aa r X i v : . [ h e p - l a t ] O c t The Phase Diagram of Two Color QCD
Simon Hands , Seamus Cotter , Pietro Giudice andJon-Ivar Skullerud Physics Department, College of Science, Swansea University, Singleton Park,Swansea SA2 8PP, UK Department of Mathematical Physics, National University of Ireland Maynooth,Maynooth, County Kildare, IrelandE-mail: [email protected]
Abstract.
I present recent results from lattice simulations of SU(2) gauge theory with N f = 2Wilson quark flavors, at non-zero quark chemical potential µ . The thermodynamic equation ofstate is discussed along with the nature of the high density matter which forms. It is conjecturedthat 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 N f = 2quark flavors on cold lattices with quark chemical potential µ = 0 [1, 2, 3]. Because of the SignProblem, QC D offers currently the best prospect of using lattice simulations to study gaugetheories at non-zero baryon charge density. In particular, we can explore the systematics ofthe lattice approach in this unexplored physical regime. There is good news: unlike the case ofhot QCD, it is possible to perform a scan in µ at fixed cutoff, and the primary thermodynamicobservable, the quark number density n q ≡ − ∂f /∂µ , as a component of a conserved currenthas no quantum corrections. The bad news is that both UV and IR artifacts are significant andcomplicated; only recently have we begun to make progress in disentangling them [3].A second, more theoretical motivation is that QC D with µ = 0 offers the chance to explorecolor deconfinement in a new physical r´egime, complementary to the more usually studiedtransition at T >
0. There are interesting differences which will challenge us to refine thelanguage used to discuss this essential feature of non-abelian gauge theories.
2. The Simulation
We define our lattice theory with the Wilson plaquette action and unimproved Wilson fermionswith hopping parameter κ . Chemical potential is introduced via the standard prescription ofweighting forward/backward temporal hops by e ± µa . Reality of the fermion measure followsfrom det M = det τ M ∗ τ where the Pauli matrix τ acts on color. Our only innovation [1]is the introduction of a scalar, isoscalar and gauge-invariant diquark source term jqq ≡ jκ ( ψ tr Cγ τ ψ − ¯ ψ Cγ τ ¯ ψ tr ), where subscripts denote flavor. Setting j = 0 mitigates theIR fluctuations due to Goldstone modes associated with superfluidity at non-zero quark density.Indeed, j is nothing but a Majorana mass for the quarks. Fig. 1 shows how the computationaleffort, measured by both the number of congrad iterations and the HMC timestep dt required N c g j=0.04j=0.04 V=16³x24j=0.03j=0.02 µ a d t A cce p t a n ce Figure 1.
Computational effort needed forQC D. µ (GeV) n q /n SB p/p SB Figure 2. n q /n SBq and p/p SB for ja = 0 . × µ and j . Despite the absence of a Sign Problem, studies ofdense 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 8 ×
16 lattice with a =0 . ×
24 lattice with a = 0 . . Both lattices have temperature T ≈ m π /m ρ ≈ .
8— up to now we have not considered the chiral limit to be so important, because effects dueto m = 0 could naively be expected to lie deep at the bottom of the Fermi Sea. In our mostrecent simulations [3] we have explored the systematics both of varying the diquark source ja = 0 . , . . . , .
02 and varying T , with N τ = 24 , . . . , T = 47 , . . . ,
3. Equation of State
Fig. 2 plots n q /n SBq as a function of µa , calculated on the fine lattice with ja = 0 .
04 [2]. Here n SBq denotes the “Stefan-Boltzmann” (SB) result evaluated for free Wilson fermions on the samevolume; taking the ratio is a first step towards correcting for discretisation and finite volumeartifacts. These are considerable, as shown by the departures from unity in Fig. 5. For µa greater than unity, the dominant error is due to UV artifacts, but for smaller µ the disparitybetween the two curves points to the IR; indeed, the oscillatory behaviour is due to departuresof the Fermi surface from sphericity as T → j → N f N c ( µ /π + µT ) /
3. The apparent peak at µa ≈ .
4, previouslyidentified as a signal of Bose-Einstein condensation (BEC) of tightly bound diquarks [1, 2], ismost likely an IR artifact due to the dip in the dashed red free fermion curve of Fig. 5. On theother hand the difference in vertical scales shows that UV artifacts still need to be corrected.Another previously neglected factor is the j → j = 0 enhances their tendency to form pairs, ineffect promoting BEC formation which could deform the results. Taking all factors into account,we conclude the approximate plateau in the range 0 . < µa < . n q /n SBq ≈ p = R µ n q dµ ; Fig. 6 shows the result of three different approaches to correctingfor lattice artifacts at T = 47MeV [3].Figs. 3,4 and 6 support an interpretation in terms of four distinct regimes separated by three .2 0.4 0.6 0.8 1 µ a n q / n S B l a t N τ =24 N τ =12 N τ =8 Figure 3. n q /n SBq for j → µ a n q / n S B c on t Figure 4. n q /n SBq for j → µ a n l a t S B / n S B c on t x2416 x24 Figure 5. π n SBq /N f N c µ vs. µa on twodifferent lattice volumes. µ a (p/p SB ) (p/p SB ) _I (p/p SB ) II Figure 6. p/p SB on 12 ×
24 for j → µ o = m π / ≈ . a − ≈ n q >
0. In mean field theory thisis predicted to be a second order phase transition [5], unlike the first order transition expectedin QCD. The non-monotonic behaviour of n q /n SBq above onset due to a BEC, predicted inmean field-theory, and reported in [1, 2] is, however, no longer a robust feature of the numericalresults; it may well be that simulations far closer to the chiral limit are needed to expose thisbehaviour. Next, at µ Q ≈ . a − ≈ n q and p areapproximately equal to their SB values, suggestive that a degenerate system of quarks with awell-defined Fermi momentum k F ∝ n / q and Fermi energy E F ≈ µ has formed. For reasons thatwill be elaborated we refer to this as the quarkyonic r´egime. Finally at µ d ≈ . a − ≈ n q and p start to climb above the SB values. If for the moment we assume the degeneratequark description still holds, then for µ Q < µ < µ d we have E F ≈ k F as expected for weaklyinteracting massless quarks, but E F < k F for µ > µ d , implying that the quark matter at verylarge densities has a large negative correction to the kinetic energy, ie. it is strongly self-bound.
