constraints from electric dipole moments on chargino baryogenesis in the minimal supersymmetric...

PHYSICAL REVIEW D 66, 116008 ~2002!

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

Darwin ChangNCTS and Physics Department, National Tsing-Hua University, Hsinchu 30043, Taiwan, Republic of China

and Theory Group, Lawrence Berkeley Lab, Berkeley, California 94720

We-Fu ChangNCTS 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 KeungNCTS 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 2002!

A commonly accepted mechanism of generating baryon asymmetry in the minimal supersymmetric standardmodel ~MSSM! depends on theCP violating relative phase between the gaugino mass and the Higgsinomterm. The direct constraint on this phase comes from the limit of electric dipole moments~EDM’s! of variouslight fermions. To avoid such a constraint, a scheme which assumes that the first two generation sfermions arevery heavy is usually evoked to suppress the one-loop EDM contributions. We point out that under such ascheme the most severe constraint may come from a new contribution to the electric dipole moment of theelectron, the neutron, or atoms via the chargino sector at the two-loop level. As a result, the allowed parameterspace for baryogenesis in the MSSM is severely constrained, independent of the masses of the first twogeneration sfermions.

DOI: 10.1103/PhysRevD.66.116008 PACS number~s!: 11.30.Er, 11.30.Fs, 12.60.Jv, 98.80.Cq

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INTRODUCTION

While the standard model of particle physics continuesaccurately describe a wide array of experimental tests mphysicists suspect that the next generation of a unified fitheory will be supersymmetric. This supersymmetric thein its simplest form, the minimal supersymmetric standamodel~MSSM! @1#, may help to solve many of the outstaning problems in the standard model. Two examples of tsort are the coupling-constant-unification problem andobserved baryon asymmetry of the universe~BAU!. It is thelatter of these two that will be discussed in this paper.

It has been demonstrated that the SM is insufficientgenerating a large enough BAU@2#. Particles lighter in massbut stronger in coupling are needed to make the electrowtransition more first order. Additionally, a newCP violatingphase is required to generate enough BAU. It is very apping that the MSSM naturally provides a solution to borequirements@3#.

The top-quark partner, the top squark, which is naturalighter than the other squarks, can make the transition mfirst order, while there are plenty of newCP violating phasesat our disposal in the soft supersymmetry~SUSY! breakingsector. In particular, it has been shown that the most likscenario is to make use of the relative phase between theSUSY breaking gaugino mass and them term of theHiggsino sector@3#. In this case, the BAU is generatethrough the scattering of the charginos on the bubble wTheCP violation is provided by the chargino mixing. It turnout that in most parameter space of the MSSM a nemaximal CP violating phase is needed to generate enou

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BAU. One immediate question is whether or not such a nsource ofCP violation is already severely experimentalconstrained. It is not surprising that the most severe cstraints are provided by the current experimental limits ofelectric dipole moments~EDM’s! of the electron (de) andthe neutron (dn).

Fortunately, the lowest order~one-loop! contributions tovarious EDM’s through chargino mixing can be easily supressed by demanding that the first two generations of smions be heavier than the third one@4,5#. For example, ifone requires these sfermions to be heavier than 10 TeV,one-loop induced EDM’s will be safely small@6#. In fact,such a scenario can even be generated naturally in a mbasic scheme referred to as the more minimal SUSY mo@7#. However, despite the enlarged parameter space ofMSSM, thanks to all the intricate limits provided by accmulated data from various collider experiments, there is oa small region of parameters left within the MSSM for subaryogenesis to work@3#.

In this article we wish to point out that even if sfermionof the first two generations are assumed to be very hethere are important contributions to the EDM of the electrat the two-loop level via the chargino sector that stronconstrain the chargino sector as the source for BAU inMSSM. Similar contributions to the quark EDM also exibut the resulting constraint turns out to be relatively weakWhile this is not the first time that two-loop contributionhave been found to be more important than the one-lones@8–12#, this chargino contribution and its relevanceBAU was never treated fully.

