Interesting theories with short range interactions include QCD in the hadronic phase and cold atom systems. The scattering length in two-to-two elastic scattering process captures the most elementary features of the interactions, such as whether they are attractive or repulsive. However, even this basic quantity is notoriously difficult to compute from first principles in strongly coupled theories. We present a method to compute the two-to-two amplitudes and the scattering length using the holographic duality. Our method is based on the identification of the residues of Green's functions in the gravity dual with the amplitudes in the field theory. To illustrate the method we compute a contribution to the scattering length in a hard wall model with a quartic potential and find a constraint on the scaling dimension of a scalar operator ∆ > d/4. For d < 4 this is more stringent than the unitarity constraint and may be applicable to an extended family of large-N theories with a discrete spectrum of massive states. We also argue that for scalar potentials with polynomial terms of order K, a constraint more restrictive than the unitarity bound will appear for d < 2K/(K − 2).

The low-energy effective theory description of a confining theory, such as QCD, is constructed, including local interactions between hadrons organized in a derivative expansion. This kind of approach also applies more generically to theories with a mass gap, once the relevant low-energy degrees of freedom are identified. The strength of local interactions in the effective theory is determined by the low-momentum expansion of scattering amplitudes, with the scattering length capturing the leading order. We compute the main contribution to the scattering length between two spin-zero particles in strongly coupled theories using the gauge/gravity duality. We study two different theories with a mass gap: a massive deformation of N ¼ 4 super-Yang-Mills theory (N ¼ 1 Ã) and a nonsupersymmetric five-dimensional theory compactified on a circle. These cases have a different realization of the mass gap in the dual gravity description: the former is the well-known GPPZ singular solution and the latter a smooth AdS 6 soliton geometry. We find that the scattering lengths have similar functional dependences on the masses of the particles and on the conformal dimension of the operators that create them in both theories. Assuming these similarities hold more generally, they could be used to constrain the effective description of gapped strongly coupled theories beyond symmetry considerations.

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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 twopion scattering amplitude, which we compute directly in the holographic dual through tree-level Witten diagrams. We compare our results with the existing standard procedure of constructing the chiral action with radial modes in the gravity dual. While the pion decay constant agrees, we point out that the values of the other coefficients have not been correctly identified. This suggests that past derivations of effective actions from holographic models may have to be revisited and future derivations more carefully considered.