Theoretical and Experimental Constraints in the $\mu$--$\tau$ Four-Lepton Sector of the SMEFT: implications to neutrino self interactions

This paper analyzes theoretical and experimental constraints on $\mu$--$\tau$ four-lepton SMEFT operators, finding that current bounds exclude heavy-mediator UV completions for strong neutrino self-interactions motivated by the Hubble tension while leaving light-mediator scenarios unconstrained.

The Gist

The Big Picture: The "Cosmic Traffic Jam"

Imagine the universe is a giant highway. For a long time, physicists thought neutrinos (tiny, ghost-like particles) were like invisible motorcycles that never touched each other; they just zipped past one another without interacting.

However, some recent measurements of the universe's expansion (the "Hubble tension") suggest that maybe, just maybe, these neutrinos do bump into each other. If they did, it would be like a massive traffic jam in the early universe that slowed everything down. To explain this, scientists proposed a theory where neutrinos have a "super-strong" self-interaction, millions of times stronger than their normal, weak interactions.

This paper asks a simple question: Is this "super-strong interaction" actually possible?

The Detective Work: The "Rulebook" (SMEFT)

The authors use a "rulebook" called the Standard Model Effective Field Theory (SMEFT). Think of this rulebook as a set of mathematical laws that describe how particles interact at different energy levels.

Specifically, they are looking at the "Mu-Tau" sector. In the world of particles, there are three "flavors" of leptons (like electrons, muons, and taus). The authors are focusing on the interactions between Muons and Taus (and their neutrino cousins).

They are checking three specific "rules" (mathematical coefficients) that would allow these particles to interact. They want to see if the universe allows for the "super-strong" interaction needed to solve the cosmic traffic jam, or if the rules forbid it.

The Three Ways They Checked the Rules

To see if the "super-strong interaction" is allowed, the authors used three different methods to set limits on how strong these interactions can be:

1. The "Speed Limit" (Perturbative Unitarity):
   Imagine driving a car. If you go too fast, the car breaks apart. In physics, if particles interact too strongly at high energies, the math breaks down (it becomes "non-unitary"). The authors calculated the "speed limit" for these interactions. If the interaction is too strong, the theory itself collapses. This gives them a theoretical ceiling on how strong the force can be.

2. The "Sign of the Force" (Positivity Sum Rules):
   Think of a spring. When you push it, it pushes back. In quantum physics, the way particles scatter can tell you if the "force carrier" (the thing passing the energy between them) is like a scalar (a simple point) or a vector (like a spinning top). The authors used mathematical "sum rules" to check if the signs of the interactions match what we expect from known physics. This helps them figure out what kind of "messenger" particle might be causing the interaction.

3. The "Real-World Test" (Experiments):
   This is the most direct check. They looked at data from two main sources:

   • Global Fits: A massive collection of data from many different experiments (like the LEP collider) that have already measured these particles.
   • NA64µ Experiment: A specific experiment at CERN that shoots muon beams at a target to look for "missing energy." If neutrinos were interacting strongly, they would carry away energy in a way that this experiment can detect.

The Findings: The "No-Go" Zone

The authors compared their theoretical "speed limits" and "sign checks" against the real-world experimental data. Here is what they found:

• The "Super-Strong" Interaction is Ruled Out: The data shows that the interaction strength between muon-neutrinos and tau-neutrinos is many, many orders of magnitude weaker than what is needed to solve the "Hubble tension" (the cosmic traffic jam).
• Heavy vs. Light Messengers:
  • Heavy Messengers: If the force is carried by a heavy particle (like a heavy Z' boson), the math says it cannot be strong enough to cause the cosmic traffic jam. The "heavy mediator" idea is dead for this specific problem.
  • Light Messengers: The paper notes that "light mediator" scenarios (where the force carrier is very light, almost massless) are not ruled out by this analysis. Why? Because the math used in this paper (the "contact interaction" model) only works for heavy messengers. If the messenger is light, the rules of the game change, and this specific test doesn't apply.

The "Z'" Connection

The paper also looked at a specific popular theory called the $L_\mu - L_\tau$ model. This theory suggests a new particle (a $Z'$ boson) exists that only talks to muons and taus.

• The authors found that the limits they set match perfectly with other dedicated studies of this $Z'$ particle.
• They confirmed that if this $Z'$ particle is heavy, it cannot be the solution to the Hubble tension because it simply isn't strong enough.

