Neutrino mass hierarchy and δ CP investigation within the biprobability ( P – P T ) plane

δ CP in the context of CP trajectory diagrams in the biprobability plane. The separation between the normal mass hierarchy (NH) and inverted mass hierarchy (IH) CP trajectory ellipses in the P – P T plane seems to be very promising as a means of investigating the mass hierarchy. An illustration of the separation between the two hierarchy ellipses in the E – L plane is very helpful to cover all the desired baselines and beam energies and also to analyze the beneﬁts and drawbacks in one step. If we know the mass hierarchy, then, from the large sizes of CP trajectory ellipses that are possible at appropriately long baselines ( L ) and at speciﬁc values of beam energy ( E ), it becomes possible to investigate at least narrow ranges of the CP/T-violating phase δ CP . The possibility of more than one set of ( θ 13 , δ CP ) parameters corresponding to any chosen coordinate in the P – P T plane, known as parameter degeneracy, may hinder the exact determination of the mass hierarchy as well as the δ CP value. To circumvent this degeneracy in the ( θ 13 , δ CP ) parameter space, in the case of opposite-sign solutions corresponding to the NH and IH cases, sufﬁciently long baselines are needed, so as to separate the opposite hierarchy ellipses to create an observable separation; in the case of same-sign solutions corresponding to either NH or IH, we need to choose an experimental conﬁguration with L (cid:3) 2535 km, E (cid:3) 5 GeV for the n = 1 scenario.


Introduction
During the past few decades, neutrino oscillation experiments [1][2][3][4][5][6][7][8] have had great success in revealing the structure of the lepton-flavor-mixing matrix, but there still remain several unanswered questions. Investigation of whether the third neutrino is heavier (lighter) than the two other neutrinos, whose mass-square difference is responsible for atmospheric neutrino oscillations, i.e., m 2 31 m 2 32 ≥ 0 ( m 2 31 m 2 32 ≤ 0), known as normal (inverted) mass hierarchy, is so far one of the most intriguing unanswered questions. An investigation of the mass hierarchy would be very helpful to disclose the theoretical phenomenology describing the neutrino mass. Another tantalizing question that current and future oscillation experiments have to uncover is the determination of the leptonic CP-violation strength, which depends on the CP/T-violation phase δ CP or δ T . This will not only shed light on the theoretical structure behind the complete parallelism between quarks and leptons, but may also be very useful in understanding the matter-antimatter asymmetry arising due to baryon number violation in the universe [30].
The expected small value of the θ 13 mixing angle is the major problem, as it obscures the sensitivity of both the CP-violating phase δ CP and the mass hierarchy effects [31], as is evident from Eqs. (1) and PTEP 2016, 083B02 M. Singh (5). Recent confirmation of the moderately large value of the reactor mixing angle (θ 13 ∼ 9 • ) [37][38][39][40] has opened the door to the possibility of precise determination of the leptonic T/CP phase δ CP .
Due to the fact that P T eμ = P T μτ = P T τ e = − P T μe = − P T τ μ = − P T eτ [32,33], which is true for the constant matter density approximation, sizes and separation between ellipses in the P-P T plane will be the same for all appearance channels in the three flavor oscillations; hence, it is enough to study any one suitable channel. Relatively long-lived charged particles, π ± mesons, can be easily stored in accelerators and can be accelerated to the desired energies, making them the most easily available source of ν μ and ν e beams. Due to this reason, the current oscillation experiments operate with the ν e ↔ ν μ channel, as will most future ones; we therefore choose this channel for discussion in this paper. In order to complete our investigation toward the investigation of optimized experimental configurations (i.e., baseline L and beam energy E), we will consider various experiments, namely, Brookhaven-Cornell (L = 350 km), CERN-Gran Sasso (L = 730 km), JHF-Seoul (L = 1200 km), JHF-Beijing (L = 2100 km), and Fermilab-SLAC (L = 2900 km). We will restrict ourselves to baselines ≤ 3000 km to fulfill the constant matter density approximation, as below this baseline value we remain confined to the region of the Earth's crust layer only. Experiments with the shortest baseline L = 350 km assume the approximation of vacuum oscillations, because of very small matter effects. At long baselines, matter contamination of the oscillation effects becomes significantly large, which fakes the signal arising due to the true/intrinsic T-violation phase in such experiments. Although contamination of matter effects hinders the precise direct measurement of the CP-violation phase, it does separate the ellipses corresponding to normal mass hierarchy (NH) and inverted mass hierarchy (IH) far apart, which makes the hierarchy investigation possible. This effect is illustrated in Fig. 3. Matter contamination at long baselines also increases the size of the ellipse at a specific beam energy, which in turn allows the determination of a narrow range for the δ CP phase; this is illustrated in Fig. 5. In this article, whenever we study any oscillation effect at a specific value of beam energy, we will consider it as the average beam energy.
