Symmetries, Sum Rules and Constraints on Effective Field Theories

TL;DR

The paper develops a universal framework of dispersion-relations-based sum rules for 2→2 forward scattering of states in arbitrary symmetry representations, linking IR EFT coefficients to UV cross sections via a crossing matrix X. By decomposing amplitudes into irreps and projecting with P±, it derives independent sum rules and, for even subtractions, positivity constraints that form a convex cone in the space of amplitude coefficients; once-subtracted rules provide UV-constrained relations among EFT low-energy constants. The authors illustrate the machinery with extensive group-theory examples (SO(N), SU(N) adjoints, and SU(N)L×SU(N)R cosets) and apply it to longitudinal WW scattering, showing how gauge contributions cancel in the forward limit and how small-g' effects recover GB-like bounds. The results yield concrete EFT constraints relevant for custodial symmetry, composite Higgs scenarios, and chiral perturbation theory, and they point to future directions such as bounds on dimension-6 operators and extensions to other spacetime or symmetry structures.

Abstract

Using unitarity, analyticity and crossing symmetry, we derive universal sum rules for scattering amplitudes in theories invariant under an arbitrary symmetry group. The sum rules relate the coefficients of the energy expansion of the scattering amplitudes in the IR to total cross sections integrated all the way up to the UV. Exploiting the group structure of the symmetry, we systematically determine all the independent sum rules and positivity conditions on the expansion coefficients. For effective field theories the amplitudes in the IR are calculable and hence the sum rules set constraints on the parameters of the effective Lagrangian. We clarify the impact of gauging on the sum rules for Goldstone bosons in spontaneously broken gauge theories. We discuss explicit examples that are relevant for WW-scattering, composite Higgs models, and chiral perturbation theory. Certain sum rules based on custodial symmetry and its extensions provide constraints on the Higgs boson coupling to the electroweak gauge bosons.

Figures (5)

• Figure 2.1: Analytic structure of the amplitude $\mathcal{A}(s)$ in the $(\text{Re}\,s,\text{Im}\,s)$ plane. The contour $\mathcal{C}$ corresponding to the Cauchy integral formula \ref{['eq:Cauchy']}, encloses the point $\mu^{2}$ around which the amplitude is expanded and the poles at $s=s_{i}$ (red points) corresponding to propagating particles with masses lighter than $4m^{2}$. The analytic structure of $\mathcal{A}(s)$ is symmetric under reflection around $2m^{2}$.
• Figure 2.2: The contour $\mathcal{C}$ of figure \ref{['fig:path']} deformed to a path along the branch cuts plus a big circle at $s=\Lambda^{2}$.
• Figure 3.1: Example of a convex polyhedral cone with six faces in three dimensions. All the edges of the cone and the orange faces lie on one of the faces of the first quadrant $\mathbb{R}^{3}_+$, while the yellow faces are internal to the quadrant. When the ambient space is generalized to $\hat{m}$ dimensions and the convex polyhedral cone to an $\hat{m}_{+}<\hat{m}$ dimensional one with $q\geq \hat{m}_{+}$ edges, the $q$ edge vectors lying on the faces of the $\mathbb{R}^{\hat{m}}_+$ space represent the unique choice of basis vectors that can generate the entire cone through linear combinations with only positive coefficients. We call the positivity constraints represented by these vectors the strongest positivity constraints.
• Figure 4.1: Two dimensional convex polyhedral cone in three dimensions (i.e. planar cone or sector of a plane) generated by $\sum_I \mathbf{v}^A_I \mathcal{G}_{IJ}$ which are the coefficients of the strongest positivity constraints for $SO(N\neq4)$ for $N=3,5,6,\ldots$
• Figure B.1: Analytic structure in the presence of light particles of mass $m_{l}$.