Searching for heavy neutrinos in $e^+ e^- \to W^+ W^-$: it is all about unitarity

This paper demonstrates that using a linearized mixing approximation to study heavy neutrino contributions in $e^+e^- \to W^+W^-$ yields physically incorrect results due to unitarity violations, whereas an exact unitary mixing scheme provides reliable, experimentally testable signatures for future colliders.

The Gist

Imagine you are a detective trying to solve a mystery at a high-speed particle collision. The crime scene is a future particle collider (like a super-powered version of the Large Hadron Collider, but for electrons and positrons). The goal? To find "Heavy Neutrinos"—ghostly, massive particles that might explain why the universe has so much matter and so little antimatter.

The paper you are asking about is a theoretical investigation into how we should look for these ghosts. The authors, Chachava and Godunov, argue that the way physicists have been calculating the search so far is fundamentally flawed, like trying to navigate a storm with a broken compass.

Here is the story of their discovery, broken down into simple concepts and analogies.

1. The Setup: The "W-W" Dance

In the Standard Model of physics, when an electron and a positron smash together, they can turn into two heavy particles called W bosons ($W^+W^-$). This process is like a perfectly choreographed dance.

• The Problem: If you calculate the energy of this dance without any special rules, the math says the energy should go to infinity as the collision gets faster. That's impossible in the real world.
• The Fix: Nature has a "safety valve" called Unitarity. It's like a bouncer at a club who ensures the party doesn't get out of hand. In this dance, two different "paths" (diagrams) the particles can take cancel each other out perfectly, keeping the energy finite and the laws of physics intact.

2. The Suspect: Heavy Neutrinos

Now, imagine we introduce a new character: a Heavy Neutrino. It's a heavy, invisible partner that can mix with the light, familiar neutrinos.

The paper asks: If we add this heavy partner to the dance, does the safety valve (Unitarity) still work?

The authors found that there are two ways physicists have been trying to add this heavy partner to the math:

Method A: The "Linearized" Approach (The Flawed Compass)

This is the popular method used in many current models (like the "HeavyN" model).

• The Analogy: Imagine you are adding a new ingredient to a soup. The linearized approach is like saying, "Let's just sprinkle a little bit of this heavy spice in, but let's pretend the rest of the soup recipe stays exactly the same."
• The Result: At low energies, this works fine. But as the collision energy gets higher, the math breaks. The "safety valve" stops working. The calculated probability of the event starts growing infinitely, like a balloon that never stops inflating until it pops.
• The Verdict: The authors say this method is physically incorrect for high-energy collisions. It gives us false hope or false alarms because it violates the fundamental laws of the universe.

Method B: The "Exact Unitary" Approach (The Perfect Recipe)

This is the method the authors advocate for.

• The Analogy: This time, when you add the heavy spice, you realize that adding it changes the flavor balance of the entire soup. You have to adjust the other ingredients (the light neutrinos) to compensate. You respect the "recipe" of the universe completely.
• The Result: The safety valve (Unitarity) stays intact. Even at super high energies, the math behaves correctly. The probability doesn't explode; it settles down.
• The Surprise: Because the math is adjusted correctly, this method predicts something the other method misses: The dance can actually slow down. In certain energy ranges, the presence of the heavy neutrino causes the collision rate to drop below the standard prediction before rising again.

3. The "Goldilocks" Zone

The authors discovered a specific "Goldilocks" zone where the heavy neutrino leaves a fingerprint:

• Too Light: If the heavy neutrino is too light, it's already been found or ruled out.
• Too Heavy: If it's too heavy, it's too hard to create.
• Just Right: There is a sweet spot where the collision energy is high enough to feel the heavy neutrino's presence, but not so high that the signal is washed out.

In this zone, the Exact Unitary method predicts a unique signature:

1. A Dip: First, the number of collisions drops slightly (a "dip" in the data).
2. A Rise: Then, as energy increases further, the number of collisions shoots up significantly.

The Linearized method (the flawed one) misses the "Dip" entirely. It only sees the rise, and it predicts the rise happens at the wrong energy levels.

