Heavy quark potential in the instanton liquid model
Search for light-to-heavy quark flavor changing neutral currents in $nu _{mu}N$ and $bar{nu
a rXiv:h ep-e x /759v131Jul2Search for light-to-heavy quark flavor changing neutral currents in νµN and ¯νµN scattering at the Tevatron A.Alton,T.Adams,T.Bolton,J.Goldman,M.Goncharov,D.Naples ∗Kansas State University,Manhattan,KS,66506R.A.Johnson,M.Vakili,†N.Suwonjandee University of Cincinnati,Cincinnati,OH,45221J.Conrad,B.T.Fleming,J.Formaggio,J.H.Kim,§S.Koutsoliotas,‡C.McNulty,A.Romosan,∗∗M.H.Shaevitz,P.Spentzouris,††E.G.Stern,A.Vaitaitis,E.D.Zimmerman Columbia University,New York,NY,10027R.H.Bernstein,L.Bugel,mm,W.Marsh,P.Nienaber,‡‡J.Yu Fermi National Accelerator Laboratory,Batavia,IL,60510L.de Barbaro,D.Buchholz,H.Schellman,G.P.Zeller Northwestern University,Evanston,IL,60208J.Brau,R.B.Drucker,R.Frey,D.MasonUniversity of Oregon,Eugene,OR,97403S.Avvakumov,P.de Barbaro,A.Bodek,H.Budd,D.A.Harris,††K.S.McFarland,W.K.Sakumoto,U.K.YangUniversity of Rochester,Rochester,NY,14627Version3.0.0(February7,2008)AbstractWe report on a search forflavor-changing neutral-currents(FCNC)in theproduction of heavy quarks in deep inelasticνµN and¯νµN scattering bythe NuTeV experiment at the Fermilab Tevatron.This measurement,madepossible by the high-purity NuTeV sign-selected beams,probes for FCNC inheavyflavors at the quark level and is uniquely sensitive to neutrino couplingsof potential FCNC mediators.All searches are consistent with zero,and limitson the effective mixing strengths|V uc|2,|V db|2,and|V sb|2are obtained.Typeset using REVT E XI.INTRODUCTIONFlavor-changing neutral current(FCNC)interactions of c-and b-quarks appear in a num-ber of extensions to the Standard Model(SM)of particle physics including extra quark gen-erations[?,?],technicolor[?,?,?,?,?],multiple Higgs sectors(as in supersymmetry)[?,?,?,?], left-right symmetric models[?],and leptoquarks[?,?].Evidence for FCNC effects in the heavy quark sector beyond higher order SM processes has not yet been observed.Present limits on FCNC result from searches for rare decays of charm[?]and beauty mesons [?,?,?],in particular decays of the type D→ℓ+ℓ−X and B→ℓ+ℓ−X,whereℓ=e or µ,D= D+,D0,D+S ,B=(B+,B0,B0S);and X is nothing,a pseudoscalar,or a vectormeson.The c→u transitions are particularly sensitive to new physics since loop level SM-FCNC decays are severely suppressed by the Cabibbo-Kobayashi-Maskawa(CKM)matrix. While experimental signatures for FCNC in D and B decays are clear,their interpretation is ambiguous.Meson decay rates depend on one or more incalculable hadronic form factors. In addition,experimentally attractivefinal states such as D0→e+e−and B0→µ+µ−are helicity-suppressed,which obscures dynamical roles played by particular FCNC models.This article presents an alternative search for FCNC processes in the DIS data of the NuTeV experiment;where either neutrinos or anti-neutrinos interact with a massive iron target.Flavor changing effects can be sought in the reactionsνµN→νµcX,c→µ+X′,(1)νµN→νµ¯bX,¯b→µ+X′,(2)νµN→νµbX,b→cX′,c→µ+X′′,(3) and their charge-conjugates.The experimental signature in the detector is a muon of oppo-site lepton number from the beam neutrino.It is possible to isolate thisfinal state because NuTeV ran with a high purity sign-selected beam in which the¯νµ/νµevent ratio in neutrino mode and theνµ/¯νµratio in anti-neutrino mode were0.8×10−3and4.8×10−3,respectively. Because of the semi-inclusive character of the measurement,FCNC effects in neutrino scat-tering can be probed at the quark,rather than the hadron level.Furthermore,neutrino scattering is particularly sensitive to any FCNC process mediated by an intermediate neu-tral object that couples more strongly to neutrinos than to charged leptons(e.g.,a Z0-like coupling).II.EXPERIMENTAL APPARATUS AND BEAMThe NuTeV(Fermilab-E815)neutrino experiment collected data during the1996-97fixed target run with the refurbished Lab E neutrino detector and a newly installed Sign-Selected Quadrupole Train(SSQT)neutrino beamline.The sign-selection optics of the SSQT pick the charge of secondary pions and kaons which determines whetherνµor¯νµare predominantly produced.During NuTeV’s run the primary production target received1.13×1018and 1.41×1018protons-on-target in neutrino and anti-neutrino modes,respectively.The SSQT and its performance are described in detail elsewhere[?].The Lab E detector[?],consists of two major parts;a target calorimeter and an iron toroid spectrometer.The target calorimeter contains690tons of steel sampled at10cm inter-vals by843m×3m scintillator counters and at20cm intervals by423m×3m drift chambers. The toroid spectrometer consists of four stations of drift chambers separated by iron toroid magnets.Precision hadron and muon calibration beams monitored the calorimeter and spectrometer performance throughout the course of data taking.The calorimeter achievesa sampling-dominated hadronic energy resolution ofσEHAD /E HAD=2.4%⊕87%/√A.Introduction and Data SelectionThe analysis technique consists of comparing the distributions of y V IS= E HAD/(E HAD+Eµ)measured in theνµand¯νµwrong sign muon(WSM)data samples to a Monte Carlo(MC)simulation containing all known conventional WSM sources and a possible FCNC signal.The FCNC signal peaks at high y V IS because the decay muon from the heavyflavor hadron is usually much less energetic than the hadron energy produced in the NC interaction.The largest backgrounds,from beam impurities,are concentrated at low y V IS inνµand distributed evenly across y V IS in¯νµmode due to the respective(1−y)2 and uniform-in-y characteristics of the CC interactions of wrong-flavor beam backgrounds.Events in the WSM sample must satisfy a number of selection criteria(“cuts”).The fiducial volume cut requires that event vertices be reconstructed at least25cm-Fe(cm of iron-equivalent)from the outer edges of the detector in the transverse directions,at least35 cm-Fe downstream of the upstream face of the detector,and at least200cm-Fe upstream of the toroid.Events must possess a hadronic energy of at least10GeV,and exactly one track(the muon)must be found.The muon is required to be well-reconstructed and to pass within the understood regions of the toroid’s magneticfield.The muon’s energy must be between10and150GeV,and its charge must be consistent with having the opposite lepton number as the primary beam component.Requiring that the muon energy reconstructed in different longitudinal sections of the toroid agree within25%of the value measured using the full toroid reduces charge mis-identification backgrounds to the2×10−5level.Finally,for the purposes of thefinal FCNCfit,the reconstructed y V IS is required to be larger than0.5. With these cuts there are207ν-mode and127¯ν-mode WSM events remaining in NuTeV’s nearly2million single muon sample.B.Source and Background SimulationsConventional WSM sources arise from beam impurities,right-flavor charged current(CC) events where the charge of the muon is mis-reconstructed,CC and NC events where aπor K decays in the hadron shower,charged current(CC)charm production where the primary muon is not reconstructed or the charm quark is produced via aνe interaction,and neutral current(NC)c¯c pair production.Single charm CC production and NC c¯c pair production background sources produce broad peaks at high y V IS and must be handled with care. Table??gives the fractional contribution of each background component.The relatively large beam impurity background consists of contributions from hadrons(including charm) that decay before the sign-selecting dipoles in the SSQT,neutral kaon decays,muon decays, decay of hadrons produced by secondary interactions in the SSQT(“scraping”),and from decay of wrong-sign pions produced in kaon decays.Table??summarizes the relative contributions of each beam source.A complete GEANT[?]simulation of the SSQT is used to model beam impurities. This simulation uses Malensek’s[?,?]parameterization for hadron production from the pri-mary target.Scraping contributions are modeled by GHEISHA[?].Production of K0L is handled by extending Malensek’s charged kaon parameterizations using the quark count-ing relation K0L=(3K−+K+)/4.Charm production is parametrized using available data from800GeV proton beams[?,?].GEANT properly handles cascade decays such as K±→π±π±π∓,π∓→µ∓¯νµ(νµ)andπ±→µ±¯νµ(νµ),µ±→e±¯νµ(νµ)νe(¯νe).The NuTeV detector is likewise modeled with a GEANT-based hit-level MC simulation.Wrong-sign muons generated from theflux simulation are propagated through the detector MC and then reconstructed using the same package that is used for data reconstruction.A fast parametric MC is also used to compare the high statistics right-signflux simulation to data inνµand¯νµ.These comparisons showed that the SSQT dipoles required a downward shift of-2.5%from their nominal values.The right-sign comparisons after these shifts,are shown in Fig.??and indicate agreement between predictedflux and data at roughly the2%level.The high density target-calorimeter suppresses WSM contributions fromπ/K decay in the hadron shower;their contribution is estimated from a previous measurement of µ-production in hadron showers using the same detector[?].The small charge mis-identification contribution is estimated by passing a large sample of simulated events through the full detector MC and event reconstruction.After impurities,the next largest WSM source comes from CC production of charm in which the charm quark decays semi-muonically(dimuon)and its decay muon is picked up in the spectrometer while the primary lepton is either an electron or a muon which exits from or ranges out in the calorimeter.Theνe beam fraction is1.9(1.3)%inν(¯ν)-mode, and22%of the CC charm events which pass WSM cuts originate from aνe.The CC charm background is simulated using a leading-order QCD charm production model with produc-tion,fragmentation,and charm decay parameters tuned on neutrino dimuon data collected by NuTeV[?]and a previous experiment using the same detector[?].Overall normalization of the source is obtained from the measured charm-to-total CC cross section ratio and the single muon right-sign data sample.Simulated dimuon events are passed through the full GEANT simulation of the detector.Fig.??provides a check of the modeling of this source through a comparison of the distribution of y′V IS=E HAD/(E HAD+Eµ2),where Eµ2is the energy of the WSM in the event,between data and MC for dimuon events in which both muons are reconstructed by the spectrometer.This distribution should closely mimic the expected background to the y V IS distribution in the WSM sample.Aχ2comparison test between data and model yields a value of19for17degrees of freedom.Finally,NC c¯c production produces a WSM when the c(¯c)decays semi-muonically in νµ(¯νµ)mode.An excess over other sources at high y V IS indicates that this source is present in the data;its analysis[?]will appear in a forthcoming publication.For the FCNC search, NC charm production is simulated at production level by a Z0−gluon fusion model[?]with charm mass parameter m c=1.70±0.19GeV/c2taken from a next-to-leading(NLO)order QCD analysis of CC charm production[?]and using the GRV94HO[?]gluon parton distribution function(PDF).The NLO charm mass is used because it is influenced in partby contributions from W−gluon fusion diagrams similar to the Z0−gluon process.Note that the value of m c used is larger than that obtained in LO analyses of CC charm production. This choice tends to reduce the NC charm contribution to the WSM sample and results in more conservative limits on FCNC production.The NC charm quarks are fragmented and decayed using procedures adapted from the CC charm simulation,and the resulting WSM events are then simulated with the full MC.IV.RESULTS AND INTERPRETATIONA.FCNC ProductionThe neutrino FCNC u→c cross section can be parameterized to LO in QCD asdσ(νµu→νµc;c→µ+)V cd 2 cos2β+sin2β(1−y)(1−xy/ξ)dξdy.(4)Here V cd is the c→d CKM matrix element,V uc1represents a possible u→c coupling, sin2βgives the fraction of right-handed coupling of the c−quark to the FCNC,y is the inelasticity,andξ≃x(1+m2c/Q2)is the fraction of the nucleon’s momentum carried by the struck u−quark,with x the Bjorken scaling variable,Q2the squared momentum trans-fer,and m c the effective charm quark mass.The d→c charged current cross section dσ(νµd→νµc;c→µ+)/dξdy is measured in the same experiment[?,?].Since the u and d quark distributions are identical in an isoscalar target,the FCNC cross section should experience the same charm mass suppression as the analogous CC charm production.Frag-mentation of subsequent semi-muonic decays of charmed mesons should also be identical forFCNC and CC-charm production.One therefore expects the extracted V uc to have little model dependence.For FCNC bottom production there is as yet no measured CC analogfinal state.There-fore,the explicit LO QCD cross section,dσ νµN→νµ¯bX π cos2β′(1−y)(1−xy/ξ)+sin2β′(1−y+xy/ξ) (5)× ¯u ξ′,Q2 +¯d ξ′,Q2 ,where M is the nucleon mass and E is the neutrino energy,must be convolved with b-quark fragmentation functions for mesons of type B i(D i b)and B i meson decay distribution functions(∆i B)multiplied by appropriate branching fractions(F i B)to yield a WSM cross section:dσ νµN→νµ¯b;¯b→µ+ dξ′dy.