Net-baryon number fluctuations
C. Schmidt, J. Goswami, G. Nicotra, F. Ziesché, P. Dimopoulos, F. Di Renzo, S. Singh, K. Zambello
NNet-baryon number fluctuations ∗ C. Schmidt, J. Goswami, G. Nicotra, F. Ziesch´e
Universit¨at Bielefeld, Fakult¨at f¨ur Physik, D-33615 Bielefeld, Germnay
P. Dimopoulos, F. Di Renzo, S. Singh, K. Zambello
Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Universit`a diParma and INFN, Gruppo Collegato di Parma I-43100 Parma, ItalyThe appearance of large, none-Gaussian cumulants of the baryon num-ber distribution is commonly discussed as a signal for the QCD criticalpoint. We review the status of the Taylor expansion of cumulant ratiosof baryon number fluctuations along the freeze-out line and also compareQCD results with the corresponding proton number fluctuations as mea-sured by the STAR Collaboration at RHIC. To further constrain the loca-tion of a possible QCD critical point we discuss poles of the baryon numberfluctuations in the complex plane. Here we use not only the Taylor coeffi-cients obtained at zero chemical potential but perform also calculations ofTaylor expansion coefficients of the pressure at purely imaginary chemicalpotentials.PACS numbers: 12.38.Mh, 25.75.Nq
1. Introduction
The phase diagram of Quantum Chromodynamics (QCD) is currently in-vestigated with large efforts by means of heavy ion experiments at LHC andRHIC, as well as by numerical calculations of lattice regularized QCD. Whilelattice calculations at vanishing chemical potential made great progress inthe last decades, they are still harmed by the infamous sign problem atnonzero chemical potential. The two main methods that are currently usedto infer on the QCD phase diagram at nonzero baryon chemical potential( µ B ) are indirect, they rely on Taylor expansions of observables at µ B = 0,or analytical continuations from simulations at imaginary chemical poten-tial ( µ B = iµ I ). Methods that allow for a direct sampling of the oscillatorypath integral at ( µ B ) > ∗ Presented at the Workshop “Criticality in QCD and the Hadron Resonance Gas”,29-31 July 2020, Online. (1) a r X i v : . [ h e p - l a t ] J a n C. Schmidt et al.
The two principles that are guiding our understanding of the QCD phasediagram are spontaneous chiral symmetry breaking and – linked to it – thephenomena of quark confinement. Our knowledge on the (2+1)-flavor QCDphase diagram based on recent lattice results is summarized in Fig. 1 (left).The variables assigned to the three axes are temperature ( T ), the baryon T µ B m u , d
5) MeV [6] and T c = 132 +3 − MeV [7]. The variation of T pc with µ B , as indicated by adashed line, has also been calculated by lattice QCD. We obtain T pc ( µ B ) T pc (0) = 1 − κ (cid:16) µ B T (cid:17) + O (cid:16) µ B T (cid:17) , (1)where κ B = 0 . O ( µ B ) correction that vanishes within errors[6]. Similar results have been obtained recently in Ref. [8].In Fig. 1 (right) we compare the pseudo-critical line with freeze-outtemperatures and chemical potentials obtained from hadron yields measuredby STAR [5] and ALICE [4]. The hadron yields have been fitted (after feed-down corrections) to the hadron resonance gas (HRG) model. In its simplestnone-interacting version, this model is based on the mass spectrum of allstable particles and resonances listed in the particle data booklet, which aretaken as an ideal gas in thermal equilibrium at a common temperature T f ,chemical potential µ f , and volume V f . As these parameters refer to the timein the expansion of the fireball from when on its chemical composition doesnot change anymore, they are called chemical freeze-out parameters. We seefrom Fig. 1 (right), that the freeze-out parameters agree well with the chiralcrossover line obtained from lattice QCD. We note, that in order to meetconditions that are found in heavy ion collisions we have determined ourvalues for the electric µ Q ≡ µ Q ( µ B ) and strangeness chemical potentials µ S ≡ µ S ( µ B ), such that the following conditions for the net-numbers ofconserved charges in the system, (cid:104) n Q /n B (cid:105) = 0 . (cid:104) n S (cid:105) = 0, are fulfilled.However, the freeze-out parameters are still model based. Hence, in thefollowing we want to follow a procedure proposed in [9], that allow for thedetermination of the freeze-out parameters by a direct comparison of latticeQCD to experiment.
