Holomorphic scales and decoupling a heavy flavor

Making a quark very heavy changes which fields appear at low energy. It also changes how fast the gauge coupling runs. If the microscopic theory was labeled by one strong scale, which scale labels the theory after the quark disappears?

The answer retains the complex phase of the quark mass. We will derive why. The preparation is holomorphy and effective actions and SQCD and its classical vacua.

We work in flat four-dimensional Lorentzian $\mathcal N=1$ SQCD, with four real supercharges, all manifest in that superspace. The gauge group is $SU(N_c)$, with rank $N_c-1$, and initially there are $N_f$ fundamental/antifundamental flavor pairs. Take $1\leq N_f<3N_c$, no soft breaking, and no gravitational background. A supersymmetric mass will remove one pair without Higgsing the gauge group.

Put the coupling in a chiral parameter

For the gauge kinetic term it is convenient to absorb the coupling into the connection, writing $\mathcal A_\mu=g_{\rm c}A_\mu^At^A$ relative to a canonically normalized vector field. Then $D_\mu=\partial_\mu-i\mathcal A_\mu$, and the gauge kinetic coefficient sits outside the field strength. Keep the ordinary fundamental trace $\operatorname{Tr}(t^At^B)=\delta^{AB}/2$.

Define the holomorphic Wilsonian coupling

$$\tau_{\rm h}(\mu)=\frac{\theta}{2\pi} +\frac{4\pi i}{g_{\rm h}^2(\mu)}.$$

It is the complex coefficient of a local chiral gauge kinetic term in a holomorphy-preserving Wilsonian prescription. Our normalization is fixed by the weight of an instanton of topological charge one:

$$q_{\rm h}(\mu)=e^{2\pi i\tau_{\rm h}(\mu)} =\exp\!\left[-\frac{8\pi^2}{g_{\rm h}^2(\mu)}+i\theta\right].$$

The topological orientation is chosen so that this charge has phase $e^{i\theta}$, and $\theta$ has period $2\pi$. Thus a shift $\tau_{\rm h}\mapsto\tau_{\rm h}+1$ leaves $q_{\rm h}$ unchanged. Intriligator–Seiberg, (2.2)–(2.3)

The subscript matters. Canonical normalization of gauge and matter fields involves real kinetic coefficients and anomalous path-integral Jacobians. It need not preserve holomorphic dependence on chiral parameters. Accordingly, the beta function of a canonically normalized coupling is not obtained by treating every statement about $g_{\rm h}$ as a statement about $g_{\rm c}$.

Here “Wilsonian” also means that light infrared fluctuations remain to be integrated over. A massless zero-momentum 1PI vertex can have infrared dependence that is absent from this local coefficient. Seiberg, §3 explains why that distinction matters.

Count the running contribution of one flavor

Use the beta-function sign convention

$$\mu\frac{dg_{\rm h}}{d\mu} =-\frac{b}{16\pi^2}g_{\rm h}^3,\qquad \frac{d}{d\log\mu}\frac{8\pi^2}{g_{\rm h}^2}=b.$$

At one loop, a gauge boson contributes $11C_2(G)/3$, a Weyl fermion contributes $-2T(\mathcal R)/3$, and a complex scalar contributes $-T(\mathcal R)/3$. An adjoint gaugino combines with the gauge boson to give $3N_c$. A chiral multiplet contributes $-T(\mathcal R)$; one fundamental plus one antifundamental contributes $-1$. Therefore

$$b=3N_c-N_f.$$

These are the conventional one-loop vacuum-polarization coefficients; the supersymmetric grouping and index normalization are the calculation here. Tong, (6.1)

In the holomorphic Wilsonian prescription this running is perturbatively one-loop exact. The reason is stronger than a cancellation of a few diagrams: holomorphy and $\theta$ periodicity constrain further weak-coupling dependence to powers of $e^{2\pi i\tau_{\rm h}}$, which are nonperturbative. This does not prohibit instanton effects in the effective action. Nor does it eliminate higher-loop running after canonical normalization. Tong, (6.4)–(6.5) and the following discussion

