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Transcript of Rocio BERMUDEZ (U Michoácan); Chen CHEN (ANL, IIT, USTC); Xiomara GUTIERREZ-GUERRERO (U...
Confinement contains Condensates
Rocio BERMUDEZ (U Michoácan);Chen CHEN (ANL, IIT, USTC);Xiomara GUTIERREZ-GUERRERO (U Michoácan);Trang NGUYEN (KSU);Si-xue QIN (PKU);Hannes ROBERTS (ANL, FZJ, UBerkeley);Lei CHANG (ANL, FZJ, PKU); Huan CHEN (BIHEP);Ian CLOËT (UAdelaide);Bruno EL-BENNICH (São Paulo);David WILSON (ANL);Adnan BASHIR (U Michoácan);Stan BRODSKY (SLAC);Gastão KREIN (São Paulo)Roy HOLT (ANL);Mikhail IVANOV (Dubna);Yu-xin LIU (PKU);Robert SHROCK (Stony Brook);Peter TANDY (KSU)
Craig Roberts
Physics Division
StudentsEarly-career scientists
Published collaborations: 2010-present
2
Wholly contained
within hadronsSP, 7-8/05/12: Perspectives in Non-P QCD - 74pgs
Craig Roberts: Confinement contains Condensates
Craig Roberts: Confinement contains Condensates
3
Some Relevant References arXiv:1202.2376
Confinement contains condensatesStanley J. Brodsky, Craig D. Roberts, Robert Shrock, Peter C. Tandy
arXiv:1109.2903 [nucl-th], Phys. Rev. C85 (2012) 012201(RapCom), Expanding the concept of in-hadron condensatesLei Chang, Craig D. Roberts and Peter C. Tandy
arXiv:1005.4610 [nucl-th], Phys. Rev. C82 (2010) 022201(RapCom.) New perspectives on the quark condensate, Brodsky, Roberts, Shrock, Tandy
arXiv:0905.1151 [hep-th], PNAS 108, 45 (2011) Condensates in Quantum Chromodynamics and the Cosmological Constant , Brodsky and Shrock,
hep-th/0012253 The Quantum vacuum and the cosmological constant problem, Svend Erik Rugh and Henrik Zinkernagel.
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Confinement
Craig Roberts: Confinement contains Condensates
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Confinement
Gluon and Quark Confinement– No coloured states have yet been observed to reach a detector
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.
X
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Colour singlets
Craig Roberts: Confinement contains Condensates
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Confinement Confinement is expressed through a dramatic 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);
Roberts, Williams & Krein (1992); Tandy (1994); …
complex-P2 complex-P2
o Real-axis mass-pole splits, moving into pair(s) of complex conjugate poles or branch pointso Spectral density no longer positive semidefinite & hence state cannot exist in observable spectrum
Normal particle Confined particle
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timelike axis: P2<0
Craig Roberts: Confinement contains Condensates
7
Dressed-gluon propagator
Gluon propagator satisfies a Dyson-Schwinger Equation
Plausible possibilities for the solution
DSE and lattice-QCDagree on the result– Confined gluon– IR-massive but UV-massless– mG ≈ 2-4 ΛQCD
perturbative, massless gluon
massive , unconfined gluon
IR-massive but UV-massless, confined gluon
A.C. Aguilar et al., Phys.Rev. D80 (2009) 085018
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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 = 8-15 • α(mG
2)/π = 2-4
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Qin et al., Phys. Rev. C 84 042202(R) (2011)Rainbow-ladder truncation
Running gluon mass– Gluon is massless in ultraviolet
in agreement with pQCD– Massive in infrared
• mG(0) = 0.67-0.81 GeV• mG(mG
2) = 0.53-0.64 GeV
9
Dynamical Chiral Symmetry Breaking
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Craig Roberts: Confinement contains Condensates
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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.
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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.
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C.D. Roberts, Prog. Part. Nucl. Phys. 61 (2008) 50M. Bhagwat & P.C. Tandy, AIP Conf.Proc. 842 (2006) 225-227
Craig Roberts: Confinement contains Condensates
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.
