1 Transverse spin physics Piet Mulders ‘Transverse spin physics’ RIKEN Spinfest, June 28 & 29, 2007 mulders@few.vu.nl.

Transverse spin physics

Piet Mulders

‘Transverse spin physics’RIKEN Spinfest, June 28 & 29, 2007

mulders@few.vu.nl

Abstract

QCD is the theory underlying the strong interactions and the structure of hadrons. The properties of hadrons and their response in scattering processes provide in principle a large number of observables. For comparison with theory (lattice calculations or models), it is convenient if these observables can be identified with well-defined correlators, hadronic matrix elements that involve only one hadron and known local or nonlocal combinations of quark and gluon operators. Well-known examples are static properties, such as mass or charge, form factors and parton distribution and fragmentation functions. For the partonic structure, accessible in high-energy (hard) scattering processes, a lot of information can be obtained, in particular if one finds ways to probe the ‘transverse structure’ (momenta and spins) of partons. Relevant scattering experiments to extract such correlations usually require polarized beams and targets and measurements of azimuthal asymmetries. Among these, single spin asymmetries are special because of their particular time-reversal behavior. The strength of single spin asymmetries depends on the flow of color in the hard scattering process, which affects the nonlocal structure of quark and gluon field operators in the correlators.

Content

• Lecture 1: – Partonic structure of hadrons– correlators: distribution/fragmentation

• Lecture 2:– Correlators: parameterization, interpretation, sum

rules– Orbital angular momentum?

• Lecture 3:– Including transverse momentum dependence– Single spin asymmetries

• Lecture 4:– Hadronic scattering processes– Theoretical issues on universality and factorization

3 colors

Valence structure of hadrons: global properties of nucleons

proton

• mass• charge• spin• magnetic moment• isospin, strangeness• baryon number

Mp Mn 940 MeV Qp = 1, Qn = 0 s = ½ gp 5.59, gn -3.83 I = ½: (p,n) S = 0 B = 1

Quarks as constituents

A real look at the proton

+ N ….

Nucleon excitation spectrumE ~ 1/R ~ 200 MeVR ~ 1 fm

A (weak) look at the nucleon

n p + e +

= 900 s Axial charge GA(0) = 1.26

A virtual look at the proton

N N + N N_

Local – forward and off-forward m.e.

2( ) (' | ) | )( i x GP O tP e t Gx i

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

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

Local operators (coordinate space densities):

P P’

Static properties:

Form factors

Examples:(axial) chargemassspinmagnetic momentangular momentum

Nucleon densities from virtual look

proton

neutron

• charge density 0• u more central than d?• role of antiquarks?• n = n0 + p+ … ?

( ) ( )i iG t x

Quark and gluon operators

Given the QCD framework, the operators are known quarkic or gluonic currents such as

probed in specific combinationsby photons, Z- or W-bosons( ) 2 1 1

3 3 3 ...u d sJ V V V

( ) 21 42 3 sin ...Z u u u

WJ V A V

( ) ' ' ...W ud udJ V A

'5( ) ( ) '( )q qA x q x q x

( ) ( ) ( )qV x q x q x

(axial) vector currents

{ }( ) ~ ( ) ( )qT x q x D q x

( ) ~ ( ) ( )GT x G x G x

energy-momentum currents

probed by gravitons

Towards the quarks themselves

• The current provides the densities but only in specific combinations, e.g. quarks minus antiquarks and only flavor weighted

• No information about their correlations, (effectively) pions, or …

• Can we go beyond these global observables (which correspond to local operators)?

• Yes, in high energy (semi-)inclusive measurements we will have access to non-local operators!

• LQCD (quarks, gluons) known!

Deep inelasticexperiments

fragmenting quark

scattered electron

proton remnants

Results directly reflect quark, antiquark and gluon distributions in the proton

QCD & Standard Model

• QCD framework (including electroweak theory) provides the machinery to calculate cross sections, e.g. *q q, qq *, * qq, qq qq, qg qg, etc.

