Implications of Flavor Symmetries for Baryon Number Violation

This paper systematically classifies dimension-six baryon-number-violating SMEFT operators under various flavor-symmetry assumptions, demonstrating how their interplay with neutrino masses can allow for multi-TeV BNV scales consistent with proton decay constraints, while also identifying UV completions where the effective field theory description may be insufficient.

The Gist

Imagine the universe is a massive, complex orchestra. For decades, physicists have been trying to figure out the rules of the music. The "Standard Model" is the sheet music they currently have, but it has some missing notes and strange gaps. One of the biggest mysteries is why the particles (the musicians) have such different weights (masses) and why they mix together in specific ways (flavor).

This paper is like a detective story where the authors try to solve a crime: Why hasn't the proton (a stable building block of matter) decayed yet?

Here is the breakdown of their investigation using simple analogies:

1. The Crime Scene: The Proton

In our current understanding of physics, protons are supposed to be immortal. They hold atoms together, and if they fall apart, the universe as we know it would dissolve. However, many theories suggest that protons should eventually decay, just very slowly.

If we ever catch a proton decaying, it would be the "smoking gun" proving that there is new physics beyond what we currently know. So far, giant detectors deep underground (like Super-Kamiokande) have been watching for this crime for decades, and the proton is still holding up. This puts a huge "speed limit" on how fast new physics can be happening.

2. The Suspects: Flavor Symmetries

The authors ask: What if the reason the proton is so stable isn't just luck, but because of a hidden rulebook called "Flavor Symmetry"?

Think of "Flavor" like the different instruments in the orchestra. You have violins (first generation), violas (second), and cellos (third). In the Standard Model, the rules for how they play together are messy and unexplained.

• The Old Theory (Anarchy): If there were no rules, the proton would decay almost instantly. Since it doesn't, the "new physics" causing the decay must be incredibly heavy and far away (like a distant galaxy).
• The New Theory (MFV - Minimal Flavor Violation): The authors propose a stricter rulebook. They say, "New physics can only break the rules if it does so in a way that respects the existing hierarchy of the instruments." It's like saying a new composer can write a song, but they must use the same scale and rhythm as the old masters.

3. The Twist: The Tiny Neutrino Connection

Here is the clever part of the paper. The authors realized that the "rulebook" for the proton is secretly linked to neutrinos (ghostly, tiny particles that barely interact with anything).

• The Analogy: Imagine the proton is a fortress. The "Flavor Symmetry" is the moat. The authors found that the depth of the moat depends on how heavy the neutrinos are.
• The Discovery: Because neutrinos are incredibly light (almost massless), the "moat" around the proton becomes very deep. This means the new physics causing the proton to decay can be much closer to us than we thought.

Instead of needing new physics to be at the edge of the universe (trillions of miles away), this paper suggests it could be right here in the "neighborhood" (the multi-TeV scale, which is the energy range of our largest particle colliders, like the LHC).

4. The Investigation: Four Types of Suspects

The authors looked at four specific ways the proton could decay (four different "criminal methods"). They ran the numbers for each one under different flavor rulebooks:

• The "Strict" Rulebook (Extended MFV): If the new physics follows the strict flavor rules and is linked to neutrino masses, the proton is very stable. However, for three of the four methods, the new physics could still be light enough to be found soon.
• The "Loose" Rulebook (Reduced Symmetry): If the rules are slightly different (favoring the heaviest particles, like the top quark), the proton becomes much more fragile. In this scenario, the new physics must be extremely heavy and far away, or we would have seen the proton decay by now.

5. The "UV Completions": Who is the Mastermind?

The paper also looks at the "Masterminds" behind the scenes. In physics, we often describe effects using "Effective Field Theory" (EFT), which is like describing a car crash by looking at the crumpled metal without seeing the driver.

The authors tried to identify the actual "drivers" (new particles like Leptoquarks) that could cause these crashes. They found that:

• If the driver is a "Flavor Singlet" (doesn't care about the instrument hierarchy), the crash happens fast, and we'd have seen it.
• If the driver is a "Flavor Transformer" (respects the hierarchy), the crash is delayed, allowing the proton to survive longer.

The Bottom Line

This paper is a roadmap for future experiments. It tells us:

1. Don't give up on finding new physics: The fact that the proton hasn't decayed yet doesn't mean new physics is impossibly far away. It might just be hiding behind the "neutrino curtain."
2. Look for specific patterns: If we do see a proton decay, the paper predicts exactly how it will happen.
   • If it decays into a muon (a heavy electron), it points to one type of new physics.
   • If it decays into an electron, it points to another.
   • If it decays into a neutrino, it points to a third.
3. The Connection: The stability of the proton and the tiny mass of neutrinos are two sides of the same coin. Solving one helps solve the other.

