CP Violation in neutrino oscillation and leptogenesis

CP Violation in neutrino oscillation and leptogenesis

We study the correlation between CP violation in neutrino oscillations and leptogenesis in the framework with two heavy Majorana neutrinos and three light neutrinos. Among three unremovable CP phases, a heavy Majorana phase contributes to leptogenesis. We show how the heavy Majorana phase contributes to Jarlskog determinant J as well as neutrinoless double beta decay by identifying a low energy CP violating phase which signals the CP violating phase for leptogenesis. For some specific cases of the Dirac mass term of neutrinos, a direct relation between lepton number asymmetry and J is obtained. For the most general case of the framework, we study the effect on J coming from the phases which are not related to leptogenesis, and also show how the correlation can be lost in the presence of those phases.

Finding any relation between baryogenesis via leptogenesis and low energy CP violation observed in the laboratory is a very interesting issue. The CP violation required for leptogenesis stems from the CP phases in the heavy Majorana sector, whereas CP violation measurable from the neutrino oscillations can be described by the neutrino mixing matrix. One interesting question concerned with the low energy leptonic CP violation is whether it can be affected by the CP violating phases responsible for leptogenesis. Several people have already discussed some potential connections between low energy CP violation and leptogenesis by using some ansatz, but it is still unclear how large the former can affect the latter in general. The major difficulty to quantify such a connection occurs due to lack of the available low energy data to fix parameters of the seesaw model.

The purpose of this paper is to examine in a rather general framework how leptogenesis can be related to the low energy CP violation by determining the parameters as many as possible from available low energy experimental results and cosmological observations. In order to make a quantitative analysis of the connection between low energy leptonic CP violation and leptogenesis, we consider the minimal CP violating seesaw model which has two heavy Majorana neutrinos and three light left-handed neutrinos; three two seesaw model. As will be shown later, to break CP symmetry, the required minimal number of singlet heavy Majorana neutrino is two in the seesaw model with three light lepton doublets. This three two seesaw model is consistent with recent data of neutrino oscillations and contains eight real parameters and three CP violating phases in the neutrino sectors which make this model more constrained and predictive compared with the general three three seesaw model with eighteen parameters. We will show that while all three CP violating phases contribute to low energy leptonic CP violation, only a single CP violating phase contributes to leptogenesis. We will also investigate how large the CP phase responsible for leptogenesis contributes to low energy CP violation by determining the independent parameters from available experimental results and cosmological observations. Finally, we will discuss the potential implication of CP violation measurable from neutrino oscillations on leptogenesis.

Let us begin our study by considering the leptonic sector of the three two seesaw model. In a basis where both heavy Majorana and charged lepton mass matrices are real diagonal, the Lagrangian is given by:

L equals negative VilmiliR minus VIimDijNRJ minus five NR CM; NRj, one where i equals one, two, three, j equals one, two and the Dirac mass term mp is three by two matrix. Here, we remark that the Dirac mass matrix mp contains three N minus three unremovable CP phases if we take N singlet heavy Majorana neutrinos in this basis. Thus, one can easily see that at least two singlet heavy Majorana neutrinos are required to break CP symmetry in the seesaw model with three lepton SU two doublets. The three by two matrix mp can be generally parameterized as:

mp equals UL zero m two zero zero m three

VR, two

UL equals O two three zero L two three U one three zero L one three eight L O one two zero L one two P Y L,

minus sine OR cosine OR minus minus, three zero where Oij and Uij denote the rotations of i, j plane, P Y L equals diag. one, e to the negative i Y L over two, e to the i Y L over two, and m two and m three are real and positive. Without loss of generality, we can choose M two less than or equal to M three. The allowed range for the angles and the phases is ceiling negative pi, pi. There are three CP violating phases, gamma R which appears in V R, gamma L and delta L in U L. In a different basis with complex M i, gamma R can be interpreted as a heavy Majorana phase. The lepton number asymmetry for the lightest heavy Majorana neutrino N one decays into l to the power of mp phi to the power of plus or minus is given by;

epsilon one equals the fraction with numerator Gamma one minus bar Gamma one over denominator Gamma one plus bar Gamma one equals negative the fraction with numerator three M one over denominator two M two V squared the fraction with numerator I m left brace left parenthesis m D dagger m D right parenthesis one two squared right brace over denominator left parenthesis m D dagger m D right parenthesis one one,

where V equals square root four pi v with v equals two four six gigaelectronvolts, and

I m left brace left parenthesis m D dagger m D right parenthesis one two squared right brace equals left parenthesis the fraction with numerator m two squared minus m three squared over denominator two right parenthesis squared sine squared two theta R sine two gamma R.

