A Quarkyonic Phase in Dense Two Color Matter? Simon Hands Department of Physics, Swansea University, Singleton Park, Swansea SA2 8PP, U.K. Seyong Kim Department of Physics, Sejong University, Seoul 143-747, Korea. Jon-Ivar Skullerud Department of Mathematical Physics, National University of Ireland Maynooth, Maynooth, County Kildare, Ireland. We present results from simulations of Two Color QCD with two Wilson quark flavors in the presence of a quark chemical potential µ at two different lattice spacings. The equation of state, conformal anomaly, superfluid order parameter and Polyakov line are all discussed. Our results suggest that the transition from hadronic to quark matter, and that from confined to deconfined matter occur at distinct values of µ , consistent with the existence of a quarkyonic phase in this model. PACS numbers: 11.15.Ha,12.38.Aw,21.65.Qr A quantitative understanding of cold dense baryonic matter, ie. the region of the ( T, µ ) plane convention- ally placed at the lower right of the QCD phase diagram (here µ is the quark chemical potential), remains an out- standing problem in theoretical physics. There has been much discussion of possible exotic color superconduct- ing (CSC) phases in this region, where color-carrying de- grees of freedom such as quarks or gluons are all gapped via a Higgs-Meissner mechanism, and which may also be superfluid or even crystalline [1]. This picture has its firmest theoretical support at weak gauge coupling, where superconductivity is understood as being due to the BCS condensation of diquark Cooper pairs. More recently, an alternative scenario based on large- N$_{c}$ ar- guments has emerged, whose ground state at sufficiently high density has a restored chiral symmetry, so that ther- modynamic properties are well-described by a degenerate relativistic Fermi sea of quarks characterised by a Fermi momentum k$_{F}$ , but which remains in a confined “quarky- onic” phase so that all excitations necessarily carry color- singlet quantum numbers [2]. It remains unclear whether the CSC and quarkyonic pictures are truly distinct, or are in some sense comple- mentary. However, it is salutary to recall that even basic questions involving dense matter, such as the maximum stable mass of a neutron star, need quantitative input about the equation of state of ultradense matter regard- less of the nature of the ground state; this requires a controlled non-perturbative calculation. The most reliable source of such information, lattice QCD, is in general inoperable in this regime for the fol- lowing reason. In Euclidean metric the QCD Lagrangian density for quarks reads It is straightforward to show γ 5 M ( µ ) γ 5 ≡ M $^{†}$( − µ ), im- plying det M ( µ ) = (det M ( − µ )) $^{∗}$, and therefore that the path integral measure is not real and positive for µ = 0. Monte Carlo importance sampling, the mainstay of nu- merical lattice QCD, is ineffective. It is helpful to con- sider what goes wrong if the real positive measure factor det M $^{†}$M as implemented in, eg. the hybrid Monte Carlo (HMC) algorithm, is used. In QCD, while M describes a color triplet of quark fields q , the M † factor describes color antitriplet “conjugate quarks” q $^{c}$. Gauge singlet bound states of the form qq c resemble mesons, but carry non-zero baryon charge B > 0. The lightest such state is degenerate with the pseudo-Goldstone π -meson; hence HMC simulations with µ = 0 predict an unphysical “on- set” transition from the vacuum to a state with quark density n$_{q}$ > 0 at µ$_{o}$ ≃ 1 $_{2}$m$_{π}$ . The resulting ground state is a Bose-Einstein condensate (BEC) of diquark baryons, bearing no resemblance to nuclear matter, which phe- nomenologically we know forms at µ$_{o}$ ≈ 1 N $_{c}$m$_{nucleon}$ . The physical transition can only be found if the correct complex path integral measure det $^{2}$M is used, and must result from extremely non-trivial cancellations between configurations with differing phases – this has come to be known as the Silver Blaze problem [3]. In this paper we consider an alternative strongly- interacting theory, Two Color QCD (QC 2 D), in which the gauge group is SU(2). Since q and ¯ q live in equiva- lent representations of SU(2), it follows that det M ( µ ) ≡ det τ 2 M $^{∗}$( µ ) τ 2 is real and therefore the theory has a positive measure for an even number N$_{f}$ of quark fla- vors [4]. Physically this is expressed through both q ¯ q mesons and qq , ¯ q ¯ q baryons falling in the same hadron mulitplets. For sufficiently light quarks the scale hierar- chy m$_{π}$ ≪ m$_{ρ}$ permits the use of chiral perturbation the- ory ( χ PT) in studying the response of the lightest multi- plet to µ = 0 [5]. The key result is that for µ ≥ µ$_{o}$ ≡ 1 $_{2}$m$_{π}$ a non-zero baryon charge density n$_{q}$ > 0 does develop, along with a gauge-invariant superfluid order parameter which for N$_{f}$ = 2 reads 〈 qq 〉 ∼ 〈 ψ $^{tr}$Cγ 5 τ 2 ϵ$_{ab}$ψ 〉 = 0, L QCD = ¯ ψ ( D / [ A ] + µγ 0 + m ) ψ ≡ ¯ ψMψ. (1) arXiv:1001.1682v2 [hep-lat] 5 May 2010