document: the theoretical description of hadron production at large transverse momentum ( @xmath2 ) in either hadronic or nuclear collisions at high energies is traditionally framed in a two - step process that involves first a hard scattering of partons , followed by the fragmentation of the scattered parton to the detected hadron @xcite . 
   the first part is calculable in perturbation qcd , while the second part makes use of fragmentation functions that are determined phenomenologically . 
   such a production mechanism has recently been found to be inadequate for the production of particles at intermediate @xmath2 in heavy - ion collisions @xcite . instead of fragmentation 
   it is the recombination of partons that is shown to be the more appropriate hadronization process , especially when the soft partons are involved . 
   although at extremely high @xmath2 fragmentation is still dominant , it is desirable to have a universal description that can be applied to any @xmath2 , based on the same hadronization scheme . to achieve that goal 
   it is necessary that the fragmentation process can be treated as the result of recombination of shower partons in a jet . 
   the purpose of this paper is to take that first step , namely : to introduce the notion of shower partons and to determine their distributions in order to represent the phenomenological fragmentation functions in terms of recombination . 
   the subject matter of this work is primarily of interest only to high - energy nuclear collisions because hadronization in such processes is always in the environment of soft partons . 
   semi - hard shower partons initiated by a hard parton can either recombine among themselves or recombine with soft partons in the environment . in the former case the fragmentation function is reproduced , and nothing new is achieved . 
   it is in the latter case that a very new component emerges in heavy - ion collisions , one that has escaped theoretical attention thus far . 
   it should be an important hadronization process in the intermediate @xmath2 region . 
   our main objective here is to quantify the properties of shower partons and to illustrate the importance of their recombination with thermal partons . 
   the actual application of the shower parton distributions ( spd ) developed here to heavy - ion collisions will be considered elsewhere @xcite . 
   the concept of shower partrons is not new , since attempts have been made to generate such partons in pqcd processes as far as is permitted by the validity of the procedure . 
   two notable examples of such attempts are the work of marchesini and webber @xcite and geiger @xcite . however , since pqcd can not be used down to the hadronization scale , the branching or cascading processes terminate at the formation of color - singlet pre - hadronic clusters , which can not easily be related to our shower partons and their hadronization . 
   we shall discuss in more detail at the end of secs . 
   iii and iv the similarities and differences in the various approaches . 
   the fragmentation of a parton to a hadron is not a process that can be calculated in pqcd , although the @xmath1 evolution of the fragmentation function ( ff ) is calculable . 
   the ff s are usually parameterized by fitting the data from @xmath3 annihilations @xcite as well as from @xmath4 and @xmath5 collisions @xcite . 
   although the qcd processes of generating a parton shower by gluon radiation and pair creation can not be tracked by perturbative methods down to low virtuality , we can determine the spd s phenomenologically in much the same way that the ff s themselves are , except that we fit the ff s , whereas the ff s are determined by fitting the data . 
   an important difference is that both the shower partons and their distributions are defined in the context of the recombination model , which is the key link between the shower partons ( inside the black box called ff ) and the observed hadron ( outside the black box ) .    in the recombination model 
   the generic formula for a hadronization process is @xcite @xmath6 where @xmath7 is the joint distribution of a quark @xmath8 at momentum fraction @xmath9 and an antiquark @xmath10 at @xmath11 , and @xmath12 is the recombination function ( rf ) for the formation of a meson at @xmath13 . 
   we have written the lhs of eq . 
   ( [ 1 ] ) as @xmath14 , the invariant ff , but the rhs would have the same form if the equation were written for the inclusive distribution , @xmath15 , of a meson produced in a collisional process . in the former case of fragmentation , @xmath16 refers to the shower partons initiated by a hard parton . in the latter case of inclusive production 
   , @xmath16 refers to the @xmath8 and @xmath10 that are produced by the collision and are to recombine in forming the detected meson . 
   the equations for the two cases are similar because the physics of recombination is the same . in either case 
   the major task is in the determination of the distribution @xmath16 . 
   we now focus on the fragmentation problem and regard eq . 
