Craig Roberts, Physics Division
Problems in Theory
www.nature.com/news/scientific-methoddefend-the-integrity-of-physics-1.16535
 Faced with difficulties in applying fundamental theories to the
observed Universe, some researchers have called for a
change in how theoretical physics is done.
 They begin to argue — explicitly — that if a theory is
sufficiently elegant and explanatory, it need not be tested
experimentally ...
 Chief among the “elegance will suffice” advocates are some
string theorists and cosmologists …
 This is NOT science …
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Scandal in Academia
 “I have no data yet.
It is a capital mistake to theorize
before one has data.
Insensibly one begins to twist facts
to suit theories, instead of theories
to suit facts.”
Sherlock Holmes
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Thomas Jefferson National Accelerator Facility (JLab)
 Driving distance:
Washington DC to JLab
≈ 270km
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Thomas Jefferson National Accelerator Facility (JLab)
 1984 … DoE provided initial funding for research,
development and design
 1987 … Construction begins on Continuous Electron Beam
Accelerator Facility (CEBAF) - February 13
 1994 … Accelerator reaches design energy of 4 GeV
 Construction cost in $FY14 ≈ $1-Billion
 Goal … Write the book about the strongest force in
nature – the force that holds nuclei together – and
determine how that force can be explained in terms of
the quarks and gluons of quantum chromodynamics
(QCD).
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Thomas Jefferson National Accelerator Facility (JLab)
e.g. S. J. Brodsky and G. R. Farrar,
Phys. Rev. Lett. 31, 1153 (1973)
 One of the primary reasons for building CEBAF/Jlab …
Prediction: at energy-scales greater than some a priori unknown
minimum value, Q0, cross-sections and form factors behave as
Parton
model
scaling
QCD scaling
violations
power = ( number valence-quarks – 1 + Δλ )
Δλ=0,1, depending on whether helicity is conserved
or flipped
… prediction of 1/k2 vector-boson exchange
logarithm = distinctive feature & concrete prediction of QCD
 Initially imagined that Q0 = 1GeV!
 So, JLab was initially built to reach 4GeV.
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Thomas Jefferson National Accelerator Facility (JLab)
 1994 – 2004
o An enormous number of
fascinating experimental results
o Including an empirical
demonstration that the
distribution of charge and
magnetisation within the proton
are completely different,
o Suggesting that quark-quark
correlations play a crucial role in
nucleon structure
But no sign of parton model scaling
and certainly not of scaling
violations
Particle physics paradigm
Particle physics paradigm
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Thomas Jefferson National Accelerator Facility (JLab)
 2004 … Mission Need Agreed on upgrade of CEBAF (JLab's
accelerator) to 12GeV
 2015 … 12GeV commissioning beams
now being delivered to
the experimental halls
 Final cost of upgrade is
approximately $370-Million
 Physics of JLab at 12GeV
arXiv:1208.1244 [hep-ex]
 QCD and Hadron Physics
arXiv:1502.05728
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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 Search for exotic hadrons – impossible in quark model
– Discovery would force dramatic reassessment of the
distinction between the notions of matter fields and
force fields
 Exploit opportunities provided by new data on
hadron elastic and transition form factors
– Chart infrared evolution of QCD’s coupling and
dressed-masses
– Reveal correlations that are key to nucleon structure
– Expose the facts or fallacies in contemporray
descriptions of hadron structure
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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 Precision experimental study of valence-quark
region, and theoretical computation of distribution
functions and distribution amplitudes
– Computation is critical
– Without it, no amount of data will reveal anything
about the theory underlying the phenomena of strong
interaction physics
 Explore and exploit opportunities to use precisionQCD as a probe for physics beyond the Standard
Model
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Dyson-Schwinger
Equations
 Well suited to Relativistic Quantum Field Theory
 Simplest level: Generating Tool for Perturbation
Theory . . . Materially Reduces ModelDependence … Statement about long-range
behaviour of quark-quark interaction
 NonPerturbative, Continuum approach to QCD
 Hadrons as Composites of Quarks and Gluons
 Qualitative and Quantitative Importance of:
 Dynamical Chiral Symmetry Breaking
– Generation of fermion mass from nothing
 Quark & Gluon Confinement
– Coloured objects not detected,
Not detectable?
