WPCF Aug. 1-3, Conservation Laws in low-multiplicity collisions Zbigniew Chajęcki and Michael A. Lisa The Ohio State University

WPCF Aug. 1-3, Outline  Introduction / Motivation –Non-femtoscopic correlations in low-multiplicity collisions : OPAL, NA22, STAR, … * data features not under control: Energy-momentum conservation?  Analytic calculation of Energy and Momentum Conservation Induced Correlations for –single particle spectra –two-particle correlations Experimentalists’ recipe: Fitting correlation functions –M inv correlation function & background subtraction –V 2 –Two-particle correlations –Resonance contribution to non-femtoscopic correlations - (π +,π - ) –(π +,π - ) correlations in p+p(  p) at 200 GeV collisions from PYTHIA  Conclusion

WPCF Aug. 1-3, Non-femtoscopic correlations : OPAL OPAL, CERN-PH-EP/ (submitted to Eur. Phys. J. C.) 1D projections of 3D CF Femtoscopic correlations should go to the constant number at large Q (no directional dependence!) Q x <0.2 GeV/c

WPCF Aug. 1-3, Non-femtoscopic correlations : NA22 NA22, Z. Phys. C71 (1996) 405 1D projections of 3D CF

WPCF Aug. 1-3, Non-femtoscopic correlations : STAR d+Au: peripheral collisions STAR preliminary Non-femtoscopic q-anisotropic behaviour at large |q| does this structure affect femtoscopic region as well? Q x <0.12 GeV/c STAR, NPA 774 (2006) 599 Clear interpretation clouded by data features

WPCF Aug. 1-3, Spherical harmonic decomposition Q OUT   Q SIDE Q LONG Q  : [0,2  ]  : [0,  ] Z.Ch., Gutierrez, Lisa, Lopez-Noriega, nucl-ex/

WPCF Aug. 1-3, Non-femtoscopic correlations : STAR Baseline problem is increasing with decreasing multiplicity STAR preliminary

WPCF Aug. 1-3, MC simulations ‘ad-hoc’ parameterizations OPAL, NA22, … Common approaches to „remove” non-femtoscopic correlations A possibility: energy-momentum conservation? –must be there somewhere! –but how to calculate / model ? (Upon consideration, non-trivial...) “zeta-beta” fit by STAR [parameterization of non-femtoscopic correlations in A lm ’s]

WPCF Aug. 1-3, GenBod Phase-Space Event Generator

WPCF Aug. 1-3, GenBod: Phase-space sampling with energy/momentum conservation F. James, Monte Carlo Phase Space CERN REPORT (1 May 1968) Sampling a parent phasespace, conserves energy & momentum explicitly –no other correlations between particles ! Events generated randomly, but each has an Event Weight WT ~ probability of event to occur P  conservation Induces “trivial” correlations (i.e. even for M=1) Energy-momentum conservation in n-body system

WPCF Aug. 1-3, N=9,  K =0.5 GeV, LCMS Frame - no cuts The shape of the CF is sensitive to: kinematic cuts frame particle multiplicity total energy : √s

WPCF Aug. 1-3, Findings Energy and Momentum Conservation Induced Correlations (EMCICs) “resemble” our data so, EMCICs... on the right track... But what to do with that? –Sensitivity to  s, multiplicity of particles of interest and other particles –will depend on p 1 and p 2 of particles forming pairs in |Q| bins  risky to “correct” data with Genbod... Solution: calculate EMCICs using data!! –Danielewicz et al, PRC (1988) –Borghini, Dinh, & Ollitraut PRC (2000) we generalize their 2D p T considerations to 4-vectors

WPCF Aug. 1-3, k-particle distributions w/ phase-space constraints single-particle distribution w/o P.S. restriction k-particle distribution (k<N) with P.S. restriction observed P - total 4-momentum

WPCF Aug. 1-3, Central Limit Theorem k-particle distribution in N-particle system For simplicity we will assume that all particles are identical (e.g. pions) and that they share the same parent distribution (same RMS of energy/momentum) Then, we can apply CLT (the distribution of averages from any distribution approaches Gaussian with increase of N) Can we assume that E and p are not correlated ?

