Vector Boson Production at Hadron Colliders Results from HERWIG and Resummed Calculations

关于科学家的作文100字英文回答:Scientists play a crucial role in advancing human knowledge and improving our lives. They are curious individuals who constantly ask questions and seek answers through systematic investigation and experimentation.Scientists use the scientific method to conduct their research. They formulate hypotheses, design experiments, collect and analyze data, and draw conclusions based on evidence. This rigorous approach ensures that their findings are reliable and can be replicated by other scientists.Science covers a wide range of disciplines, including physics, chemistry, biology, and astronomy. Each field of study has its own set of methods and tools. For example, physicists use mathematical equations and complex instruments to understand the fundamental laws of theuniverse, while biologists study living organisms and their interactions with the environment.Scientists also collaborate with colleagues from different countries and backgrounds. This international collaboration allows them to share knowledge, resources, and expertise, leading to breakthrough discoveries and innovations. For instance, the discovery of the Higgs boson particle at the Large Hadron Collider involved scientists from around the world working together.Furthermore, scientists have a responsibility to communicate their findings to the public. They often publish their research in scientific journals and present their work at conferences. By sharing their knowledge, scientists contribute to the collective understanding of the world and inspire future generations to pursuescientific careers.中文回答:科学家在推动人类知识的发展和改善我们的生活方面起着至关重要的作用。

a rXiv:h ep-ph/43194v29Oct24ANL-HEP-PR-04-4NSF-KITP-04-13hep-ph/0403194Higgs Boson Production in Weak Boson Fusion at Next-to-Leading Order Edmond L.Berger 1,∗and John Campbell 1,†1High Energy Physics Division,Argonne National Laboratory,Argonne,IL 60439(Dated:March 22,2004)Abstract The weak boson fusion process for neutral Higgs boson production is investigated with particular attention to the accuracy with which the Higgs boson coupling to weak bosons can be determined at CERN Large Hadron Collider (LHC)energies in final states that contain a Higgs boson plus at least two ing fully differential perturbative matrix elements for the weak boson fusion signal process and for the QCD background processes,we generate events in which a Higgs boson is produced along with two jets that carry large transverse momentum.The effectiveness of different prescriptions to enhance the signal to background ratio is studied,and the expected signal purities are calculated in each case.We find that a simple cut on the rapidity of one final-state jet works well.We determine that an accuracy of δg/g ∼10%on the effective coupling g may be possible after ∼200fb −1of integrated luminosity is accumulated at the LHC.PACS numbers:14.80.Bn,12.38.Bx,13.85.-t,12.38.QkI.INTRODUCTIONFollowing the discovery of the neutral Higgs boson H at the CERN Large Hadron Col-lider(LHC),attention will focus on the measurement of its couplings to gauge bosons and fermions.A promising reaction from which to extract some of these couplings,particularly the HW W coupling,is the weak-boson fusion(WBF)process[1]–[14],where the Higgs boson H is produced via fusion of the weak bosons W and Z:W W,ZZ→H,and is accom-panied in thefinal state by two jets that carry large transverse momentum p T.To extract the couplings reliably,a good understanding is required of the production processes and the background processes that lead to the samefinal state.Many strong interactions subpro-cesses also generate Higgs boson-plus-two-jet(H+2jet)final states.These background subprocesses can be computed with the techniques of perturbative quantum chromodynam-ics(QCD).They supply an irreducible background that may be reduced to some extent by judicious selections on thefinal state event topology.In the analysis presented here,we have in mind a situation in which the Higgs boson has been discovered and a sample of events exists containing a Higgs boson and two or more jets.This set of events will contain backgrounds of two types:real H+2jet events produced by QCD mechanisms other than WBF,and events which contain jets and particles that are present in typical Higgs boson decay modes,but without an explicit Higgs boson.Within the full event sample,we discuss the simulation of the real WBF signal and the irreducible QCD H+2jet background.We do not address the second type of contamination,such as events from the QCD Z+2jets process where the Z decay imitates a Higgs boson decay. Our concern is to estimate the expected signal purity,by which we mean the fraction of real Higgs boson events produced by weak boson fusion.The WBF H+2jet signal region is characterized by jets that carry large transverse momentum and large rapidity.Because the jets carry large transverse momentum,it is necessary to use hard QCD matrix elements in order to represent the signal and the H+2jet background reliably.A parton shower approach,for example,would be unlikely to provide a correct estimate of the momentum distribution of the jets in the region of phase space of interest.Next-to-leading order(NLO)QCD corrections to the total WBF production cross section have been known for some time[15],and the corresponding corrections were calculated recently in a fully differential way[16].In this paper,we use an independentcalculation to verify the results of Ref.[16]and to examine in more detail the effects of the WBF selection cuts on the NLO QCD corrections.We also use perturbative QCD expressions for the background H+2jet matrix elements.At present,the fully differential H+2jet background distributions are known only at leading order.In addition to our NLO study of the signal process,we nevertheless provide two estimates of the NLO enhancement of the QCD H+2jet background process,in order to better assess the viability of the WBF channel for measuring the coupling strength of the Higgs boson to vector bosons.Our calculations are fully differential at the partonic level.One limitation of the fact that we omit showering is that forward beam jets,which likely have low p T,are ignored.Since the WBF channel is most interesting for a Higgs boson in the mass range m H= 115−200GeV,we perform calculations with the two extremal values of this range.We compute differential cross sections in rapidity and transverse momentum at a pp collider with √method for defining WBF events with two other popular methods found in the literature: a selection on the difference in rapidities between two tagging jets in thefinal state,and a selection on the invariant mass of a pair of tagging jets.The alternative prescriptions yield some increase in signal purity with respect to our definition,but the gain is sensitive to the cut in transverse momentum used to specify the trigger jets,and it is accompanied by loss of event rate.For values of the jet cut p T>40GeV,our prescription appears to work about as well as the other methods.Relatively high luminosity will be needed for a precise determination of the uncertaintyδg/g.Our simpler definition of the WBF sample in terms of a selection on the rapidity of only one jet offers advantages in a high luminosity environment where a large value of the transverse momentum cut is appropriate and multiple events per crossing may be an issue.We provide two methods for estimating the size of next-to-leading order corrections to the H+2jet background in Sec.V.One of these relies on similarity with the Z+2jet process for which fully differential NLO results are known.The other method is an extrapolation from the known next-to-next-to-leading order(NNLO)results for the fully inclusive Higgs boson production process.The substantially different estimates for the NLO enhancement provided by these two methods show the level of uncertainty of the LO background calculation.A fully differential NLO calculation of the H+2jet background applicable in the region of interest for WBF investigations is needed in order to improve our computations of signal purity and of the expected uncertainty inδg/g.A summary of our conclusions may be found in Sec.VI.II.PRODUCTION CROSS SECTIONSExamples of the WBF diagrams that must be calculated are shown in Fig.1.The basic leading order process is shown in(a),where the exchanged bosons may be either W’s or Z’s, and one or both quark lines may be reversed,yielding qq,q¯q and¯q¯q initial states.The virtual NLO corrections are obtained by adding a gluon loop to either qqV vertex,as illustrated in (b).The remaining real NLO corrections are shown in(c),where either an additional gluon is radiated in thefinal state or a gluon from the proton splits into a q¯q pair.Calculation of the necessary loop diagrams is straightforward,providing a couple of simplifying assumptions are made.First,we ignore contributions of the form q¯q′→V⋆→V H,where V=W,Z.FIG.1:Representative diagrams for the production of a Higgs boson via weak boson fusion:(a)at lowest order;(b),(c)at NLO.Further diagrams can be obtained by crossing incoming and outgoing lines in all cases.All of the virtual corrections are of the vertex correction form,as shown in (b).There are two types of real corrections depicted in(c).Thefirst set corresponds to the emission of a gluon in all possible positions on the quark lines(left-hand diagram)and the second set corresponds to the crossing where a gluon is present in the initial state(right-hand side).Second,we neglect any interference effects from identicalflavor quarks in thefinal state.We checked that both of these approximations have little effect on the calculated cross sections at leading order,particularly in the region of phase space that we consider.The NLO calculation is embedded in the general purpose Monte Carlo program MCFM[17],which uses the dipole subtraction method[19].We use the default set of parameters in this program,in whichα=1/128.89,M W=80.419GeV,M Z=91.188GeV, and sin2θw=0.2285.For the parton distribution functions,we use CTEQ6L1for lowest order and CTEQ6M at NLO[20].In these sets of parton densities,αLO s(M Z)=0.130and (M Z)=0.118.In this paper,we choose the reference valueµ=m H for the renormal-αNLOsization and factorization scales.In Sec.III C we investigate the uncertainty of the signal and of the background associated with variation of the scale over the interval2m H>µ>m H/2.A.Generic cutsThe hallmark of WBF events is a Higgs boson accompanied by two“tagging”jets having large p T and large rapidity.In real events and in computations at NLO,there are generally more than two jets,and the goal is to pick out a clean signal.To simplify our study and to demonstrate the robust character of the WBF process,it is desirable to make as few selections(cuts)as possible on the events.We begin with a basic set of cuts,exactly as in Ref.[16].Jets obtained from the Monte Carlo runs are clustered according to the k T algorithm with p jet T>20GeV,jet pseudo-rapidity|ηjet|<4.5,and jet separation∆R jj=FIG.2:Representative diagrams for the production of a Higgs boson and two jets at lowest order, calculated in the heavy top-quark limit of the Hgg effective coupling.In Sec.V we estimate the NLO corrections to the lowest order result for the H+2jets background process by comparison with the similar Z+2jets process,calculated at NLO in Ref.[17,23],and by extrapolation from NNLO calculations of the fully inclusive process pp→HX[24]–[30],and NLO calculations of pp→H+1jet+X[31].III.RESULTSThe cuts mentioned in the previous section are a generic set of cuts.They do not exploit the kinematic structure of WBF events,where the jets tend to be produced very forward in pseudo-rapidity.In this subsection,we presentfirst the cross sections for the signal and background processes after application of the generic cuts.Without further cuts,the WBF events would be lost in the QCD continuum background.We then apply one additional constraint which defines our WBF sample,and we show results for kinematic distributions, event rates,and signal purities.A.Basic cutsWe examine the effects of the generic cuts in terms of their effects on the WBF signal and the H+2jet background.These cross sections–without any further cuts–are shown in Table I as a function of the minimum jet p T.The WBF signal process is calculated at NLO and,at this point,the H+2jet background at LO.We remark that the effects of the NLO corrections on the WBF process are rather small in this region,corresponding to K-factors between0.95and1.1,depending on the Higgs boson mass and p T cut.The values in the table show that(without consideration yet of NLO effects in the back-ground)the rates for the signal and background are comparable for p min=20GeV,and theTp T cut[GeV]40Signal(m H=115)1081Bkg743118916695896TABLE I:Cross sections in fb for the WBF signal(calculated at NLO)and H+2jet back-ground(LO),as a function of the minimum jet p T.Only the minimal set of cuts of Sec.II is applied.signal-to-background ratio improves as the p T cut is increased.The WBF signal lies above the H+2jet cross section if p minT≥40GeV.B.WBF cutsIn an attempt to exploit the WBF event structure,a popular cut invokes a separation inpseudo-rapidity between the two tagging jets,for instance|ηj1−ηj2|>4.In this paper,we define a slightly different and simpler cut,motivated by our examination of the distributions of the absolute jet pseudo-rapidities shown in Fig.3.In thesefigures,each tagging jet enters with weight one-half and cross sections have been converted to event rates with an integrated luminosity of1fb−1.The area under each curve is equal to the total number of events in that channel.The plots in Fig.3show that the shape of the distribution depends little on either the Higgs boson mass or the jet p T,but–as expected–is very different in the signal process, compared to the H+2jet background.In each case,the WBF events peak at values of |η|≈3,although there is a slight movement to lower values of|η|as the p T cut is increased. The width of the peak also tends to decrease,but the full width at half-maximum is fairly constant at approximately3units of rapidity.In contrast,the rate of background events falls offfairly sharply beyond|η|≈2.Motivated by the comparison of rapidity spectra in Fig.3,and erring on the side of simplicity,we choose a uniform cut that ensures at least one jet lies within the ly,ηpeak−ηwidth/2<|ηj|<ηpeak+ηwidth/2,(1)FIG.3:Dependence of the tagging jet pseudo-rapidities on the minimum jet p T used,for the two cases m H =115GeV (left)and m H =200GeV (right).Each of the two tagging jets in the event is entered in these plots,with weight one-half,and the rates assume an integrated luminosity of 1fb −1.The signal (solid)is calculated at NLO and the background (dashed)at LO.We show results for three different selections of the minimum jet transverse momentum,p T >20,40,and 80GeV.p T cut[GeV]40Signal(m H=115)789Bkg382Purity0.67928121534460.630.72TABLE II:Event rates for the Hjj WBF signal(NLO)and Hjj background(LO),including our WBF requirement that at least one jet carry large|η|,as defined by Eq.(1).We assume1fb−1 of integrated luminosity.Purity is defined as P=S/(S+B),where S stands for the number of signal events and B for the number of background events.for j=j1or j=j2,whereηpeak=3andηwidth=2.8.Equation(1),along with the generic cuts specified above,constitutes our definition of weak boson fusion cuts.The effects of the pseudo-rapidity restriction on the jets are shown in Table II,where we have assumed1fb−1of integrated luminosity.The rates in this table should be contrasted with those in Table I.In Table II,we include values for the signal purity,defined as P=S/(S+B),where S stands for the number of signal events and B for the number of background events.The number of events as a function of the minimum jet p T is also plotted in paring the tables,one can see that the signal rate is diminished only slightly,by about20–30%.On the other hand,the background is shrunk considerably,by about a factor of two.A p T cut of20GeV is barely sufficient to distinguish the WBF signal above the QCD LO Hjj background for m H=115GeV.However,the signal S to background B ratio improves to about2for p cut T≥40GeV.At m H=200GeV,the situation is better,with S/B of about1.7when the p T cut is20GeV,and rising to ∼3for p cut T≥40GeV.A p T cut of40GeV yields a prominent effect across the range of interesting masses,m H=115–200GeV.It is instructive to examine the origin of the different rapidity spectra for the signal and the background.Since H+2jet events are generated in both cases with identical cuts on the transverse momenta of the jets,the different rapidity spectra must originate from dynamics.FIG.4:Numbers of events for the WBF signal and the QCD background as a function of the minimum jet p T,for an integrated luminosity of1fb−1.No branching ratios for the Higgs boson decay have been applied.The pseudo-rapidity restriction on one of the jets has been enforced,as in Table II.The solid line is the NLO signal and the dashed is the LO background.Comparing the LO production diagrams in Figs.1and2,we note that gg and qg initial states contribute to the QCD background but not to the WBF signal.The gluon parton density is notably softer than the quark parton density,suggesting a plausible reason for the differences in the rapidity spectra of thefinal state jets in the two cases.This reasoning is supported by the results shown in Fig.5.The shape of the background rapidity spectrum from the qq, q¯q,and¯q¯q contributions is very similar to that of the signal,albeit with a slight shift of the peak to smaller|η|.The very different rapidity spectra of the signal and the background evident in Fig.3results therefore primarily from the gg and qg initial state contributions. The results shown in Fig.5imply that there is a basic upper limit to the purity one can achieve for the WBF event sample,regardless of which prescription one adopts to define the WBF sample.The qq,q¯q,plus¯q¯q component of the QCD background process generates afinal state event topology essentially identical to the WBF signal process.Values of the purity are listed Table III;there is not much variation with m H or the value of the cut in p T.Our results suggest that purity is bounded from above by at most P<0.95at LHC energies.C.Scale dependence studyTo examine further the effects of NLO corrections,we consider variation of the renormal-ization and factorization scale.A range of values m H/2<µ<2m H is used conventionallyFIG.5:Dependence of the tagging jet pseudo-rapidities for jet p T >40GeV,for the two cases m H =115GeV (left)and m H =200GeV (right).For the background,we show the full result with all contributions included and,for comparison of shapes,the background obtained if only the qq ,q ¯q ,and ¯q ¯q initial state contributions are used.The magnitude of the separate component is multiplied by 20.p T cut [GeV]40Signal (m H =115)789Bkg (qq ,q ¯q ,¯q ¯q )45Purity (qq ,q ¯q ,¯q ¯q )0.959281214780.950.94TABLE III:Event rates for the Hjj WBF signal(NLO)and for the part of the Hjj back-ground(LO)that arises from the qq ,q ¯q ,and ¯q ¯q initial-state terms.to estimate the theoretical uncertainty at a given order in perturbation theory.As a rep-resentative case,we show results for a minimum jet p T of 40GeV and both Higgs boson masses.In Fig.6,we show the tagging jet pseudo-rapidity distributions for the signal and background for a range of values of the common renormalization and factorization scale µ.The signal process shows very little variation with µ,a shift of less than ±2%when |ηj |∼2in the WBF signal region.In contrast,the H +2jet background at LO is enhancedFIG.6:Tagging jet pseudo-rapidity distributions in Hjj events,calculated with a range of values of the renormalization and factorization scale,µ.The signal (solid)is calculated at NLO and the background (dashed)at LO.by approximately 70%when the scale choice µ=m H /2is made,and reduced by 40%for µ=2m H .A fully differential NLO calculation of the H +2jet background process is required to reduce the large uncertainty associated with µvariation apparent in Fig.6.D.Uncertainty on the CouplingsThe signal and background events both include a real Higgs boson along with two jets.We may define a signal “purity”as the ratio S/(S +B ),where S denotes the number of signal events and B the number of background events.The purity as defined here does not improve with greater luminosity nor does it depend on the Higgs boson decay mode considered.Of interest to us is the effect of signal purity on the accuracy of the determination from data of the Higgs boson couplings g W W and g ZZ to the W W and ZZ channels.The WBF crosssection is proportional to a combination of g 2W W and g 2ZZ ,and their relative contributionchanges somewhat with the value of the cut on p T .In this paper,we discuss only an effective coupling strength g .We remark also that in our discussion of the expected accuracy on g ,we limit ourselves to uncertainties at the level of production of the Higgs boson.We set aside uncertainties associated with the fact that the Higgs boson is observed only in specific final states and that all the final states cannot be observed above backgrounds.To derive the uncertainty δg/g on the coupling,we begin with the observed number ofevents N=S+B.We define the ratio r=g2observed/g2predicted.Then,under the assumption that any deviation in the expected total number of events arises from the effective coupling, we obtain r=(N−B)/S.Taking the total derivative,we obtain an expression for the uncertainty in r.