Progresses with Transverse Momentum Distributions

N. Isgur Fellowship

Alessandro Bacchetta

Outline

Outline• Some words about the relevance of

TMDs

Outline• Some words about the relevance of

TMDs

• Some theory

Outline• Some words about the relevance of

TMDs

• Some theory

• Unpolarized TMDs

Outline• Some words about the relevance of

TMDs

• Some theory

• Unpolarized TMDs

• Sivers function (as an example of all other TMDs)

Outline• Some words about the relevance of

TMDs

• Some theory

• Unpolarized TMDs

• Sivers function (as an example of all other TMDs)

• Other TMDs

Some of EIC goals

Some of EIC goals

• Understand CONFINEMENT

Some of EIC goals

• Understand CONFINEMENT

• Study the structure of the proton (3D STRUCTURE, SPIN, FLAVOR...)

Some of EIC goals

• Understand CONFINEMENT

• Study the structure of the proton (3D STRUCTURE, SPIN, FLAVOR...)

• Test QCD in all its aspects (FACTORIZATION, EVOLUTION, LATTICE,...)

Some of EIC goals

• Understand CONFINEMENT

• Study the structure of the proton (3D STRUCTURE, SPIN, FLAVOR...)

• Test QCD in all its aspects (FACTORIZATION, EVOLUTION, LATTICE,...)

TMDs physics touches all this points

3D structure of the nucleon

3D structure of the nucleon• 3D structure = GPDs in impact parameter

space

3D structure of the nucleon• 3D structure = GPDs in impact parameter

space

• In general, parton distributions are 6 dimensional (Wigner distributions)

3D structure of the nucleon• 3D structure = GPDs in impact parameter

space

• In general, parton distributions are 6 dimensional (Wigner distributions)

• 3 dim. in coordinate space (GPDs)

3D structure of the nucleon• 3D structure = GPDs in impact parameter

space

• In general, parton distributions are 6 dimensional (Wigner distributions)

• 3 dim. in coordinate space (GPDs)

• 3 dim. in momentum space (TMDs)

X. Ji, PRL 91 (03), see talk by M. Schlegelfor even more dim. (8), see Collins, Rogers, Stasto, PRD77 (08)

• 3D structure = GPDs in impact parameter space

• In general, parton distributions are 6 dimensional (Wigner distributions)

• 3 dim. in coordinate space (GPDs)

• 3 dim. in momentum space (TMDs)

X. Ji, PRL 91 (03), see talk by M. Schlegelfor even more dim. (8), see Collins, Rogers, Stasto, PRD77 (08)

6D structure of the nucleon

New dimensions

!0.6 !0.4 !0.2 0 0.2 0.4 0.6pX !GeV"c#

!0.6

!0.4

!0.2

0

0.2

0.4

0.6

p Y!GeV

"c#

fu"p" !x#0.1#

3.58

0

QCDSF/UKQCD, PRL 98 (07) A.B., F. Conti, M. Radici, in preparation

Transverse position Transverse momentum

!0.2 !0.1 0 0.1 0.2pX !GeV"c#

!0.2

!0.1

0

0.1

0.2

p Y!GeV

"c#

fd"p" !x#0.1#

11.26

0

Momentum distributions

Vos, McCarthy, Am. J. Phys. 65 (97)

Momentum distributions

In scattering experiments, measuring momentum distributions is the closest we get to “imaging” a quantum object

Vos, McCarthy, Am. J. Phys. 65 (97)

Theory background

SIDIS

y

z

x

hadron plane

lepton plane

l!l ST

Ph

Ph"!h

!S

SIDIS cross sectiond!

dx dy d"S dz d"h dP 2h!

