Seesaw Models

Prerequisites: Standard Model of Particle Physics, Effective Field Theory. Full path (37 pages)

Intuitive
Concrete
Abstract
Why is it interesting?

Intuitive

Neutrinos have mass, but their masses are tiny: at least a million times smaller than the mass of the electron, the lightest charged particle. In the Standard Model the other matter particles get their masses from their coupling to the Higgs field, and nothing explains why the neutrino couplings would be so small.

A seesaw model explains the small number as a ratio of two scales. Imagine a seesaw on a playground with a very heavy child on one end. The heavier that child, the higher the other end is lifted. In the seesaw mechanism a new very heavy particle sits on one end. It mixes weakly with the ordinary neutrino, and the result is that the ordinary neutrino ends up light. The heavier the new particle, the lighter the neutrino. Roughly, the neutrino mass is the square of the ordinary mass scale divided by the heavy mass scale.

If the ordinary mass scale is similar to the masses of the other Standard Model particles, about 100 GeV, then the observed neutrino masses point to heavy particles with masses around a hundred thousand billion GeV. This is far beyond what any collider can produce, but not far below the scale where the forces might unify in grand unified theories. A consequence of most seesaw models is that neutrinos are their own antiparticles (Majorana particles) and that lepton number is not conserved. Experiments that search for neutrinoless double beta decay test this.

Concrete

Seesaw scenarios can be understood systematically by looking at the Weinberg operator. The Weinberg operator $\frac{1}{\Lambda} LL \phi \phi$ is a (non-renormalizable) dimension-5 operator that yields nonzero masses for the neutrinos. Since it is an dimension-5 operator, it is suppressed by some large mass scale $\Lambda$ . The origin of the Weinberg operator can be understood in terms of concrete seesaw scenarios (UV-completions).

There are three seesaw scenarios that yield the Weinberg operator in the low-energy limit when we integrate out all heavy fields and thus use an effective field theory approach. In particular, we can distinguish different scenarios depending on how we understand the contractions of our $SU(2)_L$ multiplets:

In these scenarios, $N_R$ denotes a heavy right-handed ( $SU(2)_L$ singlet) neutrino, $\Delta^0$ the neutral component of an $SU(2)_L$ scalar triplet and $\Sigma_R^0$ , the neutral component of a fermionic $SU(2)_L$ triplet.

Great reviews are:

Low energy effects of neutrino masses by A. Abada, C.Biggio, F. Bonnet, M.B. Gavela, T. Hambye
The Effective Lagrangian for the Seesaw Model of Neutrino Mass and Leptogenesis by A. Broncano, M.B. Gavela, E. Jenkins

Abstract

The Weinberg operator. Treat the Standard Model as an effective field theory below some scale $\Lambda$ . With the Standard Model field content, the only gauge invariant operator of mass dimension 5 is, up to flavour indices and hermitian conjugation,

$\mathcal{L}_5 = \frac{c_{\alpha\beta}}{\Lambda} \left(\overline{L^c_\alpha}\, \tilde\phi^*\right)\left(\tilde\phi^\dagger L_\beta\right) + \text{h.c.}, \qquad \tilde\phi = i\sigma_2 \phi^* ,$

with $c_{\alpha\beta}$ a complex symmetric matrix in flavour space. It violates lepton number by two units. After electroweak symmetry breaking, $\langle\phi\rangle = (0, v/\sqrt{2})^T$ with $v \approx 246$ GeV, it produces a Majorana mass matrix for the left-handed neutrinos of size $m_\nu \sim c\, v^2/\Lambda$ (the numerical factor depends on the normalisation of $c$ ). The mass matrix is complex symmetric and is brought to diagonal form by a unitary matrix, $m_\nu = U^* \text{diag}(m_1,m_2,m_3)\, U^\dagger$ , where $U$ is the PMNS matrix in the basis where the charged lepton masses are diagonal.

Tree-level completions. At tree level the operator $LL\phi\phi$ can be generated by exchanging a single heavy field between two pairs of the four external legs. Each pair transforms as $2 \otimes 2 = 1 \oplus 3$ under $SU(2)_L$ , and the heavy field must carry the quantum numbers of the pair. This gives exactly three options:

Type I: a fermion singlet $N_R \sim (1,1)_0$ couples to $\overline{L}\tilde\phi$ .
Type II: a scalar triplet $\Delta \sim (1,3)_1$ couples to the pair $LL$ and to the pair $\phi\phi$ .
Type III: a fermion triplet $\Sigma \sim (1,3)_0$ couples to $\overline{L}\tilde\phi$ in the triplet channel.

Type I in detail. With $\mathcal{L} \supset -\overline{L}\, \tilde\phi\, y\, N_R - \frac{1}{2}\overline{N_R^c} M N_R + \text{h.c.}$ and $m_D = y v/\sqrt{2}$ , the neutral fermion mass matrix in the basis $(\nu_L, N_R^c)$ is, up to an overall complex conjugation that depends on how the Majorana mass term is written, the complex symmetric matrix

$\mathcal{M} = \begin{pmatrix} 0 & m_D \\ m_D^T & M \end{pmatrix}.$

If all eigenvalues of $M$ are much larger than the entries of $m_D$ , a unitary transformation that is block diagonal up to corrections of order $\theta = m_D M^{-1}$ separates the light and heavy states, with

$m_\nu \simeq - m_D M^{-1} m_D^T , \qquad M_{\text{heavy}} \simeq M ,$

up to corrections of relative order $\theta^2$ . This is the seesaw: $m_\nu$ falls as $M$ grows. For $m_D \sim 100$ GeV and $m_\nu \sim 0.05$ eV one needs $M \sim 10^{14}$ GeV. The light mass eigenstates contain admixtures of $N_R^c$ of order $\theta$ , which makes the $3\times 3$ mixing matrix of the light states slightly non-unitary; constraints on this non-unitarity are among the low-energy effects discussed in the reviews listed under Concrete. Given the light masses, the PMNS matrix and $M$ , the Yukawa matrix is not fixed: the Casas-Ibarra parametrisation writes the remaining freedom as an arbitrary complex orthogonal matrix.

Why is it interesting?

Seesaw models are extensions of the Standard Model of particle physics that aim to explain why neutrino masses are so tiny.

Seesaw Models

Intuitive

Concrete

Abstract

Why is it interesting?

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