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New plots and parameter degeneracies in neutrino oscillations


hep-ph/0405005

New plots and parameter degeneracies in neutrino oscillations
Osamu Yasuda? Department of Physics, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan
(Dated: February 1, 2008)

arXiv:hep-ph/0405005v1 2 May 2004

Abstract
It is shown that eightfold degeneracy in neutrino oscillations is easily seen by plotting constant probabilities in the (sin2 2θ13 , 1/s2 23 ) plane. Using this plot, we discuss how an additional long baseline measurement resolves degeneracies after the JPARC experiment measures the oscillation probabilities P (ν? → νe ) and P (? ν? → ν ?e ) at |?m2 31 |L/4E = π/2. By measuring P (ν? → νe ) 2 or P (? ν? → ν ?e ), the sgn(?m31 ) ambiguity is resolved better at longer baselines and the δ ? π ? δ ambiguity is resolved better when |?m2 31 |L/4E ? π/2 is larger. The θ23 ambiguity may 2 be resolved as a byproduct if |?m31 |L/4E ? π is small and the CP phase δ turns out to satisfy cos(δ + |?m2 31 |L/4E ) ? 1. It is pointed out that the low energy option (E ?1GeV) at the o?axis NuMI experiment may be useful in resolving these ambiguities. The νe → ντ channel o?ers a promising possibility which may potentially resolve all the ambiguities.
PACS numbers: 12.15.Ff,14.60.Pq,25.30.Pt

?

Electronic address: E-mail: yasuda@phys.metro-u.ac.jp

Typeset by REVTEX

1

I.

INTRODUCTION

From the recent experiments on atmospheric [1] and solar [2], and reactor [3, 4] neutrinos, we now know approximately the values of the mixing angles and the mass squared di?erences of the atmospheric and solar neutrino oscillations:
2 ?5 (sin2 2θ12 , ?m2 21 ) ? (0.8, 7 × 10 eV ) 2 ?3 (sin2 2θ23 , |?m2 31 |) ? (1.0, 2 × 10 eV )

for the solar neutrino for the atmospheric neutrino,

2 and the case of ?m2 31 > 0 (?m31 < 0) corresponds to the normal (inverted) mass hierarchy, as is shown in Fig. 1. In the three ?avor framework of neutrino oscillations, the oscillation parameters which are still unknown to date are the third mixing angle θ13 , the sign of the mass squared di?erence ?m2 31 of the atmospheric neutrino oscillation, and the CP phase δ . It is expected that long baseline experiments in the future will determine these three quantities. Since the work of [6], it has been known that even if the values of the oscillation probabilities P (ν? → νe ) and P (? ν? → ν ?e ) are exactly given we cannot determine uniquely the values of the oscillation parameters due to parameter degeneracies. There are three kinds of param2 eter degeneracies: the intrinsic (θ13 , δ ) degeneracy [6], the degeneracy of ?m2 31 ? ??m31 [7], and the degeneracy of θ23 ? π/2 ? θ23 [8, 9]. The intrinsic degeneracy is exact when 2 2 ?m2 21 /?m31 √is exactly zero. The sgn(?m31 ) degeneracy is exact when AL is exactly zero, where A ≡ 2GF Ne and L stand for the matter e?ect and the baseline, respectively (GF is the Fermi constant and Ne is the electron density in matter). The θ23 degeneracy is exact when cos 2θ23 is exactly zero. Each degeneracy gives a twofold solution, so in total we have an eightfold solution if all the degeneracies are exact. In this case prediction for physics is the same for all the degenerated solutions and there is no problem. However, these degeneracies are lifted slightly in long baseline experiments 1 , and there are in general eight di?erent solutions [9]. When we try to determine the oscillation parameters, ambiguities arise because the values of the oscillation parameters are slightly di?erent for each solution. In particular, this causes a serious problem in measurement of CP violation, which is expected to be small e?ect in the long baseline experiments, and we could mistake a fake e?ect due to the ambiguities for nonvanishing CP violation if we do not treat the ambiguities carefully. In the references [6, 7, 9] in the past, various diagrams have been given to visualize how degeneracies are lifted in the parameter space. To see how the eightfold degeneracy is lifted, it is necessary for the plot to give eight di?erent points for di?erent eight solutions. An e?ort was made in [11] to visualize the eight di?erent points by plotting the trajectories of constant probabilities in the (sin2 2θ13 , s2 23 ) plane. In the present paper we propose a plot

where we use the standard parametrization [5] of the MNS mixing matrix ? ? c12 c13 s12 c13 s13 e?iδ U = ? ?s12 c23 ? c12 s23 s13 eiδ c12 c23 ? s12 s23 s13 eiδ s23 c13 ? , s12 s23 ? c12 c23 s13 eiδ ?c12 s23 ? s12 c23 s13 eiδ c23 c13

1

cos 2θ23 may be exactly zero, but the present atmospheric neutrino data [10] allow the possibility of cos 2θ23 = 0, so we will assume cos 2θ23 = 0 in general in the following discussions.

2

in the (sin2 2θ13 , 1/s2 23 ) plane, which o?ers the simplest way to visualize how the eightfold degeneracy is lifted. As a byproduct, we show how the third measurement of ν? → νe , ν ?? → ν ?e or νe → ντ resolves the ambiguities, after the JPARC experiment [12] measures the oscillation probabilities P (ν? → νe ) and P (? ν? → ν ?e ) at the oscillation maximum, i.e., at 2 |?m31 |L/4E = π/2. 2 In the following discussions we assume that |?m2 31 |, ?m21 and θ12 are su?ciently precisely known. This is justi?ed because the correlation between these parameters and the CP phase δ is not so strong in the case of JPARC [13], and we can safely ignore the uncertainty of these parameters to discuss the ambiguities in δ due to parameter degeneracies.
II. PLOTS IN THE (sin2 2θ13 , 1/s2 23 ) PLANE

As in Ref. [14], let us discuss the ambiguities due to degeneracies step by step in the 2 order (θ23 ? π/4 = 0, ?m2 21 = 0, A = 0) → (θ23 ? π/4 = 0, ?m21 = 0, A = 0) → (θ23 ? π/4 = 2 2 0, ?m21 = 0, A = 0) → (θ23 ? π/4 = 0, ?m21 = 0, A = 0).
A.
2 cos 2θ23 = 0, ?m2 21 /?m31 = 0, AL = 0

In this case the oscillation probabilities P (ν? → νe ) and P (? ν? → ν ?e ) are equal and are given by
2 2 P (ν? → νe ) = P (? ν? → ν ?e ) = s2 23 sin 2θ13 sin ?,

where we have introduced the notation ?≡ |?m31 |L . 4E

To plot the line P (ν? → νe ) = P (? ν? → ν ?e )=const. in the (sin2 2θ13 , 1/s2 23 ) plane, let us introduce the variables X ≡ sin2 2θ13 , 1 Y ≡ 2 . s23 Then
2 2 P = s2 23 sin 2θ13 sin ?

give a straight line Y = sin2 ? X P (1)

in the (X , Y ) plane, where P and sin2 ? are constant. The intersection of Eq. (1) and 2 2 Y ≡ 1/s2 23 = 2 in the (sin 2θ13 , 1/s23 ) plane is a unique point, which corresponds to a solution with eightfold degeneracy. The solution is depicted in Fig. 2(a).

