Transcript The Phenomenology of T-odd Transversity Distributions in

Transversity, Transversity-odd
Distributions and Asymmetries
in Drell-Yan Processes
Gary R. Goldstein
Tufts University
Leonard P. Gamberg
Penn State-Berks Lehigh Valley College
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Abstract
Drell-Yan unpolarized processes display large azimuthal
asymmetries.
One such asymmetry, cos(2), is directly related to the leading
twist transversity distribution h1(x,kT).
We use a model developed for semi-inclusive deep inelastic
scattering that determines the “Sivers function” f1T (x,kT) to
predict the Drell-Yan asymmetry  as a function of q2, qT and
either x or xF or a new variable, .
The resulting predictions include a non-leading twist contribution
from spin-averaged distributions that measurably effect lower
energy results.
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Outline
• Transversity
 Short history
 Helicity flip, chirality, phases & k
• SIDIS
 Asymmetries: SSA & azimuthal
 Rescattering & leading twist contributions
• Drell-Yan
 Transversity-odd distribution functions
 N (& ) distributions
 cos2 asymmetry
• Conclusions
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Transversity - some history
• 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.
Especially for relating to single spin asymmetries - only S·n
GRG & M.J.Moravcsik, Ann.Phys. 1976
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Transversity & simplicity
• 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 on-shell photoproduction dynamics
• Transversity amps in NNNN have phase simplicity
(many observables!)
GRG & Moravcsik & Arash 1980’s
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Phases & SSA
• Single Spin Asymmetries (SSA) in 2-body
• Parity allows only <S·n> non-zero for any
single spinning
n  p1 p2
particle. Requires
p2
some helicity flip

p1
or chirality flip for
m=0 quarks & phase.
<S·n>f*ab,cd[·n]dd’fab,cd’  Im[f*ab,c+ fab,c-] for particle D’s SSA
n requires some p2 transverse to p1 (at quark level? m=0 &
PQCD - no SSA)
• Inclusive A+B->X+D: sum&∫ over all C particles &
relate to A+B+anti-D forward elastic. GRG & J.F.Owens (76)
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Azimuthal asymmetries - kinematics
• Why similar to spin asymmetries?
 Need plane established (P1P2)  transverse P
 Need azimuthal angle relative to 1st plane, i.e. 2nd
plane
• via fragmentation or decay or pair production
• How does orientation information get transferred from 1st
plane to 2nd plane? Dynamical question. Polarized
intermediate particles &/or plane dependent observables leading twist (&kinematic mass/Q) or non-leading
• SIDIS & Drell-Yan involve off-shell photons - like massive
vector particles with  (& longitudinal) polarization
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SSA & AzAs dynamics:
require loops & k
• <S·n> f*AB,CD[·n]DD’fAB,CD’ helicity basis
or in transversity basis:
{|fAB,C(+)|2 - |fAB,C(-)|2}
• Imaginary part or phase requires beyond tree level in
field theory
 PQCD efforts to explain polarized hyperons via s-quark polzn
 What is tree level in soft physics &/or “effective” field theory?
 Mixing PQCD & soft physics
• Helicity or chirality flip requires a flipping interaction
(m≠0,…) & non-zero transverse momentum of
participants or k’s
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Brodsky, Hwang & Schmidt provided non-trivial model calculation
Final state corrections to tree-level DIS-> f1T(x,p2) Sivers & SSA
LG&GG: same FSI contribute to other SSA’s
& f1T(x,p2) = + h1(x,p2) Boer-Mulders
Need SIDIS or D-Y to make functions experimentally accessible in
asymmetry or polarization
Brodsky, Hwang, Schmidt PLB 2002
Collins PLB 2002; Ji & Yuan PLB 2002
Goldstein & Gamberg ICHEP 2002
Gamberg, Goldstein, Oganessyan PRD 2003 &hep-ph
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h 1
• h1(x,p2) is “Transversity-odd” distribution -(BoerMulders) probability of finding quark with non-zero
transversity in unpolarized hadron (it is P-even)
• Vanishes at tree level in T-conserving models, as
in spectator diquark model e.g. N
quark+diquark where q is struck quark (like
ordinary decay amps - final state interactions are
essential - no T violation)
• Simple model is starting point for getting at
properties to expect & observable consequences
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SIDIS kinematics
In spectator model yellow inclusive blob becomes
diquark - scalar for simplicity (ud flavor)
leaving u-quark being struck by q
1+ diquarks include uu (& dd) allowing for d-quark struck
Note: diquark actually has structure also
alternate method
*+Nq+diq SSA
CM helicity amps
then light-cone limit
Dharmaratna&GG (92)
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Brodsky,Hwang,Schmidt rescattering
y
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r2
r2
(xM  m)kx (k  )  (k  ) 

ln 
r2
r2

2
[(xM  m)  k  ] k 
 (0) 
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Interpreting rescattering
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Model calculation
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Calculating h1(x,kT)
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Distribution definitions
fj/A(0) (x) is integral over kT of j/A with gauge link added
to insure gauge invariance
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Expanding distributions
Feynman rules obtained with intermediate states inserted
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Ingredients for h1

