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Higgs Boson Theory and Phenomenology

Precision electroweak data presently favors a weakly-coupled Higgs sector as the mechanism responsible for electroweak symmetry breaking. Low-energy supersymmetry provides a natural framework for weakly-coupled elementary scalars. In this review, we summarize the theoretical properties of the Standard Model (SM) Higgs boson and the Higgs sector of the minimal supersymmetric extension of the Standard Model (MSSM). We then survey the phenomenology of the SM and MSSM Higgs bosons at the Tevatron, LHC and a future e+e- linear collider. We focus on the Higgs discovery potential of present and future colliders and stress the importance of precision measurements of Higgs boson properties.


Introduction-Origin of Electroweak Symmetry Breaking
Deciphering the mechanism that breaks the electroweak symmetry and generates the masses of the known fundamental particles is one of the central challenges of particle physics. The Higgs mechanism [1] in its most general form can be used to explain the observed masses of the W ± and Z bosons as a consequence of three Goldstone bosons (G ± and G 0 ) that end up as the longitudinal components of the gauge bosons. These Goldstone bosons are generated by the underlying dynamics responsible for electroweak symmetry breaking. However, the fundamental nature of this dynamics is still unknown. Two broad classes of electroweak symmetry breaking mechanisms have been pursued theoretically. In one class of theories, electroweak symmetry breaking dynamics is weakly-coupled, while in the second class of theories the dynamics is strongly-coupled.
The electroweak symmetry breaking dynamics that is employed by the Standard Model posits a self-interacting complex doublet of scalar fields, which consists of four real degrees of freedom [2].
Renormalizable interactions are arranged in such a way that the neutral component of the scalar doublet acquires a vacuum expectation value, v = 246 GeV, which sets the scale of electroweak symmetry breaking. Consequently, three massless Goldstone bosons are generated, while the fourth scalar degree of freedom that remains in the physical spectrum is the CP-even neutral Higgs boson (h SM ) of the Standard Model. It is further assumed in the Standard Model that the scalar doublet also couples to fermions through Yukawa interactions. After electroweak symmetry breaking, these interactions are responsible for the generation of quark and charged lepton masses. This approach is an example of weak electroweak symmetry breaking. Assuming that m h SM < ∼ 200 GeV, all fields remain weakly interacting at energies up to the Planck scale. In the weakly-coupled approach to electroweak symmetry breaking, the Standard Model is very likely embedded in a supersymmetric theory [3] in order to stabilize the large gap between the electroweak and the Planck scales in a natural way [4,5]. These theories predict a spectrum of Higgs scalars [6], with the properties of the lightest Higgs scalar often resembling that of the Standard Model (SM) Higgs boson.
Alternatively, strong breaking of electroweak symmetry is accomplished by new strong interactions near the TeV scale [7]. More recently, so-called "little Higgs models" have been proposed in which the scale of the new strong interactions is pushed up above 10 TeV [8], and the lightest Higgs scalar resembles the weakly-coupled SM Higgs boson. In a more speculative direction, a new approach to electroweak symmetry breaking has been explored in which extra space dimensions beyond the usual 3 + 1 dimensional spacetime are introduced [9] with characteristic sizes of order (TeV) −1 . In such scenarios, the mechanisms for electroweak symmetry breaking are inherently extra-dimensional, and the resulting phenomenology may be significantly different from the usual approaches mentioned above. All these alternative approaches lie outside the scope of this review.
Although there is as yet no direct evidence for the nature of electroweak symmetry breaking dynamics, present data can be used to discriminate among the different approaches. For example, precision electroweak data, accumulated in the past decade at LEP, SLC, the Tevatron and elsewhere, strongly support the Standard Model with a weakly-coupled Higgs boson [10]. Moreover, the contribution of new physics, which can enter through W ± and Z boson vacuum polarization corrections, is severely constrained. This fact has already served to rule out several models of strongly-coupled electroweak symmetry breaking dynamics. The Higgs boson contributes to the W ± and Z boson vacuum polarization through loop effects, and so a global Standard Model fit to the electroweak data yields information about the Higgs mass. The results of the LEP Electroweak Working Group analysis shown in fig. 1(a) yield [10]: m h SM = 81 +52 −33 GeV, and provides a 95% CL upper limit of m h SM < 193 GeV. These results reflect the logarithmic sensitivity to the Higgs mass via the virtual Higgs loop contributions to the various electroweak observables. The 95% CL upper limit is consistent with the direct searches at LEP [11] that show no conclusive evidence for the Higgs boson, and imply that m h SM > 114.4 GeV at 95% CL. Fig. 1(b) exhibits the most probable range of values for the SM Higgs mass [12]. This mass range is consistent with a weakly-coupled Higgs scalar that is expected to emerge from the Standard Model scalar dynamics. The "blueband plot" shows ∆χ 2 ≡ χ 2 − χ 2 min as a function of the Standard Model Higgs mass [10]. The solid line is a result of a global fit using all data; the band represents the theoretical error due to missing higher order corrections. The rectangular shaded region shows the 95% CL exclusion limit on the Higgs mass from direct searches at LEP [11]. (b) Probability distribution function for the Higgs boson mass, including all available direct and indirect data [12]. The probability is shown for 1 GeV bins. The shaded and unshaded regions each correspond to an integrated probability of 50% There are some loopholes that can be exploited to circumvent this conclusion. It is possible to construct models of new physics where the goodness of the global Standard Model fit to precision electroweak data is not compromised while the strong upper limit on the Higgs mass is relaxed. In particular, one can construct effective operators [13,14] or specific models [15] of new physics where the Higgs mass is significantly larger, but the new physics contributions to the W ± and Z vacuum polarizations, parameterized by the Peskin-Takeuchi [16] parameters S and T , are still consistent with the experimental data. In addition, some have argued that the global Standard Model fit exhibits possible internal inconsistencies [17], which would suggest that systematic uncertainties have been underestimated and/or new physics beyond the Standard Model is required. Thus, although weaklycoupled electroweak symmetry breaking seems to be favored by the strong upper limit on the Higgs mass, one cannot definitively rule out all other approaches.
Nevertheless, one additional piece of data is very suggestive. Within the supersymmetric extension of the Standard Model, grand unification of the electromagnetic, the weak and the strong gauge interac-tions can be achieved in a consistent way, strongly supported by the prediction of the electroweak mixing angle at low energy scales with an accuracy at the percent level [18,19]. The significance of this prediction is not easily matched by other approaches. For example, in strongly-coupled electroweak symmetry breaking models, unification of couplings is not addressed per se, whereas in extra-dimensional models it is often achieved by introducing new structures at intermediate energy scales. Unless one is willing to regard the apparent gauge coupling unification as a coincidence, it is tempting to conclude that weak electroweak symmetry breaking with low-energy supersymmetry is the preferred mechanism, leading to an expected mass of the lightest Higgs boson below 200 GeV (less than 135 GeV in the simplest supersymmetric models), and a possible spectrum of additional neutral and charged Higgs bosons with masses up to of order 1 TeV.
Henceforth, we shall assume that the dynamics of electroweak symmetry breaking is a result of a weakly-coupled scalar sector. The Standard Model is an effective field theory and provides a very good description of the physics of elementary particles and their interactions at an energy scale of O(100) GeV and below. However, there must exist some energy scale, Λ, at which the Standard Model breaks down. That is, the Standard Model is no longer adequate for describing the theory above Λ, and degrees of freedom associated with new physics become relevant. In particular, we know that Λ ≤ M PL , since at an energy scale above the Planck scale, M PL ≃ 10 19 GeV, quantum gravitational effects become significant and the Standard Model must be replaced by a more fundamental theory that incorporates gravity. (Similar conclusions also apply to recently proposed extra-dimensional theories in which quantum gravitational effects can become significant at energies scales as low as O(1 TeV) [9].) Of course, it is possible that new physics beyond the Standard Model exists at an energy scale between the electroweak and Planck scale, in which case the value of Λ might lie significantly below M PL . 1 The value of the Higgs mass itself can provide an important constraint on the value of Λ. If m h SM is too large, then the Higgs self-coupling blows up at some scale Λ below the Planck scale [20]. If m h SM is too small, then the Higgs potential develops a second (global) minimum at a large value of the scalar field of order Λ [21]. Thus new physics must enter at a scale Λ or below in order that the global minimum of the theory correspond to the observed SU(2)×U(1) broken vacuum with v = 246 GeV. Thus, given a value of Λ, one can compute the minimum and maximum Higgs mass allowed. The results of this computation (with shaded bands indicating the theoretical uncertainty of the result) are illustrated in fig. 2(a) [22]. Consequently, a Higgs mass range 130 GeV < ∼ m h SM < ∼ 180 GeV is consistent with an effective Standard Model that survives all the way to the Planck scale. 2 However, the survival of the Standard Model as an effective theory all the way up to the Planck 1 For example, the recent experimental evidence for neutrino masses of order 10 −2 eV or below cannot be strictly explained in the Standard Model. Yet, one can easily write down a dimension-5 operator responsible for neutrino masses that is suppressed by v/Λ. If m ν ∼ 10 −2 eV, then one obtains as a rough estimate Λ < ∼ 10 15 GeV. 2 The constraint on Λ due to vacuum stability in fig. 2 is less stringent if one allows for the electroweak vacuum to be metastable, with a lifetime greater than the age of the universe. An analysis of ref. [23] finds that for a sufficiently long-lived electroweak vacuum, the Higgs mass lower limit of 130 GeV just quoted is reduced to about 115 GeV.  [20] and the lower [21] Higgs mass bounds as a function of the energy scale Λ at which the Standard Model breaks down, assuming M t = 175 GeV and α s (m Z ) = 0.118, taken from ref. [22]. The shaded areas above reflect the theoretical uncertainties in the calculations of the Higgs mass bounds. (b) Following ref. [14], a reconsideration of the Λ vs. Higgs mass plot with a focus on Λ < 100 TeV. Precision electroweak measurements restrict the parameter space to lie below the dashed line, based on a 95% CL fit that allows for nonzero values of S and T and the existence of higher dimensional operators suppressed by v 2 /Λ 2 . The unshaded area has less than one part in ten fine-tuning.
scale is unlikely based on the following "naturalness" [4] argument. In an effective field theory, particle masses and dimensionless couplings of the low-energy theory are calculable in terms of parameters of a more fundamental theory that describes physics at the energy scale Λ. All low-energy couplings and fermion masses are logarithmically sensitive to Λ. In contrast, scalar squared-masses are quadratically sensitive to Λ. Thus, in this framework, the observed Higgs mass (at one-loop) has the following form: where (m h ) 0 is a parameter of the fundamental theory, g is an electroweak coupling and k is a constant, presumably of O(1), that is calculable within the low-energy effective theory. Because these two contributions arise from independent sources, it is very unlikely that the magnitude of m 2 h SM is significantly smaller than either of the two terms. That is, the "natural" value for the physical scalar squared-mass is at least of order g 2 Λ 2 /16π 2 . In order for this value to be consistent with the requirement that the Higgs mass is of order the electroweak symmetry breaking scale (as required from unitarity constraints [24,25]), the value of Λ must satisfy If Λ is significantly larger than 1 TeV (often called the hierarchy problem in the literature), then the only way to generate a Higgs mass of O(m Z ) is to have an "unnatural" cancellation between the two terms of eq. (1). This seems highly unlikely given that the two terms of eq. (1) have completely different origins. The requirement of Λ ∼ O(1 TeV) as a condition for the absence of fine-tuning of the Higgs mass parameter is nicely illustrated in fig. 2(b), taken from ref. [14]. A viable theoretical framework that incorporates weakly-coupled Higgs bosons and satisfies the constraint of eq. (2) is that of "low-energy" or "weak-scale" supersymmetry [3]. If supersymmetry (which relates fermion and boson masses and interactions) is exact, then boson masses must exhibit the same logarithmically sensitivity to Λ as do the fermion masses. Since no supersymmetric partners of the Standard Model particles have been found, it follows that supersymmetry is not an exact symmetry of the fundamental particle interactions. Hence, in the framework of low-energy supersymmetry, Λ should be identified with the energy scale of supersymmetry-breaking. The naturalness constraint of eq. (2) is still relevant, which implies that the scale of supersymmetry breaking should not be much larger than a few TeV in order that the naturalness of scalar masses be preserved. Moreover, low-energy supersymmetry with a supersymmetry-breaking scale of O(1 TeV) is precisely what is needed to explain the observed gauge coupling unification as previously noted. We conclude that a suitable replacement for the Standard Model is a supersymmetric extension of the Standard Model as the effective field theory of the TeV scale. One good feature of the supersymmetric approach is that the effective low-energy supersymmetric theory can be valid all the way up to the Planck scale, while still being natural! The physics of the Higgs bosons will be explored by experiments now underway at the upgraded proton-antiproton Tevatron collider at Fermilab and in the near future at the Large Hadron Collider (LHC) at CERN. Once evidence for electroweak symmetry breaking dynamics is obtained, a more complete understanding of the mechanism involved will require experimentation at a future e + e − linear collider (LC) now under development. In this review we focus primarily on the theory and phenomenology of the Standard Model Higgs boson and the Higgs bosons of low-energy supersymmetry. In Section 2, we review the theoretical properties of the Standard Model Higgs boson, and exhibit its main branching ratio and production rates at hadron colliders and at the LC. The main Higgs boson search techniques at the Tevatron, LHC and the LC are described. In Section 3, we examine the Higgs bosons of the minimal supersymmetric Standard Model (MSSM). We summarize the tree-level properties of the MSSM Higgs sector and describe the most significant effects of the radiative corrections to the computation of the Higgs masses and couplings. We then exhibit the main branching ratios and production rates of the MSSM Higgs bosons and survey the phenomenology of the MSSM Higgs sector at the Tevatron, LHC and LC. A brief summary concludes this review in Section 4.

The Standard Model Higgs Boson
In the Standard Model, the Higgs mass is given by: m 2 h SM = 1 2 λv 2 , where λ is the Higgs self-coupling parameter. Since λ is unknown at present, the value of the Standard Model Higgs mass is not predicted. However, other theoretical considerations, discussed in Section 1, place constraints on the Higgs mass as exhibited in fig. 2. In contrast, the Higgs couplings to fermions [bosons] are predicted by the theory to be proportional to the corresponding particle masses [squared-masses]. In particular, the SM Higgs boson is a CP-even scalar, and its couplings to gauge bosons, Higgs bosons and fermions are given by: 3 where h ≡ h SM , V = W or Z and v = 2m W /g = 246 GeV. In Higgs production and decay processes, the dominant mechanisms involve the coupling of the Higgs boson to the W ± , Z and/or the third generation quarks and leptons. Note that a h SM gg coupling (g=gluon) is induced by virtue of a one-loop graph in which the Higgs boson couples to a virtual tt pair. Likewise, a h SM γγ coupling is generated, although in this case the one-loop graph in which the Higgs boson couples to a virtual W + W − pair is the dominant contribution. Further details of the SM Higgs boson properties are given in ref. [2]. A review of the SM Higgs properties and its phenomenology, with an emphasis on the impact of loop corrections to the Higgs decay rates and cross-sections can be found in ref. [26].

