Confinement contains Condensates - PowerPoint Presentation

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Confinement contains Condensates

Adnan BASHIR (U Michoacan);R. BERMUDEZ (U Michoacan);Stan BRODSKY (SLAC);Lei CHANG (ANL & PKU); Huan CHEN (BIHEP);Ian CLOËT (UW);Bruno EL-BENNICH (São Paulo);Gastão KREIN (São 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)

Craig Roberts

Physics Division

StudentsEarly-career scientists

Published collaborations: 2010-present

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Relevant References arXiv:1202.2376  Confinement contains condensates Stanley J. Brodsky, Craig D. Roberts, Robert Shrock, Peter C. TandyarXiv:1109.2903 [nucl-th], Phys. Rev. C85 (2012) 012201(RapCom), Expanding the concept of in-hadron

condensatesLei Chang, Craig D. Roberts and Peter C. TandyarXiv: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. Strong Interactions Beyond the SM - 66pgsCraig Roberts: Confinement contains Condensates2Slide3

QCD’s ChallengesDynamical 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 from apparently simple rules.Craig Roberts: Confinement contains Condensates3 Quark and Gluon ConfinementNo matter how hard one strikes the proton,

one cannot liberate an individual gluon or quark

Understand emergent phenomena

Strong Interactions Beyond the SM - 66pgsSlide4

Dyson-SchwingerEquationsWell suited to Relativistic Quantum Field TheorySimplest level: Generating Tool for Perturbation Theory . . . Materially Reduces Model-Dependence … Statement about long-range behaviour of quark-quark interactionNonPerturbative, Continuum approach to QCD

Hadrons as Composites of Quarks and GluonsQualitative and Quantitative Importance of:Dynamical Chiral Symmetry Breaking – Generation of fermion mass from nothingQuark & Gluon Confinement – Coloured objects not detected, Not detectable?Craig Roberts: Confinement contains Condensates4Approach yields

Schwinger functions; i.e.,

propagators and verticesCross-Sections built from Schwinger FunctionsHence, 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

Strong Interactions Beyond the SM - 66pgsSlide5

Strong Interactions Beyond the SM - 66pgsCraig Roberts: Confinement contains Condensates5

ConfinementSlide6

ConfinementGluon and Quark ConfinementNo coloured states have yet been observed to reach a detectorEmpirical fact. HoweverThere is no agreed, theoretical definition of light-quark confinementStatic-quark confinement is irrelevant to real-world QCDThere are no long-lived, very-massive quarksConfinement entails quark-hadron duality; i.e., that

all observable consequences of QCD can, in principle, be computed using an hadronic basis. Craig Roberts: Confinement contains Condensates6XStrong Interactions Beyond the SM - 66pgsColour singletsSlide7

ConfinementConfinement 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-propagatorGribov (1978); Munczek (1983); Stingl (1984); Cahill (1989); Roberts, Williams & Krein (1992); Tandy (1994); …Craig Roberts: Confinement contains Condensates7

complex-P2complex-P2

Real-axis mass-pole splits, moving into pair(s) of complex conjugate poles or branch points

Spectral density no longer positive

semidefinite

& hence state cannot exist in observable spectrum

Normal particle

Confined particle

Strong Interactions Beyond the SM - 66pgs

timelike

axis: P

2

<0Slide8

Dressed-gluon propagatorGluon propagator satisfies a Dyson-Schwinger EquationPlausible possibilities for the solutionDSE and lattice-QCD agree on the resultConfined gluonIR-massive but UV-

masslessmG ≈ 2-4 ΛQCD Craig Roberts: Confinement contains Condensates8perturbative, massless gluonmassive , unconfined gluonIR-massive but UV-massless, confined gluonA.C. Aguilar et al., Phys.Rev. D80 (2009) 085018Strong Interactions Beyond the SM - 66pgsSlide9

DSE Studies – Phenomenology of gluonWide-ranging study of π & ρ propertiesEffective couplingAgrees with pQCD in ultraviolet

