Neutrino Masses and the Seesaw

The Standard Model as originally written has massless neutrinos — there is no right-handed neutrino and no gauge-invariant renormalizable mass term for the left-handed one. Yet neutrino oscillations prove neutrinos have (tiny) masses. This is the cleanest evidence for physics beyond the Standard Model, and this page works through how masses can be added — Dirac vs. Majorana, the Weinberg operator, and the seesaw mechanism — as an application of the EFT viewpoint.

Conventions: $\hslash =c=1$; $\Phi$ = Higgs doublet, $v=246\mathrm{GeV}$.

Why the SM has massless neutrinos

A Dirac mass $m\bar{\psi }\psi =m({\bar{\psi }}_L{\psi }_R+\text{h.c.})$ requires both chiralities. The SM contains ${\nu }_L$ (inside the lepton doublet) but no ${\nu }_R$, so the Higgs–Yukawa route that gives every other fermion its mass has nothing to pair ${\nu }_L$ with. Nor is a Majorana mass $m\overline{{\nu }_L^c}{\nu }_L$ allowed: it carries weak isospin and hypercharge, violating gauge invariance. So at the renormalizable level neutrinos are exactly massless — a genuine SM prediction, now falsified.

The experimental fact: oscillations

Neutrinos produced in one flavor eigenstate (${\nu }_e,{\nu }_{\mu },{\nu }_{\tau }$) are detected as another — neutrino oscillation, observed in solar, atmospheric, reactor, and accelerator experiments (Super-Kamiokande, SNO; Nobel Prize 2015). Oscillation occurs precisely because flavor eigenstates are superpositions of mass eigenstates with different masses, related by the PMNS matrix (the lepton analogue of CKM). The oscillation probability depends on mass-squared differences:

$$P({\nu }_{\alpha }\to {\nu }_{\beta })\sim {\sin }^2\left(\frac{\Delta m_{ij}^{2}L}{4E}\right),\Delta m_{21}^2\approx 7.5\times 10^{-5}{\mathrm{eV}}^2,|\Delta m_{31}^2|\approx 2.5\times 10^{-3}{\mathrm{eV}}^2.$$

Oscillations measure only differences, not absolute masses, but they prove at least two neutrinos are massive, with $m_{\nu }\lesssim 0.1\mathrm{eV}$ — at least $10^9$ times lighter than the electron.

Two ways to add mass

Dirac neutrinos

Add three right-handed neutrinos ${\nu }_R$ as gauge singlets. Then the usual Yukawa coupling $y_{\nu }{\bar{L}}_L\tilde{\Phi }{\nu }_R$ gives a Dirac mass $m_{\nu }=y_{\nu }v/\sqrt{2}$. This works, but requires the Yukawa to be absurdly small, $y_{\nu }\sim 10^{-12}$ — twelve orders of magnitude below the top Yukawa — with no explanation for the hierarchy. It also leaves lepton number conserved and ${\nu }_R$ otherwise undetectable (a "sterile" state).

Majorana neutrinos and the Weinberg operator

Because ${\nu }_R$ is a gauge singlet, it can have a Majorana mass $M\overline{{\nu }_R^c}{\nu }_R$ that violates lepton number — and $M$ is not tied to the electroweak scale, so it can be huge. Integrating out this heavy ${\nu }_R$ (the EFT procedure) leaves a single dimension-5 operator on the SM fields — the unique non-renormalizable operator allowed by the SM gauge symmetry, the Weinberg operator:

$$\boxed{{\mathcal{L}}_5=\frac{c}{\Lambda }\left({\bar{L}}_L^c{\Phi }^{\ast }\right)\left({\Phi }^{†}{L}_L\right)\stackrel{⟨\Phi ⟩}{\to }{m}_{\nu }\sim \frac{v^2}{\Lambda }.}$$

This is the lowest-dimension window on new physics above the SM — its very existence signals a scale $\Lambda$ where lepton number is broken.

The seesaw mechanism

The seesaw explains the tininess of $m_{\nu }$ naturally. With both a Dirac mass $m_D=y_{\nu }v/\sqrt{2}$ (electroweak scale) and a large Majorana mass $M\gg m_D$ for ${\nu }_R$, the neutral-lepton mass matrix

$$\left(\begin{array}{cc}0& m_D\\ m_D& M\end{array}\right)$$

has eigenvalues $\approx M$ (a heavy, mostly-sterile state) and

$$\boxed{m_{\nu }\approx \frac{m_D^2}{M}}$$

(a light, mostly-active neutrino). The light mass is suppressed by the heavy scale — the "seesaw": as $M$ goes up, $m_{\nu }$ goes down. Taking $m_D$ at the electroweak scale and $m_{\nu }\sim 0.05\mathrm{eV}$ points to $M\sim 10^{14}$–$10^{15}\mathrm{GeV}$ — tantalizingly near the grand-unification scale, suggesting neutrino masses are a low-energy shadow of GUT-scale physics. This connection, and the lepton-number violation it entails, also underlies leptogenesis as a route to the matter–antimatter asymmetry.

Consequences and open questions

• Dirac or Majorana? — decided by neutrinoless double-beta decay ($0\nu \beta \beta$): observation would prove neutrinos are their own antiparticles (Majorana) and lepton number is violated. Not yet seen.
• Absolute mass scale — oscillations give only $\Delta m^2$; the absolute scale comes from $\beta$-decay endpoints (KATRIN, $m_{\nu }<0.8\mathrm{eV}$) and cosmology ($\sum m_{\nu }\lesssim 0.1\mathrm{eV}$).
• Mass ordering and the PMNS CP phase — active experimental targets (DUNE, Hyper-K), the leptonic analogue of CKM CP violation.

Summary

• The SM predicts massless neutrinos; oscillations prove them massive — the cleanest BSM evidence.
• Mass requires new fields: Dirac (${\nu }_R$ + tiny Yukawa) or Majorana (the dimension-5 Weinberg operator).
• The seesaw $m_{\nu }\approx m_D^2/M$ naturally explains the tiny masses via a heavy scale $M\sim 10^{14}\mathrm{GeV}$ — a link to grand unification.

Where this fits

• The theory: Standard Model and its neutrino-mass caveat.
• The EFT logic: effective field theory (Weinberg operator, integrating out ${\nu }_R$).
• The flavor companion: Flavor, CP Violation and Precision Fits.

References

• Weinberg, Phys. Rev. Lett. 43, 1566 (1979) (dimension-5 operator).
• Minkowski (1977); Gell-Mann, Ramond & Slansky; Yanagida; Mohapatra & Senjanović (seesaw).
• Particle Data Group, Review of Particle Physics, "Neutrino masses, mixing".