Modified Stochastic Branching Model for Supersymmetric Particle Branching process

Transcription

1 Introduction QCD jets as Branching Process Supersymmetric Particle Branching Summary Modified Stochastic Branching Model for Supersymmetric Particle Branching process Supervisor: Prof Chan Aik Hui, Prof Oh Choo Hiap Master (Research) Zhang Yuanyuan National University of Singapore April 18, /40

2 Introduction Particle Physics Multiplicity Distribution Hadronic Collisions Overview QCD jets as Branching Process Jet calculus to Branching Properties Quark and Gluon Initiated Jets Supersymmetric Particle Branching Supersymmetry Pure SUSY Initiated Jets SUSY plus Ordinary Jets Summary 2/40

3 Introduction QCD jets as Branching Process Supersymmetric Particle Branching Summary Introduction 3/40

4 Particle Physics Theory : Standard Model Experiments : High energy particle collisions, Ultra high energy cosmic ray detections Comparison of observables from experiments and theory validation or beyond Standard Model Figure: Elementary Particles 4/40

5 Multiplicity Distribution straightforward observable, the probability p n for producing n particle in a collision process Probability Distribution contains information of dynamical mechanism and statistics of multi-particle production Figure: pp collision charged particle distributions at s =09, 236, 7TeV with CMS detector 5/40

6 Hadronic Collisions Hadrons are made up of partons (quarks and gluons) Collision Process Before Scattering: Initial state parton shower Scattering: hard scattering + semi-hard, soft scattering (underlying events) After Scattering: final state parton shower and hadronization Approximation In high energy collision, ignore the initial state radiation The hadronization : Local Parton-Hadron Duality (LPHD) Specta of Partons before hadronization Spectra of Hadrons 6/40

7 Introduction QCD jets as Branching Process Supersymmetric Particle Branching Summary QCD jets as Branching Process 7/40

8 Introduction QCD jets as Branching Process Supersymmetric Particle Branching Summary Jet calculus to Branching Properties Jets High energy quark or gluon (carry color charge), create particles around them, form a spray of collimated(parallel) particles Figure: Jets 8/40

9 Jet calculus to Branching Properties [Konishi,1979] establishajet Calculus algorithm based on tree diagram understanding of partons evolution equation Tree diagram, in the Leading Logarithm approximation (LLA), as branching process Virtuality (Q 2 = p µ p µ m 2 0 ) strongly ordered Figure: Tree Diagram 9/40

10 Branching Equation [Konishi,1979] get two differential equation for single Quark- and Gluon- jets multiplicity distribution generating function G and Q G = A(G 2 G)+B(Q 2 G) Y Q = Ã(QG Q) Y G = n,m=0 xn g x m q p g nm the p g nm is the probability for one gluon branch into n gluons and m quarks, same for Q Three basic branching process g g + g, q q + g and g q + q, with branching probability A, Ã and B respectively Evolution parameter Y act as time parameter 10 / 40

11 Introduction QCD jets as Branching Process Supersymmetric Particle Branching Summary Generating function The multiplicity distribution p n (t) can be collected into one expression, the generating function: P (x, t) =p 0 + p 1 x + p 2 x p n x n + = n=0 p n (t)x n One can get multiplicity distribution from GF by Taylor expansion 11 / 40

12 Approximation for Branching Equation Following [Giovannini,2008] way to calculate branching probability A = C A ϵ, à = C F ϵ A = C A ϵ, à = C F >B= N f ϵ 3 and B = N f 3 The ϵ small,, omit process B : g q + q Only consider process A : g g + g and à : q q + g In A : g g + g and à : q q + g, quark number do not change Only consider gluon number n for multiplicity contribution 12 / 40

13 Stochastic Branching Process Here we introduce two branching process Simple Birth process: For whole system, the probability of producing a new individual is λn t, with current population n Example g g + g with λ = A Poisson process: For whole system, the probability of producing a new individual is ν t, this probability do not depend on current population n Example q q + g with ν = mã (m is the quark number), when only counting the gluon number, the quark can be seen as immigrants for gluon system 13 / 40