There exists a fourfold degeneracy in the (θ 13 , δ CP ) parameter space for any given coordinate in the P-P T plane [34][35][36]42]. The values of more than one θ 13 -related solution, especially in the case of short baselines, may lie in the 2σ or 3σ range of this parameter, which makes exact determination of the phase δ CP impossible. The final section discusses the details of the sources, and how to circumvent this degeneracy within the currently available 3σ ranges of mixing parameters [43].

T-violation phenomenology
The neutrino oscillation pattern in vacuum can be modified significantly during the passage of neutrinos through matter because of the effect of coherent forward scattering of the neutrinos on the nucleons, as has been pointed out by Wolfenstein [10,11] and Mikheyev and Smirnov [12][13][14]. We can find the neutrino transition probability in matter by starting from the neutrino oscillation in vacuum with consideration of the evolution of the neutrino state vector described by the Schrödinger equation; for examples of this in the literature, see Refs. [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29]. In the literature, the results of neutrino oscillation experiments have usually been analyzed under the simplest assumption of oscillations between two neutrino types. We can formulate the transition of the neutrino flavor ν e to the flavor ν μ [9], also known as the golden channel, in the form where the upper sign corresponds to neutrinos and the lower sign to antineutrinos, such that In the case of a symmetric density profile, we have P μe = P eμ (δ CP → −δ CP ). Thus, from Eq. (1), we can write If we compare Eqs. (1) and (3), we can define the following relations: where the coefficients have been defined as η3 ± = 2α s 13 sin 2θ 12 sin 2θ 23 The coefficients η2 ± and η3 ± are proportional to the major-minor axis [41] of the ellipse in the P-P T plane with P = P eμ and P T = P μe .
We can define a new parameter A h eμ at the CP-violation phase δ CP = 180 • , which enables us to find the separation between nearby edges of NH and IH ellipses/circles centered along the diagonal of the P-P T plane, as illustrated in Fig. 1. At certain values of the beam energy, two hierarchy ellipses for a given baseline may overlap, as is evident from the right-hand panel in the figure. If we bring two hierarchy ellipses close to each other, these always touch first at the δ CP = 180 • point; hence, this is the point of nearest edges. It is also evident from the left-hand panel that the horizontal nearest edge separation (i.e., A h eμ ) and vertical nearest edge separation (i.   Table 1. We can calculate an analytic expression for A h eμ by making use of Eqs. (2a) and (5) in the following form: As soon as A h eμ or A h μe attains a negative value, the two ellipses corresponding to NH and IH will start overlapping. This parameter is illustrated in Fig. 2 as a function of beam energy E and in Fig. 3 in the E-L plane, known as an oscillogram.
It is evident from Fig. 2 that the value of parameter A h eμ increases with increasing baseline length. One more interesting thing is that, with increasing baseline length, the position of the first oscillation maximum shifts to higher beam energies. Shifting to higher values of beam energies leads to an increased detection cross section and broader maxima (or FWHM), which are beneficial for better energy resolutions. It is advantageous to choose an average beam energy for the incident neutrino  Table 1 have been used for the mixing parameters.   Table 1 have been used for the mixing angles and mass-square differences and the matter density ρ = 3.5 g/cm 3 . beam around this oscillation maximum, as it has a large value for the parameter A h eμ along with a sufficiently large value for the FWHM, which enables us to accurately measure the probability parameter.
• At 730 km, we expect overlapping between NH and IH ellipses at E ≤ 0.8 GeV. At this baseline in the appropriate beam energy range, there is an expected separation of (1-5) units.
