Higgsless Simulations in High-Energy Physics

• Higgsless simulations are modeling approaches that generate mass and restore unitarity without relying on a fundamental Higgs field.
• They employ methods such as QCD-induced symmetry breaking, deconstructed extra dimensions, and topological superconductivity to explore alternative mass generation.
• Advanced computational and quantum simulation frameworks enable detailed studies of collider signals, gravitational waves, and nonlocal mass generation phenomena.

A Higgsless simulation refers to the numerical or analytic modeling of physical systems—primarily in high-energy physics and cosmology—where a fundamental scalar Higgs field plays no direct role in mass generation or symmetry breaking. Higgsless approaches span QCD-induced electroweak symmetry breaking, deconstructed extra-dimensional models, geometric (Kaluza–Klein with torsion) frameworks, topological models for superconductivity, and boundary-condition or energy-injection methods in cosmological phase transition fluid dynamics. Across these diverse fields, the commonality lies in identifying or simulating mechanisms for mass generation, unitarity restoration, or collective phenomena without spontaneous breaking via a local order parameter or scalar potential. Higgsless simulations have emerged as powerful theoretical and computational laboratories for probing both fundamental limitations and alternative realizations of mass and symmetry breaking.

1. Higgsless Models in Particle Physics

Several classes of Higgsless models can be constructed by omitting fundamental scalar fields from the Lagrangian and instead employing alternative mechanisms for electroweak symmetry breaking (EWSB):

• QCD-induced EWSB models: Spontaneous breaking of chiral symmetry in QCD (⟨q̄q⟩ condensates) induces mass generation for gauge bosons. In the Standard Model (SM) without a Higgs, pions (would-be Goldstone bosons) are absorbed by $W^\pm$ and $Z$, with their masses set by $f_\pi$ (the pion decay constant), i.e., $M_W \approx gf_\pi/2$ (0901.3958).
• Deconstructed Higgsless models: By “deconstruction” of extra dimensions, gauge symmetry is broken via link fields (nonlinear sigma model fields $\Sigma_j$) between gauge “sites.” The resulting mass matrices for gauge bosons arise from kinetic and link terms, with longitudinal polarizations sourced by “eaten” Goldstone components (details in (0906.2474, Speckner, 2010)).
• Geometric Kaluza–Klein/torsion models: Mass emerges from pure geometry via nonvanishing torsion in higher-dimensional vacua, leading to modified connections and a simultaneously generated kinetic and mass term for gauge fields. The gauge boson masses arise as eigenvalues of a mass matrix $M_{IJ}$ determined by contorsion, reproducing $m_W = (L_1\sin\theta_w)^{-1}$ and $m_Z = (L_1\sin\theta_w\cos\theta_w)^{-1}$ (Batakis, 2011).
• Topological superconductivity scenarios: Superconducting states without a Higgs scalar are realized by the condensation of topological defects in compact BF theories. A gap arises for all but a specific, collective gapless mode via emergent statistical gauge fields, and photon mass is acquired by the topological BF mechanism ($m = eg/T$) (Diamantini et al., 2014, Diamantini et al., 2017).
• Superconducting quantum simulation: Higgsless quantum circuits (Rabi-type models with qubits) simulate spontaneous SUSY breaking and radiative mass-enhancement without scalar Higgs potentials. Here, the qubit replaces the Higgs, and radiative corrections are mimicked via quadratic interaction terms (Hirokawa, 2022).

The defining characteristic across these models is the decoupling of mass and order parameter from a fundamental scalar, with alternatives ranging from strong dynamics (QCD), geometric deformation, or nonlocal/topological field dynamics.

2. Implementation and Simulation Frameworks

Higgsless models necessitate specialized approaches for their theoretical and computational simulation:

