Running of neutrino mass parameters in the Zee model

This paper employs effective field theory techniques to derive 1-loop matching conditions and calculate quantum corrections to neutrino mass parameters within the Zee model, demonstrating through four benchmark scenarios the specific conditions under which these corrections must be included in theoretical analyses.

The Gist

Imagine the universe as a giant, multi-story building. The ground floor represents the world we can see and touch right now (everyday particles like electrons). The upper floors represent a hidden, high-energy world where new, heavy particles live.

This paper is about a specific blueprint for that building called the Zee model. This model tries to explain a mysterious property of tiny particles called neutrinos: why they have mass. In the standard rules of physics, they shouldn't have any mass at all. The Zee model suggests they get their mass through a "loop" of interactions involving new, heavy particles living on the upper floors.

Here is the simple breakdown of what the authors did, using some everyday analogies:

1. The Problem: The "Long Distance" Mess

Imagine you are trying to calculate the price of a house, but you have to account for a massive tax that only applies if you live 1,000 miles away. If you try to do the math all at once from your front door, the numbers get messy, huge, and unreliable. The "distance" in physics is the difference in energy between the heavy new particles (the upper floors) and the light particles we see (the ground floor).

In the Zee model, if you try to calculate neutrino mass directly using the full theory, you get a "large logarithm." Think of this as a giant, messy number that makes your calculation shaky and hard to trust. It's like trying to measure a grain of sand with a ruler meant for measuring mountains.

2. The Solution: The "Effective Field Theory" Elevator

To fix this, the authors used a technique called Effective Field Theory (EFT). Think of this as taking an elevator down from the top floor to the ground floor, stopping at every major landing to tidy up the math.

• Step 1 (The Top Floor): They start at the very top with the heavy new particles.
• Step 2 (The Middle Floor): They "integrate out" (remove) the heaviest particle. This is like closing a door on the top floor and leaving a note on the middle floor that says, "Hey, the heavy stuff is gone, but it left a little bit of influence here." This note is a mathematical "matching condition."
• Step 3 (The Ground Floor): They move down to the next heavy particle, close that door, and leave another note.
• Step 4 (The Result): Finally, they reach the ground floor (our current energy scale) with a clean, manageable set of rules to calculate the neutrino mass.

3. The Secret Ingredient: The "Running"

The most important discovery in this paper is about Renormalization Group (RG) running.

Imagine you are walking down a long hallway (the energy scale). As you walk, the rules of the game change slightly at every step. The "coupling constants" (which are like the strength of the interactions between particles) are not static; they run or evolve as you move from high energy to low energy.

The authors found that in the Zee model, this "running" is not a tiny, boring detail. It is the main event.

• The Analogy: Imagine you are baking a cake. You might think the flavor comes from the ingredients you mix in the bowl (the initial setup). But the authors found that the baking process itself (the running) is actually what creates the flavor. If you ignore the baking process and just look at the raw ingredients, you get the wrong cake.
• The Finding: In the Zee model, the neutrino mass is almost entirely generated by these changes as you move down the energy ladder. If you ignore this "running," your prediction for neutrino mass is wrong.

4. The Test Drive: Benchmark Scenarios

To prove this, the authors didn't just do abstract math; they ran four different "test drives" (benchmark scenarios). They changed the settings of the model (like how heavy the new particles are or how strongly they interact) to see how the "running" affected the final result.

• The Result: They found that even if you change the high-energy settings by a tiny amount (like 1%), the "running" amplifies this change significantly by the time it reaches the ground floor.
• The Consequence: Future experiments (like the JUNO experiment mentioned in the paper) are becoming incredibly precise. They will be able to measure neutrino properties with such accuracy that if scientists ignore this "running" effect, their predictions will be off by more than the experimental error. It's like trying to hit a bullseye with a bow and arrow, but ignoring the wind.

Summary

This paper argues that to understand how neutrinos get their mass in the Zee model, you cannot just look at the starting point. You must account for the journey. The "journey" (the renormalization group running) is where the magic happens.

If scientists want to match the incredible precision of upcoming neutrino experiments, they must include these quantum corrections. Ignoring them is like trying to navigate a ship without accounting for the currents; you might start in the right direction, but you'll end up far off course.

Key Takeaway: The "running" of particle properties from high energy to low energy is not a small correction; it is the dominant force shaping the neutrino mass in this model, and it must be included to make accurate predictions for the future of physics.

Technical Summary

Technical Summary: Running of Neutrino Mass Parameters in the Zee Model

Problem Statement
The Standard Model (SM) fails to explain neutrino masses, which are established experimentally via oscillation data. Radiative neutrino mass models, such as the Zee model, generate neutrino masses at the loop level. However, these models typically predict parameters at a high energy scale (UV), while experimental data is available at the electroweak (EW) scale. Connecting these scales requires Renormalisation Group (RG) evolution.

In the Zee model, the calculation of the neutrino mass matrix in the full theory involves a potentially large logarithm of the ratio of new physics mass scales ($\ln(m_h^2/m_{H_2}^2)$). When the mass hierarchy between the new scalar particles is significant, this logarithm can undermine the validity of perturbation theory. Furthermore, standard phenomenological studies often neglect RG corrections, which may become critical as next-generation experiments (e.g., JUNO, Hyper-Kamiokande, DUNE) aim for sub-percent precision on mixing parameters. The paper addresses whether these quantum corrections are negligible or if they must be included to match the precision of upcoming data.

