Polarized Heavy Quarks

a r X i v :h e p -p h /0002081v 1 8 F eb 2000ADP-99-53/T389IU/NTC 00-02Lambda Polarization in Polarized Proton-Proton Collisions atRHICC.Boros,J.T.Londergan 1and A.W.ThomasDepartment of Physics and Mathematical Physics,and Special Research Center for the Subatomic Structure of Matter,University of Adelaide,Adelaide 5005,Australia 1Department of Physics and Nuclear Theory Center,Indiana University,Bloomington,IN 47408,USA (February 1,2008)Abstract We discuss Lambda polarization in semi-inclusive proton-proton collisions,with one of the protons longitudinally polarized.The hyperfine interaction responsible for the ∆-N and Σ-Λmass splittings gives rise to flavor asym-metric fragmentation functions and to sizable polarized non-strange fragmen-tation functions.We predict large positive Lambda polarization in polarized proton-proton collisions at large rapidities of the produced Lambda,while other models,based on SU (3)flavor symmetric fragmentation functions,pre-dict zero or negative Lambda polarization.The effect of Σ0and Σ∗decays is also discussed.Forthcoming experiments at RHIC will be able to differentiate between these predictions.I.INTRODUCTIONMeasurements of the polarization dependent structure function,g1,in deep inelastic scattering[1]have inspired considerable experimental and theoretical effort to understand the spin structure of baryons.While most of these studies concern the spin structure of the nucleons,it has become clear that similar measurements involving other baryons would provide helpful,complementary information[2–9].The Lambda baryon plays a special role in this respect.It is an ideal testing ground for spin studies since it has a rather simple spin structure in the naive quark parton model.Furthermore,its self-analyzing decay makes polarization measurements experimentally feasible.Forthcoming experiments at RHIC could measure the polarization of Lambda hyperons produced in proton-proton collisions with one of the protons longitudinally polarized,p↑p→Λ↑X.The polarization dependent fragmentation function of quarks and gluons into Lambda hyperons can be extracted from such experiments.These fragmentation functions contain information on how the spin of the polarized quarks and gluons is transferred to thefinal state Lambda.The advantage of proton proton collisions,as opposed to e+e−annihilation, whereΛproduction and polarization is dominated by strange quark fragmentation,is that Lambdas at large positive rapidity are mainly fragmentation products of up and down valence quarks of the polarized projectile.Thus,the important question,intimately related to our understanding of the spin structure of baryons,of whether polarized up and down quarks can transfer polarization to the Lambda can be tested at RHIC[5].In a previous publication,we have shown that the hyperfine interaction,responsible for the∆-N andΣ0-Λmass splittings leads to non-zero polarized non-strange quark fragmen-tation functions[14].These non-zero polarized up and down quark fragmentation functions give rise to sizeable positiveΛpolarization in experiments where the strange quark fragmen-tation is suppressed.On the other hand,predictions based either on the naive quark model or on SU(3)flavor symmetry predict zero or negative Lambda polarization[2].In section II,we briefly discuss fragmentation functions and show how the hyperfine interaction leads to polarized non-strange fragmentation functions.Wefix the parameters of the model byfitting the data onΛproduction in e+e−annihilation.In section III,we discussΛproduction in pp collisons at RHIC energies.We point out that the production ofΛ’s at high rapidities is dominated by the fragmentation of valence up and down quarks of the polarized projectile,and is ideally suited to test whether non-strange quarks transfer their polarization to thefinal stateΛ.We predict significant positiveΛ-polarization at large rapidities of the producedΛ.II.FRAGMENTATION FUNCTIONSFragmentation functions can be defined as light-cone Fourier transforms of matrix ele-ments of quark operators[12,13]14 n dξ−ψ(ξ−)|0 },(1)where,Γis the appropriate Dirac matrix;P and p n refer to the momentum of the produced Λand of the intermediate system n;S is the spin of the Lambda and the plus projectionsof the momenta are defined by P+≡12(P0+P3).z is the plus momentum fraction of the quark carried by the producedΛ.Translating the matrix elements,using the integral representation of the delta function and projecting out the light-cone plus and helicity±components we obtain12√2γ−γ+1√z D±¯qΛ(z)=12nδ[(1/z−1)P+−p+n]| 0|ψ†±+(0)|Λ(P S );n(p n) |2.(3)D±qΛcan be interpreted as the probability that a quark with positive/negative helicity frag-ments into aΛwith positive helicity and similarly for antiquarks.The operatorψ+(ψ†+)either destroys a quark(an antiquark)or it creates an antiquark (quark)when acting on theΛon the right hand side in the matrix elements.Thus,whereas, in the case of quark fragmentation,the intermediate state can be either an anti-diquark state,¯q¯q,or a four-quark-antiquark state,q¯q¯q¯q,in the case of antiquark fragmentation,only four-antiquark states,¯q¯q¯q¯q,are possible assuming that there are no antiquarks in theΛ. (Production ofΛ’s through coupling to higher Fock states of theΛis more complicated and involves higher number of quarks in the intermediate states.As a result it would lead to contributions at lower z values.)Thus,we have•(1a)q→qqq+¯q¯q=Λ+¯q¯q•(1b)q→qqq+q¯q¯q¯q=Λ+q¯q¯q¯q,for the quark fragmentation and•(2)¯q→qqq+¯q¯q¯q¯q=Λ+¯q¯q¯q¯q,for the antiquark fragmentation.While,in case(1a),the initial fragmenting quark is contained in the produced Lambda, in case(1b)and(2),the Lambda is mainly produced by quarks created in the fragmentation process.Therefore,we not only expect that Lambdas produced through(1a)usually have larger momenta than those produced through(1b)or(2)but also that Lambdas produced through(1a)are much more sensitive to theflavor spin quantum numbers of the fragment-ing quark than those produced through(1b)and(2).In the following we assume that(1b) and(2)lead to approximately the same fragmentation functions.In this case,the differ-ence,D qΛ−D¯qΛ,responsible for leading particle production,is given by the fragmentation functions associated with process(1a).Similar observations also follow from energy-momentum conservation built in Eqs.(2) and(3).The delta function implies that the function,D q(z)/z,peaks at[14]z max≈MHere,M and M n are the mass of the produced particle and the produced system,n,and wework in the rest frame of the produced particle.We see that the location of the maxima of the fragmentation function depends on the mass of the system n.While the high z region isdominated by the fragmentation of a quark into thefinal particle and a small mass system, large mass systems contribute to the fragmentation at lower z values.The maxima of thefragmentation functions from process(1a)are given by the mass of the intermediate diquarkstate and that of the the fragmentation functions from the processes(1b)and(2b)by the masses of intermediate four quark states.Thus,the contribution from process(1a)is harderthan those from(1b)and(2).Energy-momentum conservation also requires that the fragmentation functions are not flavor symmetric.While the assertion of isospin symmetry,D uΛ=D dΛ,is well justified,SU(3)flavor symmetry is broken not only by the strange quark mass but also by the hyperfine interaction.Let us discuss the fragmentation of a u(or d)quark and that of an s quark intoa Lambda through process(1a).While the intermediate diquark state is always a scalarin the strange quark fragmentation,it can be either a vector or a scalar diquark in the fragmentation of the non-strange quarks.The masses of the scalar and vector non-strangediquarks follow from the mass difference between the nucleon and the Delta[10],while those of the scalar and vector diquark containing a strange quark can be deduced from the massdifference betweenΣandΛ[11].They are roughly m s≈650MeV and m v≈850MeV for the scalar and vector non-strange diquarks,and m′s≈890MeV and m′v≈1010MeV for scalar and vector diquarks containing strange quarks,respectively[11,14].According toEq.