Signatures of the genuine and matter-induced components of the CP violation asymmetry in neutrino oscillations

CP asymmetries for neutrino oscillations in matter can be disentangled into the matter-induced CPT-odd (T-invariant) component and the genuine T-odd (CPT-invariant) component. For their understanding in terms of the relevant ingredients, we develop a new perturbative expansion in both $\Delta m^2_{21},\, |a| \ll |\Delta m^2_{31}|$ without any assumptions between $\Delta m^2_{21}$ and $a$, and study the subtleties of the vacuum limit in the two terms of the CP asymmetry, moving from the CPT-invariant vacuum limit $a \to 0$ to the T-invariant limit $\Delta m^2_{21} \to 0$. In the experimental region of terrestrial accelerator neutrinos, we calculate their approximate expressions from which we prove that, at medium baselines, the CPT-odd component is small and nearly $\delta$-independent, so it can be subtracted from the experimental CP asymmetry as a theoretical background, provided the hierarchy is known. At long baselines, on the other hand, we find that (i) a Hierarchy-odd term in the CPT-odd component dominates the CP asymmetry for energies above the first oscillation node, and (ii) the CPT-odd term vanishes, independent of the CP phase $\delta$, at $E =0.92~\mathrm{GeV}\,(L/1300~\mathrm{km})$ near the second oscillation maximum, where the T-odd term is almost maximal and proportional to $\sin\delta$. A measurement of the CP asymmetry in these energy regions would thus provide separate information on (i) the neutrino mass ordering, and (ii) direct evidence of genuine CP violation in the lepton sector.


Introduction
One of the most important open questions in fundamental physics is the existence of CP symmetry breaking in the lepton sector. A positive answer could open the door to understand the matter-antimatter asymmetry of the Universe through Leptogenesis [1] at higher energy scales. Next generation neutrino flavor oscillation experiments, like T2HK [2] and DUNE [3], have this challenge as their priority aim. But the neutrino propagation from terrestrial accelerator facilities is not taking place in vacuum and there are matter effects [4,5] induced by the CP-asymmetric interaction with Earth. In the quest for a direct evidence of CP violation in the lepton sector, we have recently derived a theorem [6] for the observable CP asymmetry in neutrino oscillations propagating in matter where P αβ is the probability for any flavor transition α → β. Its achievement consists in providing the disentanglement of the genuine A T αβ and matter-induced A CPT αβ components of CP-violating (CPV) A CP αβ based on the concept that they have different properties under the other discrete symmetries Time Reversal T and CPT: A CPT αβ is CPT-odd T-invariant, whereas A T αβ is T-odd CPT-invariant. These two components are distinctly identified by their behavior as functions of the baseline L: whereas the matter-induced component A CPT αβ is an even function of L, the genuine component A T αβ is odd in L. For a description of the effective Hamiltonian in which the matter-induced term is generated by the parameter a = 2EV of the matter potential V due to charged current interactions between electron neutrinos and electrons and the genuine CPV term is generated by a phase δ in the threefamily PMNS mixing matrix, the A CPT αβ and A T αβ components of the CP asymmetry have the correct behavior, for any neutrino energy E and baseline L, in these parameters: whereas A CPT αβ in odd in a ∀δ, A T αβ is odd in sin δ ∀a. In addition, the change from a Normal Hierarchy in the ordering of the neutrino mass spectrum to an Inverted Hierarchy leads to the result that A T αβ remains invariant and, for energies above the first oscillation node, A CPT αβ changes its sign.
The planned experiments T2HK and DUNE consider the golden transition ν µ L − − → ν e for neutrinos propagating in the Earth mantle with fixed L and a continuum energy spectrum E. In the search of interesting experimental signatures able to separate the A CPT µe and A T µe components of A CP µe , we study in this paper the characteristic energy dependencies of the two terms. In doing so, we develop appropriate analytical relations between the quantities in matter and the quantities in vacuum in order to provide guiding paths in our scrutiny of the peculiar behavior of the separate A CPT µe and A T µe components with energy. The paper is organized as follows. Section 2 identifies the two components, matterinduced A CPT αβ and genuine A T αβ , of the CP asymmetry A CP αβ by means of rephasing-invariant mixings and neutrino masses in matter. The conceptual basis of the Disentanglement Theorem is provided by the different behavior under T and CPT symmetries. In Section 3 we build a consistent perturbative expansion of the relevant ingredients in terms of the vacuum parameters with ∆m 2 21 ∆m 2 31 and |U e3 | 2 1, with the interaction parameter a moving from above to below ∆m 2 21 . This approach will allow us to understand the intricacies of the ordering of the two limits a → 0 and ∆m 2 21 → 0. In Section 4 we check that our analytic expansions are an excellent approximation to the exact unique rephasing-invariant CP-odd mixing appearing in the genuine A T µe component of the CP asymmetry, leading in fact to A T µe as in vacuum by an interesting compensation of matter effects between the mixing and oscillation factors. In Section 5 we discuss the dependence of these observables on the neutrino mass hierarchy, proving that it is determined by the sign of A CPT µe at high energies, whereas A T µe is blind to it. Section 6 makes a scan of the energy dependencies of the two A CPT µe and A T µe components, identifying zeros and extremal values of these functions. The occurrence of magic energies, in which A CPT µe vanishesindependent of δ-and A T µe is maximal, will be understood. In Section 7 we discuss our conclusions and outlook.

