Exact Formula of Probability and CP Violation for Neutrino Oscillations in Matter

Transcription

1 Exact Formula of Probability and CP Violation for Neutrino Oscillations in Matter K. Kimura,A.Takamura,,andH.Yokomakura Department of Physics, Nagoya University, Nagoya, 6-86, Japan Department of Mathematics, Toyota National Collage of Technology Eisei-cho -, Toyota-shi, 7-855, Japan Abstract Within the framework of the standard three neutrino scenarios, we derive an exact and simple formula of the oscillation probability P (ν e ν µ ) in constant matter by using a new method. From this formula, it is found that the matter effects can be separated from the pure CP violation effects. Furthermore, the oscillation probability can be written in the form, P (ν e ν µ ) = A cos δ + B sin δ + C, in the standard parametrization without any approximations. We also demonstrate that the approximate formula in high-energy can be easily reproduced from this as an example. Introduction It has been shown that neutrinos have the finite mass and the finite mixing like the quark sector from the atmospheric neutrino experiments [] and the solar neutrino experiments []. In this situation, it is extremely interesting to investigate the CP phase in the lepton sector. Fortunately, recent report from SNO experiment [] favors the LMA MSW solutions to the solar neutrino problem. This means that the measurements of CP phase may be possible because of the large - mixing angle and the large - mass differences. As one of the purpose to measure CP phase, the long-baseline experiments such as the JHF experiments [] and the neutrino factory experiments [5] are planned in near future. As the main approaches to the measurements of CP phase δ, the asymmetries P CP = P (ν α ν β ) P ( ν α ν β )and P T = P (ν α ν β ) P (ν β ν α ) has been considered [6, 7, 8, 9]. These are the method to measure the direct CP violation term dependent on sin δ. However, the measurements of P CP are not directly related to the discovery of CP phase because of the fake CP violation effects from the earth matter. On the other hand, it has some experimental difficulties for the measurements of P T. So, another approach has been recently considered in [,,,, ]. This is the attempt to obtain the information on the CP phase totally address:kimukei@eken.phys.nagoya-u.ac.jp address:takamura@eken.phys.nagoya-u.ac.jp address:yoko@eken.phys.nagoya-u.ac.jp

2 from the probabilities itself, not only the direct CP violation term but also the indirect CP violation term dependent on cos δ. In these papers the oscillation probability is written in the form, P (ν e ν µ ) A cos δ + B sin δ + C, approximately. It is discussed how we can obtain the information on CP phase by in this approximate formula. In order to obtain the more precise information we need to use the exact formula. Some attempts at the exact formula have so far been made in the context of three neutrino scenarios [5, 6, 7, 8]. These formula are useful for numerical calculation. However, the CP dependence has not investigated sufficiently [6]. We obtain the hint to learn the CP dependence from the work by Naumov [9] and Harrison- Scott [6]. As we stated below, P T is expressed strictly linear in sin δ from their works. They notice that the Hamiltonian H in matter is related to H in vacuum as H = H + diag(a,, ), () E where a G F N e E, G F is Fermi constant and N e is the electron density in matter. In particular, taking the products of non-diagonal elements, Im( H eµ Hµτ Hτe )=Im(H eµ H µτ H τe ), () one obtains the following identity, which we call Naumov-Harrison-Scott identity, J = J, () in CP-odd part, where m i m j, J ImJ eµ is Jarlskog factor [], Jαβ U αiuβi U αj U βj and U is the lepton mixing matrix, so-called, Maki-Nakagawa-Sakata (MNS) matrix []. Here the quantities expressed by the tilde is the ones in matter. From this identity, J canbeexpressedby the effective masses and the parameters in vacuum. As the effective masses shown in [5, 6, 7] do not depend on the CP phase, J can be completely expressed by the linear term in sin δ. The reason why the CP dependence becomes simple is to calculate J, the products of four Ũ s. It suggests that the complicated matter effects included in a Ũ are partially canceled by them included in other Ũ. In this letter, we calculate ŨŨ without directly calculating single Ũ. Our method is that we use some matter invariant identities such as Naumov-Harrison-Scott identity and express not only J but also Re J eµ with the effective masses and the parameters in vacuum. The exact formula obtained in this method is very simple and the matter effects are only included in the effective masses. For the sake of this form Re J eµ has only the linear term in cos δ up to the constant. That is, the oscillation probability in matter can be written in the form P (ν e ν µ )= A cos δ + B sin δ + C exactly. Another merit of our result is that the exact formula immediately reduces to the well-known approximate formula both in low-energy [7] and in high-energy [, ]. We demonstrate that the approximate formula in high-energy can be easily reproduced from our exact formula as an example. Finally, we numerically calculate the coefficients A, B and C changing with the sign of the mass differences. The calculation of a Ũ is performed by diagonalizing H in Ref. [7]. The expression of ŨŨ has been also given as the result of the calculation of e i HL in Ref. [8] although the CP phase has not been considered.

