Searches for the leptophilic Z' boson at the International Linear Collider and Linear Collider Facility

This paper presents sensitivity studies for detecting a leptophilic Z' boson at the International Linear Collider and the Linear Collider Facility at CERN, utilizing WHIZARD simulations to evaluate signal and background in the di-muon channel and establish expected limits on Z' couplings to Standard Model leptons at center-of-mass energies of 250 GeV and 550 GeV.

The Gist

Imagine the universe as a giant, complex machine. For decades, physicists have been trying to understand how it works using a rulebook called the Standard Model. But this rulebook has holes in it. It can't explain gravity, why there's more matter than antimatter, or what "dark matter" is.

To fix these holes, scientists are looking for new, hidden parts of the machine. One of the most promising candidates for a new part is a particle called a Leptophilic Z' boson.

Here is a simple breakdown of what this paper is about, using some everyday analogies.

1. The Mystery Guest: The "Leptophilic" Z'

Think of the Standard Model particles as guests at a massive party.

• Quarks are the loud, rowdy dancers.
• Leptons (like electrons and muons) are the quiet guests sitting at the tables.
• The Z' Boson is a new, invisible guest who only talks to the quiet guests (the leptons). It ignores the rowdy dancers entirely.

Because this new guest ignores the loud dancers, it is very hard to spot at big particle smashers like the LHC (which smashes protons, made of rowdy quarks). It's like trying to find a shy person at a mosh pit; they just get lost in the noise.

2. The Perfect Venue: The Linear Collider

To find this shy guest, we need a quieter, more controlled environment. The paper looks at two future machines: the ILC (International Linear Collider) and the LCF (Linear Collider Facility).

• Instead of smashing two messy bags of marbles (protons) together, these machines shoot two clean, precise streams of electrons and positrons (like two perfectly aimed streams of water) at each other.
• Because the collision is so clean, if our shy Z' guest shows up, it leaves a very clear footprint.

3. The Detective Work: Finding the "Ghost"

The Z' boson is unstable; it appears and disappears instantly. We can't see it directly. Instead, we look for what it leaves behind.

• The Clue: The Z' decays into a pair of muons (a heavier cousin of the electron).
• The Trick: To create the Z', the electron and positron usually have to "radiate" a photon (a particle of light) to lose some energy first. This is called "radiative return."
• The Problem: In the past, scientists tried to spot the Z' by looking for the muons and the photon. But in the real world, the photon often flies off in a straight line down the "beam pipe" (a tunnel) and gets lost. It's like trying to find a suspect by looking for their hat, but the hat blew away before you could see it.

4. The New Strategy: "Feeling" the Missing Hat

The authors of this paper came up with a clever new way to catch the Z' without needing to see the lost photon.

• The Analogy: Imagine a person on a skateboard (the Z') throwing a heavy ball (the photon) forward. Even if you can't see the ball, you know the skateboarder must have recoiled backward to conserve momentum.
• The Method: The scientists look at the muons (the skateboarder). They measure how fast the muons are moving and in what direction. By doing some math, they can calculate how much "recoil" happened. If the math adds up perfectly to a specific energy, they know a Z' was created, even if the photon is missing.

5. The Results: How Good Are We?

The team used powerful computer simulations (like a video game engine for physics) to test this idea. They simulated billions of collisions at different energy levels (250 GeV and 550 GeV).

• The Findings: Their new method is much better than the old ones. It allows them to spot the Z' boson even when the photon is lost.
• The Limits: They calculated how small the "shyness" (coupling strength) of the Z' could be before they would miss it.
  • If the Z' exists, the ILC and LCF will likely find it or rule it out very quickly.
  • The results show that these linear colliders will be much more sensitive than the current Large Hadron Collider (LHC) or even the proposed FCC (a giant circular collider).

Why This Matters

This paper is essentially a "proof of concept" for a future physics experiment. It says:

"If we build these linear colliders and use this specific 'missing photon' detective trick, we will be able to find this elusive Z' boson. If we don't find it, we will know exactly how weak it must be, which helps us rewrite the rulebook of the universe."

It's a roadmap for how to catch a ghost that only talks to the quiet people at the party, using a machine that is far more precise than anything we have today.

Technical Summary

1. Problem Statement and Motivation

The Standard Model (SM) is incomplete, failing to explain phenomena such as dark matter, neutrino oscillations, and matter-antimatter asymmetry. A prominent Beyond the Standard Model (BSM) scenario involves a leptophilic $Z'$ boson associated with a $U(1)_{L_e - L_\mu}$ gauge group. This boson couples only to electrons, muons, and their corresponding neutrinos.

• Challenge at Hadron Colliders: At the LHC, searching for this specific $Z'$ is difficult because production is limited to associated processes, and projected High-Luminosity LHC (HL-LHC) bounds are not expected to surpass existing, relatively weak LEP2 limits.
• Opportunity at Lepton Colliders: Future $e^+e^-$ colliders offer superior sensitivity. However, prior to this work, there were no dedicated studies for the International Linear Collider (ILC) or the proposed Linear Collider Facility (LCF) at CERN regarding this specific benchmark scenario.
• Goal: To evaluate the sensitivity of the ILC and LCF to the leptophilic $Z'$ boson, specifically focusing on the di-muon decay channel ($\mu^+\mu^-$), which offers the highest sensitivity.

