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at present the top quark has not yet been conclusively discovered at the fermilab tevatron , even though there are events which look similar to those expected from top quark decays . the experimental situation is presently confusing because both the cdf @xcite and d0 @xcite collaborations only have limited statistics . as more events are collected one expects that the situation will be clarified .    at the tevatron , the top quark should be mainly produced through @xmath2 pair production from the light mass quarks and gluons in the colliding proton and antiproton . both the top quark and the top antiquark then decay to @xmath3 pairs , and each @xmath4 boson can decay either hadronically or leptonically . the @xmath5-quark becomes an on - mass - shell @xmath6-hadron which subsequently decays into leptons and ( charmed ) hadrons . a large effort is being made to reconstruct the top quark mass from the measured particles in the decay , which is complicated by the fact that the neutrinos are never detected . also there are additional jets so it is not clear which ones to choose to recombine . the best channel for this mass reconstruction is where both @xmath4 bosons decay leptonically , one to a @xmath7 pair , the other to a @xmath8 pair ( a dilepton event ) because the backgrounds in this channel are small . when only a single lepton is detected then it is necessary to identify the @xmath5 quark in the decay to remove large backgrounds from the production of @xmath9 jets @xcite . in all cases the reconstruction of the particles in the final state involves both the details of the production of the top quark - antiquark pair as well as the knowledge of their fragmentation and decay products . the cdf collaboration @xcite have reported two events with dilepton final states , six events with a single lepton and a @xmath5-quark identified by a secondary vertex , and seven single lepton events with the @xmath5-quark identified by a semileptonic decay . the cdf collaboration then constructed a likelihood function for the invariant mass @xcite and quoted the value @xmath10 gev@xmath11 . the top quark cross section quoted by the cdf collaboration is @xmath12 pb . the d0 collaboration @xcite have reported on nine events with an expected background of @xmath13 . if the excess is due to @xmath14 production and if the top quark mass is 180 gev@xmath15 , then the top quark cross section is @xmath16 pb .    the values for the top quark production cross section as a function of the top quark mass used by the cdf and d0 collaborations contain both the nlo qcd corrections and an extension to include the resummation of initial state soft partons to all orders in perturbation theory @xcite . a recent summary of the theoretical predictions has been presented in @xcite . in the laenen paper @xcite the dis factorization scheme was used with the mrsd@xmath17 parton distributions @xcite , the two - loop running coupling constant with five active flavors , and @xmath18 gev .    in the analysis of the decay distributions one needs knowledge of the inclusive differential distributions of the heavy quarks in transverse momentum @xmath19 and rapidity @xmath20 . these distributions are known in nlo . what we would like to discuss in this paper is an updating of the resummation effects on the inclusive transverse momentum distribution of the top quark . in the original paper @xcite it was not known which mass to choose whereas now we can assume that the mass is @xmath21 gev@xmath15 . also we discuss here how the resummation effects modify the rapidity distribution of the top quark . since there have been suggestions of using the mass and angular distributions in top quark production to look for physics beyond the standard model @xcite it is very important to know the normal qcd predictions for these quantities . we first summarize what is known on the top quark cross section . if the top quark mass is 175 gev@xmath15 then the dominant production channel is @xmath22 . in lowest order qcd perturbation theory it contributes about 90 % of the total cross section with the reaction @xmath23 making up the remaining 10% . one notes that the nlo corrections in the @xmath24 channel are small , whereas those in the @xmath25 channel are more than 80% . at this large top quark mass the @xmath26 and @xmath27 channels give negligible contributions so we do not consider them . even though the @xmath25 channel contribution is small in born approximation it can be significant in nlo due to multiple soft parton radiation . these large corrections are predominantly from the threshold region for heavy quark production . it was shown previously @xcite that initial state gluon bremsstrahlung ( isgb ) is responsible for