Bled Workshops in Physics Vol. 14, No. 2
A
Proceedings to the 16th Workshop What Comes Beyond ... (p. 82) Bled, Slovenia, July 14-21, 2013

8 Neutrino Masses and Mixing Within a SU(3) Family Symmetry Model With One or Two Light 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. We report a global fit of parameters for fermion masses and mixing, including light sterile neutrinos, within a local vector SU(3) family symmetry model. 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. The Nl,r sterile neutrinos allow the implementation of a 8 x 8 general tree level See-saw Majorana neutrino mass matrix with four massless eigenvalues. Hence, light fermions, including light neutrinos obtain masses from one loop radiative corrections mediated by the massive SU(3) gauge bosons. This BSM model is able to accommodate the known spectrum of quark masses and mixing in a 4x4 non-unitary Vckm as well as the charged lepton masses. The explored parameter space region provide the vector-like fermion masses: MD ra 914.365 GeV, Mu ra 1.5 TeV, Me ra 5.97TeV, SU(3) family gauge boson masses of O(1 — 10 )TeV, the neutrino masses (m-|, m2, m3, m4, m5, m6, m7, m8) = (0, 0.0085,0.049,0.22,3.21, 1749.96,1 x 108,1 x 109 ) eV, with the squared neutrino mass differences: m| — m2 ra 7.23 x 10-5 eV2, m3 — ml ra 2.4 x 10-3 eV2, m4 — ml ra 0.049 eV2, mf — ml ra 10.3 eV2. We also show the corresponding Upmns lepton mixing matrix. However, the neutrino mixing angles are extremely sensitive to parameter space region, and an improved and detailed analysis is in progress.
Povzetek. V modelu, kjer druZinsko kvantno stevilo doloca grupa SU [3], poskrbi za maso obeh tezkih kvarkov (t in b) ter za maso težkega leptona (tau) tako imenovani ,,Diracov sea-saw" mehanizem ze na drevesnem nivoju. To dosezem tako, da predpostavim eksistenco novih kvarkov U in D ter novih leptonov E in N. Vsi so levorocni in brez sibkega naboja ter nosijo tripletni naboj SU(2). Nevtrini NL,R, ki ne nosijo nobenega naboja, določajo na drevesnem nivoju masno matriko 8x8 Majoraninega tipa, ki ima stiri lastne vrednosti enake nic. Za maso lahkih fermionov, vkljucno z nevtrini, poskrbijo v popravkih z eno zanko masivni umeritveni bozoni druzinskega kvantnega stevila grupe SU(3). S primerno izbiro parametrov dosezem, da se lastnosti kvarkov in (z elektromagnetnim nabojem) nabitih leptonov ujemajo z izmerjenimi. Kvarkovska mesalna matrika 4 x 4 ni unitarna. Nova druzina kvarkov in nabitih leptonov ima mase okoli TeV ali vec (Md ra 914.365 GeV, MU ra 1.5 TeV, Me ra 5.97TeV), mase umeritvenih bozonov z druzzinskim kvantnimi stevili grupe SU(3) pa so O(1 —10 )TeV. Za mase nevtrinov najdem (m1, m2, m3, m4, m5, m6, m7, m8) = (0, 0.0085, 0.049,0.22,3.21,1749.96,1 x 108,1 x 109 ) eV, za kvadrate razlik njihovih mas
* e-mail: albino@esfm.ipn.mx
pa: m| - m2 ~ 7.23 x 10-5 eV2, m2 - m2 ~ 2.4 x 10-3 eV2, m4 - m? « 0.049 eV2, m2 — m2 « 10.3 eV2. Izracunam tudi leptonsko meSalno matriko. Ker je le-ta mocno odvisna od izbire parametrov, bo vanjo potrebno vloZiti se nekaj truda. Delo je v teku.
8.1 Introduction
The standard picture of three flavor neutrinos has been successful to account for most of the neutrino oscillation data. However, several experiments have reported new experimental results, on neutrino mixing[1], on large 613 mixing from Daya Bay[2], T2K[3], MINOS[4], DOUBLE CHOOZ[5], and RENO[6], implying a deviation from TBM[7] scenario. In addition, the recent experimental results from the LSND and MiniBooNe short-baseline neutrino oscillation experiments, provide indications in favor of the existence of light sterile neutrinos in the eV-scale, in order to explain the tension in the interpretation of these data[8,9].
The strong hierarchy of quark and charged lepton masses and quark mixing have suggested to many model building theorists that light fermion masses could be generated from radiative corrections[10], 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 . This symmetry may be discrete [11], or continuous, [12]. The radiative generation of the light fermions may be mediated by scalar particles as it is proposed, for instance, in references [13,14] and the author in [15], or also through vectorial bosons as it happens for instance in "Dynamical Symmetry Breaking" (DSB) and theories like " Extended Technicolor " [16].
