Unity of CP and T Violation in Neutrino Oscillations

In a previous work a simultaneous P- CP[P] and P- T[P] bi-probability plot was proposed as a useful tool for unified graphical description of CP and T violation in neutrino oscillation. The ``baseball diamond'' structure of the plot is understood as a consequence of the approximate CP-CP and the T-CP relations obeyed by the oscillation probabilities. In this paper, we make a step forward toward deeper understanding of the unified graphical representation by showing that these two relations are identical in its content, suggesting a truly unifying view of CP and T violation in neutrino oscillations. We suspect that the unity reflects the underlying CPT theorem. We also present calculation of corrections to the CP-CP and the T-CP relations to leading order in Delta m^2_{21} / Delta m^2_{31} and s^2_{13}.


I. INTRODUCTION
Exploring leptonic CP and T violation is one of the most challenging endeavors in particle physics. Confirming (or refuting) unsuppressed CP violation analogous to that in the quark sector must shed light on deeper understanding of lepton-quark correspondence, the concept whose importance was recognized early in sixties [1]. We should note, however, that it is only after the KamLAND experiment [2] which confirmed the MSW large mixing angle (LMA) solution [3,4] of the solar neutrino problem that we can talk about detecting CP or T violation in an experimantally realistic setting. An almost maximal mixing of θ 23 discovered by the atmospheric neutrino observation by Super-Kamiokande [5], which broke new ground in the field of research, also greatly encourages attempts toward measuring the leptonic Kobayashi-Maskawa phase δ.
Yet, we might have the last impasse to observing leptonic CP violation, a too small value of θ 13 , which lives in the unique unexplored (1-3) sector of the MNS matrix [6]. Currently, it is bounded from above by a modest constraint sin 2 2θ 13 ≤ 0.15 − 0.25 obtained by the Chooz reactor experiment [7]. Toward removing the last impasse, two different methods for measuring θ 13 are proposed and materialized into a number of concrete experimental programs. The first is the measurement of appearance probability P (ν µ → ν e ) in longbaseline (LBL) experiments using accelerator neutrino beam, being and to be performed by the ongoing [8,9] and the next generation projects [10,11,12]. The second is the reactor measurement of θ 13 . It is a pure measurement of θ 13 independent of other oscillation parameters, δ and θ 23 , and thus will play a rôle complementary to the LBL experiments [13]. This property is expected to help resolving the parameter degeneracy [14,15,16,17,18,19] related to θ 23 [13]. A spur of experimental projects that occurred over the globe for the relatively new opportunity are now summarized in the White Paper Report [20].
If such challenges are blessed by nature we will be able to proceed to measuring the leptonic CP or T violating phase δ. The relatively large value of θ 13 will allow us to measure it via ν e andν e appearance measurement using conventional superbeam experiments, whose idea may be traced back to [21]. Feasible experimental programs for such appearance measurement with upgraded beams as well as detectors are proposed. See e.g., [10,22] for the JPARC-Hyper-Kamiokande project and [11] for NOνA. It is also proposed that a fast search for CP violation can be performed by combining neutrino mode operation of such experiments with high statistics reactor measurement of θ 13 [23].
If θ 13 is too small to be seen in the above experiments an entirely new strategy is called for. We will probably need more aggressive approach with ambitious beam technologies, neutrino factory [24] or beta beam [25] or even both. Here also, vigorous world-wide activities for developing beam and target technologies as well as studying physics capabilities are underway [26,27,28]. Intense neutrino beam from a muon storage ring and the clean background for wrong sign muon detection are expected to lead to an enormous sensitivity of θ 13 up to ∼ 1 degeree. Enriched by golden (ν e → ν µ ) [29] as well as silver (ν e → ν τ ) [30] channels, it will be able to resolve all the parameter degeneracies, as claimed in [31]. See [32] for a review of old and new ideas on how to measure leptonic CP violation.
How does T violation measurement fit into the scene? To our understanding it will probably come later than CP violation measurement because the measurement is more difficult to carry out. In neutrino factory it requires electron charge identification which is highly nontrivial, if not impossible. The beta beam, if build, would give us an ideal apparatus because it can deliver a pure ν e beam which comes from decaying radioactive nuclei. By combining with superbeam (or neutrino factory) measurement of P (ν µ → ν e ) it will provides us a unique opportunity for exploring leptonic T violation.
Keeping in mind the scope of experimental realization of CP and T violation measurement in the future, we discuss in this article a unified view of leptonic CP and T violation, one of the most fundamental problems in particle physics. We hope that our discussion is illuminating and contributes to deeper understanding of the problem. In this paper we use, except for in Appendix, the standard notation of the MNS matrix [33].

