Preprint typeset in JHEP style. - HYPER VERSION

hep-ph/yymmnnn

Can a light Higgs boson explain the recent BNL measurement of the muon anomalous magnetic moment ?

Athanasios Dedes

Physikalisches Institut der Universität Bonn,

Nußallee 12, D-53115 Bonn, Germany

  E-mail: dedes@th.physik.uni-bonn.de

Howard E. Haber

Santa Cruz Institute for Particle Physics

University of California, Santa Cruz, CA, 95064 USA

  E-mail: haber@scipp.ucsc.edu

Abstract: A light CP-even Higgs boson with m_{h} ∼ 10 GeV could explain the recent BNL measurement of the muon g − 2 anomalous magnetic moment, in the framework of a general two-Higgs-doublet extension of the Standard Model. However, the allowed Higgs mass window is quite small and the corresponding model parameters are very constrained. The Higgs sector can contribute significantly to the observed BNL result of g − 2 only if the hZZ coupling (approximately) vanishes and M_{Υ} < ∼ m_{h} < ∼ 2m_{B}.

Keywords: Beyond Standard Model.

Contents

1. Introduction 1

2. Higgs boson corrections to the muon anomalous magnetic moment 4 2.1 Non-decoupling limit: sin(β − α) = 0 5 2.2 Decoupling Limit: cos(β − α) = 0 6

3. CUSB and LEP I constraints on the light Higgs boson 8

4. Final Results and Conclusions 9

1. Introduction

A new experimental value of the muon anomalous magnetic moment, a_{μ} = \frac{1}{2}(g −2)_{μ}, measured at BNL, was recently reported in ref. [1]. Comparing the measured value to its predicted value in the Standard Model, ref. [1] reported that

a_{μ}^{exp} − a_{μ}^{SM} = 426 ± 165 × 10^{−11} ,  (1.1)

Ref. [2] has reviewed the Standard Model computation of a_{μ} and concludes that if deviation of eq. (1.1) can be attributed to new physics [δa_{μ}^{NP}], then at 90% CL, δa_{μ}^{NP} must lie in the range

215 × 10^{−11} < ∼ δa_{μ}^{NP} < ∼ 637 × 10^{−11} .  (1.2)

This contribution is positive, and is of the order of the electroweak corrections to a_{μ}. More precisely, the “new Physics” needed, has to be of the order of G_{F} m_{μ}^{2}/(4π^{2}\sqrt{2}), where G_{F} = 1.16637(1)×10^{−5} GeV^{−2} is Fermi’s constant. In this paper, we consider the possibility that δa_{μ}^{NP} arises entirely from the Higgs sector. In the SM, the Higgs boson contribution to a_{μ} is further suppressed (relative to the main electroweak contribution) by a factor of m_{μ}^{2}/m_{h}^{2}. In light of the recent SM Higgs mass limit, m_{h} > ∼ 113.5 GeV obtained at the LEP collider [3], the SM Higgs contribution to a_{μ} is clearly negligible.

However, the Higgs sector contribution to a_{μ} could be considerably enhanced in a two-Higgs-doublet extension of the Standard Model (2HDM). This enhancement can arise from two different effects. First, an enhanced hμ^{+}μ^{−} coupling proportional to the ratio of Higgs vacuum expectation values, tan β, yields a Higgs contribution to δa_{μ}^{NP} proportional to tan β^{2}. Second, a suppressed hZZ coupling, proportional to sin(β − α) [using notation reviewed below], can permit the existence of a CP-even Higgs boson mass substantially below the present SM Higgs mass limit. In units of G_{F} m_{μ}^{2}/(4π^{2}\sqrt{2}), the overall enhancement is of order

\frac{m_{μ}^{2}}{m_{h}^{2}} × tan^{2} β × F (\frac{m_{μ}^{2}}{m_{h}^{2}} ) ≃ 1—10 ,  (1.3)

where F (x) is a loop factor which involves logarithms of the form ln(m_{h}^{2}/m_{μ}^{2}) ∼ O(10). A light CP-even Higgs boson with a mass m_{h} ≃ 10 GeV and 30 < ∼ tan β < ∼ 44, predicts a muon anomalous magnetic moment in agreement with the current BNL experimental measurement.

