An accurate analytic description of neutrino oscillations in matter
aa r X i v : . [ h e p - ph ] D ec An accurate analytic description of neutrinooscillations in matter
E. Kh. Akhmedov a,b ∗ and Viviana Niro a † a Max–Planck–Institut f¨ur Kernphysik, Postfach 103980D–69029 Heidelberg, Germany b National Research Centre Kurchatov InstituteMoscow, Russia
October 29, 2018
Abstract
A simple closed-form analytic expression for the probability of two-flavour neutrinooscillations in a matter with an arbitrary density profile is derived. Our formula isbased on a perturbative expansion and allows an easy calculation of higher order cor-rections. The expansion parameter is small when the density changes relatively slowlyalong the neutrino path and/or neutrino energy is not very close to the Mikheyev-Smirnov-Wolfenstein (MSW) resonance energy. Our approximation is not equivalentto the adiabatic approximation and actually goes beyond it. We demonstrate the va-lidity of our results using a few model density profiles, including the PREM densityprofile of the Earth. It is shown that by combining the results obtained from theexpansions valid below and above the MSW resonance one can obtain a very gooddescription of neutrino oscillations in matter in the entire energy range, includingthe resonance region. ∗ email: [email protected] † email: [email protected] Introduction
In most neutrino oscillation experiments neutrinos propagate substantial distances in mat-ter before reaching a detector, and therefore an accurate description of neutrino oscillationsin matter [1, 2] is an important ingredient of the analyses of the data. For a matter of anarbitrary density profile the neutrino evolution equation admits no closed-form solution,and one usually has to resort to numerical methods. While numerical integration of theevolution equation usually poses no problem, it is still highly desirable to have approximateanalytic solutions, which may provide a significant insight into the physics of neutrino os-cillations in matter, clarify the dependence of the oscillation probabilities on the neutrinoparameters and in many cases help save the CPU time. To this end, a number of analyticsolutions of the neutrino evolution equation in matter, based on various approximations,has been developed (for recent studies, see e.g. [3, 4, 5, 6, 7, 8, 9, 10]).In this paper we derive a simple analytic expression for the two-flavour oscillation prob-ability valid for an arbitrary matter density profile. We employ a perturbative approachbased on the expansion in a parameter which is small when the density changes relativelyslowly along the neutrino path and/or neutrino energy is not very close to the Mikheyev-Smirnov-Wolfenstein (MSW) [1, 2] resonance energy. Our approximation is not equivalentto the adiabatic approximation and actually goes beyond it. We demonstrate the validityof our results using a few model density profiles, including the important PREM profile[11], which gives a realistic description of matter density distribution inside the Earth. Wealso show that, by combining the results obtained for the energies below and above theMSW resonance ones, one can obtain an excellent description of neutrino oscillations inmatter in the entire energy range. The simple form of our result and the wide range of itsapplicability are the two main advantages of this approach.An approach similar to ours has been employed in [12, 7]. Unlike in those publications,in the present work we do not confine ourselves to the leading approximation, but alsocalculate the first and second order corrections and show that this improves the accuracyof the approximation drastically.The paper is organized as follows. In Sec. 2 we present the formalism used to derive ouranalytic solution. In Sec. 3 we apply this method to the case of a parabolic and a powerlaw matter potentials. In Sec. 4 we present the results obtained in the case of the realisticPREM Earth’s density profile. We discuss our results and conclude in Sec. 5.