• At 350 km, there is overlapping between the ellipses over the whole beam energy range under consideration except for (0.5-1.0) GeV, with an expected separation of (1-3) units.
We can conclude from Figs. 2 and 3 that, at L = 2900 and 2100 km, the two hierarchy ellipses are well separated, so we expect to have a clean signal for the hierarchy investigation. At L = 1200 km, there is still a detectable separation but, for a baseline of L = 730 km, the separation becomes small; at L = 350 km, this separation reduces to a smaller value, which makes it difficult to investigate the mass hierarchy with currently available spreads in beam energy sources. Figure 4 shows an elliptic CP trajectory corresponding to the 0-2π variation in δ CP phase at a given baseline L and beam energy E in the P + eμ -P + μe plane. With the help of Eqs. (2a) and (3), we can find the coordinates of the required points on the ellipse as O(η + 1 , η + 1 ), A(η + 1 + η + 2 , η + 1 + η + 2 ), and 6/14 . Now it is easy to find the widths of the major and minor axes of the ellipse as where AA for AA > BB is the major and for AA < BB is the minor axis of the ellipse and vice versa. These two axes of the ellipse are shown in Fig. 5 as contour plots in the E-L plane. In Fig. 5, we expect to investigate narrow ranges of the CP-violating phase δ CP with large ellipses in the energy region where two hierarchy ellipses may overlap, shown in Fig. 3, mentioned as the excluded energy region in the analysis of Fig. 3 above. However, due to fast oscillations in these excluded energy ranges and large spreads in the currently available beam energy sources, it is currently difficult to carry out any experimental investigation in these regions. Thus, if Fig. 3 helps to determine the optimal experimental configuration while we want to investigate mass hierarchy, then Fig. 5 is very useful in deciding an optimal experiment, when we need large ellipses in order to investigate the CP/T-violation phase δ CP /δ T . Hence, if we become sure of the mass hierarchy with the help of Fig. 3, one can make use of Fig. 5 to fix at least a narrow range for the phase δ CP .
From the above discussion, we can conclude that T-violation biprobability plots are very promising, especially for the investigation of mass hierarchy.

Parameter degeneracy in (θ 13 , δ CP ) parameter space
Degeneracy in the parameter space could create a problem for accurate investigation of the mass hierarchy and T-violation phase δ CP , discussed in the above section. If we do not study the parameter degeneracy, our whole discussion will remain incomplete. In the reactor mixing angle θ 13 and δ CP parameter space, it is possible that, if (θ 13 , δ CP ) is the solution of Eqs. (1) and (3), then (θ 13 , δ CP ) may simultaneously be another solution. This is known as the problem of parameter degeneracy. We will further find possible simultaneous solutions for both the hierarchies and will discuss how to get rid of the degeneracy problem. A similar type of problem has already been discussed in Ref. [42] and references therein.
In the limit when θ 13 1 so that sin θ 13 ≈ θ 13 ≡ θ (say), we can rewrite Eq. (1) in the form Here the L = L 2 and coefficients read where the upper "+" sign corresponds to a positive sign for m 2 31 , i.e., the NH case, and the lower "−" sign to a negative sign for m 2 31 , i.e., the IH case. Now for any given point (P, P T ) in the P eμ -P μe plane, one can write solution to Eqs. (8a) and (8b) as It is possible that, if an ellipse (θ 1 , δ CP 1 ) corresponds to the point (P, P T ), then another ellipse, say (θ 2 , δ CP 2 ), may also correspond to the same point. Note that δ CP i denotes different values of CP-violating phase δ CP . Thus, we can write the difference as In this equation, we have θ T i = θ i with i = 1, 2, as both belong to the same ellipse in the P eμ -P μe plane. This entails the degeneracy that, if (θ 1 , δ CP 1 ) is the solution, so is Thus, in total, we have four degenerate solutions as It is clear that same-sign solutions are well connected; i.e., if we know one solution, we can find the other, but not opposite-sign ones. To find the relation between opposite-sign solutions, we have from Eq. (10) In the above equation, we have chosen Z + = Z − ≡ Z (say), as is evident from Eq. (9). Now θ 1 − θ T 1 = 0 gives us the following relation: and θ 3 − θ T 3 = 0 gives the following relation: By making use of the identity [42] √ we can find, by making use of Eqs. (16) and (17), the following relation: Thus one can choose, without loss of generality, δ CP 3 = π − δ CP 1 . Now, by making use of this value of δ CP 3 , from Eqs. (14) and (15) we can write The above equation can be solved further by making use of Eq. (18), giving the following form: We can further solve the above equation with the help of Eq. (19) to get Now, from Eqs. (12), (13), and (22), we can write the four degenerate solutions corresponding to Eqs. (8a) and (8b) explicitly as: for (θ 1 , δ CP 1 ) chosen In the P + eμ -P + μe plane, the physically allowed region is restricted by the condition sin 2 δ CP i ≤ 1, from which, with the help of Eqs. (16) and (17), we can formulate the following inequality: This physical region is shown by a green curve enclosing the ellipses in Fig. 7. In this figure, we note that, at a short baseline, L = 350 km (left-hand panel), four solutions lie in the 3σ range (7.7 • ≤ θ 13 ≤ 9.9 • ) [43], except for θ 4 = 10.21 • . Similarly, at L = 730 km (right-hand panel), two solutions θ 3 = 17.17 • and θ 4 = 14.5 • lie outside the 3σ range of the third mixing angle, which suggests that the inverted mass hierarchy signal is well distinguishable even at the 3σ level for this experimental configuration. Thus we conclude by saying that, if there is opposite-sign degeneracy at short baselines, it will get circumvented at longer baselines for the same other experimental inputs. This is due to the fact that, as matter effects in long baselines increase, this pulls the two different hierarchy CP trajectory ellipses further apart, as we can confirm from Fig. 6. Opposite-sign degeneracy has a physical manifestation as long as the two ellipses corresponding to NH and IH are overlapping. We can avoid the overlap of opposite-sign ellipses by appropriately choosing a longer baseline L and specific beam energy E. To find the optimized L and E configuration, Fig. 3 is very helpful.   Table 1 have been used for the mixing parameters.  Table 2. Separation between the nearest edges of NH and IH ellipses (i.e., A h eμ given in Eq. (6)) for the n = 1 and n = 2 scenarios at possible L and E values given by Eq. (25b). The last column shows the possible separation between ellipses for a beam spread of 1 GeV, at a given beam energy E. Possible configurations of experiments for the n = 1 and n = 2 scenarios are given in Table 2.

Conclusions and perspectives
To explore the optimal setup in order to investigate mass hierarchy, separation between two opposite hierarchy ellipses in the biprobability (P-P T ) plane serves as an optimization parameter. It is evident from Figs. 3 and 6 that, at longer baselines, hierarchy ellipses separate to a large extent as compared to shorter baselines, which suggests that longer baselines are useful for the investigation of mass hierarchy in the P-P T plane. We come to know from Figs. 2 and 3 that, at a given baseline below a specific value of beam energy, the oscillations become very fast and the separation between hierarchy ellipses becomes so small that they may overlap to large extent too. This value of beam energy has values of E ≤ 3, ≤ 2, ≤ 1.3, ≤ 0.8 Gev and 0.1 ≥ E ≥ 1.0 GeV, respectively, for baselines of 2900, 2100, 1200, 730 and 350 km. The hierarchy investigation in the context of the biprobability plane is an elegant technique. The major drawback that could hinder precise hierarchy measurement is that there always exists a fourfold degeneracy for any coordinate in the P-P T plane within the (θ 13 , δ CP ) parameter space. The oppositesign degeneracy has a physical existence in the case when two hierarchy ellipses overlap. To get rid of this degeneracy, we need to observe the results at suitable long baselines and beam energies. The same-sign degeneracy disappears as soon as cos L = 0, which is fulfilled for L 507 E (GeV) km for the n = 1 and for L 1507 E (GeV) km for the n = 2 scenarios. In particular, we recommend use of Fig. 3 in order to choose suitable L and E configurations at which both the opposite-and samesign degeneracies can be circumvented. Also, we can analyze from Table 2 that the experimental configuration L = 2535 km, E = 5 GeV for the n = 1 scenario is the most suitable one that fulfills the criteria just mentioned.