The required 2HDM is not compatable with the Higgs sector of the minimal supersymmetric extension of the Standard Model (MSSM). The reason is that one cannot have a very light h with suppressed hZZ couplings without an observable rate for Z → hA, in conflict with LEP data [4].^{1} Moreover, the MSSM provides additional mechanisms for generating significant contributions to δa_{μ}^{NP}. A number of recent papers [5, 6, 7, 8, 9] have shown that the recent BNL measurement is compatable with supersymmetric contributions to δa_{μ}^{NP} involving chargino and neutralino exchange, over an interesting region of MSSM parameter space.

In this paper, we focus on the possibility that the new physics contribution to a_{μ} arises solely from the Higgs sector. The two-doublet Higgs sector [10] contains eight scalar degrees of freedom. It is convenient to distinguish between the two-doublets by employing one complex Y = −1 doublet, Φ_{d}= (Φ_{d}^{0}, Φ_{d}^{−}) and one complex Y = +1 doublet, Φ_{u}= (Φ_{u}^{+}, Φ_{u}^{0}). To avoid Higgs-mediated flavor changing neutral currents, we do not allow the most general Higgs-fermion interaction. Instead, we impose discrete symmetries (which may be softly-broken by mass terms), and consider two possible models. In Model I, Φ_{d}^{0} couples to both up-type and down-type quark and lepton pairs, while the coupling of Φ_{u}^{0} to fermion pairs is absent.^{2} In Model II, Φ_{d}^{0} [Φ_{u}^{0}] couples exclusively to down-type [up-type] fermion pairs. When the Higgs potential is minimized, the neutral components of the Higgs fields acquire vacuum expectation values:^{3}

1A possible loophole arises in the MSSM when m_{A} is large, tan β is large, and there exists very strong mixing in the top-squark sector (beyond so-called maximal mixing). In this case, radiative corrections can drive m_{h} very light while keeping | sin(β − α)| ≪ 1. In such a model, the Model II results of this paper apply. However, the required MSSM parameter regime is rather fine-tuned and probably exhibits other theoretical problems such as the existence of charge and color-breaking minima.

2One can just as well assume that Φ_{u}^{0} couples to both up-type and down-type quark and lepton pairs, while the coupling of Φ_{d}^{0} to fermion pairs is absent. In this case, the all the results of this paper would apply simply by replacing tan β with cot β.

⟨Φ_{d}⟩ = \frac{1}{\sqrt{2}} ( \begin{matrix}v_{d} \\ 0 \end{matrix} ) ,  ⟨Φ_{u}⟩ = \frac{1}{\sqrt{2}} ( \begin{matrix}0 \\ v_{u} \end{matrix} ) ,  (1.4)

where the normalization has been chosen such that v^{2} ≡ v_{d}^{2} + v_{u}^{2} = 4m_{W}^{2} /g^{2} = (246 GeV)^{2}, while the ratio tan β ≡ v_{u}/v_{d} is a free parameter of the model. The physical Higgs spectrum consists of a charged Higgs pair

H^{±} = Φ_{d}^{±} sin β + Φ_{u}^{±} cos β ,  (1.5)

one CP-odd scalar

A = \sqrt{2} (Im Φ_{d}^{0} sin β + Im Φ_{u}^{0} cos β) ,  (1.6)

and two CP-even scalars:

h = −(\sqrt{2} Re Φ_{d}^{0} − v_{d}) sin α + (\sqrt{2} Re Φ_{u}^{0} − v_{u}) cos α ,
H = (\sqrt{2} Re Φ_{d}^{0} − v_{d}) cos α + (\sqrt{2} Re Φ_{u}^{0} − v_{u}) sin α ,  (1.7)

(with m_{h} ≤ m_{H}). The angle α arises when the CP-even Higgs squared-mass matrix (in the Φ_{d}^{0}—Φ_{u}^{0} basis) is diagonalized to obtain the physical CP-even Higgs states.