δr/r=[(δS/S)2+((δN)2+(δB)2)/(N−B)2].(3)With purity P=S/(S+B),we deriveδg/g=1/2FIG.7:The predicted uncertaintyδg/g in the coupling of the Higgs boson to a pair of W bosons is shown as a function of signal purity P=S/(S+B)for expected statistical accuraciesδN/N of 10%and2%The uncertainties in knowledge of the signal S and background B are assumed to be 5%and30%respectively.choice presupposes that the20%µvariation and5%PDF uncertainty of the fully inclusive NLO cross section for Higgs boson production may also apply to the NLO calculation of H+2jet production in the WBF region of phase-space,once this calculation is done.Based on a study of conventional backgrounds[32],a minimum of roughly10fb−1of integrated luminosity is needed to discover the Higgs boson in the WBF process.This figure would be achieved after one year of LHC operation at a luminosity of1033cm−2s−1. Using the numbers in our Table II,we expect a WBF sample(S+B)of∼12000events for m H=115GeV and p T cut=40GeV,and∼7000events for m H=200GeV and p T cut=40GeV.To translate these event rates into statements about statistical significance, we must specify a Higgs boson decay mode and approximate tagging efficiencies for the decay products.For m H=115GeV,we choose the decay H→τ+τ−,with oneτdecaying leptonically and the other hadronically[8].These choices yield a branching ratio ofBr(H→ττ)×Br(τ→leptons)×Br(τ→hadrons)=0.073×0.7×0.65=0.033.For the efficiency for tagging hadronicτdecays,we take thefigure0.26from Ref.[33]as an optimistic upper bound.1The true value for this efficiency will be known only after analysis of data from LHC experiments.While our choice of tagging efficiency may seem large,it is relatively easy to scale ourfinal results if a different value is preferred.The combination of branching fraction and efficiency results in a reduction in the number events by a factor ǫ=0.033×0.26≈0.01.For m H=200GeV,the decay H→W+W−is prominent,and we select the case in which both W’s decay leptonically[11].2We obtainǫ=Br(H→W W)×Br(W→leptons)2=0.74×0.222=0.036.Using these numbers,we compute expected statistical uncertainties ofδN/N∼10%and ∼6%at m H=115and200GeV,respectively.With statistical accuracyδN/N of10%,δS/S=5%andδB/B=30%,we obtainδg/g≃10%for purity P=0.7when m H= 115GeV,andδg/g≃8%when m H=200GeV.After5years of LHC operation,we can anticipate an integrated luminosity of200fb−1 will have been accumulated.This increase allows us to reduce our estimates ofδN/N to ∼2%and∼1.5%at m H=115and200GeV,respectively,andδg/g∼7%for P=0.7.We remark that the uncertainties in the signal S and background B dominate the uncer-tainty in g.If P=0.7andδN/N=2%,then the uncertaintiesδS/S andδB/B would have to be reduced to3%and6%,respectively,before the statistical uncertainty would control the answer.Even if P=1,δg/g is controlled byδS/S untilδS/S≤δN/N.In Fig.7,we show numerical predictions for the uncertainty as a function of purity,for two choices of the statistical uncertainty.Signal purities of0.65or greater permit determinations ofδg/g of10%or better after200fb−1have been accumulated.As shown in Table II, P>0.65is obtained for p T cut>40GeV at m H=115GeV,and p T cut>20GeV at m H=200GeV.The curves indicate to us that there is not much to gain from purities greater than70%.FIG.8:The predicted uncertaintyδg/g in the coupling of the Higgs boson to a pair of W bosons is shown as a function of signal purity P=S/(S+B)for expected statistical accuraciesδN/N of 10%and2%The uncertainties in knowledge of the signal S and background B are assumed to be 4%and20%respectively.Somewhat smaller values ofδS/S andδB/B are chosen in another recent investigation of anticipated uncertainties in the couplings[18].These values areδS/S=4%andδB/B= 20%.Although the scope of that study is quite different from ours,we may compare our estimates with theirs.In Fig.8,we show the uncertainty as a function of purity for these new estimates ofδS/S andδB/B.For P=0.7,we nowfindδg/g∼9%and∼5%for the low-and high-luminosity data samples.This new lower value ofδg/g is similar to that obtained in Ref.[18]at comparable luminosity3.IV.ALTERNATIVE DEFINITIONS OF THE WBF SAMPLEIn the previous section,we define the WBF sample by a simple selection on the rapidity of one jet in an event in which there is a Higgs boson and two jets each carrying p T greater than a specified minimum value.Other definitions have been used in the literature,and we wish to compare our signal rates and purities with those obtained if we use these alternatives. We examine the traditional cut on rapidity separation between the two trigger jets and a cut on the invariant mass of the pair of trigger jets.A.Rapidity separation cutIn Fig.9,we illustrate the expected event rates as a function of the difference in rapidities between the forward and backward tagging jets.There is a clear separation in the locations of the peaks of the WBF signal and the background,not unlike that seen in our Fig.3. Distributions such as these may motivate the choice of a cut on rapidity separation,|ηj1−ηj2|>4,as in Refs.[1]–[14],and[16].Signal and background rates for WBF events selected in this fashion are shown in Table parison of Tables II and IV shows that the signal rate is diminished somewhat and that the purity is greater when the rapidity separation selection is made.As shown in Figs.7and8,a gain in purity reduces the uncertainty in δg/g.The quantitative shift from P=0.67to P=0.78at M=115GeV and p T>40GeV is an improvement of only3%inδg/g,and this reduction is offset somewhat by the loss in statistical accuracy.At lowest order in perturbation theory,one might expect naively that our simple rapidity selection and the rapidity separation cut are close to identical since there are only two jets in the event,tending to be located in opposite hemispheres.However,thefinite rapidity carried by the Higgs boson introduces differences.Our preference for the simple rapidity cut is based on a few considerations.In data(and at yet-higher orders in perturbation theory),there will be many jets,and the simple specification of events that satisfy Eq.(1) will be easier to implement.Second,in a high luminosity environment with more than one event per beam crossing,a selection on only one jet(in addition to the Higgs boson)reduces the chance that jets from different events are used.Finally,in our study of NLO event topologies with three jets in thefinal state,wefind that a gluon jet,rather than a quark jetFIG.9:The difference between the two tagging jet pseudo-rapidities for a minimum jet p T of 20GeV,for the two cases m H =115GeV (left)and m H =200GeV (right).The rates assume an integrated luminosity of 1fb −1.The signal (solid)is calculated at NLO and the background (dashed)at LO.p T cut [GeV]40Signal (m H =115)718Bkg207Purity0.78911106349200.720.84TABLE IV:Event rates for the Hjj WBF signal(NLO)and Hjj background(LO),without our WBF definition and instead with the rapidity separation cut |ηj 1−ηj 2|>4.We assume 1fb −1of luminosity.The purity and significance are as defined before.is sometimes one of the two jets with largest p T .For example,with a jet cut of 20GeV and m H =115GeV,a gluon is a tagging jet about 25%of the time when we use our definition of WBF events.B.Invariant mass cutAs an alternative to the rapidity separation cut,one might consider a cut on the invariant mass of the two trigger jets.In Fig.10and in Table V,we display the the effects of the mass cut M jj>parison of Figs.10and3shows a decided improvement in the signal to background ratio,an effect that is borne out in the purity numbers shown in Tables V and II.However,the significant gain in purity is true only for the smaller values of the p T cut and is accompanied by a substantial loss of signal rate.Since the smallest value of the p T could be employed only with low-luminosity data samples,it is not evident that the price in loss of signal rate is affordable.The combination of the mass cut and our simple forward jet cut improves purity only slightly and reduces the signal rate further.FIG.10:The tagging jet pseudo-rapidity distribution for a minimum jet p T of20GeV,for the two cases m H=115GeV(left)and m H=200GeV(right)–with a minimum dijet invariant mass, M jj>800GeV.Each of the two tagging jets in the event is entered in these plots,with weight one-half,and the rates assume an integrated luminosity of1fb−1.The signal(solid)is calculated at NLO and the background(dashed)at LO.Using the invariant mass cut to define the WBF sample,we note that the number of events at m H=115GeV with a cut on p T of20GeV is very similar to what we obtain with our WBF prescription but with a cut on p T=40GeV.The purities are also nearly the same.This comparison would seem to favor our simple prescription:a larger value of the cut on p T is more appropriate in a high rate environment and more effective at reducing backgrounds not considered here.We conclude this section with the remark that。

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arXiv:hep-ph/9805375v1 19 May 1998
for Mathematics, 630090 Novosibirsk, Russia f¨ ur Physik komplexer Systeme, Bayreuther Straße 40, 01069 Dresden, Germany‡‡ 3 Institut f¨ ur Theoretische Physik, TU Dresden, 01062 Dresden, Germany 4 D. I. Mendeleev Institute for Metrology, 198005 St. Petersburg, Russia 5 Novosibirsk State University, 630090 Novosibirsk, Russia
The production rate and the lifetime of the dimuonic atoms are both proportional to this value. The lifetimes of low lying states are of the order of τ ∼ 10−12 s and are summarized in Table II. A brief discussion of the evaluation of the lifetime of parastates is given in Appendix A. Main decay channels are the annihilation processes PM → γγ , OM → e+ e− . (4)
2
∼ 0.04 ,
<
(1)
where r 2 is the mean square radius of the charge distribution of the nucleus, and the mass of the dimuonic atom is mµµ ≈ 2 mµ = 211 MeV. Therefore, in all cases under consideration the multiphoton processes set limits on the accuracy on the level of 5 %. 2

a r X i v :h e p -p h /9804328v 2 21 A p r 1998SINGLE TOP QUARK AND LIGHT HIGGS BOSONAT TEV ATRONA.Belyaevs =2TeV and up to about 120GeV at√1.1SignalWe concentrated on the following processes of single top quark production:1.p¯p→tq¯b+X (W-gluon fusion),2.p¯p→t¯b+X,3.p¯p→tq+ X where q is a light quark and X stands for remnants of the proton and antiproton.Feyn-man diagrams for the above processes are shown in Fig.1.We refer to the paper[2]for de-tails.All analytical and almost all numeri-Figure1:Diagrams for single top production cal calculations have been done by means of CompHEP package[3].FORTRAN codes of matrix elements generated by CompHEP have been used in MC generators implemented as external processes of PYTHIA program.For t-quark production we chose Q2scale equal to top mass squared.For a180GeV top quark the total single top plus anti-top cross section is1.76+0.26−0.18pb(√∆φ2jj+∆η2jj).For Wjj background Q2=M2W has been taken.For calculation of jb¯b and jjb¯b processes we choose invariant b¯bmass for Q2scale.Total cross section of W+2jets(Wjj)“glu-onic”background(1000pb)is about2orders of magnitude higher than that of the signal. The specific feature of single top production is highly energetic b quark in thefinal state and one additional b quark for W-gluon fu-sion process.The b quark content of the Wjj processes is small(less then1%)and efficient b-tagging gives us chance to extract the signal from Wjj overwhelming background.We as-sume50%b-tagging efficiency hereafter.For initial cuts mentioned above the total cross section for W+b¯b process(gluon splitting)is 7.2pb.Contribution from Wjj background due to b quark misidentification should also be taken into account.In our study we chose1%misiden-tification probability.Complete set of Feynman diagrams for W±b¯b background is shown in Fig.2.The main con-tions for jb¯b and jjb¯b are204105and66733pb, respectively.process CS(pb)process CS(pb)gg→gb¯b120180gq→qb¯b82290q¯q→gb¯b1635gg→ggb¯b36212q¯q→q¯q b¯b2269q¯q→ggb¯b288gq→gqb¯b26166gg→q¯q b¯b1798 Table1:jb¯b and jjb¯b cross sections.processes1.3Signal and background study Even after b-tagging procedure the signal is more than one order lower than the background. This fact requires special kinematic analysis in order to reduce the background and leave the signal intact as much as possible.The main difference between kinematic dis-tributions for background and signal is that jets from Wjj,jb¯b and jjb¯b processes are much softer and less central than those for the signal with one very hard jet coming from top and another softer jet,accompanying top quark. Among the kinematic variables for separation of the signal and background the most effec-tive are:p⊥of leading jet(Fig.3a)invariantmass of the system√accompanying cuts Top W b¯b W jj j(j)b¯bˆs>180GeV495743375.p T W>30GeV455039376.jj m>25GeV454838377.H T>100GeV44463636 Table2:Signal and background events;ǫb=50%,m top=180GeV,L=1000pb−1of signal and background kinematics the opti-mal set of cuts for the background suppression has been worked out.The effect of consequent application of these cuts is presented in Ta-ble2.As a result the signal-to-background ra-tio becomes equal to1/3and the signal exceeds the background by3standard deviations.pTjetmax[GeV]Numberofevents10002000300040005000600070008000900020406080100120140160180200√s∧ [GeV]Numberofevents2004006008001000120014001600150200250300350400450500550600———background-----200·signala)b)c)d)HT[GeV]Numberofevents20004000600080001000050100150200250300350400di-jet mass [GeV]Numberofevents100020003000400050006000700050100150200250300350400 Figure3:Distributions for signal and back-ground for some most spectacular variables 1.4Wtb coupling and V tbSince the top quark is rather heavy one could expect that new physics might be revealed at the scale of its mass(see refs.65-71in[2]). We examine the effects of a deviation in the Wtb coupling from the standard model struc-ture,and study sensitivity of measurement of the CKM matrix element V tb.Experimental studies of this type are among the main goals of the single top physics.As an example of a deviation from the stan-dard model Wtb coupling,we introduce addi-tional contribution from the nonstandard(V+ A)structure with arbitrary parameter A r by:Γ=eV tb2sinθW[γµ(1−γ5)+A rγµ(1+γ5)] The production rate varies almost quadrati-cally with A r,and is nearly symmetric around A r=0.The cross section rises from2.44pb when A r=0to4.68pb when A r=−1and to 4.73pb when A r=+1.We have calculated the region in the(V tb,A r)plane where future single top measurements are expected to be sensitive. The error of V tb measurement will be half of the error of the single top cross section,since the cross sections for all single top processes are proportional to|V tb|2.This results in an errorfor V tb and A r around12-19%(L=2fb−1)[2].2Higgs bosonLuminosity upgrade of TEVATRON and in-stallation of the efficient b-tagging system opens also opportunities for the Higgs boson search. This task is crucial since the Higgs boson is the last particle(in the frame of SM)which has not been discovered yet.The most promising will be the Higgs bo-son search in association with the electroweak W and Z bosons[4,5]:Here we concentrated on detailed study of Higgs p¯p→W±H+Xp¯p→ZH+Xs=2TeV the Higgs production cross section is about0.03-0.3pb depending on the Higgs mass and therefore one can expect30-300signal events for integrated luminosity L= 1000pb−1.However the total reaction rates are about two orders of magnitude larger that the Higgs signal.That is why one has to make a detailed analysis of different4-fermionfinal state distributions tofind out whether it is pos-sible tofind a set of cuts to suppress the huge background strongly enough.For the cross sec-tion calculation and event simulation the same procedure has been done as for the single top quark study,including the effect of initial andfinal state radiation.2.2Background studyFrom the complete set of diagrams for back-ground the dominant contribution comes from several subprocesses:1)QCD2→3type mainbackground subprocesses:q¯q′→W±b¯b,q¯q→Zb¯b,gg→Zb¯b;2)2→2electroweak subpro-cesses:q¯q′→W±Z,q¯q→(Z/γ∗)(Z/γ∗);3) single top quark production q¯q′→t¯b or¯t b. Single top background has not been taken into account in the previous studies.This contri-bution becomes more important with growth of the invariant b¯b-pair mass.Signal and background kinematical distri-butions differ mainly due to concentration of events of QCD backgrounds mostly in the re-gion of small p⊥.We study b quark and charged lepton distributions over transverse momentum p⊥(b)and p⊥(ℓ)as well as b¯b distributions over transverse momentum p⊥(b¯b)and miss-ing energy/E⊥.It turned out that pair charac-teristic p⊥(b¯b)related to high transverse mo-menta of Higgs boson is very important for background suppression.After detailed com-parison of the signal and the background distri-butions appropriate set of cuts has been found:1)p⊥(b),p⊥(b¯b)>20GeV;p⊥(ℓ)>15GeV;2)typical cuts for TEVATRON detectors:/E⊥>20GeV;|ηb|<1.5;|ηℓ|<2;|∆R¯b|>0.7; |∆R bℓ|>0.7;3)−2∆M¯b+M¯b<M¯b<2∆M¯b+ M¯b.This set of cuts reduces the total back-ground more than40times.The number of events corresponding to signal and background after the cuts application is presented in Ta-ble3.Requiring3-standard deviation criteria one canfind Higgs up to the mass≃100GeV at√s=4TeV.These numbers are obtained under assumption of50%efficiency of double b-tagging.One can see that with the decreasing Higgs boson mass the contribution of the single top processes to the total background becomes more significant.√M HW ±HZHW b ¯b +W Z +b ¯t Zb ¯b +ZZs =4TeVTable 3:Number of events for Higgs bosonsignal and background after cuts N u m b e r o f e v e n t s√s = 2 TeV pp → (l ν– + l l – +ν ν– ) + bb – + X M H = 80 GeVall cuts absentall cuts appliedInvariant bb –mass distribution[ GeV ]N u m b e r o f e v e n t s250500750100012501500175020002040608010012014010203040506020406080100120140Figure 4:Invariant b ¯b mass distribution be-fore and after cutsIt should be noted also that Wbb background increases approximately two times with the in-creasing of√s =2TeV without any cuts andwith cuts applied revealing clear peak from Higgs boson.ConclusionsStudy of single top quark and Higgs boson pro-duction at the upgraded Tevatron is not only important but also convenient since they have very similar signatures.It was shown that the signal from the sin-gle top quark can be extracted and therefore Wtb vertex and V tb CKM parameter at theupgraded TEVATRON can be measured.Themain backgrounds have been studied:W ±b ¯b,Wjj,jb ¯b +jjb ¯b and the set of kinematical cuts on p ⊥(jet),√s =2TeV and up to about 120GeVat√。

calculations have also been performed for vector boson fusion tt¯ production the picture is far from complete. To the full calculations of these processes exist in the literature. The total cross section for e+e− → ℓℓ¯tt¯, using
which → tt¯)
corresponds to t√he LET. in the 2MW,Z / s = 0
We give approxi-
mation and show how this approximation effects the results.
