Transcript Slides - Indico

Recent progress in
understanding the spin
structure of the nucleon
Xiangdong Ji
U. Maryland/上海交通大学
RIKEN, July 29, 2013
PHENIX Workshop on Physics Prospects with
Detector and Accelerator Upgrades
Nucleon spin structure is still
an unsolved problem！
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The nucleon spin structure continues
motivating new theoretical ideas and
experiment measurements
Experimental efforts
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….
Polarized PP collision at RHIC (This workshop)
COMPASS
JLAB and 12 GeV upgrade
EIC (A. Deshpande)
The canonical (or “local”) approach
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The nucleon spin structure can be studied
through QCD AM operator.
Explicit gauge-invariant and local form,
yields the following spin sum rule
where OAM can be measured through GPD
sum rules.
Pros and cons
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Pros
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Explicitly gauge invariant
Local
The sum rule is maximally frame independent
Cons
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Physical interpretation of the terms does not allow
free-particle picture
Has no explicit connection to partons, in particular,
it does not naturally involve the gluon helicity
contribution in the infinite momentum frame
(IMF).
IMF picture of the nucleon
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The nucleon is mostly probed in high-energy
scattering.
Relative to the probes, the nucleon travels at
the speed of light. We learn light-cone wave
functions, or parton physics.
According to QCD factorizations, certain
observables of the nucleon can be explained in
terms of free partons A fast moving proton
is a beam of free quarks and gluons!
Can one describe the spin structure
completely in terms of partons?
Gluon helicity distribution
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The gluon partons have well-defined helicity
± 1 and densities g±(x) in wavelength
1/2
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+1 or -1
Gluon helicity distribution is
g(x) = g+(x) – g-(x) and
G = ʃdx g(x) is the fraction of the proton helicity
carried in the gluon.
Implication for the spin structure
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ΔG is an obvious contribution to the spin of
the proton.
Can contribute to the quark helicity through
axial anomaly (Altarelli & Ross, Carlitz, Collins,
& Mueller,…), several units of h-har?
Its contribution to the spin grows like 1/αS
However, there are a number of puzzles
associate with this quantity…
QCD expression
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The total gluon helicity ΔG is gauge invariant
quantity, and has a complicated expression in
QCD factorization (Manohar, 1991)
It does not look anything like gluon spin or
helicity! Not in any textbook!
Don’t know how to calculate
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ΔG involves explicit light-cone correlation or
real time. No one knows how to calculate this
in lattice QCD (Models: RL Jaffe, Chen & Ji)
One can consider A+=0 gauge, but no one
knows how to fix this gauge in lattice QCD
Thus there is no way to confront theory with
experiment: G = ʃdx g(x)
Is there a large contribution from small x?
Light-cone gauge
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In light-cone gauge A+=0, the above
expression reduces to a simple form
which is the spin of the photon (gluon)
(J. D. Jackson, CED),
but is not gauge-symmetric: There is no gauge
symmetry notion of the gluon spin!
(J. D. Jackson, L. Landau & Lifshitz).
ALL from RHIC 2009
p0 p (GeV/c)
T
0
5
10
15
2
15
Q = 10 GeV
2
DSSV++
Dc
2
PHENIX Prelim. p , Run 2005-2009
0
PHENIX shift uncertainty
DSSV++ for p 0
0.04
10
STAR Prelim. jet, Run 2009
Dc = 2% in DSSV analysis
2
STAR shift uncertainty
A LL
DSSV++ for jet
0.02
5
0
0
DSSV
DSSV+
PHENIX / STAR scale uncertainty 6.7% / 8.8% from pol. not shown
0
10
20
-0.1
ò
30
0.05
Jet p (GeV/c)
T
11
0.2
0.05
ò
Dg(x) = 0.1±
0 0.2
0.06
0.07
0.1
Dg(x,Q ) dx
2
0.2
Jaffe-Manohar Spin Sum Rule
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Consider the free-field form of the QCD AM
operator
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Every term has a simple interpretation, but
Except the first, others are not gauge invariant
Take it to the IMF and fix the light-cone
gauge, one gets the partonic AM sum rule,
Two important questions
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Although the gauge and IMF make the parton
picture clear for the spin, but why these
choices are physically relevant?
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Recall that there is a naturally gauge-invariant
formulation for parton physics in QCD. This
naturalness is the key result of OPE and
factorization theorems.
How to measure and calculate the relevant
quantities in the J-M sum rule?
A new development
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A “non-local” but gauge-invariant
decomposition (x. s. Chen et al, 2009, PRL)
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Decompose the gauge potential 𝐴 into 𝐴⊥ + 𝐴∥
where 𝐴⊥ has no covarint divergence and 𝐴∥ does not
produce chromo-magnetic field.
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𝐴⊥ transforms covariantly under gauge
transformation.