The CP Asymmetry Disentanglement Theorem
Neutrino oscillations in matter are described through the effective Hamiltonian in the flavor basis [4,[7][8][9][10][11]  where the first term describes neutrino oscillations in vacuum and the second one accounts for matter effects. The a parameter is given by a = 2EV , with V the interaction potential with matter and E the relativistic neutrino energy. For antineutrinos, U → U * , originating a genuine CP violation effect through a CP phase δ in U PMNS , as well as a → −a, originating matter-induced CP violation. In this description, the genuine CPV observable has to be odd in sin δ, whereas the matter-induced CPV effect has to be odd in a. All neutrino masses (M 2 ) and mixings (Ũ ) in matter, i.e. eigenvalues and eigenstates of H, can be calculated in terms of the parameters in the vacuum Hamiltonian (M 2 , U ) and a, as studied in Section 3. The exact Hamiltonian leads to the flavor oscillation probabilities for any α → β transition . If CPT holds, as assumed in vacuum, necessarily ∆m 2 ij = ∆m 2 ij andJ ij αβ = (J ij αβ ) * . Therefore, all L-even terms will cancel out in A CP αβ , proving they are CPT-violating. On the other hand, the absence of genuine CP violation leads to realJ ij αβ andJ ij αβ , so all transition probabilities P αβ are L-even functions. This result shows that L-odd terms in A CP αβ are T-violating. From the different behavior of each of these terms under the discrete T and CPT symmetry transformations, one derives the Asymmetry Disentanglement Theorem [6] by separating the observable CP asymmetry in any flavor transition into L-even (CPTviolating) and L-odd (T-violating) functions, Let us emphasize that not only A CPT αβ is CPT-violating and A T αβ is T-violating, we also find that A CPT αβ is T-invariant and A T αβ is CPT-invariant. In this sense the two terms are truly disentangled. To prove these properties, we analyze both CPT and T transformations.
Under CPT: αβ is T-invariant (even in sin δ) and CPT-odd in a, whereas A T αβ is CPT-invariant (even in a) and T-odd in sin δ. As a consequence, A CPT αβ vanishes for a = 0 ∀δ and A T αβ vanishes for δ = 0, π ∀a. These complementary behaviors of the two components of the experimental CP asymmetry identify the CPV component A T αβ as CPT-invariant and thus a fully genuine CPV observable, whereas the CPV component A CPT αβ is T-invariant and thus a fully fake CPV observable.

Analytic perturbation expansions
To understand the behavior of A CPT αβ and A T αβ in Eqs. (2.3) required by the CPT and T symmetries, as proved in the previous Section, we proceed to their analytic study for neutrino oscillations in matter of constant density. Notice that the formal description of the system is equivalent to neutrino oscillations in vacuum, if one parametrizes matter effects as a redefinition of neutrino masses and mixings. However, this redefinition is strongly dependent on the neutrino energy, so it does not provide a clear insight into the intrinsic properties of the system. A useful description should write all observables in matter, relevant to our two components of the experimentally accessible CP asymmetry, as functions of the vacuum parameters and the matter potential a. A similar methodology is being applied to calculations of the T [12] and CPT [13] asymmetries in matter.
The search of these formulae unavoidably finds the same issue: an exact description of the matter effects in neutrino oscillations leads to cumbersome expressions which do not provide a clear understanding [14]. The way to simplify the results is to treat perturbatively the small parameters of the system, namely ∆m 2 21 ∆m 2 31 and |U e3 | 2 1. The most important drawback of this procedure is that, in perturbing in ∆m 2 21 , the implicit relation ∆m 2 21 |a| is also assumed, so one should not expect to reproduce the right vacuum limit a → 0 for all matter ingredients. Even so, this perturbation theory leads to compact and percent-level precise expressions for neutrino oscillation probabilities written in terms of vacuum parameters only [15], as well as more precise relations mapping the mixings in matter to the quantities in vacuum [16,17].
We develop a new perturbative expansion in both ∆m 2 21 , |a| ∆m 2 31 without assumptions between ∆m 2 21 and a, similar to Ref. [18], oriented to the understanding of masses, mixings and the separate behavior of A CPT αβ and A T αβ as functions of the different variables. In doing so, we can check from our analytic expressions both the vacuum limit a → 0 and the T-invariant limit ∆m 2 21 → 0. The expansion in |a| ∆m 2 31 holds for energies below a few GeV taking into account the definite a-parity of each component of the CP asymmetry. This way, we find the most simple expressions for A CPT µe and A T µe at the energies accessible by accelerator experiments, which are accurate enough to let the reader clearly understand their behavior.
We emphasize that any desired precision can be achieved using a numerical computation of the neutrino propagation. Our aim is not finding very precise expansions, but precise enough to identify and understand the distinct characteristic patterns of the energy behavior of the two components A CPT αβ and A T αβ , with the objective of serving as a guide for experimental signatures.

3.1
The crucial role of the referencem 2 0 in matter Since a diagonal m 2 1 1 in the Hamiltonian H in Eq. (2.1) leads to a global phase in time evolution, which is unobservable, the equivalent Hamiltonian 2E ∆H ≡ ∆H = U ∆M 2 U † +a P e , where ∆M 2 = diag(0, ∆m 2 21 , ∆m 2 31 ) and P e = diag(1, 0, 0) is the e-flavor projector, is widely used.
