Impact of CP Phases on a Light Sbottom and Gluino Sector

Physics Letters B 579 (2004) 371–376 www.elsevier.com/locate/physletb

Impact of CP phases on a light sbottom and gluino sector

Chuan-Hung Chen

Department of Physics, National Cheng-Kung University, Tainan 701, Taiwan, ROC Received 25 September 2003; received in revised form 3 November 2003; accepted 5 November 2003 Editor: T. Yanagida

Abstract We study a scenario in which light bottom squarks and light gluinos with masses in the range 2–5.5 GeV and 12–16 GeV, respectively, can coexist in the MSSM, without being in conflict with flavor-conserving low-energy observables. We find that in − such a scenario, the anomalous magnetic moment of a muon could be as large as 10 9, if the theory conserves CP. However, if the theory violates CP, we conclude that not both, the gluino and bottom squark, can be light at the same time, after the neutron electric dipole moment constraint on Weinberg’s 3-gluon operator has been taken into account.  2003 Elsevier B.V. All rights reserved.

Inspired by the unforeseen excess of the bottom- decaying to a pair of light bottom squarks [6], radiative → ˜∗ ˜ + ¯ ˜ ˜ quark production observed at the hadronic collider of B meson decays [7], the decays Z bb1g bb1g Fermilab [1], Berger et al. [2] proposed a solution followed by g˜ → bb˜∗/b¯b˜ and e+e− → qq¯g˜g˜ [8], as ˜ 1 1 with light sbottom b (m ˜ 2.0–5.5 GeV) and light 1 b1 well as the running of strong coupling constant αs [9]. gluino g˜ (mg˜ 12–16 GeV). Interestingly, various To further explore more impacts on other processes, problems can easily be avoided, such as by adopting it is necessary to investigate different systems instead ˜ ˜ the proper mixing angle of two sbottoms bL and bR, of those in which the final states are directly associated the Z-peak constraint can be evaded; and also, the with the light sbottom and gluino. It is known that Rb contribution, arisen from sbottom-gluino loop, can supersymmetric models not only supply an elegant ˜ be suppressed by considering the second sbottom b2 mechanism for the electroweak symmetry breaking being lighter than 180 GeV for the CP conserved and a solution to the hierarchy problem, but also ˜ case [3], while the mass of b2 can be heavier in guarantee the unification of gauge couplings at the the CP violating one [4]. The amazing thing is that scale of GUTs [10]. Therefore, besides the effects such light supersymmetric particles, so far, have not mentioned above, other contributions will also appear been excluded by experiments, even after including when considering different phenomena. Inevitably, the data of precise measurements [5]. Moreover, for new parameters will come out. To avoid introducing searching the signals of the light sbottom and gluino, the irrelevant parameters, such as the mixing angles many testable proposals are raised, e.g., the rate of χb among different flavors of squark, we have to consider the processes in which the dependent parameters are still concentrated on the minimal set. The best E-mail address: [email protected] (C.-H. Chen). candidates are the flavor-conserving processes.

