0 Simonetta Liuti University of Virginia Structure of Nucleons and Nuclei Workshop Como, June 10 th-...

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Simonetta LiutiUniversity of Virginia

Structure of Nucleons and Nuclei Workshop

Como, June 10th-14th, 2013

N&N-StructureSimonetta Liuti

Generalized TMDs

presented by Osvaldo Gonzalez Hernandez - Torino

N&N-StructureSimonetta Liuti 2

In collaboration with:

Aurore Courtoy (Liege U.)Gary Goldstein (Tufts U.)Osvaldo Gonzalez Hernandez (INFN Torino since Fall 2012)

Graduate StudentsKunal Kathuria (U.Va.)Evan Askanazi (U.Va.)Abha Rajan (U.Va.)

Question of what components do Lq and measure(Ji, Xiong, Yuan, 2011, 2012, Burkardt 2012)


A Sum Rule was constructed which identified components of the Energy Momentum Tensor (EMT) with the Angular Momentum carried by quarks and gluons. (Jaffe&Manohar (JM))

A Saga in Several Episodes: Developing a Sum Rule for Angular Momentum

1990



Partonic picture:work directly in A+=0 gauge quark and gluon spin components are identified with the n=1 moments of spin dependent structure functions from DIS, ΔΣ and ΔG.

ΔGΔΣ

New Relation (X.Ji)1997

q

p’+= p –Δ

q'=q+Δ

p+q

P’

e

p+

New processes (DVCS …) were thought of, whose structure functions – the GPDs - admit n=2 moments that were identified with the (spin+OAM) quark and gluon components of the SR P

H,E



Lots of work followed in a series of papers by Wakamatsu, Leader and collaborators, Chen et al., Hatta, and Burkardt to understand the origin of this discrepancy.


Conceptual issues: “What type of information can one obtain?”The goal is to give a partonic interpretation of the observables

Once a scheme, whichever, is defined to connect with experiment, we stick to it. Corollary: If Lcan is not observable, does it exist?

Practical issues: “how can one extract information from experiment”?Example: GPDs are hard to extract because they need to be “deconvoluted” from Compton Form Factors…

Both points pose important theoretical problems.

We would like to do some phenomenology


p’p

x,kT’

bin

GPD

bout

x,kT


zT

p’p

bout

bin

TMD

x,kT

x,kT

p’=p, kT’=kT


(using constraints from Jlab flavor separated form factor data, G. Cates et al, 2011)

Transverse coordinate density distributions

PauliDirac

dd

u

u

GPD

What do we obtain (or wish to obtain) from experiment


TMDs

Calculations done using reggeized diquark model,O. Gonzalez et al.,arXiv:1206.1876

What do we obtain (or wish to obtain) from experiment

Bacchetta, Conti Radici

Gonzalez et al.

Qo2≈ 0.3 GeV2

d

u

This needs to be evolved to the scale of the data(Collins, Rogers, Boglione, Prokudin, …)


Unintegrated GPDs GTMDs

What partonic configurations do they correspond to?

Fourier Transform wrt ΔT Wigner Distributions

Notice! Two transverse momenta, simultaneously present


zT

p’p

GTMD

b

GTMDs correlate partonic configurations with both:• a shift in transverse position from the initial to final state zT

• an average transverse position b.

averageshift

When can these configurations exist in the “impulse approximation”, and when do we need to introduce FSI?


How would we observe GTMDs?

q, Λγ

k’= k –Δ. λ’

q’=q+Δ, Λγ’

k, λ

p’= k-Δ, Λ’

p, Λ

p, Λ

u-channel two body scattering

k’= k –Δ. λ’ k, λ

ϑ


k’CM= kCM –Δ. λ’

kCM, λ

p’CM= kCM-Δ, Λ’

pCM, Λ

ϑ This angle determined by ΔT

z

y

x

CoM motion determined by

Transverse Plane


In CoM the scattering happens in one plane therefore there is only one transverse direction defined by ΔT

In CoM frame ΔT and individual partons kT are parallel to one another

Average kT describes the motion of the CoM


λ'λ

same as GPDs

“New” terms can exist

GTMDs in Cartesian BasisMeissner, Metz, Schlegel, JHEP 2009


Spin(Helicity/Transversity) Basis

Quark-proton helicity amplitude


Chiral Even Sector

G14

F11

2F13-F11

2G13-G14

Us Helicity amps contentMetz


Can the new GTMDs F14 and G11 be defined at leading twist?

