New data on the anomalous magnetic moment a_mu of the muon together with the
b->s gamma decay rate are considered within the supergravity inspired
constrained minimal supersymmetric model. We perform a global statistical chi^2
analysis 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_mu. The newest b->s gamma
data deviate 1.7 sigma from recent SM calculations and prefer a similar
parameter region as the 2.6 sigma deviation from a_mu.Comment: 12 pages, 7 figs. Refs. update

Abbiendi

Abreu

Acciarri

Ahmed

Arnowitt

Baer

Baltz

Barate

Blazek

Brown

C. Sander

Chattopadhyay

Choi

Czarnecki

D.I. Kazakov

de Boer

Degrassi

Ellis

Everett

Feng

Gambino

Haber

Ibrahim

Igo-Kemenes

Komine

M. Huber

Misiak

W. de Boer

Yndurain

Physics Letters B

English

arXiv

AbstractNew data on the anomalous magnetic moment of the muon together with the b→Xsγ decay rate are considered within the supergravity inspired constrained minimal supersymmetric model. We perform a global statistical χ2 analysis 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 aμ. The newest b→Xsγ data deviate 1.7σ from recent SM calculations and prefer a similar parameter region as the 2.6σ deviation from aμ

de Boer, W.

Huber, M.

Sander, C.

Kazakov, D.I.

