Constraints from Electric Dipole Moments on Chargino Baryogenesis in the Minimal Supersymmetric Standard Model

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provided by National Tsing Hua University Institutional Repository PHYSICAL REVIEW D 66, 116008 ͑2002͒

Constraints from electric dipole moments on chargino baryogenesis in the minimal supersymmetric standard model

Darwin Chang NCTS and Physics Department, National Tsing-Hua University, Hsinchu 30043, Taiwan, Republic of China and Theory Group, Lawrence Berkeley Lab, Berkeley, California 94720

We-Fu Chang NCTS and Physics Department, National Tsing-Hua University, Hsinchu 30043, Taiwan, Republic of China and TRIUMF Theory Group, Vancouver, British Columbia, Canada V6T 2A3

Wai-Yee Keung NCTS and Physics Department, National Tsing-Hua University, Hsinchu 30043, Taiwan, Republic of China and Physics Department, University of Illinois at Chicago, Chicago, Illinois 60607-7059 ͑Received 9 May 2002; revised 13 September 2002; published 27 December 2002͒ A commonly accepted mechanism of generating baryon asymmetry in the minimal supersymmetric standard model ͑MSSM͒ depends on the CP violating relative phase between the gaugino mass and the Higgsino ␮ term. The direct constraint on this phase comes from the limit of electric dipole moments ͑EDM’s͒ of various light fermions. To avoid such a constraint, a scheme which assumes that the ﬁrst two generation sfermions are very heavy is usually evoked to suppress the one-loop EDM contributions. We point out that under such a scheme the most severe constraint may come from a new contribution to the electric dipole moment of the electron, the neutron, or atoms via the chargino sector at the two-loop level. As a result, the allowed parameter space for baryogenesis in the MSSM is severely constrained, independent of the masses of the ﬁrst two generation sfermions.

DOI: 10.1103/PhysRevD.66.116008 PACS number͑s͒: 11.30.Er, 11.30.Fs, 12.60.Jv, 98.80.Cq

INTRODUCTION BAU. One immediate question is whether or not such a new source of CP violation is already severely experimentally While the standard model of particle physics continues to constrained. It is not surprising that the most severe con- accurately describe a wide array of experimental tests many straints are provided by the current experimental limits of the ͑ ͒ physicists suspect that the next generation of a unified field electric dipole moments EDM’s of the electron (de) and theory will be supersymmetric. This supersymmetric theory the neutron (dn). in its simplest form, the minimal supersymmetric standard Fortunately, the lowest order ͑one-loop͒ contributions to model ͑MSSM͓͒1͔, may help to solve many of the outstand- various EDM’s through chargino mixing can be easily sup- ing problems in the standard model. Two examples of this pressed by demanding that the first two generations of sfer- sort are the coupling-constant-unification problem and the mions be heavier than the third one ͓4,5͔. For example, if observed baryon asymmetry of the universe ͑BAU͒.Itisthe one requires these sfermions to be heavier than 10 TeV, the latter of these two that will be discussed in this paper. one-loop induced EDM’s will be safely small ͓6͔. In fact, It has been demonstrated that the SM is insufficient in such a scenario can even be generated naturally in a more generating a large enough BAU ͓2͔. Particles lighter in mass basic scheme referred to as the more minimal SUSY model but stronger in coupling are needed to make the electroweak ͓7͔. However, despite the enlarged parameter space of the transition more first order. Additionally, a new CP violating MSSM, thanks to all the intricate limits provided by accu- phase is required to generate enough BAU. It is very appeal- mulated data from various collider experiments, there is only ing that the MSSM naturally provides a solution to both a small region of parameters left within the MSSM for such requirements ͓3͔. baryogenesis to work ͓3͔. The top-quark partner, the top squark, which is naturally In this article we wish to point out that even if sfermions lighter than the other squarks, can make the transition more of the first two generations are assumed to be very heavy, first order, while there are plenty of new CP violating phases there are important contributions to the EDM of the electron at our disposal in the soft supersymmetry ͑SUSY͒ breaking at the two-loop level via the chargino sector that strongly sector. In particular, it has been shown that the most likely constrain the chargino sector as the source for BAU in the scenario is to make use of the relative phase between the soft MSSM. Similar contributions to the quark EDM also exist SUSY breaking gaugino mass and the ␮ term of the but the resulting constraint turns out to be relatively weaker. Higgsino sector ͓3͔. In this case, the BAU is generated While this is not the first time that two-loop contributions through the scattering of the charginos on the bubble wall. have been found to be more important than the one-loop The CP violation is provided by the chargino mixing. It turns ones ͓8–12͔, this chargino contribution and its relevance to out that in most parameter space of the MSSM a nearly BAU was never treated fully. maximal CP violating phase is needed to generate enough In the case of chargino contributions, the two-loop contri-

