Table Of ContentReport No: IFIC/08-53, FTUV-08-1024, LPT-08-81,
ANL-HEP-PR-08-64, LPTA/08-059
9
Radiative Υ decays and a light pseudoscalar Higgs in the NMSSM
0
0
2
n
Florian Domingoa, Ulrich Ellwangera, Esteban Fullanab, Cyril Hugoniec and
a
J Miguel-Angel Sanchis-Lozanod
5
]
h
p
-
p a Laboratoire de Physique Th´eorique1, Universit´e de Paris–Sud, F–91405 Orsay, France
e
h b High Energy Physics Division, Argonne National Laboratory, Argonne, IL 60439, USA
[ c LPTA2, Universit´e de Montpellier II, 34095 Montpellier, France
3 d Instituto de F´ısica Corpuscular (IFIC) and Departamento de F´ısica Teo´rica
v
Centro Mixto Universitat de Val`encia-CSIC, Dr. Moliner 50, E-46100 Burjassot, Valencia, Spain
6
3
7 Abstract
4
WestudypossibleeffectsofalightCP-oddHiggsbosononradiativeΥdecaysintheNext-to-
.
0 Minimal Supersymmetric Standard Model. Recent constraints from CLEO on radiative Υ(1S)
1
decays are translated into constraints on the parameter space of CP-odd Higgs boson masses
8
and couplings, and comparedto constraintsfrom B physics andthe muon anomalousmagnetic
0
: moment. Possible Higgs-ηb(nS) mixing effects are discussed, notably in the light of the recent
v
i measurement of the ηb(1S) mass by Babar: The somewhat large Υ(1S) - ηb(1S) hyperfine
X splitting could easily be explained by the presence of a CP-odd Higgs boson with a mass in
r the range 9.4 - 10.5 GeV. Then, tests of lepton universality in inclusive radiative Υ decays can
a
provide a visible signal in forthcoming experimental data.
1Unit´e mixtedeRecherche– CNRS – UMR 8627
2Unit´e mixtedeRecherche– CNRS – UMR 5207
1 Introduction
TheNext-to-Minimal SupersymmetricStandardModel(NMSSM)[1]providesthesimplestsolution
totheµproblemoftheMSSM[2]. ItsphenomenologycandifferinvariousrespectsfromtheMSSM.
Notably, as emphasized in [3], a light CP-odd scalar A can appear in the Higgs spectrum.
1
In the case where the CP-even Higgs boson H decays dominantly into a pair of CP-odd
scalars [3–10], LEP constraints on CP-even Higgs masses [11] are alleviated considerably [5–10].
For m < 10.5 GeV, where the A decay into BB¯ is forbidden, this scenario could even explain
A1 1
the2.3σ e∼xcess inthee+e− Z+2b channelforM 100 GeV[6](wherethetwobquarkswould
2b
result from a CP-even Higg→s H with M 100 GeV∼and a branching ratio H b¯b 0.08,
H
∼ B → ∼
but (H A A ) 0.9). Also at hadron colliders the search for CP-even scalars would be par-
1 1
B → ∼ (cid:0) (cid:1)
ticularly difficult [4,6,8–10] if they decay dominantly into A A with m below the BB¯ threshold.
1 1 A1
In this case, however, the A can have important effects on Υ decays [12–18]. Notably a Super
1
B factory can then play an important and complementary role [19] via its potential sensitivity to
Υ γA decays.
1
→
Whereas a light A is also possible in the MSSM with a CP-violating Higgs sector [20,21],
1
scenarios with more than one gauge singlet [22], little Higgs models and non-supersymmetric two
Higgsdoubletmodels(see[23]foranoverview), weconcentratesubsequentlyonthesimplestversion
of the NMSSM with a scale invariant superpotential.
