Testing the unitarity of the light neutrino mixing matrix

This paper proposes a novel collider-based test for the unitarity of the PMNS mixing matrix by exploiting anomalous cross-section growth from incomplete cancellation between $t$-channel neutrino and $s$-channel gauge-boson exchanges, deriving model-independent bounds from LEP-II data and projecting future sensitivities for lepton and hadron colliders.

The Gist

Imagine the universe is a giant, complex dance floor. In the Standard Model of physics (our current best rulebook for how particles behave), there are three main dancers: the electron, the muon, and the tau. They are the "light neutrinos."

For decades, physicists have believed these three dancers move in perfect harmony, following a strict rule called unitarity. Think of unitarity like a perfect accounting system: if you add up the probability of a dancer being in any possible state, it must equal exactly 100% (or 1.0). Nothing is lost, nothing is created out of thin air.

However, there's a nagging suspicion that this accounting system might be broken. What if these three light dancers are actually secretly holding hands with three (or more) "heavy" dancers we can't see? If they are mixing with these invisible heavy partners, the math for the three light ones would no longer add up to 100%. The "light" dancers would be missing a piece of their identity.

This paper proposes a clever way to catch these missing pieces using giant particle colliders (like the LHC or future machines).

The Analogy: The Perfect Cancellation

To understand the test, imagine two opposing forces trying to cancel each other out perfectly, like two people pushing a car from opposite sides with equal strength. The car doesn't move.

• The Setup: In particle physics, when we smash particles together to create pairs of "W bosons" (heavy force carriers), there are two main ways this happens:
  1. The "s-channel" (The Direct Hit): Two particles collide head-on and create a heavy boson that splits into Ws.
  2. The "t-channel" (The Exchange): The particles swap a neutrino back and forth to create the Ws.

In a perfect, unitary world, these two processes are perfectly balanced. As the energy of the collision gets higher and higher, the "push" from the direct hit and the "push" from the exchange cancel each other out exactly. The result is a smooth, predictable number of W bosons produced.

The Glitch: The Missing Partner

Now, imagine the "light" dancers are actually holding hands with invisible "heavy" partners. Because the heavy partners are too heavy to be created in our current collisions, they don't participate in the "exchange" dance (the t-channel).

This breaks the balance!

• The "direct hit" (s-channel) still happens normally.
• But the "exchange" (t-channel) is weaker than it should be because the light neutrinos are "diluted" by their mixing with the heavy ones.

The Result: The two forces no longer cancel out perfectly. Instead of staying smooth, the number of W bosons produced starts to explode as you increase the energy. It's like the car suddenly lurching forward because one of the pushers stopped pushing.

The Experiment: Catching the Explosion

The authors of this paper say: "Let's look for this explosion."

1. The Test: They propose smashing electrons and positrons (or protons) together at incredibly high speeds.
2. The Signal: If the PMNS matrix (the dance card for neutrinos) is perfect, the number of W bosons produced will follow a flat, predictable line. If it's not perfect (because of the heavy hidden partners), the number of W bosons will shoot up anomalously as the energy increases.
3. The Scale: This effect only happens if the energy is high enough to feel the "missing" weight, but not so high that we actually create the heavy particles themselves. It's a "Goldilocks" zone.

What They Found (and What They Hope For)

The team ran the numbers for existing data and future machines:

• LEP II (Old Data): They looked at data from the 1990s. They found that the "accounting" is still pretty good, but there's a little wiggle room left. They set a limit: the missing piece can't be bigger than about 1.3%.
• Future Machines (The Big Guns): They looked at future colliders like the FCC-ee, ILC, CLIC, and a Muon Collider.
  • These machines are like high-definition microscopes. Because they can run for years and produce billions of collisions, they can spot even the tiniest imbalance.
  • They predict that these future machines could detect a missing piece as small as 0.01% (or even smaller).

