Building the Lagrangian of QCD

1. Classical part

Consider quarks - spin-1/2 fermions. They exist in 6 different flavours and can be arranged in 3 generations (check The Quark Model)

Starting with the Dirac Lagrangian, we can write a kinetic and mass term for the quarks,

${\mathcal{L}}_{\psi }={\sum }_q{\bar{\psi }}_q^i(i/\partial -m_q){\delta }^{ij}{\psi }_q^j$

Here, q = (u,d,s,…) are the quark flavours and i = r, g, b are the colour indices. The quarks live in the fundamental representation of SU(Nc).

The Lagrangian has a global SU(Nc) symmetry. If U is matrix in SU(Nc), then the quark and anti-quark fields transform as

${\psi }_q^i\to U_{ij}{\psi }_q^j$

${\bar{\psi }}_q^i\to {\bar{\psi }}_q^j{U}_{ji}^{†}$

Since SU(Nc) is an exact symmetry, we can “gauge” it by requiring a gauge invariance. To do this, we allow the group elements to depend on the position in space-time, by allowing the real parameters to depend on space-time. Then,

${\psi }_q^i(x)\to U_{ij}(x){\psi }_q^j(x)$

${\bar{\psi }}_q^i(x)\to {\bar{\psi }}_q^j(x)(U^{†}(x){)}_{ji}$

But, the first term in the Dirac Lagrangian is not locally gauge invariant then!

$\bar{\psi }(x)U^{†}(x)/\partial [U(x)\psi (x)]=\bar{\psi }(x)U^{†}(x)/\partial [U(x)]\psi (x)+\bar{\psi }(x)U^{†}(x)U(x)/\partial [\psi (x)]$

$=\bar{\psi }(x)U^{†}(x)/\partial [U(x)]\psi (x)+\bar{\psi }(x)/\partial \psi (x)\ne \bar{\psi }(x)/\partial \psi (x)$

So, we introduce a covariant derivative D in place of the partial derivatives, with the property,

$D_{\mu }\to D_{\mu }^{\prime }=U(x)D_{\mu }{U}^{†}(x)$

To achieve this property, we introduce the gluon field A and defining

$A_{\mu }\equiv A_{\mu }^a{t}^a$

$D_{\mu }\equiv {\partial }_{\mu }+i{g}_s\underset{A_{\mu }}{\underbrace{A_{\mu }^a{t}^a}}$

We will require that the gluons transform according to the adjoint representation of SU(Nc) under global gauge transformations. We also demand that the gluon field has the following local transformation property

$A_{\mu }(x)\to A_{\mu }^{\prime }(x)=U(x)A_{\mu }(x)U^{†}(x)+\frac{i}{g_s}({\partial }_{\mu }U(x))U^{†}(x)$

Analogous to Quantum Electrodynamics, we can give dynamics by introducing a field strength tensor

$F^{\mu \nu }=\frac{-i}{g_s}[D_{\mu },D_{\nu }]={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }+i{g}_s[A_{\mu },A_{\nu }]$

which transforms as

$F_{\mu \nu }\to F_{\mu \nu }^{\prime }=U{F}_{\mu \nu }{U}^{†}$

$F_{\mu \nu }\equiv F_{\mu \nu }^a{t}^a$

$⟹F_{\mu \nu }^a={\partial }_{\mu }{A}_{\nu }^a-{\partial }_{\nu }{A}_{\mu }^a-g_s{f}^{abc}{A}_{\mu }^b{A}_{\nu }^c$

The third term is not present in QED and arises because the gluon field is non-abelian (gives rise to gluon self-interaction).

Finally, we add to our Lagrangian a kinetic term for the gluon field

${\mathcal{L}}_A=-\frac{1}{4}{F}_{\mu \nu }^a{F}^{a,\mu \nu }$

This new term is Lorentz and gauge invariant. It introduces new gluon self-interactions!

Hence, our final classical Lagrangian is given by

${\mathcal{L}}_{classical}={\sum }_q{\bar{\psi }}_q(i/D-m_q)\psi -\frac{1}{4}{F}_{\mu \nu }^a{F}^{a,\mu \nu }$

2. Gauge fixing and Ghost terms

The equation of motion for the gluon can be derived from Euler-Lagrange equation

${\partial }_{\mu }(\frac{\partial \mathcal{L}}{\partial ({\partial }_{\mu }X)})-\frac{\partial \mathcal{L}}{\partial X}=0\text{ with }X=A_{\nu }^a$