3 Exact Formula of the Oscillation Probability The flavor and mass eigenstates are related by MNS matrix Ũαi in matter, where α = e, µ, τ is the flavor index, i =,, is mass index. The amplitude for ν e to ν µ is given by A eµ = e ipl i= Ũ ei e i λ i E L Ũ µi, () and the oscillation probability is also given by P eµ = A eµ, (5) from the amplitude, where L stands for the baseline length and P (ν e ν µ ) is expressed as P eµ for the simplicity. We note that the amplitude is constituted by the products ŨeiŨ µi as in (). One of the important points in this letter is that the products can be easily calculated from the identities which we derive below. From the unitarity relation and the other two relations, H eµ = H eµ = p/(e), (6) H eτ Hτµ H eµ Hττ = H eτ H τµ H eµ H ττ = q/(e), (7) three identities about the products ŨeiŨ µi can be obtained as follows, (k) Ũ ei Ũµi = i= λ i Ũ ei Ũµi = i= λ j λ k Ũ ei Ũ µi = U ei Uµi =, (8) i= m i U eiuµi = p, (9) i= (k) m j m k U eiuµi = q, () where we use the relation ŨeiŨµj ŨµiŨej = Ũ τk for the derivation of (), and the sum is over (k) = (), (), (). These are invariant under matter effects. The right hand side of (8), (9) and () are all constant determined by the parameters in vacuum. Solving the simultaneous equations for the products ŨeiŨ µi,weobtain Ũ ei Ũ µi = pλ i + q ji ki, ()

4 J eµ = ŨeiŨ µi (ŨejŨ µj ), the exact for- where (k) takes (), (), (). From the definition mula of the oscillation probability is presented by P eµ = Re J eµ sin ( ) L ( ) L J sin, () E where the sum is over = (), (), () and Re J eµ = p λ i λ j + q +Re(pq )(λ i + λ j ), () J = Im(pq ). () We find that the matter effects are included only in the effective masses and are completely separated from the constants p and q determined by the vacuum parameters. We can obtain the probability for antineutrinos, ν e ν µ, by the exchange a a and δ δ in and J eµ of Eq. (). In the end of this section, let us comment on the relation with other works. The second identity (9) is also given in Ref. []. The third identity () is new and play an important role to derive our exact formula. The similar expression to () is given in Ref. [8]. Our derivation of () is different from their method and the CP phase is included in our formulation. Next, Im(pq )in () are rewritten as Im(pq )=/(E) Im(H eµ H µτ H τe )= J, (5) from (6) and (7). Naumov-Harrison-Scott identity is reconstructed by the substitution (5) for (). Separation of CP odd/even Part In this section, we give more concrete expression for the oscillation probability and then, we study the dependence of the oscillation probability on CP phase. First let us consider the constants p and q. We use the standard parametrization U αi = c c c s s e iδ c s s s c e iδ c c s s s e iδ s c, (6) s s c s c e iδ s c c s s e iδ c c where sin θ = s,cosθ = c. In addition, as the oscillation probabilities in neutrino oscillations do not depend on the mass itself but the mass square differences, we take m =,m = and m = without loss of generality. So, p and q are given by p = p e iδ + p, q = q e iδ + q, (7)