4. Why This Matters for the Future

We are building new, massive colliders (like the ILC, CLIC, or FCC-ee) to hunt for these particles.

• If we use the old, flawed math (Linearized): We might look for the heavy neutrino in the wrong place, or we might misinterpret a "dip" in the data as a glitch rather than a discovery. We might think we see a signal when it's just a mathematical error.
• If we use the new, correct math (Exact Unitary): We know exactly where to look. We know to watch for that specific "dip" followed by a "rise." This gives us a much better chance of actually finding the heavy neutrino or setting strict limits on where it can't be.

The Takeaway

The paper is essentially a warning label and a new map.

• Warning: "Stop using the old, simplified math for high-energy collisions; it breaks the laws of physics."
• New Map: "Use this new, rigorous math. It tells us that heavy neutrinos will leave a very specific, detectable signature (a dip and then a rise) in the data. If we look carefully, we might finally catch these elusive ghosts."

In short: To find the heavy neutrino, we must respect the rules of the universe (Unitarity) completely, or we will never find it.

Technical Summary

1. Problem Statement

The paper investigates the process $e^+e^- \to W^+W^-$ at future lepton colliders (such as ILC, CLIC, FCC-ee, and muon colliders) as a probe for Heavy Neutral Leptons (HNLs) within the Type-I Seesaw mechanism.

The core problem addressed is the theoretical inconsistency arising from different implementations of heavy-light neutrino mixing:

• Linearized Mixing: A popular approximation (used in models like HeavyN) where the PMNS matrix is kept unitary, and heavy neutrinos are added as perturbations.
• Exact Unitary Mixing: A rigorous approach where the full mixing matrix (including light and heavy states) is unitary, implying the light neutrino mixing matrix is non-unitary.

The authors argue that the linearized approximation, while valid for many processes, leads to physically incorrect results (specifically, violations of $S$-matrix unitarity) for the $e^+e^- \to W^+W^-$ process at high energies.

2. Methodology

The study employs a combination of analytical calculations and numerical simulations:

• Toy Model Analysis: The authors first construct a simplified "toy model" with $SU(2)$ gauge symmetry, massless electrons, and degenerate $W/Z$ boson masses. This allows for an analytical derivation of the scattering amplitude and cross-section, isolating the interplay between $s$-channel ($Z$ exchange) and $t$-channel (neutrino exchange) diagrams.
• Two Mixing Schemes: They derive the differential cross-sections ($d\sigma/d\cos\theta$) for both:
  1. Exact Unitary Mixing: Where the coupling of light neutrinos is modified by a factor $(1 - |V_{eN}|^2)$ to preserve unitarity.
  2. Linearized Mixing: Where the light neutrino coupling remains unchanged, and the heavy neutrino contribution is simply added.
• Standard Model Extension: The toy model results are validated against the full Standard Model (SM) extended with Type-I Seesaw. This includes:
  • Analytical treatment of the sum of amplitudes for multiple heavy neutrinos ($\mathcal{N}=3$).
  • Numerical simulations using MadGraph with a custom FeynRules model (HeavyNU) that implements exact unitary mixing for three heavy neutrinos.
• Parameter Space: The study scans heavy neutrino masses ($M_N$) from 500 GeV to 50 TeV and mixing parameters ($|V_{eN}|^2$) up to current experimental bounds ($\sim 10^{-3}$).

3. Key Contributions

• Demonstration of Unitarity Violation: The paper proves that the linearized mixing approximation leads to a cross-section that grows linearly with energy ($\sigma \propto s$) at high energies ($\sqrt{s} \gg M_N$), violating the unitarity bound ($\sigma \propto 1/s$). This occurs because the cancellation between $s$- and $t$-channel amplitudes is broken when the light neutrino coupling is not adjusted.
• Analytical Formulas: The authors provide explicit analytical expressions for the differential cross-sections in both mixing schemes, identifying the specific energy regimes where deviations from the SM occur.
• New Experimental Signatures: They identify a unique signature of exact unitary mixing: a suppression of the cross-section (up to ~18%) at intermediate energies before a potential enhancement at higher energies, a feature absent in linearized mixing.
• FeynRules Model (HeavyNU): The authors release a model file compatible with FeynRules and MadGraph that implements exact unitary mixing for three heavy neutrinos, facilitating future phenomenological studies.

