Weak boson probes of Higgs unitarity restoration at 10 TeV parton colliders

This paper estimates and compares the discovery potential of a 100 TeV hadron collider and a 10 TeV muon collider to detect and resolve unitarity violations caused by sub-percent Higgs coupling deviations, demonstrating that both facilities can probe mass scales up to approximately 6 TeV, with intermediate precision measurements from FCC-ee providing corroborating evidence.

The Gist

Imagine the universe is built on a set of delicate rules that keep everything from falling apart. One of the most important rules is called unitarity. In simple terms, this is the universe's way of saying, "Probabilities must add up to 100%." If you calculate the odds of particles smashing into each other, the math shouldn't result in a probability of 200% or -50%. If the math breaks down at high speeds, the theory is broken.

In our current understanding of physics (the Standard Model), the Higgs boson acts like a safety valve. When particles move too fast and start to break these rules, the Higgs steps in to "fix" the math, keeping the universe stable.

The Problem: A Slight Glitch?

Scientists at the Large Hadron Collider (LHC) are currently measuring how the Higgs boson interacts with other particles. They are looking for tiny deviations. Imagine the Higgs is a key that fits a lock perfectly. If the LHC finds that the key is slightly bent (even by just 1% or 2%), it means the "safety valve" isn't working quite right.

If the key is bent, the universe's safety net is compromised. To prevent the laws of physics from breaking at high energies, something new must appear to take over the job of fixing the math. This "something new" would be heavy, new particles (resonances) that act as a backup safety valve.

The Big Question: Where Do We Look?

The paper asks: If we find this slight bend in the Higgs key, which future machine is best at finding the new backup particles?

The authors compare two giant contenders:

1. The FCC-hh: A massive proton-proton collider (like a super-charged LHC) that smashes protons together at 100 TeV. Think of this as a demolition derby. You throw two heavy trucks (protons) at each other at incredible speed. It's chaotic, creates a lot of dust and debris (background noise), but you have a huge amount of raw energy.
2. The Muon Collider: A machine that smashes muons (a heavier cousin of the electron) together at 10 TeV. Think of this as a precision surgery. You are aiming two very specific, clean needles at each other. There is much less dust and noise, and you can see the results very clearly, even if the total energy is lower than the demolition derby.

The Experiment: Weak Boson Fusion

The paper focuses on a specific way to find these new particles called Weak Boson Fusion (WBF).

• The Analogy: Imagine two people (particles) throwing balls (weak bosons) at each other. Usually, they just bounce off. But if a new, heavy particle exists, the balls might hit it, causing it to vibrate or "resonate" before breaking apart.
• The researchers simulated this process for both the "Demolition Derby" (FCC-hh) and the "Precision Surgery" (Muon Collider).

The Results: A Surprising Tie

The paper's main finding is a "no-lose theorem" for the next generation of colliders. If the Higgs is slightly off, both machines are expected to find the new particles, but they do it in different ways:

• The Heavy Hitters (FCC-hh): Because it has so much raw energy, it can create these new heavy particles easily. However, because it's a messy environment (lots of proton debris), it's hard to see the new particle clearly. It's like trying to spot a specific shiny coin in a pile of gravel.
• The Clean Scanners (Muon Collider): It has less total energy, but the environment is incredibly clean. When the new particle appears, it stands out like a diamond in a glass case. The researchers found that the Muon Collider can see these new particles just as well as the FCC-hh, even though it's "smaller," because the background noise is so low.

The Reach: Both machines are expected to be able to find these new particles if they weigh up to about 6 TeV (roughly 6,000 times heavier than a proton).

The "Fermion" Twist

The paper also looked at a complication: What if these new particles also talk to heavy things like top quarks?

• If the new particles are "shy" and only talk to force-carrying particles, both machines find them easily.
• If they are "social" and talk to heavy matter (fermions) too, they might decay in messy ways that hide them. In this case, the Muon Collider still has a slight edge because its clean environment helps separate the signal from the noise, though the search becomes harder for both.

The Role of the "Middleman" (FCC-ee)

The paper mentions a third machine, the FCC-ee, which would run before the big ones. Think of this as a calibration lab. It wouldn't smash things at high energy to find new particles directly. Instead, it would measure the Higgs key with extreme precision. If the FCC-ee confirms the key is bent, it gives the green light for the big machines (FCC-hh and Muon Collider) to go hunting for the backup safety valves.

Summary

The paper argues that if the Higgs boson isn't behaving exactly as predicted, nature must have a backup plan involving new, heavy particles. Whether we build a massive proton collider or a cleaner muon collider, we have a very good chance of finding these new particles. The "no-lose" part is that if the Higgs is slightly wrong, the universe forces us to find the solution at these next-generation facilities.

Technical Summary

Technical Summary: Weak boson probes of Higgs unitarity restoration at 10 TeV parton colliders

Problem Statement
The Standard Model (SM) relies on the Higgs boson to perturbatively unitarize massive particle scattering processes, specifically the scattering of longitudinal gauge bosons ($W_L, Z_L$). While the discovery of the 125 GeV Higgs boson confirmed this mechanism, the High-Luminosity LHC (HL-LHC) is expected to probe Higgs couplings at the $\sim 2\%$ level. A potential outcome of the HL-LHC era is a slight tension between measured Higgs couplings and SM expectations (specifically $\mu_H < 1$). Such a deviation would imply that the SM Higgs alone is insufficient to restore unitarity at high energies, necessitating new physics (NP) in the form of additional resonances (scalars or vectors) to cure the high-energy growth of scattering amplitudes. The central question addressed is which future collider concept—a 100 TeV hadron collider (FCC-hh) or a 10 TeV muon collider ($\mu$ coll)—offers superior sensitivity to discover these unitarity-restoring resonances, given a specific deviation in Higgs couplings.