Standard Model Higgs Boson Decay Modes
The branching ratios for the main decay modes of a SM Higgs boson are shown as a function of Higgs boson mass in fig. 3 and 4(a), based on the results obtained using the HDECAY program [27]. For Higgs boson masses below 135 GeV, the decay h SM → bb dominates, whereas above 135 GeV, the dominant decay mode is h SM → W W ( * ) (below W + W − threshold, one of the W bosons is virtual as indicated by the star). Above tt threshold, the branching ratio into top-quark pairs increases rapidly as a function of Higgs mass, reaching a maximum of about 20% at m h SM ∼ 450 GeV. The total Higgs width is obtained by summing all the Higgs partial widths and is displayed as a function of Higgs mass in fig. 4(b). The leading effects of the QCD corrections to the Higgs decay to quark pairs [28] 4 can be taken into account by using the tree-level formula for the Higgs partial width (which depends on the quark mass), and identifying the quark mass with the running quark mass evaluated at the Higgs mass, m Q (m h SM ). The running quark mass, m Q (m h SM ) is obtained from the MS mass, m Q (M Q ) [where M Q is the corresponding quark pole mass], by renormalization group evolution. The MS quark masses are obtained from fits to experimental data [33]. Note that the large decrease in the charm quark mass due to QCD running is responsible for suppressing BR(cc) relative to BR(τ + τ − ), in spite of the color enhancement of the former, thereby reversing the naively expected hierarchy. Below the corresponding two-body thresholds, the W W ( * ) , ZZ ( * ) and t ( * )t decay modes (where the asterisk indicates an off-shell particle) are still relevant as shown in fig. 4. The h SM gg, h SM γγ and h SM Zγ vertices are generated at one-loop. The partial width for h SM → gg is primarily of interest because it determines the gg → h SM production cross-section. The h SM γγ vertex is especially relevant both for the h SM → γγ discovery mode at the LHC and for the γγ → h SM production mode at the LC operating as a γγ collider.
Combining these Higgs production mechanisms with the decays discussed in Section 2.1, one obtains the most promising signatures. Due to the large luminosity of gluons at high energy hadron colliders, gg → h SM is the Higgs production mechanism with the largest cross-section at the Tevatron and the LHC [37,38]. The twoloop, next-to-leading order (NLO) QCD corrections enhance the gluon fusion cross-section by about a factor of two [38,39]. The corresponding NLO differential cross-section (as a function of the Higgs boson p T and rapidity) has also been obtained [40]. Recently, the next-to-NLO (NNLO) QCD corrections have been evaluated [41], and show a further enhancement of about 10% to 30% depending on the Higgs mass and center-of-mass energy of the collider. The remaining scale dependence and the effects of higher order terms not yet computed are estimated to give a theoretical uncertainty of 10-20%. The dependence of the gluon fusion cross-section on different parton densities yields roughly an additional uncertainty of order 10%.
The cross-section for qq → W ± h SM (summed over both W charge states) is the second largest Higgs cross-section at the Tevatron for m h SM < ∼ 175 GeV. At the LHC, the W ± h SM cross-section is not as prominent over the Higgs mass range of interest. The corresponding qq → Zh SM cross-section is roughly a factor of two lower than the corresponding W ± h SM cross-section. The QCD corrections to σ(qq → V h SM ) [V = W or Z] coincide with those of the Drell-Yan process and increase the crosssections by about 30% [42,43,44]. The theoretical uncertainty is estimated to be about 15% from the remaining scale dependence. The dependence on different sets of parton densities is rather weak and also leads to a variation of the production cross-sections by about 15%.
Vector boson fusion is a shorthand notation for the full qq → qqh SM process, where the quark and anti-quark both radiate virtual vector bosons (V * ) which then annihilate to produce the Higgs boson. Vector boson fusion via ud → duh SM (and its charge-conjugate process) is also possible. In figs. 5 and 6, all contributing processes are included, and the sum of all such contributions is labeled qq → qqh SM for simplicity. The QCD corrections enhance the cross-section by about 10% [44,45]. The vector boson fusion process is the second largest Higgs cross-section at the LHC; its cross-section approaches the gg → h SM cross-section for m h SM ∼ 1 TeV.
The cross-sections for gg, qq → tth SM at the Tevatron and the LHC are displayed in figs. 5 and 6. The NLO QCD corrections to qq → tth SM have recently been computed in refs. [46] and [47], and the corrections to gg → tth SM have been obtained in ref. [47]. Fig. 5 includes the complete NLO QCD corrections from ref. [47] (at Tevatron energies, tth SM production is dominated by the qq → tth SM subprocess). The size of the QCD corrections depends sensitively on the choice of scale, µ, employed in the running coupling constant and parton distribution functions. Changes in µ can significantly modify the tree-level cross-sections, whereas the NLO-corrected cross-sections are rather insensitive to reasonable changes in µ. In fig. 6, the tree-level tth SM cross-section is shown for µ 2 =ŝ (the square of the partonic center-of-mass energy), and is roughly a factor of two smaller than the corresponding cross-section with µ = m t . With respect to the latter choice, the NLO-corrections of ref. [47] are rather small, typically of order 10-20% depending on the precise choice of µ.
The tree-level gg, qq → bbh SM cross-section (as a function of m h SM ) shown in fig. 5 has been computed by fixing the scales of the parton distribution functions, the running coupling α s and the running Higgs-bottom-quark Yukawa coupling (or equivalently, the running b-quark mass) at the value of the corresponding h SM mass. In fig. 6, the b-quark pole mass is employed, 5 while the scales for α s and the parton distribution functions were set equal to the partonic center-of-mass energy.
By using the running b quark mass, one implicitly resums large logarithms associated with the QCDcorrected Yukawa coupling. Thus, the tree-level bbh SM cross-section displayed in fig. 5    , for the various production mechanisms as a function of the Higgs mass, taken primarily from refs. [32] and [35]. The full NLO QCD-corrected results are employed for the gluon fusion gg → h SM , vector boson fusion qq → V * V * qq → h SM qq (here, qq refers to both ud and qq scattering), Higgs-strahlung processes qq → V * → V h SM (where V = W ± , Z), bb → h SM (taken from ref. [36]), and gg, qq → h SM tt. Tree-level cross-sections are exhibited for gg, qq → h SM bb. In the latter case, the cross-section has been computed with a running b-quark mass, α s and the parton distribution functions all evaluated at the corresponding Higgs mass. a part of the QCD corrections to the full inclusive cross-section. However, the most significant effect of the QCD corrections to bbh SM production arises from the kinematical region where the b quarks are emitted near the forward direction. In fact, large logarithms arising in this region spoil the convergence of the QCD perturbation series since α s ln( . These large logarithms (already present at lowest order) must be resummed to all orders, and this resummation is accomplished by the generation of the b-quark distribution function [48,49]. Thus, the QCD-corrected fully inclusive bbh SM cross-section can be approximated by bb → h SM and its QCD corrections. 6 The latter is also exhibited in fig. 5 and is , for the various production mechanisms as a function of the Higgs mass, taken from ref. [44]. The cross-section curves for gg, qq → h SM tt (which has not been updated to include the NLO calculation of ref. [47]) and gg, qq → h SM bb are based on a tree-level calculation with t-quark and b-quark pole masses and α s and the parton distribution functions evaluated at the partonic center-of-mass energy.
seen to be roughly an order of magnitude larger than the tree-level bbh SM cross-section [36]. Of course, this result is not very relevant for the searches at hadron colliders in which transverse momentum cuts on the b-jets are employed. Ultimately, one needs the QCD-corrected differential cross-section for bbh SM (as a function of the final state b-quark transverse momentum) in order to do realistic simulations of the Higgs signal in this channel. However, if only one b-quark jet is tagged, it may be sufficient to consider the process bg → bh SM . The cross-section for bg → bh SM at lowest order can be found in ref. [51]; the NLO QCD-corrected cross-section has been recently obtained in ref. [52]. For example, assuming that p T > 15 GeV, and the pseudorapidity |η| < 2 for the observed b-quark jet, the NLO cross-section at the Tevatron ranges from about 6 fb to 0.25 fb for 100 GeV ≤ m h SM ≤ 200 GeV. Increasing the cuts to p T > 30 GeV and |η| < 2.5 at the LHC yields a range of cross-sections from about 200 fb to 1.2 fb for 100 GeV ≤ m h SM ≤ 500 GeV.
Not shown in figs. 5 and 6 is the cross-section for inclusive double Higgs production (h SM h SM + X).
Double Higgs production is not observable at the Tevatron, but may be possible to detect at the LHC given sufficient luminosity. The main contributions to double Higgs production in order of importance are: The gluon-gluon fusion cross-section dominates by at least an order of magnitude, so we focus on this subprocess [53,54,55,56]. Including NLO QCD corrections, typical cross-sections for pp → h SM h SM + X at √ s = 14 TeV range from about 40 fb to 10 fb for 100 < m h SM < 200 GeV [55,56]. There are two classes of diagrams that contribute to h SM h SM production via gluon fusion: gg → h * SM → h SM h SM (via the top-quark triangle diagram) which is sensitive to the triple-Higgs vertex, and gg → t * t * → h SM h SM (via the top-quark box diagram) which is independent of the triple-Higgs vertex. Due to the relatively low cross-sections, it will be very challenging to extract information on the Higgs self-coupling parameter from LHC data.
Finally, assuming that very forward protons can be tagged, diffractive production of Higgs bosons may provide a viable signal at the LHC (the corresponding cross-sections at the Tevatron are probably too small) [57,58]. 7 Such events are characterized by rapidity gaps, i.e., the absence of particle production between the forward protons and the centrally produced Higgs boson. In particular, for m h SM ∼ 130 GeV, the exclusive process pp → pph SM (with h SM → bb) can be used at the LHC to obtain the Higgs mass with an accuracy of about 1% by measuring the invariant mass recoiling against the final state protons [59].

Standard Model Higgs Boson Searches at the Tevatron
In the mass region of interest to the Tevatron Higgs search (100 GeV < ∼ m h SM < ∼ 200 GeV), the SM Higgs boson is produced most copiously via gg fusion, with a cross-section from about 1.0-0.1 pb. For m h SM < ∼ 135 GeV, the Higgs boson decays dominantly to bb. Since the cross-section for the QCD production of bb dijet events is orders of magnitude larger than the Higgs production cross-section, the gg → h SM → bb channel is not a promising channel. For m h SM > ∼ 135 GeV, the Higgs boson decays dominantly to W W ( * ) (where W * is a virtual W ), and the channel gg → h SM → W W ( * ) is accessible to the Tevatron Higgs search [60].
Given sufficient luminosity, the most promising SM Higgs discovery mechanism at the Tevatron for m h SM < ∼ 135 GeV consists of qq annihilation into a virtual V * (V = W or Z), where V * → V h SM followed by a leptonic decay of the V and h SM → bb [61]. The sum of the W h SM and Zh SM cross-sections is about 0.2-0.5 pb in the mass region of interest (100 < ∼ m h SM < ∼ 135 GeV), in which the dominant Higgs decay is h SM → bb. These processes lead to three main final states, ℓνbb, ννbb and ℓ + ℓ − bb, that exhibit distinctive signatures on which the experiments can trigger (high p T leptons and/or missing E T ). The backgrounds are manageable and are typically dominated by vector-boson pair production, tt production and QCD dijet production. The signal efficiencies and backgrounds have all been estimated 7 A critical comparison of various theoretical approaches to diffractive Higgs production can be found in ref. [58].
with both the CDF Run 1 detector simulation and with the simple SHW simulation [62]. Monte Carlo estimates have been used for the backgrounds everywhere except in the ννbb channel, where there is a significant contribution from QCD bb dijet production. To be conservative, in ref. [62] the unknown QCD bb dijet background to the ννbb channel has been taken to be equal in size to the sum of all other contributing background processes. In addition, the separation of signal from background was optimized using neural network techniques [63], resulting in a demonstrable gain in the significance of the Higgs signal for the ℓνbb and ννbb channels [62]. The b-tagging efficiencies and the bb mass resolution play a key role in determining the ultimate efficiency and background rejection. Much work remains, using real data studies, to optimize the performance in both these areas.
For larger Higgs masses (m h SM > ∼ 135 GeV) it is possible to exploit the distinct signatures present when the Higgs boson decay branching ratio to W W ( * ) becomes appreciable. In this case, there are final states with W W (from the gluon-fusion production of a single Higgs boson), and W W W and ZW W arising from associated vector boson-Higgs boson production. Three search channels were identified in ref. [62] as potentially sensitive at these high Higgs masses: like-sign dilepton plus jets (ℓ ± ℓ ± jj) events, high-p T lepton pairs plus missing E T (ℓ + ℓ − νν), and trilepton (ℓ ± ℓ ′± ℓ ∓ ) events. Of these, the first two were found to be most sensitive. The strong angular correlations of the final state leptons resulting from W W * is one of the crucial ingredients for these discovery channels [64,65,60].
The integrated luminosity required per Tevatron experiment as a function of Higgs mass to either exclude the SM Higgs boson at 95% CL or discover it at the 3σ or 5σ level of significance, for the SHW analyses with neural net selection (see ref. [62] for details), is shown in Fig. 7(a). These results are based on the combined statistical power of both the CDF and DØ experiments. The bands extend from the neural net result on the low side upward in required integrated luminosity by 30% to the high side, as an indication of the range of uncertainty in the b-tagging efficiency, bb mass resolution and background uncertainties. As the plots show, the required integrated luminosity increases rapidly with Higgs mass to 140 GeV, beyond which the high-mass channels play the dominant role. If m h SM = 115 GeV, which lies just above the 95% CL exclusion limit achieved by LEP [11], then 5 fb −1 of integrated luminosity per experiment would provide sufficient data to see a 3σ excess above background. With 15 fb −1 of integrated luminosity per experiment, a 5σ discovery of the Higgs boson would be possible.
The final result shows that for an integrated luminosity of 10 fb −1 , if the SM Higgs boson mass lies beyond the discovery reach of the Tevatron, then one can attain a 95% CL exclusion for masses up to about 180 GeV. Moreover, if the SM Higgs happens to be sufficiently light (m h SM < ∼ 125 GeV), then a tantalizing 3σ effect will be visible with the same integrated luminosity. With about 25 fb −1 of data, 3σ evidence for the Higgs boson can be obtained for the entire Higgs mass range up to 180 GeV. However, the discovery reach is considerably more limited for a 5σ Higgs boson signal. With 30 fb −1 integrated luminosity delivered per detector, a 5σ Higgs boson discovery may be possible for Higgs masses up to about 130 GeV, a significant extension of the LEP Standard Model Higgs search. The latter figure of merit is particularly significant when applied to the search for the lightest Higgs bosons of the MSSM. We address this case in Section 3.5.3.   Figure 7: (a) The integrated luminosity required per Tevatron experiment, to either exclude a SM Higgs boson at 95% CL or observe it at the 3σ or 5σ level, as a function of the Higgs mass [62]. (b) Expected 5σ discovery luminosity requirements for the SM Higgs boson at the LHC for one experiment, based on a study performed with CMS fast detector simulation, assuming statistical errors only [72]. The gg and W + W − fusion processes are indicated respectively by the solid and dotted lines.
Other Higgs signatures could help improve the sensitivity of the Higgs search at the Tevatron. In ref. [62], channels containing the h SM → τ + τ − decay mode have not been studied, as the small branching ratio (less than 8%) makes the corresponding signal rates small. Still, a significant improvement of τlepton identification could lead to a viable Higgs signal in the Higgs mass region 120 GeV < ∼ m h SM < ∼ 140 GeV [66]. Another possibility which has been explored is the detection of the Higgs boson via tth SM production (the Higgs boson is radiated off the top-quark), followed by h SM → bb. Initial studies [67] suggested that this channel could be observable at the upgraded Tevatron for m h SM < ∼ 140 GeV, with a statistical significance comparable to the Higgs signals in the W h SM and Zh SM channels.
If a Higgs boson is discovered at the Tevatron, one can begin to measure some of its properties. The Higgs mass can be measured with an accuracy of about 2 GeV [68]. However, the determination of the Higgs couplings to W and Z bosons and to bb will be model-dependent and rather crude. To improve and expand the possible Higgs measurements and determine its phenomenological profile will require Higgs studies at the LHC.