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DSE prediction of DCSB confirmed
Mass from nothing!
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12GeVThe 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.
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Jlab 12GeV: Scanned by 2<Q2<9 GeV2 elastic & transition form factors.
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Gluon & quark mass-scales
mg(0) and M(0) – dynamically generated mass scales for gluons and quarks – are insensitive to changes in the current-quark mass in the neighbourhood of the physical value
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Persistent Challenge
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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
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Invaluable check on practical truncation schemes
17
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-Takahashi identity– Statement of chiral symmetry and the pattern by which it’s broken in
quantum field theory
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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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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
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quark-antiquark scattering kernel
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Persistent challenge- truncation scheme
Happily, that has changed and there are now two nonperturbative & symmetry preserving truncation schemes1. 1995 – 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 Equations1996 – A. Bender, C.D. Roberts and L. von Smekal, Phys.Lett. B 380 (1996) 7, Goldstone Theorem and Diquark Confinement Beyond Rainbow Ladder Approximation
2. 2009 – Lei Chang and C.D. Roberts, Phys. Rev. Lett. 103 (2009) 081601, 0903.5461 [nucl-th], Sketching the Bethe-Salpeter kernel
Enables proof of numerous exact resultsSP, 7-8/05/12: Perspectives in Non-P QCD - 74pgs
Craig Roberts: Confinement contains Condensates
Dichotomy of the pion
Craig Roberts: Confinement contains Condensates
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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) This is impossible in quantum mechanics, for which one
always finds:
mm 2
tconstituenstatebound mm
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Dichotomy of the pionGoldstone mode and bound-
state The correct understanding of pion observables; e.g. mass,
decay constant and form factors, requires an approach to contain a– well-defined and valid chiral limit;– and an accurate realisation of dynamical chiral symmetry
breaking.
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HIGHLY NONTRIVIALImpossible in quantum mechanicsOnly possible in asymptotically-free gauge theories
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Some of many
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Exact Results
Pion’s Goldberger-Treiman relation
Craig Roberts: Confinement contains Condensates
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Pion’s Bethe-Salpeter amplitudeSolution of the Bethe-Salpeter equation
Dressed-quark propagator
Axial-vector Ward-Takahashi identity entails
Pseudovector componentsnecessarily nonzero.
Cannot be ignored!
Exact inChiral QCD
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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
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Dichotomy of the pionGoldstone mode and bound-state
Goldstone’s theorem has a pointwise expression in QCD;
Namely, in the chiral limit the wave-function for the two-body bound-state Goldstone mode is intimately connected with, and almost completely specified by, the fully-dressed one-body propagator of its characteristic constituent • The one-body momentum is equated with the relative
momentum of the two-body system
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fπ Eπ(p2) = B(p2)
25
Dichotomy of the pionMass Formula for 0— Mesons
Mass-squared of the pseudscalar hadron Sum of the current-quark masses of the constituents;
e.g., pion = muς + md
ς , where “ς” is the renormalisation point
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Dichotomy of the pionMass Formula for 0— Mesons
Pseudovector projection of the Bethe-Salpeter wave function onto the origin in configuration space– Namely, the pseudoscalar meson’s leptonic decay constant, which is
the strong interaction contribution to the strength of the meson’s weak interaction
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Dichotomy of the pionMass Formula for 0— Mesons
Pseudoscalar projection of the Bethe-Salpeter wave function onto the origin in configuration space– Namely, a pseudoscalar analogue of the meson’s leptonic decay
constant
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Dichotomy of the pionMass Formula for 0— Mesons
Consider the case of light quarks; namely, mq ≈ 0– If chiral symmetry is dynamically broken, then
• fH5 → fH50 ≠ 0
• ρH5 → – < q-bar q> / fH50 ≠ 0
both of which are independent of mq
Hence, one arrives at the corollary
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Gell-Mann, Oakes, Renner relation1968mm 2
The so-called “vacuum quark condensate.” More later about this.
