Ising Field Theory in a magnetic field: ${\phi }^3$ coupling at $T>T_c$

Abstract

We study the "three particle coupling" ${\Gamma }_{11}^1(\xi )$, in $2d$ Ising Field Theory in a magnetic field, as the function of the scaling parameter $\xi :=h/(-m{)}^{15/8}$, where $m\sim T_c-T$ and $h\sim H$ are scaled deviation from the critical temperature and scaled external field, respectively. The "${\phi }^3$ coupling" ${\Gamma }_{11}^1$ is defined in terms of the residue of the $2\to 2$ 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": $({\Gamma }_{11}^1{)}^2$, as the function of $u:={\xi }^2$, is analytic in the whole complex $u$-plane except for the branch cut from $-\infty$ to $-u_0\approx -0.03585$, the latter branching point $-u_0$ being associated with the Yang-Lee edge singularity. Under this assumption, the values of ${\Gamma }_{11}^1$ 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 ${\Gamma }_{11}^1$ near the Yang-Lee branching point, as well as its known asymptotic at $u\to +\infty$. 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.

27 pages, 10 figures