WPCF Aug. 1-3, E - p correlations?

WPCF Aug. 1-3, EMCICs in single-particle distribution What if all events had the same “parent” distribution f(p), and all multiplicity (centrality) dependence of spectra was due just to loosening of P.S. restrictions as N increased?

WPCF Aug. 1-3, EMCIC’s in spectra For N 

WPCF Aug. 1-3, EMCICs: Ratio of particle spectra Phys. Rev. D 74 (2006) 200GeV, STAR p T spectra from GenBod Simulations: Ratio of p T spectra for N=9 and N=18. Ratio of p T spectra in for the lowest and the highest multiplicity events

WPCF Aug. 1-3, k-particle correlation function Dependence on “parent” distrib f vanishes, except for energy/momentum means and RMS 2-particle correlation function (1 st term in 1/N expansion) 2-particle correlation function

WPCF Aug. 1-3, particle CF (1st term in 1/N expansion) “The p T term” “The p Z term” “The E term” Names used in the following plots

WPCF Aug. 1-3, EMCICs An example of EMCICs: Effect of varying multiplicity Same plots as before, but now we look at: p T ( ), p z (  ) and E (  ) first-order terms full (  ) versus first-order (  ) calculation simulation ( ) versus first-order (  ) calculation

WPCF Aug. 1-3, N=9,  K =0.9 GeV, LabCMS Frame - no cuts

WPCF Aug. 1-3, N=18,  K =0.9 GeV, LabCMS Frame - no cuts

WPCF Aug. 1-3, Findings  CF from GenBod (as well as EMCICs) depends on –multiplicity –frame –energy of the collisions  first-order and full calculations agree well for N>9 –will be important for “experimentalist’s recipe”  Non-trivial competition/cooperation between p T, p z, E terms –all three important  p T1 p T2 term does affect “out-versus-side” (A 22 )  p z term has finite contribution to A 22 (“out-versus-side”)  calculations come close to reproducing simulation for reasonable (N-2) and energy

WPCF Aug. 1-3, N  =12,N K =3,N p =3,  K =0.9 GeV, LCMS Frame - no cuts

WPCF Aug. 1-3, The Experimentalist’s Recipe Fitting formula: - average of X over # of pairs for each Q-bin

WPCF Aug. 1-3, EMCIC’s FIT: N=18,  K =0.9GeV, LCMS

WPCF Aug. 1-3, The Complete Experimentalist’s Recipe or any other parameterization of CF 9 fit parameters - 4 femtoscopic - normalization - 4 EMCICs Fit this …. or image this …

WPCF Aug. 1-3, M inv distribution w/ background subtraction N=18

WPCF Aug. 1-3, EMCICs contribution to v 2 for v 2 n=2 no contribution to v 2 from 1/N term contribution to v 2 from 1/N 2 term contribution to v 2 from 1/N 3 term

WPCF Aug. 1-3, Non-id correlations (Resonance contrib.)

WPCF Aug. 1-3, Non-id correlations GeV)

WPCF Aug. 1-3, Summary understanding particle spectra, two-particle correlations, v 2, resonances in small systems –important physics-wise –should not be attempted until data fully under control Restricted P.S. due to energy-momentum conservation –sampled by GenBod event generator –generates EMCICs [femtoscopy : quantified by A lm ’s] –stronger effects for small multiplicities and/or  s Analytic calculation of EMCICs –k-th order CF given by ratio of correction factors –“parent” only relevant in momentum variances –first-order expansion works well for N>9 –non-trivial interaction b/t p T, p z, E conservation effects Physically correct “recipe” to fit/remove EMCICs [femtoscopy] –4 new parameters, large |Q|

WPCF Aug. 1-3, Thanks to: Alexy Stavinsky & Konstantin Mikhaylov (Moscow) [suggestion to use Genbod] Jean-Yves Ollitrault (Saclay) & Nicolas Borghini (Bielefeld) [original correlation formula] Adam Kisiel (Warsaw) [don’t forget energy conservation] Ulrich Heinz (Columbus) [validating energy constraint in CLT] Mark Baker (BNL) [local momentum conservation] Dariusz Miskowiec (GSI) [multiply (don’t add) correlations]