Bled Workshops in Physics Vol. 16, No. 2
A
Proceedings to the 1 8th Workshop What Comes Beyond ... (p. 47) Bled, Slovenia, July 11-19, 2015

5 Charged Fermion Masses and Mixing from a SU(3) Family Symmetry Model
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 Beyond Standard Model augmented with a local SU(3) family symmetry, we report an updated fit of parameters, which account for the known spectrum of quarks and charged lepton masses, and the quark mixing in a 4 x 4 non-unitary Vckm. 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 See-saw Majorana neutrino mass matrix with four massless eigenvalues at tree level. Hence, light fermions, including light neutrinos obtain masses from loop radiative corrections mediated by the massive SU(3) gauge bosons. SU(3) family symmetry is broken spontaneously in two stages, whose hierarchy of scales yield an approximate SU(2) global symmetry associated with the Zi, Y± gauge boson masses of the order of 2 TeV. A global fit of parameters to include neutrino masses and lepton mixing is in progress.
Povzetek. Avtor poroca o prilagajanju vrednosti parametrov v razsirjenem standardnem modelu z dodano družinsko simetrijo SU(3), s katerim mu uspe pojasniti izmerjeni masni spekter kvarkov in leptonov ter neunitarni mesalni matriki za kvarke in leptone. V svojem scenariju doda običajnim fermionom se fermione (U, D, E, N), ki so sibki singleti SU(2)L z vektorskim znacajem. Tezki fermioni postanejo masivni ze na drevesnem nivoju z Dira-covim mehanizmom "see-saw". Sterilni nevtrini NL,R poskrbijo v nevtrinski masni matriki 8 x 8, na drevesnem nivoju, da so stiri lastne vrednosti enake 0. Maso lahkih kvarkov in leptonov, vkljucno z nevtrini, dolocajo bozonska polja z druzinskimi kvantnimi stevili v popravkih visjih redov. Avtor predvidi spontano zlomitev druzinske simetrije SU(3) v dveh korakih tako, da so mase Z1, Y± umeritvenih bozonov SU(2) reda 2 TeV.
5.1 Introduction
The origen of the hierarchy of fermion masses and mixing is one of the most important open problems in particle physics. Any attempt to account for this hierarchy introduce a mass generation mechanism which distinguish among the different Standard Model (SM) quarks and leptons.
After the discovery of the scalar Higgs boson on 2012, LHC has not found a conclusive evidence of new physics. However, there are theoretical motivations
* E-mail: albino@esfm.ipn.mx
48 A. Hernandez-Galeana
to look for new particles in order to answer some open questions like; neutrino ossccillations, dark matter, stability of the Higgs mass against radiative correc-tions,,etc.
In this article, we address the problem of charged fermion masses and quark mixing within the framework of an extension of the SM introduced by the author in [1]. This Beyond Standard Model (BSM) proposal include a vector gauged SU(3) family symmetry1 commuting with the SM group and introduce a hierarchical massgeneration mechanism in which the light fermions obtain masses through loop radiative corrections, mediated by the massive bosons associated to the SU(3) family symmetry that is spontaneously broken, while the masses of the top and bottom quarks as well as for the tau lepton, are generated at tree level from "Dirac See-saw"[3] mechanisms implemented by the introduction of a new set of SU(2)L weak singlets U, D, E and N vector-like fermions. Due to the fact that these vectorlike quarks do not couple to the W boson, the mixing of U and D vector-like quarks with the SM quarks gives rise to and extended 4 x 4 non-unitary CKM quark mixing matrix [4].
5.2 Model with SU(3) flavor symmetry 5.2.1 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 < SU(3)dR < SU(3)il <g> SU(3)Vr < SU(3)eR
D SU(3) qL+uR + dR + lL
+ dR + lL + eR + vR
= SU(3) (5.1)
We define the gauge symmetry group
G = SU(3) < SU(3)C < SU(2)L < U(1 )Y
(5.2)
where SU(3) is the gaged family symmetry among families, eq.(5.1), and GSM is the "Standard Model" gauge group, with gH, gs, g and g' the corresponding coupling constants. The content of fermions assumes the ordinary quarks and leptons assigned under G as:
Ordinary Fermions: q°L =

iL
dO
'q?L
— (3,3,2, -)l = ( q2OL
l° — , liL —
, Q — T3L + 1Y
¥° — (3,1,2, —1 )l — (l°L
See [1,2] and references therein for some other SU(3) family symmetry model proposals.
