Bled Workshops in Physics Vol. 15, No. 2
A
Proceedings to the 17th Workshop What Comes Beyond ... (p. 93) Bled, Slovenia, July 20-28, 2014

7 General Majorana Neutrino Mass Matrix from a Low Energy SU(3) Family Symmetry with Sterile Neutrinos
A. Hernández-Galeana*
Departamento de Física, Escuela Superior de Física y Matemáticas, I.P.N., U. P. "Adolfo Lopez Mateos". C. P. 07738, Mexico, D.F., Mexico
Abstract. Within the framework of a local SU(3) family symmetry model, we report a general analysis of the mechanism for neutrino mass generation and mixing, including light sterile neutrinos. In this scenario, ordinary heavy fermions, top and bottom quarks and tau lepton, become massive at tree level from Dirac See-saw mechanisms implemented by the introduction of a new set of SU(2) L weak singlet vector-like fermions, U, D, E, N, with N a sterile neutrino. Right-handed and the NL,R sterile neutrinos allow the implementation of a 8 x 8 general Majorana neutrino mass matrix with four or five massless neutrinos at tree level. Hence, light fermions, including light neutrinos get masses from radiative corrections mediated by the massive SU(3) gauge bosons. We report the corresponding Majorana neutrino mass matrix up to one loop. Previous numerical analysis of the free parameters show out solutions for quarks and charged lepton masses within a parameter space region where the vector-like fermion masses MU , MD , Me, and the SU(3) family gauge boson masses lie in the low energy region of 0(1 — 20) TeV, with light neutrinos within the correct order of square neutrino mass differences: mf — mf ra 7 x 10-5 eV2, mj — mf ra 2 x 10-3 eV2, and at least one sterile neutrino of the order ra 0.5 eV. A more precise fit of the parameters is still needed to account also for the quark and lepton mixing.
Povzetek. Avtor pojasnjuje pojav družin pri leptonih tako, da uporabi za opis družin model z lokalno simetrijo SU(3). Trem družinam kvarkov in leptonov doda se družinski triplet desnorocnih nevtrinov, ki nosi samo druzinski naboj, levorocni in desnorocni U in prav tak D kvark, ki nosijo poleg barve le hiper naboj, levorocni in desnorocni nevtrino, ki ne nosita nobenega naboja, ter levorocni in desnorocni elektron s hipernabojem (—2). Vsi ti novi delci so masivni. Novi fermioni poskrbijo na drevesnem nivoju samo za maso tretje druzine kvarkov in leptonov. Lahkim fermionom, tudi lahkim nevtrinom, priskrbijo maso popravki v naslednjih redih pri interakciji z masivnimi bozoni, ki nosijo druzinsko kvantno stevilo. Avtor izračuna masno matriko 8x8 za Majoranine nevtrine do prvega reda. Proste parametre modela doloci z izmerjenimi masami in mesalnimi matrikami. Po dosedanjih izracunih so primerne vrednosti za mase fermionov MU , MD , ME in za maso druzinskega tripleta umeritvenega bozona v intervalu 0(1 — 20) TeV, za izmerjene masne razlike lahkih nevtrinov m22 — mi 2 ra 7 x 10-5 eV2, m32 — mi 2 ra 2 x 10-3 eV2 lahko avtor poskrbi s se vsaj enim sterilnim nevtrinom, ki ima maso ra 0.5 eV. Avtor pricakuje, da bo z bolj natancnimi izracuni lahko s pomocjo tega modela pojasnil mesalne matrike kvarkov in leptonov.
* E-mail: albino@esfm.ipn.mx
94 A. Hernandez-Galeana
7.1	Introduction
Although the standard picture with three light flavor neutrinos has been successful to describe the neutrino oscillation data. On the other hand, there have been recent hints from the LSND and MiniBooNe short-baseline neutrino oscillation experiments[1,2] on the possible existence of at least one light sterile neutrino in the eV scale, which mix with the active neutrinos. On the other hand, an explanation of the strong hierarchy of quark and charged lepton masses is still a big challenge in particle physics. This hierarchy have suggested to many model building theorists that light fermion masses could be generated from radiative corrections, while those of the top and bottom quarks and the tau lepton are generated at tree level. This may be understood as the breaking of a symmetry among families , a horizontal symmetry.
In this report we update the general features of a "Beyond the Standard Model"(BSM) proposal which introduces a SU(3) [3] gauged family symmetry 1 commuting with the Standard Model group. Previous reports[4] within this scenario showed that this model has the features and particle content to account for a realistic spectrum of charged fermion masses and quark mixing. This BSM model introduce a hierarchical mass generation mechanism in which the light fermions obtain masses through one loop radiative corrections, mediated by the massive bosons associated to the SU(3) family symmetry that is spontaneously broken, while the masses for the top and bottom quarks as well as for the tau lepton, are generated at tree level from "Dirac See-saw"[5] mechanisms implemented by the introduction of a new generation of SU(2)L weak singlets vector-like fermions.
The SU(3) family symmetry model allows one to address the problem of quark and lepton masses and mixing, including active and light sterile neutrinos.