Analogously, one could argue thatm 2 1 is unobservable in neutrino oscillations in matter. Even though this is true, one must take into account that either m 2 1 orm 2 1 can be chosen as origin of phases, but not both of them at the same time when connecting the parameters in matter to those in vacuum. Indeed, one can easily check that the Hamiltonian ∆H has three non-vanishing eigenvalues, so choosing m 2 1 = 0 automatically leads to allm i = 0, despite one of them being unobservable.
On the following, we callm 2 0 the mass squared in matter leading to the relative phase shift between the unobservable global phases in vacuum and matter, writing the Hamiltonian as ∆H = U ∆M 2 U † + a P e =m 2 0 1 +Ũ ∆M 2Ũ † . In this notation, the three eigenvalues of ∆H will be the reference scale in matterm 2 0 and the two observable mass squared differences in matter ∆m 2 ij . As proposed in Ref. [14], we choose to diagonalize the Hamiltonian in the vacuum eigenbasis, ∆H ij = ∆M 2 +a U † P e U =m 2 0 1+V ∆M 2 V † . Since the eigenvalues are basis independent and real, this cleanly shows thatm 2 0 and ∆m 2 ij can only depend on (∆m 2 ij , |U ei |, a). Moreover, this factorizes the mixing matrix in matter intoŨ = U V , where U is the (vacuum) PMNS matrix and V is the change of basis between vacuum and matter eigenstates, which must go to the identity when a → 0.
The Hamiltonian being a 3 × 3 matrix leads to the characteristic equation where p(λ) is the characteristic polynomial of ∆H and its roots provide the three neutrino squared masses in matter λ i , and the observable ∆m 2 ij = λ i − λ j . The three invariants can be easily calculated, From this straightforward setup of the problem we find a fundamental result. Since at least one of the eigenvalues of ∆H will always lie in the range [0, ∆m 2 21 ]. All ∆m 2 ij are known to be nonbound by ∆m 2 21 , as will be shown in Fig. 1, so we find that 0 ≤m 2 0 ≤ ∆m 2 21 . Although the inequalities in (3.3a) and (3.3b) have been written for Normal Hierarchy neutrinos, the reader may check that the argument is also valid for the Inverted Hierarchy and antineutrinos.
Given that physically ∆m 2 21 ∆m 2 31 , this result shows thatm 2 0 is also a good perturbative parameter. Therefore, we focus in this Subsection on writing the two observable ∆m 2 ij exactly as functions of (∆m 2 ij ,m 2 0 , |U ei |, a), which gives enough information to calculate all observables of neutrino oscillations in matter.
A simple way to calculate the physical ∆m 2 ij is the diagonalization of the displaced Hamiltonian ∆H ij −m 2 0 1 = ∆M 2 + a U † P e U −m 2 0 1 = V ∆M 2 V † . By construction, one of its eigenvalues is zero, so det ∆H −m 2 0 1 = 0 and its two non-vanishing eigenvalues are given by a quadratic equation, From this definition it is clear that ∆m 2 + > ∆m 2 − , but notice that ∆m 2 + > ∆m 2 − for Normal Hierarchy, whereas ∆m 2 − > ∆m 2 + for Inverted Hierarchy. This expression for ∆m 2 ± is a good starting point from which one can easily derive approximate formulae in the limitm 2 0 ≤ ∆m 2 21 ∆m 2 31 . In order to write ∆m 2 ± as functions of vacuum parameters only, this same limit can be used directly in Eq. (3.1) to findm 2 0 perturbatively, as we will do in the following Subsection.
We finish this Subsection writing explicitly the eigenstates of ∆H ij in the canonical basis of mass eigenstates in vacuum, where the normalization factor N i is needed to ensure ν i |ν i = 1, and its phase must be chosen so that lim The eigenvalues λ i , labeled according to λ 1 < λ 2 < λ 3 (λ 3 < λ 1 < λ 2 ) if the hierarchy is normal (inverted), are given bym 2 0 and ∆m 2 ± as shown in Table 1 for neutrinos and antineutrinos. The reason why λ 1 =m 2 0 whereasλ 2 =m 2 0 will be explained analytically when exploring the vacuum limit. This fact is also shown in Fig. 1 by a numerical determination of the evolution of the eigenvalues of ∆H with the matter parameter a. All results are produced using the best-fit values in Ref. [19] for Normal Hierarchy. To show the theoretical implications of a change in mass hierarchy, we compute the Inverted Hierarchy case using ∆m 2 31 IH = − ∆m 2 32 NH , which is physically consistent since it keeps the absolute value of the largest mass splitting unchanged.