5 Charged Fermion Masses and Mixing from a SU(3) Family Symmetry Model
49
n =(3,3,1,4 )r = JcRRj , = (3,3,1, -2 )R = |sR
¥ = (3,1,1, -2)R = UR \tr/
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:
Ae
Right Handed Neutrinos: ¥VR = (3,1,1, 0)r = I
and the SU (2) L weak singlet vector-like fermions
Sterile Neutrinos: N°,NR = (1,1,1,0),
eo
R
The Vector Like quarks:
UL,UR = (1,3,1,4 ) , DL,DR = (1,3,1, - 3 )	(5.3)
and
The Vector Like electrons: EL, Eg = (1,1,1, -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. ,	(5.4)
and
mD NL NR + mL NL (NL)c + mg NR (NR)c + h.c	(5.5)
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[5]. 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.
5.3 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,
50 A. Hernandez-Galeana
KA
m = (3,i,i,0)= I n?2 ) , i = 2,3
Wa/
and acquiring the "Vacuum ExpectationValues" (VEV's): <n2>T = (0,A2,0) , (na)T = (0,0, As) .
(5.6)
The above scalar fields and VEV's break completely the SU (3) flavor symmetry. The corresponding SU (3) gauge bosons are defined in Eq.(5.20) through their couplings to fermions. Thus, the contribution to the horizontal gauge boson masses from Eq.(5.6) read
n2 : n3 :
^ (Y+Y- + Y+ Y-) + (Z1 + f - 2Zi )
2 . 2 2 ^ (Y+Y- + Y+ Y-)+ gHH3 A3Z2
These two scalars in the fundamental representation is the minimal set ofscalars to break down completely the SU (3) family symmetry. Therefore, neglecting tiny contributions from electroweak symmetry breaking, we obtain the gauge boson mass terms.
m2 y+y- + m2 y+y— + (m2 + m2) y3+y3- + 2m3 z3 + 2m3 +34m3 z3
- 1(M3) -23 Zi Z3 (5.7)
M3 = ^
M3 = ^
y =
M3 m3
A3
A3
(5.8)
Zi
Z3
M3 —M
' v3
Table 5.1. Z1 — Z3 mixing mass matrix
Diagonalization of the Zi — Z2 squared mass matrix yield the eigenvalues
M- = 2 (M2 + M3 — 0M3 — M3)3 + M3M3^	(5.9)
m+ = 3 (M3 + M3 ^(M3 — M3)3 + M2M3^	(5.10)
Z
Z
3
M 2 m2+4m3
3
5 Charged Fermion Masses and Mixing from a SU(3) Family Symmetry Model
51
72	72
M 2 Y+Y- + M2 Y+Y- + (M2 + m2) Y+Y- + M- — + M+	(5.11)
72 72
m2 Y+Y- + m2 y2 Y+Y— + M2 (1 + y2) Y3+Y3- + M2 y - + M2 y + (5.12)
where
Z1 \ _ (cos $ — sin $\ (Z_ Zj I sin $ cos $ ) lZ+
(5.13)
cos $ sin $ _ ——
a/3	M2
4 + m3(M2 — M2)
Due to the Zi — Z2 mixing, we diagonalize the propagators involving Zi and Z2 gauge bosons according to Eq.(5.13)
Zi = cos ^ Z_ — sin ^ Z+ , Z2 = sin ^ Z_ + cos ^ Z+
(Zi Zi > _ cos2 $ (Z_Z_) + sin2 $ (Z+Z+)
(Z2Z2) _ sin2 $ (Z-Z-> + cos2 $ (Z+Z+)
(Z1Z2) _ sin $ cos $ ((Z-Z-) — (Z+Z+)) So, in the one loop diagrams contribution:
Fz, _ cos2 $ F(M_) + sin2 $ F(M+) , Fz2 _ sin2 $ F(M_) + cos2 $ F(M+) Therefore, in the tree level single exchange diagrams
1 cos2 $ + sin2 $	1 sin2 $ + cos2 $
M2 M2 M+ ' M2 M2 M+
Zi	2	+	Z2	-	+
Notice that in the limit y = M > 1, sin ^ —» 0, cos ^ —» 1, and there exist a SU(2) global symmetry for the Z1, Y1± degenerated gauge boson masses.