One of the mysteries in the standard model (SM) is to CPV. Since the results must be proportional to the ˜ ˜ whether the Higgs mechanism plays an essential role mixing of bL and bR, for generality, we describe the for the symmetry breaking and the resultant of Higgs relationship between weak and physical eigenstates as particle can be captured in future colliders. Based ˜ ˜ bL = 10 cos θb sin θb b1 on the same philosophy of the symmetry breaking, ˜ iδ ˜ , b 0 e b − sin θb cosθb b the minimal supersymmetric standard model (MSSM) R 2 needs the second Higgs doublet field to balance the (1) anomaly of quantum corrections, i.e., there are three where δb is the CP violating phase which could arise ∗ − neutral Higgs particles in MSSM, one of them is CP- from the off-diagonal mass matrix element mb(Ab ˜ ˜ odd (A0) and the remains are CP-even (h and H ). It µ tanβ). To suppress the coupling Zb1b1,weset is obvious that besides the mixing angle of sbottoms sin 2θb = 0.76 in our discussions. In the following and the masses of sbottom (gluino), the essential analyses, we separate the problem into CPC and CPV parameters in MSSM are At(b), the trilinear SUSY soft cases. breaking terms, µ, the mixing parameter of two Higgs 1. CPC (δb = 0) superfields, mA0(h,H), the masses of corresponding Unlike non-SUSY two-Higgs doublet models, it Higgses, and tan β, defined by the ratio of vu to vd in is well known that besides the enhancement of large with vu(d) being the vacuum expectation of the Higgs tan β or 1/ cosβ, there exists another enhanced factor field that couples to up (down) type quarks. And also, Ab in the couplings of S = h and H to bottom squarks unlike the non-SUSY two-Higgs-doublet model, the [11]. As described early, some novel consequences mixing angle of two neutral Higgs fields, denoted by based on both large factors have been displayed on α, is not an independent parameter and can be related the Higgs hunting. With these factors, we examine to tan β [11]. the implication on aµ by considering the case of a The activity of searching for the Higgs particle and light sbottom. In addition, for convenience, we adopt studying its properties is proceeding continuously [12, ≈ the decoupling limit with mA0 mZ so that tan 2α 13]. In particular, the remarkable results with a large tan 2β. tan β have been investigated enormously because the The effective interaction hγ γ is induced from the exclusive characteristic can give the unification of bot- radiative effects in which sbottoms are the internal tom and tau Yukawa couplings and the realization of particles of the loop, illustrated in Fig. 1(a), and the the top to the bottom mass ratio in GUTs [14]. It has gauge invariant form of the coupling can be obtained been found that if the SUSY soft breaking terms carry the explicit CP violating phases, due to the enhancement of large At(b) and µ, radiative effects can in- duce a sizable mixing between scalar and pseudoscalar such that the lightest Higgs boson could be 60–70 GeV and thus escape the detection of detectors [15]. More- over, with the requirements satisfied with electroweak baryogenesis, a novel prediction on the muon electric −24 dipole moment (EDM) of 10 e cm can be reached (a) by the proposed experiment [16]. In sum, it will be more exciting that if the light sbottom and gluino can be compatible with the Higgs physics with or without CP violation (CPV). In order to further pursue the implications of the light sbottom and gluino, in this Letter, we concentrate on two flavor-conserving processes: one is anomalous (b) magnetic moment of muon, aµ, and the other is Fig. 1. (a) Diagram which induces the effective coupling S–γ –γ , EDMs of muon and neutron. The former corresponds where S can be the lightest scalar boson h or pseudo-scalar boson 0 to CP conservation (CPC) while the latter is related A ; (b) loop for aµ. C.-H. Chen / Physics Letters B 579 (2004) 371–376 373 as [tanβ = 50] in Fig. 2(a) and (b) [(c) and (d)], where the solid, dashed and dashed-dotted lines denote the γ = γ 2 · − iΓµν iA (q ) (q k)gµν qµkν , results of Mh = 100, 120 and 140 GeV, respectively. 2 Unlike non-SUSY models, which require the neutral γ 2 αemQbmbAb sin α A (q ) = Nc sin 2θb Higgs to be a few GeV to fit a [11], the lightest 2πvcosβ µ Higgs boson in SUSY models could be the values 1 + x(1 − x) constrained by the current experiments. Interestingly, × (−1)i 1 dx , 2 − 2 − due to the light sbottom, a significant contribution = M ˜ q x(1 x) i 1,2 0 bi to anomalous magnetic moment of muon without (2) extremely tuning Ab is shown up [17]. where Nc = 3 is the color number, αem is the fine structure constant, Qb =−1/3 is the charge of sbot- 2. CPV (δb = 0) tom, and mb(M ˜ ) is the mass of bottom quark bi CP problem has been investigated thoroughly in (squarks). Because we concentrate on the light sbot- the kaon system since it was discovered in 1964 [18]. tom case, we do not discuss the contributions of stops Now, CPV has been confirmed by Belle [19] and by setting their masses being heavy. Since h couples ˜ ˜ Babar [20] with high accuracy in the B system. Al- to different sbottoms bL and bR but γ couples to the though the mechanism of Kobayashi–Maskawa (KM) same sbottoms, it is clearly inevitable to introduce the [21] phase in the SM is consistent with the CP mea- mixing angle θb. In Eq. (2), we already neglect the surements, the requirement of the Higgs mass of smaller contribution related to µ cosα/cosβ.Byus- 60 GeV for the condition of the matter–antimatter ing the result of Eq. (2), via the calculation of the asymmetry has been excluded by LEP. One of candi- loop in Fig. 1(b), the anomalous magnetic moment dates to deal with the baryogenesis is to use the SUSY of muon from two-loop Higgs-sbottom-sbottom dia- theory. As known, any CP-violating models will face grams is given by [17] the serious low energy constraints from EDMs of lep- 2 2 2 ton and neutron, which are T and P violating observ- αemQb mµmbAb sin α aµ = Nc sin 2θb ables and at elementary particle level usually are de- 16π3 v2M2 cos2 β ¯ µν µν h fined by df fσµν γ5fF , with F being the electro- M2 magnetic (EM) field tensor. We note that not only the + b˜ × (−1)i 1F i , EM field but also the chromoelectric dipole moment M2 i=1,2 h (CEDM) of gluon for colored fermions will contribute. 1 Therefore, we have to examine the implication of the x(1 − x) x(1 − x) light sbottom on EDMs while δ is nonzero, i.e., A F(z)= dx ln . (3) b b z − x(1 − x) z and µ are complex. 0 Inspired by the previous mechanism for aµ,the Immediately, we see that although this is a two- similar effects with pseudoscalar A0 instead of scalar loop effect and there appears one suppressed factor h, shown in Fig. 1(a), will also contribute the EDMs 2 (mb/v)(mu/Mh) ,theaµ could be enhanced in of leptons and quarks. Since the effects have been 2 terms of (A /v) tan β.DuetoM M ˜ , one finds analyzed by Refs. [16,22], we directly summarize the b h b1 2 2 ≈ + formalisms for the fermion EDM and CEDM as that F(M˜ /Mh) 2(1 ln M ˜ /Mh).Ifwetake b1 b1 M ˜ = 180 GeV and assume the lightest Higgs boson b2 2 2 ≈ 100

