SOME COMMENTS ON HIGH-PRECISION STUDY OF NEUTRINO OSCILLATIONS

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1 Ó³ Ÿ , º 4(195).. 720Ä732 ˆ ˆŠ Œ ˆ ˆ Œ ƒ Ÿ. ˆŸ SOME COMMENTS ON HIGH-PRECISION STUDY OF NEUTRINO OSCILLATIONS S. M. Bilenky 1 Joint Institute for Nuclear Research, Dubna Some problems connected with the high-precision study of neutrino oscillations are discussed. In the general case of n-neutrino mixing, a convenient expression for transition probability, in which only independent terms ( mass-squared differences) enter, is derived. For three-neutrino mixing, a problem of a deˇnition of a large (atmospheric) neutrino mass-squared difference is discussed. The possibilities to reveal the character of neutrino mass spectrum in future reactor neutrino experiments are commented on as well. Ê ÕÉ Ö μ ² ³Ò, Ö Ò ÊÎ ³ μ Í ²²ÖÍ É μ Ô± ³ É Ì Ò- μ±μ ÉμÎ μ ÉÓÕ. μ Ð ³ ²ÊÎ ³ Ï Ö É μ μ²êî μ Ê μ μ Ò ²Ö μöé- μ É Ìμ, ±μéμ μ Ìμ ÖÉ Éμ²Ó±μ ³Ò β Ò ( ±²ÕÎ Ö μ É ± Éμ ³ ). ²Ö ²ÊÎ Ö ³ Ï Ö É Ì É μ μ Ê É Ö μ ² ³ μ ² Ö μ²óïμ ( ɳμ Ë - μ ) μ É ± Éμ ³. ³ É ÕÉ Ö μ ³μ μ É ÊÎ Ö ±É ³ É μ Ê ÊÐ Ì ±Éμ ÒÌ É ÒÌ Ô± ³ É Ì. PACS: Pq INTRODUCTION The observation of neutrino oscillations in the atmospheric Super-Kamioke [1], solar SNO [2], reactor KamLAND [3], solar neutrino oscillation experiments [4Ä6] is one of the most important recent discoveries in particle physics. Small neutrino masses, many orders of magnitude smaller than those of other fundamental fermions, are an evidence of a beyond the Stard Model physics. One of plausible scenarios which allows one to explain the smallness of neutrino masses is based on the assumption that small (Majorana) neutrino masses are generated by the lepton-number violating the dimension of ˇve effective Lagrangians [7]. In this case neutrino masses are suppressed with respect to masses of leptons quarks by the ratio of the electroweak scale v =( 2G F ) 1/2 246 GeV a scale Λ v of a new lepton-number violating physics. Neutrino oscillation data can be described by the three-neutrino mixing 3 ν ll (x) = U li ν il (x) (l = e, μ, τ). (1) i=1 Here ν i (x) is the ˇeld of neutrinos (Dirac or Majorana) with mass m i,u is the unitary 3 3 PMNS [8Ä10] mixing matrix. 1 bilenky@gmail.com

2 Some Comments on High-Precision Study of Neutrino Oscillations 721 In the framework of the three-neutrino mixing, neutrino oscillations are characterized by two neutrino mass-squared differences Δm 2 23 Δm2 12, three mixing angles θ 12, θ 23, θ 13, one CP phase δ. From the analysis of the data of neutrino oscillation experiments, it was established that Δm 2 12 Δm2 23, mixing angles θ 23 θ 12 are large mixing angle θ 13 is small. The ˇrst information about the angle θ 13 was obtained from the reactor CHOOZ experiment [11], in which only the upper bound sin 2 2θ was found. The ˇrst data of neutrino oscillation experiments were described by expressions for neutrino transition probabilities in the leading approximation which was based on the assumption that sin 2 θ 13 =0. In this approximation, oscillations in atmospheric KamLAND (solar) regions are decoupled (see [12]): in the atmospheric region (atmospheric long-baseline accelerator neutrino oscillation experiments), neutrino oscillations are two-neutrino ν μ ν τ oscillations; in the solar region (the reactor KamLAND experiment), neutrino oscillations are ν e ν μ,τ oscillations. From analysis of the atmospheric long-baseline accelerator oscillation experiments, parameters Δm 2 23 sin2 2θ 23 were determined. From analysis of the data of the KamLAND solar experiments, the other two neutrino oscillation parameters Δm 2 12 sin2 2θ 12 were inferred. In the leading approximation, the character of the neutrino mass spectrum such an important effect of the three-neutrino mixing as CP violation