Statistical Fragmentation in pp & ep & ee Collisions Karoly - PowerPoint PPT Presentation

Statistical Fragmentation in pp & ep & ee Collisions Karoly Urmossy 1,3 , Zhangbu Xu 1,2 , T. S. Biró 3 , G. G. Barnaföldi 3 1, 3, 2, Phys. Depat.,BNL, USA RCP, Hungary August 2016, USTC, Hefei Anhui China 1: e-mail: karoly.uermoessy@cern.ch

Motivation ● Goal Hadronisation inside fat jets ● Proposed model Statistical Model ● Suggestion Parametrise fragmentation functions as 2 ] D [ x = 2 P μ jet p h μ 2 = M jet , Q 2 M jet Energy fraction the Fragmentation hadron takes away scale: jet mass in the frame co-moving with the jet

Outline ● 3D Statistical Jet fragmentation model hadron distributions in jets in e + e - , ep, pp collisions ● Applications - Transverse momentum spectra in pp collisions from a pQCD parton model calculation - Spectra & anisotropy of hadrons in heavy-ion collisions

K. Urmossy – Hadronisation @ LEP, RHIC & LHC e + e - annihilations in the factorized picture Ideal world: 2 identical jets: width: q q 2 ≈ 0 ≪ M jet ∼ √ p q q, ̄ q =( √ s / 2 , 0,0, ± √ s / 2 ) p μ M ∼[ 0.1 − 0.5 ] √ s Problem: P 2 ~ 0 quark produces a heavy jet of mass 0 ● energy fraction of the p h x = hadron takes away from √ s / 2 the energy of the jet: Q ∼ √ s ● fragmentation scale:

K. Urmossy – Hadronisation @ LEP, RHIC & LHC Real world: the 2 jets are not identical heavy Energy-momentum conservation: light μ =( P 0 , 0,0, ∣ P ∣ ) P 1 M 1 M 2 μ =( √ s − P 0 , 0,0, − ∣ P ∣ ) P 2 ⃗ −⃗ P P Problems: 0 x = p h P 0 ≠ ( √ s / 2 ) ● the energy of a jet , so is no longer the √ s / 2 energy fraction , the hadron takes away from the energy of the jet. √ s / 2 ● fragmentation scale is no longer

K. Urmossy – Hadronisation @ LEP, RHIC & LHC Real world: the 2 jets are not identical heavy Energy-momentum conservation: light μ =( P 0 , 0,0, ∣ P ∣ ) P 1 M 1 M 2 μ =( √ s − P 0 , 0,0, − ∣ P ∣ ) P 2 ⃗ −⃗ P P We propose to use: μ P μ ● the real energy fraction the hadron jet x = 2 p h takes away from the energy of the jet 2 M jet in the frame co-moving with jet: Q ∼ M jet ● the jet mass as fragmentation scale :

These new variables, x and M jet emergy naturally in a Statistical Fragmentation Model

K. Urmossy – Hadronisation @ LEP, RHIC & LHC Statistical jet-fragmentation The cross-section of the creation of hadrons h 1 , … , h N in a jet of N hadrons μ − P tot h 1 , … ,h N = ∣ M ∣ ( ∑ ) d Ω h 1 , … ,h N 2 δ ( 4 ) μ d σ p h i i If |M| ≈ constans , we arrive at a microcanonical ensemble : μ − P tot n − 2 = M h 1 , … ,h n ∼ δ ( ∑ ) d Ω h 1 , … ,h n μ μ ) 2n − 4 d σ p h i ∝ ( P μ P i Thus, the haron distribution in a jet of n hadron is x = P μ p μ n = fix ∝ Ω n − 1 ( P μ − p μ ) 0 d σ n − 3 , ∝ ( 1 − x ) p 3 p Ω n ( P μ ) 2 / 2 d M Energy of the hadron in the co-moving frame

K. Urmossy – Hadronisation @ LEP, RHIC & LHC The haron distribution in a jet of n hadron with total momentum P p T ⃗ P / 2 μ n = fix x = P μ p p 0 d σ n − 3 , M ∝ ( 1 − x ) p Z 3 p 2 / 2 d M E Problems P ( n )= ( r − 1 ) ̃ n + r − 1 n ( 1 − ̃ r p ) p ● The hadron multiplicity in a jet fluctuates pp → jets @ 7 TeV e - e + → h ± Refs.: Urmossy et.al., PLB , 701 : 111-116 (2011) Urmossy et. al., PLB , 718 , 125-129, (2012)