=#2

x y Q2

y2

2 (1! $)

!FUU,T + $ FUU,L +

"2 $(1 + $) cos "h F cos !h

UU + $ cos(2"h) F cos 2!h

UU

+ %e

"2 $(1! $) sin "h F sin !h

LU + SL

#"

2 $(1 + $) sin "h F sin !h

UL + $ sin(2"h)F sin 2!h

UL

$

+ SL %e

#"

1! $2 FLL +"

2 $(1! $) cos "h F cos !h

LL

$

+ ST

#sin("h ! "S)

%F sin(!h"!S)

UT,T + $ F sin(!h"!S)UT,L

&+ $ sin("h + "S) F sin(!h+!S)

UT

+ $ sin(3"h ! "S) F sin(3!h"!S)UT +

"2 $(1 + $) sin "S F sin !S

UT

+"

2 $(1 + $) sin(2"h ! "S) F sin(2!h"!S)UT

$+ ST %e

#"

1! $2 cos("h ! "S) F cos(!h"!S)LT

+"

2 $(1! $) cos "S F cos !S

LT +"

2 $(1! $) cos(2"h ! "S) F cos(2!h"!S)LT

$'

SIDIS cross sectiond!

dx dy d"S dz d"h dP 2h!

=#2

x y Q2

y2

2 (1! $)

!FUU,T + $ FUU,L +

"2 $(1 + $) cos "h F cos !h

UU + $ cos(2"h) F cos 2!h

UU

+ %e

"2 $(1! $) sin "h F sin !h

LU + SL

#"

2 $(1 + $) sin "h F sin !h

UL + $ sin(2"h)F sin 2!h

UL

$

+ SL %e

#"

1! $2 FLL +"

2 $(1! $) cos "h F cos !h

LL

$

+ ST

#sin("h ! "S)

%F sin(!h"!S)

UT,T + $ F sin(!h"!S)UT,L

&+ $ sin("h + "S) F sin(!h+!S)

UT

+ $ sin(3"h ! "S) F sin(3!h"!S)UT +

"2 $(1 + $) sin "S F sin !S

UT

+"

2 $(1 + $) sin(2"h ! "S) F sin(2!h"!S)UT

$+ ST %e

#"

1! $2 cos("h ! "S) F cos(!h"!S)LT

+"

2 $(1! $) cos "S F cos !S

LT +"

2 $(1! $) cos(2"h ! "S) F cos(2!h"!S)LT

$'

FUU,T (x, z, P 2h!, Q2)

High and low transverse momentum

M2 Q2

Q = photon virtualityM = hadron mass

Ph! = hadron transverse momentum q2T ! P 2

h!/z

q2T

Low

High and low transverse momentum

q2T ! Q2

M2 Q2

Q = photon virtualityM = hadron mass

Ph! = hadron transverse momentum q2T ! P 2

h!/z

q2T

Low

High and low transverse momentum

q2T ! Q2

M2 Q2

Q = photon virtualityM = hadron mass

Ph! = hadron transverse momentum q2T ! P 2

h!/z

q2T

Low

High and low transverse momentum

q2T ! Q2

M2 Q2

Q = photon virtualityM = hadron mass

Ph! = hadron transverse momentum q2T ! P 2

h!/z

TMDs and FFs

kT factorizationq2T

High

High and low transverse momentum

M2 ! q2T

M2 Q2

Q = photon virtualityM = hadron mass

Ph! = hadron transverse momentum q2T ! P 2

h!/z

High

High and low transverse momentum

M2 ! q2T

M2 Q2

Q = photon virtualityM = hadron mass

Ph! = hadron transverse momentum q2T ! P 2

h!/z

Collinear PDFs and FFs

collinear factorization

Low High

High and low transverse momentum

q2T ! Q2 M2 ! q2

T

M2 Q2

Q = photon virtualityM = hadron mass

Ph! = hadron transverse momentum q2T ! P 2

h!/z

Intermediate

M2 ! q2T ! Q2

Low High

High and low transverse momentum

q2T ! Q2 M2 ! q2

T

M2 Q2

Q = photon virtualityM = hadron mass

Ph! = hadron transverse momentum q2T ! P 2

h!/z

Intermediate

M2 ! q2T ! Q2

Need a machine with Q2 >> M2

Predictions in intermediate region

?

?

??

???

