# Experimental proof for isospin conservation of strong interaction

I am having a surprisingly hard time to find direct experimental results for the (approx.) conservation of isospin in strong interactions. The canonical examples seem to be $$\omega \! \to 3\pi^0$$ and $$\rho^0 \! \to 2\pi^0$$. But the former is also forbidden by charge conjugation and I failed to find a measured limit for the latter. In fact, I would prefer measurements that approximately reproduce expected branching ratios, such as $$\Delta^0 \! \to p \pi^-$$ vs. $$\Delta^0 \! \to n \pi^0$$, but again, I don't find any direct measurements. Do you have good examples and can point me to some published results?

• Second class current decays are suppressed by dint of isospin/G-parity. Go to the PDG, – Cosmas Zachos Sep 28 at 10:48
• Thanks for your help. I searched the PDG. But I failed to find an upper limit for the $\eta$ decay, as well as branching ratios for the $\Delta$ decays. Could you be a bit more precise and / or point me to a specific PDG page / link? – avitase Sep 28 at 10:52
• Tau to eta pi nu is second class. Don’t be stymied by cutting out weak interactions. Weinberg’s book details such. – Cosmas Zachos Sep 28 at 11:45
• Sorry for the confusion. There was a typo. I wanted to refer to $\rho^0 \! \to 2\pi^0$ instead of $\eta$. You are right, $\eta \!\to 2\pi^0$ is indeed measured. – avitase Sep 29 at 14:42
• Could you explain what you mean with the $\tau \!\to \eta \pi^0 \nu_\tau$ decay? How does isospin conservation enters here? – avitase Sep 29 at 15:00

Consider J/ψ ⟶ Λ $$\bar Σ^0$$ + cc , where the J/ψ and Λ are isosinglets and $$Σ^0$$ is centrally inside an isotriplet. It is squarely in the PDG listings, $$\Gamma_{195}$$, 28 parts per million.

No external photons are involved, so a bona fide strong isospin violation.

As suggesteded in the comments, from G-violation in 2nd-class weak decays, it is well-understood that the dimensionless violation parameter in the decay amps which enters in the BRs through its square is $$\kappa \approx 10^{-2}$$, so, unfortunately, comparable to its EM competitor α, in such decays. So the violation is suppressed by crudely fourish orders of magnitude... put in the high mass products and you are there.

Theoretically, this is interesting, since $$\kappa \sim (m_d-m_u)/M$$, where the current quark mass difference is 2.5 MeV, while the hadronic scale M is $$\Lambda_{QCD}$$ or, equivalently, a constituent light quark mass, entering through the currents, so a factor 100 larger.

(This is a major takeaway in understanding why isospin works when it does: it is not that the u and d quarks are degenerate; hell, one is twice the mass of the other. It is that the masses of both are a small perturbation on the hadronic scales of QCD, so, then, the corresponding baryons p,n are almost degenerate.)

So, completely segregating electromagnetism from strong isospin violation is real messy...

• Thanks a lot for these in-depth explanations, much appreciated! Glad that you mention the $J/\psi$ decay, since I never understood the measured branching ratio $J/\psi \to 2\Lambda$ vs $J/\psi \to 2 \Sigma^0$ of ~60%. CG coeffs. give me 1/3 for QCD and even QED (since QED contributes only one diquark, which can then form either $I=1$ or $I=0$, all other quarks are iso-singlets and since $2 \Sigma^0$ can be either $I=2$ or $I=0$, but not $I=1$ QED is in that sense also (accidently) isospin conserving). I cannot belief, that this factor of 2 comes from isospin breaking QCD effects? – avitase Sep 29 at 15:09
• Besides the fact, that your explanation is very helpful, I am still looking for a measurement, where Clebsch-Gordan coeffs. predict roughly the correct branching ratio, similar to my example of the $\Delta$ decays, where I don't find references though, which is quite unfortunate, since I expect QED effects to be quite small here (?). – avitase Sep 29 at 15:14
• I thought that the squared CG coeff, i.e $3$, not $\sqrt{3}$, has to be considered when looking at branching ratios? – avitase Sep 29 at 15:55
• Oh, fair enough, yeah, a factor of 0.6 is not that big of a deal.... – Cosmas Zachos Sep 29 at 16:17
• It is the difference between 60% (measured) and 1/3 (CG), i.e. 2x more $\Sigma^0$ than $\Lambda$ than predicted from isospin conservation. I think this is quite large? – avitase Sep 29 at 21:11

There is plenty of evidence from nuclear structure. E1 transitions with $$\Delta T=0$$ are forbidden in $$N=Z$$ nuclei. We observe isospin multiplets of energy levels, with differences in energy that are small and can be explained from the Coulomb interaction. Direct tests have been made using nucleon-nucleon scattering: https://arxiv.org/abs/nucl-th/0011057

I found two nice examples: The Particle Data Group provides a nice collection of various measurements of the total collision cross-section of $$p\pi^\mp$$. The $$\Delta^0$$ and $$\Delta^{++}$$ peaks are in good agreement with the expected ratio of $$1/3$$. The other example is the decay of the $$K^+_1(1270)$$ into $$K^+ \omega$$ and $$K^+ \rho^0$$. Again, the expected ratio is $$1/3$$, which is in good agreement with the measurements: $$\frac{\mathcal{B} \left(K^+_1(1270) \! \to K^+\rho^0 \right)}{\mathcal{B} \left(K^+_1(1270) \!\to K^+\omega \right)} = \frac{0.42 \pm 0.06}{0.11 \pm 0.02} = 3.8 \pm 0.9$$