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Dichotomy of the pionMass Formula for 0— Mesons
Consider a different case; namely, one quark mass fixed and the other becoming very large, so that mq /mQ << 1
Then – fH5 1/√m∝ H5
– ρH5 √m∝ H5
and one arrives at
mH5 m∝ Q
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ProvidesQCD proof of
potential model result
Ivanov, Kalinovsky, RobertsPhys. Rev. D 60, 034018 (1999) [17 pages]
30
Dynamical Chiral Symmetry Breaking
Vacuum Condensates?
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Dichotomy of the pionMass Formula for 0— Mesons
Consider the case of light quarks; namely, mq ≈ 0– If chiral symmetry is dynamically broken, then
• fH5 → fH50 ≠ 0
• ρH5 → – < q-bar q> / fH50 ≠ 0
both of which are independent of mq
Hence, one arrives at the corollary
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Craig Roberts: Confinement contains Condensates
Gell-Mann, Oakes, Renner relation1968mm 2
The so-called “vacuum quark condensate.” More later about this.
Maris, Roberts and Tandynucl-th/9707003, Phys.Lett. B420 (1998) 267-273
We now have sufficient information to address the question of just what is this so-called “vacuum quark condensate.”
Spontaneous(Dynamical)Chiral Symmetry Breaking
The 2008 Nobel Prize in Physics was divided, one half awarded to Yoichiro Nambu
"for the discovery of the mechanism of spontaneous broken symmetry in subatomic physics"
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Nambu – Jona-LasinioModel
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Treats a chirally-invariant four-fermion Lagrangian & solves the gap equation in Hartree-Fock approximation (analogous to rainbow truncation)
Possibility of dynamical generation of nucleon mass is elucidated Essentially inequivalent vacuum states are identified (Wigner and
Nambu states) & demonstration thatthere are infinitely many, degenerate but distinct Nambu vacua, related by a chiral rotation
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Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity. I
Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 (1961) 345–358 Dynamical Model Of Elementary Particles
Based On An Analogy With Superconductivity. IIY. Nambu, G. Jona-Lasinio, Phys.Rev. 124 (1961) 246-254
Gell-Mann – Oakes – RennerRelation
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This paper derives a relation between mπ
2 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
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Behavior of current divergences under SU(3) x SU(3).Murray Gell-Mann, R.J. Oakes , B. Renner Phys.Rev. 175 (1968) 2195-2199
Gell-Mann – Oakes – RennerRelation
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PCAC hypothesis; viz., pion field dominates the divergence of the axial-vector current
Soft-pion theorem
In QCD, this is and one therefore has
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Behavior of current divergences under SU(3) x SU(3).Murray Gell-Mann, R.J. Oakes , B. Renner Phys.Rev. 175 (1968) 2195-2199
Commutator is chiral rotationTherefore, isolates explicit chiral-symmetry breaking term in the putative Hamiltonian
qqm
Zhou Guangzhao 周光召Born 1929 Changsha, Hunan province
Gell-Mann – Oakes – RennerRelation
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Theoretical physics at its best. 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 matrix element, there is no problem.
Problem arises if one over-interprets this number, which textbooks have been doing for a VERY LONG TIME.
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- (0.25GeV)3
Note of Warning
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Chiral Magnetism (or Magnetohadrochironics)A. Casher and L. Susskind, Phys. Rev. D9 (1974) 436
These authors argue that dynamical chiral-symmetry breaking can be realised as aproperty of hadrons, instead of via a nontrivial vacuum exterior to the measurable degrees of freedom
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.
QCD Sum Rules
Introduction of the gluon vacuum condensate
and development of “sum rules” relating properties of low-lying hadronic states to vacuum condensates
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QCD and Resonance Physics. Sum Rules.M.A. Shifman, A.I. Vainshtein, and V.I. Zakharov Nucl.Phys. B147 (1979) 385-447; citations: 3713
QCD Sum Rules
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: Confinement contains Condensates
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QCD and Resonance Physics. Sum Rules.M.A. Shifman, A.I. Vainshtein, and V.I. Zakharov Nucl.Phys. B147 (1979) 385-447; citations: 3781
“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 QCDJ Finger, JE Mandula… - Physics Letters B, 1980
No references to this phrase before 1980Craig Roberts: Confinement contains Condensates
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7330+ REFERENCES TO THIS PHRASE SINCE 1980
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.”
