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As far as I understand, an axial $U(1)$ transformation transforms a two-component spinor like $$ \psi \to \psi'=\text e^{\text i\epsilon \gamma^5 }\psi,\qquad \psi=\begin{pmatrix}\psi_1\\\psi_2\end{pmatrix}, $$ which means for the components $\psi_{1,2}$ of the spinor that they transform like \begin{align} \psi_1 \to \psi_1' &= \text e^{\text i\epsilon}\, \psi_1, \\ \psi_2 \to \psi_2' &= \text e^{-\text i\epsilon}\, \psi_2. \end{align} If the symmetry is realized, then the divergence of the axial current $j_A^{\,\mu} = \overline\psi\gamma^\mu\gamma^5\psi$ vanishes: $\partial_\mu \,j_A^{\,\mu}=0$. If there appears an anomaly, then $\partial_\mu\, j_A^{\,\mu} =\mathcal A \neq 0$ and the action transforms as $$ S\to S+\delta_\epsilon S = S+\int \text d^2x\,\, \epsilon\, \mathcal A .$$

How do I calculate this anomaly $\mathcal A$?

From this paper $[1]$ Eq. (2.4), this paper $[2]$ Eq. (14) and the Wiki article on the Fujikawa method $[3]$, I deduce that the divergence of the axial current in $d=1+1$ dimensions is proportional to the field strength tensor: $$ \partial_\mu \,j_A^{\,\mu} \propto \epsilon_{\mu\nu}F^{\mu\nu}. $$ Despite the references listed above, I am not able to reproduce this result. Here are my thoughts for each reference:

$\bullet$ Ref. $[1]$ motivates the result by some kind of momentum shift that I cannot connect to the axial transformation.

$\bullet$ Ref. $[2]$ gets the result by calculating some kind of vacuum polarization (why?).

$\bullet$ Ref. $[3]$ has a complete calculation of the Fujiwara method (looking at the path integral measure $\mathcal D\psi$ and how it transforms. This is done in $d$ dimensions, which I cannot "simplify" to the case $d=2$.

Any help is much appreciated!

Update: I found another paper which presents the result in Eq. (2.9). Though I do not understand the factor $\pi^{-1}$ there.

Srednicki also covers the axial anomaly in his QFT textbook (Ch. 77), although I have not been able to reproduce it in 2D yet.

My choice for the two-dimensional gamma matrices: $$ \gamma^0=\begin{pmatrix}0&1\\1&0\end{pmatrix}, \quad \gamma^1=\begin{pmatrix}0&-1\\1&0\end{pmatrix}, \quad \gamma^5=\begin{pmatrix}1&0\\0&-1\end{pmatrix}. $$

As far as I understand, an axial $U(1)$ transformation transforms a two-component spinor like $$ \psi \to \psi'=\text e^{\text i\epsilon \gamma^5 }\psi,\qquad \psi=\begin{pmatrix}\psi_1\\\psi_2\end{pmatrix}, $$ which means for the components $\psi_{1,2}$ of the spinor that they transform like \begin{align} \psi_1 \to \psi_1' &= \text e^{\text i\epsilon}\, \psi_1, \\ \psi_2 \to \psi_2' &= \text e^{-\text i\epsilon}\, \psi_2. \end{align} If the symmetry is realized, then the divergence of the axial current $j_A^{\,\mu} = \overline\psi\gamma^\mu\gamma^5\psi$ vanishes: $\partial_\mu \,j_A^{\,\mu}=0$. If there appears an anomaly, then $\partial_\mu\, j_A^{\,\mu} =\mathcal A \neq 0$ and the action transforms as $$ S\to S+\delta_\epsilon S = S+\int \text d^2x\,\, \epsilon\, \mathcal A .$$

How do I calculate this anomaly $\mathcal A$?

