Bled Workshops in Physics Vol. 18, No. 2
A
Proceedings to the 20th Workshop What Comes Beyond ... (p. 56) Bled, Slovenia, July 9-20, 2017

5 AF = 2 in Neutral Mesons From a Gauged SU(3)F Family Symmetry
A. Hernandez-Galeana *
Departamento de Física, ESFM - Instituto Politécnico Nacional. U. P. "Adolfo Lopez Mateos". C. P. 07738, Ciudad de Mexico, Mexico
Abstract. Within a broken local gauge vector-like SU(3)F family symmetry, we study some AF = 2 processes induced by the tree level exchange of the new massive horizontal gauge bosons, which introduce flavor-changing couplings. We find out that some of the dangerous FCNC processes, like for instance; Ko — Ko , Do — Do mixing, may be properly suppressed if the first stage of the Spontaneous Symmetry Breaking (SSB), SU(3)F —> SU(2)F, occurs at a high scale A ~ 1011 GeV, with the SU(2)F gauge bosons acting on the light families. We provide a parameter space region where this framework can accommodate the hierarchical spectrum of quark masses and mixing and simultaneously suppress properly the contribution to Ko — Ko mixing as well as the OLL and Oeffective operators for the AC = 2 processes.
Povzetek. Avtor obravnava procese, pri katerih se druZinsko kvantno stevilo spremeni za 2. Uporabi model, v katerem opise druzinsko kvantno stevilo kvarkov in leptonov z grupo SU3, lokalna umeritvena polja grupe SU3 pa poskrbijo za interakcijo med fermioni, ki nosijo ustrezna kvantna stevila. Masivni umeritveni bozoni dopusčajo sicer nevtralne prehode (FCNC) med fermioni iste družine, vendar so taki prehodi, kot primer navaja mesanje Ko — Ko ter Do — Do, dovolj malo verjetni, ce le pride do spontane zlomitve druzinske simetrije SU(3)F —> SU(2)F pri energiji A ~ 1011 GeV. Poisče območje parametrov, v katerem imajo kvarki opazljive lastnosti.
Keywords: Quark and lepton masses and mixing, Flavor symmetry, AF = 2 Processes.
Pacs: 14.60.Pq, 12.15.Ff, 12.60.-i 5.1 Introduction
Flavor physics and rare processes play an important role to test any Beyond Standard Model(BSM) physics proposal, and hence, it is crucial to explore the possibility to suppress properly these type of flavor violating processes.
Within the framework of a vector-like gauged SU(3)F family symmetry model[1,2], we study the contribution to AF = 2 processes[3]-[6] in neutral mesons
* E-mail: albino@esfm.ipn.mx
5 AF = 2 in Neutral Mesons From a Gauged SU(3)F Family Symmetry
57
at tree level exchange diagrams mediated by the gauge bosons with masses of the order of some TeV's, corresponding to the lower scale of the SU(3)F family symmetry breaking.
The reported analysis is performed in a scenario where light fermions obtain masses from radiative corrections mediated by the massive bosons associated to the broken SU(3)F family symmetry, while the heavy fermions; top and bottom quarks and tau lepton become massive from tree level See-saw mechanisms. Previous theories addressing the problem of quark and lepton masses and mixing with spontaneously broken SU(3) gauge symmetry of generations include the ones with chiral local SU(3)H family symmetry as well as other SU(3) family symmetries. See for instance [7]-[20] and references therein.
5.2 SU(3)f flavor symmetry model
The model is based on the gauge symmetry
G = SU(3)f <g> SU(3)c < SU(2)l < U(1)y	(5.1)
where SU(3)F is a completely vector-like and universal gauged family symmetry. That is, the corresponding gauge bosons couple equally to Left and Right Handed ordinary Quarks and Leptons, with gH, gs, g and g' the corresponding coupling constants. The content of fermions assumes the standard model quarks and leptons:
= (3,3,2,3 )l , ^ = (3,1,2, —1 )l
=(3,3,1,-)R , ¥^(3,3,1, - 2 )R , ¥0 = (3,1,1, -2)f
3)r , V°(3,3,1, —2) where the last entry is the hypercharge Y, with the electric charge defined by
Q = T3L + 2 Y.
