Craig Roberts Physics Division Students Early-career scientists Published collaborations: 2010-present 1.Rocio BERMUDEZ (U Michoácan); 2.Chen CHEN (ANL,

Craig Roberts Physics Division Students Early-career scientists Published collaborations: 2010-present 1.Rocio BERMUDEZ (U Michoácan); 2.Chen CHEN (ANL, IIT, USTC); 3.Xiomara GUTIERREZ-GUERRERO (U Michoácan); 4.Trang NGUYEN (KSU); 5.Si-xue QIN (PKU); 6.Hannes ROBERTS (ANL, FZJ, UBerkeley); 7.Chien-Yeah SENG (UW-Mad) 8.Kun-lun WANG (PKU); 9.Lei CHANG (ANL, FZJ, PKU); 10.Huan CHEN (BIHEP); 11.Ian CLOËT (UAdelaide); 12.Bruno EL-BENNICH (São Paulo); 13.Mario PITSCHMANN (ANL & UW-Mad) 14.David WILSON (ANL); 15.Adnan BASHIR (U Michoácan); 16.Stan BRODSKY (SLAC); 17.Gastão KREIN (São Paulo) 18.Roy HOLT (ANL); 19.Mikhail IVANOV (Dubna); 20.Yu-xin LIU (PKU); 21.Michael RAMSEY-MUSOLF (UW-Mad) 22.Sebastian SCHMIDT (IAS-FZJ & JARA); 23.Robert SHROCK (Stony Brook); 24.Peter TANDY (KSU) 25.Shaolong WAN (USTC)

U.Complutense, Madrid: 27/09/12 - 78pgs Craig Roberts: Confinement contains Condensates 2

Some Relevant References  arXiv:1202.2376, Phys. Rev. C85 (2012) 065202arXiv:1202.2376Phys. Rev. C85 (2012) 065202 Confinement contains condensates Stanley 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 condensates Lei Chang, Craig D. Roberts and Peter C. Tandy arXiv:1109.2903 [nucl-th]Phys. Rev. C85 (2012) 012201(RapCom)  arXiv:1005.4610 [nucl-th], Phys. Rev. C82 (2010) 022201(RapCom.) 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) arXiv:0905.1151 [hep-th]PNAS 108, 45 (2011) Condensates in Quantum Chromodynamics and the Cosmological Constant, Brodsky and Shrock,  hep-th/0012253 hep-th/0012253 The Quantum vacuum and the cosmological constant problem, Svend Erik Rugh and Henrik Zinkernagel. U.Complutense, Madrid: 27/09/12 - 78pgs Craig Roberts: Confinement contains Condensates 3

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. Craig Roberts: Confinement contains Condensates 5 X U.Complutense, Madrid: 27/09/12 - 78pgs Colour singlets

Confinement  Infinitely heavy-quarks plus 2 flavours with mass = m s –Lattice spacing = 0.083fm –String collapses within one lattice time-step R = 1.24 … 1.32 fm –Energy stored in string at collapse E c sb = 2 m s –(mpg made via linear interpolation)  No flux tube between light-quarks Craig Roberts: Confinement contains Condensates 6 G. Bali et al., PoS LAT2005 (2006) 308PoS LAT2005 (2006) 308 BsBs anti -B s “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.” U.Complutense, Madrid: 27/09/12 - 78pgs

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); … Craig Roberts: Confinement contains Condensates 7 complex-P 2 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 Normal particle Confined particle U.Complutense, Madrid: 27/09/12 - 78pgs timelike axis: P 2 <0

Dressed-gluon propagator  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 –m G ≈ 2-4 Λ QCD Craig Roberts: Confinement contains Condensates 8 perturbative, massless gluon massive, unconfined gluon IR-massive but UV-massless, confined gluon A.C. Aguilar et al., Phys.Rev. D80 (2009) 085018Phys.Rev. D80 (2009) 085018 U.Complutense, Madrid: 27/09/12 - 78pgs

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(m G 2 )/π = 2-4 U.Complutense, Madrid: 27/09/12 - 78pgs Craig Roberts: Confinement contains Condensates 9 Qin et al., Phys. Rev. C 84 042202(R) (2011)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 m G (0) = 0.67-0.81 GeV m G (m G 2 ) = 0.53-0.64 GeV

