Renormalization-group-improved constraints on dimension-7 baryon-number-violating operators

This paper demonstrates that incorporating full renormalization-group running effects, particularly through Yukawa mixings, enables stringent constraints on all 297 dimension-7 baryon-number-violating SMEFT Wilson coefficients from nucleon decay data, significantly extending the reach beyond the first and second fermion generations compared to tree-level analyses.

The Gist

Imagine the universe is a giant, complex machine built from tiny Lego bricks. For decades, physicists have been trying to figure out if there are any "glitches" in the machine's design that allow it to break its own rules. One of the most sacred rules is Baryon Number Conservation. In simple terms, this rule says that the total number of protons and neutrons (the stuff that makes up you, me, and stars) should never change. If this rule were broken, protons could simply vanish and turn into other particles, like a proton decaying into a positron and a pion.

If protons decay, the universe as we know it would eventually dissolve. But we haven't seen it happen yet. So, physicists are on a massive hunt to find the tiniest hint of this decay.

The Detective Work: SMEFT

Since we don't know exactly what new physics is causing this (if anything), the authors of this paper use a tool called SMEFT (Standard Model Effective Field Theory). Think of SMEFT as a giant "menu" of possible new interactions. It lists every conceivable way protons could decay, organized by how "heavy" or "complex" the new physics would be.

The authors are looking at a specific section of this menu: Dimension-7 operators.

• Dimension-6 operators are the "main courses"—the most likely ways protons could decay.
• Dimension-7 operators are the "appetizers" or "desserts"—slightly more complex and harder to detect, but still possible.

There are 297 different recipes (called Wilson Coefficients) on this menu. The goal of the paper is to see which of these 297 recipes are actually allowed by nature, based on the fact that we haven't seen protons decay yet.

The Problem: The "Tree-Level" Blind Spot

In the past, physicists looked at these recipes using a "Tree-Level" analysis. Imagine looking at a forest from a distance. You can clearly see the big, tall trees (operators involving the first two generations of particles, like up/down quarks and electrons). But the forest is dense, and you can't see the smaller, hidden bushes in the back (operators involving heavier particles like the top quark or the tau lepton).

Because the heavy particles are so massive, they don't show up directly in low-energy experiments like proton decay searches. So, previous studies said, "We can't constrain the recipes involving the top quark because they don't seem to affect the proton."

The Solution: The "Renormalization Group" (RG) Run

The authors of this paper decided to look closer. They used a technique called Renormalization Group (RG) running.

The Analogy: The Flavor Mixing Smoothie
Imagine you have a smoothie made of heavy fruits (top quarks, tau leptons) and light fruits (up quarks, electrons).

• The Old View (Tree-Level): You only taste the light fruits at the bottom of the cup. You think the heavy fruits aren't there.
• The New View (RG Running): As you let the smoothie sit and swirl (evolve from high energy down to low energy), the flavors start to mix. The heavy fruit flavors "leak" into the light fruit layer.

In physics, this "swirling" is driven by Yukawa couplings (interactions that give particles their mass). As the energy scale drops from the massive "New Physics" scale down to the energy of a proton, the heavy particles mix with the light ones. This means that a rule involving a heavy top quark at the high-energy scale can "migrate" down and influence the behavior of a light proton.

What They Found

By running this "flavor mixing" simulation all the way down to the proton's energy level, the authors did something amazing:

1. They saw the hidden bushes: They were able to set strict limits on all 297 recipes, not just the easy ones. Even the recipes involving the heaviest particles (top quarks and tau leptons) are now constrained.
2. The Indirect Power: Even though the heavy particles don't decay directly into protons, the "mixing" effect means that if those heavy particles were breaking the rules, we would see the evidence in the proton's decay.
3. The Results: They found that for many of these heavy-particle recipes, the universe is much stricter than we thought. The "New Physics" scale (how heavy the new particles must be) has to be incredibly high—trillions of times heavier than a proton—to avoid breaking the rules we see today.

