Strong & Electroweak Matter 2000

Proceedings of the SEWM2000 Meeting

S T R O N G AND ELECTROWEAK MATTER 2 0 0 0

i , ' -*''

if

Editor

C . P. Korthals Altes World Scientific

S T R O N G AND ELECTROWEAK MATTER 2 0 0 0

Proceedings of the SEWM2000 Meeting

STRONG AND ELECTROWEAK MATTER 2 0 0 0 Marseille, France

13-17 June 2000

Editor

C. P. Korthals Altes Centre Physique Theorique au CNRS Luminy F13288, Marseille, France

V|S* World Scientific « •

Sinqapore* L Singapore • NewJersey New Jersey •London • Hong Kong

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PREFACE

These proceedings contain the talks given at "Strong and Electroweak Matter 2000", a meeting organized at the Centre Physique Theorique and the Centre de Physique des Particules de Marseille, from 13 to 17 J u n e 2000. This meeting is held at a roughly two year basis. T h e first took place in Sintra (Portugal) in 1994, the second in Eger (Hungary) in 1997, the third at the Niels Bohr Institute (Copenhagen, Denmark) in 1998. T h e next meeting will take place in Heidelberg in June 2002. T h e conference is the meeting point of researchers in the p l a s m a phases of particle physics, and in cosmology. T h u s it combines physics on laboratory scale, such as the heavy ion experiments at C E R N (Geneva) and at RHIC (Brookhaven, USA), and on astronomical scales. T h e m a i n body of the talks are on theoretical aspects. Many are analytic in approach, m a n y numerical and a few experimental. T h e meeting had some four discussion sessions, each led by two moderators. Their crafty insistence on getting the audience involved was quite successful. We are sure t h a t the proceedings carry beneficial effects of these sessions. This format implied t h a t many short contributions had to be given as posters, and we thank the participants for their cooperation. They appear in this volume on the same basis as the contributed talks. T h e underlying volume has been divided into main talks, and contributed talks. T h e latter have been subdivided into Equilibrium Systems, Systems out of Equilibrium, and Miscellaneous. T h e talks on Equilibrium Systems have been subdivided into High Density and High Temperature systems. T h e International Committee consisted of J. Ambjorn (Copenhagen), P. Arnold (Virginia), Z. Fodor (Budapest), W. Buchmueller (Hamburg), K. Kaj a n t i e (Helsinki), F. Karsch (Bielefeld), L. McLerran (Brookhaven), A. Patkos (Budapest), A. Rebhan (Vienna), K. Rummukainen (Copenhagen), M. Shaposhnikov (Lausanne), M. Schmidt (Heidelberg), J. Smit ( A m s t e r d a m ) . They provided us with practical and indispensable advice on the format of the conference and otherwise. T h e conference has been sponsored by the Centre National de Recherche Scientifique, the Universite de la Mediterranee, the Ministere des Affaires Etrangeres, by the Conseil General of the Provence-Cote d'Azur and the Ville de Marseille, and we want to thank them for their support. We want to mention the help of the personnel of the C P T and the C P P M with the many forseen and unforseen problems. In particular use of the audi-

v

vi torium of the C P P M was due to the generosity of its director, Elie Aslanides. SEWM2000 participants had to go through severe physical hardships before they got to the conference banquet. Fortunately the afterdinner speech was quite relaxing, and we thank the speaker Prof Bobo Bizarski (who insisted on staying anonymous) for his emotionally and otherwise arousing performance. T h e unflinching assistance of Antonia Sueur, the conference secretary, m a d e the meeting into a success. Last but not least we thank Pierre Giovannangeli for his crucial help with the Proceedings. Chris P. Korthals Altes

CONTENTS

Preface

v M A I N TALKS

Color superconductivity in compact stars M. Alford, J. Bowers and K. Rajagopal

1

The colored glass condensate and extreme QCD E. Iancu

19

Aspects of parity, CP and time reversal violation in hot QCD D. Kharzeev, R. Pisarski and M. Tytgat

32

The deconfining phase transition, the Z/v domain walls and the magnetic Z/v symmetry A. Kovner

46

Electroweak phase transition beyond the Standard Model M. Laine

58

Aspects of semi-classical transport theory for QCD D. Litim

70

Do we understand the sphaleron rate? G. Moore

82

Debye screening in the QCD plasma 0. Philipsen

95

Why the Quark-Gluon plasma isn't a plasma R. Pisarski

107

Isospin matter D. Son and M. Stephanov

118

Non-perturbative dynamics of hot non-abelian gauge fields

129

L. Yaffe VII

VIII

C O N T R I B U T E D TALKS SYSTEMS IN EQUILIBRIUM High temperature systems The effective potential for composite operators in the scalar model at finite temperature G. Ananos and N. Svaiter

