b quark Electric Dipole moment in the general two Higgs Doublet and three Higgs Doublet models

Transcription

1 quark Electric Dipole moment in the general two Higgs Doulet and three Higgs Doulet models arxiv:hep-ph/993433v 1 Sep E. O. Iltan Physics Department, Middle East Technical University Ankara, Turkey Astract We study the Electric Dipole moment of quark in the general two Higgs Doulet model (model III) and three Higgs Doulet model with O() symmetry in the Higgs sector. We analyse the dependency of this quantity to the new phase coming from the complex Yukawa couplings and masses of charged and neutral Higgs osons. We see that the Electric Dipole moment of quark is at the order of 1 ecm, which is an extremely large value compared to one calculated in the SM and also two Higgs Doulet model (model II) with real Yukawa couplings. address: eiltan@heraklit.physics.metu.edu.tr

2 1 Introduction The study of CP violating effects provides comprehensive informations in the determination of free parameters of the various theoretical models. Non-zero Electric Dipole Moment (EDM) for elementary particles is the sign of such violation. Neutron EDM has a special interest and the experimental upper ound has een found as d N < e cm [1]. EDM of electron and muon have een measured experimentally as d e = (.7 ± 8.3)1 7 [] and d µ = (3.7 ± 3.4)1 19 [3]. These measurements can also give powerful clues aout the internal structure of the particles, if it exists. The source of CP violation in the SM is complex Caio-Koayashi- Maskawa (CKM) matrix elements. The quark EDM vanishes at one loop order in the SM, since moduli of matrix element is involved in the relevant expression. Further, it also vanishes at two loop order after sum over internal flavours [4, 5]. When QCD corrections are taken into account, nonzero EDM exists [6]. However, EDM for quarks is very small in the SM and need to e enhanced with the insertion of new models eyond. In the literature, quark EDM is calculated in the multi Higgs doulet models [7, 8, 9]. In [8], EDM due to the neutral Higgs oson effects is otained in the two Higgs doulet model (HDM) and [9] discusses the necessity of more scalar fields than just two Higgs doulets for non-zero EDM when only the charged Higgs oson effects are taken into account. In [1] and [11] the contriutions of charged and neutral Higgs effects to the quark-edm are studied respectively. Further, quark EDM is calculated in the HDM if the CP violating effects come from CKM matrix elements [1] and it is thought that H ± particles also mediate CP violation esides W ± osons, however, at the two loop order these new contriutions vanish. In our work, we study EDM of -quark in the general HDM (model III) and the general 3HDM with O() symmetry in the Higgs sector (3HDM(O )) [13]. In this case, CP violation comes from complex Yukawa couplings and even at one loop, it is possile to get extremely large EDM, at the order of 1 e cm. Since the theoretical values of quark EDM s are negligile in the SM and also in the model II HDM with real Yukawa couplings, model III and 3HDM(O ) results can give important information aout the new parameters, namely masses of charged and neutral Higgs particles, Yukawa couplings, eyond the SM. Note that, non-zero EDM can e otained due to the charged Higgs oson effects even in the HDM in this case. The paper is organized as follows: In Section, we present EDM for quark in the framework of model III and 3HDM(O ). Section 3 is devoted to discussion and our conclusions. 1