 Approach yields
Schwinger functions; i.e.,
propagators and vertices
 Cross-Sections built from
Schwinger Functions
 Hence, method connects
observables with longrange behaviour of the
running coupling
 Experiment ↔ Theory
comparison leads to an
understanding of longrange behaviour of
strong running-coupling
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Significant Progress
 Novel understanding of gluon and quark
confinement and its consequences
is emerging from quantum field theory
 Arriving at a clear picture of how hadron masses emerge
dynamically in a universe with light quarks
Dynamical Chiral Symmetry Breaking (DCSB)
 Realistic computations of ground-state hadron wave
functions with a direct connection to QCD are now available
 Quark-quark correlations are crucial in hadron structure
 Accumulating empirical evidence in support of this
prediction
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Light quarks & Confinement
 Folklore … Hall-D Conceptual Design Report(5)
“The color field lines between a quark and an anti-quark form flux tubes.
A unit area placed midway
between the quarks and
perpendicular to the line
connecting them intercepts
a constant number of field
lines, independent of the
distance between the
quarks.
This leads to a constant
force between the quarks –
and a large force at that,
equal to about 16 metric
tons.”
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Light quarks & Confinement
Problem:
16 tonnes of force
makes a lot of pions.
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Light quarks & Confinement
Problem:
16 tonnes of force
makes a lot of pions.
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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G. Bali et al., PoS LAT2005 (2006) 308
Light quarks & Confinement
 In the presence of
light quarks, pair
creation seems to
occur non-localized
and instantaneously
 No flux tube in a
theory with lightquarks.
 Flux-tube is not the
correct paradigm for
confinement in
hadron physics
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
 QFT Paradigm:
Confinement
– Confinement is expressed through a dramatic
change in the analytic structure of propagators
for coloured states
– It can be read from a plot of the dressedpropagator for a coloured state
Confined particle
Normal particle
Propagation described by rapidly
damped wave & hence state cannot
exist in observable spectrum
σ ≈ 1/Im(m)
≈ 1/2ΛQCD ≈ ½fm
Real-axis mass-pole splits,
moving into pair(s) of complex conjugate singularities, (or qualitatively
analogous structures chracterised by a dynamically generated mass-scale)
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 Feynman propagator for
a fermion describes a
Plane Wave
 A fermion begins to
propagate
 It can proceed a long
way before undergoing
any qualitative changes
meson
meson
meson
Baryon
meson
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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 A quark begins to
propagate
 But after each “step” of
length σ, on average, an
interaction occurs, so
that the quark loses its
identity, sharing it with
other partons
 Finally, a cloud of
partons is produced,
which coalesces into
colour-singlet final states
σ
meson
meson
meson
Baryon
meson
Confinement is a
dynamical phenomenon!
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Dynamical Chiral
Symmetry Breaking
DCSB is a fact in QCD
– Dynamical, not spontaneous
• Add nothing to QCD ,
No Higgs field, nothing!
Effect achieved purely through
quark+gluon dynamics.
– It’s the most important
mass generating mechanism
for visible matter in the Universe.
• Responsible for ≈98% of the proton’s mass.
• Higgs mechanism is (almost) irrelevant to light-quarks.
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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 Deceptively simply picture
Where does the
mass come from?
 Corresponds to the sum of a countable infinity of diagrams.
NB. QED has 12,672 α5 diagrams
 Impossible to compute this in perturbation theory.
