A formula I was looking for

As usual I put in this blog some useful formulas to work out computations in quantum field theory. My aim in these days is to compute the width of the \sigma resonance. This is a major aim in QCD as the nature of this particle is hotly debated. Some authors think that it is a tetraquark or molecular state while others as Narison, Ochs, Minkowski and Mennessier point out the gluonic nature of this resonance. We have expressed our view in some posts (see here and here) and our results heavily show that this resonance is a glueball in agreement with the spectrum we have found for a pure Yang-Mills theory.

Our next step is to understand the role of this resonance in QCD. Indeed, we have shown in our recent paper (see here) that, once the gluon propagator is known, it is possible to derive a Nambu-Jona-Lasinio model from QCD with all parameters properly fixed. We have obtained the following:

S_{NJL} \approx \int d^4x\left[\sum_q \bar q(x)(i\gamma\cdot\partial-m_q)q(x)\right.

-\frac{1}{2}g^2\sum_{q,q'}\bar q(x)\frac{\lambda^a}{2}\gamma^\mu q(x)\times

\left.\int d^4yG(x-y)\bar q'(y)\frac{\lambda^a}{2}\gamma_\mu q'(y)\right]

being G(x-y) the gluon propagator with all indexes in color and space-time already saturated. This in turn means that we can use the following formula (see my paper here and here):

e^{\frac{i}{2}\int d^4xd^4yj(x)G(x-y)j(y)}\approx {\cal N}\int [d\sigma]e^{-i\int d^4x\left[\sigma\left(\frac{1}{2}\partial^2\sigma+\frac{Ng^2}{4}\sigma^3\right)-j\sigma\right]}

being again G(x-y) the gluon propagator for SU(N) and {\cal N} a normalization factor. This formula does hold only for infrared limit, that is when the theory is strongly coupled. We plan to extract physical results from this formula and define in this way the role of \sigma resonance.

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