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QCD
How should one approach this problem, understand it, within Quantum ChromoDynamics?
1) Are the quark and gluon “condensates” theoretically well-defined?
2) Is there a physical meaning to this quantity or is it merely just a mass-dimensioned parameter in a theoretical computation procedure?
Craig Roberts: Confinement contains Condensates
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0||0 qq 1973-1974
QCD
Why does it matter?
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0||0 qq 1973-1974
“Dark Energy”
Two pieces of evidence for an accelerating universe1) Observations of type Ia supernovae
→ the rate of expansion of the Universe is growing2) Measurements of the composition of the Universe point to a
missing energy component with negative pressure: CMB anisotropy measurements indicate that the Universe is at
Ω0 = 1 ⁺⁄₋ 0.04. In a flat Universe, the matter density and energy density must sum to the critical density. However, matter only contributes about ⅓ of the critical density,
ΩM = 0.33 ⁺⁄₋ 0.04. Thus, ⅔ of the critical density is missing.
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“Dark Energy”
In order not to interfere with the formation of structure (by inhibiting the growth of density perturbations) the energy density in this component must change more slowly than matter (so that it was subdominant in the past).
Accelerated expansion can be accommodated in General Relativity through the Cosmological Constant, Λ. Einstein introduced the repulsive effect of the cosmological
constant in order to balance the attractive gravity of matter so that a static universe was possible. He promptly discarded it after the discovery of the expansion of the Universe.
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In order to have escaped detection, the missing energy must be smoothly distributed.
412 )10(8
GeVG
obs
Contemporary cosmological observations mean:
“Dark Energy”
The only possible covariant form for the energy of the (quantum) vacuum; viz.,
is mathematically equivalent to the cosmological constant.
“It is a perfect fluid and precisely spatially uniform”“Vacuum energy is almost the perfect candidate for
dark energy.”
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“The advent of quantum field theory made consideration of the cosmological constant obligatory not optional.”Michael Turner, “Dark Energy and the New Cosmology”
obsQCD 4610
“Dark Energy”
QCD vacuum contributionIf chiral symmetry breaking is expressed in a nonzero
expectation value of the quark bilinear, then the energy difference between the symmetric and broken phases is of order
MQCD≈0.3 GeVOne obtains therefrom:
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“The biggest embarrassment in theoretical physics.”
Mass-scale generated by spacetime-independent condensate
Enormous and even greater contribution from Higgs VEV!
48
Resolution?SP, 7-8/05/12: Perspectives in Non-P QCD - 74pgs
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QCD
Are the condensates real? Is there a physical meaning to the vacuum quark condensate
(and others)? Or is it merely just a mass-dimensioned parameter in a
theoretical computation procedure?
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0||0 qq 1973-1974
What is measurable?
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S. Weinberg, Physica 96A (1979)Elements of truth in this perspective
51
Dichotomy of the pionMass Formula for 0— Mesons
Consider the case of light quarks; namely, mq ≈ 0– If chiral symmetry is dynamically broken, then
• fH5 → fH50 ≠ 0
• ρH5 → – < q-bar q> / fH50 ≠ 0
both of which are independent of mq
Hence, one arrives at the corollary
SP, 7-8/05/12: Perspectives in Non-P QCD - 74pgs
Craig Roberts: Confinement contains Condensates
Gell-Mann, Oakes, Renner relation1968mm 2
The so-called “vacuum quark condensate.” More later about this.
Maris, Roberts and Tandynucl-th/9707003, Phys.Lett. B420 (1998) 267-273
We now have sufficient information to address the question of just what is this so-called “vacuum quark condensate.”
