Generalized positivity bounds on chiral perturbation theory

TL;DR

This work extends positivity bounds to include arbitrary $t$-derivatives to constrain chiral perturbation theory at NNLO. By applying the generalized (Y) bounds, including improved subtractions, the authors obtain stronger, convex constraints on the chiral LECs $ar{l}_i$ and the $b_i$ combinations, compared to traditional bounds. The improved bounds reveal a breakdown scale near $490$ MeV, consistent with the onset of the $f_0(500)$ resonance, and demonstrate that Padé unitarization does not improve the amplitudes' analyticity. Overall, the method provides robust, theory-driven limits on EFT parameters, with potential applicability to other symmetry-breaking EFTs.

Abstract

Recently, a new set of positivity bounds with $t$ derivatives have been discovered. We explore the generic features of these generalized positivity bounds with loop amplitudes and apply these bounds to constrain the parameters in chiral perturbation theory up to the next-to-next-to-leading order. We show that the generalized positivity bounds give rise to stronger constraints on the $\bar l_i$ constants, compared to the existing axiomatic bounds. The parameter space of the $b_i$ constants is constrained by the generalized positivity bounds to be a convex region that is enclosed for many sections of the total space. We also show that the improved version of these positivity bounds can further enhance the constraints on the parameters. The often used Padé unitarization method however does not improve the analyticity of the amplitudes in the chiral perturbation theory at low energies.

Figures (15)

• Figure 1: Distances between the bound plane and a fiducial point of $b_i$ of in the $(b_1, b_2,b_3,b_4,b_5,b_6)$ space for the cases of $\eta=0,1$. The bound plane is depicted by $Y^{(2N,M)}(t)=a_0+\sum_{i=1}^6 a_i b_i=0$ for a given set of $\{\eta, t, N, M\}$. The fiducial point is taken to be the fitted values of $b_i$ in Eq. \ref{['expri data1']}. For most bounds, the distance varies monotonically with $t$, except for $\{\eta=0,N=2,M=1\}$ and $\{\eta=1, N=1,M=0\}$, but in all of these cases the shortest distances are always at $t=0$ or $4$.
• Figure 2: The positivity constraints on the parameter space of $\{b_4,b_6\}$ with other $b_i$ set to the central values of the fit \ref{['expri data1']}. The yellow region is ruled out by the $Y$ bounds with $\eta=0,t=0,4$, and is smaller than the blue region that is ruled out by the $Y$ bounds with $\eta=0$ and $13$ values of $t$.
• Figure 3: Rescaled values of $Y^{(2N,M)}(t)$ for $\eta=0$ and different $\{N,M\}$ at $t=4,3.5,2,0.4$ (from top to bottom). The rescaled value of $Y^{(2N,M)}(t)$ is defined as $Y^{(2N,M)}(t)/a_0=1+\sum_{i=1}^6 a'_i b_i$. The strongest bounds are given by small $N$ and $M$.
• Figure 4: Similar to Figure \ref{['fig:NMeta=0']} but for the case of $\eta=1$.
• Figure 5: Comparison of our positivity bounds on $\bar{l}_1$ and $\bar{l}_2$ with those of Manohar and Mateu Manohar:2008tc for the $\pi\pi$ scattering to one loop. See Eqs. (\ref{['manohar cons']}) to (\ref{['Yl2ndbound']}) for the error estimates for these bounds. The rectangles GL, ABT, GKMS and the small ellipse inside it are the ranges of the fitted values of $\bar{l}_1$ and $\bar{l}_2$ given in Refs. Gasser:1983yg, Girlanda:1997ed, Amoros:2000mc and Colangelo:2001df respectively.