The standard algebraic manipulation
αS23
tools are just inadequate
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Persistent challenge
in application of DSEs
 Infinitely many coupled equations:
Kernel of the equation for the quark self-energy involves:
– Dμν(k) – dressed-gluon propagator
– Γν(q,p) – dressed-quark-gluon vertex
each of which satisfies its own DSE, etc…
 Coupling between equations necessitates a truncation
Invaluable check on
– Weak coupling expansion
practical truncation
⇒ produces every diagram in perturbation theory
schemes
– Otherwise useless
for the nonperturbative problems in which we’re interested
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Relationship must be preserved by any truncation
Highly nontrivial constraint
Persistent challenge
FAILURE has an extremely high cost
– loss of any connection with QCD - truncation scheme
 Symmetries associated with conservation of vector and axial-vector
currents are critical in arriving at a veracious understanding of
hadron structure and interactions
 Example: axial-vector Ward-Green-Takahashi identity
– Statement of chiral symmetry and the pattern by which it’s broken in
quantum field theory
Quark
propagator
satisfies a
gap equation
Axial-Vector vertex
Satisfies an inhomogeneous
Bethe-Salpeter equation
Kernels of these equations are completely different
But they must be intimately related
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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 Longitudinal WGT identity
expresses properties of the
divergence of the vertex
 Transverse identities relate
to its curl (as Faraday’s law
of induction involves an
electric field)
 The last two terms in each identity arise in computing the momentum
space expression of a nonlocal axial-vector/vector vertex, whose definition
involves a gauge-field-dependent line integral
 But … practical progress can be made without knowing their precise forms
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Persistent challenge
- truncation scheme
 These observations show that symmetries relate the kernel of the
gap equation – nominally a one-body problem, with that of the
Bethe-Salpeter equation – considered to be a two-body problem
 Until 1995/1996 people had
quark-antiquark
no idea what to do
scattering kernel
 Equations were truncated,
sometimes with good
phenomenological results,
sometimes with poor results
 Neither good nor bad
could be explained
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Persistent challenge
- truncation scheme
 Happily, that changed, and there is now at least one systematic,
nonperturbative and symmetry preserving truncation scheme
– H.J. Munczek, Phys. Rev. D 52 (1995) 4736, Dynamical chiral symmetry
breaking, Goldstone’s theorem and the consistency of the SchwingerDyson and Bethe-Salpeter Equations
– A. Bender, C.D. Roberts and L. von Smekal, Phys.Lett. B 380 (1996) 7,
Goldstone Theorem and Diquark Confinement Beyond Rainbow Ladder
Approximation
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Modified skeleton expansion in
which the propagators are
fully-dressed but the vertices
are constructed term-by-term
Cutting scheme
 The procedure generates a Bethe-Salpeter kernel from the kernel
of any gap equation whose diagrammatic content is known
– That this is possible and
achievable systematically is
necessary and sufficient to
prove some exact results
in QCD
 The procedure also enables the
formulation of practical
phenomenological models that
can be used to illustrate the
exact results and provide
predictions for experiment with
readily quantifiable errors.
Craig Roberts: (2) The Ins and Outs of Mesons
dressed propagators
gap eq.
Leading-order:
rainbow- ladder truncation
BS kernel
bare vertices
In gap eq., add
1-loop vertex correction
Then BS kernel has
3 new terms at this order
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Bethe-Salpeter Equation
Bound-State DSE
 K(q,k;P) – fully amputated, two-particle irreducible,
quark-antiquark scattering kernel
 Textbook material.
 Compact. Visually appealing. Correct
Blocked progress for more than 60 years.
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Bethe-Salpeter Equation
Lei Chang and C.D. Roberts
General
Form
0903.5461 [nucl-th]
Phys. Rev. Lett. 103 (2009) 081601
 Equivalent exact bound-state equation but in this form
K(q,k;P) → Λ(q,k;P)
which is completely determined by dressed-quark self-energy
 Enables derivation of a Ward-Takahashi identity for Λ(q,k;P)
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Ward-Takahashi Identity
Lei Chang and C.D. Roberts Bethe-Salpeter Kernel
0903.5461 [nucl-th]
Phys. Rev. Lett. 103 (2009) 081601
iγ5
iγ5
 Now, for first time, it’s possible to formulate an Ansatz for
Bethe-Salpeter kernel given any form for the dressed-quark-gluon
vertex by using this identity
 This enables the identification and elucidation of a wide range
of novel consequences of DCSB
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Non-perturbative comparison of QCD effective
charges, A.C. Aguilar, D. Binosi, J. Papavassiliou and J.
