## Energy-momentum invariant problem

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agelessdrifter
Posts: 225
Joined: Mon Oct 05, 2009 8:10 pm UTC

### Energy-momentum invariant problem

I'm trying to apply this idea,

s2=(E1+E2)2-c2(p1+p2)2=constant

to a simple homework problem in modern physics, and I'm not exactly sure how. There've been several problems so far that have applied this, but generally I've been able to work in the center-of-mass frame and rule out momenta.

However, this problem: "What is the minimum energy that a gamma ray must have in order to produce the reaction p+gamma--->p+pi0, when p (a proton) is at rest. Note that mpc2=940MeV and mpic2=140MeV."

Requires that I not use that approach. So what I've got down on my paper is

s2=(mpc2)2-c2(pgamma)=(Ep+Epi)2-c2(....)2

So cpgamma is what I'm solving for, but I have no idea what to put in the parentheses on the right-hand-side with the ellipses. I've got the solution for pgammac: 150MeV, and using that to solve backwards for the sum of the momenta of p and pi0, I'm coming up with numbers that I can't seem to reason out of the given information by another method.

Aiwendil
Posts: 313
Joined: Thu Apr 07, 2011 8:53 pm UTC
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### Re: Energy-momentum invariant problem

Well, what would go in the parentheses at the right would be the sum of the proton and pion momenta after the interaction. But it sounds like these are not constrained, if I understand the problem correctly. The problem is asking you to minimize pgamma, and the final momenta of the proton and pion are the free parameters that you can adjust to achieve that minimum. So . . . what values for the final momenta allow you to minimize pgamma?

SU3SU2U1
Posts: 396
Joined: Sun Nov 25, 2007 4:15 am UTC

### Re: Energy-momentum invariant problem

So to start, your equation for s squared is incorrect, you've neglected that the photon carries energy. Consider (using [imath]p[/imath] for four momenta),

$s = p_\gamma+p_p \rightarrow s^2 = \left(p_\gamma+p_p\right)^2 = p_p^2 + 2p_\gamma \cdot p_p +p_\gamma^2 = m_p^2 +2E_\gamma m_p$

In the last step, we used the masslessness of photons, and the fact that the proton is stationary. I used units where [imath]c=1[/imath]. Now, the right side we can write as

$s^2 = \left(p_p+p_\pi\right)^2 = m_p^2 + m_\pi^2 + 2p_\pi\cdot p_p$

Now, on the right hand side, can we make any approximations for [imath]p_\pi\cdot p_p[/imath]? If we are minimizing the photon momentum, how should the 3-momenta of the final pion and proton compare to their energy?

mfb
Posts: 950
Joined: Thu Jan 08, 2009 7:48 pm UTC

### Re: Energy-momentum invariant problem

Again, it is easier to look at it in the CMS (center-of-mass system) after the collision: What is the minimal energy there to have a proton and a pion? What is their momentum in that system - with minimal energy?
The CMS energy (squared) s^2 is the same in all systems, and you already have a relation between the photon energy and the CMS energy.

agelessdrifter
Posts: 225
Joined: Mon Oct 05, 2009 8:10 pm UTC

### Re: Energy-momentum invariant problem

Oh, that seems like it should have been really obvious now that you mention it. Thanks, everyone.