Intermittency study from the ATLAS data in proton-proton and ion-ion collisions

1. Intermittency study from the ATLAS data in proton-proton and ion-ion collisions to search possible hidden dynamical processes François Vazeille 15 November 2010  Introduction  Scaled factorial moments  Intermittency • A new use of intermittency in ATLAS • from the proton-proton and lead-lead collisions • First works in the M2/M1 context • and/or next studies 1/21

2. Introduction • Use of a mathematical method based on multiplicity measures, characterized by 3 properties: - It does not request to measure total multiplicities: a part of phase space is enough. - It is fully insensible to statistical fluctuations. - The method sensitivity is increased by increasing the segmentation: for example, a double segmentation ( ) is more efficient than a single one () or () . • What is the intermittency ? • Well known in hydrodynamics to explain the intermittent behavior • of oscillators (Y. Pomeaux and P. Manneville) •  regular oscillations interrupted by chaotic bursts. • (Seen in convection experiments in liquid helium with turbulence effects, • T. Bergé et M. Dubois from CEA). 2/21

3. Relatively new in quantum mechanics and high energy physics: • A. Bialas and R. Peschanski [Nuclear Physics B273 (1986) 703] •  explanation of 2 cosmic single events of the JACEE collaboration (balloon) • having a very high multiplicity [Phys. Rev. Lett. 50 (1983) 2062] Centre of mass  distribution of charged particles recorded in a stack of nuclear emulsions with plastic targets in between. Collision Si on Ag or Br 5 TeV/nucleon 1010 ± 30 charged particles Collision Ca on C 200 TeV/nucleon 760 ± 30 charged particles 3/21

4.  Classical analysis: 2 models were used to interpret the data: - MCM : Multi Chain Model (Cumulative superposition of each elementary collision). - WNM: Wounded-Nucleon Model (Linear superposition of meson productions from wounded nucleons)  MCM would be better since it does reproduce rather well the distributions, but other distributions (High Pt, etc.) are not very well explained.  What is seen ? a large number of particles produced in a finite volume of the collision  That suggests extreme conditions, like a large density of energy during the collision, and why not some exotic processes such as the Quark Gluon Plasma ? A new approach was proposed by Peschanski and Bialas to analyze the distributions in large multiplicity events, from the investigation of cascading phenomena and turbulent behavior, showing intermittent bursts over pure statistical fluctuations.  The study of scaled factorial moments as function of the resolution , showing bin-bin correlations at various scales of the data, while simulations are not sensitive to these parameters. 4/21

5. Example of the Si-Ag/Br event, where we have just seen one simulation fitting rather well the distributions, BUT it is no longer the case of factorial moments (that we will define later), here, at the order 5 : ●and ◦: Cosmic events for 2  ranges x : Simulation with bins smaller and smaller. ●The true data exhibit a genuine behavior in the rapidity distribution as shown by the factorial moments, but not the MCM simulation. ● The slope of this fit is the ″Intermittency index″. 5/21

6. Scaled factorial moments  Interest and definition. km Multiparticle event (s) Bin y = y/M N = ∑ km Total number of events in the phase space. M m=1 Bin number m 1 2 M y Phase space Classical problem of probabilities: frequency km/N  probability pm or particle density The law The measures ? N   Q(k1, k2, … kM)  P(p1, p2, … pM) Daughter distribution of frequencies Mother distribution of probabilities 6/21

7. Let us remind:  Bernoulli law for 1 parameter A discrete law of probability: 1 for a success, O for a failure. p if x=1 q=1-p if x=0 0 in other cases B(X=x) = E(X) = p Variance = p(1-p) • Moment at the order i i>0, X = random variable • mi(X) = [Xi] Expectation value or mean • mi(f) = ∫xi f(x) dx where f(x) is a probability density. 7/21