136

Three-loop free energy using screened perturbation theory J. Andersen

142

Hot scalar theory in large N: Bose-Einstein condensation P. Arnold and B. Tomdsik

148

The electrical conductivity in high temperature QED L. Bettencourt and E. Mottola

153

Random cluster model, percolation and deconfinement transition in quark-gluon plasma P. Blanchard and D. Gandolfo

160

Screening in hot SU(2) gauge theory and propagators in 3D adjoint Higgs model A. Cucchieri, F. Karsch and P. Petreczky

165

The effect of primordial temperature fluctuations on the QCD transition J. Ignatius and D. Schwarz

170

Dynamical resummation and damping in the O(N) model A. Jakovdc

175

QCD thermodynamics with two and three quark flavors F. Karsch, E. Laermann, A. Peikert, Ch. Schmidt and S. Stickan

180

Hadron correlators in the deconfined phase F. Karsch, M. Mustafa and M. Thoma

186

Testing MEM with diquark and thermal meson correlation functions F. Karsch and I. Wetzorke Improved resummations for the thermodynamics of the quark-gluon plasma A. Rebhan Phase transition in QCD H. Satz

193

199

204

High density systems Dimensional regularization and Mellin summation in high-temperature calculations D. Bedingham

226

Time dependent effective actions at finite temperature T. Evans

231

Scalar fields at finite densities: A delta expansion approach D. Winder

236

SYSTEMS OUT OF EQUILIBRIUM Nonequilibrium fields: Exact and truncated dynamics G. Aarts

241

A new field-theoretical path integral for out-of-equilibrium matter R. Alvarez-Estrada

247

Nonequilibrium dynamics in gauge theories J. Baacke and K. Heitmann

252

Solution of the Boltzmann equation for gluons after a heavy ion collision J. Bjoraker and R. Venugopalan MSSM electroweak phase transition on 4D lattices F. Csikor, Z. Fodor, P. Hegediis, A.jakovdc and A. Piroth

257

263

A new collision operator in hot QCD F. Guerin Baryogenesis at the electroweak phase transition for a SUSY model with a gauge singlet S. Huber and M. Schmidt

268

272

No spontaneous CP violation at finite temperature in the MSSM? P. John

279

Bubble wall velocity in the MSSM P. John and M. Schmidt

284

The quantum mechanics of the slow roll in the linear delta expansion H. Jones

289

Fermion damping rate effects in cold dense matter C. Manuel

294

Electrogenesis in a scalar field dominated epoch T. Prokopec

299

Electroweak baryogenesis M. Quiros

304

Preheating and phase transitions in gauge theories A. Rajantie

316

Scalar field dynamics: Classical, quantum and in between M. Salle, J. Smit and J. Vink

321

Twin peaks M. Salle, J. Smit and J. Vink

327

Exploiting duality in a toy model of QCD at non-zero T and /^: Massive Thirring model, sine-Gordon model and Coulomb gases D. Steer, A. Gomez Nicola, T. Evans and R. Rivers

332

XI

MISCELLANEOUS New physics in the charged relativistic Bose gas using zeta-function regularization? A. Filippi

337

Sphalerons with two Higgs doublets M. Hindmarsh and J, Grant

343

Q-ball collisions in the MSSM T. Multamaki

348

Rho meson properties in nuclear matter from QCD sum rules

353

A. Nyffeler Exact topological density in the lattice Skyrme model B. Svetitsky and A. Schramm

359

Classical order parameter dynamics and the decay of a metastable vacuum state Z. Szep

364

List of Participants

369

COLOR S U P E R C O N D U C T I V I T Y IN C O M P A C T STARS MARK ALFORD Dept. of Physics and Astronomy,

University of Glasgow, G12 8QQ, U.K.

JEFFREY BOWERS AND KRISHNA RAJAGOPAL* Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA, USA 02139 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.

