Nonperturbative Superpotentials and Exactness Ledgers

A nonperturbative superpotential is exact only after several independent constraints agree: the Wilsonian object and holomorphic variables are fixed; dimensions, ordinary symmetries, and anomalous spurions determine the possible form; zero modes or another controlled regime establish that the term is generated; and decoupling fixes the residual normalization. The Affleck–Dine–Seiberg superpotential provides the canonical complete example.

Required background. Holomorphic couplings and background superfields supplies the analytic-source argument. Instanton zero modes and selection rules supplies the semiclassical existence test.

Helpful background. Holomorphic and canonical couplings distinguishes the scale used below from a canonically normalized physical threshold.

The SQCD problem and its hypotheses

Consider four-dimensional $\mathcal N=1$ $SU(N_c)$ gauge theory with

$$0<N_f<N_c$$

pairs $Q^i,\widetilde Q_i$, no tree superpotential, and holomorphic scale

$$\Lambda_h^{b_0}=\mu^{b_0}e^{2\pi i\tau_h(\mu)}, \qquad b_0=3N_c-N_f.$$

On the mesonic patch of moduli space define

$$M^i{}_j=Q^i\widetilde Q_j, \qquad \det M\neq0.$$

The statement to be derived is about the local Wilsonian superpotential expressed in these holomorphic composite coordinates. It assumes a supersymmetric regulator, the scale convention above, and a branch of every fractional power. It does not determine the Kähler potential, canonically normalized masses, or the behavior at the locus where $M$ ceases to be a complete low-energy coordinate.

Functional form from exact constraints

Flavor symmetry $SU(N_f)_L\times SU(N_f)_R$ permits dependence on $M$ through $\det M$. The baryon number $U(1)_B$ is automatic because $M$ is neutral. Under the anomalous axial transformation with charge $1$ for both $Q$ and $\widetilde Q$,

$$q_A(\det M)=2N_f, \qquad q_A(\Lambda_h^{3N_c-N_f})=2N_f.$$

Thus their ratio is spurionically invariant. The anomaly-free R-charge is

$$R(Q)=R(\widetilde Q)=1-\frac{N_c}{N_f},$$

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

$$\left[\frac{\Lambda_h^{3N_c-N_f}}{\det M}\right] =3(N_c-N_f).$$

A holomorphic superpotential of dimension three and R-charge two is therefore restricted to

$$W_{\mathrm{np}} =C_{N_c,N_f} \left( \frac{\Lambda_h^{3N_c-N_f}}{\det M} \right)^{1/(N_c-N_f)}.$$

For this field content there is no independent neutral, dimensionless holomorphic invariant on the stated patch, so no arbitrary function remains. The coefficient $C_{N_c,N_f}$, the fact that it is nonzero, and the branch are still undetermined. This separation between formal constraint and dynamical input is essential.

Fixing the coefficient

When $N_f=N_c-1$, a generic large meson expectation value completely Higgses the gauge group. A one-instanton configuration is weakly coupled, its size integral is regulated by the Higgs scale, and gauge-Yukawa interactions lift every fermion zero mode except the universal pair. In the conventional holomorphic scale normalization, the collective-coordinate calculation gives

$$C_{N_c,N_c-1}=1.$$

For $N_f<N_c-1$, the original one-instanton has too many unlifted gaugino modes and an unbroken $SU(N_c-N_f)$ subgroup. The coefficient is instead propagated from the controlled case by holomorphic decoupling. Requiring consistency when one flavor receives a large mass yields

$$C_{N_c,N_f}=N_c-N_f.$$

The exact result is therefore

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

The original semiclassical calculation is due to Affleck, Dine, and Seiberg Affleck, Dine, and Seiberg 1984, pp. 493–534. The general Wilsonian strategy—holomorphy, symmetry, limits, and decoupling—is systematized in Intriligator, Leigh, and Seiberg 1994, §§ I–III.

The coefficient is convention dependent: a finite rescaling of $\Lambda_h$ changes its numerical value. What is exact is the paired statement consisting of the scale definition and the displayed coefficient. A detailed SQCD derivation and its controlled limits are collected in Intriligator and Seiberg 1996, §§ 3.1–3.2.

