4 April 1996

EISEVIER

PHYSICS LETTERS S

Physics Letters B 372 (1996) 106-112

A two Higgs doublet model for the top quark* Ashok Das, Chung Kao ’

Department of Physics and Astronomy, University of Rochester: Rochester; NY 14627, USA

Received 14 November 1995; revised manuscript received 18 December 1995 Editor: H. Georgi

Abstract

A two Higgs doublet model with special Yukawa interactions for the top quark and a softly broken discrete symmetry in the Higgs potential is proposed. In this model, the top quark is much heavier than the other quarks and leptons because it couples to a Higgs doublet with a much larger vacuum expectation value. The electric dipole moment (EDM) of the electron is evaluated with loop diagrams of the third generation fermions as well as the charm quark. The Do - Do mixing generated from flavor changing neutral Higgs interactions is estimated. The electron EDM and the Do - b” mixing can be significantly enhanced for a large tan/3 G [Q\//Q~.

1. Introduction

Recently, the top quark with a large mass has been observed at the Fermilab Tevatron [ 1,2]. Since the top quark is much heavier than all other known fermions, it might provide some clue to unravel the mystery of electroweak symmetry breaking. In the Standard Model (SM) of electroweak interactions, only one Higgs doublet is required to generate masses for fermions and gauge bosons. A neutral CP-even Higgs boson (Ho> remains after spontaneous symmetry breaking. The mass of a fermion is given by its Yukawa coupling with the Higgs boson times the vacuum expectation value (VEV) of the Higgs field.

A two Higgs doublet model [3] has doublets 41 and 42 with VEVs UI/~ and UZ/&. There remain five ‘Higgs bosons’ after symmetry breaking: a pair of singly charged Higgs bosons H*, two neutral CP-even scalars H1 and Hz, and a neutral CP-odd pseudoscalar A. Several two Higgs doublet models have been suggested with different Yukawa interactions for fermions and spin-0 bosons. In Model I [ 41, the different mass scales of the fermions and the gauge bosons are set by different Higgs VEVs. In Model II [ $61, one Higgs doublet couples to down-type quarks and charged leptons while another doublet couples to up-type quarks and neutrinos. A more recent model [7] was proposed to explain why mu/md is so much smaller than melms and mr/mb.

We propose that the top quark is much heavier than the other quarks and leptons, because, in the three known fetmion generations, it is the only elementary fermion getting a mass from a much larger VEV of a second

* Research supported in part by the U.S. Department of Energy grant DE-FG02-91BR40685. ’ Internet Address: Kao@Urhep.pas.rochester.edu.

0370-2693/96/$12.00 @ 1996 Elsevier Science B.V. All rights reserved PIISO370-2693(96)00031-7

A. Das, C. Kao/Physics Letters B 372 (1996) 106-112 107

Higgs doublet. This model has a few interesting features: (1) The ratio of the Higgs VEVs, tan@ = juzj/lull, is chosen to be large, which highly enhances the Yukawa couplings of the lighter fermions. (2) The top quark is then expected to be heavier than the other quarks and the leptons. (3) There are flavor changing neutral

Higgs (FCNH) interactions.

A significant electric dipole moment (EDM) for rhe electron can be generated if Higgs boson exchange

generates CP violation [8-IO]. As a first application to phenomenology, the electron EDM is evaluated with contributions from fermion loops including the t, the b, the 7, and the c. In addition, the Do - Do mixing

generated from FCNH interactions is estimated.

2. Yukawa interactions

We choose the Lagrangian density of Yukawa interactions to be of the following form

where

m.n=l l?l,?l=l a=1 m=l In=1

and

L2’= (:>:, Q;= (;)l, m=l,2,3-

I”‘, d”‘, and u”’ are the leptons, the down-type quarks and the up-type quarks

Lagrangian respects a discrete symmetry,

(2)

(3)

in the gauge eigenstates. This

(4)

with m = 1,2,3 and LY = 1,2. In this model, only the top quark has bilinear couplings to the doublet 42, while all other quarks and leptons have bilinear couplings to the doublet 41. The Yukawa interactions of the down-type quarks and leptons with neutral Higgs bosons are the same as those in Model II.

