About SSA in hadronic interactions

Mauro Anselmino: The transverse spin structure of the nucleon - III

About SSA in hadronic interactions

TMDs and SSA in inclusive hadronic interactions

TMDs and SSA in Drell-Yan processes

Drell-Yan processes, the transversity golden channel

Alternative ways to transversity A fundamental QCD test

) ˆ (

d )

( )

(

d _{/ /}

, , , , ,

/ x f x D z

f _{a b p b} ^{ab cd} _c

g q q d c b a

p

a  

    ^ 



PDF FF

pQCD elementary interactions

(collinear configurations)

X pp  

factorization theorem

a

b c



D

X

X

 ^ˆ

f

f

p

p

TMDs and SSAs in hadronic collisions

Latest PHENIX data on unpolarized cross

section

X p

p  

RHIC,

GeV

 200 s

and A

stays at high energies ….

STAR-RHIC √s = 200 GeV 1.2 < p_T < 2.8

SSA in hadronic processes: intrinsic k_┴, factorization?

Two main different (?) approaches (W. Vogelsang) Generalization of collinear scheme

(assuming factorization)

a b

c



X

D

X

 ^ˆ

f f

p p

) ,

( )

ˆ ( d ) (

) (

d _{/ /}

, , , , ,

/   

 _{ } ^ _{  }













g q q d c b a

p a

first proposed by Field-Feynman

Mulders -

Boer ty

transversi

Collins ty

transversi effect Sivers



 (1)

(2) (3)

suppressed by phases

(1)

(2) (3) Possible sources of SSA, simple approach

(one k_┴ at a time)

General formalism with helicity amplitudes

) ,

( )

ˆ ( ˆ

) (

) (

d

, , ,

; , ,

; ,

, / ,

/ ,

/ ,

/ ,

,

' '

' '

' '

 













 













 



































p k

k

k k

z D

, M

M

, x f

, x f

C C c c cd

ab

b a d c cd

ab

b a d c

B B

b A b

A a a B

A

b a

b b

S B b S B b a

a S

A a S A a X

C S B S A

non planar process, plenty of phases

main remaining contribution to SSA from Sivers effect

) ,

( )

ˆ ( d

) (

) (

d

/ / /

,











































p k

k

k k

z D

,

, x f

, x f

c b

a cd

ab

b b

p b a

a

q q p

N X

p S p

M.A., M. Boglione, U. D’Alesio, E. Leader, S. Melis, F. Murgia, PR D71, 014002 (2005), PR D73, 014020 (2006)

Computation of helicity amplitudes

) , ( )

, ( ) , ( )

,

(p₃ ₃  ^u p₁ ₁ u p₄ ₄  _u p₂ ₂ u

M  p2

p3

p4

p1





i i i

i i

i p _i

p

u(  ) ⁰ 1  _ (ˆ ) ˆ  sin cos ,sin sin ,cos





  p p

,





i p

p  ⁰ ,p



 



 

 



 



 

_ ^ _/2^/2 _ ^ _/2^/2

) 2 / sin(

) 2 / ) sin(

(ˆ

)

2 / sin(

) 2 / ) cos(

(ˆ

i i i

i

i i

i i i

i i

i i i

e e e

e















 _p

p

if scattering is not planar all Φ_i are different and many phases remain in the amplitudes; they strongly suppress the results of integrations over k_┴

Dirac-Pauli helicity spinors

8 leading-twist spin-k_┴ dependent distribution functions

Courtesy of Aram Kotzinian

E704 data STAR data

U. D’Alesio, F. Murgia

fit prediction

For a somewhat different approach see lectures by W. Vogelsang, great progresses in the last months

p p Q² = M²

q_T q_L

l⁺

l^–

 *

TMDs and SSAs in Drell-Yan processes



 



 



Y

D f x Q f x Q





d ₁ k ² ₂ k ²

factorization holds, two scales, M², and

q

3 planes:

plane

plane,

* ^  ^

 l l

p  ^{plane  to polarization vectors ,}



 



   

 

     







 1 cos sin cos 2 ^{sin cos2}

3 1 4

3

1 _{2 2 2}

φ θ

d θ d

Unpolarized cross section already very interesting

Collins-Soper frame

For example, the cos 2φ term can be originated by the Boer-Mulders effect

ˆ ) ( ˆ

) ,

( )

( d

d 

 

 h

x

 h

x

k

 d 

 d 

ˆ ) ( ˆ

) ,

( )

, (

d   h

x

k

 h

x

k

 d 

 d 

Polarized D-Y processes with intrinsic k_┴ have a rich structure, similar to SIDIS

SSA in D-Y has a contribution from the coupling of the transversity distribution to B-M function

ˆ ) ( ˆ

) ,

2 ( ) 1 ,

2 ( ) 1 ,

( _{/ 1}

/

,    



   s  pk

s k q

q p

q p

q h x k

M k k

x f

x f q

B-M

 2 cos

Sivers effect in D-Y processes By looking at the cross section

one can single out the Sivers

effect in D-Y processes

d q d

4 4







 d ( , ) ( ) ^ˆ

d



 

f

x

k

 f

x

 d

   











































 

 



q q q q q q T q A q q q B q q

q q A q q q B q q

N T

q q

q q

q N

x f

x f

d d

e

x f

x f

d d

e

d d

d A d

) ,

( )

, ( )

( 2

) ,

( )

, ( )