3

B.

2 cos 2θ23 = 0, ?m2 21 /?m31 = 0, AL = 0

At present the Superkamiokande atmospheric neutrino data gives the allowed region 0.90 < sin 2θ23 ≤ 1.0 at 90%CL [10], and sin2 2θ23 can be in general di?erent from 1.0. If sin2 2θ23 , which is more accurately determined from the oscillation probability P (ν? → ν? ) in the future long baseline experiments, deviates from 1, then we have two solutions for Y ≡ 1/s2 23 : Y+ = Y? = 2 1? 1+ 1 ? sin2 2θ23 2 1 ? sin2 2θ23 .

In this case there are two solutions, the one given by Eq. (1) and Y = Y+ and another given by Eq. (1) and Y = Y? . These are two solutions with fourfold degeneracy. The two solutions in the (sin2 2θ13 , 1/s2 23 ) plane are shown in Fig. 2(b). From this we see that even if we know precisely the values of P (ν? → νe ), P (? ν? → ν ?e ) and P (ν? → ν? ), there are two sets of solutions, and this is the ambiguity due to the θ23 ? π ? θ23 degeneracy.
C.
2 cos 2θ23 = 0, ?m2 21 /?m31 = 0, AL = 0

If we turn on the e?ect of non-zero ?m2 21 in addition to non-zero cos 2θ23 , then the 2 oscillation probabilities are P (ν? → νe ) P (? ν? → ν ?e ) = x2 sin2 ? + 2xy ? sin ? cos (δ ± ?) + y 2 ?2 ,

2 which are correct to the second order in the small parameters |?m2 21 /?m31 | and sin 2θ13 , where

x ≡ s23 sin 2θ13 , ?m2 21 c23 sin 2θ12 . y ≡ ?m2 31

(2)

? , where P and P ? are In this case, the trajectory of P (ν? → νe ) = P , P (? ν? → ν ?e ) = P 2 2 constant, in the (X ≡ sin 2θ13 , Y ≡ 1/s23 ) plane is given by a quadratic curve: 16C0 X (Y ? 1)?2 sin2 ? 1 ? )2 Y 2 + 1 ? ? 2C0 )(Y ? 1) + P + P ? ? 2X sin2 ? 2 , (P ? P = (P + P 2 2 cos ? sin ? where C0 ≡
2

(3)

?m2 21 ?m2 31

2

?2 sin2 2θ12 .

This is obtained by taking the limit A ≡

√ 2GF Ne → 0 in Eq. (16) in Ref. [15].

4

Eq. (3) becomes a hyperbola for most of the range of ?, but it becomes an ellipse for some region ? ? π . When sin2 2θ23 = 1, there are two solutions for the intersection of Y = 2 and Eq. (3). This indicates that even if we know the precise values of P (ν? → νe ), P (? ν? → ν ?e ) and P (ν? → ν? ), there are two sets of solutions for (θ13 , θ23 , δ ) with fourfold degeneracy when sin2 2θ23 = 1, as is depicted in Fig. 3(a). This is the ambiguity due to the intrinsic (θ13 , δ ) degeneracy. When sin2 2θ23 = 1, there are four sets of solutions with twofold degeneracy, as is depicted in Fig. 3(b).
D.
2 cos 2θ23 = 0, ?m2 21 /?m31 = 0, AL = 0

Furthermore, if we turn on the matter e?ect AL, then the oscillation probabilities are given by [9, 15] P (ν? → νe ) = x2 f 2 + 2xyf g cos (δ + ?) + y 2g 2 , ?2 + 2xy fg ? cos (δ ? ?) + y 2 g 2 , P (? ν? → ν ?e ) = x2 f for the normal hierarchy, while ?2 ? 2xy fg ? cos (δ ? ?) + y 2 g 2 , P (ν? → νe ) = x2 f P (? ν? → ν ?e ) = x2 f 2 ? 2xyf g cos (δ + ?) + y 2g 2 , for the inverted hierarchy, where x and y are given by Eq. (2), and f ? f sin (? ? AL/2) , (1 ? AL/2?) sin (AL/2) . g ≡ AL/2? ≡ (6) (7)

(4)

(5)

2 Eqs. (4) and (5) are correct up to the second order in |?m2 21 /?m31 | and sin 2θ13 , and all ? , where P and P ? are orders in AL. The trajectory of P (ν? → νe ) = P , P (? ν? → ν ?e ) = P 2 2 constant, in the (X ≡ sin 2θ13 , Y ≡ 1/s23 ) plane is again a quadratic curve for either of the mass hierarchies:

1 16CX (Y ? 1) = cos2 ?

??C ? 2 P P ?C P P ? + (Y ? 1) ? (f + f )X + + ? ? f f f f ??C ? P ?C P 1 ?)X + P ? P ( Y ? 1) ? ( f ? f + 2 ? ? ? f f f f sin ?

2

(8)

for the normal hierarchy, and 1 16CX (Y ? 1) = cos2 ? ??C ? 2 P P ?C P P ? (Y ? 1) ? (f + f )X + ? + ? + f f f f ??C ? P ?C P 1 ?)X + P ? P ( Y ? 1) ? ( f ? f ? + 2 ? ? f f sin ? f f

2

(9)

5

for the inverted hierarchy, where C ≡ ?m2 21 ?m2 31
2

sin(AL/2) AL/2?

2

sin2 2θ12 .