• E.g. qg qg

• Calculations work for plane waves

) .( ( )0 , ( , )s ipi ip s u p s e

Confinement in QCD

• Confinement limits us to hadrons as ‘quark sources’ or ‘targets’

• These involve nucleon states• At high energies interference terms between different hadrons

disappear as 1/P1.P2

• Thus, the theoretical description/calculation involves for hard processes, a forward matrix element of the form

( ) .( ) ,s ii

pX P S e

1 1( ) ( ). .( ) ( ,) isi

i p p pX P S eA

3( , ) | | | | ( )

(2) 0)

2(0j i

d Pp P P X X P P P p

1( , ) | |

)0) )i p

i ij jp P d e P P

quarkmomentum

Partonic structure of hadrons

• Hard (high energy) processes– Inclusive leptoproduction– 1-particle inclusive

leptoproduction– Drell-Yan – 1-particle inclusive

hadroproduction

• Elementary hard processes

• Universal (?) soft parts- distribution functions f- fragmentation functions D

PHH Ph

distributioncorrelator

fragmentationcorrelator

hard process

Need PH.Ph ~ s (large) to get separation of soft and hard parts

Allows d… = d(p.P)…

(x, pT)

(z, kT)

Intrinsic transverse momenta

• Hard processes: Sudakov decomposition for momenta:

p = xPH + pT + n

• zero: pT.PH = n2 = pT.n

large: PH.n ~ s

hadronic: pT2 ~ PH

2 = MH2

small: ~ (p.PH,p2,MH2)/s

• Parton virtuality enters in and is integrated out Hq(x,pT) describing quark distributions

• Integrating pT collinear Hq(x)

• Lightlike vector n enters in (x,pT), but is irrelevant in cross sections

Similarly for quark fragmentation: k = zKh + kT + ’ n’

correlator qh(z,kT)

PHH Kh

(calculation of) cross section in DIS

Full calculation

+LEADING (in 1/Q)

x = xB = q2/P.q

Lightcone dominance in DIS

Parametrization of lightcone correlator

Jaffe & Ji NP B 375 (1992) 527Jaffe & Ji PRL 71 (1993) 2547

leading part

• M/P+ parts appear as M/Q terms in cross section• T-reversal applies to (x) no T-odd functions

Basis of

partons

‘Good part’ of Dirac space is 2-dimensional

Interpretation of DF’s

unpolarized quarkdistribution

helicity or chiralitydistribution

transverse spin distr.or transversity

Off-diagonal elements (RL or LR) are chiral-odd functions Chiral-odd soft parts must appear with partner in e.g. SIDIS, DY

Matrix representationfor M = [(x)+]T

Quark production matrix, directly related to thehelicity formalism

Anselmino et al.

Bacchetta, Boglione, Henneman & MuldersPRL 85 (2000) 712

Results for deep inelastic processes

** 1 ˆN qN X q qf

* *1 ˆN q

N X q qg

This has resulted in a good knowledge of u(x) = f1

pu(x), d(x), u(x), d(x) and (through evolution equations) also G(x)

For example in proton d > u, in neutron u > d (naturally explained by a p component in neutron, providing additional insight beyond GE)

Polarized experiments (double spin asymmetries) have provided spin densities u(x) = g1

pu(x), etc.

End lecture 1

Lecture 2

Local – forward and off-forward

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

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

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

P P’

Static properties:

Examples:(axial) chargemassspinmagnetic momentangular momentum

Form factors

Nonlocal - forward

†0, 0 ...O

| , | | 0, |P O P P O P

2. †| 0 | | 0 | ( )ipd e P P P p P f p

Nonlocal forward operators (correlators):

Specifically useful: ‘squares’

Momentum space densities of -ons:

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

Selectivity at high energies: q = p

Quark number

• Quark distribution and quark number

• Sum rule:

• Next higher moment gives ‘momentum sum rule’

| (0) (0) | ( )2P P n n P .

1( . ) | (0) ( ) | | ( ) 2i pLCd P e P P f x P

( ) ( ) ( )q q qq qdx f x dx f x dx f x n n

{ } { }| (0) (0) | ( )2q qP P P P

( ) ( ) ( ) ( )q q qq qdx xf x dx xf x dx xf x

p n px

Quark axial charge/spin sum rule

• Quark chirality distribution and quark spin/axial charge

• Sum rule:

• This is one part of the spin sum rule

5, | (0) (0) | , 2AP S P S g M S .