A.B., Boer, Diehl, Mulders, 0803.0227

kT factorization

Collins, Soper, NPB 193 (81)Ji, Ma, Yuan, PRD 71 (05)

FUU,T (x, z, P 2h!, Q2) = C

!f1D1

"

=#

d2pT d2kT d2lT !(2)$pT ! kT + lT ! P h!/z

%

x&

a

e2a fa

1 (x, p2T , µ2)Da

1(z, k2T , µ2) U(l2T , µ2)H(Q2, µ2)

kT factorization

Collins, Soper, NPB 193 (81)Ji, Ma, Yuan, PRD 71 (05)

FUU,T (x, z, P 2h!, Q2) = C

!f1D1

"

=#

d2pT d2kT d2lT !(2)$pT ! kT + lT ! P h!/z

%

x&

a

e2a fa

1 (x, p2T , µ2)Da

1(z, k2T , µ2) U(l2T , µ2)H(Q2, µ2)

TMD PDF TMD FF Soft factor Hard part

Collins--Soper evolution equations

Collins, Soper, Sterman, talk at Fermilab Workshop on Drell-Yan Process, Batavia, Ill., Oct 7-8, 1982

Collins, Soper, NPB 193 (81)Ji, Ma, Yuan, PRD 70 (04)

kT space

b space

Collins--Soper evolution equations

Collins, Soper, Sterman, talk at Fermilab Workshop on Drell-Yan Process, Batavia, Ill., Oct 7-8, 1982

Collins, Soper, NPB 193 (81)Ji, Ma, Yuan, PRD 70 (04)

Evolution equations for TMDs are

NOT standard DGLAP

kT space

b space

Resummation

Here we test pQCDHere we measure

hadronic properties

The calculation automatically fulfills Collins-Soper evolution and requires: •collinear PDFs •calculable pQCD contributions •nonperturbative part of TMDs

FUU,T (x, z, b, Q2) = x!

a

e2a (f i

1 ! Cia) (Caj !Dj1) e!Snonpert.e!Spert.

Resummation results

http://hep.pa.msu.edu/resum/index.html

Resummation results

http://hep.pa.msu.edu/resum/index.html

Resummation results

http://hep.pa.msu.edu/resum/index.html

Resummation results

http://hep.pa.msu.edu/resum/index.html

Resummation results

http://hep.pa.msu.edu/resum/index.html

Resummation results

http://hep.pa.msu.edu/resum/index.html

Most studies done for Drell--Yan

Theoretical activity

Theoretical activity• Issues related to kT-factorization in pp

collisionsCollins, Qiu, PRD75 (07)Qiu,Vogelsang, Yuan, PRD76 (07)Bomhof, Mulders, Pijlman, A.B., PRD72 (05)

Theoretical activity• Issues related to kT-factorization in pp

collisionsCollins, Qiu, PRD75 (07)Qiu,Vogelsang, Yuan, PRD76 (07)Bomhof, Mulders, Pijlman, A.B., PRD72 (05)

• Issues related to resummation in other structure functionsBoer, Vogelsang, PRD74 (06)Berger, Qiu, Rodriguez-Pedraza, PRD76 (07)A.B., Boer, Diehl, Mulders, 0803.0227

Theoretical activity• Issues related to kT-factorization in pp

collisionsCollins, Qiu, PRD75 (07)Qiu,Vogelsang, Yuan, PRD76 (07)Bomhof, Mulders, Pijlman, A.B., PRD72 (05)

• Issues related to resummation in other structure functionsBoer, Vogelsang, PRD74 (06)Berger, Qiu, Rodriguez-Pedraza, PRD76 (07)A.B., Boer, Diehl, Mulders, 0803.0227

Can have impact on LHC physics

Unpolarized TMDs

Nonperturbative transverse momentm

The nonperturbative part has to be fitted to dataUsually assumed to be Gaussian

FUU,T (x, z, b, Q2) = x!

a

e2a (f i

1 ! Cia) (Caj !Dj1) e!Snonpert.e!Spert.

Available extractions

•In b space

Brock, Landry, Nadolsky, Yuan, PRD67 (03)

111 data points(Drell-Yan)

Is it sufficient?