As seen, the eigenvalues that fulfill 0 ≤m 2 0 ≤ ∆m 2 21 are λ 1 andλ 2 , independently of whether the hierarchy is normal or inverted. To distinguish these two functions, from now on we will call them λ 1 ≡m 2 0 andλ 2 ≡m 2 0 . Notice that, even if bothm 2 0 andm 2 0 are bounded by ∆m 2 21 , they are necessarily different functions, as seen by their different vacuum limits lim  Table 1. Relation between the eigenvalues λ i , with the convention λ i a→0 − −−− → ∆m 2 i1 , and the quantitiesm 2 0 and ∆m 2 ± as calculated from Eq. (3.4) with the corresponding sign of a for ν/ν and the sign and value of ∆m 2 31 for NH/IH. According to hierarchy, the eigenvalues are ordered from larger to smaller. The observable ∆m 2 ij = λ i − λ j can be read from the table. Figure 1. Eigenvalues λ i of the mass matrix in matter in units of ∆m 2 21 , for both neutrinos (a > 0) and antineutrinos (a < 0). The horizontal axis shows both the evolution of the matter parameter a at fixed energy (lower labels), i.e. changing the matter density, and as function of the energy if the constant density is chosen as that of the Earth mantle (upper labels). Both the exact (dashed) and the analytical (solid) results from Eqs. (3.6) are shown to illustrate the excellence of the analytic approximation. Normal Hierarchy (λ 1 < λ 2 < λ 3 ) in the left pannel, Inverted Hierarchy (λ 3 < λ 1 < λ 2 ) in the right pannel.

The way to the vacuum limit at fixed (E, L)
The perturbation theory used in the literature to make profit of the experimental relation ∆m 2 21 ∆m 2 31 also assumes ∆m 2 21 |a|. In order to ensure that all our expressions will reproduce the right vacuum limit, which is crucial to study the CPT-invariant limit on A CP αβ , we expand ∆m 2 21 ∆m 2 31 without any assumption between ∆m 2 21 and a. Up to first order in this regime, Eqs. (3.1) and (3.4) reduce to These analytical results are shown in Fig. 1 for neutrinos (a > 0) and antineutrinos (a < 0), as well as Normal (∆m 2 31 > 0) and Inverted (∆m 2 31 < 0) Hierarchies, and they match perfectly the exact numerical values. Notice that the two solutions of Eq. (3.6a) in vacuum are 0, ∆m 2 21 . From the discussion in the previous Subsection we know that the first one corresponds tom 2 0 , which is bound by ∆m 2 21 when a > 0, whereas the second one,m 2 0 , is bound by ∆m 2 21 when a < 0. The appropriate expansion for ∆m 2 ± in bothm 2 0 ≤ ∆m 2 21 ∆m 2 31 and |a| ∆m 2 The same expressions apply to antineutrinos changing a → −a,m 2 0 →m 2 0 , and the dependence on the Hierarchy is implicit in sign(∆m 2 31 ), that accounts for the interchange of the expressions ∆m 2 ± | NH ↔ ∆m 2 ∓ | IH . The approximation ofm 2 0 andm 2 0 , on the other hand, comes from neglecting ∆m 2 31independent terms in Eq. (3.6a), where the −(+) sign corresponds tom 2 0 (m 2 0 ). In order to compare their behavior above and below ∆m 2 21 , one can further expand Eq. (3.8) in the two regions adequate when looking at the CPT-invariant (vacuum) limit or the T-invariant limit, respectively. Notice that bothm 2 0 andm 2 0 converge to the same asymptotic limit in |a| above ∆m 2 21 .
The eigenstates in Eq. (3.5) up to leading order, together with the eigenvalues in Eqs. (3.7), reduce to the simple expressions valid for both hierarchies. Antineutrino eigenstates are given by (3.13) These eigenstates determine the columns of the V mixing matrix between matter and vacuum mass eigenstates, which allows us to writeŨ = U V as Since matter effects do not depend on all elements of U PMNS but only U ei , the above expressions are simpler in the α = e case. In particular, notice that both vanish if ∆m 2 21 = 0 for all a. This fact originates in the transmutation [20] of masses in vacuum to mixings in matter, leading to the absence of genuine CP violation in matter if ∆m 2 21 = 0, even though there are three non-degenerate neutrino masses.
These expressions reproduce the right vacuum limit, as seen by developing |a| A surprising result, however, appears when assuming ∆m 2 21 |a|, showing that lim ! This is strongly illustrated in the case lim e2 = 0 ∀a. The reason behind this subtlety is the following.
Setting ∆m 2 21 = 0 in vacuum means that ν 1 and ν 2 are degenerate. Therefore, any two independent linear combinations of them can be chosen as basis states, which in the language of the standard parametrization would mean that θ 12 is nonphysical. Adding the matter potential to this system breaks the degeneracy: the arbitrariness in θ 12 is lost in favor of the eigenstates of the perturbation. Since the matter term in the neutrino Hamiltonian adds a > 0 to the e-flavor component, this fact results inν 1 andν 2 such thatν 2 is mainly ν e , forcing theŨ e1 = 0 we obtained. The change of sign in a for the antineutrino case forces analogouslyν 1 to be mainlyν e , explaining the limitŨ e2 = 0.
