Models of neutrino mass

Almost all extensions of the standard model lead to nonzero neutrino masses at some level, often in the observable ($10^{-5}-10$ eV) range. One should therefore view neutrino mass as top-down physics. e.g., one can hope to compare the predictions of a specific superstring, GUT, or other model with the observed spectra, but it is hard to work backwards and infer the underlying theory from the observations. There are large numbers of models of neutrino mass. Some of the principle classes and general issues are:

• A triplet majorana mass $m_T$ can be generated by the VEV $v_T$ of a Higgs triplet field. Then, $m_T = h_T v_T$, where $h_T$ is the relevant Yukawa coupling. Small values of $m_T$ could be due to a small scale $v_T$, although that introduces a new hierarchy problem. The simplest implementation is the Gelmini-Roncadelli (GR) model [3], in which lepton number is spontaneously broken by $v_T$. The original GR model is now excluded by the LEP data on the $Z$ width.

• A very different class of models are those in which the neutrino masses are zero at the tree level (typically because no sterile neutrino or elementary Higgs triplets are introduced), but only generated by loops [4], i.e., radiative generation. Such models generally require the ad hoc introduction of new scalar particles at the TeV scale with nonstandard electroweak quantum numbers and lepton number-violating couplings. They have also been introduced in an attempt to generate large electric or magnetic dipole moments.

• In the seesaw models [5], a small Majorana mass is induced by mixing between an active neutrino and a very heavy Majorana sterile neutrino $M_N$. The light (essentially active) state has a naturally small mass
  

  $$m_\nu \sim \frac{m_D^2}{M_N} \ll m_D.$$ (7)

  
  
  There are literally hundreds of seesaw models, which differ in the scale $M_N$ for the heavy neutrino (ranging from the TeV scale to grand unification scale), the Dirac mass $m_D$ which connects the ordinary and sterile states and induces the mixing (e.g., $m_D \sim m_u$ in most grand unified theory (GUT) models, or $\sim m_e$ in left-right symmetric models), the patterns of $m_D$ and $M_N$ in three family generalizations, etc.

• There are many mechanisms for neutrino mass generation in supersymmetric models with $R$ parity breaking [6]. Breaking induced by bilinear terms connecting Higgs and lepton doublets in the superpotential or by the expectation values of scalar neutrinos leads to seesaw-type mixings of neutrinos with heavy neutralinos. Cubic $R$ parity violating terms lead to loop-induced neutrino masses.

• Grand unified theories are excellent candidates for seesaw models, with $M_N$ at or a few orders of magnitude below the unification scale. In addition to the gauge and Yukawa unification, realistic models for the quark and charged-lepton masses generally involve complicated Higgs structures and additional family symmetries to constrain the Yukawa couplings (fermion textures), often leading to interesting predictions for the neutrino spectrum. Detailed $SO(10)$ models were described by Raby [7]. The most predictive versions are excluded, while generalizations have less predictive power. It is difficult to embed such structures into superstring models.

• Heterotic superstring models often predict the existence of higher-dimensional (nonrenormalizable) operators (NRO) such as
  

  $$-L_{\rm eff} = \bar{\psi}_L H \left(\frac{S}{\mbox{$M_{\rm str}$}}\right)^P \psi_R + H.C.,$$ (8)

  
  
  where $H$ is the ordinary Higgs doublet, $S$ is a new scalar field which is a singlet under the standard model gauge group, and $\mbox{$M_{\rm str}$}\sim 10^{18}$ GeV is the string scale. In many cases $S$ will acquire an intermediate scale VEV (e.g., $10^{12}$ GeV), leading to an effective Yukawa coupling
  

  $$h_{\rm eff} \sim v \left(\frac{\langle S \rangle}{\mbox{$M_{\rm str}$}}\right)^P \ll v.$$ (9)

  
  
  Depending on the dimensions $P$ of the various operators and on the scale $\langle S \rangle$, it may be possible to generate an interesting hierarchy for the quark and charged lepton masses and to obtain naturally small Dirac neutrino masses [8]. Similarly, one may obtain triplet and singlet Majorana neutrino masses, $m_T$ and $m_N$ by analogous higher-dimensional operators. The former are generally too small to be relevant. Depending on the operators the latter ($m_N$) may be absent, implying small Dirac masses; small, leading to the possibility of significant mixing between ordinary and sterile neutrinos [9]; or large, allowing a conventional seesaw.

• There are many models in which large extra dimensions affect the neutrino masses, generally involving singlet neutrinos propagating in the bulk and/or coupling with Kaluza-Klein excitations. These couplings could even lead to oscillations without neutrino masses [10,11].

• Realistic models for three or more neutrinos often involve fermion textures [2,12,13], i.e., particular forms for the fermion mass matrices involving zeroes or hierarchies of non-zero entries. Such textures are usually assumed to be due to broken family symmetries. However, they could also be associated with unknown dynamics, such as string selection rules in an underlying theory.

• It is often assumed that small neutrino masses are most likely Majorana. This is expected in the simplest seesaw models. However, the possibility of small Dirac masses should not be excluded. These can easily come about in (string-motivated) intermediate scale models, and possibly in loop-induced scenarios.

• Mixed models, in which comparable Majorana and Dirac mass terms are both present, will be further discussed in the next section.