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Thermalization and dynamical spectral properties in the quark-meson model

Linda Shen, Jürgen Berges, Jan M. Pawlowski, and Alexander Rothkopf
Phys. Rev. D 102, 016012 – Published 16 July 2020
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Abstract
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Article Text
  • INTRODUCTION
  • THE QUARK-MESON MODEL
  • SPECTRAL FUNCTIONS
  • THE MACROSCOPIC FIELD
  • SUMMARY AND CONCLUSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We investigate the nonequilibrium evolution of the quark-meson model using two-particle irreducible effective action techniques. Our numerical simulations, which include the full dynamics of the order parameter of chiral symmetry, show how the model thermalizes into different regions of its phase diagram. In particular, by studying quark and meson spectral functions, we shed light on the real-time dynamics approaching the crossover transition, revealing, e.g., the emergence of light effective fermionic degrees of freedom in the infrared. At late times in the evolution, the fluctuation-dissipation relation emerges naturally among both meson and quark degrees of freedom, confirming that the simulation approaches thermal equilibrium.

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    • Received 14 March 2020
    • Accepted 8 June 2020

    DOI:https://doi.org/10.1103/PhysRevD.102.016012

    Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

    Published by the American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Effective field theoryFinite temperature field theorySpontaneous symmetry breaking
    Particles & Fields

    Authors & Affiliations

    Linda Shen1,*, Jürgen Berges1, Jan M. Pawlowski1, and Alexander Rothkopf2

    • 1Heidelberg University, Institute for Theoretical Physics, Philosophenweg 16, 69120 Heidelberg, Germany
    • 2University of Stavanger, Faculty of Science and Technology, Kristine Bonnevies vei 22, 4036 Stavanger, Norway

    • *shen@thphys.uni-heidelberg.de

    Article Text

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    Vol. 102, Iss. 1 — 1 July 2020

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    Images

    • Figure 1
      Figure 1

      2PI diagrams at NLO in 1/N and g. Full lines represent boson propagators, crossed circles macroscopic field insertions, and dashed lines fermion propagators. The first two-loop diagram in the first row corresponds to the leading-order contribution in 1/N. The last diagram in the second row shows the fermion boson loop. The other diagrams in the first and second rows depict the infinite series of NLO diagrams in 1/N.

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    • Figure 2
      Figure 2

      Sketch of the setup deployed in this study. We consider the real-time evolution from nonequilibrium initial states characterized by an energy density sourced either through a finite σ field expectation value (blue circle) or a nonzero occupancy of fermionic modes (orange triangle). Depending on the initial energy contained in the system, one of three discernible final states, the chiral broken phase, the crossover regime, or the (almost) symmetric phase is approached.

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    • Figure 3
      Figure 3

      We show the time evolution of the effective particle number defined in (16) for bosonic and fermionic components (rows) and two different momenta (left and right columns). At late times (red curve), the effective particle number becomes time and momentum independent and approaches the shape of a Bose-Einstein and Fermi-Dirac distribution, respectively. The shown data are interpolated using a cubic spline. Dimensionful quantities are given in units of the pseudocritical temperature Tpc (cf. Fig. 19).

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    • Figure 4
      Figure 4

      Left: the generalized Boltzmann exponents defined in (17) shown as a function of frequency ω at a given momentum |p| for bosonic and fermionic components. For better visibility, only every 39th data point is shown. Using a linear fit, one can determine the slope β and hence the temperature T for each component. The temperature T indicated in the plot is averaged over all momenta and the three components. Right: the relative deviation from the thermalization temperature Δ=(Ti−T)/T shown for all three components as a function of momentum. The results for the bosonic and fermionic sectors agree very well. In both plots, dimensionful quantities are given in units of the pseudocritical temperature Tpc (cf. Fig. 19).

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    • Figure 5
      Figure 5

      A representative selection of spectral functions from the fermion vector-zero channel in the infrared (top row) and the UV (bottom row) in three different regimes labeled by the temperatures of their final state. Each panel contains four curves indicating different snapshots along the thermalization trajectory. All three simulations employ field dominated initial conditions, i.e., σ0>0 and n0=0. Dimensionful quantities are given in units of the pseudocritical temperature Tpc (cf. Fig. 19).

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    • Figure 6
      Figure 6

      Time evolution of the pion spectral and statistical functions shown for two different initial conditions at the smallest available momentum |p|=0.012. The left column shows a simulation deploying field dominated initial conditions with σ0=1.36, the right column fluctuation dominated initial conditions with n0=0.8. Both simulations lead to thermal states at temperatures where chiral symmetry is restored. Dimensionful quantities are given in units of the pseudocritical temperature Tpc (cf. Fig. 19).

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    • Figure 7
      Figure 7

      Time evolution of the dispersion relation and the momentum-dependent width of the pion. The inset shows the time evolution of pion mass obtained from fits of the dispersion relation to Z|p|2+mπ2 at various times τ, where Z=1.07 is obtained for all times analyzed. The data are shown for field dominated initial conditions with σ0=1.24 and n0=0. Dimensionful quantities are given in units of the pseudocritical temperature Tpc (cf. Fig. 19).

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    • Figure 8
      Figure 8

      Time evolution of the pion mass and the pion width in the limit |p|→0. Results are shown for field dominated initial conditions with σ0=1.36 and n0=0 (blue dots) as well as fluctuation dominated initial conditions with σ0=0 and n0=0.8 (orange triangles). Dimensionful quantities are given in units of the pseudocritical temperature Tpc (cf. Fig. 19).

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    • Figure 9
      Figure 9

      Time evolution of the vector component quark spectral and statistical functions shown for the same initial conditions as in Fig. 6 at momentum |p|=0.016. Dimensionful quantities are given in units of the pseudocritical temperature Tpc (cf. Fig. 19).

