gary r. goldstein - int.washington.edu · gary r. goldstein tufts university, leonard p. gamberg...

8/13/03 T-Odd & Drell-Yan -- G.R.Goldstein 1

Gary R. GoldsteinTufts University,

Leonard P. GambergPenn State-Berks Lehigh Valley College,

Karo A. OganessyanINFN-Laboratori Nazionali di Frascati and DESY


Transversity is reviewed and related to nucleon spin distributions. Non-trivial transversity distributions involve transverse momenta, helicity flip and phases. Rescattering is a source for these requirements; we evaluate such distributions using a rescattering model that includes dampened quark intrinsic k

�in the quark-nucleon-spectator vertex and the quark-

pion-spectator vertex. T-odd hadron distributions are exemplars of this scheme and contribute to unpolarized and polarized observables in SIDIS, Drell-Yan and inclusive hadronic processes. This leads to rich phenomenology that may describe a large azimuthal asymmetry in pp�� X and a large cos(2�) asymmetry in the unpolarized Drell-Yan cross section. We calculate T-odd, k

�dependent distribution functions (at

leading twist) and relate them to single spin asymmetries in SIDIS and azimuthal asymmetries in unpolarized Drell-Yan processes.


• Transversity– Short history– Helicity flip, chirality, phases & k

�

– Quark distribution functions: T-even &T-odd– Fragmentation functions: T-even &T-odd

• SIDIS– Asymmetries: SSA & azimuthal– Rescattering & leading twist contributions

• Drell-Yan– N & � distributions– cos2� asymmetry

• Conclusions


• 2-body scattering amps - Exclusive hadronic– fa,b;c,d(s,t) with spin projections a,b;c,d

• What spin frame leads to simplest description of theory or data? Amps to observables?– helicity has easy relativistic covariance - theory

– states of S·p, e.g. |+1/2� , |-1/2� , etc.

– transversity: eigenstates of S·(p1�p2)|� 1/2 )T = {|+1/2� �(i) |-1/2�}/√2 for spin 1/2, etc.

Goldstein & Moravcsik, Ann.Phys. 1976


• states of {S·(p1�p2)} or {S·(p1� p2)} are transversity normal to or parallel to scattering plane

• Spin 1: | �1)T = {|+1> ��2 |0> + |-1>}/2| 0)T = {|+1> - |-1>}/ �2

• photon: | �1)T = {|+1> + |-1>}/�2 linear polzn normal to plane| 0)T = {|+1> - |-1>}/�2 linear polzn parallel to plane

useful in photoproduction dynamics

• Transversity amps in NN�NN have phase simplicity(many observables!) Goldstein & Moravcsik & Arash 1980’s


• Single Spin Asymmetries (SSA) in 2-body• Parity requires only <S·n> non-zero for any

single spinning particle. Requiressome helicity flipor chirality flip for m=0 quarks & phase.

<S·n>��f*ab,cd[�·n]dd’fab,cd’ �� Im[f*

ab,c+ fab,c-]

n � p1� p2

p1

p2�


�

• <S·n>�� f*AB,CD[�·n]DD’fAB,CD’

in transversity basis:��{|fAB,C(+�)|2 - |fAB,C(-�)|2}

• Imaginary part or phase requires beyond tree level in any field theory

• Helicity or chirality flip requires flipping interaction & non-zero transverse momentum of participants or k�’s


Tensor Charge 2001 Gamberg&Goldstein


• f1(x,Q2) � q(x, Q2) – unpolarized N&q - measured in DIS

• g1(x,Q2) � �q(x, Q2) – longitudinally polarized N&q – or helicity transfer - polarized DIS

• h1(x,Q2) � �q(x, Q2)– “transversely” polarized N & q– or transversity transfer - need SIDIS or DY


DIS vs. Drell-Yan vs. SIDISfor h1(x) or �q(x)

Figures from R. Jaffe


Final state corrections to tree-level DISContribute to h1(x) & f1T

�(x,p�

2), h1�(x,p

�2)

Need SIDIS or D-Y to make functions accessible in asymmetry or polarization Brodsky, Hwang, Schmidt PLB 2002

Collins PLB 2002; Ji & Yuan PLB 2002Goldstein & Gamberg ICHEP 2002

Gamberg, Goldstein, Oganessyan PRD 2003 &hep-ph


h1�

• h1�(x,p

�

2) is T-odd distribution - probability of finding quark with non-zero transversity in unpolarized hadron

• Vanishes at tree level in models, as in spectator diquark model e.g. N�

quark+diquark where q is struck quark


SIDIS kinematicsIn spectator model yellow inclusive blobbecomes diquark - scalar for simplicity


Generalized quark distribution function

��

X� (0)P,S � e�bk�

2U(P,S)

gaussian approximation to simulate intrinsictransverse momentum distribution & toregularize integrations

��(x,k�) � 1

2d��d2�

�

2�� 3�

X� e�i � �k� �

��

��

k �� P��(�� ,��

) X

X �ig d��A�(��,0)0

�

��

��

��

�� (0,0

�)P

� ��0 + h.c.


