The QCD Phase Diagram in the Quark Mass Plane
Prof. Zoltan Fodor
University of Wuppertal
Local Project ID:
HPC Platform used:
SuperMUC-NG at LRZ
Quarks are the constituents of the massive basic building blocks of visible matter. These building blocks are the hadrons, more precisely protons and neutrons, which are about 2,000 times heavier than electrons. The most important of these quarks are the up and the down quarks, whereas the strange quark also plays some role in these hadrons. A proton contains two up and a down quarks, whereas the neutron consists of one up and two down quarks (so-called valence quarks). In normal conditions, as on earth or even in the sun, the quarks are confined in hadrons, they cannot get out. The confining force is called the strong force, the underlying theory is the theory of the strong interactions or in other words Quantum Chromo Dynamics, or in short QCD.
It is important to note, that quark masses are constants of nature. They are what they are and experiments cannot change them. In nature the up and down quark masses are just a few per-mill of the proton's mass, whereas the strange quark mass is about one tenth of the proton's mass.
In the early universe, at very high temperatures, the quark ingredients of protons and neutrons were freed. We call this state the deconfined phase. They were not confined in protons or neutrons, in some way these hadrons were boiled and the ingredients got liberated. As the universe expanded and cooled down the quarks got confined in hadrons. We call this state the confined phase. It is an interesting question to tell the order of the transition between these two -- confined and deconfined -- phases. Is it a first order phase transition as in the case of boiling water? Or is it a second order phase transition? Or it is not a phase transition at all, just an analytic transition as in the case of melting butter? It was believed for a long time that the transition between free quarks and confined quarks is a first order phase transition. It is easy to understand why physicists conjectured this sort of transition type. When we boil water the attractive forces between the water molecules are not strong enough to keep them together and the water molecules get liberated. Similarly, it was believed that the qualitative difference between the confined and liberated quarks is so huge that it must be accompanied by a first order phase transition.
What would it mean for the early and even for the present universe? A first order phase transition is always a place for dramatic changes. Bubbles or droplets appear, they grow and finally fill out the whole space. This happens for the water vapor phase transition and this would be the case for a first order QCD phase transition, too. Bubbles would appear, they would collide and the low temperature confining phase would fill the whole universe. During the course of the phase transition various remnants would appear, such as quark nuggets or primordial black holes, just to mention two of them. These remnants could have cosmological consequences or they could be even detectable by experiments. Despite the efforts of three decades no such remnants were detected. Are our experiments not sensitive enough (this is the more plausible scenario)? Or even worse: is the underlying picture false (this seemed to be a less plausible scenario)?
It was a heroic effort to determine the nature of the phase transition for physical quark masses, which our group carried out in 2006 and published in Nature  (this publication turned out to be the most cited lattice paper of the present century). This finding has fundamental consequences for the early universe and for the possible remnants we might detect even today. Note however, the result has not only relevance for the early universe (Big Bang) but also for heavy ion collisions (Little Bang), which are carried out at the RHIC (Brookhaven, USA) and LHC (Geneva, Switzerland) accelerators.
Since the nature of the transition is an extremely crucial, fundamental information and all of the quantitative features (transition temperature, equation of state etc.) depend on it, it is very important to pin down the transition's nature for various points on a hypothetical light quark versus strange quark mass plane. This will embed the finding for the physical point and gives it further support. Here it is of particular importance to locate two characteristic points. The small mass and the large mass degenerate cases. For zero mass we expect a first order phase transition, which becomes second order if we increased the masses after which it is an analytic cross over probably all the way to the physical point. For infinitely large masses the transition is known to be first order. Again reducing the masses all the way down to the physical point with analytic cross over one should find a mass value with a second order phase transition. These cases are illustrated on the attached figure. Our goals are threefold. a) Determine the nature of the transition at the physical point with chiral fermions and with very fine lattices using so-called staggered formalism; b) locate the small mass second order phase transition point (this task needs probably chiral fermions, too, but the first steps can be done with staggered fermions); c) locate the large mass second order phase transition point (in this case chiral symmetry is not important staggered fermions can be used).
The diagram shown here is a modern version of what was first sketched by Columbia group in Ref. . Nature with its given quark masses represents a single point in this representation. This physical point is in the region where the confined and plasma phases are not separated by a real transition. Those quark masses where the transition is to be expected to be second order are marked red in the diagram. Our project aims to calculate these lines, since the position of these are, at present, unknown in the continuum limit.
Our first preliminary results concerning the upper right corner of the Columbia plot can be found in Ref . Since real phase transitions can only appear in infinite large systems, we perform simulations with different spatial lattice extensions to calculate the infinite volume extrapolation. Thus we use staggered fermions on lattices of the size: 243x8, 283x8, 323x8, 403x8, 483x8 and 643x8.
If the transition is a real phase transition and not an analytic crossover, the susceptibility of the order parameter is expected to diverge in the infinite volume limit. In the case of a second order transition this behavior is accompanied by a critical exponent.
The plot shows the inverse susceptibility of the order parameter as a function of the quark masses in lattice units. The simulated systems seem to get more and more critical as the quark masses are increased since the inverse susceptibility vanishes according to a power law. The results fit well to previous findings based on different methods such as the matrix model  or the Binder cumulant analysis .
Our goal is to calculate the corresponding critical mass on finer lattice discretizations in order to achieve a reliable continuum limit. The result will be the critical point along the diagonal of the Columbia plot located on the 2nd order line in the upper right corner.
Thus we make simulations on LRZ's SUPERMUC-NG system on wide range of lattice resolutions and sizes. The order of the transition we determine by monitoring the fluctuations of the order parameter in a finite volume scaling analysis.
Through the study of the phase structure of the strongly interacting matter in this enlarged parameter space
we will also learn something on QCD with Nature's choice of the quark masses. We'll be able to make connection to the phase structure of the strongly interacting matter in actual collider experiments. For this we'll quantify the relation of the critical quark masses (red lines in the diagram above) and the quark densities in high energy accelerator experiments.
 Y. Aoki et al, Nature 443 (2006)
 F. R. Brown et al, Phys.Rev.Lett. 65 (1990)
 S. Borsányi et al, PoS 38th Int. Symposium on lattice field theory (2021) arXiv:2112.04192
 K. Kashiwa et al, Phys. Rev. D (2012)
 F. Cuteri et al, Phys. Rev. D (2021)