This question is related with Seiberg duality for $SU(N)$ gauge theory which states a duality between electric theory, $SU(N_c)$ gauge theory with $N_f$ flavors is dual to its magnetic theory, $SU(N_f-N_c)$ theory with $N_f$ flavors with an additional gauge invaraint massless field.

This duality valid for conformal window $\frac{3N_c}{2} < N_f < 3N_c$ where both electric and magnetic theory have interacting IR-fixed point.

Here i have question about $IR$ fixed points.

The beta function of electric theory(SQCD : $SU(N_c)$ gauge theory) is written as \begin{align} \beta(g) = -\frac{g^3}{16\pi^2} \frac{3N_c - N_f + N_f \gamma(g^2)}{1- N_c \frac{g^2}{8\pi^2}} \end{align} Thus we see $\beta>0$ for $N_f > 3 N_c$ for electric theory. And using above results for magnetic theory, $\beta>0$ for $\frac{3N_c}{2} > N_f$.

  1. I want to know how we choose proper conformal window, and how we know that these window have interacting IR-fixed point rather than UV fixed point.

I know $\beta<0$ theory is asymptotically free theory which is a key properties of describing non-abelain gauge theories. And UV is related with for High energy, and IR is related with for Low energy dynamics.

  1. (These might be part of question 1) And can anyone explain the difference of UV fixed point and IR fixed point?

I'll answer (2) and then (1).

2) In the renormalization group, physics is a function of energy scale. Typically one starts at a UV fixed point, at extremely high energies where lowering the energy scale a little bit results in extremely small changes in the physics. At the end of the day, at very low energies, one hopefully reaches an IR fixed point, where the physics again changes very slowly with energy scale. Such scale invariance combined with the Lorentz group typically produces the conformal group. Hence, the fixed points are described by conformal field theories.

There are different kinds of fixed points. Deep in the IR, one may find nothing at all -- a trivial theory. There is a mass gap, and there is simply not enough energy to excite any degrees of freedom. Gauge theories with a mass gap in the deep IR we usually call confining. Or the theory may become free; the massless particles stop interacting with each other. QCD is asymptotically free, and so at extremely high energies, it has a free UV fixed point. Or, the most interesting case, one may find an interacting fixed point that is scale invariant. Such a fixed point is believed to exist in the IR in the conformal window you describe above.

1) If the theory is asymptotically free, that means the coupling gets weak at high energies, in the ultraviolet (UV), but it also means that the coupling gets strong at low energies, in the infrared (IR). You have pretty much answered your own question (1) above, by showing that the theory flows to strong coupling in the IR in the window ${3N_c \over 2} < N_f < 3N_c$ and correspondingly weak coupling in the UV, in both the magnetic and electric descriptions of the theory. There is much more to discuss here. The lecture notes seem to be a standard reference.


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