Solar neutrinos and the MSW effect

Transcription

1 Chapter 12 Solar neutrinos and the MSW effect The vacuum neutrino oscillations described in the previous section could in principle account for the depressed flux of solar neutrinos detected on Earth. But this solution requires a large mixing angle to suppress the electron neutrino flux sufficiently. Quark-sector mixing angles are relatively small, so theoretical prejudice (but no evidence) initially favored a small mixing angle in the neutrino sector. However, there is another issue: the neutrinos also have to transit out of the Sun. Electron neutrinos couple more strongly to normal matter than do other neutrinos (because electron neutrinos and the particles of normal matter all reside in the first generation of the Standard Model). Thus we must also ask how interaction with solar material will influence neutrino oscillations. 413

2 414 CHAPTER 12. SOLAR NEUTRINOS AND THE MSW EFFECT The Mass Matrix Let us first introduce an alternative formulation of the neutrino vacuum oscillation problem. Again taking two-neutrino mixing for simplicity, a neutrino in a momentum state p propagating in the vacuum can be written as a time-dependent linear combination of flavor eigenstates ν(t) =ν e (t) ν e +ν µ (t) ν µ, where the time-dependent amplitudes ν e (t) and ν µ (t) obey the matrix equation ( ) ( ) i d νe (t) νe (t) = M 0. dt ν µ (t) ν µ (t) The mass matrix in vacuum, M 0, is given by ( ) E1 cos 2 θ + E 2 sin 2 θ (E 2 E 1 )sinθ cosθ M 0 = (E 1 E 2 )sinθ cosθ E 1 sin 2, θ + E 2 cos 2 θ where θ is the mixing angle and assuming that E i >> m i c 2. E i =(p 2 + m 2 i )1/2 p+ m2 i 2p, After subtracting a multiple of the unit matrix (which will not influence the flavor probabilities), the vacuum mass matrix can be cast in the form ( ) M 0 = π cos2θ sin2θ, L sin2θ cos2θ where L is the vacuum oscillation length introduced earlier.

3 Propagation of Neutrinos in Matter So far, this is just a reformulation of our previous equations for vacuum oscillation. Now, following the insight of Mikheyev, Smirnov, and Wolfenstein (MSW), we consider the additional influence that interaction with matter may have on the neutrino oscillation. When electron neutrinos scatter elastically from electrons in the Sun, they may do so through either the charged weak current or the neutral weak current.

4 416 CHAPTER 12. SOLAR NEUTRINOS AND THE MSW EFFECT Feynman Diagrams In quantum field theory we use a pictorial representation of interaction matrix elements called Feynman diagrams. They are highly intuitive: given a Feynman diagram one can write the corresponding matrix element and given the matrix element one can sketch the Feynman diagram. Here are some weak-interaction Feynman diagrams: p W e e W + ν e n (a) ν e ν e (b) e e Z 0 ν n,p,a Z 0 ν e (c) ν n,p,a (d) ν The solid lines represent (fermion) matter fields and the wiggly lines represent exchanged virtual gauge bosons. Each diagram can represent several related processes, depending on the direction in which it is read. For example, diagram (a) read from the bottom: 1. A neutron (n) exchanges a virtual W intermediate vector boson with an electron neutrino (ν e ). 2. This converts n p and ν e e. Absence of flavor indices on neutrinos in diagrams (c) and (d) indicates that the neutral current is flavor blind. The symbol A in diagram (d) stands for a composite nucleus.

5 417 Charged current Neutral current e ν e e W + Z 0 ν e e ν e e e Z 0 ν µ ν µ Neutral current e ν e (b) Diagram contributing to ν µ scattering from electrons (a) Diagrams contributing to ν e scattering from electrons Figure 12.1: Feynman diagrams responsible for neutrino electron scattering in the MSW effect.

6 418 CHAPTER 12. SOLAR NEUTRINOS AND THE MSW EFFECT Figure 12.1 illustrates Feynman diagrams relevant to neutrino scattering in the Sun. Both the neutral and charged current contribute to electron neutrino interactions (left two diagrams). Only the neutral current contributes to the muon neutrino interactions. Neutral current contributes to both ν e and ν µ scattering, so neglect this common contribution for this discussion. Vacuum neutrino oscillations will be modified in matter because of the charged-current (W + ) diagram in Fig contributing to ν e scattering but not to ν µ.

