Convergence properties of Taylor expansions of observables, which are also
used in lattice QCD calculations at non-zero chemical potential, are analyzed
in an effective N_f = 2+1 flavor Polyakov-quark-meson model. A recently
developed algorithmic technique allows the calculation of higher-order Taylor
expansion coefficients in functional approaches. This novel technique is for
the first time applied to an effective N_f = 2+1 flavor Polyakov-quark-meson
model and the findings are compared with the full model solution at finite
densities. The results are used to discuss prospects for locating the QCD phase
boundary and a possible critical endpoint in the phase diagram.Comment: 11 pages, 6 figures; minor clarifying changes, version to be
  published in Phys. Lett. 

Ali Khan

Allton

Aoki

B.-J. Schaefer

Baker

Bazavov

Bernard

Borsanyi

Cheng

de Forcrand

DeTar

F. Karsch

Fukushima

Gavai

Hatta

Herbst

J. Wambach

Karsch

Kogut

M. Wagner

Miao

Nakano

Philipsen

Rajagopal

Rößner

Schaefer

Schmidt

Skokov

Stephanov

Wagner

Wambach

Yang

Physics Letters B

English

arXiv

Karsch F, Schaefer B-J, Wagner M, Wambach J. Towards finite density QCD with Taylor expansions. In:  PoS. 2011: 219.We analyze general convergence properties of the Taylor expansion of observables to finite chemical potential in the framework of an effective 2+1 flavor Polyakov-quark-meson model. To compute the required higher order coefficients a novel technique based on algorithmic differentiation has been developed. Results for thermodynamic observables as well as the phase structure obtained through the series expansion up to 24th order are compared to the full model solution at finite chemical potential. The available higher order coefficients also allow for resummations, e.g. Pade series, which improve the convergence behavior. In view of our results we discuss the prospects for locating the QCD phase boundary and a possible critical endpoint with the Taylor expansion method

Karsch, Frithjof

Schaefer, Bernd-Jochen

Wagner, Mathias

Wambach, Jochen

Publications at Bielefeld University

Towards finite density QCD with Taylor expansions

Karsch, F.

Schaefer, B.-J.

Wagner, M.

Wambach, J.

GSI Repository

Karsch F, Schaefer B-J, Wagner M, Wambach J. Towards finite density QCD with Taylor expansions. Physics Letters B. 2011;698(3):256-264

We analyze general convergence properties of the Taylor expansion of observables to finite chemical potential in the framework of an effective 2+1 flavor Polyakov-quark-meson model. To compute the required higher order coefficients a novel technique based on algorithmic differentiation has been developed. Results for thermodynamic observables as well as the phase structure obtained through the series expansion up to 24th order are compared to the full model solution at finite chemical potential. The available higher order coefficients also allow for resummations, e.g. Padé series, which improve the convergence behavior. In view of our results we discuss the prospects for locating the QCD phase boundary and a possible critical endpoint with the Taylor expansion method

