Single spin asymmetries in pp scattering

Piet Mulders

TrentoJuly 2-6, 2006

mulders@few.vu.nl

_

Content

Single Spin Asymmetries (SSA) in pp scattering

• Introduction: what are we after?• SSA and time reversal invariance• Transverse momentum dependence (TMD)

Through TMD distribution and fragmentation functions to transverse moments and gluonic poles

• Electroweak processes (SIDIS, Drell-Yan and annihilation)• Hadron-hadron scattering processes• Gluonic pole cross sections• What can pp add?• Conclusions

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Introduction: what are we after?The partonic structure of hadrons

For (semi-)inclusive measurements, cross sections in hard scattering processes factorize into a hard squared amplitude and distribution and fragmentation functions entering in forward matrix elements of nonlocal combinations of quark and gluon field operators ( or G)

2. †

3 . 0

( . )( , ) (0) ( )

(2 )i pT

T n

d P dx p e P P

. †

. 0

( . )( ) (0) ( )

(2 ) T

i p

n

d Px e P P

lightcone

lightfrontTMD

2.

.T

p P xM

P np xP p n

2. †

3 . 0

( . )( , ) 0 (0) , , ( ) 0

(2 )i kT

T n

d P dz k e P X P X

FF

The partonic structure of hadrons

• Quark distribution functions (DF) and fragmentation functions (FF)– unpolarized

q(x) = f1q(x) and D(z) = D1(z)

– Polarization/polarimetryq(x) = g1

q(x) and q(x) = h1q(x)

– Azimuthal asymmetriesg1T(x,pT) and h1L(x,pT)

– Single spin asymmetriesh1

(x,pT) and f1T(x,pT); H1(z,kT) and

D1T(z,kT)

• Form factors

• Generalized parton distributions

FORWARDmatrix elements

x section

one hadron in inclusive or semi-inclusive scattering

NONLOCAL lightcone

OFF-FORWARD

Amplitude

Exclusive

NONLOCAL lightfront

LOCAL

NONLOCAL lightcone

pictures?

appendix

SSA and time reversal invariance

• QCD is invariant under time reversal (T)• Single spin asymmetries (SSA) are T-odd

observables, but they are not forbidden!• For distribution functions a simple distinction

between T-even and T-odd DF’s can be made– Plane wave states (DF) are T-invariant– Operator combinations can be classified

according to their T-behavior (T-even or T-odd)

• Single spin asymmetries involve an odd number (i.e. at least one) of T-odd function(s)

• The hard process at tree-level is T-even; higher order s is required to get T-odd contributions

• Leading T-odd distribution functions are TMD functions

Intrinsic transverse momenta

• In a hard process one probes partons (quarks and gluons)• Momenta fixed by kinematics (external momenta)

DIS x = xB = Q2/2P.q

SIDIS z = zh = P.Kh/P.q • Also possible for transverse momenta

SIDIS qT = kT – pT

= q + xBP – Kh/zh Kh/zh

2-particle inclusive hadron-hadron scattering qT = p1T + p2T – k1T – k2T

= K1/z1+ K2/z2x1P1x2P2 K1/z1+ K2/z2

• Sensitivity for transverse momenta requires 3 momentaSIDIS: * + H h + XDY: H1 + H2 * + X

e+e-: * h1 + h2 + X

hadronproduction: H1 + H2 h + X

h1 + h2 + X

p x P + pT

k z-1 K + kT

K2

K1

pp-scattering

TMD correlation functions (unpolarized hadrons)

(x, pT)

2 21 1

1

2

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

2T

T T T

i p Px p f x p P h x p

M

2(1) 2 21 12( ) ( , )

2T

T Th x d p h x pM

p

Transverse moment

• T-odd• Transversely polarized quarks

quark correlator

1212

( ) ( , ) ( )T Tx d p x p f x P 12 (1)

14( ) ( , ) ( )[ , ]T T Tx d p p x p i h x P

In collinear cross section

In azimuthal asymmetries

pictures?

Color gauge invariance

• Nonlocal combinations of colored fields must be joined by a gauge link:

• Gauge link structure is calculated from collinear A.n gluons exchanged between soft and hard part

• Link structure for TMD functions depends on the hard process!

SIDIS [U+] = [+]

DY [U-] = []

0

(0, ) exp ig ds AU

P

DIS [U]

(0) ( ) (0) (0, ) ( )U

Integrating [±](x,pT) [±](x)

[ ] . †[0, ] . 0

( . )( ) (0) ( )

(2 ) T

i p n

n

d Px e P U P

collinear correlator

2[ ] . †

[0, ] [0 , ] [ , ]3 . 0

( . )( , ) (0) ( )

(2 ) T T

i p n T nTT n

d P dx p e P U U U P

2[ ] 2 . †

[0, ] [0 , ] [ , ]3 . 0

( . )( ) (0) ( )

(2 ) T T

i p n T nTT T n

d P dx d p e P U i U U P

Integrating [±](x,pT) [±](x)

[ ] . †[0, ]

( . )( ) (0) ( )

(2 )[i p n

T

d Px e P U iD P

[0, ] [ , ] [ , ](0) ( . ) ( ) ( ) ]n n n nLCP U d P U gG U P

[ ]

1 11

( ) ( ) ( , )D G

i

x ix x dx x x x

[ ] 2 [ ]( ) ( , )T T Tx d p p x p transver

se moment

G(p,pp1)

1 11

( ) ( , ) ( , )

( )

D G G

i

xx dx P x x x x x

x

T-even T-odd

Gluonic poles

• Thus

[±](x) = (x) + CG

[±] G(x,x)

• CG[±] = ±1

• with universal functions in gluonic pole m.e. (T-odd for distributions)

