Determining $\tan\beta$ with Neutral and Charged Higgs Bosons at a Future $e^+e^-$ Linear Collider

The ratio of neutral Higgs field vacuum expectation values, tan(beta), is one of the most important parameters to determine in either the Minimal Supersymmetric Standard Model (MSSM) or a general type-II Two-Higgs Doublet Model (2HDM). Assuming an energy and integrated luminosity of sqrts=500 GeV and L=2000 fb-1 at a future linear collider (LC), we show that a very accurate determination of tan(beta) will be possible for low and high tan(beta) values by measuring the production rates of Higgs bosons and reconstructing Higgs boson decays. In particular, based on a TESLA simulation, and assuming no other light Higgs bosons and 100<=mA<=200 GeV, we find that the rate for the process e+e- -->bbA -->bbbb provides a good determination of tan(beta) at high tan(beta). In the MSSM Higgs sector, in the sample case of mA = 200 GeV, we find that the rates for e+e- -->bbA+bbH -->bbbb and for e+e- -->HA -->bbbb provide a good determination of tan(beta) at high and low tan(beta), respectively. We also show that the direct measurement of the average total widths of the H and A in e+e- -->HA -->bbbb events provides an excellent determination of tan(beta) at large values. In addition, the charged Higgs boson process e+e- -->H+H- -->tbtb has been studied. The sensitivity to tan(beta) at the LHC obtained directly from heavy Higgs boson production is briefly compared to the LC results.


I. INTRODUCTION
Theories beyond the Standard Model (SM) that resolve the hierarchy and fine-tuning problems typically involve extensions of its single-doublet Higgs sector to at least a two-doublet Higgs sector (2HDM) [1]. The most attractive such model is the Minimal Supersymmetric Standard Model (MSSM), which contains a constrained two-Higgs-doublet sector [2]. In other cases, the effective theory below some energy scale is equivalent to a 2HDM extension of the SM with no other new physics. Searching for the Higgs particles and studying their properties have high priority for both theoretical and experimental activities in high energy physics.
Among other new parameters in 2HDM and SUSY theories, one is of particular importance: the ratio of the vacuum expectation values of the two Higgs fields, commonly denoted as tan β = v 2 /v 1 . It characterizes the relative fraction that the two Higgs doublets contribute to the electroweak symmetry breaking v 2 = v 2 1 + v 2 2 , where v ≈ 246 GeV. The five physical Higgs states couple to the fermions at tree-level [1,2] as where α is the mixing angle in the CP-even sector, and the approximation indicates the decoupling limit for m A ≫ m Z in the MSSM [3,4], in which the couplings of the light Higgs boson h become SM-like. Eqs. (2)- (4) show that tan β governs the coupling strength of Yukawa interactions between the fermions and the heavy Higgs bosons. In fact, heavy Higgs boson measurements sensitive to their Yukawa couplings are far and away the most direct way to probe the structure of the vacuum state of the model as characterized by the ratio of vacuum expectation values that defines tan β. The parameter tan β enters all other sectors of the theory in a less direct way [1]. For instance, in supersymmetric theories the interactions of the SUSY particles have tan β dependence. In addition, the relations of SUSY particle masses to the soft SUSY breaking parameters of supersymmetry involve tan β. The renormalization group evolution of the Yukawa couplings from the unification scale to the electroweak scale is sensitive to the value of tan β. The large top quark mass can be naturally explained with m b − m τ unification as a quasi-infrared fixed point of the top Yukawa coupling if tan β ≃ 2 or tan β ≃ 56 [5]. The possibility of SO (10) Yukawa unification requires high tan β solutions [6]. The predicted mass of the lightest SUSY Higgs boson also depends on tan β, with a higher mass at larger tan β [7]. It will be very important to compare the measurements of and constraints on tan β from these other sectors of the theory to the direct determination of tan β coming from the heavy Higgs boson measurements that depend fundamentally on tan β through the Yukawa couplings.
Currently, some regions of the MSSM parameter space have been excluded at LEP due to the lower bound on the lightest Higgs boson mass. (A review of LEP-1 Higgs results shows possible signatures for all neutral and charged Higgs boson search channels [8].) Particularly collisions [32,33,34], and improved sensitivity to tan β was obtained by considering H ± tb [35] and A/Hbb, A/Htt [36]. If a muon collider becomes available to produce a heavy Higgs boson in the s-channel, its coupling could be measured with very excellent precision [37].
In this paper, we perform a comprehensive analysis of tan β determination via heavy Higgs boson production and decay at an e + e − linear collider with √ s = 500 GeV and an integrated luminosity of 2000 fb −1 . We amplify upon the results for the heavy neutral Higgs bosons obtained during the last Snowmass workshop (as summarized in [38]) and extend our study to include the charged Higgs boson. We show how various Higgs boson measurements can be used to determine tan β. The different types of measurements we consider are complementary in that some provide good precision at low tan β and others at high tan β; combined, a good determination of tan β is possible throughout its whole range. We include background simulations and realistic b-tagging efficiencies. In Sec. II, we focus on the heavy Higgs bremsstrahlung process bbA → bbbb, the production rate for which is directly proportional to tan 2 β. We then include the bbH → bbbb process in the context of the MSSM and estimate the accuracy with which tan β can be determined by combining experimental results for both processes. In Sec. III, we examine the pair production of a CP-even Higgs boson (h or H) and the CP-odd A, followed by decay of the Higgs bosons to the bbbb final state. In particular, at large tan β, the total decay widths of the heavy Higgs bosons can be broad since these widths are proportional to tan 2 β. The resulting accuracy for the tan β determination is obtained. We extend these studies to charged Higgs in Sec. IV. In Sec. V, we briefly summarize the LHC sensitivity to tan β deriving from heavy Higgs production. Finally, we summarize our results in Sec. VI. Before proceeding with our analysis, we would like to point out that we are taking a phenomenological approach to the tan β determination. Namely, we only consider tan β as an effective way of specifying the coupling for the Higgs bosons and fermions through the usual tree-level relations and explore the extent to which this coupling can be experimentally determined at the linear collider experiments. After including radiative corrections, the relation of tan β to the various Yukawa couplings becomes process-dependent. For a recent theoretical discussion on the issue of the gauge dependence of these relations, see Ref. [39]. In this context, our results should be viewed as giving the accuracy with which the actual Yukawa couplings can be measured.

