Page 1
overview on leading-twist chiral-odd TMDschiral-odd TMDs in πN Drell-Yan
chiral-odd TMDs in single polarized pp Drell-YanTwist-3 TMDs in Drell-Yan processes
chiral-odd TMDs at twist 2 (and beyond) from(single-/un)polarized Drell-Yan asymmetries
Zhun Lu
Department of Physics, Southeast University, Nanjing, China
Zhun Lu chiral-odd TMDs at twist 2 (and beyond) from (single-/un)polarized Drell-Y
Page 2
overview on leading-twist chiral-odd TMDschiral-odd TMDs in πN Drell-Yan
chiral-odd TMDs in single polarized pp Drell-YanTwist-3 TMDs in Drell-Yan processes
1 overview on leading-twist chiral-odd TMDsBoer-Mulders functionT-even chiral-odd TMDs: h1T , h
⊥1T , h
⊥1L
2 chiral-odd TMDs in πN Drell-Yan
3 chiral-odd TMDs in single polarized pp Drell-Yan
4 Twist-3 TMDs in Drell-Yan processes
Zhun Lu chiral-odd TMDs at twist 2 (and beyond) from (single-/un)polarized Drell-Y
Page 3
overview on leading-twist chiral-odd TMDschiral-odd TMDs in πN Drell-Yan
chiral-odd TMDs in single polarized pp Drell-YanTwist-3 TMDs in Drell-Yan processes
Boer-Mulders functionT-even chiral-odd TMDs: h1T , h⊥
1T , h⊥1L
The TMD correlator for the nucleon (Goeke et.al, 05; Bacchetta et.al,06):
Φ(x, pT ) =1
2
f1 n/+ − f⊥1T
ǫρσT pTρSTσ
Mn/+ + g1sγ5 n/+
+ h1T
[
S/T , n/+]
γ5
2+ h⊥
1s
[
p/T , n/+]
γ5
2M
+ i h⊥1
[
p/T , n/+]
2M
h⊥1s = SLh
⊥1L − pT · ST
Mh⊥1T
The TMD correlator for the pion (twist-2):
Φ(x, pT ) =1
2
f1 n/+ + i h⊥1
[
p/T , n/+]
2M
Zhun Lu chiral-odd TMDs at twist 2 (and beyond) from (single-/un)polarized Drell-Y
Page 4
overview on leading-twist chiral-odd TMDschiral-odd TMDs in πN Drell-Yan
chiral-odd TMDs in single polarized pp Drell-YanTwist-3 TMDs in Drell-Yan processes
Boer-Mulders functionT-even chiral-odd TMDs: h1T , h⊥
1T , h⊥1L
describe helicity-flipped quark structure.
handbag diagram forbidden by chirality conservation
chiral-odd distributions participate in the processes involving at leasttwo hadrons (SIDIS, Drell-Yan)
chiral-odd TMDs appear as pair in high energy process
Zhun Lu chiral-odd TMDs at twist 2 (and beyond) from (single-/un)polarized Drell-Y
Page 5
overview on leading-twist chiral-odd TMDschiral-odd TMDs in πN Drell-Yan
chiral-odd TMDs in single polarized pp Drell-YanTwist-3 TMDs in Drell-Yan processes
Boer-Mulders functionT-even chiral-odd TMDs: h1T , h⊥
1T , h⊥1L
Zhun Lu chiral-odd TMDs at twist 2 (and beyond) from (single-/un)polarized Drell-Y
Page 6
overview on leading-twist chiral-odd TMDschiral-odd TMDs in πN Drell-Yan
chiral-odd TMDs in single polarized pp Drell-YanTwist-3 TMDs in Drell-Yan processes
Boer-Mulders functionT-even chiral-odd TMDs: h1T , h⊥
1T , h⊥1L
Boer-Mulders functions: model calculations
T-odd: h⊥1
∣
∣
DY= −h⊥
1
∣
∣
SIDIS. (Collins 02)
model calculations for the nucleon:
spectator model: Gamberg, Goldstein 02; Bacchetta, Schaefer,Yang 03; Gamberg, Goldstein, Schlegel 07; Bacchetta, Conti,Radici 08.constituent (light-cone) quark model: Courtoy, Scopetta, Vento 09;Pasquini and Yuan 10.bag model: Yuan 03; Courtoy, Scopetta, Vento 09baryon-meson fluctuation model for the sea quarks: ZL, Ma,Schmidt 07.
model calculations for the pion: ZL, Ma 04; Meissner, Metz,Schlegel, Goeke 08; Gamberg, Schlegel 10.
Zhun Lu chiral-odd TMDs at twist 2 (and beyond) from (single-/un)polarized Drell-Y
Page 7
overview on leading-twist chiral-odd TMDschiral-odd TMDs in πN Drell-Yan
chiral-odd TMDs in single polarized pp Drell-YanTwist-3 TMDs in Drell-Yan processes
Boer-Mulders functionT-even chiral-odd TMDs: h1T , h⊥
1T , h⊥1L
Boer-Mulders functions: model calculations
Figure: From arxiv:1108.1713
solid curve: Pasquini and Yuan 10.
dashed curve: Bacchetta, Conti, Radici 08.
dotted curve: Courtoy, Scopetta, Vento 09.
dashed-dotted curve: parametrization by Barone, Melis, Prokudin 10.
Zhun Lu chiral-odd TMDs at twist 2 (and beyond) from (single-/un)polarized Drell-Y
Page 8
overview on leading-twist chiral-odd TMDschiral-odd TMDs in πN Drell-Yan
chiral-odd TMDs in single polarized pp Drell-YanTwist-3 TMDs in Drell-Yan processes
Boer-Mulders functionT-even chiral-odd TMDs: h1T , h⊥
1T , h⊥1L
Theoretical status on the Boer-Mulders functions of the proton:
Theoretical approaches and model calculations suggest h⊥u1 and h⊥d
1have the same sign (negative in SIDIS).
