⚛️ phenomenology

Neutrinoless Double-Beta Decays from Operator Mixing

This paper presents a complete one-loop Renormalization Group analysis within the Standard Model Effective Field Theory, demonstrating that operator mixing induces sizable contributions to neutrinoless double-beta decays from operators that are inactive at tree level, thereby establishing the most stringent constraints on numerous dimension-seven operators and highlighting the critical role of higher-order logarithmic effects.

Original authors: J. de Vries, S. Fajfer, L. P. S. Leal, O. Sumensari, R. Zukanovich Funchal

Published 2026-08-11

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: J. de Vries, S. Fajfer, L. P. S. Leal, O. Sumensari, R. Zukanovich Funchal

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, cosmic kitchen where particles are the ingredients and forces are the recipes. For decades, scientists have been trying to figure out why neutrinos—the ghostly, nearly massless particles that zip through everything—are so incredibly light. In this kitchen, the "Standard Model" is the master cookbook that explains almost everything we see, but it has a glaring hole: it says neutrinos should have no mass at all. Since we know they do (thanks to experiments showing they can change flavors like a chameleon), there must be a secret ingredient or a hidden recipe we haven't found yet.

One of the most exciting ways to hunt for this secret is by looking for a very rare event called "neutrinoless double-beta decay." Picture two neutrons in an atomic nucleus deciding to turn into two protons and two electrons all at once. In the normal world, this happens, but it spits out two sneaky neutrinos to balance the books. In this special, forbidden version, the neutrinos never show up; they cancel each other out. If we ever catch this happening, it would prove that neutrinos are their own antiparticles and reveal the hidden recipe that gives them mass. It's like finding a magic trick where the rabbit disappears without a trace, proving there's a secret door in the hat we didn't know about.

Now, enter the team of physicists behind this new study. They are playing the role of cosmic detectives, but instead of looking for the rabbit, they are looking at the "mixing" of the recipes. In the world of particle physics, there's a concept called the "Effective Field Theory," which is like a simplified menu for the kitchen. It lists all the possible ways particles can interact at different energy levels. The scientists knew that some of these interactions (called "operators") were too heavy or complex to show up directly in the double-beta decay experiment. They thought, "If these heavy recipes don't show up on the menu, they can't affect our magic trick, right?"

Wrong.

The authors of this paper performed a massive, one-loop "Renormalization Group" analysis. Think of this as tracking how a recipe changes as you move it from a high-heat oven (the high-energy universe) down to a cool countertop (the low-energy lab). They discovered that even if a heavy recipe doesn't contribute to the magic trick directly, it can "mix" with other recipes as it cools down. It's like a heavy, complex sauce that, as it simmers, slowly transforms into a lighter, simpler sauce that does end up on the menu.

The team found that this "mixing" is a game-changer. Many of the heavy, dimension-seven operators (which are like the complex, high-energy recipes) that were previously thought to be harmless or irrelevant actually generate significant effects through this mixing process. In fact, for many of these heavy operators, the constraints (the rules about how big the secret ingredient can be) derived from this mixing are much stricter than the limits we get from looking at other particle decay experiments, like those involving mesons (which are like unstable, short-lived particle sandwiches).

The researchers didn't just stop at the first level of mixing. They dug deeper, looking at "second leading-logarithm" effects. Imagine a recipe that doesn't change much after one hour of simmering, but after two hours, it transforms completely. For certain specific combinations of heavy quarks (the heavy ingredients), the paper shows that these double-step mixing effects are actually the most important ones. This means that ignoring these subtle, multi-step transformations would be like missing the main flavor of the dish.

Using the latest experimental data from the KamLAND-Zen experiment, which has set a record for how long a nucleus can wait before decaying (over $3.8 \times 10^{26}$ years), the team calculated the new limits. They found that by accounting for these mixing effects, they can now probe energy scales as high as 30 TeV (tera-electronvolts) for certain types of heavy quarks. To put that in perspective, that's like being able to see details in a recipe that are 30 times smaller than what we could see before.

The paper explicitly rules out the idea that we can ignore these heavy operators just because they don't show up at the "tree level" (the direct, first-glance interaction). They argue that for specific flavor combinations, the loop-induced mixing is not just a tiny correction; it is the dominant effect. They also caution that while their results are robust within their framework, they rely on specific assumptions about how the universe's "flavor" works (specifically, that the mixing is driven by the known properties of quarks).

In short, this paper tells us that the universe is a lot more interconnected than we thought. The heavy, hidden rules of the high-energy world don't just sit in the background; they actively seep into the low-energy world, changing the rules of the game. By understanding how these rules mix and evolve, we can set much tighter bounds on where the secret recipe for neutrino mass might be hiding. It's a reminder that in the cosmic kitchen, nothing is ever truly isolated; even the heaviest ingredients leave a trace in the lightest dishes.

Technical Summary: Neutrinoless Double-Beta Decays from Operator Mixing

Problem Statement
The observation of neutrino oscillations confirms that neutrinos possess mass, necessitating physics beyond the Standard Model (SM). Within the framework of the Standard Model Effective Field Theory (SMEFT), neutrino masses are typically attributed to the dimension-five Weinberg operator ( $O^{(5)}_{LH}$ ), which induces Lepton Number Violating (LNV) processes such as neutrinoless double-beta decay ( $0\nu\beta\beta$ ). However, in various ultraviolet (UV) scenarios, particularly radiative models, dimension-seven operators can generate dominant contributions to $0\nu\beta\beta$ decays.

