New plots and parameter degeneracies in neutrino oscillations

It is shown that eightfold degeneracy in neutrino oscillations is easily seen by plotting constant probabilities in the (sin2 2θ13, 1/s232) plane. Using this plot, we discuss how an additional long baseline measurement resolves degeneracies after the JPARC experiment measures the oscillation probabilities P(νμ→νe) and at |Δm312|L/4E = π/2. By measuring P(νμ→νe) or , the sgn(Δm312) ambiguity is resolved better at longer baselines and the δ↔π−δ ambiguity is resolved better when ||Δm312|L/4E − π/2| is larger. The θ23 ambiguity may be resolved as a by-product if ||Δm312|L/4E − π| is small and the CP phase δ turns out to satisfy |cos(δ+|Δm312|L/4E)| ∼ 1. It is pointed out that the low-energy option (E∼1 GeV) at the off-axis NuMI experiment may be useful in resolving these ambiguities. The νe→ντ channel offers a promising possibility that it would potentially resolve all the ambiguities.


Introduction
From recent experiments on atmospheric [1], solar [2] and reactor [3,4] neutrinos, we now know the approximately correct values of the mixing angles and mass squared differences for atmospheric and solar neutrino oscillations: squared difference m 2 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.
From the work of Burguet-Castell et al [6], it is known that even if the values of the oscillation probabilities P(ν µ → ν e ) and P(ν µ →ν e ) are exactly given, we cannot determine uniquely the values of the oscillation parameters owing to parameter degeneracies. There are three kinds of parameter degeneracies: intrinsic (θ 13 , δ) degeneracy [6], the degeneracy of m 2 31 ↔ − m 2 31 [7] and the degeneracy of θ 23 ↔ π/2 − θ 23 [8,9]. Intrinsic degeneracy is exact when m 2 21 / m 2 31 is exactly zero. The sgn( m 2 31 ) degeneracy is exact when AL is exactly zero, where A (≡ √ 2G F N e ) and L stand for the matter effect and the baseline, respectively (G F is the Fermi constant and N e the electron density in matter). The θ 23 degeneracy is exact when cos 2θ 23 is exactly zero. Each degeneracy gives a twofold solution, so that, in total, we will 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 different solutions [9]. When we try to determine the oscillation parameters, ambiguities arise since the values of the oscillation parameters are different for each solution. In particular, this causes a serious problem in the measurement of CP violation, which is expected to be a small effect in the long baseline experiments, and we could mistake a fake effect because of the ambiguities for nonvanishing CP violation if we do not treat the ambiguities carefully.
In previous studies [6,7,9], 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 different points for the eight different solutions. An effort was made by Minakata et al [11] to visualize the eight different points by plotting the trajectories of constant probabilities in the (sin 2 2θ 13 , s 2 23 ) plane. In the present paper, we propose a plot in the (sin 2 2θ 13 , 1/s 2 23 ) plane, which offers the simplest way to visualize how the eightfold degeneracy is lifted. As a by-product, we show how the third measurement of ν µ → ν e ,ν µ →ν e or ν e → ν τ resolves the ambiguities, after the JPARC experiment [12] measures the oscillation probabilities P(ν µ → ν e ) and P(ν µ →ν e ) at the oscillation maximum, i.e. at | m 2 31 |L/4E = π/2. Unlike the work of Burguet-Castell et al [6], statistical and systematic errors are not taken into account in this paper, and we hope that the present formalism offers a way to understand intuitively how ambiguities appear and how they are resolved by combining other experiments.
In the following discussions, we assume that | m 2 31 |, m 2 21 and θ 12 are sufficiently precisely known. This is justified because the correlation between these parameters and the CP phase δ is not so strong for JPARC [13], and we can safely ignore the uncertainty of these parameters to discuss the ambiguities in δ due to parameter degeneracies.
