I have an equation for two particle collisions (the equation is just energy-momentum conservation):
$$k_{A,x}+k_{B,x} = p_{1,x}+p_{2,x}$$ $$k_{A,y}+k_{B,y} = p_{1,y}+p_{2,y}$$ $$k_{A,z}+k_{B,z} = p_{1,z}+p_{2,z}$$ $$\sqrt{m_{1}^{2}+\mathbf{k}_{A}^{2}}+\sqrt{m_{2}^{2}+\mathbf{k}_{B}^{2}} = \sqrt{m_{1}^{2}+\mathbf{p}_{1}^{2}}+\sqrt{m_{2}^{2}+\mathbf{p}_{2}^{2}}.$$
where, $k_{A,x}\in[p_{c,x}-3\sigma,p_{c,x}+3\sigma]$, $k_{B,x}=-k_{A,x}$ and $k_{A,y},k_{B,y},k_{A,z},k_{B,z}\in[-3\sigma,3\sigma]$, $p_{c,x}$ and $\sigma$ is a positive number, and $ p_{c,x} > 3\sigma $.
The first three equations will give $\mathbf{p}_{2}$ in terms of $\mathbf{k}_{A},\mathbf{k}_{B},\mathbf{p}_{1}$.
I would like to know the minimum and maximum value of $|\mathbf{p}_{1}|$ if we already know $|\mathbf{k}_{A}|_{min},|\mathbf{k}_{A}|_{max},|\mathbf{k}_{B}|_{min},|\mathbf{k}_{B}|_{max}$. Or the boundary for $|\mathbf{p}_{1}|$ in terms of $p_{c,x}$ and $\sigma$,i.e., I would like to know the minimum and maximum value of $|\mathbf{p}_{1}|$ in terms of $p_{c,x}$ and $\sigma$.
Here is a solution I found for this question, which I can not persuade myself, but it may help clear the question.
Since $\mathbf{k}_{A}$ and $\mathbf{k}_{B}$ denoting the input momentum of two incoming particles, if $|\mathbf{k}_{A}|$ and $|\mathbf{k}_{B}|$ all take the minimum values, i.e., $\mathbf{k}_{A}=(p_{c,x}-3\sigma,0,0)$ and $\mathbf{k}_{B}=(-p_{c,x}+3\sigma,0,0)$ hence $\mathbf{k}_{A}+\mathbf{k}_{B}=\mathbf{p}_{1}+\mathbf{p}_{2}$, the outcoming particles would have the minimum values. Thus we have $$\sqrt{m_{1}^{2}+\mathbf{k}_{A}^{2}}+\sqrt{m_{2}^{2}+\mathbf{k}_{A}^{2}} = \sqrt{m_{1}^{2}+\mathbf{p}_{1}^{2}}+\sqrt{m_{2}^{2}+\mathbf{p}_{1}^{2}}.$$ Thus one can easily see that the minimum value for $|\mathbf{p}_{1}|$ is $|\mathbf{k}_{A}|_{min}$
This is somewhat reluctant.
Does anyone have once encountered such problem or how to make my proof more convinsing?
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Or equivalently, if $\delta_{1},\delta_{2},\delta_{3},\delta_{4}\in[-3\sigma,3\sigma]$ and $p_{c,x}$ and $\sigma$ are positive numbers, and $ p_{c,x} > 3\sigma $., how to prove the following inequality to be wrong: $$\sqrt{m_{1}^{2}+(-p_{c,x}+\delta_{1})^{2}+\delta_{2}^{2}}+\sqrt{m_{2}^{2}+(p_{c,x}+\delta_{3})^{2}+\delta_{4}^{2}} < \sqrt{m_{1}^{2}+(p_{c,x}-3\sigma)^{2}}+\sqrt{m_{2}^{2}+(p_{c,x}-3\sigma)^{2}+(\delta_{1}+\delta_{3})^{2}+(\delta_{2}+\delta_{4})^{2}}$$