Υ γA decays havebeeninvestigated intheNMSSMbeforein[16]form <9.2GeV,where
→ 1 A1
the signal relies on a narrow peak in the photon spectrum. Recent results from C∼LEO on radiative
Υ(1S) decays [24] (assuming a A width < 10 MeV) constrain this domain of m strongly. One
1 A1
of our aims is to translate the CLEO resul∼ts into constraints on the NMSSM parameter space (see
also [25]) as X , the (reduced) coupling of A to b quarks, and to compare them with constraints
d 1
from B physics [26–28] and the anomalous magnetic moment of the muon [25,29].
For m > 9 GeV, various corrections to the Υ γA decay rate become relatively large and
A1 → 1
uncertain [30∼], which makes it difficult to translate experimental constraints on the decay rate into
constraints on X . Consequently, this region of m is hardly constrained by CLEO results.
d A1
Very recently, a CP-odd state with a mass of about 9389 MeV has been observed in Υ(3S)
decays by BaBar [31], showing up as a peak with a significance of 10 standard deviations in the
photon energy spectrum. At first sight, this state can be interpreted as the long-awaited η (1S).
b
However, in the presence of a CP-odd Higgs with a mass in the same region, the observed mass has
to be interpreted as an eigenvalue of a 2 2 mixing matrix, and would differ correspondingly from
×
m , the mass of the η (1S) in the absence of a CP-odd Higgs (m2 is now one of the diagonal
ηb0 b ηb0
entries of the massmatrix) [12]. Inthis case, a second peak in thephoton spectrumshould possibly
bevisible; however, thesearchforsuchasecondpeakwouldrequireadedicatedconsiderationofthe
various background contributions (notably from the ISR and χ (2P)), which should be performed
bJ
in the future.
First, this mixing effect could explain the fact that the observed mass is somewhat lower than
expected, if m is somewhat above m . Second, the off-diagonal element of the mass matrix
A1 ηb0
can be estimated and turns out to be proportional to X [12,17]. Assuming a reasonable range for
d
m , the observed value of 9389 MeV for one of the eigenvalues implies an upper bound on X
ηb0 ∼ d
as a function of m , which is, however, particularly strong only for m 9389 MeV and will be
A1 A1 ∼
derived below.
At present, a direct detection of a CP-odd Higgs with a mass in the particularly interesting
region 9.2 GeV < m < 10.5 GeV via a peak in the photon spectrum seems to be quite difficult.
A1
Fortunately, an a∼lternati∼ve signal for an A state below the BB¯ threshold can be a breakdown of
1
1
lepton universality (LU) in Υ (γ)l+l− decays (via an intermediate A state), since A would
1 1
decay practically exclusively in→to τ+τ− [13,15,17]. Note that, to this end, the photon does not
have to be detected. Present tests of lepton universality in Υ(1S), Υ(2S) and Υ(3S) decays [32]
have error bars in the 5–10% range. Remarkably, however, a general trend (at the 1σ level) seems
to point towards a slight excess of the τ+τ− branching ratios, as expected in the presence of a A
1
state. Another aim of the present paper is to investigate corresponding sensitivities of forthcoming
experimental data, assuming a possible reduction of the errors to the 2% range.
The layout of this paper is as follows: In section 2 we review the domains of the NMSSM
parameter space which lead to a light CP-odd Higgs with strong couplings to down type quarks
(and leptons). In section 3 we derive constraints on the NMSSM parameter space from recent
CLEO results, using quite conservative estimates for the corrections to the Υ γA decay rate
1
→
which lead to quite conservative upper bounds on X as a function of m . (These upper bounds
d A1
on X will be included in future versions of the code NMSSMTools [33].) In section 4, we discuss
d
the mixing effects of A with η (nS) following [12,13,17]. In section 5, we derive constraints on
1 b
X from the measured η mass by BaBar and from (conservative) assumptions on m , and
d bobs ηb0
discuss quantitatively the possible mixing-induced shift of the measured η mass. In section 6
bobs
we compare these CLEO and BaBar constraints with constraints from LEP, B physics and the
muon anomalous magnetic moment. This analysis is performed with the help of the updated
NMSSMTools package. In section 7 we reconsider A masses between 9.2 and 10.5 GeV and show
1
that (for less conservative estimates of the corrections to the Υ γA decay rate) a breakdown of
1
lepton universality inΥ (γ)l+l− decays can becomean impor→tantobservablefor thedetection of
→
a CP-oddHiggs inthis mass range. We presentformulas for therelevant branchingratios including
possibleA η (nS)mixings,andstudyfuturesensitivitiesonX fromleptonuniversalitybreaking.