Rodriguez-Quintero, Phys. Rev. D80 (2009) 085018
Gluons, too,
have a gap equation
 Pinch-technique + background field method … reordering of
diagrammatic summations in the self-energy – Πμν – ensures that
subclusters are individually transverse and gluon-loop and ghostloop contributions are separately transverse
 STIs → WGTIs
 Enables systematic analysis and evaluation of truncations and
straightforward comparison of results with those of lQCD
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
41
Bridging a gap between continuum-QCD and ab initio
predictions of hadron observables, D. Binosi, L. Chang,
J.Papavassiliou, C.D. Roberts, arXiv:1412.4782 [nucl-th],
Phys. Lett. B742 (2015) 183-188
 Input to the DSE analysis
= lQCD result for the ghost
dressing function at a given
renormalisation scale, ζ
 Solve ghost gap equation
self-consistently such that
αS(ζ) reproduces lQCD result
 Gluon-ghost vertex in ghost
gap equation is computed
from its own DSE in the
one-loop dressed
approximation.
 Continuum- and lattice-QCD
solutions agree on solution
for the gluon self energy
RGI
Running Interaction
d̂(k2)= α(ζ) Δ̂(k2; ζ)
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
42
Bridging a gap between continuum-QCD and ab initio
predictions of hadron observables, D. Binosi, L. Chang,
J.Papavassiliou, C.D. Roberts, arXiv:1412.4782 [nucl-th],
Phys. Lett. B742 (2015) 183-188
 Running gluon mass
In QCD: Gluons also
become massive!
mg2 (k 2 ) 
 Gluons are cannibals
– a particle species
whose members
become massive by
eating each other!
g4
 k
2
g
2
Gluon mass-squared function
Power-law suppressed in
ultraviolet, so invisible in
perturbation theory
Interaction model for the gap equation, S.-x.Qin,
L.Chang, Y-x.Liu, C.D.Roberts and D. J. Wilson,
arXiv:1108.0603 [nucl-th], Phys. Rev. C 84 (2011)
042202(R) [5 pages]
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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 Gauge boson cannibalism
… a new physics frontier … within the Standard Model
 Asymptotic freedom means
… ultraviolet behaviour of QCD is controllable
 Dynamically generated masses for gluons and quarks means
that QCD dynamically generates its own infrared cutoffs
– Gluons and quarks with
wavelength λ > 2/mass ≈ 1 fm
decouple from the dynamics … Confinement?!
 How does that affect observables?
– It will have an impact in
any continuum study
– Must play a role in gluon saturation ...
In fact, perhaps it’s a harbinger of gluon saturation?
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
44
Bridging a gap between continuum-QCD
& ab initio predictions of hadron observables
D. Binosi (Italy), L. Chang (Australia), J. Papavassiliou (Spain),
C. D. Roberts (US), arXiv:1412.4782 [nucl-th] , Phys. Lett. B 742 (2015) 183
 Top-down approach – ab initio
computation of the interaction via
direct analysis of the gauge-sector gap
equations
 Bottom-up scheme – infer interaction
by fitting data within a well-defined
truncation of the matter sector DSEs
that are relevant to bound-state
properties.
 Serendipitous collaboration, conceived
at one-week ECT* Workshop on DSEs
in Mathematics and Physics, has
united these two approaches
Top down & Bottom up
“Maris-Tandy” interaction. Developed at ANL & KSU in
1997-1998. More-than 600 citations – but quantitative
disagreement with gauge-sector solution.
Modern kernels and interaction,
developed at ANL and Peking U.
One parameter, fitted to groundstate properties without reference
to gauge-sector studies.
Modern top-down and bottom-up
results agree within 3% !
