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The Ins and Outs of Mesons
Craig Roberts, Physics Division
Craig Roberts: (2) The Ins and Outs of Mesons
2
Problems in Theory
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 …
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
www.nature.com/news/scientific-method-defend-the-integrity-of-physics-1.16535
Craig Roberts: (2) The Ins and Outs of Mesons
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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
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
4
Physics is an
empirical science
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)Craig Roberts: (2) The Ins and Outs of Mesons
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It’s not physics unless it can be tested
empirically.
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)Craig Roberts: (2) The Ins and Outs of Mesons
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It’s not proven
unless it’s verified
experimentally.Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
Craig Roberts: (2) The Ins and Outs of Mesons
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Jefferson LabHadron Physics XIII: 22-27 Mar. 2015 (85pp)
Craig Roberts: (2) The Ins and Outs of Mesons
Craig Roberts: (2) The Ins and Outs of Mesons
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Thomas Jefferson National Accelerator Facility (JLab)
Driving distance: Washington DC to JLab ≈ 270km
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
Craig Roberts: (2) The Ins and Outs of Mesons
9
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).
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
Craig Roberts: (2) The Ins and Outs of Mesons
10
Thomas Jefferson National Accelerator Facility (JLab) 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
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.
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
Parton model scaling
QCD scaling violations
e.g. S. J. Brodsky and G. R. Farrar, Phys. Rev. Lett. 31, 1153 (1973)
Craig Roberts: (2) The Ins and Outs of Mesons
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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
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
Particle physics paradigm
Particle physics paradigm
Craig Roberts: (2) The Ins and Outs of Mesons
12
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 isapproximately $370-Million
Physics of JLab at 12GeV arXiv:1208.1244 [hep-ex]
QCD and Hadron Physics arXiv:1502.05728
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
Craig Roberts: (2) The Ins and Outs of Mesons
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Selected Science Challenges for the coming decade
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 structureHadron Physics XIII: 22-27 Mar. 2015 (85pp)
Craig Roberts: (2) The Ins and Outs of Mesons
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Selected Science Challenges for the coming decade
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 precision-QCD as a probe for physics beyond the Standard Model
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
Craig Roberts: (2) The Ins and Outs of Mesons
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Overarching Science Challenges for the coming decade
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
Discover meaning of confinement and its relationship to DCSB – the Origin of Visible Mass
Craig Roberts: (2) The Ins and Outs of Mesons
Dyson-SchwingerEquations
Well suited to Relativistic Quantum Field Theory Simplest level: Generating Tool for Perturbation
Theory . . . Materially Reduces Model-Dependence … 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?
16
Approach yields Schwinger functions; i.e., propagators and vertices Cross-Sections built from Schwinger Functions Hence, method connects observables with long- range behaviour of the running coupling Experiment ↔ Theory comparison leads to an understanding of long- range behaviour of strong running-coupling
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
Craig Roberts: (2) The Ins and Outs of Mesons
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Significant Progress
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
Novel understanding of gluon and quarkconfinement and its consequencesis 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
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What is Confinement?
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
19
Light quarks & Confinement
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.”
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)Craig Roberts: (2) The Ins and Outs of Mesons
Folklore … Hall-D Conceptual Design Report(5) “The color field lines between a quark and an anti-quark form flux tubes.
20
Light quarks & Confinement
Problem: 16 tonnes of force makes a lot of pions.
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)Craig Roberts: (2) The Ins and Outs of Mesons
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Light quarks & Confinement
Problem: 16 tonnes of force makes a lot of pions.
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)Craig Roberts: (2) The Ins and Outs of Mesons
22
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 light-quarks.
Flux-tube is not the correct paradigm for confinement in hadron physics
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)Craig Roberts: (2) The Ins and Outs of Mesons
G. Bali et al., PoS LAT2005 (2006) 308
Confinement QFT Paradigm: – 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 dressed-propagator for a coloured state
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)Craig Roberts: (2) The Ins and Outs of Mesons
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Normal particleConfined particle
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)
Propagation described by rapidly damped wave & hence state cannot exist in observable spectrum
Craig Roberts: (2) The Ins and Outs of Mesons
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Plane wave propagation 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
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
meson
meson
meson
meson
Baryon
Craig Roberts: (2) The Ins and Outs of Mesons
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Quark Fragmentation 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
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
meson
meson
meson
meson
Baryon
σConfinement is a dynamical phenomenon!
Craig Roberts: (2) The Ins and Outs of Mesons
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Quark Gap EquationHadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Dynamical Chiral Symmetry Breaking
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
Craig Roberts: (2) The Ins and Outs of Mesons
DCSB is a fact in QCD– Dynamical, not spontaneous
• Add nothing to QCD , No Higgs field, nothing! Effect achieved purely throughquark+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.