(

/ /

2 2

2 2

/ / 2

2 2

2

k k

q k

k k

k

k k

q k

k k

k













s s d d u u

q  , , , , ,

Predictions for A_N at RHIC,

p

p  l

l

X

M. A., M. Boglione, U. D’Alesio, A. Kotzinian, F. Murgia, A. Prokudin

See later for crucial issue concerning Sivers function in D-Y





1 9

4 ₂

2 2

1 2

2 2

1 2

2 1

2 2 2

2

Q x q Q x q Q

x q Q x q x e

x s M dx

dM d

a a

a a

a a F

 

 ^





s q

x s

M x

x x

x

x_F  ₁  _{2 1 2}  ² /  _F  2 _L /

Simple partonic cross section at collinear level

p p

Q² = M²

q_T q_L

l⁺

l^–

 *

range of x₁, x₂ explored depends on



 

 

 



 







q q

q q q q q q

TT

TT

e q x q x q x q x

x h

x h

x h

x h

a e

A ( ) ( ) ( ) ( )

) (

) ( )

( )

ˆ ( d

d

d d

2 1

2 1

2

2 1

1 1

2 1

1 1

2









) cos(2 cos

1 sin d ˆ

d ˆ

d ˆ d ˆ

ˆ ² ₂ 













 



 _^  _^

aTT

Direct access to transversity from double transverse spin asymmetry

Possible access transversity: Drell-Yan processes











  

 l l p l l p p l l p

p ,  ,

RHIC energies:

small x₁ and/or x₂

h_1q (x, Q²) evolution much slower than Δq(x, Q²) and q(x, Q²) at small x A_TT at RHIC is very small

smaller s would help Martin, Schäfer, Stratmann, Vogelsang Barone, Calarco, Drago

2

2 100 GeV

GeV

200 

 M

s

10 3

2 ^



A_TT for Drell-Yan processes at RHIC

upgrades in luminosity expected

h₁ from

 





) ( )

( )

( )

ˆ (

2 1

2 1

1 1

2 1

2 1

2

2 1

1 1

2 1

1 1

2

x u x u

x h

x a h

x q x q x

q x q e

x h

x h

x h

x h

a e

A _TT ^{u u}

q q

q q q q q q

TT

TT 



 

 

2 2 2 GeV GeV

210

30  

 M

s

GSI energies: ^{large x1,x2}

one measures h₁ in the quark valence region: A_TT is estimated to be large, between 0.2 and 0.4

at GSI

X l l p

p^{ }  ^{ }

PAX proposal: hep-ex/0505054

Energy for Drell-Yan processes

"safe region": M  M_{J /}

s M J 

 ^{2 /}



QCD corrections might be very large at smaller values of M:

yes, for cross-sections, not for A_TT K-factor almost spin-independent

Fermilab E866 800 GeV/c

H. Shimizu, G. Sterman, W. Vogelsang and H. Yokoya

M. Guzzi,V. Barone, A. Cafarella,C. Corianò and P.G. Ratcliffe

s=30 GeV² s=45 GeV²

s=210 GeV² s=900 GeV²

s=30 GeV² s=45 GeV²

s=210 GeV² s=900 GeV²

data from CERN WA39, πN processes, s = 80 GeV²

H. Shimizu, G. Sterman, W. Vogelsang and H. Yokoya

Some alternative accesses to transversity

Inclusive Λ production and measure of Λ polarization

COMPASS analysis in progress

need to know transverse fragmentation function _T D  D_q^_^  D_q^_^

the Λ polarization vector can be measured by looking at the proton angular distribution as resulting from Λ → πp decay in the Λ helicity rest frame

X



l p

similar result in p p^  ^X

collinear configuration, no need for intrinsic k_┴

Two hadron production in SIDIS Di-hadron Fragmentation Function (DiFF)

Chiral odd fragmentation function of a transversely polarized quark into two hadrons (interference between s and p wave)

Bacchetta, Boer, Jaffe, Jakob, Radici ^…

 







 

 _^  ^_{ }

q q q

q

DiFF q

q S

R

UT e f D

H h

e d

d

d

A d sin( )

1 2

1 1

2









TMDs measurements at BELLE (spin physics without polarization)

e⁺

_

1

P 

P 

_ e^-

e⁺

thrust-axis

z y

x

1

p

2

p 

q and polarization are strongly correlated; spin q effect in quark fragmentation visible event by event

first ever results on transversity via









q h N

q Φ

Φ

UT

h D

A

1 / )

sin( in SIDIS

(M. Boglione seminar)

A similar procedure could be followed with DiFF (accessible also in SSA in ) It requires detection of many particles (low statistics)

X h h AB  _{1 2}

access to

( ) (

)

1 /

/ ₂

1

z D z

D

N q

h N









and obtain information on Collins function

A fundamental QCD test

The Sivers function, being related to gauge links, is not universal, but it changes from process to process

+ –

q q

diquark diquark

or, rather, it enters different processes with different numerical factors …

Y D q SIDIS T

q

T

f

f





 

1 J. Collins

Following example taken from a talk prepared by M. Grosse- Perdekamp and delivered by L. Bland, Trento 2007

Simple QED example:

DIS: attractive Drell-Yan: repulsive Same in QCD:

As a result:

Non-universality of Sivers Asymmetries:

Unique Prediction of Gauge Theory !

Not all spin problems have been solved, but enormous progresses have been made

The spin-orbiting structure of quarks in nucleons begins to emerge

Theory. Unintegrated PDF and FF play a crucial role; their Q² evolution is needed. Factorization

and universality issues must be clarified, … Experiment. New data from COMPASS (proton target), JLab, RHIC, and GSI. D-Y processes very

promising …

Thanks and see you again!