SHAREv2: fluctuations and a comprehensive treatment of decay
Contents
1. Redlich Phys Rev C 60 054908 1999 H Heiselberg Phys Rept 351 161 2001 Z Koba H B Nielsen and P Olesen Nucl Phys B 40 317 1972 M I Gorenstein Yad Fiz 31 1980 1630 S Mrowezynski Z Phys C 27 131 1985 R Hagedorn Z Phys C 17 265 1983 See any advanced undergraduate or graduate level book on statistical mechanics e g K Huang John Wiley and Sons C Pruneau S Gavin and S Voloshin Phys Rev C 66 044904 2002 M Gazdzicki and S Mrowczynski Z Phys C 54 127 1992 S Das STAR Collaboration Event by event fluctuation in K pi ratio at RHIC arXiv nucl ex 0503023 O Barannikova STAR Collaboration Probing collision dynamics at RHIC arXiv nucl ex 0403014 C Adler et al STAR Collaboration Phys Rev Lett 89 092301 2002 S Wheaton and J Cleymans arXiv hep ph 0407174 J Rafelski J Letessier and G Torrieri Centrality dependence of bulk fireball properties at RHIC Phys Rev C 72 024905 2005 nucl th 0412072 J Letessier and J Rafelski arXiv nucl th 0506044 J Cleymans H Oeschler K Redlich and S Wheaton arXiv hep ph 0511094 24 48 M Gazdzicki arXiv nucl ex 0512034 49 J Rafelski J Letessier and G Torrieri Phys Rev C 64 054907 2001 Erratum ibid C 65 069902 2002 arXiv nucl th 0104042 50 A Kisiel T Taluc W Broniowski and W Florkowski arXiv nucl th 0504047 51 W Florkowski arXiv nucl th 02. calculate the fluctuation of a ratio particlel should be substituted by the data point number where the ratio is defined For instance if the 5th data point from the top is a K7 a ratio than it s fluctuation is given by 05 fluct_yld data Astat Asyst fit See section 5 2 of this paper for a more general treatment of this data point referencing SHAREv2 implements most definitions of dynamical fluctuations used to date by experimental collaborations These are implemented as additional tags of fluct_xxx type where xxx refers to different ways the experimental measurement is presented The possible types of data points are fluct_dyn To calculate Gayn o 02 aS measured in 41 2 as suggested in 9 stat fluct_dnr To calculate Gayn 3 Decay feed down and particle yields 3 1 Particle decay acceptance data files As shown in section 1 decay feed down is a fundamental component of the statistical hadronization model However the limited coverage of most detectors means that the feed down coefficients will acquire an experimental correction corresponding to the probability that the decay products of a given resonance formed within the detector acceptance region will also be in that region As shown in section 1 6 these corrections need to be considered when calculating both fluctuations and yields Weak decays such as A pr most protons at RHIC are in fact given by feed down from hyperons are part
3. contributions to particle yields is therefore nec essary Specifically there should be an easy way to allow for any arbitrary decay reaction con tributing to any data point SHAREv2 provides such a possibility through user defined decay feed down files In data file containing the experimental results to be fitted see 1 section 3 4 3 a weak decay control file is now signaled by a statement of the type Weakdecay File feed where File feed is a 9 letter filename The program then obtains the decay acceptance weights from File feed an ASCII file in a format similar to the decay tree files described in 1 section 3 Fig 1 and the attached input files provided with the SHARE package show how to implement the weak decay acceptance coefficients While many weak feed down files might be involved in the same analysis generally they are experiment specific and hence can be kept track of in a systematic way Alternatively all weak feed down files and experimental data files can be combined in a single large file using the methods described in section 5 1 of this paper In more detail a typical line in a feed down file will be Parent Daughter Daughter all 1st 2nd cor coeff or for 3 body decays Parent Daughter Daughter Daughter all 1st 2nd 3rd cor coeff The switch all 1st 2nd 3rd refers to the daughter to which the decay coefficient applies all means that the decay coefficient is the same for all daughters while 1st 2nd 3rd means
4. file format f READ THERM_INI th_neq data READ THERM_INI th_neq HERE content of thermal file starts temp 0 14 accu 0 01 ok content of thermal file ends READ TOTALDATA tot200mix data READ TOTALDATA tot200mix HERE content of experiment file starts pi0139plu prt_yield 286 4 24 2 0 1 lt within the tot200mix data file gt a weakdecay star feed weakdecay star HERE content of weak decay file starts Ka492sht pi0139plu pi0139min all 0 7 x content of weak decay file ends x content of experimental file ends CALC FITRATIOS fitnw20M _neq CALC FITRATIOS fitnw20M _neq Figure 2 Left sharerun data calling other input files Right One file format only difference is that a symbol on a new line has to be present at the point where the separate file would have ended When the program encounters the symbol it switches back to reading the earlier file that is prior to the insert HERE See Fig 2 and the provided file sharerun data_onefile for an example of how this works 5 2 Combining data points SHAREv2 gives the possibility to refer to a different data point within the given fit and or combine two data points in order to fit the sum or a product of two particles This feature was described in the case of fluctuations of ratios in section 2 of this paper The referring data point consists of one or two for a combination numbers correspond
5. in file fortrat is 77 L usr local cern pro lib o sharev2 1 exe sharev2 1 f lmathlib lkernlib lpacklib C which assumes that the CERN libraries are in directory usr local cern pro lib If this is not true on your system fortrat should be changed accordingly Once the directory is unpacked the program should be compiled with fortrat After this typing sharev2 1 exe should produce a correct run with a detailed output which shows the program s capability A copy of these files produced on our computer system is also included in sharev2 1 tar gz within a directory called samplefit200 for comparison with the files produced by the installed program Several output files are produced with the following names as default The contents of each file are explained in detail in 1 section 4 fit out Fit output files graph Fit output graphics experiment fitted values calculated values prof y profiles and correlation functions for the various fits See 1 about more details about these files s contents 8 Status conclusions and future plans One of the areas of current intense interest in the field of high energy heavy ion reactions is the understanding of the mechanisms of soft hadron production chemical freeze out that is the study of how the energy confined in the central fireball turns into matter in a multi particle production process The SHARE suite of programs is an analysis tool of particle yields addressin
