CP odd weak basis invariants in minimal see-saw model and Leptogenesis

In this paper, we derive the relationship between the weak basis invariants (WB) related to CP violation responsible for leptogenesis and CP violation relevant at low energy. We examine all the experimental viable cases of Frampton-Glashow and Yanagida (FGY) model, in order to construct the WB invariants in terms of left handed Majorana neutrino mass matrix elements, and thus finding the necessary and sufficient condition for CP conservation. Further for all the viable FGY texture zeros, we have shown the explicit dependence of WB invariants on Dirac type and Majorana type CP violating phases. In the end, we discuss the implication of such interrelationships on leptogenesis.


Introduction
The origin of CP violation is one of the outstanding challenges in the fermion sector. In the Standard model (SM) [1] CP violation is related to the mixing between the flavor and mass eigen states, also known as Cabibo-Kobayashi-Maskawa mechanism (CKM) [2] in the scenario of three families of quarks and non-degenerate masses, and is well established in K 0 − K 0 system. On the other hand, in the lepton sector, neutrinos are exactly massless Wely particle and lepton flavor mixing does not exist, implying that there is no CP violation in the lepton sector. However, several neutrino oscillation experiments [3][4][5][6] provide us with very strong evidence regarding the nonzero neutrino masses as well as mixings. This, in consequence, provides the first sign to search for new physics and necessitates to look beyond the Standard model. In any extended model of SM, which incorporates neutrino masses and mixing, CP violation naturally appears in the leptonic sector. In the leptonic sector, CP violation have profound implication in cosmology, playing a pivotal role in the generation of matterantimatter asymmetry of the universe via leptogenesis [7]. In this regard, seesaw mechanism [8], is widely considered to be the most plausible candidate, which, not only, explain the smallness of neutrino masses in a natural way but also provides the origin of baryon asymmetry of the Universe. The seesaw mechanism, in fact, connects the small neutrino masses to very heavy right-handed neutrino masses. In general it contains more physical parameters than can be measured at low energies.
In an attempt to reduce the number of seesaw parameters, several theoretical ideas have been proposed either by introducing the texture zeros in Yukawa coupling Dirac neutrino matrix or by reducing the right handed heavy Majorana neutrinos . Among them, the most economical is the imposition of two zeros in Dirac neutrino mass matrix in the scheme of minimal seesaw model [9,10], popularly known as Frampton-Glashow-Yanagida (FGY) model [11]. However, the introduction of zeros in any specific model are not weak basis(WB) invariants, implying that a given set of texture zeros which exist in a certain WB may not be present or may appear in different entries in another WB, while leading to the same physics. This, in turn, brings forward a question of how to recognize the same texture zero model written in different bases where symmetry (or special texture zero) is not apparent. In such a scenario, CP odd weak basis invariants (WB) is considered to be an invaluable tool, and widely followed in the literature. The WB invariants were first used in [12] to study the CP violation in the quark sector. Similarly, leptonic WB invariants were presented for studying the CP conditions at low energy [13][14][15][16]. To investigate the CP violation at high energies one requires to establish a connection between the low energy physics and physics at high energies, for instance leptogenesis [15,16], [17][18][19], and the imposition of texture zeros in the scenario of minimal seesaw model (MSM) may serve the purpose in this regard. This makes the study of CP-odd WB invariants relevant for the model under consideration. In addition, it is crucial to examine the interrelationships between the CP-odd invariants which are required to vanish as a necessary and sufficient condition for CP conservation. The present paper aims to study the implication of CP odd invariants for FGY ansätze. To this end, we first of all construct the CP-odd WB invariants relevant for leptogenesis (at high energies) in terms of left handed Majorana mass matrix elements at low energies for viable ansätze, and then find the necessary and sufficient condition of CP conservation. Further we derive an analytical relations showing an explicit dependence of the CP-odd invariants on Dirac/Majorana CP violating phase. In the end we re-investigate the implications of such interrelationships on leptogenesis for each ansätz.

