hep-ph/0109125

CERN–TH/2001-236

KEK-TH-779

CP Violation in the Minimal Supersymmetric Seesaw Model

John Ellis,
Junji Hisano,
Smaragda Lola and
Martti Raidal

CERN, Geneva 23, CH-1211, Switzerland

Theory Group, KEK, Oho 1-1, Tsukuba, Ibaraki 305-0801, Japan

National Institute of Chemical Physics and Biophysics,

Tallinn 10143, Estonia

Abstract

We study CP violation in the lepton sector of the supersymmetric extension of the Standard Model with three generations of massive singlet neutrinos with Yukawa couplings to lepton doublets, in a minimal seesaw model for light neutrino masses and mixing. This model contains six physical CP-violating parameters, namely the phase observable in oscillations between light neutrino species, two Majorana phases that affect decays, and three independent phases appearing in , that control the rate of leptogenesis. Renormalization of the soft supersymmetry-breaking parameters induces observable CP violation at low energies, including T-odd asymmetries in polarized and decays, as well as lepton electric dipole moments. In the leading-logarithmic approximation in which the massive singlet neutrinos are treated as degenerate, these low-energy observables are sensitive via to just one combination of the leptogenesis and light-neutrino phases. We present numerical results for the T-odd asymmetry in polarized decay, which may be accessible to experiment, but the lepton electric dipole moments are very small in this approximation. To the extent that the massive singlet neutrinos are not degenerate, low-energy observables become sensitive also to two other combinations of leptogenesis and light-neutrino phases, in this minimal supersymmetric seesaw model.

CERN–TH/2001-236

September 2001

## 1 Introduction

The solar [1, 2] and atmospheric [3] neutrino anomalies, which imply the existence of non-zero masses for the light neutrinos, provide the first experimental evidence for the existence of physics beyond the Standard Model (SM). A minimal extension of the SM includes three very heavy singlet neutrinos , whose Yukawa couplings to the light neutrinos explain naturally the smallness of their masses, via the seesaw mechanism [4]. At the same time, the electroweak scale must be stabilized against large radiative corrections. In particular, after introducing right-handed neutrinos, a quadratically-divergent contribution to the Higgs boson mass proportional to has to be cancelled. This is most commonly achieved by supersymmetrizing the theory, leading to the minimal supersymmetric extension of the Standard Model (MSSM) with singlet neutrinos.

Neutrino-flavour mixing originates from off-diagonal components in the Yukawa interaction , in a basis where the charged-lepton and singlet-neutrino mass matrices are real and diagonal. Renormalization effects due to this interaction also induce flavour mixings in the soft supersymmetry-breaking slepton mass terms [5]. This may lead to observable rates for charged-lepton flavour-violating (LFV) processes such as , - conversion in nuclei, and [6, 7, 8, 9], where denotes a generic light charged lepton. LFV is also observable in principle in rare kaon decays, but at rates that are likely to be far below the current bounds [10].

In general, is complex, leading to CP violation in neutrino oscillations and in the induced rare LFV processes, as well as in Majorana phases for the light neutrinos and in electric dipole moments for the charged leptons. The existence of CP violation in is also required if the observed baryon asymmetry in the Universe originated in leptogenesis [11]. The purpose of this paper is to clarify the relations between these different manifestations of CP violation in the lepton sector, and to present numerical estimates of the T-odd CP-violating asymmetry in decay, the electric dipole moments of the electron and muon. We argue that measurements of CP violation using charged leptons, combined with CP violation in the light-neutrino sector, in principle enable the leptogenesis phases to be extracted - within the framework of the minimal supersymmetric seesaw model.

If the solar-neutrino mass-squared difference and the element of the Maki-Nakagawa-Sakita (MNS) neutrino-mixing matrix are not too small, the CP-violating phase in , which is analogous to the Cabbibo-Kobayashi-Maskawa (CKM) phase in the quark sector, can be measured via CP- and T-violating [12] observables in neutrino oscillations using a neutrino factory or possibly a low-energy neutrino superbeam. The recent SNO result [2] encourages this possibility, since it further favours the large-mixing-angle (LMA) solution to the solar-neutrino deficit [13].

