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3D Integral to 4D integral in Width Calculation

  1. Jul 23, 2011 #1

    Hepth

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    Gold Member

    While this is more a mathematics problem, it really belongs here for those theorists who have experience with QFT calculations.
    This is sort of a generic question, but can I turn a 1->3 decay width integral into a single 4-D integral over one of the momenta.
    Such as:

    [tex]
    \Gamma = \frac{1}{2 M_X} (2 \pi)^{4-9} \int \frac{d^3 p_1}{2 E_1}\frac{d^3 p_2}{2 E_2}\frac{d^3 k}{2 E_k} \delta^4 (P-p_1-p_2-k) |M|^2
    [/tex]

    into the form:

    [tex]
    \Gamma = (...) \int \frac{d^4 k}{(2 \pi)^4} |M|^2
    [/tex]

    without any delta functions in the k integral.

    I know I can replace the p2 integral times the delta with a 1-dimensional delta

    [tex]
    \int \frac{d^3 p_2}{2 E_2} \delta^4(P-p_1-p_2-k) = \delta((P-p_1-k)^2-m_2^2)
    [/tex]

    using

    [tex]
    d^4 k \theta(E_k) \delta(k^2-\mu^2) = \int \frac{d^3 k}{2 E_k}
    [/tex]

    but still I'm left with a delta function for at least the k_0 piece. I know how to ACTUALLY calculate the integrals, but I'm trying to get them into this form so I can plug them into FeynCalc/PHi to get the integrals in terms of the Passarino-Veltman integrals.

    Or is there another, BETTER way of getting tree level calculations that diverge into the Passarino-Veltman integral form to cancel divergences from 1-loop corrections?

    Again, I know there are many other ways of doing this (DREG/cutoffs/PV/OpticalThm) but I'm trying to automate some divergence cancellation with Mathematica specifically using the Pass-Velt functions (A0,B0,C0,D0).

    Thanks!
     
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