# Upper bounds on phase space momenta

Suppose I wish to calculate the phase space volume for the process $$\overline{X}X \to A_1 A_2 A_3 A_4 A_5$$ in the CM frame of $$\overline{X}, X$$ so that $$\sqrt{s} = 2m_X$$. The volume is given by

$$V \propto \int d^3 p_1 \int d^3 p_2 \int d^3 p_3 \int d^3 p_4 \int d^3 p_5 \frac{1}{E_1} \frac{1}{E_2} \frac{1}{E_3} \frac{1}{E_4} \frac{1}{E_5} \delta[2m_X - (\sum_{f=1}^5 E_f)]\ \delta^{(3)}(\sum_{f=1}^5 \mathbf{p}_f).$$

Define $$\mathbf{P} = \sum_{f=1}^5 \mathbf{p}_i$$. Integrating out the 3D delta function we obtain

$$V= (2\pi) \int d^3 p_1 \int d^3 p_2 \int d^3 p_3 \int d p_4 d(\cos\theta_{34}) \frac{1}{E_1} \frac{1}{E_2} \frac{1}{E_3} \frac{p_4^2}{E_4} \frac{1}{E_5(\mathbf{P} = 0)} \frac{\delta(\cos\theta_{34} -z_0)}{|\partial_{\cos\theta_{34}} E_5(\mathbf{P}=0)|_{z_0} }$$

where $$\cos\theta_{34} := \mathbf{p_3\cdot p_4}/p_3p_4$$ and $$z_0$$ is the value of $$\cos\theta_{34}$$ such that

$$2m_X - E_1 - E_2 - E_3 -E_4 - E_5(\mathbf{P}=0) = 0$$

and the factor of $$1/{|\partial_{\cos\theta_{34}} E_5(\mathbf{P}=0)|_{z_0} }$$ is just the Jacobian we pick up from the identity

$$\delta(g(x)) = \sum_{x_0}\frac{\delta(x-x_0)}{|g'(x_0)|}$$

Now, after integrating out the delta function we must enforce the fact that the integral is to be $$0$$ whenever $$|z_0|>1$$. This amounts to forcing the conservation of energy on the integrand.

Define the region as phase space $$\Xi$$ as

$$\Xi(p_1, p_2, p_3, p_4, \cos\theta_{14}, \cos\theta_{12}, \cos\theta_{23}) = \begin{cases} 1 \ \ , \ \ -1 \leq z_0 \leq 1 \\ 0 \ \ , \ \ \text{otherwise} \end{cases}$$

Therefore, an equivalent expression for the phase space volume $$V$$ is given by

\begin{align} V &= (2\pi) {\int_{\mathbb{R}^3} d^3 p_1 \int_{\mathbb{R}^3} d^3 p_2 \int_{\mathbb{R}^3} d^3 p_3 \int_0^\infty d p_4 }\\ & \Xi \cdot \frac{1}{E_1} \frac{1}{E_2} \frac{1}{E_3} \frac{p_4^2}{E_4} \frac{1}{E_5(\mathbf{P} = 0)_{z_0}} \frac{1}{|\partial_{\cos\theta_{34}} E_5(\mathbf{P}=0)|_{z_0} }. \end{align}

Doing this integral numerically proves to be very difficult since I need to integrate over all of the space. Hence every numerical integration scheme will be fooled into thinking this is zero since it doesn't know that all the action is happening in the regions where $$\Xi =1$$.

Question: Is there a finite upper bound on the limit of integration with respect to $$dp_i$$, such that I may entirely cover the region $$\Xi = 1$$?

My Conjecture: My reasoning is that, yes, after enforcing conservation of energy via the $$1D$$-energy delta function the momenta $$p_i$$ are bound above by $$p_i^{max} \leq \sqrt{(2 m_X )^2 - m_i^2}$$. This leads me to propose the following conjecture that

\begin{align} \frac{dV}{d\Omega_1 d\Omega_2 d\Omega_3} &= (2\pi) {\int_0^{p_1^{max}} d p_1 \int_0^{p_2^{max}} d p_2 \int_0^{p_3^{max}} d p_3 \int_0^{p_4^{max}} d p_4 }\\ & \Xi \cdot \frac{p_1^2}{E_1} \frac{p_2^2}{E_2} \frac{p_3^2}{E_3} \frac{p_4^2}{E_4} \frac{1}{E_5(\mathbf{P} = 0)_{z_0}} \frac{ 1}{|\partial_{\cos\theta_{34}} E_5(\mathbf{P}=0)|_{z_0} }. \end{align}

• I don't understand your question. First, the notation is not defined. Secondly, we do not normally calculate the phase space volume of a "process". We calculate the partition function of a "state", which is the volume of phase space weighted by the probability density in the ensemble where the calculation is done. Commented Jul 26, 2022 at 15:23