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Casimir trick in e+e->H->ffbar

  1. Jun 16, 2012 #1
    Casimir trick in e+e-->H->ffbar

    1. The problem statement, all variables and given/known data
    I have the process:

    [itex]e^+e^-\to H\to f\bar{f}[/itex]

    I have calculated the amplitude and it's conjugate, and now I want to find the averaged, unpolarized square of the invariant amplitude [itex]\langle|M|^2\rangle[/itex].

    I average over the initial spins and sum over the final and usually in some simple processes like Moller scattering, I would play with Casimir trick and traces. But here I have:

    [itex]\langle|M|^2\rangle=\frac{1}{2}\frac{1}{2}\left( \frac{g_w^2}{4m_w^2} m_e m_f\right)^2\sum_{spins} \bar{u}_4v_2\bar{v}_1u_3\bar{v}_2u_4\bar{u}_3v_1[/itex]

    Where [itex]\bar{v}_1[/itex] is the incoming positron with impulse p_1 and spin s_1, [itex]u_3[/itex] is the incoming electron, [itex]v_2[/itex] is the outgoing anti fermion, and [itex]\bar{u}_4[/itex] is the outgoing fermion.

    If I look at the spinor components, I can arrange them into pairs and use the relations:

    [itex]\sum_{s_1}u_{1\delta}\bar{u}_{1\alpha}=({\not} p_1+m_1)_{\delta\alpha}[/itex] and [itex]\sum_{s_2}v_{2\beta}\bar{v}_{2\gamma}=({\not} p_2-m_2)_{\beta\gamma}[/itex]

    But I'm not getting any trace out of this :\

    What am I doing wrong?
  2. jcsd
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