5 1( . ) , | (0) ( ) | , | ( )2i pLCd P e P S P S g x M S

( ) ( ) ( )q q q q qA Adx g x dx g x dx g x g g q q

2 2 Q GL G L

Full spin sum rule

• The angular momentum operators in this spin sum rule

• The off-forward matrix elements of the (symmetric) energy momentum tensor give access to JQ and JG

2 2 Q GL G L

Q GT T T

M x T x T x x

Local – forward and off-forward

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

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

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

P P’

Static properties:

Form factors

Nonlocal - forward

†0, 0 ...O

| , | | 0, |P O P P O P

2. †| 0 | | 0 | ( )ipd e P P P p P f p

Nonlocal forward operators (correlators):

Specifically useful: ‘squares’

Momentum space densities of -ons:

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

Selectivity at high energies: q = p

Nonlocal – off-forward

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

. †1| | (0, )ip xdx e P y y x P f p

. †2| | (0, )ip xdx e P y y y x 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’s

Selectivity q = p

Quark tensor charge

• Quark chirality distribution and quark spin/axial charge

• Sum rule:

• Note that this is not a ‘spin’ measure, even if h1(x) is the distribution of transversely polarized quarks in a transversely polarized nucleon!

5, | (0) (0) | , 2T T TP S P S g P S .

5 1( . ) , | (0) ( ) | , | ( )2i pT LC Td P e P S P S h x P S

( ) ( ) ( )q q q q qT Tdx h x dx h x dx h x g g q q

( ) ( )q qdx h x dx h x q q ‘Transverse spin’ ~

(no decent operator!)

A transverse spin rule

• One can write down a ‘transverse spin’ sum rule• It was first discussed by Teryaev and Ratcliffe, but

it involves the twist-3 funtion gT=g1+g2

(Burkhardt-Cottingham sumrule)• … and a similar gluon sumrule

• It does not involve the ‘transverse spin’. This appears in the Bakker-Leader-Trueman sumrule (which involves the assumption of having ‘free’ quarks).

( ) ( )q qTdx g x dx g x q q

(my version of Trieste meeting)

End lecture 2

Lecture 3

Issues

• Knowledge of partonic structure can be extended by looking at the ‘transverse structure’

• Time reversal invariance provides a nice discriminator for ‘special effects’

• Example is the color flow in hard processes, which is reflected in the nonlocal structure of matrix elements and shows up in single spin asymmetries

• Single spin asymmetries are being measured (HERMES@DESY, JLAB, COMPASS@CERN, KEK, RHIC@Brookhaven)

The partonic structure of hadrons

The cross section can be expressed in hard squared QCD-amplitudes and distribution and fragmentation functions entering in forward matrix elements of nonlocal combinations of quark and gluon field operators ( or G)

2. †

( . )( , ) (0) ( )

(2 )i pT

d P dx p e P P

( . )( ) (0) ( )

(2 ) T

d Px e P P

lightcone

lightfront: = 0TMD

p P xM

P np xP p n

2. †

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

(2 )i kT

d P dz k e P X P X

PHH Ph

distributioncorrelator

fragmentationcorrelator

hard process

Need PH.Ph ~ s (large) to get separation of soft and hard parts

Allows d… = d(p.P)…

(x, pT)

(z, kT)

(calculation of) cross section in SIDIS

Full calculation

+LEADING (in 1/Q)

Lightfront dominance in SIDIS

Three external momentaP Ph q

transverse directions relevantqT = q + xB P – Ph/zh

orqT = -Ph/zh

Gauge link in DIS

• In limit of large Q2 the resultof ‘handbag diagram’ survives

• … + contributions from A+ gluonsensuring color gauge invariance

A+ gluons gauge link

Ellis, Furmanski, PetronzioEfremov, Radyushkin

Distribution

From AT() m.e.

including the gauge link (in SIDIS)

One needs also AT

G+ = +AT

AT()= AT

(∞) + d G+

Belitsky, Ji, Yuan, hep-ph/0208038Boer, M, Pijlman, hep-ph/0303034

Parametrization of (x,pT)

• Link dependence allows also T-odd distribution functions since T U[0,] T† = U[0,-]

• Functions h1 and f1T

(Sivers) nonzero!

