A theorem about two-body decay and its application for a doubly-charged boson $H^{\pm\pm}$ going to $τ^{\pm}τ^{\pm}$
Li-Gang Xia


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In a general decay chain A→ B_1B_2→ C_1C_2…, we prove that the angular correlation function I(θ_1,θ_2,ϕ_+) in the decay of B_1,2 is irrelevant to the polarization of the mother particle A at production. This guarantees that we can use these angular distributions to determine the spin-parity nature of A without knowing its production details. As an example, we investigate the decay of a potential doubly-charged boson H^±± going to same-sign τ lepton pair. 


A theorem about two-body decay and its application for a doubly-charged boson H^±± going to τ^±τ^±
     Li-Gang Xia
 Department of Physics, Tsinghua University, Beijing 100084, People's Republic of China
    December 30, 2023
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§ INTRODUCTION

After the discovery of the higgs boson h(125) <cit.>, we are more and more interested in searching for high-mass particles, such as doubly-charged higgs bosons <cit.>, denoted by H^±±. Once we observe any unknown particle, it is crucial to determine its spin-parity (J^P) nature to discriminate different theoretic models. A good means is to study the angular distributions in a decay chain where the unknown particle is involved <cit.>. For the Standard Model (SM) higgs, its spin-parity nature can be probed in the decay modes h(125)→ W^+W^-/ZZ/ <cit.>. The validity of this method relies on that the correlation of the decay planes of W/Z/τ does not depend upon the polarization of h(125) at production. This is proved in a general case in this paper.  As an example, we also investigate the decay →, where the spin-statistic relation provides more interesting constraints as the final state is two identical fermions. 



§ PROOF OF THE THEOREM

Let us consider a general decay chain A→ B_1B_2 with B_1→ C_1X_1 and B_2 → C_2 X_2, where B_1 and B_2 can be different particles and C_1X_1 and C_2X_2 can be different decay modes even if B_1 and B_2 are identical particles. Here we prove a theorem, which states that the angular correlation function I(θ_1,θ_2,ϕ_+) (defined in Eq. <ref>) in the decay of the daughter particles B_1,2 is independent upon the polarization of the mother particle A. Let ϕ_+ denote the angle between two decay planes B_i→ C_iX_i (i=1,2). Therefore, we can measure the ϕ_+ distribution to determine the spin-parity nature of the mother particle A without knowing its production details [After finishing this work, I was informed that the same statement had been verified in Ref. <cit.> in the case that B_1,2 are spin-1 particles and C_1,2 and X_1,2 are spin- particles. I also admit that it is of no difficulty to generalize it to any allowed spin values for B, C and X as shown in this work.]. 

Before calculating the amplitude, we introduce the definition of the coordinate system to describe the decay chain as illustrated in Fig. <ref>. For the decay A→ B_1B_2, we take the flight direction of A as the +z axis (if it is still, we take its spin direction as the +z direction), denoted by ẑ(A). θ and ϕ are the polar angle and azimuthal angle of B_1 in the center-of-mass (c.m.) frame of A. For the decay B_1→ C_1X_1, we take the flight direction of B_1 in the c.m. frame of A as the +z axis, denoted by ẑ(B_1) and the direction of ẑ(A)×ẑ(B_1) as the +y axis, denoted by ŷ(B_1). The +x axis in this decay system is then defined as ŷ(B_1)×ẑ(B_1). θ_1 and ϕ_1 are the polar angle and azimuthal angle of C_1 in the c.m. frame of B_1. The same set of definitions holds for the decay B_2→ C_2X_2. ϕ_+ is defined in Eq. <ref>. It represents the angle between the two decay planes of B_i→ C_iX_i (i=1,2). Here ϕ_1, ϕ_2 and ϕ_+ are constrained in the range [0,2π).

    ϕ_+ ≡{[      ϕ_1+ϕ_2  , if ϕ_1+ϕ_2 < 2π;    ϕ_1+ϕ_2-2π , if ϕ_1+ϕ_2 > 2π;                 ].


