Weinberg Operator: Neutrino Mass & QCD CP Violation

• The Weinberg operator is a unique higher-dimensional operator in SMEFT, featuring a dimension-5 term for Majorana neutrino masses and a dimension-6 term for gluonic CP violation.
• It employs gauge invariance and is generated via UV completions like seesaw mechanisms and CP-violating BSM loops, linking neutrino mass generation with hadronic EDM observables.
• Its renormalization group evolution and lattice QCD studies offer practical insights into connecting high-scale new physics with low-energy phenomenology.

The Weinberg operator refers primarily to the unique dimension-five, lepton-number-violating, gauge-invariant operator of the Standard Model Effective Field Theory (SMEFT) that induces Majorana neutrino masses. In parallel, "Weinberg operator" can also denote the dimension-six, purely gluonic, CP-violating operator in QCD responsible for hadronic electric dipole moments. Both types play pivotal roles in fundamental theories of neutrino and hadronic CP violation, with critical implications for particle phenomenology, cosmology, and searches for physics beyond the Standard Model.

1. Definition and Structure

1.1 Dimension-5 Weinberg Operator (Leptonic Sector)

The dimension-5 Weinberg operator is the only gauge- and Lorentz-invariant operator of its kind in the SMEFT. It takes the schematic form

$${\mathcal{O}}_W = \frac{c}{\Lambda_W} (L_i H^j)(L^k H^l) \epsilon_{ik} \epsilon_{jl},$$

where:

• $L$ is the SM lepton doublet,
• $H$ is the Higgs doublet,
• $\Lambda_W$ is the heavy suppression scale,
• $c$ is a dimensionless Wilson coefficient,
• Indices $i,\,j$ label SU(2) components.

When $H$ acquires its VEV, the operator yields a Majorana mass for the light neutrinos: $m_\nu = c v^2 / \Lambda_W$ with $v \simeq 246$ GeV. This is the leading source of neutrino mass in the absence of renormalizable Majorana mass terms (Seo, 5 Jan 2025, Ibarra et al., 2024).

1.2 Dimension-6 Weinberg Operator (QCD/Gluonic Sector)

In QCD, the Weinberg operator is the only dimension-six, purely gluonic, CP-violating interaction: $${\cal O}_W = \frac{w}{6} f^{abc} \epsilon^{\alpha\beta\gamma\delta} G^a_{\mu\alpha} G^b_{\beta\gamma} G^{c,\,\mu}_{\;\;\;\delta},$$ with:

• $L$0 the gluon field strength,
• $L$1 the SU(3) structure constants,
• $L$2 the (dimension -2) Wilson coefficient.

Under CP, this operator is odd and mediates CP-odd phenomena such as electric dipole moments (EDMs) of nucleons and nuclei (Yamanaka et al., 2020, Yamanaka, 2021, Abe et al., 2017).

2. Origins and UV Completion Mechanisms

2.1 Neutrino Mass: UV Realizations

The dimension-5 operator is generated in the UV for instance via:

• Type-I seesaw: Exchange of heavy right-handed neutrinos ($L$3) with $L$4.
• Type-II seesaw: Exchange of an $L$5 Higgs triplet.
• Type-III seesaw: Exchange of fermion $L$6 triplets.
• Variants: Vector-like doublet mediated inverse/linear seesaw, new scalar multiplet completions, and stringy D-instanton-induced mechanisms (Giarnetti et al., 2023, Gu, 2020, Cvetič et al., 2010, Hernandez-Garcia et al., 2019, Sierra et al., 2014).

Some UV completions can forbid the standard operator but allow new "Weinberg-like" operators, e.g., with two Higgs fields of opposite hypercharge in two-Higgs-doublet or U(1)$L$7-extended models. The presence of new scalars with non-negligible VEVs at the TeV scale can induce additional effective Weinberg operators contributing to neutrino masses (Giarnetti et al., 2023).

2.2 Gluonic Weinberg Operator from BSM Physics

The QCD Weinberg operator is induced at two loops in theories with new heavy scalars/fermions endowed with CP-violating couplings, e.g.:

• Two-Higgs Doublet Models (Barr–Zee diagrams),
• Minimal Supersymmetric Standard Model (gluino–squark loops),
• General renormalizable BSM sectors: the Wilson coefficient is calculable from group-theory invariants and two-loop Feynman integrals (Abe et al., 2017, Yamanaka, 2021).

The matching can be performed for arbitrary representations, accounting for the dominant BSM contributions to $L$8.

3. Renormalization Group Evolution and Mixing

3.1 Weinberg Operator Running (Leptonic)

For the dimension-5 operator, the full two-loop renormalization group equation (RGE) in the SM was computed, yielding (Ibarra et al., 2024): $L$9 The two-loop RGE contains nontrivial "rank-increasing" terms, allowing for radiative induction of a vanishing neutrino mass eigenvalue at high scale, and renormalization-induced corrections to mixing angles and phases. Explicit two-loop diagrams with chains of charged lepton Yukawa insertions can break massless-neutrino textures generated at the matching scale (Ibarra et al., 2024).