II. THE bbA → bbbb BREMSSTRAHLUNG PROCESS
Searches for bbA and bbh were performed in the four-jet channel using LEP data taken at the Z resonance [40,41,42,43]. In this section, we consider a linear collider with a center-of-mass energy of 500 GeV or higher, and begin by focusing just on the bbA production process that probes the direct coupling of the CP-odd Higgs A to bb. Our analysis will employ cuts designed to eliminate Higgs pair resonant production, which, when kinematically accessible, dominates A production before cuts but is less sensitive to tan β. The challenge of this study is the low expected production rate and the large irreducible background for a four-jet final state, as discussed in a previous study [44]. A LC analysis has been performed using event generators for the signal process e + e − → bbA → bbbb [45] and the e + e − → eW ν, e + e − Z, W W, ZZ, qq (q = u, d, s, c, b), tt, hA background processes [46] that include initial-state radiation and beamstrahlung.
For a 100 GeV CP-odd Higgs boson and tan β = 50, the signal cross section is about 2 fb [47,48,49]. The generated events were passed through the fast detector simulation SGV [50]. The detector properties closely follow the TESLA detector Conceptual Design Report [51]. The simulation of the b-tagging performance is very important for this analysis. The efficiency versus purity distribution for the simulated b-tagging performance is shown in Fig. 1 for the hadronic event sample e + e − → qq for 5 flavors, where efficiency is the ratio of simulated bb events with the selection cuts to all simulated bb events, and purity is the ratio of simulated bb events with the selection cuts to all selected qq events. Details of the event selection and background reduction are described elsewhere [44].
For m A = 100 GeV in the context of the MSSM, the SM-like Higgs boson is the H while the light h is decoupled from W W, ZZ [cos(β − α) ∼ 1 and sin(β − α) ∼ 0]. The bbh coupling is essentially equal (in magnitude) to the bbA coupling (∝ tan β at the tree level) and m h ∼ m A , implying that the signal would be doubled from bbA and bbh. Also important will be hA pair production, which is proportional to cos(β − α) and will have full strength in this particular situation; HA production will be strongly suppressed. We focus first on bbA → bbbb.
The expected background rate for a given bbA → bbbb signal efficiency is shown in Fig. 2. One component of the background is hA → bbbb since it has rather weak dependence on tan β. Our selection procedures are, in part, designed to reduce this piece of the background as much as possible. Nonetheless, it may lead to significant systematic error in the determination of tan β due to interference with the signal, as discussed below. For the bbA → bbbb signal, the sensitivity S/ √ B for m A = 100 GeV is almost independent of the working point choice of signal efficiency in the range ǫ sel = 5% to 50%. For a working point choice of 10% efficiency, the total simulated background of about 16 million events is reduced to 100 background events with an equal number of signal events at tan β = 50. We estimate the error on determining tan β by ∆ tan 2 β/ tan 2 β = ∆S/S = √ S + B/S.
If this were the only contributing process, then we would have √ S + B/S ≈ 0.14, resulting in an error on tan β = 50 of 7%. For smaller values of tan β, the sensitivity decreases rapidly. A 5σ signal detection is still possible for tan β = 35. In the MSSM context, the bbh signal would essentially double the number of signal events and have exactly the same tan β dependence, yielding ∆ tan 2 β/ tan 2 β ∼ √ 300/200 ∼ 0.085 for tan β = 50. Although the number of hA background events is very small compared to the other background reactions after the event selection, interference between the signal (bbA → bbbb plus bbh → bbbb) and the background (hA → bbbb) could be important. At the working point of 10% signal efficiency, and after applying the selection procedures, the expected rate for the latter is 2 ± 1 events for L = 500 fb −1 . To assess the effect of the interference, let us momentarily retain only the bbA signal and the hA background. We first calculate the cross sections σ(e + e − → bbA → bbbb), σ(e + e − → hA → bbbb), and σ(e + e − → bbA + hA → bbbb) with CompHEP [52] before selections. We define the interference as For the default value m b = 4.62 GeV, at tan β = 50 we obtain σ bbA = 1.83 ± 0.01 fb, σ hA = 36.85 ± 0.10 fb, σ bbA+hA = 39.23 ± 0.12 fb, and thus σ interf = 0.55 ± 0.16 fb. We observe a constructive interference similar in size to the signal. Thus, more signal events are expected than simulated and the statistical error estimate is conservative. After selection cuts, we have found 100 signal events versus two hA background events. The maximum interference magnitude arises if the interference events are signal-like, yielding an interference excess of (10 + √ 2) 2 − 100 − 2 ∼ 28, a percentage (∼ 30%) similar to the ratio obtained before selection cuts. If the events from the interference are background-like, the resulting systematic error will be small, since the hA background is only a small part of the total background. In the MSSM context we have an exact prediction as a function of tan β for the combined contribution of hA → bbbb and bbA → bbbb (plus bbh → bbbb), including all interferences, and this exact prediction can be compared to the data. In order to test this exact prediction, it may be helpful to compare theory and experiment for several different event selection procedures, including ones that give more emphasis to the hA process. Of course, this exact prediction depends somewhat on other MSSM parameters, especially if decays of the h or H to pairs of supersymmetric particles are allowed or ratios of certain MSSM parameters are relatively large [53]. If this type of uncertainty exists, the systematic error on tan β can still be controlled by simultaneously simulating all sources of bbbb events for various tan β values and fitting the complete data set (assuming that the other MSSM parameters are known sufficiently well). Another possible theoretical systematic uncertainty derives from higher-order corrections. The full next-to-leading order (NLO) QCD corrections are given in [54,55]. There it is found that using the running b-quark mass incorporates the bulk of the NLO corrections. and errors are computed in the MSSM context using the running