(10)

Again these quadratic curves become hyperbolas for most of the region of ?, but they become ellipses for some ? ? π . If sin2 2θ23 = 1, then there are four solutions with twofold degeneracy, as is shown in Fig. 4(a). If we know for some reason (e.g., from reactor experiments) which solution is selected for each mass hierarchy, there are only two solutions. This is the ambiguity due 2 to the sgn(?m2 31 ) degeneracy. If sin 2θ23 = 1 and if we do not know which solution is favored with respect to the intrinsic degeneracy for each hierarchy, and if we do not know sgn(?m2 31 ), then there are eight solutions without any degeneracy, as is depicted in Fig. 4(b). The advantage of our plot is that all the eight solutions for (θ13 , θ23 ) give di?erent points, and all the lines in the (sin2 2θ13 , 1/s2 23 ) plane are described by (at most) quadratic curves so that their behaviors are easy to see.
E. Oscillation maximum

Finally, let us consider the case where experiments are done at the oscillation maximum, i.e., when the neutrino energyE satis?es ? ≡ |?m2 31 |L/4E = π/2. In this case, the probabilities become P (ν? → νe ) = x2 f 2 ? 2xyf g sin δ + y 2 g 2 , ?2 + 2xy fg ? sin δ + y 2g 2 , P (? ν? → ν ?e ) = x2 f for the normal hierarchy, and ?2 ? 2xy fg ? sin δ + y 2 g 2 , P (ν? → νe ) = x2 f P (? ν? → ν ?e ) = x2 f 2 + 2xyf g sin δ + y 2g 2 , (13) (14) (11) (12)

?, g in Eqs. (6), (7) for the inverted hierarchy, where x and y are given by Eq. (2), and f , f become f ? f =± cos(AL/2) sin (AL/2) , g ≡ 1 ? AL/π AL/π

? in the (X ≡ sin2 2θ13 , for ? = π/2. The trajectory of P (ν? → νe ) = P , P (? ν? → ν ?e ) = P 2 Y ≡ 1/s23 ) plane becomes a straight line and is given by Y = ? f +f ? ? C (1/f + 1/f ?) ? /f P/f + P ? f +f ?+ P ?) ? /f ? C (1/f + 1/f P/f X? C ? ff (15)

for the normal hierarchy, and Y = X? C ? ff (16)

for the inverted hierarchy, where C is given in Eq. (10). The straight lines (15) and (16) are extremely close to each other in relatively short long baseline experiments such 6

as JPARC, where the matter e?ect is small. As is shown in Appendix B, (15) and (16) have the minimum values in Y ≡ 1/s2 23 which is larger than the naive value 1 for either of the mass hierarchies. Since Eqs. (15) and (16) are linear in X , there is only one solution between them and Y =const. Thus the ambiguity due to the intrinsic degeneracy is solved by performing experiments at the oscillation maximum, although it is then transformed into another ambiguity due to the δ ? π ? δ degeneracy. If sin2 2θ23 ? 1, then all the four solutions are basically close to each other in the (sin2 2θ13 , 1/s2 23 ) plane, and the ambiguity due to degeneracies are not serious as far as θ13 and θ23 are concerned (See Fig. 5(a)). On the other hand, if sin2 2θ23 deviates fairly from 1, then the solutions are separated into two groups, those for θ23 > π/4 and those for θ23 < π/4 in the (sin2 2θ13 , 1/s2 23 ) plane, as is shown in Fig. 5(b). In this case resolution of the θ23 ? π/2 ? θ23 ambiguity is necessary to determine θ13 , θ23 and δ .
F. 1. Fake e?ects on CP violation due to degeneracies sin2 2θ23 ? 1

If the JPARC experiment ?nds out from the measurement of the disappearance probability P (ν? → ν? ) = P that sin2 2θ23 ? 1.0 with a good approximation, then we would not have to worry very much about parameter degeneracy as far as θ13 and θ23 are concerned, since the values of θ13 and θ23 for all the di?erent solutions are close to each other. On the other hand, when it comes to the value of the CP phase phase δ , we have to be careful. From Ref. [9] the true value δ and the fake value δ ′ for the CP phase satisfy the following: x′ sin δ ′ = x sin δ ?2 ? f f ? ? ?2 f ? f f2 + f x2 f 2 + f ? , ? ? sin ? f f 2yg ff (17)

?, g are given in Eqs. (6) and (7), and x′ is de?ned by where x, y are given in Eq. (2), f , f x′2 = ?2 ? f f ?) ? 2yg (f ? f ?)x sin δ sin ? x2 (f 2 + f . ? ff

Eq. (17) indicates that even if sin δ = 0 we have nonvanishing fake CP violating e?ect sin δ ′ = ?x ?2 f ? f ? f2 + f ? 2yg sin ? ff ? ff ?2 ? f f ?, f2 + f (18)

if we fail to identify the correct sign of ?m2 31 . In the case of the JPARC experiment, Eq. (18) implies sin δ ′ ? ?2.2 sin θ13 , which is not negligible unless sin2 2θ13 ? 10?2 . Therefore we have to know the sign of ?m2 31 to determine the CP phase to good precision.

7

2.

sin2 2θ23 < 1

As was explained in Sect. II E, if sin2 2θ23 deviates fairly from 1, then we have to resolve the ambiguity due to the θ23 degeneracy to determine the values of θ13 and θ23 . As for the value of the CP phase δ , we can estimate how serious the e?ect of the θ23 ambiguity on the value of δ could be. If the true value δ is zero, then the CP phase δ ′ for the fake solution can be estimated as [9]
′ sin 2θ13 sin δ ′ =

? ?m2 21 g (f ? f ) sin 2θ12 cot 2θ23 , 2 ? ?m31 sin ? ff

where sin
2 ′ 2θ13

= sin 2θ13 tan θ23 +

2

2

?m2 21 ?m2 31

2

g 2 sin2 2θ12 (1 ? tan2 θ23 ), ? ff

?, g are de?ned in Eqs. (6) and (7). In the case of JPARC, we have and f , f |sin δ ′ | ? 1 | cot 2θ23 | 1 < 1 ? 200 t23 sin 2θ13 500 1 sin2 2θ13 , (19)

where we have used the bound 0.90 ≤ sin2 2θ23 ≤ 1.0 from the atmospheric neutrino data in the second inequality, so that we see that the ambiguity due to the θ23 does not cause a > 10?2 . It should be stressed, however, serious problem on determination of δ for sin2 2θ13 ? 2 that the e?ect on CP violation due to the sgn(?m31 ) ambiguity is also serious in this case.
III. RESOLUTION OF AMBIGUITIES BY THE THIRD MEASUREMENT AFTER JPARC

In this section, assuming that the JPARC experiment, which is expected to be the ?rst superbeam experiment, measures P (ν? → νe ) and P (? ν? → ν ?e ) at the oscillation maximum 2 ? ≡ ?m31 L/4E = π/2, we will discuss how the third measurement after JPARC can resolve the ambiguities by using the plot in the (sin2 2θ13 , 1/s2 23 ) plane. Resolution of the θ23 ambiguity has been discussed using the disappearance measurement of P (? νe → ν ?e ) at reactors [8, 11, 16, 17, 18] and the silver channel P (νe → ντ ) at neutrino factories [19], but it has not been discussed much using the channel ν? → νe 3 . Here we take the following reference values for the oscillation parameters: sin2 2θ12 = 0.8, sin2 2θ13 = 0.05, sin2 2θ23 = 0.96, 2 2 ?5 2 ?3 ?m2 21 = 7 × 10 eV , ?m31 = 2.5 × 10 eV > 0, δ = π/4.
3

(20)

There have been a lot of works [20] on how to resolve parameter degeneracies, but they discussed mainly the intrinsic and sgn(?m2 31 ) degeneracies, and the present scenario, in which the third experiment follows the JPARC results on P (ν? → νe ) plus P (? ν? → ν ?e ) which are measured at the oscillation maximum, has not been considered.