As mentioned above, processes that violate charged-lepton flavour can provide important complementary information on the leptonic CP-violating phases. These may be measured using intense sources of stopped muons. The SINDRUM II experiment is designed to be sensitive to [14], and the MECO project would be sensitive to [15]. The experiment with the sesitivity is proposed at PSI [16]. The PRISM project [6, 17] and the front ends of neutrino factories now under consideration at CERN [18] and elsewhere will provide beams of low-energy muons that are more intense by several orders of magnitude than any of the present facilities. This will enable the construction of stopped-muon experiments able to probe LFV processes with sensitivities , . The latter sensitivity opens the way to measuring the T-odd, CP-violating asymmetry .

A measurement of the CP-violating electric dipole moment (EDM) of the muon
with a sensitivity e cm would also be
possible [18]. However, because the Yukawa coupling constants
appear in the renormalization-group equations (RGEs) only in the
Hermitian combination , CP-violating phases are
induced only in the off-diagonal terms of the slepton masses. This
implies suppression of the EDMs of the electron and muon, whereas CP
violation may occur in full strength in charged LFV processes, such as
^{1}^{1}1In the light of very stringent constraints from
electron, neutron and mercury EDMs [19, 20, 21], we neglect the
possible phases in diagonal soft supersymmetry-breaking terms throughout
this paper. This is natural in mechanisms which generate only real soft
terms, such as gravity- [22], gauge- [23],
anomaly- [24], gaugino- [25] and
radion-mediation [26] mechanisms..

Another arena to probe LFV is provided by rare decays. There has been some discussion in the literature of decays [27], and one could in principle hope to measure CP-violating asymmetries in the various decays. Another possibility is to search for LFV in sparticle decays [28, 29], e.g., , where CP-violating asymmetries analogous to can also be defined in principle. However, we do not investigate these possibilities further in this paper.

We concentrate here on CP-violating observables in the sector, assuming that the only sources of LFV and CP violation are the interactions with heavy singlet neutrinos. We start by discussing general parametrisations of the Yukawa matrix in terms of the high- as well as the low-energy observables, paying particular attention to the counting of physical degrees of freedom and their relations to CP-violating observables. Subsequently, we analyse the renormalization-group running of soft supersymmetry-breaking terms, assuming universal boundary conditions at the GUT scale. Our first objective in this analysis is to demonstrate in principle the complementarity of the different observables, to see how all the CP-violating phases of the minimal seesaw model come into play, and to clarify the relationship of the observable phases to the phases appearing in leptogenesis. In the leading-logarithmic approximation, in which the heavy singlet neutrinos are treated as degenerate, this renormalization is sensitive to just one combination of the leptogenesis and light-neutrino phases, but two other combinations contribute beyond this approximation. We illustrate our results in a simple two-generation model. We then present numerical estimates of , and , taking into account the present knowledge of neutrino mixings and masses as well as bounds on sparticle masses. We find that the magnitude of is in general anti-correlated with the rate of , and may be large in some models compatible with the experimental upper limit on decay. If a cancellation occurs between different contributions to the vertex, so that the box and penguin diagrams contributing to become comparable in magnitude, the T-odd asymmetry in may be as large as , while remains appreciable. However, the EDMs of the and are rather small in the minimal seesaw model.

It is important to note that the neutrino-oscillation phase and the Majorana phases are completely independent of the three physical phases in the quantity that enters in leptogenesis calculations. On the other hand, and the other renormalization-induced observables depend on mixtures of the light-neutrino and leptogenesis phases. Thus, neutrino factories and LFV measurements provide complementary information on the leptonic CP-violating phases. In particular, observation of is possible even if CP violation in neutrino oscillations is unobservable, i.e., if either or the solar-neutrino deficit is not explained by the LMA solution. However, in the minimal supersymmetric seesaw model, it is possible that a combination of CP-violating observables in the neutrino and charged-lepton sectors may provide constraints on the angles and phases responsible for leptogenesis.

Our work is organized as follows. In Section 2, we consider general parameterisations of the neutrino Yukawa couplings and discuss CP violation in the minimal supersymmetric seesaw model. In Section 3, we give general formulae for the EDMs of the charged leptons, , and , including the latter’s T-odd asymmetry . We present the results of the numerical analysis in Section 4. Finally, Section 5 is devoted to a discussion and our conclusions concerning the observability of the CP-violating phases in the minimal supersymmetric seesaw model.

