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发布时间:2014-02-13 16:04:16  

Spin effects in fragmentation processes at RHIC
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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.

2004-8-13, CCAST

Qinghua Xu, Shandong University

1

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.

W (cos? ) ? 1 ? ?P cos?
2004-8-13, CCAST Qinghua Xu, Shandong University 2

Previous study of polarized fragmentation function

fragmentation

e?

q0 f

(Boros, Liang,PRD’98)

SU(6)

e

?

Z0

q f0

DIS

0 e ?e ? ? Z 0 ? q0 q f f ?H? X

E.g., at the Z0 pole, <Pf>= -0.94 for f=s,d,b; <Pf>= -0.67 for f=u,c.
2004-8-13, CCAST Qinghua Xu, Shandong University 3

? 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).
2004-8-13, CCAST Qinghua Xu, Shandong University 4

PH in high pT jets at RHIC
? Hyperon production with high pT:
? ( ab ? cd ) H d 2? pp? HX 2 2 d? ? ? ? dxa dxb dzc f a (xa , ? ) f b ( xb , ? ) Dc ( zc , ? 2 ) dpT d? dpT d? abcd
? ( ab?cd ) d 2 ?? 2 2 d?? ? ? ? dxa dxb dzc ?f a ( xa , ? ) f b ( xb , ? ) ?DcH ( zc , ? 2 ) dpT d? abcd dpT d?

helicity distribution function

longitudinally polarized fragmentation function ? 2 H? 2 ?D ? DcH ( z , ? ) ? D ( z , ? ) ? c?
? pp ? HX
2004-8-13, CCAST Qinghua Xu, Shandong University 5

Contribution of gluon and quark jet(s) to ? production
? Different subprocess: qq?qq, qg?qg, gg?gg ……, but :

From quark fragmentation

PYTHIA

From gluon fragmentation

? Quark jets dominate the
Hyperon production with high PT !

2004-8-13, CCAST

Qinghua Xu, Shandong University

6

2 ( z , ? ) polarized fragmentation function ?D H f

? ? We divide the produced Hi’s in q f ? Hi ? X into four groups:

(a) directly produced and contain the qf ; (b) decay products of polarized heavier hyperons ; (c) dire

ctly 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!
H ? Hence : ?D f ( z ) ?
i

? ?

? a ,b

H i ,? F ,? tH ? D (z ) f i,f

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

F ,? i (? ) i (? ) tH ? ? D ( z ) / D ( z ) : spin transfer factor f f i,f H H

F ,a t ? Hi , f : fraction of spin carried by qf in Hi (qf….) ? SU(6), DIS
F ,b D F ,a t ? t t ? Hi ,H j , f for (b) qf ?Hj +X Hi ,H j Hi , f
2004-8-13, CCAST

decay spin transfer factor in H ?Hi +X’. Qinghua Xu, Shandong University j

7

Results of hyperon polarization ( pT ? 13GeVand s ? 500GeV) ? The decay contribution of ? is very large, while origin of ?+ is much cleaner!
SU(6), set I(GRSV00)

SU(6), set I SU(6), set II

DIS, set I

DIS, set I
DIS, set II(GRSV96)

DIS, set II

SU(6), set II

(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) !

Characteristic of ?+ production in pp??+X ? Contribution of directly produced
and contain a fragmenting u play the dominant role! ? In the limiting case that there is ?? ? only this contribution in pp ? ? X :
containing fragmenting u containing fragmenting s

P

? ? (lim) L ,T

F u ? t? P ? , u L ,T

decay contribution

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

pp ? ? ? X

( pT ? 13 GeV and s ? 500GeV)

F ? Different models only differ in the constant t ? ? ,u,

? Hence , we can study p.d.f. from P?+ in pp reactions.
2004-8-13, CCAST Qinghua Xu, Shandong University 9

?? ? P?+ in pp ? ?? X and ?g(x)
? Subprocesses involved,

qg ? ?q (dominant) qq ? ?g (negligible)
? P?+

?? ? in pp ? ?? X :
? ? pp ? ?HX

P ? d?? / d? ??
d?? ?
min pT

2 2 dp dx dx dz ? g ( x , ? ) q ( x , ? ) a b b ? T ?? a b f ? ? (g ? d?? qf ? ? ?q f ) 2 ?D H ( z , ? ) ? (g ? q f ) f

