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Contents

1. Introduction 1

2. SO(IO) Unification in six dimensions 2

3. CP violation in the quark sector 8

4. CP violation in the leptonic sector 11 4.1 Seesaw Mechanism and Effective Mass Matrix 12 4.2 Neutrinoless Double Beta Decay (Ov^) 15 4.3 CP Violation in Neutrino Oscillations 16 4.4 Leptogenesis 18

5. Conclusions 20

A. Mass matrices 22 A.l Down Quarks and Charged Leptons 23 A.2 Neutrinos 26

B. CP Violation and Weak Basis Invariants 35

1. Introduction

Grand unified theories (GUTs) appear to be the most promising framework [1,2] to address the still challenging question of quark and lepton masses and mixings. During the past years new results from neutrino physics have shed new light on this problem, and the large differences between the mass hierarchies and mixing angles of quarks, charged leptons and neutrinos impose strong constraints on unified extensions of the Standard Model (SM) [3,4]. Massive neutrinos are most easily incorporated in theories with right-handed neutrinos, which leads to SO(IO) as preferred GUT gauge group [5,6]. Higher-dimensional theories offer new possibilities to describe gauge symmetry breaking, the notorious doublet-triplet splitting and also fermion masses. A simple and elegant scheme is provided by orbifold compactifications which have recently been considered for GUT models in five and six dimensions [7-12]. In this paper we analyse in detail the connection between quark and lepton mass matrices in the

1 six-dimensional (6D) GUT model suggested in [13], for which also proton decay [14], supersymmetry breaking [15] and gauge coupling unification [16] have been studied. An alternative SO (10) model in five and six dimensions has previously been studied in [17]. For a recent discussion of CP violation in a 5D orbifold GUT model, see [18]. An important ingredient of orbifold GUTs is the presence of split bulk multiplets whose mixings with complete GUT multiplets, localised at the fixed points, can sig ­ nificantly modify ordinary GUT mass relations. This extends the known mechanism of mixing with vectorlike multiplets [19-21]. Such models have a large mixing of left-handed leptons and right-handed down quarks, while small mixings of the left­ handed down quarks. In this way large mixings in the leptonic charged current are naturally reconciled with small CKM mixings in the quark current. Our model of quark and lepton masses and mixings relates different orders of magnitude whereas factors O(1) remain undetermined. Hence, we can only discuss qualitative features of quark and lepton mass matrices. Recently, orbifold compact- ihcations of the heterotic string have been constructed which can account for the standard model in four dimensions and which have a six-dimensional GUT structure as intermediate step very similar to familiar orbifold GUT models [22-24]. In such models the currently unknown O(1) factors are in principle calculable, which would then allow for quantitative predictions. The goal of the present paper is twofold: As a typical example, we first study the model [13] in more detail and explicitly compute the mass eigenstates, masses and mixing angles. Second, we investigate the question of CP violation, both in the quark and lepton sector and possible connections between the two. In previous studies, CP violation has mostly been neglected assuming that, barring fortunate cancellations, the phases and mixings are practically independent. Nevertheless this question and the Savour structure are strongly interconnected, and we will see that a specific pattern of mass matrices can give a distinct signature also in the CP violation invariants. This paper is organised as follows: In Section 2 we describe the 6D orbifold GUT model and the diagonalisation of the mass matrices defining the low energy SM fermions. In Section 3 we discuss the CP violation in the quark sector, whereas Section 4 is devoted to the CP violation in the leptonic sector. Conclusions are given in Section 5. Two appendices provide details to the computation of the mass eigenstates and CP violation in extensions of the SM.

2. SO(IO) Unification in six dimensions

We study an SO (10) GUT model in 6D with N =1 supersymmetry compactihed on the orbifold T2/(Z2 x Z^ x ) [11,12]. The theo^ has four fixed points, Oi, Ops, Oca and Og, located at the four corners of a 'pillow' corresponding to the two compact dimensions (cf. Fig. 1). The extended supersymmetry is broken at all fixed

2 Ofl [Gfl]

Oi [80(10)] Figure 1: The three SO(IO) subgroups at the eorresponding fixed points (braucs) of the orbifold x x points; in addition, the gauge group SO(IO) is broken to its three subgroups Gps = SU(4) x SU(2) x SU(2); Gee = SU(5) x U(%; and hipped SU(5), Gn = SU(5)' x U(l)% at Ops, Oee and On, respeetively. The interseetion of all these GUT groups yields the standard model group with an additional U(l) factor, G Sm' = SU(3) x SU(2) x U(l)y x U(l)^„, as unbroken gauge symmetry below the compactihcation scale. The held content of the theory is strongly constrained by imposing the cancclla- tion of irreducible bulk and brane anomalies 1251. The model proposed in Ref. 1131 contains three spinors 0^(16), % = 1... 3, as branc hclds as well as six vectorial hclds Hj(10), j = 1.. .6, and two pairs of spinors, $(16) + $ c(16) and 0(16) + 0C(16) as bulk hypermultiplets. The massless zero modes #($) and #"($") acquire vacuum expectation values (vevs), r'Af = (Y) = (A^), breaking B — A and thus G^p to Gsw- The breaking scale is close to the compactihcation scale so that n^/M* ^ 10^ GeV, where M* is the cutoff of the 6D theory. At the weak scale, the doublets Hd(Hi) and HU(H2) acquire vevs, vi = (Hd) and v2 = (Hu), breaking the electroweak symmetry. The three sequential 16-plets are located on the three branes where SO(IO) is broken to its three GUT subgroups; in particular, we place 0i at Ooo, 02 at and "03 at Ops. The parities of 0, and arc chosen such that their zero modes,

have the quantum numbers of a lepton doublet and antidoublet as well as anti-down and down-quark singlets, respectively. Both A(0) and fR(0^) arc SU(2)^ doublets. Together these zero modes act as a fourth vectorial generation of down quarks and leptons. The three 'families' 0^ arc separated by distances large compared to the cutoff scale M*. Hence, they can only have diagonal Yukawa couplings with the bulk Higgs hclds; direct mixings arc exponentially suppressed. The branc hclds, however, can mix with the bulk zero modes for which we expect no suppression. These mixings

3 take place only among left-handed leptons and right-handed down quarks, leading to a characteristic pattern of mass matrices [13,14]. The mass terms assume the characteristic form,

W + d*m^ + e^m^ m^ (2.2) where latin indices only span 1, 2, 3, while greak indices include the forth generation states. The up quark and Majorana neutrino mass matrices, mu and mN, are diagonal 3 x 3 matrices,

0 0 \ /h 0 0 \ 0 ) ’ 11 M, mN = 0 0 (2.3a) m hU2V2 '^22 M, 0 hggV 2/ 0 0

Since vC is pa^ of an SU(2)L doublet, it cannot couple to the other SM singlets in ^ via the B — L breaking held. Furthermore, there is no other coupling giving it a direct Majorana mass. The Dirac mass matrices of down quarks, charged leptons and neutrinos, md, me and mD, respectively, are 4 x 4 matrices instead, due to the mixing with the bulk held zero modes,

o o 0 hggfi o md (2.3b) 0 0 hggfi \flVN fsVN y

^ h

4 This corresponds to different strengths of the Yukawa couplings at the different fixed points of the orbifold. The fourth row, proportional to Md, M' and is of order the unification scale and, we assume, non-hierarchical. The mass matrices md, m^ and mD are of the common form

0 0 0 ^2 0 /%2 m (2.4) 0 0 ^3 /%3 \Mi M2 M3 M4/ where = ^(u^) and Mi = O(^^ur). This matrix can be diagonalised using the unitary matrices

m (2.5) where the matrices U* and V4 single out the heavy mass eigenstate, that can then be integrated away, while U3 and V3 act only on the SM flavour indices and perform the final diagonalisation also in the 3 x 3 subspace. The explicit expressions for the mixing matrices and the mass eigenstates are given in Appendix A. The parameters in the matrix Eq. (2.4) are generally complex; however, we can absorb seven phases with appropriate held redefinitions and choose the remaining three physical phases to be contained into the diagonal parameters

%i| e^1 0 0 0 W e^2 0 0 0 W e#3 V Mi M2 M3 M4/ This is the maximal number of physical phases for four generations of Dirac fermions, given as usual by (n — 1)(n — 2)/2 for n generations, so our texture above does not reduce the CP violation from the typical n = 4 case. We will see that the phases survive in the low energy parameters, but that only one combination defines the single phase characteristic of three generations. With this choice, the matrix V4 is real, while U4 contains complex parameters; however, the imaginary part is suppressed by |^%| /M so that their effect on the low energy CP violation is negligible as long as the mass of the heavy eigenstate is large compared to the electroweak scale. From the unification of the gauge couplings, we expect indeed M to be of the order of the GUT scale [16]. Then the discussion of the low energy CP violation, which would in general be characterised by many CP invariants [26,27], reduces to the case of three light generations (see Appendix B). The effective mass matrix is given by m, the 3 x 3 part of

/ ^i(V4)ij + Ati(V4)4 A m' = U^mV4 m = I ^2(V4)2j + /^2(V4)4j I ; (2-7) 0 M/ \^3(V4)3j + ^3(^)4j)

5 in terms of the parameters in Eq. (2.4), it reads

_ jf Miy/M2 +M| _ ~ M4v /M22+M|\ (, Mi Mi 0

M2\]mI+mI _ ~ Mo /mI+mI m = ^M^M2+M2 ^M^M2+A^

Mi Mg ~ Mj\fm|+m| -^3 V i/M2+M2 V^2+^3

As any matrix, r/ can be transformed into upper triangular form just by basis redef­ inition on the right,

m = m %, = o //2 C# . (2.8 )

