Scalar-involved three-point Green functions and their phenomenology

We analyze within the framework of resonance chiral theory the $⟨S{A}_{\mu }{A}_{\nu }⟩$ and $⟨S{V}_{\mu }{V}_{\nu }⟩$ three-point Green functions, where $S$, $A_{\mu }$ and $V_{\mu }$ are short for scalar, axial-vector and vector $SU(3)$ hadronic currents. We construct the necessary Lagrangian such that the Green functions fulfill the asymptotic constraints, at large momenta, imposed by QCD at leading order. We study the implications of our results on the spectrum of scalars in the $\mathrm{large}\text{−}{N}_C$ limit, and analyze their decays.

Figure 1

Identification of momenta for the ${\mathrm{\Pi }}_{123}^{ijk}$ Green function. Here $q=p_1+p_2$.

Figure 2

Contribution to $⟨S{A}_{\mu }{A}_{\nu }⟩$ from the chiral Lagrangian ${\mathcal{L}}_{(2)}^{\mathrm{ChPT}}$.

Figure 3

Goldstone boson contributions to the $⟨S{A}_{\mu }{A}_{\nu }⟩$ Green function from the higher-order GB chiral Lagrangian at $\mathcal{O}(p^4)$ and $\mathcal{O}(p^6)$.

Figure 4

Diagrams contributing to the $⟨S{A}_{\mu }{A}_{\nu }⟩$ Green function in RChT. Goldstone bosons and resonance states are represented by single and double lines, respectively.

Figure 5

Diagrams contributing to the $⟨S{V}_{\mu }{V}_{\nu }⟩$ Green function in RChT. Goldstone bosons and resonance states are represented by single and double lines, respectively.