3 Electric Dipole moment of quark in the general two Higgs Doulet and three Higgs Doulet models In this section, we calculate -quark EDM in the model III and extend our results to the general 3HDM. Since nonvanishing EDM is the sign for the CP violation, we assume that there exist complex Yukawa couplings which are possile sources of such violations. We start with the Yukawa interaction in the model III L Y = η U ij Q il φ1 U jr η D ij Q il φ 1 D jr ξ U ij Q il φ U jr ξ D ij Q il φ D jr h.c., (1) where L and R denote chiral projections L(R) = 1/(1 γ 5 ), φ i for i = 1,, are the two scalar doulets. The Yukawa matrices η U,D ij are chosen as φ 1 = 1 [( v H ) and ξ U,D ij have in general complex entries. Here φ 1 and φ ( χ iχ )] ; φ = 1 ( H H 1 ih ). () with the vacuum expectation values, < φ 1 >= 1 ( v ) ; < φ >=, (3) to collect SM particles in the first doulet and new particles in the second one. The Flavor Changing (FC) part of the interaction can e written as L Y,FC = ξ U ij Q il φ U jr ξ D ij Q il φ D jr h.c., (4) where the couplings ξ U,D for the FC charged interactions are ξ U ch = ξ U N V CKM, and ξ U,D N is defined y the expression (for more details see [14]) Note that the index N in ξ U,D N ξ U,D N ξ D ch = V CKM ξ D N, (5) = (V U,D L ) 1 ξ U,D V U,D R. (6) denotes the word neutral. The effective EDM interaction for -quark is defined as L EDM = id γ 5 σ µν F µν, (7) where F µν is the electromagnetic field tensor and d is EDM of -quark. Here, d is a real numer y hermiticity. In model III, the charged H ± and neutral Higgs osons h, A can

4 induce CP violating interactions which can create EDM at loop level. We present the necessary 1-loop diagrams due to charged and neutral Higgs particles in Figs. 1 and. Since, in the on-shell renormalization scheme, the self energy (p) can e written as (p) = (ˆp m ) (p)(ˆp m ), (8) diagrams a, in Figs. 1 and vanish when -quark is on-shell. However the vertex diagrams c, d in Fig. 1 and c in Fig. give non-zero contriutions. The most general vertex operator for on-shell -quark can e written as Γ µ = F 1 (q ) γ µ F (q ) σ µν q ν F 3 (q ) σ µν γ 5 q ν (9) where q ν is photon 4-vector and q dependent form factors F 1 (q ) and F (q ) are proportional to the charge and anamolous magnetic moment of -quark respectively. CP violated interaction exists with the non-zero value of F 3 (q ) and it is proportional to EDM of -quark. By extracting the CP violating part of the vertex, -quark EDM d (see eq.(7)) is calculated as a sum of contriutions coming from charged and neutral Higgs osons, d = d H± d h d A, (1) where d H±, d h and d A are 1 1 d H± = 3π ξu m N,tt Im( ξ N, D ) V t y (( 1 Q t ( 3 y) y)(y 1) (Q t y) ln y), t (y 1) 3 d h = 1 Q Im( ξ 16π m N,) D Re( ξ N,) D (1 r 1 (r 1 ) r 1 (r 1 4) ln r1 r r 1 ln r 1 ), d A = d h (r 1 r ), (11) with r 1 = m h /m, r = m A /m and y = m t m H ±. Here Q and Q t are charges of and t quarks respectively and ξ U(D) N,ij = 4 G F ξu(d) N,ij. In eq. (11) we take into account only internal t-quark contriution for charged Higgs and internal -quark contriution for neutral Higgs interactions since we assume that the Yukawa couplings ξ U i, i = u, c, ξ D N,j, j = d, s and ξ U N,tc compared to ξ U N,tt and ξ D N, (see [15]). Further, we choose ξ U N,tt real and ξ D N, complex, are negligile ξ D N, = ξ D N, e iθ. (1) Finally, using the parametrization eq.(1), we get the EDM of -quark in model III as 3

