Day-night asymmetries in active-sterile solar neutrino oscillations are discussed in the general mixing framework with three active and sterile neutrinos. Analytical expressions of the probability of neutrino flavor transitions in the Earth in the perturbative approximation and in the slab approximation are presented and the effects of active-sterile mixing and of the CP-violating phases are discussed. The accuracy of the analytical approximations and the properties of the day-night asymmetries are illustrated numerically in the 3+1 neutrino mixing framework.
Day-Night Asymmetries in Active-Sterile Solar Neutrino Oscillations
H.W. Long 111E-mail: firstname.lastname@example.org
Department of Modern Physics, University of Science and
Technology of China, Hefei, Anhui 230026, China
Y.F. Li 222E-mail: email@example.com
Institute of High Energy Physics, Chinese Academy of
Sciences, Beijing 100049, China
C. Giunti 333E-mail: firstname.lastname@example.org
INFN, Sezione di Torino, Via P. Giuria 1, I–10125 Torino, Italy
After the measurement of non-zero in recent reactor [1, 3, 2] and accelerator [4, 5] neutrino experiments, we have a well-established standard framework of three-neutrino oscillations, which explains solar, atmospheric, reactor and accelerator neutrino oscillation data  by two distinct mass-squared differences (i.e., and ) and three non-zero mixing angles (, and ). However, there are some short base-line (SBL) neutrino oscillation anomalies that cannot be explained in the three-neutrino mixing framework: the anomalies found in the LSND  and MiniBooNE  accelerator experiments, the reactor antineutrino anomaly  and the Gallium anomaly [10, 11]. A neutrino oscillation explanation of these anomalies implies the existence of at least one extra mass-squared difference , such that , and a small mixing of the three active neutrinos with extra sterile neutrino states [11, 12, 13, 14]. Moreover, analyses [15, 16, 17] of the data from cosmological observations and Big-Bang Nucleosynthesis may point to the existence of additional dark radiation in the Universe, for which light sterile neutrinos are one of the most promising candidates.
The existence of light sterile neutrinos and their mixing with the standard active neutrinos can be tested also by studying their effects in solar [18, 19, 20, 21, 22] and atmospheric  neutrino oscillations, as well as in the processes of the beta decay [24, 25] and neutrino-less double beta decay . In this paper we discuss the possible effects of active-sterile mixing on the day-night asymmetries of solar neutrino oscillations.
Solar neutrinos are produced in the core of the Sun and undergo the large-mixing-angle (LMA) Mikheev-Smirnov-Wolfenstein (MSW) mechanism [27, 28] before they propagate out of the solar surface. This mechanism was established about ten years ago after the decisive evidences in favor of solar neutrino oscillations obtained in the SNO  and KamLAND  experiments. Since then, testing the MSW mechanism with low energy solar neutrinos  and searching for sub-leading effects [18, 19, 20, 21] beyond the standard three-neutrino oscillation have become the central concern of solar neutrino searches. Moreover, the terrestrial matter effect in terms of the regeneration effects and day-night asymmetries  of solar neutrino oscillations can give direct and independent tests of the MSW mechanism and constrain different kinds of sub-leading effects. Different calculations of the day-night asymmetries in the three-neutrino mixing scheme have been discussed in the recent literature [33, 34, 35, 36]. In this work we extend the discussion to the general framework with three active and sterile neutrinos (the scheme). In this study, we derive the analytical expressions of the day-night asymmetries in the mixing scheme without any constraint on the mixing elements, assuming only a realistic hierarchy on the mass-squared differences. Calculations using both the perturbative approximation and the slab approximation are checked by comparing their results with those of accurate numerical calculations.
The outline of this work is planned as follows. In Section 2 we review briefly the general framework of neutrino evolution in matter and present the neutrino oscillation probabilities relevant to the experimental results in the daytime . In Section 3 we calculate the oscillation probabilities in the nighttime for solar neutrinos propagating inside the Earth before being detected, and we discuss the calculation of the day-night asymmetries by using different approximation methods. In Section 4 we test the accuracy of our analytical calculations in different approximations comparing the results with those obtained with an accurate numerical solution of the solar neutrino evolution inside the Earth. We also illustrate the properties of the day-night asymmetries using the analytical approximations. Finally, we conclude in Section 5 and we present in the Appendices additional details of the calculations and the parametrization of the four-neutrino mixing matrix that we use in the examples.
