Tensor structureTensor structure

in high-energy proton-deuteron reactionsin high-energy proton-deuteron reactions

September 25, 2009September 25, 2009

shunzo.kumano@kek.jpshunzo.kumano@kek.jphttp://research.kek.jp/people/kumanos/http://research.kek.jp/people/kumanos/

KEK theory center workshop KEK theory center workshop on Short-range correlations and tensor structure at J-PARCon Short-range correlations and tensor structure at J-PARC

Shunzo KumanoShunzo Kumano

High Energy Accelerator Research Organization (KEK)High Energy Accelerator Research Organization (KEK)

Graduate University for Advanced Studies (GUAS)Graduate University for Advanced Studies (GUAS)

Based on works with Based on works with F. E. Close, S. Hino, T.-Y. Kimura, M. MiyamaF. E. Close, S. Hino, T.-Y. Kimura, M. Miyama

ContentsContents

Introduction to tensor structure functionsIntroduction to tensor structure functions Tensor structure functions (Tensor structure functions (bb11, …, , …, bb44))

Sum rule, Projection to each structure function from WSum rule, Projection to each structure function from W

Polarized proton-deuteron Drell-Yan processesPolarized proton-deuteron Drell-Yan processes

Note: The proton does not have to be polarized for Note: The proton does not have to be polarized for bb11..

If a polarized-deuteron target will becomes available at J-PARC or any other facilities, it is a possible experiment.

Introduction to Introduction to Tensor Structure FunctionsTensor Structure Functions

References on tensor structure function References on tensor structure function bb11

Theoretical formalism for polarized electron-deuteron Theoretical formalism for polarized electron-deuteron deep inelastic scatteringdeep inelastic scattering P. Hoodbhoy, R. L. Jaffe, and A. Manohar, NP B312 (1989) 571. [ L. L. Frankfurt and M. I. Strikman, NP A405 (1983) 557. ]

HERMES experimental resultHERMES experimental result A. Airapetian et al., Phys. Rev. Lett. 95 (2005) 242001.

Our worksOur works

Sum rule for Sum rule for bb11

F. E. Close and SK, Phys. Rev. D 42 (1990) 2377.

Projections to Projections to bb11, …, , …, bb44 from from WW

T.-Y. Kimura and SK, Phys. Rev. D 78 (2008) 117505.

MotivationMotivation

Spin structure of the spin-1/2 nucleonSpin structure of the spin-1/2 nucleon Nucleon spin puzzle:Nucleon spin puzzle: This issue is not solved yet, This issue is not solved yet,

but it is rather well studied theoretically and experimentally.but it is rather well studied theoretically and experimentally.

Spin-1 hadrons (e.g. deuteron)Spin-1 hadrons (e.g. deuteron) There are some theoretical studies especially on tensor structThere are some theoretical studies especially on tensor struct

ure ure in electron-deuteron deep inelastic scattering.in electron-deuteron deep inelastic scattering.

HERMES experimental resultsHERMES experimental results

A few investigations have been done for polarizedA few investigations have been done for polarized proton-deuteron processes. proton-deuteron processes.

J-PARC experiment ?J-PARC experiment ?

Structure function Structure function bb11 in a simple example in a simple example

Spin-1 particles (deuteron, mesons)Spin-1 particles (deuteron, mesons)

only in S-waveonly in S-wave

bb11 == 00

The The bb11 probes a dynamical aspect of hadron structure probes a dynamical aspect of hadron structure beyond simple expectations of a naive quark model.beyond simple expectations of a naive quark model. Description of tensor structure Description of tensor structure by quark-gluon degrees of freedomby quark-gluon degrees of freedom

Electron scattering from a spin-1 hadron Electron scattering from a spin-1 hadron

W =−F1g + F2

p p

+ g1

iελσqλsσ + g2

i 2 ελσqλ p⋅qsσ −s⋅qpσ( )

−b1r +16

b2 s + t +u( ) +12

b3 s −u( ) +12

b4 s −t( )

r =1 2 q⋅E∗q⋅E −

13 2κ⎛

⎝⎜⎞⎠⎟

g ,s =2 2 q⋅E∗q⋅E −

13 2κ⎛

⎝⎜⎞⎠⎟

p p

t =1

2 2 q⋅E∗p E +q⋅E∗p E +q⋅Ep E∗+q⋅Ep E

∗ −43 p p

⎛⎝⎜

⎞⎠⎟

u =1

E∗E + E

∗E +23

M 2g −23

p p⎛⎝⎜

⎞⎠⎟

=p ⋅q, κ = 1 + M 2 Q2 ν 2 , E2 = −M 2 , sσ = −i

M 2 ε σαβτ Eα∗Eβ pτ

P. Hoodbhoy, R. L. Jaffe, and A. Manohar, NP B312 (1989) 571.[ L. L. Frankfurt and M. I. Strikman, NP A405 (1983) 557. ]

Note: Obvious factors from qW =qW =0arenotexplicitlywritten.