It is worth to emphasize that the hierarchy of scales in the SSB yields an approximate SU(2) global symmetry in the spectrum of SU(3) gauge boson masses. Actually this approximate SU(2) symmetry plays the role of a custodial symmetry to suppress properly the tree level AF = 2 processes mediated by the M1 lower scale Zi , Yii , Yi2 horizontal gauge bosons.
52 A. Hernandez-Galeana
5.4 Electroweak symmetry breaking
Recently ATLAS[6] and CMS[7] 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, namely;
with the VEV?s
where
The contributions from (®u) and (®d) yield the W and Z gauge boson masses and mixing with the SU (3) gauge bosons
5 Charged Fermion Masses and Mixing from a SU(3) Family Symmetry Model
53
£ (vU + vd) W+W- + <¿±^2 (vU + vd) Zi
+ ^\/g2 + g'2 9h Zo [(v2u - v2u - v2d + v2d) Zi
ljg2 + g'2 gu7„ |(-2 -2 -2 -2
2 2 2 2 2 2 Z2 + (viu + v2u - 2v3u - vid - v2d + 2v3d) —=
"t/ . Y-, ~+ Y- _ . Y+ ~+ Y-+2 (viuvzu -vidv2d)-^--+ 2 (viuv3u - vidv3d)-
V2	V2
Y+ + Y-
+2 (v2uv3u - v2dv3d)
V2
nH Mi 2 . 2 . 2 . 2 \ -j2 1 , 2 . 2 „2 . 2 . 2 „2 i Z2
+ \ o (viu + v2u + vid + v;2d) Zi + - (viu + v2u + 4v3u + vi d + v^ + 4v3d) —
nH
4 I 2
+ (viu + vL + vid + v2d) -+-- + (v2u + vL + vid + v2d) Y+Y— + (v2u + vL + v2d + v2d) -+--
+ (viu - v2u + vid - v2d) Zi —2 + (v2uv3u + v2dv3d) (Y+ Y2 + Y- Y+)
V3
+ (viuv2u + vidv2d) (Y+Y- + Y-Y+) + (viuv3u + vidv3d) (Y+Y+ + Y-Y-)
Z2 Y+ + Y-	Z?, Y+ + Y-
+2 (viuv2u + vidv2d) —2 -2--+ (viuv3u + vidv3d) (Zi--l) ■
Z Y+ + Y-
-(v2uv3u + v2dv3d) (Zi + —) 3 + 3 I (5.14)
V3 V2 ' v—33u ,d 3' V3 V2
Zl) -3++ -3) -2
vu = v2u +v2u+v2u, vd = v2d +v2d +v2d.Hence, ifwe define as usual Mw = ? gv, we may write v = ^Jvu + v^ « 246 GeV.
i Y+ + Y-	, Yj t iY?
y2 = ^ , Y± = yLTTl	(5.15)
The mixing of Zo neutral gauge boson with the SU(3) gauge bosons modify the couplings of the standard model Z boson with the ordinary quarks and leptons
5.5 Fermion masses 5.5.1 Dirac See-saw mechanisms
Now we describe briefly the procedure to get the masses for fermions. The analysis is presented explicitly for the charged lepton sector, with a completely analogous procedure for the u and d quarks and Dirac neutrinos. With the fields of particles introduced in the model, we may write the gauge invariant Yukawa couplings, as
u
54 A. Hernandez-Galeana
h^ ®d ER
+ h2 n2 EL + h3 c n3 e° + m E° er + h.c
(5.16)
where M is a free mass parameter ( because its mass term is gauge invariant) and h, h2 and h3 are Yukawa coupling constants. When the involved scalar fields acquire VEV's we get, in the gauge basis ^L RT = (e°, |x°, t0, E°)L,R, the mass terms	+ h.c, where
M° =
/0 0 0 hvi\ 0 0 0 hv2
0 0 0 h v 3
\0 h2A2 h3A3 M J
/0 0 0 ai\ 0 0 0 a2 0 0 0 a3 \0 b2 b3 Mj
(5.17)
Notice that M° has the same structure of a See-saw mass matrix, here for Dirac fermion masses. So, we call M° a "Dirac See-saw" mass matrix. M° is diagonal-ized by applying a biunitary transformation ^Rr = V R xL,R. The orthogonal matrices V° and V° are obtained explicitly in the Appendix 5.9 A. From V° and V°, and using the relationships defined in this Appendix, one computes
V°TM° VR = Diag(0,0,-A3,A4)
(5.18)
V° 1 M°M°
V° = V°' M° 1 M° V° = Diag(0,0,A|,A2) .