We see that CP violation concerned with leptogenesis can be possible only if the mixing angle theta sub R and CP violating phases gamma sub R for the heavy Majorana neutrinos are non-zero. For our purpose, let us now study how the phase gamma sub R contributes to CP violation in the neutrino oscillations which is usually described in terms of the MNS neutrino mixing matrix. The effective mass matrix for light neutrinos is given by m sub eff equals negative m sub D times one over M times m sub D transposed equals negative U sub L m V sub R times one over M times V sub R transposed m transposed U sub L transposed, and is diagonalized by the MNS mixing matrix as U sub MNS dagger m sub eff U sub MNS star equals diag [n sub one, n sub two, n sub three]. Then, the MNS mixing matrix is decomposed into two mixing matrices as follows;

U sub MNS equals U sub L K sub R.

where K sub R is a unitary matrix diagonalizing the matrix Z equivalent to negative m V sub R times one over M times V sub R transposed m transposed and parameterized by,

K sub R equals left parenthesis one, zero, zero; zero, cosine theta, sine theta e to the power of negative i phi; zero, negative sine theta e to the power of i phi, cosine theta right parenthesis left parenthesis one, zero, zero; zero, e to the power of i alpha, zero; zero, zero, e to the power of negative i alpha right parenthesis.

Then, negative K sub R dagger m V sub R times one over M times V sub R transposed m transposed K sub R star equals diag [zero, n sub two, n sub three]. Note that the three, two seesaw model predicts one massless neutrino. In addition, theta, phi and alpha are determined as:

phi equals Arg left parenthesis Z sub two two star Z sub two three plus Z sub two three star Z sub three three right parenthesis, tangent two theta equals the fraction two times the absolute value of Z sub two two star Z sub two three plus Z sub two three star Z sub three three over the absolute value of Z sub three three squared minus the absolute value of Z sub two two squared, two alpha equals Arg left parenthesis cosine squared theta Z sub two two plus sine squared theta Z sub three three e to the power of negative two i phi minus sine two theta Z sub two three e to the power of negative i phi.

We remark that the mixing angle theta sub R and CP violating phase gamma sub R have been transferred to theta, phi and alpha. As one can see from the above formulae, leptogenesis occurs only if the mixing angle theta sub R and CP violating phase gamma sub R are non-zero, which in turn implies non-vanishing phi, alpha, theta via Equation eight. As we will see below, the CP phase phi contributes to CP violation in neutrino oscillations, so that it is anticipated that there is correlation between CP violation generated from neutrino mixings and leptogenesis.

To see this concretely, let us compute Jarlskog determinant J equals Im left parenthesis U sub MNS e one U sub MNS e two star U sub MNS mu one star U sub MNS mu two right parenthesis, which is proportional to the CP asymmetry in neutrino oscillation, Delta P equals P left parenthesis nu sub mu to nu sub e right parenthesis minus bar P left parenthesis nu sub mu to bar nu sub e right parenthesis equals four J left parenthesis sine left parenthesis the fraction Delta m sub one two squared L over two E right parenthesis plus sine left parenthesis the fraction Delta m sub two three squared L over two E right parenthesis plus sine left parenthesis the fraction Delta m sub three one squared L over two E right parenthesis right parenthesis. By using Equation six, we obtain:

J equals the fraction one over eight sin two theta subscript L one two sin two theta subscript L one three left bracket c subscript L one three cos two theta sin delta subscript L sin two theta subscript L two three right bracket plus c subscript L one two sin two theta sin left parenthesis delta subscript L minus gamma subscript L minus phi right parenthesis cos two theta subscript L two three left bracket minus the fraction one over two s subscript L one two s subscript L one three sin two theta sin two theta subscript L two three sin left parenthesis two delta subscript L minus gamma subscript L minus phi right parenthesis right bracket plus the fraction one over eight sin two theta sin two theta subscript L two three sin left parenthesis gamma subscript L plus phi right parenthesis times left parenthesis sin two theta subscript L one two c subscript L one three s subscript L one two minus sin two theta subscript L one three s subscript L one three c subscript L one two right parenthesis. Nine

