Formalism Of Thermal Field Theory Research Articles

Building imaginary-time thermal field theory with artificial neural networks* *The work of this research is supported by the National Natural Science Foundation of China (12375131(YJ), 12375136(LH)), the CUHK-Shenzhen university development Fund (UDF01003041) and the BMBF funded KISS consortium (05D23RI1) in the ErUM-Data action plan (KZ). L. Wang also thanks the National Natural Science Foundation of China (12147101) for supporting his

In this paper, we introduce a novel approach in quantum field theories to estimate actions using artificial neural networks (ANNs). The actions are estimated by learning system configurations governed by the Boltzmann factor, , at different temperatures within the imaginary time formalism of thermal field theory. Specifically, we focus on the 0+1 dimensional quantum field with kink/anti-kink configurations to demonstrate the feasibility of the method. Continuous-mixture autoregressive networks (CANs) enable the construction of accurate effective actions with tractable probability density estimation. Our numerical results demonstrate that this methodology not only facilitates the construction of effective actions at specified temperatures but also adeptly estimates the action at intermediate temperatures using data from both lower and higher temperature ensembles. This capability is especially valuable for detailed exploration of phase diagrams.

Mass and spectral function of scalar and pseudoscalar mesons in a hot and chirally imbalanced medium using the two-flavor NJL model

We explore the properties of neutral mesons within the context of a chirally imbalanced medium, employing the two-flavor Nambu–Jona-Lasinio model. The temperature dependence of the constituent quark mass at finite values of the chiral chemical potential (CCP) demonstrates the well-established phenomena of chiral catalysis at lower temperatures and inverse chiral catalysis at higher temperatures. The polarization functions in both the scalar (σ) and pseudoscalar (π0) channels have been evaluated using the real-time formalism of thermal field theory. These have been used to determine the masses and spectral functions of σ and π mesons. Detailed investigation of the analytic structure of the imaginary part of the polarization function for σ and π mesons results in the emergence of nontrivial Landau-cut contributions due to the presence of chiral imbalance. The multiple solutions for the mass of the π meson for specific values of CCP have been analyzed on the basis of their residue at the pole. Furthermore, we have observed abrupt changes in the masses of both scalar and pseudoscalar mesons at finite CCP values, particularly at higher temperatures. A decreasing trend in the Mott transition temperature is seen with the increase in CCP. Published by the American Physical Society 2024

Non-Abelian electric field correlator at NLO for dark matter relic abundance and quarkonium transport

We perform a complete next-to-leading order calculation of the non-Abelian electric field correlator in a SU(Nc) plasma, which encodes properties of the plasma relevant for heavy particle bound state formation and dissociation, and is different from the correlator for the heavy quark diffusion coefficient. The calculation is carried out in the real-time formalism of thermal field theory and includes both vacuum and finite temperature contributions. By working in the Rξ gauge, we explicitly show the results are gauge independent, infrared and collinear safe. The renormalization group equation of this electric field correlator is determined by that of the strong coupling constant. Our next-to-leading order calculation can be directly applied to any dipole singlet-adjoint transition of heavy particle pairs. For example, it can be used to describe dissociation and (re)generation of heavy quarkonia inside the quark-gluon plasma well below the melting temperature, as well as heavy dark matter pairs (or charged co-annihilating partners) in the early universe.

Electromagnetic spectral function and dilepton rate in a hot magnetized QCD medium

The dilepton production rate in hot QCD medium is studied within a effective description of the medium in the presence of magnetic field. This could be done by obtaining the one-loop self energy of photon due to the effective (quasi-) quark loop at finite temperature under an arbitrary external magnetic field while employing the real time formalism of Thermal Field Theory. The effective quarks and gluons encode hot QCD medium effective in terms of their respective effective fugacities. The magnetic field enters in the form of landau level quantization, in the matter sector (quarks, antiquarks). The full Schwinger proper time propagator including all the Landau levels is considered for the quasi quarks while calculating the photon self energy. The electromagnetic Debye screening (in terms of the self-energy) has seen to be influenced both by the hot QCD medium effects and magnetic field. Analogous results are also obtained from the semi classical transport theory. The imaginary part of the photon self energy function is obtained from the discontinuities of the self energy across the Unitary cuts which are also present at zero magnetic field and the Landau cuts which are purely due to the magnetic field. The dilepton production rate is then obtained in terms of the product of electromagnetic spectral functions due to quark loop and lepton loop. The modifications of both the quarks/antiquarks as well as leptons in presence of an arbitrary external magnetic field have been considered in the formalism. Significant enhancement of the low invariant mass dileptons due the appearance of the Landau cuts in the electromagnetic spectral function at finite external magnetic field has been observed. A substantial enhancement of dilepton rate is also found when the EOS effects are considered through the effective quarks/antiquraks.