Dynamical Chiral Symmetry Breaking SStrong-interaction: Q CD CConfinement –E–Empirical feature –M–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 –S–Still circumstantial, no proof yet of confinement OOn the other hand, DCSB is a fact in QCD –I–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: Confinement contains Condensates 11 U.Complutense, Madrid: 27/09/12 - 78pgs

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 Condensates 12 U.Complutense, Madrid: 27/09/12 - 78pgs C.D. Roberts, Prog. Part. Nucl. Phys. 61 (2008) 50Prog. Part. Nucl. Phys. 61 (2008) 50 M. Bhagwat & P.C. Tandy, AIP Conf.Proc. 842 (2006) 225-227AIP Conf.Proc. 842 (2006) 225-227

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 Condensates 13 DSE prediction of DCSB confirmed Mass from nothing! U.Complutense, Madrid: 27/09/12 - 78pgs

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. Craig Roberts: Confinement contains Condensates 14 Jlab 12GeV: Scanned by 2<Q 2 <9 GeV 2 elastic & transition form factors. U.Complutense, Madrid: 27/09/12 - 78pgs

Gluon & quark mass-scales  m g (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 U.Complutense, Madrid: 27/09/12 - 78pgs Craig Roberts: Confinement contains Condensates 15

 Infinitely many coupled equations: Kernel of the equation for the quark self-energy involves: –D μν (k) – dressed-gluon propagator –Γ ν (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 U.Complutense, Madrid: 27/09/12 - 78pgs Craig Roberts: Confinement contains Condensates 17 Invaluable check on practical truncation schemes

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 U.Complutense, Madrid: 27/09/12 - 78pgs Craig Roberts: Confinement contains Condensates 18 Axial-Vector vertex Satisfies an inhomogeneous Bethe-Salpeter equation Quark propagator satisfies a gap equation Kernels of these equations are completely different But they must be intimately related Relationship must be preserved by any truncation Highly nontrivial constraint FAILURE has an extremely high cost – loss of any connection with QCD

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 good phenomenological results, sometimes with poor results  Neither good nor bad could be explained U.Complutense, Madrid: 27/09/12 - 78pgs Craig Roberts: Confinement contains Condensates 19 quark-antiquark scattering kernel

Persistent challenge - truncation scheme  Happily, that has changed and there are now two nonperturbative & symmetry preserving truncation schemes 1.1995 – 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 EquationsPhys. Rev. D 52 (1995) 4736 1996 – A. Bender, C.D. Roberts and L. von Smekal, Phys.Lett. B 380 (1996) 7, Goldstone Theorem and Diquark Confinement Beyond Rainbow Ladder ApproximationPhys.Lett. B 380 (1996) 7 2.2009 – Lei Chang and C.D. Roberts, Phys. Rev. Lett. 103 (2009) 081601, 0903.5461 [nucl-th], Sketching the Bethe-Salpeter kernelPhys. Rev. Lett. 103 (2009) 081601 0903.5461 [nucl-th]  Enables proof of numerous exact results U.Complutense, Madrid: 27/09/12 - 78pgs Craig Roberts: Confinement contains Condensates 20

Dichotomy of the pion Craig Roberts: Confinement contains Condensates 21  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: U.Complutense, Madrid: 27/09/12 - 78pgs

Dichotomy of the pion Goldstone 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. U.Complutense, Madrid: 27/09/12 - 78pgs Craig Roberts: Confinement contains Condensates 22

Pion’s Goldberger -Treiman relation Craig Roberts: Confinement contains Condensates 24  Pion’s Bethe-Salpeter amplitude Solution of the Bethe-Salpeter equation  Dressed-quark propagator  Axial-vector Ward-Takahashi identity entails Pseudovector components necessarily nonzero. Cannot be ignored! Exact in Chiral QCD U.Complutense, Madrid: 27/09/12 - 78pgs Miracle: two body problem solved, almost completely, once solution of one body problem is known Maris, Roberts and Tandy nucl-th/9707003nucl-th/9707003, Phys.Lett. B420 (1998) 267-273

Dichotomy of the pion Goldstone 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 U.Complutense, Madrid: 27/09/12 - 78pgs Craig Roberts: Confinement contains Condensates 25