   ( [ 1 ] ) as the basis of the recombination model for fragmentation . 
   the lhs is the ff , known from the parameterization that fits the data . 
   the rhs has the rf that is known from previous studies of the recombination model @xcite and will be specified in the next section . 
   thus it is possible to determine the properties of @xmath16 from eq.([1 ] ) . to facilitate that determination 
   we shall assume that @xmath16 is factorizable except for kinematic constraints , i.e. , in schematic form we write it as @xmath17 where @xmath18 denotes the distribution of shower parton @xmath8 with momentum fraction @xmath9 in a shower initiated by a hard parton @xmath19 . 
   the exact form with proper kinematic constraints will be described in detail in the next section . here 
   we remark on the general implications of eqs . 
   ( [ 1 ] ) and ( [ 2 ] ) . 
   the important point to emphasize is that we are introducing the notion of shower partons and their momentum distributions @xmath20 . 
   the significance of the spd is not to be found in problems that involve only the collisions of leptons and hadrons , for which the fragmentation of partons is known to be an adequate approach , and the recombination of shower partons merely reproduces what is already known . 
   the knowledge about the spd becomes crucial when the shower partons recombine with other partons that are not in the jet but are in the ambient environment . 
   we shall illustrate this important point later . 
   it should be recognized that the spd that we shall determine through the use of eqs . 
   ( [ 1 ] ) and ( [ 2 ] ) depends on the specific form of @xmath12 , which in turn depends on the wave function of the meson produced . 
   it would be inconsistent to use our @xmath21 given below in conjunction with some approximation of the rf that differs significantly from our @xmath22 . 
   the recombination of two shower partons must recover the ff from which the spd s are obtained . 
   finally , we remark that @xmath21 should in principle depend on @xmath1 at which the @xmath23 is used for its determination , since @xmath1 evolution affects both . 
   it is outside the scope of this paper to treat the @xmath1 dependence of @xmath21 . 
   our aim here is to show how @xmath21 can be determined phenomenologically , and how it can be applied , when @xmath1 is fixed . 
   the same method can be used to determine @xmath21 at other values of @xmath1 . in practice , 
   the @xmath1 dependence of @xmath21 is not as important as the inclusion of the role of the shower partons in the first place at any reasonably approximate @xmath1 in heavy - ion collisions where hard partons are produced in a range of transverse momentum . 
   in order to solve eqs . 
   ( [ 1 ] ) and ( [ 2 ] ) for @xmath21 , we first point out that there are various @xmath24 functions corresponding to various fragmentation processes 
   . we shall select five of them , from which we can determine five spd s . 
   three of them form a closed set that involves no strange quarks or mesons . 
   let us start with those three . 
   consider the light quarks @xmath25 , @xmath26 , @xmath27 , @xmath28 , and gluon @xmath29 . 
   they can all fragment into pions . to reduce them to three essential ff s 
   , we consider the three basic types @xmath30 , @xmath31 and @xmath32 , that correspond to valence , sea and gluon fragmentation , respectively . 
   if the fragmenting quark has the same flavor as that of a valence quark in @xmath33 , then the valence part of the fragmentation is described by @xmath30 , e.g. , @xmath34 , @xmath35 , @xmath36 , the subscript @xmath37 referring to the valence component . 
   all other cases of quark fragmentation are described by @xmath31 , e.g. , @xmath38 , @xmath39 , @xmath40 . 
   if the initiating parton is a gluon , then we have @xmath32 for any state of @xmath33 . 
   those ff s are given by ref.@xcite in parametric form . 
   we shall use them even though they are older than the more recent ones @xcite , which do not give the @xmath30 and @xmath31 explicitly . 
   our emphasis here is not on accuracy , but on the feasibility of extracting the spd s from the ff s of the type discussed above . 