8

A.

ν? → νe

Let us discuss the case in which another long baseline experiment measures P (ν? → νe ). From the measurements of P (ν? → νe ) and P (? ν? → ν ?e ) by JPARC at the oscillation maximum we can deduce the value of δ , up to the eightfold ambiguity (δ ? π ? δ , θ23 ? 2 4 π/2 ? θ23 , ?m2 As is depicted in Fig. 6, depending on whether s2 31 ? ??m31 ). 23 ? 1/2 is positive or negative, we assign the subscript ±, and depending on whether our ansatz for sgn(?m2 31 ) is correct or wrong, we assign the subscript c or w. Thus the eight possible values of δ are given by δ+w , δ+c , δ?w , δ?c , π ? δ+w , π ? δ+c , π ? δ?w , π ? δ?c . (21)

Now suppose that the third measurement gives the value P for the oscillation probability P (ν? → νe ). Then there are in general eight lines in the (X ≡ sin2 2θ13 , Y ≡ 1/s2 23 ) plane given by f 2 X = P ? C + 2C cos2 (δ + ?) (Y ? 1) + P ?2 cos(δ + ?) for the normal hierarchy, and ?2 X = P ? C + 2C cos2 (δ ? ?) (Y ? 1) + P f ?2 cos(δ ? ?) C (Y ? 1) P ? C sin2 (δ ? ?) (Y ? 1) + P (23) C (Y ? 1) P ? C sin2 (δ + ?) (Y ? 1) + P (22)

for the inverted hierarchy, where C is de?ned in Eq. (10), ? ≡ |?m2 31 |L/4E is de?ned for the third measurement, and δ takes one of the eight values given in Eq. (21). The derivation of (22) and (23) is given in Appendix A. It turns out that the solutions (22) and (23) are hyperbola if cos2 (δ ± ?) > (C ? P )/P , where + and ? refer to the normal and inverted hierarchy, and ellipses if cos2 (δ ± ?) < (C ? P )/P . In practice, however, the di?erence between hyperbola and ellipses is not so important for the present discussions, because we are only interested in the behaviors of these curves in the region 1.52 < Y ≡ 1/s2 23 < 2.92 which comes from the 90%CL allowed region of the Superkamiokande atmospheric neutrino data for sin2 2θ23 . Here let us look at three typical cases: L=295km, L=730km, L=3000km, each of which corresponds to JPARC, o?-axis NuMI [21], and a neutrino factory [22] 5
4 5

I thank Hiroaki Sugiyama for pointing this out to me. For L=3000km the density of the matter may not be treated as constant, and the probability formulae (4) and (5) may no longer be valid. It turns out, however, that the approximation of the formulae becomes L good if we replace AL by AL → 0 A(x)dx everywhere in the formula. In the following discussions, the L replacement AL → 0 A(x)dx is always understood in the case of the baseline L=3000km. It should be mentioned that the neutrino energy spectrum at neutrino factories is continuous and it is assumed here that we take one particular energy bin whose energy range can be made relatively small. It should be also noted that neutrino factories actually measure the probabilities P (νe → ν? ) or P (? νe → ν ?? ), instead of P (ν? → νe ) or P (? ν? → ν ?e ). Here we discuss for simplicity the trajectory of P (ν? → νe ) whose feature is the same as that of P (νe → ν? ).

9

Figs. 7,8,9 show the trajectories of P (ν? → νe ) obtained in the third measurement together with the constraint of P (ν? → νe ), P (? ν? → ν ?e ) and P (ν? → ν? ) by JPARC, for L=295km, L=730km, L=3000km, respectively, where ? ≡ |?m2 31 |L/4E takes the values ? = jπ/8 (j = 1, · · · , 7). The purple (light blue) blob stands for the true (fake) solution given by the JPARC results on P (ν? → νe ), P (? ν? → ν ?e ) and P (ν? → ν? ). For the correct (wrong) guess on the mass hierarchy, there are in general four red (blue) curves because the CP phase δ , which is deduced from the JPARC results on P (ν? → νe ), P (? ν? → ν ?e ) and P (ν? → ν? ), is fourfold: (δ+c , δ?c , π ? δ+c , π ? δ?c ) for the correct assumption on the hierarchy and (δ+w , δ?w , π ? δ+w , π ? δ?w ) for the wrong assumption. In most cases the four (red or blue) curves are separated into two pairs of curves. As we will see later, the large split is due to the δ ? π ? δ ambiguity, while the small split is due to the θ23 ? π/2 ? θ23 ambiguity. The reason that the latter splitting is small is because the di?erence of the values in the CP phases is small, as is seen from Eq. (19). In some of the ?gures in Figs. 7,8,9 the number of the red or blue curves is less than four because not all values of δ give consistent solutions for a set of the oscillation parameters. Let us discuss each ambiguity one by one.
1. δ ? π ? δ ambiguity

As was mentioned above, the large splitting of four (red or blue) lines into two pair of lines is due to the δ ? π ? δ ambiguity. From Eqs. (22) and (23) we see that the only di?erence of the solutions with δ and with π ? δ appears in cos(δ ± ?) or sin(δ ± ?). If ? = π/2 (i.e., the oscillation maximum), we have cos(δ + ?) = ? sin δ , cos(π ? δ + ?) = ? sin δ , so that the values of X with δ and with π ? δ are the same, i.e., at oscillation maximum there is exact δ ? π ? δ degeneracy. On the other hand, if ? = π/2, we have cos(δ + ?) = cos(π ? δ + ?), and the values of X with δ and with π ? δ are di?erent. Thus, to resolve the δ ? π ? δ ambiguity it is advantageous to perform an experiment at ? which is farther away from π/2. Deviation of ? from π/2 implies either high energy or low energy. In general the number of events increases for high energy because both the cross section and the neutrino ?ux increase, so the high energy option is preferred to resolve the δ ? π ? δ ambiguity 6 .
2.
2 ? m2 31 ? ??m31 ambiguity

