Download a 1 - SMU Physics

Published collaborations in 2010/2011
Craig Roberts
Physics Division
Adnan BASHIR (U Michoacan);
Students
Stan BRODSKY (SLAC);
Early-career
scientists
Lei CHANG (ANL & PKU);
Huan CHEN (BIHEP);
Ian CLOËT (UW);
Bruno EL-BENNICH (Sao Paulo);
Xiomara GUTIERREZ-GUERRERO (U Michoacan);
Roy HOLT (ANL);
Mikhail IVANOV (Dubna);
Yu-xin LIU (PKU);
Trang NGUYEN (KSU);
Si-xue QIN (PKU);
Hannes ROBERTS (ANL, FZJ, UBerkeley);
Robert SHROCK (Stony Brook);
Peter TANDY (KSU);
David WILSON (ANL)
QCD’s Challenges
Understand emergent phenomena
 Quark and Gluon Confinement
No matter how hard one strikes the proton,
one cannot liberate an individual quark or gluon
 Dynamical Chiral Symmetry Breaking
Very unnatural pattern of bound state masses;
e.g., Lagrangian (pQCD) quark mass is small but
. . . no degeneracy between JP=+ and JP=− (parity partners)
 Neither of these phenomena is apparent in QCD’s Lagrangian
Yet they are the dominant determining characteristics of
real-world QCD.
 QCD
– Complex behaviour arises from apparently simple rules.
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
LightCone 2011, SMU 23-27 May - 77pgs
2
Universal
Truths
 Hadron spectrum, and elastic and transition form factors provide
unique information about long-range interaction between lightquarks and distribution of hadron's characterising properties amongst
its QCD constituents.
 Dynamical Chiral Symmetry Breaking (DCSB) is most important mass
generating mechanism for visible matter in the Universe.
Higgs mechanism is (almost) irrelevant to light-quarks.
 Running of quark mass entails that calculations at even modest Q2
require a Poincaré-covariant approach.
Covariance + M(p2) require existence of quark orbital angular
momentum in hadron's rest-frame wave function.
 Confinement is expressed through a violent change of the
propagators for coloured particles & can almost be read from a plot
of a states’ dressed-propagator.
It is intimately connected with DCSB.
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
LightCone 2011, SMU 23-27 May - 77pgs
3
Strong-interaction: QCD
 Asymptotically free
– Perturbation theory is valid and
accurate tool at large-Q2
– Hence chiral limit is defined
 Essentially nonperturbative
for Q2 < 2 GeV2
 Nature’s only example of truly nonperturbative,
fundamental theory
 A-priori, no idea as to what such a theory
can produce
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
LightCone 2011, SMU 23-27 May - 77pgs
4
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
LightCone 2011, SMU 23-27 May - 77pgs
5
X
Confinement
 Quark and Gluon Confinement
– No matter how hard one strikes the proton, or any other
hadron, one cannot liberate an individual quark or gluon
 Empirical fact. However
– There is no agreed, theoretical definition of light-quark
confinement
– Static-quark confinement is irrelevant to real-world QCD
• There are no long-lived, very-massive quarks
 Confinement entails quark-hadron duality; i.e., that all
observable consequences of QCD can, in principle, be
computed using an hadronic basis.
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
LightCone 2011, SMU 23-27 May - 77pgs
6
G. Bali et al., PoS LAT2005 (2006) 308
“Note that the time is not a linear function of the distance
but dilated within the string breaking region. On a linear time
scale string breaking takes place rather rapidly. […] light pair
creation seems to occur non-localized and instantaneously.”
Confinement
 Infinitely heavy-quarks plus 2 flavours with mass = ms
– Lattice spacing = 0.083fm
– String collapses
within one lattice time-step
R = 1.24 … 1.32 fm
– Energy stored in string at
collapse Ecsb = 2 ms
– (mpg made via
linear interpolation)
 No flux tube between
light-quarks
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
Bs
anti-Bs
LightCone 2011, SMU 23-27 May - 77pgs
7
Confinement
 Confinement is expressed through a violent change in
the analytic structure of propagators for coloured
particles & can almost be read from a plot of a states’
dressed-propagator
– Gribov (1978); Munczek (1983); Stingl (1984); Cahill (1989);
Krein, Roberts & Williams (1992); Tandy (1994); …
Confined particle
Normal particle
complex-P2
complex-P2
timelike axis: P2<0
o Real-axis mass-pole splits, moving into pair(s) of complex conjugate poles or branch points
o Spectral density no longer positive semidefinite
& hence state cannot exist in observable spectrum
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
LightCone 2011, SMU 23-27 May - 77pgs
8
Dressed-gluon propagator
A.C. Aguilar et al., Phys.Rev. D80 (2009) 085018
 Gluon propagator satisfies
a Dyson-Schwinger Equation
 Plausible possibilities
for the solution
 DSE and lattice-QCD
agree on the result
– Confined gluon
– IR-massive but UV-massless
– mG ≈ 2-4 ΛQCD
IR-massive but UV-massless, confined gluon
perturbative, massless gluon
massive , unconfined gluon
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
LightCone 2011, SMU 23-27 May - 77pgs
9
Charting the interaction
between light-quarks
This is a well-posed problem whose solution is
an elemental goal of modern hadron physics.