I. INTRODUCTION
Understanding the mechanism responsible for electroweak symmetry breaking is one of the most pressing problems in particle physics. Quite generally, there are two scenarios. In the first, the Higgs boson is light and at high energies the weak sector remains weakly coupled. In this scenario the Higgs boson should be observed at one of the high energy colliders under consideration. In the second scenario the Higgs boson is heavy and at high energies the weak sector becomes strongly interacting. If this were the case a new spectroscopy is likely to manifest itself at energies beyond a TeV. In our view this is a very intriguing possibility. Unfortunately, it will be very difficult to detect and even more difficult to understand. Considerable work has been devoted to understanding this problem using vector boson scattering at high energy colliders with questionable success. However, it has been shown that tt¯production via vector boson fusion at high energy colliders also provides a potentially powerful tool for understanding a strongly interacting weak sector [2] and this subject has attracted growing interest [3, 4]. This is simply a manifestation of the equivalence theorem where longitudinal bosons take on the couplings of the scalars from which they acquire mass and a consequence of the fact that the Higgs boson couples most strongly to the most massive particle available.

ar X i v :h e p -p h /0005010v 1 1 M a y 2000FIAN/TD-04/2000BNL-NT-00/12ON TRANSVERSE ENERGY PRODUCTION IN HADRON COLLISIONS Andrei Leonidov (a ),(b )(a )Physics Department,Brookhaven National Laboratory Upton,NY 11973,USA (b )Theoretical Physics Department,P.N.Lebedev Physics Institute 117924Leninsky pr.53,Moscow,Russia 1Abstract The transverse energy spectrum in the unit rapidity window in p ¯p collisions at √1Permanent address1In this note we calculate to the next-to-leading(NLO)order accuracy the transverse energy spectrum in the central rapidity window within perturbative QCD and compare it to experimental data obtained by UA2collaboration[2].The NLO calculation of a generic jet cross section requires using a so-called jet defining algorithm specifying the resolution for the jet to be observed,for example,the angular size of the jet-defining cone,see e.g.[5],The cross section in question is calculated by integrating the differential one over the phase space,with the integration domain restricted by the jet characteristicsfixed by the jet-defining alogorithm.Schematically the NLO distribution of the transverse energy produced into a given rapidity interval y a<y<y b is given,to the O(α3s)order,by dσδ(E⊥−2 i=1|p⊥i|θ(y min<y i<y max))d4p1d4p2+ D3P S dσs= 540GeV measured by UA2Collaboration[2].We see that the perturbative LO+NLO calculations start merging with the experimental data only around quite a large scale E⊥∼60GeV.It is interesting to note,that it is precisely around this energy,that the spaceacurrently available3.In practical terms this means that additional model assumptions are needed to achieve agreement with experimental data strongly indicating that higher order corrections and higher twist effects have to be taken into account(in a necessarily model-dependent way) in order to describe them.In the popular Monte-Carlo generators such as PYTHIA[3] and HIJING[4]such effects as multiple binary parton-parton collisions,initial andfinal state radiation and transverse energy production during hadronization are included.In Fig.2we compare the same experimental data by UA(2)[2]with the spectrum calcuated with HIJING event generator.To show the relative importance of different dynamical mechanims,in Fig.2we plot the contributions from the hard parton scattering without initial andfinal state radiation,full partonic contribution and,finally,the transverse energy spectrum offinal hadrons.We see,that taking into account additional partonic sources such as,e.g.,initial andfinal state radiation,allows to reproduce the(exponential) form of the spectrum,but still not the magnitude.The remaining gap isfilled in by soft contributions due to transverse energy production from decaying stretched hadronic strings.The results confirm those earlier communicated to the author by M.Gyulassy [10].Let us note,that the spectrum calculated in HIJING is somewhat steeper than the experimental one.Additionalfine-tuning can be achieved by probing different structure functions.The above results clearly demonstrate that in order to reproduce the experimentally observed transverse energy spectrum,one has to account for complicated mechanisms of parton production,such as initial andfinal state radiation accompanying hard parton-parton scattering,production of gluonic kinks by strings,as well as for nonperturbative transverse energy production at hadronization stage.This statement is a calorimetric analog of of the well-known importance of the minijet component in describing the tails of the multiplicity distributions,[11]and[4](see,however,the alternative explanation based on accounting for multipomeron contributions described in[12]).Let us note that the result has straightforward implications for describing the early stages of heavy ion collisions.In most of the existing dynamical models of nucleus-nucleus collisions they are described as an incoherent superposition of nucleon-nucleon ones.As we have seen,to correctly describe the partonic configuration underlying the observed transverse energyflow in nucleon-nucleon collisions,mechanisms beyond conventional collinear factorization are necessary.In particular this indicates that the results obtained within minijet approach based on colinearly factorized QCD,see e.g.the recent review [13]and references therein,must be taken with care.(((G s G (^ P E *H 9(^ *H9Figure 2:Transverse energy spectrum in p ¯p collisions calculated in HIJING vs the exper-imental data by UA2collaboration [2]AcknowledgementsI am grateful to Yu.Dokshitzer,M.Gyulassy,Yu.Kovchegov,L.McLerran and D.Soper for useful discussions.I am also grateful to Theoretical Physics Institute,University of Minnesota,where this work was started,and Brookhaven National Laboratory,where it was completed,for kind hospitality.This manuscript has been authored under Contract No.DE-AC02-98CH10886with the U.S.Department of Energy.This work was partially supported by RFBR Grant 00-02-161015References[1]A.Leonidov and D.Ostrovsky,Eur.Phys.Journ.C11(1999),495[2]UA2Collaboration,M.Banner et.al.,Phys.Lett.B118(1982),203[3]T.Sjostrand,mun.82(1994),74;[4]X.-N.Wang and M.Guylassy,Phys.Rev.D44(1991),3501;D45(1992),844;mun.83(1994),307[5]D.Soper,”Jet observables in theory and reality”[hep-ph/9706320][6]S.D.Ellis,Z.Kunszt and D.E.Soper,Phys.Rev.Lett.64(1990),2121;Phys.Rev.D40(1992),2188[7]Z.Kunszt and D.E.Soper,Phys.Rev.D46(1992),192[8]K.J.Eskola and K.Tuominen,”Production of Transverse Energy from Minijets inNext-to-Leading Order Perturbative QCD”,[hep-ph/000208][9]G.Pancheri and Y.Srivastava,Phys.Lett.B128(1983),433;”Jets in Minimum Bias Physics”,in Proc.5th Topical Workshop on Proton-Antiproton Collider Physics,Saint Vincent,Italy,1985,ed.M.Greco,World Sci-entific,1985;C.T.H.Davies and B.R.Webber,Zeit.Phys.C24(1984),133M.Greco,Phys.Lett.B121(1983),360;Nucl.Phys.B2501985,450[10]M.Gyulassy,private communication.[11]T.Sjostrand and M.van der Zijl,Phys.Rev.D36(1987),2019[12]S.G.Matinyan and W.D.Walker,Phys.Rev.D59(1999),034022[13]K.J.Eskola,”High-Energy Nuclear Collisions”,[hep-ph/9911350]6。

q q renormalization constants δmq , δZL and δZR are then defined as
mq0 = mq + δmq ,
q 2 q 2 ψq0 = (1 + δZL ) ψqL + (1 + δZR ) ψqR .
1 1
(2) (3)
After calculating the self-energy diagram in Fig. 1, we obtain the explicit expressions of all the renormalization constants as follows: δmq = − αs 2 2 2 2 2 2 2 2 mq [B1 (m2 ˜ sin 2θq ˜[B0 (mq , mg q , mg ˜ , mq ˜1 ) + B1 (mq , mg ˜ , mq ˜2 )] + mg ˜ , mq ˜1 ) (4) 3π 2 2 −B0 (m2 q , mg ˜ , mq ˜2 )] , 2αs ˜ 1 2 2 2 ˙1 ˙ 2 mg ˙1 ˙2 cos2 θq sin 2θq (5) ˜B1 + sin θq ˜B1 + mq [B1 + B1 + ˜(B0 − B0 )] , 3π mq 2αs ˜ 1 2 2 2 ˙1 ˙ 2 mg ˙1 ˙2 sin2 θq sin 2θq (6) ˜B1 + cos θq ˜B1 + mq [B1 + B1 + ˜(B0 − B0 )] , 3π mq
B.
Analytical Results
We consider the associated production of W + H − from the collision of the two protons with momentum P1 and P2 at the LHC. First, we define the Mandelstam variables of the subprocess b(p1 )¯ b(p2 ) → W + (p4 )H − (p3 ) as s = (p1 + p2 )2 = (p3 + p4 )2 , t = (p1 − p3 )2 = (p2 − p4 )2 , u = (p1 − p4 )2 = (p2 − p3 )2 .

