Bled Workshops in Physics Vol. 19, No. 2 JLV Proceedings to the 21 st Workshop What Comes Beyond ... (p. 299) Bled, Slovenia, June 23-July 1, 2018 14 Ko - Ko, Do - Do in a Local SU(3) 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 SU(3) gauged family symmetry, we report the analysis of AF = 2 processes induced by the tree level exchange of the new massive horizontal gauge bosons, which introduce flavor-changing couplings. We provide a parameter space region where this framework can accommodate the hierarchical spectrum of quark masses and mixing and simultaneously suppress within current experimental limits the contributions to Ko — Ko and Do — Do mixing. In addition we find out that the mass of the SU(2)L weak singlet vector-like D quark introduced in this BSM, may be of the orden of 10 TeV. Povzetek. Avtor v okviru svojega predloga teorije z zlomljeno družinsko simetrijo SU(3) analizira procese tipa AF = 2, ki jih inducira izmenjava novih masivnih horizontalnih umeritvenih bozonov na drevesnem nivoju, kar privede do sklopitev, ki spremenijo okus. Najde območje v prostoru parametrov, ki dovoljuje izmerjeni masni spekter kvarkov ter njihovo mesalno matriko, pri tem pa so prispevki mesanja Ko — Ko in D o — D1o pod trenutnimi eksperimentalnimi mejami. Maso napovedanega kvarka D, ki je sibki singlet vektorskega tipa SU(2)l, oceni na ~ 10 TeV. Keywords: Quark and lepton masses and mixing, Flavor symmetry, AF = 2 Processes. PACS: 14.60.Pq, 12.15.Ff, 12.60.-i 14.1 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) family symmetry model[1,2], we study the contribution to AF = 2 processes[3]-[6] in neutral mesons 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) family symmetry breaking. * E-mail: albino@esfm.ipn.mx 300 A. Hernandez-Galeana 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) 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]-[14] and references therein. 14.2 SU(3) family symmetry model The model is based on the gauge symmetry G = SU(3)f SU(3)c < SU(2)l U(1 )Y (14.1) where SU(3) 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: Y = (3,3,2, 3)l , = (3,1,2, —1 )l (14.2) YU = (3,3,1, 3)R , Yd(3,3,1, —2)R , Y° = (3,1,1, — 2)r (14.3) 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: = (3,1,1,0)R, introduced to cancel anomalies [7], and a new family of SU(2)L weak singlet vector-like fermions: Vector like quarks U°,UR = (1,3,1, 3) and DL,DR = (1,3,1, — 2), Vector Like electrons: EL,ER = (1,1,1,—2), and New Sterile Neutrinos: N°,NR = (1,1,1,0). The particle content and gauge symmetry assignments are summarized in Table 14.1. Notice that all SU(3) non-singlet fields transform as the fundamental representation under the SU(3) symmetry. 14.3 SU(3) family symmetry breaking To implement the SSB of SU(3), we introduce two flavon scalar fields: /n°A nt = (3,1,1,0)= I n°2 ) , i = 1,2 (14.4) \n°3/ 14 Ko - K° , Do - D° in a Local SU(3) Family Symmetry 301 SU(3) SU(3)c SU(2)l U(1)y cq 3 3 2 1 3 Cur 3 3 1 4 3 CdR 3 3 1 2 - 3 co 3 1 2 -1 3 1 1 -2 Cr 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 E°,r 1 1 1 -2 nl,r 1 1 1 0 Table 14.1. Particle content and charges under the gauge symmetry with the "Vacuum ExpectationValues" (VEV's): T = (A1}0,0) , : ^(Yi+Yr + Y2+Y2-) + ^(Z1 + f + 2Zi ) 2 „2 A 2 2 Y2 : ^(Y+Y- + Y+Y-) + g^f :z2 v2~ Z2 The "Spontaneous Symmetry Breaking" (SSB) of SU(3) occurs in two stages SU(3) x Gsm ^ SU(2) ? x GSM —1 GSM FCNC ? 