2.2. cos 2θ 23 = 0, m 2 21 / m 2 31 = 0, AL = 0 At present, the Superkamiokande atmospheric neutrino data give the allowed region 0.90 < sin 2θ 23 1.0 at 90% CL [10], and sin 2 2θ 23 can be, in general, different from 1.0. If sin 2 2θ 23 , which is more accurately determined from the oscillation probability P(ν µ → ν µ ) in the future long baseline experiments, deviates from 1, then we have two solutions for Y ≡ 1/s 2 23 : In this case, there are two solutions, one given by equation (1) and Y = Y + and another given by (1) and Y = Y − . These are two solutions with fourfold degeneracy. The two solutions in the (sin 2 2θ 13 , 1/s 2 23 ) plane are shown in figure 2(b). From this, we see that even if we know precisely the values of P(ν µ → ν e ), P(ν µ →ν e ) and P(ν µ → ν µ ), there are two sets of solutions, and this represents the ambiguity due to the θ 23 ↔ π − θ 23 degeneracy.
Equation (3) becomes a hyperbola for most of the region of , but it becomes an ellipse for the region π. When sin 2 2θ 23 = 1, there are two solutions for the intersection of Y = 2 and equation (3). This indicates that even if we know the precise values of P(ν µ → ν e ), P(ν µ →ν e ) and P(ν µ → ν µ ), there are two sets of solutions for (θ 13 , θ 23 , δ) with fourfold degeneracy when sin 2 2θ 23 = 1, as shown in figure 3(a). This represents the ambiguity due to intrinsic (θ 13 , δ) degeneracy. When sin 2 2θ 23 = 1, there are four sets of solutions with twofold degeneracy, as depicted in figure 3(b). 2.4. cos 2θ 23 = 0, m 2 21 / m 2 31 = 0, AL = 0 Furthermore, if we turn on the matter effect AL, then the oscillation probabilities are given by [9,15] P(ν µ → ν e ) = x 2 f 2 + 2xyfg cos(δ + ) + y 2 g 2 , P(ν µ →ν e ) = x 2f 2 + 2xyf g cos(δ − ) + y 2 g 2 (4) for the normal hierarchy and by for the inverted hierarchy, where x and y are given by equation (2) and Equations (4) and (5) are correct up to the second-order in | m 2 21 / m 2 31 | and sin 2θ 13 , and all orders in AL. The trajectory of P(ν µ → ν e ) = P, P(ν µ →ν e ) =P (where P andP are constant) in the (X ≡ sin 2 2θ 13 , Y ≡ 1/s 2 23 ) plane is again a quadratic curve for either of the mass hierarchies: for the normal hierarchy, and for the inverted hierarchy, where Again, these quadratic curves become hyperbolas for most of the region of , but they become ellipses for some π. If sin 2 2θ 23 = 1, there are four solutions with twofold degeneracy, as shown in figure 4(a). If we know for some reason (e.g. from reactor experiments) which solution is selected for each mass hierarchy, then there are only two solutions. This is the ambiguity due to the sgn( m 2 31 ) degeneracy. If sin 2 2θ 23 = 1 and if we do not know which solution is favoured with respect to the intrinsic degeneracy for each hierarchy, and if we do not know sgn( m 2 31 ), then there are eight solutions without any degeneracy, as depicted in figure 4(b). The advantage of our plot is that all the eight solutions for (θ 13 , θ 23 ) give different points, and all the lines in the (sin 2 2θ 13 , 1/s 2 23 ) plane are described by (at most) quadratic curves so that their behaviours are easy to see.