Integrating the holomorphic running between two real positive reference scales gives

$$\tau_{\rm h}(\mu_2)-\tau_{\rm h}(\mu_1) =-\frac{b}{2\pi i}\log\frac{\mu_2}{\mu_1}.$$

Define the scale parameter in this prescription by

$$\mathcal L\equiv\Lambda^b =\mu^b e^{2\pi i\tau_{\rm h}(\mu)}.$$

Indeed, differentiating its logarithm gives $d\log\mathcal L/d\log\mu=b-b=0$. $\mathcal L$ has mass dimension $b$ and retains the theta angle. It is often safer to use $\Lambda^b$ than a chosen $b$th root $\Lambda$: the root has $b$ phase choices, while $\mathcal L$ is unchanged under $\theta\mapsto\theta+2\pi$.

For $b>0$, $|\Lambda|=\mu\exp[-8\pi^2/(b g_{\rm h}^2(\mu))]$ marks the scale where an extrapolated weak-coupling description loses control, up to its scheme normalization. It is not a measured particle mass, and the divergence of the extrapolated coupling is not itself a description of strong-coupling physics.

Remove one flavor at a controlled threshold

Call the original theory H and the lower-energy theory L. Add

$$W_{\rm mass}=m\,\widetilde Q^{\,N_f}Q_{N_f} =m\,M^{N_f}{}_{N_f},\qquad m\ne0.$$

$m$ is a holomorphic superpotential parameter of dimension one. Treat it as a constant background chiral coupling with zero background fermion and auxiliary components. Removing this massive pair leaves $SU(N_c)$ with $N_f-1$ flavors, so

$$b_{\rm H}=b=3N_c-N_f,\qquad b_{\rm L}=b+1.$$

Begin with real $m>0$, a weakly coupled threshold, and matter normalized canonically at that matching scale. Choose a consistent finite subtraction convention in which $\tau_{\rm L}(m)=\tau_{\rm H}(m)$. This fixes a threshold normalization; it is not an observable continuity condition independent of conventions.

Run each coefficient to a common reference $\mu$:

$$\begin{aligned} \tau_{\rm L}(\mu)&=\tau_{\rm H}(m) -\frac{b+1}{2\pi i}\log\frac{\mu}{m},\\ \tau_{\rm H}(\mu)&=\tau_{\rm H}(m) -\frac{b}{2\pi i}\log\frac{\mu}{m}. \end{aligned}$$

When $\mu<m$, the second expression is an extrapolation of the high-theory coefficient, not an assertion that the massive flavor remains active. Subtracting gives

$$\tau_{\rm L}(\mu)-\tau_{\rm H}(\mu) =\frac1{2\pi i}\log\frac{m}{\mu}.$$

Exponentiate before taking any fractional root:

$$\begin{aligned} \Lambda_{\rm L}^{b+1} &=\mu^{b+1}e^{2\pi i\tau_{\rm L}(\mu)}\\ &=\mu^{b+1}e^{2\pi i\tau_{\rm H}(\mu)} \frac{m}{\mu} =m\Lambda_{\rm H}^{b}. \end{aligned}$$

Both sides have dimension $b+1$. The same derivation at any intermediate matching scale works if the logarithmic threshold correction is included there. Matching the couplings without that correction at an arbitrary scale would produce an incorrect scale-dependent answer.

This is the heavy-flavor relation in Seiberg, (2.4). Intriligator–Seiberg, the paragraph before (4.4) explicitly identifies its finite scheme-dependent threshold factor.

