$\tau \rightarrow \rho \nu $ relation with $ \rho \rightarrow e^+ e^-$

Question

I'm reading "Okun, Leptons and quarks.", specifically subsection 13.3 "Semihadronic decays. General remarks." Okun says "The amplitude of the $\tau \rightarrow \rho \nu $ decay is related directly to that of the $ \rho \rightarrow e^+ e^-$, owing to the isotopic properties of the ud current."

I understand that we have three ud isovector currents: $\bar{u} \gamma_{\mu} (1+\gamma_5) d$, $\bar{d} \gamma_{\mu} (1+\gamma_5) u$ and $\frac{1}{\sqrt{2}}\left[\bar{u} \gamma_{\mu} (1+\gamma_5) u - \bar{d} \gamma_{\mu} (1+\gamma_5) d \right]$. But I don't understand how it's connected with electroweak interaction, if electroweak doesn't respect isospin symmetry (I mean, for example, $\bar{u}d$ current interacts with $W$ bosons and $\bar{u}u$ current interacts with $Z$ and $\gamma$ bosons. So amplitudes should be different.).

And it's only part of the problem. I understand how you can interchange $\bar{\tau} \nu_{\tau} $ with $ \bar{e} \nu_{e} $ (they're both charged currents, and GSW model allows to interchange them). But it seems that Okun then interchanges $\bar{e} \nu_{e}$ current with $\bar{e} e$. And I completely don't understand, why it is allowed. Or maybe there is some other hidden logic behind this that I don't understand.

Answer 1

Yes, there is a central logic to these amps, although not hidden at all: it is in plain sight in the following supremely elegant, explicit , and clear section 13.5, where Okun explains exactly what he means. Most teachers would give their right arm to be able to write a section as clear and satisfying as that one!

The juxtaposition of the EM and EW amplitudes is only at the level of the ρ-quark coupling: the phenomenological parameter $g_ρ$ (or, equivalently, γ) he is determining from Fig 3.15:

The currents with the suitable chiral structure you are writing only describe the right vertices in these loops yielding the couplings to the W in a) or the photon in b),c). He computes those in gory detail in 13.5. In effect, the photon couples to both left and right-handed quark components, while the W only "uses" the left-handed ones.

• However, the phenomenological, left vertex of the loops summarizes how the ρ resolves to quark-antiquark pairs, which is strictly strong interactions, poorly understood, except respecting isospin! That's what the undetermined $g_ρ$ quantifies.

That strong vertex is thus beautifully constrained by isospin, and he demonstrates, as an introductory exercise!, how it connects to the EM current γ, so $g_ρ=\sqrt{2} m_ρ^2/γ$. What is done past the transition to quarks is computed explicitly, with no assumptions about isospin. The transition from a hadron state to quarks, the only obscure part of the amp is just papered over by these correlated experimental parameters. And it works quite well...