Controlled Families of Interacting CFTs

An interacting fixed point is quantitatively controlled when its existence conditions, small parameter or nonperturbative assumptions, invariant observables, and error estimates are all explicit. No single method covers every family: weak-coupling expansions, $1/N$ expansions, numerical bootstrap, and regulated statistical systems constrain different aspects and have different limiting procedures. Agreement becomes persuasive only after the quantities and normalizations have been translated into the same CFT data.

Required background. Perturbative CFT data near free fixed points supplies fixed-point observables and truncation tests. Ultraviolet and infrared fixed points supplies the RG endpoint logic. Helpful background. Complete lattice error budgets explains continuum, volume, autocorrelation, and statistical errors in regulator-based evidence.

What constitutes a controlled family

Suppose a family is described by parameters $p$ and a proposed fixed point $g_*(p)$. A quantitative claim should answer six questions.

1. Existence: Does a real RG zero or a consistent CFT solution exist in the stated domain? Is the statement perturbative, numerical and assumption-dependent, or constructive?
2. Symmetry and unitarity: Which spacetime and internal symmetries are imposed, and is reflection positivity assumed, tested, or absent?
3. Control: What dimensionless parameter suppresses omitted terms—$\epsilon$, $1/N$, a weak fixed-point coupling, inverse charge, lattice spacing over correlation length, or a numerical truncation parameter?
4. Invariant output: Which scaling dimensions, normalized OPE coefficients, current normalizations, sphere free energies, or critical exponents are compared?
5. Error: Is the uncertainty a proven bound, a statistical interval, an extrapolation estimate, or an asymptotic truncation estimate?
6. Failure: What observation would invalidate the approximation, the assumed spectrum, or the universality identification?

Coupling coordinates and beta-function coefficients are generally scheme-dependent. Fixed-point scaling dimensions, normalized OPE coefficients, and eigenvalues of the linearized RG flow are the proper comparison targets. Even those require matched operator labels and normalizations; two numbers called $C_T$ can differ by the free-scalar convention used to define them.

Four complementary routes

Expansion from a solvable endpoint

The Wilson–Fisher family begins at the free scalar in $d=4$ and is expanded in $\epsilon=4-d$. The fixed point, multiplet recombination, and mixing are calculable order by order. Its strongest direct claim is near $\epsilon=0$; a three-dimensional number obtained at $\epsilon=1$ additionally depends on resummation and on how the remainder is estimated. The original fixed-point construction is Wilson and Fisher 1972, pp. 240–243.

Weak gauge, Yukawa, and scalar fixed points follow the same logic when every fixed-point coupling is parametrically small and the scalar potential is stable. A zero of a truncated beta function is not sufficient: it must persist under a consistent increase in loop order and give physical, scheme-independent observables stable to the claimed accuracy.

Large N

Vector and matrix families may be accessible for fixed spacetime dimension as $N\to\infty$. The expansion controls correlators at fixed operator quantum numbers, but it need not be uniform at large spin, large excitation number, or near a dimension where another mode becomes light. The large-N data page gives the vector-model normalization and a first correction. Agreement between an $\epsilon$ expansion and a $1/N$ expansion in an overlap region is particularly useful because their diagrammatic organizations are different.

Numerical crossing constraints

The numerical bootstrap imposes crossing and unitarity directly on CFT data. Exclusion bounds follow from the stated correlators and numerical certification; small allowed regions require extra spectral assumptions such as gaps, uniqueness, or symmetry assignments. Increasing derivative order and numerical precision tests convergence, but it does not prove that every point in an allowed region is realized by a CFT. The convex-optimization setup, spectrum extraction, and applications to three-dimensional models are reviewed in Poland, Rychkov, and Vichi 2019, §§IV–V.

Regulated critical systems

A lattice simulation begins from a microscopic regulator, tunes to a continuous transition, and uses finite-size scaling to infer universal exponents. One must take the thermodynamic and continuum scaling limits, model irrelevant corrections, control autocorrelation and sampling errors, and justify the universality-class identification. Agreement between two different microscopic models with suppressed leading corrections is stronger than a single fit because regulator details differ.

