Will A Next Generation Collider See A Sphaleron?

Overview

A new pair of preprints, whose abstracts and citations are set forth below, examine the sphaleron energy threshold with newly updated experimental values of Standard Model physical constants and rigorous calculations, and its other properties. This tells experimentalists very specifically where to look and what to look for when trying to observe a Standard Model sphaleron interaction.

The sphaleron interaction is the only time in the Standard Model of Particle Physics that baryon number and lepton number are not simultaneously conserved. 

But it requires extremely high collider energies to form in detectable numers (an order of magnitude greater collider energy than the nominal energy levels is required because the energy of the interaction must be confined so compactly, so a collider energy of something on the order of 91 TeV is needed to confidently assert that they have been discovered).

Not Yet Observed Standard Model Phenomena

The two biggest predictions of the Standard Model of Particle Physics that haven't been observed yet are the sphaleron and the failure to definitively observe pure glueballs (a strong force bound hadron with no quarks) and certain other hadrons predicted to exist in Standard Model quantum chromodynamics (QCD). 

Standard Model Hadron Predictions Not Yet Seen

Current experiments have more than enough energy to produce glueballs, which are predicted to have masses on the order of 0.5 GeV to 3 GeV in their ground states (while the LHC can create energies up to 14,000 GeV), at some experimentally observed mesons have been provisionally identified as likely glueballs. 

But identifying them definitively against other possible explanations of glueball-like resonances is challenging, since they are electromagnetically neutral, color charge neutral, don't interact via the weak force at the tree-level, and are bosons with integer spins shared by mesons in overlapping mass ranges. Glueballs have a natural tendency to blend into mesons with the same quantum numbers, resulting in mixed hadron resonances.

A similar issue applies to some of the heaviest hadrons predicted by the Standard Model but not yet definitively identified with resonances observed at sufficient statistical significance. But the most massive of these have ground states with masses of 20 GeV or less, and a few new ones are identified every year these days. Observing the last few is mostly just a matter of time.

Similarly, the project of identifying the underlying structure of hadron resonances other than simple pseudo-scalar valence quark-antiquark mesons and simple three valence quark baryons, is progressing one hard won resonance identification at a time with no sweeping explanations for large groups of resonances whose structures do not have a consensus explanation. 

There are even hints of exceeding improbable, short lived, and rare ultrahigh energy toponium (i.e. a meson made up of a top quark and anti-top quark with a mass on the order of 340 to 350 GeV), which would also be enhanced at a next generation higher energy particle collider, with a 73 TeV collider energy with a very large number of collisions, being a key threshold to detect this highly improbable resonance. It is rare not just because it takes high energies, but also because of the high risk that the valence top quark and valence anti-top quark necessary to form one would decay or annihilate with each other, before the quarks could hadronize into the most massive theoretically possible simple quark-antiquark meson. Toponium is almost guaranteed to be seen at a next generation collider along the lines of the LHC but more powerful.

Sphalerons

A sphaleron, in contrast, is basically the only major Standard Model prediction that is not yet confirmed, with an energy scale about 730 times that of a Higgs boson, that requires a bigger accelerator to confirm or rule out, even though the nominal sphaleron energy of 9.1 TeV is less than the 14 TeV peak energy of the LHC.

Analysis

To be clear, this study does not alter the long standing conclusion that sphaleron interactions cannot explain the baryon asymmetry of the universe (i.e. the extreme excess of matter over anti-matter in quarks, which is also true of charged leptons), although it does tweak estimates of what percentage of baryon asymmetry this interaction can explain with Standard Model physics.

I put even odds on whether sphalerons actually exist or not. A mathematically consistent modification of the Standard Model that would make baryon number and lepton number conserved symmetries of the Standard Model, which would make sphalerons impossible, would have almost no important phenomenological consequences for anything other than (i) baryogenesis and leptogenesis in the first few seconds after the Big Bang, which we don't really understand yet anyway and which could be explained without any baryon number and lepton number violations in other ways, such as a mirror universe model, and (ii) the presence or absence of sphaleron decay signatures in ultrahigh energy collider experiments. 

Non-detection of sphalerons would also disfavor a wide variety of grand unified theories (GUTs) and Theories of Everything (TOEs), which usually permit violations of baryon number and lepton number, which makes phenomena like proton decay, which is forbidden in the Standard Model of Particle Physics, rare but possible.

But, detecting sphalerons as predicted would also be a triumph for the Standard Model taken to the extremes of its domain of applicability.

The Standard Model prediction is that there would be a desert of new physics (within or beyond the Standard Model) at energies above those where a sphaleron are observed.

The Papers

The electroweak sphaleron is a static, unstable solution of the Standard Model classical field equations, representing the energy barrier between topologically distinct vacua. 

In this work, we present a comprehensive updated analysis of the sphaleron using current Standard Model parameters with the physical Higgs boson mass of m(H)=125.1 GeV and m(W)=80.4 GeV, rather than the m(H)=m(W) approximation common in earlier studies. The study includes: (i) a complete derivation of the SU(2)×U(1) electroweak Lagrangian and field equations without gauge fixing constraints, (ii) high-precision numerical solutions for the static sphaleron configuration yielding a sphaleron energy E(sph)≃9.1 TeV, (iii) an analysis of the minimum energy path in field space connecting the sphaleron to the vacuum (a 1D potential barrier as a function of Chern-Simons number), and (iv) calculation of the sphaleron single unstable mode with negative eigenvalue ω^2=−2.7m(W)^2, providing analytical fits for its eigenfunction. 

We find that using the measured Higgs mass modifies the unstable mode frequency, with important implications for baryon number violation rates in both early universe cosmology and potential high-energy collider signatures. These results provide essential input for accurate lattice simulations of sphaleron transitions and precision calculations of baryon number violation processes.

Konstantin T. Matchev, Sarunas Verner, "The Electroweak Sphaleron Revisited: I. Static Solutions, Energy Barrier, and Unstable Modes" arXiv:2505.05607 (May 8, 2025).

We present a comprehensive analysis of electroweak sphaleron decay dynamics, employing both analytical techniques and high-resolution numerical simulations. 

Using a spherically symmetric ansatz, we reformulate the system as a (1+1)-dimensional problem and analyze its stability properties with current Standard Model parameters (m(H)=125.1 GeV, m(W)=80.4 GeV). We identify precisely one unstable mode with eigenvalue ω^2 ≃ −2.7m(W)^2 and numerically evolve the full non-linear field equations under various initial conditions. Through spectral decomposition, we quantify the particle production resulting from the sphaleron decay. 

Our results demonstrate that the decay process is dominated by transverse gauge bosons, which constitute approximately 80% of the total energy and multiplicity, while Higgs bosons account for only 7-8%. On average, the sphaleron decays into 49 W bosons and 4 Higgs bosons. The particle spectra consistently peak at momenta k ∼ 1−1.5m(W), reflecting the characteristic size of the sphaleron. 

Remarkably, these properties remain robust across different decay scenarios, suggesting that the fundamental structure of the sphaleron, rather than specific triggering mechanisms, determines the decay outcomes. These findings provide distinctive experimental signatures of non-perturbative topological transitions in the electroweak theory, with significant implications for baryon number violation in the early universe and potentially for high-energy collider physics.

Konstantin T. Matchev, Sarunas Verner, "The Electroweak Sphaleron Revisited: II. Study of Decay Dynamics" arXiv:2505.05608 (May 8, 2025).