Flavour Dynamics

$\texttt{HEPfit}$ has been used to address the claimed "anomaly" in the angular distribution of the four body $B \to K^* \mu^+\mu^-$ decays. The perturbative short distance physics are calculable to $\mathcal{O}(1\%)$ precision. Estimation of the form factors are available from QCD sum rules calculations or from Lattice QCD and are shrouded by $\mathcal{O}(10\%)$ errors. The non-factorizable corrections, over which not much theoretical handle can be exercised, bring forth the leading source of theoretical errors. We extract the non-factorizable hadronic contributions from available data and show the non-factorizable contribution necessary to resolve the anomaly is within reasonable theoretical expectations. In $\texttt{HEPfit}$ these are parameterized as: \begin{align} h_-(q^2) &= -\frac{m_b}{8\pi^2 m_B} \widetilde T_{L -}(q^2) h_-^{(0)} -\frac{\widetilde V_{L -}(q^2)}{16\pi^2 m_B^2} h_-^{(1)} q^2 + h_-^{(2)} q^4+{\cal O}(q^6)\,, \\ % h_+(q^2) &= -\frac{m_b}{8\pi^2 m_B} \widetilde T_{L +}(q^2) h_-^{(0)} -\frac{\widetilde V_{L +}(q^2)}{16\pi^2 m_B^2} h_-^{(1)} q^2 + h_+^{(0)} + h_+^{(1)}q^2 + h_+^{(2)} q^4+{\cal O}(q^6)\,,\\ % h_0(q^2) &= -\frac{m_b}{8\pi^2 m_B} \widetilde T_{L 0}(q^2) h_-^{(0)} -\frac{\widetilde V_{L 0}(q^2)}{16\pi^2 m_B^2} h_-^{(1)} q^2 + h_0^{(0)}\sqrt{q^2} + h_0^{(1)}(q^2)^\frac{3}{2} +{\cal O}((q^2)^\frac{5}{2})\,. \end{align}
This choice allows us to write the helicity amplitudes $H_V^\lambda$ as \begin{eqnarray} H_V^{-} \propto \bigg\{\left(C_9^{\rm SM} + h_-^{(1)}\right)\widetilde V_{L -} + \frac{m_B^2}{q^2} && \left[ \frac{2m_b}{m_B}\left(C_7^{\rm SM} + h_-^{(0)} \right) \widetilde T_{L -} - 16\pi^2 h_-^{(2)}\, q^4 \right]\bigg\}\,, \\ % H_V^{+} \propto \bigg\{\left(C_9^{\rm SM} + h_-^{(1)}\right)\widetilde V_{L +} + \frac{m_B^2}{q^2} && \left[\frac{2m_b}{m_B}\left(C_7^{\rm SM} + h_-^{(0)} \right) \widetilde T_{L +} - 16\pi^2\left(h_{+}^{(0)} + h_{+}^{(1)}\, q^2 + h_{+}^{(2)}\, q^4\right) \right]\bigg\}\,,\\ % H_V^{0} \propto \bigg\{\left(C_9^{\rm SM} + h_-^{(1)}\right) \widetilde{V}_{L0} + \frac{m_B^2}{q^2} && \left[\frac{2m_b}{m_B} \left(C_7^{\rm SM} + h_-^{(0)} \right) \widetilde{T}_{L0} - 16\pi^2\sqrt{q^2}\left({h}_{0}^{(0)} + {h}_{0}^{(1)}\, q^2\right)\right]\bigg\}\,. \end{eqnarray}
With this definition for the $h_\lambda$-coefficients, it is manifest that $h_-^{(0)}$ and $h_-^{(1)}$ can be considered as constant shifts to the Wilson coefficients $C_{7,9}^{\rm SM}$, hence indistinguishable from NP contributions to $Q_{7\gamma,9V}$. Consequently, it is not possible to extract $h_-^{(0)}$ and $h_-^{(1)}$ directly from data unless one assumes the absence of NP effects. On the other hand, it is also not possible to ascertain the presence of NP without a theory input for these hadronic effects. The advantage of our parameterization becomes clear when any of the remaining $h_\lambda$-coefficients turns out to be non-vanishing, since they likely spot purely hadronic contributions.
Lepton universality testing ratios like $R_K$ and $R_{K^*}$ that have been recently measured by LHCb and Belle are strictly close to unity within the Standard Model and any significant deviation from this would herald the discovery of new dynamics. We addressed different scenarios of physics beyond the Standard Model in the language of effective field theories (EFT) that can resolve such anomalies both in the presence and absence of large hadronic contributions.

The LFV module in $\texttt{HEPfit}$ calculates the decay rates for various lepton flavour violating observables in a given new physics model. The expression for various $\Delta F = 1$ and $\Delta F = 0$ processes are calculated in terms of the Wilson coefficients of the responsible operators. Therefore, given any new physics model, once the user provides the Wilson coefficients generated in the model, $\texttt{HEPfit}$ will calculate the decay rates for all the lepton flavour violating observables generated by those operators. This enables the user to include any new physics model to $\texttt{HEPfit}$ and analyze the lepton flavour violating processes efficiently. For example, the effective weak Hamiltonian for the $\ell_i \to \ell_j\, \gamma$ process is given by \begin{align} \mathcal{H}_{\rm eff}^{\Delta F=1} &= \mathcal{C}_{7} \mathcal{O}_{7} + \mathcal{C}_{7}^{\prime} \mathcal{O}_{7}^{\prime} , \nonumber \end{align} where \begin{align} \mathcal{O}_{7} &= e\, m_{l_i} \overline{\ell}_j \sigma_{\mu \nu} P_R \ell_i F^{\mu \nu}\ , \notag \\ \mathcal{O}_{7}^{\prime} &= e\, m_{l_i} \overline{\ell}_j \sigma_{\mu \nu} P_L \ell_i F^{\mu \nu}\ , \nonumber \end{align}
$\sigma_{\mu \nu} = \frac{i}{2}[\gamma^{\mu},\gamma^{\nu}]$, $P_{R,L} = \frac{1}{2} (1\pm\gamma_5)$, and $F_{\mu\nu} = (\partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu})$. $m_{l_i}$ is the mass of the initial state fermion. The decay rate for the process is \begin{align} \Gamma (\ell_i \to \ell_j\, \gamma) &= \frac{e^2}{4\pi} m_{l_i}^5 \bigg(|\mathcal{C}_{7}|^2 + |\mathcal{C}_{7}^\prime|^2\bigg) \ . \nonumber \end{align} In the present version of the $\texttt{HEPfit}$, all the Wilson coefficients for the processes $\ell_i \to \ell_j\, \gamma$, $\ell_i \to 3\ell$, $\mu \to e$ conversion as well as $(g-2)_{\mu}$ at two-loop level is included in the general MSSM.