Ising field theory in a magnetic field: φ 3 coupling at T > Tc

Abstract

Abstract We study the “three particle coupling” Γ 11 1 ξ $$ {\Gamma}_{11}^1\left(\xi \right) $$ , in 2d Ising Field Theory in a magnetic field, as the function of the scaling parameter ξ := h/(−m)15/8, where m ∼ T c − T and h ∼ H are scaled deviation from the critical temperature and scaled external field, respectively. The “φ 3 coupling” Γ 11 1 $$ {\Gamma}_{11}^1 $$ is defined in terms of the residue of the 2 → 2 elastic scattering amplitude at its pole associated with the lightest particle itself. We limit attention to the High-Temperature domain, so that m is negative. We suggest “standard analyticity”: Γ 11 1 2 $$ {\left({\Gamma}_{11}^1\right)}^2 $$ , as the function of u := ξ 2, is analytic in the whole complex u-plane except for the branch cut from – ∞ to – u 0 ≈ – 0.03585, the latter branching point – u 0 being associated with the Yang-Lee edge singularity. Under this assumption, the values of Γ 11 1 $$ {\Gamma}_{11}^1 $$ at any complex u are expressed through the discontinuity across the branch cut. We suggest approximation for this discontinuity which accounts for singular expansion of Γ 11 1 $$ {\Gamma}_{11}^1 $$ near the Yang-Lee branching point, as well as its known asymptotic at u → +∞. The resulting dispersion relation agrees well with known exact data, and with numerics obtained via Truncated Free Fermion Space Approach. This work is part of extended project of studying the S-matrix of the Ising Field Theory in a magnetic field.

Keywords

• Field Theories in Lower Dimensions
• Integrable Field Theories
• Non-Perturbative Renormalization
• Renormalization Group