Parameter degeneracies in neutrino oscillation measurement of leptonic CP and T violation

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Parameter degeneracies in neutrino oscillation measurement of leptonic CP and T violation

Hisakazu Minakata*

Department of Physics, Tokyo Metropolitan University, 1-1 Minami-Osawa, Hachioji, Tokyo 192-0397, Japan Hiroshi Nunokawa†

Instituto de Fı´sica Teo´rica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900 Sa˜o Paulo, SP Brazil Stephen Parke‡

Theoretical Physics Department, Fermi National Accelerator Laboratory, P.O. Box 500, Batavia, Illinois 60510

共Received 20 August 2002; published 27 November 2002兲

The measurement of the mixing angle␪13, sign of⌬m13 2

, and the C P or T violating phase␦is fraught with ambiguities in neutrino oscillation. In this paper we give an analytic treatment of the paramater degeneracies associated with measuring the␯_␮→␯eprobability and its C P and/or T conjugates. For C P violation, we give

explicit solutions to allow us to obtain the regions where there exist twofold and fourfold degeneracies. We calculate the fractional differences, (⌬␪/␪¯), between the allowed solutions which may be used to compare with the expected sensitivities of the experiments. For T violation we show that there is always a complete degeneracy between solutions with positive and negative⌬m₁₃2 which arises due to a symmetry and cannot be removed by observing one neutrino oscillation probability and its T conjugate. Thus there is always a fourfold parameter degeneracy apart from exceptional points. Explicit solutions are also given and the fractional dif-ferences are computed. The biprobability C P/T trajectory diagrams are extensively used to illuminate the nature of the degeneracies.

DOI: 10.1103/PhysRevD.66.093012 PACS number共s兲: 14.60.Pq, 25.30.Pt

I. INTRODUCTION

The discovery of neutrino oscillation in atmospheric neu-trino observation in Super-Kamiokande 关1兴 and the recent accumulating evidence for solar neutrino oscillations 关2兴 naturally suggests neutrino masses and lepton flavor mixing. It is also consistent with the result of the first man-made beam long-baseline accelerator experiment K2K 关3兴. Given the new realm of lepton flavor mixing whose door has just been opened, it is natural to seek a program of exploring systematically the whole structure of neutrino masses and lepton flavor mixing.

Most probably, the most difficult task in determining the structure of the lepton mixing matrix, the Maki-Nakagawa-Sakata 共MNS兲 matrix 关4兴, is the determination of the CP violating phase␦ and the simultaneous 共or preceding兲 mea-surement of兩Ue3兩⫽sin␪13. We use in this paper the standard notation for the MNS matrix with⌬m_{i j}2⬅m_j2⫺m_i2. The fact that the most recent analyses of the solar neutrino data 关5兴 strongly favor the large mixing angle 共LMA兲 Mikheyev-Smirnov-Wolfenstein 共MSW兲 solution 关6兴 is certainly en-couraging for any attempts to measure leptonic C P violation. Since we know that␪13is small, sin22␪13ⱗ0.1, due to the constraint imposed by the CHOOZ reactor experiment 关7兴 and we do not know how small it is, there are two different possibilities. Namely,共A兲␪13 is determined prior to the

ex-perimental search for leptonic C P violation, or共B兲 it is not. Case共A兲 is desirable experimentally. To determine unknown quantities one by one, if possible, is the most sensible way to proceed with minimal danger of picking up fake effects. But since there is no guarantee that case共A兲 is the case, we must prepare for case 共B兲. Even in case 共A兲, experimental deter-mination of␪₁₃always comes with errors, and one must face a similar problem in case B within the experimental uncer-tainties. Moreover, it is known that determination of ␪13 in low energy conventional superbeam type experiments suffers from additional intrinsic uncertainty, the one coming from unknown C P violating phase ␦. See Ref. 关8兴 for further explanation and a possible way of circumventing the prob-lem. Therefore the determination of␦ and␪13are inherently coupled with one another.

Even more seriously, it was noticed by Burguet-Castell

et al.关9兴 that there exist two sets of degenerate solutions (␦i, ␪13

i

) (i⫽1,2兲 even if oscillation probabilities of P(␯_␮

→␯e)⬅P(␯) and its C P conjugate, C P关P(␯)兴⬅P(¯␯_␮

→¯␯_e_{), is accurately measured. They presented an}

approxi-mate but transparent framework of analyzing the degeneracy problem, which we follow in this paper. It was then recog-nized in Ref.关10兴 that the unknown sign of ⌬m13

2

leads to a duplication of the ambiguity, which entails maximal fourfold degeneracy共see below兲. It was noticed by Barger et al. 关11兴 that the fourfold degeneracy is further multiplied by an am-biguity due to approximate invariance of the oscillation probability under the transformation␪₂₃→␲/2⫺␪₂₃. A spe-cial feature of the degeneracy problem at the oscillation maximum was noted and analyzed in some detail关8,11兴. Re-cently, the first discussion of the problem of parameter de-generacy in T violation measurement was given in Ref.关12兴. *Email address: minakata@phys.metro-u.ac.jp