6. of both the stable particles and the hadronic reso nances are set according to a statistical prescription calculated via a series of Bessel functions using CERN library programs We also have the option of including finite particle widths of the resonances A x minimization algorithm also from the CERN library programs is used to perform and analyze the fit Please see 1 for more details on these Purpose The vast amount of high quality soft hadron production data from experiments running at the SPS RHIC in past at the AGS and in the near future at the LHC offers the opportunity for statistical particle production model falsification This task has turned out to be difficult when considering solely particle yields addressed in the context of SHAREv1 x 1 For this reason physical conditions at freeze out remain contested Inclusion in the analysis of event by event fluctuations appears to resolve this issue Similarly a thorough analysis including both fluctuations and average multiplicities gives a way to explore the presence and strength of interactions following hadronization when hadrons form ending with thermal freeze out when all interactions cease SHAREv2 with fluctuations will also help determine which statistical ensemble if any e g canonical or grand canonical is more physically appropriate for analyzing a given system Together with resonances fluctuations can also be used for a direct estimate of the extent the sys
7. 1l39min 2nd 0 Lmlli5zer ne0Q939zer pi0135zer 2nd 0 Lm1115zrb prO0938plb pi0139plu 2nd 0 Lmlli5zrb ne0939zrb pi0135zer 2nd 0 Y opo O G ue statement is met Two special case exist for which no File feed file is needed 14 Figure 1 An example of the SHAREv2 weak feed down acceptance coefficient implementation group of data points are subject to the same set of weak decay yield contribution e g in general all data points from the same experiment should have the same weak decay file The way to do this is the same as in v1 x 1 section 3 4 3 when the program reads a weakdecay statement it assigns the current decay pattern to each data point encountered until a new weakdecay weakdecay UNCORRECT uncorrected from the perspective of experimental data set means that all weak decays contributions to particle yields are fully accepted by SHAREv 2 weakdecay NOWK_FEED means that all particle yields are computed without contributions from weak decays from the perspective of experimental data this means that either all weak decay products are not accepted and or have been all corrected for in experimental yields as e g applies to some NA49 results When fluctuations are considered it is important to deal carefully with experimental correc tions which are neither close to total close 100 accepted nor null close to 0 accepted Weak decay corrections of the daughter particles are usua
8. 509039 Invited talk at QM2005 Budapest 25
9. ameter The syntax of SNSPROFIL is the same as DATPROFIL in 1 section 4 The two com mands operate in the same way all parameters except the one on the abscissa are minimized at each point in the profile Thus the command CALC SNSPROFIL temp 0 1 0 2 100 5 will calculate a sensitivity profile for the temperature going from 0 1 to 0 2 GeV with 100 points of the fifth data point within the experimental data file 5 3 4 Additional output in y and statistical significance profiles x profile commands now output the following files name log A fit output for each point in the y profile in the same format as the usual fit output file 1 Fig 4 name chi2 name stsg Commands SNSPROFIL and DATPROFIL also output the x pro file extension chi2 and Prue profile extension stsg 5 3 5 Improved treatment of fit errors SHAREv2 automatically runs the MINOS algorithm 2 if the fit can not get a robust estimate of the errors This results in a considerable improvement of error handling This update entails no changes in the user interface or output format 19 6 Comparison with previous versions 6 1 Testing SHAREv2 SHAREv2 was extensively tested for programming and physics errors e SHM Calculations and fit results for SPS and RHIC energies were verified to be equal between SHAREv2 and SHAREv1 x reference results e SHAREv2 reads SHAREv1 x weak decay input The equivalence between the two treat ments when weak decay
10. ave been recognized to be the physical observable capable to con strain particle production models Therefore consideration of event by event fluctuations is re quired for a decisive falsification or constraining of variants of particle production models based on grand micro canonical statistical mechanics phase space the so called statistical hadroniza tion models SHM As in the case of particle yields to properly compare model calculations to data it is necessary to consistently take into account resonance decays However event by event fluctuations are more sensitive than particle yields to experimental acceptance issues and a range of techniques needs to be implemented to extract physical fluctuations from an experimental event by event measurement Method of solving the problem The techniques used within the SHARE suite of programs 1 are updated and extended to fluc tuations A full particle data table decay tree and set of experimental feed down coefficients are provided Unlike SHAREv1 x experimental acceptance feed down coefficients can be entered for any resonance decay SHAREv2 can calculate yields fluctuations and bulk properties of the fireball from provided thermal parameters alternatively parameters can be obtained from fits to experimental data via the MINUIT fitting algorithm 2 Fits can also be analyzed for significance parameter and data point sensitivity Averages and fluctuations at freeze out
11. ber b strangeness s 5 Use of the GCE for at least some conserved charges is required by the experimental observa tion of a significant fluctuation in those charges 16 18 This fluctuation has been found to be compatible with Poisson scaling will be given by AN N 4 which is approximately followed by the GCE fluctuations This is not the only scaling known to be present in this area of physics Elementary reaction systems have been observed to follow a non Poissonian scaling 33 34 w r t multiplicity averages AN N e N 5 where c is a constant As has been argued previously 35 37 it is possible to describe this scaling by considering an extension of the Grand Canonical ensemble variously referred to as Isobaric or Pressure ensemble where system volume is also allowed to fluctuate In SHAREv2 we consider only GCE yields and fluctuations and search to explore whether the grand canonical statistical hadronization model can quantitatively reproduce fluctuations in the same way as it was shown to reproduce particle yields in heavy ion A A reactions 1 1 Evaluation of yields and fluctuations In GCE particle yields and fluctuations can be calculated by a textbook method 38 For a hadron with an energy Ep yp m the energy state occupancy is 1 ni Ep TIETE 6 where the upper sign is for fermions and the lower sign is for bosons The chemical fugacity Y will be considered