Wagner, Marc

Hochschulschriftenserver - Universität Frankfurt am Main

PoS(Lattice 2011)219BI-TP 2011/33Towards ﬁnite density QCD with Taylor expansionsF. KarschPhysics Department, Brookhaven National Laboratory, Upton, NY 11973, USAFakultät für Physik, Universität Bielefeld, D-33615 Bielefeld, GermanyE-mail: karsch@physik.uni-bielefeld.de, karsch@bnl.govB.-J. SchaeferInstitut für Physik, Karl-Franzens-Universität Graz, A-8010 Graz, AustriaInstitut für Theoretische Physik, Justus-Liebig-Universität Gießen, D-35392 Gießen, GermanyE-mail: bernd-jochen.schaefer@uni-graz.atM. WagnerFakultät für Physik, Universität Bielefeld, D-33615 Bielefeld, GermanyE-mail: mwagner@physik.uni-bielefeld.deJ. WambachInstitut für Kernphysik, TU Darmstadt, D-64289 Darmstadt, GermanyGesellschaft für Schwerionenforschung GSI, D-64291 Darmstadt, GermanyE-Mail: jochen.wambach@physik.tu-darmstadt.deWe analyze general convergence properties of the Taylor expansion of observables to ﬁnite chem-ical potential in the framework of an effective 2+1 ﬂavor Polyakov-quark-meson model. To com-pute the required higher order coefﬁcients a novel technique based on algorithmic differentiationhas been developed. Results for thermodynamic observables as well as the phase structure ob-tained through the series expansion up to 24th order are compared to the full model solution atﬁnite chemical potential. The available higher order coefﬁcients also allow for resummations,e.g. Padé series, which improve the convergence behavior. In view of our results we discuss theprospects for locating the QCD phase boundary and a possible critical endpoint with the Taylorexpansion method.XXIX International Symposium on Lattice Field TheoryJuly 10-16, 2011Squaw Valley, Lake Tahoe, CaliforniaSpeaker.c  Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/PoS(Lattice 2011)219Towards ﬁnite density QCD with Taylor expansions M. Wagner1. IntroductionThe phase structure of strongly-interacting matter at non-vanishing temperature and densitiesandinparticularthepossibleexistenceofacriticalendpoint(CEP)iscurrentlyaveryactivefrontierboth theoretically and experimentally. Our present knowledge of the QCD phase diagram in thenon-perturbative regime rests upon effective descriptions since ﬁrst-principle approaches such asQCD lattice simulations mostly fail at large densities, for recent developments see e.g. [1]. Severalextrapolation methods have been proposed to overcome the fermion sign problem. All of theseapproximationshavetheirownproblemsandtheirreliabilityisstillunderinvestigations, seee.g.[2]for reviews. Therefore, it would stand to reason to combine the approximation schemes with QCD-like effective model calculations. A comparison of the analytical model results with the latticecalculations will shed light on their principal applicabilities. Here we focus on the Taylor expansionmethod [3] for small chemical potentials, see also [4].It is based on an expansion of the pressure p in powers of x  m=T around vanishing quarkchemical potential m:p(m=T)T4 =¥ån=0cn(T)mTnwith cn(T) =1n!¶n p(T;m)=T4¶ (m=T)n  m=0: (1.1)The relevant Taylor coefﬁcients cn(T) can be calculated with standard techniques at m =0 [5]. Theconvergence of the series may be improved by a resummation based on a Padé approximation[L=M]  RL;M(x) =p(x)q(x)=p0+ p1x++ pLxL1+q1x++qMxM ; (1.2)where the Padé coefﬁcients pi and qj can be extracted from the Taylor coefﬁcients cn up to theorder L+M. Hence, no further input is required for the application of a Padé resummation. Bymeans of the Padé resummation often an extended convergence range and more stable results withfewer coefﬁcients can be obtained, in particular in the presence of singularities.2. A model analysisAs an effective QCD-like model framework we chose the Nf = 2+1 ﬂavor Polyakov quark-meson (PQM) model which exhibits a chiral crossover at vanishing densities at Tc  206MeV anda CEP around (Tc;mc)  (185;167)MeV. Its thermodynamics at m = 0 is in agreement with recentlattice simulations [6]. The PQM model Lagrangian consist of a quark-meson partLPQM = ¯ q(iD = hf5)q+Lm U (F; ¯ F) ; (2.1)where the Nf quark ﬁelds q are coupled via a Yukawa interaction h to the scalar and pseudoscalarmeson nonets f and to the Polyakov loop via the covariant derivative. The purely mesonic contri-bution readsLm = Tr ¶mf†¶mf m2Tr(f†f) l1Tr(f†f)2 l2Tr f†f2+c det(f)+det(f†)+TrH(f +f†): (2.2)2PoS(Lattice 2011)219Towards ﬁnite density QCD with Taylor expansions M. Wagner 0 50 100 150 200 250 300 350 400 0  5  