Elsevier - Publisher Connector 

30 August 2001
Physics Letters B 515 (2001) 283–290
www.elsevier.com/locate/npe
A global fit to the anomalous magnetic moment, b→Xsγ and
Higgs limits in the constrained MSSM
W. de Boer a, M. Huber a, C. Sander a, D.I. Kazakov b
a Institut für Experimentelle Kernphysik, University of Karlsruhe, Postfach 6980, D-76128 Karlsruhe, Germany
b Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141 980 Dubna, Moscow Region, Russia
Received 2 July 2001; received in revised form 13 July 2001; accepted 13 July 2001
Editor: G. Rolandi
Abstract
New data on the anomalous magnetic moment of the muon together with the b→ Xsγ decay rate are considered within
the supergravity inspired constrained minimal supersymmetric model. We perform a global statistical χ2 analysis 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 aµ. The newest b→Xsγ data deviate 1.7σ from recent SM cal-
culations and prefer a similar parameter region as the 2.6σ deviation from aµ.
 2001 Elsevier Science B.V.
1. Introduction
Recently a new measurement of the anomalous
magnetic moment of the muon became available,
which suggests a possible 2.6 standard deviation from
the Standard Model (SM) expectation [1]: 	aµ =
a
exp
µ − athµ = (43± 16)× 10−10. The theoretical pre-
diction depends on the uncertainties in the vacuum po-
larization and the light-by-light scattering, see, e.g.,
the discussion in [2]. However, even with a conserva-
tive estimate of the theoretical errors, one has a pos-
itive difference 	aµ of the order of the weak con-
tribution to the anomalous magnetic moment, which
opens a window for “new physics”. The most popular
explanation is given in the framework of SUSY the-
ories [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 ex-
plain the desired difference	aµ. It requires the Higgs
E-mail address: wim.de.boer@cern.ch (W. de Boer).
mixing parameter to be positive [4] and the sparticles
contributing to the chargino–sneutrino (χ˜±–ν˜µ) and
neutralino–smuon (χ˜0–µ˜) loop diagrams to be rela-
tively light [3].
The positive sign of µ0 is also preferred by the
branching ratio of the b-quark decaying radiatively
into an s-quark — b→ Xsγ — [13]. Last year the
observed value of b → Xsγ was 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→Xsγ .
However, it was recently suggested that in the
theoretical calculation one should use the running
c-quark mass in the ratio mc/mb, which reduces
this ratio from 0.29 to 0.22 [14]. The SM value
for b → Xsγ increases from (3.35 ± 0.30) × 10−4
to (3.73 ± 0.30) × 10−4 in this case. This value is
1.7σ above 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
0370-2693/01  2001 Elsevier Science B.V.
PII: S0370-2693(01) 00 89 5- 4
Open access under CC BY license. 
Open access under CC BY license. 
284 W. de Boer et al. / Physics Letters B 515 (2001) 283–290
((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 aµ and b→ Xsγ require now very similar
mass spectra for the sparticles.
In the Constrained Minimal Supersymmetric Model
(CMSSM) with supergravity mediated breaking terms
all sparticle masses are related by the usually assumed
GUT scale boundary conditions of a common mass m0
for the squarks and sleptons and a common mass m1/2
for the gauginos. The region of overlap in the GUT
scale parameter space, where both aµ and b→ Xsγ
are within errors consistent with the data, is most
easily determined by a global statistical analysis, in
which the GUT scale parameters are constrained to the
low energy data by a χ2 minimization.
In this Letter 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 [18] 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 full 1-loop contribution of all particles and
sparticles. For details we refer to previous publications
[19,20].
In principle, one can also require b–τ Yukawa
coupling unification, 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 = 〈H 02 〉/〈H 01 〉 [19,20]. From Fig. 1 one
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 for
µ < 0 at tanβ ≈ 40, while for µ > 0 they never
become equal, although the difference between the
Yukawa couplings is less than a factor three. Since
µ > 0 is required by 	aµ > 0 (see below), we do
not insist on Yukawa coupling unification and consider
tanβ to be a free parameter, except for the fact that
the present Higgs limit of 113.5 GeV from LEP [21]
requires tanβ > 4.3 in the CMSSM [13].
We found that the allowed area of overlap between
b→ Xsγ and aµ can be increased considerably for
positive values of the common trilinear coupling A0
Fig. 1. The dependence of the third generation Yukawa couplings at
the GUT scale as function of tanβ for µ0 > 0 and µ0 < 0, obtained
by fitting them to the low energy masses of the top, bottom and tau
mass. The results are for a common mass m0 = m1/2 = 500 GeV,
but for different masses the curves look very similar, except that
