Dual EFT Bootstrap: QCD flux tubes

Abstract

We develop a bootstrap approach to Effective Field Theories (EFTs) based on the concept of duality in optimisation theory. As a first application, we consider the fascinating set of EFTs for confining flux tubes. The outcome of our analysis are optimal bounds on the scattering amplitude of Goldstone excitations of the flux tube, which in turn translate into bounds on the Wilson coefficients of the EFT action. Finally, we comment on how our approach compares to EFT positivity bounds.

QFT, EFT and GFT

“…The results we get at n = 3 (90) 13 The results have been obtained by discretising a between − 8 9 to 16 9 with step size 1 211 . For smaller step sizes the results don't change for the first 2 significant digits (Except w 01 and w n0 which were bounded a priori and the full range is realized for them) so we quote these values.…”

“…If so, how do we go about showing this? This question has been investigated by several groups starting with the seminal works [1][2][3][4] followed more recently by [5][6][7][8][9][10][11][12][13]. The typical starting point is to use a fixed-t dispersion relation and examine the constraints imposed by crossing symmetry.…”

“…13 are: −0.089 ≤ w 02 ≤ 0.039, − 0.1318 ≤ w 11 ≤ 0.1054, 0 ≤ w 20 ≤ 0.140625, −0.5625 ≤ w 01 ≤ 1.125, −0.019 ≤ w 12 ≤ 0.011, − 0.026 ≤ w 21 ≤ 0.0105, −0.0059 ≤ w 03 ≤ 0.0071, 0 ≤ w 30 ≤ 0.01977 . (88) As a comparison, if we used −3 as the lower bound, rather than −1, which follows from the Bieberbach conjecture, we would find the following weaker bounds for w 12 , w 21 , w 03 : −0.028 ≤ w 12 ≤ 0.022, −0.026 ≤ w 21 ≤ 0.022, −0.011 ≤ w 03 ≤ 0.0089.…”

Rigorous bounds on the Analytic $S$-matrix

“…The integrand on the right hand side is positive since T v (v, t 0 ) ≥ 0 for any t 0 ≥ 0. From (28) it is evident that maximizing the coupling is equivalent to minimizing the imaginary part (or the total cross-section). On the other hand, when we minimize g 0 the optimal solution will have a total cross-section that is as big as possible compatibly with unitarity.…”

“…A similar -albeit nonlinear -formulation was pioneered long ago in a series of papers [22][23][24][25][26] and used to put rigorous bounds on the π 0 π 0 scattering amplitude in four dimensions. Moreover, an alternative dual formulation has been constructed recently using the Mandelstam representation [27] -see also [9,11,28] for previous examples in two dimensions.…”