Conformal Technicolor Bounds

• Conformal technicolor bounds are nonperturbative constraints that restrict scaling dimensions and operator spectra in nearly conformal or walking gauge theories.
• The approach combines gauge theory, lattice simulations, and conformal bootstrap techniques to rigorously determine operator bounds and probe IR dynamics.
• These bounds guide model building by ensuring radiative stability, controlling flavor physics, and satisfying collider constraints in extended technicolor scenarios.

Conformal technicolor bounds are theoretical and nonperturbative constraints on the scaling dimensions and properties of operators in candidate conformal or nearly conformal (walking) strongly coupled theories, motivated by the need to realize viable extensions of the Standard Model in which electroweak symmetry breaking arises dynamically. These bounds are formulated in the framework of gauge theories, lattice simulations, and the conformal bootstrap, and address the interplay of conformality, chiral symmetry breaking, operator algebra, and phenomenological constraints from flavor and electroweak data. They have shaped the development of realistic technicolor scenarios by identifying which operator spectra and dynamical behaviors are compatible with radiative stability, flavor physics, and collider phenomenology.

1. Theoretical Framework: Conformal Versus Walking Behavior

The central question in technicolor and related models is whether the infrared (IR) dynamics of the theory is controlled by an IR fixed point (yielding a true conformal field theory), or whether chiral symmetry breaking occurs before the RG flow reaches such a fixed point, resulting in a walking regime. The running of the gauge coupling $g$ is governed by the two-loop Callan–Symanzik beta function,

$$\beta(g) = \mu\, \frac{\partial g}{\partial \mu} = -\frac{\beta_0 g^3}{(4\pi)^2} - \frac{\beta_1 g^5}{(4\pi)^4} + \cdots$$

with the sign and magnitude of $\beta_0, \beta_1$ determined by the matter content. If $\beta_0 > 0$ and $\beta_1 < 0$, a second zero appears at two-loop, indicating a possible IR fixed point. In conformal scenarios, the coupling approaches $g_*$ ($\beta(g_*) = 0$), and the theory is scale-invariant in the IR. If chiral symmetry breaking happens at a scale where $g > g_*$, the flow never reaches conformality: instead, the coupling "walks" (remains nearly constant) over a wide energy interval before symmetry breaking triggers confinement and mass generation (Kogut et al., 2010, Sinclair et al., 2010).

Lattice studies with color-sextet quarks (Kogut et al., 2010, Sinclair et al., 2010) provide direct evidence of this distinction, finding two well-separated transition scales: deconfinement (indicated by an abrupt rise in the Polyakov loop) occurs at one value $\beta_d = 6/g^2$, and chiral symmetry restoration (vanishing of the chiral condensate $\langle \bar\psi \psi \rangle$) at a much larger $\beta_\chi$; the significant separation is not observed in QCD with fundamental quarks. The running coupling in these regimes supports a slow evolution (walking), with no clear IR fixed point realized before chiral symmetry breaking.

2. Operator Scaling and Bootstrapped Bounds

The conformal bootstrap provides rigorous, nonperturbative upper bounds on the scaling dimension $\Delta_S$ of the lowest-dimension scalar singlet appearing in the operator product expansion (OPE) of a primary scalar (interpreted, in conformal technicolor, as the composite Higgs operator $H$) with its conjugate: $H \times H^\dagger \supset 1 + |H|^2 + \cdots$ A phenomenologically viable model demands that $\Delta_{|H|^2}$ is sufficiently large (preferably $\gtrsim 4$) so that the operator $|H|^2$ is irrelevant: this ensures radiative stability of the electroweak scale and avoids the naturalness problem. At weak coupling or large $N$, one might expect $\Delta_{|H|^2} = 2\Delta_H$, but robust bootstrap constraints show that this is not generally realized at small $N$ and strong coupling (Rattazzi et al., 2010, Rychkov, 2 Sep 2025).