Saturates in infraredα(0)/π = 8-15 α(mG2)/π = 2-4Strong Interactions Beyond the SM - 66pgsCraig Roberts: Confinement contains Condensates9Qin et al., Phys. Rev. C 84 042202(R) (2011)Rainbow-ladder truncationRunning gluon massGluon is massless in ultraviolet in agreement with pQCD

Massive in infrared

m

G

(0) = 0.67-0.81 GeV

m

G

(m

G

2

) = 0.53-0.64

GeVSlide10

Dynamical Chiral

Symmetry BreakingStrong Interactions Beyond the SM - 66pgsCraig Roberts: Confinement contains Condensates10Slide11

Dynamical Chiral Symmetry BreakingStrong-interaction: QCDConfinementEmpirical featureModern theory and lattice-QCD support conjecture that light-quark confinement is a factassociated with violation of reflection positivity; i.e., novel analytic structure for propagators and verticesStill circumstantial, no proof yet of confinementOn the other hand, DCSB is a fact in QC

DIt 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: Confinement contains Condensates11Strong Interactions Beyond the SM - 66pgsSlide12

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: Confinement contains Condensates12Strong Interactions Beyond the SM - 66pgsC.D. Roberts, Prog. Part. Nucl. Phys. 61 (2008) 50M. Bhagwat & P.C. Tandy, AIP Conf.Proc. 842 (2006) 225-227Slide13

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: Confinement contains Condensates13DSE prediction of DCSB confirmedMass from nothing!Strong Interactions Beyond the SM - 66pgsSlide14

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: Confinement contains Condensates14Hint of lattice-QCD support for DSE prediction of violation of reflection positivity Strong Interactions Beyond the SM - 66pgsSlide15

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.Craig Roberts: Confinement contains Condensates15Jlab 12GeV: Scanned by 2<Q2<9 GeV2 elastic & transition form factors. Strong Interactions Beyond the SM - 66pgsSlide16

Gluon & quark mass-scalesmg(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 valueStrong Interactions Beyond the SM - 66pgsCraig Roberts: Confinement contains Condensates

16Slide17

Persistent ChallengeStrong Interactions Beyond the SM - 66pgs

Craig Roberts: Confinement contains Condensates17TruncationSlide18

Infinitely many coupled equations: Kernel of the equation for the quark self-energy involves:Dμν(k) – dressed-gluon propagatorΓν(q,p) – dressed-quark-gluon vertex each of which satisfies its own DSE, etc…Coupling between equations necessitates a truncationWeak coupling expansion ⇒ produces every diagram in perturbation theory

Otherwise useless for the nonperturbative problems in which we’re interestedPersistent challenge in application of DSEsStrong Interactions Beyond the SM - 66pgsCraig Roberts: Confinement contains Condensates18Invaluable check on practical truncation schemesSlide19

Persistent challenge- truncation schemeSymmetries associated with conservation of vector and axial-vector currents are critical in arriving at a veracious understanding of hadron structure and interactionsExample: axial-vector Ward-Takahashi identityStatement of chiral symmetry and the pattern by which it’s broken in quantum field theoryStrong Interactions Beyond the SM - 66pgsCraig Roberts: Confinement contains Condensates19

Axial-Vector vertex Satisfies an inhomogeneous Bethe-Salpeter equationQuark propagator satisfies a gap equation

Kernels of these equations are completely different

But they must be intimately related

Relationship must be preserved by any truncationHighly nontrivial constraint

FAILURE has an extremely high cost – loss of any connection with

QCDSlide20

Persistent challenge- truncation schemeThese 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 problemUntil 1995/1996 people had no idea what to doEquations were truncated, sometimes with good phenomenological results, sometimes with poor resultsNeither good nor bad could be explainedStrong Interactions Beyond the SM - 66pgs

Craig Roberts: Confinement contains Condensates20quark-antiquark scattering kernelSlide21

Persistent challenge- truncation schemeHappily, that changed, and there is now at least one systematic, nonperturbative and symmetry preserving truncation schemeH.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 EquationsA. Bender, C.D. Roberts and L. von Smekal, Phys.Lett. B