2. Methodology

Model Definition

The study assumes a vector $Z'$ boson coupling to SM leptons.

• Decay Modes: Approximately 1/3 of decays are invisible ($Z' \to \nu\bar{\nu}$), while the rest are visible ($Z' \to e^+e^-, \mu^+\mu^-$).
• Production Mechanism: For $m_{Z'} < \sqrt{s}$, the dominant production mode is radiative return in $e^+e^-$ annihilation: $e^+e^- \to Z'\gamma$.

Event Simulation

• Tools: Events were generated using WHIZARD 3.1.6 and processed through DELPHES with the ILCgen model for fast detector simulation.
• Signal Modeling:
  • The study moves beyond Leading Order (LO) approximations. It includes Initial State Radiation (ISR) and multi-photon emission (up to 3 photons in the matrix element).
  • ISR-ME Matching: Proper matching between the matrix element and soft/collinear ISR structure functions was applied to avoid double-counting.
  • Impact: These higher-order effects increase the signal cross-section by up to 40% compared to LO approximations and significantly distort the reconstructed energy distributions.
• Background Modeling:
  • Simulated SM backgrounds include radiative muon and tau pair production ($\mu^+\mu^-(\gamma)$, $\tau^+\tau^-(\gamma)$), four-fermion processes, and quasi-real photon interactions (Equivalent Photon Approximation).
  • Beam polarization effects were included (80% electron polarization; 30% and 60% positron polarization for 250 GeV and 550 GeV runs, respectively).

Event Selection Strategy

A novel selection strategy was developed to overcome the inefficiency of traditional photon-reconstruction cuts (since ~85-90% of radiative photons are lost in the beam pipe).

• Kinematic Variable: Instead of relying on detecting the photon, the analysis utilizes the longitudinal momentum of the di-muon system, $|P_Z^{(\mu\mu)}|$, which balances the momentum of the lost photon.
• Primary Cut: A cut on the sum of the di-muon energy and absolute longitudinal momentum: $E^{(\mu\mu)} + |P_Z^{(\mu\mu)}|$.
• Secondary Cut: A cut on the total transverse momentum, $P_T^{(tot)}$, to suppress backgrounds involving neutrinos.
• Final State: Events with exactly two reconstructed muons and an arbitrary number of photons.

3. Key Contributions

1. First Dedicated ILC/LCF Study: This is the first rapid analysis specifically targeting the leptophilic $Z'$ benchmark for the ILC and LCF, filling a gap in the European Strategy for Particle Physics (ESPP) Physics Briefing Book.
2. Advanced Simulation: The inclusion of multi-photon emission and proper ISR-ME matching provides a more realistic and robust estimation of signal cross-sections and kinematic distributions than previous LO-only studies.
3. Novel Selection Algorithm: The introduction of a photon-independent selection method based on the kinematics of the di-muon system ($E + |P_Z|$) significantly improves signal efficiency without requiring the reconstruction of the radiated photon.
4. Comprehensive Scope: The study covers ILC at 250 GeV, LCF at 250 GeV, and LCF at 550 GeV, providing a comparative sensitivity analysis across different energy and luminosity configurations.

4. Results

Sensitivity and Limits

• Reconstruction Resolution: The di-muon invariant mass resolution is estimated at 0.7 GeV for 250 GeV running and degrades to 2.0 GeV for 550 GeV running due to higher muon momenta.
• Signal Signature: The $Z'$ signal appears as a narrow resonant peak (intrinsic width $\sim$10 keV, negligible compared to resolution) on top of a smooth SM background.
• Exclusion Limits (95% CL):
  • ILC (250 GeV, 2 ab$^{-1}$): Sets competitive limits on the $Z'$ coupling ($g_{Z'}$).
  • LCF (250 GeV, 3 ab$^{-1}$): Despite lower integrated luminosity than some circular collider proposals, LCF achieves better limits than FCC-ee at 240 GeV.
  • LCF (550 GeV, 8 ab$^{-1}$): Extends the mass reach significantly, probing $Z'$ masses up to ~400-500 GeV.

Comparative Advantage

The study highlights why linear colliders (ILC/LCF) outperform circular colliders (like FCC-ee) for this specific search:

1. Beam Polarization: The ability to polarize beams enhances the signal-to-background ratio.
2. Detector Magnetic Fields: Linear collider detectors can utilize higher magnetic fields (unlike circular colliders where high fields disrupt the beam), resulting in superior muon momentum resolution.
3. Analysis Strategy: The improved selection criteria and higher-order cross-section estimates contribute to tighter constraints.

5. Significance

This work establishes the di-muon channel as the primary discovery mode for leptophilic $Z'$ bosons at future lepton colliders. By demonstrating that the ILC and LCF can surpass existing LEP2 limits and outperform projected HL-LHC and FCC-ee sensitivities, the paper validates the $Z'$ search as a high-priority benchmark for the 2026 update of the European Strategy for Particle Physics. The methodology developed, particularly the photon-independent kinematic selection, offers a robust framework for future BSM searches in radiative return scenarios.