In two-body scattering these terms are Parity violating(valid in CM frame, demonstration in general frame hinges on appropriate Lorentz Transformation)

In a quark-target model only H, H are present~


Why are they important: the idea has been emerging that canonical OAM can be related to F14

Ji, Xiong and Yuan, 2012

Lorce and Pasquini, 2011


However F14 drops out of the observables

Not because it contains “even more information on the structure of the proton than what can be measured”, but for a good reason…


The demonstration is rather technical

Out of all the functions that parameterize the generalized correlator (MMS’09)…

… the matrix elements for the “type 3” ones transform the opposite way underParity. These define F14


Summary so far…

We confirm that canonical AM cannot be measured if defined as

because it corresponds – in two body scattering – to a combination of helicity amplitudes that violates Parity


Consider now the angular momentum sum rule including twist 3 contributions Belitsky and Mueller (2000), Polyakov et al. (2000), Hatta (2011)

-Jq Sq-Lq

Twist 3 decomposition of hadronic tensor

OAM is a twist three contribution


In order to describe GTMDs including partonic interactions, i.e. moving out of a 2 body scattering picture, we need a situation whereeven in the CoM, one can define two independent transverse vectors as if there were a third particle, so that the additional combinations giving rise to “LU”-type contributions can exist


Twist 3 GTMDs

spin flip

spin non flipMMS’09


Finally…a prerequisite for all these calculations is that we use a model whose parameters are constrained by a quantitative fit to all data which are relevant for GPDs (Form Factors, DIS and DVCS)


Sections of Wigner distributions: Completely Unpolarized Case

kT

u quark


All together: integrated over x and kT (this is a fit to real data: DVCS+G.Cates et al)

d quarku quark

N&N-StructureSimonetta Liuti 33 APS Denver 4/12/13Simonetta Liuti 33

Sections of Wigner distributions: Transversely Polarized Case

kT

u quark


All together: integrated over x and kT

Δ1Eu - Δ1Ed


Twist 3 component G2


Summary of this part and Open QuestionsG2 can be measured directly, and it does have a partonic interpretation, if one views it as a twist 3 quantity.

We can distinguish in models a WW part defining OAM, and a genuine twist 3 part.

Canonical angular momentum cannot be measured directly, but it can be linked to G2 through M. Burkardt’s “force-type” correction

We need to devise experiments that are sensitive to this contributions (they might be already in the available data!)

Question of L vs. T interpretation

We focused mainly on quarks, question of gluon component

One way to shed light on these questions is to look at deuteron


Spin 1 systems, due to

1)The presence of additional L components (D-waves) 2)Isoscalarity

provide a crucial test the working of the angular momentum sum rules

Nucleon DVCS

Deuteron

HISO=Hu+Hd EISO=Eu+Ed

λN

λq

λ

z=pN+/P+

D

What are the quark and gluon angular momenta in the deuteron?


Sum Rules in Deuteron (K.Kathuria)

OAM

Momentum

Spin 1/2

Spin 1/2

Form factors from the respective Energy Momentum tensors, Tμν


If f++(z) = f+0(z)=δ(1-z) then H2=H+E

F1+F2= GM GM

Differently from the nucleon, in the transverse case, we are not finding the same relation (other GPDs describing charge and tensor component enter…). More details later…

Longitudinal

Transverse Deuteron

DeuteronNucleon


How does Ji sum rule differ from JM in the deuteron?

Effect of evolution

Models are important Model calculations of L with w.f.s’ seem to lead to similar conclusions as M.Burkardt, more to explore here… avenue to compare different schemes?

Using GPDs from Goldstein, Gonzalez, SL, PRD84


Nuclear effect much larger than in unpolarized scattering

Needs to be treated systematically…N&N-StructureSimonetta Liuti 42

Observables: DVCS from deuteron

subleading

Can the deuteron help us understand the role of gluon OAM?(Brodsky, Gardner, 2006)

By connecting Lg to SSA in

≈ 0

Both L q and Lg contribute! Since Lq disappears because of isospinsymmetry, if AUT

π is 0 then Lg is 0


Two ways to go beyond the two body scattering scenario:

312

1

2 3

Our model for fracture functions (work in progress…)

scattered quark

outgoing qqqq-bar

Initial proton

h

q-bar frag.

BB

AA


GG, S. Liuti and O. Gonzalez, GPDS, THEIR RELATIONSHIPS WITH TMDS & RELATED TOPICSProceedings of QCD-N12, Bilbao, Spain - 2012


ConclusionsWe conclude that the distribution of an unpolarized quark in a longitudinally polarized nucleon does indeed measure OAM

We provide a theoretical basis for the “simple moment” of Wigner distributions, connecting to twist 3 GPDs Belitsky, Mueller, Polyakov, Hatta,…

Now one can think seriously of observables (DV processes including twist 3 GPDs)

Our argument stresses the fact that in order to measure OAM one has to go beyond a two body scattering picture

Transverse vs. Longitudinal sum rule? Gluon components? we should look into Spin 1 systems also


G. Goldstein, O. Gonzalez Hernandez, S.Liuti, J.Phys. G, 39 (2012) 11500G. Goldstein, O. Gonzalez Hernandez, S.Liuti, Phys. Rev. D84, 034007 (2011)S.K. Taneja, K. Kathuria, S.Liuti, G. Goldstein, Phys. Rev. D86 (2012)G. Goldstein, O. Gonzalez Hernandez, S.Liuti, K. Kathuria, arXiv:1206.1876, subm. PRCG. Goldstein, O. Gonzalez Hernandez, S.Liuti, K. Kathuria, “Proceedings of 4th QCD Evolution Workshop”, Jefferson Lab, May (2012)


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Sections of Wigner distributions: Mixed Transverse/Longitudinal Basis

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