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 Higgs boson with mass of arround ∼ 10 GeV, could explain the recent BNL measurement on muon g − 2 anomalous magnetic moment, in the framework of the minimal Standard Model with two Higgs doublets. Such a light Higgs boson should have a mass bounded from below from the Upsilon mass and from above from the bottonium mass.

Keywords: Beyond Standard Model.

Contents

1. Introduction 1

2. Higgs boson corrections to the muon g − 2 anomalous magnetic moment 5 2.1 Non-Decoupling limit: sin(β − α) = 0 6 2.2 Decoupling Limit: cos(β − α) = 0 8

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

4. Combined Results and Conclusions 10

1. Introduction

The recent reported 2.6σ [1] deviation

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

of the experimental value of the muon anomalous magnetic moment versus the Standard Model (SM) predicted one, is very exciting. It has also been reported [2] that at 90% CL a “New Physics” contribution of

215 × 10^{−11} ∼^{<} δa_{μ}(New Physics) ∼^{<} 637 × 10^{−11} ,  (1.2)

beyond the SM, is needed. This contribution is positive, and of the order of the electroweak corrections to the muon anomalous magnetic moment. More precisely, the “New Physics” needed, has to be of the order of \frac{G_{F} m_{μ}^{2}}{4\sqrt{2}π^{2}} , where G_{F} = 1.16637(1)× 10^{−5} GeV^{−2} is the Fermi constant and m_{μ} = 0.105 GeV is the mass of the muon. Furthermore, one could think that these corrections could directly be originated by the perturbative electroweak symmetry breaking mechanism or in other words by the Higgs Boson particle. However, in the SM, the Higgs boson contribution (in units of \frac{G_{F} m_{μ}^{2}}{4\sqrt{2}π^{2}} ) to the muon anomalous magnetic moment is suppressed by a factor of m_{μ}^{2}/M_{h}^{2}. Regarding the smallness of the mass of the muon and the largeness of the Higgs mass (∼^{>} 115 GeV) the SM Higgs contribution to δa_{μ} is negligible.

The above picture is not in general valid in a straightforward extension of the SM with two Higgs doublets. In this framework, δa_{μ} could be enhanced in two ways : the first one is that δa_{μ} is proportional^{1} to tan^{2} β in the case where sin(β − α) = 0, and the second, it is in that case where the light Higgs boson can be as light as 10 GeV without conflict with experiment. In terms of \frac{G_{F} m_{μ}^{2}}{4\sqrt{2}π^{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(\frac{m_{μ}^{2}}{M_{h}^{2}} ) of order O(−10). Such a light Higgs boson with mass M_{h} ≃ 10 GeV and 30 ∼^{<} tan β ∼^{<} 44, predicts a muon anomalous magnetic moment in agreement with the current BNL experimental value.

This idea cannot take place in the MSSM simply because the Higgs bosons are constraint from certain mass sum rules^{2}. Although in the exact Supersymmetric limit there are no contributions [3] to the muon g − 2, this is not the case for the softly broken N=1 supersymmetric Standard model where the neutralino/chargino contributions are proportional to \frac{m_{μ}^{2}M_{2} tan β}{M_{SUSY}^{4}} F in units of \frac{G_{F} m_{μ}^{2}}{4\sqrt{2}π^{2}} . Thus, for moderate and large tan β values one recovers the g − 2 deviation (see the corresponding Refs in [1]). Although, the SUSY solution is quite elegant, we feel that the scenario we propose, is far more generic and minimal with respect to its field condent.

The two-doublet Higgs sector [4] contains eight scalar degrees of freedom: 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}). The notation reflects the form of the MSSM Higgs sector coupling to fermions: 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}

⟨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}. Spontaneous electroweak symmetry breaking results in three Goldstone bosons, which are absorbed and become the longitudinal components of the W^{±} and Z. The remaining five physical Higgs particles consist of a charged Higgs pair

1Definitions are given below.