Bag modelconstituent (light-cone) quark modelAxia-diquark spectator modelLarge Nc limit (Pobylitsa 03)Lattice calculation (QCDSF/UKQCD 06; Musch et.al 11)GPD approach (Burkardt 05; Pasquini, Boffi 07; Burkardt, Hannafiou08)
h⊥u1 and h⊥d
1 are expected to have the same order of magnitude
The size of Boer-Mulders function is comparable to that of the otherT-odd TMD, the Sivers function
Zhun Lu chiral-odd TMDs at twist 2 (and beyond) from (single-/un)polarized Drell-Y
Page 9
overview on leading-twist chiral-odd TMDschiral-odd TMDs in πN Drell-Yan
chiral-odd TMDs in single polarized pp Drell-YanTwist-3 TMDs in Drell-Yan processes
Boer-Mulders functionT-even chiral-odd TMDs: h1T , h⊥
1T , h⊥1L
Boer-Mulders function vs Sivers function:
0.2 0.4 0.6 0.8 1
x
-0.2
-0.1
0
0.1
0.2
xf(x)
xf1T(1/2,u)
xf1T (1/2,d)
xh1⊥(1/2,u)
xh1⊥(1/2,d)
κ = - 0.333
0.2 0.4 0.6 0.8 1
x-0.2
-0.1
0
0.1
0.2
xf(x)
xf1T⊥ (1,u)
xf1T⊥ (1,d)
xh1⊥(1,u)
xh1⊥(1,d)
κ = 1.0
Figure: from Gamberg, Goldstein, Schlegel 2007
X
-0.075
-0.05
-0.025
0
0.025
0.05
0.075
0 0.25 0.5 0.75 1 X
-0.075
-0.05
-0.025
0
0.025
0.05
0.075
0 0.25 0.5 0.75 1
Figure: from Courtoy, Scopetta, Vento 2009
Zhun Lu chiral-odd TMDs at twist 2 (and beyond) from (single-/un)polarized Drell-Y
Page 10
overview on leading-twist chiral-odd TMDschiral-odd TMDs in πN Drell-Yan
chiral-odd TMDs in single polarized pp Drell-YanTwist-3 TMDs in Drell-Yan processes
Boer-Mulders functionT-even chiral-odd TMDs: h1T , h⊥
1T , h⊥1L
Boer-Mulders functions: extractions from data
parameterizations for the nucleon:
from anomalous tensor magnetic moment κqT : Barone, Ma, prokudin
08;from pp and pd Drell-Yan data (E866/Nusea): Zhang, ZL, Ma,Schmidt 08 (valence and sea quarks); ZL, Schmidt 09; Barone,Melis, and Prokudin 10 (sea quarks).from SIDIS data (COMPASS, HERMES): Barone, Melis, andProkudin 10.
no existing parametrization for the pion.
Zhun Lu chiral-odd TMDs at twist 2 (and beyond) from (single-/un)polarized Drell-Y
Page 11
overview on leading-twist chiral-odd TMDschiral-odd TMDs in πN Drell-Yan
chiral-odd TMDs in single polarized pp Drell-YanTwist-3 TMDs in Drell-Yan processes
Boer-Mulders functionT-even chiral-odd TMDs: h1T , h⊥
1T , h⊥1L
Boer-Mulders functions: extraction from DY data
Parameterizations by Zhang, ZL, Ma, Schmidt 08; ZL, Schmidt 10:
h⊥ q1 (x,p2
T ) = h⊥ q1 (x)
1
πp2bmexp
(
− p2T
p2bm
)
.
h⊥ q1 (x) = ωHq x
cq (1− x)b f q1 (x); h⊥ q
1 (x) =1
ωHq x
cq (1− x)b f q1 (x).
0.0 0.2 0.4 0.6 0.8 1.0-0.10
-0.05
0.00
X
xh (1)u1 (x)
0.0 0.2 0.4 0.6 0.8 1.0-0.10
-0.05
0.00xh (1)d
1 (x)
X
The shadow is the region allowed by the positivity bounds (correspondingto 0.48 < ω < 2.1).
Zhun Lu chiral-odd TMDs at twist 2 (and beyond) from (single-/un)polarized Drell-Y
Page 12
overview on leading-twist chiral-odd TMDschiral-odd TMDs in πN Drell-Yan
chiral-odd TMDs in single polarized pp Drell-YanTwist-3 TMDs in Drell-Yan processes
Boer-Mulders functionT-even chiral-odd TMDs: h1T , h⊥
1T , h⊥1L
parameterizations confront data (qT ≤ 2 GeV).
data from L.Y. Zhu, et,al. (E866/NuSea) 2006 (pd data), 2008 (pp data)
1
σ
dσ
dΩ=
3
4π
1
λ+ 3(1 + λ cos2 θ + µ sin 2 θ cos φ+
ν
2sin2 θ cos 2φ).
ν(x1, x2, qT ) =
F[((2h · p1T h · p2T )− (p1T · p2T ))
h⊥ q1 h
⊥ q1
M1M2
]
F [f q1 f q
1 ]
TMD factorization is valid in low qT region. At higher qT , ν is subject to pQCDeffects (Mirkes, Ohnemus 94; Boer, Vogelsang, 06; Berger, Qiu,Rodriguez-Pedraza 07).
Product h⊥ q1 h⊥ q
1 brings uncertainties on the absolute normalization for eachflavor, which is constrained by the positivity bound (Bacchetta 99).