Oscillation maximum
Finally, consider the case where experiments are done at the oscillation maximum, i.e. when the neutrino energy E satisfies ≡ | m 2 31 |L/4E = π/2. In this case, the probabilities become P(ν µ →ν e ) = x 2f 2 + 2xyf g sin δ + y 2 g 2 (12) for the normal hierarchy, and for the inverted hierarchy, where x and y are given by equation (2), and f ,f , g in equations (6) and (7) become plane becomes a straight line and is given by for the normal hierarchy, and for the inverted hierarchy, where C is given by equation (10). The straight lines (15) and (16) are very close to each other in relatively short long-baseline experiments such as JPARC, where the matter effect is small. As shown in appendix B, (15) and (16) have the minimum values in Y ≡ 1/s 2 23 , which is larger than the naive value 1 for either of the mass hierarchies. Since equations (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 sin 2 2θ 23 1, all the four solutions are basically close to each other in the (sin 2 2θ 13 , 1/s 2 23 ) plane, and the ambiguity due to degeneracies are not serious as far as θ 13 and θ 23 are concerned (see figure 5(a)). On the other hand, if sin 2 2θ 23 deviates fairly from 1, then the solutions are separated into two groups, those for θ 23 > π/4 and those for θ 23 < π/4 in the (sin 2 2θ 13 , 1/s 2 23 ) plane, as shown in figure 5(b). In this case, resolution of the θ 23 ↔ π/2 − θ 23 ambiguity is required to determine θ 13 , θ 23 and δ.
2.6. Fake effects on CP violation due to degeneracies 2.6.1. sin 2 2θ 23 1. If the JPARC experiment reveals from the measurement of the disappearance probability P(ν µ → ν µ ) = P that sin 2 2θ 23 1.0 with a good approximation, then we would not have to worry very much about parameter degeneracy as far as θ 13 and On the other hand, when it comes to the value of the CP phase δ, we have to be careful. From [9], the true value δ and the fake value δ for the CP phase satisfy the following: where x, y are given by equation (2), f,f , g are given by (6) and (7) and x is defined by Equation (17) indicates that, even if sin δ = 0, we have nonvanishing fake CP violating effect if we fail to identify the correct sign of m 2 31 . For the JPARC experiment, equation (18) implies that which is not negligible unless sin 2 2θ 13 10 −2 . Therefore we have to know the sign of m 2 31 to determine the CP phase to good precision.
2.6.2. sin 2 2θ 23 < 1. As explained in section 2.5, if sin 2 2θ 23 deviates fairly from 1, we need to resolve the ambiguity due to the θ 23 degeneracy to determine the values of θ 13 and θ 23 . As for the value of the CP phase δ, we can estimate how serious the effect of the θ 23 ambiguity on the value of δ could be. If the true value of δ is zero, the CP phase δ for the fake solution can be estimated as [9] sin 2θ 13 where we have used the bound 0.90 sin 2 2θ 23 1.0 from the atmospheric neutrino data in the second inequality. Hence, we see that the ambiguity due to the θ 23 does not cause a serious problem on determination of δ for sin 2 2θ 13 10 −2 . However, it should be stressed that the effect on CP violation due to the sgn( m 2 31 ) ambiguity is also serious in this case.

Resolution of ambiguities by the third measurement after JPARC
Assuming that the JPARC experiment, which is expected to be the first superbeam experiment, measures P(ν µ → ν e ) and P(ν µ →ν e ) at the oscillation maximum ≡ | m 2 31 |L/4E = π/2, we will discuss in this section how the third measurement after JPARC can resolve the ambiguities by using the plot in the (sin 2 2θ 13 , 1/s 2 23 ) plane. Resolution of the θ 23 ambiguity has been discussed previously using the disappearance measurement of P(ν e →ν e ) at reactors [8,11], [16]- [18], the silver channel ν e → ν τ at neutrino factories [19] and the ν µ → ν e channel [21,22]. 3 Here, we take the following reference values for the oscillation parameters:

ν µ → ν e