   we shall determine @xmath21 from the bkk parameterization @xcite with @xmath1 fixed at 100 gev@xmath41 and demonstrate that the use of shower partons is important in heavy - ion collisions . 
   for the spd s we use the notation @xmath42 and @xmath43 for valence and sea - quark distributions , respectively , in a shower initiated by a quark or antiquark , and @xmath44 for any light quark distribution in a gluon - initiated shower . that is , for example , @xmath42 @xmath45 , @xmath46 , @xmath47 . it should be recognized that @xmath43 also describes the sea quarks of the same flavor , such as @xmath48 , so that the overall distribution of shower quark that has the same flavor as the initiating quark ( e.g. @xmath49 ) is given by @xmath50    it is evident from the above discussion that there is a closed relationship that is independent of other unknowns . 
   it follows from eq . 
   ( [ 1 ] ) when restricted to sea - quark fragmentation : @xmath51 the sea - spd @xmath52 can be determined from this equation alone . in eq.([4 ] ) we have exhibited the argument of the second @xmath43 function that reflects the momentum constraint , i.e. , if one shower parton has momentum fraction @xmath9 , then the momentum fraction of the other recombining shower parton can not exceed @xmath53 , and can only be a fraction of the balance @xmath54 . 
   symmetrization of @xmath9 and @xmath11 is automatic by virtue of the invariance of @xmath55 under the exchange of @xmath9 and @xmath11 . 
   after @xmath52 is determined from eq . 
   ( [ 4 ] ) , we next can obtain @xmath42 from @xmath56 where the curly brackets define the symmetrization of the leading parton momentum    @xmath57 \quad . \label{6}\end{aligned}\ ] ]    finally , we have the closed equation for the gluon - initiated shower @xmath58 in this non - strange sector we have 3 spd s ( @xmath43 , @xmath42 and @xmath44 ) to be determined from the 3 phenomenological ff s ( @xmath31 , @xmath30 and @xmath59 ) .    in extending the consideration to the strange sector , we must make use of @xmath43 and @xmath44 determined in the above set and two new ff s @xmath60 and @xmath61 that describe the fragmentation of a non - strange quark and gluon , respectively , to a kaon . 
   that is , we have @xmath62 @xmath63 where @xmath64 and @xmath65 are two additional spd s specifying the strange quark distributions in showers initiated by non - strange and gluon partons , respectively . 
   @xmath66 is the rf for kaon .    to complete the description of the integral equations , 
   we now specify the rf s . 
   they depend on the square of the wave functions of the mesons , @xmath33 and @xmath67 , whose structures in momentum space have been quantified in the valon model @xcite . unlike the case of the proton , whose structure is well studied by deep inelastic scattering so that the valon distribution can be obtained from the parton distribution functions @xcite , the rf for the pion relies on the parton distribution of the pion probed by drell - yan process @xcite . 
   the derivation of the rf s for both @xmath33 and @xmath67 is given in @xcite ; they are @xmath68 @xmath69 the @xmath70 functions guarantee the momentum conservation of the recombining quarks and antiquarks , which are dressed and become the valons of the produced hadrons . 
   since the recombination process involves the quarks and antiquarks , one may question the fate of the gluons . this problem has been treated in the formulation of the recombination model @xcite , where gluons are converted to quark - antiquark pairs in the sea before hadronization . 
   that is , the sea is saturated by the conversion to carry all the momentum , save the valence parton . 
   such a procedure has been shown to give the correct normalization of the inclusive cross section of hadronic collisions @xcite . in the present problem of parton fragmentation 
   we implement the recombination process in the same framework , although gluon conversion is done only implicitly . what is explicit is that the gluon degree of freedom is not included in the list of shower partons . 
   it means that in the equations for @xmath30 , @xmath31 , and @xmath32 ( and likewise in the strange sector ) only @xmath42 , @xmath43 and @xmath44 appear ; they are the spd s of quarks and antiquarks that are to recombine . 
   those quarks and antiquarks must include the converted sea , since they are responsible for reproducing the ff s through eqs . 
