= ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ
NEUTRINO MASSES, MIXINGS, AND FCNC'S IN AN S3 FLAVOR SYMMETRIC EXTENSION OF THE STANDARD MODEL
©2011 A. Mondragon1)*, M. Mondragon1), E. Peinado2)
Received December 17, 2010
By introducing three Higgs fields that are SU(2) doublets and a flavor permutational symmetry, S3, in the theory, we extend the concepts of flavor and generations to the Higgs sector and formulate a Minimal S3-Invariant Extension of the Standard Model. The mass matrices of the neutrinos and charged leptons are re-parameterized in terms of their eigenvalues, then the neutrino mixing matrix, VPMNS, is computed and exact, explicit analytical expressions for the neutrino mixing angles as functions of the masses of neutrinos and charged leptons are obtained in excellent agreement with the latest experimental data. We also compute the branching ratios of some selected flavor-changing neutral current (FCNC) processes, as well as the contribution of the exchange of neutral flavor-changing scalars to the anomaly of the magnetic moment of the muon, as functions of the masses of charged leptons and the neutral Higgs bosons. We find that the S3 x Z2 flavor symmetry and the strong mass hierarchy of the charged leptons strongly suppress the FCNC processes in the leptonic sector, well below the present experimental bounds by many orders of magnitude. The contribution of FCNC's to the anomaly of the muon's magnetic moment is small, but not negligible.
1. INTRODUCTION
The observation of flavor oscillations of solar, atmospheric, reactor, and accelerator neutrinos established that they have nonvanishing masses and mix among themselves, much like the quarks do [1 — 20]. This discovery brought out very forcibly the need of extending the Standard Model (SM) in order to accomodate in the theory the new data on neutrino physics in a consistent way that would allow a unified and systematic treatment of the observed hierarchy of masses and mixings of all fermions. At the same time, the number of free parameters in the extended form of the SM had to be drastically reduced in order to give predictive power to the theory. These two seemingly contradictory demands are met by a flavor symmetry under which the families transfom in a nontrivial fashion.
In the Minimal S3-Invariant Extension of the SM [21—27], the concept of flavor and generations is extended to the Higgs sector in such a way that all the matter fields — Higgs, quarks, and lepton fileds, including the right-handed neutrino fields — have three species and therefore transform under the flavor symmetry group as the three-dimensional representation 1 © 2 of the permutational group S3. A model with more than one Higgs SU(2) doublet
'■'Institute de Fisica, Universidad Nacional Auto noma de Me xico, Me xico.
2)Institut de Fisica Corpuscular—CSCI, Universitat de Valencia, Spain.
E-mail: mondra@fisica.unam.mx
has tree level flavor changing neutral currents whose exchange may give rise to lepton flavor violating processes and may also contribute to the anomalous magnetic moment of the muon. An effective test of the phenomenological success of the model is obtained by verifying that all flavor changing neutral current processes and the magnetic anomaly of the muon, computed in the S3-Invariant extended form of the SM, agree with the experimental values.
2. THE MINIMAL S3-INVARIANT EXTENSION OF THE STANDARD MODEL
In the SM analogous fermions in different generations have identical couplings to all gauge bosons of the strong, weak, and electromagnetic interactions. Prior to the introduction of the Higgs boson and mass terms, the Lagrangian is chiral and invariant with respect to permutations of the left and right fermionic fields.
The six possible permutations of three objects (/i, f2, f3) are elements of the permutational group S3. This is the discrete, non-Abelian group with the smallest number of elements. The three-dimensional real representation is not an irreducible representation of S3. It can be decomposed into the direct sum of a doublet fD and a singlet fs, where
/, = ^(/l+/2 + /3), (1)
1075
7*
The direct product of two doublets pD = (pD1,pD2) and qD = (qD1,qD2) may be decomposed into the direct sum of two singlets rs and ry, and one doublet
r^, where
QT = (UL,db),UR ,dR,
LY = LYD + LYr, + LYT? + L
Yu
-Ye
where
and
к =
0 1 1 0.
n =
1 0 01
LYd and LYu have similar expressions to LYe and LYv , respectively.
Furthermore, we add to the Lagrangian the Majorana mass terms for the right-handed neutrinos
rs = PDlQDl + PD2QD2, (2)
ry = PDlQD2 - PD2QD1,
rTD = (rD1,rD2)= (3)
= (PDiqD2 + PD2qD1,PDiqD1 - PD2qD2).
The antisymmetric singlet ry is not invariant under S3.
Since the SM has only one Higgs SU(2)L doublet, which can only be an S3 singlet, it can only give mass to the quark or charged lepton in the S3 singlet representation, one in each family, without breaking the S3 symmetry.
Hence, in order to impose S3 as a fundamental symmetry, unbroken at the Fermi scale, we are led to extend the Higgs sector of the theory. The quark, lepton, and Higgs fields are
(4)
Lm = -MivjRCvm - M3VRCV3R. (9)
The extended Higgs sector has three SU(2) doublets, in a reducible representation 1S © 2 of the flavor group S3. The Higgs potential, invariant under S3, has an additional reflection symmetry R: Hs ^ -Hs, and an accidental permutational symmetry S'2: H1 ^ ^ H2. Hence, (H\) = (H2). Then the Yukawa interactions yield mass matrices for all fermions in the theory, of the general form [21]
M
+№2 №2
V
№2
\
№5
№1 - №2 №5
№4
(10)
The Majorana mass for the left-handed neutrinos vL is generated by the see—saw mechanism. The corresponding mass matrix is given by
(11)
Mv = Mvd M-1(MvD )T,
LT =(vL,eL),eR,vr, and H,
in an obvious notation. All of these fields have three species, and we assume that each one forms a reducible representation 1S © 2. The doublets carry capital indices I and J, which run from 1 to 2, and the singlets are denoted by Q3, U3R, d:iR, L3, e3R, vr, and HS. Note that the subscript 3 denotes the singlet representation and not the third generation. The most general renormalizable Yukawa interactions of this model are given by
where M = diag(Mi,Mi,M3).
