Higher order extensions of the Gaussian effective potential

Abstract

A variational method is discussed, extending the Gaussian effective potential to higher orders. The single variational parameter is replaced by trial unknown two-point functions, with infinite variational parameters to be optimized by the solution of a set of integral equations. These stationary conditions are derived by the self-energy without having to write the effective potential, making use of a general relation between self-energy and functional derivatives of the potential. This connection is proven to any order and verified up to second order by an explicit calculation for the scalar theory. Among several variational strategies, the methods of minimal sensitivity and of minimal variance are discussed in some detail. For the scalar theory, at variance with other post-Gaussian approaches, the pole of the second-order propagator is shown to satisfy the simple first-order gap equation that seems to be more robust than expected. By the method of minimal variance, nontrivial results are found for gauge theories containing fermions, where the first-order Gaussian approximation is known to be useless.Comment: Last version: minor changes in the text, and reference updat

Analytic structure of QCD propagators in Minkowski space

“…In more detail, the data for N f = 2 (two light quarks) have been divided by 0.18, while the data for N f = 2 + 1 + 1 (two light quarks and two heavy quarks) are divided by 0.24 and increased by adding the constant 0.45, which is equivalent to a different choice for the constant F 0 in Eq. (28). It is remarkable that the data collapse on the curve predicted by Eqs.…”

“…It is remarkable that the data collapse on the curve predicted by Eqs. (28), (44) with N f = 2 (two chiral quarks) and m = 0.8 GeV. The plot is not very sensitive to the choice of the mass M and a slight change of M is compensated by a minor change of the constant F 0 in Eq.…”

“…[2] and m = 0.73 GeV in Eq. (28). Some deviation occurs for −p 2 > 4 GeV 2 because of the large logs that require a resummation by RG equations in the UV.…”

“…Thus the one-loop ghost dressing function of QCD is the same as that of pure Yang-Mills theory and is given by Eq. (28). The weight of higher order loops can be estimated by comparing the data of lattice simulations in the presence of quenched and unquenched quarks, as the difference can only arise from terms that are neglected in the expansion.…”

“…Most of the non-perturbative approaches that have been developed so far rely on numerical calculations in the Euclidean space, where synergic studies by lattice simulations [2][3][4][5][6][7][8][9][10], Schwinger-Dyson equations [11][12][13][14][15][16][17][18][19][20][21] and variational methods [22][23][24][25][26][27][28][29][30][31][32] have drawn a clear picture for the propagators of QCD deep in the IR.…”