Twist Three Generalized Parton Distributions For

Orbital Angular Momentum

Abha RajanUniversity of Virginia

May 30, 2015

Simonetta Liuti,Aurore Courtoy, Michael Engelhardt

Proton Spin Crisis

● Quark and gluon spin contribution is smaller than ½

● Need to look at other Sources of spin → Orbital Angular Momentum

Helicity Projection Operator

Measured by EMC experiment in 1980s to be only 33% of total !!

Spin Crisis !!!

Jaffe Manohar and Ji Decompositions of Proton Spin

● Differ in their definition of Orbital Angular Momentum

● Covariant Derivative (Ji) : Description gives access to Total Angular Momentum → Spin + OAM

● Jaffe Manohar: Spin and OAM contributions are separate

Form Factors and PDFsElectron Current● Spin ½ point Particle ● No internal Structure

?

?

Unknown Proton Current● Parametrized using

form factors

Elastic Scattering

Express cross section in terms of Parton Distribution Functions

Deep Inelastic Scattering

Elastic Scattering

GPDs and GTMDs

● Generalized Parton Distributions : Off Forward PDFs

DVCS

● Enter at amplitude level

● Generalised Transverse Momentum Distributions : Off forward TMDs

Ji Sum Rule : partonic angular momentum !

Xiangdong Ji, PRL 78.610,1997

Functions ofMeissner Metz and Schlegel, JHEP 0908 (2009)

Helicity Amplitudes

● Helicity structure of the quark quark correlator (soft part)

Helicity Flip treated as Transverse Polarization (transversity basis)

Projects Helicity

Twist 3● The examples so far were at leading order

What happens if we include a gluon on one side?

● Dynamical Twist → Suppressed by

● Genuine Twist Three → Quark gluon quark correlator

Good Component

Bad Component (implicitly includes a gluon – higher twist, dynamically dependent on good components)

Projection operators determine whether good components contribute or bad

Twist 3

Genuine Tw 3 → Explicit includes a GluonMulders, Tangerman

Jaffe, Ji Nucl Phys B 375, 1992

A correlator with an explicit gluon

● These twist 3 distribution functions from the quark quark correlator are connected to quark gluon quark correlator !(The gluon dependence is integrated over)

Or look at relations derived using equations of motion

Twist 3 Helicity Amplitudes

The quark helicity is opposite to the helicity of the quark gluon combination

Use these Relations to work out Helicity Amplitudes

Why is Twist 3 interesting ?

● We are getting access to phenomenon that involve three particle interactions

● Orbital Angular Momentum : nucleon spin, confinement → Twist 3 GPD G2

● d2 : Color force● TMD e

Orbital Angular Momentum● Polyakov Sum Rule → Twist three GPD G2 gives partonic

Orbital Angular Momentum

● Measuring G2 → Connection to an observable Courtoy, Liuti, Goldstein, Gonzalez, Rajan Phys Lett B731(2014)

Kiptily, Polyakov Eur Phys J C 37 (2004); Hatta and Yoshida, JHEP (1210), 2012

Spin Contribution Total Angular Momentum → J

The color force : d2

● d2 → measure of color electric and magnetic force on quarks ; third moment of genuine twist 3 part of distributions

Burkardt PRD88 (2013)

Posik et al, PRL 111 (2013)

Burkhardt Cottingham Sum Rule

Dynamic Tw 3 KT squared moment in Tw 2

Genuine Tw 3 : Quark gluon quark correlator

What is this sum rule about? What does it say ? → Signifies that there is a twist 2 part and a twist 3 part

Unintegrated correlator

Connecting Jaffe and Ji OAM

Off forward correlator

Twist 3 GPD KT squared moment of Twist 2 GTMD

Using Equation of Motion to get to OAM

Integrate over KT

The correlator will now expand into GTMDs

OAM : G2(x)Take off forward case of unintegrated correlator expansion :

Diquark Model Calculation : Using Helicity Amplitudes

● The proton splits into a quark and a diquark structure. While the active quark interacts with the photon, the diquark acts as the 'spectator'

