The solar neutrino problem
Introduction
First observed by Ray Davis in 1968, the solar neutrino problem is the apparent discrepancy
between solar neutrino flux measured in experiments versus the theoretical calculations.
Many experiments were constructed after this to identify the cause. The solar neutrino
problem was widely accepted to be solved in 2001 after scientists at the Sudbury Neutrino
Observatory first published their results. In this paper, we will journey through the mystery
that puzzled scientists for over 30 years, interpreting experimental and theoretical
publications, in addition to explaining and analysing proposed solutions to the solar neutrino
problem.
Solar neutrinos
Nicknamed “ghost particles”, neutrinos are possibly the most unsociable particles.
Interacting only via the weak force and gravity, they pass straight through most regular
matter. Neutrinos have a very small mass, at least one million times smaller than the mass
of an electron [1], however were once thought to be massless. Solar neutrinos are produced
in the sun through complex nuclear fusion reactions in the sun’s core, most are produced
during the proton-proton reaction. Most neutrinos passing through the Earth originate from
the sun, totalling 70 billion solar neutrinos penetrating one centimetre squared every second
[2]. Neutrinos come in 3 flavours: electron, muon, and tau. However, all neutrinos produced
by the sun are electron neutrinos [3].
The Homestake experiment
In 1967 the Homestake experiment commenced. Its goal was to measure how many
neutrinos are produced by the sun. A 380 cubic metre tank filled with tetrachloroethylene
was constructed 1478 metres underground to prevent cosmic rays from interfering with the
detection of solar neutrinos [4]. Tetrachloroethylene contains chlorine which interacts with an
electron neutrino according to:
νe + 37Cl ⟶ 37Ar + e-.
Then, the argon could be collected and counted, which relates directly to the number of
electron neutrinos captured in the tank. Previously, in 1964 Bahcall had calculated the
expected count to be 20 solar neutrino units or SNU (captures per second per target atom).
Davis, in the first few months, measured only
3 SNU, approximately a factor of 7 below what
would be expected from solar-model
calculations [5]. In the next years, Davis
improved his equipment and methodology and
Bahcall improved the accuracy of his
calculations. In 1988 Bahcall and Ulrich
Fig. 1. The argon production rate obtained by the
Homestake experiment from 1970 to 1995 [6].
improved the predicted count to be 7.93 SNU.
The Homestake experiment ran for over 25
years; between 1970 and 1995 the average
production of solar neutrinos was 2.56 SNU, approximately one third of what was predicted
[6]. This experiment marked the start of the solar neutrino problem, as scientists could not
understand why there was a deficit of solar neutrinos.
Further experimentation
After the publication of the chlorine detection method, many other methods were proposed,
and other observatories were built. The Kamiokande-II detector was a large tank of ultrapure
water surrounded by photomultiplier tubes that could count neutrinos by measuring
the Čerenkov radiation given off by the elastic collision between a neutrino and an electron.
It was sensitive to mostly electron neutrinos. In 1989 Kamiokande-II measured solar
neutrinos at 46% of the value predicted by the standard model; this value was determined to
be consistent with the Homestake experiment, confirming the solar neutrino problem [7].
Another method of detecting solar neutrinos used gallium:
νe + 71Ga ⟶ 71Ge + e-.
Gallium detectors were attractive because they could detect lower energy neutrinos, for
example, those produced by the proton-proton reaction. So far, all experiments could only
detect relatively high energy neutrinos from other reactions in the sun [8]. In 1989 the first
gallium measurements were taken in SAGE (Soviet-American Gallium Experiment), then
GALLEX (GALLium EXperiment) in 1991. Both observatories reported around 50% of the
predicted number of solar neutrinos [9].
Proposed solutions
The solar neutrino problem could be explained by either a problem with the theory or a
problem with the experiments. After so many neutrino observatories recorded a deficit of
solar neutrinos, it became unlikely that experiments were unreliable. So was the theory of
the sun wrong, or does the standard model need changing?
