Quantum effects on neutrino parameters from a flavored gauge boson

This paper demonstrates that introducing a massive gauge boson with family-dependent couplings to left-handed leptons allows quantum effects to increase the rank of the neutrino mass matrix at the one-loop level, thereby enabling the dynamical generation of observed neutrino mass differences and mixing angles.

The Gist

The Big Picture: Why Do Neutrinos Have Mass?

Imagine the universe is a giant, complex machine. For a long time, physicists thought the smallest particles in this machine, called neutrinos, were weightless ghosts. But experiments showed they actually have a tiny bit of weight (mass).

The standard explanation is that these particles get their mass from a "hidden factory" operating at incredibly high energies (far beyond what our particle colliders can reach). This factory produces a "recipe" (called the Weinberg operator) that tells neutrinos how heavy to be.

Usually, physicists assume that once this recipe is written at the top of the universe's energy scale, it stays mostly the same as it travels down to the energy levels we can measure today. However, this new paper argues that there is a secret ingredient in the machine that changes the recipe along the way.

The New Ingredient: The "Flavored" Force

In the Standard Model of physics, there are forces like electromagnetism (which affects all electric charges equally) and the weak force. This paper introduces a new, hypothetical force carried by a particle called a $Z'$ boson.

Think of the $Z'$ boson as a strict bouncer at a club.

• In the old model, the bouncer treated everyone the same.
• In this new model, the bouncer is picky. He treats the "Electron" family, the "Muon" family, and the "Tau" family differently. He lets some in, charges others, and ignores the rest.

Because this bouncer treats the different "families" of neutrinos differently, it creates a unique kind of friction or interaction as the universe cools down from the high-energy "factory" to our low-energy world.

The Main Discovery: Turning a Zero into a One

Here is the most exciting part of the paper, explained with a simple analogy:

The "Rank" of the Mass Matrix:
Imagine the neutrino mass recipe as a 3x3 grid of numbers.

• If a number in the grid is 0, it means that specific neutrino has no mass.
• If the grid has a lot of zeros, the "rank" is low.
• If the grid is full of non-zero numbers, the "rank" is high.

The Old Rule (Standard Model):
In the standard universe, quantum effects (tiny jitters in the fabric of reality) are like a gentle rain. If you start with a grid that has a zero (a massless neutrino), the gentle rain of the Standard Model can't turn that zero into a number. It takes a double-strength storm (two-loop effects) to finally fill in that zero.

The New Rule (This Paper):
The authors found that the picky $Z'$ bouncer acts like a powerful tornado.

• Because the bouncer treats the families differently, the "tornado" of quantum effects is much stronger.
• Result: A neutrino that was perfectly massless (a zero in the grid) at the high-energy factory can suddenly gain mass (become a non-zero number) just by passing through the $Z'$ field.
• Significance: This happens at the one-loop level (the first level of quantum effects), which is much faster and more powerful than the old two-loop rule. It means the universe can dynamically create a mass for a neutrino that started with none.

How This Explains What We See

The paper suggests that this mechanism could explain why we see the specific patterns of neutrino masses and mixing angles we measure in experiments today.

1. The "Desert" Scenario: Imagine a vast desert between the high-energy factory and our current world. Usually, we think nothing interesting happens there. This paper says, "What if there's a hidden oasis (the $Z'$ boson) in that desert?"
2. Dynamic Generation: Even if the universe started with a very simple, boring recipe (where two neutrinos had the exact same mass, or one had zero mass), the interaction with the $Z'$ boson as the universe cooled would scramble the numbers.
3. The Result: This scrambling naturally creates the "mass splitting" (the difference in weight between the three neutrinos) and the "mixing angles" (how they switch flavors) that we observe in real life, without needing to fine-tune the initial recipe perfectly.

The "Fixed Point" Analogy

The paper also discusses "quasi-fixed points." Imagine you are rolling a ball down a bumpy hill.