   ( [ 4 ] ) , ( [ 5 ] ) and ( [ 7 ] ) without gluons . 
   thus the shower partons whose momentum distributions we calculate are defined by those equations that have no gluon component for recombination , and would not be the same as what one would conceptually get ( if it were possible ) in a pqcd calculation that inevitably has both quarks and gluons . 
   it should be noted that our procedure of converting gluons to @xmath71 pairs is essentially the same as what is done in @xcite , whose branching processes terminate at the threshold of the non - perturbative regime . in that approach 
   nearby quarks and antiquarks that are the products of the conversion from different gluons form color - singlet clusters of various invariant masses that subsequently decay ( or fragment as in strings ) sequentially through resonances to the lowest lying hadron states @xcite . 
   similar but not identical approach is taken in @xcite , where gluons are not directly converted to @xmath71 pairs , but are either absorbed or annihilated by @xmath72 born - diagram processes . 
   we now proceed to solve the integral equations for the five ff s , which are known from ref . 
   those equations relate them to the five unknown spd s : @xmath42 , @xmath43 , @xmath44 , @xmath64 and @xmath65 . 
   if those equations were algebraic , we obviously could solve them for the unknowns . 
   being integral equations , they can nevertheless be `` solved '' by a fitting procedure that should not be regarded as being unsatisfactory for lack of mathematical rigor , since the ff s themselves are obtained by fitting the experimental data in some similar manner . 
   indeed , the ff s in the next - to - leading order are given in parameterized forms @xcite @xmath73 where the parameters for @xmath74 gev@xmath41 are given in table i for @xmath75 and @xmath76 . 
   .parameters in eq . 
   ( [ 12 ] ) for @xmath77gev@xmath41 . [ cols="<,^,^,^,^",options="header " , ]         it is evident from fig . 
   1 that all the fits are very good , except in the low @xmath13 region of @xmath78 . in the latter case 
   we are constrained by the condition @xmath79 that is imposed by the requirement that there can be only one valence quark in the shower partons . 
   however , the fit for @xmath80 is excellent , and that is the important region for the determination of @xmath81 . in application to @xmath82 , say , the @xmath25 quark in the shower must have both valence and sea quarks so the shower distribution for the @xmath25 quark is always the sum : @xmath83 . since @xmath52 is large at small @xmath84 , and is accurately determined , the net result for @xmath85 should be quite satisfactory . 
   it is remarkable how well the ff s in fig . 
   1 are reproduced in the recombination model . 
   the corresponding spd s that make possible the good fit are shown in fig . 
   they have very reasonable properties , namely : ( a ) valence quark is harder ( b ) sea quarks are softer , ( c ) gluon jet has higher density of shower partons , and ( d ) the density of produced @xmath86 quarks is lower than that of the light quarks .     for valence+sea quark ( solid line ) , sea quark ( dash - dot line ) and thermal partons ( dashed line).,scaledwidth=45.0% ]    it is appropriate at this point to relate our approach to those of marchesini - webber @xcite and geiger @xcite , which are serious attempts to incorporate the qcd dynamics in their description of the branching and collision processes . 
   the former is done in the momentum space only , whereas the latter is formulated in space - time as well as in momentum space . 
   the parton cascade model of geiger is a very ambitious program that treats a large variety of processes ranging from @xmath3 annihilation @xcite to deep inelastic scattering @xcite to hadronic and nuclear collisions @xcite . 
   the evolution of partons is tracked by use of relativistic transport equations with gain and loss terms . 
   cluster formation takes into account the invariant distance between near - neighbor partons . 
   cluster decay makes use of the hagedorn spectrum and the particle data table . because of the complexity of the problems both qcd models are implemented by monte carlo codes 
   the predictive power of the models is exhibited as numerical outputs that can not easily be adapted for comparison with our results on the spd s . 