They also looked at the Large Hadron Collider (LHC) and a future 100 TeV collider. While these are messier (like trying to find a needle in a haystack made of other needles), they offer a unique advantage: they can test the Tau lepton. Current electron/muon colliders can't easily test the Tau sector, but the LHC can.

The Bottom Line

This paper is a proposal for a new "smoking gun" test. Instead of looking for the heavy, invisible particles directly (which might be too heavy for our machines to create), we look for the shadow they cast on the light particles.

If the number of W bosons starts growing weirdly fast as we turn up the energy, we will know for sure that the neutrino mixing matrix is broken. This would be a massive discovery, proving that there are new, heavy particles hiding in the shadows of the Standard Model, fundamentally changing our understanding of the universe's building blocks.

In short: They are checking the universe's math by seeing if the numbers add up when the lights get really bright. If the numbers go haywire, we've found new physics.

Technical Summary

1. Problem Statement

The Standard Model (SM) is often extended to explain the smallness of neutrino masses by introducing heavy neutral fermions (e.g., in Seesaw mechanisms). In such extensions, the $3 \times 3$ Pontecorvo–Maki–Nakagawa–Sakata (PMNS) mixing matrix describing light neutrinos is no longer unitary; it becomes a sub-block of a larger $(3+n) \times (3+n)$ unitary matrix.

• Current Constraints: Flavor-off-diagonal non-unitarity (lepton flavor violation, e.g., $\mu \to e\gamma$) is tightly constrained ($|\epsilon| \lesssim 10^{-5}$). However, flavor-diagonal non-unitarity parameters ($\delta_\alpha$, where $\alpha = e, \mu, \tau$) are less constrained ($\delta \lesssim 10^{-3} - 10^{-4}$) by global electroweak fits.
• The Gap: There is a need for a direct, model-independent collider-based test to probe these diagonal non-unitarity effects, particularly for the $\tau$ sector, which is inaccessible to pure lepton collider initial states.

2. Methodology

The authors propose a novel test based on the anomalous energy growth of cross-sections in processes involving $W$ boson pair production. The core physical mechanism relies on the incomplete cancellation between $s$-channel and $t$-channel amplitudes when PMNS unitarity is violated.

A. Theoretical Mechanism

• Process: Lepton-antilepton annihilation $\ell^+_\alpha \ell^-_\alpha \to W^+ W^-$ (at lepton colliders) and the inverse process $pp \to \ell^+_\alpha \ell^-_\alpha jj$ via $W$-fusion (at hadron colliders).
• Amplitude Cancellation: In the SM, gauge invariance ensures that terms growing with energy ($\propto s$) in the $s$-channel (mediated by $\gamma, Z, H$) and $t$-channel (mediated by neutrino exchange) cancel exactly.
• Unitarity Violation: If the PMNS matrix is non-unitary, the sum of the squared mixing elements for light neutrinos is $\sum |U_{\nu}|^2 = 1 - \delta_\alpha < 1$.
  • The $t$-channel amplitude is proportional to this sum.
  • The $s$-channel remains unchanged (SM-like).
  • Result: The cancellation is incomplete. The cross-section acquires terms proportional to $\delta_\alpha s$ and $\delta_\alpha^2 s^2$. While the linear term ($\delta_\alpha s$) suppresses the cross-section at low energies, the quadratic term ($\delta_\alpha^2 s^2$) causes an anomalous growth at high energies (below the mass threshold of the heavy neutrinos).

B. Model Context

The paper reviews how $\delta_\alpha$ arises in various neutrino mass models:

• Type-I Seesaw: Requires large mixing parameters (fine-tuning) to reach observable $\delta_\alpha \sim 10^{-4}$.
• Inverse/Double Seesaw & Linear Seesaw: These models naturally allow for larger $\delta_\alpha$ (up to $10^{-4}$) with heavy neutrinos at the TeV scale, making them prime targets for this test.

