document: measurements at lep , sld , and the tevatron have been used extensively to limit models with physics beyond that of the standard model ( sm)@xcite . by performing global fits to a series of precision measurements , information about the parameters of new models 
   can be inferred@xcite . 
   the simplest example of this approach is the prediction of the @xmath3 boson mass . in the standard model , the @xmath3- boson mass , @xmath1 , 
   can be predicted in terms of other parameters of the theory . 
   the predicted @xmath3 boson mass is strongly correlated with the experimentally measured value of the top quark mass , @xmath4 , and increases quadratically as the top quark mass is increased . 
   this strong correlation between @xmath1 and @xmath4 in the standard model can be used to limit the allowed region for the higgs boson mass@xcite .    in a model with higgs particles in representations 
   other than @xmath5 doublets and singlets , there are more parameters in the gauge / higgs sector than in the standard model . 
   the sm tree level relation , @xmath6 no longer holds and when the theory is renormalized at one loop , models of this type will require extra input parameters@xcite . models with new physics are often written in terms of the sm lagrangian , @xmath7 plus an extra contribution , @xmath8 where @xmath9 represents contributions from new physics beyond the sm . 
   phenomenological studies have then considered the contributions of @xmath7 at one - loop , plus the tree level contributions of @xmath9 . in this note , we give two specific examples with @xmath0 at tree level , where we demonstrate that this procedure is incorrect . 
   we discuss in detail what happens in these models when the scale of the new physics becomes much larger than the electroweak scale and demonstrate explicitly that the sm is not recovered .    the possibility of a heavy higgs boson which is consistent with precision electroweak data has been considered by chivukula , hoelbling and evans@xcite and by peskin and wells@xcite in the context of oblique corrections . in terms of the @xmath10 , @xmath11 and @xmath12 parameters@xcite , 
   a large contribution to isospin violation , @xmath13 , can offset the contribution of a heavy higgs boson to electroweak observables such as the @xmath3 boson mass . 
   the triplet model considered in this paper provides an explicit realization of this mechanism . 
   the oblique parameter formulation neglects contributions to observables from vertex and box diagrams , which are numerically important in the example discussed here .    in section  [ renorm ] 
   , we review the important features of the sm for our analysis . 
   we discuss two examples in sections  [ higgstrip ] and appendix  [ lrmodel ] where the new physics does not decouple from the sm at one - loop . for simplicity , we consider only the dependence of the @xmath3 boson mass on the top quark mass and demonstrate that a correct renormalization scheme gives very different results from the sm result in these models . 
   section  [ higgstrip ] contains a discussion of the sm augmented by a real scalar triplet , and appendix  [ lrmodel ] contains a discussion of a left - right @xmath14 symmetric model . in section  [ nondecoupling ] , we show that the dependence on scalar masses in the w - boson mass is quadratic and demonstrate that the triplet is non - decoupling . 
   our major results are summarized in eq . 
   [ cc1]-[cc3 ] . 
   these results are novel and have not been discussed in the literature before . 
   section  [ results ] contains our numerical results and section  [ conc ] concludes this paper . 
   similar results in the context of the littlest higgs model have previously been found in ref .  . 
   the one - loop renormalization of the sm has been extensively studied@xcite and we present only a brief summary here , in order to set the stage for sections  [ higgstrip ] and appendix [ lrmodel ] . in the electroweak sector of the sm , the gauge sector has three fundamental parameters , the @xmath15 gauge coupling constants , @xmath16 and @xmath17 , as well as the vacuum expectation ( vev ) of the higgs boson , @xmath18 . 
   once these three parameters are fixed , all other physical quantities in the gauge sector can be derived in terms of these three parameters and their counter terms . 
   we can equivalently choose the muon decay constant , @xmath19 , the z - boson mass , @xmath20 , and the fine structure constant evaluated at zero momentum , @xmath21 , as our input parameters . 
   experimentally , the measured values for these input parameters are@xcite , @xmath22    the w - boson mass then can be defined through muon decay@xcite , @xmath23\ ] ] where @xmath24 summarizes the radiative corrections , @xmath25 where @xmath26 , @xmath27 and @xmath28 is the weak mixing angle . 
   the sm satisfies @xmath29 at tree level , @xmath30 in eq . 
   ( [ rhodef ] ) , @xmath1 and @xmath31 are the physical gauge boson masses , and so our definition of the weak mixing angle , @xmath32 , corresponds to the on - shell scheme@xcite . 
   it is important to note that in the sm , @xmath32 is not a free parameter , but is derived from @xmath33    the counterterms of eq . 
   ( [ drdef ] ) are given by@xcite , @xmath34 where @xmath35 , for @xmath36 , are the gauge boson 2-point functions ; @xmath37 is defined as @xmath38 . 
   the term @xmath39 contains the box and vertex contributions to the renormalization of @xmath40@xcite . 
   the counterterm for @xmath41 can be derived from eq . 
   ( [ rhodef ] ) , @xmath42 = \frac{\overline{c}_{\theta}^{2}}{\overline{s}_{\theta}^{2 } } \biggl [   \frac{\pi_{zz}(m_{z}^{2})}{m_{z}^{2 } }   - \frac{\pi_{ww}(m_{w}^{2})}{m_{w}^{2 } } \biggr]\quad . 
   \label{stdef}\ ] ] putting these contributions together we obtain , @xmath43\quad .\nonumber\end{aligned}\ ] ] these gauge boson self - energies can be found in ref .   and and we note that the fermion and scalar contributions to the two - point function @xmath44 vanish . 
   the dominant contributions to @xmath24 is from the top quark , and the contributions of the top and bottom quarks to the gauge boson self - energies are given in [ loop ] . 
   the @xmath45 dependence in @xmath46 and @xmath47 exactly cancel , thus the difference , @xmath48 , depends on @xmath49 only logarithmically . 
   the second term , @xmath50 , also depends on @xmath49 logarithmically . 
   however , the quadratic @xmath45 dependence in @xmath51 and @xmath52 do not cancel . 
   thus @xmath24 depends on @xmath49 quadratically , and is given by the well known result , keeping only the two - point functions that contain a quadratic dependence on @xmath49@xcite , @xmath53 where @xmath54 is the number of colors and the superscript @xmath55 denotes that we have included only the top quark contributions , in which the dominant contribution is quadratic . the complete contribution to @xmath24 
   can be approximated , @xmath56 the first term in eq . 