Top-down result = gauge-sector prediction
Top-down result = gauge-sector prediction
– Interaction predicted by modern analyses of QCD's gauge sector coincides with
that required to describe ground-state observables using the sophisticated mattersector ANL-PKU DSE truncation
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
45
Bridging a gap between continuum-QCD
& ab initio predictions of hadron observables
D. Binosi (Italy), L. Chang (Australia), J. Papavassiliou (Spain),
C. D. Roberts (US), arXiv:1412.4782 [nucl-th] , Phys. Lett. B 742 (2015) 183
 Top-down approach – ab initio
computation of the interaction via
direct analysis of the gauge-sector gap
equations
 Bottom-up scheme – infer interaction
by fitting data within a well-defined
truncation of the matter sector DSEs
that are relevant to bound-state
properties.
 Serendipitous collaboration, conceived
at one-week ECT* Workshop on DSEs
in Mathematics and Physics, has
united these two approaches
Top down & Bottom up
Modern kernels and
interaction, developed at
ANL and Peking U.
One parameter, fitted to
ground-state properties
without reference to gaugesector studies.
Modern top-down and
bottom-up results agree
within 3% !
Top-down result = gauge-sector prediction
– Interaction predicted by modern analyses of QCD's gauge sector coincides with
that required to describe ground-state observables using the sophisticated mattersector ANL-PKU DSE truncation
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
46
Bridging a gap between continuum-QCD
& ab initio predictions of hadron observables
D. Binosi (Italy), L. Chang (Australia), J. Papavassiliou (Spain),
C. D. Roberts (US), arXiv:1412.4782 [nucl-th] , Phys. Lett. B 742 (2015) 183
 Top-down approach – ab initio
computation of the interaction via
direct analysis of the gauge-sector gap
equations
 Bottom-up scheme – infer interaction
by fitting data within a well-defined
truncation of the matter sector DSEs
that are relevant to bound-state
properties.
 Serendipitous collaboration, conceived
at one-week ECT* Workshop on DSEs
in Mathematics and Physics, has
united these two approaches
Top down & Bottom up
Modern kernels and
interaction, developed at
ANL and Peking U.
One parameter, fitted to
ground-state properties
without reference to gaugesector studies.
Modern top-down and
bottom-up results agree
within 3% !
Significant steps toward
parameter-free prediction
of hadron properties
Top-down result = gauge-sector prediction
– Interaction predicted by modern analyses of QCD's gauge sector coincides with
that required to describe ground-state observables using the sophisticated mattersector ANL-PKU DSE truncation
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
47
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
48
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Maris, Roberts and Tandy
nucl-th/9707003, Phys.Lett. B420 (1998) 267-273
Pion’s Goldberger
-Treiman relation
 Pion’s Bethe-Salpeter amplitude
Solution of the Bethe-Salpeter equation
 Dressed-quark propagator
 Axial-vector Ward-Takahashi identity entails
B(k2)
Owing to DCSB
& Exact in
Chiral QCD
Miracle: two body problem solved,
almost completely, once solution of
one body problem is known
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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This identity is why the pion is
massless in the chiral limit
 The quark level Goldberger-Treiman relation shows that DCSB has a
very deep and far reaching impact on physics within the strong
interaction sector of the Standard Model; viz.,
Goldstone's theorem is fundamentally an expression of equivalence
between the one-body problem and the two-body problem in the
pseudoscalar channel.
 This emphasises that Goldstone's theorem has a pointwise
expression in QCD
 Hence, pion properties are an almost direct measure of
the dressed-quark mass function.
 Thus, enigmatically, the properties of the massless pion
are the cleanest expression of the mechanism that is
responsible for almost all the visible mass in the universe.
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Universal
Conventions
 Wikipedia: (http://en.wikipedia.org/wiki/QCD_vacuum)
“The QCD vacuum is the vacuum state of quantum
chromodynamics (QCD). It is an example of a nonperturbative vacuum state, characterized by many nonvanishing condensates such as the gluon condensate or
the quark condensate. These condensates characterize the
normal phase or the confined phase of quark matter.”
