# Solar Neutrino Observables Sensitive to Matter Effects

Authors: H. Minakata; C. Peña-Garay
Journal: Advances in High Energy Physics
 (2012)

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## Abstract

We discuss constraints on the coefficientAMSW which is introduced to simulate the effect of weaker or stronger matter potential for electron neutrinos with the current and future solar neutrino data. The currently available solar neutrino data leads to a bound AMSW=1.47+0.54−0.42(+1.88−0.82) at 1σ (3σ) CL, which is consistent with the Standard Model prediction AMSW=1. For weaker matter potential (AMSW<1), the constraint which comes from the flat 8B neutrino spectrum is already very tight, indicating the evidence for matter effects. However for stronger matter potential (AMSW>1), the bound is milder and is dominated by the day-night asymmetry of 8B neutrino flux recently observed by Super-Kamiokande. Among the list of observables of ongoing and future solar neutrino experiments, we find that (1) an improved precision of the day-night asymmetry of 8B neutrinos, (2) precision measurements of the low-energy quasi-monoenergetic neutrinos, and (3) the detection of the upturn of the 8B neutrino spectrum at low energies are the best choices to improve the bound on AMSW.


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## Body

## 1. Introduction

Neutrino propagation in matter is described by the Mikheyev-Smirnov-Wolfenstein (MSW) theory [1–4]. It was successfully applied to solve the solar neutrino problem [5], the discrepancy between the data [6–15], and the theoretical prediction of solar neutrino flux [16], which blossomed into the solution of the puzzle, the large-mixing-angle (LMA) MSW solution. The solution is in perfect agreement with the result obtained by KamLAND [17] detector which measured antineutrinos from nuclear reactors, where the flavor conversion corresponds to vacuum oscillations with subpercent corrections due to matter effects.The MSW theory relies on neutrino interaction with matter dictated by the standard electroweak theory and the standard treatment of refraction which is well-founded in the theory of refraction of light. Therefore, it is believed to be on a firm basis. On the observational side it predicts a severer reduction of the solar neutrino flux at high energies due to the adiabatic flavor transition in matter than at low energies where the vacuum oscillation effect dominates. Globally, the behavior is indeed seen in the experiments observing8B solar neutrinos at high energies [11–14] and in radiochemical experiments detecting low-energy pp and 7Be neutrinos [6–10] and more recently by the direct measurement of 7Be neutrinos by Borexino [15]. For a summary plot of the current status of high- and low-energy solar neutrinos, see the review of solar neutrinos in this series. Therefore, one can say that the MSW theory is successfully confronted with the available experimental data.Nevertheless, we believe that further test of the MSW theory is worth pursuing. First of all, it is testing the charged current (CC) contribution to the index of refraction of neutrinos of the Standard Model (SM), which could not be tested anywhere else. Furthermore, in analyses of future experiments to determineδ and the mass hierarchy, the MSW theory is usually assumed to disentangle the genuine effect of CP phase δ from the matter effect. Therefore, to prove it to the accuracy required by measurement of δ is highly desirable to make discovery of CP violation robust in such experiments that could have matter effect contamination. This reasoning was spelled out in [18]. Since the survival probability Pee does not depend on δ solar neutrinos provide us with a clean environment for testing the theory of neutrino propagation in matter.We notice that in solar neutrinos, the transition from low- to high-energy behaviors mentioned above has not been clearly seen in a single experiment in a solar-model-independent manner. The Borexino and KamLAND experiments tried to fill the gap by observing8B neutrinos at relatively low energies [19, 20]. SNO published the results of analyses with lower threshold energy of 3.5 MeV [21, 22], and the similar challenge is being undertaken by the Super-Kamiokande (SK) group [23]. In addition to 7Be, a new low-energy neutrino line, pep neutrinos, was observed by Borexino [24]. Recently, the SK group announced their first detection of the day-night asymmetry of 8B neutrinos [23]. As we will see in Section 3 it gives a significant impact on our discussions. With these new experimental inputs, as well as all the aforementioned ones, it is now quite timely to revisit the question of how large deviation from the MSW theory is allowed by data.In this paper, we perform such a test of the theory of neutrino propagation in the environments of solar and earth matter. For this purpose, we need to specify the framework of how deviation from the MSW theory is parametrized. We introduce, following [25, 26], the parameter AMSW defined as the ratio of the effective coupling of weak interactions measured with coherent neutrino matter interactions in the forward direction to the Fermi coupling constant GF. We first analyze the currently available solar neutrino data to obtain the constraints on AMSW and find that the features of the constraints differ depending upon AMSW<1 or AMSW>1. We will discuss interpretations of the obtained constraints including this feature and provide a simple qualitative model to explain the bound at AMSW>1, more nontrivial one. We then discuss the question of to what extent the constraints on AMSW can be made stringent by various future solar neutrino observables.Our framework of testing the theory of neutrino propagation in matter requires comments. It actually involves the three different ingredients: (1) non-SM weak interactions in the forward direction parametrized asAMSWGF, (2) refraction theory of neutrino propagation in matter which includes the resonant enhancement of neutrino flavor conversion [1–4] and (3) electron number densities in the sun and in the earth. However, on ground of well-founded refraction theory and because no problem can be arguably raised in the formulation of the MSW mechanism we do not question the validity of (2). We also note that the electron number density in the sun is reliably calculated by the standard solar model (SSM), and the result is cross-checked by helioseismology to an accuracy much better than the one discussed here. We can also take the earth matter density and chemical compositions calculated by the preliminary reference earth model (PREM) [27] as granted. It is the case because the earth matter-dependent observable, the day-night variation of solar neutrino flux, is insensitive to the precise profile of the earth matter density. Therefore, we assume that our test primarily examines the aspect (1), namely, whether neutrino matter coupling in the forward direction receives additional contribution beyond those of SM.Are non-SM weak interactions parametrized asAMSWGF general enough? Most probably not because in many models with new non-SM interactions they have flavor structure. Flavor-dependent new neutral current interactions have been discussed in the framework of nonstandard interactions (NSI) of neutrinos [28–33], and constraints on effective neutrino matter coupling were obtained with this setting, for example, in [34, 35]. With solar neutrinos see [36] for discussion of NSI. If we denote the elements of NSI as εαβ (αβ=e,μ,τ), our AMSW may be interpreted as AMSW=1+εee, assuming that εαβ≪εee for α≠e, β≠e. To deal with the fully generic case, however, we probably have to enlarge the framework of constraining the NSI parameters by including other neutrino sources, in particular, the accelerator and atmospheric neutrinos. See Section 4 for more comments.


