The Phase Diagram of Two Color QCD
Abstract
I present recent results from lattice simulations of SU(2) gauge theory with Wilson quark flavors, at nonzero quark chemical potential . The thermodynamic equation of state is discussed along with the nature of the high density matter which forms. It is conjectured that deconfinement may mean different things for bulk and Fermi surface phenomena.
1 Why Two Colors?
In this talk I will give an update on our project to simulate SU(2) gauge theory with quark flavors on cold lattices with quark chemical potential [1, 2, 3]. Because of the Sign Problem, QCD offers currently the best prospect of using lattice simulations to study gauge theories at nonzero baryon charge density. In particular, we can explore the systematics of the lattice approach in this unexplored physical regime. There is good news: unlike the case of hot QCD, it is possible to perform a scan in at fixed cutoff, and the primary thermodynamic observable, the quark number density , as a component of a conserved current has no quantum corrections. The bad news is that both UV and IR artifacts are significant and complicated; only recently have we begun to make progress in disentangling them [3].
A second, more theoretical motivation is that QCD with offers the chance to explore color deconfinement in a new physical régime, complementary to the more usually studied transition at . There are interesting differences which will challenge us to refine the language used to discuss this essential feature of nonabelian gauge theories.
2 The Simulation
We define our lattice theory with the Wilson plaquette action and unimproved Wilson fermions with hopping parameter . Chemical potential is introduced via the standard prescription of weighting forward/backward temporal hops by . Reality of the fermion measure follows from where the Pauli matrix acts on color. Our only innovation [1] is the introduction of a scalar, isoscalar and gaugeinvariant diquark source term , where subscripts denote flavor. Setting mitigates the IR fluctuations due to Goldstone modes associated with superfluidity at nonzero quark density. Indeed, is nothing but a Majorana mass for the quarks.
Fig. 2 shows how the computational effort, measured by both the number of congrad iterations and the HMC timestep required for decent acceptance varies with and . Despite the absence of a Sign Problem, studies of dense matter still require an order of magnitude more cpu than that needed for vacuum QCD.
By now we have accumulated matched ensembles on a coarse lattice with fm [1] and a finer lattice with fm [2], with scale set by assuming the two color string tension is (440MeV). Both lattices have temperature MeV, making them “cold” in QCD terms. The quark mass is relatively heavy, corresponding to — up to now we have not considered the chiral limit to be so important, because effects due to could naively be expected to lie deep at the bottom of the Fermi Sea. In our most recent simulations [3] we have explored the systematics both of varying the diquark source and varying , with corresponding to MeV.
3 Equation of State
Fig. 2 plots as a function of , calculated on the fine lattice with [2]. Here denotes the “StefanBoltzmann” (SB) result evaluated for free Wilson fermions on the same volume; taking the ratio is a first step towards correcting for discretisation and finite volume artifacts. These are considerable, as shown by the departures from unity in Fig. 6. For greater than unity, the dominant error is due to UV artifacts, but for smaller the disparity between the two curves points to the IR; indeed, the oscillatory behaviour is due to departures of the Fermi surface from sphericity as due to the discretisation of momentum space [4].
Fig. 4 illustrates the difficulty of disentangling these two effects, updating Fig. 2 for a range of temperatures with the limit taken. Fig. 4 presents the same data normalised by the continuum SB result . The apparent peak at , previously identified as a signal of BoseEinstein condensation (BEC) of tightly bound diquarks [1, 2], is most likely an IR artifact due to the dip in the dashed red free fermion curve of Fig. 6. On the other hand the difference in vertical scales shows that UV artifacts still need to be corrected.
Another previously neglected factor is the limit, which appears to be a much more important effect for interacting quarks; setting enhances their tendency to form pairs, in effect promoting BEC formation which could deform the results. Taking all factors into account, we conclude the approximate plateau in the range is consistent with , rather than 1.4 as suggested by Fig. 2. The same gross features are also apparent in a calculation of the pressure ; Fig. 6 shows the result of three different approaches to correcting for lattice artifacts at MeV [3].
Figs. 4,4 and 6 support an interpretation in terms of four distinct regimes separated by three thresholds. The first is the onset threshold, taking place at MeV, separating the vacuum state from a nonzero quark density . In mean field theory this is predicted to be a second order phase transition [5], unlike the first order transition expected in QCD. The nonmonotonic behaviour of above onset due to a BEC, predicted in mean fieldtheory, and reported in [1, 2] is, however, no longer a robust feature of the numerical results; it may well be that simulations far closer to the chiral limit are needed to expose this behaviour. Next, at MeV is a crossover to a régime where both and are approximately equal to their SB values, suggestive that a degenerate system of quarks with a welldefined Fermi momentum and Fermi energy has formed. For reasons that will be elaborated we refer to this as the quarkyonic régime. Finally at MeV both and start to climb above the SB values. If for the moment we assume the degenerate quark description still holds, then for we have as expected for weakly interacting massless quarks, but for , implying that the quark matter at very large densities has a large negative correction to the kinetic energy, ie. it is strongly selfbound.