Standard Model Higgs Boson Searches at the LHC
Production rates for the Higgs boson in the Standard Model are significantly larger at the LHC [44,69]. The dominant Higgs production process, gluon fusion, can be exploited in conjunction with a variety of other channels, e.g., W W/ZZ fusion of the Higgs boson and Higgs radiation off top quarks [70,71,72,73,74,75]. Integrated luminosities between 30 and 100 fb −1 , achievable within the first few years of LHC operation, will be sufficient to cover the entire canonical Higgs mass range of the Standard Model up to values close to 1 TeV with a significance greater than 5σ. The required LHC luminosities for a Higgs discovery in various channels are shown in fig. 7(b) [72]. Thus, there is no escape route for the SM Higgs boson at the LHC.
The properties of the SM Higgs boson can be determined with some precision at the LHC. The h SM → ZZ ( * ) → ℓ + ℓ − ℓ + ℓ − channel allows for an accurate Higgs mass determination of about 0.1% for 120 GeV < ∼ m h SM < ∼ 400 GeV, assuming an integrated luminosity of 300 fb −1 [76]. For larger Higgs masses, the precision in the Higgs mass measurement deteriorates due to the effect of the increasing Higgs width; nevertheless a 1% Higgs mass measurement is possible for m h SM ≃ 700 GeV. The Higgs width can be extracted with a precision of 5 to 6% over the mass range 300-700 GeV from the Breit-Wigner shape of the Higgs resonance [76]. Below 300 GeV, the instrumental resolution becomes larger than the Higgs width, and the accuracy of the Higgs width measurement degrades. For example, the four-lepton invariant mass spectrum from h SM → ZZ yields a precision of about 25% at m h SM = 240 GeV [68]. For lower Higgs masses, indirect methods must be employed to measure the Higgs width.
For Higgs masses below 200 GeV, a number of different Higgs decay channels can be studied at the LHC. The most relevant processes are where V = W or Z. The gluon-gluon fusion mechanism is the dominant Higgs production mechanism at the LHC, yielding a total cross-section of about 30 pb [15 pb] for m h SM = 120 GeV [m h SM = 200 GeV]. One also has appreciable Higgs production via V V electroweak gauge boson fusion, with a total crosssection of about 6 pb [3 pb] for the Higgs masses quoted above. The electroweak gauge boson fusion mechanism can be separated from the gluon fusion process by employing a forward jet tag and central jet vetoing techniques. Note that for 2m W < ∼ m h SM < ∼ 2m Z , the Higgs branching ratio to ZZ * is quite suppressed with respect to W W (since one of the Z bosons is off-shell). Hence, in this mass window, h SM → W + W − → ℓ + νℓ −ν is the main Higgs discovery channel [65], as exhibited in fig. 7(b).
The cross-section for tth SM production can be significant for Higgs masses in the intermediate mass range [47], 0.8 pb [0.2 pb] at m h SM = 120 GeV [m h SM = 200 GeV], although this cross-section falls faster with Higgs mass as compared to the gluon and gauge boson fusion mechanisms. Finally, we note that the preferred channel at the Tevatron, qq → W h SM → ℓνbb, is not a discovery mode at the LHC due in part to the larger background cross-sections at √ s = 14 TeV. Nevertheless, with 300 fb −1 of data, it may be possible to observe a SM Higgs signal (with S/ √ B ≥ 5) if m h SM < ∼ 125 GeV [77]. The measurements of Higgs decay branching ratios can be used to infer the ratios of the Higgs couplings, and provide an important first step in clarifying the nature of the Higgs boson [78,79,80]. These can be extracted from a variety of Higgs signals that are observable over a limited range of Higgs masses. In the mass range 110 GeV < ∼ m h SM < ∼ 150 GeV, the Higgs boson can be detected [with 100 fb −1 of data] in the γγ and the τ + τ − channels indicated above. (The µ + µ − channel was considered in ref. [80]. With 300 fb −1 of data, a 3σ excess above background may be possible for 110 GeV < ∼ m h SM < ∼ 140 GeV.) For m h SM > ∼ 130 GeV, the Higgs boson can also be detected in gluon-gluon fusion through its decay to W W ( * ) , with both final gauge bosons decaying leptonically [81], and to ZZ ( * ) in the four-lepton decay mode [70,74]. There is additional sensitivity to Higgs production via V V fusion followed by its decay to W W ( * ) for m h SM > ∼ 120 GeV. These data can be used to extract the ratios of the Higgs partial widths to gluon pairs, photon pairs, τ + τ − , and W + W − [78,79]. The expected accuracies in Higgs width ratios, partial widths, and the total Higgs width are exhibited in fig. 8. These results are obtained under the assumption that the partial Higgs widths to W + W − and ZZ are fixed by electroweak gauge invariance, and the ratio of the partial Higgs widths to bb and τ + τ − are fixed by the universality of Higgs couplings to down-type fermions. One can then extract the total Higgs width under the assumption that all other unobserved modes, in the Standard Model and beyond, possess small branching ratios of order 1%. Finally, we note that the specific Lorentz structure predicted for the h SM W + W − coupling by the Higgs mechanism can be tested in angular correlations between the spectator jets in W W fusion of the Higgs boson at the LHC [79].
With an integrated luminosity of 100 fb −1 per experiment, the relative accuracy expected at the LHC for various ratios of Higgs partial widths Γ i (or equivalently, ratios of Higgs squared-couplings) range from 10% to 30%, as shown in fig. 8. The ratio Γ τ /Γ W measures the coupling of down-type fermions relative to the Higgs couplings to gauge bosons. To the extent that the one-loop h SM γγ amplitude is dominated by the W -loop, the partial width ratio Γ τ /Γ γ probes the same relationship. In contrast, under the usual assumption that the one-loop h SM gg amplitude is dominated by the top-quark loop, the ratio Γ g /Γ W probes the coupling of up-type fermions relative to the h SM W W coupling. In addition, one can measure the h SM tt coupling directly by making use of the tth SM production mode at the LHC. Recent studies suggest that for an integrated luminosity of 100 fb −1 , this signal is viable for the h SM → bb [82,83] and h SM → τ + τ − [84] decay modes if m h SM < ∼ 130-140 GeV, and for the h SM → W W ( * ) decay mode for 130 < ∼ m h SM < ∼ 200 GeV [85]. In this way, one expects to be able to measure the Higgs-top quark Yukawa coupling with a relative accuracy in the range of 10-20%.
Finally, the measurement of the triple Higgs self-coupling with an accuracy of order 25% can be obtained, in a very limited Higgs mass window, with 3000 fb −1 of data [86] (which requires at least a factor-of-ten luminosity upgrade of the LHC [87]).

Standard Model Higgs Boson Searches at the LC
The next generation of high energy e + e − linear colliders is expected to operate at energies from 300 GeV up to about 1 TeV (JLC, NLC, TESLA), henceforth referred to as the LC [88,89,90]. The possibility of a multi-TeV linear collider operating in an energy range of 3-5 TeV (CLIC) is also under study [91]. With the expected high luminosities up to 1 ab −1 , accumulated within a few years in a clean experimental environment, these colliders are ideal instruments for reconstructing the mechanism of electroweak symmetry breaking in a comprehensive and conclusive form.
Weakly-coupled electroweak symmetry breaking dynamics involving an elementary scalar Higgs field can be established experimentally in three steps. First, the Higgs boson must be observed clearly and unambiguously, and its basic properties-mass, width, spin and C and P quantum numbers-must be determined. Second, the couplings of the Higgs boson to the W ± and Z bosons and to leptons and quarks must be measured. Demonstrating that these couplings scale with the mass of the corresponding particle would provide critical support for the Higgs mechanism based on scalar dynamics as the agent responsible for generating the masses of the fundamental particles. Finally, the Higgs potential must be reconstructed by measuring the self-coupling of the Higgs field. The specific form of the potential shifts the ground state to a non-zero value, thereby providing the mechanism for electroweak symmetry breaking based on the self-interactions of scalar fields. Essential elements of this program can be realized at a high-luminosity e + e − linear collider, and high-precision analyses of the Higgs boson are possible in these machines [92,93,94].
The main production mechanisms of the SM Higgs boson at the LC are the Higgs-strahlung process [95,25], e + e − → Zh SM , and the W W fusion process [96] e + e − →ν e ν e W * W * →ν e ν e h SM . With an accumulated luminosity of 500 fb −1 , about 10 5 Higgs bosons can be produced by Higgs-strahlung in the theoretically preferred intermediate mass range below 200 GeV. As √ s is increased, the cross-section for the Higgs-strahlung process decreases as s −1 and is dominant at low energies, while the cross-section for the W W fusion process grows as ln(s/m 2 h SM ) and dominates at high energies (see ref. [97] for a convenient form for the corresponding cross-sections), as shown in fig. 9(a). 8 The ZZ fusion mechanism, e + e − → e + e − Z * Z * → e + e − h SM , also contributes to Higgs production, with a cross-section suppressed with respect to that of W W fusion by a factor of 16 cos 4 θ W /[1 + (1 − 4 sin 2 θ W ) 2 ] ≃ 9.4. The crosssections for the Higgs-strahlung and the W W fusion processes are shown in fig. 9(a) for three values of √ s. The Higgs-strahlung process, e + e − → Zh SM , with Z → ℓ + ℓ − , offers a very distinctive signature.

For
√ s = 350 and 500 GeV and an integrated luminosity of 500 fb −1 , this ensures the observation of the SM Higgs boson up to the production kinematical limit independently of its decay [92]. At √ s = 500 GeV, the Higgs-strahlung and the W W fusion processes have approximately the same cross-sections, O(50 fb), for 100 GeV < ∼ m h SM < ∼ 200 GeV. At √ s = 800 GeV with 500 fb −1 of data, the analysis of ref. [100] suggests that a Higgs boson with mass up to about 650 GeV will be observable at the LC. Finally, the process e + e − → tth SM [101] yields a distinctive signature consisting of two W bosons and four b-quark jets, and can be observed at the LC given sufficient energy and luminosity if the Higgs mass is not too large. The QCD-corrected cross-sections for this process [102] for √ s = 500 GeV and 800 GeV are shown in fig. 9(b). The phenomenological profile of the Higgs boson can be determined by precision measurements. For example, consider the case of m h SM = 120 GeV at the LC with √ s = 350 GeV and 500 fb −1 of data [92]. The spin and parity of the Higgs boson can be determined unambiguously from the steep onset of the excitation curve in Higgs-strahlung near the threshold (see fig. 10(a) [103]) and the angular correlations in this process [104]. By measuring final state angular distributions and various angular and polarization asymmetries, one can check whether the Higgs boson is a state of definite CP, or whether it exhibits CP-violating behavior in its production and/or decays [105]. The Higgs mass can be measured to an accuracy of 40 MeV by reconstructing the Higgs boson in Zh SM production and combining the results from the various final state channels. The Higgs width can be inferred in a model-independent way, with an accuracy of about 6%, by combining the partial width to W + W − , accessible in the vector boson fusion process, with the W W * decay branching ratio. Similar results (with precisions within a factor of two of those quoted above) are obtained for larger Higgs masses in the intermediate mass regime. The h SM ZZ, h SM Zγ and h SM tt couplings can be determined (with some sensitivity to possible anomalous couplings, if present) by the optimal-observable method [106], which makes optimal use of the polarized angular distributions and asymmetries for Higgs-strahlung and h SM tt production. Higgs decay branching ratios can be measured very precisely for m h SM < ∼ 150 GeV [107,108,109,110]. When such measurements are combined with measurements of Higgs production cross-sections, the absolute values of the Higgs couplings to the W ± and Z gauge bosons and the Yukawa couplings to leptons and quarks can be determined to a few percent in a model-independent way. In addition, the Higgs-top quark Yukawa coupling can be inferred from the cross-section for Higgs emission off tt pairs [102,111]. As an example, Table 1 exhibits the anticipated fractional uncertainties in the measurements of Higgs branching ratios for m h SM = 120 GeV at the LC. 9 Using this data, a program Higgs coupling δBR/BR δg/g hW W 5.1%  [107]. Results for hγγ [108], htt [107], hµµ [109] and hhh [114,115] are based on 1 ab −1 of data at √ s = 500 GeV (for γγ and hh) and √ s = 800 GeV (for tt and µµ), respectively.
HFITTER was developed in ref. [107] to perform a Standard Model global fit based on the measurements of the Zh SM , ννh SM and tth SM cross-sections and the Higgs branching ratios listed in Table 1. The output of the program is a set of Higgs couplings along with their fractional uncertainties (which are also exhibited in Table 1). These results should be considered representative of what can eventually be achieved at the LC. For example, a comprehensive analysis of LC Higgs data will have to take into account the relevant electroweak loop corrections (which are not presently included in HFITTER).
One should also note that theoretical uncertainties for the predicted Higgs couplings have not been taken into account in this analysis. The theoretical uncertainty in g h SM cc is the most significant among the channels listed in Table 1, due to the uncertainties in c quark-mass and in α s (which governs the running of the quark masses from the quark mass to the Higgs mass). Ref. [112] estimates a theoretical fractional uncertainty in g hcc of about 12%, significantly greater than the experimental uncertainty listed in Table 1. In contrast, the theoretical fractional uncertainty in g hbb is about 1.8% due to the uncertainty of the b-quark mass. Although this is less than the anticipated experimental uncertainty, it should not be neglected in the determination of the overall g h SM bb uncertainty. The theoretical uncertainties in the other channels listed above are not significant compared to the quoted experimental uncertainties. The measurement of the Higgs self-couplings is a very ambitious task that requires the highest luminosities possible at the LC, which possess unique capabilities for addressing this question. The trilinear Higgs self-coupling can be measured in double Higgs-strahlung, in which a virtual Higgs boson splits into two real Higgs particles in the final state [113]. A simulation based on 1 ab −1 of data is exhibited in fig. 10(b) [114]. In this way, for m h SM = 120 GeV, the cubic term of the scalar potential can be established at the LC with a precision of about 20% [114,115]. Such a measurement is a prerequisite for determining the form of the Higgs potential that is responsible for spontaneous electroweak symmetry breaking generated by scalar sector dynamics. Finally, the total SM Higgs width can be obtained indirectly by using Γ tot = Γ h SM W W /BR(h SM → W W * ). The partial width is proportional to g 2 h SM W W , so the fractional uncertainty in Γ tot can be obtained from the results of Table 1. For m h SM < ∼ 150 GeV, an accuracy in the range of 5-10% can be achieved for the total Higgs width [116].
If the SM Higgs mass is above 150 GeV, then the precision determination of Higgs couplings will have to be reconsidered. The Higgs branching ratios into cc, gg and τ + τ − are now too small to be accurately measured. Due to the growing importance of the W W and ZZ modes, one can perform a precision measurement of the Higgs branching ratios to W W and ZZ, while the precision of the bb branching ratio is significantly reduced as m h SM increases to 200 GeV and beyond [117]. Moreover, due to the rapid decline of tth SM production cross-section with increasing m h SM , the Higgs-top quark Yukawa coupling cannot be extracted until m h SM > ∼ 2m t , at which point the tth SM coupling can be obtained by observing Higgs bosons produced by vector boson fusion which subsequently decay to tt. The analysis of Ref. [118] finds that at the LC with √ s = 800 GeV and 1 ab −1 of data, the tth SM Yukawa coupling can be determined with an accuracy of about 10% for a Higgs mass in the range 350-500 GeV. The total Higgs width can be obtained directly from measuring the Higgs boson line-shape if m h SM > ∼ 200 GeV.
The e + e − linear collider with center-of-mass energy √ s can also be designed to operate in a γγ col-lision mode. This is achieved by using Compton backscattered photons in the scattering of intense laser photons on the initial polarized e ± beams [119,120]. The resulting γγ center of mass energy is peaked for proper choices of machine parameters at about 0.8 √ s. The luminosity achievable as a function of the photon beam energy depends strongly on the machine parameters (in particular, the choice of laser polarizations). The γγ collider provides additional opportunities for Higgs physics [120,121,122,123,124]. The Higgs boson can be produced as an s-channel resonance in γγ collisions, and one can perform independent measurements of various Higgs couplings. For example, the product Γ(h SM → γγ)BR(h SM → bb) can be measured with a statistical accuracy of about 2-10% for 120 GeV < ∼ m h SM < ∼ 160 GeV with about 50 fb −1 of data [122,123,124]. In order to reach such a precision, it is critical to control the overwhelming two-jet background (with efficient b-tagging) and overcome the irreducible γγ → bb background by optimal use of the polarization of the photon beams and by judicious kinematic cuts. Knowledge of the QCD corrections to signal and background processes is essential for this task [122,125].
Using values for BR(h SM → bb) and BR(h SM → γγ) measured at the e + e − linear collider, one can obtain a value for the total Higgs width with an error dominated by the expected error in BR(h SM → γγ). For heavier Higgs bosons, m h SM > ∼ 200 GeV, the total Higgs width can in principle be measured directly by tuning the collider to scan across the Higgs resonance. One can also use the polarization of the photon beams to measure various asymmetries in Higgs production and decay, which are sensitive to the CP quantum numbers of the Higgs boson [123].

The Higgs Bosons of Low-Energy Supersymmetry
Electroweak symmetry breaking dynamics driven by a weakly-coupled elementary scalar sector requires a mechanism for the stability of the electroweak symmetry breaking scale with respect to the Planck scale [5]. Supersymmetry-breaking effects, whose origins may lie at energy scales much larger than 1 TeV, can induce a radiative breaking of the electroweak symmetry due to the effects of the large Higgs-top quark Yukawa coupling [126]. In this way, the origin of the electroweak symmetry breaking scale is intimately tied to the mechanism of supersymmetry breaking. Thus, supersymmetry provides an explanation for the stability of the hierarchy of scales, provided that supersymmetry-breaking masses in the low-energy effective electroweak theory are of O(1 TeV) or less [5].
A fundamental theory of supersymmetry-breaking is presently unknown. Nevertheless, one can parameterize the low-energy theory in terms of the most general set of soft-supersymmetry-breaking terms [127]. The simplest realistic model of low-energy supersymmetry is a minimal supersymmetric extension of the Standard Model (MSSM), which employs the minimal supersymmetric particle spectrum. However, even in this minimal model with the most general set of soft-supersymmetry-breaking terms, more than 100 new supersymmetric parameters are introduced [128]. Fortunately, most of these parameters have no impact on Higgs phenomenology. Thus, we will focus primarily on the Higgs sector of the MSSM and identify the parameters that govern the main properties of the Higgs bosons.

