Leptonic CP Violation Phases, Quark-Lepton Similarity and Seesaw Mechanism

We explore generic features of the leptonic CP violation in the framework of the seesaw type I mechanism with similarity of the Dirac lepton and quarks mass matrices $m_D$. For this, we elaborate on the standard parametrization conditions which allow to simultaneously obtain the Dirac and Majorana phases. If the only origin of CP violation is the left-handed (LH) transformation which diagonalizes $m_D$ (similar to quarks), the leptonic CP violation is suppressed and the Dirac phase is close to $\pi$ or to $0$ with $\sin \delta_{CP} \approx (\sin \theta_{13}^q /\sin \theta_{13}) \cos \theta_{23} \sin \delta_q \sim \lambda^2 \sin \delta_q$. Here $\lambda \sim \theta_C$, is the Cabibbo mixing angle, and $\theta_{13}^q$ and $\theta_{13}$ are the 1-3 mixing angles of quarks and leptons respectively. The Majorana phases $\beta_1$ and $\beta_2$ are suppressed as $\lambda^3\sin\delta_q$. For Majorana neutrinos implied by seesaw, the right-handed (RH) transformations are important. We explore the simplest extension inspired by Left-Right (L-R) symmetry with small CKM-type CP violation. In this case, seesaw enhancement of the CP violation occurs due to strong hierarchy of the eigenvalues of $m_D$ leading to $\delta_{CP} \sim 1$. The enhancement is absent under the phase factorization conditions which require certain relations between parameters of the Majorana mass matrix of RH neutrinos.


I. INTRODUCTION
Establishing the leptonic CP violation is one of the major experimental frontiers in neutrino physics. The Dirac and Majorana CP phases are among the few yet unknown parameters for which a prediction may still be made. So, we need to understand what particular values or intervals of the CP phases will imply for fundamental theory.
Indeed, there are numerous predictions of the phases which are based on broad spectrum of ideas, approaches, and models [1]. Some approaches that have been employed are (i) Neutrino and charged lepton mass matrices with certain properties such as -textures [2], symmetries, and symmetry violations, e.g., µ − τ reflection or generalized symmetry [3]; (ii) Models with discrete flavor symmetries [4], which can realize geometric origins of the phases, the CP violation due to group structure or complex Clebsch-Gordan coefficients [5], or connect the phases and the mixing angles [6], etc.; (iii) Grand unification with seesaw type I and type II [7] 1 ; (vi) Radiative generation of CP violation [9]; (vii) Relating the leptonic CP phase to other physics, e.g., a solution to the strong CP problem wherein δ CP = 0 or π is predicted [10]. Many efforts have been devoted to obtain maximal CP violation, i.e., δ CP = π/2 [11], although other values essentially from 0 to π have also been found.
Can we really predict the leptonic CP phase, given that even in the quark sector, where all parameters are known, there is no unique and convincing explanation of the value of CP phase?
Moreover, in the lepton sector the situation is expected to be more complicated due to presence of additional structures which are responsible for the smallness of neutrino masses. Can the lepton and quark CP phases be equal, or connected in some way? To address these questions it is instructive to represent the lepton mixing matrix in the form where U L is somehow related to the quark CKM-mixing matrix and U X reflects new physics responsible for smallness of neutrino mass and large mixing angles [12], [13], [14], [15], [16], [17], [18]. Here U L and U X can follow from diagonalization of mass matrices of the charged leptons, U L = U † l , and neutrinos, U X = U ν , respectively. Origins of CP violation can be in U l [19] and/or U ν . The assumption U l ∼ V CKM corresponds to the Quark-Lepton Complementarity [13], so that U P M N S = V † CKM U X . This possibility has been explored for U X = U BM (bimaximal mixing matrix) [14] and U X = U T BM (tribimaximal mixing matrix) referred as Cabibbo-TBM [20]. In these cases the origin of CP could be in V CKM or in the diagonal phase matrix attached to U X . In [14] the "correlation matrix" U X has been taken in the form U X = P (φ l )U BM Q(φ i ), where P (φ l ) and Q(φ i ) are diagonal phase matrices. It was noticed that if φ l = 0, the Jarlskog invariant is very small [14], [17], [15]: J P M N S = sin θ q 13 sin δ CKM , i.e., too small to be measured in future experiments.
The ansatz (1) can be naturally realized in the seesaw type I mechanism [21] which is the simplest and the most natural explanation of smallness of neutrino masses as well as large lepton mixing [22]. It is simplest because only RH neutrinos are added to the theory. It is natural in the sense that it allows to explain smallness of neutrino mass and the substantial difference between lepton mixing and quark mixing, while at the same time maximally implementing the quark-lepton similarity. The latter, in turn, is expected, e.g., in Grand unified theories. Seesaw type I mechanism with similar Dirac mass matrices for neutrinos and quarks defines the canonical seesaw mechanism.
In this paper we consider the leptonic CP phases that can arise from this canonical seesaw mechanism, which provides the closest possible connection of the quark and lepton sectors. We will further generalize the relation (1) assuming that U L has similar to V † CKM structure but in general does not coincide with V † CKM . For the matrix U X we will not assume any special structure but restrict it only by the condition that the product (1) reproduces the experimentally observed values of the mixing angles. We will find the phases in the standard parametrization of the PMNS matrix. For this we formulate and use the standard parametrization conditions which allow us to obtain simultaneously both the Dirac and Majorana CP phases.
We find that if the only source of CP violation is the Kobayashi-Maskawa (KM) -type phase in U L , it leads to a small δ CP . In the seesaw mechanism due to the Majorana nature of neutrinos the CP violation in the RH sector become relevant for the PMNS CP phases. That includes the phases in the RH rotation U R that diagonalizes the Dirac mass matrix m D as well as in the Majorana mass matrix of RH neutrinos, M R . We find that generically the seesaw mechanism enhances CP violation that appears in U R , so that δ CP = O(1). Such an enhancement is absent and the CP phases are small (or close to π) if parameters of M R satisfy certain (phase factorization) relations. We find relations between the Dirac and Majorana phases which can be used to test these scenarios. An observation of (large) leptonic CP violation in oscillation experiments and/or neutrinoless double beta decay would therefore be a signature that there is a new source of CP violation, beyond the leptonic analogue of KM-phase and coming from the RH sector, or that neutrino masses do not arise from a canonical seesaw.
We will argue that specific values of the leptonic CP phases are possible since the contribution of the Kobayashi-Maskawa type phase turns out to be suppressed or be close to δ CP ∼ π and the main contribution comes from the RH sector which can obey specific symmetries.
The paper is organized as follows: In Sec. II we present the formalism for calculating the CP phases. In Sec. III we derive the expressions for the CP phases when the only source of CP violation is a KM-like phase in the left-handed transformation that diagonalizes the Dirac mass matrix. The general case with CP violation in the RH sector is considered in Sec. IV. In Sec. V we explore a specific case of CP violation in the RH sector, which corresponds approximately to a L-R symmetry of the theory. In Sec. V B we consider special conditions where the resulting CP phase is still small. We then show that in general the seesaw enhancement of CP violation occurs which leads to δ CP ∼ O(1), even if CP violation in U R is of KM-type. We present some phenomenological consequences in Sec. VI and conclude in Sec. VII.