1 b d
−
Section 8 contains conclusions and an outlook.
2 A light CP-odd Higgs in the NMSSM
In this section we show that the parameter space of the NMSSM can accomodate a light CP-odd
Higgs, which is strongly coupled to down-quarks and leptons (see also [3,6,7]). We consider the
simplest version of the NMSSM with a scale invariant superpotential
1
W = λSH H + κS3+... (1)
u d
3
and associated soft trilinear couplings
1
V =(λA SH H + κA S3)+h.c.+... (2)
soft λ u d κ
3
in the conventions of [33]. A vev of the singlet field s S generates an effective µ-term, and it
≡ h i
is convenient to define also an effective B-term:
µ = λs, B = A +κs . (3)
eff eff λ
The Higgs sector of the NMSSM contains six independent parameters, which can be chosen as
λ, κ, A , A , tanβ, µ . (4)
λ κ eff
In the NMSSM, two physical pseudoscalar states appear in the spectrum, which are superposi-
tions of the MSSM-like state A (the remaining SU(2) doublet after omitting the Goldstone
MSSM
2
boson) and the singlet-like state A . In the basis (A , A ), the 2 2 mass square matrix for
S MSSM S
×
the CP-odd Higgs bosons has the following matrix elements [33]
2µ B
M2 = eff eff, M2 = λv(A 2κs)
11 sin2β 12 λ−
λ2v2sin2β
M2 = (A +4κs) 3κsA (5)
22 2µ λ − κ
eff
where v2 = 1/(2√2G ). The masses of the CP-odd eigenstates A are
F 1,2
1
m2 = [M2 +M2 ∆M2] (6)
A1,2 2 11 22∓
with ∆M2 = (M2 M2 )2+4(M2 )2.
11− 22 12
The lighter CP-odd state A can be decomposed into (A , A ) according to
p 1 MSSM S
A = cosθ A +sinθ A , (7)
1 A MSSM A S
where the mixing angle θ is
A
M2 M2
cos2θ = 22 − 11 . (8)
A ∆M2
To a good approximation (for moderate A , small A and large tanβ), the mass of the lightest
λ κ
CP-odd Higgs boson and cosθ can be written as [3,7]
A
3λ2v2A sin2β
m2 3κs λ A , (9)
A1 ≃ 2µ B 3κsA sin2β − κ
(cid:18) eff eff − κ (cid:19)
λv(A 2κs)sin2β
λ
cosθ − . (10)
A
≃ −2µ B +3κsA sin2β
eff eff κ
(The approximate equation for cosθ ceases to be valid if the second term in the denominator is
A
large compared to the first one.)
The reduced coupling X of the light physical A Higgs boson to down-type quarks and leptons
d 1
(normalized with respect to the coupling of the CP-even Higgs boson of the Standard Model) is
given by
X = cosθ tanβ . (11)
d A
Interesting phenomena in the Υ-system take place for large values of X , i.e. large values of
d
tanβ without cosθ being too small. (A possible enhancement of X [18] can occur due to the
A d
radiatively generated tanβ-enhanced Higgs-singlet Yukawa couplings [34]. However, in the case of
a sizable value of cosθ already at tree level as considered below, this effect is small.)