Zhun Lu chiral-odd TMDs at twist 2 (and beyond) from (single-/un)polarized Drell-Y
Page 13
overview on leading-twist chiral-odd TMDschiral-odd TMDs in πN Drell-Yan
chiral-odd TMDs in single polarized pp Drell-YanTwist-3 TMDs in Drell-Yan processes
Boer-Mulders functionT-even chiral-odd TMDs: h1T , h⊥
1T , h⊥1L
Boer-Mulders functions: extraction from SIDIS data
Parameterizations by Barone, Melis, and Prokudin 10:
h⊥q1 (x, k2T ) = λq f
⊥q1T (x, k2T ) = λq ρq(x) η(kT ) f
q1 (x,k
2T ) ,
Extracted from cos 2φ asymmetry in SIDIS (HERMES preliminary 09;COMPASS preliminary 08, 09)
∫
dσ cos 2φ =4πα2
ems
Q4
∫
∑
a
e2a x(1 − y) A[fa1 , D
a1 ] +
1
2B[h⊥a
1 , H⊥a1 ]
0
0.02
0.04
0.06
0.08
0.1
10-2 10-1 1
x |h
1u⊥(1
) (x)|
x
0
0.02
0.04
0.06
0.08
0.1
10-2 10-1 1
x |h
1d⊥(1
) (x)|
x
Knowledge on the Cahn effect A[fa1 , D
a1 ] (Boglinoe, Melis, Prokudin
11)
Zhun Lu chiral-odd TMDs at twist 2 (and beyond) from (single-/un)polarized Drell-Y
Page 14
overview on leading-twist chiral-odd TMDschiral-odd TMDs in πN Drell-Yan
chiral-odd TMDs in single polarized pp Drell-YanTwist-3 TMDs in Drell-Yan processes
Boer-Mulders functionT-even chiral-odd TMDs: h1T , h⊥
1T , h⊥1L
T-even chiral-odd TMDs: model calculations
spectator model: Jakob, Mulders, Rodrigues, 97; Bacchetta, Conti,Radici 08
constituent quark model: Pasquini and Yuan 10.
bag model: Avakian, Efremov, Schweitzer, Yuan 10.
light-cone diquark (spectator) model: She, Zhu, Ma 09; Zhu, Ma 11.
Zhun Lu chiral-odd TMDs at twist 2 (and beyond) from (single-/un)polarized Drell-Y
Page 15
overview on leading-twist chiral-odd TMDschiral-odd TMDs in πN Drell-Yan
chiral-odd TMDs in single polarized pp Drell-YanTwist-3 TMDs in Drell-Yan processes
Boer-Mulders functionT-even chiral-odd TMDs: h1T , h⊥
1T , h⊥1L
T-even chiral-odd TMDs: model calculations
0
2
4
0 0.25 0.5 0.75 x
hu1
-0.4
-0.2
0
0 0.25 0.5 0.75 x
h ⊥ (1)u1L
-0.6
-0.4
-0.2
0
0 0.25 0.5 0.75 x
h ⊥ (1)u1T
-0.75
-0.5
-0.25
0
0 0.25 0.5 0.75 x
hd1
0
0.1
0.2
0 0.25 0.5 0.75 x
h ⊥ (1)d1L
0
0.1
0.2
0.3
0 0.25 0.5 0.75 x
h ⊥ (1)d1T
Figure: From arxiv:1108.1713
solid curve: Pasquini and Yuan 10.
dashed curve: Jakob, P. J. Mulders, and J. Rodrigues, 97
dotted curve: Avakian, Efremov, Schweitzer, Yuan 10.
Different model results qualitatively agree with each other.
Zhun Lu chiral-odd TMDs at twist 2 (and beyond) from (single-/un)polarized Drell-Y
Page 16
overview on leading-twist chiral-odd TMDschiral-odd TMDs in πN Drell-Yan
chiral-odd TMDs in single polarized pp Drell-YanTwist-3 TMDs in Drell-Yan processes
Boer-Mulders functionT-even chiral-odd TMDs: h1T , h⊥
1T , h⊥1L
T-even chiral-odd TMDs: model calculations
light-cone diquark model: She, Zhu, Ma 09; Zhu, Ma 11:
juv (x, k2T ) =[
fuv
1 (x, k2T )−1
2fdv
1 (x, k2T )]
W jS(x, k
2T )
− 1
6fdv
1 (x, k2T )WjV (x, k
2T ),
jdv(x, k2T ) =− 1
3fdv
1 (x, k2T )WjV (x, k
2T ), j = h1, h
⊥1T , h
⊥1L.
0.0
0.2
0.4
0.0 0.2 0.4 0.6 0.8 x
x huv1
-0.4
-0.2
0.0
0.0 0.2 0.4 0.6 0.8 x
x hdv1
-0.2
-0.1
0.0
0.0 0.2 0.4 0.6 0.8 x
x h⊥ (1)uv1T
0.0
0.1
0.2
0.0 0.2 0.4 0.6 0.8 x
x h⊥ (1)dv1T
-0.2
-0.1
0.0
0.0 0.2 0.4 0.6 0.8 x
x h⊥ (1)uv1L
0.0
0.1
0.2
0.0 0.2 0.4 0.6 0.8 x
x h⊥ (1)dv1L
Zhun Lu chiral-odd TMDs at twist 2 (and beyond) from (single-/un)polarized Drell-Y
Page 17
overview on leading-twist chiral-odd TMDschiral-odd TMDs in πN Drell-Yan
chiral-odd TMDs in single polarized pp Drell-YanTwist-3 TMDs in Drell-Yan processes
Boer-Mulders functionT-even chiral-odd TMDs: h1T , h⊥
1T , h⊥1L
T-even chiral-odd TMDs: extraction from data
most recent extraction of transversity: Anselmino et.al 08 (fromHERMES and COMPASS SIDIS data on Collins asymmetry)
d(x
)T∆
x u
(x)
T∆x
) d
(x, k
T∆x
) u
(x, k
T∆x
x (GeV)k
−0.1
0
0.1
0.2
0.3
0.4
0.2 0.4 0.6 0.8 1−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
−0.1
0
0.1
0.2
0.3
0.4
x = 0.1
0 0.2 0.4 0.6 0.8 1−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
x = 0.1
Asin(φh+φS)UT = 2
∫
dφS dφh [dσ↑ − dσ↓] sin(φh + φS)
∫
dφS dφh [dσ↑ + dσ↓]
∝∑
q e2q ∆T q(x, k⊥)⊗∆NDh/q↑(z, p⊥)
∑
q e2q fq/p(x, k⊥)⊗Dh/q(z, p⊥)
.
no extractions of pretzelosity and h⊥1L.
