Abstract
We use the numerical S-matrix bootstrap method to obtain bounds on the two leading Wilson coefficients (or low energy constants) of the chiral lagrangian controlling the low-energy dynamics of massless pions. This provides a proof of concept that the numerical S-matrix bootstrap can be used to derive non-perturbative bounds on massless EFTs in more than two spacetime dimensions.
Highlights
For each Nmax we bound all the partial waves up to spin Lmax = 90 such that the spin cutoff dependence is negligible
This involves a choice of renormalization scheme and for this reason on we will always refer to the parameters α and β which are defined by the equation above in terms of the physical scattering amplitude
This makes transparent the fact that the coefficients of the logs in (1.3) are fixed in terms of fπ and the polynomial part involves free parameters. We do this exercise to order in appendix A.2. This gives the explicit form of the pion scattering amplitude up to six powers of momenta, in perfect agreement with the chiral limit of the 2-loop computation in [17, 18], see appendix B
Summary
A big part of the setup, amplitude ansatz and numerics for massless pions follows almost verbatin the massive pion amplitude case studied in [25] which in turn was strongly based on the general ρ series parametrization of higher dimensional scattering amplitudes proposed in [26]. At this point the rules of the game are pretty much as in [26] and [25]: we pick the ansatz (2.4) with a cut-off Nmax in the sums, construct the partial amplitudes Sl(I) through (2.6) from spin 0 up to some large spin Lmax and pick an s grid with Mgrid points where to impose (2.7) for each spin l and isospin I With these many parameters and constraints we explore the allowed (α, β) space as introduced in (1.3). This is a exciting exercise in the case of pions where we have real world experimental data for their phase shifts [30,31,32,33,34,35,36].9 The outcome of this exploration is depicted in figure 4 where we highlight two interesting regions along the boundary where the numerics do resemble nicely experimental data. Perhaps with this extra physical constraint the S-matrices at the boundary might approach those observed in experiment? Will they have rich even spin structures arising? We leave those numerical explorations to the future
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