Goldstein, Gonzalez, Liuti PRD 84 (2011)

Calculating F_14

● Ji → Straight Gauge link

● Jaffe Manohar → Staple Link

● The difference is the torque

Unpolarized Quark in a longitudinally polarized proton

Bacchetta, Conti, Radici (2008)

Burkardt (2013)

qJM Lq

J i d2zTdz

2 3 P ', ' (z) (g) dyUz

z1G1(y ) z2G

2 (y ) U (z) P, z 0

F_14 with Eikonal Quark

Longitudinally Polarized Proton and Off forward

Transversely Polarized Proton

Calculating Twist Three Amplitudes: Diquark Spectator Model

● The bad component is a quark gluon combination with spin opposite to that of the quark

Goldstein, Gonzalez, Liuti PRD 84 (2011)

Courtoy, Liuti, Goldstein, Gonzalez, Rajan Phys Lett B731(2014)

Helicity Amplitudes in Diquark Model

Helicity Amplitudes in Diquark Model

Helicity Amplitudes For G2

G2(x) vs x

x(H+E)

Htilde

G2

F14

­1

­0.8

­0.6

­0.4

­0.2

 0

 0.2

 0.4

 0.6

 0.8

 0  0.2  0.4  0.6  0.8  1

x

x(H+E)H tilde

G2F14

e(x) vs x

Goldstein, Gonzalez, Liuti PRD 84 (2011)Using parameters from GGL.

Using Twist 3 amplitudes in the forward case

Summary and Conclusions● GPDs, TMDs and GTMDs carry a wealth of information

about partonic structure of the nucleon

● Calculation of twist 3 amplitudes allows us access to a whole new range of phenomenon

– Partonic Orbital Angular Momentum : GPD G2

– Color Electric and Magnetic forces : d2

– Confinement● Further work in separating the Wandzura Wilczek and

genuine twist three (explicit gluon) contributions

● Understand the connection between Jaffe Manohar and Ji OAM

Twist 3

Bacchetta, Diehl, Goeke et al JHEP 0702 (2007) Jakob, Mulders and Rodrigues, Nucl Phys A626 (1997)

Quark quark Correlator

Transverse Momentum Distributions● Transverse Momentum Distributions are like

Parton Distribution Functions but, also include transverse momentum of quarks

● Measured using SIDIS (need to observe atleast one product (apart from the electron) after scattering to fix transverse momentum)

TMDs → Unintegrated PDFs

● Projection Operator : Access different properties of the partonic structure of nucleon with different projection operators

Boglione, Mulders Phys Rev D60 (1999)

Boglione, Mulders Phys Rev D60 (1999)Jakob, Mulders and Rodrigues, Nucl Phys A626 (1997)

Semi Inclusive DeepInelastic Scattering

Bacchetta et al, JHEP, 2007

The Spin Structure of the Proton

● Spin is a form of angular momentum

● The proton is a fermion, it has spin ½

● The proton is not a point particle : it has internal structure

● How is the spin ½ distributed among the quarks and gluons (partons) that constitute the proton?

Probing Internal Structure : Deep Inelastic Scattering

● Access parton distribution functions :

Density of partons as a function of the fraction of proton momentum

● Take different combinations of beam and target polarizations to get different parton distributions

Momentum fraction

Measuring Quark Contribution to Proton Spin

● Expansion of correlator / cross section using distribution functions is similar to writing the unknown proton current using Form Factors

● Take opposite beam polarizations and take difference of cross section

Helicity Projection Operator

Measured by EMC experiment in 1980s to be only 30% of total !!

Spin Crisis !!!

Generalised Parton Distributions

● Accessed via Deeply Virtual Compton Scattering

● Include Momentum transfer in description

● When momentum transfer is zero we get back to parton distributions

Momentum Transfer

Xiangdong Ji, PRL 78.610,1997

Total Angular Momentum and GPDs

Spin Orbital Angular Momentum

Ji's Sum Rule

Measurement of GPDs gives us total angular momentum of partons !

