RAPID COMMUNICATIONS

PHYSICAL REVIEW D, VOLUME 62, 071901~R!

CP violating supersymmetric contributions to the electroweakr parameter

Sin Kyu Kang*School of Physics, Korea Institute for Advanced Study, Seoul 130-012, Korea

Jung Dae Kim†

School of Physics, Korea Institute for Advanced Study, Seoul 130-012, Koreaand Physics Department and IPAP, Yonsei University, Seoul 120-749, Korea

~Received 20 June 2000; published 12 September 2000!

The effects ofCP violation on the supersymmetric electroweak correction to ther parameter are investi-gated. To avoid the EDM constraints, we require that arg(m),1022 and the nonuniversal trilinear couplingsAf5(0,0,A0) and also assume that gluinos are heavier than 400 GeV. TheCP phasef t5arg(A0) leads to a

large enhancement of the relative mass splittings betweent 2 and bL( t 1), which in turn reduces the one-loopcontribution of the top squark and bottom squark toDr. For small tanb, such aCP violating effect isprominent. We also study how much the two-loop gluon and gluino contributions are affected by theCPphase. The possible contributions to ther parameter arising from the Higgs sector withCP violation arediscussed.

PACS number~s!: 11.30.Er, 12.15.Lk, 12.60.Jv

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The minimal supersymmetric standard model~MSSM! isthe best motivated extension of the standard model~SM!. Asis well known, in the MSSM, there are many new sourcesCP violation beyond the Cabibbo-Kobayashi-Maskaw~CKM! phase, arising from the complex soft supersymme~SUSY! breaking terms, i.e., the Majorana gaugino masMi , the trilinear scalarA terms, the bilinear scalarB terms,as well as from them parameter which is the bilinear mixinof the two Higgs chiral superfields in the superpotentWhile those supersymmetricCP violating ~CPV! phasescould give large contributions to the neutron or electron eltric dipole moments~EDMs!, they are strongly constraineby the current experimental measurements of the neutroelectron EDMs, which for the neutron isdn,1.1310225e cm @1# and for the electron is de,4.3310227e cm @2#. To resolve this problem, several possibities have been suggested. The first one is to make all phsmall „O(1022)… @3#. But, such small phases would requiresignificant amount of fine-tuning and are thus undesiraThe second is to take the SUSY spectrum heavy in theeral TeV range which lies outside the reach of the acceltors @4#. Another option suggested recently is to use the ccellations among the various contributions to the neutronelectron EDMs, allowing for large CPV phases and a supsymmetric spectrum that is still within the reach of the acelerators@5#. Finally, an interesting possibility is to takeslightly nonuniversal scenario for the trilinear couplingsAf@6,7#. As shown in Ref.@7#, requiring that arg(m),1022 andAf5(0,0,A0), and assuming that gluinos are heavier thabout 400 GeV@5#, one may significantly reduce the sizethe neutron or electron EDMs due to Weinberg’s three-gluoperator@8# well below the current experimental limit.

Another CP violation associated with the Higgs bososector in the MSSM can come from the one-loop correcti

*Email address: skkang@ns.kias.re.kr†Email address: jdkim@theory.yonsei.ac.kr

0556-2821/2000/62~7!/071901~5!/$15.00 62 0719

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of the MSSM Higgs potential through soft CPV Yukawinteractions involving squarks, i.e.,A terms. As shown inRef. @9#, an immediate consequence of Higgs sectorCP vio-lation in the MSSM is the generation of mixing mass termbetween theCP-even andCP-odd Higgs fields. Thanks tothose scalar-pseudoscalar mixing mass terms, the smalllevel mass difference between the heaviest Higgs bosonthe CP-odd scalar may be lifted considerably. The phenoenological consequences of Higgs sectorCP violation havebeen studied@9,10#.