8. 3 tricks : 1st trick: use of the multinomial law of Bernoulli B ●The condition km/N  pm is never satisfied even though we have very high N, and the more as we want using reasonably small rapidity gap. N   Q = P B ● But, it is possible to show that Q(k1, k2, … kM) = ∫ P(p1, p2, … pM) dp1 dp2, … dpM x B(k1, k2, … kM │p1, p2, … pM) M M M km with B(…│…) = N! ∏ pm /km! and normalizations ∑ pm = 1 ∑ km = 1 m=1 m=1 m=1 (easy to see with M=1) ″smearing″ effect of B, in particular for small N B k k1 km kM p1 pm pM with statistical noise without statistical noise 8/21

9. 2nd trick: use of scaled factorial moments Instead of using the means, in a complicated multidimensional phase space  km = ∫ P(p1, p2, … pM) dp1 dp2, … dpM We are interested to deviations with respect to the means, by taking the moments at the order i. ● Scaled moments Ci of the distribution P(p1, p2, … pM) M Ci   (M pm)i  = ∫ dp1 dp2, … dpM P(p1, p2, … pM) (1/M) ∑ (M pm)i m=1 not calculable unknown law The Ci have problems because of the unknown law P(p1, p2, … pM). ● Scaled factorial moments Fi of the distribution Q(k1, k2, … kM) Fi = Mi-1∑ km (km- 1) … (km-i+1) / N (N-1) … (N-i+1) F1 = 1  M Fi = 1 for M=  i Fi= Mi-1∑ km (km- 1) … (km-i+1) / N (N-1) … (N-i+1)  where the brackets … denote the averaging over a selected class of events, there, the calculation becomes possible. 9/21

10. 3rd trick: coming back to Q = P B • Fi    Ci  • Daughter Mother we obtain The scaled factorial moments  Fi  are a good estimator of the  Ci , besides, they wash the statistical noise. 10/21

11.  Practical formulae Setting Gi,m = km (km- 1) … (km-i+1) Horizontal analysis: ∑ bins Event 1 ∑ Gi,m/M M Fih = (1/N0) ∑ N/Mi N0 N = ∑ km over y N = ∑ N/N0 M Vertical analysis: ∑ events Event 2 N0 ∑ Gi,m/N0 N0 Fiv = (1/M) ∑ kmi M … km = ∑ km/N0 N0 N0 = Nb of events In principle: Fih  Fiv 11/21

12. Intermittency Benoît Mandelbrot (20 November 1924, 14 October 2010) • Theoretical approach • Several possible approaches: choice of the Mandelbrot intermittency • [B. Mandelbrot J. Fluid. Mech. 62 (1974) 331] •  Here, intermittency at high energies: rapidity fluctuations at different scales, • but without occupying the full available phase space. Cascade like a Cayley tree structure, with branches: ● Each vertex has a weight Wn. ● The path m (in red) is uniquely associated to the bin m. 1 2 3 m 1 M Multiplicity density profile m = m ∏ W(n) Path m 3 2 1 W = random number following a probability law r(W) 12/21

13. Example:  model of the random cascade. - 1 1 + - n steps - - - + 2 - - - - + + - + 1 M 1 m Limit of the microscopic zone: after there are statistical fluctuations. ● At each step, the range is divided in l parts (here: l = 2). ● r(W) is the probability to have: - a dynamical increase W+ > 1 of the density, - a dynamical decrease W-< 1 of the density, but in average W = ∫ r(W) W dW = 1. 13/21

14. After n steps  1: M = ln bins with a width y = y/M Ci   (Mpm)i fi y y  Fi  = ∏ W(n) n with fi = Log(∫r(W) wi dw)/log l (where the mutual independence of the W’s was used). Comment: the result of this toy-model can be generalized. fi is called the intermittency index. • Property of self-similarity • typical of systems having fractal or multi-fractal properties  the scale • with Di = 1 – fi / (i-1) is the mean fractal dimension. • Result independent from the number n of steps and from the law r(w). • Approximate relation in between the fi • 2 fi / i(i-1)  constant 14/21