−9 Fig. 2. aµ (in units of 10 ) for (a) [(c)] Ab = 0.5 TeV and (b) [(d)] Ab = 1 TeV with tan β = mt /mb [tan β = 50], where the solid, dashed and dashed-dotted lines denote the results of Mh = 100, 120 and 140 GeV, respectively.

= f =− = respectively, with Rf tan β(cotβ) for T3 1/2 respond to Mh 100, 120 and 140 GeV, respectively. (1/2) and We note that the origin of CP violation comes from iδ −1 Im(Abe b )/|Ab| and it is set to be O(10 ).Fromthe iδ sin 2θb mb Im(Abe b ) figure, we see clearly that without fine tuning the CP ξb = . v2 sin β cosβ phase to be tiny, the EDM of electron can be lower For simplicity, we set the effects of stops to be neg- than the experimental bound and the CEDM of neu- ligible. According to Eq. (4), we see that the lepton tron is also not too far away from current limit. The γ effects become testable in experiments. EDM of d is proportion to the m2. Due to the ra- 2 It seems that so far the scenario of the light sbottom tio mµ/me ≈ 205, the strictest limit comes from the with the mass of few GeV could give interesting re- EDM of electron, with the current upper bound being γ − C 1.6 × 10 27 e cm [23]. Although the renormalization sults in aµ, de and dN . However, what we question 32/23 is whether both light sbottom and gluino can coexist factor (gs (MW )/gs (Λ)) fortheEDMofneutron 74/23 when the CP phases are involved. We find that the pos- is around one order larger than (gs (mW )/gs (Λ)) 2 sibility encounters fierce resistance of the Weinberg’s for the CEDM contribution, due to αs and 1/(QbQq ) enhancements, the CEDM of quark is much larger 3-gluon operator, defined by [24] than the EDM of quark. Hence, only the effects of the 1 G ρ µνλσ CEDM on neutron are considered. Since the unknown O =− d fαβγ GαµρGβν Gγλσε parameters for the EDM of electron and CEDM of 6 neutron are the same, the values of parameters are with Gαµρ and fαβγ being the gluon field-strength taken to satisfy with the bound of the electron EDM. tensor and antisymmetric Gell-Mann coefficient, re- We present the results as a function of M ˜ with spectively. With the bottom, sbottom and gluino be- b1 tan β = 10 (20) and Ab = 500 (200) GeV in Fig. 3, in ing the internal particles of the two-loop mechanism, which the solid, dashed and dashed-dotted lines cor- the contribution of the Weinberg’s operator is given C.-H. Chen / Physics Letters B 579 (2004) 371–376 375