In the case of chargino contributions, the two-loop con

©2002 The American Physical Society08-1

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CHANG, CHANG, AND KEUNG PHYSICAL REVIEW D66, 116008 ~2002!

bution is dominant because the one-loop contribution is spressed when the sfermions are heavy. This aspect is simto those in Refs.@8,12#. In addition, the present case oflargeCP violating phase in the chargino mixing and the ligHiggs scalar, which is necessary to obtain a large barasymmetry, is also the same cause of the large EDM. Thfore, the resulting severe EDM constraint is very difficultavoid in the mechanism of chargino baryogenesis by tunparameters.

THE MODEL AND COUPLINGS

Before we outline the physics of the chargino mixingsupersymmetric models we will set forth our conventioWe assume the minimal set of two Higgs doublets. LetsuperfieldFd(Y521) couple to thed-type field andFu(Y51) to theu-type ~see Ref.@11# for our convention!. Thechargino fields are combinations of those of theW-ino(vL,R

1 ) and the Higgsino (huL,dR1 ). DenotecL5(vL

1 ,huL1 )T

and c R5(vR1,hdR

1 ). The chargino mass term,2L MC

5c RMCcL in our convention, becomes

MC5S M2 A2MWsinb

A2MWcosb meif D , ~1!

whereM2 is the SUL(2) gaugino mass. Note that we chooa CP violating complex Higgsino massmeif. The scalarcomponentsHu ,Hd of Fu ,Fd have real vacuum expectatiovaluesvu /A2,vd /A2, respectively, and tanb5vu /vd .

We use the biunitary transformation to obtain the diagomass matrixMD5U8MCU† with eigenvaluesmx1

,mx2for

the eigenfieldsx1 ,x2. TheCP violating chargino mixing cancontribute to the fermion EDM through the charginsfermion loop. Detailed analyses of such contributionsbe found in the literature@5#. As noted in the Introductionsuch contributions can be tuned to be small by makingsfermions heavy@6# ~typically of 10 TeV or larger!. Here weare interested in contributions to the EDM of a fermion thare still important even with very heavy sfermions. For thwe find that the leading contribution is from diagrams of ttype in Fig. 1.

To evaluate the diagram, we examine gauge couplingthe Higgs bosons,Hq

05(vq1wq)/A2,

LY5g

A2(i j

x iR@Uiv8 Uh j† wu

0* 1Uih8 Uv j† wd

0* #x jL1H.c.

~2!

Only the diagonal couplings in the chargino basis areevant to the simple diagrams in Fig. 1 mediated by an innal photon. Therefore we define

giwu[giu

S 1 igiuP 5

g

A2Uiv8 Uih* ,

giwd[gid

S 1 igidP 5

g

A2Uih8 Uiv* . ~3!

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The complex mixing amplitudes are written in terms of treal couplingsgS and gP. In the same spirit, the compleneutral Higgs fields are decomposed into the real and imnary componentswq

05hq01 iaq

0 (q5u,d). Note thathd0 and

hu0 mix in a CP conserving fashion at the tree level, and so

au0 andad

0 :

S h0

H0D 5RS hu0

hd0D , S G0

A0 D 5SS au0

ad0D , ~4!

R5S cosa 2sina

sina cosa D , S5S sinb 2cosb

cosb sinb D . ~5!

The EDM calculation involves the Higgs boson propagatowhich are defined as

^wqwq8† &p25 i(

sZ1,s

q,q8/~p22Ms2 !,

^wqwq8&p25 i(s

Z2,sq,q8/~p22Ms

2 !. ~6!

The Z factors can be shown to be real at the leading orwith the explicit forms

Z6,Hd,d 5Z6,h

u,u 5cos2a, Z6,Gd,d 5Z6,A

u,u 56cos2b,

Z6,hd,d 5Z6,H

u,u 5sin2a, Z6,Ad,d 5Z6,G

u,u 56sin2b,

Z6,Hu,d 5

1

2sin 2a52Z6,h

u,d , Z6,Au,d 56

1

2sin 2b52Z6,G

u,d ,

Z6,sd,u 5Z6,s

u,d for s5h,H,A,G.

For completeness, our list includes the unphysical GoldstbosonG0, which does not contribute to the EDM. Other surules are

(s5hHAG

Zs,sq,q852dq,q8ds,1 . ~7!

