recombination and fragmentation of hadrons from a dense parton phase

Rainer J. Fries Recombination & Fragmentation #1

Rainer J. FriesUniversity of Minnesota

Recombination and Fragmentation of Hadrons from a Dense Parton Phase

Recombination and Fragmentation of Hadrons from a Dense Parton Phase

R.J. Fries, C. Nonaka, B. Müller & S.A. Bass, PRL 90, 202303 (2003) R.J. Fries, C. Nonaka, B. Müller & S.A. Bass, nucl-th/0305079, JPG t.a. R.J. Fries, C. Nonaka, B. Müller & S.A. Bass, PRC 68, 044902 (2003) C. Nonaka, R.J. Fries & S.A. Bass, nucl-th/0308051, submitted to PLB

Talk at the RIKEN Workshop on Flow and Collective Phenomena

BNL, November 19, 2003


OutlineOutline

• Motivation: hadron spectra, ratios and flow at RHIC

• The recombination idea• Calculations using recombination +

fragmentation

• v2 scaling


Jet quenching: suppression of hard particle production

Jet quenching: suppression of hard particle production

• Central Au+Aucollisions: suppression of pions by a factor ~5• Suppression of hard (pQCD) hadron production


Baryon enhancement at high pt

Baryon enhancement at high pt

where does the large proton over pion ratio at high pt come from?

Why do mesons differ from hadrons?

• For pt>2 GeV, protons are as abundant as pions and kaons!

• hadron production via fragmentation yields p/π ratio of ~0.1


Elliptic flow of K0 and Elliptic flow of K0 and

• hyperon v2 saturates later and higher than kaon v2.

• same effect observed for protons and pions.

•what drives the different pT scales for KS and Λ v2?

novel mechanism of baryon formation?

Sorensen SQM 2003


A possible solution to the puzzle:parton recombination

Where is pQCD?


Recombination vs Fragmentation

Recombination vs Fragmentation

• for exponential parton spectrum, recombination is more effective than fragmentation• baryons are shifted to higher pt than mesons, for same quark distribution understand behavior of protons!

recombining partons:p1+p2=ph

fragmenting parton:ph = z p, z<1

1

hh3 2 3

0

( )/adNdN dz E

E D zd P z z d P z Fragmentation:


The recombination ideaThe recombination idea

basic assumptions:

• at low pt, quarks and antiquarks recombine into hadrons on a hadronization hypersurface:

• hadron momentum P is much larger than masses and momentum scales of the wave function of the hadron; features of the parton spectrum are shifted to higher pt

in the hadron spectrum

• parton spectrum has thermal part (effective quarks) and a power law tail (quarks and gluons) from pQCD.

qq M qqq B


The nine lives of recombinationThe nine lives of recombinationHigh Energy Physics Phenomenology:• K.P. Das & R.C. Hwa, Phys. Lett. B68, 459 (1977)

Quark-Antiquark Recombination in the Fragmentation Region description of leading particle effect (field of recent activity!)Heavy-Ion Phenomenology:• T. S. Biro, P. Levai & J. Zimanyi, Phys. Lett. B347, 6 (1995)

ALCOR: a dynamical model for hadronization yields and ratios via counting of constituent quarks• R.C. Hwa & C.B. Yang, PRC66, 025205 (2002) • R. Fries, B. Mueller, C. Nonaka & S.A. Bass, Phys. Rev. Lett. 90• V. Greco, C.M. Ko and P. Levai, Phys. Rev. Lett. 90Anisotropic flow:• S. Voloshin, QM2002, nucl-ex/020014• Z.W. Lin & C.M. Ko, Phys. Rev. Lett 89, 202302 (2002)• D. Molnar & S. Voloshin, nucl-th/0302014


Recombination formalism IRecombination formalism I

PMPMPd

NM ;;)2( 3

3

• Express number of mesons by the quark density matrix .

),()2

,2

;2

,2

()2()2( 3

33

,3

3

3qrq

PrRq

PrRW

rqddRd

Pd

dNMab

ba

M

• Introduce 2-quark and meson Wigner functions W, .