The bootstrap approach leverages crossing symmetry and unitarity of four-point functions to generate "vectorial sum rules" for OPE coefficients and scaling dimensions. The explicit form,

$$1 = \sum_{\Delta, \ell} p_{\Delta,\ell}\, F_{\Delta_H,\Delta,\ell}(z,\bar z), \quad p_{\Delta,\ell} = \lambda_{\Delta,\ell}^2$$

is subject to positivity (from unitarity), and the allowed region of operator dimensions is determined by searching for separating functionals. The bound on $\Delta_{|H|^2}$ as a function of $\Delta_H$ is nontrivial, especially near the region $\Delta_H = 1 + \varepsilon$ for small $\varepsilon$, and excludes large regions of parameter space for composite Higgs candidates (Rychkov, 2 Sep 2025). As $\Delta_H \to 1$, the bound approaches the free-field value $\Delta_{|H|^2} \to 2$.

3. Four-Fermion Interactions and Conformal Window Reduction

In extended technicolor (ETC) models, additional four-fermion interactions are inevitable and influence the phase diagram of strong gauge theories. The inclusion of such terms—formally, via Lagrangian additions of the type

$$\mathcal{L}_{4f} = \frac{G}{N_f d[r]}\left[ (\bar\psi \psi)^2 + (\bar\psi\, i\gamma_5 T^a \psi)^2 \right]$$

with $G$ the four-fermion coupling—lowers the critical value for gauge coupling $\alpha$ at which spontaneous chiral symmetry breaking (S$\chi$SB) occurs, thereby shrinking the conformal window (Fukano et al., 2010). The critical number of flavors $N_f^{\text{crit}}$ required for conformality, for sufficiently strong four-fermion coupling, becomes

$$N_f^\text{crit}(N,g) = \frac{34N(\sqrt{g} - g) + 33 C_2(r)}{20N(\sqrt{g} - g) + 12[1 + (\sqrt{g} - g)]C_2(r)}\, \frac{N}{C(r)}$$

where $g$ is a dimensionless strength associated to $G$, and $C_2(r)$, $C(r)$ are group invariants.

Further, the mass anomalous dimension $\gamma_m$ is enhanced,

$$\gamma_m(g) = 1 + \omega, \quad \omega = \sqrt{1 - (\alpha/\alpha^0)}$$

with $\alpha^0$ the critical gauge coupling in the absence of four-fermion terms. In practical model building this enhancement is crucial: values of $\gamma_m \gtrsim 1$ are necessary to produce observed quark masses at high ETC scales while keeping flavor-changing neutral currents (FCNCs) in check.

4. Lattice Studies: Infrared Conformality Versus Confinement

Systematic lattice investigations of candidate walking and conformal technicolor models provide quantitative support for theoretical expectations. Studies of SU(2) gauge theory with two adjoint fermions (the minimal walking technicolor model) probe the presence of an IR fixed point by correlating the scaling behavior of mesonic and gluonic observables (Debbio et al., 2010). In these lattices, the mass ratios of mesons to gluonic excitations (e.g., $M_{\text{PS}}/\sqrt{\sigma}$) remain finite and nonzero as the fermion mass vanishes, and universal power-law scaling is observed,

$M \propto m^{1/(1+\gamma_*)}$

where $M$ is any physical mass and $\gamma_*$ is the mass anomalous dimension at the fixed point (estimated in the range $0.16$–$0.28$). Such behavior is incompatible with chiral symmetry breaking (where the pseudo-scalar mass would vanish more rapidly than gluonic masses), supporting IR conformality.

Topological observables, such as the susceptibility $\chi$ and instanton size distribution, display further signs of IR conformality: in mass-deformed minimal walking technicolor, the susceptibility measured is nearly identical to the corresponding pure Yang–Mills theory, indicating the decoupling of fermion dynamics in the IR (Bennett et al., 2012). The scaling law

$$\chi \propto m^{4/(1+\gamma_*)}, \qquad \langle \rho \rangle \propto m^{-1/(1+\gamma_*)}$$

may be used to extract $\gamma_*$, providing an important nonperturbative check on model viability.