(4),these numbers lead to soft up and down quark fragmentation functions and to hardstrange quark fragmentation functions.Energy-momentum conservation,together with the splitting of vector and scalar diquarkmasses,has the further important consequence that polarized non-strange quarks can trans-fer polarization to thefinal state Lambda.To see this we note that the probabilities for the intermediate state to be a scalar or vector diquark state in the fragmentation of an up or down quark with parallel or anti-parallel spin to the spin of the Lambda can be obtained from the SU(6)wave function of theΛΛ↑=13[2s↑(ud)0,0+√2u↓(ds)1,1+u↑(ds)1,0−u↑(ds)0,0].(5)While the u or d quarks with spin anti-parallel to the spin of theΛare always associated with a vector diquark,u and d quarks with parallel spin have equal probabilities to be accompanied by a vector or scalar diquark.The fragmentation functions of non-strange quarks with spin parallel to theΛspin are harder than the corresponding fragmentation functions with anti-parallel spins.Thus,∆D uΛis positive for large z values and negative for small z.Their total contribution to polarized Lambda production might be zero or very small.Nevertheless,∆D uΛand∆D dΛcan be sizable for large z values,since both D uΛand ∆D uΛare dominated by the spin-zero component in the large z limit.Furthermore,they will dominate polarized Lambda production whenever the production from strange quarks is suppressed.The matrix elements can be calculated using model wave functions at the scale relevant to the specific model and the resulting fragmentation functions can be evolved to a higher scaleto compare them to experiments.In a previous paper[14],we calculated the fragmentation functions in the MIT bag model and showed that the resulting fragmentation functionsgive a very reasonable description of the data in e+e−annihilation.Since the mass of the intermediate states containing more than two quarks are not known we only calculate thecontributions of the diquark intermediate states in the bag model.The other contributions have been determined by performing a globalfit to the e+e−data.For this,we used the simple functional formD¯qΛ(z)=N¯q zα(1−z)β(6) to parameterize D¯qΛ=D¯uΛ=D¯dΛ=...D¯bΛand also set D gΛ=0at the initial scale,µ=0.25GeV.The fragmentation functions have to be evolved to the scale of the experiment,µ.The evolution of the non-singlet fragmentation functions in LO is given by[15,16]dz′P qq(zd lnµ2 q D qΛ(z,µ2)= 1zdz′z′) q D qΛ(z,µ2)+2n f P gq(zd lnµ2D gΛ(z,µ2)= 1z dz′z′) q D qΛ(z,µ2)+P gg(zso that one has to select certain kinematic regions to suppress the unwanted contributions.In particular,in order to test whether polarized up and down quarks do fragment into polarized Lambdas the rapidity of the produced Lambda has to be large,since at high rapidity,Λ’s are mainly produced through valence up and down quarks.(We count positive rapidity in the direction of the polarized proton beam.)The difference of the cross sections to produce a Lambda with positive helicity through the scattering of a proton with positive/negative helicity on an unpolarized proton is givenin leading order perturbative QCD(LO pQCD)by1E C ∆dσd3p C(A↑B→C↑+X)−E Cdσπz2c∆dσse−y/z c,ˆu=−x b p⊥√s is the total center of mass energy.The summation in Eq.(10)runs over all possible parton-parton combinations,qq′→qq′,qg→qg,q¯q→q¯q...The elementary unpolarized and polarized cross sections can be found in Refs.[24,25].Performing the integration in Eq.(10)over z c one obtainsE C∆dσπz c∆dσ2x b e−y+x⊥2x a−x⊥e y,x amin=x⊥e y1Since the relevant spin dependent cross sections on the parton level are only known in LO we perform a LO calculation here.√where x⊥=2p⊥/2While there is only one integration variable,x a,in inclusive jet production,once p⊥and y are fixed,both x a and x b have to be integrated over the allowed kinematic region in inclusive Lambda production,since the producedΛcarries only a fraction of the parton’s momentum.predicted Lambda polarization is shown in Fig5a.It is positive at large rapidities where the contributions of polarized up and down quarks dominates the production process.At smaller rapidities,where x a is small,strange quarks also contribute.However,since the ratios of the polarized to the unpolarized parton distributions are small at small x a the Lambda polarization is suppressed.The result also depends on the parameterization of the polarized quark distributions.In particular,the polarized gluon distribution is not well con-strained.However,it is clear from the kinematics that the ambiguity associated with the polarized gluon distributions only effects the results at lower rapidities.This can be seen in Fig.5b where we plot the contribution from gluons,up plus down quarks and strange quarks to the Lambda polarization.Next,we contrast our prediction with the predictions of various SU(3)flavor symmetric models which useD uΛ=D dΛ=D sΛ.(14) Wefitted the cross sections in e+e−annihilation using Eq.(14)and the functional form given in Eq.(6).For the polarized fragmentation functions,we discuss two different scenarios: The model,SU(3)A(c.f.Fig.5a),corresponds to the expectations of the naive quark model that only polarized strange quarks can fragment into polarized Lambdas∆D uΛ=∆D dΛ=0∆D sΛ=D sΛ.(15) It gives essentially zero polarization because the strange quarks contribute at low rapidities where the polarization is suppressed.Model,SU(3)B(c.f.Fig.5a),which was proposed in Ref.[2],is based on DIS data,and sets∆D uΛ=∆D dΛ=−0.20D uΛ∆D sΛ=0.60D sΛ.(16) This model predicts negative Lambda polarization.Finally,we address the problem of Lambdas produced through the decay of other hyper-ons,such asΣ0andΣ∗.In order to estimate the contribution of hyperon decays we assume, in the following,that(1)theΛ’s produced through hyperon decay inherit the momentum of the parent hyperon(2)and that the total probability to produceΛ,Σ0orΣ∗from a certain uds state is given by the SU(6)wave function and is independent of the mass of the produced hyperon.Further,in order to estimate the polarization transfer in the decay process we use the constituent quark model.The polarization can be obtained by noting that the boson emitted in both theΣ0→Λγand theΣ∗→Λπdecay changes the angular momentum of the nonstrange diquark from J=1to J=0,while the polarization of the spectator strange quark is unchanged.Then,the polarization of theΛis determined by the polarization of the strange quark in the parent hyperon,since the polarization of theΛis exclusively carried by the strange quark in the naive quark model.First,let us discuss the case when the parent hyperon is produced by a strange quark. Since the strange quark is always accompanied by a vector ud diquark,in bothΣ0andΣ∗the fragmenation functions of strange quarks into these hyperons are much softer than the corresponding fragmentation function into aΛ.Thus,in the high z limit,the contributions from the processes,s→Σ0→Λand s→Σ∗→Λ,are negligible compared to the directproduction,s→Λ.Furthermore,both channels,s→Σ0→Λand s→Σ∗→Λ,enhance the already positive polarization from the direct channel,s→Λ.This is different in the case when the parent hyperon is produced by an up or down quark. BothΛandΣ0can be produced by an up(down)quark and a scalar ds(us)diquark—a process which dominates in the large z limit.