Dichotomy of the pion Mass Formula for 0 — Mesons  Mass-squared of the pseudscalar hadron  Sum of the current-quark masses of the constituents; e.g., pion = m u ς + m d ς, where “ς” is the renormalisation point U.Complutense, Madrid: 27/09/12 - 78pgs Craig Roberts: Confinement contains Condensates 26 Maris, Roberts and Tandy nucl-th/9707003nucl-th/9707003, Phys.Lett. B420 (1998) 267-273

Dichotomy of the pion Mass 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 U.Complutense, Madrid: 27/09/12 - 78pgs Craig Roberts: Confinement contains Condensates 27 Maris, Roberts and Tandy nucl-th/9707003nucl-th/9707003, Phys.Lett. B420 (1998) 267-273

Dichotomy of the pion Mass 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 U.Complutense, Madrid: 27/09/12 - 78pgs Craig Roberts: Confinement contains Condensates 28 Maris, Roberts and Tandy nucl-th/9707003nucl-th/9707003, Phys.Lett. B420 (1998) 267-273

Dichotomy of the pion Mass Formula for 0 — Mesons  Consider the case of light quarks; namely, m q ≈ 0 –If chiral symmetry is dynamically broken, then f H5 → f H5 0 ≠ 0 ρ H5 → – / f H5 0 ≠ 0 both of which are independent of m q  Hence, one arrives at the corollary U.Complutense, Madrid: 27/09/12 - 78pgs Craig Roberts: Confinement contains Condensates 29 Gell-Mann, Oakes, Renner relation 1968 The so-called “vacuum quark condensate.” More later about this. Maris, Roberts and Tandy nucl-th/9707003nucl-th/9707003, Phys.Lett. B420 (1998) 267-273

Dichotomy of the pion Mass Formula for 0 — Mesons  Consider a different case; namely, one quark mass fixed and the other becoming very large, so that m q /m Q << 1  Then –f H5 ∝ 1/√m H5 –ρ H5 ∝ √m H5 and one arrives at m H5 ∝ m Q U.Complutense, Madrid: 27/09/12 - 78pgs Craig Roberts: Confinement contains Condensates 30 Maris, Roberts and Tandy nucl-th/9707003nucl-th/9707003, Phys.Lett. B420 (1998) 267-273 Provides QCD proof of potential model result Ivanov, Kalinovsky, Roberts Phys. Rev. D 60, 034018 (1999) [17 pages]

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 Condensates 33 U.Complutense, Madrid: 27/09/12 - 78pgs

Nambu – Jona-Lasinio Model Craig Roberts: Confinement contains Condensates 34  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 rotation  Nontrivial Vacuum is “Born” U.Complutense, Madrid: 27/09/12 - 78pgs 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. II Y. Nambu, G. Jona-Lasinio, Phys.Rev. 124 (1961) 246-254

Gell-Mann – Oakes – Renner Relation Craig Roberts: Confinement contains Condensates 35  This paper derives a relation between m π 2 and the expectation-value, where u o 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 u 0 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=m π 2 → t=0 to take → … in-pion → in-vacuum U.Complutense, Madrid: 27/09/12 - 78pgs 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 – Renner Relation Craig Roberts: Confinement contains Condensates 36  PCAC hypothesis; viz., pion field dominates the divergence of the axial-vector current  Soft-pion theorem  In QCD, this is and one therefore has U.Complutense, Madrid: 27/09/12 - 78pgs 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 rotation Therefore, isolates explicit chiral-symmetry breaking term in the putative Hamiltonian Zhou Guangzhao 周光召 Born 1929 Changsha, Hunan province

Gell-Mann – Oakes – Renner Relation Craig Roberts: Confinement contains Condensates 37  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. U.Complutense, Madrid: 27/09/12 - 78pgs - (0.25GeV) 3

Note of Warning Craig Roberts: Confinement contains Condensates 38 U.Complutense, Madrid: 27/09/12 - 78pgs 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 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.

