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1、9991raM021v1503099/th-lcu:nviXraAPH N.S., Heavy Ion Physics () HEAVY IONPHYSICSc Akad emiai Kiad oRelation of hard and total cross sections to centralityR. VogtNuclear Science Division, Lawrence Berkeley National LaboratoryUniversity of California, Berkeley, California 94720andPhysics Department,Uni
2、versity of California, Davis, California, 95616ReceivedFebruary 24, 1999Abstract. We compare the fractions of the hard and geometric cross sectionsas a function of impact parameter. For a given definition of central collisions,we calculate the corresponding impact parameter and the fraction of the h
3、ardcross section contained within this cut. We use charm quark production as adefinite example.In this note, revised from Ref. 1, standard nuclear density distributions aredescribed and the resulting geometrical overlap in nuclear collisions is calculated.We then compare the hard process cross secti
4、on to the total geometric cross sectionas a function of impact parameter and discuss how they are related to the collisioncentrality.A threeparameter WoodsSaxon shape is used to describe the nuclear densitydistribution,A(r) = 1 + (r/R0A)22RA(fm)162740631101972080.5130.5190.5860.5860.5350.5350.5490.1
5、6540.17390.16990.17010.15770.16930.1600R. VogtTable 1.Nuclear shape parameters taken from Ref. 2.where 1 when no nuclear effects are included. However, central collisions are ofthe greatest interest since it is there that high energy density effects are most likelyto appear. Central collisions contr
6、ibute larger than average values of ETto thesystem, in the tail of the ETdistribution, d/dET. We would like to determinewhich impact parameters are important in the high ETtail, i.e. what range of bmay be considered central. We now define the central fraction of the hard crosssection, Eq. (2), and t
7、he central fraction of the geometric cross section. We thendiscuss how the two are related.Considering only geometry with no nuclear effects, = 1 in Eq. (2), the inclu-sive production cross section of hard probes increases ashardharddAB= ppTAB(b)d2b(3)and the averagenumber of hard probes produced at
8、 impact parameter b isABbcbdbTAB(b) ,(6)0Relation to centrality3AbcbdbTA(b)(7)0would be used. Fig. 2, taken from Ref. 5, shows the increase of fABwith bcforseveral symmetric, AA, systems. Note that fAA 1 when bc 2RA. For example,hardif wechoose central= 0.1AB, this corresponds to bc= 2.05 fm in Au+A
9、u collisionshardand bc= 1.05 fm in O+O collisions. If we instead chose central= 0.01ABthenbc= 0.52 fm in Au+Au and bc= 0.33 fm in O+O collisions.Note however that fABis not the fraction of the geometric cross section whichincludes both hard and soft contributions. The geometric cross section in cent
10、ralcollisions is found by integrating the interaction probability over impact parameterup to bc,geo(bc) = 2bc0bdb1 exp(TABNN) .(8)The nucleon-nucleon inelastic cross section, NN, is 32 mb at SPS energies andgrows with energy. It is expected to be 60 mb at LHC energies. The fraction of4R. Vogt.(9)geo
11、In central collisions, where TABis large, the impact parameter dependence is simple,geo(bc) b2c. However, in peripheral collisions where the nuclear overlap becomessmall, geo(bc) deviates from the trivial b2occur until bsymmetric systems.cscaling. Deviations from this scaling donotc 2RAinFig. 2.The
12、central fraction of the hard cross section as a function of impactparameter cut bcfor several symmetric systems.Figure 3 shows the numerical result, Eq. (8), relative to the integral wherebcnegligible, for thedifferencesame systemsin the mostas in Fig.peripheral2. Wecollisionshave usedcanNN= 32 mb i
13、n Eq. (8).Abe expected if 60 mbwere used instead.The growth of the fraction of the geometric cross section isslower than that of the hard fraction, fAB. Indeed at bc 2RA, fgeoThe total geometrical cross section for a variety of colliding nuclei 0.75.is givenin Table 2.We have also calculated the imp
14、act parameter bccorresponding tofgeo= 0.05, 0.1, and 0.2 or the central 5%, 10% and 20% of all collisions. Theimpact parameter corresponding to fgeo= 0.2 is bc 1.04RAwhen symmetricsystems are considered. In asymmetric collisions, bc RAwhen fgeo= 0.2. Ifsmallercentrality cuts are imposed, the impact
15、parameters are reduced by factorsofRelation to centrality5geo(b)16+1616+2716+4016+6316+11016+19716+20827+2727+4027+6327+11027+19727+20840+4040+6340+11040+19740+20863+6363+11063+19763+208110+110110+197110+208197+197197+208208+2081.371.521.671.852.072.332.391.671.822.012.222.492.551.982.162.382.642.70
16、2.342.562.832.892.783.043.103.313.373.43fgeo= 0.12.743.043.343.704.144.664.783.353.654.014.454.985.093.954.324.765.285.404.695.135.665.785.566.096.216.626.746.86Table 2.Values of the geometric cross section and the impact parameter at whichfgeo= 0.05, 0.1 and 0.2 respectively for several colliding s
17、ystems.6R. VogtRelation to centrality7TAB(0) (mb1)16+1616+2716+4016+6316+11016+19716+20827+2727+4027+6327+11027+19727+20840+4040+6340+11040+19740+20863+6363+11063+19763+208110+110110+197110+208197+197197+208208+2080.1660.1780.1800.1850.1740.1670.1640.1910.1950.2010.1910.1840.1810.2020.2100.2040.1980
18、.1950.2200.2180.2130.2110.2220.2230.2210.2290.2290.229fgeo= 0.10.5220.5500.5580.5760.5650.5600.5550.5800.5910.6060.5990.5950.5910.6040.6230.6190.6180.6130.6420.6430.6440.6400.6480.6520.6490.6630.6620.664Table3.Valuesof TAB(0) and the fraction of the hard cross section for fgeo= 0.05,0.1 and 0.2 resp
19、ectively in several colliding systems.8R. VogthardNAB(0) = ppTAB(0) ,hard(10)hardwhere ppis the total hard process production cross section in pp interactions.The rate in the impact parameter interval 0 b bcis the ratio of the hard togeometric cross sections integrated over b,geo(bc)=hardppfgeo,(11)
20、using Eqs. (3), (6), (8), and (9). In Fig. 6, the ratiohardRAB(bc) hardpp=1fgeo(12)is shown for the same set of symmetric systems as in Figs. 24 as a function of bc.As a specific example, the average number of cNccppTAB(0) .(13)Relation to centrality9s = 200 GeV, with MRS Dparton distributions, cc pairs per Au+Au collision at b = 0. With a10% centrality cut, the average number of cNcppc0.1.(14)Since the central 10% of the geometric cross section corresponds to 40% of the hardcross section, there are 7.7 c10R. Vogt