380 (1996) 7, Goldstone Theorem and Diquark Confinement Beyond Rainbow Ladder ApproximationEnables proof of numerous exact resultsStrong Interactions Beyond the SM - 66pgsCraig Roberts: Confinement contains Condensates21Slide22

Dichotomy of the pionCraig Roberts: Confinement contains Condensates22How 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 mistakeHowever: current-algebra (1968) This is impossible in quantum mechanics, for which one always finds: Strong Interactions Beyond the SM - 66pgsSlide23

Dichotomy of the pionGoldstone mode and bound-stateThe correct understanding of pion observables; e.g. mass, decay constant and form factors, requires an approach to contain awell-defined and valid chiral limit;and an accurate realisation of dynamical chiral symmetry breaking.

Strong Interactions Beyond the SM - 66pgsCraig Roberts: Confinement contains Condensates23HIGHLY NONTRIVIALImpossible in quantum mechanicsOnly possible in asymptotically-free gauge theoriesSlide24

Some of many

Strong Interactions Beyond the SM - 66pgsCraig Roberts: Confinement contains Condensates24Exact ResultsSlide25

Pion’s Goldberger-Treiman relationCraig Roberts: Confinement contains Condensates25Pion’s Bethe-Salpeter amplitude

Solution of the Bethe-Salpeter equationDressed-quark propagatorAxial-vector Ward-Takahashi identity entailsPseudovector componentsnecessarily nonzero. Cannot be ignored!

Exact in

Chiral

QCD

Strong Interactions Beyond the SM - 66pgs

Miracle

:

two body problem solved, almost completely, once solution of one body problem is known

Maris, Roberts and Tandy

nucl-th

/9707003

,

Phys.Lett

. B

420

(1998) 267-273 Slide26

Dichotomy of the pionGoldstone mode and bound-stateGoldstone’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

Strong Interactions Beyond the SM - 66pgsCraig Roberts: Confinement contains Condensates26fπ Eπ(p2) = B(p2)Slide27

Dichotomy of the pionMass Formula for 0— MesonsMass-squared of the pseudscalar hadronSum of the current-quark masses of the constituents; e.g., pion = muς + mdς , where “ς

” is the renormalisation pointStrong Interactions Beyond the SM - 66pgsCraig Roberts: Confinement contains Condensates27Maris, Roberts and Tandynucl-th/9707003, Phys.Lett. B420 (1998) 267-273 Slide28

Dichotomy of the pionMass Formula for 0— MesonsPseudovector projection of the Bethe-Salpeter wave function onto the origin in configuration spaceNamely, the pseudoscalar meson’s leptonic decay constant, which is the strong interaction contribution to the strength of the meson’s weak interaction

Strong Interactions Beyond the SM - 66pgsCraig Roberts: Confinement contains Condensates28

Maris, Roberts and Tandy

nucl-th

/9707003

, Phys.Lett. B420

(1998) 267-273 Slide29

Dichotomy of the pionMass Formula for 0— MesonsPseudoscalar projection of the Bethe-Salpeter wave function onto the origin in configuration spaceNamely, a pseudoscalar analogue of the meson’s leptonic decay constant

Strong Interactions Beyond the SM - 66pgsCraig Roberts: Confinement contains Condensates29

Maris, Roberts and Tandy

nucl-th

/9707003

, Phys.Lett

. B420 (1998) 267-273 Slide30

Dichotomy of the pionMass Formula for 0— MesonsConsider the case of light quarks; namely, mq ≈ 0If chiral symmetry is dynamically broken, then fH5 → fH50 ≠ 0ρH5 → – < q-bar q> / fH5

0 ≠ 0 both of which are independent of mqHence, one arrives at the corollaryStrong Interactions Beyond the SM - 66pgsCraig Roberts: Confinement contains Condensates30Gell-Mann, Oakes, Renner relation1968The so-called “vacuum quark condensate.” More later about this.Maris, Roberts and Tandynucl-th/9707003, Phys.Lett. B420 (1998) 267-273 Slide31

Dichotomy of the pionMass Formula for 0— MesonsConsider a different case; namely, one quark mass fixed and the other becoming very large, so that mq /mQ << 1Then fH5 ∝ 1/√mH5ρH5 ∝ √m

