The conformal window and exact R-charges

The electric meson $\mathcal M=\widetilde Q Q$ has engineering dimension two. Its elementary magnetic representative $M$ has engineering dimension one. Can these describe the same operator? They can if the renormalization-group flows reach a common infrared fixed point: engineering dimensions need not equal the scaling dimensions there. For our five-flavor example, both descriptions give the physical meson dimension $6/5$.

This lesson builds on the duality checks and positive supercharge norms and shortening. We will identify the R-generator entering the four-dimensional superconformal algebra, then use a physical descendant norm to test where the proposed description can hold.

Specify the vacuum and the fixed-point claim

Consider massless four-dimensional $\mathcal N=1$ $SU(N_c)$ SQCD, with $N_f$ fundamental–antifundamental chiral pairs, no electric superpotential, and $N_c\geq2$. There are four real Poincaré supercharges. We sit at the origin of the moduli space, with vanishing meson and baryon expectation values and unbroken flavor symmetry. Moving to a generic vacuum, adding masses, or adding a superpotential changes the question.

The standard Seiberg-duality picture assigns an interacting infrared superconformal field theory, or SCFT, to

$$\boxed{\frac32N_c<N_f<3N_c.} \tag{Q11.1}$$

This is the conformal window. The infrared theory has four additional real conformal supercharges; this does not turn it into a Poincaré $\mathcal N=2$ theory.

The existence claim uses dynamical evidence: a controlled weak-coupling fixed point near the upper edge, a magnetic weak-coupling description near the lower edge, and consistency across duality and deformations. A negative-norm test can exclude a proposed continuation, but cannot establish an interacting fixed point everywhere the test passes. Seiberg makes this distinction explicitly in §2, following (2.4). We use the usual SQCD phase assignment, not a proof of its existence from anomalies alone.

For $N_c=3,N_f=5$,

$$\frac92<5<9,$$

so the example lies inside the window. Neither integer five nor the distance from $9/2$ supplies a small perturbative parameter. The dimensions below are exact fixed-point statements, not a claim that either Lagrangian is weakly coupled for these ranks.

Which R-symmetry belongs to the conformal algebra?

Let $r$ be the common scalar R-charge of $Q$ and $\widetilde Q$. Their Weyl fermions have charge $r-1$, while the gaugino has charge one. With $T(\mathbf{fund})=1/2$ and $T(\mathrm{adj})=N_c$, the gauge–gauge–R anomaly is

$$k_{SU(N_c)^2R} =N_c+\frac{N_f}{2}(r-1)+\frac{N_f}{2}(r-1).$$

A conserved quantum R-current requires this coefficient to vanish, giving

$$R_0(Q)=R_0(\widetilde Q)=1-\frac{N_c}{N_f}. \tag{Q11.2}$$

This is not yet a general method for identifying a superconformal current. Baryon number $B$, with charges $+1,-1$, permits the equally anomaly-free assignments

$$R_t=R_0+tB,\qquad R_t(Q)=1-\frac{N_c}{N_f}+t,\qquad R_t(\widetilde Q)=1-\frac{N_c}{N_f}-t.$$

There are two further constraints in ordinary symmetric SQCD. The superconformal R-generator commutes with the full non-Abelian flavor symmetry, so it cannot contain a preferred flavor Cartan generator. The theory and this vacuum also preserve charge conjugation, which exchanges quarks and antiquarks and reverses $B$. The superconformal current must respect that exchange. Consequently $t=-t$, hence $t=0$.

This identifies $R_{\rm SC}=R_0$ provided there is no additional accidental current in the infrared. An accidental symmetry is an infrared symmetry absent from the finite-coupling ultraviolet description. Symmetry and charge conjugation settle this particular SQCD problem under the stated assumption; they do not settle theories with adjoint matter, asymmetric interactions, or emergent free sectors. The precise SQCD argument and its limitation appear in Intriligator–Wecht v3, introduction, printed pp. 2–3. The general method of a-maximization belongs to the next lesson.

Notice a useful intermediate fact: $\mathcal M$ is baryon neutral. Its charge $2(1-N_c/N_f)$ is independent of $t$ even before the exchange argument. Baryon dimensions, unlike the meson dimension, would distinguish nonzero $t$.

Turn the identified charge into a physical dimension

In radial quantization a scalar superconformal primary $|\mathcal O\rangle$ is annihilated by all conformal supercharges. The radial adjoint of a Poincaré supercharge is a conformal supercharge. In four-dimensional $\mathcal N=1$, the scalar matrix element of the relevant anticommutator has the form

$$\langle\mathcal O| \{(\overline Q_{\dot\alpha})^\dagger,\overline Q_{\dot\beta}\} |\mathcal O\rangle =c\,\delta_{\dot\alpha\dot\beta} \left(\Delta-\frac32r_{\mathcal O}\right), \qquad c>0, \tag{Q11.3}$$

for a unit-normalized state; $c$ depends only on supercharge normalization. Rotation generators have zero matrix element on the scalar primary. Here $\overline Q$ has R-charge $+1$, whereas $Q$ has charge $-1$. These are radial adjoints, not the Poincaré adjoints used for a particle mass bound.

