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Thermal Field Theory
Abstract
Now in paperback, this text introduces the theoretical framework for describing the quark-gluon plasma, an important new state of matter. The first part of this book is a self-contained introduction to relativistic thermal field theory. Topics include the path integral approach, the real and the imaginary time formalisms, fermion fields and gauge fields at finite temperature. Useful techniques such as the evaluation of frequency sums or the use of cutting rules are illustrated on various examples. The second part of the book is devoted to recent developments, giving a detailed account of collective excitations (bosonic and fermionic), and showing how they give rise to energy scales which imply a reorganization of perturbation theory. The relation with kinetic theory is also explained. Applications to processes which occur in heavy ion collisions and in astrophysics are worked out in detail. Each chapter ends with exercises and a guide to the literature.
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“A summary of the results of these efforts mainly through the Pb + Pb experiments at SPS-CERN provided compelling evidence in favor of the existence of a new state of matter[23].”
From Phase Transitions in the Early and Present Universe (2006) · cites this paper as [23]
In the citing paper
“The excess of dileptons in the small invariant mass region cannot be explained by charged pion annihilation [23].”
From Phase Transitions in the Early and Present Universe (2006) · cites this paper as [23]
In the citing paper
“We summarize the observables that have been proposed as telltales of a QGP and the data gathered by the SPS from Pb + Pb and Pb + Au collisions, which taken together provide a hint of evidence [22, 23, 120, 129] for a QGP, although many of them could have alternative explanations.”
From Phase Transitions in the Early and Present Universe (2006) · cites this paper as [23]
In the citing paper
“The current understanding of these features is that the medium effects result in a shift in the ρ meson mass as well as a change in its width[23, 120, 135].”
From Phase Transitions in the Early and Present Universe (2006) · cites this paper as [23]
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Items 201–300 of 899
Vacuum polarization in thermal QED with an external magnetic field
Alexandre, J. (2001) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.63.073010 · Cited by 34
“We can note for the purpose of consistency that the integrand in (10) is an even function of the parameter v since Wl(−v) = −Wn−l−1(v) and therefore∑ l e −uW 2l is even in v, which ensures the symetry between the proper-times s and σ.”
BackgroundReal-time nonequilibrium dynamics in hot QED plasmas: Dynamical renormalization group approach
Wang, S.-Y., Boyanovsky, D., de Vega, H. J., & Lee, D.-S. (2000) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.62.105026 · Cited by 34
“[23] in the structure of the solution, in particular the prefactor 1/s in Eq.”
BackgroundMethodInfluentialThe second order hydrodynamic transport coefficient κ for the gluon plasma from the lattice
Philipsen, O., & Schäfer, C. (2014) · Journal of High Energy Physics · doi:10.1007/jhep02(2014)003 · Cited by 33
Holographic photon production in heavy ion collisions
Iatrakis, I., KIRITSIS, E. B., Shen, C., et al. (2017) · Journal of High Energy Physics · doi:10.1007/jhep04(2017)035 · Cited by 33
“[5]), dΓ = − d 3k (2π)3 e(2)nb(|k|) |k| Im [ tr ( ηG μν )]”
Equation of state of cold and dense QCD matter in resummed perturbation theory
Fujimoto, Y., & Fukushima, K. (2022) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.105.014025 · Cited by 33
“[26] for a standard textbook).”
Gerhold, A., & Rebhan, A. K. (2005) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.71.085010 · Cited by 33
Mesonic excitations of QGP: Study with an effective model
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Mukherjee, A., Ghosh, S., Mandal, M., Sarkar, S., & Roy, P. K. (2018) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.98.056024 · Cited by 32
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Chiral Mass-Gap in Curved Space
Flachi, A., & Fukushima, K. (2014) · Physical Review Letters · doi:10.1103/physrevlett.113.091102 · Cited by 38
Stochastic string motion above and below the world sheet horizon
Casalderrey-Solana, J., Keun-young Kim, A., & Teaney, D. A. (2009) · Journal of High Energy Physics · doi:10.1088/1126-6708/2009/12/066 · Cited by 32
Bubbling the false vacuum away
Gleiser, M., Rogers, B. N., & Thorarinson, J. (2008) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.77.023513 · Cited by 36
Lattice calculations of meson correlators and spectral functions at finite temperature
Petreczky, P. (2003) · Journal of Physics G Nuclear and Particle Physics · doi:10.1088/0954-3899/30/1/051 · Cited by 32
“e−Ĥ/T 〉T denotes the thermal average [12].”