Unified formalism for Thermal Quantum Field Theories: A geometric viewpoint

In this paper we study a unified formalism for Thermal Quantum Field Theories, i.e., for the Matsubara approach, Thermo Field Dynamics and the Path Ordered Method. To do so, we employ a mechanism akin to the Hawking effect which explores a relationship between the concept of temperature and spacetimes endowed with event-horizons. In particular, we consider an eight dimensional static spacetime, the so-calledη–ξ spacetime, which we show to form an appropriate geometric background for generic Thermal Quantum Field Theories. Within this framework, the different formalisms of Thermal Field Theory are unified in a very natural way via various analytical continuations and the set of time-paths used in the Path Ordered Method is interpreted in geometric terms. We also explain reported inconsistencies inherent in the Thermo Field Dynamics through the appearance of horizons (and ensuing loss of information) in theη–ξ spacetime.

Thermal effects on ρ meson properties in an external magnetic field

A detailed study of the analytic structure of 1-loop self energy graphs for neutral and charged $\rho$ mesons is presented at finite temperature and arbitrary magnetic field using the real time formalism of thermal field theory. The imaginary part of the self energy is obtained from the discontinuities of these graphs across the Unitary and Landau cuts, which is seen to be different for $\rho^0$ and $\rho^\pm$. The magnetic field dependent vacuum contribution to the real part of the self energy, which is usually ignored, is found to be appreciable. A significant effect of temperature and magnetic field is seen in the self energy, spectral function, effective mass and dispersion relation of $\rho^0$ as well as of $\rho^\pm$ relative to its trivial Landau shift. However, for charged $\rho$ mesons, on account of the dominance of the Landau term, the effective mass appears to be independent of temperature. The trivial coupling of magnetic moment of $\rho^\pm$ with external magnetic field, when incorporated in the calculation, makes the $\rho^\pm$ to condense at high magnetic field.

Mass modification of hot pions in a magnetized dense medium

A phenomenological pion-nucleon interaction is used to obtain pionic mass modification in presence of constant homogeneous magnetic field background at finite temperature and chemical potential in the real time formalism of thermal field theory. The magnetically modified propagator in its complete form is used to obtain the one loop self-energy for pions. For charged pions we find that the effective mass increases with the magnetic field at given temperature and chemical potential. Since the transverse momentum of charged pion is quantized and its contribution to Dyson-Schwinger Equation is large compared to the loop correction, the charged pion mass remains constant with both temperature and chemical potential for a given landau level. In order to unveil the role of the real part of the self-energy, we also calculate the effective mass neglecting the trivial shift. The effective mass for charged pions shows an oscillatory behavior which is attributed to the thermal contribution of the self-energy. It is argued that the magnetic field dependent vacuum contribution to the self-energy influences the behavior of the effective mass both qualitatively and quantitatively. We also find that very large field is necessary for neutral pions to condense.

Δself-energy at finite temperature and density and theπNcross section

The self energy of $\Delta$-baryon is evaluated at finite temperature and density using the real time formalism of thermal field theory. The Dyson-Schwinger equation is used to get the exact thermal propagator followed by the spectral function of $\Delta$. The $\pi N$ scattering cross section obtained using explicit $\Delta$ exchange is normalized to the experimental data in vacuum and its medium modification is implemented by means of the exact thermal propagator. A significant suppression of the peak of the cross-section is observed at higher temperature and baryon density. Effects on the mean relaxation time of nucleons and the temperature dependence of the shear viscosity of a pion nucleon gas are demonstrated.