• In the old model, where the ball ends up depends entirely on where you dropped it.
• In this new model, the $Z'$ boson acts like a magnet on the hill. No matter where you drop the ball (even if you drop it in a weird, degenerate spot), the magnet pulls it toward a specific, stable valley.
• This means the universe doesn't need to be "fine-tuned" to get the right neutrino properties; the laws of physics naturally push the system toward the values we observe.

Summary

• The Problem: We don't fully understand why neutrinos have the specific masses and mixing patterns they do.
• The New Idea: A new, family-picky force carrier ($Z'$) exists between the high-energy creation of neutrinos and our current low-energy world.
• The Magic: This force is so strong and specific that it can turn a "zero mass" neutrino into a "real mass" neutrino very quickly (at the one-loop level).
• The Payoff: This mechanism naturally explains the complex patterns of neutrino behavior we see today, suggesting that the universe's "recipe" for neutrinos is written dynamically as the universe cools, rather than being set in stone at the beginning.

In short: The universe has a picky bouncer ($Z'$) that forces the neutrinos to change their weights as they travel through time, creating the diversity we see today.

Technical Summary

1. Problem Statement

The paper addresses the evolution of neutrino mass parameters from high-energy scales (where they are generated) down to low-energy scales (where they are measured).

• Context: In the Standard Model (SM) extended by the dimension-5 Weinberg operator (generating Majorana neutrino masses), the Renormalization Group Equations (RGEs) describe how the Wilson coefficient ($\kappa$) of this operator evolves.
• The Gap: It is a well-established result that in the pure SM (or SM with right-handed neutrinos), one-loop RGE effects cannot increase the rank of the neutrino mass matrix. Rank increases (generating mass for initially massless neutrinos) only occur at the two-loop level.
• The Question: The authors investigate whether introducing new physics in the "desert" between the electroweak scale and the scale of new physics (e.g., right-handed neutrinos) can alter this behavior. Specifically, they ask if a flavor-dependent gauge boson ($Z'$) can induce rank-increasing effects at the one-loop level.

2. Methodology

The authors employ a combination of model building and perturbative quantum field theory calculations.

• Model Construction:

  • They extend the SM gauge group with a local $U(1)_{L_\mu - L_\tau}$ symmetry.
  • The model includes three right-handed neutrinos ($N_1, N_2, N_3$) and complex scalar singlets ($S_0, S_1, S_2$) charged under $U(1)_{L_\mu - L_\tau}$ to ensure anomaly cancellation and spontaneous symmetry breaking.
  • The $Z'$ boson acquires a mass ($M_{Z'}$) via the vacuum expectation values (VEVs) of the scalars.
  • The scenario assumes a hierarchy where $M_{Z'} < M_1$ (the mass of the lightest right-handed neutrino). This creates an effective field theory (EFT) regime containing the Weinberg operator and the massive $Z'$ boson.

• RGE Calculation:

  • The authors derive the one-loop RGE for the Weinberg operator coefficient $\kappa$ in the energy range $M_{Z'} < \mu < M_1$.
  • The RGE takes the form:
    $$16\pi^2 \frac{d\kappa}{dt} = \alpha \kappa + P^T \kappa + \kappa P + G^T \kappa G$$
    where $t = \ln(\mu/\mu_0)$.
  • Key Term: The term $G^T \kappa G$ is the novel contribution induced by the $Z'$ boson. Here, $G = \sqrt{6}g' Q'$, where $Q'$ is the diagonal charge matrix of the $U(1)_{L_\mu - L_\tau}$ symmetry.
  • They decompose $\kappa$ into eigenvalues and the leptonic mixing matrix ($U$) to analyze the running of masses and mixing angles.

• Analytical and Numerical Analysis:

  • They solve the RGEs analytically for simplified cases (2-generation and 3-generation with real parameters).
  • They perform numerical scans to study the running of eigenvalues and mixing angles for Normal Ordering (NO) and Inverted Ordering (IO) scenarios, including cases with degenerate or vanishing initial masses.

3. Key Contributions

The primary theoretical breakthrough of this work is the demonstration that flavor-dependent gauge interactions can increase the rank of the neutrino mass matrix at the one-loop level.

