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 massive chiral sector has a supersymmetric holomorphic mass source $m$, and the low-energy theory has coefficient $b_L>b_H$. Choose one finite holomorphic subtraction convention above and below the threshold, with unit finite matching constant. Comparing the analytic gauge-kinetic coefficients 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. The physical threshold lies near a real scale set by $|m|$ after canonical normalization, but writing “$\mu=m$” for complex $m$ would conflate that real matching scale with the holomorphic source. Taking absolute values in the holomorphic equation destroys its anomalous 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 solutions are branch labelled,

$$A_\ell= \left( \frac{m\Lambda_f^{b_f}}{\det\widehat M} \right)^{1/(k+1)} e^{2\pi i\ell/(k+1)}, \qquad X_\ell=\frac{A_\ell}{m}, \qquad \ell=0,\ldots,k.$$

The on-shell superpotential is

$$W_{f-1,\ell}=kA_\ell+mX_\ell=(k+1)A_\ell.$$

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

$$W_{f-1,\ell} =(N_c-N_f+1) \left( \frac{\Lambda_{f-1}^{3N_c-N_f+1}} {\det\widehat M} \right)^{1/(N_c-N_f+1)} e^{2\pi i\ell/(N_c-N_f+1)}.$$

This is exactly the Affleck–Dine–Seiberg superpotential with one fewer flavor. Analytic continuation of the phase of $m$ permutes the $k+1$ branches. 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, §§ 2–3.2, pp. 1094–1101 and Intriligator and Seiberg 1996, § 4.1, pp. 12–15.

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 $g_a$. Express every low-energy scale in terms of the fixed high-energy holomorphic scale $\Lambda_H$, $m$, and the remaining sources before differentiating. If $m$ couples linearly to a chiral operator $X$ in the higher-energy description, the integrating-in prescription uses

$$X=\left( \frac{\partial W_{\mathrm{low}}}{\partial m} \right)_{\Lambda_H,g_a}.$$

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, § 2, Eqs. (2.1)–(2.7), pp. 2–3.

The required linearity principle and simple-threshold principle are hypotheses to test in each model, not general theorems. Additional source-independent terms or non-simple thresholds can invalidate the reconstruction even when a formal Legendre transform exists.

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.