The model includes two types of extra fermions:
•	Right Handed Neutrinos: ¥VR = (3,1,1, 0)r , introduced to cancel anomalies [21],
•	and a new family of SU(2)L weak singlet vector-like fermions:
UL,UR = (1,3,1,-) , D0,DR = (1,3,1, -2 )	(5.2)
3) , D0,DR = (1,3,1,-3
Vector Like electrons: E°, ER = (1,1,1, -2)
and
New Sterile Neutrinos: NR,NR = (1,1,1,0) ,
58 A. Hernandez-Galeana
The particle content and gauge symmetry assignments are summarized in Table 5.1. Notice that all SU(3)F non-singlet fields transform as the fundamental
representation under the SU(3)F symmetry.
	SU(3)f	SU(3)c	SU(2)l	U(1)y
	3	3	2	1 3
	3	3	1	4 3
^dR	3	3	1	2 - 3
	3	1	2	-1
Cr	3	1	1	-2
^vr	3	1	1	0
	3	1	2	-1
od	3	1	2	+1
ni	3	1	1	0
Ul,r	1	3	1	4 3
Dl,r	1	3	1	2 - 3
El,r	1	1	1	-2
nl,r	1	1	1	0
Table 5.1. Particle content and charges under the gauge symmetry
5.3 SU(3)f family symmetry breaking
To implement the SSB of SU(3)F, we introduce the flavon scalar fields: ni, i = 2,3,
KA
ni = (3,1,1,0)= I n?2) , i = 2,3 Wj
with the "Vacuum ExpectationValues" (VEV's):
(n2)T = (0,A2,0) , (na)T = (0,0, As).	(5.3)
It is worth to mention that these two scalars in the fundamental representation is the minimal set of scalars to break down completely the SU(3)F family symmetry. The interaction Lagrangian of the SU(3)F gauge bosons to the SM massless fermions is
iLint,SU(3)F = gH (f0 f° f3y
/7M-	7 M	v+M
/	I Z2	Y__'2 \
'2 + 2^3 V2	^2

V2
\ 2
V V2
7M	7M v+M
Z1 I Z2 ' 3
2 + 2^3 ^2
V2
/f?\
fo f 2



ZM Z2
3
5 AF = 2 in Neutral Mesons From a Gauged SU(3)F Family Symmetry	59
i Y1 + iY2
where gH is the SU(3)F coupling constant, Zï , Z2 and Y± = ' ^ ' , j = 1,2,3 are the eight gauge bosons.
Thus, the contribution to the horizontal gauge boson masses from the VEV's in Eq.(5.3) read
•<ni> : ^(Y+Y- + Y+Y—) + ^(Z1 + Z2 -2ZTZ3)
2 2 2
• <n3> : ^(Y+Y- + Y+Y-) + gH3Al^^
The "Spontaneous Symmetry Breaking" (SSB) of SU(3)F occurs in two stages
SU(3)f x GSM ——^ SU(2)F ? x GSM ——^ GSM FCNC ?
A3: 5 very heavy boson masses (> 100 TeV's)
A2: 3 heavy boson masses (may be a few TeV's).
Notice that the hierarchy of scales A3 ^ A2 define an "approximate SU(2) global symmetry" in the spectrum of SU(2)F gauge boson masses. To suppress properly the FCNC like, for instance: ^ —> ey, ^ —> e e e, Ko — Ko, and Do — Do, it is crucial to choose properly the SU(2)F symmetry at the lower scale.
Therefore, neglecting tiny contributions from electroweak symmetry breaking, we obtain the gauge boson mass terms.