10  15  20  25t [ms]highest derivative degree dADDDFigure 1: Runtime comparison for the DD and AD methodsThe pure gauge sector is encoded (in the logarithmic version of) the Polyakov loop potential [7]UlogT4 =  a(T)2¯ FF+b(T)lnh1 6 ¯ FF+4 F3+ ¯ F3 3  ¯ FF2i(2.3)whose parameters are ﬁtted to lattice data. The matter back-reaction to the pure Yang-Mills sys-tem [9] is not considered in this work. The remaining model parameters are chosen to reproducemeson masses and decay constants in the vacuum [8]. The grand potential in mean-ﬁeld approx-imation is obtained by integrating the quark loop which yields a function of the (non-strange andstrange) chiral order parameters (sx,sy) and the Polyakov-loop variables (F; ¯ F)W =U (sx;sy)+W¯ qq sx;sy;F; ¯ F+U F; ¯ F:Their temperature and chemical potential dependence is found by minimizing the grand potential¶W¶sx=¶W¶sy=¶W¶F=¶W¶ ¯ F  min= 0 ; (2.4)where min labels the global minimum of the potential. This introduces an only numerically invert-ible implicit m-dependence in the pressure p(T;m) =   W(T;m)jmin.An analytic evaluation of the Taylor coefﬁcients is not only cumbersome due to the exponen-tially increasing number of terms but becomes ﬁnally impossible by the implicit m-dependence.Standard numerical derivate techniques such as divided differences (DD) fail at higher orders dueto the increasing numerical errors. By means of the novel derivative technique, based on algorith-mic differentiation (AD), higher order derivatives can be calculated with extremely high precision,essentially limited only by machine precision. This works even in the case of only numericallysolvable implicit dependences, such as in Eq. (2.4). Furthermore, also the performance of the ADtechnique is superior to the DD method as demonstrated in Fig. 1 where the time to calculate thed-th order derivatives is shown as a function of d. With the DD method one needs at least d +1function evaluations for a d-th order derivatives and a linear d-dependence is obtained in contrastto the logarithmic dependence for the AD method, see [10] for further details. High-order Taylorcoefﬁcients calculated with the AD technique in the PQM model are given in [11]. The method isneither restricted to vanishing m nor mean-ﬁeld approximation and can be used to search for signalsof the CEP in higher moments [14].3PoS(Lattice 2011)219Towards ﬁnite density QCD with Taylor expansions M. Wagner 0 5 10 15 20 0.89  0.9  0.91  0.92  0.93  0.94χ/T2T/TχPQMTaylor 8thTaylor 16thPade [4/4]Pade [8/8](a) m=T = 0:8 0 5 10 15 20 0.88  0.89  0.9  0.91  0.92χ/T2T/TχPQMTaylor 8thTaylor 16thPade [4/4]Pade [8/8](b) m=T = 0:9Figure 2: Comparison of the quark number susceptibility in the mean-ﬁeld model (PQM) with differentorders of the extrapolation methods in the crossover region (left panel) and at the CEP (right panel).2.1 ThermodynamicsThe Taylor expansion can be applied straightforwardly to extrapolate thermodynamic quanti-ties such as the quark-number susceptibility c = ¶2p=(¶m2) to ﬁnite densities. In Fig. 2 the scaledsusceptibility c=T2 calculated in the PQM model is compared to results obtained with the Tay-lor expansion and the Padé resummation. It diverges exactly at the CEP which is located for ourchosen parameters at (Tc;mc)  (185;167)MeV, correspondingly mc=Tc  0:9 (right panel) and isﬁnite otherwise. Both, the Taylor expansion and the Padé resummation cannot capture this diver-gence and yield ﬁnite values around Tc. For comparison the left panel shows c in the crossoverregime (m=T = 0:8). In the chirally broken phase, i.e., for temperatures smaller than the one of thec peak, the agreement improves with increasing expansion orders. At larger temperatures T  Tcthe extrapolations exhibit some unphysical peaks which are not present in the model calculation.These peaks, which are also seen in extrapolations where lattice simulation coefﬁcients have beenused (see e.g. [12]), are solely an artifact of the extrapolation. However, an important observationis that in both cases the [4=4] Padé approximation yields nearly the same result as the 16-th orderTaylor expansion although it requires only the 8-th order Taylor coefﬁcients. Hence, in our casethe Padé approximation reduces the number of required derivatives roughly by a factor of two.2.2 Determining the phase boundary and locating the CEPThe Taylor expansion is limited by the closest singularity in the complex m-plane. Its distanceto the expansion point can be obtained from the convergence radius of the series:r = limn!