BackgroundInfluentialShear viscosity from thermal fluctuations in relativistic conformal fluid dynamics
Peralta-Ramos, J., & Calzetta, E. (2012) · Journal of High Energy Physics · doi:10.1007/jhep02(2012)085 · Cited by 31
Effect of a spectral modification of the ρ meson on the shear viscosity of a pion gas
Mitra, S., Ghosh, S., & Sarkar, S. (2012) · Physical Review C · doi:10.1103/physrevc.85.064917 · Cited by 31
Thermal effects on ρ meson properties in an external magnetic field
Ghosh, S., Mukherjee, A., Mandal, M., Sarkar, S., & Roy, P. K. (2017) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.96.116020 · Cited by 31
“ρ SELF ENERGY IN THE MEDIUM In the real time formalism of thermal field theory, the thermal propagators as well as the self energies become 2×2 matrices [30, 38].”
BackgroundImpact of quark quasiparticles on transport coefficients in hot QCD
Mykhaylova, V., & Sasaki, C. (2021) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.103.014007 · Cited by 31
“All the scaled effective masses vanish logarithmically at high temperatures, matching the perturbative expectation for particles with thermal momenta p ∼ T [68].”
BackgroundMedium effects on the viscosities of a pion gas
Mitra, S., & Sarkar, S. (2013) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.87.094026 · Cited by 30
Bose-Einstein condensation and Silver Blaze property from the two-loop Φ -derivable approximation
Markó, G., Reinosa, U., & Szép, Z. (2014) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.90.125021 · Cited by 30
Spectral properties of massless and massive quarks coupled with massive boson at finite temperature
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Sánchez, Á., Ayala, A., & Piccinelli, G. (2007) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.75.043004 · Cited by 30
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Enhanced photon production from quark-gluon plasma: Finite-lifetime effect
Wang, S.-Y., & Boyanovsky, D. (2001) · Physical Review D · doi:10.1103/physrevd.63.051702 · Cited by 29
Hard-thermal-loop corrections in leptogenesis I:CP-asymmetries
Kießig, C. P., & Plümacher, M. (2012) · Journal of Cosmology and Astroparticle Physics · doi:10.1088/1475-7516/2012/07/014 · Cited by 35
Relaxion fluctuations (self-stopping relaxion) and overview of relaxion stopping mechanisms
Fonseca, N., Morgante, E., Sato, R., et al. (2020) · Journal of High Energy Physics · doi:10.1007/jhep05(2020)080 · Cited by 29
“Taking this effect into account, the timescale for photon production is estimated to be ∆tγ ∼ T f 3 γ/(φ̇) [29].”
Residues of correlators in the strongly coupled N=4 plasma
Amado, I., Landsteiner, K., Montero, S., & Hoyos, C. (2008) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.77.065004 · Cited by 29
“Both types of poles can be distinguished by the behaviour of their residues at short wavelengths: the residues of the particle poles scale like q−1 with the momentum q whereas the collective excitations show exponentially decaying residues of the form exp(−αq2/g2T 2) with a mode dependent constant α [12].”
