Based on the external perturbation that disturbs the system only slightly from its equilibrium position we make the Taylor expansion of the pressure of a quark gas. It turns out that the first term was used in the literature to construct a Hard Thermal Loop perturbation theory (HTLpt) within the variation principle of the lowest order of the thermal mass parameter. Various thermodynamic quantities within the 1-loop HTLpt encountered overcounting of the leading order (LO) contribution and also required a separation scale for soft and hard momenta. Using same variational principle we reconstruct the HTLpt at the first derivative level of the pressure that takes into account the effect of the variation of the external source through the conserved density fluctuation. This modification markedly improves those quantities in 1-loop HTLpt in a simple way instead of pushing the calculation to a considerably more complicated 2-loop HTLpt. Moreover, the results also agree with those obtained in the 2-loop approximately self-consistent Φ-derivable Hard Thermal Loop resummation. We also discuss how this formalism can be extended for the higher order contributions
We calculate the thermodynamic functions of a quark-gluon plasma for general N_c and N_f to three-loop order using hard-thermal-loop perturbation theory. At this order, all the ultraviolet divergences can be absorbed into renormalizations of the vacuum, the HTL mass parameters, and the strong coupling constant.We show that at three loops, the results for the pressure and trace anomaly are in very good agreement with recent lattice data down to temperatures T~2T_c.
We consider an AdS/QCD model at finite temperature with a dilaton field that we call thermal because, in addition to depending on the holographic coordinate, it also depends on temperature. We study two thermal dilatons in this work such that at $T=0$ they are reduced to the dilatons used by some authors. With these thermal dilatons it is possible to obtain melting temperatures for mesons close to $180$ MeV and also make predictions for other hadrons. We use a procedure based on the analysis of the holographic potential related to the E.O.M. for modes dual to hadrons to calculate melting tempe…
I review recent developments in QCD thermodynamics and collective excitations from the hard-thermal-loop effective theory. I begin by motivating the discussion with open questions from heavy-ion collisions. I then discuss a finite-temperature and -density calculation of QCD thermodynamics at NNLO from the hard-thermal-loop perturbation theory. Finally I discuss a recent exploration of generalizing the hard-thermal-loop framework to the (chromo)magnetic scale $g^2T$, from which a novel massless mode is uncovered.
The static and dynamic chiral susceptibilities in the quark-gluon plasma are calculated within the lowest order perturbative QCD at finite temperature and the Hard Thermal Loop resummation technique using an effective quark propagator. After regularisation of ultraviolet divergences, the Hard Thermal Loop results are compared to QCD lattice simulations.
We calculate the free energy of a hot gluon plasma to leading order in hard-thermal-loop perturbation theory. Effects associated with screening, gluon quasiparticles, and Landau damping are resummed to all orders. The ultraviolet divergences generated by the hard-thermal-loop propagator corrections can be cancelled by a temperature-independent counterterm. The deviation of the hard-thermal-loop free energy from lattice QCD results for T > 2 T_c has the correct sign and roughly the correct magnitude to be accounted for by next-to-leading order corrections.
We calculate the quark number susceptibility in the deconfined phase of QCD using the hard thermal loop (HTL) approximation for the quark propagator. This improved perturbation theory takes into account important medium effects such as thermal quark masses and Landau damping in the quark-gluon plasma. We explicitly show that the Landau damping part in the quark propagator for spacelike quark momenta does not contribute to the quark number susceptibility due to the quark number conservation. We find that the quark number susceptibility only due to the collective quark modes deviates from that o…
We use the recently obtained two-loop hard thermal loop perturbation theory thermodynamics functions of a plasma of quarks and gluons to compute the diagonal second- and fourth-order quark number susceptibilities. The two-loop hard thermal loop perturbation theory thermodynamic functions used are reliable in the limit that the ratio of the quark chemical potential to temperature is small. Using this result, we are able to obtain (semi-)analytic expressions for the quark number susceptibilities at leading- and next-to-leading-order in hard thermal loop perturbation theory. We compare the hard t…
In the framework of the Hard Thermal Loop effective theory, we calculate the two-loop contributions to hard lepton pair production in a quark-gluon plasma. We show that the result is free of any infrared and collinear singularity. We also recover the known fact that perturbation theory leads to integrable singularities at the location of the threshold for $q\bar{q}\toγ^*$. It appears that the process calculated here significantly enhances the rate of low mass hard dileptons.