A common-observable comparison

The three-dimensional Ising universality class illustrates the translation. With a $\mathbb Z_2$-odd scalar $\sigma$ and leading even scalar $\varepsilon$,

$$\Delta_\sigma=\frac{d-2+\eta}{2}, \qquad \Delta_\varepsilon=d-\frac1\nu.$$

A finite-size study reported $\nu=0.63002(10)$ and $\eta=0.03627(10)$, which translate to $\Delta_\sigma=0.518135(50)$ and $\Delta_\varepsilon=1.41275(25)$ after linearly propagating the quoted uncertainties Hasenbusch 2010, Abstract. A mixed-correlator bootstrap study reported $\Delta_\sigma=0.5181489(10)$ and $\Delta_\varepsilon=1.412625(10)$ under its crossing, unitarity, symmetry, gap, and uniqueness assumptions Kos, Poland, Simmons-Duffin, and Vichi 2016, Abstract and §§2–4. The intervals differ in meaning: the lattice values include a fit and regulator extrapolation, whereas the bootstrap island is conditional on its spectral assumptions and numerical truncation. Their agreement is a cross-method consistency test, not permission to replace either error analysis by the smaller interval.

The following semantic table compares the main controlled regimes used across this chapter. “Remainder” names what must still be bounded or estimated; it is not automatically a rigorous uncertainty.

Regime and target	Dimension or sector	Control parameter	Invariant observable	Computed order or numerical control	Error or extrapolation	Independent check	Evidence basis	Known failure
Wilson–Fisher $O(N)$ data	$d=4-\epsilon$; continuation toward a stated target $d$	$\epsilon$	$\Delta_\phi$, singlet dimensions, normalized OPE data, RG eigenvalues	finite loop or series order declared with the result	omitted powers and resummation dependence at finite $\epsilon$	large $N$, fixed-dimension bootstrap, critical-system data	analytic asymptotic expansion about a Gaussian fixed point	precision at $\epsilon=1$ from one or two raw terms
Critical vector-model data	fixed $2<d<4$ and fixed operator quantum numbers	$1/N$	singlet and nonsinglet dimensions, OPE scaling, $C_J$, $C_T$	finite order in $1/N$	higher orders and nonuniform spin or dimension limits	$4-d$ and $d-2$ expansions, bootstrap	tuned saddle plus diagrammatic $1/N$ expansion	reliable small-$N$ values without convergence evidence
Weak gauge/Yukawa/scalar data	declared spacetime dimension and stable fixed-point branch	loop-counting combinations of all $g_*^I$	anomalous-dimension eigenvalues and normalized correlators	stated loop order with consistent fixed-point substitution	loop truncation and competing roots	nonsingular scheme change and independent loop organization	perturbative simultaneous RG zero plus stability conditions	existence from one scheme-dependent root alone
Lowest fixed-charge dimension	one charge sector of a fixed CFT in $d>2$	$Q^{-1/(d-1)}$	$\Delta(Q)$ coefficients and excitation gaps	declared derivative and Goldstone-loop order	higher derivatives, loops, and possible extra light modes	several charges and stable finite-$Q$ fits	sectoral EFT about a homogeneous finite-density saddle	statements about neutral or arbitrary excited sectors
Nearly conserved higher-spin data	fixed spin in a declared large-$N$ spectrum	nonconservation strength, often $N^{-1/2}$	current anomalous dimensions and constrained three-point data	order matched between divergence, norms, and factorization	operator mixing and higher breaking order	descendant norms and crossing	approximate Ward identities plus large-$N$ factorization	exact higher-spin symmetry at finite breaking
Bootstrap dimensions and OPE data	fixed $d$, symmetry, correlator set, gaps, and uniqueness assumptions	derivative order, spin treatment, arithmetic precision	certified exclusions or conditional allowed intervals	finite functional space and numerical precision	convergence and assumption dependence	altered correlator systems and regulator results	crossing and unitarity optimization under declared spectral assumptions	proof that every allowed point is an existing CFT
Critical-system exponents	finite regulator with $\xi/a\gg1$ and $L/\xi$ controlled	$a/\xi$, $\xi/L$, statistics	exponents mapped to CFT dimensions and universal ratios	finite-size and continuum fit ansatz	regulator, volume, irrelevant-operator, autocorrelation, and sampling errors	distinct microscopic actions and bootstrap	regulated statistical inference plus universality identification	an exact continuum CFT from one finite lattice size
Collision and walking data	local center manifold near a fixed-point pair	distance $\lambda$ from collision	RG eigenvalues, walking scale, complex-conjugate data	normal form through declared nonlinear order	higher beta terms and analytic-continuation ambiguity	several observables and finite-size drift	bifurcation analysis, optionally supplemented by complex-CFT perturbation theory	a real fixed point merely from slow running

