Seesaw Heavy Neutrino Mass Calculator
Understanding the Type-I Seesaw
The purpose of this calculator is simple: it estimates the heavy Majorana neutrino mass scale implied by the type-I seesaw mechanism once you choose a light neutrino mass and a Yukawa coupling. In neutrino physics, this relation is important because it gives a compact explanation for why the observed neutrino masses are so tiny compared with the masses of charged leptons and quarks. Instead of requiring the neutrino Yukawa coupling to be unimaginably small in every model, the seesaw picture allows the light neutrino mass to be suppressed by a very large heavy scale.
Neutrino oscillation experiments show that neutrinos change flavor as they travel, and that behavior requires at least two neutrino mass splittings to be nonzero. The Standard Model by itself does not naturally provide masses for neutrinos in the same way it does for the charged fermions, because it contains only left-handed neutrino fields. A common extension is to add right-handed neutrino states. Once those states are present, one can write a Dirac mass term connecting left- and right-handed neutrinos, and also a Majorana mass term for the right-handed neutrino itself. The type-I seesaw mechanism studies what happens when that Majorana mass is much larger than the electroweak scale.
In the simplest one-flavor picture, the neutrino mass matrix mixes the active neutrino with a heavy sterile state. The result is one very light eigenstate and one very heavy eigenstate. The light state is what low-energy experiments mostly observe, while the heavy state can sit at scales far above direct experimental reach. This calculator focuses on that simplest relation so you can build intuition quickly without needing to solve a full three-flavor model.
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
Introduction
This page is designed for students, researchers, and curious readers who want a quick numerical estimate of the heavy neutrino scale in a type-I seesaw model. You enter the light neutrino mass in electronvolts and the Dirac Yukawa coupling as a dimensionless number. The calculator then returns two quantities: the Dirac mass mD = yv in GeV and the heavy scale M in GeV. It also labels the result with a broad interpretation such as collider-scale, intermediate, or GUT-scale so the number is easier to place in context.
That context matters because the seesaw relation spans an enormous range of energies. A light neutrino mass around 0.05 eV with a Yukawa coupling of order one points to a heavy scale near 1014 to 1015 GeV, which is far beyond direct collider reach and close to scales often discussed in grand unification. On the other hand, if the Yukawa coupling is tiny, the heavy scale can drop dramatically. The same formula therefore helps connect low-energy neutrino data to very different kinds of model building, from ultra-high-scale theories to lower-scale sterile-neutrino scenarios.
The calculator intentionally uses the absolute value of the Yukawa coupling when computing the mass scale. That is because the sign of a single effective Yukawa parameter does not change the magnitude of the Dirac mass in this simplified estimate. The result should be read as an order-of-magnitude guide, not as a full reconstruction of a realistic neutrino sector with flavor mixing, CP phases, and multiple heavy states.
How to Use
Using the calculator is straightforward. First, enter a positive value for the light neutrino mass mν in eV. Typical benchmark values are around 0.01 eV to 0.1 eV, which are broadly consistent with the scale suggested by oscillation data and cosmological limits. Second, enter the Yukawa coupling y. This quantity is dimensionless. Values near 1 correspond to a Dirac mass near the electroweak scale, while much smaller values correspond to lighter Dirac masses.
After you click the compute button, the page displays the Dirac mass and the heavy Majorana scale. The result is written in scientific notation because the heavy scale can become extremely large. If a valid result is shown, the copy button becomes available so you can quickly save the output for notes, homework, or model comparisons.
Here is a practical way to think about the inputs:
The light neutrino mass mν sets the observed low-energy scale you want to reproduce. The Yukawa coupling y controls how strongly the neutrino couples to the Higgs field, which in turn sets the Dirac mass mD. Once those are chosen, the seesaw relation tells you what heavy scale M is needed to make the light neutrino mass come out correctly. If you increase y while keeping mν fixed, the required heavy scale rises quadratically. If you increase mν while keeping y fixed, the required heavy scale falls.
For accessibility and interpretation, it helps to keep the units clear. The input mν is in electronvolts, but the output masses are in GeV. The script handles the conversion internally by using 1 eV = 10−9 GeV. That conversion is essential because the seesaw formula mixes a tiny neutrino mass with electroweak-scale quantities.
Formula
The calculator implements the standard one-flavor type-I seesaw approximation. Start from the Dirac mass relation mD = yv, where v ≈ 174 GeV is the Higgs vacuum expectation value used in this convention. In the limit where the heavy Majorana mass is much larger than the Dirac mass, the light neutrino mass is approximately mν ≈ mD2/M. Solving for M gives the expression used in the calculator.
Written in words, the heavy scale equals the square of the Yukawa coupling times the square of the Higgs vacuum expectation value, divided by the light neutrino mass after converting that mass into GeV. This is why even a modest Yukawa coupling can imply an enormous heavy scale: the numerator contains electroweak-scale quantities squared, while the denominator contains a mass that is tiny on particle-physics scales.