In principle, all entries in the mass matrices can be complex since there is no restriction coming from the flavor symmetry S3. The mass matrices are diagonalized by bi-unitary transformations as
Ud(u,e)LMd(u,e) Ud(u,e)R = = diag(md( u,e))
UV Mv Uv = diag(mvi ,mV2 ,mV3).
The entries in the diagonal matrices may be complex, so the physical masses are their absolute values.
The mixing matrices are, by definition,
vckm = UlLUdL, vpmns = uIlUV K, (13)
where K is the diagonal matrix of the Majorana phase factors.
(12)
(5)
(6)
Lye = -YfLjHseiR - V3eL3Hse3R -- Yi[LikijHiejR + LinijЩет] -- Y4eL3H1 eiR - YgLiHie3R + h.c.,
Ly„ = -YÏLi(io2)H*sviR - (7)
- YVL3(102)HSV3R - Y2V[Likij(iG2)H*lvjR + + Linij(i02)H2vjR - Y4L3(102)Hjvir -- Y5VLi(i02)Hjv3R + h.c.,
(8)
3. THE MASS MATRICES IN THE LEPTONIC SECTOR AND Z2 SYMMETRY
A further reduction of the number of parameters in the leptonic sector may be achieved by means of an Abelian Z2 symmetry. A possible set of charge assignments of Z2 compatible with the experimental data on masses and mixings in the leptonic sector is given in Table 1. These Z2 assignments forbid the following Yukawa couplings Yf, Y3f, YV, and Y5V. Therefore, the corresponding entries in the mass matrices vanish, i.e., nl = = 0 and ¡i\ = nV = 0.
Y
v
3.1. The Mass Matrix of the Charged Leptons readily be expressed in terms of the charged lepton
The remaining three parameters in the mass ma- masses [23]- The resulting expression for Me, written trix of the charged leptons \ ß2\, \ß41, and \ß5\ may to order (mßme/m2)2 and x4 = (me/mM)4 is
Me
mT
i
X^/l+x^-r
1
V2 Vl+x2
1 "V
\/2 Vl+x2
rhe(l+x2)
V2 Vl+x2
1 rhu
V2V1+X2
rhe(l+x2) eiSe ^/l+ic2 —mj
1 y/2
1
V2
1+x2—m,u \
1+x2
1 -j-x2—m2^
1+x2
(14)
0
This approximation is numerically exact up to order matrix VPmns may be written in the polar form as
10-9 in units of the r mass Notice that this matrix UeL = PeLOeL [24], where PeL is a diagonal matrix of has no tree parameters other than the Dirac phase oe.
rpi , . TT , ,. ,. „ ,,( phases and the orthogonal matrix OeL can be written 1 he unitary matrix UeL that diagonalizes MeMe
and enters in the definition of the neutrino mixing as Me, as follows
/
O
eL
(1+2 m2 +Ax2+mfi+2ml)
{l-2fhl+fhl-2fhl)
V2 ^1+m2+5ik2—m4 —m®+m2+12iK4 \f2 ^J1—4mjJ-j-x2 —4m® — 5m2
1 (1+4x2-m U—2m 2)
--7Z X
V2 ^l+m2+5£2-m4-m®+m2+12ic4 yJl-\-2x2 —rh^ — m2(l+m2 -j-x2 —2m,2) ^/l+m2 -j-5x2 — m4 — m® +m2+12ic4
(1-2 ml+mp
1
a/2 1
a/2
x
a/2 yj 1—4m2 -j-x2 +6m4 — 4m® — 5m2 (1+a;2 -m2 - 2m2) ^/l+2a;2 -m2 -m2' Vl+^m^m^ 1—4m2 -j-x2 +6m4 —4m® —5m2 ^/l+ic2 —m2 J
(15)
where, as before, mß = mß/mT, rße = me/mT, and x = me/mß.
3.2. The Mass Matrix of the Neutrinos
According to the Z2 selection rule, the mass matrix of the Dirac neutrinos takes the form
M
VD
tä -j V o
(16)
Then, the mass matrix for the left-handed Majorana neutrinos, Mv, obtained from the see—saw mechanism, Mv = MVD MM-1 (MvD )T, is
Mv
2(p2)2
0
2p2 p4 2) 0 \2p2 p4 0 2(PV )2 + (fV)2)
0 2(p^2
where pV = (¡1V)/Ml/2, p4 = (¡V)/M-12, and p\ =
= (¡4)/Ml/2; Mi and M3 are the masses of the right-handed neutrinos appearing in (9).
The non-Hermitian, complex, symmetric neutrino mass matrix Mv may be brought to a diagonal form by a unitary transformation, as
UT Mv Uv = (18)
= diag (\mVl \e^, \mv2\e^2, \m„3\'
where UV is the matrix that diagonalizes the matrix
mM .
Table 1. Z2 assignment in the leptonic sector
(17)
- +
Hs, R Hi, L3, LI, e^R, eiR, VIR
As in the case of the charged leptons, the matrices MV and UV can be reparametrized in terms of the complex neutrino masses. Then [23, 24]
(
Mv =
m
V3
0
mV3 - mVl)(m„2 - mV3)e iSv 0 (mUl + mV2 - mV3)e
0 л/{mv3 - mvi){mv2 - mV3)e
—iSv\
m
V3
0
— 2i&
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