This behavior shows that the vacuum connection should be analyzed in the regime where |a| ∆m 2 21 ∆m 2 31 . The definite a-parity of the two components of the CP asymmetry defined in the previous Section forces the leading-order term in A T αβ to be independent of a, whereas A CPT αβ is linear. To provide a precise description of A CPT αβ in this region, we keep |U e3 | 2 terms in the leading order, as well as all linear terms in a/∆m 2 21 and a/∆m 2 31 in both the mass squared differences, 18) and the mixings,Ũ The CP asymmetry components A CPT µe and A T µe computed using these expressions are represented in Fig. 2, compared with the exact results, for both hierarchies at fixed E and L as functions of the matter potential. The analytic approximations for constant A T µe and linear A CPT µe work well at low matter densities, as they should, but their range of validity is much larger than expected. For the values used in the Figure, the point a = ∆m 2 21 corresponds to ρ = 0.44ρ E , so the previous expansions should only work for ρ 0.44ρ E . The fact that they work reasonably well even above ρ E hints that higher-order corrections are dominated by (a/∆m 2 31 ) 2 . This surprising feature stems from the fact that corrections (a/∆m 2 21 ) 2 are inoperative in the region |a| ∆m 2 21 ∆m 2 31 for the A CPT µe and A T µe observables. This behavior is explained by peculiar dependence on the mixings and masses of the oscillation probabilities, as can be understood from the matter-vacuum invariants we will exploit in the following Sections for both ImJ ij αβ (Section 4) and ReJ ij αβ (Section 5). As will be discussed, they lead to dependencies in the oscillation probabilities in Eq. (2.2) on the phases associated to the small quatities a and ∆m 2 21 of the form 1 ∆ sin ∆, which cancel out if both of them are small, independently of whether |a| ∆m 2 21 or ∆m 2 21 |a|. This cancellation will happen as long as ∆ = L 4E 1, for = a, ∆m 2 21 . This peculiar dependence in the oscillation probabilities is responsible for the restoration of the commutability of the limits a → 0 and ∆m 2 21 → 0 at this level, even though they do not commute at the mixings level.

Actual experiments: fixed L in the Earth mantle and variable E
In the previous Subsection we discussed the way to obtain analytic approximated expressions for neutrino oscillations in matter that reproduce the right vacuum limit, i.e. the limit when the matter parameter a → 0 at fixed energy due to the matter density going to zero. In the following we consider the constant value of the matter density in the Earth mantle [21], and discuss dependencies in a as dependencies in the neutrino energy in ν µ → ν e transitions. In fact, the actual best-fit value [19] for ∆m 2 21 shows that the relation between a and ∆m 2 21 is given by |a| ≈ 3(E/GeV)∆m 2 21 , so we can use Eqs. (3.14) from the previous Subsection expanding up to second order in ∆m 2 21 /a, with errors only ∼ 3% around 1 GeV.
As in Eq. (2.2), all observable quantities can be written in terms of the rephasinginvariant mixingsJ ij αβ . SinceŨ e1 in Eq. (3.15) is a first order quantity, as we discussed, and expanding up to second order also in |U e3 | 1, which is of the same size than ∆m 2 21 /a, we find that allJ ij eα can be calculated at second order in these two quantities using our first-orderŨ αi in Eqs. (3.17), However, as discussed in the previous Subsection, the definite odd a-parity of the CPT component of the CP asymmetry implies that linear terms in a/∆m 2 31 are relevant to describe A CPT αβ , so we must keep them as well. These linear terms can be easily calculated setting ∆m 2 21 → 0 in the eigenstates in Eq. (3.5). Analogously, we obtain linear corrections in ∆m 2 21 /∆m 2 31 to the previousJ ij αβ setting a → 0 in the eigenstates. The resulting rephasing-invariant mixings, written in the standard parametrization for α = µ, which is the relevant transition for accelerator experiments, arẽ . Notice that these are the same results found in Ref. [15] after further expanding in |a| ∆m 2 31 , as expected. Since ∆m 2 31 ≈ 33∆m 2 21 , it turns out that ∆m 2 31 ≈ 11|a|/(E/GeV), so expanding in |a| ∆m 2 31 around the E ∼ GeV region is as reasonable as expanding in |U e3 | 1. AllJ ij µe are already second order in ∆m 2 21 and |U e3 |, so we can neglect them in the oscillation arguments,  This value is not particularly small at long baselines, but we remind the reader that both A CPT µe and A T µe have definite parity in a, the first one being odd and the second one even, as we proved in Section 2. This means that corrections to the leading order in each component of the CP asymmetry will be quadratic in a, and so we can also expand up to leading order.
In summary, the expansion quantities used are the phase A and, up to second order,  Taking into account the rephasing-invariant mixings (3.21), with the symmetry prop-ertyJ ij µe =J ji eµ , and the mass differences in matter (3.22), we find where S ≡ c 2 13 s 2 13 s 2 23 , J r ≡ c 12 c 2 13 c 23 s 12 s 13 s 23 , A ≡ aL 4E ∝ L and the two ∆ ij ≡ ∆m 2 ij L 4E ∝ L/E. From these expressions, which are precise enough to provide understanding of the physics behind these observables, we find that A T µe in matter is well described by its vacuum value. Since ∆ 21 is small, this means that A T µe oscillates as 1 E sin 2 ∆ 31 . A CPT µe , which vanishes when a → 0, is very well described by its leading (first) order in a. Fig. 3, which makes clear that, even if the value of the asymmetries in the maxima are a bit off, their position and the general behavior are well reproduced. Therefore, Eqs.