0 0.2 0.4 0.6 0.8pT2 !GeV"c#

0.5

1

1.5

2

2.5

3

3.5

f1u !x, pT2#

x!0.02

x!0.2x!0.5

0 0.2 0.4 0.6 0.8pT2 !GeV"c#

0

2

4

6

8

f1d !x, pT2#

x!0.02

x!0.2x!0.5

Simple model calculation suggests•flavor dependence•deviation from a simple Gaussian (required also by orbital angular momentum)

EIC tasks

EIC tasks• Gain better knowledge of the

nonperturbative part (also for fragmentation functions)

EIC tasks• Gain better knowledge of the

nonperturbative part (also for fragmentation functions)

• Flavor separation (also for fragmentation functions)

EIC tasks• Gain better knowledge of the

nonperturbative part (also for fragmentation functions)

• Flavor separation (also for fragmentation functions)

• Unintegrated gluon distribution function

EIC tasks• Gain better knowledge of the

nonperturbative part (also for fragmentation functions)

• Flavor separation (also for fragmentation functions)

• Unintegrated gluon distribution function

• Global fits with unintegrated PDFs

EIC tasks• Gain better knowledge of the

nonperturbative part (also for fragmentation functions)

• Flavor separation (also for fragmentation functions)

• Unintegrated gluon distribution function

• Global fits with unintegrated PDFs

• Cleaner information from jet-SIDIS?

Longitudinal spinEverything can be done also with helicity TMDs

0 0.2 0.4 0.6 0.8pT2 !GeV"c#

0

2

4

6

8

10

down, x!0.02

f1"g1

f1#g1

0 0.2 0.4 0.6 0.8pT2 !GeV"c#

1

2

3

4

up, x!0.02

f1"g1

f1#g1

Longitudinal spinEverything can be done also with helicity TMDs

0 0.2 0.4 0.6 0.8pT2 !GeV"c#

0

2

4

6

8

10

down, x!0.02

f1"g1

f1#g1

0 0.2 0.4 0.6 0.8pT2 !GeV"c#

1

2

3

4

up, x!0.02

f1"g1

f1#g1

Signs of orbital ang. mom.

Longitudinal spinEverything can be done also with helicity TMDs

0 0.2 0.4 0.6 0.8pT2 !GeV"c#

0

2

4

6

8

10

down, x!0.02

f1"g1

f1#g1

0 0.2 0.4 0.6 0.8pT2 !GeV"c#

1

2

3

4

up, x!0.02

f1"g1

f1#g1

Impossible to reproduce using simple Gaussians

Signs of orbital ang. mom.

Sivers function

Formalism

F sin(!h!!S)UT,T = C

!!pT · h

Mf"1T (x, p2

T , µ2)D1(z, k2T , µ2)

"

Formalism

F sin(!h!!S)UT,T = C

!!pT · h

Mf"1T (x, p2

T , µ2)D1(z, k2T , µ2)

"

• Collins--Soper evolution always required. Not only for the Sivers function, but also for D1

Formalism

F sin(!h!!S)UT,T = C

!!pT · h

Mf"1T (x, p2

T , µ2)D1(z, k2T , µ2)

"

• Collins--Soper evolution always required. Not only for the Sivers function, but also for D1

• We are far from the sofistication reached for FUU

Weighted asymmetries!

Ph!z

sin(!h ! !S)"

= !x#

a

e2a f!(1)

1T (x, µ2)D1(z, µ2)

Weighted asymmetries!

Ph!z

sin(!h ! !S)"

= !x#

a

e2a f!(1)

1T (x, µ2)D1(z, µ2)

• Not clear if it’s possible to recover DGLAP-like evolution equations for Sivers, but certainly for D1

Weighted asymmetries!

Ph!z

sin(!h ! !S)"

= !x#

a

e2a f!(1)

1T (x, µ2)D1(z, µ2)

• Not clear if it’s possible to recover DGLAP-like evolution equations for Sivers, but certainly for D1

• It’s important that the structure function falls fast enough with transv. mom.