From this paper $[1]$ Eq. (2.4), this paper $[2]$ Eq. (14) and the Wiki article on the Fujikawa method $[3]$, I deduce that the divergence of the axial current in $d=1+1$ dimensions is proportional to the field strength tensor: $$ \partial_\mu \,j_A^{\,\mu} \propto \epsilon_{\mu\nu}F^{\mu\nu}. $$ Despite the references listed above, I am not able to reproduce this result. Here are my thoughts for each reference:

$\bullet$ Ref. $[1]$ motivates the result by some kind of momentum shift that I cannot connect to the axial transformation.

$\bullet$ Ref. $[2]$ gets the result by calculating some kind of vacuum polarization (why?).

$\bullet$ Ref. $[3]$ has a complete calculation of the Fujikawa method (looking at the path integral measure $\mathcal D\psi$ and how it transforms. This is done in $d$ dimensions, which I cannot "simplify" to the case $d=2$.

Any help is much appreciated!

Update: I found another paper which presents the result in Eq. (2.9). Though I do not understand the factor $\pi^{-1}$ there.

Srednicki also covers the axial anomaly in his QFT textbook (Ch. 77), although I have not been able to reproduce it in 2D yet.

My choice for the two-dimensional gamma matrices: $$ \gamma^0=\begin{pmatrix}0&1\\1&0\end{pmatrix}, \quad \gamma^1=\begin{pmatrix}0&-1\\1&0\end{pmatrix}, \quad \gamma^5=\begin{pmatrix}1&0\\0&-1\end{pmatrix}. $$

As far as I understand, an axial $U(1)$ transformation transforms a two-component spinor like $$ \psi \to \psi'=\text e^{\text i\epsilon \gamma^5 }\psi,\qquad \psi=\begin{pmatrix}\psi_1\\\psi_2\end{pmatrix}, $$ which means for the components $\psi_{1,2}$ of the spinor that they transform like \begin{align} \psi_1 \to \psi_1' &= \text e^{\text i\epsilon}\, \psi_1, \\ \psi_2 \to \psi_2' &= \text e^{-\text i\epsilon}\, \psi_2. \end{align} If the symmetry is realized, then the divergence of the axial current $j_A^{\,\mu} = \overline\psi\gamma^\mu\gamma^5\psi$ vanishes: $\partial_\mu \,j_A^{\,\mu}=0$. If there appears an anomaly, then $\partial_\mu\, j_A^{\,\mu} =\mathcal A \neq 0$ and the action transforms as $$ S\to S+\delta_\epsilon S = S+\int \text d^2x\,\, \epsilon\, \mathcal A .$$

How do I calculate this anomaly $\mathcal A$?

From this paper $[1]$ Eq. (2.4), this paper $[2]$ Eq. (14) and the Wiki article on the Fujikawa method $[3]$, I deduce that the divergence of the axial current in $d=1+1$ dimensions is proportional to the field strength tensor: $$ \partial_\mu \,j_A^{\,\mu} \propto \epsilon_{\mu\nu}F^{\mu\nu}. $$ Despite the references listed above, I am not able to reproduce this result. Here are my thoughts for each reference:

$\bullet$ Ref. $[1]$ motivates the result by some kind of momentum shift that I cannot connect to the axial transformation.

$\bullet$ Ref. $[2]$ gets the result by calculating some kind of vacuum polarization (why?).

$\bullet$ Ref. $[3]$ has a complete calculation of the Fujiwara method (looking at the path integral measure $\mathcal D\psi$ and how it transforms. This is done in $d$ dimensions, which I cannot "simplify" to the case $d=2$.

Any help is much appreciated!

Update: I found another paper which presents the result in Eq. (2.9). Though I do not understand the factor $\pi^{-1}$ there.