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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“Orthodox Vacuum”
 Vacuum = “frothing sea”
u
 Hadrons = bubbles in that “sea”,
d
u
containing nothing but quarks & gluons
interacting perturbatively, unless they’re
near the bubble’s boundary, whereat they feel they’re
trapped!
ud
u
u
u
d
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Historically, DCSB has
come to be associated
with the presumed
existence of spacetimeindependent condensates
that permeate the
Universe.
However, just like gluons and quarks, and
for the same reasons:
Condensates are confined within hadrons.
There are no vacuum condensates.
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
57
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
58
Expanding the concept of in-hadron condensates
Lei Chang, Craig D. Roberts and Peter C. Tandy
arXiv:1109.2903 [nucl-th], Phys. Rev. C85 (2012) 012201(R)
GMOR Relation
 Valuable to highlight the precise form of the Gell-Mann–
Oakes–Renner (GMOR) relation: Eq. (3.4) in Phys.Rev. 175
(1968) 2195
o mπ is the pion’s mass
o Hχsb is that part of the hadronic Hamiltonian density which
explicitly breaks chiral symmetry.
 The operator expectation value in this equation is evaluated
between pion states.
 Un-approximated form of the GMOR relation doesn’t make
any reference to a vacuum condensate
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
60
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
61
Gell-Mann Oakes Renner Relation
 Demonstrated algebraically that the so-called Gell-Mann –
Oakes – Renner relation is the following statement
Namely, the mass of the pion is completely determined by the
pion’s scalar form factor at zero momentum transfer Q2 = 0.
viz., by the pion’s scalar charge
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
62
Hadron
Charges
 Matrix elements associated with hadron form factors
 Scalar charge of a hadron is an intrinsic property of
that hadron … no more a property of the vacuum
than the hadron’s electric charge, axial charge,
tensor charge, etc. …
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
63
“Orthodox Vacuum”
 Vacuum = “frothing sea”
u
 Hadrons = bubbles in that “sea”,
d
u
containing nothing but quarks & gluons
interacting perturbatively, unless they’re
near the bubble’s boundary, whereat they feel they’re
trapped!
ud
u
u
u
d
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
64
 Vacuum = perturbative hadronic fluctuations
but no nonperturbative condensates
 Hadrons = complex, interacting systems
within which perturbative behaviour is
restricted to just 2% of the interior
u
d
u
ud
u
u
u
d
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
65
Paradigm shift:
In-Hadron Condensates
“Void that is truly empty
solves dark energy puzzle”
“The biggest
embarrassment in
Rachel Courtland, New Scientist 4th Sept. 2010
theoretical physics.”
“EMPTY space may really be empty. Though quantum theory suggests that a
vacuum should be fizzing with particle
it turns out that this paradoxical
activity,
QCD
46
picture
of
nothingness
may
not
be
needed.
A
calmer
of the vacuum would
QCD condensates  8 GN
 10 view
* experiment
2
also help resolve a nagging inconsistency
dark energy, the elusive force
3Hwith
0
thought to be speeding up the expansion of the universe.”

4
Cosmological Constant:
Putting QCD condensates back into hadrons reduces the
mismatch between experiment and theory by a factor of 1046
Possibly by far more, if technicolour-like theories are the correct
paradigm for extending the Standard Model
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
66
Valence
quarks
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
67
Parton Structure of Hadrons
 Valence-quark structure of hadrons
– Definitive of a hadron.
After all, it’s how we distinguish a proton from a neutron
– Expresses charge; flavour; baryon number; and other Poincaréinvariant macroscopic quantum numbers
– Via evolution, determines background at LHC
 Foreseeable future will bring precision experimental study of (far)
valence region, and theoretical computation of distribution
functions and distribution amplitudes
– Computation is critical
– Without it, no amount of data will reveal anything about the theory
underlying the phenomena of strong interaction physics
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
68
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
69
Light-front Quantisation
 Hamiltonian formulation of quantum field theory.