NPB194(1982)445
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Integration results
Spin independent tree level:
Transversity T-odd TMD: ( .. k.j factor on both sides)
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Regularization
• 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
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Transverse momentum hadronic tensor
X  (0)P,S  e
 bk 2
U(P,S)
gaussian approximation to simulate intrinsic
transverse momentum distribution & to
regularize integrations
1
d d 2 i  k    k  


 (x,k )   
e
P

(

, ) X
3
2 X
2
     
X ig  d A ( ,0) (0,0  ) P
 0

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  0
+ h.c.
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h1(x,k) calculation with Gaussian


b
 k  2  (0) 

(m

xM)(1
x)
1



h1 (x,k  ) 
e
(k 2 )
k 2
h
 0,b(0)   0,b(k 2 )
2
2

m


2
2
2
where (k  )  k   x(1 x) M 



x 1 x 
and (0,z) is incom plete gamma function
1/(m2-k2)=(1-x)/(kT2) result of p->q+diq kinematics
h1(x,k)= f1T (x,k)
in diquark model
Gamberg, Goldstein, Oganessyan
PRD 2003
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Drell-Yan coordinates
P1
l’
P2

l
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lepton CM frame
defines plane tilted at
 rel.t. hadron plane
of P1 &P2
y x

Coordinates?
z is direction of q
z
in initial CM frame
or x is direction along qT
from initial cm boost
(Collins-Soper frame)
or …
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Drell-Yan Cross Sections
1 d
3 1 
 2

2
2

1  cos    sin  cos   sin  cos 2

 d 4   3
2
see early papers ‘70’s
Collins&Soper (1977) effects of transverse
momenta -> ,, non-zero
Boer, Mulders, Teryaev (1998), Boer (1999) &
D.Boer, S.J. Brodsky & D.S. Hwang (2002)
Unpolarized pair of hadrons  l + l’ + X
 involves transversity at leading twist
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D-Y angular dependent 
1 d
3 1 
 2

2
2

1  cos    sin  cos   sin  cos 2
 d 4   3 
2
How are angular asymmetries calculated? q+anti-q annihilation (&
q+anti-q + gluons).
Cross sections require convoluting hadron->u with hadron->anti-u
distributions.  is related to T-odd distributions at leading twist (D.
r
r 2
Boer).
r
r 2 a
2
2
2 r
a
F  f f    d p d k ( p  k  q ) f (x1 , p ) f (x2 , k )
for flavor a quark annihilating flavor a antiquark
Then the azimuthal asymmetry term becomes
 
r


h
r
2
1 h1
2 ea F (2 p x k x  p  k )

M
M
a,a

1
2 
=
2
e
 a F[ f1 f1 ]
a,a
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AzAs: h1(1) •H1(1) cos2
Both distribution & fragmentation calculated in
spectator models with gaussian k
π
π
+h.c.
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From SIDIS to Drell-Yan - analogous calculations
Beam (π, p, p, …)
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Azimuthal asymmetry
d
4 2
2

e
F[ f f ]

a
2
2
4
dq dyd qT
3Q a
F f f   d 2 pd 2 k  2 (p  k   q ) f a (x1 , p2 ) f a (x 2 ,k 2 )
for flavora quark annihilat ing flavor
a antiquark
T hen the azimuthal asymmetry term becomes
 


h
2
1 h1
2 ea F (2 px k x  p  k )

M
M
a,a

1
2 
=
2
e
 a F[ f1 f1]
a,a
Integrate over all quark transverse momenta. +pl+l- X is
in process; p+anti-p is calculated for various s. x direction is
QT direction
Notation of Boer, Mulders, Teryaev & Boer, Brodsky, Hwang
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QT dependences
General form: expectation of a hadronic tensor with distributions
from quark model of incoming particles
asymmetry~

tensor luminosity of spinning quarks
luminosity of unpolarized quarks
•Asymmetry must vanish as QT0 ; no 1st plane orientation in forward limit
of initial state.
•What is role of quark spin?
•In lepton rest frame or q+q (CM) “fat” photon produced.
•Whether q & q polarized or not, photon’s spin tensor (T & L) is fixed by
QED.
•Unpolarized q+q defines a plane via QT & tensor behaves ~ (QT2 / Q2)2 .
• Transversely polarized q+q have ST1 ST2 tensor structure to combine
with kT & pT (2 planes)
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Non-leading contributions
Spin dependent leading part ~ QT2 for small QT2 / Q2
Non-leading, spin independent part ~ extra QT2
k