12. cs with reactions occurring at large energy a study of fluctuations within a narrow momentum rapidity acceptance window provides for the division between system and bath with the bath being the unobserved rapidity domain In all exper iments currently capable to measure fluctuations detector acceptance is limited typically to the central rapidity phase space coverage Such an acceptance domain in the boost invariant denoted below as subscript b i limit is equivalent to a configuration space sub volume 32 and thus for both particle ratios and particle yield width fluctuation 23 we have Nidec dNi dy pii 1 Ni ac dN dy pi l 1The rapidity y defined b 1 n 422 where E is the particle energy and p the particle momentum yy yy 3 Ep 8y component parallel to the collision axis is a variable additive under Lorentz transformations parallel to this axis Here N is the event by event average of particle i dN dy is the average number of particles in an element of rapidity at central rapidity Similarly barring acceptance effects discussed in sections 1 5 and 1 6 of this work as well as in 23 the scaled variance defined as g AX y XP xy at ie aay a do 2 Ni dy Ns me We conclude that in experiments with limited central rapidity acceptance both yields and fluc tuations should be evaluated in the GCE with respect to the conserved quantum numbers charge Q baryon num
13. ences and Engineering research council of Canada the Fonds Nature et Technologies of Quebec G T thanks the Tomlinson foundation for support S J thanks RIKEN BNL Center and U S Department of Energy DE AC02 98CH10886 for providing facilities essential for the completion of this work The authors wish to express their gratitude to Lucy Carruthers for invaluable assistance in debugging and adapting the software described in this work t LPTHE Univ Paris 6 et 7 is Unit mixte de Recherche du CNRS UMR7589 Contents 1 Introduction 1 1 1 2 1 3 1 4 1 5 1 6 Evaluation of yields and fluctuations o oa a a a 4 ee a A we ales Chemical potential and chemical non equilibrium Resonance decays 2 4 622 42 442 83 bee ee et eae ek we hae ee Volume fluctuations and fluctuations of ratios 0 22004 Fluctuations and detector acceptance 2 00000 ee eee nee Correlations resonances and detector acceptance n o o ooo Implementation of GCE fluctuations in SHAREv2 Decay feed down and particle yields 3 1 3 2 Particle decay acceptance data files 0 0 0a a a Compatibility with SHAREv1 x experimental data files Quark chemistry 4 1 Charm mesons s rs maa ack Sa 2 eet A a N aA ba A e ee fo aes User Interface files and new commands 5 1 5 2 5 3 New single file control aosa ea atoa aaie bak Gee ent a AE g ee a AG Combining data Ponts z ga e fod a fold et
14. enough to resolve the question about the nature of SHM as seen in the recent references 29 30 45 47 and the possible relation to the onset of a phase transition 48 SHAREv2 offers the additional analysis feature the fluctuation in hadron yields and ratios which will help to settle these issues should precise particle yield experimental data not become available Moreover SHAREv2 through the study of the consistency of particle yield and fluctuation can test the SHM in depth The development of phenomenological tools capable of falsifying statistical hadronization mod els is of course far from over Possible extensions of chemical freeze out model in future version of SHARE might include a canonical ensemble module allowing to test SHM chemical non equilibrium in small physical systems and the introduction of an opacity parameter to correct the yields of observed resonances 49 Another possible future development would entail extending SHARE towards a detailed description of momentum distributions This would be somewhat dif ferent from THERMINATOR 50 model which relies on scaling in rapidity and thus only applies to ultra high energy collisions We note that particle momentum distributions are dependent in addition to the physics incorporated in SHAREv2 also on the dynamical evolution of the emitting source and on the degree of resonance rescattering following on the chemical freeze out Strategies how these two interwoven effect
15. ent events 16 Since tracks from different events have no correlations or quantum corrections a as is determined solely by a trivial Poisson contribution as well as detector acceptance effects Within the statistical hadronization model on stat 1 19 For particle ratios in mixed events the correlation term AN AN 2 in Eq 17 vanishes while ANj2 follow Poisson scaling Hence Eq 17 reduces to 1 1 ON N stat TN T Ny 20 The dynamical fluctuation ofyn 9 39 41 corresponds to the difference between the raw total fluctuation o and the fake event fluctuation oun y o Piat 21 In certain limits Oym can be shown 39 to be independent of detector acceptance Hence com paring SHM estimates of Cayn to experimental measurements is more reliable than using o 1 6 Correlations resonances and detector acceptance Because mixed event tracks are uncorrelated mixed event techniques cannot account for detector acceptance effects within particle correlations Thus the branching ratios appearing in Eqs 14 and 15 need to be modified Bj gt a B and in addition Eq 17 needs to be updated wa KAND AN ga ANMAN 2 PETN E Ni Na here amp j refers to the probability that particle will end up in the detector s acceptance region given particle j is in the region while aj measures the probability that both decay products will be inside this region For a boo
16. experiments had much less than full acceptance for these decays other experiments e g NA49 have removed X p feed down via Monte Carlo simulations accounting for the experimental acceptance of the decay products with yields obtained from the observed A yields Working with patched SHAREv1 x we realized that decay issue mattered in that some fits got better allowing for a modified weak decay pattern Issues such as this one prompted us to introduce a more general treatment of weak decay particle yield contributions in SHAREv 2 20 7 Installation The SHAREv 2 1 program code and input files are contained in a tar gz archive The file sharev2 1 tar gz is available at http www physics arizona edu torrieri SHARE share html To unpack it create a SHARE directory put the archive in it and execute the following com mands gunzip sharev2 1 tar tar xf sharev2 1 tar The following files will then be created enough for a complete representative run of SHARE decays data The complete Particle Data Group decay tree section 3 3 in 1 dec_no data An empty decays file useful for testing the program calculations abundancies re duce to modified Bessel functions as well as studying the role of resonances in stable particle ratios fortrat A shell script compiling in f77 the FORTRAN code which should be modified depending on location of FORTRAN g77 or f77 or 95 and CERN library of programs particles data Par