(6) The struck quark momentum fractionξ′becomesξ′≃x(1+m2b/Q2),with m b=4.8GeV/c2 the effective b-quark mass.It is also possible for FCNC b-production to form a WSM muon signal through the cascade b→c→µ+.This mode offers the advantages of the larger and higherξvalence d-quark PDF at the cost of reduced acceptance for the softer c-decay muon.A similar expression holds for FCNC s→b transitions with the replacements u(ξ′,Q2)+d(ξ′,Q2)→2s(ξ′,Q2),|V bd|2→|V bs|2,and sin2β′→sin2β′′.Production cross sections for both c-and b-FCNC sources are computed from the GRV94LO PDF set[?]for several choices of right-left coupling admixtures.Acceptance for a charm FCNC-WSM signal is calculated using a fragmentation-decay model tuned to NuTeV and CCFR dimuon data[?].For FCNC-WSM from b-quarks,fragmentation and decays are handled with the Lund string fragmentation model[?].Detector response is simulated with the full hit-level MC.B.Fits to DataBinned likelihoodfits are performed to the y V IS distributions of the data using a model consisting of all conventional WSM sources described above and an FCNC source.The fit varies the level,but not the shape,of the FCNC signal contribution.The NC charm contribution is also varied in shape and level by allowing m c tofloat within its errors.The three FCNC sources(u→c,d→b,and s→b)are treated separately.Only neutrino data is used for the u→c,but both modes are used for FCNC bottom production to exploit the possibility of a cascade decays to charm.A series offits are performed for each FCNC source,corresponding to different mixtures of right and left-handed FCNC couplings to the quarks;a typical result is shown in Fig.??.In all cases,the signal for FCNC is within±2.0σof zero,and limits are set accordingly. Since Gaussian statistics apply,the90%confidence level upper limit is set by adding1.64σto the best-fit value if the best-fit value is positive,or1.64σto zero if the bestfit is negative. Here,σconsists of the statistical error from thefit added in quadrature to the estimated systematic error described in the next section.Table??summarizes thefit results.C.Systematic ErrorsThe dominant systematic errors result from modeling the rejection of CC charm events, and the overall normalization of CC charm events.Estimates of systematic uncertainties are obtained by varying the event selection procedure as well as parameters characterizing the detector response and physics models.Errors are assumed to be independent.Charged current charm events are removed by requiring that exactly one track be found and reconstructed by the NuTeV tracking software.Another independent way to remove dimuons is to use calorimeter information.The stop parameter is thefirst of three con-secutive counters downstream of the interaction,each with less than1.5MIPs.The stop cut requires that the distance between the interaction and the stop counter be less than15counters.Replacing the tracking cut with the stop cut gives the systematic errors listed in Table??.The next largest systematic error is due to the normalization of CC charm events.Nor-malization of these events is obtained from the right-sign muon CC sample.One can also normalize CC charm events with only one reconstructed track,to those with both tracks found.These normalizations disagree by3%resulting in the systematic errors listed in Ta-ble??.Systematic errors due to the beam normalization,detector calibration,and other sources are small.parison to Limits from DecaysFor comparison purposes,the following expressions are used to relate FCNC heavyflavor meson decay branching fractions(BF)to the parameter V uc:BF D0→ℓ+ℓ− =2 V uc m2µBF D+S→µ+νµ ,(7)BF D+→π+ℓ+ℓ− = V ucV cs 2BF D+S→ηℓ+νℓ .(9) For estimates of V db and V sb from B decays,it is assumed thatBF B0→ℓ+ℓ− =2 V bd m2µBF B+→µ+νµ ,(10)BF B+→π+ℓ+ℓ− = V bdV ub 2m2ℓV cb 2BF B+→D0ℓ+νℓ .(13) Measured values[?]are used for the branching fractions on the right hand side except for the leptonic decay B+→µ+νµ,for which it is assumed thatBF B+→µ+νµ =2.2×10−6(f B/200MeV)2,(14)with f B=200MeV,the B decay constant.Table??summarizes the limits on|V uc|2,|V bd|2,and|V sb|2from meson decays.We note that our overall limits from neutrino scattering,which would approximately correspond to decay searches of the type D→νµ¯νµX and B→νµ¯νµX,are generally weaker than the decay search limits.Our result for V db is competitive,and we have effectively added new modes to the search that do not depend on specific mechanisms for heavy meson decay.V.CONCLUSIONIn this paper we have established a new method for probing FCNC processes in deep inelastic neutrino scattering.Our experiment tests for FCNC at the inclusive quark level, and we are particularly sensitive to any FCNC process in which the mediatingfield couples more strongly to neutrinos than to charged leptons.We observe no evidence for FCNC interactions,and we set limits on the effective mixing elements|V uc|2,|V bd|2,and|V bs|2at the10−3level.ACKNOWLEDGMENTSWe would like to thank the staffs of the Fermilab Particle Physics and Beams Divisions for their contributions to the construction and operation of the NuTeV beamlines.We would also like to thank the staffs of our home institutions for their help throughout the running and analysis of NuTeV.This work has been supported by the U.S.Department of Energy and the National Science Foundation.REFERENCES[1]V.Barger,M.S.Berger,and R.J.N.Phillips,Phys.Rev.D53(1995)1663.[2]K.S.Babu,et al.,Phys.Lett.B205(1988)540.[3]S.Dimopolous,H.Georgi,and S.Raby,Phys.Lett.B127(1983)101.[4]V.A.Miransky,S.Peris,and S.Raby,Phys.Rev.D47(1993)2058.[5]C.D.Carone and R.T.Hamilton,Phys.Lett.B301(1993)162.[6]E.Eichten and ne,Phys.Lett.B90(1980)125.[7]R.S.Chikula,H.Georgi,and L.Randall,Nucl.Phys.B292(1987)93.[8]M.Luke and M.J.Savage,Phys.Lett.B307(1993)387.[9]J.L.Diaz-Cruz and G.L.Castro,Phys.Lett.B301(1993)405.[10]K.Koike,Prog.Theor.Phys.91(1994)161.[11]J.G.Koner,A.Pilaftsis,and K.Schilcher,Phys.Rev.D47(1994)1080.[12]F.J.Gilman,K.Kleinknecht,and B.Renk,Phys.Rev.D50(1994)1315.[13]W.Buchmuller,D.Wyler,Phys.Lett.B177(1986)377.[14]S.Davidson,D.Bailey,and B.A.Campbell,Z.Phys.C61(1994)313.[15]E.M.Aitala et al.(Fermilab E791Collaboration),Phys.Lett.B462(1999)401.[16]T.Affolder,“Search for the Flavor-Changing Neutral Current Decays B+→µ+µ−K+ and B0→µ+µ−K∗0”,hep-ex/9905004.[17]F.Abe et al.[CDF Collaboration],Phys.Rev.D57,3811(1998).[18]A.Weir,Phys.Rev.D41,1384(1990)[19]C.Caso et al.,Eur.Phys.J.C3,1(1998).[20]M.Gluck,E.Reya,and A.Vogt,Z.Phys.C67,433(1995).[21]NuTeV Collaboration,J.Yu,et al.,Report No.FERMILAB-TM-2040,1998.[22]D.A.Harris et al.[NuTeV Collaboration],Nucl.Instrum.Meth.A447,377(2000) [hep-ex/9908056].B.King,et al.,ibid.302,254(1991).[23]M.Gluck,R.M.Godbole,and E.Reya,Z.Phys.C38,441(1988).[24]CERN CN/ASD,“GEANT Detector Description and Simulation Library”(1998).[25]A.J.Malensek,“Empirical Formula For Thick Target Particle Production,”FERMILAB-FN-0341.[26]H.W.Atherton et al.,CERN-80-07.[27]H.Fesefeldt,PITHA-85-02.[28]R.Ammar et al.,Phys.Rev.Lett.61,2185(1988).[29]K.Kodama et al.[Fermilab E653Collaboration],Phys.Lett.B263,573(1991).[30]M.Goncharov et al,to be published in Proceedings of the34th Rencontres de Moriond, QCD and Hadronic Interactions(2000).[31]B.Anderson et al.,Phys.Rep.97,1983.[32]P.Sandler,Ph.D.Thesis,Univ.of Wisconsin,1992.[33]A.Alton,Ph.D.Thesis,Kansas State University,2000.[34]A.O.Bazarko et al.[CCFR Collaboration],Z.Phys.C65,189(1995)[35]N.Witchey,Ph.D.Thesis,Ohio State University,1996.[36]R.Ammar et al.[CLEO Collaboration],Phys.Rev.D49,5701(1994).[37]P.Avery et al.[CLEO Collaboration],Phys.Lett.B183,429(1987).[38]M.Acciarri et al.[L3Collaboration],Phys.Lett.B391,474(1997).FIGURESparison ofνand¯νCC energy spectrum for data(pluses)to MC(histogram)using the GEANT-basedflux.parison of data to MC of E had/(E had+Eµ2)for dimuon events with two toroid-analyzed muons.parison of y V IS distributions of data(pluses)to predictions of all Standard Model sources(solid)of WSM’s and the FCNC signal(dashed).The plot on the left is neutrino mode, while that on the right is anti-neutrino mode.TABLESSourceν-mode(%)¯ν-mode(%)TABLE I.Percentage of WSM’s for each source in a given mode.ν-mode¯ν-modeEν>20GeV Eν>20GeVother prompt9%22%muon decay11%11%K→π→µ5%2%u→c0.0(1.1±1.5±0.5)×10−3 3.7×10−30.10(1.2±1.7±0.9)×10−3 4.4×10−30.35(1.6±2.2±2.6)×10−37.2×10−30.65(2.5±3.6±5.4)×10−313.1×10−30.90(4.1±7.9±7.9)×10−322.4×10−31.00(4.4±13.7±8.7)×10−334.5×10−3 d→b0.00(.3±1.3±0.7)×10−32.7×10−30.10(-0.15±1.2±0.7)×10−3 2.3×10−30.35(-1.2±1.2±0.7)×10−3 2.2×10−30.65(-1.6±0.90±0.6)×10−3 1.8×10−30.90(-1.4±0.79±0.6)×10−3 1.6×10−31.00(-1.3±0.68±0.6)×10−3 1.5×10−3 s→b0.0(-17.3±17.3±3.51)×10−329×10−30.10(-13.6±6.6±3.3)×10−312×10−30.35(-3.6±1.9±2.7)×10−3 5.4×10−30.65(-1.9±1.0±2.0)×10−3 3.7×10−30.90(-1.4±0.7±1.5)×10−3 2.7×10−31.00(-1.3±0.7±1.2)×10−32.3×10−3TABLE III.Results of the FCNCfits.u→c L0.30×10−30.32×10−30.23×10−30.01×10−30.50×10−3 u→c R8.08×10−3 3.22×10−30.24×10−30.02×10−38.70×10−3s→b L 2.37×10−30.21×10−30.91×10−32.42×10−23.51×10−2 s→b R 1.08×10−30.06×10−30.20×10−30.54×10−21.22×10−2D±→π±µ±µ∓ 1.7×10−5D±→π0l±νl 2.3×10−4 2.7×10−4[?]B±→K±e±e∓ 3.9×10−5B0→D0l±νl 2.4×10−5 2.1×10−5[?] TABLE V.Limits on FCNC couplings from meson decay searches,with|V|2=|V uc|2,|V db|2,or |V sb|2as appropriate.Equations??-??relate branching fraction(BF)limits to the|V|2limits in the table.。
How Much do Heavy Quarks Thermalize in a Heavy Ion Collision
arXiv:hep-ph/0412346v3 16 Mar 2005
Derek Teaney Department of Physics & Astronomy, SUNY at Stony Brook, Stony Brook, NY 11764, USA
(Dated: February 2, 2008)
it may be difficult to distinguish these two cases [19, 22]. Medium modifications of the heavy quark spectrum RAA flow will be studied experimentally this year and will provide an experimental estimate of this relaxation time [23, 24, 25, 26, 27]. Most recent studies of the medium modifications of the charm spectrum have computed the energy loss of a heavy quark by gluon bremsstrahlung [28, 29, 30, 31]. In weak coupling (which is the framework in which all calculations have been performed), bremsstrahlung is the dominant energy loss mechanism only if the heavy quark is ultra-relativistic, γv ≫ 1/g . (Similarly, for an electron traversing a hydrogen target, bremsstrahlung losses first exceed ionization losses when γv ≃ 700 [32].) For much of the measured momentum range, the heavy quark is not ultra-relativistic, γv < ∼ 4, and in this case it is far from clear that radiative energy loss dominates over collisional energy loss. About two thirds of all heavy quarks are produced with p < ∼ M , and therefore radiative energy loss should be neglected when studying bulk thermalization. When γv > ∼ 4, calculations do suggest that radiation dominates the average energy loss rate [33, 34]. However, as has been repeatedly emphasized [35, 36], the average energy loss is insufficient to describe the medium modifications of the spectrum RAA . Collisions have a different fluctuation spectrum than radiation and therefore might contribute more to the suppression factor than was at first anticipated [37]. Since we are primarily interested here in heavy quarks with typical momenta γv ∼ 1, we will concentrate exclusively on elastic collisions. Considering these points, we will re-examine collisional energy loss of a heavy quark in the perturbative QGP. Our tools are Hard Thermal Loop (HTL) perturbation theory and a heavy quark expansion (M ≫ T ). The average energy loss rate was first computed by Braaten and Thoma [38] and we will independently verify their results. (Recently this calculation was extended to anisotropic plasmas by Romatschke and Strickland [39].) We will also compute the rates of longitudinal and transverse momentum broadening which are essential to a complete calculation of the modification factor RAA . We will relate all of these rates to the diffusion coefficient which we will compute. (In principle the diffusion coefficient of a heavy quark could have been gleaned from the results of Braaten and Thoma [38] and Svetitsky [40].) In perturbation theory, we can compare the diffusion coefficient of the heavy quark to the hydrodynamic time scale η/(e + p) which was calculated previously [41, 42, 43]. Many of the ambiguities of perturbation theory cancel in the ratio of transport coefficients and we therefore hope to be able to extrapolate smoothly into the non-perturbative domain. Following this ideology, we express all of our phenomenological results in terms of the diffusion coefficient which may ultimately be determined from lattice QCD calculations. With these transport properties in hand, we adopt a Langevin model for the equilibration of heavy quarks in heavy ion collisions. The Langevin equations correctly describe the kinetics of a heavy particle in a thermal medium and therefore naturally interpolate between a hydrodynamic regime at small momentum and a kinetic regime at large momentum. The model is similar to old work by Svetitsky [40] and was later used without fluctuations to estimate RAA [34]. The model is strictly valid for non-relativistic quarks and for all velocities to leading logarithm in T /mD . The corresponding Fokker-Planck equation is solved analytically in Section V for a Bjorken expansion. The solution provides a simple estimate for the modification factor RAA and further elucidates the dynamics of equilibration. Then we solve the Langevin equations numerically in a hydrodynamic simulation of the heavy ion reaction. The results of the simulation are the medium modification factor RAA and the corresponding elliptic flow v2 (pT ) as a function of the diffusion coefficient. 3
Two lectures on heavy quark production in hadronic collisions
Abstract These lectures present a pedagogical introduction to the physics of heavy-flavour production in hadronic collisions. The first lecture gives the theoretical background, with a discussion of leading-order calculations and of the effects of next-to-leading-order corrections. The origin and implications of the large logarithmic corrections appearing at this order are presented in an elementary way. The second lecture provides a survey of current experimental data on charm and bottom production, and describes their comparison with theoretical predictions. We emphasize the role played by some non-perturbative effects in the determination of charm distributions, and study the theoretical systematic uncertainties which affect our predictions.