II. SEESAW TYPE I, CP VIOLATION, AND STANDARD PARAMETRIZATION
A. U X matrix in seesaw type I We introduce the Dirac matrix, m D , in the flavor basis and Majorana mass matrix, M R , according to the mass terms of the Lagrangian Integrating out the RH neutrinos we obtain L mass = −ν L m νν T L + h.c., where the matrix of light neutrinos in the flavor basis equals The Dirac mass matrix can be represented in the flavor basis as where U L and U R are the transformations The light neutrino mass matrix in the flavor basis is where are the light neutrino flavor states and m diag ν = diag(m 1 , m 2 , m 3 ) is the diagonal matrix of real and positive neutrino mass eigenvalues.
Inserting (3) and (2) into the seesaw expression (2) we obtain The relationship in (4) can be re-expressed as where It is the structure of the matrix M X that produces the difference in masses and mixing of quarks and leptons.
Since U P M N S and U L are unitary matrices, the eigenvalues on both sides of (5) should coincide.
Therefore M X can be rewritten as, and the mixing matrix U X is obtained by diagonalization of (6). From (5) and (7) we obtain which can be satisfied if and only if the matrix U L U X coincides with U P M N S up to a diagonal matrix where n, m, k are integers, which is the symmetry transformation of a generic diagonal Majorana mass matrix. Therefore, U P M N S = U L U X D. In what follows, we will absorb D into the phase matrix of U X .
Thus, within the seesaw paradigm we arrive at the relationship (1) with U X being the matrix which diagonalizes M X (6). Notice that U L would be the lepton mixing matrix, if M X is diagonal or there are no Majorana mass terms. Whereas, U X encodes information about the eigenstates of the Dirac and Majorana mass matrices, as well as about mismatch of the ν R transformations which diagonalize m D and M R . The matrix M R can be written as In what follows we will explore the relationship expressed in (1) to derive predictions for the physical CP violating phases in U P M N S in terms of the relevant parameters of the RH sector and U L . Results of this section are general and can be applied to any mechanism which reproduces (1).

