γγ satisfying the orthogonality conditions, k1 Γµ; α,β (p, k) = k2 Γµ; α,β (p, k) = 0, with µ denoting collectively J µ µ , the projection factors, η± = [1 ( 1) ]/2, and two momentum combinations, p = k + k and 1 · · · J ± − 1 2 k = k k , which are symmetric and antisymmetric under the interchange of two massless particles, i.e. 1 − 2 k k and α β, respectively. The antisymmetric tensor αβpk = ε pρkσ in terms of the totally 1 ↔ 2 ↔ h i αβρσ antisymmetric Levi-Civita tensor with the sign convention ε0123 = +1. For the sake of notation, the following orthogonal tensors are introduced,

g = g 2 k k /m2 , ⊥αβ αβ − 2α 1β g = g 2 p k /m2 , ⊥αµi αµi − α 1µi g = g 2 p k /m2 , (17) ⊥βµi βµi − β 2µi µ1···µ with i = 1, , J. The totally symmetric wave tensor ǫ J (p,σ) to be coupled to the triple vertex (16) · · · ± ± guarantees the automatic symmetrization of all the xb and yb 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),

§ ǫµ While the on-shell conditions (15) are maintained, the wave vector i of a spin-1 massless particle can be adjusted by adding kµ ~k any term proportional to i , guaranteeing that its spatial part orthogonal to i remains as two physical degrees of freedom.

5 • + The Y1,2 terms survive for non-negative even integers, J = 0, 2, 4, and so on. • The Y + term survives for positive even integers, J = 2, 4, and so on, satisfying J 2. 3 ≥ • − The Y1 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 xb and yb terms vanish with η = 0 for J = 1 and the y− term surviving only for odd J contribute to the vertex only for J 3. + b ≥ 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

S+ = g , (18) αiβi ⊥αiβi − 2 S = ı αiβipk /m , (19) αiβi h i + 2 T = g⊥α µ2 −1 g⊥β µ + g⊥β µ − g⊥α µ2 g⊥α β kµ2 −1 kµ2 /m , (20) αiβi,µ2i−1µ2j i i i 2i i 2i 1 i i − i i i j − 2 T = ı (g⊥α µ2 −1 βi µ2ipk + g⊥β µ − αiµ2ipk )/m , (21) αiβi,µ2i−1µ2i i i h i i 2i 1 h i with i = 1, ,s. The general form of the vertex tensor is then cast into a compact form as · · · X → bb + + − − + + J Γ (p, k)= η+ x S + x S S S kµ1 kµ /m µ; α,β b α1β1 b α1β1 α2β2 · · · αsβs · · · J + + + + J−2s + η+ θ(J 2s) y T T T kµ2 +1 kµ /m − b α1β1,µ1µ2 α2β2,µ3µ4 · · · αsβs,µ2s−1µ2s s · · · J − − + + J−2s + η− θ(J 2s) y T T T kµ2 +1 kµ /m , (22) − b α1β1,µ1µ2 α2β2,µ3µ4 · · · αsβs,µ2s−1µ2s s · · · J J with η± = [1 ( 1) ]/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 µ , α = α α and β = β β . One immediate 1 · · · J 1 · · · s 1 · · · s 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.

3.2 Massless particles of half-integer spin An on-shell massless particle or antiparticle of a half-integer spin s = n + 1/2 (n = 0, 1, 2, ) and · · · momentum k may be described by a product of a u or v spinor and n spin-1 wave vectors as

uα(k, s) = ǫα1 (k, ) ǫαn (k, ) u(k, ) , (23) ± ± · · · ± ± vα(k, s) = ǫ∗α1 (k, ) ǫ∗αn (k, ) v(k, ) , (24) ± ± · · · ± ± that are traceless, symmetric and divergence-free in the indices α = α α and the u and v spinor tensors 1 · · · n satisfy

γ uα1···αi···αn (k, s) = γ vα1···αi···αn (k, s) = 0 , (25) αi ± αi ± k uα1···αi···αn (k, s) = k vα1···αi···αn (k, s) = 0 , (26) 6 ± 6 ± with k = k γµ. We adopt the chiral representation for the Dirac gamma matrices γµ (µ = 0, 1, 2, 3), whose 6 µ expressions are listed in Appendix B. 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 ff with a massless fermion M = f as → ˜ X→ff =u ¯β(k , λ ) Γµ (p, k) vα(k , λ ) ǫ (p,σ) Mσ;λ1,λ2 2 2 2 β,α − 1 1 1 µ = vT α(k , λ ) ΓµT (p, k)u ¯Tβ(k , λ ) ǫ (p,σ) (27) 1 1 1 β,α − 2 2 2 µ with the superscript T denoting the transpose of the matrix. Introducing the charge-conjugation operator C satisfying C† = C−1 and CT = C relating the v spinor to the u spinor as − vα(k, λ)= Cu¯T α(k, λ) , (28)

