Sept 2006WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects1 Fitted HBT radii versus space-time variances in flow-dominated models Mike Lisa Ohio State University Frodermann, Heinz, MAL, PRC (2006); nucl-th/

Sept 2006WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects2 Outline  motivation: possible problems in comparing models to data  new formula for “fitting” model calculations  application to two common models  conclusions

Sept 2006WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects3 The many estimates of length scale  HBT radii : parameters of Gaussian fits  3D fit to 3D CF R  experimental procedure  1D fit to projections of 3D CF  R 1D (and 3 ’s)  questionable shortcut  FWHM of 1D projections R*  Space-time variances R-hat  quick to calculate if S P (x) Gaussian, then C(q) Gaussian* and R = R 1D = R* = R-hat * Coulomb ignored throughout

Sept 2006WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects4  But neither S(x) nor C(q) is “ever” Gaussian * Coulomb ignored throughout STAR Phys. Rev. C 71 (2005) dN/dx Retiere & MAL PRC (2004) Kisiel, Florkowski, Broniowski, Pluta PRC (2006) if S P (x) Gaussian, then C(q) Gaussian* and R = R 1D = R* = R-hat The many estimates of length scale

Sept 2006WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects5  But neither S(x) nor C(q) is “ever” Gaussian * Coulomb ignored throughout if S P (x) Gaussian, then C(q) Gaussian* and R = R 1D = R* = R-hat What do experimentalists do? STAR Phys. Rev. C 71 (2005) Fit with ad-hoc alternate forms ? what to do with the parameters? Paic and Skowronski J. Phys. G (2005) R o (fm) 4 6 R s (fm) 4 6 R l (fm) 4 6 q max (GeV/c) “fit-range study”  syst. err. “typical” study from STAR surely the way of the future... imaging

Sept 2006WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects6  But neither S(x) nor C(q) is “ever” Gaussian  How much does this (rather than physics) dominate model comparisons? hydro Hirano:R 1D Soff:R-hat ZschiescheR* Heinz:R-hat if S P (x) Gaussian, then C(q) Gaussian* and R = R 1D = R* = R-hat What do theorists do? cascade AMPTR MPCR-hat RQMDR HRMR

Sept 2006WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects7 It can matter (how much, is model-dependent)  AMPT, RQMD, HRM reproduce HBT radii best.  Only these use “right” method  coincidence? Hardtke & Voloshin PRC (2000) RQMD - some difference  R-hat  R AMPT - huge difference Lin, Ko, Pal PRL (2002)

Sept 2006WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects8 Our plan  Examine two popular models which have published R-hat  Blast-wave  Heinz/Kolb B.I. hydro  Compare R versus R 1D versus R-hat  for fits (R and R 1D ), perform experimentalist’s “fit-range study”  But first... an explanation of our “fit” procedure...

Sept 2006WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects9 The “data” to be “fit”  Straight-forward to calculate CF outside long hydro CE EOSBlastwave

Sept 2006WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects10 Analytic calculation of radii (“fit”) 3D functional form: only good for C>1 not for noisy data F.O.M. to minimize:

Sept 2006WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects11  non-homogeneous linear equations  invertable to find parameters P  as per data, we take  = fixed (not  ´)  (its value does not matter) Analytic calculation of radii (“fit”) 3D

Sept 2006WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects12  rather than one 4x4 set of equations for 4 parameters...  3 sets of 2x2 equations for 6 parameters  similar technique used by Wiedemann, others Analytic calculation of radii (“fit”) 1D Similarly, for R 1D...

Sept 2006WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects13 BW projections - approximately Gaussian k T =0k T =0.3 GeV/c projection of 3D fit projection of 3D CF L projection appears least Gaussian

Sept 2006WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects14 BW - 1D studies Transverse radii: R 1D  R-hat Longitudinal R 1D  R-hat signif. fit-range systematic p T =0.1 p T =0.9 “HBT radii” from variances radii from ‘fit’ using various q-ranges STAR 200 GeV 0-5% Phys. Rev. C 71 (2005) RoRo RsRs RLRL o s L RoRo RsRs RLRL o s L q max (GeV/c) K T (GeV/c)

Sept 2006WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects15 BW - 3D studies -coupling / 3D structure  Ro fit range systematic still, BW agreement w/data persists “HBT radii” from variances radii from ‘fit’ using various q-ranges STAR 200 GeV 0-5% Phys. Rev. C 71 (2005) q max (GeV/c) K T (GeV/c) RoRo RsRs RLRL RoRo RsRs RLRL

Sept 2006WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects16 CE Hydro projections - Gaussian fits “look bad” k T =0.3 GeV/ck T =0.6 GeV/c CF projections appear Gaussian projections of 3D Gaussian fit match poorly  (unseen) 3D q structure of CF drives fit

Sept 2006WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects17 CE Hydro - 3D studies larger fit-range systematic (side is least affected, despite “looking” worst in projections) significant difference b/t R, R-hat “fitted” R agree better with data “HBT radii” from variances radii from ‘fit’ using various q-ranges STAR 200 GeV 0-5% Phys. Rev. C 71 (2005) q max (GeV/c) RoRo RsRs RLRL K T (GeV/c) RoRo RsRs RLRL

Sept 2006WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects18 Hydro using 2 EoS  similar non-Gaussian effects  NCE always compared better to data, for R-hat and (by construction) for yields.  apples::apples comparison further improves agreement K T (GeV/c) RoRo RsRs RLRL K T (GeV/c) RoRo RsRs RLRL “CE” EoS assuming Chem. Equilib until FO - original publications - More realistic “NCE” EoS STAR data  Variance  3D “fit”

Sept 2006WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects19 BW & Hydro  Qualitatively sim non-Gauss effects  magnitude much smaller for BW  conclusions about BW agreement ~same (still “good” but  will increase)  hydro agreement (for R o, R l ) improves in apples::apples comparison K T (GeV/c) RoRo RsRs RLRL “CE” EoS K T (GeV/c) RoRo RsRs RLRL “NCE” EoS Blast-wave K T (GeV/c) RoRo RsRs RLRL

Sept 2006WPCF 2006, Sao Paulo Brazil - lisa Non-gaussian effects20 Summary / Conclusions  Variety of length-scale estimators are compared to experimental HBT radii  danger of apples::oranges comparison  magnitude of difference is model-dependent  analytic calculation of “fit” parameters in models  R versus R 1D versus R-hat  non-Gaussian features generate differences, fit-range systematic  R≠R 1D : importance of global 3D fit (as experimentally done)  R < R-hat in temporal components (long & out)  agreement w/hydro much improved in apples::apples  impact on “puzzles”  effect significantly smaller for BW