 

According to the helicity formalism developed by Jacob and Wick <cit.>, the amplitude is 

    =∑_λ_1,λ_2 F_λ_1λ_2^JD_M, λ_1-λ_2^J*(Ω)× G_ρ_1σ_1^j_1D_λ_1,ρ_1-σ_1^j_1*(Ω_1)  
          × G_ρ_2σ_2^j_2 D_λ_2,ρ_2-σ_2^j_2*(Ω_2)  .

Here the spin of A, B_1 and B_2 is J, j_1 and j_2 respectively. M is the third spin-component of  A. The indices λ_1,2, ρ_1,2 and σ_1,2 denote the helicity of B_1,2, C_1,2 and X_1,2 respectively. D_mn^J(Ω) ≡ D_mn^J(ϕ, θ, 0)=e^-imϕd_mn^J(θ) and D_mn^J (d_mn^J) is the Wigner D (d) function. F_λ_1λ_2^J is the helicity amplitude for A→ B_1B_2 and defined  as

    F_λ_1λ_2^J ≡⟨ JM;λ_1,λ_2|ℳ|JM⟩ ,

with ℳ being the transition matrix derived from  the S matrix. It is worthwhile to note that F_λ_1λ_2^J does not rely on M because ℳ is rotation-invariant. Similarly, G_ρ_iσ_i^j_i is the  helicity amplitude for B_i→ C_iX_i (i=1,2). 

Taking the absolute square of  and summing over all possible initial and final states, the differential cross section can be written as  

    dσ/dΩ dΩ_1dΩ_2∝∑_M,λ_1,λ_1^',λ_2,λ_2^'F_λ_1λ_2^JF_λ_1^'λ_2^'^J* e^i((λ_1-λ_1^')ϕ_1+(λ_2-λ_2^')ϕ_2)
          × d_M,λ_1-λ_2^J(θ)d_M,λ_1^'-λ_2^ '^J(θ) f_λ_1λ_1^';λ_2λ_2^'^j_1,j_2(θ_1,θ_2)   ,

with 

    f_λ_1λ_1^';λ_2λ_2^'^j_1,j_2(θ_1,θ_2) 
    ≡      ∑_ρ_1,σ_1,ρ_2,σ_2 |G_ρ_1σ_1^j_1|^2|G_ρ_2σ_2
    ^j_2|^2 d_λ_1,ρ_1-σ_1^j_1(θ_1)d_λ_1^',ρ_1-σ_1^j_1(θ_1)
          × d_λ_2,ρ_2-σ_2^j_2
    (θ_2)d_λ_2^',ρ_2-σ_2^j_2(θ_2)   .

Here the summation on M is over the polarization state of A at production. If we do not know the detailed production information, the summation cannot be performed. 

Defining δλ^(')≡λ_1^(')-λ_2^('), the exponential term in Eq. <ref> is equivalent to e^i[(λ_1-λ_1^')ϕ_+-(δλ-δλ^')ϕ_2]. Performing the integration on ϕ_2 and using the definition of ϕ_+, we have (keeping only the terms related with ϕ_2) 

    ∫_0^2πdϕ_2 e^i((λ_1-λ_1^')ϕ_1+(λ_2-λ_2^')ϕ_2)
    
    =      ∫_0^ϕ_+dϕ_2e^i[(λ_1-λ_1^')ϕ_+-(δλ-δλ^')ϕ_2]
          +∫_ϕ_+^2πdϕ_2e^i[(λ_1-λ_1^')(ϕ_++2π)-(δλ-δλ^')ϕ_2] .

Noting that (λ_1-λ_1^'), δλ and δλ^' are integers, the integration gives the requirement δλ = δλ^'. Then the differential cross section in terms of λ_1^('), δλ^(') and ϕ_+ is  

    ∑_λ_1,λ_1^',δλF_λ_1,λ_1-δλ^JF_λ_1^',λ_1^'-δλ^J*e^i(λ_1-λ_1^')ϕ_+
          ×∑_M d_M,δλ^J(θ)^2 f_λ_1λ_1^';λ_1-δλ,λ_1^'-δλ^j_1,j_2(θ_1,θ_2)  .