QCD Sum Rules  Introduction of the gluon vacuum condensate and development of “sum rules” relating properties of low-lying hadronic states to vacuum condensates Craig Roberts: Confinement contains Condensates 39 U.Complutense, Madrid: 27/09/12 - 78pgs 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 40 U.Complutense, Madrid: 27/09/12 - 78pgs 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, Instantons in non-perturbative QCD vacuum MA Shifman, AI Vainshtein… - Nuclear Physics B, 1980  Instanton density in a theory with massless quarks, Instanton density in a theory with massless quarks MA Shifman, AI Vainshtein… - Nuclear Physics B, 1980  Exotic new quarks and dynamical symmetry breaking, Exotic new quarks and dynamical symmetry breaking WJ Marciano - Physical Review D, 1980  The pion in QCD The pion in QCD J Finger, JE Mandula… - Physics Letters B, 1980 No references to this phrase before 1980 Craig Roberts: Confinement contains Condensates 41 U.Complutense, Madrid: 27/09/12 - 78pgs

Universal Conventions  Wikipedia: (http://en.wikipedia.org/wiki/QCD_vacuum)(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 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.” U.Complutense, Madrid: 27/09/12 - 78pgs Craig Roberts: Confinement contains Condensates 42

“Orthodox Vacuum”  Vacuum = “frothing sea”  Hadrons = bubbles in that “sea”, containing nothing but quarks & gluons interacting perturbatively, unless they’re near the bubble’s boundary, whereat they feel they’re trapped! U.Complutense, Madrid: 27/09/12 - 78pgs Craig Roberts: Confinement contains Condensates 43 u u u d u u d d u

Background  Worth noting that nonzero vacuum expectation values of local operators in QCD—the so-called vacuum condensates—are phenomenological parameters, which were introduced at a time of limited computational resources in order to assist with the theoretical estimation of essentially nonperturbative strong- interaction matrix elements.  A universality of these condensates was assumed, namely, that the properties of all hadrons could be expanded in terms of the same condensates. While this helps to retard proliferation, there are nevertheless infinitely many of them.  As qualities associated with an unmeasurable state (the vacuum), such condensates do not admit direct measurement. Practitioners have attempted to assign values to them via an internally consistent treatment of many separate empirical observables.  However, only one, the so-called quark condensate, is attributed a value with any confidence. U.Complutense, Madrid: 27/09/12 - 78pgs Craig Roberts: Confinement contains Condensates 44

QCDQCD  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 45 U.Complutense, Madrid: 27/09/12 - 78pgs 1973-1974

QCDQCD Why does it matter? Craig Roberts: Confinement contains Condensates 46 U.Complutense, Madrid: 27/09/12 - 78pgs 1973-1974

“Dark Energy”  Two pieces of evidence for an accelerating universe 1)Observations of type Ia supernovae → the rate of expansion of the Universe is growing 2)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 Ω 0 = 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, Ω M = 0.33 ⁺⁄₋ 0.04. Thus, ⅔ of the critical density is missing. Craig Roberts: Confinement contains Condensates 47 U.Complutense, Madrid: 27/09/12 - 78pgs

“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 Condensates 48 U.Complutense, Madrid: 27/09/12 - 78pgs  In order to have escaped detection, the missing energy must be smoothly distributed.  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.” Craig Roberts: Confinement contains Condensates 49 U.Complutense, Madrid: 27/09/12 - 78pgs “The advent of quantum field theory made consideration of the cosmological constant obligatory not optional.” Michael Turner, “Dark Energy and the New Cosmology”

“Dark Energy”  QCD vacuum contribution  If 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 GeV  One obtains therefrom: Craig Roberts: Confinement contains Condensates 50 U.Complutense, Madrid: 27/09/12 - 78pgs “The biggest embarrassment in theoretical physics.” Mass-scale generated by spacetime-independent condensate Enormous and even greater contribution from Higgs VEV!

QCDQCD 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 52 U.Complutense, Madrid: 27/09/12 - 78pgs 1973-1974

What is measurable? Craig Roberts: Confinement contains Condensates 53 U.Complutense, Madrid: 27/09/12 - 78pgs S. Weinberg, Physica 96A (1979) Elements of truth in this perspective

In-meson condensate Craig Roberts: Confinement contains Condensates 55 Maris & Roberts nucl-th/9708029  Pseudoscalar projection of pion’s Bethe-Salpeter wave- function onto the origin in configuration space: |Ψ π 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 U.Complutense, Madrid: 27/09/12 - 78pgs

In-meson condensate Craig Roberts: Confinement contains Condensates 56  Pseudovector projection of pion’s Bethe-Salpeter wave- function onto the origin in configuration space: |Ψ π 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 U.Complutense, Madrid: 27/09/12 - 78pgs Maris & Roberts nucl-th/9708029