Recover the mass phase

For $m=|m|e^{i\varphi}$, holomorphy continues the threshold logarithm locally on the punctured mass plane:

$$\tau_{\rm L}-\tau_{\rm H} =\frac1{2\pi i}\log\frac{m}{\mu} =\frac{\varphi}{2\pi} -\frac{i}{2\pi}\log\frac{|m|}{\mu}.$$

Consequently

$$\theta_{\rm L}=\theta_{\rm H}+\arg m\pmod{2\pi}, \qquad \Lambda_{\rm L}^{b+1}=m\Lambda_{\rm H}^{b}.$$

There is an independent anomaly check. Rotate just the heavy pair, $Q_{N_f}\mapsto e^{i\alpha}Q_{N_f}$ and $\widetilde Q^{\,N_f}\mapsto e^{i\alpha}\widetilde Q^{\,N_f}$. As a spurionic transformation preserving the mass term, $m\mapsto e^{-2i\alpha}m$. Its two Weyl fermions have total quadratic gauge index one. With the topological convention above, the compensating anomalous shift is $\theta_{\rm H}\mapsto\theta_{\rm H}+2\alpha$. Thus $\Lambda_{\rm H}^b\mapsto e^{2i\alpha}\Lambda_{\rm H}^b$ and the product $m\Lambda_{\rm H}^b$ is invariant.

Choosing $\alpha=\varphi/2$ makes the mass real and moves its original phase into $\theta_{\rm H}$. That gives precisely the displayed low-energy theta angle. Replacing $m$ by $|m|$ would fail this check. The factor two follows from the instanton index $2T(\mathcal R)$ per Weyl fermion; our mixed-anomaly convention uses $T(\mathbf{N_c})=1/2$. Tong, §6.2.1

On taking $m$ once around zero, the logarithm changes by $2\pi i$ and $\tau_{\rm L}$ by one. The exponentiated scale parameter returns to itself. Phase choices of a root $\Lambda_{\rm L}$ do not by themselves count physical vacua.

What fixes the form, and what fixes the constant?

The weak-coupling calculation fixes a normalization. A selection rule explains why the holomorphic parameter relation has this form beyond that matching region.

Under the heavy-pair rotation, the dimensions and formal charges are

Parameter	Mass dimension	Heavy-pair axial charge
$m$	$1$	$-2$
$\mathcal L_{\rm H}=\Lambda_{\rm H}^b$	$b$	$+2$
$\mathcal L_{\rm L}=\Lambda_{\rm L}^{b+1}$	$b+1$	$0$

There is no remaining heavy field to transform in the low-energy theory. A holomorphic monomial $m^u\mathcal L_{\rm H}^{\,v}$ with the required charge and dimension satisfies

$$-2u+2v=0,\qquad u+bv=b+1,$$

so $u=v=1$. A putative dimensionless, charge-neutral combination instead satisfies the same first equation and $u+bv=0$, giving $u=v=0$. Thus there is no additional invariant function of these parameters to multiply the result:

$$\mathcal L_{\rm L}=C\,m\mathcal L_{\rm H}.$$

This argument concerns the field-independent holomorphic scale map in massless SQCD with one mass and no other couplings or background expectation values. It does not classify all field-dependent terms in the low-energy action. Holomorphic continuation also does not turn a strongly coupled threshold into a weakly coupled particle calculation.

Our matching prescription sets $C=1$. If finite constant redefinitions preserving the chosen one-loop running give $\mathcal L_{\rm H}'=a_{\rm H}\mathcal L_{\rm H}$ and $\mathcal L_{\rm L}'=a_{\rm L}\mathcal L_{\rm L}$, then

$$C'=\frac{a_{\rm L}}{a_{\rm H}}\,C.$$

Numerical factors in definitions of strong scales must therefore be translated before comparing formulas. The exponents, dimensions, and anomalous phase relation do not depend on that choice.

As another check, rotate all flavor pairs axially. Now $\mathcal L_{\rm H}$ has charge $2N_f$ and $m$ has charge $-2$, so the product has charge $2(N_f-1)$, exactly the low-theory value. These are transformations of a parameterized family; the anomalous axial current is not an exact conserved symmetry with theta and the mass held fixed.