spin-1/2, spin-1

spin-1 only

2xb1 =b2 inthescalinglimit~O(1)

b3 , b4 =twist-4~M 2

Q2

E =polarizationvector

b1 ,⋅⋅⋅,b4 temsaredefinedsothattheyvanishbyspinaverage.

b1 , b2 tems are defined to satisfy

2xb1 =b2 intheBjorkenscalinglimit.

Projections to Projections to FF11, , FF22, …, , …, bb44 from W from W Calculate W inhadronmodels→ needtoextractstructurefunctionsb1 ,b2 ,⋅⋅⋅ProjectionoperatorsareneededtoextractthemfromthecalculatedW .ForF1 andF2 ,theyarewellknown:

F1 =−12

g −κ −1κ

p p

M 2

⎛

⎝⎜⎞

⎠⎟W , F2 =−

xκ

g −κ −1κ

3p p

M 2

⎛

⎝⎜⎞

⎠⎟W , κ =1+Q2 2

QuickTime˛ Ç∆ êLí£ÉvÉçÉOÉâÉÄ

ǙDZÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅB

Try to obtain projections

in a spin-1 hadron by combinations of

g , p p

M 2 , ε αβqαsβ , ...

Results on a spin-1 hadron

Bjorken scaling limit

F1 =12x

F2 =−12

g 13δλ f λi

Wλ f λi

g1 =−i2

ε αβqαsβδλ f 1δλi 1

Wλ f λi

b1 =12x

b2 =12

g δλ f 1δλi 1

−δλ f 0δλi 0( )W

λ f λi T.-Y. Kimura and SK, PRD 78 (2008) 117505.

StructureStructureFunctionsFunctions

PartonPartonModelModel

F1 ∝ dσ

g1 ∝ dσ ↑,+1( ) −dσ ↑,−1( )

b1 ∝ dσ 0( ) −dσ +1( ) + dσ −1( )

2

q↑H x,Q2( )⎡⎣ ⎤⎦

F1 =12

ei2

i∑ qi + qi( ) qi =

13

qi+1 + qi

0 + qi−1( )

g1 =12

ei2 Δqi + Δqi( )

i∑ Δqi = qi↑

+1 − qi↓+1

b1 =12

ei2 δqi +δqi( )

i∑ δqi = qi

0 −qi

+1 + qi−1

2

note: σ (0)−σ (+1) + σ (−1)

2=3 σ −

32

σ (+1) + σ (−1)[ ]

dx b1(x) =dimensionless∫ : QM 2???

M =hadron mass Q =quadrupolemoment

↔

dx b1D (x)∫ = dx

49

δuD +δuD( ) +19

δdD +δdD +δsD +δsD( )⎡⎣⎢

⎤⎦⎥∫

=59

dx δuv(x) + δuv(x)[ ]∫ +19

δQ+δQ( )sea

ΓH , H = p, H J0 (0) p, H = ei dx qi↑H + qi↓

H − qi↑H − qi↓

H⎡⎣ ⎤⎦∫i

∑1

2Γ0,0 −

12

Γ1,1 + Γ−1,−1( )⎡⎣⎢

⎤⎦⎥= ei dx δqD −δqD

⎡⎣ ⎤⎦∫i∑ =

13

dx δuv(x) + δdv(x)[ ]∫

Sum rule for bSum rule for b11

dx

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟∫

q→ 0

δQ + δ Q( )sea

= dx 5 δu + δu + δdD + δ d D( ) + 2 δsD + δ sD( )⎡⎣

⎤⎦∫ sea

Elastic amplitude in a parton modelElastic amplitude in a parton model

F.E.Close and SK, PRD42, 2377 (1990).

dx b1D (x) =

56

Γ0,0 −12

Γ1,1 + Γ−1,−1( )⎡⎣⎢

⎤⎦⎥+19

δQ+δQ( )sea∫

Γ0,0 = limt→0

Fc (t) −t

3M 2FQ(t)⎡

⎣⎢⎤⎦⎥

Γ+1,+1 = Γ−1,−1 = limt→0

Fc (t) −t

6M 2FQ(t)⎡

⎣⎢⎤⎦⎥

dx∫ b1D(x) =lim

t→ 0−

512

tM 2 FQ(t) +

19

δQ+δQ( )sea

⇒ limt→ 0

−512

tM 2 FQ(t)

dx

xF2

p (x)−F2n(x)⎡⎣ ⎤⎦∫ =

13

dx uv −dv[ ]∫ +23

dx u−d⎡⎣ ⎤⎦∫

MacroscopicallyMacroscopically t → 0

If the sum-rule violationis shown by experiment,it suggests antiquark tensor polarization.

