New Physics from U (3)-Family Nonet Higgs Boson Scenario

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Yoshio KOIDE‡ Department of Physics, University of Shizuoka 52-1 Yada, Shizuoka 422, JAPAN

Abstract

Being inspired by a phenomenological success of a charged lepton mass formula, a model with U(3)-family nonet Higgs bosons is proposed. Here, the Higgs bosons φL (φR) couple only between light fermions (quarks and leptons) fL (fR) and super- heavy vector-like fermions FR (FL), so that the model leads to a seesaw-type mass 1 matrix M m M − m for quarks and leptons f = u, d, ν and e. Lower bounds f ≃ L F R of the physical Higgs boson masses are deduced from the present experimental data and possible new physics from the present scenario is speculated.

1 Motives

One of my dissatisfactions with the standard model is that for the explanation of the mass f spectra of quarks and leptons, we must choose the coefficients yij in the Yukawa coupling i f f φ0 (f = ν,e,u,d, and i, j are family indices) “by hand”. In order to reduce f i,j L jRh i thisP P dissatisfaction, for example, let us suppose U(3)family nonet Higgs fields which couple i with fermions as f φ0j f . Unfortunately, we know that the mass spectra of up- f i,j Lh i i jR and down-quarksP andP charged leptons are not identical and the Kobayashi-Maskawa [1] (KM) matrix is not a unit matrix. Moreover, we know that in such multi-Higgs models, in general, flavor changing neutral currents (FCNC) appear unfavorably. Nevertheless, I would like to dare to challenge to a model with U(3)family nonet Higgs bosons which leads to a seesaw-type quark and lepton mass matrix

1 M m M − m . (1) f ≃ L F R My motives are as follows. One of the motives is a phenomenological success of a charged lepton mass relation [2]

2 2 me + mµ + mτ = (√me + √mµ + √mτ ) , (2) 3 ‡E-mail: [email protected]

1 which predicts m = 1776.96927 0.00052 0.00005 MeV for the input values [3] of τ ± ± m = 0.51099906 0.00000015 MeV and m = 105.658389 0.000034 MeV (the ﬁrst e ± µ ± and second errors in (1.2) come from the errors of mµ and me, respectively). Recent +0.18+0.20 measurements [4] of tau lepton mass mτ = 1776.96 0.19 0.16 MeV excellently satisﬁes the charged lepton mass relation (2). An attempt to derive− the− mass relation (2) from a Higgs j model has been tried [5]: We assumed U(3)family nonet Higgs bosons φi (i, j = 1, 2, 3), whose potential is given by

2 1 2 V (φ)= µ Tr(φφ†)+ λ Tr(φφ†) + ηφ φ∗Tr(φ φ† ) . (3) 2 s s oct oct h i

Here, for simplicity, the SU(2)L structure of φ has been neglected, and we have expressed the nonet Higgs bosons φj by the form of 3 3 matrix, i × 1 φ = φoct + φs 1 , (4) √3 where φoct is the octet part of φ, i.e., Tr(φoct) = 0, and 1 is a 3 3 unit matrix. For 2 × µ < 0, conditions for minimizing the potential (3) lead to the relation

vs∗vs = Tr voct† voct , (5) together with v = v†, where v = φ , v = φ and v = φ , so that we obtain the h i oct h octi s h si relation 2 Tr v2 = [Tr(v)]2 . (6) 3 1 If we assume a seesaw-like mechanism for charged lepton mass matrix Me, Me mME− m, with m v and heavy lepton mass matrix M 1, we can obtain the mass relation≃ (2). ∝ E ∝ Another motives is a phenomenological success [6] of quark mass matrices with a seesaw-type form (1), where

√me 0 0 1/2 mL mR M 0 √mµ 0 , (7) ∝ ∝ e ≡   0 0 √mτ     1 0 0 1 1 1 iβF iβF MF 1 + bF e 3X  0 1 0  + bF e  1 1 1  . (8) ∝ ≡ 0 0 1 1 1 1         The model can successfully provide predictions for quark mass ratios (not only the ratios mu/mc, mc/mt, md/ms and ms/mb, but also mu/md, mc/ms and mt/mb) and KM matrix parameters. These phenomenological successes can be reasons why the model with a U(3)family nonet Higgs bosons, which leads to a seesaw-type mass matrix (1), should be taken seri- ously.

