Download Lessons from large–Nc model field theories

PDFs as momentum densities:
Lessons from large–Nc model field theories
C. Weiss (JLab), MCFP Workshop on Lattice Parton Physics, U. Maryland, 31–Mar–14
• Model field theory and nucleon
Light cone
correlation
function
Dynamics from spontaneous χSB
Solution in 1/Nc expansion
• Parton distributions
LC correlation functions
Momentum densities at P → ∞
P
Momentum
density
Equivalence
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• UV regularization
Power divergences of moments
Large–x tails and anomalies
Equivalence, physical requirements
studied in simpler dynamical setting!
Quantum field theory: UV divergences,
particle number fluctuations, ...
Completeness of states and causality!
Physical regularization by Pauli–Villars cutoff
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Dynamics: Chiral symmetry breaking
• Chiral symmetry breaking in QCD
Non-pert. gluon fields can flip chirality
Topological gauge fields, instantons
L
..
.
pert.
L
L
R
R
R
Condensate of q q̄ pairs hψ̄L ψR + ψ̄R ψL i,
pion as collective excitation
non−pert.
Order parameter, Goldstone boson
L
R
ρ ~ 0.3 fm
Dynamical mass generation:
Constituent quarks, hadron structure
Euclidean correlation functions → Lattice, analytic methods
Range of χSB iteractions ρ ≪ 1 fm
New dynamical scale: Shuryak; Diakonov, Petrov 80’s
...
...
...
condensate
Gauge–invariant measure of q q̄ pair size
hψ̄∇2ψi/hψ̄ψi ∼ 1 GeV2
Lattice: Teper 87, Doi 02, Chiu 03. Instantons: Polyakov, CW 96
dynamical mass
• Dynamical model of chirally broken phase
Valid at momenta peucl . ρ−1
Based on large–Nc limit of QCD
2
Dynamics: Field-theoretical model
3
• Effective description of χSB
Diakonov, Eides 83; Diakonov, Petrov 86
Quark field acquires dynamical
mass M ∼ 0.3-0.4 GeV
chiral field
...
.
.
.
Coupled to chiral field (Goldstone boson)
with eff. coupling M/fπ = 3–4 strong!
Valid up to χSB scale ρ−2,
implemented as UV cutoff
M dynamical mass
• Solved non–perturbatively
using 1/Nc expansion
iγ5 τ π/fπ
Leff = ψ̄ i∂/ − M e
Z=
Z
Z
[dπ] [dψ̄ dψ] e−Seff
ψ
Correlation functions of
composite operators
Functional integral evaluated in
saddle point approximation Seff ∼ Nc
Semiclassical approximation
Parametric approximation,
defined accuracy
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Dynamics: Nucleon
• Nucleon correlation function
...
Diakonov, Petrov, Pobylitsa 88
Classical chiral field — soliton
.. N
.
T
c
“Hedgehog” π k r in rest frame, cf. skyrmion
Binds Nc quarks, distorts chiral vacuum
Relativistic mean–field approximation
classical field
Rest frame: Quark Hamiltonian,
single–particle levels, discrete + continuum
En
...
Quantization of (iso)rotational
zero modes: N, ∆ quantum numbers
hN |Ô|N i from 3–point functions
M
Nc
0
−M
discrete
level
...
Dirac
continuum
• Field–theoretical description!
Completeness of single–particle levels
P
†
(3)
(x − y)
n Φn (x)Φn (y) = δ
No Fock space truncation
→ PDFs, sea quarks
Relativistically covariant
Dynamics: Matrix elements
..
.
• hN |Ô|N i from 3–point functions
...
Calculated in 1/Nc expansion
Connected and disconnected quark
diagrams in classical chiral field
“Valence” and “sea quark” contribution
.. .
• Sums over single–particle levels
..
.
...
X
hn|Ô|ni
n
n = discrete + continuum
• Rich phenomenology: Charges,
form factors, SU (3) extension...
Review Christov et al., Prog. Part. Nucl. Phys. 37, 91 (1996)
5
6
PDFs: Light-cone correlation function
• PDF in model as LC correlation function
Diakonov, Petrov, Pobylitsa, Polyakov, Weiss NPB 480 (1996) 341;
PRD 56 (1997) 4069; see also Wakamatsu et al. 97+
f (x) =
Z
−
dξ
iP + z − /2
e
hN |ψ̄(0)γ + ψ(ξ)|N iξ+ =0
8π
=
(
Calculate moments or x–dependence; model respects analyticity
q(x)
x>0
−q̄(−x)
<0
preserved by UV cutoff → later
Parametrically x = O(1/Nc )
• Evaluate in rest frame
f
u+d
(x) = NcMN
X Z
n
d3 p
3
† 0 +
δ(p
+
E
−
xM
)
Φ
(p)
γ γ Φn(p)
n
N
n
(2π)3
Sum over quark single–particle states, n = discrete level + continuum
Evaluated numercially
Alt. method: Green function, analytic approximations
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PDFs: Positivity and sum rules
• Positivity
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Diakonov et al. NPB 480 (1996) 341
− antiquarks
quarks
Discrete level give negative antiquarks
Continuum gives positive antiquarks
f u+d (x)
5
Total quark/antiquark PDFs positive!