   our approach makes no attempt to treat the qcd dynamics ; however , the spd s obtained are guaranteed to reproduce the ff s on the one hand , and are conveniently parameterized for use in other context that goes beyond fragmentation , as we shall show in the next section . from the way the color - singlet clusters are treated in the qcd models , it is clear that our shower partons do not correspond to the partons of those models at the end of their evolution processes , except in the special case when the cluster consists of only one particle . in our approach the non - perturbative part of how the shower partons dress themselves and recombine to form hadrons with the proper momentum - fraction distributions 
   is contained in the rf s . 
   such shower partons that are ready to hadronize are sufficiently far from other shower partons as to be independent from them . in general , they can not be identified with the @xmath8 and @xmath87 that form the color - singlet clusters in the qcd models , but are more closely related to the constituents of the final hadrons , as in the case of quarkonium formation @xcite . 
   the distribution of those constituents in a hard - parton shower can not be displayed in the qcd models , but are determined by us by solving eqs . 
   ( [ 4 ] ) and ( [ 9 ] ) 
   as we have stated in the introduction , the purpose of determining the spd s is for their application to problems where the ff s are insufficient to describe the physics involved . 
   we consider in this section two such problems as illustrations of the usefulness of the spd s . 
   the first is when a hard parton is produced in the environment of thermal partons , as in heavy - ion collisions . 
   the second is the determination of two - pion distribution in a jet . 
   let us suppose that a @xmath25 quark is produced at @xmath88 gev / c in a background of thermal partons whose invariant @xmath89 distribution is @xmath90 let the parameters @xmath91 and @xmath92 be chosen to correspond to a typical situation in au+au collisions at @xmath93 gev @xcite @xmath94 the high-@xmath89 @xmath25 quark generates a shower of partons with various flavors . consider specifically @xmath25 and @xmath28 in that shower . 
   the valence quark distribution is given by @xmath95 , while the @xmath28 sea - quark distribution ( including the ones converted from the gluons ) is given by @xmath96 . in fig . 
   3 we plot @xmath97 for ( a ) @xmath25 quark ( valence and sea ) in solid line , ( b ) @xmath28 sea antiquark in dash - dot line , and ( c ) @xmath28 thermal antiquark in dashed line . 
   they correspond to @xmath98 ( invariant distributions @xmath99 , @xmath43 , and @xmath100 , respectively ) , in which @xmath101 and @xmath96 are evaluated at @xmath102 , with @xmath103  gev / c . 
   note that the thermal distribution is higher than the shower parton distributions for @xmath104 gev / c . 
   that makes a crucial difference in the recombination of those partons . 
   such a thermal distribution is absent in @xmath105 collisions , whose soft partons are at least two orders of magnitude lower . in @xmath3 annihilation 
   there are , of course , no soft partons at all .     in @xmath2 arising from thermal - shower recombination ( solid line ) and shower - shower recombination , i.e. fragmentation ( dash - dot line).,scaledwidth=45.0% ] 
   we now calculate the production of @xmath106 from the assemblage of @xmath25 and @xmath28 partons . 
   the thermal - shower ( @xmath107 ) recombination gives rise to @xmath108 where eq . 
   ( [ 10 ] ) has been used in an equation such as eq.([1 ] ) for @xmath109 , but expressed for @xmath110 . using eqs . 
   ( [ 3 ] ) , ( [ 15 ] ) and the parametrizations given in table ii , the integral in eq . 
   ( [ 17 ] ) can readily be evaluated . 
   the result is shown by the solid line in fig . 
   4 . it is to be compared with the @xmath2 distribution from the fragmentation of the @xmath25 quark to @xmath106 , which is @xmath111 since this is just retracing the path in which we obtained @xmath67 and @xmath43 from the @xmath112 function in the first place , eq . 