2 2 Outline of the model

The model is based on SU(2) SU(2) U(1) U(3) [7] symmetries. These sym- L× R× Y × family metries except for U(3)family are gauged. The prototype of this model was investigated by Fusaoka and the author [8]. However, their Higgs potential leads to massless physical Higgs bosons, so that it brings some troubles into the theory. In the present model, the global symmetry U(3)family will be broken explicitly, and not spontaneously, so that massless physical Higgs bosons will not appear. The quantum numbers of our fermions and Higgs bosons are summarized in Table I.

Table I. Quantum numbers of fermions and Higgs bosons

Y SU(2)L SU(2)R U(3)family Y = 1 Y =1/3 fL (ν, e)L − , (u, d)L 2 1 3 Y = 1 Y =1/3 fR (ν, e)R − , (u, d)R 1 2 3 Y =0 Y = 2 Y =4/3 Y = 2/3 FL NL , EL − , UL , DL − 1 1 3 Y =0 Y = 2 Y =4/3 Y = 2/3 FR NR , ER − , UR , DR − 1 1 3 + 0 Y =1 φL (φ ,φ )L 2 1 8+1 + 0 Y =1 φR (φ ,φ )R 1 2 8+1 Y =0 Y =0 ΦF Φ0 , ΦX 1 1 1, 8

Note that in our model there is no Higgs boson which belongs to (2, 2) of SU(2) SU(2) . L× R This guarantees that we obtain a seesaw-type mass matrix (2) by diagonalization of a 6 6 mass matrix for fermions (f, F ): ×

0 m M 0 L = f , (9) mR MF ! ⇒ 0 MF′ !

1 where M m M − m and M ′ M for M m , m . (See Fig. 1.) f ≃ − L F R F ≃ F F ≫ L R

(2, 1, 8+1) (1, 1, 1) (1, 2, 8+1) 0 0 φ ΦF φ h Li h i h Ri × × ×

✲s ✲ s ✲ s ✲ L R gf gF gf fL FR FL fR (2, 1, 3) (1, 1, 3) (1, 1, 3) (1, 2, 3)

1 Fig. 1. Mass generation mechanism of M m M − m . f ≃ L F R

3 3 Higgs potential and “nonet” ansatz

We assume that φ φ , i.e., each term in V (φ ) takes the coeﬃcient which is exactly h Ri∝h Li R proportional to the corresponding term in V (φL). This assumption means that there is a kind of “conspiracy” between V (φR) and V (φL). However, in the present stage, we will not go into this problem moreover. Hereafter, we will drop the index L in φL. The potential V (φ) is given by

V (φ)= Vnonet + VOct Singl + VSB , (10) · where Vnonet is a part of V (φ) which satisﬁes a “nonet” ansatz stated below, VOct Singl · is a part which violates the “nonet” ansatz, and VSB is a term which breaks U(3)family explicitly. The “nonet” ansatz is as follows: the octet component φoct and singlet component φs of the Higgs scalar ﬁelds φL (φR) always appear with the combination of (4) in the La- grangian. Under the “nonet” ansatz, the SU(2)L invariant (and also U(3)family invariant) potential Vnonet is, in general, given by

2 1 j i l k V = µ Tr(φφ)+ λ1(φ φ )(φ φ ) nonet 2 i j k l

1 j l k i 1 j l i k 1 j k l i + λ2(φ φ )(φ φ )+ λ3(φ φ )(φ φ )+ λ4(φ φ )(φ φ ) 2 i k l j 2 i k j l 2 i j k l

