In this letter, we construct the recursion relations for one-loop planar integrands in the SU(N) non-linear sigma model. This generalizes the soft recursions for tree-level amplitudes in a variety of quantum field theories with special soft limits. The main ingredient is the definition of the one-loop planar integrand, which is fixed by cuts in the sense of generalized unitarity and by requiring the Adler zero on all external legs. We show that this does not uniquely fix the integrand, so additional constraints on the soft behavior of the loop momentum have to be imposed. Our work is the first step in extending modern amplitudes methods for loop amplitudes to effective field theories with special soft limits.

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Recursion relations for one-loop Goldstone boson amplitudes

A bstract We present positivity bounds, derived from the principles of analyticity, unitarity and crossing symmetry, that constrain the low-energy constants of chiral perturbation theory. Bounds are produced for 2, 3 or more flavours in meson-meson scattering with equal meson masses, up to and including next-to-next-to-leading order (NNLO), using the second and higher derivatives of the amplitude. We enhance the bounds by using the most general isospin combinations posible (or higher-flavour counterparts thereof) and by analytically integrating the low-energy range of the discontinuities. In addition, we present a powerful and general mathematical framework for efficiently managing large numbers of positivity bounds. 

/pdf/nnlo-positivity-bounds-on-chiral-perturbation-theory-for-a-29wv74xh.pdf

NNLO positivity bounds on chiral perturbation theory for a general number of flavours

Recently, the formalism needed to relate the finite-volume spectrum of systems of nondegenerate spinless particles has been derived. In this work we discuss a range of issues that arise when implementing this formalism in practice, provide further theoretical results that can be used to check the implementation, and make available codes for implementing the three-particle quantization condition. Specifically, we discuss the need to modify the upper limit of the cutoff function due to the fact that the left-hand cut in the scattering amplitudes for two nondegenerate particles moves closer to threshold; we describe the decomposition of the three-particle amplitude $\mathcal{K}_\text{df,3}$ into the matrix basis used in the quantization condition, including both $s$ and $p$ waves, with the latter arising in the amplitude for two nondegenerate particles; we derive the threshold expansion for the lightest three-particle state in the rest frame up to $\mathcal{O}(1/L^5)$; and we calculate the leading-order predictions in chiral perturbation theory for $\mathcal{K}_\text{df,3}$ in the $\pi^+\pi^+K^+$ and $\pi^+K^+K^+$ systems. We focus mainly on systems with two identical particles plus a third that is different ("2+1" systems). We describe the formalism in full detail, and present numerical explorations in toy models, in particular checking that the results agree with the threshold expansion, and making a prediction for the spectrum of $\pi^+\pi^+K^+$ levels using the two- and three-particle interactions predicted by chiral perturbation theory. 

/pdf/implementing-the-three-particle-quantization-condition-for-p-2mfahjxs.pdf

Implementing the three-particle quantization condition for π+π+K+ and related systems

We calculate the relativistic six-meson scattering amplitude at low energy within the framework of QCD-like theories with $n$ degenerate quark flavors at next-to-leading order in the chiral counting. We discuss the cases of complex, real and pseudoreal representations, i.e. with global symmetry and breaking patterns $\mathrm{SU}(n)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(n)/\mathrm{SU}(n)$ (extending the QCD case), $\mathrm{SU}(2n)/\mathrm{SO}(2n)$, and $\mathrm{SU}(2n)/\mathrm{Sp}(2n)$. In case of the one-particle-irreducible part, we obtain analytical expressions in terms of ten six-meson subamplitudes based on the flavor and group structures. We extend on our previous results obtained within the framework of the $\mathrm{O}(N+1)/\mathrm{O}(N)$ nonlinear sigma model, with $N$ being the number of meson flavors. This work allows for studying a number of properties of six-particle amplitudes at one-loop level. 

/pdf/six-meson-amplitude-in-qcd-like-theories-2qn19quy.pdf

Six-meson amplitude in QCD-like theories

In this work, higher-derivative corrections of the non-linear sigma model of both even and odd intrinsic-parity sectors are systematically studied, focusing on ordered amplitudes of flavor scalars in massless limit. It should correspond to a theory known as Chiral perturbation theory (ChPT) without external sources and with only single-trace operators. We briefly overview its formal development and apply new S-matrix methods to its amplitude constructions. The bottom-up analysis of the tree-level amplitudes of different orders and multiplicities focuses on the formal structure of general ChPT. Possible theoretical simplifications based on the Kleiss-Kuijf and Bern-Carrasco-Johansson relations are presented. Finally, in the same context, the comparison with the so-called Z-function, which is connected with string theory, is also discussed.

The ChPT: top-down and bottom-up

Within the framework of the massive $\mathrm{O}(N)$ nonlinear sigma model extended to the next-to-leading order in the chiral counting [for $N=3$ corresponding to the two(-quark)-flavor Chiral Perturbation Theory], we calculate the relativistic six-pion scattering amplitude at low energy up to and including terms $\mathcal{O}({p}^{4})$. Results for the pion mass, decay constant, and four-pion amplitude in the case of $N$ (meson) flavors at $\mathcal{O}({p}^{4})$ are also presented.

Tomáš Husek

Papers

Six-pion amplitude