The answer provides QCD’s running coupling.
 Confinement can be related to the analytic properties of
QCD's Schwinger functions.
 Question of light-quark confinement can be translated into
the challenge of charting the infrared behavior
of QCD's universal β-function
– This function may depend on the scheme chosen to renormalise
the quantum field theory but it is unique within a given scheme.
– Of course, the behaviour of the β-function on the
perturbative domain is well known.
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
LightCone 2011, SMU 23-27 May - 77pgs
10
Charting the interaction
between light-quarks
 Through QCD's Dyson-Schwinger equations (DSEs) the pointwise
behaviour of the β-function determines the pattern of chiral
symmetry breaking.
 DSEs connect β-function to experimental observables. Hence,
comparison between computations and observations of
o Hadron mass spectrum
o Elastic and transition form factors
can be used to chart β-function’s long-range behaviour.
 Extant studies show that the properties of hadron excited states
are a great deal more sensitive to the long-range behaviour of the
β-function than those of the ground states.
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
LightCone 2011, SMU 23-27 May - 77pgs
11
Maris & Tandy, Phys.Rev. C60 (1999) 055214
DSE Studies
– Phenomenology of gluon
 Wide-ranging study of π & ρ properties
 Effective coupling
– Agrees with pQCD in ultraviolet
– Saturates in infrared
• α(0)/π = 3.2
• α(1 GeV2)/π = 0.35
 Running gluon mass
– Gluon is massless in ultraviolet
in agreement with pQCD
– Massive in infrared
• mG(0) = 0.76 GeV
• mG(1 GeV2) = 0.46 GeV
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
LightCone 2011, SMU 23-27 May - 77pgs
12
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
LightCone 2011, SMU 23-27 May - 77pgs
13
Dynamical Chiral Symmetry Breaking
 Strong-interaction: QCD
 Confinement
– Empirical feature
– Modern theory and lattice-QCD support conjecture
• that light-quark confinement is a fact
• associated with violation of reflection positivity; i.e., novel analytic
structure for propagators and vertices
– Still circumstantial, no proof yet of confinement
 On the other hand, DCSB is a fact in QCD
– It is the most important mass generating mechanism for visible
matter in the Universe.
Responsible for approximately 98% of the proton’s mass.
Higgs mechanism is (almost) irrelevant to light-quarks.
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
LightCone 2011, SMU 23-27 May - 77pgs
14
Frontiers of Nuclear Science:
Theoretical Advances
In QCD a quark's effective mass
depends on its momentum. The
function describing this can be
calculated and is depicted here.
Numerical simulations of lattice
QCD (data, at two different bare
masses) have confirmed model
predictions (solid curves) that the
vast bulk of the constituent mass
of a light quark comes from a
cloud of gluons that are dragged
along by the quark as it
propagates. In this way, a quark
that appears to be absolutely
massless at high energies (m =0,
red curve) acquires a large
constituent mass at low energies.
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
C.D. Roberts, Prog. Part. Nucl. Phys. 61 (2008) 50
M. Bhagwat & P.C. Tandy, AIP Conf.Proc. 842 (2006) 225-227
LightCone 2011, SMU 23-27 May - 77pgs
15
Frontiers of Nuclear Science:
Theoretical Advances
In QCD a quark's effective mass
depends on its momentum. The
function describing this can be
calculated and is depicted here.
Numerical simulations of lattice
QCD (data, at two different bare
masses) have confirmed model
predictions (solid curves) that the
vast bulk of the constituent mass
of a light quark comes from a
cloud of gluons that are dragged
along by the quark as it
propagates. In this way, a quark
that appears to be absolutely
massless at high energies (m =0,
red curve) acquires a large
constituent mass at low energies.
Mass from nothing!
DSE prediction of DCSB confirmed
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
LightCone 2011, SMU 23-27 May - 77pgs
16
Frontiers of Nuclear Science:
Theoretical Advances
In QCD a quark's effective mass
depends on its momentum. The
function describing this can be
calculated and is depicted here.
Numerical simulations of lattice
QCD (data, at two different bare
masses) have confirmed model
predictions (solid curves) that the
vast bulk of the constituent mass
of a light quark comes from a
cloud of gluons that are dragged
along by the quark as it
propagates. In this way, a quark
that appears to be absolutely
massless at high energies (m =0,
red curve) acquires a large
constituent mass at low energies.
Hint of lattice-QCD support
for DSE prediction of violation of reflection positivity
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
LightCone 2011, SMU 23-27 May - 77pgs
17
12GeV
The Future of JLab
Numerical simulations of lattice
QCD (data, at two different bare
masses) have confirmed model
predictions (solid curves) that the
vast bulk of the constituent mass
of a light quark comes from a
cloud of gluons that are dragged
along by the quark as it
propagates. In this way, a quark
that appears to be absolutely
massless at high energies (m =0,
red curve) acquires a large
constituent mass at low energies.
Jlab 12GeV: Scanned by 2<Q2<9 GeV2
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
elastic & transition form factors.