One proposed solution was that the sun was colder than originally thought. Since the
rate of fusion in the sun is a function of temperature, decreasing the temperature would
decrease the neutrino flux. However, there were two problems with this explanation. Firstly,
the Kamiokande-II experiment required a decrease of the sun’s temperature by 4%, but the
Homestake experiment required a decrease of 8%. This is a difference of 4 standard
deviations, so it is implausable [10]. Secondly, the stellar model had been successful for
many years, and the proposed decrease had a confidence level of less than 1%. This is too
low to make the case for changing the physics of the sun [11], therefore this solution can be
confidently rejected.
Another proposed solution was that the neutrino had a large magnetic moment. A
large magnetic moment would allow the neutrino to flip from having left-handed spin to righthanded spin in the presence of the solar magnetic field. This would explain the solar neutrino
problem since the neutrino detectors are only sensitive to left-handed neutrinos [12][13]. This
‘helicity flip’ could also explain the recorded anticorrelation with sunspots because there are
more sunspots during periods of increased magnetic activity. A large magnetic moment,
however, would require a large neutrino mass. The neutrino was not widely believed to have
mass, let alone a large mass [14]. Furthermore, right-handed neutrinos have never been
observed. Consequently, this solution can also be rejected.
Bruno Pontecorvo, in 1957, presented the idea of neutrino oscillation. He asserted
that neutrinos could transition to and from their antiparticle counterparts. This theory
advanced over the next decades to include oscillations between the 3 flavours of neutrino
[15]. This would explain the solar neutrino problem because the experiments so far were
only sensitive to one flavour of neutrino. In 1998 the Super-Kamiokande, measuring
atmospheric neutrinos, found compelling evidence of neutrino oscillation. Super-Kamiokande
was special because it could measure the direction of neutrinos in real time. Researchers
found that muon neutrinos travelling up were occurring at half the rate of muon neutrinos
travelling down, this could be explained if the muon neutrinos were transforming into another
neutrino that could not be detected since in each direction the neutrino has travelled a
different distance [16]. The confirmation of neutrino oscillation also requires the neutrino to
have mass. To fully confirm this theory, neutrino oscillation must be observed in solar
neutrinos.
The Sudbury Neutrino Observatory
In 1985 Herbert Chen recognised that the reactions
νe + d ⟶ e- + p + p
(charged current)
ν+ d ⟶ν+p+ n
(neutral current)
could be used to measure both the total neutrino flux and the electron neutrino flux [17]. In
these equations, d is a deuterium nucleus (one neutron and one proton bound together).
The Sudbury Neutrino Experiment (SNO) used 1000 tonnes of heavy water, water that has a
deuterium instead of a hydrogen (D2O), to detect all flavours of neutrino. Like the
Kamiokande experiments, SNO used photomultiplier tubes to detect Čerenkov radiation to
measure the neutrino flux.
In the SNO, there are now three interactions that produce Čerenkov radiation:
charged current, neutral current, and elastic scattering which was the interaction also used in
Kamiokande. To differentiate the three interactions, statistical separation was performed
using the inclusion of three more variables: electron kinetic energy, the angle between the
sun and the path the electron takes, and the position in the tank where the interaction took
place. Using these three parameters, researchers could calculate the proportion of electron
neutrinos versus other types of neutrinos. The observatory started recording results in 1999,
and first fully published these results in 2002.
Fig. 2 shows the flux of high energy solar
neutrinos, muon or tau neutrinos versus
electron neutrinos. The three coloured bands
are the fluxes measured with the three
different reactions. The dashed lines
represent the flux predicted by the standard
solar model. The intersection of the bands is
the measured final flux of electron neutrinos
versus other kinds. This shows that 34% of
Fig. 2. SNO’s measured solar neutrino flux [18]
the neutrino flux is attributed to electron
neutrinos and that the total flux is consistent
with the standard model. The only explanation for this is neutrino oscillation, which causes
neutrinos to change flavour on their journey to Earth [18][19].
Neutrino oscillation
The discovery of neutrino oscillation implies that neutrinos have mass, more specifically a
mixture of three mass eigenstates (the characteristic state of some quantum object).