In this report, we address the problem of fermion masses and quark mixing within an extension of the SM introduced by the author in [17], which includes a vector gauged SU(3)[18] family symmetry commuting with the SM group. In previous reports[19] we showed that this model has the properties to accommodate a realistic spectrum of charged fermion masses and quark mixing. We 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"[20] mechanisms implemented by the introduction of a new generation of SU(2)L weak singlets vector-like fermions.
Recently, some authors have pointed out interesting features regarding the possibility of the existence of vector-like matter, both from theory and current experiments[23]. From the fact that the vector-like quarks do not couple to the W boson, the mixing of U and D vector-like quarks with the SM quarks gives rise to an extended 4 x 4 non-unitary CKM quark mixing matrix. It has pointed out for some authors that these vector-like fermions are weakly constrained from Electroweak Precision Data (EWPD) because they do not break directly the custodial symmetry, then main experimental constraints on the vector-like matter come from the direct production bounds, and their implications on flavor physics. See the ref. [23]
for further details on constraints for vector-like fermions. Theories and models with extra matter may also provide attractive scenarios for present cosmological problems, such as candidates for the nature of the Dark Matter ([21],[22]).
In this article, we report for the first time a global fit of the free parameters of the SU(3) family symmetry model to accommodate quark and lepton masses and mixing, including light sterile neutrinos.
8.2 Model with SU(3) flavor symmetry 8.2.1 Fermion content
We define the gauge group symmetry G = SU(3) < GSM , where SU(3) is a flavor symmetry among families and GSM = SU(3)C < SU(2)L < U(1 )Y is the "Standard Model" gauge group, with gs, g and g' the corresponding coupling constants. The content of fermions assumes the ordinary quarks and leptons assigned under G as:
^ = (3,3,2,1 )L ,	=(3,3,1,4 , ^ = (3,3,1, - 3 )r
= (3,1,2, —1 )l , = (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, and the SU(2)L singlet vector-like fermions
UL,R = (1,3,1,4) , Dl,r = (1,3,1, —2)	(8.1)
= (1,1,1,0) , E0,r = (1,1,1, —2)	(8.2)
The transformation of these vector-like fermions allows the mass invariant mass terms
Mu UL UR + Md D0 DR + Me EL ER + h.c. ,	(8.3)
and
mD INL NR + mL IN0 (NL)c + mR INR (NR)c + h.c	(8.4)
The above fermion content make the model anomaly free. After the definition of the gauge symmetry group and the assignment of the ordinary fermions in the usual form under the standard model group and in the fundamental 3-representation under the SU(3) family symmetry, the introduction of the right-handed neutrinos is required to cancel anomalies[24]. The 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. These vector like fermions play a crucial role to implement a hierarchical spectrum for quarks and charged lepton masses, together with the radiative corrections.
8.3 SU(3) family symmetry breaking
To implement a hierarchical spectrum for charged fermion masses, and simultaneously to achieve the SSB of SU(3), we introduce the flavon scalar fields: ni, i = 2,3, transforming under the gauge group as (3,1,1,0) and taking the "Vacuum Expectation Values" (VEV's):
(ns>T = (0,0, As) , (nl)T = (0,A2,0) .	(8.5)
The above scalar fields and VEV's break completely the SU(3) flavor symmetry. The corresponding SU(3) gauge bosons are defined in Eq.(8.29) through their couplings to fermions. Thus, the contribution to the horizontal gauge boson masses from Eq.(8.5) read
2 . 2 2 • ns : ^(Y++Y- + Y+Y-) + 92H3AlZ2
• n2 : ^ (Y+Y- + Y3+Y3-) +	(Z? + - 2Z1 ^)
Therefore, neglecting tiny contributions from electroweak symmetry breaking, we obtain the gauge boson mass terms
Ml 4 1 1 Zl Ml M? Y+ Y- + M Z? + (3M2 + 3M1) ^ - M Zi Z2
+ Ml Y+Y- + (M? + Ml) Y+Y- (8.6)
9 2 a2	gl A2
Mi = 2 2 , Ml = h2 3 , Ml = Mi + Ml (8.7)
From the diagonalization of the Z1 — Z2 squared mass matrix, we obtain the eigenvalues
M- = 2 (m2 + Ml — ^(M2 — M2)2 + MfMl^ ,
M+ = 3 (M? + M2 + ^(M2 — M2)2 + M2M2^ (8.8) Z2 Z2
M? Y+ Y- + M- — + M+ Z23 + Ml Y+Y- + (Ml + Ml) Y+Y- (8.9)
where
(Z;)=(cm:—";)£)
. . . /3 1
cos : sin : = ——
4 L m2 , m2
with the hierarchy M;, M2 > MW.
