On the Day-Night Effect and CC to NC Event Rate Ratio Predictions for the SNO Detector

Detailed predictions for the D-N asymmetry for the Super-Kamiokande and SNO experiments, as well as for the ratio of the CC and NC event rates measured by SNO, in the cases of the LMA MSW and of the LOW solutions of the solar neutrino problem, are derived. The possibilities to further constrain the regions of the LMA MSW and LOW solutions of the solar neutrino problem by using the forthcoming SNO data on the D-N asymmetry and on the CC to NC event rate ratio are also discussed.


Introduction
The recent SNO results [1], combined with the data from the Super-Kamiokande experiment [2], clearly demonstrate the presence of ν µ ( ν τ ) component in the flux of solar neutrinos reaching the Earth 2 . This represents a compelling evidence for oscillations and/or transitions of the solar neutrinos.
The SNO experiment measured the rate of the charged current (CC) reaction ν e +D → e − +p+p for T e ≥ 6.75 MeV, T e being the (effective) kinetic energy of the final state electron [1]. The reaction is due to the flux of solar ν e from 8 B decay having energy of E ∼ > 8. 2 MeV. Assuming that the 8 B neutrino energy spectrum is not substantially modified by the solar neutrino oscillations, the SNO collaboration obtained the following value of the solar ν e flux: Φ CC (ν e ) = (1.75 ± 0.15) × 10 6 cm −2 s −1 , where we have added the statistical and systematic errors and the estimated theoretical uncertainty (due to the uncertainty in the CC reaction cross section) given in [1] in quadrature. Utilizing the data on Φ CC (ν e ) and the data on the solar neutrino flux obtained by the Super-Kamiokande experiment, it is possible to deduce [1] (see also [3]) the value of the non-electron neutrino component in the flux of solar neutrinos measured by the Super-Kamiokande collaboration: Φ(ν µ,τ ) = (3.69 ± 1.13) × 10 6 cm −2 s −1 .
The best fit values of ∆m 2 and sin 2 2θ for the LOW solution, derived, e.g., in [3,8,9,10,11,14] are compatible with each other and are all approximately given by (∆m 2 ) BF V ∼ = 10 −7 eV 2 and (sin 2 2θ) BF V ∼ = (0.94 − 0.97). A substantially different value of (∆m 2 ) BF V was found in [16]: (∆m 2 ) BF V ∼ = 5.5 × 10 −8 eV 2 and (sin 2 2θ) BF V ∼ = 0.99. The analyses [3,8,9,10,11,12,16] were based, in particular, on the standard solar model (SSM) predictions of ref. [18] (BP2000) for the different components of the solar neutrino flux (pp, pep, 7 Be, 8 B, CN O, hep, 17 F). In [3,8,9,10,11,12,14] the published Super-Kamiokande data on the day-night (D -N) asymmetry [2] were used as input in the analyses, while in [16] the latest (preliminary) results on the D-N asymmetry, obtained from the analysis of all currently available Super-Kamiokande solar neutrino data was utilized (see further). The authors of ref. [14] have used in their analysis a new value of the 8 B neutrino flux which is suggested by the results of the latest (and more precise) experimental measurement [19] of the cross section of the reaction p+ 7 Be → 8 B +γ. According to the SSM, the 8 B is produced in the Sun in the indicated reaction and the β + −decay of 8 B in the central part of the Sun gives rise to the solar 8 B neutrino flux. The results obtained in [19] give a larger p− 7 Be reaction cross-section (with smaller uncertainty), and correspondingly -a larger astrophysical factor S 17 (see, e.g., [14]) than the one used in [18], which implies, in particular, a larger value of the 8 B neutrino flux than the value predicted 3 in [18]. In the global Bayesian analysis performed in [15] the SSM predictions for the solar neutrino fluxes were not used: both the values of the fluxes and of the oscillation parameters were derived from the data.
The best fit values of ∆m 2 found in [3,8,9,10,11] differ from that derived in [16] essentially due to the difference in the Super-Kamiokande data on the D-N asymmetry used as input in the corresponding analyses: in [16] the latest (preliminary) Super-Kamiokande result implying a smaller mean value of the D-N asymmetry than the published one in [2] was utilized. The smaller possible D-N asymmetry drives (∆m 2 ) BF V to larger (smaller) value in the LMA MSW (LOW) solution region [16]. Although the data on the D-N asymmetry used in [3,8,9,10,11] and in [14] are the same, the best fit value of ∆m 2 in the LMA MSW solution region found in [14] is smaller than those found in [3,8,9,10,11] because of the difference between the values of the astrophysical factor S 17 , and thus of the 8 B neutrino flux, used in [14] and in 4 [3,8,9,10,11].