In-meson condensate Craig Roberts: Confinement contains Condensates 57  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. U.Complutense, Madrid: 27/09/12 - 78pgs Chiral limit Maris & Roberts nucl-th/9708029 |Ψ π PS (0)|*|Ψ π 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 Craig Roberts: Confinement contains Condensates 58 U.Complutense, Madrid: 27/09/12 - 78pgs Langeld, Roberts et al. nucl-th/0301024nucl-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., |Ψ π PS (0)|*|Ψ π 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. U.Complutense, Madrid: 27/09/12 - 78pgs Craig Roberts: Confinement contains Condensates 59 QCDQCD Brodsky, Roberts, Shrock, Tandy, Phys. Rev. C82 (Rapid Comm.) (2010) 022201Phys. Rev. C82 (Rapid Comm.) (2010) 022201 Brodsky and Shrock, PNAS 108, 45 (2011)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 f π and ρ π U.Complutense, Madrid: 27/09/12 - 78pgs Craig Roberts: Confinement contains Condensates 60 Brodsky, Roberts, Shrock, Tandy, Phys. Rev. C82 (Rapid Comm.) (2010) 022201Phys. Rev. C82 (Rapid Comm.) (2010) 022201 Brodsky and Shrock, PNAS 108, 45 (2011)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 f π and ρ π –And U.Complutense, Madrid: 27/09/12 - 78pgs Craig Roberts: Confinement contains Condensates 61 Chiral limit Brodsky, Roberts, Shrock, Tandy, Phys. Rev. C82 (Rapid Comm.) (2010) 022201Phys. Rev. C82 (Rapid Comm.) (2010) 022201 Brodsky and Shrock, PNAS 108, 45 (2011)PNAS 108, 45 (2011)

Topological charge of “vacuum”  Wikipedia : Instanton effects are important in understanding the formation of condensates in the vacuum of quantum chromodynamics (QCD)quantum chromodynamics  Wikipedia : The difference between the mass of the η and that of the η' is larger than the quark model can naturally explain.quark model This “η-η' puzzle” is resolved by instantons.η-η' puzzleinstantons  Claimed that some lattice simulations demonstrate nontrivial topological structures in QCD vacuum  Now illustrate new paradigm perspective … U.Complutense, Madrid: 27/09/12 - 78pgs Craig Roberts: Confinement contains Condensates 62

 AVWTI ⇒ QCD mass formulae for all pseudoscalar mesons, including those which are charge-neutral  Consider the limit of a U(N f )-symmetric mass matrix, then this formula yields: Topological charge density: Q(x) = i(α s /4π) tr C ε μνρσ F μν F ρσ  Plainly, the η – η’ mass splitting is nonzero in the chiral limit so long as ν η’ ≠ 0 … viz., so long as the topological content of the η’ is nonzero! Charge-neutral pseudoscalar mesons U.Complutense, Madrid: 27/09/12 - 78pgs Craig Roberts: Confinement contains Condensates 63 Bhagwat, Chang, Liu, Roberts, Tandy Phys.Rev. C76 (2007) 045203 Algebraic result. Very different than requiring QCD’s vacuum to possess nontrivial topological structure Qualitatively the same as f π, a property of the bound- state

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 U.Complutense, Madrid: 27/09/12 - 78pgs Craig Roberts: Confinement contains Condensates 64 Bhagwat, Chang, Liu, Roberts, Tandy Phys.Rev. C76 (2007) 045203

GMOR Relation  Valuable to highlight the precise form of the Gell-Mann–Oakes– Renner (GMOR) relation: Eq. (3.4) in Phys.Rev. 175 (1968) 2195Phys.Rev. 175 (1968) 2195 o m π is the pion’s mass o H χsb 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. U.Complutense, Madrid: 27/09/12 - 78pgs Craig Roberts: Confinement contains Condensates 66 Expanding the concept of in-hadron condensates Lei Chang, Craig D. Roberts and Peter C. Tandy arXiv:1109.2903 [nucl-th], Phys. Rev. C85 (2012) 012201(R) arXiv:1109.2903 [nucl-th]Phys. Rev. C85 (2012) 012201(R)