(5.19)
where A3 and A4 are the nonzero eigenvalues defined in Eqs.(5.53-5.54), A4 being the fourth heavy fermion mass, and A3 of the order of the top, bottom and tau mass for u, d and e fermions, respectively. We see from Eqs.(5.18,5.19) that at tree level the See-saw mechanism yields two massless eigenvalues associated to the light fermions:
5.6 One loop contribution to fermion masses
Subsequently, the masses for the light fermions arise through one loop radiative corrections. After the breakdown of the electroweak symmetry we can construct the generic one loop mass diagram of Fig. 5.1. Internal fermion line in this diagram represent the Dirac see-saw mechanism implemented by the couplings in Eq.(5.16). The vertices read from the SU(3) flavor symmetry interaction Lagrangian
iLnt = gH (e~°^e° -	Z^ + 2— (e"°Y^e° +	- 2T"°Y^T°) Z£
+ -2 (e"°Y,Y+ + e-°Y^T°Y+ + ^t°Y3+ + h.c.) , (5.20)
where gH is the SU(3) coupling constant, Z1, Z2 and Y? , i = 1,2,3 , j = 1,2 are the eight gauge bosons. The crosses in the internal fermion line mean tree level mixing, and the mass M generated by the Yukawa couplings in Eq.(5.16) after the scalar
T
5 Charged Fermion Masses and Mixing from a SU(3) Family Symmetry Model	55
Y
o iL
ejR ekR ; EL	ER j efL
I
< nk >	< >
Fig. 5.1. Generic one loop diagram contribution to the mass term m^ e°Le°R
fields get VEV's. The one loop diagram of Fig. 1 gives the generic contribution to
the mass term m^ e"L e"R
CyY m" (V0)ik(V0)jkf(MY,m") n ^—
k=3,4

where MY is the gauge boson mass, cY is a factor coupling constant, Eq.(5.20),
A| and m^ = A4 are the See-saw mass eigenvalues, Eq.(5.18), and
f(x,y) = x2x-y2 ln yr. Using the results of Appendix 5.9, we compute
X mk (VO)ik(VO)jkf(MY,mk) =	F(My) ,	(5.22)
. , .	A4 A3
k=3,4
MM 2	mm 2	mm 2	mm 2
i = 1,2,3 , j = 2,3, and F(MY) — —2 —a2 ln — — —a2 ln . Adding up all the
one loop SU(3) gauge boson contributions, we get the mass terms	^R + h.c.,
M" =
/Du D12 D13 0\
0 D22 D23 0 0 D32 D33 0
V 0 0 00/
Oh
n
(5.23)
D11 = mil + "pf" + Fm) + 1 (P22Fl + I33F2)
4
12
D12 = P12(——f + F¡f ) D13 = ——Z2 + Fm)
D22 = |22( —ZL + ^y — —m) + ^ (W1—1 + I33F3)
D23 = —P23(FZ2 — Fm) b

D32 = —— —m) D33 = m-3^+1 (mi—2 + IJ.22F3) ,
Fz
2
56 A. Hernandez-Galeana Here,
Ft = F(My,) , F2 = F(My2) , F3 = F(My3) , Fz, = F(Mz, ) , Fz2 = F(Mz2 )
My, - M2 , My2 - M2 , My3 - M2 + M3
cos ^ sin ^
Fm =-[F(M-) - F(M+) ]
with M2, M3, Mz, and MZ2 the horizontal boson masses, Eqs.(5.8-5.10),
Mdj
at bj M ai bj A2 - A3 = ab
A3 Ca Cp )
(5.24)
and ca = cos a , Cp = cos |3 , sa = sin a , sp = sin p, as defined in the Appendix 5.9, Eq.(5.55). Therefore, up to one loop corrections we obtain the fermion masses
+ tfOM? ¥R = XL M XR ,
with M = Diag(0,0, -A3,A4)+ V° 1 M° V° .
