3.1 Renormalization

Recall the one-loop renormalization of the gauge coupling in a general Lagrangian field theory:

E d dEg = g3 (4π)2 11 3 C(adj) 2 3C(Rf) 1 3C(Rs) . (3.1.1)

Here, E is the energy scale at which g is measured, and we use the convention that all fermions are written in terms of left-handed Weyl fermions. Then Rf and Rs are the representations of the gauge group to which the Weyl fermions and the complex scalars belong, respectively. The quantity C(ρ) is defined so that

trρ(Ta)ρ(Tb) = C(ρ)δab (3.1.2)

where Ta are the generators of the gauge algebra and ρ(Ta) is the matrix in the representation ρ, normalized so that C(adj) is equal to the dual Coxeter number. For SU(N), we have

C(adj) = N,C(fund) = 1 2. (3.1.3)

In an 𝒩=1 gauge theory, the equation simplifies to

E d dEg = g3 (4π)2 3C(adj) C(R) (3.1.4)

or equivalently

E d dE 8π2 g2 = 3C(adj) C(R), (3.1.5)

where R is the representation of the chiral multiplet. In an 𝒩=2 gauge theory, one adjoint chiral multiplet Φ is considered to be a part of the vector multiplet. Then we have

E d dE 8π2 g2 = 2C(adj) C(R), (3.1.6)

where R is now the representation of the 𝒩=1 chiral multiplets describing the hypermultiplets of the system. If one has one adjoint hypermultiplet, consisting of two 𝒩=1 chiral multiplets A and B, we have zero one-loop beta function. When the mass terms for A, B are zero, the system in fact has a further enlarged supersymmetry, and is the 𝒩=4 super Yang-Mills. When the mass term is nonzero, it is called the 𝒩=2 theory.

In a supersymmetric theory, the coupling g is combined with the theta angle 𝜃 and enters in the Lagrangian as

d2𝜃 i 8π τtrWαWα + cc. (3.1.7)

where τ is given by

τ = 4πi g2 + 𝜃 2π. (3.1.8)

We call this τ the complexified gauge coupling.

We can consider τ to be an expectation value of a background chiral superfield. There is a renormalization scheme where the superpotential remains a holomorphic function of the chiral superfields, including background fields whose vevs are the gauge and superpotential couplings [36]. We call it Seiberg’s holomorphy principle.

In this scheme, the one-loop running coupling at the energy scale E can be expressed as

τ(E) = τUV b 2πi log E ΛUV + (3.1.9)

where b is the rational number appearing on the right hand side of (3.1.5) or (3.1.6). Note that the coupling τ starts from 1g2, and therefore the n loop diagram would have the dependence g2(n 1). The constant shift as in the imaginary part in (3.1.9) is then a one-loop effect.

Perturbation theory is independent of the 𝜃 angle, since FμνF̃μν is a total derivative, although of a gauge-dependent quantity. Therefore the n loop effect is a function of (Imτ)1 n, which is not holomorphic unless n = 1. We conclude that the running (3.1.9) is one-loop exact in the holomorphic scheme. We find that the combination

Λb = Ebe2πiτ(E) (3.1.10)

is invariant to all orders in perturbation theory. We call this Λ the complexified dynamical scale of the theory.5 Note that Λ is a complex quantity, and can be considered as a vev of a background chiral superfield.

This one-loop exactness does not necessarily mean that the physical gauge coupling, which controls the scattering process for example, is one-loop exact. In the holomorphic scheme in generic 𝒩=1 supersymmetric theories, we have nontrivial wave-function renormalization factors Zij

d4𝜃Zīj(E)QīeV Qj (3.1.11)

which need to be taken into account by a further field redefinition to compute physical scattering amplitudes. This is known to produce further perturbative contributions to the physical running of the gauge coupling. For more on this point, see e.g. [37].

For 𝒩=2 supersymmetric theories, however, one can make a stronger statement. We assume that there is a holomorphic renormalization scheme which is compatible with the existence of SU(2)R symmetry. Then, the structure of the Lagrangian is restricted to be of the form (2.1.11) for the vector multiplets and of the form (2.1.14) for the hypermultiplets. We consider τ as the vev of a background field. Then, on the vector multiplet side, one finds that we cannot have nontrivial wavefunction renormalization factors Zīj as in (3.1.11) in the vector multiplet Lagrangian (2.1.11). On the hypermultiplet side, the coefficient c in (2.1.14) is not renormalized in the holomorphic scheme. Since c = c, the Kähler potential is not renormalized. Therefore, there is no renormalization in the hypermultiplet Lagrangian (2.1.14).

Then, in particular when b = 0, the beta function is zero to all orders in perturbation theory. This makes the system conformal, and the value of τ becomes an exactly marginal coupling parameter. The non-perturbative corrections will induce finite renormalization, but are not thought to introduce any additional infinite renormalization.

For example, the 𝒩=4 super Yang-Mills is automatically superconformal, with one exactly marginal coupling. Another example with b = 0 is 𝒩=2 supersymmetric SU(N) gauge theory with 2N hypermultiplets in the fundamental representation. Indeed, in (3.1.6), we have C(adj) = N and C(R) = 2 2N 12.