LightCone 2011, SMU 23-27 May - 77pgs
18
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
LightCone 2011, SMU 23-27 May - 77pgs
19
Dichotomy of the pion
 How does one make an almost massless particle from two
massive constituent-quarks?
 Naturally, one could always tune a potential in quantum
mechanics so that the ground-state is massless
– but some are still making this mistake
 However:
current-algebra (1968)
m  m
2
 This is impossible in quantum mechanics, for which one
always finds: mbound state  mconstituent
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
LightCone 2011, SMU 23-27 May - 77pgs
20
Gell-Mann – Oakes – Renner
Behavior of current divergences under SU(3) x SU(3).
Relation
Murray Gell-Mann, R.J. Oakes , B. Renner
Phys.Rev. 175 (1968) 2195-2199
 This paper derives a relation between
mπ2 and the expectation-value < π|u0|π>,
where uo is an operator that is linear in the putative
Hamiltonian’s explicit chiral-symmetry breaking term
 NB. QCD’s current-quarks were not yet invented, so u0 was not
expressed in terms of current-quark fields
 PCAC-hypothesis (partial conservation of axial current) is used in
the derivation
 Subsequently, the concepts of soft-pion theory
 Operator expectation values do not change as t=mπ2 → t=0
to take < π|u0|π> → < 0|u0|0> … in-pion → in-vacuum
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
LightCone 2011, SMU 23-27 May - 77pgs
21
Gell-Mann – Oakes – Renner
Behavior of current divergences under SU(3) x SU(3).
Relation
Murray Gell-Mann, R.J. Oakes , B. Renner
Phys.Rev. 175 (1968) 2195-2199
 PCAC hypothesis; viz., pion field dominates the divergence of
Zhou Guangzhao 周光召
the axial-vector current
Born 1929 Changsha, Hunan province
 Soft-pion theorem
Commutator is chiral rotation
Therefore, isolates explicit
chiral-symmetry breaking term
in the putative Hamiltonian
 In QCD,
this is
m qq
and one therefore has
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
LightCone 2011, SMU 23-27 May - 77pgs
22
Gell-Mann – Oakes – Renner
Relation
 Theoretical physics at its best.
- (0.25GeV)3
 But no one is thinking about how properly to consider or
define what will come to be called the
vacuum quark condensate
 So long as the condensate is just a
mass-dimensioned constant,
which approximates another
well-defined transition matrix element,
there is no problem.
 Problem arises if one
over-interprets this number,
which textbooks have been doing
for a VERY LONG TIME.
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
LightCone 2011, SMU 23-27 May - 77pgs
23
Note of Warning
Chiral Magnetism (or Magnetohadrochironics)
A. Casher and L. Susskind, Phys. Rev. D9 (1974) 436
 These authors argue
that dynamical chiralsymmetry breaking
can be realised as a
property of hadrons,
instead of via a
nontrivial vacuum exterior to
the measurable degrees of
freedom
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
The essential ingredient required for a
spontaneous symmetry breakdown in a
composite system is the existence of a
divergent number of constituents
– DIS provided evidence for divergent sea of
low-momentum partons – parton model.
LightCone 2011, SMU 23-27 May - 77pgs
24
QCD Sum Rules
QCD and Resonance Physics. Sum Rules.
M.A. Shifman, A.I. Vainshtein, and V.I. Zakharov
Nucl.Phys. B147 (1979) 385-447; citations: 3713
 Introduction of the gluon vacuum condensate
and development of “sum rules” relating properties of
low-lying hadronic states to vacuum condensates
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
LightCone 2011, SMU 23-27 May - 77pgs
25
QCD Sum Rules
QCD and Resonance Physics. Sum Rules.
M.A. Shifman, A.I. Vainshtein, and V.I. Zakharov
Nucl.Phys. B147 (1979) 385-447; citations: 3781
 Introduction of the gluon vacuum condensate
and development of “sum rules” relating properties of
low-lying hadronic states to vacuum condensates
 At this point (1979), the cat was out of the bag: a
physical reality was seriously attributed to a plethora
of vacuum condensates
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
LightCone 2011, SMU 23-27 May - 77pgs
26
“quark condensate”
1960-1980
 Instantons in non-perturbative QCD vacuum,
MA Shifman, AI Vainshtein… - Nuclear Physics B, 1980
 Instanton density in a theory with massless quarks,
MA Shifman, AI Vainshtein… - Nuclear Physics B, 1980
 Exotic new quarks and dynamical symmetry breaking,
WJ Marciano - Physical Review D, 1980
 The pion in QCD
J Finger, JE Mandula… - Physics Letters B, 1980
No references to this phrase before 1980
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
LightCone 2011, SMU 23-27 May - 77pgs
27
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: Dyson-Schwinger Equations and Continuum QCD.
LightCone 2011, SMU 23-27 May - 77pgs
28
Universal
Misapprehensions
 Since 1979, DCSB has commonly been associated literally with a spacetimeindependent mass-dimension-three “vacuum condensate.”