   ( [ drsm ] ) results from the scaling of @xmath57 from zero momentum to @xmath20@xcite . in the numerical results , the complete contributions to @xmath24 from top and bottom quarks , the higgs boson as well as the gauge bosons 
   are included , as given in eq . 
   ( [ drsm1 ] ) . 
   in this section , we consider the sm with an additional higgs boson transforming as a real triplet ( y=0 ) under the @xmath59 gauge symmetry . 
   hereafter we will call this the triplet model ( tm)@xcite . this model has been considered at one - loop by blank and hollik@xcite and we have checked that our numerical codes are correct by reproducing their results . in addition , we derive the scalar mass dependence in this model and show that the triplet is non - decoupling by investigating various scalar mass limits . 
   we also find the conditions under which the lightest neutral higgs can be as heavy as a tev , which has new important implications on higgs searches . 
   these results concerning the scalar fields are presented in the next section . 
   the @xmath58 higgs doublet in terms of its component fields is given by , @xmath60 with @xmath61 being the goldstone boson corresponding to the longitudinal component of the @xmath62 gauge boson . 
   a real @xmath59 triplet , @xmath63 , can be written as @xmath64 , @xmath65 there are thus four physical higgs fields in the spectrum : there are two neutral higgs bosons , @xmath66 and @xmath67 , @xmath68 and the mixing between the two neutral higgses is described by the angle @xmath69 . 
   the charged higgses @xmath70 are linear combinations of the charged components in the doublet and the triplet , with a mixing angle @xmath71 , @xmath72 where @xmath73 are the goldstone bosons corresponding to the longitudinal components of @xmath74 . 
   the masses of these four physical scalar fields , @xmath75 , @xmath76 and @xmath77 , respectively , are free parameters in the model . 
   the @xmath3 boson mass is given by , @xmath78 where @xmath79 is the vev of the neutral component of the @xmath59 higgs boson and @xmath80 is the vacuum expectation value of the additional scalar , leading to the relationship @xmath81 . 
   a real triplet does not contribute to @xmath31 , leading to @xmath82 the main result of this section is to show that the renormalization of a theory with @xmath83 at tree level is fundamentally different from that of the sm . 
   due to the presence of the @xmath58 triplet higgs , the gauge sector now has four fundamental parameters , the additional parameter being the vev of the @xmath58 triplet higgs , @xmath84 . 
   a consistent renormalization scheme thus requires a fourth input parameter@xcite . 
   we choose the fourth input parameter to be the effective leptonic mixing angle , @xmath85 , which is defined as the ratio of the vector to axial vector parts of the @xmath86 coupling , @xmath87 with @xmath88 and @xmath89 . 
   this leads to the definition of @xmath85 , @xmath90 the measured value from lep is given by @xmath91@xcite .    as usual 
   the @xmath3 boson mass is defined through muon decay@xcite , @xmath92 where we have chosen @xmath93 instead of @xmath94 as an input parameter . 
   the contribution to @xmath95 is similar to that of the sm , @xmath96 where the counter term @xmath97 is defined through @xmath98 , @xmath20 and @xmath85 as , @xmath99 unlike in the sm case where @xmath100 is defined through @xmath98 and @xmath20 as given in eq . 
   ( [ stdef ] ) , now @xmath85 is an independent parameter , and its counter term is given by@xcite , @xmath101\ ; \biggr ]   \ ; , \nonumber \\ & \equiv & \biggl(\frac{\hat{c}_{\theta}}{\hat{s}_{\theta } } \biggr ) \frac{re\biggl(\pi^{\gamma z}(m_{z}^{2})\biggr)}{m_{z}^{2 } }   + \delta_{v - b}^{\prime } \ ; , \end{aligned}\ ] ] where @xmath102 is the axial part of the electron self - energy , @xmath103 and @xmath104 are the vector and axial - vector form factors of the vertex corrections to the @xmath105 coupling . 
   these effects have been included in our numerical results@xcite . the total correction to @xmath95 in this case 
   is then given by , @xmath106 where @xmath39 summarizes the vertex and box corrections in the tm model , and it is given by@xcite , @xmath107 , \qquad   r \equiv m_{w}^{2 } / m_{z}^{2 } \ ; , \ ] ] where we show only the finite contributions in the above equation . 
   keeping only the top quark contribution , @xmath108 where @xmath109 is the momentum cutoff in dimensional regularization and the definition of the function @xmath110 can be found in appendix a. as @xmath111 is logarithmic , the @xmath49 dependence of @xmath98 is now logarithmic . 
   we note that this much softer relation between @xmath98 and @xmath49 is independent of the choice of the fourth input parameter . 
   this will become clear in our second example , the left - right symmetric model , given in [ lrmodel ] . in our numerical results , we have included in @xmath95 the complete contributions , which are summarized in [ scalar ] , from the top and bottom quarks and the four scalar fields , as well as the gauge bosons , and the complete set of vertex and box corrections . 
   as shown in [ scalar ] , @xmath95 depends on scalar masses quadratically . 
   this has important implications for models with triplets , such as the littlest higgs model@xcite . 
   the two point function @xmath44 does not have any scalar dependence , while @xmath112 and @xmath113 depend on scalar masses logarithmically . 
   the quadratic dependence thus comes solely from the function @xmath114 . 