0556-2821/2002/66͑11͒/116008͑8͒/$20.0066 116008-1 ©2002 The American Physical Society CHANG, CHANG, AND KEUNG PHYSICAL REVIEW D 66, 116008 ͑2002͒ bution is dominant because the one-loop contribution is suppressed when the sfermions are heavy. This aspect is similar to those in Refs. ͓8,12͔. In addition, the present case of a large CP violating phase in the chargino mixing and the light Higgs scalar, which is necessary to obtain a large baryon asymmetry, is also the same cause of the large EDM. There- fore, the resulting severe EDM constraint is very difﬁcult to avoid in the mechanism of chargino baryogenesis by tuning parameters.

THE MODEL AND COUPLINGS Before we outline the physics of the chargino mixing in supersymmetric models we will set forth our conventions. We assume the minimal set of two Higgs doublets. Let the FIG. 1. A two-loop diagram of the EDM of the electron, or ⌽ ϭϪ ⌽ superfield d(Y 1) couple to the d-type field and u(Y quarks. The chargino runs in the inner loop. ϭ1) to the u-type ͑see Ref. ͓11͔ for our convention͒. The chargino fields are combinations of those of the W-ino The complex mixing amplitudes are written in terms of the ␻ϩ ϩ ␺ ϭ ␻ϩ ϩ T S P ( L,R) and the Higgsino (huL,dR). Denote L ( L ,huL) real couplings g and g . In the same spirit, the complex ϩ ϩ and ¯␺ ϭ(¯␻ ¯,h ). The chargino mass term, ϪL C neutral Higgs fields are decomposed into the real and imagi- R R dR M ␸0ϭ 0ϩ 0 ϭ 0 ϭ¯␺ ␺ nary components q hq iaq (q u,d). Note that hd and RM C L in our convention, becomes 0 hu mix in a CP conserving fashion at the tree level, and so do ͱ ␤ a0 and a0 : M 2 2M Wsin u d M ϭͩ ͪ , ͑1͒ C ͱ ␤ ␮ i␾ 0 0 0 0 2M Wcos e h hu G au ͩ ͪ ϭRͩ ͪ , ͩ ͪ ϭSͩ ͪ , ͑4͒ H0 h0 A0 a0 where M 2 is the SUL(2) gaugino mass. Note that we choose d d a CP violating complex Higgsino mass ␮ei␾. The scalar ⌽ ⌽ cos ␣ Ϫsin ␣ sin ␤ Ϫcos ␤ components Hu ,Hd of u , d have real vacuum expectation Rϭͩ ͪ Sϭͩ ͪ ͑ ͒ ͱ ͱ ␤ϭ ␣ ␣ , ␤ ␤ . 5 values vu / 2,vd / 2, respectively, and tan vu /vd . sin cos cos sin We use the biunitary transformation to obtain the diagonal D † mass matrix M ϭUЈM U with eigenvalues m␹ ,m␹ for The EDM calculation involves the Higgs boson propagators, C 1 2 ␹ ␹ which are defined as the eigenfields 1 , 2. The CP violating chargino mixing can contribute to the fermion EDM through the chargino- ͗␸ ␸† ͘ ϭ q,qЈ ͑ 2Ϫ 2 ͒ sfermion loop. Detailed analyses of such contributions can q p2 i͚ Zϩ,␴ / p M ␴ , qЈ ␴ be found in the literature ͓5͔. As noted in the Introduction, such contributions can be tuned to be small by making the sfermions heavy ͓6͔͑typically of 10 TeV or larger͒. Here we ͗␸ ␸ ͘ ϭ q,qЈ ͑ 2Ϫ 2 ͒ ͑ ͒ q qЈ p2 i͚ ZϪ,␴ / p M ␴ . 6 are interested in contributions to the EDM of a fermion that ␴ are still important even with very heavy sfermions. For this we find that the leading contribution is from diagrams of the The Z factors can be shown to be real at the leading order type in Fig. 1. with the explicit forms To evaluate the diagram, we examine gauge couplings of d,d u,u d,d u,u Z ϭZ ϭcos2␣, Z ϭZ ϭϮcos2␤, 0ϭ ϩ␸ ͱ Ϯ,H Ϯ,h Ϯ,G Ϯ,A the Higgs bosons, Hq (vq q)/ 2, d,d u,u 2 d,d u,u 2 ZϮ ϭZϮ ϭsin ␣, ZϮ ϭZϮ ϭϮsin ␤, g ,h ,H ,A ,G L ϭ ¯␹ ͓ Ј † ␸0 ϩ Ј † ␸0 ͔␹ ϩ Y ͚ iR Ui␻Uhj u* UihU␻ j d* jL H.c. ͱ2 ij 1 1 Zu,d ϭ sin 2␣ϭϪZu,d , Zu,d ϭϮ sin 2␤ϭϪZu,d , ͑2͒ Ϯ,H 2 Ϯ,h Ϯ,A 2 Ϯ,G

