# Symmetries of the QCD

An important feature of the QCD is the conspicuous amount of symmetries of its lagrangian.

First and foremost, its lagrangian is invariant under local gauge transformations, i.e. one can redefine the quark fields indipendently at every point in the space-time, without changing the physical content of the theory.
This determines a number of implications in the dynamics of the theory.
Furthermore, strong interactions do not depend on the quark flavour and if the masses of quarks are identical, the QCD Lagrangian is invariant under arbitrary flavour rotations of the quark fields.
This flavour symmetry introduces the symmetry under isospin transformations.

If one requires that all the quark masses are equal to zero, the flavour symmetry can be enlarged.
Fields in the QCD Lagrangian can be decomposed in left-handed and right-handed quark fields

$q_{L,R}=\frac{1}{2}\left(1\pm\gamma_5\right)q$.

These are eigenstates of the Dirac chirality operator $\gamma_5$ with eigenvalues $\pm1$.
For massless free quarks the chirality coincides with the helicity $\vec{\sigma}\cdot \hat{p}$.
The helicity is defined as the projection of the spin $\vec{\sigma}$ on the direction of the momentum $\hat{p}=\frac{\vec{p}}{|p|}$.
So, in the chiral limit ($m_i\rightarrow 0$) chiral symmetry is an exact symmetry; otherwise it is an approximate symmetry.

Then, the QCD Lagrangian density (*) becomes

$\mathcal{L}_{(m_q\rightarrow0)}=-\frac{1}{4}G^{\mu\nu}G_{\mu\nu} +i\bar q_L \mathcal{D}\kern-6.5pt\slash\ q_L +i\bar q_R \mathcal{D}\kern-6.5pt\slash\ q_R$

and it is symmetric for the group

$U(1)_V\otimes U(1)_A\otimes SU(n_f)_L\otimes SU(n_f)_R\otimes SU(3)_C$

where $U(1)_V$ is the vector baryon conservation symmetry group and its axial counterpart is $U(1)_A$.

The lagrangian of a physical system is chirally symmetric if it is invariant under the global $SU(n_f)_L \otimes SU(n_f)_R$ transformations

$SU(n_f)_L:\ q_L\rightarrow e^{i\theta^L_a\frac{\lambda^a}{2}}q_R$
$SU(n_f)_R:\ q_R\rightarrow e^{i\theta^R_a\frac{\lambda^a}{2}}q_L$

where $\theta_a$ are the generators of the $SU(n_f)$ group.

Requiring this symmetry is equivalent to require the one that leaves the lagrangian invariant under global vector and axial vector transformation on the chiral symmetry group $SU(n_f)_V\otimes SU(n_f)_A$.

Spontaneous breakdown of the $SU(n_f)_V$ group due to the quark masses produces the breaking of the entire group symmetry.

In relativistic quantum field theories (RQFTs), the spontaneous breaking of exact continuous global symmetries implies the existence of massless Goldstone bosons.
In the case of chiral symmetry and under chiral limit, they can be identified with the pion triplet ($\pi=\bar qq$, $n_f^2-1=3$).

If chiral symmetry would be an exact symmetry of QCD the pions should be massless. Instead, due to masses of quarks it is an approximate symmetry and pions are expected to have finite – even though relatively small – masses.
In fact, the mass of pions is $\sim 140\ MeV$, quite small if compared, for example, with the proton mass that is $\sim 940\ MeV$.

The spontaneous symmetry breaking of chiral symmetry corresponds to the existence of a non-zero vacuum expectation value of the axial symmetry.
This has non-perturbative origin and it is related to the existence of a non-zero chiral quark condensate $\langle \bar{q}(x) q(x)\rangle$.

This condensate is a measure of spontaneous chiral symmetry breaking.
The connection between spontaneous chiral symmetry breaking and non-vanishing chiral condensate can be highlighted by introducing a pseudoscalar operator $P(x)\equiv \bar q(x) \gamma_5 q(x)$.
The pseudo-scalar operator $P$ applied to the ground state returns a pion, that is the massless Goldstone pseudoscalar boson: $P|0\rangle = |\pi\rangle$.
If $f_\pi$ indicates the pion decay constant, the chiral condensate expectation value can be derived from the GOR (Gell-Mann, Oakes, Renner) relation

$m^2_{\pi}f_\pi^2=-(m_u+m_d)\langle\bar{q}(x)q(x)\rangle$,

that gives $\langle\bar{q}(x)q(x)\rangle=-(240\ MeV)^3$.

The chiral condensate is calculated as a function of the temperature $T$ and of the density $\rho$. In the limit for low temperatures and low densities it is

$\frac{\langle\bar{q}q\rangle_{T,\rho}}{\langle\bar{q}q\rangle_0} \sim1-aT^{-2}-b\rho$

where $\langle\bar{q}q\rangle_0$ is the vacuum condensate at $T=0$ and $\rho=0$, with $a,\ b$ being constant.
As it can be seen from the latter equation, in high temperature and high density regime, the chiral condensate approaches to zero.

Although for the ordinary hadronic matter the chiral symmetry is broken at low temperature, it is expected that in conditions created by heavy-ion collisions the quarks are deconfined and the chiral symmetry becomes restored.

This suggests that the chiral symmetry can be restored in the QGP phase.