C. Statistical Analysis

The authors employ $\chi^2$ tests to derive bounds on $\delta_\alpha$ at 95% confidence level (CL):

• LEP II: Re-analysis of existing $e^+e^- \to W^+W^-$ data.
• Future Colliders: Projections based on projected luminosities and center-of-mass energies ($\sqrt{s}$) for FCC-ee, ILC, CLIC, Muon Colliders, HL-LHC, and FCC-hh.
• Hadron Colliders: Utilization of the Effective Vector Boson Approximation (EVBA) to calculate $pp \to \ell^+\ell^- jj$ cross-sections, distinguishing between longitudinal ($LL$) and transverse ($TT$)$W$ polarizations.

3. Key Contributions

1. Novel Observable: Proposes using the energy scaling of $W^+W^-$ production cross-sections as a direct probe of PMNS unitarity, distinct from traditional precision electroweak fits or rare decay searches.
2. Model Independence: The bounds derived are largely model-independent, relying only on the assumption that heavy neutrino masses are above the collider energy scale.
3. $\tau$-Sector Accessibility: Demonstrates that hadron colliders ($pp$) can probe non-unitarity in the $\tau$ sector ($\delta_\tau$) via the process $pp \to \tau^+\tau^- jj$, a channel inaccessible to $e^+e^-$ or $\mu^+\mu^-$ colliders.
4. Angular Analysis: Shows that angular cuts (specifically around $\Theta = \pi/2$) can enhance the sensitivity to unitarity-violating terms at high energies.

4. Results

The paper provides projected 95% CL upper bounds on the non-unitarity parameters $\delta_\alpha$:

Collider	$\sqrt{s}$ (TeV)	Luminosity ($ab^{-1}$)	Bound on $\delta$
LEP II (Data)	0.183–0.202	-	$\delta_e \lesssim 1.35 \times 10^{-2}$
FCC-ee	0.35	1.8	$\delta_e \lesssim 1.6 \times 10^{-4}$
ILC	1.0	8.0	$\delta_e \lesssim 9.6 \times 10^{-5}$
CLIC	3.0	5.0	$\delta_e \lesssim 9.1 \times 10^{-5}$
Muon Collider (BM1)	3.0	1.0	$\delta_\mu \lesssim 2.2 \times 10^{-4}$
Muon Collider (BM2)	10.0	10.0	$\delta_\mu \lesssim 3.1 \times 10^{-5}$
HL-LHC	13.0	3.0	$\delta_{e,\mu,\tau} \lesssim 1.0 \times 10^{-3}$
FCC-hh	100.0	30.0	$\delta_{e,\mu,\tau} \lesssim 4.4 \times 10^{-5}$

• Comparison: The proposed method at future high-energy colliders (ILC, CLIC, FCC-hh) is projected to outperform current global electroweak fit constraints, particularly for the electron and tau sectors.
• Hadron vs. Lepton: While lepton colliders offer cleaner environments, hadron colliders provide unique access to the $\tau$ flavor and, at very high energies (FCC-hh), can achieve sensitivities comparable to lepton colliders.

5. Significance

• Complementarity: This method offers a complementary approach to testing the origin of neutrino mass. While precision fits constrain parameters globally, this collider test directly probes the perturbative unitarity of the scattering amplitude, providing a "smoking gun" for heavy neutrino mixing.
• Discovery Potential: If heavy neutrinos exist at the TeV scale (as suggested by Inverse/Linear Seesaw models), this method could detect their indirect effects even if the heavy states themselves are too heavy to be produced directly.
• Future Physics: The results underscore the critical role of future high-luminosity and high-energy colliders (FCC-ee, ILC, CLIC, Muon Collider, FCC-hh) in resolving the nature of neutrino mixing and the structure of the PMNS matrix.

Note on Limitations: The authors acknowledge that the hadron collider projections currently neglect specific background processes (e.g., $Z$-mediated diagrams) and detector effects. A dedicated study including these factors is required for realistic experimental implementation, but the theoretical potential remains robust.