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the higgs potential of the standard model ( sm ) , which is crucial in implementing the mechanism of spontaneous symmetry breaking , contains the unknown quartic coupling of the higgs field . as a consequence , the mass of the only higgs boson in the sm , which is determined by this quartic coupling , is not known  @xcite . if a higgs boson is discovered and its mass measured , the higgs potential of the standard model can be uniquely determined . on the other hand , supersymmetry is at present the only known framework in which the higgs sector of the standard model ( sm ) , so crucial for its internal consistency , is natural  @xcite . the minimal version of the supersymmetric standard model ( mssm ) contains two higgs doublets @xmath19 with opposite hypercharges : @xmath20 , @xmath21 , so as to generate masses for up- and down - type quarks ( and leptons ) , and to cancel gauge anomalies . after spontaneous symmetry breaking induced by the neutral components of @xmath22 and @xmath23 obtaining vacuum expectation values , @xmath24 , @xmath25 , @xmath26 , the mssm contains two neutral @xmath0-even - even higgs particles as @xmath1 and @xmath3 . ] ( @xmath1 , @xmath3 ) , one neutral @xmath0-odd ( @xmath5 ) , and two charged ( @xmath27 ) higgs bosons @xcite . although gauge invariance and supersymmetry fix the quartic couplings of the higgs bosons in the mssm in terms of @xmath28 and @xmath29 gauge couplings , @xmath30 and @xmath31 , respectively , there still remain two independent parameters that describe the higgs sector of the mssm . these are usually chosen to be @xmath32 and @xmath33 , the mass of the @xmath0-odd higgs boson . all the higgs masses and the higgs couplings in the mssm can be described ( at tree level ) in terms of these two parameters .    in particular , all the trilinear self - couplings of the physical higgs particles can be predicted theoretically ( at the tree level ) in terms of @xmath33 and @xmath32 . once a light higgs boson is discovered , the measurement of these trilinear couplings can be used to reconstruct the higgs potential of the mssm . this will go a long way toward establishing the higgs mechanism as the basic mechanism of spontaneous symmetry breaking in gauge theories . although the measurement of all the higgs couplings in the mssm is a difficult task , preliminary theoretical investigations by plehn , spira and zerwas @xcite , and by djouadi , haber and zerwas @xcite ( referred to as ` dhz ' in the following ) , of the measurement of these couplings at the lhc and at a high - energy @xmath11 linear collider , respectively , are encouraging .    in this paper we consider in detail the question of possible measurements of the trilinear higgs couplings of the mssm at a high - energy @xmath11 linear collider . we assume that such a facility will operate at an energy of 500  gev with an integrated luminosity per year of @xmath34 @xcite . ( this is a factor of 10 more than the earlier estimate . ) in a later phase one may envisage an upgrade to an energy of 1.5  tev . since the ` interesting ' cross sections fall off like @xmath35 , the luminosity should increase by a corresponding factor . an earlier estimated luminosity of @xmath36 at 1.5  tev may turn out to be too conservative . the trilinear higgs couplings that are of interest are @xmath9 , @xmath10 , and @xmath37 , involving both the @xmath0-even ( @xmath1 , @xmath3 ) and @xmath0-odd ( @xmath5 ) higgs bosons . the couplings @xmath9 and @xmath10 are rather small with respect to the corresponding trilinear coupling @xmath38 in the sm ( for a given mass of the lightest higgs boson @xmath39 ) , unless @xmath39 is close to the upper value ( decoupling limit ) . the coupling @xmath37 remains small for all parameters .    throughout , we include one - loop radiative corrections @xcite to the higgs sector in the effective potential approximation . in particular , we take into account the parameters @xmath5 and @xmath40 , the soft supersymmetry breaking trilinear parameter and the bilinear higgs(ino ) parameter in the superpotential , respectively , and as a consequence the left  right mixing in the squark sector , in our calculations . we thus include all the relevant parameters of the mssm in our study , which is more detailed than the preliminary one of dhz .    for a given value of @xmath39 , the values of these couplings significantly depend on the soft supersymmetry - breaking trilinear parameter @xmath5 , as well as on @xmath40 , and thus on the resulting mixing in the squark sector . since the trilinear couplings tend to be small , and depend on several parameters , their effects are somewhat difficult to estimate . the plan of the paper is as follows . in section 2 we review the higgs sector of the mssm , including the radiative corrections to the masses . the trilinear couplings are presented in section 3 . in section 4 we review the possible production mechanisms for the multiple production of higgs bosons through which the trilinear higgs couplings can be measured at an @xmath11 linear collider . in section 5 we consider the dominant source of the multiple production of the higgs ( @xmath1 ) boson through higgs - strahlung of @xmath3 , and through production of @xmath3 in association with the @xmath0-odd higgs boson ( @xmath5 ) , and the background to these processes . this source of multiple production can be used to extract the trilinear higgs coupling @xmath9 . section 6 deals with a detailed calculation of the cross section for the double higgs - strahlung process @xmath41 . this process involves the trilinear couplings @xmath9 and @xmath10 of the @xmath0-even higgs bosons ( @xmath1 , @xmath3 ) . in section 7 we consider the different fusion mechanisms for multiple @xmath1 production , especially the non - resonant process @xmath42 , for which we present a detailed calculation of the cross section in the ` effective @xmath43 approximation ' . this process also involves the two trilinear higgs couplings , @xmath9 and @xmath10 , and is the most useful one for extracting the coupling @xmath10 . in section 8 we present , based on our calculations , the regions of the mssm parameter space in which the trilinear couplings @xmath9 and @xmath10 could be measured ; finally , in section 9 we present a summary of our results and conclusions . in this section we review the higgs sector of the minimal supersymmetric standard model in order to set the notation and to describe the approximations we use in our calculations . as mentioned in the introduction , we shall include the dependence on the parameters @xmath5 and @xmath40 through mixing in the squark sector . where there is an overlap , our notation and approach closely follow those of ref . @xcite .    at the tree level , the higgs sector of the mssm is described by two parameters , which can be conveniently chosen as @xmath33 and @xmath32 @xcite . there are , however , substantial radiative corrections to the @xmath0-even neutral higgs masses and couplings  @xcite . in the one - loop effective potential approximation , the radiatively corrected squared - mass matrix for the @xmath0-even higgs bosons can be written as @xcite @xmath44 \nonumber \\ & & + \frac{3 g^2}{16 \pi^{2 } m_w^2 } \left [ \begin{array}{cc } \delta_{11 } & \delta_{12}\\ \delta_{12 } & \delta_{22 } \end{array } \right ] , \end{aligned}\ ] ] where the second matrix represents the radiative corrections . the functions @xmath45 depend , besides the top- and bottom - quark masses , on the higgs bilinear parameter @xmath40 in the superpotential , the soft supersymmetry - breaking trilinear couplings ( @xmath46 , @xmath47 ) and soft scalar masses ( @xmath48 , @xmath49 , @xmath50 ) , as well as on @xmath32 . we shall ignore the @xmath51-quark mass effects in @xmath45 in our calculations , which is a reasonable approximation for moderate values of @xmath5230 . furthermore , we shall assume , as is often done , @xmath53    with these approximations we can write ( @xmath54 is the top quark mass ) @xcite : @xmath55 where @xmath56 and @xmath57 are squared stop masses given by @xmath58 ( we have ignored the small @xmath59-term contributions to the stop masses ) and @xmath60    the one - loop radiatively corrected masses ( @xmath61 , @xmath62 ; @xmath63 ) of the @xmath0-even higgs bosons ( @xmath1 , @xmath3 ) can be obtained by diagonalizing the @xmath64 mass matrix in eq . ( [ eq : m2 ] ) . the radiative corrections are , in general , positive , and they shift the mass of the lightest higgs boson upwards from its tree - level value . we show in fig . [ fig : masses ] the resulting mass of the lightest higgs boson , @xmath39 , as a function of @xmath40 and @xmath32 , for two values of @xmath5 and two values of @xmath33 , and for @xmath65  tev . with a wider range of parameter values , or when the squark mass scale is taken to be smaller , the dependence on @xmath40 and @xmath32 can be more dramatic @xcite . the higgs mass falls rapidly at small values of @xmath32 . since the lep experiments are obtaining lower bounds on the mass of the lightest higgs boson , they are beginning to rule out significant parts of the small-@xmath32 parameter space , depending on the model assumptions . for @xmath66 , aleph finds @xmath67 at 95% c.l . gev , irrespective of @xmath32 , and a limit of @xmath68  gev for @xmath69 @xcite . ] in our calculations , we shall therefore take @xmath70 to be a representative value . [ for a recent discussion on how the lower allowed value of @xmath32 depends on some of the model parameters , see ref .  @xcite . ] the trilinear higgs couplings that are of interest can be written  @xcite as a sum of the tree - level coupling and one - loop radiative corrections : @xmath71 in units of @xmath72 , the tree - level couplings are given by @xmath73 with @xmath74 the mixing angle in the @xmath0-even higgs sector , which can be calculated in terms of the parameters appearing in the @xmath0-even higgs mass matrix ( [ eq : m2 ] ) . the one - loop radiative corrections in ( [ eq : lambda - hhh])([eq : lambda - haa ] ) are ( in the above units ) : @xmath75 \left[1+(m_{\tilde t_1}^2 - m_{\tilde t_2}^2)c_t f_t\right]^2 \right . \nonumber \\ & & + \left . \left.\frac{m_t^2}{m_{\tilde t_2}^2 } \left[1-(m_{\tilde t_1}^2 - m_{\tilde t_2}^2)c_t e_t\right ] \left[1-(m_{\tilde t_1}^2 - m_{\tilde t_2}^2)c_t f_t\right]^2 - 2 \right )   \right ] , \label{17 }   \\ \delta \lambda_{hhh } & = & \left ( \frac{3g^2 \cos^2\theta_w}{16 \pi^2 } \frac{m_t^4}{m_w^4 } \frac{\cos^3\alpha}{\sin^3\beta } \right )   \nonumber \\ & \times & \left [ 3\log\frac{m_{\tilde t_1}^2 m_{\tilde t_2}^2}{m_t^4 } + 3 ( m_{\tilde t_1}^2 - m_{\tilde t_2}^2)c_t f_t     \log\frac{m_{\tilde t_1}^2}{m_{\tilde t_2}^2 } \right . \nonumber\\   & & + \left . 