 Under this assumption, “condensates” couple directly to gravity in general
relativity and make an enormous contribution to the cosmological constant
QCDcondensates  8 GN
 Experimentally, the answer is

4
QCD
2
0
3H
 10
46
Ωcosm. const. = 0.76
 This mismatch is a bit of a problem.
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
LightCone 2011, SMU 23-27 May - 77pgs
29
Resolution?
 Quantum Healing Central:
 “KSU physics professor [Peter Tandy] publishes groundbreaking
research on inconsistency in Einstein theory.”
 Paranormal Psychic Forums:
 “Now Stanley Brodsky of the SLAC National
Accelerator Laboratory in Menlo Park,
California, and colleagues have found a way
to get rid of the discrepancy. “People have
just been taking it on faith that this quark
condensate is present throughout the
vacuum,” says Brodsky.
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
LightCone 2011, SMU 23-27 May - 77pgs
30
Paradigm shift:
In-Hadron Condensates
Brodsky, Roberts, Shrock, Tandy, Phys. Rev. C82 (Rapid Comm.) (2010) 022201
Brodsky and Shrock, PNAS 108, 45 (2011)
M. Burkardt, Phys.Rev. D58 (1998) 096015
 Resolution
– Whereas it might sometimes be convenient in computational
truncation schemes to imagine otherwise, “condensates” do not
exist as spacetime-independent mass-scales that fill all spacetime.
– So-called vacuum condensates can be understood as a property of
hadrons themselves, which is expressed, for example,
in their Bethe-Salpeter or
light-front wavefunctions.
– GMOR
cf.
QCD
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
LightCone 2011, SMU 23-27 May - 77pgs
31
Paradigm shift:
In-Hadron Condensates
Brodsky, Roberts, Shrock, Tandy, Phys. Rev. C82 (Rapid Comm.) (2010) 022201
Brodsky and Shrock, PNAS 108, 45 (2011)
 Resolution
– Whereas it might sometimes be convenient in computational
truncation schemes to imagine otherwise, “condensates” do not
exist as spacetime-independent mass-scales that fill all spacetime.
– So-called vacuum condensates can be understood as a property of
hadrons themselves, which is expressed, for example,
in their Bethe-Salpeter or
light-front wavefunctions.
– No qualitative difference
between fπ and ρπ
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
LightCone 2011, SMU 23-27 May - 77pgs
32
Paradigm shift:
In-Hadron Condensates
Brodsky, Roberts, Shrock, Tandy, Phys. Rev. C82 (Rapid Comm.) (2010) 022201
Brodsky and Shrock, PNAS 108, 45 (2011)
 Resolution
– Whereas it might sometimes be convenient in computational
truncation schemes to imagine otherwise, “condensates” do not
exist as spacetime-independent mass-scales that fill all spacetime.
– So-called vacuum condensates can be understood as a property of
hadrons themselves, which is expressed, for example,
in their Bethe-Salpeter or
light-front wavefunctions.
Chiral limit
– No qualitative difference
 (0;  )    qq 0
between fπ and ρπ
– And
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
LightCone 2011, SMU 23-27 May - 77pgs
33
Paradigm shift:
In-Hadron Condensates
“Void that is truly empty
solves dark energy puzzle”
Rachel Courtland, New Scientist 4th Sept. 2010
“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
QCD
46
picture of nothingness
may
not
be
needed.
A
calmer
view
of
the
vacuum
would
QCDcondensates  8 GN
 10
2
also help resolve a nagging inconsistency with dark
energy,
the elusive force
3H
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: Dyson-Schwinger Equations and Continuum QCD.
LightCone 2011, SMU 23-27 May - 77pgs
34
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
LightCone 2011, SMU 23-27 May - 77pgs
35
Strong-interaction: QCD
Dressed-quark-gluon vertex
 Gluons and quarks acquire momentum-dependent masses
– characterised by an infrared mass-scale m ≈ 2-4 ΛQCD
 Significant body of work, stretching back to 1980, which shows
that, in the presence of DCSB, the dressed-fermion-photon vertex is
materially altered from the bare form: γμ.
– Obvious, because with
A(p2) ≠ 1 and B(p2) ≠ constant,
the bare vertex cannot satisfy the Ward-Takahashi identity; viz.,
 Number of contributors is too numerous to list completely (300
citations to 1st J.S. Ball paper), but prominent contributions by:
J.S. Ball, C.J. Burden, C.Roberts, R. Delbourgo, A.G. Williams, H.J. Munczek,
M.R. Pennington, A. Bashir, A. Kizilersu, P.Tandy, L. Chang, Y.-X. Liu …
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
LightCone 2011, SMU 23-27 May - 77pgs
36
Dressedquark-gluon vertex
 Single most important feature
– Perturbative vertex is helicity-conserving:
• Cannot cause spin-flip transitions
– However, DCSB introduces nonperturbatively generated
structures that very strongly break helicity conservation
– These contributions
• Are large when the dressed-quark mass-function is large
– Therefore vanish in the ultraviolet;
i.e., on the perturbative domain
– Exact form of the contributions is still the subject of
debate but their existence is model-independent - a fact.
