Unitarity and Unitarization

Abstract

This article reviews unitarization methods essential for extending Effective Field Theories (EFTs) beyond their perturbative limits, particularly in hadronic and electroweak (EW) sectors. Perturbative EFTs, like Chiral Perturbation Theory (ChPT), often violate unitarity bounds at higher energies, a breakdown observed in phenomena such as $ππ$ scattering resonances. To overcome this, non-perturbative techniques including the Inverse Amplitude Method (IAM), $K$-matrix formalism, and N/D approach are detailed. The IAM and the N/D methods resum perturbative series while preserving fundamental $S$-matrix principles: unitarity, analyticity, and causality, dynamically generating resonant behavior. The article emphasizes the unique role of dispersive frameworks, especially the Roy equations, which rigorously incorporate analyticity and crossing symmetry. It highlights their potential for future application in the electroweak sector, offering a powerful tool to constrain the Standard Model and interpret collider data.

Figures (6)

• Figure 1: Representation of a scattering process where two incoming particles with momentum $p_1$ and $p_2$ collide producing $n$ outgoing particles travelling freely with momentum $k_i$ each.
• Figure 2: Mandelstam plane for two identical particles.
• Figure 3: A resonance using the IAM extended to the complex $s$ plane, with the parameters, in the notation of Delgado:2015kxa, $a=0.95$, $a_4=-2.5\cdot 10^{-4}$ and $a_5=-1.75\cdot 10^{-4}$. The first plot shows the first Riemann sheet, cut along the real axis ${\rm Im}(s)=0$, although it does not leave a pole in this first sheet, it is seen to saturate unitarity (${\rm Im}(s)\simeq 1$ for a certain real $s$ (the scale does not allow to visualize the cut). The plot in the bottom line is the extension to the second Riemann sheet and clearly features a pole for negative ${\rm Im} (s)<0$.
• Figure 4: Analytic structure of the $2\to2$ particle scattering amplitude $T(s,t,u)$ and the contour $C$ for its dispersion relation. Figure from RuizdeElviraCarrascal:2013tix.
• Figure 5: Analytic structure of elastic scattering partial waves for scalars of mass $m$ and the contour $C$ in the complex-$s$ plane that will be used to write a dispersion relation. The red lines represent the discontinuity cuts in the partial wave amplitude. The red crosses additionally represent the $n$-th order pole at $z=\epsilon$ and the simple pole at $z=s+i\epsilon$ (with $s>4m^2$) coming from the denominators in eq. (\ref{['I']}).