B. Standard parametrization conditions
Motivated by its widespread use, we will consider the CP phases that appear in the standard parametrization of the PMNS matrix U std P M N S [23]: where Γ δ ≡ diag(1, 1, e iδ CP ) and δ CP is the Dirac CP violating phase. Usually to find the CP phase one computes the Jarlskog invariant of U li , and uses the mixing parameters in the standard parametrization. We find that a more instructive and transparent way to find CP phases is to compute the mixing matrix directly and reduce it to the standard parametrization form by rephasing.
In general, the PMNS matrix can be written as Here D(φ) ≡ diag(e iφe , e iφµ , e iφτ ) is the matrix of phases which can be eventually absorbed into the wave functions of charged leptons, and is the matrix of the Majorana phases 2 . We will use the standard parametrization also for the matrices on the RH side of (1): where D(ψ) ≡ diag(e iψe , e iψµ , e iψτ ), etc., U std L and U std X are the matrices in the standard parametrizations which contain a single CP phase each. Then the product of matrices in Eq. (1) equals Clearly, introduction of the two separate matrices D(χ) and D(y) is irrelevant for the light neutrino mass matrix but it does matter for the structure of the RH sector.
Inserting expressions (8) and (10) into (1), and multiplying it by D(φ) * and Γ * M from the left and the right sides respectively, we obtain The phases should be selected in such a way that they bring the RHS of (11) to the standard parametrization form.
The conditions, that the matrix on the RH side of (11) is in the standard parametrization, are given by the following 5 equations These conditions which we call the Standard Parametrization (SP) conditions fix 5 phases γ e , γ µ , γ τ and η 1 , η 2 . Notice that conditions (12) determine the phases of the mixing matrix up to a rephasing: 2 An alternate parametrization of ΓM is diag(1, e i α 21 2 , e i α 31 2 ), and we can recover these Majorana phases, α21 = 2(β2 − β1) and α31 = −2β1, by an overall rephasing of UP M N S from the right side by e −iβ 1 .
. This allows, in particular, to eliminate the phase of the 1-3 element. It is the condition (13) that fixes Φ and removes the ambiguity.
Once the SP-conditions are satisfied the phase of the 1-3 element of the obtained matrix will and the Majorana phases equal C. Quark-lepton similarity and general expression for the Dirac CP phase The main assumption in this paper is that the Dirac mass matrix of neutrinos has similar structure to that of quarks: m D ∼ m u or ∼ m d , as can be motivated by the Grand unification or common flavor symmetry with the same charge assignments. Consequently, the mixing in leptonic sector which follows from the Dirac matrices is similar to that in the quark sector: Essentially, we will only assume that mixing matrix U L has a hierarchical structure of elements, as the CKM matrix, i.e., V ud V cd V td , etc., and express the smallness of these quantities by referring to the Wolfenstein parameter λ. We make no use on any other details of this similarity.
In particular, the parameter λ does not have to be exactly the same as in the quark sector.
According to (14), we will suggestively denote the elements (U std L ) li by the elements of V * ul , where the charged lepton index l = (e, µ, τ ) here corresponds to the down quarks (d, s, b) in V CKM and the neutrino index i = (1, 2, 3) corresponds to up-quarks (u, c, t). Denoting the elements of the matrix U std X by X li we obtain for the matrix elements on the RHS of (11) where l = e, µ, τ and j = 1, 2, 3. We remind that in V we replace e → d, µ → s, τ → b.
Introducing ξ lj -the phases of the expressions in the brackets of (15), we can rewrite the elements of the PMNS matrix (15) as The phases γ l and η j should be determined from the conditions of the standard parametrization.
The elements V ud , V us , V cb , and V tb are real. The elements V cd = −|V cd |e iφ cd and V ts = −|V ts |e iφts have an overall negative sign, so that the phases φ cd and φ ts are of order λ 4 and λ 2 , respectively.
The other phases are defined as usual, V ub = |V ub |e iφ ub , V td = |V td |e iφ td , and V cs = |V cs |e iφcs . The phases φ ub and φ td are O(1), while φ cs is of order λ 6 and can be neglected. All these phases are known in terms of the quark CP violating phase δ q .
Consider the element U e3 which contains the Dirac CP phase: Modulus and argument of U e3 determine θ 13 and δ CP , respectively. Since |U e3 | = s 13 , from (16) we obtain Recall that the phases α l and δ X parametrize the CP violation which originates from the RH sector. The phase γ e is fixed by the standard parametrization conditions: γ e = γ e (α l , δ X , δ q ). The phase η 3 = 0, as z 3 and β 3 can be chosen to be zero, and the above expressions do not explicitly depend on η. The important feature of the result (17) is that contribution of α τ to δ CP is always suppressed by V td /s 13 ∼ λ 2 , δ X is suppressed by X e3 , whereas the contributions of α e and α µ are unsuppressed.