¥r2n = limn!¥  c2nc2n+2  1=2: (2.5)If the limiting singularity lies on the real axis the convergence radius corresponds to the locationof a critical point. As the limit n ! ¥ cannot be carried out in a numerical study the estimators ofthe convergence radius at ﬁnite n may depend on the considered observable and approach the trueconvergence radius differently. Therefore we also consider the convergence radius for the quark4PoS(Lattice 2011)219Towards ﬁnite density QCD with Taylor expansions M. Wagner 0 0.2 0.4 0.6 0.8 1 0  0.5  1  1.5  2T/Tχµ/Tχ2n= 82n=162n=24(a) 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 2  4  6  8  10  12  14  16  18  20  22µ/T2nr2np [n+1/n+1]p [n+2/n]p [n+3/n-1]rχ2n-2χ [n/n]χ [n+1/n-1](b)Figure 3: Left: Estimates for the radius of convergence obtained from r2n (solid lines) and rc2n 2 (dashedlines) for different orders of the Taylor expansion. Also shown is the chiral phase boundary (red line; dashed:crossover, solid: ﬁrst order). The black dot indicates the CEP. Right: Estimate of the phase boundary atT = 190MeV  0:92 Tc with r2n and rc2n and poles in the Padé approximation of the pressure and quarknumber susceptibility as a function of the order of the Taylor expansion 2n. The used highest coefﬁcient isc2n+2. The dotted horizontal line indicates the phase boundary calculated directly at ﬁnite m.number susceptibility rc2n which is related to the one of the pressure viarc2n =  cc2ncc2n+2  1=2=(2n+2)(2n+1)(2n+3)(2n+4)1=2r2n+2 : (2.6)In Fig. 3(a) we show the results for both estimators for three different truncation orders(2n = 8;16;24) in the (T;m) phase diagram. Note that the difference in the radii of convergence,estimated from r2n and rc2n 2 is only caused by the prefactor in Eq. (2.6) as the same Taylor coef-ﬁcients contribute to both estimators. Close to Tc the oscillations in the Taylor coefﬁcients causedby the imaginary part of the limiting singularity entail oscillations in the radii of convergence anddo not allow for a stable estimate of the phase boundary there. For somewhat smaller temperaturesthe radii of convergence approach the phase boundary from above with increasing truncation ordern. The observed agreement in the crossover region suggests that the singularity limiting the Taylorexpansion is close to the real m-axis and the small imaginary part is negligible. In particular, thisis still valid for the region in the vicinity of the CEP.The determination of the radius of convergence or the phase boundary with the Padé seriesis more involved. For the [N=2] case the Padé approximant has a pole at x = pcN=cN+2 whichcoincides with the estimator r2N, see Eq. (2.5). For a general [L=M] Padé approximant the polestructure is more elaborated. In the general case we use the ﬁrst pole at real and positive m inorder to estimate the phase boundary. Since all Taylor coefﬁcients with their corresponding errorenter in the Padé approximant the error propagation is more involved here in contrast to the pre-vious discussion where only two error sources enter. In this context, the application of the Padéapproximation in lattice simulations is much more involved. However, in the model analysis thecoefﬁcients, obtained with the AD technique, exhibit extremely small numerical errors and result ina stable and reliable Padé approximation. In Fig. 3(b) we show for a ﬁxed temperature T = 0:92 Tcthe estimates of the phase boundary extracted from Padé approximations for the pressure (p) and5PoS(Lattice 2011)219Towards ﬁnite density QCD with Taylor expansions M. Wagner 185 190 195 200 205 210 6  8  10  12  14  16  18  20Tc,n [MeV]nFigure 4: Temperature Tc;n at which the Taylor coefﬁcients cn becomes negative for different truncationorders n. The solid horizontal line denotes the critical temperature of the CEP.the quark number susceptibility (c) for different truncation orders n. The chosen temperature isslightly above the temperature of the CEP in the phase diagram. In addition, the estimates forthe radii of convergence (r2n and rc2n) and the chiral crossover chemical potential (dashed hori-zontal line) are also shown. For small truncation orders 2n  8 the [N=2] Padé approximationscoincide with the radii of convergence as expected. For 2n > 8 we observe that the Padé approx-imation converges faster for the pressure as well as for the susceptibility compared to the radii ofconvergence of the Taylor series. We conclude that the Padé approximation converges faster atintermediate truncation orders in particular