Zhun Lu chiral-odd TMDs at twist 2 (and beyond) from (single-/un)polarized Drell-Y
Page 18
overview on leading-twist chiral-odd TMDschiral-odd TMDs in πN Drell-Yan
chiral-odd TMDs in single polarized pp Drell-YanTwist-3 TMDs in Drell-Yan processes
1 overview on leading-twist chiral-odd TMDsBoer-Mulders functionT-even chiral-odd TMDs: h1T , h
⊥1T , h
⊥1L
2 chiral-odd TMDs in πN Drell-Yan
3 chiral-odd TMDs in single polarized pp Drell-Yan
4 Twist-3 TMDs in Drell-Yan processes
Zhun Lu chiral-odd TMDs at twist 2 (and beyond) from (single-/un)polarized Drell-Y
Page 19
overview on leading-twist chiral-odd TMDschiral-odd TMDs in πN Drell-Yan
chiral-odd TMDs in single polarized pp Drell-YanTwist-3 TMDs in Drell-Yan processes
h1(P1) + h2(P2) → γ∗(q) +X → ℓ(l) + ℓ(l′) +X.
Figure: kinematics of Drell-Yan process in the Collins-Soper frame
Left: Definition of φ and φS by Boer (99).
right: Definition of φ and φS by Arnold, Metz, Schlegel (08).
Two conventions are connected by φ ↔ −φ and φS ↔ φS − φ
Zhun Lu chiral-odd TMDs at twist 2 (and beyond) from (single-/un)polarized Drell-Y
Page 20
overview on leading-twist chiral-odd TMDschiral-odd TMDs in πN Drell-Yan
chiral-odd TMDs in single polarized pp Drell-YanTwist-3 TMDs in Drell-Yan processes
cross-section contributed by Chiral-odd TMDs: (Boer 99; Arnold,Metz, Schlegel 09)
dσ(h1h2 → l+l−X)
dx1 dx2 d2qT dΩ=
α2em
12Q2
sin2 θ cos 2φF cos 2φUU + SLsin
2 θ sin 2φF sin 2φLU
+|ST |sin2 θ[
sin(2φ+ φS)Fsin(2φ+φS)TU + sin(2φ− φS)F
sin(2φ−φS)TU
]
+ · · ·
.
F cos 2φUU = C
[2(h · p1T )(h · p2T )− p1T · p2T
M1M2h⊥1 h
⊥1
]
,
Fsin(2φ−φS)TU = C
[h · p1T
M1h1h
⊥1
]
,
Fsin(2φ+φS)TU = C
[2(h · p1T )[2(h · p1T )(h · p2T )− p1T · p2T ]− p21T (h · p2T )
2M1M22
×h⊥1T h
⊥1
]
,
F sin 2φLU = C
[2(h · p1T )(h · p2T )− p1T · p2T
M1M2h⊥1Lh
⊥1
]
.
Zhun Lu chiral-odd TMDs at twist 2 (and beyond) from (single-/un)polarized Drell-Y
Page 21
overview on leading-twist chiral-odd TMDschiral-odd TMDs in πN Drell-Yan
chiral-odd TMDs in single polarized pp Drell-YanTwist-3 TMDs in Drell-Yan processes
Asymmetries in πN DY at COMPASS
The azimuthal asymmetries are defined as
Asin(2φ±φS)UT =
2∫ 2π
0dφ sin(2φ± φS) [dσ
↑ − dσ↓]∫ 2π
0 dφ[dσ↑ + dσ↓]
for transversely polarized target, and
Asin(2φ)UL =
2∫ 2π
0 dφ sin 2φ [dσ⇒ − dσ⇐]∫ 2π
0dφ[dσ⇒ + dσ⇐]
.
for longitudinally polarized target.
Fsin(2φ−φS)UT (h1, h2) = −F
sin(2φ−φS)TU (h2, h1)
Fsin(2φ+φS)UT (h1, h2) = −F
sin(2φ+φS)TU (h2, h1)
F sin 2φUL (h1, h2) = −F sin 2φ
LU (h2, h1)
Zhun Lu chiral-odd TMDs at twist 2 (and beyond) from (single-/un)polarized Drell-Y
Page 22
overview on leading-twist chiral-odd TMDschiral-odd TMDs in πN Drell-Yan
chiral-odd TMDs in single polarized pp Drell-YanTwist-3 TMDs in Drell-Yan processes
Asymmetries in πN DY at COMPASS
Kinematics:
√s = 18.9 GeV, 0.1 < x1 < 1, 0.05 < x2 < 0.5,
4 6 M 6 8.5 GeV, 0 6 qT 6 2 GeV.
190 GeV pion beams collide on nucleon target
isospin symmetry and charge conjugation(Φq [Γ] = Φq [Γ]for γµ, iσµνγ5) ⇒ simple flavor structure of the pion:
f u/π−
= fd/π−
= f d/π+
= fu/π+
, f = f1 or h⊥1
ideal in unraveling the flavor content of nucleon TMDs:
f1 ⊗ f⊥1T ; h⊥
1 ⊗ h⊥1 ; h⊥
1 ⊗ h⊥1T ; h⊥
1 ⊗ h1; h⊥1 ⊗ h⊥
1L
Zhun Lu chiral-odd TMDs at twist 2 (and beyond) from (single-/un)polarized Drell-Y
Page 23
overview on leading-twist chiral-odd TMDschiral-odd TMDs in πN Drell-Yan
chiral-odd TMDs in single polarized pp Drell-YanTwist-3 TMDs in Drell-Yan processes
Asymmetries in πp↑ DY at COMPASS by h⊥1T
4 6 8 0.5 1.0 1.5 2.00.2 0.4 0.6 0.8-0.06
-0.04
-0.02
0.00
0.02
0.04
0.06
h 1 h
1T
qT/GeV
xF
Asin(
2S
UT
-
+
- with qT cut
+ with qT cut
M/GeV
Asin(2φ+φS)UT in π−p DY: positive. (−h
⊥u/π−
1 ⊗ h⊥u1T )
Asin(2φ+φS)UT in π+p DY: negative, consistent with zero
(−h⊥d/π+
1 ⊗ h⊥d1T ). Suppressed by the smaller size of h⊥d
1T
Blue curves: qT is integrated from 1 GeV to 2 GeV, increase theasymmetries.
