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Cern-Ph-Th/2008-221
The curvature of the critical surface :

a progress report

###### Abstract

At zero chemical potential , the order of the temperature-driven quark-hadron transition depends on the quark masses and . Along a critical line bounding the region of first-order chiral transitions in the plane, this transition is second order. When the chemical potential is turned on, this critical line spans a surface, whose curvature at can be determined without any sign or overlap problem. Our past measurements on lattices suggest that the region of quark masses for which the transition is first order shrinks when is turned on, which makes a QCD chiral critical point at small unlikely. We present results from two complementary methods, which can be combined to yield information on higher-order terms. It turns out that the term reinforces the effect of the leading term, and there is strong evidence that the and terms do as well. We also report on simulations underway, where the strange quark is given its physical mass, and where the lattice spacing is reduced.

Curvature of critical surface
\FullConferenceThe XXVI International Symposium on Lattice Field Theory

July 14 - 19, 2008

Williamsburg, Virginia, USA

## 1 Introduction

The fundamental importance of the phase diagram of QCD, as a function of temperature
and quark chemical potential , makes it the object of several current lattice investigations.
It depends sensitively on the quark masses. At , Fig. 1 (Left) summarizes
the prevalent understanding of the order of the finite-temperature quark-hadron transition
as a function of and .
The physical point lies in the crossover region, separated from the chiral, first-order region
by a second-order chiral critical line. While the situation is far from settled, it
can in principle be resolved by manageable increases in computer resources. When ,
the complex nature of the fermion determinant makes the matter much worse. While finite-
results, including the location of the QCD critical point, have been obtained by reweighting
data [3], assessing the reliability of these results is a challenge in itself [4].
It appears that the only information that can be obtained reliably (i.e. performing thermodynamic
and continuum extrapolations) in principle, barring an algorithmic breakthrough, is the
Taylor expansion of thermodynamic observables in about . This makes the
detection of a finite- critical point, characterized by a singularity in the free energy,
particularly difficult.

To circumvent this problem, our strategy consists of Taylor-expanding the surface swept by the
chiral critical line of Fig. 1 (Left). The Taylor expansion of a generic quark mass
on the chiral critical surface, and the associated transition temperature , can be written as:

(1) | |||||

(2) |

The sign of governs the small- behaviour, as illustrated Fig. 1. Our first results [1], for the theory on an lattice, favored a negative value for . In [5], we presented a new numerical method to obtain the ’s. Here, we combine the two methods and report on our progress towards determining and higher Taylor coefficients (i) on larger lattices; (ii) for the theory with physical ; (iii) for the theory on a finer, , lattice.

## 2 Extracting the -dependence of the critical point

On the lattice, the Taylor expansion (2) is replaced by that of dimensionless observables:

(3) | |||||

(4) |

To differentiate between crossover, second- and first-order transitions, we monitor the Binder cumulant of the quark condensate:

(5) |

when . On the chiral critical surface, takes value 1.604 as dictated by the Ising universality class. It can be expanded as:

(6) |

with coefficients satisfying the scaling behaviour for large . Having measured the first few ’s by the methods of Sec. 3, we can reconstruct the ’s eq.(4) as:

(7) | |||||

(8) |

and finally and as:

(9) | |||||

(10) |

## 3 Two methods to measure derivatives

varies steeply with the quark mass, and in eq.(6) can be
obtained straightforwardly from fits of measured at for different quark masses [1].
Measuring the variation of with is another matter: is a noisy quantity,
its variation is small, and simulating at non-zero (real) is not feasible. We have used two
different, complementary methods to bypass these difficulties [5]:

1. We perform simulations at several imaginary values , where the sign problem is
absent, and fit our measurements of with a truncated Taylor series in .

2. We perform simulations at , reweight to small values , and measure the
finite difference quotients , with

(11) |

A comparison between the two methods is provided Fig. 2 (Left), on an lattice for . The error band is the fit to the finite- data (method 1). The data points are the finite-difference quotients (method 2). Consistency between the two methods is observed. The second method is clearly more efficient, since the statistics is only 1/4 of the other. This efficiency can be traced to the strong cancellation of statistical fluctuations when measuring on the and the reweighted ensemble. Reweighting itself is done stochastically with a Gaussian-distributed vector , since the reweighting factor is

(12) |

Note the small values of in Fig. 2 (Left): they guarantee a good overlap between the Monte Carlo ensemble and the reweighted ensemble, and small fluctuations in .

Since our lattice is not very large (), we performed a finite-size scaling check by comparing with a lattice. Fig. 2 (Right) shows nice consistency with the expected large volume universal behaviour, not only for the -axis intercept yielding , but also for the slope yielding . The result ( like ) reinforces the finding that the transition weakens and turns into a crossover (i.e. increases) as is turned on (see eq. (6)).

Finally, we can combine the data from our two methods, since the simulations were performed independently and cover different ranges of . A combined fit of the data Fig. 3 shows that is an alternating series in [7]. The fit gives

(13) |

with a /d.o.f. of 0.57. The large values of higher-order coefficients indicate that higher-order terms become important when . However, after rotation to real , they all tend to increase , pushing the system deeper in the crossover region. This only increases the validity of the exotic scenario Fig. 1 (Right) up to larger values of . Conservatively, we trust only the and terms. After continuum conversion following eqs.(7-10), our final result for on coarse, , lattices reads [6]:

(14) |

## 4 Towards the continuum limit

We are currently investigating two reasons why our result eq.(14) could change qualitatively as we consider real QCD. The sign of the curvature could change as we move along the critical line away from the degenerate case. It could also change as we take the continuum limit.

The first possibility appears unlikely given our current results Fig. 4 (Left), where is given its physical value on the critical line determined in [1] (see Fig. 4 (Right)). Since our pions are lighter than in nature, large lattices are required and thereby large computer resources. This is achieved, like for the , method 2 case above, by dispatching our simulations over the computing Grid. Many independent Monte Carlo runs are performed, all at , over a range of temperatures near , using prioritized scheduling. Current statistics reach 600k thermalized configurations.

The effect of a finer lattice is studied by simulating lattices with degenerate flavors. The current results, Fig. 5 (Left), give opposite signs for using a leading or subleading order fit. While the sign of the curvature is consequently not clear, one can already say that is not large, or less. Thus, the critical surface is almost vertical.

In addition, another qualitative effect takes place: the critical line, and thereby the whole chiral critical surface, moves towards the origin as . For instance, the pion mass on the critical line drops from to going from to lattices [5]. The first-order region, in physical units, shrinks dramatically as . To compensate this effect and maintain a critical point for real QCD at small chemical potentials , a large positive curvature would be needed. We presently do not see it.

Finally, we note that effective models like PNJL [8] or linear sigma model [9], with simple modifications, can reproduce the qualitative features of the chiral critical surface which we observe. Nevertheless, let us stress again that our study concerns only the chiral critical surface, swept by the chiral critical line as the chemical potential is turned on. Our results do not preclude other phase transitions, not connected to the chiral one.

## Acknowledgements:

This work is partially supported by the German BMBF, project Hot Nuclear Matter from Heavy Ion Collisions and its Understanding from QCD, No. 06MS254. We thank the Minnesota Supercomputer Institute for providing computer resources, and the CERN IT/GS group for their invaluable assistance and collaboration using the EGEE Grid for part of this project. We acknowledge the usage of EGEE resources (EU project under contracts EU031688 and EU222667). Computing resources have been contributed by a number of collaborating computer centers, most notably HLRS Stuttgart (GER), NIKHEF (NL), CYFRONET (PL), CSCS (CH) and CERN.

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