My choice for the two-dimensional gamma matrices is $$ \gamma^0=\begin{pmatrix}0&1\\1&0\end{pmatrix}, \quad \gamma^1=\begin{pmatrix}0&-1\\1&0\end{pmatrix}, \quad \gamma^5=\begin{pmatrix}1&0\\0&-1\end{pmatrix}. $$

As far as I understand, an axial $U(1)$ transformation transforms a two-component spinor like $$ \psi \to \psi'=\text e^{\text i\epsilon \gamma^5 }\psi,\qquad \psi=\begin{pmatrix}\psi_1\\\psi_2\end{pmatrix}, $$ which means for the components $\psi_{1,2}$ of the spinor that they transform like \begin{align} \psi_1 \to \psi_1' &= \text e^{\text i\epsilon}\, \psi_1, \\ \psi_2 \to \psi_2' &= \text e^{-\text i\epsilon}\, \psi_2. \end{align} If the symmetry is realized, then the divergence of the axial current $j_A^{\,\mu} = \overline\psi\gamma^\mu\gamma^5\psi$ vanishes: $\partial_\mu \,j_A^{\,\mu}=0$. If there appears an anomaly, then $\partial_\mu\, j_A^{\,\mu} =\mathcal A \neq 0$ and the action transforms as $$ S\to S+\delta_\epsilon S = S+\int \text d^2x\,\, \epsilon\, \mathcal A .$$

How do I calculate this anomaly $\mathcal A$?

From this paper $[1]$ Eq. (2.4), this paper $[2]$ Eq. (14) and the Wiki article on the Fujikawa method $[3]$, I deduce that the divergence of the axial current in $d=1+1$ dimensions is proportional to the field strength tensor: $$ \partial_\mu \,j_A^{\,\mu} \propto \epsilon_{\mu\nu}F^{\mu\nu}. $$ Despite the references listed above, I am not able to reproduce this result. Here are my thoughts for each reference:

$\bullet$ Ref. $[1]$ motivates the result by some kind of momentum shift that I cannot connect to the axial transformation.

$\bullet$ Ref. $[2]$ gets the result by calculating some kind of vacuum polarization (why?).

$\bullet$ Ref. $[3]$ has a complete calculation of the Fujiwara method (looking at the path integral measure $\mathcal D\psi$ and how it transforms. This is done in $d$ dimensions, which I cannot "simplify" to the case $d=2$.

Any help is much appreciated!

Update: I found another paper which presents the result in Eq. (2.9). Though I do not understand the factor $\pi^{-1}$ there.

Srednicki also covers the axial anomaly in his QFT textbook (Ch. 77), although I have not been able to reproduce it in 2D yet.

My choice for the two-dimensional gamma matrices: $$ \gamma^0=\begin{pmatrix}0&1\\1&0\end{pmatrix}, \quad \gamma^1=\begin{pmatrix}0&-1\\1&0\end{pmatrix}, \quad \gamma^5=\begin{pmatrix}1&0\\0&-1\end{pmatrix}. $$

As far as I understand, an axial $U(1)$ transformation transforms a two-component spinor like $$ \psi \to \psi'=\text e^{\text i\epsilon \gamma^5 }\psi,\qquad \psi=\begin{pmatrix}\psi_1\\\psi_2\end{pmatrix}, $$ which means for the components $\psi_{1,2}$ of the spinor that they transform like \begin{align} \psi_1 \to \psi_1' &= \text e^{\text i\epsilon}\, \psi_1, \\ \psi_2 \to \psi_2' &= \text e^{-\text i\epsilon}\, \psi_2. \end{align} If the symmetry is realized, then the divergence of the axial current $j_A^{\,\mu} = \overline\psi\gamma^\mu\gamma^5\psi$ vanishes: $\partial_\mu \,j_A^{\,\mu}=0$. If there appears an anomaly, then $\partial_\mu\, j_A^{\,\mu} =\mathcal A \neq 0$ and the action transforms as $$ S\to S+\delta_\epsilon S = S+\int \text d^2x\,\, \epsilon\, \mathcal A .$$

How do I calculate this anomaly $\mathcal A$?