5 where p i and q i are real numbers as p =( s )s s c, p = s c c c, (8) q = c s s c, q = s c c c. (9) Here we separate the terms which depend on the CP phase. Then, we have p = p + p +p p cos δ, () q = q + q +q q cos δ, () Re(pq ) = p q + p q +(p q + q p )cosδ, () Im(pq ) = (p q p q )sinδ. () Therefore, the oscillation probability can be written in the form P eµ = A cos δ + B sin δ + C, () from ()-(). Here A = B = C = ( ) A sin L, (5) ( ) B L sin, (6) E ( ) C sin L, (7) are expressed by the products of the oscillation part dependent on L and A,B and C. And then, A,B and C are given by A = [p p λ i λ j +q q +(p q + q p )(λ i + λ j )], (8) B = (p q p q ), (9) C = [(p + p )λ iλ j +(q + q )+(p q + q p )(λ i + λ j )], () as the function of the masses and mixing angles. As the effective masses λ i shown in [5, 6, 7] do not depend on CP phase δ, it is find that A, B and C do not depend on the CP phase completely and the oscillation probability is expressed only by the linear terms in cos δ and sin δ up to constant. 5

6 μp ff a Y X b? P Figure : An example of CP trajectory We take P for the horizontal axis and P for the vertical axis. The value of δ changes from to π. This is one of our main results in this letter. Thus, the CP dependence of the exact form of P eµ becomes also simple although it seems to be complicated from the result in Ref. [6]. It is useful to determine the CP phase together with the graphical method which is proposed in []. In this method, the trajectory becomes ellipse in bi-probability space when δ changes from to π. One can determine the value of CP phase by the position of experimental value numerically. Let us here propose the complementary method of the above graphical method to determine the value of CP phase. We can obtain the analytic expressions by solving () for sin δ and cos δ as sin δ = B(P C) ± A A + B (P C) A + B, () cos δ = A(P C) B A + B (P C) A + B. () Thus, we can determine the value of CP phase except for the ambiguity of the sign from the measurement of the probability in an energy and a baseline. The ambiguity of the sign is understood as follows. Typical CP trajectory in bi-probability space is shown in Fig.. If we measure the probability of the neutrino at fixed energy and baseline, we find the solutions on a line a. As in Fig., there are two intersections X and Y of line a with CP trajectory. This is the reason why the ambiguity due to the sign is appeared in the analytic solutions () and (). In order to resolve the ambiguity of the sign, we need to measure more than two kinds of the probabilities, for example neutrino and antineutrino. We denote P and P of the oscillation probabilities for neutrino and antineutrino respectively as Then, CP phase can be determined by P = A cos δ + B sin δ + C, () P = Ā cos δ + B sin δ + C. () sin δ = (ĀP A P ) (ĀC A C) ĀB A B, (5) 6

7 cos δ = ( BP B P ) ( BC B C), (6) BA BĀ without the ambiguity of the sign. This means that the solution is exist on line a, line b and CP trajectory, namely X, in Fig.. In the end of this section, we comment on some ambiguities in determining the value of CP phase. Although the value of CP phase is determined in principle in (5) and (6), there remain other ambiguities included in A, B, C and Ā, B, C. The methods resolving these ambiguities are discussed in the references for example [,, 5]. We discuss the ambiguities due to the sign of mass squared differences in Sec. 5. Simple Derivation of Approximate Formula In the previous section, we have shown that the exact formula of the oscillation probability can be expressed as P eµ = A cos δ +B sin δ +C. In this section, we demonstrate that the approximate formula seen in [, ] is easily derived as an example in the case of m <m m. One obtains the approximate formula for other patterns of mass hierarchy in the same way. Below, we perform the calculation with three steps. First, it is shown that A and B expressedinthe form of sum are rewritten in the form of the product. Second, we put A, B and C in order about. Third, we neglect the terms including λ, which is the smallest in the effective masses. Let us first consider the coefficient B of sin δ. B is expressed in the form of the sum as (6). Under the condition x + y + z =, the identity sin x +siny +sinz = sinx sin y sin z, (7) holds and B is rewritten in the form of product as B = B sin ( ) L E (8) ( ) ( ) ( ) = B L L L sin sin sin. (9) This is a well known procedure. Next, let us consider the coefficient A of cos δ. Under the same condition as B, the identity sin x = (sin x sin y cos z +sinxcos y sin z). () holds and A is also rewritten as A = ( ) A sin L () cyclic ( ) ( ) ( ) L jk L ki L = (A jk + A ki )cos sin sin. () (k) 7

