Transcript 一般化パートン分布と核子のスピン構造

一般化パートン分布と核子のスピン構造
with Y. Nakakoji, H. Tsujimoto (Osaka Univ.)
1. Introduction
Nucleon Spin Puzzle
:
( EMC measurement, 1988 )
- ハドロン物理の基本的な未解決問題 If intrinsic quark spin carries little of total nucleon spin
What carries the rest of nucleon spin ?
初期の解答案
(I) グルオン・スピン説
(A) 素朴な説
なら OK
なぜ
か
という疑問は残る！
(B) QCD の axial-anomaly 起源説 （説明後述）
(II) クォーク軌道角運動量説
が大きいから
が小さ
い！
(III) グルオン軌道角運動量説
これに関しては手がかりがないので真面目に考えられたことな
し！
もう少し注意深い考察
(I) 核子のスピン・コンテンツのスケール依存性
観測のエネルギー・スケールに言及せずに核子の
スピン・コンテンツを論じることには意味がない ！
grows rapidly as

increases, even though it is
small at low energy scale
decreases rapidly to partially compensate the increase

of
もし、非常に素朴に核子のスピン・コンテンツに言及するときには
非摂動論的QCDを特徴づける低エネルギー・スケールでのそれ
と考えるべきである。
(II) スピン・コンテンツの factorization scheme 依存性
QCD の axial-anomaly の扱い方
2 popular factorization schemes for longitudinally polarized PDF
関係
weakly scale-dependent beyond L.O.
２つの factorization scheme に基づく empirical PDF fits
MSbar scheme
AB scheme
not incompatible with Naïve Quark Model ?
しかし、 後者のシナリオは非常に大きなグルオン偏極
を必要とし、最近の COMPASS 実験と矛盾 !
実際, COMPASS グループが指摘したように関係式
より 、
で
incompatible with the recent COMPASS data !

“Gluon polarization in the nucleon from quasi-real
photoproduction of high-pT hadron pairs, hep-ex / 0511028.
について ?
“Evidence for the Absence of Gluon Orbital Angular Momentum
in the Nucleon”, S.J. Brodsky and S. Gardner, hep-ph / 0608219.

The Sivers mechanism for the single-spin asymmetry in the
unpolarized lepton scattering from a transversely polarized nucleon
is driven by the orbital angular momentum of quarks and gluons.

They argued that small single-spin asymmetry on the deuteron
target measured by the COMPASS collaboration is an
indication of small gluon OAM !.
残るは
のみ ？
クォーク軌道角運動量の重要性
- chiral soliton picture of the nucleon 
Skyrme model
(Ellis-Karliner-Brodsky, 1988)

Chiral Quark Soliton Model (Wakamatsu-Yoshiki, 1991)
特に後者は、有効クォーク理論なので
回転するハリネズミ型平均場
中のクォークの集団運動

CQSM predicts at model energy scale around

CQSM well reproduces
and
at
at the leading order pQCD
assuming
• SU(2) : M. W. and T. Kubota, Phys. Rev. D60 (1999) 034022
• SU(3) : M. Wakamatsu, Phys. Rev. D67 (2003) 034005
New compass data (2005)
New COMPASS QCD fits at NLO
New HERMES QCD fits at NLO
s-quark polarization
ＣＱＳＭ
最近の著しい成果のまとめ :
全くわかっていないのは
と
!
もう一つの進展
possibility of direct measurement of
through Generalized Parton Distributions (GPDs)
appearing in high-energy DVCS & DVMP processes
Ji の角運動量和則
2. Deeply Virtual Compton Scatterings (DVCS) and GPDs
深部非弾性散乱の運動学的条件
pQCD が適用可能
DIS processes
factorization
DVCS scattering amplitude dominant in Bjorken limit
Handbag diagram
lower part of Handbag Diagram contains information on
nonpertubative quark-gluon structure of the nucleon, parametrized
by 4 GPDs depending on 3 kinematical variables
Physical meaning of 3 kinematical variables,
Handbag diagram
4-momentum-transfer square of the nucleon
average longitudinal momentum fraction of the
struck parton in the initial and final state
- generalized Bjorken variable difference between the above two longitudinal
momentum fractions of the struck parton
- skewdness parameter -
3. Unpolarized GPD の基本的性質
通常の unpolarized PDF の場の理論的定義 （gauge link 省略）
forward
unpolarized GPD の場の理論的定義
off-forward
特に前方極限（
）で
GPD の n 次のモーメント
gauge link
Light-cone 演算子は twist-2 の局所演算子のタワーと等価
クォークの運ぶ電磁流
クォークの運ぶエネルギー
運動量テンソル
高次項
局所演算子群の非前方核子行列要素
Dirac F.F.
Pauli F.F.
electromagnetic current coupled to photon
energy momentum tensor coupled to graviton
GPD のモーメントとの関係
electromagnetic F.F.
特に前方極限（
第３式の右辺はクォーク
）で
が運ぶ全角運動量
と同定できる。
ー Ji の角運動量和則 ー
Ji の角運動量和則の起源
核子行列要素
QCD の角運動量演算子の定義
energy-momentum tensor
ー Ji の角運動量和則 （積分
形） ー
更に
を一般化した演算子のタワーを考えれば
を一般化した次の関係式が証明できる。
これは分布関数の関係式に焼き直すと
ー Ji の角運動量和則 （非積分形） ー
クォークのヘリシティ分布
を通じて
は既に測定済みだから
の測定
がわかれば
により核子中のクォークの軌道角運動量分布も純粋に実験的に決められる！
が鍵を握る物理量 ！
既に述べた以下の和則より
magnetic moment desity
in Feynman x-space
canonical part
anomalous part
quark number dist.
angular momentum density
in Feynman x-space
canonical part
momentum dist.
anomalous part
4. CQSM analyses of unpolarized GPDs
natural spin decomposition in Breit frame
corresponds to Sachs decomposition of electromagnetic F.F.
Generalized Form Factors as Mellin moments of GPDs
(1) 1st moments
(2) 2nd moments
more generally
(A) Isovector channel
anomalous magnetic moment & anomalous gravitomagnetic moment
where
The forward limit of GPDs
already calculated in CQSM (M.W. and H. Tsujimoto)
magnetic moment density
in Feynman x-space
a prominent feature of CQSM prediction for