for the quark number susceptibility. The estimate of thephase boundary from the Padé approximation becomes comparable to the one obtained from theTaylor expansion coefﬁcients only at signiﬁcantly larger truncation order. For example, in Fig. 3(b)the distance of the Padé estimate to the horizontal line at 2n = 12 is achieved with Taylor coefﬁ-cients only for 2n  20. However, the lower truncation order in the Padé approximation induces amore involved error propagation which might hamper lattice simulations but not calculations withthe AD-techniques.From the discussion given so far and from Fig. 3(b) one may conclude that at ﬁxed temperaturethe radius of convergence can be estimated with an accuracy of about 15%-20% from a O(m12)series expansion. This is an acceptable uncertainty in view of the difﬁculties in current QCDcalculations at non-zero chemical potential. When Tc is known the CEP could be determine frommc = r(Tc) = limn!¥rn(Tc) : (2.7)However, the determination of Tc is still a non-trivial task. A criterion is needed, that allows toestimate for a given expansion order the temperature regime where all expansion coefﬁcients maystay positive. In this case the singularity limiting the convergence is located on the real axis in thecomplex m-plane and hence corresponds to a second-order phase transition [13]. In Fig. 4 we plotthe temperature Tc;n where the Taylor coefﬁcient cn(T) becomes negative. This yields an upperbound for the critical temperature. Of course, in the limit n ! ¥ the temperature series shouldapproach the critical temperature of the CEP, i.e.,Tc = limn!¥Tc;n : (2.8)6PoS(Lattice 2011)219Towards ﬁnite density QCD with Taylor expansions M. WagnerHowever, we found in our model analysis only a very slow convergence of the estimators forthe critical temperature of the CEP. Even at the 20-th truncation order there is still a temperaturedifference of the order of 20 MeV which is why we do not apply Eq. (2.7).3. Summary and ConclusionIn this talk we discussed convergence properties of the Taylor expansion towards ﬁnite den-sity QCD in a three ﬂavor PQM model framework. A newly developed differentiation techniqueallowed for the calculation of high order Taylor coefﬁcients with high precision. With a Padé re-summation the number of required Taylor coefﬁcients can be reduced by a factor of two to obtaina similar convergence radius. For temperatures above the critical endpoint a good agreement of theradius of convergence with the phase boundary is found, in particular for the estimator of the quarknumber susceptibility. With a 12-th order Padé approximation a satisfying estimate of the phaseboundary could be achieved. However, a determination of the critical temperature including thelocation of the CEP is hampered by the slow convergence of the estimators Tc;n.Support by the BMBF Grant 06BI9001 and by the Helmholtz-University Young Investigator GrantNo. VH-NG-332 is acknowledged.References[1] L. Levkova, these proceedings, PoS LAT2011, (2011).[2] O. Philipsen, PoS LAT2005, 016 (2006); C. Schmidt, PoS LAT2006, 021 (2006); P. de Forcrand, PoSLAT2009, 010 (2009).[3] C. R. Allton et al. Phys. Rev. D66, 074507 (2002); Phys. Rev. D68, 014507 (2003); Phys. Rev. D71,054508 (2005). R. V. Gavai and S. Gupta, Phys. Rev. D68, 034506 (2003); Phys. Rev. D71, 114014(2005); Phys. Rev. D78, 114503 (2008).[4] F. Karsch, B.-J. Schaefer, M. Wagner, and J. Wambach, Phys.Lett. B698, 256 (2011).[5] C. W. Bernard et al. (MILC Collaboration), Phys.Rev. D55, 6861 (1997); F. Karsch, E. Laermann, andA. Peikert, Phys.Lett. B478, 447 (2000); A. Ali Khan et al. (CP-PACS), Phys. Rev. D64, 074510(2001).[6] B.-J. Schaefer, M. Wagner, and J. Wambach, Phys. Rev. D81, 074013 (2010).[7] S. Rößner, C. Ratti, and W. Weise, Phys. Rev. D75, 034007 (2007).[8] B.-J. Schaefer and M. Wagner, Phys. Rev. D79, 014018 (2009).[9] B.-J. Schaefer et al., Phys. Rev. D76, 074023 (2007); T.K. Herbst et al., Phys. Lett. B696 , 58 (2011).[10] M. Wagner, A. Walther, and B.-J. Schaefer, Comp. Phys. Commun. 181, 756 (2010).[11] B.-J. Schaefer, M. Wagner, and J. Wambach, PoS CPOD2009, 017 (2009); J. Wambach, B.-J.Schaefer, and M. Wagner, Acta Phys. Pol. B Proc. Suppl. 3, 691 (2010).[12] C. Schmidt, PoS CPOD2009, 024 (2009).[13] M. A. Stephanov, Phys. Rev. D73, 094508(2006); C.-N. Yang and T. Lee, Phys. Rev. 87, 404 (1952).[14] B.-J. Schaefer and M. Wagner, in preparation (2011).7

Towards finite density QCD with Taylor expansions

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