Unitarity bounds with subthreshold and anomalous cuts for $b$-hadron decays

TL;DR

The paper addresses the limitation of the Boyd–Grinstein–Lebed (BGL) unitarity framework when hadronic form factors (FFs) in $b$-hadron decays exhibit subthreshold or anomalous branch cuts. It develops a general, unitarity-bounded parametrization by using a subtracted dispersion relation for correlators and an operator product expansion to bound the relevant integrals, combined with conformal mappings and outer functions to map the analytic domain to the unit disk. For subthreshold cuts, the authors derive explicit parametrizations for $f_+^{BK}$ and $f_0^{BK}$ with $z$-maps based on $s_\Gamma$ and show that the coefficients obey $\sum_n |c_{\lambda,n}|^2<1$, while handling poles via pole subtraction. For anomalous cuts in non-local FFs, they introduce a Schwarz–Christoffel–based conformal map $\hat{z}$ to accommodate the complex-cut geometry and outline how to construct a unitarity-bounded expansion, with the remaining challenge of bounding the anomalous-contribution $\Delta\chi^J$ and a pathway to numerical implementation. The resulting framework extends the BGL program to a broader class of FFs, enabling first-principles, model-independent analyses with controlled truncation uncertainties and improving the reliability of SM tests in rare $B$ decays.

Abstract

We derive a generalisation of the Boyd-Grinstein-Lebed (BGL) parametrization. Most form factors (FFs) in $b$-hadron decays exhibit additional branch cuts -- namely subthreshold and anomalous branch cuts -- beyond the ``standard'' unitarity cut. These additional cuts cannot be adequately accounted for by the BGL parametrization. For instance, these cuts arise in the FFs for $B\to D^{(*)}$, $B\to K^{(*)}$, and $Λ_b\to Λ$ processes, which are particularly relevant from a phenomenological standpoint. We demonstrate how to parametrize such FFs and derive unitarity bounds in the presence of subthreshold and/or anomalous branch cuts. Our work paves the way for a wide range of new FF analyses based solely on first principles, thereby minimising systematic uncertainties.

Figures (2)

• Figure 1: Illustration of the $z(q^2, s_+)$ and $z(q^2, s_\Gamma)$ mappings. The magenta line represents the subthreshold cut, while the blue line represents the standard branch cut starting at $s_+$.
• Figure 2: Illustration of the conformal mapping $\hat{z}$ with $s_0\!=\!0$. We use the same colour scheme as in Fig. \ref{['fig:1']} with the anomalous branch cut represented in green.