7.2	SU(3) flavor symmetry model 7.2.1 Fermion content
Before "Electroweak Symmetry Breaking"(EWSB) all ordinary, "Standard Model"(SM) fermions remain massless, and the global symmetry in this limit of all quarks and leptons massless, including R-handed neutrinos, is:
SU(3)qL <g> SU(3)uR <8> SU(3)dR <g> SU(3)1l <g> SU(3)Vr <8> SU(3)eR (7.1)
D SU(3)qL+uR + dR + lL + eR + vR = SU(3)	(7.2)
We define the gauge group symmetry G = SU(3) < GSM, where Eq.(7.2) defines the SU(3) gauged family symmetry, and GSM = SU(3)C < SU(2)L < U(1)Y is the "Standard Model" gauge group, with gH, gs, g and g' the corresponding
1 See [3,4] and references therein for some SU(3) family symmetry models.
7 General Majorana Neutrino Mass Matrix from a Low Energy SU(3) Family... 95
coupling constants. The content of fermions assumes the ordinary quarks and leptons assigned under G as:
^ = (3,3,2,1)l , K = (3,3,1,4)r , ^ = (3,3,1,-3)R
= (3,1,2, —1 )L , C = (3,1,1, -2)R ,
where the last entry corresponds to the hypercharge Y, and the electric charge is defined by Q = T3L + 2 Y. The model also includes two types of extra fermions:
•	Right handed neutrinos = (3,1,1,0) R required to cancel anomalies[6], and
•	the SU(2)l singlet vector-like fermions:
UL,R = (1,3,1,4) , DLR = (1,3,1, — 2)	(7.3)
N°,R = (1,1,1,0) , E°,R = (1,1,1, —2),	(7.4)
which conserve the previous anomaly cancellation. The transformation of these vector-like fermions allows the mass invariant mass terms
Mu UL UR + Md DL DR + Me E° ER + h.c. ,	(7.5)
and
mD NL NR + mL INL (NL)c + mR NR (NR)c + h.c	(7.6)
These SU (2) L weak singlets vector-like fermions have been introduced to give masses at tree level only to the third family of known fermions through Dirac See-saw mechanisms. MU , MD , ME play a crucial role to implement a hierarchical spectrum for quarks and charged lepton masses and mixing, meanwhile mD , mL , mR play a similar role for neutrino masses and lepton mixing, all together with the radiative corrections.
7.3 SU(3) family symmetry breaking
The corresponding SU(3) gauge bosons are defined through their couplings to fermions as
iLint = gH (f°Yf ° — W°°) + 2^3 (f°Yf ° + f°Yf ° — 2f0Yf°)
+ ¡2 (f°Yf 0 Y+ + f°Yf° Y+ + f°Y^f° Y+ + h.c.) (7.7)
f° = u°,d°,e°,v° , f° = c°,s°,|j°,v° and f° = t°,b°,T°,v°. To implement a hierarchical spectrum for charged fermion masses, and simultaneously to achieve
96 A. Hernandez-Galeana
the SSB of SU(3), we introduce the flavon scalar fields: n = (3,1,1,0), i = 1,2,3, transforming as the fundamental representation under SU(3) and being standard model singlets, with the "Vacuum Expectation Values" (VEV's):
(m)T = (At,0,0) , <n2>T = (0,A2,0) , (na)T = (0,0,As) .
(7.8)
Actually, let us point out here that only two scalar flavons in the fundamental representation are needed to completely break down the SU(3) symmetry. The most convenient way to accomplish the spontaneous breaking of the SU(3) family symmetry is under current study. Thus, the contribution to the horizontal gauge boson masses from Eq.(7.8) read
^ (Y+Y- + Y+ Y-) + ^(Z2 + f + 2Zi )
ni n2 n3
9h, A2 2
9H2 A2
2 (Y+Y- + Y+ Y-) +
9h2 a2
(z2+Z2 - 2zI a? )
Zi )
as)
^ (Y+Y- + Y+ Y-) + gHH3 ASZi
Therefore, neglecting tiny contributions from electroweak symmetry breaking, we obtain the gauge boson mass terms
(M2 + m2) Y+Y- + (M2 + M3) Y+Y- + (M2 + M3) Y+Y-
2 _i_ 1\/12! V+A
12 _i_ 1\/12! V+'\
i -, -, -, 1 m2 + m2 + 4m2 i i i i 2 +1 (m? + m2) z2 + + M2 + 4M3 Z2 +1 (Ml - m2) ^ zT Z2 (7.9)
3
V3
M? =
gH, a?
m2 = "H2
gH, a2
M23 =
gH A3
(7.10)
Z2
Zi Z2
m2 + M2
M, +M, +4M3
75	3
Table 7.1. Zt — Z2 mixing mass matrix
From the diagonalization of the Zi — Z2 squared mass matrix, we obtain the
eigenvalues
m- = 2 (m? + m2 + m3 — ^(m2 — m?)2 + (m2 — m?)(m3 — m2^ M+ = 3 (m? + m2 + m3 + \j(m2 — m?)2 + (m3 — m2)(m2 — m2^