a rXiv:h ep-th/04121v12O ct24SLAC–PUB–10739,IPPP/04/59,DCPT/04/118,UCLA/04/TEP/40Saclay/SPhT–T04/116,hep-th/0410021October,2004N =4Super-Yang-Mills Theory,QCD and Collider Physics Z.Bern a L.J.Dixon b 1 D.A.Kosower c a Department of Physics &Astronomy,UCLA,Los Angeles,CA 90095-1547,USA b SLAC,Stanford University,Stanford,CA 94309,USA,and IPPP,University of Durham,Durham DH13LE,England c Service de Physique Th´e orique,CEA–Saclay,F-91191Gif-sur-Yvette cedex,France1Introduction and Collider Physics MotivationMaximally supersymmetric (N =4)Yang-Mills theory (MSYM)is unique in many ways.Its properties are uniquely specified by the gauge group,say SU(N c ),and the value of the gauge coupling g .It is conformally invariant for any value of g .Although gravity is not present in its usual formulation,MSYMis connected to gravity and string theory through the AdS/CFT correspon-dence[1].Because this correspondence is a weak-strong coupling duality,it is difficult to verify quantitatively for general observables.On the other hand, such checks are possible and have been remarkably successful for quantities protected by supersymmetry such as BPS operators[2],or when an additional expansion parameter is available,such as the number offields in sequences of composite,large R-charge operators[3,4,5,6,7,8].It is interesting to study even more observables in perturbative MSYM,in order to see how the simplicity of the strong coupling limit is reflected in the structure of the weak coupling expansion.The strong coupling limit should be even simpler when the large-N c limit is taken simultaneously,as it corresponds to a weakly-coupled supergravity theory in a background with a large radius of curvature.There are different ways to study perturbative MSYM.One approach is via computation of the anomalous dimensions of composite,gauge invariant operators[1,3,4,5,6,7,8].Another possibility[9],discussed here,is to study the scattering amplitudes for(regulated)plane-wave elementaryfield excitations such as gluons and gluinos.One of the virtues of the latter approach is that perturbative MSYM scat-tering amplitudes share many qualitative properties with QCD amplitudes in the regime probed at high-energy colliders.Yet the results and the computa-tions(when organized in the right way)are typically significantly simpler.In this way,MSYM serves as a testing ground for many aspects of perturbative QCD.MSYM loop amplitudes can be considered as components of QCD loop amplitudes.Depending on one’s point of view,they can be considered either “the simplest pieces”(in terms of the rank of the loop momentum tensors in the numerator of the amplitude)[10,11],or“the most complicated pieces”in terms of the degree of transcendentality(see section6)of the special functions entering thefinal results[12].As discussed in section6,the latter interpreta-tion links recent three-loop anomalous dimension results in QCD[13]to those in the spin-chain approach to MSYM[5].The most direct experimental probes of short-distance physics are collider experiments at the energy frontier.For the next decade,that frontier is at hadron colliders—Run II of the Fermilab Tevatron now,followed by startup of the CERN Large Hadron Collider in2007.New physics at colliders always contends with Standard Model backgrounds.At hadron colliders,all physics processes—signals and backgrounds—are inherently QCD processes.Hence it is important to be able to predict them theoretically as precisely as possi-ble.The cross section for a“hard,”or short-distance-dominated processes,can be factorized[14]into a partonic cross section,which can be computed order by order in perturbative QCD,convoluted with nonperturbative but measur-able parton distribution functions(pdfs).For example,the cross section for producing a pair of jets(plus anything else)in a p¯p collision is given byσp¯p→jjX(s)= a,b1 0dx1dx2f a(x1;µF)¯f b(x2;µF)׈σab→jjX(sx1x2;µF,µR;αs(µR)),(1)where s is the squared center-of-mass energy,x1,2are the longitudinal(light-cone)fractions of the p,¯p momentum carried by partons a,b,which may be quarks,anti-quarks or gluons.The experimental definition of a jet is an in-volved one which need not concern us here.The pdf f a(x,µF)gives the prob-ability forfinding parton a with momentum fraction x inside the proton; similarly¯f b is the probability forfinding parton b in the antiproton.The pdfs depend logarithmically on the factorization scaleµF,or transverse resolution with which the proton is examined.The Mellin moments of f a(x,µF)are for-ward matrix elements of leading-twist operators in the proton,renormalized at the scaleµF.The quark distribution function q(x,µ),for example,obeys 10dx x j q(x,µ)= p|[¯qγ+∂j+q](µ)|p .2Ingredients for a NNLO CalculationMany hadron collider measurements can benefit from predictions that are accurate to next-to-next-to-leading order(NNLO)in QCD.Three separate ingredients enter such an NNLO computation;only the third depends on the process:(1)The experimental value of the QCD couplingαs(µR)must be determinedat one value of the renormalization scaleµR(for example m Z),and its evolution inµR computed using the3-loopβ-function,which has been known since1980[15].(2)The experimental values for the pdfs f a(x,µF)must be determined,ide-ally using predictions at the NNLO level,as are available for deep-inelastic scattering[16]and more recently Drell-Yan production[17].The evolu-tion of pdfs inµF to NNLO accuracy has very recently been completed, after a multi-year effort by Moch,Vermaseren and Vogt[13](previously, approximations to the NNLO kernel were available[18]).(3)The NNLO terms in the expansion of the partonic cross sections must becomputed for the hadronic process in question.For example,the parton cross sections for jet production has the expansion,ˆσab→jjX=α2s(A+αs B+α2s C+...).(2)The quantities A and B have been known for over a decade[19],but C has not yet been computed.Figure 1.LHC Z production [22].•real ×real:וvirtual ×real:וvirtual ×virtual:וdoubly-virtual ×real:×Figure 2.Purely gluonic contributionsto ˆσgg →jjX at NNLO.Indeed,the NNLO terms are unknown for all but a handful of collider puting a wide range of processes at NNLO is the goal of a large amount of recent effort in perturbative QCD [20].As an example of the im-proved precision that could result from this program,consider the production of a virtual photon,W or Z boson via the Drell-Yan process at the Tevatron or LHC.The total cross section for this process was first computed at NNLO in 1991[21].Last year,the rapidity distribution of the vector boson also be-came available at this order [17,22],as shown in fig.1.The rapidity is defined in terms of the energy E and longitudinal momentum p z of the vector boson in the center-of-mass frame,Y ≡1E −p z .It determines where the vector boson decays within the detector,or outside its acceptance.The rapidity is sensitive to the x values of the incoming partons.At leading order in QCD,x 1=e Y m V /√s ,where m V is the vector boson mass.The LHC will produce roughly 100million W s and 10million Z s per year in detectable (leptonic)decay modes.LHC experiments will be able to map out the curve in fig.1with exquisite precision,and use it to constrain the parton distributions —in the same detectors that are being used to search for new physics in other channels,often with similar q ¯q initial states.By taking ratios of the other processes to the “calibration”processes of single W and Z production,many experimental uncertainties,including those associated with the initial state parton distributions,drop out.Thus fig.1plays a role as a “partonic luminosity monitor”[23].To get the full benefit of the remarkable experimental precision,though,the theory uncertainty must approach the 1%level.As seen from the uncertainty bands in the figure,this precision is only achievable at NNLO.The bands are estimated by varying the arbitrary renormalization and factorization scales µR and µF (set to a common value µ)from m V /2to 2m V .A computation to all orders in αs would have no dependence on µ.Hence the µ-dependence of a fixed order computation is related to the size of the missing higher-order terms in the series.Althoughsub-1%uncertainties may be special to W and Z production at the LHC, similar qualitative improvements in precision will be achieved for many other processes,such as di-jet production,as the NNLO terms are completed.Even within the NNLO terms in the partonic cross section,there are several types of ingredients.This feature is illustrated infig.2for the purely gluonic contributions to di-jet production,ˆσgg→jjX.In thefigure,individual Feynman graphs stand for full amplitudes interfered(×)with other amplitudes,in order to produce contributions to a cross section.There may be2,3,or4partons in thefinal state.Just as in QED it is impossible to define an outgoing electron with no accompanying cloud of soft photons,also in QCD sensible observables require sums overfinal states with different numbers of partons.Jets,for example,are defined by a certain amount of energy into a certain conical region.At leading order,that energy typically comes from a single parton, but at NLO there may be two partons,and at NNLO three partons,within the jet cone.Each line infig.2results in a cross-section contribution containing severe infrared divergences,which are traditionally regulated by dimensional regula-tion with D=4−2ǫ.Note that this regulation breaks the classical conformal invariance of QCD,and the classical and quantum conformal invariance of N=4super-Yang-Mills theory.Each contribution contains poles inǫranging from1/ǫ4to1/ǫ.The poles in the real contributions come from regions ofphase-space where the emitted gluons are soft and/or collinear.The poles in the virtual contributions come from similar regions of virtual loop integra-tion.The virtual×real contribution obviously has a mixture of the two.The Kinoshita-Lee-Nauenberg theorem[24]guarantees that the poles all cancel in the sum,for properly-defined,short-distance observables,after renormal-izing the coupling constant and removing initial-state collinear singularities associated with renormalization of the pdfs.A critical ingredient in any NNLO prediction is the set of two-loop ampli-tudes,which enter the doubly-virtual×real interference infig.2.Such ampli-tudes require dimensionally-regulated all-massless two-loop integrals depend-ing on at least one dimensionless ratio,which were only computed beginning in 1999[25,26,27].They also receive contributions from many Feynman diagrams, with lots of gauge-dependent cancellations between them.It is of interest to develop more efficient,manifestly gauge-invariant methods for combining di-agrams,such as the unitarity or cut-based method successfully applied at one loop[10]and in the initial two-loop computations[28].i,ij+ i iFigure3.Illustration of soft-collinear(left)and pure-collinear(right)one-loop di-vergences.3N=4Super-Yang-Mills Theory as a Testing Ground for QCDN=4super-Yang-Mills theory serves an excellent testing ground for pertur-bative QCD methods.For n-gluon scattering at tree level,the two theories in fact give identical predictions.(The extra fermions and scalars of MSYM can only be produced in pairs;hence they only appear in an n-gluon ampli-tude at loop level.)Therefore any consequence of N=4supersymmetry,such as Ward identities among scattering amplitudes[29],automatically applies to tree-level gluonic scattering in QCD[30].Similarly,at tree level Witten’s topological string[31]produces MSYM,but implies twistor-space localization properties for QCD tree amplitudes.(Amplitudes with quarks can be related to supersymmetric amplitudes with gluinos using simple color manipulations.)3.1Pole Structure at One and Two LoopsAt the loop-level,MSYM becomes progressively more removed from QCD. However,it can still illuminate general properties of scattering amplitudes,in a calculationally simpler arena.Consider the infrared singularities of one-loop massless gauge theory amplitudes.In dimensional regularization,the leading singularity is1/ǫ2,arising from virtual gluons which are both soft and collinear with respect to a second gluon or another massless particle.It can be char-acterized by attaching a gluon to any pair of external legs of the tree-level amplitude,as in the left graph infig.3.Up to color factors,this leading diver-gence is the same for MSYM and QCD.There are also purely collinear terms associated with individual external lines,as shown in the right graph infig.3. The pure-collinear terms have a simpler form than the soft terms,because there is less tangling of color indices,but they do differ from theory to theory.The full result for one-loop divergences can be expressed as an operator I(1)(ǫ) which acts on the color indices of the tree amplitude[32].Treating the L-loop amplitude as a vector in color space,|A(L)n ,the one-loop result is|A(1)n =I(1)(ǫ)|A(0)n +|A(1),finn ,(3)where |A (1),fin nis finite as ǫ→0,and I (1)(ǫ)=1Γ(1−ǫ)n i =1n j =i T i ·T j 1T 2i 1−s ij ǫ,(4)where γis Euler’s constant and s ij =(k i +k j )2is a Mandelstam invariant.The color operator T i ·T j =T a i T a j and factor of (µ2R /(−s ij ))ǫarise from softgluons exchanged between legs i and j ,as in the left graph in fig.3.The pure 1/ǫpoles terms proportional to γi have been written in a symmetric fashion,which slightly obscures the fact that the color structure is actually simpler.We can use the equation which represents color conservation in the color-space notation, n j =1T j =0,to simplify the result.At order 1/ǫwe may neglect the (µ2R /(−s ij ))ǫfactor in the γi terms,and we have n j =i T i ·T j γi /T 2i =−γi .So the color structure of the pure 1/ǫterm is actually trivial.For an n -gluon amplitude,the factor γi is set equal to its value for gluons,which turns out to be γg =b 0,the one-loop coefficient in the β-function.Hence the pure-collinear contribution vanishes for MSYM,but not for QCD.The divergences of two-loop amplitudes can be described in the same for-malism [32].The relation to soft-collinear factorization has been made more transparent by Sterman and Tejeda-Yeomans,who also predicted the three-loop behavior [33].Decompose the two-loop amplitude |A (2)n as|A (2)n =I (2)(ǫ)|A (0)n +I (1)(ǫ)|A (1)n +|A (2),fin n,(5)where |A (2),fin n is finite as ǫ→0and I (2)(ǫ)=−1ǫ+e −ǫγΓ(1−2ǫ)ǫ+K I (1)(2ǫ)+e ǫγT 2i µ22C 2A ,(8)where C A =N c is the adjoint Casimir value.The quantity ˆH(2)has non-trivial,but purely subleading-in-N c ,color structure.It is associated with soft,rather than collinear,momenta [37,33],so it is theory-independent,up to color factors.An ansatz for it for general n has been presented recently [38].3.2Recycling Cuts in MSYMAn efficient way to compute loop amplitudes,particularly in theories with a great deal of supersymmetry,is to use unitarity and reconstruct the am-plitude from its cuts [10,38].For the four-gluon amplitude in MSYM,the two-loop structure,and much of the higher-loop structure,follows from a sim-ple property of the one-loop two-particle cut in this theory.For simplicity,we strip the color indices offof the four-point amplitude A (0)4,by decomposing it into color-ordered amplitudes A (0)4,whose coefficients are traces of SU(N c )generator matrices (Chan-Paton factors),A (0)4(k 1,a 1;k 2,a 2;k 3,a 3;k 4,a 4)=g 2 ρ∈S 4/Z 4Tr(T a ρ(1)T a ρ(2)T a ρ(3)T a ρ(4))×A (0)4(k ρ(1),k ρ(2),k ρ(3),k ρ(4)).(9)The two-particle cut can be written as a product of two four-point color-ordered amplitudes,summed over the pair of intermediate N =4states S,S ′crossing the cut,which evaluates toS,S ′∈N =4A (0)4(k 1,k 2,ℓS ,−ℓ′S ′)×A (0)4(ℓ′S ′,−ℓS ,k 3,k 4)=is 12s 23A (0)4(k 1,k 2,k 3,k 4)×1(ℓ−k 3)2,(10)where ℓ′=ℓ−k 1−k 2.This equation is also shown in fig.4.The scalar propagator factors in eq.(10)are depicted as solid vertical lines in the figure.The dashed line indicates the cut.Thus the cut reduces to the cut of a scalar box integral,defined byI D =4−2ǫ4≡ d 4−2ǫℓℓ2(ℓ−k 1)2(ℓ−k 1−k 2)2(ℓ+k 4)2.(11)One of the virtues of eq.(10)is that it is valid for arbitrary external states in the N =4multiplet,although only external gluons are shown in fig.4.Therefore it can be re-used at higher loop order,for example by attaching yet another tree to the left.N =41234=i s 12s 231234Figure 4.The one-loop two-particle cuts for the four-point amplitude in MSYM reduce to the tree amplitude multiplied by a cut scalar box integral (for any set of four external states).i 2s 12s121234+s 121234+perms Figure 5.The two-loop gg →gg amplitude in MSYM [11,39].The blob on theright represents the color-ordered tree amplitude A (0)4.(The quantity s 12s 23A (0)4transforms symmetrically under gluon interchange.)In the the brackets,black linesare kinematic 1/p 2propagators,with scalar (φ3)vertices.Green lines are color δab propagators,with structure constant (f abc )vertices.The permutation sum is over the three cyclic permutations of legs 2,3,4,and makes the amplitude Bose symmetric.At two loops,the simplicity of eq.(10)made it possible to compute the two-loop gg →gg scattering amplitude in that theory (in terms of specific loop integrals)in 1997[11],four years before the analogous computations in QCD [36,37].All of the loop momenta in the numerators of the Feynman di-agrams can be factored out,and only two independent loop integrals appear,the planar and nonplanar scalar double box integrals.The result can be writ-ten in an appealing diagrammatic form,fig.5,where the color algebra has the same form as the kinematics of the loop integrals [39].At higher loops,eq.(10)leads to a “rung rule”[11]for generating a class of (L +1)-loop contributions from L -loop contributions.The rule states that one can insert into a L -loop contribution a rung,i.e.a scalar propagator,transverse to two parallel lines carrying momentum ℓ1+ℓ2,along with a factor of i (ℓ1+ℓ2)2in the numerator,as shown in fiing this rule,one can construct recursively the external and loop-momentum-containing numerators factors associated with every φ3-type diagram that can be reduced to trees by a sequence of two-particle cuts,such as the diagram in fig.7a .Such diagrams can be termed “iterated 2-particle cut-constructible,”although a more compact notation might be ‘Mondrian’diagrams,given their resemblance to Mondrian’s paintings.Not all diagrams can be computed in this way.The diagram in fig.7b is not in the ‘Mondrian’class,so it cannot be determined from two-particle cuts.Instead,evaluation of the three-particle cuts shows that it appears with a non-vanishing coefficient in the subleading-color contributions to the three-loop MSYM amplitude.