   when there is a large hierarchy among the three scalar masses ( case ( c ) in [ scalar ] and its generalization ) , all contributions are of the same sign , and are proportional to the scalar mass squared , @xmath115 \nonumber\\ & & -s_{\delta}^{2}\frac{m_{h^{0}}^{2 } m_{h^{\pm}}^{2}}{2m_{w}^{4 } }   % \biggl ( 1 + \ln \biggl(\frac{m_{h^\pm}^{2}}{m_{h^{0}}^{2}}\biggr ) % \biggr ) - c_{\delta}^{2 } \frac{2m_{k^{0}}^{2}m_{h^{\pm}}^{2}}{m_{w}^{4 } }   % \biggl(1 + \ln \biggl(\frac{m_{h^\pm}^{2}}{m_{k^{0}}^{2}}\biggr)%\biggr ) \biggr\ }    \ ; , \nonumber\end{aligned}\ ] ] for @xmath116 . thus the scalar contribution to @xmath95 in this case is very large , and it grows with the scalar masses . on the other hand , when the mass splitting between either pair of the three scalar masses is small ( case ( a ) and ( b ) and their generalization ) , the scalar contributions grow with the mass splitting@xcite , @xmath117 \nonumber\\ & & \qquad + \frac{5}{72}\biggl [ s_{\delta}^{2}\frac { \bigl(m_{h^{\pm}}^{2}-m_{h^{0}}^{2}\bigr)}{m_{h^{0}}^{2 } }   + 4c_{\delta}^{2 } \frac {   \bigl(m_{h^{\pm}}^{2 } - m_{k^{0}}^{2}\bigr)}{m_{k^{0}}^{2 } } \biggr ] \biggr\ }   \nonumber \ ; , \end{aligned}\ ] ] for @xmath118 , and , @xmath119 \nonumber\\ & & -s_{\delta}^{2}\frac{m_{h^{0}}^{2}m_{h^{\pm}}^{2}}{2m_{w}^{4 } }   % \biggl ( 1 + \ln \biggl(\frac{m_{h^\pm}^{2}}{m_{h^{0}}^{2}}\biggr ) % \biggr )   + \frac{5}{18}c_{\delta}^{2 }   \frac{\bigl(m_{h^{\pm}}^{2 } - m_{k^{0}}^{2}\bigr)}{m_{k^{0}}^{2 } } \biggr\ } \nonumber \ ; , \end{aligned}\ ] ] for @xmath120 . cancellations can occur in this case among contributions from different scalar fields , leading to the viability of a heavier neutral higgs boson than is allowed in the sm@xcite . 
   the non - decoupling property of the triplet is seen in eq . 
   ( [ cc1 ] ) , ( [ cc2 ] ) and ( [ cc3 ] ) . because @xmath95 depends quadratically on the scalar masses@xcite , the scalars must be included in any effective field theory analysis of low energy physics . 
   the scalar potential of the model with a @xmath58 triplet and an @xmath58 doublet is given by the following@xcite : @xmath121 where @xmath122 denotes the pauli matrices , and @xmath123 from the minimization conditions ( see [ minimize ] ) , @xmath124 we obtain the following conditions , @xmath125 the two mixing angles , @xmath69 and @xmath71 , in the neutral and charged higgs sectors defined in eqs . 
   ( [ neutralmix ] ) and ( [ chargedmix ] ) , are solutions to the following two equations@xcite , @xmath126 \nonumber \label{cond1}\\ 0 & = & -\lambda_{4}v + \lambda_{3}vv^{\prime}+\tan\gamma   \biggl [ \mu_{1}^{2 } - \mu_{2}^{2 } + 3 \lambda_{1 } v^{2 } - \frac{1}{2 } \lambda_{3 } v^{2 }   -\lambda_{4}v^{\prime } -3\lambda_{2 } v^{\prime 2 } \\ & & \qquad + \frac{1}{2 } \lambda_{3 } v^{\prime 2 }   + \lambda_{4 } v \tan\gamma - \lambda_{3 } vv^{\prime}\tan\gamma \biggr ] \ ; , \label{cond2 } \nonumber\end{aligned}\ ] ] which are obtained by minimizing the scalar potential . in terms of the parameters in the scalar potential , 
   the masses of the four scalar fields are given by@xcite , @xmath127 this model has six parameters in the scalar sector , @xmath128 . 
   equivalently , we can choose @xmath129 as the independent parameters . 
   two of these six parameters , @xmath18 and @xmath130 , contribute to the gauge boson masses . 
   when turning off the couplings between the doublet and the triplet in the scalar potential , @xmath131 , the triplet could still acquire a vev , @xmath132 , provided that @xmath133 is negative . since we have not observed any light scalar experimentally up to the ew scale , the triplet mass which is roughly of order @xmath134 has to be at least of the ew scale , @xmath135 . 
   this is problematic because the vev of a real triplet only contributes to @xmath98 but not to @xmath31 , which then results in a contribution of order @xmath136 to the @xmath137 parameter , due to the relation , @xmath138 . 
   for @xmath134 greater than @xmath18 , the ew symmetry is broken at a high scale . in order to avoid these problems , 
   the parameter @xmath133 thus has to be positive so that the triplet does not acquire a vev via this mass term when @xmath139 is turned off . 
   once the coupling @xmath140 is turned on while keeping @xmath141 , the term @xmath142 effectively plays the role of the mass term for @xmath63 and for @xmath143 . 
   thus , similar to the reasoning given above , for @xmath144 , the coupling @xmath140 has to be positive so that it does not induce a large triplet vev . for simplicity , consider the case when there is no mixing in the neutral higgs sector , @xmath145 . in this case , when the mixing in the charged sector approaches zero , @xmath146 , the masses @xmath76 and @xmath77 approach infinity , and their difference @xmath147 approaches zero . 
   the contribution due to the new scalars thus vanishes , and only the lightest neutral higgs contributes to @xmath95 . even though the contribution due to the new scalars vanishes , @xmath95 does not approach @xmath24 . 
   this is because in the tm case , four input parameters are needed , while in the sm case three inputs are needed . 
   there is no continuous limit that takes one case to the other@xcite . 
   one way to achieve the @xmath146 limit is to take the mass parameter @xmath148 while keeping the parameter @xmath139 finite . 
   ( [ cond1 ] ) then dictates that @xmath149 . 
   however , satisfying eq . 
   ( [ cond2 ] ) requires that @xmath150 . 
   as @xmath151 , this condition implies that the dimensionless coupling constant @xmath140 has to scale as @xmath152 , which approaches infinity as @xmath146 . 
   this can also be seen from eq . 