1 j i l k 1 j k i l 1 j l k i + λ5(φ φ )(φ φ )+ λ6(φ φ )(φ φ )+ λ7(φ φ )(φ φ ) , (11) 2 i l k j 2 i j l k 2 i k j l + 0 0 where (φφ)= φ−φ + φ φ . On the other hand, the “nonet ansatz” violation terms VOct Singl are given by · j i VOct Singl = η1(φsφs)Tr(φoctφoct)+ η2 φs(φoct)i (φoct)jφs · j i j i +η3 φs(φoct)i φs(φoct)j + η3∗ (φoct)i φs (φoct)jφs . (12) 2 For a time, we neglect the term VSB in (10). For µ < 0, conditions for minimizing the potential (10) lead to the relation 2 2 2 µ vs = Tr(voct)= − , (13) 2(λ1 + λ2 + λ3)+(η1 + η2 +2η3) under the conditions λ4 + λ5 +2(λ6 + λ7) = 0, and v = v†, where we have put η3 = η3∗ for simplicity. Hereafter, we choose the family basis as

v1 0 0 v =  0 v2 0  . (14) 0 0 v3     For convenience, we deﬁne the parameters zi as

e vi mi zi = , (15) ≡ v0 sme + mµ + mτ

4 where 2 2 2 1/2 v0 =(v1 + v2 + v3 ) , (16) so that (z1, z2, z3)=(0.016473, 0.23687, 0.97140). We deﬁne two independent diagonal elements of φoct as 1 2 3 φx = x1φ1 + x2φ2 + x3φ3 , 1 2 3 (17) φy = y1φ1 + y2φ2 + y3φ3 , where the coeﬃcients xi and yi are given by x = √2z 1/√3 , (18) i i − (y1,y2,y3)=(x2 x3, x3 x1, x1 x2)/√3 . (19) − − − Then, the replacement φ0 φ0 + v means that φ0 φ0 + v ; φ0 φ0 + v ; φ0 φ0; → s → s s x → x x y → y (φ0)j (φ0)j (i = j), where v = v /√3+ x v . This means that even if we add a term i → i 6 i s i x j V = ξ φ φ + φ φi , (20) SB  y y i j i=j X6   in the potential Vnonet + VOct Singl, the relation (13) are still unchanged. · 4 Physical Higgs boson masses

For convenience, we deﬁne: φ+ i√2 χ+ = 1 , (21) φ0 √2 H0 iχ0 ! − ! z1 z2 z3 1 φ1 φ1 2 2 2 2  φ2  =  z1 3 z2 3 z3 3   φ2  . (22) − − − 3 φ3 2 q 2 q 2 q φ    3 (z2 z3) 3 (z3 z1) 3 (z1 z2)   3     − − −    Then, we obtain masses ofq these Higgs bosonsq which areq sumalized in Table II. 2 2 Table II. Higgs boson masses squared in unit of v0 = (174 GeV) , where 2 ξ = ξ/v0. For simplicity, the case of λ4 = λ5 = λ6 = λ7 = 0 are tabled.

0 0 φ H χ χ± 2 2(λ1 + λ2 + λ3) m (φ1) 0 0 +η1 + η2 +2η3 2 m (φ2) (η1 + η2 +2η3) 2(λ3 +2η3) (λ2 + λ3 + η2 +2η3) − − − 2 1 m (φ3) ξ ξ 2(λ3 + η3) ξ [λ2 + λ3 + (η2 +2η3)] − − 2 2 j 2 0 2 0 2 m (φi ) = m (H3 ) = m (χ3) = m (χ3±)

0 0 The massless states χ1± and χ1 are eaten by weak bosons W ± and Z , so that they are 2 2 2 not physical bosons. The mass of W ± is given by mW = g v0/2, so that the value of v0 deﬁned by (16) is v0 = 174 GeV.

5 5 Interactions of the Higgs bosons

(A) Interactions with gauge bosons µ Interactions of φL with gauge bosons are calculated from the kinetic term Tr(DµφLD φL). The results are as follows :

1 ↔ µ + 1 0 ↔ µ 0 HEW =+i eAµ + gz cos2θW Zµ Tr(χ− ∂ χ )+ gzZµTr(χ ∂ H ) 2 2 1 + ↔ µ 0 ↔ µ + + g Wµ [Tr(χ− ∂ H ) i(χ− ∂ χ ) + h.c. 2 − 1 +µ µ 0 + 2gm W −W + g m Z Z H , (23) 2 W µ z Z µ 1 0 where gz = g/ cos θW and χ1± = χ1 = 0. 0 0 Note that the interactions of H1 are exactly same as that of H in the standard model. (B) Three-body interactions among Higgs bosons