A possible way to probe SUSY is to search for the virtueffects of SUSY particles which would be sizable enoughbe detected in the present experiments. In most experimtally accessible processes, the heavy SUSY particlescouple from the low energy electroweak observables. Hoever, similar to the SM, if there happens large ‘‘custodiaSU(2)V breaking, the electroweak observables which arerametrized by ther parameter@11#, defined as the ratio othe strengths of neutral and charged currents at vanismomentum transfer, can severely be affected. In the MSSthe main potential source of SU(2)V breaking is the splittingbetween the top squark and bottom squark masses. In ation, there can be large splitting within a supersymmetmultiplet, which also affects ther parameter. From the numerical analysis, it is well known that the leading contribtion of the squark loops, in particular top squark and bottsquark loops, to ther parameter can reach at the level offew 1023 which lies within the range of the experimentsensitivity@12#. There are also small additional contributioncoming from the exchange of the additional Higgs bosothe charginos and neutralinos. Some discussions of susymmetric contributions to ther parameter already exist inthe literature @12,13#. Moreover, higher order correctionhave also been studied so as to treat the SUSY loop cobutions to the electroweak observables at the same leveaccuracy as the SM contribution@14#. However, so far, theanalyses have been done within the context of theCP con-

©2000 The American Physical Society01-1

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RAPID COMMUNICATIONS

SIN KYU KANG AND JUNG DAE KIM PHYSICAL REVIEW D 62 071901~R!

serving ~CPC! MSSM. Since new CPV phases may affethe SUSY spectrum and generate additional couplings whdo not appear in the CPC MSSM, it would be quite intereing to study the effects of the CPV SUSY phases onelectroweak observables via ther parameter.

In this respect, the purpose of this Rapid Communicatis to examine how much ther parameter can be affected bthe nonvanishingCP phases in the MSSM. In this analysiwe will impose the universal conditions on the soft SUSbreaking terms which provide three complex parameters,universal gaugino massm1/2, the trilinear soft termA0, andthe bilinear soft termB0. In addition, there is a complemass parameterm. However, not all phases of the four complex parameters are physical@15#. It is always possible tomake a phase transformation on the gaugino fields so amakem1/2 real. From minimization conditions on the Higgpotential, one can make the phase ofB0 equal to the phasedifference of the two Higgs doublets in the MSSM. Asresult, there are only two independent CPV phases inMSSM with universal soft SUSY breaking terms, which ausually chosen to be arg(A0) and arg(m). To satisfy theEDM constraints, we will simply take arg(m) to be zero andthe trilinear coupling of the Higgs bosons to the squarksthe first and second generation to be much smaller thanone of the third generation, as mentioned above, and assthe gluino mass to be larger than 400 GeV. Thanks tononvanishing phase arg(A0), as will be shown later, the topsquark spectrum and their mixing angle are changed, whlead in turn to the shift of the contributions toDr. Also,since the neutral Higgs boson mass eigenstates in theMSSM are mixtures ofCP-even andCP-odd states, newgauge-Higgs couplings are induced at the tree level throthose mixings. These couplings can generate the additiradiative corrections to the gauge boson self-energies,may thus affect ther parameter. However, the contributioncoming from the gaugino and higgsino exchanges will notaffected by the nonvanishing CPV phase arg(A0) in the lead-ing order because they are affected by only the CPV pharg(m) which is neglected in this analysis. Thus, we shall nconsider those contributions. In addition, we shall study hmuch the dominant two-loop contributions via gluon agluino exchanges can be shifted by the effect of the Cphases. The numerical calculation will be done with the hof the program packageFEYNHIGGS @16# which is modifiedso as to calculate the CPV SUSY contributions to therparameter.

The r parameter is represented by

r51

12Dr, Dr5

PZZ~0!

MZ2

2PWW~0!

MW2

, ~1!

where PWW(ZZ)(0) denotes the transverse parts of tW/Z-boson self-energies at zero momentum transfer. InSM, the main contribution comes from the large splittibetween the top quark and the bottom quark masses angiven by

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e

is

Dr0SM5

NcGF

8A2p2F0~mt

2 ,mb2!, ~2!

whereNc is the color factor and the functionF0 is given by

F0~x,y!5x1y22xy

x2yln

x

y. ~3!

In the limit of large mass splitting it becomes proportionalthe heavy quark mass squared, i.e.,F0(mt

2 ,mb2).mt

2 . In-cluding QCD corrections, the SM contribution ofDr isDr1

SM52Dr0SM(2as/3p)(11p2/3) @17#. For as(MZ)

.0.12, the QCD correction reduces the one-loop resultabout 10% and shiftsmt upwards by about 10 GeV@18#.

