---
address: |
    Department of Physics, University of California\
    Riverside, CA 92521, USA
author:
- ERNEST MA
title: |
    MODELS OF NEUTRINO MASS AND INTERACTIONS\
    FOR NEUTRINO OSCILLATIONS
---

\#1\#2\#3\#4[[\#1]{} [**\#2**]{}, \#3 (\#4)]{}

Neutrino Masses
===============

In the minimal standard model, under the gauge group $SU(3)_C \times SU(2)_L 
\times U(1)_Y$, the leptons transform as: $$\left[ \begin{array} {c} \nu_e \\ e \end{array} \right]_L, 
\left[ \begin{array} {c} \nu_\mu \\ \mu \end{array} \right]_L, 
\left[ \begin{array} {c} \nu_\tau \\ \tau \end{array} \right]_L 
\sim (1, 2, -1/2); ~~~ e_R, ~ \mu_R, ~ \tau_R \sim (1, 1, -1).$$ There is also the Higgs scalar doublet $(\phi^+, \phi^0) \sim (1, 2, 1/2)$ whose nonzero vacuum expectation value $\langle \phi^0 \rangle = v$ breaks $SU(2)_L \times U(1)_Y$ to $U(1)_Q$. Whereas charged leptons acquire masses proportional to $v$, the absence of $\nu_R$ implies that $m_{\nu_i} = 0$. If nonzero neutrino masses are desired (which are of course necessary for neutrino oscillations), then we must ask “What is the nature of this mass?" and “What new physics goes with it?"

If $\nu_R$ does not exist, one way to have $m_\nu \neq 0$ is to add a Higgs triplet $(\xi^{++}, \xi^+, \xi^0)$. Each $\nu_L$ then gets a Majorana mass. However, $\langle \xi^0 \rangle$ must be very small, and if the lepton number being carried by $\xi$ is spontaneously violated [@1], the decay of $Z$ to the associated massless Goldstone boson (the triplet Majoron) and its partner would count as two extra neutrinos. Since the effective number of light neutrinos in $Z$ decay is now measured [@2] to be $2.989 \pm 0.012$, the triplet Majoron model is clearly ruled out.

If one $\nu_R \sim (1, 1, 0)$ exists for each $\nu_L$, the most general $2 \times 2$ neutrino mass matrix linking $(\bar \nu_L, \bar \nu_R^c)$ to $(\nu_L^c, \nu_R)$ is given by $${\cal M} = \left[ \begin{array} {c@{\quad}c} m_L & m_D \\ m_D & m_R \end{array} 
\right].$$ If $m_L = 0$ and $m_D << m_R$, we get the famous seesaw mechanism [@3] $$m_\nu \sim {m_D^2 \over m_R}.$$ Here, $\nu_L - \nu_R^c$ mixing is $m_D/m_R$ and $m_R$ is the scale of new physics. In this minimal scenario, new physics enters only through $m_R$, hence there is no other observable effect except for a nonzero $m_\nu$. Actually, $m_D/m_R$ is in principle observable but it is in practice far too small.

In general, the mass matrix of Eq. (2) yields two nondegenerate interacting Majorana neutrinos (unless $m_L = m_R = 0$ is maintained exactly). If both eigenvalues are small, the effective number of neutrinos counted in Big Bang Nucleosynthesis may be as high as six, instead of the usual three, depending on the mass splitting and mixing in each case [@4].

The smallness of neutrino masses may be indicative of their radiative origin. Many papers have been written on the subject. For a brief review, see Ref. 5. There are three one-loop mechanisms: the exchange of two scalar bosons with one fermion mass insertion; the exchange of one scalar boson with three fermion mass insertions; and the coupling to a scalar boson which gets a radiative vacuum expectation value through a fermion loop with five mass insertions. A prime example of the first mechanism is the Zee model [@6]. Here the minimal standard model is extended to include a charged scalar singlet $\chi^+$ and a second scalar doublet $(\eta^+, \eta^0)$. We then have the coupling $$f_{ij} \chi^+ (\nu_i l_j - l_i \nu_j),$$ which by itself would require $\chi^+$ to have lepton number $-2$. However, this model also allows the cubic scalar coupling $$\chi^- (\phi^+ \eta^0 - \phi^0 \eta^+),$$ hence lepton number is broken explicitly. A radiative Majorana mass matrix is thus obtained through the exchange and mixing of $\chi^+$ and the physical linear combination of $\phi^+$ and $\eta^+$. Since $f_{ij}$ of Eq. (4) is zero for $i = j$ and $\phi^+$ couples $\nu_i$ to $l_i$ with strength proportional to $m_{l_i}$ which is also the one fermion mass insertion required, the $3 \times 3$ neutrino mass matrix for $\nu_e$, $\nu_\mu$ and $\nu_\tau$ is of the form $${\cal M}_\nu \propto \left[ \begin{array} {c@{\quad}c@{\quad}c} 0 & 0 & 
f_{e \tau} m_\tau^2 \\ 0 & 0 & f_{\mu \tau} m_\tau^2 \\ f_{e \tau} m_\tau^2 
& f_{\mu \tau} m_\tau^2 & 0 \end{array} \right] + {\cal O} (m_\mu^2).$$ This means that $\nu_\tau$ is almost degenerate with a linear combination of $\nu_\mu$ and $\nu_e$ in this model. This may have a practical application in present neutrino-oscillation phenomenology [@7].

