8.1 Generalities

In this section and next, we consider SU(2) gauge theory with Nf flavors, with Nf = 2, 3, 4. In terms of 𝒩=1 chiral multiplets, we have (Qi,Q̃i) for i = 1,,Nf with the superpotential

i QiΦQ̃i + μiQiQ̃i (8.1.1)

where μi are bare mass terms. With all μi are the same, there is a U(Nf) symmetry acting on the indices i of Qi and Q̃i. On the Coulomb branch with Φ = diag(a,a), the physical masses of the hypermultiplets are given by

|± a ± μi|. (8.1.2)

With μi = 0, we can combine Qi and Q̃i into

(qIa)I=1,2,,2Nf = (Q1a,,QNfa,𝜖abQ̃b1,,𝜖abQ̃bNf) (8.1.3)

with SO(2Nf) symmetry. In this notation the superpotential is

ηIJqIaΦabqJb,whereη = 0 1Nf ̲ ̲ 1Nf 0 . (8.1.4)

Since ηIJ is a symmetric matrix, the flavor symmetry acting on the indices I,J is SO(2Nf). Equivalently, we have 2Nf half-hypermultiplets in the doublet representation of SU(2).

Classically, introducing an odd number of half-hypermultiplets in the doublet of SU(2) is all right, with SO(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 SU(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||μi| is

τ(a) = 2τUV 2(4 Nf) 2πi log a ΛUV + (8.1.5)

which can further be rewritten as, when Nf4,

= 2(4 Nf) 2πi log a ΛwhereΛ4 Nf = ΛUV 4 Nfe2πiτUV . (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 Nf = 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 Nf = 2 and Nf = 3. The case Nf = 4 opens up a whole new field, to which Sec. 9 is dedicated.