BackgroundThe linear sigma model at a finite isospin chemical potential
Mao, H., Petropoulos, N., Shu, S., & Zhao, W.-Q. (2006) · Journal of Physics G Nuclear and Particle Physics · doi:10.1088/0954-3899/32/11/012 · Cited by 29
Kubo formulas for thermodynamic transport coefficients
Kovtun, P. K., & Shukla, A. (2018) · Journal of High Energy Physics · doi:10.1007/jhep10(2018)007 · Cited by 43
“The diagram can be evaluated by the standard methods of equilibrium thermal field theory in flat space [26, 27].”
MethodAbreu, L. M., Corrêa, E. B. S., Linhares, C. A., et al. (2019) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.99.076001 · Cited by 29
Right-handed neutrino production in dense and hot plasmas
Ayala, A., D’Olivo, J. C., & Torres, M. (2000) · Nuclear Physics B · doi:10.1016/s0550-3213(99)00642-2 · Cited by 29
Thermal dileptons from π - ρ interactions in a hot pion gas
Baier, R., Dirks, M., & Redlich, K. (1997) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.55.4344 · Cited by 28
“comparable to the hard thermal loop expansion for QCD [22, 23], is implemented in our approach, which maybe compared with an analogous fixed-order calculation for dilepton production from a quark-gluon plasma [24].”
BackgroundMethodThermal Green functions in coordinate space for massless particles of any spin
Weldon, H. A. (2000) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.62.056010 · Cited by 30
Late Time Correlations in Hydrodynamics: Beyond Constitutive Relations
Jain, A., & Kovtun, P. K. (2022) · Physical Review Letters · doi:10.1103/physrevlett.128.071601 · Cited by 28
Flavor diagonal and off-diagonal susceptibilities in a quasiparticle model of the quark-gluon plasma
Bluhm, M., & Kämpfer, B. (2008) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.77.114016 · Cited by 28
Plasmons in anisotropic quark-gluon plasma
Carrington, M. E., Deja, K., & Mrówczyński, S. (2014) · Physical Review C · doi:10.1103/physrevc.90.034913 · Cited by 28
Quantum Sensors for the Generating Functional of Interacting Quantum Field Theories
Bermúdez, A., Gert Aarts, A., & Müller, M. (2017) · Physical Review X · doi:10.1103/physrevx.7.041012 · Cited by 28
“Even though they are related [82] to standard Euclidean correlation functions computable in lattice field theory, the analytical continuation from Euclidean to real time (or from Matsubara to real frequency) is a non-trivial process.”
BackgroundMore on thermal probes of a strongly coupled anisotropic plasma
Jahnke, V., Luna, A., Patiño, L., & Trancanelli, D. (2014) · Journal of High Energy Physics · doi:10.1007/jhep01(2014)149 · Cited by 27
“The spectral density above can then be used to compute the dilepton production rate by means of the expression [46] dΓ` ̄̀ dk = e(2)e` 2 (2π)46π|k|5 Φ(k) Θ(k0)Θ(−k(2) − 4m`(2))(−k(2) − 4m`(2))(−k(2) + 2m`(2))η χμν(k) , (2.”
Quark-flavor dependence of the shear viscosity in a quasiparticle model
Mykhaylova, V., Bluhm, M., Redlich, K., et al. (2019) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.100.034002 · Cited by 27
Evolution of the Primordial Axial Charge across Cosmic Times
Boyarsky, A. M., Cheianov, V., Oleg Ruchayskiy, C., & Sobol, O. O. (2021) · Physical Review Letters · doi:10.1103/physrevlett.126.021801 · Cited by 28
“[46–52].”
