S-matrix bootstrap in 3+1 dimensions: regularization and dual convex problem

Abstract

The S-matrix bootstrap maps out the space of S-matrices allowed by analyticity, crossing, unitarity, and other constraints. For the 2 → 2 scattering matrix S2→2 such space is an infinite dimensional convex space whose boundary can be determined by maximizing linear functionals. On the boundary interesting theories can be found, many times at vertices of the space. Here we consider 3 + 1 dimensional theories and focus on the equivalent dual convex minimization problem that provides strict upper bounds for the regularized primal problem and has interesting practical and physical advantages over the primal problem. Its variables are dual partial waves kℓ(s) that are free variables, namely they do not have to obey any crossing, unitarity or other constraints. Nevertheless they are directly related to the partial waves fℓ(s), for which all crossing, unitarity and symmetry properties result from the minimization. Numerically, it requires only a few dual partial waves, much as one wants to possibly match experimental results. We consider the case of scalar fields which is related to pion physics.

Bootstrapping 2d ϕ4 theory with Hamiltonian truncation data

“…LCT can be applied to theories with more general field content in 2d, including gauge fields and fermions, and 2d QCD at finite N c would be a particularly interesting application. 18 Our approach here is similar in spirit to that of [26], which studied Ising Field theory with both a σ and deformation using TFFSA and Luscher's method [62,63], but it would be interesting to see if any more mileage could be gained by also including form factors and spectral densities in a generalized unitarity condition as we did in this paper. More ambitiously, our method in principle can be applied to higher dimensions, the main challenge being that it is difficult to obtain the input data.…”

“…In this paper we will refer to it as the numerical S-matrix bootstrap. The S-matrix bootstrap gained further attention in recent years, see [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. The recent work [23,24] made a concrete proposal for how to extended the S-matrix bootstrap to accommodate form factors and spectral densities.…”

Dual EFT bootstrap: QCD flux tubes

“…When the minimal value found from the Primal approach and the maximal of Dual approach touch each other, indicated with a dashed line above, the duality gap is closed. The concept of duality in optimisation theory has been successfully applied to bound the space of O(N ) models [37] and the couplings of bound states [38] in two spacetime dimensions, and quartic couplings in four spacetime dimensions [39,40]. 2 The logic of the dual S-Matrix bootstrap approach resembles that of the CFT bootstrap [45], were kinks and island are found [46][47][48] after excluding allowed values of the operator's scaling dimensions.…”

Positivity and geometric function theory constraints on pion scattering

“…• It will be appealing to see the implications of our bounds and sum rules (2.21) to S-matrix bootstrap for pion amplitudes [36,37,54] and to the dual the S-matrix bootstrap [55][56][57][58]. n,r,m (µ, δ), χ (1) n,r,m (µ, δ), χ (2) n,r,m (µ, δ) First, we take the combinations…”