302 A. Hernandez-Galeana Notice that the hierarchy of scales A2 > Ai yield an "approximate SU(2) global symmetry" in the spectrum of SU(2) gauge boson masses. Therefore, neglecting tiny contributions from electroweak symmetry breaking, we obtain the gauge boson mass terms. (m2 + M2) Y+Y- + M2 Y+Y- + M2 Y+Y- + -M? Z? + 2 M2 = g2HA2 1M2 + 4M2 Z2 + -(M2)^ Zi Z2 (14.7) 2 3 M2 = ^ 2 'V3' (14.8) M2 M Ml 73 M2 M2+4M 3+4M2 73 Table 14.2. Z-| — Z2 mixing mass matrix Diagonalization of the Z? — Z2 squared mass matrix yield the eigenvalues M- = ; ^M2 + M2 — ^(M2 — M2)2 + M?M2 M+ = 2 (M2 + M2 + ^(M2 — M?)2 + M?M2 3 2 and finally (14.9) (14.10) Z2 Z2 (M? + M2) Y+Y- + M2 Y+Y- + M2 Y+Y- + M- — + M+ Z+ . where cos § sin §\ (Z - sin § cos §/ \Z . • , V3_Mi_ cos § sin § = —----' - 4 y'M? + M2(M2 — M2) (14.11) (14.12) (14.13) Z Z 2 Z Z 2 3 2 14 Ko - K° , Do - D° in a Local SU(3) Family Symmetry 303 14.4 Electroweak symmetry breaking For electroweak symmetry breaking we introduce two triplets of SU(2)L Higgs doublets, namely; with the VEV's where OU =(3,1,2,-1) W>N (OU> = | (®u> (o^ = _L (Oi> Od = (3,1,2, +1), '(®d>N (Od> = | (®d> | , ,<®3d>, = ¿Cr (14.14) (14.15) (14.16) The contributions from (OU> and (O d> generate the W and Zo SM gauge boson masses (vU + vdd) W+W- + -i9-^-1 (vU + vdd) ZO + tiny contribution to the SU(3) gauge boson masses and mixing with Zo , (14.17) (14.18) (14.19) vU = v-u+v-u + v3u ' v3 = v-d + v-d + v3d. So'if mw = - gv we may write v = \jvU + vd « 246 GeV. 14.5 Fermion masses 14.5.1 Dirac See-saw mechanisms The scalars and fermion content allow for quarks the gauge invariant Yukawa couplings Hu UR + hiu ^Ur ni UL + Mu UL UR + h.c (14.20) Hd ^q Od DR + hid ^3r ni DL + Md DL DR + h.c (14.21) Mu , Md are free mass parameters and Hu, Hd , hiu, hid are Yukawa coupling constants. When the involved scalar fields acquire VEV's, we get in the gauge basis rt = (e°, m°,t°, E°)l,r, the mass terms + h.c, where 304 A. Hernandez-Galeana M° = ( 0 0 0hv^ 0 0 0hv2 0 0 0hv3 Vh At h2A2 0 M J ( 0 0 0 a,\ 0 0 0 a2 0 0 0 a3 \b1 b2 0 Mj (14.22) M° is diagonalized by applying a biunitary transformation = V° R xl,r. V°TM° VR = Diag(0,0,-A3,A4) (14.23) V° 1 M°M° 1 V° = V°' M° 1 M° V° = Diag(0,0,A2,A4) , (14.24) where A3 and A4 are the nonzero eigenvalues, 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.(14.23,14.24) that from tree level the See-saw mechanism yields two massless eigenvalues associated to the light fermions: 14.6 One loop contribution to fermion masses The one loop diagram of Fig. 1 gives the generic contribution to the mass term mij e°Le°R, < Vk > < > Fig. 14.1. Generic one loop diagram contribution to the mass term my e°Le°R 2 mij = cy^ Y mO (VO)ik(VR)jkf(My,m£) , «h = , (14.25) n i— k=3,4 My being the mass of the gauge boson, cy is a factor coupling constant, Eq.