• Rank Increase Mechanism: In the SM, the one-loop term is proportional to $\kappa$ itself (or commutes with it in a way that preserves rank). However, the $Z'$ term $G^T \kappa G$ involves the charge matrix $Q'$. Since $Q'$ is not proportional to the identity (due to flavor dependence), this term mixes the eigenstates.
• Contrast with SM: This contrasts sharply with the SM, where rank generation requires two-loop effects. The presence of the $Z'$ effectively promotes a two-loop-like effect to the one-loop level.
• Generalization: The authors provide the generalization of these results to non-Abelian flavor gauge groups in Appendix A, showing the structure of the RGEs for charged and neutral gauge bosons.

4. Key Results

A. Generation of Mass for Massless Neutrinos

• Scenario: Consider a scenario where the lightest neutrino is massless at the cutoff scale ($\kappa_1 = 0$).
• Result: At the scale $M_{Z'}$, the quantum corrections generate a non-zero mass for the lightest neutrino.
• Magnitude: The generated hierarchy is approximately:
  $$\frac{\kappa_1}{\kappa_3} \bigg|_{M_{Z'}} \sim -\frac{6g'^2}{16\pi^2} \log\left(\frac{M_1}{M_{Z'}}\right) \sin^2(2\theta)$$
  For $g' \sim 1$, this yields a hierarchy of $\mathcal{O}(10^{-2})$, which is phenomenologically significant.

B. Quasi-Fixed Points in Mixing Angles

• Degenerate Eigenvalues: If two eigenvalues are degenerate at the high scale (e.g., $\kappa_1 = \kappa_2$), the RGEs drive the mixing angles toward specific "quasi-fixed points" in the infrared.
• Normal Ordering ($\kappa_1 = \kappa_2 \ll \kappa_3$): The mixing angles evolve to satisfy a condition like $W_{31}W_{32} \approx 0$ (where $W$ relates to the mixing matrix elements). This leads to specific predictions for the reactor angle $\theta_{13}$ based on solar and atmospheric angles.
  • Note: In the real case, the fixed point predicts $\theta_{13} \approx 23^\circ$, which conflicts with data. However, the authors note that in the complex case (with CP and Majorana phases), the condition becomes $\text{Re}[W_{31}W_{32}] = 0$, allowing the phases to adjust the fixed point to match experimental values.
• Inverted Ordering ($\kappa_3 \ll \kappa_1 = \kappa_2$): Similar fixed point behavior is observed, potentially splitting the degenerate heavy states and generating the solar mass splitting dynamically.

C. Dynamical Generation of Mass Splittings

• The paper demonstrates that even if the neutrino mass spectrum is highly hierarchical or degenerate at the high scale, the $Z'$-induced RGE running can naturally generate the observed mass splittings ($\Delta m^2_{21}$ and $\Delta m^2_{31}$) at low energies.
• The splitting is driven by the off-diagonal elements of the transformed charge matrix $\tilde{G} = U G U^\dagger$.

5. Significance

• Theoretical Impact: This work challenges the standard assumption that rank generation in neutrino mass matrices is strictly a two-loop phenomenon. It establishes that new flavor-dependent gauge symmetries can fundamentally alter the perturbative structure of neutrino mass evolution.
• Phenomenological Implications:
  • It offers a mechanism to explain the origin of neutrino mass hierarchies and mixing angles without fine-tuning the initial conditions at the high scale.
  • It suggests that the existence of a light $Z'$ boson (lighter than the right-handed neutrinos) could leave distinct imprints on the neutrino sector, potentially testable via precision measurements of mixing angles and mass splittings.
  • It provides a framework for "radiative" neutrino mass generation where the mass hierarchy is a consequence of renormalization group running rather than the underlying high-scale mass matrix structure.
• Model Building: The results are applicable to a broad class of models involving $U(1)_{L_\mu - L_\tau}$ or other flavor-dependent gauge extensions, offering a new tool for constructing viable neutrino mass models.

In summary, the paper proves that flavor-dependent gauge bosons act as a catalyst at the one-loop level, capable of generating neutrino masses from zero and driving mixing angles to fixed points, thereby providing a robust mechanism for the dynamical generation of observed neutrino parameters.