   ( [ 18 ] ) can more directly be identified with @xmath113 . 
   \label{19}\end{aligned}\ ] ] the result is shown by the dash - dot line in fig . 
   evidently , the contribution from the thermal - shower recombination is much more important than that from fragmentation in the range of @xmath2 shown . despite the fact that @xmath114 is lower than @xmath115 for @xmath116 gev / c , 
   its dominance at @xmath117 gev / c is enough to result in the @xmath118 recombination to dominate over the @xmath119 recombination for all @xmath120 gev / c . 
   this example demonstrates the necessity of knowing the spd s in a jet , since @xmath101 is used in eq . 
   ( [ 17 ] ) . 
   if @xmath121 recombination is the only important contribution as in @xmath105 collisions , then fragmentation as in eq . 
   ( [ 19 ] ) is all that is needed , and the search for spd s plays no crucial role . in realistic problems 
   the hard - parton momentum @xmath122 has to be integrated over the weight of the jet cross section . 
   however , for our illustrative purpose here , that is beside the point . 
   our next example is the study of the dihadron distribution in a jet . 
   we need only carry out the investigation here for a jet in vacuum , since the replacement of a shower parton by a thermal parton for a jet in a medium is trivial , having seen how that is done in the replacement of eq . 
   ( [ 18 ] ) by ( [ 17 ] ) in the case of the single - particle distribution . 
   consider the joint distribution of two @xmath106 in a jet initiated by a hard @xmath25 quark . 
   as we shall work in the momentum fraction variables , the value of the momentum of the initiating @xmath25 quark is irrelevant , except that it should be high . 
   let @xmath123 and @xmath124 denote the momentum fractions of the two @xmath106 , and @xmath125 denotes that of the @xmath19th parton , @xmath126 . then , since only one @xmath25 quark can be valence , the other three quarks being in the sea , we have one @xmath67 , three @xmath43 , and two @xmath22 functions . 
   combinatorial complications arise when we impose the condition that @xmath127 for @xmath128 . 
   there are two methods to keep the accounting of the different orderings of the four @xmath125 . 
   _ method 1 . 
   _    let one ordering be @xmath129 there are 4 ! 
   ways to rearrange the four @xmath125 in all orders . 
   however , they are to be convoluted with @xmath130 , which is symmetric in @xmath131 , and similarly with @xmath132 . thus there are @xmath133 independent terms . since @xmath67 can appear at any one of the four positions in eq . 
   ( [ 20 ] ) , we have altogether 24 terms . thus we have @xmath134 r_{\pi}(x_1,x_2,x_1 ) r_{\pi}(x_3,x_4,x_2 ) , \label{21}\end{aligned}\ ] ] where @xmath135 symbolizes the permutation of all @xmath125 and summing over all four positions of @xmath67 , but eliminating redundant terms that are symmetric under the interchanges of @xmath131 and @xmath136 . 
   _ method 2 . 
   _    let us fix the ordering in eq . 
   ( [ 20 ] ) but permute the contributing @xmath125 to @xmath123 and @xmath124 . 
   there are six arrangements of @xmath125 and @xmath137 in @xmath138 , while counting in @xmath139 and @xmath140 is unnecessary 
   . let us denote the summation over them by @xmath141 . 
   thus we have @xmath142 \left [ { 1 \over 6}\sum_q r_{\pi}(x_i , x_j , x_1 ) r_{\pi}(x_{i'},x_{j'},x_2)\right ] \label{22}\end{aligned}\ ] ] where @xmath143 denotes summing over the four positions of @xmath67 . equation ( [ 22 ] ) is equivalent to ( [ 21 ] ) . 
   it should be noted that not all terms in these equations can be expressed in the form factorizable ff s . 
