1. Remarks on statistical model fits Remarks on statistical model fits Outline Ratios vs multiplicities The meaning of minimal c 2 Conclusions F. Becattini, University and INFN Florence Critical Point and Onset of Deconfinement – Florence

2. Introduction Many statistical model fits have been using ratios of particle yields to determine temperature and baryon-chemical potential The goal: to get rid of the normalization volume or any normalization constant In these analyses ratios have been calculated a posteriori from published experimental yields, both at midrapidity and in full phase space. Only in few cases, ratios have been quoted by experiments themselves. This fitting procedure is not well founded and leads to a bias in the determination of temperature and baryon-chemical potential

3. Introduction(2) N yields (either n or dn/dy) - Can form N(N-1)/2 ratios out of them Of course, it would be wrong to make an uncorrelated  2 fit using all of them. The number of degrees of freedom would be artificially enhanced. So, some authors chose N-1 ratios out of N(N-1)/2. This in fact keeps the dof constant But how choose ratios? If this procedure were correct, common sense says (hopes): any chosen set including all measured particles at least once should be equivalent. This is what is tacitly assumed (hoped). THIS IS NOT THE CASE

4. The simplest example: weighted average A = 1.2  0.2 B = 0.8  0.2 C = 0.8  0.2 D= 0.8  0.2 The model: a constant Solution: weighted average = 0.9  0.1 c 2 /dof = 1.0 Excellent! Now, I want to test whether these numbers are consistent from being the measurements of a constant and I want to use ratios. Expected outcome: 1 Choose 3 of them including all measurements: 1 st set : A/B, A/C, A/D ratios = 1.5, 1.5, 1.5 2 nd set: B/C, C/D, B/A ratios = 1.0, 1.0, 0.66 The result of the test will clearly differ for the two sets: they are NOT equivalent

5. A second example: simple fit x=1; 2; 3 y=2.3  0.1; 2.8  0.13; 4.3  0.08 The model: y(x) = x + c 1 st fit (correct): 2 nd fit (wrong): Redefine data as R 1 = y 1 /y 3 and R 2 = y 2 /y 3 Minimum  2 differs Parameter best fit value differs Parameter error differs

6. A third example: exponential fit x=0; 1; 2 y=1.8  0.1; 2.71  0.13; 6.5  0.08 The model: y(x) = a exp(b x) 1 st fit (correct): 2 nd fit (wrong): Redefine data as R 1 = y 1 /y 3 and R 2 = y 2 /y 3 OR R 1 = y 1 /y 2 and R 2 = y 3 /y 2

7. Minimum  2 differs Parameter best fit value differs Parameter error differs Results will coincide only for “overperfect” fit, i.e.  2 = 0

8. Statistical model fits with ratios Some papers ignore correlations. Ratios are formed and fitted with: This entails a further bias Quite intuitive example: perform “normal” fit choose heaviest particles X 1 with largest negative residual and lightest particle X 2 with largest positive residual for any particle Y: if Y lighter than X 1, form ratio Y/X 1 ; or, if Y heavier than X 2 Y/X 2 this will artificially lower temperature

9. Fit to RHIC dN/dy yields and ratios at 130 GeV Data from STAR collected by J. Manninen: p, p, K, L, X, W and antiparticles N-1 ratios: X/X See more in Jaakko's talk

10. T (GeV)=0.1675 – 0.1583 m 2 J. Cleymans et al., PRC 71, 054901 (2005) F.B., J. Manninen, M. Gazdzicki, Phys. Rev. C 73 (2006) 044905 This is error-free

11.  2 minimization Some (same) authors take c2 minimum too seriously In general: you expect c2 to be “perfect” (say 1) only for “perfect” theories Statistical model fits are not expected to be perfect because of side-assumptions which are unavoidable even if the model held true Therefore, you cannot expect 1 and you cannot draw excessive conclusions if c2 ~ 2-4. In case of “imperfect” c2 there are generally accepted methods to increase the uncertainties on fitted parameters (PDG)

12. Conclusions Don't use ratios in the statistical model fits unless quoted by the experiment Even introducing correlations, the method is conceptually wrong Statistical model fits are not expected to be perfect