14 Quark initiated and Gluon initiated Jets Quark initiated (initial number of quarks m, gluon 0) : Poisson process à : q q + g plus simple birth process A : g g + g Initial condition P Q (x, Y = 0) = p 0 =1 P Q (x, Y ) =A(x 2 x) P Q(x, Y ) + Y x mã(x 1)P Q(x, Y ) [ ] 1 k P Q (x, Y )= (1 e AY )x + e AY ; k = mã A Gluon initiated (initial number of gluon a, quark 0): Simple birth process A : g g + g Initial condition P G (x, Y = 0) = p a x a = x a P G (x, Y ) = A(x 2 x) P G(x, Y ) Y [ x ] x a P G (x, Y )= (1 e AY )x + e AY 14 / 40

15 Generalized Multiplicity Distribution [Chew, 1987] propose a general distribution for initial quark number m and gluon number a, initial condition P GMD (x, Y = 0) = p a x a = x a P GMD (x, Y ) Y = A(x 2 x) P GMD(x, Y ) x The generating function is as follow x a + mã(x 1)P GMD(x, Y ) P GMD (x, Y )= [(1 e AY )x + e AY ] k+a [ 1 x =[ (1 e AY )x + e AY ]k (1 e AY )x + e AY =P Q (x, Y ) P G (x, Y ) ] a The generalized multiplicity distribution(gmd) is the convolution of gluon initiated jets MD and quark initiated jets MD 15 / 40

16 Evolution Parameter Y The evolution parameter act as time parameter Y = 1 2πb ln ln(q2 /Λ 2 ) ln(q 2 0 /Λ2 ) The primary parton with large virtuality Q 2 max (Y = Y max ), branch out particles The virtuality of particles decrease until reach hadronization threshold Q 2 0 (Y =0) Notice Y max Y =0with initial m quarks and a gluons Prove system with initial m quarks and a gluons evolve from Y max Y =0, have same MD as system with same initial condition from Y =0 Y max Notice all initial quarks and gluons have same virtuality t max or Y max (total evolution time) 16 / 40

17 Introduction QCD jets as Branching Process Supersymmetric Particle Branching Summary Supersymmetric Particle Branching 17 / 40

18 Introduction QCD jets as Branching Process Supersymmetric Particle Branching Summary Supersymmetry New symmetry between fermions and bosons, all ordinary parton has its super-parterners, differ by half spin Spontaneous broken symmetry, mass difference between normal particle and SUSY RparityR P =( 1) 3(B L)+2s conserve, process involve even number of SUSY particles Lightest Supersymmetric Particle(LSP), can not decay 18 / 40

19 SUSY Branching Process Probability Following the same way [Giovannini,2008] of ordinary parton to calculate SUSY branching probability, involving squark q and gluino g Approximation, only choose terms with 1 ϵ : I = C F ϵ : q qg J = C A ϵ : g gg SUSY branching q q + g C F /2 q q + g C F /ϵ q q + g C F g q + q N f /3 g g + g 2C A /3 g g + g C A /ϵ g q + q N f /2 Table: SUSY Branching Probability A ba 0 19 / 40

20 Pure SUSY Initiated Jets Branching Three basic processes I : q qg, J : g gg and A : g gg The SUSY particle branching consist of two phases [Berezinsky,2001]: Phase One: SUSY particle start branching with high virtuality Q 2 max (or Y max ), until their virtuality reach the mass scale of SUSY particle M 2 SUSY (Y = Y SUSY) The SUSY particles on-shell decay into ordinary parton and LSP χ 0 1 q q + χ 0 1 g q + q + χ 0 1 Phase Two: The ordinary partons decayed from SUSY particles continue branching until hadronization threshold (Q 2 0 and Y =0) 20 / 40

21 Modified Stochastic Model for SUSY Branching Two phase simple birth and Poisson process can describe SUSY parton branching, Poisson process parameter ν change Phase One [Y =0,Y 1 = Y max Y SUSY ] Initial squark q number l, gluino g number p (of the same Y max ), gluon number 0, Initial condition P SUSY (x, Y )=p 0 =1 P SUSY Y = A(x 2 x) P SUSY x +(li + pj)(x 1)P SUSY Phase Two [Y 1,Y max ] The phase one equation gives generating function P SUSY (x, Y 1 ) at Y = Y 1, it is the initial condition for phase two The l squarks and p gluinos change to l +2p quarks P SUSY Y = A(x 2 x) P SUSY x +(l +2p)Ã(x 1)P SUSY 21 / 40

