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Homework Statement
A muon decays to an electron, an electron neutrino and a muon neutrino. ## \mu \rightarrow e \ \nu_\mu \ \nu_e## .The matrix element of the process is ## |\mathcal{M}|^2 = G^2_F (m^2-2mE)mE## with ##m## being the mass of the muon and ##E## the energy of the resulting electron neutrino.
I need to show that decay rate is $$ \Gamma = \frac{G^2_F m^5 }{192 \pi ^3}$$
Homework Equations
$$ \Gamma = \frac{1}{2 E_1} |\mathcal{M}|^2 d\Pi_{LIPS} $$
with ##d\Pi_{LIPS}## being an integral over this entitiy
$$ d\Pi = \prod_{final \ states j} \frac{d^3 p_j}{(2 \pi )^3}\frac{1}{2 E_{p_j}}
(2 \pi )^4 \delta ^4(\sum p_i^\mu - \sum p_f^\mu)$$
The delta function shows energy-momentum conservation with ##p_i^\mu## related to initial particles' momenta and ##p_f^\mu## related to final particles' momenta.
The Attempt at a Solution
$$d\Pi = \frac{d^3p_e}{(2\pi)^3} \frac{d^3p_{\nu \mu}}{(2\pi)^3} \frac{d^3p_{\nu_e}}{(2\pi)^3} (2 \pi )^4 \delta (E + E_{\nu \mu} + E_e - m) \delta ^3( \vec{p_\mu} - \vec{p_e} - \vec{p_{\nu_{mu}}} - \vec{\nu_e}) \frac{1}{2E_e} \frac{1}{2E_{\nu \mu}} \frac{1}{2E_{\nu _e}}$$
Integrating over three four-momenta and using delta function over momenta and the fact that we can use a frame in which the initial particle is not moving, I wrote:
$$d\Pi_{LISP} = \frac{1}{(2 \pi)^5 2^3} \int d^3p_e \int d^3p_{\nu \mu} \delta (E + E_{\nu \mu} + E_e - m) \frac{1}{E_e E_{\nu \mu} E } $$
This time we have ## \vec{p_e} = -\vec{p_{\nu_{mu}}} - \vec{\nu_e}##.
Changing the integral variable so that we have delta function of some variable like x we have:
$$ x= E + E_{\nu \mu} + E_e - m$$
$$ E_{\nu \mu} = P_{\nu \mu} \quad $$
$$E_e = \sqrt{p_e^2 + m_e} = \sqrt{|\vec{p_{\nu_{\mu}}} + \vec{\nu_e}| + m_e} = \sqrt{p_{\nu_{\mu}}^2 + p_e^2 -2 p_{\nu_{\mu}} p_e \cos \theta}$$
$$ \frac{dx}{dp_{\nu \mu}}=1 + \frac{2 p_{\nu \mu}}{E}$$
Using this new variable in integral we have
$$d\Pi_{LISP} = \frac{1}{(2 \pi)^5 2^3} \int d^3p_e \int_{E + E_{\nu \mu} + E_e - m}^{\infty} dx \delta(x) \frac{p_{\nu \mu}^2}{E_e E_{\nu \mu} E } (1 + \frac{2 p_{\nu \mu}}{E})$$
$$d\Pi_{LISP} = \frac{1}{(2 \pi)^5 2^3} \int d^3p_e \frac{p_{\nu \mu}^2}{E_e E_{\nu \mu} E } (1 + \frac{2 p_{\nu \mu}}{E}) \Theta(E + E_{\nu \mu} + E_e - m)$$
Now I can not reach the result that the problem want ! Please help me!
This problem is problem number 3 of chapter 5 from the book "Quantum Field Theory and Standard model" by Matthew D. Schwartz
I really appreciate your help and patience.
Thankyou