Why This Matters

This paper is like upgrading from a pair of binoculars to a high-powered telescope.

• Before: We could only say, "We don't see protons decaying, so the simple rules are safe."
• Now: We can say, "We don't see protons decaying, which means even the most complex, heavy-particle rules must be incredibly suppressed."

It tells us that if there is a new, heavy world of physics waiting to be discovered, it is hiding very well. The "mixing" effects (RG running) are the key to unlocking these secrets, proving that what happens at the highest energy scales in the universe inevitably leaves a fingerprint on the low-energy world we live in.

In a nutshell: The authors used a mathematical "time machine" to trace how heavy, invisible particles influence the everyday proton. They found that the universe is holding its breath very tightly, forbidding almost all possible ways for protons to decay, even the sneaky ones involving the heaviest particles.

Technical Summary

1. Problem Statement

Baryon number violation (BNV) is a crucial prediction of many Beyond the Standard Model (BSM) scenarios, including Grand Unified Theories (GUTs) and leptoquark models. While dimension-6 BNV operators in the Standard Model Effective Field Theory (SMEFT) are well-studied, dimension-7 (dim-7) BNV operators provide the next leading contribution, particularly when dim-6 operators are suppressed or forbidden.

Previous analyses of dim-7 operators were largely limited to tree-level approximations. These approaches could only constrain Wilson Coefficients (WCs) involving the first and second fermion generations because they relied on direct matching to nucleon decay processes. Operators involving the third generation (top quark, $\tau$ lepton) or specific second-generation combinations were often unconstrained or weakly constrained in tree-level analyses, as they do not directly induce nucleon decays at leading order. The authors identify a critical gap: the lack of a comprehensive analysis incorporating Renormalization Group (RG) running effects from the new physics scale ($\Lambda_{NP}$) down to the electroweak scale ($\Lambda_{EW}$), which can mix heavy-flavor operators into light-flavor ones via Yukawa couplings.

2. Methodology

The authors perform a comprehensive, RG-improved analysis of two-body nucleon decays ($N \to \ell M$, where $\ell$ is a lepton and $M$ is a meson) induced by dim-7 SMEFT BNV operators.

• Operator Basis: They consider the full set of 297 independent dim-7 SMEFT BNV operators (accounting for all generation indices but excluding hermitian conjugates). These operators are categorized into classes involving four fermions and one Higgs ($\psi^4 H$) or four fermions and one derivative ($\psi^4 D$).
• RG Evolution:
  • They utilize the complete one-loop RG equations for dim-7 operators derived in previous works [23, 34].
  • The evolution is performed using the numerical code D7RGESolver [35].
  • The analysis is conducted in both the up-quark and down-quark flavor bases to ensure consistency with matching conditions.
  • The new physics scale is set to $\Lambda_{NP} = 10^8$ GeV, evolved down to $\Lambda_{EW} \approx 246$ GeV.
• Matching to LEFT:
  • At $\Lambda_{EW}$, the SMEFT operators are matched onto the Low-Energy Effective Field Theory (LEFT).
  • This matching accounts for the vacuum expectation value (VEV) of the Higgs field and CKM matrix elements.
  • Crucially, the authors derive matching relations for operators involving derivatives (e.g., $O_{\bar{L}QdDd}$) by using integration by parts, equations of motion, and Fierz identities to transfer derivatives to lepton fields or modify Higgs-coupled operators.
• Chiral Perturbation Theory (ChPT):
  • The LEFT operators are matched to the hadronic level using ChPT at the scale $\Lambda_\chi \sim 1.2$ GeV.
  • They employ the spurion field method to map quark-level interactions to baryon-meson-lepton vertices.
  • Decay widths are calculated for specific channels (e.g., $p \to \nu K^+$, $n \to e^- K^+$) using lattice QCD inputs for low-energy constants (LECs).
• Constraint Derivation:
  • Experimental bounds on nucleon lifetimes (from Super-Kamiokande, IMB, Frejus, etc.) are used to set limits on the inverse decay widths.
  • By equating the theoretical decay rates (functions of the WCs at $\Lambda_{NP}$) to experimental limits, they derive constraints on all 297 WCs.