17. files are designed to reproduce SHAREv1 x format was shown to all decimal places e Fluctuations of conserved quantities such as AQ were compared before and after resonance decays The conservation of this quantity implies that the enhanced fluctuations after all resonances decayed are exactly balanced by multiplicity correlations between the resonance decay products This holds true to two decimal places up to two step correlations arising from decays such as 1600 A 892 Krr These correlations are not tracked by SHARE but their contribution is below 1 6 2 SHAREv1 x bugs found While developing and testing SHAREv2 several minor bugs and choice issues were found in the previous version SHAREv1 x The most noteworthy issues which lead to sometimes noticeable beyond line width changes in the results are SHAREv1 1 v1 2 The Bessel function series was incorrectly truncated for large 7 close to pion B E condensation SHAREv1 3 Quark flavor mixing error in calculation of mesons such as and for Yq 4 Ys SHAREv1 1 v1 3 The most relevant issue is actually not an error but lack of versatility in the handling of pr decays particles decay weakly like the As and the s However unlike A and decays are not experimentally reconstructible since at least one of decay products is neutral In general SHAREv1 x particles from these decays were included in the yield count However it turned out that while some
18. fines y3 cancels out in 7 Importantly the evolution of quark coalesced hadrons into final quan cleanest anes hadrons like the oscillation of neutral kaons into Ks and Kz means that the source QGP quark content will not in general be equal to the final hadron quark content 1 3 Resonance decays Eqs 7 and 8 can be used to calculate the event by event averages and fluctuations of all hadrons at hadronization This however is quite different from the observed averages and fluctuations since most hadrons are strong resonances unstable states which decay after freeze out either to stable particles or to other resonances The final state particle yields can be computed by taking the effect of these feed downs into account 6 The ensemble average of the total yield N is Nijtota NG divect ye Bi i N 14 j i B i i is the probability branching ratio for the decay products of j to include i The fluctuation after resonance feed down is given by AN a Ba NEB a NG Bp AND 15 The second term corresponds to the fluctuation in the yield of resonances The first term in the number of j i decays given the branching ratio b Nj is the number of particles type i produced in the decay so that 5 By Njai Nj 1 for nearly all decays of nearly all resonances The most significant exception are decays to multiple 7 s such as n 37 1 4 Volume fluctuations and fluct
19. g the following questions e What is the chemical freeze out temperature chemical potentials and volume e What are the physical properties of the fireball which hadronizes e Is the hadron system in chemical equilibrium at freeze out e What is the degree of re interaction between hadronization and freeze out The need for SHARE arises from recognition that the book keeping task involved in the correct application of the statistical hadronization model is considerable often transcending the resources 22 available to individual researchers The current SHAREv2 program follows on SHAREv1 x 1 adding three significant novel features a flexible handling of particle decay feed down b fluctu ations and c complete u d s c flavor content treatment We note that since SHAREv1 x was released another analysis package appeared THERMUS 44 THERMUS has more convenient setup for the experimental data analysis environment On the other hand THERMUS is not addressing many of the features SHARE offers including chemical non equilibrium and now in the current SHAREv2 fluctuations and light flavor details SHARE allows an analysis of experimental data that can address indirectly questions related to the dynamics of the fireball evolution since chemical non equilibrium implies a fast hadronization However the analysis of particle yields cannot give results of greater precision than is inherent in the data it treats and this is not yet good
20. h ee a oe a a Miscellaneous fis cc ae e Gira a Ba oh ee ee ae ee eee A 5 3 1 Expanded parameter set Satis wm We dnd amp Be ee a eo ee ee ee 5 3 2 Data point sensitivity analysis oo a 0000 5 3 3 Data point sensitivity profiles 2 fae a See e Eas 2 Eee EA 5 3 4 Additional output in y and statistical significance profiles 5 3 5 Improved treatment of fit errors 6 en amp Gok Gene et eed ek a Comparison with previous versions 6 1 Testine SHAREV2 Seg 8 lo 26 8 a Rag eee BES e ie ah abe ie te Le Ue He ae ee ea 62 SHARE Wis bugs found sroti 4 sara i hada ba SA DESRS DHS Installation Status conclusions and future plans Program Summary Title of the program SHAREv2 April 2006 Computer PC Pentium III 512MB RAM not hardware dependent Operating system Linux RedHat 6 1 7 2 FEDORA etc not system dependent Programming language FORTRAN77 Size of the package 167 KB directory without libraries see http wwwasdoc web cern ch wwwasdoc minuit minmain html http wwwasd web cern ch wwwasd cernlib html for details on library requirements Distribution format tar gzip file Number of lines in distributed program tar file including test data etc 26017 Keywords fluctuations relativistic heavy ion collisions particle production statistical models decays of resonances Computer Any computer with an f77 compiler Nature of the physical problem Event by event fluctuations h
21. icularly susceptible to experimental acceptance as they occur at a macroscopic distance from the primary vertex Hence weak experimental feed down corrections include a geometrical as well as a momentum space component Since the parent particles are not always directly observed SHARE must be able to compute final hadron multiplicities including experimental feed down coefficients for all decays where this effect is non negligible 12 SHAREv1 x allowed the user to input experimental weak feed down contributions to pro duced particle yields via four acceptance coefficients Ks anything Kz anything Y Mesons Y baryons see 1 section 3 4 1 It turns out this approach was not sufficiently flexible for instance undetected amp Na feed down can be treated very differently from A pz corrections considering the difference in lifespan and vertex acceptance cuts applied Moreover the experimental acceptance of different hyperon nucleon weak decays such as An as compared to A pz is likely to be considerably different Finally different weak decays of the same hadron can have varying acceptances compare Kz 37 with Kz rev and with Kz muv A similar acceptance problem may arise in special cases involving strong decay chains when the acceptance region is particularly narrow and or the particles rapidity distribution is not well understood A more flexible way of treating weak decay