M 2 Y+ Y- + M2 Y+Y- + (M2 + M2 ) Y+Y- + Z2
+ 2
1M2 + 4M2 Z2 - 1 (M2)^ Z, Z2
3
2_
'73'
(5.4)
M2 =
92hA2
m2 =
92hA3
y =
M3 = Aa M2 A2
(5.5)
2
2
	Zi	Z2
Zi	m2	M2 — vf
Z2	M2 —"Tf	m2+4M§ 3
Table 5.2. Zi — Z2 mixing mass matrix
Diagonalization of the Zi — Z2 squared mass matrix yield the eigenvalues
60 A. Hernandez-Galeana
M- = 2 + Ml - \\(M2 - M2)2 + M2M3 M+ = 2 (M2 + M2 + \(M2 - M2)2 + M^M2
and finally
(5.6)
(5.7)
72	72
m2 Y+Y- + M2 Y+Y- + (M2 + M2) Y+Y- + M- — + M+ —+
where
—_ /'cos * — sin (— —2
sin * cos * ) V —
cos * sin * = ——
m2
4 + m2(M2 - M2)
—1 = cos * —- - sin * —+ , — 2 = sin * — _ + cos * —+
(5.8)
(5.9)
(5.10)
5.4 Electroweak symmetry breaking
For electroweak symmetry breaking we introduction two triplets of SU(2)L Higgs doublets, namely;
and the VEV?s
where
= (3,1,2,-1)
= 1 (ou)
= — fVui ^ ) V2y°
®d = (3,1,2, +1),
(®d) = ( (®2d) I ,

The contributions from (®u) and (® d) yield the W and — o gauge boson masses and mixing with the SU(3)F gauge bosons
g2 (vU + vdd) W+W- + fc^ (vU + vdd) —0
8
+ tiny contribution to the SU(3)F gauge boson masses and mixing with the gauge boson —o ,
5 AF = 2 in Neutral Mesons From a Gauged SU(3)F Family Symmetry 61
vU = vL + v2u + v3u ,v d = v2d + v2d + v3d • So, if we defme Mw = 1 gv we may write v = \JvU + vd « 246 GeV.
5.5 Fermion masses 5.5.1 Dirac See-saw mechanisms
The scalars and fermion content allow the gauge invariant Yukawa couplings
Hu UR + hiu lUR ni UO + Mu UL UR + h.c	(5.11)
Hd ®d DR + hid ni DO + Md DO DR + h.c	(5.12)
He if ®d ER + hie ifR ni E° + Me E° ER + h.c	(5.13)
Hv if NR + hiv iRni Nf + Mnd N° NR + h.c	(5.14)
hL	(NL)c + mL NR (N°)c + h.c	(5.15)
hiR ^V
vr
ni (NR)c
+ mR NR (NR)c + h.c
(5.16)
MU , Md , Me , Mnd , mL , mR are free mass parameters and Hu, Hd, He, Hv, hiu, hid, hie, hiv, hL, hiR are Yukawa coupling constants. When the involved scalar fields acquire VEV's, we get in the gauge basis 1L RT = (e°, m°,t°, EO)L,R, the mass terms 10M°1R + h.c, where
M0 =
/0 0 0 hvi\ 0 0 0 hv2 0 0 0 hv3 \0 h2A2 h3A3 M J
/0 0 0 ai\ 0 0 0 a2 0 0 0 a3
V0 b2 b3 Mj
(5.17)
M0 is diagonalized by applying a biunitary transformation ^LR = V0 R xL,R. Using the possible parametrizations for the orthogonal matrices and V£ are written explicitly in the Appendix A, Using one obtains
y°TM0 V° = Diag(0,0, -A3,A4) (5.18)
VltM°M°t V° = VrtM°tM° VR = Diag(0,0,A3,A4) . (5.19)
where A3 and A4 are the nonzero eigenvalues defined in Eqs.(5.56-5.58), A4 being the fourth heavy fermion mass, and A3 of the order of the top, bottom and tau
62 A. Hernandez-Galeana
mass for u, d and e fermions, respectively. We see from Eqs.(5.18,5.19) that from tree level the See-saw mechanism yields two massless eigenvalues associated to the light fermions:
it is worth to mention that the Yukawa couplings in Eqs.5.11-5.16 are invariant under the global symmetry U(1 )B x U(1 )Y x U(1 )a x U(1 , where B is the baryon number, Y is the hypercharge, and U(1 )a, U(1 are two additional symmetries, and one of them could play the role of a Peceei-Quinn symmetry to address the strong CP problem[22].
5.6 One loop contribution to fermion masses
After tree level contributions the first two generations remain massless. Therefore, in this scenario light fermion masses, including neutrinos, may get small masses from radiative corrections mediated by the SU(3)F heavy gauge bosons.