According to the orthogonality relations of the Wigner D functions, we obtain

    ∫ d_mn^J(θ)^2 dcosθ = 2/2J+1 ,

which is independent upon the indices m, n. Using this property, we find that integration over θ of the terms related with M in Eq. <ref> only  provides a constant factor ∑_M2/2J+2, which is irrelevant to the normalized angular distributions in the B_1,2 decays. So we finalize the proof of this theorem in Eq. <ref>.

    I(θ_1,θ_2,ϕ_+) ≡1/σdσ/dcosθ_1dcosθ_2dϕ_+
          ∝∑_λ_1,λ_1^',δλ  
    F_λ_1,λ_1-δλ^JF_λ_1^',λ_1^'-δλ^J*
          × e^i(λ_1-λ_1^')ϕ_+  
    f_λ_1λ_1^';λ_1-δλ,λ_1^'-δλ^j_1,j_2(θ_1,θ_2)  .


Experimentally, we are interested in the ϕ_+ distribution, which can be used to measure the spin-parity nature of A. We integrate out θ_1 and θ_2 and rewrite F_mn^J≡ R_mn^Je^i φ_mn^J, where R_mn^J and φ_mn^J are real. The ϕ_+ distribution turns out to be

    dσ/σ dϕ_+∝∑_λ_1,δλR_λ_1,λ_1-δλ^J^2F_λ_1λ_1;λ_1-δλ,λ_1-δλ^j_1,j_2
           +∑_λ_1≠λ_1^'∑_δλR_λ_1,λ_1-δλ^JR_λ_1^',λ_1^'-δλ^J F_λ_1λ_1^';λ_1-δλ,λ_1^'-δλ^j_1,j_2
          ×cos[(λ_1-λ_1^')ϕ_++ (φ_λ_1,λ_1-δλ^J-φ_λ_1^',λ_1^'-δλ^J)]   ,

with

    F_λ_1λ_1^';λ_2,λ_2^'^j_1,j_2≡∫ f_λ_1λ_1^';λ_2,λ_2^'^j_1,j_2(θ_1,θ_2) dcosθ_1dcosθ_2   .

Here the second term in Eq. <ref> is obtained using the fact that the summation is invariant with the exchange λ_1 ↔λ_1^'. 

If the parity is conserved in the decay A→ B_1B_2 (namely, ^-1ℳ = ℳ with  being the parity operator), we have 

    R_mn^J =P_A P_B_1P_B_2(-1)^J-j_1-j_2R_-m,-n^J  , 
          φ_mn^J=φ_-m,-n^J   ,

where P_A/B_1/B_2 is the parity of A/B_1/B_2 and the factor -1 is absorbed in R_mn^J (namely, we require 0≤φ_mn^J<π). Noting that the second summation in Eq. <ref> is invariant with the index exchange (λ_1,λ_1',δλ) ↔ (-λ_1, -λ_1', -δλ), thus we have

    ∑_λ_1≠λ_1^'∑_δλ⋯ = 1/2∑_λ_1≠λ_1^'∑_δλ⋯ + 1/2∑_-λ_1≠-λ_1^'∑_-δλ⋯  .

Using the symmetry relation in Eq. <ref>, this summation turns out to be 

    1/2∑_λ_1≠λ_1^'∑_δλR_λ_1,λ_1-δλ^JR_λ_1^',λ_1^'-δλ^J×{F_λ_1λ_1^';λ_1-δλ,λ_1^'-δλ^j_1,j_2. 
          ×cos[(λ_1-λ_1^')ϕ_++ (φ_λ_1,λ_1-δλ^J-φ_λ_1^',λ_1^'-δλ^J)]  
          + F_-λ_1,-λ_1^';-λ_1+δλ,-λ_1^'+δλ^j_1,j_2
          .×cos[(λ_1-λ_1^')ϕ_+- (φ_λ_1,λ_1-δλ^J-φ_λ_1^',λ_1^'-δλ^J)]  } .