the ‘triple’ unification point for µ0 < 0 shifts between 42 and 48,
if the common mass is shifted from 200 to 1000 GeV. Clearly, for
mu0 > 0 no b–τ Yukawa unification can be obtained within this
CMSSM model.
at the GUT scale. For A0 > 0 the present Higgs limit
becomes more stringent than for the no-scale models
with A0 = 0, as will be shown.
2. aµ and b→Xsγ in the CMSSM
The contributions to the anomalous magnetic mo-
ment 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 contri-
W. de Boer et al. / Physics Letters B 515 (2001) 283–290 285
bution to aµ can be approximated by [3]
∣∣aSUSYµ ∣∣
 α(MZ)8π sin2 θW
m2µ
m2SUSY
× tanβ
(
1− 4α
π
ln
mSUSY
mµ
)
(1)
 140× 10−11
(
100 GeV
mSUSY
)2
tanβ,
where mµ is the muon mass, mSUSY is an average
mass of supersymmetric particles in the loop (essen-
tially the chargino mass). In our calculations we use
the complete one-loop SUSY contributions from [4]
with zero phase factors and the additional logarithmic
suppression factor as in Eq. (1). The calculated value
of aµ is shown in Fig. 2 as function of tanβ . Clearly, it
is approximately proportional to tanβ and its sign de-
pends on the sign of µ0. 1 Only positive values of µ0
are allowed for the positive deviation from the SM and
in addition the sparticles have to be rather light. How-
Fig. 2. The dependence of aSUSYµ versus tanβ for various values of
the SUSY breaking parameters m0 and m1/2. The horizontal band
shows the discrepancy between the experimental data and the SM
estimate. Good agreement with the data is only achieved at large
tanβ and for light sparticles. Clearly, the fit allows only the positive
sign of µ.
1 Our sign conventions are as in Ref. [22].
ever, light sparticles contribute also substantially to the
b→Xsγ decay rate. In the past this posed a conflict.
However, if one uses in the b→Xsγ calculations the
running mass for the charm quark, as suggested re-
cently by Gambino and Misiak, the SM prediction is
increased by 11%. In this case the newest world aver-
age on b→Xsγ is 1.7σ below the SM, as mentioned
in the introduction. Such a deviation is most easily
obtained for large tanβ and not too heavy sparticles,
as shown in Fig. 3. In the upper part the scale uncer-
tainty of the low energy scale µb is displayed by the
width of the theoretical curves, while in the lower part
the dependence on the trilinear coupling A0 is shown.
The scale µb was varied between 0.5mb and 2mb . For
tanβ ≈ 40 only positive values of the Higgs mixing
parameter at the GUT scale µ0 are allowed in agree-
ment with the preferred sign of µ0 by the anomalous
magnetic moment. For intermediate sparticle masses
and µ0 > 0 large values of A0 and small values of
the low energy scale (µb ≈ 0.5mb) bring the calcu-
lated values of b→ Xsγ closest 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 ef-
fect 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. 4 shows the values of b → Xsγ and aSUSYµ
as function of m0 and m1/2 for tanβ = 35. For
b → Xsγ the ratio mc(µ)/mpoleb = 0.22 was used,
while for the NLO QCD contributions the formulae
from Ref. [23] were used. The calculated values
have to be compared with the experimental values
BR(b→ Xsγ ) = (2.96 ± 0.46) × 10−4 [15–17] and
	aµ = (43 ± 16) × 10−10 [1], which shows once
more that b→Xsγ and aSUSYµ prefer a relatively light
supersymmetric spectrum.
To find out the allowed regions in the parameter
space of the CMSSM, we fitted both the b→ Xsγ
and aµ data simultaneously. The fit includes the fol-
lowing constraints: (i) the unification of the gauge
couplings, (ii) radiative electroweak symmetry break-
ing, (iii) the masses of the third generation particles,
(iv) b → Xsγ and 	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–τ unification, since it
prefers µ0 < 0, as shown in Fig. 1, while 	aµ re-
quires µ0 > 0, as shown in Fig. 2. Yukawa unification
286 W. de Boer et al. / Physics Letters B 515 (2001) 283–290
Fig. 3. The upper picture shows the dependence of the b→Xsγ rate on tanβ for A0 = 0 and m0 = 600 (1000) GeV, m1/2 = 400 (1000) GeV
at the left (right). For each value of tanβ a fit was made to bring the predicted b→ Xsγ rate (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 between 0.5mb
and 2mb . The bottom picture shows the same dependence but for a fixed renormalization scale of 1mb . The width of the band is given by the
variation of A0 between −3m0 and 3m0.
for µ0 > 0 can only be obtained by relaxed unification
of the gauge couplings and nonuniversality of the soft
terms in the Higgs sector [24].
The χ2 contributions of b→ Xsγ and the anom-
alous magnetic moment aµ in the global fit are shown
in Fig. 5 for A0 = 0 and tanβ = 35. As expected, the
χ2 contribution from b→ Xsγ is smallest for heavy
sparticles, if b→ Xsγ is calculated with mc/mb =
0.29, while the minimum χ2 is obtained for interme-
diate sparticles, if mc/mb = 0.22 is used. With the
