Real-time Thermal Field Theory Research Articles

From spin chains to real-time thermal field theory using tensor networks

One of the most interesting directions in theoretical high-energy and condensed-matter physics is understanding dynamical properties of collective states of quantum field theories. The most elementary tool in this quest is retarded equilibrium correlators governing the linear response theory. In this article we examine tensor networks as a way of determining them in a fully ab initio way in a class of (1+1)-dimensional quantum field theories arising as infrared descriptions of quantum Ising chains.We show that, complemented with signal analysis using the Prony method, tensor network calculations for intermediate times provide a powerful way to explore the structure of singularities of the correlator in the complex frequency plane and to make predictions about the thermal response to perturbations in a class of nonintegrable interacting quantum field theories.

Effective field theory for chaotic CFTs

We derive an effective field theory for general chaotic two-dimensional conformal field theories with a large central charge. The theory is a specific and calculable instance of a more general framework recently proposed in [1]. We discuss the gauge symmetries of the model and how they relate to the Lyapunov behaviour of certain correlators. We calculate the out-of-time-ordered correlators diagnosing quantum chaos, as well as certain more fine-grained higher-point generalizations, using our Lorentzian effective field theory. We comment on potential future applications of the effective theory to real-time thermal physics and conformal field theory.

Shear and bulk viscosities of quark matter from quark-meson fluctuations in the Nambu–Jona-Lasinio model

We have calculated the temperature dependence of shear $\eta$ and bulk $\zeta$ viscosities of quark matter due to quark-meson fluctuations. The quark thermal width originating from quantum fluctuations of quark-$\pi$ and quark-$\sigma$ loops at finite temperature is calculated with the formalism of real-time thermal field theory. Temperature-dependent constituent-quark and meson masses, and quark-meson couplings are obtained in the Nambu--Jona-Lasinio model. We found a non-trivial influence of the temperature-dependent masses and couplings on the Landau-cut structure of the quark self-energy. Our results for the ratios $\eta/s$ and $\zeta/s$, where $s$ is the entropy density (also determined in the Nambu--Jona-Lasinio model in the quasi-particle approximation), are in fair agreement with results of the literature obtained from different models and techniques. In particular, our result for $\eta/s$ has a minimum very close to the conjectured AdS/CFT lower bound, $\eta/s = 1/4\pi$.

Shear Viscosity of Pionic and Nucleonic Components from Their Different Possible Mesonic and Baryonic Thermal Fluctuations

Owing to the Kubo relation, the shear viscosities of pionic and nucleonic components have been evaluated from their corresponding retarded correlators of viscous stress tensor in the static limit, which become non-divergent only for the non-zero thermal widths of the constituent particles. In the real-time thermal field theory, the pion and nucleon thermal widths have respectively been obtained from the pion self-energy for different meson, baryon loops and the nucleon self-energy for different pion-baryon loops. We have found a non-monotonic momentum distributions of pion and nucleon thermal widths, which have been integrated out by their respective Bose-enhanced and Pauli-blocked phase space factors during evaluation of their shear viscosities. The viscosity to entropy density ratio for this mixed gas of pion-nucleon system decreases and approaches toward its lower bound as the temperature and baryon chemical potential increase within the relevant domain of hadronic matter.

A Comparative Analysis of In-Medium Spectral Functions for N(940) and N ∗(1535) in Real-Time Thermal Field Theory

In the real-time thermal field theory, the nucleon self-energy at finite temperature and density is evaluated where an extensive set of pion-baryon ($\pi B$) loops are considered. On the other side the in-medium self-energy of $N^*(1535)$ for $\pi N$ and $\eta N$ loops is also determined in the same framework. The detail branch cut structures for these different $\pi B$ loops for nucleon $N(940)$ and $\pi N$, $\eta N$ loops for $N^*(1535)$ are addressed. Using the total self-energy of $N(940)$ and $N^*(1535)$, which contain the contributions of their corresponding loop diagrams, the complete structures of their in-medium spectral functions have been obtained. The Landau and unitary cut contributions provide two separate peak structures in the nucleon spectral function while $N^*(1535)$ has single peak structure in its unitary cuts. At high temperature, the peak structures of both at their individual poles are attenuated while at high density Landau peak structure of nucleon is completely suppressed and its unitary peak structure is tending to be shifted towards the melted peak of $N^*(1535)$. The non-trivial modifications of these chiral partners may indicate some association of chiral symmetry restoration.

