Transcript Xiangdong Ji

Parton Physics
on a Bjorken-frame lattice
Xiangdong Ji
University of Maryland
Shanghai Jiao Tong University
July 1, 2013
Knowledge of parton
distributions is data-driven
─── Paul Reimer from the prevous talk of this
workshop
Outline
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Review of Bjorken frame and parton physics
Why parton physics is hard to calculate?
A new proposal, resource requirement, and
applicability
Gluon polarization: its physics and calculation
Outlooks
High-energy scattering
and Bjorken frame
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In high-energy scattering, the nucleon has a large
momentum relative to the probes.
In the Bjorken frame, the probes (electron or
virtual photon) may have the smallest momentum,
but the proton has a large momentum (infinite
momentum frame, IMF) relative to the observer
and travels at near the speed of light.
This frame has been used frequently in the old
literature, but giving away to rest-frame lightfront quantization in recent years.
Electron scattering in Bjorken frame
4-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.
Bjorken frame and Parton physics
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The interactions between particles are Lorentzdilated, and thus the system appears as if
interaction-free: the proton is probed as free
partons.
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In QCD, parton physics emerges when working in lightcone gauge. A+=0.
In field theory, parton physics is cut-off dependent.
This is only true to a certain degree: leading twist.
The so-called higher-twist contributions are
sensitive to parton off-shellness, transverse
momentum and correlations.
Quark and gluon parton distributions
The Feynman momentum is, in the Bjorken frame,
fraction of the longitudinal momentum carried by quarks:
x = kz/Pz , 0<x<1
Parton Physics
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Light-cone wave function, ψn(xi, k⊥i)
Distributions amplitudes, ψn(xi)
Parton distributions, f(x)
Transverse momentum dependent (TMD)
parton distributions, f(x, k⊥)
Generalized parton distributions, F(x,𝜉,r⊥)
Wigner distributions, W(x, k⊥, r⊥)
Fragmentation functions…
Frame-independent formulation
of parton physics
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Over the years, the parton physics have been
formulated in a boost-invariant way. In
particular it can be described as the physics
in the rest frame.
In this frame, the probe appears a lightfront (light-like) correlation.
Thus light-cone quantization is the essential
tool (S. Brodsky)!
Light-front quantization
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Unique role of lattice QCD (1974)
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Lattice is the only non-perturbative approach
to solve QCD
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Light-front quantization: many years of efforts but
hard for 3+1 physics
AdS/CFT: no exact correspondence can be
established, a model.
An intrinsically Euclidean approach
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“time” is Eucliean 𝜏=i t, no real time
A4 = iA0 is real (as oppose to A0 is real)
No direct implementation of physical time.
Ken Wilson (1936-2013)
Don’t know how to calculate!
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Parton physics? Light-like correlations
𝜉-
𝜉0
𝜉+
𝜉3
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For parton distributions & distribution
amplitudes: moments are ME of local
operators, 2-3 moments. Very difficult
beyond that…
For parton physics that cannot be reduced to
local operators, there is no way to calculate!
A Euclidean distribution
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Consider space correlation in a large
momentum P in the z-direction.
𝜉0
0
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Z
𝜉3
Quark fields separated along the z-direction
The gauge-link along the z-direction
The matrix element depends on the momentum P.
This distribution can be calculated using standard
lattice method.
Taking the limit P-> ∞ first

After renormalizing all the UV divergences,
one has the standard quark distribution!
One can prove this using the standard OPE
 One can also see this by writing
|P˃ = U(Λ(p)) |p=0>
and applying the boost operator on the gauge link

