“Time-reversal-odd” distribution functionsin chiral models

with vector mesons

Alessandro Drago

University of Ferrara

Outline

• T-odd distributions in QCD

• Chiral models with vector mesons as dynamical gauge bosons

• Large Nc expansion and the relation

),(),( 00 kk xfxf dT

uT

Brodsky, Hwang and Schmidt mechanism

Gauge link and factorization in Drell-Yan and in DIS (Collins 2002)

2†

03( , , , ) | (0, , ) (0) |

(2 ) 2T Tixp y ik yT

T T T y

dy d yP x k s e p y y W W p

From QCD to chiral lagrangians

In QCD the two main ingredients are:

Gauge theory Wilson lines

Factorization violation of universality

How to compute “T-odd” distributions in chiral models?

If we are not using a gauge theory we have no necessity to

introduce Wilson lines…

Chiral lagrangians with a hidden local symmetryBando, Kugo, Uehara, Yamawaki, Yanogida 1985

A theorem:

“Any nonlinear sigma model based on the manifold G/H is gauge equivalent to a linear model with a Gglobal Hlocal symmetry and the gauge bosons corresponding to the hidden local symmetry Hlocal

are composite gauge bosons”

For instance:

G = SU(2)L SU(2)R

H = SU(2)V

mesons

as hidden gauge fields

What is a hidden symmetry?

Kinetic term of a nonlinear sigma model:

Under the global SU(2)L X SU(2)R symmetry:

We rewrite U(x) in terms of two auxiliary variables:

The transformation rules under [SU(2)L X SU(2)R ]global X [SU(2)V]local are:

2 †( / 4) ( )

( ) exp[2 ( ) / ]

L f Tr U U

U x i x f

†( ) ( )L RU x g U x g

†

†

† †

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

L L L

R R R

x h x x g

x h x x g

V x ih x h x h x V x h x

†( ) ( ) ( )L RU x x x

where V is the gauge field

2 † † 2

2 † † 2

( / 4) ( )

( / 4) ( )

V L L R R

A L L R R

L f Tr D D

L f Tr D D

, ,

the covariant derivative is

( ) [ ( )] ( )L R L RD x iV x x

LV and LA are both invariant under [SU(2)L X SU(2)R ]global X [SU(2)V]local

Any linear combination of LV and LA is equivalent to the original kinetic term.

V

†L

A

V

For instance, fixing the [SU(2) ] gauge by

exp( / )

L is identical to the original lagrangian

L vanishes substituting the solution of field eqs.

local

R i f

About the mesons

So far we have given no dynamics to V

“For simplicity” we add a kinetic term:

The meson acquires a mass via spontaneous breaking of the hidden local SU(2)V symmetry

g = g universality

M2 = 2 g

2 f 2 KSRF relation

2

1

2

aaVf

2

2

1

4F

g

Introducing matter fields

linear representation of Hlocal

singlet of Gglobal

0

0( ( ))h x

matter ( )L i iV … is the simplest term

{

After the gauge fixing matter fields transform non-linearlybecause h(x) has to be restricted to the dependent form.

Again about matter fields (quarks)

Take a representation of G whose restriction to H contains 0

Defining so that g)we have twotypes of quarks:

“constituent quarks” (singlet of Gglobal)

“current quarks” (singlet of Hlocal )

Vector mesons are non-singlet of Hlocal and therefore they are

“composed” of constituent quarks .

Sivers function in models

constituent quarks

Almost identical to the QCD diagram.

has a physical mass.

The exchange of a meson can also be included

The 1/Nc expansion

At leading order (1/Nc)0 hedgehog solution(the problem of time reversal has already been solved by the link operators…)

(| | ) / 2

ˆ( ) ( )

( ) ( )

( ) ( )

a a

a ika ki

u d

r r r

r r r

r r

Grand Spin

hedgehog: 0

G J I

G

T0

c

The isospin dependence of ( , ) is dictated by the

symmetry of the single quark wave function.

Therefore, at leading order in 1/N :

f x k

),(),( 00 kk xfxf dT

uT

Conclusions

“T-odd” distributions can be computed also in chiral models, at least if vector mesons are introduced as gauge bosons.

At leading order in 1/Nc :

Work to be done: to compute explicitely these distributions.

Open problem: if one does not introduce a dynamics associated with the vector mesons, T-odd distributions can still be computed?

),(),( 00 kk xfxf dT

uT