Helicity Amplitudes

● The Soft part can also be described using Helicity Amplitudes

Transversely polarized nucleon (transversity basis)

2

Matthias Burkardt, PRD 66 (2002)

Goldstein, Gonzalez, Liuti PRD 84 (2011)

Twist 3

● The analysis so far was at leading order

What happens if we include a gluon on one side? Equivalent to using a different set of projection operators

Good Component

Bad Component (implicitly includes a gluon – higher twist)

R

Twist 3 GPDs

G2 and E have the same set of helicity amplitudes !! (at different twist)

Courtoy, Liuti, Rajan, Goldstein, Gonzalez Phys Lett B731(2014)

Extending to Twist Three

● As a first check we used the twist three helicity amplitudes to calculate the twist three transverse momentum distribution 'e'

Includes transverse momentum of quarks

Jakob, Mulders and Rodrigues (Nucl Phys A, 1997)

Bacchetta et al (JHEP, 2007)

e(x) vs x

Goldstein, Gonzalez, Liuti PRD 84 (2011)Using parameters from GGL.

Next need to extend to calculating G2 to get access to Twist Three contribution to Orbital Angular Momentum!

Summary and Conclusions

● The twist three GPD G2 gives us access to Orbital Angular Momentum (Experimental Observable!)

● Present results only for twist two, what is happening at genuine twist three?

● Helicity Amplitude structure at twist three successfully used to calculate TMD 'e'

● Need to extend analysis to get Helicity Amplitude Structure of G2 (ongoing)

Thank you !

Density in transverse plane for transversely polarized proton

Uniform in all directions

The derivative causes a shift

Fourier Transform

Matthias Burkardt, PRD 66 (2002)

With momentum transfer

b

k

Proton Spin b X k

Matthias Burkardt, PRD 66 (2002)

● The quark density is not centered at the origin anymore

● The quarks are moving along z direction● b_y X k_z leads to OAM along x

Courtoy, Liuti, Rajan, Goldstein, Gonzalez Phys Lett B731(2014)

Courtoy, Liuti, Rajan, Goldstein, Gonzalez Phys Lett B731(2014)

Connected to an observable!!

Belitsy Mueller and Kirchner, NuclPhys B 629 (2002)

Observing OAM Experimentally

Unpolarized beamLongitudinally Polarized Target

Sensitive to G2

Goldstein, Gonzalez, Liuti PRD 84 (2011)

GTMDs and OAM

● GTMDs are like GPDs but, they also include transverse momentum of quarks

Meissner Metz and Schlegel, JHEP 0908 (2009)

Proposed to be connected to OAMLorce and Pasquini, PRD 84 (2011)

Quark Target Model

● Treat the proton like a free quark

Brodsky Diehl Hwang, Nucl Phys B 596 (2001)

Bag ModelThe proton is treated like three free quarks confined to a given volume. Only one quark interacts with the lepton(ongoing with Dr Courtoy)

OAM Sum RuleGenuine Twist 3 contribution

GPD corresponding to g1 → spin contribution

Hatta and Yoshida, JHEP (1210), 2012

Courtoy, Goldstein, Gonzalez, Liuti, Rajan Phys Lett B731(2014)

Twist 3 GPDs consist of a twist 2 contribution (Wandzura Wilczek) and a genuine twist 3 contribution (explicitly includes a gluon)

Kiptily, PolyakovEur Phys J C 37 (2004)

M. Penttinen, M.V. Polyakov, A.G. Schuvaev, M.I. Strikman, Phys. Lett. B 491, 96 (2000)

Twist 3

Dirac Equation only gives us the bad components in terms of the good components at fixed

Bad components are not an independent dynamical field

Light front time like variable

This is the definition of Twist through OPE (dimensional analysis)

Expansion in terms of local operators

Jaffe, Spin Review - Erice, 1996

Non local

Mass Dimension Connected to Twist

Projection of identityJakob, Hab Thesis

e

● Contribution from finite quark masses to mass of nucleon ; second moment in x gives the number of quarks of that flavor