Dynamical Supersymmetry Breaking and Calculability

Dynamical supersymmetry breaking (DSB) means that a theory with no explicit supersymmetry-breaking parameter generates a strictly positive vacuum energy through quantum gauge dynamics. The strongest demonstrations combine an exact holomorphic obstruction to solving all $F$ - and $D$ -term equations with a tunable limit in which the vacuum lies at weak coupling. The classic $SU(3)\times SU(2)$ “3–2 model” has both features.

Required background. Chiral-theory anomaly constraints supplies the microscopic consistency checks; Nelson–Seiberg-type criteria supplies the $R$ -symmetry hypotheses; and quantum chiral rings and Konishi anomalies explains exact operator relations. Helpful background. The Witten index and its failure modes explains why a vanishing or ill-defined index is not a proof of breaking.

What a convincing DSB argument must exclude

A positive-energy stationary point is not enough. A stable DSB conclusion must address:

all gauge and global anomalies, including discrete or global gauge anomalies;
every invariant branch and every supersymmetric solution at finite field values;
runaways on which $V\to0$ only at infinite distance;
the Kähler metric in the region where the vacuum is claimed;
extra vacua that enter from infinity when a deformation is varied;
whether an $R$ symmetry is exact, accidental, explicitly broken, or spontaneously broken.

Holomorphy can establish an incompatible set of $F$ -term equations, but a vacuum energy depends on the nonholomorphic Kähler metric. “Calculable DSB” therefore requires the vacuum to sit where the gauge couplings and Kähler corrections are parametrically controlled.

Theory card for the 3–2 model

Take the simply connected product gauge group

G=SU(3)\times SU(2)

and chiral fields

Q:({\bf3},{\bf2}),\quad \overline U:(\overline{\bf3},{\bf1}),\quad \overline D:(\overline{\bf3},{\bf1}),\quad L:({\bf1},{\bf2}),

with

W_{\mathrm{tree}}=\lambda\,\epsilon^{\alpha\beta} Q^a_{\alpha}\overline D_aL_\beta.

No nontrivial subgroup of the $\mathbb Z_3\times\mathbb Z_2$ gauge center acts trivially on all four fields, so there is no quotient ambiguity for this matter content and no surviving electric one-form symmetry.

The $SU(3)^3$ anomaly cancels because the two $SU(2)$ components of $Q$ give two fundamentals against the two antifundamentals. The $SU(2)$ factor has four left-handed doublets—three colors of $Q$ plus $L$ —so its mod-two global gauge anomaly also vanishes. The one-loop coefficients are

\begin{aligned} b_3&=3(3)-\left[2\left(\frac12\right) +\frac12+\frac12\right]=7,\\ b_2&=3(2)-\left[3\left(\frac12\right)+\frac12\right]=4. \end{aligned}

Thus the two holomorphic scales are $\Lambda_3^7$ and $\Lambda_2^4$ .

One anomaly-free ordinary symmetry and one anomaly-free $R$ symmetry can be represented by the scalar charges

\begin{array}{c|rrrr} &Q&\overline U&\overline D&L\\ \hline X&1&-4&2&-3\\ R&-1&0&0&3 \end{array}

The tree term has $(X,R)=(0,2)$ . Direct substitution gives zero for both $SU(3)^2X$ and $SU(2)^2X$ , and for each mixed gauge– $R$ anomaly after including the charge-one gaugino. These checks ensure that the nonperturbative term below has the right selection rules; they do not generate it by themselves.

The exact obstruction from the SU(3) node

Choose a hierarchy $\Lambda_3\gg\Lambda_2$ and eventually take $\lambda\ll1$ . Viewed as an $SU(3)$ theory, $Q$ provides two fundamentals and $(\overline U,\overline D)$ two antifundamentals, so $N_f=2=N_c-1$ . The exact ADS term is

W_3=\frac{\Lambda_3^7} {\det(Q\overline Q)}, \qquad \overline Q=(\overline U,\overline D).

The full superpotential in this regime is

W_{\mathrm{eff}} =\lambda Q\overline D L +\frac{\Lambda_3^7}{\det(Q\overline Q)}.

If all $F$ terms vanished at a finite point, $F_L=0$ would require $Q\overline D=0$ . That makes the second column of the $2\times2$ meson matrix vanish, hence $\det(Q\overline Q)=0$ , precisely where the ADS term is singular. The equations therefore have no simultaneous finite solution. This is an exact holomorphic incompatibility, first exploited in Affleck, Dine, and Seiberg 1984, pp. 1678–1680.