2 Oscillations from the Sun to Earth
In this Section, we review the general framework in Ref.  of the neutrino evolution in matter and the solar neutrino oscillation probabilities in the daytime.
where is the flavor transition amplitudes, is the neutrino mixing matrix, and
where is the neutrino energy and are the mass-squared differences. The charge-current (CC) and neutral-current (NC) neutrino matter potentials in Eq. (3) are defined as
where is the Fermi constant, and are the electron and neutron number densities respectively, and is the Avogadro’s number. In a neutral medium, we can use the electron fraction
to define the ratio of matter potentials as
In the vacuum mass basis , the flavor transitions generated by are decoupled from those generated by the larger mass-squared differences, because of the hierarchy
for solar and terrestrial matter densities. Therefore, the full neutrino evolution equation in Eq. (1) can be truncated to a evolution equation
All the other amplitudes evolve independently as
The vacuum oscillation wave number , the effective potential , the matter-dependent mixing angle and the complex phase are defined respectively as
We can define an effective mass basis in matter for the two-neutrino framework in Eq. (8) as
where is the amplitude vector in the effective mass basis, and
is the unitary matrix which diagonalizes the Hamiltonian in Eq. (9). The evolution equation in the effective mass basis in matter is
where the Hamiltonian is given by
Note that we have decomposed the Hamiltonian into the adiabatic (ad) and non-adiabatic (na) parts in Eq. (21), which take into account the dependence of and on the variable matter density along the neutrino propagation path.
To proceed, we define a parametrization  of the neutrino mixing matrix as
and a formal solution of the averaged amplitudes for solar neutrino evolution as
where is the coordinate of the detector on the Earth, is the level-crossing probability generated by the off-diagonal non-adiabatic terms in Eq. (21) between and during their propagation inside the Sun.
Finally, we obtain the averaged solar neutrino oscillation probabilities in the daytime (see Ref.  for a careful derivation) as
and all the parameters with a ‘0’ superscript are defined at the neutrino production point.
3 Day-Night Asymmetries
In Section 2 we have briefly reviewed the derivation of the solar neutrino oscillation probabilities (26) in the daytime. However, solar neutrino fluxes might be affected by the Earth’s matter if they are detected at night and travel inside the Earth before detection (the so-called “regeneration effect”). Therefore, there can be measurable differences of the oscillation probabilities in the daytime and nighttime which can be quantified by defining suitable day-night asymmetries. In the standard framework of three-neutrino mixing, there is only a single day-night asymmetry for the electron neutrino survival probability [33, 34, 35, 36]. In the general mixing scheme with three active and sterile neutrinos which we consider there is an additional asymmetry for the electron-to-sterile neutrino transition probability, which can be revealed by precise neutral-current measurements.
Because of the spread of the solar neutrino production areas and finite detector energy resolution, the contributions of the different mass eigenstates to the oscillation probabilities are incoherent . Therefore, the oscillation probabilities without [crossing only the Sun (S)] and with [crossing both the Sun (S) and the Earth (E)] the terrestrial matter effects are given, respectively, by111 Some aspects of the terrestrial matter effects in the case of four-neutrino mixing have been already discussed in Refs. [38, 39, 20].
is the transition probability between and in vacuum. From Eq. (10), we obtain
and due to the unitarity of oscillation probabilities, we have the following relations
By using the above relations and the probabilities in Eq. (26), we obtain the neutrino oscillation probabilities including the terrestrial matter effects as follows:
Using Eqs. (25), (26) and (27), this expression can be written as222From the discussion in Ref. , we know that is negligibly small in the Sun. Therefore, we neglect it in the numerical discussion in Section 4.
are the regeneration factors of active-sterile solar neutrino oscillations.
It is convenient to define the day-night asymmetries of the oscillation probabilities
which depend on the mixing parameter , on the solar oscillations quantities and , and on the regeneration factors in the Earth.