\ 0 0 ^ /

This form is particularly suitable in the case of the down quarks, where / acts on the right-handed quarks and disappears from the low energy Lagrangian due to the absence of right-handed current interactions. Note that we can reshuffle the phases, reabsorbing three of them into the unitary transformation P3, but we are still left with three complex parameters. We can exploit this freedom to obtain real diagonal elements jiZg, ^3 and q^i, while a, /), and /7i remain complex. On the other hand, we can still redefine two phases on the left-hand side, keeping an overall phase free, with a diagonal matrix

PL3 = diag (e-iZl ,e-iZ2, l). (2.9)

This transformation allows ns to shift the phase of /7i into Y, which will be convenient later in the limit where Y vanishes. Again, such a phase shift does not reduce the number of complex parameters in the down quark matrix, which remains three. Moreover, this reparametrisation does not change the CKM matrix, since the up quark mass matrix is diagonal and so such phase transformation can be compensated by an identical one for both u% and uC. The matrix V3 di&rs from the upper 3 x 3 part of the diagonalising matrix V3 = V3V; however, they are very similar in the hierarchical case. The relation between these two can be found in Appendix A, together with the general expression for U3, the 3 x 3 part of which is the CKM matrix. For the leptons, it is the matrix V4V3 that acts on the left-handed states, so the mismatch between the charged leptons and neutrinos (see Eq. (2.3c)) basis appears in the charged current interaction and the definition of the Savour neutrino eigenstates. However, the matrix V4 which contains large mixing angles and rotates away the heavy eigenstate is the same for charged leptons and neutrinos since the heavy state is an SU(2)L doublet. Therefore the PMNS matrix will be given only by the mismatch between the V3 ^ V3 matrices for charged leptons and neutrinos.

6 The complete expressions for the parameters in m are given in Appendix A; in this section, we will only consider the limit of small ^ ^ well as small pq and/or ^2 For p1 = j = 0, the hrst row simply vanishes, whereas for p1 = p2 = 0, the two hrst rows of the mass matrix are aligned (see Eq. (2.6)). Therefore both cases correspond to vanishing down-quark and electron mass. Since gives VUa, we focus on the case p1 = p2 = 0, where^

_ n __ P2 (p3 M4 T3M3 + j3M4 Y = 0 , m y /%3 M fZgM

jh = Th = Irnl VMi + m2 (2.10) 7%1 P2 p3 M

The eigenvalues of the heavier states are given by

7^ = 7^, (2.11a) Tn2s = n\ + \nif = n\ (1 + ~ p2 , where ~ Vus (2.11b) V j2 / j2 In this limit, only one single physical CP violating phase survives, even in the 4 x 4 picture; it is contained in p3 and so in a and ^ (see Eq. (2.10)). We will see, however, that this single phase is not sufficient to have low-energy CP violation. The down-quark mass is indeed very small, so we will use these expressions as the order zero approximation, together with the corrections proportional to |p2| /p2, which determine the masses of the down-quark and the electron. Our expansion parameter will therefore be of the order of the mass ratio of the down and strange- qnark, md/mg. In fact, for |p1|

md = 7jui ~ ~ |^2| JM ~ Vus\jjL2\ , (2.11c) P2 /t3 P2 so our expansion parameter is

0.23 (2.12) /^2 Wig* us

The mass ratio of electron and muon is much smaller than the ratio of down and strange quark. This implies (p2/%1/7%2)e (p27%1/7%2)d- Assuming that the difference is due to the smallest matrix elements, this indicates (p2)e/(j2)d 1 and/or (7%1)e/(/%1)d

^ As mentioned above, it is instructive to choose the basis in which pi is real and the vanishing parameter ^pi complex. Then it is obvious that we are left with only two complex parameters in m, namely a and /), containing the same phase.

7 brane leads to different values of ^2 for the down quarks and charged leptons and the parameter pq stems from different couplings in the superpotential. While we derived the fermion mass matrices (2.3) within a specific model, they can also arise in other models, where additional matter is present at the GUT (or compactihcation) scale. Thus we could take these matrices as a starting point for the following discussion, leaving open the question of their origin.

3. CP violation in the quark sector

We will first consider the CP violation in the quark sector. As we have seen in the previous section, our effective 3 x 3 down quark mass matrix contains three phases as a remnant of the original 4 x 4 matrix, with the dominant complex element being a/Z2. We will now derive the combination of the three phases, which plays the role of the CKM phase. To describe CP violation for three generations, as is the case in the SM, it is convenient to use the Jarlskog invariant [28], Jq, which is given by

6 i AM,2 AMd Jq = tr [Hu, Hd]3 = 6 Imtr (H^H^HHd) , (3.1) where H = m m^ and

AM2 = (m3 - m2) (m3 - m^ (m2 - m^ ; (3.2) note that AM2 has mass-dimension six. In our model, the up quark mass matrix is diagonal, as is H, Then the invariant strongly simplifies and reads

Im(HfHfHp Jq (3.3) AV

It is clear from this expression, that any diagonal phase transformation of m on the left does not have any effect on the Jarlskog invariant. As discussed in Appendix B, we can use the effective 3 x 3 mass matrix Hj^ = By means of Eq. (2.8), we obtain

(1 + |/3|" + |7|") (1 + %*/3) m/WA Hf = (1 + a/3*) /l2 (1 + kl') , (3-4) \ /%/?* m/w* ^ y where //2 and //g are real parameters, as displayed in Eqs. (A.4). Then we have

(Hf)12 (Hf)''e@) 23 M = Ijuil^T^hnrx/r (1 + a*/3) Im d ^)31 (3.5) = Ijuil^slma/r

8 We see that the Jarlskog invariant is always independent of the argument of Y and it vanishes in the limit //i, E 0 such that E = 0. As we might expect, E vanishes for a = 0 ^ well, i.e., in the limit m1, m2 ^ 0. So the presence of a single phase in a is not sufficient to give CP violation in the low energy: this phase cancels out in the Jarlskog invariant. This effect stems from the alignment of the vectors in Savour space; however, even in the case of vanishing first generation mass, the corresponding eigenvector does not decouple from the other two and the mixing matrix does not reduce to the two-generational case. In fact, the CKM matrix is given by (see Appendix A)^

( 1 E j~l2 /ts ^KM 0 ^KM (md = 0) - B 1 1^2 g U3 (3.6) j~l2 3 0 1 1 J 0 V 3 Hence, we cannot conclude that the CP effects disappear due to the reduction of the system to two generations, nor to the mass degeneracy between quarks. Instead the absence of low energy CP violation is caused by the particular texture of m in exactly the same basis for the left-handed quark doublet, where the up quark matrix is diagonal. This feature is similar to the absence of CP violation in 4D 80(10) constructions, where a single ten-dimensional Higgs held generates fermionic masses, yielding a trivial CKM matrix. Note that there is still some CP violation effect arising from the dominant phase E in ^3, but it is only apparent in the mixings involving the fourth heavy state. Now, the down quark is not massless and the real physical case corresponds to non-zero mid juq. ^om the up quark phenomenology, we know that m : m2 is similar to the mass ratio of up and charm-quark [13]; in addition, E : E is fixed by the Cabibbo angle. We will therefore focus on the linear terms in m2 and keep m - 0. As is apparent in Eq. (3.5), contributions to J come from the complex quantities «E, /Ei, and E; however, /Ei is independent of /t2 (see Eq. (A.4)),

M4 /13M4 T //3M3 /El (3.7) M EM

The first order terms are

M2 H3M4 + EM3 M E-M

r_ _ — E /^3 M2 //3M3 — EM4 0 jJj\ _9 (3.8) m ^3 m jism

^We can exploit the phase transformation P31, (2.9 ) to absorb the phases of a and make all elements of the CKM matrix real showing explicitly that the CP violation disappears.

9 and the Jarlskog invariant reads

,’ A-

We see that vanish^ if either /2 or |3 vanish, so two complex quantities are needed to obtain CP violation at low energies. It is instructive to calculate Hj^ also from the matrix m, Eq. (2.7). Here we notice that the off-diagonal elements of such matrix are relatively simple since we can exploit the unitarity of the matrix V*, which gives ^3=1(V4)ik(V4)jk = % — (V4)i4(V4)j4- So we have for i = j

eg\ ij ImM4 + fiiMi (Hd (l — aiaj) , ai = (3.10) from which we get the simple expression

Im (H?)reffX 12 ({Hfr Tje&\ 23 (H( ueS\dT 31 J = ~2~2~2 £ (1 + H2) Im (aja* ) • (3.11) cyd. perm ijk

In the limit of vanishing we see that ai = M^/M; thus for /1 = /2 = 0, the expression simplifies to

M* 12 (Hf ^23 (Hf ^31 ■~--'2 /~'~'2 /~'~'2 *221 1 ^4 Im (H Ld ; Ld ; |1|2|3 1 + Im M2 M M

For /1 = 0 but /2 = 0, we then obtain

■—-'2 -M2M3 / /^3//2 Im (Hf)12 (Hf)23 (Hd^)= (3.12) I1I2I3 \/%3/%2 M2 M2

M3M4 \fi3\ M2M1 w! ,m ( Im M2 /%3 ^2 M2 l2 \/3/

The complete expression for J is displayed in Eq. (A. 19 ); the dominant terms are exactly those given in Eq. (3.9 ). For degenerate heavy masses M, the result simplifies to

4 = ^ (3 ^ I™ W4) + |//3|^ Im (//2) ) . (3.13)

Note that the numerical factor, is minimal for degenerate M. Due to the hierarchy of the down quarks, A^^ So we hnally obtain, substituting the order of magnitude of the parameters, with |3 ^ |/3|,

^ _ 02) _|_ sin6* 2) 10^ (3 sin (6*3 - #2) + sin6* 2). 4\/2 (3.14)

10 This is the right order of magnitude; the current experimental value is J = 3 x 10-5 [29]. From Eq. (3.14) we can conclude that a single complex parameter, with the other two vanishing, is not enough to have low-energy CP violation in the quark sector and that the CKM phase is a combination of the high-energy phases ^ weighted by mass hierarchies. Moreover, maximal phases seem to be needed to give the large low-energy phase observed.

