I would like to understand the distinction between an axial anomaly in QCD (Theta Vacuum: axion -> 2 gluons) and an axial anomaly in QCD of current (Chern–Simons term: pion->two photons, photon->three pions, ...). A more specific question: is the current axial anomaly related to the topological properties of the theory like "internal" axial anomaly?

If you don't understand question there are clarifications about "internal" axial anomaly and current axial anomaly (as I see it):

1) First, in quantum chromodynamics, a violation of the axial group $U_{A}(1)$ leads to a nonconservation of the axial current: \begin{gather}\label{nonconservation of curent} \partial^{\mu}J_{5,\mu}=2\,i \, \bar q \, \hat m_{q} \gamma_{5} \, q+\frac{N_{f}\,g^{2}}{8\pi^{2}}\epsilon_{\mu\nu\alpha\beta}\,tr(G_{\mu\nu}G_{\alpha\beta}) \;\ , \end{gather}

where $ G_{\mu\nu}$ - gluon field strength tensor. The violation of the axial group is connected with the fact that the vacuum of quantum chromodynamics has a complex topological structure, and this eventually leads to an additional term in the Lagrangian: \begin{gather}\label{theta term} \mathcal{L}_{\theta}=\theta\frac{g^{2}}{16 \pi^{2}}\epsilon_{\mu\nu\alpha\beta}\,tr(G_{\mu\nu}G_{\alpha\beta}) \end{gather}

2) Second, in addition to the "internal", anomaly of chromodynamics written above, there are external anomalies in the chromodynamics of external currents, the simplest of which corresponds to the process $\pi_{0}\rightarrow\gamma\gamma$: \begin{gather}\label{nonconservation of curent in algebra curents} \partial^{\mu}J^{em}_{5,\mu}=2m(\bar q \, \gamma_{5} \, \tau_{3}\, q)+\frac{e^{2}}{16\pi^{2}}\epsilon_{\mu\nu\alpha\beta}F_{\mu\nu}F_{\alpha\beta} \;\ , \end{gather} where $q$ - quark field, $F_{\mu\nu}$ - electromagnetic field strength. The corresponding Lagrangian for the anomaly has the form ($\bar q \, \gamma_{5} \, \tau_{3}\, q=f_{\pi}m^{2}\pi_{0}$): \begin{gather} \mathcal{L}_{em}=-\frac{N_{c}\,e^{2}}{96 \pi^{2}f_{\pi}}\epsilon_{\mu\nu\alpha\beta}F_{\mu\nu}F_{\alpha\beta}\,\pi_{0}\, \end{gather}

I think this violation is not related to the topological properties of the theory.

In addition to this anomaly, there is a huge number of others, for example an anomaly corresponding to the process $\gamma\rightarrow\pi\pi\pi$. In order to describe all the anomalies, the Wess-Zumino-Witten action is used. This is possible due to the following statement: any non-Abelian anomaly in four-dimensionality can be represented through the action of Wess-Zumino-Witten in five-dimension (Chern-Simons term) (for further information please refer to Can the effective vertex for $\gamma\to3\pi$ be derived directly from the anomaly?, Chiral anomaly in odd spacetime dimensions). \begin{align} W &=-\frac{iN_{c}}{96\pi^{2}}\int^{1}_{0}dx_{5}\int d^{4}x \epsilon^{\,\mu\nu\sigma\lambda\rho}\,Tr \Bigl[-j^{-}_{\mu}F^{\mathcal{L}}_{\nu\sigma}F^{\mathcal{L}}_{\lambda\rho}-j^{+}_{\mu}F^{\mathcal{R}}_{\nu\sigma}F^{\mathcal{R}}_{\lambda\rho} \nonumber \\ &-\frac{1}{2}\,j^{+}_{\mu}F^{\mathcal{L}}_{\nu\sigma}\,U(x_{5})F^{\mathcal{R}}_{\lambda\rho}\,U^{\dagger}\!(x_{5}) -\frac{1}{2}j^{+}_{\mu}F^{\mathcal{R}}_{\nu\sigma}\,U^{\dagger}\!(x_{5})F^{\mathcal{L}}_{\lambda\rho}\,U(x_{5}) \nonumber \\ &+i F^{\mathcal{L}}_{\mu\nu}\,j^{-}_{\sigma}j^{-}_{\lambda}j^{-}_{\rho}+i F^{\mathcal{R}}_{\mu\nu}\,j^{+}_{\sigma}j^{+}_{\lambda}j^{+}_{\rho} +\frac{2}{5}j^{-}_{\mu}j^{-}_{\nu}j^{-}_{\sigma}j^{-}_{\lambda}j^{-}_{\rho}\Bigl] \;\ \label{wzw} \end{align}