The electron EDM via Fig. 1 is given by

FIG. 1. A two-loop diagram of the EDM of the electron, oquarks. The chargino runs in the inner loop.

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CONSTRAINTS FROM ELECTRIC DIPOLE MOMENTS ON . . . PHYSICAL REVIEW D66, 116008 ~2002!

S de

e D5a

16p3

gme

MWcosb (i ,q

gi ,qP

mx i

FgS mx i

2

Mh2D Z1,h

q,d

1gS mx i

2

MH2 D Z1,H

q,d 1 f S mx i

2

MA2 D Z1,A

q,d G . ~8!

Here the Barr-Zee@9# functions are defined as

Kn~z!5z

2E0

1ynln@y~12y!/z#

y~12y!2zdy,

~9!f ~z!5K0~z!22K1~z!12K2~z!, g~z!5K0~z!.

For the EDM of the down quark, we simply use thcharge ratio1

3 to give (dd /e)5 13 (de /e)(md /me), while for

the EDM of the up quark, we need to replaceZq,d→Zq,u inEq. ~8! as well as the obvious charge ratio2 2

3 and replace-ment ofme→mu . In the Appendix, we offer a more compaanalytic form of these results together with additional detawhich include the radiative correction to the Higgs bosmass in the simplified form suggested in Ref.@13#.

Since the charginos do not couple to the gluon, there ischromo-EDM generated@11#. Note that if one wishes to include the contribution with the internal photon replacedthe Z boson, it is necessary to include the off-diagonchargino couplings of theZ and the Higgs bosons. We ignorsuch contributions here because they are expected tomuch smaller than that of the photon which was confirmedprevious similar two loop calculations@10#. In particular, theelectron EDM viaZ is highly suppressed by the small valuof theZ vectorial coupling to the electron due to the appromate relation sin2uW'1

4. There are other two-loop diagramwith CP violation originating from the same phase suchthe ones with a chargino-neutralino loop mediatedgH1W2

effective vertex orgW1W2 (W EDM! effective vertex. Wedo not include them here because these contributionsexpected to be smaller~by roughly an order of magnitude! assuggested by previous two-loop calculations@11,12#. In anycase, these additional diagrams form a separate gaugependent set.

Because the imaginary parts of the off-diagonal entriesMC are zero in our convention, we obtained the followisum rules:

(i

gi ,uP mx i

52g

A2Im~U8†MDU !vh50, (

igi ,d

P mx i50.

~10!

Therefore,g2,qP 52g1,q

P (mx1/mx2

). It is easy to see that in

the case of degenerate massesmx15mx2

, perfect cancella-tion occurs, yielding a zero EDM.

Based upon another fact, that the diagonal scalar coupof x iG

0x i is zero, we can show that sinbgi,uP 5cosbgi,d

P .Therefore, each of the fourCP violating coefficientsgi ,q

P canbe simply related to one of them, sayg1,u

P , which againdepends on the fundamental MSSM parame

11600

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r

tanb,meif,MA2 ,M2. The usual SUSY breaking terms in

clude the last two parameters as well as the trilinear sfermcoupling, theA term, which is not relevant in our analysbecause it does not participate directly in this particumechanism of baryogenesis@3#. If we replace charginos bytop squarks in the inner loop, the effect of the relative phof A andm can contribute to the two-loop EDM as studiedRef. @8#. The top squark loop effect can be small ifAt issmall, if At is in phase withm, or if the left-handed topsquark is very heavy but the right-handed top squark is ralight. This last scenario is preferred by BAU. Such a larmass gap will suppress top squark mixing and kill the EDcontribution via the top squark loop. In addition, it has beconcluded by many groups@3# that usingCP violating mix-ing of the top squark to generate BAU is much more difficthan using that of the chargino.

NUMERICAL ANALYSIS AND BARYOGENESIS

To our current knowledge, the experimental constraintthe electron EDM has become very restrictive:

udeu,1.6310227 e cm ~90% C.L., Ref. @14# !. ~11!