   one example that can is @xmath144 r_{\pi}(x_1,x_2,x_1)\nonumber\\ & & \times l\left ( { x_3 \over 1-x_1-x_2 } \right)l\left ( { x_4 \over 1-x_1-x_2-x_3 } \right)r_{\pi}(x_3,x_4,x_2)\nonumber\\ & & = d^{\pi^+}_u(x_1 ) d^{\pi^+}_s\left(x_2/(1-x_1)\right ) .\label{23}\end{aligned}\ ] ]    because of the presence of terms that can not be written in factorizable form , the two - particle distribution can not be adequately represented by the ff s only .     correlated distribution in a @xmath25-quark initiated jet.,scaledwidth=45.0% ]    using the spd s obtained in the previous section , we get the results shown in fig 
   .  5 , which exhibits the @xmath124 distribution for four fixed values of @xmath123 . this type of correlation in parton fragmentation has never been calculated before . 
   although the shapes of the @xmath124 distributions look similar in the log scale in fig .  5 , 
   there is significant attenuation as @xmath145 for each value of @xmath123 . thus the effective slope becomes steeper for larger @xmath123 . 
   recent experiments at rhic have begun to measure the distribution of particles associated with triggers restricted to a small interval . 
   the extension of our calculation here to such problems will need the input of jet cross sections for all hard partons in heavy - ion collisions and the participation of thermal partons in the recombination . 
   here we only demonstrate the utility of the spd s in the study of dihadron correlation . 
   we have described the fragmentation process in the framework of recombination . 
   the shower parton distributions obtained are shown to be useful in problems where the knowledge of the fragmentation functions alone is not sufficient to provide answers to questions concerning the interaction between a jet and its surrounding medium or between particles within a jet . 
   such questions arise mainly in nuclear collisions at high energies .    in our view 
   the basic hadronization process is recombination , even for fragmentation in vacuum . 
   since the recombination process can only be formulated in the framework of a model , the shower parton distributions obtained are indeed model dependent . 
   that is a price that must be paid for the study of hadrons produced at intermediate @xmath2 where the interaction between soft and semi - hard partons can not be ignored , and where perturbative qcd is not reliable . 
   once recombination is adopted for treating hadronization in that @xmath2 range , the extension to higher @xmath2 can remain in the recombination framework , since the fragmentation process is recovered by the recombination of two shower partons . for hadron production in heavy - ion collisions at super high energies , such as at lhc 
   , then the high density of hard partons produced will require the consideration of recombination of hard partons from overlapping jets . 
   thus it is sensible to remain in the recombination mode for all @xmath2 . 
   we have shown in this paper how the spd s can be determined from the ff s . 
   although we have determined the spd s at only one value of @xmath1 for the ff s , it is clear that the same procedure can be followed for other value of @xmath1 . 
   the formal description of how the @xmath1 dependences of the ff s can be transferred to the @xmath1 dependences of the spd s is a problem that is worth dedicated attention . 
   while the numerical accuracy of the spd s obtained here can still be improved , especially at lower @xmath1 , for the purpose of phenomenological applications the availability of the parametrizations given in table ii is far more important than not taking into account at all the shower partons and their interactions with the medium in the environment . 
   the @xmath1 evolution of the spd s may have to undergo a long process of investigatory evolution of its own just as what has happened to the ff s . that can proceed in parallel to the rich phenomenology that can now be pursued in the application of the role of shower partons to heavy - ion collisions . 
   we are grateful to s.  kretzer for a helpful communication . 
   this work was supported , in part , by the u.  s.  department of energy under grant no . 
   de - fg03 - 96er40972 and by the ministry of education of china under grant no .
summary: we develop the notion of shower partons and determine their distributions in jets in the framework of the recombination model . 
   the shower parton distributions obtained render a good fit of the fragmentation functions . 
   we then illustrate the usefulness of the distributions in a problem where a jet is produced in the environment of thermal partons as in heavy - ion collisions . 
   the recombination of shower and thermal partons is shown to be more important than fragmentation . 
   application to the study of two - particle correlation in a jet is also carried out . 
   @xmath0 2@xmath1