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Chromohydrodynamical instabilities induced by relativistic jets
Mannarelli, M., & Manuel, C. (2007) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.76.094007 · Cited by 27
“The dispersion laws in the limit k ≪ ω for the longitudinal and transverse modes are given by [40]”
MethodIslam, C. A., Majumder, S., Haque, N., & Mustafa, M. G. (2015) · Journal of High Energy Physics · doi:10.1007/jhep02(2015)011 · Cited by 27
Nucleons near the QCD deconfinement transition
Datta, S., Gupta, S., Padmanath, M., et al. (2013) · Journal of High Energy Physics · doi:10.1007/jhep02(2013)145 · Cited by 27
“Information about thermal change of properties of hadrons, on the other hand, is easier to obtain from the real-time retarded correlator [33]”
BackgroundPhoton and axion oscillation in a magnetized medium: A general treatment
Ganguly, A. K., Jain, P., & Mandal, S. (2009) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.79.115014 · Cited by 27
Running coupling constant from lattice data and bulk viscosity of strange quark matter
Xiaoping, Z., Kang, M., Xuewen, L., et al. (2005) · Physical Review C · doi:10.1103/physrevc.72.025809 · Cited by 27
“The equation of state (EOS) of strange stars has also been studied as the base of studying the viscosity.”
BackgroundHard-Thermal-Loop Resummed Pressure of a Degenerate Quark-Gluon Plasma
Baier, R., & Redlich, K. (2000) · Physical Review Letters · doi:10.1103/physrevlett.84.2100 · Cited by 27
“This allows to perform the Matsubara frequency sum and to take the T = 0 limit in the standard way [9].”
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Three-loop Φ -derivable approximation in QED
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Second Order Theory of Excitations in Trapped Bose Condensates at Finite Temperatures
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Fermion damping in a fermion-scalar plasma
Boyanovsky, D., de Vega, H. J., Lee, D.-S., Ng, Y. J., & Wang, S.-Y. (1999) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.59.105001 · Cited by 27
Generalized Pions in Dense QCDPreprint
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Disentangling longitudinal and transverse modes of the ϕ meson through dilepton and kaon decays
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Hard-thermal-loop corrections in leptogenesis II: solving the Boltzmann equations
Kießig, C. P., & Michael Plümacher, A. (2012) · Journal of Cosmology and Astroparticle Physics · doi:10.1088/1475-7516/2012/09/012 · Cited by 26
“2 Propagators at Finite Temperature When going to finite temperature [26], one has to employ ensemble weighted expectation values of operators rather than the vacuum expectation values, so for an operator  we get 〈0|Â|0〉 → 〈Â〉ρ ≡ tr(ρÂ) .”
MethodInfluentialSatapathy, S., Ghosh, S., & Ghosh, S. (2021) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.104.056030 · Cited by 26
Renormalization group optimized perturbation theory at finite temperatures
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Effective CP violation in the Standard Model
Smit, J. (2004) · Journal of High Energy Physics · doi:10.1088/1126-6708/2004/09/067 · Cited by 26
Axion models with high scale inflation
Moroi, T., Mukaida, K., Nakayama, K., et al. (2014) · Journal of High Energy Physics · doi:10.1007/jhep11(2014)151 · Cited by 26
Fading of symmetry nonrestoration at finite temperature
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“Some linear combinations of these tensors are usually chosen as the standard basis set [28], denoted A µν , B µν , C µν and D µν and defined in Appendix A.”
Thermal charm production by massive gluons and quarks
Lévai, P., & Vogt, R. (1997) · Physical Review C · doi:10.1103/physrevc.56.2707 · Cited by 26
Gluon self-energy in a two-flavor color superconductor
Rischke, D. H. (2001) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.64.094003 · Cited by 26
The role of fluctuation-dissipation dynamics in setting initial conditions for inflation
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The role of fluctuation-dissipation dynamics in setting initial conditions for inflation
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Scale-Invariant Resummed Perturbation at Finite Temperatures
Kneur, J.-L., & Pinto, M. B. (2016) · Physical Review Letters · doi:10.1103/physrevlett.116.031601 · Cited by 25
Long range dark matter self-interactions and plasma instabilities
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“where Πνμ is the leading order response function (the ‘self-energy’, in the language of thermal field theory [30]).”
BackgroundMedium effects on the thermal conductivity of a hot pion gas
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Critical line forHsuperfluidity in strange quark matter?