4. CP violation in the leptonic sector

The charged lepton and Dirac neutrino mass matrices can be transformed like the down quark mass matrix. The heavy state is an SU(2)^ doublet, so V4 singles out the same state for charged leptons and neutrinos. The effective 3 x 3-matrices read (cf. Eq. (2.7))

/ |1(V4)1j + /1(V4)4j \ /p1(V4)1j + p1 (V4)4j \ m6 = I |2(V4)2j + |2(V4)4j I , mD = I p2(V4)2j + p2(V4)4j I • (4.1) \/3(V4)3j + |3(V4)4j/ \p3 (V4)3j + p3 (V4)4j/

Within our model we assume the hierarchical patterns of ^ and # ^ well as |i and pi (i = 1..3) to be the same as for down quarks. The precise values, however, can be different since they originate from different Yukawa couplings, see Eqs. (2.3c). Again, we choose the couplings between the brane states, and pi, complex. Although some of the charged lepton and down quark parameters, namely /1 and /3, are related by GUT symmetries, the corresponding phases after the redef­ inition leading to Eq. (2.6) are completely uncorrelated. Thus, there is no direct relation between the CP violation in the leptonic and in the hadronic observables, even though, barring cancellations, we expect the leptonic CP violation to be large as well. Furthermore, we will see that different combinations of the phases determine the experimental observables. Thus even if there were relations between the phases in the quark and lepton sector, these would not be observable. Some correlations, however, could survive between charged and neutral leptons. As in the quark sector, we expect similar suppression for the CP violation due to the specific mass texture in our model. The discussion of the charged lepton masses closely follows the discussion of the down quarks in the previous section. The parameters are chosen such that they match the observed hierarchy, as described in Appendix A.l. The light neutrino masses, however, result from the seesaw mechanism, since we have heavy Majorana masses for the right-handed neutrinos. This Majorana matrix is diagonal, but can have complex entries (cf. Eq. (2.3a)),

/M1e2i^1 0 0 \ /M1 e2i^13 0 0 \ mN = I 0 M2e2i^ 0 I = e2i^3 I 0 M^^s 0 I , (4.2) \ 0 0 M3e2i^/ \ 0 0 M3/

11 where A^j = ^ — j Altogether, we have nine independent phases in the lep­ ton sector; in the limit of small ^ and p1, they reduce to seven. Since neutrinos are Major ana, we have less freedom in the phase reshuffling. However, except for electroweak breaking effects in U4, the heavy state is effectively an SU(2)-doublet of Dirac fermions. This allows ns to absorb some phases in the Dirac mass matrix and reduce the system to three generations for both charged and neutral leptons at the same time. In the following, we will neglect any effect of this heavy fourth generation doublet and concentrate on the three light generations including the right-handed neutrinos. We expect this approximation to be valid as long as M ^ Msur is much larger than the Major ana masses Mi [16].

4.1 Seesaw Mechanism and Effective Mass Matrix

In the case of the leptons, neither m6 nor mD is diagonal and therefore we will change the basis in order to simplify the discussion of the CP violation. Luckily, the large rotations of type V3, which bring the Dirac matrices into triangular form, are similar for charged leptons and neutrinos, thanks to the same hierarchical structure. To distinguish the Savour of the light neutrinos, we first act on the neutrino Dirac mass matrix with exactly the same V3 that transforms the charged lepton mass matrix into the upper triangular form, see Eq. (2.8), and obtain

(4.3)

At this stage the charged lepton mass matrix is not yet diagonal, but not very far from it: the complete diagonalisation can be obtained by applying another nearly diagonal rotation matrix on the right, corresponding to the mismatch between V3 and kg, and a CKM-like rotation U3 on the left as described in Appendix A. Note that such a transformation from the left, as in this case acts on the right-handed fields and leaves both # = frA m and the light neutrino Majorana mass matrix,

(4.4) unchanged. In fact ^ts in ve^ good approximation as the unity matrix on mN up to terms O(v2/M2), while U3 just cancels out. So apart for the small rotation on the right needed to diagonalise H, which affects the CP violation in the neutrino oscillation only weakly (see Section 4.3), the neutrino masses and mixings can be obtained from Eq. (4.4), in the form

/ C2£3 + B2£2 + A2g! CF03 + BE02 + AD01 C£3 + B£2 + AgA meff = — I CF£3 + BE£2 + AD£1 F2£3 + E2£2 + D2£1 F£3 + E£2 + D£1 I , \ C £3 + B£2 + A£1 F £3 + E£2 + D£1 £3 + £2 + £1 / (4.5)

12 where = e_^p^/M;. Note that the determinant of the (23)-submatrix of is not of order p^; instep it reads p3p2 (F — E)2 + p3p1 (F — D)2 + p2p1 (E — D)2, allowing a large solar mixing angle [30]. The leading part of the light neutrino mass matrix (4.5) is obtained in the limit p1, p1 ^ 0. From the general expressions (A.24) one obtains

1 1 Pi Pl- p3M123 — p3M3M4 //sM^ L 1 1 r_ P2 //3M'l23 — P3 M3M4 P2 M2 P3M4 + p^jMs 7^3 M2 1 1 P3 P3 //3M'l23 — ^3 M3^4 P3 P3 M3M4 — P3M'l24 (4.6) 7/3 M2 where we have introduced M%pY ^ V In our model, the Dirac neutrino mass matrix has a hierarchical structure similar to the one of down quarks and charged leptons. The three smallest elements, however, have a considerable uncertainty. Since m6 = md, these elements cannot be equal for the three matrices. Inspection of md sugg^ts for /% the range between md and ms the difference is a factor O(1). In the following we shall consider the case of small T/L. For large /i ^ that |p11 > |p31, in the following discussion we should interchange 7)3, gs with pi, gi and consider it as the dominant scale.

We here assume p3 : P2 : Pi ^ pg : P2 : Pi ^ m& : which yields [13]

IP2I m^m* |pi| p^Mg m^ 1—r ~ I9 T7 ~ —9 — ~ U.2 , -—- ~ 3^—— ~ —9 -~ U.2 , (4,7) |pg| p3 Af2 me |ps| p3 Mi m% such that £1 ^ £2 < p3. Hence, in this model, the weak hierarchy in the neutrino sector can be traced back to the nearly perfect compensation between down and up quark hierarchies.

The relation p1 ^ p2 impli^ for the two small neutrino masses |m11 ^ |m2| barring cancellations or small parameters. As computed in Appendix A, the masses at leading order assuming p3 to dominate are given by

m3 — £3 (3 + |F|2 + |C, |(F - E){A -B) + (D- E){B - (7)|s |m2m11 I £2 £1 (4.8) l + |F|2 + |Cf

The light neutrino mass spectrum has normal hierarchy, and the ratio m^/m3 can be identified with Am^i/Am^, which is indeed consistent with observations within the theoretical uncertainties.

13 The coefficients A.. F of the neutrino mass matrix m^g become in the limit p1, p1 ^ 0,

_ p1 p2 p3 M1 Pi p2 p3 M

p2p2 — P2P2 p3 Ml B P2P2 p3 M

P3p3 — P3p3 P2 Mi C = P" 3P" 3 p2 M P1 1 1 1 D P2 |P31 p2p3M2 |^p3M3 — p3M4 Pi P2 P3 p1 P2 11 1 E = D P2P3M2M3 + p2 fp3M'lg — p3M3M4 P1P2P2P3M2 1 1111 F (p3p3 — p3P3) p2p3M12 + p2M2 ( p3M3 — p3M4 ) (4.9) P3 p2 p3 Note that B, C F vanish in the limiting case of equal hierarchy in the neutrino and charged lepton Dirac mass matrix, i.e., for pi/pi = pi/p^, and A is in this case proportional to qpi. In fact, if the neutrino and charged lepton vectors are perfectly aligned in Savour space the neutrino Dirac matrix becomes triangular at the same time as the charged lepton one and we cannot reproduce large neutrino mixing. There is though no reason to expect such alignment since the parameters pi, pi are not related by any GUT relation, as can be seen in Eq. (2.3c). So the large neutrino mixing angles are not generated simply by the large LH rotation contained in the charged lepton's but from its misalignment with the neutrinos. Using the relations between p%, p% and p%, and p%, and due to the hierarchical structure of the mass matrices in our model, one obtains the simple expressions,

r, p2 n p2 p2 A (_y ~ --- JD ------— ------D ~ E ~ F ~ 1 (4.10) p2 P2 p2 The mixing angles are computed in Appendix A.2; in the case the parameters A C are small, they are given by

tanP23 - |F| ,

|B| tan 012 ~ /1 + |F|2 , |E — F |

C B (EF + 1) |g 2 sin P13 ~ (4.11) a /1 + 1*1 ^ (1 + 1^3

The atmospheric mixing angle 023 is naturally large; the current best ht [29, 31] restricts the parameter F as 0.7 ^ |F| < 1.^ ^^mal. Note that F > 0.7 can naturally be obtained even for |p3| // ^ |p3| /p3, as discussed in Appendix A.2.