In-meson condensate
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Maris & Robertsnucl-th/9708029
Pseudoscalar projection of pion’s Bethe-Salpeter wave-function onto the origin in configuration space: |Ψπ
PS(0)|
– or the pseudoscalar pion-to-vacuum matrix element
Rigorously defined in QCD – gauge-independent, cutoff-independent, etc. For arbitrary current-quark masses For any pseudoscalar meson
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In-meson condensate
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Pseudovector projection of pion’s Bethe-Salpeter wave-function onto the origin in configuration space: |Ψπ
AV(0)|
– or the pseudoscalar pion-to-vacuum matrix element – or the pion’s leptonic decay constant
Rigorously defined in QCD – gauge-independent, cutoff-independent, etc. For arbitrary current-quark masses For any pseudoscalar meson
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In-meson condensate
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Define
Then, using the pion Goldberger-Treiman relations (equivalence of 1- and 2-body problems), one derives, in the chiral limit
Namely, the so-called vacuum quark condensate is the chiral-limit value of the in-pion condensate
The in-pion condensate is the only well-defined function of current-quark mass in QCD that is smoothly connected to the vacuum quark condensate.
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0);0( qq
Chiral limit
Maris & Robertsnucl-th/9708029
|ΨπPS(0)|*|Ψπ
AV(0)|
I. Casher Banks formula:
II. Constant in the Operator Product Expansion:
III. Trace of the dressed-quark propagator:
There is only one condensate
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Langeld, Roberts et al.nucl-th/0301024,Phys.Rev. C67 (2003) 065206
m→0
Density of eigenvalues of Dirac operator
Algebraic proof that these are all the same. So, no matter how one chooses to calculate it, one is always calculating the same thing; viz.,
|ΨπPS(0)|*|Ψπ
AV(0)|
Paradigm shift:In-Hadron Condensates
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.
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QCD
Brodsky, Roberts, Shrock, Tandy, Phys. Rev. C82 (Rapid Comm.) (2010) 022201Brodsky and Shrock, PNAS 108, 45 (2011)
Paradigm shift:In-Hadron Condensates
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
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Brodsky, Roberts, Shrock, Tandy, Phys. Rev. C82 (Rapid Comm.) (2010) 022201Brodsky and Shrock, PNAS 108, 45 (2011)
Paradigm shift:In-Hadron Condensates
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
– And
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0);0( qq
Chiral limit
Brodsky, Roberts, Shrock, Tandy, Phys. Rev. C82 (Rapid Comm.) (2010) 022201Brodsky and Shrock, PNAS 108, 45 (2011)
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Topological charge of “vacuum”
Wikipedia: Instanton effects are important in understanding the formation of condensates in the vacuum of quantum chromodynamics (QCD)
Wikipedia: The difference between the mass of the η and that of the η' is larger than the quark model can naturally explain. This “η-η' puzzle” is resolved by instantons.
Claimed that some lattice simulations demonstrate nontrivial topological structures in QCD vacuum
Now illustrate new paradigm perspective …
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AVWTI ⇒ QCD mass formulae for all pseudoscalar mesons, including those which are charge-neutral
Consider the limit of a U(Nf)-symmetric mass matrix, then this formula yields:
Topological charge density: Q(x) = i(αs/4π) trC εμνρσ Fμν Fρσ
Plainly, the η – η’ mass splitting is nonzero in the chiral limit so long as νη’ ≠ 0 … viz., so long as the topological content of the η’ is nonzero!
Charge-neutral pseudoscalar mesons
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Bhagwat, Chang, Liu, Roberts, TandyPhys.Rev. C76 (2007) 045203
Algebraic result.Very different than requiring QCD’s vacuum to possess nontrivial topological structure
Qualitatively the same as fπ, a property of the bound-state
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Topology and the “condensate”
Exact result in QCD, algebraic proof:
“chiral condensate” = in-pion condensate the the zeroth moment of a mixed vacuum polarisation– connecting topological charge with the pseudoscalar quark
operator
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Bhagwat, Chang, Liu, Roberts, TandyPhys.Rev. C76 (2007) 045203
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GMOR Relation
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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. Crucial to observe that the operator expectation value in this
equation is evaluated between pion states. Moreover, the virtual low-energy limit expressed in the equation is
purely formal. It does not describe an achievable empirical situation.
