A numerical scan of a particle-physics model has found simulated phase-transition histories whose gravitational-wave signals could be detectable by LISA under the study’s assumptions. The strongest result came from the second transition in an inverted-hierarchy, two-step history: its maximum three-year signal-to-noise ratio, or SNR, was 88. For normal-hierarchy one-step histories, the maximum reported SNR was 17.
Those figures describe the top of a simulated range, not a prediction that LISA will see a signal. The spectra in the scan were dominated by sound waves, and the SNR calculation assumed a three-year observation. The size of the forecast also depended on choices for the renormalization scale, transition temperature and bubble-wall velocity.
The work is a numerical exploration of possible vacuum histories. ScannerS generated parameter samples compatible with the stated theoretical and experimental constraints, while BSMPT computed phase-transition histories and gravitational-wave spectra; the configurations were then run with the CalcGW executable. The scan contained about 1.6 million valid configurations for each of the normal and inverted mass hierarchies.
Several routes through the vacuum
Across that search, the model admitted one-, two-, three- and four-step transitions, as well as more unusual histories that passed through intermediate CP-breaking or charge-breaking phases. The result is a range of possible paths through the model’s vacuum structure, rather than a single transition pattern.
The study classified a first-order transition as strong when ξp exceeded 1 at percolation, using ξp as its transition-strength measure. That threshold sets how the scan sorted its simulated points; it is not, by itself, an observation.
The strongest cases split by hierarchy
For the normal hierarchy, the scan found one-, two- and three-step transitions. The largest reported values occurred in one-step cases: α reached 0.52 and ξp reached 6.1, while the percolation temperature Tp fell as low as 40 GeV. These are extremes from the scan, not typical values for every allowed configuration.
The inverted hierarchy produced one- and two-step transitions. Its most forceful reported case was the second step of a two-step history, where α reached 2.5 and ξp reached 11, with the transition temperature T⋆ as low as 22 GeV. That step also supplied the maximum reported SNR of 88.
Possible clues in collider data
The scan points to different collider regions for the two hierarchies. Among the largest-ξp normal-hierarchy one-step points, two mass patterns centered on roughly 190 GeV—where the H2 and charged H± masses were similar—and 420 GeV—where the A and H± masses were similar—came with associated A and H± decay signatures. The authors present these patterns as possible indicators, not unique fingerprints.
In the inverted hierarchy, strong one-step points had A and H± masses in the 220–440 GeV range. Strong two-step points covered a wider span, with A masses from 60 to 350 GeV and H± masses from 120 to 350 GeV. The study describes associated A and H± decay signatures for these scan regions, which are associations rather than a single collider prediction.
Self-coupling patterns do not line up neatly
Another possible experimental link is the Higgs self-coupling. In the normal hierarchy, the effective SM-like trilinear coupling modifier κλ ranged from 1.1 to 2.1 for one-step histories, 1.15 to 1.6 for two-step histories and 1.2 to 1.5 for three-step histories. Larger mass gaps correlated with larger enhancements in the reported scan.
The inverted hierarchy showed a less uniform relationship. κλ ranged from 1.1 to 2 for one-step histories and 1.0 to 1.7 for two-step histories, and strong two-step points could lie close to the Standard Model value. An enhanced trilinear coupling was therefore not required by every strong transition in the scan.
Why the headline numbers are provisional
The uncertainty checks make the gravitational-wave forecasts much less settled. At one benchmark point, varying the renormalization scale produced SNR values from 0.045 to 5.7, compared with 0.33 at the default scale. The reported range is a sensitivity test, not a statistical confidence interval.
Temperature choices had a similar effect. When the benchmark transition temperature was varied from 52 to 55 GeV, with 53.2 GeV as the benchmark, the SNR moved from 2.5 × 10−5 to 0.36 and the spectral peak shifted. The manual variation was not fully coupled to the vacuum evolution and related thermal quantities.
The assumed bubble-wall speed also mattered. For the benchmark, changing the hand-set wall velocity produced SNR values from 6.2 × 10−4 to 2.5 and changed the peak properties; the peak height decreased toward wall velocities below 0.6. The calculation did not derive that speed from a self-consistent plasma and out-of-equilibrium treatment.
Taken together, the scan offers a map of possibilities rather than a settled route from model parameters to an observed signal. The authors argue that self-consistent thermal dynamics and lower theoretical uncertainty are needed before underlying parameters can be inferred from a gravitational-wave or collider pattern. The work is an arXiv version 1 preprint dated 26 August 2026.
Paper data and sources
Original title: A Comprehensive Analysis of the R2HDM Vacuum Evolution and the Induced GW and Collider Phenomenology
Authors: Lisa Biermann, Christoph Borschensky, Rafael Boto et al.
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text