Periphery - jlab.org · Periphery Charge and Current Transverse Densities, Energy Momentum Tensor and Orbital Angular Momentum Carlos Granados ... Infinite Momentum Frame Light-Cone

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]

Documents

Periphery - jlab.org · Periphery Charge and Current Transverse Densities, Energy Momentum Tensor and Orbital Angular Momentum Carlos Granados ... Infinite Momentum Frame Light-Cone