2There is one exception here : It could be the case that the light Higgs boson in the MSSM decays “invisibly” into LSPs (ussually neutralinos). Although there are strict bounds on sin(β − α) − M_{h} plane from LEP searches, an “invisible” Higgs boson with a mass of few GeV’s and sin(β − α) = 0 is allowed. Such a “supelight” higgs boson contributes to the g − 2 muon anomalous magnetic, in exactly the same way as we describe here. However the Higgs boson should decay to very light particles and one has propably to consider R-parity violation or to depart from the minimal field content by adding a singlet superfield (NMSSM). This scenario would be highly fine-tuned.

3The phases of the Higgs fields can be chosen such that the vacuum expectation values are real and positive. That is, the tree-level MSSM Higgs sector conserves CP, which implies that the neutral Higgs mass eigenstates possess definite CP quantum numbers.

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. The angle β is defined as tan^{−1}(\frac{v_{u}}{v_{d}}).

The phenomenology of the Higgs sector depends in detail on the various couplings of the Higgs bosons to gauge bosons, Higgs bosons and fermions. The couplings of the two CP-even 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)
g_{HV V} = g_{V} m_{V} cos(β − α) ,  (1.8)

where

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

There are no tree-level couplings of A or H^{±} to V V. The couplings of one gauge boson to two neutral Higgs bosons:

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

There are two versions of the SM extended with two Higgs doublets called here Model I and Model II. The Model I is the case where quarks and leptons do not couple to the first Higgs doublet (denoted as Φ_{d} ) but couple to the second doublet (denoted as Φ_{u}) in a manner analogous to the minimal Standard Model. The gauge invariant Model I Yukawa interactions for the muons can be read from

−L_{Yukawa} = h_{μ} [μ̄P_{L}μΦ_{u}^{0} − μ̄P_{L}ν_{μ}Φ_{u}^{−}] + h.c. ,  (1.11)

where P_{L} ≡ \frac{1}{2}(1 − γ_{5}) is the left-handed projection operator. In Model II, the Higgs tree-level couplings to fermions obey the following property: Φ_{d}^{0} couples exclusively to down-type fermion pairs and Φ_{u}^{0} couples exclusively to up-type fermion pairs. The gauge-invariant Type-II Yukawa interactions for the muons are given by:

−L_{Yukawa} = h_{μ} [μ̄P_{L}μΦ_{d}^{0} − μ̄P_{L}ν_{μ}Φ_{d}^{−}] + h.c. ,  (1.12)

Fermion masses are generated when the neutral Higgs components acquire vacuum expectation values. Inserting eq. (1.4) into eq. (1.12) yields a relation between the muon mass and its Yukawa coupling:

h_{μ} = \frac{\sqrt{2} m_{μ}}{v_{u}} = \frac{\sqrt{2} m_{μ}}{v sin β} , Model I ,  (1.13)
h_{μ} = \frac{\sqrt{2} m_{μ}}{v_{d}} = \frac{\sqrt{2} m_{μ}}{v cos β} , Model II .  (1.14)

The couplings of the physical Higgs bosons to the muon is obtained from eqs. (1.11)– (1.12) by using eqs. (1.5)–(1.7). In particular, the couplings of the neutral Higgs bosons to μμ̄ relative to the Standard Model value, gm_{f}/2m_{W}, are given by the following Table :

Model I Model II
g_{hμ^{+}μ^{−}} −\frac{cos α}{sin β} \frac{sin α}{cos β}
g_{Hμ^{+}μ^{−}} −\frac{sin α}{sin β}  −\frac{cos α}{cos β}
g_{Aμ^{+}μ^{−}} −γ_{5} cot β γ_{5} tan β

(1.15)

(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

Model I  Model II
g_{H^{−}μ^{+}ν_{μ}} −\frac{g cot β}{\sqrt{2}M_{W}} m_{μ}P_{L} \frac{g tan β}{\sqrt{2}M_{W}} m_{μ}P_{L} (1.16)

Some very useful trigonometric relations are in order :

−\frac{sin α}{cos β} = sin(β − α) − tan β cos(β − α) ,
\frac{cos α}{sin β} = sin(β − α) + cot β cos(β − α) ,
\frac{cos α}{cos β} = cos(β − α) + tan β sin(β − α) ,
\frac{sin α}{sin β} = cos(β − α) − cot β sin(β − α) .  (1.17)