A massive check and the Nc branches

Add a full-rank mass matrix,

$$W_{\mathrm{tot}}=W_{\mathrm{ADS}}+\operatorname{tr}(mM).$$

Define on a chosen branch

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

Differentiating gives

$$\frac{\partial W_{\mathrm{tot}}}{\partial M^i{}_j} =-A(M^{-1})^j{}_i+m^j{}_i.$$

The F-term equation is

$$mM=A\mathbf 1_{N_f}.$$

Taking determinants and using the definition of $A$ gives

$$A^{N_c}=\Lambda_h^{3N_c-N_f}\det m.$$

Hence there are $N_c$ solutions,

$$A_\ell= \left(\Lambda_h^{3N_c-N_f}\det m\right)^{1/N_c} e^{2\pi i\ell/N_c}, \qquad \ell=0,\ldots,N_c-1.$$

On shell,

$$W_\ell=N_cA_\ell.$$

Holomorphic scale matching identifies

$$\Lambda_{\mathrm{SYM}}^{3N_c} =\Lambda_h^{3N_c-N_f}\det m,$$

so $W_\ell=N_c\Lambda_{\mathrm{SYM},\ell}^3$, the expected $N_c$ branches of pure super-Yang–Mills. This calculation checks the coefficient, phase dependence, and branch count simultaneously.

The massless theory runs away

With $m=0$, the derivative of $W_{\mathrm{ADS}}$ is proportional to

$$-A(M^{-1})^j{}_i.$$

It cannot vanish at finite invertible $M$. The potential can instead approach zero along directions with $M\to\infty$, where $A\to0$. Thus the exact Wilsonian superpotential implies a runaway rather than a supersymmetric vacuum at finite meson expectation value.

This conclusion assumes the Kähler metric is nonsingular enough along the asymptotic direction to interpret the F-term potential in the usual way. The existence of the runaway is robust, but its detailed metric distance and time evolution are not determined by $W$ alone.

The pole at $\det M=0$ does not mean that the microscopic theory has an infinite fundamental interaction there. It signals that the meson-only effective description is invalid where additional nonabelian gauge degrees of freedom become light. Exactness on one holomorphic patch is not global regularity in inappropriate coordinates.

Evidence and remaining obligations

Ingredient	Role in the ADS result	Limitation
holomorphy and locality	restrict a Wilsonian F-term	do not apply unchanged to a massless nonlocal 1PI action
flavor, axial-spurion, and R-charges	fix allowed dependence on $M$ and $\Lambda_h$	do not prove a nonzero coefficient
dimension	fixes the fractional exponent	does not choose a branch
$N_f=N_c-1$ instanton	establishes generation and normalization in a controlled regime	does not directly describe smaller $N_f$
holomorphic decoupling	transports the coefficient between flavor numbers	assumes the threshold branch and scale matching
massive pure-SYM limit	checks the $N_c$ vacua and phase dependence	imports the pure-SYM condensate normalization

This table is a useful template for other theories. If a row lacks evidence, the conclusion should be weakened accordingly—for example, from an exact term to an allowed functional form.

Common pitfalls

Symmetry proves the coefficient is nonzero. It does not. A semiclassical calculation or other controlled dynamical input is required.

Every fractional power is one multivalued observable. A local branch describes one vacuum. Theta-angle monodromy permutes branches, and a global statement must include the whole set.

A singular effective superpotential is inconsistent. The singularity can mark the breakdown of the chosen low-energy coordinates. One must identify the additional light degrees of freedom before judging it.

Exercises

For $SU(3)$ with $N_f=1$, write the exact superpotential and add a mass $mM$. Find the three supersymmetric solutions.

Solution

Here $b_0=8$ and $N_c-N_f=2$, so

$$W=2\left(\frac{\Lambda_h^8}{M}\right)^{1/2}+mM.$$

The general equation above gives $mM=A$ and $A^3=m\Lambda_h^8$. Thus

$$A_\ell=(m\Lambda_h^8)^{1/3}e^{2\pi i\ell/3}, \qquad M_\ell=\frac{A_\ell}{m}, \qquad \ell=0,1,2.$$

Explain why multiplying $W_{\mathrm{ADS}}$ by a holomorphic function of $\det M/\Lambda_h^{2N_f}$ is not allowed.

Solution

The proposed argument is dimensionless, but it is not neutral. Under the anomaly-free R-symmetry, $\det M$ has charge $2(N_f-N_c)$ whereas the holomorphic scale is neutral; under the axial spurion symmetry, the two factors also have different charges in general. Solving dimension, axial, and R constraints simultaneously leaves no nontrivial neutral dimensionless invariant for this field content, so any nonconstant factor would violate at least one constraint.

References

• Ian Affleck, Michael Dine, and Nathan Seiberg, “Dynamical Supersymmetry Breaking in Supersymmetric QCD,” Nuclear Physics B 241 (1984), 493–534, DOI.
• Kenneth A. Intriligator, Robert G. Leigh, and Nathan Seiberg, “Exact Superpotentials in Four Dimensions,” Physical Review D 50 (1994), 1092–1104, 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, §§ 3.1–3.2, arXiv, DOI.