The "JUNO" Side Note

The paper briefly mentions a future experiment called JUNO (a giant neutrino detector in China). They did a similar analysis for the "electron-muon" sector using projected data from JUNO. They found that JUNO will be able to set very tight limits on interactions involving electrons, but the "muon-tau" sector remains the hardest to test because we have fewer experiments shooting muon beams.

The Bottom Line

In simple terms:
The authors acted like detectives checking if a specific "super-power" (strong neutrino self-interaction) exists. They used math rules, speed limits, and real-world data from CERN.

Their verdict: The "super-power" does not exist in the way proposed by heavy-particle theories. The interactions are too weak to explain the mystery of the universe's expansion. However, if the "messenger" particle is very light and behaves differently, this specific investigation didn't catch it, leaving that door slightly open.

What this means for the "Hubble Tension":
If you were hoping that heavy, new particles were the key to fixing the universe's expansion puzzle, this paper says "No." The key must be something else (perhaps light particles or a completely different mechanism), because the heavy ones are too weak to do the job.

Technical Summary

1. Problem Statement

The paper addresses the lack of direct experimental constraints on neutrino self-interactions (NSI) in the muon-tau ($\mu-\tau$) flavor sector.

• Cosmological Motivation: Strongly self-interacting neutrinos (with effective coupling $G_{\text{eff}} \sim 10^7 - 10^9 G_F$) have been proposed to resolve the Hubble tension ($H_0$ discrepancy) and the $\sigma_8$ tension. These interactions delay neutrino free-streaming in the early universe.
• Experimental Gap: While the electron-muon ($e-\mu$) and electron-tau ($e-\tau$) sectors are tightly constrained by $e^+e^-$ collider data (LEP) and global fits, the $\mu-\tau$ sector remains largely unconstrained. Standard future $e^+e^-$ colliders (FCC-ee, CEPC, ILC) offer no improvement for $\mu-\tau$ operators because these operators do not contribute to $e^+e^- \to f\bar{f}$ at tree level.
• Theoretical Challenge: It is unclear whether the "strongly interacting" regime required for cosmology is compatible with the Standard Model Effective Field Theory (SMEFT) framework, which assumes heavy mediators and contact interactions.

2. Methodology

The authors analyze dimension-six SMEFT four-lepton operators in the Warsaw basis, specifically focusing on the $\mu-\tau$ sector:

• Operators Studied:
  • $[O_{\ell\ell}]_{2222} = (\bar{\ell}_2\gamma^\mu\ell_2)(\bar{\ell}_2\gamma^\mu\ell_2)$
  • $[O_{\ell\ell}]_{2233} = (\bar{\ell}_2\gamma^\mu\ell_2)(\bar{\ell}_3\gamma^\mu\ell_3)$
  • $[O_{\ell\ell}]_{2332} = (\bar{\ell}_2\gamma^\mu\ell_3)(\bar{\ell}_3\gamma^\mu\ell_2)$
    (Where indices 2 and 3 correspond to muon and tau flavors, respectively.)

The study employs a three-pronged approach to constrain the Wilson coefficients ($C_{\ell\ell}$):

1. Experimental Constraints:

   • Global SMEFT Fits: Utilizing data from LEP, $\tau$ decays, and parity-violation experiments (Ref. [1]).
   • NA64$\mu$ Experiment: A CERN experiment using muon beams on a fixed target to search for missing energy signatures ($\mu N \to \mu N \nu\bar{\nu}$). This provides the first direct bounds on the $\mu-\tau$ mixed operators.
   • JUNO Projection: A complementary analysis for the $e-\mu$ sector using projected reactor neutrino sensitivities.

2. Theoretical Constraints:

   • Perturbative Unitarity: Applying $2 \to 2$ partial-wave analysis ($J=0$) to determine the energy scale ($\sqrt{s}^*$) where the EFT description breaks down.
   • Spin-Summing Positivity Sum Rules: Using forward-scattering dispersion relations to determine the sign of Wilson coefficients. This distinguishes between UV completions dominated by scalar ($J=0$) vs. vector ($J=1$) intermediate states.

3. Renormalization Group (RG) Running:

   • Evolving Wilson coefficients from 1 GeV to 30 TeV to check for significant modifications.