There are also three two-loop mechanisms: the exchange of three scalar bosons which are tied together by a cubic coupling; the exchange of two $W$ bosons; and the exchange of $W_L$ and $W_R$ which mix at the one-loop level. The second mechanism [@8] is unique in that it requires only one additional $\nu_R$ beyond the standard model. In this specific case, one $\nu_L$ gets a seesaw mass and the other two get two-loop masses proportional to this mass and as functions of the charged-lepton masses with double GIM suppression [@9]. A detailed analytical and numerical study of this mechanism has been made [@10].

Finally let me return to the triplet-Higgs mechanism. If lepton number is violated explicitly by the coupling of $\xi$ to the scalar doublet $\phi$, then one may let $\xi$ be very heavy and integrate it out to obtain the following effective nonrenormalizable interaction: $${1 \over M} [\phi^0 \phi^0 \nu_i \nu_j + \phi^+ \phi^0 (\nu_i l_j + l_i \nu_j) 
+ \phi^+ \phi^+ l_i l_j] + h.c.$$ For $M \sim 10^{13}$ GeV, one gets $m_\nu \sim$ few eV. This is the most economical solution and could also be a realistic model of leptogenesis [@11] in the early universe which gets converted at the electroweak phase transition into the present observed baryon asymmetry.

Neutrino Oscillations
=====================

Present experimental evidence for neutrino oscillations [@12] includes the solar $\nu_e$ deficit which requires $\Delta m^2$ of around $10^{-5}$ eV$^2$ for the MSW explanation or $10^{-10}$ eV$^2$ for the vacuum-oscillation solution, the atmospheric neutrino deficit in the ratio $\nu_\mu + \bar \nu_\mu / \nu_e + \bar \nu_e$ which implies a $\Delta m^2$ of around $10^{-2}$ eV$^2$, and the LSND experiment which indicates a $\Delta m^2$ of around 1 eV$^2$. Three different $\Delta m^2$ necessitate four neutrinos, but the invisible width of the $Z$ boson as well as Big Bang Nucleosynthesis allow only three. If all of the above-mentioned experiments are interpreted correctly as due to neutrino oscillations, we are faced with a theoretical challenge in trying to understand how three can equal four. I will focus on addressing this issue rather than trying to review the many theoretical models for the three known neutrinos.

Three Neutrinos and One Light Singlet
=====================================

One possibility is that there is a light singlet neutrino $\nu_S$ in addition to the three known doublet neutrinos $\nu_e$, $\nu_\mu$, and $\nu_\tau$. This is necessary so that it is not counted in the effective number of neutrinos in $Z$ decay [@2]. On the other hand, it has to mix with the doublet neutrinos for it to be relevant to oscillation experiments. Hence it is also contrained [@4] by Big Bang Nucleosynthesis. Using all available data, a model-independent analysis [@13] shows that the $4 \times 4$ neutrino mass matrix must separate approximately into two blocks: one for $\nu_e - \nu_S$ and the other for $\nu_\mu - \nu_\tau$, the latter with large mixing.

