Triangle and box diagrams in coupled-channel systems from the chiral Lagrangian

We perform an analysis of triangle- and box-loop contributions to the generalized potential in the scattering of Goldstone bosons off the $J^P=0^{-}$ and $1^{-}$ charmed mesons. Particular emphasis is put on the use of on-shell mass parameters in such contributions in terms of a renormalization scheme that ensures the absence of power-counting violating terms. This is achieved with a systematically extended set of Passarino-Veltman basis functions that leads to manifest power-counting conserving one-loop expressions and avoids the occurrence of superficial kinematical singularities. Compact expressions to chiral order three and four are presented that are particularly useful in coding such coupled-channel systems. Our formal results are generic and prepare analogous computations for other systems, like meson-baryon scattering from the chiral Lagrangian.

Figure 1

Dashed lines stand for pion, kaon, or eta mesons and solid lines for charmed mesons with $J^P=0^{-}$. The vertices are from [37].

Figure 2

Dashed lines stand for pion, kaon, or eta mesons and solid and double-solid lines for charmed mesons with $J^P=0^{-}$ and $J^P=1^{-}$, respectively. The vertices are from [37].

Figure 3

Dashed lines stand for pion, kaon, or eta mesons and solid and double-solid lines for charmed mesons with $J^P=0^{-}$ and $J^P=1^{-}$, respectively. The vertices are from [37].

Figure 4

Dashed lines stand for pion, kaon, or eta mesons and solid and double-solid lines for charmed mesons with $J^P=0^{-}$ and $J^P=1^{-}$, respectively. The vertices are from [37].

Figure 5

Dashed lines stand for pion, kaon, or eta mesons and solid lines and double-solid lines for charmed mesons with $J^P=0^{-}$ and $J^P=1^{-}$, respectively. The vertices are from [37].

Figure 6

Dashed lines stand for pion, kaon, or eta mesons and solid and double-solid lines for charmed mesons with $J^P=0^{-}$ and $J^P=1^{-}$, respectively. The vertices are from [37].

Figure 7

Dashed lines stand for pion, kaon, or eta mesons and solid and double-solid lines for charmed mesons with $J^P=0^{-}$ and $J^P=1^{-}$, respectively. The vertices are from [37].