Strong and Electroweak Matter 2000

The following sections are included:

• PREFACE

• CONTENTS

After a brief review of the phenomena expected in cold dense quark matter, color superconductivity and color-flavor locking, we sketch some implications of recent developments in our understanding of cold dense quark matter for the physics of compact stars. We give a more detailed summary of our recent work on crystalline color superconductivity and the consequent realization that (some) pulsar glitches may originate in quark matter.

The high energy limit of QCD is controlled by the small-x part of a hadron wavefunction. I argue that this part is universal to all hadrons and is composed of a new form of matter: a Colored Glass Condensate. This matter is weakly interacting at very small x, but is non-perturbative because of the highly occupied boson states which compose the condensate. Such a matter might be studied in high energy lepton-hadron or hadron-hadron interactions.

We discuss various aspects of parity, CP, and time reversal invariances in QCD. In particular, we focus attention on the previously proposed possibility that these experimentally established symmetries of strong interactions may be broken at finite temperature and/or density. This would have dramatic signatures in relativistic heavy ion collisions; we describe some of the most promising signals.

We discuss the relation between the deconfining phase transition in gauge theories and the realization of the magnetic Z_N symmetry. At low temperature the Z_N symmetry is spontaneously broken while above the phase transition it is restored. This is intimately related to the change of behaviour of the spatial 't Hooft loop operator V(C). In the confining phase V has a perimeter law behaviour V(C) ∝ exp{-mP(C)}, while in the deconfined phase it has the area law behaviour V(C) ∝ exp{-αS(C)}. We also show that the "dual string tension" α is equal to the "wall tension" of the Z_N domain walls previously calculated in the literature.

Standard theories of electroweak interactions are based on the concept of a gauge symmetry broken by the Higgs mechanism. If they are placed in an environment with a sufficiently high temperature, the symmetry gets restored. It turns out that the characteristics of the symmetry restoring phase transition, such as its order, are important for cosmological applications, such as baryon asymmetry generation. We first briefly review how, by a combination of analytic and numerical methods, the properties of the phase transition can be systematically resolved for any given type of a (weakly interacting) Higgs sector. We then summarise the numerical results available for the Standard Model, and present a generic model independent statement as to how the Higgs sector should differ from the Standard Model for the properties of the transition to be very different. As an explicit example, we discuss the possibilities available for a strong transition in the experimentally allowed parameter region of the Minimal Supersymmetric Standard Model.

We discuss some aspects of a recently proposed semi-classical transport theory for QCD plasmas based on coloured point particles. This includes the derivation of effective transport equations for mean fields and fluctuations which relies on the Gibbs ensemble average. Correlators of fluctuations are interpreted as collision integrals for the effective Boltzmann equation. The approach yields a recipe to integrate-out fluctuations. Systematic approximations (first moment, second moment, polarisation approximation) based on a small plasma parameter are discussed as well. Finally, the application to a hot non-Abelian plasma close to thermal equilibrium is considered and the consistency with the fluctuation-dissipation theorem established.

I begin by answering a different question, "Do we know the sphaleron rate?" and conclude that we do. Then I discuss a crude but purely analytic picture which provides an estimate of the sphaleron rate within the context of Bödeker's effective theory. The estimate, which comes surprisingly close to the numerically determined sphaleron rate, gives a physical picture of baryon number violation in the hot phase, and provides a conjecture of the N_c dependence of the sphaleron rate in SU(N_c) gauge theory.

Various definitions for the QCD Debye mass and its evaluation are reviewed in a non-perturabtive framework for the study of screening of general static sources. While it is possible to perturbatively integrate over scales ~ T and thus construct a 3d effective theory, the softer scales ~ gT and ~ g²T are strongly coupled for temperatures ≲ 10⁷ GeV and require lattice simulations. Within the effective theory, a lattice treatment of screening at finite quark densities µ ≲ 4/T is also possible.

An alternate picture of the deconfined phase of gauge theories is described. Instead of a plasma, the theory is viewed as a condensate of Polyakov lines. The pressure is determined by an elementary mean field theory.

QCD at finite isospin chemical potential µ_I is studied. This theory has no fermion sign problem and can be simulated on the lattice using present-day techniques. We solve this theory analytically in two limits: low µ_I where chiral perturbation theory is applicable, and asymptotically high µ_I where perturbative QCD is at work. At low isospin density the ground state is a superfluid pion condensate. At very high density it is a Fermi liquid with Cooper pairing. The pairs carry the same quantum numbers as the pions. Motivated by this observation, we put forward a conjecture that the transition from hadron to quark matter is smooth. The conjecture passes several nontrivial tests. Our results imply a nontrivial phase diagram in the space of temperature and chemical potentials of isospin and baryon number.

