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over the past few years rapidity gaps , i.e. pseudorapidity regions without hadronic activity , have been observed in hadronic collisions at both the hera @xmath7 collider  @xcite and in @xmath8 collisions at the fermilab tevatron  @xcite . such rapidity gaps are widely attributed to the exchange of color singlet quanta between incident partons  @xcite , the exchange of two gluons in a color singlet state being the simplest such model  @xcite . at the tevatron , a fraction @xmath9 of all dijet events with jet transverse energies @xmath10  gev and jet separations of more than three units of pseudorapidity exhibit rapidity gaps between the jets . this observation is particularly striking since it demonstrates that color singlet exchange effects in qcd events are relevant at momentum transfers of order 1,000  gev@xmath6 , raising the hope that perturbative methods can be used for quantitative descriptions . a gap fraction of order one percent was in fact predicted by bjorken  @xcite , in terms of a fraction @xmath11 of dijet events which are due to @xmath0-channel color - singlet exchange and a survival probability @xmath12 of rapidity gaps of order 10%  @xcite , [ eq : ps ] f_gap = f_s p_s . here the survival probability estimates the fraction of hard dijet events without an underlying event , i.e. without soft interactions between the other partons in the scattering hadrons . such multiple interactions would fill the rapidity gap produced in the hard scattering process . for @xmath13 elastic scattering , bjorken estimated the color - singlet fraction @xmath14 in terms of the imaginary part of the two - gluon @xmath0-channel exchange amplitude , which is known to dominate the forward scattering amplitude for @xmath0-channel color - singlet exchange . in impact parameter space , at impact parameters small compared to @xmath15 , the result is f_s^impact & = & 29    & & 12 |433 - 2n_f|^2 = 0.15 . [ eq : fbjorken ] here 2/9 is the relative color factor of the two - gluon color - singlet to the one - gluon color - octet exchange cross section and @xmath16 is an infrared cutoff parameter which regularizes the two - gluon loop - integral . this model for the calulation of the color singlet fraction @xmath14 , with the two gluon - exchange amplitude replaced by its imaginary part , will be called the two - gluon exchange model in the following .    in this model , the color singlet fraction grows with the color charge of the scattered partons . for @xmath2 and @xmath3 elastic scattering @xmath14 would be larger by factors @xmath17 and @xmath18 , respectively  @xcite . this results in a substantial decrease of the observable gap fraction as the contribution from gluon induced dijet events is reduced , _ e.g. _ by increasing the average transverse momentum of the observed jets and thereby the feynman-@xmath19 values of the incident partons . such measurements have recently been reported by both the cdf  @xcite and the d0  @xcite collaborations , and no such effect is observed . in fact , the d0 data are compatible with a slight increase of the gap fraction with increasing jet @xmath20 , casting doubt on the validity of the two - gluon exchange model  @xcite .    in this paper we reconsider the basic ideas behind the two - gluon exchange model . we demonstrate its limitations and show that , even when starting from this perturbative picture of rapidity gap formation , the determination of the color singlet exchange fraction @xmath14 is essentially nonperturbative . we start from a basic feature of the two - gluon exchange model : unitarity fixes the imaginary part of the @xmath0-channel two - gluon exchange amplitude in terms of the born amplitude and this imaginary part dominates @xmath0-channel color singlet exchange  @xcite . rewriting this relationship in terms of phase shifts , the one- and two - gluon exchange amplitudes are found to be too large to be compatible with unitarity . phase shift unitarization leads to a more realistic description , in which the total differential cross section remains unchanged compared to the born result , but with @xmath0-channel color singlet exchange fractions which differ substantially from the expectations of the two - gluon exchange model . these features are demonstrated analytically for fixed values of the strong coupling constant , @xmath21 , in section  [ sec2 ] . in section  [ sec3 ] we then perform a numerical analysis for running @xmath21 , showing that the key properties of the fixed-@xmath21 results