The contribution of deformed Dirac sea quarks has a large
and sharp peak around

Since this large Dirac-sea contribution to
is nearly symmetric with respect to
significant contribution to the 1st moment
but no contribution to the 2nd moment
, it gives a
Interpretation of sharp peak of

Since partons with
around
are at rest in the longitudinal direction
its large contribution to the first moment
must come from transverse motion of quarks and antiquarks
If one remembers the important role of pion clouds in the
isovector magnetic moment of the nucleon, the above
transverse motion can be interpreted as simulating
pionic quark-antiquark excitation with long-range tail
validity of claimed picture may be confirmed by investigating
dependence of
Further support may be obtained by investigating the canonical part
of
NMC observation
(B) Isoscalar channel
anomalous magnetic moment & anomalous gravitomagnetic moment
where
The forward limit of GPDs
first calculated in CQSM by J. Ossmann et. al. (2005)
and its canonical part in CQSM
positivity of antiquark dist.
and its canonical part in CQSM
anomalous part
no Dirac sea contribution
Dirac sea contribution
exactly
valence contribution
cancel !
to
small valence contribution
Sivers function と anomalous magnetic moment density の密接な関係 ?
• Z. Lu and I. Schmidt, hep-ph / 0611158
Sivers function
と その lowest
-moment
diquark 模型の light-front formalism の枠組みで
主な仮定
anomalous magnetic moment density
• スカラー diquark
• Sivers mechanism に必要な終状態相互作用は one-gluon-exchange
5. 核子スピン・コンテンツに対する模型に依らない予言
important observation (it is a sound fact)
total nucleon AGM identically vanishes !
It follows from
two possibilities
Reasoning to show smallness of
small
!
probably even
smaller !
Remember that
anomalous gravitomagnetic form factor
Lattice QCD
CQSM
LHPC
model independet prediction for nucleon spin contents

Not only CQSM but also LHPC & QCDSF lattice simulations
indicate smallness of quark AGM

In the following, we assume smallness of
and set them 0, for simplicity.
We are then led to surprisingly simple proportionality relations :
Important observation
(I) The quark- and gluon-momentum fractions,
and
are empirically fairly precisely determined.
In fact, MRST2004 & CTEQ5 QCD fits give almost the same
numbers for those below
,
(II) The above proportionality relations holds scale-independently,
since the evolution equations for
and
are exactly the same !
[Reason] forming spatial moments of
and
does not
change the short-distance singularity of the operators !
The above evolution equations at NLO may be used to estimate
and
at lower energy scales !
MRST2004 evolved down to
New COMPASS & HERMES QCD fits
Since the scale dependence of
is weak, we can estimate
at
and
are still uncertain, but we can say that

Gluons carry about 20% of linear and total angular momentum
fraction at this low energy scale of nonperturbative QCD !

We conjecture that this comes from gluon OAM not from
!

This statement is not inconsistent with the recent observation by
Brodsky and Gardener, since what would be related to Sivers
mechanism is the anomalous part of gluon OAM.

On the other hand, our postulated identity,
implies
that gluon OAM comes totally from its canonical orbital motion,
not from the anomalous contribution related to GPD
6. Summary and Conclusion
: long-lasting dispute over this issue.
もっともらしい理論的仮定

smallness of net quark contribution to AGM :
と、empirical PDF fits (MRST2004 + new COMPASS) のみを基にして
model independent conclusion for
MRST2004 &
COMPASS (D)
HERMES
COMPASS (A)
SMC
核子スピン・コンテンツのスケール依存性
cross over around

勿論、より確実な結論を得るためには、unpolarized spin-flip
GPD (そのforward limit) の実験的な引き出しが不可欠である。

は Feynman momentum x-space
における anomalous magnetic moment density と解釈できる
それ自身非常に興味深い量である。

上の anomalous magnetic moment density はまた、semi-inclusive
reaction の single spin asymmetry の起源の一つを与えると考え
られる Sivers function と密接な関係を有するかもしれない？
相対論的複合粒子の AMM & AGM & OAM の起源？
[Appendix]
Request for future Lattice QCD studies

More refined check of the relation
A) Larger lattice space, higher statistics, etc.
B) Stability againt the variation of pion mass, ……

More reliable evaluation of
A) Simulation with smaller pion mass
or
B) Reliable chiral extrapolation ( ex., by using chiral PT )
物理量のπ中間子質量 依存性について
Basic model laglangian
with
and
のいくつかの値に対して self-consistent soliton solutions を
求めた後に、問題となる核子の観測量を計算する