Nuclear Astrophysics Physics and Astronomy Department at the University of Washington


Solar Neutrinos

3.1 Solar neutrino detectors

Careful analyses of the experiments that will be described below indicate that the observed solar neutrino fluxes differ substantially from standard solar model (SSM) expectations.





This pattern is difficult to reproduce in a solar model because of the temperature dependences of the neutrino fluxes

(These results come from our standard formula, but with the constraint imposed that the solar luminosity be correctly reproduced. This means that the ppI cycle production must go up at the temperature goes down in order to produce the desired luminosity.) A reduced B neutrino flux can be produced by lowering the central temperature of the sun somewhat. However, such adjustments, either by varying the parameters of the SSM or by adopting some nonstandard physics, tend to push the Be)/B) ratio to higher values rather than the low one above,

Thus the observations seem difficult to reconcile with plausible solar model variations.

Five solar neutrino experiments have now provided data, the Homestake Cl experiment, the gallium experiments SAGE and GALLEX, Kamiokande, and SuperKamiokande The first three detectors are radiochemical, while Kamiokande and SuperKamiokande record neutrino-electron elastic scattering event-by-event.

The Homestake Experiment
Detection of neutrinos by the reaction Cl(,e)Ar was suggested independently by Pontecorvo (1946) and by Alvarez (1949). Davis's efforts to mount a 0.61 kiloton experiment using perchloroethylene (CCl) were greatly helped by Bahcall's demonstration that transitions to excited states in Ar, particularly the Fermi transition to the analog state at 4.99 MeV, increased the B cross section by a factor of 40. This suggested that Davis's detector would have the requisite sensitivity to detect B neutrinos, thereby accurately determining the central temperature of the sun. The experiment was mounted in the Homestake Gold Mine, Lead, South Dakota, in a cavity constructed approximately 4850 feet underground [4900 meters water equivalent (m.w.e.)]. It has operated continuously since 1967 apart from a 17 month hiatus in 1985/86 caused by the failure of the circulation pumps, and a recent hiatus due to funding shortfalls. The result of 25 years of measurement is

which can be prepared to two recent standard solar model predictions of 8.0 1.0 SNU and 6.4 1.4 SNU, all with 1 errors. The B and Be contributions account for about 75% and 16% of the total.

The experiment depends on the special properties of Ar: as a noble gas, it can be removed readily from perchloroethylene, while its half life ( = 35 days) allows both a reasonable exposure time and counting of the gas as it decays back to Cl. Argon is removed from the tank by a helium purge, and the gas then circulated through a condensor, a molecular sieve, and a charcoal trap cooled to the temperature of liquid nitrogen. Typically 95% of the argon in the tank is captured in the trap. (The efficiency is determined each run from the recovery results for a known amount of carrier gas, Ar or Ar, introduced into the tank at the start of the run.) When the extraction is completed, the trap is heated and swept by He. The extracted gas is passed through a hot titanium filter to remove reactive gases, and then other noble gases are separated by gas chromatography. The purified argon is loaded into a small proportional counter along with tritium-free methane, which serves as a counting gas. Since the electron capture decay of Ar leads to the ground state of Cl, the only signal for the decay is the 2.82 keV Auger electron produced as the atomic electrons in Cl adjust to fill the K-shell vacancy. The counting of the gas typically continues for about one year ( 10 half lives).

The measured cosmic ray-induced background in the Homestake detector is 0.06 Ar atoms/day while neutron-induced backgrounds are estimated to be below 0.03 atoms/day. A signal of 0.48 0.04 atoms/day is attributed to solar neutrinos. When detector efficiencies, Ar decays occurring in the tank, etc., are taken into account, the number of Ar atoms counted is about 25/year.

The Kamiokande and SuperKamiokande Experiments
The Kamiokande experiment used a 4.5 kiloton cylindrical imaging water Cerenkov detector originally designed for proton decay searches, but later reinstrumented to detect low energy neutrinos. It detected neutrinos by the Cerenkov light produced by recoiling electrons in the reaction

Both and heavy flavor neutrinos contribute, with 7. The light was detected by photomultiplier tubes that viewed the inner volume of the detector. Kamiokande had an inner fiducial volume of 0.68 kilotons. Its successor, SuperKamiokande, has been running for about 2.5 years. SuperKamiokande has a much larger fiducial volume of 22.5 kilotons.