CERN-TH/97-328 hep-ph/9711337
Test of the heavy quark-light diquark approximation for baryons with a heavy quark
The set of coordinates more adequate for a heavy quark-light diquark description are the Jacobi coordinates (See Fig. 1)
PACS numbers: 12.39.Jh,14.20.Lq,14.20.Mr
I. INTRODUCTION
Heavy quark symmetry [1, 2, 3, 4, 5, 6] (HQS) predicts that in baryons with a heavy quark, and up to corrections in the inverse of the heavy quark mass, the light degrees of freedom quantum numbers are well defined, in particular the total spin of the light degrees of freedom is well defined. This prediction has been taken in different calculations as the basis for treating the light quark subsystem as a diquark, and the baryon as a heavy quark-light diquark (HQLD) system [7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. This HQS prediction does not imply though that the orbital motion of the two light quarks is not affected by the presence of the heavy quark as it seems to be implicit in the HQLD approximation 1. Very recently the diquark structure of heavy baryons have been analyzed in Λc production in heavy ion collisions [17] where its enhanced yield is seen as a signal for the existence of light diquark correlations both in the quark gluon plasma and the heavy baryon.
庞加莱 相对论
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潮汐锁定英语作文
潮汐锁定英语作文The Tidal LockThe concept of tidal locking is a fascinating phenomenon that has captivated the minds of scientists and astronomers for centuries. This unique process occurs when a celestial body's rotational period is equal to its orbital period around another body, resulting in one side of the object permanently facing the other. This phenomenon is particularly prevalent in binary star systems and planets with moons, where the gravitational interactions between the bodies lead to this remarkable synchronization.One of the most well-known examples of tidal locking is the relationship between the Earth and the Moon. The Moon's rotation period is exactly the same as its orbital period around the Earth, causing the same side of the Moon to always face our planet. This tidal lock has had a profound impact on the Earth-Moon system, shaping the dynamics and evolution of both bodies.The tidal locking process is driven by the gravitational forces exerted by the larger body on the smaller one. As the smaller body orbits the larger one, the uneven distribution of mass within the smaller bodycreates a gravitational imbalance. This imbalance causes a slight bulge on the side of the smaller body closest to the larger one, and a smaller bulge on the opposite side. These bulges, known as tidal bulges, create a torque that acts to slow down the smaller body's rotation until it matches its orbital period.Over time, as the smaller body's rotation slows down, the tidal bulges become more pronounced, further reinforcing the tidal locking process. This feedback loop continues until the smallerbody's rotation period is exactly equal to its orbital period, resulting in the permanent synchronization of the two bodies.The consequences of tidal locking are far-reaching and have significant implications for the habitability and evolution of celestial bodies. In the case of the Earth-Moon system, the tidal locking has led to the stabilization of the Earth's axial tilt, which is crucial for the maintenance of a stable climate and the development of complex life. The Moon's gravitational influence also plays a crucial role in the generation of tides, which have shaped the coastlines and influenced the evolution of marine life on Earth.Beyond the Earth-Moon system, tidal locking is observed in many other celestial bodies throughout the universe. For example, many of the moons of Jupiter and Saturn are tidally locked to their parent planets, and some exoplanets in binary star systems are also believedto be tidally locked to their host stars.The study of tidal locking has also led to important insights into the formation and evolution of planetary systems. By understanding the dynamics of tidal locking, scientists can better model the long-term stability and habitability of exoplanetary systems, as well as the potential for the development of life on other worlds.In conclusion, the tidal locking phenomenon is a remarkable example of the complex and intricate processes that shape the universe around us. From the Earth-Moon system to the most distant exoplanets, this fundamental principle of gravitational interactions continues to captivate and inspire scientists, offering new insights into the nature of our cosmos and the potential for life beyond our own planet.。
Nonperturbative Flow Equations and Low--Energy QCD
∗
1
dynamics can be encoded in the masses and interactions of mesons. Any attempt to deal with this situation analytically and to predict the meson properties from the short distance physics (as functions of the strong gauge coupling αs and the current quark masses mq ) has to bridge the gap between two qualitatively different effective descriptions. Two basic problems have to be mastered for an extrapolation from short distance QCD to mesonic length scales: • The effective couplings change with scale. This does not only concern the running gauge coupling, but also the coefficients of non–renormalizable operators as, for example, four quark operators. Typically, these non–renormalizable terms become important in the momentum range where αs is strong and deviate substantially from their perturbative values. Consider the four–point function which obtains after integrating out the gluons. For heavy quarks it contains the information about the shape of the heavy quark potential whereas for light quarks the complicated spectrum of light mesons and chiral symmetry breaking are encoded in it. At distance scales around 1fm one expects that the effective action resembles very little the form of the classical QCD action which is relevant at short distances. • Not only the couplings, but even the relevant variables or degrees of freedom are different for long distance and short distance QCD. It seems forbiddingly difficult to describe the low–energy scattering of two mesons in a language of quarks and gluons only. An appropriate analytical field theoretical method should be capable of introducing field variables for composite objects such as mesons. A conceptually very appealing idea for our task is the block–spin action [1, 2]. It realizes that physics with a given characteristic length scale l is conveniently described by a functional integral with an ultraviolet (UV) cutoff Λ for the momenta. Here Λ should be larger than W l−1 but not necessarily by a large factor. The Wilsonian effective action SΛ replaces then the classical action in the functional integral. It obtains by integrating out the fluctuations with 2 W momenta q 2 > ∼ Λ . An exact renormalization group equation [2]–[6] describes how SΛ changes with the UV cutoff Λ. We will use here the somewhat different but related concept of the effective average action [7] Γk which, in the language of statistical physics, is a coarse grained free energy with coarse graining scale k . The effective average action is based on the quantum field theoretical concept of the effective action [8] Γ which obtains by integrating out all quantum fluctuations. The effective action contains all information about masses, couplings, form factors and so on, since it is the generating functional of the 1P I Green functions. The field equations derived from Γ are exact including all quantum effects. For a field theoretical description of thermal equilibrium this concept is easily generalized to a temperature dependent effective action which includes now also the thermal fluctuations. In statistical physics Γ describes the free energy as a functional of some convenient (space dependent) order parameter, for instance the magnetization. In particular, the behavior of Γ for a constant order parameter (the effective potential) specifies the equation of state. The effective average action Γk is a simple generalization of the effective 2 action, with the distinction that only quantum fluctuations with momenta q 2 > ∼ k are included. This can be achieved by introducing in the functional integral defining the partition function (or 2
Production of (super)heavy quarkonia and new Higgs physics at hadron colliders
G.A. Kozlov, A.N. Sissakian, J.I. Khubua, G. Arabidze and G. Khoriauli Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region, Russia e-mail: kozlov@thsun1.jinr.ru and T. Morii Div. of Sciences for Natural Environment, Faculty of Human Development, Kobe University, Kobe, Japan e-mail: morii@kobe-u.ac.jp
arXiv:hep-ph/0305006v2 6 May 2003
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¯ ) composed of a (super)heavy quark Q and an antiquark A study on quarkonia T (QQ ¯ Q (a possible and interesting candidate of Q is the up(U )- and/or down(D )-quarks in the fourth generation family) is required in current particle physics for testing the standard model (SM) and/or searching for signals for physics beyond the SM. It is nowadays one of the most interesting subjects since the subject can be studied, with high priority, in the forthcoming experiments at high energy hadron colliders, i.e. the Tevatron and LHC. In particular, no theoretical arguments are seen to rule out the (super)heavy quarks and the (super)heavy quarkonium states with the masses around hundred GeV or even a few TeV (see, e.g. [1]). As the recent example, in one of the extended models, the little Higgs model [2], there could be an additional heavy quark q ˜ with a mass of order O(1 TeV) to promote the quark doublet q to quark triplet under the global SU(3), Q = (q, q ˜). By preserving the global symmetry of the coupling, the one-loop quadratic divergence to the top quark is removed. In fact, the SM and its extensions, e.g., the Minimal Supersymmetric Standard Model (MSSM), do not explain the family (generation) structure of the quark masses. Each quark has an arbitrary Yukawa coupling and hence is independent of the family to which it belongs. It is required to explain the family structure and the Cabibbo-Kobayashi-Maskawa (CKM) matrix for the quark sector in any extension of the SM or even in the SM. 1
The QCD heavy-quark potential to order v^2 one loop matching conditions
(4)
In Eq. (10), the fields ψ and χ annihilate a quark and an antiquark, respectively. The fields are labelled by a momentum p, and a greek index for their color and spin. The operator in Eq. (4) is local on the scale x ∼ 1/mv 2 , but non-local on the scale p ∼ mv . The potential V is computed as a matching coefficient at the scale µ = m between QCD and an effective theory for non-relativistic QCD valid below the scale m. The effective theory is constructed to have the same infrared structure as perturbative QCD for the two heavy quark system. Therefore, defining the potential as a matching coefficient provides an infrared safe definition. For instance, Ref. [17] shows how the three-loop matching potential V is infrared safe, despite the divergence in the QCD potential of Appelquist, Dine and Muzinich [3]. Although several different formulations of the effective theory for non-relativistic QCD are currently in use [5–16,18–20], certain universal features have emerged. The on-shell degrees of freedom in the effective theory include quarks with energy E ∼ mv 2 and momentum p ∼ mv , soft gluons with E ∼ p ∼ mv , and ultrasoft gluons with E ∼ p ∼ mv 2 . The soft and ultrasoft modes are distinct, for instance a consistent power counting in v demands that the ultrasoft gluon interactions are multipole expanded [6,9], while soft gluon interactions are not. The soft gluons are essential to correctly reproduce the beta function in the effective theory [14], and run between the scales m and mv . Other massless on-shell fields, such as light quarks, will have ultrasoft and soft components too. There are also important off-shell field components, such as the exchange of gluons with E ∼ mv 2 and p ∼ mv that build up the potential. Soft heavy quarks with E ∼ p ∼ mv are also off-shell relative to the heavy quark states of interest. These off-shell field components can be integrated out of the Lagrangian in the effective theory. Doing this leaves a Lagrangian that is non-local at the scale mv , but local at mv 2 . This procedure, which treats the potential components as four quark operators, was first seriously investigated in Ref. [11], and the resulting effective theory is referred to as pNRQCD. In Ref. [11] it was proposed that the matching onto effective theories should take place in two stages: at µ = m one matches QCD onto NRQCD as originally defined in Ref. [5], and then matches NRQCD onto pNRQCD at the scale µ = mv . The matching of four-quark operators at m was considered in Ref. [12], following the proposal in Ref. [8] that the matching procedure should be similar to that in HQET. How3
Instanton-Induced Interactions in Finite Density QCD