II. CP AND T VIOLATION IN NEUTRINO OSCILLATION
It has been known for a long time that CP and T conservation are intimately related to each other by the CPT theorem. For neutrino oscillation in vacuum the invariance leads to a relation between neutrino and antineutrino oscillation probabilities Then, the question might be "if there exists analogous relation in neutrino oscillation in matter?". It was shown in [34] that indeed there exists such a relationship, which comes from the classical time reversal and the complex conjugate of the neutrino evolution equation assuming that the matter profile is symmetric about the mid-point between production and detection. Let us call (2) the CPT relation in matter. Here, a = 2 √ 2G F N e E is the fundamental quantity which is related to neutrino's index of refraction in matter [3] with G F being the Fermi constant, E neutrino energy, and N e (x) an electron number density in the earth. The mass squared difference of neutrinos is defined as ∆m 2 ij ≡ m 2 i − m 2 j where m i is the mass of the ith eigenstate.
There is an immediate consequence of the CPT relation in matter, Eq. (2). If we define ∆P CP T as then, ∆P CP T is an odd function of a. The property may be used to formulate the method for detecting extrinsic CPT violation in neutrino oscillation due to matter effect [35]. Do the CPT relation and various other relationships between oscillation probabilities give a unified picture of CP and T violation in neutrino oscillation in matter? In this article we argue that the answer is indeed yes. Although our argument in this paper is based on the line of thought in [34], we believe that we made a step forward from the previous work.

III. UNIFIED GRAPHICAL REPRESENTATION OF CP AND T VIOLATION
Toward the goal of this paper, let us introduce a graphical representation of the characteristic features of neutrino oscillations relevant for leptonic CP violation [15]. For simplicity of notations let us define the symbols for CP and T conjugate probabilities, CP [P ] ≡ P (ν α →ν β ) and T [P ] ≡ P (ν β → ν α ), for a given probability P (ν α → ν β ). It is the CP trajectory diagram in the P -CP [P ] bi-probability space, which can be extended to incorporate the P -T [P ] bi-probability plot [34]. bi-probability plot with experimental parameters corresponding to the baseline distance and about twice the optimal energy corresponding to maximal exhancement of T violating effect [36]. Notice the difference between movement of the direction in P -T [P ] and P -CP [P ] plot; they are orthogonal to each other for reasons explained in the text. The figure is the same as Fig.1 of [34] apart from that we have changed the convention of δ to the standard one used in the text of the paper. The convention is employed by almost everybody who works in the field (see e. g., [14,18]) including all our previous works, [15,19,34,37], but is different from that of [33]. Although the convention of the MNS matrix is the same, the latter takes a convention such that U in the neutrino evolution equation (A1) is replaced by U * .
Given two observables P and CP [P ], you can draw a dot in P -CP [P ] space, and it becomes a closed ellipse when δ is varied. In Fig. 1 the ellipses labeled V ± are the ones for vacuum oscillation probabilities where the subscripts ± denote the sign of ∆m 2 31 . When the matter effect is turned on they split into two ellipses labeled CP ± in the CP bi-probability plot and into T ± in the T bi-probability plot, both of which are simultaneously depicted in Fig. 1.
Let us first focus on the P -CP [P ] bi-probability plot. We first note that the oscillation probability P (ν α → ν β ) can be written on very general ground (for α = e, β = µ, τ , or vice versa) 1 as [38,39] where A, B, and C are functions of ∆m 2 31 , ∆m 2 21 and a. (Of course, the previously obtained approximate formulas do have such form. See, e.g., [29,40,41].) It is nothing but (4) that guarantees the elliptic nature of the trajectories. Then, one can show that the lengths of major and minor axes (the "polar" and "radial" thickness of the ellipses) represent the size of the sin δ and the cos δ terms, respectively, whereas the distance between two ellipses with positive and negative ∆m 2 31 displays the size of the matter effect [15]. Finally, the distance to the center of the ellipse from the origin is essentially given by sin 2 2θ 13 . Notice that all the features of the bi-probability plot except for distance between ∆m 2 31 = ± ellipses are essentially determined by the vacuum parameters in setting of E and L relevant for the superbeam experiments. Therefore, one can easily guess how it looks like in the other experimental settings. As indicated in Fig. 1 the CP violating and CP conserving effects of δ are comparable in size with the matter effect even at such high energy and long baseline.
We notice that in the P -T [P ] bi-probability plot the matter effect splits the vacuum ellipses V ± in quite a different way from the P -CP [P ] bi-probability plot. It is because the T violating measure ∆P T , which is given by for symmetric matter profiles, vanishes at δ = 0. Therefore, the T (or CP) conserving point must remain on the diagonal line in P -T [P ] bi-probability plot. Equation (5) stems from the fact that the coefficients except for B are symmetric under the interchange α ↔ β. Therefore, if ∆P T = 0, then δ = 0 even in matter. The matter effect cannot create a fake T violation for symmetric matter profiles [42]. For modifications which occur for asymmetric matter profiles, see e.g., [43,44,45].
Notice that the matter effect cannot create fake T violation, it does modify the coefficient B in Eq. (5), whose feature is made transparent in [36]. Among other things, it was shown in [36] that the matter effect can enhance the T asymmetry up to a factor of 1.5. Other earlier references on T violation in neutrino oscillation include [46,47,48,49].
These relations are meant to be valid in leading order in ∆m 2 21 /∆m 2 31 , i.e., in zeroth order in δ-independent and to first order in δ-dependent terms, respectively.
Roughly speaking, the CP-CP relation guarantees that the locations of the first and the third bases are approximately symmetric under reflection with respect to the diagonal line in P -CP [P ] space, whereas the T-CP relation guarantees that the ordinates of the T ± ellipse are approximately the same as those of CP ∓ . Of course, one has to specify the values of the CP phase δ to make the relationship precise, and that is why the change in δ is involved between the RHS and the LHS of Eqs.(6) and (7).
For the T-CP relation the first equality in (7) can be derived by using (8) in the CPT relation in matter (2). Then, the second approximate equality holds for small ∆m 2 21 /∆m 2 31 with the same adjustment of the phase δ.