−27 −26 Fig. 3. (a) [(b)] EDM of electron (in units of 10 ) and (c) [(d)] CEDM of neutron (in units of 10 )fortanβ = 10 and Ab = 500 GeV [tan β = 20 and Ab = 200 GeV] with δb = 0.1, where the solid, dashed and dashed-dotted lines denote the results of MA = 150, 200 and 250 GeV, respectively.

Table 1 −25 Neutron EDM (in units of 10 e cm) from the Weinberg’s operator with some taken values for M ˜ and M ˜ and ﬁxing M ˜ = 180 GeV and b1 g b2 tan β = 10

(M ˜ ,M˜ ) (5, 16)(5, 1000)(100, 16)(100, 1000) b1 g G × 4 × 3 dN / sin δb 2.2 10 8.21.7 10 8.0 by [25] dG. It is worth mentioning that the similar tendency 3 can be also found in Ref. [26], in which the considered G 3αs mb gs situation is via the gluino−stop−top-quark two-loop. d =− sin 2θb 2 4π For an illustration, we present the results in Table 1 − z1 z2 with some given values for Mb˜ and Mg˜ and fixing × H(z1,z2,zb) sin δb (5) 1 3 M ˜ = 180 GeV and tan β = 10. From the table, we Mg˜ b2 conclude that unless the CP violating phase δb is tuned − − where function H is a two-loop integration, zi = to be of O(10 4–10 3), the scenario of light sbottom 2 2 = 2 2 M ˜ /M ˜ and zb M /M ˜ . Here, we have rotated and gluino with CPV will encounter the problem of bi g b g away the phase of gluino. With naive dimensional naturalness. analysis, the neutron EDM could be estimated by In summary, we have extended the scenario of light G = G ∼ dN (eµX/4π)ηGd , with µX 1.19 GeV and sbottom to the flavor-conserved low energy physics ηG ∼ 3.4 being the chiral symmetry breaking scale and shown that by choosing proper values of tan β, and renormalization factor of the Weinberg’s operator, Ab and Mh(A), the predictions of aµ and EDMs of respectively. Although the estimation still has a large electron and neutron could satisfy with experiments. theoretical uncertainty, our concern is on the problem We have also found that except extreme fine tuning, of order of magnitude. Due to the result being propor- the light sbottom and gluino cannot coexist after 3 tional to 1/Mg˜ , we see that the lighter Mg˜ is, the larger the neutron electric dipole moment constraint on 376 C.-H. Chen / Physics Letters B 579 (2004) 371–376 the Weinberg’s 3-gluon operator has been taken into P. Langacker, M. Luo, Phys. Rev. D 44 (1991) 817. account. We note that it is possible that there exist [11] J. Gunion, et al., The Higgs Hunter’s Guide, Addison–Wesley, some cancellations while we consider all possible Reading, MA, 1990. [12] M. Carena, et al., Nucl. Phys. B 625 (2002) 345; contributions to the neutron EDM [27]. Nevertheless, M. Carena, et al., Nucl. Phys. B 659 (2003) 145; it will depend on how to fine tuning the involved B.E. Cox, et al., Phys. Rev. D 68 (2003) 075004. parameters and that is beyond our scope of the current [13] J. Gunion, et al., Phys. Lett. B 565 (2003) 