– Fields are specified on a particular initial surface:
Light front x+ = x0 + x3 = 0
 Using LF quantisation:
 quantum mechanics-like wave
functions can be defined;
 quantum-mechanics-like
expectation values can be defined
and evaluated
 Parton distributions are correlation
functions at equal LF-time x+ ;
namely, within the initial surface
x+ = 0, and can thus be expressed
directly in terms of ground state LF
wavefunctions
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Imaging dynamical chiral symmetry breaking: pion wave function
on the light front, Lei Chang, et al., arXiv:1301.0324 [nucl-th],
Phys. Rev. Lett. 110 (2013) 132001 (2013) [5 pages].
Pion’s valence-quark
Distribution Amplitude
 Following a workshop in Caraguatatuba (2012), methods were
developed that enable direct computation of the pion’s light-front
wave function
 φπ(x) = twist-two parton distribution amplitude = projection of the
pion’s Poincaré-covariant wave-function onto the light-front
 Results have been obtained with rainbow-ladder DSE kernel,
simplest symmetry preserving form; and the best DCSB-improved
kernel that is currently available.
xα (1-x)α, with α≈0.5
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
72
Imaging dynamical chiral symmetry breaking: pion wave function
on the light front, Lei Chang, et al., arXiv:1301.0324 [nucl-th],
Phys. Rev. Lett. 110 (2013) 132001 (2013) [5 pages].
Pion’s valence-quark
Distribution Amplitude
 Both kernels agree: marked broadening of φπ(x), which owes to DCSB
 This may be claimed because
PDA is computed at a low
renormalisation scale in the
chiral limit, whereat the
quark mass function owes
entirely to DCSB.
 Difference between RL and
DB results is readily
understood: B(p2) is more
slowly varying with DB kernel
and hence a more balanced
result
Asymptotic
DB
RL
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
73
Imaging dynamical chiral symmetry breaking: pion wave function
on the light front, Lei Chang, et al., arXiv:1301.0324 [nucl-th],
Phys. Rev. Lett. 110 (2013) 132001 (2013) [5 pages].
Pion’s valence-quark
Distribution Amplitude
 Both kernels agree: marked broadening of φπ(x), which owes to DCSB
 This may be claimed because
PDA is computed at a low
renormalisation scale in the
chiral limit, whereat the
quark mass function owes
entirely to DCSB.
 Difference between RL and
DB results is readily
understood: B(p2) is more
slowly varying with DB kernel
and hence a more balanced
result
Asymptotic
DB
RL
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
74
arXiv:1301.0324 [nucl-th], arXiv:1306.2645 [nucl-th],
arXiv:1311.1390 [nucl-th], arXiv:1405.0289 [nucl-th],
arXiv:1406:3353 [nucl-th]
Features of
Ground-state PDAs
 A diverse array of studies since Caraguatatuba (2012) have shown
that ground-state meson PDAs are broad, concave functions
Concave function: no
line segment lies above
any point on the graph
 Camel-humped distributions – popular for many years – are
physically unreasonable because they correspond to bound-state
amplitudes that disfavour equal momentum partitioning between
valence-quark degrees of freedom
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
75
Hard Exclusive Processes
& PDAs
 In the theory of strong interactions, the cross-sections for many
hard exclusive hadronic reactions can be expressed in terms of the
PDAs of the hadrons involved
 Example: pseudoscalar-meson elastic electromagnetic form factor
o αS(Q2) is the strong running coupling,
o φπ(u) is the meson’s twist-two valence-quark PDA
o fP is the meson's leptonic decay constant
It was promised that
JLab would verify this
fundamental
prediction
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
76
Pion electromagnetic
form factor
 In 2001 – seven years after
beginning operations,
Jefferson Lab provided the
first high precision pion
electroproduction data for
Fπ between Q2 values of 0.6
and 1.6 (GeV/c)2.