III. A CKM-TYPE ORIGIN OF THE LEPTONIC CP VIOLATION
Suppose that the only source of CP violation is U std L (δ L ) ≈ V CKM (δ q ), i.e., the matrix of transformation of the LH neutrino components that diagonalizes m D . This is a direct analogy to the Kobayashi-Maskawa mechanism in the quark sector, as previously considered e.g. in [14]. It corresponds to U X being a real matrix, so that The matrix in front of U std L (δ L ) can be absorbed into the phases of the charged leptons. This can be thought of as the minimal CP violation that we expect for leptons if their Dirac masses are similar to quarks. In the context of the seesaw mechanism such a situation can be realized if both U R and M R are real, and the diagonal phase matrices vanish or cancel with each other. The cancellation can be due certain symmetries for RH neutrino components. In this case according to (16) and (17): and where φ cd has been neglected. The absolute value of U e3 according to (18) equals Thus, the CP phase is determined essentially by γ e which we find (see Appendix A for details) by imposing the SP conditions (12,13) to be where we used the result (40) and |V td | sin φ td ≡ Im V td = s q 13 sin δ q . Since s q 13 = λ 3 the expression (20) shows that sin γ e = O(λ 3 ).
Let us express the elements X li in terms of the elements of U std P M N S . Using the relations (15), at O(1) we have X lj ≈ |U lj |/(V CKM ) ll , while X e3 turns out to be of the order λ: X e3 = ±s 13 /V ud + s 23 |V cd |/(V ud |V cs |). With these expressions for X li and γ e , we obtain from (19) Similarly according to (39) and (41), the Majorana phases are The following comments are in order. (21) is of the order λ 3 /s 13 ∼ λ 2 , that is, suppressed by ∼ λ 2 . This agrees with results obtained previously (e.g., [14], [17]). At leading order (21) can be rewritten as

The main term in
or Im U e3 = −c 23 Im V ub . So, the Dirac CP phase in the leptonic sector is suppressed because the mixing is relatively large, compared to quark mixing.
2. The subleading term in the Dirac CP phase is of the order λ 3 , and it is proportional to deviation of the 2-3 mixing from maximal.
4. The Majorana phases are smaller and suppressed as λ 3 . Numerically one finds that β 1 ≈ 0.01 and β 2 ≈ −0.005. Notice that these are the "induced" phases by the Dirac quark phase δ q and SP conditions. Indeed, the phase δ q appears in a mixing matrix that is not in the standard form, and β i are the phases obtained in rephasing procedure to bring this matrix to the standard form.
5. As we remarked before, the Dirac phase can be obtained from the Jarlskog invariant in the standard parametrization: sin 2θ 13 sin 2θ 13 sin 2θ 13 cos θ 13 sin δ CP .
Using expressions (15) for the elements in the LHS of this equality taken for all zero phases but δ q we obtain in the lowest order Expressions (47) and (48)  Finally, inserting this into LHS of eq. (22) we find sin δ CP = −c 23 (s q 13 /s 13 ) sin δ q which coincides with the lowest order term in eq. (21).
6. The results obtained in this section do not actually depend on mechanism of neutrino mass generation. They are based on a general parametrization of the PMNS matrix (1), with the assumption that U L ∼ V † CKM is the only source of the CP violation and requirement that the product (1) reproduces the observed lepton mixing angles. Although we have motivated this ansatz in the context of seesaw type I, any model that satisfies U L ∼ V † CKM and has no other source of CP violation leads to the same result.