   ( [ neutralmix1 ] ) . 
   as there is no mixing in the neutral sector , @xmath153 the condition @xmath154 then follows . 
   so , in the absence of the neutral mixing , @xmath145 , in order to take the charged mixing angle @xmath71 to zero while holding @xmath139 fixed , one has to take @xmath140 to infinity . 
   in other words , for the triplet to decouple requires a dimensionless coupling constant @xmath140 to become strong , leading to the breakdown of the perturbation theory . 
   alternatively , the neutral mixing angle @xmath69 can approach zero by taking @xmath155 while keeping @xmath140 and @xmath139 fixed . in this case , the minimization condition , @xmath156 where @xmath157 , implies that the charged mixing angle @xmath71 has to approach zero . 
   this again corresponds to the case where the custodial symmetry is restored , by which we mean that the triplet vev vanishes , @xmath158 . in this case , severe fine - tuning is needed in order to satisfied the condition given in eq . 
   ( [ cond2 ] ) . 
   another way to get @xmath146 is to have @xmath159 , which trivially satisfies eq . 
   ( [ cond1 ] ) . 
   this can also be seen from eq . 
   [ chargemix2 ] , @xmath160 eq .  ( [ cond2 ] ) then gives @xmath161 . 
   so for small @xmath140 , the masses of these additional scalar fields are of the weak scale , @xmath162 . 
   this corresponds to a case when the custodial symmetry is restored . 
   so unless one imposes by hand such symmetry to forbid @xmath139 , four input parameters are always needed in the renormalization . if there is a symmetry which makes @xmath163 ( to all orders ) , say , @xmath164 , then there are only three input parameters needed . 
   so the existence of such a symmetry is crucial when one - loop radiative corrections are concerned . 
   mass as a function of the top quark mass in the sm , tm and the lr model . 
   the data point represents the experimental values with @xmath165 error bars@xcite . for the sm , 
   we include the complete contributions from top and bottom quarks , the sm higgs boson with @xmath166 gev , and the gauge bosons . for the tm and the lr model , 
   we include only the top quark contribution and the absolute normalization is fixed so that the curves intersect the data point . 
   the @xmath167 boson mass is chosen to be @xmath168 tev in the lr model . ] 
   mass in the tm as a function of the top quark mass for scalar masses @xmath169 gev , with @xmath93 and @xmath85 varying within their @xmath165 limits@xcite @xmath170 and @xmath171 . 
   the solid curve indicates the prediction with @xmath93 and @xmath85 taking the experimental central values , @xmath172 and @xmath173 . 
   the area bounded by the short dashed ( dotted ) curves indicates the prediction with @xmath174 ( @xmath175 ) and @xmath176 ( @xmath177 ) varying with its @xmath165 limits . 
   the data point represents the experimental values with @xmath165 error bars@xcite . 
   ]     mass in the tm as a function of the top quark mass for scalar masses , @xmath75 , @xmath76 and @xmath77 , varying independently between ( a ) @xmath178 tev , ( b ) @xmath179 gev , and ( c ) @xmath180 gev . 
   the data point represents the experimental values with @xmath165 error bars@xcite.,title="fig : " ]   mass in the tm as a function of the top quark mass for scalar masses , @xmath75 , @xmath76 and @xmath77 , varying independently between ( a ) @xmath178 tev , ( b ) @xmath179 gev , and ( c ) @xmath180 gev . 
   the data point represents the experimental values with @xmath165 error bars@xcite.,title="fig : " ]   mass in the tm as a function of the top quark mass for scalar masses , @xmath75 , @xmath76 and @xmath77 , varying independently between ( a ) @xmath178 tev , ( b ) @xmath179 gev , and ( c ) @xmath180 gev . 
   the data point represents the experimental values with @xmath165 error bars@xcite.,title="fig : " ]     mass in the tm as a function of the lightest neutral higgs boson mass , @xmath75 , for various values of @xmath76 and @xmath77 . 
   the area bounded by the two horizontal lines is the @xmath165 allowed region for @xmath98@xcite.,title="fig : " ]   mass in the tm as a function of the lightest neutral higgs boson mass , @xmath75 , for various values of @xmath76 and @xmath77 . 
   the area bounded by the two horizontal lines is the @xmath165 allowed region for @xmath98@xcite.,title="fig : " ]   mass in the tm as a function of the lightest neutral higgs boson mass , @xmath75 , for various values of @xmath76 and @xmath77 . 
   the area bounded by the two horizontal lines is the @xmath165 allowed region for @xmath98@xcite.,title="fig : " ] 
   the previous section has presented analytic results for the triplet model , demonstrating that the dependence of the @xmath3 mass on the top quark mass is logarithmic , while the dependence on the scalar masses is quadratic . a dramatic change in the behavior of the @xmath3 mass 
   is also observed in the @xmath181 model@xcite . for comparison with the triplet model , we summarize the results of the left - right model in appendix  [ lrmodel ] . in this case , the dependence of the @xmath3 mass on the top quark mass is weakened from that of the sm since it depends on @xmath182 , where @xmath183 is the heavy charged gauge boson mass of the left - right model . 
   the dependence of the @xmath3 mass on the top quark mass , @xmath49 , in the case of the sm , the model with a triplet higgs , and the minimal left - right model , are shown in fig . 
   [ fg : mwmt ] . for the sm 
   , we include the complete contributions from top and bottom quarks , the sm higgs boson with @xmath166 gev , and the gauge bosons . in this case 
   , the @xmath49 dependence in the prediction for @xmath1 is quadratic . 
   the range of values for the input parameter @xmath49 that give a prediction for @xmath98 consistent with the experimental @xmath165 limits@xcite , @xmath184 gev , is very narrow . 
   it coincides with the current experimental limits@xcite , @xmath185 gev . for the triplet model and the lr model , we include only the top quark contribution . as we have shown in sec . 