4
2
2
Z
7 General Majorana Neutrino Mass Matrix from a Low Energy SU(3) Family... 
97
Z2
2 Z+
my, Y+Y- + My2 y+y- + My3 y+y- + Ml — + M+
2
where
(7.11)
mY = m2 + m2 , mY2 = m1 + m2 , mY3 = m2 + m3 (7.12)
cos $ — sin f Z
sin $ cos $
Z
(7.13)
A • A ^	M1 - M2
cos 9 sin 9 = —----' 2	-,
4 v/(m2 -m2)2 + (m2 - m2)(m2 - m2)
with the hierarchy M', M2 ^ MW2. Due to the Z' - Z2 mixing we diagonalize the propagators involving Zi and Z2 gauge bosons according to Eq.(7.13):
Zi = cos $ Z_ — sin $ Z+ , Z2 = sin $ Z_ + cos $ Z
2
(Zi Zi > = cos2 $ (Z_Z_> + sin2 $ (Z+Z+) (Z2Z2) = sin2 $ (Z_Z_> + cos2 $ (Z+Z+)
(Z1Z2) = cos 9 sin9 ((Z_Z_) - (Z+Z+)) 7.4 Electroweak symmetry breaking
Recently ATLAS[7] and CMS[8] at the Large Hadron Collider announced the discovery of a Higgs-like particle, whose properties, couplings to fermions and gauge bosons will determine whether it is the SM Higgs or a member of an extended Higgs sector associated to a BSM theory. The electroweak symmetry breaking in the SU(3) family symmetry model involves the introduction of two triplets of SU(2)l Higgs doublets.
To achieve the spontaneous breaking of the electroweak symmetry to U(1 )q, we introduce the scalars: =(3,1,2, -1) and ® d = (3,1,2, +1), with the VEVs:
W>N (®u> = | (®U>
'(®2>N (®d> = | (®2> ,(®2>,
(7.14)
/®u\ = -L
<-2> = ¿(v!
(7.15)
I2; sin ^ = 0, cos ^ = 1
98 A. Hernandez-Galeana
contribute to the W and Z boson masses:
g2 (vU + vd) W+W- +	(vU + vd) ZO
vU = vUi + vu2 + vUa z vd = vdi + vd2 + vd3- Hence, if we define Mw = i g v, we may write v = wvU + vd « 246 GeV.
7.5 Tree level neutrino masses
Now we describe briefly the procedure to get the masses for ordinary fermions. The analysis for quarks and charged leptons has already discussed in [4]. Here, we introduce the procedure for neutrinos.
7.5.1 Tree level Dirac neutrino masses
With the fields of particles introduced in the model, we may write the Dirac type gauge invariant Yukawa couplings
ho v? NR + hi ni NL + h2n2 nl + h3vvn3 nl
+ Mo NL NR + h.c (7.16)
ho, h1, h2 and h3 are Yukawa couplings, and Mo a Dirac type, invariant neutrino mass for the sterile neutrinos N? R. After electroweak symmetry breaking, we obtain in the interaction basis	= (v?,v°,v?, No)L,R, the mass terms
ho [vi vOL + v2 v°L + v3 vOj NR + [hiAi vOR + h2A2 v°R + haAa vOR NR
+ Mo N R NR + h.c. (7.17)
7.5.2 Tree level Majorana masses:
Since NRr, Eq.(7.4), are completely sterile neutrinos, we may also write the left and right handed Majorana type couplings
hLPR ®u(NR)c + mL INR (NR)c + h.c	(7.18)
and
hiRPni (NR)c + h2RPVn2 (NR)c + h3RPVn3 (NR)c
+ mR NR (NR)c + h.c , (7.19)
respectively. After spontaneous symmetry breaking, we also get the left handed and right handed Majorana mass terms
7 General Majorana Neutrino Mass Matrix from a Low Energy SU(3) Family... 99 hL [vi vOl + V2 V°L + V3 V0L] (NL)c + mL NO (N£)c + h.c.,	(7.20)
+ [hiR Ai vOr + h2R A2 V°r + h3R A3 vOr] (NR)c + mR NR (NR)c + h.c., (7.21)
	(v»l)c	Kl)c	(v»l)c	(NL)C	V»R			NR
V»L	0	0	0	hL Vi	0	0	0	ho vi
	0	0	0	hL V2	0	0	0	ho V2
	0	0	0	hL V3	0	0	0	ho V3
Nl	hL Vi	hL V2	hL V3	mL	hi Ai	h2 A2	h3 A3	mo
(V»r)c	0	0	0	hi Ai	0	0	0	hiR Ai
(V°r)c	0	0	0	h2 A2	0	0	0	h2R A2
(v»r)c	0	0	0	h3 A3	0	0	0	h3R A3
(NR)C	ho vi	ho V2	ho V3	mo	hiR Ai h2R A2 h3R A3			mR
Table 7.2. Tree Level Majorana masses
Thus, in the basis
w»
1 V
= (-
eL
VL'
,V»L ,N» , (V»R)c , (V°R)C , (V»R)c , (NR)C ) ,
(7.22)
the Generic 8 x 8 tree level Majorana mass matrix for neutrinos MV, from Table 7.2, ¥"V MV (¥V)c, read
MV =
ML MD
<MDt MR,
(7.23)
where
ML =
/ 0	0 0 ai \
0	0 0 a2
0	0 0 a3
\ai	a2 a3 mL/
MR =
( 0 0 0 ßi\ 0 0 0 ß2 0 0 0 ß3 Vßi ß2 ß3 mR/
(7.24)
O
o
and
100 A. Hernandez-Galeana
MD —
0	0	0 a1
0	0	0 af
0	0	0 a3
b1	bf	b3 mD
(7.25)
a — hL vt , at — hD vt , bt — hi At , pi — htR At	(7.26)
Diagonalization of MVo), Eq.(7.23), yields four zero eigenvalues, associated to the
neutrino fields:
^ - ^ v°L , ^ ^ + a;2-a3 - aP v°L, (7.27) ap	ap	ap a eL ap a	a TL
b2 ° b °	bi ba ° b2 ba ° bT
bpv°R	, b^V°R + b^V^ "b
ap — y af + a2 , bp — ^bf + bf , a — ^ af + af + a2 , b — ^bf + bf + bf . Assuming for simplicity — j^t — hf" ^ Cr' that is at Hl	pt htR
- — Z— — cL , 7— — ~T~ — cR ,
at hD	bt ht
the Characteristic Polynomial for the nonzero eigenvalues of MV reduce to the one of the matrix m43, Eq.(7.29), where
m4
/0 a 0 a \ a mL b mD 0 b 0 p Va mD p mRJ
a — \ I af + af + a3
U4 —
/Ufi U12 U13 UM\
U21 U22 U23 U24 U31 U32 U33 U34 VU41 U42 U43 U44J
p — ^p2 + pf + P3 .
