Strong Interactions and quarks

Quantum chromodynamics (QCD) is the gauge theory describing strong interactions between quarks.

It constitutes a fundamental theory within the Standard Model $U(1)_Y\otimes SU(2)_L\otimes SU(3)_C$ and it is responsible for the $SU(3)_C$ portion.

Weak and electromagnetic interactions are considered unified in the single $U(1)_Y\otimes SU(2)_L$ symmetry group where $Y$, the weak hypercharge, is the generator of the group that is given by the Gell-Mann–Nishijima relation between electrical charge and isospin $Q=\frac{Y}{2}+I_3$.
The subscript $L$ is used to denote left-handed spinors recording the vector-axial nature of the charged currents and the subscript $C$ refers to the charge of strong interactions given by the colour.

In quantum field theories, charges point out any generator of continuous symmetries in a physical system. In correspondence of a symmetry the existence of a conserved current is implied according to the Noether’s theorem.
For example, the electric charge is the generator of the $U(1)$ symmetry group of electromagnetism and the corresponding conserved current is the electric current.
The charge is then the generator of the local symmetry group.
A gauge field is associated to each charge and the request of the local gauge invariance of the lagrangian leads to the introduction of new vector fields (as many as the generators of the group, $N^2-1$ for $SU(N)$), when the field is quantized.
In the QCD, these fields are associated to 8 ($N_C^2-1=8$, for $SU(3)_C$) gauge bosons which take the name of gluons and are the intermediate bosons of the strong interactions corresponding to the eight Gell-Mann generators of the $SU(3)_C$ group.

The QCD is a complex non-abelian quantum field theory: gluons carry colour charge (they are said coloured) and thus they can interact with themselves.
Quantum electrodynamics (QED), instead, is a linear theory.
Photons cannot interact with themselves and they do not carry electrical charge.

Quarks ($q$) and corresponding antiquarks ($\bar q$) occur in six different flavours: $u$, $d$, $c$, $s$, $t$, $b$ They are spin $\frac{1}{2}$ fermions and they carry both electrical and colour charges.
Quarks have a fractional — with respect to the electron charge $e$, taken as unity — electric charge $Q$ and they are classified according to the flavour quantum number as: up ($u$), charm ($c$), top ($t$), with $Q = +\frac{2}{3}|e|$, and down ($d$), strange ($s$), bottom ($b$), with $Q = -\frac{1}{3}|e|$.
Antiquarks present electric charges of the same module but opposite in sign.
The colour is the charge of the strong interactions and quarks and antiquarks are the only fermions endowed with colour quantum numbers.
The colour quantum number is linked to the colour charges by the colour isospin $I^C_3\equiv \lambda_3$ and the colour hypercharge $Y^C\equiv \frac{1}{\sqrt{3}}\lambda_8$, where $\lambda_i$ indicate the Gell-Mann matrices.
While quarks present colours, i.e. red ($r$), green ($g$), blue ($b$), antiquarks have anticolours: cyan ($\bar r$), magenta ($\bar g$), yellow ($\bar b$).

With colour, quarks gain an additional degree of freedom that has been decisive to solve the problem of the particle proliferation.
Thus, quarks take a threefold variety of colours (red, green, blue), e.g. an up quark is presented as a colour triplet ($u_r$, $u_g$, $u_b$), and so on.
One might therefore expect each hadron to exist in a multitude of versions due to all possible colour combinations, instead hadrons occur in nature only as neutral colour combination particles. This is known as colour confinement and it is equivalent to say that free quarks do not exist in Nature.

There are two kinds of hadrons, or, in other words, two ways to achieve colour neutrality: mesons and baryons. Mesons are composed of a $q\bar q$ pair that carry opposite signs of the same colour while baryons are composed by a triplet $qqq$ of three different coloured quarks.
Such quarks determine the properties of a hadron and they are named valence quarks.
An exception could be represented by the $t$ quark that presents a too short lifetime to make a bound state.

Thus, the density of the QCD Lagrangian is

$\mathcal{L}_{QCD}=\bar{q}_i^\alpha\left(i\gamma^{\mu}\mathcal{D}_\mu-\hat{m} \right)^{\alpha \beta}_{ij} q^\beta_j -\frac{1}{4}G^\alpha_{\mu\nu}G^{\mu \nu}_{\alpha}$

where:

• $\gamma$ are the Dirac matrices;
• $q^\alpha_i(x)$ denotes the quark fields of colour ($i=1\div3$) and of flavour
($\alpha=1\div 6=u,d,c,s,t,b$);
• the mass matrix is colour independent and it is diagonal in the flavour space:
$\hat m= diag_f(m_{1\div6})$;
• the covariant derivative has the form of a
$3\times 3$ matrix and it is defined as$(\mathcal{D}_\mu)_{\alpha\beta}\equiv\delta_{\alpha\beta}\partial_\mu - ig \left(\frac{\lambda^\alpha}{2}\right)_{\alpha\beta}A^\alpha_\mu$, where:

• $A^\alpha_\mu$ is the gluon field, with $\alpha=1\div 8=N^2_C-1$,
• $g$ is the bare coupling constant,
• $\lambda_\alpha$ denotes the Gell-Mann matrices of the gauge $SU(3)_C$ group;
• the gluonic field tensor is
$G^\alpha_{\mu\nu}=\partial_\mu A^\alpha_\nu-\partial_\nu A^\alpha_\mu+gf^{abc}A^b_\mu A^c_\nu$,
where $f_{abc}$ is a totally antisymmetric tensor and its elements are the structure constants of the $SU(3)$ group.

The bilinear term in the gluonic fields $A^\alpha_\mu+gf^{abc}A^b_\mu A^c_\nu$ generates self-interaction terms with vertices of three and four gluonic propagators. This expresses the non-abelianity of the theory.

A large body of experimental evidence of QCD has been collected, since its formulation.
This brought attention to the two fundamental properties of QCD: colour confinement and asymptotic freedom.
All interactions contained in the QCD lagrangian depend on a single gauge coupling constant $g=g_s$ for strong interactions. This is known as universality.