FGY ansätze in minimal seesaw model
In the present analysis, we take into account a most simple and economical see saw model [9,10], which incorporates the two heavy right handed neutrinos N 1,2 having strong hierarchical pattern (i.e. M 2 > M 1 ), and keep the Lagrangian of electroweak interactions invariant under the SU(2) L × U(1) Y gauge transformation [11]. After the spontaneous electroweak symmetry breaking, this simple but interesting model leads to the following neutrino mass term: where N c i ≡ CN T i (i = 1, 2) with C being the charge-conjugation operator; and (ν e , ν µ , ν τ ) denote the left-handed neutrinos. M D and M denote a 3 × 2 Dirac neutrino mass matrix, and 2 × 2 symmetric Majorana neutrino mass matrix, respectively. The scale of M D is characterized by the electroweak scale v = 174 GeV. In contrast, the scale of M can be much higher than v, because N 1 and N 2 are SU(2) L singlets and their corresponding mass term is not subject to the scale of gauge symmetry breaking. Then one may obtain the effective (light and left-handed) neutrino mass matrix M ν via the well-known seesaw mechanism [8] Without loss of generality, both heavy right-handed Majorana neutrino mass matrix M, and the charged lepton mass matrix M l are assumed to be diagonal, real and positive; i.e., where M 1,2 denotes the masses of two heavy Majorana neutrinos. The choice of this specific basis implies that one of the light (left-handed) Majorana neutrinos must be zero. On the other hand, M D is a complex 3 × 2 rectangular matrix, and can be given as where, a 1 , a 2 , b 1 , b 2 , c 1 , c 2 denote the complex entries. The minimal seesaw model itself has no restriction on the structure of M D . Frampton, Glashow and Yanagida [11] first introduce the two zeros, with a aim to restrict the structure of M D , whose origin comes from an underlying horizontal flavor symmetry. Such ansätze have been investigated by many authors, while taking into account both strongly hierarchical (i.e. M 1 << M 2 ) [20][21][22][23] as well as nearly degenerate (i.e. M 1 ≃ M 2 ) [24,25] neutrino spectrum of heavy right-handed Majorana neutrinos. Among the fifteen different possibilities of Eq.(4), only four are found to be compatible with neutrino oscillation data for inverted mas ordering, while same are ruled out for normal mass ordering [24]. The four viable FGY ansätze are given below: It is worthwhile to note here that in the MSM, the low-energy phenomenological implications are driven by M ν , while cosmological baryon number asymmetry is associated with M D via the leptogenesis mechanism.

Parameterization of lepton mass matrices in MSM
Before proceeding further, we briefly go through the different parameterizations used for effective Majorana neutrino mass matrix (M ν ) and Yukawa coupling Dirac neutrino mass matrix (M D ), respectively. These may be useful for deriving the relationship between CP odd invariants related to CP violation at high energies and CP violation at low energies. As mentioned earlier, the lightest neutrino in the MSM must be massless, therefore we are then left with either m 1 = 0 (normal mass ordering) or m 3 = 0(inverted mass ordering). Since normal mass ordering is ruled out for all the FGY ansätze, therefore we restrict our analysis for inverted mass ordering. In the basis of diagonal M l , M ν can be parameterized as follows  Here, c ij = cos θ ij , s ij = sin θ ij for i, j = 1, 2, 3, and δ, σ denote the Dirac and Majorana CP violating phase, respectively. From Eqs.(6) and (7), it is obvious that M ν depends on seven low energy physical parameters: two neutrino masses (m 1 , m 2 ), three mixing angles (θ 12 , θ 23 , θ 13 ), two CP violating phases (δ, σ), therefore, one can trivially derive each element of M ν in terms of these parameters. The number of available parameters here, is lesser than that found in M D , which reduces to nine after eliminating the three trivial phases by rephasing the charged-lepton field in the chosen basis. To account this difference, Casas-Ibarra-Ross [9, 10] introduce a orthogonal complex matrix R for m 1 = 0, and for m 3 =0 is The complex parameter z encodes the two hidden parameters viz. a real parameter and one phase, which are required to match the total number of parameters at high energies and low energies in the MSM model. Using Eq. (2,8,9), one can now parameterize the M D in terms of V, M ν , M, and z as Using Eq. (10), for m 3 = 0, one obtain where, V e1 , V e2 , V e3 denote the first row elements of neutrino mixing matrix given in Eq. (7). The remaining elements of M D can be expressed in the same manner following the generic relations used in [9,10].

