Remarks on analyticity and unitarity in the presence of a Strongly Interacting Light Higgs

TL;DR

The article derives a general IR/UV sum rule for Strongly Interacting Light Higgs scenarios within the SILH framework by enforcing Lorentz invariance, analyticity, and unitarity on Goldstone/Higgs scattering. Through a dispersion relation and SU(2)$_{ m L} imes$SU(2)$_{ m R}$ decomposition, the IR amplitude proportional to the low-energy coefficient $c_H$ is related to a UV integral over forward cross sections in specific chiral channels, yielding $c_H = rac{f^2}{4\,\pi} \,\int_0^{\infty} rac{ds}{s} [ \sigma^{ m tot}_{00}(s) + 2 \,\sigma^{ m tot}_{ m LR}(s) - 3 \,\sigma^{ m tot}_{11}(s) ]$, with generalizations when left–right symmetry is broken. This links low-energy Higgs coupling deviations to the resonance content of the UV completion, distinguishing compact from non-compact cosets (e.g., $SO(5)/SO(4)$ vs $SO(4,1)/SO(4)$) and informing expectations for resonances like $oldsymbol{\eta}$, $oldsymbol{ ho}$, and $oldsymbol{\Delta}$ in high-energy scattering. The work further clarifies how current Higgs coupling measurements constrain $c_H$ and discusses caveats related to resonance widths and couplings in determining their impact on the sum rule. Overall, the paper provides a model-independent diagnostic connecting UV strong dynamics to measurable low-energy Higgs phenomenology.

Abstract

Applying the three axiomatic criteria of Lorentz invariance, analyticity and unitarity to scattering amplitudes involving the Goldstone bosons and the Higgs boson, we derive a general sum rule for the Strongly Interacting Light Higgs Lagrangian. This sum rule connects the IR coefficient $c_H$ to the UV properties of the theory, and can be used, for instance, to capture the role of resonances in processes like $V_{\rm L}V_{\rm L}\to hh$ and $V_{\rm L}V_{\rm L}\to V_{\rm L}V_{\rm L}$, with $V=W^{\pm},Z$.

Figures (3)

• Figure 1: Contour of integration $\mathcal{C}$ (counterclockwise, blue solid line) in the complex s-plane according to Eq. (\ref{['eq:MasterIntegral']}), decomposed into the four contributions (I-IV) surrounding the cuts (red zigzag line), and the contribution from the big circle at infinity $\mathcal{C}_{\infty}$. The green dashed line pictorially represents the crossing transformation relating the s- and u-channel amplitudes in the forward limit [see Eqs. (\ref{['eq:CUTS']}, \ref{['eq:MasterCrossing']}) and Appendix \ref{['App:Crossing']}].
• Figure 2: $\chi^2-\chi^2_{\rm min}$ as a function of $\xi c_H$ obtained from a fit to the Higgs data at the LHC. The blue solid line (red dashed line) is obtained marginalizing over $\xi c_{f=t,b,\tau}$ (setting $\xi c_{f=t,b,\tau}=0$).
• Figure 3: Confidence regions (68%, 95%, 99% C.L.) for the $S$ and $T$ oblique parameters ($U=0$) obtained from the fit of LEP-I and LEP-II data.