Dispersive Analysis of $D$- and $B$-Meson Form Factors with Chiral and Heavy-Quark Constraints

Abstract

We analyze the isovector vector form factors of $D$, $D^*$, $B$, and $B^*$ mesons at low energies. We employ all constraints due to chiral and heavy-quark symmetry, and include the physics of resonant pion-pion rescattering in a model-independent way, using dispersion theory. Special attention is paid to the analytic properties of these form factors, which include anomalous thresholds due to triangle diagrams that are located on the physical Riemann sheets in some of the form factors. We extract the couplings of the $ρ(770)$ resonance to all these heavy mesons by determining the appropriate pole residues.

Figures (13)

• Figure 1: Diagrammatic representation of the form factor unitarity relation in Eq. \ref{['eq:FF_unitarity']}.
• Figure 2: Diagrammatic representation of the amplitudes for $M^{(*)}\bar{M}^{(*)}\to\pi\pi$ as a sum of a contact term and an $M^{\prime(*)}$ left-hand cut.
• Figure 3: Diagrammatic representation of the unitarized $M^{(*)}\bar{M}^{(*)}\to\pi\pi$$P$-wave amplitudes $T(s)$ as the sum of the Born term $K(s)$, the MO term $M(s)\,\Omega(s)$, and the subtraction term $P(s)\,\Omega(s)$. Notice the triangle topology in the MO term.
• Figure 4: General triangle loop diagram.
• Figure 5: Left: Positions of $s_\text{thr}$ and $s_+$ in the complex $s$-plane before the analytic continuation in $M_V^2$. The integration path is indicated by the thick solid line. Middle: The path traced by $s_+$ during the analytic continuation is indicated by the thin solid line with arrows. When $s_+$ moves through the unitarity cut, the integration path needs to be deformed, picking up the additional red integration path. Right: Deformed integration path after the analytic continuation enclosing the anomalous branch cut, depicted by the zig-zag line.