1. Transversity in D_Y
4/5/2019
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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2. Transversity2005 Como G.R.Goldstein
Transversity in D_Y
4/5/2019
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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3. Transversity2005 Como G.R.Goldstein
Transversity in D_Y
4/5/2019
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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4. Transversity - some history
Transversity in D_Y
4/5/2019
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
GRG & M.J.Moravcsik, Ann.Phys. 1976
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5. Transversity & simplicity
Transversity in D_Y
4/5/2019
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 on-shell photoproduction dynamics
Transversity amps in NNNN have phase simplicity (many observables!)
GRG & Moravcsik & Arash 1980’s
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6. Transversity2005 Como G.R.Goldstein
Transversity in D_Y
4/5/2019
Phases & SSA
Single Spin Asymmetries (SSA) in 2-body
Parity allows 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-] 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)
n  p1 p2 p2  p1
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7. Azimuthal asymmetries - kinematics
Transversity in D_Y
4/5/2019
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. 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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8. SSA & AzAs dynamics: require loops & k
Transversity in D_Y
4/5/2019
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 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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9. Transversity2005 Como G.R.Goldstein
Transversity in D_Y
4/5/2019
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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10. Transversity2005 Como G.R.Goldstein
Transversity in D_Y
4/5/2019
h1
h1(x,p2) is “Transversity-odd” distribution -(Boer-Mulders) 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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11. Transversity2005 Como G.R.Goldstein
Transversity in D_Y
4/5/2019
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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12. Brodsky,Hwang,Schmidt rescattering
Transversity in D_Y
4/5/2019
Brodsky,Hwang,Schmidt rescattering
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13. Interpreting rescattering
Transversity in D_Y
4/5/2019
Interpreting rescattering
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14. Transversity2005 Como G.R.Goldstein
Transversity in D_Y
4/5/2019
Model calculation
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15. Transversity2005 Como G.R.Goldstein
Transversity in D_Y
4/5/2019
Calculating h1(x,kT)
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16. Distribution definitions
Transversity in D_Y
4/5/2019
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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17. Expanding distributions
Transversity in D_Y
4/5/2019
Expanding distributions
Feynman rules obtained with intermediate states inserted
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18. Transversity2005 Como G.R.Goldstein
Transversity in D_Y
4/5/2019
Ingredients for h1
NPB194(1982)445
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19. Transversity2005 Como G.R.Goldstein
Transversity in D_Y
4/5/2019
Integration results
Spin independent tree level:
Transversity T-odd TMD: ( .. k.j factor on both sides)
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20. Transversity2005 Como G.R.Goldstein
Transversity in D_Y
4/5/2019
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
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21. Transverse momentum hadronic tensor
Transversity in D_Y
4/5/2019
Transverse momentum hadronic tensor
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22. Transversity2005 Como G.R.Goldstein
Transversity in D_Y
4/5/2019
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)
in diquark model
Gamberg, Goldstein, Oganessyan
PRD 2003
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23. Drell-Yan coordinates
Transversity in D_Y
4/5/2019
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 CM frame
or x is direction along qT
from initial cm boost
(Collins-Soper frame)
or … y x l’ P2  P1 z  l
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24. Drell-Yan Cross Sections
Transversity in D_Y
4/5/2019
Drell-Yan Cross Sections
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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25. D-Y angular dependent 
Transversity in D_Y
4/5/2019
D-Y angular dependent 
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. Boer).
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26. Transversity2005 Como G.R.Goldstein
Transversity in D_Y
4/5/2019
AzAs: h1(1) •H1(1) cos2
Both distribution & fragmentation calculated in
spectator models with gaussian k π π
+h.c.
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27. Transversity2005 Como G.R.Goldstein
Transversity in D_Y
4/5/2019
From SIDIS to Drell-Yan - analogous calculations
Beam (π, p, p, …)
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28. Transversity2005 Como G.R.Goldstein
Transversity in D_Y
4/5/2019
Azimuthal asymmetry
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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29. Transversity2005 Como G.R.Goldstein
Transversity in D_Y
4/5/2019
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)
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30. Non-leading contributions
Transversity in D_Y
4/5/2019
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
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31. Transversity2005 Como G.R.Goldstein
Transversity in D_Y
4/5/2019
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.
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32. Transversity2005 Como G.R.Goldstein
Transversity in D_Y
4/5/2019
Drell-Yan kinematics
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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33. Transversity2005 Como G.R.Goldstein
Transversity in D_Y
4/5/2019
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) .
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34. (QT2) leading h1 contribution
Transversity in D_Y
4/5/2019
(QT2) leading h1 contribution
Calculated 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.
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35. (m) leading h1 contribution (s=50 GeV2)
Transversity in D_Y
4/5/2019
(m) leading h1 contribution (s=50 GeV2)
Data for
π-p at s=500 GeV2
E615
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36. Transversity2005 Como G.R.Goldstein
Transversity in D_Y
4/5/2019
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
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37. Transversity2005 Como G.R.Goldstein
Transversity in D_Y
4/5/2019
() s=50 GeV2
()
Blue -leading
Red - with non-leading 
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38. Transversity2005 Como G.R.Goldstein
Transversity in D_Y
4/5/2019
 versus x1 s=50 GeV2
Leading twist only
Including non-leading
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39. Transversity2005 Como G.R.Goldstein
Transversity in D_Y
4/5/2019
 vs. xF s=50 GeV2
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40. Transversity2005 Como G.R.Goldstein
Transversity in D_Y
4/5/2019
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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41. Transversity2005 Como G.R.Goldstein
Transversity in D_Y
4/5/2019
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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