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I have encountered myself with the following definition for $\pi$-fields as quark bilinears: $$ \pi^a = i\bar{q}\tau^a \gamma_5 q \ ,\quad\text{with }\ q = \left(\begin{array}{c}u\\d\end{array}\right) \ . $$ I understand the motivation for the appearence of the Pauli matrices $\tau^a$ and the $\gamma_5$ matrix for pseudo-scalar parity, so I have no problem with the expression itself. My problem is related to the dimensionality of the expression, since the Pauli matrices are $2\times 2$ matrices while $\gamma_5$ is $4\times 4$.

I guess that my problem comes from the fact maybe some indices are implicit in the expression. Using greek symbols for spinor indices, I think that the whole bilinear would read like $$ \pi^a = i\bar{q}^{b\mu} (\tau^a)_{bc} (\gamma_5)_{\mu\nu} q^{c\nu} \ . $$

I would appreciate if someone could confirm that this was the case of hidden implicit indices that solve the issue with the dimensions of the matrices.

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    $\begingroup$ Your answer given to your own question is correct, explaining at the same time the reason for preferring the compact notation used in your first formula. $\endgroup$
    – Hyperon
    Commented Apr 8 at 6:47
  • $\begingroup$ @Hyperon gave you as close to the perfect answer as it comes. Beyond that, however, and conceptually completely independently, your dimensional analysis is egregiously wrong in your crude schematic interpolating field expression: the l.h.s. has energy dimension 1, but the r.h.s. has energy dimension 3. You need to match those with two powers of $f_\pi$. $\endgroup$
    – Cosmas Zachos
    Commented Apr 8 at 19:43

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