a r X i v :h e p -p h /9908314v 1 10 A u g 19991Instanton-Induced Interactions in Finite Density QCDG.W.Carter a and D.Diakonov baThe Niels Bohr Institute,Blegdamsvej 17,DK-2100Copenhagen,Denmark bNORDITA,Blegdamsvej 17,DK-2100Copenhagen,DenmarkWe consider the finite density,zero-temperature behaviour of quark matter in the in-stanton picture.Since the instanton-induced interactions are attractive in both ¯q q and qq channels,a competition ensues between phases of matter with condensation in either or both.It results in chiral symmetry restoration due to the onset of diquark condensation,a ‘colour superconductor’,at finite density.1.IntroductionDue to a lack of lattice QCD techniques for implementing quark chemical potential,the finite density properties of strongly-interacting matter remain unresolved.To date,model studies suggest not only chiral symmetry restoration but also the possibility of Cooper pairing of quarks at high density,via an attractive qq interaction,similar to superconducting electrons.The analogy has been extended to nomenclature,with the QCD version called colour superconductivity.It has been known for some years that perturbative,single gluon exchange between quarks is attractive and will generate a pairing gap around the Fermi surface [1].More re-cently,it was suggested that colour superconductivity might also arise by non-perturbative means at moderate quark density [2].Since then,more detailed studies using models in-spired by that of Nambu and Jona-Lasinio [3]and instantons [4]have supported this idea.This talk describes how diquark formation restores broken chiral symmetry in the con-text of the QCD instanton vacuum,an approach which has accounted for many hadronic observables through the use of fundamental degrees of freedom (quarks and gluons)in a microscopic approximation.2.Quark Effective ActionThe derivation of an effective action for chiral quarks in N f flavours has been discussed in detail in other publications.Here we concentrate on the two flavour case,which is often adequate for low energy phenomenology.Growing quark chemical potential naturally makes the strange quark more relevant,as has been studied in other models [5].These authors conclude that the two-flavour superconducting state is likely to be present at moderate values of the quark chemical potential for a realistic strange mass.The basic idea is to replace the partition function of QCD with an effective form which divides the low and high energy contributions.The high momentum part is taken to be2SS;FFigure 1.Schwinger-Dyson-Gorkov diagrams to first order in λ.perturbative and as such the gluons here are assumed to be small corrections to the stable,low-energy configurations of the gauge fields –the instantons.Each (anti-)instanton in turn induces a quark zero mode of (right)left chirality,and averaging over all possible instanton backgrounds results in a delocalization of the zero modes which spontaneously breaks chiral symmetry.This picture of the vacuum is supported by various lattice studies and has a long history of successful phenomenology.Following this procedure,the expected ’t Hooft interaction is obtained and one has an effective quark action which is suitable for practical calculations.We have reformulated this effective action for finite quark chemical potential.The result can be expressed asS [ψ,ψ†]=−d 4p ψ†(p /+iµγ4)ψ+λdUN ff(d 4p f d 4k f )3combine in the physical quantities:M 1=(5−4/N c )g 1+(2N c −5+2/N c )g 2,M 2=2(2−1/N c )g 1+2(N c −2)g 2,and ∆=(1+1/N c )f .The M 1,2are measures of chiral symmetry breaking and act as an effective mass.Meanwhile the diquark loop 2∆plays the role of twice the single-quark energy gap formed around the Fermi surface.The solution of the gap equations depends on the vertex coupling constant,λ,which itself is determined by balancing the instanton background with the condensates through its saddle-point value.This minimization of the partition function leads to [4]N/V =λ Y ++Y − =4(N 2c −1)[2g 1M 1+(N c −2)g 2M 2+4f ∆]/λ.(2)This joins the gap equations to close a system of equations,numerically solvable.Once this is done,the chiral condensate proper may be computed as an integral over the resummed propagator.For any given chemical potential,multiple solutions can be obtained for the gaps.These correspond to different phases of quark matter,and they are summarized as follows:(0)Free massless quarks:g 1=g 2=f =0;(1)Pure chiral symmetry breaking:g 1=g 2=0,f =0;(2)Pure diquark condensation:g 1=g 2=0,f =0;and (3)Mixed symmetry breaking:g 1=g 2=0,f =0.The free energy,calculated to first order in λ,is minimized in order to resolve the stable solution.The phase corresponding to the lowest coupling λis the thermodynamically favoured [4].No solutions were found matching Phase (0),and the Phase (3)solution obtained disap-pears at relatively low chemical potential (µ≈80MeV)and is never thermodynamically competitive [4].The remaining phase competition is then between Phases (1)and (2).In the vacuum,where µ=0,one finds Phase (1)preferred –this is the standard picture.However,at a critical chemical potential µc ,defined by the ratio of superconductive gap to chiral effective masses ∆/M = 3/4,a first-order phase transition occurs.With the standard instanton parameters N/V =1fm −4and ¯ρ=0.33fm,we find µc ≃340MeV.The first-order nature of the phase transition is clearly seen in Fig.2.Physically,the quark density is more relevant than the chemical potential.As an intermediate step and in order to demonstrate the microscopic differences between the two phases,we have calculated the occupation number density for quarks.This is nontrivial50100150200250300350010*******400500600700800(M e V )µ(MeV)M∆−<ψψ>1/3Figure 2.Condensates for N c =3as a function of µ.0.020.040.060.080.10.120.1400.20.40.60.81n ρ3µρFigure 3.The quark density n q vs.µ.424681012140120240360480600720840n (|p |)|p| (MeV)Figure 4.Occupation number n (p )vs.p for Phase (1)for µ=1/¯ρ=600MeV.024*********120240360480600720840n (|p |)|p| (MeV)Figure 5.Occupation number n (p )vs.pfor Phase (2)for µ=1/¯ρ=600MeV.and here we present only numerical results in Figs.4and 5.In Phase (1),there isclearly an effective mass brought about by spontaneous symmetry breaking,indicated by the reduced Fermi radius.We stress that,despite the complicated four-momentum dependence of the interaction,the resulting occupation number density appears as a perfect Fermi step function.Cooper pairing,however,smears the Fermi surface,and this is evinced in the second plot.The residual discontinuity at | p |=µis the contribution from the free,colour-3quarks which do not participate in the diquark.Integrating over momenta and recalling the critical chemical potential,the quark density profile as a function of chemical potential is plotted in Fig.3for the equilibrium states.We see a discontinuity at the phase transition,where the horizontal line signifies the quark density of stable nuclear matter.The phase transition occurs at an extremely low quark density,which remains a conceptual conundrum.4.ConclusionsBeginning from the instanton picture of the QCD vacuum,we have extended the model for finite density and found chiral symmetry restoration due to the onset of colour super-conductivity.This phase transition is strongly first order and in agreement with other quark-based approaches.REFERENCES1. D.Bailin and A.Love,Phys.Rep.107(1984)325.2.M.Alford,K.Rajagopal and F.Wilczek,Phys.Lett.B422(1998)247;R.Rapp,T.Sch¨a fer,E.V.Shuryak and M.Velkovsky,Phys.Rev.Lett.81(1998)53.3.J.Berges and K.Rajagopal,Nucl.Phys.B538(1999)215.4.G.W.Carter and D.Diakonov,Nucl.Phys.A642(1998)78c;Phys.Rev.D60(1999)016004;R.Rapp,T.Sch¨a fer,E.V.Shuryak and M.Velkovsky,hep-ph/9904353.5.M.Alford,J.Berges and K.Rajagopal,hep-ph/9903502;T.Sch¨a fer and F.Wilczek,hep-ph/9903503.。
趁热打铁勿失良机英语作文
趁热打铁勿失良机英语作文英文回答：Capitalizing on Momentum: Seize the Moment and Exploit Opportunities.In the ever-changing landscape of business and life, the ability to recognize and capitalize on momentum is crucial for success. Momentum acts as a force that propels us forward, creating a sense of progress and driving us towards our goals. When momentum is present, it isessential to seize the moment and exploit the opportunities that arise.There are several benefits associated with riding the wave of momentum. First, it provides a sense of confidence and motivation. When we are making progress towards our goals, we feel more confident in our abilities and more likely to persevere. Second, momentum helps us overcome obstacles and challenges. When we are moving forward with apositive trajectory, we are less likely to be deterred by setbacks. Third, momentum can create a virtuous cycle, leading to even greater opportunities and success.To effectively capitalize on momentum, it is important to identify the specific opportunities that arise during this period. These opportunities may come in the form of new partnerships, customer acquisitions, or market trends. It is essential to be proactive and take advantage of these opportunities when they present themselves. Additionally, it is important to maintain the focus and drive that created the momentum in the first place. This may involve setting clear goals, delegating tasks effectively, and staying motivated throughout the process.While capitalizing on momentum is crucial, it is equally important to recognize when it is time to shift gears. Momentum can be a powerful force, but it can also become unsustainable if it is not managed correctly. As circumstances change, it may be necessary to adjust our approach or redirect our efforts to ensure continued success.Overall, the ability to seize the moment and exploit opportunities when momentum is present is a valuable skill that can drive success in business and life. By understanding the benefits of momentum, identifying opportunities, and maintaining focus, individuals and organizations can maximize their potential and achieve their goals.中文回答：趁热打铁，抓住机遇。
Effect of heavy-quark energy loss on the muon differential production cross section in Pb-P
a r X i v :0712.0051v 3 [h e p -p h ] 13 M a r 2008Effect of heavy-quark energy loss on the muondifferential production cross section in Pb–Pb collisions at√Abstract We study the nuclear modification factors R AA and R CP of the high transverse momentum (5<p t <60GeV /c )distribution of muons in Pb–Pb collisions at LHC energies.We consider two pseudo-rapidity ranges covered by the LHC experiments:|η|<2.5and 2.5<η<4.Muons from semi-leptonic decays of heavy quarks (c and b)and from leptonic decays of weak gauge bosons (W and Z)are the main contributions to the muon p t distribution above a few GeV /c .We compute the heavy quark contributions using available pQCD-based programs.We include the nuclear shadowing modification of the parton distribution functions and the in-medium radiative energy loss for heavy quarks,using the mass-dependent BDMPS quenching weights.Muons from W and Z leptonic decays,that dominate the yield at high p t ,can be used as a medium-blind reference to observe the medium-induced suppression of beauty quarks.Key words:Quark Gluon Plasma,Relativistic Heavy Ion Collisions,HeavyQuarks,Weak Gauge Bosons,MuonsPACS:24.85.+p,25.75.Dw,25.75.-q1Now at LLR (CNRS/IN2P3-Ecole Polytechnique)Palaiseau,Francethe initial hard-scattering processes and they may subsequently interact with the medium itself.At the Relativistic Heavy Ion Collider(RHIC),a significant suppression of the so called‘non-photonic electrons’,expected to be produced in the semi-leptonic decay of charm and beauty hadrons,has been measured in central Au–Au collisions at centre-of-mass energy√d2N AA/d p t dηN collmuons at central pseudo-rapidity,|η|<∼2.5,with a larger cutoff,p t>3–4GeV/c.As we will quantify in section4,the high-p t reach for the measure-ment of the inclusive muon production spectrum is expected to extend well into the region where muons from W and Z decays become dominant over muons from beauty decays.In this work,we study the effect of heavy-quark energy loss on the transverse momentum distribution of muons in Pb–Pb collisions at√p collisions at the √Tevatron(there could be a dependence of their production cross section on the isospin of the input channel.In nucleus–nucleus collisions this dependence can be mim-icked by a weighted cocktail of proton–proton(pp),neutron–neutron(nn), proton–neutron(pn)and neutron–proton(np)collisions.The cross section per nucleon–nucleon binary collision can be expressed asd2σNNA2×d2σppA2×d2σnnA2×d2σpn d p t d y ,(2)where A and Z are the mass number and the atomic number of the colliding nu-clei.CTEQ4L[17]parton distributions functions(PDFs)are used,and nuclear shadowing is accounted for via the EKS98parametrization[18].The resulting p t distributions are normalized to the cross sections obtained from Refs.