†_{Email address: nunokawa@ift.unesp.br} ‡_{Email address: parke@fnal.gov}

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Meanwhile, there were some technological progresses in analyzing the interplay between the genuine C P phase and the matter effects in measuring leptonic C P or T violation in neutrino oscillation, the issue much-discussed but still un-settled 关13,14兴. The authors of Refs. 关10兴 and 关12兴 intro-duced, respectively, the ‘‘C P and T trajectory diagrams in bi-probability space’’ for pictorial representation of

C P-violating and C P-conserving phase effects as well as the

matter effect in neutrino oscillations. They showed that when these two types of trajectory diagrams are combined it gives a unified graphical representation of neutrino oscillations in matter关12兴. We demonstrate in this paper that they provide a powerful tool for understanding and analyzing the problem of parameter degeneracy, as partly exhibited in Refs. 关10,11,15兴.

It is the purpose of this paper to give a completely general treatment of the problem of parameter degeneracy in neu-trino oscillations associated with C P and T violation mea-surements. We elucidate the nature of the degeneracy and determine the region where it occurs, namely, the regions in the P-C P关P兴共and P-T关P兴) biprobability space in which the same-⌬m132 -sign and/or the mixed-⌬m₁₃2-sign degeneracies take place. While partial treatment of the parameter degen-eracy has been attempted for C P measurement before 关8–11兴, such general treatment is still lacking. We believe that it is worthwhile to have such an overview of the param-eter degeneracy issue to uncover ways of resolving this prob-lem. See关16–19兴 for recent discussions.

We present the first systematic discussion of parameter degeneracy in T measurement following our previous paper in which we set up the problem 关12兴. We uncover a new feature of the degeneracy in T measurement. Namely, we show that for a given T trajectory diagram there always ex-ists another T diagram which completely degenerates with the original one and has the opposite sign of⌬m132 . It means that for any given values of P(␯) and T关P(␯)兴 there are two degenerate solutions of (␦, ␪₁₃) with a differing sign of ⌬m13

2

. It should be noticed that this is true no matter how large the matter effect, quite contrary to the case of C P measurement. Therefore determination of the signs of⌬m132 is impossible in a single T measurement experiment unless one of the following conditions is met: 共a兲 one of the solu-tions is excluded, for example, by the CHOOZ constraint, or 共b兲 some additional information, such as energy distribution of the appearance electrons, is added.

We emphasize that a complete understanding of the struc-ture of the parameter degeneracy should be helpful to one who wants to pursue the solution of the ambiguity problem in an experimentally realistic setting. We, however, do not attempt to discuss the ␪23↔␲/2⫺␪23 degeneracy. We also do not try to solve the problem of parameter degeneracy exactly though it is in principle possible by using an exact but reasonably compact expression of the oscillation prob-ability obtained by Kimura, Takamura, and Yokomakura 关20兴. Instead, we restrict ourselves into the treatment with the approximation introduced by Burguet-Castell et al. 关9兴 in which the approximate formula for the oscillation probability derived by Cervera et al. 关21兴 was employed. Though not

exact, it gives us a much more transparent overview of the problem of parameter degeneracy.

II. PROBLEM OF PARAMETER AMBIGUITY IN CP AND T VIOLATION MEASUREMENT

We define the ‘‘C P共T兲 parameter ambiguity’’ as the prob-lem of having multiple solutions of (␦,␪13) and the sign of ⌬m23

2

for a given set of measured values of oscillation prob-abilities of P(␯_␮→␯e) and its C P 共T兲 conjugate, C P关P(␯)兴⬅P(¯␯_␮→¯␯e) „T关P(␯)兴⬅P(␯e→␯␮)…. We

con-centrate in this paper on this channel because precise mea-surement is much harder in other channels, e.g., in ␯_e

→␯␶. Our use of␯␮→␯eand its C P conjugate is due to our primary concern on conventional superbeam type experi-ments关22兴. The reader should keep this difference in mind if they try to compare our equations with those in Refs. 关9,12兴 in which they use ␯e→␯␮ and its C P conjugate, a natural

choice for neutrino factories关21,23兴. It should also be noted that the neutrino factories and the superbeam experiments are studying processes which are T conjugates.

In this section we utilize the C P and the T trajectory diagrams introduced in关10兴 and 关12兴, respectively, to explain what is the problem of parameter ambiguity and to achieve qualitative understanding of the solutions without using equations. But before entering into the details we want to justify, at least partly, our setting, i.e., prior determination of all the remaining parameters besides␦ and␪13.