The Tree-Level Higgs Sector of the MSSM
Both hypercharge Y = −1 and Y = +1 complex Higgs doublets are required in any Higgs sector of an anomaly-free supersymmetric extension of the Standard Model. The supersymmetric structure of the theory also requires (at least) two Higgs doublets to generate mass for both "up"-type and "down"-type quarks (and charged leptons) [6]. Thus, the MSSM contains the particle spectrum of a two-Higgs-doublet extension of the Standard Model and the corresponding supersymmetric partners.
The two-doublet Higgs sector [129] contains eight scalar degrees of freedom: where the normalization has been chosen such that Spontaneous electroweak symmetry breaking results in three Goldstone bosons, which are absorbed and become the longitudinal components of the W ± and Z. The remaining five physical Higgs particles consist of a charged Higgs pair one CP-odd scalar and two CP-even scalars: (with m h ≤ m H ). The angle α arises when the CP-even Higgs squared-mass matrix (in the Φ 0 d -Φ 0 u basis) is diagonalized to obtain the physical CP-even Higgs states (explicit formulae will be given below). The supersymmetric structure of the theory imposes constraints on the Higgs sector of the model. For example, the Higgs self-interactions are not independent parameters; they can be expressed in terms of the electroweak gauge coupling constants. As a result, all Higgs sector parameters at tree-level are determined by two free parameters: the ratio of the two neutral Higgs field vacuum expectation values, and one Higgs mass, conveniently chosen to be m A . In particular, and the CP-even Higgs bosons h and H are eigenstates of the following squared-mass matrix The eigenvalues of M 2 0 are the squared-masses of the two CP-even Higgs scalars and α is the angle that diagonalizes the CP-even Higgs squared-mass matrix. From the above results, one obtains: In the convention where tan β is positive (i.e., 0 ≤ β ≤ π/2), the angle α lies in the range −π/2 ≤ α ≤ 0. An important consequence of eq. (12) is that there is an upper bound to the mass of the light CP-even Higgs boson, h. One finds that: This is in marked contrast to the Standard Model, in which the theory does not constrain the value of m h SM at tree-level. The origin of this difference is easy to ascertain. In the Standard Model, m 2 h SM = 1 2 λv 2 is proportional to the Higgs self-coupling λ, which is a free parameter. On the other hand, all Higgs self-coupling parameters of the MSSM are related to the squares of the electroweak gauge couplings.
Note that the Higgs mass inequality [eq. (14)] is saturated in the limit of large m A . In the limit of m A ≫ m Z , the expressions for the Higgs masses and mixing angle simplify and one finds Two consequences are immediately apparent. First, . This limit is known as the decoupling limit [130] because when m A is large, there exists an effective low-energy theory below the scale of m A in which the effective Higgs sector consists only of one CP-even Higgs boson, h. As we shall demonstrate below, the tree-level couplings of h are precisely those of the Standard Model Higgs boson when cos(β − α) = 0. From eq. (18), one can also derive: This result will prove useful in evaluating the CP-even Higgs boson couplings to fermion pairs in the decoupling limit. The phenomenology of the Higgs sector depends in detail on the various couplings of the Higgs bosons to gauge bosons, Higgs bosons and fermions. The couplings of the two CP-even Higgs bosons to W and Z pairs are given in terms of the angles α and β by where The couplings of V to two neutral Higgs bosons (which must have opposite CP-quantum numbers) are given by g φAZ (p φ − p A ), where φ = h or H and the momenta p φ and p A point into the vertex, and From the expressions above, we see that the following sum rules must hold separately for V = W and Z: Similar considerations also hold for the coupling of h and H to W ± H ∓ . Four-point couplings of vector bosons and Higgs bosons can be found in ref. [2]. The properties of the three-point and four-point Higgs boson-vector boson couplings are conveniently summarized by listing the couplings that are proportional to either sin(β − α) or cos(β − α), and the couplings that are independent of α and β [2]: where φ = h or H and V V = W + W − , ZZ, Zγ or γγ. Note in particular that all vertices in the theory that contain at least one vector boson and exactly one non-minimal Higgs boson state (H, A or H ± ) are proportional to cos(β − α). This can be understood as a consequence of unitarity sum rules which must be satisfied by the tree-level amplitudes of the theory [24,25,131,132].
In the MSSM, the tree-level Higgs couplings to fermions obey the following property: Φ 0 d couples exclusively to down-type fermion pairs and Φ 0 u couples exclusively to up-type fermion pairs. This pattern of Higgs-fermion couplings defines the Type-II two-Higgs-doublet model [133,2]. The gauge-invariant Type-II Yukawa interactions (using 3rd family notation) are given by: where .] Fermion masses are generated when the neutral Higgs components acquire vacuum expectation values. Inserting eq. (5) into eq. (26) yields a relation between the quark masses and the Yukawa couplings: Similarly, one can define the Yukawa coupling of the Higgs boson to τ -leptons (the latter is a down-type fermion). The couplings of the physical Higgs bosons to the third generation fermions is obtained from eq. (26) by using eqs. (6)- (8). In particular, the couplings of the neutral Higgs bosons to ff relative to the Standard Model value, gm f /2m W , are given by htt : Hbb (or Hτ + τ − ) : Htt : Abb (or Aτ + τ − ) : Att : (the γ 5 indicates a pseudoscalar coupling), and the charged Higgs boson couplings to fermion pairs, with all particles pointing into the vertex, are given by We next examine the behavior of the Higgs couplings at large tan β. This limit is of particular interest since at large tan β, some of the Higgs couplings to down-type fermions can be significantly enhanced. 11 Consider two large tan β regions of interest: (i) If m A ≫ m Z , then the decoupling limit is reached, in which | cos(β −α)| ≪ 1 and m H ≃ m A . From eqs. (18)- (33), it follows that the bbH and bbA couplings have equal strength and are significantly enhanced (by a factor of tan β) relative to the bbh SM coupling, whereas the V V H coupling is negligibly small. In contrast, the values of the V V h and bbh couplings are equal to the corresponding couplings of the Standard Model Higgs boson. To show that the value of the bbh coupling [eq. (28)] reduces to that of bbh SM in the decoupling limit, note that eq. (18) implies that | tan β cos(β − α)| ≪ 1 when m A ≫ m Z even when tan β ≫ 1. Indeed, h is a SM-like Higgs boson. (ii) If m A < ∼ m Z and tan β ≫ 1, then | cos(β − α)| ∼ 1 [see fig. 11] and m h ≃ m A . In this case, the bbh and bbA couplings have equal strength and are significantly enhanced (by a factor of tan β) relative to the bbh SM coupling, while the V V h coupling is negligibly small. Using eq. (22) it follows that the V V H coupling is equal in strength to the V V h SM coupling. In this case, it is conventional to refer to H as a SM-like Higgs boson. However, this nomenclature is somewhat inaccurate, since the value of the bbH coupling can differ from the corresponding bbh SM coupling when tan β ≫ 1 [since in case (ii), where | sin(β − α)| ≪ 1, the product tan β sin(β − α) need not be particularly small]. Note that in both cases (i) and ii) above, only two of the three neutral Higgs bosons have enhanced couplings to bb.
The decoupling limit of m A ≫ m Z is effective for all values of tan β. It is easy to check that the pattern of all Higgs couplings displayed in eqs. (20)-(33) respect the decoupling limit. That is, in the , which means that the h couplings to Standard Model particles approach values corresponding precisely to the couplings of the SM Higgs boson. There is a significant region of MSSM Higgs sector parameter space in which the decoupling limit applies, because cos(β − α) approaches zero quite rapidly once m A is larger than about 200 GeV, as shown in fig. 11. As a result, over a significant region of the MSSM parameter space, the search for the lightest CP-even Higgs boson of the MSSM is equivalent to the search for the Standard Model Higgs boson. This result is more general; in many theories of non-minimal Higgs sectors, there is a significant portion of the parameter space that approximates the decoupling limit. Consequently, simulations of the Standard Model Higgs signal are also relevant for exploring the more general Higgs sector.

Radiatively-Corrected MSSM Higgs Masses
The discussion of Section 3.1 was based on a tree-level analysis of the Higgs sector. However, radiative corrections can have a significant impact on the predicted values of Higgs masses and couplings. The radiative corrections involve both loops of Standard Model particles and loops of supersymmetric part-ners. The dominant effects arise from loops involving the third generation quarks and squarks and are proportional to the corresponding Yukawa couplings. Thus, we first review the parameters that control the masses and mixing of the third-generation squarks. (We shall neglect intergenerational mixing effects, which have little impact on the discussion that follows.) For each left-handed and right-handed quark of fixed flavor, q, there is a corresponding supersymmetric partner q L and q R , respectively. These are the so-called interaction eigenstates, which mix according to the squark squared-mass matrix. The mixing angle that diagonalizes the squark mass matrix will be denoted by θ q . The squark mass eigenstates, denoted by q 1 and q 2 , are obtained by diagonalizing the following 2 × 2 matrix where In addition, f = t, M R ≡ M U , e t = 2/3 and T 3f = 1/2 for the top-squark squared-mass matrix, and f = b, M R ≡ M D , e b = −1/3 and T 3f = −1/2 for the bottom-squark squared-mass matrix. The squark mixing parameters are given by Thus, the top-squark and bottom-squark masses and mixing angles depend on the supersymmetric Higgsino mass parameter µ and the soft-supersymmetry-breaking parameters: For simplicity, we shall initially assume that A t , A b and µ are real parameters. That is, we neglect possible CP-violating effects that can enter the MSSM Higgs sector via radiative corrections. The impact on new MSSM sources of CP-violation on the Higgs sector will be addressed in Section 3.2.3.

Radiatively-corrected Higgs masses in the CP-conserving MSSM
The radiative corrections to the Higgs squared-masses have been computed by a number of techniques, and using a variety of approximations such as the effective potential at one-loop [134,135,136] and two-loops [137,138,139,140] , and diagrammatic methods [141,142,143,144,145,146]. Complete oneloop diagrammatic computations of the MSSM Higgs masses have been presented by a number of groups [143,144]; and partial two-loop diagrammatic results are also known [145,146]. These include the O(m 2 t h 2 t α s ) contributions to the neutral CP-even Higgs boson squared-masses in the on-shell scheme [146]. Finally, renormalization group methods (to be discussed further below) provide a powerful technique for identifying many of the most important contributions to the radiatively corrected Higgs masses [147,148,149,150]. Typical results for the radiatively corrected value of m h as a function of the relevant supersymmetric parameters are shown in fig. 12.
One of the most striking effects of the radiative corrections to the MSSM Higgs sector is the modification of the upper bound of the light CP-even Higgs mass, as first noted in refs. [134] and [141]. Consider the region of parameter space where tan β is large and m A ≫ m Z . In this limit, the tree-level prediction for m h corresponds to its theoretical upper bound, m h = m Z . Including radiative corrections, the theoretical upper bound is increased. The dominant effect arises from an incomplete cancellation 12 of the top-quark and top-squark loops (these effects cancel in the exact supersymmetric limit). The qualitative behavior of the radiative corrections can be most easily seen in the large top squark mass limit, where in addition, the splitting of the two diagonal entries and the off-diagonal entry of the top-squark squared-mass matrix are both small in comparison to the average of the two top-squark squared-masses: In this case, the upper bound on the lightest CP-even Higgs mass is approximately given by The more complete treatments of the radiative corrections cited above show that eq. (39) somewhat overestimates the true upper bound of m h . Nevertheless, eq. (39) correctly reflects some noteworthy features of the more precise result. First, the increase of the light CP-even Higgs mass bound beyond m Z can be significant. This is a consequence of the m 4 t enhancement of the one-loop radiative correction. Second, the dependence of the light Higgs mass on the top-squark mixing parameter X t implies that (for a given value of M S ) the upper bound of the light Higgs mass initially increases with X t and reaches its maximal value for X t = √ 6M S . This point is referred to as the maximal mixing case (whereas X t = 0 corresponds to the minimal mixing case). In a more complete computation that includes both two-loop logarithmic and non-logarithmic corrections, the X t values corresponding to maximal and minimal mixing are shifted and exhibit an asymmetry under X t → −X t as shown in fig. 12. In the numerical analysis presented in this and subsequent figures in this section, we assume for simplicity that the third generation diagonal soft-supersymmetry-breaking squark squared-masses are degenerate: M SUSY ≡ M Q = M U = M D , which defines the parameter M SUSY . 13 Third, note the logarithmic sensitivity to the top-squark masses. Naturalness arguments that underlie low-energy supersymmetry imply that the supersymmetric particle masses should not be larger than a few TeV. Still, the precise upper bound on the light Higgs mass depends on the specific choice for the upper limit of the top-squark masses. The dependence of the light Higgs mass obtained by the more complete computation as a function of M SUSY is shown in fig. 13. 14 As noted above, the largest contribution to the one-loop radiative corrections is enhanced by a factor of m 4 t and grows logarithmically with the top squark mass. Thus, higher order radiative corrections can be non-negligible for large top squark masses, in which case the large logarithms must be resummed. Renormalization group (RG) techniques for resumming the leading logarithms have been developed by a number of authors [147,148,149]. The computation of the RG-improved one-loop corrections requires numerical integration of a coupled set of RG equations [148]. Although this procedure has been carried out in the literature, the analysis is unwieldy and not easily amenable to large-scale Monte-Carlo studies. It turns out that over most of the parameter range, it is sufficient to include the leading and sub-leading logarithms at two-loop order. (Some additional non-logarithmic terms, which cannot be ascertained by the renormalization group method, must also be included [151].) Compact analytic expressions have been obtained for the dominant one and two-loop contributions to the matrix elements of the radiatively-corrected CP-even Higgs squared-mass matrix: where the tree-level contribution M 2 0 was given in eq. (11) and δM 2 is the contribution from the 13 We also assume that M SUSY ≫ m t , in which case it follows that M 2 S ≃ M 2 SUSY up to corrections of O(m 2 t /M 2 SUSY ). 14 The flattening of the curves in fig. 13 as a function of M SUSY in the maximal mixing scenario is due to the squarkmixing contributions at two-loops which partially cancel the contributions that grow logarithmically with M SUSY .
where s β ≡ sin β, c β ≡ cos β, and the coefficients c ij are: is the average squared top squark mass. 15 The δM 2 ij also depend on the MSSM parameters A t , A b and µ that enter the off-diagonal top-squark and bottom-squark squared-mass matrices. We employ the following notation: Diagonalizing the CP-even Higgs squared-mass matrix yields radiatively-corrected values for m 2 h , m 2 H and the mixing angle α. 16 The end result is a prediction for the Higgs mass in terms of running parameters in the MS scheme. It is a simple matter to relate these parameters to the corresponding on-shell parameters used in the diagrammatic calculations [138,151]. Additional non-logarithmic two-loop contributions, which can generate a non-negligible shift in the Higgs mass (of a few GeV), must also be included. 17 A compact analytical expression that incorporates these effects at O(m 2 t h 2 t α s ) was given in ref. [154] (with further refinements provided by ref. [155] to take into account the possibility of arbitrary top-squark splitting), and the corresponding corrections proportional to h 2 b α s can be found in ref. [140]. An important source of such contributions are the one-loop supersymmetric threshold corrections to the relation between the Higgs-top-quark and Higgs-bottomquark Yukawa couplings and the corresponding quark masses [eqs. (56) and (57)]. These generate a non-logarithmic two-loop shift of the radiatively corrected Higgs mass proportional to the corresponding squark mixing parameters. One consequence of these contributions [151] is the asymmetry in the predicted value of m h under X t → −X t as noted in fig. 12(a). Recently, the computation of m h has been further refined by the inclusion of genuine two-loop corrections of O(m 2 t h 4 t ) [139], and estimates of the two-loop corrections proportional to h 2 b h 2 t and h 4 b [140] (which can be numerically relevant for values of tan β > ∼ m t /m b ). These non-logarithmic corrections, which depend on the third generation squark mixing parameters, can slightly increase the value of the radiatively-corrected Higgs mass.
The numerical results displayed in figs. 11-14 are based on the calculations of refs. [149] and [150], with improvements as described in refs. [146] and [151]. The supersymmetric parameters in the maximal and minimal mixing cases have been chosen according to the first two benchmark scenarios of ref. [156]. Of particular interest is the upper bound for the lightest CP-even Higgs mass (m h ). At fixed tan β, the maximal value of m h is reached for m A ≫ m Z (see fig. 14). Taking m A large, fig. 12(b) illustrates that the maximal value of the lightest CP-even Higgs mass bound is realized at large tan β in the case of 15 Eqs. (41)- (43) have been derived under the assumption that |M 2 The approximate forms of eqs. (41)- (43) are sufficient to provide insight on the dependence of the radiatively-corrected Higgs masses and couplings on the MSSM parameters, although our numerical work is based on more exact forms for these expressions. 16 Although M 2 12 is negative at tree level (implying that −π/2 ≤ α ≤ 0), it is possible that radiative corrections flip the sign of M 2 12 . Thus, the range of the radiatively corrected angle α can be taken to be −π/2 ≤ α ≤ π/2. 17 An improved procedure for computing the radiatively-corrected neutral Higgs mass matrix and the charged Higgs mass in a self-consistent way (including possible CP-violating effects), which incorporates one-loop supersymmetric threshold corrections to the Higgs-top-quark and Higgs-bottom-quark Yukawa couplings, can be found in ref. [153].
In practice, parameters leading to maximal mixing are not expected in typical models of supersymmetry breaking. Thus, in general, the upper bound on the lightest Higgs boson mass is expected to be somewhere between the two extreme limits quoted above. Cross-checks among various programs [157] and rough estimates of higher order corrections not yet computed suggest that the results for Higgs masses should be accurate to within about 2 to 3 GeV over the parameter ranges displayed in figs. 12-14.
In fig. 14, we exhibit the masses of the CP-even neutral and the charged Higgs masses as a function of m A . The squared-masses of the lighter and heavier neutral CP-even Higgs are related by 18 It is interesting to consider the behavior of the CP-even Higgs masses in the large tan β regime.