H5and one arrives at mH5 ∝ mQStrong Interactions Beyond the SM - 66pgsCraig Roberts: Confinement contains Condensates31Maris, Roberts and Tandynucl-th/9707003, Phys.Lett. B420 (1998) 267-273 ProvidesQCD proof ofpotential model resultIvanov, Kalinovsky, RobertsPhys. Rev. D

60, 034018 (1999) [17 pages]Slide32

Dynamical Chiral

Symmetry BreakingVacuum Condensates?Strong Interactions Beyond the SM - 66pgsCraig Roberts: Confinement contains Condensates32Slide33

Dichotomy of the pionMass Formula for 0— MesonsConsider the case of light quarks; namely, mq ≈ 0If chiral symmetry is dynamically broken, then fH5 → fH50 ≠ 0ρH5 → – < q-bar q> / fH5

0 ≠ 0 both of which are independent of mqHence, one arrives at the corollaryStrong Interactions Beyond the SM - 66pgsCraig Roberts: Confinement contains Condensates33Gell-Mann, Oakes, Renner relation1968The 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.”Slide34

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"

Craig Roberts: Confinement contains Condensates34Strong Interactions Beyond the SM - 66pgsSlide35

Nambu – Jona-LasinioModelCraig Roberts: Confinement contains Condensates35

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 that there are infinitely many, degenerate but distinct Nambu vacua, related by a chiral rotationNontrivial Vacuum is “Born”Strong Interactions Beyond the SM - 66pgsDynamical Model of Elementary Particles Based on an Analogy with Superconductivity. IY. 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) 246-254Slide36

Gell-Mann – Oakes – RennerRelationCraig Roberts: Confinement contains Condensates36This paper derives a relation between mπ2 and the expectation-value < π|u

0|π>, where uo is an operator that is linear in the putative Hamiltonian’s explicit chiral-symmetry breaking termNB. QCD’s current-quarks were not yet invented, so u0 was not expressed in terms of current-quark fieldsPCAC-hypothesis (partial conservation of axial current) is used in the derivationSubsequently, the concepts of soft-pion theoryOperator expectation values do not change as t=mπ2 → t=0to take < π|u0|π> → < 0|u0|0> … in-pion → in-vacuumStrong Interactions Beyond the SM - 66pgsBehavior of current divergences under SU(3) x SU(3).Murray Gell-Mann, R.J. Oakes , B. Renner Phys.Rev. 175 (1968) 2195-2199Slide37

Gell-Mann – Oakes – RennerRelationCraig Roberts: Confinement contains Condensates37PCAC hypothesis; viz., pion field dominates the divergence of the axial-vector current

Soft-pion theoremIn QCD, this is and one therefore has Strong Interactions Beyond the SM - 66pgsBehavior of current divergences under SU(3) x SU(3).Murray Gell-Mann, R.J. Oakes , B. Renner Phys.Rev. 175 (1968) 2195-2199Commutator is chiral rotationTherefore, isolates explicit chiral-symmetry breaking term in the putative Hamiltonian

Zhou

Guangzhao

周光召

Born 1929 Changsha, Hunan provinceSlide38

Gell-Mann – Oakes – RennerRelationCraig Roberts: Confinement contains Condensates38Theoretical 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 condensateSo 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. Strong Interactions Beyond the SM - 66pgs- (0.25GeV)3Slide39

Note of WarningCraig Roberts: Confinement contains Condensates39Strong Interactions Beyond the SM - 66pgsChiral Magnetism (or

Magnetohadrochironics)A. Casher and L. Susskind, Phys. Rev. D9 (1974) 436These authors argue that dynamical chiral- symmetry breaking can be realised as a property 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.Slide40

QCDHow should one approach this problem, understand it, within Quantum ChromoDynamics?