ZL, Ma and She, 11
Zhun Lu chiral-odd TMDs at twist 2 (and beyond) from (single-/un)polarized Drell-Y
Page 24
overview on leading-twist chiral-odd TMDschiral-odd TMDs in πN Drell-Yan
chiral-odd TMDs in single polarized pp Drell-YanTwist-3 TMDs in Drell-Yan processes
Asymmetries in πp↑ DY at COMPASS by h1
4 6 8 0.5 1.0 1.5 2.00.2 0.4 0.6 0.8
-0.12
-0.08
-0.04
0.00
0.04
0.08
0.12 h 1 h
1
qT/GeV
xF
Asin(
2S
UT
-
+
M/GeV
Asin(2φ−φS)UT in π−p DY: negative (− h
⊥u/π−
1 ⊗ hu1 ).
Asin(2φ−φS)UT in π+p DY: positive (− h
⊥d/π+
1 ⊗ hd1).
Zhun Lu chiral-odd TMDs at twist 2 (and beyond) from (single-/un)polarized Drell-Y
Page 25
overview on leading-twist chiral-odd TMDschiral-odd TMDs in πN Drell-Yan
chiral-odd TMDs in single polarized pp Drell-YanTwist-3 TMDs in Drell-Yan processes
Asymmetries in πp⇒ DY at COMPASS by h⊥1L
4 6 8 0.5 1.0 1.5 2.00.2 0.4 0.6 0.8
-0.12
-0.08
-0.04
0.00
0.04
0.08
0.12 h 1 h1L
qT/GeV
xF
Asin2
UL
-
+
M/GeV
Asin 2φUL in π−p DY: positive (− h
⊥u/π−
1 ⊗ h⊥u1L ).
Asin 2φUL in π+p DY: negative (− h
⊥d/π+
1 ⊗ h⊥d1L ).
Zhun Lu chiral-odd TMDs at twist 2 (and beyond) from (single-/un)polarized Drell-Y
Page 26
overview on leading-twist chiral-odd TMDschiral-odd TMDs in πN Drell-Yan
chiral-odd TMDs in single polarized pp Drell-YanTwist-3 TMDs in Drell-Yan processes
Asymmetries in πD⇒ DY at COMPASS by h⊥1L
4 6 8 0.5 1.0 1.5 2.00.2 0.4 0.6 0.8
-0.12
-0.08
-0.04
0.00
0.04
0.08
0.12 h 1 h1L
qT/GeV
xF
Asin2
UL
-
+
M/GeV
Asin(2φ)UL in π−D DY: positive (−h
⊥u/π−
1 ⊗ (h⊥u1L + h⊥d
1L )).
Asin(2φ)UL in π+D DY: positive (−h
⊥d/π+
1 ⊗ (h⊥u1L + h⊥d
1L )).
Asymmetries for the π− and π+ beams are similar.
Zhun Lu chiral-odd TMDs at twist 2 (and beyond) from (single-/un)polarized Drell-Y
Page 27
overview on leading-twist chiral-odd TMDschiral-odd TMDs in πN Drell-Yan
chiral-odd TMDs in single polarized pp Drell-YanTwist-3 TMDs in Drell-Yan processes
qT weighted asymmetries
Bacchetta, Conti, Radici, Guagnelli 10 :
AqT sin(2φ−φS2
)
UT= 2
⟨qTM1
sin(2φ − φS2)⟩
UT
〈1〉UU
= −2B(y)
A(y)
∑a e2a x1h
⊥(1)a1 (x1)x2ha
1(x2)∑a e2a x1fa
1 (x1)x2fa1 (x2)
,
Aq3T sin(2φ+φS2
)
UT= 2
⟨q3T
6M1M22
sin(2φ + φS2)⟩
UT
〈1〉UU
=− 2B(y)
A(y)
∑a e2a x1h
⊥(1)a1 (x1)x2h
⊥(2)a1T (x2)∑
a e2a x1fa1 (x1)x2fa
1 (x2),
AqT sin 2φUL
= 2
⟨q2T
4M1M2sin 2φ
⟩
UL
〈1〉UU
= −2B(y)
A(y)
∑a e2a x1h
⊥(1)a1 (x1)x2h
⊥(1)a1L (x2)∑
a e2a x1fa1 (x1)x2fa
1 (x2),
ZL, Ma, Schmidt 06:
Aq2T cos 2φ
UU= 2
⟨q2T
4M1M2cos(2φ)
⟩
UU
〈1〉UU
=2B(y)
A(y)
∑a e2a x1h
⊥(1)a1 (x1)x2h
⊥(1)a1 (x2)∑
a e2a x1fa1 (x1)x2fa
1 (x2),
Zhun Lu chiral-odd TMDs at twist 2 (and beyond) from (single-/un)polarized Drell-Y
Page 28
overview on leading-twist chiral-odd TMDschiral-odd TMDs in πN Drell-Yan
chiral-odd TMDs in single polarized pp Drell-YanTwist-3 TMDs in Drell-Yan processes
A(y) and B(y) are defined as:
A(y) =1
2− y + y2
cm=
1
4(1 + cos2 θ), B(y) = y(1− y)
cm=
1
4sin2 θ.
In the region where x1 and x2 are not small (ZL, Ma, Schmidt 06):
Aq2T cos 2φ
UU,π−D
Aq2T
cos 2φ
UU,π−p
≈1 +
h⊥(1)d1 (x2)
h⊥(1)u1 (x2)
1 +fd1 (x2)
fu1 (x2)
;A
q2T cos 2φ
UU,π+p
Aq2T
cos 2φ
UU,π−p
≈h⊥(1)d1 (x2)
h⊥(1)u1 (x2)
fd1 (x2)
fu1 (x2)
.
similarly:
AqT sin(2φ−φS2)
UT,π−D
AqT sin(2φ−φS2)
UT,π−p
≈1 +
hd1(x2)
hu1 (x2)
1 +fd1 (x2)
fu1 (x2)
;A
qT sin(2φ−φS2)
UT,π+p
AqT sin(2φ−φS2)
UT,π−p
≈hd1(x2)
hu1 (x2)
fd1 (x2)
fu1 (x2)
.