1

Color Superconductivity and Color-Flavor Locking

Because QCD is asymptotically free, its high temperature and high baryon density phases are more simply and more appropriately described in terms of quarks and gluons as degrees of freedom, rather than hadrons. The chiral symmetry breaking condensate which characterizes the vacuum melts away. At high temperatures, in the resulting quark-gluon plasma phase all of the symmetries of the QCD Lagrangian are unbroken and the excitations have the quantum numbers of quarks and gluons. At high densities, on the other hand, quarks form Cooper pairs and new condensates develop. The formation of such superconducting phases 1 ' 2 ' 3,4 ' 5 ' 6 requires only weak attractive interactions; these phases may nevertheless break chiral symmetry 5 and have excitations with the same quantum numbers as those in a confined phase. 5 ' 7 ' 8 ' 9 These cold dense quark matter phases may arise in the cores of neutron stars; understanding this region of the QCD phase diagram requires an interplay between QCD and neutron star phenomenology. The relevant degrees of freedom in cold dense quark matter are those which involve quarks with momenta near the Fermi surface. At high density, where the quark number chemical potential \i (and hence the quark Fermi momentum) is large, the QCD gauge coupling g((j.) is small. However, because of the infinite degeneracy among pairs of quarks with equal and opposite momenta at the Fermi surface, even an arbitrarily weak attraction between quarks renders the Fermi surface unstable to the formation of a condensate of •SPEAKER. 1

2

quark Cooper pairs. Creating a pair costs no free energy at the Fermi surface and the attractive interaction results in a free energy benefit. Pairs of quarks cannot be color singlets, and in QCD with two flavors of massless quarks, they form in the (attractive) color 3 channel in which the quarks in a Cooper pair are color-antisymmetric 1 ' 2 ' 3 ' 4 . The resulting condensate creates a gap A at the Fermi surfaces of quarks with two out of three colors, but quarks of the third color remain gapless. Five gluons get a Meissner mass by the AndersonHiggs mechanism; 10 a 5L^(2)coi0r subgroup remains unbroken. The Cooper pairs are flavor singlets and no flavor symmetries are broken. There is also an unbroken global symmetry which plays the role of U(1)B- Thus, no global symmetries are broken in this 2SC phase. There must therefore be a phase transition between the hadronic and 2SC phases at which chiral symmetry is restored. This phase transition is first order 3 ' 11 ' 12 ' 13 since it involves a competition between chiral condensation and diquark condensation. 11 ' 13 In QCD with three flavors of massless quarks, the Cooper pairs cannot be flavor singlets, and both color and flavor symmetries are necessarily broken. The symmetries of the phase which results have been analyzed in Refs. 5 ' 7 . The attractive channel favored by one-gluon exchange exhibits "color-flavor locking." A condensate of the form (V'gVf) oc Aea0AeabA

(1)

involving left-handed quarks alone, with a, ft color indices and a, b flavor indices, locks SU(3)L flavor rotations to SU(3)coior: the condensate is not symmetric under either alone, but is symmetric under the simultaneous SU(3)L+COIOT rotations." A condensate involving right-handed quarks alone locks SU(3)R flavor rotations to SU(3)C0\0V. Because color is vectorial, the combined effect of the LL and RR condensates is to lock SU(3)L to SU(3)R, breaking chiral symmetry. 6 Thus, in quark matter with three massless quarks, the 5J7(3)coior x SU(3)L X SU(3)R X U(1)B symmetry is broken down to the global diagonal SU(3)CO\OT+L+R group. A gauged U(l) subgroup of the original symmetry group — a linear combination of one color generator and electromagnetism, which lives within SU(3)L X SU(3)R — also remains unbroken. All nine quarks have a gap. All eight gluons get a mass. 5 ' 15 There °It turns out 5 that condensation in the color 3 channel induces a condensate in the color 6 channel because this breaks no further symmetries. 8 The resulting condensates can be written in terms of M and K2 where ( V £ a ^ O ~ niSaaSl3b+ K2Sab5l3a. Here, the Kronecker 1 8 ; pairing in this theory is simpler to analyze because quark Cooper pairs are color singlets. The Nc —> oo limit of QCD is often one in which hard problems become tractable. However, the ground state of Nc = oo QCD is a chiral density wave, not a color

