Six-pion amplitude

Within the framework of the massive $\mathrm{O}(N)$ nonlinear sigma model extended to the next-to-leading order in the chiral counting [for $N=3$ corresponding to the two(-quark)-flavor Chiral Perturbation Theory], we calculate the relativistic six-pion scattering amplitude at low energy up to and including terms $\mathcal{O}(p^4)$. Results for the pion mass, decay constant, and four-pion amplitude in the case of $N$ (meson) flavors at $\mathcal{O}(p^4)$ are also presented.

Figure 1

Leading-order contribution to the four-pion amplitude, with the vertex stemming from ${\mathcal{L}}_{\mathrm{nL}\sigma \mathrm{M}}$.

Figure 2

NLO topologies relevant for the calculation of the four-pion amplitude. The multiplicities of the respective diagrams are quoted as subcaptions. The standard dotlike vertices come from ${\mathcal{L}}_{\mathrm{nL}\sigma \mathrm{M}}$, while the square vertex stems from the terms proportional to $l_i$s in Eq. (4). As indicated in the main text, there are other diagrams not explicitly depicted here, relevant at NLO for the field renormalization, and mass and decay-constant redefinitions.

Figure 3

Leading-order contributions to the six-pion amplitude. The topology depicted on the right in the subfigure (b) manifests itself with ten contributions with different permutations of the external legs.

Figure 4

NLO topologies relevant for the six-pion amplitude. The multiplicities of the respective diagrams are quoted as subcaptions.

Figure 5

The six-pion subamplitude in the kinematic configuration of Eq. (36). Plotted are the lowest-order result $A^{(2)}(p_1,p_2,\dots ,p_6)$ of Eq. (34) and the next-to-leading-order result $A^{(4)}(p_1,p_2,\dots ,p_6)$ of Eq. (35) together with several groups of its individual constituents. We show only the real parts. To enhance readability, the order in the key is consistent with how the curves appear in the plot. Following Eq. (35), the solid curve representing the NLO result is the sum of all the curves below it. The lines corresponding to $A_{C_3}$ and $A_{C_{11}}+{\sum }_i{A}_C^{(i)}$ are added for completeness; in this plot, they lie very close to zero and basically overlap.