6 withu ¯ = u†γ0, we can rewrite the amplitude as ˜ X→ff = u¯β(k , λ ) C ΓµT (p, k) C−1 vα(k , λ ) ǫ (p,σ) . (29) Mσ;λ1,λ2 − 1 1 1 β,α − 2 2 2 µ Since Fermi statistics requires ˜ = , the triple vertex tensor must satisfy the relation M −M C ΓµT (p, k) C−1 = Γµ (p, k) , (30) β,α − α,β 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 γT C−1 = γ with µ − µ a unitary matrix C. Repeatedly using the basic relation, we can derive +1 for Γ=1, γ , γ γ c T −1 5 µ 5 Γ C Γ C = ǫC Γ with ǫC = , (31) ≡ ( 1 for Γ= γµ − There are no further independent terms as any other operator can be replaced by a linear combination of 1, γ5, γµ, and γµγ5 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 γµ can be contracted only with the wave vector ǫµ(p,σ) of the decaying particle X with the helicity σ = 1, 0 = , 0. Eﬀectively we can make the ± ± following replacements S− [1, γ ] S+ [γ , 1] , (32) αβ 5 ←→ − αβ 5 γ [1, γ ] T − γ [γ , 1] T + , (33) µ1 5 αβ,µ2µ3 ←→ µ1 5 αβ,µ2µ3 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 Xff triple vertex tensor can be written as

X → ff + − + + J Γ (p, k)= η+ (x + x γ5) S S kµ1 kµ /m µ; α,β f f α1β1 · · · αnβn · · · J + + + J−2s−1 + η+ θ(J 2s) y γµ1 T T kµ2 +2 kµ /m − f α1β1,µ2µ3 · · · αnβn,µ2nµ2n+1 n · · · J − + + J−2s−1 + η− θ(J 2s) y γµ1 γ5 T T kµ2 +2 kµ /m , (34) − f α1β1,µ2µ3 · · · αnβn,µ2nµ2n+1 n · · · J in terms of four independent parameters, x± and y±, with n = s 1/2. The scalar part proportional to f f − g k k of T + in Eq. (20) can be discarded because its contribution vanishes in the opposite ⊥αiβi µ2i µ2i+1 αiβi,µ2iµ2i+1 helicity case with λ = λ enforced by the helicity-preserving (axial-)vector currents. It is straightforward 1 − 2 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 γµγ5 current as can be checked with the last expression in Eq. (34).

4 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 deﬁned in Eq. (3) both in the integer-spin Cλ1,λ2 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 → ± ± J/2 2 J! + − ±,± = (x x ) , (35) C (2J)! b ± b + − surviving only for even-integer J. If parity isp preserved, the term, xb or xb , 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 ± ∓ ≥

J/2 (J + 2s)! (J 2s)! + ±,∓ = 2 − yb , (36) C s (2J)!

7 for even-integer spin J 2s and even X intrinsic parity and ≥

J/2 (J + 2s)! (J 2s)! − ±,∓ = 2 − yb , (37) C ∓ s (2J)! 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 ff with the same helicities ( , ) read → ± ± J/2 2 J! + − ±,± = (x x ) , (38) C (2J)! f ∓ f

p + − surviving only for even-integer J. If parity is preserved, the xf and xf 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 ≥

2s J/2 (J + 2s)! (J 2s)! + ±,∓ = ( 1) 2 − yf , (39) C − s (2J)! for even-integer spin J 2s and odd X intrinsic parity η = , and ≥ −

2s J/2 (J + 2s)! (J 2s)! − ±,∓ = ( 1) 2 − yf , (40) C ∓ − s (2J)! for odd-integer spin J 2s and even X intrinsic parity η = +. ≥

5 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:

n [J, s]ζ = 1 [0, 0]+ , 1 [0,s> 0]± , 1 [2k 1, 0 k]± , (41) ≤ ≤ 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 doubly- charged 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.

