In many books and papers (e.g. Refs. 1 & 2) we can find information that in the Standard Model there are only first-class currents. Why is it said that the second-class currents are beyond the SM?
References:
''Theoretical Nuclear And Subnuclear Physics, 2nd Edition'' by John Walecka page 441
Phys. Rev. D 98, 033005, Fatima et al. https://arxiv.org/abs/1806.08597 page 4.
Second class currents is a 1958 term introduced by Weinberg for naturally suppressed non-strange G-parity violating effects in the hadronization of the charged weak current coupling to the Ws, not fully understood at that time. (Second class $J^{PG}: 0^{+-}, 0^{-+}, 1^{++}, 1^{--}$.) Nowadays, people try chiral perturbation theory for this sort of thing, with whose detailed effects on it I am not fully familiar (cf last ref).
Crudely, second class currents have the "forbidden" effective tensor structure $J^\mu\propto \bar u_+ i\gamma_5 [\gamma^\mu , ~ q\!\!/~ ]u_0 $ which has an inaccessible C structure in coupling with the hadronic weak current. It is basically a hallmark of G-parity violation in the weak interactions, which amounts to isospin violation normalized by the QCD hadronic scale.
The quick diagnostic for isovector decays is that first class corresponds to positive (allowed) $GP(-)^J$, like the $\pi; ~~\rho; ~~a_1;~~ f_0;...$ (The weak current coupling to the W hadronizes to such $J^{PG}: 0^{--}, 1^{-+}, 1^{+-}, 0^{++}, ...$ states).
But second class corresponds to negative signature for the above, like the $a_0; ~~b_1;..$ and is suppressed, albeit allowed in principle. ($\eta \pi$ is dominated by $a_0$.)
$$G\equiv C e^{-i\pi I_2}, \leadsto \\ \frac{\Gamma(\tau\to\eta \pi \nu)}{\Gamma(\tau\to \pi \nu)} < 10^{-4}. $$ The numerator is second class, and the denominator first class (allowed). The suppression is of the order of the square of the isospin violation amplitude parameter $(m_d-m_u)/\Lambda_{QCD} \sim 10^{-2}$, so a hadronic suppression effect superposed on a semileptonic decay.
Recall isospin is a good hadronization symmetry not because the masses of the down and up quarks are close (hell, their ratio is 2!), but because both are so much smaller than the scale of QCD which dictates their hadronization.
There might be usable recent reviews, but I don't know of any. Weinberg’s vII text and the classic Dynamics of the Standard Model (Cambridge 2014, ISBN-13: 978-0521768672) by J Donoghue, E Golowich, & B Holstein, p 167, cover the waterfront.
If you really want to get serious, try P Langacker (1977) "General treatment of second-class currents in field theory", Physical Review D15 (8), p 2386. Also, R Escribano, S Gonzalez-Solis, & P Roig, (2016) "Predictions on the second-class current decays τ− → π− η ν", Physical Review D94 (3), p034008.