K. Urmossy – Hadronisation @ LEP, RHIC & LHC Averaging over n fluctuations The distribution in a jet with fix n x = P μ p μ n = fix 0 d σ n − 3 , ∝ ( 1 − x ) p 3 p 2 / 2 d M The multiplicity distribution P ( n )= ( r − 1 ) ̃ n + r − 1 p n ( 1 − ̃ p ) r The n-averaged distribution 1 −̃ p = A [ 1 + q − 1 x ] − 1 /( q − 1 ) 0 d σ τ = p p ( r + 3 ) ̃ τ d 3 p q = 1 + 1 r + 3

K. Urmossy – Hadronisation @ LEP, RHIC & LHC Jet mass fluctuations ⃗ P e+e- → 2 jet: both E and of the jets fluctuate heavy light M 1 M 2 ⃗ −⃗ P P ⃗ P pp collisions: jet is measured, E, M fluctuates K.U, Z. Xu, arXiv:1605.06876 thin fat jets

K. Urmossy – Hadronisation @ LEP, RHIC & LHC Problems ep → 2 jets of approximately same E nergy ρ( M ) and large rapidity gap: data? E 1 ≈ E 2 P 1 ≈− ⃗ ⃗ P 2 M 1 Jet 1 ⃗ M P 1 proton Target fragmen- rapidity gap tation ⃗ P 2 ∣⃗ P ∣ still fluctuates! M 2 Jet 2

K. Urmossy – Hadronisation @ LEP, RHIC & LHC μ x = P μ p We have a haron distribution, which depends on 2 / 2 M but, in case of available data, the jet E or P fluctuate: ⃗ P - pp collisions: is measured, E fluctuates ⃗ P - e + e - → 2 jet: both E and of the jets fluctuate ⃗ P - e + p → 2 jet: of the jets fluctuate So, we fit a characteristic/average jet mass and extract the scale dependence of the parameters of the model

Results

K. Urmossy – Hadronisation @ LEP, RHIC & LHC e + P → 2 jets → charged hadrons with large rapidity gap ∼ x p [ 1 + q − 1 x p ] − 1 /( q − 1 ) M 2 JET = E 1 + E 2 d σ τ dx p 2 E 1 = 1 ± 0.2 x p = 2p / M 2 JET E 2 Urmossy, Z. Xu, arXiv:1606.03208

K. Urmossy – Hadronisation @ LEP, RHIC & LHC Fitted average characteristic jet mass 2 E JET sin (ϑ cone ) 〈 M JET 〉 = M 0 + E JET / E 0 fitted Fitted average jet mass is of the order of that used in DGLAP calcs. 〈 M JET 〉 ∼ 2 E JET sin (ϑ cone ) Urmossy, Z. Xu, arXiv:1606.03208

K. Urmossy – Hadronisation @ LEP, RHIC & LHC Scale evolution of the fit parameters Urmossy, Z. Xu, arXiv:1606.03208

K. Urmossy – Hadronisation @ LEP, RHIC & LHC PP → jet → charged hadrons ⃗ P jet ⃗ P jet ϑ c T p h ∥ p h jet p μ ∼ [ 1 + q − 1 x ] − 1 /( q − 1 ) x = 2P μ p 0 d σ d 3 p τ 2 M jet Urmossy, Z. Xu, proc. of conf.: DIS2016 , arXiv:1605.06876

What we have: ● an approximate formula for the fragmentation function which does not solve DGLAP D ( x ) ∼ [ 1 + q − 1 x ] − 1 /( q − 1 ) τ ● Let us use this ansatz with scale dependent parameters q, T → q ( t ) , T ( t ) ● along with some other conjectures First step: in the Φ 3 theory

