Dispersion relations for unphysical particles

Abstract

Abstract. Generalized dispersion relations are discussed for unphysical particles, e.g. confined degrees of freedom that are not present in the physical spectra but can give rise to observable bound states. While in general the propagator of the unphysical particles can have complex poles and cannot be reconstructed from the knowledge of the imaginary part, under reasonable assumptions the missing piece of information is shown to be in the rational function that contains the poles and must be added to the integral representation. For pure Yang-Mills theory, the rational part and the spectral term are identified in the explicit analytical expressions provided by the massive expansion of the gluon propagator. The multi particle spectral term turns out to be very small and the simple rational part provides, from first principles, an approximate propagator that is equivalent to the tree-level result of simple phenomenological models like the refined Gribov-Zwanziger model.In many interacting theories and, notably, in non-Abelian gauge theories, some of the quantum fields describe confined particles that are not present in real spectra. They can be regarded as internal degrees of freedom of the theory and we can call them unphysical particles in this note, even when they play a very important role in the description of the phenomenology. For instance, gluons and quarks are believed to be confined but their color singlet bound states are observed in physical spectra. More generally, we are interested in the physical class of unphysical particles that give rise to observable bound states [1][2][3].The propagators of these unphysical particles are usually studied in the Euclidean space where the correlators emerge by lattice simulations [4][5][6][7][8][9][10][11][12][13] or by numerical solution of a coupled set of integral equations [14][15][16][17][18][19][20][21][22][23]. They are well interpolated by regular functions on the negative real axis of the squared momentum p 2 = −p 2 E . However, their analytic continuation to the complex plane z = p 2 is not a trivial task at all, because of the ill-defined problem of continuing a limited set of data points [24]. Moreover, if the particle does not appear in the spectra, the general Källen-Lehmann representation does not hold and, in the most studied case of QCD, the numerical data were shown to be not compatible with the standard positivity constraints of the spectral functions. While that is usually regarded as an indirect evidence for confinement, the same argument can question the whole existence of a spectral representation and the meaning of the spectral functions. Actually, on general grounds, the usual dispersion relation between real and imaginary part of the propagator does not hold for a generic unphysical particle, making even more ardue any guess of the analytic properties in Minkowski space.On the other hand, under general physical assumptions, an extension of the standard dispersion relation can be proven, with a rational part that replaces the discrete spe...

Complex poles and spectral function of Yang-Mills theory

“…Since the two-point vertex function has no pole, we conclude that the number of unphysical poles is two. N P = 2 is consistent with [25], which reports that the gluon propagator has a pair of complex conjugate poles.…”

Gluon propagator in linear covariant Rξ gauges

“…[64], even if the simple rational part ∆ P was used in that work for a fit of the lattice data, ignoring the corrections which are included in the present optimized one-loop propagator. In fact, the corrections are gauge dependent and very small below 1 GeV, as already shown in the Landau gauge by a direct evaluation of the spectral function [41,65]. The Schwinger function ∆(t) can be evaluated by a numerical integration, as a function of the Euclidean time t, according to its definition…”

Calculation of the nonperturbative strong coupling from first principles

“…A great advantage of the screened expansion is that, when optimized, it provides explicit analytical expressions which can be continued to the whole complex plane, gaining access to information that cannot be currently obtained by other theoretical tools [58][59][60][61]. At variance with the Curci-Ferrari model which was studied in Refs.…”