Holomorphic and Canonical Couplings and the NSVZ Relation

The holomorphic gauge coupling and the canonically normalized gauge coupling are different coordinates. In a holomorphic Wilsonian normalization the perturbative running is one-loop exact; anomalous Jacobians generated while canonically normalizing the vector and matter fields convert that simple flow into the NSVZ relation. The displayed rational beta function is exact only in a scheme that preserves this coupling relation.

Required background. R-symmetry, anomalies, and the holomorphic scale fixes $\tau$, $b_0$, and $\Lambda_h$. Beta functions and anomalous dimensions fixes the RG definitions. Gauge–matter actions and F- and D-potentials supplies the superspace kinetic terms.

Helpful background. Operator mixing and renormalization matrices explains why anomalous dimensions depend on a basis when fields mix.

Two gauge couplings

For a simple gauge group $G$, let

$$\tau_h=\frac{\theta}{2\pi}+\frac{4\pi i}{g_h^2}$$

multiply the chiral gauge kinetic term in a holomorphic Wilsonian action. Matter kinetic terms have real coefficients $Z_i$,

$$\int d^4\theta\;Z_i(\mu)\, \Phi_{h,i}^\dagger e^V\Phi_{h,i}.$$

Canonical matter fields are $\Phi_{c,i}=Z_i^{1/2}\Phi_{h,i}$. Canonically normalizing the vector multiplet similarly defines $g_c$ through the interaction vertices. These changes of variables are not holomorphic functions of the chiral coupling: $Z_i$ and $g_c$ are real. Their regulated functional Jacobians shift the coefficient of $W^{a\alpha}W^a_\alpha$.

We choose

$$\gamma_i=-\frac{d\ln Z_i}{d\ln\mu}, \qquad \beta_c(g_c)=\frac{dg_c}{d\ln\mu}.$$

If fields mix, $Z$ and $\gamma=-Z^{-1}dZ/d\ln\mu$ are matrices, and the matter contribution below is the appropriately traced representation-weighted matrix expression. The scalar formula assumes a basis in which the relevant $Z_i$ are diagonal.

The anomalous rescaling relation

In an NSVZ normalization, the real parts of the holomorphic and canonical couplings obey

$$\frac{8\pi^2}{g_h^2} =\frac{8\pi^2}{g_c^2} +T(G)\ln g_c^2 +\sum_iT(R_i)\ln Z_i +C,$$

where $C$ is a scale-independent convention constant. The $T(G)\ln g_c^2$ term comes from canonically normalizing the vector multiplet; the matter terms are Konishi rescaling anomalies. If instead $\gamma_i'=d\ln Z_i/d\ln\mu=-\gamma_i$ while $Z_i$ retains the same definition, the sign of the $\ln Z_i$ term does not change; the NSVZ numerator is written $3T(G)-\sum_iT(R_i)(1+\gamma_i')$. A source displays the opposite $\ln Z_i$ sign only when it defines the wavefunction factor inversely. The invariant check is the beta function derived from all paired definitions, not one isolated sign.

This relation is not a classical field redefinition. Its logarithms are the finite remnants of regulated Jacobians. Arkani-Hamed and Murayama give a Wilsonian cutoff derivation in which holomorphy and these Jacobians are manifest Arkani-Hamed and Murayama 2000, §§ 2–4.

The holomorphic coupling satisfies

$$\frac{d}{d\ln\mu}\left(\frac{8\pi^2}{g_h^2}\right) =b_0, \qquad b_0=3T(G)-\sum_iT(R_i).$$

Differentiate the anomalous rescaling relation:

$$\begin{aligned} b_0 &=-\frac{16\pi^2}{g_c^3}\beta_c +\frac{2T(G)}{g_c}\beta_c -\sum_iT(R_i)\gamma_i\\ &=-\frac{16\pi^2}{g_c^3} \left(1-\frac{T(G)g_c^2}{8\pi^2}\right)\beta_c -\sum_iT(R_i)\gamma_i. \end{aligned}$$

Solving gives the NSVZ relation

$$\beta_c(g_c)= -\frac{g_c^3}{16\pi^2} \frac{3T(G)-\sum_iT(R_i)\bigl(1-\gamma_i(g_c,\lambda)\bigr)} {1-T(G)g_c^2/(8\pi^2)}.$$

The numerator contains the matter anomalous dimensions, which also depend on superpotential couplings $\lambda$. The pure-gauge instanton-based origin of this exact-beta-function program was developed by Novikov, Shifman, Vainshtein, and Zakharov Novikov et al. 1983, pp. 381–393. The matter-dependent relation displayed here follows from the rescaling-anomaly derivation cited above.

What “exact” means here

The equation is exact as a relation among RG functions in an NSVZ scheme. A finite redefinition

$$g_c' = g_c+a g_c^3+O(g_c^5), \qquad \Phi_i'=F_i(g_c,\lambda)\Phi_i,$$

changes $\beta$, $\gamma_i$, and generally the visible rational form beyond the universal low-loop data. Dimensional reduction with minimal subtraction does not automatically coincide with the NSVZ scheme at every order; finite redefinitions can relate schemes order by order Jack, Jones, and North 1997, pp. 479–499.