• FeynRules and Monte Carlo frameworks: In collider phenomenology, Lagrangian-level definitions in Mathematica (FeynRules) are used to derive Feynman rules, which are then automatically exported to event generators (MadGraph, CalcHEP, Sherpa, WHIZARD) for cross-section and distribution simulations. The entire workflow enables automated sanity checks, model validation, and cross-code consistency (agreement to $\leq$1% in cross sections for the Three-Site Higgsless Model) (0906.2474).
• Boundary-condition or energy-injection methods (cosmological phase transitions): For simulating gravitational wave (GW) production by first-order phase transitions, Higgs degrees of freedom are integrated out; the latent heat is injected via spatially dependent equation-of-state changes (e.g., local bag constant $\epsilon(x,t)$) without resolving Higgs wall thickness. Relativistic fluid equations are solved numerically (often using Kurganov–Tadmor schemes) in three dimensions, efficiently capturing the bubble and sound shell evolution (Jinno et al., 2022, Caprini et al., 5 Sep 2024, Stomberg et al., 6 Aug 2025).
• Quantum simulation (SUSY-breaking models): Hamiltonians invariant under $\mathcal{N}=2$ SUSY are constructed in trapped-ion or superconducting circuits using a combination of a single bosonic mode and a qubit, including “A$^2$” quadratic corrections to simulate mass enhancement and (pseudo)spontaneous SUSY-breaking transitions without a local Higgs field (Hirokawa, 2022).
• Topological lattice gauge simulations: Mechanisms for Higgsless superconductivity are amenable to lattice simulation frameworks, where emergent gauge fields and the condensation dynamics of topological defects can be studied without the need for local order parameters (Diamantini et al., 2014, Diamantini et al., 2017).

The crucial advantage of these frameworks is their capacity to isolate the macroscopic physics of mass generation, symmetry breaking, and phase transition—without the computational burden or conceptual dependence on the Higgs sector.

3. Physical Consequences and Phenomenology

The physical signatures, parameter dependencies, and phenomenological consequences in Higgsless simulations are profound and diverse:

• Altered weak interaction strength and mass spectra: With $f_\pi\ll v_\text{EW}$, the $W$ and $Z$ masses in QCD-induced Higgsless worlds are $\sim28$ MeV and $32$ MeV, respectively—orders of magnitude below SM values. Weak and strong interactions become comparable, dramatically modifying beta decay, nuclear forces, and mass splittings (0901.3958).
• Resonance structure and LHC discovery potential: Deconstructed models and extra-site extensions (e.g., four-site models) predict multiple heavy vector bosons with substantial coupling to SM fermions, enabling discovery via Drell–Yan channels. For example, four-site models avoid imposed fermiophobicity, yielding accessible $Z'$/$W'$ signals up to $\sim2$ TeV for $\mathcal{O}(1)$ couplings (Accomando et al., 2010, Accomando et al., 2011).
• Nondecoupling and flavor observables: KK excitations in deconstructed models affect rare decays such as $b\rightarrow s\gamma$ even at large mass, leading to measurable $7$–$8\%$ shifts in branching ratios as mixing parameters ($\epsilon_{tR}$) range over electroweak-allowed values. This sensitivity persists due to $O(1/x)$ vertex enhancements and incomplete decoupling (Kurachi et al., 2010).
• GW spectrum in cosmological phase transitions: Higgsless fluid dynamics predicts the GW spectrum scale, slope, and peak as a function of bubble wall velocity, phase transition strength ($\alpha$), nucleation rate ($\beta$), and source duration. Crucially, for weak transitions, the kinetic energy driving the GW source remains stationary, matching the “sound shell” model, whereas non-linear decay (with index $b>0$) for strong/intermediate transitions yields a suppressed GW amplitude and shallower spectral slopes. Vorticity production and the transition from compressional to turbulent regimes are observed in simulation and alter both spectral behavior and GW amplitude growth (Jinno et al., 2022, Caprini et al., 5 Sep 2024, Stomberg et al., 6 Aug 2025).

The table below summarizes key dependence of GW spectral features on phase transition parameters (values as reported in (Caprini et al., 5 Sep 2024, Stomberg et al., 6 Aug 2025)):

Parameter	Effect on Spectrum	Regime
Wall velocity $v_w$	Shifts peak, alters power	High $v_w$ increases amplitude
Strength $\alpha$	Controls amplitude, decay	Strong $\alpha$ induces decay
Decay exponent $b$	Suppresses amplitude	$b=0$ (stable), $b>0$ (decaying)
Nucleation rate $\beta$	Sets characteristic $R_*$	Affects spectral knee

4. Analytical Structures and Key Formulas

Multiple analytical structures undergird Higgsless simulation frameworks:

• Gauge boson masses from chiral or geometric breaking:

$$M_W \sim g f_\pi/2 \qquad m_W = (L_1 \sin\theta_w)^{-1}$$

for QCD and torsion-induced cases, respectively (0901.3958, Batakis, 2011).

