O(N) universality and the chiral phase transition in QCD
Abstract
We discuss universal scaling properties of (2+1)flavor QCD in the vicinity of the chiral phase transition at vanishing as well as nonvanishing light quark chemical potential (). We provide evidence for scaling of the chiral order parameter in (2+1)flavor QCD and show that the scaling analysis of its derivative with respect to the light quark chemical potential provides a unique approach to the determination of the curvature of the chiral phase transition line in the vicinity of .
1 Introduction
It is well understood that the phase structure of QCD at nonzero temperature crucially depends on the quark mass values [1, 2]. While in the case of three degenerate quark masses (3flavor QCD) a first order transition occurs for sufficiently small values of the quark mass, it is expected that the transition in 2flavor QCD is continuous and belongs to the universality class of symmetric, 3dimensional spin models. Several studies of the QCD phase diagram as function of two degenerate light () and a strange () quark mass suggest that the region around the 3flavor chiral limit, where the QCD transition becomes first order, is indeed small and does not include the point of physical light and strange quark masses. Our
current understanding of the location of the region of order transitions that is separated by a line of order transitions from the crossover region and its positioning relative to the physical point is presented in the Columbia plot shown in Fig. 1.
While numerical calculations in 3flavor QCD gave evidence for the existence of a first order transition, many of the details of the transition in 2 or (2+1)flavor QCD with light up and down quarks are still poorly constrained through lattice calculations. Earlier attempts to verify scaling in numerical studies of 2flavor QCD using standard staggered fermions were not very successful in establishing the expected universal scaling properties [3].
We present here a new analysis of scaling properties using improved staggered fermion and gauge actions [4]. We perform this analysis for (2+1)flavor QCD. Implicitly we thus also analyze whether the region of first order chiral transitions ends in a tricritical point for a strange quark mass value below its physical value or above. We will use the scaling analysis of the chiral order parameter to determine the curvature of the chiral phase transition line at nonvanishing values of the light quark chemical potential , i.e. the first (Taylor) expansion coefficient of the transition line in powers of around .
2 O(N) scaling of the chiral condensate
In the vicinity of a critical point regular contributions to the logarithm of the partition function become negligible. The universal critical behavior of the order parameter of, e.g. 3dimensional spin models, is then controlled by a scaling function that arises from the singular part of the logarithm of the partition function,
(1) 
with and scaling variables and that are related to the temperature, , and the symmetry breaking (magnetic) field, ,
(2) 
Here and are critical exponents, unique for the universality class of the second order phase transition which the system undergoes in the limit . The form of the scaling function is well known from numerical simulations of 3dimensional symmetric spin models [5].
In QCD symmetry breaking arises due to a nonvanishing light quark mass, , and the corresponding order parameter is the chiral condensate, which we write as
(3) 
where denotes the strange quark mass in lattice units and is the temporal extent of the 4dimensional lattice, . One may improve the operator by subtracting a fraction of the strange quark condensate [4]. This eliminates additive linear divergent terms proportional to the light quark masses. Such a subtraction is, in fact, mandatory, if one wants to take the continuum limit at finite quark mass before taking the chiral limit. This ordering of limits, indeed is needed in order to recover the correct scaling behavior from calculations performed with staggered fermions, which only preserve a global symmetry for any nonzero value of the lattice spacing. We will be less ambitious here and discuss the chiral limit on lattices with fixed temporal extent, . In this case, we can only expect to find rather than scaling behavior. However, the scaling functions, , are very similar for both universality classes and it thus will be difficult to distinguish and scaling through an analysis of the order parameter alone. Moreover, given the large scaling violations observed in earlier studies with staggered fermions[3], already the observation of scaling in terms of a generic O(N) scaling function at nonzero values of the lattice cutoff is a major step forward.
We show in Fig. 2(left) results from a calculation of chiral condensates in (2+1)flavor QCD. The bare strange quark mass () has been chosen such that the physical value of the strange pseudoscalar mass, , is reproduced. The light quark mass has been varied in a range , which for the light pseudoscalar Goldstone meson corresponds to a regime . The lattice size has been varied from for the heavier quark masses to for the lightest quark masses [4]. This insures that finite volume effects remain small in the entire light quark mass regime, i.e. in units of the spatial extent we always have .
From Fig. 2(left) it is obvious that the chiral condensate scales with the square root of the quark mass in the low temperature, chiral symmetry broken phase,
(4) 
This is characteristic for Goldstonemodes in three dimensional symmetric spin models. In fact, this also is the dominant term characterizing the scaling function in the symmetry broken phase, i.e. for [4].
Results for the chiral condensate may be put on the universal scaling curve by using the reduced temperature and rescaled symmetry breaking field introduced in Eq. 2. Of course, this is expected to be possible only close to criticality where contributions from regular terms and corrections to scaling are small. The scaling analysis shown in Fig. 2(right) has therefore only been performed for the three lightest quark mass values, and for temperatures close to . From this one determines the three free parameters, and . As expected results for heavier quarks, also shown in Fig. 2(right), show deviations from the universal scaling behavior. Contributions from corrections to scaling become significant for , i.e. MeV. This is in contrast to calculations with Wilson fermions, where indications for scaling have been reported for even large values of the pseudoscalar meson mass [6].
The scaling analysis of the order parameter provides two nonuniversal parameters that are unique for QCD, the chiral phase transition temperature, , and the scale parameter . In the continuum limit both quantities are functions of the strange quark mass only. Of course, as well as are cutoff dependent and a proper continuum extrapolation is needed to extract their values in the continuum limit. From our analysis on lattices with temporal extent we find . A preliminary analysis on lattices with temporal extent suggests that this value drops by almost a factor 2 [7]. A more detailed analysis of the approach to the continuum limit thus is needed. We stress, however, that is a physical parameter of QCD. It gives the slope of the quark mass dependence of the pseudocritical temperature, which can be determined from the location of a peak in the chiral susceptibility,
(5)  
(6) 
The scaling functions and are shown in the right hand part of Fig. 4. The scaling function has a maximum at . The dependence of the pseudocritical temperature, , on the quark mass is given by the condition that , i.e.
(7) 
For the 3d universality class the peak in the chiral susceptibility is located at . Using this and expressing the symmetry breaking field in terms of pion and kaon masses rather than quark masses, , we find
(8) 
This allows to estimate the phase transition temperature in the chiral limit. With one finds with our current estimates for that the transition temperture in the chiral limit is about smaller than the crossover temperature at physical values of the quark masses. Of course, this still needs to be analyzed closer to the continuum limit.
3 Curvature of the critical line in the  phase diagram
For nonvanishing light quark chemical potential, , the second order chiral phase transition persists to exist in the  plane. For small values of the chemical potential, , the curvature of the phase transition line can be determined by making use of the scaling analysis of the order parameter. In this case the reduced temperature variable, , also depends on the chemical potential as it couples to the quark number operator, which does not break chiral symmetry. To leading order it contributes quadratically,
(9) 
The condition for criticality, , fixes the shape of the transition line for small values of . In the scaling regime the order parameter depends on temperature and quark mass only through the scaling variable . The derivative of with respect to thus is, up to a constant, identical to the second derivative of with respect to the chemical potential . This derivative too is related to the scaling function which we have introduced in Eq. 6 and which is shown in Fig. 4(right),
(10) 
We note that has properties similar to the chiral susceptibility. It diverges in the chiral limit at and the peak position in can be used to define a pseudocritical temperature at nonzero values of the light quark mass.
Once the scale parameters , needed to project the chiral order parameter onto the scaling curve, are known, we can use this information to determine the curvature of the critical line, , by calculating and by matching the scaling curve to the data using an appropriate scaling factor . Such a scaling analysis is shown in Fig. 4(left). From this we find for the curvature of the phase transition line in the chiral limit the preliminary result [8], which is in good agreement with earlier determinations of the curvature of the pseudocritical line at nonzero values of the quark mass performed with different numbers of flavors [9]. We note that the current analysis suggests that the curvature of the transition line, expressed in terms of^{1}^{1}1Some caution is needed when translating to and comparing lattice results with experimental findings. Lattice calculations have been performed for vanishing strange quark chemical potential (in some cases even for 2flavor QCD), while the freezeout conditions in a heavy ion collision refer to a system with [10]. , is smaller than the experimentally determined freezeout curve, which is well parametrized by [10],
(11) 
4 Conclusions
We have discussed universal properties of the chiral condensate of (2+1)flavor QCD in the limit of vanishing light quark masses. It agrees well with expected scaling predictions. We showed how this can be used to calculate the curvature of the second order chiral phase transition line in the vicinity of .
At present this analysis is limited to rather coarse lattices. Calculations closer to the continuum limit are needed to get control over the scale parameter and the phase transition temperature in the chiral limit.
Acknowledgements
We thank the organizers of the workshop ’New Frontiers in QCD 2010’ at the Yukawa Institute for Theoretical Physics, Kyoto, for a very stimulating workshop program and for support. This work has been supported in part by contract DEAC0298CH10886 with the U.S. Department of Energy.
References
 [1] R. Pisarski and F. Wilczek, Phys. Rev. D 29, 338 (1984).
 [2] F. R. Brown et al., Phys. Rev. Lett. 65, 2491 (1990).