A. Problem of parameter degeneracy; setup of the problem

We assume that at the time that an experiment for mea-suring (␦,␪₁₃) is carried out all the remaining parameters in the MNS matrix, ␪23, 兩⌬m23

2 _兩, _␪

12, and ⌬m12 2

, are deter-mined accurately. The discussion on how the experimental uncertainties affect the problem of parameter degeneracy is important, but is beyond the scope of this paper. It should be more or less true, because␪23and⌬m23

2

will be determined quite accurately by the future long-baseline experiments 关24–26兴 up to the ␪23↔␲/2⫺␪23 ambiguity. Most notably, the expected sensitivities in the JHF-SK experiment in its phase I are 关24兴

␦共sin2₂_␪_23兲⯝10⫺2_,

␦共兩⌬m232_{兩兲⯝共4⫺6兲⫻10}_⫺5 _eV2 _共1兲 at around兩⌬m232兩⫽3⫻10⫺3 eV2. On the other hand, the best place for accurate determination of␪₁₂and⌬m122 , assuming the LMA MSW solar neutrino solution, is most probably the KamLAND experiment; the expected sensitivities are关27兴

␦共tan2_␪_12兲⯝10%,

␦共⌬m12

2_兲⯝10% _共2兲

at around the LMA best fit parameters. Therefore we feel that our setting, prior determination of all the mixing angles and ⌬m2_{’s besides}_␦_,_␪

13, and the sign of⌬m23

2 _{, is a reasonable} one at least in the first approximation.

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B. Pictorial representation of parameter ambiguities

In this section we use C P and the T trajectory diagrams 关10,12兴 to explain intuitively what is the problem of param-eter degeneracy, and to achieve qualitative understanding of the solutions without using equations. In Fig. 1 we display four C P trajectories in the P-C P关P兴 biprobability space which all have intersections at P(␯)⫽1.9% and CP关P(␯)兴 ⫽2.6%. The four CP trajectories are drawn with four dif-ferent values of ␪13, sin22␪13⫽0.055, 0.050, 0.586, and 0.472, and the former 共latter兲 two trajectories correspond to positive 共negative兲 ⌬m132 . The analytic expressions of the four degenerate solutions will be derived in Sec. IV. The neutrino energy E and the baseline length L are taken as E ⫽250 MeV and L⫽130 km, respectively.1 _{The setting} an-ticipates an application to the CERN-Frejius project 关28兴, where the regions with parameter ambiguities are widest. The values of all the remaining mixing parameters are given in the caption of Fig. 1.

Figure 1 demonstrates that there is a fourfold degeneracy in the determination of (␦, ␪13) for a given set of P(␯) and

C P关P(␯)兴. The region between the solid lines and the

re-gion between the dashed lines in Fig. 1 are the rere-gions where twofold degeneracies exist in the solutions of (␦, ␪13) for positive and negative ⌬m132 sectors, respectively. 关See Eq. 共48兲 in Sec. IV.兴 It is intuitively understandable that the re-gion where degenerate solutions exists is the rere-gion swept over by the C P trajectories when the parameter␪13is varied, while keeping other mixing parameters and the experimental conditions fixed. The shaded region is the region where the full fourfold degeneracy exists.

Now we turn to the T measurement. In Fig. 2 we display four T trajectories in the P-T关P兴 biprobability space which all have intersections at P(␯)⫽1.7% and T关P(␯)兴⫽2.5%. The four T trajectories are drawn with four different values of ␪13, sin22␪13⫽0.05, 0.0427, 0.575, and 0.490, and the former共latter兲 two trajectories correspond to positive 共nega-tive兲 ⌬m132 . Matter effects split the positive and negative ⌬m13

2

trajectories, see 关12兴, thus different values of ␪ are required for mixed-sign trajectories to overlap. The remain-ing mixremain-ing parameters and the experimental settremain-ing are the same as in Fig. 1.

A clear and interesting difference from the C P diagram manifests itself already at this level, reflecting the highly symmetric nature of the T conjugate probabilities as will be made explicit in Eq. 共3兲. Namely, the two different-⌬m13

2_{-sign diagrams} _{共the first and the third, and the second} and fourth兲 completely overlap with each other. In the next section we will make it clear that the complete degeneracy 1_{In all the figures of this paper we do not average over neutrino}

energy distributions as was performed in Refs. 关10,12兴 but use a fixed neutrino energy specified for each figure.

FIG. 1. An example of the degenerate solutions for the CERN-Frejius project in the P(␯)⬅P(␯_␮→␯e) versus C P关P(␯)兴

⬅P(␯¯␮→␯¯e) plane. Between the solid共dashed兲 lines is the allowed

region for positive共negative兲 ⌬m₁₃2 and the shaded region is where solutions for both signs are allowed. The solid共dashed兲 ellipses are for positive 共negative兲 ⌬m13

2

and they all meet at a single point. This is the C P parameter degeneracy problem. We have used a fixed neutrino energy of 250 MeV and a baseline of 130 km. The mixing parameters are fixed to be兩⌬m13

2 兩⫽3⫻10⫺3_eV2_{, sin}2₂_␪ 23 ⫽1.0, ⌬m12 2_⫽⫹5⫻10⫺5 _eV2 , sin22␪12⫽0.8, and Ye␳ ⫽1.5 g cm⫺3_.