4. UV Completion Mapping:

   • Translating SMEFT bounds onto a specific UV model: a leptophilic $L_\mu - L_\tau$ $Z'$ boson.

3. Key Contributions and Results

A. Hierarchy of Constraints

The paper establishes a clear hierarchy of constraints for the $\mu-\tau$ sector:

• $[C_{\ell\ell}]_{2222}$ and $[C_{\ell\ell}]_{2332}$: The global fit dominates, providing bounds orders of magnitude tighter than perturbative unitarity limits.
• $[C_{\ell\ell}]_{2233}$: This coefficient is not constrained by the global fit or $e^+e^-$ colliders. NA64$\mu$ provides the leading (and currently only) direct experimental bound.
  • Current NA64$\mu$ bound: $|C_{\ell\ell}|_{2233}/\Lambda^2 \lesssim 10^4 \text{ TeV}^{-2}$.
  • Projected NA64$\mu$ bound: $|C_{\ell\ell}|_{2233}/\Lambda^2 \lesssim 1.4 \times 10^2 \text{ TeV}^{-2}$.
  • The projected sensitivity intersects the perturbative unitarity line at $\sqrt{s} \approx 212$ GeV.

B. Implications for Neutrino Self-Interactions

The authors translate the Wilson coefficient bounds into the effective four-neutrino coupling, $G_{\text{eff}}$:

• Magnitude: The derived bounds are $G_{\text{eff}} \lesssim 10^{-4} - 10^{-2} \text{ MeV}^{-2}$ (roughly $10^3 - 10^4$ times $G_F$).
• Exclusion of Strong Interactions: These values are many orders of magnitude smaller than the $G_{\text{eff}} \sim 10^7 - 10^9 G_F$ regime required to solve the Hubble tension via heavy mediators.
• Conclusion: Within the validity of the dimension-six SMEFT (heavy mediator assumption), heavy-mediator UV completions of strong $\nu_\mu-\nu_\mu$ and $\nu_\mu-\nu_\tau$ self-interactions are excluded.
• Light Mediators: The analysis does not constrain light-mediator scenarios (where $m_{\text{med}} \lesssim \text{momentum transfer}$), as these fall outside the contact-interaction approximation of SMEFT.

C. UV Model Analysis ($L_\mu - L_\tau$ $Z'$)

• The paper maps the SMEFT bounds to the parameter space $(g', M_{Z'})$ of a $L_\mu - L_\tau$ $Z'$ model.
• Spin-Sum Rules: The analysis confirms that the $Z'$ model (vector exchange) generates negative diagonal coefficients, consistent with the vector-dominated region predicted by positivity sum rules.
• Consistency: The SMEFT-derived bounds reproduce existing dedicated analyses for $M_{Z'}$ above the experimental momentum transfer scale.

D. RG Running

• Running between 1 GeV and 30 TeV modifies the coefficients by $\lesssim 10\%$. This is considered a subleading correction that does not alter the qualitative conclusions regarding the interplay between unitarity and experiment.

4. Significance

• Closing the Experimental Gap: The paper highlights that the $\mu-\tau$ sector is the "weakest link" in lepton flavor constraints and identifies NA64$\mu$ and future muon colliders as the only viable paths to close this gap.
• Ruling out Heavy Mediators for Hubble Tension: It provides a rigorous theoretical and experimental argument that the "strongly interacting neutrino" solution to the Hubble tension cannot be realized via heavy mediators described by dimension-six SMEFT operators.
• Methodological Framework: It demonstrates how to combine global fits, dedicated fixed-target experiments, and fundamental theoretical constraints (unitarity and positivity) to constrain specific flavor sectors of the SMEFT.
• Future Directions: The work motivates dedicated searches for light mediators (which remain viable for cosmology) at facilities like NA64$\mu$, FASER, and SHiP, and suggests that a 10 TeV muon collider could probe these operators at much higher energy scales.

5. Conclusion

The study concludes that while the $\mu-\tau$ four-lepton sector was previously unconstrained, the combination of NA64$\mu$ data and global SMEFT fits now places stringent limits on contact interactions. These limits effectively rule out heavy-mediator models for strong neutrino self-interactions, leaving light-mediator scenarios as the only remaining viable explanation for cosmological tensions within this framework. The analysis also validates the use of spin-summing rules to distinguish between scalar and vector UV completions.