Revisiting the chiral effective action in holographic models

We obtain the pion decay constant and coefficients of fourth derivative terms in the chiral Lagrangian for massless quarks in the Witten-Sakai-Sugimoto model. We extract these quantities from the two-pion scattering amplitude, which we compute directly in the holographic dual through tree-level Witten diagrams. Identification of the low energy coefficients in the chiral action is subtle as their values will be shifted when the tower of massive vector bosons are integrated out. Indeed, by a direct comparison with the existing standard procedure of constructing the chiral action with radial modes in the gravity dual, we explicitly show that there are finite ’t Hooft coupling corrections that have been missed. This suggests that past derivations of effective actions from holographic models may have to be revisited and future derivations more carefully considered.

Figure 1

The three kinematically nonidentical 2-to-2 elastic scattering processes. All propagating particles are pions, hence all legs are represented by dashed lines. Left: $s$-channel; middle: $t$-channel; and right: $u$-channel.

Figure 2

Embedding of the D8-branes in the Witten background. We only consider massless pions in the zero temperature confining phase, which corresponds to configurations where D8- and $\overline{\mathrm{D}8}$-branes are fixed at antipodal points of the $\tau$ direction in the asymptotic boundary and they join at a holographic radial position $U_{\mathrm{KK}}$ where the geometry smoothly closes off.

Figure 3

Four-point vertex (left) and exchange (right) Witten diagrams used to compute the one-point function of the axial current. Lines ending at the boundary represent bulk-to-boundary propagators and introduce factors proportional to the source, while lines connecting points in the interior represent bulk propagators.