Observing the proton
Off the light-cone
Xiangdong Ji
University of Maryland/SJTU
JLab Physics Colloquium,
April 23, 2013
Collaborators and papers
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Feng Yuan, LBL
Jianhui Zhang, Shanghai Jiao Tong University
Xiaonu Xiong, Peking University
Yong Zhao, U. Maryland
1. Phys. Rev. Lett. 109, 152005 (2012)
2. Phys.Lett. B717 (2012) 214-218
3. Parton picture for longitudinal polarization, Phys. Rev. D
4. Physics of gluon helicity
5. parton distribution from Euclidean lattice
Outline
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Frame dependence in relativistic quantum
theories
Two special frames for understanding the
proton structure: the lab frame and infinite
momentum frame (IMF)
The proton spin structure in IMF
Lattice QCD calculation of parton physics in
IMF
Conclusions
Important mission
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An important mission of Jlab 12 GeV upgrade
and EIC is to study the internal structure of
proton and neutron at a new level.
A complex system of the quarks and gluons
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Strong interactions, relativistic
Test the fundamental theory: QCD
Constraints of Relativity
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The proton wave function is a frame
dependent concept
Boost operators, Ki, are interactiondependent
|P˃ = U(Λ(p)) |p=0>
U is not just kinematical, it is dynamical!
Wave functions in Relativistic QM
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Wave function is not a Lorentz invariant
concept as it is defined by observations of
different space points in a fixed time
(simultaneously) at a particular frame.
Simultaneity for two events in one frame does
not mean simultaneity in a different frame.
In general, WF has not been very a popular
concept in field theory. However, it is an
important and intuitive concept.
Two special frames
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Lab frame: where the proton is at rest, or low
momentum, relatively speaking
Probe frame: in which the electron or virtual
photon has the smallest momentum.
In high-energy limit, the probe frame reaches
the infinite momentum frame (IMF), i.e., the
proton is probed as it travels at the speed of
light!
Electron scattering
four momentum transfer qµ = (v, q) is a space
like vector v2-q2 < 0 and fixed.
smallest momentum happens when v=0, Q2=q2
Pq = P3Q = Q2/2x, thus P3 = Q/2x.
In the scaling limit, P3 -> infinity.
Reconcile physics in two frames
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Physics shall be frame-independent!
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In the rest frame, one probes the light-cone, timedependent correlations (light-front)
In the IMF, one probes the static correlations.
The two different correlations are related by
Lorentz boost.
If one uses the light-front quantization in the
rest frame (an effective theory), one gets
the same physics as IMF!
Proton WF in the IMF is what high-energy
scattering study.
IMF and parton physics
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In the IMF, the interactions between
particles are Lorentz-dilated, and thus the
systems appear as if interaction free: the
proton is made of free partons.
This is only true to a certain degree: leading
twist. The so-called higher-twist
contributions are sensitive to parton offshellness, transverse momentum and parton
correlations.
Partons provide a useful language to
understand the proton structure.
Spin of the proton in QCD
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The spin of the proton is generated from the
angular momentum of the internal quarks and
gluons
From general consideration,
one can write down,
J  1/ 2  J q ( )  J g ()
The quark orbital contribution
can be probed through GPDs
These are frame-independent
results.
The spin structure in IMF
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How is the spin of the proton generated from
the AM of the underlying partons?
Longitudinal polarization
Transverse polarization
Longitudinal polarization
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Quark orbital AM:
intrinsically twist-three
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Orbital AM is nominally a leading twist,
however, it is actually a high-twist
contribution because it involves the parton
transverse momentum.
The total contribution can be probed through
the GPD sum rule (L=J-ΔΣ. However, the
individual parton contributions do not yet
have a simple parton picture.
OAM parton distribution can be probed
through twist-three GPDs. (Ji, xiong, & yuan)
Gluon polarization
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In IMF, however, the gluon partons have welldefined helicity ± 1 and densities g±(x)
+1 or -1
1/2
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Gluon helicity distribution is
g(x) = g+(x) – g-(x) and
G = ʃdx g(x)
Total gluon helicity
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The total gluon helicity ΔG is gauge invariant,
and has a complicated expression on the rest
frame
In light-cone gauge A+=0, it reduces to a
simple expression
, which is the spin of
the gluon, but has no gauge-symmetric notion
(Laudau, Jackson).
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
17
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
Perspective on Delta G
0.2
Q2 = 10 GeV 2
RHIC 200 GeV
0.1
1
1
in units of h
xg
ò g(x,Q2) dx
xmin
0.8
2
Q = 10 GeV
2
0.6
0.4
DSSV++
0
DSSV
RHIC
200 GeV
0.2
DSSV+
-0.1
DSSV++
10
18
-2
10
-1
forward rapidity
10
-3
10
x
0.2
0.05
ò
500 GeV
0
Dg(x) = 0.1±
0.06
0.07
-2
10
-1
xmin
Total gluon helicity in the IMF
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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,
One can define the gauge-symmetric gluon
spin as ExA┴ (X.Chen et al, 2009)
It be shown that ΔG is the matrix element of
about operator in the IMF. (Ji, Zhang, Zhao)
Reproducing the light-cone gauge
result by boost
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In fact, transforming to the rest frame
Which in the light-cone gauge, reduces to the
standard expression ExA.
Transverse polarization
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Transverse polarization in term of the
polarization vector S is of subleading, thus
does not seem to have a simple parton picture
However, this is incorrect.
There are many misconceptions and pitfalls
about transverse pol.
The best language is not the transverse AM,
rather, it shall be transverse Pauli-Lubanski
spin. [Transverse AM does not commute with
the hamiltonian]
Transverse polarization
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There is a leading twist-contribution, the
subleading contributions either are cancelled
or related to the leading one by symmetry.
The leading contribution has a partonic
interpretation.
First pointed out by Burkardt (2005), but the
connection to the spin sum rule was not
rigourous.
A plane-wave derivation of transversespin parton sum rule
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Consider parton momentum density
It has a “distribution” term depending on
coordinate ξ,
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Calculating its contribution to the transverse
spin density
Parton distributions
The Feynman momentum is easier to understand in the
infinite momentum frame: fraction of the longitudinal
momentum carried by quarks x = kz/Pz, 0<x<1
Lattice QCD calculation
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Current approach is based on the rest frame
formalism.
To compute light-cone correlation difficult
because the lattice cannot handle real time
correlations
One can calculate the moments (2-3) of
parton distributions: local operators.
Lattice QCD calculations
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Parton distributions are calculated using the
rest-frame formalism, the corresponding
operators are light-cone correlations
Since lattice cannot handle real-time
dependence, only moments are calculated on
lattice.
Usually only a few moments (2-3) are
calculated.
A new proposal
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Using the IMF formalism.
Start with static correlation in the zdirection.
X. Ji, to be published
Comments
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This can be proved using the local matrix
elements formulation
Approaching the light-cone through boost
One-loop calculation demonstrates how this
work.
One-loop calculation
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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In high-energy scattering, one probes parton
physics, which can be understood both in the
rest frame and IMF formulation
Understanding the spin structure of the
proton in term of partons needs more
theoretical and experimental efforts
Parton physics is best calculated in lattice
QCD using IMF formalism.