A Two Higgs Doublet Model for the Top Quark

UR 1446 November 1995 A Two Higgs Doublet Model for the Top Quark Ashok Das and Chung Kao 1 Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA 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 electron EDM is significantly enhanced for a naturally large tanβ v 2 / v 1. 1 Internet Address: Kao@Urhep.pas.rochester.edu

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 as well as gauge bosons. A neutral CP-even Higgs boson (H 0 ) remains after spontaneous symmetry breaking. The mass of a quark or a lepton is given by its Yukawa coupling constant with the Higgs boson times the vacuum expectation value (VEV) of the Higgs field. A two Higgs doublet model [3] has doublets φ 1 and φ 2 with VEVs v 1 / 2andv 2 / 2. There remain five Higgs bosons after symmetry breaking: a pair of singly charged Higgs bosons H ±, two neutral CP-even scalars H 1 and H 2, 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 [4], the different mass scales of the fermions and the gauge bosons are set by different Higgs VEVs. In Model II [5, 6], 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 m u /m d is so much smaller than m c /m s and m t /m b. We propose that the top quark is much heavier than the other quarks and leptons, because, in the three known fermion generations, it is the only elementary fermion getting a mass from a much larger VEV of a second Higgs doublet. This model has a few interesting features: (1) The top quark is naturally heavier than other quarks and leptons. (2) The ratio of the Higgs VEVs, tan β v 2 / v 1, is naturally large in this model, which highly enhances the Yukawa couplings of the lighter quarks and leptons with the Higgs bosons. (3) There are flavor changing neutral Higgs (FCNH) interactions. 1

A significant electric dipole moment (EDM) for the electron and the neutron can be generated if Higgs boson exchange generates CP violation [8]-[10]. In this paper, we discuss the effect of a large tan β with CP violation from the Higgs exchange. The phenomenology of FCNH interactions as well as Yukawa interactions for the charged Higgs boson will be presented in the near future. In Section II, we present the Lagrangian density for the Yukawa interactions of this model. As a first application to phenomenology, the electron EDM is evaluated in Section III, with contributions from fermion loops including t, b, τ, andc,andcp violation generated from neutral Higgs exchange. Promising conclusions are drawn in Section IV. 2 Yukawa Interactions We choose the Lagrangian density of Yukawa interactions to be of the following form 3 3 L Y = L m L φ 1 E mn lr n m,n=1 m,n=1 2 3 3 Q m L φ 1 G mα u α R α=1 m=1 m=1 Q m L φ 1 F mn d n R Q m L φ 2 G m3 u 3 R +H.c. (1) where and φ α = φ α = ( φ + α v α+φ 0 α 2 ( v α +φ 0 α 2 φ α ) ), φ α= φ + α,α=1,2 (2) L m L = Q m L = ( νl l ( u d ) m L ) m L,, m =1,2,3 (3) 2

l m, d m,andu m are the leptons, the down-type quarks and the up-type quarks in the gauge eigenstates. This Lagrangian respects a discrete symmetry, φ 1 φ 1,φ 2 +φ 2, l m R l m R, d m R d m R, u α R u α R, L m L +L m L, Q m L +Q m L,u 3 R +u 3 R, (4) with m =1,2,3andα=1,2. In this model, only the top quark has bilinear couplings to the doublet φ 2, while all other quarks and leptons have bilinear couplings to the doublet φ 1. 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 φ s have developed VEVs, <φ 1 >= v 1 / 2and<φ 2 >=v 2 / 2, which can both be complex. We propose that v 2 v 1 and tan β v 2 / v 1 is close to m t /m b,sothatm t is much larger than m b. The mass terms for the up quarks are 3 L U M = ū m M L U u n R, (5) m,n=1 where M U = v 1 2 G 11 G 12 (v2/v 1)G 13 G 21 G 22 (v2/v 1)G 23 G 31 G 32 (v2/v 1)G 33 (6) and u m, m =1,2,3 are the gauge eigenstates. Let us introduce unitary transformations u 1 u u 2 = U L,R c (7) u 3 t L,R L,R such that U M L U U R = U M R UU L = m u 0 0 0 m c 0 0 0 m t (8) 3

where u, c, and tare the mass eigenstates. The neutral Yukawa interactions of the up quarks are L U N = m u ū L u φ0 1 R ( φ ū a u=u,c,t v1 L Σ abu b 0 2 R φ0 ) 1, ab v2 v1 +H.c. (9) where u a,b = u, c, t and Σ = = m u 0 0 0 m c 0 0 0 m t U R 0 0 0 0 0 0 0 0 1 m u z 31 2 m u z31z 32 m u z31z 33 m c z32z 31 m c z 32 2 m c z32z 33 m t z33z 31 m t z33z 32 m t z 33 2 U R (10) where U R = z 11 z 12 z 13 z 21 z 22 z 23 z 31 z 32 z 33 (11) To a good approximation, the unitary matrix U R has the following form U R = cos φ sin φ cos φɛ 1 +sinφɛ 2 sin φ cos φ sin φɛ 1 cos φɛ 2 ɛ 1 ɛ 2 1 (12) We have introduced two small parameters 2 ɛ 1 = ɛ 1 e iδ 1 and ɛ 2 = ɛ 2 e iδ 2,with ɛ 1 m u /m t and ɛ 2 m c /m t. We will keep terms only to the first order in the ɛ s in our analysis. Introducing a transformation, which takes the two Higgs doublets to the gauge eigenstates (Φ 1 and Φ 2 ), such that < Φ 1 >= v/ 2, < Φ 2 >=0,wehave φ 1 = (cosβφ 1 sin βφ 2 ), φ 2 = (sinβφ 1 +cosβφ 2 )e iθ, (13) ( ) Φ 1 =, G + v+h 1 +ig 0 2 2 This is similar to what was also suggested for the U L in a recent model [12] with Topcolor dynamics. 4