A
At first sight eq. (10) seems to imply (from sin2β 2/tanβ for large tanβ) that cosθ
A
∼
decreases indeed with tanβ – this would be the case in the PQ-symmetry-limit (κ 0) or
→
R-symmetry-limit (A , A 0), where the second term in the denominator of (10) tends to
κ λ
→
zero. On the other hand, it follows from the minimization equations of the scalar potential of the
NMSSM (as in the MSSM), for fixed soft Higgs mass terms and µ , that tanβ is proportional to
eff
1/µ B for large tanβ [35], hence large values of tanβ are associated to small values of B
eff eff eff
| | | |
(since µ > 100 GeV from the lower bound on chargino masses). It is useful to replace B by
eff eff
| | | |
the paramet∼er
2µ B
M2 M2 = eff eff , (12)
A ≡ 11 sin2β
3
whichsetsthescaleforthemassesofthecompleteSU(2)multipletofHiggsstatesincludingascalar,
a pseudoscalar and a charged Higgs as in the MSSM (in our case, the corresponding pseudoscalar
is the heavier one A ). In terms of M2, X can be written approximately as
2 A d
λv(A 2κs)
λ
X − tanβ , (13)
d ≃ −M2 +3κsA ×
A κ
and it is reasonable to examine the large tanβ region keeping M fixed.
A
It follows from eq. (9) that there exist always values of A of the same sign as A (typically
κ λ
bothnegative) wherem issmall[9], whilecosθ 0.1 0.6andhenceX isnotsuppressed. This
A1 A ≃ − d
requires a moderate fine-tuning of A (or M ); on the other hand the authors of Ref. [16] stress
κ A
that such values for A , which allow a light SM-like Higgs to decay into two A with m < 2m ,
κ 1 A1 b
correspond to the smallest degree of fine-tuning in the entire parameter space of the NMSSM. For
correspondingvalues of A , the denominator of X in (13) is dominated by M2. For M2 < κA s ,
κ d A A | κ |
even larger values of cosθ 0.6 1.0 are possible while m remains small. In this re∼gime, the
A ≃ − A1
approximations leading to eqs. (9), (10) and (13) are no longer valid, however.
To summarize, the following conditions can be fulfilled simultaneously in the NMSSM, which
yield possibly observable effects in Υ decays:
m < 10.5 GeV from, e.g., appropriate values of A ;
• A1 κ
∼
a large value of X , if tanβ is large while M in the denominator of (13) remains moderate.
d A
•
The numerical results in section 6 confirm the analytical estimates above.
3 Constraints from CLEO
Recently, the CLEO collaboration presented results on Higgs searches from Υ(1S) decays [24].
21.5 106 Υ(1S) decays had been collected and, for the Υ(1S) γ +(A τ+τ−) search, the
1
· → ±→ ±
photon energy spectrum in events with missing energy and one identified µ or e (allegedly from
τ eνν or τ µνν) had been examined. For the A µ+µ− search, both muons were identified.
1
→ → →
No narrow peaks (of a width below 10 MeV) in the photon energy spectrum are observed
(except for Υ(1S) γJ/Ψ γµ+µ−), w∼hich allows to place stringent upper limits between 10−4
and 10−5 on the br→anching →ratio (Υ(1S) γ(A τ+τ−/µ+µ−)) for m < 9.2 GeV [24].
B → 1 → A1
The (Υ(1S) γA ) is given by the Wilczek formula [36,37] ∼
1
B →
(Υ(1S) γA ) G m2X2 m2
B → 1 = F b d 1 A1 F (14)
(Υ(1S) µ+µ−) √2πα − m2 ×
B → (cid:18) Υ(1S)(cid:19)
where α denotes the fine structure constant and X is given in (11). F is a correction factor, which
d
includes three kinds of corrections to the leading-order Wilczek formula (the relevant formulas are
summarized in [30]): bound state, QCD and relativistic corrections. Bound state effects have a
quite different behaviour for a scalar or a pseudoscalar Higgs, increasing the ratio (14) by 20%
∼
in the latter case [38–40]. QCD corrections reduce the ratio (14) by a similar amount [41,42].