22. in section 1 2 The yield average is obtained by multiplying the occupancy number Eq 6 by the density of states where V is volume and g degeneracy d p ny ov aE 7 The fluctuation in this number is found to be AND TEL a f eas Ml LF mE 8 Eqs 7 8 can be evaluated to any desired accuracy by converting them into an expansion of Bessel function terms 5 gVT 1 iY pam ae Ni ym 2 n T E Tem lt 1 9 oan gVTF 41 11 24 n 1 nm AN Qn a ae n Ware au where W x x K x see 1 section 2 for the technical details required in doing these calcu lations as well as a discussion of particles with finite width 1 2 Chemical potential and chemical non equilibrium The particle fugacity Y can be obtained from the quark number content of the particle as well as the fugacities and phase space occupancies of the individual quark flavors If a particle has N Ni Nj Ni and Nj light up down strange and charm quarks Ni Ni N Ni and N antiquarks and isospin Iz the fugacity Y or equivalently the associated particle chemical potential u will be given by Yi e P dau Aaa 4 Aste Aete ara ara 4 sts ere Ui where A and 7 y The individual u d light quark variables are related to the SHARE q and I variables Ag VAyAaq and r Au Aa see Ref 1 and similarly yq YuVa and y3 Yu Ya See Eq 23 below The condition of chemical e
23. ing to the position in the input file of the point s being referred to 17 Two numbers united by an operation sign X will add subtract multiply and divide two data points For instance if the first data points from the top of the file see 1 section 3 4 for a detailed explanation of the format are Lm11l5zer pi0139min Data Astat Ags Fit Lm1115zrb pi0139plu Data Astat Asyst Fit then 01X02 prt_yield Data Asat syst Fit will fit AA rtr while 01 fluct_dyn Data Asat Asyst Fit will fit the dynamical A a fluctuation as described in section 2 To fit A A a 77 but NOT the separate yields the input file will read Lm111l 5zer prt_yield Data Astat Asyst Lm11l zrb prt yield Data Astat Asyst pi0139plu prt_yield Data Astat Asyst pi0139min prt_yield Data Asta Asyst 01 02 03 04 Data Astat Asyst NOTE SHARE was written in FORTRAN77 Feature mentioned in this subsection use implicitly recursive code SHAREv2 has been tested on several compilers and platforms and found to work However compilers and operating systems vary we would like to know if and when you experience problems rPo ocooco 5 3 Miscellaneous The following small modifications were made in SHAREv2 compared to SHAREv1 5 3 1 Expanded parameter set The expanded parameter set includes as noted before Eq 23 gam3 which allows to incorporate a different u d flavor phase space occupancy A further new variable dvol desc
24. ions is useful as a test of new physics a quantitative analysis including both average multiplicities and their event by event fluctuations constitutes a powerful probe of hadronization conditions 20 23 In particular the following questions can be addressed when both yields and fluctuations are considered in the same model framework e SHM can be falsified if and when fluctuations do not scale w r t averages as expected in statistical physics Moreover only if the same set of thermal parameters gives good description of experimentally measured yields and fluctuations can we claim that the SHM fit is physically sound e As has recently been shown 24 28 the value to which the scaled variance ow see Eq 2 for a single particle converges in the thermodynamic limit varies by as much as an order of magnitude when different statistical ensembles are considered Thus fluctuations can help decide if and when certain particle yields should be studied in grand canonical or canonical ensembles e SHM fits containing both the average particle multiplicity and the fluctuation de correlate the hadronization temperature T and light quark phase space occupancy Yq see Eq 11 and 1 typical of fits when only the average multiplicities are fitted 20 Therefore the study of both fluctuations and yields can help to experimentally distinguish between the chemical equilibrium freeze out model T 170 MeV 4 1 29 or the best fit with chemical no
25. lly correlated with each other in momentum space so the straight forward application of Eq 8 will not be a good description of fluctuations with a non trivial detector acceptance function 13 In this case it is better to use dynamical rather than total fluctuations as discussed in section 1 5 3 2 Compatibility with SHAREv1 x experimental data files The improved weak decay treatment does not impair compatibility of experimental input files be tween SHAREv2 and SHAREv1 x SHAREv 2 will read a SHAREv1 x experimental data file and automatically calculate applicable contributions for each weak decay based on the information con tained in the SHAREv1 x weakdecay statement A line will be printed within the sharerun out output file that signals a SHAREv1 x format weakdecay statement was encountered In addition an output v2 weakdecay file called weak v1 x where refers to the data point number is automatically generated translating the vl x weak decay information into v2 format The user is advised to eventually change all weakdecay lines to weakdecay weak v1 x as the v2 format is considerably more powerful and less amenable to systematic error stemming from an incomplete understanding of weak decays 4 Quark chemistry SHAREv1 x input files listed particle chemical content by total isospin J and its third component I3 as well as the number of light q either u or d strange s and charm c quarks See section 3 2 of 1 In SHAREv2
26. n Nucl Phys A 698 611 2002 J Adams et al STAR Collaboration Phys Rev C 68 044905 2003 K Adcox et al PHENIX Collaboration Phys Rev Lett 89 082301 2002 C Alt et al NA49 Collaboration Phys Rev C 70 064903 2004 G Torrieri S Jeon and J Rafelski arXiv nucl th 0510024 G Torrieri S Jeon and J Rafelski arXiv nucl th 0509077 G Torrieri S Jeon and J Rafelski arXiv nucl th 0509067 G Torrieri S Jeon and J Rafelski Accepted for Publication Physical Review C arXiv nucl th 0503026 V V Begun M I Gorenstein A P Kostyuk and O S Zozulya Phys Rev C 71 054904 2005 V V Begun M I Gorenstein and O S Zozulya Phys Rev C 72 014902 2005 V V Begun M Gazdzicki M I Gorenstein and O S Zozulya Phys Rev C 70 034901 2004 arXiv nucl th 0404056 F Becattini A Keranen L Ferroni and T Gabbriellini Phys Rev C 72 064904 2005 arXiv nucl th 0507039 J Cleymans M Stankiewicz P Steinberg and S Wheaton The origin of the difference between multiplicities in e e annihilation and heavy ion collisions arXiv nucl th 0506027 F Becattini J Manninen and M Gazdzicki arXiv hep ph 0511092 J Letessier and J Rafelski arXiv nucl th 0504028 K Hagiwara et al Particle Data Group Collaboration Phys Rev D 66 010001 2002 see also earlier versions note that the MC identification scheme for most hadrons was last presented in 1996 J Cleymans K