Let us discuss the case in which another long-baseline experiment measures P(ν µ → ν e ). From the measurements of P(ν µ → ν e ) and P(ν µ →ν e ) by JPARC at the oscillation maximum, we can deduce the value of δ up to the eightfold ambiguity (δ ↔ π − δ, θ 23 ↔ π/2 − θ 23 , m 2 31 ↔ − m 2 31 ). 4 As depicted in figure 6, depending on whether s 2 23 − 1/2 is positive or negative, we assign subscript + or −, and depending on whether our ansatz for sgn( m 2 31 ) is correct or wrong, 3 There have been a number of studies [20] on how to resolve parameter degeneracies using the ν µ → ν e channel, and they have discussed mainly the intrinsic and sgn( m 2 31 ) degeneracies. In [21,22], the θ 23 ambiguity as well as others using the ν µ → ν e channel and its combination with ν e → ν τ are discussed. The present scenario, in which the third experiment follows the JPARC results on P(ν µ → ν e ) plus P(ν µ →ν e ), measured at the oscillation maximum, has been discussed in [23] from a different viewpoint. 4 I thank Hiroaki Sugiyama for pointing this out to me. we assign subscript c or w. Thus, the eight possible values of δ are given by Now, suppose that the third measurement gives the value P for the oscillation probability P(ν µ → ν e ). Then, there are in general eight lines in the (X ≡ sin 2 2θ 13 , Y ≡ 1/s 2 23 ) plane, given by for the normal hierarchy and for the inverted hierarchy. Here, C is defined by equation (10), (≡| m 2 31 |L/4E) is defined for the third measurement and δ takes one of the eight values given in equation (21). The derivation of (22) and (23) is given in appendix A. It turns out that the solutions (22) and (23) are hyperbola if cos 2 (δ ± ) > (C − P)/P, where + and − refer to the normal and inverted hierarchy respectively, and ellipse if cos 2 (δ ± ) < (C − P)/P. In practice, however, the difference between hyperbola and ellipse is not so important for the present discussions, because we are only interested in the behaviours of these curves in the region 1.52 < Y ≡ 1/s 2 23 < 2.92, which comes from the 90% CL allowed region of the Superkamiokande atmospheric neutrino data for sin 2 2θ 23 .
In this context, let us look at three typical cases: L = 295, 730 and 3000 km, each of which corresponds to JPARC, off-axis NuMI [24] and a neutrino factory [25]. 5 Figures 7-9 show the trajectories of P(ν µ → ν e ) obtained in the third measurement, together with the constraint of P(ν µ → ν e ), P(ν µ →ν e ) and P(ν µ → ν µ ), by JPARC for L = 295, 730 and 3000 km, respectively, where ≡ | m 2 31 |L/4E takes the values = jπ/8 (j = 1, . . . ,7, j = 4). The purple (light blue) blob stands for the true (fake) solution given by the JPARC results on P(ν µ → ν e ), P(ν µ →ν e ) and P(ν µ → ν µ ). For the correct (wrong) guess on the mass hierarchy, there are in general four red (blue) curves, owing to the fact that the CP phase δ, which is deduced from the JPARC results on P(ν µ → ν e ), P(ν µ →ν e ) and P(ν µ → ν µ ), 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 5 For L = 3000 km, 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 good if we replace AL by AL → L 0 A(x) dx everywhere in the formula. In the following discussions, the replacement AL → L 0 A(x) dx is always understood for the baseline L = 3000 km. 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 also be noted that neutrino factories actually measure the probability P(ν e → ν µ ) or P(ν e →ν µ ), instead of P(ν µ → ν e ) or P(ν µ →ν e ). Here, we discuss for simplicity the trajectory of P(ν µ → ν e ), whose feature is the same as that of P(ν e → ν µ ).  (20). The green line is the JPARC result obtained by P(ν µ → ν e ) and P(ν µ →ν e ) at the oscillation maximum. The red (blue) lines are the trajectories of P(ν µ → ν e ) given by the third experiment assuming the normal (inverted) hierarchy, where δ takes four values for each mass hierarchy.  (20). For (3/8)π, the blue curves (with the wrong assumption for the mass hierarchy) are not seen in the figure because they are far to the right. of curves. As we will see later, the large split is due to the δ ↔ π − δ ambiguity, whereas the small split is due to the θ 23 ↔ π/2 − θ 23 ambiguity. The reason that the latter splitting is small is because the difference in values for the CP phases is small, as can be seen from (19). In some of the figures in figures 7-9, the number of red or blue curves is less than 4 since not all values of δ give consistent solutions for a set of oscillation parameters.
Let us study the ambiguities one by one.