b-quark mass. Higher-order corrections of all kinds will be better known by the time the Linear Collider (LC) is constructed and data is taken and thus should not be a significant source of systematic uncertainty. An experimental challenge is associated with knowing the exact efficiency of the event selection procedure. At the working point of an efficiency ǫ sel = 10%, to achieve ∆ tan β/ tan β < 0.05 requires ∆ǫ sel /ǫ sel < 0.1, equivalent to ∆ǫ sel < 1%. In addition to the hA Higgs boson background, two other Higgs boson processes could lead to a bbbb topology. First, the process e + e − → HZ can give a bbbb final state. In fact, for large tan β the HZ cross section is maximal and similar in size to the hA cross section. Nonetheless, its contribution to the background is much smaller because the HZ → bbbb branching is below 10% compared to about 80% for hA → bbbb. Since the hA process contributed only 2% of the total background, the contribution to the background from the HZ process can be neglected. The second Higgs boson process leading to a bbbb topology is that already discussed, e + e − → bbh. The only distinction between this and the e + e − → bbA process is a small difference in the angular distribution due to the different production matrix elements. Thus, the selection efficiency is almost identical. The production rate of the bbA process is proportional to tan 2 β while the bbh production rate is proportional to sin 2 α/ cos 2 β. In the MSSM context, this latter factor is ∼ tan 2 β for m A ≥ 100 GeV and large tan β (assuming M SUSY ∼ 1 TeV). In the general 2HDM, since tan β ≈ 1/ cos β at large tan β, the expected rate depends mostly on sin α and the h mass. In this more general case where m h ≈ m A but the MSSM expectation of α ∼ −β ∼ −π/2 does not hold, the enhancement of the bbA signal by the bbh addition would only allow a determination of | sin α| as a function of the presumed value of tan β (using the constraint that one must obtain the observed number of bbh + bbA events). Independent measurements of the HZ and hA production rates would then be needed to determine the value of β − α and only then could α and β be measured separately.
It is essential for the tan β determination that a very high integrated luminosity can be accumulated (we assume L = 2000 fb −1 after several years of data-taking). Fig. 3 shows the expected statistical error on tan β for m A = 100, 150 and 200 GeV, assuming that the only measured process is bbA with the help of our selection cuts. At the two higher m A values, in the MSSM context it is the H that would be decoupled and have mass m H ∼ m A and the h would be SM-like. Consequently, the bbH rate would be essentially identical to the bbA rate and, assuming that one could verify the MSSM Higgs context by independent means, would lead to still smaller tan β statistical errors than plotted, the exact decrease depending upon the signal-to-background ratio. For m A = 150 and 200 GeV, the HA process (like the hA process at m A = 100 GeV) would have to be computed in a specific model context or its relative weight fitted by studying bbbb production in greater detail in order to minimize any systematic error from this source. Results obtained in the case of the MSSM will be given in the following section.

III. HA PRODUCTION: DECAY BRANCHING RATIOS AND TOTAL WIDTHS
The branching ratios for H, A and H ± decay to various allowed modes vary rapidly with tan β in the MSSM when tan β is in the low to moderate range, roughly below 20. Consequently, if these branching ratios can be measured accurately, tan β can be determined with good precision in this range. Measurement of the branching fractions is most easily accomplished using HA and H + H − pair production. In particular, the pair production processes are nearly independent of tan β so that the rate in a given channel provides a fairly direct probe of the branching ratio for that channel. That tan β could be accurately determined using Higgs branching ratios measured in pair production was first demonstrated in [33,34]. Refs. [33,34] consider a number of models for which SUSY decays of the H, A and H ± are kinematically allowed. It was found that by measuring all available ratios of branching ratios it was possible to determine tan β to better (often much better) than 10% for tan β values ranging from 2 up to as high as 25 to 30 for m A in the 200-400 GeV range, assuming √ s = 1 TeV and an effective luminosity (defined as the total luminosity times the selection efficiency of the cuts required to isolate the pair production process) of L eff = 80 fb −1 (equivalent, for example, to L = 2000 fb −1 for a selection efficiency of 4%). A more recent analysis using a few specific points in MSSM parameter space, focusing on the bbbb event rate and including a study at √ s = 500 GeV, is given in [36]. This latter study uses a selection efficiency of 13% and negligible background for detection of e + e − → hA → bbbb (relevant for m A ≤ 100 GeV) or e + e − → HA → bbbb (relevant for m A ≥ 150 GeV) and finds small errors for tan β at lower tan β values. Both [33,34] and [36] assume MSSM scenarios in which there are significant decays of the A and H to pairs of SUSY particles, in particular neutralinos and charginos. These decays remain non-negligible up to fairly high tan β values. As a result, the bb branching fractions of the A and H increase more markedly as tan β increases than if SUSY decays are absent. Indeed, in the absence of SUSY decays, the bbbb rate asymptotes quickly to a fixed value as tan β increases. As we shall see, this means that much smaller errors for the tan β determination using the HA → bbbb rate are achieved if SUSY decays are present. We now examine the errors on tan β that could be achieved using Higgs pair production, following procedures related to those of [33,34,36], but using updated luminosity expectations and somewhat more realistic experimental assumptions and analysis techniques. We restrict the analysis to the process e + e − → HA → bbbb, ignoring possible additional sensitivity through ratios relative to other final states. With both Higgs bosons reconstructed in their bb final state as two back-to-back clusters of similar mass, backgrounds are expected to be negligible. All the results of this section are obtained using version 2.0 of HDECAY [56] for computing the branching ratios and total widths of the Higgs bosons.