22 Generating Function and Multiplicity Distribution for SUSY Jets Solve the two phase equation, we get generating function P SUSY (x, Y )= [(1 eλ(y Y 1) )x + e λ(y Y 1) ] k 1 k 2 [(1 e λy )x + e λy ] k 1 with λ = A, k 1 = distribution li + pj A and k 2 = (l +2p)Ã,andmultiplicity A p SUSY (n, Y )= e λ(y Y 1)(k 1 k 2 ) (1 e λy ) n (e λy ) k 1 (n + k 1 1)! (k 1 1)!n! 2 F 1 (k 2 k 1, n, 1 k 1 n, eλy e λy 1 e λy 1 ) 2F 1 (a, b, c, d) is the hypergeometric function 22 / 40

23 SUSY plus Ordinary Jets More general case, intial partons :SUSY and ordinary partons The I : q qg, J : g gg, Ã : q qg and A : g gg four process happen The q, g and q(initial number l, p and m), undergoes a two phase Poisson plus simple birth process, its generating function P Q GQ (x, Y )= [(1 eλ(y Y 1) )x + e λ(y Y 1) ] k 1 k 2 [(1 e λy )x + e λy ] k 1 li + pj + mã (l +2p + m)ã with different k 1 = and k 2 = A A The gluon (initial number a) branchingdescribedby [ ] x a P G (x, Y )= (1 e AY )x + e AY 23 / 40

24 Generating Function for SUSY plus Ordinary Jets Following the GMD generating function composition: P GMD (x, Y )=P G (x, Y ) P Q (x, Y ) We know the generating function for mix initial partons (SUSY and ordinary partons) P mix (x) =P Q GQ (x, Y ) P G (x, Y ) = [(1 eλ(y Y 1) )x + e λ(y Y 1) ] k 1 k 2 [(1 e λy )x + e λy ] k 1 [ x (1 e AY )x + e AY ] a due to the independence of all individual jets 24 / 40

25 Multiplicity Distribution for SUSY plus Ordinary Jets We can get the SUSY included multiplicity distribution (SIMD) by Taylor expansion of generating function p SIMD (n, Y )=e λ(y Y 1)(k 1 k 2 ) (e λy ) k 1+a (1 e λy ) n a (n + k 1 1)! (k 1 + a 1)!(n a)! 2 F 1 [k 2 k 1, n + a, 1 k 1 n, eλy e λy1 e λy 1 ] Consistency check: 1 When initial quark number m =0gluon number a =0,this distribution reduced to pure SUSY MD 2 When Y = Y 1 or k 1 = k 2 = k, whichmeansnosusyparticle involved, this distribution reduced to GMD 25 / 40

26 Data Fitting Simply the expression for SIMD, with m = e λy 1 l = e λy p SIMD (n, Y )= ( l m )k 1 k 2 (1 1 l )n a ( 1 l )k 1 (n + k 1 1)! (k 1 + a 1)!(n a)! 2 F 1 (k 2 k 1, n + a, 1 k 1 n, l m l 1 ) Constraints: 1 Evolution parameter ordering 0 <Y 1 <Y 2 In SUSY-QCD, C F = C A = N c (N c number of colors), I = J = A = Ã, k li + pj + mã 1 = = l + p + m, A k 2 = l +2p + m, weknowk 1 <k 2 26 / 40

27 Fitting Parameter Table Table: SUSY included Multiplicity Distribution best fit parameters and χ 2 /dof for CMS s =09, 236, 7TeV data at different pseudorapidity intervals η < 05, 10, 15, 20, 24 s(tev) ηc m l k 1 k 2 a χ 2 /dof(dof) (17) (34) (51) (54) (62) (17) (34) (44) (54) (64) (35) (64) (89) (109) (121) 27 / 40

28 Fitting Plots s =09TeV P(n) 3 10 η < CMS 09 TeV η < η < η <10 10 η < n Figure: Charged particle multiplicity distribution measures by CMS at s =09TeV, fitted with SUSY included model 28 / 40

29 Fitting Plots s =236TeV P(n) 3 10 η < CMS 236 TeV η < η < η <10 10 η < n Figure: Charged particle multiplicity distribution measures by CMS at s =236TeV, fitted with SUSY included model 29 / 40

30 Fitting Plots s =7TeV P(n) 3 10 η < CMS 7 TeV η < η < η <10 10 η < n Figure: Charged particle multiplicity distribution measures by CMS at s =7TeV, fitted with SUSY included model 30 / 40

