## QCD with two colors

Having an understanding of Yang-Mills theory grants the possibility to make QCD truly manageable and amenable to a perturbation treatment also in the infrared limit. A very easy example of this can be obtained working out the equations of QCD with two colors. In this case the gauge group is SU(2) and algebra is not too much involved. The relevant simplification is obtained via the “mapping theorem” (see my paper here). This theorem grants the existence of a leading order solution of Yang-Mills theory to do perturbation computations in the infrared limit by mapping it on a quartic massless scalar field through the so-called Smilga’s choice (see here). In turn this implies that in all QCD computations we have to manage just a scalar field making things simpler. In QCD with two colors we are reduced to the following action

$S=\int d^4x[\sum_q \bar q(i\gamma\cdot\partial+\frac{g}{2}\phi\Sigma-m_q)q+\frac{3}{2}(\partial\phi)^2-3\frac{2g^2}{4}\phi^4]$

being $\Sigma=\sigma_1\gamma^1+\sigma_2\gamma^2+\sigma_3\gamma^3$ with $\sigma_i$ Pauli matrices and $\gamma^i$ Dirac matrices. So, classical equations of motion are

$(i\gamma\cdot\partial+\frac{g}{2}\phi\Sigma-m_q)q=0$

$\partial^2\phi+2g^2\phi^3=\frac{g}{6}\bar q\Sigma q$

and we can do a strong coupling expansion by rescaling time as $\tau=\sqrt{2}gt$ leaving us with the non-trivial leading order equations

$i\gamma^0\partial_\tau q_0+\frac{1}{2\sqrt{2}}\phi_0\Sigma q_0=0$

$\partial_\tau^2\phi_0+\phi_0^3=0$

and this set of equations is easily solved. We observe that at the leading order quarks can be considered massless (chiral simmetry) and the spectrum of the Yang-Mills theory is part of observational QCD. Finally, a discrete spectrum for quarks is also obtained whose ground state is zero, an expression of chiral symmetry. So, at this order, we expect pion mass to be zero and glueballs having no decay.

These are quite interesting results but higher order corrections should be exploited to have a clear understanding of all physics. Essentially, we would like to compute the pion mass, i.e. to see the breaking of the chiral symmetry seen at the leading order, and the decay width of the lowest glueball state. I will exploit computations to higher orders to reach such aims.

### 3 Responses to QCD with two colors

1. […] of two color QCD In a preceding post (see here) I have shown how SU(2) QCD can be reduced to a theory easily amenable to perturbation treatment. […]

2. dorigo says:

Marco, these are very interesting posts, and I wish I had more time to study them with care. Thank you for the content you publish here! I think this is a great resource.

Cheers,
T.

3. mfrasca says:

Tommaso,

Thank you very much for your comment. This is a great encouragement for my work and you know how this is important for us.

Best,

Marco

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