Conserved charge fluctuations in the chiral limit
Mugdha Sarkar, Olaf Kaczmarek, Frithjof Karsch, Anirban Lahiri, Christian Schmidt
CConserved charge fluctuations in the chiral limit ∗ Mugdha Sarkar † , Olaf Kaczmarek, Frithjof Karsch, AnirbanLahiri, Christian Schmidt Fakult¨at f¨ur Physik, Universit¨at Bielefeld, D-33615 Bielefeld, Germany.We study the signs of criticality in conserved charge fluctuations andrelated observables of finite temperature QCD at vanishing chemical poten-tial, as we approach the chiral limit of two light quarks. Our calculationshave been performed on gauge ensembles generated using Highly ImprovedStaggered Quark (HISQ) fermion action, with pion masses ranging from140 MeV to 55 MeV.PACS numbers: 11.10.Wx, 11.15.Ha, 12.38.Aw, 12.38.Gc, 12.38.Mh, 24.60.Ky,25.75.Nq
1. Introduction
Understanding the phase diagram of QCD in the plane of temperatureand various chemical potentials is one of the primary goals of lattice QCDcalculations and the heavy-ion collision experiments at RHIC and LHC.The temperature variation at zero chemical potential is being explored atLHC and studied extensively using lattice techniques due to the absenceof the infamous sign problem. At physical quark masses, there exists achiral crossover at a temperature T pc around 157 MeV [1, 2]. A schematicphase diagram of QCD with an additional axis for degenerate light (up anddown) quark masses is shown in Fig. 1 [3]. In the figure, the dotted linesalong the horizontal and vertical axes correspond to the plane of physicalquark masses and the dashed line is the chiral crossover line which ends ata critical end point T cep , which is being actively pursued in experiments.The chiral limit, however, is not accessible to experiments and can onlybe studied via theoretical techniques. The spontaneous symmetry break-ing of the exact SU (2) × SU (2) symmetry in the chiral limit of two light ∗ Presented at workshop on Criticality in QCD and the Hadron Resonance Gas; 29-31July 2020, Wroclaw, Poland. † Speaker (1) a r X i v : . [ h e p - l a t ] O c t Mugdha Sarkar et al quarks is expected to be a phase transition belonging to the universalityclass of 3 d O (4) spin model [4]. There exists an alternative scenario wherethe chiral crossover turns into a first order transition as we move towards T µ B m u , d
2. Critical behavior in the chiral limit
According to Wilson’s renormalization group (RG) theory, the effectiveHamiltonian of a theory in the space of all possible couplings consists of“energy-like” terms which respect the symmetry and “magnetic-like” termswhich break the symmetry. In QCD, the temperature T and chemical po-tentials µ X for different conserved charges X = baryon number B , electriccharge Q , strangeness S, . . . would thus be energy-like couplings with re-spect to the chiral phase transition whereas the light quark mass m l ≡ m u,d would be the magnetic-like coupling.In the vicinity of a phase transition, the imprint of the criticality inthermodynamic quantities can be expressed as singular or non-analytic uni-versal contributions. The starting point of the discussion is writing downthe logarithm of the partition function i.e. the free energy density or the onserved charge fluctuations in the chiral limit -1.6-1.4-1.2-1-0.8-0.6-0.4-0.2 0 -5 -4 -3 -2 -1 0 1 2 3 4 5 -( h h ) − α / β δ f f ( ) ( z ) z (T-T c )/T c h h=10.50.250.050.0 -0.8-0.6-0.4-0.2 0 0.2 -5 -4 -3 -2 -1 0 1 2 3 4 5 -( h h ) − ( + α ) / β δ f f ( ) ( z ) z (T-T c )/T c h h=10.50.250.150.0 Fig. 2. Scaling behavior of the scaled singular parts of fourth (left) and sixth(right)order fluctuations as a function of the reduced temperature scaled by z ≡ h /βδ /t .The figure has been taken from Ref. [10] pressure, close to a phase transition using generalized scaling laws [11],1 T f ( T, (cid:126)µ, m l ) = 1 V T ln Z ( T, (cid:126)µ, m l ) = h (2 − α ) /βδ f f ( z ) + f r ( T, (cid:126)µ, m l ) , (1)where V is the system volume and f r denotes the regular non-critical con-tributions which are particular to the theory, QCD in our case. The non-analytic contribution is expressed in terms of a generalized energy-like cou-pling, the reduced temperature t and a magnetic-like coupling h , writtenupto leading order near the critical