Phase transitions in dense 2-colour QCD
Tamer Boz, Seamus Cotter, Leonard Fister, Jon-Ivar Skullerud
aa r X i v : . [ h e p - l a t ] N ov Phase transitions in dense 2-colour QCD
Tamer Boz, Seamus Cotter, Leonard Fister, Jon-Ivar Skullerud ∗ Department of Mathematical Physics, National University of Ireland Maynooth, Maynooth,County Kildare, IrelandE-mail: [email protected]
We investigate 2-colour QCD with 2 flavours of Wilson fermion at nonzero temperature T andquark chemical potential m , with a pion mass of 700 MeV ( m p / m r = . m we find that the critical temperature for the superfluid to normal transition dependsonly very weakly on m above the onset chemical potential, while the deconfinement crossovertemperature is clearly decreasing with m . We also present results for the Landau-gauge gluonpropagator in the hot and dense medium. ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ hase transitions in QC D Jon-Ivar Skullerud
1. Introduction
Despite intensive theoretical efforts over the past decade and more, we do still not have a quan-titative understanding of QCD at large baryon density. This is primarily due to the sign problempreventing first-principles Monte Carlo simulations in this régime. One way of circumventing thisis to study QCD-like theories without a sign problem, and use these to provide a benchmark formodel studies and other methods which do not suffer from the sign problem. The simplest suchtheory, which shares with QCD the properties of confinement and dynamical symmetry breaking,is 2-colour QCD (QC D).In a series of papers [1, 2, 3, 4] we have studied QC D with 2 flavours of Wilson fermion atnonzero baryon chemical potential m and temperature T , culminating in a tentative mapping out ofthe phase diagram in the ( m , T ) plane [3, 4]. Here we will report on the determination of the phasetransition lines [4] and present new results for the gluon propagator at nonzero m and T . Updatedresults for the equation of state are presented in a separate talk [5].We use a standard Wilson gauge and fermion action augmented with a diquark source termto lift low-lying eigenvalues in the superfluid phase. The lattice spacing is a = . ( ) fm and m p / m r =0.8, with am p = . ( ) [3]. We have performed simulations at four fixed tempera-tures, T = , ,
94 and 141 MeV, corresponding to N t = , ,
12 and 8 respectively, for arange of chemical potentials m a = m a = . , . , . × N t lattices with N t = j we have used ja = . , .
04 in order to allow an extrapolation to the physical j =
2. Superfluid to normal transition
Figure 1 shows the order parameter for superfluidity, the (unrenormalised) diquark condensate h qq i , as a function of the temperature T , for m a = . , . , . j =
0. We can clearly observe a transition from a superfluidphase, characterised by h qq i 6 =
0, at low temperature, to a normal phase with h qq i = . . Ta . .
12 for all four values of m .We have estimated the critical temperatures T s for the superfluid to normal transition by deter-mining the inflection points for h qq i at ja = .
02 and 0.04, and extrapolated the resulting values to j = T s is remarkably constantover the whole range of m -values considered. The indications are that the transition happens at asomewhat lower temperature at m a = .
35, but this point is already very close to the onset fromvacuum to superfluid at T = m o a = m p a / = .
32, suggesting that T s ( m ) rises very rapidly fromzero at m = m o before suddenly flattening off.
3. Deconfinement transition
The Polyakov loop h L i serves as the traditional order parameter for deconfinement in gaugetheories, with h L i 6 = h L i isnever zero in a theory with dynamical fermions, but it typically increases with temperature from2 hase transitions in QC D Jon-Ivar Skullerud < qq > Ta Ta j a =0.05j a =0.04j a =0.03j a =0.02j=0 m a =0.60 m a =0.50 m a =0.40 m a =0.35 Figure 1:
Diquark condensate h qq i as a function of temperature T for chemical potential m a = . , . , . , . j = ja ≤ .
04; the shaded circles denote the results of a linear extrapolation using ja = . , .
03 only. a very small value in a fairly narrow region, which may be identified with the deconfinementtransition region. Unlike the diquark condensate, the renormalisation of the Polyakov loop dependson temperature; specifically, the relation between the bare Polyakov loop L and the renormalisedPolyakov loop L R is given by L R ( T , m ) = Z N t L L (( aN t ) − , m ) . In order to investigate the sensitivityof our results to the renormalisation scheme, we have used two different conditions to determine theconstant Z L [4], L R ( T = T , m = ) = c , with T = a − and c = c = . h L i evaluated in both schemes, as a function of temperature. The Scheme Bdata have been multiplied by 2 to ease the comparison with the Scheme A data. Also shown arecubic spline interpolations of the data and the derivative of these interpolations, with solid linescorresponding to Scheme A and dotted lines to Scheme B.At all m , we see a transition from a low-temperature confined region to a high-temperaturedeconfined region. In contrast to the diquark condensate, we see a clear, systematic shift in thetransition region towards lower temperatures as the chemical potential increases. For all four m -values, the Polyakov loop shows a nearly linear rise as a function of temperature in a broad region,suggesting that the transition is a smooth crossover rather than a true phase transition. This isreinforced by the difference between Scheme A and Scheme B, with the crossover occuring athigher temperatures in Scheme B. At m =
0, the difference between the two schemes is small, butincreases with increasing m , suggesting a broadening of the crossover.Because of the smaller value of Z L , our results for Scheme B are considerably less noisy than3 hase transitions in QC D Jon-Ivar Skullerud Ta < L > m a =0.35 m a =0.40 m a =0.50 m a =0.60 Figure 2:
The renormalised Polyakov loop h L i as a function of temperature T for ja = .
04 and m a = . , . , . , .
6, with two different renormalisation schemes: Scheme A (solid symbols) and Scheme B(open symbols), see text for details. The solid (dashed) lines are the derivatives of cubic spline interpolationsof the data points for Scheme A (B). The smaller, shaded symbols are results for ja = .
