We present results on the QCD phase diagram for small densities without
reweighting. Our simulations are performed with an imaginary chemical potential
mu_I for which the fermion determinant is positive. On an 8^3x4 lattice with 2
flavors of staggered quarks, we map out the pseudo-critical temperature
T_c(mu_I). For mu_I/T < pi/3, this is an analytic function whose Taylor
expansion converges rapidly, with truncation errors smaller than statistical
ones. The result is analytically continued to give the location of the
pseudo-critical line for real mu_B<500 MeV.Comment: Lattice2002(nonzerot), 3 pp, 5 figure

de Forcrand, P.

Philipsen, O.

English

arXiv

We present results on the QCD phase diagram for small densities without reweighting. Our simulations are performed with an imaginary chemical potential mu_I for which the fermion determinant is positive. On an 8^3x4 lattice with 2 flavors of staggered quarks, we map out the pseudo-critical temperature T_c(mu_I). For mu_I/T < pi/3, this is an analytic function whose Taylor expansion converges rapidly, with truncation errors smaller than statistical ones. The result is analytically continued to give the location of the pseudo-critical line for real mu_B<500 MeV.We present results on the QCD phase diagram for small densities without reweighting. Our simulations are performed with an imaginary chemical potential mu_I for which the fermion determinant is positive. On an 8^3x4 lattice with 2 flavors of staggered quarks, we map out the pseudo-critical temperature T_c(mu_I). For mu_I/T < pi/3, this is an analytic function whose Taylor expansion converges rapidly, with truncation errors smaller than statistical ones. The result is analytically continued to give the location of the pseudo-critical line for real mu_B<500 MeV

