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


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)


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


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



• 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



)(Im)(0 tFbtK


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 >


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


• 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


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


24

0 0Im1

2)(

M

itFπ

btKdt

b


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


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.


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


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


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


IMF

Equal time xo

Equal LF time x+


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


•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),(


''

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


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

()' )(


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


Angular Momentum


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


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]

periphery - jlab.org · periphery charge and current transverse densities, energy momentum tensor...

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