The dynamics of high temperature gauge fields, on scales relevant for non-perturbative phenomena such as electroweak baryogenesis, may be described by a remarkably simple effective theory. This theory, which takes the form of a local, stochastic, classical Yang-Mills theory, depends on a single parameter, the non-Abelian (or "color") conductivity. This effective theory has recently been shown to be valid to next-to-leading-log order (NLLO), provided one uses an improved NLLO value for the non-Abelian conductivity. Comparisons of numerical simulations using this NLLO effective theory and a more microscopic effective theory agree surprisingly well.

We discuss the φ⁴ and φ⁶ theory defined in a flat D-dimensional space-time. Assuming that the system is in equilibrium with a thermal bath at temperature β^-1. To obtain a non-perturbative result, the 1/N expansion is used. We use the method of the composite operator (CJT) for summing a large set of Feynman graphs.

The conventional weak-coupling expansion for the pressure of a hot plasma shows no sign of convergence unless the coupling constant g is tiny. In this talk, I discuss screened perturbation theory (SPT) which is a reorganization of the perturbative expansion by adding and subtracting a local mass term in the Lagrangian. We consider several different mass prescriptions, and compute the pressure to three-loop order. The SPT-improved approximations appear to converge for rather large values of the coupling constant.

I review the Bose-Einstein condensation phase transition of dilute gases of cold atoms, for particle theorists acquainted with methods of field theory at finite temperature. I then discuss how the dependence of the phase transition temperature on the interaction strength can be computed in the large N approximation.

The calculation of the electrical conductivity of a high temperature e⁺e^- plasma from first principles of QED is outlined. The principal feature of our approach is a non-trivial resummation of perturbation theory beyond hard thermal loops, which involves truncation of the Schwinger-Dyson hierarchy to include multiple scattering effects, in a manner consistent with gauge invariance.

We report on results concerning the partition function of SU(2) gauge theory in the strong coupling limit, which we analyze as a continuous spin model through a Fortuin-Kasteleyn transformation. The properties of the corresponding cluster distribution are investigated to show that the thermal and geometrical phase transitions indeed coincide supporting the belief that deconfinement in QCD can be characterized as percolation of Polyakov loop clusters.

We investigate the large distance behavior of the electric and magnetic propagators of hot SU(2) gauge theory in different gauges using lattice simulations of the full 4d theory and the effective, dimensionally reduced 3d theory. A comparison of the 3d and 4d data for the propagators suggests that dimensional reduction works surprisingly well down to temperatures T ~ 2T_c. A detailed study of the volume dependence of magnetic propagators is performed. The electric propagators show exponential decay at large distances in all gauges considered and a possible gauge dependence of the electric screening mass turns out to be statistically insignificant.

We analyse a new mechanism for the cosmological QCD first-order phase transition: inhomogeneous nucleation. The primordial temperature fluctuations are larger than the tiny temperature interval, in which bubbles would form in the standard picture of homogeneous nucleation. Thus the bubbles nucleate at cold spots. We find the typical distance between bubble centers to be a few meters. This exceeds the estimates from homogeneous nucleation by two orders of magnitude. The resulting baryon inhomogeneities may affect primordial nucleosynthesis.

In this talk I summarize the one loop and higher loop calculations of the effective equations of motion of the O(N) symmetric scalar model in the linear response approximation. At one loop one finds essential difference in long time behavior for the fields below and above a dynamically generated length scale. A partial resummation assuming quasi-particle propagation seems to cancel the relevance of this scale.

We discuss the flavor dependence of the pressure and critical temperature calculated in QCD with 2, 2+1 and 3 flavors using improved gauge and staggered fermion actions on lattices with temporal extent N_τ = 4. For T ≳ 2T_c we find that bulk thermodynamics of QCD with 2 light and a heavier strange quark is well described by 3-flavor QCD while the transition temperature is closer to that of 2-flavor QCD. Furthermore, we present evidence that the chiral critical point of 3-flavor QCD, i.e. the second order endpoint of the line of first order chiral phase transitions, belongs to the universality class of the 3d Ising model.

Temporal meson correlators and their spectral functions are calculated in the deconfined phase using the hard thermal loop resummation technique. The spectral functions exhibit strong medium effects coming from the hard thermal loop approximation for the quark propagator. The correlators, on the other hand, do not differ significantly from free correlators, for which bare quark propagators are used. This is in contrast to lattice calculations showing a clear deviation from the free correlations functions.