Kamiokande was (SuperKamiokande is) sensitive to the high energy portion of the B neutrino spectrum. Between December, 1985, and July, 1993, Kamiokande accumulated 1667 live detector days of data. Under the assumption that the incident neutrinos are s with an undistorted B decay spectrum, the Kamiokande data gave

The total number of detected solar neutrino events was 476.

The corresponding result from SuperKamiokande obtained in the first 504 effective days of running is

Note that SuperKamiokande has already substantially surpassed Kamiokande in accuracy.

These experiments are remarkable in several respects. They are the first detectors to measure solar neutrinos in real time. Essential to the method is the sharp peaking of the electron angular distribution in the direction of the incident neutrino: this forward peaking allows the experimenters to separate solar neutrino events from an isotropic background. The unambiguous observation of a peak in the cross section correlated with the position of the sun is the first direct demonstration that the sun produces neutrinos as a byproduct of fusion.

The SAGE and GALLEX Experiments
Two radiochemical gallium experiments exploiting the reaction Ga(,e)Ge, SAGE and GALLEX, began solar neutrino measurements in January, 1990, and May, 1991, respectively. SAGE operated in the Baksan Neutrino Observatory, under 4700 m.w.e. of shielding from Mount Andyrchi in the Caucasus, while GALLEX was housed in the Gran Sasso Laboratory at a depth of 3300 m.w.e. These experiments are sensitive primarily to the low-energy pp neutrinos, the flux of which is sharply constrained by the solar luminosity in any steady-state model of the sun. The gallium experiment was first suggested by Kuzmin in 1966. In 1974 Ray Davis and collaborators began work to develop a practical experimental scheme. Their efforts, in which both GaCl solutions and Ga metal targets were explored, culminated with the 1.3-ton Brookhaven/Heidelberg/Rehovot/Princeton pilot experiment in 1980-82 that demonstrated the procedures later used by GALLEX. SAGE used a liquid metal target.

SAGE began operations with 30 tons of gallium, and later increased to 55 tons. The result is

The corresponding result from GALLEX is

Both detectors were tested with neutrino sources, marking the first time such calibrations of solar neutrino detectors had been made.

The nuclear physics of the reaction Ga(,e)Ge accounts for its sensitivity to low-energy neutrinos. As the threshold is 233 keV, the ground state (and first excited state) can be excited by pp neutrinos. The ground-state cross section can be determined from the measured electron capture lifetime of Ge, and is quite strong. The low-energy pp neutrinos account for about 55% of the capture rate. Because of this strong pp neutrino contribution, there exists a minimal astronomical counting rate of 79 SNU for the Ga detector that assumes only a steady-state sun and standard model weak interaction physics. This minimum value corresponds to a sun that produces the observed luminosity entirely through the ppI cycle. The rates found by SAGE and GALLEX are quite close to this bound.

3.2 Neutrino masses and vacuum neutrino oscillations

One odd feature of particle physics is that neutrinos, which are not required by any symmetry to be massless, nevertheless must be much lighter than any of the other known fermions. For instance, the current limit on the mass is 5 eV. The standard model requires neutrinos to be massless, but the reasons are not fundamental. Dirac mass terms , analogous to the mass terms for other fermions, cannot be constructed because the model contains no right-handed neutrino fields. Neutrinos can also have Majorana mass terms

where the subscripts L and R denote left- and right-handed projections of the neutrino field , and the superscript c denotes charge conjugation. The first term above is constructed from left-handed fields, but can only arise as a nonrenormalizable effective interaction when one is constrained to generate with the doublet scalar field of the standard model. The second term is absent from the standard model because there are no right-handed neutrino fields.

None of these standard model arguments carries over to the more general, unified theories that theorists believe will supplant the standard model. In the enlarged multiplets of extended models it is natural to characterize the fermions of a single family, e.g., , e, u, d, by the same mass scale . Small neutrino masses are then frequently explained as a result of the Majorana neutrino masses. In the seesaw mechanism,

Diagonalization of the mass matrix produces one light neutrino, , and one unobservably heavy, . The factor (/) is the needed small parameter that accounts for the distinct scale of neutrino masses. The masses for the , and are then related to the squares of the corresponding quark masses , , and . Taking GeV, a typical grand unification scale for models built on groups like SO(10), the seesaw mechanism gives the crude relation

The fact that solar neutrino experiments can probe small neutrino masses, and thus provide insight into possible new mass scales that are far beyond the reach of direct accelerator measurements, has been an important theme of the field.