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Just one of the terms that are summed in a solution of the simplest, reasonable truncation of QCD’s gap equation
Where does the mass come from?
Deceptively simply picture 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 tools are just inadequate
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)Craig Roberts: (2) The Ins and Outs of Mesons
αS23
29
Persistent Challenge
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
Craig Roberts: (2) The Ins and Outs of Mesons
Truncation
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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
– Weak coupling expansion ⇒ produces every diagram in perturbation theory
– Otherwise useless for the nonperturbative problems in which we’re interested
Persistent challenge in application of DSEs
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)Craig Roberts: (2) The Ins and Outs of Mesons
Invaluable check on practical truncation schemes
31
Ward-Green-
Takahashi Identities
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)Craig Roberts: (2) The Ins and Outs of Mesons
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Persistent challenge- 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
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)Craig Roberts: (2) The Ins and Outs of Mesons
Axial-Vector vertex Satisfies an inhomogeneous Bethe-Salpeter equation
Quark propagator satisfies a gap equation
Kernels of these equations are completely differentBut they must be intimately related
Relationship must be preserved by any truncationHighly nontrivial constraintFAILURE has an extremely high cost
– loss of any connection with QCD
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Transverse Ward-Green-
Takahashi IdentitiesHadron Physics XIII: 22-27 Mar. 2015 (85pp)
Craig Roberts: (2) The Ins and Outs of Mesons
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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)
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)Craig Roberts: (2) The Ins and Outs of Mesons
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
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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 no idea what to do
Equations were truncated,sometimes with goodphenomenological results,sometimes with poor results
Neither good nor badcould be explained
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)Craig Roberts: (2) The Ins and Outs of Mesons
quark-antiquark scattering kernel
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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 Schwinger-Dyson 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
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)Craig Roberts: (2) The Ins and Outs of Mesons
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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
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)Craig Roberts: (2) The Ins and Outs of Mesons
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.
Leading-order:rainbow- ladder truncation
dressed propagators
bare vertices
Modified skeleton expansion in which the propagators are fully-dressed but the vertices are constructed term-by-term
gap eq.
BS kernel
In gap eq., add1-loop vertex correction
Then BS kernel has3 new terms at this order
Bethe-Salpeter EquationBound-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
38Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
Craig Roberts: (2) The Ins and Outs of Mesons
Bethe-Salpeter EquationGeneral Form
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)
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Lei Chang and C.D. Roberts0903.5461 [nucl-th]Phys. Rev. Lett. 103 (2009) 081601
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
Craig Roberts: (2) The Ins and Outs of Mesons
Ward-Takahashi IdentityBethe-Salpeter Kernel
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
40
Lei Chang and C.D. Roberts0903.5461 [nucl-th]Phys. Rev. Lett. 103 (2009) 081601
iγ5 iγ5
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
Craig Roberts: (2) The Ins and Outs of Mesons
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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 ghost-loop contributions are separately transverse
STIs → WGTIs Enables systematic analysis and evaluation of truncations and
straightforward comparison of results with those of lQCD
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
Non-perturbative comparison of QCD effective charges, A.C. Aguilar, D. Binosi, J. Papavassiliou and J. Rodriguez-Quintero, Phys. Rev. D80 (2009) 085018
Craig Roberts: (2) The Ins and Outs of Mesons
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RGI Running Interaction 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
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
(kd ̂ 2)= α(ζ) (kΔ̂ 2; ζ)
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
43
In QCD: Gluons alsobecome massive! Running gluon mass
Gluons are cannibals – a particle species whose members become massive by eating each other!
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)Craig Roberts: (2) The Ins and Outs of Mesons
22
422 )(
kkm
g
gg
Power-law suppressed in ultraviolet, so invisible in perturbation theory
Gluon mass-squared function
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]
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
Craig Roberts: (2) The Ins and Outs of Mesons
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Massive Gauge Bosons! 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?Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
Craig Roberts: (2) The Ins and Outs of Mesons
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Top down & Bottom up
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
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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 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 matter-sector ANL-PKU DSE truncation
“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 ground-state properties without reference to gauge-sector studies. Modern top-down and bottom-up results agree within 3% !
Top-down result = gauge-sector prediction
Craig Roberts: (2) The Ins and Outs of Mesons
46
Top down & Bottom up
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
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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 result = gauge-sector prediction
Modern kernels and interaction, developed at ANL and Peking U.
One parameter, fitted to ground-state properties without reference to gauge-sector studies. Modern top-down and bottom-up results agree within 3% !
– Interaction predicted by modern analyses of QCD's gauge sector coincides with that required to describe ground-state observables using the sophisticated matter-sector ANL-PKU DSE truncation
Craig Roberts: (2) The Ins and Outs of Mesons
47
Top down & Bottom up
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
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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 result = gauge-sector prediction
Modern kernels and interaction, developed at ANL and Peking U.
