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Constraining the Two-Higgs-Doublet-Model parameter space

We confront the Two-Higgs-Doublet Model with a variety of experimental constraints as well as theoretical consistency conditions. The most constraining data are the \bar B\to X_s\gamma decay rate (at low values of M_{H^\pm}), and \Delta\rho (at both low and high M_{H^\pm}). We also take into account the B\bar B oscillation rate and R_b, or the width \Gamma(Z\to b\bar b) (both of which restrict the model at low values of \tan\beta), and the B^-\to\tau\nu_\tau decay rate, which restricts the model at high \tan\beta and low M_{H^\pm}. Furthermore, the LEP2 non-discovery of a light, neutral Higgs boson is considered, as well as the muon anomalous magnetic moment. Since perturbative unitarity excludes high values of \tan\beta, the model turns out to be very constrained. We outline the remaining allowed regions in the \tan\beta-M_{H^\pm} plane for different values of the masses of the two lightest neutral Higgs bosons, and describe some of their properties.


I. INTRODUCTION AND NOTATION
As compared with the Standard Model (SM), the Two-Higgs-Doublet Model (2HDM) allows for an additional mechanism for CP violation [1,2,3,4]. This is welcome, in view of baryogenesis [5,6], and one of the main reasons for continued interest in the model.
Several experimental constraints restrict it. The B −B oscillations and branching ratio R b exclude low values of tan β, whereas theB → X s γ rate excludes low values of the charged-Higgs mass, M H ± . The precise measurements at LEP of the ρ parameter constrain the mass splitting in the Higgs sector, and force the masses to be not far from the Z mass scale [7]. These individual constraints are all well-known, but we are not aware of any dedicated attempt to combine them, other than those of [8,9]. The present study aims to go beyond that of [9], by using more complete and more up-to-date experimental results, as well as more accurate theoretical predictions for the above quantities.
From the theoretical point of view, there are also various consistency conditions. The potential has to be positive for large values of the fields [10,11]. We also require the tree-level Higgs-Higgs scattering amplitudes to be unitary [12,13,14]. Together, these constraints dramatically reduce the allowed parameter space of the model.
The present study is limited to the 2HDM (II), which is defined by having one Higgs doublet (Φ 2 ) couple to the up-type quarks, and the other (Φ 1 ) to the down-type quarks [15].
We write the general 2HDM potential as: Thus, the Z 2 symmetry will be respected by the quartic terms, and Flavour-Changing Neutral Currents are constrained [16]. We shall refer to this model (without the λ 6 and λ 7 terms) as the 2HDM 5 . The more general model, with also λ 6 and λ 7 couplings, will be discussed elsewhere.
We allow for CP violation, i.e., λ 5 and m 2 12 may be complex. Thus, the neutral sector will be governed by a 3 × 3 mixing matrix, parametrized in terms of the angles α 1 , α 2 and α 3 as in [4,17]: where c 1 = cos α 1 , s 1 = sin α 1 , etc., and (In ref. [17], the angles are denoted asα = α 1 , α b = α 2 , α c = α 3 .) For a discussion of this parameter space, including the CP-nonviolating limits, see [19]. We will use the terminology "general 2HDM" as a reminder that CP violation is allowed. The present study extends that of [9] also in this respect. Rather than taking the parameters of the potential (1.1) to describe the model, we take the two lightest neutral Higgs boson masses, M 1 and M 2 , together with the mixing angles (1.3), the charged Higgs boson mass, M H ± , and tan β as our basic parameters. (The third neutral Higgs boson mass, M 3 , is then a derived quantity.) It is convenient to split the constraints into three categories: (i) Theoretical consistency constraints: positivity of the potential [10,11] and perturbative unitarity [12,13,14], (ii) Experimental constraints on the charged-Higgs sector. These all come from B-physics, and are due to b → sγ, B-B oscillations, and B → τ ν τ . They are all independent of the neutral sector.
(iii) Experimental constraints on the neutral sector. These are predominantly due to the precise measurements of R b , non-observation of a neutral Higgs boson at LEP2, ∆ρ, and a µ = 1 2 (g − 2) µ . The first and third categories of constraints will depend on the neutral sector, i.e., the neutral Higgs masses and the mixing matrix. The second category is due to physical effects of the charged-Higgs Yukawa coupling in the B-physics sector. These are "general" in the sense that they do not depend on the spectrum of neutral Higgs bosons, i.e., they do not depend on the mixing (and possible CP violation) in the neutral sector.
When considering the different experimental constraints, our basic approach will be that they are all in agreement with the Standard Model, and simply let the experimental or theoretical uncertainty restrict possible 2HDM contributions (this procedure yields lower bounds on the charged-Higgs mass, possibly also other constraints). An alternative approach would be to actually fit the 2HDM to the data. This will not be discussed in the present paper.
This paper is organized as follows. In Sec. II we discuss the general constraints of positivity of the Higgs potential, together with tree-level unitarity of Higgs-Higgs scattering amplitudes. The impact of these constraints is displayed in the tan β-M H ± plane for a few representative values of neutral Higgs boson masses. Next, in Sec. III, we discuss the constraints coming from the B-physics experiments, in particular theB → X s γ, B → τν τ and B-B oscillations. Sec. IV is devoted to various experimental constraints that depend on details of the neutral-Higgs sector. In Sec. V, we combine all the constraints and in Sec. VI we give some characteristics of the surviving parameter space. In Sec. VII we speculate on possible future experimental constraints, and then summarize in Sec. VIII.