Relativistic corrections can generate an important reduction, and were calculated in [43].
These relativistic corrections depend quite strongly on the b quark mass m , and become unre-
b
liable at least for Higgs masses m above 8 GeV [43] where they can generate a vanishing (or even
A1
negative) correction factor F. Frequently, the approximation F 0.5 for all m is employed in
∼ A1
the literature [26,37]. However, in order to derive conservative bounds on the NMSSM parameters
4
0.7
0.6
0.5
0.4
0.3
F
0.2
0.1
0
-0.1
-0.2
0 1 2 3 4 5 6 7 8 9 10
m [GeV]
A
1
Figure 1: F(m )includingtheboundstate and QCDcorrections, anda naive extrapolation of the
A1
relativistic corrections computed for m m at larger values of m . Black curve: assuming
A1 ≪ Υ A1
m = 4.9 GeV (as used later), green curve: assuming m = 5.3 GeV.
b b
from CLEO results, we use in this section the smaller values of F(m ) for larger m , which are
A1 A1
obtained by a naive extrapolation of the relativistic corrections [43]. Using the quark model value
m = 4.9 GeV, the resulting behaviour of F(m ) (including also the bound state and QCD cor-
b A1
rections) is shown in Fig. 1, according to which F vanishes (and even becomes negative, in which
case we take F = 0) for m > 8.8 GeV. Correspondingly, the CLEO bounds on the NMSSM
A1
parameters disappear for m >∼ 8.8 GeV. (For larger values of m as 5.3 GeV, F would vanish
A1 b
only for m 9.4 GeV m ∼as also indicated in Fig. 1.)
A1 ∼ ∼ Υ
Next, in order to translate the CLEO bounds into bounds on X (m ) using eq. (14), the
d A1
branching ratios (A τ+τ−/µ+µ−) have to be known, which depend essentially on tanβ. For
1
m above 2m ,B (A→ τ+τ−) varies from 70% for tanβ = 1.5 to 95% for tanβ = 50,
whAe1reas (A τ µB+µ−1)→is always below 10% ev∼en for m below 2m (whi∼ch implies to reconsider
B 1 → A1 τ
the estimates of the CLEO reach in [44]). Using the code NMSSMTools [33] for the determination
of the (A τ+τ−/µ+µ−) and an interpolation of the CLEO bounds [24], we show our resulting
1
B →
upper limits on X as a function of m for two extreme values of tanβ = 1.5 and 50 in Fig. 2.
d A1
Actually, in Ref. [24] the total decay width of A , Γ , is assumed to be below 10 MeV.
1 A1
Although we do not believe that the CLEO bounds disappear completely in the case where Γ
A1
(which increases with X and m ) is larger than 10 MeV, we indicate in Fig. 2 also the region at
d A1
large X and m > 3.5 GeV where Γ exceeds 10 MeV (depending also slightly on tanβ). In
d A1 A1
the updated version∼2.1 of the NMSSMTools package [33] these bounds are included.
5
100
10
X
d
1
0.1
1 2 3 4 5 6 7 8 9 10
m [GeV]
A
1
Figure 2: Upper bounds on X as a function of m for two extreme values of tanβ = 1.5 (red
d A1
curve) and tanβ = 50 (black curve) using results from CLEO [24]. We also indicate as dashed
lines the region at large X and m > 3.5 GeV where Γ exceeds 10 MeV (same colour code for
d A1 A1
tanβ = 1.5 and 50).
4 Mixing of A with the η (nS) resonances
1 b
In the presence of a pseudoscalar Higgs boson with a mass close to one of the different η (nS)
b
resonances, a significant mixing between these states can occur [12].