Accordingly:

• the one-loop coefficient $b_0$ is universal in ordinary mass-independent schemes;
• the holomorphic one-loop flow refers to $g_h$, not the physical canonical coupling;
• the denominator’s pole is a coordinate feature of this NSVZ coupling and is not, by itself, evidence for a physical singularity; and
• a zero of the numerator is a candidate fixed point only if the chosen coupling coordinates are regular and all other beta functions vanish.

At a superconformal fixed point with finite denominator,

$$3T(G)-\sum_iT(R_i)(1-\gamma_i^*)=0.$$

This condition is invariantly related to the anomaly-free superconformal R-symmetry, but solving for $\gamma_i^*$ still requires dynamics and the superpotential constraints.

Thresholds and the meaning of a mass

The same distinction is essential at a supersymmetric threshold. In a convention with unit finite holomorphic matching constant, integrating out matter with chiral mass source $m_h$ gives

$$\Lambda_{h,L}^{b_L} =m_h^{b_L-b_H}\Lambda_{h,H}^{b_H}.$$

This is a holomorphic equation: the phase of $m_h$ is matched with the theta angle. If a term $m_h\Phi_i\Phi_j$ is written using holomorphic fields, then the corresponding canonically normalized running mass is

$$m_c(\mu)=\frac{m_h}{[Z_i(\mu)Z_j(\mu)]^{1/2}}.$$

The physical decoupling threshold is controlled by a pole or other specified 1PI mass and may include further finite matching corrections. Substituting that real mass directly for $m_h$ in the holomorphic scale relation mixes schemes. The detailed branch-sensitive recursion is derived on Holomorphic decoupling and scale matching; the SQCD normalization is reviewed in Intriligator and Seiberg 1996, § 4.1, pp. 12–15.

SQCD translation

For $SU(N_c)$ SQCD with $N_f$ pairs and equal anomalous dimensions $\gamma_Q=\gamma_{\widetilde Q}=\gamma$, the representation sum is $N_f$. Hence

$$\beta_c(g_c)= -\frac{g_c^3}{16\pi^2} \frac{3N_c-N_f(1-\gamma)} {1-N_cg_c^2/(8\pi^2)}.$$

If the theory reaches an interacting fixed point in a regime where this description is valid, the numerator condition gives

$$\gamma^*=1-\frac{3N_c}{N_f}.$$

This is consistent with $\Delta(Q)=1+\gamma^*/2$ and the superconformal relation $\Delta=3R/2$ for $R(Q)=1-N_c/N_f$. The agreement checks the signs and group factors; it does not prove that the fixed point exists for every $N_f$.

The holomorphic scale remains

$$\Lambda_h^{3N_c-N_f} =\mu^{3N_c-N_f}e^{2\pi i\tau_h(\mu)}.$$

Trying to replace $g_h$ by $g_c$ in this formula without the $g_c$ and $Z_i$ Jacobian factors destroys RG invariance. Exact superpotentials are naturally written using the holomorphic scale; observable thresholds are obtained only after canonical normalization and the relevant 1PI pole condition.

Common pitfalls

Calling either coordinate an observable by itself. The holomorphic coordinate makes supersymmetry transparent, while $g_c$ canonically normalizes vertices. A scattering amplitude also contains momentum-dependent 1PI corrections and a declared subtraction prescription.

Quoting NSVZ without a gamma convention. The sign of $\gamma$ must be paired with the definition of $Z$. Reversing only the definition of $\gamma$ changes the sign multiplying $\gamma$ in the numerator, not the rescaling relation itself.

Treating the denominator pole as a phase transition. A finite coupling redefinition moves or removes such a coordinate pole. Only scheme-invariant observables can diagnose a physical singularity.

Exercises

Starting from the anomalous rescaling relation, reproduce the NSVZ beta function and identify the origin of its numerator and denominator.

Solution

Differentiate with respect to $\ln\mu$. The one-loop holomorphic derivative supplies $b_0$. The derivatives of $\ln Z_i$ give $-\gamma_i$, producing $b_0+\sum_iT(R_i)\gamma_i$ in the numerator. The derivative of $T(G)\ln g_c^2$ combines with that of $8\pi^2/g_c^2$ to give $1-T(G)g_c^2/(8\pi^2)$ in the denominator.

Show that the SQCD fixed-point value of $\gamma$ agrees with the anomaly-free R-charge.

Solution

The numerator condition gives $\gamma^*=1-3N_c/N_f$. Therefore

$$\Delta(Q)=1+\frac12\gamma^* =\frac32\left(1-\frac{N_c}{N_f}\right) =\frac32R(Q).$$

This is the required chiral-primary relation.

References

• Nima Arkani-Hamed and Hitoshi Murayama, “Holomorphy, Rescaling Anomalies and Exact Beta Functions in Supersymmetric Gauge Theories,” Journal of High Energy Physics 2000(06), 030, arXiv, DOI.
• Kenneth A. Intriligator and Nathan Seiberg, “Lectures on Supersymmetric Gauge Theories and Electric–Magnetic Duality,” Nuclear Physics B Proceedings Supplements 45BC (1996), 1–28, § 4.1, arXiv, DOI.
• I. Jack, D. R. T. Jones, and C. G. North, “Scheme Dependence and the NSVZ Beta Function,” Nuclear Physics B 486 (1997), 479–499, arXiv, DOI.
• V. A. Novikov, M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, “Exact Gell-Mann–Low Function of Supersymmetric Yang–Mills Theories from Instanton Calculus,” Nuclear Physics B 229 (1983), 381–393, DOI.