## 2 CP Violation in the Lepton Sector of the Minimal Supersymmetric Seesaw Model

We consider the MSSM with three additional heavy singlet-neutrino superfields , constituting the minimal supersymmetric seesaw model. The relevant leptonic part of its superpotential is

(1) |

where the indices run over three generations and is the heavy singlet-neutrino mass matrix. Taking account of the possible field redefinitions, this minimal supersymmetric seesaw model contains 21 parameters: 3 charged-lepton masses , 3 light-neutrino masses , 3 heavy Majorana neutrino masses , 3 light-neutrino mixing angles , 3 CP-violating light-neutrino mixing phases (the MNS phase and two Majorana phases), and 3 additional mixing angles and 3 more phases associated with the heavy-neutrino sector.

### 2.1 High-Energy Parametrization

In order to clarify the appearance and rôles of these parameters, we first analyze (1) in a basis where the charged leptons and the heavy neutrinos both have real and diagonal mass matrices:

(2) |

where . A priori, the neutrino Yukawa-coupling matrix has nine phases, which can be exposed by writing it in the form: where is diagonal and . However, in the basis (2) one may redefine the left-handed lepton fields , and thus rotate away the three phases in , which are unphysical. Thus the Yukawa-coupling matrix may be written in the form:

(3) |

The matrix is the analogue in the lepton sector of the quark CKM matrix, and thus it has only one physical phase. On the other hand, we can always write in the form

(4) |

where is a CKM-type matrix with three real mixing angles and one physical phase, and are diagonal matrices containing two phases each. Thus has 5 physical phases to add to that in , and all six real mixing angles and six phase parameters in this basis are physical observables.

We now study the combination of the Yukawa couplings, which governs leptogenesis in this minimal seesaw model. It is straightforward to see from (3) that

(5) |

which depends on just three of the CP-violating phases, namely the two phases in and the single residual phase in , as well as the three real mixing angles in . This is consistent with the observation that, since the overall lepton number involves a sum over the light-lepton species (both charged leptons and light neutrinos), one would not expect leptogenesis to depend on the 6 MNS angles and phases.

On the other hand, as we discuss in more detail below, mixing and CP violation in the slepton sector of this minimal supersymmetric seesaw model is controlled by the combination of the neutrino Yukawa couplings, in the leading-logarithmic approximation where . It is again straightforward to see from (3) that

(6) |

Therefore, in this approximation, CP violation in charged LFV processes arises only from the one physical phase in the diagonalizing matrix .

### 2.2 Low-Energy Parametrization

We now reconsider leptonic CP violation from a more familiar point of view [30], namely that of the effective low-energy theory obtained after the heavy neutrinos are decoupled. In this energy range, physics is described by the following effective superpotential:

(7) |

where the effective light-neutrino masses are given in the basis (2) by

(8) |

where GeV and as usual . The mass matrix can be diagonalized by a unitary matrix :

(9) |

where Since is a symmetric matrix and contains in general six phases, must also have 6 phases. It can be expressed in the form

(10) |

where and is the MNS matrix written in the CKM form:

(11) |

The phases in (10) can be removed by redefinition of the fields, leading to a new basis in which

(12) |

This differs from the basis (2) by the phase rotation . The new basis is appropriate if one works with the effective low-energy observables in the effective superpotential (7), e.g., for studying neutrino oscillations. Indeed, the mixing angles , whose we denote by , are measurable in neutrino-oscillation experiments, as is the CP-violating MNS phase . One combination of two CP-violating Majorana phases is in principle measurable in experiments.

The physical interpretation of the Yukawa couplings in (1) is made more transparent in the basis (2), which does not contain the unphysical low-energy phases in that we rotated away in the previous paragraph. Note that one must change if one works in the basis (12).