(The component with a vector diquark can be neglected in this limit).Furthermore,the up and down fragmentation function of theΣ∗are as important as those of theΛandΣ0in the large z limit.This is because the u fragmentation function ofΣ∗peaks at about1385/(1010+1385)≈0.58which is almost the same as the peak of the scalar components of theΛandΣ,which are1115/(890+1115)≈0.57and 1190/(890+1190)≈0.57,respectively.Thus,for the up and down quark fragmentation,it is important to include theΛ’s from these decay processes.The relevant probabilities to produce aΛwith positive and negative polarization from a fragmenting up quark with positive polarization and an ds diquark are shown in Table III. We assumed that all spin states of the ds diquark are produced with equal probabilities. Thefinal weights which are relevant in the large z limit are set in bold.Wefind that if we include all channels,which survive in the large z limit,the polarization of theΛis reduced by a factor of10/27compared to the case where only the directly producedΛ’s are included. Since theΣ∗decay is a strong decay it is sometimes included in the fragmentation function of theΛ.Including onlyΣ∗,the suppression factor we obtain is49/81.(Note that our model predicts that u↑(ds)0,0→Λ,u↑(ds)0,0→Σ0and u↑(ds)1,0→Σ∗have approximately the same z dependence and are approximately equal(up to the Clebsch-Gordon factors)since the ratios,M/(M+M n),have roughly the same numerical values.Thus,the effect of the Σ0andΣ∗decays can be taken into account by a multiplicative factor.) In order to illustrate the effect of these decays on thefinalΛpolarization,we multiplied our results with these factors.The results are shown in Fig.6b as dotted lines.We note that our implementation of this correction relies on the assumptions that the producedΛcarries all the momentum of the parent hyperon and that all states are produced with equal probabilities.Since neither of these assumptions is strictly valid,we tend to overestimate the importance of hyperon decays.Note also that the inclusion ofΣ0decay in the SU(3) symmetric models makes the resulting polarization more negative.As a result,even if effects ofΣdecays are included,large discrepancies still persist between our predictions and those of SU(3)symmetric models.IV.CONCLUSIONSMeasurements of the Lambda polarization at RHIC would provide a clear answer to the question of whether polarized up and down quarks can transfer polarization to thefinal state Λ.We predict positive Lambda polarization at high rapidities,in contrast with models based on SU(3)flavor symmetry and DIS which predict zero or negative Lambda polarization.Our prediction is based on the same physics which led to harder up than down quark distributions in the proton and to the∆-N andΣ-Λmass splittings.We also estimated the importance ofΣ0andΣ∗decays which tend to reduce the predictedΛpolarization.ACKNOWLEDGMENTSThis work was partly supported by the Australian Research Council.One of the authors [JTL]was supported in part by National Science Foundation research contract PHY-9722706. One author[CB]wishes to thank the Indiana University Nuclear Theory Center for its hospitality during the time part of this work was carried out.REFERENCES[1]EMC Collaboration,J.Ashman et al.,Phys.Lett B206,364(1988).[2]M.Burkardt and R.L.Jaffe,Phys.Rev.Lett.70,2537(1993).[3]ar and P.Hoodbhoy,Phys.Rev.D51,32(1995).[4]J.Ellis,D.Kharzeev and A.Kotzinian,Z.Phys.C69,467(1996).[5]D.de Florian,M.Stratmann and W.Vogelsang,Phys.Rev.Lett.81,530(1998).[6]C.Boros and A.W.Thomas,Phys.Rev.D60(1999)074017.[7]B.Ma,I.Schmidt and J.Yang,Phys.Rev.D61,034017(2000).[8]D.Ashery and H.J.Lipkin,Phys.Lett.B469,263(1999).[9]C.Boros and Liang Zuo-tang,Phys.Rev.D,4491(1998).[10]F.E.Close and A.W.Thomas,Phys.Lett.B212,227(1988).[11]M.Alberg et al.,Phys.Lett.B389,367(1996).[12]J.C.Collins and D.E.Soper,Nucl.Phys.B194,445(1982).[13]R.L.Jaffe and X.Ji,Phys.Rev.Lett71,2547(1993).[14]C.Boros,J.T.Londergan and A.W.Thomas,Phys.Rev.D61,014007(2000).[15]J.F.Owens,Phys.Lett.76B,85(1978).[16]T.Uematsu,Phys.Lett.79B,97(1978).[17]M.Miyama and S.Kumano,.94,(1996)185and M.Hirai,S.Ku-mano,and M.Miyama .108,38(1998).[18]K.Abe et al.,SLD Collaboration,Phys.Rev.D59,052001(1999).[19]D.Buskulic et al.,Aleph Collaboration,Z.Phys.C64,361,1994.[20]Abreu et al.,Delphi Collaboration,Phys.Lett.318B,249(1993).[21]M.Acciarri et al.,L3Collaboration,Phys.Lett.B407,389(1997).[22]G.Alexander et al.,Opal Collaboration,Z.Phys.C73,569(1997).[23]For a compilation of the TASSO,HRS and CELLO data see fferty,P.I.Reevesand M.R.Whalley,J.Phys.G21,A1(1995).[24]J.F.Owens,Rev.Mod.Phys.59,465(1987).[25]M.Stratmann and W.Vogelsang,Phys.Lett.B295,277(1992).[26]M.Gluck et al.Phys.Rev.D53,4775(1996).[27]i et al.Phys.Rev.D55,1280(1997).TABLESTABLE I.Fit parameters obtained byfitting the e+e−data.We also parametrized D qΛ−D¯qΛand∆D qΛ−∆D¯qΛ,calculated in the bag.D sΛ−D¯sΛD¯qΛ∆D uΛ−∆D¯uΛ5.81×10999.76−6.25×101021.551.2532.4813.6011.6027.72——0.52TABLE II.Fit parameters obtained byfitting the e+e−data and asumming that the fragmen-tation fucntions areflavor symmetric.D qΛ−D¯qΛN99.767.47β11.60TABLE III.Different channels for the production ofΛhyperons from a positively polarized up quark and a ds diquark.It is assumed that all spin states of the ds diquark are produced with the same probabilities.Σ∗↑andΣ∗⇑stand for the1/2and3/2spin component of theΣ∗.See text for further details.u(ds)states u↑(ds)0,0u↑(ds)1,1u↑(ds)1,0u↑(ds)1,−14141productsΛ↑Σ0↑Σ∗0↑Σ∗0⇑Λ↑Σ0↑Σ∗0↑Λ↓Σ0↓Σ∗0↓43413161decay productsΛ↑Λ↑Λ↓Λ↑Λ↓Λ↑Λ↓Λ↑Λ↓3301012333312final weights116801010119188113618FIGURES(1/σh ) (d σ/d x E )1010.10.011010.11010.11010.11010.1x EFIG.1.Inclusive Lambda production in e +e −annihilation.The solid lines are the result of the global fit.They contain two parts,the fixed contributions from D q Λ−D ¯q Λcalculated in the bag (dashed line only shown for the Aleph data)and D ¯q Λobtained from the fit (dash-dotted line).x Eis defined as x E =2E Λ/√s is the total center of mass enery.z FIG.2.Fragmentation functions.The solid and dashed lines stand for the calculated fragmen-tation functions of up and strange quarks into Lambda baryons through production of a Lambdaand an anti-diquark and correspond to D uΛ−D¯uΛand D sΛ−D¯sΛ,respectively.The dash-dotted line represents the contributions from higher intermediate states,and is obtained byfitting thee+e−data and corresponds to D¯qΛ.The short dashed line is the gluon fragmentation function. The light and heavy lines are the fragmentation functions at the scales Q2=µ2and Q2=M2Z,respectively.Note that D gΛ=0at Q2=µ2.FIG.3.z c as a function of x a and y for two different transverse momenta,p⊥=10GeV(left) and p⊥=30GeV(right)and for two different values of x b,x b=x bmin+0.01(top)and and x b=x bmin+0.1(bottom).10101010101010E d 3σ/d p 3(m b /G e V 2)(a)→ qq’→qq –→ qq –→ qg –→ gg→ qq –→ gg 1010101010101010E d 3σ/d p 3(m b /G e V 2)(b)FIG.4.Contributions from the various channels (a)to the inclusive Lambda production cross section (pp →Λ+X )and (b)to the inclusive jet production cross section (pp →jet +X )atp ⊥=10GeV (left)and p ⊥=30GeV (right)at√y mbda polarization at RHIC.(a)The solid line represents our prediction.The pre-dictions of SU(3)symmetric fragmentation models are shown for comparision.The model labeled as SU(3)A is based on the quark model expectation that only the polarized strange quark may fragment into polarized Lambdas,while SU(3)B,is based on DIS data.(b)Contributions of dif-ferentflavors to theΛ-polarization.The light dashed,dash-dotted and heavy dashed lines stand for the contributions from up plus down,from strange and from gluon fragmentation,respectively, as calculated here.The estimated polarization including bothΣ0andΣ∗(lower dotted line)andonlyΣ∗(upper dotted line)decays are also shown.See text for further details.。