   [ higgstrip ] , the prediction for @xmath1 in the triplet model depends on @xmath49 only logarithmically . in our numerical result for the left - right model , 
   we have used @xmath186 in the gauge sector , in addition to @xmath49 in the fermion sector , to predict @xmath98 . 
   here we have identified @xmath187 and @xmath188 as the @xmath3 and @xmath62 bosons in the sm and consequently @xmath189 and @xmath190 . in this case , the @xmath49 dependence in the prediction for @xmath98 is similarly softer because the top quark contributions are suppressed by a heavy scale , @xmath183 . in the triplet model and the left - right model , 
   the range of @xmath49 that gives a prediction for @xmath98 consistent with the experimental value is thus much larger , ranging from @xmath191 to @xmath192 gev . 
   the presence of the triplet higgs thus dramatically changes the @xmath49 dependence in @xmath98 . 
   this is clearly demonstrated in fig . 
   [ fg : mwmt ] by the almost flat curves of the triplet and left - right symmetric models , contrary to that of the sm , which is very sensitive to @xmath49 . in fig . 
   [ fg : mwmterr ] , we show the prediction for @xmath98 as a function of @xmath49 in the triplet model , with @xmath93 and @xmath85 varying within the @xmath165 limits@xcite , @xmath170 and @xmath171 . 
   we find that the prediction for @xmath98 is very sensitive to the input parameters @xmath177 and @xmath85 .    the complete contributions from the top and bottom quarks and the sm gauge bosons , as well as all four scalar fields in the triplet model are included in fig . 
   [ fg : mwmt1 ] and [ fg : mwmh ] . 
   we have also included the box and vertex corrections . in fig . 
   [ fg : mwmt1 ] , we show the prediction in the triplet model for @xmath98 as a function of @xmath49 , allowing @xmath75 , @xmath77 and @xmath76 to vary independently between @xmath178 tev , @xmath179 gev and @xmath180 gev . 
   interestingly , for all scalar masses in the range of @xmath178 tev , the prediction for @xmath98 in the tm model still agrees with the experimental @xmath165 limits . 
   [ fg : mwmh ] shows the prediction for @xmath98 as a function of @xmath75 for various values of @xmath76 and @xmath77 . for small @xmath147 , the lightest neutral higgs boson mass can range from @xmath193 gev to a tev and still satisfy the experimental prediction for @xmath98 
   this agrees with our conclusion in sec . 
   [ higgstrip ] that to minimize the scalar contribution to @xmath95 , the mass splitting @xmath147 has to be small and that when the mass splitting is small , cancellations can occur between the contributions of the lightest neutral higgs and those of the additional scalar fields . 
   this has new important implications for the higgs searches . 
   we have considered the top quark contribution to muon decay at one loop in the sm and in two models with @xmath0 at tree level : the sm with an addition real scalar triplet and the minimal left - right model . in these new models , because the @xmath137 parameter is no longer equal to one at the tree level , a fourth input parameter is required in a consistent renormalization scheme . 
   these models illustrate a general feature that the @xmath49 dependence in the radiative corrections @xmath95 becomes logarithmic , contrary to the case of the sm where @xmath24 depends on @xmath4 quadratically . 
   one therefore loses the prediction for @xmath49 from radiative corrections . on the other hand , due to cancellations between the contributions to the radiative corrections from the sm higgs and the triplet , 
   a higgs mass @xmath75 as large as a few tev is allowed by the @xmath3 mass measurement . 
   we emphasize that by taking the triplet mass to infinity , one does not recover the sm . 
   this is due to the fact that the triplet scalar field is non - decoupling , and it implies that the one - loop electroweak results can not be split into a sm contribution plus a piece which vanishes as the scale of new physics becomes much larger than the weak scale . 
   this fact has been overlooked by most analyses in the littlest higgs model@xcite , and correctly including the effects of the triplet can dramatically change the conclusion on the viability of the model@xcite . such non - decoupling effect has been pointed out in the two higgs doublet model@xcite , left - right symmetric model@xcite , and the littlest higgs model@xcite . 
   it has _ not _ been discussed before in the model with a triplet . 
   we comment that the non - decoupling observed in these examples do not contradict with the common knowledge that in gut models heavy scalars decouple . 
   these two cases are fundamentally different because in gut models , heavy scalar fields do not acquire vev that break the ew symmetry , while in cases where non - decoupling is observed , heavy scalar fields _ do _ acquire vev that breaks the ew symmetry . the quadratic dependence in scalar masses in the triplet model can be easily understood physically . in sm with only the higgs doublet present , the quadratic scalar mass contribution is protected by the tree level custodial symmetry , and thus the higgs mass contribution is logarithmic at one - loop . 
   this is the well - known screening theorem by veltman@xcite . 
   as the custodial symmetry is broken in the sm at one - loop due to the mass splitting between the top and bottom quarks , the two - loop higgs contribution is quadratic . in models with a triplet higgs , 
   as the custodial symmetry is broken already at the tree level , there is no screening theorem that protects the quadratic scalar mass dependence from appearing . 
   our results demonstrate the importance of performing the renormalization correctly according to the ew structure of the new models . 
   this manuscript has been authored by brookhaven science associates , llc under contract no . 
   de - ac02 - 76ch1 - 886 with the u.s . 
   department of energy . 
   the united states government retains , and the publisher , by accepting the article for publication , acknowledges , a world - wide license to publish or reproduce the published form of this manuscript , or allow others to do so , for the united states government purpose . 
   thanks w. marciano for useful discussions . 
   we summarize below the leading contributions due to the sm top loop to the self - energies of the gauge bosons@xcite , where the definitions of the passarino - veltman functions utilized below are given in@xcite . 