From the expression of J, it is obvious that all three CP violating phases delta subscript L, gamma subscript L and phi contribute to CP violation in the neutrino oscillations, and that the CP phase gamma subscript I always hangs around phi. Since only phi is closely related to leptogenesis, in order to investigate the interplay between CP violation for leptogenesis and low energy leptonic CP violation, we should determine the contributions of the phases left parenthesis delta subscript L, gamma subscript L right parenthesis and phi separately as well as to fix the parameters theta prime s. Before discussing the correlation between both CP violations, let us study how we can get some information on the mixing angles and CP phases from the available experimental and cosmological results. The mixing angles and CP phases can be classified into two categories, one contains theta, phi and alpha which are related to phenomena at high energy and the other contains parameters in U subscript L. First of all, we show how we can estimate the allowed values of CP violating phase phi and mixing angle theta. The information on phi and alpha may come from the constraints on light neutrino mass spectra as well as cosmological condition for leptogenesis. To see this, we first present the parameters gamma subscript R, theta subscript R, m subscript two, m subscript three and lepton number asymmetry epsilon subscript one in terms of some physical quantities which will be taken as inputs in numerical calculation. Here, we choose the heavy Majorana neutrino masses left parenthesis M subscript one, M subscript two right parenthesis, their decay widths left parenthesis Gamma subscript one, Gamma subscript two right parenthesis, and light neutrino masses left parenthesis m subscript two, m subscript three right parenthesis as the physical input parameters. As will be clear later, it is convenient to define two parameters x subscript i left parenthesis i equals one, two right parenthesis x subscript i equals the fraction left parenthesis m subscript D dagger m subscript D right parenthesis subscript i i over M subscript i equals Gamma subscript i left parenthesis the fraction V over M subscript i right parenthesis squared. Ten

Then, by considering the light neutrino mass eigenvalue equation, det left parenthesis m subscript eff m subscript eff dagger minus bar n squared right parenthesis equals zero, the lepton number asymmetry epsilon subscript one and the phase gamma subscript R can be written in terms x subscript one, x subscript two, m subscript two, and

N sub three, epsilon sub one equals negative three M sub one divided by four x sub one V squared square root of open parenthesis n sub minus squared minus x sub minus squared close parenthesis open parenthesis x sub plus squared minus n sub plus squared close parenthesis, cosine two gamma sub R equals n sub two squared plus n sub three squared minus x sub one squared minus x sub two squared divided by two open parenthesis x sub one x sub two minus n sub two n sub three close parenthesis.

where n plus equals n sub three plus or minus n sub two and x plus equals x sub one minus x sub two. There are two solutions of equation twelve leading to negative epsilon sub one; Y sub R and Y sub R minus T for zero less than Y sub R less than T over two, which in turn gives positive baryon number via sphaleron process. Next, let us present the parameters, O sub R, m sub two and m sub three in terms of the above six physical quantities. From the eigenvalue equations for V R mom P V R t we can express O sub R, m sub two, and m sub three as follows;

open parenthesis m sub two, m sub three close parenthesis equals V M sub one M sub two open parenthesis zero plus minus P, zero plus plus P close parenthesis.

open parenthesis cosine O sub R, sine O sub R close parenthesis equals o sub plus P two P

where o plus equals x sub two L x sub R two V R apostrophe equals one open parenthesis x sub one x sub two minus n sub two n sub three close parenthesis plus o squared, and R equals M sub one over M sub two. We also determine omega, alpha, and zero with a given P set of parameters open parenthesis x sub one, x sub two, R, n sub two, n sub three close parenthesis by using the same procedure given in equations seven and eight. We take R equals zero point one. In order to determine the values of omega, zero and alpha, it is necessary to determine those of x i open parenthesis i equals one, two close parenthesis. Let us now show how the variables x i can be constrained. From the neutrino mass eigenvalue equation, it follows that open parenthesis x sub one minus x sub two close parenthesis less than n sub three minus twelve, n sub three plus n sub two less than or equal to x sub one plus twelve.

From the experimental results for the neutrino oscillation, let us take n sub three equals V A m prime at m approximately five times ten to the negative two electron volts and n sub two equals V A m two o l a r equals seven times ten to the negative three electron volts. From equation fifteen, the lower bound on x sub one is zero point zero zero seven electron volts. By solving the Boltzmann equation, we can obtain a value of Y sub L equals comma i.e., lepton number density n sub L normalized by entropy density s. When solving the Boltzmann equation, we need the value of epsilon sub one. For fixed x sub one, one can get the maximum value of negative epsilon sub one which gives the maximum negative Y sub L via Boltzmann equation. In Figure one, we plot the maximum lepton number density negative Y sub L predicted from equation eleven as a function of x sub one for several fixed M sub one. We set the initial conditions for Boltzmann equation at ten to the sixteen GeV and we take the distribution of the heavy majorana particle N sub one in thermal equilibrium and Y sub L equals zero at the temperature.

The allowed values of negative Y sub L consistent with baryogenesis are presented by shaded band in Figure one. Thus, we can obtain the allowed region of x sub one for a fixed M sub one. However, there is no allowed value of x sub one for a rather lower value of M sub one less than one point zero times ten to the eleven GeV, which in turn leads to the lower bound on M sub one. By using the allowed region for x sub one as given in the above, we can estimate the allowed region of x sub two via equation eleven again. Figure two shows how we can get the allowed region of x sub two. For example, for a given set M sub one equals two times ten to the eleven GeV, and x sub one equals zero point zero three electron volts, we obtain the allowed range for x sub two as zero point zero three less than x sub two less than zero point zero seven electron volts.

