Transcript Document

BNL
December 2003
Universality of T-odd effects in single
spin azimuthal asymmetries
Universality of T-odd effects in single spin and azimuthal asymmetries,
D. Boer, PM and F. Pijlman, NP B667 (2003) 201-241; hep-ph/0303034
P.J. Mulders
Vrije Universiteit
Amsterdam
[email protected]
Content



Soft parts in hard processes
 twist expansion
 gauge link
 Illustrated in DIS
Two or more (separated) hadrons
 transverse momentum dependence
 T-odd phenomena
 Illustrated in SIDIS and DY
Universality


Items relevant for other processes
Illustrated in high pT hadroproduction
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Soft physics in inclusive deep
inelastic leptoproduction
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(calculation of)
cross section
DIS
Full calculation
+
PARTON
MODEL
+
+
+…
Lightcone dominance in DIS
Leading order DIS


In limit of large Q2 the result
of ‘handbag diagram’ survives
… + contributions from A+ gluons
A+
Ellis, Furmanski, Petronzio
Efremov, Radyushkin
A+ gluons
 gauge link
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Color gauge link in correlator
Matrix elements <yA+y>
produce the gauge link
U(0,x) in leading quark
lightcone correlator
A+
Distribution
functions
Soper
Jaffe & Ji NP B 375 (1992) 527
Parametrization
consistent with:
Hermiticity, Parity
& Time-reversal
Distribution
functions
 M/P+ parts appear
as M/Q terms in s
 T-odd part vanishes
for distributions but is
important for fragmentation
leading part
Jaffe & Ji
Jaffe & Ji
NP B 375 (1992) 527
PRL 71 (1993) 2547
Distribution
functions
Selection via
specific probing
operators
(e.g. appearing in
leading order DIS,
SIDIS or DY)
Jaffe & Ji
NP B 375 (1992) 527
Lightcone correlator
momentum density
Sum over
lightcone wf
squared
Basis for
partons
 ‘Good part’ of Dirac
space is 2-dimensional
 Interpretation of DF’s
unpolarized quark
distribution
helicity or chirality
distribution
transverse spin distr.
or transversity
Matrix representation
Bacchetta, Boglione, Henneman & Mulders
PRL 85 (2000) 712
Related to the
helicity formalism
Anselmino et al.
 Off-diagonal elements (RL or LR) are chiral-odd functions
 Chiral-odd soft parts must appear with partner in e.g. SIDIS, DY
Summarizing DIS

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
Structure functions (observables) are
identified with distribution functions
(lightcone quark-quark correlators)
DF’s are quark densities that are directly
linked to lightcone wave functions squared
There are three DF’s
f1q(x) = q(x), g1q(x) =Dq(x), h1q(x) =dq(x)
Longitudinal gluons (A+, not seen in LC
gauge) are absorbed in DF’s
Transverse gluons appear at 1/Q and are
contained in (higher twist) qqG-correlators
Perturbative QCD  evolution
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Hard processes
with two or more hadrons
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SIDIS
cross section
 variables
 hadron tensor
(calculation of)
cross section
SIDIS
Full calculation
+
PARTON
MODEL
+
+
+…
Lightfront dominance in SIDIS
Lightfront dominance in SIDIS
Three external momenta
P Ph q
transverse directions relevant
qT = q + xB P – Ph/zh
or
qT = -Ph^/zh
Leading order SIDIS


In limit of large Q2 only result
of ‘handbag diagram’ survives
Isolating parts encoding soft physics
?
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Lightfront
correlator
(distribution)
+
Lightfront correlator (fragmentation)
Collins & Soper
NP B 194 (1982) 445
no T-constraint
T|Ph,X>out = |Ph,X>in
Jaffe & Ji,
PRL 71 (1993) 2547;
PRD 57 (1998) 3057
Distribution
A+
including the gauge link (in SIDIS)
One needs also AT
G+a =  +ATa