FIG. 2. An example of the degenerate solutions using the energy and path length of the CERN-Frejius project in the P(␯)⬅P(␯_␮

→␯e) versus T关P(␯)兴⬅P(␯e→␯␮) plane. For this figure there is

complete overlap in the region 共shaded兲 that allows solutions for either sign of ⌬m₁₃2 . The solid 共dashed兲 ellipses are for positive

共negative兲 ⌬m13 2

and they all meet at the ‘‘measured point,’’ ( P, PT₎_{⫽(1.7,2.5)%. This is the T-parameter degeneracy problem.}

Notice that for the each ellipse with positive⌬m13 2

there is a com-pletely degenerate ellipse with negative⌬m13

2

. This feature will be explained in Sec. III C. Parameters are the same as in Fig. 1.

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originates from a symmetry. Therefore discrimination of the sign of⌬m132 is impossible in a single T violation measure-ment experimeasure-ment unless one of the solutions is excluded by another experiment. We will demonstrate in Sec. III that the degeneracy is not an accidental one specific to this particular case, but its existence is generic. There is always a fourfold degeneracy in T measurement.

III. PARAMETER DEGENERACY IN T VIOLATION MEASUREMENTS

We start by presenting an analytic treatment of the prob-lem of parameter degeneracy in T violation measurements primarily because it is simpler and instructive. To do this we generalize the formalism developed by Burguet-Castell et al. 关9兴 by treating the cases of positive and negative ⌬m132 si-multaneously. It provides the basis of our treatment of the degeneracy between the same as well as across the alternat-ing ⌬m132 -sign probabilities. It will become clear from the following discussions that the treatment of the mixed-sign degenerate solutions, for both C P and T measurements, can be done as a straightforward generalization of the same-sign degeneracy case by simply taking account of duplication due to the alternating sign of ⌬m132 关10兴.

There are four basic equations satisfied by the T conjugate probabilities in the case of T measurement for small sin␪13:

⫹P䉺, 共3兲 where X_⫾ and Y_⫾ are given in the Appendix, P_䉺 indicates the term which is related with solar neutrino oscillations, and ⌬13⬅兩⌬m132 _{兩L/2E. Note that} _{⫾ here refers to the sign of} ⌬m13

2 _and _␪ _{is an abbreviation of}_␪_13⯝sin_␪

13. In the next sections we discuss the possible solutions for ␪ and␦ for a given measurement of both P and PT for both the positive and negative sign of⌬m132 .

A. The same-sign degeneracy; T measurement

The treatment in this section applies for two overlapping

T trajectories with the same sign of ⌬m132 . The degeneracy associated with alternating-sign trajectories will be explored in the next section.

There are two sets of approximate solutions of Eq.共3兲,␪i

and␦i, where (i⫽1,2) and (i⫽3,4) denotes the solutions in

the positive and negative ⌬m₁₃2 sectors, respectively. They are ␪i⫽

冑

P⫺P_X 䉺 ⫾ ⫺ Y_⫾ 2X_⫾cos

冉

␦i⫾ ⌬13 2

冊

, ␪i⫽

冑

P T_⫺P 䉺 X_⫾ ⫺ Y_⫾ 2X_⫾cos

冉

␦i⫿ ⌬13 2

冊

, 共4兲

where ⫾ correspond to solutions in positive and negative ⌬m13

2

sectors.2 We then obtain, e.g., for the positive ⌬m132 sector

␪2⫺␪1⫽⫺ Y⫹

2X_⫹

冋

共cos␦2⫺cos␦1兲cos

冉

⌬13

2

冊

⫺共sin␦2⫺sin␦1兲sin

冉

⌬13 2

冊册

,

␪2⫺␪1⫽⫺_2XY⫹

⫹

冋

共cos␦2⫺cos␦1兲cos

冉

⌬13

2

冊

⫹共sin␦2⫺sin␦1兲sin

冉

⌬13

2

冊册

, 共5兲

which entails the degeneracy that if (␪1,␦1) is a solution so is ␪2⫽␪1⫹ Y_⫹ X_⫹cos␦1cos

冉

⌬13 2

冊

and ␦2⫽␲⫺␦1, 共6兲 in addition to the trivial solution. A similar degeneracy also holds for the negative ⌬m132 sector, that is, if (␪₃,␦₃) is a solution so is

␪4⫽␪3⫹Y_X⫺

⫺cos␦3cos

冉

⌬13

2

冊

and ␦4⫽␲⫺␦3. 共7兲 Both of these same sign ⌬m13

2

degeneracies are in matter though they look like the vacuum degeneracies as discussed in关10兴. Notice that if the experimental setup is chosen such that cos(⌬13/2)⫽0 关8兴 or nature has chosen cos␦⫽0 then the same-sign degeneracies are removed.

