Transcript The proton spin sum rules

Physics of gluon polarization
Xiangdong Ji
University of Maryland/SJTU
Jlab, May 9, 2013
Gluon polarization
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Ever since the EMC “spin crisis”, the gluon
polarization has been one of the most
important pursuits of hadron physics
community
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HERMES
COMPASS
RHIC SPIN
“HERA-N”
“EIC…”
Physics arguments
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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,…)
Its contribution to the spin grows like 1/αS
However, there are a number of theoretical
puzzles about it!
Axial Anomaly
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It was argued that Δq probed by DIS is not
entirely due to the quark contribution. There
is a gluon anomaly contribution. This
contribution is proportional to (αS /2π)ΔG
For the anomaly to have a large contribution
ΔG must be on the order few unit of hbar.
Thus, ΔG could be large even at nonperturbative scale.
Physics: Feynman parton picture
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A fast moving proton is a beam of free quarks
and gluons.
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.
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!
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).
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?
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
0.04
DSSV++ for p 0
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
9
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
Electric field of a charge
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A moving charge
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Gauge potential
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Observations
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Although the transverse part of the vector
potential is gauge invariant, the separately E┴
does not transform properly, under Loretez
transformation, and is not a physical
observable (X. Chen et al, x. Ji, PRL)
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E ┴ generated from E ║ from Lorentz boost.
A lorentz-transformed E has different
decomposition E = E┴ + E║ in different frames.
There is no charge that separately responds to E┴
and E║
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)
Thus gauge-invariant A┴ appears to be now
physical which generates the E ┴ & B.
Gauge invariant photon helicity
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X. Chen et al (PRL, 09’) proposed that a gauge
invariant photon angular momentum can be
defined as
ExA┴
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This is not an observable when the system move at
finite momentum because this is only a part of the
contribution which cannot be measured separately.
However, it becomes an observable in the
IMF when Weizsacker-William’s picture is
true!
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 anomalous dimension.
Subtlety of the limiting procedures
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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
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.
x-dependence?
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X-dependence of a parton distribution has
been very difficult to calculate in the past.
The only approach is through the local
moments.
However, it is very difficult to calculate
higher moments numerically.
It will be nice to find a way to directly
calculate the x-dependence on lattice
arXiv1305.1539
observation
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Usual parton distribution
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Consider instead
Relationship
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The matching condition is perturbative
The correction is power-suppressed. For
practical calculation, a momentum of 5 GeV
might be good enough.
The extension of the approach
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GPDs
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TMDs
The extension of the approach
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Wigner distribution
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Light-cone amplitudes
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Light-cone wave functions
Higher-twists….
Conclusions
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We find the gluon helicity measured in highenergy scattering is just EXA┴ in the large
momentum limit,
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This gives the gauge invariant and physically
manifest notion of the total gluon helicity
This gives practical way to calculate ΔG
We find a practical way how to calculate
light-cone distributions:
PDFs, TMDs, GPDs, HTs, LCWFs, LCDAs, etc…
ten years from now there will be a lot of
lattice result.