4. Key Results

A. Behavior of Cross-Sections

• Exact Unitary Mixing:

  • At low energies ($\sqrt{s} \ll M_N$), the cross-section exhibits a linear growth ($\sigma \propto s$) in a specific window defined by $M_W/\sqrt{3}|V_{eN}| \lesssim \sqrt{s} \lesssim M_N$.
  • At high energies ($\sqrt{s} \gg M_N$), the cross-section correctly falls off as $1/s$, preserving unitarity. This is due to the cancellation between the $s$-channel and the modified $t$-channel (where the heavy neutrino contribution effectively subtracts from the light neutrino term).
  • Suppression: For specific mixing values, the interference between $s$- and $t$-channels causes the cross-section to dip below the SM prediction by up to 18% (ratio $\approx 0.82$).

• Linearized Mixing:

  • The cross-section grows cubically ($\sigma \propto s^3$) at intermediate energies and linearly ($\sigma \propto s$) at high energies.
  • It never drops below the SM prediction; it only enhances the cross-section.
  • This behavior is physically inconsistent with $S$-matrix unitarity.

B. Impact of Heavy Neutrino Mass ($M_N$)

• Light HNLs ($M_N \sim 500$ GeV): The effects are negligible across the energy ranges of proposed colliders because the condition for significant deviation ($M_N \gtrsim M_W/\sqrt{3}|V_{eN}|$) is not met.
• Heavy HNLs ($M_N \sim 3-50$ TeV): Significant deviations appear.
  • For $M_N = 3$ TeV and large mixing ($|V_{eN}|^2 \approx 2.25 \times 10^{-2}$), the cross-section can be several times larger than the SM prediction at $\sqrt{s} \approx 3$ TeV.
  • The transition from suppression to enhancement depends critically on the ratio of $\sqrt{s}$ to $M_N$.

C. Multi-Neutrino Scenarios

• In a scenario with three heavy neutrinos ($\mathcal{N}=3$) and a hierarchical mass spectrum ($M_1 \ll M_2 \ll M_3$), the physics is dominated by the second neutrino ($N_2$) at intermediate energies.
• The contribution of the heaviest neutrino ($N_3$) only becomes relevant at extremely high energies ($\sqrt{s} \sim M_3$), which may be beyond the reach of current collider plans.
• In the linearized scheme, the contribution of $N_3$ is negligible at all energies due to the small mixing parameter $|V_{e3}|^2$.

5. Significance and Conclusion

• Theoretical Consistency: The paper establishes that exact unitary mixing is the only theoretically consistent framework for analyzing $e^+e^- \to W^+W^-$ in the presence of heavy neutrinos. The linearized approximation, while convenient, yields unphysical results for this specific process.
• Experimental Strategy:
  • Future colliders can probe heavy neutrinos not just by looking for an excess of events, but by searching for cross-section suppression in the central angular region ($\cos\theta \approx 0$).
  • A measurement precision of ~5% could reveal the existence of heavy neutrinos even if $M_N$ is very large, provided the mixing parameter is within the detectable range.
  • The distinct energy dependence (suppression followed by enhancement) in the unitary case offers a clear discriminant against background and other new physics models.
• Comparison with Literature: The authors note that while a recent paper [37] considered the limit $M_N \to \infty$, their work provides the necessary conditions on $M_N$ and mixing parameters to observe sizable corrections at finite collider energies, highlighting the importance of the mass-dependent transitions.

In summary, the paper argues that searching for heavy neutrinos in $W^+W^-$ production requires a rigorous unitary treatment of mixing. This approach reveals unique signatures (suppression and specific energy scaling) that are missed by standard linearized approximations, offering a robust path for discovery or exclusion at future high-energy lepton colliders.