IV. GENERAL CASE OF CP VIOLATION
In general the assumption made in the previous section, that the left transformation is the only source of CP violation, is not valid for Majorana neutrinos implied by seesaw. In the case of Majorana neutrinos, phases of the RH sector become important for PMNS mixing. In particular, the CP phase in the right matrix U R will contribute to δ CP . The CP violation in RH sector doesn't affect the CP violation in the CKM matrix because quarks do not have a Majorana mass term. In this sense, the analogy between the lepton and quark sector cannot be exact even if Dirac matrices are the same -the matrix U R has physical consequences for neutrinos.
Consider the most general possibility, when CP violating parameters exist in both the Dirac and Majorana mass matrices involved in the seesaw. Neglecting terms of the order λ 3 we obtain from (17) sin In the leading order in λ the conditions of standard parametrization (12) give and the 5th condition reads: (Notice that r ≈ −2, because U X is close to being U T BM ). The only solution of this system is the following: η 1 = η 2 ≡ η from (24), then γ µ = −η from (25), and then α µ = η, γ e + α e = −η.
Inserting these expressions into (23) we obtain All three phases δ X , α e , and α µ are free parameters and one can obtain any value of the CP phase. In specific cases, some of these phases can be removed or fixed resulting in a more precise prediction, e.g., if X e3 = 0, we get sin δ CP ≈ − sin α e . For α e = α µ = δ X = 0 we obtain δ CP = 0, in agreement with our consideration in Sec. III at this order.
If α µ = 0 and α e = α τ = 0, we obtain by using the standard parametrization conditions According to this expression δ CP can be of the order 1 if α µ is unsuppressed.
The Majorana phases equal which gives β 1 − β 2 = z 1 − z 2 , where z i are also unknown parameters, which can be fixed once M R is determined. So, in general, all leptonic CP phases are unconstrained and can be large.

A. CP phases in the Left-Right symmetric case
Here we explore the minimal extension of the CKM case that includes effect of the RH sector.
In the spirit of L-R symmetric models we assume that and there is no CP violation in M R in the L-R symmetry basis. So, where now M X is a complex symmetric matrix. The CP violation in U R ∼ U L is very small, being suppressed by λ 3 .
To elucidate the role of CP violation from U R and effect of seesaw we assume that M R has the following form: where V 0 CKM = V CKM (δ q = 0) is the CKM-like matrix with zero value of the CP phase and M T BM ≈ M T BM . The latter ensures that matrix U X is close to U T BM , which leads to the observed PMNS mixing angles. (26) we can represent M X as

Inserting expression (27) into
is the correction matrix that captures the effect of a non-zero CP phase. Indeed, for δ q = 0, K = I the above would provide M X ≈ −M T BM .
Computing explicitly, we find and V td ≈ λ 3 (1−e iδ ) ≡ λ 3 ξ. Let us take m D = m 3D diag(λ m , λ n , 1). We can also include coefficients of order one here, but they will not change final conclusion. Then For m ≥ 4, the 3-1 element of the correction matrix is large, i.e., enhanced as ≥ ξλ −1 . It is this factor, related to the strong hierarchy of the eigenvalues of the Dirac matrix, that can lead to enhancement of the CP violation. Note that the correction in (28) does not depend on the second eigenvalue λ n .
We takeM where a, b, g, h are real coefficients of the order 1. Then using the correction matrix (28) we obtain The only possibility to have M X be an approximate TBM mass matrix is m ≤ 4 and p ≥ 2. That is, the hierarchy of the Dirac mass matrix is strongly restricted by the condition that correct PMNS mixing is reproduced. If the hierarchy of the eigenstates of the Dirac mass matrix is too strong, i.e., m > 4, no solution which gives correct mixing angles exists in the presence of a CP violating phase. At the same time a solution always exists for arbitrarily strong hierarchies if there is no CP phases in U R .
Taking m = 4 and p = 2, we obtain and corrections to all the elements are suppressed by at least λ 2 . Now the problem is to find phases of the matrix U X (y l , δ X , z i ) that diagonalizes (30).

B. Factorization of phases
The phases of U X can be found immediately if the phases are factorized from M X . Under the conditions Arg (aξ + f ) = Arg (bξ + g) = 1 2 Arg (aξ 2 + 2f ξ + h) ≡ φ F , which we will call the phases factorizations conditions, the matrix (30) can be written as where D(φ F ) = (1, 1, e iφ F ) and Here F ≡ |F |e iφ F ≡ ξ + f /a.

