In this paper we explore practicable ways for self-consistent calculations of spectral functions from analytically continued functional renormalization group (aFRG) flow equations. As a particularly straightforward one we propose to include parametrizations of self-energies based on explicit analytic one-loop expressions. To exemplify this scheme we calculate the spectral functions of pion and sigma meson of the <math><mrow><mi>O</mi><mo>(</mo><mn>4</mn><mo>)</mo></mrow></math> model at vanishing temperature in the broken phase. Comparing the results with those from previous aFRG calculations, we explicitly demonstrate how self-consistency at all momenta fixes the tight relation between particle masses and decay thresholds. In addition, the two-point functions from our new semianalytic FRG scheme have the desired domain of holomorphy built in and can readily be studied in the entire cut-complex frequency plane, on physical as well as other Riemann sheets. This is very illustrative and allows, for example, to trace the flow of the resonance pole of the sigma meson across an unphysical sheet. In order to assess the limitations due to the underlying one-loop structure, we also introduce a fully self-consistent numerical scheme based on spectral representations with scale-dependent spectral functions. The most notable improvement of this numerically involved calculation is that it describes the three-particle resonance decay of an off-shell pion, <math><msup><mi>π</mi><mo>*</mo></msup><mo>→</mo><mi>σ</mi><mi>π</mi><mo>→</mo><mn>3</mn><mi>π</mi></math>. Apart from this further conceptual improvement, overall agreement with the results from the considerably simpler semianalytic one-loop scheme is very encouraging, however. The latter can therefore provide a sound and practicable basis for self-consistent calculations of spectral functions in more realistic effective theories for warm and dense matter.

Germany

Christopher Jung

Jan-Hendrik Otto

Austria

Ralf-Arno Tripolt

Lorenz von Smekal

Self-consistent <math><mi>O</mi><mo>(</mo><mn>4</mn><mo>)</mo></math> model spectral functions from analytically continued functional renormalization group flows

As a first step towards a realistic phenomenological description of vector and axial-vector mesons in nuclear matter, we calculate the spectral functions of the <math><mi>ρ</mi></math> and the <math><msub><mi>a</mi><mn>1</mn></msub></math> meson in a chiral baryon-meson model as a low-energy effective realization of QCD, taking into account the effects of fluctuations from scalar mesons, nucleons, and vector mesons within the functional renormalization group (FRG) approach. The phase diagram of the effective hadronic theory exhibits a nuclear liquid-gas phase transition as well as a chiral phase transition at a higher baryon-chemical potential. The in-medium <math><mi>ρ</mi></math> and <math><msub><mi>a</mi><mn>1</mn></msub></math> spectral functions are calculated by using the previously introduced analytically-continued FRG (aFRG) method. Our results show strong modifications of the spectral functions—in particular near the critical endpoints of both phase transitions—which may well be of relevance for electromagnetic rates in heavy-ion collisions or neutrino emissivities in neutron-star merger events.

Jochen Wambach

Vector and axial-vector mesons in nuclear matter

In this work we study contributions due to vector and axial-vector meson fluctuations to their in-medium spectral functions in an effective low-energy theory inspired by the gauged linear sigma model. In particular, we show how to describe these fluctuations in the effective theory by massive (axial-)vector fields in agreement with the known structure of analogous single-particle or resonance contributions to the corresponding conserved currents. The vector and axial-vector meson spectral functions are then computed by numerically solving the analytically continued functional renormalization group flow equations for their retarded two-point functions at finite temperature and density in the effective theory. We identify the new contributions that arise due to the (axial-)vector meson fluctuations, and assess their influence on possible signatures of a QCD critical end point and the restoration of chiral symmetry in thermal dilepton spectra.