K. Urmossy – Hadronisation @ LEP, RHIC & LHC 1. The Φ 3 theory case Resummation of branchings with DGLAP 1 dz d g 2 = 1 /(β 0 t ) dt D ( x ,t ) = g 2 ∫ t = ln ( Q 2 / Q 0 2 ) , z P ( z ) D ( x / z ,t ) , x P ( z ) = z ( 1 − z )− 1 12 δ( 1 − z ) with LO splitting function : M. Grazzini, Nucl. Phys. Proc. Suppl. Let the non-perturbative input at starting scale Q 0 be: 64: 147-151, 1998 D 0 ( x ) = ( 1 + q 0 − 1 x ) − 1 /( q 0 − 1 ) τ 0 1 dz D ( x ,t ) = ∫ The full solution is z f ( z,t ) D 0 ( x / z ) x j! ( k − 1 − j ) ! x ln k − 1 − j [ x ] [ (− 1 ) j +(− 1 ) k x ] k − 1 k ∞ ( k − 1 + j ) ! b 1 f ( x ) ∼ δ( 1 − x ) + ∑ k ! ( k − 1 ) ! ∑ with k = 1 j = 0 − 1 ln ( t / t 0 ) b = β 0 Urmossy, Z. Xu, arXiv:1606.03208

K. Urmossy – Hadronisation @ LEP, RHIC & LHC 1. Approximations Let the FF preserve its form: D apx ( x ,t ) = ( 1 + q ( t )− 1 D ( x , 0 ) = ( 1 + q 0 − 1 x ) − 1 /( q ( t )− 1 ) x ) − 1 /( q 0 − 1 ) with τ 0 τ( t ) From DGLAP: D ( s,t ) = ̃ ̃ D ( s, 0 ) exp { b ( t ) ̃ − 1 ln ( t / t 0 ) b ( t ) = β 0 P ( s )} with Let us prescribe the approximations: a1 −α 2 ( t / t 0 ) − a2 q ( t )=α 1 ( t / t 0 ) ∫ D apx ( x ,t ) = ∫ D ( x ,t ) α 3 ( t / t 0 ) a1 −α 4 ( t / t 0 ) − a2 ∫ x D apx ( x ,t ) = ∫ x D ( x ,t ) = 1 τ 0 τ( t )= (by definition) α 4 ( t / t 0 ) − a2 −α 3 ( t / t 0 ) a1 ∫ x 2 D apx ( x ,t ) = ∫ x 2 D ( x ,t ) a 1 = ̃ a 2 = ̃ P ( 1 )/β 0 , P ( 3 )/β 0 Urmossy, Z. Xu, arXiv:1606.03208

K. Urmossy – Hadronisation @ LEP, RHIC & LHC Scale evolution of the fit parameters a1 −α 2 ( t / t 0 ) − a2 τ 0 q ( t )=α 1 ( t / t 0 ) τ( t )= α 4 ( t / t 0 ) − a2 −α 3 ( t / t 0 ) a1 α 3 ( t / t 0 ) a1 −α 4 ( t / t 0 ) − a2 2 ) t = ln ( M jet 2 / Λ Urmossy, Z. Xu, arXiv:1606.03208

K. Urmossy – Hadronisation @ LEP, RHIC & LHC 2. pp & ee collisions pp → jets @ LHC ( pT = 25–500 GeV/c) p Jet dN − b d z ∝ [ 1 − a ln ( 1 − z ) ] p Urmossy et.al. Phys. Lett. B , 718 , 125-129, (2012) e + e - annihilation @ LEP (√ s = 14–200 GeV) e + e – Urmossy et.al., Urmossy et. al., T. S. Biró et.al., Acta Phys. Polon. Phys. Lett. B , 701 , Acta Phys. Polon. B, Supp. 5 (2012) 363-368 111-116 (2011) 43 (2012) 811-820

K. Urmossy – Hadronisation @ LEP, RHIC & LHC 2. Application in a pQCD calculation π + spectrum in pp --> π + X @ √s=7 TeV (NLO pQCD) AKK vs. Tsallis as Frag. Func. + ( z )∼( 1 +( q i − 1 ) z / T i ) π − 1 /( q i − 1 ) D p i Barnaföldi et. al., Proceedings of the Workshop Gribov '80 (2010)

K. Urmossy – Hadronisation @ LEP, RHIC & LHC 3. How about the soft part? The power of the spectrum changes drastically at pT ~ 6 GeV/c. − 6.08 ∼ p T soft − 13.7 Tsallis ∼ p T hard Boltzmann T = 293 MeV A hard + soft model: hard soft E dN = E dN + E dN 3 p 3 p 3 p d d d