Alternative cosmologies, based on extensions of General Relativity, predict modified thermal histories in the Early Universe during the pre Big Bang Nucleosynthesis (BBN) era, epoch which is not directly constrained by cosmological observations. When the expansion rate is enhanced with respect to the standard case, thermal relics typically decouple with larger relic abundances. The correct value of the relic abundance is therefore obtained for larger annihilation cross--sections, as compared to standard cosmology. A direct consequence is that indirect detection rates are enhanced. Extending pr…
The quark number susceptibility, associated with the conserved quark number density, is closely related to the baryon and charge fluctuations in the quark-gluon plasma, which might serve as signature for the quark-gluon plasma formation in ultrarelativistic heavy-ion collisions. Beside QCD lattice simulations, the quark number susceptibility has been calculated recently using a resummed perturbation theory (Hard Thermal Loop resummation). In the present work we show, based on general arguments, that the computation of this quantity neglecting Hard Thermal Loop vertices contradicts the Ward ide…
We develop a discrete lattice implementation of the hard thermal loop effective action by the method of added auxiliary fields. We use the resulting model to measure the sphaleron rate (topological susceptibility) of Yang-Mills theory at weak coupling. Our results give parametric behavior in accord with the arguments of Arnold, Son, and Yaffe, and are in quantitative agreement with the results of Moore, Hu, and Muller.
We compute the simplest hard thermal loops for a spatial 't Hooft loop in the deconfined phase of a SU(N) gauge theory. We expand to quadratic order about a constant background field A_0 = Q/g, where Q is a diagonal, color matrix and g is the gauge coupling constant. We analyze the problem in sufficient generality that the techniques developed can be applied to compute transport properties in a "semi"-Quark Gluon Plasma. Notably, computations are done using the double line notation at finite N. The quark self-energy is a Q-dependent thermal mass squared, of order g^2T^2, where T is the tempera…
We propose a minimal extension of the standard model of particle physics to accommodate cosmic inflation, dark matter and light neutrino masses. While the inflationary phase is obtained from a modified chaotic inflation scenario, consistent with latest cosmology data, the dark matter particle is a fermion singlet which remains out of equilibrium in the early universe. The scalar field which revives the chaotic inflation scenario by suitable modification also assists in generating tiny couplings of dark matter with its mother particle, naturally realizing the non-thermal or freeze-in type dark …
We propose a formulation of the long-distance dynamics of gauge theories at finite temperature on a lattice in Minkowski space, including the effects of hard thermal loops on the dynamics of the long wavelength modes. Our approach is based on the dual classical limits of quantum fields as waves and particles in the infrared and ultraviolet limits, respectively. It exhibits manifest invariance under space-dependent lattice gauge transformations and conserves Gauss' law.
Based on the hard-thermal-loop resummed improved ladder Dyson-Schwinger quation for the fermion mass function, we study how we can get the gauge invariant solution in the sense it satisfies the Ward identity. Properties of the ``gauge-invariant'' solutions are discussed.
Naively resummed perturbative approximations to the thermodynamic functions of QCD do not converge at phenomenologically relevant temperatures. Here we review recent results of a three-loop hard-thermal-loop perturbation theory calculation of the thermodynamic functions of a quark-gluon plasma for general N_c and N_f. We show comparisons of our recent results with lattice data from both the hotQCD and Wuppertal-Budapest groups. We demonstrate that the three-loop hard-thermal-loop perturbation result for QCD thermodynamics agrees with lattice data down to temperatures T ~ 2 T_c.
It is shown how the hard thermal loop approximation can be used in chiral perturbation theory to study some thermal properties of Goldstone bosons. Hard thermal effects are first studied in the non-linear sigma model. Then those results are used to obtain the thermal corrections to the transverse and longitudinal gauge field masses in the electroweak theory in the limit of a strongly interacting Higgs boson.
Classical transport theory for colored particles is investigated and employed to derive the hard thermal loops of QCD. A formal construction of phase-space for color degrees of freedom is presented. The gauge invariance of the non-Abelian Vlasov equations is verified and used as a guiding principle in our approximation scheme.We then derive the generating functional of hard thermal loops from a constraint satisfied at leading-order by the color current. This derivation is more direct than alternative ones based on perturbative quantum field theory, and shows that hard thermal effects in hot QC…
Hard thermal loop effective actions furnish the building blocks of resummed thermal perturbation theory, which is expected to work as long as the quantities under consideration are not sensitive to the nonperturbative (chromo-)magnetostatic sector. A different breakdown of perturbation theory occurs whenever external momenta are light-like, because the hard thermal loops themselves develop collinear singularities. By additionally resumming asymptotic thermal masses for hard modes these singularities are removed while leaving the gauge invariance of the effective action intact.