Methodology
The authors employ a model-independent framework grounded in perturbative unitarity and sum rules derived from the high-energy behavior of Weak Boson Fusion (WBF) scattering ($WW \to WW, WZ, ZZ$).

• Theoretical Framework: The analysis utilizes sum rules (Eqs. 1a–1d) that relate the Higgs coupling modification ($\kappa_H = \sqrt{\mu_H}$) to the couplings and masses of new resonances. Two benchmark scenarios are studied:
  1. Scalar Sector Extension: A heavy scalar partner $H'$ restores unitarity.
  2. Vector Sector Extension: Iso-triplet vector resonances ($W', Z'$) restore unitarity.
     The couplings of these new states are fixed by the requirement to cancel the $s^2$ and $s$ divergences in longitudinal scattering amplitudes, given a specific $\mu_H < 1$.
• Simulation: Event generation is performed using MadGraph5_aMC@NLO with custom models implemented via FeynRules. Both resonant (signal) and non-resonant (SM background) WBF processes are simulated, including interference effects.
• Collider Comparison:
  • FCC-hh: $\sqrt{s} = 100$ TeV, $\mathcal{L} = 30$ ab$^{-1}$. Analysis focuses on WBF topologies (two forward tagging jets, large rapidity gap). Backgrounds include non-resonant SM WBF and $V+$jets (scaled from ATLAS data).
  • Muon Collider: $\sqrt{s} = 10$ TeV, $\mathcal{L} = 10$ ab$^{-1}$. Analysis utilizes inclusive final states, leveraging the absence of large QCD multijet backgrounds to reconstruct fully hadronic decays ($V \to jj$) with high efficiency.
• Selections: Signal regions are defined by diboson invariant mass ($m_{VV}$) for fully reconstructed states and generalized transverse mass ($m_T$) for states with neutrinos. Systematic uncertainties are not included; significance is estimated via $S/\sqrt{B}$.

Key Contributions

1. Direct Comparison of Discovery Potentials: The paper provides a quantitative comparison of the discovery reach for unitarity-restoring resonances at a 100 TeV hadron collider versus a 10 TeV muon collider, specifically within the context of WBF production.
2. Coupling-Dependent Sensitivity: The study demonstrates how the sensitivity to new physics scales with the magnitude of the Higgs coupling deviation ($\mu_H$). It highlights that the "no-lose theorem" for the next generation of colliders is contingent on the persistence of Higgs coupling deviations observed at the HL-LHC.
3. Role of Fermion Couplings: The analysis explicitly addresses the impact of fermion couplings on vector resonance searches. It shows that if $Z'$ resonances couple significantly to top quarks ($Z' \to t\bar{t}$), the branching ratio to $WW$ is suppressed, drastically reducing sensitivity in WBF channels for both colliders, though $t\bar{t}$ searches would become the primary discovery mode.

Results

• Scalar Resonances ($H'$): Both colliders exhibit comparable sensitivity to scalar resonances with masses up to $\mathcal{O}(6)$ TeV for narrow widths ($\Gamma/m \sim 2.5\% - 10\%$).
  • FCC-hh: Relies on large production cross-sections but suffers from background contamination in fully leptonic channels due to missing neutrinos. Semi-leptonic channels do not improve sensitivity due to overwhelming $V+$jets backgrounds.
  • Muon Collider: Exploits fully hadronic decays ($V \to jj$) with high tagging efficiency (80%) and negligible QCD backgrounds, allowing direct reconstruction of the narrow resonance peak.
• Vector Resonances ($W', Z'$):
  • For large coupling deviations ($\mu_H = 0.95$), both colliders probe masses up to $\sim 7$ TeV ($Z'$) and $\sim 4$ TeV ($W'$).
  • For small deviations ($\mu_H = 0.99$), the muon collider retains sensitivity up to $\sim 5.5$ TeV for $Z'$, whereas the FCC-hh loses sensitivity above $\sim 2$ TeV. This is attributed to the FCC-hh probing kinematic endpoints with lower statistical yield compared to the muon collider's cleaner environment.
• Fermiophobic vs. Fermiophilic: If vector resonances are fermiophobic, both machines show strong sensitivity. If they couple to fermions (specifically $t\bar{t}$), the $Z' \to WW$ rate is suppressed, reducing WBF discovery reach significantly at both facilities.

Significance and Claims
The paper posits a "modern incarnation of the no-lose theorem": if Higgs coupling deviations persist (as potentially hinted by HL-LHC data), multi-TeV resonances required for unitarity restoration are expected to become accessible at next-generation facilities.

• The authors claim that both the FCC-hh and a 10 TeV muon collider offer excellent discovery potential for these resonances, with sensitivities being relatively similar in the parameter space of narrow resonance searches.
• The study emphasizes that an intermediate precision program at an $e^+e^-$ machine (FCC-ee) is crucial to corroborate any Higgs coupling deviations observed at the HL-LHC, thereby validating the motivation for the high-energy resonance searches detailed in this work.
• The work remains modest regarding specific experimental details, acknowledging that many parameters for future colliders are not yet fixed, and systematic uncertainties are not fully modeled. The results are presented as indicative of the relative performance of these two machine concepts in the specific context of WBF resonance searches.