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

[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 anyone 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 ?

 

[AndyW]

Hello,

Thank you for your question. I am happy to assist you in calculating the matrix element amplitude for the $b→s l^+l^−$ penguin diagram mediated by Z boson or the photon.

First, let's address your question about the choice of gauge. In general, the choice of gauge should not affect the final result of the calculation. However, the Feynman gauge is often preferred for its simplicity and convenience. You can certainly use the Feynman gauge for your calculations.

Moving on to the matrix element amplitude, it seems like you have made a mistake in your calculation. The correct expression for the matrix element amplitude should be:

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

where the momentum flow is given by:

$b(p) ~\to~ s(q) ~+~ l^+(k) ~+~ l^-(k')$

In this expression, the first term in the parentheses is the contribution from the W boson propagator, and the second term is from the Z boson propagator. Also, there is a typo in the second term of the momentum flow,