[9,10], that is a cross section per nucleon–nucleon collision of6.56(7.34)nb for the W and0.63(0.68)nb for the Z in Pb–Pb(pp)collisions at5.5TeV,including the muonic branching ratios(10.6%for W and3.4%for Z[19]).Notice that the Z production cross section is about ten times smaller than that of the W[20,21]. The uncertainty due to neglecting higher order corrections(next-to-next-to-leading order)was quantified as1–2%in Ref.[9]by varying the values of the factorization and renormalization scales.The uncertainty due to the errors on the PDFs was quantified as about10%in Ref.[22].2.2Heavy-quark decay muonsWithin the pQCD collinear factorization framework,the expression for the production cross section of heavy-flavoured hadrons in the collision of two hadrons A and B can be schematically written as:d2σAB→hno mediumQ Xz2,(3)where f i/A(x i/A)and f j/B(x j/B)are the parton distribution functions,the dif-ferential probabilities for the partons i and j to carry momentum fractions x i/A and x j/B of their respective nucleons.ˆσij is the cross section of the partonic process ij→Qs dependence ofˆσij and the renormalization/factorization scale dependences of f i/A(j/B),ˆσij and D h/Q(the squares of scales are normally of the order of the momentumtransfer Q2∼p2t,Q of the hard scattering).We use the NLO pQCD calculation implemented in the HVQMNR program[11] to obtain the heavy-quark p t–y double-differential cross sections,with the fol-lowing parameters values:for charm,m c=1.2GeV/c2and factorization andrenormalization scalesµF=µR=2µ0,whereµ0≡ Q)/2;for beauty,m b=4.75GeV/c2andµF=µR=µ0.CTEQ4M[17]parton distribu-tion functions are used,and nuclear shadowing is taken into account with the EKS98parametrization[18].For the b quark,the perturbative uncertainty was quantified,by varying the scales,in about30%for p t>30GeV/c[23]. Starting from the heavy-quark double-differential cross sections at NLO,we obtain the muon-level cross sections using the following Monte Carlo proce-dure.We sample p t and y of a c(or b)quark according to the shape of the NLO cross section and fragment it to a hadron using the Peterson[13]fragmenta-tion function following the parameterization obtained in a recent analysis of e+e−data from LEP[23].Finally,we decay the hadron into a muon accord-ing to the spectator model[24].In the spectator model,the heavy quark in a meson is considered to be independent of the light quark and is decayed as a free particle according to the V−A weak interaction[14].Thus,we as-sume the momentum of the hadron to be entirely carried by the constituent heavy quark and we perform the heavy-quark three-body decay,c→sµνµor b→cµνµ,to obtain the muon transverse momentum and rapidity.The muon production cross sections per nucleon–nucleon collision from charm(beauty) at√loss of heavy quarks by medium-induced gluon radiation,we used the quench-ing weights in the multiple soft scattering approximation,which were derived in Ref.[12]in the framework of the BDMPS formalism[26].Schematically, energy loss is introduced by modifying Eq.(3)to:d2σAB→hmediumQ X(p t,Q+∆E),(4)z2where E is the heavy-quark energy and∆E is the radiated energy.The quench-ing weight,represented by P(∆E,ˆq,L,m Q/E),is the probability for a heavy quark with mass m Q and energy E to lose an energy∆E while propagating over a path length L inside a medium with transport coefficientˆq.The latter is defined as k2t /λ,the average transverse momentum,k t,squared transferred from the medium to the parton per unit mean free pathλ.It is expected to be proportional to the volume density d N g/d V of gluons in the medium(thus, to its energy density)and to the typical momentum transfer per scattering. We calculate the energy loss∆E following the Monte Carlo approach intro-duced in Ref.[27]for light quarks and gluons,and adapted for heavy quarks in Refs.[12,28].We start by sampling the heavy-quark kinematics,p t and y,according to the NLO double-differential cross section.Then,we sample the parton production point in the transverse plane(x,y)according to the Glauber-model[29]densityρcoll(x,y)of binary collisions,and we sample the azimuthal parton propagation direction.We calculate the path length L and the value ofˆq,which is a mean value of the local time-averaged(ˆq(x,y)along the path of the parton.We do not include the expan-sion of the medium in the longitudinal and transverse directions.However,it has been shown in Ref.[30]that,numerically,the effects of a time-dependent medium on parton energy loss can be accounted for by an equivalent static medium,specified in terms of the time-averaged transport coefficientside the range of interest for the present study.We assume the heavy-quark rapidity to stay constant during the process of energy loss,since heavy quarks co-move with the longitudinally-expanding medium and any modification of the initial heavy-quark rapidity should remain small.Finally,we apply frag-mentation and decay as described in section2.2.In Ref.[27]the localˆq transverse profile at central rapidity(y=0)is as-sumed to be proportional to the density of binary collisions.Since we want to study muon production in a broad rapidity range(|y|<∼4),we introduce a dependence of the localˆq on the pseudo-rapidityηin order to account for the reduced medium density in the forward ly,we assume the transport coefficientˆq to scale as a function of pseudo-rapidity according to the gluon pseudo-rapidity density of the medium d N g/dη,this choice being justified by the fact that d N g/d V scales inηaccording to d N g/dη.We as-sume the pseudo-rapidity density of charged particles and the pseudo-rapidity density of gluons to have the same dependence onηand we write[33]:(η) d N chdη=5.5TeV by scalingsNNthe values extracted from an analysis of the light-flavour hadrons suppression at RHIC[27,34].The corresponding values for the parton-averaged( )and time-averaged(ˆq =25and100GeV2/fm.For the pseudo-rapidity dependence we use the pseudo-rapidity distribution of charged particles predicted for Pb–Pb collisions at the LHC in Ref.[35].The effect of this dependence on the suppression of muons from heavy-quark decays is expected to be small,because d N ch/dηis predicted to have only a modest variation in the range|η|<4(for illustration,forη=3it would be reduced by about15%with respect toη=0).Before presenting our results,we point out that,in addition to radiative energy loss,there are other possible medium-induced effects(e.g.:collisional energy loss of heavy quarks[36],in-medium hadronization and dissociation of heavy-flavour hadrons[37],formation of bound states and hadronization of heavy quarks via coalescence[38])that could noticeably affect the muon spectrum for p t<∼10GeV/c.These effects have not been considered in the present study.And one has to keep in mind that all current jet quenching models do not describe the suppression of single non-photonic electron p t spectra at RHIC equally well as they do describe the suppression of pion spectra.It is with this caveat that one should view the predictions for LHC based on these models.Fig.1.Muons in central(0–10%)Pb–Pb collisions at√=5.5TeV,when only nuclear shadowing is included.sNNThe contributions from charm,beauty and weak gauge bosons are shown sepa-rately.Muons from charged pion and kaon decays and from Drell-Yan processes (qdecay in minimum-bias Pb–Pb collisions at√C shad is the mean value of C shad).The x values were obtained using the PYTHIA event generator for the leading order processes gg→Qx1x2= q→W/Z.For these two processes,we have,qualitatively,sNNQ2≈4(p2t,µ+m2Q)and s NN x1x2=Q2≈m2W(Z)≈4p2t,µ,respectively;and, for both,x1≈2p t,µexp(+yµ)/√s NN.Weak gauge boson decays(long-dashed line in Fig.1)probe the quarks nuclear shadowing, whichfluctuates around its mean value of0.9.The overall nuclear modification factor of muons(solid line in Fig.1)increases rapidly with p t up to a value of about1.1and then decreases to about0.9.In order to explore the uncertainties due to the limited knowledge of the nuclear PDFs in the kinematic region relevant to our study,we considered two other parametrizations of the nuclear parton distribution:nDS[39]and HKN07[40].The values of the corresponding shadowing factors are reported in table1.We note that the shadowing factor uncertainty for heavy quarks at high p t is7%and for weak gauge boson is less than5%.For heavy quarks at low p t,it becomes as large as20%for charm and10%for beauty.Note that the study of proton–nucleus collisions at RHIC energies has been absolutely necessary to disentangle cold and hot nuclear matter effects.Cold nuclear matter effects at LHC energies remain relatively unknown.Therefore,p–Pb runs at the LHC will be needed in order to fully understand the nuclear modification factors measured in Pb–Pb.We now include in the calculation the in-medium energy loss for heavy quarks. We start by considering the nuclear modification factor R AA(p t)of muonsTable1Qualitative estimation of Bjorken-x values and shadowing factors(according to EKS98[18],nDS(NLO)[39]and HKN07(NLO)[40])for heavy quarks and W/Z bosons produced in Pb–Pb collisions at√y p t x1x2C EKSshad C nDSshadC HKNshadc 004·10−40.500.650.810.920.740.87 0301·10−2 1.10 1.090.93309·10−32·10−50.570.830.743302·10−15·10−40.910.96 1.09W 0all1·10−20.890.890.880.830.830.84 3all3·10−17·10−40.760.770.85from beauty decays in central(0–10%)Pb–Pb collisions,in order to studythe effects of the b quark mass and of the dependence of the transport coef-ficientˆq onηaccording to d N ch/dη.The latter is relevant only in the large pseudo-rapidity range.The result is shown in Fig.2.The shaded band rep-resents our baseline result for theˆq range25–100GeV2/fm,with mass effect (m b=4.75GeV)and with pseudorapidity dependence ofˆq.The suppressionobtained at high p t,where R AA becomes independent of p t,is about a factor 5(10)forˆq=25(100)GeV2/fm.The suppression for electrons from beauty decays was calculated within the same framework(except for the treatment ofhadronization and decay)in Ref.[12],for central rapidity and p t<∼15GeV/c. For the same p t range,we obtain similar values for R AA.With reference toFig.2,by comparing the thick solid line(m b=4.75GeV)and the long-dash line(m b=0),we notice that the quark mass effect in parton energy loss in-creases R AA by up to a factor of three for p t∼5GeV/c,and that some effect persists even at15GeV/c.When going to p t>∼20GeV/c,the quark mass dependence becomes negligible since the quark mass becomes negligible with respect to the momentum.By comparing the thick and thin solid lines withˆq=100GeV2/fm,which are with and withoutηdependence ofˆq respectively,Fig.2.Nuclear modification factor of muons from beauty decays with/without mass effect in the energy loss,and with/without pseudo-rapidity dependence ofˆq(d N/dηdependence),in the central(0–10%)Pb–Pb collisions at√=5.5TeV in the two pseudo-rapidity domains.The results sNNwith transport coefficient valuesˆq=0,25and100GeV2/fm are reported, whereˆq=0corresponds to the case of no energy loss.With reference to the upper panels of Fig3,note that,since muons from W and Z decays are un-affected by the energy loss,the crossing point in transverse momentum of the distributions of b-quark and W-boson decay muons shifts down by≈5GeV/c at large rapidities and by≈7GeV/c at mid-rapidity,when energy loss is in-cluded.Concerning the muon R AA(p t)(lower panels of Fig.3),at large pseudo-rapidities(on the left),with heavy-quark energy loss,the overall muon yield is suppressed by about a factor of2–5in the range2<p t<20GeV/c,where the beauty contribution dominates.For higher p t,R AA increases rapidly in the 20<∼p t<∼30GeV/c range andflattens at around0.8above≈30GeV/c.The ˆq-independence of the R AA of overall muons at large p t is due to the fact that the W/Z boson contribution to the yield becomes dominant(see upper panels of Fig.3).At mid-rapidity the behaviour is similar(lower-right panel).The small difference of the R AA shape at different rapidities is due to the different proportion of heavy-quark and W/Z boson decays.For the same reason theFig.3.Differential cross section per nucleon–nucleon collision and nuclear modifica-tion factor of muons from W,Z,c and b decays in central(0–10%)Pb–Pb collisions at√s NN=5.5TeV.The left-hand panel shows the results for2.5<η<4,the right-hand panel the results for|η|<2.5.nuclear modification factor without energy loss(dot-dashed lines in the lower panels in Fig.3)differs slightly from that obtained without acceptance cuts (solid line in Fig.1).Besides the Pb–Pb-to-pp nuclear modification factor R AA,also the central-to-peripheral nuclear modification factor R CP will provide information on the medium-induced suppression of b quarks.R CP is defined as:R CP(p t)= N AA coll P d2N PAA /d p t d y,(6)where the index C(P)stands for central(peripheral)collisions.From the ex-perimental point of view,the R CP measurement will be more straight-forward than the R AA measurement,for the following two reasons.1)The measure-ments in pp and in Pb–Pb will be affected by different systematic errors(espe-cially for the cross section normalization),which will add up in the R AA uncer-tainty.2)pp collisions at the LHC will have different