Generalities

8.1 Generalities

In this section and next, we consider $S U (2)$ gauge theory with $N_{f}$ ﬂavors, with $N_{f} = 2, 3, 4$ . In terms of $𝒩 = 1$ chiral multiplets, we have $(Q_{i}, {\tilde{Q}}^{i})$ for $i = 1, \dots, N_{f}$ with the superpotential

\sum_{i} (Q_{i} Φ {\tilde{Q}}^{i} + μ_{i} Q_{i} {\tilde{Q}}^{i})

(8.1.1)

where $μ_{i}$ are bare mass terms. With all $μ_{i}$ are the same, there is a $U (N_{f})$ symmetry acting on the indices $i$ of $Q_{i}$ and ${\tilde{Q}}^{i}$ . On the Coulomb branch with $Φ = d i a g (a, - a)$ , the physical masses of the hypermultiplets are given by

| \pm a \pm μ_{i} | .

(8.1.2)

With $μ_{i} = 0$ , we can combine $Q_{i}$ and ${\tilde{Q}}^{i}$ into

{(q_{I}^{a})}_{I = 1, 2, \dots, 2 N_{f}} = (Q_{1}^{a}, \dots, Q_{N_{f}}^{a}, 𝜖^{a b} {\tilde{Q}}_{b}^{1}, \dots, 𝜖^{a b} {\tilde{Q}}_{b}^{N_{f}})

(8.1.3)

with $S O (2 N_{f})$ symmetry. In this notation the superpotential is

\propto η^{I J} q_{I}^{a} Φ_{a b} q_{J}^{b}, where η = (\begin{matrix} 0 & 1_{N_{f}} \\ ̲ & ̲ \\ 1_{N_{f}} & 0 \end{matrix}) .

(8.1.4)

Since $η^{I J}$ is a symmetric matrix, the ﬂavor symmetry acting on the indices $I,$ J is $S O (2 N_{f})$ . Equivalently, we have $2 N_{f}$ half-hypermultiplets in the doublet representation of $S U (2)$ .

Classically, introducing an odd number of half-hypermultiplets in the doublet of $S U (2)$ is all right, with $S O (odd)$ ﬂavor symmetry. However, such a theory would have odd number of Weyl fermions in the doublet, and is plagued quantum mechanically by Witten’s global anomaly, as reviewed in Sec. 3.2.1. Therefore, for $S U (2)$ gauge group, we can only consider an even number of half-hypermultiplets in the doublet, or equivalently, an integral number of full-hypermultiplets in the doublet.

The one-loop running of this theory in the ultraviolet region $| a | ≫ | μ_{i} |$ is

τ (a) = 2 τ_{U V} - \frac{2 (4 - N_{f})}{2 π i} log \frac{a}{Λ_{U V}} + \dots

(8.1.5)

which can further be rewritten as, when $N_{f} \neq 4$ ,

= - \frac{2 (4 - N_{f})}{2 π i} log \frac{a}{Λ} where Λ^{4 - N_{f}} = Λ_{U V}^{4 - N_{f}} e^{2 π i τ_{U V}} .

(8.1.6)

We guessed the form of the curves of these theories in Sec. 6.4.4. The results were given in (6.4.6), (6.4.11), (6.4.12) for $N_{f} = 2, 3, 4$ respectively. The aim of this section and the next section is to perform various checks that they do reproduce expected properties, and to study strong coupling dynamics using them. In this section we deal with $N_{f} = 2$ and $N_{f} = 3$ . The case $N_{f} = 4$ opens up a whole new ﬁeld, to which Sec. 9 is dedicated.

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