QFT, EFT and GFT

Abstract

We explore the correspondence between geometric function theory (GFT) and quantum field theory (QFT). The crossing symmetric dispersion relation provides the necessary tool to examine the connection between GFT, QFT, and effective field theories (EFTs), enabling us to connect with the crossing-symmetric EFT-hedron. Several existing mathematical bounds on the Taylor coefficients of Typically Real functions are summarized and shown to be of enormous use in bounding Wilson coefficients in the context of 2-2 scattering. We prove that two-sided bounds on Wilson coefficients are guaranteed to exist quite generally for the fully crossing symmetric situation. Numerical implementation of the GFT constraints (Bieberbach-Rogosinski inequalities) is straightforward and allows a systematic exploration. A comparison of our findings obtained using GFT techniques and other results in the literature is made. We study both the three-channel as well as the two-channel crossing-symmetric cases, the latter having some crucial differences.We also consider bound state poles as well as massless poles in EFTs. Finally, we consider nonlinear constraints arising from the positivity of certain Toeplitz determinants, which occur in the trigonometric moment problem.

Crossing Symmetric Spinning S-matrix Bootstrap: EFT bounds

“…Recently, this representation was explored in [28] in the context of EFT bootstrap. In this section, we will review this dispersion relation and its multi-faceted consequences, which were explored recently in [24,28,43] 8 for scalar amplitudes. We will present the discussion in such a fashion which generalizes naturally to helicity amplitudes that we will be considering in the present work for dealing with spinning particles.…”

“…We also show that the dispersive part of the amplitude can be written as a Typically Real function leading to bounds on the range of the variable a. In general, for massless theories the lower bound on a = a min is zero [24], which only leads to one-sided bounds. We observe that the Wigner-d functions, d m,n ( ξ(s 1 , a)), are positive for all spins when its argument ξ(s 1 , a) is greater than 1.…”

“…These null constraints lead to two-sided bounds on Wilson coefficients. In [23,24] a different consideration was put forth, which makes use of powerful techniques and theorems in an area of mathematics called Geometric Function Theory (GFT) [25,26]. The starting point in this approach makes use of a crossing symmetric dispersion relation (CSDR) [27][28][29].…”

“…The amplitude for identical scalars then has unobvious and interesting properties in terms of the function in the complex z plane. As shown in [24], for a suitable range of the parameter a, for pion scattering, the amplitude is, in the parlance used in Geometric Function Theory, typically real. In other words, in this range of a, it satisfies the condition…”

“…inside the unit disk |z| < 1, which in turn imposes the Bieberbach-Rogosinski (BR) two-sided bounds on the Taylor expansion coefficients of f (z). In terms of Wilson coefficients, an argument based on the Markov brothers' inequality, as shown in [24], leads to two-sided bounds on the ratios of Wilson coefficients. It is known that there is a connection between typically real functions in geometric function theory (GFT) and quantum field theory (QFT) dating back to [30][31][32][33][34].…”