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1QCD phase diagram for small densities from simulations at imaginary µP. de Forcranda and O. Philipsenb aETH, CH-8093 Zu¨rich, Switzerland and CERN Theory Division, CH-1211 Geneva 23, SwitzerlandbCenter for Theoretical Physics, MIT, Cambridge, MA 02139-4307, USAWe present results on the QCD phase diagram for small densities without reweighting. Our simulations areperformed with an imaginary chemical potential µI for which the fermion determinant is positive. On an 83 × 4lattice with 2 flavors of staggered quarks, we map out the pseudo-critical temperature Tc(µI). For µI/T ≤ pi/3, thisis an analytic function whose Taylor expansion converges rapidly, with truncation errors smaller than statisticalones. The result is analytically continued to give the location of the pseudo-critical line for real µB <∼ 500 MeV.1. INTRODUCTIONIn view of heavy ion collision experiments apressing task for lattice QCD is to map out thedeconnement transition in the (µ, T )-plane. Thecomplexity of the fermion determinant at nitebaryon density renders standard Monte Carlotechniques impossible [1]. One way of circum-venting this ‘sign problem’ are multi-dimensionalreweighting techniques, and a phase diagram hasbeen presented last year [2]. Reweighting gener-ally breaks down for large volumes and/or densi-ties, and one diculty of the approach is to knowwhen that happens. In [3] both reweighting fac-tor and observables were expanded in a Taylorseries in µ/T , and the rst coecients were mea-sured [3]. This method allows to simulate largervolumes, but a priori nothing is known about theerror introduced by truncating the series.Here we present an alternative method avoidingreweighting altogether. We simulate at imaginaryµ for which there is no sign problem and arbitrar-ily large volumes are feasible. Observables aretted by truncated Taylor series in µ/T , keepingfull control over the associated systematic error.The series may then be continued to real µ, pro-vided the observable is analytic [4]. This strategyhas been successfully tested for screening masses[5], here we extend it to the critical line itself. Adetailed account of this work is given in [6].∗Talk given by O. P.0 0.5 13µΙ/(2piT)TFigure 1. Phase diagram in the (µI , T )-plane.The vertical line marks a Z(3) transition crossingthe deconnement transition. For T > Tc theformer is of rst order.2. QCD AT IMAGINARY µFinite density QCD is symmetric under reflec-tion of µ, Z(µ) = Z(−µ), for both real and imag-inary µ. For imaginary µ = iµI it is moreoverperiodic, Z(µI) = Z(µI + 2pi/3) [7]. Hence, cer-tain shifts in µI are equivalent to Z(3) transfor-mations to dierent vacua. Consequently, Z(3)transitions occur at the critical values (µ/T )c =(k +1/2)2pi/3. The Z(3) sectors can be identiedby monitoring the phase of the Polyakov loop.Ref. [7] predicts these transitions to be of rst or-2-1 -0.5 0 0.5 1ϕ/pi00.10.20.30.40.5p(ϕ)β=5.26β=5.286β=5.30β=5.313β=5.34β=5.353β=5.366Figure 2. Probability distribution of the phaseof the Polyakov loop for the critical value aµcI =pi/12.der for T > Tc and crossover for T < Tc, with thecritical deconnement temperature Tc. Schemat-ically, the phase diagram is shown in Fig. 1, whichis periodically repeated for larger µI .Our simulations support this picture. Fig. 2shows the phase of the Polyakov loop at (µ/T )cfor various lattice couplings/temperatures. Asmooth distribution is found for low tempera-tures, whereas at high temperatures the two-statedistribution signals a rst order phase transition.Similar ndings are reported for four flavors in[8]. The existence of a Z(3) transition with itsendpoint joining on the deconnement transitionimplies that the critical line of the latter be-comes non-analytic at that point, and hence onlyµI  pi/3 can be continued to real µ.Note that the deconnement line as well as allobservables even under reflections µ ! −µ aresymmetric around the Z(3) transition. This isa feature that is not reproduced by reweightingfrom a Monte Carlo ensemble generated at µ = 0,since such an ensemble does not suciently probethe other Z(3) sectors [6,2].3. CONTINUATION OF THE DECON-FINEMENT LINEOne method to locate a phase transition is tolook for the peak χmax = χ(µc, βc) of the suscep-tibilities χ = V Nt〈(O − hOi)2, where we haveused the plaquette, the chiral condensate and thePolyakov loop as operators O(x). Since simula-tions are always performed on nite volumes, sus-5.24 5.26 5.28 5.3 5.32 5.34 5.36β00.00050.0010.00150.0020.00250.003χ |P|aµI=0.0aµI=0.15aµI=0.23Figure 3. Susceptibilities of the modulus of thePolyakov loop for various µI .ceptibilities are analytic functions over the wholeparameter space of the theory. Non-analyticitiesassociated with phase transitions develop only inthe thermodynamic limit.For a given nite volume, the pseudo-criticalline is then found by locating χmax for every µ,∂χ∂ββc= 0∂2χ∂β2βc< 0. (1)This is an implicit denition of the critical cou-pling βc(µ). For analytic χ(µ, β) the implicitfunction theorem implies that also βc(µ) is ana-lytic for all µ. Moreover, the symmetry consider-ations of the last section ensure βc(µ) = βc(−µ),so that we can expand in µ/T = aNtµ to obtainβc(µ) =Xncn(aµ)2n. (2)As long as, for imaginary µ = iµI , the non-perturbative data are well described by a few co-ecients in this series, analytic continuation toreal µ simply proceeds by the operation µI ! iµI .This provides us with the location of the pseudo-critical line on a nite volume.If there is a rst order deconnement transitionending in a critical point, then, as the volumeis increased, βc(µ) approaches the line of phasetransitions, while remaining pseudo-critical in acrossover regime. In principle, the nature of theline can be determined from the way the thermo-dynamic limit is approached, (βc(V )− βc(1)) V −σ, with σ = 1 or 0 < σ < 1 for a rst or sec-ond order transition, respectively, and σ = 0 for30 0.02 0.04 0.06 0.08 0.1(aµΙ)25.285.295.35.315.325.335.34βfit O(aµΙ)2fit O(aµΙ)4Figure 4. Location of the pseudo-critical decon-nement line as determined from plaquette sus-ceptibilities. The lines show ts of the form Eq. 2to quadratic and quartic order.a crossover. Our rst simulations are on a singlevolume and this question is postponed to futurework.Fig.3 shows a sequence of susceptibilities forthe modulus of the Polyakov loop, with similarpictures for the plaquette and the chiral conden-sate. All three observables give statistically com-patible values βc(µI). Note that the peak is mov-ing to larger βc for growing µI , as one would ex-pect for imaginary µ, cf. Eq. 2 and Fig. 1.Fig. 4 shows βc for dierent values of µI . Thedata perfectly reproduce the expected symmetryof the partition function about the rst Z(3) tran-sition point. The lines in the gure show ts of aquadratic and a quartic polynomial to the data,with χ2/dof 0.6 and 0.7, respectively. The co-ecient of the quartic term is found to be zerowithin errors, and hence the two ts are statis-tically compatible: up to µI/T = pi/3, and forthe quark flavors and masses under consideration,the critical line is well described by the quadraticterm only.After analytic continuation the result is trans-formed into physical units by means of the pertur-bative two-loop expression for the scale parameteraL and using Tc(µ = 0) from the literature [9]to set the physical scale. We thus arrive at thephase diagram of Fig. 5, with the pseudo-criticalline given by (µB = 3µ)Tc(µB)Tc(µB = 0)= 1− 0.00563(38)µBT2. (3)0 100 200 300 400 500 600µB/MeV160165170175180T/MeVFigure 5. Location of the deconnement transi-tion. The error bar gives the uncertainty in Tc(0)used to set the scale, the dotted lines reflect theerror on c1, Eq. 2.This result is consistent with those obtained bythe reweighting approaches [2,3], for which it pro-vides a crucial, controlled check. An importanttask ahead is to perform a nite size scaling anal-ysis of the continued critical line, in order to de-termine the order of the transition as well as tolocate a possible critical point.REFERENCES1. For reviews and refs. see, e.g., S. Hands,Nucl. Phys. Proc. Suppl. 106 (2002) 142.O. Philipsen, Nucl. Phys. Proc. Suppl. 94(2001) 49.2. Z. Fodor and S. D. Katz, JHEP 0203 (2002)014; Phys. Lett. B 534 (2002) 87.3. C. R. Allton et al., hep-lat/0204010.4. M. P. Lombardo, Nucl. Phys. Proc. Suppl. 83(2000) 375.5. A. Hart, M. Laine and O. Philipsen, Phys.Lett. B 505 (2001) 141. Nucl. Phys. B 586(2000) 443.6. P. de Forcrand and O. Philipsen, hep-lat/0205016, to appear in Nucl. Phys. B.7. A. Roberge and N. Weiss, Nucl. Phys. B 275(1986) 734.8. M. D’Elia and M. P. Lombardo, hep-lat/0205022.9. F. Karsch, E. Laermann and A. Peikert, Nucl.Phys. B 605 (2001) 579.

QCD phase diagram for small densities from simulations at imaginary mu

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