To understand the sensitivity to the presence of SUSY decays of the heavy Higgs bosons, two different MSSM scenarios are considered: In scenario (I), SUSY decays of the H and A are kinematically forbidden. Scenario (II) is taken from [36] in which SUSY decays (mainly to χ 0 1 χ 0 1 ) are allowed. We will assume that appropriate event selection criteria can be found such that for an event selection efficiency of 10% there will be negligible background. The resulting HA → bbbb event rates (per 2000 fb −1 of integrated luminosity) are plotted for √ s = 500 GeV in Fig. 4 as a function of tan β. The difference in the dependence of the event rates on tan β is apparent. In more detail: in scenario (I) the bbbb event rates, after 10% selection efficiency, are 8, 77, 464, 1762, and 1859 at tan β = 1, 2, 3, 10, and 40, respectively. The corresponding event rates in scenario (II) are 1, 5, 34, 1415 and 1842. These differing tan β dependencies imply significant sensitivity of the tan β errors to the scenario choice, with worse errors for scenario (I). Finally, we note that for tan β > 2 the above event numbers are such that backgrounds are indeed negligible after 10% efficient selection cuts; for tan β ∼ 1, backgrounds might become an issue.
To determine the 1σ statistical errors of the tan β determination, we compute, for each choice of tan β, the 1σ upper and lower bounds on the expected event number as N(bbbb) ± N(bbbb). These upper and lower bounds are also shown in Fig. 4. The upper (lower) event rate numbers are required to be ≥ 10 to set an upper (lower) tan β bound, respectively. Since the event number increases monotonically with tan β for both MSSM scenarios, we can then use the given MSSM model scenario to determine the tan β value for which the number of events is equal to the 1σ upper (lower) bound. These values define the 1σ upper (lower) bound on tan β, respectively. The resulting fractional upper and lower limit errors ∆ tan β/ tan β are plotted for MSSM scenarios (I) and (II) in Fig. 5. This procedure assumes that other measurements of SUSY particle production at the LHC and the LC will have fixed the MSSM scenario.
Let us discuss in more detail the tan β errors from the HA → bbbb rate in scenario (II) as compared to scenario (I). From Fig. 4 we see that in scenario (I) once tan β reaches 10 to 12 the bbbb rate will not change much if tan β is increased further since the H → bb and A → bb branching ratios approach constant values. In contrast, if tan β is decreased the bbbb rate declines significantly as other decay channels come into play. Thus, meaningful lower bounds on tan β are retained out to relatively substantial tan β values whereas upper bounds on tan β disappear for tan β > ∼ 10 −12. In scenario (II), for reasons explained below, we have not plotted upper bounds on tan β for tan β > ∼ 30. In fact, our numerical results indicate that the upper bound on ∆ tan β/ tan β decreases again as tan β increases beyond 30. We have traced this to the fact that HDECAY predicts that m H decreases (at fixed m A = 200 GeV) as tan β increases beyond 30. This results in an increase of the HA production cross section with increasing tan β. This, in turn, implies that the bbbb rate increases (as shown in Fig. 4) and that we can for cuts, acceptance and tagging. The upper and lower 1σ bounds for Γ R H ± include an additional efficiency factor of 0.75 for keeping only events in the central mass peak and assume the estimated mass resolution of Γ res = 5 GeV, including 10% systematic uncertainty. Results for Γ R H,A in SUSY scenario case (II) are very similar to those plotted for case (I). HDECAY [56] is used to compute the H and A widths and branching ratios.  7) for the H and A as determined in e + e − → HA → bbbb events. For the rates, results for SUSY scenarios (I) and (II) differ significantly, as shown. For bbA + bbH → bbbb and Γ R H,A we show only the results for MSSM scenario (I). Results for scenario (II) are essentially identical. Upper and lower curves of a given type give the upper and lower 1σ bounds, respectively, obtained using a given process as shown in the figure legend. We include running b-quark mass effects and employ HDECAY [56].
obtain an upper bound on tan β despite the fact that the HA → bbbb final state branching ratio approaches a constant value. However, since this predicted decrease of m H at high tan β is somewhat peculiar to the precise parameters chosen for scenario (II), we do not regard this result as representative. For this reason, we have chosen not to show the scenario (II) upper limit curve beyond tan β = 30. Had we plotted the region above tan β = 30, one would see a slowly declining upper limit on tan β.
The above results can be compared to the tan β determination based on the bbH +bbA → bbbb rate using the procedures of Sec. II applied in the MSSM model context. For the computation of this rate, our calculation of the bbH and bbA cross sections includes the dominant radiative corrections as incorporated via b-quark mass running starting with m b (m b ) = 4.62 GeV. The H and A branching ratios and widths are computed using HDECAY. Since there is little sensitivity of this rate to the MSSM scenario (for the high tan β values for which this means of determining tan β is useful) we only present results for scenario (I); where plotted, errors for tan β from the bbH + bbA → bbbb rate are essentially independent of the MSSM scenario choice. The errors on tan β resulting from the rate for bbH + bbA → bbbb quickly become far smaller than those based on HA → bbbb once tan β > ∼ 30. This is illustrated in Fig. 5, which compares the results for ∆ tan β/ tan β obtained using the e + e − → HA → bbbb rate to those based on the bbH + bbA → bbbb rate. This comparison shows the natural complementarity between these two techniques for measuring tan β. However, with these two techniques alone, there is always a range of intermediate-size tan β values for which a good determination of tan β is not possible.
This "gap" can be partly filled, and the error on tan β at high tan β can be greatly reduced, by using the intrinsic total widths of the H and A to determine tan β. However, it is only for tan β > 10 that the intrinsic widths can provide a tan β determination. This is because (a) the widths are only > 5 GeV (the detector resolution discussed below) for tan β > 10 and (b) the number of events in the bbbb final state becomes maximal once tan β > 10.