The left side is the Gram matrix of the $\overline Q$ descendants because $(\overline Q)^\dagger|\mathcal O\rangle=0$. A chiral primary also obeys $\overline Q_{\dot\alpha}|\mathcal O\rangle=0$ for both components. Therefore

$$\boxed{\Delta(\mathcal O)=\frac32R_{\rm SC}(\mathcal O).} \tag{Q11.4}$$

The factor is $3/2$ in four dimensions, rather than the three-dimensional factor one. This is the scalar chiral relation in Seiberg, (2.2), and Intriligator–Wecht, (1.1). It applies to nonzero chiral primary classes. A chiral expression that is a supersymmetry descendant or vanishes in the chiral ring is not an additional primary with a dimension obtained this way.

Charges add in the nonzero chiral products defining mesons and baryons. Thus

$$\begin{aligned} R_{\rm SC}(\mathcal M)&=2\left(1-\frac{N_c}{N_f}\right), & \Delta_{\mathcal M}&=3\left(1-\frac{N_c}{N_f}\right),\\ R_{\rm SC}(B)&=R_{\rm SC}(\widetilde B) =N_c\left(1-\frac{N_c}{N_f}\right), & \Delta_B&=\Delta_{\widetilde B} =\frac{3N_c}{2}\left(1-\frac{N_c}{N_f}\right). \end{aligned} \tag{Q11.5}$$

For the recurring example these are

Object	R-charge	Physical scalar-primary dimension
Electric meson $\mathcal M$, or magnetic singlet $M$	$4/5$	$6/5$
Electric baryon or antibaryon	$6/5$	$9/5$
Corresponding magnetic two-quark baryon	$2(3/5)=6/5$	$9/5$

The elementary electric charge is $R(Q)=2/5$. The number $3R(Q)/2=3/5$ sometimes serves as shorthand for the corresponding quark wavefunction exponent. It is not the dimension of a gauge-invariant local primary state. Applying the scalar physical unitarity bound to it is invalid. The same caution applies to the magnetic quark value $3R(q)/2=9/10$. Mesons and color-contracted baryons are the physical local operators in this calculation.

A norm test exposes the lower edge

A nonidentity scalar primary in a unitary four-dimensional CFT satisfies $\Delta\geq1$. One way to see the obstruction is the trace-descendant norm, derived from the conformal algebra in the operators lesson:

$$\|P^2|\mathcal O\rangle\|^2=32\Delta(\Delta-1). \tag{Q11.6}$$

Positivity of the first descendants first gives $\Delta>0$ for a nonidentity scalar. The displayed norm then excludes $0<\Delta<1$. At $\Delta=1$, the null relation $P^2|\mathcal O\rangle=0$ is the free-field equation at separated insertions: a scalar saturating this bound is a free sector, not an interacting meson with the same shortening.

For our meson,

$$\|P^2|\mathcal M\rangle\|^2 =32\frac65\frac15=\frac{192}{25}>0.$$

Its normalized two-point function scales as $|x|^{-12/5}$, and its cylinder energy gap on a sphere of radius $L$ is $6/(5L)$. These are observable consequences of $\Delta_{\mathcal M}$; neither determines the two-point normalization in a separately chosen ultraviolet field convention.

Applying the norm requirement to (Q11.5) gives

$$3\left(1-\frac{N_c}{N_f}\right)\geq1 \quad\Longleftrightarrow\quad N_f\geq\frac32N_c.$$

At equality the meson is free. Below equality, continuing the same interacting R-identification would produce a negative-norm descendant. The conclusion is that this identification and interacting description must change. It does not exclude the underlying unitary gauge theory.

Check the weak edge without treating it as a global proof

Define $\gamma$ by the convention that the quark wavefunction exponent is $1+\gamma/2$. In an NSVZ coupling scheme the canonical gauge beta function is

$$\beta_g=-\frac{g^3}{16\pi^2} \frac{3N_c-N_f+N_f\gamma(g)} {1-N_cg^2/(8\pi^2)},\qquad \gamma(g)=-\frac{g^2}{8\pi^2}\frac{N_c^2-1}{N_c} +O(g^4). \tag{Q11.7}$$

This sourced beta-function input is Seiberg, (2.1); its denominator and coupling value are scheme dependent. At weak coupling $\gamma\to0$, so asymptotic freedom requires $N_f<3N_c$, giving the upper edge. At a nonzero fixed point in a nonsingular coordinate patch, the numerator gives $\gamma_*=1-3N_c/N_f$, consistently yielding $\Delta_{\mathcal M}=2+\gamma_*$. An unknown function $\gamma(g)$ is not an existence proof.