In short: The authors used the "rules of flavor" to show that the universe might be more accessible than we thought. The "forbidden" new physics might not be at the edge of the universe, but potentially within reach of our next generation of particle detectors, provided we look at the right decay channels and understand the secret link to neutrino masses.

Technical Summary

1. Problem Statement

In the Standard Model (SM), baryon number ($B$) is an accidental symmetry. Its violation (BNV) is a hallmark of new physics, with proton decay being the most prominent experimental signature. Current experimental limits (e.g., from Super-Kamiokande) constrain the proton lifetime to $\tau_p \gtrsim 10^{34}$ years, which, within the Standard Model Effective Field Theory (SMEFT), implies that the scale of BNV physics ($\Lambda_B$) must be extremely high ($\Lambda_B \sim 10^{16}$ GeV) in an "anarchic" flavor scenario.

However, the SM exhibits a highly structured flavor pattern (masses and mixings). The Minimal Flavor Violation (MFV) hypothesis posits that all flavor and CP violation arises solely from the SM Yukawa couplings. This paper investigates whether embedding BNV operators within flavor-symmetry frameworks (specifically extended MFV and reduced symmetries) can relax the stringent bounds on $\Lambda_B$. Specifically, the authors explore the interplay between BNV and Lepton Number Violation (LNV), mediated by neutrino masses, to determine if BNV scales could be accessible at the multi-TeV range.

2. Methodology

The authors employ a systematic Effective Field Theory (EFT) approach combined with Ultraviolet (UV) completion analysis:

• Extended MFV Framework: They generalize the MFV framework to include neutrino masses. The flavor symmetry group $G_F = U(3)^5$ is broken by three spurions: the charged-lepton Yukawa matrix ($Y_e$), the up/down quark Yukawa matrices ($Y_u, Y_d$), and a new symmetric spurion $\Upsilon_\nu$ associated with the dimension-5 Weinberg operator (generating Majorana neutrino masses).
• SMEFT Operator Analysis: They focus on the four independent dimension-six BNV operators in SMEFT: $O_{qqq\ell}$, $O_{duq\ell}$, $O_{qque}$, and $O_{duue}$.
  • They construct flavor-invariant combinations by inserting the minimal number of spurions required to restore $G_F$ invariance.
  • They identify the least-suppressed structures for each operator, determining the leading-order contributions to proton decay.
• Phenomenological Calculation:
  • Matching: They perform tree-level matching from SMEFT to the Low-Energy Effective Theory (LEFT) at the electroweak scale.
  • Renormalization Group (RG) Evolution: They evolve the Wilson coefficients from the BNV scale ($\Lambda_B$) down to the hadronic scale ($\sim 2$ GeV) using DsixTools, accounting for operator mixing. This is crucial as some operators generate proton decay only at the one-loop level.
  • Hadronic Matrix Elements: They match the quark-level operators to nucleon decay rates using Baryon Chiral Perturbation Theory (B$\chi$PT), utilizing lattice QCD inputs for low-energy constants.
• UV Completions: They analyze single-particle UV completions (specifically Leptoquarks) that generate these operators at tree level, checking if the UV dynamics align with the EFT spurion counting.
• Reduced Symmetries: They explore alternative flavor frameworks ($U(2)^5$, $U(2)^3 \times U(3)^2$, etc.) where the symmetry is partially reduced to allow preferential coupling to the third generation.

3. Key Contributions

• Systematic Classification: The paper provides the first complete classification of dimension-six BNV SMEFT operators under the Extended MFV hypothesis, explicitly incorporating the neutrino mass spurion $\Upsilon_\nu$.
• Quantitative Link between BNV and LNV: They establish a quantitative relationship between the BNV scale ($\Lambda_B$) and the LNV scale ($\Lambda_L$). They demonstrate that in specific flavor scenarios, the suppression of proton decay is proportional to neutrino masses ($m_\nu$), allowing $\Lambda_B$ to be significantly lower than the GUT scale.
• Operator-Specific Phenomenology: They distinguish the phenomenological behavior of the four BNV operators:
  • $O_{qqq\ell}$: Dominated by tree-level contributions, leading to the strongest constraints.
  • $O_{duq\ell}, O_{qque}, O_{duue}$: Often suppressed by additional spurion insertions or generated radiatively (via operator mixing), allowing for much lower $\Lambda_B$.
• UV Sensitivity: They demonstrate that the EFT spurion counting can be modified by specific UV mediator representations (e.g., bifundamental leptoquarks), which can introduce additional suppression factors beyond the leading-order EFT expectation.