The mixing between a CP-odd Higgs and a single η (nS) (n = 1,2 or 3) resonance can be
b
described by the introduction of off-diagonal elements denoted by δm2 in the mass matrix [12,17]
n
′ ′
(here and below we neglect possible induced η (nS) η (nS) mixings for n = n)
b b
− 6
m2 im Γ δm2
2 = A10 − A10 A10 n (15)
Mn (cid:18) δm2n m2ηb0(nS)−imηb0(nS)Γηb0(nS) (cid:19)
where the subindex ’0’ indicates unmixed states: m and m (Γ and Γ ) denote the
A10 ηb0(nS) A10 ηb0(nS)
masses (widths) of the pseudoscalar Higgs boson and η (nS) states, respectively, before mixing.
b0
In Ref. [17] only the mixing of the Higgs with the η (1S) resonance was taken into account. In
b0
this paper, we extend the analysis by considering the possible mixing between the Higgs and any
of the three η (nS) (n = 1,2 or 3) states. Thus three mixing angles have to be defined; however,
b0
only the contribution from the closest η (nS) state to the hypothetical A mass will be assumed to
b 1
be significant for the mixing, i.e. only one among the three mixing angles will deviate significantly
from zero. The generally complex mixing angle α between the pseudoscalar Higgs A and an
n 10
η (nS) state is given by [17]
b0
sin2α = δm2/∆2 (16)
n n n
where
∆2 = [D2 +(δm2)2]1/2 (17)
n n n
6
with
D = (m2 m2 im Γ +im Γ )/2 . (18)
n A10 − ηb0(nS)− A10 A10 ηb0(nS) ηb0(nS)
The off-diagonal element δm2 can be computed within the framework of a non-relativistic quark
n
potential model as
3m3 1/2
δm2 = ηb(nS) R (0) X . (19)
n 8πv2 | ηb(nS) |× d
(cid:18) (cid:19)
In anon-relativistic approximation to the bottomonium boundstates, the radialwave functions
at the origin can be considered as identical for vector and pseudocalar states, i.e. R (0)
Υ(nS)
R (0), and can therefore be determined from the measured Υ e+e− decay widths: ≃
ηb(nS) →
9m2 16α (m2)
R (0) 2 Γ[Υ(nS) e+e−] Υ(nS) 1+ s Υ (20)
| Υ(nS) | ≃ → × 4α2 3π
(cid:20) (cid:21)
Substituting recent values for the dielectron widths from [32] we obtain R (0)2 = 6.60 GeV3,
| ηb(1S) |
R (0)2 = 3.02 GeV3 and R (0)2 = 2.18 GeV3, leading to3
| ηb(2S) | | ηb(3S) |
δm2 = 0.14 GeV2 X , δm2 = 0.11 GeV2 X , δm2 = 0.10 GeV2 X . (21)
1 × d 2 × d 3 × d
The A and η (nS) physical (mixed) states can be written as
1 b
A = cosα A + sinα η (nS) ,
1 n 10 n b0
η (nS) = cosα η (nS) sinα A (22)
b n b0 n 10
−
assumingcos2α +sin2α 1,i.e. neglectingtheimaginarycomponentsofα . (Hereandbelowwe
n n n
≃
usethenotation A andη (nS)forthemixedstatesinordertoindicatetheirdominantcomponents
1 b
for small mixing angles. Clearly, for α 90o, their dominant components would be reversed.)
n
∼
The full widths Γ and Γ of the A and η (nS) physical states can be expressed in terms
A1 ηb(nS) 1 b
of the widths of the unmixed states according to [17]
Γ cos2α Γ + sin2α Γ ,
A1 ≃ n A10 n ηb0(nS)
Γ cos2α Γ + sin2α Γ . (23)
ηb ≃ n ηb0(nS) n A10
Finally, let us recall that the mixing of the A with η states should lead to mass shifts which
10 b0
can be sizable [12,17]. These mass shifts might have spectroscopic consequences concerning the
hyperfineη (nS) Υ(nS)splitting [12,17,46] whosepredictions within theSM are reviewed in [47],
b
−
and with respect to which the BaBar result [31] on the η (1S) Υ(1S) hyperfine splitting – in the
b
−
absence of a light CP-odd Higgs – would be somewhat large (see the next section).