14 For (//2//We ^ (^2/^2)^ ^ 0.1 one then obtains |C| ^ 0.1 and a value for 013 close to the current upper bound. In this case though, p1 has to be suppressed with respect to the down quark case in order to give a consistently small me. The large solar mixing 012 can then be achieved for B ^ 0.1 — 1 with moderate tuning of E — F. Another possibility is that a very small p2 is called for to explain the smallness of the electron mass. In this case, we have naturally |A|, |C| ^ 0.01 and the reactor angle is dominated by the second term in Eq. (4.11). Then the angles 012 and 013 depend on the same parameter B, but for the second one there is a suppression by g 2/g 3 So in the c^e of hierarchical g i, both a large and small angle can be explained even with relatively large B. Such value for B is not unnatural, even for small p2, if we accept p2 > (p2)e. In this case we have sin 013 < 0.1 correlated with the mass eigenvalues m1 ^ m2 ^ m3. Note that in general, if all parameters A B, and C are smaller than one, we obtain the prediction m1 < m2, while for B ^ 1 the two lowest eigenvalues are nearly degenerate. The largest of the heavy neutrino masses is given by M3 ^ ^ 1015 GeV. For the lightest heavy Majorana state the model provides the rough estimate M1 ^ M3mu/mt ^ 1010 GeV.

4.2 Neutrinoless Double Beta Decay (Gv^^)

The simultaneous decay of two neutrons may result in nentrinoless double beta decay, e.g., ^ + 2e. This process is currently most promising to prove the Majorana nature of neutrinos. The decay width can be expressed as

r = G|M2||mee|2, (4.12) where G is a ph^e space factor, M the unclear 0v,0,0 matrix element, and mee is the (ll)-element of the light neutrino mass matrix. Since the electron mass is very small, the charged lepton mass matrix in trian­ gular form has nearly a vanishing first row. Then the left-handed electron is already singled out; the remaining rotation mostly affects the (23)-block. Therefore we can already make an estimate of mgg from the elective neutrino Majorana matrix, m^. From Eq. (4.5), we read off

|mee| = IC2g 3 + B2g 2 + A2^11 , (4.13) where the last term can be negleted. This result has the same form as the standard formula in the case of hierarchical neutrinos [32],

| mee | — Vsilf #i3e^3 ^ ^sin^ 6»i2 cos^ 6»i3 (4.14) where mid ^2 are the two Majorana phases in the conventional parametrization of neutrino mass matrix (A.34).

15 We can estimate the size of |mee| in our model using

W ^ m, Iml ^3, 3^ - 1, |gs| ^ \ZA™atm, |g 2| ^ V Am^ , (4.15) P2 * which gives

,,2 -______n2 i------+ 0\/a ^4 mee ^ (4-16) p2 p2 Clearly, the last term dominates, yielding the familiar result for hierarchical neutrinos |m«.| i/Am^ ^ Q.01 eV if /12///2 4C P2/P2.

4.3 CP Violation in Neutrino Oscillations

Leptonic CP violation at low energies can be detected via neutrino oscillations, which are sensitive to the Dirac phase of the light neutrino mass matrix. For a diagonal charged lepton mass matrix, the strength of Dirac-type CP violation is obtained from the invariant [27]

tr , he]3 = 6i AM2 Im [(hv)12 (hv)23 (hv)31 ] , (4.17) where hv = (mv )^ m^ mid AMe2 is the product of the mass squared differences of the charged leptons, cf. Eq. (3.2). This quantity is connected to the leptonic equivalent of the Jarlskog invariant through

■h = ~ Im [(ft")12 (ft")23 (ft")31] , (4.18) where

AM2 = (m3 — m^) (m3 — m1) (m2 — m1) = ^m^^m^, - |g2 | W (4-19) is now the product of the light neutrino mass squared differences, In the standard parametrisation given in Eq. (A.34),

J( = Im [(%/)u(%/)22(%/)L(%/)2i] = § COS #13 sin 2013 sin 2012 sin 2#23 sin (4.20) where # is the CP violating Dirac phase in the SM with massive neutrinos. The expressions (4.18 ) and (4.20) assume that the charged lepton mass matrix is diagonal. In our case, this matrix is nearly diagonal after the F3 rotation, as the electron mass is very small; in fact, the remaining rotation V3 deviates from a unit matrix only in the 23 sector and at order O (mf[/m^)

Je = Im [(ft".)12 (ft".)23 (ft",)31] , (4.21)

16 with now = (m^g)^ m^g. We compute the hrst few terms and obtain

12 C*F |m3|2 (% )"= + C*E (1+F*E +C*B) g2g3 +B*F (1+F*E +C*B )* g 3 , 1 + |F |2+ |C |2

(A«)23= (1+ F-E+C-B) mel + E- (1 + F-E+C-BY

v \ 31 C I'”31 2 + B (l + F'E+C'B) S2sl + C (1 + F4E+C4B j‘&q3 . (h^r) 1 + |F |2 + |C |2 (4.22)

The leading contribution in the cyclic product of %, which is x |m3|6, is real and does not contribute to J^; that is to be expected since it corresponds to the limit of two massless neutrinos where no physical Dirac phase can be defined. In general the hrst non-trivial terms are of order |g 3|4 |g 2|2, as AMV*, so that we expect | J| < 1. We obtain in fact

(l + |B|2 + |E|2) (|F|2 - |F|2 + \b |C |2) Im[C*F(F - E)* (B - C)] . (1 + |F|2 + |C|Y

(4.23)

Note that the imaginary part vanishes for E = F or B = C, when the havour eigen ­ vectors are partially aligned. Furthermore, the contribution disappears for C = 0, so it is suppressed by the small reactor angle as expected. Due to the unknown param­ eters O(1), no useful upper bound on J can be derived in the general case, but we see that the Dirac CP phase is given by a combination of the phases of the neutrino Dirac mass coefficients B, C, E and F, derived from the complex parameters p3, p2, p3, p2. No dependence arises from the heavy neutrino Majorana phases ^3,2 since they cancel out in |g 3| |g2 | - In the limit p2 ^ 0, where A = C = 0, but with B of order unity, the dominant contribution to J comes from higher order terms. We can obtain it from

Kg )12 = B*F (1 + F*E )* + B*E n1 + |B|2 + |E|2) |g 2|2 ,

Kg )23 = F* |g3 |2 n1 + |F|2) + F* (1 + F*E) g2g3 + E* (1 + F*E )* g^g3 ,

(&Vg )31 = B (1 + F*E ) g2g3 + B n1 + |B|2 + |E|2) |g 212 . (4.24)

Note that the leading term, proportional to |B|2 |g 314 |g 2|2, is real, and in fact we did not have any |B|2 contributions at that order above. Hence, we consider the next terms,

Kg )12 Kg )23 Kg )31 x (1 + F*E) F* (KiE + K2F) g 2g 3

+ (1 + F*E)* F (K1F* + K,E*) ^3 , (4.25)

17 where we defined the real parameters

K1 = (1 + |B|2 + |E|^ (1 + |F|2)

K2 = |1 + E*F |2 . (4.26)

Note again that the two terms in Eq. (4.25) are exactly conjugate to each other for E = F when the two hea^ eigenstates are nearly aligned. In this limit tan 012 becomes maximal. Therefore, if B gives the dominant contribution, the Dirac type CP violation is suppressed for maximal solar angle. The CP invariant vanishes as well if B = 0 ^ the system e&ctively reduces to two generations and sin 013 = 0 (recall that we are already in the limit A = C = 0). We then obtain

with

n = (1 + FF*) F* (F - F) — which yields

J ~ — |B| (K1 — k2) Im (Q). (4.28)

Comparison with Eq. (4.20) shows then that in this case the standard Dirac phase # is a complicated function of the phases of p3, p3, p2 in the leptonic Dirac mass matrices, the difference between two of the Majorana phases A^32 and neutrino masses. It is suppressed by the ratio |g 2| / |g 31, ^ is sin013. Whenever only few of the parameters in the Dirac neutrino mass matrix mat­ ter, we expect correlations between the lightest eigenvalue, the mixing angles and the maximal value for J^. In Appends A.2, we consider the simple case where B dominates and the lightest eigenvalue m1 vanishes; then all the observables are only function of B, E, F, g 2/g 3 and we show relations among them. In this specific case, even allowing for the uncertainty on the phases, upper bounds can be obtained for sin 013, mee and J^. In the more general case, subleading terms and other parameters become important and relax any such bounds.

4.4 Leptogenesis

The ont-of-eqnilibrinm decays of heavy Majorana neutrinos is a natural source of the cosmological matter-antimatter asymmetry [33]. In recent years this leptogenesis mechanism has been studied in great detail. The main ingredients are CP asymmetry and washout processes, which depend on neutrino masses and mixings. It is convenient to work with a diagonal and real matrix for the right-handed neutrinos, which is obtained from mN by the phase transformation

PM = diag (e-^1 ,e-^ ,e-^) (4.29)

18 For hierarchical heaw neutrinos the generated barvon asvmmetrv is dominated bv decays of the lightest state Ni. In supersymmetric models the corresponding CP asymmetry is [34]

Im (M^) Mi £i M = PM , (4.30) Muv2a Mi where the matrix elements are given, analogously to Eq. (3.10),

PiM4 + Pi Mi Mij = e^'piPj. (1 - bib*) , bi = (4.31) Pi M The terms involving one index 1 simplify for pi = 0 as

M^ Mii = Pi | 1 -

Mi Pj M4 T p* Mj Mij = e^' pipj ( 1 - PhPb (4.32) M p,M

The result then reads -i 3 Mi M^ 5,1' (4.33) M^ where

Mi PjMi + p*jMj Pj = — Im e^'1 I 1 - (4.34) M p,M

Since p^Ms/^Mg) ^ 0.2, the CP asymmetry is dominated by the intermedi­ ate state #3, i.e., ei 3/(8 ?r) Mi Am^m/In any case, the phases involved, A^i3, and the phases of P3,P2, are completely independent of the low-energy CP violating phase in the quark sector and also not so directly connected to that in neutrino oscillations (even if they can contribute to it). For Mi — 10i0GeV, one obtains ei — 10-^ with a ba^ogenesis temperature TB — Mi — 10i0 GeV. These are typical parameters of thermal leptogenesis [35,36]. The strength of the washout processes crucially depends on the effective neutrino mass

Mn Pi_ M^ m i 1 — gi < 0.01 eV . (4.35) Mi Mi M2

With the efficiency factor [37] i.i 0.01 eV Kf - 10-2 - 10-2 , (4.36) i

19 one obtains for the baryon asymmetry

na - 10-26K/ - 10-8 Kf - 10-i0 , (4.37) consistent with observation. So for successful leptogenesis we need a non vanishing Pi,^ and in particular g i - g 2. In such case a zero neutrino eigenvalue is only possible due to alignment. In the above estimate of the baryon asymmetry we have summed over the lepton Savours in the final state. In general, the CP asymmetries as well as the washout pro­ cesses depend on the lepton Savour, which can lead to a considerable enhancement of the generated baryon asymmetry [38,39]. The neutrino masses Mi - 10i0 GeV, mi - 0.01 eV lie in the 'fully Savoured regime' where these effects can indeed be im­ portant [40]. Hence, depending on the CP violating phases the generated asymmetry may be signiScantly larger than the estimate (4.37).

