Holomorphic Decoupling, Integrating In and Out, and Scale Matching

Holomorphic decoupling compares exact Wilsonian descriptions across a supersymmetric mass threshold. Matching the complex gauge coupling fixes the relation between holomorphic scales; solving the heavy field’s F-term then transports exact superpotentials and their coefficients. The procedure is branch-sensitive and scheme-sensitive, but within one declared convention it provides a stringent consistency test.

Required background. Holomorphic and canonical couplings fixes the gauge-coupling scheme. Nonperturbative superpotentials supplies the exact term to be matched.

Helpful background. Decoupling theorems and threshold corrections supplies the general EFT logic behind integrating out a massive field.

Matching a holomorphic threshold

Let a high-energy theory have one-loop holomorphic coefficient $b_H$ and scale

$$\Lambda_H^{b_H}=\mu^{b_H}e^{2\pi i\tau_H(\mu)}.$$

Suppose a chiral multiplet or vectorlike pair has a supersymmetric holomorphic mass $m$, and the low-energy theory has coefficient $b_L>b_H$. In a matching convention with no additional finite threshold constant, continuity of the holomorphic coupling at $\mu=m$ gives

$$\Lambda_L^{b_L} =m^{,b_L-b_H}\Lambda_H^{b_H}.$$

This equation includes phases: $m$ is complex, and an anomalous rotation of its phase is compensated by the theta angle. Taking absolute values prematurely destroys that covariance.

The formula assumes that $m$ is large compared with the low-energy strong scale, that the threshold is supersymmetric, and that the same holomorphic normalization is used above and below. A finite redefinition of either coupling multiplies the right-hand side by a convention constant. Canonical pole masses also contain wavefunction factors and should not be inserted into a holomorphic matching equation without translating schemes.

Removing one SQCD flavor

Begin with $SU(N_c)$ SQCD and $N_f<N_c$ flavors. Write

$$b_f=3N_c-N_f, \qquad k=N_c-N_f,$$

and give the last flavor a mass,

$$W_f=k\left(\frac{\Lambda_f^{b_f}}{\det M}\right)^{1/k} +mX, \qquad X=M^{N_f}{}_{N_f}.$$

On the flavor-symmetric branch relevant to decoupling, the off-diagonal heavy–light mesons vanish and

$$\det M=X\det\widehat M,$$

where $\widehat M$ is the $(N_f-1)\times(N_f-1)$ light meson matrix. Define

$$A=\left( \frac{\Lambda_f^{b_f}} {X\det\widehat M} \right)^{1/k}.$$

The heavy F-term equation is

$$0=\frac{\partial W_f}{\partial X} =-\frac{A}{X}+m,$$

so $X=A/m$. Substituting this into the definition of $A$ gives

$$A^{k+1} =\frac{m\Lambda_f^{b_f}}{\det\widehat M}.$$

The on-shell superpotential is

$$W_{f-1}=kA+mX=(k+1)A.$$

The low-energy theory has

$$b_{f-1}=b_f+1, \qquad N_c-(N_f-1)=k+1.$$

Using the holomorphic scale match

$$\Lambda_{f-1}^{,3N_c-N_f+1} =m\Lambda_f^{,3N_c-N_f},$$

we obtain

$$W_{f-1} =(N_c-N_f+1) \left( \frac{\Lambda_{f-1}^{,3N_c-N_f+1}} {\det\widehat M} \right)^{1/(N_c-N_f+1)}.$$

This is exactly the Affleck–Dine–Seiberg superpotential with one fewer flavor. The calculation checks the exponent, coefficient, mass phase, and branch recursion. Repeating it transports the controlled $N_f=N_c-1$ instanton normalization to every $N_f<N_c$; the general exact-superpotential and SQCD treatments are Intriligator, Leigh, and Seiberg 1994, §§ I–III and Intriligator and Seiberg 1996, §§ 3.1–3.2.

Why the order of operations matters

The decoupling limit is not obtained by simply deleting a row and column of $M$ while holding $\Lambda_f$ fixed. The meaningful limit is

$$m\to\infty, \qquad \Lambda_f\to0, \qquad m\Lambda_f^{b_f}=\Lambda_{f-1}^{b_f+1} \quad\text{fixed}.$$

Then the high-energy theory is weak near the threshold while the low-energy strong scale remains finite. The heavy field is eliminated by its holomorphic F-term on a chosen branch before the limit is interpreted.