## 2. Simple Analytic Treatment of Matter Effect Dependences

In this section, we give a simple analytic description of how various solar neutrino observables depend upon the matter effect. It should serve for intuitive understanding of the characteristic features which we will see in the later sections. The reader will find a physics discussion in the flavor conversion review of this series. In the following, we denote the matter densities inside the sun and in the earth asρS and ρE, respectively. Solar neutrino survival or appearance probabilities depend on three oscillation parameters: the solar oscillation parameters (θ12, Δm212≡m22-m12) and θ13. Smallness of the recently measured value of θ13 [37–41] and its small error greatly restrict the uncertainty introduced by this parameter on the determination of matter effects.To quantify possible deviation from the MSW theory, we introduce the parameterAMSW by replacing the Fermi coupling constant GF by AMSWGF [25, 26]. The underlying assumption behind such simplified framework is that the deviation from the Fermi coupling constant is universal over fermions, in particular up and down quarks.The survival probability in the absence of the earth matter effect, that is, during the day, is well described by [42–44]:
(2.1)PeeD=cos4θ13(12+12·cos2θS·cos2θ12)+sin4θ13.
Here θS is the mixing angle at the production point inside the sun:
(2.2)cos2θS≡cos2θm(ρS),
where θm(ρ) is the mixing angle in matter of density ρS,
(2.3)cos2θS=cos2θ12-ξS(1-2ξScos2θ12+ξS2)1/2.
In (2.4), ξS is defined as the ratio of the neutrino oscillation length in vacuum, lν, to the refraction length in matter, l0:
(2.4)ξS≡lνl0=22AMSWGFρSYecos2θ13mNEΔm2=0.203×AMSWcos2θ13(E1MeV)(ρSYe100gcm-3),
where
(2.5)lν≡4πEΔm2,l0≡2πmN2AMSWGFρSYecos2θ13.
In (2.4) and (2.5), ρS is the matter density, Ye is the number of electrons per nucleon, and mN is the nucleon mass. In the last term we have used the best fit of the global analysis Δm212=7.5×10-5 eV2. The average electron number densities ρSYe at the production point of various solar neutrino fluxes are tabulated in Table 1. These numbers serve to show the differences in solar densities probed by the different sources of neutrinos, but the precise calculations are correctly done by averaging the survival probability with the production point distribution of the corresponding source [5, 16, 45].Table 1
Average electron density at the neutrino production region and energy of the relevant pp solar neutrinos fluxes. Last column shows the ratio of the electron neutrino elastic scattering with electrons cross-section to theμ (or τ) neutrino one. For this calculation, we have assumed a measured electron kinetic energy range of [0.05,0.4], [1,1.4], [0,0.8], and [5,16] MeV for the pp, pep, 7Be, and 8B, respectively.
Source
ρ
S
Y
e
(
g
c
m
-
3
)
Energy (MeV)
σ
μ
/
σ
e
pp
67.9
≤0.42
0.284
pep
73.8
1.44
0.203
7
Be
86.5
0.86
0.221
8
B
92.5
≤16
0.155We observe thatPeeD in (2.1) depends on neutrino energy E and AMSW in the particular combination AMSWE. The property may have the following implications to constraints on AMSW: (1) since shifting AMSW is equivalent to shifting E our analysis which calculates χ2 as a function of AMSW is inevitably affected by the whole spectrum. (2) Nonetheless, we generically expect that the constraint at AMSW<1 (AMSW>1) principally comes from neutrino spectrum at high (low) energies. It appears that the apparently contradictory remarks are both true in view of the results in Section 3.


### 2.1. Energy Spectrum

Solar neutrino observables taken in a single experiment have not shown an energy dependence yet. The neutrino oscillation parameters are such that we cannot expect strong energy dependences. At low neutrino energies, smallξS, (2.1) can be approximated by
(2.6)PeeD=cos4θ13[1-12sin22θ12(1+cos2θ12ξS)]+sin4θ13,
whereas at high energies, small 1/ξS, the oscillation probability (2.1) can be approximated, keeping only the first energy-dependent term as
(2.7)PeeD=cos4θ13[sin2θ12+14sin22θ12cos2θ12(1ξS)2]+sin4θ13.
Notice that the correction to the asymptotic behavior is linear in AMSW at low energies while it is quadratic in AMSW-1 at high energies. It may mean that the energy spectrum at low energies could be more advantageous in tightening up the constraint on AMSW provided that these formulas with leading order corrections are valid.It is well known that, in the LMA MSW mechanism,8B neutrino spectrum must show an upturn from the asymptotic high energy (E≫10MeV) to lower energies. The behavior is described by the correction term in (2.7) but only at a qualitative level. It indicates that the upturn component in the spectrum is a decreasing function of AMSW. On the other hand, at low energies populated by pp, 7Be, and pep neutrinos, the solar neutrino energy spectrum displays vacuum-averaged oscillations or decoherence, (2.6). The deviation from this asymptotic low energy limit can be described by the correction term in (2.6) again at the (better) qualitative level. The term depends upon AMSW linearly so that the correction term is an increasing function of AMSW. Because of the negative sign in the correction term in (2.6), larger values of AMSW lead to smaller absolute values of Pee in both low- and high-energy regions. (The simpler way to reach the same conclusion is to use the property PeeD(E,sAMSW)=PeeD(sE,AMSW) mentioned earlier. Then, for larger AMSW (s>1) PeeD corresponds to the one at higher energy. Since PeeD is a monotonically decreasing function of E, larger the AMSW, smaller the PeeD.)To see how accurate is the behavior predicted by the above approximate analytic expressions, we have computed numerically (using the PREM profile) the average〈[(1-rμ/e)Pee+rμ/e](Ee,i)〉 as a function of electron energy. Here, 〈O〉 means taking average of Pee over neutrino energies with neutrino fluxes times the differential cross-sections integrated over the true electron energy with response function. In the above expression, rμ/e≡σμ/σe with σe and σμ being the cross-sections of νee and νμe scattering, respectively. The computed results confirm qualitatively the behavior discussed above based on our analytic approximations. Thus, the energy spectrum of solar neutrinos at low and high energies can constrain AMSW in this way, as will be shown quantitatively in Section 3.


