I decide to begin writing some back-of-the-envelope stuff in this blog; mostly very basic things.
Consider Dirac fermion coupled to U(1) field.
\mathcal{L}=\bar{\psi}\gamma^\mu (\partial_\mu-iA_\mu) \psi - \frac{1}{e^2} F_{\mu\nu}F^{\mu\nu}.
In d spacetime dimensions, the first part in the kinetic term implies [\partial]+2[\psi]=d, leading to [\psi]=(d-1)/2. The second part leads to 2[\psi]+[A]=d. Plugging in the value for [\psi], we have [A]=1, then [F]=2.
Using this and plug back in the Maxwell term 2[F]-[e^2]=d, we arrive at [e^2]=4-d.
When we are in 3+1d, the gauge coupling is thus dimensionless. For 2+1d, the gauge coupling has dimension [e^2]=1, so this theory is strongly coupled in the infrared. Also, the potential energy between two charges increases logarithmically with respect to the distance between them, which is a very mild form of confinement.
References:
[1] David Tong, Lectures on gauge theory, chapter 8. http://www.damtp.cam.ac.uk/user/tong/gaugetheory.html
[2] Chat with Hao-Yu Sun.
Comments
Post a Comment