B. The mixed-sign degeneracy; T measurement

Let us now examine the problem of parameter degeneracy which involves positive and negative⌬m₁₃2 . The basic equa-tions 共3兲 can be approximately solved for the mixed-sign situation as

2_{The above solutions are exact solutions to the system of Eq.}_{共3兲 if} we were to add terms Y_⫾2cos2(␦⫾⌬13/2)/(4X⫾) to the equations,

␦3⫹ ⌬13 2

冊

. 共9兲

P T_⫺P 䉺兲共11兲 and cos␦iis given by cos␦i⫽⫾冑1⫺sin2␦i. Using these cos␦i the values of␪ are given by

derivable under the Cervera et al. approximation.3 Then, it follows that

sin␦1⫽sin␦3, 共cos␦1⫹cos␦3兲cos⌬13₂ ⫽⫺2

冑

X⫹

Y_⫹ 共冑X⫹␪1⫺

冑

X⫺␪3兲.

共15兲 One can choose without loss of generality ␦3⫽␲⫺␦1 as a solution of Eq. 共15兲. Then, for a given P and PT

i j ⬅ ␪i⫺␪j 共␪i⫹␪j兲/2 , 共18兲

to quantify how different the two degenerate solutions are. In fact, one can obtain simple expressions for the various frac-tional differences:

冉

⌬␪ ␪ ¯

冊

12 ⫽

冉

⌬␪ ␪ ¯

冊

34 ⫽

⫺2Y_⫹cos␦₁cos

冉

⌬13 2

冊

In Figs. 3–5 we have plotted the differences in the

al-lowed ␪ solutions divided by half the sum for the CERN-Frejius, JHF-SK, and Fermilab-NUMI 关29兴 possible experi-ments using ␯_␮→␯e and its T conjugate ␯e→␯␮. The

regions where this fractional difference is small are regions where the parameter degeneracy inherent in such measure-ments is only important once the experimental resolution on

␪ for a fixed solution is of the same size or smaller. Notice that near the boundaries on the allowed region the fractional differences are small for the same-sign solutions. For the mixed sign, either the 共1,4兲 or 共3,2兲 fractional difference plots have a line for which the fractional difference is zero. This line can be understood as follows: for a given small value of ␪ the positive and negative⌬m₁₃2 ellipses overlap and intersect at two points. As ␪ varies these intersection points give us this line with zero mixed-sign fractional dif-ference.

C. Symmetry between the two alternating-⌬m13 2

-sign T diagrams

The observant reader will notice that there is in a general one-to-one correspondence between the solutions with posi-tive ⌬m₁₃2 , labeled 1 and 2, and those solutions with nega-tive⌬m132 , labeled 3 and 4, in Eq.共16兲 by using the identity, Eq. 共14兲. In fact this correspondence applies not only to the solutions but to the complete T diagram. For a given T tra-jectory with positive⌬m132 there always exists a T trajectory with negative ⌬m132 with a different value of␪, which nev-ertheless completely overlaps the positive trajectory, see Fig. 2. This surprising phenomenon occurs because there exists a symmetry in the T probability system defined in Eq. 共3兲.

One can observe from Eq.共3兲 that a positive ⌬m₁₃2 trajec-tory which is defined by the first two equations of Eq.共3兲 can

FIG. 3. The isofractional differences, as a percent, for the allowed solutions for the mixing angle␪13in the P(␯)⬅P(␯␮→␯e) versus

T关P(␯)兴⬅P(␯e→␯␮) plane for the CERN-Frejius project. The fractional differences for solutions共3,4兲 is identical to that for 共1,2兲 and the

fractional difference for共3,2兲 equals the fractional difference for 共1,4兲 plus or minus a constant, see Eqs. 共19兲–共22兲. In this case the fractional difference共3,2兲 has a zero contour. The parameters are the same as in Fig. 1. The ellipses are labeled T_sin2₂_␪

13

(⫾)

to show the relevant size of sin2₂_␪

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be used to generate a negative ⌬m132 trajectory which com-pletely overlaps with the original one by the transformation

␦→␲⫺␦,

␪→

冑

X_X⫺

⫹␪. 共24兲

This means that for a measure set of P and PT _{there are}

always two sets of solutions with a different sign of ⌬m132 . There is no way to resolve this ambiguity because the two T trajectories are completely degenerate. It should be noticed that this situation occurs no matter how large a matter effect at a much longer baseline. In such a case, two T trajectories with the same ␪13 but opposite sign of⌬m13