Focusing on the expressions of Eq. <ref> and Eq. <ref>, we are able to show that

    F_λ_1λ_1^';λ_1-δλ,λ_1^'-δλ^j_1,j_2 = F_-λ_1,-λ_1^';-λ_1+δλ,-λ_1^'+δλ^j_1,j_2  ,

using the following property of the Wigner d function 

    d_mn^j(π-θ) = (-1)^j-n d_-m,n^j(θ)   .

With Eq. <ref> and Eq. <ref>, Eq. <ref> can be simplified as 

    dσ/σ dϕ_+∝∑_λ_1,δλR_λ_1,λ_1-δλ^J^2F_λ_1λ_1;λ_1-δλ,λ_1-δλ^j_1j_2
           +∑_λ_1≠λ_1^'∑_δλR_λ_1,λ_1-δλ ^JR_λ_1^',λ_1^'-δλ^JF_λ_1λ_1^';λ_1-δλ,λ_1^'-δλ^j_1j_2
          ×cos(φ_λ_1,λ_1-δλ^J-φ_λ_1^',λ_1^'-δλ^J)cos[(λ_1-λ_1^  ')ϕ_+] 
     ,

This expression is actually the Fourier series for a 2π-periodic even function. Comparing Eq. <ref> and Eq. <ref>, we can see that the terms which are odd with respective to ϕ_+ are forbidden due to parity conservation in the decay A→ B_1B_2.







Now we consider the special case that B_1 and B_2 are identical particles and B_1,2 decay to the same final state, for example, we will study a doubly charged boson decay H^++→→ν̅_τν̅_τ. For identical particles, the state with the spin J and the third component M is 

    |JM;λ_1λ_2⟩_S = |JM;λ_1λ_2⟩ +(-1)^J |JM;λ_2λ_1⟩  ,

which satisfies the spin-statistics relation. Here the normalization factor is omitted. The helicity amplitude F_λ_1λ_2^J = _S⟨ JM;λ_1λ_2|ℳ|JM⟩ has the symmetryF_λ_1λ_2^J = (-1)^JF_λ_2λ_1^J. This symmetry relation will further constrain the helicity states, namely, the indices λ_1, λ_1^' and δλ in the summation in Eq. <ref>, <ref> and  <ref>.  











































§ STUDY OF →→Ν̅_ΤΝ̅_Τ

Ref. <cit.> is an example of the application of this theorem. It studies the decay Z^'→ ZZ → l^+l^-l^+l^-, where B_1,2 are identical bosons. Here we consider the decay chain →→ν̅_τν̅_τ. For two spin- identical fermions, we write down all states explicitly. The helicity index λ=+ () is denoted by R  (L). 

    |JM;LL⟩_S =       (1+(-1)^J)|JM;LL⟩
          |JM;LL⟩_S = -|JM;RR⟩_S 
    |JM;RR
    ⟩_S =       (1+(-1)^J)|JM;RR⟩
          |JM;RR⟩_S = -|JM;LL⟩_S 
    |JM;LR⟩_S =       |JM;LR⟩+(-1)^J|JM;RL⟩
          |JM;LR⟩_S = -|JM;LR⟩_S


The third state is already a parity eigenstate. The first two states can be combined to have a definite parity.

    (1+(-1)^J)(|JM;LL⟩±|JM;RR⟩)  ,     P = ∓ 1

In addition, the angular momentum conservation requires |λ_1- λ_2|≤ J. Now we can give the selection rules, which are summarized in  Table <ref>. We can see that the states with odd spin and even parity are forbidden. For comparison, the selection rules for a neutral particle decaying to spin-  fermion anti-fermion pair are summarized in Table <ref>. 

In future electron-electron colliders, H^– may be produced in the process e^-e^- → H^–. However,  the reaction rate for a spin-1 H^– will be highly suppressed because the vector coupling requires that both electrons have the same handness while the only allowed state is |LR  ⟩-|RL⟩. Similarly, the production rate for a scalar H^– is also highly suppressed. This is called “helicity suppression”.




