R-Symmetry, Anomalies, and the Holomorphic Scale

An anomalous chiral or R-symmetry remains an exact bookkeeping tool when the complexified gauge coupling is transformed with the regulated measure. The resulting charge of the renormalization-group invariant combination $\Lambda_h^{b_0}$ is what allows holomorphy to constrain strong-coupling terms. This page fixes the sign, index, global-form, theta-periodicity, and branch conventions explicitly and applies them to four-dimensional $\mathcal N=1$ SQCD.

Required background. Holomorphic couplings and background superfields supplies spurionic covariance. Regulated Jacobians and measure variation supplies the Fujikawa logic behind the anomaly coefficient.

Helpful background. Theta terms, periodicity, and vacuum sectors explains why the phase of the complex coupling is periodic and why roots of the holomorphic scale require branch data.

Mixed gauge–R anomalies

For the displayed formulas, let $G$ be compact, connected, simple, and simply connected, take finite-action fields on Euclidean $\mathbb R^4$ with the usual compactification to $S^4$, and assume that perturbative gauge anomalies cancel. Normalize the generators by

$$\operatorname{tr}_{R}(T^aT^b)=T(R)\delta^{ab}.$$

With this global form and normalization, the instanton number is $k\in\mathbb Z$ and $\theta\sim\theta+2\pi$. For a left-handed Weyl fermion of $U(1)_X$ charge $q$ in representation $R$, the mixed $G^2U(1)_X$ coefficient is $qT(R)$. In an $\mathcal N=1$ theory, a chiral superfield $\Phi_i$ of R-charge $r_i$ contains a fermion of charge $r_i-1$, while the gaugino has R-charge $1$. Therefore

$$\mathcal A_{G^2R} =T(G)+\sum_iT(R_i)(r_i-1).$$

This coefficient is local and perturbatively one-loop exact in the anomaly equation, although its interpretation inside a full supercurrent multiplet can depend on operator definitions. The only claim needed here is the regulated Jacobian of the chiral change of variables.

The supersymmetric anomaly and instanton-measure normalizations used here are reviewed in Shifman 2022, §§ 10.7 and 10.16.

We use the following sign convention. In an instanton sector of charge $k$, the Euclidean weight contains $e^{i\theta k}$. Under

$$\psi_i\longmapsto e^{iq_i\alpha}\psi_i,$$

the fermion zero-mode measure transforms by

$$\mathcal D\psi\longmapsto e^{-2i\alpha k\sum_iq_iT(R_i)}\mathcal D\psi.$$

The family of theories is therefore covariant if the background angle transforms as

$$\theta\longmapsto\theta+2\mathcal A\alpha, \qquad \mathcal A=\sum_iq_iT(R_i),$$

including the adjoint gaugino when appropriate. Reversing the sign used for the topological term reverses both displayed signs and leaves invariant conclusions unchanged.

Changing only the global form does not create fractional sectors on the smooth $S^4$ domain declared above: $H^2(S^4,\Gamma)=0$, so a $G/\Gamma$ bundle there has no obstruction to lifting to $G$. Fractional instanton number can instead occur for a quotient group on a general four-manifold with a nonzero lifting obstruction, or on $\mathbb R^4$ after line defects or boundary data change the topology. The theta identification then also depends on the line-operator and discrete-theta choice. For example, on a spin four-manifold a $2\pi$ theta shift exchanges the two $SO(3)_\pm$ theories, while either fixed theory has a $4\pi$ period. The fixed-theta remnant must therefore be recomputed for the complete global theory rather than imported from its Lie algebra Aharony, Seiberg, and Tachikawa 2013, §§ 1.1–1.2, pp. 2–5.

The holomorphic scale as a charged source

Define

$$\tau(\mu)=\frac{\theta}{2\pi}+\frac{4\pi i}{g_h^2(\mu)}, \qquad b_0=3T(G)-\sum_iT(R_i),$$

and

$$\Lambda_h^{b_0}=\mu^{b_0}e^{2\pi i\tau(\mu)}.$$

Perturbative holomorphic running is

$$\mu\frac{d\tau}{d\mu}=\frac{i b_0}{2\pi},$$

so $\Lambda_h^{b_0}$ is independent of $\mu$. Under the anomalous transformation above,

$$\tau\longmapsto\tau+\frac{\mathcal A\alpha}{\pi}, \qquad \Lambda_h^{b_0}\longmapsto e^{2i\mathcal A\alpha}\Lambda_h^{b_0}.$$

Thus $\Lambda_h^{b_0}$ has spurionic $U(1)_X$ charge $2\mathcal A$. This is the invariant datum. Writing a charge for $\Lambda_h$ itself requires choosing a $b_0$th root and hence a branch.

A finite holomorphic redefinition $\tau\mapsto\tau+c$ rescales $\Lambda_h^{b_0}$ by $e^{2\pi ic}$. Consequently an overall coefficient written in terms of $\Lambda_h$ is meaningful only with the scale convention. Monodromy under $\theta\mapsto\theta+2\pi$ can permute the branches of condensates even though $\Lambda_h^{b_0}$ is single-valued.

The discrete remnant at fixed theta

The continuous spurionic transformation above moves through a family of theories by shifting $\theta$. If instead $\theta$ is held fixed, invariance of every integer-$k$ sector requires

$$2\mathcal A\alpha\in2\pi\mathbb Z.$$

This leaves a discrete subgroup whose faithful order depends on the charge normalization and on identifications with gauge transformations. For pure $SU(N_c)$ super-Yang–Mills, $\mathcal A=N_c$ for the gaugino rotation, so the classical $U(1)_R$ is reduced to $\mathbb Z_{2N_c}$. The gaugino bilinear has charge two; a nonzero condensate can further break this group to $\mathbb Z_2$ and produce $N_c$ branches. This last step is a dynamical statement, not a consequence of the Jacobian alone.

