Aspects of the low-energy constants in the chiral Lagrangian for charmed mesons

We investigate the numerical values of the low-energy constants in the chiral effective Lagrangian for the interactions between the charmed mesons and the lightest pseudoscalar mesons, the Goldstone bosons of the spontaneous breaking of chiral symmetry for QCD. This problem is tackled from two sides: estimates using the resonance-exchange model and positivity constraints from the general properties of the $S$-matrix including analyticity, crossing symmetry, and unitarity. These estimates and constraints are compared with the values determined from fits to lattice data of the scattering lengths. Tensions are found, and possible reasons are discussed. We conclude that more data from lattice calculations and experiments are necessary to fix these constants better. As a byproduct, we also estimate the coupling constant $g_{DD{a}_2}$, with $a_2$ the light tensor meson, via the QCD sum rule approach.

Figure 1

The Mandelstam plane. The Mandelstam triangle is the region bounded by the thick lines: $s=(M_D+M_{\varphi }{)}^2$, $u=(M_D+M_{\varphi }{)}^2$, and $t=4M_{\varphi }^2$. The upper part of the Mandelstam triangle is marked in dark gray, which is surrounded by the previous three lines and the one corresponding to $t=0$. The physical regions are marked in light gray.

Figure 2

Diagrams for the resonance-exchange contribution to $D\varphi$ scattering.

Figure 3

Comparison of the NLO positivity bounds for $h_4^{\prime }$ and $h_5^{\prime }$ with their values obtained from fitting to the lattice data using unitarized ChPT at NLO. The positivity-bound region is depicted in light yellow bounded by the lines $h_4^{\prime }=0$ and $h_4^{\prime }-h_5^{\prime }=0$. The area in light blue denotes the region where the bound $h_4^{\prime }-h_5^{\prime }\ge 0$ is respected, while $h_4^{\prime }\ge 0$ is violated. The green dot-dashed and magenta dashed ellipses represent the $1\text{−}\sigma$ regions for $h_4^{\prime }$ and $h_5^{\prime }$ from the five- and four-parameter fits in Ref. [25], respectively. The red dot and blue square with error bars, denoted by AGW-HQS and AGW-$\chi \mathrm{SU}(3)$, respectively, are taken from Ref. [36].

Figure 4

Comparison of the positivity bounds for $h_4^{\prime }$ and $h_5^{\prime }$ with their six-channel NNLO fit values. The graphs in the first, second, and third column correspond to the case in which $g_3^{\prime }$ is fixed at its lowest, central, and largest values, respectively. The blue dots with error bars represent the fitting values of $h_4^{\prime }$ and $h_5^{\prime }$ from different fits: UChPT-6(a), UChPT-6(b), UChPT-6($a^{\prime }$), and UChPT-6($b^{\prime }$); see Ref. [37]. The NNLO positivity-bound region is in light yellow bounded by the lines $h_4^{\prime }=0.55$ and $h_4^{\prime }-h_5^{\prime }=g(g_3^{\prime })$. The area in light blue denotes the region where the bound $h_4^{\prime }-h_5^{\prime }\ge g(g_3^{\prime })$ is respected, while $h_4^{\prime }\ge 0.55$ is violated.

Figure 5

Momentum dependence of the $DD{a}_2$ form factor (for off-shell $a_2$). The dots give the results from QCD sum rules, and the solid line gives the extrapolation.