Let us move to the other category of the parameters, zero L twelve, zero L twenty-three, zero L thirteen, Y sub L, omega L in U L, which are not determined from high energy phenomena, but must be related to the low energy M N S mixing matrix. Thus two of them can be determined from the neutrino oscillation experimental results. For simplicity, we focus on the case with the small mixing angles zero L thirteen and zero, which is consistent with Chooz experiment. In this case, the M N S mixing matrix is given, in the leading order, by

U M N S approximately equal

C L twelve S L twelve S L thirteen e to the negative i omega L plus S L twelve P e to the negative i D prime S L twenty-three

S L twelve S L twenty-three minus C L twelve S L twenty-three C L twenty-three minus S L twelve C L twenty-three C L twelve C L twenty-three x P open parenthesis a prime, minus a prime close parenthesis.

where a prime equals omega plus Y sub L, a prime equals a minus and P open parenthesis a prime, minus a prime close parenthesis equals diagonal open bracket one, e to the i a prime, e to the negative i a prime close bracket. Note that we do not present the sub-leading contributions in open parenthesis U M N S close parenthesis i j, open parenthesis e three close parenthesis, which are comparable to open parenthesis U M N S close parenthesis e three. Taking zero twenty-three approximately equal to zero twelve approximately equal to hash which lead to bi-large mixing pattern, in this approximation, open parenthesis U M N S close parenthesis e three, J and absolute value open parenthesis m sub e f close parenthesis e e absolute value are given by:

absolute value open parenthesis U M N S close parenthesis e three approximately equal to absolute value S L thirteen cosine omega L plus five cosine D prime, J equals S L thirteen sine omega L plus five sine D prime,

open parenthesis m sub e f close parenthesis e e approximately equal to twelve e t i D prime plus n three open parenthesis S L thirteen cosine omega L plus five cosine D prime close parenthesis squared.

In principle, we are able to fix three unknown parameters; Y sub L, S L thirteen and D prime once the left-hand sides of equation seventeen are measured. It is then possible to quantitatively see whether the low energy CP violation denoted by J is dominated by leptogenesis phase phi or by the CP violating phases gamma sub L and delta sub L which are not related to leptogenesis. First of all, let us study the interesting case of theta sub L one three equals zero, which makes the analysis more predictive because a CP violating phase delta sub L is simultaneously suppressed. This can be understood as the extreme case of Sin theta sub L one three much less than Sin theta. Interestingly enough, this case dictates that the origin of absolute value of (U sub M N S) sub e three may come from the mixing angle theta which is related to heavy Majorana neutrino sector. Jarlskog determinant is then simply given by,

<LATEX>J = \sin 2 \theta \sin \phi ^ { \prime } \frac { 1 } { 8 \sqrt { 2 } } .</LATEX> (18)

Only gamma sub L is a completely arbitrary parameter in this case and thus we can easily investigate how J can be affected by gamma sub L. In other words, gamma sub L can be estimated through J in this case. If gamma sub L is turned out to be much smaller than phi, the measurement of CP violation in low energy experiment may directly indicate leptogenesis.

In Figure three, we show the correlation between lepton number asymmetry epsilon sub one and J in Equation eighteen with gamma sub L equals zero. In each contour, M sub one and x sub one are fixed and x sub two is varied. For x sub one equals zero point zero one electron volt, we obtain absolute value of (U sub M N S) sub e three equals Sin theta divided by square root two approximately equal to zero point zero six six and absolute value of (meff) sub ee equals zero point zero zero eight electron volt, while for x sub one equals zero point zero three electron volt, absolute value of (U sub M N S) sub e three is in the range [zero point one five, zero point two] and absolute value of (meff) sub ee is in the range [zero point zero one four, zero point zero one eight] electron volt. On the other hand, the case with Sin theta much less than Sin theta sub L one three, we see from Equation seventeen that J mainly depends on delta sub L, which has nothing to do with leptogenesis.

We have studied how CP violation responsible for baryogenesis manifests itself in MNS matrix and Jarlskog determinant which signals low energy CP violation in neutrino oscillation. We have obtained cosmological constraints on CP violation and mixing which originate from high energy phenomena. Then using the low energy constraints we have showed it is possible to estimate the size and sign of baryon number in the most general case once absolute value of (U sub M N S) sub e three, neutrinoless double beta decay and CP

violation of neutrino oscillation are measured. In a specific case of the framework, a correlation between CP violation in neutrino oscillation and leptogenesis has been studied and the size of J has been estimated.