20.07.2005 17:51
schwinger model
I have been fighting with now for some days, trying to understand what is going on in this model. So I will just try to write down, how I believe it works. This is just a kind of a very brief memo.
The massless one flavour model,
![\mathcal{L}[\bar{\psi}, \psi, A_{\mu}] = \bar{\psi}(x)
\left[i\gamma_{\mu}\partial^{\mu} - g_0\gamma_{\mu}A^{\mu}(x) - m_0\right]\psi(x)
- \frac{1}{4}F_{\mu \nu}F^{\mu \nu} \hspace{1cm} \mu, \nu = 0,1](/texer/images/?format=gif&fontsize=12pt&color=white&bgcolor=%23ce4521&tex=%0D%0A%5Cmathcal%7BL%7D%5B%5Cbar%7B%5Cpsi%7D%2C+%5Cpsi%2C+A_%7B%5Cmu%7D%5D++%3D+%5Cbar%7B%5Cpsi%7D%28x%29%0D%0A%5Cleft%5Bi%5Cgamma_%7B%5Cmu%7D%5Cpartial%5E%7B%5Cmu%7D+-+g_0%5Cgamma_%7B%5Cmu%7DA%5E%7B%5Cmu%7D%28x%29+-++m_0%5Cright%5D%5Cpsi%28x%29+%0D%0A-+%5Cfrac%7B1%7D%7B4%7DF_%7B%5Cmu+%5Cnu%7DF%5E%7B%5Cmu+%5Cnu%7D+%5Chspace%7B1cm%7D+%5Cmu%2C+%5Cnu+%3D+0%2C1%0D%0A)
i.e. at
has 
and a 
gauge symmetry.
Since the gauge field 
has only two degrees of freedom, it can be cancelled by a gauge trafo. Such that we arrive at the Lagrangian of a massless free fermion. But since the is anomolous, this is only true at tree level. At a one loop level we get the non trivial Jacobian which can be expressed by a massiv Boson field. So we have a massless meson, which can be expressed as a Boson field through the bosonization thingy and the massiv one from the anomaly.
At
the massless meson gets a mass through the explicit breaking of the chiral symmetry by the mass term.
For the two flavour model, the chiral anomaly appears only in the singlet, so the singlet remains massiv even in the chiral limit, where as the triplet becomes massless. It's in fact completely analogous to the
case.
I don't measure the singlet. In fact I never saw any signal in the singlet channel, that's why we gave it up. This anomaly term, at least as you have it in 4d, is not gauge invariant. Since all I do in the code is gauge invariant (by the
), doesn't it make sense that is isn't there? Shouldn't it then just be 0 in the gauge average. So it would then be necessary to fix the gauge to calculate it???? =%-$
Ick weess nich, this is for people who understand something of lattice gauge theories not for me.
