Mass That ClicksEpisode 5 ／ From here on, to "the floor's minimum value"

The smallest known mass ── its outlandish lightness is a letter from the largest scale in the universe

The Smallest Mass, UV Side:
Why Is the Neutrino So Light The smallest known nonzero mass is less than one ten-millionth of the electron.
Solve that lightness with the "seesaw" and you begin to see that the smallest mass points to the largest scale in the universe (grand unification).

Tools you'll need: the Higgs equation from Episode 3, chiral symmetry from Episode 2, division Hook equation: \(m_\nu\sim v^2/M\)

Through Episode 4 we finished seeing "how a finite floor wells up," on both the Higgs and confinement sides. From here we head toward the floor's minimum value. The smallest nonzero mass among known elementary particles ── that is the neutrino. It is so light: less than one ten-millionth of the electron. Why is it so light? The reveal is the seesaw mechanism, and from it an unexpected panorama opens up: the smallest mass points to the largest scale in the universe.

01First, just how light is it?

Look at the ladder of masses (in \(\text{eV}\)). The top quark \(1.7\times10^{11}\), the proton \(9.4\times10^8\), the electron \(5.1\times10^5\) ── and the neutrino is \(0.05\) or less. Even counting from the electron it is seven orders of magnitude down; the ladder collapses.

In terms of Episode 3's Higgs equation \(m_f=y_f v/\sqrt2\), if the neutrino too took its mass straightforwardly from the Higgs (Dirac type), the Yukawa coupling would be \(y_\nu\sim3\times10^{-13}\). That is 12 orders of magnitude below the top (\(y\approx1\)). Thanks to Episode 2's chiral symmetry it is "technically natural even if small," but ── why is just this one so small? A straightforward Higgs is no answer.

02The clue ── only the neutrino is allowed a different way of being written

There is a decisive feature. The right-handed neutrino \(\nu_R\) is a completely neutral particle carrying none of the Standard Model's gauge charges. That means it can be written with a Majorana mass (a type of mass where particle and antiparticle are the same), which is forbidden for the other fermions.

The neutrino's exclusive privilege

\(\nu_R\) is completely neutral → a Majorana mass \(M\) is allowed.
And \(M\) is not tied to the electroweak scale \(v\) ── it may naturally be enormous (the grand-unification scale).

This is where it pays off. Higgs-derived masses (Episode 3) were tied to \(v\approx246\) GeV. But nothing stops the Majorana mass \(M\). So \(M\) can naturally take values far above any accelerator ── like \(10^{14\text{–}15}\) GeV.

03The seesaw mechanism ── smallness is the flip side of largeness

When the electroweak-scale Dirac mass \(m_D\) (Higgs-derived) and the enormous Majorana mass \(M\) coexist, the "balance beam" of mass looks like this. Diagonalize it (= find the actual masses) and two eigenvalues come out.

The seesaw ── raise one side, the other side sinks
$$m_{\text{light}}\approx\frac{m_D^{2}}{M},\qquad m_{\text{heavy}}\approx M$$
Let's try it ── get the neutrino's weight

Substitute \(m_D\sim100\) GeV (the natural value for a Yukawa \(\sim1\)) and \(M\sim10^{15}\) GeV

$$m_\nu\sim\frac{(100\ \text{GeV})^2}{10^{15}\ \text{GeV}}=10^{-11}\ \text{GeV}=0.01\ \text{eV}$$

Right on target with observation. No unnatural Yukawa of \(10^{-13}\) is needed. Just using a natural O(1) coupling and a naturally enormous \(M\), the neutrino's outlandish smallness comes out automatically as a "ratio" ── the cause of the smallness was the largeness of \(M\).

Figure: the seesaw. The pivot is the electroweak scale \(m_D\) (fixed). Raise the heavy partner's mass \(M\) with the slider and the neutrino \(m_\nu\) drops. On a log scale the two are symmetric about the pivot ── exactly a seesaw.
Move M and the neutrino's weight changes.
neutrino m_ν (light) heavy partner N (mass M)

04So the smallest mass points to the largest scale

Now read that equation in reverse. From the observed \(m_\nu\), find the hidden \(M\).

The smallest mass tells you the largest scale
$$M\sim\frac{v^2}{m_\nu}\sim\frac{(100\ \text{GeV})^2}{0.05\ \text{eV}}\sim2\times10^{14}\ \text{GeV}$$

The grand-unification scale \(10^{14\text{–}15}\) GeV, which accelerators can never reach, can be read off from the weight of the lightest particle in the universe. The neutrino's lightness was a letter from the largest scale in the universe. Whereas Episode 4 was "dimensionless running generates its own scale (UV self-containment)," this time the smallest mass becomes a window peering into far higher energies. Even for the same "mass," the direction it points is utterly different.

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05So how do we measure it? ── four handles

The absolute value is still not fully known. Four independent handles each look at something different.