2

are far apart,

T关P(␯)兴⬅P(␯e→␯_␮) plane for the Fermilab-NUMI project. The fractional differences for solutions共3,4兲 is identical to that for 共1,2兲 and

the fractional difference for 共3,2兲 equals the fractional difference for 共1,4兲 plus or minus a constant, see Eqs. 共19兲–共22兲. At very small probability the共1,4兲 fractional difference has a zero contour. The parameters are the same as in Fig. 1. The ellipses are labeled as in Fig. 3. FIG. 4. The isofractional differences, as a percent, for the allowed solutions for the mixing angle␪13in the P(␯)⬅P(␯␮→␯e) versus

T关P(␯)兴⬅P(␯e→␯␮) plane for the JHK-SK project. The fractional differences for solutions 共3,4兲 is identical to that for 共1,2兲 and the

fractional difference for 共3,2兲 equals the fractional difference for 共1,4兲 plus or minus a constant, see Eqs. 共19兲–共22兲. The zero contour appearing in the共1,4兲 fractional difference is explained in the text. The parameters are the same as in Fig. 1. The ellipses are labeled as in Fig. 3.

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and one would have expected that there is no ambiguity in the determination of the sign of ⌬m132 . Hence there is a ‘‘no-go theorem’’ for the determination of the sign of ⌬m132 by a single T violation measurement. The possible cases in which the ‘‘theorem’’ is circumvented are, as mentioned in the Introduction, 共a兲 one of the solutions is excluded, for example, by the CHOOZ constraint, or 共b兲 some additional information, such as energy distribution of the appearance electrons or another T violation measurement with different parameters, is added.

IV. PARAMETER DEGENERACY IN CP-VIOLATION MEASUREMENTS

We now turn to the analytic treatment of the parameter degeneracy in C P-violation measurements. We proceed in an analogous way to the analytic treatment of T violation given in Sec. III.

⫹P䉺 where X_⫾and Y_⫾are given in the Appendix. As before, P_䉺 indicates the term which is related with solar neutrino oscil-lations, ⌬13⬅兩⌬m₁₃2兩L/2E, the ⫾ here refers to the sign of ⌬m13

2

and␪ is an abbreviation of ␪13⯝s13.

Note that there exist relations among coefficients;

X_⫾⬅X共⫾⌬m132 ,a兲, 共26兲

FIG. 6. The isofractional differences, as a percent, for the allowed solutions for the mixing angle␪13in the P(␯)⬅P(␯_␮→␯e) versus

C P关P(␯)兴⬅P(␯¯_␮→␯¯e) plane for the CERN-Frejius project. The parameters are the same as in Fig. 1 with⌬m12

2_⫽5⫻10⫺5_eV2

. The dashed ellipses are labeled C P_sin2₂_␪

13

(⫾)

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X ¯

⫾⫽X共⫾⌬m132 ,⫺a兲. 共27兲

In leading order in⌬m122 /⌬m132 , there exist further relations,

␦3⫹ ⌬13 2

冊

. 共33兲 The solutions of these equations are

sin␦1,2⫽ 1 D

冋

⫺C (⫹)_sin⌬13 2 ⌬P⫹ ⫾C(⫺)_cos⌬13 2

冑

D⫺共⌬P⫹兲 2

册

_, _共34兲 sin␦3,4⫽ 1 D

冋

⫺C (⫹)_sin⌬13 2 ⌬P⫺ ⫿C(⫺)_cos⌬13 2

冑

D⫺共⌬P⫺兲 2

册

_, _共35兲

C P关P(␯)兴⬅P(␯¯_␮→␯¯e) plane for the JHF-SK project. The parameters are the same as in Fig. 1 with⌬m12 2

⫽5⫻10⫺5_eV2_{. The ellipses are}

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␪1,2⫽ 1 2D(⫺)

冋

冑共P⫺P

䉺兲X⫹ Y_⫹ ⫹

冑共P¯⫺P

䉺兲X⫺ Y_⫺ ⫹sin␦1,2sin ⌬13 2

册

, 共36兲 ␪3,4⫽ 1 2D(⫺)

P⫺P䉺 X_⫾ ⫺

冑

P ¯_⫺P_䉺 X_⫿ . 共41兲

The sign in Eq.共35兲 is determined relative to Eq. 共34兲 so that it reproduces the pair of degenerate solutions in the case of a precisely determined value of ␪13 关12兴. It should be noticed that provided

冑

D⫺(⌬P_⫾)2is real the constraint兩sin␦i兩⭐1 is satisfied automatically in Eqs. 共34兲 and 共35兲.