Note: FQ (t) in the unit of 1

M 2

HERMES results on bHERMES results on b11

deuteronpositron

27.6 GeV/c

Ä , 0

b1 measurement in the kinematical region

0.01 < x < 0.45,0.5GeV 2 < Q2 < 5GeV 2

b1 sum rule

dx0.002

0.85

∫ b1(x) = 1.05 ±0.34(stat) ±0.35(sys)[ ] ×10−2

atQ2 =5GeV 2

In the restricted Q2 range Q2 > 1GeV 2

dx0.02

0.85

∫ b1(x) = 0.35 ±0.10(stat) ±0.18(sys)[ ] ×10−2

atQ2 =5GeV 2

dx∫ b1D (x) =lim

t→ 0−

512

tM 2 FQ(t) +

19

δQ+δQ( )sea =0?

dx

xF2

p (x)−F2n(x)⎡⎣ ⎤⎦∫ =

13

dx uv −dv[ ]∫ +23

dx u−d⎡⎣ ⎤⎦∫ ≠1 / 3

A. Airapetian et al. (HERMES), PRL 95 (2005) 242001.

Drell-Yan experiments probethese antiquark distributions.

E866

E906

J-PARCDrell-Yan: p + p→ +− + X, p+ d→ +− + X

σ DY (pd)2σ DY (pp)

≈12

1+d(x2 )u(x2 )

⎡

⎣⎢

⎤

⎦⎥

Actual experimental proposals at J-PARC: P04, P24Actual experimental proposals at J-PARC: P04, P24

E866: existing measurements by the Fermilab-E866

E906: expected measurements by the Fermilab-E906

(from 2010)

J-PARC: proposal stage

It should be possible to use polarized proton-deuteron Drell-Yan processeIt should be possible to use polarized proton-deuteron Drell-Yan processess to measure the tensor polarized distributions.to measure the tensor polarized distributions.

+

–

q

qAntiquarkAntiquarkdistributionsdistributions

Polarized proton-deuteronPolarized proton-deuteronDrell-Yan process Drell-Yan process

rp +

rd→ +− + X

(p+rd→ +− + Xisenoughfortensorstructure)

ReferencesReferences • • General formalism for polarized Drell-Yan processesGeneral formalism for polarized Drell-Yan processes

with spin-1/2 and spin-1 hadronswith spin-1/2 and spin-1 hadrons M. Hino and SK, Phys. Rev. D59 (1999) 094026.

• • Parton-model analysis of polarized Drell-Yan processesParton-model analysis of polarized Drell-Yan processes with spin-1/2 and spin-1 hadronswith spin-1/2 and spin-1 hadrons M. Hino and SK, Phys. Rev. D60 (1999) 054018.

• • An application: Possible extraction of polarized An application: Possible extraction of polarized light-antiquark distributions from Drell-Yanlight-antiquark distributions from Drell-Yan SK and M. Miyama, Phys. Lett. B497 (2000) 149.

Comments on the situationComments on the situation

• There was a feasibility study for polarized deuteron beam at RHIC: E. D. Courant, BNL-report (1998).

• No acutual experimental progress with hadron facilities.

• Future: J-PARC, COMPASS, U70, GSI-FAIR, RHIC, …

Purposes of studying polarized deuteron reactions Purposes of studying polarized deuteron reactions (1) Neutron information

• PolarizedPDFsintheneutron(Note:Therearecontributionsfromb1tolongitudinalspinasymmetryA1

ed.)(2)Newstructurefunctions

• Tensorstructurefunctionb1

→ (1)Testofourhadrondescriptioninanotherspin

(2)Descriptionoftensorstructurebyquark-gluondegreesoffreedom

(3)Asymmetriesinpolarizedlight-antiquarkdistributions

• Δu / Δd, ΔTu / ΔTd

StatusStatus

• re +

rd → ′e + X

Theoretical studies: some Experimental measurements: HERMES

• rp +

rd → μ +μ − + X

Theoretical studies: a few papers Experimental measurements: none (J-PARC, ...)