As one can easily imagine, the sgn(?m2 is resolved better with longer base31 ) ambiguity √ lines, since the dimensionless quantity AL ≡ 2GF Ne L ? (L/1900km)(ρ/2.7g·cm?3 ) be>1000km. On the other hand, from Figs. 8 and 9, we observe comes of order one for L ? that the split of the curves with the di?erent mass hierarchies (the red vs blue curves) is larger for lower energy. Naively this appears to be counterintuitive, because at low energy the matter e?ect is expected to be less important (|?m2 31 |L/4E ? AL). However, this is 2 not the case because we are dealing with the value of sin 2θ13 which is obtained for a given value of P (ν? → νe ). To see this, let us consider for simplicity the the value of X ≡ sin2 2θ13 2 at Y ≡ 1/s2 23 = 1, i.e., the X-intercept of the quadratic curves at Y = 1. (sin 2θ13 )n ((sin2 2θ13 )i ) at Y = 1 for the normal (inverted) hierarchy is given by x2 by putting y = 0
6

Resolution of δ ? π ? δ ambiguity at neutrino factories was discussed in [13]

10

in Eq. (4) (Eq. (5)): (sin2 2θ13 )n = P f2 for s2 23 = 1 P (sin2 2θ13 )i = ?2 . f The ratio of these two quantities is given for small AL by sin2 (? ? AL/2) 1 + AL/2? (sin2 2θ13 )n f2 = = ?2 f (sin2 2θ13 )i sin2 (? + AL/2) 1 ? AL/2? 1 1 ? 1 + 2AL , ? ? tan ?
2

so that the larger ? is (the smaller the neutrino energy is), the larger this ratio becomes, as long as ? does not exceed π . This phenomenon suggests that it is potentially possible to enhance the matter e?ect by performing an experiment at low energy (? > π/2) even with L=730km, and it may enable us to determine the sign of ?m2 31 at the o?-axis NuMI experiment. While the neutrino ?ux decreases for low energy at the o?-axis NuMI experiment, the cross section at E ?1GeV is not particularly small compared to higher energy, so the low energy possibility at the o?-axis NuMI experiment deserves serious study.
3. θ23 ? π/2 ? θ23 ambiguity

Figs. 7,8,9, which are plotted for δ = π/4, suggest that there is a tendency in which the slope of the red curve which goes through the true point (the purple blob) is almost the same for high energy as that of the straight green line obtained by JPARC, while for the low energy the slope of the red curve is smaller than that of the JPARC green line. Here we will discuss the X -intercept at Y = 1 instead of calculating the slope itself, because it is easier to consider the X -intercept and because the di?erence in the X -intercepts inevitably implies the di?erent slopes for the two lines, as almost all the curves are approximately straight lines. In the case of JPARC, the matter e?ect is small (AL ? 0.08) so that we can ? ? 1. From Eq. (15) we have the X -intercept at Y = 1 put f ? f XJPARC = ? P +P ? /f ? P/f + P ? x2 , ? ? 2 f +f (24)

where the term g 2 y 2 has been ignored for simplicity. On the other hand, for the third measurement, from Eq. (22) we have X3rd = P g ? x2 + 2 xy cos(δ + ?), 2 f f (25)

where the term g 2 y 2 has been ignored again for simplicity. Eq. (25) indicates that it is the second term in Eq. (25) that deviates the intercept X3rd of the red line from the intercept XJPARC of the JPARC green line. In order for the di?erence between XJPARC and X3rd to be large, f has to be small and | cos(δ + ?)| has to be large. When AL is small, in order for 11

f to be small, ||?m2 31 |L/4E ? π | has to be small. This is one of the conditions to resolve the θ23 ambiguity. Here we are using the reference value δ = π/4, so the deviation becomes maximal if |δ + ?| = |π/4 + ?| ? π . In real experiments, however, nobody knows the value of the true δ in advance, so it is di?cult to design a long baseline experiment to resolve the θ23 ? π/2 ? θ23 ambiguity. If δ turns out to satisfy | cos(δ + ?)| ? 1 in the result of the third experiment, then we may be able to resolve the θ23 ambiguity as a byproduct.
B. ν ?? → ν ?e

It turns out that the situation does not change very much even if we use the ν ?? → ν ?e channel in the third experiment. Typical curves are given for ν ?? → ν ?e in Fig. 10, which are similar to those in Fig. 7,8,9. Thus the conclusions drawn on resolution of the ambiguities hold qualitatively in the case of ν ?? → ν ?e channel.
C. νe → ντ

The experiment with the channel νe → ντ requires intense νe beams and it is expected that such measurements can be done at neutrino factories or at beta beam experiments [23]. The oscillation probability P (νe → ντ ) is given by P (νe → ντ ) = x ?2 f 2 + 2f g x ?y ? cos(δ + ?) + y ?2 g 2 , where x ? ≡ c23 sin 2θ23 ?m2 21 y ? ≡ s23 sin 2θ12 , ?m2 31 and f , g are given in Eqs. (6) and (7). The solution for P (νe → ντ ) = Q, where Q is constant, is given by X = Q f2 ? 1+ 2 cos2 (δ + ?) 1 ? C/Q +1 1 ? C/Q Y ?1 1+ cos2 (δ + ?) 1 ? C/Q +1 , 1 ? C/Q Y ?1 (26)

2 cos(δ + ?) 1 ? C/Q

where X ≡ sin2 2θ13 , Y ≡ 1/s2 23 as before and C is given in Eq. (10). Eqs. (26) is plotted in Fig. 11 in the case of L=2810km. From Fig. 11 we see that the curve P (νe → ντ ) = Q intersects with the JPARC green line almost perpendicularly, and it is experimentally advantageous. Namely, in real experiments all the measured quantities have errors and the curves become thick. In this case the allowed region is small area around the true solution in the (sin2 2θ13 , 1/s2 23 ) plane and one expects that the fake solution with respect to the θ23 ambiguity can be excluded. This is in contrast to the case of the ν? → νe and ν ?? → ν ?e channels, in which the slope of the red curves is almost the same as that of the JPARC green line and the allowed region can easily contain both the true and fake solutions, so that it becomes di?cult to distinguish the true point from the fake one. 12

As in the case of the ν? → νe channel, the δ ? π ? δ ambiguity is expected to be resolved more likely for the larger value of |? ? π/2|, and the sgn(?m2 31 ) ambiguity is resolved easily for larger baseline L (e.g., L ?3000km). Thus the measurement of the νe → ντ channel is a promising possibility as a potentially powerful candidate to resolve parameter degeneracies in the future.
IV. DISCUSSION AND CONCLUSION