ℓ1ℓ2−→i (ℓ1+ℓ2)2ℓ1ℓ2Figure 6.The rung rule for MSYM.(a)(b)Figure 7.(a)Example of a ‘Mondrian’diagram which can be determined re-cursively from the rung rule.(b)Thefirst non-vanishing,non-Mondrian dia-grams appear at three loops in nonplanar,subleading-color contributions.4Iterative Relation in N =4Super-Yang-Mills TheoryAlthough the two-loop gg →gg amplitude in MSYM was expressed in terms of scalar integrals in 1997[11],and the integrals themselves were computed as a Laurent expansion about D =4in 1999[25,26],the expansion of the N =4amplitude was not inspected until last fall [9],considerably after similar investigations for QCD and N =1super-Yang-Mills theory [36,37].It was found to have a quite interesting “iterative”relation,when expressed in terms of the one-loop amplitude and its square.At leading color,the L -loop gg →gg amplitude has the same single-trace color decomposition as the tree amplitude,eq.(9).Let M (L )4be the ratio of this leading-color,color-ordered amplitude to the corresponding tree amplitude,omitting also several conventional factors,A (L ),N =4planar 4= 2e −ǫγg 2N c2 M (1)4(ǫ) 2+f (ǫ)M (1)4(2ǫ)−12(ζ2)2is replaced by approximately sixpages of formulas (!),including a plethora of polylogarithms,logarithms and=+f(ǫ)−12(ζ2)2+O(ǫ)f(ǫ)=−(ζ2+ǫζ3+ǫ2ζ4+...)Figure8.Schematic depiction of the iterative relation(13)between two-loop and one-loop MSYM amplitudes.polynomials in ratios of invariants s/t,s/u and t/u[37].The polylogarithm is defined byLi m(x)=∞i=1x i t Li m−1(t),Li1(x)=−ln(1−x).(14)It appears with degree m up to4at thefinite,orderǫ0,level;and up to degree4−i in the O(ǫ−i)terms.In the case of MSYM,identities relating these polylogarithms are needed to establish eq.(13).Although the O(ǫ0)term in eq.(13)is miraculously simple,as noted above the behavior of the pole terms is not a miracle.It is dictated in general terms by the cancellation of infrared divergences between virtual corrections and real emission[24].Roughly speaking,for this cancellation to take place,the virtual terms must resemble lower-loop amplitudes,and the real terms must resemble lower-point amplitudes,in the soft and collinear regions of loop or phase-space integration.At the level of thefinite terms,the iterative relation(13)can be understood in the Regge/BFKL limit where s≫t,because it then corresponds to expo-nentiation of large logarithms of s/t[40].For general values of s/t,however, there is no such argument.The relation is special to D=4,where the theory is conformally invariant. That is,the O(ǫ1)remainder terms cannot be simplified significantly.For ex-ample,the two-loop amplitude M(2)4(ǫ)contains at O(ǫ1)all three independent Li5functions,Li5(−s/u),Li5(−t/u)and Li5(−s/t),yet[M(1)4(ǫ)]2has only the first two of these[9].The relation is also special to the planar,leading-color limit.The subleading color-components of thefinite remainder|A(2),finn defined by eq.(5)show no significant simplification at all.For planar amplitudes in the D→4limit,however,there is evidence that an identical relation also holds for an arbitrary number n of external legs, at least for certain“maximally helicity-violating”(MHV)helicity amplitudes. This evidence comes from studying the limits of two-loop amplitudes as two of the n gluon momenta become collinear[9,38,41].(Indeed,it was by analyzing these limits that the relation for n=4wasfirst uncovered.)The collinear limits turn out to be consistent with the same eq.(13)with M4replaced by M n everywhere[9],i.e.M(2)n(ǫ)=12(ζ2)2+O(ǫ).(15)The collinear consistency does not constitute a proof of eq.(15),but in light of the remarkable properties of MSYM,it would be surprising if it were not true in the MHV case.Because the direct computation of two-loop amplitudes for n>4seems rather difficult,it would be quite interesting to try to examine the twistor-space properties of eq.(15),along the lines of refs.[31,42].(The right-hand-side of eq.(15)is not completely specified at order1/ǫandǫ0for n>4.The reason is that the orderǫandǫ2terms in M(1)n(ǫ),which contribute to thefirst term in eq.(15)at order1/ǫandǫ0,contain the D=6−2ǫpentagon integral[43],which is not known in closed form.On the other hand, the differential equations this integral satisfies may suffice to test the twistor-space behavior.Or one may examine just thefinite remainder M(L),finn definedvia eq.(5).)It may soon be possible to test whether an iterative relation for planar MSYM amplitudes extends to three loops.An ansatz for the three-loop planar gg→gg amplitude,shown infig.9,was provided at the same time as the two-loop re-sult,in1997[11].The ansatz is based on the“rung-rule”evaluation of the iterated2-particle cuts,plus the3-particle cuts with intermediate states in D=4;the4-particle cuts have not yet been verified.Two integrals,each be-ginning at O(ǫ−6),are required to evaluate the ansatz in a Laurent expansion about D=4.(The other two integrals are related by s↔t.)The triple ladder integral on the top line offig.9was evaluated last year by Smirnov,all the way through O(ǫ0)[44].Evaluation of the remaining integral,which contains a factor of(ℓ+k4)2in the numerator,is in progress[45];all the terms through O(ǫ−2)agree with predictions[33],up to a couple of minor corrections.5Significance of Iterative Behavior?It is not yet entirely clear why the two-loop four-point amplitude,and prob-ably also the n-point amplitudes,have the iterative structure(15).However, one can speculate that it is from the need for the perturbative series to=i3s12s212+s223+2s12(ℓ+k4)+2s23(ℓ+k1)21Figure9.Graphical representation of the three-loop amplitude for MSYM in the planar limit.be summable into something which becomes“simple”in the planar strong-coupling limit,since that corresponds,via AdS/CFT,to a weakly-coupled supergravity theory.The fact that the relation is special to the conformal limit D→4,and to the planar limit,backs up this speculation.Obviously it would be nice to have some more information at three loops.There have been other hints of an iterative structure in the four-point correlation func-tions of chiral primary(BPS)composite operators[46],but here also the exact structure is not yet clear.Integrability has played a key role in recent higher-loop computations of non-BPS spin-chain anomalous dimensions[4,5,6,8].By imposing regularity of the BMN‘continuum’limit[3],a piece of the anoma-lous dimension matrix has even been summed to all orders in g2N c in terms of hypergeometric functions[7].The quantities we considered here—gauge-invariant,but dimensionally regularized,scattering amplitudes of color non-singlet states—are quite different from the composite color-singlet operators usually treated.Yet there should be some underlying connection between the different perturbative series.6Aside:Anomalous Dimensions in QCD and MSYMAs mentioned previously,the set of anomalous dimensions for leading-twist operators was recently computed at NNLO in QCD,as the culmination of a multi-year effort[13]which is central to performing precise computations of hadron collider cross sections.Shortly after the Moch,Vermaseren and Vogt computation,the anomalous dimensions in MSYM were extracted from this result by Kotikov,Lipatov,Onishchenko and Velizhanin[12].(The MSYM anomalous dimensions are universal;supersymmetry implies that there is only one independent one for each Mellin moment j.)This extraction was non-trivial,because MSYM contains scalars,interacting through both gauge and Yukawa interactions,whereas QCD does not.However,Kotikov et al.noticed, from comparing NLO computations in both leading-twist anomalous dimen-sions and BFKL evolution,that the“most complicated terms”in the QCDcomputation always coincide with the MSYM result,once the gauge group representation of the fermions is shifted from the fundamental to the adjoint representation.One can define the“most complicated terms”in the x-space representation of the anomalous dimensions—i.e.the splitting kernels—as follows:Assign a logarithm or factor ofπa transcendentality of1,and a polylogarithm Li m or factor ofζm=Li m(1)a transcendentality of m.Then the most complicated terms are those with leading transcendentality.For the NNLO anomalous dimensions,this turns out to be transcendentality4.(This rule for extracting the MSYM terms from QCD has also been found to hold directly at NNLO,for the doubly-virtual contributions[38].)Strikingly,the NNLO MSYM anomalous dimension obtained for j=4by this procedure agrees with a previous result derived by assuming an integrable structure for the planar three-loop contribution to the dilatation operator[5].7Conclusions and OutlookN=4super-Yang-Mills theory is an excellent testing ground for techniques for computing,and understanding the structure of,QCD scattering amplitudes which are needed for precise theoretical predictions at high-energy colliders. One can even learn something about the structure of N=4super-Yang-Mills theory in the process,although clearly there is much more to be understood. Some open questions include:Is there any AdS/CFT“dictionary”for color non-singlet states,like plane-wave gluons?Can one recover composite operator correlation functions from any limits of multi-point scattering amplitudes?Is there a better way to infrared regulate N=4supersymmetric scattering amplitudes,that might be more convenient for approaching the AdS/CFT correspondence,such as compactification on a three-sphere,use of twistor-space,or use of coherent external states?Further investigations of this arena will surely be fruitful.AcknowledgementsWe are grateful to the organizers of Strings04for putting together such a stim-ulating meeting.This research was supported by the US Department of En-ergy under contracts DE-FG03-91ER40662(Z.B.)and DE-AC02-76SF00515 (L.J.D.),and by the Direction des Sciences de la Mati`e re of the Commissariat `a l’Energie Atomique of France(D.A.K.).。

The Standard Model which describes within an unprecendental scale of energies and distances the strong and electroweak interactions of elementary particles relays on a few basic principles - the renormalizability, the gauge invariance and the spontaneous breaking of the underlying gauge symmetry. The principle of the renormalizability which is considered often as something lying beyond the limits of experimental test is in fact one of the most important (if not the major) ingredients of the quantum field theory. The requirement of renormalizabilty which content and deep meaning were uncovered in the fundamental textbook by N.N.Bogolyubov and D.V.Shirkov [1] plays the central role in the construction and classification of the field theoretic models. They split in general on two classes. In the renormalizable models the ultraviolet divergences of the radiative corrections are under mathematically rigorous control due to the famous Bogolyubov-Parasiuk theorem [2]. These models which preserve their locality in all orders of the perturbation theory are characterised by a finite number of relevant dimensionless coupling constants whose dependence on an arbitrary dimensional normalization parameter is described by the renormalization group [1]. These so-called the “running” coupling constants depending on the model may have or the asymptotic freedom behaviour at large momenta (as for non-abelian gauge theories) or like in quantum electrodynamics with an abelian gauge symmetry reveal the growth of the effective coupling constant in the ultraviolet region. The second class of field theoretical models - the non-renormalizable models have a very serious drawback which makes them useless for description of particle interactions at the present level of knowledge. First of all, the non-renormalizable models have infinite number of divergent matrix elements that requires as a consequence an introduction of 3

二十一世纪的新发现作文英文回答：In the 21st century, there have been numerous groundbreaking discoveries in various fields, from science and technology to medicine and space exploration. One of the most significant discoveries is the detection of gravitational waves, which was first announced in 2016. This discovery confirmed a major prediction of Albert Einstein's general theory of relativity and opened up a new window to observe the universe.Another remarkable discovery in the 21st century is the development of CRISPR gene-editing technology. This revolutionary tool allows scientists to precisely edit DNA, leading to potential breakthroughs in treating genetic diseases and creating genetically modified organisms.Furthermore, the discovery of exoplanets outside of our solar system has expanded our understanding of the universeand the possibility of extraterrestrial life. With the advancement of telescopes and space exploration technology, astronomers have identified thousands of exoplanets, someof which may have the conditions to support life.In the field of medicine, the development of immunotherapy has transformed cancer treatment byharnessing the body's immune system to target and destroy cancer cells. This has led to significant improvements inthe survival rates of patients with certain types of cancer.In addition, the discovery of the Higgs boson particleat the Large Hadron Collider in 2012 provided crucial evidence for the mechanism of mass generation in the universe, as proposed by the Standard Model of particle physics.These discoveries have not only advanced human knowledge and understanding of the world around us, butthey have also opened up new possibilities for scientific and technological advancements in the future.中文回答：在21世纪，各个领域都取得了许多突破性的发现，从科学技术到医学和太空探索。

学科学爱科学的作文600到1千字英文回答:I love science because it allows me to explore the wonders of the world and understand how things work. It satisfies my curiosity and fuels my desire to learn.Science is like a language that helps us communicate with nature and unravel its secrets.Science is not just about memorizing facts and formulas. It is a way of thinking and problem-solving. It teaches usto observe, experiment, and analyze data to draw logical conclusions. For example, when I was studying biology, I learned about the process of photosynthesis. By understanding the chemical reactions involved, I could explain why plants need sunlight to grow and produce oxygen.Moreover, science has practical applications that improve our lives. It has led to the development of new technologies and medicines that save lives and make ourdaily tasks easier. For instance, the invention of the internet has revolutionized communication, allowing us to connect with people from all over the world in an instant.Science also encourages critical thinking and creativity. It challenges us to question existing theories and come up with innovative solutions. One famous example is Albert Einstein's theory of relativity, which revolutionized our understanding of space and time. His groundbreaking ideas were initially met with skepticism, but they eventually changed the course of physics.In addition, science fosters collaboration and global cooperation. Scientists from different countries work together to tackle complex problems and share knowledge. This international collaboration has led to major breakthroughs in various fields, such as the discovery of the Higgs boson particle at the Large Hadron Collider.中文回答：我热爱科学，因为它让我能够探索世界的奇迹，理解事物的运作方式。

Eri Asakawaa 1, Oliver Breinb 2 and Shinya Kanemurac 3
a Theory Group KEK, Tsukuba, Ibaraki, 305-0801, Japan b Institut fu¨r Theoretische Physik E, RWTH Aachen D-52056 Aachen, Germany c Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan
The Two Higgs Doublet Model (THDM) is the model with the minimal extension of the SM Higgs sector, which leads to new phenomena in the matter and gauge field sector4. In the THDM, tree level FCNC can be eliminated by imposing a discrete symmetry, under which there are two possibilities depending on its charge assignment [7, 8]; i.e., (Type I) only one of the Higgs doublets gives mass to the fermions, and (Type II) one gives mass to the up-type quarks and the other to the down-type quarks and charged leptons. The Higgs sector of the minimal supersymmetric extension of the SM (MSSM) is a special case of

