A gobal fit to the anomalous magnetic moment, b-s gamma and Higgs limits in the constrained

a r X i v :h e p -p h /0106311v 2 29 J u n 2001

IEKP-KA/2001-14hep-ph/0106311

A global ?t to the anomalous magnetic moment,b →X s γand

Higgs limits in the constrained MSSM

W.de Boer,M.Huber,C.Sander

Institut f¨u r Experimentelle Kernphysik,University of Karlsruhe

Postfach 6980,D-76128Karlsruhe,Germany

D.I.Kazakov

Bogoliubov Laboratory of Theoretical Physics,Joint Institute for Nuclear Research,

141980Dubna,Moscow Region,Russian Federation

Abstract

New data on the anomalous magnetic moment of the muon together with the b →X s γdecay rate are considered within the supergravity inspired constrained minimal supersymmetric model.We perform a global statistical χ2analysis of these data and show that the allowed region of parameter space is bounded from below by the Higgs limit,which depends on the trilinear coupling and from above by the anomalous magnetic moment a μ.The newest b →X s γdata deviate 1.7σfrom recent SM calculations and prefer a similar parameter region as the 2.6σdeviation from a μ.

1Introduction

Recently a new measurement of the anomalous magnetic moment of the muon became available,which suggests a possible 2.6standard deviation from the Standard Model (SM)

expectation[1]:?a μ=a exp μ?a th μ=(43±16)·10

?10

.The theoretical prediction depends on the uncertainties in the vacuum polarization and the light-by-light scattering,see e.g.the discussion in [2].However,even with a conservative estimate of the theoretical errors,one has a positive di?erence ?a μof the order of the weak contribution to the anomalous magnetic moment,which opens a window for ”new physics”.The most popular explanation is given in the framework of SUSY theories [3]-[12],since the contribution of superpartners to the anomalous magnetic moment of the muon is of the order of the weak contribution and allows to explain the desired di?erence ?a μ.It requires the Higgs mixing parameter to be positive[4]and the sparticles contributing to the chargino-sneutrino (?χ±??νμ)and neutralino-smuon (?χ0??μ)loop diagrams to be relatively light[3].

The positive sign of μ0is also preferred by the branching ratio of the b-quark decaying radiatively into an s-quark -b →X s γ-[13].Last year the observed value of b →X s γwas close to the SM expectation,so in this case the sparticles contributing to the chargino-squark (?χ±??q )and charged Higgs-squark (H ±??q )loops have to be rather heavy in order not to contribute to b →X s γ.

10

-7

10

-6

10

-5

10

-4

10

-3

10

-2

10

-1

1

10

Y t ,b ,τ(0)

tan β

Y t

Y b

Y τ

μ > 0

μ < 0

Figure 1:The dependence of the third generation Yukawa couplings at the GUT scale as function of tan βfor μ0>0and μ0<0,obtained by ?tting them to the low energy masses of the top,bottom and tau mass.The results are for a common mass m 0=m 1/2=500GeV,but for di?erent masses the curves look very similar,except that the ’triple’uni?cation point for μ0<0shifts between 42and 48,if the common mass is shifted from 200to 1000GeV.

However,it was recently suggested that in the theoretical calculation one should use the running c-quark mass in the ratio m c /m b ,which reduces this ratio from 0.29to 0.22[14].The SM value for b →X s γincreases from (3.35±0.30)×10?4to (3.73±0.30)×10?4in this case.This value is 1.7σabove the most recent world average of (2.96±0.46)×10?4,which is the average from CLEO ((2.85±0.35stat ±0.22sys )×10?4)[15],ALEPH ((3.11±

0.80stat ±0.72sys )×10?4)[16]and BELLE ((3.36±0.53stat ±0.42sys (±0.500.54)model )×10?4

)[17].For the error of the world average we added all errors in quadrature.

As will be shown,the small deviations from the SM for both a μand b →X s γrequire now very similar mass spectra for the sparticles.

In the Constrained Minimal Supersymmetric Model (CMSSM)with supergravity me-

diated breaking terms all sparticles masses are related by the usually assumed GUT scale boundary conditions of a common mass m0for the squarks and sleptons and a common mass m1/2for the gauginos.The region of overlap in the GUT scale parameter space, where both aμand b→X sγare within errors consistent with the data,is most eas-ily determined by a global statistical analysis,in which the GUT scale parameters are constrained to the low energy data by aχ2minimization.

In this paper we present such an analysis within the CMSSM assuming common scalar and gaugino masses and radiatively induced electroweak symmetry breaking.We use the full NLO renormalization group equations to calculate the low energy values of the gauge and Yukawa couplings and the one-loop RGE equations for the sparticle masses with decoupling of the contribution to the running of the coupling constants at threshold.For the Higgs potential we use the full1-loop contribution of all particles and sparticles.For details we refer to previous publications[18,19].