Relevant to our analysis, we note the tree-level couplings of lightest CP-even Higgs boson, h, to gauge bosons and Higgs bosons: The couplings of the two CPeven Higgs bosons to W and Z pairs are given in terms of the angles α and β by

g_{hV V} = g_{V} m_{V} sin(β − α)  (1.8)

where

g_{V} ≡ \{ \begin{matrix}g, & V = W , \\ g/ cos θ_{W} , & V = Z, \end{matrix}  (1.9)

and

g_{hAZ} = \frac{g cos(β − α)}{2 cos θ_{W}} .  (1.10)

The pattern of couplings of the Higgs bosons to fermions depends on the choice of model. However, in this paper we are only concerned with the coupling of down-type fermions to Higgs bosons. As a result, the distinction between Model I and Model II is not required. For our analysis, the relevant couplings of the neutral Higgs bosons to bb̄ or μ^{+}μ^{−} relative to the Standard Model value, gm_{f}/2m_{W} [f= b or μ], are given by

hbb̄ (or hμ^{+}μ^{−}) : − \frac{sin α}{cos β} = sin(β − α) − tan β cos(β − α) ,
Hbb̄ (or Hμ^{+}μ^{−}) : \frac{cos α}{cos β} = cos(β − α) + tan β sin(β − α) ,
Abb̄ (or Aμ^{+}μ^{−}) : γ_{5} tan β ,  (1.11)

3In this paper, we neglect the possibility of significant CP-violation in the Higgs sector. In this case, the phases of the Higgs fields can be chosen such that the vacuum expectation values are real and positive.

(the γ_{5} indicates a pseudoscalar coupling), and the charged Higgs boson couplings to muon pairs (with all particles pointing into the vertex) is given by

g_{H^{−}μ^{+}ν} = \frac{gm_{μ}}{\sqrt{2}m_{W}} tan β P_{L} ,  (1.12)

where P_{L} ≡ \frac{1}{2}(1 − γ_{5}).

We have noted above that only light Higgs bosons with enhanced couplings to down-type fermions can contribute appreciably to δa_{μ}^{NP}. To avoid the LEP Higgs mass limit, such a light Higgs boson should be almost decoupled from ZZ. This implies that either h is light, with | sin(β − α)| ≪ 1 [see eq. (1.8)] or A is light (since A has no tree-level coupling to vector boson pairs). In the next section, we will show that a light A makes a negative contribution to δa_{μ}^{NP} and thus is not compatible with the recent BNL measurement. Hence, we focus on a 2HDM model in which only h is light and sin(β − α) ≃ 0. From eq. (1.11), we see that if sin(β − α) ≃ 0, then the coupling of h to down-type fermions is proportioanl tan β. Thus, in the region of large tan β and small sin(β − α), the contribution of a light CP-even Higgs boson of the 2HDM may yield a significant correction to δa_{μ}^{NP} without being in conflict with the LEP Higgs search.

Finally, we note that in the parameter region cited above, the heavier Higgs bosons, H, A, H^{±}, cannot be arbitrarily heavy. If one attempts to take such a limit, one finds that there must be some Higgs quartic self-couplings that become significantly larger than 1. That is, this model does not possess a decoupling limit. However, the model stays weakly coupled as long as the heavier Higgs states are not too much larger than v = 246 GeV. In contrast, in the limit of cos(β − α) = 0, the coupling of h reduce to those of the Standard Model Higgs boson. This decoupling limit can be formally reached by taking the masses of H, A, H^{±} to be arbitrarily large. The resulting low-energy effective theory is just the SM with one Higgs doublet. Of course, as we have noted above, in this limit, the contribution of h to δa_{μ}^{NP} is negligible.^{4}

2. Higgs boson corrections to the muon anomalous magnetic moment

The first calculation of the electroweak corrections to the muon anomalous magnetic moment was done by Weinberg and Jackiw [11] and by Fujikawa, Lee and Sanda [12]. A very nice overview of these corrections in a general field theory framework can be found in Ref.[13]. In a two-Higgs-doublet model, these corrections can be obtained from Refs. [14] and [15]^{5}

4We demonstrate in the next section that in the decoupling limit [where cos(β − α) ≃ 0], if the masses of H, A and H^{±} are large, but not too large, and tan β ≫ 1, then the heavy Higgs states dominate the Higgs sector contribution to δa_{μ}^{NP}. Nevertheless, the overall Higgs contribution is significantly smaller than the electroweak contribution and thus could not be a source of the new physics contribution to a_{μ}.