(M2 — 1)2 + M2
( M2 1) + M2
	Zi	Z2
Zi	Ml	Mf - "Tf
Z2	Mf - VT	(4 M2 +1M2)
8.4	Electroweak symmetry breaking
Recently ATLAS[25] and CMS[26] 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:
(®U)T = ((®u), (®u), (®u)) , (®d)T = «®d>, (®d), <®3d», where T means transpose, and
W) = ^ ( o ) , ^ = £ (V0, )•
The contributions from (®u) and (®d) generate the W and Z gauge boson masses
(vU+v3) w+w- +	(vU+vd) zo	(8.12)
vU = v2 + v2 + v|, vd = V2 + V| + V2. Hence, if we define as usual MW = 2 9v, we may write v = ^JvU + vd « 246 GeV.
8.5	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 [19]. Here, we introduce the procedure for neutrinos.
Before "Electroweak Symmetry Breaking"(EWSB) all ordinary, "Standard Model"(SM) fermions remain massless, and the quarks and leptons global symmetry is:
SU(3)qL <g> SU(3)ur <8> SU(3)dR << SU(3)1l << SU(3)Vr << SU(3)eR (8.13)
8.5.1	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°0 NR + h2VV n2 NL + haVVna N + MD N° NR + h.c (8.14)
hD, h2 and ha are Yukawa couplings, and MD a Dirac type, invariant neutrino mass for the sterile neutrinos NO R. After electroweak symmetry breaking, we obtain in the interaction basis VVL R = (v0, v°, v0, No)L,R, the mass terms
ho [vi vOl + V2 v°L + va vOl] NR + [h2 A2 V°r + ha Aa vOr] NL
+ Mo Nj° NR + h.c., (8.15)
8.5.2	Tree level Majorana masses:
Since NL R, Eq.(8.2), are completely sterile neutrinos, we may also write the left and right handed Majorana type couplings
hL V°0 (N°°)c + mL N0 (NL)c	(8.16)
and
h2R VV n2 (NR)c + haR VV na (NR)c + mR NR (NR)c + h.c, (8.17)
respectively. After spontaneous symmetry breaking, we also get the left handed and right handed Majorana mass terms
hL [vi vOl + V2 v°L + va vOl] (NL)c + mL NO (N0)c + h.c., (8.18)
+ [h2R A2 v°R + haR Aa v0R (NR)c +
mR NR (NR)c
+ h.c.,
(8.19)
Thus, in the basis VVT = ( vol , v°l , vol , (vor)c , (v°r)c , (vor)c , NO , (NR)c ) the Generic 8 x 8 tree level Majorana mass matrix for neutrinos MV, from Table 8.1, VV MV (VV)c + h.c., read
M0
/0	0	0	00	0	ai	ai
0	0	0	00	0	a2	a2
0	0	0	00	0	aa	aa
0	0	0	00	0	0	0
0	0	0	00	0	b2	ß2
0	0	0	00	0	ba	ßa
ai	a2	aa	0 b2	ba	mL	mo
(8.20)
\ai a2 aa 0 ß2 ßa mo mR/
	(vol)c	Kl)c	(vol)c	vor	V°R	vor	(NL)c	NR
	0	0	0	0	0	0	hLVl	hovi
	0	0	0	0	0	0	hLV2	hoV2
	0	0	0	0	0	0	hLV3	hDV3
(Vor)c	0	0	0	0	0	0	0	0
	0	0	0	0	0	0	h2A2 h2RA2	
(Vor)c	0	0	0	0	0	0	hsAs h3RA3	
	hLVl	hLV2	hLV3	0	h2A2	hsAs	mL	MD
(NR)c	hovi	hoV2	hDV3	0	^2rA2	HbrAb	Mo	mR
Table 8.1. Tree Level Majorana masses
Diagonalization of the MV , Eq.(8.20), yields four zero eigenvalues, associated to the neutrino fields: ap = ^Ja2 + a|
ap eL ap
'eR
a-Pa3 V°L + aapa3 V»L
ap a	ap a ^
b3 ,o	b2 o

VR
'tr
Assuming for simplicity, hh^ = h3, the Characteristic Polynomial for the
nonzero eigenvalues of MV reduce to the one of the matrix m4, Eq.(8.21), where
m4
/0 0 a a \ 0 0b ß a b mL mD Va ß mo mRj
U4 =
/UH U12 U13 UM\
U21 u22 u23 u24 U31 U32 U33 U34 \u41 U42 U43 U44J
(8.21)
a = J a2 + a2 + a3, a = \ / a| + a2 + a2 ,
22
b = >2 + b3 , ß = J ß2 + ß3
o
b
b
U4 m4 U4 = Diag(mo,mo,mo,mo) = d4 , m4 = U4 d4 U{ (8.22)
Eq.(8.22) impose the constrains
u?! mg + u22 mg + u13 mg + u?4 mg = 0	(8.23)
u21 mg + u|2 mg + u|3 mg + u24 mg = 0	(8.24)
uiiu2i mg + ui2u22 mg + ui3u23 mg + ui4u24 mg = 0, (8.25) corresponding to the (m4)11 = (m4)22 = (m4)12 = 0 zero entries, respectively.