2. Cumulants of net-baryon number
Higher order cumulants of the net-baryon number are obtained as deriva-tives of the logarithm of the QCD partition functions with respect to thedimension less parameter ˆ µ B = µ B /T , χ Bn ( T, µ B , µ Q , µ S ) = 1 V T ∂ n ln Z ( T, µ B , µ Q , µ S ) ∂ ˆ µ nB , (2)where µ Q an µ S are the electric charge and strangeness chemical potentials.In the same way, we can can also calculate derivatives with respect to µ Q and µ S , which we denote as χ Qn and χ Sn , respectively. C. Schmidt et al.
Aiming on the comparison with the experimental results, we furtherintroduce ratios of cumulants of baryon number fluctuations as R Bnm = χ Bn χ Bm . (3)By using these ratios, the leading order dependence on the freeze-out vol-ume ( V f ) is removed. However, among other things fluctuations of theexperimentally observed freeze-out volume might still hinder a comparisonto lattice QCD. The first ratio we discuss is R B , which is shown in Fig. 2 and R B (T, µ B ) µ B /TT=152 MeV155 MeV158 MeV161 MeV R B (T pc , µ B ) µ B [MeV]QCDHRG Fig. 2. Left: Continuum estimate of the cumulant ratio R B as function of thechemical potential, for different temperatures. Right: The same ratio along thepseudo critical line. Shown are the QCD and HRG model results. can be interpreted as the mean of the net-baryon number, normalized by thevariance of the baryon number fluctuations. The presented HotQCD results[10] are obtained from high statistics lattice QCD calculations on 32 × ×
12 lattices, with (2+1)-flavor of highly improved staggered quarks(HISQ) at physical light and strange quark masses. The values in the range0 < ˆ µ B (cid:46) . µ B = 0 to 8 th order in ˆ µ B . As it is evident from the contin-uum estimate shown in Fig. 2 (left), the leading order of R B is linear in µ B .We further notice that the ratio is rather independent under the variationof temperature. Therefore the ratio has been termed a baryometer [9].The same ratio is shown in Fig. 2 (right), now plotted along the pseudocritical line as defined in Eq. (1). Here we compare the QCD result with thecorresponding calculation of a Hadron Resonance Gas (HRG). We see thatthe HRG model deviates from QCD only for µ B (cid:38)
150 MeV. We thus notethat for small µ B the HRG can be used to analyse the differences betweennet-baryon number and net-proton number fluctuations. The latter is thequantity which is directly accessible by heavy ion experiments. On theother hand, this also means that we do not see any indication of a divergingbaryon number fluctuation ( χ B ) in the range where we trust our Taylor et-baryon number fluctuations R Bnm (T pc ) R B (T pc ) dashed lines:joint fi t toSTAR data forR P31 , R
P42 s NN [GeV]: STAR preliminary: open symbolsLO, T=150 MeV
NNLO, R
B31 (T pc )R B42 (T pc )STAR 2020: R p R p Fig. 3. The cumulant ratios (bands) R B and R B versus R B on the pseudo-criticalline, calculated from a NNLO Taylor series. Data are results on cumulant ratiosof net-proton number fluctuations obtained by the STAR Collaboration [11]. Alsoshown are preliminary results obtained at √ s NN = 54 . expansion, which we would expect in QCD close to a critical point. In thiscase the ratio R B would decrease and approach zero at the critical point.As higher order cumulants are expected to diverge more rapidly whenapproaching a critical point, it is tempting to discuss also the ratios R B and R B along the pseudo critical line, which are shown in Fig. 3 as a functionof R B [10]. Since R B is still a monotonous function of µ B in the plottedrange, it is a measure for the baryon density and enables us to compare withthe experiment in a model free way. We see that the over all agreement withthe corresponding net-proton number cumulants R P and R P from STAR[11, 12] is very good. We conclude that a high freeze-out temperature of T f >