(7.29)
U4 m4 U4 — Diag(m°,mo,m°,m°) = d4 , m4 — U4 d4 U{	(7.30) Eq.(7.30) impose the constrains
u11 m° + Uff m° + u13 m° + u?4 m° — 0	(7.31)
u31 m° + Uff m° + u33 m° + u34 m° — 0	(7.32)
U11U31 m° + U12U32 m° + U13U33 m° + U14U34 m° — 0,	(7.33)
3 The relation a b — a p would yield five massless neutrinos at tree level.
7 General Majorana Neutrino Mass Matrix from a Low Energy SU(3) Family... 101 corresponding to the (m4)n = (m4)33 = (m4)i3 = 0 zero entries, respectively. Therefore, MV is diagonalized by the orthogonal matrix
/ ai ai a3 ap a ap
o ai un ai uuai ui3 afui4\

ai a2 a3 o ap a ap
a
o aiuii aiu^a2ui3aiui
ai
a
a 2 .
a
o a3uii aiui2a3u^a^M
0
0
0
Uo
0 0 0
0 0
00
u2i u22 u23 u24
b2 b1 b3 b,, b^, b^, b^. bp Tbbp3 bb"u3i ""u32 ""u33 "" u34
bl bibs biu3i "bu32 u33 ""u34

bp b bp b
0 -b"P "3u3i "3u32 "3u33 "b3u34
00
u4i
(7.34)
u42 u43 u44 /
(UV)T MV UV = Diag(0,0,0,0,mO,mO,m?,mg)	(7.35)
7.6 One loop neutrino masses
After tree level contributions the fermion global symmetry, Eq.(7.1), is broken
down to
SU(2)qL <g> SU(2)uR < SU(2)dR < SU(2)lL << SU(2)Vr < SU(2)eR . (7.36)
Therefore, in this scenario light neutrinos may get extremely small masses from radiative corrections mediated by the SU(3) heavy gauge bosons.
7.6.1 One loop Dirac Neutrino masses
After the breakdown of the electroweak symmetry, neutrinos may get tiny Dirac mass terms from the generic one loop diagram in Fig. 7.1, The internal fermion line in this diagram represent the tree level see-saw mechanisms, Eqs.(7.16-7.21). The vertices read from the SU(3) family symmetry interaction Lagrangian
iLint = gH (VOYivO -	Z|+(VOY^O + V°Y|V| - 2v°Y|V?) Z|
+ -2 (VOY|V| Y+ + v"^Y|VO Y+ + v|Y|VO Y3+ + h.c.) (7.37)
102 A. Hernandez-Galeana
The contribution from these diagrams may be written as
Cy-mv(My )ij
n
«h
gH
4n '
mv (My)ij = X mk U?kU?k f (My, mg)
k=5,6,7,8
f(My '
m

k) - M2
l Mr.
mo2 lnmo7
oT Y-mk
(7.38)
(7.39)

Y
< nk >
< >
V
iL
Fig. 7.1. Generic one loop diagram contribution to the Dirac mass term m^ v°Lv°R. M — Md, mL, mR
o A
o
	vor	V°R	V?R	NR
V ol	Dv 11	D v 12	Dv13	0
V°L	Dv 21	Dv22	Dv23	0
V ol	Dv 31	Dv32	Dv33	0
N L	0	0	0	0
Table 7.3. One loop Dirac mass terms Dv ^ v?L V°R
mv(My )i,4+j —	Fv(My )
J ab
(7.40)
Fv(My) — U,,U3, mO f(My,mg) + UT2U32 mg f(My,m§)
+ U13U33 mO f(My,m°) + U14U34 mg f(My,mg) (7.41)