5 1 d = 3π ξ D N, sin θ{ 1 ξu m N,tt V t y (( 1 Q t ( 3 y) y)(y 1) (Q t y) ln y) t (y 1) 3 Q ξ N, D m cos θ( r 1 (r 1 ) r 1 (r 1 4) ln r1 r r 1 ln r 1 r (r ) r (r 4) ln r r 4 1 r ln r )}. (13) Now we would like to extend our result to the general 3HDM(O ) [13]. The new general Yukawa interaction is L Y = η U ij Q il φ1 U jr η D ij Q il φ 1 D jr ξ U ij Q il φ U jr ξ D ij Q il φ D jr ρ U ij Q il φ3 U jr ρ D ij Q il φ 3 D jr h.c., (14) where φ i for i = 1,, 3, are three scalar doulets and η U,D ij having complex entries, in general. With the choice φ 1 = 1 [( φ = 1 ( H H 1 ih ) v H ), ξ U,D ij, ρ U,D ij ( χ iχ, φ 3 = 1 ( F H 3 ih 4 )] ) are the Yukawa matrices,, (15) and the vacuum expectation values, < φ 1 >= 1 ( v ) ; < φ >= ; < φ 3 >= (16) the information aout new physics eyond the SM is carried y the last two doulets φ and φ 3, while the first doulet φ 1 descries only the SM part. The Yukawa interaction for the Flavor Changing (FC) part is L Y,FC = ξij U Q il φ U jr ξij D Q il φ D jr ρ U Q ij il φ3 U jr ρ D Q ij il φ 3 D jr h.c., (17) where, the charged couplings ξ U,D ch and ρ U,D ch are ξch U = ξ N V CKM, ξch D = V CKM ξ N, ρ U ch = ρ N V CKM, ρ D ch = V CKM ρ N, (18) 4

6 and ξ U,D N ρ U,D N = (V U,D L ) 1 ξ U,D V U,D R, = (V U,D L ) 1 ρ U,D V U,D R, (19) In the 3HDM, there are additional charged Higgs particles,f ±, and neutral Higgs osons h, A (see [13]). These particles ring new part to EDM of -quark and y taking only couplings ξ U N,tt, ξ D N,, ρ U N,tt and ρ D N, into account, it can e written as where d F ±, d h and d A are d = d H± d h d A d F ± d h d A, () d F 1 1 ± = ρ U 3π N,tt m Im( ρd N, ) V t y (( 1 Q t ( 3 y ) y )(y 1) (Q t y ) ln y ), t (y 1) 3 d h = 1 Q Im( ρ D 16π N, m ) Re( ρd N, ) (1 r 1 (r 1 ) r r 1 (r 1 4) ln 1 r r 1 ln r 1 ), d A = d h (r 1 r ), (1) with r 1 = m h /m, r = m A /m and y = m t m F ±. The expressions for d H±, d h and d A are defined in eq.(11). Further, y introducing O() symmetry in the Higgs sector, the masses of new Higgs osons F ±, h and A ecome the same as the masses of model III Higgs osons, H ±, h and A respectively (see [13]). Therefore, the final result for EDM of -quark in the 3HDM(O ) is d = 1 3π { 1 m t ( ξ U N,tt Im( ξ D N, ) ρu N,tt Im( ρd N, )) V t y (( 1 Q t ( 3 y) y)(y 1) (Q t y) ln y) (y 1) 3 Q (Im( ξ m N,) D Re( ξ N,) D Im( ρ D N,) Re( ρ D N,)) ( r 1 (r 1 ) r 1 (r 1 4) ln r1 r r 1 ln r 1 r (r ) r (r 4) ln r r 4 1 r ln r )}. () New O() symmetry also permits us to parametrize the Yukawa matrices ξ U(D) and ρ U(D) as [13] ξ U(D) = ǫ U(D) cos θ, ρ U = ǫ U sin θ, ρ D = iǫ D sin θ, (3) 5