   @xmath194 \nonumber\\ \pi^{ww } ( 0 ) & = & -\frac{3 \alpha}{16\pi s_{\theta}^{2 } }   \ , \cdot \ , m_{t}^{2 } \biggl [ 1 + 2 \ln \biggl ( \frac{q^{2}}{m_{t}^{2 } } \biggr ) \biggr ]   \\ \pi^{zz } ( m_{z}^{2 } ) & = & -\frac{3 \alpha}{8\pi s_{\theta}^{2 }   c_{\theta}^{2 } }   \biggl [   \biggl(\biggl ( \frac{1}{2 } - \frac{4}{3}s_{\theta}^{2 } \biggr)^{2 }   + \frac{1}{4 } \biggr )    h_{1}(m_{t}^{2})\\ & & -\frac{8}{3 } s_{\theta}^{2 } \biggl (   1-\frac{4}{3}s_{\theta}^{2 } \biggr ) h_{2}(m_{t}^{2 } ) \biggr]\nonumber \\ 
   \pi_{\gamma\gamma}^{\prime}(0 ) & = & \frac{4\alpha}{9\pi }   \ln \biggl(\frac{q^{2}}{m_{t}^{2}}\biggr ) \\ \pi^{\gamma z } ( m_{z}^{2 } ) & = & -\frac{\alpha}{\pi s_{\theta}c_{\theta } } \biggl ( \frac{1}{2 } - \frac{4}{3 } s_{\theta}^{2 } \biggr ) m_{z}^{2 } \biggl [ \frac{1}{3 }   \ln \biggl(\frac{q^{2}}{m_{t}^{2}}\biggr ) -2 i_{3}\biggl(\frac{m_{z}^{2}}{m_{t}^{2}}\biggr)\biggr]\end{aligned}\ ] ] where @xmath195 \\ h_{2}(m_{t}^{2 } ) & = & m_{t}^{2 }   \biggl [ i_{1}\biggl(\frac{m_{z}^{2}}{m_{t}^{2}}\biggr )   - \ln \biggl(\frac{q^{2}}{m_{t}^{2}}\biggr ) \biggr]\quad .\end{aligned}\ ] ] the integrals are defined as , @xmath196 here @xmath197 is defined in the on - shell scheme ( eq .  ( [ swos ] ) ) for the sm and as the effective weak mixing angle ( eq .  ( [ swdef ] ) ) for the tm and lr model . 
   the complete contributions to various two - point functions that appear in @xmath95 are given below , where the scalar and fermion contributions are given in ref .  , and we have taken the sm gauge boson contributions from ref .  . 
   @xmath198 \\ & & + \frac{3 \alpha}{4\pi \hat{s}_{\theta}^{2 } } \biggl [ a_{0}(m_{t}^{2 } ) +   m_{b}^{2 } b_{0}(m_{w}^{2},m_{b}^{2},m_{t}^{2 } ) \nonumber\\ & & \qquad   -m_{w}^{2 } b_{1}(m_{w}^{2},m_{b}^{2},m_{t}^{2 } ) - 2 b_{22}(m_{w}^{2},m_{b}^{2},m_{t}^{2 } ) \biggr ] \nonumber\\ & & + { \alpha\over 4\pi \hat{s}_{\theta}^{2}}\biggl\ { s_\delta^2h(m_{h^0 } , m_{h^\pm})+c_\delta^2h(m_{h^0 } , m_w)\nonumber \\ & & \qquad + 4c_\delta^2h(m_{k^0 } , m_{h^\pm})+4s_\delta^2h(m_{k^0 } , m_w)\nonumber \\ & & \qquad + s_\delta^2h(m_z , m_{h^\pm})+c_\delta^2h(m_z , m_w)\biggr\ } \nonumber\\ & & + { \alpha\over 4 \pi \hat{s}_{\theta}^{2}}m_w^2 \biggl [ { s_\delta^2 c_\delta^2\over \hat{c}_{\theta}^{2 } }   \biggl(b_0(0,m_z , m_{h^\pm})- b_0(m_{w},m_z , m_{h^\pm } ) \biggr ) \nonumber\\ & & \qquad   + { ( s_\delta^2-\hat{s}_\theta^2)^2\over \hat{c}_{\theta}^{2 } } \biggl ( b_0(0,m_z , m_w ) - b_0(m_{w},m_z , m_w ) \biggr ) \nonumber \\   & & \qquad   + \hat{s}_\theta^2   \biggl(b_0(0,0,m_w ) - b_0(m_w,0,m_w ) \biggr ) \nonumber\\ & & \qquad   + c_\delta^2   \biggl ( b_0(0,m_{h^0 } , m_w ) - b_0(m_w , m_{h^0 } , m_w ) \biggr ) \nonumber\\ & & \qquad   + 4 s_\delta^2   \biggl(b_0(0,m_{k^0},m_w ) - b_0(m_w , m_{k^0},m_w ) \biggr )   \biggr ] \nonumber \\ & &   + \frac{\alpha}{4\pi \hat{s}_{\theta}^{2 } } \biggl [ \hat{c}_{\theta}^{2 } \biggl ( a_{1}(0,m_{z},m_{w } ) - a_{1}(m_w , m_z , m_w ) \biggr ) \nonumber\\ & & \qquad   + \hat{s}_{\theta}^{2 } \biggl ( a_{1}(0,0,m_w ) - a_{1}(m_w,0,m_w ) \biggr ) \nonumber\\ & & \qquad   - 2\hat{c}_{\theta}^{2 } h(m_z , m_w )   - 2\hat{s}_{\theta}^{2 } h(0,m_w ) \biggr]\nonumber \ ; , \\ & & \nonumber \\&&\nonumber \\&&\nonumber \\&&\nonumber \\&&\nonumber \\&&\nonumber \\&&\nonumber \\&&\nonumber \\&&\nonumber \\&&\nonumber\end{aligned}\ ] ]    @xmath199 \\ & & +   \frac{\alpha}{4\pi \hat{s}_{\theta}\hat{c}_{\theta } } \biggl [ 2(c_{\delta}^{2}-\hat{s}_{\theta}^{2 }   + \hat{c}_{\theta}^{2})b_{22}(m_z , m_{h^\pm},m_{h^\pm } ) \nonumber\\ & & \qquad + 2 ( s_{\delta}^2-\hat{s}_{\theta}^2+\hat{c}_{\theta}^2 ) b_{22}(m_z , m_w , m_w )   \nonumber\\ & & \qquad   + ( \hat{s}_{\theta}^2-\hat{c}_{\theta}^2-c_{\delta}^2)a(m_{h^\pm } )   + ( \hat{s}_{\theta}^2-\hat{c}_{\theta}^2-s_{\delta}^2)a(m_{w } ) \biggr ] \nonumber\\ & & +   \frac{\alpha}{4\pi}\bigl(2m_{w}^{2 } ) \frac{\hat{s}_{\theta}^2-s_{\delta}^2 } { \hat{s}_{\theta}\hat{c}_{\theta } } b_{0}(m_z , m_w , m_w ) \nonumber\\ & &   -\frac{\alpha}{4\pi \hat{s}_{\theta}^{2 } }   \biggl [ \hat{s}_{\theta}\hat{c}_{\theta } a_{1}(m_z , m_w , m_w ) + 2 \hat{c}_{\theta}\hat{s}_{\theta } a_{2}(m_w ) \nonumber\\ & & \qquad + 2 \hat{s}_{\theta}\hat{c}_{\theta } b_{22}(m_z , m_w , m_w ) \biggr ] \ ; , \nonumber % % % \end{aligned}\ ] ]    @xmath200 \ ; , % % % \end{aligned}\ ] ]    @xmath201 \ ; , \nonumber\end{aligned}\ ] ]    where @xmath202 and @xmath100 is defined in eq . 