in the lepton sector cannot be revealed. With the measurement of the mixing angle θ 13 in the reactor Daya Bay [13], RENO [14], Double CHOOZ [15] experiments, the situation with the study of neutrino oscillations drastically changed. The investigation of neutrino oscillations entered into a high-precision era, the era of measurements of small, beyond the leading approximation effects, which could allow one to determine the character of the neutrino mass spectrum to measure CP phase δ. In this paper, for the general case of the n-neutrino mixing, we derive a convenient expression for the neutrino transition probability in vacuum, in which only independent terms ( mass-squared differences) enter. In various papers on neutrino oscillations large (atmospheric) neutrino mass-squared difference is determined differently. Difference between these deˇnitions is small (a few percent), but in the era of precision measurements, apparently, it is desirable to have one uniˇed deˇnition. The expression for transition probability presented here provides a natural framework for introduction of two independent neutrino mass-squared differences in the case of the three-neutrino mixing. Determination of the neutrino mass spectrum character is one of the major aims of future reactor neutrino experiments JUNO [16] RENO-50 [17]. On the basis of the proposed expression for the transition probability, we comment on this possibility. 1. GENERAL EXPRESSION FOR NEUTRINO TRANSITION PROBABILITY IN VACUUM For the general case of the neutrino mixing ν αl (x) = 3+n s i=1 U αi ν il (x) (α = e, μ, τ, s 1,...,s ns ), (2)

3 722 Bilenky S. M. we derive here an expression for ν α ν α transition probability alternative to the stard one. Here n s is the number of sterile neutrino ˇelds, U is an unitary (3 + n s ) (3 + n s ) mixing matrix, ν i (x) is the ˇeld of neutrino with mass m i. From (2) Heisenberg uncertainty relation it follows that normalized states of avor ν e,ν μ,ν τ sterile ν s1,ν s2,... neutrinos are described by coherent superpositions of the states of neutrinos with deˇnite masses (see, for example, [12, 18, 19]): ν α = n Uαi ν i. (3) i=1 Here ν i is the state of the left-hed neutrino with mass m i,momentum p, energy E i = p 2 + m 2 i E + m2 i /2E (E = p is the energy of neutrino at m i 0). If at t =0avor neutrino ν α is produced, at the time t we have ν α t = ν α ν α e ih0t ν α = ( ) ν α U α i, e ieit Uαi, (4) α α where H 0 is the free Hamiltonian. From (4) for the normalized probability of the ν α ν α transition, we ˇnd the following expression: 2 P (ν α ν α )= U α i, e ieit Uαi = i = i U α i 2 U αi 2 +2 i>k i Re (U α iu αiu α ku αk e 2iΔ ki ). (5) Here Δ ki = Δm2 ki L 4E, (6) where Δm 2 ki = m2 i m2 k L t is the neutrino source-detector distance. Taking into account the unitarity of the mixing matrix U for the ˇrst term of the probability (5), we have U α i 2 U αi 2 = δ α α 2 Re (U α iuαi U α k U αk). (7) i i>k From (5) (7) for the ( ) ν α ( ) expression (see [20Ä22]): ν α transition probability, we obtain the following stard P ( ν α ν α )=δ α α 4 i>k Re (U α iu αi U α k U αk)sin 2 Δ ki ± ± 2 Im (U α iuαi U α k U αk)sin2δ ki. (8) i>k Let us stress that not all quantities in (8) are independent. For example, in the case of the three-neutrino mixing, three mass-squared differences in (8) are connected by the relation Δm 2 13 =Δm Δm For α α the quantities in the last term of (8) are connected