5. Phenomenological and Collider Bounds

A central phenomenological constraint arises from the precision electroweak S parameter. Calculations near the upper end of the conformal window yield an exact lower bound (in normalized units),

$$\lim_{q^2/m^2 \to 0} \frac{6\pi S}{\mathcal{N}} = 1 + \frac{17}{72} \gamma_m(\alpha_*)$$

where $\mathcal{N}$ is the effective number of electroweak doublets (Sannino, 2010). This leads to $S_\text{norm} \geq 1$, typically in conflict with experimental limits ($S \lesssim 0.2$). In weakly coupled holographic models, the combination of Kaluza–Klein spectrum sum rules with unitarity and cutoff arguments yields $S \gg 0.2$, which is strongly disfavored by data (Levkov et al., 2012).

Collider searches for technirho resonances in diboson channels provide further distinctive signatures. In models implementing the conformal barrier and hidden local symmetry, decay channels such as $\rho \to WH,\, ZH$ are forbidden, while branching ratios into $WW,\, WZ,\, ZZ$ remain sizable. The observed widths are narrow (e.g., $\Gamma/M_\rho \sim 70$ GeV/$2$ TeV), and the absence of $VH$ channels in LHC Run II would strongly suggest that scale invariance is operative and the Higgs is a (pseudo-)dilaton (Fukano et al., 2015).

The role of condensate enhancement is also critical. In flavor physics, particularly with ETC scales constrained to $\Lambda_\text{ETC} \gtrsim 1000$ TeV by $D$–meson mixing, the requirement is that the technifermion condensate (renormalized at the ETC scale) satisfies

$$\langle \bar U U \rangle_\text{ETC} \sim (\Lambda_\text{ETC}/\Lambda_\text{TC})^{\gamma_m} \langle \bar U U \rangle_\text{TC}$$

where large $\gamma_m$ is needed to successfully generate observed quark masses (Chivukula et al., 2010).

6. Emergence and Evolution of Conformal Technicolor Bounds via the Bootstrap

The modern numerical conformal bootstrap, beginning with the work of Rattazzi et al. and its application to conformal technicolor, provided the first rigorous nonperturbative exclusion bounds on the dimensions of relevant scalar operators in 4D CFTs. By analyzing crossing symmetry of four-point functions and transforming the problem into a linear programming one, upper bounds such as

$\Delta_{|H|^2} \leq \Delta_*(\Delta_H)$

were established, with the bound approaching $\Delta_{|H|^2}\to 2$ as $\Delta_H\to 1$ (free field limit), but rising rapidly for $\Delta_H > 1$. This method enabled the exclusion of large swathes of parameter space previously thought accessible to conformal technicolor scenarios (Rychkov, 2 Sep 2025, Rattazzi et al., 2010).

The evolution of these methods—from discrete sampling to high-order derivative expansion at symmetric kinematic points and ultimately to semidefinite programming—strengthened the reliability and sharpness of the bounds, culminating in the so-called "islands" in operator dimension space with possible identification with known CFTs.

7. Implications for Model Building and Future Directions

Conformal technicolor bounds circumscribe the space of viable models for dynamical electroweak symmetry breaking, ensuring compatibility with radiative stability, flavor physics, and precision data. The requirements $\Delta_{|H|^2}\gtrsim 4$ and large $\gamma_m$ for condensate enhancement, as well as suppressed $S$ parameter contribution, are interconnected theoretical and phenomenological constraints. Walking models, especially those implementing strong ETC-induced four-fermion interactions, are forced to lie near or inside the conformal window for viability (Fukano et al., 2010).

Numerical lattice simulations and bootstrap-based conformal bounds are indispensable for refining candidate models, integrating topological, spectroscopic, and scattering observables to conclusively determine IR behavior.

Continued refinement in numerical bootstrap techniques, larger-scale lattice studies extending to more exotic representations and new gauge groups, and further collider data, notably on diboson resonance searches, are necessary for closing the remaining gaps in understanding and for testing which, if any, conformal technicolor scenarios remain experimentally tenable.