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

1. Jan 23, 2017

### Safinaz

I'd like to calculate the matrix element amplitude for$b→s l^+l^−$ penguin diagram mediated by Z boson or the photon , like :

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:

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. Feb 12, 2017