7 General Majorana Neutrino Mass Matrix from a Low Energy SU(3) Family... 
103
Dvii =
mbi ab
1 Fy(Mz, ) + 12Fv(Mz2 ) + Fv,m
+ ■
a2b2 Fv(MYl) + a3b3Fv(MY2 )
ab
ab
Dv12 =
aib2 ab
-4Fy(Mz,) + 12Fv(Mz2 )
Dv13 =
aib3
ab

Fv(Mz2) - F,
Dv 21 =
ab
ab

4Fv(Mz,) + 12F,(Mz2 )
D v 22 =
a2b2 ab
1 Fv(Mz,) + 12Fv(Mz2 ) - Fv,m
1 + 2
ai b
1 Fv(My1 ) + ^Fv(MY3 ) ab	1 ab	3
D v 23 =
a2b3 ab
Fv(Mz2) + Fv
Dv 31 =
a3bi
ab

Fv(Mz2) - Fv
D v 32 =
a3b2 ab
Fv(Mz2) + Fv
D v 33 =	Fv(Mz2 ) + 2
3 ab
aiblFv(My2 ) + ^Fv(My3) 2 ab	3
ab
Fv(Mz,) = cos2 $Fv(M-) + sin2 $Fv(M+)
Fv(Mz2) = sin2 $Fv(M_) + cos2 $Fv(M+)
Fv,m = cos $ sin $ [Fv(M-) - Fv (M+)] , 2V3
(7.42)
7.6.2 One loop L-handed Majorana masses
Neutrinos also obtain one loop corrections to L-handed and R-handed Majorana masses from the diagrams of Fig. 7.2 and Fig. 7.3, respectively. A similar procedure as for Dirac Neutrino masses leads to the one loop Majorana mass terms
104 A. Hernandez-Galeana
mv(My )i,j = ^ Gv(MY )
(7.43)
Gv(My) = mO u?? f(My,mg) + mO u?2 f(My,mO)+ mO u?3 f(My,m?)
+ mg u?4 f(My,mg) (7.44)
VjL
Y
< >
< >
v
iL
Fig. 7.2. Generic one loop diagram contribution to the L-handed Majorana mass term mij VPL(YPL)T. M = Md , mL,mR
o
	Vol	V°L		NL
vol	Lvii	Lv 12	Lv 13	0L
V°L	Lv 12	Lv 22	Lv 23	0
	Lv 13	Lv 23	Lv 33	0
NL	0	0	0	0
Table 7.4. One loop L-handed Majorana mass terms OH Lvij vgL (v°L)T
7 General Majorana Neutrino Mass Matrix from a Low Energy SU(3) Family... 105
T	Ul
Lv 11 = 2
1 Gv(Mz,) + I2Gv(Mz2) + Gv,m
T	a2
Lv 22 = —r
n 2
1 Gv(Mz, ) + 12Gv(Mz2 )- Gv,m
1 a2
Lv 33 = 1"f Gv(Mz2 ) ,
3 a2
ai a2
Lv 12 = -2—
a2
-1 Gv (Mz,) + 12Gv (Mz2 ) + 1 Gv (Mi)
Lv 13 =
ai a3
a
2
6Gv(Mz2) + 2Gv(M2)- Gv,m
Lv 23 =
a2a3
-1 Gv(Mz2 ) + 1 Gv(M3)+ Gv,m b 2
Gv(Mz,) = cos2 $ Gv(M_) + sin2 $ Gv(M+)
Gv(Mz2) = sin2 $ Gv(M_) + cos2 $ Gv(M+)
Gv,m = ^ cos $ sin $ [Gv(M-) - Gv(M+)].
(7.45)
7.6.3 One loop R-handed Majorana masses
mv(My )4+i,4+j = ^bj1 Hv(My)
(7.46)
Hv(My) = mO u31 f(My,m0) + m£ u32 f(My,m§) + mO u33 f(My,m?)
_ o \ 1 o ., 2
o ^ 1 -v.-. o .. 2
+ mo u34 f (My,mo) (7.47)
2
2
2
a
106 A. Hernandez-Galeana
Y

v
iR
< nk >	< ns >
Fig. 7.3. Generic one loop diagram contribution to the R-handed Majorana mass term
tij V°R(v°Rf
mij v°r(v?r)t. M — Mo,mL,mR
	Vor	V°R	V O R	NR
vor	Rvii	Rv 12	Rv 13	0
V°R	Rv 12	Rv 22	Rv 23	0
V?R	Rvi3	Rv 23	Rv 33	0
NR	0	0	0	0
Table 7.5. One loop R-handed Majorana mass terms OH Rvij v°R (v°R)T
R - b1
1 hv(Mz, ) + 12hv(Mz2 )+
rv22 - b2
1 Hv(Mz, ) + 12Hv(Mz2 )- Hv,m
1 b2
Rv 33 — 3 b3Hv(MZ2 ) ,
Rv 12 —
Rv 13 —
Rv 23 —
bib2 b2
bib3 b2
b2b3 b2

-4Hv(Mz, ) + 12Hv(Mz2 ) + 1 Hv(Mi )
1 Hv(Mz2 ) + 1 Hv(M2)- Hv
4Hv(Mz2 ) + 1 Hv(M3)+ Hv,m b 2
o
2
7 General Majorana Neutrino Mass Matrix from a Low Energy SU(3) Family... 
107
Hv(Mz, ) = cos2 ^Hv(M-) + sin2 ^Hv(M+)
Hv(Mz2) = sin2 ^Hv(M-) + cos2 ^Hv(M+)
^v.m —
2V3
cos^ sin^ [Hv(M_) -Hv(M+)] ,
(7.48)
where Fv,m , Gv,m and Hv,m, Eqs.(7.42,7.45,7.48), come from Z1 — Z2 mixing diagram contributions.