2 0 2 0 1 m (H1 ) 0 0 0 1 m (H2 ) 0 0 0 0 0 0 Hφφφ = H1 Tr(H H )+ H1 H2 H2 H1 H1 H2 + . (24) 2√2 v0 2√2 v0 − ··· The full expression will be given elsewhere. (C) Interactions with fermions Our Higgs particles φL do not have interactions with light fermions f at tree level, and they can couple only between light fermions f and heavy fermions F . However, when the 6 6 fermion mass matrix is diagonalized as (9), the interactions of φL with the physical fermion× states (mass eigenstates) become

0 Γ Γ11 Γ12 L = , (25) 0 0 ! ⇒ Γ21 Γ22 ! where ΓL = yf φL, and f 1 f Γ11 U φ v− U †D . (26) ≃ L L L f e 0 For charged leptons, since UL = 1, the interactions of φL are given by

lepton 1 0 j 0 j H = ei(aij bijγ5)ej(H ) + iei(bij aijγ5)ej(χ ) , (27) Y ukawa 2√2 − i − i Xi,j h i mi mj mi mj aij = + , bij = . (28) vi vj vi − vj

Therefore, in the pure leptonic modes, the exchange of φL cannot cause family-number non-conservation. q 0 For quarks, in spite of UL = 1, the Higgs boson H1 still couples with quarks qi diagonally: 6 q quark 1 mi 0 HY ukawa = (qiqi)H1 + . (29) √2 v0 ··· Xi j However, the dotted parts which are interaction terms of φ2, φ3 and φ (i = j) cause i 6 family-number non-conservation.

6 6 Family-number changing and conserving neutral currents

(A) Family-number changing neutral currents 0 In general, the Higgs boson H1 do not contribute to ﬂavor-changing neutral currents 0 (FCNC), and only the other bosons contribute to P -P 0 mixing. The present experimental 10 1 0 values [3] ∆mK = m(KL) m(KS)=(0.5333 0.0027) 10 h¯s− , ∆mD = m(D1) 0 10 1 − ± × | 12| |1 − m(D2) < 20 10 h¯s− , ∆mB = m(DH ) m(DL) = (0.51 0.06) 10 h¯s− , and so | × − 0 0 ±5 × on, give the lower bound of Higgs bosons m(H2 ), m(χ2) > 10 GeV. For the special case of m(H)= m(χ), we obtain the eﬀective Hamiltonian

1 1 1 mimj 1 zk zl zm HFCNC = 2 0 2 0 2 2 + − − 3 m (H2 ) − m (H3 )! i=j v0 k zk z1z2z3 ! X6 X

k k 2 2 2 (U U ∗) (f f ) (f γ5f ) , (30) × i j i j − i j h i 0 where (k,l,m) are cyclic indices of (1, 2, 3), so that the bound can reduce to m(H2 ) = 0 m(χ ) > a few TeV. Note that FCNC can highly be suppressed if m(H2) m(H3). 2 ≃ (B) Family-number conserving neutral currents The strictest restriction on the lower bound of the Higgs boson masses comes from

4 B(KL e±µ∓) v0 6 → 1.94 10− . (31) B(K π0ℓ ν) ≃ m × × L → ± H 11 The present data [3] B(K e±µ∓) < 3.3 10− leads to the lower bound m 3/v0 > L → exp × H 12, i.e., mH3 > 2.1 TeV.

7 Productions and decays of the Higgs bosons

0 As stated already, as far as our Higgs boson H1 is concerned, it is hard to distinguish it from H0 in the standard model. We discuss what is a new physics expected concerned with the other Higgs bosons. (A) Productions Unfortunately, since masses of our Higgs bosons φ2 and φ3 are of the order of a few TeV, it is hard to observe a production