Off-diagonal mesons also have F-term equations. Setting them to zero is consistent on the branch used above, but in a more general superpotential they can mix with light fields and must be solved rather than silently discarded.

If $m$ crosses zero, the effective field content changes and the decoupled description fails. Analytic continuation around $m=0$ can permute fractional-power branches. Matching is holomorphic on the punctured mass plane, not a claim of regularity through the massless threshold.

Integrating out and integrating in

Suppose a low-energy Wilsonian superpotential $W_{\mathrm{low}}(m,g_a)$ is known as a holomorphic function of a source $m$ and other couplings $g_a$. If $m$ couples linearly to a chiral operator $X$ in the higher-energy description, then

$$X=\frac{\partial W_{\mathrm{low}}}{\partial m}.$$

When this relation can be inverted locally for $m=m(X,g_a)$, define the Legendre transform

$$W_{\mathrm{high}}(X,g_a) =W_{\mathrm{low}}(m,g_a)-mX.$$

Adding $mX$ back and extremizing with respect to $X$ returns $W_{\mathrm{low}}$. This is “integrating in.” It can reconstruct highly nontrivial exact terms from a simpler theory Intriligator 1994, §§ 1–2.

The transform has hypotheses:

• $m$ must genuinely be the chiral source for $X$ with the stated normalization;
• the Wilsonian superpotential must be linear in that source before $X$ is eliminated;
• the $m\leftrightarrow X$ relation must be locally invertible on the selected branch;
• threshold scales and anomalous charges must match; and
• no extra holomorphic term independent of $m$ may be dropped without separate evidence.

Failure of invertibility often signals a singular locus or additional light fields. Integrating in is then not a globally valid change of coordinates.

A pure-SYM consistency check

Give all $N_f$ SQCD flavors a full-rank mass matrix $m$. Repeated threshold matching yields

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

Solving the meson F-terms in the massive ADS superpotential gives

$$W_\ell=N_c \left(\Lambda_{N_f}^{3N_c-N_f}\det m\right)^{1/N_c} e^{2\pi i\ell/N_c},$$

with $\ell=0,\ldots,N_c-1$. This equals $N_c\Lambda_{\mathrm{SYM},\ell}^3$. The determinant, rather than $|\det m|$, is forced by holomorphy and reproduces the anomalous phase dependence.

Common pitfalls

Matching at a canonical pole mass in a holomorphic formula. The canonical mass contains $Z$ factors. Either use the chiral mass source throughout or translate every coupling and wavefunction consistently.

Holding both strong scales fixed as $m\to\infty$. They are related by the threshold equation. The controlled decoupling limit fixes the low-energy scale while the high-energy scale changes.

Treating a Legendre transform as global. Multiple solutions for $m(X)$ are distinct branches, and the map can fail at discriminant loci.

Exercises

Starting with $SU(N_c)$ SQCD with two massive flavors of diagonal masses $m_1,m_2$, integrate them out sequentially and show that the final scale is independent of the order.

Solution

Each pair changes $b$ by one. In either order,

$$\Lambda_{N_f-2}^{3N_c-N_f+2} =m_2\Lambda_{N_f-1}^{3N_c-N_f+1} =m_2m_1\Lambda_{N_f}^{3N_c-N_f}.$$

Because the masses are holomorphic numbers, multiplication commutes. For a nondiagonal full-rank mass matrix, the invariant generalization is the determinant of the heavy mass block.

Check that the scale-matching equation has the same mass dimension and anomalous axial charge on both sides.

Solution

Both sides have dimension $b_f+1$. Under the axial symmetry of the high-energy theory, $q_A(m)=-2$ while $q_A(\Lambda_f^{b_f})=2N_f$, so the right side has charge $2(N_f-1)$, equal to the charge of $\Lambda_{f-1}^{b_f+1}$ in the low-energy theory.

References

• Kenneth A. Intriligator, “Integrating In and Exact Superpotentials in Four Dimensions,” Physics Letters B 336 (1994), 409–414, arXiv, 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, arXiv, DOI.