(5.25)
Using VO, VO from Eqs.(5.51-5.52) we get the mass matrix:
M -
( mn mi2 Cp mi3	sp mi3 \
m2i m22 cp m23	sp m23
Ca m.31 Ca m32 (-A3 + CaCp m33)	Casp m33
W m3i sa m32 saCp m.33	(A4 + sasp m.33)/
(5.26)
where
1	a2
mii - - — ni
2	a'
1	ai a3
m21 - ^-:-ni
2	a'a
mi2
-iaFbT (n2 - 6H22Fm)
m31
-	iaini
—	2 a 111
mi3 --	[n2 + 2(2b^ - 1)H22Fm]
2 a'b	b2
m22 - ~
1 a3 b3
2 ab
^ (n2 - 6H22Fm) + ^ (n3 + A) a' 2 22 m a3b3 3
(5.27)
(5.28)
(5.29)
(5.30)
5 Charged Fermion Masses and Mixing from a SU(3) Family Symmetry Model
57
m-23 = 2"
1 Q3b3
ab
ab2 (n + 2(2bf - Wm)-^ (ns
a'b3	b2	a3
b2 b2 bf A + 2bf ^33 Fm
2

(5.31)
m32 = x
1 a3b3
2 ab
a b a a2 -(n2 - 6^22Fm) - TT (ns--2 A - 2-2 ^33Fm)
a3	b3	a2	a3
(5.32)
m33 = -
1 a3b3
2 ab
a2b2 a3b3
(n2 - 2^22Fm)+ ns +
a/2b2
a3b2 ,2
A - -
1 a2 b
22
3 a3 b3
H33Fz
2
2
b2 a2 a
+2( d + 2-l - ^) m-33ft t3 a3 a3
(5.33)
2
ni = H22Fi + H33F2 , n2 = H22Fz, + M-33F3 ns = H22Fs + M-33FZ2 , A = 21^33 (Fz2 - Fz, )	(5.34)
Notice that the m^ mass terms depend just on the ratio a1 and b1 of the tree level parameters.
a/ = yla1 + a2 , a = ^a/2 + a3 , b =	,	(5.35)
The diagonalization of M, Eq.(5.26) gives the physical masses for u, d, e and v fermions. Using a new biunitary transformation xl,r = V^R ¥l,r; Xl M Xr = ?l VL1 )TM VR1' ¥r, with ¥l,rt = (fi, f2, f3, F)l,r the mass eigenfields, that is
vL1'TMMt vL1' = vR1'TMt M vR1' = Diag(m?,m2,m3,MF) , (5.36)
m2 = m^, m2 = m2, m3 = m^ and Mj2 = M| for charged leptons. Therefore, the transformation from massless to mass fermions eigenfields in this scenario reads
^L = VO VL1'	and ^R = V£ vR1' ¥r	(5.37)
58 A. Hernandez-Galeana
5.6.1 Quark (Vckm)4x4 and Lepton (Upmns)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
^L = VLo VL15	and ^R = VR vr1 5 ¥R ,
Recall now that vector like quarks, Eq.(5.3), 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; f^LT = (uo,co,to)L and f°LT = (do,so,bo)L, to the W charged gauge boson is
^g2fouLY^f°dLW+^ = -g2^PuL [(Vol VU1L))3x4lT (V^l <^3x4 Y^°l W+^ ,
(5.38)
g is the SU(2)L gauge coupling. Hence, the non-unitary VCKM of dimension 4 x 4 is identified as
(VCKM)4X4 = [(Vol V01L))3X4]T (V^l V°L))3x4	(5.39)
5.7 Numerical results
To illustrate the spectrum of masses and mixing, let us consider the following fit of space parameters at the MZ scale [8]
Taking the input values
M1 = 2 TeV , M2 = 2000 TeV , — = 0.2
n
for the M1, M2 horizontal boson masses, Eq.(5.8), and the SU(3) coupling constant, respectively, and the ratio of the electroweak VEV's: vio from and vid from
V1u = 0 ,	= 0.1
V3u
— = 0.23257 , — = 0.08373
V2°	V3°
5.7.1 Quark masses and mixing
u-quarks:
Tree level see-saw mass matrix:
mo =
0	0	0	0.
0	0	0	29834.
0	0	0	298340.
MeV,
(5.40)
\0 1.49495 x 107 -730572. 1.58511 x 107/
5 Charged Fermion Masses and Mixing from a SU(3) Family Symmetry Model
59
the mass matrix up to one loop corrections: 11.38 0.	0.
Mu =
0.
\
0. -532.587 -2587.14 -2442.42 0. 7064.64 -172017. 31927.1 V 0. 70.6499 338.204 2.18023 x 107/
MeV
(5.41)
and the u-quark masses
( mu , mc , mt , MU ) = ( 1.38 , 638.22 , 172181 , 2.18023 x 107 ) MeV (5.42)
d-quarks:
0 0 0
13375.7 \
57510.3
686796.