Apply different beams/targets to explore the flavor contents ofTMDs
Zhun Lu chiral-odd TMDs at twist 2 (and beyond) from (single-/un)polarized Drell-Y
Page 29
overview on leading-twist chiral-odd TMDschiral-odd TMDs in πN Drell-Yan
chiral-odd TMDs in single polarized pp Drell-YanTwist-3 TMDs in Drell-Yan processes
1 overview on leading-twist chiral-odd TMDsBoer-Mulders functionT-even chiral-odd TMDs: h1T , h
⊥1T , h
⊥1L
2 chiral-odd TMDs in πN Drell-Yan
3 chiral-odd TMDs in single polarized pp Drell-Yan
4 Twist-3 TMDs in Drell-Yan processes
Zhun Lu chiral-odd TMDs at twist 2 (and beyond) from (single-/un)polarized Drell-Y
Page 30
overview on leading-twist chiral-odd TMDschiral-odd TMDs in πN Drell-Yan
chiral-odd TMDs in single polarized pp Drell-YanTwist-3 TMDs in Drell-Yan processes
p↑/→(P1) + p(P2) → γ∗(q) +X → ℓ(l) + ℓ(l′) +X.
Figure: kinematics of Drell-Yan process in the Collins-Soper frame
current/planned polarized pp program
c.m. energy (GeV) mode
RHIC 200, 510 colliderRHIC 22 fixed-target
J-PARC 10 fixed-targetE906 15 fixed-targetNICA 12÷ 27 collider
SPASCHARM 10.7 fixed-target
Zhun Lu chiral-odd TMDs at twist 2 (and beyond) from (single-/un)polarized Drell-Y
Page 31
overview on leading-twist chiral-odd TMDschiral-odd TMDs in πN Drell-Yan
chiral-odd TMDs in single polarized pp Drell-YanTwist-3 TMDs in Drell-Yan processes
In PRD84 074036 (ZL, Ma, Zhu 11), the asymmetries is defined as
Asin(2φ−φS)TU (x1, x2, qT ) =
Fsin(2φ−φS)TU
F 1UU
,
Asin(2φ+φS)TU (x1, x2, qT ) =
Fsin(2φ+φS)TU
F 1UU
,
Asin 2φLU (x1, x2, qT ) =
F sin 2φLU
F 1UU
.
In an analogy with the definition of the cos 2φ: ν = 2F cos 2φUU /F 1
UU
AW (φ,φS)PU =
2A(y)∫ 2π
0dφW (φ, φS)[dσ
↑/⇒ − dσ↑/⇐]
B(y)∫ 2π
0dφ[dσ↑/⇒ + dσ↑/⇐]
, P = L, T
Zhun Lu chiral-odd TMDs at twist 2 (and beyond) from (single-/un)polarized Drell-Y
Page 32
overview on leading-twist chiral-odd TMDschiral-odd TMDs in πN Drell-Yan
chiral-odd TMDs in single polarized pp Drell-YanTwist-3 TMDs in Drell-Yan processes
-0.2
0.0
0.2
0.4
ATUsin(2φ - φS) ∼ h1⊗ h⊥
1
ω = 0.5 ω = 0.5
-0.2
0.0
0.2
0.4 ω = 1.0 ω = 1.0
-0.2
0.0
0.2
0.4
0.0 0.2 0.4 0.6 y
ω = 2.0
4.0 5.0 6.0 7.0 Q
ω = 2.0
-0.2
0.0
0.2
ATUsin(2φ + φS) ∼ h⊥
1T⊗ h⊥1
ω = 0.5 ω = 0.5
-0.2
0.0
0.2 ω = 1.0 ω = 1.0
-0.2
0.0
0.2
0.0 0.2 0.4 0.6 y
ω = 2.0
4.0 5.0 6.0 7.0 Q
ω = 2.0
-0.2
0.0
0.2
ALUsin2φ ∼ h⊥
1L⊗ h⊥1
ω = 0.5 ω = 0.5
-0.2
0.0
0.2 ω = 1.0 ω = 1.0
-0.2
0.0
0.2
0.0 0.2 0.4 0.6 y
ω = 2.0
4.0 5.0 6.0 7.0 Q
ω = 2.0
h⊥
1 from the parametrization by ZL, Schmidt 09; T-even chiral-odd TMDs fromlight-cone diquark model (She, Zhu, Ma 09; Zhu, Ma 11)
Dashed line in the left panels: calculated from the parameterization for h1 byAnselmino et.al., 08
Shaded regions: ranges of the asymmetries by considering the additional
contribution from hq1, h
⊥q1T and h⊥q
1L√s = 15 GeV, 0.3 < x1 < 0.7, 0.1 < x2 < 0.3,
0 < qT < 1 GeV, 4 GeV < Q < 7 GeV,
Zhun Lu chiral-odd TMDs at twist 2 (and beyond) from (single-/un)polarized Drell-Y
Page 33
overview on leading-twist chiral-odd TMDschiral-odd TMDs in πN Drell-Yan
chiral-odd TMDs in single polarized pp Drell-YanTwist-3 TMDs in Drell-Yan processes
