single transverse spin asymmetries

June 1, 2005 Jianwei Qiu, ISU 1

Single Transverse Spin Asymmetries

Jianwei QiuIowa State University

RBRC Workshop onSingle Spin Asymmetries

June 1 – 3, 2005Brookhaven National Lab, Upton, NY


Single spin asymmetry – definition

Single spin asymmetry before QCD

Single spin asymmetry within the collinear

factorization – high twist matrix elements

KT- factorization – Sivers and Collins effects

Connection between high twist matrix elements

and Sivers and Collins functions

Open questions

Outline


Single Spin Asymmetry – definition

Spin-avg X-section:

Spin-dep X-section:

Single spin asymmetry:

single longitudinal spin asymmetry:

single transverse spin asymmetry:


Symmetry and Single Spin Asymmetry

Even though cross section is finite,

single

spin asymmetries can vanish if the

polarized cross

sections are independent of spin directions

due to

the fundamental symmetries of the

interactions

AL vanishes for Parity conserved interactions:

AL ≠ 0 for W+ production

good for parton flavor separation


Single transverse spin asymmetry before QCD

Almost 40 years ago, AN in inclusive DIS was proposed by Christ and Lee to test the Time-Reversal invariance:

In the approximation of one-photon exchange,

AN of inclusive DIS vanishes if Time-Reversal

is invariant for EM and Strong interactions

N. Christ and T.D. Lee, Phys. Rev. 143, 1310 (1966)


AN = 0 for inclusive DIS

DIS cross section:

Leptionic tensor is symmetric: Lμν = Lνμ

Hadronic tensor:

Polarized cross section:

P and T invariance:


Large AN in hadronic collisionsprocess: only one hadron is transversely polarized:

Large asymmetries AN

observed in hadron

collisions: decay of production of ’s

FNAL - E704

PT=0.5~2 GeV/cs=23.5 GeV


Single spin asymmetry corresponds

to a T-odd triple product


Single transverse spin asymmetry – AN

in the parton model

the operator for δq has even γ’s quark mass term

the phase requires an imaginary part loop diagram

transverse spin information at leading twist – transversity:

q x = Chiral-odd helicity-flip density

PQCD calculable partonic parts should not depend on light current quark mass – singals a high twist effect in pQCD


Single transverse spin asymmetry – AN

in collinear factorization approach Leading twist PDF with transverse hadron spin:

5

( , )q x s

need an even number ’s:

Twist-3 matrix elements

An extra transverse index extra gluon (its polarization) extra vector direction


Spin flip from interference between a quark state and a quark-gluon composite state Observed hadron momentum provides the 3rd vector

( )NA i s p

Unpinched pole to give the phase:

Q: pQCD factorization beyond leading twist?

High twist contribution to AN

x


Multiple scales in hadronic collisions

(3) (2) (1)

(1) Hard collision: Q >> ΛQCD

(2) Gluon shower: <kT>dynamic ~ F[Q2,log(s/Q2)]

(3) Hadron wave function: <kT>intrinsic ~ 1/fm ~ ΛQCD

PQCD factorization:

(1) Perturbative IR safe single hard

partonic “cross section”

(2) Leading log DGLAP evolution of PDFs

(3) Nonperturbative PDFs


Perturbative QCD factorization – (I)

42 2

0

1 1 perturbati

1T( , vely( , ) ) H

rQd k k k

k i k i

Perturbative pinch poles:

Perturbative factorization:

22 2

0

2 2H( , 0) 1 1

( 1

T , )T dkdxd k k

i ikQ

rk

kx

2 2

2T

T

k kk x

xp n k

p n

Short-distance

Nonperturbative matrix element

T

H(Q)k k

p p

Parton state of momentum, k, lives much longer than the time of hard collision: |k2|<<|Q2|

Remove soft interactions between different time scales


Perturbative QCD factorization (II)

2 ~ TQ n k kxp

Collinear factorization:

Provides systematic ways to quantify high order corrections

KT –factorization 2 Tp nQ x k k /

2 2ˆ ( ,) ), (h iphy

iTs

hTx k T x k

No all-order proof of kT – factorization for an arbitrary kT

Factorization fails beyond the next-to-leading powers in hadronic collisions

Leading Twist

Power corrections

perturbative

Long lived parton state

/2 2

/33

/4

2

22 4

2

ˆ [1 ...] ( )

ˆ + [1 ...] ( )

ˆ + [1 ...] ( )

+ ...

physh

s s

s s

s

i i h

ii

is

h

ih

T

TQ

TQ

x

x

x


Factorization:

1 2

1 2

,

,

F

D

T x x F

T x x D

Normal twist-2 distributions

Twist-3 correlation functions:

have different properties under the P and T transformation

1 2 1 2an, ,dF DT x x T x x

1 2,DT x x does not contribute to the AN

1 2,FT x x is universal, x1=x2 for AN due to the pole

AN from polarized twist-3 correlations


Single spin asymmetry within the

collinear factorization Generic twist-3 factorized contributions

Even ’s !

Even ’s !

Provides hadron spin dependence

transversity

(1) Calculated by Qiu and Sterman, Phys. Rev. D, 1999

(2) Calculated by Kanazawa and Koike, Phys. Lett. B, 2000


Leading twist-3 contribution to AN

Minimal approach (within the collinear factorization):

if ( , ) ( ) (1 )1N F

nA T x x q x xxu

n


Predictive power of the factorization approach


What is the T(3)(x)?

Represents a fundamental quantum correlationbetween quark and gluon inside a hadron


What the T(3) (x) tries to tells us?