An example of a specific model of this kind already exists [@14]. The neutrino interaction eigenstates are related to the mass eigenstates by $$\left[ \begin{array} {c} \nu_S \\ \nu_e \\ \nu_\mu \\ \nu_\tau \end{array} 
\right] = \left[ \begin{array} {c@{\quad}c@{\quad}c@{\quad}c} -s & c & 
s''/\sqrt 2 & s''/\sqrt 2 \\ c & s & -s'/\sqrt 2 & s'/\sqrt 2 \\ -s' & 0 & 
-1/\sqrt 2 & 1/\sqrt 2 \\ 0 & -s'' & 1/\sqrt 2 & 1/\sqrt 2 \end{array} 
\right] \left[ \begin{array} {c} \nu_1 \\ \nu_2 \\ \nu_3 \\ \nu_4 \end{array} 
\right],$$ where $m_1 = 0$, $m_2 \sim 2.5 \times 10^{-3}$ eV, $m_3 \sim m_4 \sim 2.5$ eV, with $\Delta m_{34}^2 \sim 1.8 \times 10^{-2}$ eV$^2$; $s \sim s' \sim 
0.04$, but $s''$ is undetermined. Note that $m_{\nu_e} < m_{\nu_S}$ is necessary for the MSW solution [@15] of the solar neutrino deficit. Note also that $\nu_\mu$ and $\nu_\tau$ are pseudo-Dirac partners, hence the mixing angle for atmospheric neutrino oscillations is 45 degrees.

What is the nature of this light singlet? and how does it mix with the usual neutrinos? There have been some discussions on these questions in the past two or three years. One idea [@16] is that it is the fermion partner of the massless Goldstone boson of a sponatneously broken global symmetry, such as lepton number (hence a Majorino) in supersymmetry. Another [@17] is that it is the fermion partner of a scalar field corresponding to a flat direction (hence a modulino) in the supersymmetric Higgs potential. If the standard model is extended to include a mirror $[SU(2) \times U(1)]'$ sector, then $\nu_S$ may be identified as a mirror neutrino, either in a theory where the mirror symmetry is broken [@18] or one where it is exact [@19]. In the latter case, maximal mixing between the three known neutrinos and their mirror counterparts would occur and Big Bang Nucleosynthesis would count six neutrinos under normal conditions.

Both questions can be answered naturally in a model [@20] based on $E_6$ inspired by superstring theory. In the fundamental [**27**]{} representation of $E_6$, outside the 15 fields belonging to the minimal standard model, there are 2 neutral singlets. One ($N$) is identifiable with the right-handed neutrino because it is a member of the [**16**]{} representation of $SO(10)$; the other ($S$) is a singlet also under $SO(10)$. In the reduction of $E_6$ to $SU(3)_C \times SU(2)_L \times U(1)_Y$, an extra U(1) gauge factor may survive down to the TeV energy scale. It could be chosen such that $N$ is trivial under it, but $S$ is not. This means that $N$ is allowed to have a large Majorana mass so that the usual seesaw mechanism works for the three doublet neutrinos. At the same time, $S$ is protected from having a mass by the extra U(1) gauge symmetry, which I call $U(1)_N$. However, it does acquire a small mass from an analog of the usual seesaw mechanism because it can couple to doublet neutral fermions which are present in the [**27**]{} of $E_6$ outside the [**16**]{} of $SO(10)$. Renaming $S$ as $\nu_S$, the $3 \times 3$ mass matrix spanning $\nu_S$, $\nu_E$, and $N_E^c$ is given by $${\cal M} = \left[ \begin{array} {c@{\quad}c@{\quad}c} 0 & m_1 & m_2 \\ m_1 & 
0 & m_E \\ m_2 & m_E & 0 \end{array} \right].$$ Hence $m_{\nu_S} \sim 2 m_1 m_2/m_E$, which is a singlet-doublet seesaw rather than the usual doublet-singlet seesaw. Furthermore, the mixing of $\nu_S$ with the doublet neutrinos is also possible through these extra doublet neutral fermions. The spontaneous breaking of $U(1)_N$ is not possible without also breaking the supersymmetry, hence both are assumed to occur at the TeV energy scale, resulting in a rich $Z'$ and Higgs phenomenology [@21].

Three Neutrinos and One Anomalous Interaction
=============================================

If one insists on keeping only the usual three neutrinos and yet try to accommodate all present data, how far can one go? It has been pointed out by many authors [@22] that both solar and LSND data can be explained, as well as most of the atmospheric data except for the zenith-angle dependence. It is thus worthwhile to consider the following scenario [@23] whereby a possible anomalously large $\nu_\tau$-quark interaction may mimic the observed zenith-angle dependence of the atmospheric data. Consider first the following approximate mass eigenstates: $$\begin{aligned}
\nu_1 &\sim& \nu_e ~~~ {\rm with} ~ m_1 \sim 0, \\ \nu_2 &\sim& c_0 \nu_\mu 
+ s_0 \nu_\tau ~~~ {\rm with} ~ m_2 \sim 10^{-2} ~{\rm eV}, \\ \nu_3 &\sim& 
-s_0 \nu_\mu + c_0 \nu_\tau ~~~ {\rm with} ~ m_3 \sim 0.5 ~{\rm eV},\end{aligned}$$ where $c_0 \equiv \cos \theta_0$, $s_0 \equiv \sin \theta_0$, and $\theta_0$ is not small. Allow $\nu_1$ to mix with $\nu_3$ with a small angle $\theta '$ and the new $\nu_1$ to mix with $\nu_2$ with a small angle $\theta$, then the LSND data can be explained with $\Delta m^2 \sim 0.25$ eV$^2$ and $2 s_0 s' c' \sim 0.19$ and the solar data can be understood as follows.

