⚛️ phenomenology

The Plan B Model: $Z^{\prime}$ collider phenomenology and discovery prospects

This paper constrains the parameter space of the Plan B Model, which introduces a TeV-scale $Z^{\prime}$ to explain fermion mass features and $b \rightarrow s$ transitions, by re-analyzing LHC direct searches and differential cross-section measurements to identify excluded regions and project the discovery potential of the High-Luminosity LHC.

Original authors: Ben Allanach, Hannah Banks, Jon Butterworth

Published 2026-07-22

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Ben Allanach, Hannah Banks, Jon Butterworth

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, cosmic puzzle. For decades, scientists have been trying to fit the pieces together using a rulebook called the "Standard Model." This rulebook explains how tiny particles like electrons and quarks interact, kind of like a manual for the universe's most basic building blocks. But there's a problem: the manual has some missing pages. It can't quite explain why some particles are heavy, some are light, or why certain rare particle transformations happen more often than the math predicts. It's like having a recipe that works perfectly for a cake but fails miserably when you try to make a soufflé.

To fix these gaps, physicists propose adding new "secret ingredients" to the recipe. One popular idea is a new, invisible force carrier called a Z′ boson (pronounced "Z-prime"). Think of the Standard Model as a band playing a familiar song, and the Z′ as a new, hidden instrument that only shows up when the music gets really loud and energetic. If this instrument exists, it would explain the weird "off-key" notes we've been hearing in particle experiments, particularly those involving heavy particles called "bottom quarks." The big question is: Is this new instrument real, or is it just a figment of our imagination?

The "Plan B" Search for a New Cosmic Instrument

In this paper, a team of physicists introduces a specific recipe for this new instrument called the "Plan B Model." They suggest that the Z′ boson isn't just a random addition; it's a carefully tuned part of the universe that helps explain why the "flavor" of particles (like why a top quark is so heavy while an electron is so light) looks the way it does. This model predicts that the Z′ boson has a mass in the TeV (tera-electronvolt) range, which is a fancy way of saying it's incredibly heavy—about a thousand times heavier than a proton.

The authors of this paper didn't just build the model; they went on a digital treasure hunt to see if the Large Hadron Collider (LHC)—the world's biggest particle-smashing machine—has already found it. They treated the LHC data like a massive library of crime scene photos, looking for the "smoking gun" of a Z′ boson.

The Hunt: Looking for a "Bump" in the Data

When particles smash together at the LHC, they usually create a smooth, predictable spray of debris. But if a heavy, new particle like the Z′ exists, it would appear as a sudden "bump" or spike in the data, like a sudden peak in a mountain range where there should only be a gentle hill.

The researchers used a clever strategy called "re-casting." Instead of waiting for the LHC experiments (ATLAS and CMS) to run a new search specifically for the Plan B Model, they took existing search results and asked: "If our Plan B Model were true, would we have seen it in these old results?" They simulated millions of collisions on their computers, injecting the Plan B Z′ into the mix, and then checked if the resulting "bumps" would have been caught by the LHC's detectors.

They looked at several different "channels" or ways the Z′ could show up:

The "Di-lepton" Channel: The Z′ decaying into pairs of electrons or muons (heavy cousins of electrons). This is the cleanest signal, like looking for a bright flash in a dark room.
The "B-Jet" Channel: The Z′ decaying into pairs of bottom quarks (which turn into jets of particles). This is messier, like trying to find a specific type of leaf in a pile of autumn debris.
The "B-Tagged" Channel: A special search looking for a Z′ that decays into muons plus extra bottom quarks. This is like looking for a specific suspect who is always seen with a particular accomplice.

The Findings: The Ghost is Still Hiding

The results of their hunt are a mix of "good news" and "bad news" for the Plan B Model.

First, the bad news: While the LHC has ruled out many generic versions of new physics, the researchers found that within the specific "sweet spot" where the Plan B Model works best to explain the flavor puzzles, there is currently no 95% confidence bound excluding the Z′ boson. In other words, the current LHC data has not yet been able to say for sure that the particle doesn't exist in the most theoretically favored region. The "bump" would have been too subtle to ignore if the particle were in a different, less favored spot, but in the best-fit area, it remains hidden.