δa_{μ}^{h} = \frac{G_{F} m_{μ}^{2}}{4π^{2}\sqrt{2}} (\frac{sin α}{cos β} )^{2} R_{h} F_{h}(R_{h})  (2.1)
δa_{μ}^{H} = \frac{G_{F} m_{μ}^{2}}{4π^{2}\sqrt{2}} (\frac{cos α}{cos β})^{2} R_{H} F_{H}(R_{H})  (2.2)
δa_{μ}^{A} = \frac{G_{F} m_{μ}^{2}}{4π^{2}\sqrt{2}} tan^{2} β R_{A} F_{A}(R_{A})  (2.3)
δa_{μ}^{H^{±}} = \frac{G_{F} m_{μ}^{2}}{4π^{2}\sqrt{2}} tan^{2} β R_{H^{±}} F_{H^{±}}(R_{H^{±}}, R_{ν})  (2.4)

where R_{h,H,A,H^{±}} ≡ m_{μ}^{2}/m_{h,H,A,H^{±}}^{2}, R_{ν} ≡ m_{ν}^{2}/m_{H^{±}}^{2} and

F_{h,H}(R_{h,H}) = ∫_{0}^{1} dx \frac{2x^{2} − x^{3}}{R_{h,H}x^{2} − x + 1} ,  (2.5)
F_{A}(R_{A}) = ∫_{0}^{1} dx \frac{−x^{3}}{R_{A}x^{2} − x + 1} ,  (2.6)
F_{H^{±}}(R_{H^{±}}, R_{ν}) = ∫_{0}^{1} dx \frac{x^{2} − x^{3}}{R_{H^{±}}x^{2} + (1 − R_{H^{±}} − R_{ν})x + R_{ν}} ,  (2.7)

The neutrino mass is negligible, so henceforth we set R_{ν} = 0. Since R_{h,H,A,H^{±}} ≪ 1, one can easily expand the above integrals in the corresponding small parameter. In the next two subsections, we write out the leading terms in this expansion, which are very accurate in the Higgs mass range of interest.^{6}

2.1 Non-decoupling limit: sin(β − α) = 0

In section 1, we argued that the most significant Higgs contribution to δa_{μ}^{NP} (consistent with the LEP Higgs search data) arises in the parameter regime in which sin(β − α) ≃ 0 and tan β ≫ 1. Setting sin(β − α) = 0 and keeping only the leading terms in R, the total Higgs sector contribution to a_{μ} is given by:

δa_{μ}^{Higgs} = δa_{μ}^{h} + δa_{μ}^{H} + δa_{μ}^{A} + δa_{μ}^{H^{±}}

≃ \frac{G_{F} m_{μ}^{2}}{4π^{2}\sqrt{2}} tan^{2} β\{\frac{m_{μ}^{2}}{m_{h}^{2}} [ln (\frac{m_{h}^{2}}{m_{μ}^{2}} ) − \frac{7}{6}] − \frac{m_{μ}^{2}}{m_{A}^{2}} [ln (\frac{m_{A}^{2}}{m_{μ}^{2}} ) − \frac{11}{6}] − \frac{m_{μ}^{2}}{6m_{H^{±}}^{2}} \} .

(2.8)

Note that the logarithms appearing in eq. (2.8) always dominate the corresponding constant terms when the Higgs masses are larger than 1 GeV. It is then clear that A and H^{±} exchange contribute a negative value to δa_{μ}^{NP}. Since our goal is to explain the BNL g − 2 measurement which suggests a positive value for δa_{μ}^{NP}, we should take m_{A} and m_{H^{±}} large (masses above 100 GeV are sufficient) in order that the corresponding A and H^{±} negative contributions are neglibly small.^{7} If δa_{μ}^{NP} is to be a consequence of the Higgs sector, it must be entirely due to the contribution of the light CP-even Higgs boson. Note that the heavier CP-even Higgs, H, does not give a contribution proportional to tan β (as shown in section 1); thus its contribution to δa_{μ}^{NP} can be neglected in eq. (2.8). Thus, to a good approximation,

5We correct a small error in the expression for the charged Higgs contribution given in ref. [14].

6The plots shown in this paper are based on the exact values of the above integrals.