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
LightCone 2011, SMU 23-27 May - 77pgs
37
Gap Equation
General Form
 Dμν(k) – dressed-gluon propagator
 Γν(q,p) – dressed-quark-gluon vertex
 Until 2009, all studies of other hadron phenomena used
the leading-order term in a symmetry-preserving
truncation scheme; viz.,
Bender, Roberts & von Smekal
– Dμν(k) = dressed, as described previously
– Γν(q,p) = γμ
Phys.Lett. B380 (1996) 7-12
• … plainly, key nonperturbative effects are missed and cannot be
recovered through any step-by-step improvement procedure
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
LightCone 2011, SMU 23-27 May - 77pgs
38
Gap Equation
General Form
 Dμν(k) – dressed-gluon propagator
 good deal of information available
 Γν(q,p) – dressed-quark-gluon vertex
 Information accumulating
If kernels of Bethe-Salpeter
and gap equations don’t match,
one won’t even get
right charge for the pion.
 Suppose one has in hand – from anywhere – the exact
form of the dressed-quark-gluon vertex
What is the associated symmetrypreserving Bethe-Salpeter kernel?!
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
LightCone 2011, SMU 23-27 May - 77pgs
39
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: Dyson-Schwinger Equations and Continuum QCD.
LightCone 2011, SMU 23-27 May - 77pgs
40
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: Dyson-Schwinger Equations and Continuum QCD.
LightCone 2011, SMU 23-27 May - 77pgs
41
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: Dyson-Schwinger Equations and Continuum QCD.
LightCone 2011, SMU 23-27 May - 77pgs
42
Relativistic quantum mechanics
 Dirac equation (1928):
Pointlike, massive fermion interacting with electromagnetic field
Spin Operator
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
LightCone 2011, SMU 23-27 May - 77pgs
43
Massive point-fermion
Anomalous magnetic moment
 Dirac’s prediction held true for the electron until improvements in
experimental techniques enabled the discovery of a small
deviation: H. M. Foley and P. Kusch, Phys. Rev. 73, 412 (1948).
– Moment increased by a multiplicative factor: 1.001 19 ± 0.000 05.
 This correction was explained by the first systematic computation
using renormalized quantum electrodynamics (QED):
J.S. Schwinger, Phys. Rev. 73, 416 (1948),
0.001 16
– vertex correction
 The agreement with
e
experiment established
e
quantum electrodynamics as a valid tool.
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
LightCone 2011, SMU 23-27 May - 77pgs
44
Fermion electromagnetic current
– General structure
with k = pf - pi
 F1(k2) – Dirac form factor; and F2(k2) – Pauli form factor
– Dirac equation:
• F1(k2) = 1
• F2(k2) = 0
– Schwinger:
• F1(k2) = 1
• F2(k2=0) = α /[2 π]
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
LightCone 2011, SMU 23-27 May - 77pgs
45
Magnetic moment
of a massless fermion?
 Plainly, can’t simply take the limit m → 0.
 Standard QED interaction, generated by minimal substitution:
 Magnetic moment is described by interaction term:
– Invariant under local U(1) gauge transformations
– but is not generated by minimal substitution in the action for a free
Dirac field.
 Transformation properties under chiral rotations?
– Ψ(x) → exp(iθγ5) Ψ(x)
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
LightCone 2011, SMU 23-27 May - 77pgs
46
Magnetic moment
of a massless fermion?
 Standard QED interaction, generated by minimal substitution:
– Unchanged under chiral rotation
– Follows that QED without a fermion mass term is helicity conserving
 Magnetic moment interaction is described by interaction term:
– NOT invariant
– picks up a phase-factor exp(2iθγ5)
 Magnetic moment interaction is forbidden in a theory with manifest
chiral symmetry
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
LightCone 2011, SMU 23-27 May - 77pgs
47
Schwinger’s result?
 One-loop calculation:
 Plainly, one obtains Schwinger’s result for me2 ≠ 0
 However,
e
e
F2(k2) = 0 when me2 = 0
 There is no Gordon identity:
m=0
So, no mixing
γμ ↔ σμν
 Results are unchanged at every order in perturbation theory …
owing to symmetry … magnetic moment interaction is forbidden in
a theory with manifest chiral symmetry
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
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QCD and dressed-quark
anomalous magnetic moments
 Schwinger’s result for QED:
 pQCD: two diagrams
o (a) is QED-like
o (b) is only possible in QCD – involves 3-gluon vertex
 Analyse (a) and (b)
o (b) vanishes identically: the 3-gluon vertex does not contribute to
a quark’s anomalous chromomag. moment at leading-order
o (a) Produces a finite result: “ – ⅙ αs/2π ”
~ (– ⅙) QED-result
 But, in QED and QCD, the anomalous chromo- and electromagnetic moments vanish identically in the chiral limit!
