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Edited to add new information

I have seen the relation between cross sections postulated:

\begin{align} \sigma(\nu+p\to\mu^- + \Delta^{++})=9\sigma(\nu+n\to\mu^- + \Delta^{+}) \end{align}

Motivated by isospin symmetry.

So what I want to do is understand the isospin rules for calculating matrix elements of the form:

\begin{align} \langle N^*|O|N\rangle \end{align}

Between a nucleon $N$ and some excited state $N^*$.

Looking in Electroweak Interactions by Renton, the isospin rules for transitions between nucleons can be determined considering the electromagnetic current first:

enter image description here

Which seems completely reasonable. The weak charged currents between protons and neutrons belong to the isovector part so they couple with strength $a-b$.

Returning to my problem, how do you approach this when the initial and final particles belong to different representations of SU(2) ?

There were two things I tried a few days ago:

1.

Interpret the W boson involved as a raising operator $\sigma_+$ in isospin space such that:

\begin{align} \sigma_+|p\rangle &=\alpha|\Delta^{++}\rangle \\ \sigma_+|n\rangle &=\beta|\Delta^{+}\rangle \\ \end{align}

Which should lead to $\alpha/\beta=3$ to give the ratio of cross sections as 9.

However if $|p\rangle=|1/2,1/2\rangle$, then $\sigma_+|p\rangle=0$.

2.

Interpret the W boson as a state $|1,1\rangle$ such that the ratio arises from calculating:

\begin{align} \frac{\left(\langle 1/2,1/2|\otimes\langle 1,1|\right)|3/2,3/2\rangle}{\left(\langle 1/2,-1/2|\otimes\langle 1,1|\right)|3/2,1/2\rangle} = \sqrt{3} \end{align}

Ratio of $\sqrt{3}$ according to Wolfram Alpha's CG coefficient calculator, ie predicting the ratio of cross sections as 3 not 9.

I'd like to know the correct way to derive this relation, especially as I've seen more complicated arguments like this to derive relations between cross sections for processes involving strangeness as well.

If people have recommendations for lecture notes/books that explore the role of group/representation theory in particle physics that would be appreciated. My teaching never went beyond very simple branching ratios and I find the level that papers typically work at is very far above where I am.

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  • $\begingroup$ Yes, but of isovector current only the left-chiral piece enters the weak interactions (couples to the Ws) thereby violating parity. $\endgroup$ – Cosmas Zachos Oct 20 at 0:37
  • $\begingroup$ The left chiral piece in the case is? In the weak interaction there will be a copy of eqn 8.31 with a $\gamma_5$ added that gives rise to the parity violation, but its not true V-A because the couplings in the vector and axial parts are not equal. $\endgroup$ – CT1234 Oct 20 at 1:01
  • $\begingroup$ sure thing. Not a bland Clebsching though... $\endgroup$ – Cosmas Zachos Oct 20 at 2:47

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