a r X i v :n u c l -t h /0309040v 2 16 S e p 2003Heavy Quark Energy Loss in Nuclear MediumBen-Wei Zhang a ,Enke Wang a and Xin-Nian Wang b,caInstitute of Particle Physics,Huazhong Normal University,Wuhan 430079,ChinabNuclear Science Division,MS 70R0319,Lawrence Berkeley National Laboratory,Berkeley,CA 94720USAcDepartment of Physics,Shandong University,Jinan 250100,China(September 5,2003)Multiple scattering,modified fragmentation functions and radiative energy loss of a heavy quark propagating in a nuclear medium are investigated in perturbative QCD.Because of the quark mass dependence of the gluon formation time,the medium size dependence of heavy quark energy loss is found to change from a linear to a quadratic form when the initial energy and momentum scale are increased relative to the quark mass.The radiative energy loss is also significantly suppressed relative to a light quark due to the suppression of collinear gluon emission by a heavy quark.An energetic parton propagating in a dense medium suffers a large amount of energy loss due to multiple scattering and induced gluon bremsstrahlung [1].In a static medium,the total energy loss of a massless parton (light quark or gluon)is found to have a quadratic de-pendence on the medium size [2–6]due to non-Abelian Landau-Pomeranchuk-Migdal (LPM)interference effect.In an expanding medium,the total energy loss can be cast into a line integral weighted with local gluon density along the parton propagation path [7–9].Therefore,the measurement of parton energy loss can be used to study properties of the medium similar to the technique of com-puted tomography.Recent experimental measurements [10,11]of centrality dependence of high-p T hadron sup-pression agree very well [12]with such a parton energy loss mechanism.Because of the large mass of the heavy quark with a velocity v ≈1−M 2/2E 2,the formation time of gluon radiation,τf ∼1/(ωg M 2/2E 2+ℓ2T /2ωg )is reduced rel-ative to a light quark.One should then expect the LPM effect to be significantly reduced for intermediate energy heavy quarks.In addition,the heavy quark mass also suppresses gluon radiation amplitude at angles smaller than the ratio of the quark mass to its energy [13]rela-tive to the gluon radiation offa light quark.Both mass effects will lead to a heavy quark energy loss different from a light quark propagating in a dense medium.This might explain why one has not observed significant heavy quark energy loss from the PHENIX [14]measurement of the single electron spectrum from charm production in Au +Au collisions at√d 3L 2d 3ℓH=G 2Fd 3ℓH.(1)Here L 1and L 2are the four momenta of the incominglepton and the outgoing neutrino,ℓH the observed heavyquark meson momentum,p =[p +,m 2N /2p +,0⊥]is the momentum per nucleon in the nucleus,and s =(p +L 1)2.G F is the four-fermion coupling constant and q =L 2−L 1=[−Q 2/2q −,q −,0⊥]the momentum transfer via the exchange of a W -boson.The charge-current leptonic ten-sor is given by L cc µν=1/2Tr(L 1γµ(1−γ5)L 2(1−γ5)γν).We assume Q 2≪M 2W .The semi-inclusive hadronic ten-sor is defined as,E ℓH dW µν2 X A |J +µ|X,H X,H |J +†ν|A×2πδ4(q +p −p X −ℓH )(2)where X runs over all possible final states and J +µ=¯c γµ(1−γ5)s θis the hadronic charged current.Here,s θ=s cos θC −d sin θC and θC is the Cabibbo angle.To the leading-twist in collinear approximation,the semi-inclusive cross section factorizes into the product of quarkdistribution f As θ(x B +x M ),the heavy quark fragmenta-tion function D Q →H (z H )(z H =ℓ−H /ℓ−Q )and the hard partonic part H (0)µν(k,q,M )[17].Here,x B =Q 2/2p +q −is the Bjorken variable and x M =M 2/2p +q −.Similar to the case of light quark propagation in nu-clear medium [6],the generalized factorization of multiple scattering processes [16]will be employed.We will only consider double parton scattering.The leading contri-butions are the twist-four terms that are enhanced by the nuclear medium in a collinear expansion,assuming a 1small expansion parameter αs A 1/3/Q 2.The evaluation of 23cut diagrams are similar to the case of a light quark [17].The dominant contribution comes from the central cut diagram,giving the semi-inclusive tensor for heavy quark fragmentation from double quark-gluon scattering,W D µνzD Q →H (z H2π1+z2[ℓ2T +(1−z )2M 2]4ℓ4T2παs2dy −1+z 2M 21+z 2M 421−(1−z )(2z 3−5z +8z −1)C A (1−z )3 M 2(1+z 2)+2C F (1+z 2)M 4C A1−8z (1−z )2ℓ2T−(1−z )4(z 2−4z +1)ℓ4T,(8)where, x L ≡x L +(1−z )x M /z is the additional frac-tional momentum of the initial quark or gluon in therescattering that is required for gluon radiation,andx L =ℓ2T /2p +q −z (1−z ).The contribution from gluon fragmentation is similar to that from quark fragmenta-tion with z →1−z .The virtual correction can be ob-tained via unitarity constraint.One can recover the re-sults for light quark rescattering [18]by setting M =0in the above equations.Notice that we have embedded the phase factors from the LPM interference in the effectivetwist-four parton matrix T A,Cqg (x,x L ,M 2).Rewriting the sum of single and double scattering contributions in a factorized form for the semi-inclusive hadronic tensor,one can define a modified effective frag-mentation function DQ →H (z H ,µ2)as DQ →H (z H ,µ2)≡D Q →H (z H ,µ2)+µ2dℓ2T2π1z hdzz)+µ2dℓ2T2π1z hdzz),(9)where D Q →H (z H ,µ2)and D g →H (z H ,µ2)are the leading-twist fragmentation functions of the heavy quark andgluon.The modified splitting functions are given as∆γq →qg (z )=1+z 2[ℓ2T +(1−z )2M 2]3N c f A q (x ),(10)∆T A,C qg (x,ℓ2T ,M 2)≡1dz p + x L=2z (1−z )q −xA f Aq (x )(1−e − x 2L /x 2A)× c 1(z,ℓ2T ,M 2)−c 2(z,ℓ2T,M 2) +c 3(z,ℓ2T ,M 2)with the initial intrinsic transverse momentum.The co-efficient C≡2Cx T f N g(x T)should in principle depend on Q2.With this simplified form of twist-four matrix,one can then calculate the heavy quark energy loss,defined as the fractional energy carried by the radiated gluon,∆z Q g (x B,Q2)=αsℓ2T +z2M2z= CC Aα2s x B z(1−z) xµ x M d x L( x L− x M)2 2c3(z,ℓ2T,M2)+ 1−e− x2L/x2A× c1(z,ℓ2T,M2)−c2(z,ℓ2T,M2) ,(14) where x M=(1−z)x M/z and xµ=µ2/2p+q−z(1−z)+ x M.Note that x L/x A=L−A/τf with L−A=R A m N/p+ the nuclear size in the chosen frame.The LPM interfer-ence is clearly contained in the second term of the inte-grand that has a suppression factor1−e− x2L/x2A.Thefirst term that is proportional to c3(z,ℓ2T,M2)corresponds to afinite contribution in the factorization limit.We have neglected such a term in the study of light quark propa-gation since it is proportional to R A,as compared to the R2A dependence from thefirst term due to LPM effect. We have to keep thefirst term for heavy quark propa-gation since the second term will have a similar nuclear dependence when the mass dependence of the gluon for-mation time is important.Since x L/x A∼x B M2/x A Q2,there are two distinct limiting behaviors of the energy loss for different values of x B/Q2relative to x A/M2.When x B/Q2≫x A/M2for small quark energy(large x B)or small Q2,the formation time of gluon radiation offa heavy quark is always smaller than the nuclear size.In this case,1−exp(− x2L/x2A)≃1, so that there is no destructive LPM interference.The in-tegral in Eq.(14)is independent of R A,and the heavy quark energy loss∆z Q g ∼C A Cα2s x A Q2(15)is linear in nuclear size R A.In the opposite limit, x B/Q2≪x A/M2,for large quark energy(small x B)or large Q2,the quark mass becomes negligible.The gluon formation time could still be much larger than the nu-clear size.The LPM suppression factor1−exp(− x2L/x2A) will limit the available phase space for gluon radia-tion.The integral in Eq.(14)will be proportional to d x L[1−exp(− x2L/x2A)]/ x2L∼1/x A.The heavy quark energy loss∆z Q g ∼C A Cα2s x2A Q2(16)now has a quadratic dependence on the nuclear size simi-lar to the light quark energy loss.Shown in Fig.1are thenumerical results of the R A dependence of charm quark energy loss,rescaled by C(Q2)C Aα2s(Q2)/N C,for differ-ent values of x B and Q2.One can clearly see that the R A dependence is quadratic for large values of Q2or small x B.The dependence becomes almost linear for small Q2 or large x B.The charm quark mass is set at M=1.5 GeV in the numericalcalculation.24681012141618FIG.1.The nuclear size,R A,of charm quark energy loss for different values of Q2and x B.0.10.20.30.40.50.60.70.80.910102030405060708090100FIG.2.The Q2dependence of the ratio between charm quark and light quark energy loss in a large nucleus.Another mass effect on the induced gluon radiation is the“dead-cone”phenomenon[13]which suppresses the the small angle gluon radiation.Such a“dead-cone”ef-fect is manifested in Eq.