We remind the reader, that we need the Higgs couplings to muons to be enhanced by tan β (or cot β for tan β < 1) and a very light Higgs boson which does not couple to the Z-Gauge boson and thus almost undetectable at LEP. From the above tables and the trigonometric relations it is obvious that this is satisfied if and only if,

sin(β − α) = 0 .  (1.18)

In that case from eq.(1.8) we obtain, g_{hZZ} = 0 and

g_{hμ^{+}μ^{−}} , g_{Aμ^{+}μ^{−}} , g_{H^{−}μ^{+}ν_{μ}} ∝ tan β : Model I  (1.19)
g_{hμ^{+}μ^{−}} , g_{Aμ^{+}μ^{−}} , g_{H^{−}μ^{+}ν_{μ}} ∝ cot β : Model II  (1.20)

and |g_{Hμ^{+}μ^{−}}| = 1 in both models.

We would like to comment here on the other extreme limit where cos(β −α) = 0. In this case it is

g_{Hμ^{+}μ^{−}} , g_{Aμ^{+}μ^{−}} , g_{H^{−}μ^{+}ν_{μ}} ∝ tan β : Model I  (1.21)
g_{Hμ^{+}μ^{−}} , g_{Aμ^{+}μ^{−}} , g_{H^{−}μ^{+}ν_{μ}} ∝ cot β : Model II  (1.22)

and |g_{hμ^{+}μ^{−}}| = 1. An interesting aspect of both models is that as far as the higgs coupling to the muons is concerned, they are dual to one each other

Model I tan β ↔ cot β Model II  (1.23)

Thus, investigating one of the two models is enough. We follow the notation of Model II.

2. Higgs boson corrections to the muon g − 2 anomalous magnetic moment

μ  h, H, A  μ μ  μ γ  ^{μ H^{−} γ ν H^{−} μ}

The first calculation of the electroweak corrections to the g − 2 muon anomalous magnetic moment was done by Weinberg and Jackiw [5] and by Fujikawa, Lee and Sanda [6]. A very nice overview of these corrections in a general field theory framework can be found in Ref.[7]. In the Type II 2HDM framework these corrections can be read from [8]

δa_{μ}^{h} = \frac{G_{F} m_{μ}^{2}}{4\sqrt{2}π^{2}} (\frac{sin α}{cos β} )^{2} R_{h} F_{h}(R_{h})  (2.1)

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

where R_{h,H,A,H^{±}} = \frac{m_{μ}^{2}}{M_{h,H,A,H^{±}}^{2}} , R_{ν} = \frac{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)

Although we find approximate analytical solutions for the above integrals, we solve them numerically in the figures presented. The small neutrino mass does not affect our results and we set it zero for simplicity.

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

Expanding the integrals in powers of R, the leading contribution of the Higgs sector for sin(β − α) = 0 is given by

δa_{μ}(higgs) = δa_{μ}^{h} + δa_{μ}^{H} + δa_{μ}^{A} + δa_{μ}^{H^{±}} =

−\frac{G_{F} m_{μ}^{2}}{4\sqrt{2}π^{2}} tan^{2} β\{\frac{m_{μ}^{2}}{M_{h}^{2}} [\frac{7}{6} + ln(\frac{m_{μ}^{2}}{M_{h}^{2}} )] − \frac{m_{μ}^{2}}{M_{A}^{2}} [\frac{11}{6} + ln(\frac{m_{μ}^{2}}{M_{A}^{2}} )] + \frac{1}{6} \frac{m_{μ}^{2}}{M_{H^{±}}^{2}} \}

(2.8)

Note the minus sign out of Eq.(2.8). Also note that the logarithms encountered are negative and always larger than the constant term when the Higgs masses are larger than 1 GeV. Having that in our mind, it is obvious that a possible light CP-Odd Higgs does not help since it gives negative δa_{μ}. The cituation of both A and h being light does not help either, since the logarithms associated with them cancel out. Furthermore, the charged Higgs boson enters^{4} in Eq.(2.8) without a logarithm and

δa_{μ}^{H^{±}} = \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)

4Actually Grifols and Pascua [9] have found that the two-loop corrections, are bigger than the 1-loop ones. But again a very light charged higgs of the order of few GeV’s is required and this case is experimentally excluded.