II. GENERAL THEORY CONSTRAINTS
In the general CP-non-conserving case, the neutral sector is conveniently described by the three mixing angles, together with two masses (M 1 , M 2 ), tan β and We shall here project these constraints from the multi-dimensional parameter space onto the tan β-M H ± plane. Such a projection of information from a multidimensional space onto a point in the tan β-M H ± plane can be done in a variety of ways, all of which will lead to some loss of information. However, we feel that this loss of detailed information can be compensated for by the "overview" obtained by the following procedure: 1. Pick a set of neutral-Higgs-boson masses, (M 1 , M 2 ) together with µ 2 .
2. Scan an N = n 1 × n 2 × n 3 grid in the α 1 -α 2 -α 3 space, and count the number j of these points that give a viable model. (Alternatively, one could scan over N random points in this space.) 3. The ratio is then a figure of merit, a measure of "how allowed" the point is, in the tan β-M H ± plane. If Q = 0, no sampled point in the α α α = (α 1 , α 2 , α 3 ) space is allowed, if Q = 1, they are all allowed. An alternative measure counts in the denominator only those points N + for which positivity is satisfied.
Of course the 2HDM, if realized in nature, would only exist at one point in this parameter space. However, we think the above quantities Q and Q + give meaningful measures of how "likely" different parameters are.

A. Reference masses
We shall impose the conditions of positivity, unitarity and experimental constraints on the model, for the different "reference" mass sets given in Table I (and variations around these). For each of these mass sets we scan the model properties in the α α α = (α 1 , α 2 , α 3 ) space. (All scans have been performed over a 200 × 200 × 100 grid. We note that other scanning procedures might be more efficient [18].) From these reference masses, some trends will emerge, allowing us to draw more general conclusions.  In the 2HDM 5 , with Im λ 5 = 0, the two input masses will together with α α α and tan β determine M 3 [17]. Specifying also M H ± and µ 2 , all the λ's can be determined. (For explicit formulas, see [19].)

B. Positivity and unitarity
Let us first discuss the effect of imposing positivity. Actually, we will use the term "positivity" to refer to the non-trivial conditions M 2 3 > 0 and M 2 ≤ M 3 together with V (Φ 1 , Φ 2 ) > 0 as |Φ 1 |, |Φ 2 | → ∞. We shall henceforth refer to the set of points in the α α α space where positivity is satisfied, as α α α + . In Table II of [19] we report percentages Q of points in α α α space for which positivity is satisfied. For the mass parameters of Table I, the fraction is around 30%. For small and negative values of µ 2 , "most" of the exclusion provided by the positivity constraint is due to the conditions M 2 3 > 0 and M 3 > M 2 , without the explicit conditions on the λ's discussed, for example, in Appendix A of [11]. Percentage of points Q+ in the α α α+ space that satisfy unitarity. Four sets of (M1, M2) values are considered, as indicated. The contours show Q+ = 0, 20%, 40% and 60% (upper panels) and 0, 5%, 10% and 15% (lower panels). All panels: µ 2 = 0. Yellow region: Q+ > 0. Also shown, are the 90% and 95% C.L. exclusion contours from Fig. 2.
In Fig. 1, for µ 2 = 0, we study the effects of imposing unitarity [12,13,14]. This has a rather dramatic effect at "large" values of tan β and M H ± . While the general constraints on the charged-Higgs sector, to be discussed in Sec. III, exclude low values of tan β and M H ± , the constraints of unitarity exclude high values of these same parameters. Only some region in the middle remains not excluded. For (M 1 , M 2 ) = (100, 300) GeV and µ 2 = 0, unitarity excludes everything above tan β ∼ 5 (for any value of M H ± ), and above M H ± ∼ 650 GeV (for any value of tan β).
We shall refer to the set of α α α values for which unitarity as well as positivity are satisfied asα α α ∈ α α α + . For M 2 = 300 GeV (upper panels), the percentage of points in α α α + space for which unitarity is satisfied, reaches (at low tan β and low M H ± ) beyond 60%, whereas for M 2 = 500 GeV (lower panels), it only reaches values of the order of 15-20%.
The domains in which Q + > 0 depend on µ 2 : For negative values of µ 2 , the region typically shrinks to lower values of tan β, for positive values of µ 2 it extends to larger values of tan β. When µ 2 ≃ 0, and tan β ≤ 5, significant fractions of theα α α space are allowed. However, for large positive values of µ 2 , when also large values of tan β are allowed, only small domains inα α α remain allowed, as discussed in [19]. They can actually be hard to find in a regular, equidistant scan overα α α.
The unitarity constraints are conveniently formulated in terms of the different weak isospin and hypercharge channels [14]. At large values of tan β it turns out to be the isospin-zero, hypercharge-zero channel that is most constraining.

III. GENERAL CONSTRAINTS FROM THE H± SECTOR
The precise B-physics experiments provide severe constraints on the charged-Higgs sector, excluding low values of M H ± and tan β. We shall here discuss the three most severe constraints of this kind that are independent of the neutral sector.

