Mass Deformations, Higgsing, and Dual RG Flows

Relevant deformations are among the sharpest tests of Seiberg duality because the same infrared endpoint is reached by different microscopic mechanisms. An electric quark mass becomes a linear magnetic meson term and forces magnetic Higgsing. An electric Higgs expectation value becomes a magnetic quark mass. Scale matching, vacuum choice, and the surviving global sectors must agree along both paths.

Required background. The operator and anomaly dictionary fixes the deformation map, while holomorphic decoupling and scale matching fixes threshold conventions. Helpful background. General duality flows explains the complete endpoint test.

One electric mass and magnetic Higgsing

Begin with the $SU(N_c)$ electric theory with $N_f$ flavors and add

$$W_{\mathrm{el}}=mQ^{N_f}\widetilde Q_{N_f}.$$

At energies $E\ll|m|$, the heavy flavor decouples. If

$$b=3N_c-N_f,$$

holomorphic matching in a fixed scheme gives

$$\Lambda_{N_f-1}^{,b+1} =m\Lambda_{N_f}^{,b}.$$

On the magnetic side the meson map gives

$$W_{\mathrm{mag}} =\frac{1}{\mu}M^i{}_j q_i\widetilde q^j +mM^{N_f}{}_{N_f}.$$

The $M^{N_f}{}_{N_f}$ F-term is

$$q_{N_f}\widetilde q^{N_f}=-m\mu.$$

Choose a D-flat representative with equal magnitudes for the two expectation values. It breaks

$$SU(\widetilde N_c)\longrightarrow SU(\widetilde N_c-1), \qquad \widetilde N_c=N_f-N_c.$$

The broken vector multiplets and appropriate quark components become massive. The remaining light theory has $N_f-1$ flavors and magnetic rank

$$\widetilde N_c-1=(N_f-1)-N_c,$$

exactly as required by the lower-flavor dual pair.

Magnetic threshold matching

Let

$$\widetilde b=3\widetilde N_c-N_f.$$

Higgsing at $v^2=-m\mu$ yields, up to the already fixed phase convention,

$$\widetilde\Lambda_{N_f-1}^{,\widetilde b-2} =\frac{\widetilde\Lambda_{N_f}^{,\widetilde b}} {-m\mu}.$$

Combine this with the original relation

$$\Lambda_{N_f}^{,b} \widetilde\Lambda_{N_f}^{,\widetilde b} =(-1)^{\widetilde N_c}\mu^{N_f}.$$

Then

$$\Lambda_{N_f-1}^{,b+1} \widetilde\Lambda_{N_f-1}^{,\widetilde b-2} =(-1)^{\widetilde N_c-1}\mu^{N_f-1},$$

which is the scale relation for the new pair because its magnetic rank is $\widetilde N_c-1$. This calculation checks the otherwise easy-to-miss sign and power of $\mu$. The mass–Higgs flow and holomorphic decoupling relation are derived in Seiberg 1995, §4 and reviewed in Intriligator and Seiberg 1996, §5.5.

The exact numerical coefficients depend on the normalization of $M$ and the holomorphic scales, but the two threshold equations and the lower-rank relation must transform together. Changing only one convention creates a spurious mismatch.

Matching the light fields and anomalies

The deformation breaks the flavor symmetry to a subgroup acting on the first $N_f-1$ flavors, together with anomaly-free abelian combinations. Heavy fermions can shift background contact terms, so the endpoint comparison should use the same mass phase on both sides.

After Higgsing, magnetic components organize into:

• an $SU(\widetilde N_c-1)$ vector multiplet;
• $N_f-1$ light magnetic flavors;
• the $(N_f-1)\times(N_f-1)$ meson block;
• massive fields paired through the superpotential or eaten by the Higgs mechanism;
• gauge singlets that are removed by F-term masses rather than silently discarded.

Recomputing $SU(N_f-1)^3$, mixed baryon, and R-anomalies gives the same formulas as the original pair with $N_f\to N_f-1$. This is stronger than a rank count because it tests the complete surviving multiplet content.