MeV
\0 723708. -37338.1 6.89219 x 107/
(5.43)
Md
( 2.82461 0.0338487 -0.656039 -0.00689715 \
Md =
0.65453 -25.1814 -217.369 0.0562685 423.166 -2820.62 0.000562713 4.23187 44.2671
-2.28527 46.5371 6.89291 x 107
MeV
(5.44)
( md , ms , mb , MD ) = ( 2.82368 , 57.0005 , 2860 , 6.89291 x 107 ) MeV
and the quark mixing
(5.45)
V,
CKM
0.97362 -0.226684 0.0260403 -0.000234396
-
0.225277 0.973105 0.0481125 0.000826552
0.0362485 0.000194044 -0.040988 -0.000310055 0.998387 -0.00999333 -0.011432 0.000114632 J
(5.46)
5.7.2 Charged leptons:
MO =
0	0	0	37956.9
0	0	0	189784.
0	0	0	1.93543 x 106
0	548257.	-30307.4	1.94497 x 10s /
MeV
(5.47)
60 A. Hernandez-Galeana
Me =
( -0.486368 -0.00536888 0.0971221	0.000274163 \ 0.0967909 -34.7536 -250.305 -0.706579 0.0096786 485.768 -1661.27 -0.0000967909 4.85792 38.2989
—i
10.8107 1.94507 x 108/
MeV (5.48)
fit the charged lepton masses:
(me , m^, mT, ME) = (0.486095, 102.7, 1746.17, 3.15956 x 108 ) MeV and the charged lepton mixing

0.973942 -0.226798 -2.90427 x 10-6 V 2.62189 x 10-7
0.221206 0.050052 0.000194 \ 0.949931 0.214927 0.0008342 -0.220675 0.975296 0.009963 0.0013632 -0.009906 0.99995
(5.49)
5.8 Conclusions
We reported recent numerical analysis on charged fermion masses and mixing within a BSM with a local SU(3) family symmetry, which combines tree level "Dirac See-saw" mechanisms and radiative corrections to implement a successful hierarchical mass generation mechanism for quarks and charged leptons.
In section 5.7 we show a parameter space region where this scenario account for the hierarchical spectrum of ordinary quarks and charged lepton masses, and the quark mixing in a non-unitary (VCKM)4x4 within allowed values2 reported in PDG 2014 [9].
Let me point out here that the solutions for fermion masses and mixing reported in section 5.7 suggest that the dominant contribution to Electroweak Symmetry Breaking comes from the weak doublets which couple to the third family.
It is worth to comment here that the symmetries and the transformation of the fermion 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, 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.(5.46), may be small enough to suppress properly the FCNC mediated by the scalar fields within this scenario.
2 except (Vckm)i3 and (Vckm)31
5 Charged Fermion Masses and Mixing from a SU(3) Family Symmetry Model
61
5.9 Appendix: Diagonalization of the generic Dirac See-saw mass matrix
Mo =
/0 0 0 a,\
0 0 0 a2 0 0 0 a3
\0b2 b3 cj
(5.50)
Using a biunitary transformation ^L = VR xl and ^R = VR Xr to diagonalize Mo, the orthogonal matrices VO and VR may be written explicitly as
I	21 cos a 21 sin a\
VO
Vo VR
- ai a'	a2 a3 a'a	Oa2 cos a	Oo2 sin a
0	a' a	a3 cos a	oO3 sin a
0	0	sin a	cos a
/1	0	0	0 \
0	b3 b2 b b	cos 3 b2	sin (
0	b2 b3 b b	cos 3 bb3	sin 3
0	0	sin ( cos (	
(5.51)
(5.52)
A3 = ^ (B - VB2 - 4D) , A2 = ^ (B WB2 - 4D are the nonzero eigenvalues of MOMoT (MoTMO), and
(5.53)
B = a2 + b2 + c2 = A3 + A4 , D = a2b2 = A2a2 ,	(5.54)
cos a ^
cos (3 =
/A2 - a2
A2 - A3
A - b2 A4 - A3
sin a =
sin ( =
la2 - A3
A2 - A3
lb2 - A3 A4 - A3
(5.55)
62 A. Hernandez-Galeana
Acknowledgements
It is my pleasure to thank the organizers N.S. Mankoc Borstnik, H.B. Nielsen, M. Yu. 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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