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Transcript of Periphery - jlab.org · Periphery Charge and Current Transverse Densities, Energy Momentum Tensor...
Partonic Structure of the Nucleon’s Chiral
Periphery Charge and Current Transverse Densities, Energy Momentum Tensor and
Orbital Angular Momentum
Carlos Granados
in collaboration with Christian Weiss
Jefferson Lab, Virginia , US
QCD Evolution Workshop
Jefferson Lab
May 10, 2013
Aim and Context
• Spatial representation of hadrons as relativistic
systems
– GPDs, Transverse Densities
• Universality in large distance dynamics:
– Chiral symmetry breaking, effective field
theory
• Study of chiral periphery of transverse nucleon
structure
– Charge and current densities, EM form
factors
– Matter density and angular momentum,
energy momentum tensor and GPDs
Motivation
• Methodology
– Reveal spatial structure of χEFT:
Mπ-1 vs. short distance contributions
– Explore different formulations of orbital angular momentum in
field theory applied to a πN system
• Practical
– Calculate model independent chiral components of the nucleon
structure
– Constrain form factors, peripheral GPDs
• Experiment
– Form factors measurements in the low Q2 region
JLab E12-11-106 Q2 ~ 10-2 - 10-4 GeV2
– Connect chiral dynamics with Peripheral Processes in High Energy
ep and pp Reactions: EIC, LHC
Methodology
χEFT
L (N,π)
Infinite
Momentum
Frame
Light-Cone WF
Ψ(x,kt)
GPDs
ρ(x,b)H(x,0,t)
TRANSVERSE
DENSITIES
ρ(b) Spectral
Functions
Im F(t)
Invariant
Formulation
EQUIVALENT
• Fourier Transform in Transverse Momentum
• From Electromagnetic (EM) Form Factors (FF)
• From Energy-Momentum Tensor (EMT) FF
Transverse Charge and Current Densities
Definition
12122
UM
iUNJN
2
2
2
1 ΔFΔF
1
2
2122
UMgM
g
MiPPUNN N
22
2
(
2
)
ΔC~
ΔC
ΔBΔA )(
0
2
02
)( tFbJd
b
2
2
2
2)( T
iT Fed
b T b
2
2
2
2)( T
iT Fed
b T b
Transverse Charge and Current Densities
Definition
• Charge distribution in
transverse plane:
Proper densities, Relativistic
Systems
• Proper d
bρ...,,,0)x(0,J,,
bρ2,0)x(0,J
2311T
3
22
112
2
11T
0
22
21
21
xe
M
SxPxP
δ)x(xPδ,σP,x,σP,x
N
TT
σσTT
)(
TT
1
2
2
2
1212 ΔF2
ΔF NM
iNNJN
Parton current
picture
G.Miller, PRL99(2007)
Transverse Charge and Current Densities
Dispersion Representation, Spectral Functions
• Form Factors
– Analytic Continuation,
• Dispersion Relation : For Ret<0,
Branch cut 4Mπ
2
Imt
Ret
)(tF
24
)0'(Im
'
')(
M
itF
tt
dttF
l p
k
Or any t- channel
exchange
Transverse Charge and Current Densities
Dispersion Representation, Spectral Functions and
Analytic Structure Near Threshold
24
)0'(Im
'
')(
M
itF
tt
dttF
0
2
02
)( tFbJd
b
24
0 0Im1
2)(
M
itFπ
btKdt
b
b=1/Mπ-1
b=1.5/Mπ-1
b=0.75/Mπ-1
Transverse Charge and Current Densities
Dispersion Representation, Spectral Functions
• As b grows, F(t) is sampled closer to threshold (tthr=4Mπ2)
24
0 0Im1
2)(
M
itFπ
btKdt
b
2
102
bt
ebtK
bt
• For large b, Transverse densities are dominated by near
threshold values of spectral functions
Transverse Charge and Current Densities
Dispersion Representation, Spectral Functions
)(Im)(0 tFbtK
Transverse Charge and Current Densities
Analytic Structure Near Threshold
• Sub-threshold singularity
– End-point singularity
– Intermediate nucleon on-shell
– Limits convergence of expansion near threshold
– Controls large b(~MN2Mπ
-3) behavior of transverse densities