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. 7 shows the superfluid order parameter h qq i as a µ a < qq > / µ T a= a =1/12T a =1/8 Figure 7. h qq i /µ for j → T . µ a < L > N τ =24 N τ =16 N τ =12 N τ = 8 Figure 8.
Polyakov line L ( µ ) for various T .function of µ , extrapolated to j = 0, for three different temperatures. For a degenerate systemsuperfluidity arises through Cooper pair condensation of diquark pairs at the Fermi surface;hence h qq i should scale as the area of the Fermi surface k F . The plot shows h qq i /µ , andindeed it is remarkably constant within the quarkyonic r´egime, although clearly T -sensitive.The temperature sensitivity becomes more marked once µ > µ d , and at the highest temperaturestudied the order parameter vanishes everywhere, implying restoration of the “normal” phase.We also studied the Polyakov loop L as a function of both µ and T . Since this involvescomparison of data with different N τ , it has been necessary to renormalise L via multiplicationby a factor Z N τ L [6], determined at µ = 0 and normalised so that L ( T = 1 / a ) ≡
1. The resultsare shown in Fig. 8; the inset shows the unrenormalised data. First consider the data fromthe lowest temperature with N τ = 24. Although in the presence of fundamental matter L isnot an exact order parameter for global Z N c center symmetry, its behaviour strongly suggeststhat the transition at µ d is for all practical purposes identical with deconfinement, ie. the freeenergy for a static fundamental color source becomes finite once µ > µ d . The quark densityat deconfinement is 16 – 32 fm − (the uncertainty arises from the difficulty in handling latticeartifacts discussed in the previous section), some 30 – 60 times that of nuclear matter. µ q (MeV) T ( M e V ) µ a τ BEC BCS
Quarkyonic?
QGP
Figure 9.
Tentative QC D phase diagram. µ a χ q / χ S B c on t Figure 10. χ q /χ SBq for ja = 0 .
04, various T .It’s then interesting that µ d defined via L falls rapidly as T rises. A pragmatic definitionof µ d ( T ) is the value at which L ( µ, T ) ≈ L (0 , T d ). The resulting tentative phase diagram isshown in Fig. 9. There are at least three distinct regions/phases: a normal hadronic phase with qq i = 0, L ≈ T and µ ; the quarkyonic region with h qq i ∝ µ and L ≈ T andintermediate µ ; and a deconfined, normal phase with h qq i = 0, L > T and/or large µ .At this stage we cannot exclude a deconfined superfluid phase at large µ and small/intermediate T . The “quarkyonic” nomenclature is now justified; this phase has the thermodynamic bulkscaling of a weakly-interacting degenerate quarks, but remains confined. It thus has two of thefeatures of dense baryonic matter originally proposed on the basis of large- N c arguments [7].
5. Quark Number Susceptibility and Deconfinement
We have also recently calculated the quark number susceptibility χ q ≡ ∂n q /∂µ [8, 3]. Whilstnaively χ q is expected to reflect local fluctuations of the n q operator, it turns out that at large µ the dominant term comes from a connected “hairpin” diagram. Fig. 10 shows χ q divided by thecontinuum SB result. Once again, we note a large range over which the ratio is approximatelyconstant; indeed it is compatible with one as j →
0, though more sensitive to the quark massvalue used for the free fermions than other bulk observables [3].The most striking feature of Fig. 10, however, is the absence of T -dependence at all butthe highest temperature studied (141 MeV), is in stark contrast to the behaviour of L inFig. 8. Despite our intuition from the thermal transition in QCD [9], χ q cannot be regardedas a proxy for L once µ/T ≫
1. We conjecture that the change in behaviour of the bulkthermodynamic quantities n q , p and χ q observed at µ d is a transition from short-ranged binaryconfining interactions to longer-ranged interactions among several quarks within the medium,and that this corresponds with the transition from weak to strong self-binding noted in Sec. 3.The T -dependent behaviour of L , in contrast, must be due to degrees of freedom close to theFermi surface which can be thermally excited; a correlation between gapless excitations and L >
6. Summary QC D offers an accessible theoretical laboratory for the study of dense baryonic matter. For low T at least three physical regions can be identified: the vacuum for µ < µ o ; a confined quarkyonicsuperfluid for µ Q < µ < µ d ; and a deconfined phase for µ > µ d . The most recent simulationshave only made our findings in the quarkyonic r´egime more robust. Not discussed here are newresults for renormalised energy density, conformal anomaly, and chiral symmetry restoration [3]. Acknowledgments
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