Overlap and noncommuting limits

Two methods overlap only if their validity domains overlap. For example, a comparison of $1/N$ and $\epsilon$ expansions should keep both $N$ large and $\epsilon$ small before extrapolating. A term of the form $\epsilon^2N$, $1/[N(4-d)]$, or $\ell/N$ reveals that taking one limit first can erase effects retained by the other. The correct response is a double-scaling analysis or a restriction of the claim, not an average of incompatible truncations.

Cross-method comparisons should use a small set of shared invariants and show the transformation explicitly. Critical exponents map to operator dimensions as above; susceptibility-amplitude ratios or finite-volume observables may require additional universal relations; raw bare couplings and regulator masses should not be compared.

Failure tests

Existence test. Follow the fixed point as the control parameter varies. A root that becomes complex, violates stability, or merges with another root changes the physical conclusion.

Unitarity test. Check norm positivity or the assumed bootstrap positivity conditions. Analytic continuation in $N$ or $d$ can invalidate them even when formulas remain finite.

Extrapolation test. Remove the smallest-$N$, lowest-order, smallest-volume, or lowest-derivative data in turn. A conclusion that moves beyond its quoted uncertainty is not stable.

Independence test. Identify shared inputs between methods. Two resummations of the same perturbative coefficients are not fully independent evidence, and two lattice actions can share the same finite-size ansatz.

Exercises

Propagate the quoted Hasenbusch uncertainties to $\Delta_\sigma$ and $\Delta_\varepsilon$, treating $\nu$ and $\eta$ as uncorrelated.

Solution

$\delta\Delta_\sigma=\delta\eta/2=0.00005$. Since $\Delta_\varepsilon=3-1/\nu$, linear propagation gives $\delta\Delta_\varepsilon=\delta\nu/\nu^2\simeq0.000252$. Correlations and systematic errors would require the original covariance and fit analysis.

Why is agreement of $\Delta_\sigma$ insufficient to identify two theories as the same CFT?

Solution

One dimension is not a complete invariant. Distinct CFTs can have nearby leading dimensions. The comparison should include symmetry, several operator dimensions, normalized OPE coefficients or current data, and compatible RG or universality information.

References

• Hasenbusch, M. (2010), “A finite size scaling study of lattice models in the three-dimensional Ising universality class,” Physical Review B 82, 174433. doi:10.1103/PhysRevB.82.174433. Open PDF
• Kos, F., Poland, D., Simmons-Duffin, D., and Vichi, A. (2016), “Precision islands in the Ising and $O(N)$ models,” Journal of High Energy Physics 2016(08), 036. doi:10.1007/JHEP08(2016)036. Open PDF
• Poland, D., Rychkov, S., and Vichi, A. (2019), “The conformal bootstrap: Theory, numerical techniques, and applications,” Reviews of Modern Physics 91, 015002. doi:10.1103/RevModPhys.91.015002. Open PDF
• Wilson, K. G., and Fisher, M. E. (1972), “Critical exponents in 3.99 dimensions,” Physical Review Letters 28, 240–243. doi:10.1103/PhysRevLett.28.240