The quark number susceptibility near the QCD critical end-point (CEP), the tricritical point (TCP) and the O(4) critical line at finite temperature and quark chemical potential is investigated. Based on the universality argument and numerical model calculations we propose a possibility that the hidden tricritical point strongly affects the critical phenomena around the critical endpoint. We made a semi-quantitative study of the quark number susceptibility near CEP/TCP for several quark masses on the basis of the Cornwall-Jackiw-Tomboulis (CJT) potential for QCD in the improved-ladder approximation. The results show that the susceptibility is enhanced in a wide region around CEP inside which the critical exponent gradually changes from that of CEP to that of TCP, indicating a crossover of different universality classes.

We discuss how to extract renormalized from bare Polyakov loops in SU (N ) lattice gauge theories at nonzero temperature. Single loops in an irreducible representation are multiplicatively renormalized, without mixing, through mass renormalization. The values of renormalized loops in the four lowest representations of SU (3) were measured numerically on small, coarse lattices. We find that in magnitude, condensates for the sextet and octet loops are approximately the square of the triplet loop. This agrees with a large N expansion, where factorization implies that the expectation values of loops in adjoint and higher representations are powers of fundamental and anti-fundamental loops. The corrections to the large N relations at three colors are greatest for the sextet loop, ∼ 1/N , and are found to be ≤ 25%. The values of the renormalized triplet loop can be described by a matrix model, with an effective action dominated by the triplet loop: the deconfining phase transition for N = 3 is close to the Gross-Witten point at N = ∞.

We argue that the event-by-event fluctuation of the proton number is a meaningful and promising observable for the purpose of detecting the QCD critical end-point in heavy-ion collision experiments. The long range fluctuation of the order parameter induces a characteristic correlation between protons which can be measured. The proton fluctuation also manifests itself as anomalous enhancement of charge fluctuations near the end-point, which might be already seen in existing data.The event-by-event fluctuations in heavy-ion collisions carry information about the degrees of freedom of the created system and their correlations [1]. In particular, thermodynamic properties of QCD can be inferred from event-by-event fluctuation measurements [2,3,4,5,6,7].Of particular interest are fluctuations originating from the QCD critical end-point [3,4,8,9,10,11,12]. Since the fluctuation of the order parameter induces characteristic correlations among particles, in particular pions, it is expected that the end-point affects the event-by-event fluctuations of certain observables in a nontrivial way [3,4,13].Here we discuss a new observable which may serve as a signal of the end-point; the event-by-event fluctuation of the net proton number, i.e., the number of the protons minus the number of antiprotons observed [30].Our starting point is the fact that the baryon number susceptibility χ B [14,15,16,17,18,19] diverges at the critical end-point [3,9,12,20]. χ B is related to the average magnitude of the fluctuation δB of the baryon number:where V and T are the volume and the temperature. This letter is devoted to clarifying where, in the observed quantities, the divergence occurs and advocating the proton number fluctuation as a sensible and promising observable for the search of the critical point in the heavy-ion experiments.In this work, we confine ourselves to equilibrium thermodynamic fluctuations. Various important issues such as the nonequilibrium evolution of the fluctuations will be (and some already have been) studied separately.For simplicity and clarity we shall work in QCD with exact isospin invariance. The relevant corrections due to isospin breaking are small as we discuss below. Let us first show that in this case the isospin number susceptibility, χ I , is finite at the end-point. The proof is based on the fact that the singular behavior of thermodynamic quantities near the critical point is due to the divergence of a certain correlation length. It is the correlation length in the σ-channel, the channel with quantum numbers of the chiral condensate ψ ψ [3, 9]. A densitydensity correlator, such as χ I = (1/T ) d 3 x V 0 (x)V 0 (0) can diverge only if the density can mix with the σ field. For the isospin density this mixing is strictly forbidden by the SU(2) V (isospin) symmetry. The isospin density, V 0 (x), transforms as a triplet, 3. On the other hand, σ is a singlet. The mixing is forbidden and there is no singular contribution in χ I . [32] Small explicit breaking of the SU(2) V symmetry by the quark mass differ...