Sivers function - TorinoFree fit“Symmetric sea”

Anselmino et al., 0805.2677

Sivers function - Bochum

-0.04

-0.02

0

0.02

0.04

0 0.5

(a)

x

x f1T !(1)a(x)

u-u

-0.04

-0.02

0

0.02

0.04

0 0.5

(b)

x

x f1T !(1)a(x)

d-d

-0.04

-0.02

0

0.02

0.04

0 0.5

(c)

x

x f1T !(1)a(x)

s- s

Arnold et al. , 0805.2137

Limits of the analyses

Limits of the analyses

• Evolution equations neglected (will be very relevant at EIC)

Limits of the analyses

• Evolution equations neglected (will be very relevant at EIC)

• Limited x range (EIC can improve on both sides)

Limits of the analyses

• Evolution equations neglected (will be very relevant at EIC)

• Limited x range (EIC can improve on both sides)

• 173 data points (cf. 467 points in Δq fits)

Gluon Sivers function

M. Burkardt, PRD69 (04)

Gluon Sivers function

• Based on the Burkardt sum rule, the gluon Sivers function is claimed to be small

M. Burkardt, PRD69 (04)

Gluon Sivers function

• Based on the Burkardt sum rule, the gluon Sivers function is claimed to be small

• Limited x range

M. Burkardt, PRD69 (04)

Gluon Sivers function

• Based on the Burkardt sum rule, the gluon Sivers function is claimed to be small

• Limited x range

• There could be nodes in the function

M. Burkardt, PRD69 (04)

Gluon Sivers function

• Based on the Burkardt sum rule, the gluon Sivers function is claimed to be small

• Limited x range

• There could be nodes in the function

• Connection with orbital angular momentum is not straightforward

M. Burkardt, PRD69 (04)

EIC tasks

EIC tasks

• Provide more data

EIC tasks

• Provide more data

• Measure weighted asymmetries

EIC tasks

• Provide more data

• Measure weighted asymmetries

• Extend x, Q range

EIC tasks

• Provide more data

• Measure weighted asymmetries

• Extend x, Q range

• Demand theory improvements

EIC tasks

• Provide more data

• Measure weighted asymmetries

• Extend x, Q range

• Demand theory improvements

• Explore gluon Sivers function

Connection with orbital angular momentum

Connection with orbital angular momentum

• There is no direct connection between Sivers function and OAM

Connection with orbital angular momentum

• There is no direct connection between Sivers function and OAM

• Several TMD observables can help constrain models with OAM see also next talk by H. Avakian

Connection with orbital angular momentum

• There is no direct connection between Sivers function and OAM

• Several TMD observables can help constrain models with OAM see also next talk by H. Avakian

• Global effort to put together GPDs and TMDs information is required

Other observables

Leading-twist functions

Leading-twist functions

• There are 8 leading-twist TMDs

Leading-twist functions

• There are 8 leading-twist TMDs

• There are 8 leading-twist structure functions where the PDFs are connected with either D1 or Collins function

Leading-twist functions

• There are 8 leading-twist TMDs

• There are 8 leading-twist structure functions where the PDFs are connected with either D1 or Collins function

• 4 can be studies also through jet-SIDIS without fragmentation functions

EIC tasks

EIC tasks

• It’s difficult to choose the most relevant measurements at the moment

EIC tasks

• It’s difficult to choose the most relevant measurements at the moment

• Shall EIC measure them all? see COMPASS Coll. 0705.2402 see also next talks

Post Scriptum

• I did not mention the issue of AdS/QCD correspondence see work of Brodsky, de Teramond

• EIC can map a new dimension: the distribution of parton in transverse momentum space

• EIC can map a new dimension: the distribution of parton in transverse momentum space

• It’s a new dimension also in terms of theoretical complexity and offers new tests of QCD

• EIC can map a new dimension: the distribution of parton in transverse momentum space

• It’s a new dimension also in terms of theoretical complexity and offers new tests of QCD

• With present experiments and phenomenology, we just scratched the surface

• EIC can map a new dimension: the distribution of parton in transverse momentum space

• It’s a new dimension also in terms of theoretical complexity and offers new tests of QCD

• With present experiments and phenomenology, we just scratched the surface

• EIC will be a precision machine for TMDs