Note that, taking into account the unitarity of probabilities, the measurable event rates in solar experiments depend on two independent regeneration factors: and . To count the number of relevant mixing angles and CP-violating phases in the regeneration factors, we can employ the same arguments presented in Ref. . Because of the freedom in the complex rotations of the - sector and among the sterile neutrino flavors, the probability depends on 3+2 mixing angles and 2 Dirac CP-violating phases. In order to express the regeneration factors and the oscillation probabilities in terms of this minimal number of oscillation parameters, it is necessary to choose a convenient parameterization of the mixing matrix. However, considering for example the 3+1 mixing scheme, it is interesting to study the information that solar neutrino measurements can give on the elements , and of the mixing matrix, which quantify the mixing of active and sterile neutrinos. Therefore, it is convenient to use the parametrization of in Eq. (87), in which , and are proportional, respectively, to the sines of the mixing angles , and and and are directly related, respectively, to the CP-violating phases and [see Eqs. (90) and (95)]. In this case, the regeneration factors and the oscillation probabilities depend on all the six mixing angles and the three CP-violating phases in Appendix B333 The mixing angles and CP-violating phases of the appropriate parameterization in which the regeneration factors and the oscillation probabilities depend on only five mixing angles and two CP-violating phases are complicated functions of the mixing parameters in Appendix B. .
where the amplitudes in the vacuum mass basis with can be calculated using the evolution equation inside the Earth.
The most accurate calculation of the oscillation probabilities can be obtained with a numerical solution of the evolution equation (8), using a precise density profile of matter in the Earth. However, such a numerical solution is too time-consuming if one wants to explore a large volume of the space of the mixing-parameters and often one needs analytic expressions for the oscillation probabilities in order to study their properties. Therefore, in the following two subsections, we describe two different approximations which allow us to obtain analytical expressions of . We discuss the accuracy of these approximations in Section 4.
3.1 Perturbative Approximation
For the density profile inside the Earth, the matrix elements of the non-adiabatic Hamiltonian in Eq. (21) are much smaller than those of the adiabatic Hamiltonian and can be treated as a perturbation. This is due to the fact that the effective potential itself can be treated as a perturbation, because . Therefore, both and are small, even at the boundaries of two adjacent shells in the Earth where there are sudden changes of the matter density.
In the -matrix formalism, the neutrino evolution can be written formally as
where the superscript denotes the mass basis in vacuum () or in matter (), and and are the initial and final points of the neutrino trajectory. The two -matrices in the vacuum and matter mass bases are connected by the transformation
with given in Eq. (18).
Since the matter effects at the boundaries (i.e., and ) of the neutrino path inside the Earth are negligible, we have and . Therefore, as shown in Appendix A.1, the transition probability can be approximated to the first order of and as
where . Notice that is invariant under the rephasing transformation of , and gives the intrinsic phase-dependence of the day-night asymmetries. Furthermore, as shown in Appendix A.1, in the approximation of a constant electron fraction  (i.e., ), the transition probability can be further simplified to the first order of as
In Appendix A.3 we derive the limit of this expression for in the case of vanishing active-sterile mixing in order to show that it agrees with that presented in the case of standard three-neutrino mixing in Ref. .
In Section 4 we will show that the perturbative approximation gives an accurate description of neutrino evolution inside the Earth. However, since the integrations in Eqs. (48) and (49) are very time-consuming, it is useful to investigate if there are other approximate methods which give accurate and more rapid solutions of the evolution equation. In the next subsection we discuss the analytical calculation of in the slab approximation.
3.2 Slab Approximation
In the slab approximation, the radial symmetric profile of the matter density inside the Earth is divided into shells (, where is the innermost shell), such that the density variation is very small within each shell and sudden changes happen at the boundaries (the biggest change happens at the mantle-core boundary).
When a solar neutrino travels inside the Earth, the neutrino trajectory crosses the outer shells () of the Earth and contains segments with constant density. The number and the lengths of the segments depend on the nadir angle of the trajectory, which ends at the detector. We denote the coordinates of the segment boundaries as , starting from the beginning of the first segment at where the neutrino enters the Earth and following the neutrino path until it reaches the detector at . For convenience, we have define a point at the middle point of the trajectory, which artificially splits in two equal parts the segment of the trajectory in the ()-th shell. In this way, there are two segments with equal length in each shell: in the ()-th shell, with , the two segments have boundaries and .
In each segment of the trajectory the electron fraction , the CC potential and the induced parameters , , , and are constant. Due to the radial symmetry of the Earth, we have for . Since the matrix elements of the non-adiabatic Hamiltonian of Eq. (21) vanish in each segment and is constant, we obtain
where and the subscript indicates the ()-th shell. Since , the evolution inside the Earth is given recursively by
Note that there are two in Eq. (53) because we have divided the innermost segment into two connected ones with the central point . By using Eqs. (40), (52) and (53), we can calculate the transition probability in a self-consistent way. This method is usually adopted in the data analysis of solar neutrino experiments, because it is much more rapid and efficient than the perturbation method.