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𝐷∥ = −𝜕 + 𝑖𝑔𝐴∥ is also gauge covariant.
Gauge-symmetric but non-physical
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Although the transverse part of the vector
potential is gauge invariant, the separately
𝜕 𝐴⊥
−
𝜕𝑡
𝐸⊥ =
does not transform properly, under
Lorentez transformation, and is not a physical
observable ( X. Ji, PRL, comments)
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E ┴ can generated from E ║ from Lorentz boost.
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A Lorentz-transformed 𝐸 has different
decomposition E = E┴ + E║ in different frames.
There is no charge that separately responds to E┴
and E║
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A new observation
(X.Ji, Y. Zhao,J.Zhang, 2013)
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In the IMF, the individual terms become
physical.
Therefore the spin decomposition by X. Chen
et al makes sense only when considering the
nucleon moving at the speed of light.
Considering the contribution 𝐸 × 𝐴⊥ , which
becomes the gluon helicity Δ𝐺 in the IMF
Electric field of a charge
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A moving charge
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Gauge potential
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Large momentum limit
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As the charge velocity approaches the speed
of light, E┴ >>E║, B ~ E┴, thus
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E┴ become physically meaningul
The E┴ & B fields appear to be that of the free
radiation
Weizsacker-William equivalent photon
approximation (J. D. Jackson)
Gauge-invariant A┴ appears to be now
physical, which generates the E ┴ & B.
Theorem
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Thus, one would expect that the total gluon
helicity ΔG must be the matrix element of
ExA┴ in a large momentum nucleon.
X. Ji, J. Zhang, and Y. Zhao (arXiv:1304.6708)
is just the IMF limit of the matrix element
of ExA┴
QCD case
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A gauge potential can be decomposed into
longitudinal and transverse parts (R.P.
Treat,1972),
The transverse part is gauge covariant,
In the IMF, the gauge-invariant gluon spin
becomes
One-loop example
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The result is frame-dependent, with log
dependences on the external momentum
Anomalous dimension coincides with X. Chen et al.
Taking large P limit
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If one takes P-> ∞ first before the loop
integral, one finds
This is exactly photon (gluon) helicity
calculated in QCD factorization! Has the
correct scale evolution.
Subtlety of limiting procedure
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There are two possible limits,
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Taking 𝑃 → ∞ before UV regularization (physical
case, light-cone)
Taking UV regularization before 𝑃 → ∞ (practical
calculation, time-independent)
Two limits get the same IR physics
One can get one limit from the other by a
perturbative matching condition, Z.
Lattice QCD
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ExA┴ is perfectly fit for lattice QCD
calculation of ΔG!
To get large momentum nucleon, one has to
have a fine lattice in the z-direction:
P ~ 1/a
To separate excited states of the moving
nucleon, one also needs fine lattice spacing in
the time direction.
322X642
γ=4
x,y
z
What a lattice calculation of ΔG
implies?
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Settles if axial anomaly plays an important
role in the quark helicity measurement, by
determining how large is ΔG
Since the experimental data says,
0.2
0.05
ò
Dg(x) = 0.1±
0.06
0.07
how much ΔG sits at very small x?
 How much the gluon helicity contributes to
the proton helicity at small scale.
Orbital angular momentum
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Recall AM operator by Chen et al,
Each individual term goes to IM limit, and in
the 𝐴+ = 0 gauge, they recover the JaffeManohar sum rule.
This observation allows on to calculate the
OAM as well (it was not possible before)
Detailed matching condition will appear soon.
Matching condition for OAM
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Calculate OAM at finite momentum
Extract MS-bar matrix elements by solving
the matching conditions
Measuring orbital contribution
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Parton OAM contribution can be related
twist-three GPD’s
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Y. Hatta, Phys. Lett. B708 (2012) 186-190; Y. Hatta
and S. Yoshida, JHEP 1210 (2012) 080
Ji, Xiong, Yuan, Phys. Rev. Lett. 109, 152005
(2012); to appear in PRD，2013
Need some DVCS like process (Xiong et al, to
be published)
X-dependence
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X-dependence of Δ𝑔 𝑥 can be calculated on
lattice in principle. (euclidean lattice)
X. Ji, PRL 110, 262002 (2013)
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Many other parton properties, including
parton distributions, TMDs, GPDs, light-cone
wave functions, etc can now be formulated on
lattice a strong support for the EIC
project. (K computer?)
An example...
Ken Wilson (1936-2013)
Conclusions
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The gluon helicity measured in high-energy
scattering is just 𝐸 × 𝐴⊥ in the large
momentum limit.
The finding can be used to justify a simple
partonic sum rule for the proton spin (J-M
sum rule), providing a practical way to
calculate the OAM on lattice as well.
Parton OAM can also be measured
experimentally.
More exciting times for spin!