Only the diagonal couplings in the chargino basis are rel- d,u u,d ZϮ ␴ϭZϮ ␴ for ␴ϭh,H,A,G. evant to the simple diagrams in Fig. 1 mediated by an inter- , , nal photon. Therefore we deﬁne For completeness, our list includes the unphysical Goldstone boson G0, which does not contribute to the EDM. Other sum ␸ g uϵ S ϩ P ϭ Ј rules are g giu igiu Ui␻Uih* , i ͱ2 q,qЈϭ ␦q,qЈ␦ ͑ ͒ ͚ Zs,␴ 2 s,ϩ . 7 ␴ϭhHAG ␸ g dϵ S ϩ P ϭ Ј ͑ ͒ g gid igid UihUi*␻ . 3 i ͱ2 The electron EDM via Fig. 1 is given by

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2 ␤ ␮ i␾ 2 d ␣ gm g P m␹ tan , e ,M A ,M 2. The usual SUSY breaking terms in- ͩ e ͪ ϭ e ͚ i,q ͫ ͩ i ͪ q,d clude the last two parameters as well as the trilinear sfermion g 2 Zϩ,h e ␲3 M cos ␤ i,q m␹ 16 W i M h coupling, the A term, which is not relevant in our analysis

2 2 because it does not participate directly in this particular m␹ m␹ mechanism of baryogenesis ͓3͔. If we replace charginos by ϩ ͩ i ͪ q,d ϩ ͩ i ͪ q,d ͬ ͑ ͒ g Zϩ,H f Zϩ,A . 8 top squarks in the inner loop, the effect of the relative phase M 2 M 2 H A of A and ␮ can contribute to the two-loop EDM as studied in ͓ ͔ Here the Barr-Zee ͓9͔ functions are defined as Ref. 8 . The top squark loop effect can be small if At is ␮ small, if At is in phase with , or if the left-handed top z 1 y nln͓y͑1Ϫy͒/z͔ squark is very heavy but the right-handed top squark is rather K ͑z͒ϭ ͵ dy, light. This last scenario is preferred by BAU. Such a large n 2 y͑1Ϫy͒Ϫz 0 mass gap will suppress top squark mixing and kill the EDM ͑9͒ contribution via the top squark loop. In addition, it has been ͑ ͒ϭ ͑ ͒Ϫ ͑ ͒ϩ ͑ ͒ ͑ ͒ϭ ͑ ͒ f z K0 z 2K1 z 2K2 z , g z K0 z . concluded by many groups ͓3͔ that using CP violating mixing of the top squark to generate BAU is much more difficult For the EDM of the down quark, we simply use the than using that of the chargino. 1 ϭ 1 charge ratio 3 to give (dd /e) 3 (de /e)(md /me), while for q,d→ q,u the EDM of the up quark, we need to replace Z Z in NUMERICAL ANALYSIS AND BARYOGENESIS ͑ ͒ Ϫ 2 Eq. 8 as well as the obvious charge ratio 3 and replace- → ment of me mu . In the Appendix, we offer a more compact To our current knowledge, the experimental constraint on analytic form of these results together with additional details the electron EDM has become very restrictive: which include the radiative correction to the Higgs boson Ϫ ͉d ͉Ͻ1.6ϫ10 27 e cm ͑90% C.L., Ref. ͓14͔͒. ͑11͒ mass in the simplified form suggested in Ref. ͓13͔. e Since the charginos do not couple to the gluon, there is no Since the tree-level Higgs boson mass relation ͓1͔ predicts a ͓ ͔ Ͻ chromo-EDM generated 11 . Note that if one wishes to in- light Higgs boson mh0 mZ , which has already been ruled clude the contribution with the internal photon replaced by out by experimental searches at the CERN eϩeϪ collider the Z boson, it is necessary to include the off-diagonal LEP II, our analysis has included the leading mass correction chargino couplings of the Z and the Higgs bosons. We ignore ͓13͔ at the one-loop level. For completeness, the resulting such contributions here because they are expected to be Higgs boson mass dependence on tan ␤ in this scheme