5 Upper bounds on X from the measured η mass, and the
d b
mixing-induced η mass shift
b
The observation of an η -like state with a mass of 9.389 GeV by BaBar [31], allows to obtain
b
≃
upperlimits onthe reducedcouplingX as afunction of thelightest CP-oddHiggs mass parameter
d
m , if m is near 9.39 GeV. This follows from the fact that the measured mass squared has
A10 A10
3Similar values can beobtained from a Buchmuller-Tye potential [45].
7
now to be considered as (the real part of) the eigenvalue of the matrix 2 (15), corresponding
M1
algebraic relations and an estimate of hadronic parameters as m .
ηb0(1S)
Subsequently we denote the “η ” mass as measured by BaBar by m , and the state η (1S)
b obs b0
by η . The observed state has now to be considered as a superposition of A and η . Then the
b0 10 b0
following algebraic identity holds (where δm2 is the off-diagonal element of the matrix 2 (15)):
1 M1
γ2
δm2 2 = ∆ ∆ 1+ (24)
1 A η" (∆A+∆η)2#
(cid:0) (cid:1)
where
∆ = m2 m2 , ∆ = m2 m2 (25)
A A10 − obs η ηb0 − obs
and
γ = m Γ m Γ . (26)
A10 A10 − ηb0 ηb0
Note that ∆ and ∆ must have the same sign, which follows already from properties of
A η
eigenvalues of real 2 2 matrices.
×
Now, if we use estimates for the parameters m and γ, eq. (24) allows to obtain an upper
ηb0
bound on X as a function of m . First, for γ we can assume γ . m 20 MeV (from Γ ,
d A10 | | obs × A10
Γ . 20 MeV) with the result that the term γ2 in (24) is relevant only for m very close to
ηb0 ∼ A10
m .
obs
For (m , m ) m (but (m , m ) m larger than a few MeV such that the term
A10 ηb0 ∼ obs A10 ηb0 − obs
γ2 can be neglected), eq. (24) can be simplified further with the result
∼
δm2 2 4m2 (m m )(m m ) (27)
1 ≃ obs A10 − obs ηb0 − obs
and hence, from (21), (cid:0) (cid:1)
X 125 (m m )(m m ) GeV−1 . (28)
d ≃ A10 − obs ηb0 − obs
q
To proceed further, we have to estimate m . Most of previous estimates for m correspond
ηb0 ηb0
actually to m m > 0 [47,48], but subsequently we allow for
ηb0 − obs
m m = 30 ... +40 MeV . (29)
ηb0 − obs −
Next we have to treat the cases m m > 0 and m m < 0 separately. Starting with
A10− obs A10− obs
m m < 0, the maximally possible value for X from (28) is assumed for the lowest estimate
A10− obs d
of m , with the result
ηb0
Xmax(m ) 22 m m GeV−1/2 . (30)
d A10 ∼ obs − A10
For m m > 0, on the other hand, thepmaximally possible value for X is assumed for the
A10 − obs d
largest estimate of m , with the result
ηb0
Xmax(m ) 25 m m GeV−1/2 . (31)
d A10 ∼ A10 − obs
These analytic expressions for Xmax(m p) are fairly good approximations to the numerical
d A10
upper bounds on X (m ) which can be derived from (24) without the approximation (27), apart
d A10
from the region where m m is less than about 0.5 MeV (where (30) and (31) would
| A10 − obs|
imply Xmax(m ) 0). In fact, with γ m 20 MeV, one obtains Xmax(m ) 0.6 for
d A10 → | | ∼ obs × d A10 ∼
m m < 0.5 MeV (see Fig. 3 below).