the large corrections at nlo near threshold . in @xcite the dominant logarithms from isgb , which are the cause of the large corrections near threshold , were carefully examined . such logarithms have been studied previously in drell - yan ( dy ) @xcite production at fixed target energies ( again near threshold ) where they are responsible for correspondingly large corrections . the analogy between dy and heavy quark production cross sections was exploited in @xcite and a formula to resum the leading and next - to - leading logarithms in pqcd to all orders was proposed . since the contributions due to these logarithms are positive ( when all scales @xmath28 are set equal to the heavy quark mass @xmath29 ) , the effect of summing the higher order corrections increases the top quark production cross section over that predicted in @xmath1 . this sum , which will be identified as @xmath30 , depends on a nonperturbative parameter @xmath31 . the reason that a new parameter has to be introduced is that the resummation is sensitive to the scale at which pqcd breaks down . as we approach the threshold region other , nonperturbative , physics plays a role ( higher twist , bound states , etc . ) indicated by a dramatic increase in @xmath32 and in the resummed cross section . this is commonly called the effect of the infrared renormalon or landau pole @xcite . we chose to simply cut off the resummation at a specific scale @xmath31 where @xmath33 since it is not obvious how to incorporate the nonperturbative effects . note that our resummed corrections diverge for small @xmath31 but this is _ not _ physical since they should be joined smoothly onto some nonperturbative prescription and the total cross section will be finite . another way to make it finite would be to avoid the infrared renormalon by a specific continuation around it , i.e. the principal value resummation method . however , at the moment our total resummed corrections depend on the parameter @xmath31 for which we can only make a rough estimate . see @xcite for more details . to make this paper self - contained we list some relevant formulae . the partonic processes under discussion will be denoted by @xmath34 where @xmath35 . the kinematical variables @xmath36 are introduced in the calculation of the corrections to the single particle inclusive differential distributions of the heavy ( anti)quark . we do not distinguish in the text between the heavy quark and heavy antiquark since the distributions are essentially identical in our calculations . here @xmath37 is the square of the parton - parton c.m . energy and the heavy quark transverse momentum is given by @xmath38 . the rapidity variable is defined by @xmath39 . the born approximation differential cross sections can be expressed by @xmath40 with @xmath41\,,\ ] ] and @xmath42 \nonumber \\ & & \times\big [ \frac{t_1}{u_1 } + \frac{u_1}{t_1 } + \frac{4m^2s}{t_1u_1 } \big(1 - \frac{m^2s}{t_1u_1}\big ) \big ] \,.\end{aligned}\ ] ] here the color factors are @xmath43 and @xmath44 . the color average factors are @xmath45 and @xmath46 . the parameter @xmath28 denotes the renormalization scale . in @xcite the inclusive cross section was examined near threshold ( @xmath47 ) where the contributions from the radiation of soft and collinear gluons are large . a variable @xmath48 was defined , where @xmath49 and @xmath50 are inelastic variables in the channel @xmath51 . the variable @xmath52 now depends on the four momentum of the extra parton(s ) emitted in the reaction . in the born approximation there are no additional partons so @xmath53 . in @xcite the nlo contributions were examined in the soft region ( where @xmath54 ) and it was found that the dominant contribution to the nlo cross section in this region had a similar form to the nlo correction in the drell - yan process . as the latter correction is known exactly in nnlo @xcite this correspondence was used to write the differential cross section in order @xmath55 as follows @xmath56 \sigma^b_{ij}(s , t_1,u_1 ) \,.\end{aligned}\ ] ] here a small parameter @xmath57 has been introduced to allow us to distinguish between the soft ( @xmath58 ) and the hard ( @xmath59 ) regions in phase space . the quantities @xmath60 contain terms involving the qcd @xmath61-functions and color factors . the variables @xmath62 and @xmath63 were then mapped onto the variables @xmath64 and @xmath65 , where @xmath66 is the parton - parton c.m . scattering angle . after explicit integration over the angle @xmath66 , the resulting series was exponentiated by the introduction of the @xmath64 variable into the argument of the running coupling constant .    