In a number of important cases the full three-flavour neutrino oscillations can to a verygood accuracy be reduced to effective two-flavour ones. These include ν e ↔ ν µ ( ν τ ) os-cillations either in the limit of vanishingly small 1-3 mixing, when the oscillations areessentially driven by the “solar” parameters ∆ m and θ , or at sufficiently high energies( E & m and θ . For definiteness, in our numerical exampleswe will concentrate on the second case, though our general discussion will be valid in bothsituations.Two-flavour oscillations of neutrinos in matter are described by the Schr¨odinger-likeevolution equation [1, 2] i (cid:18) ˙ ξ ˙ η (cid:19) = (cid:18) − A BB A (cid:19) (cid:18) ξη (cid:19) , (1)where the overdot denotes the differentiation with respect to the coordinate, and ξ and η are respectively the probability amplitudes to find ν e and ν a , the latter being a linearcombination of ν µ and ν τ . In the limit when the 1-3 mixing vanishes, θ →
0, one has ν a =cos θ ν µ − sin θ ν τ , whereas in the situations when the solar parameters play practically norole (e.g. for oscillations of high-energy neutrinos in the Earth), ν a = sin θ ν µ + cos θ ν τ .The quantities A and B in Eq. (1) are B = δ sin 2 θ ,A ( x ) = δ cos 2 θ − V ( x ) / . (2)Here the function A ( x ) depends on the electron number density N e ( x ) through the Wolfen-stein potential V ( x ) defined as V ( x ) = √ G F N e ( x ) ∼ = 7 . × − Y e ( x ) ρ ( x )(g / cm ) eV , where G F is the Fermi constant, ρ ( x ) is the mass density of matter and Y e ( x ) is the numberof electrons per nucleon. The parameter δ is defined as δ ≡ ∆ m / E , and θ is the relevantmixing angle in vacuum. In the limit θ → m = ∆ m , θ = θ , and the ν e ↔ ν µ ( ν τ ) oscillation probabilities are given by P ( ν e → ν µ ; x ) = P ( ν µ → ν e ; x ) = cos θ P ( x ) , (3) P ( ν e → ν τ ; x ) = P ( ν τ → ν e ; x ) = sin θ P ( x ) . (4)Here P ( x ) is the effective two-flavour oscillation probability: P ( x ) = P ( ν e → ν a ; x ) ≡ | η ( x ) | (5)(we assume the initial conditions ξ (0) = 1, η (0) = 0). For oscillations of high-energyneutrinos in the Earth one has ∆ m = ∆ m , θ = θ , and the ν e ↔ ν µ ( ν τ ) oscillationprobabilities are P ( ν e → ν µ ; x ) = P ( ν µ → ν e ; x ) = sin θ P ( x ) , (6) P ( ν e → ν τ ; x ) = P ( ν τ → ν e ; x ) = cos θ P ( x ) , (7)where, as before, P ( x ) is given by Eq. (5). 3ifferentiating Eq. (1), one can find decoupled second order differential equations for ξ ( x ) and η ( x ) [13, 14]. The equation for the transition amplitude η ( x ) reads¨ η + ( ω + i ˙ A ) η = 0 , (8)where we have defined the function ω ( x ) as ω ( x ) = A ( x ) + B . (9)Note that the instantaneous eigenvalues of the effective Hamiltonian in Eq. (1) are ± ω ( x ).The equation for ξ ( x ) differs from Eq. (8) by the sign of the ˙ A term.It will be convenient for our purposes to rewrite Eq. (8) in the following form:¨ η + ( ω − i ˙ ω ) η = ( − i ˙∆) η , (10)where we have introduced the notation ˙∆ ≡ ˙ A + ˙ ω . (11)Eq. (10) cannot in general be solved exactly, but, as we shall see, it admits a simpleperturbative solution. To show that, let us first notice that, for energies (or densities)above the MSW resonance one, the quantity ˙∆ on the right hand side of Eq. (10) is small.Indeed, from Eqs. (9) and (2) it follows that for V / − cos 2 θ δ ≫ sin 2 θ δ (i.e. for − A ≫ B ) one has ˙ ω ≃ − ˙ A , so that ˙∆ ≃
0. The smallness of the parameter ˙∆ allows oneto solve Eq. (10) perturbatively, order by order. Expanding in powers of ˙∆, we find theequation for the n th order transition amplitude η n (with n > η n + ( ω − i ˙ ω ) η n = ( − i ˙∆) η n − . (12)The zero order transition amplitude η satisfies the equation with the vanishing right handside: ¨ η + ( ω − i ˙ ω ) η = 0 . (13)Its solution for an arbitrary functional dependence of ω ( x ) on the coordinate can be readilyfound by considering the quantity X ≡ ˙ η − iωη , which, as follows from (13), satisfiesthe first-order equation ˙ X + iωX = 0. Taking into account that the initial conditions ξ (0) = 1, η (0) = 0 also imply, through Eq. (1), ˙ η (0) = − iB , one finds η ( x ) = − i B e iφ ( x ) Z x dx e − iφ ( x ) , (14)where φ ( x ) ≡ Z x ω ( x ′ ) dx ′ . (15)This yields the zero-order solution for