27. n equilibrium at typically lower T 30 e Considering the directly detectable resonance decays fluctuations of particle yield ratios offer a way to quantitatively gauge the effect of hadronic re interactions between formation and thermal freeze out 21 To investigate these questions it is necessary in evaluation of both particle yields and fluctu ations to e Incorporate all particles resonance decay trees 31 in the program structure e Obtain particle yields and fluctuations for a given set of thermodynamic parameters e i Check if the parameters obtained by fitting particle yields are consistent with observed fluctuations ii Once all corrections to fluctuations due to experimental setup are understood incorporate the fluctuations along with yields into the chemical freeze out fitting procedure SHAREv2 comprises a framework that addresses these challenges As implied above event by event particle yield fluctuations are subject to many subtle exper imental effects which need to be understood and kept under control for a joint yield fluctuation analysis to proceed Further there is the choice of statistical model ensemble in computation of the phase space volume 1 Evaluation with exact energy and discrete quantum number conservation micro canonical en semble MCE 2 In the canonical ensemble CE statistical energy fluctuations are allowed conserving discrete quantum number s exactly 3 In the grand canonical e
28. nsemble GCE statistical fluctuations of all conserved quantities occur there are also mixed CE GCE ensembles where some particle yields are conserved and other fluctuate Clearly the fluctuations of particle yields are most constrained in MCE and least constrained in GCE Thus although in the three ensembles the first moments of any observable distribution i e expectation values coincide in the thermodynamic limit this will not be the case for the fluctuations 24 26 The choice of appropriate ensemble in the situation considered has to be made based on evaluation of prevailing physical conditions In study of total particle yields in the physical context of heavy ion collisions the electrical charge and baryon number are fixed and in these variables we have to consider the CE or MCE if and when we are observing all particles On the other hand if we only observe a sub volume of the system which is exchanging energy and particles with an unobserved bath consisting of the remainder of the reaction system then also conserved quantum numbers must be allowed to fluctuate which implies use of the GCE for all observables However when the totality of the produced particles carrying a conserved quantum number falls within the detector acceptance region occasional detection of one such particle implies presence of the corresponding anti particle and thus in that case CE or MCE must be applied Within the context of heavy ion physi
29. only the 1st 2nd 3rd daughter will be removed from the experimental yield For example in the A rp decay in STAR 42 43 STAR accepts the nucleon from the A decay but not the 7 and this fine tuning of the decay is clearly quite important as a relatively large fraction of all nucleons comes from weak A decays cor refers to the fractional contribution of the acceptance to the two particle correlation Daughter Daughters induced by a common resonance decay from parent denoted as aj in Eq 22 section 1 6 SHARE will renormalize the decay Parent all 1st 2nd 3rd cor by the coefficient coeff when calculating all data points after the given weakdecay statement It is therefore possible to assign a different weak decay file to each data point or assume that a 13 Example of STAR data points Experimental data file weakdecay star feed Ka0492plu pid01l39plu 0 156 Ka0492min pi0139min O25 PHENIX data points weakdecay phen feed Example of weak feed down file K_L gt pi lepton 0 0208 0 02 K_L corrections Hyperon corrections Ka04921ng pi0139min el0000plu nue000zer Ka04921ng pi0139plu el0000min nue000zer Ka04921ng pi0139min mu0000plu num000zer Ka04921ng pi0139plu mu0000min numO000zer Ko L gt Bpi Ka04921ng pi0139plu pi0139min pi0135zer K_S corrections Kad492sht pid0139plu pi0139min all 0 Lm1115zer pr0938plu pi0
30. phase space occupancy Ye described in 1 section 3 1 5 User Interface files and new commands 5 1 New single file control As described in detail in 1 section 3 SHARE relies on quite a few input files e The run file sharerun data e The particles list e The particles decay tree e The initial values of the thermodynamic parameters e The experimental data points e Initialization for each fit parameter This structure makes it easy to quickly explore regions of parameter space within an analysis in progress However this system makes it easy to mistakenly lose a successfully completed and saved analysis since a change in each of the files could considerably alter the end result The introduction of weak decay correction file see section 3 aggravates this problem SHAREv2 therefore makes it possible to combine some or all input files into a single file Once the user found an optimum analysis all input files involved in it can be combined into one large sharerun data file which can be easily kept for future reproduction and modification This is done by changing the extension data or feed of the filename into HERE If the program encounters a filename ending in HERE it assumes the relevant input is immediately following the given line within the currently read file The subsequent format is assumed to be unchanged from what it would have been had a separate file been opened comments etc The 16 Many files format One
31. quilibrium for a flavor f imposes ys 1 38 The assumption of chemical equilibrium is not automatic in a dynamically expanding system with a possible phase transition and in fact good theoretical arguments have been proposed for 74 5 4 1 for a range of energies see 30 and references therein However it is difficult using fits to particle yields to distinguish between two models based on different temperatures and y values For instance models based on both a higher freeze out temperature and yg 1 29 or a lower freeze out temperature and yz gt 1 30 have been used to fit SPS and RHIC data As shown in 23 this ambiguity is resolved when both yields and fluctuations can be considered A complication arises for hadrons such as the 7 or the 7 which are in a flavor superposition state If yu A 1 the yield of the hadron with fractional flavor content is considerably altered by the mixing For a meson of fractional quark number structure i gt a ut gt ag dd gt a s3 gt a2 o2 o0 1 12 the fugacity comprises the chemical yield fugacities as follows Ti 7302 Yaz ya 13 Fractional flavor content has non negligible influence on the abundances of 7 and 7 and their decay products in fits which allow for chemical non equilibrium factor ys The same remarks applies when 73 Yu z7_ 1 to 7 p etc Thus y3 1 can considerably enhance 7 73 737 yield while maintaining 7 yields symmetry