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
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L. Chang, Y. –X. Liu and C.D. Roberts
arXiv:1009.3458 [nucl-th]
Phys. Rev. Lett. 106 (2011) 072001
Dressed-quark anomalous
magnetic moments
 DCSB
Three strongly-dressed and essentiallynonperturbative contributions to dressed-quark-gluon vertex:
Ball-Chiu term
•Vanishes if no DCSB
•Appearance driven by STI
Anom. chrom. mag. mom.
contribution to vertex
•Similar properties to BC term
•Strength commensurate with lattice-QCD
Skullerud, Bowman, Kizilersu et al.
hep-ph/0303176
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
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 Lattice-QCD
Dressed-quark anomalous
– m = 115 MeV
chromomagnetic moment
 Nonperturbative result is two orders-of-magnitude larger
than the perturbative computation
– This level of
Quenched
Skullerud, Kizilersu et al.
magnification is
lattice-QCD
JHEP 0304 (2003) 047
typical of DCSB
– cf.
Quark mass function:
M(p2=0)= 400MeV
M(p2=10GeV2)=4 MeV
Prediction from
perturbative QCD
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
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L. Chang, Y. –X. Liu and C.D. Roberts
arXiv:1009.3458 [nucl-th]
Phys. Rev. Lett. 106 (2011) 072001
Dressed-quark anomalous
magnetic moments
 DCSB
Three strongly-dressed and essentiallynonperturbative contributions to dressed-quark-gluon vertex:
Ball-Chiu term
•Vanishes if no DCSB
•Appearance driven by STI
Anom. chrom. mag. mom.
contribution to vertex
•Similar properties to BC term
•Strength commensurate with lattice-QCD
Skullerud, Bowman, Kizilersu et al.
hep-ph/0303176
Role and importance is
novel discovery
•Essential to recover pQCD
•Constructive interference with Γ5
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
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L. Chang, Y. –X. Liu and C.D. Roberts
arXiv:1009.3458 [nucl-th]
Phys. Rev. Lett. 106 (2011) 072001
Dressed-quark anomalous
magnetic moments
Formulated and solved general
Bethe-Salpeter equation
Obtained dressed
electromagnetic vertex
Confined quarks
don’t have a mass-shell
Factor of 10
magnification
o Can’t unambiguously define
magnetic moments
o But can define
magnetic moment distribution
 AEM
is opposite in sign but of
roughly equal magnitude
as ACM
Full vertex
ME
κACM
κAEM
0.44
-0.22
0.45
0
0.048
Rainbow-ladder 0.35
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
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L. Chang, Y. –X. Liu and C.D. Roberts
arXiv:1009.3458 [nucl-th]
Phys. Rev. Lett. 106 (2011) 072001
Dressed-quark anomalous
magnetic moments
Formulated and solved general
Bethe-Salpeter equation
Obtained dressed
electromagnetic vertex
Confined quarks
don’t have a mass-shell
Factor of 10
magnification
o Can’t unambiguously define
magnetic moments
o But can define
magnetic moment distribution
Contemporary theoretical estimates:
1 – 10 x 10-10
Largest value reduces discrepancy
expt.↔theory from 3.3σ to below 2σ.
 Potentially important for elastic and transition form factors, etc.
 Significantly, also quite possibly for muon g-2 – via Box diagram,
which is not constrained by extant data. (I.C. Cloët et al.)
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
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Location of zero
marks –m2meson
Dressed Vertex
& Meson Spectrum
Experiment Rainbow- One-loop
ladder
corrected
a1
1230
759
885
ρ
770
644
764
Mass splitting
455
115
121
Ball-Chiu
Full vertex
 Splitting known experimentally for more than 35 years
 Hitherto, no explanation
 Systematic symmetry-preserving, Poincaré-covariant DSE
truncation scheme of nucl-th/9602012.
o Never better than ∼ ⅟₄ of splitting
 Constructing kernel skeleton-diagram-by-diagram,
DCSB cannot be faithfully expressed:
Full impact of M(p2)
cannot be realised!
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
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BC: zero
moves deeper
Full
vertex:
for both
ρ &deeper
a1
zero
moves
for a1
Bothshallower
masses grow
but
for ρ
Problem solved
Dressed Vertex
& Meson Spectrum
Experiment Rainbow- One-loop
ladder
corrected
Ball-Chiu
Full vertex
a1
1230
759
885
1066
1128
1270
ρ
770
644
764
924
919
790
Mass splitting
455
115
121
142
209
480
 Fully consistent treatment of Ball-Chiu vertex
o Retain λ3 – term but ignore Γ4 & Γ5
o Some effects of DCSB built into vertex & Bethe-Salpeter kernel
 Big impact on σ – π complex
 But, clearly, not the complete answer.
 Fully-consistent treatment of complete vertex Ansatz
Lei Chang & C.D. Roberts,
 Promise of 1st reliable prediction
arXiv:1104.4821 [nucl-th]
Tracing massess of ground-state
of light-quark hadron spectrum
light-quark mesons
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
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Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
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Unification of
Meson & Baryon Properties
 Correlate the masses of meson and baryon ground- and excited-states
within a single, symmetry-preserving framework
 Symmetry-preserving means:
Poincaré-covariant & satisfy relevant Ward-Takahashi identities
 Constituent-quark model has hitherto been the most widely applied
spectroscopic tool; whilst its weaknesses are emphasized by critics and
acknowledged by proponents, it is of continuing value because there is
nothing better that is yet providing a bigger picture.