Figure 1: Contour plot of the predicted 1-loop Higgs contributions to the muon g − 2 anomalous magnetic moment, δα_{μ}(higgs) (in units of 10^{−11}), on M_{h} − tan β plane, in the 2HDM(II) scenario. We present results with sin(β − α) = 0 as this is motivated in the text. The other Higgs masses have been set to M_{H^{±}} = M_{A} = M_{H}=200 GeV but their variation for above 100 GeV do not affect the results (see Eq.(2.8)). The dashed line represents the central difference between the experimental value and the SM predicted one, δ_{α_{μ}} = 426 ± 165 × 10^{−11}, as reported from BNL [1]. Contours of 90% CL 215 × 10^{−11} ≤ (New Physics) ≤ 637 × 10^{−11} [2] need to recover the current experimental value, are shown. The region on the left side and behind the blue line is excluded by CUSB Collaboration [10]. The region above the solid red line is experimentally excluded by DELPHI LEP I searches [11].

a light charged Higgs boson less than 45 GeV is excluded by LEP I searches. The other CP-even Higgs does not give contributions proportional to tan β (see previous section) and thus its contribution to δa_{μ} is negligible.

The only possibility left is a very light Higgs boson which we advocate here. In that case, its contribution to the muon g − 2 is given in a very good approximation by

δa_{μ}(higgs) ≃ δa_{μ}^{h} = −\frac{G_{F} m_{μ}^{2}}{4\sqrt{2}π^{2}} (\frac{m_{μ}^{2}}{M_{h}^{2}} ) tan^{2} β[\frac{7}{6} + ln(\frac{m_{μ}^{2}}{M_{h}^{2}} )] .  (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 actually the new physics we need in Eq.(1.2). The full numerical 1-loop light Higgs boson contribution to muon g − 2, δa_{μ}^{h}(higgs) (in units of 10^{−11}), is presented in contour lines on the plane M_{h} − tan β in Fig.1. The other Higgs masses have been set to M_{H^{±}} = M_{A} = M_{H}=200 GeV but their variation for above 100 GeV do not affect the results. The experimental bounds are also presented in the figure (see discussion below). As it is clear, a light Higgs boson of 10 GeV mass and 30 ∼^{<} tan β ∼^{<} 44 is capable to give the neccesary contributions to the currently measured value of the muon anomalous magnetic moment. We will turn back into this point in the last section where the mass region where such a light Higgs might exist, will be set.

2.2 Decoupling Limit: cos(β − α) = 0

The couling of the Z-gauge boson to the light Higgs boson, g_{ZZh} is maximal and the lower LEP II bound^{5} on the light Higgs mass, M_{h} ∼^{>} 115 GeV, is applicable. The same approximate formula as in Eq.(2.8) is applied, with the light Higgs Boson h being replaced by the heavy, H, one. In the limit where all the other Higgses are equal, M_{A} = M_{H} = M_{H^{±}}, we obtain

δa_{μ}(higgs) ≃ \frac{G_{F} m_{μ}^{2}}{4\sqrt{2}π^{2}} (\frac{m_{μ}^{2}}{M_{A}^{2}} ) tan^{2} β[\frac{1}{2} + 2(\frac{m_{μ}^{2}}{M_{A}^{2}} ) ln(\frac{m_{μ}^{2}}{M_{A}^{2}} )] .  (2.11)

What is interesting here is that even though the Higgs masses are quite large, M_{A} → ∞, there is still an enhancement of δa_{μ} with tan^{2} β (note that the logarithmic contribution is suppressed). The situation is even clearer if we look at Fig.2 where contour lines of the muon g − 2 (in units of 10^{−14}) are plotted on a plane of M_{A} − tan β. The conributions to δa_{μ} are three orders of magnitude smaller than those in the non-decoupling case and thus negligible (for reasonable values of tan β). For fixed tan β, δa_{μ}(higgs) dies out with a rate ∼ \frac{m_{μ}^{2}}{M_{A}^{2}} and the decoupling is manifest, i.e., we recover the SM-case.

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

As we have already discussed, the light higgs does not couple to the Z-boson in the case of sin(β − α) = 0 and thus the SM LEP bound of 115 GeV does not apply. In principle, such a Higgs boson can be very light, or in other words : it has been escaped detection from the past Higgs searches. However, there are two major constraints on a possible Higgs mass, of arround 10 GeV, we quickly review here.