Formalism of Formalism of pdpd Drell-Yan process Drell-Yan process

proton-protonproton-proton proton-deuteronproton-deuteron

Number ofNumber ofstructure functionsstructure functions 4848 108108

c.m.frame

rQT

After integration over rQT

(or rQT → 0) 1111 2222

See Ref. PRD59 (1999) 094026.

I will briefly explainI will briefly explainin the following.in the following.

In parton modelIn parton model 33 44

Additional structureAdditional structurefunctions due tofunctions due totensor structuretensor structure

Drell-Yan cross section and hadron tensorDrell-Yan cross section and hadron tensor

dσ =1

4 PA ⋅PB( )2 −MA

2 MB2

2π( )4 δ 4

X∑

Sl−

Sl+

∑ PA + PB −kl+−k

l−−PX( ) l+l−X T AB

2 d3kl+

2π( )3 2El+

d3kl−

2π( )3 2El−

l+l−X T AB =u kl−,λ

l−( )eυ kl+,λ

l+( )g

kl++ k

l−( )2 X eJ 0( ) AB

dσd4QdΩ

=α 2

2sQ4 LW

W ≡d4ξ2π( )4

eiQ⋅ξ PASAPBSB J 0( ) J ξ( )∫ PASAPBSB

Possible vectors to expand W

• X μ = PAμQ2Z ⋅PB − PB

μQ2Z ⋅PA

+ Qμ Q ⋅PB Z ⋅PA − Q ⋅PA Z ⋅PB( )

• Yμ = ε μαβγ PAα PBβ Qγ

• Zμ = PAμQ ⋅PB − PB

μQ ⋅PA

W ( )QT =0

=−g A−ZZ

Z2 ′B + Z{TA }C + Z{TB

}D+ Z{SAT } E + Z{SBT

} F −SBT SBT

′G −SAT{SBT

} ′H

+ TA{SBT

} ′I + SBT{TB

} J + QQ K + Q{Z }L + Q{SAT } M + Q{SBT

} N + Q{TA }O+ Q{TB

}P

Expand W by possible combinations

Q =photonmomentum

QT %<

1R

= hardscale, R =hadronsize

AsQT → 0,X =Y → 0

T =ε αβSαZβQ Q{Z } ≡QZ + QZ

Use current conservation: QW =0

W ( )QT =0

=− g −QQ

Q2

⎛

⎝⎜⎞

⎠⎟A−

ZZ

Z2 −13

g −QQ

Q2

⎛

⎝⎜⎞

⎠⎟⎡

⎣⎢

⎤

⎦⎥B+ Z{TA

}C + Z{TB }D+ Z{SAT

} E + Z{SBT } F

− SBT SBT

−12

SBT ⋅SBT g −QQ

Q2 −ZZ

Z2

⎛

⎝⎜⎞

⎠⎟⎡

⎣⎢

⎤

⎦⎥G− SAT

{ SBT } −SAT ⋅SBT g −

QQ

Q2 −ZZ

Z2

⎛

⎝⎜⎞

⎠⎟⎡

⎣⎢

⎤

⎦⎥H

+ TA{SBT

} −TA ⋅SBT g −QQ

Q2 −ZZ

Z2

⎛

⎝⎜⎞

⎠⎟⎡

⎣⎢

⎤

⎦⎥I + SBT

{TB } J

A = ′A1 +MAMB

sZ2 Z⋅SA Z⋅SBA2 −SAT ⋅SBT A3 +8MB

2 Z⋅SB( )2

s2 Q⋅PB( )2 ′A4 +

MB

Z2Q⋅PB

Z⋅SB TA ⋅SBT A5

=A1 +14λAλBA2 +

rSAT

rSBT cos φA −φB( )A3 +

23

2rSBT

2−λB

2( )A4 + λB

rSAT

rSBT sin φA −φB( )A5

•rZ = 0,0,

rZ( ) •

rk =

rk sinθ cosφ, sinθ sinφ, cosθ( )

•rSAT =

rSAT cosφA , sinφA ,0( ) •

rSBT =

rSBT cosφB , sinφB ,0( )

•rTA = Q

rZ

rSAT sinφA ,− cosφA ,0( ) •

rTB = Q

rZ

rSBT sinφB ,− cosφB ,0( )

• SAμ = λ A PA

μ MA + SATμ − δ−

μ λ AMA PA+( )

• SBμ = λ B PB

μ MB + SBTμ − δ+

μ λ BMB PB−( )

• STμ = gμν −

QμQν

Q2−

Zμ Zν

Z2

⎛

⎝⎜⎞

⎠⎟Sν

• T μ = ε μαβγ Sα Zβ Qγ

a =[a−, a+ ,

raT ], a± =(a0 + a3 ) / 2

δ+ =[0, 1,

r0T ], δ−

μ = [1, 0, r0T ]

The coefficients A, B, ... still contain spinfactors in scalar and pseudoscalar forms.

