Connecting Light and Heavy Neutrino Masses Through the Type-I Seesaw
The observation of neutrino oscillations has firmly established that at least two neutrino species possess nonzero masses. In the Standard Model, neutrinos are massless because only left-handed fields are present; a Dirac mass term requires right-handed partners, while a Majorana mass term violates lepton number. The seesaw mechanism elegantly explains the smallness of neutrino masses by introducing heavy right-handed Majorana neutrinos. When these heavy states are integrated out, an effective dimension-five operator emerges that yields tiny masses for the active neutrinos. The calculator on this page allows users to explore the quantitative relationship between the light neutrino masses, the Yukawa couplings that set the Dirac mass scale, and the heavy Majorana mass scale M implied by the type-I seesaw framework.
In its simplest single-flavor form, the seesaw mechanism involves a Dirac mass term mD = y v and a Majorana mass term M for the right-handed neutrino N. The relevant mass matrix in the (νL, Nc) basis is
Diagonalizing this matrix yields one light and one heavy mass eigenvalue. In the regime where M ≫ mD, the light mass is approximately
and the heavy mass is essentially M. The smallness of mν thus results from the suppression by the heavy scale. Expressing mD in terms of the Yukawa coupling y and the electroweak Higgs vacuum expectation value v ≈ 174 GeV gives the celebrated relation
This formula is implemented in the calculator. Users input a light neutrino mass mν in electronvolts and a dimensionless Yukawa coupling y. The tool outputs the heavy mass scale M in GeV along with the intermediate Dirac mass mD. Because current oscillation data suggest mν ≲ 0.1 eV, even moderate values of y lead to extremely high M, often near grand unified theory (GUT) scales.
The seesaw mechanism carries profound implications for both particle physics and cosmology. If M resides near 1014 GeV, it hints at connections to unification and inflationary physics. Alternatively, if y is very small, M could fall in the keV–TeV range, opening possibilities for direct collider searches or contributions to warm dark matter. The calculator includes a simple classification scheme: heavy scales below 1 TeV are labeled "Collider-scale," between 1 TeV and 1012 GeV "Intermediate," and above that "GUT-scale." These boundaries are not rigid but help contextualize the numerical output.
The table below offers example outputs:
| mν (eV) | y | mD (GeV) | M (GeV) | Classification |
|---|---|---|---|---|
| 0.05 | 1 | 174 | 6.0×1014 | GUT-scale |
| 0.1 | 10−3 | 0.174 | 3.0×108 | Intermediate |
These examples demonstrate how dramatically the heavy scale responds to the Yukawa coupling. A coupling of order unity pushes M near the unification scale, whereas a tiny coupling lowers M to a range potentially accessible through leptogenesis or low-energy phenomenology. The Dirac mass column also illustrates the connection to charged fermion masses; for y comparable to the electron Yukawa coupling (~2×10−6), mD would be of order MeV.
Beyond its role in generating masses, the seesaw mechanism underpins models of baryogenesis via leptogenesis. In such scenarios, the CP-violating decays of the heavy neutrinos create a lepton asymmetry that electroweak sphalerons partly convert into a baryon asymmetry. The magnitude of M and the structure of the Yukawa couplings influence the viability of thermal leptogenesis, resonant leptogenesis, or low-scale mechanisms. By rapidly estimating M, this calculator aids in judging whether a given parameter choice can support one of these baryogenesis pathways.
The seesaw also leaves imprints on low-energy observables. If the heavy neutrinos are not exceedingly heavy, they can induce deviations in electroweak precision tests, contribute to neutrinoless double beta decay amplitudes, or modify neutrino oscillation phenomenology through mixing with sterile states. The calculator's output therefore serves as a starting point for more detailed model studies, including the impact on flavor symmetries, renormalization group running, and potential links to grand unified theories.
From a pedagogical perspective, the seesaw mechanism showcases how effective field theory captures the influence of heavy states at low energies. Integrating out the heavy Majorana neutrinos yields the dimension-five Weinberg operator (LH)(LH)/Λ. After electroweak symmetry breaking, this operator produces the light neutrino masses. The relationship M ≈ y²v²/mν succinctly encapsulates this chain of reasoning. Students can explore how varying mν and y changes the suppression scale Λ, deepening their appreciation for the interplay between high-scale physics and low-energy observables.
Finally, it is worth noting that the type-I seesaw is only one among several variants. Type-II seesaw introduces scalar triplets, while type-III utilizes fermionic triplets. Each variant offers distinct phenomenology and symmetry structures. Moreover, extended frameworks such as inverse seesaw, linear seesaw, and radiative seesaw attempt to lower the heavy scale while preserving small neutrino masses. Although the calculator focuses on the simplest type-I scenario, the core insight—small masses arising from large scales—remains a pervasive theme across neutrino model building.
By providing a straightforward mapping between light neutrino masses, Yukawa couplings, and heavy scales, this tool invites users to experiment with parameter choices and develop intuition about neutrino mass generation. Whether exploring theoretical models, assessing the feasibility of leptogenesis, or simply satisfying curiosity about how a feeble coupling can imply a colossal mass, the calculator underscores the explanatory power of the seesaw mechanism in modern particle physics.