lQCD only provides access to
this small domain, already
well-mapped by experiments
JLab Data
Result imagined by many
to be QCD prediction
Evaluated with φπ = 6x(1-x)
 2006 & 2007 – new result, at Q2=2.45 (GeV/c)2
 Authors of the publications stated: “still far from the transition
to the Q2 region where the pion looks like a simple quarkantiquark pair”
 disappointment and surprise
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
77
Pion electromagnetic
form factor
 Year 2000 prediction for
Fπ(Q2)
– P.Maris & P.C. Tandy,
Phys.Rev. C62 (2000)
055204
Factor of three
discrepancy
JLab Data
Result imagined by many
to be QCD prediction
Evaluated with φπ = 6x(1-x)
 Problem … used brute-force
computational method …
unable to compute for
Q2>4GeV2
 Shape of prediction suggested to many that one might never see
parton model scaling and QCD scaling violations
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
78
Pion electromagnetic
form factor
 Plans were made and an
experiment approved that
use the higher-energy
electron beam at the 12
GeV Upgrade at Jefferson
Lab.
 The Upgrade will allow an
extension of the
Fπ measurement up to a
value of Q2 of about 6
(GeV/c)2, which will probe
the pion at double the
resolution.
Projected JLab reach
Result imagined by many
to be QCD prediction
Evaluated with φπ = 6x(1-x)
 Will there be any hint of a trend
toward the asymptotic pQCD
prediction?
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
80
Pion electromagnetic form factor
 Solution – Part 1
– Compare data with the
real QCD prediction; i.e.
the result calculated using
the broad pion PDA
predicted by modern
analyses of continuum
QCD
Real QCD prediction – obtained
withResult
realistic,
computed
PDA
imagined
by many
to be QCD prediction
Evaluated with φπ = 6x(1-x)
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
81
Pion electromagnetic form factor
 Solution – Part 1
– Compare data with the
real QCD prediction; i.e.
the result calculated using
the broad pion PDA
predicted by modern
analyses of continuum
QCD
Agreement within 15%
maximum
Real QCD prediction – obtained
with realistic, computed PDA
 Solution – Part 2
– Algorithm used to
compute the PDA can also
be employed to compute
Fπ(Q2) directly, to
arbitrarily large Q2
 Predictions:
 JLab will see maximum
 Experiments to 8GeV2 will see
parton model scaling and QCD
Pion electromagnetic form factor at spacelike momenta
scaling violations for the first
L. Chang, I. C. Cloët, C. D. Roberts, S. M. Schmidt and P. C. Tandy,
time in a hadron form factor
arXiv:1307.0026 [nucl-th], Phys. Rev. Lett. 111, 141802 (2013)
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
82
Implications
 Verify the theory of factorisation in hard exclusive processes,
with dominance of hard contributions to the pion form factor
for Q2>8GeV2.
 Notwithstanding that, normalisation of Fπ(Q2) is fixed by a
pion wave-function whose dilation with respect to
φπasy(x)=6x(1-x) is a definitive signature of DCSB
– Empirical measurement of the strength of DCSB in the
Standard Model – the origin of visible mass
 Close the book on a story that began thirty-five years ago
 Paves the way for a dramatic reassessment of pictures of
proton & neutron structure, which is already well underway
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
84
I.C. Cloët, C.D. Roberts, A.W. Thomas:
arXiv:1304.0855 [nucl-th], Phys. Rev. Lett. 111 (2013) 101803
 DCSB →equivalence between
quark 1-body and pseudoscalar 2-body problem
– Complete understanding of pion … no model dependence …
the dressed-quark mass function is known
– Hence, continuum-QCD theory is on the threshold of a
complete picture of the pion’s light-front valence-quark wave
function
 In parallel, continuum-QCD is providing predictions of the lightfront structure of other light-quark hadrons
 Near-term Aim
– Unified QCD-based and quantum mechanical picture of the lightquark meson sector of the Standard Model
 Developments in baryon sector
are equally dramatic: e.g.,
is
understood
Craig Roberts: (2) The Ins and Outs of Mesons
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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