These lectures give an introduction to thermal perturbation theory, hard thermal loops, and their use in a nonperturbative, approximately self-consistent resummation of the thermodynamical potentials of quantum chromodynamics.
We compare our previously proposed hard-thermal-loop (HTL) resummed calculation of quark number susceptibilities using a self-consistent two-loop approximation to the quark density with a recent calculation of the same quantity at the one-loop level in a variant of HTL-screened perturbation theory. Besides pointing out conceptual problems with the latter approach, we show that it severely over-includes the leading-order interaction effects while including none of the plasmon term which after all is the reason to construct improved resummation schemes.
The recent status of hard thermal photon production in relativistic heavy ion collisions is reviewed and the current rates are presented with emphasis on corrected bremsstrahlung processes in the quark-gluon plasma (QGP) and quark-hadron duality. Employing Bjorken hydrodynamics with an EOS supporting the phase transition from QGP to hot hadron gas (HHG), thermal photon spectra are computed. For SPS 158 GeV Pb+Pb collisions, comparison with other theoretical results and the WA98 direct photon data indicates significant contributions due to prompt photons. Extrapolating the presented approach to…
In hot gauge theories, perturbation theory at the scale of the Debye screening mass requires the resummation of the so-called hard thermal loops, which corresponds to using an effective action obtained by integrating out the modes with momentum of order of the temperature. As is well-known, quantities which are sensitive to the nonperturbative magnetic screening mass still remain incalculable in this resummed perturbation theory. A different breakdown of the latter occurs whenever external momenta are light-like, because the hard thermal loops themselves develop collinear singularities. Howeve…
We calculate all those QCD $N$-point functions which are relevant for a three-loop QCD thermodynamics calculation with finite quark masses within the hard the thermal loop approximation. Using the effective quark propagator, we also calculate second order quark and baryon number susceptibilities within the hard thermal loop approximation and compare the results with available lattice data.
Chiral symmetry is used as the guiding principle to derive hard thermal loop effects in chiral perturbation theory. This is done by using a chiral invariant background field method for the non-linear sigma model and the Wess-Zumino-Witten lagrangian, with and without external vector and axial vector sources. It is then shown that the n-point hard thermal loop is the leading thermal correction for the Green function of n point vector soft quark currents.
The generating functional for hard thermal loops in QCD is rewritten in terms of a gauged WZNW action by introducing an auxiliary field. This shows in a simple way that the contribution of hard thermal loops to the energy of the quark-gluon plasma is positive.
Classical transport theory for colored particles is reviewed and used to derive the hard thermal loops of QCD. A perturbative study of the non-Abelian transport equations that preserves their gauge symmetry is used to compute the induced color current in a hot quark-gluon plasma. From this approach the effective action of hard thermal loops can be derived. This derivation is more direct than alternative ones based on perturbative quantum field theory, and shows that hard thermal effects in hot QCD are essentially classical.
The drag and diffusion coefficients of heavy quarks propagating through quark gluon plasma (QGP) have been evaluated using Hard Thermal Loop (HTL) approximations. The HTL corrections to the relevant propagators and vertices have been considered. It is observed that the magnitudes of both the transport coefficients are changed significantly from values obtained by earlier approaches where either (i) the $t$ channel divergence in T=0 pQCD matrix element is shielded simply by Debye mass. or (ii) only HTL resummed propagator is used ignoring the HTL corrections at the interaction vertices. The imp…
Thermal freeze-out or freeze-in during a period of early matter domination can give rise to the correct dark matter abundance for $\langle σ_{\rm ann} v \rangle_{\rm f} < 3 \times 10^{-26}$ cm$^3$ s$^{-1}$. In the standard scenario, a single field that behaves like matter drives the early matter dominated era. However, in realistic models, this epoch may involve more than one field. In this paper, we study the effect of such a modification on the production of dark matter during early matter domination. We show that even a subdominant second field that decays much faster than the dominant one …
We derive a thermal effective action for soft fields in the broken phase of the electroweak theory in the limit of a strongly interacting Higgs sector. This action is just the proper generalization of the hard thermal loop effective action of a Yang-Mills theory when there is a Higgs mechanism and for a heavy Higgs boson. One can obtain from this action the thermal corrections to the masses of the W, Z and the photon.