The one loop diagram of Fig. 1 gives the generic contribution to the mass term mtj ë?Le?R, where
Y
i i
< nk >	< >
Fig. 5.1. Generic one loop diagram contribution to the mass term my e?Le°R
2
mij = cy^ Y m° (V0)ik(V0)jkf(My,mk) , <xh = , (5.20) n i—
k=3,4
My being the mass of the gauge boson, cy is a factor coupling constant, Eq.(5.3), —A3 and m ln , and
Y mk (VL°)ik(VRO)jkf(My,mk) = ^ F(My) ,	(5.21)
m° = —A3 and m° = A4 are the See-saw mass eigenvalues, Eq.(5.18), f(x,y) =
2 2 x2 —y2 ln y2 ,
x2-y2 y
,on/oï rw^	^ _ ai M
k=3,4	A2 - A2
ll2
i = 1,2,3 , j = 2,3, and F(My) = ^r—^ ln -^j"1 - M2 —ln "^r. Adding up all possible the one loop contributions, we get the mass terms "[MO "R + h.c.,
5 AF = 2 in Neutral Mesons From a Gauged SU(3)F Family Symmetry
63
MO =
/Du D12 D13 0\
0 D22 D23 0
0 D32 D33 0
V 0 0 00/
oh n
(5.22)
D11 = 1 (H22Fl + H33F2) , D12 = W2(-F|1 + ^y) , Dl3 ="W3(F|2 + Fm) ,
D22 = H22(^f1 + - Fm) + 2^33F3
D23 = M-23 ( ^l2 - Fm)
Fz2
F| 1
D32 = M-32 ( 12 - Fm) , D33 = M-33^2 + 2M22F3 ,
aH = , Fi = F(Mv, ) , F2 = F(My2 ) , F3 = F(M^ )
4 n
Fz, = cos2 ^ F(M_) + sin2 ^ F(M+) , Fz2 = sin2 ^ F(M_) + cos2 ^ F(M+)
cos ^ sin ^
Fm =-2V3 [F(M_) - F(M+)] •
FZ, , FZ2 are the contributions from the diagrams mediated by the Z1 , Z2 gauge bosons, Fm comes from the Z1 - Z2 mixing diagrams, with M2, M3, M_, M+ the horizontal boson masses, Eqs.(5.5-5.7),
at bj M at bj
^ = .2 .2 =	A3 Ca Cp ,
A2 - A3
(5.23)
with ca = cos a, cp = cos |3, sa = sin a, sp = sin |3 the mixing angles coming from the diagonalization of Mo. Therefore, up to one loop corrections the fermion masses are
^r + ^M? ^r = xl M Xr ,
where = Vl,r xl,r, and M = Diag(0,0, -A3,A4)+ V° 1 M° VR written as:
(5.24) can be
M =
/ mii mi2	Cp mi3	Sp mi3 \
m21 m22	Cp m23	Sp m23
Ca m3i Ca m32	(-A3 + CaCp m33)	CaSp m33
Vsa m3i Sa m32	SaCp m.33	(A4 + SaSp m33)/
(5.25)
64 A. Hernandez-Galeana
The explicit expression for the m^ mass terms depends on the used parametriza-tion for V°, VR.