(14.6), 2 2 mO = -A3 and mO = A4, and f(x,y) = x2-y2 ln yr, k=3,4 X mk (VL°)ik(VR)jkf(MY,mk) = F(My) , (14.26) A24 - A23 M2 m2 M2 M2 i = 1,2,3 , j = 1,2, and F(My) = M2_TA2 ln M - M2_TA2 ln M. Adding up all possible the one loop diagramss, we get the contribution ^R + h.c., my -a2 LmI 14 Ko - K° , Do - D° in a Local SU(3) Family Symmetry 305 M° = /Du D12 0 0\ D2I D22 0 0 D31 D32 D33 0 V 0 0 0 0/ oh n (14.27) Dl1 = mi ("f1 + "^h + Fm) + 1 M-22^1 D12 = m2(-- Fm) D21 = M-21 (--12 - Fm) D31 = ^31(-Ff1 + "f2 ) D33 = 2 (W1F2 + M-22 F3 ) gH D22 = 2 M-11F1 + 31J.22FZ2 D32 = ^32(-"f2 + Fm) «h = , F1 = F(Mv, ) , F2 = F(MY2 ) , F3 = F(MY3 ) 4 n Fz, = cos2 ^ F(M_) + sin2 ^ F(M+) Fz2 = sin2 ^ F(M_) + cos2 ^ F(M+) cos ^ sin ^ Fm =-2^3 [F(M+) - F(M_) ] . (14.28) (14.29) (14.30) (14.31) FZ, , FZ2 are the contributions from the diagrams mediated by the Z1 , Z2 gauge bosons, Fm comes from the Z1 - Z2 mixing diagrams, with M1, M2, M_, M+ the horizontal boson masses, Eqs.(7-11), at bj M at bj Mij = ,2 ,2 = A3 Ca Cp , A2 - A3 (14.32) ca = cos a, cp = cos |3, sa = sin a, sp = sin | are the mixing angles from the diagonalization of Mo. Therefore, up to one loop corrections the fermion masses are ^°M° ^R + ^°M° ^R = xl M XR , (14.33) where = VL,R Xl,r, and M= Diag(0,0, -A3,A4)+ V° 1 M° V° maybe written as: ( mn m12 m21 m22 M = Cp m13 Cp m23 sp m13 \ sp m23 Ca m31 Ca m32 (-A3 + CaCp m33) CaSp m33 Vsa m31 Sa m32 SaCp m33 (A4 + SaSp m.33)/ (14.34) 306 A. Hernandez-Galeana The diagonalization of M, Eq.(14.34) gives the physical masses for u and d quarks, e charged leptons and v Dirac neutrino masses. Using a new biunitary transformationxl,r = V^R YL,R; Xl M Xr =^lVR1' M V1R1)Vr, with Yl,rt = (f1, f2, f3, F)l,r the mass eigenfields, that is VL1)TMMt VL1' = V(1)TMt M vR1' = Diag(m?,m2,m|,M2) , (14.35) ml = m;|, mj = m2, m| = mT and Mj = M| for charged leptons. So, the rotations from massless to mass fermions eigenfields in this scenario reads ^L = VLo VL1' ¥l and ^R = Vr° vR1' ¥r (14.36) 14.6.1 Quark Mixing Matrix Vckm We recall that vector like quarks are SU(2)L weak singlets, and hence the in- teraction of L-handed up and down quarks; f£L = (uL,cL,tL)L and fdL = (dL, sL, bL)L, to the W charged gauge boson is ^2 f"LuLY2fdLW+2 = (VCKM)4x4 Y^dL W+2 , (14.37) where the non-unitary quark mixing matrix Vckm of dimension 4 x 4 is (VCKM)4x4 = TO V^x^ (VLL V^^ (14.38) 14.7 Numerical results for quark masses and mixing As an example of the possible spectrum of quark masses and mixing from this scenario, we show up the following fit of parameters at the MZ scale [15] Using the input values for the horizontal boson masses, Eq.(8), and the coupling constant of the SU(3) symmetry: M1 = 3.3 x 103 TeV , M2 = 3.3 x 105 TeV , — = 0.05 , (14.39) n we write the tree level Mq, and up to one loop corrections Mq quark mass matrices, as well as the corresponding mass eigenvalues and mixing: d-quarks: Tree level see-saw mass matrix: 14 Ko - K° , Do - D° in a Local SU(3) Family Symmetry 307 Ml 0 0 906.643 0 0 5984.81 0 0 8139.76 MeV, \3.00124 x 106 -670943. 