point, as t = 1 t (cid:18) T − T c T c + κ X (cid:16) µ X T (cid:17) (cid:19) , h = 1 h m l m s . (2)These dimensionless couplings are defined such that the phase transitionoccurs at t = h = 0. Since we are interested in the chiral phase transitionat chemical potential µ X = 0, this corresponds to temperature T = T c andlight quark mass m l = 0. The strange quark mass m s is used in h to getrid of multiplicative mass renormalization factors in order to obtain a well-defined scaling field in the continuum limit. It is important to note that m s does not break the two-flavor chiral symmetry group and in principle,can also be included in the definition of t . The dimensionless t and h arenon-universal constants. In the singular term in Eq. 1, f f ( z ) is a universalscaling function with the scaling variable z being a particular combinationof the t and h couplings, z ≡ t/h /βδ . Depending on the universality class,the critical exponents α, β and δ determine the singular behavior in thechiral limit, h → Mugdha Sarkar et al
The conserved charge fluctuations are obtained as derivatives of the freeenergy density f w.r.t. the corresponding chemical potentials and are there-fore, energy-like observables. At zero chemical potential, the expressions forthe singular parts in these cumulants can be obtained from Eq. 1 as χ X n = − ∂ n f /T ∂ ( µ X /T ) n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) µ X =0 ∼ − (2 κ X ) n h (2 − α − n ) /βδ f ( n ) f ( z ) , (3)where f ( n ) f ( z ) is the n th derivative w.r.t the scaling variable z . Note thatthe odd order cumulants are zero. From Eq. 2, it is easy to see that a sin-gle temperature derivative yields a singular part same as the one obtainedafter a double chemical potential derivative. Therefore, if the regular con-tributions are small enough, the second and fourth order conserved chargefluctuations would behave like energy density and specific heat, respectively.As one expects the chiral phase transition to belong to the O (4) universalityclass, one can plug in the O (4) critical exponents in the above-mentionedcumulants. Since α is negative one finds that the second and fourth orderfluctuations remain finite in the chiral limit whereas the sixth and higher or-der fluctuations diverge. The scaling behavior of the universal non-analyticcontributions of χ and χ obtained from 3 d O (4) spin model calculations,are shown in Fig. 2, which have been taken from Ref. [10]. The plots showthe variation of the singular parts w.r.t. the reduced temperature for dif-ferent h . It can be clearly seen in the left plot that the singular part of χ does not diverge in the chiral limit h → T c with a characteristic spike. On the other hand, the sixth order cumulant χ has a positive and negative peak which diverge in the chiral limit. We willconfront these expectations with our numerical findings.It is also interesting to look at the “mixed” observables which are deriva-tives of the free energy with respect to both the energy-like and magnetic-like couplings. In contrast to energy-like observables, mixed quantities aredivergent already at second order. For example, consider the second orderconserved charge fluctuation, χ X = − Aκ X H (1 − α ) /βδ f (cid:48) f ( z ) + χ X , reg , (4)where we move the factor of non-universal constant h into A by defining H ≡ m l /m s = h h and χ X , reg denotes the regular terms. Taking a H -derivative, one obtains the second order mixed susceptibility ∂χ X ∂H = Aκ X H ( β − /βδ f (cid:48) G ( z ) + ∂χ X , reg ∂H (5) The fourth order cumulant would diverge for the Z (2) universality class as α ispositive. onserved charge fluctuations in the chiral limit χ B2 T [MeV] H o t Q C D p r e li m i n a r y N τ =8 m s /27m s /40m s /80m s /160 χ B2 (T c ) (m l /m s ) (1- α )/ βδ HotQCD preliminaryN τ =8 Fig. 3. (
Left ) Second order net baryon-number fluctuations as a function of tem-perature for different H values. ( Right ) Linear fit of χ B ( T N τ =8 c ) as a function of H (1 − α ) /βδ using O (2) critical exponents. where f G is a universal scaling function related to f f as f G ( z ) = − (1 +1 /δ ) f f ( z ) + ( z/βδ ) f (cid:48) f ( z ). The observable has a moderate divergence, i.e., H ( β − /βδ = H − . in the chiral limit.