4 Some Comments on High-Precision Study of Neutrino Oscillations 723 by the relations Im (U α 2Uα2 U α 1 U α1) =Im(U α 3Uα3 U α 2 U α2) = Im (U α 3Uα3 U α 1 U α1) which follow from the unitarity of the mixing matrix (see [21, 22]). We obtain here a simple expression for the neutrino transition probability in vacuum, in which we take into account that there is one arbitrary common phase in the transition amplitude; we use the unitarity of the mixing matrix in the transition amplitude. We have 2 P (ν α ν α )= U α i, e i(ei Ep)t Uαi = δ α α + 2 U α i(e 2iΔpi 1) Uαi = i p where p is an arbitrary ˇxed index. From (9) we ˇnd i = δ α α 2i U α i Uαi e iδpi sin Δ pi 2, (9) i p P (ν α ν α )=δ α α 4 U αi 2 (δ α α U α i 2 )sin 2 Δ pi + i p +8 Re (U α iuαi U α k U αk e i(δpi Δpk) )sinδ pi sin Δ pk. (10) i>k;i,k p Finally, we obtain the following general expression for ( ) ν α ( ) ν α transition probability [23]: ν α ( ) ν α )=δ α α 4 U αi 2 (δ α α U α i 2 )sin 2 Δ pi + i p +8 i>k;i,k p ± 8 Re (U α iu αiu α ku αk )cos(δ pi Δ pk )sinδ pi sin Δ pk ± i>k;i,k p Im (U α iu αiu α ku αk )sin(δ pi Δ pk )sinδ pi sin Δ pk, (11) where sign + ( ) refers to ν α ν α ( ν α ν α ) transition. In (11) only independent terms ( mass-squared differences) enter. For example, for the three-neutrino mixing there are only two independent mass-squared differences one i>kterm in the transition probability (because i, k p). 2. THREE-NEUTRINO OSCILLATIONS 2.1. Atmospheric Neutrino Mass-Squared Difference. Flavor Neutrino Transition Probability. From analysis of the neutrino oscillation data it follows that one mass-squared difference (atmospheric) is much larger than the other one (solar). Two neutrino mass spectra are possible in such a situation 1 : 1 Usually neutrinos with small mass-squared difference are called ν 1 ν 2. It is also assumed that m 2 >m 1, i.e., Δm 2 12 > 0.

5 724 Bilenky S. M. 1. Neutrino spectrum with small mass-squared difference between the lightest neutrinos (normal spectrum, NS) m 1 <m 2 <m 3, Δm 2 12 Δm 2 23; 2. Neutrino spectrum with small mass-squared difference between the heaviest neutrinos (inverted spectrum, IS) m 3 <m 1 <m 2, Δm 2 12 Δm There are only two possibilities to introduce small (solar) Δm 2 S large (atmospheric) Δm 2 A mass-squared differences in the framework of the approach advocated here1 : 1. NS. Δm 2 21 = Δm 2 S, Δm 2 23 = Δm 2 A (p =2), (12) IS. Δm 2 12 = Δm 2 S, Δm 2 13 = Δm 2 A (p =1); (13) 2. NS. Δm 2 12 = Δm2 S, Δm2 13 = Δm2 A IS. Δm 2 21 = Δm2 S, Δm2 23 = Δm2 A (p =1), (14) (p =2). (15) In all papers on neutrino oscillations mixing angles, CP phase solar mass-squared difference are determined in the same way. However, atmospheric mass-squared difference in various papers is determined differently. For example, (in terms of parameters introduced in 1 above) 1. The Bari group determines large neutrino mass-squared difference as follows (see [24]): Δm 2 = 1 2 Δm2 13 +Δm2 23 =Δm2 A Δm2 S. (16) 2. The NuFit group determines the atmospheric mass-squared difference as in 2 (see [25]): Δm 2 13 =Δm2 A +Δm2 S (NS), Δm2 23 = (Δm2 A +Δm2 S ) (IS). (17) 3. In the T2K paper [26], the atmospheric mass-squared difference is determined as in In the MINOS paper [27], large mass-squared difference is determined as Δm 2 23 for both mass spectra. It is obvious, however, that Δm 2 23 for NS Δm2 23 for IS are different quantities. The difference between different atmospheric neutrino mass-squared differencesª is a few percent. It is determined by the ratio Δm 2 S /Δm2 A cannot be neglected in the precision era. Apparently, one deˇnition is desirable. We choose here the option 1. From (11) in the case of normal inverted neutrino mass spectra, we have, respectively, P NS ( ( ) ν l ( ) ν l )=δ l l 4 U l3 2 (δ l l U l 3 2 )sin 2 Δ A 4 U l1 2 (δ l l U l 1 2 )sin 2 Δ S 8Re(U l 3Ul3 U l 1 U l1)cos(δ A +Δ S )sinδ A sin Δ S 8Im(U l 3Ul3 U l 1 U l1)sin(δ A +Δ S )sinδ A sin Δ S (18) 1 Notice that the ˇrst option corresponds to extraction of the phase connected with the intermediate neutrino mass in the expression (9) the second one, to extraction of the phase connected with the last mass.