   ( [ swdef ] ) .    to extract the dependence on the masses of the lightest neutral higgs , @xmath75 , and the extra scalar fields , @xmath203 and @xmath204 , we first note 
   that , in the limit @xmath205 , @xmath206 -\frac{1}{12 } p^{2 } \ln \biggl ( \frac{q^{2}}{m_{1}^{2 } } \biggr ) -\frac{1}{72}\frac{p^{4}}{m_{1}^{2 } } \\ & & + \biggl [ \frac{1}{4 } \ln \biggl(\frac{q^{2}}{m_{1}^{2 } } \biggr )   + \frac{5}{72}\frac{p^{2}}{m_{1}^{2}}\biggr ] \delta m^{2 }   + \mathcal{o}\biggl ( ( \delta m^{2})^{2 } , \ ;   \biggl(\frac{p^{2}}{m_{1}^{2}}\biggr)^{2 } \biggr)\nonumber \\ & &   + ( \mbox{terms with no scalar dependence})\nonumber \nonumber \ ; , \end{aligned}\ ] ] where we have defined @xmath207 and assumed that @xmath208 . using these relations 
   , we then have , @xmath209 @xmath210 on the other hand , in the limit @xmath211 , we get , @xmath212 which gives , @xmath213 % -\frac{1}{12}p^{2 } \ln q^{2 } + \mathcal{o } \biggl ( \biggl(\frac{m_{2}^{2}}{m_{1}^{2}}\biggr ) ,   \ ;   \biggl(\frac{p^{2}}{m_{1}^{2}}\biggr ) \biggr )   \\ & & +   \ ; ( \mbox{terms with no scalar dependence } ) \nonumber\end{aligned}\ ] ] @xmath214    the two - point function @xmath44 does not have any scalar dependence , and the function @xmath215 depends on the scalar mass only logarithmically , @xmath216 the scalar dependence in the function @xmath113 is , @xmath217 \\ 
   & = &   \frac{\alpha}{4\pi \hat{s}_{\theta } \hat{c}_{\theta } }   ( \hat{s}_{\theta}^{2 } -\hat{c}_{\theta}^{2}-c_{\delta}^{2 } ) m_{z}^{2 } \biggl [   \frac{1}{12 } \ln \biggl ( \frac{m_{h^{\pm}}^{2}}{q^{2}}\biggr )   - \frac{1}{72 } \frac{m_{z}^{2}}{m_{h^{\pm}}^{2}}\biggr ] \ ; ,   \nonumber\end{aligned}\ ] ] thus the dependence is also logarithmic . on the other hand , in the function @xmath114 , the scalar dependence is given by , @xmath218 \ ; . 
   \nonumber\end{aligned}\ ] ] from eqs . 
   ( [ b : eq ] ) and ( [ b0:eq2 ] ) , we know that the contributions from the terms in the square brackets of eq . 
   ( [ square ] ) are logarithmic . 
   thus the only possible quadratic dependence comes from terms in the curly brackets . 
   we consider the following three limits , assuming that all scalar masses are much larger than @xmath98 and @xmath20 :    * @xmath118 : in this case , the leading order scalar dependence is given by , @xmath219 \nonumber\\ & & + \frac{5}{72}\biggl [ s_{\delta}^{2}\frac{m_{w}^{2}}{m_{h^{0}}^{2 } }   \bigl(m_{h^{\pm}}^{2}-m_{h^{0}}^{2}\bigr ) + 4c_{\delta}^{2 } \frac{m_{w}^{2}}{m_{k^{0}}^{2 } }   \bigl(m_{h^{\pm}}^{2 } - m_{k^{0}}^{2}\bigr ) \biggr ] \biggr\ } \nonumber \ ; .\end{aligned}\ ] ] so the dominant scalar contribution to @xmath220 in this case is given by , @xmath221 \nonumber\\ & & \qquad + \frac{5}{72}\biggl [ s_{\delta}^{2}\frac { \bigl(m_{h^{\pm}}^{2}-m_{h^{0}}^{2}\bigr)}{m_{h^{0}}^{2 } }   + 4c_{\delta}^{2 } \frac {   \bigl(m_{h^{\pm}}^{2 } - m_{k^{0}}^{2}\bigr)}{m_{k^{0}}^{2 } } \biggr ] \biggr\ }   \nonumber \ ; .\end{aligned}\ ] ] * @xmath222 : in this limit , the leading scalar dependence becomes , @xmath223 \nonumber\\ & & -s_{\delta}^{2}\frac{m_{h^{0}}^{2}}{2m_{w}^{2 } }   % \biggl ( 1 + \ln \biggl(\frac{m_{h^\pm}^{2}}{m_{h^{0}}^{2}}\biggr ) % \biggr )   m_{h^{\pm}}^{2 } + \frac{5}{18}c_{\delta}^{2 } \frac{m_{w}^{2}}{m_{k^{0}}^{2 } }   \bigl(m_{h^{\pm}}^{2 } - m_{k^{0}}^{2}\bigr ) \biggr\ } \nonumber \ ; .\end{aligned}\ ] ] the leading scalar contribution to @xmath220 is , @xmath224 \nonumber\\ & & -s_{\delta}^{2}\frac{m_{h^{0}}^{2}m_{h^{\pm}}^{2}}{2m_{w}^{4 } }   % \biggl ( 1 + \ln \biggl(\frac{m_{h^\pm}^{2}}{m_{h^{0}}^{2}}\biggr ) % \biggr )   + \frac{5}{18}c_{\delta}^{2 }   \frac{\bigl(m_{h^{\pm}}^{2 } - m_{k^{0}}^{2}\bigr)}{m_{k^{0}}^{2 } } \biggr\ } \nonumber \ ; .\end{aligned}\ ] ] * @xmath225 : in this limit , the leading scalar dependence becomes , @xmath226 \nonumber\\ & & -s_{\delta}^{2}\frac{m_{h^{0}}^{2}}{2m_{w}^{2 } }   % \biggl ( 1 + \ln \biggl(\frac{m_{h^\pm}^{2}}{m_{h^{0}}^{2}}\biggr ) % \biggr )   m_{h^{\pm}}^{2 } - c_{\delta}^{2 } \frac{2m_{k^{0}}^{2}}{m_{w}^{2 } }   % \biggl(1 + \ln \biggl(\frac{m_{h^\pm}^{2}}{m_{k^{0}}^{2}}\biggr)%\biggr ) m_{h^{\pm}}^{2 } \biggr\ } \ ; , \nonumber\end{aligned}\ ] ] the leading scalar contribution to @xmath220 is thus given by , @xmath227 \nonumber\\ & & -s_{\delta}^{2}\frac{m_{h^{0}}^{2 } m_{h^{\pm}}^{2}}{2m_{w}^{4 } }   % \biggl ( 1 + \ln \biggl(\frac{m_{h^\pm}^{2}}{m_{h^{0}}^{2}}\biggr ) % 
   \biggr ) - c_{\delta}^{2 } \frac{2m_{k^{0}}^{2}m_{h^{\pm}}^{2}}{m_{w}^{4 } }   % 
   \biggl(1 + \ln \biggl(\frac{m_{h^\pm}^{2}}{m_{k^{0}}^{2}}\biggr)%\biggr ) \biggr\ }    \ ; . \nonumber\end{aligned}\ ] ] for the case @xmath228 , make the replacement , @xmath229 . 