A.B → Xsγ
TheB → X s γ branching rate was early found to constrain the allowed charged-Higgs-boson masses, but also to be very sensitive to QCD effects. At leading logarithmic order (LO), it is given by [20,21,22]: where the first factor is a ratio of CKM matrix elements, g(z = m 2 c /m 2 b ) is a phase space factor, and C  i (µ 0 ), i = 2, 7, 8 [23], where the effects of the 2HDM enter. Certain linear combinations are denoted "effective" coefficients, they are defined such that e.g. C (µ) includes all one-loop contributions to b → sγ, i.e., also those of four-quark operators [23]. The effective Wilson coefficient was early found to be quite sensitive to the scale relevant to B-meson decay, changing by ±25% if the scale µ b is varied by a factor of two in either direction around m b ≃ 5 GeV [23,24]. At the NLO, this scale sensitivity is however significantly reduced [25,26,27,28,29,30,31,32,33].
The additional contributions due to the 2HDM can at the weak scale be described by diagrams involving H ± exchange, and depend on this mass, M H ± , as well as on the Yukawa couplings, i.e., on tan β. At leading logarithmic order in QCD, they are discussed in [21,34]. Some of these additional terms can be enhanced by factors cot 2 β from the H ± Yukawa coupling squared. They all vanish linearly in m 2 t /M 2 H ± , i.e., for M 2 H ± ≫ m 2 t . The NLO results for the 2HDM have been studied by many authors, see [30,31,35,36]. As compared with the LO calculation, it has been found that the NLO effects weaken the constraints on the allowed region in the tan β-M H ± plane [23,35,36], the bound on M H ± is significantly relaxed.
Apart from minor effects, this calculation has for the SM been carried to the next-to-next-to-leading order (NNLO) [37]. At the NNLO, the branching ratio can be written as [31,38]: where P and N denote perturbative and non-perturbative effects that both depend on the photon lower cut-off energy E 0 . In the LO limit, the term P reduces to the square of the effective Wilson coefficient C , whereas NLO and NNLO contributions include effects due to gluon exchange and emission, and require the summation over a bilinear expression involving also other Wilson coefficients. The factor C accounts for m c -dependence associated with the semileptonic decaȳ B → X c eν e [31].
The SM prediction of Misiak et al. is (3.15 ± 0.23) × 10 −4 for E 0 = 1.6 GeV [37,38], if all errors are added in quadrature. This is to be compared with the recent experimental results, which are averaged to 3.55 × 10 −4 [40], with an uncertainty of 7-7.5%, again with statistical and systematic errors added in quadrature. Andersen and Gardi [41] advocate a different approach to the resummation of the perturbation series, by "Dressed Gluon Exponentiation", which accounts for multiple and soft collinear radiation. These effects are particularly important for large photon energies. Their approach yields (3.47 ± 0.48) × 10 −4 , i.e., they find a 14% uncertainty. Becher and Neubert also introduce further corrections to the calculation of the photon spectrum, and find a rather low value, (2.98 ± 0.26) × 10 −4 [42], leaving more room for new physics.
The 2HDM contribution is positive, a finite value for the charged Higgs mass would thus bring the results of [37,42] in closer agreement with the experiment. We shall however take the attitude that these numbers are compatible and compare the uncertainty in the experimental result and the SM prediction with the 2HDM contribution.
In the NNLO, the perturbative contribution P is obtained via the following three steps [38]: 1. Evaluation of the Wilson coefficients at the "high" (electroweak) scale, µ 0 [43,44]. These coefficients are expanded to second order in α s and rotated to "effective" Wilson coefficients C eff i (µ 0 ) [23,45]. The 2HDM effects enter at this stage, at lowest order in the Wilson coefficients C 7 (µ 0 ) and C 8 (µ 0 ).
2. Evaluation of the "running" and mixing of these operators, from the high scale to the "low" (B-meson) scale. This is where the main QCD effects enter via a matrix U that is given in terms of powers of [23,28,45,46].
3. Evaluation of matrix elements at the low scale [38,47], which amounts to constructing . We adopt the scale parameters of [37]: Actually, our treatment of the higher-order effects has been simplified compared to that described in [37,38], in the sense that: (i) We determine the contribution P Furthermore, while the "matching" in the SM is performed to second order in α s , we include, in addition to the dominant, lowest order, 2HDM effects, also the firstorder (in α s (µ 0 )) contributions. In fact, we include the latter at the level of "effective" Wilson coefficients, following [36], and take their "matching scale" µ W as µ 0 . Characteristic relative magnitudes of the LO and NLO 2HDM contributions are given in Table II, for tan β = 1 and 10, and M H ± = 300 GeV and 600 GeV. The LO 2HDM contribution is measured with respect to the full SM value, whereas the NLO 2HDM contribution (according to [36]) is measured relative to the SM value plus the LO 2HDM contribution. The missing O(α 2 s (µ 0 )) corrections appear unimportant. As input parameters for (3.2), in addition to (3.3), we take the CKM ratio to be 0.9676 [49], B(B → X c eν e ) = 0.1061 [50] (see also [51] For the non-perturbative part, N (E 0 ), we follow refs. [38] and [53,54]. Our result for the branching ratio, Eq. (3.2), in the SM limit (M H ± → ∞), is 3.12 × 10 −4 . We collect in Table III  We define a χ 2 measure of the amount by which the 2HDM would violate the agreement with the SM value, Here, B(B → X s γ) 2HDM denotes the 2HDM prediction. As noted above, it will depend on tan β and M H ± , whereas B(B → X s γ) ref denotes a reference value, taken to be the averaged experimental value, 3.55 × 10 −4 [40].
For the uncertainty that enters in (3.4), we adopt the value which corresponds to the experimental and (SM) theoretical uncertainties [37] added in quadrature. For the 2HDM, the theoretical studies have not been carried to the same level of precision, but we recall that the 2HDM-specific NLO-level contributions only modify the over-all branching ratios by O(5%). These numbers give χ 2 min = 1.52.
With the above choice, we can determine exclusion regions in the tan β-M H ± plane. Such regions are shown in Fig. 2 for χ 2 = 4.61 and 5.99, corresponding to confidence levels of 90% and 95%, respectively, for 2 degrees of freedom (tan β and M H ± ). The obtained bound differs from that given by Misiak et al. [37] even though the branching ratio is reproduced to a reasonable precision. The reason is as follows. For compatibility with the treatment of the other experimental constraints, we consider a two-dimensional constraint, whereas they consider a single-sided one-dimensional constraint. 1 By adopting the approximate SM-value P (2)rem 2 = 5, we only include part of the 2HDM-specific contributions to P (2) . A more well-defined procedure would be to leave out all 2HDM-specific contributions to P (2) . 2 However, that procedure leads to an exclusion limit about 30 GeV higher in M H ± , i.e., more of the parameter space would be excluded. We have chosen the more conservative approach, which leads to less exclusion. has been obtained [55] at the 2σ level. This bound, which would exclude a corner to the lower right outside the region shown in Fig. 2 Interpreted in the framework of the 2HDM, one finds [55] (3.10) We take m B = 5.28 GeV, and formulate a χ 2 measure as It follows that two sectors at large values of tan β and low values of M H ± will be excluded. This bound is stronger than that of (3.7). It has some relevance in the lower right of Fig. 2, when all effects are added (see Sec. III D).

