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Momentum density of states for pion decay.

  1. Mar 3, 2016 #1
    1. The problem statement, all variables and given/known data

    I am trying to calculate the ratio of the density of states factor, ##\rho(p)##, for the two decays:
    $$\pi^+\rightarrow e^++\nu_e~~$$ and $$\pi^+\rightarrow \mu^++\nu_{\mu}~~$$

    2. Relevant equations

    ##\rho(p)~dp=\frac{V}{(2\pi\hbar)^3}p^2~dp~d\Omega##

    Which is the number of states with momentum between ##p## and ##dp## and lie within a small solid angle ##d\Omega##. ##V## is an arbitrary volume to which we confine the system.

    Also, ##\rho_{Total}=\rho_1(p_1)\rho_2(p_2)...\rho_n(p_n)##

    3. The attempt at a solution

    Using the above equation:

    ##\rho_n(p_n)=\frac{V}{(2\pi\hbar)^3}p_n^2~d\Omega##

    The ratio R should be:

    ##R=\frac{p^2(e^+)p^2(\nu_e)}{p^2(\mu^+)p^2(\nu_{\mu})}##

    The only way I can think to proceed is:

    ##M_{\pi^+}^2=(P_e+P_{\mu_e})^2~~\text{ where } P_x \text{ is the 4-momentum of particle } x##
    After assuming the mass of the decay products is negligible when compared to its momentum, and that the angle between the two products is 180 degrees, I arrive at: ##p^2(e)p^2(\mu_e)=\frac{1}{16}M_{\pi^+}^4##.
    But I will just get the same expression for the second decay, so I feel I am doing it wrong. Any suggestions?
     
  2. jcsd
  3. Mar 3, 2016 #2

    mfb

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    2016 Award

    Staff: Mentor

    If you neglect masses, the decays are completely identical.
    The muon mass is not small compared to the pion mass.
     
  4. Mar 4, 2016 #3
    Let the sub script one mean electron and the subscript two mean electron neutrino, then:

    ##M_{\pi^+}^2=(P_1+P_2)^2)##
    ##~~~~~~~=E_1^2+E_2^2+2E_1E_2-p_1^2-p_2^2-2\vec{p_1}\cdot\vec{p_2}##
    Using ##E_i^2=m_i^2+p_i^2##
    ##~~~~~~~=m_1^2+m_2^2+2p_1p_2+2\sqrt{(m_1^2+p_1^2)(m_2^2+p_2^2)}## assumes angle between ##p_1## and ##p_2 ## is 180 degrees
    Where can I go from here? Maybe I need to make some sort of approximation.. Do I use the fact that ##\vec{p_1}=-\vec{p_2}## in the pions rest frame ?
     
    Last edited: Mar 4, 2016
  5. Mar 4, 2016 #4

    mfb

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    2016 Award

    Staff: Mentor

    I would certainly work in the pion rest frame, yes. You can approximate the neutrino masses with 0, and if you don't care about the amplitude of the decay process I guess the same works for the electron.
     
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