8 Next, we put them in order of. Substituting (8) and (9) for p and q in (8)-(), A, B and C are rewritten with the masses and the mixings as A = (k) 8J r [ λ k (λ k )+A () k ] ( L jk cos ki ) ( ) ( ) jk L ki L sin sin, () B = 8 ( ) ( ) ( ) L L L J r sin sin sin, () C = [s (s c λ iλ j + C () + C (a) )+C (b) ] sin ( ) L, (5) where J r = s c s c s c,and A () k = { λ k (c s )+λ k s c }, (6) C () = { λ i (λ j s + c ) λ j(λ i s + c )}s c, (7) C (a) = (λ i s + c )(λ j s + c )s c, (8) C (b) = (λ i )(λ j )s c c c. (9) Note that these are the expressions without any approximations. The superscripts of A and C stand for the degrees of,and(a) represents the term proportional to s and (b) isthe term independent of s. Finally, we obtain the well known approximate formula by neglecting the smallest effective mass. In the high energy the smallest effective mass is λ and almost equivalent to. Other effective masses λ and λ, correspond to a or. Accordingly, A, B and C are approximated by A = 8J ( ) ( ) r a( a) cos L al sin sin B = 8J ( ) ( ) r a( a) sin L al sin sin ( ( a)l ( ( a)l ), (5) ), (5) C = ( a) s s c sin ( ( a)l ), (5) under the condition / <s. When s is smaller than ( / ), the term C (b) independent of s becomes the dominant term. Although the approximate formula derived here is in agreement with the ones seen in [, ], the derivation is rather simple. Moreover, one can also reproduce the approximate formula in low-energy [7] although it is not discussed in this letter. 8

9 5 Numerical Analysis of CP odd/even Part In this section, we investigate how the values of the coefficients A, B and C change by the differences of neutrino-antineutrino, the signs of and using the exact expressions. In this numerical analysis, we take θ = π/, = ev, θ = π/ and = ev to be consistent with the LMA MSW solution to the solar neutrino problem [, ] and the zenith-angle dependences of atmospheric neutrinos []. We also take θ =.5 within the upper bound of CHOOZ experiment [7]. The baseline length is typically taken to be L = 9km and the matter density is taken to be.g/cm. In Fig. we show the coefficients A, B and C changing with the energy E. We observe that the sign of A is opposite for example in Fig. (a) and (d). We also observe that A and B have the opposite sign but C has the same sign comparing Fig. (a) with (e). In addition, some peaks are appeared in all graphs of Fig. with the change of energy. In the case of >, the peaks around 6 GeV in Fig. (a) for neutrinos are enhanced compared with those in Fig. (b) for antineutrinos. Inversely, in the case of <, the peaks in Fig. (d) for antineutrinos are enhanced compared with those in Fig. (c) for neutrinos. These features are understood qualitatively from the approximate formula (5)-(5). First let us consider the sign of A, B and C. As we found from (5)-(5), when the signs of both and a change, the sign of A becomes opposite and the signs of B and C do not change. On the other hand, when the sign of changes, the signs of both A and B change while the sign of C does not change. Next, let us consider the magnitude of the peak around 6 GeV. These are strongly affected by the denominator ( a). Since the signs of and a are opposite in Fig. (a) and (d), the denominator ( a) becomes small and the magnitude of the peaks are enhanced. On the other hand, since the signs of and a are the same in Fig. (b) and (c), the peaks are suppressed. Finally, let us explain the position of the peak in Fig. (a) and (d) around 6 GeV. Roughly, the peak energy is determined by the following, sin [.7 ( )( )( ) ] a L E ev E 6 GeV (at L = 9 km). (5) km GeV As pointed out by Parke and Weiler [8], and Lipari [], the peak energy is lower than the energy of - MSW resonance since the baseline length is short compared with the earth diameter. The above discussions on Fig. (a)-(e) can be applied to other figures. We have studied how the magnitude of A, B and C change due to the sign of the mass squared differences. In the case of m <m m, the coefficients have been investigated in Ref. [] by using the approximate formula. These correspond to Fig. (a) and (b). The sign of is determined from the leading term C as pointed out by many authors (for example see []). On the other hand, the sign of is determined from next leading terms A or B. This means that the sign of is simultaneously determined in addition to the CP phase. It may be interesting as the first observation of the sign of using artificial neutrino beam. 9