MSSM Higgs mass limits after LEP
No significant evidence for a Higgs signal has been detected at LEP [161]. As a result, one can obtain bounds on the possible MSSM Higgs parameters. These limits are often displayed in the m A -tan β plane, although there is additional dependence on various MSSM parameters that effect the radiative corrections to the Higgs masses as discussed above. In representative scans of the MSSM parameters, the LEP Higgs Working Group [161] finds that m h > 91.0 GeV and m A > 91.9 GeV at 95% CL. These limits actually correspond to the large tan β region in which Zh production is suppressed, as shown in fig. 15. In this case, the quoted Higgs limits arise as a result of the non-observation of hA and HA production. As tan β is lowered, the limits on m h and m A become more stringent. In this regime, the hA production is suppressed while the Zh production rate approaches its SM value. Thus, in this case, the SM Higgs limit applies (m h > ∼ 114 GeV) as shown in fig. 15(a). The precise region of MSSM Higgs parameter space that is excluded depends on the values of the MSSM parameters that control the Higgs mass radiative corrections. For example, a conservative exclusion limit is obtained in the maximal mixing scenario, since in this case the predicted value of m h as a function of m A and tan β is maximal (with respect to changes in the other MSSM parameters). The excluded regions of the MSSM Higgs parameter space based on the maximal mixing benchmark scenario of ref. [156], are shown in fig. 15, and correspond to the exclusion of the range 0.5 < tan β < 2.4 at the 95% CL. However, the tan β exclusion region can still be significantly reduced (even to the point of allowing all tan β values) by, e.g., taking M SUSY = 2 TeV and m t = 180 GeV (which still lies within the error bars of the experimentally measured value), and allowing for the theoretical uncertainty in the prediction of m max h [162]. 18 At tree level, eq. (46) is a consequence of the sum rule: n m 2 φn g 2 φnZZ = 4 3 m 4 Z g GGGG , first derived in ref. [131], where the φ n are neutral Higgs bosons of a multi-Higgs-doublet model and g GGGG is the quartic coupling of neutral Goldstone bosons. (In the MSSM, g GGGG = 3 4 g 2 cos 2 2β/ cos 2 θ W [2].) A general discussion of related tree-level Higgs mass sum rules and bounds can be found in ref. [158]. In fact, eq. (46) is more general and applies to the radiatively-corrected MSSM Higgs sector [159,160] in the approximation where the renormalized α is determined as discussed in Section 3.3.1. No evidence for the charged Higgs boson has yet been found. The LEP Higgs Working Group quotes a limit of m H ± > 78.6 GeV at 95% CL [163], which holds for a more general non-supersymmetric two-Higgs doublet model and assumes only that the H + decays dominantly into τ + ν τ and/or cs. Although the MSSM tree-level bound m H ± ≥ m W can be relaxed somewhat by radiative corrections, the LEP bound quoted above provides no useful additional constraints on the MSSM Higgs sector.

Effect of explicit CP-violation on the radiatively-corrected MSSM Higgs masses
In the Standard Model, CP-violation is due to the existence of phases in the Yukawa couplings of the quarks to the Higgs field, which results in one non-trivial phase in the CKM mixing matrix. In the MSSM, there are additional sources of CP-violation, due to phases in the various supersymmetric mass parameters. In particular, the gaugino mass parameters (M i , i = 1, 2, 3), the Higgsino mass parameter, µ, the bilinear Higgs squared-mass parameter, m 2 12 , and the trilinear couplings of the squark and slepton fields ( f ) to the Higgs fields, A f , may carry non-trivial phases. The existence of these CP phases can significantly affect the MSSM Higgs sector through one-loop radiative corrections [164,165,166,153].
Note that if one sets µ = M i = A f = m 2 12 , then the MSSM Lagrangian possesses two independent global U(1) symmetries-a Peccei-Quinn (PQ) symmetry and an R symmetry. (The quantum numbers of the MSSM fields with respect to U(1) PQ and U(1) R can be found in refs. [167] and [166].) Consequently, in the MSSM with nonzero values for the above parameters, there are two independent phase redefinitions of the fields that can be used to remove two phases from µ, M i , A f and m 2 12 . However, certain combinations of these parameters remain invariant under such phase redefinitions. The simplest way to determine these combinations is to treat the aforementioned parameters as spurions with quantum numbers under the U(1) PQ and U(1) R symmetries chosen such that the full MSSM Lagrangian is invariant. One can then easily check that the phases of the parameter combinations, arg[µA f (m 2 12 ) * ] and arg[µM i (m 2 12 ) * ], are indeed invariant under the U(1) PQ and U(1) R phase redefinitions of the MSSM fields [167,166]. Therefore, if one of these two quantities is different from zero (modulo π), one should expect new CP-violating effects induced by the production or exchange of supersymmetric particles.
We have already noted that the tree-level Higgs sector is CP-conserving. This is a consequence of the fact that m 2 12 is the only possible complex parameter that appears in the tree-level Higgs potential. Thus the phase of m 2 12 can be rotated away by redefining the phases of the complex Higgs doublets appearing in the Lagrangian. The same field redefinition implies that one can choose the vacuum expectation values of the two Higgs fields to be real and positive. However, at the one loop-level, the Higgs potential acquires a dependence on the parameters µA t and µM i through loops of third generation squarks and weak gauginos, respectively, which induce non-trivial CP-violating effects. The most important of these CP-violating effects is the generation of mixing between the neutral CP-odd and CP-even Higgs boson states. Therefore, the physical neutral Higgs bosons are no longer CP-eigenstates and the CP-odd Higgs boson mass m A is no longer a physical parameter. The charged Higgs mass is still physical and can be used as an input for the computation of the neutral Higgs spectrum of the theory [153]. The Higgs mass spectrum can therefore be quite different from the CP-conserving case. For example, a large splitting between the masses of the next-to-lightest and the heaviest neutral Higgs bosons is possible if the charged Higgs boson is not too heavy.
For large values of the charged Higgs mass, the decoupling limit applies, and the properties of the lightest neutral Higgs boson state approach those of the SM Higgs boson. That is, for m H ± ≫ m W , the lightest neutral Higgs boson is approximately a CP-even state, with CP-violating couplings that are suppressed by terms of O(m 2 W /m 2 H ± ) [168]. In particular, the upper-bound on the lightest neutral Higgs boson mass, which is reached in the decoupling limit, takes the same value as in the CP-conserving case [166]. Nevertheless, there still can be significant mixing between the two heavier neutral mass eigenstates. Quantitatively, the leading contribution to the squared-mass terms that mix CP-even and CP-odd eigenstates, M 2 SP (in a convention where m 2 12 is real) is of order Under the reasonable assumption that |µA t | < 10M 2 S , it is clear that the mixing effects between the lightest neutral Higgs boson and the heavier Higgs states are small if the masses of the heavy Higgs bosons are larger than 2m t . In this limit, the two heavier states are highly degenerate in mass, and the CP-violating effects may still lead to non-trivial mixing of the two heavier CP-eigenstates. For a detailed study of the Higgs mass spectrum and parametric dependence of the Higgs mass radiative corrections, see ref. [153].

Radiatively-Corrected MSSM Higgs couplings 3.3.1 Renormalization of cos(β − α)
Radiative corrections also significantly modify the tree-level values of the Higgs boson couplings to fermion pairs and to vector boson pairs. As discussed in Section 3.1, the tree-level Higgs couplings depend crucially on the value of cos(β − α). In first approximation, when radiative corrections of the Higgs squared-mass matrix are computed, the diagonalizing angle α is shifted from its tree-level value. Thus, one may compute a "radiatively-corrected" value for cos(β − α). This provides one important source of the radiative corrections of the Higgs couplings. In fig. 11, we show the effect of radiative corrections on the value of cos(β − α) as a function of m A for different values of the squark mixing parameters and tan β. One can then simply insert the radiatively corrected value of α into eqs. (20), (21) and (28)- (33) to obtain radiatively-improved couplings of Higgs bosons to vector bosons and to fermions.
The mixing angle α which diagonalizes the mass matrix in eq. (40) can be expressed as: where s α ≡ sin α and c α ≡ cos α. Note that if M 2 12 → 0, then either sin α → 0 (if M 2 11 > M 2 22 ) or cos α → 0 (if M 2 11 < M 2 22 ). At tree level, M 2 12 is small for small m A and/or large tan β, but it cannot vanish. However, radiative corrections to M 2 12 ≡ −(m 2 A + m 2 Z )s β c β + δM 2 12 can be of the same order as its tree level value for small values of m A and large tan β. Hence, it is possible for the oneloop contribution to approximately cancel the tree-level result (with two-loop corrections to M 2 12 small compared to the corresponding one-loop result). For moderate or large values of tan β, the vanishing of M 2 12 [see eq. (42)] leads to the approximate numerical relation [152]: where h t , α s and the weak gauge couplings have been replaced by their approximate numerical values at the the electroweak scale. For low values of m A or large values of the squark mixing parameters, a cancellation can easily take place.
If M 2 12 ≃ 0 and tan β is large (values of tan β > ∼ 5 are sufficient), the resulting pattern of Higgs couplings is easy to understand. In this limit, M 2 11 ≃ m 2 A and M 2 22 ≃ m max h , as noted at the end of Section 3.2.1. Two cases must be treated separately depending on the value of m A . First, if m A < m max h , then sin α ≃ −1, cos α ≃ 0 and sin β ≃ − cos(β −α) ≃ 1. In this case, the lighter CP-even Higgs boson h is roughly aligned along the Φ 0 d direction and the heavier CP-even Higgs boson H is roughly aligned along the Φ 0 u direction [see eq. (8)]. In particular, the coupling of H to bb and τ + τ − is significantly diminished (since down-type fermions couple to Φ 0 d ), while the HV V couplings [eq. (20)] are approximately equal to those of the Standard Model [since cos 2 (β − α) ≃ 1]. Consequently, the branching ratios of H into gg, γγ, cc, and W + W − can be greatly enhanced over Standard Model expectations [152,160,169,170]. Second, if m A ≫ m max h then sin α ≃ 0 and sin β ≃ cos α ≃ sin(β−α) ≃ 1 and the previous considerations for H apply now to h.
Although it is difficult to have an exact cancellation of the off-diagonal element M 2 12 , in many regions of the MSSM parameter space, a significant suppression of M 2 12 may be present. Generically, the leading radiative corrections to M 2 12 depend strongly on the sign of the product µX t (A t ≃ X t for large tan β and moderate µ) and on the value of |A t |. For the same value of X t , a change in the sign of µ can lead to observable variations in the branching ratio for the Higgs boson decay into bottom quarks.

The decoupling limit revisited
Radiative corrections can also significantly affect the onset of the decoupling limit. Recall that at tree level [see eq. (18)], | cos(β − α)| ≪ 1 for m A ≫ m Z , in which case the couplings of h are nearly identical to those of the SM Higgs boson. Including the effects of δM 2 , we use eq. (48) to obtain Since in the limit of m A ≫ m Z , where Eq. (51) exhibits the expected decoupling behavior for m A ≫ m Z . However, eq. (50) illustrates another way in which cos(β − α) = 0 can be achieved-simply choose the MSSM parameters (that govern the Higgs mass radiative corrections) such that the numerator of eq. (50) vanishes. That is, Note that eq. (53) is independent of the value of m A . For a typical choice of MSSM parameters, eq. (53) yields a solution at large tan β. That is, by approximating tan 2β ≃ − sin 2β ≃ −2/ tan β, one can determine the value of β at which the decoupling occurs: The explicit expressions for δM 2 ij quoted in eq. (42) confirm that the assumption of tan β ≫ 1 used to derive this result is a consistent approximation because δM 2 12 is typically small. We conclude that for the value of tan β specified in eq. (54), cos(β − α) = 0 independently of the value of m A . We shall refer to this phenomenon as m A -independent decoupling. From eq. (42), it follows that explicit solutions to eq. (53) depend on ratios of MSSM parameters and are thus insensitive to the overall supersymmetric mass scale, modulo a mild logarithmic dependence on M S /m t .

Corrections to tree-level Higgs-fermion Yukawa couplings
We have seen in Section 3.3.1 that Higgs couplings are modified at one loop due to the renormalization of the CP-even Higgs mixing angle α. Additional contributions from the one-loop vertex corrections to tree-level Higgs couplings must also be considered [171,172,173,174]. These corrections are typically small and therefore do not alter significantly the pattern of Higgs couplings. However, at large tan β, the corrections to Higgs-fermion Yukawa couplings can be enhanced, and thus require a careful analysis.
In the supersymmetric limit, bottom quarks only couple to Φ 0 d and top quarks only couple to Φ 0 u . However, supersymmetry is broken and a small coupling of the bottom quark [top quark] to Φ 0 u [Φ 0 d ] will be generated from the one-loop Yukawa vertex corrections. These results can be summarized by an effective Lagrangian that describes the coupling of the Higgs bosons to the third generation quarks: 19 19 Due to weak isospin breaking, one should allow for different radiatively induced couplings to charged and neutral Higgs bosons. For example, one should write ∆h bbR b L Φ 0 * u + ∆h bbR t L Φ − u in place of ∆h bbR Q k L Φ u k * , etc. To the extent that weak isospin breaking effects are small in the loop diagrams that generate ∆h b and ∆h b , it follows that ∆h b ≈ ∆h b (and similarly for the other radiatively generated coefficients), and we may use eq. (55) as written.
implying a modification of the tree-level relations between h t , h b and m t , m b as follows [175,176,172,144]: The dominant contributions to ∆ b are tan β-enhanced, with ∆ b ≃ (∆h b /h b ) tan β; for tan β ≫ 1, δh b /h b provides a small correction to ∆ b . In the same limit, ∆ t ≃ δh t /h t , with the additional contribution of (∆h t /h t ) cot β providing a small correction. 20 Explicitly, one finds that for M SUSY ≫ m Z (where M SUSY represents a typical supersymmetric mass that appears in the loops) and for tan β ≫ 1 [175,176,144], where α s ≡ g 2 3 /4π, Mg is the gluino mass, Mb 1,2 are the bottom squark masses, and smaller electroweak corrections have been ignored. The loop integral I(a 2 , b 2 , c 2 ) is given by and is of order 1/max(a 2 , b 2 , c 2 ) when at least one of its arguments is large compared to m 2 Z . Note that the Higgs coupling proportional to ∆h b is a manifestation of the broken supersymmetry in the low energy theory; hence, ∆ b does not decouple in the limit of large values of the supersymmetry breaking masses. Indeed, if all supersymmetry breaking mass parameters (and µ) are scaled by a common factor, the correction ∆ b remains constant.
Similarly to the case of the bottom quark, the relation between m τ and the Higgs-tau-lepton Yukawa coupling h τ is modified: The correction ∆ τ contains a contribution from a tau slepton-neutralino loop (depending on the two tau-slepton masses Mτ 1 and Mτ 2 and the mass parameter of the B component of the neutralino, M 1 ) and a tau sneutrino-chargino loop (depending on the tau sneutrino mass Mν τ , the mass parameter of the W ± component of the chargino, M 2 , and µ). It is given by [176,144]: where α 2 ≡ g 2 /4π and α 1 ≡ g ′ 2 /4π are the electroweak gauge couplings. Since corrections to h τ are proportional to α 1 and α 2 , they are expected to be smaller than the corrections to h b . From eq. (55) we can obtain the couplings of the physical Higgs bosons to third generation fermions. The resulting interaction Lagrangian is of the form: Using eqs. (56) and (57), one obtains: and the τ couplings are obtained from the above equations by replacing m b , ∆ b and δh b with m τ , ∆ τ and δh τ , respectively. In addition, one must employ the renormalized value of α in the above formulae to incorporate the radiative corrections discussed in section 3.3.1. In writing out the Higgstop quark couplings above, we found it convenient to express the results in terms of ∆ t and ∆h t /h t , since ∆ t ≃ δh t /h t and the corresponding contribution of ∆h t /h t is tan β suppressed [eq. (57)]. Alternatively, eqs. (64)- (69) can be rewritten in a more symmetrical form by using eqs. (56) and (57) to eliminate ∆ b and ∆ t from the numerators of the corresponding expressions.
At large tan β, terms involving ∆ b ∝ tan β [eq. (58)] provide the dominant corrections to the neutral Higgs couplings to bb. The corrections proportional to δh b /h b [see eqs. (55) and (56) and the discussion that follows] are never tan β-enhanced and are therefore numerically unimportant. The sign of ∆ b is governed by the sign of Mgµ, since the bottom-squark gluino loop gives the dominant contribution to eq. (58). Thus, in a convention where Mg > 0, the radiatively corrected coupling g Abb is suppressed (enhanced) with respect to its tree level value for µ > 0 (µ < 0). In contrast, the radiative corrections to g hbb and g Hbb have a more complicated dependence on the supersymmetric parameters due to the dependence on the CP-even mixing angle α. Since α and ∆ b are governed by different combinations of the supersymmetry breaking parameters, it is difficult to exhibit in a simple way the behavior of the radiatively corrected couplings of the CP-even Higgs bosons to the bottom quarks as a function of the MSSM parameters.
One can check [using eq. (19)] that in the decoupling limit, g hqq = g h SM qq = m q /v. Away from the decoupling limit, the Higgs couplings to bottom-type fermions can deviate significantly from their tree-level values due to enhanced radiative corrections at large tan β [where ∆ b ≃ O(1)]. In particular, because ∆ b ∝ tan β, the leading one-loop radiative correction to g hbb is of O(m 2 Z tan β/m 2 A ), which formally decouples only when m 2 A ≫ m 2 Z tan β. This behavior is called delayed decoupling in ref. [174]. In addition, there are regions of MSSM parameter space in which there is a strong suppression of the Higgs coupling to bb (or τ + τ − ) as compared to its tree-level value. As a result, there can be significant corrections to the tree-level relation g hbb /g hτ + τ − = m b /m τ [169]. In particular, in some parameter regimes, the τ + τ − decay mode can be the dominant h decay channel, a result that would be fatal to certain Higgs search strategies which assume that h → bb is the dominant decay mode.
Assuming that weak isospin breaking effects in the loop corrections to the charged Higgs fermion Yukawa couplings are small (see footnote 19), then g H − tb [defined in eq. (63)] is given by with a similar form for g H − ντ τ + with the replacements noted below eq. (69).