42. Letter. [14] M. Olechowski, S. Pokorski, Phys. Lett. B 214 (1988) 393; B. Anantharayan, G. Lazarides, Q. Shafi, Phys. Rev. D 44 (1991) 1613; G. Anderson, S. Dimopoulos, L.J. Hall, S. Raby, Phys. Rev. Acknowledgements D 47 (1993) 3702; L.J. Hall, R. Rattazzi, U. Sarid, Phys. Rev. D 50 (1994) 7048; We thank C.Q. Geng, Otto C.W. Kong, C. Kao, M. Carena, M. Olechowski, S. Pokorski, C.E.M. Wagner, Nucl. K. Cheung and W.Y. Keung for their useful discus- Phys. B 426 (1994) 269; sions. This work was supported in part by the National P. Langacker, N. Polonsky, Phys. Rev. D 49 (1994) 1454. [15] A. Pilaftsis, Phys. Lett. B 435 (1998) 88; Science Council of the Republic of China under Con- A. Pilaftsis, Phys. Rev. D 58 (1998) 096010; tract No. NSC-91-2112-M-001-053. A. Pilaftsis, C.E.M. Wagner, Nucl. Phys. B 553 (1999) 3; M. Carena, et al., Nucl. Phys. B 586 (2000) 92; M. Carena, et al., Phys. Lett. B 495 (2000) 155. References [16] A. Pilaftsis, Nucl. Phys. B 644 (2002) 263. [17] C.H. Chen, C.Q. Geng, Phys. Lett. B 511 (2001) 77; A. Arhrib, S. Baek, Phys. Rev. D 65 (2002) 075002. [1] CDF Collaboration, F. Abe, et al., Phys. Rev. Lett. 71 (1993) [18] J.H. Christenson, J.W. Cronin, V.L. Fitch, R. Turly, Phys. Rev. 500; Lett. 13 (1964) 138. CDF Collaboration, F. Abe, et al., Phys. Rev. Lett. 79 (1997) [19] Belle Collaboration, A. Abashian, et al., Phys. Rev. Lett. 86 572; (2001) 2509. D0 Collaboration, B. Abbott, et al., Phys. Lett. B 487 (2000) [20] BaBar Collaboration, B. Aubert, et al., Phys. Rev. Lett. 86 264; (2001) 2515. D0 Collaboration, B. Abbott, et al., Phys. Rev. Lett. 85 (2000) [21] M. Kobayashi, T. Maskawa, Prog. Theor. Phys. 49 (1973) 652. 5068. [22] A. Pilaftsis, Phys. Lett. B 471 (1999) 174; [2] E.L. Berger, et al., Phys. Rev. Lett. 86 (2001) 4231. D. Chang, W.Y. Keung, A. Pilaftsis, Phys. Rev. Lett. 82 (1999) [3] J. Cao, Z. Xiong, J. Yang, Phys. Rev. Lett. 88 (2002) 111802; 900. G.C. Cho, Phys. Rev. Lett. 89 (2002) 091801. [23] B.C. Regan, et al., Phys. Rev. Lett. 88 (2002) 071805. [4] S. Baek, Phys. Lett. B 541 (2002) 161. [24] S. Weinberg, Phys. Rev. Lett. 63 (1989) 2333. [5] A. Dedes, H.K. Dreiner, JHEP 0106 (2001) 006. [25] J. Dai, H. Dykstra, R.G. Leigh, S. Paban, D.A. Dicus, Phys. [6] E. Berger, J. Lee, Phys. Rev. D 65 (2002) 114003. Lett. B 237 (1990) 216; [7] T. Becher, et al., Phys. Lett. B 540 (2002) 278. T. Ibrahim, P. Nath, Phys. Rev. D 57 (1998) 478. [8] K. Cheung, W.Y. Keung, Phys. Rev. Lett. 89 (2002) 221801; [26] D. Demir, M. Pospelov, A. Ritz, Phys. Rev. D 67 (2003) K. Cheung, W.Y. Keung, Phys. Rev. D 67 (2003) 015005; 015007. R. Malhotra, D.A. Dicus, Phys. Rev. D 67 (2003) 097703; [27] M. Brhlik, G.J. Good, G.L. Kane, Phys. Rev. D 59 (1999) P. Janot, Phys. Lett. B 564 (2003) 183. 115004. [9] C.W. Chiang, et al., Phys. Rev. D 67 (2003) 035008. [10] J. Ellis, et al., Phys. Lett. B 260 (1991) 131;