Further tests of lepton flavour universality from the charged lepton energy distribution in $b\to c$ semileptonic decays: The case of $\Lambda_b\to \Lambda_c \ell \bar\nu_\ell$

Abstract

In a general framework, valid for any $H\to H'\ell^-\bar\nu_\ell$ semileptonic decay, we analyze the $d^2\Gamma/(d\omega d\cos\theta_\ell)$ and $d^2\Gamma/ (d\omega dE_\ell)$ distributions, with $\omega$ being the product of the hadron four-velocities, $\theta_\ell$ the angle made by the three-momenta of the charged lepton and the final hadron in the $W^-$ center of mass frame and $E_\ell$ the charged lepton energy in the decaying hadron rest frame. Within the Standard Model (SM), $ d^2\Gamma/(d\omega dE_\ell)\propto \left(c_0(\omega)+c_1(\omega)E_\ell/M+c_2(\omega)E^2_\ell/M^2\right)$, with $M$ the initial hadron mass. We find that $c_2(\omega)$ is independent of the lepton flavor and thus it is an ideal candidate to look for lepton flavor universality (LFU) violations. We also find a correlation between the $a_2(\omega)$ structure function, that governs the $(\cos\theta_\ell)^2$ dependence of $d^2\Gamma/(d\omega d\cos\theta_\ell)$, and $c_2(\omega)$. Apart from trivial kinematical and mass factors, the ratio of $a_2(\omega)/c_2(\omega)$ is a universal function that can be measured in any semileptonic decay, involving not only $b\to c$ transitions. These two SM predictions can be used as new tests in the present search for signatures of LFU violations. We also generalize the formalism to account for some new physics (NP) terms. Finally, in order to illustrate our findings, we apply our general framework to the $\Lambda_b\to \Lambda_c \ell \bar\nu_\ell$ decay. We show that a measurement of $c_2$ (or $a_2$) for $\tau$ decay would not only be a direct measurement of the possible existence of NP, but it would also allow to distinguish from NP fits to $b\to c\tau\bar\nu_\tau$ anomalies in the meson sector, that otherwise give the same total and differential $d\Gamma/d\omega$ widths.