7 The situation is similar to two coupled oscillators where the frequency of one oscillator is modified more by coupling to its surroundings than the other. Such a modification will influence the nature of the coupling between the two oscillators. 419

8 420 CHAPTER 12. SOLAR NEUTRINOS AND THE MSW EFFECT The charged-current Feynman diagram contributes an additional term to the vacuum-scattering mass matrix that may be expressed as M = 2G F n e ν e ν e, where G F is the weak coupling constant, n e is the local electron number density, and ν e ν e is a projection operator that selects the electron neutrino eigenstate (it causes M applied to a wavefunction to give zero unless that wavefunction corresponds to a ν e eigenstate). With this additional contribution, the mass matrix in the presence of matter becomes ( ) M = π cos2θ+ L/lm sin2θ. L sin2θ cos2θ L/l m The additional matter contribution to the oscillation lengthl m is given by 2π l m = = meters G F n e ρ e where ρ e is the electron number density in units of Avogadro s number per cm 3.

9 421 m 2 m 2 ν e m 2 ν (vac) µ m 2 ν µ m 2 ν (vac) e 0 n e Figure 12.2: Effective mass-squared of electron neutrinos and muon neutrinos as a function of electron number density ne, neglecting flavor mixing. Because the ν µ does not couple to the charged weak current its m 2 does not depend on ne but the effective m 2 of νe increases linearly with the electron density. The order of states in the m 2 spectrum in vacuum (left side) can become inverted in matter (right side). Fig shows that the charged-current changes the effective mass of an electron neutrino in medium. An electron neutrino less massive than a muon neutrino in vacuum will become effectively more massive in matter if the electron density is high enough: the m 2 spectrum can become inverted. Gaining an effective mass through interaction with a medium is common for particles in many contexts. Example: A superconductor expels a magnetic field because a photon gains an effective mass inside it (Meissner effect).

10 422 CHAPTER 12. SOLAR NEUTRINOS AND THE MSW EFFECT The time-evolved states in matter, ν1 m and νm 2, may be obtained by diagonalizing ( ) M = π cos2θ+ L/lm sin2θ. L sin2θ cos2θ L/l m Then the electron neutrino state after a time t becomes ν(t) =(cos 2 θ m e ie 1t + sin 2 θ m e ie 2t ) ν e +sinθ m cosθ m ( e ie 1t + e ie 2t ) ν µ, which is analogous to the corresponding vacuum equation, but The vacuum mixing angle θ v is replaced by the matter mixing angle θ m, tan2θ m = sin2θ v cos2θ v ± L/l m (with + for m 1 > m 2 and for m 1 < m 2 ), The vacuum oscillation length L is replaced by the oscillation length in matter L L m = 1+ 2L l m cos2θ v + L2 l 2 m Thus, for a fixed n e the flavor probabilities are given by the vacuum equations with the replacements θ v θ m and L L m. P(ν e ν e,r)=1 sin 2 (2θ m )sin 2 (πr/l m ) P(ν e ν µ,r)=sin 2 (2θ m )sin 2 (πr/l m ). Generally, θ m and L m will vary with the solar depth since they depend on the number density n e..

11 The MSW Resonance Condition From P(ν e ν e,r)=1 sin 2 (2θ m )sin 2 (πr/l m ) P(ν e ν µ,r)=sin 2 (2θ m )sin 2 (πr/l m ). we see that maximal flavor mixing occurs when sin 2 (2θ m ) = 1 or θ m = π 4. The most significant property of tan2θ m = sin2θ v cos2θ v ± L/l m relative to the vacuum solution is that if m 2 and L are positive (which requires that m 1 < m 2 and selects the negative sign in the above equation), When cos2θ v = L l m, tan2θ m and thus θ m π 4 Because 2π l m = G F n e L= 4πE m 2, this occurs when the electron density satisfies n e = cos2θ v m 2 2 2G F p nr e.