0
• Partonic sum rules
discrete level
continuum
total
-5
-1
-0.5
0
x
Completeness of quark
single–particle states!
0.5
1
Z
dx f
Z
dx x f
Z
dx g u+ū−d−d̄
u−ū+d−d̄
= Nc
u+ū+d+d¯
=1
= gA
satisfied within model!
baryon nr
momentum
Bjorken
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PDFs: Interpretation
• Interpretation of model PDFs
Scale
QCD
_
Model uses effective degrees of freedom:
Massive quarks and antiquarks, no gluons
q, q , g
−2
ρ
~ 0.4 GeV
2
_
Momentum sum rule saturated by
effective degrees of freedom
q eff , q eff
Effective
chiral
dynamics
Matching with QCD quarks, antiquarks,
and gluons at χSB scale ρ−2
PDF fits at µ2 ∼ 0.5 GeV2 show 30% of
nucleon momentum carried by gluons
1.4
“Accuracy” of matching
FNAL E866 analysis
χQSM + DGLAP
1.2
2
2
Q = 54 GeV
1
• Flavor asymmetries
0.6
Model describes well measured d¯ − ū
f1
-d − -u
(x)
0.8
0.4
Pobylitsa, Polyakov, Goeke, Watabe, Weiss, PRD59 (1999) 034024
0.2
Predicts sizable asymmetry ∆ū − ∆d¯
0
Diakonov et al. NPB 480 (1996) 341; PRD 56 (1997) 4069.
Hints seen in DSSV global fits, RHIC W data
-0.2
0
0.1
0.2
x
0.3
0.4
PDFs: Momentum density at P → ∞
f (x) =
Z
3
3
d p
δ
(2π)3
x−
p
P
!
†
hP |a a(p)|P i
• PDF in model as momentum
density of quarks/antiquarks
Diakonov et al. PRD 56 (1997) 4069
†
f¯(x) = ...
b b(p)
“Parton model” definition
• Nucleon state at P → ∞
...
Nc quarks
P
fast−moving
chiral field
Model dynamics is
relativistically invariant
Boosted chiral field and quark
single–particle wave funtions
• Equivalence of PDF formulations within model
P
†
(3)
(x − y)
Uses completeness of quark single–particle states
n Φn (x)Φn (y) = δ
Ensures causality
ψ̄(x), ψ(y) + = 0 if (x − y)2 < 0 spacelike separation
Preserved by UV cutoff
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UV regularization: Physical requirement
• UV cutoff physical ingredient of model
Represents χSB scale
Form determined from physical considerations: symmetries, conservation laws,
• PDFs UV divergent, require cutoff
Logarithmic divergence: dominant contribution from momenta p ≪ cutoff
• Physical requirements
Diakonov et al. NPB 480 (1996) 341; PRD 56 (1997) 4069
Causality and spectral properties of LC correlation function
→ Preserve completeness of quark single–particle states!
→ Preserve analyticity properties
• Pauli–Villars subtraction
f (x)reg
M2
= f (x|M ) −
f (x|Mreg )
2
Mreg
10
UV regularization: Artifacts of unphysical scheme
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• Energy cutoff |En| < Ω
Used in numerical calculations in finite model space
Violates completeness of single–particle states!
• Power–like divergence of moments
Mn ≡
Z
n−1
dx x
f (x) ∼ Ω
n−2
(n > 2)
2Ω
<x<0
• Large–x tail of distribution, extends over range −
MN
• Anomaly phenomenon
X
occ
+
X
non−occ
!
f (x)tail
6= 0
even in limit Ω → ∞
• All artifacts disappear with Pauli–Villars subtraction – physical regularization!
Diakonov et al. PRD 56 (1997) 4069
Summary
• Equivalence of “light–cone correlation function” and “momentum density”
demonstrated in field–theoretical model
• Causality/completeness as essential prerequisites
Lessons for QCD calculations
• Causality/completeness–violating regularization can produce unphysical
tails in x–distribution
• Regularization artifacts may persist even in limit cutoff → ∞ (“anomalies”)
• Pauli–Villars subtraction as physical regularization
Extensions and applications
• Quasi–PDFs and 1/P corrections in large–Nc model
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