C. B-B oscillations
The precisely measured B d -B d oscillations are at lowest order given by the formula [57] (3.13) The contributions proportional to diagrams with the exchange of one or two charged-Higgs-bosons are denoted S W H and S HH . They are proportional to cot 2 β and cot 4 β, and vanish when M H ± ≫ m t , as (m 2 t /M 2 H ± ) log(m 2 t /M 2 H ± ) and m 2 t /M 2 H ± , respectively. At the NLO, the corresponding result has also been obtained [59]: B B and η (denoted η 2 in [59]) receive corrections of order α s . In particular, the O(α s ) contribution to η 2 becomes a non-trivial function of tan β and M H ± . A few comments are here in order: (i) The factor of 2 for the second term in (3.13) has been adopted to follow the notation of [59]. (ii) There is a book-keeping problem with the expression for L (i,H) in Eq. (A.20) of [59]. Since the last term in that expression, proportional to a quantity denoted HH, has an explicit coefficient 1/ tan 4 β, the S HH in Eqs. (A.21) and (A.24) for HH (i) should be replaced by tan 4 β × S HH . (iii) There is a discrepancy in a quantity denoted 2W W (8) tu , between Eq. (A.16) in [59] and the later PhD thesis of the same author. We have chosen to take the formula given in the thesis. At the level of η 2 , it amounts to a difference of the order of 2%. 3 The O(α s ) corrections to η introduce a variation of η (or η 2 ) from 0.334 at tan β = 0.5 and M H ± = 200 GeV to 0.552 at tan β = 50. (We have adopted the overall normalization of η such that it agrees with the SM value 0.552 [60] at tan β = 50.) However, the product η × S 2HDM varies by a factor of 2.7 over this same range, as compared with a factor of 4.5 for S 2HDM itself. Thus, the inclusion of the O(α s ) QCD corrections reduce the sensitivity of ∆m B to charged-Higgs contributions. In other words, the inclusion of NLO corrections weakens the constraints on the 2HDM at low values of tan β.
Recently, also the B s -B s oscillation parameter ∆m Bs has been measured [61]. It is given by an expression similar to (3.12), except that the CKM matrix elements are different:  .12) and (3.14) are determined from lattice QCD studies, and rather uncertain. We shall use the values adopted recently by Ball and Fleischer [60], based on unquenched calculations. In fact, these uncertainties are the dominant ones. For each of these observables, one can form a χ 2 measure of the deviation from the SM: We adopt the attitude that the measurements of ∆m B d and ∆m Bs furnish determinations of the CKM matrix elements |V td | and |V ts | that are compatible with the SM, and simply require that the additional 2HDM contributions do not spoil this consistency. With the assumption that the uncertainties in (3.15) are the dominant ones, we have However, with the theory error (3.15) being the dominant one, we can not claim that the measurements of ∆m B d and ∆m Bs furnish independent constraints on the model. Therefore, we consider only the χ 2 contribution from the more constraining one of these two measurements, namely ∆m B d with a 21% uncertainty: Contours of χ 2 B−B are shown in Fig. 2. As mentioned above, they are more generous than the corresponding bounds based on the LO theory.

D. Combining general constraints from the H ± sector
In order to quantify the extent to which a particular point in the tan β-M H ± plane is forbidden, we form a χ 2 as follows: The subscript "general" refers to the fact that constraints depending on the neutral sector are not yet taken into account. Yet, no choice of neutral-sector parameters can avoid these constraints. Figure 2 shows excluded regions in the tan β-M H ± plane, due to the constraints ofB → X s γ, B − → τν τ and B-B oscillations at 90% (dashed) and 95% C.L. (solid), as well as combined (indicated in yellow and red at 90% and 95% C.L., respectively). Earlier versions of such exclusion plots can be found in [8,9,31].