5. Conclusions

We have studied in detail a speciSc pattern of quark and lepton mass matrices ob­ tained from a six-dimensional GUT model compactiSed on an orbifold. Up quarks and right-handed neutrinos have diagonal 3 x 3 matrices with the same hierarchy whereas down quarks, charged leptons and Dirac neutrino mass terms are described by 4 x 4 matric^ which have one large eigenvalue O(McuT). The origin of this pat­ tern are diagonal mass terms for three ordinary quark-lepton families together with large mixings O(^^uT) with a pah of SU(5) (5 + 5) plets. This vectorial fourth generation though is made of different split multiplets allowing for a relaxation of GUT relations. The six mass parameters of the model in the quark sector can be fixed by the up and down quark masses. This pattern of mass matrices has several remarkable features: The CKM matrix is correctly predicted and the electron mass is naturally different from the down quark mass. The mismatch between down and up quark mass hierarchies leads, via the seesaw mechanism, to three light neutrino masses with a much milder hierarchy. Left-handed leptons and right-handed quarks have large mixings. This leads to large neutrino mixings and to small CKM mixings of the left-handed down quarks in agreement with observation. Factors O(1) of the mass matrices are unknown, and the predictive power of the model is therefore limited. The neutrino mixings sin 023 - 1 and sin 0i3 < 0.1 are naturally accommodated. The corresponding neutrino masses are mi ^ m2 - VAmL < ?7% ~ \/and |m^| - < 0.01 eV. The elements of the mass matrices arise from a large number of different oper­ ators. Hence, most of the CP violating high-energy phases are unrelated. We find that the measured CP violation in the quark sector can be obtained, even if the CP

20 invariant is suppressed by the alignment between the two lightest mass eigenstates. Due to the uncertainties of O(1) factors no useful upper bound on the CP violation in neutrino oscillations is obtained in general. Some constraints can be given in the limited case where the number of dominant parameters is reduced, as it happens if the parameters A C in the neutrino Dirac mass matrix are suppressed by the small­ ness of the electron mass. It is indeed intriguing that in our setting the smallness of the reactor angle can be connected to the lightness of the electron. The model is consistent with thermal leptogenesis, with a possible enhancement of the baryon asymmetry by Savour effects. We conclude that mixings O(MQur) of three sequential quark-lepton families with vectorial split multiplets, a pair of lepton doublets and right-handed down quarks, can account simultaneously for small quark mixings and large neutrino mix ­ ings in the charged weak current and, correspondingly, for hierarchical quark masses together with almost degenerate neutrino masses. The CP phases in the quark sec­ tor, neutrino oscillations and leptogenesis are unrelated. Quantitative predictions for the lightest neutrino mass ^ and sin 0^ require currently unknown O(1) factors in more specific GUT models.

Acknowledgements

We would like to thank R. Fleischer and S. Willenbrock for helpful discussions. SW thanks the SLAG Theoretical Physics Group, the DESY Theory Group, and the CERN Theory Division for hospitality during various stages of this work. LG acknowledges the support of the "Impuls- and Vernetznngsfond" of the Helmholtz Association, contract number VH-NG-006. SW is supported in part by the U. S. De­ partment of Energy under contract No. DE-FG02-91ER40677. The work of DEC is presently supported by a CFTP-FCT UNIT 777 fellowship and through the projects POCTI/FNU/44409/2002, PDCT/FP/63914/2005, PDCT/FP/63912/2005 (Funda- gao para a Ciencia e a Tecnologia - FCT, Portugal).

21 A. Mass matrices

We will discuss here the mass eigenvalues and the mixing matrices for the low energy theory in relation to the high energy parameters. Given a general matrix of the form as in Eq. (2.4),

^/i 0 0

_ 0 ^2 0 /%2 m = 0 0 / m , \Mi M2 M3 M^ where = O(v1,2^ mid = O(McuTmatrices mid V4 that single out the heavy state can be given as [14]

( 1 0 0 _M^ - /12M2+/12M4 0 1 0 _M^ _ U4 — (A.l) 0 0 1 143M3+143M4 M2 V- iUiMi+jUiM4 U.2M2+H2M4 _113M3+113M4 2 M2 f2 f2

M1V/M|+M| Mi\ ( m4 i/M2+M2 m^/mI+mI m M2^Ml+Ml 0 Ms yMf+Mf Mi/m2+m| m V4 (A.2) M3\J+m| m3 0 M2 1/Mf+M2 m 1/ Mf +Mf M M4y M|+M| m4 Mi 0 V sJmI+mI Mi/M^+M^ M with M ^ M2 lo general mixings, while U4 is approximately the unity matrix, up to terms O (v/M^xt, mid V3 = I/3V/ diagonalise

/ m U4 mV4 0 f)+°(&)•

so both U3 and ^ mm-trivial 3 x 3 part only. In the following we will use the symbols U3, V3 both for the that non-trivial upper-left comers and the full 4 x 4 matrices obtained adding a row and column of zeros and a diagonal 1 to those. The effective mass matrix m can be brought into the upper triangular form by a unitary matrix V3 ^ V3 such that

wi m PP A m = m 1^ 0 JI2 aji2 I • . 0 0 jn3 /

22 With vi = (fn^ 1,fni2,fni3), the new basis is given by

V3 % 63-^2^ e3 e2 |%| % ^ el = e2 x e3 . (A.3) %

Note that V3 corresponds to a large angle rotation for the right-handed quark helds. While /7g and //2 are real by construction, we have the freedom to choose any entry of the first row to be real. For concrete calculations, it is convenient to have real or even use the parameters as given in the basis (A.3); however, vanishes in the limit //2 —» 0, so for a general discussion, it is more appropriate to have /li real. Here, we list the entries of m with real in a general form,

2 H3 — |%| — \l |/^31 + I/V.3I - i /3M3 + /3M4

HzH3 /2M2 + /2M4 /3M3 + /3M4 0//2 H 3 M //3M

2 - |2 H 2 \HZ12 + 1/^2 |2 - /2M2 + /2M4

/1/3 /1M1 + /1M4 /3M3 + /3M4 Hs M

_ , /^2 */^3 fi\M\ + fiiM^ M4 //%2 */%3 Hi — Hi \ ------OL — ,/^2 /^3 M M \ Hz H3 ^2 M H3 M

2 7 Hi I - \I\Hi\ + \Hi\ ~ /1M1 + /U1M4 — \hi \ (l + |/3| )• A.4)

In particular, we find as well the simple expressions

CK//2 H3M4 + H3M3 Hi Mi Hz M2 (A.5) Hz Hi //3M Hi M 7^2 M

~ /2 /3 M1 jl2 /3 M2 Hz H3M3 — H3M4 IHi (A.6) A<2 Hs M Hz Hs M 7^2 /7gM

These expressions vanish trivially in the limit /1, /2 ^ 0 and then we obtain the limiting case discussed in Section 2. As already discussed in Section 3, is inde­ pendent of /2.

A.l Down Quarks and Charged Leptons

Mass Eigenvalues and Eigenvectors. Now take the matrix m as a starting point and compute the eigenvalues, eigenvectors and mixing matrices. For making things simpler, consider for the moment all the parameters as complex, even if actually ^3, 7^2, 77h, or jiZs, ju2, T^i can be chosen real absorbing the phases into F3. To compute

23 the eigenvalues, it is better to consider the hermitian matrices Wm or The hrst option simply gives

|2U,|2 iilV |2Y*^ (A.7) + Iml^* Iml2 + Iml'kl" + lml2l/)|2

Then the determinant is simply

121 — 121 — 12 I — 12 det (m)rh) = |det (to) |2 | TO1| TO2| TO3| (A.8 ) and is only non-vanishing if YAq 0. The eigenvalue equation is a cubic equation; to obtain the dominant terms, we expand around Y = 0. In this case the equation reduces to a quadratic one with the solutions

A2/3 (A.9)

- |2 2 ± 3 - a 2) - - l^l2^l)] 2 M OL

So in this limit, we have eigenvalues at lowest order

A3 = l^l2 + lml2k|" + |/7i| W + ^ ,

Ag = |/^212 + |^i|2 — O , Ai = 0 . (A,10)

We can also compute the hrst correction to the zero eigenvalue simply as

det(m^m) |2|^i|2lml2 iwkiaw 121— 12 A1 i i^1i (A.ll) A2 A3

This means that for vanishing ^1 we have

2 md ^ (A.12)

Using the eigenvalues, we can also solve for the mixing matrices at lowest order,

(l 0 0 \ I77~|2^,_U77-, |2 r 0 1 lAWW+lAWW Vi _ _ I AW (A.13) n Ia W2«*+ia WW* 1 \M ) where we must recall that we had already acted on the mass matrix with a large angle rotation V3, so the V above is just a small correction to it.