Replacing A, B_1,2 and C_1,2 by ,  and  respectively in Eq. <ref>, the amplitude is 

    =       G_0^ G_0^ e^iMϕ[F_RR^Jd_M0^J(θ)e^i(ϕ_1+ ϕ_2)sinθ_1/2sinθ_2/2. 
          +.F_LL^Jd_M0^J(θ)e^-i(ϕ_1+ϕ_2)cosθ_1/2cosθ_2/2. 
          -.F_LR^Jd_M,-1^J(θ)e^i(-ϕ_1+ϕ_2)cosθ_1/2sinθ_2/2. 
          -.F_RL^Jd_M,1^J(θ)e^i(ϕ_1-ϕ_2)sinθ_1/2cosθ_2/2]   .

 Here we have only one decay helicity amplitude, G_0^, for the  decay. This is because  is a pseudo-scalar and ν̅_τ is right-handed.
 
 The angular correlation function is  

    I(θ_1,θ_2,ϕ_+)  ∝       1+ cosθ_1cosθ_2  ,   for odd  J 
    
    I(θ_1,θ_2,ϕ_+) ∝       1+a_J^2 + (1-a_J^2)cosθ_1cosθ_2 
           -P_Hsinθ_1sinθ_2cosϕ_+  ,   for even  J

Here for even J, a_J is defined as a_J≡|F_LR^J|/|F_RR^J|. P_H is the parity of . We can see that the polarization information of  does not appear in the angular distributions. The ϕ_+ distribution is   

    dσ/σ dϕ_+∝{[                           1                  for odd  J; 1 -P_Hπ^2/161/1+a_J^2cosϕ_+                 for even  J;                             ].   .

 The ϕ_+ distributions for different J^Ps are shown in Fig. <ref>, where a_J=1 is assumed for illustration.  
 

Here are a few conclusions.
   
 
  * The ϕ_+ distribution is uniform for odd J. 
 
  * For J=0, the 
helicity amplitudes F_LR^J and  F_RL^J are forbidden due to angular momentum conservation. Thus a_J=0 and the ϕ_+ distribution becomes
 
    dσ/σ dϕ_+∝ 1-P_Hπ^2/16cosϕ_+  ,

  which is the same as that in the decay h(125)→τ^+τ^-→ν_τν̅
_τ.  

  * For nonzero even J, the ϕ_+ distribution depends upon J through the amplitude ratio a_J.
  
 
 Experimentally,  it is difficult to reconstruct the τ lepton information due to the invisible neutrinos <cit.>. But we are able to obtain the decay plane angle ϕ_+ in some ways (see a  most recent review Ref. <cit.> and references therein). The so-called impact parameter method <cit.> is suitable for the decay →ν̅_τ studied here. It requires that final s have significant impact parameters, which condition can be satisfied at high-energy colliders such as the Large Hadron Collider (LHC). 
 
 

§ CONCLUSIONS

 In summary, for  a general decay chain A→ B_1B_2→ C_1C_2…, we have proved that the angular correlation function I(θ_1,θ_2,ϕ_+) in the decay of the daughter particles B_1,2 is independent upon the polarization of the mother particle A at production. It guarantees that the spin-parity nature of the mother particle  A can be determined by measuring the angular correlation of the two decay planes B_i→ C_i… (i=1,2) without knowing its production details. This theorem has a simple form if the parity is conserved in the decay A→ B_1B_2. Taking a potential doubly-charged particle decay → as example, we present the selection rules for various spin-parity combinations. It is found that this decay is forbidden for the  with odd spin and even parity. Furthermore, we show that the angle between the two τ  decay plans is an effective observable to determine the spin-parity nature of . 
 
 

§ ACKNOWLEDGEMENT

 Li-Gang Xia would like to thank Fang Dai for many helpful discussions. The author is also indebted to Yuan-Ning Gao for enlightening discussions. This work is supported by the General Financial Grant from the China Postdoctoral Science Foundation (Grant No. 2015M581062).   
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