We now discuss the experimental issues in determining the Higgs boson width. The expected precision of the SM Higgs boson width determination at the LHC and at a LC was studied in [57]. As described in [57], a simple estimate (based on a detector energy flow resolution of ∆E/E = 0.3/ √ E for each of the two b-jets) yields an expected detector resolution of Γ res = 5 GeV for m A ∼ 200 GeV. However, an overall fit to the bb mass distribution similar to the one in the study of [57] would give a Higgs boson resonance peak width which is about 2σ larger than that expected from the convolution of the 5 GeV resolution with the intrinsic Higgs width. This can be traced to the fact that the overall fit includes wings of the mass distribution that are present due to wrong pairings of the b-jets. The mass distribution contains about 400 di-jet masses (2 entries per HA event), of which about 300 are in a central peak. If one fits only the central peak, the width is close to that expected based on simply convoluting the 5 GeV resolution with the intrinsic Higgs width. This indicates that about 25% of the time wrong jetpairings are made and contribute to the wings of the mass distribution. Therefore, our estimates of the error on the determination of the Higgs width will be based on the assumption that only 3/4 of the events (i.e. those in the central peak) retained after our basic event selection cuts (with assumed selection efficiency of 10%) can be used in the statistics computation. The m bb for each of the bb pairs identified with the H or A is binned in a single mass distribution. This is appropriate since the H and A are highly degenerate for the large tan β values being considered so that the resolution of 5 GeV is typically substantially larger than the mass splitting. Our effective observable is then the resolved average width defined by: The resolved average width, Γ R H,A , for SUSY scenario (I) (including m A = 200 GeV) is plotted in Fig. 4 as a function of tan β. The results for scenario (II) are indistinguishable.
In order to extract the implied tan β bounds, we must account for the fact that the detector resolution will not be precisely determined. There will be a certain level of systematic uncertainty which we have estimated at 10% of Γ res , i.e. ∆Γ sys res = 0.5 GeV. This systematic uncertainty considerably weakens our ability to determine tan β at the lower values of tan β for which Γ H tot and Γ A tot are smaller than Γ res . This systematic uncertainty should be carefully studied as part of any eventual experimental analysis. Given Γ res , ∆Γ sys res and the number of selected bbbb events, N(bbbb), we compute the useful number of entries in the bbbb mass distribution for determining Γ R H,A as N entries = 2 × 0.75 × N(bbbb). The factor of 2 is because each bbbb event results in two entries, one for the H and one for the A, and the factor of 0.75 is that for retaining only the central peak of the distribution. The error for Γ R H,A is then computed (following the procedure of [20]) as The resulting upper and lower 1σ bounds on Γ R H,A are plotted in Fig. 4. The upper and lower limits on tan β are then obtained as the values tan β ± ∆ tan β for which the central prediction (the solid curve of Fig. 4

) agrees with the values Γ R
H,A ± ∆Γ R H,A . In computing ∆Γ R H,A we have assumed a selection efficiency of 10% for computing N(bbbb). These errors are for L = 2000 fb −1 and √ s = 500 GeV. That an excellent determination of tan β will be possible at high tan β is apparent. The resulting accuracy for tan β obtained from measuring the average (resolved) H/A width is shown in Fig. 5. We see that good accuracy is already achieved for tan β as low as 25 with extraordinary accuracy predicted for very large tan β. The sharp deterioration in the lower bound on tan β for tan β < ∼ 24 occurs because the width falls below Γ res as tan β is taken below the input value and sensitivity to tan β is lost. If there were no systematic error in Γ res , this sharp fall off would occur instead at tan β < ∼ 14. To understand these effects in a bit more detail, we again give some numbers for scenario (II). At tan β = 50, 55 and 60, Γ H tot , Γ A tot ∼ 10.4, 12.5 and 14.9 GeV, respectively. After including the detector resolution, the effective average widths become 11.5, 13.4 and 15.7 GeV, respectively, whereas the total error in the measurement of the average width, including systematic error, is ∼ 0.54 GeV. Therefore, tan β can be determined to about ±1, or to better than ±2%. This high-tan β situation can be contrasted with tan β = 15 and 20, for which Γ H tot , Γ A tot = 0.935 and 1.64 GeV, respectively, which become 5.09 and 5.26 GeV after including detector resolution. Meanwhile, the total error, including the statistical error and the systematic uncertainty for Γ res , is about 0.57 GeV and no sensitivity to tan β is obtained.
The accuracies from the width measurement are somewhat better than those achieved using the bbA + bbH → bbbb rate measurement. However, both of these high-tan β methods for determining tan β are important because they are beautifully complementary in that they rely on very different experimental observables. Further, both methods are nicely complementary in their tan β coverage to the tan β determination based on the HA → bbbb rate, which comes in at lower tan β. In fact, the width measurement can provide a decent tan β determination even in the previously identified "gap" region where neither the HA → bbbb nor the bbA + bbH → bbbb rates were able to provide such a determination. In particular, in the case of MSSM scenario (II), combining the HA → bbbb rate and the width measurements implies that the worst tan β lower bound is ∆ tan β/ tan β ∼ −0.25 at tan β ∼ 28 and the worst tan β upper bound is ∆ tan β/ tan β ∼ +0.30 at tan β ∼ 20. However, in the case of MSSM scenario (I) a good upper bound on tan β is not possible if tan β ∼ 12 − 15, even after including the width measurement. Overall, there is a window, 10 < ∼ tan β < ∼ 25 in scenario (I) or 20 < ∼ tan β < ∼ 25 in scenario (II), for which an accurate determination of tan β (∆ tan β/ tan β < 0.2) using just the bbbb final state processes will not be possible. This window expands rapidly as m A increases (keeping √ s fixed). Indeed, as m A increases above 250 GeV, HA pair production becomes kinematically forbidden at √ s = 500 GeV and, in addition, detection of the bbH + bbA processes at the LC (or the LHC) requires [58] increasingly large values of tan β. This difficulty persists even for √ s ∼ 1 TeV and above; if m A > √ s/2, the H and A cannot be pair-produced and yet the rate for bbH + bbA production is undetectably small for moderate tan β values.