DESY 98-159 hep-ph/9810289 October 1998Higgs Boson Production and Decay at the Tevatron∗arXiv:hep-ph/9810289v1 8 Oct 1998Michael SpiraII. Institut f¨r Theoretische Physik† , Universit¨t Hamburg, Luruper Chaussee 149, u a D–22761 Hamburg, GermanyAbstractThe theoretical status of Higgs boson production and decay at the Tevatron within the Standard Model and its minimal supersymmetric extension is reviewed. The focus will be on the evaluation of higher-order corrections to the production cross sections and their phenomenological implications.Contribution to the Workshop Physics at Run II – Supersymmetry/Higgs, 1998, Fermilab, USA Supported by Bundesministerium f¨ r Bildung und Forschung (BMBF), Bonn, Germany, under Conu tract 05 7 HH 92P (5), and by EU Program Human Capital and Mobility through Network Physics at High Energy Colliders under Contract CHRX–CT93–0357 (DG12 COMA).†∗1IntroductionThe search for Higgs bosons is of primary interest for present and future experiments. In the Standard Model [SM] the existence of a single Higgs particle is predicted as a consequence of electroweak symmetry breaking by means of the Higgs mechanism [1]. The direct search at the LEP experiments leads to a lower bound of ∼ 90 GeV [2] on the value of the Higgs mass, while triviality and unitarity constraints require the Higgs mass to be smaller than O(1 TeV). At the upgraded Tevatron experiment, a p¯ collider with p √ a c.m. energy s = 2 TeV, SM Higgs bosons will mainly be produced via gluon fusion gg → H and the Drell–Yan like production q q → W ∗ → W H. Since an intermediate mass ¯ ¯ pairs, the QCD background of b¯ production Higgs boson will dominantly decay into bb b will be too large to allow for a detection of the Higgs boson produced in the gluon fusion process. Recently it has been shown that a detection of the Higgs boson from W fusion W W → H seems to be impossible due to the overwhelming QCD background [3]. Thus, the primary possibility to find the SM Higgs boson at the Tevatron will be via the Drell– Yan like process. Careful studies have shown that a discovery of the SM Higgs boson at the upgraded Tevatron might be possible for Higgs masses up to about 120 GeV [4]. Apart from the dominant b¯ decay [4] it turned out that a discovery may also be feasible via the b + − H → τ τ decay [5] in H + W/Z production, while the gold-plated mode of the LHC, gg → H → ZZ ∗ → 4ℓ± , is hopeless at the Tevatron [4]. Recently it has been found that there is also the possibility to detect the processes gg → H → W ∗ W ∗ → ℓνjj, ℓ+ ℓ− ν ν by ¯ using the strong angular correlations among the final state leptons [6]. The SM Higgs mass is unstable against quantum fluctuations in the context of grand unified theories, which force the Higgs mass to be of the order of the GUT scale MGU T ∼ 1016 GeV. The most attractive solution to this hierarchy problem is provided by the introduction of supersymmetry. The minimal supersymmetric extension of the SM [MSSM] requires two Higgs doublets, leading to the existence of 5 elementary Higgs particle, two neutral CP-even (h, H), one neutral CP-odd (A) and two charged ones (H ± ). At treelevel the mass of the light scalar Higgs boson h is restricted to be smaller than the Z mass MZ . However, radiative corrections to the MSSM Higgs sector are large, since they increase with the fourth power of the large top quark mass mt . They enhance the upper < bound on the light scalar Higgs mass to Mh ∼ 135 GeV [7]. For the discovery at the Tevatron the light scalar Higgs boson h will mainly be produced via q q → W/Z + h analogously to the SM case. However, for large tgβ the associ¯ ated production meachanisms q q , gg → b¯ + h/A become competitive due to the enhanced ¯ b Yukawa couplings to b quarks. Finally, similar to the LHC the light scalar Higgs may be detectable in SUSY particle production process via the decay χ0 → χ0 h in the final state ˜2 ˜1 cascade decays [8]. Charged Higgs bosons can also be searched for at the Tevatron [9]. They will be produced in top quark decays t → bH + , if their masses are light anough. At the Tevatron ¯ the process p¯ → tt with t → bH + is sensitive to charged Higgs bosons for large tgβ. The p Drell–Yan like charged Higgs pair production p¯ → H + H − and gluon fusion gg → H + H − p processes seem to be difficult to detect, while the analysis of the associated production 2p¯ → W ± H ∓ requires a more careful background study in order to investigate its relevance p for charged Higgs searches at the Tevatron. Finally gluon-gluon fusion gg → t¯ − , bH − ′ ± gluon-bottom fusion gb → tH and quark-bottom fusion qb → q bH provide additional possibilities to search for charged Higgs bosons at the Tevatron.2Higgs Boson Decays¯ φ → ffIn the intermediate Higgs mass range the SM Higgs decay H → b¯ is dominating, while the b decay H → τ + τ − reaches a branching ratio of O(10%), see Fig. 1. In the past the QCD1Standard Model BR(HSM)10-1bb WW* *_ττ cc_+ −gg10-2ZZ* *100105110115120125130135140MH [GeV]Figure 1: Branching ratios of the dominant decay modes of the SM Higgs particle. All relevant higher order corrections are taken into account. The shaded bands represent the 2 variations due to the uncertainties in the input parameters: αs (MZ ) = 0.120 ± 0.003, mb (Mb ) = (4.22 ± 0.05) GeV, mc (Mc ) = (1.22 ± 0.06) GeV, Mt = (174 ± 5) GeV. corrections have been evaluated [10]. They turn out to be moderate for the decay mode ¯ H → tt in the threshold region, while they are large for the H → b¯ c¯ decays due to large b, c logarithmic contributions. These can be absorbed in the running quark mass by replacing the scale by the Higgs mass. In order to gain a consistent prediction of the partial decay widths one has to use direct fits of the MS masses mQ (MQ ) to experimental data. The evolution of mQ (MQ ) to mQ (MH ) is controlled by the renormalization group equations b for the running MS masses. As a result the QCD corrections reduce the H → b¯ decay width by ∼ 50% and the H → c¯ width by ∼ 75% relative to the Born term expressed in c terms of the quark pole masses MQ . In the MSSM the decay modes h, H, A → b¯ τ + τ − dominate the neutral Higgs decay b, < modes for large tgβ, while for small tgβ they are important for Mh,H,A ∼ 150 GeV as can be inferred from Figs. 2a–c. The dominant decay modes of the charged Higgs particles are H + → τ + ντ , t¯ [see Fig. 2d]. Analogous to the SM case the QCD corrections reduce b the partial decay widths into b, c quarks by about 50–75% as a result of the running quark masses, while they are moderate for decays into top quarks. 31 BR(h) tgβ = 3 10-1bb_1bb BR(h) tgβ = 30_1_ bb (tgβ=30) BR(H) _ bb tgβ=3 τ τ (tgβ=30)+ −1hhhhtt BR(H) tgβ=3ττ+ −10-1ττ+ −WW→←gg ←cc_WW 10-110-1ZZ10-2gg cc_WW10-2ττ ←ZZ gg ←γγ 10 90 100 110 Mh [GeV] 120-2+ −gg ZA _ cc 100 200 300 500 MH [GeV] 1000 10-2γγ 10-310 85 90 95 Mh [GeV] 100 105-38080100200 300 500 MH [GeV]1000Fig. 2a1 bb BR(A) tgβ = 3 10-1_Fig. 2bbb_tt Zh_1 BR(A) tgβ = 30 10-11τνtb1 BR(H ) tgβ = 3±τνtbBR(H ) tgβ = 30±ττ+ −ττ+ −10-1Wh10-1WA 10-210-210 tt_-2cb cs µν 10cb-2cs µνgg 10-3gg 200 500 MA [GeV] 1000 10-3100100200 500 MA [GeV]100010-3100200 300 MH± [GeV]50010-3100200 300 MH± [GeV]500Fig. 2cFig. 2dFigure 2: Branching ratios of the MSSM Higgs bosons h(a), H(b), A(c), H ± (d) for nonSUSY decay modes as a function of their masses for two values of tgβ = 3, 30 and vanishing mixing. The common squark mass has been chosen as MS = 1 TeV. Below the corresponding thresholds the decays A → t∗ t and H + → t∗¯ into off-shell b top quarks turn out to be important, since they reach branching ratios of O(1%) already far below the thresholds for on-shell top quarks [11]. H → W W, ZZ> In the SM the decays H → W W, ZZ are dominant for MH ∼ 140 GeV, since they increase with the third power of the Higgs mass for large Higgs masses, see Fig. 1. Decays into > off-shell W, Z bosons H → W ∗ W ∗ , Z ∗ Z ∗ are sizeable already for Higgs masses MH ∼ 100 GeV, i.e. significantly below the W W/ZZ thresholds [11, 12]. In the MSSM the decays h, H → W W, ZZ are suppressed by kinematics and, in general, by SUSY couplings and are thus less important than in the SM. Their branching ratios turn out to be sizeable only for small tgβ or in the decoupling regime, where the light scalar Higgs particle h reaches the upper bound of its mass. The gluonic (photonic) decays of the Higgs bosons h, H, A → gg(γγ) reach branching ratios of ∼ 10% (∼ 10−3 ) in the SM and MSSM and are unimportant at the Tevatron. 4The decay mode H → hh is dominant in the MSSM for small tgβ in a wide range of ¯ heavy scalar Higgs masses MH below the tt threshold, see Fig. 2b. The dominant radiative corrections to this decay arise from the corrections to the self-interaction λHhh in the MSSM and are large [7]. The decay mode H → AA is only important at the lower end of the heavy scalar Higgs mass range. φ → φ′ + W/ZH → hh, AA¯ The decay modes H → AZ and A → hZ are important for small tgβ below the tt thresh+ ∗ (∗) old, see Figs. 2b,c. Similarly the decays H → W A, W h are sizeable for small tgβ and MH + < mt + mb , see Fig. 2d. The dominant higher-order corrections can be absorbed into the couplings and masses of the Higgs sector. Below the corresponding thresholds decays into off-shell Higgs and gauge bosons turn out to be important especially for small tgβ [11], see Figs. 2b–d. Decays into SUSY particles Higgs decays into charginos, neutralinos and third-generation sfermions can become important, once they are kinematically allowed [13] as can be inferred from Fig. 3, which shows the branching ratios of the heavy Higgs bosons into gauginos and squarks as functions of their masses for a special SUSY scenario. Thus they could complicate the Higgs search at the Tevatron, since the decay into the LSP will be invisible.1 bb BR(h) tgβ = 30* _χ0χ0 1 1 WW*10-1ττ+ −ggZZ-2* *cc_108090100 Mh [GeV]110120130Figure 3: Branching ratios of the light scalar MSSM Higgs boson h decays into charginos/neutralinos as a function of its mass for tgβ = 30. The mixing parameters have been chosen as µ = 150 GeV, At = 2.45 TeV, Ab = 0 and the squark masses of the first two generations as MQ = 1000 GeV. The gaugino mass parameter has been set to M2 = 136 GeV. The masses of the lightest gauginos are mχ0 = 56.5 GeV and mχ± = 94.1 ˜1 ˜1 GeV.53Higgs Boson ProductionThe most relevant SM Higgs production mechanism at the Tevatron is Higgs-strahlung off W, Z bosons q q → W ∗ /Z ∗ → W/Z + H. The cross section reaches values of 10−1 – ¯ < 1 pb in the relevant Higgs mass range MH ∼ 120 GeV, where this production process may be visible at the Tevatron [4], see Fig. 4. The QCD corrections coincide with those of the Drell-Yan process and increase the cross sections by about 30% [14, 15]. The theoretical uncertainty can be estimated as ∼ 15% from the remaining scale dependence. The dependence on different sets of parton densities is rather weak and leads to a variation of the production cross sections by about 15%.10 2 σ(pp→H+X) [pb] √s = 2 TeV Mt = 175 GeV gg→H 1-1_ _q q → V ∗ → V H [V = W, Z] ¯10CTEQ4M10qq→Hqq-2qq’→HW qq→HZ gg,qq→Htt_ _ _ _1010-310-4gg,qq→Hbb 80 100 120 140 MH [GeV] 160 180 200_√ Figure 4: Higgs production cross sections at the Tevatron [ s = 2 TeV] for the various production mechanisms as a function of the Higgs mass. The full QCD-corrected results for the gluon fusion gg → H, vector boson fusion qq → V V qq → Hqq, Higgs-strahlung ¯ q q → V ∗ → HV and associated production gg, q q → Htt, Hb¯ are shown. The QCD ¯ ¯ b corrections to the last process are unknown and thus not included. In the MSSM the Higgs-strahlung processes are in general suppressed by the SUSY couplings. However, the process q q → W ∗ /Z ∗ → W/Z + h can be important in the ¯ decoupling regime, where the light scalar Higgs particle h exhibits SM Higgs properties, see Fig. 5a,b. ¯ b q q , gg → φtt, φb¯ ¯In the SM both processes of Higgs radiation off top and bottom quarks are unimportant due to the small event rates, see Fig. 4. However, in the MSSM Higgs radiation off bottom quarks becomes important for large tgβ with cross sections exceeding 10 pb for the light scalar (h) and the pseudoscalar (A) Higgs particles, see Fig. 5. Thus, the theoretical prediction is crucial for the large tgβ regime in the MSSM. 610 2σ(pp→h/H+X) [pb] √s = 2 TeV Mt = 175 GeV gg→h CTEQ4 tgβ = 3_101-110hW hZ hqq HW hbb _ htt_gg→H10-2Htt_10-3Hqq h H 80 100 120 H H Hbb_HZ 160 180 20010-4140 Mh/H [GeV]Fig. 5a10 4 103σ(pp→h/H+X) [pb] √s = 2 TeV gg→h__10 2 10 1 10 10 10 10-1 -2 -3 -4Mt = 175 GeV CTEQ4 tgβ = 30hbb gg→H Hbb hW hZ hqq h H 80_ _htt 100HW Htt_H H HZ 140 Mh/H [GeV]←Hqq 160 180 200120Fig. 5b √ Figure 5: Neutral MSSM Higgs production cross sections at the Tevatron [ s = 2 TeV] for gluon fusion gg → Φ, vector-boson fusion qq → qqV V → qqh/qqH, vector-boson ¯ bremsstrahlung q q → V ∗ → hV /HV and the associated production gg, q q → Φb¯ ¯ ¯ b/Φtt including all known QCD corrections. (a) h, H production for tgβ = 3, (b) h, H production for tgβ = 30, (c) A production for tgβ = 3, (d) A production for tgβ = 30.7σ(pp→A+X) [pb] 10 √s = 2 TeV Mt = 175 GeV 1-1_CTEQ4 gg→A tgβ = 31010-2gg,qq→Abb__10-310-480100120140 MA [GeV]160180200Fig. 5cσ(pp→A+X) [pb] 10 3 √s = 2 TeV Mt = 175 GeV CTEQ4 102_gg→Atgβ = 3010 gg,qq→Abb 1_ _10-180100120140 MA [GeV]160180200Fig. 5d Figure 5: Continued.8Until now the full QCD corrections are unknown so that the cross sections are only known within about a factor of 2. However, the QCD corrections are known in the limit ¯ of light Higgs particles compared with the heavy quark mass, which is applicable for tt + h ¯ production [16]. In this limit the cross section factorizes into the production of a tt pair, which is convoluted with a splitting function for Higgs radiation t → t + h. This reults in an increase of the cross section by about 20–60%. However, since this equivalent Higgs approxiamtion is only valid within a factor of 2 the result may not be sufficiently reliable. Moreover, it is not valid for bottom quarks, which are more relevant for the Tevatron. 2 2 In the opposite limit of large Higgs masses MH ≫ MQ large logarithms log MH /MQ arise due to quasi-on-shell t-channel quark exchanges, which can be resummed by absorbing them into the heavy quark parton densities. After adding the processes q q , gg → b¯ ¯ b+φ, gb → b + φ and b¯ → φ, after the logarithms have been subtracted accordingly, the final b result turns out to be about a factor of 2 larger than the pure q q , gg → b¯ + φ processes ¯ b [17]. However, there are additional sources of large logarithms, e.g. Sudakov logarithms from soft gluon radiation and large logarithms related to the Yukawa coupling, which will appear in the NLO cross section. Thus, a full NLO calculation is needed in order to gain a satisfactory theoretical prediction of these production processes. In this analysis the scales of the parton densities have been identified with the invariant mass of the final state ¯ QQφ triplet and the bottom Yukawa coupling in terms of the bottom pole mass Mb = 5 GeV. gg → φThe gluon fusion processes are mediated by heavy top and bottom quark triangle loops and in the MSSM by squark loops in addition [18–20]. Gluon fusion is the dominant neutral Higgs production mechanism at the Tevatron, even for large tgβ, with cross sections of b 1–103 pb, see Figs. 4, 5. However, the dominant b¯ final states are overwhelmed by the huge ¯ production and thus hopeless for a detection of the Higgs bosons QCD background of bb via gluon fusion. Only the τ + τ − decay modes may be promising for large tgβ, especially if the Higgs bosons are produced in association with a jet [21]. Moreover, similar to the LHC it may be possible to detect the H → W ∗ W ∗ decay mode in a significant Higgs mass range due to the strong angular correlations of the final state leptons [6]. The two-loop QCD corrections enhance the gluon fusion cross sections by about 60– 100% and are thus large [19]. They are dominated by soft and collinear gluon radiation in the SM and for small tgβ in the MSSM [22]. The remaining scale dependence yields an estimate of ∼ 15% for the theoretical uncertainty. The dependence of the gluon fusion cross section on different parton densities amounts to about 15%. The K factor remains nearly the same after including squark loops, since the dominant soft and collinear gluon effects are universal, thus suppressing the (s)quark mass dependence [20]. Recently the analytical QCD corrections to Higgs + jet production have been evaluated in the limit of heavy top quarks, but there is no numerical analysis so far [23].9Vector boson fusion W W/ZZ → H in the SM leads to a sizeable cross section at the Tevatron, see Fig. 4. Since there are two forward jets in the full process qq → W W/ZZ + qq → H + qq, one may hope to be able to suppress the QCD backgrounds by appropriate cuts. Unfortunatly it turned out that this is not possible [3]. The QCD corrections can easily be obtained in the structure function approach from deep inelastic scattering. They are small enhancing the cross section by about 10% [15, 24]. In the MSSM vector boson fusion is additionally suppressed by SUSY couplings and thus unimportant at the Tevatron. Higgs pairs Light scalar Higgs pair production gg → hh yields a sizeable cross section at the Tevatron > with σ ∼ 10f b [25, 26]. The cross section for q q , gg → hA is of similar size in some regions ¯ of the MSSM paramater space [see Fig. 6]. Since the process gg → H → hh is sensitive to the trilinear coupling λHhh it is important for a partial reconstruction of the Higgs potential. One may hope that the dominant b¯ ¯ b¯ + τ − final states can be extracted bbb, bτ from the QCD backgrounds due to the different event topologies. The two-loop QCD corrections have recently been calculated [for gg initial states in the limit of heavy top quarks, thus leading to a reliable result for small tgβ]. They enhance the gg → hh, hA production cross sections by about 70–90% and the Drell–Yan-like q q → hA cross section ¯ by about 30% [26].123.5 67.1 103W W/ZZ → H131.6 77.7_143.3 85.0157.9 89.6174.4 92.4192.0 94.3210.3 MH [GeV] 95.5 Mh [GeV]10 2σ(pp → hh/hA + X) [fb] √s=2 TeVmt = 175 GeV tgβ = 310 qq,gg → hA 1-1_gg → hh1010-280100120140 MA [GeV]160180200√ Figure 6: QCD corrected production cross sections of hh, hA pairs at the Tevatron [ s = 2 TeV] as a function of the pseudoscalar Higgs mass for tgβ = 3. The secondary axis exhibits the corresponding values of the light and heavy scalar Higgs masses Mh , MH . The bump in the gg → hh cross section originates from resonance gg → H → hh production.104ConclusionsAll relevant NLO QCD corrections to Higgs boson decays within the SM and MSSM areknown so that the theoretical uncertainties are sufficiently small.At the Tevatron thedecay modesφ0→b¯b,τ+τ−,W∗W∗are relevant for the Higgs boson search in Run II.All corrections beyond LO are contained in the program HDECAY1[15,27],which calculatesthe branching ratios of SM and MSSM Higgs bosons.For neutral Higgs boson production most QCD corrections are known leading to anearly complete theoretical status.The only processes,which are known at LO,are Higgsradiation offtop and bottom quarks,the latter being important for large tgβin the MSSM.The known corrections to the important processes are moderate and thus well under control.The remaining theoretical uncertainties are less than∼15%2. 