as noted in the previous paper @xcite in addition to the total cross section we can also derive the resummed heavy ( anti)quark inclusive @xmath19 ( and below the @xmath20 ) distributions . the transverse momentum @xmath19 of the heavy quark is related to our previous variables by @xmath67^{1/2}\big\}\,,\ ] ] @xmath68^{1/2}\big\}\,,\ ] ] with @xmath69 the double differential cross section is therefore @xmath70^{1/2 } % \frac{d^2\sigma_{ij}(s , s_4 , p_t^2)}{dp_t^2ds_4 } \ , , \ ] ] with the boundaries @xmath71 the @xmath72 contribution to the inclusive transverse momentum distribution @xmath73 is given by @xmath74^{1/2 } } \sigma^b_{ij}(s , s_4,p_t^2 ) %   \ , , \end{aligned}\ ] ] where we have inserted an extra factor of two so that @xmath75 . after some algebra we can rewrite this result as @xmath76\,,\end{aligned}\ ] ] with the definition @xmath77^{1/2 } } \sigma_{ij}^b(s , s_4,p_t^2)\,,\ ] ] where @xmath78 again represents the born differential @xmath19 distribution . for the @xmath79 and @xmath25 subprocesses we have the explicit results @xmath80^{1/2 } } % \nonumber \\ & & % \times \big[\frac{(s - s_4)^2 - 2sp_t^2}{s^2}\big ] \,,\end{aligned}\ ] ] and @xmath81^{1/2 } } % \nonumber \\ & & % \times\big[c_f - c_a \frac{m_t^2}{s}\big ] % \nonumber \\ & & % \times \big[\frac{(s - s_4)^2 - 2sm_t^2}{sm_t^2 } + % \frac{4m^2}{m_t^2 } \big ( 1 -\frac{m^2}{m_t^2}\big)\big ] \,.\end{aligned}\ ] ] since the above formulae are symmetric under the interchange @xmath82 the heavy quark and heavy antiquark inclusive @xmath19 distributions are identical . note that ( 2.12 ) is basically the integral of a plus distribution together with a surface term .    the corresponding formula to ( 2.12 ) for the rapidity @xmath20 of the heavy quark is obtained by using @xmath83 @xmath84 the double differential cross section is therefore @xmath85 with the boundaries @xmath86 where @xmath87 . the @xmath72 contribution to the inclusive rapidity distribution @xmath88 is given by @xmath89 after some algebra we can rewrite this result as @xmath90\,,\end{aligned}\ ] ] with the definition @xmath91 where @xmath92 again represents the born differential @xmath20 distribution . for the @xmath79 and @xmath25 subprocesses we have the explicit formulae @xmath93 \,,\end{aligned}\ ] ] and @xmath94 % \times \big[\cosh^2 y + \sinh^2 y % \nonumber \\ & & % + \frac{8m^2s \cosh^2 y}{(s - s_4)^2 } \big ( 1 -\frac{4m^2s\cosh^2 y}{(s - s_4)^2 } % \big)\big ] \,.\end{aligned}\ ] ] since the above formulae are symmetric under the interchange @xmath82 the heavy quark and heavy antiquark inclusive @xmath20-distributions are identical . also ( 2.21 ) is again of the form of a plus distribution together with a surface term . finally , we note that the terms in ( 2.12 ) and ( 2.21 ) are all finite . the resummed contribution to the top quark cross section can be written as @xcite @xmath95 \sigma_{ij}^b(s , t_1,u_1),\ ] ] where @xmath96^{\eta}}{\gamma(1+\eta ) } % \exp(-\eta\gamma_e).\end{aligned}\ ] ] the straightforward expansion of the exponential plus the change of the argument in @xmath97 via the renormalization group equations generates the corresponding leading logarithmic terms written expicitly in @xcite . the scheme dependent @xmath98 and @xmath97 in the above expression are given by @xmath99 in the @xmath100 scheme , and @xmath101 in the dis scheme , where @xmath102 is the lowest order coefficient of the qcd @xmath61-function . the color factors @xmath103 are defined by @xmath104 and @xmath105 , and @xmath106 is the euler constant . the quantity @xmath107 is given by @xmath108    following the procedure in @xcite for the resummation of the order @xmath109 contributions to the @xmath19 distribution we have @xmath110 note that we now have cut off the lower limit of the @xmath64 integration at @xmath111 because @xmath97 in ( 3.2 ) diverges as @xmath54 . this parameter @xmath112 must satisfy the conditions @xmath113 and @xmath114 . it is convenient to rewrite @xmath112 in terms of the scale @xmath28 as    @xmath115    @xmath116    here @xmath31 is a nonperturbative parameter @xcite satisfying @xmath117 . the derivative of @xmath118 is obtained from ( 3.2 ) . it is equal to @xmath119^\eta}{\gamma(1+\eta ) } % \nonumber \\ & & \times \exp(-\eta\gamma_e)\ , , \end{aligned}\ ] ] where we have neglected terms which are higher order in @xmath97 . the analogous formula for the rapidity distribution is @xmath120 with the conditions @xmath121 and @xmath114 . following the notation in @xcite the total hadron - hadron cross section in order @xmath122 is @xmath123 where @xmath124 is the square of the hadron - hadron c.m . energy and @xmath125 run over @xmath126 and @xmath127 . the parton flux @xmath128 is defined via @xmath129 and @xmath130 is a product of the scale - dependent parton distribution functions @xmath131 , where @xmath132 stands for the hadron which is the source of the parton @xmath133 @xmath134 the mass