Perturbative and non-perturbative infrared behavior of ...

8.3. SQCD with singlets: SSQCD
drives the RG flow away from the original fixed point to a new family of conformal field theories
in the infrared.
The global symmetry of the theory is SU(Nf) × SU(Nf) × SU(N ′ f ) × SU(N ′ f ) × U(1)B ×
U(1)B ′ ×U(1)F ×U(1)R enhanced to SU(Nf +N ′ f )×SU(Nf +N ′ f )×U(1)B ×U(1)R when h = 0.
The magnetic dual of the theory can be obtained by deforming the Seiberg dual of SQCD with a
mass term which pairs up the gauge singlets S with the N ′2
f mesons M ′ ∼ Q ′ ˜ Q ′ and integrating
them out. We are left with an SU(N = Nf + N ′ f − Nc) gauge theory with the magnetic quarks
q ′ i , ˜q′ i , qi ′ and ˜qi ′ and N2 f gauge singlets Mij and 2NfN ′ f singlets Kij ′ and Lij ′ with superpotential
W = Mq ′ ˜q ′ + Kq ′ ˜q + L˜q ′ q (8.3.57)
The first term is similar to the superpotential of the electric theory. The additional terms distinguish
the magnetic duals from the original electric theory.
The duality maps mesons and singlet fields as
Q ˜ Q → M Q ˜ Q ′ → K Q ′ ˜ Q → L (8.3.58)
As a consequence, at the infrared fixed point the superconformal R-charges of the electric and
magnetic descriptions should satisfy
2R(Q) = R(M) R(S) = 2R(q) R(Q) + R(Q ′ ) = R(P) (8.3.59)
together with the constraints from anomaly freedom
Nc + Nf(R(Q) − 1) + N ′ f (R(Q′ ) − 1) = 0 R(S) + 2R(Q ′ ) = 2 (8.3.60)
which follow from the vanishing of the NSVZ β-function and the condition that the superpotential
(8.3.56) is exactly marginal. Thus, we also have
R(q ′ ) = 1 − R(Q) R(q) = 1 − R(Q ′ ) (8.3.61)
Note that in the above formulas we made use of R(Q) = R( ˜ Q) and R(Q ′ ) = R( ˜ Q ′ ) and similarly
for the magnetic quarks, because of the global symmetries. The U(1)F factor can mix with the
canonical U(1)R R-symmetry at the superconformal fixed point and thus its presence allows R(Q)
and R(Q ′ ) to differ.
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