A critical gap in previous analyses is the treatment of Renormalization Group (RG) evolution. While simple power counting suggests that higher-dimensional operators are suppressed, loop-induced mixing between operators can generate sizable contributions to $0\nu\beta\beta$ even for operators that do not contribute at tree level. Specifically, operators involving heavy quark flavors (second and third generations) may not directly mediate $0\nu\beta\beta$ at tree level but can mix into operators involving valence quarks via RG evolution. Previous studies have often focused on diagonal RG evolution or specific operators, potentially underestimating constraints on the full set of dimension-seven SMEFT operators.

Methodology
The authors perform a complete one-loop RG analysis of $0\nu\beta\beta$ decays within the SMEFT, considering the full set of dimension-seven LNV operators defined in the basis of Ref. [19]. The methodology involves:

RG Evolution: Solving the RG equations for dimension-seven operators from the New Physics scale ( $\Lambda$ ) down to the electroweak scale ( $\mu_{ew} \simeq m_W$ ). The evolution is governed by the one-loop anomalous dimension matrix $\gamma^{(7)}$ computed in Ref. [23].
Logarithmic Expansion: The solution is expanded in powers of $\log(\mu_{ew}/\Lambda)$ . The authors explicitly account for contributions beyond the first leading logarithm (single-log), including second leading-logarithm (double-log) terms. This is crucial for cases where the direct one-loop mixing vanishes or is suppressed, requiring a two-step mixing process (e.g., $b \to c \to a$ ) to generate non-zero effects.
Flavor Structure: The analysis assumes a down-aligned flavor basis for quarks and includes the full CKM matrix structure. The mixing effects are driven primarily by Yukawa couplings, particularly those of the top and bottom quarks, introducing an irreducible source of quark flavor violation.
Phenomenological Matching: The evolved SMEFT Wilson coefficients are matched onto the Low-Energy EFT (LEFT) at the electroweak scale. This includes dimension-six and dimension-seven LEFT operators relevant for $0\nu\beta\beta$ .
Constraint Comparison: The derived constraints on the Wilson coefficients are compared against:
- Experimental limits on $0\nu\beta\beta$ half-lives (using data from KamLAND-Zen for $^{136}\text{Xe}$ ).
- Constraints from meson decays, specifically leptonic decays ( $P \to \ell\nu$ ), three-body LNV decays ( $P \to P' \ell^- \ell^-$ ), and flavor-changing neutral currents ( $P \to P' \nu\nu$ ).
Computational Tools: The nuclear matrix elements and phase space factors are handled using the automated tool $\nu$ DoBe [43], which implements the "Master Formula" for $0\nu\beta\beta$ rates.

Key Contributions

Complete Operator Mixing: The paper provides the first systematic investigation of the impact of full one-loop RG mixing on all dimension-seven SMEFT operators relevant for $0\nu\beta\beta$ , extending beyond previous studies limited to diagonal evolution or specific subsets.
Beyond Leading Logarithm: The authors highlight the phenomenological necessity of including second leading-logarithm contributions. For specific flavor combinations (e.g., operators involving heavy quarks mixing into light quarks), the single-log contribution may be suppressed by small Yukawa couplings, making the double-log contribution the dominant effect.
Heavy Quark Constraints: The study demonstrates that operators involving heavy quarks (which do not contribute at tree level to $0\nu\beta\beta$ ) can generate significant constraints through RG mixing into operators involving valence quarks.

Results

Stringent Bounds: For numerous dimension-seven operators, particularly those of the $\psi^4H$ and $\psi^4D$ types involving second and third-generation quarks, the constraints derived from $0\nu\beta\beta$ decays (including RG effects) are generally more stringent than those derived from meson decays.
Probing Scales: Current $0\nu\beta\beta$ limits allow the probing of New Physics scales up to $\Lambda \sim 30$ TeV for specific operator-flavor combinations via RG effects. This sensitivity is expected to improve by a factor of two with next-generation experiments.
Specific Operator Cases:
- For operators like $O^{(7)}_{\bar{d}LueH}$ with third-generation quark indices, RG mixing into $O^{(7)}_{LeHD}$ (modifying $W$ -boson couplings) generates constraints comparable to tree-level limits on valence quarks, despite loop suppression, due to chiral enhancement.
- For operators like $O^{(7)}_{\bar{d}uLLD}$ with indices $(i,j)=(3,1)$ or $(2,1)$ , the single-log mixing is suppressed, and the double-log mixing provides the leading contribution, testing scales up to $\mathcal{O}(1)$ TeV.
Comparison with Meson Decays: While meson decays (specifically $K \to \pi + \text{inv}$ ) provide the strongest constraints for specific scalar coefficients, testing scales up to $\Lambda \sim 40$ TeV, the $0\nu\beta\beta$ channel generally offers superior or competitive sensitivity for the broader set of dimension-seven operators once RG mixing is included.
Discrepancy with Other Works: The authors note a discrepancy with Ref. [58] regarding operators that mix into the Weinberg operator. They attribute this to Ref. [58] using unrealistic neutrino mass values in their calculation, whereas this work enforces consistency with experimental bounds ( $m_\nu \lesssim 0.1$ eV).

Significance
The paper establishes that RG-induced operator mixing is a critical component in the phenomenological analysis of $0\nu\beta\beta$ decays. Ignoring these effects, particularly the mixing of heavy-flavor operators into light-flavor ones and the inclusion of higher-order logarithmic terms, leads to an incomplete and potentially weaker understanding of the constraints on New Physics. The results demonstrate that $0\nu\beta\beta$ experiments serve as powerful probes for dimension-seven operators, even those that do not directly mediate the decay at tree level, thereby providing the most stringent bounds on a wide class of SMEFT operators. The work underscores the importance of consistent two-loop anomalous dimension calculations for fully capturing these effects in future precision studies.

Technical Summary: Neutrinoless Double-Beta Decays from Operator Mixing

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