 Nevertheless,
 no connection with quantum field theory & therefore not with QCD
 not symmetry-preserving & therefore cannot veraciously connect
meson and baryon properties
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
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R.T. Cahill et al.,
Austral. J. Phys. 42 (1989) 129-145
Baryons
 Dynamical chiral symmetry breaking (DCSB)
– has enormous impact on meson properties.
 Must be included in description and prediction of baryon
properties.
 DCSB is essentially a quantum field theoretical effect.
In quantum field theory
 Meson appears as pole in four-point quark-antiquark Green function
→ Bethe-Salpeter Equation
 Nucleon appears as a pole in a six-point quark Green function
→ Faddeev Equation.
 Poincaré covariant Faddeev equation sums all possible exchanges
and interactions that can take place between three dressed-quarks
 Tractable equation is based on observation that an interaction
which describes colour-singlet mesons also generates nonpointlike
quark-quark (diquark) correlations in the colour-antitriplet channel
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
rqq ≈ rπ
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R.T. Cahill et al.,
Austral. J. Phys. 42 (1989) 129-145
Faddeev Equation
quark exchange
ensures Pauli statistics
quark
 Linear, Homogeneous Matrix equation
diquark composed of stronglydressed quarks bound
by dressed-gluons
 Yields wave function (Poincaré Covariant Faddeev Amplitude)
that describes quark-diquark relative motion within the nucleon
 Scalar and Axial-Vector Diquarks . . .
 Both have “correct” parity and “right” masses
 In Nucleon’s Rest Frame Amplitude has
s−, p− & d−wave correlations
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
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Diquarks in QCD
Masses of ground and excited-state hadrons
Hannes L.L. Roberts, Lei Chang, Ian C. Cloët
and Craig D. Roberts, arXiv:1101.4244 [nucl-th]
Few Body Systems (2011) pp. 1-25
 “Spectrum” of nonpointlike
quark-quark correlations
 Observed in
H.L.L. Roberts et al., 1102.4376 [nucl-th]
Phys. Rev. C in press
– DSE studies in QCD
– 0+ & 1+ in Lattice-QCD
 Scalar diquark form factor
– r0+ ≈ rπ
 Axial-vector diquarks
– r1+ ≈ rρ
BOTH
essential
CraigZero
relation with old notion of pointlike constituent-like diquarks
Roberts: Dyson-Schwinger Equations and Continuum QCD.
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Masses of ground and excited-state hadrons
Hannes L.L. Roberts, Lei Chang, Ian C. Cloët
and Craig D. Roberts, arXiv:1101.4244 [nucl-th]
Few Body Systems (2011) pp. 1-25
Baryons & diquarks
 Provided numerous insights into baryon structure; e.g.,
 There is a causal connection between mΔ - mN & m1+- m0+
mΔ
mN
Physical splitting grows
rapidly with increasing
diquark mass difference
mΔ - mN
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
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 Provided numerous insights
into baryon structure; e.g.,
 mN ≈ 3 M & mΔ ≈ M+m1+
Baryons & diquarks
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
LightCone 2011, SMU 23-27 May - 77pgs
63
Vertex contains dressed-quark
anomalous magnetic moment
Photon-nucleon current
 Composite nucleon
must interact with
photon via nontrivial
current constrained by
Ward-Takahashi
identities
 DSE, BSE, Faddeev
equation, current →
nucleon form factors
Oettel, Pichowsky, Smekal
Eur.Phys.J. A8 (2000) 251-281
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
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Survey of nucleon electromagnetic form factors
I.C. Cloët et al, arXiv:0812.0416 [nucl-th],
Few Body Syst. 46 (2009) pp. 1-36
Nucleon Form Factors
Unification of meson
and nucleon form
factors.
Very good
description.
Quark’s momentumdependent
anomalous magnetic
moment has
observable impact &
materially improves
agreement in all
cases.
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
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I.C. Cloët, C.D. Roberts, et al.
arXiv:0812.0416 [nucl-th]
I.C. Cloët, C.D. Roberts, et al.
In progress
 pG (Q )
p
E
p
M
2
2
G (Q )
DSE result Dec 08
DSE result
– including the
anomalous
magnetic
moment distribution
Highlights again the
critical importance of
DCSB in explanation of
real-world observables.
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
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I.C. Cloët, C.D. Roberts, et al.
In progress
Quark anomalous magnetic moment has
big impact on proton ratio
nG (Q )
n
E
n
M
2
2
G (Q )
But little impact on the
neutron ratio
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
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I.C. Cloët, C.D. Roberts, et al.
arXiv:0812.0416 [nucl-th]
nG (Q )
n
E
n
M
2
2
G (Q )
 New JLab data:
S. Riordan et al.,
arXiv:1008.1738 [nucl-ex]
Phys.Rev.Lett. 105 (2010) 262302
 DSE-prediction
 This evolution is
very sensitive to
momentum-dependence
of dressed-quark
propagator
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
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Legend:
Particle Data Group
H.L.L. Roberts et al.