(3)for the induced gluon spectra from a heavy quark which is suppressed by a factorf Q/q= ℓ2Tθ2 −4,(17)relative to that of a light quark for small angle radiation.Hereθ0=M/q−andθ=ℓT/q−z.This will lead toa reduced radiative energy loss of a heavy quark,amid 3other mass dependence as contained in c i (z,ℓ2T ,M 2)in Eqs.(6)-(8).Setting M =0in Eq.(14),we recover the energy loss for light quarks as in our previous study [18].To illustrate the mass suppression of radiative en-ergy loss imposed by the “dead-cone”,we plot the ratio∆z Q g (x B ,Q 2)/ ∆z q g (x B ,Q 2)of charm quark and light quark energy loss as functions of Q 2and x B in Figs.2and3.00.10.20.30.40.50.60.70.80.9100.020.040.060.080.10.120.140.160.18FIG.3.The x B dependence the ratio between charm quark and light quark energy loss in a large nucleus.Apparently,the heavy quark energy loss induced by gluon radiation is significantly suppressed as compared to a light quark when the momentum scale Q or the quark initial energy q −is not too large as compared to the quark mass.Only in the limit M ≪Q,q −,is the mass effect negligible.Then the energy loss approaches that of a light quark.In summary,we have calculated medium modification of fragmentation and energy loss of heavy quarks in DIS in the twist expansion approach.We demonstrated that heavy quark mass not only suppresses small angle gluon radiation due to the “dead-cone”effect but also reduces the gluon formation time.This leads to a reduced radia-tive energy loss as well as a different medium size depen-dence (close to linear),as compared to a light quark when the quark energy and the momentum scale Q are of the same order of magnitude as the quark mass.The result approaches that for a light quark when the quark mass is negligible as compared to the quark energy and the mo-mentum scale Q .Similar to the case of light quark prop-agation,the result can be easily extended to a hot and dense medium,which will have practical consequences for heavy quark production and suppression in heavy ion collisions.ACKNOWLEDGEMENTSThis work was supported by the U.S.Department of Energy under Contract No.DE-AC03-76SF00098andby NSFC under project Nos.19928511and 10135030.E.Wang and B.Zhang thank the Physics Department of Shandong University for its hospitality during the com-pletion of this work.。

a rXiv:h ep-ph/945231v15May1994Institute of Physics,Acad.of Sci.of the Czech Rep.PRA–HEP–94/3and hep-ph/9405231Nuclear Centre,Charles University May 5,1994Prague Feasibility of Beauty Baryon Polarization Measurement in Λ0J/ψdecay channel by ATLAS-LHC Julius Hˇr ivn´a ˇc ,Richard Lednick´y and M´a ria Smiˇz ansk´a Institute of Physics AS CR Prague,Czech Republic submitted to Zeitschrift f¨u r Physik CAbstractThe possibility of beauty baryon polarization measurement by cascade decay angu-lar distribution analysis in the channel Λ0J/ψ→pπ−l +l −is demonstrated.The error analysis shows that in the proposed LHC experiment ATLAS at the luminosity 104pb −1the polarization can be measured with the statistical precision better than δ=0.010for Λ0b and δ=0.17for Ξ0b .IntroductionThe study of polarization effects in multiparticle production provides an important infor-mation on spin-dependence of the quark confinement.Thus substantial polarization of the hyperons produced in nucleon fragmentation processes[1,2]as well as the data onthe hadron polarization asymmetry were qualitatively described by recombination quark models taking into account the leading effect due to the valence hadron constituents[3−6].Although these models correctly predict practically zero polarization ofΛandΩ−,they fail to explain the large polarization of antihyperonsΣ−recently discovered in Fermilab[7,8].The problem of quark polarization effects could be clarified in polarization measure-ments involving heavy quarks.In particular,an information about the quark mass de-pendence of these effects could be obtained[4,9].The polarization is expected to be proportional to the quark mass if it arises due to scattering on a colour charge[10−12]. The opposite dependence takes place if the quark becomes polarized due to the interac-tion with an”external”confiningfield,e.g.,due to the effect of spontaneous radiation polarization[13].The decrease of the polarization with increasing quark mass is expected also in the model of ref.[14].In QCD the polarization might be expected to vanish with the quark mass due tovector character of the quark-gluon coupling[10].It was shown however in Ref.[15] that the quark mass should be effectively replaced by the hadron mass M so that even the polarization of ordinary hadrons can be large.The polarization is predicted to be independent of energy and to vanish in the limit of both low and high hadron transverse momentum p t.The maximal polarization P max(x F)is reached at p t≈M and depends on the Feynman variable x F.Its magnitude(and in particular its mass dependence)is determined by two quark-gluon correlators which are not predicted by perturbative QCD.The polarization of charm baryons in hadronic reactions is still unmeasured due to the lack of sufficient statistics.Only some indications on a nonzero polarization were reported[16,17].For beauty physics the future experiments on LHC or HERA give an opportunity to obtain large statistical samples of beauty baryon(Λ0b,Ξ0b)decays intoΛ0J/ψ→pπ−l+l−,which is favorable mode to detect experimentally.Dedicated triggers for CP-violation effects in b-decays,like the high-p t one-muon trigger(LHC)[18] or the J/ψtrigger(HERA)[19]are selective also for this channel.Below we consider the possibility of polarization measurement of beauty baryonsΛ0b andΞ0b with the help of cascade decay angular distributions in the channelΛ0b(Ξ0b)→Λ0J/ψ→pπ−l+l−.1Polarization measurement method and an estimation of the statistical error.In the case of parity nonconserving beauty baryon(B b)decay the polarization causes the asymmetry of the distribution of the cosine of the angelθbetween the beauty baryon decay and production analyzers:w(cosθ)=1| p inc× p B b|,where p inc and p Bbare momenta of incident particle and B b in c.m.system.The asymmetry parameterαb characterizes parity nonconservation in a weak de-cay of B b and depends on the choice of the decay analyzer.In the two-body decay B b→Λ0J/ψit is natural to choose this analyzer oriented in the direction ofΛ0momentum pΛ0in the B b rest system.The considered decay can be described by4helicity ampli-tudes A(λ1,λ2)normalized to unity:a+=A(1/2,0),a−=A(−1/2,0),b+=A(−1/2,1) and b−=A(1/2,−1),|a+|2+|a−|2+|b+|2+|b−|2=1.(2) The difference ofΛ0and J/ψhelicitiesλ1-λ2is just a projection of B b spin onto the decay analyzer.The decay asymmetry parameterαb is expressed through these amplitudes in the formαb=|a+|2−|a−|2+|b+|2−|b−|2.(3) If P-parity in B b decay were conserved,then|a+|2=|a−|2,|b+|2=|b−|2so thatαb would be0.In the case of known and sufficiently nonzero value ofαb the beauty baryon polarization could be simply measured with the help of angular distribution(1)(see, e.g.,[20]).Due to lack of experimental information and rather uncertain theoretical estimates ofαb for the decayΛ0b→Λ0J/ψ[21]both the polarization andαb(or the decay amplitudes)should be determined simultaneously.This can be achieved with the2help of information onΛ0and J/ψdecays.Though it complicates the analysis,it shouldbe stressed that the measurement of the beauty baryon decay amplitudes could give valuable constrains on various theoretical models.Generally,such a measurement canbe done provided that at least one of the secondary decays is asymmetric and its decayasymmetry parameter is known[9].In our case it is the decayΛ0→pπ−with the asymmetry parameterαΛ=0.642.The angular distribution in the cascade decay B b→Λ0J/ψ→pπ−l+l−follows di-rectly from Eq.