2004-8-13, CCASTQinghua Xu, Shandong University1 Spin effects in fragmentation processes at RHIC Qinghua Xu ( 徐庆华 ) Physics department, Shandong University

2004-8-13, CCAST Qinghua Xu, Shandong University 1

Spin effects in fragmentation processes at RHIC

Qinghua Xu ( 徐庆华 )

Physics department, Shandong University

References:

Xu Q.H., Liu C.X., and Liang Z.T., Phys. Rev. D65, 114008 (2002);

Xu Qing-hua, and Liang Zuo-tang, Phys. Rev. D67, 114013 (2003);

Xu Qing-hua, and Liang Zuo-tang, hep-ph/0406119.


Introduction

• Spin-dependent fragmentation function(spin effect in frag.):

(study the polarization and/or momentum distribution of hadrons

produced in the fragmentation of a polarized quark.)

hadronization mechanism;

to study spin structure of nucleon, as filters for exotic p.d.f, e.g., g(x) and q(x)!• Hyperon polarizations (, +) are widely used to study

polarized fragmentation function, since their polarization

can be measured through their self-analyzing decay.

cosP1 )(cos W


Previous study of polarized fragmentation function

SU(6)

DIS

(Boros, Liang,PRD’98)

e

e0Z

0fq

0fq fr

agm

enta

tion

XHqqZee ff 000

E.g., at the Z0 pole,

<Pf>= -0.94 for f=s,d,b;

<Pf>= -0.67 for f=u,c.


• The DIS data available:

HERMES(PRD01), E665(EPC00) ----- large error !

NOMAD(NPB00), better statistics, but energy too low !

• RHIC spin program (pp collision) is promising:

(a) high statistics (high luminosity)

(b) high energy (200-500GeV).

• Available theoretical studies in pp collisions:

D.de Florian,M.Stratmann, W.Vogelsang, Phys. Rev. Lett. 81(1998);

C. Boros, J.T.Londergan, A.W.Thomas, Phys. Rev. D62 (2000);

B.Q. Ma, I.Schmidt, J.Soffer, J.J.Yang, Nucl. Phys. A703 (2002);

Xu Q.H., Liu C.X, Liang Z.T., Phys. Rev. D65, 114008 (2002).


PH in high pT jets at RHIC

• Hyperon production with high pT:

HXpp

),(ˆ

),(),()( 222

2

c

Hc

T

cdab

bbaabcd

acbaT

HXpp

zDddp

dxfxfdzdxdx

ddp

d

),(ˆ

),(),()(

2222

cHc

T

cdab

bbaabcd

acbaT

zDddp

dxfxfdzdxdx

ddp

d

helicity distribution function

longitudinally polarized fragmentation function

),(),( 22 zDzDD Hc

Hc


Contribution of gluon and quark jet(s) to production

Quark jets dominate the

Hyperon production with high PT !

From gluon fragmentation

From quark fragmentation

PYTHIA

• Different subprocess: qqqq, qgqg, gggg ……, but :


polarized fragmentation function

We divide the produced Hi’s in into four groups:

(a) directly produced and contain the qf ;

(b) decay products of polarized heavier hyperons ; (c) directly produced but don’t contain the qf ;

(d) decay products of unpolarized heavier hyperons. Those from (a) and (b) can be polarized, (c) and (d) are not! Hence :

XHq if

: spin transfer factor)(/)( )()(,, zDzDt ii

i

H

f

H

fF

fH

(z))( ,,,

,

i

i

i Hf

FfH

ba

Hf DtzD

),( 2zDHf

• : fraction of spin carried by qf in Hi (qf….) SU(6), DIS

• for (b) qf Hj +X

aFfH i

t ,,

aFfH

DHH

bFfHH ijiji

ttt ,,,

,,,

decay spin transfer factor in HjHi +X’.

can be obtained from generator (e.g.PYTHIA)

Results of hyperon polarization

SU(6), set II

SU(6), set I(GRSV00)

DIS, set I

DIS, set II(GRSV96)

SU(6), set I

DIS, set I

SU(6), set II

DIS, set II

GeV) andGeV ( 50013 spT

• The decay contribution of is very large, while origin of + is much cleaner!

(1) P is very small .

(2) Difference between results of SU(6) and DIS is small.(3) P has a strong dependence of

different sets of f (x) !

• P+ is large .

• Difference between results of SU(6) and DIS is large.

• P+ has a weak dependence

of different sets of f (x) !


containing fragmenting u

containing fragmenting s

decay contribution

GeV) 500 and GeV 13( spT

Contribution of directly produced and contain a fragmenting u play the dominant role!

In the limiting case that there is only this contribution in :

Different models only differ in the constant ,

Hence , we can study p.d.f. from P+ in pp reactions.

F

ut

,

Xpp

Characteristic of + production in pp+X

uTL

F

uTL PtP ,,

(lim),

Polarization of fragmenting u, related to polarized p.d.f.