Since the tree-level Higgs boson mass relation@1# predicts alight Higgs bosonmh0,mZ , which has already been ruleout by experimental searches at the CERNe1e2 colliderLEP II, our analysis has included the leading mass correc@13# at the one-loop level. For completeness, the resultHiggs boson mass dependence on tanb in this scheme isillustrated in Fig. 2. Figure 3 shows the tanb dependence ofthe predicted value of the electron EDM from different cotributions due to the Higgs bosons,A0, H0, andh0. We showthe case of maximalCP violation whenf5p/2, as requiredby baryogenesis@15#, with masses at the electroweak scaMA5150 GeV, M25m5200 GeV. Note that, in this casethe h contribution dominates until about tanb'3. The Hcontribution becomes dominant for tanb.5.4. When tanbbecomes large, the increase of the Yukawa coupling ofelectron overwhelms the reduction ofCP violation in thechargino sector. This gives the increase of the electron Eas tanb increases. The same effect happens to the EDMthed quark, but not theu quark. Figure 4 shows the electroEDM contour plot versusM2 andm for the case tanb53,MA5100 GeV, andf5p/2. In the many calculations oBAU in the MSSM@3# the largest uncertainty seems to comfrom the calculation of the source term for the diffusioequations that couples to the left-handed quarks@15,16#. Us-ing the latest summary of the situation in Ref.@17# as areference point, large BAU@2<h10[(nB2nB)/ng31010

<3# requires tanb<3 with the wall velocity and the wallwidth close to their optimal valuesvw.0.02, l w.6/T, m.M2, and theCP phase sinf close to 1. Note that a smalletanb gives a larger BAU; however, it tends to give a smlightest Higgs boson mass which violates the LEP II limunless the left top squark is much heavier than 1 TeV. Usthe SUSY parameters in the above range, the numeranalysis in our figures indicates that the predicted valuethe electron EDM is more than a factor of 5 to 10 bigger ththe experimental limit on the electron EDM in most of th

8-3

ses


FIG. 2. The mass of the light Higgs bosonh0

versus tanb. The lower set of curves correspondto the tree-level result. The upper set of curv

includes the leading one-loop (t, t ) effect, formt L

51 TeV andmt R5150 GeV. Curves within

each set are in the order of casesmA

5150,200,250,300 GeV, from bottom to top.

tro

DMhee

BAU preferred parameter range. In fact, if sinf51 andtanb53, then the parameter space allowed by the elecEDM limit is limited to a narrow strip withm.M2 and mhas to be as large as 600 GeV in order to satisfy this Econstraint. The ranges of values form andM2 ~both smaller

11600

nthan 250 GeV! presented in Ref.@17# are all ruled out. Un-less the numerical constraint on BAU in Ref.@17# is relaxedby an order of magnitude, it seems to be very difficult for tchargino mechanism for BAU to be compatible with thelectron EDM constraint.

n
FIG. 3. The predicted value of the electroEDM versus tanb from different contributionsdue to the Higgs bosonsh0,A0, and H0, at themaximalCP violation whenf5p/2. Masses areset at the electroweak scale,MA5150 GeV,M2
5m5200 GeV.

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On the other hand, for the neutron EDM, our analyindicates that the current experimental limit in Eq.~12! givesonly a marginal constraint on the MSSM parameters requfor chargino BAU.

With the quark EDM, one uses the quark model to predthe neutron EDM. A new limit@18# udnu,6.3310226 e cm~95% C.L.! for the neutron EDM has been reported baseda combination of the recent data of low statistical accurand the earlier measurement@19#. This combination of theold and the new results has been criticized in Ref.@20#. Asshown in the contour plot of Fig. 5, using the parametsuggested by the chargino baryogenesis mechanism, ourdicted EDM value is around the size of the more consertive experimental limit udnu&12310226 e cm, recom-mended in Ref.@20#. Due to large theoretical uncertaintiesthe relation between the quark EDM and the neutron EDthe constraint from the neutron EDM on the parameter spcannot be as important as that from the electron EDM evethe more stringent limit is used.

Note, however, that the uncertainties in the calculationthe nonequilibrium electroweak baryogenesis process arefrom settled. For example, in the latest review by the groin Ref. @21# a small CP violating phase of 1022 may besufficient to generate BAU. In that case even the larger vaof tanb is allowed. For this purpose, in Fig. 6, we also pthe electron EDM for tanb up to 50.