Methodology
The authors employ Effective Field Theory (EFT) techniques to systematically handle the separation of scales and resum large logarithms. The methodology involves a sequence of EFTs:

1. Full Theory (Zee Model): Contains the SM, a second Higgs doublet ($H_2$), and a singly charged scalar ($h$).
2. Intermediate EFT (2HDM EFT): Obtained by integrating out the heavy charged singlet $h$ at scale $m_h$. This generates dimension-5 Weinberg-like operators ($\kappa_{ij}$).
3. Low-Energy EFT (SMEFT): Obtained by integrating out the heavy second Higgs doublet $H_2$ at scale $m_{H_2}$. This generates the standard dimension-5 Weinberg operator ($C_{\alpha\beta}$).

The calculation proceeds in three main steps:

• 1-Loop Matching: The authors derive 1-loop matching conditions for the transition from the Zee model to the 2HDM EFT and from the 2HDM EFT to the SMEFT. They utilize the Matchete package and verify results diagrammatically.
• RG Running: They solve the 1-loop Renormalisation Group Equations (RGEs) for the Wilson coefficients ($\kappa_{ij}$ and $C_{\alpha\beta}$) and the relevant couplings (Yukawa matrices $Y_e, Y_{u,d}$ and quartic couplings $\lambda_i$) in the 2HDM EFT and SMEFT. The running is performed between the high scale $m_h$ and the intermediate scale $m_{H_2}$, and then down to the EW scale.
• Numerical Analysis: Four benchmark scenarios are constructed to test the sensitivity of neutrino mixing parameters to UV model parameters. These benchmarks vary the hierarchy of Yukawa couplings ($Y_e^1$ vs $Y_e^2$), the alignment parameter $\alpha$ (relating quark Yukawas), and the mass scales (ranging from TeV to $10^4$ TeV). The authors fit the UV parameters to reproduce low-energy neutrino data and then analyze how scaling these UV parameters affects the final predictions when running is included versus when it is ignored.

Key Contributions

• Derivation of Matching Conditions: The paper provides explicit 1-loop matching conditions for the Zee model to the 2HDM EFT and subsequently to the SMEFT. This includes the matching of the Weinberg-like coefficients $\kappa_{ij}$ and the quartic couplings.
• Identification of Running Mechanisms: The analysis reveals that the neutrino mass matrix in the SMEFT is generated primarily by the discrepancy in the RG running of different components of the matching conditions. Specifically, the tree-level contribution to $\kappa_{12}$ and the 1-loop running of $\kappa_{11}$ do not cancel perfectly when evolved to the low scale. The running of $\kappa_{11}$ is driven by terms involving $\kappa_{12}$ and Yukawa couplings, leading to a linear dependence on the logarithm of the scale, which dominates over the exponential scaling of other terms.
• Quantification of Quantum Corrections: Through numerical benchmarks, the authors demonstrate that RG corrections can induce shifts in neutrino mixing parameters (angles and mass squared differences) that exceed the projected experimental uncertainties of future facilities.
• UV Sensitivity Analysis: The paper shows that the EFT framework is sensitive to the specific choice of UV parameters (e.g., the relative magnitude of $Y_e^1$ and $Y_e^2$) in a way that the full theory calculation is not. Scaling UV parameters that leave the full theory invariant results in non-invariant predictions in the EFT calculation due to the running, highlighting the necessity of the EFT approach for precision.

Results

• Magnitude of Corrections: In the benchmark scenarios, a 1% variation in high-scale parameters (which leaves the full theory prediction invariant) leads to corrections in the neutrino mass parameters that can exceed 10% in the EFT calculation. These corrections are often larger than the current and future experimental uncertainties (e.g., JUNO's projected 0.6% precision).
• Dependence on Yukawa Couplings: The running effects are most significant when the Yukawa couplings of the second Higgs doublet ($Y_e^2$) are large or comparable to the first doublet's couplings ($Y_e^1$). The running of the charged lepton Yukawa matrix $Y_e^1$ is a dominant driver of the running of the neutrino mass matrix.
• Scale Independence of Qualitative Behavior: The qualitative behavior of the running effects is robust across different mass scales, from the TeV scale up to $10^4$ TeV. Even for relatively light new physics, the RG corrections remain significant.
• Benchmark Specifics:
  • Benchmark 1 (Minimally constrained): Shows extreme non-linearity in the running of $\kappa_{11}$ due to the rapid change in the scale of $Y_e^1$.
  • Benchmark 2 ($Y_e^1 \sim Y_e^2$): Shows that even with comparable Yukawas, significant running occurs, and the effects persist at lower mass scales (1 TeV).
  • Benchmark 3 ($\alpha=0$): Demonstrates that suppressing the quark Yukawa contribution to the running (via $\alpha=0$) reduces the running of $Y_e^1$ but does not eliminate the running of the neutrino mass matrix, which is then driven by other terms.

Significance and Claims
The authors claim that their work demonstrates the imperative need to include RG corrections in phenomenological studies of the Zee model and similar radiative neutrino mass models. They argue that:

1. Precision Matching: To match the precision of next-generation neutrino oscillation experiments, theoretical predictions must account for RG running. Neglecting these corrections introduces theoretical uncertainties that exceed experimental ones.
2. EFT Validity: The EFT framework is not just a tool for handling large logarithms but is essential for correctly capturing the generation of neutrino masses in radiative models where the mass matrix arises from the running of Wilson coefficients rather than direct tree-level matching.
3. General Applicability: While focused on the Zee model, the authors expect similar large running effects in other radiative models (e.g., the scotogenic model), suggesting that a shift towards EFT-based analyses is necessary for the broader field of neutrino physics.

The paper concludes that future statistical surveys of new physics parameter spaces must utilize EFT frameworks to fully exploit the information gained from upcoming high-precision experiments.