   as our second example to show that new physics does not decouple from the sm at one - loop , we consider the left - right ( lr ) symmetric model@xcite which is defined by the gauge group , @xmath230 the minimal left - right symmetric model contains a scalar bi - doublet , @xmath63 , and two @xmath5 triplets , @xmath231 and @xmath232 . 
   we assume that the scalar potential is arranged such that the higgs fields obtain the following vev s : @xmath233 where the quantum numbers of these higgs fields under @xmath58 , @xmath234 and @xmath235 are given inside the parentheses . 
   the vev @xmath236 breaks the @xmath237 symmetry down to @xmath238 of the sm , while the bi - doublet vev s @xmath239 and @xmath240 break the electroweak symmetry ; the vev @xmath241 may be relevant for generating neutrino masses@xcite . 
   after the symmetry breaking , there are two charged gauge bosons , @xmath242 and @xmath243 , two heavy neutral gauge bosons , @xmath244 and @xmath245 , and the massless photon . 
   we will assume that @xmath242 and @xmath244 are the lighter gauge bosons and obtain roughly their sm values after the symmetry breaking . 
   turning off the @xmath58 triplet vev , @xmath246 , and assuming for simplicity that the @xmath247 gauge coupling constants satisfy @xmath248 , there are five fundamental parameters in the gauge / higgs sector , @xmath249 we can equivalently choose @xmath250 as input parameters . the counter term for the weak mixing angle is then defined through these parameters and their counter terms . assuming that the heavy gauge bosons are much heavier than the sm gauge bosons , @xmath251 , then to leading order @xmath252 , the counterterm @xmath253 is given as follows  @xcite , @xmath254 where the effective weak mixing angle , @xmath85 , is defined as in eq . 
   ( [ swdef ] ) . to go from the first to the second step in the above equation , we have used the following relation , @xmath255 . in the third line of eq . 
   ( [ ds ] ) , we include only the leading top quark mass dependence . 
   when the limit @xmath256 is taken , @xmath257 approaches zero , and thus the sm result , @xmath258 is _ not _ recovered , which is not what one would naively expect . 
   one way to understand this is that in the left - right model , four input parameters are held fixed , while in the sm three input parameters are fixed . 
   there is thus no continuous limit which takes one from one case ( @xmath83 at tree level ) to the other ( @xmath259 at tree level ) . 
   this discontinuity , which has been pointed out previously@xcite , is closely tied to the fact that the triplet higgs boson is non - decoupling@xcite . due to this non - decoupling effect , even if the triplet vev is extremely small , as long as it is non - vanishing , there is the need for the fourth input parameter . 
   the only exception to this is if there is a custodial symmetry which forces @xmath260 : in this case only the usual three input parameters are necessary . 
   we also note that the contribution of the lightest neutral higgs in this case is given by@xcite , @xmath261 which depends on @xmath262 quadratically , and is suppressed by the heavy gauge boson masses , @xmath263 and @xmath264 . 
   the contributions of the remaining scalar fields also have a similar structure . 
   in this section , we summarize our results on minimization of the scalar potential in the model with a triplet higgs . from the minimization conditions , @xmath265 
   , we obtain the following conditions , @xmath266 we ues the short hand notaion , @xmath267 . 
   the second derivatives are , @xmath268 if @xmath269 , then there is no mixing between the doublet and the triplet . 
   this requires @xmath270 . 
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summary: electroweak precision data have been extensively used to constrain models containing physics beyond that of the standard model . 
   when the model contains higgs scalars in representations other than singlets or doublets , and hence @xmath0 at tree level , a correct renormalization scheme requires more inputs than the three commonly used for the standard model case . in such cases , 
   the one loop electroweak results can not be split into a standard model contribution plus a piece which vanishes as the scale of new physics becomes much larger than @xmath1 . 
   we illustrate our results by presenting the dependence of @xmath1 on the top quark mass in a model with a higgs triplet and in the @xmath2 left - right symmetric model . in these models , the allowed range for the lightest neutral higgs mass can be as large as a few tev .