4Mπ2
Imt
Ret
)(tF
2
N
4
π
M
M
)(coscos)()(
)cos,(Im
1
1
d
tiBtA
tfF
cos)()(
1
0
1~
22 tiBtAiMl N
l
θ
2
224)()(
N
subsubsubM
MMttiBtA
tsub
p
k
l
0
• Chiral Region
– Δb~
• Molecular Region
– Δb >
Transverse Charge and Current Densities
Parametric Regions
fm5.1
1~
12/1
Mt
fm90
156
12
2
MMM
M N
Peripheral Densities from Invariant χPT
...4
1
2 25 cbabcaaaA
FF
g
intL
1
2
2
2
1212 ΔF2
ΔF UM
iUNJN
• Chiral EFT Lagrangian
– Relativistic formulation of pion-nucleon dynamics
– Axial Vector coupling and contact terms
• EM current. Leading π contributions to isovector spectral
functions:
12 NJN N
Peripheral Densities from Invariant χPT
• Find Spectral Functions (ImF(t))
instead,
– Cutkosky Rules
Pion on mass-shell
)3(
2
22
2
)2(
2
22)1(2
21
8
81
2
IF
gMF
IF
gMIg
FF
AN
ANA
)3,2(
212
4)3,2(
)1(
212
4)1(
2
2
NlDkDkDkd
iI
NkDkDkd
iI
N
2
2
2
2
2
2
2)3(
2
2
2
2
2
2
2
2
2
2
2
2)2(
2
2
2)1(
32
1
32
1
2
1
3
1
P
Mk
P
kPkN
P
Mk
P
kPk
k
P
kPkN
kkN
N
N
)(21 22
22
2,1
2,1
mkiimk
DN(l)
)()()(
2)( 212
4
kkDkDkd
itI
cmcm kkd
t
ktI k,0cos
16)(Im
1 01
12
3
2
2
21
21
ppP
kkk
2
4M
tkcm
• Spectral Functions
• Sub-threshold singularity at
x(tsub)=±i
Peripheral Densities from Invariant χPT
xxx
tP
Mt
gF
F
Mtgxx
tMxx
t
tP
Mt
gF
N
ANN
3)arctan(3
4
2Im
46
41)arctan(
8)arctan(
84
22
Im
2
2
522
2
2
22
2
2
322
22
2
522
2
2
21
2
22
2
442
Mt
Mtt
M
xN
2
224
N
subM
MMt
ImF2
ImF1
Strikman, Weiss, PRC82 (2010) 042201
Peripheral Densities from Invariant χPT
24
0 0Im1
2)(
M
itFπ
btKdt
b
Peripheral Densities from Invariant χPT
Chiral vs. Non-Chiral
ρ(b)~e-2Mπb
ρ(b)~e-Mρb
π π
ρ
Miller, Strikman, Weiss, PRC84 (2011) 045205
Chiral component dominant only at b >> 2 fm
Peripheral Densities from Invariant χPT
Heavy Baryon Expansion
)(~,1 0
2
OM
t
M
M
N
8
1
4
964
1
1
2
133
2
2
1
2
3
1
1
246
22
35
21
24
20
C
C
C
02
22
24
4
2
1)(
1
i
i
iAHB bMfεF
Mgb
232
1352
3
1
2
0
2
1
2
20
)(2
1)(
,2
1
2
1,
2
53,
2
93
2)(
)(16
1)(
8
1)(
16
1)(
Kf
f
KKKf
02
22
24)(Im
1
i
i
iAHB
M
tC
F
MgtF
ImF1(t)
%Diff.
Peripheral Densities from Invariant χPT
Heavy Baryon Expansion
• Consistency with QCD in the Large Nc Limit
– At Large Nc
• Mπ~ Nc0, MN,Δ~ Nc
1
• gπNN~Nc3/2, gπNΔ~Nc
3/2
• ρ~ Nc0
Peripheral Densities from Invariant χPT
Contributions from Δ and Large Nc Limit
• But in the Large Nc Limit,
considering only nucleonic
intermediate state χPT contributions
to F(t) lead to
0
)( cNcNNN NBNA
• Contributions from Δ-
intermediate states remedy this
discrepancy ,
0
)( ccNN NBNA
0
)()(
)( cN
NNNN
NBB
N Δ
Peripheral Densities from Invariant χPT
Contributions from Δ and Large Nc Limit
• FNπ Δ comparable to
FNπN at b< 2fm
• Cancelation of leading
Nc component
Peripheral Densities in Light-Front χPT
• Develop a partonic formulation of chiral dynamics
Charge and current of pions in the chiral periphery
Orbital angular momentum decomposition of chiral π-N
LCWF
• Demonstrate equivalence with invariant formalism
• Connect to GPD formalism
Compute model–independent χGPDs
Peripheral Densities in Light-Front χPT
IMF
Equal time xo
Equal LF time x+
Peripheral Densities in Light-Front χPT
UiUyMk
yyky NNTN
'22 )(
)1()',,,(
Light-Cone Wave-function
Zero mode
P
• Form factors from the infinite
momentum frame
– LCWF from NπN pseudo
scalar coupling
– In impact parameter space
)',,,()',,',()1()2(
22
1)(()',,,()',,',()1()2(
2
'2
2
2
2
'2
2
1
TNTNL
ATNTN
