QCD at Nonzero Baryon Density
Frithjof Karsch, Krishna Rajagopal, Jacobus Verbaarschot and
Uwe-Jens Wiese

We propose a three-month program for the study of QCD at nonzero baryon density at the INT during the year 2000. The behavior of matter at high quark density is interesting in itself and is relevant to phenomena in heavy ion collisions and in neutron stars. The temperature axis of the phase diagram for QCD as a function of temperature and chemical potential is relatively well understood: from extensive lattice QCD simulations it is clear that chiral symmetry is approximately restored and confinement ceases to be relevant above a temperature of about 150 MeV. On the other hand, we have little quantitative information about the phases and phase transitions which may occur at nonzero chemical potential. In a Euclidean formulation, the fermion determinant is complex in this regime, and this has so far foiled standard simulation methods. In particular, the evaluation of any physical quantity using standard techniques requires an exponentially large number of gauge field configurations, because of sign cancellations. The usual approximation of simply ignoring the fermion determinant fails dramatically in this case. Instead of resembling nuclear matter, the lowest density non-vacuum phase is dominated by unphysical hadrons which have a mass equal to that of the pion but which carry baryon number. These results (eg these failures) have recently been clearly understood within the context of a random matrix model with the chiral symmetries of the QCD partition function.

Because of the failure of Monte Carlo simulations, our understanding of the phase diagram derives from QCD-based models and qualitative arguments, rather than from controlled calculations in the full theory. During the past decade many studies have been performed within the context of various Nambu--Jona-Lasinio (NJL) models, which describe quarks interacting via four-fermion interactions. These models and many others support the long-held expectation that above some chemical potential, hadronic matter gives way to a phase in which chiral symmetry is restored. It was realized some time ago that this phase must be a color superconductor, if there is any channel in which the interaction between pairs of quarks is attractive. NJL models and instanton liquid models have both been used profitably within the last six months to yield some (perhaps semi-quantitative) insight into the symmetry breaking patterns and physical phenomena associated with this phase, in which a condensate of quark pairs develops, yielding a gap at the Fermi surface. Many questions remain, however. It is sobering to realize that even the symmetry properties of the ground state of matter at high density are still in doubt, as evidenced by the fact that the proposed diquark condensates break chiral symmetry in three-flavor QCD, but not in two-flavor QCD.

The combination of fascinating new proposed phenomena and a much clearer understanding of why and how standard lattice techniques fail make this an opportune time for a 3-month program, in which we propose to mount a concerted theoretical assault on these problems. A brief workshop in Bielefeld this year was very beneficial, and we think that all there would agree that a longer program is called for. Short workshops are planned for Brookhaven and Heidelberg during the coming year. The INT seems the perfect venue for a longer program.

One part of our effort would be further discussion of the problems with lattice simulations, and possible directions toward solutions. These problems can be emulated by a random matrix model which can be solved analytically. Numerical control of the phase of the fermion determinant of this much simpler model might be achievable during this meeting. During this meeting we should also critically evaluate different algorithms that have been tried. For example, the failure of the quenched approximation might not be complete and it might be possible to extract physical information from such calculations. Closely related to this is the use of an imaginary chemical potential, which can yield information about real densities via a Fourier transform. It is also intriguing to note that cluster algorithms have been used to achieve an exponential speed-up of Monte Carlo simulations of some scalar field theories with complex actions. This first of all suggests that a model including pions and baryons treated as skyrmions may be amenable to lattice simulation at finite chemical potential. Second, if the recently proposed reformulation of gauge theories as quantum link models were to yield a cluster algorithm for QCD, as is hoped, this could greatly ameliorate the sign problem in QCD itself. We therefore feel that progress during the program we propose is guaranteed (for example by the random matrix approach) and if we are lucky, by bringing together people coming at it from different directions we might even crack this hard problem wide open.

The second part of our program is complementary to the first, in that it does not rely on lattice simulations of the full theory. We will seek and address questions about the QCD phase diagram at finite temperature and density which can be answered based on general considerations alone, or upon making a small number of assumptions testable in the future. We will bring together people who have used several different model approaches and will discuss which predictions are robust and can be translated into unambiguous signatures in heavy ion experiments or neutron star astrophysics. As examples we mention recent developments concerning the tricritical point in the phase diagram and its signatures and some of the insights gained from the work on color superconductivity.

In addition to the core participants, whom we list below, we plan to invite a few people from each of several other communities, for parts of the program. We would want experimentalists to share SPS and RHIC results, and to help focus our investigation of signatures for heavy ion experiments. Some suggestions are: (list of names)

It will also be helpful to have one period of time during which we invite several experts on neutron star interiors. Some suggestions: (list of names)

Finally, there are many overlaps with condensed matter physics --- for example in the study of superconductivity, critical phenomena, and percolation. Also, the Hubbard model suffers from a sign problem which is similar to that in QCD. Suggestions for people in these areas: (list of names)

We turn now to the core participants. Our aim is to bring together a group of people working on QCD who have the expertise necessary to make progress at finite chemical potential. A tentative list of potential participants is: (list of names)

Our email addresses:
karsch@physik.uni-bielefeld.de
krishna@ctp.mit.edu
wiese@mitlns.mit.edu
verbaarschot@nuclear.physics.sunysb.edu


Effective Field Theories and Effective Interactions
Wick Haxton, David Kaplan, Peter Lepage

This proposal came out of discussions among the proposers about the intellectual connections between recent effective field theory treatments of the nucleon-nucleon interaction and the classical nuclear physics problem of solving the Schroedinger equation for A point nucleons interacting through a nonrelativistic potential. We believe that interactions among experts in these two areas could produce some interesting new ideas. A few of the specific motivations include:

1)Effective field theory approaches provide a systematic procedure for incorporating relativity, chiral symmetry, and many-body interactions into calculations. Furthermore, the formalism is easily extended to account for inelastic processes.