is much smaller than that of the photon which was confirmed in illustrated in Fig. 2. Figure 3 shows the tan ␤ dependence of previous similar two loop calculations ͓10͔. In particular, the the predicted value of the electron EDM from different con- electron EDM via Z is highly suppressed by the small value tributions due to the Higgs bosons, A0, H0, and h0. We show of the Z vectorial coupling to the electron due to the approxi- the case of maximal CP violation when ␾ϭ␲/2, as required 2␪ Ϸ 1 mate relation sin W 4. There are other two-loop diagrams by baryogenesis ͓15͔, with masses at the electroweak scale, with CP violation originating from the same phase such as ϭ ϭ␮ϭ ϩ Ϫ M A 150 GeV, M 2 200 GeV. Note that, in this case, the ones with a chargino-neutralino loop mediated ␥H W ␤Ϸ ϩ Ϫ the h contribution dominates until about tan 3. The H effective vertex or ␥W W (W EDM͒ effective vertex. We contribution becomes dominant for tan ␤Ͼ5.4. When tan ␤ do not include them here because these contributions are becomes large, the increase of the Yukawa coupling of the expected to be smaller ͑by roughly an order of magnitude͒ as electron overwhelms the reduction of CP violation in the suggested by previous two-loop calculations ͓11,12͔.Inany chargino sector. This gives the increase of the electron EDM case, these additional diagrams form a separate gauge inde- as tan ␤ increases. The same effect happens to the EDM of pendent set. the d quark, but not the u quark. Figure 4 shows the electron Because the imaginary parts of the off-diagonal entries in ␮ ␤ϭ EDM contour plot versus M 2 and for the case tan 3, M are zero in our convention, we obtained the following ϭ ␾ϭ␲ C M A 100 GeV, and /2. In the many calculations of sum rules: BAU in the MSSM ͓3͔ the largest uncertainty seems to come from the calculation of the source term for the diffusion g ͓ ͔ P ϭϪ ͑ Ј† ͒ ϭ P ϭ equations that couples to the left-handed quarks 15,16 . Us- ͚ gi,um␹ Im U M DU ␻h 0, ͚ gi,dm␹ 0. ͓ ͔ i i ͱ2 i i ing the latest summary of the situation in Ref. 17 as a ͓ р␩ ϵ Ϫ ϫ 10 ͑10͒ reference point, large BAU 2 10 (nB n¯B)/n␥ 10 р3͔ requires tan ␤р3 with the wall velocity and the wall P ϭϪ P width close to their optimal values v Ӎ0.02, l Ӎ6/T, ␮ Therefore, g2,q g1,q(m␹ /m␹ ). It is easy to see that in w w 1 2 ӍM , and the CP phase sin ␾ close to 1. Note that a smaller the case of degenerate masses m␹ ϭm␹ , perfect cancella- 2 1 2 tan ␤ gives a larger BAU; however, it tends to give a small tion occurs, yielding a zero EDM. lightest Higgs boson mass which violates the LEP II limit Based upon another fact, that the diagonal scalar coupling unless the left top squark is much heavier than 1 TeV. Using ¯␹ 0␹ ␤ P ϭ ␤ P of iG i is zero, we can show that sin gi,u cos gi,d . the SUSY parameters in the above range, the numerical P Therefore, each of the four CP violating coefficients gi,q can analysis in our figures indicates that the predicted value of P be simply related to one of them, say g1,u , which again the electron EDM is more than a factor of 5 to 10 bigger than depends on the fundamental MSSM parameter the experimental limit on the electron EDM in most of the