-24- B

For the left-handed quark fields, we have instead at leading order

( 1 B£\

U3 1^3 (A.14) (a* - ^ *) 1 y

Since the up quark mass matrix is already diagonal, this last mixing matrix cor­ responds to the CKM matrix. From m F^ = m^, we get F^KM = B, so for o = we have the prediction F^d = (a* — /)*) 7^i/^3 = 0 at leading order, and the

CP violation vanishes! On the other hand, FU& Vhas 2 the right order of magnitude as we thought. 1 ElB Quark Masses and Mixing Angles. We can reproduce the observed quark mass eigenvalues and mixing, that satisfy the relations

mu : mc : m* — A7 : A3 : 1 ,

md : ms : mb — A4 : A2 : 1 , (A.15) where A — F^ ^ 0.22 is the Cabibbo angle. In fact, if we assume

pi : P2 : As — A7 : A3 : 1 ,

pq : /%2 : As — A3 : A2 : 1 , (A.16) it gives correctly

It/- I 1/^1 I IbI \ \Vus\ ~ T^-T ~ T^-T ~ A (A.17) w Iml

N Iml I A*2 | | A*21 ,2 |Fub| A3 , |FCb| A2 ; iBl IBI Iml Iml moreover,

hi ,I— | 1/^21 |/^11 , , 3 x 4 md — mi\ ~ —r^— mb ~ AA mb ~ A mb . (A.18) x/T a M //3

Again F^ is suppressed by the difference of «*—/)* ^2/^2; ^ is the Jarlskog invariant, Jq.

Low-energy CP violation As discussed in the following Appendix, we can ex ­ press the low-energy CP violation in the quark section via an effective Jarlskog invariant. We calculate this invariant, using Eqs. (A.4). The dominant terms are

25 displayed in Eq. (3.9 ); the complete expression reads

Q Q Q 2 KKK M2M3 jm M3M4 |p3 Jq 1 - Im (A.19) A.//J M2 M^ PsP2 M2 PS P2 2 M% M3 MA | p Im^ M4 P2 As 2 Mi M3 M^ jm P3Pi M3M4 |p3 Im^l 1 - M^ M2 PsPi M 2 As Pi MS |ps| 2 Mi Mg P2P1 + 1 - + Im M2 M2 M 2 PS P2P1

M1M| |p M4

M 3 P2 M Pi M PsPi

M2 |pi MsM M2 M4 P2 M£hlmlizg 'im^- —-—Im — M 2 Pi M 2 Ps M 2 P2 M2 PsP2

Charged Leptons. The charged leptons show a different hierarchy than the down quarks, we have in fact

mg : m^ : m — A5 6 : A2 : 1

md : ms : m& — A4 : A2 : 1 . (A.20)

The discrepancy can be solved with a smaller value for (P^^e, compared to (P2/l1)d. As an example, we choose P% — A4 and pi — As-4 such that

|Y e |p2| |Pi .5-6 |pi| ~ 73—mT ~ A A mT ~ A m? (A.21) xA" a |P2| P3 Regarding the rotations, the large F4 rotation acts now on the left-handed fields, but it has to act on both the charged leptons and the neutrinos, so it has not a large effect in the charged current. There is, however, an effect coming from the mismatch between the two FS's in the charged leptons and neutrino cases.

A.2 Neutrinos

The charged lepton mass matrix is eventually diagonalised via FS = FSFS and Us as the down quark matrix. For the light neutrino Majorana mass matrix, given by

m^ = — (mD(mN) 1 mD, (A.22) we can neglect the rotation Us of the right-handed fields as this transformation cancels out. U4 does in principle rotate the RH states, but its effect is suppressed as long as Mi < M. Regard!ng FS, we do not expect it to be the same for both charged

26 and neutral leptons, so the mismatch between the two provides Savour mixing in the neutrino sector. The neutrino Dirac mass matrix can be written after the large rotation FS that bring the charged lepton mass matrix into triangular form as

_ /Api Dpi pA ^ %, = Dp2 Dps P2 , (A.23) \Cp3 Dps P3/ where

1 1 Pi Pi Ps M22s PS Ms M4 Pi Mi Ps M4 + PgMs 7^3 1 1 P2 P2 Ps ML PS Ms M4 P2M2 PsM4 + Pg Ms 7^3 M2 1 1 Ps M^23 P3 PS PS Ms M4 PSMS M4 — PsM'l24 7^3 M2 Ps and, using the notation M^g Ma+M^,

1111 A i — Pi P2PSM2 + P2 ( PsMS — PsM4 Pi P'2 P3 M P2P2 ~ P2P2 P3 Ml B P2P2 P3 M P3P3 ~ P3P3 P2 Mi C PsPs" " p2 M 1111 D Pi P2 |PS| Ml2 + Pg PsM2 ( PsMS — P3M4 Pi P2 P3 M2 2 + PiMi P2PS ( PsM4 — PSMA + p2M2 (p2 + |PS|

1111 E P2 P2 |PS| Ml2 + p2PsM2 ( PsMS — pSM4 P2 P2 P3 M2

P2 P2PSM2 I PSMS — P SM4

p 2 (p2mi2s — 2 |P S| PSMSM4 COS Ps + |ps|2 M2

1111 F (PsPs — PsPs) P2PsMI2 + p2M2 ( PsMS — pSM4 ) (A.24) P3 P2 P3 M2

Note that we are here projecting the neutrino Savour states into the basis deSned by the charged leptons as in Eq. (A.3). So we can immediately see that if the neutrino Savour vectors are aligned with the charged leptons B, C, F should vanish and the neutrino mass matrix would become triangular as well. This corresponds to having

27 exactly the same hierarchy in the rows of the charged and neutral lepton Dirac mass matrices, i.e. We do not expect such alignment since the parameters Pi, Pi are generated by different operators and not related by any GUT relation, as can be seen from Eq. (2.3c). We will consider in the following the case where the neutrino hierarchies are similar to those of the down quark matrix, while the charged leptons differ due to the lighter electron mass. Of course even more involved scenarios are possible. In the following we neglect as well corrections coming from the final diagonalisation, since the entries of are suppressed by (^2/^2)^ < 0.01.

Mass eigenvalues and eigenvectors. We need to compute the eigenvalues of the neutrino mass matrix and the first step is again to compute the determinant of the matrix mVg. Note that this is a symmetric matrix, but not real. Therefore the eigenvalues are in general complex and the matrix is diagonalised using a unitary matrix FV as

m^FV = diag(mi,m2,ms). (A.25)

Consider for the moment just the absolute value of the eigenvalues and then see that we have the relation^

(det det (mV@) (A.26) det (m^)

The last determinant is simply the product of the heavy neutrino masses, while the first one is given by

det(nA) = pipgm [(B - B)(A - B) + (B - B)(B - C)]. (A.27)

In order to have three non-vanishing eigenvalues, we need all p% 0 and at least one of A B, and C different from zero. Also the three vectors corresponding to the rows of the Dirac matrix must not be aligned with each other. So we obtain

mim2ms — gi g 2gs [(F — E)(A — B) + (D — E)(B — C)]

Pi P2 Ps 1 1 1 Mi ' Q1Q2Q3 — — — _o —0 (A.28) pi p2 ps P2 Ps M2 M

x < P2 | P S| Mi2 + 2 |P 2| P 2 |p S| M2 PsMS cos (P2 Ps) - PsM cos P2 12 + |p21 P3M"lS — 2 |PS| PsMSM4 cos PS + |P S^ Mi4 for pi = 0, where = e ^'p^/A^.

^Note that for a n x n matr#, the rnnus sign on the r.h.s. gives a (—1)n contribution.

28 Singling out the heaviest mass eigenstate. In the case when g s ^ g 2,i, it is easy to single out the heaviest eigenstate:

1 (C*,F *, 1), (A.29) yi + i*i 2 + \cf and the mass eigenvalue to lowest order is given by

mS = — g s (l- + |F|2 + |C|2) . (A.30)

Then up to a rotation in the 12 submatrix, at lowest order the mixing matrix can be written as

/ Vl+\F\2 Q C* \ ^fl+\F\2+\C\2 \Jl+\F\2 + \C\2 F° -CF* 1 F* . (A.31) VWf+jcFVWf VWf \?l+\F\2 + \C\2 ’ -C -F 1 XVWf+jcFVWf Vwf \/l+\F\2 + \C\2 / this is the basis which gives decoupling of the hrst eigenstate in the limit of vanishing C. From this matrix, we can directly read off the dominant part of the mixing angles with the heavy eigenstate, P2S and PiS. The charged lepton mass matrix is nearly diagonal, so we can actually relate with good accuracy the hrst row to the electron neutrino flavour. The left-handed charged lepton flavour eigenstates are given as a function of the mass eigenstates by

if = (FsFsf (A.32) and therefore the neutrino flavour eigenstates correspond to

Vf = (FSFsA FSFVVi = (FS)^ FVVi , (A.33) where FS cancels out as it acts equally on the whole lepton doublet; moreover, as we have seen, F' is limited to the 23 corner and does not modify the electron entry. We use here the convention of [32], and define the PMNS matrix as

/ CiSCi2 cissi2 sis \ A 0 0 \ FV = I — sI2C2S — CI2s2SsIS A Ci2C2s — sI2s2SsISe^ ciSs2Se^ I I 0 e*^ 2/2 0 I \ SI2S2S — Ci2c2ssisei^ —Ci2s2s — SI2C2Ssisei^ cisC2sei^/ \0 0 ei^3/2/

Z1 0 0 \Z cis 0 sise-^\ / ci2 si2 0\/1 0 0 \ = I 0 C2S S2S I I 0 1 0 I I —Si2 Ci2 0 I I 0 ei?2/2 0 I , \0 —S2S C2J \—sisei^ 0 ci^ \ 0 0 1/ \0 0 ei(^+^3/2)/

(A.34) where Cj = cos Pij sij- = sinPij ^ phase and ^i,2 are the Majorana phases.