• Similar functions (of course) exist as fragmentation functions (no T-constraints) H1

(Collins) and D1T

Interpretation

unpolarized quarkdistribution

helicity or chiralitydistribution

transverse spin distr.or transversity

need pT

pT-dependent functions

T-odd: g1T g1T – i f1T and h1L

h1L + i h1

(imaginary parts)

Matrix representationfor M = [±](x,pT)+]T

Bacchetta, Boglione, Henneman & MuldersPRL 85 (2000) 712

T-odd single spin asymmetry

• with time reversal constraint only even-spin asymmetries• the time reversal constraint cannot be applied in DY or in 1-particle

inclusive DIS or ee

• In those cases single spin asymmetries can be used to measure T-odd quantities (such as T-odd distribution or fragmentation functions)

W(q;P,S;Ph,Sh) = W(q;P,S;Ph,Sh)

W(q;P,S;Ph,Sh) = W(q;P, S;Ph, Sh)

W(q;P,S;Ph,Sh) = W(q;P,S;Ph,Sh)

_ ____

__ _time

reversal

symmetrystructure

parity

hermiticity

Lepto-production of pions

H1 is T-odd

and chiral-odd

End lecture 3

Lecture 4

Quarks

2. [ ]

[0, ]3 . 0(0) ( )

( . )( , ; , )

(2 )q i p CTij T i nj

d P dx p n C e P U P

. [ ][0, ] . 0

( . )( ; )

(2 )(0) ( )ij

q i p n

d Px n e P U P

[ ][0, ]

expCig ds AU

• Integration over = P allows ‘twist’ expansion• Gauge link essential for color gauge invariance

• Arises from all ‘leading’ matrix elements containing

Sensitivity to 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 = q + xBP – Kh/zh Kh/zh

= kT – pT 2-particle inclusive hadron-hadron scattering qT = K1/z1+ K2/z2x1P1x2P2 K1/z1+ K2/z2

= p1T + p2T – k1T – k2T • Sensitivity for transverse momenta requires 3

momentaSIDIS: * + H h + XDY: H1 + H2 * + X

e+e-: * h1 + h2 + X

hadronproduction: H1 + H2 h1 + h2 + X h + X (?)

p x P + pT

k z-1 K + kT

pp-scattering

Knowledge of hard process(es)!

Generic hard processes

C. Bomhof, P.J. Mulders and F. Pijlman, PLB 596 (2004) 277 [hep-ph/0406099]; EPJ C 47 (2006) 147 [hep-ph/0601171]

Link structure for fields in correlator 1

• E.g. qq-scattering as hard subprocess

• Matrix elements involving parton 1 and additional gluon(s) A+ = A.n appear at same (leading) order in ‘twist’ expansion

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

• this leads for correlator involving parton 1 to gauge links to lightcone infinity

SIDIS [U+] = [+]

DY [U-] = []

A 2 2 hard processes: qq qq

• 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)

Gluons

2. [ ] [ ']

[ ,0] [0, ]3 . 0

( . )( , ; , ')

)) ( )

2i p C CT n

d P dx p C C e P U PF FU

• Using 3x3 matrix representation for U, one finds in gluon correlator appearance of two links, possibly with different paths.

• Note that standard field displacement involves C = C’

[ ] [ ][ , ] [ , ]( ) ( )C CF U F U

Integrating [±](x,pT) [±](x)

[ ] . †[0, ] . 0

( . )( ) (0) ( )

(2 ) T

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

d Px e P U iD P

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

( ) ( ) ( , )D G

x ix x dx x x x

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

se moment

G(p,pp1)

( ) ( , ) ( , )

xx dx P x x x x x

T-even T-odd

Gluonic poles

• Thus: [U](x) = (x)

[U](x) =

(x) + CG[U] G

• Universal gluonic pole m.e. (T-odd for distributions)

• G(x) contains the weighted T-odd functions h1(1)(x)

[Boer-Mulders] and (for transversely polarized hadrons) the function f1T

(1)(x) [Sivers]

• (x) contains the T-even functions h1L(1)(x) and g1T

(1)(x)

• For SIDIS/DY links: CG[U±] = ±1

• In other hard processes one encounters different factors:

CG[U□ U+] = 3, CG

[Tr(U□)U+] = Nc

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

A 2 2 hard processes: qq qq

• 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)

examples: qqqq in pp

2 2[( ) ] [ ]

c cq G

2 2 2[( ) ] [ ]

c c cq G

CG [D1]