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FIG. 2. The mass of the light Higgs boson h0 versus tan ␤. The lower set of curves corresponds to the tree-level result. The upper set of curves includes the leading one-loop (t,˜t ) effect, for m˜ ϭ1 TeV and m˜ ϭ150 GeV. Curves within t L t R each set are in the order of cases mA ϭ150,200,250,300 GeV, from bottom to top.

BAU preferred parameter range. In fact, if sin ␾ϭ1 and than 250 GeV͒ presented in Ref. ͓17͔ are all ruled out. Un- tan ␤ϭ3, then the parameter space allowed by the electron less the numerical constraint on BAU in Ref. ͓17͔ is relaxed ␮Ӎ ␮ EDM limit is limited to a narrow strip with M 2 and by an order of magnitude, it seems to be very difﬁcult for the has to be as large as 600 GeV in order to satisfy this EDM chargino mechanism for BAU to be compatible with the ␮ ͑ constraint. The ranges of values for and M 2 both smaller electron EDM constraint.

FIG. 3. The predicted value of the electron EDM versus tan ␤ from different contributions due to the Higgs bosons h0,A0, and H0, at the maximal CP violation when ␾ϭ␲/2. Masses are ϭ set at the electroweak scale, M A 150 GeV, M 2 ϭ␮ϭ200 GeV.

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␮ ␮ FIG. 4. The electron EDM contour plot versus M 2 and for the FIG. 5. The neutron EDM contour plot versus M 2 and for the ␤ϭ ϭ ␾ϭ␲ ␤ϭ ϭ ␾ϭ␲ case tan 3, M A 100 GeV, and /2. case tan 3, M A 100 GeV, and /2.