Thus, in the basis, Eq.(7.22), we may write the one loop contribution for neutrinos as	C^V)c,
MOv —
fLvii	Lv i2	Lv i3	0	Dv 11	Dv12	Dv13	0
Lv 12	Lv 22	Lv 23	0	Dv 21	Dv22	Dv23	0
Lv 13	Lv 23	Lv 33	0	Dv 31	Dv32	Dv33	0
0	0	0	0	0	0	0	0
Dvii	Dv2i	Dv 31	0	Rv11	Rv 12	Rv 13	0
D v 12	Dv22	Dv32	0	Rv 12	Rv 22	Rv 23	0
D v 13	Dv23	Dv33	0	Rv 13	Rv 23	Rv 33	0
V 0	0	0	0	0	0	0	0
(7.49)
n
7.6.4 Neutrino mass matrix up to one loop
Finally, we obtain the general symmetric Majorana mass matrix for neutrinos up
to one loop
Mv = (UV)T M?v UV + Diag(0,0,0,0,mO,mO,mO,mO), (7.50) where explicitly
10S A. Hernandez-Galeana
(UV )T MOv UV =
/Nii Nl2 Nl3 Nl4 Nl5 Nl6 Nl7 Ni8\
Nl2 N22 N23 N24 N25 N26 N27 N28 Ni3 N23 N33 N34 N35 N36 N37 N38 Ni4 N24 N34 N44 N45 N46 N47 N48 N15 N25 N35 N45 N55 N56 N57 N58 Ni6 N26 N36 N46 N56 N66 N67 N68 N17 N27 N37 N47 N57 N67 N77 N78 \Nl8 N28 N38 N48 N58 N68 N78 N88/
(7.51)
Majorana L-handed:
22
N11 = a2 a2
-p-2
(Gz, - Gi )
N12 =
22 ai a2a3 ra2 - af
la3
-(Gz, - Gi ) + G2 - Gs - 6Gm]
(7.52)
(7.53)
N22 = -2
1 (a2 - a2)2
4
2 2—(Gz, - Gi ) + "-I (G2 - Gs) 2 2 , 2
a2p a
a2	a2 - a2
(Gi + 3Gz2 - 4G2) - 3 2 2 i Gm 4 a2	2	a2
(7.54)
Dirac:
Nib =
1
lap bp a b
{(ai bi + a2b2)Fi + aaba (a2b2F2 + aibiFs)
+laibia2b2Fz, j (7.55)
a
2
2
Ni4 = iapbp abl3 {bib2(a2 - a2)Fi + a3bs(a2biF2 - a^Fs)
+ai a2 (b2 - b2)Fz, + 6ai a2 bp2 F m} (7.56)
7 General Majorana Neutrino Mass Matrix from a Low Energy SU(3) Family... 109
N23 = 2apbp ab"cf iaia2(b2 - b2)Fi + a3b3(aib2F2 - a2biF3)
+b!b2(af - a2)Fz, + 6bib2 ap2F,} (7.57)
N24 = 1
ap bp a2 b2
1	3
4 (a2 - a2)(b2 - b2)Fz, + 4
+1(a3b2 + ap2 bp2)(aibiF2 + a2b2F3) + 3a3b3(afb2 - a^F
13 a3 b3[aibia2b2Fi + 7K - a2)(b2 - b2)Fz, + 7 ap bp Fz2 ]
Majorana R-handed:
N34 =
b2b2
N33 = ^(Hz, - Hi) bib2b3 [ b^ (Hz, - Hi)+ H2 - H3 - 6Hm]
2b3
bp
(7.58)
(7.59)
N44 = b|
1 (b2 - b2)2 4 2b2 b2i (Hz, - Hi) + b2 (H2 - H3)
b2,
bp
+^(Hi + 3Hz2 - 4H2)- 3
b2 - b2 b2
,
(7.60)
110 A. Hernández-Gáleana Majorana L-handed and Dirac:
Nis = Gis Uii + mi3 U31 ; N?6 = Gis Ui? + mi3 U3? (7.61) Ni/ = Gis Ui3 + mi3 U33 ; Nia = Gis U14 + mi3 U34 (7.62)
ai a?