3. Key Contributions

• First Full RG-Improved Analysis: This is the first study to systematically constrain all 297 dim-7 SMEFT BNV Wilson coefficients using full RG running effects, moving beyond the limitations of tree-level analyses.
• Indirect Probing of Heavy Flavors: The paper demonstrates that RG running, driven primarily by Yukawa mixings (specifically the large top quark Yukawa coupling), allows operators involving the third generation (top quark, $\tau$ lepton) to mix into operators involving light quarks ($u, d, s$). This enables indirect constraints on heavy-flavor operators that would otherwise be invisible to nucleon decay searches.
• Flavor Basis Dependence: The authors highlight that constraints on operators involving left-handed quark doublets ($O_{\bar{L}dQQ\tilde{H}}$, $O_{\bar{e}Qdd\tilde{H}}$, $O_{\bar{L}QdDd}$) are sensitive to the choice of quark flavor basis (up vs. down), a nuance often overlooked in simpler treatments.
• Derivative Operator Matching: They provide explicit matching relations for derivative-type operators ($O_{\bar{L}QdDd}$ and $O_{\bar{e}ddDd}$), showing how they contribute to nucleon decays either directly or through RG-induced mixing.

4. Key Results

• Stringent Bounds on Heavy Generations:
  • For operators involving the third generation (e.g., those with top quarks or $\tau$ leptons), the RG-improved constraints are significantly stronger than direct limits from $\tau$ or top quark decays at colliders.
  • Effective scales ($\Lambda_{eff}$) for these operators are constrained up to $O(10^5 - 10^{10}$ GeV), depending on the specific generation combination.
  • For example, an operator involving a $\tau$ lepton and third-generation quarks, which cannot induce nucleon decay at tree level, induces a non-zero decay width at $\Lambda_{EW}$ via RG mixing, leading to a bound of $\Lambda_{eff} \sim 10^5$ GeV.
• Comparison with Tree-Level:
  • For operators involving only first-generation fermions, the RG-improved results are consistent with previous tree-level analyses (differing by $O(1)$ factors).
  • For operators involving second and third generations, the RG-improved bounds are orders of magnitude stronger than direct limits from heavy-flavor processes.
• Specific Operator Constraints:
  • The paper provides a comprehensive table (Table V) of dimensionless Wilson coefficients ($c_7$) for all 297 combinations.
  • Operators like $O_{\bar{L}dQQ\tilde{H}}$ with third-generation indices are constrained to $c_7 \lesssim 10^{-9}$, whereas tree-level analysis would suggest no constraint.
• Future Sensitivity: The authors note that future experiments like Hyper-Kamiokande, DUNE, and JUNO could improve these bounds by a factor of $\sim 2$.

5. Significance

This work fundamentally changes the landscape of BNV searches in the SMEFT framework:

1. Completeness: It establishes that nucleon decay experiments are sensitive to all dim-7 BNV operators, not just those involving light fermions.
2. Indirect Probes: It validates nucleon decay as a powerful indirect probe for high-scale physics involving the top quark and $\tau$ lepton, complementing direct searches at the LHC.
3. Theoretical Rigor: It underscores the necessity of including RG evolution in phenomenological studies of BNV. Ignoring RG mixing leads to a severe underestimation of constraints on heavy-flavor operators and a misinterpretation of the allowed parameter space for BSM models.
4. Model Building: The results provide robust, model-independent constraints that any UV-complete theory generating dim-7 BNV operators must satisfy, particularly those involving heavy fermions.

In conclusion, the paper demonstrates that Renormalization Group effects are not merely a correction but a dominant mechanism for constraining BNV physics involving heavy generations, significantly tightening the limits on new physics scales compared to previous tree-level analyses.