In deriving the axial anomaly Peskin and Schroeder use dimensional regularization, continuing loop momenta to $ 4 - \epsilon $ dimenstions. The loop momenta can now be split into pieces ``parallel'' to $ d = 0,1,2,3 $ and those "perpendicular" to $ d = 0,1,2,3 $, $$ \ell = \ell _{ \parallel } + \ell _{ \perp } $$ Furthermore, they define $ \gamma ^5 $ as, $$ \gamma ^5 \equiv i \gamma ^0 \gamma ^1 \gamma ^2 \gamma ^3 $$ with this definition $ \gamma ^5 $ commutes with $ \gamma ^{ \mu } $ in the extra dimensions and so, $$ \require{cancel} P _L {\cancel \ell} _{ \parallel } = {\cancel \ell} _{ \parallel } P _R \hspace{1.5cm}P _L {\cancel \ell} _{ \perp } = {\cancel \ell} _{ \perp } P _L $$ I have no problem with the results in this section which are presented when trying to compute the axial anomaly for a vector theory.
However, if I try to apply this technique to computing the chiral gauge anomaly for three left-handed vertices I am getting the wrong answer. Peskin and Schroeder only remark on the the final result here. The issue arises from the triangle diagram in figure 19.9 will be proportional to, $$ \require{cancel} \text{Tr} \left[ \gamma ^\mu P _L ( {\cancel \ell} - {\cancel k} ) \gamma ^\lambda P _L {\cancel \ell} \gamma ^\nu P _L ( {\cancel \ell} + {\cancel p} ) \right] $$ which after splitting the loop momenta gives, $$ \require{cancel} \text{Tr} \left[ \gamma ^\mu P _L ( {\cancel \ell} _{ \parallel } + {\cancel \ell} _\perp - {\cancel k} ) \gamma ^\lambda P _L ( {\cancel \ell} _{ \parallel } + {\cancel \ell} _{ \perp } ) \gamma ^\nu P _L ( {\cancel \ell} _{ \parallel } + {\cancel \ell} _{ \perp } + {\cancel p} ) \right] $$ Now if we carry through the projectors since $ P _L {\cancel \ell} _\perp P _R = 0 $ there won't be terms left with two factors of $ \ell _{ \perp } $, which are necessary to generate the anomaly. Since this should be anomalous something has gone wrong!
The strategy employed here is somewhat different then what the book suggests to do for this computation below equation 19.130:
The gauge boson vertices also contain factors of $ ( 1 - \gamma ^5 ) /2$. The three projectors can be moved together into a single factor. Then, if we regularize this diagram as in Section 19.2, the term containing a $ \gamma ^5 $ has an axial anomaly.
Thus the book proposes to move around the projectors and only THEN employ the regularization procedure. Indeed in doing this I can get the correct result. However, this is a strange approach since it suggests working with an ill-defined integral, massaging the expression with strategies which are wrong with the later chosen regularization scheme, and then regularizing the expression.
Why is this second approach the correct approach?