-0.2
0.0
0.2
0.4
ATUsin(2φ - φS) ∼ h1⊗ h⊥
1
ω = 0.5 ω = 0.5
-0.2
0.0
0.2
0.4 ω = 1.0 ω = 1.0
-0.2
0.0
0.2
0.4
0.0 0.2 0.4 0.6 y
ω = 2.0
4.0 4.2 4.4 4.6 4.8 5.0 Q
ω = 2.0
-0.2
0.0
0.2
ATUsin(2φ + φS) ∼ h⊥
1T⊗ h⊥1
ω = 0.5 ω = 0.5
-0.2
0.0
0.2 ω = 1.0 ω = 1.0
-0.2
0.0
0.2
0.0 0.2 0.4 0.6 y
ω = 2.0
4.0 4.2 4.4 4.6 4.8 5.0 Q
ω = 2.0
-0.2
0.0
0.2
ALUsin2φ ∼ h⊥
1L⊗ h⊥1
ω = 0.5 ω = 0.5
-0.2
0.0
0.2 ω = 1.0 ω = 1.0
-0.2
0.0
0.2
0.0 0.2 0.4 0.6 y
ω = 2.0
4.0 4.2 4.4 4.6 4.8 5.0 Q
ω = 2.0
kinematical cuts at J-PARC:
4 GeV < Q < 5 GeV, 0 < qT < 1 GeV,
0.5 < x1 < 0.9,
Zhun Lu chiral-odd TMDs at twist 2 (and beyond) from (single-/un)polarized Drell-Y
Page 34
overview on leading-twist chiral-odd TMDschiral-odd TMDs in πN Drell-Yan
chiral-odd TMDs in single polarized pp Drell-YanTwist-3 TMDs in Drell-Yan processes
-0.2
0.0
0.2
0.4
ATUsin(2φ - φS) ∼ h1⊗ h⊥
1
ω = 0.5 ω = 0.5
-0.2
0.0
0.2
0.4 ω = 1.0 ω = 1.0
-0.2
0.0
0.2
0.4
-1.0 -0.5 0.0 0.5 1.0 y
ω = 2.0
4.0 5.0 6.0 7.0 8.0 9.0 Q
ω = 2.0
-0.2
0.0
0.2
ATUsin(2φ + φS) ∼ h⊥
1T⊗ h⊥1
ω = 0.5 ω = 0.5
-0.2
0.0
0.2 ω = 1.0 ω = 1.0
-0.2
0.0
0.2
-1.0 -0.5 0.0 0.5 1.0 y
ω = 2.0
4.0 5.0 6.0 7.0 8.0 9.0 Q
ω = 2.0
-0.2
0.0
0.2
ALUsin2φ ∼ h⊥
1L⊗ h⊥1
ω = 0.5 ω = 0.5
-0.2
0.0
0.2 ω = 1.0 ω = 1.0
-0.2
0.0
0.2
-1.0 -0.5 0.0 0.5 1.0 y
ω = 2.0
4.0 5.0 6.0 7.0 8.0 9.0 Q
ω = 2.0
The kinematics cuts at NICA:√s = 27 GeV, 4 GeV < Q < 9 GeV,
0 < qT < 1 GeV, 0.1 < x1 < 0.8,
Zhun Lu chiral-odd TMDs at twist 2 (and beyond) from (single-/un)polarized Drell-Y
Page 35
overview on leading-twist chiral-odd TMDschiral-odd TMDs in πN Drell-Yan
chiral-odd TMDs in single polarized pp Drell-YanTwist-3 TMDs in Drell-Yan processes
-0.2
0.0
0.2
0.4
ATUsin(2φ - φS) ∼ h1⊗ h⊥
1
ω = 0.5 ω = 0.5
-0.2
0.0
0.2
0.4 ω = 1.0 ω = 1.0
-0.2
0.0
0.2
0.4
-0.6 -0.3 0.0 0.3 0.6 0.9 y
ω = 2.0
5.0 6.0 7.0 8.0 Q
ω = 2.0
-0.2
0.0
0.2
ATUsin(2φ + φS) ∼ h⊥
1T⊗ h⊥1
ω = 0.5 ω = 0.5
-0.2
0.0
0.2 ω = 1.0 ω = 1.0
-0.2
0.0
0.2
-0.6 -0.3 0.0 0.3 0.6 0.9 y
ω = 2.0
5.0 6.0 7.0 8.0 Q
ω = 2.0
-0.2
0.0
0.2
ALUsin2φ ∼ h⊥
1L⊗ h⊥1
ω = 0.5 ω = 0.5
-0.2
0.0
0.2 ω = 1.0 ω = 1.0
-0.2
0.0
0.2
-0.6 -0.3 0.0 0.3 0.6 0.9 y
ω = 2.0
5.0 6.0 7.0 8.0 Q
ω = 2.0
RHIC kinematics for the fixed-target experiment:
√s = 22 GeV, 4.5 GeV < Q < 8 GeV,
0 < qT < 1 GeV, 0.2 < x1 < 0.6,
Zhun Lu chiral-odd TMDs at twist 2 (and beyond) from (single-/un)polarized Drell-Y
Page 36
overview on leading-twist chiral-odd TMDschiral-odd TMDs in πN Drell-Yan
chiral-odd TMDs in single polarized pp Drell-YanTwist-3 TMDs in Drell-Yan processes
-0.2
0.0
0.2
0.4
ATUsin(2φ - φS) ∼ h1⊗ h⊥
1
ω = 0.5 ω = 0.5
-0.2
0.0
0.2
0.4 ω = 1.0 ω = 1.0
-0.2
0.0
0.2
0.4
-1.0 0.0 1.0 2.0 y
ω = 2.0
4.0 5.0 6.0 7.0 8.0 9.0 Q
ω = 2.0
-0.2
0.0
0.2
ATUsin(2φ + φS) ∼ h⊥
1T⊗ h⊥1
ω = 0.5 ω = 0.5
-0.2
0.0
0.2 ω = 1.0 ω = 1.0
-0.2
0.0
0.2
-1.0 0.0 1.0 2.0 y
ω = 2.0
4.0 5.0 6.0 7.0 8.0 9.0 Q
ω = 2.0
-0.2
0.0
0.2
ALUsin2φ ∼ h⊥
1L⊗ h⊥1
ω = 0.5 ω = 0.5
-0.2
0.0
0.2 ω = 1.0 ω = 1.0
-0.2
0.0
0.2
-1.0 0.0 1.0 2.0 y
ω = 2.0
4.0 5.0 6.0 7.0 8.0 9.0 Q
ω = 2.0
kinematics for collider experiment at RHIC-STAR:
√s = 200 GeV, 4 GeV < Q < 9 GeV,
0 < qT < 1 GeV, − 1 < y < 2.