Oscillation
only the "differences" of mass are visible\(\sqrt{\Delta m^2_{\rm sol}}\approx0.0086\) eV, \(\sqrt{\Delta m^2_{\rm atm}}\approx0.050\) eV. → The heaviest is at least 0.05 eV. But the absolute value and ordering are unknown.
Cosmology
the degree to which structure formation is suppressedFrom Planck + galaxy distributions, \(\Sigma m_\nu\lesssim0.07\text{–}0.12\) eV. Strongest, but ΛCDM-dependent.
KATRIN
the endpoint of β decay (direct, model-independent)From the kinematics of tritium decay alone, \(m_\beta<0.45\) eV. Weak, but assumption-free.
0νββ
the double β decay that occurs if it is MajoranaNot yet found (\(m_{\beta\beta}\lesssim0.03\text{–}0.2\) eV). If detected, Majorana is confirmed = corroboration of the seesaw.

The honest current state: at least one is 0.05 eV or more, the total is 0.1 eV or less, the lightest may be 0, and Dirac vs. Majorana is undecided. The seesaw is the leading "story," but it is not yet demonstrated. Its trigger is 0νββ.

A bonus ── so that matter exists If the seesaw's heavy partner \(N\) decays while violating CP in the early universe, a lepton-number asymmetry is born, which is converted into a baryon asymmetry (leptogenesis). In other words, the very mechanism behind the neutrino's lightness may connect to "why matter, not antimatter, remained in the universe = why we exist." The smallest mass may point all the way to the reason for existence.
An honest line

This time we do not put the watchword \(c\cdot t=\text{constant}\) front and center. The neutrino's lightness is about the internal structure of elementary particles (the UV side) and has nothing to do with the expansion of the universe. The "smallest floor" set by the universe's size (the IR side) comes next time, in Episode 6, where \(c\cdot t\) legitimately works. That this episode (UV) and Episode 6 (IR) close in on the same neighborhood of meV from opposite directions is a setup for the finale.

The numbers are representative values, and the order of magnitude shifts with how you choose \(m_D\) and \(M\). There are several kinds of seesaw (Type I/II/III); here we depict the most basic, Type I.

Practice problems (solvable with today's material)
  1. In one phrase, why is a Majorana mass allowed only for the neutrino?
    See the answer
    Because the right-handed neutrino \(\nu_R\) is a completely neutral particle carrying not a single Standard-Model gauge charge. For the other fermions, which carry electric charge and so on, a Majorana mass (treating particle = antiparticle) is forbidden by symmetry.
  2. In the seesaw \(m_\nu\approx m_D^2/M\), with \(m_D\sim100\) GeV and \(M\sim10^{15}\) GeV, what is \(m_\nu\)?
    See the answer
    \((100)^2/10^{15}\) GeV \(=10^{-11}\) GeV \(=10^{-2}\) eV \(=0.01\) eV. Same order as the observed neutrino mass. It comes out without using an unnaturally small Yukawa.
  3. What does "the smallest mass points to the largest scale" mean?
    See the answer
    Solving the seesaw in reverse gives \(M\sim v^2/m_\nu\sim10^{14}\) GeV. From the lightness of the lightest particle (the neutrino), the grand-unification scale that accelerators cannot reach can be read off. Smallness becomes a window peering into enormousness.

Episode 5 summarySmallness was the flip side of largeness

The neutrino is the smallest known mass (less than one ten-millionth of the electron). A straightforward Higgs would require an unnatural Yukawa of \(10^{-13}\). But because \(\nu_R\) is completely neutral, a Majorana mass \(M\) (not tied to \(v\), free to be enormous) is allowed, and the seesaw \(m_\nu\approx m_D^2/M\) makes it naturally light. Read in reverse, \(M\sim v^2/m_\nu\sim10^{14}\) GeV ── the smallest mass points to the largest scale in the universe (grand unification).

This is the answer to "the floor's minimum value" from the particle side (UV). The ways to measure it are four ── oscillation, cosmology, KATRIN, 0νββ ── and the absolute value and Dirac/Majorana are still undecided. Next we close in on the same smallest floor from a completely different direction ── the size of the universe.

This document is Episode 5 of the "Mass That Clicks" series, a reading piece for physics-loving high-schoolers and undergraduates. The outlandish smallness of the neutrino mass, the allowance of a Majorana mass because the right-handed neutrino is a gauge singlet (neutral), the Type I seesaw mechanism \(m_\nu\approx m_D^2/M\), the \(M\sim v^2/m_\nu\sim10^{14\text{–}15}\) GeV (near grand unification) obtained by inverting it, the lower bound from mass differences (\(\sqrt{\Delta m^2_{\rm atm}}\approx0.05\) eV), the cosmological upper bound (\(\Sigma m_\nu\lesssim0.07\text{–}0.12\) eV), KATRIN's direct upper bound, 0νββ and the Majorana nature, and leptogenesis are all established physics / current research topics. The absolute mass, the mass ordering, and Dirac vs. Majorana are unsettled. The numbers are representative values and depend on how \(m_D, M\) are chosen. The figure schematizes the seesaw relation \(\log m_\nu + \log m_N = 2\log m_D\) on a log scale. ── To print, use your browser's "Print" and "Save as PDF" (in the print version, the slider and answers are static and hidden).

Print / save as PDF: Ctrl+P (⌘+P on Mac). On screen, raising the heavy partner M with the slider drops the neutrino on the seesaw. "See the answer" opens each solution.