Our objective in this paper is to study CP-violating observables which are sensitive to different physical phases. For this purpose, we need a proper parametrization of the input parameters of the model. The most straightforward choice is to work in the basis (2) and to choose the physical observables in (2) and (3) as the input parameters. In this case, the physics is entirely transparent. However, the present experiments do not measure heavy-neutrino masses, their Yukawa-coupling and mixings directly. All the information we have on neutrinos comes from the low-energy neutrino-oscillation and experiments. If we choose the input parameters from (2), (3), we have to check every time that the induced in (8) agrees with the experimental data. Instead, one can attempt to use the effective low-energy observables as an input.

To this end, we first rewrite the seesaw mechanism in the different form:

(13) |

which is equivalent to (8). Starting with any given and as input parameters, we obtain as outputs the seesaw-induced low-energy parameters and , and an auxiliary complex orthogonal matrix . It is possible to choose different parameter sets for and that give the same low energy effective and , but lead to different values for .

One can turn the argument around [8], and parameterize the neutrino Yukawa-coupling matrix in terms of an arbitrary complex orthogonal matrix as follows:

(14) |

We emphasize that the output in this parametrization is in the low-energy basis (12), and therefore contains unphysical phases. If one wants to use the induced to parametrize the superpotential (1), one should be careful to count correctly the physical degrees of freedom.

We now form the combinations and out of (14). In the first case, we obtain

(15) |

which contains three independent physical phases that are given entirely in terms of the parameters in the orthogonal matrix This is consistent with (5), and the new parametrization therefore has not changed the counting of phases in On the other hand, we also obtain from (14)

(16) |

This expression also appears to contain three phases, which are combinations of all the parameters in and .

However, according to (6), is supposed to contain only one physical phase. What has happened? The answer is that physics has not changed, and thus two out of three phases in are unphysical. This is the case because we are working in the low-energy basis (12), and not in the basis (2). The three phases in , which were rotated away in defining , appear now in . Instead of (6), we now have . One overall phase is irrelevant, and the two unphysical relative phases in explain the faulty phase counting in (16).

In the following, we show explicitly that the unphysical phases in cancel out in the Jarlskog invariants which can be constructed using . Therefore, in the leading-logarithmic approximation, all the CP-violating LFV observables depend only on the one physical phase in (16), which is a combination of the phases in and Henceforward, we omit the superscript , but one must still be careful to distinguish between the different bases.

### 2.3 Relations to CP-Violating Observables

So far we have only considered the parametrization of the input neutrino parameters, which in general are complex, and the 6 resulting independent CP-violating phases. We now consider how physical observables depend on these various phases.

#### 2.3.1 Leptogenesis

At present, our only experimental knowledge on CP violation in the lepton sector may be obtained from the baryon asymmetry of the Universe, assuming that this originated from leptogenesis. In leptogenesis scenarios, initial asymmetries appeared in decays of the heavy neutrinos in the early Universe, as results of interferences between the tree-level and one-loop amplitudes for decays. The asymmetry in the decay of an individual species is given in the supersymmetric case [31] by

(17) | |||||

where and both triangular and self-energy type loop diagrams are taken into account. This asymmetry is converted into the observed baryon asymmetry by sphalerons acting before the electroweak phase transition. It is clear from (17) that the generated asymmetry depends only on the phases in . Hence, according to the parametrization (14), the only phases entering in the calculation of the baryon asymmetry of the Universe are those in . In order to demonstrate the feasibility of leptogenesis, it would be necessary to prove that at least one of the phases in is non-zero. Moreover, as we shall see, at least one of the real part of the mixing angles in must also be non-zero, and one would need to control other parameters, such as the heavy-neutrino mass spectrum, before being able to calculate the baryon asymmetry in terms of , or vice versa.

#### 2.3.2 CP Violation in Neutrino Oscillations

Measuring this is one of the main motivations for building neutrino factories. We assume that the real MNS mixing angles and the mass-squared differences are all non-vanishing, in which case the the MNS phase in (11) is in principle observable. It is, realistically, observable in long-baseline neutrino factory experiments if the LMA solution of the solar neutrino problem is correct. The Majorana phases do not affect neutrino oscillations at observable energies, but do affect decay. The conventional nuclear experiments measure one combination of the light-neutrino masses and the Majorana phases . As in the CKM case, one can introduce a Jarlskog invariant that characterizes the strength of CP violation in neutrino oscillations:

(18) | |||||

One sees explicitly that the Majorana phases cancel out in .