32. r s resnsnsnnssnsnennsmnssnncennsmmmn arXiv nucl th 0603026v2 16 Mar 2007 SHAREv2 fluctuations and a comprehensive treatment of decay feed down t G Torrieri S Jeon J Letessier and J Rafelski Department of Physics McGill University Montreal QC H8A 2T8 Canada RIKEN BNL Research Center Upton NY 11973 USA Laboratoire de Physique Th orique et Hautes Energies Universit Paris 7 2 place Jussieu F 75251 Cedex 05 France 4 Department of Physics University of Arizona Tucson AZ 85721 USA Abstract This the user s manual for SHARE version 2 SHARE 1 Statistical Hadronization with Resonances is a collection of programs designed for the statistical analysis of particle production in relativistic heavy ion collisions While the structure of the program remains similar to vl x v2 provides several new features such as evaluation of statistical fluctuations of particle yields and a greater versatility in particular regarding decay feed down and in put output structure This article describes all the new features with emphasis on statistical fluctuations Keywords Heavy ion collisions Statistical models Fluctuations PACS 24 10 Pa 24 60 k 25 75 Dw 24 60 Ky 25 75 Nq 12 38 Mh Updates available from http www physics arizona edu torrieri SHARE share html and the authors upon request f Work supported by U S Department of Energy grant DE FG02 04ER41318 the Natural Sci
33. r each flavor separately Misc Many new commands and features have been introduced and added to the basic user interface For example it is possible to study combinations of particles and their ratios It is also possible to combine all the input files into one file SHARE Compatibility and Manual This write up is an update and extension of 1 The user should consult Ref 1 regarding the principles of user interface and for all particle yield related physics and program instructions other than the parameter additions and minor changes described here SHAREv2 is downward com patible for the changes of the user interface offering the user of SHAREv1 a computer generated revised input files compatible with SHAREv 2 1 Introduction The statistical hadronization model 3 6 SHM assumes particles are created according to their phase space weight given the locally available energy and quantum numbers Such a reaction model implies that the underlying dynamics of strong interactions saturates the strength of each particle production quantum matrix element This approach can be used to calculate the event by event average as well as fluctuation distribution width and higher cumulants of any soft observable Event by event particle fluctu ations have been subject to intense current theoretical 7 15 and experimental interest 16 19 SHAREv 2 will offer a standardized framework to evaluate these While qualitative study of fluctuat
34. ribes statistical pressure ensemble fluctuations in volume Section 1 Eq 16 The provided input file sets and fixes dvol to zero and gam38 to unity since experimental measurements sensitive to these param eters have not as yet been published All details about how to configure these parameters and fix or relax them in the context of fits to experimental data are unchanged w r t vl x described in 1 sections 3 1 3 6 and 3 7 5 3 2 Data point sensitivity analysis Command DFIT within the file sharerun data can turn on and off the given data point as a point to be fitted The syntax for this command is DFIT Datapoint n Fit 0 1 where Datapoint n refers to the data point s position in the experimental data file from the 18 top while Fit 0 1 turns this point on 1 or off 0 as a point to be fitted For instance the following input in sharerun data READ TOTALDATA tot200mix data DFIT 51 CALC FITRATIOS fitnw20M kpi DFIT 50 CALC FITRATIOS fitnw20M nkpi performs two fits The first saved in file fitnw20M kpi uses the 5th data point in tot200mix data when calculating the x to be minimized The second one saved in file fitnw20M nkpi does not 5 3 3 Data point sensitivity profiles Command SNSPROFIL calculates the data point sensitivity The sensitivity is defined as the ra tio between the data point s SHM prediction for a given statistical parameter and SHM prediction at the best fit value for that par
35. s can be disentangled are being developed For further discussion of these and related questions we refer to a recent review 51 References 1 G Torrieri S Steinke W Broniowski W Florkowski J Letessier and J Rafelski Comput Phys Commun 167 229 2005 F James and M Roos Comput Phys Commun 10 343 1975 E Fermi Prog Theor Phys 5 570 1950 I Pomeranchuk Proc USSR Academy of Sciences in Russian 43 889 1951 L D Landau Izv Akad Nauk Ser Fiz 17 1953 51 R Hagedorn Suppl Nuovo Cimento 2 147 1965 S Jeon and V Koch Event by event fluctuations arXiv hep ph 0304012 In Hwa R C ed et al Quark gluon plasma Singapore 2004 pp 430 490 S Jeon V Koch K Redlich and X N Wang Nucl Phys A 697 546 2002 S Jeon and V Koch Phys Rev Lett 83 5435 1999 10 S Jeon and V Koch Phys Rev Lett 85 2076 2000 NOD OR W DY ied 23 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 M Asakawa U W Heinz and B Muller Phys Rev Lett 85 2072 2000 S Mrowczynski Phys Rev C 57 1518 1998 C Pruneau S Gavin and S Voloshin Phys Rev C 66 044904 2002 J Zaranek Phys Rev C 66 024905 2002 Q H Zhang V Topor Pop S Jeon and C Gale Phys Rev C 66 014909 2002 J G Reid STAR Collaboratio
36. st invariant azimuthally complete system a _ 1 since particles leaving the detector acceptance region will be balanced by particles entering However in general aj lt 1 since if a resonance is outside the detector acceptance region both particles can not be inside it and the intrinsic particle decay momentum adds a rapidity scale to the system breaking boost invariance 20 See 20 for an illustration of how to calculate ay While such a comprehensive calculation is outside the scope of the current version of the program we offer the user the possibility of entering an aj for any resonance decay as an input parameter See section 3 on how to do this In practice this should only be necessary for a few most frequent and energetic resonance decays such as p gt mr and K gt Kr 11 2 Implementation of GCE fluctuations in SHAREv2 Experimental event by event fluctuation data points were implemented in the SHARE interface in a similar manner as yield and ratio data points see 1 section 3 4 The tag which denotes that a fluctuation is being calculated is fluct_yld A statement such as particlel fluct yld data Astat Asyst fit will calculate o of particle1 defined in Eq 2 If fit is set to 1 this data point is used within a fit together with the experimentally measured value data and the statistical Astat and systematic Asyst error The format of the data line is exactly the same as in SHAREv1 x 1 section 3 4 To