However, there is still hope. The paper shows that there is a "safe zone" left over. If the Z′ boson is heavier than the current reach, or if it interacts with matter in a very specific way, it might still be hiding in the shadows. The current LHC data hasn't looked deep enough into this high-energy territory to say for sure that the particle doesn't exist in the preferred region.

The authors also looked ahead to the High-Luminosity LHC (HL-LHC), a future upgrade that will smash particles together with much more intensity (3000 fb⁻¹ of data). They simulated what would happen with this super-powerful machine. Their calculations suggest that the HL-LHC could push the search limit up to a mass of about 2.4 TeV, but they explicitly note that this is a conservative estimate. Because the HL-LHC will likely operate at a slightly higher energy than the 13 TeV used in their simulations, the actual sensitivity could extend even higher, potentially reaching 2.5 TeV or more. This means the "safe zone" will get smaller, but it won't disappear completely. The HL-LHC might finally be the instrument loud enough to hear the Z′ if it's there.

The Verdict: Not Proven, But Not Dead

The paper doesn't claim to have discovered the Z′ boson. In fact, it explicitly states that no convincing evidence of such a resonance has been found yet. Instead, the paper acts as a map, drawing a clear line around the areas where the Plan B Model is no longer a valid option, while highlighting the specific region where it still survives.

The authors emphasize that while the "bump-hunting" method (looking for simple spikes) has been very effective so far, the future of particle physics might need to change its tune. As the LHC gets more powerful but doesn't necessarily get much faster (in terms of energy), the "bumps" might become harder to find. The paper suggests that in the future, scientists might need to look for subtler clues—tiny, precise differences in how particles scatter, rather than just big spikes. This is like listening for a whisper in a noisy room rather than waiting for a shout.

In summary, the Plan B Model is a clever idea that explains some of the universe's weirdest quirks, but the LHC has not yet kicked it out of the neighborhood in its most favored form. If the Plan B Z′ is real, it's likely a heavyweight champion hiding just beyond our current reach, waiting for the High-Luminosity LHC to shine a brighter light on the darkness.

Technical Summary: The Plan B Model: Z′ collider phenomenology and discovery prospects

Problem and Motivation
The Plan B Model, originally proposed in Ref. [1], is an extension of the Standard Model (SM) designed to explain gross features of the fermion mass spectrum and address anomalies in $b \to s$ quark-flavour transitions. The model introduces an additional anomaly-free $U(1)_X$ gauge symmetry, where fermion charges are defined by $X = 3B_3 - L_1 - 2L_2$ (three times the third-family baryon number minus electron number minus twice muon number). This symmetry is spontaneously broken by a TeV-scale flavon field, generating a massive, electrically neutral spin-1 boson, $Z'$ . While the model successfully improves the fit to hundreds of flavour and LEP2 measurements (reducing $\chi^2$ by 34 units), it predicts a new particle that must be constrained by direct collider searches. The primary challenge addressed in this work is to rigorously delineate the viable parameter space of the Plan B Model ( $M_{Z'}$ and $g_{Z'}$ ) by re-interpreting existing LHC Run II data and estimating the future reach of the High-Luminosity LHC (HL-LHC), moving beyond simple "best-fit" points to explore the full 95% confidence level (CL) preferred region.

Methodology
The authors employ a "re-casting" methodology to constrain the model using direct resonance searches and differential cross-section measurements:

Theoretical Framework: The model assumes the chiral fermion content of the SM plus three right-handed neutrinos. The $Z'$ couplings to mass eigenstates are derived from the gauge eigenbasis via specific mixing matrices ( $V_{dL}$ , $V_{uL}$ , etc.), resulting in dominant couplings to the third generation and specific flavour-changing neutral currents (FCNCs) involving $b \to s$ . The production cross-section at the LHC is dominated by $b\bar{b} \to Z'$ , with sub-dominant contributions from $s\bar{b}$ fusion and associated production with a $b$ -quark.
Direct Search Re-interpretation: The authors re-analyze four specific LHC search channels using Run II data ( $\sqrt{s}=13$ $s = 13$ TeV) and simulate events using MadGraph5_aMC@NLO, Pythia 8, and Delphes. The channels include:
- ATLAS di-lepton ( $e^+e^-, \mu^+\mu^-$ ) resonance searches.
- ATLAS di-lepton searches with specific $b$ -jet requirements (zero or one $b$ -tagged jet).
- CMS di-lepton resonance searches.
- CMS di-muon resonance searches in association with additional $b$ -jets (specifically designed for models like Plan B).
- For each channel, the authors compute the ratio $\mu$ of the experimental 95% CL upper bound to the theoretical prediction. Exclusion contours are defined where $\mu = 1$ .
Differential Cross-Section Constraints: Using the Contur tool, the authors test the model against a broad spectrum of unfolded LHC differential cross-section measurements (including di-jets, top production, and Drell-Yan). This involves injecting the $Z'$ contribution into SM predictions and evaluating the likelihood ratio using the CLs method.
HL-LHC Projections: Projected sensitivities for 3000 fb $^{-1}$ are estimated by scaling signal and background uncertainties, assuming a conservative center-of-mass energy of 13 TeV (acknowledging that the actual HL-LHC may operate at higher energies, which would improve sensitivity).

Key Contributions

Comprehensive Parameter Space Scan: Unlike previous studies that focused on the "best-fit" indirect measurement parameters (IMPs), this work explores the full 95% CL preferred region of the Plan B Model.
Multi-Channel Analysis: The study incorporates constraints from channels with $b$ -jet vetoes and $b$ -tagging, specifically leveraging the CMS di-muon plus $b$ -jets search, which is highly sensitive to the $Z'$ 's coupling to $b$ -quarks.
Perturbativity and Kinetic Mixing: The authors explicitly delineate the region where the $Z'$ width-to-mass ratio $\Gamma_{Z'}/M_{Z'} \leq 0.1$ , ensuring the validity of their perturbative calculations. They also discuss the potential impact of kinetic mixing ( $\chi$ ) but assume it is negligible for the primary analysis, noting that significant mixing would require re-evaluation of bounds.
Integration of Indirect and Direct Constraints: The paper combines direct resonance searches with differential cross-section constraints to provide a holistic view of the model's viability.

Results

Current Exclusions: A significant fraction of the 95% CL preferred region for the Plan B Model is already excluded by existing LHC Run II data. The di-muon channel generally provides the strongest constraints across the mass range, followed by the di-electron channel.
Role of $b$ -jets: For $M_{Z'} < 3500$ GeV, searches involving $b$ -tagging or $b$ -vetoes provide the leading bounds. Specifically, the ATLAS search with a $b$ -veto outperforms the $b$ -tag search due to reduced top-quark background.
HL-LHC Reach: The HL-LHC is projected to probe the remaining viable parameter space up to $M_{Z'} \approx 2.5$ TeV. However, the authors note that the gain in sensitivity from increased luminosity is modest compared to the gains from direct resonance searches in the past, suggesting a shift in strategy may be necessary.
Differential Measurements: The Contur analysis confirms that direct resonance searches are currently more constraining than differential cross-section measurements for the Plan B Model. The differential constraints are weaker, particularly at high masses, due to statistical limitations in the high-mass tails of the distributions.
Model Viability: While much of the parameter space is excluded, a region remains viable for $M_{Z'}$ above a few TeV, depending on the coupling strength $g_{Z'}$ .

Significance and Claims
The paper claims that the Plan B Model is under significant pressure from current LHC data, with a large portion of its preferred parameter space already excluded. The authors argue that while the HL-LHC will extend the reach to higher masses, the marginal gains from increased luminosity in "bump-hunt" searches suggest diminishing returns. Consequently, the paper advocates for a strategic shift in experimental analysis for the HL-LHC era: moving away from purely inclusive resonance searches toward high-precision differential cross-section measurements in channels motivated by direct searches. This approach, the authors suggest, could uncover subtle hints of new physics or provide robust monolithic constraints on Beyond the Standard Model (BSM) theories, similar to the legacy of LEP. The work emphasizes that measuring multiple channels (e.g., di-leptons with and without $b$ -jets) is crucial for model testing, as the relative rates are predicted by the theory at every point in parameter space.

The "Plan B" Search for a New Cosmic Instrument

The Hunt: Looking for a "Bump" in the Data

The Findings: The Ghost is Still Hiding

The Verdict: Not Proven, But Not Dead

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