A closer look at the genuine CPV component
The last term in Eq. (2.1) indicates that the Hamiltonian of our problem in the flavor basis is proportional to the hermitian mass matrix squared in matter In such a basis, the necessary and sufficient condition for CP invariance is [22] Im[H eµ H µτ H τ e ] = 0 . where J is the rephasing-invariant CPV quantity in vacuum [24], J = c 12 c 2 13 c 23 s 12 s 13 s 23 sin δ. The proportionality ofJ andJ to ∆m 2 21 explains the absence of genuine CP violation in matter in the limit of vanishing ∆m 2 21 , even in the presence of three non-degenerate neutrinos and antineutrinos in matter. The vanishing ofJ andJ in this limit comes from the transmutation of masses in vacuum to mixings in matter calculated in Section 3.2, leading toŨ e1 = 0 andŨ e2 = 0. To leading order in ∆m 2 21 , the non-vanishingJ andJ differ by linear terms in the matter potential a present in the neutrino masses in matter.
Using the analytic perturbation expansion of Section 3 for the connection between quantities in matter and in vacuum, we can writẽ Notice that the proportionality factors in Eqs. (4.4) are neutrino energy dependent through a, as shown in Fig. 4. The behavior at low/high energies can be easily understood using the expansions at leading order ofm 2 0 andm 2 0 in Eqs. (3.9,3.10). Indeed, at low energies both of them decrease roughly as 1/a, and changing the sign(∆m 2 31 ) is equivalent to changing the sign(a), which explains why the two plots in Fig. 4 seem to be symmetrical.
The small value of theJ /J ratio at high energies does not necessarily mean that genuine CP violation is unobservable at these energies, since the genuine CPV component of the CP asymmetry contains this energy-dependent factor together with the matterdependent oscillation function -odd in L-that depends on both energy and baseline. The effects of the baseline are shown in Fig. 5, comparing the whole A T µe as function of the energy for T2HK L = 295 km and DUNE L = 1300 km, where it is seen that the oscillation amplitude of each of them (fixed L) decreases as 1/E, as expected, and a higher baseline (at fixed E) enhances the values of A T µe . This behavior is understood with the perturbation expansion in |U e3 | 2 1 in the energy regime between the two MSW resonances, ∆m 2 21 |a| ∆m 2 31 , that we performed in the previous Section. The 1/E dependence inJ is changed by the approximated oscillating functions into L/E, producing genuine CPV components of the same size at the spectrum peak of both experiments. In fact, the matter effects inJ and oscillating phases just compensate to generate in this approximation a genuine CPV asymmetry equal to that in vacuum, i.e. Eq. (3.25b). As such, it is odd in L/E, independent of a and the Hierarchy, and proportional to sin δ.

Neutrino mass ordering discrimination
Last Section has demonstrated that the genuine A T µe component of the CP asymmetry in matter is, to a good approximation for energies -as planned in accelerator facilitiesbetween the two resonances ∆m 2 21 |a| ∆m 2 31 , given by the vacuum CP asymmetry. Its information content is then crucial to identify experimental signatures of genuine CPV. On the other hand, it has nothing to say about the neutrino mass ordering: it is invariant under the change of sign in ∆m 2 31 . This simple change of sign, without changing the absolute value ∆m 2 31 , is in fact the only effect of changing the hierarchy under the approximations leading to Eqs. (3.25).
This Section discusses the information on the neutrino mass ordering contained in A CPT µe , which is even in L and sin δ and odd in a. Propagation in matter is needed to generate effects of the change of hierarchy and our A CPT µe is able to separate out this information, going beyond studies of its influence on transition probabilities [25].
There is no simple matter-vacuum relation such as Eq. (4.3) to easily write ReJ ij αβ as function of the vacuum ReJ ij αβ -the most compact result following this idea is [26,27] ∆m 2 12 ∆m 2 23 ∆m 2 31 ∆m 2 ij ReJ ij αβ = K ij αβ + ∆m 2 12 ∆m 2 23 ∆m 2 31 ∆m 2 ij ReJ ij αβ , where all K ij αβ vanish in vacuum. This relation explains the dependence of all L-even terms in the oscillation probabilities in each of the∆ ij phases as 1 ij , which is the reason why the vacuum limit a → 0 is restored in these observables even after taking ∆m 2 21 |a|, as discussed in Section 3. However, the K ij αβ are complicated functions of the vacuum quantities, and do not provide a clear insight into the behavior of A CPT αβ , so we will use Eq. (3.25a) instead.