5For large tan β values this lower bound becomes weaker (∼^{>} 91 GeV) but it does not make any difference in the results since it is already too heavy and its contribution to δa_{μ} is not enhanced by tan^{2} β.

Figure 2: Contour plot of the Higgs 1-loop contributions to the muon g − 2 anomalous magnetic moment δ_{α_{μ}}(higgs) (in units of 10^{−14}), on a plane M_{A} − tan β, in the decoupling limit, i.e., cos(β − α) = 0.. The light Higgs boson mass has been fixed to its current experimental bound, M_{h} = 115 GeV. The heavy CP-even and the charged Higgses are taken to be equal to M_{A}. Note the residual enhancement with the tan β even in the region of large M_{A}.

The first lower bound comes from CUSB Collaboration [10], where an extensive search for a monochromatic photon in radiative decays Υ(1S) → hγ where

\frac{Γ(Υ → hγ)}{Γ(Υ → μ^{+}μ^{−})} = \frac{G_{F} m_{b}^{2} tan^{2} β(M_{Υ}^{2} − M_{h}^{2})}{4\sqrt{2}πα}|\frac{T (Υ → hγ)}{T (Γ(Υ → μ^{+}μ^{−})}|^{2}
×(1 − \frac{4α_{s}}{3π} a_{H}(z) + O(a_{s}^{2}))  (3.1)

had led to a lower bound,

600 MeV ∼^{<} M_{h} ∼^{<} 5 GeV , for tan β = 1 .  (3.2)

The first term [12, 13, 14, 15] in Eq.(3.1) contains the matrix elements T (six diagrams in time order perturbation theory [13]) and the second [16, 17] the QCD corrections with α_{H}(z) being a loop factor with z = 1 − \frac{M_{h}^{2}}{4m_{b}^{2}} . The QCD corrections reduce the rate by 84% for M_{h} << M_{Υ(1S)}. Some comments are in order here. In the first calculation, done by Wilczek [12], one of the two dominant diagrams was considered. In fact, as Faldt, Osland and Wu [13] had pointed out, strong cancellations can occur between the two diagrams leading to a suppresion factor to the rate of about 14 for M_{h} = 8.5 GeV (the suppression factor has a tendency of being increased while M_{h} → M_{Υ}). Their calculation is valid only when the Higgs mass is arround the Upsilon mass (7.5 GeV ∼^{<} M_{h} ∼^{<} M_{Υ}). These supressions and the large QCD corrections and even problems with renormalization scheme dependencies had led to a cloudy situation, and no reliable SM bound on the Higgs mass had arised from CUSB. However, as the reader might have already noticed, in the 2HDM we have an extra huge enhancement proportional to tan^{2} β. We feel like, whatever large theoretical uncertainties^{6} involved in the calculations of the radiative Upsilon decays, the ratio (3.1) cannot be suppressed by a factor of tan^{2} β ∼ 10^{2} − 30^{2} (!), which is the case we are dealing with. We conclude that the lower bound on the Higgs boson mass is just at the Upsilon threshold

M_{h} ∼^{>} M_{Υ(1S)} = 9.46 GeV for tan β ∼^{>} 10 .  (3.3)

This bound is denoted with the left area behind the yellow line in Fig.(1).

The second lower bound on the Higgs mass arises from (DELPHI) LEP I searches [11]. The only process which is capable to constraint our very light Higgs boson is the Yukawa process ^{7}. Preliminary limits on the Yukawa production (Z → ff̄h) of light neutral Higgs above the 2m_{τ} threshold were set : 2m_{τ} < M_{h} < 2m_{B} from 4τ (see left corner bounded from the red line in our fig.(1)) topologies and M_{h} > 2m_{B} from 4b topologies (see right corner above the red line in fig.(1)). It turns out that LEP I might have already produced the higgs boson without seeing it. Unfortunately, LEP searches cannot ruled out or confirm the existence of such a light Higgs boson and only a bound,

M_{h} ∼^{>} 2m_{B} = 10.6 GeV for tan β ∼^{>} 28
M_{h} ∼^{>} m_{Υ} for tan β ∼^{>} 44 ,  (3.4)

can be set.