Xpp


P+ in and g(x) Xpp

XHpp

• Subprocesses involved,

)(),(ˆ

),(),(

2)(

22

min

fHfqqg

bbaf

ba

p

T

qgzDd

xqxgdzdxdxdpd

ff

T

qqg gqq

• P+ in

ddP /

: Xpp

qqg qgq

(dominant)

(negligible)


P+ in at GeV 200s Xpp

BB02 g

GRSV00 g

AAC g

)01(

)ph/0406119-hep Liang, (Xu,


Transverse polarization of hyperons with high pT and q(x)

)(

)(

)()(

)()(

HXpp

XHppT

XHppXHpp

XHppXHppHT d

d

dd

ddP

d

dzDxfxfdzdxdxdp

d

d dcbaTH

cTbbaabcd

aba

p

T

XHppT

T

)(222

)( ˆ),(),(),(

min

• PH in transversely polarized pp collisions:

• For , similar as the longitudinally polarized case,

)(

)()(

)()(

, zD

zDT H

c

HcTF

cH

),( 2)( zDHcT

, (z))( ,,,

,

i

i

i Hc

FcH

ba

HcT DTzD

(z) ,,

, , ,

( )i iH Hc c

a b c d

D z D


P+ with pT>13 GeV in at GeV. 500sXpp

Numerical results for + polarization

)(xq

(J.Soffer)

(B.Q.Ma et al)

take

for a simple assumption!

FfH

FfH ii

tT ,,

Spin alignment of vector meson

• The data at LEP:

• K*0

spz /2

]cos)()[()W(cosθ 20000 131

4

3• Polarization of V:

00 : Prob. of being in h= 0 state.

Spin alignment exists in fragmentation of a longitudinally polarized quark !


• To study what the data tell us, we made the following calculations

(1) We divided the produced vector mesons into two groups:(A) contain the initial quark qf

0 ; (B) do not contain the qf0 .

n(z|A,f), n(z|A): number density of V from (A) and (B), calculable using a MC event generator such as JETSET; V(A,f) and V(B): spin density matrix of V from (A) and (B).

)(

)|()(

)(

),|(),()(

zn

BznB

zn

fAznfAz V

f

VV

(2) We calculate V(A,f) and V(B) in the following way:(i) Vector mesons in group (B) are taken as unpolarized, i.e., 00(B)=1/3

(ii) V(A,f) is calculated according to the assumption:

Spin of V(qf0q-bar) = Spin of qf

0 + Spin of q-bar


• transforming it to the coupled basis and obtain:

)1)(1()(2

)1(0

)(2

)1()1()(

2

)1(

0)(2

)1()1)(1(

3

1),(

zfyxf

yxf

zfyxf

yxf

zf

zf

V

PPiPPP

iPPP

PPiPPP

iPPP

PP

PPfA

• Hence:

qPPPP zyx for ),,(

• The spin density matrix of V can be obtained from the product:

zyx

yxz

f

fqqqq

PiPP

iPPP

P

Pff

1

1

10

01

4

100

zf

zfV

PP

PPfA

3

1),(00

Calculation of V(A,f)

To get V00(A,f ) > 1/3,

signs of Pf and Pz should be opposite!

Pf is the polarization of qf0 ,


• Fitting the data, we obtained that: 5.0 , fz PP

XKZee 0*0

PRD(2001). Liang, Liu, Xu,



Spin alignment of V in pp-->VX

(a) spin alignment of V in

at s=500 GeV and pT>13 GeV.

VXpp

fz PP

• Is this relation universal for the fragmentation of a longitudinally polarized quark ?

• Applying it to other process

to make further tests !

DIS, pp collisions…PRD(2003). Liang, Xu,


Fig: Spin alignment of V in pp reaction at s=500 GeV and pT>13 GeV.

(b) in helicity frame

Spin alignment in transversely polarized pp collisions

(a) in transversity frame

Light cone model

Soffer inequality

• Further extents to the transversely polarized case! .qTTq PP


Summary

• Spin dependent fragmentation function is studied from

PH in high pT jets in polarized pp collisions. We find that,

+ is more suitable than for this purpose.

• + polarization in can provide important information on g(x). + polarization in can provide useful information on q(x). Numerical results are given.

• Spin alignment of vector meson in polarized pp collisions are studied at RHIC energy.

Xpp

Xpp

Thank you!


20

20

20

20

||||||

||||||

sXsX

sXsXh

XqhXqh

XqhXqhP

ss

ss

s s

s s

X X ss

X X ss

hXqhXq

hXqhXqQ

20

20

20

20

0||||||

||||||

0 0( )q h q X x • Polarization of h(q0x) from

0QPh

• Fraction of spin carried by q0 in a hadron h


zqi

qT

iqTzq

iqT

iqTh

qhq

hqq PeP

ePP

eP

eP

1

1

1

1

4

1

2

2

00 3

1)(

qT

qTh

P

PA

3100 /

.qTTq PP z-distribution of type (a) +


dfH

cfH

bfH

afH

f j

bHHH

DHH

afHf

FfH

H

iiii

jijjiii

i nnnn

nPtnPtP

,,,,

,,,,

),(ˆ),(),( 2)(22min

c

Hc

cdabbba

abcdacbap TH zDdxfxfdzdxdxdpn

Ti

• :independent of polarization properties, and can

be obtained by hadronization model or event generator (PYTHIA …)!

• Hyperon polarization can be given by:

: decay spin transfer factor in HjHi +X’.DHH ji

t ,

aFfH

DHH

bFfHH ijiji

ttt ,,,

,,,

• For hyperons of group (b), qf Hj +X

Hi +X ’

ˆ

ˆ

),(

),(2

2

d

d

xq

xqPf

( , , , )iH

n a b c d


P+ in Xpp

BB02

GRSV00

GeV. 8 and GeV 200 Tps

AAC00


Fig: P+ with 0<<1.5 in at GeV. 500s Xpp

Documents

2004-8-13, CCASTQinghua Xu, Shandong University1 Spin effects in fragmentation processes at RHIC Qinghua Xu ( 徐庆华 ) Physics department, Shandong University