(6)of[9],taking into account that the only nonzero multipole parameters.It can be written in the formin the decay J/ψ→l+l−are T00=1and T20=110w(Ω,Ω1,Ω2)=1formula(4)integrated over the azimuthal anglesφ1,φ2would be in principle sufficient [9].In this case the number of free parameters is reduced to4(the phases don’t enter) and only a3-dimensionalfit is required.We will see,however,that the information on these angles may substantially increase the precision of the P b determination.To simplify the error analysis,we follow ref.[9]and consider here only the most unfavourable situation,when the parameters P2b,|a+|2−|a−|2and|b+|2−|b−|2are much smaller thanα2Λ.In this case the moments<F i>can be considered to be independent, having the diagonal error matrixW=13,19,115,145,16135,16135,2135,2135,245,245).(7)Here N is a number of B b events(assuming that the background can be neglected,see next section).The error matrix V of the vector a of the parameters a j,j=1,..7defined in (6)isV( a)=(A T W−1A)−1,(8) where the elements of the matrix A are A ij=d(f1i.f2i)V11=δ0N,(9)δ0=1α2Λ.[(2r0−1)2180+4r2015+(1−r0)(1+coshχ)10.(10)Hereδ0depends only on the relative contribution r0of the decay amplitudes with helicityλ2=0and on the relative phaseχ(Figs.1a,b).The maximal error on P b is δmax=4.7Nand it corresponds to the case when r0=1√√√,Σ∗b →Λ0b πand the electromagnetic decays Ξ0′b →Ξ0b γor Ξ0∗b →Ξ0b γ.The observable polarization P obs depends on the polarizations P B b of direct beauty baryons and their production fractions b B b (i.e.probabilities of the b-quark to hadronize to certain baryons B b ).In considered decays the beauty baryon Λ0b or Ξ0b retains −13)of the polarizationof a parent with spin 12+)(see Appendix).For P obs we get:P obs =b Λ0b P Λ0b + i (−13b Σ∗bi P Σ∗bi )3b Ξ0′b P Ξ0′b+1b Ξ0b +b Ξ0′b +b Ξ0∗b.(12)The summation goes over positive,negative and neutral Σb and Σ∗b .Assuming the polar-ization of the heavier states to be similar in magnitude to that of directly produced Λ0b or Ξ0b (P Λ0b or P Ξ0b )we may expect the observed polarization in an interval of (0.34−0.67)P Λ0bfor Λ0b and (0.69−0.84)P Ξ0b for Ξ0b.The polarization can be measured for Λ0b and Ξ0b baryons and for their antiparticles.Λ0b (Ξ0b )are unambigously distinquishable from their antiparticles by effective mass of pπ−system from Λ0→pπ−decay.The wrong assignment of antiproton and pion masses gives the kinematical reflection ofΞ0b is governed by b→dcc.HoweverΛ0fromΞ0b→Ξ0J/ψorΞ−b→Ξ−J/ψis produced in a weak hyperon decay,so this background can be efficiently reduced by the cut on the minimal distance d between J/ψandΛ0.A conservative cut d<1.5mm reduces this background by a factor≈0.05(Fig.3b).The background from B0d→J/ψK0when one ofπmesons is considered as a proton is negligible after the effective mass cuts on(pπ)and(pπJ/ψ)systems.Background from fake J/ψ′s,as it has been shown in[18],can be reduced to a low level by cuts on the distance between the primary vertex and the production point of the J/ψcandidate and the distance of closest approach between the two particles from the decay.These cuts also suppress the background from real J/ψ′s comming directly from hadronization.The number of producedΛ0b andΞ0b is calculated for the cross section of pp→busing the last segment of the hadron calorimeter by its minimum ionizing signature.-ForΛ0J/ψ→pπ−e+e−decay both electrons are required to have p e⊥>1GeV.The low threshold for electrons is possible,because of electron identification in the transition radiation tracker(TRT)[24].The events are required to contain one muon with a pµ⊥> 6GeV and|η|<1.6The second set of cuts corresponds to’offline’analysis cuts.The same cuts as for B0d→J/ψK0reconstruction[18]can be used(the only exception is the mass requirement forΛ0candidate,see the last of the next cuts):-The two charged hadrons fromΛ0decay are required to be within the tracking volume |η|<2.5,and transverse momenta of both to be greater than0.5GeV.-Λ0decay length in the transverse plane with respect to the beam axis was required to be greater than1cm and less than50cm.The upper limit ensures that the charged tracks fromΛ0decay start before the inner radius of TRT,and that there is a space point from the innermost layer of the outer silicon tracker.The lower limit reduces the combinatorial background from particles originating from the primary vertex.-The distance of closest approach between the two muon(electron)candidates forming the J/ψwas required to be less than320µm(450µm),giving an acceptance for signal of 0.94.-The proper time of theΛ0b decay,measured from the distance between the primary vertex and the production point of the J/ψin the transverse plane and the reconstructed p⊥ofΛ0b,was required to be greater than0.5ps.This cut is used to reduce the combina-torial background,giving the acceptance for signal events0.68.-The reconstructedΛ0and J/ψmasses were required to be within two standart de-viations of nominal values.The results on expectedΛ0b andΞ0b statistics and the errors of their polarization mea-surement are summarized in Table2.For both channels the statistics of reconstructed events at the luminosity104pb−1will be790000(220000)Λ0b and2600(720)Ξ0b,where the values are derived using UA1(CDF)results.For this statistics the maximal value of the statistical error on the polarization mea-surement,calculated from formulae(9)and(10),will be0.005(0.01)forΛ0b and0.09(0.17) forΞ0b.7ConclusionAt LHC luminosity104pb−1the beauty baryonsΛ0b andΞ0b polarizations can be measured with the help of angular distributions in the cascade decaysΛ0J/ψ→pπ−µ+µ−and Λ0J/ψ→pπ−e+e−with the statistical precision better than0.010forΛ0b and0.17for Ξ0b.AppendixThe polarization transfered toΛ0b,which was produced indirectly in strongΣb andΣ∗b decays,depends on the ratio∆| p inc× pΣb|,where p inc and pΣb are momenta of incident particle andΣb in c.m.system.Ω1=(θ1,φ1)are the polar and the azimuthal angles ofΛ0inΛ0b rest frame with the axes defined as z1↑↑ pΛ0b,y1↑↑ n× pΛ0b.After the transformation ofΩ1→Ω′1ofΛ0angles from the helicity frame x1,y1,z1to the canonical frame x,y,z with z↑↑ n and the integration over cosθandφ′1we get the distribution of the cosine of the angle between theΛ0momentum vector(Λ0b decay analyzer)and theΣb orΣ∗b production normal(which can be considered coinciding with theΛ0b production normal due to a small energy release in theΣb orΣ∗b decays):w(cosθ′1)∼1∓13(13(1References[1]K.Heller,Proceedings of the VII-th Int.Symp.on High Energy SpinPhysics,Protvino,1986vol.I,p.81.[2]L.Pondrom,Phys.Rep.122(19985)57.[3]B.Andersson et al.,Phys.Lett.85B(1979)417.[4]T.A.De Grand,H.I.Miettinen,Phys.Rev.D24(1981)2419.[5]B.V.Struminsky,Yad.Fiz.34(1981)1954.[6]R.Lednicky,Czech.J.Phys.B33(1983)1177;Z.Phys.C26(1985)531.[7]P.M.Ho et al.,Phys.Rev.Lett.65(1990)1713.[8]A.Morelos et al.,FERMILAB-Pub-93/167-E.[9]R.Lednicky,Yad.Fiz.43(1986)1275(Sov.J.Nucl.Phys.43(1986),817).[10]G.Kane,Y.P.Yao,Nucl.Phys.B137(1978)313.[11]J.Szwed,Phys.Lett.105B(1981)403.[12]W.G.D.Dharmaratna,Gary R.Goldstein,Phys.Rev.D41(1990)1731.[13]B.V.Batyunya et al.,Czech.J.Phys.B31(1981)11.[14]C.M.Troshin,H.E.Tyurin,Yad.Fiz.38(1983)1065.[15]A.V.Efremov,O.V.Teryaev,Phys.Lett.B150(1985)383.[16]A.N.Aleev et al.,Yad.Fiz.43(1986)619.[17]P.Chauvatet et al.,Phys.Lett.199B(1987)304.[18]The ATLAS Collaboration,CERN/LHCC/93-53,Oct.1993.[19]W.Hoffmann,DESY93-026(1993).[20]H.Albrecht et al.,DESY93-156(1993).9[21]A.H.Ball et al.,J.Phys.G:Nucl.Part.Phys.18(1992)1703.[22]UA1Collaboration,Phys.Lett.273B(1991)544.[23]CDF Collaboration,Phys.Rev.D47(1993)R2639.[24]I.Gavrilenko,ATLAS Internal Note INDET-NO-016,1992.[25]A.F.Falk and M.E.Peskin,SLAC-PUB-6311,1993.