Analytic structure of QCD and Yang-Lee edge singularity

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Uncovering the structure of the phase diagram of Quantum Chromodynamics at nonzero tem- perature and density has been the central goal for both the theoretical and the experimental nuclear physic community (see e.g. [1]). The transition from hadronic matter to a quark gluon plasma along the finite temperature axis in the QCD phase diagram is known to be a smooth crossover. As the baryon doping is increased, this crossover may turn into a first-order phase transition at a critical endpoint. Experimental searches for critical behavior have focused on a possible non-monotonic dependence of fluctuation observables, such as the cumulants of conserved charges, on the beam energy [2]. Due to complexity of the experimental measurement and theoretical interpretation of the complex dynamics of the collision, the experimental search has not yet led to a conclusive result. On the other side, model independent non- perturbative theoretical understanding of the QCD phase diagram is hampered by the so-called sign problem. Aside from attempts to entirely circumvent the sign problem, e.g. by using methods that do not have to rely on importance sampling (such as functional methods, see e.g. [3, 4]), common strategies are 1) to expand the path integral about μ = 0 or 2) to compute thermodynamic quantities at purely imaginary chemical potential. The first approach yields a power series in μ, where each coefficient can be computed from the path integral as an operator expectation value with the μ = 0 weight. As the results of both methods, the information at finite μ can be obtained through an analytical continuation from μ = 0 or purely imaginary μ. On the one side, a major obstacle for schemes based on analytical expansions is that they are bound by the analytical constraints of the underlying theory [5]. For example, it is well known that singularities in the complex plane determine the radius of convergence of analytical expansions. This significantly limits what we can learn at least from a naive analytical continuation strategies. On the other side, the information about the singularities in the complex plane can be used to learn the location of the critical point. The apparent smoothness of the QCD crossover obscures non-trivial critical behavior at complex values of the baryon chemical potential, where the remnants of the critical point reside, known by the name of Yang-Lee edge (YLE) singularities [6, 7]. These singularities are continuously connected to the associated critical points: when two YLE singularities pinch a physical axis of the corresponding thermodynamic variable (for the case of the chiral critical point, the baryon chemical potential), the critical point with corresponding critical scaling emerges. Thus, locating and especially tracking the Yang- Lee edge singularities as a function of temperature may reveal the existence and the location of the QCD critical point. This strategy has been recently implemented in Refs. [8, 9] both of which use the Taylor series expansion coefficients calculated on the lattice. Ref. [8] includes the imaginary chemical potential data as well. Despite this difference, both studied led to a similar results for the location of the critical point. Using the information regarding the singularity structure of the equation of state to extract the physics of critical phenomenon opens a new avenue for studying the QCD phase diagram. Interestingly, QCD exhibits spontaneous symmetry breaking in various limiting cases of the μсep FIG. 1. Schematic diagram of the YLE scaling in the complex μ/T plane. Indicated are the lines on which the YLES associated with different critical points are in their corresponding scaling regimes with varying temperatures. control parameters. Besides the above described QCD critical point (cep) the chiral transition occurs in the massless limit of QCD. Another critical point resides in the imaginary chemical potential domain as a remnant of the center-symmetry of the gauge group SU(3), known as the Roberge- Weiss (RW) transition. As the scaling fields of these transitions differ, they indicate different scaling of the YLE. The expected temperature scaling of the YLE in the complex chemical potential plane is indicated in Fig. 1 (the figure is to be provided separately, as the web form does not accept it). For temperatures close to the RW-transition the scaling of the LEY could be observed [10, 11]. Also in other systems, such as the 2d-Ising model, it could be demonstrated that tracing the LYE is a valid and efficient method to precisely determine the location of the phase transition and to investigate the related universal scaling [12]. The methods to determine the YLE in numerical simulations are, however, still under development. Besides the Padé and multi-point Padé meth- ods, Fourier coefficients of the free energy [13-16], as well as the baryon number density [17-20] are discussed as tools for the YLE determination.

Currently, the major obstacles are the limited numerical precision of the data and possible finite-size effects. For a recent discussion see, e.g., the talks at Lattice 2024 [21-24].