6 Some Comments on High-Precision Study of Neutrino Oscillations 725 P IS ( ( ) ν l ( ) ν l )=δ l l 4 U l3 2 (δ l l U l 3 2 )sin 2 Δ A 4 U l2 2 (δ l l U l 2 2 )sin 2 Δ S 8Re(U l 3Ul3 U l 2 U l2)cos(δ A +Δ S )sinδ A sin Δ S ± ± 8Im(U l 3Ul3U l 2U l2 )sin(δ A +Δ S )sinδ A sin Δ S. (19) Here Δ A,S = Δm2 A,S L 4E. (20) Thus, transition probabilities depend on extreme valuesª of the elements of neutrino mixing matrix: U l 1(3) U l1(3) in the NS case (p =2); U l 2(3) U l2(3) in the IS case (p =1). Difference in signs of the last terms of (18) (19) is connected with signs in (12) (13). If CP is violated in the lepton sector, we have Let us determine the CP asymmetry P (ν l ν l ) P ( ν l ν l ) (l l). (21) l l = P (ν l ν l ) P ( ν l ν l ). (22) The CP asymmetry satisˇes the following general conditions: l l The ˇrst condition follows from the relation l = ll (23) l l =0. (24) P (ν l ν l )=P ( ν l ν l ), (25) which is a consequence of the CPT invariance. The second condition follows from the conservation of the probability l P (ν l ν l )= l P ( ν l ν l )=1. (26) From (23) (24) it follows that in the case of the three-neutrino mixing CP asymmetries in different avor channels are connected by the following relations [28]: From (18) in the case NS, we have For IS from (19) we ˇnd μe = eτ = μτ. (27) l l = 16 Im U l 3U l3u l 1U l1 sin (Δ A +Δ S )sinδ A sin Δ S. (28) l l =16ImU l 3U l3u l 2U l2 sin (Δ A +Δ S )sinδ A sin Δ S. (29)

7 726 Bilenky S. M. In the next subsections, we present expressions for transition probabilities which are of experimental interest. For that we use the stard parameterization of the PMNS mixing matrix U = c 13 c 12 c 13 s 12 s 13 e iδ c 23 s 12 s 23 c 12 s 13 e iδ c 23 c 12 s 23 s 12 s 13 e iδ c 13 s 23 s 23 s 12 c 23 c 12 s 13 e iδ s 23 c 12 c 23 s 12 s 13 e iδ c 13 c 23. (30) Here c 12 =cosθ 12, s 12 =sinθ 12,etc ν e ν e Survival Probability. Expressions for the three-neutrino ν e survival probabilities are important for the analysis of the data of the reactor neutrino experiments. From (18) (19) for normal inverted mass ordering, we have, respectively, P NS ( ν e ν e )=1 4 U e3 2 (1 U e3 2 )sin 2 Δ A 4 U e1 2 (1 U e1 2 )sin 2 Δ S 8 U e3 2 U e1 2 cos (Δ A +Δ S )sinδ A sin Δ S (31) P IS ( ν e ν e )=1 4 U e3 2 (1 U e3 2 )sin 2 Δ A 4 U e2 2 (1 U e2 2 )sin 2 Δ S 8 U e3 2 U e2 2 cos (Δ A +Δ S )sinδ A sin Δ S. (32) Using the stard parameterization of the PMNS mixing matrix (30), from (18) (19) for NS IS, we have, respectively, P NS ( ν e ν e )=1 sin 2 2θ 13 sin 2 Δ A (sin 2 2θ 12 c sin 2 2θ 13 c 4 12) sin 2 Δ S 2sin 2 2θ 13 c 2 12 cos (Δ A +Δ S )sinδ A sin Δ S (33) P IS ( ν e ν e )=1 sin 2 2θ 13 sin 2 Δ A (sin 2 2θ 12 c sin2 2θ 13 s 4 12 )sin2 Δ S 2sin 2 2θ 13 s 2 12 cos (Δ A +Δ S )sinδ A sin Δ S. (34) Notice that P IS ( ν e ν e ) can be obtained from P NS ( ν e ν e ) by the change c 2 12 s ν μ ν e ( ν μ ν e ) Appearance Probability. Vacuum three-neutrino expressions for ( ) ν e transition probabilities are important for the analysis of the data of long-baseline accelerator experiments, in which matter effects are negligible. From (18) (19), we have P NS ( ( ) ν e )=4 U e3 2 U μ3 2 sin 2 Δ A +4 U e1 2 U μ1 2 sin 2 Δ S 8Re(U e3 Uμ3U e1u μ1 )cos(δ A +Δ S )sinδ A sin Δ S 8Im(U e3 Uμ3U e1u μ1 )sin(δ A +Δ S )sinδ A sin Δ S (35) P IS ( ( ) ν e )=4 U e3 2 U μ3 2 sin 2 Δ A +4 U e2 2 U μ2 2 sin 2 Δ S 8Re(U e3 Uμ3 U e2 U μ2) cos(δ A +Δ S )sinδ A sin Δ S ± ± 8Im(U e3 Uμ3U e2u μ2 )sin(δ A +Δ S )sinδ A sin Δ S. (36)