7 where ǫ U(D) are real matrices satisfy the equation ( ξ U(D) ) ξ U(D) ( ρ U(D) ) ρ U(D) = (ǫ U(D) ) T ǫ U(D) (4) Here T denotes transpose operation. In eq. (3), we take ρ D complex to carry all CP violating effects on the third Higgs scalar. Using the parametrization in eq. (3), it is easy to see that only charged Higgs osons contriute to EDM of -quark ut not the neutral Higgs ones, since ρ D N, is a pure imaginary numer. Finally, -quark EDM reads as d = 1 3π 1 m t V t ǫ U N,tt ǫd N, sin θ y (( 1 Q t ( 3 y) y)(y 1) (Q t y) ln y) (y 1) 3 (5) 3 Discussion In this section, we study dependencies of -quark EDM on the masses of charged and neutral Higgs osons and the CP violating parameter θ. In the analysis, we use the CLEO measurement[16] to find a constraint region for our free parameters. The procedure is to restrict the Wilson coefficient C eff 7 in the region.57 C eff where upper and lower limits were calculated using the CLEO measurement and all possile uncertainities in the calculation of C eff 7 [17]. This restriction allows us to find a region for the parameters ξ N,tt U, ξ N D, θ in the model III and ǫ U Ntt, ǫ D N, θ in the general 3HDM(O ). In our numerical calculations, we also respect the constraint for the angle θ due to the experimental upper limit of neutron electric dipole moment, d n < e cm, which leads to m tm Im( ξ N,tt U D ξ N, ) < 1. for M H ± GeV 1 [18] in the model III and m tm ( ǫ U N,tt ǫ D N,) sin θ < 1. in 3HDM(O ). Further, we take only ξ Ntt U and ξ N, D ( ǫu N,tt and ǫd N, ) nonzero in the model III (3HDM(O )) and neglect all other couplings. Note that h is assumed as the lighest Higgs oson in our calculations. In Fig. 3 we plot EDM d with respect to the ratio R neutr = m h m A for sin θ =.5, m A = 8 GeV, m H ± = 4 GeV, ξ N, D = 4 m and r t = ξ N,tt U < 1 in the model III. Here d lies in ξ N, D the region ounded y solid lines for C eff 7 > and y dashed lines for C eff 7 <. It is oserved that -quark EDM is strongly sensitive to the ratio R neutr and this dependence increases with decreasing masses of neutral Higgs osons. If the ratio R neutr ecomes small the considerale enhancement of d will e otained and therefore the mass difference of h and A should not e large. In the case of degenerate masses of h and A, the contriution of the neutral Higgs sector to -quark EDM vanishes. Fig. 4 (5) is devoted to sin θ dependence of d for m h = 7 GeV, m A = 8 GeV (m h = m A = 8 GeV ), m H ± = 4 GeV, ξ D N, = 4 m and r t < 1. If the value of sin θ decreases, 6

8 d ecomes small as expected and the restricted region ecomes narrower, for oth C eff 7 > and C eff 7 <. d is positive for C eff 7 >, however, for C eff 7 <, it can also take negative values when m h reaches to m A. This is an intersting result which can e used in the determination of the sign of C eff 7. In Fig. 6, we plot d with respect to the charged Higgs mass m H ± for sin θ =.5, m h = 7 GeV, m A = 8 GeV, ξd N, = 4 m and r t < 1. This figure shows that d is weakly sensitive to the charged Higgs mass m H ± for m H ± 4 GeV, especially in the case C eff 7 <. The restriction region ecomes narrower with increasing m H ±. In 3HDM(O ), the possile choice of the parametrization for Yukawa couplings (eq. (3)) causes to vanish the contriution of the neutral Higgs sector to the -quark EDM (see eq. (5)) and therefore only the charged Higgs sector contriutes. Fig. 7 is devoted to sin θ dependence of d for m H ± = 4 GeV, ξ D N, = 4 m and r t < 1 in 3HDM(O ). The ehaviour of EDM is similar to the model III case with degenerate masses m h and m A, however, d is more sensitive to sin θ, since it is proportional to sin θ (see eq. (5)). Further, d is not sensitive to the charged Higgs mass m H ± for m H ± 4 GeV, especially in the case C eff 7 <, similar to the model III. (Fig. 8) Now we would like to summarize the main points of our results: It is interesting that EDM is generated y the one loop diagrams. This is due to the freedom to choose the Yukawa couplings as complex numers in the models under consideration. Since -quark EDM d is strongly sensitive to the ratio R neutr, in model III, the mass difference of h and A should not e large. For the 3HDM under consideration, there is no need to restrict the neutral Higgs masses ecause they do not contriute to d for the given parametrization of Yukawa couplings. If d is positive, C eff 7 can have oth signs. However, if it is negative, C eff 7 must e negative for oth models. This is an important oservation which is useful in the determination of the sign of C eff 7. d is not sensitive to the mass of charged Higgs oson for its large values in oth models. EDM of quark is at the order of 1 e cm in oth model III and 3HDM(O ). The neutral Higgs oson effects are strong in the model III and there is an enhancement, even one order (d (1 19 ), if the masses of neutral Higgs osons, m h and m A, are far from degeneracy. 7