Electric Higgsing and magnetic decoupling

Now give an electric quark pair a D-flat expectation value of rank one,

$$Q^{N_f}_{N_c}=\widetilde Q^{N_c}_{N_f}=v.$$

Generically this breaks

$$SU(N_c)\longrightarrow SU(N_c-1)$$

and leaves $N_f-1$ light flavors. The low-energy electric scale satisfies

$$\Lambda_{N_c-1,N_f-1}^{,b-2} =\frac{\Lambda_{N_c,N_f}^{,b}}{v^2}.$$

In magnetic variables, the expectation value maps to

$$\langle M^{N_f}{}_{N_f}\rangle=v^2.$$

The term $M q\widetilde q/\mu$ gives one magnetic flavor mass $v^2/\mu$. Integrating it out leaves the same magnetic gauge rank,

$$\widetilde N_c=N_f-N_c=(N_f-1)-(N_c-1),$$

and its scale obeys the ordinary massive-flavor threshold relation. Thus electric Higgsing maps to magnetic decoupling, the converse pattern of the mass flow.

The comparison must be made on corresponding moduli strata. A higher-rank expectation value repeats the operation until stabilizers or special-rank dynamics change the description.

Superpotential deformations

An electric meson polynomial maps directly to a magnetic singlet polynomial. For example,

$$\delta W_{\mathrm{el}} =\frac{\kappa}{2}(Q\widetilde Q)^2$$

becomes

$$\delta W_{\mathrm{mag}} =\frac{\kappa}{2}M^2.$$

When $M$ is massive, its equation of motion gives

$$M=-\frac{1}{\kappa\mu}q\widetilde q,$$

and substituting back generates a magnetic quartic interaction with coefficient proportional to $-1/(\kappa\mu^2)$. This illustrates a general pattern: an elementary deformation on one side can become a multi-trace composite interaction on the other.

Baryonic deformations can select branches with qualitatively different Higgs patterns. Their flavor epsilon tensors and scale factors must be included; a schematic $B\leftrightarrow b$ is insufficient for numerical matching.

When operations fail to commute

Consider the sequence “take the infrared limit, then add a mass” versus “add the mass in the ultraviolet, then flow.” They agree only if the operator remains identifiable and no accidental sector changes the endpoint. Near the bottom of the conformal window, the meson can become free, so a mass for $M$ couples to an accidental sector and must be treated in the corrected fixed-point description.

Compactification adds another noncommuting operation: real masses can induce three-dimensional Chern–Simons contact terms, while circle monopoles generate superpotentials. A four-dimensional mass-flow check does not automatically prove the reduced three-dimensional pair.

A complete flow comparison

For each path, record:

1. the normalized deformation and its operator image;
2. the selected F- and D-flat vacuum;
3. the unbroken gauge and faithful global groups;
4. every heavy, eaten, light, free, and topological sector;
5. electric and magnetic holomorphic threshold relations;
6. anomalies and chiral-ring relations of the endpoint;
7. the energy hierarchy that justifies integrating out fields.

If the square closes only after adding a decoupled field or TQFT, that factor is part of the result.

Common pitfalls

Deleting a magnetic flavor by hand. An electric mass forces a magnetic quark expectation value and lowers the magnetic rank; simple deletion gives the wrong theory.

Matching ranks but not scales. The phase and powers in the scale relation are necessary for repeated decoupling to remain consistent.

Ignoring the chosen vacuum. The linear meson term has several gauge-related representatives and can meet special strata; the low-energy spectrum must be computed about the actual D-flat orbit.

Exercises

Take $N_c=3$, $N_f=6$. Add a mass to one electric flavor.

1. Give the magnetic gauge groups before and after the deformation.
2. Find the powers of the electric and magnetic scales before and after.
3. Check the new magnetic rank formula.

Solution

Initially $\widetilde N_c=3$, so the magnetic group is $SU(3)$. The mass forces a rank-one magnetic Higgs expectation value, leaving $SU(2)$. The electric exponent changes from $b=3N_c-N_f=3$ to $b+1=4$. The magnetic exponent starts at $\widetilde b=3\widetilde N_c-N_f=3$ and becomes $\widetilde b-2=1$. Finally, $(N_f-1)-N_c=5-3=2$, agreeing with the $SU(2)$ endpoint.

References

• Intriligator, Kenneth, and Nathan Seiberg. “Lectures on Supersymmetric Gauge Theories and Electric–Magnetic Duality.” Nuclear Physics B Proceedings Supplements 45BC (1996): 1–28. arXiv:hep-th/9509066.
• Seiberg, Nathan. “Electric–Magnetic Duality in Supersymmetric Non-Abelian Gauge Theories.” Nuclear Physics B 435 (1995): 129–146. arXiv:hep-th/9411149.