Are the quark and gluon “condensates” theoretically well-defined?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 Condensates40Strong Interactions Beyond the SM - 66pgs1973-1974Slide41

QCD Why does it matter?Craig Roberts: Confinement contains Condensates

41Strong Interactions Beyond the SM - 66pgs1973-1974Slide42

“Dark Energy”Two pieces of evidence for an accelerating universeObservations of type Ia supernovae → the rate of expansion of the Universe is growingMeasurements 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. Craig Roberts: Confinement contains Condensates42Strong Interactions Beyond the SM - 66pgsSlide43

“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.

Craig Roberts: Confinement contains Condensates43Strong Interactions Beyond the SM - 66pgsIn order to have escaped detection, the missing energy must be smoothly distributed. Contemporary cosmological observations mean:Slide44

“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.”

Craig Roberts: Confinement contains Condensates44Strong Interactions Beyond the SM - 66pgs“The advent of quantum field theory made consideration of the cosmological constant obligatory not optional.”Michael Turner, “Dark Energy and the New Cosmology”Slide45

“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 M

QCD≈0.3 GeVOne obtains therefrom: Craig Roberts: Confinement contains Condensates45Strong Interactions Beyond the SM - 66pgs“The biggest embarrassment in theoretical physics.”Mass-scale generated by spacetime-independent condensateEnormous and even greater contribution from Higgs VEV!Slide46

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. Strong Interactions Beyond the SM - 66pgsCraig Roberts: Confinement contains Condensates46Slide47

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?Craig Roberts: Confinement contains Condensates

47Strong Interactions Beyond the SM - 66pgs1973-1974Slide48

What is measurable? Craig Roberts: Confinement contains Condensates48Strong Interactions Beyond the SM - 66pgsS. Weinberg,

Physica 96A (1979)Elements of truth in this perspectiveSlide49

Dichotomy of the pionMass Formula for 0— MesonsConsider the case of light quarks; namely, mq ≈ 0If chiral symmetry is dynamically broken, then fH5 → fH50 ≠ 0ρH5 → – < q-bar q> / fH5

0 ≠ 0 both of which are independent of mqHence, one arrives at the corollaryStrong Interactions Beyond the SM - 66pgsCraig Roberts: Confinement contains Condensates49Gell-Mann, Oakes, Renner relation1968The 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.”Slide50

In-meson condensateCraig Roberts: Confinement contains Condensates50Maris & Robertsnucl-th/9708029Pseudoscalar projection of pion’s Bethe-Salpeter wave-function onto the origin in configuration space:

|ΨπPS(0)| – or the pseudoscalar pion-to-vacuum matrix elementRigorously defined in QCD – gauge-independent, cutoff-independent, etc.For arbitrary current-quark massesFor any pseudoscalar mesonStrong Interactions Beyond the SM - 66pgsSlide51

In-meson condensateCraig Roberts: Confinement contains Condensates51Pseudovector 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 constantRigorously defined in QCD – gauge-independent, cutoff-independent, etc.For arbitrary current-quark massesFor any pseudoscalar mesonStrong Interactions Beyond the SM - 66pgsMaris & Robertsnucl-th/9708029Slide52

In-meson condensateCraig Roberts: Confinement contains Condensates52Define 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 condensateThe in-pion condensate is the only well-defined function of current-quark mass in QCD that is smoothly connected to the vacuum quark condensate.Strong Interactions Beyond the SM - 66pgs

Chiral limit

Maris & Robertsnucl-th/9708029

|

Ψ

π

PS(0)|*|Ψ

π

AV

(0)|Slide53

Casher Banks formula: Constant in the Operator Product Expansion:Trace of the dressed-quark propagator:There is only one condensateCraig Roberts: Confinement contains Condensates

53Strong Interactions Beyond the SM - 66pgsLangeld, Roberts et al.nucl-th/0301024,Phys.Rev. C67 (2003) 065206m→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.,

|

Ψ

π

PS

(0)|*|

Ψ

π

AV

(0)|Slide54

Paradigm shift:In-Hadron CondensatesResolutionWhereas 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.Strong Interactions Beyond the SM - 66pgsCraig Roberts: Confinement contains Condensates54QCDBrodsky, Roberts, Shrock, Tandy, Phys. Rev. C82 (Rapid Comm.) (2010) 022201Brodsky and Shrock, PNAS 108, 45 (2011)Slide55