8 Some Comments on High-Precision Study of Neutrino Oscillations 727 Using the stard parameterization of the PMNS mixing matrix in the case of NS, we have P NS ( ( ) ν e )=sin 2 2θ 13 s 2 23 sin 2 Δ A + +(sin 2 2θ 12 c 2 13 c2 23 +sin2 2θ 13 c 4 12 s Kc2 12 cos δ) sin2 Δ S + +(2sin 2 2θ 13 s 2 23c K cos δ) cos(δ A +Δ S )sinδ A sin Δ S 8J CP sin (Δ A +Δ S )sinδ A sin Δ S. (37) Here K =sin2θ 12 sin 2θ 13 sin 2θ 23 c 13, (38) J CP = 1 K sin δ (39) 8 is the Jarlskog invariant [29]. In the case of the inverted neutrino mass spectrum, we ˇnd P IS ( ( ) ν e )=sin 2 2θ 13 s 2 23 sin2 Δ A + +(sin 2 2θ 12 c 2 13c sin 2 2θ 13 s 4 12s 2 23 Ks 2 12 cos δ) sin 2 Δ S + +(2sin 2 2θ 13 s 2 23 s2 12 K cos δ) cos(δ A +Δ S )sinδ A sin Δ S 8J CP sin (Δ A +Δ S )sinδ A sin Δ S. (40) For the CP asymmetry in the case of NS (IS), we have eμ = 16J CP sin (Δ A +Δ S )sinδ A sin Δ S. (41) 2.4. ν μ ν μ ( ν μ ν μ ) Survival Probability. From (11) for ( ) ν μ survival probability in the case of the normal inverted mass ordering, we have, correspondingly, P NS ( ( ) ν μ )=1 4 U μ3 2 (1 U μ3 2 )sin 2 Δ A 4 U μ1 2 (1 U μ1 2 )sin 2 Δ S 8 U μ3 2 U μ1 2 cos (Δ A +Δ S )sinδ A sin Δ S (42) P IS ( ( ) ν μ )=1 4 U μ3 2 (1 U μ3 2 )sin 2 Δ A 4 U μ2 2 (1 U μ2 2 )sin 2 Δ S 8 U μ3 2 U μ2 2 cos (Δ A +Δ S )sinδ A sin Δ S. (43) Using stard parameterization of the PMNS matrix, we ˇnd P NS ( ( ) ν μ )=1 (sin 2 2θ 23 c sin 2 2θ 13 s 4 23) sin 2 Δ A ( 4 c 2 23 s s2 23 c2 12 s K cos δ )( 4c 2 1 c 2 23 s2 12 s2 23 c2 12 s2 13 K cos δ ) 13 4c 2 sin 2 Δ S 13 2(sin 2 2θ 23 c 2 13s sin 2 2θ 13 c 2 12s Ks 2 23 cos δ) cos(δ A +Δ A )sinδ A sin Δ S (44)

9 728 Bilenky S. M. P IS ( ( ) ν μ )=1 (sin 2 2θ 23 c sin 2 2θ 13 s 4 23)sin 2 Δ A ( 4 c 2 23 c s2 23 s2 12 s2 13 K cos δ )( 4c 2 1 c 2 23 c2 12 s2 23 s2 12 s K cos δ ) 13 4c 2 sin 2 Δ S 13 2(sin 2 2θ 23 c 2 13c sin 2 2θ 13 s 2 12s 4 23 Ks 2 23 cos δ) cos(δ A +Δ A )sinδ A sin Δ S, (45) where K is given by the relation (38). Notice that in the case of long-baseline experiments with Δ A 1 (MINOS, T2K), the term proportional to sin 2 Δ S gives a very small contribution to the probability (sin 2 Δ S 10 3 ) A Comment on the Possibility to Reveal the Character of Neutrino Mass Spectrum in Future Reactor Experiments. Dependence on the neutrino mass ordering of the probability of reactor ν e 's to survive was noticed in [30], in which reactor CHOOZ data were analyzed in the framework of three-neutrino mixing. A reactor experiment with reactordetector distance of 20Ä30 km, which could reveal the character of neutrino mass spectrum, was proposed in [31, 32]. Later in numerous papers a possibility to determine the neutrino mass ordering in an intermediate-baseline reactor experiment ( 50 km) was analyzed in detail (see [33] references therein). Two reactor experiments JUNO [16] RENO-50 [17], in which the neutrino mass ordering is planned to be determined, are under preparation at present. The ν e ν e survival probability (expressions (18) (19)) can be written in the form P ( ν e ν e )=1 sin 2 2θ 13 sin 2 Δ A 4X(1 X) sin 2 Δ S 8sin 2 θ 13 X cos (Δ A +Δ S )sinδ A sin Δ S. (46) In the case of the normal inverted mass spectra, we have, respectively, X = X NS =cos 2 θ 13 cos 2 θ 12 (47) X = X IS =cos 2 θ 13 sin 2 θ 12. (48) From the ˇt of the data that will be obtained in the reactor JUNO experiment after six years of data taking, the parameters Δm 2 S, Δm2 A,sin2 2θ 12 will be determined with accuracy better than 1% (see, for example, [34]). In the Daya Bay experiment the parameter sin 2 θ 13 can be determined with accuracy 4%. Such a precision will, apparently, allow one to distinguish the value X (NS) from the value X (IS) (we used best-ˇt values sin 2 θ 12 =0.302, sin 2 θ 13 =0.0227). 3. TRANSITIONS OF FLAVOR NEUTRINOS INTO STERILE STATES Data of atmospheric, solar, reactor, accelerator neutrino oscillation experiments are described by the three-neutrino mixing with two-neutrino mass-squared differences Δm 2 S ev 2 Δm 2 A ev 2. There exist, however, indications in favor of neutrino oscillations with mass-squared difference(s) about 1 ev 2. These indications were

10 Some Comments on High-Precision Study of Neutrino Oscillations 729 obtained in the following short-baseline neutrino experiments (with L ranging from a few meters to about 500 m): 1. In the LSND experiment [35]. In this experiment neutrinos were produced in decays of π + 's μ + 's. Appearance of ν e 's (presumably produced in the transition ν μ ν e ) was detected. In the MiniBooNE experiment [36,37]. In this experiment an excess of low energy ν e 's ( ν es) was observed. 2. In the old reactor neutrino experiments. Data of these experiments were reanalyzed in [38]. In this new analysis recent calculations of the reactor neutrino ux [39, 40] were used. 3. In the calibration experiments, performed with radioactive sources by the GALLEX [41] SAGE [42] collaborations. In these experiments a deˇcit of ν e 's was observed. In order to interpret these data in terms of neutrino oscillations, it is necessary to assume, that, in addition to the avor neutrinos ν e,ν μ,ν τ, sterile neutrinos exist as well. Let us ˇrst consider 3+1 scheme with three close neutrino masses m i (i =1, 2, 3) the forth mass m 4 separated from m i by about 1 ev gap. We ( choose p =1. ) In the Δm 2 region of L/E sensitive to large neutrino mass-squared difference 14 L 4E 1,wehave Δ 12 Δ From (11) we ˇnd in this case ν α ( ) ν α )=δ α α 4 U α4 2 (δ α α U α 4 2 )sin 2 Δ 14. (49) From this expression for ( ) ν e appearance probability, ( ) ν e ( ) ν e ( ) ν μ disappearance probabilities, we have, respectively, the following expressions: Here ν e )=sin 2 2θ eμ sin 2 Δ 14, (50) ν e ( ) ν e )=1 sin 2 2θ ee sin 2 Δ 14, (51) ν μ )=1 sin 2 2θ μμ sin 2 Δ 14. (52) sin 2 2θ eμ =4 U e4 2 U μ4 2, sin 2 2θ ee =4 U e4 2 (1 U e4 2 ), (53) sin 2 2θ μμ =4 U μ4 2 (1 U μ4 2 ). Notice that the global analysis of all short-baseline neutrino data [43, 44] revealed the