Brodsky,Hwang,Schmidt rescattering


Interpreting rescattering


Model calculation


Spin independent tree level:

Transversity T-odd GPD:


• SSA’s & asymmetries involve moments of distribution & fragmentation functions

e.g. h1�(1)(x) = ∫ d2k

�k�

2 h1�(x, k

�

2)which would diverge without k

�

2

damping


h1�(x,k

�) calculation with Gaussian

h1� (x,k� )�

(m � xM)(1� x)�(k�

2 )1

k�2 e

�b k�2 ��(0)��

��

��

� � 0,b�(0)� �� 0,b�(k�2)� ��

where �(k�2 ) � k�

2 � x(1� x) �M 2 �m 2

x�

�2

1� x��

��

��

��

and �(0,z) is incomplete gamma function

h1�(x,k

�)= f1T

� (x,k�)

in diquark modelGamberg, Goldstein, Oganessyan

PRD 2003 & hep-ph


1�(1)

1�(1) �

π π

+h.c.

Both distribution & fragmentation calculated in spectator modelswith gaussian k

�


�

Ph�2

MMh

cos2� UU �

d2Ph�

Ph�2

MMh

cos(2�) d��

d2Ph� d��

�eq

2h1�(1)(x) � z 2

�H1�(1)(

q� z)

eq2 f1(x) �D1(z)

q�

Ignoring 1/Q2 T-even contributionBoer & Mulders PRD 1998Gamberg, Goldstein, Oganessyan PRD 2003 & hep-ph


<cos2�>vs. xfor SIDISwith spectatorfor distribution& fragmentation


<cos2�>vs. zfor SIDISwith spectatorfor distribution& fragmentation


Sivers asymmetry in SIDIS vs. x


lepton CM framedefines plane tilted at� rel.t. hadron planeof p1 &p2

z is direction of q in initial frame

p1

p2 �

l

l’

� z

xy


1�

d�d�

�3

4�1

� � 31� � cos2

� � �sin2� cos� �

�

2sin2

� cos2��

��

see D. Boer PRD60 &D.Boer, S.J. Brodsky & D.S. Hwang

hep-ph/0211110

Unpolarized pair of hadrons � l + l’ + X� involves transversity at leading twist


Upper hadron can be � or p or anti-p

Can use � � anti-q + “q” to relate to h1� for pion


��

F f f � �� d2 p�� d2k

��

2(��

p ��

k ��

q �

) f a(x1,��

p �

2 ) f a (x2,��

k �

2 )

for flavor a quark annihilating flavor a antiquarkThen the azimuthal asymmetry term becomes

� =2 ea

2F (2 p�x k�x �

��

p ��

k �)

h1�h 1

�

M1M2

��

��

�

��

a,a �

ea2F[ f1 f 1]

a,a �

Integrate over all quark transverse momenta. �+p�l+l- X isin process along with p+anti-p.

Notation of Boer, Mulders, Teryaev & Boer, Brodsky, Hwang


• Transversity is important for full understanding of hadron spin composition. Accessed via SIDIS & Drell-Yan with SSA’s. Require flips & loops.

• GPD’s are new interesting distributions for accessing spin & transversity M. Diehl’s talk

• BHS rescattering is mechanism for generating GDP’s at leading twist that can be measured via SSA’s D. Boer’s talk

• quark-diquark (S=0) model allows simple calculation to demonstrate existence of interesting GDP’s & SSA’s, e.g.

T-odd, Sivers fcn, cos2� in SIDIS & DY. • Extended to Generalized Fragmentation Functions; q�hadron+X,

q�π+X (X�q’) & Collins fcn. L.Gamberg’s talk• Does data support T-odd GDP’s?• Improvements: S=1 diquark; flavors; better starting model (2-body

constrants are limiting).• Many workers, much work. U. d’Alesio, F. Yuan, X-D. Ji, D.S. Hwang, D. Boer, M.Diehl,

A. Balitsky + P. Mulders, M. Anselmino, S. Brodsky, …G.G.O.

gary r. goldstein - int.washington.edu · gary r. goldstein tufts university, leonard p. gamberg...

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