Nucleon thermal width owing to pion-baryon loops and its contributions to shear viscosity

In real-time thermal field theory, the standard expression of shear viscosity for nucleonic constituents is derived from the two-point function of nucleonic viscous stress tensors at finite temperature and density. The finite thermal width or Landau damping is traditionally included in the nucleon propagators. This thermal width is calculated from the in-medium self-energy of nucleons for different possible pion-baryon loops. The dynamical part of nucleon-pion-baryon interactions are accounted for by the effective Lagrangian densities of standard hadronic model. The shear viscosity to entropy density ratio of the nucleonic component decreases with the temperature and increases with the nucleon chemical potential. However, adding the contribution of the pionic component, the total viscosity to entropy density ratio also reduces with the nucleon chemical potential when the mixing effect between pion and nucleon components in the mixed gas is considered. Within the hadronic domain, the viscosity to entropy density ratio of the nuclear matter gradually reduces as temperature and nucleon chemical potential increase and therefore the nuclear matter is approaching the (nearly) perfect-fluid state.

Low mass enhanced probability of pion in hadronic matter due to its Landau cut contributions

In the real-time thermal field theory, the pion self-energy at finite temperature and density is evaluated where the different mesonic and baryonic loops are considered. The interactions of pion with the other mesons and baryons in the medium are governed by the effective hadronic Lagrangian densities whose effective strength of coupling constants have been determined from the experimental decay widths of the mesons and baryons. The detail branch cut structures of these different mesonic and baryonic loops are analyzed. The Landau cut contributions of different baryon and meson loops become only relevant around the pion pole and it is completely appeared in presence of medium. The in-medium spectral function of pion has been plotted for different values of temperature, baryon chemical potential as well as three momentum of the pion. A noticeable low mass probability in pion spectral function promise to contribute in the low mass dilepton enhancement via indirect modification of ρ self-energy for loop.

Shear viscosity due to Landau damping from the quark-pion interaction

We have calculated the shear viscosity coefficient $\ensuremath{\eta}$ of the strongly interacting matter in the relaxation time approximation, where a quasiparticle description of quarks with their dynamical masses is considered from the Nambu-Jona-Lasinio model. Due to the thermodynamic scattering of quarks with pseudoscalar-type condensate (i.e., pion), a nonzero Landau damping will be acquired by the propagating quarks. This Landau damping may be obtained from the Landau cut contribution of the in-medium self-energy of a quark-pion loop, which is evaluated in the framework of real-time thermal field theory.

NOISE AND ULTRAVIOLET DIVERGENCES IN SIMULATIONS OF GINZBURG–LANDAU–LANGEVIN TYPE OF EQUATIONS

The time evolution of an order parameter towards equilibrium can be described by nonlinear Ginzburg–Landau (GL) type of equations, also known as time-dependent nonlinear Schrödinger equations. Environmental effects of random nature are usually taken into account by noise sources, turning the GL equations into stochastic equations. Noise sources give rise to lattice-spacing dependence of the solutions of the stochastic equations. We present a systematic method to renormalize the equations on a spatial lattice to obtain lattice-spacing independent solutions. We illustrate the method in approximation schemes designed to treat nonlinear and nonlocal GL equations that appear in real time thermal field theory and stochastic quantization.

Relativistic spectral function of nucleons in hot nuclear matter

We present a simple calculation of the nucleon self-energy in nuclear matter at finite temperature in a relativistic framework, using the real time thermal field theory. The imaginary parts of one-loop graphs are identified with discontinuities across the unitary and the Landau cuts. We find that in general both the cuts contribute significantly to the spectral function in the region of (virtual) nucleon mass usually considered, even though the unitary cut is ignored in the literature. Also our relativistic spectral function differs from the one in non-relativistic approximation, used in some earlier calculations.

Stochastic quantization of real-time thermal field theory

We use the stochastic quantization method to obtain the free scalar propagator of a finite temperature field theory formulated in the Minkowski space-time. First, we use the Markovian stochastic quantization approach to present the two-point function of the theory. Second, we assume a Langevin equation with a memory kernel and a colored noise. The convergence of the Markovian and non-Markovian stochastic processes in the asymptotic limit of the fictitious time is obtained. Our formalism can be the starting point to discuss systems at finite temperature out of equilibrium.

Chemical potentials in real-time thermal field theory

In the functional integral formulation of real-time thermal field theory, a time-dependent canonical transformation of the integration variables can remove the chemical potential from the action. The transformation eliminates the chemical potential from the differential equation satisfied by the propagator, but the chemical potential appears in the transformed boundary conditions and the final result for the perturbative Green's functions is unchanged.

Rigorous theory of nuclear fusion rates in a plasma

Real-time thermal field theory is used to reveal the structure of plasma corrections to nuclear reactions. Previous results are recovered in a fashion that clarifies their nature, and new extensions are made. Brown and Yaffe have introduced the methods of effective quantum field theory into plasma physics. They are used here to treat the interesting limiting case of dilute but very highly charged particles reacting in a dilute, one-component plasma. The highly charged particles are very strongly coupled to this background plasma. The effective field theory proves that this mean field solution plus the one-loop term dominate; higher loop corrections are negligible even though the problem involves strong coupling. Such analytic results for very strong coupling are rarely available, and they can serve as benchmarks for testing computer models.

