1. Transversity in D_Y
4/17/2019
Transversity Distributions and T-odd Asymmetries in Drell-Yan Processes
Gary R. Goldstein
Tufts University
Leonard P. Gamberg
Penn State-Berks Lehigh Valley College
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein

2. SIR2005: T-Odd & D-Y G.R.Goldstein
Transversity in D_Y
4/17/2019
Abstract
Drell-Yan unpolarized processes display 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.
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein

3. SIR2005: T-Odd & D-Y G.R.Goldstein
Transversity in D_Y
4/17/2019
Outline
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
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein

4. Transversity - some history
Transversity in D_Y
4/17/2019
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
Goldstein & Moravcsik, Ann.Phys. 1976
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein

5. Transversity & simplicity
Transversity in D_Y
4/17/2019
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 photoproduction dynamics
Transversity amps in NNNN have phase simplicity (many observables!)
Goldstein & Moravcsik & Arash 1980’s
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein

6. SIR2005: T-Odd & D-Y G.R.Goldstein
Transversity in D_Y
4/17/2019
Phases & SSA
Single Spin Asymmetries (SSA) in 2-body
Parity requires only <S·n> non-zero for any
single spinning
particle. Requires
some helicity flip
or chirality flip for m=0 quarks & phase.
<S·n>f*ab,cd[·n]dd’fab,cd’  Im[f*ab,c+ fab,c-]
n requires some p2 transverse to p1
(at quark level?)
n  p1 p2 p2  p1
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein

7. Azimuthal asymmetries - kinematics
Transversity in D_Y
4/17/2019
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.
SIDIS & Drell-Yan involve off-shell photons - like massive vector particles with  & longitudinal polarization
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein

8. SSA & AzAs dynamics: require loops & k
Transversity in D_Y
4/17/2019
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 any field theory
What is tree level in “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
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein

9. SIR2005: T-Odd & D-Y G.R.Goldstein
Transversity in D_Y
4/17/2019
What reactions to look at? DIS vs. Drell-Yan vs. SIDIS for h1(x)=q(x) & for T-odd
Figures from R. Jaffe
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein

10. SIR2005: T-Odd & D-Y G.R.Goldstein
Transversity in D_Y
4/17/2019
Brodsky, Hwang & Schmidt provided non-trivial model calculation
Final state corrections to tree-level DIS-> f1T(x,p2) & SSA
G&G: contributes to h1(x) & f1T(x,p2) = ± h1(x,p2)
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
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein

11. SIR2005: T-Odd & D-Y G.R.Goldstein
Transversity in D_Y
4/17/2019
h1
h1(x,p2) is “T-odd” distribution - 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)
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein

12. SIR2005: T-Odd & D-Y G.R.Goldstein
Transversity in D_Y
4/17/2019
SIDIS kinematics
In spectator model yellow inclusive blob
becomes diquark - scalar for simplicity
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein

13. Brodsky,Hwang,Schmidt rescattering
Transversity in D_Y
4/17/2019
Brodsky,Hwang,Schmidt rescattering
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein

14. Interpreting rescattering
Transversity in D_Y
4/17/2019
Interpreting rescattering
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein

15. SIR2005: T-Odd & D-Y G.R.Goldstein
Transversity in D_Y
4/17/2019
Model calculation
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein

16. SIR2005: T-Odd & D-Y G.R.Goldstein
Transversity in D_Y
4/17/2019
Calculating h1(x,kT)
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein

17. Distribution definitions
Transversity in D_Y
4/17/2019
Distribution definitions
fj/A(0) (x) is integral over kT of j/A with gauge link added
to insure gauge invariance
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein

18. Expanding distributions
Transversity in D_Y
4/17/2019
Expanding distributions
Feynman rules obtained with intermediate states inserted
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein

19. SIR2005: T-Odd & D-Y G.R.Goldstein
Transversity in D_Y
4/17/2019
Ingredients for h1
NPB194(1982)445
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein

20. SIR2005: T-Odd & D-Y G.R.Goldstein
Transversity in D_Y
4/17/2019
Integration results
Spin independent tree level:
Transversity T-odd GPD: ( .. k.j factor on both sides)
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein

21. SIR2005: T-Odd & D-Y G.R.Goldstein
Transversity in D_Y
4/17/2019
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
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein

22. Transverse momentum hadronic tensor
Transversity in D_Y
4/17/2019
Transverse momentum hadronic tensor
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein

23. SIR2005: T-Odd & D-Y G.R.Goldstein
Transversity in D_Y
4/17/2019
h1(x,k) calculation with Gaussian h
1/(m2-k2)=(1-x)/(kT2) result of p->q+diq kinematics
h1(x,k)= f1T (x,k) (sign+)
in diquark model
Gamberg, Goldstein, Oganessyan
PRD 2003 & hep-ph
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein

24. Drell-Yan coordinates
Transversity in D_Y
4/17/2019
Drell-Yan coordinates
lepton CM frame
defines plane tilted at
 rel.t. hadron plane
of p1 &p2
Coordinates?
z is direction of q
in initial frame
or x is direction along qT
from initial cm boost
(Collins-Soper frame)
or … y x l’ p2  p1 z  l
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein

25. Drell-Yan Cross Sections
Transversity in D_Y
4/17/2019
Drell-Yan Cross Sections
see early papers ‘70’s
Collins&Soper 1977 effects of transverse
momenta -> ,, non-zero
D. Boer PRD60 &
D.Boer, S.J. Brodsky & D.S. Hwang
hep-ph/
Unpolarized pair of hadrons  l + l’ + X
 involves transversity at leading twist
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein

26. D-Y angular dependent 
Transversity in D_Y
4/17/2019
D-Y angular dependent 
How are angular asymmetries calculated?  is related to
T-odd distributions at leading twist (D. Boer).
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein

27. SIR2005: T-Odd & D-Y G.R.Goldstein
Transversity in D_Y
4/17/2019
AzAs: h1(1) •H1(1) cos2
Both distribution & fragmentation calculated in
spectator models with gaussian k π π
+h.c.
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein

28. SIR2005: T-Odd & D-Y G.R.Goldstein
Transversity in D_Y
4/17/2019
From SIDIS to Drell-Yan - analogous calculations
Beam (π, p, p, …)
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein

29. SIR2005: T-Odd & D-Y G.R.Goldstein
Transversity in D_Y
4/17/2019
Azimuthal asymmetry
Integrate over all quark transverse momenta. +pl+l- X is
in process; p+anti-p is calculated for s at fixed target Fermilab. x direction is QT direction
Notation of Boer, Mulders, Teryaev & Boer, Brodsky, Hwang
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein

30. SIR2005: T-Odd & D-Y G.R.Goldstein
Transversity in D_Y
4/17/2019
QT dependences
General form: expectation of a hadronic tensor with distributions from quark model of incoming particles
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)
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein

31. Non-leading contributions
Transversity in D_Y
4/17/2019
Non-leading contributions
Spin dependent leading part ~ QT2 for small QT2 / Q2
Non-leading, spin independent part ~ extra QT2
Collins & Soper ‘77
defined tensor A2 =
B = (2kTx pTx  kTpT) / M2
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein

32. SIR2005: T-Odd & D-Y G.R.Goldstein
Transversity in D_Y
4/17/2019
convolutions
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.
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein

33. SIR2005: T-Odd & D-Y G.R.Goldstein
Transversity in D_Y
4/17/2019
Drell-Yan kinematics
Asymmetry  is function of 3 variables: x, √q2 =m ,qT
Want to obtain  integrated over 2 variables.
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
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein

34. SIR2005: T-Odd & D-Y G.R.Goldstein
Transversity in D_Y
4/17/2019
AzAs function
Insert convolutions into asymmetry expression:
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) .
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein

35. (QT2) leading h1 contribution
Transversity in D_Y
4/17/2019
(QT2) leading h1 contribution
Calculated for same s50 Gev2 - lower kinematic range than Conway, et al. E615.
Antiproton beam vs. π
Very similar to Boer,
Brodsky, Hwang
But gaussian supressed
Data for
π-p at s=500 GeV2
E615
f f part is at most
10%
At higher s 500 Gev2 with comparable range curve decreases a bit
f f part can be % of this for some values of 3 variables.
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein

36. (m) leading h1 contribution (s=50 GeV2)
Transversity in D_Y
4/17/2019
(m) leading h1 contribution (s=50 GeV2)
Data for
π-p at s=500 GeV2
E615
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein

37. SIR2005: T-Odd & D-Y G.R.Goldstein
Transversity in D_Y
4/17/2019
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
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein

38. SIR2005: T-Odd & D-Y G.R.Goldstein
Transversity in D_Y
4/17/2019
() s=50 GeV2
()
Blue -leading
Red - with non-leading 
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein

39. SIR2005: T-Odd & D-Y G.R.Goldstein
Transversity in D_Y
4/17/2019
 versus x1 s=50 GeV2
Leading twist only
Including non-leading
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein

40. SIR2005: T-Odd & D-Y G.R.Goldstein
Transversity in D_Y
4/17/2019
Summary& Conclusions
Transversity is important for full understanding of hadron spin composition. Accessed via SIDIS & Drell-Yan with SSA’s & azimuthal asymmetries. Require flips & loops. Probing applicability of models of factorized soft-hadronic & PQCD.
TMD’s are important distributions for accessing spin & transversity content of hadrons.
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 of interesting TMD’s and thus SSA’s & AzAs’s.
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein

41. SIR2005: T-Odd & D-Y G.R.Goldstein
Transversity in D_Y
4/17/2019
Summary (cont’d)
Example considered: “T-odd” contribution to cos2 in D-Y compared to “T-even” non-leading spin independent piece.
Does data support “T-odd” TMD? Large effect in π+p at hi s makes this very plausible. Need AzAs data on anti-p+p.
Improvements:
S=1 diquark is I=1 uu flavor  p->d+diq
better starting model (2-body constraints are limiting)
Questions:
How do “T-odd” TMD’s evolve?
Are Sudakov effects important for low qT in AzAs’s?
Many workers, much work to be done. Everyone here + experiments!
Transversity has arrived!
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein

42. SIR2005: T-Odd & D-Y G.R.Goldstein
Transversity in D_Y
4/17/2019
Fitting f1(x)
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein

43. cos2 asymmetry in SIDIS
Transversity in D_Y
4/17/2019
cos2 asymmetry in SIDIS
Ignoring 1/Q2 T-even contribution
Boer & Mulders PRD 1998
Gamberg, Goldstein, Oganessyan PRD 2003 & hep-ph
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein

44. SIR2005: T-Odd & D-Y G.R.Goldstein
Transversity in D_Y
4/17/2019
(x) for s = 50 GeV2
Including non-leading
Leading twist only
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein

45. (QT2) leading h1 contribution
Transversity in D_Y
4/17/2019
(QT2) leading h1 contribution
preliminary
Calculated for same s500Gev2 kinematic range as Conway, et al. E615 for  beam
they get larger
asymmetry that grows.
Very similar to Boer,
Brodsky, Hwang
f f part is only
1 or 2% of this.
At lower s 50Gev2 with comparable range curve increases
f f part can be % of this for some values of 3 variables.
4/17/2019
SIR2005: T-Odd & D-Y G.R.Goldstein
SIR Goldstein