δa_{μ}^{Higgs} ≃ δa_{μ}^{h} ≃ \frac{G_{F} m_{μ}^{2}}{4π^{2}\sqrt{2}} (\frac{m_{μ}^{2}}{m_{h}^{2}} ) tan^{2} β[ ln (\frac{m_{h}^{2}}{m_{μ}^{2}} ) − \frac{7}{6}] .  (2.10)

One can easily check that a light Higgs boson with a mass of arround 10 GeV and with tan β = 35 gives δa_{μ}^{Higgs} ≃ 280 × 10^{−11} which is precisely the new physics contribution we need in Eq.(1.2). Contour lines corresponding to a full numerical evaluation of the Higgs sector one-loop contribution to δa_{μ}^{Higgs} [in units of 10^{−11}], is presented in Fig. 1, for sin(β − α) = 0 and m_{H} = m_{A} = m_{H^{±}} = 200 GeV. The results are insensitive to the value of the heavy Higgs masses above 100 GeV. The relevant experimental bounds are also displayed in the figure (see discussion below). Clearly, a light Higgs boson of 10 GeV mass and 30 < ∼ tan β < ∼ 45 will yield a value for δa_{μ}^{NP} within the 90% CL allowed range [eq. (1.2)]. In section 3, we review the experimental limits for light CP-even masses and show that if sin(β − α) ≃ 0, and tan β in the range required to explain the BNL measurement of a_{μ}, then m_{h} must lie in a very narrow range centered around 10 GeV.

2.2 Decoupling Limit: cos(β − α) = 0

In the decoupling limit, where cos(β − α) ≃ 0 and m_{A} ≫ m_{Z}, the couplings of the light Higgs boson, h, are (nearly) identical to those of the one-Higgs-doublet Standard Model. As a result, the LEP Higgs mass bound of m_{h} > ∼ 113.5 GeV applies. In this limit, the H couplings to fermions are enhanced by tan β [see eq. (1.11)]. Thus, the Higgs sector contribution to δa_{μ}^{NP} is given by eq. (2.8), with m_{h} replaced by m_{H}. In the decoupling limit, m_{H} ≃ m_{A} ≃ m_{H^{±}} (the mass diferences are of O(m_{Z}^{2}/m_{A})). Setting cos(β − α) = 0 and m_{H} = m_{A} = m_{H^{±}}, we find

δa_{μ}^{Higgs} ≃ \frac{G_{F} m_{μ}^{2}}{4π^{2}\sqrt{2}} (\frac{m_{μ}^{2}}{m_{A}^{2}} ) tan^{2} β[\frac{1}{2} − (\frac{2m_{μ}^{2}}{m_{A}^{2}} ) ln (\frac{m_{A}^{2}}{m_{μ}^{2}} )] .  (2.11)
δa_{μ}^{H^{±}} = a_{μ}^{H^{±}}(1–loop) + \frac{1}{180} (\frac{α}{π})^{2} (\frac{m_{μ}}{m_{H^{±}}} )^{2} + O[(\frac{m_{μ}}{m_{H^{±}}} )^{4} ln (\frac{m_{μ}}{m_{H^{±}}} )] .  (2.9)

7Grifols and Pascual [16] found that for a very light charged Higgs boson, the two-loop contribution to a_{μ} is positive and can be larger in magnitude than the one-loop result given in eq. (2.4):

However, the LEP bound on the charged Higgs mass ref. [4], m_{H^{±}} > 80.5 GeV, implies that both the one and two-loop charged Higgs contribution to δa_{μ}^{NP} are negligible.

Figure 1: Contour plot of the predicted one-loop Higgs sector contribution to the muon anomalous magnetic moment, δα_{μ}^{Higgs} (in units of 10^{−11}). in the 2HDM with sin(β−α) = 0, and m_{H} = m_{A} = m_{H^{±}} = 200 GeV. (There is little sensitivity to the heavier Higgs masses.) The dashed line contour corresponds to the central value of δa_{μ} ≡ a_{μ}^{exp} − a_{μ}^{SM}, as reported in ref. [1]. The contours lines marked 215 and 637 correspond to 90% CL limits for the contrubution of new physics to a_{μ} [2]. The dark-shaded (red) region is excluded by CUSB Collaboration [18]. The light-shaded (yellow) region is experimentally excluded by DELPHI LEP I search for e^{+}e^{−} → bb̄h [19]. In the small hatched region (green) nestled between the two excluded shaded regions centered around m_{h} ≃ 10 GeV, the Higgs sector contributes a value for δa_{μ}^{NP} within the 90% CL allowed range [eq. (1.2)].