Φ 2 = v = ( H + H 2 +ia 2 ), (14) v 1 2 + v 2 2, (15) where G ± and G 0 are Goldstone bosons, H ± are singly charged Higgs bosons, H 1 and H 2 are CP-even scalars, and A is a CP-odd pseudoscalar. Without loss of generality, we will take <φ 1 >=v 1 / 2and<φ 2 >=v 2 e iθ / 2, with v 1 and v 2 real and tan β v 2 /v 1. The neutral Yukawa interactions of the quarks now become L N Y = d=d,s,b m d d=d,s,b m d dd u=u,c,t m u ūu v dd(h 1 tan βh 2 ) i d=d,s,b m d v dγ 5 d(g 0 tan βa) m u v ūu[h 1 tan βh 2 ] m c v cc[h 1 tan βh 2 ] m t v tt[h 1 +cotβh 2 ] +i m u v ūγ 5u[G 0 tan βa]+i m c v cγ 5c[G 0 tan βa] +i m t v tγ 5 t[g 0 +cotβa] +L FCNH (16) where L FCNH are the terms that will generate flavor changing neutral Higgs interactions, L FCNH = {ɛ 1ū[ (m u + m t )+(m t m u )γ 5 ]t +ɛ 1 t[ (m u +m t ) (m t m u )γ 5 ]u +ɛ 2 c[ (m c +m t )+(m t m c )γ 5 ]t H 2 +ɛ 2 t[ (m c +m t ) (m t m c )γ 5 ]c} ( vsin 2β ) +i{ɛ 1ū[ (m t m u )+(m u +m t )γ 5 ]t +ɛ 1 t[+(m t m u )+(m u +m t )γ 5 ]u +ɛ 2 c[ (m t m c )+(m c +m t )γ 5 ]t A +ɛ 2 t[+(m t m c )+(m c +m t )γ 5 ]c} ( ). (17) vsin 2β 5

There is no FCNH interactions between the up and the charm quarks. 3 The Electron Electric Dipole Moment The experimental bound on the EDM of the electron is d e = ( 2.7 ± 8.3) 10 27 e cm [13]. The electron EDM (d e ) in the SM, generated from the Cabibbo-Kobayashi- Maskawa (CKM) phase, has been found to be extremely small [14]. It is too small to be observed. In a multi-higgs-doublet model, for flavor symmetry to be conserved naturally to a good degree, a discrete symmetry [11] is usually required. In a model with two Higgs doublets only, there is no CP violation from the Higgs sector if the discrete symmetry enforcing the natural flavor conservation were exact. By letting this symmetry be broken by soft terms, CP violation can be introduced while the flavor changing interaction can still be kept at an acceptably low level [15, 16]. In two Higgs doublet models, a significant electron EDM can be generated if Higgs boson exchange mediates CP violation. There are contributions from two-loop diagrams with the top quark [10], the gauge bosons [10]-[19] and the charged Higgs boson [20]. The contributions from the b quark and the τ dominate for tan β larger than about 10 with the same Yukawa interactions as those of the Model II. In our model, in addition, even the c quark loop produces a large electron EDM for a large tan β. Adopting Weinberg s parameterization [9] and applying the identity v 2 =( 2G F ) 1, we can write the following neutral Higgs exchange propagators as sin 2β ImZ 0n <H 1 A> q = 2 n q 2 m 2 n = 1 cos 2 β cot βim Z 1n +sin 2 βtan βim Z 2n 2 n q 2 m 2 n <H 2 A> q = 1 cos 2βImZ 0n Im Z 0n 2 n q 2 m 2 n 6