V. EQUIVALENCE BETWEEN THE CP-CP AND THE T-CP RELATIONS
We now point out that the the CP-CP and the T-CP relations are equivalent to each other in their physics contents. Roughly speaking, the T-CP relation is "T conjugate" of the CP-CP relation. It reflects the relationships among various neutrino oscillation probabilities discussed in the previous section. Their equivalence again testifies for the unity of CP and T violation in neutrino oscillations.
In passing, we note the following: It was noted in [15] that there exists an approximate symmetry in the vacuum oscillation probability under the simultaneous transformations ∆m 2 31 → −∆m 2 31 and δ → π − δ which explains almost overlap of V + and V − trajectories as in Fig. 1. A generalization of the approximate symmetry into the case with matter effect has been obtained [34], from which the CP-CP and the T-CP relations also follow. Clearly, the correction to the approximate symmetry is also related to ∆P CP −CP . To show this, we define the difference ∆P f lip between the LHS and the RHS of (16), Using the frequently used identity, one can show that ∆P f lip = P (ν α → ν β ; ∆m 2 31 , ∆m 2 21 , δ, a) − P (ν α → ν β ; +∆m 2 31 , −∆m 2 21 , π + δ, a). (18) Thus, ∆P f lip = ∆P CP −CP ; they are identical.

VI. LEADING-ORDER CORRECTIONS TO THE CP-CP AND THE T-CP RE-LATIONS
We now compute the leading-order corrections to the CP-CP and the T-CP relations. During the course of the computation, we will give an explicit proof of these relations. We start from the Kimura-Takamura-Yokomakura (KTY) formula [38] of the oscillation probability, Eq. (4). We note that the coefficients A, B, and C are functions of ∆m 2 21 , ∆m 2 31 , and the matter coefficient a, but we here suppress dependences on the latter two quantities. We also note that A and B start from first order in ∆m 2 21 , so that we can write A(x) = xα(x) and B(x) = xβ(x). Using the fact that sin(π + δ) = − sin(δ) and cos(π + δ) = − cos(δ), we obtain Therefore, we have shown that the RHS of (19) is of order ǫ ≡ ∆m 2 21 ∆m 2 31 (C term), or ǫ 2 (A and B terms). This is an explicit proof of the CP-CP relation, and hence also the T-CP relation.
We are now left with the computation of the first-order terms of α, β, and C. The exact form of these coefficients are calculated in [38]. 2 Therefore, it is straightforward to compute the RHS of (19). It reads where J r ≡ c 12 s 12 c 23 s 23 c 2 13 s 13 . ∆P T −CP can be obtained by replacing δ by 2π−δ in ∆P CP −CP , as dictated in (15). We have kept in the expression of the oscillation probability the terms up to order O(ǫs 2 13 ) and O(s 4 13 ) in C, and to O(ǫ 2 s 13 ) and O(ǫs 3 13 ) in A and B. But the contributions from terms higher order in s 13 cancel in (20).
The feature that the coefficients A and B start with first-order terms of ∆m 2 21 played an important rôle in proving the CP-CP relation to leading order. It comes from the fact that they vanish in the two flavor limit ∆m 2 21 → 0 and that the probabilities allow Taylor expansion in terms of the variable. The former statement is proved in Appendix on very general ground without assuming adiabaticity or constant matter density.