In this form, we diagonalize MV by using the orthogonal matrix
/ ap	a1 a ap	0	0	ai un	ai,. ai u12	ai u13	ai ai u14
_ ai ap	a 2 a 3 a ap	0	0	aa2 un	a2 u12	u13	aa2 u14
0	_ ap a	0	0	03 un	a3 u12	a3 u13	u14
U° =
0 0 0
0 0
0 10 0 0 0	0 0 0 u21 u22 u23 Yu24 0 0 - u21 u22 u23 u24
0 0 0	u31	u32	u33	u34
000	u41	u42	u43	u44 y
(UV)T MV UV = Diag (0,0,0,0, mg, mg, mg, mg)	(8.27)
Notice that the first four columns in UV correspond to the four massless eigenvectors. Hence, the tree level mixing, UV, depends on the ordering we define for these four degenerated massless eigenvectors. However, it turns out that the final mixing product UV UV, as well as the final mass eigenvalues are independent of the choice of this ordering.
8.6 One loop neutrino masses
After tree level contributions the fermion global symmetry is broken down to:
SU(2)qL <g> SU(2)uR <8> SU(2)dR <8> SU(2)k << SU(2)vr << SU(2)eR (8.28)
Therefore, in this scenario light neutrinos may get extremely small masses from radiative corrections mediated by the SU(3) heavy gauge bosons.
8.6.1 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. 8.1, The internal fermion line in this diagram represent the tree level see-saw mechanism, Eq.(8.15). The vertices read from the SU(3) flavor symmetry interaction Lagrangian
iLnt = g1 - Z° + (e"°Y°e° +	- 2t~°Y°t°) Z° + ^ (e"°y,Y+ + e"°Y°T°Y2+ + n"°Y°t°Y+ + b.c.) , (8.29)
The contribution from these diagrams may be written as
2
Cy — mT(My )ij , ai = g1,	(8.30) n
mv(My)ij = £ mk U°kU°k f(My,m°)	(8.31)
k=5,6,7,8
, „	MY i MY T M2
and fYk = Mi-mg2in—Y '
< Vk >
< >
viL
Fig. 8.1. Generic one loop diagram contribution to the Dirac mass term my v°L v°R. M : Md, mL, mR
o
	V°R	V°R	V°R	NR
V°L	Dv 14	D v 15	Dv 16	0
V°L	0	D v 25	D v 26	0
	0	D v 35	D v 36	0
NL	0	0	0	0
Table 8.2. One loop Dirac mass terms my v°L v°R
mv(Mz, )ij = cos ^mv(M_)ij — sin ^ mv(M+)ij mY(Mz2 )ij = sin ^mv(M_)ij + cos ^mv(M+)ij VoiOs 1
Gv,mij = —n— ^v! C°s^ sin^ [mv(M-)ij — mv(M+)ij]
F(My) = mg u?i fYs + mg uf2 fY6 + mg u^ fY7 + mg u^ fYg	(8.32)
G(My) = mg u2! fYs + mg u22 fY6 + mg uis fY7 + mg ^4 fYg	(8.33)
h(my ) = mg uiiu21 fYs + mg ui2u22 fY6 + m° ui3u23 fY7
+ mg ui4u24 fYg (8.34)
mv(MY) 15 = aab2h(MY) ; mv(MY)i6 = ^nrh(MY)
mv(MY)25 = ^ab2h(MY) ; mv(MY)i6 = ^l3h(MY) mv(MY)s5 = aab2h(My) ; mv(MY)s6 = ^h(MY)
Dv 14 = ~
^ h(mi)+^ h(M2)
ab	ab
Dv 15 =
aib2 ab

4h(Mz, ) + 12h(Mz2 )
D v 25 =
a2b2 ab
4h(Mz, ) + 12h(Mz2 )- H(Gv,m)
+ H(M3)
1 a2b2
1 a3b3.