0 9.10502 x 106/ (14.40) the mass matrix up to one loop corrections: / -5.64571 -11.0583 46.8646 15.829 \ Md = -29.9051 -39.4588 -11.5894 -3.91444 40.9245 -30.3588 -2859.86 130.424 V0.0409246 -0.0303588 0.386143 9.61036 x 106/ MeV , (14.41) the d-quark mass eigenvalues (md ,ms,mb,MD) = (2.97549, 51.0, 2860.72, 9.61036 x 106 ) MeV. and the product of mixing matrices: (14.42) VdL = V°r V. (i). dL d L * 0.981831 0.17522 0.0728363 0.0000922 — i 0.183881 0.783786 -0.593184 0.0005976 0.0468496 -0.5958 -0.801765 0.0008133 V-0.0000187 -6.6982 x 10-10 0.0010134 0.999999 J (14.43) VdR = V°R Vd1,^: / 0.146421 0.577678 0.803005 -0.0056951 —i 0.175219 -0.783785 0.595801 -9.0660 x 10- — 0.922135 0.312291. \ 0.217014 -0.0698145 0.0142936 4.3164 x 10-9 0.319949 0.947418 (14.44) u-quarks: MU = 0 0 673649. 0 0 5.57857 x 106 0 0 7.8041 x 106 \4.10528 x 108 -4.1775 x 107 0 1.92243 x 1010/ MeV, (14.45) Mu = /-0.47816 -0.551837 5.4868 0.117774 \ -3.21341 602.954 4467.75 95.9001 4.51209 1368.75 -173107. 714.009 \0.00225605 0.684377 16.632 1.92287 x 1010/ MeV, (14.46) 8 308 A. Hernandez-Galeana the u-quark mass eigenvalues (mu,mc,mt,MU) = (1.37677, 638.055 , 173170, 1.92287 x 1010)MeV (14.47) and the product of mixing matrices: VuL = VUL V^: { 0.996356 0.0468431 -0.0010006 -0.829224 -0.0852899 0.556949 -0.0712817 0.0000350\ -0.558915 0.0002900 -0.826155 0.0004057 \ 0 0.0000128 0.0004998 1. VuR = V°R VuR (1). 0.0003359 0.0032952 0.999995 0.0934631 -0.995394 0.0213497 \ 0.995617 0.0934386 -0.0021725 -0.0033122 0.0000265 0 \—1.4066 x 10-8 0.0001676 0.0214593 0.99977 ) (14.48) (14.49) and the quark mixing matrix VCKM: 0.974441 0.224564 — i 0.0059177 \6.3092 x 10-8 0.224613 -0.973557 0.0416636 8.2754 x 10-6 0.0035948 0.0000219 0.041928 -0.0000382 0.999114 -0.0010126 0.0004999 5.0666 x 10-7/ (14.50) 14.8 AF = 2 Processes in Neutral Mesons Here we study the tree level FCNC interactions that contribute to Ko — K0, Do — Do mixing via Z1 , Y± exchange from the depicted diagram in Fig. 2. K0 d s Fig. 14.2. Generic tree level exchange contribution to Ko gauge bosons. K0 d Ko from the SU(3) horizontal i i Y^ Tiy2 The Z1 , Y± (Y± = 2) gauge bosons become massive at the second stage of the SU(3) symmetry breaking, and have flavor changing couplings in both left-and right-handed fermions, and then contribute the AS = 2 effective operators s 14 Ko - K° , Do - D° in a Local SU(3) Family Symmetry 309 Oll = (aLY^SL)(aLY^SL ) , ORR = (dRY^sR)(aRY^sR) (14.51) olr = (aLY^SL)(aRY^sR) (14.52) The SU(3) couplings to fermions, Eq.(14.6), when written in the mass basis yield the gauge couplings Lint,Z1 = (CLZ1 d~LY|^SL + Cr z, ÎrY^Sr) Z^ (14.53) Lint,Yi = ^H (CLY1 dLY|SL + CRy! d~RY|sr) Y 1 (14.54) Lint,Y2 = f-(Cl y2 d_LY|SL + Cr y| dVy^Sr) iY21 (14.55) with the coefficients Cl Zi = Ln L12 — L31 L32 , Cr Zi = R11 R12 — R31 R32 ClyI = L12 L31 + L11 L32 , CryI = Ri2 R31 + R11 R32 (14.56) CLY| = (L12 L31 — L11 L32) , CR Y2 = (R12 R31 — R11 R32) where VL,R == V° R vL1,^, and Lij = VLij , Rij = VRij. For each gauge boson, the effective four-fermion hamiltonian at the scale of the gauge boson mass is Hzi = ^ (CL Zi oll + 2 Cl zi Cr Zi olr + CR Zi orr) (14.57) Zi 2 hyI = 4^2 (C2l Yi oll + 2 Cl yI Cr yI olr + CR yI orr) (14.58) 2 hy2 = -(CL Y2 oll + 2 Cl y| Cr y, olr + CR Y| orr) (14.59) with Myi = MY2 = Mi. Therefore, the total four-fermion hamiltonian Hsu(2) = Hz, + hyi + HY2 can be written as 310 A. Hernandez-Galeana HSU(2) = 4MH2 (CLZ, + CL yi - CL y2 )OLL + (CRzi + CR y, + CR Y? )ORR +2(CLZi CRZ, + CLY] CRY-| - CLY2 CR Y2 )OLR g2 1 1 + gf(^ - M ) [Clz, oll + Crz, orr + 2ClZi Crz, olr)] (14.60) Zi 1 From the coefficients in eq.