[26]R.Lednicky,DrSc Thesis,JINR-Dubna1990,p.174(in russian).10i f 2i011P ba +a ∗+−a −a ∗−−b +b ∗++b −b ∗−cos θ13P b αΛ−a +a ∗+−a −a ∗−+12b −b ∗−d 200(θ2)52b +b ∗+−1P b −a +a ∗++a −a ∗−−12b −b ∗−d 200(θ2)cos θ172b +b ∗+−1P b αΛ8P b αΛ3Im (a +a ∗−)sin θsin θ1sin 2θ2sin φ1102Re (b −b ∗+)sin θsin θ1sin 2θ2cos (φ1+2φ2)112Im (b −b ∗+)sin θsin θ1sin 2θ2sin (φ1+2φ2)−32Re (b −a ∗++a −b ∗+)sin θcos θ1sin θ2cos θ2cos φ213√P b αΛ−32Re (b −a ∗−+a +b ∗+)cos θsin θ1sin θ2cos θ2cos(φ1+φ2)15√P b αΛ16√P b−32Im (a −b ∗+−b −a ∗+)sin θsin θ2cos θ2sin φ218√αΛ−32Im (b −a ∗−−a +b ∗+)sin θ1sin θ2cos θ2sin(φ1+φ2)Table 1:The coefficients f 1i ,f 2i and angular functions F i in distribution (4).11Parameter Value forΛ0b CommentL[cm−2s−1]1033b(b→B b)0.08br(B b→Λ0J/ψ)2.210−2(0.610−2)J/ψ→µ+µ−0.06Λ0→pπ−0.641.110−4(0.310−3)0.060.64b)500µbN(µ+µ−pπ−)accepted1535000pµ⊥>6GeV,|η|<1.6(426000)pµ⊥>3GeV,|η|<2.5pπ,p⊥>0.5GeV,|η|<2.5740(210)2400(670)N(µeepπ−)reconstructed65000(18000)the maximum statistical error0.005on the polarization measurement(0.010)δ(P b)Table2:Summary on beauty baryon measurement possibilities for LHC experiment AT-LAS.The values in brackets correspond to the CDF result,while the analogical values without brackets to the UA1result.12Figure1:The maximal statistical error on the polarization measurementδ(P b)andδ0=N+1Figure 2:The Λ0J/ψeffective mass distribution:The peak at 5.62GeV is from Λ0b and background comes from J/ψfrom a b-hadron decay and Λ0either from the multiparticle production or from a b-hadron decay (a).The events that passed the cut on the transverse momenta (p T >0.5GeV )for p and π−from Λ0decay (b).14Figure 3:The Λ0J/ψeffective mass distribution:The peak at 5.84GeV is from Ξ0b →Λ0J/ψdecay.The background with the centre at ≈5.5GeV comes from Ξ0b →Ξ0J/ψ,Ξ0→Λ0π0and Ξ−b →Ξ−J/ψ,Ξ−→Λ0π−decays (a).The events that passed the cut on the minimal distance of J/ψand Λ0(d <1.5mm )(b).15。

a rXiv:h ep-ph/951410v22Mar1995Small-x physics and heavy quark photoproduction in the semihard approach at HERA.V.A.Saleev Samara State University,Samara 443011,Russia N.P.Zotov Nuclear Physics Institute,Moscow State University,Moscow 119899,Russia Abstract Processes of hevy quark photoproduction at HERA energies and be-yond are investigated using the semihard (k ⊥factorization)approach.The virtuality and longitudinal polarization of gluons in the photon -gluon subprocess as well as the saturation effects in the gluon distribu-tion function at small x have been taken into account.The total cross sections,rapidity and p ⊥distributions of the charm and beauty quark photoproduction have been calculated.We obtained the some differences between the predictions of the standard parton model and the semihard approach used here.1Introduction The heavy quark production at HERA is very intresting and important subject of study [1,2,3,4].Because it is dominated by photon-gluon fusion subprocess (Fig.1)one can study the gluon distribution functions G (x,Q 2)in small x region (roughly at x >10−4).Secondlythis last issue is important for physics at future colliders (such as LHC):many processes at these colliders will be determined by small-x gluondistributions.γg q ¯q γg ¯q q Fig.1The heavy quark production processes are so-called semihard processes [5]at HERA en-ergies and beyond.By definition in these processes a hard scattering scale Q (or heavy quark mass M )is large as compared to the ΛQCD parameter,but Q is much less than the total center of mass energies:ΛQCD ≪Q ≪√which are O(ln n(Q2/Λ2)).So in perturbative QCD the heavy quark photoproduction crosssection as result of photon-gluon fusion processe has the form[6]:σγg=σoγg+αsσ1γg+ ...,whereσ1γg was calculated by Ellis and Nason[7].The photon-gluon fusion cross sec-tion in low order decreases vs s at s→∞:σoγg∼M2/s ln(s/Q2).But in the same limit σ1γg→Const,because of that the heavy quark photoproduction cross section is dominated by the contribution of the gluon exchange in the t-channel.That results breakdown thestandard perturbative QCD expansion and the problem of summing up all contributions of the order(αs ln(Q2/Λ2))n,(αs ln(1/x))n and(αs ln(Q2/Λ2)ln(1/x)n in perturbative QCD appears in calculation ofσγg.It is known that summing up the terms of the order(αs ln(Q2/M2))n in leading logarithm approximation(LLA)of perturbative QCD leads to the linear DGLAP evolution equation for deep inelastic structure function[8].Resummation of the large contributions of the order of(αs ln(1/x))n leads to the BFKL evolution equation[9]and its solution gives the drastical increase of the gluon distribution:xG(x,Q2)∼x−ω0,ω0=(4N c ln2/π)αs(Q20).The speed of the growth of the parton density as x→0makes the parton-parton interactions very important,which in turn makes the QCD so-called GLR evolution equation essentially nonlinear[5].The growth of the parton density at x→0and interaction between partons induces sub-stantial screening(shadowing)corrections which restore the unitarity constrains for deep inelastic structure functions(in particular for a gluon distribution)in small x region[10]. These facts break the assumption of the standard parton model(SPM)about x and trans-verse momentum factorization for a parton distribution functions.We should deal with the transverse momentum factorization(k⊥factorization)theory[11,12].Making allowance for screening corrections at small x we have so-called semihard approach[5,10],which take into account the virtuality of gluons,their transverse momenta as well as the longitudional polarizations of initial gluons.In semihard approach[5,10]screening corrections stop the growth of the gluon distribution at x→0.This effect was interpreted as saturation of the parton(gluon)density:gluon distribution function xG(x,Q2)becames proportional to Q2R2at Q2≤q20(x)andσ∼1s,one need take into account the dependence of the photon-gluon fusion cross section on the virtuality and polar-izations of initial gluon.Thus in semihard approach the matrix elements of this subprocess differs from ones of SPM(see for example[6,12]).We would like to remark thatαs corrections in the semihard approach look quite different in comparison with usual[6,7]and k⊥factorization[11,12]ones.This corrections have been taken into account using new gluon densityΦ(x, q2⊥)(see below)which was calculated by Gribov,Levin and Ryskin[5]in LLA where all large logs(of the types ln(1/x)and ln(Q2/λ2))were summed up.The another corrections which give contributions to K−factor have been taken into account in the normalization of function xG(x,Q2)(see below also) [10].As far as the SPM calculations of the next-to-leading (NLO)cross section to the photo-production of heavy quarks that the review of its may found in the papers [2,16].The results [7]have been confirmed by Smith and Van Neerven [17].The futher results for the electro-and photoproduction of heavy quarks was obtained in Refs.[18,19].Authors in [16]notes that for beauty quark production the NLO corrections are large and various estimates of corrections lead to theoretical uncertainties of the order of a factor of 2to 3.As was estimated in [20]the total cross section for beauty quark production at HERA will be only a few tens of percent large than the one-loop results [7].For charm quark a theoretical uncertainties are even the higher.It is known that there is also the strong mass dependence of the results of calculations for charm quark production.Since the mass of the charm quark is small for perturbative QCD calculations the resumme procedure [11]is not applicable for charm quark production [20].In Refs.[21,22]we used the semihard approach in order to calculate the total and differ-ential cross sections of the heavy quarkonium,J/Ψand Υ,photoproduction.We obtained the remarkable difference between the predictions of the semihard approach and the SPM especially for p ⊥-and z -distributions of the J/Ψmesons at HERA.In present paper we investigate the open heavy quark production processes in the semihard approach,which was used early in Ref.