On the Analytic Structure of Scalar Glueball Operators

Abstract

The correlator of the square of the Yang-Mills field-strength tensor corresponds to a scalar glueball, i. e., to a bound-state formed by gluonic ingredients only. It has quantum numbers 0 ++ and its mass, as predicted by different theoretical approaches, is expected to lie between 1 and 2 GeV. Here we restrict our considerations to the Born level, that is, we consider the correlator to zeroth order in the coupling. Gluonic self-interaction is taken into account indirectly by using non-perturbative gluon propagators. The employed closed expressions are motivated by lattice and Dyson-Schwinger studies. The analytic continuation of the integrals themselves is complicated by additional obstructive structures like branch cuts and poles that are induced by the inner integral in the complex plane of the outer integration variable. We deal with this problem by deforming the outer integration contour accordingly. For different input gluon propagators we find a positive glueball spectral density which is required for physical states. Poles are, however, absent which is most likely an artifact of working at Born level.

Real time correlation functions and the functional renormalization group

“…First of all, the regulator necessarily triggers cutoffdependent shifts of the location of the poles evaluated in Section II A. This also extends to possible cuts in the complex plane, see [11,[13][14][15] for an extensive discussion, that have not been discussed in Section II A. However, if these k-dependent cuts are present for the one-loop diagrams in full propagators and vertices, the partial tderivative converts them into poles.…”

“…This includes both, imaginary frequencies or Euclidean spacetime and real frequencies or Minkowski space-time. Such an approach was put forward in the functional renormalisation group (FRG), [5][6][7][8][9][10], for recent work with Dyson Schwinger equations (DSE) see [11][12][13][14][15].…”

Gluon screening mass at finite temperature from the Landau gauge gluon propagator in lattice QCD

“…This is a nontrivial problem that would require, for example, the computation of the spectral density as discussed in [112][113][114]. The analytic structure of the gluon propagator has also been investigated within the Dyson-Schwinger approach in [115][116][117][118][119][120][121][122][123][124]. However, given the approximations involved in the calculation and the difficulty of numerical computation, the outcome of the Dyson-Schwinger equations requires an independent confirmation.…”

How to Determine the Branch Points of Correlation Functions in Euclidean Space

“…The branch point location(s) obtained from this analysis served as a check for the numerics. In [4], the case of complex conjugate masses was studied, while in [5,6] complex conjugate masses as well as a propagator featuring a cut with a singular endpoint were considered. For the complex conjugate poles, all three resulting branch points have been found by this procedure.…”

“…Thereby very general analytic structures of the loop integrands are permitted which allows to employ a large class of propagators known in closed form for perturbative calculations, as well as to extend this to non-perturbative equations which need to be solved self-consistently. In principle this is straightforward, but as we applied this technique to various problems in recent studies [4,5,6] and are using it in an ongoing non-perturbative study of the quark propagator, we will provide details of the technique here. The applicability of Cutkosky's cut rules in Euclidean space is discussed also in [7] using the inverse Stieltjes transformation.…”