In general, this matter-induced component of the CP asymmetry has no definite transformation properties under the change of sign in ∆m 2 31 . Under the approximations made in Section 3, there are two distinct terms in A CPT µe , a first one A CPT − which is an odd function of ∆m 2 31 and a second one A CPT + which is an even function of ∆m 2 31 , As seen, the information content in A CPT µe on the neutrino mass hierarchy is due to A CPT − , its dominant zeroth-order term in ∆m 2 21 , independent of the phase δ. In the limit ∆m 2 21 → 0, our results from Eq. (3.22) in Section 3.3 show that the mass spectrum in matter changes under a change of hierarchy from neutrinos to antineutrinos as whereas theJ ij αβ do not change sign, so all L-even terms in the oscillation probabilitieswhich are blind to the sign change in Eq. (5.3)-are simply interchanged between neutrinos and antineutrinos. As the CP asymmetry is a difference between neutrino and antineutrino oscillation probabilities, we discover that A CPT µe is only changing its sign under a change of hierarchy in the vanishing limit of ∆m 2 21 . The A CPT + term in Eqs. (5.2) is appreciable only at low energies, needing a non-vanishing ∆m 2 21 and then sensitive to the δ phase as a CP conserving cos δ factor. In Fig. 6 we represent these two components of A CPT µe as function of E for the baselines of T2HK and DUNE for Normal and Inverted Hierarchies. To test the neutrino mass ordering from A CPT µe , we find that imposing condition A CPT − > A CPT + in the non-oscillating (high energy) region leads to E > 1.1 E 1 st node . For these energies above the first node of the vacuum T2HK DUNE oscillation probability, the whole effect of the change of sign in ∆m 2 31 is an almost odd A CPT µe . In addition, the A CPT − term in A CPT µe dominates the whole CP asymmetry at long baselines, as seen in Fig. 3, so the measurement of the sign of A CP µe at these energies fixes the hierarchy.

Signatures of the peculiar energy dependencies
In this Section we identify those aspects of the energy distribution of the CP asymmetry that can offer an experimental signature for the separation of its genuine and matter-induced components. With experiments in which the fingerprint of the baseline dependence, L-odd and L-even functions, cannot be used, the peculiar patterns of the energy distribution provide precious information. The general trend of this dependence for L = 1300 km is given in Fig. 3, showing the appearance of oscillations in the low energy region of the spectrum with different behavior for the two components A T µe and A CPT µe , where nodes and extremal values are at different energies.
However, this rich structure is lost when the baseline is decreased to L = 295 km and a threshold energy of 300 MeV is imposed. The emerging picture under these conditions is given in Fig. 7 and the main conclusion is the relative suppression of A CPT µe with respect to A T µe , due to its proportionality to A ∝ L. In addition, this small A CPT µe is mainly the δ-independent A CPT 2), which appear slightly above the oscillation maxima ∆ osc max = (2n + 1) π 2 . This is a first fortunate fact, implying that the experimental configurations with maximal A T µe are close to those with highest statistics. A perturbative expansion of cot ∆ around ∆ osc max leads to the approximate solutions which show that the interesting (see below) second and higher maxima in A T µe are within a 3% interval above the oscillation maxima. and are graphically depicted in Fig. 8 too. As seen, they appear slightly below the oscillation maxima in f (∆) starting from the second one, with approximate values which almost coincide with the maxima ∆ T max in Eq. (6.3) of A T µe . Not only that: these zeros ∆ CPT 0 of A CPT µe are again near the oscillation maxima ∆ osc max = (2n+1) π 2 , so we conclude that there are "magic energies" at these (6.7) phase values, within a 5% interval below the corresponding oscillation maximum, in which A CPT µe vanishes and A T µe is close to a maximum. These magic points have additional bonuses: i) the zero of A CPT µe is independent of cos δ, providing no ambiguity in its position; ii) these are simple zeros, in such a way that the sign of A CPT µe is changing around them; iii) although A T µe is not exactly at its maximum value when A CPT µe = 0, the leading order deviations from ∆ osc max we calculated show that its value is above 90% A T µe max . A look into the derivative of f CPT (∆) shows that the sign-change of A CPT µe around these zeros is such that A CPT µe is always decreasing (increasing) around the relevant δ-independent zeros for Normal (Inverted) Hierarchy, and opposite around δ-dependent zeros.
Taking into account the dependence in L/E of these remarkable values of the phases, we give in Table 2 the relevant energies around the second oscillation maximum for both the baselines of the T2HK and DUNE experiments. The precise position of this energy, which is slightly above the second oscillation maximum, is proportional to L ∆m 2 31 as E = 0.92 GeV L 1300 km ∆m 2 31 2.5 × 10 −3 eV 2 , (6.8) which explains the absence of this rich oscillatory structure in Fig. 7: at the short baseline of T2HK, all interesting points lie below the threshold energy of 300 MeV. in Eq. (6.7), the second vacuum oscillation maximum and the second maximum ∆ T max in Eq. (6.3), corresponding to the highest-energy zero of A CPT µe independent of δ. For each of these three points, we show the value of the oscillation phase, which is independent of any experimental parameter; the L/E, whose value depends linearly on the inverse of ∆m 2 31 ; and the particular energy associated to this L/E for T2HK L = 295 km and DUNE L = 1300 km. Notice that these three values of the phase ∆ 31 correspond to the position of the green/black/blue dashed lines within the red ellipse in Fig. 8 This magic configuration around the second oscillation maximum is well apparent in the results presented in Fig. 9 for 1 L = 1300 km. One can observe that the uninteresting (increasing/decreasing for NH/IH) zeros in A CPT µe are strongly dependent on cos δ, and their position when cos δ = 0 is that of the δ-independent zeros in A T µe . As understood from the previous discussion, we have identified the most relevant δ-independent zeros of A CPT µe , decreasing/increasing for NH/IH, correlated to near maximal A T µe proportional to sin δ. Since A CPT µe is changing its sign around the zero, integrating statistics in a bin around this point would still result in a vanishing matter term in the experimental CP asymmetry, providing a direct test of CP violation in the lepton sector as clean as in vacuum. This result in the energy distribution of the experimental CP asymmetry provides a positive response to our search of signatures able to separate out the genuine and matter-induced components. 350 MeV, with the same energy-dependence for both components of the CP asymmetry, but a relatively smaller A CPT µe due to its proportionality to A ∝ L.