VII. CONCLUDING REMARKS
In this contribution to the Focus Issue on 'Neutrino Physics' we have presented a new unified view of the leptonic CP and T violation in neutrino oscillation. Based on the CPT relation in matter and other relations obeyed by the oscillation probabilities which are derived in [34] we were able to complete our understanding of the structure of unified description of CP and T violation in terms of bi-probability plot. Namely, the diamond shaped structure of simultaneous P -CP [P ] and P -T [P ] bi-probability plot is now understood as a consequence of a unique relation, the CP-CP (or the equivalently, the T-CP) relation. Based on this observation and relying on the KTY formula we have computed leading order corrections to the CP-CP relation. 2 Note, however, that there is an error in the sign of the term denoted as A (1) k in Eq. (44) of the first reference in [38].
We have also briefly touched upon the basic features of the T violating measure which are in contrast with those of CP violation. They include vanishing T violating measure at vanishing CP phase δ, and the enhancement of T violating asymmetry by the matter effect up to a factor of 1.5. Though measurement of T violation should give us a cleaner way of detecting genuine CP violating effects, it is not easy to carry out experimentally. We must wait for the construction of an intense electron (anti-) neutrino beam either by beta beam [25] or in neutrino factories [24].
Though it should be the case on physics ground, it is not entirely trivial to show that δ-dependence disappears from the oscillation probabilities in the two-flavor limit ∆m 2 21 → 0. We carry it out explicitly in this Appendix. It is a slight modification of the method [50] that allows us to show that δ-dependence disappears in the survival probability P (ν e → ν e ).
We write down the evolution equation of three flavor neutrinos in matter which is valid to leading order in electroweak interaction: In this Appendix we take a slightly different parametrization of the mixing matrix U = e iλ 7 θ 23 Γ δ e iλ 5 θ 13 e iλ 2 θ 12 (A2) where λ i are SU(3) Gell-Mann's matrix and Γ contains the CP violating phase We then rewrite the evolution equation (A1) in terms of the new basis defined bỹ In vanishing ∆m 2 12 limit it reads Now we observe that the CP phase δ disappears from the equation. It is due to the specific way that the matter effect comes in; a(x) only appears in (1.1) element in the Hamiltonian matrix and therefore the matter matrix diag(a, 0, 0) is invariant under rotation in 2-3 space by e iλ 7 θ 23 . Then the rotation by the phase matrix Γ does nothing. Notice thatν µ does not have time evolution due to (A5).
It is clear from (A5) that any transition amplitudes computed withν α basis is independent of the CP violating phase. Of course, it does not immediately imply that the CP violating phase δ disappears in the physical transition amplitude ν β | ν α . The latter is related with the transition amplitude defined withν α basis as where T is defined in (A4) and its explicit form in our parametrization (A2) of the mixing matrix reads One can show that the amplitude of ν e → ν µ has a pure phase factor e iδ and hence P (ν e → ν µ ) is independent of phase δ; ν µ (x) | ν e (0) = c 23 ν µ (x) |ν e (0) + s 23 e iδ ν τ (x) |ν e (0) (A8) The first term, however, vanishes because ν µ (x) |ν e (0) = ν µ (0) |ν e (0) = 0. (No evolution inν µ .) Notice that the same statement does apply to the P (ν e → ν τ ) and P (ν µ → ν τ ) as well. One can show that the same conclusion holds for different choice of the phase matrix from that in (A2).
Since absence or presence of T violation should not depend on the parametrization used, this completes the proof that the δ dependence disappears from all the oscillation probabilities in the limit ∆m 2 12 → 0.