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    • Figure 10
      Figure 10

      Time evolution of the vector-zero component quark spectral and statistical functions shown for two different initial conditions at momentum |p|=0.012. The left column shows field dominated initial conditions with σ0=0.98, the right column fluctuation dominated initial conditions with n0=0.11. Both lead to the same late-time state with T=1.04. Dimensionful quantities are given in units of the pseudocritical temperature Tpc (cf. Fig. 19).

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    • Figure 11
      Figure 11

      Time evolution of the vector component quark spectral and statistical functions shown for two different initial conditions at momentum |p|=0.012. The left column shows field dominated initial conditions with σ0=0.98, the right column fluctuation dominated initial conditions with n0=0.11. Both lead to the same late-time state with T=1.04. Dimensionful quantities are given in units of the pseudocritical temperature Tpc

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    • Figure 12
      Figure 12

      Left: temperature dependence of the characteristic decay momentum Q shown for the σ and π mesons. The inset shows examples for the momentum-dependent width at high and low temperatures. Q corresponds to the momentum at which the width Γ(τ,|p|) is maximal. Right: temperature dependence of quasiparticle masses. Restoration of chiral symmetry is reflected in identical masses of the σ and π mesons at high temperatures. The quark q quasiparticle mass is obtained from the dominant peak of the vector-zero component quark spectral function. We also plot the “plasmino” branch p obtained from the quark spectral function. In both plots, gray lines show cubic spline interpolations of the data points. Dimensionful quantities are given in units of the pseudocritical temperature Tpc (cf. Fig. 19).

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    • Figure 13
      Figure 13

      The dispersion relation of the vector-zero quark spectral function shown for three different temperatures. At low temperature, a fit to the relativistic dispersion relation Z|p|2+mq2 is shown by the black dashed line. For higher temperatures, the behavior at small and large momenta differs as the additional low-frequency peak and the main peak merge into one peak. We perform separate fits at low and high momenta, shown by the dashed and dotted black lines. Dimensionful quantities are given in units of the pseudocritical temperature Tpc (cf. Fig, 19).

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    • Figure 14
      Figure 14

      The vector-zero quark spectral function as a function of frequency ω shown for a range of spatial momenta |p|. The three plots correspond to the same three temperatures as in Fig. 13. The purple line indicates peak position of the spectral function in the |p|−ω plane and is therefore equivalent to the dispersion relation shown in Fig. 13. The spectral function reveals a narrow quasiparticle peak at low temperatures. As the temperature is increased, the light mode interferes with the low-momentum spectral function, leading to a broad peak at small momenta. At high momenta, the quasiparticle peak remains narrow. Dimensionful quantities are given in units of the pseudocritical temperature Tpc (cf. Fig. 19).

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    • Figure 15
      Figure 15

      The thermalized scalar component of the quark spectral function as a function of frequency shown for different temperatures. Dimensionful quantities are given in units of the pseudocritical temperature Tpc (cf. Fig. 19).

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    • Figure 16
      Figure 16

      Left: the time evolution of the field shown for field dominated initial conditions with different initial field values as indicated in the legend. Right: the time evolution of the field shown for field dominated initial conditions with σ0=1.98 (blue) and fluctuation dominated initial conditions with n0=0.11 (orange). The same late-time field value σ¯=0.33 is approached for both initial states. The gray line in both plots serves as a guide to the eye for ⟨σ(t)⟩=0. Dimensionful quantities are given in units of the pseudocritical temperature Tpc (cf. Fig. 19).

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    • Figure 17
      Figure 17

      The energy density at initial time ϵinit=ϵ(t=0) and at late times ϵth=ϵ¯ as a function of the initial field value. We present the classical, bosonic, and fermionic contributions to the initial energy density separately. Together, they form a bounded shape with minimum at a nonzero initial field value (gray curve). At late times, the energy density reaches the constant shape shown by ϵtherm in black. The minimum of the energy density at late times corresponds to the maximal field values found. The initial field is given in units of the pseudocritical temperature Tpc (cf. Fig. 19).

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    • Figure 18
      Figure 18

      The value of the thermalized one-point function for different initial conditions. On the right, the thermal field value σ¯ is shown for initial conditions with different field values σ0. The gray dashed line indicates σ¯=σ0. On the left, the thermal field value is shown for different initial fermion occupation numbers n0. In both plots, the black star indicates the value obtained for initial conditions with n0=0 and σ0=0. In both plots, gray lines show cubic spline interpolations of the data points. Dimensionful quantities are given in units of the pseudocritical temperature Tpc (cf. Fig. 19).

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    • Figure 19
      Figure 19

      Order parameters of the quark-meson model as a function of temperature. In the upper plot, the order parameter is given by the macroscopic field σ¯ which is the thermalized value of the one-point function. In the lower plot, the order parameter is given by the ratio of the σ-meson and pion masses. The masses are derived from the two-point functions of the corresponding bosonic fields. The gray lines show cubic spline fits to the data points. The inflection points are indicated by the black vertical lines. Dimensionful quantities are given in units of the pseudocritical temperature Tpc defined as the inflection point Tinflection of the order parameter σ¯ shown in the upper plot.

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    • Figure 20
      Figure 20

      Left: the time evolution of the field with initial value σ0=0.62 shown for three different bare fermion masses mψ. The field reaches the stationary value σ¯ at late times. Right: the asymptotic field value σ¯ shown for different bare fermion masses mψ. The green, red, and black data points correspond to the simulations shown in the left plot. The field value decreases with the fermion bare mass and approaches an asymptotic value for mψ→0. Dimensionful quantities are given in units of the pseudocritical temperature Tpc (cf. Fig. 19).

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