IV. EXPERIMENTAL CONSTRAINTS ON THE NEUTRAL SECTOR
We now turn to the constraints coming from experiments related to the neutral Higgs sector. These are not "general", they will depend on the parameters of this sector, namely neutral Higgs masses and mixing angles. We will study, as representative cases, those given in Table I. A problem with point 2 of the procedure of Sec. II, is the following. The conditions of positivity and unitarity are absolute 4 , with a certain fraction Q + of the points in the α α α + space satisfying these, whereas the experimental constraints are statistical, i.e., a certain parameter point may violate an experimental observation by 1σ, 2σ or more. There is no obvious way to combine this information.
As a first step, we will consider one of these experimental constraints on the neutral sector at a time, and display the way it excludes parts of the tan β-M H ± plane by the following simple consideration. Let χ 2 i be the contribution to χ 2 due to a particular experimental observable O i : where the "reference" value O i,ref will be either the experimental value or the SM value. We shall consider the following observables: R b , the branching ratio for Z → bb [62] (in Sec. IV A); the LEP2 non-discovery of a light neutral Higgs boson [63,64] (in Sec. IV B); ρ, the LEP determination of the relation between the Z and W masses [65] (in Sec. IV C); and a µ = 1 2 (g − 2) µ , the precise Brookhaven determination of the muon anomalous magnetic moment [66] (in Sec. IV D). Then, the criterion we will consider in this first step, can be expressed in terms of the following quantity: • For fixed tan β and M H ± takê where χ 2 i is minimized over the partα α α of the α α α + space for which positivity and also unitarity are satisfied.
The minimization overα α α finds the point where the model can most easily accommodate the particular experimental constraint, with the chosen masses (M 1 , M 2 ) and µ 2 fixed. This point inα α α will in general differ from one experimental constraint to another. We will establish 90 and 95% C.L. allowed regions in the tan β-M H ± plane, corresponding to one of these observables at a time. These will subsequently be combined. However, the over-all allowed regions will in general be less than the intersection of the individual ones, since the latter will correspond to different points inα α α.
This procedure preserves none of the probabilistic information of Fig. 1, which shows that some region of the tan β-M H ± plane contains more possible solutions than some other region. All focus is here on the one "best" point in α α α, where χ 2 is lowest.
The 2HDM-specific contributions to R b are of two kinds. At low values of tan β, the exchange of charged Higgs bosons is important, whereas at high values of tan β the exchange of neutral Higgs bosons is important [67]. For the general CP non-conserving case, the latter contributions are given in [11].
For this observable, the reference value in (4.1) is not well defined, since R b [68] is part of the electroweak observables from which a SM Higgs mass is fitted. Hence, we take where ∆R b,2HDM refers to the 2HDM-specific contributions to this quantity [11,67] and with σ(R b ) = 0.05% the experimental uncertainty [62]. We show in Fig. 3, for two sets of (M 1 , M 2 ) values, and µ = 0, how this constraint removes a sliver of low-tan β values. The allowed regions are here cut off at tan β > ∼ 7 due to the unitarity constraint discussed in Sec. II B. The neutral-Higgs-exchange contribution to R b is in this case of no importance.
Next, we show in Fig. 4, the corresponding allowed regions for µ = 200 GeV. The large-tan β region is then  less constrained by the unitarity constraints, but for the higher value of M 2 (right panel) the R b modification that is caused by neutral-Higgs exchange starts to exclude high values of tan β.