The contribution of h is not tan β–enhanced and is thus negligible. In is interesting to note that for values of m_{A} < ∼ m_{h} tan β, the heavier (“decoupled”) Higgs bosons actually dominate in the Higgs sector contribution to δa_{μ}^{NP}.^{8} However, for 100 GeV < m_{A} < 1000 GeV, and 30 < tan β < 100, the Higgs sector contribution to a_{μ} ranges from about 5 × 10^{−12} to 5 × 10^{−14}, which is three to five orders of magnitude below what is needed to explain the BNL measured value of a_{μ}.

8If we formally take m_{A} → ∞, we recover the Standard Model Higgs contribution to a_{μ}.

3. CUSB and LEP I constraints on the light Higgs boson

Let us consider a 2HDM model in which sin(β − α) = 0 and tan β is large. The tree-level hZZ coupling is absent, which implies that the LEP Higgs search based on e^{+}e^{−} → Z → Zh does not impose any significant constraints on the Higgs mass.^{9} Moreover, tan β must be large if the Higgs sector is to be the source for δa_{μ}^{NP} observed by BNL. However, there are a number of constraints on light Higgs masses that do not rely on the hV V coupling. For Higgs bosons with m_{h} < ∼ 5 GeV, the Standard Model Higgs boson was ruled out by a variety of arguments that were summarized in ref. [10]. For 5 GeV < ∼ m_{h} < ∼ 10 GeV, the relevant Higgs boson constraint can be derived from the absence of Higgs production in Υ → hγ.

An experimental search for this process by the CUSB Collaboration [18] found no candidates. The Higgs mass limit obtained from this results depends on the theoretical prediction. In addition to the tree-leve prediction of ref. [20], there are three classes of corrections that have been explored in the literature: hard O(α_{s}) QCD corrections [24, 25], relativistic corrections to the non-relativistic treatment of the bb̄ bound state [22, 23], and bound state threshold corrections [21]. The theoretical picture that emerges is confused. The hard QCD corrections are large and suggest that O(α_{s}^{2}) corrections could be significant. Second, relativistic effects enter at the same order as the O(α_{s}) corrections [both are of O(v^{2}/c^{2}), and thus the two must be treated consistently. Finally, Faldt, Osland and Wu [21] argue that strong cancellations can occur among various contributions in the threshold region, leading to an additional suppression in rate of about 14 for m_{h} = 8.5 GeV (and even a larger suppression as m_{h} → M_{Υ}). The application of the theoretical analysis of Γ(Υ → hγ) to the CUSB data suggests that values of m_{h} < ∼ 5–7 GeV can be ruled out, although a precise upper limit cannot be obtained due to the theoretical uncertainties outlined above.

The above discussion was relevant for limits on the mass of the Standard Model Higgs boson. In the 2HDM model of interest here, tan β ≫ 1, and the prediction for Γ(Υ → hγ) is enhanced by a factor of tan^{2} β. This factor dwarfs the theoretical uncertainties discussed above except for values of m_{h} very close to M_{Υ}. For values of tan β ≥ 10, the CUSB data can reliably rule out Higgs masses up to about 8 GeV; the excluded region is indicated by the dark (red) shaded region in Fig. 1. But, for Higgs masses above 8 GeV, we see that tan β > 30 if the Higgs sector contribution to δa_{μ}^{NP} lies in the 90% CL range specified in eq. (1.2). For such large values of tan β, the predicted rate for Υ → hγ is increased by at least three orders of magnitude. This factor dwarfs the theoretical uncertainties discussed above except for values of m_{h} very close to M_{Υ}. Thus, we obtain an upper bound of m_{h} > ∼ M_{Υ}.

9Presumably, radiative corrections would lead to a small effective value for sin(β − α). The LEP Higgs search yields an excluded region in the sin(β − α)–m_{h} plane, and suggest that for m_{h} ∼ 10 GeV, sin(β − α) < ∼ 0.14 [17].