CONCLUSION

The baryogenesis in the MSSM requires the lightHiggs boson to be light in order to get a strong first ordphase transition. It also requires theCP violating phase inchargino mixing to be large in order to get large enouBAU. As we discussed, both requirements imply thatpredicted values of the EDM’s of the electron and the n

FIG. 4. The electron EDM contour plot versusM2 andm for thecase tanb53, MA5100 GeV, andf5p/2.

11600

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sre--

,ceif

ffarp

e

tr

he-

tron must be large. For sinf51 and tanb53, the currentelectron EDM constraint requiresm.M2.600 GeV. Takingthe uncertainty in the calculations of BAU in the literatuinto account, it is probably still premature to claim that thparticular mechanism of baryogenesis is absolutely ruledbut it is clear that the precision measurements of the EDof fermions, especially the electron EDM, give a tight costraint on the mechanism.

Note added. While this paper was under consideration wreceived a preprint of a paper by A. Pilaftsis, Nucl. PhB644, 263 ~2002!, with calculations that overlap with oursOur numerical results agree with this later calculation.

ACKNOWLEDGMENTS

W.Y.K. is partially supported by a grant from the U.SDepartment of Energy~Grant No. DE-FG02-84ER40173!.D.C. is supported by a grant from the National ScienCouncil ~NSC! of the Republic of China~Taiwan!. We wishto thank H. Haber, H. Murayama, O. Kong, and K. Cheufor discussions. D.C. wishes to thank the theory groupsSLAC and LBL for hospitality during his visit. W.F.C. anW.Y.K. wish to thank NCTS of NSC for support.

APPENDIX: HIGGS POTENTIAL WITHRADIATIVE CORRECTIONS IN THE MSSM

AND ELECTRIC DIPOLE MOMENTS

The Higgs potential has the form

V5mHd

2 uHdu21mHu

2 uHuu21~2m122 HdHu1H.c.!

11

8~g1

21g22!~ uHdu22uHuu2!21tuHuu41•••. ~A1!

FIG. 5. The neutron EDM contour plot versusM2 andm for thecase tanb53, MA5100 GeV, andf5p/2.

8-5

n

c-


FIG. 6. The predicted value of the electroEDM versus large tanb at the maximalCP vio-lation whenf5p/2. Masses are set at the eletroweak scale,M25m5200 GeV. Curves fromtop to bottom are in the order of casesmA

5150,300,450,600 GeV.

ar

led

ealar

a-

thatof

ass

At the tree level, SUSY requires the dim54 coefficientt50. However, it arises from the large top-quark–top-squloop correction. Denote

^Hd&5Vd , ^Hu&5Vu , V2[Vd21Vu

2 ,

tanb[Vu /Vd , mW2 5 1

2 g22V2, mZ

25 12 ~g1

21g22!V2.

~A2!

We try to derive the mass matrix of theCP-even Higgsbosons, which correspond to the real part of the compfields. We use superscriptsR,I to abbreviate the real animaginary parts. The first derivatives of the potential are

~]V/]HdR!52mHd

2 HdR22m12

2 HuR1 1

2 ~g121g2

2!

3~ uHdu22uHuu2!HdR,

~]V/]HuR!52mHu

2 HuR22m12

2 HdR2 1

2 ~g121g2

2!

3~ uHdu22uHuu2!HuR14tuHuu3. ~A3!

The minimization condition can then be written as

mHd

2 2m122 tanb1 1

2 mZ2cos 2b50,

mHu

2 2m122 cotb2 1

2 mZ2cos 2b12tV2sinb50. ~A4!

Continue to obtain the second derivatives,

11600

k

x

~]2V/]HdR2!52m12

2 tanb12MZ2cb

2 ,

]2V/~]HuR]Hd

R!522m122 2mZ

2sin 2b, ~A5!

~]2V/]HuR2!52m12

2 cotb12sb2~MZ

214tV2!,

~]2V/]HdI2!52m12

2 tanb,

]2V/~]HuI ]Hd

I !52m122 , ~A6!

~]2V/]HuI2!52m12

2 cotb.

The basis defined in Eqs.~4!, ~5! agrees with that in Martin’sreview@1#. One can easily show thatG is massless as it is thunphysical Goldstone boson. The mass of the pseudoscA0 is

mA02

52m122 /sin 2b, mH6

25mA0

21mW

2 . ~A7!

