UV Completion of the Standard Model

• Ultraviolet completion of the Standard Model is defined as a framework that replaces high-energy point-like interactions with extended classical field configurations known as classicalons.
• The mechanism of classicalization employs a self-sourcing classicalizer field that dynamically forms classicalons, shifting scattering processes from hard, perturbative events to soft, geometric interactions.
• This paradigm predicts distinct experimental signatures, such as high-multiplicity soft final states and geometric cross sections, offering an alternative to conventional Wilsonian UV completions.

An ultraviolet (UV) completion of the Standard Model (SM) is a theoretical framework in which the apparent non-renormalizability or missing high-energy behavior of the SM is resolved by new dynamics or symmetries that render the theory predictive and unitary at arbitrarily high energies. Traditional Wilsonian approaches posit that new weakly-coupled degrees of freedom or new gauge symmetries enter at high scales, but alternative paradigms, such as classicalization, propose self-completion via the formation of extended classical field configurations. In the "UV-Completion by Classicalization" scenario, UV completion is realized not by adding new point-like particles, but by the non-perturbative formation of "classicalons"—extended, semiclassical field objects generated dynamically during high-energy scattering. This approach offers a qualitatively distinct resolution to the limitations of the Standard Model at high energy.

1. Classicalization: Mechanism and Concept

Classicalization refers to a dynamical process by which a high-energy, non-renormalizable theory avoids unitarity violation not by adding new weakly-coupled particles, but by forming extended classical field configurations called "classicalons." In contrast with the standard Wilsonian paradigm—where higher and higher energies probe ever-shorter distances—classicalization posits that attempts to localize energy beyond a critical scale instead "inflate" the interaction region, making it larger as the energy increases.

The key elements of the mechanism are:

• The presence of a "classicalizer" field whose interactions become strong for localized energy-momentum sources.
• Formation of classicalons with a radius $r_*$ that increases with the center-of-mass energy $\sqrt{s}$, so that hard scattering with large momentum transfer is replaced by production of extended field configurations.
• The dominance of long-distance, infrared dynamics in the UV, leading to self-unitarization and effectively "self-completing" the theory at high energies.

Formally, the general Lagrangian is of the form:

$$\mathcal{L}(\phi, J) = (\partial\phi)^2 + M_*^2 J \phi + \ldots$$

where $M_*$ is the putative unitarity violation scale, and $J$ is a source term that grows with $\sqrt{s}$ in high-energy scattering. When probed at energies much larger than $M_*$, the interaction region grows as:

$r_* \sim L_* (\sqrt{s} L_*)^\alpha$

with $L_* = 1/M_*$ and $\alpha>0$.

2. Classicalons and the Classicalizer Field

The classicalizer field $\phi$ (which may be a Nambu-Goldstone boson, the longitudinal polarization of a massive gauge boson, or a scalar coupled to $T_{\mu}^{\mu}$) interacts with localized energy sources such that attempts to probe short distances ($L\ll L_*$) are dynamically frustrated: the classical solution $\phi(r)$ extends over a large region whose size $r_*$ grows with $\sqrt{s}$. The process is non-perturbative, as the field self-sources itself strongly, and the high-energy state becomes a coherent, semi-classical configuration rather than a collection of few, high-momentum quanta.

In particular, for derivative couplings, any localization of energy sharp enough to violate perturbative unitarity triggers formation of a classicalon. The process is deeply analogous (though not identical) to the way black holes "protect" short distances in gravity by classicalizing above the Planck scale.

3. Impact on Scattering Amplitudes and Unitarity

In perturbation theory, amplitudes such as $2\to 2$ or $n\to m$ scatterings in theories with non-renormalizable couplings typically grow with energy, violating unitarity around $M_*$. Classicalization suppresses these dangerous channels:

• The "hard" (large momentum transfer) $2\to 2$ amplitude for $|t|\sim s\gg M_*^2$ is exponentially damped:

$$\mathcal{A}_{2\to 2}(\sqrt{s}, t\sim s) \sim \exp\left[-\left(\frac{\sqrt{s}}{M_*}\right)^c\right],\quad c>0$$

• The dominant scattering processes are "soft," involving small momentum transfer $\sim 1/r_*$, and the cross section becomes geometric:

$\sigma \sim (r_*)^2$

• Instead of two-particle final states, generic events at energies $\sqrt{s}\gg M_*$ are dominated by production and decay of classicalons: extended field objects that decay to high-multiplicity, low-momentum (soft) quanta.

This is a dramatic IR/UV transmutation—high-energy behavior is determined by classical, macroscopic dynamics, not by microscopic quantum effects.

4. Application to the Standard Model

A. Higgless Standard Model

In the absence of the Higgs boson, the longitudinal polarizations of massive vector bosons ($W^\pm$, $Z$)—which act as the Equivalence Theorem Nambu-Goldstone bosons—naturally classicalize. Their derivative self-interactions cause attempt at high-energy $WW$ or $WZ$ scattering to produce classicalons rather than violate unitarity.