On the other hand, for the neutron EDM, our analysis tron must be large. For sin ␾ϭ1 and tan ␤ϭ3, the current indicates that the current experimental limit in Eq. ͑12͒ gives ␮Ӎ Ӎ electron EDM constraint requires M 2 600 GeV. Taking only a marginal constraint on the MSSM parameters required the uncertainty in the calculations of BAU in the literature for chargino BAU. into account, it is probably still premature to claim that this With the quark EDM, one uses the quark model to predict particular mechanism of baryogenesis is absolutely ruled out, ͓ ͔ ͉ ͉Ͻ ϫ Ϫ26 the neutron EDM. A new limit 18 dn 6.3 10 e cm but it is clear that the precision measurements of the EDM’s ͑95% C.L.͒ for the neutron EDM has been reported based on of fermions, especially the electron EDM, give a tight con- a combination of the recent data of low statistical accuracy straint on the mechanism. and the earlier measurement ͓19͔. This combination of the Note added. While this paper was under consideration we old and the new results has been criticized in Ref. ͓20͔.As received a preprint of a paper by A. Pilaftsis, Nucl. Phys. shown in the contour plot of Fig. 5, using the parameters B644, 263 ͑2002͒, with calculations that overlap with ours. suggested by the chargino baryogenesis mechanism, our pre- Our numerical results agree with this later calculation. dicted EDM value is around the size of the more conserva- ͉ ͉Շ ϫ Ϫ26 tive experimental limit dn 12 10 e cm, recom- mended in Ref. ͓20͔. Due to large theoretical uncertainties in ACKNOWLEDGMENTS the relation between the quark EDM and the neutron EDM, W.Y.K. is partially supported by a grant from the U.S. the constraint from the neutron EDM on the parameter space Department of Energy ͑Grant No. DE-FG02-84ER40173͒. cannot be as important as that from the electron EDM even if D.C. is supported by a grant from the National Science the more stringent limit is used. Council ͑NSC͒ of the Republic of China ͑Taiwan͒. We wish Note, however, that the uncertainties in the calculation of to thank H. Haber, H. Murayama, O. Kong, and K. Cheung the nonequilibrium electroweak baryogenesis process are far for discussions. D.C. wishes to thank the theory groups at from settled. For example, in the latest review by the group SLAC and LBL for hospitality during his visit. W.F.C. and ͓ ͔ Ϫ2 in Ref. 21 a small CP violating phase of 10 may be W.Y.K. wish to thank NCTS of NSC for support. sufﬁcient to generate BAU. In that case even the larger value of tan ␤ is allowed. For this purpose, in Fig. 6, we also plot the electron EDM for tan ␤ up to 50. APPENDIX: HIGGS POTENTIAL WITH RADIATIVE CORRECTIONS IN THE MSSM AND ELECTRIC DIPOLE MOMENTS CONCLUSION The Higgs potential has the form The baryogenesis in the MSSM requires the lightest Higgs boson to be light in order to get a strong ﬁrst order Vϭm2 ͉H ͉2ϩm2 ͉H ͉2ϩ͑Ϫm2 H H ϩH.c.͒ phase transition. It also requires the CP violating phase in Hd d Hu u 12 d u chargino mixing to be large in order to get large enough 1 BAU. As we discussed, both requirements imply that the ϩ ͑g2ϩg2͉͒͑H ͉2Ϫ͉H ͉2͒2ϩ␶͉H ͉4ϩ . ͑A1͒ predicted values of the EDM’s of the electron and the neu- 8 1 2 d u u •••

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FIG. 6. The predicted value of the electron EDM versus large tan ␤ at the maximal CP violation when ␾ϭ␲/2. Masses are set at the elec- ϭ␮ϭ troweak scale, M 2 200 GeV. Curves from top to bottom are in the order of cases mA ϭ150,300,450,600 GeV.