[3]
F. Karsch and E. Laermann,
Phys. Rev. D 50, 6954 (1994);
S. Aoki et al. (JLQCD Collaboration), Phys. Rev. D 57, 3910 (1998);
C. W. Bernard et al., Phys. Rev. D 61, 054503 (2000).  [4] S. Ejiri et al., Phys. Rev. D 80, 094505 (2009)

[5]
J. Engels, S. Holtmann, T. Mendes and T. Schulze,
Phys. Lett. B 492, 219 (2000);
J. Engels and T. Mendes, Nucl. Phys. B 572, 289 (2000). 
[6]
Y. Iwasaki, K. Kanaya, S. Kaya and T. Yoshíe,
Phys. Rev. Lett. 78, 179 (1997);
A. Ali Khan et al. (CPPACS Collaboration), Phys. Rev. D 63, 034502 (2001).  [7] W. Unger (RBCBielefeld Collaboration), poster presented at Strong and Electroweak Matter 2010, Montreal, Canada, June 29July 2, 2010.
 [8] RBCBielefeld Collaboration, in preparation

[9]
for a summary of earlier results on the curvature of the transition line see:
O. Philipsen, Prog. Theor. Phys. Suppl. 174, 206 (2008).  [10] J. Cleymans, H. Oeschler, K. Redlich and S. Wheaton, Phys. Rev. C 73, 034905 (2006).