The factorization conditions can be satisfied if
Since M 0 X is real and, in general, can be diagonalized by real matrix O, we have U X = D(φ F )O. So that in the notation of (9), y e = y µ = δ X = z = 0 and y τ = φ F . Furthermore, since in the L-R symmetric case D(ξ) is irrelevant, α l = y l and α τ = y τ = φ F . Thus, D(α) = diag(1, 1, e iατ ) and the factorization phase is determined by tan α τ = sin δ q 1 + f /a − cos δ q . Furthermore, If f = −1 and a = 1 we have α τ = −δ q and |F | = 1. More generally, for the interval a = 1 − λ to 1 + λ we obtain that α τ changes from −74 • to −86 • . For f > 0 the interval for the phase α τ is 27 • − 30 • . In both cases α τ differs from φ td ≈ 50 • . Solving the SP conditions, (see Appendix A) we find Thus, in the case of factorization with only α τ = 0, the final value of the CP phase is still small, being suppressed by λ 2 . The reason is that α τ enters the expression for sin δ CP with small factor V td .
The Majorana phases (which appear as by-product of the standard parametrization conditions) equal (see Appendix B) sin β 1 ≈ sin β 2 ≈ −sign{A}2 sin φ td V ts cot 2θ 23 .
For α τ = φ td all three CP phases are equal.
Indeed, from the first and second equalities we have b|F | = −1 and from the last one: b|F | ≈ 1 − bλ ≈ 1. The deviation of M 0 X from the TBM form leads, in particular, to a non-zero 1 -3 mixing: which can be in agreement with data.

C. Seesaw enhancement of CP violation
In general in the absence of factorization the mass matrix M X will generate a non-zero α e , α µ , and δ X , and consequently a large δ CP . Expressions for phases of U X in the three generation case are very complicated and difficult to analyze. Therefore to show effect of enhancement of the CP phases we will consider the two leptonic generations. In the case of a hierarchical neutrino mass spectrum the 2 − 3 block of elements in the mass matrix is dominant, with elements of the first row and column being suppressed by m 2 /m 3 ∼ λ as in (30). Therefore we consider the second and third neutrinos. Results obtained in this approximation are expected to receive corrections of the order λ when mixing with the first neutrino is turned on.
The matrix M X can be written as where D(Φ H ) = diag(1, e iΦ H /2 ) and Here It is easy to show that selecting parameters a, b, g, f, h one can get any value of ψ from zero to O(1).
We will diagonalize M 0 where R X (θ) is a 2 × 2 rotation matrix, D(y) = diag(e iyµ , e iy 0 τ ), and D(z) = diag(e iβ 2 , 1) are the phase matrices. Then U X = , can be written as where D(∆) ≡ diag(1, e i∆ ) and ∆ ≡ y µ − y 0 τ . From (34) we obtain the relations which determine the phases y µ , y 0 τ , β 2 : The solution is very simple in the case of maximal mixing: cos 2θ = 0, when the first equation in (35) is satisfied for H = 1 and ∆ = 0, so that y µ = y 0 τ . The two other equations give From these equations we obtain and G determines the mass hierarchy: The equality (36) implies that sin 2y µ is of the order sin ψ. And since ψ can be O(1), can have a large α µ = y µ , and consequently, a large δ CP . Furthermore, by selecting G the correct mass hierarchy can be obtained.
In the case of deviation of 2-3 mixing from maximal, H = 1, one obtains in general corrections to the obtained results of the order (H − 1). In special case cos ψ ≈ 0 the corrections can be enhanced.

D. CP phases with other assumptions on M R
Similar results can be obtained with other ansatzes for M −1 R . 1) Consider withM T BM given in (29). It differs from the ansatz in Sec. V C by the absence of the rotation V 0 CKM . Taking m D = diag(λ 4 , λ 2 , 1), which the only possibility which can lead to nearly TBM mass matrix for M X , we obtain In contrast to the previous case, now it is possible to have p = 1, leading to dominance of terms with a. That is, the whole matrix at the lowest order is generated by the 1 − 1 element ofM T BM : At the lowest order (the first term) phase factorization occurs automatically and the matrix M X is close to TBM, having only one nonzero mass eigenvalue. The factorization phase equals α τ = Arg ξ = φ td , and according to (32) sin δ CP = −sign{A}2 sin φ td V ts cot 2θ 23 .
Corrections of the order λ then generate lighter masses giving naturally m 2 /m 3 = O(λ) as well as modify CP phases. Selecting g and h one can achieve phase factorization of the whole matrix.
In this case the elements of the third column become The latter gives α τ = φ td + O(λ).
2) Instead of U R = V † CKM we could use a more general expression U R = D * (κ)V † CKM D(κ), where D(κ) = diag(e iκ 1 , e iκ 2 , e iκ 3 ). We can fix κ i in such a way that the 3-1 element in the matrix CKM , which led to the seesaw enhancement, is zero. For κ 1 = κ 2 = 0 and κ 3 = δ q we obtain Through this rephasing we moved the CP phase from the 1-3 to the 2-3 element. For the correction matrix we find Notice that now the second eigenvalue of m D matters. Finally, withM T BM from (29) we obtain . M X ∼ M T BM can be obtained for n = 2. In this case The factorization is absent, in general, but it can be achieved by imposing relations g 2 = h, f /b = g.
As a result, where ξ ≡ ξ − ge iδq . If g = −1, we have ξ = 1. In this case the contribution to the CP phase from the RH sector disappears and we revert to the situation described in Sec. III with CKM origin of CP violation.
Three main results emerge from this analysis of CP violation under the assumptions that U L ≈ U R ∼ V † CKM and there is no CP violation in M R in the L-R symmetric basis: 1. The hierarchy of Dirac masses of neutrinos cannot be too strong, i.e., m 1D /m 3D ≤ λ 4 and m 2D /m 3D ≤ λ 2 . The observed mixing angles of U P M N S impose this requirement. This is significantly weaker than the mass hierarchy of up quarks.
2. The CP phases can in general be large, even if the only sources of CP violation are the Dirac phases in U std L and U std R , where the CP phase effect is suppressed by λ 3 . This enhancement originates from seesaw and the hierarchy of Dirac masses of neutrinos.
3. If parameters of M R satisfy certain relations -the phase factorization conditions (which could be a consequence of some symmetry), the phases can factor out from M X . Furthermore, the only non-vanishing phase which enters the phase factors is α τ . This is related to certain pattern of CP violation in CKM matrix. In this case no enhancement occurs and δ CP turns out of the order M X deviates from M T BM since the correction in K is relatively large: being of the order λ 3 , which is still larger than the hierarchy of masses in m D .

