Energy of neutral pion in proton's rest mass frame

• maximus123
In summary, the conversation discusses the decay of a proton into a positron and a pion, and how to determine the energy of the pion in the lab frame. The participant has attempted to find the invariant mass of the positron and pion, but is unsure of how to proceed and seeks help. They are reminded to consider the center of momentum frame for simplifying calculations.
maximus123
Hello, my problem is as follows
For the decay $p\rightarrow e^+\pi ^0$ show that the energy of the pion in the proton’s rest
frame (i.e., the lab frame) is

$E_{\pi}=\frac{m_p^2+m_{\pi}^2-m_e^2}{2m_p}$​
I've tried finding the invariant mass of the positron and pion as follows

$M^2=(E_e+E_{\pi})^2-(\mathbf{p_e}+\mathbf{p_{\pi}})\\$
$=E_e^2+E_{\pi}^2+2E_eE_{\pi}-p_e^2-p_{\pi}^2-2\mathbf{p_ep_{\pi}}\\$
$=m_e^2+m_{\pi}^2-2(E_eE_{\pi}+\mathbf{p_ep_{\pi}})\\$
$=m_e^2+m_{\pi}^2-2(E_eE_{\pi}+p_ep_{\pi}\textrm{cos}\theta)\\$
And this is presumably equal to the mass of the proton so

$m_e^2+m_{\pi}^2-2(E_eE_{\pi}+p_ep_{\pi}\textrm{cos}\theta)=m_p^2$
At this point I cannot see how to get any closer to the given answer. Any help with how to proceed from here, or advice on where I may have already gone wrong, would be greatly appreciated.

Thanks

maximus123 said:
[snips]
$m_e^2+m_{\pi}^2-2(E_eE_{\pi}+p_ep_{\pi}\textrm{cos}\theta)=m_p^2$

What is the value of theta in the lab frame? What is the total momentum in the lab frame? And also, what is E_e compared to E_pi, given the answer to the question about momentum?

Whenever you do interactions of this type (collisions, decays, scattering, etc.) the center of momentum frame (COM) is always something to keep in mind for simplifying calculations.

Hi. As pointed out by DEvens, in this kind of problem you only need energy and momentum conservation laws;
4-vector invariant is useful when going from one frame to another but here you can solve the problem by staying in the lab frame the whole time since it's also the COM frame!

What is the neutral pion's energy in the proton's rest mass frame?

The energy of a neutral pion in the proton's rest mass frame is typically around 135 MeV/c^2. This value can vary slightly depending on the specific conditions of the particle interactions.

How is the energy of a neutral pion calculated in the proton's rest mass frame?

The energy of a neutral pion in the proton's rest mass frame can be calculated using the equation E = mc^2, where m is the mass of the neutral pion and c is the speed of light. This equation is derived from Einstein's famous equation, E=mc^2, which relates energy and mass.

What is the significance of the energy of a neutral pion in the proton's rest mass frame?

The energy of a neutral pion in the proton's rest mass frame is significant because it helps us understand the nature of particle interactions and the behavior of subatomic particles. It also allows us to make predictions and calculations about the behavior of these particles in different circumstances.

How does the energy of a neutral pion in the proton's rest mass frame affect its decay?

The energy of a neutral pion in the proton's rest mass frame can affect its decay by determining the possible decay products and their energy distribution. This energy also plays a role in the probability of different decay pathways occurring.

Is the energy of a neutral pion in the proton's rest mass frame constant?

No, the energy of a neutral pion in the proton's rest mass frame can vary depending on the specific conditions of the particle interactions. However, it will always be in the range of approximately 135 MeV/c^2.

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