the two-flavour transition probability P ( x ) [12, 7]:[ P ( x )] ≡ | η ( x ) | = B (cid:12)(cid:12)(cid:12)(cid:12)Z x dx e − iφ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) . (16)4ssuming that the amplitude η n − ( x ) on the right hand side of Eq. (12) is known, onecan solve it for η n . To this end, we introduce the quantity X n = ˙ η n − iωη n , (17)in terms of which Eq. (12) can be rewritten as˙ X n + iωX n = ( − i ˙∆) η n − . (18)This can now be solved by the standard methods. First, we find the general solution ofthe homogeneous equation ˙ X n + iωX n = 0 , (19)which gives X n ( x ) = F e − iφ ( x ) (20)with F an integration constant. Next, the solution of the inhomogenous equation (18) isfound by allowing F to depend on the coordinate x and substituting Eq. (20) back intoEq. (18). Taking into account the initial condition F (0) = ˙ η (0) − iω (0) η (0) = − iB , onefinds F ( x ) = Z x dx e iφ ( x ) (cid:16) − i ˙∆ ( x ) (cid:17) η n − ( x ) − iB . (21)The solution for X n is now given by Eq. (20) with F replaced by F ( x ) from Eq. (21). Once X n is known, it is straightforward to solve Eq. (17) for η n . This yields η n ( x ) = e iφ ( x ) Z x dx e − iφ ( x ) Z x dx e iφ ( x ) (cid:16) − i ˙∆ ( x ) (cid:17) η n − ( x ) + η ( x ) , (22)where we have used Eq. (14). The corresponding n th order effective two-flavour oscillationprobability is then found as [ P ( x )] n = | η n ( x ) | .Eq. (22) represents the main result of our paper. It gives an analytic expression for theoscillation amplitude in the n th order in perturbation theory in terms of the lower-ordersolutions η n − and η . For our numerical illustrations we will consider the solutions with n = 0, 1 and 2.Eq. (22) has been derived under the assumption that ˙∆ is a small parameter. As wepointed out before, this is true for energies above the MSW resonance one. This meansthat the perturbative approach considered above should, in general, fail for energies belowthe MSW resonance one. However, a simple modification of the above procedure leads toa description of neutrino oscillations valid below the MSW resonance. In order to showthis, let us, instead of casting Eq. (8) in the form (10), rewrite it as¨ η + ( ω + i ˙ ω ) η = ( − i ˙∆) η , (23)where ˙∆ is now defined as ˙∆ = ˙ A − ˙ ω . (24)5or small vacuum mixing angles, this is a small parameter below the MSW resonance,since in that case A ≫ B and so ˙ ω ≃ ˙ A . Therefore, we can proceed with the perturbativeapproach, as before. Comparing Eqs. (23) and (24) with Eqs. (10) and (11) respectively, wesee that the two pairs of equations differ only by the sign of ω ( x ). Therefore the solutionof Eq. (23) can be obtained from Eq. (22) by simply replacing ω ( x ) by − ω ( x ). This willalso change the values of the oscillation probabilities obtained in all orders in perturbationtheory except for the zero-order probability which, as can be seen from (16), is invariantwith respect to the flip of the sign of ω ( x ). As we shall see, by combining the results validabove and below the MSW resonance one can obtain a very good description of neutrinooscillations in matter in the entire energy range.Let us now discuss the expansion parameter of our perturbative approach. We havefound that the corrections to the zero order amplitude η are proportional to ˙∆ = ˙ A ± ˙ ω ,where the upper and lower signs refer to the energies above and below the MSW resonance,respectively. These quantities can be expressed through the mixing angle in matter θ m : ˙∆ = ˙ A ± ˙ ω = − ˙ V ± cos 2 θ m ] . (25)Far above the MSW resonance one has cos 2 θ m ≃ −
1, whereas far below the resonancecos 2 θ m ≃ cos 2 θ , which is close to 1 in the case of small vacuum mixing. This demonstratesthe smallness of ˙∆ in its corresponding domains of validity. At the MSW resonance onehas cos 2 θ m = 0, and ˙∆ is only small if ˙ V is.An examination of Eq. (22) shows that the expansion parameter of our perturbativeapproach is actually ∼ | ˙∆ | /ω (see Eq. (15)). In various energy domains we have | ˙∆ | ω = | ˙ A − ˙ ω | ω ≃ | ˙ V | s δ c δ − V/ if ( c δ − V / ≫ s δ (below the resonance) | ˙ A ± ˙ ω | ω ≃ | ˙ V | s δ if | c δ − V / | ≪ s δ (near the resonance) | ˙ A + ˙ ω | ω ≃ | ˙ V | s δ V/ − c δ ) if ( V / − c δ ) ≫ s δ (above the resonance) (26)where we have used the shorthand notation c ≡ cos 2 θ , s ≡ sin 2 θ . From Eq. (26) it iseasy to see that outside the MSW resonance region the expansion parameter approximatelysatisfies | ˙∆ | ω ≃ sin θ m | ˙ V | ω = sin 2 θ m γ − , (27)where γ MSW = 4 ω / ( | ˙ V | B ) = 4 ω / ( | ˙ V | sin 2 θ m ) is the MSW adiabaticity parameter. Thus,for small mixing in matter (sin 2 θ m ≪