remain unchanged . the predicted color singlet fractions are found to very strongly depend on the regularization of gluon exchange at small momentum transfer , however , and thus can not be reliably calculated within perturbation theory . within our unitarized model the non - perturbative effects can be summarized in terms of two parameters , the survival probability of gaps , @xmath22 , and a universal coulomb phase shift , @xmath23 . implications for the formation of gaps at the tevatron are analyzed in section  [ sec4 ] . in particular we calculate how the gap fraction between two hard jets varies with jet transverse energies and jet pseudorapidity separation and then compare predicted fractions with tevatron data  @xcite . our conclusions are given in section  [ sec5 ] . consider the elastic scattering of two arbitrary partons , @xmath24 and @xmath25 , p(i_1)+p(j_1)p(i_2)+p(j_2 ) , at momentum transfer @xmath26 . here @xmath27 denote the colors of the initial and final state partons . the cross section and the partial wave amplitudes are completely dominated by the forward region , @xmath28 , where the rutherford scattering amplitude , [ eq : mborn ] m = -8_s t^at^a = 8_s f_c = m_0f_c , provides an excellent approximation . note that helicity is conserved in forward scattering , hence spin need not be considered in the following . the only process dependence arises from the color factor @xmath29 .      in order to study unitarity constraints , we need to diagonalize the amplitude in both momentum / coordinate space and in color space . the first step is most easily achieved by transforming to impact parameter space , [ eq : fourier ] t(*b * ) = ( * q * ) e^-i*q * . neglecting multi - parton production processes , _ i.e. _ inelastic channels , unitarity of the @xmath30-matrix implies the relation [ eq : unitarity1 ] t(*b * ) =    @xmath31 . ( [ eq : unitarity1 ] ) represents a matrix relation in color space . more fully it can be written as t(*b*)_i_2 j_2 , i_1 j_1 = _ i , j t(*b*)_i_2 j_2 , i j t^(*b*)_i j , i_1 j_1 , [ eq : unitarity2 ] where the sum runs over the dimension of the color space , @xmath32 for @xmath33 and @xmath13 scattering and @xmath34 ( 64 ) for @xmath2 ( @xmath3 ) elastic scattering .    since the color factors can be written as hermitian matrices , the right - hand side of eq . ( [ eq : unitarity2 ] ) represents a simple matrix product of the color matrices . this product is easily diagonalized by decomposing the color factors @xmath35 into a linear combination of projection operators onto the irreducible color representations which are accessible in the @xmath36-channel , [ eq : col.factor ] f_c = ( f_c)_i_2j_2,i_1j_1 = _ k f_k ( p_k)_i_2j_2,i_1j_1 = _ k f_k p_k . for the case of quark - antiquark elastic scattering , for example , with color decomposition @xmath37 , the color factor can be written in terms of gell - mann matrices as ( f_c)_i_2j_2,i_1j_1 = ( ^a2 ) _ i_2i_1 ( ^a2 ) _ j_1j_2 = 49_j_1i_1_i_2j_2- 13(^a2 ) _ j_1i_1 ( ^a2 ) _ i_2j_2 = 43p_1 - 16 p_8 . for all cases , @xmath13 , @xmath33 , @xmath2 and @xmath3 elastic scattering , the decomposition into @xmath36-channel projectors is summarized in table  [ tab : colorop ] . this color decomposition , combined with the transformation to impact parameter space , diagonalizes the unitarity relation for elastic scattering amplitudes . .representations and color operators for qcd elastic scattering . the indices of the projection operators @xmath38 in the last column represent the dimensionalities of the irreducible color representations in the @xmath36-channel . results for the @xmath39 decomposition are taken from ref . @xcite . [ cols="^,<,<,^",options="header " , ] both the d0  @xcite and the cdf  @xcite collaborations at the tevatron have analyzed the fraction of dijet events with rapidity gaps , as a function of both the transverse energy , @xmath20 , and the pseudorapidity separation , @xmath40 , of the two jets . as these phase space variables change , the composition of dijet events varies , from mostly gluon initiated processes at small @xmath20 and @xmath41 ( and , hence , small feynman-@xmath19 , @xmath42 ) to @xmath13 scattering at large values . a dependence of the gap fraction on the color structure of the scattering partons would thus be reflected in a variation with @xmath20 and @xmath41 . bjorken s two gluon exchange model , which is equivalent to the small @xmath43 region in our analysis , predicts a larger fraction of color singlet exchange events for gluon initiated processes  @xcite ( see fig .  [ fig : fs.vs.psi ] for @xmath44 ) . the gap fraction should thus decrease with increasing @xmath20 or @xmath41 . the opposite behavior is expected in statistical models of color rearrangement  @xcite . here the eight color degrees of freedom for gluons , as compared to three for quarks , make it less likely for gluon initiated processes that @xmath0-channel color singlet exchange is achieved by random color rearrangement . this would lead to a smaller gap fraction at small @xmath42 and therefore small @xmath20 or @xmath41 .    in the unitarized rpwe framework , the dependence on the regularization parameters is sufficiently strong to encompass both scenarios . this is demonstrated in figs . [ fig : fgap.exp.d0 ] and [ fig : fgap.exp.cdf ] , where the results of the running coupling analysis for three choices of the regularization parameters are compared with tevatron data , taken at @xmath45  gev . the data correspond to dijet events with two opposite hemisphere jets of @xmath46  gev , @xmath47 ( cdf ) or @xmath48  gev , @xmath49 ( d0 ) . d0 data are taken from ref . @xcite and show the fraction of dijet events with rapidity gaps . cdf  @xcite shows the ratio of gap fractions in individual @xmath41 and @xmath20 bins to the overall gap fraction in the acceptance region . for comparing our calculation with the data we fix the survival probability @xmath22 in eq . ( [ eq : ps ] ) to reproduce the overall gap fraction in the acceptance region , which was measured as @xmath50 for the d0 sample and @xmath51 for the cdf sample . required survival probabilities strongly depend on regularization parameters and vary between 1.9% and 5.5% , which is on the low side of previous estimates  @xcite . for a given choice of regularization parameters , predictions for the @xmath20 or @xmath41 dependence of the gap fraction are quite similar for the cdf and d0 cuts . to the extent that the two data sets are consistent within errors , it is not yet possible to discriminate between different choices of regularization parameters , i.e. to obtain sensitivity to the non - perturbative dynamics . d0 data somewhat favor color singlet fractions which grow with @xmath42 and which are more in line with expectations from color evaporation models  @xcite . note that our unitarized gluon exchange model , with @xmath52  gev and @xmath53  gev is able to describe this trend , even though it is an extension of the two - gluon exchange model . cdf data slightly prefer a gap fraction which decreases with increasing @xmath41 and , hence , with larger @xmath42 . the unitarized gluon exchange model , with @xmath54  gev and @xmath55  gev describes such a situation . comparison with fig . [ fig : fs.reg.dep ] shows that this set of parameters predicts a much smaller gap fraction for @xmath13 scattering than for gluon initiated processes , which is qualitatively similar to the two - gluon exchange model  @xcite . indeed , the shape of the gap fraction for the two gluon exchange model is very similar to the long - dashed curves in fig .  [ fig : fgap.exp.cdf ] . clearly , the data are not yet precise enough to unambiguously distinguish between these different scenarios . on the theoretical side , the variation of the rpwe predictions with model parameters highlights the limitations of a perturbative approach to the color singlet exchange probability . taking the phase @xmath23 and the survival probability @xmath22 as free parameters , the unitarized two - gluon exchange model is clearly capable of fitting the present tevatron data , however . the formation of rapidity gaps in hadronic scattering events is a common occurrence , and its ubiquity asks for a theoretical explanation within qcd . the formation of gaps between two hard jets at the tevatron is particularly intriguing and is commonly being explained in terms of color singlet exchange in the @xmath0-channel , be it via an effective color singlet object like the `` pomeron '' or via a statistical color rearrangement , in terms of multiple soft gluon exchange . `` pomeron '' exchange models build on the observation that color singlet exchange in the @xmath0-channel can be achieved in qcd via the exchange of two gluons , with compensating colors  @xcite . when trying to build a quantitative model for the formation of rapidity gaps  @xcite , one encounters infrared divergences in the color singlet hard scattering amplitude , which in a full treatment would be regularized by the finite size and the color singlet nature of physical hadrons  @xcite . in turn , this indicates that