Krishna Rajagopal

QCD at finite baryon density: Nucleon droplets and color superconductivity

Abstract We use a variational procedure to study finite density QCD in an approximation in which the interaction between quarks is modelled by that induced by instantons. We find that uniform states with conventional chiral symmetry breaking have negative pressure with respect to empty space at all but the lowest densities, and are therefore unstable. This is a precisely defined phenomenon which motivates the basic picture of hadrons assumed in the MIT bag model, with nucleons as droplets of chiral symmetry restored phase. At all densities high enough that the chirally symmetric phase fills space, we find that color symmetry is broken by the formation of a 〈 qq 〉 condensate of quark Cooper pairs. A plausible ordering scheme leads to a substantial gap in a Lorentz scalar channel involving quarks of two colors, and a much smaller gap in an axial vector channel involving quarks of the third color.

Color superconductivity in dense quark matter

Matter at high density and low temperature is expected to be a color superconductor, which is a degenerate Fermi gas of quarks with a condensate of Cooper pairs near the Fermi surface that induces color Meissner effects. At the highest densities, where the QCD coupling is weak, rigorous calculations are possible, and the ground state is a particularly symmetric state, the color-flavor locked (CFL) phase. The CFL phase is a superfluid, an electromagnetic insulator, and breaks chiral symmetry. The effective theory of the low-energy excitations in the CFL phase is known and can be used, even at more moderate densities, to describe its physical properties. At lower densities the CFL phase may be disfavored by stresses that seek to separate the Fermi surfaces of the different flavors, and comparison with the competing alternative phases, which may break translation and/or rotation invariance, is done using phenomenological models. We review the calculations that underlie these results and then discuss transport properties of several color-superconducting phases and their consequences for signatures of color superconductivity in neutron stars.

Color-flavor locking and chiral symmetry breaking in high density QCD

Abstract We propose a symmetry breaking scheme for QCD with three massless quarks at high baryon density wherein the color and flavor SU(3) color × SU(3) L × SU(3) R symmetries are broken down to the diagonal subgroup SU(3) color+ L + R by the formation of a condensate of quark Cooper pairs. We discuss general properties that follow from this hypothesis, including the existence of gaps for quark and gluon excitations, the existence of Nambu-Goldstone bosons which are excitations of the diquark condensate, and the existence of a modified electromagnetic gauge interaction which is unbroken and which assigns integral charge to the elementary excitations. We present mean-field results for a Hamiltonian in which the interaction between quarks is modeled by that induced by single-gluon exchange. We find gaps of order 10–100 MeV for plausible values of the coupling. We discuss the effects of non-zero temperature, non-zero quark masses and instanton-induced interactions on our results.

Signatures of the Tricritical Point in QCD

Several approaches to QCD with two {ital massless} quarks at finite temperature T and baryon chemical potential {mu} suggest the existence of a tricritical point on the boundary of the phase with spontaneously broken chiral symmetry. In QCD with {ital massive} quarks there is then a critical point at the end of a first order transition line. We discuss possible experimental signatures of this point, which provide information about its location and properties. We propose a combination of event-by-event observables, including suppressed fluctuations in T and {mu} and, simultaneously, enhanced fluctuations in the multiplicity of soft pions. {copyright} {ital 1998} {ital The American Physical Society }

The Condensed matter physics of QCD

Important progress in understanding the behavior of hadronic matter at high density has been achieved recently, by adapting the techniques of condensed matter theory. At asymptotic densities, the combination of asymptotic freedom and BCS theory make a rigorous analysis possible. New phases of matter with remarkable properties are predicted. They provide a theoretical laboratory within which chiral symmetry breaking and confinement can be studied at weak coupling. They may also play a role in the description of neutron star interiors. We discuss the phase diagram of QCD as a function of temperature and density, and close with a look at possible astrophysical signatures.

