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Interpret constraints on scattering amplitude

  1. Nov 26, 2017 #1
    1. The problem statement, all variables and given/known data
    For the following theory: ##\mathcal{L}=\frac{1}{2}[(\partial \phi)^2-m^2\phi^2+(\partial\Phi)^2-M^2\Phi^2]+g\phi^2 \Phi^2##

    Compute s-channel amplitude for process ##\phi\phi \rightarrow \phi\phi##. Interpret result for ##M>2m##.
    2. Relevant equations
    Scattering amplitude: ##\mathcal{iA}=(ig)^2\frac{i}{s-M^2}##
    ##s=(p_1+p_2)^2##
    3. The attempt at a solution
    Choosing to work in center-of-mass frame: ##\vec{p_1}+\vec{p_2}=0##.
    ##s=(E_1+E_2)^2## in CoM.
    $$E_1=E_2$$, because ##|\vec{p_1}|=|\vec{p_2}|## and masses are same.
    Then ##s=4E^2##

    We get in CoM:##\mathcal{A}=(ig)^2\frac{1}{4E^2-M^2}=(ig)^2\frac{1}{4(\vec{p}^2+m^2)-M^2}##
    Applying ##M>2m## we get ##\mathcal{A}> (ig)^2\frac{1}{4|\vec{p}|^2}##

    So far it is hard for me to interpret this result. Could anyone give me some hints how to think about this constraint?
     
    Last edited: Nov 26, 2017
  2. jcsd
  3. Nov 27, 2017 #2

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    What can happen with the denominator that cannot happen for M<2m?
     
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