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以探索为题的英语作文全文共3篇示例，供读者参考篇1Exploration has always been a fundamental aspect of human nature. From the early days of our ancestors setting out to discover new lands to the modern-day exploration of outer space, the drive to explore and push the boundaries of what is known has been a driving force in our development as a species.One of the most famous examples of exploration in history is the Age of Exploration, when European sailors set out to discover new trade routes, lands, and resources. This period, which lasted from the 15th to the 17th centuries, led to the discovery of the Americas, Australia, and numerous other lands previously unknown to Europeans. The explorers of this era faced numerous challenges, from treacherous seas to hostile natives, but their efforts ultimately reshaped the world and laid the foundations for the globalized world we live in today.In more recent times, exploration has taken on new forms. Space exploration, for example, has captured the imagination of people around the world. From the first moon landing in 1969 tothe ongoing exploration of Mars and beyond, humanity's quest to explore the wonders of the universe has led to countless discoveries and advancements in technology.But exploration is not only about physical journeys to unknown places. It can also involve the exploration of ideas, emotions, and experiences. Artists, writers, and thinkers throughout history have sought to explore the depths of the human experience through their work, pushing the boundaries of what is known and challenging societal norms and beliefs.In a more personal sense, exploration can also refer to the journey of self-discovery. Through exploring our own thoughts, feelings, and beliefs, we can better understand ourselves and our place in the world. This process of introspection can lead to personal growth and development, as we confront our fears, challenges, and limitations head-on.In conclusion, exploration is a fundamental aspect of what it means to be human. Whether it is the physical exploration of new lands, the intellectual exploration of new ideas, or the personal exploration of self-discovery, the drive to explore and discover is a powerful force that has shaped our history and will continue to shape our future. So let us embrace the spirit ofexploration and venture forth into the unknown, for it is through our efforts to explore that we truly come alive.篇2Exploration has been a fundamental aspect of human history, driving us to discover new territories, technologies, and ideas. From the great explorers of the past who traversed uncharted lands in search of riches and knowledge, to the modern-day space explorers pushing the boundaries of our solar system and beyond, the spirit of exploration continues to shape and define our world.One of the most iconic explorers in history is Christopher Columbus, whose voyage across the Atlantic in 1492 opened up a new era of exploration and colonization. His discovery of the Americas not only changed the course of history but also sparked the Age of Exploration, as European nations raced to stake their claim on new lands and resources. The quest for new trade routes, exotic goods, and scientific knowledge fueled this era of exploration, leading to the discovery of new continents, oceans, and cultures.In more recent times, exploration has taken on new dimensions with the advent of space travel. The race to the moonin the 1960s captured the imagination of the world and symbolized the pinnacle of human achievement. Since then, space agencies like NASA and SpaceX have continued to push the boundaries of exploration, sending probes to distant planets, launching satellites into orbit, and planning manned missions to Mars and beyond. The quest to understand the universe and our place within it drives these explorers to new heights of ingenuity and bravery.But exploration is not limited to the distant reaches of space or the uncharted depths of the ocean. It can also be a personal journey of self-discovery and growth. Whether it's venturing into unknown territories, trying new experiences, or challenging ourselves to overcome our fears and limitations, exploration can lead to new insights, opportunities, and perspectives. By stepping outside our comfort zones and embracing the unknown, we can unlock our full potential and become the architects of our own destinies.In conclusion, exploration is a timeless and essential aspect of human nature. It drives us to seek out new horizons, expand our knowledge, and transcend our limitations. Whether it's the intrepid explorers of the past, the daring astronauts of today, or the everyday adventurers seeking to discover themselves, thespirit of exploration continues to inspire and motivate us to push the boundaries of what is possible. So let us embrace the unknown, embark on new journeys of discovery, and chart a course towards a brighter and more enlightened future.篇3Exploration is a fundamental part of human nature. From the earliest days of mankind, humans have been driven by a curiosity to discover and explore the world around them. Whether it be the depths of the ocean, the vastness of space, or the intricate workings of the human body, exploration has shaped and defined our understanding of the world.One of the most famous explorers in history was Christopher Columbus. In 1492, Columbus set sail from Spain with the goal of finding a new trade route to Asia. Instead, he stumbled upon the Americas, completely unknown to Europeans at the time. Columbus's voyage marked the beginning of European exploration and colonization of the New World, forever changing the course of history.Exploration has also played a vital role in scientific discovery. From Charles Darwin's voyage on the HMS Beagle to the discovery of the Higgs boson particle at the Large HadronCollider, scientists have always pushed the boundaries of human knowledge through exploration and experimentation. Without curiosity and a willingness to take risks, these groundbreaking discoveries would not have been possible.In modern times, exploration has taken on new forms with the rise of space exploration and deep-sea exploration. Countries around the world are investing billions of dollars into space programs in the hopes of one day colonizing other planets. Meanwhile, deep-sea explorers are discovering new species and ecosystems thousands of feet below the surface of the ocean.However, exploration is not without its risks. Countless explorers throughout history have lost their lives in the pursuit of discovery. From the ill-fated Franklin Expedition to the Challenger disaster, exploration has always carried with it the potential for danger. Yet, despite these risks, humans continue to push the boundaries of what is possible in search of new knowledge and understanding.In conclusion, exploration is a fundamental part of human nature that has shaped our understanding of the world and led to countless discoveries. From the depths of the ocean to the far reaches of space, humans have always been driven by a curiosity to explore and discover. While the risks of exploration are great,the rewards are even greater, as each new discovery brings us closer to understanding the complexities of the world around us.。

a r X i v :0801.4256v 1 [h e p -p h ] 28 J a n 2008The Phase Diagram of Strongly-Interacting MatterP.Braun-Munzinger 1,2and J.Wambach 1,21Gesellschaft f¨u r Schwerionenforschung mbH,Planckstr,1,D64291Darmstadt,Germany 2Technical University Darmstadt,Schlossgartenstr.9,D64287Darmstadt,GermanyA fundamental question of physics is what ultimately happens to matter as it is heated or com-pressed.In the realm of very high temperature and density the fundamental degrees of freedom of the strong interaction,quarks and gluons,come into play and a transition from matter consisting of confined baryons and mesons to a state with ’liberated’quarks and gluons is expected.The study of the possible phases of strongly-interacting matter is at the focus of many research activi-ties worldwide.In this article we discuss physical aspects of the phase diagram,its relation to the evolution of the early universe as well as the inner core of neutron stars.We also summarize recent progress in the experimental study of hadronic or quark-gluon matter under extreme conditions with ultrarelativistic nucleus-nucleus collisions.PACS numbers:21.60.Cs,24.60.Lz,21.10.Hw,24.60.KyContentsI.Introduction1II.Strongly-Interacting Matter under ExtremeConditions2A.Quantum Chromodynamics 2B.Models of the phase diagram 4III.Results from Lattice QCD6IV.Experiments with heavy ions8A.Opaque fireballs and the ideal liquid scenario 8B.Hadro-Chemistry9C.Medium modifications of vector mesons 11D.Quarkonia-messengers of deconfinement 14V.Phases at high baryon densities16A.Color Superconductivity 16VI.Summary and Conclusions17Acknowledgments 18References18I.INTRODUCTIONMatter that surrounds us comes in a variety of phases,which can be transformed into each other by a change of external conditions such as temperature,pressure,com-position etc.Transitions from one phase to another are often accompanied by drastic changes in the physical properties of a material,such as its elastic properties,light transmission or electrical conductivity.A good ex-ample is water,whose phases are (partly)accessible to everyday experience.Changes in external pressure and temperature result in a rich phase diagram which,be-sides the familiar liquid and gaseous phases,features a variety of solid (ice)phases in which the H 20molecules arrange themselves in spatial lattices of certain symme-tries (Fig.1).Twelve of such crystalline (ice)phases are known at present.In addition,three amorphous (glass)phasesFIG.1The phase diagram of H 20(Chaplin,2007).Besides the liquid and gaseous phase a variety of crystalline and amor-phous phases occur.Of special importance in the context of strongly-interacting matter is the critical endpoint between the vapor and liquid phase.have been identified.Famous points in the phase dia-gram are the triple point where the solid-liquid-and gas phases coexist and the critical endpoint at which there is no distinction between the liquid and gas phase.This is the endpoint of a line of first-order liquid-gas transitions and is of second order.Under sufficient heating water and,for that matter any other substance,goes over into a new state,a ’plasma’,consisting of ions and free electrons.This transition is mediated by molecular or atomic collisions.It is continuous 1and hence not a phase transition in the strict thermodynamic sense.On the other hand,the2plasma exhibits new collective phenomena such as screen-ing and ’plasma oscillations’(Mrowczynski and Thoma,2007).Plasma states can alsobeinduced byhighcompression,whereelectronsare delocalizedfrom their orbitals and form a conducting ’degenerate’quantum plasma.In contrast to a hot plasma there exists in this case a true phase transition,the ’metal-insulator’transition (Gebhard,1997;Mott,1968).A good exam-ple are white dwarfs,stars at the end of their evolution which are stabilized by the degeneracy pressure of free electrons (Chandrasekhar,1931;Shapiro and Teukolsky,1983).One may ask what ultimately happens when matter is heated and compressed.This is not a purely academic question but is of relevance for the early stages of the universe as we go backwards in time in the cosmic evolu-tion.Also,the properties of dense matter are important for our understanding of the composition and properties of the inner core of neutron stars,the densest cosmic objects.Here,the main players are no longer forces of electromagnetic origin but the strong interaction,which is responsible for the binding of protons and neutrons into nuclei and of quarks and gluons into hadrons.In the Standard Model of particles physics the strong in-teraction is described in the framework of a relativistic quantum field theory called Quantum Chromodynamics (QCD),where point-like quarks and gluons are the ele-mentary constituents.The question of the fate of matter at very high tem-perature was first addressed by Hagedorn in a seminal paper in 1965(Hagedorn,1965)and later elaborated by Frautschi (Frautschi,1971).The analysis was based on the (pre-QCD)’bootstrap model’in which strongly-interacting particles (hadrons)were viewed as composite ’resonances’of lighter hadrons.A natural consequence of this model is the exponential growth in the density of mass statesρ(M h )∝M −5/2he M h /T H.(1)This is well verified by summing up the hadronic states listed by the Particle Data Group (Yao et al.,2006).A fit to the data yields T H ∼160−180MeV.It is then easy to see that logarithm of the partition function of such a ’resonance gas’ln ZRG(T )=iln Z RG i +κ∞m 0dM h ρ(M h )M 3/2h e −M h /T(2)and,hence,all thermodynamic quantities diverge when T =T H ,which implies that matter cannot be heated be-yond this limiting ’Hagedorn temperature’.Here,ln Z i is the logarithm of the partition function for all well iso-lated particles with mass m i .Above a certain mass m 0all particles start to overlap and from that point on the sum is converted into an integral over the mass density ρ(m )and all particles can be treated in Boltzmann ap-proximation.For the present argument the explicit valueof the constant κis immaterial.The energy that is sup-plied is used entirely for the production of new particles.This is of course at variance with our present understand-ing of the big bang in which the temperature is set by the Planck scale T ∼M Planck =2In all formulas below we use ¯h =c =13This term was coined by Edward Shuryak (Shuryak,1978a)3of electrons with photons is given byL QED=−14G aµνGµνa+¯qγµ(i∂µ−g sλa4Quarks form a fundamental representation of the Lie group SU(3).which now includes a non-linear term.Its form is en-tirely dictated by the gauge group(which is now SU(3) rather than U(1))through its’structure constants’f abc5. The group structure is also reflected in the quark-gluon coupling through the’Gell-Mann’matricesλa which are the analog of the SU(2)Pauli matrices.The more elab-orate group structure renders QCD much more compli-cated than QED even at the classical level of Maxwell’s equations.6In any relativisticfield theory the vacuum itself be-haves,due to quantumfluctuations,like a polarizable medium.In QED the photon,although uncharged, can create virtual electron-positron pairs,causing partial screening of the charge of a test electron.This implies that the dielectric constant of the QED vacuum obeys7ǫ0>1.On the other hand,because of Lorentz invari-ance,ǫ0µ0=1,i.e.the magnetic permeabilityµ0is smaller than one.Thus the QED vacuum behaves like a diamagnetic medium.In QCD,however,the gluons carry color charge as well as spin.In addition to vir-tual quark-antiquark pairs,which screen a color charge and thus would make the vacuum diamagnetic,the self-interaction of gluons can cause a color magnetization of the vacuum and make it paramagnetic.This effect ac-tually overcomes the diamagnetic contribution from¯q q pairs such thatµc0>1.The situation is somewhat simi-lar to the paramagnetism of the electron gas,where the intrinsic spin alignment of electrons overwhelms the dia-magnetism of orbital motion.Sinceµc0>1it follows thatǫc0<1,so that the color-electric interaction be-tween charged objects becomes stronger as their separa-tion grows(’infrared slavery’).In this sense the QCD vac-uum is an’antiscreening’medium.As the distance r→0, on the other hand,µc0andǫc0→1,and the interaction be-comes weaker(’asymptotic freedom’).This gives rise to a pronounced variation(’running’)of the strong’fine struc-ture constant’αs=g2s/4πwith(space-time)distance or momentum transfer Q.Its mathematical form to lead-ing order was worked out in1973by Gross and Wilczeck and independently by Politzer(Gross and Wilczek,1973; Politzer,1973)and yieldsαs(Q2)=12π5Gauge groups other than U(1)wherefirst discussed by Yang and Mills in1954(Yang and Mills,1954)in the context of SU(2)and the correspondingfield theories are therefore called’Yang-Mills theories’.Since the generators of SU(N)do not commute such theories are also called’non-abelian’.6For instance,the wave equation for the vector potentials A aµis non-linear and its solutions in Euclidean space-time include solitons called’instantons’.7Provided the distance r is large enough so that the virtual cloud around the test charge is not penetrated.This distance is ex-tremely small.scale parameter.As indicated in Fig.2the running ofαs8The situation is analogous to the case of a cavity in a perfectconductor(superconductor)withµ=0,ǫ=∞except that therole ofµandǫare interchanged.baryons or quark-antiquark pairs for mesons and impos-ing appropriate boundary conditions on the quark wavefunctions to prevent leakage of color currents across theboundary,B can be determined from afit to knownhadron masses.For the quark-hadron transition the MIT-Bag modelprovides the following picture:when matter is heated,nuclei eventually dissolve into protons and neutrons(nu-cleons).At the same time light hadrons(preferentiallypions)are created thermally,which increasinglyfill thespace between the nucleons.Because of theirfinitespatial extent the pions and other thermally producedhadrons begin to overlap with each other and with thebags of the original nucleons such that a network of zoneswith quarks,antiquarks and gluons is formed.At a cer-tain critical temperature T c these zonesfill the entire vol-ume in a’percolation’transition.This new state of mat-ter is the quark-gluon plasma(QGP).The vacuum be-comes trivial and the elementary constituents are weaklyinteracting sinceµc0=ǫc0=1everywhere.There is,how-ever,a fundamental difference to ordinary electromag-netic plasmas in which the transition is caused by ioniza-tion and therefore gradual.Because of confinement therecan be no liberation of quarks and radiation of gluonsbelow the critical temperature.Thus a relatively sharptransition is expected.A similar picture emerges whenmatter is strongly compressed.In this case the nucleonsoverlap at a critical number density n c and form a colddegenerate QGP consisting mostly of quarks.This statecould be realized in the inner core of neutron stars andits properties will be discussed later.In the MIT-Bag model thermodynamic quantities suchas energy density and pressure can be calculated as afunction of temperature and quark chemical potential9µqand the phase transition is inferred via the Gibbs con-struction of the phase boundary.Under the simplifyingassumption of a free gas of massless quarks,antiquarksand gluons in the QGP atfixed T andµq one obtains thepressurep QGP(T,µq)=37π22π2−B.