12 424 CHAPTER 12. SOLAR NEUTRINOS AND THE MSW EFFECT From P(ν e ν e,r)=1 sin 2 (2θ m )sin 2 (πr/l m ) P(ν e ν µ,r)=sin 2 (2θ m )sin 2 (πr/l m ). the resonance condition θ m 4 π leads to maximal mixing between ν e and ν µ, with a survival probability ( ) πr P(ν e ν e,r)=1 sin 2 (Resonance), L m and an oscillation length L m = L sin2θ v (Resonance). This is the Mikheyev Smirnov Wolfenstein (MSW) resonance.

13 425 8 (a) θ = 33.5 ο (b) θ = 5 ο sin 2 2θ m sin 2 2θ m tan 2θ m 4 2 Resonance Resonance 0.5 sin 2 2θ m 0 tan 2θ m tan 2θ m L/l m L/l m Figure 12.3: The MSW resonance condition for two values of the vacuum mixing angle θv. At the resonance the flavor conversion probability sin 2 2θm attains its maximum value and almost complete flavor conversion can be obtained for any non-vanishing vacuum oscillation angle θv. No matter how small the vacuum mixing angle θ v, as long as it is not zero there is some value of the electron density n R e where the MSW resonance condition occurs and maximal flavor mixing occurs at that density. Note that if the mixing angle θ v is small the corresponding oscillation length L m = L sin2θ v. will be large.

14 426 CHAPTER 12. SOLAR NEUTRINOS AND THE MSW EFFECT If the condition m 1 < m 2 is not satisfied there is no resonance for electron neutrinos because then the positive sign would be chosen in tan2θ m = sin2θ v cos2θ v ± L/l m and the denominator will not go to zero. In that case one can show that there is a corresponding resonance condition instead for the electron antineutrino ν e. This would be very interesting physics but could not be used to solve the problem of an observed deficit of solar ν e in detectors on Earth.

15 Resonance Flavor Conversion If, for example, m 1 < m 2 and θ v is small so that ν e ν1 m, and the electron density in the central part of the Sun where the neutrino is produced satisfies n e > n R e, a neutrino leaving the Sun will inevitably encounter the MSW resonance while on its way out of the Sun. If the change in density is sufficiently slow (adiabatic condition), the ν e flux produced in the core can be almost entirely converted to ν µ by the MSW resonance near the radius where the resonance condition is satisfied.

16 428 CHAPTER 12. SOLAR NEUTRINOS AND THE MSW EFFECT The MSW resonance conversion of flavors can be viewed as an adiabatic level crossing m 2 Surface of Sun MSW resonance νe m 2 (n e ) m 2 2 νµ νµ m 2 vac νe n e R Center of Sun m 2 1 n e If the level crossing is adiabatic, a neutrino that starts out near the center as a ν e (corresponding to high density on the right side of the figure) changes adiabatically into a ν µ by the time it exits the Sun (corresponding to low density in the left side of the figure). That is, the neutrino follows the upper curved trajectory though the resonance in the level-crossing region, as indicated by the arrows. Therefore, the neutrino can emerge from the Sun in a completely different flavor state than the one in which it was created.

17 429 Mass 2 (ev 2 ) (a) θ = 1 ο λ MSW λ (b) θ = 5 ο + MSW λ + λ ξ = 0.25 ξ = 6.2 Mass 2 (ev 2 ) (c) θ = 15 ο λ + λ MSW ξ = ρ (g cm -3 ) (d) θ = 33.5 ο MSW λ + λ ξ = ρ (g cm -3 ) Figure 12.4: Solutions λ ± of the MSW eigenvalue problem as a function of mass density. Each case corresponds to the choices m 2 = ev 2 and E = 10 MeV, but to different values of the vacuum mixing angle θv. The individual neutrino masses are presently unknown but for purposes of illustration m 2 1 = ev 2 has been assumed in vacuum, so that m 2 2 = m2 1 + m2 = ev 2. The critical density leading to the MSW resonance (corresponding to minimum splitting between the eigenvalues) and the value of the adiabaticity parameter ξ = δr R /L R m are indicated for each case. Realistic conditions in the Sun are expected to imply the very adiabatic crossing exhibited in case (d). From conditions in the Sun inferred from the Standard Solar Model, the strongly-avoided level crossing in Fig. (d) above is expected to apply for the Sun.