10 a b c d e f g h Figure : A, B, C at L = 9 km The graphs of the left and right side correspond to the neutrino and the antineutrino respectively. The solid lines, the dotted lines and the dashed lines are for A, B and C in all graphs. And from top to bottom, ( >, > ), ( <, > ), ( >, < ) and ( <, < ) cases.

11 6 Summary We have studied the neutrino oscillations in constant matter within the framework of the three neutrino scenario. We summarize the results obtained in this letter. (i) We have derived an exact expression of the oscillation probability by using new method. We have calculate ŨŨ from the identities without directly calculating single Ũ. Not only the derivation but also the result becomes simple and the matter effects are separated completely. (ii) We have obtained the CP dependence of the oscillation probability exactly by using the standard parametrization. It has been shown that the oscillation probability is in the form, P eµ = A cos δ + B sin δ + C. We have also demonstrated that the approximate formula in high-energy can be easily reproduced from our result. Finally, let us comment on the oscillation probabilities for other channels. The CP dependence of these probabilities are easily derived in the same way as P eµ. However, they have different forms from P eµ a little. We will present these results in the next paper [8]. References [] Super-Kamiokande Collaboration, Y. Fukuda et al., Phys. Rev. Lett. 8, (998) 56; Phys. Lett. B, (998) 9; Phys. Lett. B6, (998) ; Phys. Rev. Lett. 8, (999) 6. [] GALLEX Collaboration, W. Hampel et al., Phys. Lett. B7, (999) 7; SAGE Collaboration, J. N. Abdurashitov et al., Phys. Rev. C6 (999) 558; Homestake Collaboration, B. T. Cleveland et al., Astrophys. J. 96, (998) 55; Super-Kamiokande Collaboration, Y. Fukuda et al., Phys. Rev. Lett. 8, (999) 8, ibid. 8, (999). [] SNO Collaboration, Q. R. Ahmad et al., Phys. Rev. Lett. 87, () 7. [] Y. Itow et al., hep-ex/69; N. Okamura, M. Aoki, K. Hagiwara, Y. Hayato, T. Kobayashi, T. Nakaya and K. Nishikawa, hep-ph/; hep-ph/8. [5] S. H. Geer, Phys. Rev. D57, (998) [6] P. F. Harrison and W. G. Scott, Phys. Lett. B76, () 9; hep-ph/. [7] J. Arafune, M. Koike and J. Sato, Phys. Rev. D56, (997) 9; Erratum ibid., D6, (999) 995. [8] S. J. Parke and T. J. Weiler, Phys. Lett. B5, () 6. [9] For example J. Arafune and J. Sato, Phys. Rev. D55, (997) 65; H. Minakata and H. Nunokawa, Phys. Lett. B, (997) 69, Phys. Rev. D57, (998) ;

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