In the above study, we have not made use of other decay channels of the H and A, such as H → W W, ZZ, H → hh, A → Zh and H, A → SUSY. The theoretical studies of [33,34] indicate that their inclusion could improve the precision with which tan β is measured at low to moderate tan β values. A determination of Γ R H,A is also possible using the bbA + bbH → bbbb events. To estimate how well tan β can be determined in this way, let us assume that 50% of the events selected in the analysis of Sec. II can be used for a fit of the average width and that (as in the HA → bbbb study) a resolution of 5 GeV can be achieved, based on a detector energy flow resolution of ∆E/E = 0.3/ √ E. If we again assume that 10% systematic error for the width measurement can be achieved, the resulting tan β errors would be similar to those obtained from the bbA + bbH → bbbb event rate for tan β > 30 (see Fig. 5), i.e. not as small as the tan β errors obtained from the measurement of Γ R H,A in HA → bbbb events. A complete analysis that takes into account the significant background and the broad energy spectrum of the radiated H and A is needed to reliably assess the tan β errors that would be obtained by measuring Γ R H,A in bbA + bbH → bbbb events. If tan β is large enough, bbA + bbH → bbbb would be observable in the MSSM even when m A > √ s/2, or, for a more general model, whenever HA pair production is kinematically forbidden but bbA → bbbb and/or bbH → bbbb production is allowed. Then, the event rate for, and width measurements from bbA and/or bbH production would allow a determination of tan β. We have not attempted a quantitative study of this situation. Let us briefly return to the interpretation of these measurements in terms of tan β. As stated in the introduction, we are using tan β as a tree-level mnemonic to characterize the bb Yukawa coupling of the Higgs bosons. For the soft-SUSY-breaking parameters for MSSM scenarios (I) and (II), the one-loop corrections to the bb couplings of the H and A and the stop/sbottom mixing present in the one-loop corrections to the Higgs mass matrix [53] are small. More generally, however, substantial ambiguity can arise, especially if the sign and magnitude of µ is not fixed. Assuming that these parameters are known, the errors for the Yukawa coupling obtained from these measurements can be related to any given definition of tan β and, except in very unusual cases, the resulting error on tan β would be fairly insensitive to the precise scenario. For example, one possible definition of tan β would be that the µ + µ − A coupling should be precisely given by −(m b /v) tan β, see Eq (4). This is a convenient definition since the µ + µ − A coupling will have very modest higher-order corrections relative to the tree-level and any such corrections can then be sensibly absorbed using the above definition of tan β. Given this definition of tan β, the Hbb and Abb couplings can be computed to any desired order once the necessary MSSM parameters are known. In this way, all the probes of heavy Higgs Yukawa couplings discussed in this paper can be related to this common definition of tan β.

IV. H + H − PRODUCTION: DECAY BRANCHING RATIOS AND TOTAL WIDTH
In this section, we extend our study to include charged Higgs boson production processes. Existing analyses of e + e − → H + H − production indicate that the absolute event rates and ratios of branching ratios in various H + H − final state channels will allow a relatively accurate determination of tan β at low tan β [33,34]. The process e + e − → H ± tb can also be sensitive to tan β [35]. Here, we focus on an experimentally based analysis of the determination of tan β using the H + H − → tb tb event rate. As anticipated on the basis of the earlier work referenced above, we find that good accuracy can be achieved at low tan β. We also demonstrate that the total width of the H ± measured in the tb decay channel using H + H − → tb tb production will allow a fairly precise determination of tan β at high tan β. Since these two techniques for determining tan β are statistically independent of one another and of the tan β measurements that employ neutral Higgs production, they will increase the overall accuracy with which tan β can be measured at both low and high tan β.
The reaction e + e − → H + H − → tb tb can be observed at a LC [59], and recent highluminosity simulations [60] show that the cross section times branching ratio can be measured precisely. As soon as the charged Higgs boson decay into tb is allowed this decay mode is dominant. Nonetheless, BR(H ± → tb) varies significantly with tan β, especially for small values of tan β where the tb mode competes with the τ ν mode. The H + → tb branching ratio and width are sensitive to tan β in the form As in Sec. III, we will use HDECAY (which incorporates running of the b-quark mass) to evaluate the charged Higgs boson branching ratios and decay width. It is useful to note that the above form results in a minimum in the tb partial width and branching ratio in the vicinity of tan β ∼ 6 − 8. The depth of the minimum in the branching ratio depends upon the extent to which the tb channel is competing against other modes. In contrast, the cross section for e + e − → H + H − production is independent of tan β at tree-level. (The one-loop corrections [61] result in a 10% variation of the cross section with tan β which must be taken into account when the data are taken; we do not include them in our study.) The net result is that the rate for e + e − → H + H − → tb tb has significant dependence upon tan β, coming mainly from the variation in the branching ratio. Our procedures for estimating errors for the tb tb rate and for the total width are similar to those given earlier for HA production rates and width in the bbbb channel. We base our efficiency for the tb tb final state on the study of [60]. For m H ± = 300 GeV at √ s = 800 GeV, this study finds that the tb tb final state can be isolated with an efficiency of 2.2%. The reason for the much lower efficiency as compared to 10% efficiency for the bbbb final state of HA production is the difficulty of assigning the non-b jets from t-decays to the correct top mass cluster. For √ s = 500 GeV and m H ± = 200 GeV, we have adopted the same 2.2% efficiency, for which we assume little or no background after cuts. For the total width determination, we assume that we keep only 75% of the events after cuts (i.e. a fraction 0.75 × 0.022 of the raw event number), corresponding to throwing away wings to the mass peaks, and each tb tb event is counted twice since we can look at both the H + and the H − decay. We define a resolved width which incorporates the intrinsic resolution for the width determination, taken to be Γ res = 5 GeV: Estimated errors based on the width measurement will assume a 10% systematic error in our knowledge of Γ res , i.e. ∆Γ sys res = 0.5 GeV as for the H, A case. We employ Eq. (8), with the replacement Γ R H,A → Γ R H ± , to compute ∆Γ R H ± . In this case, N entries = 2 × 0.75 × N(tb tb), where N(tb tb) is computed using the above-noted selection efficiency of 0.022. Figure 6 shows the resulting tb tb final state rate, N(tb tb), for MSSM scenarios (I) and (II) and the resolved width (Γ R H ± ) for scenario (I). Also shown are the corresponding 1σ upper and lower bounds on the rate and resolved width. These are then used in exactly the same manner as described in the HA case to determine the upper and lower bounds on tan β.