4

magnitude of the gaps in quark matter at large but accessible /J, has been estimated using two broad strategies. The first class of estimates are done within the context of models whose parameters are chosen to reproduce zero density physics.3,4,11,5,8,9,13,14,23,24,25,22 -J-J^ s e c o n c i strategy for estimating gaps and critical temperatures is to use fi = oo physics as a guide. At asymptotically large /i, models with short-range interactions are bound to fail because the dominant interaction is due to the long-range magnetic interaction coming from single-gluon exchange. 12 ' 26 The collinear infrared divergence in small angle scattering via one-gluon exchange (which is regulated by dynamical screening26) results in a gap which is parametrically larger at fj, —> oo than it would be for any point-like four-fermion interaction. 26 Weak coupling estimates of the gap26'27'28'29-30'31'32>33>34>35,36,37,38 a r e v a j i d a t a S y m p totically high densities, with chemical potentials /x > 108 MeV. 37 Neither class of methods can be trusted quantitatively for quark number chemical potentials /j, ~ 400 — 500 MeV, as appropriate for the quark matter which may occur in the cores of neutron stars. It is nevertheless satisfying that two very different approaches, one using zero density phenomenology to normalize models, the other using weak-coupling methods valid at asymptotically high density, yield predictions for the gaps and critical temperatures at accessible densities which are in good agreement: the gaps at the Fermi surface are of order tens to 100 MeV, with critical temperatures about half as large. Tc ~ 50 MeV is much larger relative to the Fermi momentum than in low temperature superconductivity in metals. This reflects the fact that color superconductivity is induced by an attraction due to the primary, strong, interaction in the theory, rather than having to rely on much weaker secondary interactions, as in phonon mediated superconductivity in metals. Quark matter is a high-Tc superconductor by any reasonable definition. Its Tc is nevertheless low enough that it is unlikely the phenomenon can be realized in heavy ion collisions. 2

Color Superconductivity in Compact Stars

Our current understanding of the color superconducting state of quark matter leads us to believe that it may occur naturally in compact stars. The critical superconductor. 1 9 At asymptotically high densities color superconductivity persists up to Nc's of order thousands 2 0 , 2 1 before being supplanted by the phase described in Ref. 1 9 . At any finite Nc, color superconductivity occurs at arbitrarily weak coupling whereas the chiral density wave does not. For Nc = 3, color superconductivity is still favored over the chiral density wave (although not by much) even if the interaction is so strong that the color superconductivity gap is ~ ft/2.22

5

temperature Tc below which quark matter is a color superconductor is high enough that any quark matter which occurs within neutron stars that are more than a few seconds old is in a color superconducting state. In the absence of lattice simulations, present theoretical methods are not accurate enough to determine whether neutron star cores are made of hadronic matter or quark matter. They also cannot determine whether any quark matter which arises will be in the CFL or 2SC phase: the difference between the u, d and s Fermi momenta will be a few tens of MeV which is comparable to estimates of the gap A; the CFL phase occurs when A is large compared to all differences between Fermi momenta. Just as the higher temperature regions of the QCD phase diagram are being mapped out in heavy ion collisions, we need to learn how to use neutron star phenomena to determine whether they feature cores made of 2SC quark matter, CFL quark matter or hadronic matter, thus teaching us about the high density region of the QCD phase diagram. It is therefore important to look for astrophysical consequences of color superconductivity. 2.1

Equation of State

Much of the work on the consequences of quark matter within a compact star has focussed on the effects of quark matter on the equation of state, and hence on the radius of the star. As a Fermi surface phenomenon, color superconductivity has little effect on the equation of state: the pressure is an integral over the whole Fermi volume. Color superconductivity modifies the equation of state at the ~ (A//i) 2 level, typically by a few percent. 3 Such small effects can be neglected in present calculations, and for this reason we will not attempt to survey the many ways in which observations of neutron stars are being used to constrain the equation of state. 39 We will describe one current idea, however. As a neutron star in a low mass X-ray binary (LMXB) is spun up by accretion from its companion, it becomes more oblate and its central density decreases. If it contains a quark matter core, the volume fraction occupied by this core decreases, the star expands, and its moment of inertia increases. This raises the possibility 40 of a period during the spin-up history of an LMXB when the neutron star is gaining angular momentum via accretion, but is gaining sufficient moment of inertia that its angular frequency is hardly increasing. In their modelling of this effect, Glendenning and Weber 40 discover that LMXB's should spend a significant fraction of their history with a frequency of around 200 Hz, while their quark cores are being spun out of existence, before eventually spinning up to higher frequencies. This may explain the observation that LMXB frequencies are clustered around 250-350 Hz, 41 which is otherwise puzzling in

6

that it is thought that LMXB's provide the link between canonical pulsars and millisecond pulsars, which have frequencies as large as 600 Hz.42 It will be interesting to see how robust the result of Ref. 40 is to changes in model assumptions and also how its predictions fare when compared to those of other explanations which posit upper bounds on LMXB frequencies,43 rather than a most probable frequency range with no associated upper bound. 40 We note here that because Glendenning and Weber's effect depends only on the equation of state and not on other properties of quark matter, the fact that the quark matter must in fact be a color superconductor will not affect the results in any significant way. If Glendenning and Weber's explanation for the observed clustering of LMXB frequencies proves robust, it would imply that pulsars with lower rotational frequencies feature quark matter cores. 2.2