### 2.2. Day-Night Variation

Theνe survival probability at night during which solar neutrinos pass through the earth can be written, assuming adiabaticity, as [46, 47]
(2.8)PeeN=PeeD-cos2θScos2θ13〈freg〉zenith,
where PeeD is the one given in (2.1). freg denotes the regeneration effect in the earth and is given as freg=P2e-sin2θ12cos2θ13, where P2e is the transition probability of second mass eigenstate to νe. Under the constant density approximation in the earth, freg is given by [46, 47]
(2.9)freg=ξEcos2θ13sin22θEsin2[AMSWaEcos2θ13(1-2ξE-1cos2θ12+ξE-2)1/2(L2)]
for passage of distance L, where we have introduced aE≡2GFNeearth=2GFρEYeE/mN. In (2.9), θE and ξE stand for the mixing angle and the ξ parameter (see (2.4)) with matter density ρE in the earth. Within the range of neutrino parameters allowed by the solar neutrino data, the oscillatory term averages to 1/2 in good approximation when integrated over zenith angle. Then, the equation simplifies to
(2.10)〈freg〉zenith=12cos2θ13ξEsin22θE.
At E=7MeV, which is a typical energy for 8B neutrinos, ξE=3.98×10-2 and sin2θE=0.940 for the average density ρ-E=5.6g/cm3 and the electron fraction YeE=0.5 in the earth. Then, 〈freg〉zenith is given as 〈freg〉zenith=1.72×10-2 for AMSW=1 and sin22θ13=0.089. This result is in reasonable agreement with more detailed estimate using the PREM profile [27] for the earth matter density.We now give a simple estimate of the day-night asymmetryADN assuming constant matter density approximation in the earth and its AMSW dependence. Under the approximation of small regeneration effect freg≪1, the day-night asymmetry ADN for the CC number of counts NCC measurement is approximately given by
(2.11)ADNCC≡NCCN-NCCD(1/2)[NCCN+NCCD]≈-2cos2θS1+cos2θ12cos2θS〈freg〉zenith,
where in the right-hand side we have approximated ADNCC by the asymmetry of survival probabilities in day and in night at an appropriate neutrino energy and ignored the terms of order 〈freg〉zenith2. Notice that the effects of the solar and the earth matter densities are contained only in cos2θS and 〈freg〉zenith, respectively.AtE=7MeV, ξS=1.31, cos2θ12=0.377, cos2θS=-0.710, and hence ADNCC=3.41×10-2AMSWcos4θ13, about 3% day-night asymmetry for AMSW=1. Note that cos4θ13=0.95 for sin22θ13=0.1, so that the impact of θ13 on ADNCC gives only a minor modification. Though based on crude approximations, the value of ADNCC at AMSW=1 obtained above is in excellent agreement with the one evaluated numerically for SNO CC measurement.SNO and SK observe the day-night asymmetry by measurement of CC reactions and elastic scattering (CC + NC), respectively. We have computedADN as a function of AMSW numerically (with PREM profile) without using analytic approximation. The result of ADN scales linearly with AMSW in a good approximation, ADNCC≈0.044AMSW. Similarly, the day-night asymmetry for elastic scattering measurement can be easily computed. Its relationship to the ADNCC can be estimated in the similar manner as in (2.11),
(2.12)ADNES≡NESN-NESD(1/2)[NESN+NESD]≈ADNCC×[1+2rμ/e(1-rμ/e)[PeeN+PeeD]]-1,
taking into account the modification due to NC scattering. Using approximate values, rμ/e=1/6 and (1/2)[PeeN+PeeD]=1/3, the factor in the square bracket can be estimated to be 5/8, giving a reasonable approximation for the ratio of ADNES to ADNCC. A better approximation to the computed results of the AMSW dependence of the asymmetry is given by ADNES=0.02AMSW.


## 2.1. Energy Spectrum

Solar neutrino observables taken in a single experiment have not shown an energy dependence yet. The neutrino oscillation parameters are such that we cannot expect strong energy dependences. At low neutrino energies, smallξS, (2.1) can be approximated by
(2.6)PeeD=cos4θ13[1-12sin22θ12(1+cos2θ12ξS)]+sin4θ13,
whereas at high energies, small 1/ξS, the oscillation probability (2.1) can be approximated, keeping only the first energy-dependent term as
(2.7)PeeD=cos4θ13[sin2θ12+14sin22θ12cos2θ12(1ξS)2]+sin4θ13.
Notice that the correction to the asymptotic behavior is linear in AMSW at low energies while it is quadratic in AMSW-1 at high energies. It may mean that the energy spectrum at low energies could be more advantageous in tightening up the constraint on AMSW provided that these formulas with leading order corrections are valid.It is well known that, in the LMA MSW mechanism,8B neutrino spectrum must show an upturn from the asymptotic high energy (E≫10MeV) to lower energies. The behavior is described by the correction term in (2.7) but only at a qualitative level. It indicates that the upturn component in the spectrum is a decreasing function of AMSW. On the other hand, at low energies populated by pp, 7Be, and pep neutrinos, the solar neutrino energy spectrum displays vacuum-averaged oscillations or decoherence, (2.6). The deviation from this asymptotic low energy limit can be described by the correction term in (2.6) again at the (better) qualitative level. The term depends upon AMSW linearly so that the correction term is an increasing function of AMSW. Because of the negative sign in the correction term in (2.6), larger values of AMSW lead to smaller absolute values of Pee in both low- and high-energy regions. (The simpler way to reach the same conclusion is to use the property PeeD(E,sAMSW)=PeeD(sE,AMSW) mentioned earlier. Then, for larger AMSW (s>1) PeeD corresponds to the one at higher energy. Since PeeD is a monotonically decreasing function of E, larger the AMSW, smaller the PeeD.)To see how accurate is the behavior predicted by the above approximate analytic expressions, we have computed numerically (using the PREM profile) the average〈[(1-rμ/e)Pee+rμ/e](Ee,i)〉 as a function of electron energy. Here, 〈O〉 means taking average of Pee over neutrino energies with neutrino fluxes times the differential cross-sections integrated over the true electron energy with response function. In the above expression, rμ/e≡σμ/σe with σe and σμ being the cross-sections of νee and νμe scattering, respectively. The computed results confirm qualitatively the behavior discussed above based on our analytic approximations. Thus, the energy spectrum of solar neutrinos at low and high energies can constrain AMSW in this way, as will be shown quantitatively in Section 3.