In principle,one can also

require

b?

τYukawa coupling uni?cation,which has a solution at low and high values of the ratio of vacuum expectation values of the neutral components of the two Higgs doublets,denoted tanβ= H02 / H01 [18,19].From Fig.1one observes that if the third generation Yukawa couplings at the GUT scale are constrained by the low energy top,bottom and tau masses,they become equal forμ<0at tanβ≈40,while for μ>0they never become equal,although the di?erence between the Yukawa couplings is less than a factor three.Sinceμ>0is required by?aμ>0(see below),we do not insist on Yukawa coupling uni?cation and consider tanβto be a free parameter,except for the fact that the present Higgs limit of113.5GeV from LEP[20]requires tanβ>4.3in the CMSSM[13].

We found that the allowed area of overlap between b→X sγand aμcan be increased considerably for positive values of the common trilinear coupling A0at the GUT scale. For A0>0the present Higgs limit becomes more stringent than for the no-scale models with A0=0,as will be shown.

2aμand b→X sγin the CMSSM

The contribution to the anomalous magnetic moment of the muon from SUSY particles are similar to that of the weak interactions after replacing the vector bosons by charginos and neutralinos.The total contribution to aμcan be approximated by[3]

|a SUSY

μ

|?

α(M Z)

m2SUSY

tanβ 1?4αmμ ?140·10?11 100GeV 1Our sign conventions are as in Ref.[21].

Figure

ing

at

and in

also

one

newest

Such

shown

on the trilinear coupling A0is shown.The scaleμb was varied between0.5m b and2m b. For tanβ≈40only positive values of the Higgsmixing parameter at the GUT scaleμ0are allowed in agreement with the preferred sign ofμ0by the anomalous magnetic moment. For intermediate sparticle masses andμ0>0large values of A0and small values of the low energy scale(μb≈0.5m b)bring the calculated values of b→X sγclosest to the data, as can be seen from the left hand side of Fig. 3.Note that for heavy sparticles(right hand side of Fig.3)the e?ect of the trilinear coupling is small,because the stop mixing

is small,if the left and right handed stops are much heavier than the top mass.

Fig.4shows the values of b→X sγand a SUSY

μ

as function of m0and m1/2for tanβ=35.

For b→X sγthe ratio m c(μ)/m pole

b =0.22was used,while for the NLO QCD contributions

the formulae from Ref.[22]were used.The calculated values have to be compared with the experimental values BR(b→X sγ)=(2.96±0.46)×10?4[15]-[17]and?aμ=

(43±16)·10?10[1],which shows once more that b→X sγand a SUSY

μprefer a relatively

light supersymmetric spectrum.

To?nd out the allowed regions in the parameter space of the CMSSM,we?tted both the b→X sγand aμdata simultaneously.The?t includes the following constraints:i) the uni?cation of the gauge couplings,ii)radiative elctroweak symmetry breaking,iii) the masses of the third generation particles,iv)b→X sγand?aμ,v)experimental limits on the SUSY masses,vi)the lightest superparticle(LSP)has to be neutral to be a viable candidate for dark matter.We do not impose b?τuni?cation,since it prefers μ0<0,as shown in Fig.1,while?aμrequiresμ0>0,as shown in Fig. 2.Yukawa uni?cation forμ0>0can only be obtained by relaxed uni?cation of the gauge couplings and non-universality of the soft terms in the Higgs sector[23].

Theχ2contributions of b→X sγand the anomalous magnetic moment aμin the global ?t are shown in Fig.5for A0=0and tanβ=35.As expected,theχ2contribution from b→X sγis smallest for heavy sparticles,if b→X sγis calculated with m c/m b=0.29, while the minimumχ2is obtained for intermediate sparticles,if m c/m b=0.22is used. With the newly calculated b→X sγvalues,one can see,that b→X sγand aμprefer a similar region of the m0,m1/2plane.Fig.6shows the combinedχ2contributions from

b→X sγand a SUSY

μin the m0,m1/2plane,both in3D and2D,for A0=0(top)and A0

free(bottom).In the latter case the lower2σcontour from b→X sγmoves to the lower left corner,but for the preferred value A0≈3m0,which is the maximum allowed value in the?t in order to avoid negative stop-or Higgs masses and colour breaking minima,the Higgs bound moves up considerably.The total allowed region is similar in both cases,as shown by the light shaded areas in the contour plots.The2σcontours from the individual contributions are in good agreement with previous calculations[6,9],but in these paper a simple scan over the parameter space was performed without calculating the combined probability.In addition,A0=0was assumed.

