Contents

Weinberg OperatorEdit

The Weinberg operator is a cornerstone of modern neutrino physics, encapsulating how tiny neutrino masses can arise in a minimal, model-independent way within the Standard Model framework. Proposed by Steven Weinberg in 1979, this operator is the leading non-renormalizable term in the Standard Model effective field theory (SMEFT) that, once the Higgs field acquires a vacuum expectation value, generates Majorana masses for the neutrinos. It provides a bridge between low-energy phenomenology and potential high-energy completions, without requiring new light particles.

In its essence, the Weinberg operator signals that the Standard Model is an effective theory valid up to some high energy scale Λ, where new physics resides. The operator is gauge-invariant, Lorentz-invariant, and lepton-number-violating by two units, and it is the unique dimension-5 operator one can write down with Standard Model fields that achieves these properties. After electroweak symmetry breaking, the operator gives neutrinos a mass term proportional to the square of the Higgs vacuum expectation value, linking the observed smallness of neutrino masses to physics at scales far beyond direct experimental reach. The operator thus captures a universal low-energy imprint of whatever heavy states lie at or above Λ, and its flavor structure encodes the pattern of neutrino masses and mixings observed in experiments such as neutrino oscillations and neutrinoless double beta decay searches.

Definition and structure

The Weinberg operator is a dimension-5, lepton-number-violating, gauge-invariant operator constructed from the lepton doublets L and the Higgs doublet H of the Standard Model. A conventional schematic form is O5 ∼ (L^i_L C L^j_L) H^k H^l ε{ik} ε{jl} / Λ, where C is the charge-conjugation matrix, ε_{ij} is the antisymmetric tensor for SU(2) indices, and Λ is the high-energy scale at which new physics appears. The exact flavor structure is encoded in a coupling matrix in flavor space, but the overall suppression by Λ is universal: the larger Λ is, the smaller the induced neutrino masses.

When the Higgs field acquires its vacuum expectation value v ≈ 246 GeV, the operator induces Majorana mass terms for the neutrinos: mν ∼ c5 v^2 / Λ, where c5 denotes the dimensionless flavor-dependent coefficient of the operator. This simple relation ties the small observed neutrino masses to physics at a high scale Λ, assuming order-one coefficients. The operator is the simplest way to realize the observed phenomenon of neutrino flavor mixing and mass splittings within the SMEFT framework, and it remains valid as long as the energies probed are well below Λ.

Links to core concepts: - The operator is embedded in the structure of the Standard Model and its effective theories, and its existence rests on the principles of effective field theory. - The lepton-number-violating character of the operator connects to processes like neutrinoless double beta decay and to the idea that neutrinos may be Majorana fermion. - The role of the Higgs field in generating masses ties to electroweak symmetry breaking and the dynamics of the Higgs boson.

Origin, UV completions, and the seesaw picture

A hallmark of the Weinberg operator is that it can arise from several ultraviolet (UV) completions, in which heavy degrees of freedom are integrated out to leave a low-energy effective description. The most discussed classes of UV completions are the various types of seesaw mechanisms, which generate the operator at tree level or loop level depending on the model.

  • Type I seesaw: This scenario introduces heavy right-handed neutrinos neutrino with Majorana masses. When these states are integrated out, they produce the Weinberg operator with Λ set by their masses and their Yukawa couplings to the lepton doublets and the Higgs. This is one of the simplest and most studied UV completions, linking the smallness of neutrino masses to very heavy new particles.

  • Type II seesaw: In this framework, a heavy scalar triplet couples directly to the lepton doublets and to the Higgs, providing a tree-level realization of the operator once the triplet is integrated out. The scale Λ is then tied to the mass of the triplet and its couplings.

  • Type III seesaw: Here, heavy fermionic triplets play the role of the heavy states that generate the Weinberg operator upon being integrated out. Like Type I, this yields a high-scale origin for neutrino masses with a distinctive collider phenomenology if the triplet states are not far above the electroweak scale.

Beyond these canonical seesaws, there are loop-induced realizations and other UV completions that can generate the Weinberg operator at lower or higher scales with different flavor structures. The essential point remains: O5 is the low-energy signature of heavy physics, and the details of Λ and c5 depend on the particular UV completion.

Phenomenology and experimental status

The Weinberg operator provides a direct mechanism for neutrinos to acquire tiny masses and, in doing so, predicts observable consequences beyond the Standard Model. The most prominent phenomenological implications include:

  • Neutrino masses and mixing: The flavor structure of the coefficient matrix c5 translates into the observed pattern of neutrino masses and the angles and phases that govern neutrino oscillations.

  • Majorana nature and lepton-number violation: Since the operator violates lepton number by two units, it implies that neutrinos may be Majorana fermions, a hypothesis probed by experiments searching for neutrinoless double beta decay.

  • Cosmological and astrophysical constraints: The sum of neutrino masses affects cosmic evolution and structure formation, leading to bounds from cosmology that complement laboratory measurements.

  • Experimental probes and limits: Direct tests include searches for 0νββ and collider or precision experiments that constrain the scale Λ and the flavor structure of c5. The absence or presence of signals informs models of UV completions and the naturalness of the neutrino sector.

See also