The generating functional for hard thermal loops in QCD is important in setting up a resummed perturbation theory. I review how this functional is related to the eikonal for a Chern-Simons theory, and using an auxiliary field, to the gauged WZNW-action. The induced current due to hard thermal loops, properly incorporating damping effects, is also briefly discussed. (Invited talk at the Third Worshop on Thermal Field Theories, Banff, Canada, August, 1993.)
The hard thermal loop effective action for Goldstone bosons is deduced by symmetry arguments from the corresponding result for gauge bosons. Pseudoscalar mesons in Chromodynamics and magnons in an antiferromagnet are discussed as special cases, including the hard thermal loop contribution to their scattering.
Dispersion relations for fermions at high temperature and in a background magnetic field are calculated in two different ways. First from a straightforward one-loop calculation where, in the weak field limit, we find an expression closely related to the standard dispersion relations in the absence of the magnetic field. Secondly, we derive the dispersion relations directly from the Hard Thermal Loop effective action, which allows for an exact solution (i.e. to all orders in the external field), up to the last numerical integrals.
We argue on a basis of a simple few mode model of SU(2) Yang-Mills theory that the color off-diagonal coupling of the soft plasmon to hard thermal excitations of the gauge field drives the collective plasma oscillations into chaotic motion despite the presence of the plasmon mass.
The conventional results for hard thermal loops, which are the building blocks of resummed perturbation theory in thermal field theories, have collinear singularities when external momenta are light-like. It is shown that by taking into account asymptotic thermal masses these singularities are removed. The thus improved hard thermal loops can be summarized by compact gauge-invariant effective actions, generalizing the ones found by Taylor and Wong, and by Braaten and Pisarski.
We compute the weak magnetic field correction to the four-point quark-gluon vertex at finite temperature and chemical potential within the hard thermal loop approximation. The resulting four-point quark-gluon vertex satisfies the Ward identity with the three-point quark-gluon vertex.
We analyze the transverse momentum distributions of pp, pPb, XeXe and PbPb collisions at different RHIC and LHC energies and centralities as well as the corresponding distributions for Higgs production decaying into $γγ$ and 4l. A simple linear relation is found between the effective thermal temperature and the hard scale, approximately valid for all processes and mainly determined by the hard scale fluctuations. In order to go further, it is shown that the whole spectrum of pp collisions can be described by a single function showing that the thermal temperature is determined solely by the har…
Self-consistent approximations allowing the calculation of the entropy and the baryon density of a quark-gluon plasma are presented. These approximations incorporate the essential physics of the hard thermal loops, involve only ultraviolet-finite quantities, and are free from overcounting ambiguities. While being nonperturbative in the strong coupling constant $g$, agreement with ordinary perturbation theory is achieved up to and including order $g^3$. It is shown how the pressure can be reconstructed from the entropy and the baryon density taking into account the scale anomaly. The results ob…
We investigate the use of the hard thermal loop (HTL) resummation technique in non-equilibrium field theory. We use the Keldysh representation of the real time formalism (RTF). We derive the HTL photon self energy and the resummed photon propagator. We show that no pinch singularities appear in the non-equilibrium HTL effective propagator. We discuss a possible regularization mechanism for these singularities at higher orders. As an example of the application of the HTL resummation method within the RTF we discuss the damping rate of a hard electron.
A review is given of the hard thermal loop resummation methods initiated by Braaten, Pisarski, and others. We describe some successes of these techniques as well as instances where modifications are necessary. Some possible directions where these modifications may lead are also discussed. [Review talk given at the 4th workshop on thermal field theories held in Dalian, China]
We consider finite temperature 1-loop diagrams with hard loop momenta and an arbitrary number of external gauge fields when the external momenta are either soft, or near the light cone and nearly collinear with the loop momentum. We obtain a recursion relation for these diagrams which we translate into an equation for their generating functional. By integrating out the soft fields while keeping two collinear ones we find an integral equation, originally due to Arnold, Moore, and Yaffe, which sums the bremsstrahlung and pair annihilation contribution to the thermal photon production rate.
I consider the calculation of the two and three-point functions for QED at finite temperature in the presence of a background plasma velocity. The final expressions are consistent with Lorentz invariance, gauge invariance and current conservation, pointing to a straightforward generalization of the hard thermal loop formalism to this physical situation. I also give the resulting expression for the effective action and identify the various terms.
The Hard Thermal Loop expansion, is an attractive theory, but it reveals difficulties when one uses it as a perturbative scheme. To illustrate this we use the HTL expansion to calculate the two loop corrections for soft virtual photons. The resulting corrections are of the same order as the 1-loop correction.