Color superconductivity and chiral symmetry restoration at non-zero baryon density and temperature ☆

We explore the phase diagram of strongly interacting matter as a function of temperature and baryon number density, using a class of models for two-flavor QCD in which the interaction between quarks is modelled by that induced by instantons. Our treatment allows us to investigate the possible simultaneous formation of condensates in the conventional quark-anti-quark channel (breaking chiral symmetry) and in a quark-quark channel leading to color superconductivity: the spontaneous breaking of color symmetry via the formation of quark Cooper pairs. At low temperatures, chiral symmetry restoration occurs via a first-order transition between a phase with low (or zero) baryon density and a high density color superconducting phase. We find color superconductivity in the high density phase for temperatures less than of order tens to 100 MeV, and find coexisting 〈qq〉 and 〈qq〉 condensates in this phase in the presence of a current quark mass. At high temperatures, the chiral phase transition is second order in the chiral limit and is a smooth crossover for non-zero current quark mass. A tricritical point separates the first-order transition at high densities from the second-order transition at high temperatures. In the presence of a current quark mass this tricritical point becomes a second-order phase transition with Ising model exponents, suggesting that a long correlation length may develop in heavy ion collisions in which the phase transition is traversed at the appropriate density.

Anti-de sitter/conformal-field-theory calculation of screening in a hot wind.

One of the challenges in relating experimental measurements of the suppression in the number of J/\psi mesons produced in heavy ion collisions to lattice QCD calculations is that whereas the lattice calculations treat J/\psi mesons at rest, in a heavy ion collision a c\bar c pair can have a significant velocity with respect to the hot fluid produced in the collision. The putative J/\psi finds itself in a hot wind. We present the first rigorous non-perturbative calculation of the consequences of a wind velocity v on the screening length L_s for a heavy quark-antiquark pair in hot N=4 supersymmetric QCD. We find L_s(v,T) = f(v)[1-v^2]^{1/4}/\pi T with f(v) only mildly dependent on v and the wind direction. This L_s(v,T)\sim L_s(0,T)/\sqrt{\gamma} velocity scaling, if realized in QCD, provides a significant additional source of J/\Psi suppression at transverse momenta which are high but within experimental reach.

Wilson loops in heavy ion collisions and their calculation in AdS/CFT

Expectation values of Wilson loops define the nonperturbative properties of the hot medium produced in heavy ion collisions that arise in the analysis of both radiative parton energy loss and quarkonium suppression. We use the AdS/CFT correspondence to calculate the expectation values of such Wilson loops in the strongly coupled plasma of N = 4 super Yang-Mills (SYM) theory, allowing for the possibility that the plasma may be moving with some collective flow velocity as is the case in heavy ion collisions. We obtain the N = 4 SYM values of the jet quenching parameter ˆ q, which describes the energy loss of a hard parton in QCD, and of the velocity-dependence of the quark-antiquark screening length for a moving dipole as a function of the angle between its velocity and its orientation. We show that if the quark-gluon plasma is flowing with velocity vf at an angle θ with respect to the trajectory of a hard parton, the jet quenching parameteris modified by a factor γf (1 − vf cos θ), and show that this result applies in QCD as in N = 4 SYM. We discuss the relevance of the lessons we are learning from all these calculations to heavy ion collisions at RHIC and at the LHC. Furthermore, we discuss the relation between our results and those obtained in other theories with gravity duals, showing in particular that the ratio betweenin any two conformal theories with gravity duals is the square root of the ratio of their central charges. This leads us to conjecture that in nonconformal theories ˆ q defines a quantity that always decreases

Crystalline color superconductivity

In any context in which color superconductivity arises in nature, it is likely to involve pairing between species of quarks with differing chemical potentials. For suitable values of the differences between chemical potentials, Cooper pairs with nonzero total momentum are favored, as was first realized by Larkin, Ovchinnikov, Fulde, and Ferrell (LOFF). Condensates of this sort spontaneously break translational and rotational invariance, leading to gaps which vary periodically in a crystalline pattern. Unlike the original LOFF state, these crystalline quark matter condensates include both spin-zero and spin-one Cooper pairs. We explore the range of parameters for which crystalline color superconductivity arises in the QCD phase diagram. If in some shell within the quark matter core of a neutron star (or within a strange quark star) the quark number densities are such that crystalline color superconductivity arises, rotational vortices may be pinned in this shell, making it a locus for glitch phenomena.

Enforced Electrical Neutrality of the Color-Flavor Locked Phase

We demonstrate that quark matter in the color-flavor locked phase of QCD is rigorously electrically neutral, despite the unequal quark masses, and even in the presence of an electron chemical potential. As long as the strange quark mass and the electron chemical potential do not preclude the color-flavor locked phase, quark matter is automatically neutral. No electrons are required and none are admitted.