kykyyy
dykdF
M
gykykyyy
dykdF ct, )
†
†
,2
,2
,2
,
1
3
22
1
3
21
PP
JPF
M
PP
JPF
L
UUyMk
yy
F
igky A
TN 5'22 )(
)1(
2)',,,(
),()2(
),()(
2
2
TN
kiTN kye
kdby T
b
Peripheral Densities in Light-Front χPT
•Transverse densities from form
factors
• Charge and current densities from
pion-nucleon light-cone wave
functions with
2
2
2
2)( T
biT Fed
b T
'3
',',
2
'
2
3
2
'
1
)1(
',',
22)(
C.T.)()1()1(
)',(
22)(
yy
bybydyiMb
b
ygyy
bydyb
R
A
i
NA
N
NA
N
eMKMMyy
F
gy
MKMyy
F
igy
bb
bb
1
0
2
2
2
1),(
2
1),(
Peripheral Densities in Light-Front χPT
''
12
2
2)( 2
1
2
2
0
2
2
2
2
2
1 bMKM
MbMKy
y
yMgdyb
N
N
• Slower pions at larger impact
parameter
y=Mπ/MN
),()( 11 bydyb
'
2
'
1)1(
)',(2,
yy
byby
Peripheral Densities in Light-Front χPT
Energy-Momentum Tensor: Matter density and Orbital
Angular Momentum
• Energy-Momentum tensor form factors
(calculable in χPT)
• Angular momentum of a pion-nucleon
system,
• Transverse densities ρA, ρB from form
factors A and B, and A+B.
• Calculated leading chiral contribution to
spectral functions (Cutkosky Rules). No
contact term diagrams!
)0()0(2
1BAJN
thr
BA itBitAπ
tbdtKb 00Im1
2
1)( 0
p
p
N
uMgM
g
MiPPuNN
22
22
ΔCΔC
ΔBΔA
~
2'
2
()' )(
Energy-Momentum Tensor: Matter density and Orbital
Angular Momentum
• EMT form factors
• Invariant integrals
• Spectral functions (from Cutkosky rules)
)(21 22
22
2,1
2,1
mkiimk
DN(l)
2
22
2
442
Mt
Mtt
M
xN
• Asymptotic behavior of ρ controled by
ImF at threshold and near threshold
)arctan(5353
4
4
2
8
3Im
)arctan(332213
4
4
2
8
3Im
23
522
3
2
2
2
2
23
2
2
522
3
2
2
xxxx
tP
Mt
gB
xxP
Mxx
P
M
tP
Mt
gA NN
...1~
NM
MO
...1~
3
2
bM
MO N
Energy-Momentum Tensor: Matter density and Orbital
Angular Momentum
Energy-Momentum Tensor: Matter density and Orbital
Angular Momentum
• EMT form factors in IMF
• Corresponding Transverse densities as
overlap of LC- wave functions
• Matter distributions in impact parameter
space: • Moments of parton distributions
'3
''
'3
2
'
)1(
,,
22
3)(
)1(
),(
22
3)(
yy
bybyy
dyiMb
b
yy
byy
dyb
B
R
A
,2
,2
,2
,
12
33
2
12
33
2
PP
PBM
PP
PA
L
ρA(y,b) = 3yρ1(y,b)/4
y=Mπ/MN
'3
2
'
)1(
)',(
2
3,
yy
byybyA
Energy-Momentum Tensor: Matter density and Orbital
Angular Momentum
''
1222
3)( 2
1
2
2
0
2
2
22
2
2
bMKM
MbMKy
y
Mygdyb
N
NA
'21222
3)( 2
0
22
2
2
bMyKy
Mygdyb N
B
Summary
• Explored Nucleonic Structure in a setting that guarantees a model
independent analysis of the dynamics governed by χ EFT.
• Derived EM and EMT transverse peripheral densities from
corresponding FF
• From spectral functions (invariant formalism)
– Distinguish parametrical regions (chiral and molecular scales)
– Accuracy of the Heavy Baryon expansion
– Consistency with the QCD Large Nc limit ( add Δ-
contribution)
• Light–front χPT (IMF)
– Equivalent to invariant formalism
– Calculated transverse densities from LC-Wave Functions
(Connection to GPD formalism)
Outlook
• Understand origin of contact term (higher mass states,
nucleon compositeness)
• Use the chiral pion –nucleon system as a toy model for
exploring the nature of orbital angular momentum OAM
and other operators in field theory (moments of GPDs,
Axial form factors)
• Test use of πN-LCWF in experimental studies at Low and
High energies
Peripheral exclusive processes in e-N scattering at EIC
[Strikman, Weiss Phys.Rev. D69 (2004) 054012; EIC White Paper
2012]