9 Therefore, the experimental investigation of the -quark EDM gives powerful informations aout the physics eyond the SM. References [1] K. F. Smith et.al, Phys. Lett. B34 (199) 191, 885; I. S. Altarev et. al, Phys. Lett. B76 (199) 4. [] K. Adullah et,al. Phys. Rev.Lett. 65 (199) 347. [3] J. Bailey et al, Journ. Phys. G4 (1978) 345; [4] E. P. Sahaalin, Sov. J. Nucl. Phys. 8 (1978) 75. [5] J. F. Donoghue, Phy. Rev. D18 (1978) 163. [6] A. Czarnecki and B. Krause, Phys. Rev. Lett. 78 (1997) [7] S, Weinerg, Phys. Rev. Lett. 58 (1976) 657; N. G. Deshpande and E. Ma, Phys. Rev.D 16 (1977) 1583; for a recent review, see H. Y. Cheng, Int. J. Mod. Phys.A 7 (199) 159. [8] G. C. Branco and M. N. Reelo,Phys. Rev. Lett. B 16 (1985) 117; J. Liu and L. Wolfenstein, Nucl. Phys..B 89 (1987) 1. [9] C. H. Alright, J. Smith and S. H. H. Tye, Phys. Rev. D. D 1 (198) 711. [1] Atwood. et. al., Phys. Rev. D 51 (1995) 134. [11] Soni and Xu, Phys. Rev. Letter 69 (199) 33. [1] Y. Liao and X. Li, Phys. Rev. D6 (1999) 734. [13] E. Iltan Phys. Rev. D61 () 541. [14] D. Atwood, L. Reina and A. Soni, Phys. Rev. D53 (1996) 119. [15] E. Iltan, Phys. Rev. D6 (1999) 343. [16] M. S. Alam Collaoration, to appear in ICHEP98 Conference (1998) [17] T. M.Aliev, E. Iltan, J. Phys. G5 (1999) 989. [18] D. B. Chao, K. Cheung and W. Y. Keung, Phys. Rev. 59 (1999)

10 Figure 1: One loop diagrams contriute to EDM of -quark due to H ± in the HDM. Wavy lines represent the electromagnetic field and dashed lines the H ± field. γ H H γ u, c, t u, c, t a γ H H u, c, t u, c, t γ c d 9

11 Figure : The same as Fig. 1, ut for neutral Higgs osons h and A. γ A, h A, h γ d, s, d, s, a A, h d, s, γ c 1

12 d (e cm) m h =m A Figure 3: -quark EDM d as a function of the ratio R neutr = m h, for m m H ± = 4 GeV, A sin θ =.5, ξ N, D = 4 m and r t = ξ N,tt U < 1, in the model III. Here d is restricted in the ξ N, D region ounded y solid lines for C eff 7 > and y dashed lines for C eff 7 <. 11

13 3.5 1 d (e cm) sin Figure 4: -quark EDM d as a function of sin θ for m ± H = 4 GeV, m h = 7 GeV, m A = 8 GeV, ξ N, D = 4 m and r t < 1, in the model III. Here d is restricted in the region ounded y solid lines for C eff 7 > and y dashed lines for C eff 7 < d (e cm) sin Figure 5: The same as Fig.4 ut for m h = m A = 8 GeV. 1

14 d (e cm) mh (GeV ) Figure 6: -quark EDM d as a function of m ± H, for sin θ =.5, m h = 7 GeV, m A = 8 GeV, ξ N, D = 4 m and r t < 1, in the model III. Here d is restricted in the region ounded y solid lines for C eff 7 > and y dashed lines for C eff 7 < d (e cm) sin Figure 7: The same as Fig. 5, ut in 3HDM(O ). 13

15 d (e cm) mh (GeV ) Figure 8: The same as Fig. 6, ut in 3HDM(O ). 14