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space is reduced to the two-body phase space by pairing a with b and further integrating over c 

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Then we find that the spectrum of the observed final state, for instance a , is obtained by integrating over x$_{b}$ . Kinematics constrains x$_{a}$ to be in the region 

where ϵ 2 a ≡ m 2 $_{a}$/Y $^{2}$. If all the final state masses are ignorable, the region is between 0 and 1. For fixed x$_{a}$ , x$_{b}$ is constrained within the region [ R − b , R + b ] 

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where the approximation is valid for small mass limit ϵ$_{a,b,c}$ ≪ 1, and the triangle function is defined as λ ( x, y, z ) = x 2 + y 2 + z 2 − 2 xy − 2 xz − 2 yz . 

The general process is S → ψ$_{1}$ψ$_{2}$φ$_{3}$ with amplitude given in Eq. (A1). In terms of x and ϵ , we can write the amplitude as follows 

which is symmetric for the subscripts 1 and 2. The distribution function of the fermion ψ$_{1}$ is obtained by paring ψ$_{1}$ and φ$_{3}$ , i.e. , setting a = ψ$_{1}$, b = φ$_{3}$, c = ψ$_{2}$, 

where N = 1 / Γ is the normalization factor. In the massless limit, it is approximated to be a simple function x 2 $_{1}$. So the fermion spectrum is very hard. The distribution of ψ$_{2}$ is got simply by replacing the subscript 1 by 2 in the above formula. Similarly, we get the distribution function of scalar φ$_{3}$ by choosing a = φ$_{3}$, b = ψ$_{1}$ , and c = ψ$_{2}$ 

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