Cooling by Neutrino

Emission

We turn now to neutron star phenomena which are affected by Fermi surface physics. For the first 10 5 ~ 6 years of its life, the cooling of a neutron star is governed by the balance between heat capacity and the loss of heat by neutrino emission. How are these quantities affected by the presence of a quark matter core? This has been addressed recently in Refs. 44>455 following earlier work in Ref. 46 . Both the specific heat Cy and the neutrino emission rate Lv are dominated by physics within T of the Fermi surface. If, as in the CFL phase, all quarks have a gap A » T then the contribution of quark quasiparticles to Cy and L„ is suppressed by ~ exp(—A/T). There may be other contributions to L„, 44 but these are also very small. The specific heat is dominated by that of the electrons, although it may also receive a small contribution from the CFL phase Goldstone bosons. Although further work is required, it is already clear that both Cy and Lv are much smaller than in the nuclear matter outside the quark matter core. This means that the total heat capacity and the total neutrino emission rate (and hence the cooling rate) of a neutron star with a CFL core will be determined completely by the nuclear matter outside the core. The quark matter core is "inert": with its small heat capacity and emission rate it has little influence on the temperature of the star as a whole. As the rest of the star emits neutrinos and cools, the core cools by conduction, because the electrons keep it in good thermal contact with the rest of the star. These qualitative expectations are nicely borne out in the calculations presented by Page et al. 45 The analysis of the cooling history of a neutron star with a quark matter core in the 2SC phase is more complicated. The red and green up and down quarks pair with a gap many orders of magnitude larger than the tempera-

7

ture, which is of order 10 keV, and are therefore inert as described above. Any strange quarks present will form a (ss) condensate with angular momentum J — 1 which locks to color in such a way that rotational invariance is not broken. 47 The resulting gap has been estimated to be of order hundreds of keV, 47 although applying results of Ref. 48 suggests a somewhat smaller gap, around 10 keV. The blue up and down quarks also pair, forming a J = 1 condensate which breaks rotational invariance.3 The related gap was estimated to be a few keV,3 but this estimate was not robust and should be revisited in light of more recent developments given its importance in the following. The critical temperature Tc above which no condensate forms is of order the zero-temperature gap A. (Tc = 0.57A for J = 0 condensates. 27 ) Therefore, if there are quarks for which A ~ T or smaller, these quarks do not pair at temperature T. Such quark quasiparticles will radiate neutrinos rapidly (via direct URCA reactions like d -> u + e + u, u -) d + e+ + v, etc.) and the quark matter core will cool rapidly and determine the cooling history of the star as a whole. 46 ' 45 The star will cool rapidly until its interior temperature is T < Tc ~ A, at which time the quark matter core will become inert and the further cooling history will be dominated by neutrino emission from the nuclear matter fraction of the star. If future data were to show that neutron stars first cool rapidly (direct URCA) and then cool more slowly, such data would allow an estimate of the smallest quark matter gap. We are unlikely to be so lucky. The simple observation of rapid cooling would not be an unambiguous discovery of quark matter with small gaps; there are other circumstances in which the direct URCA processes occur. However, if as data on neutron star temperatures improves in coming years the standard cooling scenario proves correct, indicating the absence of the direct URCA processes, this would rule out the presence of quark matter with gaps in the 10 keV range or smaller. The presence of a quark matter core in which all gaps are 3> T can never be revealed by an analysis of the cooling history.

2.3

Supernova Neutrinos

We now turn from neutrino emission from a neutron star which is many years old to that from the protoneutron star during the first seconds of a supernova. Carter and Reddy 49 have pointed out that when this protoneutron star is at its maximum temperature of order 30-50 MeV, it may have a quark matter core which is too hot for color superconductivity. As such a protoneutron star core cools over the next few seconds, this quark matter will cool through Tc, entering the color superconducting regime of the QCD phase diagram. For T ~ Tc, the specific heat rises and the cooling slows. Then, as T drops