29 So we have at lowest order for #13 that

(%% = Sin #13 ^ ' = . (A.35) yi+|F 12+ic 12

This gives ns directly a constraint on the parameter C from the upper bound on |sin #131 < 0.1:

|C| - \/l + |ff+ |Cf |sin#!3| < 0.1\/l + |F|'. (A.36)

Then since the mixing with the first Savour is small, the atmospheric mixing matrix is given simply by requiring the 23 corner of the matrix in Eq. (A.31) to give

cos #23 sin #23 e ^23\ rn, 23 (A.37) — sin #23 e^23 cos #23 /

So considering the 23 sector, we get, again at lowest order,

&3 = arg (F),

tan#23 = |F| . (A.38)

To have large mixing angle tan2#23 > ^ ^^nust restrict |F| between

0.7 < |F| < 1.4 . (A.39)

Such a value is natural in the case where p3, and ^3, /%3 are of the same order but not exactly equal, while ^2 is small. Note that even a phase difference can be important. Assuming simply ^ = e^3# and degenerate gives

2i/2(l — COSCJ3) (A.40) 3 — cos cv3 so we obtain |F| = 1 phase difference w3 = Wide |F| > 0.7 arises in the wide interval 0.26 n < ^3 < 1.73 n. Hence, a nearly maximal atmospheric angle is natural even for the most simple choice of parameters. Of course, more solutions are possible for the general case. Thus in order to reproduce the observed pattern of mixing parameters, C has to be small, while |F| is nearly unity. We can use the maximal value for |F| and the experimental bound on #13 to derive an upper limit on |C|,

|C| < 0.17 , (A.41) in agreement e.g. with the ratio ^ necessary to have a small electron mass. Note, however, that we can obtain significant corrections from g 2,1 = 0.

30 Light eigenstates and solar mixing angle. The other two eigenvalues and the correction to the heavy mass can be obtained from the trace and determinant of the matrix (m^g)^ m^g, which can be computed in any basis. Expanding both the mass matrix and the eigenvalues to first order,

mV@ = mes + mei,2 , m3 = m? + ^m^ while mi2 = #mi2 (A.42) we have then

tr [ml3mei>2] #m.3 |m1|2 + |m2|2 + |5ms|2 =tr M)* mei,2 mei,2

|m,|2|m,|2-|det<

Choosing the basis appropriately, the relations can be simplified to give

= ((v;)Tm«,,, V?). '33

|m:|2 + |m2|2 =tr — |( (V? )Tm,i,2 V?0 ) |2 mei,2 mei,2 331 Kl2 |m,|2 ~ |£,lto|2 W ~ E)(A ~ B)+ I.D - E)(.B - C)\* (A.44) (1 + |F |2 + |C |2)

We will give the result of these expressions for vanishing C and g 1 = qg 2:

(1 — FE)2 + q(1 — FD)2 =g 2- 1 + |F |2

= |g 2^ [|1 + 9 ^ + |E + qD2|2 + |B2 + qA^ tr mgi, 2 mgi, 2 +2 |BE + qAD|2 + 2 |B + qA|2 + 2 |E + qD|2]

^ ,2 |A(F — E) + B(D — F)|4 |m2| |m:| — |^21 |q| ATS) (1 + |F |2)'

Then the mass splitting which should generate the solar oscillations is given by

8mlol = \J(|mi|2 + |m2|2)2 - 4 |m2|2 |mi|2

|g 2 | (1 + |F|2)2 ^|1 + q|2 + |E2 + qD2|2 + |B2 + qA2|2 (1 + |F |2)2

+ 2 |BE + qAD|2 + 2 |B + qA|2 + 2 |E + qD|2)

— |(1 — FE)2 + q(1 — FD)2|T

4 |q|12 (1(, +. 1|F r,i|^2)22 '|A(F ' ' — — E—') + B(D—2— — F)|„s,44^} 1/2 . (A.46)

31 So the solar neutrino mass splitting can be matched even in the case q = 0 or A (F — E) + B (D — F) = 0, i.e., when the lightest neutrino is massless. However, we do not expect the hrst limit to be realised, if we assume the same hierarchies between ^ in the in the down quark sector, while for as the up quark sector. In that case we have in fact |g 2| — |g 1| and the two lighter eigenvalues are similar in scale, mi m2 — \Z#m^. On the other hand, the determinant could be suppressed by alignment, i.e., for |A (F — E) + B (D — F)|

B2g2 + A2g1 Bg2 E-F + Ag1 D-F V1+i*f Vwf mei,2(12) (E-Ff (A.47) BQ2-A#+^7#

Taking the solar mixing matrix as in Eq. (A.37) with 023,^23 ^ 012,^12 we obtain

e-i(i2 (meh2)n(meh2yn + (mei[2)22(mei[2)^ (mei,2 )12 (mei,2 )*1 + (mei,2 )22 (mgi,2

2^/l + |jf |V| 2 tan 2^12 I / \ 12 I / \ 12 | (mei,2 J221 — | (mgi,2 )n| V where, for q = g 1/g 2,

N = [B(E — F) + qA(D — F)] (B2 + qA2)* (1 + |F|2) + [(E — F)2 + q (D — F)2] [B (E — F) + qA (D — F)]* , D = ^E — F)2 + q (D — F)2^ — jB2 + qA2^ (1 + |F|2)2 .

In order to have a large solar mixing angle, either Aq or B must not be small compared to F — F and F — F. But since A, F oc we are led to the case

A = F~0, F = ^^ = F(1) . (A.48)

P2 M

Then we can neglect the terms proportional to A and we have simply

\B\2 (1 + Iff) + \E - ff + q(D - 2|B||£-F| y6 + |ff tan 26*12 \(E - Ff + q(D - Ff\2 - |B|4 (l + |F|2)2 (A.49)

32 This formula simplifies further if we neglect the q (D — F) terms as well.4 Then using general trigonometric formulae leads to the expression in Eq. (4.11),

tan#12 ~ ' (A-50)

Taking the experimental value for the solar angle, tan2 012 = 0.45 ± 0.05, gives us for |F| ~ 1 the range |B| ~ (0.45 — 0.50) |F — F|. We can also compute the corrections of order g 1,2 to the other two mixing angles, that we have discussed in the lowest order. In fact, since ^2

|B (EF +1)| |Q2 m2 sin 13 |B| -0.2 IB (A.51) (1+iFiy^ igs m3

So even for vanishing leading order, we expect the first order term to bring 013 near to the experimental bound. Note that it is the large solar angle that naturally gives 013 ^ g 2/g 3; in our model it seems pretty difficult to suppress this angle to much smaller values, apart if there is a tuned cancellation between zero and first order. The corrections to the atmospheric angle are of the same order g 2/g 3 and do not have a large effect since we need in any case large parameters in the 23 sector. This small shift can in fact be easily compensated by a small change in the value of F, especially since we do not have any particular symmetry in the model imposing F = 1.

Sum Rules for B domiu^ce and vanishing We have seen in the previous paragraph that in case of vanishing C, A and g 1, simple expressions can be obtained for all observables as functions of only few parameters B, F, F and g 3,2. Then it is possible to obtain relations between the different observables,

tan 023 |F| ,

ib^+v '1 + ifi tan 0i2

|B(gF+ 1)| |g 2| sin 013 (1+iFiy/' w

g2 | y/(i + |ff)2(1 + l+ + l+)2 ^mgoi 1 — FF |4 (A.52) £matm 0si (i+My

^Note that taking A = C = D — F = 0 gives a zero determinant for the neutrino mass matrix, so this case applies when the lightest eigenvalue is suppressed compared to the solar mass scale.

33 Now we can write the following relation,

sin#i3 $rnatm _ \E — F\ \EF + 11 tan#12 5msol ^[(1 + |F|2) (i + \E\2) + \E - F|2 tan2 #12]2 - |1 - FF|4

(A.53)

To estimate its value, we can use the fact that |F| ^ 1 and vary only |F| and the phases of F, F. We obtain then a m^mal value of the r.h.s. for FF = 1 so that

tan #12 sin 13 < ~ 0.09 (A.54) ^matm 1 + tan2 #12

Of course, the angle 013 can always be reduced by an appropriate choice of the phases and in particular for F = F, so that there is no lower bound in this type of models. The effective neutrino Majorana matrix, which is relevant for nentrinoless double beta decay, simplifies such that

|mee| = |B|2 |g2

tan2 012 |F — F|2 = Smso i (A.55) ^[(1 + lFl2) (1 + |F|2) +tan2#12|F-F|2]2— |1 —FF|4

Again varying the phases and the modulus of F, we hud the maximal value for FF = -1,

tan 012 |mee| < 0.43 . A.56) \J2 T tan 2 #i2

Moreover, we can give a simple relation between m^ and the reactor angle,

me |F — F | sin 013 tan 012 . W.57) ^matm IFF + 1

Note that the singular value for FF +1 = 0 corresponds to a vanishing reactor angle. We can even derive a maximal value for the Dirac CP violation for this case. Prom Eqs. (4.21) and (4.27) we get

|B|2 (K — k2) Im (Q) Ji = — ;A.58) (1 + |F|2)2 [(1 + |F|2)2 (1 + |F|2 + |B|^2 — |1 — FF|4

|F — F|4 tan2 #12 (1 + tan2 #12) Im (Q) l + l^l2 [(l + |F|2) (1 + |F|2) + |F-F|2tan2#12]2- |1-FF|4

|F — F|4 tan2 012 (1 + tan2 012) Im [(1 + FF*) F* (F — F) e^23] ^matm 1 + |F | 3/2 : ((1 + |F|^ (1 + |F|^ + |F—F|2tan2012) —|1 — FF| 4

34 where A23 is the phase of g 2/g^^ ^^^^tor is maximal for FF 1 and F = —F, giving ^msoi 1 + tan^ 012 Ji| < sin A23 < 0.06 . (A.59) #?7%atm 2 tan ^12 (2 + tan^ ^12)^

Here the imaginary part is only given by the phase A23, but in more general cases the phases of F and F will play a role as well. So even for the CP violation in the leptonic sector, the model displays a suppression given by the ratio of the mass eigenvalues. Contrary to the quark case, however, the CP violation is not proportional to the smallest mass eigenvalue, but it can be non-vanishing even for m1 = 0.