Now one of the most interesting possibilities for solving the solar neutrino problem has to do with neutrino masses. For simplicity we will discuss just two neutrinos. If a neutrino has a mass m, we mean that as it propagates through free space, its energy and momentum are related in the usual way for this mass. Thus if we have two neutrinos, we can label those neutrinos according to the eigenstates of the free Hamiltonian, that is, as mass eigenstates.

But neutrinos are produced by the weak interaction. In this case, we have another set of eigenstates, the flavor eigenstates. We can define a as the neutrino that accompanies the positron in decay. Likewise we label by the neutrino produced in muon decay.

Now the question: are the eigenstates of the free Hamiltonian and of the weak interaction Hamiltonian identical? Most likely the answer is no: we know this is the case with the quarks, since the different families (the analog of the mass eigenstates) do interact through the weak interaction. That is, the up quark decays not only to the down quark, but also occasionally to the strange quark. (This is why we had a in our decay amplitude: the amplitude for is proportional to .) Thus we suspect that the weak interaction and mass eigenstates, while spanning the same two-neutrino space, are not coincident: the mass eigenstates and (with masses and ) are related to the weak interaction eigenstates by



where is the (vacuum) mixing angle.

An immediate consequence is that a state produced as a or a at some time t - for example, a neutrino produced in decay - does not remain a pure flavor eigenstate as it propagates away from the source. This is because the different mass eigenstates comprising the neutrino will accumulate different phases as they propagate downstream, a phenomenon known as vacuum oscillations (vacuum because the experiment is done in free space). To see the effect, suppose we produce a neutrino in some decay where we measure the momentum of the initial nucleus, final nucleus, and positron. Thus the outgoing neutrino is a momentum eigenstate. At time t=0, then

Each eigenstate subsequently propagates with a phase

But if the neutrino mass is small compared to the neutrino momentum/energy, one can write

Thus we conclude



We see there is a common average phase (which has no physical consequence) as well as a beat phase that depends on

Now it is a simple matter to calculate the probability that our neutrino state remains a at time t



where the limit on the right is appropriate for large t. Now , where E is the neutrino energy, by our assumption that the neutrino masses are small compared to k. Thus we can reinsert the units above to write the probability in terms of the distance x of the neutrino from its source,

(When one properly describes the neutrino state as a wave packet, the large-distance behavior follows from the eventual separation of the mass eigenstates.) If the the oscillation length

is comparable to or shorter than one astronomical unit, a reduction in the solar flux would be expected in terrestrial neutrino oscillations.

The suggestion that the solar neutrino problem could be explained by neutrino oscillations was first made by Pontecorvo in 1958, who pointed out the analogy with oscillations. From the point of view of particle physics, the sun is a marvelous neutrino source. The neutrinos travel a long distance and have low energies ( 1 MeV), implying a sensitivity to

In the seesaw mechanism, , so neutrino masses as low as could be probed. In contrast, terrestrial oscillation experiments with accelerator or reactor neutrinos are typically limited to .

From the expressions above one expects vacuum oscillations to affect all neutrino species equally, if the oscillation length is small compared to an astronomical unit. This appears to contradict observation, as the pp flux may not be significantly reduced. Furthermore, the theoretical prejudice that should be small makes this an unlikely explanation of the significant discrepancies with SSM Be and B flux predictions.

The first objection, however, can be circumvented in the case of ``just so" oscillations where the oscillation length is comparable to one astronomical unit. In this case the oscillation probability becomes sharply energy dependent, and one can choose to preferentially suppress one component (e.g., the monochromatic Be neutrinos). This scenario has been explored by several groups and remains an interesting possibility. However, the requirement of large mixing angles remains.

Below we will see that stars allow us to ``get around" this problem with small mixing angles. In preparation for this, we first present the results above in a slightly more general way. The analog of the result marked Eq. A above for an initial muon neutrino () can be derived and is



Now if we compare eqs. (A) and (B) we see that they are special cases of a more general problem. Suppose we write our initial neutrino wave function in the most general form

Then eqs. (A) and (B) tell us that the subsequent propagation is described by changes in and according to (this takes a bit of algebra)

Note that the common phase has been ignored: it can be absorbed into the overall phase of the coeeficients and , and thus has no consequence. The matrix above is called the mass matrix in the flavor basis. If one were to diagonalize the mass matrix, the eigenvectors would be the mass eigenstates and the difference between the eigenvalues would be .