Expand B, C, ⋅⋅⋅inthesameway⋅⋅⋅

dσd4QdΩ

=α 2

2sQ2 2 W0,0 +14λAλBV0,0

LL +rSAT

rSBT cos φA −φB( )V0,0

TT⎡⎣⎢

⎧⎨⎩

+23

2rSBT

2−λB

2( )V0,0

UQ0 +rSAT λB

rSBT sin φA −φB( )V0,0

TQ1 ⎤⎦⎥

+13−cos2θ⎛

⎝⎜⎞⎠⎟

W2,0 +14λAλBV2,0

LL +rSAT

rSBT cos φA −φB( )V2,0

TT⎡⎣⎢

+23

2rSBT

2−λB

2( )V2,0

UQ0 +rSAT λB

rSBT sin φA −φB( )V2,0

TQ1 ⎤⎦⎥

+2sinθ cosθ sin φ−φA( )rSAT U2,1

TU +23

2rSBT

2−λB

2( )U2,1

TQ0⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢ + sin φ−φB( )

rSBT U2,1

UT +14λAλBU2,1

LQ1⎛⎝⎜

⎞⎠⎟

+ sin φ +φA −2φB( )rSAT

rSBT U2,1

TQ2 + cos φ−φA( )rSAT λBU2,1

TL +cos φ−φB( )rSBT λAU2,1

LT + λBU2,1UQ1( )⎤⎦

+sin2θ cos 2φ−2φB( )rSBT

2U2,2

UQ2⎡⎣

+ cos 2φ−φA −2φB( )rSAT

rSBT U2,2

TT

+ sin 2φ−φA −2φB( )rSAT λB

rSBT U2,2

TQ1 +sin 2φ−2φB( )λA

rSBT

2U2,2

LQ2 ⎤⎦}

spin-1/2, spin-1

spin-1 only

A1 =W0,0 , A2 =V0,0LL , A3 =V0,0

TT , A4 =V0,0UQ0 , A5 =V0,0

TQ1 ,

B1 =W2,0 , B2 =V2,0LL , B3 =V2,0

TT , B4 =V2,0UQ0 , B5 =V2,0

TQ1 ,

C1 =U2,1TU , C2 =U2,1

TQ0 , D1 =U2,1UT , D2 =U2,1

LQ1 , D3 =U2,1TQ2 ,

E1 =U2,1TL , F1 =U2,1

LT , F2 =U2,1UQ1 ,

H1 =U2,2TT , G1 =U2,2

UQ2 , I 1 =U2,2TQ1 , J 2 =U2,2

LQ2

W for unpolarized structure functions

V, U for polarized structure functions

Structure functions and cross sectionsStructure functions and cross sections

Q0 for the term 3cos2θB −1 : Y20

Q1sinθB cosθB : Y21

Q2 sin2θB: Y22

d∫ ΩYL ,Mdσ

d4QdΩ∝WL ,M

λB =

rSB cosθ B ,

rSBT =

rSB sinθ B

Possible spin asymmetriesPossible spin asymmetries

σ , ALL , ATT , ALT , AT

σ , ALL , ATT , ALT , ATL ,

AUT , ATU , AUQ0, ATQ0

, AUQ1,

ALQ1, ATQ1

, AUQ2, ALQ2

, ATQ2

pp Drell-Yan

pd Drell-Yan

The quadrupole spin asymmetries areThe quadrupole spin asymmetries arenew ones in spin-1 hadron reactions.new ones in spin-1 hadron reactions.