Effects of explicit CP-violation
In Section 3.2.3, we noted the possibility of mixing between the CP-even and CP-odd eigenstates due to CP-violating effects that enter via the one-loop radiative corrections. In this case, the neutral scalar mass eigenstates, denoted by H i (i = 1, 2, 3), are determined by diagonalizing a 3 × 3 squared-mass matrix. Thus, one can no longer parameterize the various Higgs couplings in terms of the CP-even Higgs mixing angle α. It is convenient to work in a convention where the two vacuum expectation values are real and positive (by absorbing any potential phases into the definition of the Higgs field) so that tan β = v u /v d as before. Then, eqs. (7) and (8) are replaced by where O is a 3 × 3 real orthogonal matrix.
In the CP-violating case, vector boson pairs V V (V = W or Z) couple to all three neutral Higgs mass eigenstates, H i , with [166] g Fig. 16 shows the dependence of the Higgs masses and the H i ZZ squared-couplings on the phase of A t for a particular choice of MSSM parameters [as indicated in fig. 16(a)]. Clearly, these couplings can depend sensitively on the phases of the complex supersymmetry-breaking parameters that generate the mixing of the CP-even and CP-odd scalar eigenstates through one-loop radiative effects. The couplings of V to a pair of neutral Higgs bosons are given by Using the orthogonality of O (and det O = 1), it is easy to derive the relation [166]: 21 The sum rules of eqs. (22)-(24) are then easily extended: which when combined with eq. (74) yields The couplings of a neutral Higgs boson to H − W + are given by where the momenta p H − and p H i point into the vertex, and [166] Another consequence of the CP-violating effects in the scalar sector is that all neutral Higgs scalars can couple to both scalar and pseudoscalar fermion bilinear densities (ψψ andψγ 5 ψ, respectively). The couplings of the mass eigenstate H i to fermions depend on the loop-corrected fermion Yukawa couplings, h b,t , δh b,t , ∆h b,t , and on tan β and the O ji . It is convenient to adjust the phases of the fields so that the quantities h b + δh b + ∆h b tan β and h t + δh t + ∆h t cot β are both real and positive [i.e., the physical fermion masses are still given by eqs. (56) and (57)]. The resulting expressions are a straightforward generalization of those presented above for the CP-conserving case [153]: with where the Higgs scalar couplings are normalized with respect to the corresponding SM values.
For large values of the charged Higgs boson mass and for heavy supersymmetric particles, the expressions of the lightest neutral Higgs boson coupling to fermions reduce to those of the (CP-conserving) SM Higgs boson, as expected for the decoupling limit. In contrast, the two heavy neutral Higgs bosons are still admixtures of CP-even and CP-odd eigenstates; hence, CP-violating effects are still present in the heavy neutral Higgs sector. However, due to the high degeneracy in mass of the heavy scalar sector (especially in the decoupling limit), CP-violating effects may be difficult to observe without precision measurements of the heavy neutral Higgs properties.
The couplings of the charged Higgs bosons to fermions are of the form L int =b g H − tb tH − + h.c., with One can check that for real ∆h b and ∆h t , this result is equivalent to eq. (70) [with the same caveats noted in footnote 19]. An explicit computation of the CP-violating H − tb vertex and its phenomenological implications can be found in ref. [178].

MSSM Higgs Boson Decay Modes
In the MSSM, we must consider the decay properties of three neutral Higgs bosons and one charged Higgs pair. 22 In the region of parameter space where m A ≫ m Z and the masses of supersymmetric particles are large, the decoupling limit applies, and we find that the properties of h are indistinguishable from the SM Higgs boson. If supersymmetric particles are light, then the decoupling limit does not strictly apply even in the limit of m A ≫ m Z . In particular, the h branching ratios are modified, if the decays of h into supersymmetric particles are kinematically allowed. In addition, if light superpartners exist that can couple to photons and/or gluons, then the one-loop gg and γγ decay rates would also deviate from the corresponding Standard Model Higgs decay rates due to the extra contribution of the light superpartners appearing in the loops. In both cases, the heavier Higgs states, H, A and H ± , are roughly mass degenerate, and their decay branching ratios depend crucially on tan β as shown below. For values of m A ∼ O(m Z ), all Higgs boson states lie below 200 GeV in mass. In this parameter regime, there is a significant area of the parameter space in which none of the neutral Higgs boson decay properties approximates that of the SM Higgs boson. For tan β ≫ 1, the resulting Higgs phenomenology shows marked differences from that of the SM Higgs boson [179]. In particular, radiative corrections can significantly modify the bb and/or the τ + τ − decay rates with respect to those of the SM Higgs boson, as noted in Section 3.3.3. Additionally, the Higgs bosons can decay into new channels, either containing lighter Higgs bosons or supersymmetric particles. In the following, the decays of the neutral Higgs bosons h, H and A and the decays of charged Higgs bosons are discussed with particular emphasis on differences from Standard Model expectations. In the following discussion, we exhibit results for tan β = 3 and 30 to illustrate the difference between "low" and "high" tan β. The results shown below include the effects of the dominant radiative corrections, which affect both the masses and the couplings of the Higgs sector as described in Sections 3.2 and 3.3.
In order to display results for Higgs branching ratios, we must choose a set of MSSM parameters. We fix tan β (for two representative choices) and vary m A from its LEP experimental lower bound of 90 GeV up to 1 TeV. In addition, the gluino and MSSM squark mass parameters have been chosen to be M SUSY ≡ Mg = M Q = M U = M D = 1 TeV, the squark mixing parameter X t = A t −µ cot β = 2.4M SUSY , and the gaugino mass matrix parameters, µ = M 2 ≃ 2M 1 = 1 TeV. This differs somewhat from the maximal mixing benchmark scenario of ref. [156]. Nevertheless, the value of m h is still close to maximal (for fixed m A and tan β), so we will continue to loosely refer to the above choice of MSSM parameters as a maximal mixing scenario. Our motivation for choosing the gaugino mass parameters large is to avoid possible supersymmetric decay modes for the Higgs bosons for Higgs masses below 1 TeV. We shall briefly comment on possible supersymmetric decay modes at the end of this section.
The branching ratios for h and H as a function of their masses are shown in fig. 17. As m A varies from 90 GeV to 1 TeV, with the MSSM parameters as specified above, 135 GeV < ∼ m H < ∼ 1 TeV when Figure 17: Branching ratios of the MSSM Higgs bosons h and H, with tan β = 3 and 30, respectively. Final states labeled above include the possibility of one off-shell final state particle below the corresponding two-particle decay threshold. The above plots were made under the assumption that the average top and bottom squark masses are 1 TeV and top-squark mixing is maximal. In this case, m max h ≃ 115 GeV (125.9 GeV) for tan β = 3 (30), corresponding to the limit of large m A , is indicated by the vertical line in the two left-side plots. The range of m H shown corresponds to varying m A between 90 GeV and 1 TeV, while m h > 100 GeV corresponds to m A > 139 GeV (104 GeV) for tan β = 3 (30). Other supersymmetric parameters have been chosen such that there are no supersymmetric particle decay modes in the Higgs mass ranges shown above. GeV. This reveals a detailed pattern of branching ratios that is not easily visible in fig. 17. The branching ratios for A and H + as a function of their masses are shown in fig. 19.
The total Higgs decay widths as a function of the corresponding Higgs mass are shown in fig. 18(b) for the two cases of tan β = 3 and 30 (and the other relevant MSSM parameters as described above). Note that for large values of the Higgs mass, the corresponding widths are considerably smaller than that of the SM Higgs boson. This is due to the suppressed HV V couplings at large Higgs mass and to the absence of tree-level AV V and H + W − Z couplings. One can also check that in the decoupling limit (m A ≫ m Z ), the total width of h coincides with that of h SM . This is illustrated by replotting the h (and H) widths on the same plot as the h SM width [see fig. 4(b)]. In particular, note that the dashed and dot-dashed h contours in fig. 4(b) approach the h SM contour as m h reaches its maximal value. (which corresponds to the limit of large m A at fixed tan β). It is interesting to note that in the opposite limit of small m A (especially at large tan β), cos(β − α) → 1 and it is H that assumes many of the properties of h SM . However, there can still be deviations in the Hbb coupling from the corresponding Standard Model value at large tan β, as noted below eq. (35). This explains why the H contours in fig. 4(b) Figure 19: Branching ratios of the MSSM Higgs bosons A and H + , with tan β = 3 and 30, respectively. Final states labeled above include the possibility of one off-shell final state particle below the corresponding two-particle decay threshold. The above plots were made under the assumption that the average top and bottom squark masses are 1 TeV and top-squark mixing is maximal. The range of m H ± shown corresponds to varying m A between 90 GeV and 1 TeV. Other supersymmetric parameters have been chosen such that there are no supersymmetric particle decay modes in the Higgs mass ranges shown above.
do not quite coincide with the result of the h SM contour as m H approaches its lower limit (with the discrepancy between the H and h SM contours more pronounced at large tan β).
The branching ratios and widths in figs. [17][18][19] have been computed using a modified version of the HDECAY program [180] that incorporates the leading radiative corrections to the Higgs couplings discussed in Section 3.3. 23 The decay modes h, H, A → bb, τ + τ − dominate the neutral Higgs decay modes when tan β is large for all values of the Higgs masses. For small tan β, these modes are significant for neutral Higgs masses below 2m t (although there are other competing modes in this mass range), whereas the tt decay mode dominates above the tt decay threshold. In contrast to the SM Higgs boson, the vector boson decay modes of H are strongly suppressed at large m H due to the suppressed HV V couplings in the decoupling limit. For the charged Higgs boson, H + → τ + ν dominates below tb threshold, while H + → tb dominates for large values of m H ± . Note that final states labeled in figs. 17 and 19 include the possibility of one off-shell final state particle below the corresponding two-particle decay threshold [181]. For example, for m H ± < m t + m b , the tb contour shown in fig. 19 actually corresponds to an off-shell t quark that decays to bW + . That is, in this mass region, the tb contour corresponds to the branching ratio for the three-body decay H + → W + bb. This decay mode can be especially significant at moderate values of tan β due to the large Higgs-top Yukawa coupling.
As in the Standard Model case, the partial decay widths of the neutral Higgs bosons into bb and cc are reduced by about 50-75% when QCD corrections are included (e.g., by employing running quark masses in the decay width formulae), whereas the QCD corrections are less significant for Higgs decays into tt [29]. The effects of the QCD radiative corrections on the charged Higgs branching ratios [182] are significant in the region of tan β where the cs and τ + ν τ decay modes are competitive or for large values of tan β for the decay mode H + → tb (and for H + → W + bb below tb threshold [183]). Additional supersymmetric radiative corrections discussed in Section 3.3 can also significantly affect the Higgs boson partial widths. Some of these corrections can be absorbed into the effective mixing angle α [173] as shown in Section 3.3.1. As a consequence of this universal correction, the coupling of h to bb and τ + τ − can be suppressed for small m A and large tan β. For the h → bb decay width, the supersymmetric corrections [152,171,172,173,174] proportional to the strong coupling constant α s and the Higgs-top quark Yukawa coupling h t can be significant for large values of µ and tan β. As shown in Section 3.3.3, this effect can be interpreted as a correction to the tree-level relation between m b and h b .
In addition to the decay modes of the neutral Higgs bosons into fermion and gauge boson final states, there exist new Higgs decay channels that involve scalars of the extended Higgs sector and supersymmetric final states. The unambiguous observation of these modes (as well as any decay mode of a charged scalar) would clearly constitute direct evidence of new physics beyond the Standard Model. Higgs decays into charginos, neutralinos and third-generation squarks and sleptons can become impor-tant, once they are kinematically allowed [184]. One interesting possibility is a significant branching ratio for the decay of a neutral Higgs boson to the invisible mode χ 0 1 χ 0 1 (where χ 0 1 is the lightest supersymmetric particle). In such a scenario, the discovery of this neutral Higgs boson would be difficult at a hadron collider [185]. In contrast, at lepton colliders, methods exist for detecting an invisibly decaying Higgs boson by observing a peak in the missing mass recoiling against the produced Z.