The diagonalization of M, Eq.(5.25) gives the physical masses for u and d quarks, e charged leptons and v Dirac neutrino masses. Using a new biunitary transformation
Xl,r = V[,) ¥l,r; XXl M Xr = Pl V'1 )TM V'1 5 Pr, with Pl,rt = (f1, f2, f3, F)l,r the mass eigenfields, that is
VL1)TMMt VL15 = VR1)TMt M VR15 = Diag(mf,m2,m3,MF) , (5.26)
ml = m;, m; = m2, m| = mT and M^ = M| for charged leptons. So, the rotations from massless to mass fermions eigenfields in this scenario reads
^L = VO VL15 Pl and ^R = VR vr15 Pr	(5.27)
5.6.1 Quark Mixing Matrix Vckm
We recall that vector like quarks, Eq.(5.2), are SU(2)L weak singlets, and 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 fdLT = (dO, sO, bO)l, to the W charged gauge boson is
^fVLYfW + 2 =
^UL [(VOL VU_L_5 )3x4]T (VOl Vd L) 3 x 4 Y^dL W+2 , (5.28) Hence, in this scenario the non-unitary VCKM of dimension 4 x 4 is identified as (VCKM )4x4 = [(VOl V^L )3 x4]T (V^l Vd1L))3x4	(5.29)
5.7 Numerical results for quark masses and mixing
As an example of the possible spectrum of quark masses and mixing from this scenario, we consider the following set of parameters at the MZ scale [23]
Using the input values for the horizontal boson masses, Eq.(5.5), and the coupling constant of the SU(3)F symmetry:
M2 = 6.0 TeV , M3 = 1.5 x 108 TeV , — = 0.2, (5.30)
n
we show in the interaction basis the following tree level Mq, and one loop Mq 1 quark mass matrices, and the corresponding mass eigenvalues and mixing:
5 AF = 2 in Neutral Mesons From a Gauged SU(3)F Family Symmetry
65
u-quarks:
Tree level see-saw mass matrix:
MU =
0	0	0	5573.43
0	0	0	23883.8
0	0	0	397346.
MeV,
(5.31)
\0 -1.931 x 108 5.193 x 106 2.470 x 10s/
the mass matrix up to one loop corrections:
MUi =
/1.42 -220.786 34.6742 0\ 0 -944.713 148.589 0 0 -91921.3 11631.6 0 \ 0 0 0 0/
MeV
(5.32)
the u-quark mass eigenvalues
(mu, mc , mt, MU) = (1.382, 633.289 , 172968, 313.606 x 106) MeV (5.33) and the mixing matrices:
VuL = VUL VUV:
0.973838 -0.227244 9.34208 x 10-7 V—2.14837 x 10-9
0.226464 0.970491

0.0188217 0.0000144353 -0.080663 0.0000618569 0.0828299 -0.996563 0.0011792 0.0000343726 0.00118041 0.999999
/
Vur = V°r v(1r:
/ 1.	0.0005077911.54519 x 10-
7.6088 x 10-6 0.0147444 0.787788
— /
—
0.000507462 0.0000166153
0
0.61577
0.999362 0.0316481 0.0165598 0.0325336 0.615132 0.787752/

d-quarks:
Md =
0 0 0 0 0 0	0	3102.75
0 0	0	61977.5
\0 -9.805 x 107 2.837 x 106 6.046 x 108/
MeV
ldi
2.82 0 0 0 0 -130.851 10.43 0 0 -7200.83 664.801 0 \ 0 0 0 0/
MeV
(5.34)
(5.35)
(5.36)
(5.37)
66 A. Hernandez-Galeana
the d-quark mass eigenvalues
( md , ms, mb , MD ) = ( 2.82, 52.087, 2861.96, 612.541 x 106 ) MeV (5.38) the mixing matrices:
VdL = V0L Vd1':
0
1.
0 0
V0 7.94612 x 10-6 0.000101576
0
0.991883 -0.127155
0
\
0.127155 5.03428 x 10-6 0.991883 0.000101762
1.
/
(5.39)
VdR = Vd°R VdR
1.
0
0
0
\
0 -0.127762 0.9788 -0.160083 0 0.99148 0.130175 0.00463162 0 -0.0253722 0.158128 0.987093
(5.40)
and the quark mixing matrix
Vckm =
0.97383 0.22646 0.01882
—i
—i
—
0.2254 0.97314 0.04670
0.02889 0.04124 0.99873

1.14 x 10-6\ 3.54 x 10-6 0.00010
—i
\1.44 x 10-5 8.85 x 10-5 -0.00117 1.20 x 10-7 /
(5.41)
5.8 AF = 2 Processes in Neutral Mesons
Here we study the tree level FCNC interactions that contribute to Ko — K0, Do — Dc mixing via Zi , Y1± exchange from the depicted diagram in Fig. 2.
d
K0
K0
s
d
Fig. 5.2. Generic tree level exchange contribution to Ko — Ko from the SU(3) horizontal gauge bosons.