1) our approximation is better than the adiabaticone. Close to the resonance the two approaches have comparable accuracy. Note that sin 2 θ m = B/ω , cos 2 θ m = A/ω . Two examples: parabolic and power law profiles
As a first study, we apply our formalism to two simple density distributions: a parabolicand a power law profile.For the parabolic profile, we consider the following density distribution: ρ ( x ) = ρ (cid:20) − k ( x − L/ L / (cid:21) (28)with ρ = ρ max = 8 g / cm , k = 1 − ρ min ρ max = 0 . , (29)and we take the baseline to be L = 10000 km. Note that the parabolic density profilerepresents a good approximation for the density distribution felt by neutrinos in the Earthwhen they cross only the Earth’s mantle.Next, we analyze the case of the following power-law density distribution: ρ ( x ) = ρ (cid:18) x x + x (cid:19) (30)with x = 10 km and ρ = 10 g / cm , (31)and we consider neutrino propagation over the distance L = 100 km. The profile ρ ∝ x − represents a realistic description of the density distribution inside supernovae; note, how-ever, that neutrino flavour transitions in supernovae are more adequately described bydifferent methods (see, e.g., [15]), and so we consider the profile (30) just for illustration.The results based on our perturbative analytic approach for the profiles (28) and (30)are presented in Fig. 1, where they are compared with the exact ones, obtained by directnumerical integration of the neutrino evolution equation (1). The upper panels show theoscillation probabilities for the parabolic density profile and the lower ones, for the power-law profile (30). The left panels correspond to the expansion valid for energies belowthe MSW resonance ones, whereas the right panels were obtained for the expansion validabove the resonance energies. As expected, the zero-order approximation gives a goodaccuracy only outside the MSW resonance region (i.e., outside the intervals E ∼ E ∼
30 – 50 MeV for the power-law one). The first-order perturbative results obtained using the expansion valid below the MSW resonanceextend slightly the region of good accuracy towards higher energies, closer to the MSWresonance, though in general fail for energies above the MSW resonance, whereas the first-order results found from the expansion valid above the MSW resonance extend the regionof good accuracy to lower energies, but in general fail below the MSW resonance. Thus,the first-order calculation taken in their respective domains of applicability allow to achieve Note that, since the profiles (28) and (30) (as well as the PREM profile considered in the next section)span a range of matter densities, neutrinos in an interval of energies experience the MSW resonance. | η | practicallycoincide with the corresponding exact results, irrespectively of whether they are obtainedusing the expansion valid below or above the MSW resonance. Neutrinos coming from various sources can propagate inside the Earth before reaching adetector. Examples are atmospheric neutrinos, neutrinos coming from WIMP annihilationinside the Earth or the Sun, as well as neutrinos studied in long-baseline accelerator exper-iments. We will consider here oscillations of high-energy neutrinos in the Earth, for whichwe take the matter density distribution as described by the PREM profile [11] (Fig. 3).Note that the PREM profile is symmetric with respect to the midpoint of the neutrinotrajectory, and therefore the two-flavour transition amplitude η ( x ) obtained as a solutionof Eq. (1) is pure imaginary due to the time reversal symmetry of the problem [16].In Fig. 2 we present the oscillation probability P as a function of neutrino energy E for two values of the zenith angle of the neutrino trajectory: cos θ z = −
1, when theneutrinos propagate the longest distance inside the Earth, traversing it along its diameter,and cos θ z = − .