In the present article we update our earlier predictions [20,21,22] for the D-N asymmetry for the Super-Kamiokande and SNO experiments, taking into account the recent progress in the studies of solar neutrinos. The day-night (D-N) effect -a difference between the solar neutrino event rates during the day and during the night, caused by the additional transitions of the solar neutrinos taking place at night while the neutrinos cross the Earth on the way to the detector (see, e.g., [23,24] and the references quoted therein), is a unique testable prediction of the MSW solutions of the solar neutrino problem. The experimental observation of a non-zero D-N asymmetry where R N and R D are, e.g., the one year averaged event rates in a given detector respectively during the night and the day, would be a very strong evidence in favor (if not a proof) of an MSW solution of the solar neutrino problem. Extensive predictions for the magnitude of the D-N effect for the Super-Kamiokande and SNO detectors have been obtained in [20,21,22,25,26,27,28]. High precision calculations of the D-N asymmetry in the one year averaged recoil-e − spectrum measured in the Super-Kamiokande experiment and in the energy-integrated event rates for the 3 The 8 B neutrino flux predicted in [18] reads Φ(B)BP 2000 = 5.05 × (1 +0.20 −0.16 ) × 10 6 cm −2 s −1 , while the flux utilized in the analysis performed in [14] is Φ(B)NEW = 5.93 × (1 +0.14 −0.13 ) × 10 6 cm −2 s −1 .
two experiments were performed for three event samples, Night, Core and Mantle, in [20,21,22,27]. The night fractions of these event samples are due to neutrinos which respectively cross the Earth along any trajectory, cross the Earth core, and cross only the Earth mantle (but not the core), on the way to the detector. We focus here, in particular, on providing detailed predictions for the D-N asymmetry for the LMA MSW and the LOW solutions of the solar neutrino problem, which are favored by the current solar neutrino data. We will consider in what follows the Night (or Full Night) and the Core D-N asymmetries, A N D−N and A C D−N . The current Super-Kamiokande data [2] do not contain evidence for a substantial D-N asymmetry: the latest published result on A N D−N reads [2] A N D−N (SK) = 0.033 ± 0.022 (stat.) +0.013 −0.012 (syst.), while the result of the latest analysis of all currently available Super-Kamiokande solar neutrino data gives even smaller mean value [16] A N D−N (SK) = 0.021 ± 0.022 (stat.) +0.013 −0.012 (syst.).
Adding the errors in eqs. (7) and (8) which is normalized above to the value of the same ratio in the absence of oscillations of solar neutrinos, R 0 First results on the D-N asymmetry and on the CC to NC event rate ratio R SN O CC/N C are expected to be published in the near future by the SNO collaboration. We discuss as well the possibilities to further constrain the regions of the LMA MSW and LOW-QVO solutions of the solar neutrino problem by using the forthcoming SNO data on the D-N asymmetry A N D−N and on the CC to NC event rate ratio R SN O CC/N C . Updated predictions for the Night D-N asymmetry and the average CC to NC event rate ratio for the SNO experiment were derived after the publication of the first SNO results also in [11,14]. However, our study overlaps little with those performed in [11,14]. Consider the predictions for the D-N asymmetry in the case of the LMA MSW solution. As are smaller than 1%. For given ∆m 2 ∼ < 10 −4 eV 2 and sin 2 2θ from the LMA solution region we have [22] in the indicated region is due to i) the contribution of the NC ν µ(τ ) − e − elastic scattering reaction (in addition to that due to the ν e − e − elastic scattering) to the solar neutrino event rate measured by the Super-Kamiokande experiment, and ii) to the relatively small value of the solar ν e survival probability in the Sun,P ∼ 0.3. The indicated NC contribution to the Super-Kamiokande event rate tends to diminish the D-N asymmetry. Obviously, there is no similar contribution to the SNO CC event rate.
Thus, in the case of the LMA MSW solution of the solar neutrino problem, the D-N asymmetry measured in SNO can be considerably larger than the D-N asymmetry measured in the Super-Kamiokande experiment [22]. The 2 s.d. upper limit on the D-N asymmetry, A N D−N (SK) < 8.5% (7.3%), following from the Super-Kamiokande data, eq. (7) (eq. (8)), for instance, does not exclude a D-N asymmetry in the SNO CC event rate as large as ∼ (10 − 15)%. As Fig. 2 shows, can reach a value of ∼ 20% in the 99% C.L. region of the LMA MSW solution, eq. (3). In the 95% C.L. LMA solution region of [16] one has A N D−N (SK) ∼ < 13%. In the best fit point of the LMA MSW solution, found in [3,8,9,11], we get for T e,th = 6.75 MeV ( 5.  [3,8,9,11], [14] and [16] for where r ≡ σ(ν µ(τ ) e − )/σ(ν e e − ), σ(ν l e − ) being the ν l − e − elastic scattering cross section, l = e, µ, τ , andP is the average probability of solar ν e survival in the Sun. For the solar neutrino energies of interest one has r ∼ = 0.155. For ∆m 2 and sin 2 2θ from the LMA MSW solution region, the transitions of the solar ( 8 B) neutrinos with energies E ∼ > 5.0 MeV are adiabatic and in a relatively large sub-region one findsP ∼ = sin 2 θ. We would like to emphasize that the relation (10) is not very precise and can serve only for rough estimates.