ATa(x)= ATa(J)
+dh G+a
Ji, Yuan, PLB 543 (2002) 66
Belitsky, Ji, Yuan, hep-ph/0208038
From <y(0)AT()y(x)> m.e.
Distribution
A+
including the gauge link (in SIDIS or DY)
SIDIS
A+
DY
hep-ph/0303034
SIDIS  F[-]
DY  F[+]
Distribution
 for plane waves
T|P> = |P>
 But...
T U[0, ] T = U[0,- ]
 this does affect
F[](x,pT)
 it does not affect
F(x)
  appearance of
T-odd functions
in F[](x,pT)
including the gauge link (in SIDIS or DY)
Parameterizations including pT
Ralston & Soper
NP B 152 (1979) 109
Constraints from Hermiticity & Parity
Tangerman & Mulders
PR D 51 (1995) 3357
 Dependence
on …(x, pT2)
 Without T:
h1^ and f1T^
nonzero!
T-odd functions
 Fragmentation
fD
gG
hH
 No T-constraint:
H1^ and D1T^
nonzero!
Distribution functions with pT
Ralston & Soper
NP B 152 (1979) 109
Tangerman & Mulders
PR D 51 (1995) 3357
Selection via
specific probing
operators
(e.g. appearing
in leading order
SIDIS or DY)
Lightcone correlator
Bacchetta, Boglione, Henneman & Mulders
PRL 85 (2000) 712
momentum density
Remains valid
for F(x,pT)
… and also
after inclusion
of links for
F[](x,pT)
Sum over
lightcone wf
squared
Brodsky, Hoyer, Marchal, Peigne, Sannino
PR D 65 (2002) 114025
Interpretation
unpolarized quark
distribution
need pT
helicity or chirality
distribution
need pT
transverse spin distr.
or transversity
need pT
need pT
Integrated distributions
T-odd functions only for fragmentation
Weighted distributions
Appear in azimuthal asymmetries in SIDIS or DY
These are process-dependent (through gauge link)
Matrix representation
for M = [F(x)g+]T
Collinear structure of the nucleon!
reminder
Matrix representation
for M = [F(x,pT)g+]T
 pT-dependent
functions
T-odd: g1T  g1T – i f1T^ and h1L^  h1L^ + i h1^
Bacchetta, Boglione, Henneman & Mulders
PRL 85 (2000) 712
Matrix representation
for M = [D(z,kT) g-]T
 FF’s:
fD
gG
hH
 No T-inv
constraints
H1^ and
D1T^
nonzero!
 pT-dependent
functions
Matrix representation
for M = [D(z,kT) g-]T
 R/L basis for spin 0
 Also for spin 0
a T-odd function
exist, H1^
(Collins function)
 FF’s after
kT-integration
leaves just the
ordinary D1(z)
 pT-dependent
functions
e.g. pion
Summarizing SIDIS
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Beyond just extending DIS by tagging
quarks …
Transverse momenta of partons become relevant,
appearing in azimuthal asymmetries
DF’s and FF’s depend on two variables,
F[](x,pT) and D[](z,kT)
Gauge link structure is process dependent ( [])
pT-dependent distribution functions and (in
general) fragmentation functions are not
constrained by time-reversal invariance
This allows T-odd functions h1^ and f1T^ (H1^ and
D1T^) appearing in single spin asymmetries
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T-odd effects in single
spin asymmetries
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T-odd  single spin asymmetry
 Wmn(q;P,S;Ph,Sh) = -Wnm(-q;P,S;Ph,Sh)
symmetry
structure
*
 Wmn
(q;P,S;Ph,Sh) = Wnm(q;P,S;Ph,Sh)
_
__ __
 Wmn(q;P,S;Ph,Sh) = Wmn(q;P, -S;Ph, -Sh)
___ _
_
*
 Wmn
(q;P,S;Ph,Sh) = Wmn(q;P,S;Ph,Sh)
hermiticity
_
_
Conclusion: with time reversal constraint
only even-spin asymmetries
But time reversal constraint cannot be applied
in DY or in 1-particle inclusive DIS or e+e-
parity
time
reversal
Single spin asymmetries
sOTO
 T-odd fragmentation function (Collins function)
or
 T-odd distribution function (Sivers function)
 Both of the above also appear in SSA in pp  pX
 Different asymmetries in leptoproduction!
Collins
NP B 396 (1993) 161
Sivers
PRD 1990/91
Boer & Mulders
PR D 57 (1998) 5780
Boglione & Mulders
PR D 60 (1999) 054007
Process dependence and
universality
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Difference between F[+] and F[-]
Integrate
over pT
Difference between F[+] and F[-]
integrated quark
distributions

