Abstract
The renormalized contribution of fermions to the curvature masses of vector and axial-vector mesons is derived with two different methods at leading order in the loop expansion applied to the (2+1)-flavor constituent quark-meson model. The corresponding contribution to the curvature masses of the scalar and pseudoscalar mesons, already known in the literature, is rederived in a transparent way. The temperature dependence of the curvature mass of various (axial-)vector modes obtained by decomposing the curvature mass tensor is investigated along with the (axial-)vector--(pseudo)scalar mixing. All fermionic corrections are expressed as simple integrals that involve at finite temperature only the Fermi-Dirac distribution function modified by the Polyakov-loop degrees of freedom. The renormalization of the (axial-)vector curvature mass allows us to lift a redundancy in the original Lagrangian of the globally symmetric extended linear sigma model, in which terms already generated by the covariant derivative were reincluded with different coupling constants.
Highlights
The extension of the linear sigma model with vector and axial-vector degrees of freedom has a long history
All fermionic corrections are expressed as simple integrals that involve at finite temperature only the Fermi-Dirac distribution function modified by the Polyakov-loop degrees of freedom
Such a calculation within the linear sigma model would allow for a comparison with in-medium properties of the vector mesons obtained with functional renormalization group (FRG) techniques in [12,13,14,15]
Summary
The extension of the linear sigma model with vector and axial-vector degrees of freedom has a long history (see e.g., [1,2,3]). Further investigations in the above-mentioned directions require the calculation of the mesonic and/or fermionic contribution to the (axial-) vector curvature mass matrix and its proper mode decomposition, as was done in many models dealing with the description of hot and/or dense nuclear matter [9,10,11] Such a calculation within the linear sigma model would allow for a comparison with in-medium properties of the (axial-) vector mesons obtained with functional renormalization group (FRG) techniques in [12,13,14,15]. We mention that while our focus here is on the curvature mass, the pole mass and screening mass can be obtained from the analytic expression of the self-energy calculated at nonzero momentum using the usual definitions given in Eq (6) of [18], where the relation between the pole and curvature masses of the mesons was investigated with FRG techniques within the two-flavor quark-meson model. The appendixes not mentioned here contain some further technical aspects used in the calculations
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