8

further and A increases to become greater than T, the specific heat drops rapidly. Furthermore, as the number density of quark quasiparticles becomes suppressed by exp(—A/T), the neutrino transport mean free path rapidly becomes very long. 49 This means that all the neutrinos previously trapped in the now color superconducting core are able to escape in a sudden burst. If a terrestrial neutrino detector sees thousands of neutrinos from a future supernova, Carter and Reddy's results suggest that there may be a signature of the transition to color superconductivity present in the time distribution of these neutrinos. Neutrinos from the core of the protoneutron star will lose energy as they scatter on their way out, but because they will be the last to reach the surface of last scattering, they will be the final neutrinos received at the earth. If they are released from the quark matter core in a sudden burst, they may therefore result in a bump at late times in the temporal distribution of the detected neutrinos. More detailed study remains to be done in order to understand how Carter and Reddy's signature, dramatic when the neutrinos escape from the core, is processed as the neutrinos traverse the rest of the protoneutron star and reach their surface of last scattering. 2.4

R-mode Instabilities

Another arena in which color superconductivity comes into play is the physics of r-mode instabilities. A neutron star whose angular rotation frequency Cl is large enough is unstable to the growth of r-mode oscillations which radiate away angular momentum via gravitational waves, reducing fl. What does "large enough" mean? The answer depends on the damping mechanisms which act to prevent the growth of the relevant modes. Both shear viscosity and bulk viscosity act to damp the r-modes, preventing them from going unstable. The bulk viscosity and the quark contribution to the shear viscosity both become exponentially small in quark matter with A > T and as a result, as Madsen 50 has shown, a compact star made entirely of quark matter with gaps A = 1 MeV or greater is unstable if its spin frequency is greater than tens to 100 Hz. Many compact stars spin faster than this, and Madsen therefore argues that compact stars cannot be strange quark stars unless some quarks remain ungapped. Alas, this powerful argument becomes much less powerful in the context of a neutron star with a quark matter core. First, the rmode oscillations have a wave form whose amplitude is largest at large radius, outside the core. Second, in an ordinary neutron star there is a new source of damping: friction at the boundary between the crust and the neutron superfluid "mantle" keeps the r-modes stable regardless of the properties of a quark matter core. 51,50

9

2.5

Magnetic Field Evolution

Next, we turn to the physics of magnetic fields within color superconducting neutron star cores. 52 ' 53 The interior of a conventional neutron star is a superfluid (because of neutron-neutron pairing) and is an electromagnetic superconductor (because of proton-proton pairing). Ordinary magnetic fields penetrate it only in the cores of magnetic flux tubes. A color superconductor behaves differently. At first glance, it seems that because a diquark Cooper pair has nonzero electric charge, a diquark condensate must exhibit the standard Meissner effect, expelling ordinary magnetic fields or restricting them to flux tubes within whose cores the condensate vanishes. This is not the case. 53 In both the 2SC and CFL phase a linear combination of the U(l) gauge transformation of ordinary electromagnetism and one (the eighth) color gauge transformation remains unbroken even in the presence of the condensate. This means that the ordinary photon A^ and the eighth gluon G® are replaced by new linear combinations A® = cos ao An + sin ao G® A* = - sin a0 A^ + cos a0 G®

(2)

where A® is massless and A* is massive. That is, BQ satisfies the ordinary Maxwell equations while Bx experiences a Meissner effect. The mixing angle Qo is the analogue of the Weinberg angle in electroweak theory, in which the presence of the Higgs condensate causes the A^ and the third SU{2)w gauge boson to mix to form the photon, A^, and the massive Z boson, sin(ao) is proportional to e/g and turns out to be about 1/20 in the 2SC phase and 1/40 in the CFL phase. 53 This means that the Q-photon which propagates in color superconducting quark matter is mostly photon with only a small gluon admixture. If a color superconducting neutron star core is subjected to an ordinary magnetic field, it will either expel the X component of the flux or restrict it to flux tubes, but it can (and does 53 ) admit the great majority of the flux in the form of a BQ magnetic field satisfying Maxwell's equations. The decay in time of this "free field" (i.e. not in flux tubes) is limited by the Q-conductivity of the quark matter. A color superconductor is not a Qsuperconductor — that is the whole point — but it turns out to be a very good (^-conductor due to the presence of electrons: the BQ magnetic field decays only on a time scale which is much longer than the age of the universe. 53 This means that a quark matter core within a neutron star serves as an "anchor" for the magnetic field: whereas in ordinary nuclear matter the magnetic flux tubes can be dragged outward by the neutron superfluid vortices as the star spins down, 54 the magnetic flux within the color superconducting core simply