Search for combinations of thermaln-point functions with analytic extensions

The 2^{n} different n-point functions that occur in real-time thermal field theory are Fourier transformed to real energies. Because of branch cuts in various energy variables, none of these functions can be extended analytically to complex energies. The known linear combinations that form the fully retarded and advanced functions can be extended analytically. It is proven that no other linear combinations have an analytic extension to complex energies.

Neutrino damping rate at finite temperature and density

A first principles derivation is given of the neutrino damping rate in real-time thermal field theory. Starting from the discontinuity of the neutrino self-energy at the two loop level, the damping rate can be expressed as integrals over phase space of amplitudes squared, weighted with statistical factors that account for the possibility of particle absorption or emission from the medium. Specific results for a background composed of neutrinos, leptons, protons, and neutrons are given. Additionally, for the real part of the dispersion relation we discuss the relation between the results obtained from the thermal field theory and those obtained by the thermal average of the forward scattering amplitude.

Two-Point Function Reduction of Four-Point Amputated Functions and Transformations in and RA Basis in a Real-Time Finite Temperature NJL Model**The project partly supported by National Natural Science Foundation of China and by Grant No. LWTZ-1298 of the Chinese Academy of Sciences

Based on a general analysis of Green functions in the real-time thermal field theory, we have proven that the four-point amputated functions in an NJL model in the fermion bubble diagram approximation behave like usual two-point functions. We expound the thermal transformations of the matrix propagators for a scalar bound state in the basis and in the RA basis respectively. The resulting physical causal, advanced and retarded propagators are respectively identical to corresponding ones derived in the imaginary-time formalism, and this shows once again the complete equivalence of the two formalisms of thermal field theory on the discussed problem in the NJL model.

The ω meson at high temperatures

The decays of the omega meson in a heat bath of thermally excited pions is studied within the framework of real-time thermal field theory using an appropriate effective Lagrangian. We show that the omega meson spectrum broadens considerably at temperatures T > 100 MeV, primarily because of omega pi -> pi pi reactions in the thermal environment.

Real-Time Thermal Field Theory Analyses of 2D Gross-Neveu Model**The project partially supported by National Natural Science Foundation of China and by Grant No. LWTZ-1298 of the Chinese Academy of Sciences

The discrete symmetry breaking and possible restoration at finite temperature are analyzed in 2D Gross-Neveu model by the real-time thermal field theory in the fermion bubble approximation. The dynamical fermion mass m is proven to, be scale-independent and this fact indicates the equivalence between the fermion bubble diagram approximation and the meanfield approximation used in the auxiliary scalar field approach. Reproducing of the nonzero critical temperature is the dynamical fermion mass at , shows the equivalence between the real-time and the imaginary-time thermal. field theories in this problem. However, in the real-time formalism, more results including absence of scalar bound state, the equation of criticality curve of chemical potential-temperature and the behavior of at can be easily obtained. The last one indicates the second-order phase transition feature of the symmetry restoration.

Nambu-Goldstone mechanism in real-time thermal field theory

In a one-generation fermion condensate scheme of electroweak symmetry breaking, it is proven based on Schwinger-Dyson equation in the real-time thermal field theory in the fermion bubble diagram approximation that, at finite temperature $T$ below the symmetry restoration temperature $T_c$, a massive Higgs boson and three massless Nambu-Goldstone bosons could emerge from the spontaneous breaking of electroweak group $SU_L(2)\times U_Y(1) \to U_Q(1)$ if the two fermion flavors in the one generation are mass-degenerate, thus Goldstone Theorem is rigorously valid in this case. However, if the two fermion flavors have unequal masses, owing to "thermal flactuation", the Goldstone Theorem will be true only approximately for a very large momentum cut-off $\Lambda$ in zero temperature fermion loop or for low energy scales. All possible pinch singularities are proven to cancel each other, as is expected in a real-time thermal field theory.

Discrete symmetry breaking and restoration at finite temperature in 3D Gross-Neveu model

Dynamical spontaneous breaking of some discrete symmetries including special parities and time reversal and their restoration at finite temperature T are researched in 3D Gross-Neveu model by means of Schwinger-Dyson equation in the real-time thermal field theory in the fermion bubble diagram approximation. When the momentum cut-off Λ is large enough, the equation of critical chemical potential μ c and critical temperature T c will be Λ-independent and identical to the one obtained by auxiliary scalar field approach. The dynamical fermion mass m≡ m( T, μ), as the order parameter of symmetry breaking, has the same ( T c − T) 1/2 behavior at T≲ T c as one in 4D NJL-model and this shows the second-order phase transition feature of the symmetry restoration at T> T c . It is also proven that no scalar bound state could exist in this model.