Thermal conductivity of hot pionic medium due to pion self-energy for πσ and πρ loops

The thermal conductivity of pionic medium has been evaluated with the help of its standard expression from the relaxation time approximation, where inverse of pion relaxation time or pion thermal width has been obtained from the imaginary part of pion self-energy. In the real-time formalism of thermal field theory, the finite temperature calculations of pion self-energy for πσ and πρ loops have been done. The numerical value of our thermal conductivity increases with temperature very softly, though at particular temperature, our estimation has to consider a large band of phenomenological uncertainty.

Effect of in-medium spectral density of D and mesons on the dissociation in hadronic matter

The one-loop self-energy of the D and D ⁎ mesons in a hot hadronic medium is evaluated using the real time formalism of thermal field theory. The interaction of the heavy open-charm mesons with the thermalized constituents ( π , K , η ) of the hadronic matter is treated in the covariant formalism of heavy meson chiral perturbation theory. The imaginary parts are extracted from the discontinuities of the self-energy function across the unitary and the Landau cuts. The non-zero contribution from the latter to the spectral density of D and D ⁎ mesons opens a number of subthreshold decay channels of the J / ψ leading to a significant increase in the dissociation width in hadronic matter.

Observing many-body effects on lepton pair production from low mass enhancement and flow at RHIC and LHC energies

The ρ spectral function at finite temperature calculated using the real-time formalism of thermal field theory is used to evaluate the low mass dilepton spectra. The analytic structure of the ρ propagator is studied and contributions to the dilepton yield in the region below the bare ρ peak from the different cuts in the spectral function are discussed. The space-time integrated yield shows significant enhancement in the region below the bare ρ peak in the invariant mass spectra. It is argued that the variation of the inverse slope of the transverse mass (M T ) distribution can be used as an efficient tool to predict the presence of two different phases of the matter during the evolution of the system. The sensitivities of the effective temperature obtained from the slopes of the M T spectra to the medium effects are studied.

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.

Explicit Proof of Equivalence of Two-Point Functions in the Two Formalisms of Thermal Field Theory**The project partially supported by National Natural Science Foundation of China and Grant No. LWTZ-1298 of the Chinese Academy of Sciences

We give an explicit proof of equivalence of the two-point function to one-loop order in the two formalisms of thermal theory based on the expressions in the real-time formalism and indicate that the key point of completing the proof is to separate carefully the imaginary part of the zero-temperature loop integral from relevant expressions and this fact will certainly be very useful for examination of the equivalent problem of two formalisms of thermal field theory in other theories, including the one of the propagators for scalar bound states in an NJL model.

Nambu-Goldstone mechanism at finite temperature in the imaginary-time and real-time formalism

In the imaginary-time formalism of thermal field theory, and also in the real-time formalism, but by means of some redefined physical propagators for scalar bound states by diagonalization of four-point function matrices, we reexamine the Nambu-Goldstone mechanism of electroweak symmetry breaking in a one-generation fermion condensate scheme, based on the Schwinger-Dyson equation in the fermion bubble diagram approximation, and compare the obtained results. We have reached the conclusion that in both formalisms the Goldstone theorem of spontaneous electroweak symmetry breaking is rigorously true for the case of mass-degenerate two flavors of fermions and only approximately valid at low energy scales for the mass-nondegenerate case, in spite of the existence of some differences between the two formalisms in the imaginary parts of the denominators of the propagators for scalar bound states. When the two flavors of fermions have unequal nonzero masses, the induced possible fluctuation effect for the Higgs particle is negligible if the momentum cutoff in the zero temperature loops is large enough. All the results show physical equivalence of the two formalisms in the present discussed problems.