Weak basis invariant(WB) for leptogenesis
In the seesaw mechanism, lepton number asymmetry can be generated through the decays of the heavy Majorana neutrinos M 1 , M 2 . This is called leptogenesis mechanism [7,27] and requires CP violation at high energies. Taking into account the general seesaw mechanism, it is not possible to establish a connection between leptonic CP violation at low energies and CP violation at high energies. Such a relation can only be establish in the context of flavor theory. Using the single flavor approximation for leptogenesis (i.e. in the case when wash out effects are not sensitive to the different flavors of the charged leptons into which the heavy neutrino decays), the leptogenesis can be probed using the CP odd invariants [17].
In the weak basis (WB), where M and M l are real and diagonal, there are six physical phases in M D , which can be used to characterize the CP violation in the leptonic sector . This corresponds to six possible CP-odd WB invariants relevant for leptogenesis [18]. For instance, The non-zero value of I 1 signals the CP violation in leptonic sector. Since WB invariants are basis independent. Therefore, in the chosen basis, one can express I 1 as where, k = M † D M D denotes the 3 × 3 hermitian mass matrix. Clearly, I 1 = 0 implies CP conservation in leptonic sector. This condition holds for either degenerate righthanded neutrino masses or diminishing imaginary part of k 2 ij (i = j, i=1, 2, 3) or both. The interest in I 1 stems from the dependence on the term Im (k 2 ij ), which eventually determines the strength of leptogenesis. Hence one can say that I 1 is sensitive to the CP violating phases which appear in the leptogenesis.
Following the similar WB as above, CP-odd invariants I 2 and I 3 can be expressed as and, In the MSM, one of the diagonal elements of M (i.e. M 3 = 0) is zero. Therefore, CP odd invariants in Eqs. (14,15,16) are reduced to It must be noted that Eqs. (14), (15) and (16) hold for the general case of seesaw model, where all the three heavy Majorana neutrino masses (M 1 , M 2 , M 3 ) are real, diagonal and non-zero, and M D is 3 × 3 complex matrix. Hence k turns out to be 3 × 3 hermitian matrix, while in minimal seesaw model, M is a 2 × 2 real and diagonal matrix. This implies that M D is necessarily 3 × 2 complex matrix following the see-saw mechanism in Eq. (2). Therefore, k is reduced to 2 × 2 hermitian matrix The remaining three CP odd invariants I 4 , I 5 and I 6 can be written in a similar manner by simply substituting Eqs. (21), (22) and (23) can be deduced in MSM model as where, K is 2 × 2 hermitian matrix and its elements are given below: where, m e , m µ and m τ denote the electron, muon and tau neutrinos, respectively. In the following section, we shall discuss the implications of six CP odd invariants for FGY ansätze.

Type 1
Using the seesaw mechanism in Eq.(2) and Eq. (5), one can write the expression for M ν for type 1 as in terms of Dirac neutrino matrix elements a 1 , b 1 , b 2 , c 2 . On comparing Eqs. (6) and (28), one can trivially find a 1 , b 1 , b 2 , c 2 in terms of the elements of Eq. (28), . Since I i (i = 1, 2, 3) is directly proportional to Im k 2 12 . Therefore it is sufficient to evaluate I 1 for each FGY ansätz. Using Eq.(17), one can write I 1 , for type 1, where, k 2 12 = (b * 1 b 2 ) 2 . The CP violation depends on the phase i.e. arg The vanishing of this phase implies CP conservation, and leads to following phase relation arg(m ee ) + 2arg(m µτ ) = arg(m τ τ ) + 2arg(m eµ ).
From the above equation, one can say that CP violation is brought about by the mismatch among the phases of elements m 2 eµ , m 2 µτ , m ee and m τ τ , while phase of the elements m eτ or m µµ does not have any contribution for CP violation and can be rephased away.
The type 1 and type 2 are phenomenologically related to each other via µ − τ exchange symmetry.