The fermion masses are generated when the 4s have developed VEVs, (41) = VI/& and (42) = u~/v’?. In

general, both VEVs can be complex. We propose that )uzj B ju,J and tan p = \uz\flull is close to m,/mh, so that mt - (Iu21//ull)rnb and it is much larger than mb,

The neutral Yukawa interactions of the up quarks are

R* m,$LuRT - c i-$8&b, + h-c.,

ll=u,c,t “I ab

where u”“’ = u, c, t are the mass eigenstates of the up-type quarks, and

(5)

(6)

108 A. Das, C. Kao/Physics Letters B 372 (1996) 106-112

with UR and UL being the unitary transfotmations that diagonalize the mass matrix of the up-type quarks. To a good approximation, the unitary matrix UR has the following form:

cosf$ -sinf$ - cos 4~; + sin 4.f; u, =

(

sin 4 cos 4 - sin 4~; - cos +52* . (7) Et E2 1 )

We have introduced two small parameters2 : et = ]et]eisl and ~2 = ]c2]eis2, with let] 2 1~21 Nm, mt. / Introducing a transformation, which takes the two Higgs doublets to the gauge eigenstates (@t and @2),

such that (at) = o/d and (@2) = 0, we have

41 = cos@t - sinP&, q52 = (sinP@t +cosP&)e i@ , (8)

(9)

where G* and Gc are Goldstone bosons, H* are singly charged Higgs bosons, Ht and H2 are CP-even scalars, A is a CP-odd pseudoscalar, and u = dw. Without loss of generality, we will take (4,) = ut/fi and (~$2) = u2eie/& with ut and Q real and tanp = u2/ut.

The neutral Yukawa interactions of the quarks now become

J$ = - c Tf&f(Ht - tanfl&) - i c T&d(@ - tan PA) d=d.s,b d=d,s,b

- c :iru[H~ - tan/3H2] - :fl[Ht + cotplY

U=U,C

(10)

where &CNH are the terms that will generate flavor changing neutral Higgs interactions,

LFCNH = {--E?@C[ (4, + m,)& + i(mc - m,)Al - e;iit[ (m, + m,)H2 + i(mt - m,)A]

- @t[(mc + mt)fk + i(mt - m,>Al + E;E#~~c[ (m, - m,)H2 + i(m, + m,)A]

+E;iirSr[(m,-m,)H2+i(m,+m,)A] i-&st[ (m,-m,)H2+i(mc+m,),4]} +h.c. (11)

The FCNH interactions between the u and the c quarks can generate Do - 6’ mixing, which will be discussed in Section 4.

3. The electron electric dipole moment

The experimental bound on the EDM of the electron is de = (-2.7 f 8.3) x lo-27 e-cm [ 131. The electron EDM (de) in the SM, generated from the Cabibbo-Kobayashi-Maskawa (CKM) phase, has been found to be extremely small [ 141.

In a model with multi-Higgs doublets, for flavor symmetry to be conserved naturally to a good degree, a discrete symmetry [ 111 is usually required. In a two Higgs doublet model, there is no CP violation from the

* This is similar to what was also suggested for the UL in a recent model [ 121 with topcolor dynamics.

A. Dans, C. Kao/ Physics L..eners B 372 (1996) 106-I 12 109

e e e e (a> (b)

Fig. 1. Feynman diagrams for fermion loops contributing to the electric dipole moment of the electron

Higgs sector if the discrete symmetry enforcing the natural flavor conservation were exact. If this symmetry is broken only by soft terms, CP violation can be introduced while the flavor changing interaction can still be

kept at an acceptably low level [ 15,161.

A significant electron EDM can be generated if Higgs boson exchange mediates CP violation. The electron EDM has contributions from two-loop diagrams with the top quark [lo], the gauge bosons [ 17-191 and the charged Higgs boson [ 201. In addition, there are significant contributions from the b quark and the T, for tan p

larger than about 10 with the same Yukawa interactions as those of the Model II. In our model, even the c quark loop produces a large electron EDM for a large tan /I.