The Zi , Y1± gauge bosons have flavor changing couplings in both left- and right-handed fermions, and then contribute the AS = 2 effective operators
s
5 AF = 2 in Neutral Mesons From a Gauged SU(3)F Family Symmetry
67
Oll = (dLY^sO(dLY^sO , Orr = (dRY^R^Y^ sR) (5.42)
Olr = (dLY^SL)(dRY^SR)	(5.43)
The SU(3)f couplings to fermions, Eq.5.3, when written in the mass basis yield the gauge couplings
Lint,Z , = (CLZ 1 d~LY^SL + Cr z , JrY^Sr) Z1^	(5.44)
Lint,Y 1 = (CLY,1 d_LY^SL + CR Y] d_RY|xsr) Y] ^	(5.45)
Lint,Y2 = (Cly 2 d~LY^SL + Cry 2 d~RY^Sr) iY^	(5.46)
with the coefficients
CL Z1 = L11 L12 — L21 L22 , CRZ1 = R11 R12 — R21 R22
Cly1 = L12 L21 + L11 L22 , Cry1 = R12 R21 + R11 R22	(5.47)
Cl Y2 = (L12 L21 — L11 L22) , Cr y2 = (R12 R21 — R11 R22)
where Lij = VL ij and Rij = VR ij. For each gauge boson, the effective four-fermion hamiltonian at the scale of the gauge boson mass is
HZ1 = JJt- (CL Z1 Oll + 2 Cl Z1 Cr Z1 Olr + CR Z1 Orr)	(5.48) Z1
2
Hy] = (C2l y; Oll + 2 Cl y, Cr y; Olr + CR y, Orr)	(5.49)
2
Hy2 = — 4^2 (C2l y2 Oll + 2 Cl y2 Cr y ; Olr + CR y2 Orr)	(5.50)
with My, = MY2 = M2. Therefore, the total four-fermion hamiltonian Hsu(2) = HZ, + Hy; + HY2 can be written as
HSU(2) = 4IH2 (CLZi + CLY; — CLY2 )OLL + (CRZi + CR Y; + CR Y2 )ORR
+2(clz, cr z, + cly; cr y1 — cly2 cry2 )ol
g2 1 1
+ gf(— M2) [C2lz 1 Oll + CRz, Orr + 2Clz , Crz , Olr)] (5.51)
68 A. Hernandez-Galeana
From the coefficients in Eq.5.47 we get:
CLZ , + CLY 1 — CL Y 2 = ^L , CRZ , + CR Y j — CR Yf = ^R , ,
cl,z i cr,z i + cl,y j cr y j — cl,y2 cr,y? = §r
+ 2(Ln R21 — L21 R11 )(L22 R12 — L12 R22)
and finally we can write
hsu(2) = 4M2 [5L oll + 6R orr + 62R olr]	(5.52)
+ ^ (- MM2 ) [(Lll L12 - L-21 L22)2 oll + (Rl1 R12 - R21 R22)2 orr
+2(L11L12 - L21L22)(R11R12 - R21R22) olr)] (5.53)
with
5l = L11 L12 + L21 L22 , 5r = R11 R12 + R21 R22 5lr = \J2(6l 6r + 2(Ln R21 - L21 R11 )(L22 R12 - L12 R22))
5.8.1 Ko — Ko meson mixing
The numerical fit of parameters provided in section 7 yield the mixing angles Vd 12 = Vd21 = 0 for left- and right-handed d-quarks, and then all the contributions to the effective operators, Eqs.5.42-5.43, for AS = 2 vanish.
5.8.2 Do — Do meson mixing
The reported parameter space region in section 7 generate MZl = M2 with very good approximation, and then only the four-fermion Hamiltonian in Eq.5.52
contribute. For this case we compute the numerical values
5l = -7.73804 x 10-8 ,	= -5.51894 x 107 TeV
L	' «H 5l
5r = 5.07679 x 10-4 ,	= 8411.97 TeV	(5.54)
5lr = 0.0508636	, «M2 = 83.9614 TeV
'	5LR
Accordingly to the Review "The CKM quark - mixing matrix" in PDG2016[24], the AC = 2 effective operators OLL and ORR are within suppression limits.