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“2π ρi(ω ′,p) ω − ω′ = 1 ω2 − p2 −m(2)i + 2iγiω (2) The quasiparticle pole masses for gluons and quarks are defined, inspired by the asymptotic HTL masses [13, 14], by”
Millicharged relics reveal massless dark photons
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“This corresponds to setting Γ′ = 0 and modifying M ′ 2 eff ' ω′ 2 p →M ′ 2 eff ' (3/2)ω′ 2 p [23, 24, 74], as in eq.”
Effective field theories for dark matter pairs in the early universe: cross sections and widths
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Photon and dilepton production in soft wall AdS/QCD
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ππ -scattering lengths at finite temperature
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Domain formation in finite-time quenches
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“The correlation length at this freeze-out time determines the characteristic size of ordered domains and hence the typical separation between the resultant topological defects.”
BackgroundSchwinger-Keldysh effective field theory for stable and causal relativistic hydrodynamics
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Relic density computations at NLO: infrared finiteness and thermal correction
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How to unify dark matter, dark energy, and inflation
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Color screening potential at finite density in two-flavor lattice QCD with Wilson fermions
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Properties of open and hidden charm mesons in light quark matter
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Lattice calculations of meson correlators and spectral functions at finite temperature
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“where (11(?) is the 11-component of the momentum space real time free thermal Dirac propagator whose explicit form reads [45, 46]”
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Finite-size effects on the phase structure of the Walecka model
Abreu, L. M., & Nery, E. S. (2017) · Physical Review C · doi:10.1103/physrevc.96.055204 · Cited by 22
Gu, X.-W., Duan, C.-G., & Guo, Z.-H. (2018) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.98.034007 · Cited by 21
Dark gauge boson production from neutron stars via nucleon-nucleon bremsstrahlung
Shin, C. S., & Yun, S. (2022) · Journal of High Energy Physics · doi:10.1007/jhep02(2022)133 · Cited by 22
“Moreover, we will consider the thermal field theory [45, 46] to account for the effective couplings in medium [19].”
MethodNoncommutative black-body radiation: Implications on cosmic microwave background
Fatollahi, A. H., & Hajirahimi, M. (2006) · Europhysics Letters (EPL) · doi:10.1209/epl/i2006-10149-x · Cited by 22
Effective field theory and dispersion law of the phonons of a nonrelativistic superfluid
Escobedo, M. Á., & Manuel, C. (2010) · Physical Review A · doi:10.1103/physreva.82.023614 · Cited by 22
Parity-odd and CPT -even electrodynamics of the standard model extension at finite temperature
Casana, R., Ferreira, M. M., & Silva, M. R. O. (2010) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.81.105015 · Cited by 22
QCD pressure and the trace anomaly
Drummond, I. T., Horgan, R. R., Landshoff, P. V., et al. (1999) · Physics Letters B · doi:10.1016/s0370-2693(99)00725-x · Cited by 22
Metadata
- Authors
- Le Bellac, M.
- Venue
- Cambridge University Press eBooks
- Publisher
- Cambridge University Press
- Published
- 1996-08-01
- OpenAlex
- W4210557146
- Cited by
- 897
- References
- 0
- Citations by year
- 2026: 20 · 2025: 35 · 2024: 28 · 2023: 30 · 2022: 25 · 2021: 32 · 2020: 54 · 2019: 28 · 2018: 21 · 2017: 25 · 2016: 22 · 2015: 32 · 2014: 36 · 2013: 40 · 2012: 38
- FWCI
- 13.98
- Type
- book
- Language
- en
- Open access
- closed
- Locations
- Cambridge University Press eBooks (publishedVersion)
- Topics
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › High-Energy Particle Collisions Research
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Quantum Chromodynamics and Particle Interactions
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Particle physics theoretical and experimental studies
- Keywords
- thermal field theory, quark-gluon plasma, path integral, imaginary time formalism, real-time formalism, fermion fields, gauge fields, cutting rules, collective excitations, kinetic theory, heavy-ion collisions, astrophysical plasmas
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