In this paper we have shown that the eightfold parameter degeneracy in neutrino oscillations can be easily seen by plotting the trajectory of constant probabilities in the (sin2 2θ13 , 1/s2 23 ) plane. Using this plot, we have seen that the third measurement after the JPARC > results on P (ν? → νe ) and P (? ν? → ν ?e ) may resolve the sgn(?m2 31 ) ambiguity at L ?1000km, the δ ? π ? δ ambiguity o? the oscillation maximum (|? ? π/2| ? O(1)), and the θ23 ambiguity if ||?m2 31 |L/4E ? π | is small and δ turns out to satisfy |cos(δ + ?)| ? 1. In general all these constraints on ? ≡ |?m2 31 |L/4E may be satis?ed by taking ? = π . The condition ? = π , however, actually corresponds to the oscillation minimum, and the number of events is expected to be small for a number of reasons: (1) The probability itself is small at the oscillation minimum; (2) ? = π implies low energy and the neutrino ?ux decreases at low energy; (3) The cross section is in general smaller at low energy than that at high energy. Therefore, to gain statistics, it is presumably wise to perform an experiment at π/2 < ? < π after JPARC. The o?-axis NuMI experiment with π/2 < ? < π (E ?1GeV) may have advantage to resolve these ambiguities. As is seen in Figs. 8 and 9, the experiments at the oscillation maximum does not appear to be useful after JPARC except for the sgn(?m2 31 ) ambiguity. In order to achieve other goals such as resolution of the δ ? π ? δ ambiguity and the θ23 ambiguity, it is wise to stay away from ? = π/2 in experiments after JPARC. Although only the oscillation probabilities were discussed without taking the statistical and systematic errors into account in this paper, we hope that the present work gives some insight on how the ambiguities may be resolved in the future long baseline experiments.
APPENDIX A: EXPRESSION FOR P (ν? → νe ) = P

First of all, let us derive Eqs. (22) and (23). For the normal hierarchy, the probability P (ν? → νe ) = P is given by P = P (ν? → νe ) = x2 f 2 + 2xyf g cos (δ + ?) + y 2g 2 = f2 X + 2f Y X Y C 1? 1 Y cos (δ + ?) + C 1 ? 1 Y , (A1)

where X ≡ sin2 2θ13 , Y ≡ 1/s2 23 as in the text, f is de?ned in Eq. (6), and C is given in Eq. (10). Eq. (A1) is rewritten as (P ? C )(Y ? 1) + P ? f 2 X = 2 f 2 X C (Y ? 1) cos (δ + ?) . (A2)

13

Taking the square of the both hand sides of Eq. (A2), we get (f 2 X )2 ? 2 f 2 X

7

P ? C + 2C cos2 (δ + ?) (Y ? 1) + P + [(P ? C )(Y ? 1) + P ]2 = 0.

Solving this quadratic equation, we obtain f 2 X = P ? C + 2C cos2 (δ + ?) (Y ? 1) + P = P ? C + 2C cos2 (δ + ?) (Y ? 1) + P ±2 cos (δ + ?) C (Y ? 1) ± {[P ? C + 2C cos2 (δ + ?)] (Y ? 1) + P } ? [(P ? C )(Y ? 1) + P ]2 [P ? C + C cos2 (δ + ?)] (Y ? 1) + P . (A3)

If cos (δ + ?) > 0 then from Eq. (A2) we see that (P + C )(Y ? 1) + P ? f 2 X has to be positive. On the other hand, Eq. (A3) gives (P + C )(Y ? 1) + P ? f 2 X × [P ? C + C cos2 (δ + ?)] (Y ? 1) + P ? cos (δ + ?) C (Y ? 1) C (Y ? 1) . (A4)

= ?2 cos (δ + ?)

From Eq. (A4) we conclude that we have to take the minus sign in Eq. (A3) for the right hand side of Eq. (A4) to be positive. Hence from P (ν? → νe ) = P we get f 2 X = P ? C + 2C cos2 (δ + ?) (Y ? 1) + P ?2 cos(δ + ?) C (Y ? 1) ? we have and from P (? νe → ν ?? ) = P ?2 X = P ? ? C + 2C cos2 (δ ? ?) (Y ? 1) + P ? f ?2 cos(δ ? ?) C (Y ? 1) ? ? C sin2 (δ ? ?) (Y ? 1) + P ?. P (A6) When P ? C sin2 (δ + ?) > 0, Eq. (A5) is a hyperbola, and the physical region for Y ? 1 is Y ? 1 ≥ 0. On the other hand, when P ? C sin2 (δ + ?) < 0, Eq. (A5) becomes an ellipse and the physical region for Y ? 1 is 0 ≤ Y ? 1 ≤ P/[C sin2 (δ + ?) ? P ]. Similarly, we obtain for the inverted hierarchy: ?2 X = P ? C + 2C cos2 (δ ? ?) (Y ? 1) + P f +2 cos(δ ? ?) C (Y ? 1) P ? C sin2 (δ ? ?) (Y ? 1) + P , (A7) P ? C sin2 (δ + ?) (Y ? 1) + P , (A5)

? ? C + 2C cos2 (δ + ?) (Y ? 1) + P ? f 2X = P +2 cos(δ + ?)
7

C (Y ? 1)

? ? C sin2 (δ + ?) (Y ? 1) + P ?. P

(A8)

Here we consider for simplicity the case where all the arguments of the square root are positive. After we obtain the ?nal result, we see that the ?nal formula makes sense as long as the whole product of all the arguments is positive.

14

APPENDIX B: TRAJECTORIES AT THE OSCILLATION MAXIMUM

Throughout this appendix we will assume ? = π/2 and we will assume ?m2 31 > 0 for most part of this appendix. From Eq. (A5), the condition P (ν? → νe ) = P for the neutrino mode alone gives C (Y ? 1) [P ? C cos2 δ ] (Y ? 1) + P , ? for the anti-neutrino mode alone gives while the condition P (? ν? → ν ?e ) = P ?2 X = P ? ? C + 2C sin2 δ (Y ? 1) + P ? f +2 sin δ ?2 sin δ C (Y ? 1) ? ? C cos2 δ (Y ? 1) + P ?. P f 2 X = P ? C + 2C sin2 δ (Y ? 1) + P (B1)

(B2)

When δ ranges from ?π/2 to π/2, Eq. (B1) sweeps out the inside of a hyperbola, as is depicted by the red curves in Fig. 12(a), while (B2) sweeps out the inside of another hyperbola for the anti-neutrino mode (cf. the blue curves in Fig. 12(a)). Notice that the left (right) edge of the hyperbola (B1) for the neutrino mode corresponds to δ = ?π/2 (δ = +π/2) whereas the left (right) edge of the other hyperbola (B2) for the anti-neutrino mode corresponds to δ = +π/2 (δ = ?π/2). Since the straight line (15) is the intersection of the two regions (the yellow and light blue regions in Fig. 12(b)), the lowest point in the ?2 (if P/f 2 > P ?2 ), ? /f ? /f straight line is obtained by putting δ = +π/2 (δ = ?π/2) if P/f 2 < P respectively, depending on whether the region for the anti-neutrino mode is to the right of ?2 , then putting δ = +π/2 in Eqs. (11) ? /f that for the neutrino mode. Therefore, if P/f 2 < P and (12) and assuming xf > yg , which should hold if sin2 2θ13 is not so small, we get √ P = xf ? yg = f X ? Y C 1? C 1?
?1