Adopting Weinberg’s parameterization [ 93 and applying the identity u* = (&G,) -I, we can write the neutral Higgs exchange propagators as

(H,A), = ; c s1nq;y-,2 = f c - cos* /3 cot /3 Im 21, + sin* /? tan p Im 22,

n n n q* -mi

W24, = ; c cos2/3ImZe, -Iman 1

c cos2 p Im Zt n + sin* /3 Im 22”

(12) n q* - rni = Z n q* - rni

where the summation is over all the mass eigenstates of neutral Higgs bosons. We will approximate the above expressions by assuming that the sums are dominated by a single neutral Higgs boson of mass m, and drop the sums and indices n in Eq. (12) hereafter. There are relations among the CP violation parameters:

(1) ImZ0 +ImZo = -cot*pIm&, and (2) ImZc, - Im$ = + tan* /3Im $. Weinberg has shown [ 91 that IImZi] I (1/2)]tanP](l+tan*P) I/*, and 1 Im Z2] 5 ( l/2) I cot PI ( 1 + cot* p) ‘I*, with unitarity constraints.

The Feynman diagrams of fermion loops contributing to the electron EDM are shown in Fig. 1. The diagrams with the intermediate Z boson are highly suppressed by the vector part of the Ze+e- couplings. Therefore, we consider only the diagrams involving an intermediate y. The top-loop contribution is [ lo] 3

4 ( > t-l00p 16 m,a&GF - =--

e 3 (47l)3 {[f(h) +&,)I ImZo + L&t) - f(~,)l Ima},

16 rn,ct&GF =--- 3 c4Tj3 [-gh) cot*PImZ + f(p,) tan*PIm$l,

(13)

where p, = rnf/rni and the functions f and g are defined as

’ In Eq. (2) of this reference, ImZt should be -1m Zl.

110 A. Das, C. Kao / Physics Letters B 372 (1996) 106-l 12

f(r) 5 i jdr ~~12~~j~~ In

I X(1-X)

[ 1

1 x(1 -X>

r ’ g(r) = ; J dx

X(1-X)--r In

0 0 1 1 r ’

(15)

For mo N rntr ( de)r-loop = -6.5 x 10m2? [ImZj + O.l7Im&] e+ cm. In Ref. [ 171, the fine structure constant ty was taken to be l/128, therefore, our numerical data for the t and the W loops are (1281137) times smaller. 4

In our model, the EDM generated from the b and the 7 loops is

b,s-loop me&%

= -(4KQ*tan*P) c4Tj3 [fbf) +g(pf)l(ImZo+Im&o),

mad% = +(4&Q2) c4T13 {[fbf) +gbf)l Im&},

(16)

(17)

where N, is the color factor and Q is the charge. For the b and the 7, 4N,Q2 is equal to 4/3 and 4 respectively. The electron EDM generated from the c loop is

de ( > C4OOP

16 = -( - tan2 PI

m,adGF - e 3

c4,rrj3 U(&) -gh)l(ImZo+Im$),

16 m,ahGF =+($ (4T)3 {[f(kk) - gh)l Im%},

(18)

(19)

where pf = rn;/rni and pc = rn:/rn$ The difference between the contributions from the c-loop and the b-loop comes from a relative sign between the Abii and the ACE couplings.

In Fig. 2, we present the electron EDM from heavy fermion loops (d, f-‘oop), in units of (a) Im Z, and (b) Im $, as a function of m, with tan /3 = 20, where ~JJ is the mass of the lightest physical spin-0 boson. It is

clear that for mo < 200 GeV and tan/I > 10, the fermion loops of the b and the 7 become dominant. In Fig. 3, we present the electron EDM from heavy fermion loops ( d,f-loop), in units of (a) Im & and

(b) Im &, as a function of tan p with mo = Mw W-loop [ 17 ] is d~-‘oop =

= 80 GeV. For a large tan/3, the electron EDM from the

+2.1 x 1O-26 Im Zo e . cm. For tan /3 > 20 and mo N Mw, the fermion loops of the b, and the r become dominant.