3. Numerical setup
Our calculations have been done on gauge ensembles generated by theHotQCD collaboration with the Highly Improved Staggered Quark (HISQ)fermion discretization and tree-level Symanzik-improved gauge action. Partof these ensembles have been recently used in the determination of the chiralphase transition temperature T c [6] and in studying the sensitivity of thePolyakov loop to the chiral phase transition [8]. Keeping the strange quarkmass m s fixed at its physical value, the gauge configurations have beengenerated with light quark masses m l = m s / , m s / , m s /
80 and m s / a = 1 / T , i.e., at fixed latticetemporal extent N τ = 8.
4. Results
At the outset, we mention that since our work has been done with stag-gered quarks at finite lattice spacing, the relevant universality class wouldbe that of 3 d O (2) spin models. Hence, we use the O (2) critical exponentswhich are quite close to O (4) and the qualitative conclusions should remainthe same.We start with the results of second order conserved charge fluctuations,shown in the left plots of Figs. 3, 4 and 5. In the scaling regime, the second Mugdha Sarkar et al χ Q2 T [MeV] H o t Q C D p r e li m i n a r y N τ =8 m s /27m s /40m s /80m s /160 χ Q2 (T c ) (m l /m s ) (1- α )/ βδ HotQCD preliminaryN τ =8 Fig. 4. Same as Fig. 3 but for electric charge fluctuations χ S2 T [MeV] H o t Q C D p r e li m i n a r y N τ =8 m s /27m s /40m s /80m s /160 χ S2 (T c ) (m l /m s ) (1- α )/ βδ HotQCD preliminaryN τ =8 Fig. 5. Same as Fig. 3 but for strangeness fluctuations order chemical potential derivative would behave like a single derivativew.r.t. temperature, i.e., like an energy density, if the singular part has adominant contribution. It is possible to estimate the singular contributionin these quantities by extrapolating the value of the function in the chirallimit at a given temperature [11]. Setting T = T c at any given light quarkmass in Eq. 4, we have χ X ( t = 0 , H ) = − Aκ X H (1 − α ) /βδ f (1) f (0) + const. reg. term + O ( H ) . (6)We see that the leading mass dependence is given by the critical exponentsin the singular term, followed by H . For O ( N ) and Z (2) universalityclasses, the combination (1 − α ) /βδ is positive and <
1, thus, the singularterm vanishes in the chiral limit. Using the T c value extracted in [6] for N τ = 8, we can plot χ X ( T c , H ) versus H (1 − α ) /βδ , as shown in the right plotsof Figs. 3, 4, 5. For small enough H , one would expect a linear dependence,which is what we find already from physical light quark masses within errors.From the intercept of the linear fit which denotes the constant regular term The free energy density is even in H . onserved charge fluctuations in the chiral limit Singular contribution at physical mass χ B ( T c ) ∼ χ Q ( T c ) ∼ χ S ( T c ) ∼ Table 1. Approximate singular contributions to various χ X at T = T c at physicallight quark mass m l = m s /
27 at lattice temporal extent N τ = 8. in Eq. 6, one can obtain the singular contribution at physical quark massas χ X ( T c , H = 0) − χ X ( T c , H = 1 / T N τ =8 c = 144 MeV inTable 1.From the singular contributions estimated above, one can get an ideaabout the curvature of the chiral transition line in the chiral limit. Theratios of the curvature coefficients κ X for different chemical potentials can beobtained directly from the ratio of the corresponding singular contributionsas everything else cancels out (see Eq. 6; note that f (1) f (0) is a constantuniversal number). The preliminary estimates of the ratios κ Q /κ B and κ B /κ S are 2.6 and 1.0 respectively. These values are quite close to thecorresponding ratios of the curvatures of the crossover line at physical quarkmasses, 1 . . th order