Let us focus first on the features of the same-sign degen-erate solution. The set (␪i, ␦i) with i⫽1,2 (i⫽3,4)

de-FIG. 8. The isofractional differences, as a percent, for the allowed solutions for the mixing angle␪13in the P(␯)⬅P(␯␮→␯e) versus

C P关P(␯)兴⬅P(␯¯_␮→␯¯e) plane for the Fermilab-NUMI project. The mixed-sign fractional differences (⌬␪/␪¯)13 and 14terminate at around P⫽CP关P兴⬇1.6% because above this probability the sign of ⌬m₁₃2 is determined, as discussed in the text. The critical value of P (⬇1.6%) and sin22␪13 (⬇0.033) can be calculated from Eqs. 共49兲 and 共50兲. The parameters are the same as in Fig. 1 with ⌬m12

2_⫽5

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scribes two degenerate solutions with positive 共negative兲 ⌬m13

2

for given values of P and P¯ . Of course, they reproduce the relationships obtained by Burguet-Castell et al. in关9兴:

sin␦2⫺sin␦1⫽⫺2

冉

sin␦1⫹zcos␦1

. 共44兲

Let us illuminate how the relative phases between ␦’s between these degenerate solutions can be obtained in a transparent way. Toward the goal we first calculate cos␦i.

cos␦_1,2and cos␦_3,4can be obtained from Eqs.共34兲 and 共35兲, respectively, by replacing of C(⫾)by⫿C(⫿)and sin⌬13/2 by cos⌬132 and vice versa. One can show by using these re-sults that

cos共␦1⫹␦2兲⫽cos共␦3⫹␦4兲⫽ 1⫺z2

1⫹z2, 共45兲 which implies that

␦2⫽␲⫺␦1⫹arccos 关共z2_⫺1兲/共z2_⫹1兲兴,

␦4⫽␲⫺␦3⫹arccos 关共z2_⫺1兲/共z2_⫹1兲兴. _共46兲 Thus in the allowed region of biprobability space ␦2 (␦4) differs from ␲⫺␦1 (␲⫺␦3) by a constant, arccos关(z2 ⫺1)/(z2_{⫹1)兴, which depends on the energy and path length} of the neutrino beam but not on the mixing angle␪. Near the

C P关P(␯)兴⬅P(␯¯_␮→␯¯e) plane for the CERN-Frejius project. The parameters are the same as in Fig. 1 except for the solar⌬m12 2

which is set to 1⫻10⫺4 eV2for this plot. The ellipses are labeled as in Fig. 6.

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oscillation maximum, z→⬁, this constant vanishes so that

␦2⯝␲⫺␦1 and␦3⯝␲⫺␦4 as noticed in关9兴.

For the mixed-sign degenerate solution one can show that

cos共␦1⫺␦3兲⫽ 共⌬P⫹⌬P⫺⫺

冑

D⫺⌬P⫹2

冑

D⫺⌬P⫺2兲 D , sin共␦1⫺␦3兲⫽ 共⌬P⫹

冑

D⫺⌬P⫺2⫹⌬P⫺

冑

D⫺⌬P⫹2兲 D . 共47兲 One can show, for example, cos(␦1⫺␦3)⫽⫺1 and sin(␦1 ⫺␦3)⫽0 in the P¯→P limit by noting that ⌬P⫺⫽⫺⌬P⫹ in the limit. It means that ␦3⫽␦1⫹␲ 共mod 2␲), in agreement with the result obtained in Ref.关12兴.

The conditions for existence of the same-sign solution are

D⫺共⌬P_⫹兲2⭓0 and D⫺共⌬P_⫺兲2⭓0 共48兲

for positive and negative ⌬m132 , respectively. The condition for existence of the mixed-sign solution is the intersection of the two regions which satisfy the conditions of Eq. 共48兲. An

example of the regions satisfying conditions for the existence of the same-sign as well as mixed-sign solutions are depicted in Fig. 1.

The maximum value of P and P¯ which allows mixed-sign solutions is determined by D⫺(⌬P_⫹)2_⫽D⫺(⌬P

⫺)2⫽0 with P⫽P¯. This occurs for a critical value of P given by

Pcrit⫽P_䉺⫹ X⫹X⫺D

共冑X_⫹⫺

冑

X_⫺兲2, 共49兲

which can be used to determine the critical value of␪ as

␪crit⫽

C(⫹)sin2⌬13 2

2D(⫺)

冑

D . 共50兲

There is no degeneracy in the value of␪ at this critical point, i.e.,␪1⫽␪2⫽␪3⫽␪₄. An example of this can be seen in Fig. 8. At the first peak in the oscillation probability,⌬13⫽␲, the value of the critical␪ is simply given by

C P关P(␯)兴⬅P(␯¯_␮→␯¯e) plane for the JHF-SK project. The parameters are the same as in Fig. 1 except for the solar⌬m12 2

which is set to 1⫻10⫺4eV2for this plot. The ellipses are labeled as in Fig. 6.

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␪crit共⌬13⫽␲兲⫽

_冑

兩Y⫹兩

X_⫹共

冑

X_⫹⫺冑X_⫺兲. 共51兲

As ⌬13→2␲, the critical ␪ goes to zero and so does the oscillation probabilities P and P¯ as this is the position of the first trough in the oscillation probabilities.