B. LEP2 non-discovery
The non-discovery of a neutral Higgs boson at LEP2 is relevant only for M 1 < 114.4 GeV [62]. However, it is well known that these searches do not exclude certain other light neutral Higgs bosons, if they couple more weakly to the Z boson, or if they decay to final states that are more difficult to detect and identify.
The constraint is implemented in an approximate way as follows. Following [63] (see also [64]), we consider the searches for a neutral Higgs boson that decays to bb jets or a tau pair. Thus, the quantity of interest is the product of the production cross section (proportional to the square of the ZZH 1 coupling) and the bb (or τ + τ − ) branching ratio (proportional to the square of the H 1 bb coupling). We denote the reduced sensitivity, as compared to the SM sensitivity, a "dilution factor" C 2 [63]: (4.4) where the dilution factor is the product of a factor C 2 (ZH 1 ) related to the production and another, C 2 (H 1 → bb), related to the branching ratio: For the general 2HDM, C 2 (ZH 1 → Zbb) ≡ C 2 2HDM is given by Eq. (4.3) in [11]. It depends on the neutralsector rotation angles, as well as on tan β, and is the same for bb and τ + τ − final states.  As experimental constraints, we approximate the 95% C.L. bounds obtained in [63] by linear interpolations passing through the points given in Table IV, and form the ad hoc single-sided χ 2 penalty (summed over bb and τ + τ − ): for C 2 2HDM > C 2 exp and M 1 < 114.4 GeV, and zero otherwise. The coefficient 5.99 corresponds to 95% probability in the context of our two "degrees of freedom", tan β and M H ± . We show in Fig. 5 the resulting exclusion for M 1 = 100 GeV, µ = 0 and two values of M 2 as indicated. For the higher value of M 2 , we note that some part of the otherwise allowed parameter space gets excluded. Given that C 2 2HDM is determined by tan β and the α α α parameters, on might wonder why this excluded region depends also on M H ± . The reason is of course that the subspace of α α α that is allowed by the positivity and unitarity constraints depends on M H ± , and need not overlap with the corresponding subspace of α α α for which C 2 2HDM is within the allowed range.  The ρ parameter, defined as is very sensitive to fields that couple to the W and Z [65]. Experimentally, ρ is constrained as [68] ρ exp = 1.0050 ± 0.0010. (4.10) The deviation from unity is mostly due to the top-quark one-loop contributions, but there is also a weak dependence on the SM Higgs mass. In order to extract this quantity from the data, one fits for a SM Higgs mass, the contribution of which should then be subtracted from the 2HDM prediction. The additional Higgs fields of the 2HDM can easily spoil the agreement with the SM [7]. Roughly speaking, this constraint requires the Higgs masses to be not too far from the W and Z masses, and not very much apart. For the general CP-non-conserving 2HDM, the results of [7] for the contributions to ρ, denoted ∆ρ = ρ 2HDM − ρ SM , were generalized in [11] and given by Eqs. (4.8)-(4.12) there. In order to study this constraint, we evaluate χ 2 ρ as defined by with σ(ρ) = 0.0010, subtracting the SM contribution corresponding to an SM Higgs mass M 0 = 129 GeV. In Figs. 7 and 8 we show the impact ofχ 2 ρ in constraining the parameter space for µ = 0 and µ = 200 GeV, respectively. At the higher value of M 2 , and high values of tan β, this constraint tends to exclude both low and high values of M H ± . The irregular boundaries seen in Fig. 8 are obviously due to the scanning not finding the whole allowed region. (At high µ and high tan β, only very tiny regions in α 1 and α 2 are allowed [19].) It is instructive to consider the contribution to ∆ρ in the limit of large values of tan β. The expressions (4.10) and (4.12) of [11] simplify considerably in the limit sin β → 1, and provide an understanding of features seen in Figs. 7 and 8: and with F ∆ρ (m 2 1 , m 2 2 ) found in Eq. (4.11) of [11]. This function vanishes when the two masses are equal, and grows quadratically with the bigger of the two masses. The contribution given by (4.13) is rather small, since M Z and M W are relatively close.
The only part sensitive to the charged-Higgs mass is the first sum in (4.12), which may however get significant contributions from all three neutral Higgs bosons, j = 1, 2, 3. Consider first the contribution from H 1 . Barring cancellations (see below), a viable parameter point must for large tan β and large M 2 Expressed in terms of the angles, this means that c 2 1 c 2 2 + s 2 2 ≪ 1, or |α 1 | ∼ π/2, and α 2 ∼ 0. (4.15) Note that this condition is compatible with (4.7) and corresponds to case (iii) of (4.8).
Next, we consider the case (with increasing splitting between M 2 and M 3 , the contribution to ∆ρ will increase) This condition cannot be satisfied, and low values of M H ± are thus excluded when tan β is large and M 2 is large.
For high values of M H ± , the most sensitive contribution is that involving H 1 . But unless M H ± is close to M 2 , the condition (4.15), related to the H 1 contribution, is not sufficient. The H 2 and H 3 contributions are proportional to R 2 21 + R 2 23 and R 2 31 + R 2 33 , respectively. In the limit (4.15), these are both equal to 1, i.e., when the rotation matrix is adjusted such as to cancel the H 1 contribution to the first sum in (4.12), there is no suppression of the H 2 and H 3 contributions. As a result, such high values of M H ± are forbidden.
An exception to this situation arises when µ is large compared with M 2 . Then, there can be a considerable splitting between M 2 and M 3 , M H ± and M 3 can be similar, and a cancellation between the (M 1 , M H ± ) and the (M 1 , M 3 ) terms of (4.12) is possible. As a result, for large values of µ, large values of M H ± can be allowed.
Finally, we consider the situation which applies to the "finger" protruding to the right in the right panel of Fig. 8. Then, the contributions of the two sums in Eq. (4.12) tend to cancel. For j = 1 and k = 2, 3, we get where we have used the orthogonality of the rotation matrix. Thus, when tan β is large, and M 2 ≃ M 3 ≃ M H ± , there is a cancellation among the terms in (4.12), only the (small) (4.13) part of the ∆ρ constraint is relevant.
In the 2HDM, there are additional contributions, dominated by the two-loop Barr-Zee effect [70] with a photon and a Higgs field connected to a heavy-fermion loop. For the CP-conserving case, the contribution is given by [9,71]. For the general (CP-violating) 2HDM, the top-quark contributions to a µ for the muon, is given by Eq. (4.13) in [11], whereas the b-quark contributions to the fermion loop is given by (4.22) with N c = 3 the number of colours associated with the fermion loop, α e.m. the electromagnetic finestructure constant, Q b = −1/3 and m b the b-quark charge and mass, and m µ the muon mass. The τ -loop contribution, which we also include, is given by a similar expression, with obvious substitutions for the colour factor, charge and mass. The functions f and g are given in [70].
At low values of tan β, these contributions are negligible, but the b-and τ -loop contributions can become relevant at very large values of tan β. As a measure of the possible conflict with the 2HDM, we consider where ∆a µ,2HDM is the 2HDM-specific contribution, and for the uncertainty we take the (SM) theoretical value, σ(a µ ) = 6.8 × 10 −10 , since this is larger than the experimental one. Actually, this constraint does not have any significant impact within the range of tan β considered. Let us consider its "natural value" as that contributed by one of the two terms in (4.22), with the rotation matrix element set to 1. This reaches χ 2 = O(1) for tan β of the order of 70. However, there can be significant reductions by the rotation matrix elements, and also cancellations among the two terms. At such high values of tan β the B → τ ν τ constraint (see Sec. III B) is important at low values of M H ± , and the unitarity constraint may be important at high M H ± (depending on the relative magnitude of M 2 and µ).
We recall that at high tan β, the rotation matrix is rather constrained by unitarity [19]. Let us focus on the contribution of the lightest neutral Higgs boson, H 1 , whose contributions to (4.22) are given by R 2 11 and R 2 13 . Thus, in spite of the enhancement given by the tan βdependent factors, this contribution to a µ may be small if |α 1 | ∼ π/2, and α 2 ∼ 0. (4.24) In the limit of large tan β and large M H ± , this is actually the only region allowed by the ∆ρ constraint, see Eq. (4.15). We conclude that at large tan β and large M H ± , the a µ constraint is covered by the ∆ρ constraint. But this coefficient R 2 11 + R 2 13 arises in one case from the Yukawa couplings, and in the other from the gauge-Higgs couplings. Furthermore, at large tan β and moderate M H ± values, the a µ constraint is covered by the B → τ ν τ constraint.

E. Summary on neutral-sector constraints
We will here summarize the conclusions on the neutralsector constraints, treating first the simpler case of µ = 0 (where tan β is bounded), and next comment on the less restrictive case of "large" µ (where also larger values of tan β are allowed). The R b constraint is at low values of tan β dominated by the charged-Higgs-exchange contribution. This part of the R b constraint is thus independent of the neutral sector.