4. Key Results

A. Extended MFV Results

• Correlation with Neutrino Masses: The analysis confirms that proton decay amplitudes in Extended MFV acquire a suppression factor proportional to $m_\nu^2$ (or higher powers depending on the operator).
• Multi-TeV Viability:
  • For operators $O_{duq\ell}$, $O_{qque}$, and $O_{duue}$, the proton decay constraints are compatible with $\Lambda_B$ in the multi-TeV range (e.g., $10^3 - 10^9$ TeV), provided the LNV scale $\Lambda_L$ is around $10^3 - 10^9$ TeV. This is significantly lower than the naive $10^{16}$ GeV expectation.
  • For $O_{qqq\ell}$, the constraints remain extremely strong, requiring $\Lambda_B \gtrsim 10^7$ TeV, as it generates proton decay at tree level with less suppression.
• Decay Channels:
  • $O_{qqq\ell}$ is primarily constrained by $p \to K^+ \nu$.
  • $O_{duq\ell}$ is constrained by $p \to \pi^0 e^+$ (via RG mixing) and $p \to K^+ \nu$.
  • $O_{qque}$ and $O_{duue}$ predominantly yield muonic final states ($p \to \pi^0 \mu^+$, $p \to K^0 \mu^+$) due to the structure of the leptonic spurion $T_e$.
• Ratio Predictions: The paper predicts specific ratios of decay widths (e.g., $\Gamma(p \to \pi^0 \mu^+) / \Gamma(p \to K^+ \nu)$) that depend only on $\Lambda_B$, offering a way to discriminate between different BNV operators in future experiments.

B. Reduced Flavor Symmetries

• $U(2)^5$ Framework: In this framework, where the first two generations are doublets and the third is a singlet, leading-order BNV operators can be constructed without spurion insertions (using only singlet components). This removes the neutrino-mass suppression, resulting in much stronger bounds on $\Lambda_B$ (up to $10^{11}$ TeV), effectively ruling out the multi-TeV scenario for these operators.
• Intermediate Symmetries ($U(2)^3 \times U(3)^2$): These frameworks restore a suppression pattern similar to MFV, allowing for multi-TeV $\Lambda_B$ scales, but with different relative strengths for the operators compared to the full MFV case.

C. UV Completions

• Leptoquarks: The authors identify specific Leptoquark representations (scalars $\omega_1, \omega_4, \zeta$ and vectors $Q_1, Q_5$) that generate these operators.
• Additional Suppression: They show that if Leptoquarks transform as bifundamentals under the flavor group (e.g., $(3_q, 3_\ell)$), they can introduce additional spurion insertions (e.g., extra factors of $\Upsilon_\nu$). This can further suppress the proton decay rate (e.g., $\Gamma_p \propto m_\nu^4$), making the multi-TeV scenario even more viable and linking it more tightly to the smallness of neutrino masses.

5. Significance and Conclusions

• Re-evaluating BNV Scales: The work challenges the assumption that BNV must occur at the GUT scale. It demonstrates that flavor symmetries, particularly when linked to the smallness of neutrino masses, can naturally lower the BNV scale to the multi-TeV range, making it potentially accessible to future colliders and next-generation proton decay experiments (Hyper-K, DUNE).
• Experimental Signatures: The paper provides clear, testable predictions. If proton decay is observed, the specific decay channel (e.g., muonic vs. electronic vs. neutrino modes) and the ratio of branching fractions can identify the underlying dimension-six operator and the flavor structure of the new physics.
• Model Building Guidance: By analyzing UV completions, the authors show that simple UV models (like minimal Leptoquarks) might not suffice; specific flavor representations are required to achieve the necessary suppression. This guides model builders in constructing consistent theories that satisfy both flavor constraints and proton decay limits.
• Interplay of Scales: The study highlights a deep connection between the stability of the proton and the origin of neutrino masses, suggesting that an observation of proton decay would simultaneously provide evidence for the Majorana nature of neutrinos.

In summary, this paper provides a rigorous, systematic framework for understanding how flavor symmetries protect the proton, revealing that new physics responsible for baryon number violation could be significantly lighter than previously thought, provided it respects the specific flavor patterns dictated by the Standard Model and neutrino oscillations.