One parameter, fitted to ground-state properties without reference to gauge-sector studies. Modern top-down and bottom-up results agree within 3% !
– Interaction predicted by modern analyses of QCD's gauge sector coincides with that required to describe ground-state observables using the sophisticated matter-sector ANL-PKU DSE truncation
Significant steps towardparameter-free prediction
of hadron properties
48
Enigma of MassHadron Physics XIII: 22-27 Mar. 2015 (85pp)
Craig Roberts: (2) The Ins and Outs of Mesons
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Symmetry preserving analyses in continuum
QCDHadron Physics XIII: 22-27 Mar. 2015 (85pp)Craig Roberts: (2) The Ins and Outs of Mesons
Pion’s Goldberger-Treiman relation
Craig Roberts: (2) The Ins and Outs of Mesons
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Pion’s Bethe-Salpeter amplitudeSolution of the Bethe-Salpeter equation
Dressed-quark propagator
Axial-vector Ward-Takahashi identity entails
Owing to DCSB& Exact inChiral QCD
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
Miracle: two body problem solved, almost completely, once solution of one body problem is known
Maris, Roberts and Tandynucl-th/9707003, Phys.Lett. B420 (1998) 267-273
B(k2)
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The most fundamental expression of
Goldstone’s Theorem and DCSB
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)Craig Roberts: (2) The Ins and Outs of Mesons
fπ Eπ(p2) = B(p2)
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Enigma of mass
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.
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)Craig Roberts: (2) The Ins and Outs of Mesons
fπ Eπ(p2) = B(p2)This identity is why the pion is massless in the chiral limit
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Dynamical Chiral Symmetry Breaking
Vacuum Condensates?Hadron Physics XIII: 22-27 Mar. 2015 (85pp)Craig Roberts: (2) The Ins and Outs of Mesons
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 non-perturbative vacuum state, characterized by many non-vanishing condensates such as the gluon condensate or the quark condensate. These condensates characterize the normal phase or the confined phase of quark matter.”
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)Craig Roberts: (2) The Ins and Outs of Mesons
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55
“Orthodox Vacuum” Vacuum = “frothing sea” Hadrons = bubbles in that “sea”,
containing nothing but quarks & gluonsinteracting perturbatively, unless they’re near the bubble’s boundary, whereat they feel they’re trapped!
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)Craig Roberts: (2) The Ins and Outs of Mesons
u
u
ud
u ud
du
Craig Roberts: (2) The Ins and Outs of Mesons
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However, just like gluons and quarks, and for the same reasons:Condensates are confined within hadrons. There are no vacuum condensates.
Historically, DCSB has come to be associated with the presumed existence of spacetime-independent condensates that permeate the Universe.
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Confinement contains
condensatesHadron Physics XIII: 22-27 Mar. 2015 (85pp)
Craig Roberts: (2) The Ins and Outs of Mesons
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GMOR Relation
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)Craig Roberts: (2) The Ins and Outs of Mesons
Craig Roberts: (2) The Ins and Outs of Mesons
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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 condensateHadron Physics XIII: 22-27 Mar. 2015 (85pp)
Expanding the concept of in-hadron condensatesLei Chang, Craig D. Roberts and Peter C. TandyarXiv:1109.2903 [nucl-th], Phys. Rev. C85 (2012) 012201(R)
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GMOR is synonymous with “Vacuum Quark
Condensate”Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
Craig Roberts: (2) The Ins and Outs of Mesons
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Concept of in-hadron
condensatesHadron Physics XIII: 22-27 Mar. 2015 (85pp)
Craig Roberts: (2) The Ins and Outs of Mesons
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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
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)Craig Roberts: (2) The Ins and Outs of Mesons
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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. …
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)Craig Roberts: (2) The Ins and Outs of Mesons
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“Orthodox Vacuum” Vacuum = “frothing sea” Hadrons = bubbles in that “sea”,
containing nothing but quarks & gluonsinteracting perturbatively, unless they’re near the bubble’s boundary, whereat they feel they’re trapped!
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)Craig Roberts: (2) The Ins and Outs of Mesons
u
u
ud
u ud
du
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New Paradigm Vacuum = perturbative hadronic fluctuations
but no nonperturbative condensates Hadrons = complex, interacting systems
within which perturbative behaviour is restricted to just 2% of the interior
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)Craig Roberts: (2) The Ins and Outs of Mesons
u
u
ud
u ud
du
“EMPTY space may really be empty. Though quantum theory suggests that a vacuum should be fizzing with particle activity, it turns out that this paradoxical picture of nothingness may not be needed. A calmer view of the vacuum would also help resolve a nagging inconsistency with dark energy, the elusive force thought to be speeding up the expansion of the universe.”