~~8 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 + k and k , and the combination k = k k read 1 2 1,2 1 − 2 p = m (1, 0, 0, 0) , (42) m k = (1, sin θ cos φ, sin θ sin φ, cos θ) , (43) 1 2 m k = (1, sin θ cos φ, sin θ sin φ, cos θ) , (44) 2 2 − − − k = m (0, sin θ cos φ, sin θ sin φ, cos θ) , (45)

~~For the sake of notation, we introduce three unit vectors~~

kˆ = (sin θ cos φ, sin θ sin φ, cos θ) = ~k/m , (46) θˆ = (cos θ cos φ, cos θ sin φ, sin θ) , (47) − φˆ = ( sin φ, cos φ, 0) , (48) − which are mutually orthonormal, i.e. kˆ θˆ = θˆ φˆ = φˆ kˆ = 0 and kˆ kˆ = θˆ θˆ = φˆ φˆ = 1. · · · · · · The wave vectors for the particle with momentum p and two massless particles with momenta k1,2 are given by 1 ǫ(p, ) = (0, 1, ı, 0) , (49) ± √2 ∓ − ǫ(p, 0) = (0, 0, 0, 1) , (50) 1 ±ıφ ǫ1(k1, ) = e (0, cos θ cos φ + ı sin φ, cos θ sin φ ı cos φ, sin θ) ± √2 ∓ ∓ − ± 1 = e±ıφ (0, θˆ ıφˆ) , (51) √2 ∓ − ǫ (k , ) = ǫ (k , ) = ǫ∗(k , ) = ǫ∗(k , ) , (52) 2 2 ± 1 1 ∓ − 1 1 ± − 2 2 ∓ where the last relation between two wave vectors is satisﬁed in the Jacob-Wick convention.

~~The spin-1/2 u and v 4-component spinors of the massless particles with momenta k1,2 are given in the Jacob-Wick convention by~~

~~0 u (k , +) = v (k , )= √m , (53) 1 1 1 1 ˆ − − χ+(k) !~~

~~χ−(kˆ) u1(k1, ) = v1(k1, +) = √m , (54) − − 0 ! 0 u (k , +) = v (k , )= √m , (55) 2 2 2 2 ˆ − χ−(k) !~~

~~χ+(kˆ) u2(k2, ) = v2(k2, +) = √m , (56) − 0 !~~

9 satisfying the relations u (k , ) = γ0u (k , ) and v (k , ) = γ0v (k , ) with the expression of γ0 2 2 ± 1 1 ∓ 2 2 ± − 1 1 ∓ given in Appendix B, where the 2-component spinors χ±(kˆ) are written in terms of the polar and azimuthal angles, θ and φ, as

~~cos θ e−iφ/2 ˆ iφ/2 2 χ+(k) = e θ iφ/2 , (57) sin 2 e ! sin θ e−iφ/2 ˆ −iφ/2 − 2 χ−(k) = e θ iφ/2 , (58) cos 2 e !~~

~~† being mutually orthonormal, i.e. χa(kˆ)χ (kˆ)= δ , with a, b = . b a,b ±~~

~~B Scalar, vector and tensor currents in the Jacob-Wick convention~~

Appendix B contains the list of the expressions for several scalar, vector and tensor currents in the kinematic configuration depicted in Figure 1 to be used in the main text. For explicit current calculations, the chiral representation is adopted for four anti-commutating Dirac gamma matrices γµ with µ = 0, 1, 2, 3 and γ = ıγ0γ1γ2γ3, whose expressions are given by the following 4 4 matrices 5 × 0 σµ 1 0 µ + − γ = µ and γ5 = , (59) σ− 0 ! 0 1 ! with σµ = (1, ~σ) in terms of three Pauli matrices ~σ = (σ ,σ ,σ ) defined by ± ± 1 2 3 0 1 0 ı 1 0 σ1 = , σ2 = − , σ3 = . (60) 1 0 ! ı 0 ! 0 1 ! − The metric tensor is defined to be gµν = g = diag(1, 1, 1, 1). µν − − − Firstly, the helicity-dependent scalar and pseudo-scalar currents, surviving only when two helicities are same, i.e. λ1 = λ2, read

u+ u¯ (k , λ )v (k , λ )/m = δ , (61) λ1λ2 ≡ 1 1 1 2 2 2 λ1,λ2 u+ u¯ (k , λ )γ v (k , λ )/m = λ δ , (62) λ1λ2 ≡ 1 1 1 5 2 2 2 − 1 λ1,λ2 where λ = 1 = which is two times the helicity, 1/2. Secondly, the helicity-dependent vector and 1,2 ± ± ± axial-vector currents, surviving only when two helicities are opposite, i.e. λ = λ , read 1 − 2 µ ∗µ u¯ (k , λ )γ v (k , λ ) = √2 m δ − ǫ (k , λ ) , (63) 1 1 1 2 2 2 − λ1, λ2 1 1 1 µ ∗µ u¯ (k , λ )γ γ v (k , λ ) = √2 m λ δ − ǫ (k , λ ) , (64) 1 1 1 5 2 2 2 − 1 λ1, λ2 1 1 1 with λ = 1= where the expression of the wave vector ǫµ(k , λ ) is given in Eq. (51). 1,2 ± ± 1 1 1 The contraction of the X wave vector ǫ(p,σ) with σ = , 0 and k gives ± k ǫ(p, 0) = m cos θ , (65) · − 1 k ǫ(p, ) = m sin θ e±iφ , (66) · ± ± √2 in terms of the polar and azimuthal angles, θ and φ, in the XRF. For the sake of eﬃciently calculating and deriving the (reduced) helicity amplitudes of the decay X ± ± → MM, we introduce four helicity-dependent quantities, sλ1λ2 and tλ1λ2 , deﬁned by contracting the scalar and tensor operators given in Eqs. (18), (19), (20) and (21) with the X and M wave vectors appropriately as

