Second International Summer School of the GDR PH-QCD
“Correlations between partons in nucleons”
N
Multidimensional pictures
of the nucleon (1/3)
Cédric Lorcé
IFPA Liège
June 30-July 4, 2014, LPT, Paris-Sud University, Orsay, France
Outline
Lecture 1
• Introduction
• Tour in phase space
• Galileo vs Lorentz
• Photon point of view
Introduction
1/33
Structure of matter
Atom
Nucleus
Nucleons
Quarks
u
Up
Proton
d
Down
Neutron
10-10m
Atomic
physics
10-14m
Nuclear
physics
10-15m
Hadronic
physics
10-18m
Particle
physics
Introduction
2/33
Elementary particles
Matter
Interaction
Graviton ?
Introduction
Degrees of freedom
« Resolution »
Relevant degrees of freedom
depend on typical energy scale
3/33
Introduction
4/33
Nucleon pictures
« Naive »
Realistic
3 non-relativistic
heavy quarks
Indefinite # of relativistic
light quarks and gluons
MN ~ Σ mconst
MN = Σ mconst (~2%) + Eint (~98% !)
Brout-Englert-Higgs
mechanism
QCD
Quantum Mechanics + Special Relativity = Quantum Field Theory
Introduction
5/33
Goal : understanding the nucleon internal structure
Why ?
At the energy frontier
LHC
Quark & gluon distributions ?
Proton
Proton
Introduction
6/33
Goal : understanding the nucleon internal structure
Why ?
E.g.
At the intensity frontier
Electron
Nucleon
Nucleon spin structure ?
Nucleon shape & size ?
…
Introduction
7/33
Goal : understanding the nucleon internal structure
How ?
Deep Inelastic Scattering
Elastic Scattering
Proton
« tomography »
Semi-Inclusive DIS
Deeply Virtual Compton Scattering
Phase-space
distribution
Phase space
8/33
Classical Mechanics
Momentum
State of the
system
Particles follow well-defined trajectories
Position
[Gibbs (1901)]
Phase space
9/33
Statistical Mechanics
Phase-space
density
Momentum
Position-space density
Momentum-space density
Phase-space average
Position
[Gibbs (1902)]
Phase space
10/33
Quantum Mechanics
Wigner
distribution
Momentum
Position-space density
Momentum-space density
Phase-space average
Position
[Wigner (1932)]
[Moyal (1949)]
Phase space
11/33
Quantum Mechanics
Momentum
Wigner
distribution
Position
Hermitian
(symmetric)
derivative
Fourier conjugate
variables
[Wigner (1932)]
[Moyal (1949)]
Phase space
Wigner distributions have applications in:
12/33
Harmonic oscillator
• Nuclear physics
• Quantum chemistry
• Quantum molecular dynamics
• Quantum information
• Quantum optics
• Classical optics
• Signal analysis
• Image processing
• Quark-gluon plasma
•…
Quasi-probabilistic
Heisenberg’s
uncertainty relations
Phase space
13/33
In quantum optics, Wigner distributions are « measured » using homodyne tomography
[Lvovski et al. (2001)]
[Bimbard et al. (2014)]
Idea : measuring projections of Wigner
distributions from different directions
3D
2D
2D
Binocular vision
in phase space !
Find the hidden 3D picture
Phase space
14/33
Quantum Field Theory
Covariant Wigner operator
Time ordering ?
Scalar fields
Equal-time Wigner operator
Phase-space/Wigner distribution
[Carruthers, Zachariasen (1976)]
[Ochs, Heinz (1997)]
Phase space
Nucleons are composite systems
~
15/33
Interesting
Not so interesting
internal motion
center-of-mass motion
Phase space
Localized state in momentum space
in position space
16/33
CoM
momentum
CoM
position
Phase-space compromise
Intrinsic phase-space/Wigner distribution
Breit frame
Identified with
intrinsic variables
[Ji (2003)]
[Belitsky, Ji, Yuan (2004)]
Phase space
17/33
CoM
momentum
Localized state in momentum space
in position space
CoM
position
Phase-space compromise
Intrinsic phase-space/Wigner distribution
Breit frame
Same energy !
Identified with
intrinsic variables
Time translation
[Ji (2003)]
[Belitsky, Ji, Yuan (2004)]
Galilean symmetry
All is fine as long as space-time symmetry is Galilean
Position operator can be defined
18/33
Lorentz symmetry
19/33
But in relativity, space-time symmetry is Lorentzian
Position operator is ill-defined !
No separation of CoM
and internal coordinates
Further issues :
Lorentz contraction
Creation/annihilation of pairs
Spoils (quasi-)
probabilistic
interpretation
Forms of dynamics
Space-time foliation
20/33
Light-front components
« Space »
= 3D hypersurface
« Time »
= hypersurface label
Instant-form dynamics
Light-front form dynamics
Time
Space
Energy
Momentum
[Dirac (1949)]
Instant form vs light-front form
Ordinary point of view
t1, z1
t3, z3
t2, z2
t4, z4
t5, z5
t7, z7
t6, z6
21/33
Instant form vs light-front form
Photon point of view
x+
x+
x+
x+
x+
x+
x+
22/33
Instant form vs light-front form
Initial frame
23/33
Instant form vs light-front form
Boosted frame
24/33
Instant form vs light-front form
(Quasi) infinite-momentum frame
25/33
Light-front operators
26/33
Transverse space-time symmetry is Galilean
Transverse position operator can be defined !
Longitudinal momentum plays the role of mass in the transverse plane
Transverse boost
[Kogut, Soper (1970)]
Quasi-probabilistic interpretation
27/33
What about the further issues with Special Relativity ?
Transverse boosts are Galilean
No transverse Lorentz contraction !
No sensitivity to longitudinal Lorentz contraction !
Particle number is conserved in Drell-Yan frame
Drell-Yan frame
is conserved
and positive
Conclusion : quasi-probabilistic interpretation is possible !
Non-relativistic phase space
Localized state in momentum space
in position space
28/33
CoM
momentum
CoM
position
Equal-time Wigner operator
Intrinsic phase-space/Wigner distribution
Identified with
intrinsic variables
Time translation
[Ji (2003)]
[Belitsky, Ji, Yuan (2004)]
Relativistic phase space
29/33
« CoM »
momentum
Localized state in momentum space
in position space
« CoM »
position
Equal light-front time Wigner operator
[Soper (1977)]
[Burkardt (2000)]
[Burkardt (2003)]
[Ji (2003)]
[Belitsky, Ji, Yuan (2004)]
Intrinsic relativistic phase-space/Wigner distribution
Boost invariant !
Space-time
translation
Normalization
[C.L., Pasquini (2011)]
Relativistic phase space
30/33
Our intuition is instant form and not light-front form
NB :
is invariant under light-front boosts
It can be thought as instant form phase-space/Wigner distribution in IMF !
Transverse
momentum
Longitudinal
momentum
Transverse
position
2+3D
In IMF, the nucleon looks like a pancake
[C.L., Pasquini (2011)]
Link with parton correlators
31/33
Phase-space/Wigner distributions are Fourier transforms
Space-time translation
General parton correlator
Link with parton correlators
32/33
Adding spin to the picture
Dirac matrix
~ quark polarization
Leading twist
Link with parton correlators
Adding spin and color to the picture
Wilson line
Gauge transformation
Very very very complicated object not directly measurable
Let’s look for simpler measurable correlators
33/33
Summary
Lecture 1
• Understanding nucleon internal structure is essential
• Concept of phase space can be generalized to QM and QFT
• Relativistic effects force us to abandon 1D in phase space
Transverse
momentum
Transverse
position
2+3D