3.2 Weinberg Operator Running (QCD)

The dimension-6 QCD Weinberg operator runs under QCD: $H$0 with significant two- and three-loop corrections. The full anomalous dimension at two-loop (full QCD) and three-loop (Yang-Mills) is known, with substantial corrections driven by quartic color group invariants. This impacts the evolution of $H$1 from the high BSM scale down to hadronic scales (1 GeV) (Vries et al., 2019).

4. Phenomenological Implications

4.1 Leptonic Sector: Majorana Neutrino Mass and Collider Probes

The operator sets the scale of Majorana neutrino mass: $H$2 String theory, seesaw, and other UV completions set $H$3 in the range $H$4 GeV, depending on the precise mechanism and presence of instanton suppressions (Cvetič et al., 2010). Collider searches can probe the $H$5 and $H$6 entries via same-sign dilepton plus jets events, setting lower limits on $H$7 in the multi-TeV regime, but nuclear decays remain most sensitive to the $H$8 channel (Fuks et al., 2020). The species-scale argument (in presence of Kaluza-Klein towers or large numbers of neutrino-like states) sets a lower bound on $H$9: $\Lambda_W$0 where $\Lambda_W$1 and $\Lambda_W$2 are the active and gravitational species numbers (Seo, 5 Jan 2025).

4.2 Hadronic Sector: Electric Dipole Moments

The QCD Weinberg operator is irreducibly responsible—alongside quark chromo-EDMs—for hadronic and nuclear EDMs: $\Lambda_W$3 and contributes to EDMs of light nuclei such as $\Lambda_W$4He and diamagnetic atoms. The operator also generates short-range CP-odd NN forces and, subdominantly, isovector CP-odd pion-nucleon couplings at next-to-leading order (Yamanaka et al., 2020, Yamanaka, 2021, Osamura, 2022, Osamura et al., 2022, Yamanaka et al., 2022). The irreducible three-quark contribution to the nucleon EDM is found to be subleading relative to the chiral-rotation contribution from the CP-odd nucleon mass (Yamanaka et al., 2020, Yamanaka et al., 2021).

Current EDM experiments constrain $\Lambda_W$5 GeV$\Lambda_W$6. Theoretical uncertainties (QCD sum rules, nuclear structure) are on the order of 50–100% (Osamura et al., 2022).

4.3 Leptogenesis via Weinberg Operator

A time- and space-varying Weinberg operator coupling at a phase transition (CP-violating phase transition scenario) generates a lepton–antilepton reflection asymmetry, which, after sphaleron conversion, yields the baryon asymmetry of the Universe. The magnitude of the generated baryon asymmetry depends directly on the imaginary part of the low-energy neutrino mass matrix (Pascoli et al., 2018, Turner et al., 2018).

5. Lattice QCD, Gradient Flow, and Decisive Matrix Elements

Direct nonperturbative computation of nucleon matrix elements of the Weinberg operator via lattice QCD is being pursued, employing the gradient flow to obtain renormalized, continuum-limit definitions of $\Lambda_W$7 and its mixing with the $\Lambda_W$8-term. The susceptibilities of the vacuum to $\Lambda_W$9 and $c$0 are being quantified, and the impact of $c$1 on the effective $c$2 angle in Peccei–Quinn frameworks is under investigation. Nucleon EDMs $c$3, once lattice-computed matrix elements are available, will yield decisive, model-independent bounds on $c$4 (Bhattacharya et al., 1 Feb 2025).

6. Extensions and Model-building Considerations

New scalar multiplets or additional fermion content at the TeV scale can induce further Weinberg-like operators, modifying low-energy phenomenology and offering direct collider signatures such as multiply charged scalar production and decay. Oblique electroweak parameters ($c$5, $c$6), $c$7-mass measurements, and LHC pair-production searches provide crucial constraints on the allowed scalar sector parameter space and, indirectly, on classes of UV completions generating the Weinberg operator (Giarnetti et al., 2023). In models with additional symmetry (e.g. $c$8), the standard Weinberg operator can be forbidden while mixed-Higgs operators become the leading contributors (Hernandez-Garcia et al., 2019).

7. Summary Table of Core Weinberg Operators

Name	Dimension	Formulation	Physical Effect
Leptonic Weinberg (SMEFT)	5	$c$9	Majorana $i,\,j$0 mass
Gluonic Weinberg (QCD)	6	$i,\,j$1	Nucleon/nuclear EDMs
Electroweak Weinberg (SU(2)$i,\,j$2)	6	$i,\,j$3	$i,\,j$4 EDM, EW baryogenesis

In conclusion, the Weinberg operator—both in the context of neutrino masses and of hadronic CP violation—serves as a central probe of new physics at high scales, with ongoing phenomenological consequences for EDM searches, neutrino physics, collider programs, and precision theory efforts across particle physics (Seo, 5 Jan 2025, Yamanaka et al., 2020, Ibarra et al., 2024, Fuks et al., 2020, Bhattacharya et al., 1 Feb 2025, Vries et al., 2019, Yamanaka, 2021, Osamura et al., 2022, Osamura, 2022, Giarnetti et al., 2023, Yamanaka et al., 2021, Yamanaka et al., 2022, Cvetič et al., 2010).