We compare the effective dynamics of soft fields, based on temperature field theory, with the mean field dynamics from non-Abelian kinetic theory. We derive the polarization tensor with the leading logarithmic factor $\log({gT\overμ})$ from the effective Boltzmann-Langevin equation given by Litim and Manuel. The tensor is identical with effective one-loop contributions within the hard thermal loop effective theory.
We compute the pressure of a finite density quark-gluon plasma at zero temperature to leading order in hard-thermal-loop perturbation theory, which includes the fermionic excitations and Landau damping. The result is compared with the weak-coupling expansion for finite positive chemical potential $μ$ through order $α_s^2$ and with a quasiparticle model with a mass depending on $μ$.
Novel back-to-back correlations are shown to arise for thermal ensembles of squeezed bosonic states associated with medium-modified mass-shifts. The strength of these correlations could become unexpectedly large in heavy ion collisions.
Classical transport theory is employed to analyze the hot quark-gluon plasma at the leading order in the coupling constant. A condition on the (covariantly conserved) color current is obtained. {}From this condition, the generating functional of hard thermal loops with an arbitrary number of soft external bosonic legs can be derived. Our approach, besides being more direct than alternative ones, shows that hard thermal loops are essentially classical.
We consider the hard thermal loops of QCD for a moving quark-gluon plasma. Generalizing from this we suggest a candidate for the magnetic mass term. This term may also be useful in understanding the mass gap of three-dimensional non-Abelian gauge theories.
Inspired by recent progress on classical limits of scattering amplitudes, we show that hard thermal loops can be obtained from classical limits of off-shell currents. The classicality of hard thermal loops is made manifest by associating classical wavenumbers to soft particles. We compute the classical limit of these currents in QED, QCD and gravity. Our proposal does not involve the introduction of ghosts.
In this talk I briefly discuss the thermodynamics of the quark-gluon plasma The calculation of the free energy of a hot gluon plasma to leading order in hard-thermal-loop perturbation theory is outlined. The HTL free energy is compared with the weak-coupling expansion and lattice results.
The production rate of soft real photons or soft lepton pairs by a hot QCD plasma is dominated by strong collinear divergences. As a consequence, it appears that the effective theory based on the resummation of hard thermal loops fails to handle properly these light-cone sensitive processes since some formally higher order diagrams are in reality the dominant ones.
Chiral phase transition in thermal QCD is studied by using the Dyson-Schwinger (DS) equation in the real time hard thermal loop approximation. Our results on the critical temperature and the critical coupling are significantly different from those in the preceding analyses in the ladder DS equation, showing the importance of properly taking into account the essential thermal effects, namely the Landau damping and the unstable nature of thermal quasiparticles.
The weak-coupling expansion for thermodynamic quantities in thermal field theories is poorly convergent unless the coupling constant is tiny. We discuss the calculation of the free energy for a hot gas of electrons and photons to three-loop order using hard-thermal-loop perturbation theory (HTLpt). We show that the hard-thermal-loop perturbation reorganization improves the convergence of the successive approximations to the QED free energy at large coupling, e ~ 2. The reorganization is gauge invariant by construction, and due to the cancellations among various contributions, we obtain a compl…
I outline various derivations of the non-Abelian Kubo equation, which governs the response of a quark-gluon plasma to hard thermal perturbations. In the static case, it is proven that gauge theories do not support hard thermal solitons. Explicit solutions are constructed within an SU(2) Ansatz and they are shown to support the general result. The time-dependent problem, i.e., non-Abelian plasma waves, has not been completely solved. We express and motivate the hope that the intimate relations linking the gauge-invariance condition for hard thermal loops to the equation of motion for T=0, topol…
We study the realization of dimensional reduction and the validity of the hard thermal loop expansion for lambda phi^4 theory at finite temperature, using an environmentally friendly finite-temperature renormalization group with a fiducial temperature as flow parameter. The one-loop renormalization group allows for a consistent description of the system at low and high temperatures, and in particular of the phase transition. The main results are that dimensional reduction applies, apart from a range of temperatures around the phase transition, at high temperatures (compared to the zero tempera…
We derive the hard thermal loop action for soft electromagnetic fields in the finite temperature world-line formulation at imaginary time, by first integrating out the hard fermion modes from the microscopic QED action. Further, using the finite T world-line method, we calculate all static higher order terms in the soft electromagnetic field. At high T, the leading non-linear terms are independent of the temperature and, except for a term quartic in the time component of the vector potential, they cancel exactly against the vacuum contribution. The remaining T-dependent non-linear terms become…