B. CP Violation and Weak Basis Invariants

For completeness we discuss here the CP invariants in the case of an additional vectorial state. We prove that if the additional state is much heavier than the electroweak scale, the low energy CP violation can be expressed by the Jarlskog invariant defined from an effective 3 x 3 down quark mass matrix. The transformation of a Dirac spinor T(t, X) under parity and charge conjugation is given by

P^(t,x) P-1 = np Y0#,—x) , c T(t,x) c-1 = nc cT(t,x)\ where nP,C are non-obse^able phases. The matrix C obeys the relation y^C = —CyT. Since the Lagrangian is a Lorentz scalar, it only depends on fermionic held bilinears. Thus, we deduce the CP transformation for such terms,

CP ^ (CP)-1 = TTi , CP (CP)-1 = —Ty% , cp Tiy^j (CP)-1 = —TjY^i , CP TiyVTj (CP)-1 = —Ty^y'Ti .

Note that the operator transforns under CP as ^

Quark Sector. In the Standard Model, it is easy to verify the existence of the CP symmetry in the Lagrangian, up to mass terms. In general, the quark mass terms are CP invariant if and only if it is possible to find a weak basis transformation which realises

Hu* = WL , Hd* = WLHdWL , (B.3) where HIt follows that

WL [Hu,Hd] = (B.4)

35 such that, for r odd,

tr[Hu,Hd]r = 0 (B.5) is a necessary and sufficient condition for CP invariance [42]. The case of r = 1 is trivial: the tr^e of a commutator [H^ Hd] is zero. For r = 3 and three generations, we have

7sw = tr [Hu, Hd]3 = (mj - mj!) (mj - mU) (mj! - mU) (mj - mg (mj - mg (mj - mg , where the quantity J does not depend of the mass spectrum, and can be related, up to a sign, with the CKM matrix, V, as IJ| = |Im(Vi2^^3^22^23)1- We conclude that in order to have CP violation, we need to have J = 0. This quantity is the lowest weak basis invariant which measure CP violating effects and it has mass-dimension twelve. Apart from CP violation in the strong interactions, there is no other mechanism in the SM which can generate CP violating effects if J = 0. Note that in the chiral limit, mu = md = ms = 0, we do not generate CP violation even if J = 0. In the literature, the lowest weak basis invariant is called Jarlskog determinant M,

det [Hu, Hd] = 2i (mj mj (m, - mu m„ - mu (B.7) (m2 - mJ) (’»» - md) (mJ - md) . which is equivalent to the Eq. (B.6).^ The Jarlskog determinant is only applicable to the case of three generations, in contrast to the more general invariant in Eq. (B.5).

Now let us add a down quark isosinglet. The gauge couplings to quarks and their mass terms are (i, j = 1, 2, 3 and a =1, 2, 3,4):

— —-y= + h.c.) — edg^A^

— %—— 2 sin^ dg^) (B.8a) 2 COS vw = — (ULi URj + d,Li MdT dRa + (^L4 m" dR^) + h.C. (B.8b) where the matrices M^, Md and m^ ^^mension 3x3 3 x ^ mid 1x4, respectively. The electromagnetic current is given by dg^ = — ^dyd. The most general weak basis transformation consistent with the Lagrangian of Eq. (B.8 ) is:

UL > .dULj URi -^ (UR)ij URj , dRa -^ (UR)"^ dRa . (B.9) dLi Lj

3 For any 3 x 3 tr^d^s H^mitian matrix ^ has: tr M3 = 3 |M|.

- 36 - where % and UR are 3 x 3 unitary matrices, while U# is 4 x 4. Once we diagonalise the mass terms, the Lagrangian reads

g r/LiY^uLi - dLaYg ( ) dL^ - 2 sm2 ^ Zg > 2 cos \ / a^ = - (uLi Dui + (^La D#a dR^) + h.C (B.10) where = U[U IuL is a3 x 4 matrix. The number of independent phases which are related to CP violation is, for N = 3,

%cp = H (H + 1) - (H - 1) - 2H = - 1) = 3 . (B.ll)

With the matrices as defined in Eq. (B.8b ) and HU = MUMU, H# = M#M[], and h# = M^m#, we can write down a set of weak basis invariants,

Ii = Imtr HuHdhdh# , I2 = ImtrHUH^h^h# ,

Is = Imtr HU [Hu, Hd] hdhj , I4 = ImtrHuHjhdh# ,

I5 = ImtrHUH^h^h# , Ia = ImtrHU [Hu, HU] h#h# ,

I7 = ImtrHUH^Hu H# , (B.12) representing a set of necessary and sufficient conditions for having CP invariance in the quark sector [43]. In our model, H# and h# read

^ /1 Mi + |g-i| + / /I1/I2 /^1/Z H# ^1^2 |/2| + ^2 ^2^3 h# / 2 M2 + ^2M4 (B.13) ^1 ^3 ^2 ^3 | ^ S| + ^s \/3 MS + ^SM4 J

Since H^ mid H^ real, 17 vanishes. The remaining invariants are in general different from zero; the dominant terms are

2 (^1 + ^2) ^3M4 Im / 3 , 12 = m2 /1

.6 (^1 + M4 Im /3 ,

U (^1 + ^2) (^1 + ^2 + ^3 + /3) ^3M3M4 Im /3 , ^5 — mt ^4

Z = —mt (^1 + ^2) (^1 + ^2 + ^3 + /2) ^SMSM4 Im /S • (B.14)

Hence, CP is generally violated even by the presence of a single complex parameter /3. Note that this case is not equivalent to the chiral limit because both the charm and strange masses are different from zero, mc x /2 and ms ^ ju2 (albeit /2

37 ju2). As we might expect, the invariants vanish if all quarks of the hrst and second generation are massless. Now we single out the heavy eigenstate with the rotations V4, U4. While the action of V4 leav^ the in^riants unaffected, U strongly modifies them and reshuffles terms from one to the other. In fact after this transformation, h# vanishes to lowest order and survives only at order O(vEW/M2); then in the new basis all the invariants involving h#, i.e., /1 — /6 are suppressed by /M2 and vanish for M ^ On the other hand /7 is now non-vanishing and given by 17 — Imtr HUHfH. «)2 , (B.15) where H^ — Note that U4 also changes the weak interactions,

8J^w =----7= Ui^ (t/4 — I)i4 di + di^ (u\U/± — ll'j 6?4 -\- h,c,, (B.16) V 2 v / i4 so we expect both CP violation and CKM nnitarity violation from these terms as well. However, the mass of the heavy state is O (McuT) so that the contributions to low-energy processes are suppressed by a factor MEw/Mcy? and are negligible. Hence, at the electroweak scale, we are left to consider the single invariant

17 — ImtrHlHfHuHf2 , (B.17) which corresponds to the usual Jarlskog invariant J for three generations, but com­ puted for the effective quark mass m.

Lepton Sector. As discussed above, we can ignore the heavy states for low-energy CP violation and use the effective 3 x 3 Yukawa matrices instead. In the SM, extended by right-handed neutrinos, we have three mass terms for the leptons,

m^ e# + z/a, + ^ ^ + h.c. (B.18 )

In analogy to the quark sector, invariance of the mass terms under CP transformation requires

UtmeV — m* , mDW — mD , W^m#W — —MR , (B.19 ) where U, V, and W are unita^ matric^ acting in Savour space. Defining h — mDmD and H — m#mN, we obtain

Wth W — h* , W+H W — H* . (B.20)

Now we can write down the weak basis invariants

/f — Imtr hH m# h*m N, 12 — Imtr hH2m# h*m N, 13 — Imtr hH2mN h*m N H; (B.21)

38 for the three further invariants, substitute h = for h [26]. In the basis where the right-handed neutrino mass is diagonal, one obtains

/i — Mi M2 (M2 — M2) Im h12

+ Mi M3 (M2 — M"2) Im h23 + M^a (M2 — M2) Im h^ ,

/i — Mi M2 (M4 — M4) Im hl1 + M"M3 (Mg — M"4) Im h^ + M2M3 (Mg — M|) Im h|3 , /i — M3 M3 (M2 — M2) Im hi2 + M3 M3 (M2 — M2) Im h"3 + M3 M3 (M2 — M2) Im h^ .

If none of the M% vanish and there is no degeneracy, the vanishing of /^ /2, and /3 implies the vanishing of Im h^, Im hi3, and Im hls for CP invariance. Note that in our model, mD stan^ for the effective 3 x 3 part of the Dirac neutrino mass matrix, m#, as given in Eq. (A.23). Then we obtain from Eq. (4.3),

biz = , h\s = Ap\ + Bpl + Cpl , hg 3 = . (B.22)

The coefficients A,..., F are displayed in Eqs. (A.24). They are generically complex, so we do not expect CP to be conserved. As in the quark sector, these invariants are rather general and give the necessary conditions for the presence of CP violation. On the other hand, only few of the phases remain important also in the low-energy limit. In onr case, to study the low-energy Dirac invariant, we can use the analogue of the Jarlskog invariant,

1 o

as discussed in Section 4.3. Here, h^ — (m^)t m^ and AM^ and AMe2 are the products of the mass squared differences of the light neutrinos and charged leptons, respectively.

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