## Fermi’s insight and the width of the sigma

Enrico Fermi is one of the greatest Italian physicist . He obtained a wide list of great accomplishments but a reason for him to be remembered in History is that, like the legendary Prometheus did for the fire, he brought mankind into nuclear era building the first nuclear pile in Chicago in 1942. The other reason why I am citing him here is that he uncovered weak interactions by postulating the first beta decay Lagrangian that has been since then the building block for our understanding culminating into the Standard Model of sixties and seventies. Il va sans dire that the paper Fermi wrote about and sent to Nature was rejected. This paper contains a true spectacular insight for a field theory when very few elements are known about a reaction than just the involved fields. Fermi’s insight arose from an analogy with the electromagnetic field. The concept is that when no information is given about the interaction itself the way to get a description of a particle reaction is through a contact interaction. Fermi just takes for the Green function of the interaction the following

$G(x-y)=\frac{G_F}{2}\delta^4(x-y)$

being $G_F$ the Fermi constant and the factor 2 has been introduced just for convenience. Then, if you consider e.g. the reaction $\mu^-\rightarrow e^-+\bar\nu_e+\nu_\mu$ you have just to take the product of the fields to define a Lagrangian as

$L_{int}=\frac{G_F}{2}(\bar\mu\gamma^\mu\nu_\mu)(\bar e\gamma_\mu\nu_e)+h.c.$

and you are done. At this stage you do not have to care about renormalizability (Fermi did not even know the problem at that time (1934)) as you just want a first understanding of the process. You need to fix the Fermi constant by comparison with experimental data and so, all other processes involving muons, electrons and neutrinos are fixed themselves. This is known as the “universality” of Fermi interactions. This model will fail if universality is lost for some reason.

The sigma or f0(600) is a hotly debated resonance seen in strong interactions. It has vacuum quantum numbers and its nature is the most important open question. We know from experimental data that this particle has two decay modes: $\sigma\rightarrow\pi^+\pi^-$ and $\sigma\rightarrow\gamma\gamma$. The first mode is largely dominant. From the paper of Colangelo, Caprini and Leutwyler (see here and here but other determinations exist as here and here) we know the main properties of this particle

$M_\sigma-i\Gamma_\sigma=441^{+16}_{-8}-i272^{+9}_{-12.5} MeV$

and one immediately realize that this resonance is quite broad. Using Fermi’s insight we can put down a contact interaction to compute such a width as

$L_{int}=\frac{G_{NJL}}{2}M_\sigma^3\sigma\pi^\dagger\pi$

being $G_{NJL}$ the Fermi constant for this case and we assume that we can neglect the contribution of the $\gamma\gamma$ decay being this really small. This latter process can also be interpreted as a $\pi\pi$ rescattering as done by Narison, Ochs and Mennessier here explaining in this way the smallness of this contribution. We can determine $G_{NJL}$ from QCD. We have done this here and we have obtained

$G_{NJL}=3.761402959\frac{g^2}{\sigma}$

being $g$ the coupling constant and $\sqrt{\sigma}=410\pm 20 MeV$ the square root of the string tension. We will be able to determine $g$ from our Fermi Lagrangian. It should be clear at this point that NJL stays for Nambu-Jona-Lasinio as we have obtained this constant in this context and is equivalent to a Fermi constant for QCD. A standard calculation as given e.g. here yields

$\Gamma_\sigma = \frac{G_{NJL}^2M_\sigma^5}{8\pi}\sqrt{1-\frac{4M_\pi^2}{M_\sigma^2}}$

that can be inverted to give the strong coupling constant

$\alpha_s=\frac{g^2}{4\pi}=\frac{\sigma}{3.761402959 M_\sigma^2}\sqrt{\frac{\Gamma_\sigma}{2\pi \sqrt{M_\sigma^2-4M_\pi^2}}}.$

Using Colangelo, Caprini and Leutwyler data we get the very good result

$\alpha_s\approx 0.16365$

that is physically consistent with expectations for the strong coupling at low energies.

So, we have seen how effective can be Fermi’s insight also for strong interactions. It would be interesting to extend this approach to other kind of resonances to see how far one can go with universality. At this stage we can be really satisfied with our result that justify the broadness of the sigma resonance using just QCD parameters.