1 Y 1 Y

? + yg = f ? X + ? = xf P Y which lead to the minimum value of Y
(n) Ymin

√ √ ? P )2 ??f (f P = 1? ?)2 C (f + f

for the normal hierarchy. On the other hand, for the inverted hierarchy, the corresponding values of δ for the edges for the two modes are the same as those for the normal hierarchy ?2 < P ? /f 2 , then putting δ = +π/2 in Eqs. (13) and (14) and (δ = ±π/2). Hence, if P/f ? assuming xf > yg , we obtain √ ? ? yg P = xf ? = xf + yg P which leads to the minimum value of Y
(i) Ymin

√ √ ? P ? ? f P )2 (f = 1? ?)2 C (f + f

?1

for ?m2 31 < 0. 15

ACKNOWLEDGMENTS

I would like to thank Hiroaki Sugiyama for many discussions. This work was supported in part by Grants-in-Aid for Scienti?c Research No. 16540260 and No. 16340078, Japan Ministry of Education, Culture, Sports, Science, and Technology.

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16

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m2 3

m2 2 m2 1

m2 2 m2 1 m2 3

(a)

(b)

2 FIG. 1: Two mass patterns. (a), (b) correspond to the normal (?m2 31 > 0), inverted (?m31 < 0) hierarchy, respectively.

(a)
1/s2 23 2 1 0 sin2 2θ13

(b)
1/s2 23 2 1 0 sin2 2θ13
FIG. 2: The solutions, which are marked by black blobs, for given P (ν? → νe ), P (? ν? → ν ?e ) 2 2 and P (ν? → ν? ) in the case of ?m21 /?m31 = 0, AL = 0. (a) For cos 2θ23 = 0, the intersection of Y ≡ 1/s2 23 = 2 and the trajectory of P (ν? → νe ) = P (ν? → ν? ) = const. is one point with eightfold degeneracy. (b) For cos 2θ23 = 0, the intersections are two solutions with fourfold degeneracy.

4

(a)
3 1/s2 23 2 1 0 4 sin2 2θ13

(b)
3 1/s2 23 2 1 0 sin2 2θ13
FIG. 3: The solutions, which are marked by black blobs, for given P (ν? → νe ), P (? ν? → ν ?e ) and 2 2 P (ν? → ν? ) in the case of ?m21 /?m31 = 0, AL = 0. (a) For cos 2θ23 = 0, the intersection of Y ≡ 1/s2 23 = 2 and the trajectory of P (ν? → νe ) = const., P (ν? → ν? ) = const. are two points with fourfold degeneracy. (b) For cos 2θ23 = 0, the intersections are four solutions with twofold degeneracy.

4

normal inverted

(a)
3 1/s2 23 2 1 0 sin2 2θ13 normal inverted 4

(b)
3 1/s2 23 2 1 0 sin2 2θ13
FIG. 4: The solutions, which are marked by black blobs, for given P (ν? → νe ), P (? ν? → ν ?e ) and 2 2 P (ν? → ν? ) in the case of ?m21 /?m31 = 0, AL = 0. (a) For cos 2θ23 = 0, the intersection of Y ≡ 1/s2 23 = 2 and the trajectory of P (ν? → νe ) = const., P (ν? → ν? ) = const. four points with twofold degeneracy. (b) For cos 2θ23 = 0, the intersections are eight solutions without degeneracy.

4 1/s2 23

inverted normal fake 3 true 2 1 0 0.05 sin2 2θ13 0.1

4 1/s2 23

inverted normal fake 3 true 2 1 0 0.05 sin2 2θ13 0.1

(b)

FIG. 5: The θ23 ambiguity which could arise after the JPARC measurements of P (ν? → νe ), P (? ν? → ν ?e ) and P (ν? → ν? ) at the oscillation maximum. (a) If sin2 2θ23 ? 1.0 then the values of θ13 and θ23 are close to each other for all the solutions, and the ambiguity is not serious as far as θ13 and θ23 are concerned. (b) If sin2 2θ23 < 1 then the θ23 ambiguity has to be resolved to determine θ13 and θ23 to good precision.

δ-w
1/s2 23

δ-c δ+w δ+c
sin2 2θ13

FIG. 6: Four possible values for the CP phase δ at the oscillation maximum. The red (blue) line stands for the normal (inverted) hierarchy. Since we are assuming the normal hierarchy here, the red (blue) line corresponds to the correct (wrong) assumption on the mass hierarchy. ± sign stands 1 ? sin2 2θ23 )/2 in the θ23 ambiguity, and c (w) stands for the correct for the choice of s2 23 = (1 ± (wrong) assumption on the mass hierarchy.

?= (1/8)π, E=2.39 GeV, P=0.0048
3.5 3 2.5

?= (2/8)π, E=1.19 GeV, P=0.0158
3.5 3 2.5

1/s2 23

2 1.5 1 0.02

1/s2 23
0.04 0.06 0.08 0.1 0.12

2 1.5 1 0.02

sin2 2θ13 ?= (3/8)π, E=0.80 GeV, P=0.0253
3.5 3 2.5

sin2 2θ13 ?= (5/8)π, E=0.48 GeV, P=0.0206
3.5 3 2.5

0.04

0.06

0.08

0.1

0.12

1/s2 23

2 1.5 1 0.02

1/s2 23
0.04 0.06 0.08 0.1 0.12

2 1.5 1 0.02

sin2 2θ13 ?= (6/8)π, E=0.40 GeV, P=0.0099
3.5 3 2.5

sin2 2θ13 ?= (7/8)π, E=0.34 GeV, P=0.0020
3.5 3 2.5

0.04

0.06

0.08

0.1

0.12

1/s2 23

2 1.5 1 0.02

1/s2 23
0.04 0.06 0.08 0.1 0.12

2 1.5 1 0.02

sin2 2θ13

0.04

sin2 2θ13

0.06

0.08

0.1

0.12

L = 295 km
true solution fake solution

JPARC ν+?ν correct wrong

FIG. 7: The trajectories of P (ν? → νe ) = const. of the third experiment at L=295km with ? ≡ |?m2 31 |L/4E = (j/8)π (0 ≤ j ≤ 7, j = 4) after JPARC. The true values are those in Eq. (20). The green line is the JPARC result obtained by P (ν? → νe ) and P (? ν? → ν ?e ) at the oscillation maximum. The red (blue) lines are the trajectories of P (ν? → νe ) given by the third experiment assuming the normal (inverted) hierarchy, where δ takes four values for each mass hierarchy.