inconsistency (tension) of existing short-baseline data. Let us consider a more complicated 3+2 scheme with two masses m 4 m 5 separated from three close masses m i (i =1, 2, 3) by about 1 ev gaps. We choose p =1. In the region of L/E sensitive to large neutrino mass-squared differences Δm 2 14 Δm2 15,we have Δ 12 Δ From (11) we ˇnd the following expression for ( ) ν l (l = e, μ) survival probability: ν l ( ) ν l )=1 4 U l4 2 (1 U l4 2 )sin 2 Δ 14 4 U l5 2 (1 U l5 2 )sin 2 Δ U l5 2 U l4 2 cos (Δ 15 Δ 14 )sinδ 15 sin Δ 14. (54)

11 730 Bilenky S. M. For the probability of the transitions ( ) ν l ( ) ν l, l l, weˇnd ν l ( ) ν l )=4 U l 4 2 U l4 2 sin 2 Δ U l 5 2 U l5 2 sin 2 Δ Re(U l 5Ul5U l 4U l4 )cos(δ 15 Δ 14 )sinδ 15 sin Δ 14 ± ± 8Im(U l 5Ul5 U l 4 U l4) sin(δ 15 Δ 14 )sinδ 15 sin Δ 14. (55) CONCLUSIONS Discovery of neutrino oscillations is one of the most important recent discoveries in particle physics. After the ˇrst stage of investigation of this new phenomenon, now with the measurement of the small parameter sin 2 θ the era of precision study has started. Such fundamental problems of neutrino masses mixing as what is the ordering of neutrino masses (normal or inverted), what is the value of the CP phase δ, what are precise values (with accuracies better than 1%) of other oscillation parameters, is the number of massive neutrinos equal to the number of avor neutrinos (three) or larger than three (do sterile neutrinos exist) are planned to be solved by future neutrino oscillation experiments. At the moment there is no consensus in deˇnition of the large (atmospheric) neutrino masssquared difference: in various experimental theoretical papers this parameter is deˇned differently. Today it is not so important but with future precision different atmospheric mass-squared differencesª will be distinguishable. We believe that a universal deˇnition must be accepted. In this paper, for the general case of n-neutrino mixing we propose a convenient expression for neutrino transition probability in vacuum, in which the unitarity of the mixing matrix is fully employed freedom of the common phase is used. As a result, only independent quantities (including mass-squared differences) enter into expression for the transition probability. On the basis of the proposed expression, we discuss the problem of the atmospheric neutrino mass-squared difference comment on the possibility to reveal the character of the neutrino mass spectrum in future reactor neutrino experiments. Acknowledgements. The author is grateful to A. Olshevskiy C. Giunti for useful discussions. REFERENCES 1. Fukuda Y. et al. (Super-Kamioke Collab.). Evidence for Oscillation of Atmospheric Neutrinos // Phys. Rev. Lett V. 81. P. 1562Ä1567; arxiv:hep-ex/ Ahmad Q. R. et al. (SNO Collab.). Measurement of Day Night Neutrino Energy Spectra at SNO Constraints on Neutrino Mixing Parameters // Phys. Rev. Lett V. 89. P ; arxiv:nucl-ex/ Araki T. et al. (KamLAND Collab.). Measurement of Neutrino Oscillation with KamLAND: Evidence of Spectral Distortion // Phys. Rev. Lett V. 94. P ; arxiv:hep-ex/

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