factorization scale @xmath28 is chosen to be identical with the renormalization scale in the running coupling constant . since the @xmath19 distribution in hadron - hadron collisions is not altered by the lorentz transformation along the collision axis from the parton - parton c.m . frame , we can write an analogous formula to ( 4.1 ) for the heavy - quark inclusive differential distribution in @xmath135 @xmath136 with @xmath137 . in the case of the all - order resummed expressions the lower boundaries in ( 4.1 ) and ( 4.4 ) have to be modified according to the conditions @xmath138 or @xmath139 ( see above ) . resumming the soft gluon contributions to all orders we obtain @xmath140 where @xmath141 is given in ( 3.24 ) of @xcite and @xmath142 with @xmath143 ( @xmath100 scheme ) or @xmath144 ( dis scheme ) [ see ( 3.7 ) ] . the all - order resummed differential distribution in @xmath135 is @xmath145 with @xmath73 given in ( 3.6 ) and @xmath146 the corresponding formula to ( 4.4 ) for the heavy quark inclusive differential distribution in @xmath20 is @xmath147 order by order in perturbation theory the heavy quark rapidity plots in the parton - parton c.m . frame show peaks away from @xmath148 ( see fig . 7 in @xcite ) . however , upon folding with the partonic densities the heavy quark rapidities in the hadron - hadron c.m . frame peak near @xmath149 . therefore we will assume that the plots for the resummed rapidity distribution show a similar feature . the all - order resummed differential distribution in @xmath150 is therefore taken to be @xmath151 with @xmath88 given in ( 3.9 ) and @xmath152 the hadronic heavy quark rapidity @xmath150 is related to the partonic heavy quark rapidity @xmath20 by @xmath153    we now specialize to top quark production at the fermilab tevatron where @xmath154 tev and choose the top quark mass to be @xmath155 gev@xmath15 . in the presentation of our results for the exact , approximate and resummed hadronic cross sections we use the mrsd@xmath156 parametrization for the parton distributions @xcite . note that the hadronic results only involve partonic distribution functions at moderate and large @xmath157 , where there is little difference between the various sets of parton densities . we have used the mrsd@xmath156 set 34 as given in pdflib @xcite in the dis scheme with the number of active light flavors @xmath158 and the qcd scale @xmath159 gev . we have used the two - loop corrected running coupling constant as given by pdflib . since we know the exact @xmath1 result , we can make an even better estimate of the differential distributions by calculating the perturbation theory improved @xmath19 and @xmath20 distributions . we define the improved @xmath19 distribution by @xmath160 and the improved @xmath150 distribution by @xmath161 to exploit the fact that @xmath162 and @xmath163 are known and @xmath164 and @xmath165 are included in @xmath166 and @xmath167 respectively . we note that here @xmath168 denotes the @xmath169 contribution to the differential cross section . moreover , @xmath170 denotes the exact calculated differential cross section , and @xmath171 the approximate one where only the leading soft gluon corrections are taken into account . first we present the differential @xmath19 distributions at @xmath154 tev for a top quark mass @xmath172 gev@xmath15 . for these plots the mass factorization scale is not everywhere equal to @xmath29 . we chose @xmath173 in @xmath112 , @xmath174 and @xmath175 , but @xmath176 in the mrsd@xmath17 parton distribution functions and the running coupling constant @xmath177 . we begin with the results for the @xmath24 channel in the dis scheme . in fig . 1 we show the born term @xmath178 , the first order exact result @xmath162 , the first order approximation @xmath164 , the second order approximation @xmath179 , and the resummed result @xmath166 for @xmath180 and for @xmath181 . these are the same values for @xmath31 that were used in @xcite . as we decrease @xmath31 the differential cross sections increase . in fig . 2 we show the exact @xmath1 result @xmath182 , and , for comparison , @xmath183 for @xmath180 and @xmath181 . the resummed differential cross sections were calculated with the cut @xmath184 while no such cut was imposed on the phase space for the individual terms in the perturbation series . the improved differential distributions are uniformly above the exact @xmath1 result . it is also evident from fig . 