~~± ∗α ∗β ± sλ1λ2 = ǫ1 (k1, λ1) ǫ2 (k2, λ2) Sαβ , (67) ± ∗α ∗β ± µ ν tλ1λ2 = ǫ1 (k1, λ1) ǫ2 (k2, λ2) Tαβ,µν ǫ (p, +) ǫ (p, +) , (68)~~

~~10 and two additional helicity-dependent quantities obtained by contracting the vector and axial-vector currents with the X wave vector ǫ(p, +) as~~

+ µ vλ1λ2 =u ¯1(k1, λ1) γµ v2(k2, λ2) ǫ (p, +)/m , (69) − µ vλ1λ2 =u ¯1(k1, λ1)γµγ5v2(k2, λ2) ǫ (p, +)/m , (70) with λ = 1= . Explicitly the same-helicity quantities and the opposite-helicity quantities read 1,2 ± ± + − s±± = s±± = 1 , (71) + ± 2 2 t+− = (1 + cos θ) /2 = 2 d2,2(θ) , (72) t+ = (1 cos θ)2 e4ıφ/2 = 2 d2 (θ) e4ıφ , (73) −+ − 2,−2 t− = t+ , (74) ±∓ ± ±∓ and the quantities involving the spinors in the fermionic case

u+ = u− = 1 , (75) ±± ∓ ±± + 1 v+− = (1 + cos θ)/√2 = √2 d1,1(θ) , (76) v+ = (1 cos θ) e2ıφ/√2 = √2 d1 (θ) e2ıφ , (77) −+ − 1,−1 v− = v+ , (78) ±∓ ± ±∓ in terms of the polar and azimuthal angles θ and φ in the XRF. All the quantities with other helicity combinations are zero. The validity of the replacements in Eq. (33) can be conﬁrmed with the four relations

+ ± − ∓ s±±u±± = s±±u±± , (79) + ± − − ∓ t±∓v±∓ = + t±∓v±∓ , (80) which can be checked by use of all the expressions from Eq. (71) to Eq. (78). Moreover, in the general and covariant form, we have the corresponding four identities for non-zero decay helicity amplitudes

S− ǫ∗α(k , ) ǫ∗β(k , )u ¯ (k , ) [1, γ ] v (k , ) αβ 1 1 ± 2 2 ± 1 1 ± 5 2 2 ± = S+ ǫ∗α(k , ) ǫ∗β(k , )u ¯ (k , ) [γ , 1] v (k , ) , (81) − αβ 1 1 ± 2 2 ± 1 1 ± 5 2 2 ± T − ǫ∗α(k , ) ǫ∗β(k , )u ¯ (k , ) γ [1, γ ] v (k , ) αβ,µ2µ3 1 1 ± 2 2 ∓ 1 1 ± µ1 5 2 2 ∓ = T + ǫ∗α(k , ) ǫ∗β(k , )u ¯ (k , ) γ [γ , 1] v (k , ) , (82) αβ,µ2µ3 1 1 ± 2 2 ∓ 1 1 ± µ1 5 2 2 ∓ where the last term g k k /m2 of the tensor T + defined in Eq. (20) does not contribute to the − ⊥αβ µ2 µ3 αβ,µ2µ3 expression on the right-hand side of Eq. (82) surviving only in the opposite-helicity case with λ = λ = . 1 − 2 ± 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 λ = 0, 2s for 1 − 2 ± deriving the reduced helicity amplitudes. For them, three relevant Wigner-d functions read

J J ( 1) (2J)! J dJ, 0(θ) = − J sin θ , (83) 2 p J! ( 1)J−2s (2J)! dJ (θ) = − (1 + cos θ)2s sinJ−2s θ , (84) J, 2s 2J (J + 2s)! (J 2s)! s − ( 1)J+2s (2J)! dJ (θ) = − (1 cos θ)2s sinJ−2s θ , (85) J,−2s 2J (J + 2s)! (J 2s)! − s − which are to be factored out for deriving the reduced helicity amplitudes in Section 4.

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