Abstract
We study the “three particle coupling” \( {\Gamma}_{11}^1\left(\xi \right) \), in 2d Ising Field Theory in a magnetic field, as the function of the scaling parameter ξ := h/(−m)15/8, where m ∼ Tc − T and h ∼ H are scaled deviation from the critical temperature and scaled external field, respectively. The “φ3 coupling” \( {\Gamma}_{11}^1 \) 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”: \( {\left({\Gamma}_{11}^1\right)}^2 \), as the function of u := ξ2, is analytic in the whole complex u-plane except for the branch cut from – ∞ to – u0 ≈ – 0.03585, the latter branching point – u0 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 → +∞. 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.
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Acknowledgments
AZ is grateful to V. Bazhanov and F. Smirnov for interest to this work and very useful discussions, and HLX thanks B. McCoy and R. Shrock for helpful discussions and comments. Research of AZ was supported in part by National Science Foundation under Grant PHY-1915093 and PHY-2210533.
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Xu, HL., Zamolodchikov, A. Ising field theory in a magnetic field: φ3 coupling at T > Tc. J. High Energ. Phys. 2023, 161 (2023). https://doi.org/10.1007/JHEP08(2023)161
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DOI: https://doi.org/10.1007/JHEP08(2023)161