Near the upper edge there is perturbative control. Set $x=N_f/N_c=3-\epsilon$, $a=N_cg^2/(8\pi^2)$, and $c_F=1-1/N_c^2$. At small $a$,

$$\beta_a=-\frac{a^2}{1-a} \left[\epsilon-(3-\epsilon)c_Fa+O(a^2)\right].$$

For $\epsilon\ll1$,

$$a_*=\frac{\epsilon}{3c_F}+O(\epsilon^2), \qquad \gamma_*=-\frac{\epsilon}{3}+O(\epsilon^2), \qquad \Delta_{\mathcal M}=2-\frac{\epsilon}{3}+O(\epsilon^2). \tag{Q11.8}$$

The beta-function slope is positive at this small root, so small deviations shrink toward the infrared. The exact expression $3(1-1/(3-\epsilon))$ has the same expansion. Small $\epsilon$ is most naturally controlled with large ranks and fixed ratio; it is not a continuous flavor adjustment at fixed small integer $N_c$.

The magnetic gauge group has rank parameter $\widetilde N_c=N_f-N_c$, with one-loop coefficient

$$b_{\rm m}=3\widetilde N_c-N_f=2N_f-3N_c.$$

It becomes small and positive just above the lower edge. Its weak fixed point involves both gauge and superpotential couplings. The magnetic theory with $h=0$ is a different fixed point: $M$ is then a free singlet, while the magnetic SQCD meson $\widetilde q q$ has dimension $3N_c/N_f$. The perturbation $hM\widetilde q q$ therefore has chiral dimension

$$1+\frac{3N_c}{N_f}<3$$

throughout the open window and is relevant, since $d^2\theta$ contributes dimension one. At the final fixed point $h$ is nonzero. The condition $R(W)=2$ does not by itself say that varying $h$ gives an exactly marginal physical deformation. Tong, §6.5.2, printed pp. 202–203

The boundaries require a different interpretation

Flavor range	Infrared description at the origin
$N_f=3N_c$	Gaussian electric limit, approached with logarithmic running; the meson has dimension two.
$3N_c/2<N_f<3N_c$	Interacting electric–magnetic fixed point; (Q11.5) applies.
$N_c+2\leq N_f\leq3N_c/2$	Free magnetic phase with $SU(N_f-N_c)$ variables, free singlet mesons, and couplings tending to zero.

Equality at the lower edge is possible only for suitable integer ranks. The table’s magnetic row assumes a nontrivial magnetic gauge group. The special $SU(2)$, $N_f=3$ lower-edge case instead belongs to the already discussed $N_f=N_c+1$ confining description; it has no non-Abelian magnetic gauge sector. For $SU(3)$, the integer interacting values are $N_f=5,6,7,8$; the lower endpoint $9/2$ is not a theory with an integer number of flavors.

For $N_f>3N_c$, the usual electric theory is infrared free and is not an asymptotically free ultraviolet completion. Its meson still has dimension two. Continuing (Q11.5) above the window would give a different answer, signaling a wrong R-identification at the Gaussian limit.

Similarly, in the free magnetic phase the physical meson has $\Delta_M=1$ and $R_{\rm SC}(M)=2/3$. The inherited anomaly-free ultraviolet R-current can remain meaningful without being the infrared superconformal current. Once interactions vanish, additional infrared currents permit the required mixing. Continuing the electric formula below the window misses these accidental symmetries. The phase assignment includes the running of the Yukawa coupling as well as the gauge coupling; the sign of $b_{\rm m}$ alone is not a proof of the entire statement. Seiberg, §3, printed p. 8

Finally, retain the normalization dictionary $\mathcal M=\mu_*M$, $W=hM q\widetilde q$ from the duality lesson. Here $\mu_*$ converts holomorphic ultraviolet field conventions: engineering dimensions two and one. In the infrared, renormalized representatives on both sides have the same scaling exponent. The matching parameter is not an extra mass deformation or an instruction to add one to the infrared dimension. Wavefunction factors carry the conversion between these descriptions.

Exercises

1. Follow an incorrect continuation to its contradiction

Consider $SU(5)$ with seven flavor pairs. Apply the interacting formula provisionally to the meson and calculate its trace-descendant norm. Identify the actual phase, the magnetic gauge group, and the dimensions of the meson and magnetic baryon at its Gaussian infrared limit. Explain why repairing only the meson charge does not generally identify every infrared charge.