95, when they do not cross the inner core of the Earth. As in Fig. 1,we compare the approximate solutions, up to the second order ([ P ] = | η | ), with theexact solutions found by direct numerical integration of the neutrino evolution equation.In this figure (as well as in Figs. 4 – 6 below) in the left panels we present the oscillationprobabilities obtained with the expansion valid below the MSW resonance energy, whereasthe right panels show the results found from the expansion valid above the MSW resonance.It can be seen from Fig. 2 that the zero order probability | η | reproduces accuratelythe exact one, | η | , only for energies that are outside the resonance region. Indeed, the twosolutions nearly coincide for E ≤ E > | η | valid belowthe resonance (left panels of the figure) allow an accurate description of the probabilityfor slightly higher energies than | η | does, allowing to come closer to the MSW resonancefrom below; however, they fail badly (not even being bounded by 1) above the resonance.Likewise, the solutions | η | valid above the resonance (right panels) allow to come closerto the MSW resonance from above, but fail below the resonance.At the same time, the second-order solution | η | gives quite a good approximation tothe exact probability | η | for all energies, though the solutions obtained through the ex-pansions in their corresponding domains of validity give a better accuracy in these energydomains. By combining the second-order solutions valid below and above the MSW reso-nance, one can obtain a very good description of the exact oscillation probability practicallyat all energies, including the resonance region. We have also checked that for trajectoriesthat do not cross the core of the Earth (cos θ z > -0.838), for which the matter density pro-8le seen by the neutrinos is relatively smooth, the second order solutions obtained throughboth expansions essentially coincide with the exact one for all energies.In Fig. 4 we present the oscillation probability P , obtained in different orders in per-turbation theory, as a function of the distance travelled by neutrinos inside the Earth forvertically up-going neutrinos (cos θ z = −
1) and two values of neutrino energy, E = 2 . | η | nearly concide with the exactprobability | η | along the entire neutrino path.Fig. 5 illustrates the dependence of the analytic solutions on the zenith angle for twovalues of the neutrino energy, E = 2 . E = 2 . E = 6 GeV, while thesituation is opposite in the case of the solution corresponding to the expansion valid abovethe resonance.Finally, in Fig. 6 we show the dependence of the accuracy of the analytic solutions onthe value of the vacuum mixing angle θ = θ . As one can see by comparing the upperpanels with the corresponding lower ones, with decreasing value of θ the accuracy of ourperturbative expansion improves. This is the consequence of the fact that the expansionparameter (27) decreases with decreasing θ . We have developed a perturbative approach for two-flavour neutrino oscillations in matterwith an arbitrary density profile. The zero-order oscillation amplitude η satisfies theequation which can be solved analytically for an arbitrary dependence of the matter densitydistribution on the coordinate along the neutrino path; higher order amplitudes are thenobtained from the lower-order ones through a simple perturabative procedure. We havestudied the zeroth, first and second order solutions and compared them with each otherand with the exact oscillation probability obtained by numerical integration of the neutrinoevolution equation. In all orders except the zeroth one, the expansion scheme depends onwhether the neutrino energy is above or below the MSW resonance energy, and one has toconsider these two cases separately.While the zero-order result gives a very good accuracy outside the resonance region,higher order corrections are necessary to achieve an accurate description of the oscillationprobability in the vicinity of the MSW resonance. We have demonstrated how these cor-rections, when taken in their respective energy domains of validity, improve drastically theprecision of the approximation.For the smooth density profiles that we have studied, we found that the second orderoscillation probability reproduces the exact one extremely well in the whole interval ofenergies, including the MSW resonance region, independently of whether the expansion9cheme valid below or above the resonance was used. The same is also true for the PREMdensity profiles in the case when neutrinos cross only the mantle of the Earth, since thedensity jumps experienced by neutrinos in that case are relatively small. The high accuracyof the second order approximation for smooth density profiles is related to the fact that ourexpansion parameter, Eq. (27), is proportional to | ˙ V | . This parameter is smaller than theexpansion parameter of the adiabatic approximation by the factor sin 2 θ m and thereforeour approach gives a better accuracy than the adiabatic expansion when the mixing inmatter is small. Note that a different expansion of the same evolution equation (8) wasemployed in [17].For energies above the MSW resonance, our expansion parameter is essentially | ˙∆ | ω ≃ sin θ m | ˙ V | V . (32)For the PREM density profile of the Earth, the function | ˙ V | /V is plotted in the right panelof Fig. 3. As can be seen from the figure, in most of the coordinate space the value ofthis function does not exceed 0.25. The spikes corresponding to the density jumps, thoughquite high, are very narrow; they do not destroy our approximation because their contribu-tions get suppressed due to the integrations involved in the calculation of the higher-ordercorrections to the oscillation amplitude (see Eq. (22)). Still, these contributions are notnegligible, especially for neutrinos crossing the Earth’s core. As a result, for core-crossingneutrinos with energies close to the MSW resonance ones, even the second-order oscillationprobabilities are only adequate when taken in their respective energy domains of validity.By combining the solutions valid below and above the MSW resonance one obtains a veryaccurate description of neutrino oscillations in matter in the entire energy range.To conclude, we have derived a simple closed-form analytic expression for the prob-ability of two-flavour neutrino oscillations in a matter with an arbitrary density profile.Our formula is based on a perturbative expansion and allows an easy calculation of higherorder corrections. We have applied our formalism to a number of density distributions,including the PREM density profile of the Earth, and demonstrated that the second-orderapproximation gives a very good accuracy in the entire energy interval. Acknowledgments
We thank Andreas Hohenegger for the help with numerical calculations and Michele Mal-toni for useful discussions.