𝜉-
𝜉0
𝜉+
𝜉3
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The Altarelli-Parisi evolution was derived this way!
Finite but large P
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The distribution at a finite but large P is the
most interesting because it is potentially
calculable in lattice QCD.
Since it differs from the standard PDF by
simply an infinite P limit, it shall have the
same infrared (collinear) physics.
It shall be related to the standard PDF by a
matching condition in the sense that the
latter is an effective theory of the former.
Relationship: factorization theorem
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The matching condition is perturbative
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The correction is power-suppressed.
Pictorial factorization
ZZ(P, μ)
q(x, P, μ)
q(x, μ)
One-loop example
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Pz dependence is mostly isolated in the large
logs of the loop integral.
Practical considerations
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For a fixed x, large Pz means large kz, thus, as
Pz gets larger, the valence quark distribution
in the z-direction get Lorentz contracted,
z~1/kz.
Thus one needs increasing resolution in the zdirection for a large-momentum nucleon.
Roughly speaking: aL/aT ~ γ
One needs special kinds of lattices
γ=2
x,y
z
γ=4
x,y
z
Small x partons
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The smallest x partons that one access for a
nucleon momentum P is roughly,
xmin = ΛQCD/P~ 1/3γ
small x physics needs large γ as well.
Consider x ~ 0.01, one needs a γ factor about
10~30. This means 100 lattice points along the
z-direction.
A large momentum nucleon costs considerable
resources!
Ideal lattice configurations
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Time direction also needs longer evolution
because the energy difference between
excited states and the ground state goes like
1/ γ
Thus ideal configurations for parton physics
calculations will be
242x(24γ)2 or 362x(36γ)2
There are not yet available!
Sea quarks
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The parton picture is clearest in the axial
gauge AZ =0.
In this gauge, see quarks correspond to
backward moving quarks (Pz>0, kz<0) or
forward moving antiquark, but otherwise
having arbitrary transverse momentum (with
cut-off μ) and energy (off-shellness).
In the limit of Pz->∞, the contribution does
not vanish.
Flavor structure? (Hueywen Lin’s talk)
1/P2 correction
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Two types (to be published)
The nucleon mass corrections in the traces of
the twsit-2 matrix elements can easily be
calculated.
Corrections in twist-four contributions can
also be directly calculated on lattice. The
contribution is suspected to be smaller than
the mass correction.
Higher-order corrections can similarly be
handled.
Other applications
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This approach is applicable for all parton
physics
Recipe:
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Replace the light-cone correlation by that in the zdirection.
Replace the gauge link in the light-cone direction
by that in the z-direction.
Derive factorizations of the resulting distributions
in terms of light-cone parton physics.
GPDs and TMDs
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GPDs
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TMDs
Wigner distributions and LC
amplitudes
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Wigner distribution
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Light-cone amplitudes
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Light-cone wave functions
Higher-twists….
Gluon helicity distribution
∆g(x)
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An important part of the nucleon spin
structure
Much attention has been paid to this quantity
experimentally
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DIS semi-inclusive
RHIC spin
…
In principle, it can be calculated from the
approach discussed previously. However, it is
still difficult to get ∆G, the integral.
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
36
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
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).
Two long-standing problems
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∆G does not have a gauge-invariant notion of
the gluon spin.
There is no direct way to calculate ∆G, unlike
∆∑, and orbital angular momentum.
Electric field of a charge
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A moving charge
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Gauge potential
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Suggestion by X. Chen et al
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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)
Gauge invariant photon helicity
X. Chen et al (PRL, 09’) proposed that a gauge
invariant photon angular momentum can be
defined as
ExA┴
 This is not an observable when the system
move at finite momentum because (X. Ji)

A┴
generated from A║ from Lorentz boost.
 A lorentz-transformed A has different
decomposition A = A┴ + A║ in different frames.
 There is no charge that separately responds to A┴
and A║
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.
Theorem
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The total gluon helicity ΔG shall be the matrix
element of ExA┴ in a large momentum nucleon.
We proved in the following paper
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.
Matching condition
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Taking UV regularization before p-> ∞
(practical calculation, time-independent)
One can get one limit from the other by a
perturbative matching condition, Z.
A┴ can be obtained from Coulomb gauge fixing
on lattice.
Conclusions
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Parton physics can be explored in lattice QCD
calculations using the Bjorken frame. This
opens the door for precision comparisons of
high-energy scattering data and fundamental
QCD calculations.
It will be a while before that “data driving”
era is over. However, we know how to get
there.
” He (Wilson) was decades ahead of his time
with respect to computing and networks.”
─── Paul Ginsparg, Cornell