37. tem re interacts between chemical and thermal freeze out We hope and expect that SHAREv2 will contribute to decide if any of the statistical hadroniza tion model variants has a genuine physical connection to hadron particle production Computation time survey We encounter in the FORTRAN version computation times up to seconds for evaluation of particle yields These rise by up to a factor of 300 in the process of minimization and a further factor of a few when y Npor profiles and contours with chemical non equilibrium are requested Accessibility The program is available from e The CPC program library e The following website http www physics arizona edu torrieri SHARE share html e from the authors upon request SUMMARY OF NEW FEATURES w r t SHAREv1 x Fluctuations In addition to particle yields ratios and bulk quantities SHAREv2 can calculate fit and analyze statistical fluctuations of particles and particle ratios Decays SHAREv2 has the flexibility to account for any experimental method of allowing for decay feed downs to the particle yields Charm flavor Charmed particles have been added to the decay tree allowing as an option study of statistical hadronization of J w Xe De ete Quark chemistry Chemical non equilibrium yields for both u and d flavors as opposed to gener ically light quarks q are considered 7 7 mixing etc are properly dealt with and chemical non equilibrium can be studied fo
38. ticle properties with full widths section 3 2 in 1 partnowdt data Particle properties with no widths Calculations with this input file require considerably less computational time and it suffices when there are no resonances in the fit ratioset data The FORTRAN fit input file section 3 5 in 1 sharerun data A representative run input file section 4 in 1 including an analysis of fluc tuations and yields similar to what was presented by members of our collaboration in 21 sharerun data_onefile The same file in the single file format as explained in section 5 1 samplefit200 A directory containing the output files generated by running the provided share run data as a debugging comparison standard sharev2 1 f SHAREv1 1 FORTRAN code The header contains information about bug fixes thermo data A representative thermal parameter input file section 3 1 in 1 It is set to reasonable non equilibrium fit values totbar200 data A representative data input file section 3 4 in 1 containing ratios yields and fluctuations drawn from RHIC experiments as of July 2004 see references in 21 star feed An example of a decay feed down coefficients file See section 3 This chapter is nearly identical to the READMEv2 html file found on the SHARE webpage 21 Note that SHARE requires CERN libraries to be downloaded separately from http wwwasd web cern ch wwwasd cernlib html The compiler statement
39. u and d quarks are now separately accounted for The particle listing format is name mass width spin I I3 u d s au ad as c ac MC where name is the particle s 9 character name J and I are the total and third component of the isospin u d s c are the numbers of up down strange and charm quark numbers while au ad as ac are the respective anti quark numbers The format of the table is otherwise identical to that discussed in 1 section 3 2 To check for the possibility that phase space occupancy differs for the up and down quarks a statistical model fit parameter see 1 section 3 1 gam3 73 has been introduced such that Yu W3 Ya Ya V3 23 15 The quark anti quark numbers can be fractional to account for the superposition states described at the end of section 1 2 To calculate u d s quark abundance in the statistically hadronizing QGP system different in general from the freeze out content as shown in section 1 2 new bulk variables tot_u_qgp tot_d_qgp and tot_s_qgp were introduced These can be used in the same way as other bulk variables see 1 section 3 4 2 4 1 Charm mesons Charmed particles have now been added in the files particles data and partnowdt data Their nomenclature follows the general structure as described in 1 section 3 2 Dcxxxxxxx refer to D mesons Dsxxxxxxx chixxxxcc to x states and psixxxxcc to J w states Their abundance is regulated by the chemical potential A and the
40. uations of ratios The expression 10 neglects volume fluctuations coming from centrality cuts and dynamics of system expansion These are accounted for by dividing the observed fluctuation into an extensive and an intensive part AX we Aa V a AV 16 x x can be calculated by the statistical methods described in this section It is difficult to describe the volume fluctuation coefficient AV in a model independent way The most straight forward way to deal with this problem is to choose observables insensitive to AV Any observable where x lt Ax would be a good candidate This is why the fluctuation in electromagnetic charge has long been considered to be a promising observable 10 A more general approach is to consider the event by event fluctuation of particle ratios 9 where the volume fluctuation AV is zero by construction Fluctuation of particles ratios can be calculated from the denominator s and numerator s fluctuation once the full resonance decay tree is known 9 Note that unlike in the case of particle yields resonance decays produce both fluctuations and correlations since a resonance can decay both into a numerator and a denominator particle If this is the case a high resonance admixture can considerably reduce the fluctuation of a ratio w r t Poisson expectation The formulas to be used are for the event by event fluctuation of the ratio of two particles N No for e
41. xample K7 r 9 2 _ ANI ANg ANIAN ON N N T No 2 Ni Na 3 17 The last correlation term is given by the resonance decay into both particles AN AN Ni N Ny No a By 2 N 18 a does not depend on the average system volume V since it cancels between the numerator and the denominator o IND however does acquire a dependence on V since ratio fluctua tions scale as N ia Hence an analysis incorporating fluctuations of particle ratios should also consistently account for particle yields and the system normalization thermodynamic parameter norm 1 section 3 1 should be considered as a fit parameter The equations presented in this section can be used to compute fluctuations of particles yields and ratios from given SHM parameters temperatures chemical potentials and phase space occu pancies However it has long been known 39 that fluctuations are considerably less robust than yields against systematic effects resulting from limited experimental acceptance These effects therefore have to be taken into account within the SHM The next two subsections give two such issues addressed within SHAREv2 1 5 Fluctuations and detector acceptance One way to separate detector acceptance effects from physics is to eliminate the former via mixed event techniques A static fluctuation o is measured in a sample of fake events constructed by 10 using tracks from differ
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