The argument would be incomplete if the fields could run to infinity with every auxiliary field tending to zero. In the 3–2 model, the tree interaction grows along the classical flat directions on which the ADS term decreases, while the ADS term diverges on the directions that try to set $Q\overline D=0$ . The gauge $D$ terms remove the remaining relative orientations. Their competition stabilizes the runaway at finite field values.

Why the vacuum is calculable

Let $v$ denote the common magnitude of the Higgsing expectation values, suppressing order-one ratios fixed by minimizing the $D$ terms. Dimensional analysis along the relevant branch gives

W_{\mathrm{tree}}\sim\lambda v^3, \qquad W_3\sim\frac{\Lambda_3^7}{v^4}.

Balancing their $F$ terms,

\lambda v^2\sim\frac{\Lambda_3^7}{v^5},

yields

v\sim\Lambda_3\lambda^{-1/7}, \qquad F\sim\Lambda_3^2\lambda^{5/7}, \qquad V_{\min}\sim\Lambda_3^4\lambda^{10/7}.

For $\lambda\ll1$ , $v\gg\Lambda_3\gg\Lambda_2$ . Both gauge groups are Higgsed where their couplings are weak, and corrections to the approximately canonical Kähler potential are suppressed by powers of $\Lambda_3/v\sim\lambda^{1/7}$ and weak logarithms. The order-one coefficients and the exact location require minimizing the full weak-coupling $F$ - and $D$ -term potential, but the parametric scaling and positive energy are controlled. This distinction—exact nonexistence of a supersymmetric solution versus perturbatively calculable vacuum location—is essential.

The vacuum spontaneously breaks the continuous $R$ symmetry, so its phase contains an $R$ axion in the limit that the symmetry is exact. Gauging additional interactions or adding higher-dimension terms can explicitly reduce that symmetry and change the light spectrum; those are deformations, not properties of the minimal model.

Other exact mechanisms and their limits

A different route couples singlets to composites on a quantum-modified moduli space. If the quantum constraint forbids all composites from vanishing while singlet $F$ terms demand that they do, the equations have a rank-condition obstruction. The $SU(2)$ model with four doublets and singlets developed by Intriligator and Thomas 1996, §§ 2–3, pp. 126–132 is the canonical example. Such a proof still needs Kähler control to compute the vacuum quantitatively; symmetry can sometimes locate it near a point where qualitative breaking is robust.

Two popular shortcuts are unsafe:

A zero Witten index permits breaking but does not require it. Noncompact directions and vacua arriving from infinity can invalidate deformation arguments.
The Nelson–Seiberg criterion assumes a generic superpotential in a specified set of infrared fields. Strong dynamics may add fields, constraints, or accidental symmetries, so the theory card and exact chiral relations must come first; the theorem’s hypotheses and exceptions are stated in Nelson and Seiberg 1994, pp. 46–62.

Order of limits

The controlled statement takes $0<\lambda\ll1$ at fixed nonzero $\Lambda_3$ with $\Lambda_2/\Lambda_3$ sufficiently small. Setting $\lambda=0$ first restores flat directions and sends the stabilized point to $v=\infty$ . Raising $\Lambda_2$ to compete with $\Lambda_3$ removes the sequential-node derivation, even though holomorphy may continue some protected information. Neither altered limit disproves the finite-small- $\lambda$ result; it changes the available control.

Common pitfalls

Stopping after incompatible F terms. One must also exclude $D$ -flat runaways and show that the potential has a finite minimum. The 3–2 tree interaction and ADS singularity do this together.

Calling a strong-coupling estimate calculable. Calculability here comes from $v/\Lambda_3\sim\lambda^{-1/7}\gg1$ . Without a tunable hierarchy, exact holomorphy does not determine the Kähler metric or vacuum energy.

Taking a deformation to zero in the wrong order. At $\lambda=0$ the vacuum escapes to infinity. Continuity at a fixed compact field point is therefore not a valid argument across that limit.

Exercises

1. Check both gauge factors

Verify the perturbative $SU(3)^3$ anomaly, the $SU(2)$ mod-two anomaly, and the two one-loop coefficients.

Solution

For $SU(3)^3$ , $Q$ is two fundamentals and $\overline U,\overline D$ are two antifundamentals, giving $2-1-1=0$ . The $SU(2)$ factor sees $3+1=4$ doublets, an even number. The index sums are $2$ for each gauge factor, so $b_3=9-2=7$ and $b_2=6-2=4$ .

2. Derive the small-coupling exponents

Starting only from $W_{\mathrm{tree}}\sim\lambda v^3$ and $W_3\sim\Lambda_3^7/v^4$ , recover the powers of $\lambda$ in $v,F$ , and $V$ .