= 1 cos 2 βim Z 1n +sin 2 βim Z 2n, (18) 2 n q 2 m 2 n 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 0, and drop the sums and indices n in Eq. 18 hereafter. There are interesting relations among the CP violation parameters, Weinberg has shown [9] that with unitarity constraints. ImZ 0 +Im Z 0 = cot 2 βim Z 1, ImZ 0 Im Z 0 = + tan 2 βim Z 2. (19) ImZ 1 (1/2) tan β (1 + tan 2 β) 1/2, ImZ 2 (1/2) cot β (1 + cot 2 β) 1/2, (20) The Feynman diagrams of fermion loops contributing to the electron EDM are shown in Figure 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 γ. In our analysis, we will take m t = 175 GeV, m b =4.8GeV, m τ =1.777 GeV, m c =1.4 GeV, and the fine structure constant α =1/137. The top-loop contribution is [10] 3 ( ) t loop de = 16 m e α 2G F {[f(ρ e 3 (4π) 3 t )+g(ρ t )]ImZ 0 +[g(ρ t ) f(ρ t )]Im Z 0 }, (21) = 16 m e α 2G F [ g(ρ 3 (4π) 3 t )cot 2 βim Z 1 + f(ρ t )tan 2 βim Z 2 ], (22) where ρ t = m 2 t /m 2 0 and the functions f and g are defined as f(r) r [ ] 1 1 2x(1 x) x(1 x) dx 2 0 x(1 x) r ln, r g(r) r [ ] 1 1 x(1 x) dx 2 0 x(1 x) r ln. (23) r 3 In Eq. (2) of this reference, ImZ 1 should be ImZ 1. 7

For m t m 0, (d e ) t loop 6.5 10 27 [ ImZ 0 +0.17Im Z 0 ] e cm. (24) In Ref. [17], the fine structure constant α was taken to be 1/128, therefore, our numerical data for the t and W loops are (128/137) times smaller. In our model, the EDM generated from the b and the τ loops is ( ) b,τ loop de = (4N c Q 2 tan 2 β) m eα 2G F [f(ρ e (4π) 3 f )+g(ρ f )](ImZ 0 +Im Z 0 ), (25) = +(4N c Q 2 ) m eα 2G F (4π) 3 {[f(ρ f )+g(ρ f )]Im Z 1 }, (26) The N c is the color factor and Q is the charge. For the b and the τ, 4N c Q 2 is equal to 4/3 and 4 respectively. The electron EDM generated from the c loop is ( ) c loop de = ( 16 e 3 tan2 β) m eα 2GF [f(ρ (4π) 3 c ) g(ρ c )](ImZ 0 +Im Z 0 ), (27) = +( 16 3 )m eα 2G F (4π) 3 {[f(ρ c ) g(ρ c )]Im Z 1 }, (28) where ρ f = m 2 f/m 2 0 and ρ c = m 2 c/m 2 0. The difference between the contributions from the c-loop and the b-loop comes from a relative sign between the Ab b and the Ac c couplings. In Figures 2, we present the electron EDM from heavy fermion loops (d e f loop ), in units of (a) Im Z 0 and (b) Im Z 0, as a function of m 0,withtanβ= 20, where m 0 is the mass of the lightest physical spin-0 boson. It is clear that for m 0 < 200 GeV and tan β>10, the fermion loops of the b and the τ become dominant. In Figures 3, we present the electron EDM from heavy fermion loops (d e f loop ), in units of (a) Im Z 0 and (b) Im Z 0, as a function of tan β with m 0 = M W =80GeV.For a large tan β, the electron EDM from the W -loop[17] is d W loop e =+2.1 10 26 ImZ 0 e cm (29) For tan β>20 and m 0 M W, the fermion loops of the b, and the τ become dominant. 8

There are several interesting aspects to note from the different contributions. (1) The contributions from the b, theτand the c loops are proportional to (Im Z 0 +Im Z 0 ). (2) For the the t-loop, the coefficient of the Im Z 0 is much smaller than that of the Im Z 0.(3)TheW-loop does not contribute to the Im Z 0 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 β, the c-loop contribution can be larger than that of the t-loop. 4 Conclusions The model for Yukawa interactions proposed in this paper has several interesting features: (1) The top quark is naturally heavier than other quarks and leptons. (2) The ratio of the Higgs VEVs, tan β v 2 /v 1, is naturally large, which highly enhances the Yukawa couplings of the bottom quark (b), the tau lepton (τ), and even the charm quark (c), with the Higgs bosons. (3) There are flavor changing neutral Higgs (FCNH) interactions. The electron EDM from loop diagrams of the bottom quark (b), the tau lepton (τ), and even the charm quark (c), can be significantly enhanced with a large tan β. More precise experiments for the electron EDM will set bounds on the tan β and the CP violation parameters, the ImZ 0,Im Z 0,Im Z 1,andIm Z 2. We might be able to unravel the mystery of electroweak symmetry breaking and CP violation with the same stone. Acknowledgments This research was supported in part by the U. S. Department of Energy grant DE-FG02-91ER40685. 9

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Figures 1. Feynman diagrams for fermion loops contributing to the electric dipole moment of the electron. 2. The electron EDM from fermion loops d f loop e in units of (a) Im Z 0 and (b) Im Z 0, as a function of m 0,withtanβ= 20, for the t-loop (dash), the b-loop (dashdot), the τ-loop (dot), and the c-loop (dash-dot-dot), where m 0 is the mass of the lightest physical spin-0 boson. 3. The electron EDM from fermion loops d e f loop in units of (a) Im Z 0 and (b) Im Z 0, as a function of tan β with m 0 = 80 GeV, for the t-loop (dash), the b-loop (dash-dot), the τ-loop(dot), and the c-loop (dash-dot-dot). 12