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A Calculation of ## b \to s~ l^+l^− ## penguin diagram

  1. Jan 23, 2017 #1
    I'd like to calculate the matrix element amplitude for## b→s l^+l^−## penguin diagram mediated by Z boson or the photon , like :
    Untitled.png

    These calculations are made of course from many time ago, so if any one has a good reference for such calculations, because it's my first trail in a penguin loop ?

    Any way , I started by taking the gauge boson propagator in Landau gauge , i.e.:
    ## i\Delta_{\mu\nu} = -i \frac{g_{\mu\nu}-k_\mu k_\nu/k^2}{k^2-M^2_W} ##, I don't know whether here for simplicity i can consider the Feynman gauge where
    ## i\Delta_{\mu\nu} = -i \frac{g_{\mu\nu}}{k^2-M^2_W} ## like Cheng & Li made in the calculations of ## K^0−\bar{K}^0## mixing ?

    As I made the matrix element amplitude equals:

    ## i \mathcal{M} = \Big(\frac{ig}{2\sqrt{2}}\Big)^2~~ \Big(\frac{ig}{4\cos\theta_W}\Big)^2 [1-\frac{8}{3}\sin^2\theta_W-\gamma_5][-1+4\sin^2\theta_W+\gamma_5] \sum_i~ V^*_{ib}~ V^*_{is}~\int \frac{d^4k}{(2\pi)^4} ~
    \Big( \bar{b}_l~ \gamma_\mu~ (-i \frac{g_{\mu\nu}-(p+k)_\mu (p+k)_\nu/(p+k)^2}{(p+k)^2-M^2_W}) ~\gamma_\nu~ s_l \Big) ~~ \Big( \bar{l}_l~ \gamma_\lambda~ (-i \frac{g_{\lambda\rho}-k_\lambda k_\rho/q^2}{q^2-M^2_Z}) ~ l_l \Big) \gamma_\rho~ \frac{\gamma.k+m_i}{k^2-m_i^2} \frac{\gamma.(k+q)+m_i}{(k+q)^2-m_i^2} ##

    where the momentum flow is given by:

    ee.png

    Till here i think there is some thing wrong because simply

    ## \bar{b}_l~ \gamma_\mu~ (g_{\mu\nu}-k_\mu k_\nu) ~\gamma_\nu~ s_l
    = \bar{b}~ P_R ~\gamma_\mu~ \gamma_\mu P_L~ s
    - \bar{b}~ P_R ~\gamma_\mu k_\mu k_\nu ~ \gamma_\nu~ P_L~ s = 0 ##

    Any one make like these calculations before ?
     
  2. jcsd
  3. Feb 12, 2017 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
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