Progresses with Transverse Momentum Distributions• 3D structure = GPDs in impact parameter space...
Transcript of Progresses with Transverse Momentum Distributions• 3D structure = GPDs in impact parameter space...
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