?= (1/8)π, E=5.89 GeV, P=0.0049
3.5 3 2.5

?= (2/8)π, E=2.94 GeV, P=0.0166
3.5 3 2.5

1/s2 23

2 1.5 1 0.02

1/s2 23
0.04 0.06 0.08 0.1 0.12

2 1.5 1 0.02

sin2 2θ13 ?= (3/8)π, E=1.96 GeV, P=0.0277
3.5 3 2.5

sin2 2θ13 ?= (4/8)π, E=1.47 GeV, P=0.0315
3.5 3 2.5

0.04

0.06

0.08

0.1

0.12

1/s2 23

2 1.5 1 0.02

1/s2 23
0.04 0.06 0.08 0.1 0.12

2 1.5 1 0.02

sin2 2θ13 ?= (5/8)π, E=1.18 GeV, P=0.0258
3.5 3 2.5

sin2 2θ13 ?= (6/8)π, E=0.98 GeV, P=0.0144
3.5 3 2.5

0.04

0.06

0.08

0.1

0.12

1/s2 23

2 1.5 1 0.02

1/s2 23
0.04 0.06 0.08 0.1 0.12

2 1.5 1 0.02

sin2 2θ13 ?= (7/8)π, E=0.84 GeV, P=0.0043
3.5 3 2.5

0.04

sin2 2θ13

0.06

0.08

0.1

0.12

L = 730 km
JPARC ν+?ν correct wrong true solution fake solution
0.04

1/s2 23

2 1.5 1 0.02

sin2 2θ13

0.06

0.08

0.1

0.12

FIG. 8: The trajectories of P (ν? → νe ) = const. of the third experiment at L=730km with ? ≡ |?m2 31 |L/4E = (j/8)π (0 ≤ j ≤ 7) after JPARC. The true values are those in Eq. (20).

?= (1/8)π, E=24.20 GeV, P=0.0044 ?= (2/8)π, E=12.10 GeV, P=0.0184
3.5 3 2.5 3.5 3 2.5

1/s2 23

2 1.5 1 0.02

1/s2 23
0.04 0.06 0.08 0.1 0.12

2 1.5 1 0.02

sin2 2θ13 ?= (3/8)π, E=8.07 GeV, P=0.0388
3.5 3 2.5

sin2 2θ13 ?= (4/8)π, E=6.05 GeV, P=0.0581
3.5 3 2.5

0.04

0.06

0.08

0.1

0.12

1/s2 23

2 1.5 1 0.02

1/s2 23
0.04 0.06 0.08 0.1 0.12

2 1.5 1 0.02

sin2 2θ13 ?= (5/8)π, E=4.84 GeV, P=0.0684
3.5 3 2.5

sin2 2θ13 ?= (6/8)π, E=4.03 GeV, P=0.0655
3.5 3 2.5

0.04

0.06

0.08

0.1

0.12

1/s2 23

2 1.5 1 0.02

1/s2 23
0.04 0.06 0.08 0.1 0.12

2 1.5 1 0.02

sin2 2θ13 ?= (7/8)π, E=3.46 GeV, P=0.0510
3.5 3 2.5

0.04

sin2 2θ13

0.06

0.08

0.1

0.12

L = 3000 km
JPARC ν+?ν correct wrong true solution fake solution
0.04

1/s2 23

2 1.5 1 0.02

sin2 2θ13

0.06

0.08

0.1

0.12

FIG. 9: The trajectories of P (ν? → νe ) = const. of the third experiment at L=3000km with ? ≡ |?m2 31 |L/4E = (j/8)π (0 ≤ j ≤ 7) after JPARC. The true values are those in Eq. (20). For ? ≥ (3/8)π , the blue curves (with the wrong assumption for the mass hierarchy) are not in the ?gure because they are far to the right.

L = 295 km, E=2.39 GeV, P=0.0051
3.5 3 2.5

L = 730 km, E=5.90 GeV, P=0.0049
3.5 3 2.5

1/s2 23

2 1.5 1 0.02

1/s2 23
0.04 0.06 0.08 0.1 0.12

2 1.5 1 0.02

sin2 2θ13

0.04

sin2 2θ13

0.06

0.08

0.1

0.12

L = 3000 km, E=24.26 GeV, P=0.0029
3.5 3 2.5

?ν?→?νe
JPARC ν+?ν correct wrong true solution fake solution

?=(1/8)π

1/s2 23

2 1.5 1 0.02

0.04

sin2 2θ13

0.06

0.08

0.1

0.12

FIG. 10: The trajectories of P (? ν? → ν ?e ) = const. of the third experiment with ? ≡ |?m2 31 |L/4E = π/8 after JPARC. The behaviors are almost similar to those for P (ν? → νe ) = const. The true values are those in Eq. (20).

?= (1/8)π, E=24.26 GeV, P=0.0031
3.5 3 2.5

?= (2/8)π, E=12.13 GeV, P=0.0125
3.5 3 2.5

1/s2 23

2 1.5 1 0.02

1/s2 23
0.04 0.06 0.08 0.1 0.12

2 1.5 1 0.02

sin2 2θ13

0.04

sin2 2θ13

0.06

0.08

0.1

0.12

?= (3/8)π, E=8.09 GeV, P=0.0249
3.5 3 2.5

νe→ντ
correct wrong true solution fake solution

1/s2 23

2 1.5 1 0.02

L = 2810 km JPARC ν+?ν

0.04

sin2 2θ13

0.06

0.08

0.1

0.12

FIG. 11: The trajectories of P (νe → ντ ) = const. of the third experiment at L=2810km with ? ≡ |?m2 31 |L/4E = (j/8)π (j = 1, 2, 3) after JPARC. The true values are those in Eq. (20). The curves intersect with the JPARC line perpendicularly, so this channel is advantageous to resolve the ambiguities from experimental point of view.

δ=π/2 δ=π/4 δ=0 δ=-π/4 δ=-π/2

δ=π/2 δ=π/4 δ=0 δ=-π/4 δ=-π/2

1/s2 23

(a)
sin2 2θ13

1/s2 23

(b)
P=const ?P=const ?P, P=const

sin2 2θ13
FIG. 12: The region of constant probabilities at the oscillation maximum. (a) Each red (blue) line stands for P (ν? → νe ) = const. (P (? ν? → ν ?e ) = const.) for a speci?c value of δ. The red line on the right (left) edge corresponds to δ = +π/2 (δ = ?π/2), while the blue line on the edge right (left) corresponds to δ = ?π/2 (δ = +π/2). (b) When δ varies from 0 to 2π , the line P (ν? → νe ) = const. sweeps out the yellow region, whereas the line ν ?? → ν ?e ) = const. sweeps out the light blue region. The black straight line, which is given by P (ν? → νe ) = const. and P (? ν? → ν ?e ) = const., lies in the overlapping green region.


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