When applying the maximum entropy method (MEM) to the analysis of hadron correlation functions in QCD a central issue is to understand to what extent this method can distinguish bound states, resonances and continuum contributions to spectral functions. We discuss these issues by analyzing meson and diquark correlation functions at zero temperature as well as free quark anti-quark correlators. The latter test the applicability of MEM to high temperature QCD.

Two recent attempts for overcoming the poor convergence of the perturbation expansion of the thermodynamic potentials of QCD are discussed: an HTL-adaption of "screened perturbation theory" and approximately self-consistent HTL resummations in the two-loop entropy.

At high temperatures or densities, hadronic matter shows different forms of critical behaviour: colour deconfinement, chiral symmetry restoration, and diquark condensation. I first discuss the conceptual basis of these phenomena and then consider the description of colour deconfinement in terms of symmetry breaking, through colour screening and as percolation transition.

A general method for calculating asymptotic expansions of infinite sums in thermal field theory is presented. It is shown that the Mellin summation method works elegantly with dimensional regularization. A general result is derived for a class of one-loop Feynman diagrams at finite-temperature.

I study derivative expansions of effective actions at finite temperature, illustrating how the standard methods are badly defined at finite temperature. I then show that by setting up the initial conditions at a finite time, these problems are solved.

We use an optimized hopping parameter expansion (linear δ-expansion) for the free energy to study the phase transitions at finite temperature and finite charge density in a global U(1) scalar Higgs sector in the continuum and on the lattice at large lattice couplings. We are able to plot out phase diagrams in lattice parameter space and find that the standard second-order phase transition with temperature at zero chemical potential becomes first order as the chemical potential increases.

The nonperturbative real-time evolution of quantum fields out of equilibrium is often solved using a mean-field or Hartree approximation or by applying effective action methods. In order to investigate the validity of these truncations, we implement similar methods in classical scalar field theory and compare the approximate dynamics with the full nonlinear evolution. Numerical results are shown for the early-time behaviour, the role of approximate fixed points, and thermalization.

An isolated statistical many-particle system, formed by relativistic neutral spinless massive bosons, is considered as a model of Out-of-Equilibrium Relativistic Quantum Field Theory. A possible initial non-equilibrium state (ρ_Q,in) for it is studied. Arguments are given to justify that ρ_Q,in generates the same ultraviolet behaviour as the renormalized correlation functions do in standard Equilibrium Thermal Field Theory. The time evolution of the system, as determined by ρ_Q,in, is discussed shortly.

We examine gauge theories out of equilibrium. The main purpose of our investigations concerns the problem of gauge invariance. Therefore, we discuss different gauges and analyse their special features. At the end we compare them numerically.

A non-linear Boltzmann equation¹ describing the time evolution of a partonic system in the central rapidity region after a heavy ion collision is solved numerically². A particular model of the collinear logarithmic divergences due to small angle scattering^3,4 is employed in the numerical solution. The system is followed until it reaches kinetic equilibrium where the equilibration time, temperature and chemical potential are determined for both RHIC and LHC.

We present the results of our large scale 4-dimensional (4d) lattice simulations for the MSSM electroweak phase transition (EWPT).

The following sections are included:

• The Setting

• A short path to

• The results

• References

SUSY models with a gauge singlet easily allow for a strongly first order electroweak phase transition (EWPT). We discuss the wall profile, in particular transitional CP violation during the EWPT. We calculate CP violating source terms for the charginos in the WKB approximation and solve the relevant transport equations to obtain the generated baryon asymmetry.

In order to generate the baryon asymmetry of the Universe sufficiently strong CP violation is needed. It was therefore proposed that at finite temperature there might be spontaneous (transitional) CP violation within the bubble walls at the electroweak phase transition in supersymmetric models. We investigate this question in the MSSM.

We compute the wall velocity in the MSSM with W, tops and stops contributing to the friction. In a wide range of parameters including those which fulfil the requirements of baryogenesis we find a wall velocity of order v_w ≈ 10^-2 much below the SM value.

We apply the linear delta expansion to the quantum mechanical version of the slow roll transition which is an important feature of inflationary models of the early universe. The method, which goes beyond the Gaussian approximation, gives results which stay close to the exact solution for longer than previous methods. It provides a promising basis for extension to a full field theoretic treatment.

We review the non-Fermi or marginal liquid behavior of a relativistic QED plasma. In this medium a quasiparticle has a damping rate that depends linearly on the distance between its energy and the Fermi surface. We stress that this dependence is due to the long-range character of the magnetic interactions in the medium. Finally, we study how the quark damping rate modifies the gap equation of color superconductivity, reducing the value of the gap at the Fermi surface.

In this talk I discuss models in which a homogeneous scalar field is used to modify standard cosmology above the nucleosynthesis scale to provide an explanation for the observed matter-antimatter asymmetry of the Universe.