We repeated the?ts for tanβ=20and50,as shown in Fig.7.For smaller values of tanβthe allowed region decreases,since aμbecomes too small.At larger tanβvalues the region allowed by aμand b→X sγincreases towards heavier sparticles,as expected from Eq.1,but it is cut by the region where the charged stau lepton becomes the Lightest Supersymmetric Particle(LSP),which is assumed to be stable and should be neutral.A charged stable LSP would have been observed by its electromagnetic interactions after being produced in the beginning of the universe.Furthermore,it would not be a candidate for dark matter.The increase of the LSP-excluded area is due to the larger mixing term between the left-and right handed staus at larger tanβ.

We conclude that the aμmeasurement strongly restricts the allowed region of the parameter space in the CMSSM,since it excludes theμ0

At large tanβa global?t including both b→X sγand aμas well as the present Higgs

limit of113.5GeV leaves a quite large region in the CMSSM parameter space.Here we left the trilinear coupling to be a free parameter,which a?ects both the Higgs limit constraint and the b→X sγconstraint,but in opposite ways,so that the preferred region is similar for the no-scale models with A0=0and models which leave A0free.

The95%lower limit on m1/2is300GeV(see Figs.6+7),which implies that the lightest chargino(neutralino)is above240(120)GeV.The95%upper limit on m1/2is

determined by the lower limit on a SUSY

μand therefor depends on tanβ(see Fig.2).For

tanβ=35(50)one?nds m1/2≤610(720)GeV,which implies that the lightest chargino is below500(590)GeV and the lightest neutralino is below260(310)GeV. Acknowledgements

D.K.would like to thank the Heisenberg-Landau Programme,RFBR grant#99-02-16650 and DFG grant#436/RUS/113/626for?nancial support and the Karlsruhe University for hospitality during completion of this work.

tanβtanβ

Figure3:The upper picture shows the dependence of the b→X sγrate on tanβfor A0=0and m0=600(1000)GeV,m1/2=400(1000)GeV at the left(right).For each value of tanβa?t was made to bring the predicted b→X sγrate(curved bands)as close as possible to the data(horizontal bands).The width of the predicted values shows the renormalization scale uncertainty from a scale variation between0.5m b and2m b.

The bottom picture shows the same dependence but for a?xed renormalization scale of 1m b.The width of the band is given by the variation of A0between-3m0and3m0.

02

4

2000

200

400600800V ]

1/2

(b → X s γ)*10

4

]

(a μS U S Y )*10

-11

Figure 4:The values of b →X s γand a SUSY μ

in the m 0,m 1/2plane for positive μand tan β=35to be compared with experimental data b →X s γ=(2.96±0.46)·10?4and a SUSY μ=(43±16)·10?10.One can see that both b →X s γand a SUSY μprefer relatively light sparticles.

12345678e V ]

m 1/2 [

G e V χ2 b → X s

γ

12345678m e V ]

m 1/2 [

G e V χ2 b → X s γ

12345678e V ]

m 1/2 [

G e V χ2

a μS U S Y

Figure 5:The individual contributions to χ2from b →X s γand a μin the m 0,m 1/2plane

for tan β=35,μ>0and A 0=0.On the left handside we show the old contribution from b →X s γ,as calculated with m c /m b =0.29,which has the lowest χ2for heavy supersymmetric particles.In the middle the contribution from b →X s γfor m c /m b =0.22is shown,which now has a minimum for intermediate masses.The χ2contribution from a μis shown on the right handside,which clearly prefers light sparticles.

1234567

8500

1000

m e V ]m 1/2

[G e V ]χ

2

m 1/2 [G e V ]

m 0

[GeV]

1234567

8500

1000

m e V

]

m 1/2

[G e V

]χ

2

m 1/2 [G e V ]

m 0 [GeV]

Figure 6:The upper part shows the χ2contribution (left)and its projection (right)in the m 0,m 1/2plane for A 0=0and tan β=35.The light shaded area is the region,where the combined χ2is below 4.The regions outside this shaded region are excluded at 95%C.L..The white lines correspond to the ”two-sigma”contours,i.e.χ2=4for that particular contribution.The lower row shows the same for the ?t,where A 0was left free,in which case A 0≈3m 0(its maximum allowed value in our ?t)is preferred in the region where the stop mixing is important,i.e.regions where the left and right handed stops are not very heavy compared with the top mass.One observes that with A 0as a free parameter the Higgs limit becomes the most important lower bound on the SUSY sparticles,while for the no-scale models with A 0=0(top)the b →X s γrate determines mainly the lower bound.

1234567

8500

1000

m e V ]

m 1/2

[G e V

]χ2

m 1/2 [G e V ]

m 0 [GeV]

12345678500

1000

m e V ]

m 1/2

[G e V

]χ2

m 1/2 [G e V ]

m 0 [GeV]

Figure 7:The total χ2and the allowed regions in the parameter space for μ>0and tan β=20(top)and 50(bottom),with A 0free,as in Fig.6(bottom).

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Although the data are consisten with a Higgs of115GeV,the deviation from the SM is of the order of3standard deviations and to be considered preliminary..Therefore we only consider the lower limit of113.5GeV at present.

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