QQ00QQ11 QQ22

Parton-model analysisParton-model analysis q in A( ) + q in B( ) → l+ + l−

• Φa A PASA;ka( )ij

≡d 4ξ

2π( )4 eika ⋅ξ PASA ψ j

a( ) 0( )∫ ψ ia( ) ξ( ) PASA

• Φa B PBSB;ka( )ij

≡d 4ξ

2π( )4 eika ⋅ξ PBSB ψ i

a( ) 0( )∫ ψ ja( ) ξ( ) PASA

• W μν =1

3δbaea

2 d 4ka∫a,b∑ d 4kbδ4 ka + kb − Q( )Tr Φa A PASA;ka( )γ μΦb B PBSB;kb( )γ ν⎡⎣ ⎤⎦

= −1

3δbaea

2 d 2rkaT d 2

rkbT∫a,b∑ δ 4

rkaT +

rkbT −

rQT( )

× Φa A γ +⎡⎣ ⎤⎦Φb B γ −⎡⎣ ⎤⎦+ Φa A γ +γ 5⎡⎣ ⎤⎦Φb B γ −γ 5⎡⎣ ⎤⎦( ){ gTμν

+Φa A iσ i+γ 5⎡⎣ ⎤⎦Φb B iσ j −γ 5⎡⎣ ⎤⎦ gTi{μ gTj

ν } − gTij gTμν

( )} + O1

Q

⎛⎝⎜

⎞⎠⎟

We express We express ΦΦ in terms of parton distributions. in terms of parton distributions.

The details are in PRD60 (1999) 054018.

4 ( )jk

( )li= 1 ji1lk + i5( ) ji

i5( )lk− α( )

jiα( )lk

− α5( )jiα5( )lk

⎡⎣

+12

iσαβ5( )ji

iσ αβ5( )lk

⎤⎦⎥g

+ {( )jk

}( )li+ {5( )

ji }5( )

lk+ iσ α {5( )

jiiσ }5( )

lkΦa A Γ[ ] ≡

1

2dk−Tr∫ ΓΦa A

⎡⎣

⎤⎦

Φb B Γ[ ] ≡1

2dk+Tr∫ ΓΦb B

⎡⎣

⎤⎦

QuickTime˛ Ç∆ êLí£ÉvÉçÉOÉâÉÄ

ǙDZÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅB

Φa A( )ijγ μ( ) jk Φb B( )kl

γ ν( )li

We use Fierz transformation for γ μ( ) jk γ ν( )li

so that the index summations are taken

separately in hadrons A and B.

Spin asymmetries in the parton modelSpin asymmetries in the parton modelunpolarized: qa , longitudinally polarized: Δqa,transverselypolarized:ΔT qa, tensorpolarized:δqa

dσdxAdxBdΩ

=α 2

4Q2 1 + cos2θ( )13

ea2 qa xA( )qa xB( ) + qa xA( )qa xB( )⎡⎣ ⎤⎦

a∑

Unpolarized cross section

Spin asymmetries

ALL =ea2 Δqa xA( )Δqa xB( ) + Δqa xA( )Δqa xB( )⎡⎣ ⎤⎦a∑

ea2 qa xA( )qa xB( ) + qa xA( )qa xB( )⎡⎣ ⎤⎦a∑

ATT =sin2θ cos 2φ( )1 + cos2θ

ea2 ΔTqa xA( )ΔTqa xB( ) + ΔTqa xA( )ΔTqa xB( )⎡⎣ ⎤⎦a∑

ea2 qa xA( )qa xB( ) + qa xA( )qa xB( )⎡⎣ ⎤⎦a∑

AUQ0=

ea2 qa xA( )δqa xB( ) + qa xA( )δqa xB( )⎡⎣ ⎤⎦a∑ea2 qa xA( )qa xB( ) + qa xA( )qa xB( )⎡⎣ ⎤⎦a∑ ALT =ATL =AUT =ATU =ATQ0

=AUQ1

=ALQ1=ATQ1

=AUQ2=ALQ2

=ATQ2=0

Advantage of the hadron reaction (δqmeasurement)

AUQ0large xF( ) ≈

ea2qa xA( )δqa xB( )

a∑ea2qa xA( )qa xB( )

a∑Note: δ ≠transversityinmynotation

Summary on Summary on pdpd Drell-Yan Drell-Yan

• 108 (48) structure functions exist in the pd (pp) Drell-Yan

• 22 (11) structure functions by the rQT integration or by the

rQT → 0 limit

• New polarized structure functions → associated with the tensor structure

• Tensor polarizations and spin asymmetries

• Only 4 structure functions are finite in the parton model

• The tensor distributions δq and δq can be measured by AUQ0

• The pd Drell-Yan suitable for measuring δq

• Future experimental possibilities: J-PARC, COMPASS, ⋅⋅⋅

• Numerical analysis has not been done about feasibility at J-PARC, etc.