In the Standard Model, the Higgs boson is an elementary scalar field. Its discovery at the LHC in 2012 confirmed the mechanism of electroweak symmetry breaking, but the nature of the Higgs remains open: is it truly elementary, or is it a composite object made of more fundamental constituents? Many beyond‑the‑Standard‑Model theories, such as technicolor and composite‑Higgs models, propose that the Higgs is a bound state of new strong‑dynamics fermions.
The Syntactic Token Calculus offers a different kind of compositeness: the Higgs pattern [[#] [#] [#]] can be viewed as a bound state of three photons (massless gauge bosons). Each photon is represented by the pattern [#]. The Higgs pattern is simply three such patterns placed inside a common outer enclosure. Syntactically, this is a stable normal form; it cannot be reduced further because there is no ## or [[A]] substring.
However, in quantum field theory, photons do not interact directly; they couple only via charged particles. A bound state of three photons would be extremely weakly bound, if it exists at all. The STC does not rely on quantum field theory; it is a syntactic calculus where patterns are defined by their formal properties, not by dynamical equations. The compositeness here is structural: the Higgs pattern is decomposable into three identical sub‑patterns, and that decomposition has physical consequences.
If the Higgs is composite, it should have excited states—patterns where the three constituents are arranged differently. In the STC, these excited states correspond to different normal forms that share the same charge and spin invariants but have higher syntactic complexity. They would appear as heavier scalar resonances with the same quantum numbers as the Higgs (spin 0, positive parity, zero electric charge).
Thus, the STC predicts a tower of excited Higgs resonances with masses that follow a geometric progression, reflecting the discrete scale invariance of the Bruhat‑Tits tree.
Consider the ground‑state Higgs pattern [[#] [#] [#]]. An excited state could be obtained by inserting an extra enclosure around one of the photons, e.g., [[[#]] [#] [#]]. But [[#]] reduces to #, so this pattern becomes [# [#] [#]], which is not a scalar (it has an outer enclosure containing a mark and two photons). That pattern might correspond to a different particle, perhaps a heavier lepton.
A more plausible excited Higgs pattern is [[#] [#] [#] [#]]—four photons inside an outer enclosure. This pattern is also a normal form (no ##, no [[A]]). Its charge pattern reduces to the same invariant as the Higgs (zero charge), and its spin pattern is symmetric, indicating a scalar. Thus, [[#] [#] [#] [#]] is a candidate for the first excited Higgs resonance.
More generally, the pattern with $n$ photons inside an outer enclosure, [[#] [#] … [#]] ($n$ copies), is a normal form for any $n \ge 2$. These patterns form an infinite family, each with zero charge and scalar statistics. They correspond to excited Higgs resonances with masses that scale with $n$.
In the Bruhat‑Tits tree, the depth of a pattern is related to its mass. For the Higgs family, the depth increases linearly with $n$. Because the tree is self‑similar with branching ratio $p$, masses should follow a geometric progression:
$$ mn = mH \cdot q^{\,n-3}, $$
where $m_H$ is the ground‑state Higgs mass (≈125 GeV), $q$ is the discrete scale factor (likely $q = p$, the prime underlying the tree), and $n$ is the number of photons in the pattern. For $p=2$, $q=2$, the first few resonances would appear at:
This pattern continues indefinitely, though at high masses the resonances become increasingly broad and overlapping due to decay into multiple particles.
The width of each resonance is also predicted: it should scale with the number of decay channels, which grows with $n$. The ground‑state Higgs has a narrow width (≈4 MeV) because its decays are suppressed by the small couplings to fermions and bosons. Excited states, being heavier, can decay into pairs of W/Z bosons, top quarks, and even into lower Higgs resonances, leading to larger widths.
Thus, the STC predicts a spectrum of scalar resonances at masses 250 GeV, 500 GeV, 1 TeV, 2 TeV, … with widths that increase with mass. These resonances should be produced at hadron colliders via gluon‑fusion (like the Higgs) and vector‑boson fusion, and they should decay into the same final states as the Higgs (WW, ZZ, γγ, $b\bar b$, ττ) but with larger branching ratios to boson pairs.