1 Answer 1


Any anomaly (I mean the chiral anomaly) is related to the topology. Precisely, the integrated anomaly equation $$ \partial_{\mu}J^{\mu}(x) = \text{A}(x) $$ can be interpreted as the relation between the difference $n_{+} - n_{-}$ of the numbers of left and right zero modes of the Dirac operator $\gamma_{\mu}D^{\mu}$, known as the index of the Dirac operator, $\text{Index}(\gamma^{\mu}D_{\mu})$, and the integrated Chern secondary characteristic class density $ F^{2}$ for the gauge field: $$ \text{Index}(\gamma^{\mu}D_{\mu}) = n_{L} - n_{R} = \int F^{2} $$ The statement is known as the Atyah-Singer theorem.

For the axial anomaly (the anomaly for the global current) and abelian gauge anomaly the theorem is formulated for 4D euclidean space-time, while the non-abelian gauge anomaly in 4D euclidean space-time is translated to the abelian gauge anomaly in 6D space-time (and thus the 6D index theorem is valid).

The non-trivial QCD gauge group (which is $SU_{c}(3)$) structure is another story. Since the homotopic group $\pi_{3}(SU_{c}(3))$ is non-trivial, $\pi_{3}(SU_{c}(3)) = Z$, it leads to existence of the non-trivial vacuum (see some overview here), being the sum over the vacua $|n\rangle$ carrying the given topological number $n$ with the weight $e^{in\theta}$ (the number $n$ is related to the different homotopic classes of $SU_{c}(3)$). Initially it is not really related to the anomaly, since such vacuum exists even without any fermions, in pure Yang-Mills theory. The story particularly changes when massless fermiions are added to the theory, but I hope that the main point is clear.

  • $\begingroup$ But are you sure that any anomaly (chiral anomaly) is related to the topology? Because QED anomaly is determined by the topologically trivial Abelian gauge fields of QED whereas the counterpart U(l)A QCD anomaly is determined by the topologically non-trivial non-Abelian gauge fields of QCD. $\endgroup$
    – illuminato
    Commented Oct 6, 2017 at 10:34
  • $\begingroup$ @illuminates : indeed, any chiral anomaly is related to the topology. Being integrated, the left part is just the index, while the right part is the Pontryagin number. This is true independently on whether the integral $\int F^{2}$ vanishes or not. $\endgroup$
    – Name YYY
    Commented Oct 6, 2017 at 14:52
  • $\begingroup$ @illuminates : the answer on the question whether the Pontryagin number vanishes or not depends on the topological structure of the gauge group in the given space-time. From the other side, the same Pontryagin number labels non-equivalent vacua in case of the QCD, which is related to the fact that its expression coincides with Maurer-Cartan invariant for the homotopy group $\pi_{3}(SU_{c}(3))$. More correctly, the Pontryagin number can be in some sense interpreted as the difference of the topological vacua numbers evaluated for the moments of time $t \to \pm \infty$. $\endgroup$
    – Name YYY
    Commented Oct 6, 2017 at 14:58

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