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)Craig Roberts: (2) The Ins and Outs of Mesons
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“Void that is truly empty solves dark energy puzzle”Rachel Courtland, New Scientist 4th Sept. 2010
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
experiment*103
8 4620
4
HG QCDNscondensateQCD
Paradigm shift:In-Hadron Condensates
“The biggest embarrassment in
theoretical physics.”
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Parton structure of
hadronsHadron Physics XIII: 22-27 Mar. 2015 (85pp)
Craig Roberts: (2) The Ins and Outs of Mesons
Valence quarks
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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 Poincaré-
invariant 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
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)Craig Roberts: (2) The Ins and Outs of Mesons
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Why light-front?Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
Craig Roberts: (2) The Ins and Outs of Mesons
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Light-front Quantisation Hamiltonian formulation of quantum field theory.
– Fields are specified on a particular initial surface: Light front x+ = x0 + x3 = 0
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)Craig Roberts: (2) The Ins and Outs of Mesons
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
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Pion’s Wave FunctionHadron Physics XIII: 22-27 Mar. 2015 (85pp)
Craig Roberts: (2) The Ins and Outs of Mesons
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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≈0.5Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
Craig Roberts: (2) The Ins and Outs of Mesons
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].
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Pion’s valence-quark Distribution Amplitude
Both kernels agree: marked broadening of φπ(x), which owes to DCSB
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)Craig Roberts: (2) The Ins and Outs of Mesons
Asymptotic
RL
DB
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
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].
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Pion’s valence-quark Distribution Amplitude
Both kernels agree: marked broadening of φπ(x), which owes to DCSB
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)Craig Roberts: (2) The Ins and Outs of Mesons
Asymptotic
RL
DB
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
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].
These computations are the first and only to directly expose DCSB – pointwise – on the light-front; i.e., in the infinite momentum frame.
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Features of Ground-state PDAs
A diverse array of studies since Caraguatatuba (2012) have shown that ground-state meson PDAs are broad, concave functions
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
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
Concave function: no line segment lies above any point on the graph
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]
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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 PDAo fP is the meson's leptonic decay constant
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
It was promised that JLab would verify this fundamental prediction
Craig Roberts: (2) The Ins and Outs of Mesons
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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.
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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 quark-antiquark pair” disappointment and surprise
Result imagined by many to be QCD prediction
JLab Data
Evaluated with φπ = 6x(1-x)
lQCD only provides access to this small domain, already well-mapped by experiments
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Pion electromagnetic form factor
Year 2000 prediction for Fπ(Q2) – P.Maris & P.C. Tandy,
Phys.Rev. C62 (2000) 055204
Problem … used brute-force computational method … unable to compute for Q2>4GeV2
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
Result imagined by many to be QCD prediction
JLab Data
Shape of prediction suggested to many that one might never see parton model scaling and QCD scaling violations
Factor of three discrepancy
Evaluated with φπ = 6x(1-x)
Craig Roberts: (2) The Ins and Outs of Mesons
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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.
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
Projected JLab reach
Will there be any hint of a trend toward the asymptotic pQCD prediction?
Result imagined by many to be QCD prediction
Evaluated with φπ = 6x(1-x)
Craig Roberts: (2) The Ins and Outs of Mesons
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New AlgorithmNew Insights
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
Craig Roberts: (2) The Ins and Outs of Mesons
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Result imagined by many to be QCD prediction
Evaluated with φπ = 6x(1-x)
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
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
Real QCD prediction – obtained with realistic, computed PDA
Craig Roberts: (2) The Ins and Outs of Mesons
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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
Solution – Part 2– Algorithm used to
compute the PDA can also be employed to compute Fπ(Q2) directly, to arbitrarily large Q2
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
Real QCD prediction – obtained with realistic, computed PDA
Predictions: JLab will see maximum Experiments to 8GeV2 will see
parton model scaling and QCD scaling violations for the first time in a hadron form factor
Pion electromagnetic form factor at spacelike momentaL. Chang, I. C. Cloët, C. D. Roberts, S. M. Schmidt and P. C. Tandy, arXiv:1307.0026 [nucl-th], Phys. Rev. Lett. 111, 141802 (2013)
maximum
Agreement within 15%
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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 φπ
asy(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
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
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Hadron Physics XIII: 22-27 Mar. 2015 (85pp)Craig Roberts: (2) The Ins and Outs of Mesons
Epilogue
Craig Roberts: (2) The Ins and Outs of Mesons
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Epilogue 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 light-front structure of other light-quark hadrons
Near-term Aim– Unified QCD-based and quantum mechanical picture of the light-
quark meson sector of the Standard Model Developments in baryon sector
are equally dramatic: e.g.,
Hadron Physics XIII: 22-27 Mar. 2015 (85pp)
is understood
I.C. Cloët, C.D. Roberts, A.W. Thomas:arXiv:1304.0855 [nucl-th], Phys. Rev. Lett. 111 (2013) 101803