5 AF = 2 in Neutral Mesons From a Gauged SU(3)F Family Symmetry
69
5.9 Conclusions
Horizontal gauge bosons from the local SU(3)F introduce flavor changing couplings, and in particular mediate AF = 2 processes at tree level. We reported the analytic contribution to Ko — Ko and Do — Do meson mixing from tree level exchange diagrams mediated by the SU(2)F gauge bosons Z1 , Y± with masses in the TeV region. We provide a particular parameter space region in in section 7 where this scenario can accommodate the hierarchy spectrum of quark masses and simultaneously suppress properly the Ko — Ko meson mixing, and the effective operators OLL and ORR for the AC = 2 processes.
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. The author is grateful for the warm hospitality at the APC Laboratory, Paris, France, during sabbatical staying, and the financial support of the Cosmology group to attend this conference. This work was partially supported by the "Instituto Politecnico Nacional", (Grants from EDI and COFAA) and "Sistema Nacional de Investigadores" (SNI) in Mexico.
5.10 APPENDIX: Diagonalization of the generic Dirac See-saw mass matrix
M° =
/0 0 0 ai\
0 0 0 a2 0 0 0 a3
\0b2 b3 cj
(5.55)
The tree level Mo 4 x 4 See-saw mass matrix is diagonalized by a biunitary transformation ^0 = VR xl and ^R = VR Xr- The diagonalization of MoMoT (MoTM0) yield the nonzero eigenvalues
A3 = 2 (B - Vb2 - 4D) , A2 = ! (B + VB2 - 4D)
(5.56)
and rotation mixing angles
cos a =
cos |3 =
/A2
A4 - A3
/A2

b2
A2 - A3
sin a
A3
sin | =
a2 - A3
/b2 - A3
A2 - A3
(5.57)
2
2
a
a
70 A. Hernandez-Galeana
B = a2 + b2 + c2 = A2 + A^
D = a2b2 _ A2A2 ,
(5.58)
a2 _ af + a2 + a3
b2 = bf + b2 + b2
The rotation matrices VO, VR admit several parametrizations related to the two zero mass eigenstates.
5.10.1 Parametrization P12
/ Ci C2 Si Si S2 Ca Si S2 Sa\ Vo _ -Si Ci C2 Ci S2 Ca Ci S2 Sa L	0 —S2 C2 Ca C2 Sa
\ 0 0 —Sa Ca ,
Vo VR
/10 0 0 \
0 Cr Sr Cp Sr Sp 0 —Sr Cr Cp Cr Sp \0 0 —Sp Cp J
ap _ J a2 + a2 , bp _
+ a2 , b _Jbp2 + b2
S1
p a2 b2 b2 - , Ci _ - , S2 = — , C2 = - , Sr = — , Cr = —
-	a	a	b	b
a±
ap
a2 ap
5.10.2 Parametrization P13
/ Ci —Si S2 Si C2 Ca Si C2 Sa\
0 C2 S2 Ca S2 Sa
— Si —Ci S2 Ci C2 Ca Ci C2 Sa
vo =
0
0
—Sa
v o
VR
1 0 0 0
0 Cr Sr Cp Sr Sp 0 —Sr Cr Cp Cr Sp \0 0 —Sp Cp J
an = J ai + a2 , bn _
2
an + a 2 , b =
2
a
a
2
p
C
oc
2
2
ai
a2
a2
Si _ - , Ci = - , S2 = - , C2 = - , Sr =
an	an	a	a	b
b2 b2 _ — , Cr _ — (5.59)
5.10.3 Parametrization P23
Ci 0 Si Ca Si Sa —Si S2 C2 Ci S2 Ca Ci S2 Sa
—Si C2 —S2 Ci C2 Ca Ci C2 Sa
VO =
0
0
Sa
Ca
Vo VR
1 0 0 0
0 Cr Sr Cp Sr Sp
0 —Sr Cr Cp Cr Sp \0 0 —Sp Cp J
an _ \/ a2 + a2
bn =
a = J a2 + ai
b _
Si
ai _ an a i a
S2
a2 _ 03 an ,	an
b2
b2
IT , Cr _ "¡3	(5.60)
