$\textbf{Fact 1}$ In principle, the QCD Lagrangian should contain a Lorentz invariant, gauge invariant, dimension-4 term $\sim\theta \text{Tr}[F^{\mu\nu}\tilde{F}_{\mu\nu}]$. This term, however, is usually neglected in classical physics because it is a total divergence, and therefore, cannot affect the equations of motion. However, when instanton effects are included, it turns out that the QCD action should be augmented by this term. This is explained, for example, in the book Quantum Field Theory by Mark Srednicki at the pages 598-599.

$\textbf{Fact 2}$ The part of the QCD Lagrangian with $u$ and $d$ quark, in the massless limit of $u$ and $d$, has a $U(1)$ axial anomaly $$\partial_\mu j^{\mu 5}=-\frac{g^2}{16\pi^2}\epsilon^{\alpha\beta\mu\nu}F_{\alpha\beta}^a F_{\mu\nu}^a\sim \text{Tr}[F\tilde{F}]$$ where $j^{\mu 5}=\bar{Q}\gamma^\mu\gamma^5 Q$ and $Q=\begin{pmatrix}u\\d\end{pmatrix}$. This anomaly term is exactly same in form as the term included in the QCD action by instanton effects.

Question This uncanny similarity strongly provokes me to guess that this anomaly is solely responsible for inducing the $\theta-$term $\sim\theta \text{Tr}[F^{\mu\nu}\tilde{F}_{\mu\nu}]$ to the QCD action. In other words, if the current $j^{\mu 5}$ were conserved or anomaly-free i.e., $\partial_\mu j^{\mu 5}=0$, the term $\sim \theta \text{Tr}[F^{\mu\nu}\tilde{F}_{\mu\nu}]$ to QCD action can always be dropped and instanton effects will not be present.

Is this the correct way to think about this striking correlation between fact 1 and fact 2? I'm doubtful because while discussing instantons of Yang-Mills action, in pages 590-599, Srednicki doesn't talk about anomalies at all.

  • $\begingroup$ The $\theta$ term can be present even in the absence of fermions, in which case you have no anomalies. $\endgroup$
    – user1504
    Oct 13, 2017 at 10:32
  • $\begingroup$ Actually it is the other way around -- if there is a chiral anomaly, the $\theta$-term can be dropped, see this answer. $\endgroup$ Oct 17, 2017 at 22:25

1 Answer 1


Even assuming the presence of fermions (because in other case the anomaly doesn't exist), the $\theta$-term isn't related to chiral anomaly. It appears because of:

  • non-trivial topology of the QCD gauge group, leading to the statement that there are topologically inequivalent vacua which can be parametrized by the winding number $n$

  • Cluster decomposition principle, which requires the true vacuum be the sum over all non-trivial vacua with the weight $e^{in\theta}$

See also this question. By using the expression for the winding number $n$ in terms of the integral over the gauge configurations, the weight $e^{in\theta}$ can be written as the additional term in the effective action. The latter is $\theta$-term.

The axial anomaly mentioned in your question appears because of the different reason, namely, the absence of the renormalization scheme preserving both the gauge symmetry and global axial symmetry. Fundamentally it is formulated in terms of local equation. It appears in all gauge theories, independently on whether the $\theta$-vacuum exist. However, being integrated over the space-time, the anomaly equation $$ \partial_{\mu}J^{\mu}_{5} \sim \text{tr}[F_{\mu\nu}\tilde{F}^{\mu\nu}] $$ can be represented as $$ n_{+} - n_{-} = \nu, $$ where $n_{\pm}$ are number of zero left and right modes of the Dirac operator $D = \gamma_{\mu}(i\partial^{\mu} - g_{s}G^{\mu})$, and $\nu \sim \int \text{tr}[F_{\mu\nu}\tilde{F}^{\mu\nu}]$. Since the left hand-side is integer, the right hand-side also has to be integer. This is indeed the case because it can be represented as the difference of the winding numbers for $t = +\infty$ and $t = -\infty$.

See also the similar question.

  • 1
    $\begingroup$ It's not exactly correct, because in the presence of chiral anomaly you can apply a chiral rotation, which will shift the $\theta$-term. In particular you can set $\theta$ to zero, and so $\theta$-term is not really physical in the presence of chiral anomaly. See e.g. this answer. $\endgroup$ Oct 17, 2017 at 22:23
  • $\begingroup$ @PeterKravchuk : although this is correct, the essence of the $\theta$-term - non-trivialness of the QCD vacuum - doesn't suffer from the presence of fermions. $\endgroup$
    – Name YYY
    Oct 18, 2017 at 7:24

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