$\gamma^5$ rotation of chiral fermion in (1) Peskin&Schroeder, (2) Weinberg, or (3) Srednicki The theta angle due to the chiral gamma^5 rotation of chiral fermion results in the phase alpha(x) that has different + or - sign for
(1) Peskin&Schroeder, (2) Weinberg or (3) Srednicki.
Here

*

*Peskin&Schroeder does a psi' = exp(i alpha(x)  gamma5) psi rotation, and it gives a (+) sign F F dual term in (19.79)


*Weinberg book  does a psi' = exp(i alpha(x)  gamma5) psi rotation, and it gives a (-) sign F F dual term in (23.68)


*Srednicki book does a psi' = exp(-i alpha(x) ) psi rotation, and it gives a (-) sign F F dual term in (94.4)
It looks that (1) and (3) have the same results, while (2) Weinberg gives a different result from others. Does someone know why? Who may possibly make a mistake?
p.s. Pardon me to show their results directly in images below, I believe that it is easier to show them directly so people can see from their books -- which will be more clear than I typed the equations in a partial manner.

Peskin&Schroeder



Weinberg

Srednicki

 A: The gamma matrices that Peskin & Schroeder work with are $$\gamma^0 = \begin{pmatrix}0 & 1\\ 1 & 0\end{pmatrix},\quad \gamma^i=\begin{pmatrix}0& \sigma^i\\ -\sigma^i & 0\end{pmatrix}\tag{3.25}$$
On the other hand, for Weinberg, the gamma matrices are $$\gamma^0=-i\begin{pmatrix}0 & 1 \\ 1 &0\end{pmatrix},\quad \gamma^i=-i\begin{pmatrix}0 & \sigma^i \\ -\sigma^i & 0\end{pmatrix}\tag{5.4.17}$$
In particular we see that the gamma matrices used by Weinberg, call them $\gamma^\mu_W$, are related to the gamma matrices used by Peskin & Schroeder, call them $\gamma^\mu_{PS}$, by $$\gamma^\mu_W=-i\gamma^\mu_{PS}.$$
Peskin & Schroeder define $\gamma^5_{PS}=i\gamma^0_{PS}\gamma^1_{PS}\gamma^2_{PS}\gamma^3_{PS}$, while Weinberg defines $\gamma^5_W=-i\gamma^0_W\gamma^1_W\gamma^2_W\gamma^3_W$. In that case we can compare the two and we observe that $$\gamma^5_W=-i(-i\gamma^0_{PS})(-i\gamma^1_{PS})(-i\gamma^2_{PS})(-i\gamma^3_{PS})=-i\gamma^0_{PS}\gamma^1_{PS}\gamma^2_{PS}\gamma^3_{PS}=-\gamma^5_{PS}.$$
So because of the different conventions for the gamma matrices we see that what Peskin & Schroeder define as $\gamma^5$ differs from what Weinberg defines by a minus sign. In that case when Weinberg does $\psi' = e^{i\alpha\gamma_W^5}\psi$ he is doing $\psi' = e^{-i\alpha\gamma_{PS}^5}\psi$. So for Weinberg transforming with $\alpha$ is the same as transforming with $-\alpha$ for Peskin & Schroeder. Therefore the two are really consistent. Finally it is immediate to see that Srednick is also consistent with the other two as you said yourself in the question.