Type 3
Like type 2, type 3 also leads to m eµ = 0, and using Eqs. (2) and (5) , one gets Using Eqs. (6) and (34), we obtain the following mathematical relations for the ele- mee , a 2 2 = M 2 m ee . Using these relations, one can find where, k 2 12 = (c * 1 c 2 ) 2 , and CP violation for type 3 depends on the physical phase i.e. arg The results obtained here are just the complex conjugate of the results obtained in case of type 2.
Like in type 1 and type 2, we find that type 3 and type 4 are also related via µ − τ exchange symmetry. In addition, the results obtained in type 3 are simply a complex conjugate to that in type 1, and the same is true for type 2 and type 4 texture zeros.
Similarly, one can derive the relations for I 2 and I 3 in terms of Majorana mass matrix elements using Eqs. (18,19). The CP violating phase remain similar to I 1 , while the coefficients dependence in terms of heavy right handed neutrinos are different as shown in Eqs. (17,18,19).
On the other hand, the remaining CP-odd invariants (I 4 , I 5 , I 6 ) depend on Im K 2 12 . For illustration, we shall only evaluate I 4 for type 1. Using Eq. (24), and elements of M D provided in subsection 5.1, it is trivial to find the expression for I 4 where, The above relation is similar to Eq. (29) except that I 4 depends on additional charged lepton parameter m µ . The CP invariance condition obtained here is similar to I 1 for type 1. For the sake of completion, we have tabulated all the CP-odd invariants for all the viable FGY ansätze alongwith the necessary and sufficient CP invariance condition in Table [1]. The conditions on phases can be visualized as fine tuning required to have CP conservation at high energies. From the above discussion, it is trivial to find that M D with three or more zeros leads to CP invariance in the leptonic sector. In addition, it is found that that all the CP-odd invariants strongly depend on the effective neutrino mass term |m ee | i.e. I i ∝ 1 |mee| 2 , where i = 1, 2, 3, 4, 5, 6. If |m ee | =0 , I i simply blows up. Therefore the measurement of |m ee | in neutrinoless double beta decay experiments could have serious implications on these WB invariants.

CP-odd WB invariants and low energy CP violating phases
In this section, we discuss how the CP odd invariants depend on CP violating phases (δ, σ) in an explicit manner. Using Eqs. (11) and (12), we get the following relations , for a 1 = 0 and a 2 = 0, respectively. The symbols R = m 2 m 1 , and t 12 = c 12 s 12 .
Using Eq.(41), it is trivial to find the imaginary part as where c z = cos(z). For Type 1 and Type 4, we obtain, m eτ = 0 as evident from Eqs. (28) and (37). Using this constraint, we arrive at the following relation between δ and σ Following the same procedure as in Eq.(43), we get, using Eq.(42), For Type 2 and Type 3, we obtain, m eµ = 0. Using this condition, one can easily obtain the relation On comparing Eqs. (43) and (44), we get Im[c 2 z ] = +t 12 t −1 23 c 2 12 s 13 sinδ, Since we know that imaginary part of pure imaginary number is again imaginary. Therefore, one can write I 1 using Eq.(52) where, C ≃ 2M is the coefficient of I 1 , and we have used the approximation, m 1 ≃ m 2 ≃ m. Clearly, I 1 depends on the neutrino mass m and heavy right-handed Majorana neutrino masses M 1 and M 2 . The CP violation depends on the phase of complex term s * z c z . On evaluating further using Eq.(41), we obtain The relation holds for type1 and type2. For Type3 and Type4, using Eq.(42), I 1 is given as Similarly, we can easily derive the expressions for I 2 , I 3 , I 4 , I 5 and I 6 in terms of sinσ using Eqs. (18,19,24,25,26)for each ansatz. From Eqs.(54) and (55), we conclude that sinσ=0 leads to CP conservation in leptonic sector. Taking into account the analytical relation between δ and σ in Eqs.(44),(45), one find that CP conservation holds for δ, σ = ±nπ, where n is a integer.