a
2
2
Sr=
5 AF = 2 in Neutral Mesons From a Gauged SU(3)F Family Symmetry
71
References
1.	A. Hernandez-Galeana, Rev. Mex. Fis. Vol. 50(5), (2004) 522. hep-ph/0406315.
2.	A. Hernandez-Galeana, Bled Workshops in Physics, (ISSN:1580-4992), Vol. 17, No. 2, (2016) Pag. 36; arXiv:1612.07388[hep-ph]; Vol. 16, No. 2, (2015) Pag. 47; arXiv:1602.08212[hep-ph]; Vol. 15, No. 2, (2014) Pag. 93; arXiv:1412.6708[hep-ph]; 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];
3.	E. Golowich, J. Hewett, S. Pakvasa, and A. Petrov, Phys. Rev. D 76, 095009 (2007).
4.	E. Golowich, J. Hewett, S. Pakvasa, and A. Petrov, Phys. Rev. D 79,114030 (2009).
5.	M. Kirk, A. Lenz, and T. Rauh, arXiv:1711.02100[hep-ph]; T. Jubb, M. Kirk, A. Lenz, and G. Tetlalmatzi-Xolocotzi, arXiv:1603.07770[hep-ph];
6.	C. Bobeth, A. J. Buras, A. Celis, and M. Junk, arXiv:1703.04753[hep-ph]; A. J. Buras, arXiv:1611.06206[hep-ph]; arXiv:1609.05711[hep-ph];
7.	Z. Berezhiani and M. Yu.Khlopov: Theory of broken gauge symmetry of families, Sov.J.Nucl.Phys. 51, 739 (1990).
8.	A. S. Sakharov and M. Yu. Khlopov: Horizontal unification as the phenomenology of the theory of everything, Phys.Atom.Nucl. 57, 651 (1994).
9.	M. Yu. Khlopov: Cosmoparticle physics, World Scientific, New York -London-Hong Kong - Singapore, 1999.
10.	Maxim Khlopov, Fundamentals of Cosmic Particle physics, CISP-SPRINGER, Cambridge, 2012.
11.	M. Yu. Khlopov: Fundamental Particle Structure in the Cosmological Dark Matter, J. of Mod. Phys. A 28 1330042 (60 pages) (2013).
12.	Z. Berezhiani and M. Yu.Khlopov: Physical and astrophysical consequences of breaking of the symmetry of families, Sov.J.Nucl.Phys. 51, 935 (1990).
13.	Z. Berezhiani, M. Yu.Khlopov and R. R. Khomeriki: On the possible test of quantum flavor dynamics in the searches for rare decays of heavy particles, Sov.J.Nucl.Phys. 52, 344 (1990).
14.	T. Appelquist, Y. Bai and M. Piai: SU(3) Family Gauge Symmetry and the Axion, Phys. Rev. D 75, 073005 (2007).
15.	T. Appelquist, Y. Bai and M. Piai: Neutrinos and SU(3) family gauge symmetry, Phys. Rev. D 74, 076001 (2006).
16.	T. Appelquist, Y. Bai and M. Piai: Quark mass ratios and mixing angles from SU(3) family gauge symmetry, Phys. Lett. B 637, 245 (2006).
17.	J. L. Chkareuli: Quark-Lepton families: From SU(5) to SU(8) symmetry, JETP Lett. 32, 671 (1980).
18.	Z. G. Berezhiani, J. L. Chkareuli: Mass of the T quark and the number of quark lepton generations, JETP Lett. 35, 612 1982.
19.	Z. G. Berezhiani, J. L. Chkareuli: Neutrino oscillations in grand unified models with a horizontal symmetry, JETP Lett. 37, 338 1983.
20.	Z.G. Berezhiani: The weak mixing angles in gauge models with horizontal symmetry: A new approach to quark and lepton masses, Phys. Lett. B 129, 99 (1983).
21.	T. Yanagida, Phys. Rev. D 20, 2986 (1979).
22.	A. Hernandez-Galeana and M. Yu.Khlopov, work in progress.
23.	Zhi-zhong Xing, He Zhang and Shun Zhou, Phys. Rev. D 86, 013013 (2012).
24.	The Review of Particle Physics (2016), C. Patrignani et al. (Particle Data Group), Chin. Phys. C, 40, 100001 (2016).
25.	See PDG review: The CKM quark-mixing matrix, by A. Ceccucci (CERN), Z. Ligeti (LBNL), and Y. Sakai (KEK).