conservedcharge fluctuations. The singular part of χ X would vanish at T c as h . with O (2) exponents and thus, no divergence would be present in the chirallimit. The appearance of the characteristic spike in the full quantity, as seenjust for the singular part in the left plot of Fig. 2, is therefore, dependenton the relative size of the regular term at a given light quark mass m l . Ourpreliminary results for the fourth order electric charge fluctuation χ Q as afunction of temperature is shown in the left plot of Fig. 6. With decreasinglight quark mass m l , a spike seems to be developing near T c . The plot for χ B shows similar features but is noisy and requires more statistics. Thesimilar plot for strangeness, however, is quite different with a monotoni-cally increasing behavior. This happens most likely due to a relatively largecontribution of the regular terms, as seen already in Table 1 for χ S .According to O (2) or O (4) universality, the 6 th and higher order fluctu-ations are divergent in the chiral limit and should provide a strong evidencefor criticality. Due to a diverging singular part, the regular terms shouldbe less relevant towards the chiral limit. However, these observables require Mugdha Sarkar et al N τ =8 H o t Q CD p r e li m i na r y χ Q4 T[MeV] m s /27m s /40m s /80m s /160 -1.00-0.500.000.501.00 130 140 150 160 170 180 N τ =8 HotQCD preliminary χ Q6 T[MeV] m s /27m s /40 Fig. 6. (
Left ) The fourth order cumulant of electric charge fluctuations, χ Q as afunction of temperature for different H values. ( Right ) The sixth order cumulantof electric charge fluctuations, χ Q versus temperature at two H values. large statistics and lattice calculations are increasingly expensive at smallermasses. Our preliminary result for χ Q in the right plot of Fig. 6 shows fea-tures similar to that of the universal singular term in the right plot of Fig.2. At first glance, it might look like there is no divergence with decreasingmass in the figure but one must keep in mind that firstly, the regular partscan still be appreciable at these masses and second, the ratio of the peakheights for H = 1 /
27 and H = 1 /
40 expected from O (2) singular parts isonly about 1 .
26, which is not too far from the actual data. The relativelyhigh positive part at lower T compared to the negative peak at high T for χ Q , in contrast to Fig. 2 (right), is probably due to large regular contribu-tions to the electric charge fluctuations from pions as predicted from hadronresonance gas (HRG) calculations.Finally, we discuss results from two mixed observables: ∂χ S /∂H , thelight quark mass derivative of the strangeness fluctuation χ S , and χ ls ≡ m s ∂ (cid:10) ¯ ψψ (cid:11) s /∂m l , the mass derivative of the strange quark chiral condensate(the dimensionless strange chiral condensate is defined as (cid:10) ¯ ψψ (cid:11) s = tr M − s ,where M s is the strange quark, staggered fermion matrix). Both of thesequantities have the same divergence in the singular part as discussed towardsthe end of Sec. 2 but with different non-universal factors. The regular termsin these quantities are also undoubtedly different. It is worthwhile to pointout again that the strange quark condensate does not break the 2-flavorchiral symmetry and hence is a energy-like observable. We show these twosusceptibilities in the left plots of Fig. 7 and 8. An interesting way to lookat the scaling behavior is to rescale the quantities with H − ( β − /βδ and plotit against the scaling variable z , as shown in Fig. 7 (right) and 8 (right).Apart from regular contributions, the rescaled observable is proportional tothe universal scaling function f (cid:48) G ( z ) (see Eq. 5). The data seems to fall on onserved charge fluctuations in the chiral limit N τ =8 H o t Q CD p r e li m i na r y - ∂χ S2 / ∂ H T[MeV]