D v 36 = ~ ^I2H(M3) + T ^T3H(MZ2 )
2 ab
3 ab
Dv 16 =
aib3
ab
-6 H(Mz2 )- H(Gv,m)
D v 26 =
a2b3 ab
- 6 H(MZ2 )+ H(Gv,m)
D v 35 =
a3b2 ab
-ZH(Mz2 )+ H(Gv,m) b
H(Gvm) = ^a2a3 cos ^ sin ^ [H(M-) - H(M+)] n 2y3
8.6.2 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. 8.2 and Fig. 8.3, respectively.
A similar procedure as for Dirac Neutrino masses, leads to the one loop Majorana mass terms
mv(My)ii = -1F(My) ; mT(My)i2 = ^F(My)
mv (My )i3 = -i-3 F (My)	; mv(My )22 = # F (My) a2 a
mv (My)23 = a2a3F(My)	; mv(My)33 = F(My) a2 a
2
Y
! No
< >
No ! vOL
< >
v
iL
Fig. 8.2. Generic one loop diagram contribution to the L-handed Majorana mass term
mij V°l(V?l)t. M = Mo,mL,m.R
	Vol	V°L	V?L	NL
vol	Lvii	Lv 12	Lv 13	0
V°L	Lv 12	Lv 22	Lv 23	0
V?L	Lv 13	Lv 23	Lv 33	0
NL	0	0	0	0
Table 8.3. One loop L-handed Majorana mass terms my v°L (t°l)1
o
v
T	a1
Lv 11 = —2
n 2
4F(mz, ) + 12F(Mz2 )+ F(Gv,m)
Lv 22 =
4 F (Mz, ) + 12 F (Mz2 )- F (Gv,m )
1 a2
Lv 33 = 1"f F(Mz2 ) ,
3 a2
Lv 12 =
a1 a2
a
2
-1F (Mz, ) + 1F (Mi) + 12 F M )
a1 a3
Lv 13 = -2—
a2
-6F(Mz2 ) + 2F(M2) - F(Gv,m)
2
a
Lv 23 =
a2 a3
a
2
4 F (Mz2 ) + 1F (M3)+ F (Gv,m) b 2
F (Gv,m ) —
sjaiaz 1
n
2V3
cos $ sin $ [F(M-) - F(M+)]
(8.35)
8.6.3 One loop R-handed Majorana masses
viR
Fig. 8.3. Generic one loop diagram contribution to the R-handed Majorana mass term m V°r(V°r)t. M = Mo,mL,mR
	Vor	V°R	V?R	NR
vor	0	0	0	0
V°R	0	Rv 55	Rv 56	0
V?R	0	Rv 56	Rv 66	0
NR	0	0	0	0
Table 8.4. One loop R-handed Majorana mass terms m^ v°R (v°R)T
bf
mv(My )55 — ^f G (My )
mv(My )66 — bf G (My )
mv(My )56 — bbr G (My )
Rv 55 = b2
4G(Mz, ) + 12G(Mz2)- G(Gv,m)
1 b2
Rv 66 = 3b3G(Mz2 ) ,
Rv 56 =
b2b3 b2
-1G(Mz2 ) + 1 G(M3)+ G(Gv,m) b 2
G(Gv,m) =	cos $ sin $ [G(M_) - G(M+)]
n 2V3
Thus, in the basis, we may write the one loop contribution for neutrinos as
2
l1 v
/Lvll Lv 12 Lv 13
Lv 12 Lv 22 Lv 23
Lv 13 Dv 14 Dv 15 Dv16 0
V o
Lv 23 Lv 33 00
Dv 25 Dv 35 Dv 26 Dv 36
0
0
Dv 14	Dv 15	Dv 16	0	0
0	D v 25	D v 26	0	0
0	Dv 35	Dv 36	0	0
0	0 0	0	0
0	Rv 55	Rv 56	0	0
0	Rv 56	Rv 66	0	0
0	0 0	0	0 0
0
(8.36)
0 0 0/
0
0
8.6.4 Neutrino mass matrix up to one loop
Finally, we obtain the Majorana mass matrix for neutrinos up to one loop
M, = (UV)T M?V UV + Diag(0,0,0,0,mO,mO,mO,mO), (8.37) where explicitly
Mv =
/N11 N12 N13 N14	N15	N16	N17	N18
n12 n22 n23 n24	N25	N26	N27	N28
N13 N23 O O	N35	N36	N37	N38
n14 n24 0 n44	N45	N46	N47	N48
N15 N25 N35 N45	N55 + mO	N56	N57	N58
N16 N26 N36 N46	N56	N66 + mg	N67	N68
N17 N27 N37 N47	N57	N67	N77 + mO	N78
\N 18 N28 N38 N48 a L-handed:	N58	N68	N78	N88 + mg/
(8.38)
22
N11 = (Fz, - F1)
-P-2
22
N	-1 -2-3 r -2 - -2 (F
N12 = r--r~(Fz
F1 ) + F2 - F3 - 6Fm]
(8.39)
(8.40)