A second bound on m_{h} can be derived from the non-observation of Higgs bosons at LEP in the process e^{+}e^{−} → bb̄h. The cross-section for this process depends on the hbb̄ Yukawa coupling. In the 2HDM model with sin(β − α) = 0, the hbb̄ coupling is enhanced (relative to its SM value) by tan β.^{10} A preliminary analysis of DELPHI [19]^{11} based on the search for e^{+}e^{−} → bb̄h, h → τ^{+}τ^{−}, bb̄ finds no evidence for Higgs production and excludes the light-shaded (yellow) region of Fig. 1. Note that the lower limit on tan β is weaker for m_{h} < 2m_{B}, where B is the lightest Bmeson [m_{B} = 5.279 GeV]. In this mass region, the dominant decay of h is into τ^{+}τ^{−}. For values of m_{h} > 2m_{B}, the Higgs decays primarily into bb̄, and the DELPHI limits completely exclude the region of parameter space in which the Higgs sector contribution to δa_{μ}^{NP} lies in the 90% CL range specified in eq. (1.2).

Finally, regarding other constraints, it has been shown in [26] that even in the presence of such a light higgs boson, the two Higgs doublets model can describe the electroweak data with the same precision to that given by the SM. It has also been shown [27] that the Wilczek process Z → hγ constraints tan β < ∼ 75 for M_{h} = 10 GeV. This bound is weaker than the recent (DELPHI) LEP I constraint [19] depicted in Fig.1.

4. Final Results and Conclusions

If we combine the experimental searches for a light Higgs boson, we conclude that a light Higgs boson can be responsible for the observed the 2.6σ deviation of the BNL measurement of the muon anomalous magnetic moment at the 90% CL in the framework of a two-Higgs-doublet model only if the model parameters satisfy the following requirements:

m_{Υ} < ∼ M_{h} < ∼ 2m_{B} ,
sin(β − α) ≃ 0 ,
30 < ∼ tan β < ∼ 45 .  (4.1)

One noteworthy consequence of a Higgs boson in this mass regime was discussed in Refs. [14] and [28]. Namely, there would be mixing between the h and the 0^{++} bb̄ bound states χ_{b0}(1P) and χ_{b0}(2P). As a result, the decay χ_{b0} → τ^{+}τ^{−} should be prominent. The predicted rate is roughly

\frac{Γ(χ_{b0} → τ^{+}τ^{−})}{Γ(χ_{b0} → hadrons)} = \frac{2.5 × 10^{−7} GeV^{2}}{(m_{χ} − m_{h})^{2}} tan^{4} β ,  (4.2)

10We assume that m_{A} is heavy enough so that e^{+}e^{−} → hA is not observed at LEP.

11ALEPH [?] looked at process e^{+}e^{−} → bb̄A. Non observation of the CP-Odd Higgs boson implied that for m_{A} = 10 GeV, tan β is restricted to be below 35. Although, ALEPH formally did not display the limits on h, we beleive that they would be similar since there is no reason for an aprreciable difference due to the CP-quantum number. If this turn out to be the case then δδa_{μ}^{NP} < ∼ 300 × 10^{−11} and the solution to the muon anomalous magnetic moment is less likely.

which is valid for m_{h} near m_{χ} but separated by a few Higgs widths.^{12} Due to the large tan β enhancement, the predicted branching ratio for χ_{b0} → τ^{+}τ^{−} can be substantial. Remarkably, the Particle Data book [29] provides no data on possible decay modes of the χ_{b0} other than radiative decays, χ_{b0} → Υγ, Υ′γ.

Apart from a careful study of χ_{b0} decays, the 2HDM model specified by eq. (4.1) could be confirmed or ruled out by experiments at future colliders in the future colliders [30], or by a further improvement of the BNL measurement of the muon anomalous magnetic moment.

A.D. would like to acknoweledge financial support from the Network RTN European Program HPRN-CT-2000-0014 “Physics Across the Present Energy Frontier: Probing the Origin of Mass.” H.E.H. would like to thank H.-P. Nilles and H.K. Dreiner for their hospitality during his visit to the Physikalisches Institut der Universität Bonn. H.E.H. is supported in part by a grant from the U.S. Department of Energy.

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12If the two masses are within a Higgs width, then the mixing of the two states will be close to maximal [28], and the corresponding τ^{+}τ^{−} branching ratio of both eigenstates would be close to 100% due to the large tan β enhancement.

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