A Simple Expression for Heavy to Light Meson Semileptonic Decays Form Factors

Like the two-photon and two-gluon decays of the $P$-wave $\chi_{c0,2}$ and $\chi_{b0,2}$ charmonium state for which the Born term produces a very simple decgays amplitude in terms of an effective Lagrangian with two-quark local operator, the Born term for the processes $c\bar{d} \to (\pi,K) \ell \nu$ and $b \bar{d}\to (\pi, K) \ell\nu$, could also produce the $D$ and $B$ meson semileptonic decays with the light meson $\pi, K$ in the final state. In this approach to heavy-light meson form factors with the $\pi$, K meson treated as Goldstone boson, a simple expression is found for the decays form factors, given as~: $f_{+}(0)/(1 -q^{2}/(m_{H}^{2}+ m_{\pi}^{2}))$, with $H=D,B$ for $D,B \to \pi$ form factors, and $f_{+}(0)/(1 -q^{2}/(m_{H}^{2}+ m_{K}^{2}))$ for $B,D\to K$ form factor. The purpose of this paper is to show that this expression for the form factors could describe rather well the $q^{2}$- behaviour observed in the BaBar, Belle and BESIII measurements and lattice simulation. In particular, the $D\to K$ form factors are in good agreement with the measured values in the whole range of $q^{2}$ showing evidence for $SU(3)$ breaking with the presence of $m_{K}^{2}$ term in the quark propagator, but some corrections to the Born term are needed at large $q^{2}$ for $D,B \to \pi$ form factors.