4. Combined Results and Conclusions

If we combine the experimental searches for a light Higgs boson, (3.3) and (3.4) we conclude that the only explanation for the 2.6σ deviation of the BNL measurement in the minimal two Higgs doublet SM model (Model I or II) comes from a light Higgs boson in the range

6Unfortunately the calculations towards that direction had been stopped since collider searches took over.

7The pair production process is not kinematically allowed here since as we have already explained, all the other Higgses should be above the Z-threshold in order to obtain the correct δa_{μ}.

m_{Υ} ≤ M_{h} ≤ 2m_{B} .  (4.1)

with tan β being in the region 30 ∼^{<} tan β ∼^{<} 44. Such a Higgs could be confirmed or ruled out in the future colliders [18], or by the future improvement of the BNL [1] g − 2 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”.

References

[1] R. Prigl [Muon g − 2 Collaboration], “Status of the BNL muon(g − 2) experiment,” hep-ex/0101042.

[2] A. Czarnecki and W. J. Marciano, “The muon anomalous magnetic moment: A harbinger for ’new physics’,” hep-ph/0102122, and references therein.

[3] S. Ferrara and E. Remiddi, “Absence Of The Anomalous Magnetic Moment In A Supersymmetric Abelian Gauge Theory,” Phys. Lett. B 53, 347 (1974).

[4] For a review, see p.195-233 of J. F. Gunion, H. E. Haber, G. L. Kane and S. Dawson, “The Higgs Hunter’s Guide,”, Addison-Wesley 1990.

[5] R. Jackiw and S. Weinberg, “Weak Interaction Corrections To The Muon Magnetic Moment And To Muonic Atom Energy Levels,” Phys. Rev. D 5 (1972) 2396.

[6] K. Fujikawa, B. W. Lee and A. I. Sanda, “Generalized Renormalizable Gauge Formulation Of Spontaneously Broken Gauge Theories,” Phys. Rev. D 6 (1972) 2923.

[7] J. P. Leveille, “The Second Order Weak Correction To (G - 2) Of The Muon In Arbitrary Gauge Models,” Nucl. Phys. B 137 (1978) 63.

[8] H. E. Haber, G. L. Kane and T. Sterling, “The Fermion Mass Scale And Possible Effects Of Higgs Bosons On Experimental Observables,” Nucl. Phys. B 161 (1979) 493.

[9] J. A. Grifols and R. Pascual, “Contribution Of Charged Higgs Bosons To The Anomalous Magnetic Moment Of The Muon,” Phys. Rev. D 21 (1980) 2672.

[10] P. Franzini et.al, Phys. Rev. D35, 2883 (1987).

[11] J. Kurowska, O. Grajek and P. Zalewski [DELPHI Collaboration], “Search for Yukawa production of a light neutral Higgs at LEP 1,” CERN-OPEN-99-385.

[12] F. Wilczek, “Decays Of Heavy Vector Mesons Into Higgs Particles,” Phys. Rev. Lett. 39 (1977) 1304.

[13] G. Faldt, P. Osland and T. T. Wu, “Relativistic Theory Of The Decay Of Upsilon Into Higgs + Photon,” Phys. Rev. D 38 (1988) 164.

[14] S. N. Biswas, A. Goyal and J. Pasupathy, “Radiative Decay Of Quarkonium Into Higgs Scalar,” Phys. Rev. D 32 (1985) 1844.

[15] I. Aznauryan, S. Grigoryan, and S. Matinyan, JETP Lett. 43 (1986) 646.

[16] M. I. Vysotsky, “Strong Interaction Corrections To Semiweak Decays: Calculation Of The V → H Gamma Decay Rate With Alpha-S Accuracy,” Phys. Lett. B 97 (1980) 159.

[17] P. Nason, “QCD Radiative Corrections To Upsilon Decay Into Scalar Plus Gamma And Pseudoscalar Plus Gamma,” Phys. Lett. B 175 (1986) 223.

[18] For a review see, M. Krawczyk, P. Mattig and J. Zochowski, “The light Higgs window in the 2HDM at GigaZ,” hep-ph/0009201.