• Lagrangians in deconstructed models:

$$\mathcal{L}_\text{gauge} = -\frac{1}{4}\sum_{i} F_{i}^{\mu\nu}F_{i,\mu\nu} + \frac{f^2}{4}\sum_{j}\mathrm{Tr}[(D_\mu\Sigma_j)^\dagger D^\mu\Sigma_j]$$

capturing gauge and nonlinear-sigma model dynamics (0906.2474).

• Photon mass in topological models:

$[\Box + m^2] F_{\mu\nu} = 0, \quad m = eg/T$

where $g$ is the emergent gauge coupling, $T$ is topological “charge” (Diamantini et al., 2014).

• GW spectrum with non-linear decay:

$$\frac{d\rho_\text{GW}}{d\ln k} = 3\, F_\text{GW}^0\, \tilde{\Omega}_\text{GW}\, K_\text{int, exp}^2\, \ast\, {\cal R}_\ast\, S(k{\cal R}_\ast)$$

and for a power-law kinetic energy decay $K(t) = K_0 (t/t_0)^{-b}$,

$K_\text{int, exp}^2 = \int [K(t)]^2 dt$

serving as the source normalization for the GW spectrum (Stomberg et al., 6 Aug 2025).

• Spectral shape for the GW power:

$$S(k, k_1, k_2) = S_0\, \left(\frac{k}{k_1}\right)^{n_1} [1 + (k/k_1)^{a_1}]^{(-n_1 + n_2)/a_1} [1 + (k/k_2)^{a_2}]^{(-n_2 + n_3)/a_2}$$

with spectral indices and breakpoints fit to simulations (Stomberg et al., 6 Aug 2025).

5. Broader Implications and Theoretical Significance

Higgsless simulations serve as critical laboratories for evaluating and constraining alternatives to the Higgs mechanism:

• Viability of Strong/Composite EWSB: The low-energy effective field theory with explicit vector resonances (spin-1 $\rho$-like states) provides a predictive, chiral perturbation (hidden local symmetry) framework for resonance-rich EWSB. This framework accommodates both unitarity restoration and experimental constraints, requiring spin-1 resonance masses to lie above $\sim 1.8$ TeV for consistency with $S$ parameter bounds (Kaminska, 2012, Pich et al., 2012).
• Phenomenological differentiation: Multi-resonant structures, spectral shapes, and discovery channels in collider and cosmology distinguish Higgsless from scalar Higgs and even composite Higgs scenarios, especially in observables sensitive to coupling strengths and non-decoupling effects.
• Computational and methodological efficiency: By integrating out microscopic degrees of freedom (bubble wall thickness, scalar radiative corrections), Higgsless approaches achieve numerically efficient modeling over vast dynamic ranges—crucial for cosmological applications or quantum simulation of nonlocal mass generation.
• Interpretation of topological and geometric mass generation: The geometric and topological mechanisms (BF terms, torsion-induced mass) provide paradigms for mass generation where the notion of spontaneous symmetry breaking via a scalar potential is replaced by nonlocal or global effects, offering alternative avenues in both condensed matter and high-energy contexts (Diamantini et al., 2014, Batakis, 2011).

6. Outlook and Prospects

The development and simulation of Higgsless models continue to inform multiple research frontiers:

• Experimental searches: As LHC and next-generation collider data further probe multi-resonance structures and flavor observables, detailed Higgsless simulations remain essential for mapping signatures and exclusion regions, especially for heavy spin-1 states and rare decays.
• Gravitational Wave Astronomy: Higgsless cosmological simulations deliver accurate and robust templates for the GW backgrounds generated by phase transitions, accommodating both linear and strongly nonlinear evolution, and interfacing with public packages such as CosmoGW for prediction and data analysis (Stomberg et al., 6 Aug 2025).
• Quantum simulation and analog models: Higgsless quantum circuits may yield insights into mass generation, radiative corrections, and the nature of SUSY breaking, providing experimental access to dynamical phenomena unreachable in traditional high-energy settings (Hirokawa, 2022).
• Fundamental theory development: Theoretical advances in geometric, topological, and deconstructed frameworks deepen understanding of possible alternatives to the Higgs mechanism, with implications for new physics beyond the Standard Model, the structure of the vacuum, and the origin of mass itself.

In summary, Higgsless simulations constitute a dynamic and multifaceted domain, uniting effective field theory, numerical computation, collider and cosmological phenomenology, and even experimental quantum simulation, all aimed at dissecting and realizing alternative mechanisms of mass generation and symmetry breaking.