c.m.s.energy(14TeV)with respect to Pb–Pb(5.5TeV),therefore the muon spectra measured in pp will have to be extrapolated from14TeV to5.5TeV with the guidance ofperturbative QCD calculations,introducing an additional systematic error of the order of10%on R AA[5].In our calculation,the initial-state effects are assumed to be the same in cen-tral and peripheral collisions—namely,we do not include an impact parameter dependence for shadowing—thus,they cancel out in the central-to-peripheral ratio.As a consequence,the R CP of muons from weak gauge boson decays isequal to one.The central(0–10%)to peripheral(40–70%)ratios are shown in Fig.4.In central(0–10%)collisions the yield might be reduced with re-spect to peripheral collisions(40–70%)by a factor2–3in the p t range from about2GeV/c to about13GeV/c,where the b-quark contribution dominates. When going to larger p t,the R CP of muons increases fast and thenflattensat around0.8at mid-rapidity and1.0at forward rapidity.This difference at high p t between the two pseudo-rapidity regions is due to the different relative abundances of the heavy-quarks and weak bosons components.For the samereason,the curves forˆq=25GeV2/fm and100GeV2/fm cross each other at p t≈20GeV/c at large pseudo-rapidity and at p t≈25GeV/c at mid-rapidity. We have checked that the uncertainties on the cross sections of muons from W decays and of muons from beauty decays(approximately10%and30%, respectively,as discussed in section2)translate into a variation smaller than5%of the R AA and R CP values for p t>∼35GeV/c,while they have no effect at lower p t.5ConclusionsThe effect of heavy-quark energy loss on the differential cross section of muons produced in Pb–Pb and pp collisions at LHC energies has been investigated. The most important contributions to the decay muon yield in the range5<p t<60GeV/c have been included:b(and c)quarks have been computed us-ing a NLO pQCD calculation supplemented with the BDMPS mass-dependent quenching weights;weak gauge bosons muonic decays have been computedusing the PYTHIA event generator.The heavy-quark mass-dependence of en-ergy loss reduces the suppression of muon yields from beauty decays by aboutfactor two for5<∼p t<∼15GeV/c.To account for the decrease of the mediumdensity at large pseudo-rapidity,we assumed a decrease of the transport coef-ficient proportionally to d N ch/dη,and we found that the effect on the muonp t distribution is negligible,especially at large transverse momentum.We in-vestigated the energy loss effect by means of the Pb–Pb-to-pp and of the Pb–Pb central-to-peripheral nuclear modification factors in the acceptance ofthe LHC experiments:ATLAS,ALICE,and CMS.In the p t interval below approximately20GeV/c,where the beauty component is dominant,R AA for0–10%central Pb–Pb collisions relative to pp and R CP for0–10%relativeto40–70%Pb–Pb collisions are found to be about0.2–0.4and0.3–0.5,re-spectively.Then,we observe a steep rise in the beauty/W crossover interval20–40GeV/c,up to the values R AA≈0.8and R CP≈1in the interval above 40GeV/c,dominated by W/Z decay muons.These muon nuclear modifica-tion factors could provide thefirst experimental observation of the b quarkmedium-induced suppression in Pb–Pb collisions at the LHC.The presence of a medium-blind component(muons from W and Z decays)that dominates thehigh-p t muon yield will allow an intrinsic calibration of the medium-sensitiveprobe(heavy quarks),because it will provide a handle on the strength of the initial-state effects that may alter the hard-scattering cross sections in nucleus–nucleus collisions at the unprecedented energies of the LHC.Acknowledgments.The authors,members of the ALICE Collaboration, would like to thank their ALICE Colleagues for useful exchanges during the accomplishment of the present work.In particular,we would like to thank F.Antinori,P.Crochet,A.Morsch and J.Schukraft for fruitful discussions, N.Armesto and C.A.Salgado for their helpful suggestions at the beginning of this work and the calculation of the HKN parameterized nPDFs,and to S.Ku-mano and R.Sassot for the calculation of the HKN and the nDS parameterized nPDFs,and to M.Mangano for the HVQMNR program implementation. 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夸克英语作文
夸克英语作文In the realm of particle physics, quarks are the fundamental building blocks of matter, akin to the alphabet letters that make up the diverse lexicon of our universe. These elementary particles are bound together by the strong force, one of the four fundamental forces of nature, and they are never found in isolation. The study of quarks and their interactions is not just a scientific endeavor; it's a journey into the very fabric of existence.Quarks come in six different "flavors": up, down, charm, strange, top, and bottom. Each flavor has its own unique properties, such as mass and electric charge. The lighter quarks, up and down, are the most common and form the protons and neutrons in the atomic nucleus. The heavier quarks, on the other hand, are less stable and quickly decay intolighter ones.The discovery of quarks was a monumental leap in our understanding of the universe. It was in the 1960s when physicist Murray Gell-Mann proposed the quark model, which was later confirmed by deep inelastic scattering experiments. This model revolutionized the field of particle physics and earned Gell-Mann the Nobel Prize in Physics.Quarks are held together by gluons, which are the mediator particles of the strong force. This force is so powerful that it overcomes the quarks' natural tendency todisperse. The strong force is also responsible for the phenomenon of color charge, a quantum property that quarks possess. Each quark can exist in one of three color states, and the force only allows quarks to combine in such a waythat they form a color-neutral state, like the protons and neutrons in an atom.The study of quarks has led to the development of the Standard Model of particle physics, which describes the electromagnetic, weak, and strong forces. Despite its successes, the Standard Model is not without its limitations. It does not, for example, account for gravity or the nature of dark matter, which is believed to make up a significant portion of the universe's mass.As scientists continue to probe the depths of the subatomic world, quarks remain at the forefront of research. The Large Hadron Collider (LHC) and other particle accelerators are instrumental in these investigations, providing a window into the high-energy interactions of quarks and the potential for discovering new particles or forces.In conclusion, quarks are not just esoteric particles of interest to physicists; they are the cornerstone of the matter that makes up everything we see and touch. The quest to understand their nature and the forces that govern them is a testament to human curiosity and our relentless pursuit of knowledge. As we continue to unravel the mysteries of quarks, we gain a deeper appreciation for the intricate and awe-inspiring design of the universe.。
T-matrix approach to heavy quark diffusion in the QGP
a r X i v :0808.3710v 1 [h e p -p h ] 27 A u g 2008EPJ manuscript No.(will be inserted by the editor)T-matrix approach to heavy quark diffusion in the QGPH.van Hees 1,M.Mannarelli 2,V.Greco 3,and R.Rapp 41Institut f¨u r Theoretische Physik,Justus-Liebig-Universit¨a t Giessen,Heinrich-Buff-Ring 16,D-35392Giessen,Germany 2Instituto de Ciencias del Espacio (IEEC/CSIC),E-08193Bellaterra (Barcelona),Spain 3Dipartimento di Fisica e Astronomia,Via S.Sofia 64,I-95125Catania,Italy4Cyclotron Institute and Physics Department,Texas A&M University,College Station,Texas 77843-3366,U.S.A.August 27,2008Abstract.We assess transport properties of heavy quarks in the Quark-Gluon Plasma (QGP)using static heavy-quark (HQ)potentials from lattice-QCD calculations in a Brueckner many-body T -matrix approach to evaluate elastic heavy-quark-light-quark scattering amplitudes.In the attractive meson and diquark channels resonance states are formed for temperatures up to ∼1.5T c ,increasing pertinent drag and diffu-sion coefficients for heavy-quark rescattering in the QGP beyond the expectations from perturbative-QCD calculations.We use these transport coefficients,complemented with perturbative elastic HQ gluon scat-tering,in a relativistic Langevin simulation to obtain HQ p t distributions and elliptic flow (v 2)under conditions relevant for the hot and dense medium created in ultrarelativistic heavy-ion collisions.The heavy quarks are hadronized to open-charm and -bottom mesons within a combined quark-coalescence fragmentation scheme.The resulting single-electron spectra from their semileptonic decays are confronted with recent data on “non-photonic electrons”in 200A GeV Au-Au collisions at the Relativistic Heavy-Ion Collider (RHIC).1IntroductionOne of the most interesting questions in high-energy nu-clear physics is that about the properties of the hot and dense medium created in ultra-relativistic heavy-ion col-lisions.Finite-temperature lattice-QCD (lQCD)calcula-tions of strongly-interacting matter predict a phase transi-tion from hadronic matter to a quark-gluon plasma (QGP)at a critical temperature,T c ≃180MeV [1].In the re-cent years the experimental program at the Relativistic Heavy-Ion collider has resulted in convincing evidence for the formation of such a hot and dense partonic state [2,3,4,5].The heavy charm and bottom quarks are particularly valuable probes for the properties of this medium since they are created in the primordial hard collisions of the nucleons within the colliding nuclei.Thus,they form a rather well defined initial state and interact with the hot and dense fireball during its entire evolution.Recently,measurements of the transverse-momentum distributions of “non-photonic single electrons”(e ±),which originate mainly from the semi-leptonic decays of open-charm and -bottom mesons,in 200A GeV Au-Au collisions at RHIC have found a surprisingly large suppression at high trans-verse momenta (p t )(i.e.,a small nuclear modification fac-tor,R AA )and a large elliptic-flow parameter,v 2.Both findings indicate that during the lifetime of the hot and dense fireball heavy quarks come close to thermal equilib-rium with the medium [6,7,8].The theoretical challenge is to understand the corre-sponding thermalization times of heavy quarks from the underlying microscopic scattering processes with the con-stituents of the QGP,in particular how the heavy quarks,despite their large masses,m Q ≫T c ,become part of the collective flow of the fireball.In calculations of the pertinent transport coefficients from perturbative QCD (pQCD),based on gluon-bremsstrahlung energy loss,in-cluding elastic HQ scattering,one has to artificially tune the coupling strength beyond the applicability range of perturbation theory [9,10].It has also been shown that the convergence of the perturbative series for the HQ dif-fusion coefficient is quite poor [11].Thus,non-perturba-tive approaches have to be used to explain the strong HQ couplings necessary.One suggested mechanism is the for-mation of D -and B -meson resonance excitations in the deconfined phase of QCD matter [12,13]which has lead to a quite satisfactory description of the e ±data at RHIC.This paper is organized as follows:In Sec.2we use HQ static potentials from lattice-QCD calculations at fi-nite temperature in a many-body Brueckner T -matrix ap-proach to calculate elastic HQ light-quark scattering-ma-trix elements in the medium [14,15].We show that after inclusion of a complete set of color channels,taking into account l =0,1states in the partial-wave expansion of the T -matrix,the resonance states,conjectured in the ear-lier approaches,are confirmed by these interactions,which are in principle free of tunable parameters.The resulting elastic-scattering amplitudes are used in Sec.3to calculate2H.van Hees et al.:T-matrix approach to heavy quark diffusion in the QGPdragand diffusion coefficients for a Fokker-Planck equa-tion[16,12,17],describing the rescattering of the heavy quarks within the hot and dense sQGPfireball.In the next step we employ a relativistic Langevin simulation tofind the corresponding HQ p t distributions,using a thermal-fireball parameterization,including ellipticflow for non-central heavy-ion collisions.To confront these spectra with the e±data from the PHENIX and STAR collaborations at RHIC,in Sec.4we use a combined quark-coalescence and fragmentation model to hadronize the heavy quarks to D and B mesons which then are decayed semi-leptonically leading to thefinal e±spectra which can be directly con-fronted with recent data on nonphotonic single electrons in200A GeV Au-Au collisions at RHIC.The paper closes with brief conclusions and an outlook(Sec.5).2HQ scattering in the QGPIn this Sec.we calculate in-medium matrix elements for elastic scattering of heavy quarks(Q=c,b)with light quarks q=u,d,s in a Brueckner-like many-body ap-proach,assuming that a static heavy-quark light-quark potential,V(r),can be employed as the interaction kernel. Such a model has been used in the vacuum to successfully describe D-meson spectra and decays[18,19].Further,we assume that the effective in-medium potential can be ex-tracted fromfinite-temperature lQCD calculations of the color-singlet free energy F1(r,T)[20,21]for a static¯QQ pair as the internal potential energy by the usual thermo-dynamic relation[14,22,23,24],U1(r,T)=F1(r,T)−T∂F1(r,T)8V1,V¯3=14V1,(3)which is also justified by recent lQCD calculations of the finite-T HQ free energy[25,26].This approach is in principle parameter free in the choice of the interactions,since their strength is taken Brueckner many-body scheme for the coupled system of the T-matrix based on the lQCD static internal potential energy as the interaction kernel and the HQ self-energy.fromfirst-principle lQCD simulations.However,there are considerable uncertainties in the potentials(a)between different lattice calculations and(b)in the extraction and parameterization of the corresponding free energies,par-ticularly their temperature dependence needed to subtract the entropy term in Eq.