Work through a two-color threshold

Start with the two-color, two-flavor theory of the preceding lesson and give the second pair a mass. Then $b_{\rm H}=4$ and $b_{\rm L}=5$. Let $L_0>0$ be a reference mass unit, and choose

$$\Lambda_{\rm H}^4=L_0^4e^{i\pi/5}, \qquad m=2^{20}L_0e^{i\pi/3}.$$

Matching gives

$$\Lambda_{\rm L}^5=2^{20}L_0^5e^{8\pi i/15}, \qquad |\Lambda_{\rm L}|=16L_0.$$

The physical interpretation is a hierarchy: $|\Lambda_{\rm H}|=L_0<|\Lambda_{\rm L}|=16L_0\ll |m|$. Removing screening matter makes the coupling grow faster toward low energies. More generally, in the unit-threshold convention,

$$\frac{|\Lambda_{\rm L}|}{|\Lambda_{\rm H}|} =\left(\frac{|m|}{|\Lambda_{\rm H}|}\right)^{1/(b+1)}, \qquad \frac{|\Lambda_{\rm L}|}{|m|} =\left(\frac{|\Lambda_{\rm H}|}{|m|}\right)^{b/(b+1)}.$$

Thus a sufficiently heavy weak-coupling threshold remains well above the low strong scale. In the example, $8\pi^2/g_{\rm h}^2(|m|)=4\log(2^{20})=80\log2$. The scale relation says nothing yet about a dynamically generated superpotential in the one-flavor theory. The next calculation distinguishes a symmetry-allowed interaction from the instanton input that establishes its nonzero coefficient.

Two qualifications prevent misleading limits. First, the actual perturbative mass after matter normalization is

$$|m_{\rm c}(\mu)|=\frac{|m|}{\sqrt{Z_Q(\mu)Z_{\widetilde Q}(\mu)}}.$$

The physical decoupling threshold is near that mass, with appropriate radiative corrections. The holomorphic formula uses $m$ and $\tau_{\rm h}$ in the same field convention; substituting $m_{\rm c}$ alone would omit the accompanying anomalous change of the gauge coefficient.

Second, $m\to0$ at fixed high scale removes the assumed heavy/light separation. Conversely, to take a decoupling limit $|m|\to\infty$ at fixed low theory, keep $m\Lambda_{\rm H}^b$ fixed, so $\Lambda_{\rm H}^b\to0$. These are different changes of physical parameters.

Practice

1. Match at a different reference scale

Take a real mass $m>0$ and choose an arbitrary positive reference $\mu_*$ in a perturbative overlap for matching. Compute $\Lambda_{\rm L}^{b+1}/\Lambda_{\rm H}^b$ using the threshold correction at $\mu_*$. What incorrect result follows from imposing $\tau_{\rm L}(\mu_*)=\tau_{\rm H}(\mu_*)$ instead? Finally, how does the matching constant change if the two scale parameters are multiplied by $a_{\rm H}=2$ and $a_{\rm L}=3$?

Hint 1

The ratio contains one uncanceled power of $\mu_*$ and an exponential of the difference of the two couplings.

Hint 2

Use $2\pi i(\tau_{\rm L}-\tau_{\rm H})=\log(m/\mu_*)$ before exponentiating.

Solution

The correct ratio is

$$\frac{\Lambda_{\rm L}^{b+1}}{\Lambda_{\rm H}^b} =\mu_*e^{2\pi i(\tau_{\rm L}-\tau_{\rm H})} =\mu_*\frac{m}{\mu_*}=m.$$

It is independent of the arbitrary reference. If the logarithmic correction is omitted, the answer is $\mu_*$, which agrees only at $\mu_*=m$ in the unit-threshold prescription. Varying a reference scale would then spuriously change the low theory.

After the stated finite redefinitions, $\mathcal L_{\rm L}'=(3/2)m\mathcal L_{\rm H}'$. The new constant is $3/2$; this is a changed parameter convention, not a correction to an observable by a factor $3/2$.