For the rate, we observe from Fig. 6 that upper bounds on tan β will be poor once tan β > ∼ 10 because of the very slow variation of the tb tb final rate in this region. At high tan β, for SUSY scenario (I) lower bounds will be determined by the part of the tb tb rate curve that rises rapidly when tan β falls below 10. For SUSY scenario (II), the beginning of the dip will fix the lower bounds on tan β when tan β is large, assuming that we know ahead of time from other experimental data that tan β is larger than 15. As regards the width, the main point to note is that Γ tot H ± rises only slowly with increasing tan β. As a result, the 5 GeV resolution and 10% systematic error for this resolution are significant compared to the < 10 GeV H ± width that applies throughout the tan β range studied. Note also, that for moderate tan β values, there will be no lower bound on tan β as a result of the fact that Γ R H ± never falls below Γ res = 5 GeV, while the 1σ errors are substantially lower than this. We will also assume that if tan β is large, then we will know from other experimental information (such as the HA final state) that tan β is not small and that the small rise in the width for tan β ∼ 1 is not relevant.
The resulting tan β upper and lower bounds appear in Fig. 7. Comparing to Fig. 5, we observe that for SUSY scenario (I) the tb tb rate measurement gives a tan β determination that is quite competitive with that from HA production in the bbbb final state. For SUSY scenario (II), the tb tb rate gives an even better tan β determination than does the bbbb rate. On the other hand, the width measurement from the tb tb final state of H + H − production is much poorer than that from the bbbb final state of HA production, as was to be expected from the discussion given earlier.
By combining in quadrature the tan β errors for the various individual measurements, as given in Figs. 5 and 7, we obtain the net errors on tan β shown in Fig. 8.

V. COMPARISON TO LHC DETERMINATIONS OF tan β
In this section, we will compare the LC results summarized in Figs. 5, 7 and 8 to the tan β accuracies that can be achieved at the LHC based on H, A, H ± production and decay processes. First note that there is a wedge-shaped window of moderate tan β and m A > ∼ 200 GeV for which the A, H and H ± are all unobservable (see, for example, Refs. [21,22,62]). In this wedge, the only Higgs boson that is detectable at the LHC is the light SM-like Higgs boson, h. Precision measurements of the properties of the h typically only provide weak sensitivity to tan β, and will not be considered here. The lower tan β bound of this moderate-tan β wedge is defined by the LEP-2 limits [9], which are at tan β ∼ 3 for m A ∼ 200 GeV, falling to tan β ∼ 2.5 for m A > ∼ 250 GeV, assuming the maximal mixing scenario [see SUSY scenario (I) defined earlier]. The upper tan β limit of the wedge is at tan β ∼ 7 for m A ∼ 200 GeV rising to tan β ∼ 15 at m A ∼ 500 GeV. For either smaller or larger tan β values, the heavy MSSM Higgs bosons can be detected and their production rates and properties will provide sensitivity to tan β.
We will now summarize the results currently available regarding the determination of tan β at the LHC using Higgs measurements (outside the wedge region) assuming a luminosity of Results for scenario (II) are essentially identical. Upper and lower curves of a given type give the upper and lower 1σ bounds, respectively, obtained using a given process as shown in the figure legend. We include running b-quark mass effects and employ HDECAY [56]. L = 300 fb −1 . The methods employed are those proposed in [20]. The reactions that have been studied at the LHC are the following.
The best accuracy that can be achieved at low tan β is obtained from the H → ZZ → 4ℓ rate. One finds ∆ tan β/ tan β = ±0.1 at tan β = 1 rising to > ±0.3 by tan β = 1.5 for the sample choice of m H = 300 GeV. For m H < 2m Z , tan β cannot be measured via this process. In the MSSM maximal mixing scenario, such low values of tan β are unlikely in light of LEP-2 results.
Interpolating, using Figs. 19-86 and 19-87 from [22], we estimate that at m A ∼ 200 GeV (our choice for this study) the error on tan β based on these rates would be smaller than ±0.1 for tan β > ∼ 13, asymptoting to ±0.05 at large tan β.
It is important to note that the tan β sensitivity for tan β < 20 − 30 is largely due to the loop-induced gg → A and gg → H production processes. Thus, interpreting these fully inclusive rates in terms of tan β when tan β < 20 − 30 requires significant knowledge of the particles, including SUSY particles, that go into the loops responsible for the gg → H and gg → A couplings.
The importance of including the gg → H, gg → A as well as the gg → bbH + gg → bbA processes in order to obtain observable signals for tan β values as low as 10 in the µ + µ − channels is apparent from [63]. For L = 300 fb −1 and m A = 200 GeV, they find that the bbµ + µ − final states can only be isolated for tan β > 30 whereas the inclusive µ + µ − final state from all production processes becomes detectable once tan β > 10.
The tbH ± → tbτ ν rate gives a fractional tan β uncertainty, ∆ tan β/ tan β, ranging from ±0.074 at tan β = 20 to ±0.054 at tan β = 50. This signal is somewhat cleaner to interpret in terms of a tan β measurement than the inclusive signals for the H and A summarized above, since there are no uncertainties related to SUSY loop contributions.
Sensitivity to tan β deriving from direct measurements of the decay widths has not been studied by the LHC experiments. One can expect excellent tan β accuracy at the higher tan β values for which the gg → bbµ + µ − signal for the H and A is detectable.