Paradigm shift:In-Hadron CondensatesResolutionWhereas 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 ρπStrong Interactions Beyond the SM - 66pgsCraig Roberts: Confinement contains Condensates55Brodsky, Roberts, Shrock, Tandy, Phys. Rev. C82 (Rapid Comm.) (2010) 022201Brodsky and Shrock, PNAS 108, 45 (2011)Slide56

Paradigm shift:In-Hadron CondensatesResolutionWhereas 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 ρπAnd Strong Interactions Beyond the SM - 66pgsCraig Roberts: Confinement contains Condensates56Chiral limitBrodsky, Roberts, Shrock, Tandy,

Phys. Rev. C82 (Rapid Comm.) (2010) 022201

Brodsky and Shrock, PNAS 108, 45 (2011)Slide57

GMOR Relation

Strong Interactions Beyond the SM - 66pgsCraig Roberts: Confinement contains Condensates57Slide58

GMOR RelationValuable to highlight the precise form of the Gell-Mann–Oakes–Renner (GMOR) relation: Eq. (3.4) in Phys.Rev. 175 (1968) 2195 mπ is the pion’s mass 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.Strong Interactions Beyond the SM - 66pgsCraig Roberts: Confinement contains Condensates58Expanding the concept of in-hadron condensatesLei Chang, Craig D. Roberts and Peter C. TandyarXiv:1109.2903 [nucl-th], Phys. Rev. C85 (2012) 012201(R)Slide59

GMOR RelationIn terms of QCD quantities, GMOR relation entails mudζ = muζ + mdζ … the current-quark massesS πζ(0)

is the pion’s scalar form factor at zero momentum transfer, Q2=0RHS is proportional to the pion σ-termConsequently, using the connection between the σ-term and the Feynman-Hellmann theorem, GMOR relation is actually the statementStrong Interactions Beyond the SM - 66pgsCraig Roberts: Confinement contains Condensates59Expanding the concept of in-hadron condensatesLei Chang, Craig D. Roberts and Peter C. TandyarXiv:1109.2903 [nucl-th], Phys. Rev. C85 (2012) 012201(R)Slide60

GMOR RelationUsing it follows that This equation is valid for any values of mu,d, including the neighbourhood of the chiral limit, whereinStrong Interactions Beyond the SM - 66pgsCraig Roberts: Confinement contains Condensates

60Expanding 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 Slide61

GMOR RelationConsequently, in the neighbourhood of the chiral limitThis 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. Strong Interactions Beyond the SM - 66pgs

Craig Roberts: Confinement contains Condensates61Expanding the concept of in-hadron condensatesLei Chang, Craig D. Roberts and Peter C. TandyarXiv:1109.2903 [nucl-th], Phys. Rev. C85 (2012) 012201(R)Slide62

Expanding the Concept

Strong Interactions Beyond the SM - 66pgsCraig Roberts: Confinement contains Condensates62Slide63

In-Hadron CondensatesPlainly, 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 Q

2 = 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. Strong Interactions Beyond the SM - 66pgsCraig Roberts: Confinement contains Condensates63Expanding the concept of in-hadron condensatesLei Chang, Craig D. Roberts and Peter C. TandyarXiv:1109.2903 [nucl-th], Phys. Rev. C85 (2012) 012201(R)Slide64

Strong Interactions Beyond the SM - 66pgsCraig Roberts: Confinement contains Condensates64

ConfinementSlide65

ConfinementConfinement 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.

Strong Interactions Beyond the SM - 66pgsCraig Roberts: Confinement contains Condensates65Confinement Contains CondensatesS.J. Brodsky, C.D. Roberts, R. Shrock and P.C. TandyarXiv:1202.2376 [nucl-th]Slide66

“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.”