N22 = 4
Dirac:
1 (-2 - -2)2 4 -p -2
-(Fz, - F1 ) + -2(F2 - F3)
a2	a2 - a2
+^(F1 + 3Fz2 - 4F2)- 3-2 -1 4 a2
2
Fm
(8.41)
a2 a2 b2	a3 b3
N13 = ^— (—^r H1 + -^r- H2) = q 11
2 ap ab
ab
N14 =
-1 b3 ( a2 b2
2 ap b ab
Hz, +
-3 b3
ab
H3 - 6
a2 b2 ab
Hm) = q12
a1 a3 a2 b2	a3 b3
N23 = T—— (^T" H1 + —¡T H2) = q21
2 ap a ab
ab
(8.42)
(8.43)
(8.44)
N24 =
-2 (-P b2 + a2 ^ (-2 - -2)-3 b2 b3 -H3 +--^-ÏTÏ-Hz,
2 -p a2 b2
4 -p a2 b2
3 ap a3 b2 b3 3 a2 -3 b2 b3 +--:—TTñ-hz2--:-m7ï-Hm = q22
4 a2 b2
-p a2 b2
(8.45)
2
2
8 Neutrino Masses and Mixing Within a SU(3) Family Symmetry...	97 Majorana R-handed:
b2 b2
N44 = ^bp ( Gz, + 3GZ2 - 4G3 - 12Gm )	(8.46) Majorana L-handed and Dirac:
Ni5 = -Fi5 uii + qi3 U2i ; Ni6 = -Fi5 ui2 + qi3 U22 (8.47) Ni7 = -Fi5 Ui3 + qi3 U23 ; Ni8 =-Fi5 Ui4 + qi3 U24 (8.48)
ai a2 F i5 = ~-
2ap a
2 2 2 a2 - -i (Fz, - Fi) + at (F3 - F2)
a
22 1 2 a —
+2
(2a3 - a!)
a2
ai b2
qi3 = -r
2ap b
^Hz, + 03^H3 - 2^ ^^ H
ab ' "Z1 ' ab ' "3 a b2 b m
N25 = F25 Uii + q23 U2i ; N26 = F25 Ui2 + q23 U22 (8.49)
N27 = F25 Ui3 + q23 U23 ; N28 = F25 Ui4 + q23 U24 (8.50)
F25 =
[ (a2 - ai)2 (Fz, - Fi)+ 2a2(a2 - a^) (F3 - F2) - ap (Fz2 - Fi)
4 ap a
-2 aP (a2 - aP) (Fz2 - F2) + 4 (a2 - O?) (a2 - 2aP) Fm ] a2 (a2 - ap )b2 b3u | (a2 - ai)a3 b2u | -p a3 (b2 - 2 b2)
q23 =	^p a2b2	H3 +	4-p	Hz, +	Hz
2-p a2 b2 3 4-p a2 b2 Zl	4 a2 b2 "Z2
a3 [-p b2 + a2(b2 - 2b2)]
Kt

ap a2 b2
Dirac:
N35 = q3i Uii , N36 = q3i Ui2 , N37 = q3i u?3 , N38 = q3i UM (8.51)
q	ai ( a2 b2 H + a3 b3 H )
q3i = 2a Hi + H2)
F
m
98 A. Hernandez-Galeana Dirac and Majorana R-handed:
N45 = q32 U11 + G45 U21 , N46 = q32 U12 + G45 U22	(8.52)
N47 = q32 U13 + G45 U23 , N48 = q32 U14 + G45 U24	(8.53)
o2 03 (b2 - b2^ , (o2 - oi) b2 b3u	(2o2 - op)b2 b3
q32 = 2Q2b H3 + 40b2 Hz,	4Q2b Hz2
+ Ia3 + o1— 2q2 h
(o3 + o2 - 2o2) b2 b3
o2 b2	Hm
G45 =
b2 b3
4 b2
b2 - 2b3 (G
b2
G3) + b2 (Gz, - G3)

(2b2 -b2) G ■ —b2— Gm
2
8 Neutrino Masses and Mixing Within a SU(3) Family Symmetry...	99 Majorana L-handed, Dirac and Majorana R-handed:
N55 = F55 uf, + 2 q33 U11 U21 + G55 Uf	(8.54)
N56 = F55 U11 U12 + q33 (U11 U22 + U12 U21) + G55 U21 U22	(8.55)
N57 = F55 U11 U13 + q33 (U11 U23 + U13 U21) + G55 U21 U23	(8.56)
N58 = F55 U11 U14 + q33 (U11 U24 + U14 U21) + G55 U21 U24	(8.57)
N66 = F55 U^2 + 2q33 U12 U22 + G55 U22	(8.58)
N67 = F55 U12 U13 + q33 (U13 U22 + U12 U23) + G55 U22 U23	(8.59)
N68 = F55 U12 U14 + q33 (U14 U22 + U12 U24) + G55 U22 U24	(8.60)
N77 = F55 U23 + 2 q33 U13 U23 + G55 U23	(8.61)
N78 = F55 U13 U14 + q33 (U14 U23 + U13 U24) + G55 U23 U24	(8.62)
N88 = F55 U24 + 2 q33 U14 U24 + G55 U24	(8.63)
F a2 a2 F + a2 a2 F + a2 a2 F + (a2 - a?)2 F . (2af - ap)2 F F55 = F1 + ^ F2 + ^ F3 + 4a4 Fzi + 12 a4 Fz^