## 2.2. Day-Night Variation

Theνe survival probability at night during which solar neutrinos pass through the earth can be written, assuming adiabaticity, as [46, 47]
(2.8)PeeN=PeeD-cos2θScos2θ13〈freg〉zenith,
where PeeD is the one given in (2.1). freg denotes the regeneration effect in the earth and is given as freg=P2e-sin2θ12cos2θ13, where P2e is the transition probability of second mass eigenstate to νe. Under the constant density approximation in the earth, freg is given by [46, 47]
(2.9)freg=ξEcos2θ13sin22θEsin2[AMSWaEcos2θ13(1-2ξE-1cos2θ12+ξE-2)1/2(L2)]
for passage of distance L, where we have introduced aE≡2GFNeearth=2GFρEYeE/mN. In (2.9), θE and ξE stand for the mixing angle and the ξ parameter (see (2.4)) with matter density ρE in the earth. Within the range of neutrino parameters allowed by the solar neutrino data, the oscillatory term averages to 1/2 in good approximation when integrated over zenith angle. Then, the equation simplifies to
(2.10)〈freg〉zenith=12cos2θ13ξEsin22θE.
At E=7MeV, which is a typical energy for 8B neutrinos, ξE=3.98×10-2 and sin2θE=0.940 for the average density ρ-E=5.6g/cm3 and the electron fraction YeE=0.5 in the earth. Then, 〈freg〉zenith is given as 〈freg〉zenith=1.72×10-2 for AMSW=1 and sin22θ13=0.089. This result is in reasonable agreement with more detailed estimate using the PREM profile [27] for the earth matter density.We now give a simple estimate of the day-night asymmetryADN assuming constant matter density approximation in the earth and its AMSW dependence. Under the approximation of small regeneration effect freg≪1, the day-night asymmetry ADN for the CC number of counts NCC measurement is approximately given by
(2.11)ADNCC≡NCCN-NCCD(1/2)[NCCN+NCCD]≈-2cos2θS1+cos2θ12cos2θS〈freg〉zenith,
where in the right-hand side we have approximated ADNCC by the asymmetry of survival probabilities in day and in night at an appropriate neutrino energy and ignored the terms of order 〈freg〉zenith2. Notice that the effects of the solar and the earth matter densities are contained only in cos2θS and 〈freg〉zenith, respectively.AtE=7MeV, ξS=1.31, cos2θ12=0.377, cos2θS=-0.710, and hence ADNCC=3.41×10-2AMSWcos4θ13, about 3% day-night asymmetry for AMSW=1. Note that cos4θ13=0.95 for sin22θ13=0.1, so that the impact of θ13 on ADNCC gives only a minor modification. Though based on crude approximations, the value of ADNCC at AMSW=1 obtained above is in excellent agreement with the one evaluated numerically for SNO CC measurement.SNO and SK observe the day-night asymmetry by measurement of CC reactions and elastic scattering (CC + NC), respectively. We have computedADN as a function of AMSW numerically (with PREM profile) without using analytic approximation. The result of ADN scales linearly with AMSW in a good approximation, ADNCC≈0.044AMSW. Similarly, the day-night asymmetry for elastic scattering measurement can be easily computed. Its relationship to the ADNCC can be estimated in the similar manner as in (2.11),
(2.12)ADNES≡NESN-NESD(1/2)[NESN+NESD]≈ADNCC×[1+2rμ/e(1-rμ/e)[PeeN+PeeD]]-1,
taking into account the modification due to NC scattering. Using approximate values, rμ/e=1/6 and (1/2)[PeeN+PeeD]=1/3, the factor in the square bracket can be estimated to be 5/8, giving a reasonable approximation for the ratio of ADNES to ADNCC. A better approximation to the computed results of the AMSW dependence of the asymmetry is given by ADNES=0.02AMSW.


## 3. Constraints onAMSW by Solar Neutrino Observables

In this section we investigate quantitatively to what extentAMSW can be constrained by the current and the future solar neutrino data. The results of our calculations are presented in Figure 1, supplemented with the relevant numbers in Table 2. We will discuss the results and their implications to some details in a step-by-step manner. We first discuss the constraints by the data currently available (Section 3.1). Then, we address the question of how the constraint on AMSW can be tightened with the future solar neutrino data, the spectral upturn of 8B neutrinos (Section 3.2), the low-energy 7Be and pep neutrinos (Section 3.3), and finally the day-night asymmetry of the solar neutrino flux (Section 3.4). We pay special attention to the question of how the constraints on AMSW depend upon the significance of these measurements.Table 2
TheΔχ2 minimum of AMSW, the allowed regions of AMSW at 1σ, and 3σ CL are shown in the first, second, and third columns, respectively, for the analyses with the currently available data (first row), the one with spectrum upturn of 8B neutrinos at 3σ added to the current data (second row), the one with 7Be and pep neutrinos with 5% and 3% accuracies, respectively, added to the current data (third row), the one with the new spectral information in the second and the third rows added to the current data (fourth row), and the one with day-night asymmetry of 8B neutrinos at 3σ added to the current data (fifth row). The last row presents results of global analysis with all the above data. The numbers in parentheses imply the ones obtained with improved knowledge of θ12; see text for details.
Analysis
Δ
χ
2 minimum
Allowed region (1σ)
Allowed region (3σ)
present data
A
MSW
=
1.47
1.05–2.01 (1.05–2.00)
0.65–3.35 (0.65–3.27)
+upturn (3σ)
1.34
1.02–1.79 (1.02–1.76)
0.65–3.00 (0.66–2.88)
+7Be (5%), pep (3%)
1.25
0.97–1.53 (0.97–1.52)
0.65–2.34 (0.65–2.31)
+spectral shape
1.22
0.97–1.49 (0.97–1.46)
0.65–2.23 (0.65–2.12)
+ADN (3σ)
1.17
0.96–1.43 (0.96–1.42)
0.66–1.98 (0.66–1.97)
+all
1.12
0.95–1.33 (0.95–1.32)
0.67–1.78 (0.67–1.73)Figure 1
Δ
χ
2 as a function of AMSW for the currently available solar neutrino data (shown in blue solid line) and the various solar neutrino observables expected in the near future (by color lines specified in the following). The current data include the one from SNO lower energy threshold analysis and SK I–IV. In addition to the current constraints on AMSW, we show the improved constraints when future solar neutrino data are added one by one: 3σ detection of the SK spectral upturn (magenta dashed line), low-energy solar neutrino flux measurements of 7Be at 5% and pep at 3% (red dash-dotted line), and 3σ detection of the SK day-night asymmetry (black dotted line). The red dashed line shows the improved constraints by adding future spectral information at high and low energies. Finally, the global analysis by adding all the spectral information data and the day-night data produces the solid green line.