? ? gq ? ?q
2004-8-13, CCAST Qinghua Xu, Shandong University

? ? qg ? ?q
10

?? ? P?+ in pp ? ?? X at

s ? 200GeV

BB02 ?g

(?1 ? ? ? 0)
AAC ?g

GRSV00 ?g

(Xu, Liang,he p- ph/0406119 )
2004-8-13, CCAST Qinghua Xu, Shandong University 11

Transverse polarization of hyperons with high pT and ?q(x) ? PH in transversely polarized pp collisions:
PTH ?
d?T ?
? ? ( p? p ? H ? X )

d? ? d? d? T? ? ( p p? H ? X ) ( p p? H ? X ) d? ( pp? HX ) d? ? ? d? ?

( p? p? H ? X )

( p? p? H ? X )

? ? ( p? p? H ? X )

d?

?

min pT

2 2 H 2 dp dx dx dz ? f ( x , ? ) f ( x , ? ) ? D ( z , ? ) ? T a b a a b b T c ? ? abcd

? d?T ? d?

? ? ( a? b ?c? d )

? For ?T DcH (? ) ( z, ? 2 ) , similar as the longitudinally polarized case,
?T DcHi ( z) ?
T
2004-8-13, CCAST
F (? ) H ,c

? ?

?a ,b

Hi H i ,? F ,? TH ? D (z) , D c c (z) ? i ,c

? ?

?a , b , c , d

DcHi ,? (z) ,

? T DcH (? ) ( z ) ? DcH (? ) ( z )
Qinghua Xu, Shandong University 12

Numerical results for ?+ polarization

(J.Soffer) (B.Q.Ma et al)

?q( x )

F F take TH ? t Hi , f i, f for a simple assumption!

?? ? P?+ with pT>13 GeV in p at s ? 500GeV. ? p ? ?? X
2004-8-13, CCAST Qinghua Xu, Shandong University 13

Spin alignment of vector meson
3 W (cos θ ) ? [(1 ? ? 00 ) ? (3? 00 ? 1) cos2 ? ] ? Polarization of V: 4

? The data at LEP:
? K*0

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

z ? 2p/ s

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 qf0 ;
? V ( z ) ? ? ? V ( A, f )
f

(B) do not contain the qf0 .

? n( z | A, f )? ? n( z | B)? ? ? V ( B) ? n( z )? ? n( z )?

?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).

(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 qf0 + Spin of q-bar
2004-8-13, CCAST Qinghua Xu, Shandong University 15

Calculation of ?V(A,f)
? The spin density matrix of V can be obtained from the product:
?
q0 fq

? Pf is the polarization of qf , P( Px , Py , Pz ) for q
0

0 ? ? 1 ? Pz Px ? iPy ? 1 ? 1 ? Pf ??? ? ? ? ?? ? ? ? ? ? 1 ? Pf ? ? Px ? iPy 1 ? Pz ? 4? 0 ?
q0 f q

? transforming it to the coupled basis and obtain:
(1 ? Pf ) ? ? ? (1 ? Pf )(1 ? Pz ) ? ( Px ? iPy ) 0 2 ? ? ? (1 ? Pf ) ? (1 ? Pf ) 1 ? V ( A, f ) ? ( Px ? iPy ) (1 ? Pf Pz ) ( Px ? iPy ) ? ? 3 ? Pf Pz ? 2 2 ? ( 1 ? P ) ? ? f 0 ( Px ? iPy ) (1 ? Pf )(1 ? Pz ) ? ? 2 ? ?

? Hence: ? ( A, f ) ? 3 ? P P f z
V 00

1 ? Pf Pz

To get ?V00(A,f ) > 1/3, signs of Pf and Pz should be opposite!
16

2004-8-13, CCAST

Qinghua Xu, Shandong University

? Fitting the data, we obtained that:

Pz ? ??Pf ,

? ? 0.5

e ? e ? ? Z 0 ? K *0 ? X

Xu, Liu,Liang,PRD(2001).