10

cannot decay. Even though this distinction is a qualitative one, it will be difficult to confront it with data since what is observed is the total dipole moment of the neutron star. A color superconducting core anchors those magnetic flux lines which pass through the core, while in a neutron star with no quark matter core the entire internal magnetic field can decay over time. In both cases, however, the total dipole moment can change since the magnetic flux lines which do not pass through the core can move. 3

Crystalline Color Superconductivity and Glitches in Quark Matter

The final consequence of color superconductivity we wish to discuss is the possibility that (some) glitches may originate within quark matter regions of a compact star. 48 In any context in which color superconductivity arises in nature, it is likely to involve pairing between species of quarks with differing chemical potentials. If the chemical potential difference is small enough, BCS pairing occurs as we have been discussing. If the Fermi surfaces are too far apart, no pairing between the species is possible. The transition between the BCS and unpaired states as the splitting between Fermi momenta increases has been studied in electron superconductors, 55 nuclear superfluids56 and QCD superconductors, 8,9 ' 57 assuming that no other state intervenes. However, there is good reason to think that another state can occur. This is the "LOFF" state, first explored by Larkin and Ovchinnikov58 and Fulde and Ferrell59 in the context of electron superconductivity in the presence of magnetic impurities. They found that near the unpairing transition, it is favorable to form a state in which the Cooper pairs have nonzero momentum. This is favored because it gives rise to a region of phase space where each of the two quarks in a pair can be close to its Fermi surface, and such pairs can be created at low cost in free energy. Condensates of this sort spontaneously break translational and rotational invariance, leading to gaps which vary periodically in a crystalline pattern. 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. In Ref. 4 8 , we have explored the range of parameters for which crystalline color superconductivity occurs in the QCD phase diagram, upon making various simplifying assumptions. We focus primarily on a toy model in which the quarks interact via a four-fermion interaction with the quantum numbers of single gluon exchange. Also, we only consider pairing between u and d quarks, with fid = p, + S/x and fiu — jl — S/j,, whereas we expect a LOFF state

11

wherever the difference between the Fermi momenta of any two quark flavors is near an unpairing transition, including, for example, near the unlocking phase transition between the 2SC and CFL phases. In the LOFF state, each Cooper pair carries momentum 2q with |q| « 1.2(5/x. The condensate and gap parameter vary in space with wavelength 7r/|q|. In Ref. 48 , we simplify the calculation by assuming that the condensate varies in space like a plane wave, leaving the determination of the crystal structure of the QCD LOFF phase to future work. We give an ansatz for the LOFF wave function, and by variation obtain a gap equation which allows us to solve for the gap parameter A^, the free energy and the values of the diquark condensates which characterize the LOFF state at a given 6fi and |q|. We then vary |q|, to find the preferred (lowest free energy) LOFF state at a given dfx, and compare the free energy of the LOFF state to that of the BCS state with which it competes. We show results for one choice of parameters 6 in Fig. 1(a). The LOFF state is characterized by a gap parameter A^ and a diquark condensate, but not by an energy gap in the dispersion relation: we obtain the quasiparticle dispersion relations 48 and find that they vary with the direction of the momentum, yielding gaps that vary from zero up to a maximum of A A- The condensate is dominated by the regions in momentum space in which a quark pair with total momentum 2q has both members of the pair within ~ A^ of their respective Fermi surfaces. Because it violates rotational invariance by involving Cooper pairs whose momenta are not antiparallel, the quark matter LOFF state necessarily features condensates in both the J = 0 and J = 1 channels. (Cooper pairs in the symmetric J — 1 channel are antisymmetric in color but symmetric in flavor, and are impossible in the original LOFF context of pairing between electrons, which have neither color nor flavor.) Both J = 0 and J = 1 condensates are present even if there is no interaction in the J = 1 channel, as is the case when we use a four-fermion interaction with the quantum numbers of Lorentz-invariant single gluon exchange. Because there is no interaction in the J = 1 channel, the J = 1 condensate does not affect the quasiparticle dispersion relations; that is, the J — 1 gap parameter vanishes. The LOFF state is favored for values of S^i which satisfy 6fii < 5fi < / T5T 2k+ S{x ~ £ ) x(ji(x+,t+,k±)Ai(x+,-lr+,-kj_)),

(17)

where the brackets denote the average over the hadron wavefunction. In the LC-gauge gauge, Fl+(k) = ik+Al(k), and therefore (with k+ = xP+) :

xG(x,Q2) = iJ^Q(Q2-kl)(F^(x+,k)F:+(x+,-k)).