[10]for calculation of heavy quark production rates at hadron colliders and for prediction of J/ΨandΥphotoproduction cross sections at high energies in Refs.[21,22].2QCD Cross Section for Heavy Quark ElectroproductionWe calculate the total and differential cross sections (the p ⊥and rapidity distributions)of charm and beauty quark photoproduction via the photon-gluon fusion QCD subprocess (Fig.1)in the framework of the semihard (k ⊥factorization)approach [4,10].First of all we take into account the transverse momentum of gluon q 2⊥,its virtuality q 22=− q 22⊥and the alignment of the polarization vectors along the transverse momentum such as ǫµ=q 2⊥µ/| q 2⊥|[10,12].~e (P 1)p (P 2)q 1q 2¯q (p 2)q (p 1)Fig.2Let us define Sudakov variables of the process ep →Q ¯QX(Fig.2):p 1=α1P 1+β1P 2+p 1⊥,p 2=α2P 1+β2P 2+p 2⊥,q 1=x 1P 1+q 1⊥,q 2=x 2P 2+q 2⊥,(1)wherep21=p22=M2,q21=q21⊥,q22=q22⊥,p1and p2are4-momenta of the heavy quarks,q1is4-momentum of the photon,q2is4-momentum of the gluon,p1⊥,p2⊥,q1⊥,q2⊥are transverse4-momenta of these ones.In the center of mass frame of colliding particles we can write P1=(E,0,0,E),P2=(E,0,0,−E), where E=√√√√√d2p1⊥(ep→Q¯QX)= dy∗1dy∗2d2q1⊥π|¯M|2Φe(x1,q21⊥)Φp(x2,q22⊥)d2p1⊥(γp→Q¯QX)= dy∗1d2q2⊥16π2(sx2)2α2(5)We use generalized gluon structure function of a protonΦp(x2,q22⊥)which is obtained in semihard approach.When integrated over transverse momentum q2⊥(q2⊥=(0, q2⊥,0)) of gluon up to some limit Q2it becomes the usual structure function giving the gluon momentum fraction distribution at scale Q2:Q2Φp(x,q22⊥)d q22⊥=xG p(x,Q2).(6) We use in our calculation following phenomenological parameterization[10]:Φp(x,q2⊥)=C0.05q2⊥)2, q2⊥>q20(x),(8) and q20(x)=Q20+Λ2exp(3.562π[1+(1−x1)2q4⊥].(9)The effective gluon distribution xG(x,Q2)obtained from(7)increases at not very small x (0.01<x<0.15)as x−ω0,whereω0=0.5corresponds to the BFKL Pomeron singularity [9].This increases continuously up to x=x0,where x0is a solution of the equation q20(x0)= Q2.In the region x<x0there is the saturation of the gluon distribution:xG(x,Q2)≃CQ2. The square of matrix element of partonic subprocessγ∗g∗→Q¯Q can be written as follows: |M|2=16π2e2Qαsα(x1x2s)2[1q2 1⊥q22⊥(1+α2β1sˆu−M2)2](10)For real photon and off-shell gluon it reads:|M|2=16π2e2Qαsα(x2s)2[α21+α22q2 2⊥(α1ˆt−M2)2],(11)whereα2=1−α1andˆs,ˆt,ˆu are usual Mandelstam variables of partonic subprocess ˆs=(p1+p2)2=(q1+q2)2,ˆt=(p1−q1)2=(p2−q2)2,(12)ˆu=(p1−q2)2=(p2−q1)2,ˆs+ˆt+ˆu=2M2+q21⊥+q22⊥.3Discussion of the ResultsThe results of our calculations for the total cross sections of c-and b-quark photoproduction are shown in Fig.3(solid curves).Dashed curves correspond to the SPM predictions with the GRV parametrization of the gluon distribution[24].The results of calculations for the SPM are shown without K-factor,wich have the typical value K=2for hard scattering processes.(In the semihard approach K-factor is absent[10,22]).The K-factor in the SPM may change the relation between the results of calculations in the semihard approach and SPM.But independently from it we see that the saturation effects more clearly are pronounced for charm quark photoproduction(at√sγp>200GeV.The p⊥distributions for c-and b-quark photoproduction in the semihard approach(solid curves)and in the SPM(dashed curves)are shown in Fig.4.The curves are obtained in the semihard approach for charm quark photoproduction show the saturation effects in low p⊥region(p⊥<2Gev/c).In middle p⊥region(2GeV/c<p⊥<15GeV/c)the heavy quark photoproduction p⊥distributions in the semihard approach are the higher ones of the SPM(with GRV paramatrization of gluon distribution).At high p⊥>15GeV/c we have contrary relation between p⊥distributions in the semihard approach and SPM. Thus the SPM leads to over-estimatedcross section in the low p⊥region and under-estimated one in the middle p⊥region as was noted as early as in Refs.[10,15](see also[22]).This behavior of p⊥distributions in the k⊥factorization approach is result from the offmass shell subprocess cross section[15]as well as the saturation effects of gluon structure function in semihard approach[10].Fig.5show the comparison of the results for heavy quark rapidity distribution(in the photon-proton center of mass frame)in the different models:solid curve shows the y dis-tribution in the semihard approach,dashed curve shows one in SPM.The discussed above effects are sufficiently large near the kinematic boundaries,i.e.at big value of|y∗|.We see that the difference between solid and dushed curvers can’t be degrade at all y∗via changeof normalization of SPM prediction.4ConclusionsWe shown that the semihard approach leads to the saturation effects for the total crosssection of charm quark photoproduction at available energies and predicts the remarkable difference for rapidity and transverse momentum distributions of charm and beauty quark photoproduction,which can be study already at HERA ep collider.AcknowledgementsThis research was supported by the Russian Foundation of Basic Research(Grant93-02-3545).Authors would like to thank J.Bartels,S.Catani,G.Ingelman,H.Jung,J.Lim, M.G.Ryskin and A.P.Martynenko for fruitfull discussions of the obtained results.One of us(N.Z.)gratefully acknowladges W.Buchmuller,G.Ingelman,R.Klanner,P.Zerwas and DESY directorate for hospitality and support at DESY,where part of this work was done.References[1]A.Ali et al.Proceedings of the HERA Workshop,Hamburg,ed.R.D.Peccei,1988,vol.1,p.395[2]A.Ali,D.Wyler.Proceedings of the Workshop”Physics at HERA”,eds W.Buchmullerand G.Ingelman,Hamburg,1992,vol.2,p.669;A.Ali.DESY93-105(1993)[3]M.A.G.Aivazis,J.C.Collins, F.I.Olness,Wu-Ki Tung.Southern Methodist Univer.preprint SMU-HEP/93-17.[4]M.Derrick et al.(ZEUS).Phys.Lett.B297(1992)404;B315(1993)481;B322(1994)287;B332(1994)228.T.Ahmed et al.(H1).Phys.Lett.B297(1992)205;A.De Roeck.DESY-94-005.[5]L.V.Gribov,E.M.Levin,M.G.Ryskin.Rhys.Rep.C100(1983)1[6]G.Martinnelli.Univ.of Roma preprint N842(1991)[7]R.K.Ellis,P.Nason.Nucl.Phys.B312(1989)551[8]V.N.Gribov,L.N.Lipatov.Sov.J.Nucl.Phys.15(1972)438;L.N.Lipatov.Yad.Fiz.20(1974);G.Altarelli,G.Parisi.Nucl.Phys.B126(1977)298;Yu.L.Dokshitzer.Sov.Phys.JETP46(1977)641[9]L.N.Lipatov.Sov.J.Nucl.Phys.23(1976)338;E.A.Kuraev,L.N.Lipatov,V.S.Fadin.Sov.Phys.JETP45(1977)199;Y.Y.Balitskii,L.N.Lipatov.Sov.J.Nucl.Phys.28 (1978)822;L.N.Lipatov.Sov.Phys.JETP63(1986)904[10]E.M.Levin,M.G.Ryskin,Yu.M.Shabelski,A.G.Shuvaev.Yad.Fiz.53(1991)1059;54(1991)1420[11]S.Catani,M.Chiafaloni,F.Hautmann.Nucl.Phys.B366(1991)135.[12]J.C.Collins,R.K.Ellis.Nucl.Phys.B360(1991)3.[13]E.M.Levin,M.G.Ryskin.Phys.Rep.189(1990)276;E.M.Levin.Review talk at Intern.Workshop on DIS and Related Subjects(Eilat,Israel,1994);FERMILAB-Conf-94/068-T(1994)[14]E.M.Levin,M.G.Ryskin,Yu.M.Shabelski,A.G.Shuvaev.DESY91-110(1991)[15]G.Marchesini,B.R.Webber.CERN-6495/92(1992);B.R.Webber.Proceedings of theWorkshop”Physics at HERA”,eds W.Buchmuller and G.Ingelman,Hamburg,1992, vol.1,p.285[16]P.Nason,S.Frixione,M.L.Mangano,G.Ridolfi.CERN-TH.7134/94[17]J.Smith,W.L.van Neerven.Nucl.Phys.B374(1992)36[18]enen,S.Rienersma,J.Smith,W.L.van Neerven.Nucl.Phys.B392(1993)162and229.[19]S.Frixione,M.L.Mangano,P.Nason,G.Ridolfi.CERN-TH.6921/93[20]S.Catani,M.Ciafaloni,F.Hautmann.Proceedings of the Workshop”Physics at HERA”,eds W.Buchmuller and G.Ingelman,Hamburg,1992,vol.2,p.690;CERN-TH.6398/92 [21]V.A.Saleev,N.P.Zotov.Proceedings of the Workshop”Physics at HERA”,Hamburg,1992,vol.1,p.637.[22]V.A.Saleev,N.P.Zotov.Modern Phys.Lett.A9(1994)151;Errata,A9(1994)1517[23]C.F.Weizsacker.Z.Phys.88(1934)612;E.J.Williams.Phys.Rev.45(1934)729[24]M.Gluck,E.Reya,A.Vogt.Z.Phys.C53(1992)127;Phys.Lett.B306(1993)391 Figure captions1.QCD diagrams for open heavy quark photoproduction subprocesses2.Diagram for heavy quark elecrtoproduction3.The total cross section for open charm and beauty quark photoproduction:solid curve-the semihard approach,dashed curve-the SPM4.The p⊥distribution for charm and beauty quark photoproduction at√sγp=200 GeV:curves as in Fig.3。