transverse
moments
measured in
azimuthal
asymmetries
±
Difference between F[+] and F[-]
gluonic
pole m.e.
Time reversal constraints for
distribution functions
T-odd
(imaginary)
pFG
Time reversal: F[+](x,pT)  F[-](x,pT)
F[+]
F
T-even
(real)
F[-]
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Consequences for
distribution functions
SIDIS F[+]
DY
F[-]
F[](x,pT) = F(x,pT) ± pFG
Time reversal

Distribution functions
F[](x,pT)
= F(x,pT) ± pFG
Sivers effect in SIDIS
and DY opposite in sign
Collins
hep-ph/0204004
Time reversal constraints for
fragmentation functions
T-odd
(imaginary)
pDG
Time reversal: D[+]out(z,pT)  D[-]in(z,pT)
D[+]
D
T-even
(real)
D[-]
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Time reversal constraints for
fragmentation functions
T-odd
(imaginary)
pDG out
Time reversal: D[+]out(z,pT)  D[-]in(z,pT)
D[+]out
D out
D[-]out
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Fragmentation functions
D[](x,pT)
= D(x,pT) ± pDG
Collins effect in SIDIS
and e+e- unrelated!
If pDG = 0
But at present this
seems (to me) unlikely
Time reversal
does not lead
to constraints
T-odd phenomena
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T-invariance does not constrain fragmentation
^
 T-odd FF’s (e.g. Collins function H1 )
T-invariance does constrain F(x)
 No T-odd DF’s and thus no SSA in DIS
T-invariance does not constrain F(x,pT)
 T-odd DF’s and thus SSA in SIDIS (in combination with
azimuthal asymmetries) are identified with gluonic poles that
also appear elsewhere (Qiu-Sterman, Schaefer-Teryaev)
 Sign of gluonic pole contribution process dependent
In fragmentation soft T-odd and (T-odd and T-even) gluonic pole
effects arise
 No direct comparison of Collins asymmetries in SIDIS and e+e(unless pDG = 0)
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What about hadroproduction?
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Issues in hadroproduction
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Weighted functions will appear in L-R asymmetries (pT now hard scale!)
There are various possibilities with gluons
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G(x,pT) – unpolarized gluons in unpolarized nucleon
DG(x,pT) – transversely polarized gluons in a longitudinally polarized nucleon
GT(x,pT) – unpolarized gluons in a transversely polarized nucleon (T-odd)
H^(x,pT) – longitudinally polarized gluons in an unpolarized nucleon
…
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Issues in hadroproduction

Contributions of F(x,pT) and pFG not necessarily in one combination
AN ~ … G(xa)  f1T ^(1)[-](xb)  D1 (zc) + … f1(xa)  f1T ^(1)[+](xb)  D1 (zc)
+ … f1(xa)  h1(xb)  H1^[-] (zc) + … f1(xa)  h1(xb)  H1^[+] (zc)
+ … f1(xa)  GT(xb)  D1 (zc)
Many issues to be sorted out
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Thank you for your attention
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Relations among distribution functions
1. Equations of motion
2. Define interaction dependent
functions
3. Use Lorentz invariance
Distribution functions
F[](x,pT)
= F(x,pT) ± pFG
Sivers effect in SIDIS
and DY opposite in sign
Collins
hep-ph/0204004
(omitting mass terms)
Fragmentation functions
D[](x,pT)
= D(x,pT) ± pDG
including relations
Collins effect in SIDIS
and e+e- unrelated!
Example of a single spin asymmetry
example:
sOTO in
ep  epX




example of a leading azimuthal asymmetry
T-odd fragmentation function (Collins function)
involves two chiral-odd functions
q
Best way to get transverse spin polarization h1 (x)
Collins
NP B 396 (1993) 161
Tangerman & Mulders
PL B 352 (1995) 129