Let us now compare these LHC results to the LC errors for tan β, assuming m A = 200 GeV. First, consider tan β ≤ 10. As summarized above, the LHC error on tan β is ±0.12 at tan β ∼ 10 and at tan β ∼ 1, and the error becomes very large for 1.5 < ∼ tan β < ∼ 5. Meanwhile, the LC error from Fig. 8 ranges from roughly ±0.03 to ±0.05 for 2 < ∼ tan β < ∼ 5 rising to about ±0.1 at tan β ∼ 10 [in the less favorable SUSY scenario (I)]. Therefore, for tan β < ∼ 10 the LC provides the best determination of tan β using Higgs observables related to their Yukawa couplings. (In the MSSM context, other non-Higgs LHC measurements would allow a good tan β determination at low to moderate tan β based on other kinds of couplings.) In the middle range of tan β (roughly 13 < tan β < 30 at m A ∼ 200 GeV), the heavy Higgs determination of tan β at the LHC might be superior to that obtained at the LC. This depends upon the SUSY scenario: if the heavy Higgs bosons can decay to SUSY particles, the LC will give tan β errors that are quite similar to those obtained at the LHC; if the heavy Higgs bosons do not have substantial SUSY decays, then the expected LC tan β errors are substantially larger than those predicted for the LHC. At large tan β, the LC measurement of the heavy Higgs couplings and the resulting tan β determination at the LC is numerically only slightly more accurate than that obtained at the LHC. For example, both are of order ±0.05 at tan β = 40.  [22] with those using H ± production from [24], both of which assume the standard MSSM maximal mixing scenario. All entries are approximate.  Table I. It is possible that the net LHC tan β error would be somewhat smaller than the LC error for tan β > ∼ 40 if both ATLAS and CMS can each accumulate L = 300 fb −1 of luminosity; combining the two data sets would presumably roughly double the statistics and decrease errors by a factor of order 1/ √ 2. In any case, a very small error on tan β will be achievable for all tan β by combining the results from the LC with those from the LHC. [65] VI. CONCLUSIONS A high-luminosity linear collider will provide a precise determination of the value of tan β throughout much of the large range of possible interest, 1 < tan β < 60. In this paper, we have studied the sensitivity to tan β that will result from measurements of heavy Higgs boson production processes, branching fractions and decay widths. These are all directly determined by the ratio of vacuum expectation values that defines tan β, and each can be very accurately measured at an LC over a substantial range of relevant tan β values. In particular, there are several Higgs boson observables which are likely to provide the most precise measurement of tan β when tan β is very large. In the context of the MSSM, there is a particularly large variety of complementary methods that will allow an accurate determination of tan β when m A < ∼ √ s/2 so that e + e − → HA pair production is kinematically allowed. By combining the tan β errors from all these processes in quadrature, we obtain the net errors on tan β shown in Fig. 8 by the lines [solid for SUSY scenario (I) and dashed for SUSY scenario (II)], assuming a multi-year integrated luminosity of L = 2000 fb −1 . We see that, independent of the scenario, the Higgs sector will provide an excellent determination of tan β at small and large tan β values, leading to an error on tan β of 10% or better. If SUSY decays of the H, A, H ± are significant [SUSY scenario (II)], the tan β error will be smaller than 13% even in the more difficult moderate tan β range. However, if SUSY decays are not significant [SUSY scenario (I)] there is a limited range of moderate tan β for which the error on tan β would be large, reaching about 50%.
In the preceding section, we considered how these tan β errors from the LC compared to tan β errors determined at the LHC based only on measurements involving H, A, H ± production and decay. The broad conclusions were: (i) for low tan β ( < ∼ 10) the errors on tan β from LHC Higgs measurements would be much larger than those attainable at the LC; (ii) for high tan β ( > ∼ 30) the LHC and LC tan β errors were both small and quite comparable in magnitude; and (iii) in the moderate-tan β range (13 < ∼ tan β < ∼ 30) the LHC errors on tan β would very possibly be smaller than the LC errors. However, we also noted that in this latter region some care in interpretation of the LHC results would be necessary due to the need to include loop-induced gg → H and gg → H production processes in order to obtain good sensitivity to tan β; these might be influenced by loops of SUSY particles and, possibly, other undiscovered new physics. This LHC versus LC comparison should also be viewed as highly preliminary since the LHC collaborations have not yet studied all the relevant observables. In particular, they have not looked at the tan β determination using the directly measured widths of the H and A. Regardless of the relative magnitude of the LHC versus LC tan β errors, the clean LC environment will provide an important and independent measurement that will complement any LHC determination of tan β. Different uncertainties will be associated with the determination of tan β at a hadron and an e + e − collider because of the different backgrounds. Further, the LHC and LC measurements of tan β will be highly complementary in that the systematic errors involved will be very different.
Combining all the different LC measurements as above does not fully account for the fact that the "effective" tan β value being measured in each process is only the same at tree-level. The tan β values measured via the H → bb Yukawa coupling, the A → bb Yukawa coupling and the H + → tb Yukawa coupling could all be influenced differently by the MSSM one-loop corrections. For some choices of MSSM parameters, the impact of MSSM radiative corrections on interpreting these measurements can be substantial [53]. However, if the masses of the SUSY particles are known, so that the important MSSM parameters entering these radiative corrections (other than tan β) are fairly well determined, then a uniform convention for the definition of tan β can be adopted and, in general, an excellent determination of tan β (with accuracy similar to that obtained via our tree-level procedures) will be possible using the linear collider observables considered here. Even for special SUSY parameter choices such that one of the Yukawa couplings happens to be significantly suppressed, the observables a)-e) would provide an excellent opportunity for pinning down all the Yukawa couplings and checking the consistency of the MSSM model.
Finally, it is important to note that the techniques considered here can also be employed in the case of other Higgs sector models. For example, in the general (non-SUSY) 2HDM, if the only non-SM-like Higgs boson with mass below √ s is the A [64], then a good determination of tan β will be possible at high tan β from the bbA → bbbb production rate. Similarly, in models with more than two Higgs doublet and/or triplet representations, the Yukawa couplings of the Higgs bosons, and, therefore, the analogues of the 2HDM parameter tan β, will probably be accurately determined through Higgs production observables in e + e − collisions.