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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 Relation
In terms of QCD quantities, GMOR relation entails
o mudζ = mu
ζ + mdζ … the current-quark masses
o S π
ζ(0) is the pion’s scalar form factor at zero momentum transfer, Q2=0
RHS is proportional to the pion σ-term Consequently, using the connection between the σ-term and the
Feynman-Hellmann theorem, GMOR relation is actually the statement
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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 Relation
Using
it follows that
This equation is valid for any values of mu,d, including the neighbourhood of the chiral limit, wherein
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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)
Maris, Roberts and Tandynucl-th/9707003, Phys.Lett. B420 (1998) 267-273
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GMOR Relation
Consequently, in the neighbourhood of the chiral limit
This is a QCD derivation of the commonly recognised form of the GMOR relation.
Neither PCAC nor soft-pion theorems were employed in the analysis.
Nature of each factor in the expression is abundantly clear; viz., chiral limit values of matrix elements that explicitly involve the hadron.
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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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Expanding the Concept
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In-Hadron Condensates
Plainly, the in-pseudoscalar-meson condensate can be represented through the pseudoscalar meson’s scalar form factor at zero momentum transfer Q2 = 0.
Using an exact mass formula for scalar mesons, one proves the in-scalar-meson condensate can be represented in precisely the same way.
By analogy, and with appeal to demonstrable results of heavy-quark symmetry, the Q2 = 0 values of vector- and pseudovector-meson scalar form factors also determine the in-hadron condensates in these cases.
This expression for the concept of in-hadron quark condensates is readily extended to the case of baryons.
Via the Q2 = 0 value of any hadron’s scalar form factor, one can extract the value for a quark condensate in that hadron which is a reasonable and realistic measure of dynamical chiral symmetry breaking.
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Hadron Charges
Hadron Form factor matrix elements 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. …
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Confinement
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Confinement Confinement is essential to the validity of the notion of in-hadron
condensates. Confinement makes it impossible to construct gluon or quark
quasiparticle operators that are nonperturbatively valid. So, although one can define a perturbative (bare) vacuum for QCD,
it is impossible to rigorously define a ground state for QCD upon a foundation of gluon and quark quasiparticle operators.
Likewise, it is impossible to construct an interacting vacuum – a BCS-like trial state – and hence DCSB in QCD cannot rigorously be expressed via a spacetime-independent coherent state built upon the ground state of perturbative QCD.
Whilst this does not prevent one from following this path to build practical models for use in hadron physics phenomenology, it does invalidate any claim that theoretical artifices in such models are empirical.
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Confinement Contains CondensatesS.J. Brodsky, C.D. Roberts, R. Shrock and P.C. TandyarXiv:1202.2376 [nucl-th]
“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.”
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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
4620
4
103
8
HG QCDNscondensateQCD
Paradigm shift:In-Hadron Condensates
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This is not the end
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Simulations with static quarks
Mean-field picture of gauge-field action– <Sg> is not observable– Strongly reminiscent of mean-field – bag model –
pictures of the nucleon, popular in 80s. Gauge configurations are instanton-like, and instantons have nothing to do
with confinement. What is the dynamically generated confinement length-scale in this
simulation? – This is not related in any known way to the length-scale imposed on the
simulation by choosing a value of the string tension. Infinitely heavy quarks, repositioned by hand.
– Natural size of system constituted from infinitely heavy quarks is radius=0 [ r ~ ln MQ/MQ ]
– Therefore, no dynamical information present. What is hadron spectrum associated with this simulation? There are no
quarks, so is there a quark-hadron duality? – The latter is critical to the new condensate paradigm.
Provide simulation results with realistic quark masses, then one can test the new perspective on condensates … present pictures quite likely represent features of hadron interiors.
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