(18)

If the integration over k± were unrestricted, this quantity would be manifestly gauge-invariant. But even for a finite Q2, this has a gauge-invariant meaning Of course, the Fock-space gluon distribution can be defined in any gauge; but it is only in the LC-gauge that the definition (17) can be given a gauge invariant meaning 1 , l s .

26

when evaluated on the non-Abelian Weizsacker-Williams field in Sect. 3.1 16 . In this approximation, Fi+{x+,k) « Ti+{kx_) = - ^ ( f c j . ) (cf. eq. (15)), and xG(x,Q*)*R2J

| ^ |

d

V e - ^ ^ ( < ( 0 ) < ( * x ) )

A

,

(19)

where R is the hadron radius (I have assumed homogeneity in the transverse plane, for simplicity), and the average is to be understood in the sense of eq. (13) with A+ = xP+. Thus, the r.h.s. of eq. (19) is still dependent on x, but only via the respective dependence of the weight function for p. With the Gaussian weight function (16), and the non-linear classical solution in Sect. 3.1, the gluon distribution (19) can be computed exactly 6 ' 7 : U™(0) AZ(x±)) = - ^ — \ / irasNc

-2 x±

,

(20)

where Nc is the number of colors, and Qa oc O:S£A is the saturation momentum and is a function of A + , that is, of Bjorken's x. This equation displays saturation: the vector potential never becomes larger than A1 ~ \/g. This is the maximal occupation number permitted for a classical field, since larger occupation numbers are blocked by repulsive interactions of the gluon field. This interpretation can be made sharper by going to momentum space: If N(k±) is the Fourier transform of (20) [this is the same as (dN/d?k± dr d2x±), the gluon density per unit rapidity and unit transverse phase-space], then: N(k±)

<x as(Q2Jkl)

for

k\ » Q\,

(21)

which is the normal perturbative behavior, but 6 N(k±)

<x -

In %

for

A2QCD

« k\ « Q%,

(22)

which shows a much slower increase, i.e., saturation, at low momenta. Note, however, that the above argument is not rigurous, since the local Gaussian form for W[p] in eq. (16) is valid only at sufficiently large transverse momentum scales so that the effects of high gluon density are small. It is therefore important to verify if saturation comes up similarly with a more realistic form for the weight function, as obtained after including the quantum evolution in x. This will be discussed now. e

I n relativistic heavy ion collisions, one expects 5 Q„ ~ 1 GeV at RHIC, and Qs ~ 2 - 3 GeV at LHC. Thus, at least at LHC, the kinematical window in eq. (22) should be non-negligible.

27

4

The non-linear evolution equation

Because the separation of scales is only logarithmic, the effective theory (5)~ (6) with scale A + applies only to gluon correlations at a scale p+ slightly below A + . If one is interested in correlations at the softer scale 6A+ with i « 1, then, to LLA, one has to include also the corrections of order as \n(l/b) due to the semi-fast quantum fluctuations with longitudinal momenta in the strip bA+ < \k+\ < A+.

(23)

Together with the dynamical information already contained in the effective theory at scale A + ("Theory I"), these additional corrections will determine the effective theory at the softer scale 6A+ ("Theory II"). This suggests an iterative construction of the effective theory where the quantum fluctuations are integrated out in layers of k+, down to the physical scale of interest. At each step in this procedure, the quantum corrections must be computed to leading order in as ln(l/6) (LLA), but to all orders in the strong background field A1 ~ l/g produced by the color source p at the previous step. To compute quantum corrections, one needs the quantum generalization of the McLerran-Venugopalan model 8 - 9 ' 10 . This is obtained by replacing eq. (6) with (below, the gauge condition A+ = 0 is implicit):

J

{

f VA elSlA>ri

J

which involves two functional integrals: a quantum path integral over the soft (k+ < A + ) gluon fields A11, which defines quantum expectation values at fixed p, and a classical average over p, with weight function WA[P]- (This double averaging is similar to the one performed for spin systems at finite temperature and in a random external magnetic field 1B.) Unlike eq. (6), the 2-point function given by eq. (24) is independent of the arbitrary separation scale A + : the cutoff dependence of the quantum loops cancels against the corresponding dependence of the classical weight function WA[P]. The action 5[-4,p] is chosen such as to be gauge-invariant and reproduce the classical equations of motion (5) in the saddle point approximation SS/SA" = 0. It reads 8 : S = SYM +SW, where SYM = / d 4 x { - F * j 4 ) is the Yang-Mills action, and Sw is a gauge-invariant generalization of the eikonal vertex J