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. This page first treats the generic $N_c\ge3$, $N_f\ge N_c+3$ mass step, then resolves the exceptional confinement endpoints. Scale matching, vacuum choice, anomaly generators, and every surviving or decoupled sector must agree.

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

The mass/Higgs square is summarized in the shared Seiberg-duality figure.

One electric mass and magnetic Higgsing

Begin with $SU(N_c)$ SQCD with $N_f\ge N_c+3$ and add a mass for the last flavor,

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

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

$$b=3N_c-N_f,$$

holomorphic matching in a fixed scheme gives

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

The magnetic superpotential becomes

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

so the $M^{N_f}{}_{N_f}$ equation is

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

Split the magnetic color index as $a=(\widehat a,\widetilde N_c)$ and choose the D-flat representative

$$q_{N_f}^{a}=v\,\delta^a_{\widetilde N_c}, \qquad \widetilde q_{a}^{N_f}=\widetilde v\,\delta_a^{\widetilde N_c}, \qquad v\widetilde v=-m\mu, \qquad |v|=|\widetilde v|.$$

It breaks

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

The daughter magnetic rank is therefore

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

as required. The holomorphic equation fixes only the product $v\widetilde v$; canonical masses also depend on the Kähler normalization. A weakly coupled Higgs description requires energies below the broken-vector and Yukawa masses, parametrically $E\ll g_{\mathrm{mag}}\sqrt{|m\mu|}$ and $E\ll\sqrt{|m\mu|}$.

Every magnetic field after Higgsing

The spectrum is easiest to see before abbreviating it as “one fewer flavor.” Let $i,j=1,\ldots,N_f-1$.

• The $2\widetilde N_c-1$ broken vector multiplets eat the corresponding Goldstone chiral directions in $q_{N_f}$ and $\widetilde q^{N_f}$.
• The remaining gauge-invariant radial fluctuation of $q_{N_f}\widetilde q^{N_f}$ pairs with $M^{N_f}{}_{N_f}$ and is massive.
• The singlets $M^i{}_{N_f}$ pair with the color-$\widetilde N_c$ components $q_i^{\widetilde N_c}$ through the expectation value $\widetilde v$.
• The singlets $M^{N_f}{}_j$ pair with $\widetilde q_{\widetilde N_c}^{j}$ through $v$.
• The light interacting fields are the $SU(\widetilde N_c-1)$ vector multiplet, the $N_f-1$ pairs $q_i^{\widehat a},\widetilde q_{\widehat a}^{j}$, and the meson block $M^i{}_j$ with $$W_L=\frac{1}{\mu}M^i{}_j q_i^{\widehat a}\widetilde q_{\widehat a}^{j}.$$

Thus no unexplained gauge-singlet remainder has been discarded. The paired chiral fields and eaten fields are as important to the endpoint comparison as the surviving gauge rank.

Magnetic threshold and the daughter scale relation

Let

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

Higgs matching gives

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

Combining this with

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

gives

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

This is precisely the scale relation for the daughter pair. The phase and power of $\mu$ are preserved only when the electric threshold, magnetic Higgs threshold, and baryon convention are transformed together Seiberg 1995, §4, pp. 9–14, Open PDF; Intriligator and Seiberg 1996, §5.5, pp. 24–25, eqs. (5.11)–(5.14), Open PDF.

The surviving R-current and anomalies

The daughter anomaly-free R-current is not numerically identical to the parent current. Let $X$ be the traceless flavor generator with

$$X(Q^i)=X(\widetilde Q_i)=1, \qquad X(Q^{N_f})=X(\widetilde Q_{N_f})=-(N_f-1),$$

for $i=1,\ldots,N_f-1$. Then

$$R'=R_0-\frac{N_c}{N_f(N_f-1)}X$$

has

$$R'(Q^i)=R'(\widetilde Q_i)=1-\frac{N_c}{N_f-1}, \qquad R'(Q^{N_f})=R'(\widetilde Q_{N_f})=1.$$

The corresponding magnetic Higgs fields have $R'=0$, so their expectation values preserve this current. With $N_f'=N_f-1$, the surviving electric and magnetic fields give

Anomaly	Daughter value on both sides
$SU(N_f')_L^3$	$N_c$
$SU(N_f')_L^2U(1)_B$	$N_c/2$
$SU(N_f')_L^2U(1)_{R'}$	$-N_c^2/(2N_f')$
$\operatorname{Tr}B^2R'$	$-2N_c^2$
$\operatorname{Tr}R'$	$-N_c^2-1$

The right-flavor cubic and baryon rows are $-N_c$ and $-N_c/2$, while $SU(N_f')_R^2U(1)_{R'}=-N_c^2/(2N_f')$. The heavy electric fermions have $R'_{\mathrm{fermion}}=0$ and are singlets of $SU(N_f')$; their opposite baryon charges also cancel in the remaining pure-baryon traces. This explains, rather than merely asserts, why the daughter formulas are obtained by $N_f\to N_f-1$.

The exceptional confinement ladder

The preceding gauge-theory daughter card assumes $N_f\ge N_c+3$. If the parent has $N_f=N_c+2$, then $\widetilde N_c=2$ and the same expectation value completely Higgses the magnetic $SU(2)$. The uneaten magnetic-quark components become the daughter baryons. Tree-level matching supplies $M B\widetilde B$, but the completely broken $SU(2)$ also has a one-instanton contribution. Together they give

$$W_{N_c+1} =\frac{B_iM^i{}_j\widetilde B^j-\det M} {\Lambda_{N_c+1}^{2N_c-1}}.$$

The determinant term is indispensable: without it the daughter is not the $N_f=N_c+1$ s-confining theory. The broken-group instanton and its normalization are derived in Intriligator and Seiberg 1996, §5.5, pp. 24–25, eqs. (5.15)–(5.17), Open PDF.