Gis = — ~-
2 ap a
a? — a?	a?	(2 a? — ap )
-sr1 (Gz, — Gi) + a?(Gs — G?) + 2 ( 3 ? p) Gm
n ?	n ?	n ?
mis = 2ap1ab? ^bib?(a? — a?)Fi + a3b3(a?biF? — aib?F3)
+ai a?(b? — b?)Fz, + 2ai a? (bp? — 2b?)Fm}
N?s = G?s U11 + m?3 U31 ; N?6 = G?s Ui? + m?3 U3? (7.63) N?/ = G?s U13 + m?3 U33 ; N?a = G?s U14 + m?3 U34 (7.64)
G?s = 3
4 ap a4
{ (a? — a?)? (Gz, — Gi ) + 2 a?(a3 — -£) (Gs — G?) — ap (Gz2 — Gi )
—2 -p(-3 — ap) (Gz2 — G?) + 4 (a? — a2) (a3 — 2ap) Gm }
1
m?3
ap a? b?
as [aibia?b?Fi + 1 (a? — a?)(b? — b?)Fz, + 1ap?(bp? — 2b?)Fz2 ] +1b3(a? — ap?)(aibiF? + a?b?F3) + -s [a? (3b? — b?) + a?(b? — 3b?)]Fn
7 General Majorana Neutrino Mass Matrix from a Low Energy SU(3) Family... 111 Dirac and Majorana R-handed:
N35 = m.3! un + H35 U31 , N36 — m3i + H35 U32
N37 = m.31 U13 + H35 U33 , N38 — m3i U14 + H35 U34
m3i = 2bp a2b iaia2(b2 -bi)Fi + a3b3(aib2F2 - a2biF3)
+bib2(a2 - a2)Fz, + 2bib2(ap2 - 2a3)Fm}
H35 —
bi b2 2bp b
b2 - b2 b2 -b2-(HZi - Hi ) + b2 (H3 - H2) + 2
(2b3 - bp) b2
m
N45 — m32 Uii + H45 U3i , N46 — m32 Ui2 + H45 U32
(7.65)
N47 — m.32 Ui3 + H45 U33 , N48 — m32 Ui4 + H45 U34
(7.66)
1
m32
1
bp a2 b2
b3[aibia2b2Fi + 1 (a2 - a2)(b2 - b2)Fz, + ^bp2(ap2 - 2a3)Fz2]
+^a3(b3 - bp2)(aibiF2 + a2b2F$) + b3 [b2(3af - a2) + b2(a2 - 3a2)]F
H45 —
b3
4 bp b4 ,2 \~2\2
{(b2 - b2)2 (Hz, - Hi)+ 2b2 (b3 - bp) (H3 - H2)- bp (Hz 2 - Hi)
2 bp (b2 - bp) (Hz2 - H2) + 4 (b2 - b2) (b2 - 2bp) Hm }
— /
112 A. Hernandez-Galeana
Majorana L-handed, Dirac and Majorana R-handed:
N55 _ G55 u2i IL m.33 U11 U31 IH55 u2i	(Z.6Z)
N56 _ G55 U11 U12 I m33 (uii U32 IU12 U31 ) IH55 U31 U32	(Z.68)
N57 _ G55 U11 U13 I m33 (Uii U33 I U13 U31 ) I H55 U31 U33	(Z.69)
N58 _ G55U11 U14 Im.33 (uii U34 IU14U31) IH55U31 U34	(Z.ZO)
N66 _ G55 U22 I L m33 U12 U32 I H55 U22	(Z.Z1)
N67 _ G55 U12 U13 I m.33 (Ui3 U32 IU12 U33) IH55 U32 U33	(7.72)
N68 _ G55 U12 U14 I m33 (Ui4 U32 I U12 U34) I H55 U32 U34	(7.73)
N77 _ G55 u23 IL m33 U13 U33 IH55 u23	(7.74)
N78 _ G55 U13 U14 I m33 (Ui4 U33 I U13 U34) I H55 U33 U34	(7.75)
N88 _ G55 U24 i Lm33 U14 U34 IH55 u24	(7.76)
a2 a2 + a2 a3 + a2 a2 + (a2 - a2)2 + (l"3 - ag)2
G55 _ G1 + "a^" G2 + G3 + —404— Gzi + —na4— gz2
| (a2 - a2) (L a2- ag)
+ /i
m33
a2 b2
4 (a2 - a2)(b2 - b2)Fz, + -L,
+a3b3(aibiF2 + a2b2F3) + [a2b2 - a2b2 + a3(b2 - b2) + b2(a2 - a2)]Fm}
aibi a2b2Fi + - (a2 - a2)(b2 - b2)Fz, + ^ (ap2 - La3)(bp2 - Lb2)Fz2
y _ b2 b^ + b2 b2 + b2 b2 + (b2 - b2)2 + (Lb3 - bP)2 y
Hs5 _ -iT2 Hi + "IT" H2 + ""24" H3 +-T¿4-Hzi +-ÏT^Ï-hz2
b4 "" b4 2 b4 3 4b4 zi 1Lb4
| (b2 - b2)(Lb2 - bp)