In this talk we review the actual situation concerning electroweak phase transition and baryogenesis in the minimal supersymmetric extension of the Standard Model. A strong enough phase transition requires light Higgs and stop eigenstates. For a Higgs mass in the range 110–115 GeV, there is a stop window in the range 115–135 GeV. If the Higgs is heavier than 115 GeV, stronger constrains are imposed on the space of supersymmetric parameters. A baryon-to-entropy ratio is generated by the chargino sector provided that the µ parameter has a CP-violating phase larger than ~ 0.04.

It has recently been suggested that the baryon washout problem of the standard electroweak baryogenesis scenario could be avoided if inflation ends with a period of parametric resonance at a low enough energy density. I present results of numerical simulations in which this process was studied in the Abelian Higgs model. Our results show that because of the masslessness of the gauge field, the parametric resonance takes place naturally, and that the system reaches a quasi-equilibrium state in which the long-wavelength part of the spectrum has a high effective temperature. This enhances baryon number violation and makes baryogenesis more efficient.

Using a Hartree ensemble approximation, we investigate the dynamics of the φ⁴ model in 1 + 1 dimensions. We find that the fields initially thermalize with a Bose-Einstein distribution for the fields. Gradually, however, the distribution changes towards classical equipartition. Using suitable initial conditions quantum thermalization is achieved much faster than the onset of this undesirable equipartition. We also show how the numerical efficiency of our method can be significantly improved.

The on-shell imaginary part of the retarded selfenergy of massive φ⁴ theory in 1 + 1 dimensions is logarithmically infrared divergent. This leads to a zero in the spectral function, separating its usual bump into two. The twin peaks interfere in time-dependent correlation functions, which causes oscillating modulations on top of exponential-like decay, while the usual formulas for the decay rate fail. We see similar modulations in our numerical results for a mean field correlator, using a Hartree ensemble approximation.

We focus on the massive Thirring model in 1 + 1 dimensions at finite temperature T and non-zero chemical potential µ, and comment on some parallels between this model and QCD. In QCD, calculations of physical quantities such as transport coefficients are extremely difficult. In the massive Thirring model, similar calculations are greatly simplified by exploiting the duality which exists with the sine-Gordon model and its relation, at high T, to the exactly solvable classical Coulomb gas on the line.

The multiplicative anomaly, recently introduced in QFT, plays a fundamental role in solving some mathematical inconsistencies of the widely used zeta-function regularization method. Its physical relevance is still an open question and is here analyzed in the light of a non-perturbative method. Even in this approach the "different physics" seems to hold and not to be easily removable by renormalization.

We report on work studying the properties of the sphaleron in models of the electroweak interactions with two Higgs doublets in as model-independent a way as possible: by exploring the physical parameter space described by the masses and mixing angles of the Higgs particles. If one of the Higgs particles is heavy, there can be several sphaleron solutions, distinguished by their properties under parity and the behaviour of the Higgs field at the origin. In general, these solutions are not spherically symmetric, although the departure from spherical symmetry is small.

Collisions of non-topological solitons, Q-balls, are studied in the Minimal Supersymmetric Standard Model in two different cases: where supersymmetry has been broken by a gravitationally coupled hidden sector and by a gauge mediated mechanism at a lower energy scale. Q-ball collisions are studied numerically on a two dimensional lattice for a range of Q-ball charges. Total cross-sections as well as cross-sections for fusion and charge exchange are calculated.

We study the properties of rho mesons in nuclear matter by means of QCD sum rules at finite density. For increased sensitivity, we subtract out the vacuum contributions. With the spectral function as estimated in the literature, these subtracted sum rules are found to be not well satisfied. We suppose that Landau singularities from higher resonance states in the nearby region in this channel are the cause for this failure.

We propose using the Skyrme model on a lattice as an effective field theory of meson–baryon interactions. To this end we construct a local topological density that involves the volumes of tetrahedra in the target space S³ and we make use of Coxeter's formula for the Schläfli function to implement it. We calculate the mean-square radius of a skyrmion in the three-dimensional Skyrme model, and find some surprises.

Transition of the ground state of a classical Φ⁴ theory in 2 + 1 dimensions is studied from a metastable state into the stable equilibrium. The transition occurs in the broken Z² symmetry phase and is triggered by a vanishingly small amplitude homogeneous external field h. A phenomenological theory is proposed in form of an effective equation of the order parameter which quantitatively accounts for the decay of the false vacuum. The large amplitude transition of the order parameter between the two minima displays characteristics reflecting dynamical aspects of the Maxwell construction.

The following sections are included:

• List of participants