Employing the perturbative approximation for the effective potential , we can further simplify the transition probability to the first order of as
where is the length of the trajectory inside the outer boundary of the ()-th shell:
The derivation of this expression is described in details in Appendix A.2. We also derive in Appendix A.3 the limit in the case of vanishing active-sterile mixing of the expression (54) for the transition probability , in order to show that it agrees with that derived in the case of standard three-neutrino mixing in Ref. .
Finally, let us discuss the dependence in and on the mixing parameters. From Eqs. (49) and (54), one can see that the regeneration factors are explicitly proportional to the mixing parameters and . Moreover, the effective potential , the effective mixing angle and the oscillation wave number in matter take part in the regeneration factors via the integration or summation of the contributions in the whole trajectory inside the Earth. The most distinct property in is the explicit dependence on the CP-violating phases , which contribute to the oscillating phases.
4 Numerical Discussion
In the previous Section we have obtained in Eqs. (49), (52) and (54) three approximate analytical expressions for the transition probability . We will call them, respectively, the perturbative approximation (PERT), the slab approximation (SLAB) and the slab plus perturbative approximation (SLABPERT). In this Section, we are going to test the accuracy of these analytical approximations in the case of mixing and discuss the main properties of the corresponding day-night asymmetries.
First of all, in order to calculate the neutrino evolution inside the Earth we need an accurate description of the density profile of the Earth. In our calculations, we employ a simplified version of the preliminary Earth reference model (PREM) , which contains  five shells and uses the polynomial function
for the -th shell (, where is the innermost shell) to describe the Earth’s density at the radial distance . The values of the coefficients are given in Table 1, which gives also the averaged electron density in the -th shell, which will be used in the slab approximation. The electron fractions are also provided for the core and mantle regions. The cosine values of the nadir angle corresponding to the radial boundaries of each shell are given in the form . We employ the fourth-order Runge-Kutta method for the numerical solution of the neutrino evolution equation in the Earth, using the density matrix formalism (see Appendix C of Ref.  for a detailed introduction). Finally, in Appendix B we give the explicit parametrization of and the values of the oscillation parameters used in our examples.
In Figs. 1 and 2, we check the accuracy of the three analytical approximations by drawing the regeneration factors (i.e., and ) as functions of the nadir angle, for a neutrino energy of . The three upper panels show the comparison of the result of each analytical approximation (PERT, SLAB and SLAB+PERT from top to bottom) with that of the numerical calculation. In the lowermost panels we have drawn the differences between the analytical and numerical results as functions of the nadir angle.
From Figs. 1 and 2, we observe that the PERT approximation gives an accurate description of the solar neutrino evolution inside the Earth, even for the neutrino trajectories crossing the core (i.e. for ). One can also see that the SLAB approximation gives the correct frequencies of the oscillatory behavior of and , but there are significant differences in the oscillating amplitudes with respect to those obtained with the numerical calculation for neutrinos crossing the core. However, as we will show later, these discrepancies are acceptable when the regeneration factors are averaged over the relevant ranges of the energy and/or nadir angle. Finally, one can notice that the accuracy of the SLAB+PERT approximation is low, both for the frequencies and the amplitudes of the oscillatory regeneration factors. Therefore, the SLAB+PERT approximation can be used only for a qualitative analysis and is not suitable for any realistic calculations.
The PERT approximation is the most accurate one, but it turns out to be very time-consuming because of the two-dimensional numerical integration in Eq. (49). Hence, in the data analysis of solar neutrino experiments it is convenient to employ the SLAB approximation, which gives a rapid and approximately accurate description for the terrestrial matter effect. In the following discussion, we use the SLAB approximation as our default choice.