Consider the basis $\nu_e$ and $\nu_\alpha \equiv c_0 \nu_\mu + s_0 \nu_\tau$. Then $$-i {d \over {dt}} |\nu \rangle_{e,\alpha} = \left( p + {{\cal M}^2 \over {2p}} 
\right) |\nu \rangle_{e,\alpha},$$ where $${\cal M}^2 = {\cal U} \left[ \begin{array} {c@{\quad}c} 0 & 0 \\ 0 & m_2^2 
\end{array} \right] {\cal U}^\dagger + \left[ \begin{array} {c@{\quad}c} 
A+B & 0 \\ 0 & B+C \end{array} \right].$$ In the above, $A$ comes from the charged-current interaction of $\nu_e$ with $e$, $B$ from the neutral-cuurent interaction of $\nu_e$ and $\nu_\alpha$ with the quarks and electrons, and $C$ from the assumed anomalous $\nu_\tau$-quark interaction. Let $${\cal U} = \left( \begin{array} {c@{\quad}c} c & s \\ -s & c \end{array} 
\right),$$ then the resonance condition is $$m_2^2 \cos 2 \theta - A + C = 0,$$ where [@24] $$A - C = 2 \sqrt 2 G_F (N_e - s_0^2 \epsilon'_q N_q) p.$$ In order to have a large $\epsilon'_q$ and yet satisfy the resonance condition for solar-neutrino flavor conversion, $m_2$ should be larger than its canonical value of $2.5 \times 10^{-3}$ eV, and $\epsilon'_q$ should be negative. \[If $\epsilon'_q$ comes from $R$-parity violating squark exchange, then it must be positive, in which case an inverted mass hierarchy, [*i.e.*]{} $m_2 < m_1$ would be needed. If it comes from vector exchange, it may be of either sign.\] Assuming as a crude approximation that $N_q \simeq 4 N_e$ in the sun, the usual MSW solution with $\Delta m^2 = 6 \times 10^{-6}$ eV$^2$ is reproduced here with $$s_0^2 \epsilon'_q \simeq -3.92 = -4.17 (m_2^2/10^{-4}{\rm eV}^2) + 0.25.$$ The seemingly arbitrary choice of $\Delta m_{21}^2 \sim 10^{-4}$ eV$^2$ is now sen as a reasonable value so that $\epsilon'_q$ can be large enough to be relevant for the following discussion on the atmospheric neutrino data.

Atmospheric neutrino oscillations occur between $\nu_\mu$ and $\nu_\tau$ in this model with $\Delta m^2_{32} \sim 0.25$ eV$^2$, the same as for the LSND data, but now it is large relative to the $E/L$ ratio of the experiment, hence the factor $\cos \Delta m^2 (L/2E)$ washes out and $$P_0 (\nu_\mu \rightarrow \nu_\mu) = 1 - {1 \over 2} \sin^2 2 \theta_0 
\simeq 0.66 ~~ {\rm for} ~~ s_0 \simeq 0.47.$$ In the standard model, this would hold for all zenith angles. Hence it cannot explain the present experimental evidence that the depletion is more severe for neutrinos coming upward to the detector through the earth than for neutrinos coming downward through only the atmosphere. This zenith-angle dependence appears also mostly in the multi-GeV data and not in the sub-GeV data. It is this trend which determines $\Delta m^2$ to be around $10^{-2}$ eV$^2$ in this case. As shown below, the hypothesis that $\nu_\tau$ has anomalously large interactions with quarks will mimic this zenith-angle dependence even though $\Delta m^2$ is chosen to be much larger, [*i.e.*]{} 0.25 eV$^2$.