+ (a2 - af)(2a3 - a*)	f
+ A	F
m
q a2 a3 b2 b3 H , (a2 - a2) b2 H (2a3 - a^) (b2 - 2b3) h q33 = -T^Tt- H3 +-^ - Hzi--n - H
, a2 b2 - ap b3 - a2 (b2 - 2b2) H
+	a2 b2	H
G bf bf G + b4 G + (bf - 2bf)2 G bf (bf - 2bf) G G55 -	G3 + 4^4 Gz, + -^- gz2	b4- Gm
8.6.5 (Vckm)4x4 and (Vpmns)4x8 mixing matrices
Within this SU(3) family symmetry model, the transformation from massless to physical mass fermions eigenfields for quarks and charged leptons is
^o = Vo y(i) ¥l and ^R = VR vR1 ) ¥R ,
and for neutrinos ¥V = UV UV ¥V. Recall now that vector like quarks, Eq.(8.1), 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; fULT = (uW°)L and fdLT = (d°,s°,b°)L, to the W charged gauge boson is
^fuLYfdLW^ = -g2¥uL [(VOL VU1L))3X4]T (VOL <^3x4 Y^dL W+^ ,
(8.64)
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^L )3X4	(8.65)
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 ViL^x^ (U UY)3x8	(8.66)
8.7 Numerical results
To illustrate the spectrum of masses and mixing, let us consider the following fit of space parameters at the MZ scale [27]
Using the strong hierarchy for quarks and charged leptons masses[15], here we report the fermion masses and mixing, coming out from a global fit of the parameter space.
In the approach a2 « a3 = aH, we take the input values
M1 = 10 TeV , M2 = 1 TeV , — = 0.05
n
for the Mi, M2 horizontal boson masses, Eq.(8.7), and the SU(3) coupling constant, respectively, and the ratio of electroweak VEV's: Vt from ® d, and vt from
V1	A2 + V22
y1 = 0.09981 ,	-= 0.54326
V2	V3
v1 Vv2 + v2 — = 0.1 , -i-= 0.5
v2	v3
8.7.1 Quark masses and mixing u-quarks:
Tree level see-saw mass matrix:
MU =
0	0	0	7933.76
0	0	0	79337.6
0	0	0	159467.
MeV,
(8.67)
\0 1.18613 x 106 -841128. 374542./ the mass matrix up to one loop corrections:
Mu =
{ -1.40509 0.125675 -0.062809 -0.001885
and the u-quark masses
187.442 -66.8139 -255.74 \ 1564.71 1825.79 1.502 x 106/
609.844 408.793 1197.67 -172100. 35.9461 14.3165
MeV
(8.68)
(mu , mc , mt , Mu ) =
(1.3802 , 640.801
172105 , 1.502 x 106 ) MeV (8.69)
d-quarks:
0	0	0	1740.94
0	0	0	17442.3
0	0	0	32265.8
MeV
\0 70019.9 -41383.4 910004 /
(8.70)
Md =
/3.09609 28.1593 -47.4565 -4.23475\ 0.271539 -40.5966 215.617 19.2404 0.147401 -176.235 -2846.26 37.484 \0.005900 -7.05504 16.8159 914365./
MeV
(8.71)
Ma
d
(md , ms , mb , MD ) = (2.82 , 61.9998 , 2860 , 914365) MeV and the quark mixing
VcKM =
0.974352 0.225001 0.003647 0.000410 -0.224958 0.973502 0.041031 -0.001417 0.005632 0.040662 -0.997868 -0.039994 \ 0.000576 -0.002325 0.031130 0.001251 /
.72)
.73)
8.7.2 Charged leptons:
M° =
0	0	0	28340.3
0	0	0	283940.
0	0	0	525249.