155 MeV seems to be excluded by the data. This lattice calculation isbased on an 8 th order expansion of the logarithm of the partition function.Finally we want to mention that the radius of convergence, which isinherent to the expansion of any thermodynamic observable, can in principleprovide valuable information on the phase structure of QCD. E.g. , in thecase of a second order phase transition, we expect the convergence radiusto be limited by the critical point. A simple estimator for the radius ofconvergence ˆ ρ ≡ µ crit B /T is given by the ratio estimatorˆ ρ = lim n →∞ (cid:113) ( n + 2)( n + 1) (cid:12)(cid:12) χ Bn /χ Bn +2 (cid:12)(cid:12) , (4)more advanced estimators are also discussed [13]. Unfortunately, we haveonly a limited number of expansion coefficients (cumulatns χ Bn ) at our dis-posal, which makes it difficult to draw strong conclusions with given latticedata. Especially, since the statistical and systematical error on higher ordercumulants is drastically increasing with the order n . It is however interest- C. Schmidt et al. -0.6-0.4-0.2 0 0.2 0.4 0.6 0 1 2 3 4 5 6 7Im[ µ B /T]Im[ χ ] T=201 MeVT=176 MeVT=160 MeV -0.5 0 0.5 1 1.5 0 1 2 3 4 5 6 7Im[ µ B /T]-Re[ χ ] T=201 MeVT=176 MeVT=160 MeV Fig. 4. Preliminary results of the first and second cumulant of the net-baryon num-ber, χ B , χ B , as a function of the imaginary chemical potential for three differenttemperatures, obtained from calculations on 24 × ing to note that all expansion coefficients have to be positive if the limitingsingularity lies on the real axis. Hence, we can obtain an upper bound forthe phase transition temperature T cep , as for T >
140 MeV many of the ex-pansion coefficients turn negative [14]. This estimate is in good agreementwith the statement that the temperature of the QCD critical point shallbe lower than the chiral transition temperature ( T cep < T c ) as indicated inFig. 1 (left).
3. Cumulants at imaginary chemical potential
Besides the Taylor expansion method, lattice QCD calculations can alsobe performed at purely imaginary chemical potential, followed by an ana-lytic continuation of the results. The QCD partition function is symmet-ric under the transformation ˆ µ B → ˆ µ B + 2 πi . Any simulations at imagi-nary chemical potential are thus constrained to the interval [ − iπ, iπ ] (firstRoberge-Weiss sector). We further note that even/odd order cumulantson this interval are purely real/imaginary and are even/odd functions ofIm[ˆ µ B ]. Making use of this symmetry, we thus need to simulate only inthe interval [0 , iπ ] and symmetrize/anti-symmetrize the data afterwards.We calculate the first four cumulants of the baryon number. Preliminaryresults from 24 × χ B ] develops a discontinu-ity at Im[ˆ µ B ] = π . The temperature where this is happening is called theRoberge-Weiss temperature ( T RW ), which was estimated to T RW = 201MeV [15] (for N τ = 4). In accordance with this discontinuity, we also ob-serve that the second cumulant χ B develops a divergence at T RW . Theuniversal scaling of the Polyakov-Loop (order parameter of the confine-ment transition) and chiral condensate have been investigated close to the et-baryon number fluctuations Roberge-Weiss transition [15].The periodic data on χ B can be analyzed in terms of Fourier coefficients[16, 17], which are inherently linked to the canonical partition sums. Theaim of this project is, however, to use the information of all the availablecumulants to construct a precise rational function approximation, i.e. a[ n, m ] Pad´e of χ B , χ B ≈ R mn (ˆ µ B ) = P m Q n = (cid:80) mi =0 a i ˆ µ iB (cid:80) nj =0 b j ˆ µ jB . (5)We are currently testing several methods to determine the coefficients a i , b j .Among them is a direct