• There is only one function h1(1)(x) [Boer-Mulders] and

(for transversely polarized hadrons) only one function f1T

(1)(x) [Sivers] contained in G

• These functions appear with a process-dependent sign• Situation for FF is (maybe) more complicated because

there are no T-constraints

Efremov and Teryaev 1982; Qiu and Sterman 1991Boer, Mulders, Pijlman, NPB 667 (2003) 201Metz and Collins 2005

What about other hard processes (e.g. pp and pp scattering)?_

Other hard processes

C. Bomhof, P.J. Mulders and F. Pijlman, PLB 596 (2004) 277

Link structure for fields in correlator 1

• qq-scattering as hard subprocess

• insertions of gluons collinear with parton 1 are possible at many places

• this leads for ‘external’ parton fields to a gauge link to lightcone infinity

Other hard processes

• qq-scattering as hard subprocess

• insertions of gluons collinear with parton 1 are possible at many places

• this leads for ‘external’ parton fields to a gauge link to lightcone infinity

• The correlator (x,pT) enters for each contributing term in squared amplitude with specific link

[Tr(U□)U+](x,pT)

U□ = U+U†

[U□U+](x,pT)

C. Bomhof, P.J. Mulders and F. Pijlman, PLB 596 (2004) 277

Gluonic pole cross sections

• Thus

[U](x) = (x) + CG

[U] G(x,x)

• CG[U±] = ±1

CG[U□ U+] = 3, CG

[Tr(U□)U+] = Nc

• with the same uniquely defined functions in gluonic pole matrix elements (T-odd for distributions)

examples: qqqq

2 2[( ) ] [ ]

2 2 2

1 52

1 1 1

c cq G

c c c

N N

N N N

2 2 2[( ) ] [ ]

2 2 2

2 1 3

1 1 1

c c cq G

c c c

N N N

N N N

CG [D1]

D1

= CG [D2]

= CG [D4]CG

[D3]

D2

D3

D4

( )

c

Tr U

NU

U U

Bacchetta, Bomhof, Pijlman, Mulders, PRD 72 (2005) 034030; hep-ph/0505268

Gluonic pole cross sections

• In order to absorb the factors CG[U], one can define specific

hard cross sections for gluonic poles (which will appear with the functions in transverse moments)

• for pp:

etc.

• for SIDIS:

for DY:

• Similarly for gluon processesBomhof, Mulders, Pijlman, EPJ; hep-ph/0601171

[ ]

[ ]

ˆ ˆ Dqq qqD

[ ] [ ]( )

[ ]

ˆ ˆD Dq q qq G

D

C

( )ˆ ˆq q q q

( )ˆ ˆq q qq

(gluonic pole cross section)

y

( )ˆ q q qqd

ˆqq qqd

examples: qqqq

†2 2

[( ) ] [ ]2 2 2

2 31

1 1 1

c cq G

c c c

N N

N N N

†2 2

[( ) ] [ ]2 2 2

11

1 1 1

c cq G

c c c

N N

N N N

D1

For Nc:

CG [D1] 1

(color flow as DY)

Conclusions

• Single spin asymmetries in hard processes can exist• They are T-odd observables, which can be described in

terms of T-odd distribution and fragmentation functions• For distribution functions the T-odd functions appear in

gluonic pole matrix elements• Gluonic pole matrix elements are part of the transverse

moments appearing in azimuthal asymmetries• Their strength is related to path of color gauge link in

TMD DFs which may differ per term contributing to the hard process

• The gluonic pole contributions can be written as a folding of universal (soft) DF/FF and gluonic pole cross sections

Belitsky, Ji, Yuan, NPB 656 (2003) 165Boer, Mulders, Pijlman, NPB 667 (2003) 201Bacchetta, Bomhof, Pijlman, Mulders, PRD 72 (2005) 034030Bomhof, Mulders, Pijlman, EPJ; hep-ph/0601171Eguchi, Koike, Tanaka, hep-ph/0604003Ji, Qiu, Vogelsang, Yuan, hep-ph/0604023

end

Local – forward and off-forward

.1 2' | ( ) | ( ) ( )i xP O x P e G t i G t

2t

1(0) | ( ) |G P O x P

2 (0) | ( ) |G P x O x P

Local operators (coordinate space densities):

P P’

Static properties:

Examples:(axial) chargemassspinmagnetic momentangular momentum

Form factors

Nonlocal - forward

†0, 0 ...O y y

| , | | 0, |xP O x y P P O y P

2. †| 0 | | 0 | ( )ip ydy e P y P P p P f p

Nonlocal forward operators (correlators):

Specifically useful: ‘squares’

Momentum space densities of -ons:

Selectivity at high energies: q = p

1( ) (0)dp f p GSum rules form factors

Nonlocal – off-forward

. † .1 2' | | ( , ) ( , )ip y i xdy e P x x y P e f t p i f t p

. †1| | (0, )ip ydy e P x x y P f p

. †2| | (0, )ip ydy e P x x x y P f p

Nonlocal off-forward operators (correlators AND densities):

1 1( , ) ( )dp f t p G t2 2( , ) ( )dp f t p G t

Sum rules form factors

Forward limit correlators

GPD’sb

Selectivity q = p

2t

Caveat

• We study forward matrix elements, including transverse momentum dependence (TMD), i.e. f(p||,pT) with enhanced nonlocal sensitivity!

• This is not a measurement of orbital angular momentum (OAM). Direct measurement of OAM requires off-forward matrix elements, i.e. GPD’s.

• One may at best make statements like:linear pT dependence nonzero OAM

no linear pT dependence no OAM

back

Interpretation

unpolarized quarkdistribution

helicity or chiralitydistribution

transverse spin distr.or transversity

need pT

need pT

need pT

need pT

need pT

T-odd

T-odd

back

unpolarized hadrons