Zhun Lu chiral-odd TMDs at twist 2 (and beyond) from (single-/un)polarized Drell-Y
Page 37
overview on leading-twist chiral-odd TMDschiral-odd TMDs in πN Drell-Yan
chiral-odd TMDs in single polarized pp Drell-YanTwist-3 TMDs in Drell-Yan processes
1 overview on leading-twist chiral-odd TMDsBoer-Mulders functionT-even chiral-odd TMDs: h1T , h
⊥1T , h
⊥1L
2 chiral-odd TMDs in πN Drell-Yan
3 chiral-odd TMDs in single polarized pp Drell-Yan
4 Twist-3 TMDs in Drell-Yan processes
Zhun Lu chiral-odd TMDs at twist 2 (and beyond) from (single-/un)polarized Drell-Y
Page 38
overview on leading-twist chiral-odd TMDschiral-odd TMDs in πN Drell-Yan
chiral-odd TMDs in single polarized pp Drell-YanTwist-3 TMDs in Drell-Yan processes
angular dependence of DY at O(1/Q) (Arnold, Metz, Schlegel 09, ZL, Schmidt11):
dσtwist-3
dx1dx2d2qT dΩ=α2em
3Q2sin 2θ
cos φF cosφ
UU+ S1L sinφF sinφ
LU+ S2L sinφF sinφ
UL
+ |~S1T |[sin(φ1 + φ)F
sin(φS1+φ)
TU+ sin(φS1
− φ)Fsin(φS1
−φ)
TU
]
+ |~S2T |[sin(φS2
+ φ)Fsin(φS2
+φ)
UT+ sin(φS2
− φ)Fsin(φS2
−φ)
UT
]
TMD correlator at twist-3 (Goeke et.al, 05; Bacchetta et.al, 06):
Φ(x, pT ) = . . .+M
2P+
e− i es γ5 − e⊥T
ǫρσT
pTρ
STσ
M
+ f⊥p/T
M− f ′
T ǫρσT
γρSTσ − f⊥
s
ǫρσT
γρpTσ
M
+ g′T γ5 S/T + g⊥s γ5p/T
M− g⊥γ5
ǫρσT
γρpTσ
M
+ hs[ n/+, n/−]γ5
2+ h⊥
T
[S/T , p/T
]γ5
2M+ i h
[n/+, n/−
]
2
.
hL, hT , h⊥
T, and h might be probed in Drell-Yan process
Zhun Lu chiral-odd TMDs at twist 2 (and beyond) from (single-/un)polarized Drell-Y
Page 39
overview on leading-twist chiral-odd TMDschiral-odd TMDs in πN Drell-Yan
chiral-odd TMDs in single polarized pp Drell-YanTwist-3 TMDs in Drell-Yan processes
un/single-polarized structure functions at twist-3
F cosφUU
=2
QC[(h · k1T )
(f⊥f1 − M2
M1h⊥
1 h
)− (h · k2T )
(f1 f
⊥ − M1
M2h h⊥
1
)]
F sinφLU
=2
QC[(h · k1T )
(f⊥
L f1 +M2
M1h⊥
1L h
)− (h · k2T )
(g1Lg
⊥ +M1
M2hL h
⊥
1
)]
Fsin(φS1
−φ)
TU=
1
QC[2M1 fT f1 + 2M2 h1h
+(k1T · k2T )
f⊥
1T f⊥
M1− g1T g⊥
M1− hTh⊥
1
M2+
h⊥
Th⊥
1
M2
Fsin(φS1
+φ)
TU=
1
QC
−(2 (h · k1T )2 − k2
1T
)
f⊥
Tf1
M1+
M2 h⊥
1T h
M21
+(2h · k1Th · k2T − k1T · k2T )
f⊥
1T f⊥
M1+
g1T g⊥
M1+
hTh⊥
1
M2+
h⊥
T h⊥
1
M2
f = x1
((1 − c) f + c f
), f = x2
(c f + (1− c) f
). c =
1
2: CS frame
Zhun Lu chiral-odd TMDs at twist 2 (and beyond) from (single-/un)polarized Drell-Y
Page 40
overview on leading-twist chiral-odd TMDschiral-odd TMDs in πN Drell-Yan
chiral-odd TMDs in single polarized pp Drell-YanTwist-3 TMDs in Drell-Yan processes
Accessible in DY?
lesson from SIDIS
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
LUhφ
sin
A
-0.05
0
0.05
0.10.1< x <0.2
(GeV)TP0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-0.05
0
0.05
0.1
0.3< x <0.40.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.2< x <0.3
(GeV)TP0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.4< x <0.6
Figure: sin φ asymmetry measured by CLAS (arxiv:1106.2293) for Q > 1 GeV,W 2 > 4 GeV2 and 0.4 < z < 0.7
Zhun Lu chiral-odd TMDs at twist 2 (and beyond) from (single-/un)polarized Drell-Y
Page 41
overview on leading-twist chiral-odd TMDschiral-odd TMDs in πN Drell-Yan
chiral-odd TMDs in single polarized pp Drell-YanTwist-3 TMDs in Drell-Yan processes
SIDIS at twist-3 (Bacchetta et.al,06)
F sinφh
LU =2M
QC
PT · pT
M
[
Mh
Mh⊥1
E
z+ x g⊥D1
]
−
PT · kT
Mh
[
Mh
Mf1
G⊥
z+ x eH⊥
1
]
.
Drell-Yan at low mass (2.0 < Q < 2.5GeV) region (COMPASS,SPASCHARM)?
Zhun Lu chiral-odd TMDs at twist 2 (and beyond) from (single-/un)polarized Drell-Y
Page 42
overview on leading-twist chiral-odd TMDschiral-odd TMDs in πN Drell-Yan
chiral-odd TMDs in single polarized pp Drell-YanTwist-3 TMDs in Drell-Yan processes
Summary
The πN Drell-Yan program at COMPASS provides greatopportunity to explore the flavor contents of chiral-odd TMDs
Single polarized pp Drell-Yan programme at E906, RHIC, J-PARC,NICA is vital to access h1, h1L in the valence and sea region,
through asymmetries Asin(2φ−φS)TU and Asin 2φ
LU .
Possibility for accessing twist-3 TMDs by Drell-Yan is explored.
Zhun Lu chiral-odd TMDs at twist 2 (and beyond) from (single-/un)polarized Drell-Y