solve method, where we directly solve a set ofequations that we obtain by equating the analytic expressions for R mn aswell as its first few derivatives ∂ j R mn /∂ ˆ µ jB , j = 0 , , χ Bj +1 (ˆ µ ( k ) B ) = ∂ j R mn (ˆ µ B ) ∂ ˆ µ jB (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ µ B =ˆ µ ( k ) B . (6)Here χ Bj (ˆ µ ( k ) B ) represent the numerical values of the cumulants at the simu-lation points ˆ µ ( k ) B , as obtained by our lattice calculations. A similar methodis based on a χ -fit of R mn to our cumulant data. Finally we are testing atwo step approach where in a first step a suitable interpolation of the latticedata is chosen. In the second step we are making use of the Remez algo-rithm to determine R mn until the min-max criteria is satisfied with respectto the interpolation.Having the approximation R mn at hand, we are able to integrate thebaryon density to obtain the free energy, which will also develop a cusp at T RW . However, our main interest lies in the determination of the roots of thenumerator P m and denominator Q n , which will allow us to infer informationon the singularities in the complex ˆ µ B plane. A singularity in the complexplane is the reason for a finite radius of convergence of the Taylor series andwill also indicate a true physical phase transition when it approaches thereal axis in the complex ˆ µ B -plane.There are two models that can guide our thinking about the location ofthe singularities in the complex plane. At large temperatures the thermalbranch cut singularities from the Fermi-Dirac distribution of a free quark gasis expected to pinch the imaginary axis in the complex ˆ µ B -plane. In QCDsuch a behavior is expected to happen at T RW . In fact, this is something wealready see, when we analyze the data shown in Fig. 4. How this thermalsingularity moves in the complex plane with with decreasing temperature isshown in Fig. 5 (left). C. Schmidt et al.
Re[ µ B / T] . . . . . . . . I m [ µ B / T ] T=170T=160 T=150 T=140 T=130 T=120
Fig. 5. Left: Singularity in the complex plane, associated with the branch cutsingularity of the Fermi-Dirac distribution function of the quarks. The position ofthe singularity is shown for three temperatures T = 201 ,
176 and 160 MeV fromleft to right. The results have been obtained from calculations on 24 × f f ( z ). The results are model predictions for N τ = 4 , O (2)-model. At temperatures close to the chiral transition ( T c ), we might be able tomap our results to the universal scaling behavior connected to the chiralphase transition. The scaling function of the free energy f f ( z ) will have asingularity in the complex z -plane, known as the Lee-Yang edge singular-ity. This singularity has been determined recently [18]. Given a mappingfrom QCD to the universal theory, defined by the non-universal constants t , h , T c , κ B [HotQCD, private communication], we can calculate the po-sition of the singularity in the complex ˆ µ B -plane, shown in Fig. 5 (right).Preliminary results from calculations on 36 × T = 145 MeVseem to be in rather good agreement with this prediction. It will be veryinteresting but also challenging to see if the singularity will approach thereal axis in the complex ˆ µ B -plane for even smaller temperatures. Acknowledgements
We thank all members of the HotQCD collaboration for discussionsand comments. This work is support by Deutsche Forschungsgemeinschaft(DFG, German Research Foundation) through the Collaborative ResearchCentre CRC-TR 211 “Strong-interaction matter under extreme conditions”project number 315477589 and from the European Union’s Horizon 2020research and innovation program under the Marie Sk(cid:32)lodowska-Curie grantagreement No H2020-MSCAITN-2018-813942 (EuroPLEx).
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