The case µ = 0
For µ = 0, and (M 1 , M 2 ) = (100, 300) GeV, the only neutral-sector constraint that has some impact, apart from R b at low values of tan β, is the ∆ρ constraint, which excludes the higher range of M H ± , as illustrated in the left panel of Fig. 7. However, for M 2 = 500 GeV, other constraints are also important. The LEP2 nondiscovery rules out large values of tan β and M H ± , and to some extent, also the ∆ρ constraint rules out some region of large tan β.
However, with µ = 0, high values of tan β are also excluded by the unitarity constraints, and, to some extent, the low values of M H ± are excluded by the B-physics constraints.

The case µ > M1
When µ > M 1 , large values of tan β become accessible. This parameter region is known as the decoupling region [72]. We here distinguish two cases (4.25) The R b constraint, which at low tan β is dominated by the charged-Higgs contributions, can at large tan β also exclude some region of neutral Higgs boson mass values (compare the left and right panels of Fig. 4). Furthermore, the LEP2 constraint may exclude high values of M H ± (see right panel of Fig. 6) and the ∆ρ constraint may at high tan β constrain the range of M H ± values to a band around M 2 .

V. COMBINING ALL CONSTRAINTS
Let us now combine all constraints. This is done by a dedicated scan overα α α for each point in tan β and M H ± . The value of χ 2 is determined as where χ 2 general is given by Eq. (3.19) and the sum runs over the observables R b , LEP2 non-discovery, ∆ρ and a µ , all of them evaluated at the same point inα α α. This quantity is then minimized overα α α: for fixed tan β and M H ± and allowed regions are determined. In the tan β-M H ± plane, the allowed regions will in general be less that the intersection of the regions that are allowed by the individual constraints. The reason is that the individual constraints may refer to different parts of the three-dimensional α α α space. We shall split this discussion into the two cases µ = 0 and M 1 < µ. In the former case, unitarity restricts the allowed range of tan β, as illustrated in Fig. 1, whereas in the latter case also higher values of tan β are allowed.
A. Combining all constraints for µ = 0 In Fig. 9 we display the 90 and 95% C.L. limits for (M 1 , M 2 ) = (100, 300) GeV and (100, 500) GeV, in both cases for µ = 0. For the case of moderately low M 2 = 300 GeV, we note that there is little additional exclusion, other than that due to the B-physics constraints and unitarity. The little extra is due to the ∆ρ constraint, at high values of M H ± . However, for M 2 = 500 GeV, there is a considerable reduction of the allowed parameter space at tan β > ∼ 1 − 1.5. In this range of tan β values, we see from Similarly, Fig. 10 is devoted to the case M 1 = 150 GeV. In this case, the LEP2 non-discovery plays no role, but there are of course also other differences, due to the way different parameters are correlated by the constraints. Overall, this case is less constrained than the M 1 = 100 GeV case of Fig. 8.

B. Combining all constraints for M1 < µ
When M 1 < µ, the unitarity constraints no longer restrict tan β to low and moderate values. However, we shall see that various other constraints may cause a cutoff for large tan β. For M 1 = 100 GeV and two values of M 2 , namely 300 GeV and 500 GeV, we display in  For the lower value, M 2 = 300 GeV (left panels), the allowed region in the tan β-M H ± plane is fairly extended, whereas for the higher value, M 2 = 500 GeV, it is more constrained, until µ reaches values comparable to M 2 (see the right panels of Figs. 11, 12 and 13).
For µ = 500 GeV and M 2 = 500 GeV (not shown), the exclusion of low and high values of M H ± for tan β > ∼ 2 is due to the ∆ρ constraint, whereas the LEP2 nondiscovery by itself does not exclude anything in this case. However, the simultaneous imposition of the LEP2 nondiscovery and the ∆ρ constraints yields a "forbidden finger" at tan β ∼ 1.5 and low M H ± , as well as some exclusion at low tan β and high M H ± .
In all cases, we note that simultaneously high values of both tan β and M H ± are excluded, except when M 2 < µ. This is due to the ∆ρ constraint, as discussed in Sec. IV C. For the cases shown (M 1 = 100 GeV), also the LEP2 non-discovery plays a role, but this will of course not have any impact for M 1 > 114.4 GeV.

VI. PROFILE OF SURVIVING PARAMETER SPACE
We shall here give a profile of the surviving parameter space in terms of three "hidden" parameters, α 1 , α 2 and M 3 . In discussing the surviving parameter space, we shall distinguish between low and high values of tan β, giving some details relevant to tan β < 5 and tan β > 10. At low values of tan β, both low and high values of µ lead to consistent solutions, with values of α 1 and α 2 distributed over extended regions of the parameter space, as shown in Fig. 14 for the case µ = 400 GeV, M 1 = 100 GeV and two values of M 2 , namely 300 GeV and 500 GeV. (Further plots of this kind, but taking into account only the positivity and unitarity constraints, are presented in [19]. ) We have here plotted the distributions of all α 1 and α 2 for which the total χ 2 < 5.99 (see Eq. (5.1)), i.e., in general several points in (α 1 , α 2 ) for each point in (tan β, M H ± ). At very low tan β, an important constraint is the positivity of λ 2 : and the constraint from unitarity that it does not become "large". For small µ, the M 2 3 -term cannot be too large (in order not to violate unitarity). This requires As µ becomes large, with M 2 fixed, the M 2 3 -term must compensate the µ 2 -term, with |c 1 s 3 + s 1 s 2 c 3 | = O(1). The distributions in Fig. 14 are seen to satisfy this condition.
While the allowed range of M H ± depends on the neutral Higgs boson mass M 2 , it is typically of the order of 300-700 GeV. In most of the allowed parameter space, CP is violated, but along the edges α 2 → ±π/2 there is no CP-violation. This is the limit where the lightest Higgs boson, H 1 , becomes CP odd.