2. Two mixed heavy flavors

Start with $SU(3)$ and five flavor pairs. Give two pairs the holomorphic mass matrix

$$\mathbf m=M_0\begin{pmatrix}2&1\\1&3\end{pmatrix}, \qquad M_0>0$$

in $W=\widetilde Q^{\,i}\mathbf m_{ij}Q_j$ for those two flavors. Derive the scale relation for the theory with three remaining flavors. Check it by algebraically eliminating either diagonal heavy block first. Are those intermediate holomorphic coefficients the physical mass eigenvalues?

Hint 1

For a block matrix, eliminating the first block produces the Schur complement $m_{22}-m_{21}m_{11}^{-1}m_{12}$.

Hint 2

Multiply the two resulting mass coefficients. Repeat with the opposite order; compare with $\det\mathbf m$.

Solution

Two removals increase $b$ from $4$ to $6$. Diagonal masses would give the product of the two masses, and holomorphic flavor-basis covariance expresses that product as a determinant:

$$\Lambda_{\rm L}^{6} =(\det\mathbf m)\Lambda_{\rm H}^{4} =5M_0^2\Lambda_{\rm H}^{4}.$$

Eliminating the first block uses coefficients $2M_0$ and $3M_0-M_0/2=5M_0/2$. The opposite order uses $3M_0$ and $2M_0-M_0/3=5M_0/3$. Both products are $5M_0^2$. Completing the holomorphic bilinear by triangular flavor changes of determinant one gives these Schur complements without an extra flavor-Jacobian determinant.

This is an algebraic consistency check of holomorphic matching. In the original canonical kinetic metric the positive mass eigenvalues are $(5+\sqrt5)M_0/2$ and $(5-\sqrt5)M_0/2$. They have the same product but differ from the intermediate coefficients, because the triangular changes do not preserve canonical kinetic terms. A physical threshold calculation uses those kinetic terms as well.

3. Keep track of which R symmetry is being used

For $SU(3)$ with five flavors, use the symmetric scalar R-charge $r_{\rm H}=2/5$. Treat the mass of one pair as a spurion. Find its R-charge and the charge of $\Lambda_{\rm L}^{5}=m\Lambda_{\rm H}^{4}$. Check that result against the mixed gauge anomaly of the four light pairs retaining charge $2/5$. Then find the symmetric gauge-anomaly-free R-charge of the four-flavor theory itself.

Hint 1

$R(W)=2$, the high-theory scale parameter is neutral under its anomaly-free R symmetry, and the gaugino has charge one.

Hint 2

For the retained high-theory R assignment, calculate $3+4(r_{\rm H}-1)$. A scale parameter’s compensating charge is twice that mixed-anomaly coefficient.

Solution

The mass spurion has $R(m)=2-2(2/5)=6/5$. Since $R(\Lambda_{\rm H}^4)=0$, matching assigns $R(\Lambda_{\rm L}^5)=6/5$. Using the old R action on the light fields gives

$$k_{SU(3)^2R}=3+4\left(\frac25-1\right)=\frac35,$$

whose compensating scale charge is $2k=6/5$, in agreement.

The symmetric anomaly-free R symmetry of four massless flavors instead has $r_{\rm L}=1-3/4=1/4$, so its low scale parameter is neutral. These are different transformations. One can extend the latter to the massive high theory by giving the heavy pair scalar R-charge one and the fixed mass charge zero; its gauge anomaly then also cancels.

There is no inconsistency in the two charge tables. The first transforms a family with a charged mass; the second preserves that fixed mass and acts differently on the light fields. Neither computation identifies a superconformal R symmetry or determines an operator’s infrared dimension.

What the matching calculation has established

The result is a map between holomorphic parameters of two Wilsonian theories, including the theta-angle information carried away by a heavy pair. It can be derived at weak coupling, checked by anomalous transformations, and continued holomorphically in its nonzero mass parameter with the stated scheme choice.

It does not prove that a symmetry-allowed dynamical superpotential is generated, fix its nonzero coefficient, or determine the vacuum structure. Those questions require additional dynamical input. For the two-color example, the precise input for that next calculation is now available: the one-flavor theory carries $\Lambda_{\rm L}^5=m\Lambda_{\rm H}^4$.