MeV
\0 17105.4 -11570.9 5.94752 x 106/
;.74)
Me =
/-0.499137 29.7086 -43.9181 -0.15097 \ 0.043776 -72.8148 238.953 0.821414 0.023663 -183.913 -1720.65 1.18425


\-0.002378 -18.4839 34.6241 5.977 x 106/ fit the charged lepton masses:
MeV
:.75)
(me , m^, mT, ME) = (0.486, 102.7, 1746.17 , 5.977 x 106 ) MeV and the mixing
Ve°L VeL =
{ 0.968866 0.24054 -0.0584594 0.00474112\ 0.205175 -0.912554 0.138557 0.330471 -0.00217545 0.0132348
—i
0.350561 0.929446 0.0990967


0.0475013 0.0878703 0.994987
;.76)
8.7.3 Neutrinos:
Ml = eV
53594.6 535946. 1.07 x 106 0 1.8097 x 106 y44137.2 441372. 887147. 0 1.49 x 106
53594.6 535946. 1.07 x 106 0


0 0 0 0 0 0
886604. 1.97 x 108 4. 730152. 4
44137.2 441372. 887147. 0
1.80 x 106 1.49 x 106 -886604. -730152.
x 108
x 108 7.02 x 108/
8.77)
Mv = eV
I	-0.0119	0.0527	0.0227	-0.0878	-0.0693	0.1674	-0.0016	0.0004	\
	0.0527	-0.036	0.002	0.068	0.043	-0.748	0.007	-0.002	
	0.0227	0.002	0.	0.	0.0008	0.0005	-5.2 x 10-6	1.5 x 10-	6
	-0.0878	0.068	0.	-0.125	-0.1218	1.282	-0.012	0.003	
	-0.0693	0.043	0.0008	-0.121	3.206	-0.7430	0.0074	-0.0021	
	0.1674	-0.748	0.0005	1.282	-0.7430	1749.96	0.0003	-0.0001	
	-0.0016	0.007	-5.2 x 10-6	-0.012	0.0074	0.0003	-1. x 108	1.1 x 10-	6
V	0.0004	-0.002	1.5 x 10-6	0.003	-0.0021	-0.0001	1.1 x 10-6	1. x 109	J
(8.78)
generates the neutrino mass eigenvalues
(mi,m.2, m3, m4, m5, m6, m7,m8) = eV
(0, -0.0085 , 0.049, -0.22 , 3.21 , 1749.96, -1 x 108 ,1 x 109 ) (8.79)
the squared mass differences
m2 - m2 « 0.0000723 eV2 , mj - mf « 0.0024 eV2	(8.80)
m4 - mf « 0.0492 eV2 , mj - mf « 10.3182 eV2	(8.81)
and the lepton mixing matrix
UpMNS =
/ 0.2104 0.3520 0.8658 -0.8282 0.0030 0.0186 0.0807 0.0041 0.0052 \ 0.0034 0.0003 0.0011
8.8 Conclusions
We have reported a low energy parameter space, within a local SU(3) Family symmetry model, which combines tree level "Dirac See-saw" mechanisms and radiative corrections to implement a successful hierarchical spectrum, for charged fermion masses and quark mixing. In section 8.7 we illustrated the predicted values for quark and charged lepton masses at the the MZ scale[27], and a non-unitary quark mixing matrix (Vckm)4x4 within allowed values reported in PDG 2012 [28], coming from a parameter space with the horizontal gauge boson masses within (1-10) TeV, the SU(2)L weak singlet vector-like fermion masses MD ~
-0.2861 0.0060 -0.5478 0.1038 0.0881 0.8475
0.0053 0.00005 0.00001 -0.0507 -0.0005 -0.0001 -0.5074 -0.0050 -0.0014
(8.82)
—i
0.0021 -0.0857 0.0512 0.0005 0.0001 J
914.365 GeV, MU « 1.5 TeV, ME « 5.97 TeV, the neutrino masses in Eq.(8.79), including two light sterile neutrinos, and the squared neutrino mass differences: m2 - mf « 7.23 x 10-5 eV2, m2 - mf « 2.4 x 10-3 eV2, m2 - mf « 0.049 eV2, m? - m2 « 10.3eV2.
Hence the new particles introduced in this model are within reach at the current LHC and neutrino oscillation experiments.
It is worth to comment here that the symmetries and the transformation ofthefermion and scalar fields, all together, forbid tree level Yukawa couplings between ordinary standard model fermions. Consequently, theflavon scalar fields introduced to break the symmetries: ®d, n2 and n3, couple only ordinary fermions to their corresponding vector like fermion at tree level. Thus, FCNC scalar couplings to ordinary fermions are suppressed by light-heavy mixing angles, which as is shown in (Vckm)4x4, Eq.(8.73), may be small enough to suppress properly the FCNC mediated by the scalar fields within this scenario.
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.
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