3.3 The Mikheyev-Smirnov-Wolfenstein mechanism

The view of neutrino oscillations changed radically when Mikheyev and Smirnov showed in 1985 that the density dependence of the neutrino effective mass, a phenomenon first discussed by Wolfenstein in 1978, could greatly enhance oscillation probabilities: a is adiabatically transformed into a as it traverses a critical density within the sun. It became clear that the sun was not only an excellent neutrino source, but also a natural regenerator for cleverly enhancing the effects of flavor mixing.

While the original work of Mikheyev and Smirnov was numerical, their phenomenon was soon understood analytically as a level-crossing problem. If one writes the neutrino wave function in matter in the same way we did at the end of section 3.2

where x is the coordinate along the neutrino's path, the evolution of and is governed by

where G is the weak coupling constant and the solar electron density. If = 0, this is exactly our previous result and can be trivially integrated to give the vacuum oscillation solutions of Sec. 3.2. The new contribution to the diagonal elements, , represents the effective contribution to that arises from neutrino-electron scattering. The indices of refraction of electron and muon neutrinos differ because the former scatter by charged and neutral currents, while the latter have only neutral current interactions. The difference in the forward scattering amplitudes determines the density-dependent splitting of the diagonal elements of the new matter equation.

It is helpful to rewrite this equation in a basis consisting of the light and heavy local mass eigenstates (i.e., the states that diagonalize the right-hand side of the equation),



The local mixing angle is defined by



where . Thus ranges from to as the density goes from 0 to .

If we define

the neutrino propagation can be rewritten in terms of the local mass eigenstates

with the splitting of the local mass eigenstates determined by

and with mixing of these eigenstates governed by the density gradient

The results above are quite interesting: the local mass eigenstates diagonalize the matrix if the density is constant. In such a limit, the problem is no more complicated than our original vacuum oscillation case, although our mixing angle is changed because of the matter effects. But if the density is not constant, the mass eigenstates in fact evolve as the density changes. This is the crux of the MSW effect. Note that the splitting achieves its minimum value, , at a critical density

that defines the point where the diagonal elements of the original flavor matrix cross.

Our local-mass-eigenstate form of the propagation equation can be trivially integrated if the splitting of the diagonal elements is large compared to the off-diagonal elements,

a condition that becomes particularly stringent near the crossing point,

The resulting adiabatic electron neutrino survival probability, valid when , is

where is the local mixing angle at the density where the neutrino was produced.

The physical picture behind this derivation is illustrated in the accompanying figure. One makes the usual assumption that, in vacuum, the is almost identical to the light mass eigenstate, , i.e., and 1. But as the density increases, the matter effects make the heavier than the , with as becomes large. The special property of the sun is that it produces s at high density that then propagate to the vacuum where they are measured. The adiabatic approximation tells us that if initially , the neutrino will remain on the heavy mass trajectory provided the density changes slowly. That is, if the solar density gradient is sufficiently gentle, the neutrino will emerge from the sun as the heavy vacuum eigenstate, . This guarantees nearly complete conversion of s into s, producing a flux that cannot be detected by the Homestake or SAGE/GALLEX detectors.

But this does not explain the curious pattern of partial flux suppressions coming from the various solar neutrino experiments. The key to this is the behavior when 1. Our expression for shows that the critical region for nonadiabatic behavior occurs in a narrow region (for small ) surrounding the crossing point, and that this behavior is controlled by the derivative of the density. This suggests an analytic strategy for handling nonadiabatic crossings: one can replace the true solar density by a simpler (integrable!) two-parameter form that is constrained to reproduce the true density and its derivative at the crossing point . Two convenient choices are the linear and exponential profiles. As the density derivative at governs the nonadiabatic behavior, this procedure should provide an accurate description of the hopping probability between the local mass eigenstates when the neutrino traverses the crossing point. The initial and ending points and for the artificial profile are then chosen so that is the density where the neutrino was produced in the solar core and (the solar surface), as illustrated in in the second figure. Since the adiabatic result () depends only on the local mixing angles at these points, this choice builds in that limit. But our original flavor-basis equation can then be integrated exactly for linear and exponential profiles, with the results given in terms of parabolic cylinder and Whittaker functions, respectively. This treatment, called the finite Landau-Zener approximation, has been used extensively in numerical calculations.