There are several interesting aspects to note from the different contributions. (1) The contributions from the b, the T and the c loops are proportional to (Im & + Im &). (2) For the the r-loop, the coefficient of the Im & is much smaller than that of the Im &. (3) The W-loop does not contribute to the Im & term. (4) The charm

loop has the same sign as that of the W and the charged Higgs boson loops. (5) For a large tan /3, the c-loop contribution can be larger than that of the t-loop.

4. The Do - Do mixing

In the SM, the Do - Do mixing is very small [ 21-261. Recent studies in the heavy quark effective field theory [25,26] have found the mass difference to be (A~~)HQE~ x (0.9-3.5) x lo-l7 GeV. The present experimental limit is Amo < 1.3 x lo-l5 GeV [ 271.

The FCNH interactions in Eq. ( 11) can generate Do - Do mixing with the following AC = 2 effective Hamiltonian,

4 In our analysis, we take m, = 175 GeV, ?nb = 4.8 GeV, rnr = I .777 GeV, mC = 1.4 GeV, and the fine structure constant a = I/ 137.

A. Das, C. Kao/Physics Letters B 372 (1996) 106-112 111

Fig. 2. The electron EDM from fermion loops &f-loop in units of (a) Im Zo and (b) Im .& as a function of mr~, with tan p = 20, for the t-loop (dash), the b-loop (dash-dot), the vloop (dot), and the c-loop (dash-dot-dot), where ~4 is the mass of the lightest physical spin-0 boson.

Fig. 3. The electron EDM from fermion loops &f-‘“P in units

of (a) Im & and (b) Im &, as a function of tan p with mo = 80 GeV, for the t-loop (dash), the b-loop (dash-dot), the 7-100~ (dot), and the c-loop (dash-dot-dot).

f&e =

+[tmc-m.)2 _ tm+mc12 _ 4f* 44 1 (WC) W54). (20)

For simplicity, we assume that the scalar H2 and the pseudoscalar A are the mass eigenstates with no mixing between them and 1~11 N 1~21 N m,/m,. Let us define Am; = lrn$ - m&I and take mo as the smaller mass between mA and m&.

To estimate the order of magnitude for the Do - Do mixing, let us evaluate the mass difference with the vacuum insertion approximation [ 281

Am = W’“lK~I~o) N (mclmt)4 4v2 sin2 p cos2 p ’

Cm, - muj2 D -

2mD 42 (21)

For tan/? = 20, mo = 100 GeV, fD = 0.2 GeV, and mD = 1.86 GeV, with ImA - m&I = 50 GeV and 500 GeV, IArnDl is about 2 x lo-l6 GeV and 3 x IO-l6 GeV, respectively. It can be significantly enhanced if tan/? is larger than m,/mt,.

The mass difference is greatly suppressed if mA = mH,. For mA = mHz = rnb and tan p 2 10, it becomes

5 111, 4 \AmD\ N t-1 tan2p(-_)

12 ml

For tan/3 = 20 and rn4 = 100 GeV, IAmD( is about 4 x lo-‘* GeV.

5. Conclusions

(22)

We propose that the top quark is much heavier than the other fermions because it couples to a Higgs doublet with a much larger VEV. The tan/3 is chosen to be large, which enhances the Yukawa couplings with the Higgs bosons for the b, the 7, and even the c. In addition, there are flavor changing neutral Higgs (FCNH) interactions.

112 A. Das, C. Kao/Physics Letters B 372 (1996) 106-112

In this model, with tan /3 N 20 and mc N 100 GeV, the Do - 6’ mixing generated from the FCNH interactions can be slightly larger than the contribution from the SM. The mass difference Amo can be greatly enhanced if tanp is larger than m,/mb.

The electron EDM from loop diagrams of the 6, the 7, and even the c, can be significantly enhanced with a large tan p. More precise experiments for the electron EDM will set bounds on the me and the tan p as well as the CP violation parameters, the Im .ZJ, and the Im .&, i = 0, 1,2. We might be able to unravel the mystery of electroweak symmetry breaking and CP violation with the same ‘stone’.

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