Hadronization @ RHIC: interplay of fragmentation and recombination

Steffen A. Bass Hadronization @ RHIC #1

Steffen A. Bass

Duke University& RIKEN-BNL Research Center

• Data: Protons at RHIC - the demise of pQCD?• Recombination + Fragmentation Model• Results and Predictions

Hadronization @ RHIC: interplay of fragmentation and

recombination

Hadronization @ RHIC: interplay of fragmentation and

recombination

R.J. Fries, C. Nonaka, B. Mueller & S.A. Bass, PRL 90 202303 (2003) R.J. Fries, C. Nonaka, B. Mueller & S.A. Bass, nucl-th/0306027


Some of the RHIC Puzzle(s):• protons and the demise of pQCD?

• spezies-dependent saturation of v2


The proton puzzle @ RHIC

The proton puzzle @ RHIC• where does the large proton over pion ratio at high pt come from?

• why do protons not exhibit the same suppression as pions? fragmentation yields Np/Nπ<<1 fragmentation starts with a single fast parton: energy loss affects pions and protons in the same way!

ratio of KKP fragmentation functions for p and π from u quarks


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.

• the phenomenology seems better described in mT – m0 than pT ; why (kinetic energy)?

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

novel mechanism of baryon formation?


A possible solution to the puzzle:parton recombination


Recombination+Fragmentation Model

Recombination+Fragmentation Model

basic assumptions:• at low pt, the quarks and antiquark spectrum is

thermal and they recombine into hadrons locally “at an instant”:

features of the parton spectrum are shifted to higher pt in the hadron spectrum

• at high pt, the parton spectrum is given by a pQCD power law, partons suffer jet energy loss and hadrons are formed via fragmentation of quarks and gluons

qq M qqq B


Recombination: Pro’s & Con’s

Recombination: Pro’s & Con’s

Pro’s:• 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!Con’s:

• recombination violates entropy conservation• gluons at hadronization need to be converted

recombining partons:p1+p2=ph

fragmenting parton:ph = z p, z<1


Recombination: new life for an old idea

Recombination: new life for an old idea

High 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• T. Ochiai, Prog. Theo. Phys. 75, 1184 (1986)• E. Braaten, Y. Jia & T. Mehen, Phys. Rev. Lett. 89, 122002 (2002)• R. Rapp & E.V. Shuryak, Phys. Rev. D67, 074036 (2003)

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, Phys. Rev. C66, 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. 90

Anisotropic 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: nonrelativistic formalism

Recombination: nonrelativistic formalism

3

3 31 12 2

212

3

2

3

2

(2 ) (2 )

21

(

ˆ ( )

2 )

MM

M

MM

dN V d qC w P q w P q

d P

VC

PP

Tw

q

• use thermal quark spectrum given by: w(p) = exp(-p/T)

• for a Gaussian meson wave function with momentum width ΛM, the meson spectrum is obtained as:

• similarly for baryons:

313

23 3

1 ( / )(2 )

BB Bw

dN VC O TP

d PP


Recombination: relativistic formalism

Recombination: relativistic formalism

• choose a hypersurface Σ for hadronization

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

• wa(r,p): single particle Wigner function 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 thermal quarks

Recombination of thermal quarks

( , ) ( , (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

• results are insensitive to the model used for recombination (light-cone & Wigner functions vs. non-relativistic approx.)

• important features: ph = Σ pq

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

• for a thermal distribution:

product of all Wigner functions only depends on hadron momentum! 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


pt range of parton recombination

pt range of parton recombination

• quark recombination (coalescence) may dominate for all pt < p0 . Combinatorical models (ALCOR, etc.) work well for total particle yields at SPS and RHIC

• low pt is not calculable, but calculation at moderate pt (few GeV) may be possible using hadron light-cone formalism

• transition from dense medium to dilute medium appears very rapid for fast partons (Δt=Δx/γ), validating sudden approximation

• focus on “high” pt evades problems of energy and entropy conservation in recombination: E = (p2+m2)1/2 p


• 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 fragmentation 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:


Elliptic Flow: partons at high pt

Elliptic Flow: partons at high pt

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

R Lr

22

2 20

1

2

Rt

t t t t

tdN pd Nd r

p

p dp d R p dp

21 1 cos 2exp A

t t ptA

R bp

Rp L p

•L: average thickness of the medium

•α(pt): damping function for low pt

•the unquenched parton pt distribution is modified according to:

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


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• hadron spectra• hadron ratios

• RAA

• elliptic flow


Input and Model Parameters

Input and Model Parameters

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


Hadron Spectra IHadron Spectra I


Hadron Ratios vs. ptHadron Ratios vs. pt


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: 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!


Parton Number Scaling of v2

Parton Number Scaling of v2

smoking gun for recombination

measurement of partonic v2 !

P. Soerensen, UCLA & STAR @ SQM2003

2 2

2 2

22

33

pM

p

t

B t

t

t

pv

p

v

p v

p

v

•in leading order of v2, recombination predicts:


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

issues to be addressed in the future:• entropy production• treatment of gluons• realistic space-time dynamics of parton source• need improved data of identified hadrons at high pt


The End


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


Statistical Model vs. Recombination

Statistical Model vs. Recombination

3

3

3

1

exp / 12with

i

ii

iB s I

i

i i

d NE P dd

f P u

gf P u

P u B I T

P

S

•in the Statistical Model, the hadron distribution at freeze-out is given by:

max

0

/ 22

0

3 si

cos , cos , sin , si

nh sinh with

nh

t t p t p t

ti

is p

t

d Nd d d P u

p dp dy

P m y p p m y

Pf u

•using one obtains:

• for pt , hadron ratios in SM are identical to those in recombination!

(only determined by hadron degeneracy factors & chem. pot.) recombination provides microscopic basis for SM at large pt

see, e.g. Broniowski & Florkowski: PRL 87, 272302 (2001)


Hadron Spectra IIHadron Spectra II


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

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Hadronization @ RHIC: interplay of fragmentation and recombination