4 Order Parameters and Phase Diagram
In order to characterise the high density system we have also examined two “order parameters”, one exact, the other approximate. Fig. 8 shows the superfluid order parameter as a function of , extrapolated to , for three different temperatures. For a degenerate system superfluidity arises through Cooper pair condensation of diquark pairs at the Fermi surface; hence should scale as the area of the Fermi surface . The plot shows , and indeed it is remarkably constant within the quarkyonic régime, although clearly sensitive. The temperature sensitivity becomes more marked once , and at the highest temperature studied the order parameter vanishes everywhere, implying restoration of the “normal” phase.
We also studied the Polyakov loop as a function of both and . Since this involves comparison of data with different , it has been necessary to renormalise via multiplication by a factor [6], determined at and normalised so that . The results are shown in Fig. 8; the inset shows the unrenormalised data. First consider the data from the lowest temperature with . Although in the presence of fundamental matter is not an exact order parameter for global Z center symmetry, its behaviour strongly suggests that the transition at is for all practical purposes identical with deconfinement, ie. the free energy for a static fundamental color source becomes finite once . The quark density at deconfinement is 16 – 32 fm (the uncertainty arises from the difficulty in handling lattice artifacts discussed in the previous section), some 30 – 60 times that of nuclear matter.
It’s then interesting that defined via falls rapidly as rises. A pragmatic definition of is the value at which . The resulting tentative phase diagram is shown in Fig. 10. There are at least three distinct regions/phases: a normal hadronic phase with , at low and ; the quarkyonic region with and at low and intermediate ; and a deconfined, normal phase with , at large and/or large . At this stage we cannot exclude a deconfined superfluid phase at large and small/intermediate . The “quarkyonic” nomenclature is now justified; this phase has the thermodynamic bulk scaling of a weaklyinteracting degenerate quarks, but remains confined. It thus has two of the features of dense baryonic matter originally proposed on the basis of large arguments [7].
5 Quark Number Susceptibility and Deconfinement
We have also recently calculated the quark number susceptibility [8, 3]. Whilst naively is expected to reflect local fluctuations of the operator, it turns out that at large the dominant term comes from a connected “hairpin” diagram. Fig. 10 shows divided by the continuum SB result. Once again, we note a large range over which the ratio is approximately constant; indeed it is compatible with one as , though more sensitive to the quark mass value used for the free fermions than other bulk observables [3].
The most striking feature of Fig. 10, however, is the absence of dependence at all but the highest temperature studied (141 MeV), is in stark contrast to the behaviour of in Fig. 8. Despite our intuition from the thermal transition in QCD [9], cannot be regarded as a proxy for once . We conjecture that the change in behaviour of the bulk thermodynamic quantities , and observed at is a transition from shortranged binary confining interactions to longerranged interactions among several quarks within the medium, and that this corresponds with the transition from weak to strong selfbinding noted in Sec. 3. The dependent behaviour of , in contrast, must be due to degrees of freedom close to the Fermi surface which can be thermally excited; a correlation between gapless excitations and has also been noted in analytical and numerical studies on small cold volumes [10]. The same phenomenon should inform a theory of transport properties.
6 Summary
QCD offers an accessible theoretical laboratory for the study of dense baryonic matter. For low at least three physical regions can be identified: the vacuum for ; a confined quarkyonic superfluid for ; and a deconfined phase for . The most recent simulations have only made our findings in the quarkyonic régime more robust. Not discussed here are new results for renormalised energy density, conformal anomaly, and chiral symmetry restoration [3].
This work used the DiRAC Facility jointly funded by STFC, the Large Facilities Capital Fund of BIS and Swansea University. We thank the DEISA Consortium (www.deisa.eu), funded through the EU FP7 project RI222919, for support within the DEISA Extreme Computing Initiative.
References
References
 [1] S. Hands, S. Kim and J.I. Skullerud, Eur. Phys. J. C 48 (2006) 193.
 [2] S. Hands, S. Kim and J.I. Skullerud, Phys. Rev. D 81 (2010) 091502(R).
 [3] S. Cotter, P. Giudice, S. Hands and J.I. Skullerud, Preprint arXiv:1210.4496 [heplat].
 [4] S. Hands and D.N. Walters, Phys. Lett. B 548 (2002) 196.

[5]
J.B. Kogut, M.A. Stephanov, D. Toublan, J.J.M. Verbaarschot and
A. Zhitnitsky,
Nucl. Phys. B 582 (2000) 477.  [6] S. Borsanyi et al., Phys. Lett. B 713 (2012) 342.
 [7] L. McLerran and R.D. Pisarski, Nucl. Phys. A 796 (2007) 83.
 [8] P. Giudice, S. Hands and J.I. Skullerud, PoS LATTICE 2011 (2011) 193.

[9]
Y. Aoki, Z. Fodor, S.D. Katz and K.K. Szabo,
Phys. Lett. B 643 (2006) 46;
A. Bazavov et al., Phys. Rev. D 80 (2009) 014504.  [10] S. Hands, T.J. Hollowood and J.C. Myers, JHEP 1007 (2010) 086; JHEP 1012 (2010) 057.