If the Higgs is composite, its couplings to other particles may deviate from the Standard‑Model predictions. In composite‑Higgs models, such deviations arise because the Higgs is a pseudo‑Goldstone boson of a broken global symmetry, leading to a form factor that suppresses couplings at high energies.
In the STC, the Higgs couplings are determined by the cross‑ratio with the other particle’s pattern. For a composite Higgs, the cross‑ratio may receive corrections from the internal structure of the pattern. These corrections are expected to be small for the ground‑state Higgs (consistent with current LHC measurements) but could become significant for excited resonances.
Specifically, the coupling of the Higgs to two photons (the $Hγγ$ vertex) is of special interest because it is generated via loops of charged particles. In the STC, the photon pattern [#] is a constituent of the Higgs; thus, the $Hγγ$ coupling might be enhanced relative to the Standard Model. Current measurements of the Higgs diphoton decay rate are consistent with the Standard Model within ≈10%, but future precision measurements at the HL‑LHC could detect deviations.
For excited Higgs resonances $H_n$, the coupling to two photons should be stronger because the pattern contains more photon constituents. This would lead to an enhanced diphoton decay channel, making the resonances more visible in the $γγ$ final state.
Similarly, couplings to W and Z bosons may be modified. The STC predicts that the ratio of $Hn WW$ to $Hn ZZ$ couplings should be the same as for the ground‑state Higgs (because both W and Z patterns are built from photons), but the absolute normalization could scale with $n$.
These form‑factor deviations provide additional handles to distinguish a composite Higgs from an elementary one. A combined analysis of the resonance masses, widths, and coupling patterns could confirm or rule out the STC’s composite picture.
Searching for excited Higgs resonances is a major goal of the High‑Luminosity LHC (HL‑LHC) and future colliders. The STC’s prediction of a geometric mass spectrum provides a clear target.
1. Mass range: The first excited resonance $H_4$ is expected around 250 GeV. This mass is accessible with current LHC data. Searches for a heavy scalar decaying to WW or ZZ have been performed by ATLAS and CMS up to about 1 TeV, with no significant excess yet. However, these searches typically assume a narrow resonance; the STC predicts a broader width for excited states, which could reduce sensitivity. Re‑analysing existing data with a broader width hypothesis could reveal a signal.
2. Final states: The most promising channels are $Hn → WW → ℓνℓν$ and $Hn → ZZ → 4ℓ$ (golden channel). The diphoton channel $H_n → γγ$ is also promising because of its clean signature and expected enhancement. Additionally, decays to $t\bar t$ become important for masses above 350 GeV.
3. Combined fit: A simultaneous fit to multiple mass points, assuming a geometric progression with a common scale factor, could increase sensitivity. If one resonance is found, the others should appear at predictable higher masses.
4. Future colliders: An electron‑positron collider (e.g., the proposed FCC‑ee or ILC) would provide a clean environment to scan for scalar resonances with high precision. The STC predicts that the cross section for $e^+e^- → H_n Z$ should exhibit peaks at the resonance masses.
5. Width measurements: If a resonance is discovered, measuring its width will be crucial. The STC predicts widths that grow with mass, potentially reaching tens of GeV for the 1 TeV resonance. This would be a distinctive signature.
Given the current lack of evidence for heavy scalars, the STC’s composite‑Higgs model is not yet confirmed. However, the geometric mass pattern is a sharp prediction that will be tested with upcoming data. If no resonances are found up to, say, 2 TeV, the model would be disfavored unless the scale factor $q$ is much larger than 2 (e.g., $q=3$ would place $H_4$ at 375 GeV, still within reach).
Chapter 27 has explored the composite‑Higgs interpretation within the STC. The Higgs pattern [[#] [#] [#]] can be seen as a bound state of three photons, leading to a tower of excited resonances with masses that follow a geometric progression. This prediction is testable at current and future colliders through searches for heavy scalars in WW, ZZ, and γγ final states. Deviations in Higgs couplings from Standard‑Model expectations provide additional handles.
The next chapter shifts from particle physics to neuroscience, examining the prediction of ultrametric clustering in neural data—a surprising connection between the hierarchical structure of the universe and the organization of thought.