H=1/401/801/160 N τ =8 O(2) H o t Q CD p r e li m i na r y - H (1- β )/ βδ ∂χ S2 / ∂ H z/z H=1/401/801/160
Fig. 7. (
Left ) The H -derivative of second order strangeness fluctuation χ S as afunction of temperature at different masses. ( Right ) The derivative, rescaled with H (1 − β ) /βδ , plotted as a function of z/z = H − /βδ ( T − T c ) /T c for fixed values of H . Apart from regular contributions, the rescaled quantity is proportional to theuniversal scaling function f (cid:48) G ( z ). N τ =8 H o t Q CD p r e li m i na r y χ ls T[MeV]
H=1/401/801/160 N τ =8 O(2) H o t Q CD p r e li m i na r y H (1- β )/ βδ χ ls z/z H=1/401/801/160
Fig. 8. Same as Fig. 7 but for H -derivative of strange chiral condensate, χ ls ≡ m s ∂ (cid:10) ¯ ψψ (cid:11) s /∂m l . top of each other in the figures with some deviations which can be takencare by considering regular terms. This clearly shows the universal scalingbehavior in these quantities already at physical quark masses. Althoughseemingly different, quantities like χ mP , the H -derivative of the Polyakovloop studied in Ref. [8], behave exactly same as the mixed susceptibilitiesdiscussed above, as expected from universal scaling behavior. It is possibleto do a similar analysis of the regular and singular parts of the above-mentioned mixed quantities, as has been done in Ref. [8].
5. Conclusions and Outlook
To summarize, the fluctuations of conserved charges at finite lattice spac-ing seem to be consistent with chiral phase transition belonging to O (2) uni- Mugdha Sarkar et al versality class, and therefore, to O (4) in the continuum limit. They exhibitexpected energy-like behavior with respect to chiral phase transition. Evenstrangeness fluctuations and strange quark condensate behave as energy-like quantities in the 2-flavor chiral limit. The singular contributions can beestimated for different observables and may be used to determine the cur-vature coefficients of the chiral critical line. In particular, our analysis forthe second order fluctuations at physical pion masses show a considerablesingular contribution. We intend to do a future comparison with HRG atsmaller masses to understand the interplay of singular and regular parts.A more quantitative understanding of the mixed susceptibilities and con-served charge susceptibilities will be obtained in future through a scalinganalysis following Ref. [8]. The work is under progress and more statisticsare being generated at lower masses in order to achieve proper continuumand thermodynamic limits.
6. Acknowledgments
This work was supported by the Deutsche Forschungsgemeinschaft (DFG)Proj. No. 315477589-TRR 211; the German Bundesministerium f¨ur Bil-dung und Forschung through Grant No. 05P18PBCA1 and the EU H2020-MSCA-ITN-2018-813942 (EuroPLEx). We thank the HotQCD Collabora-tion for providing access to their latest data sets and for useful discussions.REFERENCES [1] HotQCD, A. Bazavov et al. , Phys. Lett. B , 15 (2019), 1812.08235.[2] S. Borsanyi et al. , Phys. Rev. Lett. , 052001 (2020), 2002.02821.[3] F. Karsch, PoS
CORFU2018 , 163 (2019), 1905.03936.[4] R. D. Pisarski and F. Wilczek, Phys. Rev. D , 338 (1984).[5] O. Philipsen and C. Pinke, Phys. Rev. D , 114507 (2016), 1602.06129.[6] H. T. Ding et al. , Phys. Rev. Lett. , 062002 (2019), 1903.04801.[7] H.-T. Ding et al. , Nucl. Phys. A , 211 (2019), 1807.05727.[8] D. A. Clarke, O. Kaczmarek, F. Karsch, A. Lahiri, and M. Sarkar, (2020),2008.11678.[9] O. Kaczmarek, F. Karsch, A. Lahiri, and C. Schmidt, Contribution in thisProceedings (2020).[10] B. Friman, F. Karsch, K. Redlich, and V. Skokov, Eur. Phys. J. C , 1694(2011), 1103.3511.[11] J. Engels and F. Karsch, Phys. Rev. D85