Quantum field theory 2, lecture 12

Faddeev-Popov determinant and ghosts

We also need to find a way to deal with the functional Jacobi determinant \(\delta\left[G[A, a^\prime]\right]\). For the generalize Landau gauge the Jacobi matrix is \[\begin{split} N^z_{~w}(x,y)[A,a^\prime[\alpha]] = & \frac{\delta}{\delta \alpha^w(y)} G^z(x)[A, a^\prime[\alpha]] \\ = & (D^\mu[A])^z_{~u} \frac{\delta}{\delta\alpha^w(y)} \left[ a^{\prime z}_\mu(x) + \frac{1}{g}\partial_\mu \alpha^z(x) - \alpha^y(x) f_{yw}^{~~z} \left[A^w_\mu(x) + a^{\prime w}_\mu(x) \right] \right] \\ = & \frac{1}{g} (D^\mu[A])^z_{~u} (D_\mu[A+a^\prime])^u_{~w} \delta^{(d)}(x-y) \end{split}\] In order to take the determinant of this matrix into account one introduces auxilliary Grassmann values scalar fields \(\bar c_z(x)\) and \(c^w(x)\) and writes \[\text{Det}\left[ \tfrac{\delta}{\delta \alpha} G[A, a^\prime] \right] = \int D\bar c Dc \exp\left( \int_x\left\{ \bar c_z(x) (D^\mu[A])^z_{~u} (D_\mu[A+a^\prime])^u_{~w} c^w(x) \right\} \right)\] The factor \(1/g\) could be absorbed into a rescaling of the fields \(\bar c_z(x)\) and \(c^w(x)\), which are known as Fadeev-Popov ghost fields. In this way, the Jacobi determinant adds a so-called ghost term to the action, \[S_\text{ghost}[A,a^\prime,\bar c,c] = \int_x\left\{ - \bar c_z(x) (D^\mu[A])^z_{~u} (D_\mu[A+a^\prime])^u_{~w} c^w(x) \right\}.\] Ghosts are Grassmann variables that belong to the adjoint representation of the gauge group. They cannot be observed as particles.

Action for QCD with gauge fixing

Let us now combine terms for the action of QCD with gauge fixing à la Faddeev-Popov. Without quarks we get \[\begin{split} S[A,a',\bar c,c] = S_\text{Yang-Mills}[A+a']+S_\text{gauge fixing}[A,a']+S_\text{ghost}[A,a',\bar c,c]. \end{split}\] There we use the Yang-Mills action with background-fluctuation spliting \[S_\text{Yang-Mills}[A+a'] = \int_x \left\{\frac{1}{2g^2}\text{tr}\left\{\mathbf{F}^{\mu\nu}(x)\mathbf{F}_{\mu\nu}(x)\right\} \right\},\] where \[\begin{split} \mathbf{F}_{\mu\nu}(x) = & \partial_\mu\mathbf{A }_\nu(x)+\partial_\mu\mathbf{a}_\nu'(x)-\partial_\nu\mathbf{A}_\mu(x) - \partial_\nu\mathbf{a}_\mu'(x) \\ & -i[\mathbf{A}_\mu(x)+\mathbf{a}_\mu'(x),\mathbf{A}_\nu(x)+\mathbf{a}_\nu'(x)]. \end{split}\] In the presence of quarks we also need to add \[S_\text{quarks}[A, a^\prime,\bar\psi, \psi] = \int_x \left\{ - \bar \psi(x) \gamma^\mu D_\mu[A+a^\prime] \psi(x) \right\}.\]

Invariance of gauge fixed action under background field gauge transforms

Although we have fixed the gauge by the Faddeev-Popov method, there actually still is a gauge symmetry remaining, namely with respect to the background field gauge symmetry.

For a finite transformation of this type, the background gauge field expectation value transforms as \[\mathbf{A}_\mu(x) \to U(x) \mathbf{A}_\mu(x) U^\dagger(x) - i U(x)\partial_\mu U^\dagger(x),\] such that \(D_\mu[A] \to U(x)D_\mu[A] U^\dagger(x)\). In contrast, the fluctuating part of the gauge field transforms simply as \[\begin{split} \mathbf{a}_\mu^\prime(x) \to U(x) \mathbf{a}_\mu^\prime(x) U^\dagger(x), \end{split}\] and similarly, the ghost fields transform as \[\begin{split} \mathbf{c}(x) &= c^z(x) T_z\to U(x) \mathbf{c}(x)U^\dagger(x),\\ \bar{\mathbf{c}}(x) & = \bar{c}^z(x) T_z \to U(x) \bar{\mathbf{c}}(x) U^\dagger(x). \end{split}\] These are the transformation laws for matter fields in the adjoint representation of the gauge group.