If we try to calculate with the classical Lagrangian, we get

$(g^{\mu \nu }{\partial }^2-{\partial }^{\mu }{\partial }^{\nu })A_{\nu }^a=0$

This means that we cannot straightforwardly define a gluon (or photon) propagator. This is because of a gauge symmetry leading to modes with 0-eigenvalue

$A_{\nu }^a\to A_{\nu }^a+{\partial }_{\mu }\Lambda$

$⟹(g^{\mu \nu }\square -{\partial }^{\mu }{\partial }^{\nu })({\partial }_{\mu }\Lambda )\equiv 0$

The path integral over-counts the configurations for the gauge field. The solution is to impose a gauge-fixing condition $f_{\omega }^a[A_{\mu }]$ which selects a particular gauge. This is done by the Fadeev-Popov procedure.

Now, ${\mathcal{L}}_{QCD}={\mathcal{L}}_{classical}+{\mathcal{L}}_{fix}+{\mathcal{L}}_{ghost}$

${\mathcal{L}}_{fix}=-\frac{1}{2\xi }(f_{\omega }^a[A_{\mu }]{)}^2$

${\mathcal{L}}_{ghost}=-{\bar{c}}^a(x)M^{ab}(x)c^b(x)$

Where $\frac{\delta f_{\omega }^a[A_{\mu }]}{\delta {\theta }^b(y)}=M^{ab}(x)\delta (x-y)$

The gauge-fixing term breaks gauge invariance, allowing unphysical modes to propagate. The ghost field c is a scalar, anti-commuting (Grassman) and transforms in the adjoint representation of SU(Nc), and introduces unphysical modes that cancel those left by the gauge-fixing term.

Note: $\delta A_{\mu }^a=g_s{f}^{abc}\delta {\theta }^b{A}_{\mu }^c+{\partial }_{\mu }(\delta {\theta }^a)$

Covariant Gauge ($R_{\xi }$ gauge)

Defined by $f^a[A_{\mu }]={\partial }^{\mu }{A}_{\mu }^a$

Using these, we obtain

${\mathcal{L}}_{fix}=-\frac{1}{2\xi }({\partial }^{\mu }{A}_{\mu }^a{)}^2$

${\mathcal{L}}_{ghost}=-{\bar{c}}^a(x){\partial }^2(x)c^a(x)-g_s{f}^{abc}{\bar{c}}^a{\partial }^{\mu }(A_{\mu }^c{c}^b)$

The second term introduces a coupling between the ghosts and gluons (not present in QED since the ghosts decouple completely).

Hence, the equation of motion becomes

$(g^{\mu \nu }{\partial }^2-(1-\frac{1}{\xi }){\partial }^{\mu }{\partial }^{\nu })A_{\nu }^a(x)=0$

Fourier transforming to momentum space and inverting, we obtain the gluon propagator:

$G^{A_{\mu }^a{A}_{\nu }^b}(p)=\frac{-i{\delta }^{ab}}{p^2+iϵ}(g^{\mu \nu }-(1-\xi )\frac{p_{\mu }{p}_{\nu }}{p^2+iϵ})$

The ξ parameter can be 1 (Feynman Gauge), 0 (Landau Gauge), infinity (Unitary Gauge) or any arbitrary number that drops out of gauge invariant quantities.

Axial Gauge

Defined by $f^a[A_{\mu }]=n^{\mu }{A}_{\mu }^a$

where n is an arbitrary vector (hence the gauge is not covariant).

As we did for the Covariant gauge, we obtain

${\mathcal{L}}_{fix}=-\frac{1}{2\xi }(n^{\mu }{A}_{\mu }^a{)}^2$

${\mathcal{L}}_{ghost}=-n^{\mu }{\bar{c}}^a(x){\partial }_{\mu }(x)c^a(x)-g_s{f}^{abc}{n}^{\mu }{\bar{c}}^a{c}^b{A}_{\mu }^c$

The gluon propagator is given by

$G^{A_{\mu }^a{A}_{\nu }^b}(p)=\frac{i{\delta }^{ab}}{p^2+iϵ}(g^{\mu \nu }-\frac{p_{\mu }{n}_{\nu }+p_{\nu }{n}_{\mu }}{(p.n)}+\frac{(n^2+\xi p^2)p_{\mu }{p}_{\nu }}{(p.n{)}^2})$

Common choices of gauge are

• $\xi \to 0$ - axial gauge
• $n^2=0$ - light-cone gauge

In axial gauges, the ghost decouples from the theory and we can compute without including ghosts; the axial gauge is said to be ghost free. In a gauge with both choices of gauge, the third term vanishes and only the two physical modes propagate.