Recombination formalism IIRecombination formalism II

• choose a hypersurface Σ for hadronization

• use local light cone coordinates (hadron defining the + axis)

• wa(r,p): single particle distribution functions for quarks at hadronization

• ФM & ФB: light-cone wave-functions for the meson & baryon respectively

• x, x’ & (1-x): momentum fractions carried by the quarks

• integrating out transverse degrees of freedom yields:2

M

2

M3 3

,

B3 3

, ,B

( , ) ( , (1 ) )

( , ) ( , ' ) ( ,

( )

( ,

(

(1 ') '

2 )

'(2 )

) )

dN P uE d dxd P

dN P

w R xP w R x P

w R xP w R x Pu

E d dx dxd p

w x P xR x

x

x


Recombination of an exponential spectrumRecombination of an exponential spectrum

( , ) ( , (1 ) ) exp /

( , ) ( , ' ) ( , (1 ') ) exp /B

w R xP w R x P P u T

w R xP w R x P w R x x P P u T

( , ) exp( / )qw r p p u T

•important features: ph = Σ pq

d3N/dp3h (wq)n (with n=2,3)

• for an exponential distribution:

product of all distribution functions only depends on hadron momentum! results are insensitive to the model used for recombination Baryon/Meson ratio is independent of momentum, e.g.

// /B Tp pdN dN e C C

(Cp, Cπ : degeneracy factors)


Recombination vs. Fragmentation

Recombination vs. Fragmentation

1h 1

h3 3 20

( , ) ( )(2 ) z

dN P u dzE d w R P D zd P z

Fragmentation…

frag bdN P rec 2bdN P

… but it wins out at large pT, when the spectrum is a power law ~ (pT)-b :

… never competes with recombination for a thermal (exponential) spectrum:

/ ex expp( / ) ( / ) /n

w P n P u T P u zT w P z


Recombination + Fragmentation

Recombination + Fragmentation

• Fragmentation of perturbative partons dominates at high pt.

• Recombination kicks in at 4-6 GeV at RHIC energies.

• Our description of recombination fails when /PT and m/PT corrections become large (from 1-2 GeV on at RHIC).

• But: recombination will still be the dominant hadronization mechanism. Take into account binding energies, mass effects.


Results & Comparison to Data• hadron spectra• hadron ratios

• RAA


Input and Parameters IInput and Parameters I

Input for the model is the momentum distributions of constituent quarks and anti-quarks at the time of hadronization

• the quark distribution is assumed to have a low pt thermal component and a high pt pQCD mini-jet component

• the thermal component is parameterized as:

with a flavor dependent fugacity ga, temperature T,

rapidity width Δ and transverse distribution f(ρ,ф).

• the pQCD component is parameterized as:

with parameters C, B and β taken from a lo pQCD calculation

2 2/ / 2, ,Tp va agw e fp e

0 1 /a

t t y t

dN

p dp y

CK

d Bp


Input and parameters IIInput and parameters II

• Use hypersurface with t2-z2=2; = 5 fm/c. • Fix T=175 MeV

Determine:• Radial flow =0.55 c• Emission volume• Energy loss parameter• Fugacities


Hadron Spectra IHadron Spectra I


Hadron Spectra IIHadron Spectra II


Hadron Ratios vs. ptHadron Ratios vs. pt


• anisotropic or “elliptic” flow is sensitive to initial geometry

Elliptic FlowElliptic Flow

more flow in collision plane than perpendicular to it

Force P

less absorption in collision plane than perpendicular to it

E L

low pt domain: high pt domain:

r fecomb g222ra1t t ttt v pv r p v pr pp

•total elliptic flow is the sum of both contributions:

r(pt): relative weight of the recombination contribution in spectra


Elliptic Flow: partons at low pt

Elliptic Flow: partons at low pt

2

2 11 2 cos 2

2 pt t p t t

d N dN

p dp d p dv

p

2

2 10

22

0 10

sinh coshcos 2

cos 2sinh cosh

st ts s

t p

t ts

s

s s

p md I K

T Tv p

p md I K

T T

0 2

0

111ln 1 cos 2

2 1and

1 /p t p

ts t

t

st

p pp p

• azimuthal anisotropy of parton spectra is determined by elliptic flow:

• with Blastwave parametrization for parton spectra:

(Фp: azimuthal angle in p-space)

• azimuthal anisotropy is parameterized in coordinate space and is damped as a function of pt:


Parton Number Scaling of Elliptic Flow

Parton Number Scaling of Elliptic Flow

•in the recombination regime, meson and baryon v2 can be obtained from the parton v2 in the following way:

3

2

2

2

2

2 2

2

2

2

3 322

1

an

2

d

3

12

3

63

p pt tp t

Mt

p pt t

Bt

p pp v vv

p pv

p pv v

v

neglecting quadratic and cubic terms, one finds a simple scaling law:

2 2 2 2an22 3

d 3p pM tt

B tt

pv p vv

pv p


Results & Comparison to Data• elliptic flow


Elliptic Flow: InputElliptic Flow: Input

r fecomb g222ra1t t ttt v pv r p v pr pp

parton elliptic flow: relative weight of recombination:

•grey area: region of uncertainty for limiting behavior of R & F

•hadron v2 calcuated separately for R and F and superimposed via:


Flavor Dependence of Recombination

Flavor Dependence of Recombination

Recombination describes measured flavor-dependence!


Elliptic Flow: Recombination vs. Fragmentation

Elliptic Flow: Recombination vs. Fragmentation

• high pt: v2 for all hadrons merge, since v2 from energy-loss is flavor blind

charged hadron v2 for high pt shows universal & limiting fragmentation v2

quark number scaling breaks down in the fragmentation domain


Bill Zajc (DNP Tucson)

• New PHENIX Run-2 result on v2 of 0’s:• New STAR Run-2 result on v2 for ’s:• ALL hadrons measured to date

obey quark recombination systematics

PHENIX Preliminary

0

STAR Preliminary

smoking gun for recombination

measurement of partonic v2 !


New developments INew developments I

• Another test: the meson. Do we see a mass effect or the valence quark structure of hadrons?

• Reco differs from hydro!

• The deuteron and the pentaquark should have tremendous v2.

• STAR: deuteron v2 follows the scaling law!


New developments IINew developments II

• The + will be measured at RHIC. Will v2 scale with n=5?

• What about other resonances? Influence of the hadronic stage?


Summary & Outlook

Summary & Outlook

The Recombination + Fragmentation Model:• provides a natural solution to the baryon puzzle at RHIC

• describes the intermediate and high pt range of hadron ratios & spectra jet-quenching phenomena elliptic flow

• provides a microscopic basis for the Statistical Model• One universal parametrization of the parton phase can explain the data!

• v2: proof of collectivity in the parton phase

issues to be addressed in the future:• entropy & energy• resonances and influence of the hadronic phase• need improved data of identified hadrons at high pt


The End


Centrality Dependence of Spectra & Ratios

Centrality Dependence of Spectra & Ratios

•R+F model applicable over full range of centrality•deviations from SM as soon as fragmentation sets in

•low pt deviations due to neglected const. quark mass


Flavor Dependence of high-pt Suppression

Flavor Dependence of high-pt Suppression

• R+F model describes different RAA behavior of protons and pions

• Lambda’s already exhibit drop into the fragmentation region• in the fragmentation region all hadron flavors exhibit jet-quenching


Elliptic Flow: partons at high pt

Elliptic Flow: partons at high pt

• azimuthal anisotropy is driven by parton energy/momentum loss Δpt

21 1 cos 2exp A

t t ptA

R bp

Rp L p

•L: average thickness of the medium

• the unquenched parton pt distribution is shifted by Δpt.

v2 is then calculated via: 2 cos 2t pv p


pQCD approach to parton recombination

pQCD approach to parton recombination

A A

meson

double parton scattering scales:

22 22

2 412( )s

T T

dA

dp p

single parton scattering and fragmentation scales:

24/3

2 2 4( / )s

T T

d A

dp z p z

21/3 2

2

16DPS

SPSs

T

A

zp

T. Ochiai, Prog. Theor. Phys. 75 (1986) 1184


New developments IIINew developments III

• Can we distinguish production scenarios for the pentaquark?

• 5q recombination• K+N recombination &

coalescence, • K+N fragmentation &

coalescence• K+N fragmentation &

coalescene in a jet cone (= 5q fragmentation)

• Even obtain information about the structure?

recombination and fragmentation of hadrons from a dense parton phase

Documents

recombination ideacalculations

recombination vs fragmentation

quarkantiquark recombination

higher pt

hadron momentum p

effective quarks

high pt

fragmentation of hadrons