# Axial anomaly in QCD VS axial anomaly in current algebra QCD

+ 3 like - 0 dislike
74 views

I would like to know whether the next calculations are true or not.

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}

Isn't this violation 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 https://physics.stackexchange.com/questions/193568/can-the-effective-vertex-for-gamma-to3-pi-be-derived-directly-from-the-anomal and https://physics.stackexchange.com/questions/43317/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}

My questions:

1) So, do I understand everything correctly?

2) Which books clearly explain 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, ...)?

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOverf$\varnothing$owThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.