The next two mass steps continue in confined variables:

$$\begin{aligned} N_f=N_c+1\longrightarrow N_c: &\quad \Lambda_{N_c}^{2N_c}=m\Lambda_{N_c+1}^{2N_c-1}, \quad \det\widehat M-B\widetilde B=\Lambda_{N_c}^{2N_c},\\ N_f=N_c\longrightarrow N_c-1: &\quad \Lambda_{N_c-1}^{2N_c+1}=m\Lambda_{N_c}^{2N_c}, \quad W_{\mathrm{ADS}}=\frac{\Lambda_{N_c-1}^{2N_c+1}} {\det\widehat M}. \end{aligned}$$

Thus the generic magnetic Higgs rule, s-confinement, the quantum-modified constraint, and ADS dynamics form one continuous decoupling ladder, but they are not the same low-energy Lagrangian.

Electric Higgsing and magnetic decoupling

Now move along a rank-one electric D-flat direction,

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

It breaks $SU(N_c)\to SU(N_c-1)$. The interacting charged sector is $SU(N_c-1)$ with $N_f-1$ flavors and

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

This is not the entire light spectrum. The electric components $Q^i_{N_c}$ and $\widetilde Q^{N_c}_j$ give $2(N_f-1)$ gauge singlets, and the radial Higgs modulus gives one more. At sufficiently low energy these $2N_f-1$ chiral multiplets are free up to sigma-model interactions suppressed by $|u|$.

Magnetically,

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

gives $q_{N_f},\widetilde q^{N_f}$ a mass $u^2/\mu$. The magnetic gauge rank remains

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

while massive-flavor matching gives

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

Combining the two thresholds reproduces the daughter relation

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

The free magnetic singlets are exactly $M^i{}_{N_f}$, $M^{N_f}{}_j$, and the fluctuation of $M^{N_f}{}_{N_f}$: again $2N_f-1$ fields. This matches the electric Goldstone and radial coordinates. The reverse Higgs/mass flow and both scale thresholds appear in Intriligator and Seiberg 1996, §5.5, printed p. 26, Open PDF. The effective description requires $E\ll|u|$ electrically and $E\ll|u^2/\mu|$ for magnetic flavor decoupling.

Superpotential deformations

A meson polynomial maps to the same polynomial in the magnetic singlet, but its flavor contractions must be declared. For example, choose an identification of the left and right flavor spaces and add

$$\delta W_{\mathrm{el}} =\frac{\kappa}{2}\operatorname{tr}\!\left[(Q\widetilde Q)^2\right] =\frac{\kappa}{2}\operatorname{tr}(M^2).$$

This spurion preserves only the corresponding diagonal flavor subgroup, and $\kappa$ has mass dimension $-1$ in the composite normalization. The full magnetic superpotential is

$$W=\frac{1}{\mu}\operatorname{tr}(M q\widetilde q) +\frac{\kappa}{2}\operatorname{tr}(M^2).$$

Its $M$ equation and exact tree-level substitution give

$$M=-\frac{1}{\kappa\mu}q\widetilde q, \qquad W_{\mathrm{eff}} =-\frac{1}{2\kappa\mu^2}\operatorname{tr}\!\left[(q\widetilde q)^2\right].$$

Thus an elementary quadratic interaction on one side becomes a composite quartic interaction on the other. Baryonic deformations can instead select different Higgs branches; their complementary flavor epsilon tensors and the convention-fixed coefficient $C$ must be retained.

When operations fail to commute

The sequences “flow to the infrared, then deform” and “deform in the ultraviolet, then flow” agree only if the operator remains identifiable and no accidental sector changes the endpoint. At the lower edge of the conformal window the meson becomes free, and below it the magnetic Yukawa interaction is marginally irrelevant and flows logarithmically to zero. A meson deformation must therefore be applied to the corrected interacting-plus-free description, not extrapolated through an accidental threshold unchanged.

Compactification is 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 by itself establish 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, broken-group instanton term, 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 gauge group and spectrum.

Using the generic step at $N_f=N_c+2$. Complete magnetic $SU(2)$ breaking produces an instanton determinant term. Omitting it loses the s-confining daughter superpotential.

Keeping only the charged Higgs daughter. Electric Higgsing also leaves $2N_f-1$ singlet moduli, matched by free magnetic meson components.

Exercises

A generic mass step

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

1. Give the magnetic gauge groups before and after the deformation.
2. Find the electric and magnetic scale powers 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.

The exceptional next step

Now take $N_c=3$, $N_f=5$ and mass one flavor. What is missed by treating the magnetic $SU(2)\to SU(1)$ step as ordinary Higgsing?

Solution

The magnetic $SU(2)$ is completely broken. Its uneaten quark components become the baryons of the four-flavor electric daughter, but a broken-group instanton also generates the determinant term. The complete s-confining superpotential is

$$W=\frac{B_iM^i{}_j\widetilde B^j-\det M}{\Lambda_4^5}.$$

Keeping only the tree-level $MB\widetilde B$ term would give the wrong chiral-ring relations.

References

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