Hydrodynamic transport functions from quantum kinetic field theory

Starting from the quantum kinetic field theory [E. Calzetta and B. L. Hu, Phys. Rev. D 37, 2878 (1988)] constructed from the closed-time-path (CTP), two-particle-irreducible (2PI) effective action we show how to compute from first principles the shear and bulk viscosity functions in the hydrodynamic-thermodynamic regime. For a real scalar field with $\ensuremath{\lambda}{\ensuremath{\Phi}}^{4}$ self-interaction we need to include four-loop graphs in the equation of motion. This work provides a microscopic field-theoretical basis to the ``effective kinetic theory'' proposed by Jeon and Yaffe [S. Jeon and L. G. Yaffe, Phys. Rev. D 53, 5799 (1996)], while our result for the bulk viscosity reproduces their expression derived from linear-response theory and the imaginary-time formalism of thermal field theory. Though unavoidably involved in calculations of this sort, we feel that the approach using fundamental quantum kinetic field theory is conceptually clearer and methodically simpler than the effective kinetic theory approach, as the success of the latter requires a clever rendition of diagrammatic resummations which is neither straightforward nor fail-safe. Moreover, the method based on the CTP-2PI effective action illustrated here for a scalar field can be formulated entirely in terms of functional integral quantization, which makes it an appealing method for a first-principles calculation of transport functions of a thermal non-Abelian gauge theory, e.g., QCD quark-gluon plasma produced from heavy ion collisions.

Effective Two-Loop Thermodynamic Potential with Fermions in the Real-Time Formalism of Thermal Field Theory**The project supported by National Natural Science Foundation of China

Within the real-time formalism of thermal field theory, we apply the hard thermal loop resummation technique to calculate effective two-loop thermodynamic potential in quark-gluon plasma and its renormalization. The result with collective effects is obtained, which is valid for an arbitrary number of quark flavors with masses.

UL(N) × UR(N)-Invariant Four-Fermion Interactions and Nambu–Goldstone Mechanism at Finite Temperature**The project supported partially by National Natural Science Foundation of China and by Grant No. LWTZ-1298 of the Chinese Academy of Sciences

In a chiral UL(N) × UR(N) fermion model of NJL-form, we prove that, if all the fermions are assumed to have equal masses and equal chemical potentials, then at the finite temperature T below the symmetry restoration temperature Tc, there will be N2 massive scalar composite particles and N2 massless pseudoscalar composite particles (Nambu–Goldstone bosons). This shows that the Goldstone theorem at finite temperature for spontaneous symmetry breaking UL(N) × UR(N) → UL+R(N) is valid and consistent with the real-time formalism of thermal field theory in this model.

Top-Quark Condensate at Finite Temperature and Electroweak Symmetry Restoration**The project supported partially by National Natural Science Foundation of China and by Grant No. LWTZ-1298 of the Chinese Academy of Sciences

The gap equation at finite temperature in the top-quark condensate scheme of electroweak symmetry breaking is proven to have the identical form in both the imaginary and real time formalisms of thermal field theory. By means of the gap equation, combined with the basic relation to define the vacuum expectation value of the effective Higgs field, we analyze the dependence on temperature and chemical potential of the dynamical top-quark mass as the order parameter characteristic of symmetry breaking, and obtain the criticality curve for symmetry restoration. We find out that the critical temperature for and the critical chemical potential for When , the top-quark mass near has the leading behavior with an extra factor dependent on temperature and the momentum cutoff . However, it is generally argued that the symmetry restoration at is still a second-order phase transition.

Reexamination of the path-ordered approach to real time thermal field theory.

We argue that the asymptotic condition should not be applied in the derivation of the real time formalism of thermal field theory. It is shown that, contrary to popular belief, the generating functional of time-ordered Green functions does not factorize. Despite this, we then show why the normal two-component Feynman rules are sufficient for most Green function calculations. In addition, the extra Feynman rule, which is applied when calculating vacuum diagrams, is simply derived. We also clear up any doubts regarding the equivalence of the real and imaginary time formalisms. Finally, we consider these results in the case of the new real time contour.

Diagonization of Dyson series in the closed-time path formalism of thermal field theory

It is found that, similar to thermo field dynamics, there exist as well Bogoliubov matrices to diagonize the Dyson series in the closed-time path formalism of thermal field theory, contrary to the comment in recent literature that a meaningful diagonization of propagators cannot be achieved in this theory.