As a comparison of Figs. 2 and 3 indicates, for given ∆m 2 and sin 2 2θ from the LMA MSW solution region, the Core D-N asymmetry in the SNO detector is predicted to be larger than the Full Night D-N asymmetry typically by a factor of ∼ 1.2 [22]: In the best fit point of the solution's region found in [3,8,9,11] and in [14] we get for T e,th = 6.  (Fig. 1). Obviously, an observation of A N D−N (SN O) ∼ > 10% will strongly disfavor the LOW solution of the solar neutrino problem.

Predictions for R SN O CC/N C
The importance of the measurement of the CC to NC solar neutrino event rate ratio in the SNO experiment, R SN O CC/N C , for determining the correct solution of the solar neutrino problem has been widely discussed (see, e.g., [33,34] and the references quoted therein). We have performed a high precision calculations of the ratio R SN O CC/N C , in particular, for three different CC reaction crosssections which were taken from [30,31,32], and using the electron number density distribution in the Sun from [18]. The differences in the results obtained for R SN O CC/N C using the three cross sections are negligible in the regions of the LMA MSW and LOW solutions of the solar neutrino problem of interest. We have found that the effect of the SNO energy resolution function on the predictions for R SN O CC/N C is negligible as well. The SNO experiment will measure the CC and NC average event rates, R exp SN O (CC) and R exp SN O (N C). In order to compare these results with the predictions for the double ratio R SN O CC/N C , eq. (9), one has to use as a normalization factor the theoretically calculated (in the absence of solar neutrino oscillations) value of the ratio R 0 of interest is practically the same when it is calculated within a given theoretical model, ref. [31] or ref. [32], for the CC and NC reaction cross sections: for T e,th = 5.00 (6.75) MeV we find . This is not the case, however, if one calculates the ratio of interest by taking the CC reaction cross section from ref. [31] and the NC reaction cross section from ref. [32] and vice versa -the CC cross section from ref. [32] and the NC cross section from ref. [31]. One finds for T e,th = 5.00 (6.75) MeV in the two cases, respectively: Similarly, although the effect of the SNO energy resolution function on the predictions for the double ratio R SN O CC/N C is negligible, it is not negligible in the case of the ratio R 0 For T e,th = 6.75 MeV, for instance, we find that the value of R 0 using the results of ref. [31] (or of ref. [32]) assuming ideal resolution, is by a factor of 1.033 bigger than the value obtained by taking the SNO resolution function into account. Our predictions for the average ratio during the day (Day ratio), R SN O CC/N C (D), during the night (Full Night ratio), R SN O CC/N C (N ), and for the case of the CC event rate produced at night by solar neutrinos which cross the Earth core on the way to SNO (Core ratio), R SN O CC/N C (C), are shown respectively in Figs. 4 -6. Results for each of the three ratios were obtained for two values of the (effective) kinetic energy threshold of the detected e − in the CC reaction: for T e,th = 6.75 MeV (upper panels) and T e,th = 5.00 MeV (lower panels).
A comparison of Fig. 4 and Figs. 5 -6 shows that CC to NC ratio increases substantially during the night for values of ∆m 2 and sin 2 2θ from the region 2 × 10 −7 eV 2 ∼ < ∆m 2 ∼ < 2 × 10 −5 eV 2 , 10 −2 ∼ < sin 2 2θ ∼ < 0.98, which, however, is not favored by the current solar neutrino data. The increase is due to the Earth matter effect. The difference between Night and Core ratio in the indicated region is essentially caused by the Earth mantle-core interference effect [36]. In the best fit points in the LMA MSW solution region, obtained in [3,8,9,11], [14] and [16], we get, respectively, R  [8,16]. In the LOW solution best fit points found in [3,8,9,11,14] and [16], we obtain for T e,th = 6.75 MeV (5. where R SN O CC/N C is the averaged ratio over the period of SNO data-taking [1], and we have used eqs. (1) and (2). Taking into account an uncertainty corresponding to " 1 standard deviation" and to "2 standard deviations", we find from eq.

Conclusions
In the present article we have derived detailed predictions for the D-N asymmetry in the solar neutrino induced CC event rate in the SNO detector for the LMA MSW and the LOW solutions of the solar neutrino problem, which are favored by the current solar neutrino data. We have obtained results for the Night (or Full Night) and the Core D-N asymmetries for SNO, A  [2,16]. In the best fit point of the LMA MSW solution region found in [3,9,8,11] and in [14] we get for T e,th = 6.75 MeV ( 5.