(a) Squark contributions. Since the scalar partners of thlight quarks are expected to be almost degenerate in msupersymmetric theories, their contributions toDr are neg-ligible and thus only third generation will contribute. The tosquark mass matrix is given by

M t25S mt L

21mt

21DL mtmLR*

mtmLR mt R

21mt

21DRD , ~4!

where mLR5A02m* /tanb, DL5( 12 2 2

3 sin2uW)cos 2bMZ2 ,

and DR5 23 sin2uWcos 2bMZ

2 . The top squark fields (t L , t R)

are linear combinations of the mass eigenstatest 1,2 so thatt L5cosu t t12sinu te

2iwt t2, t R5sinu teiwt t11cosu t t2, where

tan 2u t52mtumLRu

mt L

22mt R

21DL2DR

, ~5!

and the mass eigenvalues of top squark fields are given

mt 1,2

25

1

2@mt L

21mt R

212mt

21DL

1DR7~@mt L

22mt R

21DL2DR#214mt

2umLRu2!1/2#,

~6!

umLRu5uA02m* /tanbu

5AuA0u21umu2/tan2b22uA0uumu/tanb cosf t,

~7!

wheref t[arg(A0). As expected, the nonvanishing phasef twill change the top squark mass spectrum which can in tlead to the shift ofDr. Neglecting the mixing inb sector, thecontribution of (t ,b) to Dr is presented at one-loop order b

Dr t b53GF

8A2p2@2sin2u tcos2u tF0~mt 1

2 ,mt 2

2!

1cos2u tF0~mt 1

2 ,mbL

2!1sin2u tF0~mt 2

2 ,mbL

2!#.

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RAPID COMMUNICATIONS

CP VIOLATING SUPERSYMMETRIC CONTRIBUTIONS . . . PHYSICAL REVIEW D 62 071901~R!

Before examining how much the value ofDr t b can beaffected by the phasef t , let us first investigate the massplittings between the two squark mass eigenstates wmay indicate to what extent the contribution of (t ,b) to Dris deviated. In Fig. 1~a! and ~b!, we present numerical estmation of the relative mass splittings between the two squmass eigenstates,umt 1

22mt 2

2 u/mq2 and umt 2

22mbL

2 u/mq2 , as a

function of the phase f t , for tanb51.6(20),mq5200 GeV,uA0u5200 GeV, andumu5100 GeV. Here wetake 0<f t<p. As the phasef t increases, both relativemass splittings increase. The effect of the maximalCP vio-lation corresponding tof t5p may lead to the enhancemeof the relative mass splittings by about 90%(40%)umt 1

22mt 2

2 u/mq2 (umt 2

22mbL

2 u/mq2) compared to the CPC cas

This is attributed to the fact that the absolute value ofmLRmonotonically increases with the increasing phasef t , whichmakes the mass splitting betweent 1(bL) and t 2 larger, andthe mixing angleu t closer top/4. Note that the mixing angleu t close top/4 is naturally obtained because the off-diagonelements of the top squark mass matrix may be larger tthe difference of the diagonal entries. As one can see fFig. 1~a!, in the case of small tanb, the splitting umt 2

2

2mbL

2 u/mq2 is larger thanumt 1

22mt 2

2 u/mq2 for f t50 ~CPC

FIG. 1. The relative mass splittings between the two squmass eigenstates,umt 1

22mt 2

2 u/mq2

~solid line! and umt 2

22mbL

2 u/mq2

~dotted line! as a function of the phasef t for tanb51.6 ~a! andtanb520 ~b!. We takemq5200 GeV, uA0u5200 GeV, andumu5100 GeV.