(14.56) we obtain: CLZ, + CL Y1 - CLY2 = 5L , CRz, + CRyi - Cry2 = 5R , , (14.61) CL,Zi CR,Zi + CL,Y] CR,Yj — CL,Y2 CR,Y2 = ^L §R + 2(L11 R31 - L31 R11 )(L32 R12 - L12 R32) , (14.62) and we can write hsu(2) = ^ [5L oll + 5R orr + 6lr olr (14.63) + ^ (M^r - ) [(L11L12 - L31L32)2 oll + (R11 R12 - R31R32)2 orr Zi 1 +2(Ln L12 - L31 L32)(R11 R12 - R31R32) olr)] with 5l = L11 L12 + L31 L32 , 6r = R11 R12 + R31 R32 (14.64) 5lr = 2(5l 6r + 2(Ln R31 - L31 R11HL32 R12 - L12 R32)) (14.65) The reported parameter space region in section 7 generate MZi « M1 with quite good approximation, and then the dominant contribution to neutral meson mixing comes from the four-fermion Hamiltonian in eq.(14.63). The suppression of the generic meson mixing couplings (q~iLY^PL,R qj)2 come out as follows 14.8.1 Ko — Ko meson mixing 9hî 1 1 5l = 0.144124 , gHTiL = 32594.5 TeV 5r = 0.452775 , gMi = 10375.2 TeV Mi — |6r| (14.66) = 0.361261 , gH Mi = 13003.4 TeV ~2 V |ÔLR 1 14 Ko - K° , Do - D° in a Local SU(3) Family Symmetry 311 14.8.2 Do — Do meson mixing 5l = -0.000829741 , aHMl = 5.66157 x 106 TeV L ' Is L | 5r = -0.00328084 , aHMl = 1.43184 x 106 TeV (14.67) JWlrÎ = 0.456165 , ^ = 10298.1 TeV v 1 LKI ' ^a/ ISlrI 2 These numerical values are within the suppression required for BSM contributions reported for instance in the review "CKM Quark - Mixing Matrix" in PDG2018[16]. 14.9 Conclusions Horizontal gauge bosons from the local SU(3) family symmetry introduce flavor changing couplings, and in particular mediate AF = 2 processes at tree level. We reported the analytic and numerical contribution to K° - K° and D° - D° meson mixing from tree level exchange diagrams mediated by the SU(2) horizontal gauge bosons Zï ,Y±. We provided in section 7 a particular parameter space region where this scenario can accommodate the hierarchy spectrum of quark masses and mixing, and simultaneously suppress properly the AS = 2 and AC = 2 processes. 2 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. This work was partially supported by the "Instituto Politecnico Nacional", (Grants from EDI and COFAA) in Mexico. 14.10 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 (14.68) 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 A2 = 2 (B - VB2 -4D) , A2 = -(B + VB2 -4D) (14.69) 312 A. Hernandez-Galeana and rotation mixing angles cos a: cos ß _ /A2 - a2 A2 - A2 /A2 - b2 A4 - A3 sin a _ sin ß _ la2 - A2 A2 - A3 lb2 - A3 A4 - A3 (14.70) B = a2 + b2 + c2 = A2 + A4 D = a2b2 _ A3A2 (14.71) a2 _ af + a2 + a3 b2 = bf + b2 + b3 (14.72) The rotation matrices V°, VR admit several parametrizations related to the two zero mass eigenstates, for instance vLo ( Ci -si S2 S1 C2 Ca Si C2 Sa\ 0 C2 S2 Ca S2 Sa -Si -Ci S2 Ci C2 Ca Ci C2 Sc 0 0 Sa Ca V o VR /10 0 0 \ 0 Cr Sr Cß Sr Sß 0 -Sr Cr Cß Cr Sß \0 0 -Sß Cß ) (14.73) ai + a 3 bn = b2 + b3 an + a 2 b _*/ bn + b2 (14.74) Si ai _ a3 an ' an S2 a2 _ an a2a ^ ' Cr _ b3 (14.75) References 1. A. Hernandez-Galeana, Rev. Mex. Fis. Vol. 50(5), (2004) 522. hep-ph/0406315. 2. A. 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