GMOR Relation  In terms of QCD quantities, GMOR relation entails o m ud ζ = m u ζ + m d ζ … the current-quark masses o S π ζ (0) is the pion’s scalar form factor at zero momentum transfer, Q 2 =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 U.Complutense, Madrid: 27/09/12 - 78pgs Craig Roberts: Confinement contains Condensates 67 Expanding the concept of in-hadron condensates Lei Chang, Craig D. Roberts and Peter C. Tandy arXiv:1109.2903 [nucl-th], Phys. Rev. C85 (2012) 012201(R) arXiv:1109.2903 [nucl-th]Phys. Rev. C85 (2012) 012201(R)

GMOR Relation  Using it follows that  This equation is valid for any values of m u,d, including the neighbourhood of the chiral limit, wherein U.Complutense, Madrid: 27/09/12 - 78pgs Craig Roberts: Confinement contains Condensates 68 Expanding the concept of in-hadron condensates Lei Chang, Craig D. Roberts and Peter C. Tandy arXiv:1109.2903 [nucl-th], Phys. Rev. C85 (2012) 012201(R) arXiv:1109.2903 [nucl-th]Phys. Rev. C85 (2012) 012201(R) Maris, Roberts and Tandy nucl-th/9707003nucl-th/9707003, Phys.Lett. B420 (1998) 267-273

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. U.Complutense, Madrid: 27/09/12 - 78pgs Craig Roberts: Confinement contains Condensates 69 Expanding the concept of in-hadron condensates Lei Chang, Craig D. Roberts and Peter C. Tandy arXiv:1109.2903 [nucl-th], Phys. Rev. C85 (2012) 012201(R) arXiv:1109.2903 [nucl-th]Phys. Rev. C85 (2012) 012201(R)

In-Hadron Condensates  Plainly, the in-pseudoscalar-meson condensate can be represented through the pseudoscalar meson’s scalar form factor at zero momentum transfer Q 2 = 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 Q 2 = 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. U.Complutense, Madrid: 27/09/12 - 78pgs Craig Roberts: Confinement contains Condensates 71 Expanding the concept of in-hadron condensates Lei Chang, Craig D. Roberts and Peter C. Tandy arXiv:1109.2903 [nucl-th], Phys. Rev. C85 (2012) 012201(R) arXiv:1109.2903 [nucl-th]Phys. Rev. C85 (2012) 012201(R)

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. … U.Complutense, Madrid: 27/09/12 - 78pgs Craig Roberts: Confinement contains Condensates 72

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. U.Complutense, Madrid: 27/09/12 - 78pgs Craig Roberts: Confinement contains Condensates 75 Confinement Contains Condensates S.J. Brodsky, C.D. Roberts, R. Shrock and P.C. Tandy arXiv:1202.2376 [nucl-th] arXiv:1202.2376 [nucl-th]

“Orthodox Vacuum”  Vacuum = “frothing sea”  Hadrons = bubbles in that “sea”, containing nothing but quarks & gluons interacting perturbatively, unless they’re near the bubble’s boundary, whereat they feel they’re trapped! U.Complutense, Madrid: 27/09/12 - 78pgs Craig Roberts: Confinement contains Condensates 76 u u u d u u d d u

New Paradigm  Vacuum = hadronic fluctuations but no condensates  Hadrons = complex, interacting systems within which perturbative behaviour is restricted to just 2% of the interior U.Complutense, Madrid: 27/09/12 - 78pgs Craig Roberts: Confinement contains Condensates 77 u u u d u u d d u

“ 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.”dark energy U.Complutense, Madrid: 27/09/12 - 78pgs Craig Roberts: Confinement contains Condensates 78 “Void that is truly empty solves dark energy puzzle” Rachel Courtland, New Scientist 4 th Sept. 2010 Cosmological Constant: Putting QCD condensates back into hadrons reduces the mismatch between experiment and theory by a factor of 10 46 Possibly by far more, if technicolour-like theories are the correct paradigm for extending the Standard Model Paradigm shift: In-Hadron Condensates

Simulations with static quarks  Mean-field picture of gauge-field action – 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 M Q /M Q ] –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. U.Complutense, Madrid: 27/09/12 - 78pgs Craig Roberts: Confinement contains Condensates 80

Craig Roberts Physics Division Students Early-career scientists Published collaborations: 2010-present 1.Rocio BERMUDEZ (U Michoácan); 2.Chen CHEN (ANL,

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Craig Roberts Physics Division Students Early-career scientists Published collaborations: 2010-present 1.Rocio BERMUDEZ (U Michoácan); 2.Chen CHEN (ANL,

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