3.1.1. δ ↔ π − δ ambiguity. As 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 (22) and (23), we see that the only difference between the solutions with δ and π − δ appears in cos(δ ± ) or sin(δ ± ). If = π/2 (i.e. the oscillation maximum), we have cos(δ + ) = − sin δ and 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 different. Thus, to resolve the δ ↔ π − δ ambiguity, it is advantageous to perform an experiment at farther away from π/2. Deviation of value from π/2 implies either high or low energy. In general, the number of events increases for high energy because both the cross-section and the neutrino flux increase. Therefore the high-energy option is preferred in resolving the δ ↔ π − δ ambiguity. 6 3.1.2. m 2 31 ↔ − m 2 31 ambiguity. As one can easily imagine, the sgn( m 2 31 ) ambiguity is resolved better with longer baselines, since the dimensionless quantity AL ≡ √ 2G F N e L ∼ (L/1900 km)(ρ/2.7 g cm −3 ) becomes of order one for L 1000 km. On the other hand, from figures 8 and 9, we observe that the split of the curves with different mass hierarchies (the red versus blue curves) is larger for lower energy. Naively, this appears to be counterintuitive, since, at low energy, the matter effect is expected to be less important (| m 2 31 |L/4E AL). However, this is not the case since we are dealing with the value of sin 2 2θ 13 obtained for a given value of P(ν µ → ν e ). To see this, let us consider for simplicity the value of X ≡ sin 2 2θ 13 at Y ≡ 1/s 2 23 = 1, i.e. the X intercept of the quadratic curves at Y = 1. (sin 2 2θ 13 ) n ((sin 2 2θ 13 ) i ) at Y = 1 for the normal (inverted) hierarchy is given by x 2 by putting y = 0 in (4) (equation (5)): The ratio of these two quantities is given, for small AL, by . 6 Resolution of δ ↔ π − δ ambiguity at neutrino factories has been discussed in [13].
Hence, the larger the (the smaller the neutrino energy), the larger the above ratio, as long as does not exceed π. This phenomenon suggests that it is potentially possible to enhance the matter effect by performing an experiment at low energy ( > π/2) even with L = 730 km, and it may enable us to determine the sign of m 2 31 at the off-axis NuMI experiment. Although the neutrino flux decreases for low energy at the off-axis NuMI experiment, the cross-section at E ∼ 1 GeV is not particularly small compared with higher energy; hence, the low energy possibility at the off-axis NuMI experiment deserves serious study.
3.1.3. θ 23 ↔ π/2 − θ 23 ambiguity. Figures 7-9, which are plotted for δ = π/4, suggest that there is a tendency that, for high energy, the slope of the red curve, which goes through the true point (the purple blob), is almost the same as the slope of the straight green line obtained by JPARC, whereas, for 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, since it is easier to consider the X intercept and since the difference in the X intercepts inevitably implies different slopes for the two lines with almost all the curves being approximately straight lines. For JPARC, the matter effect is small (AL 0.08), so that we can put f f 1. From equation (15), we have the X intercept at Y = 1 as where the term g 2 y 2 has been ignored for simplicity. On the other hand, for the third measurement, from (22), we have where the term g 2 y 2 has been ignored again for simplicity. Equation (25) indicates that it is the second term in (25) that deviates the intercept X 3rd of the red line from the intercept X JPARC of the JPARC green line. For the difference between X JPARC and X 3rd to be large, f has to be small and |cos(δ + )| has to be large. When AL is small, for f to be small, || m 2 31 |L/4E − π| has to be small. This is one of the conditions to resolve the θ 23 ambiguity. Here, we are using the reference value δ = π/4; hence, the deviation becomes maximal if |δ + | = |π/4 + | π. In real experiments, however, nobody knows the value of the true δ in advance and, therefore, it is difficult to design a long baseline experiment to resolve the θ 23 ↔ π/2 − θ 23 ambiguity. If δ turns out to satisfy |cos(δ + )| ∼ 1 in the results of the third experiment, then we may be able to resolve the θ 23 ambiguity as a by-product.

3.2.ν µ →ν e
It turns out that the situation does not change very much even if we use theν µ →ν e channel in the third experiment. Typical curves are given forν µ →ν e in figure 10, which are similar to those  (20). in figures 7-9. Thus the conclusions we drew on resolution of the ambiguities hold qualitatively for theν µ →ν e channel.

