Bootstrapping QCD Using Pion Scattering Amplitudes

Abstract

We consider the S-matrix bootstrap of four-dimensional scattering amplitudes with Oð3Þ symmetry and no bound states. We explore the allowed space of scattering lengths which parametrize the interaction strength at threshold of the various scattering channels. Next we consider an application of this formalism to pion physics. A signature of pions is that they are derivatively coupled leading to (chiral) zeros in their scattering amplitudes. In this work we explore the multidimensional space of chiral zeros positions, scattering length values, and resonance mass values. Interestingly, we encounter lakes, peninsulas, and kinks depending on which sections of this intricate multidimensional space we consider. We discuss the remarkable location where QCD seems to lie in these plots, based on various experimental and theoretical expectations.

The O(N) S-matrix monolith

“…Finally, in higher dimensions similar unitarity saturation was also observed [8,9]. It would be very interesting to develop the higher dimensional dual problem which should explain this saturation, see also [17][18][19][20][21].…”

“…This clarifies a long standing puzzle. It was thus far stated as a mystery why was unitarity saturated at the boundary of the physical S-matrix space in many different contexts [4,5,8,9,[11][12][13]. This dual problem, with its associated zero duality gap theorems, provides a clean explanation in the two dimensional examples.…”

S-matrix bootstrap: Supersymmetry, Z2 , and Z4 symmetry

“…At this point unitarity is cast in the same spirit of previous S-matrix bootstrap works [5,17]. As in those works, we are splitting the symmetry group into irreducible representations associated with the projectors P þ and P − corresponding to the fundamental and antifundamental representations of the supersymmetry algebra.…”

Dual S-matrix bootstrap. Part I. 2D theory

“…Convergence is perfect when the bound-state mass (measured in units of the lightest mass) is bigger than √ 2 and quite painful otherwise. b) The allowed chiral zeroes space of putative pion S-matrices associated to an SU(2) chiral symmetry breaking patterns draws a beautiful peninsula like object with a sharp tip [2]. 1 Convergence is great almost everywhere except close to the tip where numerics struggle.…”