Strong Interactions Beyond the SM - 66pgsCraig Roberts: Confinement contains Condensates66“Void that is truly empty solves dark energy puzzle”Rachel Courtland, New Scientist 4th Sept. 2010Cosmological Constant: Putting QCD condensates back into hadrons reduces the mismatch between experiment and theory by a factor of 1046Possibly by far more, if technicolour-like theories are the correct paradigm for extending the Standard ModelParadigm shift:

In-

Hadron

CondensatesSlide67

This is not the endStrong Interactions Beyond the SM - 66pgsCraig Roberts: Confinement contains Condensates67Slide68

Strong-interaction: QCDAsymptotically freePerturbation theory is valid and accurate tool at large-Q2 Hence chiral limit is definedEssentially

nonperturbative for Q2 < 2 GeV2Strong Interactions Beyond the SM - 66pgsCraig Roberts: Confinement contains Condensates68 Nature’s only example of truly nonperturbative, fundamental theory A-priori, no idea as to what such a theory can produceSlide69

Universal TruthsHadron spectrum, and elastic and transition form factors provide unique information about the long-range interaction between light-quarks and the distribution of hadron's characterising properties amongst its QCD constituents.Dynamical Chiral Symmetry Breaking (DCSB) is the 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: Confinement contains Condensates69Strong Interactions Beyond the SM - 66pgsSlide70

ConfinementInfinitely heavy-quarks plus 2 flavours with mass = ms Lattice spacing = 0.083fmString collapses within one lattice time-step R = 1.24 … 1.32 fmEnergy stored in string at collapse Ecsb = 2 ms

(mpg made via linear interpolation)No flux tube between light-quarksCraig Roberts: Confinement contains Condensates70G. Bali et al., PoS LAT2005 (2006) 308Bsanti-Bs“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.”Strong Interactions Beyond the SM - 66pgsSlide71

Charting the interaction between light-quarksConfinement 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-functionThis 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: Confinement contains Condensates71This is a well-posed problem whose solution is an elemental goal of modern hadron physics.The answer provides QCD’s running coupling.Strong Interactions Beyond the SM - 66pgsSlide72

Charting the interaction between light-quarksThrough 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 ofHadron mass spectrumElastic 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: Confinement contains Condensates72Strong Interactions Beyond the SM - 66pgsSlide73

QCD Sum RulesIntroduction of the gluon vacuum condensate and development of “sum rules” relating properties of low-lying hadronic states to vacuum condensatesCraig Roberts: Confinement contains Condensates

73Strong Interactions Beyond the SM - 66pgsQCD and Resonance Physics. Sum Rules.M.A. Shifman, A.I. Vainshtein, and V.I. Zakharov Nucl.Phys. B147 (1979) 385-447; citations: 3713Slide74

QCD Sum RulesIntroduction of the gluon vacuum condensate and development of “sum rules” relating properties of low-lying hadronic states to vacuum condensatesAt 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

74Strong Interactions Beyond the SM - 66pgsQCD and Resonance Physics. Sum Rules.M.A. Shifman, A.I. Vainshtein, and V.I. Zakharov Nucl.Phys. B147 (1979) 385-447; citations: 3875Slide75

“quark condensate”1960-1980Instantons 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, 1980The pion in QCD J Finger, JE Mandula… - Physics Letters B, 1980No references to this phrase before 1980Craig Roberts: Confinement contains Condensates75Strong Interactions Beyond the SM - 66pgs7170+ references to this phrase since 1980Slide76

Universal ConventionsWikipedia: (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.”

Strong Interactions Beyond the SM - 66pgsCraig Roberts: Confinement contains Condensates76Slide77

Precedent?

Strong Interactions Beyond the SM - 66pgsCraig Roberts: Confinement contains Condensates77Slide78

Precedent-Luminiferous AetherPhysics theories of the late 19th century postulated that, just as water waves must have a medium to move across (water), and audible sound waves require a medium to move through (such as air or water), so also light waves require a medium, the “luminiferous aether”. Apparently unassailable logic

Until, of course, “… the most famous failed experiment to date.” Craig Roberts: Confinement contains Condensates78Strong Interactions Beyond the SM - 66pgsPre-1887On the Relative Motion of the Earth and the Luminiferous EtherMichelson, Albert Abraham & Morley, Edward WilliamsAmerican Journal of Science 34 (1887) 333–345.Since the Earth is in motion, the flow of aether across the Earth should produce a detectable “aether wind”