2\left(\frac{m_t^2}{m_{\tilde t_1}^2 } \left[1+(m_{\tilde t_1}^2 - m_{\tilde t_2}^2)c_t f_t\right]^3 + \frac{m_t^2}{m_{\tilde t_2}^2 } \left[1-(m_{\tilde t_1}^2 - m_{\tilde t_2}^2)c_t f_t\right]^3   -2 \right ) \right ] , \label{16 } \ ] ] @xmath76 ,       \label{21}\end{aligned}\ ] ] where @xmath77 and we have ignored the contributions from @xmath51-quarks and @xmath51-squarks , which are in general small with respect to those arising from @xmath78-quarks and @xmath78-squarks . we have also adopted the simplification described in eq . ( [ 2 ] ) in writing the above results . we shall make these approximations throughout this paper . we show in figs . [ fig : lamhhh2 ] , [ fig : lamhhh2 ] and [ fig : lamhaa2 ] the couplings @xmath9 , @xmath10 and @xmath37 as functions of @xmath40 and @xmath32 , for two values of @xmath5 and two values of @xmath33 , all for @xmath65  tev . the explicit dependence on @xmath5 and @xmath40 is not dramatic , but it should be kept in mind that unless @xmath33 is rather small , @xmath39 may change considerably with @xmath5 . the trilinear couplings change significantly with @xmath33 , and thus also with @xmath39 . this is shown more explicitly in fig . [ fig : lam - mh ] , where we compare @xmath9 , @xmath10 and @xmath37 for three different values of @xmath32 , and the sm quartic coupling @xmath79 . the sm quartic coupling includes one - loop radiative corrections @xcite , and its normalization is such that at the tree - level , it coincides with the trilinear coupling .    at low values of @xmath39 , the mssm trilinear couplings are rather small . for some value of @xmath39 the couplings @xmath9 and @xmath10 start to increase in magnitude , whereas @xmath37 remains small . the values of @xmath39 at which they start becoming significant depend crucially on @xmath32 . for @xmath70 ( fig . [ fig : lam - mh]a ) this transition takes place around @xmath80100  gev , whereas for @xmath81 and 15 , the critical values of @xmath39 increase to 100110 and 120  gev , respectively ( see figs .  [ fig : lam - mh]b and c ) . in this region , the actual values of @xmath9 and @xmath10 ( for a given value of @xmath39 ) change significantly if @xmath5 becomes large and positive . a non - vanishing squark - mixing parameter @xmath5 is thus seen to be quite important . also , we note that for special values of the parameters , the couplings may vanish @xcite . see also fig .  1 of ref .  @xcite .    to sum up the behaviour of the trilinear couplings , we note that @xmath9 and @xmath10 are small ( @xmath82 ) for @xmath83120  gev , depending on the value of @xmath32 . however , as @xmath39 approaches its maximum value , which is reached rapidly as @xmath33 becomes large , @xmath84  gev , these trilinear couplings become large ( @xmath85 ) . thus , as functions of @xmath33 , the trilinear couplings @xmath9 and @xmath10 are large for most of the parameter space . we also note that , for large values of @xmath32 , @xmath9 tends to be relatively small , whereas @xmath10 becomes large , if also @xmath33 ( or , equivalently , @xmath39 ) is large . we note that for a given higgs boson mass @xmath39 , the tree level sm trilinear higgs coupling is given by @xmath86 on the other hand , for large values of @xmath33 ( the decoupling limit ) the corresponding mssm trilinear coupling , eq . ( [ eq : lambda - hhh0 ] ) , becomes @xmath87 i.e. , it approaches the sm trilinear coupling . the different mechanisms for the multiple production of the mssm higgs bosons in @xmath11 collisions have been discussed by dhz . the dominant mechanism for the production of multiple @xmath0-even light higgs bosons ( @xmath1 ) is through the production of the heavy @xmath0-even higgs boson @xmath3 , which then decays by @xmath8 . the heavy higgs boson @xmath3 can be produced by @xmath3-strahlung , in association with @xmath5 , and by the resonant @xmath43 fusion mechanism . these mechanisms for multiple production of @xmath1 @xmath88 are shown in fig . [ fig : feynman - resonant ] . we note that all the diagrams of fig . [ fig : feynman - resonant ] involve the trilinear coupling @xmath9 . a background to ( [ eq : res - hhh ] ) comes from the production of the pseudoscalar @xmath5 in association with @xmath1 and its subsequent decay to @xmath89 @xmath90 leading to @xmath91 final states [ see fig .  [ fig : feynman - nonres - zhh]d ] . a second mechanism for @xmath92 production is double higgs - strahlung in the continuum with a @xmath93 boson in the final state [ see fig .  [ fig : feynman - nonres - zhh]a  d ] , @xmath94 we note that the feynman diagram of fig . [ fig : feynman - nonres - zhh]c involves , apart from the coupling @xmath9 , the trilinear higgs coupling @xmath10 as well , whereas the other diagrams do not involve any of the trilinear higgs couplings . a third way of generating multiple higgs bosons in @xmath11 collisions is through associated production of ( @xmath92 ) with the pseudoscalar @xmath5 in the continuum [ see fig .  [ fig : feynman - nonres - ahh ] ] : @xmath95 this process will be briefly discussed in section  6 . it involves , besides @xmath9 and @xmath10 , the trilinear coupling @xmath37 as well . it is , however , difficult  @xcite to measure this coupling @xmath37 through the process ( [ eq : zstar - hha ] ) .    finally , there is a mechanism of multiple production of the lightest higgs boson through non - resonant @xmath43 ( @xmath96 ) fusion in the continuum [ see fig .  [ fig : feynman - nonres - ww ] ] : @xmath97 which will be discussed in section  7 . it is important to note that all the diagrams of fig . [ fig : feynman - resonant ] involve the trilinear coupling @xmath9 only . on the other hand , fig . [ fig : feynman - nonres - zhh]c , fig . [ fig : feynman - nonres - ahh]b and fig . [ fig : feynman - nonres - ww]c all involve both the trilinear higgs couplings @xmath9 and @xmath10 . as stated in section  4 , the dominant source for the production of multiple higgs bosons ( @xmath1 ) in @xmath11 collisions is through the production of the heavier @xmath0-even higgs boson @xmath3 either via higgs - strahlung or in association with @xmath5 @xcite , followed , if kinematically allowed , by the cascade decay @xmath8 . in terms of the @xmath93-electron couplings @xmath98 , @xmath99 , the cross sections for these processes can be written as @xcite @xmath100}{(1-m_z^2/s)^2 } , \label{eq : sigzh}\\ \sigma ( e^+e^- \rightarrow ah ) & = & \frac{g_f^2m_z^4}{96\pi s } ( v_e^2 + a_e^2)\sin^2(\beta - \alpha )   \frac{\lambda_a^{3/2}}{(1-m_z^2/s)^2 } ,   \label{eq : sigah}\end{aligned}\ ] ] where @xmath101 refers to @xmath102 , the two - body phase - space function , and is given as @xmath103    in fig . [ fig : sigma-500 - 1500 ] we plot the cross sections ( [ eq : sigzh ] ) and ( [ eq : sigah ] ) for the @xmath104 centre - of - mass energies @xmath105 and @xmath106 tev , as functions of the higgs mass @xmath107 and for @xmath108 . for large values of the mass @xmath33 of the pseudoscalar higgs boson , all the higgs bosons , except the lightest one ( @xmath1 ) , become heavy and decouple  @xcite from the rest of the spectrum . in this case @xmath109 and the associated @xmath110 production ( [ eq : sigah ] ) becomes the dominant production mechanism for @xmath3 .    at values of @xmath32 that are not too large , the trilinear @xmath111 coupling @xmath9 can be measured by the decay process @xmath8 , which has a width @xmath112 however , this is possible only if the decay is kinematically allowed , and the branching ratio is sizeable . in fig . [ fig : br - h - a-2 ] we show the branching ratios ( at @xmath70 ) for the main decay modes of the heavy @xmath0-even higgs boson as a function of the @xmath3 mass . apart from the @xmath92 decay mode , the other important decay modes are @xmath113 , @xmath114 . ( we have here disregarded decays to supersymmetric particles : charginos , stops , etc . if such particles are kinematically accessible , the @xmath115 and @xmath116 rates could be much smaller @xcite . ) we note that the couplings of @xmath3 to gauge bosons can be measured through the production cross sections for @xmath117 ; therefore the branching ratio @xmath118 can be used to measure the triple higgs coupling @xmath9 . the higgs - strahlung process [ fig . [ fig : feynman - resonant]a , eq .  ( [ eq : sigzh ] ) ] gives rise to resonant two - higgs @xmath119 $ ] final states . this is to be contrasted with the associated production process [ fig . [ fig : feynman - resonant]b , eq . ( [ eq : sigah ] ) ] , which typically yields three higgs @xmath120 $ ] final states , since the channel @xmath121 is the dominant decay mode of @xmath5 in the mass range of interest . the decay width for @xmath121 can be written as @xcite @xmath122 where the @xmath123 are phase - space factors given by eq . ( [ 27 ] ) . in fig . [ fig : br - h - a-2 ] we show the branching ratios for the pseudoscalar @xmath5 for @xmath108 . a background to the multiple production of lighter higgs bosons @xmath1 comes from @xmath92 states generated in the sequential reaction @xmath124h$ ] [ see fig .  [ fig : feynman - nonres - zhh]d ] . this is a genuine background in the sense that no higgs self - couplings are involved . but these background events are expected to be topologically very different from the signal events , since the two @xmath1 bosons do not form a resonance , whereas the @xmath125 $ ] does . the cross section for the process @xmath126 can be written as @xcite @xmath127 and is shown in fig . [ fig : sigma-500 - 1500 ] together with the signal cross sections ( [ eq : sigzh ] ) and ( [ eq : sigah ] ) . as a consequence of the decoupling theorem  @xcite , the cross section becomes small for large values of @xmath107 .    for increasing values of @xmath32 , the @xmath111 coupling gradually gets weaker ( see figs .  [ fig : lamhhh2 ] and [ fig : lam - mh ] ) , and hence the prospects for measuring @xmath9 diminish . this is indicated by fig . [ fig : br - h - a-5 ] , where we show the @xmath3 and @xmath5 branching ratios for @xmath81 . there is in fact a sizeable region in the @xmath33@xmath32 plane where the decay @xmath115 is kinematically forbidden . this is indicated in fig . [ fig : hole ] . in this figure we also display the regions where the @xmath115 branching ratio is in the range 0.10.9 . clearly , in the forbidden region , the @xmath9 can not be determined from resonant production . for small and moderate values of @xmath32 , the study of decays of the heavy @xmath0-even higgs boson @xmath3 provides a means of determining the triple - higgs coupling @xmath9 . in order to extract the coupling @xmath10 , other processes involving two - higgs ( @xmath1 ) final states must be considered . the @xmath91 final states , which can be produced in the double higgs - strahlung @xmath128 of fig . [ fig : feynman - nonres - zhh ] , could provide one possible opportunity , since it involves the coupling @xmath10 through the mechanism of fig . [ fig : feynman - nonres - zhh]c . in this section we shall study these non - resonant processes in detail . the doubly differential cross section for the process @xmath128 shown in fig . [ fig : feynman - nonres - zhh ] can be written as @xcite @xmath129 where the couplings @xmath130 and @xmath131 have been defined at the beginning of section 5 . because of some misprints in the formulas given in @xcite for the coefficient @xmath132 , we have recalculated it . following @xcite , we introduce @xmath133 for the scaled energies of the higgs particles , @xmath134 for the scaled energy of the @xmath93 boson , and @xmath135 . also , we denote by @xmath136 the scaled squared masses of various particles : @xmath137 we can express our result in a compact form as follows : @xmath138\ , g_{ab } + { { \rm re}}[b(y_1)b^*(y_2)]\,g_{bb}\right\ } + \{x_1\leftrightarrow x_2\}. \label{eq : cala}\ ] ] here , @xmath139 + \biggl[\frac{\sin^2(\beta-\alpha)}{y_1+\mu_h-\tilde\mu_z } + \frac{\sin^2(\beta-\alpha)}{y_2+\mu_h-\tilde\mu_z}\biggr ] + \frac{1}{2\mu_z } \label{eq : lca}\ ] ] represents a contribution from diagram [ fig : feynman - nonres - zhh]a , where the lepton tensor couples directly to the final - state @xmath93 polarization tensor , as well as the contributions of diagrams [ fig : feynman - nonres - zhh]b and [ fig : feynman - nonres - zhh]c . similarly , @xmath140 represents the part of diagram [ fig : feynman - nonres - zhh]a where the lepton tensor couples to the final - state @xmath93 polarization tensor indirectly via the higgs momenta @xmath141 and @xmath142 , as well as diagram [ fig : feynman - nonres - zhh]d . the tildes on @xmath143 keep track of the widths , e.g.@xmath144 . the higgs self - couplings @xmath9 and @xmath10 occur only in the function @xmath145 , eq . ( [ eq : lca ] ) . the coefficients @xmath146 and @xmath30 , which do not involve any higgs couplings , can be expressed rather compactly as @xmath147 , \nonumber \\ g_{ab } & = &   \mu_z[2(\mu_z-4\mu_h)+x_1 ^ 2+x_2(x_2+x_3 ) ] -y_1(2y_2-x_1x_3),\nonumber \\ g_{bb } & = & \mu_z^2(4\mu_h + 6 -x_1x_2 ) + 2\mu_z(\mu_z^2 + y_3 -4\mu_h )   \nonumber \\ & & + ( y_3-x_1x_2-x_3\mu_z-4\mu_h\mu_z)(2y_3-x_1x_2 - 4\mu_h+4\mu_z).\end{aligned}\ ] ] these coefficients ( we use a mixed notation , which involves both @xmath148 and @xmath149 ) correspond to those of @xcite as follows : @xmath150 . with this identification , we agree with the result given in the erratum to @xcite .    in the limit of large @xmath33 , @xmath151 , the cross section reduces to the standard model cross section with @xmath152 + \frac{1}{2\mu_z } \\ b(y)&= & \frac{1}{2\mu_z}\ , \frac{1}{y+\mu_h-\tilde\mu_z},\end{aligned}\ ] ] where at the tree level , @xmath153 , as discussed in sec .  3 . we show in fig . [ fig : sig - zll-2]a the @xmath91 cross section , as given by eqs . ( [ eq : sigzh ] ) , ( [ eq : gamhhh ] ) and ( [ eq : sigzhh ] ) , in the limit of no squark mixing , and with @xmath154 . the structure around @xmath155 is due to the vanishing and near - vanishing of the trilinear coupling . in fig . [ fig : sig - zll-2]b  d we have introduced squark mixing : @xmath156 , @xmath157 . ( for the decoupling - limit cross section , which is also shown , we use the mssm coupling , instead of the sm coupling , for the reason given in sec .  3 . )    in the case of no mixing , there is a broad minimum from @xmath158 to 90  gev , followed by an enhancement around @xmath159100  gev . this structure is due to the vanishing of the branching ratio for @xmath115 , which is kinematically forbidden in the region @xmath15890  gev , see fig . [ fig : hole ] ( this coincides with the opening up of the channel @xmath160 ) , followed by an increase of the trilinear couplings . this particular structure depends considerably on the exact mass values @xmath107 and @xmath39 . thus , it depends on details of the radiative corrections and the mixing parameters @xmath5 and @xmath40 . the @xmath161 channel contributes of the order of 20% in the region of the maximum at @xmath162100  gev .      the resonant and non - resonant production of @xmath163 [ fig . [ fig : feynman - nonres - ahh ] ] can lead to three-@xmath1 final states in the region of @xmath33 , where @xmath5 has a significant branching ratio for decaying to @xmath164 , i.e.  for @xmath33 below the @xmath165 threshold , and relatively low values of @xmath32 [ cf . [ fig : br - h - a-2 ] and [ fig : br - h - a-5 ] ] .    in principle , this channel allows for a study of the coupling @xmath37 [ cf . [ fig : feynman - nonres - ahh]a ] . however , the prospects for measuring this coupling , which is rather small [ see fig .  [ fig : lamhaa2 ] ] , was studied in ref . @xcite and found not to be very encouraging . as mentioned in section 4 , a double higgs ( @xmath92 ) final state in @xmath11 collisions can also result from the @xmath43 fusion mechanism , which can either be a resonant process as in ( [ eq : res - hhh ] ) , or a non - resonant one like ( [ eq : ww - fusion ] ) . since the neutral - current couplings are smaller than the charged - current ones , the cross section for the @xmath96 fusion mechanism in ( [ eq : res - hhh ] ) and ( [ eq : ww - fusion ] ) is an order of magnitude smaller than the @xmath43 fusion mechanism . we shall thus , in the following , ignore the @xmath96 fusion mechanism , and concentrate instead on the @xmath43 mechanism . the @xmath43 fusion mechanism provides another large cross section for the multiple production of @xmath1 bosons . the cross section for @xmath166 can be written as  @xcite @xmath167 ^ 2}\ { \cal f}(x , y)\right ] \cos^2(\beta - \alpha ) ,   \label{eq : fusion - exact}\ ] ] where @xmath168 , \\ f(x , y ) & = & \left [ \frac{2x}{y^3 } - \frac{1 + 2x}{y^2 } + \frac{2 + x}{2y } - \frac{1}{2}\right ] \left[\frac{z}{1 + z } - \log(1 + z)\right ] + \frac{x}{y^3}\frac{z^2(1 - y)}{(1 + z ) } , \\   g(x , y ) & = & \left [ -\frac{x}{y^2 }   + \frac{2 + x}{2y } - \frac{1}{2}\right ] \left[\frac{z}{1 + z } - \log(1 + z)\right],\end{aligned}\ ] ] with @xmath143 defined by eq . ( [ eq : mu ] ) and @xmath169 for @xmath170 , @xmath107 @xmath171 @xmath172 , and in the effective longitudinal @xmath173 approximation , the cross section ( [ eq : fusion - exact ] ) for @xmath166 can be written in the following simple form @xcite @xmath174\cos^2(\beta - \alpha ) . \label{32}\ ] ] however , in this approximation the cross section may be overestimated by a factor of @xmath175 for small values of masses and/or small centre - of - mass energies . for example , at @xmath176  gev the equivalent @xmath173 approximation gives a result that is twice as large as the exact cross section . therefore , we use the exact cross section ( [ eq : fusion - exact ] ) in our calculations .    the cross section ( [ eq : fusion - exact ] ) is plotted in fig . [ fig : sigma-500 - 1500 ] for centre - of - mass energies , @xmath176 gev and @xmath106 tev , and for @xmath108 , as a function of @xmath107 . the resonant fusion mechanism , which leads to @xmath119 $ ] + [ missing energy ] final states is competitive with the process @xmath177 $ ] + [ missing energy ] , particularly at high energies . since the dominant decay of @xmath1 will be into @xmath178 pairs , the @xmath3-strahlung and the fusion mechanism will give rise to final states that will predominantly include four @xmath51-quarks . on the other hand , the process @xmath6 will give rise to six @xmath51-quarks in the final state , since the @xmath110 final state typically yields three - higgs @xmath120 $ ] final states . besides the resonant @xmath43 fusion mechanism for the multiple production of @xmath1 bosons , there is also a non - resonant @xmath43 fusion mechanism : @xmath180 through which the same final state of two @xmath1 bosons can be produced . the cross section for this process , which arises through @xmath43 exchange as indicated in fig . [ fig : feynman - nonres - ww ] , can be written in the ` effective @xmath43 approximation ' as approximation from the exact result . ] @xmath181 where @xmath182 . in the above , the cross section is written as a @xmath43 cross section , at invariant energy squared @xmath183 , folded with the @xmath43 ` luminosity ' @xcite : @xmath184 where @xmath185 . the @xmath43 cross section receives contributions from several amplitudes , according to the diagrams ( a)(d ) in fig . [ fig : feynman - nonres - ww ] . we have evaluated these contributions and express the result in a form analogous to that of ref . @xcite and @xmath186 reads @xmath187 ; it should be @xmath188 . with @xmath189 , the prefactors in their eq . ( 16 ) reduce to ours . ] : @xmath190 ^ 2\,g_0 \nonumber \\ & & \phantom{\frac{2}{\beta_h } } + \frac{2}{\beta_h}\biggl [   \frac{\hat\mu_z\sin(\beta-\alpha)}{1-\hat\mu_h}\,\lambda_{hhh } + \frac{\hat\mu_z\cos(\beta-\alpha)}{1-\hat\mu_h}\,\lambda_{hhh } + 1\biggr]\nonumber \\ & & \phantom{\frac{2}{\beta_h}aaaa } \times[\sin^2(\beta-\alpha)\,g_1 + \cos^2(\beta-\alpha)\,g_2 ] \nonumber \\ & & \phantom{\frac{2}{\beta_h } } + \frac{1}{\beta_h^2 } \{\sin^4(\beta-\alpha)\,g_3+\cos^4(\beta-\alpha)\,g_4 + \sin^2[2(\beta-\alpha)]\,g_5\}\biggr\},\end{aligned}\ ] ] where we have introduced ` reduced squared masses ' @xmath191 and the higgs velocity is @xmath192 . our approach differs from that of dhz in that we do not project out the longitudinal degrees of freedom of the intermediate @xmath173 bosons . instead , we follow the approach of ref . @xcite , where transverse momenta are ignored everywhere except in the @xmath173 propagators , the integrations over which are approximated as ( here @xmath193 and @xmath194 denote electron and neutrino momenta , respectively , in the process ( [ eq : ww - nonres ] ) ) : @xmath195 ^ 2 } \simeq\frac{\pi(1-x_1)}{m_w^2},\ ] ] where @xmath196 [ and @xmath197 represents the energy of the @xmath173 . the energy squared of the subprocess is given as @xmath198 . the contributions of diagrams ( b)+(c ) , ( a ) and ( d ) are given by the terms @xmath199 , @xmath200 and @xmath201 , respectively , with @xmath202 , and @xmath203 [ 2\hat\mu_w+1 - 