Conclusions
A direct evidence of genuine CP violation means the measurement of an observable odd under the symmetry. The CP asymmetry for long baseline neutrino oscillation experiments suffers from fake effects induced by the interaction with matter. This matter effect is, however, welcome as a source of information for the ordering of the neutrino mass spectrum. Based on the different transformation properties under T and CPT of the genuine and matter-induced CP violation we have proved a Disentanglement Theorem for these two components. In order to raise this disentanglement to a phenomenological separation of the two components we have identified in this work their peculiar signatures from a detailed study in terms of the experimentally accessible variables.
For a precise-enough understanding of the problem, we have developed a new analytical perturbative expansion in both ∆m 2 21 , |a| ∆m 2 31 without any assumption between ∆m 2 21 and a, which we use to analyze each of the disentangled components of the CP asymmetry, A CP αβ = A CPT αβ + A T αβ , the first one (L-even) accounting for matter effects, the second one (L-odd) being genuine.
The two components of the CP violation asymmetry for the ν µ → ν e transition are shown in Fig. 2 as function of the interaction parameter a. They fulfill all the T and CPT symmetry requirements proved in Section 2: the CPT-odd component A CPT αβ is an odd function of a and vanishes linearly in the limit a → 0 for any value of the CP phase δ, as well as being an even function of sin δ due to T-invariance. The T-odd component A T αβ is an odd function of sin δ that vanishes, even in matter, if there is no genuine CP violation, as well as being even in a due to CPT-invariance, which means that its value is that of the CP asymmetry in vacuum up to small quadratic corrections O(a/∆m 2 31 ) 2 . By analyzing the vacuum limit a → 0 both above and below the T-invariant limit ∆m 2 21 → 0, some intricacies for the mixings in matter appear. If one assumes |a| ∆m 2 21 , the vacuum limit of the mixing matrix in matter will be the free PMNS matrix. On the other hand, ∆m 2 21 |a| will force U e1 = 0 in the vacuum limit. This different behavior stems from the fact that setting ∆m 2 21 = 0 in the vacuum Hamiltonian leads to degenerate ν 1 , ν 2 mass eigenstates. The two limits mentioned above correspond to breaking this degeneracy in favor of ∆m 2 21 or a, respectively, projecting onto different bases in the 12 subspace. At the level of oscillation probabilities and asymmetries, the matter-vacuum invariant relations studied in Sections 4 and 5, which involve both mixings and masses, show that the dependence on the phases associated to the small quantities = a, ∆m 2 21 are of the form 1 sin L 4E , which cancel out if both of them are small, independently of whether |a| ∆m 2 21 or ∆m 2 21 |a|. Therefore, the commutability of the two limits a → 0 and ∆m 2 21 → 0 is restored for the final observables. We have searched for experimental signatures in the ν µ → ν e oscillation channel assuming ∆m 2 21 |a|, valid for actual accelerator neutrino energies through the Earth mantle. The definite a-parity of each component of the CP asymmetry allows us to expand in |a| ∆m 2 31 to leading (linear in A CPT µe , constant in A T µe ) order, since corrections are quadratic. For baselines and energies such that both = ∆m 2 21 , a lead to L/4E 1, and taking |U e3 | where S ≡ c 2 13 s 2 13 s 2 23 , J r ≡ c 12 c 2 13 c 23 s 12 s 13 s 23 , A ≡ aL 4E ∝ L and the two ∆ ij ≡ ∆m 2 ij L 4E ∝ L/E. Equipped with such precise-enough analytical results, we have performed a detailed study of the different features of these quantities, focusing especially on signatures of genuine CP violation and hierarchy effects.
Since A T µe is blind to sign(∆m 2 31 ), a determination of the neutrino mass ordering must come from regions where the hierarchy-odd (and δ-independent) term of A CPT µe dominates, which can only happen at long baselines due to the proportionality of A CPT µe to A ∝ L. Our analysis at DUNE L = 1300 km shows that this is the case for energies above the first node of the vacuum oscillation, where the sign of the experimental A CP µe determines the hierarchy. The strategy towards the measurement of genuine CP violation depends on the baseline. At medium baselines such as T2HK L = 295 km, the CPT-odd component A CPT µe is small and, for energies above the first oscillation node, dominated by its δ-independent term. Therefore, it can be theoretically subtracted from the experimental A CP µe , if the hierarchy is previously known, in order to obtain the genuine component A T µe . At long baselines, both A CPT µe and A T µe are of the same order, so A CP µe will directly test genuine CP violation only when the CPT-odd component vanishes. We find a family of simple zeros of A CPT µe with decreasing/increasing slope for Normal/Inverted Hierarchy corresponding to the solutions of tan ∆ 31 = ∆ 31 . These zeros are close to the second and higher vacuum oscillation maxima sin 2 ∆ 31 = 1, implying that their position is independent of δ and corresponds to a nearly maximal A T µe proportional to sin δ. The main conclusion is thus that the magic energy around the second oscillation maximum is the ideal choice to find a direct evidence of genuine CP violation in the lepton sector. This vanishing of A CPT