2 that the resummation of the soft gluon contributions to the @xmath19 distribution modifies the exact @xmath1 result only slightly for the values of @xmath31 that have been chosen . we continue with the results for the @xmath25 channel in the @xmath100 scheme . the corresponding plots are given in figures 3 and 4 . in this case the values of @xmath31 have been chosen to be @xmath185 and @xmath186 to correspond to those in @xcite . note that @xmath31 need not be the same in the @xmath24 and @xmath25 reactions because the convergence properties of the qcd perturbation series could be different in these channels and moreover depend on the factorization scheme . the first and second order corrections in the @xmath25 channel in the @xmath100 scheme are larger than the respective ones in the @xmath24 channel in the dis scheme . in fact , for the range of @xmath19 values shown the second - order approximate correction is larger than the first - order approximation . hence , the relative difference in magnitude between the improved @xmath183 and the exact @xmath1 results is significantly larger than that in the @xmath24 channel in the dis scheme . we finish our discussion of the differential @xmath19 distributions with the results of adding the @xmath24 and @xmath25 channels . the plots appear in figures 5 and 6 . it is evident that resummation produces an enhancement of the exact @xmath1 result , with very little change in shape . now we turn to a discussion of the differential @xmath150 distributions at @xmath154 tev for a top quark mass @xmath172 gev@xmath15 . in this case we set the factorization mass scale equal to @xmath29 everywhere . we begin with the results for the @xmath24 channel in the dis scheme . in fig . 7 we show the born term @xmath187 , the first order exact result @xmath188 , the first order approximation @xmath165 , the second order approximation @xmath189 , and the resummed result @xmath167 for @xmath180 and @xmath181 . in fig . 8 we show the exact @xmath1 result @xmath190 , and , for comparison , @xmath191 for @xmath180 and @xmath181 . again , the resummed differential cross sections were calculated with the cut @xmath184 while no such cut was imposed on the phase space for the individual terms in the perturbation series . it is also evident from fig . 8 that the resummation of the soft gluon contributions to the @xmath150 distribution modifies the exact @xmath1 result only slightly for the values of @xmath31 that have been chosen . we continue with the results for the @xmath25 channel in the @xmath100 scheme . the corresponding plots are given in figures 9 and 10 . here , the values of @xmath31 are @xmath185 and @xmath186 . as in the case of the @xmath19 distributions , the first and second order corrections in this channel are larger than the respective ones in the @xmath24 channel in the dis scheme . for the range of @xmath150 values shown the second - order approximate correction is larger than the first - order approximation . again , as in the @xmath19 distributions , the relative difference in magnitude between the improved @xmath191 and the exact @xmath1 results is significantly larger than that in the @xmath24 channel in the dis scheme .    finally , we conclude our discussion of the differential @xmath150 distributions by showing the results of adding the @xmath24 and @xmath25 channels . the plots appear in figures 11 and 12 . again , it is evident that resummation produces a non - negligible modification of the exact @xmath1 result . however , the shape of the distribution is unchanged . we have shown that the resummation of soft gluon radiation produces a small difference between the perturbation improved distributions in @xmath19 and @xmath150 and the exact @xmath1 distributions in @xmath19 and @xmath150 for the @xmath24 reaction in the dis scheme for the values of @xmath31 chosen . however , for the @xmath25 channel in the @xmath100 scheme the resummation produces a large difference . the difference between the perturbation improved and the exact @xmath1 distributions depends on the mass factorization scheme ( dis or @xmath100 ) , the factorization scale @xmath28 , as well as the specific reaction under consideration ( @xmath24 or @xmath25 ) . for a mass @xmath155 gev@xmath15 the @xmath25 channel is not as important numerically as the @xmath24 channel . however , since the corrections for the @xmath25 channel are quite large , resummation produces a non - negligible difference between the perturbation improved and the exact @xmath1 distributions when adding the two channels . however , the shapes of the distributions are essentially unchanged .          99 ( cdf collaboration ) f. abe et al . d * 50 * , 2966 ( 1994 ) . 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the inclusive transverse momentum and rapidity distributions for top quark production at the fermilab tevatron are presented both in order @xmath0 in qcd and using the resummation of the leading soft gluon corrections in all orders of qcd perturbation theory . 
 the resummed results are uniformly larger than the @xmath1 results for both distributions .