KAHIBARO
Table of Contents
Tiny masses with big importance
For a long time, neutrinos were treated in the Standard Model as massless particles. That means their mass was taken to be exactly zero. This worked well in the original theory, but later experiments showed that neutrinos can change from one flavor to another while traveling. That phenomenon is called neutrino oscillation, and it is only possible if neutrinos have nonzero mass.
So, neutrino masses are one of the clearest signs that the Standard Model, in its simplest original form, is incomplete.
Neutrino oscillations imply that at least two neutrinos have nonzero mass.
Therefore, the original Standard Model assumption,
$$m_{\nu} = 0,$$
cannot be exactly correct.
Why mass matters
There are three known neutrino flavors, electron neutrino, muon neutrino, and tau neutrino. In weak interactions, neutrinos are produced and detected in these flavor states. But the states that travel through space are states of definite mass, called mass eigenstates.
If flavor states and mass states are not the same, a neutrino created as one flavor can later be measured as another flavor. This is the basic reason oscillations happen.
The important point for this chapter is simple. If all neutrino masses were exactly equal to zero, there would be no oscillation in the observed way. The discovery of oscillations therefore shows that neutrinos have mass, even though those masses are extremely small.
Flavor states and mass states
A flavor neutrino is a mixture of mass neutrinos. Symbolically, one writes
$$|\nu_\alpha\rangle = \sum_i U_{\alpha i} |\nu_i\rangle,$$
where $\alpha = e, \mu, \tau$ labels flavor, $i = 1,2,3$ labels mass states, and $U_{\alpha i}$ are elements of a mixing matrix.
You do not need all the mathematical details here. The key idea is that the neutrino produced in a reaction is not usually a particle with one single definite mass. Instead, it is a combination of several mass states, and those parts evolve differently as the neutrino moves.
That difference reveals the masses indirectly.
What experiments actually measure
Oscillation experiments do not usually measure the individual neutrino masses directly. They measure differences in the squares of the masses:
$$\Delta m_{ij}^2 = m_i^2 - m_j^2.$$
This is enough to produce oscillations. It tells us that the masses are not all the same, but it does not immediately tell us the absolute mass of each neutrino.
A simple summary is shown below.
| Quantity | What it tells us |
|---|---|
| $m_1, m_2, m_3$ | The actual masses of the three mass states |
| $\Delta m_{21}^2$ | Difference between two squared masses |
| $\Delta m_{31}^2$, $\Delta m_{32}^2$ | Other squared-mass differences |
| Mixing angles | How flavor states are built from mass states |
Oscillation experiments measure mass squared differences, not usually the full absolute masses.
Knowing
$$\Delta m_{ij}^2 \neq 0$$
proves neutrinos have mass differences, and therefore at least some neutrino masses are nonzero.
How small are neutrino masses
Neutrino masses are extremely tiny compared with the masses of other known matter particles. For example, the electron mass is about $0.511 \, \text{MeV}/c^2$, while neutrino masses are below the electronvolt scale, less than about $1 \, \text{eV}/c^2$ from present constraints.
This means neutrinos are lighter than electrons by a huge factor.
| Particle | Typical mass scale |
|---|---|
| Electron | $0.511 \, \text{MeV}/c^2$ |
| Neutrino | below about $1 \, \text{eV}/c^2$ |
Since $1 \, \text{MeV} = 10^6 \, \text{eV}$, the electron is more than hundreds of thousands of times heavier than a neutrino.
This striking smallness is one reason neutrino masses are so interesting. The Standard Model does not naturally explain why they are so tiny.
Mass ordering
We know the neutrino masses are different, but the full order is not yet completely settled in the simplest way one might want. There are two main possibilities for the arrangement of the three masses.
In the normal ordering, the third state is the heaviest:
$$m_1 < m_2 < m_3.$$
In the inverted ordering, two heavier states are close together and the third is lighter:
$$m_3 < m_1 < m_2.$$
Oscillation experiments determine the differences in $m^2$, but determining the complete ordering requires more detailed measurements.
Why neutrino mass is a problem for the Standard Model
In the original Standard Model, neutrinos were included only as left-handed neutrinos in weak interactions, and no ordinary mass term was added for them. By contrast, charged leptons and quarks get masses through their coupling to the Higgs field.
To give neutrinos mass in the same simple way, the theory would need additional ingredients, such as right-handed neutrino states. Without such additions, the original Standard Model predicts neutrinos to be massless.
So neutrino masses tell us that new physics must exist beyond the original Standard Model framework.
Neutrino mass is evidence for physics beyond the original Standard Model.
The minimal Standard Model predicts neutrinos are massless, but experiments show this is false.
Two broad ways neutrinos can get mass
There are two important ideas for neutrino mass.
The first is a Dirac mass, similar in spirit to the masses of electrons and quarks. In that case, a neutrino would need both left-handed and right-handed components.
The second is a Majorana mass. In that case, a neutrino could be its own antiparticle. This is a very special possibility because neutrinos are electrically neutral.
These ideas are deep and connect to more advanced particle physics. For beginners, the important message is that neutrino mass may come from a mechanism different from the one used by ordinary charged matter.
| Type of mass | Basic idea |
|---|---|
| Dirac mass | Similar structure to electron or quark mass, needs right-handed neutrino |
| Majorana mass | Neutrino may be its own antiparticle |
The seesaw idea
One famous explanation for tiny neutrino masses is the seesaw mechanism. The idea is that ordinary neutrinos are light because they are connected to very heavy new particles. Roughly, the heavier the new scale, the lighter the observed neutrino masses can become.
A schematic relation is often written as
$$m_\nu \sim \frac{m_D^2}{M},$$
where $m_D$ is a typical ordinary mass scale and $M$ is a very large new mass scale.
This is not a full derivation, but it shows the central idea. A very large denominator can produce a very small neutrino mass.
A common explanation for tiny neutrino masses is the seesaw relation
$$m_\nu \sim \frac{m_D^2}{M}.$$
If $M$ is extremely large, then $m_\nu$ can be extremely small.
How absolute neutrino masses are studied
Since oscillations only give mass differences, other methods are needed to learn the absolute neutrino mass scale.
One method studies the energy spectrum in beta decay. Another uses cosmology, because neutrinos affect the evolution of the universe. A third important search is for neutrinoless double beta decay, which could reveal that neutrinos are Majorana particles.
These approaches are complementary. Together they help answer three big questions. How heavy are neutrinos, what is their ordering, and are neutrinos their own antiparticles.
Why this matters for the universe
Neutrinos are incredibly abundant. Vast numbers were produced in the early universe, and they are still present today. Even though each neutrino is very light, their total contribution can affect cosmic structure formation.
Neutrino masses also connect particle physics to cosmology, and may help us understand why matter exists in such great excess over antimatter, though that topic belongs to broader discussions beyond this chapter.
The main lesson
Neutrino masses are small, but their significance is enormous. They show that the original Standard Model is incomplete. They imply mixing between flavor and mass states. They open the possibility of new particles, new mass mechanisms, and new links between particle physics and the cosmos.
Main takeaway:
Neutrinos have nonzero mass.
This fact is established by neutrino oscillations and is one of the strongest pieces of evidence that the Standard Model must be extended.