### 3.1. Current Constraint onAMSW

We include in our global analyses the KamLAND and all the available solar neutrino data [6–15, 17, 19–23]. To obtain all the results quoted in this paper we marginalize over the mixing angles θ12 and θ13, the small mass squared difference Δm212, and the solar neutrino fluxes fi [16, 48] imposing the luminosity constraint [49, 50]. We include in the analysis the θ13 dependence derived from the analysis of the atmospheric, accelerator, and reactor data included in [51] as well as the recent measurement of θ13 by [37–41]. The χ2 used is defined by
(3.1)χglobal2(AMSW)=Marg[χsolar2(Δm212,θ12,θ13,AMSW,fB,fBe,fpp,fCNO)+χKamLAND2(Δm212,θ12,θ13)+χREACTOR+ATM+ACC2(θ13)],
where Marg implies to marginalize over the parameters shown but not over AMSW. Further details of the analysis methods can be found in [48].The currently available neutrino data (blue solid line), which include SNO lower-energy threshold data [21, 22] and SK IV [23], do not allow a very precise determination of the AMSW parameter. A distinctive feature of the Δχ2 parabola shown in Figure 1 is the asymmetry between the small and large AMSW regions. At AMSW<1 the parabola is already fairly steep, and the “wall” is so stiff that can barely be changed by including the future data. While at AMSW>1 the slope is relatively gentle. More quantitatively, AMSW=1.47+0.54-0.42(+1.88-0.82) at 1σ (3σ) CL. The best fit point with the present data is significantly larger than unity, AMSW=1.47. It was 1.32 before and has driven to the larger value mostly by the new SK data which indicates a stronger matter effect than those expected by the MSW LMA region preferred by the KamLAND data. The larger best fit value could also partly be due to an artifact of the weakness of the constraint in AMSW>1 region. Notice that the Standard Model MSW theory value AMSW=1 is off from the 1σ region but only by a tiny amount, as seen in Table 2. Let us understand these characteristics.The lower bound onAMSW mostly comes from the SK and the SNO data which shows that 8B neutrino spectrum at high energies is well described by the adiabatic LMA MSW solution (AMSW=1). The energy spectrum is very close to a flat one with Pee which can be approximated by sin2θ with corrections due to the contribution of the energy-dependent term (see (2.7)). The value is inconsistent with the vacuum oscillation, and hence the point AMSW=0 is highly disfavored, showing the evidence for the matter effect.One would think that the upper bound onAMSW should come from either the low energy solar neutrino data or the deviation from the flat spectra at high energies. But, we still lack precise information on low-energy solar neutrinos, and the spectral upturn of 8B neutrinos has not been observed beyond the level in [19, 20]. Then, what is the origin of the upper bound AMSW<2 at about 1σ CL? We argue that it mainly comes from the day-night asymmetry of 8B neutrino flux which is contained in the binned data of SK and SNO. Recently, the SK collaboration reported a positive indication of the day-night asymmetry though the data is still consistent with no asymmetry at 2.3σ CL [23].To show the point, we construct a very simple model forΔχ2 for the day-night asymmetry ADNES. It is made possible by the approximate linearity of ADNES to AMSW. Let us start from the data of day-night asymmetry at SK I–IV obtained with the D/N amplitude method [23]: ADNES=(2.8±1.1±0.5)%, giving the total error 1.2% if added in quadrature. The expectation of ADNES by the LMA solution is ADNES=AMSW×2.1% for Δm212=7.6×10-5 eV2. Then, one can create an approximate model Δχ2 as Δχ2=[(ADN-2.8%)/1.2%]2=3.1(AMSW-1.3)2.Despite admittedly crude nature it seems to capture the qualitative features ofΔχ2 with the current data (blue solid line) in Figure 1 in region AMSW>1. It is true that it predicts a little too steep rise of Δχ2 and leads to Δχ2≃22 at AMSW=4, whereas Δχ2≃15 in Figure 1. In the actual numerical analysis for Figure 1, however, Δχ2 parabola can naturally become less steep because various other parameters are varied to accommodate such a large values of AMSW. Therefore, we find that about 2σ evidence of ADNES in the SK data is most likely the main cause of the sensitivity to AMSW in the region AMSW>1. The simple model cannot explain the behavior of Δχ2 in region AMSW<1 in Figure 1, because the other more powerful mechanism is at work to lead to stronger bound on AMSW, as discussed above.To what extent an improved knowledge ofθ12 affects AMSW? It was suggested that a dedicated reactor neutrino experiment can measure sin2θ12 to ≃2% accuracy [52, 53]. It is also expected that precision measurement of pp spectrum could improve the accuracy of θ12 determination to a similar extent [48]. Therefore, it is interesting to examine to what extent an improved knowledge of θ12 affects the constraint on AMSW. Therefore, we recompute the Δχ2 curves presented in Figure 1 by adding the artificial term (sin2θ12-BEST)2/0.02 in the Δχ2 assuming 2% accuracy in sin2θ12 determination. The result of this computation is given in Table 2 in parentheses. As we see, size of the effect of improved θ12 knowledge is not very significant.


### 3.2. Spectrum of Solar Neutrinos at High Energies

Evidence for the upturn of8B neutrino spectrum must contribute to constrain the larger values of AMSW because AMSW could be very large without upturn, if day-night asymmetry is ignored. We discuss the impact on AMSW of seeing the upturn in recoil electron energy spectrum with 3σ significance, which we assume to be in the region Ee≥3.5MeV. To calculate Δχ2 we assume the errors estimated by the SK collaboration [23]. Adding the simulated data to the currently available dataset produces the magenta dashed line in Figure 1. We find a 25% reduction of the 3σ allowed range, AMSW=1.34+0.45-0.32(+1.66-0.69) at 1σ (3σ) CL. We can see that it does improve the upper bound on AMSW, for which the current constraint (blue solid line) is rather weak, but the improvement in the precision of AMSW is still moderate.Some remarks are in order about the minimum point ofΔχ2. The best fit point with the present data is at AMSW>1 as we saw above. For the analysis with future data discussed in this and the subsequent subsections, we assume that the Δχ2 minimum is always at AMSW=1 for simulated data. Therefore, the analysis with the present plus simulated data tends to pull the Δχ2 minimum toward smaller values of AMSW, and at the same time make the Δχ2 parabola narrower around the minimum. By conspiracy between these two features the current constraint (blue solid line) is almost degenerate to the other lines at AMSW<1, the ones with spectral upturn (magenta dashed line) and low-energy neutrinos (red dash-dotted line). These features can be observed in Figure 1 and in Table 2.