This contrasts with conventional many-body theory, where many-body forces are added phenomenologically to correct discrepancies, where inelastic processes are difficult to relate properly to the static potential, and where relativity may or may not be handled in an appropriate way, depending on the practitioner. Thus there is some prospect that effective field theory will provide important guidance to many-body nuclear physics generally.

2) Underlying both effective field theory and conventional many-body theory is the concept of integrating out high momentum degrees of freedom, so that true interactions and operators can be replaced by effective ones which give equivalent results at low energies. Yet, in part because conventional nuclear theory developed in a time when computing was difficult, nuclear effective theories have been carried out crudely, with many uncontrolled approximations. As several groups are now reexamining this old problem, we feel the parallels with effective field theory are important to explore.

3) A great deal has been learned in conventional nuclear structure about the ingredients necessary for accurate calculations. This information should allow effective field theory practitioners to gain some intuition about the order to which they need to work to realize results of a given precision.

4) Effective field theory is being applied to few-body nuclear physics problems: NN scattering, deuteron polarizabilities such as the anapole moment, nucleon-deuteron scattering, etc.

Extending this work to few-body systems with more proximate nucleons (3H, 3He, 4He) will be one challenging goal of this program. Another will be to bridge the theoretical gap between these calculations, and effective descriptions of many-body physics.

As a new direction in nuclear structure theory, it is important to have a forum where results can be presented to the broad community working in this area. This is especially true because many of the effective field theory practitioners come from the particle physics side of subatomic physics and thus do not have traditional interactions in this area.

5) Nonperturbative QCD, especially lattice QCD, has made major strides in recent years; accurate quantitative analyses of many low-energy nonperturbative features of QCD are proliferating. (This is in part because of the application of effective field theory methods in refining the relation between discretized QCD to continuum QCD). Unfortunately, direct application of lattice QCD methods to multi-nucleon systems remains far in the future; such systems are too complicated for conventional lattice techniqes. This situation is salvaged by effective field theory which provides a rigorous and systematic connection between conventional nuclear physics and QCD. Indeed it is quite conceivable that the coupling constants in a rigorous effective pion-nucleon field theory could be determined from lattice QCD simulations, leading to nuclear physics from first principles. A detailed strategy for such analyses would be very desirable.

6) Another important agenda item is to establish the energy region over which the effective field theories are viable: at what energy does the expansion upon which the effective field theory is built diverge. At higher energies QCD probably becomes essential; at lower energies it is stupidly inefficient to use QCD directly. This energy threshold is probably of order a couple of 100 MeV; a more accurate number requires a careful comparison of the (systematic) theory with experiment.

We would like to give one specific example of the kinds of interactions that such a program might engender. Four years ago the INT program of Bruce Barrett and James Vary (Microscopic Nuclear Structure Theory) concluded with a "benchmark challenge," to develop and implement an exact effective interactions treatment of finite nuclei. A possible approach suggested at that meeting by one of us - a possibility for efficiently solving the Bloch-Horowitz equation self consistently - has been successfully carried out for the deuteron and 3He. The resulting shell-model-like theory produces Faddeev-quality energies and operator matrix elements (e.g., the M1 form factor) for realistic potentials, such as Av18. This formalism immediately applies to a nucleus like 16O, where it remains formally exact but, of course, can be implemented only through finite order. There are some lovely ideas from effective field theory that might feed into such a program. One of these is notion that the highest energy quantum fluctuations, which now consume the vast majority of the numerical effort, can be resumed very accurately into a local operator. The physics this summation exploits is that the nuclear size is much larger than the distance a high momentum cluster can propagate. This should produce, for some realistic potential like Av18, an effective potential Av18(lambda), where all propagators are replaced by delta functions, by delta functions combined with two gradients that "see" the curvature of the wave function, and so on in higher order. This is almost exactly the same physics as in the contact interactions of effective field theory treatments, as has been recognized before. There are connections to the renormalization group. We hope our program will identify many such areas of overlap, and that progress will come from the resulting interactions between the nuclear structure and effective field theory groups.

The program would welcome participants interested in effective field theories for reasons other than nuclear structure or QCD. Effective field theory methods are used throughout theoretical physics --- for example in the recent attempt to relate 10-dimensional supergravity to ordinary non-supersymmetric 4-D QCD, resulting in (very) approximate nonperturbative glueball masses. One field that has similar needs to nuclear physics is high-precision atomic physics, for both single-electron and multi-electron atoms. Effective field theories are slowly revolutionizing the way people calculate in precision atomic physics. Since the data are extremely accurate, and the physics very well understood, atomic physics is one of the main proving grounds for effective field theory methods. Some contact, perhaps a couple of seminars, with people working on this atomic physics frontier might be useful.

Potential Participants:
Effective Field Theory Practitioners (list of names)
Nuclear Structure (list of names)


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