Cross-sections at hadron colliders
The production mechanisms for the SM Higgs boson at hadron colliders can also be relevant for the production of the MSSM neutral Higgs bosons. However, we must take into account the possibility of enhanced or suppressed couplings (with respect to those of the Standard Model). For example, the HV V couplings are very suppressed in the decoupling limit, and tree-level AV V couplings are completely absent. On the other hand, at large tan β, typically two of the three neutral Higgs couplings to bottom-type quarks are enhanced. These effects can significantly modify the neutral Higgs production cross-sections. New production mechanisms must be considered for charged Higgs production.
As in the case of Higgs branching ratios, the predicted cross-sections are sensitive to the MSSM Higgs parameters. Again, we consider two representative values of tan β: a low value of tan β = 3 and a high value of tan β = 30. We then vary m A , evaluate the other Higgs masses, and compute each Higgs cross-section as a function of the corresponding Higgs mass. The Higgs masses and cross-sections depend on other MSSM parameters through radiative corrections. As in Section 3.4, we work in a maximal squark mixing scenario in which the value of m h for a fixed choice of tan β and m A is maximal. In addition, because the squark masses are assumed to be heavy (of order 1 TeV), potential supersymmetric contributions to the one-loop Higgs-gluon-gluon vertex (due to squark loops) are suppressed. Cross-sections for neutral MSSM Higgs production at the Tevatron and the LHC are shown in figs. 20 and 21 respectively. The dominant Higgs production mechanism over much of the MSSM parameter space is gluon-gluon fusion, which is mediated by heavy top and bottom quark triangle loops and the corresponding supersymmetric partners [186,187,188]. The gluon-gluon fusion results shown in figs. 20 and 21 include NLO QCD corrections [189]. 24 The cross-sections for the production of the neutral CP-even Higgs bosons (φ = h or H) via gauge boson fusion V * V * → φ (V = W or Z) [45] and via the process qq → V * → V φ [42], including firstorder QCD corrections, 25  TeV] for gluon fusion gg → φ, vector-boson fusion qq → qqV * V * → qqh, qqH, vector-boson bremsstrahlung qq → V * → hV /HV and the associated production gg, qq → φbb/φtt including all known QCD corrections, where φ = h, H or A [32,34]. As in fig. 5, in the vector boson fusion process, qq refers to both ud and qq scattering. The four panels exhibited above show (a) h, H production for tan β = 3, (b) A production for tan β = 3, (c) h, H production for tan β = 30, (d) A production for tan β = 30. TeV] for gluon fusion gg → φ, vectorboson fusion qq → qqV * V * → qqh, qqH, vector-boson bremsstrahlung qq → V * → hV /HV and the associated production gg, qq → φbb/φtt including all known QCD corrections, where φ = h, H or A [34,44]. The four panels exhibited above show the cross-section in pb vs. the Higgs mass, ranging from 90 GeV to 1 TeV, for (a) h, H production for tan β = 3, (b) A production for tan β = 3, (c) h, H production for tan β = 30, (d) A production for tan β = 30. For comparison, the cross-section for gluon-gluon fusion to a SM Higgs boson is also shown.
Higgs scalar, φ, has a mass less than or about equal to 130 GeV, and the corresponding cross-sections for V * V * → φ and qq → V * → V φ are phenomenologically relevant. The other (non-SM-like) CP-even scalar has suppressed couplings to V V , and the corresponding cross-sections are generally too small to be observed.
Higgs boson radiation off bottom quarks becomes important for large tan β in the MSSM, where the Higgs coupling to bottom-type fermions is enhanced. Thus, the theoretical predictions, including full NLO computations, are crucial for realistic simulations of the MSSM Higgs signals in these channels. 26 Moreover, as discussed in Section 3.3.3, vertex corrections to the bbφ coupling play a very important role in enhancing or suppressing (depending on the MSSM parameters) these production cross-sections at large tan β [152,160,170].
We now turn to charged Higgs production. If m H ± < m t − m b , then H ± can be produced in the decay of the top quark via t → bH + (andt →bH − ) [192]. The t → bH + decay mode can be competitive with the dominant Standard Model decay mode, t → bW + , depending on the value of tan β, as shown in fig. 22(a) for m H ± = 120 GeV. This figure, taken from ref. [193] illustrates the effects of including one-loop radiative corrections. The curved labeled BR QCD , which incorporates the one-loop QCD corrections (first computed in ref. [194]), is applicable to a more general (non-supersymmetric) Type-II two-Higgs doublet model [based on the tree-level Higgs-fermion couplings of eqs. (28)- (35)]. Note that the supersymmetric corrections can be particularly significant at large tan β due to the effect of ∆ b [see eq. (70)] depending on the choice of MSSM parameters. A full one-loop calculation of Γ(t → H + b) in the MSSM including all sources of large Yukawa couplings can be found in refs. [195] and [193]. A treatment including resummation of the leading QCD quantum effects and the dominant contributions from loop effects arising from supersymmetric particle exchange can be found in ref. [196].
For m H ± < m t − m b , the total cross-section for charged Higgs production (in the narrow-width approximation) is then given by: 27 With σ(pp → tt) ≃ 7 pb at √ s = 2 TeV and σ(pp → tt) ≃ 1 nb at √ s = 14 TeV [197], roughly 1400 tt pairs per detector will be produced per year in Run 2a of the Tevatron (assuming a yearly luminosity of 2 fb −1 ), while about 10 7 -10 8 tt pairs will be produced at the LHC (assuming a yearly luminosity of 10-100 fb −1 ). Folding in the top quark branching ratio, it is a simple matter to compute the inclusive The charged Higgs production cross-section at the LHC near the threshold for t → bH + for tan β = 3 (taken from ref. [199]). charged Higgs cross-section. For values of m H ± near m t , the width effects are important. In addition, the full 2 → 3 processes pp → H +t b+X and pp → H − tb+X must be considered. In this case eq. (85) no longer provides an accurate estimate of the charged Higgs cross-section [198], as illustrated in fig. 22(b) (taken from ref. [199]). The results of fig. 22(a) imply that for m H ± < m t − m b , the discovery of the charged Higgs boson at the Tevatron and/or LHC (given sufficient luminosity) is possible if tan β ≫ 1 or tan β < ∼ 1 (the latter is theoretically disfavored). The precise bound on tan β (as a function of m H ± ) depends somewhat on the details of the other MSSM Higgs parameters.
If m H ± > m t − m b , then charged Higgs boson production occurs mainly through radiation off a third generation quark. Single charged Higgs associated production proceeds via the 2 → 3 partonic processes gg, qq → tbH − (and the charge conjugate final state). As in the case of bbh SM production, large logarithms ln(m 2 H ± /m 2 b ) arise for m H ± ≫ m b due to quasi-on-shell t-channel quark exchanges, which can be resummed by absorbing them into the b-quark parton densities. Thus, the proper procedure for computing the charged Higgs production cross-section is to add the cross-sections for gb → tH − and gg → tbH − and subtract out the large logarithms accordingly from the calculation of the 2 → 3 Figure 23: The leading-order production cross-sections for charged Higgs production at (a) the Tevatron (pp → tbH − + X) and (b) the LHC (pp → tbH − + X) are shown as a function of m H ± for three values of tan β = 2, 10 and 50. The cross-sections are obtained by adding the contribution of the 2 → 2 processes, gb → tH − , to those of the 2 → 3 processes, gg → tbH − and qq → tbH − (suitably subtracted to avoid double counting). Renormalization and factorization scales have been both set to m t + m H ± . These results are taken from ref. [201].
process [48,200]. This procedure avoids double-counting of the large logarithms at O(α s ), and correctly resums the leading logs to all orders. In particular, the contribution to the total cross-section coming from the kinematical region of the gluon-initiated 2 → 3 process in which one of the two gluons splits into a pair of b-quarks (one of which is collinear with the initial proton or antiproton), is incorporated into the b-quark parton density. A cruder calculation would omit the contribution of the 2 → 2 process and simply include the results of the unsubtracted 2 → 3 process. The latter procedure would miss the resummed leading logs that are incorporated into the b-quark density. However, the numerical difference between the two procedures is significant only for m H ± ≫ m t .
The single inclusive charged Higgs cross-sections at the Tevatron and LHC are exhibited in fig. 23 as a function of the charged Higgs mass, for tan β = 2, 10 and 50. Note that the cross-sections shown include the region of charged Higgs mass below m H ± = m t − m b corresponding to the case discussed above where the charged Higgs cross-section is dominated by tt production followed by t → bH − . These results are based on the calculations of ref. [201] and include the contributions of the 2 → 2 process and suitably subtracted 2 → 3 process as described above. Similar results have also been obtained in ref. [202]. The impact of the leading electroweak and MSSM radiative corrections has been studied in ref. [203]. In addition, the NLO QCD corrections to the 2 → 2 process gb → H + t have recently been evaluated [204]. These corrections typically increase the tree-level cross-section by a factor of 1.3 to 1.6, depending on the value of the charged Higgs mass and tan β, with some additional dependence on the choice of renormalization and factorization scales.
Associated production of a charged Higgs boson and a W ± can occur via bb annihilation and ggfusion [205]. The contribution of bb annihilation to σ(pp → H ± W ∓ + X) [σ(pp → H ± W ∓ + X)] at the Tevatron [LHC] (both charge combinations are included) are shown as function of the charged Higgs mass for tan β = 6 and 30 in fig. 24. The loop-induced gg fusion contribution is significantly suppressed relative to the tree-level bb annihilation if tan β > ∼ 6, independently of the value of m H ± [206].
Charged Higgs bosons can also be produced in pairs via Drell-Yan qq annihilation. The dominant contribution, which arises from uū and dd annihilation into a virtual photon or Z, is independent of tan β. Some tan β dependence enters through bb annihilation via t-channel top-quark exchange, although this effect is more than one order of magnitude suppressed relative to the dominant contribution at the Tevatron. The bb annihilation is more significant at the LHC (at large tan β where the H − tb coupling is enhanced). The tree-level results for σ(pp → H + H − + X) at the Tevatron and σ(pp → H + H − + X) at the LHC are shown in fig. 24. These results are obtained [207] with the Higgs-fermion Yukawa coupling based on a fixed b-quark pole mass of M b = 4.7 GeV. The contribution of the loop-induced gg → H + H − is typically less important than that of qq annihilation [208,209], and is not included in fig. 24. However, the gg fusion contribution can become significant at large tan β, with further enhancements in some regions of MSSM parameter space in which the squarks (which appear in the loop) are light and strongly mixed [209]. Nevertheless, the inclusive H + H − cross-section lies below the cross-section for single charged Higgs associated production (c.f. figs. 23 and 24).
Finally, one can compute the cross-sections for double neutral Higgs production at hadron colliders. These include the inclusive production of hh, hH, HH, hA, HA and AA. Cross-sections can be found in refs. [54,55,56] and [210,211] (QCD corrections to these cross-sections are evaluated in ref. [55]). In general, the rates for these processes are considerably smaller than for the corresponding single Higgs production rates. However, in certain regions of supersymmetric parameter space, squark loops can enhance the cross-section for pair production of two CP-even Higgs bosons by as much as two orders of magnitude [210]. In some cases, observation of double Higgs production provides some information on three-Higgs couplings. For example, for low to moderate values of tan β, gluon fusion to a virtual Higgs boson, which splits into hh, is dominant over bb → hh. Thus, the overall rate for pp → hh + X would provide a measure of the hhh vertex.
Additional sources for Higgs boson production can arise from the decay of supersymmetric particles into final states containing one or more Higgs boson in the decay chain [212]. These processes depend in detail on the details of the supersymmetric particle spectrum and their couplings. For example, the production of h in supersymmetric particle decay followed by the decay h → bb can yield a signal above background at LHC [213]. Processes of this type provide additional channels for possible Higgs discovery and precision study, and deserve further analysis.

Benchmarks for Higgs searches
In the search for the MSSM Higgs bosons, one must first search for the lightest Higgs scalar, which is expected (in almost all cases) to be the neutral CP-even scalar, h. In the decoupling region of the MSSM Higgs parameter space (where m A ≫ m Z ), the search techniques already outlined for h SM are relevant for h, since the properties of h approximately coincide with those of the SM Higgs boson. The h discovery reach can be mapped out as a region of m A -tan β parameter space, since these two parameters (along with the MSSM parameters that determine the size of the radiative corrections) fix the value of m h . Next, it is critical to identify deviations of the properties of h from those of h SM . Positive evidence for such a deviation would signal the existence of additional scalar states of the non-minimal Higgs sector. The difficulty of this step depends on how close the model is to the decoupling limit. After the discovery of h, the Higgs boson search will focus on the non-minimal Higgs states of the model.
For values of m A ∼ m Z , all the Higgs bosons of the MSSM are of a similar order of magnitude, and the properties of h will no longer resemble those of h SM . In principle, one can then discover multiple scalar states in one experiment. Since the two CP-even scalars share the coupling to vector boson pairs [eq. (22)], one may identify the CP-even scalar whose squared-coupling to V V is larger than 0.5g 2 h SM V V . The Tevatron and LHC production cross-sections of this scalar (compared to that of h SM ) are reduced by no more than 50% (by assumption), while the Higgs branching ratio into bb is similar to that of h SM over most of the MSSM parameter space. Thus, the Tevatron and LHC SM Higgs search results also apply here modulo minor modifications (which account for the somewhat suppressed production cross-section and the effects of supersymmetric corrections to the third generation Yukawa couplings).
The general MSSM parameter space involves many a priori unknown parameters. In practice, only a small subset of these parameters govern the properties of the Higgs sector. Nevertheless, a full scan of this reduced subset is still a formidable task. However, a detailed study of a few appropriately chosen points of the parameter space can help determine the ultimate MSSM Higgs discovery reach of the Tevatron and LHC. It is convenient to choose a set of benchmark MSSM parameters that govern the Higgs radiative corrections [156,214]. These include the supersymmetric Higgs mass parameter µ, the third generation squark mixing parameters, A t and A b , the gluino mass Mg, the diagonal softsupersymmetry-breaking third generation squark squared-masses (which we take for simplicity to be degenerate and equal to M SUSY ), and the top quark mass (which is held fixed at m t = 174.3 GeV). The maximal mixing benchmark scenario, is defined as the one in which the squark mixing parameters are such that they maximize the value of the lightest CP-even Higgs boson mass for fixed values of m A , tan β and M SUSY . Here, we choose The maximal mixing scenario poses a challenge for Higgs searches, since the predicted Higgs mass takes on its maximal value for a given set of MSSM parameters. However, different regions of the MSSM Higgs parameter space pose new challenges. For example, regions of parameter space exist in which the CP-even neutral Higgs boson with SM-like couplings to the W, Z and t has suppressed couplings to bb. The benchmark scenario denoted by "suppressed V φ → V bb production" is an example of this behavior. In this case, we take µ = −A t = 1.5 TeV, A b = 0, M 2 = 200 GeV and Mg = M SUSY = 1 TeV [corresponding to m max h = 120 GeV]. The regions of strongly suppressed BR(φ → bb) correspond to a suppressed Hbb coupling at lower m A and a suppressed hbb coupling at larger m A . In particular, the suppression for large tan β extends to relatively large values of m A ∼ 300 GeV, indicating a delay in the onset of the decoupling limit. Moreover, in the suppressed V φ → V bb benchmark scenario, all the Higgs couplings to bb are generally suppressed, since 0 < ∆ b ≪ 1 and sin 2α ≃ 0. From the analytic formulae, it can be deduced that µA t < 0 and large values of |A t |, |µ| and tan β are needed.
The coverage in the m A -tan β plane by different Higgs production and decay channels can vary significantly, depending on the choice of MSSM parameters. In the last example in which the CP-even Higgs boson with the larger coupling to the W and Z has a strongly suppressed coupling to bottom quarks, the Higgs searches at the Tevatron will become more problematical, while the LHC search for Higgs production followed by its decay into photons becomes more favorable [160]. At the same time the LHC Higgs discovery reach via vector bosons fusion to Higgs production followed by its decay into τ + τ − pairs can be significant [215].

MSSM Higgs Boson searches at the Tevatron
We first consider the Tevatron MSSM Higgs search. Specifically, we make use of the Tevatron h SM search techniques, where h SM is replaced by either h or H. If tan β ≫ 1, a new search mode becomes viable, due to the possibility of enhanced couplings of the neutral Higgs boson states to bb [216,170]. Thus, we also consider the possibility of the bbφ → bbbb signature, where φ = h, H, and/or A. If tan β is large, two of the neutral Higgs boson states, φ = A and h [H] are produced with enhanced rates if m A < ∼ m Z [m A ≫ m Z ], as noted below eq. (35). We may combine the results for the various channels to provide summary plots of the MSSM Higgs discovery reach of the upgraded Tevatron collider. We consider here the results based on a generic MSSM analysis [62]; see ref. [217] for a similar analysis in the context of a variety of models of supersymmetry breaking. In the latter case, the results obtained will be somewhat more constraining than the generic analysis, since the supersymmetry-breaking parameters that control the radiative corrections to Higgs masses and couplings are no longer arbitrary.  In figs. 25 and 26, we show the regions of 95% CL Higgs exclusion and 5σ Higgs discovery on the m A -tan β plane, for two representative MSSM parameter choices, via the search of neutral Higgs bosons in the channels: qq → V φ [φ = h, H], φ → bb (shaded regions) and gg, qq → bbφ [φ = h, H, A], φ → bb (region in the upper left-hand corner bounded by the solid lines), for different integrated luminosities as indicated by the color coding. The shaded regions presented in these figures reflect the results of the SHW simulation of qq → V φ improved by neutral network techniques [62]. The two sets of lines (for a given color) bounding the regions accessible by the bbφ search correspond to the CDF and DØ simulations, respectively. The solid black line near the bottom of each plot indicates the lower limit of tan β (as a function of m A ) based on the absence of observed e + e − → Zφ events at LEP [161]. Note the importance of the complementarity between the qq → V φ and qq → bbφ channels for improving the coverage of the MSSM parameter space in the low m A region in fig. 26(a). The results of figs. 25(a) and 26(a) demonstrate that 5 fb −1 of integrated luminosity per experiment will be sufficient to cover nearly all of the MSSM Higgs parameter space at 95% CL in the benchmark scenarios specified above.
To assure discovery of a CP-even Higgs boson at the 5σ level, the luminosity requirement becomes very important. Figs. 25(b) and 26(b) show that a total integrated luminosity of about 20 fb −1 per experiment is necessary in order to assure a significant, although not exhaustive, coverage of the MSSM parameter space. In general, we observe that the complementarity between the two channels, qq → V φ and qq → bbφ, is less effective in assuring discovery of a Higgs boson as compared with a 95% CL Higgs exclusion. This is due to the much higher requirement of total integrated luminosity combined with the existence of MSSM parameter regimes which can independently suppress both Higgs production channels. Fig. 26 exhibits one of the most difficult regions of MSSM parameter space for Higgs searches at the Tevatron collider. Nevertheless, even in this case, a very high luminosity experiment can cover a significant fraction of the available MSSM parameter space.
If explicit CP violation occurs through nonzero phases of the supersymmetry breaking parameters, then the three neutral Higgs bosons are a combination of CP-even and CP-odd states and the phenomenology can become much more complicated. In particular the couplings of the neutral Higgs bosons to the W and Z bosons are now shared by the three Higgs bosons and it may well be that the lightest Higgs has such a weak coupling to the vector bosons that it would have been missed at LEP and will be elusive at the Tevatron. Fig. 16 shows an interesting example where the effects of CP violation are such that for CP-violating phases of the parameter A t of about 90 • , the lightest Higgs boson cannot be detected at the Tevatron even though its mass is below 100 GeV, but the second lightest Higgs has SM-like couplings to the W and Z and thus can be detected if sufficient luminosity is provided.