E. Enhancement of a small phase in U R
In the previous examples large δ CP has been obtained at the cost of deviation of U X from U T BM .
With decrease of δ q , correction to the matrix M X due to CP violation (given by K) decreases and M X →M T BM . So, M X can coincide with M T BM up to small corrections. (This however implies that we depart from L-R symmetry or quark-lepton similarity, assuming smaller values of δ q .) Suppose δ q = λ 2 , where | | ≤ 1. In this case ξ ≈ −iδ q = −i λ 2 and Here the correction is suppressed by λ 2 in comparison with that in (28). Let us take for definiteness the parameters ofM T BM to be a = b = f = −g = h = 1 which ensures the exact TBM mixing in the lowest order with vanishing lowest neutrino mass. Then where λ ≈ ∆m 2 21 /∆m 2 31 leads to the correct neutrino masses. The additional imaginary terms give corrections to the TBM values of the 1 − 2 and 2 − 3 mixing angles proportional to λ 2 . They also generate small 1 − 3 mixing: X e3 ≈ λ 2 and All the other induced phases are close to 0 or to π, i.e., D(y) = diag(1, −1, 1) and D(z) = diag(−1, −1, 0), with corrections as λ 2 . According to (17) this contributes to the PMNS phase as So, seesaw can convert a tiny CP phase δ q ≡ λ 2 in U R to a maximal CP phase δ X ≈ π/2 in U X .
This happens because of the large hierarchy of Dirac masses and seesaw.

VI. REMARKS ON PHENOMENOLOGY
Our results have the following phenomenological consequences: 1. For the scenarios with CKM type CP violation and in the L-R symmetric case with phase factorization the value of sin δ CP is expected to be small, and the phase is close to π or zero.
This agrees with the result of a global fit in [24]: although statistical significance of this indication is low. At a 2σ-level, δ CP is also consistent with zero because of a second local minimum at that value (in both hierarchies). The value π/2, however, is disfavored in both cases.
Observation of δ CP ∼ π would be some indication of the CKM scenario or L-R scenario with phase factorization.
2. Observation of a large value, δ CP λ 2 , in experiments will rule out these scenarios and imply that either there are other sources of CP violation besides the CKM-like phase in U L or that the considered framework (canonical seesaw) is invalid, e.g., Dirac mass matrices are non-hierarchical, or seesaw type I is not the mechanism for generating neutrino masses.
3. In our notation, the effective Majorana mass of the electron neutrino is which, for inverted mass ordering in the limit of hierarchical masses, is mainly sensitive to , no cancellation of contributions to m ee from the first two mass eigenstates is expected and m ee is expected to be relatively large. For normal ordering m ee depends mainly on the combination δ CP + β 2 . Measuring the Majorana phases (or their differences) will be challenging for scenarios described above.
4. Future precise measurements of the phases may allow to disentangle the possibilities: CP in the left rotations only and L-R symmetric case. In the former, one expects sin δ CP β 1,2 , whereas the latter predicts all three phases to be equal in the specific case of factorization.
5. If the baryon asymmetry of the Universe is generated via leptogenesis (decays of the RH neutrinos in our case), this imposes certain restrictions on structure of the RH sector of seesaw; see, e.g., [25] and [26] for recent reviews. In particular, successful leptogenesis gives the bounds on mass of the lightest RH neutrino (in most of the cases we require a strongly hierarchical spectrum) and on combinations where α = e, µ, τ is the flavor index and i, j are indices of the RH neutrino mass eigenstates.
The combinations determine the lepton asymmetries in the lepton channel α. In the case of unflavored leptogenesis a summation over α proceeds, and the dependence on U L disappears.
So, leptogenesis would require complex phases in U R and/or U M . This is not necessary in the flavored case [25].