| A10 − obs|
∼
8
We emphasize, however, that most previous estimates for m correspond to m m > 0
ηb0 ηb0 − obs
[47,48] in contrast to our more conservative assumption (29). These estimates can still be correct
within the present framework, if an additional A state with m m > 0 exists, which mixes
10 A10 − obs
strongly with the η , reducing the lower eigenvalue of the mass matrix (15). The induced mass
b0
shift m m can easily be derived from eq. (28):
ηb0 − obs
X2 1 GeV2
m m d × (32)
ηb0 − obs ≃ 1.56 104 (m m )
· · A10 − obs
which, for m m > 0, would unwittingly be interpreted as an excess of the “observed”
ηb0 − obs
hyperfine splitting m m . For instance, an induced mass shift of m m 20 MeV
Υ(1S) − obs ηb0 − obs ∼
would be generated by a CP-odd Higgs with mass m and a reduced coupling X satisfying
A10 d
X 17.7 √m m GeV−1/2 as, e.g., X 12 for m 9.85 GeV.
d ≃ × A10 − obs d ≃ A10 ≃
Note, however, that this mecanism would imply a heavier mass eigenvalue of the mixing matrix
(15)isnottoofarabovem . Thisfavoursaheavymasseigenvaluebelow10.5GeV,whichnotonly
obs
satisfies LEP constraints but could even, as mentionned in the introduction, explain an observed
excess of events at LEP.
6 Comparison of constraints from CLEO, BaBar, B physics and
the muon anomalous magnetic moment
In addition to the constraints obtained in section 3 from CLEO and in section 5 from BaBar,
the (m ,X )-plane is already constrained by processes from B physics [26,27] and the muon
A1 d
anomalous magnetic moment (g 2) [29]. (Upper limits on X have been derived by OPAL [49]
µ d
−
from Yukawa production of a light neutral Higgs Boson at LEP, which seem more restrictive than
the constraints from CLEO for m > 9.2 GeV. We believe, however, that the η (nS) A mixing,
A1 b − 1
which is relevant here, depends on an∼additional b-b-η form factor, where the initial b-quark is far
b
off-shell. Since this effect has not been considered in [49], we will not consider the corresponding
limits below.)
In the following, we will compare the different constraints in the (X ,m )- and (X ,M )-
d A1 d A
planes. (In this section, m is the CP-odd Higgs mass parameter denoted as m in the mass
A1 A10
matrix (15). However, the difference between m and m would hardly be visible in the Figures
A10 A1
below.)
For this purpose we have performed a scan over the NMSSM parameter space using the
NMHDECAY program from the NMSSMTools package [33]. NMHDECAY allows to verify si-
multaneously the phenomenological constraints from SUSY searches, Higgs searches, B physics
and (g 2) . We have varied the NMSSM parameters (4) λ, κ, A , A , µ , tanβ (the latter
µ λ κ eff
−
between 1 and 50), as well as the SUSY breaking gaugino, squark and slepton masses and trilinear
couplings, keeping only points wherem < 10.5 GeV. Then we identified regions in the parameter
A1
space which are ruled out by the various phenomenological constraints for any choice of parame-
ters. In particular, LEP constraints from Higgs searches require tanβ > 1.5 in the NMSSM, while
constraints from (g 2) lead to tanβ > 2 for m < 10.5 GeV. ∼
− µ A1
The various curves in the (X ,m )∼-plane in Fig. 3 indicate lower bounds on X from various
d A1 d
phenomenological constraints. We found that even for very large X there always exist parameter
d
choices such that no region is always excluded by either the constraints from (B X γ) or
s
B¯+ τ+ν . However, constraints from (B µ+µ−) and ∆M , q = d,s (sBhown→as a green
τ s q
B → B →
dashed line) always exclude a funnel for m M 5.3 GeV, the width of which depends on
(cid:0) (cid:1) A1 ∼ Bq ∼
9