MSSM Higgs Searches at the LHC
If no Higgs boson is discovered at the Tevatron, the LHC will cover the remaining unexplored regions of the m A -tan β plane, as shown in figs. 27 and 28. That is, in the maximal mixing scenario (and probably in most regions of MSSM Higgs parameter space), at least one of the Higgs bosons is guaranteed to be discovered at either the Tevatron and/or the LHC. A large fraction of the parameter space can be covered by the search for a neutral CP-even Higgs boson by employing the SM Higgs search techniques, where the SM Higgs boson is replaced by h or H with the appropriate re-scaling of the couplings. Moreover, in some regions of the parameter space, both h and H can be simultaneously observed, and additional Higgs search techniques can be employed to discover A, and/or H ± at the LHC.
A CP-even Higgs boson, φ, can be observed in a number of different decay modes. If m A > m max h , then φ = h is SM-like (near the decoupling limit), whereas at large tan β and m A < m max h , φ = H is the SM-like Higgs boson. 28 It is possible to observe φ → γγ when φ is produced singly via gg and V * V * fusion, or when produced in association with W ± and/or tt. A second decay mode, h → bb, can be observed in φtt production. Finally, it may be possible to observe φ → τ + τ − when φ is produced via V * V * fusion, where the forward jets are used to help reduce backgrounds [215]. Hence, by using the complementarity of the various Higgs signatures described above, one can discover φ over 28 For m A < m max h and moderate tan β values, neither CP-even Higgs boson is SM-like, although both Higgs masses lie below about 150 GeV and will appear (albeit with reduced couplings to V V ) in the Higgs searches described above. Figure 27: (a) 5σ discovery contours for MSSM Higgs boson detection in various channels in the m A -tan β plane, in the maximal mixing scenario, assuming an integrated luminosity of L = 30 fb −1 for the CMS detector [218]. (b) As in (a), but for an integrated luminosity of L = 300 fb −1 for the ATLAS detector [222].
nearly the entire MSSM parameter space, given sufficient integrated luminosity. 29 In order to illustrate the complementarity of the γγ and bb decay modes, we exhibit in fig. 29 the regions of MSSM Higgs parameter space that can be covered for the two benchmark scenarios of MSSM parameters described in Section 3.5.2. The behavior illustrated in this figure can be understood by noting that the φbb coupling can be significantly suppressed (or enhanced), depending on the impact of the radiative corrections discussed in Section 3.3. As a result, the branching ratio for φ → γγ is correspondingly larger (or smaller), with obvious implications for the φ → bb and φ → γγ searches.
We next focus on the potential for observing the heavier Higgs states (H ± , A and H). A number of recent studies [74,75,218,219,220,221] show that the following modes will be effective in searching for the heavier MSSM Higgs bosons. For the heavy neutral Higgs bosons, the most relevant decay signatures Figure 28: Regions in the m A -tan β plane in the maximal mixing scenario in which up to four Higgs boson states of the MSSM can be discovered at the LHC with 300 fb −1 of data, based on a simulation that combines data from the ATLAS and CMS detectors. Taken from ref. [87].
are: A, h → τ + τ − (where the τ is detected either via its leptonic or hadronic decay) and A, h → µ + µ − , which yield promising signals if tan β is large. The τ + τ − channel provides the largest discovery reach in the heavy Higgs mass. Other possible neutral Higgs decays: A, H → tt; H → ZZ * → 4ℓ; H → hh and A → Zh are significant in regions of the parameter space that are (nearly) ruled out by the LEP Higgs search. For the charged Higgs boson, we must again consider whether H ± can be produced in (on-shell) top-quark decays. If this decay is forbidden, the positively charged Higgs boson will be produced primarily by gb → H + t (see Section 3.5.1). In either case, the observation of the charged Higgs boson is possible if tan β ≫ 1 or tan β < ∼ O(1) [221]. For large tan β, the decays H + → τ + ν and tb (if kinematically allowed) provide the most favorable signatures. In particular, the τ ν decay mode, followed by the hadronic decay of the τ provides the largest discovery reach for large m H ± . The ultimate charged Higgs mass reach can depend significantly on the choice of MSSM parameters that control the radiative corrections to the Higgs-bottom quark Yukawa coupling [196] [see, e.g., eq. (70)].
Putting all of the above results together, it may be possible at the LHC to either exclude the entire m A -tan β plane (thereby eliminating the MSSM Higgs sector as a viable model), or achieve a Figure 29: Complementarity between the LHC searches for the decay modes φ → bb and φ → γγ, where φ = h or H corresponds to the CP-even Higgs boson with the larger coupling to V V . Two different choices of MSSM parameters are exhibited: (a) the maximal mixing scenario and (b) the suppressed V b → V bb production scenario. In both cases, the region corresponding to 5σ discovery of φ → γγ with 30 fb −1 of data is shaded with parallel horizontal lines. With 100 fb −1 of data, these regions expand to include areas shaded with diagonal parallel lines with positive slope. The region corresponding to 5σ discovery of ttφ → ttbb with 30 fb −1 of data is shaded with diagonal parallel lines with negative slope. With 100 fb −1 of data, these regions expand to include the entire m A -tan β plane excluding the blue cross-hatched region. The "unexcluded" region (where no discovery of ttφ → ttbb is possible) occupies a small region at low tan β and m A in both (a) and (b). In addition, in case (b), the excluded region also includes the two narrow wedge regions at large tan β and low m A . Taken from ref. [223].
5σ discovery of at least one of the MSSM Higgs bosons, independently of the value of tan β and m A . For example, Fig. 27 shows what can be achieved by the CMS detector with 30 fb −1 [218] and by the ATLAS detector with 300 fb −1 [222], assuming the maximal mixing scenario. Note that over a significant fraction of the MSSM Higgs parameter space, at least two Higgs bosons can be observed as shown in fig. 28. Nevertheless, there is still a sizable wedge-shaped region at moderate values of tan β opening up from about m A = 200 GeV to higher values in which the heavier Higgs bosons cannot be discovered at the LHC. In this parameter regime, only the lightest CP-even Higgs boson can be discovered, and its properties are nearly indistinguishable from those of the SM Higgs boson. Precision measurements of Higgs branching ratios and other properties will be required in order to detect deviations from SM Higgs predictions and demonstrate the existence of a non-minimal Higgs sector.
Finally, we noted at the end of Section 3.5.3 that CP-violating effects in the Higgs sector can modify the usual CP-conserving Higgs phenomenology. As a result, the LHC discovery reach of various Higgs channels discussed above may be altered in a significant way. It is therefore essential to make complementary measurements in as many Higgs channels as possible in order to cover the most general MSSM parameter space [223].

MSSM Higgs Boson Searches at the LC
The main production mechanisms for the MSSM Higgs bosons are [224] (i) e + e − → Zh , ZH via Higgs-strahlung , As in the SM Higgs search, process (i) followed by Z → ℓ + ℓ − allows the Higgs boson recoiling against the Z to be reconstructed, independently of the Higgs decay channel.
can be discovered at the LC even if it has a large branching fraction into invisible modes. 30 Processes (i) and (iii) are complementary to each other as a consequence of unitarity sum rules for tree-level Higgs couplings [131,132]. In particular, eq. (24) implies that both g 2 φZZ and g 2 φAZ (φ = h or H) cannot simultaneously vanish. If m A < ∼ m max h , then all the MSSM Higgs boson states have mass below 150 GeV, and can be cleanly reconstructed at the LC (with √ s ≥ 350 GeV) via the four production mechanisms listed above [92]. On the other hand, when m A > ∼ 200 GeV, one finds that m A ∼ m H ∼ m H ± and g HZZ ∼ g hAZ ∼ 0, and the couplings of h are nearly identical to those of h SM as a consequence of the decoupling limit. Since m h < ∼ 135 GeV, the LC with a center-of-mass energy of 300 GeV is more than sufficient to observe the h [via processes (i) and (ii)] and thus cover the entire MSSM parameter space with certainty. Moreover, the cross-sections for HZ, Hνν and hA are strongly suppressed [since | cos(β − α)| ≪ 1]. The cross-sections for HA and H + H − production are unsuppressed if kinematically allowed. 31 That is, the heavy Higgs bosons, H, A and H ± can only be observed in pair production processes where both Higgs states are heavy (and the minimum √ s required is somewhat above 2m A ). These features are evident in fig. 30, which depicts cross-sections for Higgs-strahlung [process (i)] and associated Higgs pair production [processes (iii) and (iv)] as a function of the corresponding Higgs mass for two different choices of √ s and tan β. The cross-section 30 The processes e + e − → e + e − h, e + e − H (via ZZ fusion) also allow for Higgs detection, independently of its decay channel, by reconstruction of the Higgs boson recoiling against the final state e + e − pair. However, the ZZ fusion rates are an order of magnitude smaller than the corresponding W + W − fusion rates [process (ii)]. 31   √ s = 800 GeV as a function of m H ± and m A , respectively. Taken from ref. [225].
In addition to H + H − production, there are a number of mechanisms in which the charged Higgs boson is singly produced. Charged Higgs bosons can be produced in top decays via t → b + H + if m H ± < m t − m b , as discussed previously in Section 3.5.1. The process e + e − → W ± H ∓ , which arises at one-loop [226,227], allows for the possibility of producing a charged Higgs boson with m H ± > √ s/2, when H + H − production is kinematically forbidden. With favorable MSSM parameters and moderate values of tan β, more than ten W ± H ∓ events can be produced at the LC for m H ± < ∼ 350 GeV with √ s = 500 GeV and 500 fb −1 of data, or for m H ± < ∼ 600 GeV with √ s = 1 TeV and 1 ab −1 [227]. Other single charged Higgs production mechanisms include tbH − /tbH + production [101], τ + νH − / τ −ν H + production [228], and a variety of processes in which H ± is produced in association with a one or two other gauge and/or Higgs bosons [229]. The heavier Higgs states could lie beyond the discovery reach of the LC ( √ s ≤ 1 TeV) and the LHC [cf. fig. 28]. In this case, the precision measurements of the h decay branching ratios and couplings achievable at the LC are critical for distinguishing between h SM and h of a non-minimal Higgs sector with properties close to that of the SM Higgs boson. To illustrate the challenge of probing the decoupling limit, suppose that m A > √ s/2 so that only the light Higgs boson, h, can be observed directly at the  Table 1) with √ s = 500 GeV and an integrated luminosity of 500 fb −1 . The contours correspond to 68, 90, 95, 98 and 99% confidence levels (right to left) for the observables g 2 hbb , g 2 hτ τ , and g 2 hgg . See ref. [112] for additional details.
LC. In this case, the fractional deviation of the couplings of h relative to those of the SM Higgs boson scales as m 2 Z /m 2 A . Thus, if precision measurements reveal a non-zero deviation, one could in principle derive a constraint (e.g., upper and lower bounds) on the heavy Higgs masses of the model. In the MSSM, the constraint is sensitive to the MSSM parameters that control the radiative corrections to the Higgs couplings. This is illustrated in fig. 31, where the constraints on m A are derived for two different sets of MSSM parameter choices [112]. Here, a simulation of a global fit of measured hbb, hτ τ and hgg couplings is made (based on the results of Table 1) and χ 2 contours are plotted indicating the constraints in the m A -tan β plane, assuming a deviation from SM Higgs boson couplings is seen. 32 In the maximal mixing scenario shown in fig. 31(a), the constraints on m A are significant and rather insensitive to the value of tan β. Similar results, in which deviations of BR(h → bb)/BR(h → τ + τ − ) from the SM prediction yield limits on the allowed values of m A as a function of tan β, have been obtained in ref. [230]. However in some cases, as shown in fig. 31(b), a region of tan β may yield almost no constraint on m A . This is due to the phenomenon of m A -independent decoupling noted below eq. (54), in which cos(β − α) [which controls the departure from the decoupling limit] vanishes at a particular value of tan β independently of the value of m A . Thus, one cannot extract a fully model-independent upper bound on the value of m A (beyond what can be deduced if no direct A production is observed at the LC). Of course, after supersymmetric particles are discovered, information about the MSSM spectrum can be used to obtain a more stringent bound on m A , using the techniques described above.
The e + e − linear collider running in the γγ collider mode presents additional opportunities for the study of the MSSM Higgs sector. Resonance production γγ → H and A can be used to extend the reach in Higgs masses beyond the limit set by HA pair production in the e + e − mode [231,123,124]. Typically, one can probe the heavy Higgs masses out to m A ∼ 0.8 √ s (where √ s is the center of mass energy of the LC). This expands the MSSM Higgs discovery reach further into regions of the m A -tan β parameter space for which the LHC is not sensitive in general (the so-called "blind wedge" of large m A and moderate values of tan β seen in fig. 28). As noted above, at least one Higgs boson must be observable at the LC in the MSSM. In non-minimal supersymmetric models, additional Higgs bosons appear in the spectrum, and the "no-lose" theorem of the MSSM must be reconsidered. For example, in the non-minimal supersymmetric extension of the Standard Model (the so-called NMSSM where a Higgs singlet is added to the model [232]), the lightest Higgs boson decouples from the Z boson if its wave function is dominated by the Higgs singlet component. However, in this case the second lightest neutral CP-even Higgs boson usually plays the role of h of the MSSM. That is, the mass of the second lightest Higgs boson is typically below 150 GeV with significant couplings to ZZ, so that it can be produced by the Higgs-strahlung process with an observable cross-section [233,158]. If the second lightest Higgs boson also decouples from the Z, then the third lightest Higgs boson will play the role of h of the MSSM for which the observation is ensured, and so on. Even in bizarre scenarios where all the neutral Higgs boson share equally in the coupling to ZZ (with the sum of all squared couplings constrained to equal the square of the h SM ZZ coupling [131,132]), the "no-lose" theorem still applies-Higgs production at the LC must be observable [234]. In contrast, despite significant progress, there is no complete guarantee that at least one Higgs boson of the NMSSM must be discovered at the LHC for all choices of the model parameters [235].
One of the key parameters of the MSSM Higgs sector is the value of the ratio of Higgs vacuum expectation values, tan β. In addition to providing information about the structure of the non-minimal Higgs sector, the measurement of this parameter also provides an important check of supersymmetric structure, since this parameter also enters the chargino, neutralino and third generation squark mass matrices and couplings. Thus, tan β can be measured independently using supersymmetric processes and compared to the value obtained from studying the Higgs sector. Near the decoupling limit, the properties of h are almost indistinguishable from those of h SM , and thus no information can be extracted on the value of tan β. However, the properties of the heavier Higgs bosons are tan β-dependent. Far from the decoupling limit, all Higgs bosons of the MSSM will be observable at the LC and exhibit strong tan β-dependence in their couplings. Thus, to extract a value of tan β from Higgs processes, one must observe the effects of the heavier Higgs bosons of the MSSM at the LC.
The ultimate accuracy of the tan β measurement at the LC depends on the value of tan β. In Ref. [236], it is argued that one must use a number of processes, including bbbb final states arising from bbH, bbA, and HA production, and ttbb final states arising from tbH + , btH − and H + H − production. One subtlety that arises here is that in certain processes, the determination of tan β may be sensitive to loop corrections that depend on the values of other supersymmetric parameters. One must settle on a consistent definition of tan β when loop corrections are included [237]. A comprehensive analysis of the extraction of tan β from collider data, which incorporates loop effects, has not yet been given.
The study of the properties of the heavier MSSM Higgs bosons (mass, width, branching ratios, quantum numbers, etc.) provides a number of additional challenges. For example, in the absence of CP-violation, the heavy CP-even and CP-odd Higgs bosons, H and A, are expected to be nearly massdegenerate. Their CP quantum numbers and their separation can be investigated at the same time in the γγ collider mode of the LC. If the polarization states of the two incoming linearly-polarized photons are parallel [perpendicular] then only the CP-even Higgs boson H [CP-odd Higgs boson A] will be produced [238]. Thus, the determination of the Higgs boson CP quantum numbers and the separation of the two different states can be achieved. In the case of a CP-violating Higgs sector, the observation and measurement of Higgs boson properties become much more challenging. The γγ collider can provide new opportunities to test the nature of the couplings of the Higgs neutral eigenstates (with indefinite CP quantum numbers) to gauge bosons and fermions [124,239].
Finally, once the heavy Higgs spectrum is revealed, one would like to reconstruct the two-Higgsdoublet scalar potential [240]. This is not likely to be accomplished at a first generation LC, although one can make a start if the heavy Higgs masses are not too large. To probe aspects of the Higgs potential one must observe multiple Higgs production in order to extract the Higgs self-couplings [113,114,240,241]. Ultimately, such a program would require an LC with very high energy and luminosity such as CLIC.

Conclusions
The physical origin of electroweak symmetry breaking is not yet known. In all theoretical approaches and models, the dynamics of electroweak symmetry breaking must be revealed at the TeV-scale or below. This energy scale will be thoroughly explored by hadron colliders, starting with the Tevatron and followed later in this decade by the LHC. Even though the various theoretical alternatives can only be confirmed or ruled out by future collider experiments, a straightforward interpretation of the electroweak precision data suggests that electroweak symmetry breaking dynamics is weakly-coupled, and a Higgs boson with mass between 100 and 200 GeV must exist. With the supersymmetric extension of the Standard Model, this interpretation opens the route to grand unification of all the fundamental forces, with the eventual incorporation of gravity in particle physics.
In this review we have summarized the theoretical properties of the Standard Model Higgs boson and the Higgs bosons of the MSSM, and surveyed the search strategies for discovering the Higgs boson at hadron and lepton colliders. We have assessed the Higgs boson discovery reach of present and future colliders, and described methods for measuring the various Higgs boson properties (mass, width, CP quantum numbers, branching ratios and coupling strengths).
The observation of a Higgs boson in the theoretically preferred mass range below 200 GeV may be possible at the Tevatron, whereas experiments at the LHC can discover the SM Higgs boson over the full Higgs mass range up to 1 TeV. The Tevatron can also extend the LEP search for Higgs bosons of the MSSM by either discovering the lightest CP-even MSSM Higgs boson, h (or in some special cases discovering additional Higgs scalars of the model), or by further constraining the MSSM Higgs parameter space. The LHC is sensitive to nearly the entire MSSM Higgs parameter space, in which either h alone can be discovered or multiple Higgs states can be observed. A program of Higgs measurements will be initiated at the LHC to measure Higgs partial widths with an accuracy in the range of 10-30%.
The discovery of the Higgs boson at the Tevatron and/or the LHC is a crucial first step. The measurement of Higgs properties at the LHC will begin to test the dynamics of electroweak symmetry breaking. However, a high-luminosity e + e − linear collider, now under development, is needed for a systematic program of precision Higgs measurements. For example, depending on the value of the Higgs mass, branching ratios and Higgs couplings can be determined in some cases at the level of a few percent. In this way, one can extract the properties of the Higgs sector in a comprehensive way, and establish (or refute) the existence of scalar sector dynamics as the mechanism responsible for generating the masses of the fundamental particles.