= CG [D2]

= CG [D4]CG

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

examples: qqqq in pp

†2 2

[( ) ] [ ]2 2 2

c cq G

†2 2

[( ) ] [ ]2 2 2

c cq G

For Nc:

CG [D1] 1

(color flow as DY)

Bacchetta, Bomhof, D’Alesio,Bomhof, Mulders, Murgia, hep-ph/0703153

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:

• for SIDIS:

for DY:

Bomhof, Mulders, JHEP 0702 (2007) 029 [hep-ph/0609206]

ˆ ˆ Dqq qq

[ ( )] [ ][ ]

ˆ ˆU D Dq q qq G

[ ]ˆ ˆq q q q

[ ]ˆ ˆ

q q qq

(gluonic pole cross section)

( )ˆ q q qqd

ˆqq qqd

examples: qgqin pp

2 2[( ) ] [ ]

c cq G

Only one factor, but more DY-like than SIDIS

Note: also etc.

Transverse momentum dependent collinear

examples: qgqg

2 2[( ) ] [ ]

c cq G

e.g. relevant in Bomhof, Mulders, Vogelsang, Yuan, PRD 75 (2007) 074019

examples: qgqg

2 2[( ) ] [ ]

c cq G

examples: qgqg

†2 2

[( )( ) ] [( ) ] [ ]2 2 2 2

2( 1) 2( 1) 1 1

c cq G

c c c c

N N N N

2 2[( ) ] [ ]

c cq G

†2 2

[( )( ) ] [( ) ] [ ]2 2 2 2

2( 1) 2( 1) 1 1

c cq G

c c c c

N N N N

[( )( ) ] [ ]2 2

†4 2 2

[( )( ) ] [( ) ] [ ]2 2 2 2 2 2

( 1) ( 1) 1 1

c c cq G

c c c c

N N N N

examples: qgqg

†2 2

[( )( ) ] [( ) ] [ ]2 2 2 2

2( 1) 2( 1) 1 1

c cq G

c c c c

N N N N

2 2[( ) ] [ ]

c cq G

†2 2

[( )( ) ] [( ) ] [ ]2 2 2 2

2( 1) 2( 1) 1 1

c cq G

c c c c

N N N N

[( )( ) ] [ ]2 2

†4 2 2

[( )( ) ] [( ) ] [ ]2 2 2 2 2 2

( 1) ( 1) 1 1

c c cq G

c c c c

N N N N

†2 2

[( )( ) ] [( ) ] [ ]2 2 2

2( 1) 2( 1) 1

[( )( ) ] [ ]2 2

It is also possible to group the TMD functions in a smart way into two!(nontrivial for nine diagrams/four color-flow possibilities)

But still no factorization!

‘Residual’ TMDs

• We find that we can work with basic TMD functions [±](x,pT) + ‘junk’• The ‘junk’ constitutes process-dependent residual TMDs

• The residuals satisfies int (x) = 0 and int G(x,x) = 0, i.e. cancelling kT contributions; moreover they most likely disappear for large kT

† †

†[( )( ) ]int

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

( , ) ( , ) ( , )

x p x p x p

[ ] [ ] [ ] [ ]int( , ) 2 ( , ) ( , ) ( , )T T T Tx p x p x p x p

definite T-behavior

no definiteT-behavior

Conclusions

• Appearance of single spin asymmetries in hard processes is calculable

• For integrated and weighted functions factorization is possible

• For TMDs one cannot factorize cross sections, introducing besides the normal ‘partonic cross sections’ some ‘gluonic pole cross sections’

• Opportunities: the breaking of universality can be made explicit and be attributed to specific matrix elements

Related:Qiu, Vogelsang, Yuan, hep-ph/0704.1153Collins, Qiu, hep-ph/0705.2141Qiu, Vogelsang, Yan, hep-ph/0706.1196Meissner, Metz, Goeke, hep-ph/0703176

1 Transverse spin physics Piet Mulders ‘Transverse spin physics’ RIKEN Spinfest, June 28 & 29, 2007 mulders@few.vu.nl.

proton n n n n

proton n

factorization slide

fm slide

nucleon n p e

local operators

gluon operators

properties of hadrons

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