Matrix element of electron muon scattering

  • #1
Thyrac
1
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Homework Statement:
I’m wondering what the matrix element is for electron muon scattering for right helicity particles. I got $$e^2(1 + cos(\theta))$$. Is this correct?
Relevant Equations:
$$u(p_2)\sigma^{\mu}u(p_1)\frac{e^2}{q^2}
u^{\dagger}(p_3)\sigma_{\mu}u^{\dagger}(p_4)$$
The current I have found for the incoming electron and muon is

$$u(p_2)\sigma^{\mu}u(p_1)$$ = $$E_{cm}(0, i, -1, 0)^\mu$$

The current I have found for the outgoing particles are

$$u^{\dagger}(p_3)\sigma_{\mu}u^{\dagger}(p_4)$$ = $$E_{cm}(0, i\cos(\theta), 1, -i\sin(\theta))_{\mu}$$

Multiplying these 4-vectors together give me $$E_{cm}^2(1+\cos(\theta))$$

And multiplying it by the photon propagator with vertices $$\frac{e^2}{q^2}$$ give us

$$e^2(1 + \cos(\theta))$$

(Note that again all particles involved are right helicity)

Now I haven’t found any source confirming that the matrix element for the process is what I calculated, but I checked my work and can’t see any possible errors. So my question is, is $$e^2(1 + \cos(\theta))$$ the correct matrix element of electron muon scattering?
 

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