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ch

rk

r

ln

m

case!, whereas the latter becomes larger than the formerlarge f t . Thus, the splitting originating from the left-righmixing of top squarks may be more important for the CPcase with large phase. In the case of large tanb, umt 1

2

2mt 2

2 u/mq2 is larger thanumt 2

22mbL

2 u/mq2 for all f t , and the

mass splittings are weakly dependent onf t since the termconcerned with the phasef t in umLRu is suppressed by thelarge tanb. We also observed that the mass splittingumt 1

2

2mbLu/mq

2 is much smaller than the other two. For the saminput values of the SUSY parameters, the dependenceDr t b on the phasef t is shown as solid lines in Fig. 2Although the CPV case provides larger mass splitting ththe CPC case,Dr t b is monotonically decreased with the increasingf t . The effect of the CPV phase reduces the vaof Dr t b for the CPC case by about up to 35%. The reasothat as the phase increases, the mixing angleu t gets close tothe maximal mixing andF0(mt 1

2 ,mt 2

2 ) increases faster tha

F0(mt 2

2 ,mbL

2 ) in Eq. ~8! so that they lead to a destructiv

contribution toDr t b . As uA0u increases,Dr t b tends to de-crease, whereas it tends to increase with the increasingumu.We have observed that the effect of the CPV phase is dimished as long asmq becomes larger thanuA0u andumu. This isbecausemt 1

;mt 2;mbL

in the limit of largemq .

The two-loop QCD correction toDr t b is given by Eq.~8!in Ref. @14#, and the two-loop contribution mediated bgluino exchange is also represented in Ref.@14#. Those gluonand gluino contributions add up to about 30% of the onloop value in the CPC case.

In Fig. 3, the dependence of the dominant two-loop QCcorrection and gluino contribution to ther parameter on thephasef t is plotted by the dashed line and dotted line, resptively, for tanb51.6 ~a! and tanb520 ~b!. As the phasef tincreases, the two-loop gluonic SUSY contribution dcreases, whereas the gluino contribution increases. Theson that the gluino contribution increases with the increasphasef t is that its contribution depends inversely on tmass splittings betweent 1(bL) and t 2. In particular, for

k

FIG. 2. The contribution of top squark and bottom squark toDras a function of the phasef t for the same input values of the SUSparameters as those in Fig. 1.

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SIN KYU KANG AND JUNG DAE KIM PHYSICAL REVIEW D 62 071901~R!

small tanb, the value of the gluino contribution becomelarger than that of the gluonic SUSY contribution at larf t .

(b) Higgs sector contributions. There are also possiblcontributions arising from the Higgs sector to ther param-eter. In the CPC case, it is known@12# that the Higgs bosonmasses and couplings to gauge bosons are related in sway that they lead to large cancellations in their contribtions to ther parameter, which is at most of order of 1024.In the decoupling limit where the heavy neutralCP-evenHiggs, CP-odd Higgs, and the charged Higgs bosonsnearly degenerate and their couplings to gauge bosonsto zero, the lightestCP-even Higgs boson reaches its maxmal mass value and has almost the same properties aSM Higgs boson. Then, the contribution of the Higgs secof the MSSM to ther parameter is practically the same asthe SM. However, while the contribution of the SM Higgsector gives rise to logarithmic logMh /MZ in the limit oflarge Higgs mass which may reach about 1023 order, theMSSM contribution reaches at most 1024 due to the upperbound on the mass of the lightest Higgs boson.

Now, let us take into account the contributions comifrom the neutral Higgs sector withCP violation. The decom-

FIG. 3. The contributions of the two-loop QCD correctio~dashed line!, the two-loop gluino~dotted line!, and the Higgs bo-son sector~solid line! to Dr as a function of the phasef t fortanb51.6 ~a! and tanb520 ~b!. The input values of the SUSYparameters are taken to be the same as those in Fig. 1, angluino mass and the charged Higgs boson mass to bemg

5500 GeV andMHc5200 GeV, respectively.

07190

h a-

end

ther

position of the two Higgs doublets is given by

F15S f11

1

A2~v11f11 ia1!D ,

F25eiuS f21

1

A2~v21f21 ia2!D , ~9!