ν e → ν τ
The experiment with the channel ν e → ν τ requires intense ν e beams and it is expected that such measurements can be done at neutrino factories or at beta beam experiments [26]. The oscillation probability P(ν e → ν τ ) is given by P(ν e → ν τ ) =x 2 f 2 + 2fgxỹ cos(δ + ) +ỹ 2 g 2 , wherex ≡ c 23 sin 2θ 23 (20). The curves intersect with the JPARC line perpendicularly; hence, this channel is advantageous to resolve the ambiguities from an experimental point of view. and f, g are defined in (6) and (7). The solution for P(ν e → ν τ ) = Q, where Q is constant, is given by where X ≡ sin 2 2θ 13 , Y ≡ 1/s 2 23 as before and C is given by equation (10). Equation (26) is plotted in figure 11 for L = 2810 km. From figure 11, we see that the curve P(ν e → ν τ ) = 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 a small area around the true solution in the (sin 2 2θ 13 , 1/s 2 23 ) plane and one expects that the fake solution with respect to the θ 23 ambiguity can be excluded. This contrasts with the case of the ν µ → ν e andν µ →ν e channels, where the slope of the red curve 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 difficult to distinguish the true point from the fake one.
Similar to the case of the ν µ → ν e channel, the δ ↔ π − δ ambiguity is expected to be resolved more easily for larger values of | − π/2| and the sgn( m 2 31 ) ambiguity is resolved easily for larger baseline L (e.g. L ∼ 3000 km).
Thus measurement of the ν e → ν τ channel offers a promising possibility as a potentially powerful candidate to resolve parameter degeneracies in the future. 7

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 (sin 2 2θ 13 , 1/s 2 23 ) plane. Using this plot, we have seen that the third measurement after the JPARC results on P(ν µ → ν e ) and P(ν µ →ν e ) may resolve the sgn( m 2 31 ) ambiguity at L 1000 km, the δ ↔ π − δ ambiguity off the oscillation maximum (| − π/2| ∼ O(1)) and the θ 23 ambiguity if || m 2 31 |L/4E − π| is small and δ turns out to satisfy | cos(δ + )| ∼ 1. In general, all these constraints on ≡ | m 2 31 |L/4E may be satisfied 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: (i) the probability itself is small at the oscillation minimum; (ii) = π implies low energy and the neutrino flux decreases at low energy; and (iii) the crosssection is generally smaller at low energy than at high energy. Therefore, to gain statistics, it is presumably wise to perform an experiment at π/2 < < π after JPARC. The off-axis NuMI experiment with π/2 < < π (E ∼ 1 GeV) may be advantageous to resolve these ambiguities.
As seen in figures 8 and 9, experiments at the oscillation maximum do not appear to be useful after JPARC, except for the sgn( m 2 31 ) ambiguity. To achieve other goals such as resolution of the δ ↔ π − δ and θ 23 ambiguities, it is wise to stay away from = π/2 in experiments after JPARC.
Although only oscillation probabilities were discussed here without taking the statistical and systematic errors into account, we hope that the present paper gives some insight on how the ambiguities can be resolved in future long-baseline experiments. (a) Each red (blue) line stands for P(ν µ → ν e ) = const. (P(ν µ →ν e ) = const.) for a specific value of δ. The red line on the right (left) edge corresponds to δ = +π/2 (δ = −π/2), whereas 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 µ → ν e ) = const. sweeps out the yellow region, whereas the line (ν µ →ν e ) = const. sweeps out the light blue region. The black straight line, which is given by P(ν µ → ν e ) = const. and P(ν µ →ν e ) = const., lies on the overlapping green region. hold if sin 2 2θ 13 is not very small, we get which lead to the minimum value of Y , 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 (δ = ±π/2). Hence, if P/f 2 <P/f 2 , then putting δ = +π/2 in (13) and (14) and assuming xf > yg, we obtain √ P = xf − yg, P = xf + yg, which leads to the minimum value of Y , for m 2 31 < 0.