EBAC
Jülich
Hadron Spectrum
o Symmetry-preserving unification
of the computation of meson & baryon masses
o rms-rel.err./deg-of-freedom = 13%
o PDG values (almost) uniformly overestimated in both cases
- room for the pseudoscalar meson cloud?!
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
Masses of ground and excited-state hadrons
H.L.L. Roberts et al., arXiv:1101.4244 [nucl-th]
Few Body Systems (2011) pp. 1-25
69
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Baryon Spectrum
 In connection with EBAC's analysis, dressed-quark Faddeev-equation
predictions for bare-masses agree within rms-relative-error of 14%.
 Notably, EBAC finds a dressed-quark-core for the Roper
resonance, at a mass which agrees with Faddeev Eq. prediction.
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
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N. Suzuki et al., Phys.Rev.Lett. 104 (2010) 042302
 EBAC examined the
dynamical origins of the two
poles associated with the
Roper resonance are
examined.
 Both of them, together with
the next higher resonance in
the P11 partial wave were
found to have the same
originating bare state
 Coupling to the mesonbaryon continuum induces
multiple observed resonances
from the same bare state.
 All PDG identified resonances
consist of a core state and
meson-baryon
components.
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
EBAC
& the Roper resonance
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71
Legend:
Particle Data Group
H.L.L. Roberts et al.
EBAC Now and
Jülich
Hadron Spectrum
for the foreseeable future, QCD-based theory will
provide only dressed-quark core masses;
EBAC or EBAC-like tools necessary for mesons and baryons
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
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Masses of ground and excited-state hadrons
Hannes L.L. Roberts, Lei Chang, Ian C. Cloët
and Craig D. Roberts, arXiv:1101.4244 [nucl-th]
Few Body Systems (2011) pp. 1-25
Baryon Spectrum
diquark content
Fascinating suggestion:
nucleon’s first excited state = almost entirely axial-vector diquark,
despite the dominance of scalar diquark in ground state
0+
77%
1+
23%
100%
100%
0-
51%
43%
1-
49%
57%
100%
100%
100%
Owes fundamentally to close relationship between scalar diquark and pseudoscalar meson
And this follows from dynamical chiral symmetry breaking
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
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Nucleon to Roper Transition Form Factors
 Extensive CLAS @ JLab Programme has produced the first
measurements of nucleon-to-resonance transition form factors
I. Aznauryan et al.,
Results of the N* Program at JLab
arXiv:1102.0597 [nucl-ex]
 Theory challenge is to
explain the measurements
 Notable result is zero in F2p→N*,
explanation of which is a real
challenge to theory.
 DSE study connects appearance
of zero in F2p→N* with
 axial-vector-diquark dominance in Roper resonance
 and structure of form factors of J=1 state
I.C. Cloët, C.D. Roberts, et al.
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
In progress
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Nucleon to Roper Transition Form Factors
 Tiator and Vanderhaeghen – in progress
– Empirically-inferred light-front-transverse
charge density
– Positive core plus negative annulus
 Readily explained by dominance of
axial-vector diquark
– Considering isospin and charge
– Negative d-quark twice as likely to be
delocalised from always-positive core than positive u-quark
d
2
{uu}
u
+
1
{ud}
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Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
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Epilogue
Confinement is almost
Certainly the origin of DCSB
 Dynamical chiral symmetry breaking (DCSB) – mass from
nothing for 98% of visible matter – is a reality
o Expressed in M(p2), with observable signals in experiment
 Poincaré covariance
Crucial in description of contemporary data
e.g., BaBar
 Fully-self-consistent treatment of an interaction
anomaly
Comprehensive treatment of vast body of observables
Essential if experimental data is truly to be understood.
 Dyson-Schwinger equations:
o single framework, with IR model-input turned to advantage,
“almost unique in providing unambiguous path from a defined
interaction → Confinement & DCSB → Masses → radii → form
McLerran & Pisarski
factors
→
distribution
functions
→
etc.”
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
arXiv:0706.2191 [hep-ph]
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I.C. Cloët, C.D. Roberts, et al.
arXiv:0812.0416 [nucl-th]
 New JLab data:
S. Riordan et al.,
arXiv:1008.1738 [nucl-ex]
 DSE-prediction
p ,d
2
p ,u
2
F1 (Q )
F1 (Q )
Location of zero
measures relative
strength of scalar
and axial-vector
qq-correlations
Brooks, Bodek, Budd, Arrington
fit to data: hep-ex/0602017
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I.C. Cloët, C.D. Roberts, et al.
arXiv:0812.0416 [nucl-th]
Neutron Structure
Function at high x
SU(6) symmetry
Deep inelastic scattering
– the Nobel-prize winning
quark-discovery experiments
Reviews:
 S. Brodsky et al.
NP B441 (1995)
 W. Melnitchouk & A.W.Thomas
PL B377 (1996) 11
 N. Isgur, PRD 59 (1999)
 R.J. Holt & C.D. Roberts
RMP (2010)
Craig Roberts: Dyson-Schwinger Equations and Continuum QCD.
pQCD, uncorrelated Ψ
DSE: 0+ & 1+ qq
0+ qq only
Distribution of neutron’s
momentum amongst quarks
on the valence-quark domain
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