wherev1 andv2 are the vacuum expectation values andu istheir relative phase. As shown in Ref.@19#, the Higgs quarticcouplings receive significant radiative corrections fromAfterms. The vacuum expectation valuesv1 ,v2 and the phaseuare determined from the minimization conditions on tHiggs potential of the MSSM. TheCP-odd fields are rotatedto a15cosbG02sinba anda25sinbG01cosba so that theHiggs potential shows up a flat direction with respect toGoldstone fieldG0. The nonvanishing CPV phases inAt andAb can lead to the nonvanishingu, for which the scalar-pseudoscalar mixing mass terms are generated and thetral Higgs boson mass matrix in the weak ba(G0,a,f1 ,f2) can be written by

M025S M P

2 M PS2

MSP2 MS

2 D , ~10!

where M P2 and MS

2 are 232 matrix for CPC parts in thebasis (G0,a) and (f1 ,f2), respectively, and M PS

2

5(MSP2 )T describes the CPV mixings between (G0,a) and

(f1 ,f2). From the tadpole condition, the Goldstone fieldG0

does not mix with the other neutral fields and thus becommassless. Then, the neutral Higgs mass matrixM0

2 reduces toa 333 matrixM2, which is spanned in the basis (a,f1 ,f2).The Higgs boson mass matrixM2 can be diagonalized withthe help of an orthogonal rotation matrixU:UTM2U

FIG. 4. The relative mass splittings among three neutral Hibosons:uM3

22M22u/MHc

2 ~solid line! and uM222M1

2u/MHc

2 ~dottedline!. The input values of the SUSY parameters are taken to besame as the case of Fig. 3. The lightest~heaviest! Higgs masses aredenoted byM2(M3).

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CP VIOLATING SUPERSYMMETRIC CONTRIBUTIONS . . . PHYSICAL REVIEW D 62 071901~R!

5diag(MH3

2 ,MH2

2 ,MH1

2 ).

Then, the contributions of the Higgs sector withCP vio-lation to ther parameter can be given by

DrH53a

16pcW2 (

i 51

3

U1i2 DrH

SM~MHi

2 !1a

16psW2 MW

2

3S (i 51

3

~12U1i2 !F0~MHi

2 ,MHc

2 !

21

2 (i , j

3

~U1iU2 j2U2iU1 j !2F0~MHi

2 ,MH j

2 !D ,

~11!

whereMHcdenotes the charged Higgs boson mass and

DrHSM~M2!5

a

16psW2 MW

2 @F0~M2,MW2 !2F0~M2,MZ

2!#

1a

16psW2 F M2

M22MW2

logM2

MW2

21

cW2

M2

M22MZ2log

M2

MZ2G ~12!

corresponds to the SM Higgs boson contribution toDr.In Fig. 3, we present the dependence of the Higgs bo

sector contribution toDr on the phasef t ~solid lines!. Simi-

.

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lar to the one-loop contribution of top squark toDr, theone-loop contribution of the Higgs sector is decreased wthe increasing phasef t . This is mainly because the relativmass splittings among three neutral Higgs bosons decras the phasef t increases as shown in Fig. 4. In the casesmall tanb, uM3

22M22u/MHc

2 ~solid line! is larger thanuM22

2M12u/MHc

2 ~dotted line!. For only small tanb, the contribu-

tion of the Higgs sector is larger than the two-loop contribtions mediated by the gluon and gluino exchanges, and ththree contributions are almost the same atf t5p, which in-crease the one-loop contribution of top squark and bottsquark by about 30–35%. For large tanb, the Higgs sectorcontribution toDr is negligibly small.

In summary, effects ofCP violation on the supersymmetric electroweak correction to ther parameter have been investigated. To avoid the EDM constraints, we have requinegligibly small arg(m) and the nonuniversal couplingsAf5(0,0,A0) and also assumed that the mass of gluino is larthan 400 GeV. TheCP phasef t leads to large enhancemeof the relative mass splittings betweent 2 and bL( t 1), whichin turn reduces the one-loop contribution of the top squand bottom squark to ther parameter. For small tanb, sucha CP violating effect is prominent. We have also studiehow much the two-loop gluon and gluino contributions aaffected by theCP phase. Possible contributions to therparameter arising from the Higgs sector withCP violationhave been discussed.

The work of J.D.K. is supported in part by the KoreInstitute for Advanced Study.

nd,

D.

s.

tt.

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