B. High values of tan β
We show in Fig. 15 the populated regions in the α 1α 2 plane, for µ = 400 GeV and tan β > 10. Two points are worth noting: (1) The allowed regions satisfy the constraints of (4.8). (2) The majority of points do not satisfy the condition (4.15), meaning that the M H ± ≫ M 1 case is not very relevant here. Instead, the degenerate case, µ ∼ M H ± ∼ M 2 < ∼ M 3 is important, as also reflected in Table V c). At high values of tan β, µ has to be comparable to M 2 , or higher. In particular, no solution exists for high values of tan β and µ = 0. This is due to the unitarity constraints. Also, we note from Figs. 9-13, that unless µ is comparable to, or larger than M 2 , the allowed range in tan β and M H ± can be rather limited. The range of M H ± depends on the neutral Higgs boson mass M 1 , being typically of the order of 400-600 GeV. In order to understand the large-tan β parameter space, let us review This should be positive, but not "too large". High values of µ require high M 3 and |c 1 s 2 c 3 − s 1 s 3 | = O(1).

C. Distribution of M3
We recall that with our choice of parameters, the third neutral Higgs mass is a derived quantity. The distribution of M 3 can be discussed in terms of the dimensionless ratio For three bins in ξ, this is for µ = 0, 200, 400 and 600 GeV distributed as given in Table V. For µ = 0, M 3 tends to be low, just marginally above M 2 , as seen in Table V a). This pattern is valid also for M 1 < µ, provided only that µ < M 2 , see

D. The Standard-Model-like limit
The parameters of the 2HDM can be chosen such that the ZZH 1 , bbH 1 and ttH 1 couplings all approach the corresponding SM values. This requires, in our notation [11] cos(β − α 1 ) cos α 2 ≃ 1, cos α 1 cos α 2 cos β ≃ 1, sin α 1 cos α 2 sin β ≃ 1, (6.5) which is satisfied for β ≃ α 1 and α 2 ≃ 0. There could also be a "quasi-SM-like" limit, where one or more of the above quantities approaches −1 (denoted "Solution B" in [73]). For the familiar observables, such a sign change would not have any effect. For tan β < ∼ 5, the allowed regions in the α 1 -α 2 space are rather extended, and SM-like solutions are found for a range of mass values. For tan β > ∼ 10, on the other hand, the populated parts of the α 1 -α 2 space become very localized, and have the following features: (i) for M 2 > ∼ µ, α 1 ≃ 0 and |α 2 | > 0, and (ii) for M 2 < ∼ µ, additional regions emerge for small values of |α 2 | and |α 1 | ≃ π/2. The latter, seen as small specks near the horizontal axis in the left panel of Fig. 15, correspond to the SM-like (and "quasi-SM-like") case.
For the case M 1 = 100 GeV, which is studied in most of our figures, there is of course a limit to how close we come to the SM limit, since an SM Higgs mass of this low value is excluded. Actually, the fact that for (M 1 , M 2 ) = (100, 500) GeV and µ = 0, some region of the tan β-M H ± plane is excluded by the LEP2 non-discovery constraint (see right panel of Fig. 6), means that those regions correspond to solutions near the SM limit.

VII. POSSIBLE FUTURE CONSTRAINTS
It is interesting to see how the parameter-space constraints would be modified by possible future results from the B-physics sector. We shall not consider any change to the ∆ρ or LEP2 constraints, but rather discuss the possibility that the central value for the branching ratio forB → X s γ be reduced from 3.55 × 10 −4 to 3.20 × 10 −4 , a value closer to the SM prediction. Also changing the overall uncertainty to 0.25×10 −4 , we find that the resulting constraints are significantly modified, as illustrated in Figs. 16 and 17 for µ = 0 and 400 GeV, respectively.

VIII. SUMMARY
We have shown that the B-physics results, together with the precise measurement of the ρ-parameter at LEP and the constraint of tree-level unitarity of Higgs-Higgs scattering, exclude large regions of the 2HDM (II) parameter space. High values of tan β are excluded unless both M 2 and M 3 are heavy. Furthermore, they should be reasonably close to each other. Improved precision of theB → X s γ measurement could significantly reduce the remaining part of the parameter space, but it appears unlikely that the model could be excluded other than by a negative search at the LHC.
What is the corresponding situation for supersymmetric models? While the consistency then is guaranteed by internal relations, it should be kept in mind that light charged Higgs bosons would be in conflict with the Bphysics data unless some superpartner (for example, the chargino [74]) is also light. A possibility which has received some attention, is a light chargino and a light stop [75]. It has also been shown that anomalous effects at large tan β could weaken the bound on M H ± without light superpartners [76] (see also [77]). However, the more recent data on B → τ ν τ discussed in Sec. III B would presumably close this loophole (see Fig. 2 of [78]).
Similar scans over the parameter space of the Constrained MSSM [79,80,81] differ from the present work in one major respect: they are required to yield an amount of dark matter that is compatible with the WMAP data [82]. Additionally, the a µ constraint is more severe, due to one-loop contributions involving superpartners of the muon and muon neutrino. These studies are also more focused on the high-scale parameters, like m 0 and m 1/2 , with less emphasis on M H ± and tan β. The study by [80] shows a preference for positive µ (higgsino mass parameter, not to be confused with the µ of Eq. (2.1)), a relatively light charged Higgs mass (a few hundred GeV) and rather high values of tan β (∼ 50 − 60).