(9)To the factor37=16+21,16gluonic(8×2),12quark(3×2×2)and12antiquark degrees of freedom contribute.For quarks an additional factor of7/8accounts for thedifferences in Bose-Einstein and Fermi-Dirac statistics.The temperature dependence of the pressure follows aStefan-Boltzmann law,in analogy to the black-body ra-diation of massless photons.The properties of the physi-cal vacuum are included by the bag constant B,which is a measure for the energy density of the vacuum.By con-struction,the quark-hadron transition in the MIT bag model is offirst order,implying that the phase boundary is obtained by the requirement that at constant chemi-cal potential the pressure of the QGP is equal to that in the hadronic phase.For the latter the equation of state (EoS)of hadronic matter is needed.Taking for simplic-ity a gas of massless pions with pπ(T,µq)=(3π2/90)T4, a simple phase diagram emerges in which the hadronic phase is separated from the QGP by afirst-order transi-tion line.Taking for the bag constant the original MIT fit to hadronic masses,B=57.5MeV/fm3one obtains T c∼100MeV atµq=0andµc∼300MeV at vanishing temperature(Buballa,2005).These results have a number of problems.On the one hand,the transition temperature is too small,as we know in the mean time.We will come back to this in the next section.On the other hand at3µq=µb∼M N(mass of the nucleon M N=939MeV),where homogeneous nu-clear matter consisting of interacting protons and neu-trons is formed,a cold QGP is energetically almost de-generate with normal nuclear matter.Both problems are, however,merely of quantitative nature and can be cir-cumvented by raising the value of B.More serious is the fact that,at largeµq,a gas of nucleons because of its color neutrality is always energetically preferred to the QGP. The biggest problem is,however,that QCD has a number of other symmetries besides local gauge symmetry which it shares with QED.Most notable in the present context is chiral symmetry,which is exact in the limit of vanishing quark masses.For physical up and down quark masses of only a few MeV this limit is well satisfied10.Exact chiral symmetry implies that only quarks with the same helicity or’chirality’interact11,i.e.the left-handed and right-handed world completely decouple.This means in particular that physical states of opposite parity must be degenerate in mass.Similar to a ferromagnet,where rotational symmetry is spontaneously broken at low temperatures through spin alignment,also the chiral symmetry of the strong inter-action is spontaneously broken in the QCD vacuum as a result of the strong increase ofαs at small momenta (Fig.2).Empirical evidence is the absence of parity doublets in the mass spectrum of hadrons.Since mass-less quarksflip their helicity at the bag boundary the MIT-Bag model massively violates chiral symmetry.For100%0%FIG.3Fraction of the effective quark mass generated dy-namically (light-grey)as compared to that from the Higgs mechanism in the electro-weak sector of the Standard Model (dark grey).QCD vacuum.In QCD,mesons emerge as bound states of quark-antiquark pairs with constituent mass.Because of spon-taneous chiral symmetry breaking there appears,how-ever,a peculiarity that is known from condensed matter physics and was first noted by J.Goldstone (Goldstone,1961).For vanishing (bare)quark mass there must be a massless excitation of the vacuum,known as the ’Gold-stone mode’.Such highly collective modes occur e.g.in spin systems.The ferromagnetic ground state has a spon-taneous alignment of all spins.A spin wave of infinite wavelength (λ→∞,k →0)corresponds to a simulta-neous rotation of all spins,which costs no energy 12.In strong interaction physics with two flavors this mode is identified with the pion.The fact that pions are not ex-actly massless is related to the finite bare mass of the up and down quarks.Nevertheless the pion mass with ∼140MeV is significantly smaller than that of the ρ-or the ωmeson (∼800MeV ∼2M q ).In the 1980’s and 1990’s the NJL model was used ex-tensively in theoretical studies of the phase diagram.Since it incorporates spontaneous symmetry breaking and the ensuing mass generation one can address ques-tions of chiral symmetry restoration with increasing T and µq and the corresponding medium modifications of hadron masses.The quark-antiquark condensate ¯q q serves as an order parameter for chiral symmetry break-ing,analogous to the spontaneous magnetization in a spin system.Similar to the Curie-Weiss transition,the order parameter vanishes at a critical temperature T c in the chiral limit.This is the point where chiral symmetry is restored and the quarks become massless 13.Figure 4displays a prediction for the evolution of the chiral con-densate with temperature and quark-chemical potential for physical up and down quark masses obtained in mean-7 and the statistical mechanics of a system with temper-ature T=1/τ.With this method of’lattice QCD’thepartition function of the grand canonical ensembleZ(V,T,µq)= D[A,q]e 1/T0dτ V d3x(L QCD−µq q†q)(11)can be evaluated stochastically via Monte Carlo samplingoffield configurations14.From the partition function,thethermodynamic state functions such as energy densityand pressure can be determined asε≡EV ∂ln Z V;p=T∂ln Z14For vanishingµq the integration measure is always positive defi-nite.This is no longer true forfiniteµq due to the fermion’sign problem’to the QGP.The critical energy densityǫ(T c)is700±300 MeV/fm3which is roughly5times higher than the en-ergy density in the center of a heavy nucleus like208P b. At the same time the chiral condensate ¯q q =∂p/∂m q diminishes rapidly near T c signalling therestoration of broken chiral symmetry.As indicated in Fig.5a sys-tematic discrepancy of about15%between the calcu-lated energy density(and pressure)and the free gas Stefan-Boltzmann limit is observed for T>2T c.Al-though this is roughly consistent with thefirst-order cor-rection from perturbation theory,the perturbation series is poorly convergent and resummation techniques have to be employed(Blaizot et al.,2006)for a quantitative understanding of the high-temperature EoS.The ab-initio numericalfindings support the simple model results for the existence of a QGP transition dis-cussed above.In this connection it should be mentioned, however,that most lattice calculations still have to use unrealistically large values for the light quark masses and rather small space-time volumes.With anticipated high-performance computers in the range of hundreds of Ter-aflop/s these calculations will be improved in the near future.Ultimately they will also provide definite answers concerning the nature of the transition.Among others, this is of importance for primordial nucleosynthesis,i.e. the formation of light elements,such as deuterium,he-lium and lithium.In a stronglyfirst-order quark-hadron transition bubbles form due to statisticalfluctuations, leading to significant spatial inhomogeneities.These would influence the local proton-to-neutron ratios,pro-viding inhomogeneous initial conditions for nucleosyn-thesis(Boyanovsky et al.,2006).Other consequences would be the generation of magneticfields,gravitational waves and the enhanced probability of black-hole forma-tion(Boyanovsky et al.,2006).At present,indications are that forµq=0,relevant for the early universe,the transition is a’cross over’,i.e. not a true phase transition in the thermodynamic sense (Aoki et al.,2006a).Near T c the state functions change smoothly but rapidly as in hot electromagnetic plasmas. For most of the experimental observables to be discussed below this subtlety is,however,of minor relevance.A cross over would wash out large spatialfluctuations and hence rule out inhomogeneous cosmic scenarios.Very recent studies(Aoki et al.,2006b;Cheng et al.,2006)in-dicate that the exact value of the transition temperature is still poorly known.In fact,these investigations have yielded values for T c in the range150-190MeV.This is in part due to difficulties with the necessary extrapola-tion to the thermodynamic(infinite volume)limit and in part due to the general difficulty in providing an absolute scale for the lattice calculations.Progress in this difficult area is expected with simulations on much larger lattices at the next generation computer facilities.While atµq=0the lattice results are relatively pre-cise,the ab-initio evaluation of the phase boundary in the(T,µq)-plane(Fig.4)poses major numerical dif-ficulties.This is basically related to the Fermi-Dirac8statistics of the quarks and is known in many-body physics as the ’fermion-sign problem’.For the integral (11)this implies that the integrand becomes an oscilla-tory function and hence Monte-Carlo sampling methods cease to work.Only recently new methods have been developed (Allton et al.,2003;Fodor and Katz,2002;de Forcrand and Philipsen,2002;Philipsen,2006)to go into the region of finite µq .What can be expected?Considering the phase bound-ary as a line of (nearly)constant energy density,the bag model (Braun-Munzinger and Stachel,1996a)pre-dicts that the critical temperature decreases with increas-ing µq .By construction the bag model describes a first-order phase transition for all chemical potentials.For large values of µq and low temperatures there are indi-cations from various QCD-inspired model studies,chiefly the NJL model (see Fig.4),that the (chiral)phase tran-sition is indeed first order.On the other hand,the lat-tice results discussed above seem to indicate that at very small µq the transition is a cross over.This would im-ply that there is a critical endpoint in the phase dia-gram,where the line of first-order transitions ends in a second-order transition (as in the liquid-gas transition of water).In analogy to the static magnetic suscepti-bility χM =∂M/∂H in a spin system one can define a ’chiral susceptibility’as the derivative of the in-medium chiral condensate ¯q q T,µq wrt the bare quark mass m q or equivalently as the second derivative of the pressure,χm =∂ ¯q q T,µq /∂m q =∂2p/∂m 2q .Here the quark mass m q plays the role of the external magnetic field H .In the Curie-Weiss transition χM diverges.The same should happen with χm at the CEP.On the other hand lat-tice studies and model calculations indicate that also the quark number susceptibility χn =∂n q /∂µq =∂2p/∂µ2q diverges.This implies that in the vicinity of the CEP the matter becomes very easy to compress since the isother-mal compressibility is given by κT =χn /n 2q .It is con-jectured that the critical behavior of strongly-interacting matter lies in the same universality class as the liquid-gas transition of water (Stephanov,2004).The exper-imental identification of a CEP and its location in the (T,µq )plane would be a major milestone in the study of the phase diagram.Although very difficult,there are several theoretical as well experimental efforts un-derway (Proceedings of Science,2006)to identify signals for such a point.For a recent critical discussion concern-ing the existence of a CEP in the QCD phase diagram see (Philipsen,2007).IV.EXPERIMENTS WITH HEAVY IONSThe phase diagram of strongly-interacting matter can be accessed experimentally in nucleus-nucleus collisions at ultrarelativistic energy,i.e.energies/nucleon in the center of mass (c.m.)frame that significantly ex-ceed the rest mass of a nucleon in the colliding nu-clei.After first intensive experimental programs at theBrookhaven Alternating GradientSynchrotron (AGS)and the CERN Super Proton Synchrotron (SPS)the ef-fort is at present concentrated at the Relativistic Heavy-Ion Collider (RHIC)at Brookhaven.A new era of ex-perimental quark matter research will begin in 2008with the start of the experimental program at the CERN Large Hadron Collider (LHC).Here we will not attempt to give an overview over the experimental status in this field (for recent reviews see (Braun-Munzinger and Stachel,2007;Gyulassy and McLerran,2005))but concentrate on a few areas which in our view have direct bearing on the phase diagram.Before doing so we will,however,briefly sketch two of the key results from RHIC,which have led to the discovery that quark-gluon matter in the vicinity of the phase boundary behaves more like an ideal liquid rather than a weakly-interacting plasma.A.Opaque fireballs and the ideal liquid scenarioAt RHIC,Au-Au collisions are investigated at c.m.energies of 200GeV per nucleon pair.In such colli-sions a hot fireball is created,which subsequently cools and expands until it thermally freezes out 15and free-streaming hadrons reach the detector.The spectroscopy of these hadrons (and the much rarer photons,elec-trons and muons)allow conclusions about the state of the matter inside the fireball,such as its temperature and density.The four experiments at RHIC have re-cently summarized their results (Adams et al.,2005b;Adcox et al.,2005;Arsene et al.,2005;Back et al.,2005).For a complete overview see also the proceedings of the two recent quark matter con-ferences (Proc.Quark-Matter 2005Conference,2006;Proc.Quark-Matter 2006Conference,2007).9FIG.6Geometry in the plane perpendicular to the beam di-rection of thefireball in a nucleus-nucleus collision with large impact parameter.The producedfireball has such a high density and tem-perature that apparently all partons(quarks and glu-ons)reach equilibrium very rapidly(over a time scale of less than1fm/c).Initially,the collision zone is highly anisotropic with an almond-like shape,at least for colli-sions with not too small impact parameter.The situa-tion is schematically described in Fig.6.In this equili-brated,anisotropic volume large pressure gradients exist, which determine and drive the hydrodynamic evolution of thefireball.Indeed,early observations at RHIC con-firmed that the data on theflow pattern of the mat-ter follow closely the predictions(Huovinen et al.,2001; Kolb and Heinz,2004;Teaney et al.,2002)based on the laws of ideal relativistic hydrodynamics.By Fourier anal-ysis of the distribution in azimuthal angleΦ(see Fig.6) of the momenta of produced particles the Fourier coef-ficient v2= cos(2Φ) can be determined as a function of the particles transverse momentum p t.These distri-butions can be used to determine the anisotropy of the fireball’s shape and are compared,in Fig.7for various particle species,to the predictions from hydrodynamical calculations.The observed close agreement between data and predictions,in particular concerning the mass or-dering of theflow coefficients,implies that thefireball flows collectively like an liquid with negligible shear vis-cosityη.Similar phenomena were also observed in ultra-cold atomic gases of fermions in the limit of very large scattering lengths,where it was possible,by measuring ηthroughflow data,to establish that the system is in a strongly coupled state(O’Hara et al.,2002).The appli-cation of such techniques are currently also discussed for quark-gluon matter.This liquid-likefireball is dense enough that even quarks and gluons of high momentum(jets)cannot leave without strong rescattering in the medium.This’jet quenching’manifests itself in a strong suppression(by about a factor of5)of hadrons with large momenta transverse to the beam axis compared to expectations from a superposition of binary nucleon-nucleon collisions. The interpretation is that a parton which eventually turns into a hadron must suffer a large energy loss while traversing the hot and dense collision zone.To make matters quantitative one defines the suppression factor R AA as the ratio of the number of events at a given transverse momentum p t in Au-Au collisions to that in proton-proton collisions,scaled to the Au-Au system by the number of collisions such that,in the absence of parton energy loss,R AA=1.Corresponding data are presented in Fig.8.The strong suppression observed by the PHENIX and,in fact,all RHIC collaborations (Adams et al.,2005b;Adcox et al.,2005;Arsene et al., 2005;Back et al.,2005)demonstrates the opaqueness of thefireball even for high momentum partons while pho-tons which do not participate in strong interactions can leave thefireball unscathed.Theoretical analysis ofthese FIG.7The Fourier coefficient v2for pions,kaons,protons andΛbaryons(with masses of140,495,940and1115MeV, respectively)emitted with transverse momentum p t in semi-central Au-Au collisions at RHIC.The data are from the STAR collaboration(Adams et al.,2005a).The lines cor-respond to predictions(Huovinen et al.,2001)from hydro-dynamical calculations with an equation of state based on weakly interacting quarks and gluons.data(Gyulassy and McLerran,2005;Vitev,2006)pro-vides evidence,albeit indirectly,for energy densities ex-ceeding10GeV/fm3in the center of thefireball.There is even evidence for the presence of Mach cone-like shock waves(Casalderrey-Solana et al.,2005;Stoecker,2005) caused by supersonic partons traversing the QGP.Ap-parently both elastic parton-parton collisions as well as gluon radiation contribute to the energy loss but it is fair to say that the details of this mechanism are currently not well understood.The situation is concisely summa-rized in(Gyulassy and McLerran,2005).B.Hadro-ChemistryIn ideal hydrodynamics no entropy is generated dur-ing the expansion and cooling of thefireball,i.e.the system evolves through the phase diagram(essentially) along isentropes,starting in the QGP phase.This can be experimentally verified through the production of a variety of mesons and baryons.The analysis of particle production data at AGS,SPS and RHIC en-ergies has clearly demonstrated(Andronic et al.,2006; Becattini et al.,2004;Braun-Munzinger et al.,2004a) that the measurements can understood to a high ac-curacy by a statistical ansatz,in which all hadrons are produced from a thermally and chemically equilibrated state.This hadro-chemical equilibrium is achieved dur-ing or shortly after the phase transition and leads to abundances of the measured hadron species that can be described by Bose-Einstein or Fermi-Dirac distributions n j=g j。