The regeneration factors and the day-night asymmetries depend on the nadir angle of the incoming solar neutrinos, and thus on the latitude of the experimental site. With the aim of illustrating the dependence on the site latitude, we define the annual averaged quantity
where , and the weight function represents the solar exposure of the trajectory and depends on the latitude of the experimental site. and are the possible maximal and minimal values of the nadir angle at a certain latitude (see Table II in Ref. ). In our calculation, we use the form of defined in Appendix C of Ref. , in order to illustrate the effect of the average over the nadir angle.
|Experimental Site||Latitude||Experimental Site||Latitude|
In the following we consider for illustration six experimental sites: three with ongoing neutrino experiments (Kamioka in Japan, Gran Sasso in Italy, and Sudbury in Canada) and three with planned neutrino experiments (Kaiping  in China, Pyhasalmi  in Finland, and the South Pole ). Their latitudes are listed in Table 2. The left and right panels in Fig. 3 show the behavior as a function of the neutrino energy of the annual averages during the nighttime of the regeneration factors and calculated with the SLAB approximation for the six typical examples in Table 2. The oscillation parameters are set to M1 and P1 in Eqs. (96) and (97). One can see that the annual averages of the regeneration factors increase almost monotonously with the neutrino energy, except for tiny oscillatory variations.
The behavior of the averaged regeneration factors in Fig. 3 can be understood with the help of Eq. (54). The magnitudes of the regeneration factors are proportional to the neutrino energy via the amplitude term , which contributes dominantly to the shape of the curves and determines the sizes of the regeneration factors. The residual oscillatory behavior is due to the summation of the functions for and , which oscillate as functions of the trajectory length, which is determined by the nadir angle. The averaging over the nadir angle reduces the size of the oscillation amplitudes. The size and pattern of the oscillatory behavior are different for different site latitudes, thus their effects are more significant in the right panels of Fig. 3, where the experimental sites range from the tropic area to the South Pole. In the following discussion, we consider only the Kamioka site for simplicity.
As we have illustrated in Figs. 1 and 2, the accuracy of the SLAB approximation is low for neutrinos crossing the core of the Earth when one considers precise values of the energy and nadir angle. Now we check the validity of the SLAB approximation in a more realistic case, in which the regeneration factors are averaged over bins of the nadir angle and the neutrino energy . The annual average of the regeneration factor in the (, ) bin is defined as
where and are the weight functions of the nadir angle and neutrino energy. We use the expression of given in Appendix C of Ref. . The weight function depends on the energy-dependent flux spectrum, the detection cross section and the experimental efficiency. However, for simplicity we neglect the effect of , since the energy bin-size is small compared to the variations of .
In the upper panels of Fig. 4, we show the binned average of the numerical (NUM) calculations of the regeneration factors (left) and (right) as functions of the neutrino energy and nadir angle. The lower panels illustrate the corresponding differences between the NUM and SLAB calculations. All the oscillation parameters are set to M1 and P1 in Eqs. (96) and (97), and the latitude for the Kamioka site is assumed. In our calculation, we choose equal-sized bins for the neutrino energy ( MeV) and the cosine of the nadir angle (). From the lower panels of Fig. 4, we can observe that the discrepancy between the NUM and SLAB calculations of the regeneration factors is visible only for high neutrino energies and small nadir angles. The accuracy of the SLAB calculation of the binned average is excellent in most of the parameter space, and is better than in the worst case. Thus, we conclude that the SLAB approximation is reliable for practical applications.
Next, we are going to explore the properties of the annual averages and defined in Eq. (57) of the day-night asymmetries of solar neutrino oscillations defined in Eq. (39). Although the regeneration factors are always positive in the whole energy range, can be both positive or negative, depending on the sign of the parameter . From the energy dependence of in Fig. 5 of Ref., we know that the sign of flips at MeV and thus the sign of changes accordingly. For neutrino energies lower than about MeV, both and are negative and their magnitudes are of the order of , which is too small for the detection ability of ongoing and near-future solar neutrino experiments. For neutrino energy greater than about MeV, the day-night asymmetries increase monotonously as the energy grows, giving hope for their detection.
To illustrate the different contributions of the oscillation parameters to the day-night asymmetries, we first show in Figure 5 the dependence of on the relevant oscillation parameters (left), (middle) and (right) in the three-neutrino mixing scheme for MeV. The solid and dashed lines in each panel correspond to the maximal (MAX) and minimal (MIN) values of when the other parameters (e.g. and in the right panel) are scanned in the full ranges . The short-dashed line corresponds to the best-fit values of these parameters. The three vertical lines in each panel correspond to the best-fit and values of the parameter in abscissa. One can see that the magnitude of the day-night asymmetry is of the order of . Moreover, one can see that increases with , but decreases as increases. This can be explained with the help of Eqs. (84) and (86), which show that the day-night asymmetry is proportional to . Considering the role of , higher induce lower values of , which is consistent with the fact that the asymmetry is proportional to in Eq. (86).