18 430 CHAPTER 12. SOLAR NEUTRINOS AND THE MSW EFFECT Mass 2 (ev 2 ) 2 λ MSW resonance λ + 1 Log electron density n e 2 Resonance layer ν 1 Production layer Surface ρ Center Surface R/R 0 Center Figure 12.5: Adiabatic flavor conversion by the MSW mechanism in the Sun. An electron neutrino is produced at Point 1 where the density lies above that of the MSW resonance and propagates radially outward to Point 2 where the density lies below that of the resonance. The width of the resonance layer is assumed to be much larger than the matter oscillation length in the resonance layer, justifying the adiabatic approximation. Widths of resonance and production layers are not meant to be to scale. The adiabatic conversion of neutrino flavor in the Sun is illustrated in Fig If Point 2 lies at the solar surface the classical probability to detect the neutrino as an electron neutrino when it exits the Sun is P(ν e ν e )= 1 2 [1+cos2θ v cos2θ 0 m] (at solar surface), where θ v is the vacuum mixing angle and θ 0 m θ m (t 1 ) is the matter mixing angle at the point of neutrino production.

19 (a) θ = 5 ο (b) θ = 15 ο (c) θ = 25 ο (d) θ = 35 ο _ P νµ _ P νµ _ P νµ _ P νµ _ P _ P νe _ P νe R/R R/R R/R R/R _ P νe _ P νe Figure 12.6: MSW flavor conversion vs. fraction of solar radius for four values of the vacuum mixing angle θv. Calculations are classical averages over local oscillations in adiabatic approximation using Eq. (12.0.6), assuming m 2 = ev 2 and E = 10 MeV. Neutrinos were assumed to be produced in a νe flavor state at the center of the Sun (right side of diagram at R/R = 0). Solid curves show the classical electron-neutrino probability and dashed curves show the corresponding classical muon-neutrino probability. Table 12.1: Energy dependence of solar ν flavor conversion for θ v = 35 E (MeV) P νe (surface) R R /R Flavor conversion by the MSW mechanism for a 2-flavor model in adiabatic approximation is illustrated for four different values of the vacuum mixing angle θ v in Fig The MSW resonance occurs at the radius corresponding to the intersection of the solid and dashed curves. Figure 12.6(d) approximates the situation expected for the Sun. The table gives the energy dependence of flavor conversion.

20 432 CHAPTER 12. SOLAR NEUTRINOS AND THE MSW EFFECT 12.1 Resolution of the Solar Neutrino Problem A series of experiments have in principle resolved the solar neutrino problem. These experiments demonstrate rather directly that flavor conversion of neutrinos is taking place. This, in turn then implies that at least some neutrinos have mass. Detailed comparison of these experiments indicates that solar electron neutrinos are being converted to muon neutrinos by neutrino oscillations, If all flavors of neutrinos coming from the Sun are detected the solar neutrino deficit relative to the Standard Solar Model disappears. The favored oscillation scenario is MSW resonance conversion in the Sun, but for a large vacuum mixing angle solution. Let us now describe briefly the experiments that have led to these rather remarkable conclusions.

21 12.1. RESOLUTION OF THE SOLAR NEUTRINO PROBLEM Super Kamiokande Observation of Flavor Oscillation The Super Kamiokande detector has been used to observe neutrinos produced in atmospheric cosmic ray showers. When high-energy cosmic rays hit the atmosphere, they generate showers of mesons that decay to muons, electrons, positrons, and neutrinos. Theory assuming no physics beyond the Standard Model indicates that the ratio of muon neutrinos plus antineutrinos to electron neutrinos plus antineutrinos should be 2, R ν µ+ ν µ ν e + ν e = 2 (Standard Model). Instead, Super Kamiokande confirmed that the ratio R is only 64% of what is expected. This result could be explained by neutrino flavor oscillations: if, for example, the muon neutrinos oscillate into another flavor, the observed ratio would be reduced below the expected value.