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D38 , 935 (1988).11 .0 10.05.02.0 20.03.01.5 15.07.00.00.20.40.60.81.0 E @ GeV D P sin Θ = È Η È È Η È È Η È È Η È @ GeV D P sin Θ = È Η È È Η È È Η È È Η È @ GeV D P sin Θ = È Η È È Η È È Η È È Η È @ GeV D P sin Θ = È Η È È Η È È Η È È Η È Figure 1: Oscillation probability P versus neutrino energy E in the case of the parabolic(upper plots) and power law (lower plots) density profiles. Left panels: probabilities ob-tained from the expansion valid below the MSW resonance, right panels: the same for theexpansion valid above the resonance. We have taken ∆ m = 2 . − eV and Y e = 0 . .0 10.05.02.0 20.03.01.5 15.07.00.00.20.40.60.81.0 E @ GeV D P sin Θ = cos Θ z = - È Η È È Η È È Η È È Η È @ GeV D P sin Θ = cos Θ z = - È Η È È Η È È Η È È Η È @ GeV D P sin Θ = cos Θ z = - È Η È È Η È È Η È È Η È @ GeV D P sin Θ = cos Θ z = - È Η È È Η È È Η È È Η È Figure 2: Probability P versus neutrino energy E for neutrino oscillations in the Earth(PREM density profile) for two values of the zenith angle. Left panels: probabilitiesobtained from the expansion valid below the MSW resonance, right panels: the same forthe expansion valid above the resonance. We have taken ∆ m = 2 . − eV .13 @ km D Ρ @ g (cid:144) c m D CoreCore MantleInner Outer @ km D - V (cid:144) V Figure 3: Left panel: matter density distribution inside the Earth as predicted by thePREM profile [11]. Right panel: the function − ˙ V /V as calculated with the PREM profilewith density jumps smoothed over the distance of 30 km. P sin Θ = cos Θ z = - = È Η È È Η È È Η È È Η È P sin Θ = cos Θ z = - = È Η È È Η È È Η È È Η È Figure 4: Oscillation probability P in different orders in perturbation theory versus thedistance travelled by neutrinos inside the Earth, for cos θ z = -1.0 and for two values ofneutrino energy ( E = 2.8 GeV and 6 GeV). Left panels: probabilities obtained from theexpansion valid below the MSW resonance, right panels: the same for the expansion validabove the resonance. We have taken ∆ m = 2 . − eV .14 ° ° ° ° ° ° ° - - - - - - Θ z P cos Θ z sin Θ = = È Η È È Η È È Η È È Η È ° ° ° ° ° ° ° - - - - - - Θ z P cos Θ z sin Θ = = È Η È È Η È È Η È È Η È ° ° ° ° ° ° ° - - - - - - Θ z P cos Θ z sin Θ = = È Η È È Η È È Η È È Η È ° ° ° ° ° ° ° - - - - - - Θ z P cos Θ z sin Θ = = È Η È È Η È È Η È È Η È Figure 5: Oscillation probability P versus the zenith angle θ z for neutrinos propagatinginside the Earth, for neutrino energies E = 2 . m = 2 . − eV .15 .0 10.05.02.0 20.03.01.5 15.07.00.00.20.40.60.81.0 E @ GeV D P sin Θ = cos Θ z = - È Η È È Η È È Η È È Η È @ GeV D P sin Θ = cos Θ z = - È Η È È Η È È Η È È Η È @ GeV D P sin Θ = cos Θ z = - È Η È È Η È È Η È È Η È @ GeV D P sin Θ = cos Θ z = - È Η È È Η È È Η È È Η È Figure 6: Oscillation probability P versus neutrino energy E in the case of cos θ z = − . θ . Left panels: probabilities obtained from theexpansion valid below the MSW resonance, right panels: the same for the expansion validabove the resonance. We have taken ∆ m = 2 . − eV2