In Figs. 6– 8 we have plotted the fractional difference, (⌬␪/¯ ), see Eq.␪ 共18兲, for the CERN-Frejius, JHF-SK, and Fermilab-NUMI possible experiments using␯_␮→␯eand its

C P conjugate ¯␯_␮→¯␯e. The regions where this fractional

difference is small are regions where the parameter degen-eracy inherent in such measurements is only important once the experimental resolution on␪ for a fixed solution is of the same size or smaller. Notice that near the boundaries on the allowed region the fractional differences are small for the same sign solutions. For the mixed sign, 共1,3兲, fractional difference plots there is a zero along the diagonal. This is explained by the fact that in our approximation the positive and negative⌬m132 ellipses, for a given␪, intersect along the

diagonal. Figures 9–11 are similar to Figs. 6– 8 except that the size of solar⌬m122 has been increased by a factor of 2 to 1⫻10⫺4eV2. Notice that the parameter degeneracies prob-lem is more pronounced as the solar ⌬m2 is increased. At even larger values of⌬m122 , our approximations become less reliable.

V. SUMMARY AND CONCLUSION

In this paper we have given a complete analytic treatment of the parameter degeneracy issue for␪13, sign of⌬m13

2 the and the C P, and T violating phase␦that appears in neutrino oscillations. For a given neutrino flavor transition probability and its C P or T conjugate probability we have derived the allowed values of␪13, the sign of⌬m13

2

, and the C P and T violating phase ␦. We have given explicit expressions of degenerate solutions and obtained, among other things, exact formulas for the relationship between solutions of ␦’s up to the correction of order (⌬m122/⌬m132)2.

In general there is a fourfold degeneracy, two allowed

FIG. 11. The isofractional differences, as a percent, for the allowed solutions for the mixing angle␪13in the P(␯)⬅P(␯_␮→␯e) versus

C P关P(␯)兴⬅P(␯¯_␮→␯¯e) plane for the Fermilab-NUMI project. The parameters are the same as in Fig. 1 except for the solar⌬m12 2

which is set to 1⫻10⫺4eV2_{for this plot. The ellipses are labeled as in Fig. 6.}

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values of␪₁₃for both signs of⌬m132 . This is always true for the T violation measurement whereas for the C P violation measurement the fourfold degeneracy can be reduced to two-fold degeneracy if matter effects are sufficiently large, or we live close to the region␦⬃␲/2 or 3␲/2. The significance of matter effects the dependence on the energy of the neutrino beam, the separation between the source and the detector, as well as the density of matter between them. The fractional difference of␪13between the various solutions has been cal-culated which can be compared with the experimental sensi-tivity for a given setup to determine whether or not the de-generacy issue is significant or not.

For the possible future experimental setups CERN-Frejius, JHK-SK, and Fermilab-NUMI we have given nu-merical results for the channel ␯_␮→␯e and its C P and T

conjugate. The C P conjugate being most relevant for these future superbeam experiments. For the CERN-Frejius, JHF-SK, and Fermilab-NUMI experimental setups the parameter degeneracy issue is only relevant once the experimental reso-lution on the determination of ␪₁₃ is better than 15%, 10%, and 5% respectively, assuming a transition probability near 1% and a⌬m122⫽5⫻10⫺5 eV2, see Figs. 6– 8. At larger val-ues of⌬m₁₂2 the parameter degeneracy issue becomes more important. These iso-fractional difference plots are useful for comparing the sensitivity of different experimental setups, neutrino energy, path length, and experimental sensitivity to this parameter degeneracy issue.

ACKNOWLEDGMENTS

H.M. thanks the Theoretical Physics Department of Fer-milab for warm hospitality extended to him through his stay. This work was supported by the Grant-in-Aid for Scientific Research in Priority Areas No. 12047222, Japan Ministry of Education, Culture, Sports, Science, and Technology. Fermi-lab is operated by URA under DOE contract No. DE-AC02-76CH03000.

APPENDIX

The standard flavor transition probability for neutrino os-cillations in the␯_␮→␯e channel can be written as

P共␯兲_⫾⫽X_⫾␪2⫹Y_⫾␪cos

冉

␦⫾⌬13

2

冊

⫹P䉺, 共A1兲 where the⫾ signs in X_⫾and Y_⫾refer to positive or negative values of ⌬m132 , ␪ is an abbreviation for sin␪13, and P䉺 indicates the terms related to solar neutrino oscillations. For details on the approximations used in deriving this transition probability see Ref. 关21兴. All other channels used in this paper, ␯e→␯␮ and¯␯␮→␯¯e, can also be expressed with the

aL 2

冊

共A4兲 with ⌬i j⬅ 兩⌬mi j 2_兩L

2E and B⫾⬅兩aL⫾⌬13兩, 共A5兲

where a⫽

冑

2GFNedenotes the index of refraction in matter

with G_F being the Fermi constant and N_ea constant electron number density in the earth.

Obviously from the above definitions, X_⫾and Y_⫾ satisfy the identity

Y_⫹

冑

X_⫹⫽⫺ Y_⫺

冑

X_⫺ 共A6兲

which is used throughout this paper.

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