# Lattice QCD calculations

Due to the asymptotic freedom, the coupling constant $\alpha_S$ of QCD is a diminishing function as the energy scale increases according to the following equation, already stated (*): $\alpha(|q^2|)=\frac{12\pi}{\left(11n-2f\right)ln\left(\frac{|q^2|}{\Lambda^2_{QCD}}\right)}$.

Therefore, the high-energy or equivalently the short-distance behavior can be described by a perturbative expansion, but a perturbative approach to the QCD fails at large distances where the $\alpha_S$ begins to diverge as the scale of energy decreases.

A different approach allows to better characterize the transition to the deconfined state of hadronic matter and the physical mechanisms at the origin of colour confinement.
A suitable non-perturbative approach is the numerical study of QCD on a lattice (L-QCD).

The leading idea is to outline QCD interactions as a grid in the space-time with quarks placed on nodes and gluonic fields on links.
While the size of the grid is considered infinitely large, the sites are infinitesimally close to each other.

Many progresses have been achieved on the algorithms and on the computing performances and nowadays L-QCD computation represents a notably reliable method to test QCD in the non-perturbative domain.

The computational complexity of such calculations is so high that for example the Italian National Istituf of Nuclear Physics (INFN) began to build specific SIMD
supercomputers (APE Project) to perform these simulations already in 1984.

Single Instruction Multiple Data (SIMD) computers are vector machines where a single control unit can drive several functional units which are able to execute simple operations. SIMD computers can be considered precursors of modern GPUs.

Lattice calculations present intrinsic systematic errors due to the use of a finite lattice cutoff and to the use of quark masses which become eventually infinite.
To lessen the computational load, so-called quenched calculations are introduced.
In such approximations quark fields are considered as non-dynamic “frozen” variables.
While this represented the ordinary way to perform calculations in early L-QCD computing, “dynamical” fermions are now standard.

In addition, numerical methods suffer in evaluating integrals of high oscillatory functions with a large number of variables.
This is the fermion sign problem that emerges for example when quark-chemical potentials are included, e.g. in calculations at non-zero net baryon density or when wave functions change sign due to the effects of the symmetry introduced by the Pauli’s principle.

Relatively simple models (e.g. the MIT Bag Model furnish yet a reasonable valuation for the critical temperature $T_C \sim 170\ MeV$ and the critical energy density $\epsilon_C \sim 1\ GeV/fm^3$.

Lattice QCD calculations have shown that for massless quarks at baryonic potential $\mu_B=0$ the transition to the QGP happens via a first order transition if $n_f\geq 3$ (three quarks with zero masses) and via a second order transition for $n_f=2$ (two quarks and zero masses).
Critical temperature should amount to $(173\pm15)\ MeV$, and the critical energy density to $\epsilon=(0.7\pm0.3)\ GeV/fm^3$, where the uncertainties are mainly due to the method used for its determination.

The Figure below shows a more realistic calculation that includes the mass for the $s$ quark (case “$2+1$ flavours”) and indicates that at zero chemical potential the transition appears most likely as a crossover.

If the transition was of the first order, $\epsilon$ would have a discontinuity in correspondence of the critical temperature $T_C$.

Since the crossover takes place in a small range of temperatures, the phase transition shows a rapid variation in the observables and in the Figure it can be seen that the energy density $\epsilon$ abruptly rises in just $20\ MeV$ of temperature interval.

In the previous Figure it is also visible that the saturated values of energy density at high temperatures are still under the Stefan-Boltzmann (SB) limit; this indicates residual interactions among the quarks and gluons in the QGP phase.
Even the $p/T^4$ ratio saturates under the SB limit, for temperatures $\sim 2T_c$. This suggests a non ideal behavior for the gas considered in Lattice QCD calculation.

The inclusion of lighter quarks masses in the calculations results in a significant decrease for the transition temperature, but early predictions led to significant discrepancies in the results.

Although critical temperature depends on the number of quark flavours involved in the restoring of the chiral symmetry, these differences strongly diminished in current calculations.

A reliable extrapolation of the transition temperature to the chiral limit gave
$T_C = (173 \pm 8)\ MeV,\ n_f=2$,
$T_C = (154 \pm 8)\ MeV,\ n_f=3$.

Calculations based on chiral order parameter show a crossover transition for $T_{\chi}=155\ MeV$.
In addition, even though QCD seems to give only one transition from the low temperature hadronic regime to the high temperature plasma phase, it has been speculated that two distinct phase transitions leading to deconfinement at $T_d$ and chiral symmetry restoration at $T_\chi$ could occur in QCD, with $T_d\leq T_\chi$ according to general arguments about energy scales.

Another important outcome of lattice QCD is the prediction of the restoration of the chiral symmetry that would occur in correspondence of the deconfinement transition.
It is expected, in fact, that the value of the chiral condensate after the deconfinement transition goes to zero, allowing the restoration of the chiral symmetry.

The Figure below shows a comparison between predictions in two-flavours L-QCD for the chiral condensate $\psi\bar\psi$, which is the order parameter for chiral-symmetry breaking in the chiral limit ($m_q\rightarrow0$), and the Polyakov loop, which is the order parameter for deconfinement in the pure gauge limit ($m_q\rightarrow\infty$).

It can be seen that as the temperature increases through the crossover, the value of the chiral condensate $\psi\bar\psi$ drops down and the Polyakov loop boosts.
Such variations occur at the same temperature, suggesting that deconfinement and restoration of chiral symmetry happen at the same temperature.

The corresponding susceptibilities $\chi_L\propto \left(\langle L^2\rangle - \langle L \rangle^2\right)$ and $\chi_m=\partial\langle\psi\bar\psi\rangle/\partial m$ are also showed.
Their peak occur at the same value of the coupling.

In addition, the calculation of the potential energy between two heavy quarks as a function of the temperature shows a confirmation of the deconfinement.

The last Figure shows the predicted behaviour in the three flavour QCD scenario
of the potential energy between a quark and an antiquark.
It can be seen on the left side that as the separation increases, the potential energy flattens and it becomes constant at long distances, validating the hypothesis of deconfinement.

On the right side, instead, it is shown that the separation between a quark and and antiquark decreases with the raise of the temperature

Finally, the results of lattice calculations suggest to consider the QGP as a weakly
coupled medium characterized by the coupling constant

$\alpha_S(T)\propto \frac{1}{log\left[\frac{2\pi T}{\Lambda_{QCD}}\right]}$,

confirming the evidence of deconfinement found at \SPS and the perfect fluid behavior highlighted by $\textsf{RHIC}$ data and maybe discussed later.