The calculator also reports the intermediate Dirac mass because it is physically informative. If y is close to 1, then mD is of order 102 GeV, similar to the electroweak scale. If y is much smaller, mD can instead be in the MeV, keV, or even lower range. Seeing both mD and M together helps you understand whether the small neutrino mass is being explained mainly by a huge heavy scale, a tiny Yukawa coupling, or a combination of both.
The classification labels are intentionally broad. Results below 1 TeV are marked collider-scale, results from 1 TeV up to 1012 GeV are marked intermediate, and larger values are marked GUT-scale. These labels are not strict theoretical boundaries. They are simply a convenient way to summarize where the heavy state sits relative to common energy scales discussed in phenomenology.
Example
Suppose you choose a light neutrino mass of 0.05 eV and a Yukawa coupling y = 1. The Dirac mass is then mD = yv ≈ 174 GeV. Plugging that into the seesaw relation gives a heavy scale of roughly 6.0 × 1014 GeV. The calculator will classify that result as GUT-scale. This is a classic textbook example of the seesaw idea: a neutrino mass that is tiny in the laboratory can point to physics at an extraordinarily high energy.
Now compare that with a smaller coupling, such as y = 10−3, while taking mν = 0.1 eV. The Dirac mass becomes 0.174 GeV, and the heavy scale drops to about 3.0 × 108 GeV. That is still very high, but it is many orders of magnitude below the previous example. The comparison shows how sensitive the heavy scale is to the Yukawa coupling. Because M depends on y2, reducing y by a factor of 1000 reduces M by a factor of one million when the light neutrino mass is held fixed.
The table below gives two benchmark outputs that match the calculator's conventions:
| 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 |
Worked examples like these are useful because they show the scale separation directly. They also help explain why neutrino mass models are often discussed together with leptogenesis, sterile-neutrino searches, and grand unified theories. A single low-energy neutrino mass can be compatible with very different ultraviolet pictures depending on the Yukawa structure.
Limitations and Assumptions
This calculator uses the simplest possible seesaw estimate, so it is important to understand what it does not include. Realistic neutrino physics involves three active flavors, mixing angles, mass splittings, possible CP-violating phases, and potentially more than one heavy right-handed neutrino. In a full model, the relation between the observed neutrino masses and the heavy mass matrix is generally matrix-valued rather than a single-number formula. The result shown here should therefore be interpreted as a one-parameter benchmark, not a complete fit to data.
The approximation mν ≈ mD2/M assumes the seesaw hierarchy M ≫ mD. If that hierarchy is not strong, the simple expression becomes less accurate and one should diagonalize the full mass matrix exactly. The calculator also does not include renormalization-group running, threshold corrections, flavor textures, Casas-Ibarra parameterizations, or model-dependent constraints from collider searches, cosmology, or neutrinoless double beta decay.
Another limitation is interpretive rather than mathematical. The classification labels are descriptive shortcuts, not experimental statements. A result marked collider-scale does not automatically mean the heavy neutrino is discoverable, because production rates and mixing angles matter. Likewise, a GUT-scale result does not prove grand unification; it only indicates that the inferred mass lies in a range often discussed alongside high-scale theories.
Even with those caveats, the calculator remains useful. It gives a fast, transparent estimate of how the seesaw mechanism translates tiny neutrino masses into a heavy scale. That makes it a good teaching tool, a quick consistency check for toy models, and a convenient way to build intuition before moving on to more detailed numerical studies.
Physical Interpretation
The seesaw mechanism is often described as a balancing act between a low-energy mass and a high-energy scale. The name comes from the inverse relationship: as the heavy scale rises, the light neutrino mass falls. This is one of the clearest examples in particle physics of how heavy new physics can leave a small but measurable imprint at low energies. In effective field theory language, integrating out the heavy Majorana neutrino produces the dimension-five Weinberg operator, and after electroweak symmetry breaking that operator generates the light neutrino mass.
That perspective is part of what makes the seesaw so influential. It links neutrino masses to questions about lepton number violation, the origin of matter-antimatter asymmetry, and the structure of physics beyond the Standard Model. If the heavy neutrinos decay out of equilibrium with CP violation, they can generate a lepton asymmetry that later becomes a baryon asymmetry through sphaleron processes. For that reason, the heavy scale estimated here is often used as a first pass when discussing leptogenesis scenarios.
The calculator also helps illustrate a broader lesson: a tiny measured quantity does not always require a tiny fundamental parameter. Sometimes it points instead to a ratio of scales. In the seesaw picture, the smallness of neutrino masses can be a window into very high-energy physics. That is why this simple formula continues to appear in textbooks, lectures, and research discussions across particle theory and cosmology.