+ -b4- Hm
7 General Majorana Neutrino Mass Matrix from a Low Energy SU(3) Family.
113
7.6.5 Quark (Vckm)4x4 and (Vpmns)4x8 mixing matrices
Within this SU(3) family symmetry model, the transformation from massless to physical mass fermion eigenfields for quarks and charged leptons is
^o = Vo y(D ¥l and ^R = VO VR1' ,
and for neutrinos ¥V = UV UV ¥V. Recall now that vector like quarks, Eq.(7.3), are SU(2)l weak singlets, and hence, they do not couple to W boson in the interaction basis. In this way, the interaction of L-handed up and down quarks; fOLT = (uO,cO,tO)L and fdLT = (dO, sO,bO)L, to the W charged gauge boson is
^fuLYfdLW^ = -gL^uL [(VOL VU1L))3X4]T (VOL V^'^ Y^dL ,
(7.77)
g is the SU(2)L gauge coupling. Hence, the non-unitary VCKM of dimension 4 x 4 is identified as
(VCKM)4x4 = [(VOL VU1L))3x4]T (VOL V^^	(7.78)
Similar analysis of the couplings of active L-handed neutrinos and L-handed charged leptons to W boson, leads to the lepton mixing matrix
(UPMNS)4x8 = [(VOL V(L')3x4]T (U UV)3x8	(7.79)
7.7 Conclusions
We reported an updated and general analysis for the generation of neutrino masses and mixing within the SU(3) family symmetry model. The right handed neutrinos (ve v^ vT)R, and the vector like completely sterile neutrinos NL,R, the flavon scalar fields and their VEV's introduced to break the symmetries: ®u, ® d, ni, n2 and n3, all together, yields a 8 x 8 general Majorana neutrino mass matrix with four or five massless neutrinos at tree level. Therefore, light neutrinos get tiny masses from radiative corrections mediated by the heavy SU(3) gauge bosons. Neutrino masses and mixing are extremely sensitive to the parameter space region, and a global fit for all quark masses and mixing together with neutrino masses and lepton mixing is in progress.
Acknowledgements
It is my pleasure to thank the organizers N.S. Mankoc Borstnik, H.B. Nielsen, M. Y. Khlopov, and participants for the stimulating Workshop at Bled, Slovenia. This work was partially supported by the "Instituto Politecnico Nacional", (Grants from EDI and COFAA) and "Sistema Nacional de Investigadores" (SNI) in Mexico.
114 A. Hernandez-Galeana
References
1.	MiniBooNE Collaboration, A. A. Aguilar-Arevalo et. al., A Combined	and Vn —> Ve Oscillation Analysis of the MiniBooNE Excesses., arXiv: 1207.4809.
2.	J.M. Conrad, W.C. Louis, and M.H. Shaevitz, arXiv:1306.6494; I. Girardi, A. Meroni, and S.T. Petcov, arXiv:1308.5802; M. Laveder and C. Giunti, arXiv:1310.7478; A. Palazzo, arXiv:1302.1102; O. Yasuda, arXiv:1211.7175; J. Kopp, M. Maltoni and T. Schwetz, Phys. Rev. Lett. 107, (2011) 091801; C. Giunti, arXiv:1111.1069; 1107.1452; F. Halzen, arXiv:1111.0918; Wei-Shu Hou and Fei-Fan Lee, arXiv:1004.2359; O. Yasuda, arXiv:1110.2579; Y.F. Li and Si-shuo Liu, arXiv:1110.5795; B. Bhattacharya, A. M. Tha-lapillil, and C. E. M. Wagner, arXiv:1111.4225; J. Barry, W. Rodejohann and He Zhang, arXiv:1110.6382[hep-ph]; JHEP 1107,(2011)091; F.R. Klinkhamer, arXiv:1111.4931[hep-ph].
3.	A. Hernandez-Galeana, Rev. Mex. Fis. Vol. 50(5), (2004) 522. hep-ph/0406315.
4.	A. Hernandez-Galeana, Bled Workshops in Physics, (ISSN:1580-4992),Vol. 14, No. 2, (2013) Pag. 82; arXiv:1312.3403[hep-ph]; Vol. 13, No. 2, (2012) Pag. 28; arXiv:1212.4571[hep-ph]; Vol. 12, No. 2, (2011) Pag. 41; arXiv:1111.7286[hep-ph]; Vol. 11, No. 2, (2010) Pag. 60; arXiv:1012.0224[hep-ph]; Bled Workshops in Physics,Vol. 10, No. 2, (2009) Pag. 67; arXiv:0912.4532[hep-ph];
5.	Z.G.Berezhiani and M.Yu.Khlopov, Sov.J.Nucl.Phys. 51 (1990) 739; 935; Sov.J.Nucl.Phys. 52 (1990) 60; Z.Phys.C- Particles and Fields 49 (1991) 73; Z.G.Berezhiani, M.Yu.Khlopov and R.R.Khomeriki, Sov.J.Nucl.Phys. 52 (1990) 344; A.S.Sakharov and M.Yu.Khlopov Phys.Atom.Nucl. 57 (1994) 651; M.Yu. Khlopov: Cosmoparticle physics, World Scientific, New York -London-Hong Kong - Singapore, 1999; M.Yu. Khlopov: Fundamentals of Cosmoparticle physics, CISP-Springer, Cambridge, 2011; Z.G. Berezhiani, J.K. Chkareuli, JETP Lett. 35 (1982) 612; JETP Lett. 37 1983 (338); Z.G. Berezhiani, Phys. Lett. B 129 1983 (99).
6.	T. Yanagida, Phys. Rev. D 20 (1979) 2986.
7.	G. Aad et. al., ATLAS Collaboration, Phys. Lett. B 716 (2012) 1, arXiv: 1207.7214.
8.	S. Chatrchyan et. al, CMS Collaboration, Phys. Lett. B 716 (2012) 30, arXiv: 1207.7235.