Because neutrinos evolve, this must mean they experience time. Since they experience
time, they must be travelling at a speed slower than the speed of light. According to special
relativity, only massless particles can travel at the speed of light. Therefore, the flavours of
neutrino can be thought of as each a different superposition of three neutrino mass
eigenstates when travelling, which collapses to a single flavour when observed. The wave
packets of each mass eigenstate have different masses; therefore, they propagate with
different velocities. As the wave packets propagate, their phase difference changes. As the
phase difference evolves, the probability of finding each flavour of neutrino changes. We can
write the transformation between flavour eigenstates and mass eigenstates as a matrix,
called the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix:
𝜈𝑒
𝑈𝑒1
(𝜈µ ) = (𝑈µ1
𝜈𝜏
𝑈𝜏1
𝑈𝑒2
𝑈µ2
𝑈𝜏2
𝜈1
𝑈𝑒3
𝑈µ3 ) (𝜈2 ).
𝜈3
𝑈𝜏3
This means, for example, that an electron neutrino’s wave function is
|𝜈𝑒 > = 𝑈𝑒1 |𝜈1 > + 𝑈𝑒2 |𝜈2 > + 𝑈𝑒3 |𝜈3 >.
Which is a superposition of the three mass eigenstates. The |X> notation just means the
state that this quantum object, X, is in. The mass eigenstates propagate according to a plane
wave solution, so that at a later time, t, the wave function of the neutrino has changed to
|𝛹(𝑡) > = 𝑈𝑒1 |𝜈1 > 𝑒 −𝑖𝜑1 + 𝑈𝑒2 |𝜈2 > 𝑒 −𝑖𝜑2 + 𝑈𝑒3 |𝜈3 > 𝑒 −𝑖𝜑3 [20].
Instead of using this complex notation, we can think of the changing coefficients of the mass
eigenstates as arrows that rotate at different speeds. Because they are rotating at different
speeds, a combination that initially looked like an electron neutrino could become a muon
neutrino. If you wait longer, the combination will look like an electron neutrino again and so
on as the neutrino travels through space from the sun to the earth.
|𝜈𝑒 > =
|𝜈1 > +
|𝜈2 > +
|𝜈3 >
|𝜈? > =
|𝜈1 > +
|𝜈2 > +
|𝜈3 >
|𝜈µ > =
|𝜈1 > +
|𝜈2 > +
|𝜈3 >
In between the transition, we are in a situation where there is a probability of finding an
electron neutrino and a probability of finding a muon neutrino. It is possible to calculate this
probability from the values in the PMNS matrix that are measured experimentally.
In reality, the oscillations are much more
complicated. From Fig. 3 we can see the
probability of observing each flavour of
neutrino change as it travels farther from the
source. At 11000 km/Gev there is
approximately an equal chance to observe
each flavour of neutrino. Unlike the simple,
unrealistic example above, the electron
Fig. 3. Neutrino oscillation. Black is electron neutrino,
blue is tau neutrino, and red is muon neutrino [21]
neutrino never fully transforms into another
flavour; it always has a non-zero probability.
The oscillations of tau neutrinos and muon neutrinos are similar because the tau and muon
parameters in the rows in the PMNS matrix have similar values [22]. The values in the
PMNS matrix are still not fully understood, and values are being updated all the time to
correlate better with experimental results.
Conclusion
The solar neutrino problem was a result of a misunderstanding of the standard model. The
standard model is used to, in theory, describe all fundamental forces and particles,
paramount to our understanding of the universe. After nearly 40 years, it was discovered that
neutrinos do not behave as expected due to their finite mass; a property that they were not
thought to have until 1998. Four scientists working on neutrino physics during this time were
awarded The Nobel Prize in Physics; Raymond Davis Jr. for his work to initially detect the
solar neutrino problem in the Homestake experiment and Masatoshi Koshiba for his work
with the Kamiokande detectors, awarded in 2002 [23]. And Takaaki Kajita for discovering
neutrino oscillations from atmospheric neutrinos in the Super-Kamiokande detecter, and
Arthur B. Mcdonald for discovering these oscillations in solar neutrinos using the SNO,
awarded in 2015 [24]. This discovery is a brilliant story of scientific endeavour, resulting in a
major change to how we think about the fundamental physics of our universe.
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