Abstract
We present an analytic calculation of the paramagnetic and diamagnetic contributions to the one-loop effective action in the SU(2) Higgs model. The paramagnetic contribution is produced by the gauge boson, while the diamagnetic contribution is produced by the gauge boson and the ghost. In the limit, where these particles are massless, the standard result of - 12 for the ratio of the paramagnetic to the diamagnetic contribution is reproduced. If the mass of the gauge boson and the ghost become much larger than the inverse vacuum correlation lengths of the Yang–Mills vacuum, the value of the ratio goes to - 8 . We also find that the same values of the ratio are achieved in the deconfinement phase of the model, up to the temperatures at which the dimensional reduction occurs.
Highlights
We present an analytic calculation of the paramagnetic and diamagnetic contributions to the one-loop effective action in the SU(2) Higgs model
For the effective action, we will use the known closed-form expression. It can be obtained by using either the standard covariant perturbation theory, or the world-line formalism [25,26].), a terms standing in the pre-exponent [4,25,26]: which corresponds to two Fμν hΓ[ Aμa ]i = −
In the standard Yang–Mills theory, the absolute value of the paramagnetic contribution to the one-loop effective action exceeds the diamagnetic contribution by a factor of 12, which is the origin of the factor of 11 = 12 − 1 in the one-loop coefficient of the Yang–Mills β-function
Summary
We consider the SU(2) Higgs model, whose Euclidean Lagrangian has the form. In the world-line representation for the effective action a T a appears [6,7], where T a is of a gauge boson (which is a spinning particle), an additional term ∝ Fμν a bc abc an SU(2)-generator in the adjoint representation: ( T ) = −iε This term can be recovered by acting on the Wilson loop with the area-derivative operator [8,9] δσ δ (z). For the effective action, we will use the known closed-form expression It can be obtained by using either the standard covariant perturbation theory, or the world-line formalism [25,26].), a terms standing in the pre-exponent [4,25,26]: which corresponds to two Fμν hΓ[ Aμa ]i = −. Note that Equation (5) depends on the vacuum correlation lengths 1/M and 1/M through the two-point a , whose parts contributing to the perimeter and area laws of the Wilson loop correlation function of the Fμν fall off at the distances equal to these lengths
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