R2͒ϭ 2 ␤ϩ 2 2 ץ 2Vץ͑ At the tree level, SUSY requires the dimϭ4 coefficient ␶ / Hd 2m12tan 2M Zc␤ , ϭ0. However, it arises from the large top-quark–top-squark loop correction. Denote ͒ ͑ ␤ R͒ϭϪ 2 Ϫ 2 ץR ץ͑ 2Vץ / Hu Hd 2m12 mZsin 2 , A5 ͗ ͘ϭ ͗ ͘ϭ 2ϵ 2ϩ 2 Hd Vd , Hu Vu , V Vd Vu , R2͒ϭ 2 ␤ϩ 2 ͑ 2 ϩ ␶ 2͒ ץ 2Vץ͑ / Hu 2m12cot 2s␤ M Z 4 V , ␤ϵ 2 ϭ 1 2 2 2 ϭ 1 ͑ 2ϩ 2͒ 2 tan Vu /Vd , mW 2 g2V , mZ 2 g1 g2 V . ͑A2͒ ␤ I2͒ϭ 2 ץ 2Vץ͑ / Hd 2m12tan , We try to derive the mass matrix of the CP-even Higgs ͒ ͑ I ͒ϭ 2 ץ I ץ͑ 2Vץ bosons, which correspond to the real part of the complex fields. We use superscripts R,I to abbreviate the real and / Hu Hd 2m12 , A6 imaginary parts. The first derivatives of the potential are ␤ I2͒ϭ 2 ץ 2Vץ͑ / Hu 2m12cot . R͒ϭ 2 RϪ 2 Rϩ 1 ͑ 2ϩ 2͒ ץ Vץ͑ / H 2m H 2m H 2 g g d Hd d 12 u 1 2 ͑ ͒ ͑ ͒ ϫ͉͑H ͉2Ϫ͉H ͉2͒HR , The basis defined in Eqs. 4 , 5 agrees with that in Martin’s d u d review ͓1͔. One can easily show that G is massless as it is the unphysical Goldstone boson. The mass of the pseudoscalar R͒ϭ 2 RϪ 2 RϪ 1 ͑ 2ϩ 2͒ 0 ץ Vץ͑ / H 2m H 2m H 2 g g A is u Hu u 12 d 1 2 ϫ͉͑H ͉2Ϫ͉H ͉2͒HRϩ4␶͉H ͉3. ͑A3͒ d u u u 2 ϭ 2 ␤ 2 ϭ 2 ϩ 2 ͑ ͒ mA0 2m12/sin 2 , mHϮ mA0 mW . A7 The minimization condition can then be written as 2 The coefficient m12 corresponds to the non-Hermitian qua- 2 Ϫ 2 ␤ϩ 1 2 ␤ϭ 2 m m tan 2 m cos 2 0, dratic term in the Higgs potential. If m ϭ0, the Lagrangian Hd 12 Z 12 possesses a Peccei-Quinn symmetry and it guarantees that M 0ϭ0. It is practical to express all other masses in terms of 2 Ϫ 2 ␤Ϫ 1 2 ␤ϩ ␶ 2 ␤ϭ ͑ ͒ A m m cot 2 m cos 2 2 V sin 0. A4 Hu 12 Z mA0. From the second derivatives above, the tree-level mass 0 0 matrix of the scalar Higgs bosons in the basis of hu ,hd be- Continue to obtain the second derivatives, comes

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2 2␤ϩ 2 2␤ Ϫ͑ 2 ϩ 2 ͒ ␤ ␤ mA0cos mZsin mA0 mZ sin cos 2 M ϭͩ 2 2 2 2 2 2 ͪ , ͑A8͒ 0 Ϫ͑ ϩ ͒ ␤ ␤ ␤ϩ ␤ mA0 mZ sin cos mA0sin mZcos

where the subscript 0 indicates tree-level quantities. One can P q,d ϭ P ͑ u,d ϩ ␤ d,d ͒ then prove that (m 0) рm ͉cos 2␤͉. ͚ gi,qZϩ,h gi,u Zϩ,h tan Zϩ,h h 0 Z q The leading correction from top-quark–top-squark loops is 1 ϭg P ͩ Ϫ sin 2␣ϩtan ␤ sin2␣ͪ i,u 2 10 M2 ϷM 2ϩ 2ͩ ͪ 2ϭ ␶ 2 2 1LT 0 T , T 4 V s␤ 00 1 P 2 2 2 ϭ g tan ␤͓1Ϫ͑m Ϫ4c m 2 i,u A ␤ A 3g2m4 t 2 ϭ ln͑m˜ m˜ /m ͒. ͑A9͒ Ϫm2 ϪT2͒/͑m2 Ϫm2͔͒, ͑A14͒ ␲2 2 2␤ t L t R t Z H h 8 mWsin This formula can be found in Ref. ͓13͔, where different P q,d ϭ P ͑ u,d ϩ ␤ d,d ͒ schemes of approximation were studied. As we have uncer- ͚ gi,qZϩ,H gi,u Zϩ,H tan Zϩ,H tainty from the SUSY breaking scale, it may be overboard to q use the full-ﬂedged one-loop calculation. We use this leading 1 ϭ P ͩ ␣ϩ ␤ 2␣ͪ approximation in the remaining study. The CP-even Higgs gi,u sin 2 tan cos boson mass-squared eigenvalues are then given by 2