Consider a classical (Abelian) situation:


Model for TF(x,x)

TF (x,x) tells us something about quark’s transverse motion in a transversely polarized hadron

One parameter and one sign!

It is non-perturbative, has unknown x-dependence

1

if ( , ) ( ) (1 )

N

nF

Axu

T

n

x x q x x


Numerical results – (I)(compare apples with oranges)

Qiu and StermanPhy. Rev. D, 1999


Numerical results – (II)(compare apples with oranges)

Qiu and StermanPhy. Rev. D, 1999


Numerical results – (III)(comparison with RHIC data)

Comparison with

STAR data

Too small PT value

to be comfortable

for initial state

twist-3

See Koike’s talk on

final state twist-3

New data from

STAR,

BRAHMS, PHENIX


Twist-3 vs KT approachTwist-3 contribution in collinear factorization – leading corrections from parton correlation (a minimal approach)

Effect of non-vanish parton kT (when kT ~ pT):

(3)

(3)

(

)1

(

( )

1

1

)T TA Bs p

T

p pN

x T xxAxS

x Tp x

T

x

MNon-perturbative scale, e.g., di-quark mass, …

1 1T TA Bs p

Np p TA

MS Mf

px


Sivers’ functions

Sivers’ functions: 1 ( )Tf x

1( , , ) ( , , ) ( ) T

n k sq x k s q x k s f x

M

Sivers’ functions are connected to a polarized hadron beam

Sivers’ functions are T- odd, but do not need another T- oddfunction to produce nonvanish asymmetries – the extractedproportional factor is T- odd, proportional to AN

Polarized SIDIS cross section:

1

( , , ) [ ( , , ) ( , , )] ( )

( ( ))T

x Q p q x k s q x k s D z

P n k s x zf D


Collins’ functions

Collins’ functions: 1 ( )H x

1( , , ) ( , , ) ( ) k n

D z k s D x k s H zM

Collins’ functions are connected to the unpolarized Fragmentation contributions to a hadron

Collins’ functions are T- odd, need another T- odd function to produce nonvanish asymmetries

Polarized SIDIS cross section:

1

( , , ) ( , ) [ ( , , ) ( , , )]

( (z), )

x Q p q x s D z k s D z k s

k n q x Hs


KT - Factorization

Factorization requires a separation of perturbative hard scale from nonperturbative hadronic scale a physical hard scale, Q, much larger than the kT

Q ~ pT >> kT

kT-factorization measures parton kT directly, while

twist-expansion gives integrated kT information No formal proof of kT-factorization for hadronic

collisions at kT ~ pT

kT-factorization works for semi-inclusive DIS and

Drell-Yan, or others with a large scale Q


Semi-inclusive deeply inelastic scattering

ℓℓ’

q

P

k

pk’

sT

Breit frame:

Two scale problem: Q2=-q2, pT

Fixed order pQCD:

Sudakov resummation:

q

k

k’P

QCDTQ p

TQ p

Single spin asym:

low pT data sensitive to parton kT

( ) 0 N TA s P p

If P is anti-parallel to p


ℓℓ’

q

P

k

pk’

sT

In q-P frame, if

2

2Tkk xP k nk n

T Tk p Q 2k we can neglect

in partonic part But, we cannot neglect

in partonic rt paTk

One can define kT-dependent and gauge invariant parton distributions Soft interaction between the hadrons can spoil factorization

S

Sudakov resummation (done in b- or kT-space) resums dynamical kT from gluon shower Parton orbital motion is more relevant to the intrinsic kT

Intrinsic vs dynamical kT


Open questions – (I)

kT approach Twist-3

1 ( )Tf x ( , )FT x x

1 ( )H z (3) ( , )D z z

Sivers:

Collins:

How much overlap between kT-approach at low pT

and twist expansion at high pT?

What these nonperturbative functions try to tell us?

“direct kT” vs “kT – moments”


Although there is kT-factorization in SIDIS, Drell-Yan

and the others, how to separate dynamical kT from

intrinsic kT ?

How Sivers and Collins functions are affected by

the process dependent gluon shower (resummation)?

…

Open questions – (II)

If there is no KT – factorization, how universal are

Sivers and Collins functions?


Backup transparencies


Collinear approximation is important

With collinear approximation:

Cuts

Cuts =

In general, matrix elements with different cuts are not equal:

≠

IR safe


When does the factorization lose its predictive power?

At the time when the nonperturbative functions lose their universalityFor final-state fragmentation:

Factorization breaks if the fragmentationtook place inside the hadronic medium

If the lifetime of the parton state of momentun k is shorter than the medium size

Lifetime: 02

1 22 2 fmmedium

jet

y t L rE m

2

25 GeV

GeVjetm


Asymmetries

Double longitudinal spin asymmetries:

Reduce under parity to:

Double transverse spin asymmetries:

Single longitudinal spin asymmetries:

Single transverse spin asymmetries:


Measure Sivers’ and Collins’ functions

12

Sin2

((

( ) ( )

) )qUT

q q

HeA

e q

x zq

x D z

SIDIS:

Collins functions:


Seperate Sivers’ and Collins’ functions


Initial success of RHIC pp runs

cross section measured

over 8 order of magnitude

[PRL 91, 241803 (2003)]

Good agreement with NLO

pQCD calculation at low pT

Can be used in interpretation

of spin-dependent results

9.6% normalization error not shown

single transverse spin asymmetries

Documents

qcd single spin asymmetry

transverse hadron spin

perturbative factorization

pqcd jianwei qiu

qcd pqcd factorization

nyjianwei qiu

anjianwei qiu

e704jianwei qiu