We derive a simpler (``infinite") Landau-Zener approximation by observing that the nonadiabatic region is generally confined to a narrow region around , away from the endpoints and . We can then extend the artificial profile to , as illustrated by the dashed lines in the second figure. As the neutrino propagates adiabatically in the unphysical region , the exact soluation in the physical region can be recovered by choosing the initial boundary conditions



That is will then adiabatically evolve to as x goes from to . The unphysical region can be handled similarly.

With some algebra a simple generalization of the adiabatic result emerges that is valid for all and

where P is the probability of hopping from the heavy mass trajectory to the light trajectory on traversing the crossing point. For the linear approximation to the density,

As it must by our construction, reduces to P for 1. When the crossing becomes nonadiabatic (e.g., ), the hopping probability goes to 1, allowing the neutrino to exit the sun on the light mass trajectory as a , i.e., no conversion occurs.

Thus there are two conditions for strong conversion of solar neutrinos: there must be a level crossing (that is, the solar core density must be sufficient to render when it is first produced) and the crossing must be adiabatic. The first condition requires that not be too large, and the second 1. The combination of these two constraints, illustrated in third figure, defines a triangle of interesting parameters in the plane, as Mikheyev and Smirnov found by numerically integration. A remarkable feature of this triangle is that strong conversion can occur for very small mixing angles ), unlike the vacuum case.

One can envision superimposing on this figure the spectrum of solar neutrinos, plotted as a function of for some choice of . Since Davis sees some solar neutrinos, the solutions must correspond to the boundaries of the triangle in the figure. The horizontal boundary indicates the maximum for which the sun's central density is sufficient to cause a level crossing. If a spectrum properly straddles this boundary, we obtain a result consistent with the Homestake experiment in which low energy neutrinos (large 1/E) lie above the level-crossing boundary (and thus remain 's), but the high-energy neutrinos (small 1/E) fall within the unshaded region where strong conversion takes place. Thus such a solution would mimic nonstandard solar models in that only the B neutrino flux would be strongly suppressed. The diagonal boundary separates the adiabatic and nonadiabatic regions. If the spectrum straddles this boundary, we obtain a second solution in which low energy neutrinos lie within the conversion region, but the high-energy neutrinos (small 1/E) lie below the conversion region and are characterized by at the crossing density. (Of course, the boundary is not a sharp one, but is characterized by the Landau-Zener exponential). Such a nonadiabatic solution is quite distinctive since the flux of pp neutrinos, which is strongly constrained in the standard solar model and in any steady-state nonstandard model by the solar luminosity, would now be sharply reduced. Finally, one can imagine ``hybrid" solutions where the spectrum straddles both the level-crossing (horizontal) boundary and the adiabaticity (diagonal) boundary for small , thereby reducing the Be neutrino flux more than either the pp or B fluxes.

What are the results of a careful search for MSW solutions satisfying the Homestake, Kamiokande/SuperKamiokande, and SAGE/GALLEX constraints? This has been done by many authors and yields the results in the next-to-last figure. The preferred solution (i.e., the best fit) corresponds to a region surrounding and , is the hybrid case described above. It is commonly called the small-angle solution. A second, large-angle solution exists, corresponding to and 0.6.

These solutions can be distinguished by their characteristic distortions of the solar neutrino spectrum. The survival probabilities (E) for the small- and large-angle parameters given above are shown as a function of E in the last figure.

The MSW mechanism provides a natural explanation for the pattern of observed solar neutrino fluxes. While it requires profound new physics, both massive neutrinos and neutrino mixing are expected in extended models. The preferred solutions correspond to eV, and thus are consistent with few eV. This is a typical mass in models where . On the other hand, if it is the participating in the oscillation, this gives GeV and predicts a heavy 10 eV. Such a mass is of great interest cosmologically as it would have consequences for supernova physics, the dark matter problem, and the formation of large-scale structure.

If the MSW mechanism proves not to be the solution of the solar neutrino problem, it still will have greatly enhanced the importance of solar neutrino physics: the existing experiments have ruled out large regions in the plane (corresponding to nearly complete conversion) that remain hopelessly beyond the reach of accelerator neutrino oscillation experiments.

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Last update: July 10, 1999