The coefficientm122 corresponds to the non-Hermitian qu

dratic term in the Higgs potential. Ifm122 50, the Lagrangian

possesses a Peccei-Quinn symmetry and it guaranteesMA050. It is practical to express all other masses in termsmA0. From the second derivatives above, the tree-level mmatrix of the scalar Higgs bosons in the basis ofhu

0 ,hd0 be-

comes

8-6


M 025S mA0

2 cos2b1mZ2sin2b 2~mA0

21mZ

2!sinb cosb

2~mA02

1mZ2!sinb cosb mA0

2 sin2b1mZ2cos2b D , ~A8!

c

p

cet

in

een

p

where the subscript 0 indicates tree-level quantities. Onethen prove that (mh0)0<mZucos 2bu.

The leading correction from top-quark–top-squark loois

M1LT2 'M 0

21T2S 1 0

0 0D , T254tV2sb2

53g2mt

4

8p2mW2 sin2b

ln~mt Lmt R

/mt2!. ~A9!

This formula can be found in Ref.@13#, where differentschemes of approximation were studied. As we have untainty from the SUSY breaking scale, it may be overboarduse the full-fledged one-loop calculation. We use this leadapproximation in the remaining study. TheCP-even Higgsboson mass-squared eigenvalues are then given by

mH0,h02

51

2@M 11

2 1M 222 6A@M 11

2 2M 222 #214~M 12

2 !2#.

~A10!

The mass ofh0 has been substantially raised above the trlevel prediction which is lower than the experimental costraint. The corresponding mixing anglea is given by

sin 2a52M 12

2

A@M 112 2M 22

2 #214~M 122 !2

,

~A11!

cos 2a5M 22

2 2M 112

A@M 112 2M 22

2 #214~M 122 !2

.

The eigenmasses (mH02

.mh02 ) are given by

mH02

1mh02

5mA02

1mZ21T2,

~mH02

2mh02

!25@~mA02

2mZ2!cos 2b1T2#2

1~mA21mZ

2!2 sin22b. ~A12!

In terms of these masses, the mixing anglea is determined attree level by

sin 2a

sin 2b52

mA02

1mZ2

mH02

2mh02 , cos 2a5

~mZ02

2mA2 !cos 2b2T2

mH02

2mh02 .

~A13!

From the vanishing of the diagonal scalar coupling ofxG0x,we havesbgi ,u

P 5cbgi ,dP for each mass eigenstatei. Therefore

11600

an

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r-og

--

(q

gi ,qP Z1,h

q,d 5gi ,uP ~Z1,h

u,d 1tanbZ1,hd,d !

5gi ,uP S 2

1

2sin 2a1tanb sin2a D

51

2gi ,u

P tanb@12~mA224cb

2mA2

2mZ22T2!/~mH

2 2mh2!#, ~A14!

(q

gi ,qP Z1,H

q,d 5gi ,uP ~Z1,H

u,d 1tanbZ1,Hd,d !

5gi ,uP S 1

2sin 2a1tanb cos2a D

51

2gi ,u

P tanb@11~mA224cb

2mA2

2mZ22T2!/~mH

2 2mh2!#, ~A15!

and

(q

gi ,qP Z1,A

q,d 5gi ,uP ~Z1,A

u,d 1tanbZ1,Ad,d !

5gi ,uP S 1

2sin 2b1tanb sin2b D

5gi ,uP tanb. ~A16!

The two-loop EDM of the electron with the leading one-loomass correction becomes

S de

e D5a

16p3

gme

2MWcosbg1,u

P mx1

3tanbF S 11T21MZ

21MA2~112c2b!

mH2 2mh

2 D g~mx1

2 /Mh2!

mx1

2

1S 12T21MZ

21MA2~112c2b!

mH2 2mh

2 D g~mx1

2 /MH2 !

mx1

2

12f ~mx1

2 /MA2 !

mx1

22~mx1

→mx2!G . ~A17!

8-7

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hep-ph/0208043.

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constraints from electric dipole moments on chargino baryogenesis in the minimal supersymmetric...

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