####
2.3.3 Renormalization of Soft Supersymmetry-Breaking Terms:

Flavor-Changing Processes

In the minimal supersymmetric seesaw model, renormalization induces sensitivity to the neutrino Yukawa couplings in the soft supersymmetry-breaking parameters in the slepton sector, in particular to the CP-violating phases in . These may have measurable effects on several CP-violating lepton observables, including asymmetries in LFV decays, which are observable in rare and/or decays, and electric dipole moments. The electric dipole moment as well as rare decays may be measurable using slow or stopped muons produced at the front end of a neutrino factory. In this subsection, we concentrate on the flavor-changing processes, such as asymmetries in LFV decays, and we will discuss flavor-conserving processes, such as the electric dipole moment.

The soft supersymmetry-breaking terms in the leptonic sector of the minimal supersymmetric seesaw model are

(19) | |||||

We assume that the soft supersymmetry-breaking terms have universal boundary conditions at the GUT scale GeV:

(20) |

At lower energies below and above the heavy-neutrino mass scale , which we assume to be , off-diagonal entries in generate via the renormalization-group running off-diagonal entries in the effective soft supersymmetry-breaking terms. In the leading-logarithmic approximation the flavor-dependent parts of the soft supersymmetry-breaking terms are given by

(21) |

Here, the Yukawa coupling constants are given at , and then is diagonal. This means that remains diagonal in this approximation. Below , the heavy neutrinos decouple, and the renormalization-group running is given entirely in terms of the MSSM particles and couplings, and is independent of . We use in our numerical examples full numerical solutions to the one-loop renormalization-group equations, but the approximate analytical solutions (2.3.3) are useful for a qualitative analysis.

It is important to notice that, in the leading-logarithmic approximation (2.3.3), the only combination of neutrino Yukawa couplings entering the renormalization-group equations is . This implies that CP-violating phases are induced only in the off-diagonal elements of and , and further indicates that the lepton-flavour conserving but CP-violating observables like the electric dipole moments of charged leptons are naturally suppressed [20], while CP violation in the charged LFV processes should occur in full strength. This is analogous to CP violation in the quark sector of the Standard Model, which is also directly related to flavour-changing processes. As we saw earlier (6), the combination depends on just one CP-violating phase, namely that in the matrix . Therefore, in the leading-logarithmic approximation, all slepton-induced observables are independent of the phases associated with leptogenesis, which are combinations of those in the matrices and (6), in the high energy parametrization. On the other hand, in the low-energy parametrization, depends on one combination of the phases in and , as explained in subsection 2.2.

Since depends on only one physical phase, there is only one invariant for describing the strength of CP violation in any process induced by sleptons. By analogy with the Standard Model quark sector, this can be taken to be [28]

(22) |

Additional invariants including the terms can be constructed:

(23) |

and cyclic permutations, and similar invariants with two or three factors. However, in this model they are all related to the basic invariant (22), and proportional to

(24) |

in the leading-logarithmic approximation (2.3.3). Here, .

The above analysis is modified when one includes in the renormalization-group running effects associated with the non-degeneracy of the heavy neutrinos: . In this case, in (2.3.3) is replaced as follows: , and is given by

(25) |

where is now interpreted as the geometric mean of the heavy singlet-neutrino mass eigenvalues . The first term in (25) contains the matrix factor

(26) |

which induces some dependence on phases in . In the three-generation case, there are two independent entries in the traceless diagonal matrix , so the renormalization induces in principle dependences on two new combinations of these phases, as well as the single phase in . Thus low-energy observables become sensitive to all three leptogenesis phases. However, the dependences on the two extra phases are suppressed to the extent that

####
2.3.4 Renormalization of Soft Supersymmetry-Breaking Terms:

Flavor-Conserving Processes

As mentioned above, since the CP-violating phases are in the off-diagonal components of the soft supersymmetry-breaking terms, the electric dipole moment of lepton is naturally suppressed. The following is the lowest-order combination of the Yukawa couplings and whose diagonal components have imaginary parts:

(27) | |||||

and the dominant contributions to the electric dipole moment are
proportional to it^{2}^{2}2
Similar studies for the electric dipole moment of neutron in the
MSSM are done in [32], assuming that all CP violating phases
come from the CKM matrix.
.
Since in
comes from the radiative correction to the soft supersymmetry-breaking
terms, the leading contribution to the electric dipole moment is
proportional to when
. The dependence on in comes from the radiative correction in the
soft supersymmetry-breaking terms or the tree-level mass matrix of the
charged sleptons. In this subsection, we present the Jarlskog
invariant for the soft supersymmetry-breaking terms contributing to
the
electric dipole moment in .
Also, we discuss cases where this approximation is invalid, namely
when i) , or ii)
non-degeneracy
between the heavy singlet neutrinos induces dependences of and on phases in the product .

In order to evaluate the contribution to the electric dipole moment in , we need the corrections to the soft supersymmetry-breaking terms at , which are

(28) |

Here, we neglect irrelevant terms with a trace over flavor indices, or which are flavor-independent. The Yukawa couplings are evaluated at . From these equations and (2.3.3), non-vanishing contributions to arise from the following combinations of :

(29) | |||||

(30) |

In (30), the combination arises from the tree-level mass matrix of the charged sleptons. It is found from (29,30) that the electric dipole moments depend strongly on and less on .

When , (30) is proportional to and (29) is not enhanced. On the other hand, terms such as

(31) |

are proportional to and . Thus, they may make sizeable contributions to the electric dipole moments for , even if .

If the heavy neutrinos are not degenerate in mass, they induce dependences of the soft supersymmetry-breaking terms on phases in , as mentioned in the previous Section, which then contribute to the electric dipole moments. The Jarlskog invariant depends on , and this factor suppresses the electric dipole moment when is small. In this case, the non-degeneracy of the heavy neutrinos may have a more important effect on the electric dipole moment. The corrections of

(32) |

Here, we neglect terms with factors. The
interesting
point is that the second term in can have
imaginary parts in the diagonal components, and thus can contribute to
the electric dipole moment ^{3}^{3}3
Whilst the combination has an imaginary part, it does not contribute to the electric
dipole moments, since .
.
Since phases in arise from
, we do not need three generations of leptons in order
for to have imaginary parts in the diagonal
terms. The behaviour of this contribution will be discussed in
the next subsection.

### 2.4 Two-Generation Model

We now demonstrate the interdependences of the above physical observables in a toy two-generation model. In this model, has no physical phase while there may be one phase in . We parametrize the light- and heavy-neutrino masses and as follows:

(33) |

In this model, the leptogenesis invariant is , where

(34) |

so that

(35) |

As explained above, the phase in controls leptogenesis, and the mixing angle must also be non-vanishing.

As concerns neutrino observables, we recall that there is no analogue of the MNS phase in this two-generation model. There is one CP-violating Majorana phase for the light neutrinos, but this does not contribute to leptogenesis, as we argued previously on general grounds and now see explicitly in (35).

We now consider the quantity which controls the renormalization of the soft super- symmetry-breaking terms in the leading-logarithmic approximation, in particular which has a non-zero imaginary part. For illustrational purposes, we assume that the light-neutrino mass matrix has maximal mixing:

(36) |

where is a light-neutrino Majorana phase. In this case,

(37) | |||||

(38) | |||||

We see that the imaginary part of the off-diagonal component depends both on the Majorana phase in (36) and the phase in . Even if it could be measured, and the neutrino mass eigenvalues were known, still only one combination of the angle factors entering in leptogenesis (35) would be known, and there would still be an ambiguity associated with the Majorana phase . In fact, no CP violation is induced by the renormalization (38) in this simple two-generation model, since it is not possible to define the Jarlskog invariant (22) and its analogues (23). Such invariants can be defined in a three-generation model, and CP-violating observables are demonstrably proportional to it, as we show in the next Section.

As commented in subsections 2.3.3 and 2.3.4, non-degeneracy between the heavy singlet neutrinos induces, via renormalization, dependences of the entries of and on phases in the product . In the two-generation case, this dependence is on the one phase in , since has no phases in this case. This makes changes in Arg and Arg, but these are suppressed to the extent that