22 434 CHAPTER 12. SOLAR NEUTRINOS AND THE MSW EFFECT Attribution of the anomalous ratio found for R ν µ+ ν µ ν e + ν e = 2 (Standard Model). to a flavor oscillation was strengthened by the observed dependence of the ratio R for higher-energy neutrinos on the zenith angle ϕ for the neutrinos. This observation is possible because the water Čerenkov detectors give directional information on the detected neutrinos. Detailed observations indicated that for muon neutrinos of the highest energy, R 1 for neutrinos coming directly up through the Earth and into the detector (ϕ = 180 ) but R 2 for those neutrinos coming from directly overhead (ϕ = 0 ). Similar measurements for electron neutrinos indicated no such asymmetry.

23 12.1. RESOLUTION OF THE SOLAR NEUTRINO PROBLEM 435 These results can be explained by oscillations of the muon neutrino because the zenith angle is a measure of how far the neutrino has traveled since it was produced in the atmosphere. Atmospherically-produced neutrinos coming up through the Earth have traveled more than the diameter of the Earth since they were produced in an atmospheric shower on the opposite side of the Earth Neutrinos coming from a shower directly overhead have traveled a distance that is only a fraction of the height of the atmosphere. If the oscillation length L is less than the diameter of the Earth but greater than the height of the atmosphere, neutrinos coming from below the detector would be strongly influenced by oscillation but those coming from directly overhead would have little chance to oscillate into another flavor before being detected.

24 436 CHAPTER 12. SOLAR NEUTRINOS AND THE MSW EFFECT Detailed analysis suggests that the oscillation partner of the muon neutrino is not the electron neutrino Hence these results are not directly applicable to the solar neutrino problem. ν µ is oscillating either with the tau neutrino or some other flavor of neutrino that does not undergo normal weak interactions but does participate in neutrino oscillations ( sterile neutrinos ). The best fit to the data suggests a mixing angle close to maximal (a large mixing angle solution) and a mass squared difference in the range m ev 2. The large mixing angle indicates that the mass eigenstates are approximately equal mixtures of the two weak flavor eigenstates.

25 12.1. RESOLUTION OF THE SOLAR NEUTRINO PROBLEM SNO Observation of Neutral Current Interactions The Super Kamiokande results cited above indicate conclusively the existence of neutrino oscillations and thus of physics beyond the Standard Model. However, the detected oscillations do not appear to involve the electron neutrino. Thus the Super Kamiokande results cannot be applied directly to the solar neutrino problem. But a water Čerenkov detector in Canada has yielded information about neutrino oscillations that is directly applicable to the solar neutrino problem.

26 438 CHAPTER 12. SOLAR NEUTRINOS AND THE MSW EFFECT The Sudbury Neutrino Observatory (SNO) differs from Super-K in that it contains heavy water The heavy water is important because of the deuterium that it contains. In regular water, to produce sufficient Čerenkov light the ν energy has to be greater than about 5 7 MeV. Because deuterium (d) contains a weakly-bound neutron, it can undergo a breakup reaction: Any flavor neutrino can initiate the reaction ν+ d ν+p+n (Neutral current), but only electron neutrinos can initiate ν e + d e + p+ p (Charged current). Both of these reactions have much larger cross sections than elastic neutrino electron scattering, so SNO can gather events at relatively high rates. The energy threshold can be lowered to 2.2 MeV, the binding energy of the deuteron. Because the neutral currents are flavor blind, ν + d ν+p+n gives SNO the ability to see the total neutrino flux of all flavors coming from the Sun.

27 12.1. RESOLUTION OF THE SOLAR NEUTRINO PROBLEM 439 Table 12.2: Comparison of SNO results and Standard Solar Model predictions for solar neutrino fluxes. All fluxes are in units of 10 6 cm 2 s 1. SSM ν e Flux SNO ν e Flux SNO ν e /SSM SNO all flavors SNO All/ SSM 5.05 ± ± ± Because of its energy threshold, SNO sees primarily 8 B solar neutrinos. The initial SNO results confirmed results from the pioneering solar neutrino experiments: a strong suppression of the electron neutrino flux is observed relative to that expected in the Standard Solar Model. of the ex- Specifically, SNO found that only about 1 3 pected ν e were being detected. However, SNO went further. By analyzing the flavor-blind weak neutral current events, it was possible to show that The total flux of all neutrinos in the detector was almost exactly that expected from the Standard Solar Model. Table 12.2 summarizes.