Relationship between the thermal leptogenesis and left handed Majorana neutrino mass matrix
In the thermal leptogenesis in the MSM, seesaw mechanism with only two right handed neutrinos succeeds in reproducing the observed baryon asymmetry of universe for a nearly degenerate heavy neutrino mass spectrum. In [28], seesaw mechanism with thermal leptogenesis is also tested in the context of gravitational waves. D. Croon et.al [29] have studied how the observed baryon asymmetry is realized after high scale reheating into the lightest sterile neutrino in the framework of MSM. In this choosen framework, the decays of two heavy right-handed Majorana neutrinos, N i → l + H and N i → l + H * (for i = 1, 2), are both lepton-number-violating and CP-violating [27]. The CP asymmetry ǫ i originates from the interference between the tree-level and one-loop decay amplitudes. If N 1 and N 2 have a hierarchical mass spectrum (M 1 << M 2 ), the interactions involving N 1 can be in thermal equilibrium when N 2 decays. The asymmetry term ǫ 2 is erased before N 1 decays. The CP-violating asymmetry ǫ 1 , which is produced by the out-of-equilibrium decay of N 1 , in the choosen basis where M l and M are both diagonal, can be given as In this section, we discuss the implications of FGY ansatz on leptogenesis. To this end, we find the relationship between ǫ 1 and M ν for each ansatz. Using Eq.(56) and a 1 , b 1 , b 2 , c 2 in subsection 5.1, one can easily arrive at, where, (k † k) 11 = |a 1 | 2 + |c 1 | 2 . From Eq. (57), ǫ 1 depends on physical phase i.e. arg[(m * eµ ) 2 m 2 µτ m ee m * τ τ ]. For type 2, one can obtain ǫ 1 , simply by the exchange of µ ↔ τ . Similarly, with the help of Eq. 56 and b 1 , c 1 , c 2 , a 2 in subsection 5.3, ǫ 1 can be expressed as where, (k † k) 11 = |b 1 | 2 + |c 1 | 2 . In case of type3, ǫ 1 depends on physical phase i.e. arg [(m * µτ ) 2 m 2 eτ m µµ m * ee ]. The result for type 4 can simply be obtained through µ − τ exchange symmetry. From Eqs.(57) and (58), we conclude that CP-violating asymmetry ǫ 1 requires the mismatch among the phases associated with m eµ , m µτ , m ee and m τ τ pertaining to M ν for type1 ansatz, while, for type3, same holds true for the phases associated with m eτ , m µτ , m ee and m µµ . This, in turn, lead to net lepton number asymmetry, Y L ≡ nL s = dǫ 1 g * , where g * = 106.75 corresponds to an effective number featuring the relativistic degree of freedom which contribute to the entropy s, and d is the dilution effects induced by the lepton-number-violating wash-out processes [27]. The lepton number asymmetry Y L is finally converted into a net baryon number asymmetry Y B through the nonperturbative sphaleron processes [30]: Y B ≡ nB s ≈ 0.5Y L . In addition to the phase dependence, ǫ 1 depends only on M 1 , for M 2 >> M 1 . Another careful observation reveal that ǫ 1 for all the FGY ansätze depends inversely on |m ee |. Therefore, the measurement of |m ee | through various neutrinoless double beta decay experiments is important for calculating the baryon asymmetry of Universe. In the following discussion, we shall see how ǫ 1 depends explicitly on the CP violating phases related to low energy. With the help of Eqs.(51) and (56) we can arrive at following relations or where, m 1 = v(m 1 |c z | 2 + m 2 |s z | 2 ). Using Eqs.(47,48) and (59, 60), ǫ 1 is given as for type 1 (minus) and type 4 (plus), respectively. Similarly, using Eqs.(49, 50) and (59, 60), ǫ 1 is given as for type 2(plus) and type 3(minus), respectively. These relations show the explicit dependence of lepton asymmetry on δ. From Eqs. (44) and (46), it is clear that sinδ is directly proportional to sin2σ, implying that ǫ 1 ∝ sin2σ. Therefore lepton asymmetry depends on the Majorana CP-violating phase σ. It is worthwhile to note that this phase parameter does not affect CP violation in neutrino oscillation, but it can be instrumental in the scenarios of leptogenesis due to the lepton number violating and CP violating decays of the two heavy right handed Majorana neutrinos. The discussion also remain consistent with Ref. [23].

Summary and Conclusion
In summary, we have considered the minimal seesaw model (MSM) augmented with two zero in the Dirac neutrino mass matrix. Taking into account the four experimentally viable ansätze with inverted mass ordering, we construct the weak basis invariants (WB) relevant for leptogenesis in terms of low energy effective neutrino mass matrix elements, and then find the necessary and sufficient conditions of CP conservation. It is shown that textures having three or more zeros lead to CP conservation. The CP violation at high energies for these ansätze requires that phases among the low energy effective Majorana mass matrix elements are not fine tuned and, in addition, the right handed Majorana neutrino masses M 1 and M 2 are non degenerate. To extend our analysis further, we have explicitly shown the dependence of these CP odd invariants on Majorana CP violating phase (σ) for each ansätz, and find that δ, σ = ±nπ, where n is a integer, holds for CP invariance in leptonic sector at high energy scale.
In the end we re-examine the implications of these interrelationships on leptogenesis.
In this regard, we have shown the relations for CP violating asymmetry in terms of left handed Majorana neutrino mass matrix for all ansätze. Further, it is shown that it's non-zero value depends on the mismatch among the phases associated with the elements of M ν . In addition, for all ansätze, CP violating asymmetry depends on effective neutrino mass, |m ee |, related to neutrinoless double beta decay. In future long baseline experiments and neutrinoless double beta decay experiments, the precise determination of low energy parameters e.g. CP violating phases(δ, σ), octant of θ 23 , is critical to rule in or rule out the FGY ansätze.

Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.