(1).In addition,the very notion of an“in-medium potential”is not a unique concept[27], and its identification with the internal potential energy may be seen as an upper limit in interaction strength.We use three different parameterizations of F1[23,22,14]: [Wo]of quenched lQCD[20],[SZ]of two-flavor lQCD[28],and[MR]of three-flavor lQCD[29].The resulting potentials from[Wo]and[SZ]are compara-ble to a numerical extraction from three-flavor lQCD[29], while that from[MR]is deeper than the other two for T 1.6T c,but falls offfaster at higher temperatures. The resulting uncertainty in the transport coefficients(see Sec.3)amounts to up to40%.To define the Brueckner-type many-body scheme the four-dimensional(4D)Bethe-Salpeter(BS)ladder approximation,symbolized in dia-grammatical form by the upper panel of Fig.1,has to be reduced to a3D Lippmann-Schwinger(LS)equation, neglecting antiparticle components in the quark propaga-tors,in order to implement the static potential from lQCD via Eqs.(1-3).After this reduction the LS equation in the color channel,a∈{1,¯3,6,8}reads[14]1T a(E;q′,q)=V a(q′,q)− d3k1Here and in the following all vertex and Green’s func-tions are understood as the retarded real-time quantities which can be derived as analytic continuations of the correspond-ing imaginary-time(Matsubara)quantities of thermal quan-tumfield theory.H.van Hees et al.:T-matrix approach to heavy quark diffusion in the QGP 3spectively.f F =1k 2+m 2q,Q ,(7)where for simplification we do notsolvethefullyself-consistentschemein Fig.1but use a fixed mass of m q =0.25GeV,m c =1.5GeV,and m b =4.5GeV for the light,charm,and bottom quarks,respectively.Finally,the two-particle-qQ propagator in (4)is given in terms of the Thompson-reduction scheme [30]G qQ (E ;k )=1E −(ωq k+i Σq I )−(ωQk+i ΣQ I )(8)with a quasi-particle width for both light and heavy quarksof −2Σq,QI =0.2GeV.The solution of the LS equation (4)is simplified by using a partial-wave expansion of the potential and T -matrix,V a (q ′,q )=4πl(2l +1)V a,l (q ′,q )P l [cos ∠(q ,q ′)],T a (E ;q ′,q )=4πl(2l +1)T a,l (E ;q ′,q )P l [cos ∠(q ,q ′)],(9)which leads to the 1D LS equations,T a,l (E ;q ′,q )=V a,l (q ′,q )+26k 2d k d x∂t=∂p i p j(B ij f Q ).(12)4H.van Hees et al.:T-matrix approach to heavy quark diffusion in the QGPThe drag or friction coefficient,γ,and diffusion coeffi-cients,B ij =B 0p i p jp 2,(13)are calculated fromthe invariant scattering-matrix ele-ments [16].Taking into account elastic scattering of the heavy quark with a light quark or antiquark the latter given in terms of the above calculated T -matrix by|M|2=64π2E pd 3q(2π)32E q ′d 3p ′γc|M|2(2π)4δ(4)(p +q −p ′−q ′)f q (q )X (p ′),(15)for a heavy-quark observable,X ,over the elastic scatter-ings per unit time of the heavy quark with momentum p with a light quark of momentum q ,changing their mo-menta to p ′and q ′.Here,f q is the (thermal)distribution of the light quarks in the medium.Then we can calculate the transport coefficients asγ(|p |)= 1 −p ·p ′4 p′2 − (p ·p ′)22(p ·p ′)2 4παs ,using αs =0.4.As shown in Fig.4,close to T c the equili-bration times of τeq =1/γ≃7fm /c for charm quarks are a factor of ∼4larger than the values from a corresponding pQCD calculation,reminiscent to the results based on the model,assuming the survival of D -meson like resonance states above T c [12,13].In contrast to this and other cal-culations of the HQ transport coefficients,here the drag coefficients decrease with increasing temperature because0 0.05 0.1 0.151 2 3 4 5γ (1/f m )p (GeV)T-matrix: 1.1 T c T-matrix: 1.4 T c T-matrix: 1.8 T c pQCD: 1.1 T c pQCD: 1.4 T c pQCD: 1.8 T cFig.4.(Color online)The drag coefficient,γ,as a function of HQ momentum,calculated via (16)with scattering-matrix el-ements from the non-perturbative T -matrix calculation (using the parameterization of the lQCD internal potential energies by [Wo])compared to a LO perturbative calculation based on matrix elements from [33].of the “melting”of the dynamically generated resonances at increasing temperatures due to the diminishing inter-action strength from the lQCD potentials.To solve the Fokker-Planck equation (12)under con-ditions of the sQGP medium produced in heavy-ion colli-sions,we use an isentropically expanding thermal fireball model,assuming an ideal-gas equation of state of N f =2.5effective massless light-quark flavors and gluons.The to-tal entropy is fixed by particle multiplicities at chemical freeze-out which we assume to occur at the critical tem-perature,T c =180MeV.For semi-central collisions the fireball is chosen to be of elliptic-cylindrical shape with isobars given by confocal ellipses with a perpendicular radial-flow field,scaling linearly with the distance from the center as seen in hydrodynamic calculations [34]to which also the (average)radial flow velocity and elliptic-ity,v 2,is fixed.To compare to “minimum-bias”data on e ±spectra in 200A GeV Au-Au reactions at RHIC we simulate collisions with an impact parameter of b =7fm,implying an initial spatial eccentricity of about -ing a QGP-formation time of 0.33fm /c leads to an initial temperature of 340MeV.The evolution stops after the fireball has undergone a mixed QGP-hadronic phase after about 5fm /c ,at which the radial velocity of the fireball has reached about a radial velocity of about v ⊥=0.5c at the surface and an ellipticity of v 2=5.5%.Given this description of the medium,(12)is solved with help of an equivalent relativistic Langevin simulation which is defined by the stochastic equation of motion for a heavy quark at position,x ,and momentum p :δx =p4δt,(21)H.van Hees et al.:T-matrix approach to heavy quark diffusion in the QGP50 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 012 3 4 5R A Ap T (GeV)Au-Au √s=200 GeV (central)charm quarkspQCD, αs =0.4Reso, Γ=0.4-0.75 GeVT-matrix [SZ]T-matrix [Wo] 05 10 150 1 2 3 4 5v 2 (%)p T (GeV)Au-Au √s=200 GeV (b=7 fm)charm quarkspQCD, αs =0.4Reso, Γ=0.4-0.75 GeVT-matrix [SZ]T-matrix [Wo]Fig.5.(Color Online)Nuclear modification factor,R AA ,for central (upper panel)and elliptic flow,v 2,for semicentral (lower panel)200A GeV Au-Au collisions for charm quarks from the Langevin simulation,using the T -matrix results for the transport coefficients employing the parameterizations [Wo]and [SZ]for the lQCD potentials,compared to a calcu-lation based on pQCD and the resonance-model interactions in [13]B −1denoting the inverse of the diffusion-coefficient ma-trix (13).Note that in (20)and (21)the diffusion coeffi-cients are to be evaluated at the updated momenta p +δp .This H¨a nggi-Klimontovich realization of the stochas-tic process together with the dissipation-fluctuation rela-tion (19)ensures the correct equilibrium limit in the long-time regime t ≫1/γ[35].After evaluation of the time step (20)the resulting momenta are Lorentz boosted to the laboratory frame.The initial condition for (20)is given by the phase-space distribution of the heavy quarks.The spatial dis-tribution is determined with a Glauber model for heavy-quark production.The initial p t spectrum is determined from data on p -p and d-Au collisions at RHIC as follows:The c -quark spectra are taken from a modified PYTHIA calculation to fit D and D ∗spectra in d-Au collisions [36],assuming δ-function fragmentation.After decaying this spectrum to single e ±they saturate corresponding data from p -p -and d-Au [6,37]collisions up to p t ≃3.5GeV.The missing yield at higher p t is assumed to be filled with the corresponding contributions from B mesons,leadingto a cross-section ratio of σb ¯b /σc ¯c≃5·10−3and a crossing of the c -and b -decay electron spectra at p t ≃5GeV.0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 012 3 4 5R A Ap T (GeV)Au-Au √s=200 GeV (central)bottom quarks pQCD, αs =0.4Reso, Γ=0.4-0.75 GeVT-matrix [SZ]T-matrix [Wo] 012 3 4 5 0 1 2 3 4 5v 2 (%)p T (GeV)Au-Au √s=200 GeV (b=7 fm)bottom quarks pQCD, αs =0.4Reso, Γ=0.4-0.75 GeVT-matrix [SZ]T-matrix [Wo]Fig. 6.(Color online)The same as Fig.5but for bottom quarksIn Figs.5and 6we show the nuclear modification fac-tor,R AA ,defined by R AA =P Q (t fin ,p t )/P Q (0,p t )(where t fin denotes the time at the end of the mixed QGP-hadronic phase)in central and the elliptic flow,v 2= (p 2x −p 2y )/p 2t ,(22)in semicentral 200A GeV Au-Au collisions.We compare the results from the T -matrix model with the parameteri-zations by [Wo]and [SZ]of the lQCD potentials with the those using pQCD or the resonance-model interactions of Ref.[13].While for charm quarks for the [Wo]potential the result for R AA is comparable to the upper end of the uncertainty band of the resonance-model calculation,the v 2is slightly enhanced at low p t .The reason for this be-havior is the decrease of the transport coefficients with increasing T :While the suppression of the p t spectra at high p t is due to the evolution along the whole history of the fireball,leading to comparable effects at the end of the mixed phase,the anisotropic flow is mostly devel-oped at the later stages and thus can be transferred to the heavy quarks at the end of the evolution efficiently,when the drag coefficient become larger due to the dynamical formation of the resonance states.The T -matrix result with the somewhat less attractive [SZ]potential leads to the usual ordering of the coefficients (increasing with in-creasing temperature)and thus shows weaker effects for both the R AA and v 2than the result of the resonance model.For b quarks the T -matrix calculations yield larger medium modifications of the p t spectra than the resonance6H.van Hees et al.:T-matrix approach to heavy quark diffusion in the QGPmodel which is due to the mass effect,leading to stronger binding effects for the resonances in the T -matrix calcula-tion.As to be expected,the effects of pQCD-based trans-port coefficients on the HQ spectra for both charm and bottom quarks is much weaker than the non-perturbative ones via the resonance-scattering mechanism.4Single-electron observables at RHICThe last step toward a comparison of the above described model for HQ diffusion in the QGP with the single-electron p t data from RHIC is the hadronization of the HQ spec-tra to D -and B -mesons and their subsequent semileptonic decay to e ±.Here we use the quark-coalescence model de-scribed in [38,39].In the recent years,the coalescence of quarks in the hot and dense medium created in heavy-ion collisions has been shown to provide a successful hadroni-zation mechanism to explain phenomena such as the scal-ing of hadronic elliptic-flow parameters,v 2with the num-ber of constituent quarks,v 2,h (p t )=n h v 2,q (p t /n h ),where n h =2(3)for mesons (hadrons)denotes the number of constituent quarks contained in the hadron,h ,and the large p/πratio in Au-Au compared to p -p collisions [38,40,41].Quark coalescence is most efficient in the low-p t regime where most c and b quarks combine into D and B mesons,respectively.To conserve the total HQ number,we assume that the remaining heavy quarks hadronize via (δ-function)fragmentation.As shown in Fig.7the Langevin simulation of the HQ diffusion based on transport coefficients from the lQCD static potentials,followed by the combined quark-coale-scence fragmentation description of hadronization to D and B mesons and their subsequent semileptonic decay,successfully accounts simultaneously for both the nuclear modification factor,R AA ,and the elliptic flow,v 2,of sin-gle electrons in 200A GeV Au-Au collisions [8,7]at RHIC.The uncertainty due to the two different parameteriza-tions of the potentials by [Wo]and [SZ]is not so large.However,the deviations from other parameterizations are bigger and will be demonstrated elsewhere [42].The ef-fects from the “momentum kick”of the light quarks in quark coalescence,an enhancement of both,R AA and v 2,is important for the quite good agreement of both observ-ables with the data.As can be seen from the lower panel in Fig.7,within our model the effects from the mixing of the B -meson decay contribution to the e ±spectra be-comes visible in the region of p t ≃2.5-3GeV.A closer in-spection of the time evolution of the p t spectra shows that the suppression of high-p t heavy quarks occurs mostly in the beginning of the time evolution,while the v 2is built up later at temperatures close to T c which is to be ex-pected since the v 2of the bulk medium is fully developed at later stages only.This effect is also pronounced for the [Wo]parameterization of the HQ potential since in this case due to resonance formation the transport coefficients become largest close to T c .R A Ap T [GeV]v 2R A Ap T [GeV]v 2Fig.7.