Hint 1

Compare seven with $3N_c/2$ and $N_c+2$. A magnetic baryon uses $N_f-N_c$ magnetic quarks.

Hint 2

The invalid meson continuation gives $\Delta=6/7$. At the actual free magnetic limit the scalar in each free chiral has engineering and scaling dimension one; form a gauge-invariant baryon before interpreting its dimension.

Solution

The putative meson charge is $4/7$ and its dimension is $6/7$. Its trace-descendant norm is

$$32\frac67\left(-\frac17\right)=-\frac{192}{49},$$

which cannot occur in a positive physical Hilbert space. Here $N_c+2=7<15/2$, so the theory is in the free magnetic phase with gauge group $SU(2)$ and $b_{\rm m}=6-7=-1$.

At the Gaussian magnetic limit, $\Delta_M=1$ and the gauge-invariant two-quark baryon has $\Delta_b=2$, with $R_{\rm SC}(M)=2/3$ and $R_{\rm SC}(b)=4/3$. The inherited electric formula would instead give $\Delta_B=15/7$. Thus it is not enough to replace one entry of an interacting table while retaining all its other identifications. Accidental currents at the free limit change the superconformal R-generator. No negative-norm physical quark state has been asserted: all bounds here tested gauge-invariant composites.

2. Assign a dimension to the mass source

At the $SU(3)$, five-flavor fixed point, perturb by one chiral meson component:

$$\delta S=\int d^4x\,d^2\theta\, \lambda_{\rm IR}\mathcal M_{\rm IR}+\text{complex conjugate}.$$

Find the scaling dimension of $\lambda_{\rm IR}$. At a reference energy $E_0$ where the fixed-point description is applicable, let the dimensionless source be $\widehat\lambda_0\ll1$. Estimate the energy where the deformation becomes order one. Explain why this does not contradict the engineering dimension one of the ultraviolet quark mass, or compute an exact pole mass.

Hint 1

The superspace measure has dimension $-3$. Use the meson dimension, not the dimension of its highest component twice.

Hint 2

To linear order near the fixed point, $\widehat\lambda(E)=\widehat\lambda_0(E_0/E)^{3-\Delta_{\mathcal M}}$.

Solution

The source has scaling dimension $3-6/5=9/5$, hence it is relevant. Setting its dimensionless magnitude to one gives

$$E_{\rm cross}\sim E_0|\widehat\lambda_0|^{5/9}.$$

This is the leading scaling estimate near the undeformed fixed point, up to the normalization of the source and subsequent dynamics. The ultraviolet parameter $m$ multiplying the engineering-dimension-two composite $\widetilde Q Q$ still has engineering dimension one. Its matching to $\lambda_{\rm IR}$ includes the scale and operator-renormalization factors needed to convert operator conventions. The estimate is a crossover energy, not an exact particle mass. In particular, removing one flavor leads to the distinct four-flavor dynamics already studied, rather than automatically to a completely gapped theory.

3. Test the magnetic interaction before turning it on

For the five-flavor example, begin at the fixed point of magnetic $SU(2)$ SQCD with $h=0$, together with twenty-five decoupled singlets $M$. Calculate the dimension and relevance exponent of $M\widetilde q q$. Then determine the singlet dimension at the coupled dual fixed point. Why is it wrong to retain $\Delta_M=1$ while imposing the final nonzero interaction?

Hint 1

Without $h$, the magnetic SQCD factor has $R(q)=1-2/5=3/5$. Its gauge-invariant meson is a chiral primary; the singlet is independently free.

Hint 2

At the coupled fixed point $R(M)+R(q)+R(\widetilde q)=2$. The source exponent of a chiral superpotential perturbation is three minus the bottom-component dimension.

Solution

At $h=0$, $\Delta_{\widetilde q q}=9/5$ and $\Delta_M=1$. The product in the decoupled theory is a chiral primary with dimension $14/5$. The source exponent is $3-14/5=1/5>0$, so this interaction destabilizes the decoupled fixed point toward the infrared.

At the coupled fixed point the gauge anomaly and exchange symmetry retain $R(q)=R(\widetilde q)=3/5$. The nonzero superpotential requires $R(M)=4/5$, hence $\Delta_M=6/5$. Keeping $M$ free would use the wrong infrared R-generator and the wrong theory. Moreover $\widetilde q q$ becomes a supersymmetry descendant through the $M$ equation of motion; it is no longer an independent chiral-ring primary to which the uncoupled dimension argument can be applied. The R-charge condition on the interaction does not establish an exactly marginal family.

You can now identify which R-charge gives a physical meson dimension and diagnose a failed extrapolation without declaring SQCD nonunitary. a-maximization chooses the superconformal R-generator when symmetry leaves genuine mixing parameters, with the same care about accidental currents.