10.2 Argyres-Douglas CFT from the Nf = 2 theory

Consider the curve of the Nf = 2 theory,

2Λ(x μ) z + 2Λ(x μ)z = x2 u (10.2.1)

where we set the masses of the two flavors the same. We can also use the curve of the alternative form

(x μ)2 z + 4Λ2z = x2 u. (10.2.2)

Its moduli space for generic μ was shown in Fig. 8.9.

When μ = 0, two singularities without the Higgs branch attached collide. Instead, let us tune the parameter μ so that the singularity with the Higgs branch collides with a singularity without, see Fig. 10.3. For definiteness let us use the first form of the curve. Then this collision happens when μ = 2Λ, at u = 4Λ2. The four branch points then collide at z = 1.

Figure 10.3: Argyres-Douglas point of Nf = 2 theory

We find that the monodromy around the resulting singularity is

MAD2 0 11 0 , (10.2.3)

acting on the coupling as

τ τ = 1 τ. (10.2.4)

The strong coupling value τ = i is the fixed point of this transformation, and the low-energy coupling approaches this value when we let u 4Λ2.

Expanding the variables as before,

z = 1 + δz,x = 2Λ + δx,u = 4Λ2 + δu,μ = 2Λ + δμ, (10.2.5)

we find that the curve in the limit is

(δx)2 + δu = (δz)4 + δμδz2 + Δμδz (10.2.6)

with the differential λ δxdδz. Here we reinstated a small difference Δμ = μ1 μ2 between the bare masses μ1, μ2 of two doublet hypermultiplets. Demanding λ to have scaling dimension 1, we see that

[δx] = 2 3,[δz] = 1 3. (10.2.7)

Then we find

[δu] = 4 3,[δμ] = 2 3,[Δμ] = 1. (10.2.8)

We see again that [δu] + [δμ] = 2, and therefore δμ is a deformation parameter corresponding to the operator δu. Δμ is a mass parameter for the non-Abelian flavor symmetry SU(2)F . In general, in a conformal theory, a non-Abelian flavor symmetry current Ja should have scaling dimension 3. The 𝒩=2 supersymmetry relates it to the mass term, which is given for a Lagrangian theory by the familiar term QQ̃ and has scaling dimension 2. Therefore, the non-Abelian mass parameter of 𝒩=2 superconformal theory should always have scaling dimension 1. Our computation of [Δμ] is consistent with this general argument. We call this resulting theory ADNf=2(SU(2)).

Figure 10.4: Argyres-Douglas theory ADNf=2(SU(2))

Let us study the limiting procedure of the Nf = 2 theory from the 6d point of view. Before taking the limit, the curve in the first form (10.2.1) was λ2 = ϕ(z) with two order-4 poles of ϕ(z). We collide them, and we end up with a singularity of order 8. Just as in the analysis before, we conclude that the curve in the limit is given by

λ2 = 1 + μz2 + Δμz3 + uz4 z8 dz2. (10.2.9)

We easily see that [z] = 13. Then we find the same scaling dimensions as in (10.2.8).

The curve in the second form (10.2.2), when written as λ2 = ϕ(z), had two poles of order 2, and another of order 3. At the two order-two poles, the residues of xdzz are ± (μ1 + μ2)2 and ± (μ1 μ2)2, respectively. Let us collide an order-2 pole with the residue ± (μ1 + μ2)2 and an order-3 pole to form a pole of order 5. We end up having a ϕ(z) with one pole of order 5, say at z = 0, and another pole of order 2, with the residue ± (μ1 μ2)2, see Fig. 10.4. The curve in the limit can also be easily found:

λ2 = 1 + δμz + δuz2 + (μ1μ2 2 )2z3 z5 dz2 (10.2.10)

The last coefficient was fixed by the condition at z = . Demanding λ to have scaling dimension 1, we see that [z] = 23, and

[δμ] = 2 3,[δu] = 4 3. (10.2.11)

It is reassuring to find the same answer.