31 Introduction QCD jets as Branching Process Supersymmetric Particle Branching Summary Summary 31 / 40

32 Summary Supersymmetric particle branching model established, generating function and multiplicity distribution derived General case of initial partons contain both Supersymmetric and ordinary partons are investigated The SUSY included multiplicity distribution(simd) is derived and fitted with current data Future Work: This SIMD can be used to fit with Higher energy data Ultra-high energy cosmic ray data Monte Carlo Calculation data[berezinsky,2001] 32 / 40

33 Introduction QCD jets as Branching Process Supersymmetric Particle Branching Summary Thanks for Listening! 33 / 40

34 References A Giovannini (1979) QCD JETS AS MARKOV BRANCHING PROCESSES Nuclear Physics B 2-3, K Konishi, A Ukawa, G, Veneziano (1979) Jet calculus: a simple algorithm for resolving QCD jets Nuclear Physics B 1, V Berezinsky, M Kachelriess (2001) Monte Carlo simulation for jet fragmentation in SUSY QCD Physical Review D 3, A Giovannini, R Ugoccioni (2008) Monte Coherence and Incoherence in QCD Jets Dynamics (QCD Jets and Branching Processes) String Theory and Fundamental Interactions CK Chew, D Kiang, H Zhou (1987) Ageneralizednon-scalingmultiplicitydistribution Physics Letters B 3-4, / 40

35 Introduction QCD jets as Branching Process Supersymmetric Particle Branching Summary Backups Concept Pseudorapidity η a measure of the angle of particle relative to the beam axis η = ln[tan( θ 2 )] Figure: Pseudorapidity and angle relation 35 / 40

36 Backups Leading Logarithm Approximation In Leading Logarithm Approximation, in the parton density function G a h (x, Q2 ) G a h (x, ln Q2 ) n=0 f n (x)( α s π ln Q2 ) n Only consider the terms ( α s π ln Q2 ) n is the LLA Energy conservation and momentum conservation will be considered in Next to Leading Logarithm Approximation and higher order approximation 36 / 40

37 Backups Branching Probability The branching probability A = A gg 0, à = Agq 0 and B = Aqg 0 is the zeroth moment of parton decay probability function P ba (x) A ba n = 1 0 dxx n P ba (x); P ba (x) =P (a b(x)+c(1 x)) [Giovannini,2008] propose a way to calculate branching probability, with a fixed cutoff for x, x min = ϵ and x max =1 ϵ (fix infrared divergence problem) For P gq (x) =C F 1+(1 x) 2 x à = A gq 0 = 1 ϵ ϵ P gg (x)dx with substitution ϵ =( 2lnϵ ) 1,wehaveà = C F ϵ Similarly, A = C A ϵ, B = N f 3 (C A C F color factors, N f flavor number ) 37 / 40

38 Backups Convolution Two non-negative random variable X and Y,independent with each other If we define S = X + Y S is the convolution of X and Y : For S = n, it comprises (X =0,Y = n), (X =1,Y = n 1),,(X = n, Y =0) The generating function of S is the multiplying of GF of X and Y The independence of quark and gluon jets ensure that P GMD (x, Y )=P Q (x, Y ) P G (x, Y ) 38 / 40

39 Backups Prove the system from Y = Y max to Y =0 Start from scratch, the probability of having n gluons at Y Y p n (Y Y )=p n 1 (Y )A(n 1) Y + p n (t)(1 An Y ) The differential equation: dp n(y ) dy p n (Y ) p n (Y Y ) = lim = A(n 1)p n 1 (Y ) Anp n Y 0 Y Then solve it, we get the generating function at Y =0 [ ] 1 a [ x P (x, 0) = 1 (1 x 1 )e AY = max (1 e AY max )x + e AY max ] a P (x, 0) is the same as P (x, Y max ) for system from Y =0to Y = Y max 39 / 40

40 Backups Hypergeometric Function Evaluation The biggest difficulty with our fitting is the Hypergeometric Function 2 F 1 (a, b, c, d) evaluation We use CERN-ROOT-MINUIT to do the fitting, the math library in ROOT is GSL library (open source), gives error message when d goes near 1 (its singularity) In mathematica, the hypergeometric function can be calculated fast and no error message near singularity We learn how to call mathematica from C code under ROOT environment, thanks to CERN-ROOT discussion forum! 40 / 40