The Breakdown of Resummed Perturbation Theory at High Energies

Abstract

Calculations of high-energy processes involving the production of a large number of particles in weakly-coupled quantum field theories have previously signaled the need for novel nonperturbative behavior or even new physical phenomena. In some scenarios, already tree-level computations may enter the regime of large-order perturbation theory and therefore require a careful investigation. We demonstrate that in scalar quantum field theories with a unique global minimum, where suitably resummed perturbative expansions are expected to capture all relevant physical effects, perturbation theory may still suffer from severe shortcomings in the high-energy regime. As an example, we consider the computation of multiparticle threshold amplitudes of the form 1 → n in ϕ 6 theory with a positive mass term, and show that they violate unitarity of the quantum theory for large n, even after the resummation of all leading-n quantum corrections. We further argue that this is a generic feature of scalar field theories with higher-order self-interactions beyond ϕ 4 , thereby rendering the latter unique with respect to its high-energy behavior.

Multiparticle Amplitudes in a Scalar EFT

“…In fact, the exponentiation has to be replaced by a more complicated combination of functions, all of which are generated by separate matrix elements. Evidence for this has already emerged in [27], where different complex branches of the classical background field were identified that cancel quantum contributions at certain loop orders in order to account for the difference in their generating functions.…”

“…This property can be used to simplify the calculation of quantum corrections tremendously, because we can now consider an analogue of the quantum effective action as follows. At each quantum-loop order, any tadpole diagram can be converted into a tree graph through appropriately cutting all internal propagators [11] (see also [27,35]). Any additional "external leg" obtained through this procedure can then be assigned half a power of the propagator and evaluated at tree-level, while carefully accounting for the different symmetry factors.…”

“…In order to illustrate the problem, let us consider a toy theory of a real scalar field with a sextic self-interaction, as has been used in [27],…”

“…We will also explain how this result is extended to the generic EFT case. The expression (1.1) is an immediate generalization of the purely quartic [11,26] and the purely sextic cases [27]. 3 Here, the sign of the exponent A is crucial.…”

Numerical study of multiparticle production in $ϕ^4$ theory: comparison with analytic results

“…A considerable revival of the interest in multiparticle processes occurred recently [6][7][8][9][10][11] when Ref. [12] suggested that, contrary to Eq.…”