Consider the basis $\nu_\mu$ and $\nu_\tau$. Then the analog of Eq. (13) holds with Eq. (14) replaced by $${\cal M}^2 = {\cal U}_0 \left[ \begin{array} {c@{\quad}c} 0 & 0 \\ 0 & m_3^2 
\end{array} \right] {\cal U}_0^\dagger + \left[ \begin{array} {c@{\quad}c} 
B & 0 \\ 0 & B + C \end{array} \right].$$ The resonance condition is then $$m_3^2 \cos 2 \theta_0 + C = 0,$$ where $N_q$ in $C$ now refers to the quark number density inside the earth and the factor $s_0^2$ in Eq. (17) is not there. If $C$ is large enough, the probability $P_0$ would not be the same as the one in matter. Using the estimate $N_q \sim 9 \times 10^{30}$ m$^{-3}$ and defining $$X \equiv \epsilon'_q E_\nu/(10 ~{\rm GeV}),$$ the effective mixing angles in matter are given by $$\begin{aligned}
\tan 2 \theta_m^E &=& {{\sin 2 \theta_0} \over {\cos 2 \theta_0 + 0.091 X}} 
~~ {\rm for} ~ \nu, \\ \tan 2 \bar \theta_m^E &=& {{\sin 2 \theta_0} \over 
{\cos 2 \theta_0 - 0.091 X}} ~~ {\rm for} ~ \bar \nu.\end{aligned}$$ For sub-GeV neutrinos, $X$ is small so matter effects are not very important. For multi-GeV neutrinos, $X$ may be large enough to satisfy the resonance condition of Eq. (21). Assuming adiabaticity, the neutrino and antineutrino survival probabilities are given by $$\begin{aligned}
P(\nu_\mu \rightarrow \nu_\mu) &=& {1 \over 2} (1 + \cos 2 \theta_0 \cos 2 
\theta_m^E), \\ \bar P(\bar \nu_\mu \rightarrow \bar \nu_\mu) &=& {1 \over 2} 
(1 + \cos 2 \theta_0 \cos 2 \bar \theta_m^E).\end{aligned}$$ Since $\sigma_\nu \simeq 3 \sigma_{\bar \nu}$, the observed ratio of $\nu + \bar \nu$ events is then $$P_m \simeq {{3 r P + \bar P} \over {3 r + 1}},$$ where $r$ is the ratio of the $\nu_\mu$ to $\bar \nu_\mu$ flux in the upper atmosphere. The atmospheric data are then interpreted as follows. For neutrinos coming down through only the atmosphere, $P_0 = 0.66$ applies. For neutrinos coming up through the earth, $P_m \simeq P_0 \simeq 
0.66$ as well for the sub-GeV data. However, for the multi-GeV data, if $X = -15$, then $P = 0.31$ and $\bar P = 0.76$, hence $P_m$ is lowered to 0.39 if $r = 1.5$ or 0.42 if $r = 1.0$. The apparent zenith-angle dependence of the data may be explained.

Conclusion and Outlook
======================

If all present experimental indications of neutrino oscillations turn out to be correct, then either there is at least one sterile neutrino beyond the usual $\nu_e$, $\nu_\mu$, and $\nu_\tau$, or there is an anomalously large $\nu_\tau$-quark interaction. The latter can be tested at the forthcoming Sudbury Neutrino Observatory (SNO) which has the capability of neutral-current detection. The predicted $\Delta m^2$ of 0.25 eV$^2$ in $\nu_\mu$ to $\nu_e$ and $\nu_\tau$ oscillations will also be tested at the long-baseline neutrino experiments such as Fermilab to Soudan 2 (MINOS), KEK to Super-Kamiokande (K2K), and CERN to Gran Sasso.

More immediately, the new data from Super-Kamiokande, Soudan 2, and MACRO on $\nu_\mu + \bar \nu_\mu$ events through the earth should be analyzed for such an effect. For a zenith angle near zero, the $\Delta m^2 \sim 10^{-2}$ eV$^2$ oscillation scenario should have $R \sim 1$, whereas the $\Delta m^2 
\sim 0.25$ eV$^2$ oscillation scenario (with anomalous interaction) would have $R = P_0 \sim 0.66$. Furthermore, if $\nu$ and $\bar \nu$ can be distinguished (as proposed in the HANUL experiment), then to the extent that $CP$ is conserved, matter effects can be isolated.

Neutrino physics is on the verge of major breakthroughs. New experiments in the next several years will be decisive in leading us forward in its theoretical understanding, and may even discover radically new physics beyond the standard model.

Acknowledgments {#acknowledgments .unnumbered}
===============

I thank Biswarup Mukhopadhyaya and all the other organizers for their great hospitality and a stimulating workshop. This work was supported in part by the U. S. Department of Energy under Grant No. DE-FG03-94ER40837.

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