non - perturbative information may be indispensable for a quantitative understanding of the hard color singlet exchange process . we have analyzed this question within a particular model , based on the unitarization of single gluon exchange in the @xmath0-channel . the low - nussinov model  @xcite corresponds to a truncation of the unitarization at order @xmath56 . we find that , for any reasonable range of regularization parameters , the two - gluon exchange approximation violates partial wave unitarity , and thus a fully unitarized amplitude is needed for phenomenological applications . the unitarization of hard elastic quark and gluon scattering is not unique , of course , but any acceptable method must preserve the successful description of hard dijet events by perturbative qcd . the phase shift approach used here fulfills this requirement : the unitarization does not change the born - level predictions for the color averaged differential cross sections . as a corollary , the color - inclusive dijet cross section is independent of the regularization parameters which need to be introduced for the full phase shift analysis . the situation is entirely different when considering the @xmath0-channel color singlet exchange component which is introduced by the exchange of two or more gluons or by unitarization . the @xmath0-channel color singlet exchange fraction , @xmath14 , is strongly affected by the full unitarization and deviates from the expectations of the two - gluon exchange approximation , changing even the qualitative predictions of the low - nussinov model . these strong unitarization effects are reflected by a strong dependence on the precise regularization procedure . this cutoff dependence , which parameterizes non - perturbative effects , does not allow to make quantitative predictions for the color singlet exchange fractions in particular partonic subprocesses . these limitations , which have been demonstrated here for the two - gluon exchange approximation to the pomeron , may be generic to pomeron exchange models , and should be analyzed more generally .    in spite of these limitations , we find some intriguing features of the unitarized gluon exchange amplitudes . for all partonic subprocesses and for all regularization parameters , the color singlet exchange fractions can be described in terms of a single universal phase , @xmath57 , which absorbs all non - perturbative effects . this suggests a unified phenomenological description of the rapidity gap data , via the gap survival probability @xmath22 and the phase @xmath57 . such an analysis goes beyond the transverse momentum and pseudorapidity dependence of rapidity gap fractions which have just become available , and should best be performed directly by the experimental collaborations . we would like to thank f.  halzen and j.  pumplin for useful discussions on the physics of rapidity gaps . this research was supported in part by the u.s . department of energy under grant no . de - fg02 - 95er40896 , and in part by the university of wisconsin research committee with funds granted by the wisconsin alumni research foundation . e.  berger , j.  collins , g.  sterman , and d.soper , nucl . * b286 * , 704 ( 1987 ) ; n.  nikolaev and b.  zakharov , z.  phys . * c53 * , 331 ( 1992 ) ; v.  del  duca , scientifica acta * 10 * , 91 ( 1995 ) [ hep - ph/9503226 ] ; e.  levin , proceedings of the _ gleb wataghin school on high energy phenomenology _ , campinas , brazil , july 1116 , 1994 , [ hep - ph/9503399 ] ; and references therein .    d.  zeppenfeld , in _ particles and fields _ , proceedings of the _ viiith j.  a.  swieca summer school _ , rio de janeiro , brazil , february 618 , 1995 , ed . by j.  barcelos - neto , s.  f.  novaes , and v.  o.  rivelles ( world scientific , singapore , 1996 ) , p.  78 [ hep - ph/9603315 ] .
rapidity gaps between two hard jets at the tevatron have been interpreted as being due to the exchange of two gluons which are in an overall color - singlet state . 
 we show that this simple picture involves unitarity violating amplitudes . 
 unitarizing the gluon exchange amplitude leads to qualitatively different predictions for the fraction of @xmath0-channel color singlet exchange events in forward @xmath1 , @xmath2 or @xmath3 scattering , which better fit tevatron data . 
 = cmssbx10 scaled 2 to     + r.  oeckl@xmath4 and d.  zeppenfeld@xmath5 + .5 cm @xmath5_department of physics , university of wisconsin , madison , wi 53706 , usa _ 
 + @xmath6_damtp , university of cambridge , cambridge cb3 9ew , uk _