+ 0 j 0 i e + e− Z∗ (H ) + (χ ) , → → i j (f + f ֒ f + f , (32 ֒ → i j → j i + even in e e− super linear colliders which are planning in the near future. Only a chance j of the observation of our Higgs bosons φi is in a production

u t +(φ)3 , (33) → 1

7 at a super hadron collider with several TeV beam energy, for example, at LHC, because the coupling atu (btu) is suﬃciently large:

mt mu atu + =1.029+0.002, (34) ≃ v3 v1

[c.f. a (m /v3)+(m /v1)=0.026+0.003]. bd ≃ b d (B) Decays 0 2 0 1 0 2 Dominant decay modes of (H )3 and (H )3 are hadronic ones, i.e., (H )3 tc, bs 0 1 0 1 → and (H )3 tu, bd . Only in (H )2 decay, a visible branching ratio of leptonic decay is expected: → 1 1 1 + Γ(H cu):Γ(H sd):Γ(H µ−e ) 2 → 2 → 2 → m 2 m 2 m 2 m 2 m 2 m 2 3 c + u :3 s + d : µ + e ≃ " v2 v1 # " v2 v1 # " v2 v1 # = 73.5% : 24.9%:1.6%. (35)

8 Summary

We have proposed a U(3)-family nonet Higgs boson scenario, which leads to a seesaw-type 1 quark and lepton mass matrix Mf mLMF− mR. It has been investigated what a≃ special form of the the potential V (φ) can provide the relation 2 2 me + mµ + mτ = (√me + √mµ + √mτ ) , 3 0 0 and the lower bounds on the masses of φL have been estimated from the data of P -P mixing and rare meson decays. 0 Unfortunately, the Higgs bosons, except for H1 , in the present scenario are very heavy, 0 3 i.e., mH mχ a few TeV. We expect that our Higgs boson (φ )1 will be observed ≃ ∼ 0 3 through the reaction u t +(φ )1 at LHC. The present scenario→ is not always satisfactory from the theoretical point of view: (1) A curious ansatz, the “nonet” ansatz, has been assumed. (2) The potential includes an explicitly symmetry breaking term VSB. These problems are future tasks of our scenario.

Acknowledgments Portions of this work (quark mass matrix phenomenology) were begun in collaboration with H. Fusaoka [6]. I would like to thank him for helpful conversations. The problem of the ﬂavor-changing neutral currents in the present model was pointed out by K. Hikasa. I would sincerely like to thank Professor K. Hikasa for valuable comments. An improved version of this work is in preparation in collaboration with Prof. M. Tanimoto. I am indebted to Prof. M. Tanimoto for helpful comments. I would also like to thank the organizers of this workshop, especially, Professor R. Najima for a successful and enjoyable

8 workshop. This work was supported by the Grant-in-Aid for Scientiﬁc Research, Ministry of Education, Science and Culture, Japan (No.06640407).

References and Footnote

[ 1 ] M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49, 652 (1973).

[ 2 ] Y. Koide, Lett. Nuovo Cimento 34, 201 (1982); Phys. Lett. B120, 161 (1983); Phys. Rev. D28, 252 (1983); Mod. Phys. Lett. 8, 2071 (1993).

[ 3 ] Particle data group, Phys. Rev. D50, 1173 (1994).

[ 4 ] J. R. Patterson, a talk presented at the International Conference on High Energy Physics, Glasgow, July 20 – 27, 1994.

[ 5 ] Y. Koide, Mod. Phys. Lett. A5, 2319 (1990).

[ 6 ] Y. Koide and H. Fusaoka, US-94-02, 1994 (hep-ph/9403354), (unpublished); H. Fusaoka and Y. Koide, AMUP-94-09 & US-94-08, 1994 (hep-ph/9501299), to be published in Mod. Phys. Lett. (1995). Also see, Y. Koide, Phys. Rev. D49, 2638 (1994).

[ 7 ] The “family symmetry” is also called a “horizontal symmetry”: K. Akama and H. Terazawa, Univ. of Tokyo, report No. 257 (1976) (unpublished); T. Maehara and T. Yanagida, Prog. Theor. Phys. 60, 822 (1978); F. Wilczek and A. Zee, Phys. Rev. Lett. 42, 421 (1979); A. Davidson, M. Koca and K. C. Wali, Phys. Rev. D20, 1195 (1979); J. Chakrabarti, Phys. Rev. D20, 2411 (1979).

[ 8 ] Y. Koide and H. Fusaoka, US-94-02, (1994), (hep-ph/9403354), (unpublished).