newly calculated b→ Xsγ values, one can see, that
b→ Xsγ and aµ prefer a similar region of the m0,
m1/2 plane. Fig. 6 shows the combined χ2 contri-
butions from b → Xsγ and aSUSYµ in the m0, m1/2
plane, both in 3D and 2D, for A0 = 0 (top) and A0
free (bottom). In the latter case the lower 2σ con-
tour from b→ Xsγ moves to the lower left corner,
but for the preferred value A0 ≈ 3m0, which is the
W. de Boer et al. / Physics Letters B 515 (2001) 283–290 287
Fig. 4. The values of b→ Xsγ and aSUSYµ in the m0, m1/2 plane for positive µ and tanβ = 35 to be compared with experimental data
b→ Xsγ = (2.96 ± 0.46) × 10−4 and aSUSYµ = (43 ± 16) × 10−10. One can see that both b→ Xsγ and aSUSYµ prefer relatively light
sparticles.
Fig. 5. The individual contributions to χ2 from b→Xsγ and aµ in the m0, m1/2 plane for tanβ = 35, µ> 0 and A0 = 0. On the left handside
we show the old contribution from b→Xsγ , as calculated with mc/mb = 0.29, which has the lowest χ2 for heavy supersymmetric particles.
In the middle the contribution from b→ Xsγ for mc/mb = 0.22 is shown, which now has a minimum for intermediate masses. The χ2
contribution from aµ is shown on the right handside, which clearly prefers light sparticles.
maximum allowed value in the fit in order to avoid
negative stop- or Higgs masses and colour break-
ing minima, the Higgs bound moves up consider-
ably. The total allowed region is similar in both cases,
as shown by the light shaded areas in the contour
plots. The 2σ contours from the individual contribu-
tions are in good agreement with previous calcula-
tions [6,9], but in these paper a simple scan over the
parameter space was performed without calculating
the combined probability. In addition, A0 = 0 was as-
sumed.
We repeated the fits for tanβ = 20 and 50, as
shown in Fig. 7. For smaller values of tanβ the al-
lowed region decreases, since aµ becomes too small.
At larger tanβ values the region allowed by aµ and
b → Xsγ increases towards heavier sparticles, as
288 W. de Boer et al. / Physics Letters B 515 (2001) 283–290
Fig. 6. The upper part shows the χ2 contribution (left) and its projection (right) in the m0, m1/2 plane for A0 = 0 and tanβ = 35. The light
shaded area is the region, where the combined χ2 is 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 = 4 for that particular contribution. The lower row shows the same for the fit, where A0 was
left free, in which case A0 ≈ 3m0 (its maximum allowed value in our fit) 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 A0 as a free parameter
the Higgs limit becomes the most important lower bound on the SUSY sparticles, while for the no-scale models with A0 = 0 (top) the b→Xsγ
rate determines mainly the lower bound.
expected from Eq. (1), but it is cut by the region
where the charged stau lepton becomes the Light-
est Supersymmetric Particle (LSP), which is assumed
to be stable and should be neutral. A charged sta-
ble LSP would have been observed by its electro-
magnetic interactions after being produced in the be-
ginning 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 mix-
ing term between the left- and right-handed staus at
larger tanβ .
We conclude that the aµ measurement strongly re-
stricts the allowed region of the parameter space in the
CMSSM, since it excludes the µ0 < solution, which
was the preferred one from b–τ Yukawa unification.
In addition, it prefers large tanβ with relatively light
W. de Boer et al. / Physics Letters B 515 (2001) 283–290 289
Fig. 7. The total χ2 and the allowed regions in the parameter space for µ> 0 and tanβ = 20 (top) and 50 (bottom), with A0 free, as in Fig. 6
(bottom).
sparticles, if the present deviation from the SM of 2.6σ
persists.
At large tanβ a global fit including both b→ Xsγ
and aµ as well as the present Higgs limit of 113.5 GeV
leaves a quite large region in the CMSSM parameter
space. Here we left the trilinear coupling to be a
free parameter, which affects both the Higgs limit
constraint and the b→Xsγ constraint, but in opposite
ways, so that the preferred region is similar for the no-
scale models with A0 = 0 and models which leave A0
free.
The 95% lower limit on m1/2 is 300 GeV (see
Figs. 6, 7), which implies that the lightest chargino
(neutralino) is above 240(120) GeV. The 95% upper
limit on m1/2 is determined by the lower limit on
aSUSYµ and therefor depends on tanβ (see Fig. 2).
For tanβ = 35(50) one finds m1/2  610(720) GeV,
which implies that the lightest chargino is below
500(590) GeV and the lightest neutralino is below
260(310) GeV.
Acknowledgements
D.K. would like to thank the Heisenberg–Landau
Programme, RFBR grant # 99-02-16650 and DFG
grant # 436/RUS/113/626 for financial support and the
Karlsruhe University for hospitality during completion
of this work.
290 W. de Boer et al. / Physics Letters B 515 (2001) 283–290
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