• The perturbative violation of unitarity at $\sqrt{s}\sim v$ ($v$ is the electroweak scale) is avoided.
• Two-body high-$p_T$ final states are suppressed at high energy; instead, classicalons form and decay to many $W^\pm$, $Z$, and leptons/hadrons, producing high-multiplicity, soft events.

B. Standard Model with the Higgs as a Classicalizer

If the Higgs is present, a higher-dimensional operator coupling $H^\dagger H$ to $T^\mu_\mu$:

$$\mathcal{L}_{\text{int}} \sim \frac{1}{M_*^2}(H^\dagger H) T^\mu_\mu$$

acts as a classicalizer interaction for the Higgs. Excitation of energy-momentum above $M_*$ creates a classical configuration ("Higgsion") of the Higgs field in high-energy collisions, again preventing hard scatterings above $M_*$.

• This mechanism can "self-protect" the scale $v$ and stabilize the electroweak hierarchy by damping high-momentum contributions; Higgs quantum corrections are absorbed into non-perturbative classicalon dynamics.

Both scenarios share the core prediction that the Standard Model (with or without a fundamental Higgs) self-completes via classicalon formation, rather than by introducing new, weakly-coupled BSM particles above $M_*$.

5. Experimental Phenomenology and Distinctive Signatures

Predictions for collider experiments (e.g., the LHC) are sharply distinct from standard BSM scenarios:

• Above the classicalization scale ($\sim v$, $M_*$), cross sections for relevant processes grow geometrically ($\sigma\sim r_*^2$).
• High-multiplicity, soft final states dominate—no sharp, isolated high-$p_T$ leptons or jets in classicalon-dominated channels.
• The transition region may display a series ("tower") of quantum resonances, marking the threshold where quantum excitations merge into a single classicalon.
• In the Higgs classicalizer scenario, local modifications of the Higgs expectation value are possible in regions of high energy density, potentially altering decay patterns or kinematic distributions for multi-particle states.
• Electroweak precision observables (e.g., S, T parameters) may receive contributions from the resonance tower.

Table: Key Phenomenological Discriminants

Scenario	Final State Signature	EW Observables
Higgless (classicalon)	Many $W/Z$ + high mult.	Tower of resonances, large $r_*$
Higgs as classicalizer	Many Higgs, soft $W/Z$, $v$ modulated locally	Higgs expectation value altered, $S/T$ modifications

The difference in multiplicity patterns, resonance structure, and possible deviations in Higgs-related observables can, in principle, distinguish classicalization from more conventional Wilsonian UV completions.

6. Theoretical Implications and Significance

Classicalization, as a UV completion paradigm, bypasses the need for new weakly-coupled particles or high-scale BSM physics traditionally expected in Wilsonian completions. Instead:

• The SM self-completes via its own non-linear dynamics, with classicalizer fields protecting short distances by dynamical delocalization of energy.
• The approach predicts that "hard" scattering at scales $\gg M_*$ disappears—the high-energy regime is dominated by "soft" physics determined by $r_*(s)$.
• The mechanism is a field-theoretic analog of black hole formation in gravity, and offers an alternative route to unitarity and high-energy predictivity in non-renormalizable theories, including sigma models, non-linear gauge fields, and models with derivative couplings.

From the UV completion perspective, this unitarization via classical configuration precludes the need for new light degrees of freedom above $M_*$, shifting the focus to macroscopic, multi-particle dynamics at high energy. Classicalization thus provides a qualitatively novel solution to longstanding questions about the high-energy fate of the Standard Model.

7. Relation to Other UV Completion Paradigms

The classicalization approach is fundamentally distinct from:

• Wilsonian UV completions: required BSM physics at $\Lambda \gg v$.
• Technicolor or composite Higgs models: new strongly coupled sectors at $\Lambda$ dynamically break EW symmetry.
• Supersymmetry: introduces partner fields to stabilize quantum corrections.
• Asymptotic safety: invokes a fixed-point structure to control the RG flow.

Classicalization instead predicts—and requires—no new weakly-coupled particles, but non-perturbative field configurations detectable in collider signatures and their geometric cross sections. It offers a minimalistic and non-Wilsonian self-completion that is endogenous to the infrared degrees of freedom of the Standard Model.

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In summary, ultraviolet completion of the Standard Model via classicalization proposes that high-energy unitarity and predictivity are ensured by the formation of classicalons—semiclassical, extended field configurations—rather than by weakly-coupled new physics. The theory becomes "self-protecting" at high energies, with hard scattering exponentially suppressed and the dynamics governed by the collective behavior of its own fields, manifesting in distinctive, high-multiplicity collider events and providing an alternative framework for exploring the ultimate fate of the Standard Model at energies far above the electroweak scale (1010.1415).