1 P 2 2 2 1 ϭ g tan ␤͓1ϩ͑m Ϫ4c␤m m2 ϭ ͓M 2 ϩM 2 Ϯͱ͓M 2 ϪM 2 ͔2ϩ4͑M 2 ͒2͔. 2 i,u A A H0,h0 2 11 22 11 22 12 ͑ ͒ Ϫ 2 Ϫ 2͒ ͑ 2 Ϫ 2͔͒ ͑ ͒ A10 mZ T / mH mh , A15 The mass of h0 has been substantially raised above the tree- level prediction which is lower than the experimental con- and straint. The corresponding mixing angle ␣ is given by

P q,d P u,d d,d M 2 ͚ g Zϩ ϭg ͑Zϩ ϩtan ␤Zϩ ͒ 2 12 i,q ,A i,u ,A ,A sin 2␣ϭ , q ͱ͓M 2 ϪM 2 ͔2ϩ4͑M 2 ͒2 11 22 12 1 ͑A11͒ ϭ P ͩ ␤ϩ ␤ 2␤ͪ gi,u sin 2 tan sin M 2 ϪM 2 2 22 11 cos 2␣ϭ . ͱ͓M 2 ϪM 2 ͔2ϩ ͑M 2 ͒2 ϭg P tan ␤. ͑A16͒ 11 22 4 12 i,u

2 Ͼ 2 The eigenmasses (mH0 mh0) are given by The two-loop EDM of the electron with the leading one-loop mass correction becomes 2 ϩ 2 ϭ 2 ϩ 2 ϩ 2 mH0 mh0 mA0 mZ T , ␣ ͑ 2 Ϫ 2 ͒2ϭ͓͑ 2 Ϫ 2 ͒ ␤ϩ 2͔2 de gme P m 0 m 0 m 0 m cos 2 T ͩ ͪ ϭ H h A Z g m␹ 3 ␤ 1,u 1 e 16␲ 2M Wcos ϩ͑ 2 ϩ 2 ͒2 2 ␤ ͑ ͒ mA mZ sin 2 . A12 2 2 2ϩ 2 ϩ 2 ͑ ϩ ͒ g͑m␹ /M ͒ T M Z M A 1 2c2␤ 1 h In terms of these masses, the mixing angle ␣ is determined at ϫtan ␤ͫͩ 1ϩ ͪ 2 2 2 m Ϫm m␹ tree level by H h 1

2 2 2 ͑ 2 2 ͒ 2 2 2 2 2 ϩ ϩ ͑ ϩ ͒ g m␹ /M H sin 2␣ m 0ϩm ͑m 0Ϫm ͒cos 2␤ϪT T M Z M A 1 2c2␤ 1 ϭϪ A Z ␣ϭ Z A ϩͩ 1Ϫ ͪ 2 2 , cos 2 2 2 . 2 2 2 sin 2␤ Ϫ Ϫ m Ϫm m␹ mH0 mh0 mH0 mh0 H h 1 ͑A13͒ 2 2 f ͑m␹ /M ͒ 1 A ¯␹ 0␹ ϩ2 Ϫ͑m␹ →m␹ ͒ͬ . ͑A17͒ From the vanishing of the diagonal scalar coupling of G , 2 1 2 P ϭ P m␹ we have s␤gi,u c␤gi,d for each mass eigenstate i. Therefore 1

116008-7 CHANG, CHANG, AND KEUNG PHYSICAL REVIEW D 66, 116008 ͑2002͒

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