SQCD: two useful charge assignments

Consider $SU(N_c)$ with $N_f$ pairs

$$Q^i\in\mathbf{N_c}, \qquad \widetilde Q_i\in\overline{\mathbf{N_c}}, \qquad i=1,\ldots,N_f,$$

and no tree superpotential. Since $T(\mathbf{N_c})=1/2$,

$$b_0=3N_c-N_f.$$

First take the axial transformation $U(1)_A$ with $q_A(Q)=q_A(\widetilde Q)=1$ and neutral gaugino. Its mixed anomaly is

$$\mathcal A_{G^2A} =N_f\left(\frac12+\frac12\right)=N_f.$$

Therefore $\Lambda_h^{3N_c-N_f}$ has axial charge $2N_f$. The meson

$$M^i{}_j=Q^i\widetilde Q_j$$

has charge $2$, so $\det M$ also has charge $2N_f$. The ratio

$$\frac{\Lambda_h^{3N_c-N_f}}{\det M}$$

is spurionically axial invariant. The anomalous symmetry has not disappeared; its anomaly supplies exactly the transformation needed to form the invariant.

Now seek an ordinary anomaly-free R-symmetry with equal charges $r$ for $Q$ and $\widetilde Q$. The mixed coefficient is

$$\mathcal A_{G^2R} =N_c+N_f(r-1).$$

Setting it to zero gives

$$r=1-\frac{N_c}{N_f}.$$

Then $R(M)=2(1-N_c/N_f)$, while $\Lambda_h^{b_0}$ is neutral under this anomaly-free R-symmetry. These two charge assignments, together with dimension, severely restrict a possible nonperturbative superpotential for $N_f<N_c$.

From charges to an allowed exact term

Let

$$W=C\left(\frac{\Lambda_h^{3N_c-N_f}}{\det M}\right)^p.$$

The quantity in parentheses is axially invariant and has dimension

$$(3N_c-N_f)-2N_f=3(N_c-N_f).$$

Requiring $[W]=3$ gives $p=1/(N_c-N_f)$. The anomaly-free R-charge gives the same condition because

$$R(\det M)=2(N_f-N_c).$$

Thus

$$W=C\left(\frac{\Lambda_h^{3N_c-N_f}}{\det M}\right)^{1/(N_c-N_f)}.$$

The derivation has fixed the functional form on the patch $\det M\neq0$. It has not fixed $C$, selected one of the $N_c-N_f$ branches, proved that a superpotential is dynamically generated, or established validity at the singular locus. Those obligations are discharged using zero-mode analysis and holomorphic decoupling on later pages. The anomaly and charge argument in SQCD is reviewed with its dynamical completion in Intriligator and Seiberg 1996, § 4.1, Eqs. (4.1)–(4.6), pp. 12–14, while the logic of holomorphic background couplings is isolated in Seiberg 1993, pp. 469–475.

The chapter’s spurion-to-exactness flow shows where this charge calculation stops and where independent instanton and decoupling inputs begin. Its structured description preserves the charge table, branch assumptions, and failure exits without requiring the visual diagram.

What is and is not invariant

• $\Lambda_h^{b_0}$ is RG invariant in the stated holomorphic convention; $\Lambda_h$ requires a root.
• The anomalous Jacobian coefficient is fixed by the regulated measure; a current may still mix with other currents under renormalization.
• A holomorphic scale is not automatically a pole mass or confinement scale. Canonical normalization introduces real wavefunction factors.
• A spurionic anomalous transformation relates couplings and operators across a family of theories. It is not a conserved charge acting within a fixed theory when $\theta$ is held fixed.
• Matching formulas remain covariant only when the phase of every complex mass and the theta angle are tracked together.

Common pitfalls

Dropping an anomalous symmetry. Its current is not conserved at fixed $\theta$, but its regulated Jacobian determines how the background coupling transforms. That information is indispensable.

Assigning a charge to $\Lambda$ without a branch. The single-valued object is $\Lambda^{b_0}$. Roots label local branches and can be permuted by theta-angle monodromy.

Using scalar R-charges in the anomaly sum. A chiral multiplet’s left-handed fermion has charge $r_i-1$, while the gaugino has charge $1$.

Exercises

For $SU(N_c)$ SQCD, add a mass source $m^j{}_iM^i{}_j$. Determine the axial and anomaly-free R-charges of $m$.

Solution

Since $q_A(M)=2$ and a superpotential term must be spurionically neutral, $q_A(m)=-2$. Since $R(W)=2$ and $R(M)=2(1-N_c/N_f)$,

$$R(m)=2-R(M)=\frac{2N_c}{N_f}.$$

These assignments ensure that a complex mass phase is included consistently in anomalous transformations and scale matching.

Show directly that $\Lambda_h^{b_0}$ is independent of $\mu$ using the stated beta function for $\tau$.

Solution

Taking a logarithmic derivative,

$$\frac{d}{d\ln\mu}\ln\Lambda_h^{b_0} =b_0+2\pi i\frac{d\tau}{d\ln\mu} =b_0+2\pi i\frac{i b_0}{2\pi}=0.$$

References

• Ofer Aharony, Nathan Seiberg, and Yuji Tachikawa, “Reading between the Lines of Four-Dimensional Gauge Theories,” Journal of High Energy Physics 2013(08), 115, 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, arXiv, DOI.
• Nathan Seiberg, “Naturalness versus Supersymmetric Non-renormalization Theorems,” Physics Letters B 318 (1993), 469–475, arXiv, DOI.
• Mikhail Shifman, Advanced Topics in Quantum Field Theory: A Lecture Course, 2nd ed., Cambridge University Press (2022), §§ 10.7 and 10.16, DOI.