02. The black circlesand thick lines in the bottom right panel are the m = j = those for Scheme A. For this reason, we choose to define the crossover region to be centred on theinflection point from Scheme B, with a width chosen such that it also encompasses the onset of thelinear region from Scheme A.The transition region taken from the ja = .
04 data is shown in fig. 3. From Fig. 2 we seethat at low T , the value of h L i increases as j is reduced, and at m a = .
6, the crossover region willmost likely move to smaller T in the j → ja = .
02 at low T to make any quantitative statement about this.
4. Gluon propagator
One of the main motivations for studying dense QC D on the lattice is to provide constraints onapproaches which do not suffer from the sign problem. The gluon propagator provides a key inputfor several of these approaches, in particular functional studies using the functional renormalisationgroup or Dyson–Schwinger equations. These are most often carried out in the Landau gauge.In Landau gauge only the transverse part of the vacuum propagator is non-zero. However,the external parameters break manifest Lorentz invariance, hence the gluon propagator D must be4 hase transitions in QC D Jon-Ivar Skullerud m q (MeV) T ( M e V ) m a t BEC?
BCS?
Quarkyonic
QGPHadronic
Figure 3:
Phase diagram of QC D with m p / m r = .
8. The black circles denote the superfluid to normalphase transition; the green band the deconfinement crossover. The blue diamonds are the estimates for thedeconfinement line from [3]. s =016³x24, a q s =0.8612³x16, q s =012³x16, a q s =0.90 16³x12, q s =016³x12, a q s =0.8616³x8, q s =016³x8, a q s =0.86 m a D ( q s , q ) m a Magnetic, q =0 Electric, q =0Magnetic, q =2 p T Electric, q =2 p T Figure 4:
The zeroth (top) and first (bottom) Matsubara mode of the magnetic (left) and electric (right)gluon propagator as a function of chemical potential m for selected values of the spatial momentum q s = | ~ q | ,and different temperatures. hase transitions in QC D Jon-Ivar Skullerud decomposed into chromoelectric and chromomagnetic modes, D E and D M , respectively, D mn ( q ,~ q ) = P M mn D M ( ~ q , q ) + P E mn D E ( ~ q , q ) . (4.1)The projectors on the longitudinal and transversal spatial subspaces, P E mn and P M mn , are defined by P M mn ( ~ q , q ) = (cid:0) − d m (cid:1) ( − d n ) (cid:18) d mn − q m q n ~ q (cid:19) , P E mn ( ~ q , q ) = (cid:18) d mn − q m q n q (cid:19) − P M mn ( ~ q , q ) . (4.2)In this section we extend the results presented in [4] to a wider area of the ( m , T ) plane. Wehave fixed our gauge configurations to the minimal Landau gauge using the standard overrelaxationalgorithm. The Landau gauge condition has been imposed with a precision | ¶ m A m | < − .In figure 4 we show the two lowest Matsubara modes for selected spatial momenta as a functionof chemical potential for N t = , , ,
8. The results shown are for ja = .
04, but we havefound no significant difference for ja = .
02. We have investigated the volume dependence on the N t =
24 lattices and found it to be very mild [4]. At the three lower temperatures, both the electricand magnetic form factors are roughly independent of m up to m a ≈ .
5, and become suppressed forlarge m . This changes dramatically at the highest temperature shown ( N t = m ,while the magnetic form factor for small spatial momenta has a clear enhancement at intermedate m and an enhancement at large m for larger spatial momenta. On closer inspection it is possible tosee the onset of this behaviour also for N t =
12. No qualitative differences are seen between theelectric and magnetic form factors for the first nonzero Matsubara mode. Ta D ( q s , ) Ta q s =0 a q s =0.39 a q s =0.55 a q s =0.86 a q s =1.24 a q s =1.62 Magnetic Electric
Figure 5:
Thermal behaviour of the zeroth Matsubara mode of the magnetic (left) and electric (right) prop-agators at m a = . ja = .
04 on 16 × N t lattices, for selected spatial momenta q s = | ~ q | . We now turn to the thermal behaviour of the gluon propagator at fixed chemical potential. Fig.5 shows the zeroth Matsubara modes of the propagators for m a = . ja = .
04 on 16 × N t lattices as a function of temperature. The magnetic component has a very mild enhancement at6 hase transitions in QC D Jon-Ivar Skullerud intermediate temperatures and a slight suppression at very high T . In contrast, the electric prop-agator shows a strong suppression with increasing temperature. We note that the deconfinementcrossover for this value of m happens for 0 . . Ta . .
0, and that this coincides roughly with theregion where the magnetic propagator is enhanced. In contrast to early studies in pure Yang–Millstheory, but in line with a recent study in QCD with twisted-mass Wilson fermions [6], there is noenhancement in the electric mode in the transition region.
5. Summary and outlook
We have studied the superfluid and deconfinement transition lines in QC D in the ( m , T ) plane.We find that the superfluid transition temperature is remarkably insensitive to m for the quark masswe are using, while the deconfinement temperature is clearly decreasing as m increases.At low temperature, the low-momentum modes of both the electric and magnetic Landau-gauge gluon propagator become suppressed relative to the (already infrared suppressed) vacuumpropagator at large m , with no qualitative differences between the two form factors found. At hightemperature, the static electric and magnetic propagators are found to exhibit very different be-haviours, with a strong suppression of the electric form factor and an enhancement of the magneticform factor at intermediate m .We are in the process of extending these studies to smaller quark masses as well as finer latticespacings. In a forthcoming publication we will also study the response of the quark propagatorto m and T . This will enable us to directly confront the results from functional methods for thesequantities. Acknowledgments
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