## An useful hint

24/07/2008

Dietmar Ebert is a retired professor of Humboldt University in Berlin. He did relevant work in QCD and particle physics. I have come upon a paper of him at arxiv about the question of bosonization. In a paper of mine I showed how a Nambu-Jona-Lasinio (NJL) model can be derived from QCD using recent results about gluon propagator that is the corner stone of all this construction. In order to make contact with the mesonic spectrum of QCD one needs to manage in some way quark fermionic fields of NJL model to recover bosonic degrees of freedom. In Ebert’s paper this is done through Hubbard-Stratonovich transformation that is a widely known tool to condensed matter theorists. This is a key point to prove that our recent derivation of the width of the sigma resonance given here using a Fermi’s intuition is indeed correct. Ebert obtains by a NJL-model the following bosonic Hamiltonian

$L_{int}=g_{\sigma\pi\pi}\sigma(\sigma^2+\pi^2)+g_{4\pi}(\sigma^2+\pi^2)^2$

being

$g_{\sigma\pi\pi}=\frac{m}{\sqrt{N_c I_2}}$

and

$g_{4\pi}=\frac{1}{8N_c I_2}$

being $N_c$ the number of colors,

$m=m_0+i8mG_{NJL}\int^{\Lambda}\frac{d^4k}{(2\pi)^4}\frac{1}{k^2-m^2}$

quark constituent mass and $m_0$ the quark mass assumed to be equal for u and d, and finally

$I_2=-i\int^{\Lambda}\frac{d^4k}{(2\pi)^4}\frac{1}{(k^2-m^2)^2}.$

In order to make contact with QCD, as we have shown one has

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

being $g$ the coupling constant and $\sqrt{\sigma}=410\pm 20 \ MeV$ the square root of the string tension.

Ebert’s Lagrangian gives us exactly the term we derived with Fermi’s insight plus other terms implying also the one to compute f0(980) decay rate $2g_{4\pi}\sigma^2\pi^2$. So, as it is well-known, a good idea repeats itself at different levels in the description of Nature. I would call this the “gluonic sector” of QCD. I hope to put down a paper about in the next days.