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Axial anomaly in QCD VS axial anomaly in current algebra QCD

+ 3 like - 0 dislike
66 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, ...)?

asked Oct 1 in Theoretical Physics by illuminates (15 points) [ revision history ]
edited Oct 1 by illuminates

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