28 440 CHAPTER 12. SOLAR NEUTRINOS AND THE MSW EFFECT ν µ + ν τ flux (10 6 cm -2 s -1 ) Electron scattering Charged current Neutral current (a) Standard Solar Model m 2 (ev 2 ) * (b) ν e flux (10 6 cm -2 s -1 ) tan 2 θ Figure 12.7: (a) Flux of solar neutrinos from 8 B detected for various flavors by SNO. The band widths represent one standard deviation. Bands intersect at the point indicated by the star, implying that about 2 3 of the Sun s 8 B neutrinos have changed flavor between being produced in the Sun and being detected on Earth. The Standard Solar Model band is the prediction for the 8 B flux, irrespective of flavor changes. It tracks the neutral current band, which represents detection of all flavors of neutrino coming from the Sun. (b) 2-flavor neutrino oscillation parameters. The 99%, 95% and 90% confidence-level contours are shown, with the star indicating the most likely values. The best fit corresponds to the large-angle solution. The SNO case for neutrino oscillation was strengthened by analysis of neutrino electron elastic scattering data (largely from Super K) combined with SNO data from ν e + d e + p+ p (Charged current). which estimates the total neutrino flux of all flavors. Figure 12.7 illustrates. Best fit indicates that 2 3 of the Sun s electron neutrinos have changed flavor when they reach the Earth.

29 12.1. RESOLUTION OF THE SOLAR NEUTRINO PROBLEM 441 ν µ + ν τ flux (10 6 cm -2 s -1 ) Electron scattering Charged current Neutral current (a) Standard Solar Model m 2 (ev 2 ) * (b) ν e flux (10 6 cm -2 s -1 ) tan 2 θ Assuming a two-flavor mixing model, it is common to plot confidence level contours in a two dimensional plane with m 2 on one axis and tan 2 θ on the other. The figure above right shows the best-fit confidence-level contours for parameters based on SNO data. The SNO results suggest that the solar neutrino problem is solved by neutrino oscillations between ν e and ν µ flavors, with parameters m ev 2 θ 35. This is again a large-mixing-angle solution, which implies that a ν e is actually almost an equal superposition of two mass eigenstates, probably separated by no more than a few hundredths of an ev.

30 442 CHAPTER 12. SOLAR NEUTRINOS AND THE MSW EFFECT KamLAND Constraints on Mixing Angles KamLAND is housed in the same Japanese mine cavern that housed Kamiokande, predecessor to Super-K. It uses phototubes to monitor a large container of liquid scintillator. It looks specifically for electron antineutrinos produced during nuclear power generation in a set of 22 Japanese and Korean reactors that are located within a few hundred kilometers of the detector. The antineutrinos are detected from the inverse β - decay in the scintillator: ν e + p e + + n. From power levels in the reactors, the expected antineutrino flux at KamLAND can be modeled. The experiment has detected a shortfall of antineutrinos, consistent with a large-angle neutrino oscillation solution having m 2 = ev 2 θ = 37. KamLAND results exclude at the 95% level all previously proposed alternatives to a large-angle solution.

31 12.1. RESOLUTION OF THE SOLAR NEUTRINO PROBLEM 443 Combining the solar neutrino and KamLAND results leads to a solution m 2 = 7.59± ev 2 tan 2 θ v = , implying a vacuum mixing angle θ v 34.4, which is a large mixing angle solution (recall that θ has been defined so that its largest possible value is 45 ).

32 444 CHAPTER 12. SOLAR NEUTRINOS AND THE MSW EFFECT The SNO and KamLAND results together greatly shrink the allowed parameter space for solar neutrino oscillation parameters. The large mixing angle solutions implied by SNO and KamLAND for the ν e ν µ mixing indicate that the vacuum oscillations of solar neutrinos are of secondary importance to the MSW matter oscillations in the body of the Sun itself. Because the inferred vacuum oscillation lengths for the large-angle solutions are much less than the Earth Sun distance, they would largely wash out any energy dependence of the neutrino shortfall. Since the detectors indicate that such an energy dependence exists, the MSW resonance is strongly implicated as the source of the neutrino flavor conversion responsible for the solar neutrino problem. Ironically, the MSW effect was proposed to justify a small mixing angle solution but instead seems to resolve the solar neutrino anomaly through a large mixing angle solution.