From this paper $[1]$ Eq. (2.4), this paper $[2]$ Eq. (14) and the Wiki article on the Fujikawa method $[3]$, I deduce that the divergence of the axial current in $d=1+1$ dimensions is proportional to the field strength tensor: $$ \partial_\mu \,j_A^{\,\mu} \propto \epsilon_{\mu\nu}F^{\mu\nu}. $$ Despite the references listed above, I am not able to reproduce this result. Here are my thoughts for each reference:

$\bullet$ Ref. $[1]$ motivates the result by some kind of momentum shift that I cannot connect to the axial transformation.

$\bullet$ Ref. $[2]$ gets the result by calculating some kind of vacuum polarization (why?).

$\bullet$ Ref. $[3]$ has a complete calculation of the Fujiwara method (looking at the path integral measure $\mathcal D\psi$ and how it transforms. This is done in $d$ dimensions, which I cannot "simplify" to the case $d=2$.

Any help is much appreciated!

My choice for the two-dimensional gamma matrices is $$ \gamma^0=\begin{pmatrix}0&1\\1&0\end{pmatrix}, \quad \gamma^1=\begin{pmatrix}0&-1\\1&0\end{pmatrix}, \quad \gamma^5=\begin{pmatrix}1&0\\0&-1\end{pmatrix}. $$

As far as I understand, an axial $U(1)$ transformation transforms a two-component spinor like $$ \psi \to \psi'=\text e^{\text i\epsilon \gamma^5 }\psi,\qquad \psi=\begin{pmatrix}\psi_1\\\psi_2\end{pmatrix}, $$ which means for the components $\psi_{1,2}$ of the spinor that they transform like \begin{align} \psi_1 \to \psi_1' &= \text e^{\text i\epsilon}\, \psi_1, \\ \psi_2 \to \psi_2' &= \text e^{-\text i\epsilon}\, \psi_2. \end{align} If the symmetry is realized, then the divergence of the axial current $j_A^{\,\mu} = \overline\psi\gamma^\mu\gamma^5\psi$ vanishes: $\partial_\mu \,j_A^{\,\mu}=0$. If there appears an anomaly, then $\partial_\mu\, j_A^{\,\mu} =\mathcal A \neq 0$ and the action transforms as $$ S\to S+\delta_\epsilon S = S+\int \text d^2x\,\, \epsilon\, \mathcal A .$$

How do I calculate this anomaly $\mathcal A$?

From this paper $[1]$ Eq. (2.4), this paper $[2]$ Eq. (14) and the Wiki article on the Fujikawa method $[3]$, I deduce that the divergence of the axial current in $d=1+1$ dimensions is proportional to the field strength tensor: $$ \partial_\mu \,j_A^{\,\mu} \propto \epsilon_{\mu\nu}F^{\mu\nu}. $$ Despite the references listed above, I am not able to reproduce this result. Here are my thoughts for each reference:

$\bullet$ Ref. $[1]$ motivates the result by some kind of momentum shift that I cannot connect to the axial transformation.

$\bullet$ Ref. $[2]$ gets the result by calculating some kind of vacuum polarization (why?).

$\bullet$ Ref. $[3]$ has a complete calculation of the Fujiwara method (looking at the path integral measure $\mathcal D\psi$ and how it transforms. This is done in $d$ dimensions, which I cannot "simplify" to the case $d=2$.

Any help is much appreciated!

Update: I found another paper which presents the result in Eq. (2.9). Though I do not understand the factor $\pi^{-1}$ there.

My choice for the two-dimensional gamma matrices is $$ \gamma^0=\begin{pmatrix}0&1\\1&0\end{pmatrix}, \quad \gamma^1=\begin{pmatrix}0&-1\\1&0\end{pmatrix}, \quad \gamma^5=\begin{pmatrix}1&0\\0&-1\end{pmatrix}. $$