(Color online)Upper panel:single-electron spectra from the T -matrix calculation of HQ diffusion in the QGP based on the [Wo](solid line)and the [SZ](dash-double-dotted line)parameterizations of the lQCD static HQ potential in comparison to data from the PHENIX [7]and STAR [7]collab-orations in 200A GeV Au-Au collisions at RHIC.The dashed line shows the result when only δ-function fragmentation is considered for hadronization.Lower panel:R AA and v 2,as in the upper panel for the [Wo]parameterization for the elec-trons from D (dashed line)and B mesons (dash-dotted line)separately.H.van Hees et al.:T-matrix approach to heavy quark diffusion in the QGP75Conclusions and outlookWe have used static potentials fromfinite-temperature lQCD calculations within a Brueckner-type many-body calculation,complemented by pQCD HQ-gluon elastic-scattering matrix elements,to assess drag and diffusion coefficients for c and b quarks in the QGP in a princi-pally parameter free approach,however plagued with large uncertainties in the determination of the relevant poten-tial from lattice data.The diffusion of heavy quarks in the QGP is calculated with a Langevin simulation.The medium is parameterized as an expanding thermalfire-ball(including anisotropicflow for semicentral heavy-ion collisions)with an equation of state of a massless gas of gluons and N f=2.5light-quarkflavors.To confront this model with data on non-photonic single electrons in 200A GeV Au-Au collisions at RHIC,we have used a com-bined coalescence-fragmentation model to hadronize the heavy quarks to D and B mesons which subsequently de-cay semi-leptonically.The resulting p t spectra agree with recent data on the nuclear modification factor,R AA and ellipticflow,v2quite well.In a schematic estimate from the evaluated drag and diffusion coefficients leading to this results,based on ki-netic theory using either pQCD for a weakly coupled plas-ma[43,44]or the strong-coupling limit applying AdS/CFT correspondence[45,46],wefind values for the space-diffu-sion coefficient2πT D s=4-6and a viscosity to entropy-density ratio ofη/s=2-5/(4π)(to be compared to the conjectured AdS/CFT bound(η/s)min=1/(4π)),indi-cating a strongly coupled(liquid like)quark-gluon plasma close to the phase transition[47].In future works detailed studies of the uncertainties in the potential approach is necessary,in particular about the question whether a static-potential approach is justi-fied and which in-medium potential(i.e.,free or internal potential energy or combinations thereof)should be used to describe the interactions of heavy quarks within the sQGP.First steps in this direction have been made in[27] in an effective-field theory approach for non-relativistic in-medium QCD-bound states.Also an inclusion of inelastic processes like gluon bremsstrahlung which should become effective at higher p t is mandatory for a complete picture of the in-medium behavior of heavy quarks[48].Another step,which is quite natural,given that within our model the underlying microscopic effect of the strong coupling of the heavy quarks to the medium is the forma-tion of resonance states close to T c,is to substitute the quark-coalescence model with a transport-model based resonance-recombination model for hadronization in the QGP[49]which obeys the conservation laws for energy and momentum as well as the second law of thermodynam-ics.This model shares with quark-coalescence hadroniza-tion the phenomenologically successful feature of the scal-ing of v2with the hadrons’number of constituent quarks. 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疯狂的最新消息英语作文
疯狂的最新消息英语作文In the ever-changing landscape of current events, it's not uncommon to come across stories that seem to defy belief. The latest crazy news that has caught everyone's attention is truly a testament to the unpredictable nature of our world. Let's delve into the details of this bizarre tale and explore its implications.The Unbelievable StoryRecently, a small town made headlines for an incident that sounds straight out of a science fiction novel. A local farmer reported discovering a peculiar object in his cornfield, which, upon closer inspection, turned out to be a prototype of a next-generation solar panel. The twist? The technology was unlike anything currently on the market, andit was apparently designed to harness energy from multiple renewable sources simultaneously.The ReactionThe news spread like wildfire, with experts and enthusiasts alike scrambling to understand the implications of such a discovery. The scientific community was abuzz with theories about who could have developed this technology and why it was left in such an unusual location. Conspiracy theories quicklyfollowed, with some suggesting that the object was part of a secret government experiment gone awry.The InvestigationLocal authorities, along with a team of scientists, launched an investigation to determine the origin of the mysterious device. They were particularly interested in its advanced features, which seemed to hint at a significant leap forward in clean energy technology. As the investigation continues, the town has become a hub for journalists and curiosity-seekers, all hoping to uncover the truth behind the story.The ImpactThe impact of this crazy news extends beyond the immediate fascination with the unknown. It has sparked a global conversation about the future of energy and the potential for hidden breakthroughs that could revolutionize the way we power our world. The incident has also highlighted the importance of continued investment in research and development, as well as the need for transparency inscientific advancements.The AftermathAs the dust begins to settle, the town is left with a newfound fame and a lingering question: what other secrets might be hidden in the shadows of our daily lives? The crazy news has served as a reminder that the world is full of surprises, and sometimes, the most extraordinary stories canemerge from the most ordinary places.This English composition captures the essence of a sensational news story, exploring the narrative from discovery to investigation and its broader impact on society. It encourages readers to consider the potential for unexpected advancements and the importance of staying informed in a rapidly evolving world.。
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a r X i v :h e p -l a t /9809091v 1 14 S e p 19981Heavy quark potential in the instanton liquid modelR.C.Brower a ,D.Chen b ∗,J.W.Negele b and E.Shuryak ca Department of Physics,Boston University,Boston,MA 02215bCenter for Theoretical Physics,Massachusetts Institute of Technology,Cambridge,MA 02139cDepartment of Physics and Astronomy,State University of New York,Stony Brook,NY 11794We study the heavy quark potential in the instanton liquid model by carefully measuring Wilson loops out to a distance of order 3fm .A random instanton ensemble with a fixed radius ρ=1/3fm generates a potential V (R )growing very slowly at large R .In contrast,a more realistic size distribution growing as ρ6at small ρand decaying as ρ−5at large ρ,leads to a potential which grows linearly with R .The string tension,however,is only about 1/10of the phenomenological value.1.INTRODUCTIONThere is growing evidence that instantons play a very important role in QCD.The development of the Interacting Instanton Liquid Model [1]al-lows one to calculate nonperturbatively to all or-ders in the ‘t Hooft interaction,and results show that it correctly generates the quark condensate and most salient properties of light hadrons.As reviewed at this conference [2],the essential fea-tures and parameters of this model have now been confirmed on the lattice,and there is strong lat-tice evidence that instantons and their associated zero modes play a significant role in hadron struc-ture.Motivated by a provocative result by Fuku-shima et al.[3]reporting a string tension in the instanton liquid model close to the phenomeno-logical value,we have undertaken a careful calcu-lation of the instanton induced heavy quark po-tential.2.SIMULATION DETAILSThe gauge field of a single instanton in SU (2)in the singular gauge (centered at the origin)isx 2ρ2(ρ3.50+ρ3.5)1120.0 2.04.06.0T (fm)0.000.100.200.300.40V (T ,R ) (G e V )Figure 1.V (T,R )as a function of T for the fixed instanton size distribution ρ=1/3fm at an instanton number density n =1.0fm −4.Each data set from bottom to top corresponds to R =0.4fm ,0.8fm ,1.2fm ,1.6fm and 2.0fm ,respec-tively.=(13.6fm,7.2fm,7.2fm,20fm ).Wilson loops W (R,T )=trP exp[iR ×TA µdx µ](3)are measured in a 6.4fm ×12.8fm rectangle in the central yz plane.This leaves a distance of at least 3.6fm or more than 10times the average instanton size ¯ρfrom the measured Wilson loop to the 4-d boundary so that the edge effects from the open boundary are small.The largest Wilson loop is 3.2fm ×6.4fm in size.The discretization ∆x in the path ordered integral is 0.05fm ,which is about 1/6of the average instanton size.We study three different instanton number den-sities,n =N/V =0.5fm −4,1.0fm −4and 1.5fm −4for both fixed and variable instanton size distributions.The number of gauge configu-rations studied for each different n is 1600,8000and 1600,respectively.0.02.04.0 6.0T (fm)0.000.100.200.300.40V (T ,R ) (G e V )Figure 2.V (T,R )as a function of T for a variable instanton size distribution described in Eq.(2)at an instanton number density n =1.0fm −4.Each data set from bottom to top corresponds to R =0.4fm ,0.8fm ,1.2fm ,1.6fm and 2.0fm ,respectively.3.RESULTSThe fundamental difficulty in this calculation is measuring with adequate statistical accuracy the large Wilson loops required to determine the potential at large distances.We determine the heavy quark potential V (R )from rectangular loops of different time extent as follows:V (T,R )=−1W (R,T ),(4)V (R )=lim T →∞V (T,R ).(5)To obtain the most accurate measurement of the potential,we plot V (T,R )vs.T and find the plateau corresponding to V (R ).Figures 1and 2show such plots at different R for both fixed and variable instanton size dis-tributions at an instanton number density n =1.0fm −4.For the fixed instanton size distribu-tion,ρ=1/3fm ,plateaus for V (T,R )set in at about T >(1∼2)R .We have good plateaus for all R ≤3.2fm at n =1.0fm −4.In contrast,for30.00.51.01.52.02.53.0R (fm)0.000.100.200.300.40V (R ) (G e V )N/V = 1.5 fm−4N/V = 1.0 fm−4N/V = 0.5 fm−4Figure 3.Heavy quark potential in the instanton liquid model for a fixed instanton size ρ=1/3fm .Three different instanton number densities are plotted.the variable instanton size distribution,not only do the plateaus set in at a larger T >(2−3)R ,but also the statistical fluctuations are much larger at large R ,especially for R ≥2.0fm .It is clear that this behavior of the variable size distribu-tion arises from large instantons in the tail of the distribution.Even with an ensemble as large as 8000configurations,we are only able to obtain a potential V (R )for R ≤2.0fm at n =1.0fm −4.Figures 3and 4show the heavy quark poten-tial out to the maximum distance we can reliably measure.The salient results are the following.At short distance,the slope is identical for both the fixed and the variable size distributions at a given n and is proportional to n .At the value n =1.0fm −4of the instanton liquid model,this slope is ∼0.1GeV/fm ,corresponding to 1/10of the physical string tension.We note that by scal-ing,the physical string tension would be obtained at n =1.0fm −4by increasing the mean value of ρby 101/4to 0.59fm .When R is much greater than the largest instantons in the medium,we expect V (R )to approach a constant [4].For the distribu-tion in Eq.(2),however,it is interesting that the0.00.5 1.01.52.0 2.53.0R (fm)0.000.100.200.300.40V (R ) (G e V )N/V = 1.5 fm−4N/V = 1.0 fm−4N/V = 0.5 fm−4Figure 4.Heavy quark potential for a variable in-stanton size distribution with ¯ρ=1/3fm .Three different instanton number densities are plotted.potential is essentially linear in the region of 0fm –2fm .Hence,even though the random instan-ton liquid is not strictly confining,in the region of physical interest for hadron structure,it has a significant linear component.4.ACKNOWLEDGEMENTThe authors are grateful to Sun Microsystems for providing the 24Gflops E5000SMP cluster and the Wildfire prototype system on which our 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