### 3.3. Spectrum of Solar Neutrinos at Low Energies

Now, let us turn to the low-energy solar neutrinos,7Be and pep lines. The Borexino collaboration has already measured the 7Be neutrino-electron scattering rate to an accuracy of ≃±5%  [15], which we assume throughout this section. For pep neutrinos we assume measurement with 3% precision in the future. See [24] for the first observation of pep neutrinos and its current status of the uncertainties.The measurement of the pep flux has two important advantages, when compared to the7Be flux, in determining AMSW: (a) the neutrino energy is higher, 1.44 MeV, so the importance of the solar matter effects is larger and (b) the uncertainty in the theoretical estimate is much smaller. Firstly, the ratio of the pep to the pp neutrino flux is robustly determined by the SSM calculations, so it can be determined more accurately than the individual fluxes because the ratio depends only weakly on the solar astrophysical inputs. Secondly, a very precise measurement of the 7Be flux, with all the other solar data and assuming energy conservation (luminosity constraint) leads to a very precise determination of the pp and pep flux, at the level of ~1% accuracy [48]. On the other hand, to determine 7Be flux experimentally, we have to use the SSM flux to determine the neutrino survival probability, and therefore, the uncertainties in the theoretical estimate [16] limit the precision of the 7Be flux measurement.The red dash-dotted line in Figure1 shows the result of the combined analysis of future low energy data, an improved 7Be measurement with 5% precision, and a future pep measurement with 3% precision, added to the current data. The obtained constraint on AMSW is AMSW=1.25±0.28(+1.09-0.60) at 1σ (3σ) CL. The resultant constraint on AMSW from above is much more powerful than the one obtained with spectrum upturn of high-energy 8B neutrinos at 3σ.By having solar neutrino spectrum information both at high and low energies it is tempting to ask how tight the constraint becomes if we combine them. The result of this exercise is plotted by the red dashed line in Figure1 and is also given in Table 2. The resultant constraint on AMSW is AMSW=1.22+0.27-0.25(+1.01-0.57) at 1σ (3σ) CL.


### 3.4. Day-Night Asymmetry

To have a feeling on to what extent constraint onAMSW can be tightened by possible future measurement, we extend the simple-minded model discussed in Section 3.1, but with further simplification of assuming AMSW=1 as the best fit. Let us assume that the day-night asymmetry ADNES can be determined with (2/N)% accuracy, an evidence for the day-night asymmetry at Nσ CL. Then, the appropriate model Δχ2 is given under the same approximations as in Section 3.1 as Δχ2=N2(AMSW-1)2. We boldly assume that the day-night asymmetry at 3σ CL would be a practical goal in SK. It predicts Δχ2=9(AMSW-1)2, which means that AMSW can be constrained to the accuracy of 33% uncertainty at 1σ CL.Now, we give the result based on the real simulation of data. The black dotted line in Figure1 shows the constraint on AMSW obtained by future 3σ CL measurement of the day-night asymmetry, which is added to the present solar neutrino data. As we see, the day-night asymmetry is very sensitive to the matter potential despite our modest assumption of 3σ CL measurement of ADNES. (Given the powerfulness of the day-night asymmetry for constraining AMSW, it is highly desirable to measure it at higher CL in the future. Of course, it would be a challenging task and probably requires a megaton class water Cherenkov or large volume liquid scintillator detectors with solar neutrino detection capability. They include, e.g., Hyper-Kamiokande [54], UNO [55], or the ones described in [56]). The obtained constraint on AMSW is AMSW=1.17+0.26-0.21(+0.81-0.51) at 1σ (3σ) CL. The obtained upper bound on AMSW is actually stronger than the one expected by our simple-minded model Δχ2. Apart from the shift of the bast fit to a larger value of AMSW, the behavior of Δχ2 is more like Δχ2≈14(AMSW-1)2 in the region AMSW>1. It can also be seen in Figure 1 that the upper bound on AMSW due to the day-night asymmetry at 3σ CL (black dotted line) is stronger than the one from combined analysis of all the expected measurements of the shape of the spectrum (red dashed line) discussed at the end of Section 3.3.


### 3.5. Global Analysis

We now discuss to what extent the constraint onAMSW can become stringent when all the data of various observable are combined. The solid green line in Figure 1 shows the constraint on AMSW obtained by the global analysis combining all the datasets considered in our analysis. The obtained sensitivity reads AMSW=1.12+0.21-0.17(+0.66-0.45) at 1σ (3σ) CL. Therefore, the present and the future solar neutrino data, under the assumptions of the accuracies of measurement stated before, can constrain AMSW to ≃15% (40%) at 1σ (3σ) CL from below and to ≃20% (60%) at 1σ (3σ) CL from above. If we compare this to the current constraint AMSW=1.47+0.54-0.42(+1.88-0.82) the improvement of the errors for AMSW over the current precision is, very roughly speaking, a factor of ≃1.5–2 in region AMSW<1, and it is a factor of ≃2 at AMSW>1. Noticing that the efficiency of adding more data to have tighter constraint at AMSW<1 is weakened by shift of the minimum of Δχ2, improvement of the constraint on AMSW is more significant at AMSW>1.


