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

$$E\frac{d}{dE}g=-\frac{{g}^{3}}{{\left(4\pi \right)}^{2}}\left[\frac{11}{3}C\left(\text{adj}\right)-\frac{2}{3}C\left({R}_{f}\right)-\frac{1}{3}C\left({R}_{s}\right)\right].$$ | (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 ${R}_{f}$ and ${R}_{s}$ are the representations of the gauge group to which the Weyl fermions and the complex scalars belong, respectively. The quantity $C\left(\rho \right)$ is deﬁned so that

$$tr\phantom{\rule{0.3em}{0ex}}\rho \left({T}^{a}\right)\rho \left({T}^{b}\right)=C\left(\rho \right){\delta}^{ab}$$ | (3.1.2) |

where ${T}^{a}$ are the generators of the gauge algebra and $\rho \left({T}^{a}\right)$ is the matrix in the representation $\rho $, normalized so that $C\left(\text{adj}\right)$ is equal to the dual Coxeter number. For $SU\left(N\right)$, we have

$$C\left(\text{adj}\right)=N,\phantom{\rule{2em}{0ex}}C\left(\text{fund}\right)=\frac{1}{2}.$$ | (3.1.3) |

In an $\mathcal{\mathcal{N}}=1$ gauge theory, the equation simpliﬁes to

$$E\frac{d}{dE}g=-\frac{{g}^{3}}{{\left(4\pi \right)}^{2}}\left[3C\left(\text{adj}\right)-C\left(R\right)\right]$$ | (3.1.4) |

or equivalently

$$E\frac{d}{dE}\frac{8{\pi}^{2}}{{g}^{2}}=3C\left(\text{adj}\right)-C\left(R\right),$$ | (3.1.5) |

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

$$E\frac{d}{dE}\frac{8{\pi}^{2}}{{g}^{2}}=2C\left(\text{adj}\right)-C\left(R\right),$$ | (3.1.6) |

where $R$ is now the representation of the $\mathcal{\mathcal{N}}=1$ chiral multiplets describing the hypermultiplets of the system. If one has one adjoint hypermultiplet, consisting of two $\mathcal{\mathcal{N}}=1$ 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 $\mathcal{\mathcal{N}}=4$ super Yang-Mills. When the mass term is nonzero, it is called the $\mathcal{\mathcal{N}}={2}^{\ast}$ theory.

In a supersymmetric theory, the coupling $g$ is combined with the theta angle $\mathit{\theta}$ and enters in the Lagrangian as

$$\phantom{\rule{1em}{0ex}}\int {d}^{2}\mathit{\theta}\frac{-i}{8\pi}\tau tr\phantom{\rule{0.3em}{0ex}}{W}_{\alpha}{W}^{\alpha}+cc.$$ | (3.1.7) |

where $\tau $ is given by

$$\tau =\frac{4\pi i}{{g}^{2}}+\frac{\mathit{\theta}}{2\pi}.$$ | (3.1.8) |

We call this $\tau $ the complexiﬁed gauge coupling.

We can consider $\tau $ 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

$$\tau \left(E\right)={\tau}_{UV}-\frac{b}{2\pi i}log\frac{E}{{\Lambda}_{UV}}+\cdots $$ | (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 $\tau $ starts from $1\u2215{g}^{2}$, and therefore the $n$ loop diagram would have the dependence ${g}^{2\left(n-1\right)}$. The constant shift as in the imaginary part in (3.1.9) is then a one-loop eﬀect.

Perturbation theory is independent of the $\mathit{\theta}$ angle, since ${F}_{\mu \nu}{\stackrel{\u0303}{F}}_{\mu \nu}$ is a total derivative, although of a gauge-dependent quantity. Therefore the $n$ loop eﬀect is a function of ${\left(Im\tau \right)}^{1-n}$, 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

$${\Lambda}^{b}={E}^{b}{e}^{2\pi i\tau \left(E\right)}$$ | (3.1.10) |

is invariant to all orders in perturbation theory. We call this
$\Lambda $ the complexiﬁed dynamical
scale of the theory.^{5}
Note that $\Lambda $
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 $\mathcal{\mathcal{N}}=1$ supersymmetric theories, we have nontrivial wave-function renormalization factors ${Z}_{ij}$

$$\int {d}^{4}\mathit{\theta}{Z}^{\u012bj}\left(E\right){Q}_{\u012b}^{\u2020}{e}^{V}{Q}_{j}$$ | (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 $\mathcal{\mathcal{N}}=2$ 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 $SU{\left(2\right)}_{R}$ 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 $\tau $ as the vev of a background ﬁeld. Then, on the vector multiplet side, one ﬁnds that we cannot have nontrivial wavefunction renormalization factors ${Z}_{\u012bj}$ as in (3.1.11) in the vector multiplet Lagrangian (2.1.11). On the hypermultiplet side, the coeﬃcient ${c}^{\prime}$ in (2.1.14) is not renormalized in the holomorphic scheme. Since $c={c}^{\prime}$, 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 $\tau $ 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 $\mathcal{\mathcal{N}}=4$ super Yang-Mills is automatically superconformal, with one exactly marginal coupling. Another example with $b=0$ is $\mathcal{\mathcal{N}}=2$ supersymmetric $SU\left(N\right)$ gauge theory with $2N$ hypermultiplets in the fundamental representation. Indeed, in (3.1.6), we have $C\left(\text{adj}\right)=N$ and $C\left(R\right)=2\cdot 2N\cdot 1\u22152$.