VII. CONCLUSIONS
We have studied the Dirac and Majorana CP violating phases in context of the seesaw type I mechanism with similar Dirac mass matrices for quarks and leptons. In this case a relationship We formulated the standard parametrization conditions for the mixing matrix to obtain simultaneously both the Dirac and Majorana CP phases.
Possible connections of the Dirac CP violating phases in the quark and lepton sectors have been explored.
The main results that we obtained are: 1. If the Dirac CP phase in U L is the only source of CP violation (which is similar to what happens in quark sector with Kobayashi-Maskawa mechanism), and there is no CP violation in the RH sector, the leptonic CP violation is very small sin δ CP = O(λ 2 ). The phase itself is either close to zero or to π with the deviation of the order of λ 2 . The Majorana phases are expected to be even smaller: 2. If the Dirac mass matrices are symmetric so that U L = U R ∼ V CKM (δ q ) and the Majorana mass matrix of the RH fields is real in the L-R symmetric basis, δ CP is in general enhanced by the seesaw mechanism. Furthermore, the Dirac masses of the neutrinos are constrained to be not strongly hierarchical. to reproduce the correct mixing.
3. The seesaw enhancement of phase is absent if M R has a specific form that leads to the phase factorization in M X . In this case, β 1 ≈ β 2 = O(λ 2 ) α τ = O(λ 2 ) and sin δ CP = O(λ 2 ). In particular case α τ = φ td three phases are equal and small β 1 ≈ β 2 ≈ sin δ CP = O(λ 2 ). Thus, the presence of the CP violation in the RH sector in the factorization case enhances the Majorana phases, but keeps the Dirac phase at the same order for this scenario.
4. Generic CP violation in the RH sector can lead to arbitrary and independent values of all three phases for arbitrary hierarchy of the eigenvalues of m D . We identify that the observable CP phase depends mainly on α e , α µ , and δ X , if it is measured to be large.
The formalism developed here allows to explore implications of measurements of the CP phases for the RH sector. For example, if a large CP phase is observed, the observable CP phases will mainly depend on three unknown phases in the RH sector : α e , α µ , and δ X . Thus, determination of δ CP and the Majorana phase may provide information on these parameters.
We may also get some direct hints about the flavor symmetry and quark-lepton unification, if special values of the CP violating phases are observed or if certain correlations between them are seen. Coming back to the initial question about the quark and leptonic CP phases, even in the context in which quarks and leptons are maximally related (quark-lepton symmetry, seesaw type I) one cannot expect equality of the quark and lepton Dirac phases. The phases are related but, generically, strongly different. The difference can be related to different mixing angles (especially 1-3 mixing angle) and to seesaw mechanism itself. Some results of this paper can be modified by the RGE effects. Since the light neutrino spectra we have considered are hierarchical, the renormalization correction are small and they will not affect our conclusions. The threshold effects due to possible large hierarchy of masses of the RH neutrinos are important when implications for M R are considered but this is beyond the scope of this paper. In this appendix we provide details of computations of the CP phases using the standard parametrization conditions.

CP violation from CKM only
Using explicit expressions for U e1 and U e2 in (15), we obtain from the conditions Arg {U e1 } = Arg {U e2 } = 0 that β 1 = γ e + ξ e1 , β 2 = γ e + ξ e2 , where ξ ei are given by We see that ξ ei = O(λ 3 ), which means sin(η i + γ e ) is of the order λ 3 . The reason behind this is that the CP violation originates from the Kobayashi-Maskawa phase associated with the element suppressed by λ 3 , while one of real terms in (15) is always of the order 1. Similarly, using (15), with α l = δ X = 0, and the conditions Arg {U µ3 } = Arg {U τ 3 } = 0, we find As we will show, X e3 ≤ O(λ), so that sin γ τ is also at most order λ 4 .
According to (46) effect of non-zero α τ , i.e. from the RH sector, is of the same order as the result for the CKM phase only. If X µ3 = X τ 3 , that is the 2 -3 mixing in U X is maximal β = 0, but δ CP = − |V td | √ 2 s 13 sin(α τ − φ td )V td .