## 3.1. Current Constraint onAMSW

We include in our global analyses the KamLAND and all the available solar neutrino data [6–15, 17, 19–23]. To obtain all the results quoted in this paper we marginalize over the mixing angles θ12 and θ13, the small mass squared difference Δm212, and the solar neutrino fluxes fi [16, 48] imposing the luminosity constraint [49, 50]. We include in the analysis the θ13 dependence derived from the analysis of the atmospheric, accelerator, and reactor data included in [51] as well as the recent measurement of θ13 by [37–41]. The χ2 used is defined by
(3.1)χglobal2(AMSW)=Marg[χsolar2(Δm212,θ12,θ13,AMSW,fB,fBe,fpp,fCNO)+χKamLAND2(Δm212,θ12,θ13)+χREACTOR+ATM+ACC2(θ13)],
where Marg implies to marginalize over the parameters shown but not over AMSW. Further details of the analysis methods can be found in [48].The currently available neutrino data (blue solid line), which include SNO lower-energy threshold data [21, 22] and SK IV [23], do not allow a very precise determination of the AMSW parameter. A distinctive feature of the Δχ2 parabola shown in Figure 1 is the asymmetry between the small and large AMSW regions. At AMSW<1 the parabola is already fairly steep, and the “wall” is so stiff that can barely be changed by including the future data. While at AMSW>1 the slope is relatively gentle. More quantitatively, AMSW=1.47+0.54-0.42(+1.88-0.82) at 1σ (3σ) CL. The best fit point with the present data is significantly larger than unity, AMSW=1.47. It was 1.32 before and has driven to the larger value mostly by the new SK data which indicates a stronger matter effect than those expected by the MSW LMA region preferred by the KamLAND data. The larger best fit value could also partly be due to an artifact of the weakness of the constraint in AMSW>1 region. Notice that the Standard Model MSW theory value AMSW=1 is off from the 1σ region but only by a tiny amount, as seen in Table 2. Let us understand these characteristics.The lower bound onAMSW mostly comes from the SK and the SNO data which shows that 8B neutrino spectrum at high energies is well described by the adiabatic LMA MSW solution (AMSW=1). The energy spectrum is very close to a flat one with Pee which can be approximated by sin2θ with corrections due to the contribution of the energy-dependent term (see (2.7)). The value is inconsistent with the vacuum oscillation, and hence the point AMSW=0 is highly disfavored, showing the evidence for the matter effect.One would think that the upper bound onAMSW should come from either the low energy solar neutrino data or the deviation from the flat spectra at high energies. But, we still lack precise information on low-energy solar neutrinos, and the spectral upturn of 8B neutrinos has not been observed beyond the level in [19, 20]. Then, what is the origin of the upper bound AMSW<2 at about 1σ CL? We argue that it mainly comes from the day-night asymmetry of 8B neutrino flux which is contained in the binned data of SK and SNO. Recently, the SK collaboration reported a positive indication of the day-night asymmetry though the data is still consistent with no asymmetry at 2.3σ CL [23].To show the point, we construct a very simple model forΔχ2 for the day-night asymmetry ADNES. It is made possible by the approximate linearity of ADNES to AMSW. Let us start from the data of day-night asymmetry at SK I–IV obtained with the D/N amplitude method [23]: ADNES=(2.8±1.1±0.5)%, giving the total error 1.2% if added in quadrature. The expectation of ADNES by the LMA solution is ADNES=AMSW×2.1% for Δm212=7.6×10-5 eV2. Then, one can create an approximate model Δχ2 as Δχ2=[(ADN-2.8%)/1.2%]2=3.1(AMSW-1.3)2.Despite admittedly crude nature it seems to capture the qualitative features ofΔχ2 with the current data (blue solid line) in Figure 1 in region AMSW>1. It is true that it predicts a little too steep rise of Δχ2 and leads to Δχ2≃22 at AMSW=4, whereas Δχ2≃15 in Figure 1. In the actual numerical analysis for Figure 1, however, Δχ2 parabola can naturally become less steep because various other parameters are varied to accommodate such a large values of AMSW. Therefore, we find that about 2σ evidence of ADNES in the SK data is most likely the main cause of the sensitivity to AMSW in the region AMSW>1. The simple model cannot explain the behavior of Δχ2 in region AMSW<1 in Figure 1, because the other more powerful mechanism is at work to lead to stronger bound on AMSW, as discussed above.To what extent an improved knowledge ofθ12 affects AMSW? It was suggested that a dedicated reactor neutrino experiment can measure sin2θ12 to ≃2% accuracy [52, 53]. It is also expected that precision measurement of pp spectrum could improve the accuracy of θ12 determination to a similar extent [48]. Therefore, it is interesting to examine to what extent an improved knowledge of θ12 affects the constraint on AMSW. Therefore, we recompute the Δχ2 curves presented in Figure 1 by adding the artificial term (sin2θ12-BEST)2/0.02 in the Δχ2 assuming 2% accuracy in sin2θ12 determination. The result of this computation is given in Table 2 in parentheses. As we see, size of the effect of improved θ12 knowledge is not very significant.


## 3.2. Spectrum of Solar Neutrinos at High Energies

Evidence for the upturn of8B neutrino spectrum must contribute to constrain the larger values of AMSW because AMSW could be very large without upturn, if day-night asymmetry is ignored. We discuss the impact on AMSW of seeing the upturn in recoil electron energy spectrum with 3σ significance, which we assume to be in the region Ee≥3.5MeV. To calculate Δχ2 we assume the errors estimated by the SK collaboration [23]. Adding the simulated data to the currently available dataset produces the magenta dashed line in Figure 1. We find a 25% reduction of the 3σ allowed range, AMSW=1.34+0.45-0.32(+1.66-0.69) at 1σ (3σ) CL. We can see that it does improve the upper bound on AMSW, for which the current constraint (blue solid line) is rather weak, but the improvement in the precision of AMSW is still moderate.Some remarks are in order about the minimum point ofΔχ2. The best fit point with the present data is at AMSW>1 as we saw above. For the analysis with future data discussed in this and the subsequent subsections, we assume that the Δχ2 minimum is always at AMSW=1 for simulated data. Therefore, the analysis with the present plus simulated data tends to pull the Δχ2 minimum toward smaller values of AMSW, and at the same time make the Δχ2 parabola narrower around the minimum. By conspiracy between these two features the current constraint (blue solid line) is almost degenerate to the other lines at AMSW<1, the ones with spectral upturn (magenta dashed line) and low-energy neutrinos (red dash-dotted line). These features can be observed in Figure 1 and in Table 2.


## 3.3. Spectrum of Solar Neutrinos at Low Energies

Now, let us turn to the low-energy solar neutrinos,7Be and pep lines. The Borexino collaboration has already measured the 7Be neutrino-electron scattering rate to an accuracy of ≃±5%  [15], which we assume throughout this section. For pep neutrinos we assume measurement with 3% precision in the future. See [24] for the first observation of pep neutrinos and its current status of the uncertainties.The measurement of the pep flux has two important advantages, when compared to the7Be flux, in determining AMSW: (a) the neutrino energy is higher, 1.44 MeV, so the importance of the solar matter effects is larger and (b) the uncertainty in the theoretical estimate is much smaller. Firstly, the ratio of the pep to the pp neutrino flux is robustly determined by the SSM calculations, so it can be determined more accurately than the individual fluxes because the ratio depends only weakly on the solar astrophysical inputs. Secondly, a very precise measurement of the 7Be flux, with all the other solar data and assuming energy conservation (luminosity constraint) leads to a very precise determination of the pp and pep flux, at the level of ~1% accuracy [48]. On the other hand, to determine 7Be flux experimentally, we have to use the SSM flux to determine the neutrino survival probability, and therefore, the uncertainties in the theoretical estimate [16] limit the precision of the 7Be flux measurement.The red dash-dotted line in Figure1 shows the result of the combined analysis of future low energy data, an improved 7Be measurement with 5% precision, and a future pep measurement with 3% precision, added to the current data. The obtained constraint on AMSW is AMSW=1.25±0.28(+1.09-0.60) at 1σ (3σ) CL. The resultant constraint on AMSW from above is much more powerful than the one obtained with spectrum upturn of high-energy 8B neutrinos at 3σ.By having solar neutrino spectrum information both at high and low energies it is tempting to ask how tight the constraint becomes if we combine them. The result of this exercise is plotted by the red dashed line in Figure1 and is also given in Table 2. The resultant constraint on AMSW is AMSW=1.22+0.27-0.25(+1.01-0.57) at 1σ (3σ) CL.


