periphery - jlab.org · periphery charge and current transverse densities, energy momentum tensor...
TRANSCRIPT
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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
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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
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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
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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
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• 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
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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)
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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
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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
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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
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• 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
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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
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• 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
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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
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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
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• 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
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Peripheral Densities from Invariant χPT
24
0 0Im1
2)(
M
itFπ
btKdt
b
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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
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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.
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Peripheral Densities from Invariant χPT
Heavy Baryon Expansion
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• 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
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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
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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
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Peripheral Densities in Light-Front χPT
IMF
Equal time xo
Equal LF time x+
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Peripheral Densities in Light-Front χPT
UiUyMk
yyky NNTN
'22 )(
)1()',,,(
Light-Cone Wave-function
Zero mode
P
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• 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
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•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
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''
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
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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
()' )(
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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
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• 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
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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
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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
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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)
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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]