15.  Experimental observations Log  Fi  Dynamical fluctuations = intermittency Resonance decays Plateau (except if zero …) fi - Log y 3 domains ~1 Resolution y0 … long range Correlations … short range fi characterizes the intermittency strength, with a hierarchy of slopes in y or  • firises with i. • fi(e+e-) > fi(hh) > fi(hA) > fi(A1A2) at a given s (collective effects for A1A2). • fi(hh) decreases when s rises except for fi(A1A2) and i4 if QGP is created. • No generator is able to account for intermittency. • From R. Peschanski, FRITIOF, VENUS, JETSET, PYTHIA … • are like ″CANADA DRY″. • 5. A 2-dimensional binning is more efficient than a single one • - fi(y,) > fi(y) or fi(). • - M bins y  M bins  = M bins. fi(y,) y  y   Fi  = 15/21

16. Examples: ● KLM results [Phys. Rev. Lett. 62 (1989) 733] Data Simulation ● Explanation of many phenomena: - Fractal structure of QCD jets. - Pertubative cascade in QCD. - Nuclei decays out of equilibrium. - QGP evolution versus time. - … 16/21

17.  Improvements of the formalisms ●Other indexes can be studied Fractal dimension of Renyi Di = fi/i-1 - If Di = constant i  monofractal system. - If Di rises with i  multifractal system, (in connection with chaotic phenomena out of equilibrium). Formalism of the stable law of Levy: 1 single parameter = Levy index. iµ - i f2 i -1 2µ - 2 Di = µ is the Levy index Factorial correlators … • Other definitions of scaled factorial moments (Generalized scaled factorial • moments), advices on the selections of data … •  shown in recent bibliography. 17/21

18. A new use of intermittency in ATLAS from the proton-proton and lead-lead collisions • Principle of the study: • ●To choose the multiplicity as starting variable for the study. • ●To correlate the evolution of the intermittency index (or any other index) • with a global variable in direct relation with the collision strength. • For example: ET, total multiplicity… … and also fitting my curiosity  What kind of physics? ● ″Exotic processes″ (It is the ATLAS terminology) requesting extreme energy conditions (or rather energy density) and so a priori containing short range correlations. ● If possible, study made at various s in order to show possible energy threshold effects and 2 extreme environments (p-p and Pb-Pb), taken separately (p or ion) or compared. 18/21

19. Identification of 2 exotic domains requesting extreme conditions, • already described in ATLAS Notes but never till now using the intermittency. ● In lead-lead collisions: Search of Quark Gluon Plasma (QGP)  central collision of 2 ions. Not new: very often considered at the time of the ″Intermittency-mania″ by many authors, Peschanski/Bialas and many many others  QGP possibly made of plasma droplets with cascade processes. … and also FV in several talks (CERN, Lyon, Paris, USA) Example: Quark Matter ‘91, Gatlinburg, Tennessee, USA ″Intermittency and dimuon as combined tools to search QGP″ … with the study of the evolution of the Levy index. ● In proton-proton collisions (and also lead-lead collisions): Search of gravitational effects via the formation of ″Micro black holes″ or “String balls″ or ″p-branes″  frontal collision of 2 partons (~Impact parameter < Schwarzschild radius) • - Based on the generally accepted idea that the scale of quantum gravity • could be as low as about one TeV. • Depending from the mass of black holes, • they could be directly produced or via string balls as progenitors • or not produced being replaced by string balls or p-branes. 19/21

20.  Comments about the multiplicity-transverse energy correlation ● Well established in heavy ion collisions Radius R Depending of b with respect to R: peripheral or central collision with multiplicity correlated to ET. Impact parameter b ● Could be true … or broken at the parton level because of gravitons. Bulk • If gravitons escape in the bulk • Lack of measured ET • so ET is no longer signing the strength • of the collision. 3-brane Schwarzschild radius or equivalent 20/21

21. First works in the M2/M1 context and/or next studies • Discovery of ATLAS and of the high energy physics. • First look of the corresponding exotic physics. • Old and recent bibliography on the intermittency. • Compilation and checks of formulae. • First analyses using published ATLAS data (p-p at 900 GeV). • Preparation of next steps: access to ATLAS data in p-p and Pb-Pb, • and at various collision energies. 21/21