3(\hat\mu_h-\hat\mu_w)^2]\frac{l_w}{a_w } \nonumber \\ & & + 16[2\hat\mu_w+(\hat\mu_h-\hat\mu_w)^2]^2y_w + 16\beta_h^2(1+a_w)^2 , \nonumber \\ g_4(x)&=&8\beta_h(\hat\mu_h-\hat\mu_c)^2[1 - 3(\hat\mu_h-\hat\mu_c)^2 ] \frac{l_c}{a_c } \nonumber \\ & & + 16(\hat\mu_h-\hat\mu_c)^4y_c + 16\beta_h^2(1+a_c)^2 , \ ] ] where @xmath204 and similarly @xmath205 , @xmath206 and @xmath207 , with @xmath208 replaced by @xmath209 , the latter being defined in terms of the charged higgs mass @xmath210 . the interference between diagrams ( b)+(c ) and ( a ) is given by the term @xmath211 , whereas the interferences between diagrams ( b)+(c ) and ( d ) , and between ( a ) and ( d ) are given by @xmath186 and @xmath212 , respectively . for these interference terms , we find @xmath213l_w -4\beta_h(1 + 2\hat\mu_h-2\hat\mu_w ) , \nonumber \\ g_2(x)&=&8(\hat\mu_h-\hat\mu_c)^2l_c -4\beta_h(1 + 2\hat\mu_h-2\hat\mu_c ) , \nonumber \\ g_5(x)&=&\frac{\beta_h}{4}(z_wl_w+z_cl_c)+8\beta^2_h(1+a_w)(1+a_c),\end{aligned}\ ] ] with @xmath214        + \frac{(1 - 2a_w)^2}{a_c+a_w}[8\hat\mu_w+(1 + 2a_w)^2 ] , \\ z_c&=&-\frac{(1 + 2a_c)^2}{a_c - a_w}[8\hat\mu_w+(1 + 2a_c)^2 ]         + \frac{(1 + 2a_c)^2}{a_c+a_w}[8\hat\mu_w+(1 - 2a_c)^2].\end{aligned}\ ] ]    our functions @xmath211@xmath212 correspond to those of dhz , cf . ( [ eq : sighat ] ) and their eq . ( 16 ) in @xcite . at small @xmath39 , the cross section is sensitive to small @xmath148 , where the ` effective @xmath43 approximation ' is not well defined , and our results differ from those of dhz . however , apart from the contributions from small @xmath148 , our results agree with those of dhz to a precision of 15% . we show in fig . [ fig : sig - ww-2 ] the @xmath43 fusion cross section , at @xmath215 , as given by eqs . ( [ eq : fusion - exact ] ) and ( [ eq : sigww - nonres ] ) , in the limit of no squark mixing , as well as with mixing ( as indicated ) , and with @xmath154 . the structure is very reminiscent of that of fig . [ fig : sig - zll-2 ] , and for the same reasons . however , the scale is different . following @xcite , we have indicated in the @xmath33@xmath32 plane the regions where @xmath9 and @xmath10 might be measurable , according to criteria analogous to those given there . in fig . [ fig : sensi-500 ] , we consider @xmath216 , and identify regions according to the following criteria :    * regions where @xmath9 might become measurable are identified as those where @xmath217 ( solid ) , with the simultaneous requirement of @xmath218 [ see figs .  [ fig : br - h - a-2][fig : hole ] ] . in view of the recent , more optimistic , view on the luminosity that might become available , we also give the corresponding contours for 0.05  fb ( dashed ) and 0.01  fb ( dotted ) . for @xmath219 we take the sum of ( [ eq : sigzh ] ) , ( [ eq : sigah ] ) and ( [ eq : fusion - exact ] ) . * regions where @xmath10 might become measurable are those where the _ continuum _ @xmath220 cross section [ eq . ( [ eq : sigww - nonres ] ) ] is larger than 0.1  fb ( solid ) . also included are contours at 0.05 ( dashed ) and 0.01  fb ( dotted ) . such regions are given for four cases of the mixing parameters @xmath5 and @xmath40 , as indicated . we have excluded from the plots the region where @xmath221 , according to the lep lower bound @xcite . this corresponds to low values of @xmath33 . we note that with an integrated luminosity of 500  fb@xmath222 , the contours at 0.1  fb correspond to 50 events per year . this will of course be reduced by efficiencies , but should indicate the order of magnitude that can be reached .    at @xmath216 , with a luminosity of 500  fb@xmath222 per year , the trilinear coupling @xmath9 is accessible in a considerable part of the @xmath33@xmath32 parameter space : at @xmath33 of the order of 200300  gev and @xmath32 up to the order of 5 . with increasing luminosity , the region extends somewhat to higher values of @xmath33 .    at values of @xmath33 below 100  gev , there is also a narrow band where @xmath9 is accessible . the ` steep ' edge around @xmath223 ( where increased luminosity does not help ) is determined by the vanishing of @xmath224 , see fig . [ fig : hole ] . the coupling @xmath10 is accessible in a much larger part of this parameter space , but with a moderate luminosity , ` large ' values of @xmath32 are accessible only if @xmath5 is small .    in fig . [ fig : sensi-1500 ] , we consider @xmath225  tev , and present the analogous contours . here , for the case of @xmath9 we demand @xmath226 ( solid ) and 0.1  fb ( dashed ) , and for the case of @xmath10 we require the corresponding cross section [ eq . ( [ eq : sigww - nonres ] ) ] to be larger than 0.5  fb ( solid ) and 0.1  fb ( dashed ) . if a luminosity corresponding to these cross sections becomes available at @xmath225  tev , a somewhat larger region than at @xmath227  gev is accessible in the @xmath33@xmath32 plane . it should be stressed that the requirements discussed here are necessary , but not sufficient conditions for the trilinear couplings to be measurable . we also note that there might be sizable corrections to the @xmath43 approximation , and that it would be desirable to incorporate the dominant two - loop corrections to the trilinear couplings in the calculations . we have carried out a detailed investigation of the possibility of measuring the mssm trilinear couplings @xmath9 and @xmath10 at an @xmath11 collider . where there is an overlap , we have confirmed the results of ref .  @xcite . our emphasis has been on taking into account all the parameters of the mssm higgs sector . we have studied the importance of mixing in the squark sector , as induced by the trilinear coupling @xmath5 and the bilinear coupling @xmath40 .    at moderate energies ( @xmath216 ) the range in the @xmath33@xmath32 plane that is accessible for studying @xmath9 changes quantitatively for non - zero values of the parameters @xmath5 and @xmath40 . as far as the coupling @xmath10 is concerned , however , there is a qualitative change from the case of no mixing in the squark sector . if @xmath5 is large , then high luminosity is required to reach ` high ' values of @xmath32 . at higher energies ( @xmath215 ) , the mixing parameters @xmath5 and @xmath40 change the accessible region of the parameter space only in a quantitative manner . p.  o. would like to thank the desy theory group and the cern theory division , whereas p.  n.  p. would like to thank the university of bergen , for kind hospitality while parts of this work were finished . it is also a pleasure to thank abdel djouadi , wolfgang hollik , bernd kniehl , conrad newton and peter zerwas for valuable discussions and advice . this research was supported by the research council of norway , and ( pnp ) by the university grants commission , india under project number 10 - 26/98(sr - i ) . j. ellis , g. ridolfi and f. zwirner , phys . * b257 * , 83 ( 1991 ) ; y. okada , m.  yamaguchi and t. yanagida , prog . phys .  * 85 * , 1 ( 1991 ) ; h.  e. haber and r. hempfling , phys . lett .   * 66 * , 1815 ( 1991 ) . r. hempfling and a. h. hoang , phys . lett .  * b331 * , 99 ( 1994 ) ; + m. carena , j.r . espinosa , m. quirs and c.e.m . wagner , phys . * b355 * , 209 ( 1995 ) ; + m. carena , m. quirs and c.e.m . wagner , nucl . * b461 * , 407 ( 1996 ) ; + s. heinemeyer , w. hollik and g. weiglein , phys . * d58 * , 091701 ( 1998 ) . b. kileng , p. osland and p. n. pandita , z. phys . * c71 * , 87 ( 1996 ) ; b. kileng , p.  osland and p. n. pandita , in _ xth international workshop : high energy physics and quantum field theory _ , proceedings of the conference , zvenigorod , russia , 1995 , ed . levtchenko and v.i . savrin ( moscow university press , moscow , 1996 ) , p.  167 ( hep - ph/9601284 ) ; b. kileng , p. osland and p. n. pandita , in proceedings of the international workshop on _ quantum systems : new trends and methods _ , minsk , belarus , 1996 , eds . y. s. kim , l. m. tomilchik , i. d. feranchuk and a. z. gazizov ( world scientific , singapore , 1997 ) , p.  231 ( hep - ph/9608315 ) . h. e. haber , in proceedings of the conference on _ perspectives for electroweak interactions in e@xmath228 e@xmath229 collisions _ , ringberg ( tegernsee ) , germany , 1995 ; ed . b. a. kniehl ( world scientific , singapore , 1995 ) p.  219 .    for recent references , see : a. djouadi , p. janot , j. kalinowski , p.m.  zerwas , phys . b376 * , 220 ( 1996 ) ; + a. bartl , h. eberl , k. hidaka , t. kon , w. majerotto and y. yamada , hep - ph/9709253 , in _ beyond the standard model , 5th international conference on physics beyond the standard model _ , balholm , norway , eds . g. eigen , p. osland and b. stugu ( aip , woodbury , n.y . , 1997 ) , p.  494 . r. n. cahn and s. dawson , phys . lett . * b136 * , 196 ( 1984 ) ; s. dawson , nucl . b249 * , 42 ( 1984 ) ; m. chanowitz and m. k. gaillard , phys . * b142 * , 85 ( 1984 ) ; i. kuss and h. spiesberger , phys .  rev .  * d53 * , 6078 ( 1996 ) .
we present a detailed analysis of multiple production of the lightest @xmath0-even higgs boson ( @xmath1 ) of the minimal supersymmetric standard model ( mssm ) at high - energy @xmath2 colliders . 
 we consider the production of the heavier @xmath0-even higgs boson ( @xmath3 ) via higgs - strahlung @xmath4 , in association with the @xmath0-odd higgs boson ( @xmath5 ) in @xmath6 , or via the fusion mechanism @xmath7 , with @xmath3 subsequently decaying through @xmath8 , thereby resulting in a pair of lighter higgs bosons ( @xmath1 ) in the final state . 
 these processes can enable one to measure the trilinear higgs couplings @xmath9 and @xmath10 , which can be used to theoretically reconstruct the higgs potential . 
 we delineate the regions of the mssm parameter space in which these trilinear higgs couplings could be measured at a future @xmath11 collider . in our calculations , we include in detail the radiative corrections to the higgs sector of the mssm , especially the mixing in the squark sector .    cern - th/98 - 189 + hep - ph/9806351    * measuring the trilinear couplings of mssm + neutral higgs bosons at high - energy @xmath2 colliders * 0.5 cm p. osland@xmath12 and p. n. pandita@xmath13 + @xmath14 department of physics , university of bergen , n-5007 bergen , norway@xmath15 + @xmath16 deutsches elektronen - synchrotron desy , d-22603 hamburg , germany + @xmath17 theoretical physics division , cern , ch 1211 geneva 23 , switzerland + @xmath18 department of physics , north eastern hill university , shillong 793 022 , india@xmath15 +    pacs : 14.80.cp , 12.60.jv , 13.90.+i     + * permanent addresses    cern - th/98 - 189 + june 1998