Fermions in the fundamental representation transform simply in the familiar way, \(\psi(x) \to U(x) \psi(x)\).

It is straight forward to see that the combined action is invariant undert this transformation. For the Yang-Mills term and the quark term this is clear because they are anyway gauge invariant. For the gauge fixing term it is clear when it is written as \[S_\text{gauge fixing}[A,a^\prime] = \int_x \left\{ \frac{1}{\xi} \text{tr} \left\{D^\mu[A] \mathbf{a}_\mu^{\prime }(x)D^\nu[A] \mathbf{a}^{\prime }_\nu(x) \right\} \right\}.\] It is crucial here that \(D^\mu[A] \mathbf{a}_\mu^{\prime }(x)\), with \(D_\mu[A]\) the covariant derivative in the adjoint representation, transforms again like a matter field in the adjoint representation. Similarly, the ghost term can be written as \[S_\text{ghost}[A,a^\prime,\bar c,c] = \int_x \text{tr}\left\{ - 2 \bar{\mathbf{c}}(x) D^\mu[A] D_\mu[A+a^\prime] \mathbf{c}(x) \right\},\] with both covariant derivatives in the adjoint representation.

Effective action with quantum corrections

Let us now attempt to calculate the quantum effective action \(\Gamma[A]\) to one-loop order. We concentrate on pure Yang-Mills theory, the extension to theories with fermions is straight forward.

Because of the invariance under gauge transformations on \(\mathbf{A}_\mu(x)\), only gauge invariant terms can appear in \(\Gamma[A]\). The leading term is expected to be of the form \[\Gamma_\text{1-loop}[A] = \int_x \frac{1}{2g^2}\text{tr}\left\{\mathbf{F}^{\mu\nu}(x)\mathbf{F}_{\mu\nu}(x)\right\},\] This is the form of the microscopic action but the coupling \(g\) may differ from the microscopic coupling by renormalization group running.

We will now perform a one-loop calculation, based on \[\label{eq:firstOrderExpOfEffAction} \begin{split} \Gamma[A] = S[A] + \frac{1}{2}\text{STr}\left\{\ln\left(S^{(2)}[A]\right)\right\} + \ldots. \end{split}\] The operation \(\text{STr}\) is to a trace over all indices of the fields, including momentum and frequency. The S stands for “super” and should be a reminder to add a minus sign for fermionic degrees of freedom, including ghosts.

In order to calculate the quantum correction to \(1/g^2\), we need to determine the propagator for the fluctuating fields, \(\mathbf{a}_\mu'(x)\), \(\bar{\mathbf{c}}(x)\) and \(\mathbf{c}(x)\) in the presence of a background field \(\mathbf{A}_\mu(x)\). Eventually, we will expand the left hand side and the right hand side of the above relation to at least quadratic order in \(\mathbf{A}_\mu(x)\) in order to identify the coefficient of the term \(\text{tr}\left\{\mathbf{F}_{\mu\nu}(x) \mathbf{F}^{\mu\nu}(x)\right\}\).

Yang-Mills term

Our next goal is to derive the second functional derivative \(S^{(2)}[A]\). We start by rewriting the Yang-Mills term which involves the field strength tensor for background plus fluctuation fields, \[\begin{split} \mathbf{F}_{\mu\nu}(x) = & \partial_\mu \mathbf{A}_\nu(x)-\partial_\nu \mathbf{A}_\mu(x) - i[\mathbf{A}_\mu(x), \mathbf{A}_\nu(x)] \\ & +\partial_\mu \mathbf{a}_\nu'(x) - \partial_\nu \mathbf{a}_\mu'(x)-i[\mathbf{A}_\mu(x), \mathbf{a}_\nu'(x)]-i[\mathbf{a}_\mu'(x),\mathbf{A}_\nu(x)] -i[\mathbf{a}_\mu'(x), \mathbf{a}_\nu'(x)]\\ = & \bar{\mathbf{F}}_{\mu\nu}(x) + D_\mu[A] \mathbf{a}_\nu'(x) - D_\nu[A] \mathbf{a}_\mu'(x) -i[\mathbf{a}_\mu'(x), \mathbf{a}_\nu'(x)], \end{split}\] where \(D_\mu[A]\) is the covariant derivative in the adjoint representation, \[D_\mu[A] = \partial_\mu - i [\mathbf{A}, \ldots],\] and \(\bar{\mathbf{F}}^\text{fnd}_{\mu\nu}(x) = g \bar F^z_{\mu\nu}(x) T_z\) is the background field strength in the fundamental representation.