# Eta prime decay width

1. Dec 1, 2006

### BB1974

Why does the eta prime meson have such a narrow decay width (ie long lifetime) compared to the rho and omega mesons? Is there some conservation rule that supresses its decays?

2. Dec 3, 2006

### Meir Achuz

The eta prime and the eta' have J^P 0^-, and so cannot decay into two pions.
They have positive G parity, and so cannot decay strongly into three pions.
This makes the eta very narrow (~1 keV) and the eta' much narrower
(~200 keV) than the rho which decays strongly into two pions and the omega which decays strongly into three pions. The dominant decay of the eta' into the eta and two pions is a strong decay, but the phase space is small, limiting the rate.

3. Dec 3, 2006

### arivero

Indeed it is very interesting so substract the strong decay and to study the anomaly-mediated decay of eta' in the same way than eta and pion.

EDITED: the amazing scaling of anomaly-mediated decays and more was touched in a post in Dorigo's blog, http://dorigo.wordpress.com/2006/09/14/a-mistery-behind-the-z-width/

Last edited: Dec 3, 2006
4. Dec 3, 2006

### arivero

Generically, when one finds than am allowed strong decay is, er, strongly supressed then the first suspect is OZI rule, which "roughly" asks some of the initial quark content of the decaying particle to survive in the decaying products.

Last edited: Dec 3, 2006
5. Dec 3, 2006

### arivero

To put the numbers in. The reduced decay width, $$\tilde \Gamma \equiv \Gamma / m^3$$ for the decay of eta' to two photons is
$$\tilde \Gamma (\eta' \to \gamma \gamma) = 4.89 \pm 0.17 \, TeV^{-2}$$
The energy scale of about a half TeV is a bit misleading, because this definition of the reduced width does not care of factoring out the electromagnetic coupling constant nor other fixed factors.
Consider that $$\tilde \Gamma (...)^{-1/2} \approx 452 GeV \approx \alpha^{-1} 3.2 GeV$$, which sounds as a more reasonable scale (factor in some pi and some fraction too if you wish).
To see that there is some sense on using this scaled rate we can compare it with other total reduced decay rates:
$$\tilde \Gamma (\pi^0) =3.17 \pm 0.24 \, TeV^{-2}$$
$$\tilde \Gamma (J/\Psi) =3.14 \pm 0.07 \, TeV^{-2}$$
$$\tilde \Gamma (Z^0) = 3.29 \pm 0.003 \, TeV^{-2}$$
and we see that they are all in the same order, not amazingly for the pion, but sort of surprising, for diferent reasons, in the other two.

The electromagnetic decays of eta are in the same range:
$$\tilde \Gamma (\eta \to \gamma \gamma) = 3.1 \ TeV^{-2}$$
$$\tilde \Gamma (\eta \to \gamma \pi^+ \pi^-) =0.37 \ TeV^{-2}$$
$$\tilde \Gamma (\eta \to \gamma \gamma, \to \gamma \pi^+ \pi^-)= 3.5 \ TeV^{-2}$$

(can anyone remember me about the formulae for the error of the product of two measures? Must be in the wikipedia somewhere.)

An interesting question is that eta' has those, hmm, "semielectromagnetic", decays:
$$\to \rho^0 \gamma,$$$$\to \omega \gamma,$$$$\to \mu^+ \mu^- \gamma$$.
The later is negligible, and can be included in the gamma gamma without changing the above estimate (due to the error). But the other two account for relevant sizes of decay rate, in fact we have
$$\tilde \Gamma (\eta' \to \rho^0 \gamma) = 67.9 \, TeV^{-2}$$
$$\tilde \Gamma (\eta' \to \omega \gamma) = 7.0 \, TeV^{-2}$$

Last edited: Dec 3, 2006
6. Dec 5, 2006

### BB1974

Thanks everyone. I attempted to explain that the eta' lasts about 40 times as long as the omega with a combination of OZI and phase-space arguments.

As far as strong decays go:

eta' = (dd-bar + uu-bar + ss-bar)/sqrt(3) goes to eta = (dd-bar + uu-bar - 2 ss-bar)/sqrt(6) plus 2 pions.

I try to separately consider each piece of the eta' wave function goes to each piece of the eta wave function plus two pions.

But all of those are OZI supressed except uu-bar goes to uu-bar plus 2 pions and dd-bar goes to dd-bar plus two pions. So I get a diagram times 2 over sqrt(18).

Doing the same thing for omega goes to three pions gives me a factor of 2 over sqrt(4):
omega' = (dd-bar + uu-bar)/sqrt(2) goest to pion = (uu-bar -dd-bar)/sqrt(2) plus two more pions

Squaring that ratio accounts for a factor of 4.5 between the two decays (I didn't actually calculate the QCD diagrams).

Energetics also contributes a bunch. In the omega to 3 pion decay, the decay particles have over 500 MeV of energy to work with. In the eta' to eta plus 2 pion decay, there's less than 150MeV of excess energy. The phase space goes roughly like the square of the available energy, so that contributes a factor of about 10.

So my back-of-the envelope approach roughly explains the disparity. There's obviously much more to consider here to get a more precise answer. A lot of the stuff you guys brought up is really beyond the scope of what we've learned so far. Thanks again. --BRIAN

7. Dec 5, 2006

### arivero

Indees that is the way to go I'd wish OZI suppression were more deeply understood, but the limitations of perturbative calculations for SU(3) are an slap in the face.
the stuff on the Z0 is actually unpublished because you should ask for a low energy GUT model to explain it, and such beast plainly does not exist. The stuff on J/Psi could eventually be proved, it amounts to say that all the total sum of allowed decays is "dual" to the decay via the forbidden channel. But nobody tries dualities in electroweak theory, so it will stay in the limbo too. Reduced decay widhts are actually used in some works, but for energies more or less in the same range. I am not sure if one should refine the definition by considering the running of the coupling constants (hard to do, if you only want to use experimental data, theory-independent)

Ok course you will find in the textbooks the thing about the pi0, if you have not met it already. The "decay via anomalies" is a famous theme.

Last edited: Dec 5, 2006