## 3.4. Day-Night Asymmetry

To have a feeling on to what extent constraint onAMSW can be tightened by possible future measurement, we extend the simple-minded model discussed in Section 3.1, but with further simplification of assuming AMSW=1 as the best fit. Let us assume that the day-night asymmetry ADNES can be determined with (2/N)% accuracy, an evidence for the day-night asymmetry at Nσ CL. Then, the appropriate model Δχ2 is given under the same approximations as in Section 3.1 as Δχ2=N2(AMSW-1)2. We boldly assume that the day-night asymmetry at 3σ CL would be a practical goal in SK. It predicts Δχ2=9(AMSW-1)2, which means that AMSW can be constrained to the accuracy of 33% uncertainty at 1σ CL.Now, we give the result based on the real simulation of data. The black dotted line in Figure1 shows the constraint on AMSW obtained by future 3σ CL measurement of the day-night asymmetry, which is added to the present solar neutrino data. As we see, the day-night asymmetry is very sensitive to the matter potential despite our modest assumption of 3σ CL measurement of ADNES. (Given the powerfulness of the day-night asymmetry for constraining AMSW, it is highly desirable to measure it at higher CL in the future. Of course, it would be a challenging task and probably requires a megaton class water Cherenkov or large volume liquid scintillator detectors with solar neutrino detection capability. They include, e.g., Hyper-Kamiokande [54], UNO [55], or the ones described in [56]). The obtained constraint on AMSW is AMSW=1.17+0.26-0.21(+0.81-0.51) at 1σ (3σ) CL. The obtained upper bound on AMSW is actually stronger than the one expected by our simple-minded model Δχ2. Apart from the shift of the bast fit to a larger value of AMSW, the behavior of Δχ2 is more like Δχ2≈14(AMSW-1)2 in the region AMSW>1. It can also be seen in Figure 1 that the upper bound on AMSW due to the day-night asymmetry at 3σ CL (black dotted line) is stronger than the one from combined analysis of all the expected measurements of the shape of the spectrum (red dashed line) discussed at the end of Section 3.3.


## 3.5. Global Analysis

We now discuss to what extent the constraint onAMSW can become stringent when all the data of various observable are combined. The solid green line in Figure 1 shows the constraint on AMSW obtained by the global analysis combining all the datasets considered in our analysis. The obtained sensitivity reads AMSW=1.12+0.21-0.17(+0.66-0.45) at 1σ (3σ) CL. Therefore, the present and the future solar neutrino data, under the assumptions of the accuracies of measurement stated before, can constrain AMSW to ≃15% (40%) at 1σ (3σ) CL from below and to ≃20% (60%) at 1σ (3σ) CL from above. If we compare this to the current constraint AMSW=1.47+0.54-0.42(+1.88-0.82) the improvement of the errors for AMSW over the current precision is, very roughly speaking, a factor of ≃1.5–2 in region AMSW<1, and it is a factor of ≃2 at AMSW>1. Noticing that the efficiency of adding more data to have tighter constraint at AMSW<1 is weakened by shift of the minimum of Δχ2, improvement of the constraint on AMSW is more significant at AMSW>1.


## 4. Summary

In this paper, we have discussed the question of to what extent tests of the MSW theory can be made stringent by various solar neutrino observables. First, we have updated the constraint onAMSW, the ratio of the effective coupling constant of neutrinos to GF, the Fermi coupling constant with the new data including SNO 8B spectrum and SK day-night asymmetry. Then, we have discussed in detail how and to what extent the solar neutrino observable in the future tightens the constraint on AMSW.The features of the obtained constraints can be summarized as follows.(i)
Interpretation of solar neutrino data at high energies by the vacuum oscillation is severely excluded by the SNO and SK experiments, which leads to a strong and robust lower bound onAMSW. On the other hand, the day-night asymmetry at ≃2σ level observed by SK dominates the bound at high AMSW side. We find that present data lead to AMSW=1.47+0.54-0.42(+1.88-0.82) at 1σ (3σ) CL. The Standard Model prediction AMSW=1 is outside the 1σ CL range but only by tiny amount.(ii)
We have explored the improvements that could be achieved by solar neutrinos experiments, ongoing and in construction. We discussed three observables that are sensitive enough to significantly improve the limits onAMSW, particularly in the region AMSW>1: (a) upturn of the 8B solar neutrino spectra at low energies at 3σ CL, (b) high precision measurement of monoenergetic low-energy solar neutrinos, 7Be (5% precision), and pep (3% precision) neutrinos, and (c) day-night asymmetry of the 8B solar neutrino flux at 3σ CL. They lead to the improvement of the bound as follows:
(a)
AMSW=1.34+0.45-0.32(+1.66-0.69) at 1σ (3σ) CL,(b)
AMSW=1.25±0.28(+1.09-0.60) at 1σ (3σ) CL,(c)
AMSW=1.17+0.26-0.21(+0.81-0.51) at 1σ (3σ) CL.It could be expected that future measurement by SNO+ [57, 58] and KamLAND [59] may detect spectrum modulation of B neutrinos at low energies at CL higher than 3σ.Finally, by combining all the datasets we have considered we obtainAMSW=1.12+0.21-0.17(+0.66-0.45) at 1σ (3σ) CL.(iii)
As mentioned in Section1, the issue of effective neutrino matter coupling in a wider context may be better treated in the framework of NSI. If we think about the extended setting together with accelerator and atmospheric neutrino measurement to look for effects of NSI, the off-diagonal elements εαβ (α≠β) can be better constrained by long-baseline experiments. In fact, in a perturbative treatment with small parameter ϵ≡Δm212/Δm312 with the assumption εαβ~ϵ, the terms with εeμ and εeτ are of second order in ϵ, while εee comes only at third order in ϵ [60]. The analyses show that the sensitivity to εee is indeed lower at least by an order of magnitude compared to the ones to εeμ or εeτ. See the analysis in [61] and the references cited therein. It is also known that εμτ can be severely constrained by atmospheric neutrinos [62]. Hence, we feel that the solar neutrinos are still a powerful and complementary probe for εee in such extended setting.In conclusion, testing the theory of neutrino propagation in matter deserves further endeavor. The lack of an accurate measurement of the matter potential felt by solar neutrinos reflects the fact that solar neutrino data only do not precisely determine the mass square splitting. The good match of the independently determined mass square splitting by solar neutrino data and by reactor antineutrino data will confirm the Standard Model prediction of the relative index of refraction of electron neutrinos to the other flavor neutrinos. The lack of match of both measurements would point to new physics like the one tested here.



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*Source: 349686-2012-11-05.xml*