2004-8-13, CCAST

Qinghua Xu, Shandong University

17

2004-8-13, CCAST

Qinghua Xu, Shandong University

18

Spin alignment of V in pp-->VX
? Is this relation universal for the fragmentation of a longitudinally polarized quark ?

Pz ? ??Pf
? Applying it to other pro

cess to make further tests ! DIS, pp collisions…
Xu, Liang,PRD(2003).

(a) spin alignment of V in pp ? VX at ?s=500 GeV and pT>13 GeV.
2004-8-13, CCAST Qinghua Xu, Shandong University 19

?

Spin alignment in transversely polarized pp collisions ? Further extents to the transversely polarized case!

PqT ? ??PqT .

Soffer inequality

Light cone model

(a) in transversity frame

(b) in helicity frame

Fig: Spin alignment of V in pp reaction at ?s=500 GeV and pT>13 GeV.
2004-8-13, CCAST Qinghua Xu, Shandong University 20

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 pp ? ?? X can provide important information on ?g(x). ?? ? ?+ polarization in p? p ? ? ? X 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.

2004-8-13, CCAST

Qinghua Xu, Shandong University

21

Thank you!

? Polarization of h(q0x) from q0 ? h(q0 X ) ? x
Ph

?

? ? ?

Xs Xs

? ? |? h? | q0 X s ? |2 ? ?X |? h? | q0 X s ? |2 ? ? |? h? | q0 X s ? |2 ? ?X |? h? | q0 X s ? |2
s s

? Fraction of spin carried by q0 in a hadron h
?Q0

? ? ?

Xs Xs

? ? | ? q0 X s | h? ? |2 ? ?X | ? q0 X s | h? ? |2 ? ? | ? q0 X s | h? ? |2 ? ?X | ? q0 X s | h? ? |2
s s

Ph ? ?Q0
2004-8-13, CCAST Qinghua Xu, Shandong University 23

PqT ? ??PqT .
z-distribution of type (a) ?+
h ?q q

1 1? h h ? ?q ? ?q ? ? i? P e 4? ? qT
2 1 ? ?PqT 2 3 ? ?PqT

PqT e ? i? ? ? 1 ? Pq z ??? ? ? ?P e i? 1 ? qT ? ?

? ?PqT e ? i? ? ? 1 ? Pq z ? ?

h ? 00 ( A) ?

?00 ? 1/ 3

2004-8-13, CCAST

Qinghua Xu, Shandong University

24

? For hyperons of group (b), qf ?Hj +X Hi +X ’ F ,b D F ,a tH ? t t Hi ,H j Hi , f i ,H j , f D tH : decay spin transfer factor in Hj?Hi +X’. i ,H j ? Hyperon polarization can be given by:
? q ( x , ? 2 ) d? ? ? Pf ? ? q ( x , ? 2 ) d?

PH i ?

F a D b t P ? n ? ? t P ? n ? f H i , f f H i , f ? j H i ,H j H j H i ,H j ? a b c d ? nH ? ? ? n ? ? ? n ? ? ? n Hi , f Hi , f Hi , f ? i,f

2 2 ( ab?cd ) H 2 ? ?n? ? ? dp dx dx dz f ( x , ? ) f ( x , ? ) d ? D ( z , ? ) Hi b b c c ? min T ? ? a b c a a pT abcd

? ? n? H i ? (? ? a , b, c , d ) :independent of polarization properties, and can
be obtained by hadronization model or event generator (PYTHIA …)!
2004-8-13, CCAST Qinghua Xu, Shandong University 25

?? ? P?+ in pp ? ?? X

BB02 AAC00

GRSV00

s ? 200GeV and pT ? 8 GeV.
2004-8-13, CCAST Qinghua Xu, Shandong University 26

?? ? Fig: P?+ with 0<?<1.5 in p

? p ? ?? X at
2004-8-13, CCAST Qinghua Xu, Shandong University

s ? 500GeV.
27


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