[Paper Review] Minimal Tree-Level Seesaws with a Heavy Intermediate Fermion

Kristian L. McDonald|arXiv (Cornell University)|Mar 19, 2013

Particle physics theoretical and experimental studies14 references4 citations

TL;DR

This paper systematically catalogs minimal tree-level seesaw models that generate naturally suppressed neutrino masses via a heavy intermediate fermion with a single mass insertion—either Majorana or Dirac type. It identifies only three minimal models with Majorana mass insertions (Type-I, Type-III, and quintuplet seesaws) and five with Dirac mass insertions, all arising from effective operators of dimension d=7, 9, or 11, and shows these are the only viable, non-tuned realizations under naturalness constraints.

ABSTRACT

There exists a generic minimal tree-level diagram, with two external scalars and a heavy intermediate fermion, that can generate naturally small neutrino masses via a seesaw. This diagram has a mass insertion on the internal fermion line, and the set of such diagrams can be partitioned according to whether the mass insertion is of the Majorana or Dirac type. We show that, once subjected to the demands of naturalness (i.e. precluding small scalar vacuum expectation values that require fine-tuning), this set is finite, and contains a relatively small number of elements. Some of the corresponding models have appeared in the literature. We present the remaining original models, thus generalizing the Type-I and Type-III seesaws, and apparently exhausting the list of their minimal non-tuned variants.

Motivation & Objective

• To identify all minimal, non-tuned tree-level seesaw models that generate naturally suppressed neutrino masses via a heavy intermediate fermion.
• To classify these models based on the type of lepton number violation—Majorana mass insertion versus Dirac vertex-induced violation.
• To determine which models are testable at the LHC by analyzing the dominance of tree-level over loop-induced neutrino masses.
• To exhaustively catalog viable models with effective operators of dimension d>5, ensuring natural small vacuum expectation values (VEVs).

Proposed method

• Reverse-engineer seesaw models by analyzing the generic tree-level diagram with two external scalars and a heavy fermion with a single mass insertion.
• Use group theory to classify fermion and scalar representations under SM gauge symmetry (SU(3)c × SU(2)L × U(1)Y), requiring the fermion to be a real representation with odd SU(2)L representation and zero hypercharge.
• Impose constraints: (1) the scalar VEVs must be naturally small, (2) the Yukawa couplings must be allowed by gauge invariance, and (3) the mass insertion must break lepton number.
• Derive the effective neutrino mass as mν ≈ λ²⟨S⟩²/MF, where MF is the heavy fermion mass and ⟨S⟩ is the scalar VEV.
• Assess model viability by comparing tree-level and loop-induced neutrino masses, estimating the parameter space where tree-level dominance holds.
• Use perturbative unitarity and flavor constraints to exclude models with overly large scalar multiplets.

Experimental results

Research questions

• RQ1 What are all the minimal, non-tuned tree-level seesaw models that generate naturally suppressed neutrino masses via a heavy fermion exchange?
• RQ2 How do the distinct types of lepton number violation—Majorana mass insertion versus Dirac vertex—classify and constrain the viable models?
• RQ3 Which of these models are testable at the LHC, and under what conditions do tree-level contributions dominate over loop corrections?
• RQ4 Are there any new seesaw models beyond the well-known Type-I and Type-III seesaws that satisfy naturalness and gauge invariance?
• RQ5 What is the complete list of effective operators (of dimension d>5) that can realize such minimal seesaw mechanisms?

Key findings

• Only three minimal models with Majorana mass insertions exist under naturalness: Type-I seesaw (singlet fermion), Type-III seesaw (triplet fermion), and the quintuplet seesaw from Ref. [11].
• Five additional minimal models exist with Dirac-type lepton number violation: one d=7 model, two d=9 models (one previously known), and two d=11 models.
• The d=9 models are particularly testable at the LHC, with tree-level dominance possible for heavy fermion masses below ~0.6 TeV.
• For d=11 models, tree-level dominance is only marginally possible and may already be excluded by current LHC data.
• The list of viable, non-tuned models is exhaustive and finite under the given constraints, with no further minimal realizations possible.
• Models with distinct scalars (S₁ ≠ S₂) are non-minimal and can be reduced to simpler forms; only R_F ≥ 3 allows such non-minimal realizations with natural VEVs.