Heavy meson semileptonic differential decay rate in two dimensions in the large Nc

Abstract

We study QCD in 1+1 dimensions in the large $N_c$ limit using light-front Hamiltonian perturbation theory in the $1/N_c$ expansion. We use this formalism to exactly compute hadronic transition matrix elements for arbitrary currents at leading order in $1/N_c$. We compute the semileptonic differential decay rate of a heavy meson, $dΓ/dx$, and its moments, $M_N$, using the hadronic matrix elements obtained previously. We put some emphasis in trying to understand parity invariance. We also study with special care the kinematic region where the operator product expansion ($1/N \sim 1-x \sim 1$) or non-local effective field theories ($1/N \sim 1-x \sim Λ_{QCD}/m_Q$) can be applied. We then compare with the results obtained using an effective field theory approach based on perturbative factorization, with the focus to better understand quark-hadron duality. At the end of the day, using effective field theories, we have been able to obtain expressions for the moments with relative accuracy of $O(Λ_{QCD}^2/m_Q^2)$ in the kinematic region where the operator product expansion can be applied, and with relative accuracy of $O(Λ_{QCD}/m_Q)$ in the kinematic region where non-local effective field theories can be applied. These expressions agree, within this precision, with those obtained from the hadronic result using the layer-function approximation plus Euler-McLaurin expansion. Very good numerical agreement for the moments is obtained between the exact result and the result using effective field theories.