2
Collins & Soper ‘77
defined tensor
A2 =
T
 pT
2
Q QT

2

k
2
2
T
-
 ea F A2 f1  f1 2F B h1  h1
2


2
a,a



 ea 2 F f1  f1



 pT

2
Q2

a,a
B = (2kTx pTx  kTpT) / M2
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convolutions
d k d
2

2

1

1
p (k  p  Q )h (x,k )h (x, p )
2
2
2
 use up  leaving integral overk &
2
p  k  Q  k  Q cosQ
2
2
2
There will be the tensor B and azimuthal dependence, crucial for
transmitting plane orientation information.
Integrate numerically to obtain convoluted functions depending
on x, mee, QT (and s). Note x ≈ mee2/xs for s >> mee2 >>QT2.
Convolutions of h’s have extra factors of S at appropriate scale
compared to f’s. But f’s in numerator have QT2 relative to h’s.
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Drell-Yan kinematics
q2
x1 x2 

s
x  x1 , x  x2
x1 =+   2   2
x goes from xmin
xF  x  x  2
x2     2   2
q2

to 1
s


q2 
q2 
xF from   1   to   1  
s
s


Asymmetry  is function of 3 variables: x, √Q2 =m ,qT
(after separating sin2 cos2 dependence)
Want to obtain  integrated over 2 variables to c.f. experiments.
How to do this while keeping “symmetry” x1, x2
Using xF treats x1, x2 symmetrically, but different range vs. q.
Use = xF /2(1-) from -1/2 to +1/2
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AzAs function
Insert convolutions into asymmetry expression:
2



 ea F A2 f1  f1 2F B h1  h1
a,a
 e F f
2
a
1



 f1 
a,a
Obtained for range of x, mee, QT (and s).
Choose kinematic ranges of Conway, et al. (FNAL fixed
target π p) applied to p p . Sum over their (limited) ranges
to obtain (x), (mee2), (QT) .
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(QT2) leading h1 contribution
Calculated s50 Gev2 lower kinematic range
than Conway, et al.
E615.
Antiproton beam vs. π
Data for
π-p at s=500 GeV2
E615
Very similar to Boer,
Brodsky, Hwang
But gaussian supressed
f f part is at most
10%
At higher s 500 Gev2 with comparable range curve decreases a bit
f f part can be 10 - 15% of this for some values of 3 variables.
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(m) leading h1 contribution
(s=50 GeV2)
Data for
π-p at s=500 GeV2
E615
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q vs. vs. x
x = 0.9
x = 0.8
x = 0.7
x = 0.6
x = 0.5
x = 0.4
x = 0.3
x = 0.2
x = 0.1
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()
s=50 GeV2
nu(zeta)
s50
0 .7
()
0 .6
0 .5
Blue -leading
Red - with non-leading
nu
0 .4
0 .3
0 .2
0 .1
0
- 0 .6
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- 0 .4
- 0 .2

0
0 .2
0 .4
0 .6
zet a
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 versus x1
s=50 GeV2
Leading twist only
Including non-leading
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 vs. xF
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Summary& Conclusions
• Transversity is important for full understanding of hadron
spin composition. Accessed experimentally via SIDIS &
Drell-Yan with SSA’s & azimuthal asymmetries.
• Require helicity flips & loops; combinations of factorized
soft-hadronic & PQCD.
• BHS rescattering is mechanism for generating TMD’s at
leading twist that can be measured via SSA’s & AzAs’s.
• quark-diquark (S=0) model with gaussian regulators allows
simple calculations to demonstrate existence & size of
interesting TMD’s and thus SSA’s & AzAs’s.
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Summary (cont’d)
• Example considered: “Transversity-odd” contributions to
cos2 in D-Y compared to non-leading spin independent
piece (Collins&Soper).
• Does data support “Transversity-odd” TMD? Large effect in
π+p at hi s makes this very plausible. Cleanest determination
would be AzAs data on anti-p+p (GSI - PAX?).
• Improvements:
 S=1 diquark is I=1 uu flavor  p->d+diq
 better starting model (2-body constraints are limiting)
• Questions:
 How do Transversity-odd TMD’s evolve?
 Are Sudakov effects important for low qT in AzAs’s?
 Gluon bremsstrahlung, Qui-Sterman mechanism at other kinematics
• Many workers are here, much work to be done.
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