Craig Roberts Physics Division

1. Deconstructing Hadrons in Continuum Strong QCD 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); Chien-Yeah SENG (UW-Mad) Kun-lun WANG (PKU); Lei CHANG (ANL, FZJ, PKU); Huan CHEN (BIHEP); Ian CLOËT (UAdelaide); Bruno EL-BENNICH (São Paulo); Mario PITSCHMANN (ANL & UW-Mad) David WILSON (ANL); Adnan BASHIR (U Michoácan); Stan BRODSKY (SLAC); Gastão KREIN (São Paulo) Roy HOLT (ANL); Mikhail IVANOV (Dubna); Yu-xin LIU (PKU); Michael RAMSEY-MUSOLF (UW-Mad) Sebastian SCHMIDT (IAS-FZJ & JARA); Robert SHROCK (Stony Brook); Peter TANDY (KSU) Shaolong WAN (USTC) Published collaborations: 2010-present Craig Roberts Physics Division Students Early-career scientists

2. Confinement Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD

3. X Confinement • 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 • But light-quarks are ubiquitous • Flux tubes, linear potentials and string tensions play no role in relativistic quantum field theory with light degrees of freedom. Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD • Gluon and Quark Confinement • Empirical Fact: No coloured states have yet been observed to reach a detector

4. 1993: "for elucidating the quantum structure of electroweak interactions in physics" Regge Trajectories? Phys.Rev. D 62 (2000) 016006 [9 pages] Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD MartinusVeltmann, “Facts and Mysteries in Elementary Particle Physics” (World Scientific, Singapore, 2003): In time the Regge trajectories thus became the cradle of string theory. Nowadays the Regge trajectories have largely disappeared, not in the least because these higher spin bound states are hard to find experimentally. At the peak of the Regge fashion (around 1970) theoretical physics produced many papers containing families of Regge trajectories, with the various (hypothetically straight) lines based on one or two points only!

5. Confinement Confined particle Normal particle complex-P2 complex-P2 timelike axis: P2<0 • Real-axis mass-pole splits, moving into pair(s) of complex conjugate poles or branch points, • or more complicated nonanalyticities … • Spectral density no longer positive semidefinite • & hence state cannot exist in observable spectrum Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD • QFT Paradigm: 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); …

6. Dynamical ChiralSymmetry Breaking Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD

7. Dynamical ChiralSymmetry Breaking Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD • Whilst confinement is contentious … • DCSB is a fact in QCD • Dynamical, not spontaneous • Add nothing to QCD , no Higgs field, nothing, effect achieved purely through the dynamics of gluons and quarks. • 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.

8. Frontiers of Nuclear Science:Theoretical Advances C.D. Roberts, Prog. Part. Nucl. Phys. 61 (2008) 50 M. Bhagwat & P.C. Tandy, AIP Conf.Proc. 842 (2006) 225-227 Mass from nothing! DSE prediction of DCSB confirmed Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD 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.

9. Frontiers of Nuclear Science:Theoretical Advances C.D. Roberts, Prog. Part. Nucl. Phys. 61 (2008) 50 M. Bhagwat & P.C. Tandy, AIP Conf.Proc. 842 (2006) 225-227 Hint of lattice-QCD support for DSE prediction of violation of reflection positivity Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD 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.

10. 12GeVThe Future of JLab Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD Jlab 12GeV: This region scanned by 2<Q2<9 GeV2 elastic & transition form factors.

11. π or K The Future of Drell-Yan N Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD Valence-quark PDFs and PDAs probe this critical and complementary region

12. Science Challenges for the coming decade: 2013-2022 Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD Search for exotic hadrons Exploit opportunities provided by new data on nucleon elastic and transition form factors Precision experimental study of valence region, and theoretical computation of distribution functions and distribution amplitudes Develop QCD as a probe for physics beyond the Standard Model

13. Overarching Science Challenges for the coming decade: 2013-2022 Discover meaning of confinement, and its relationship to DCSB – the origin of visible mass • Search for exotic hadrons • Exploit opportunities provided by new data on nucleon elastic and transition form factors • Precision experimental study of valence region, and theoretical computation of distribution functions and distribution amplitudes • Develop QCD as a probe for physics beyond the Standard Model Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD

14. Universal Truths • Dynamical Chiral Symmetry Breaking (DCSB) is most important mass generating mechanism for visible matter in the Universe. Higgs mechanism is (almost) irrelevant to light-quarks. • Running of quark mass entails that calculations at even modest Q2 require a Poincaré-covariant approach. Covariance requires 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: Deconstructing Hadrons in Continuum Strong QCD • Spectrum of hadrons, and hadron elastic and transition form factors provide unique information about long-range interaction between light-quarks and distribution of hadron'scharacterising properties amongst its QCD constituents.

15. Universal Conventions Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD • 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.”

16. Universal Misapprehensions Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD • Since 1979, DCSB has commonly been associated literally with a spacetime-independent mass-dimension-three “vacuum condensate.” • Under this assumption, “condensates” couple directly to gravity in general relativity and make an enormous contribution to the cosmological constant • Experimentally, the answer is Ωcosm. const. = 0.76 • This mismatch is a bit of a problem.

17. New Paradigm“in-hadron condensates” Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD

18. “Orthodox Vacuum” u d u u d u u u d Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD 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!

19. New Paradigm u d u u d u u u d Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD Vacuum = hadronic fluctuations but no condensates Hadrons = complex, interacting systems within which perturbativebehaviour is restricted to just 2% of the interior

20. Some Relevant References Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD arXiv:1202.2376, Phys. Rev. C85, 065202 (2012) [9 pages] 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 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 HenrikZinkernagel.

21. Charting the interaction between light-quarks Process-independent αS(Q2) → unified description of observables This is a well-posed problem whose solution is an elemental goal of modern hadron physics. The answer provides QCD’s running coupling. Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD • Confinement can be related to the analytic properties of QCD's Schwinger functions. • Question of light-quark confinement is thereby translated into the challenge of charting the infrared behavior of QCD's universalβ-function • Through QCD's DSEs, the pointwisebehaviour of the β-function determines the pattern of chiral symmetry breaking. • DSEs connect β-function to experimental observables. Hence, comparison between computations and observations of • Hadron spectrum, Elastic & transition form factors, Parton distribution fns can be used to chart β-function’s long-range behaviour.

22. Mass function exhibits inflexion • point at QIR ≈ mG ≈ 0.6GeV • So … pQCD is definitely invalid • for momentaQ<QIR • E.g., use of DGLAP equations • cannot be justified in QCD at • Q<QIR=0.6GeV, irrespective of order. • Distribution Functions of the Nucleon and Pion in the Valence Region, Roy J. Holt and Craig D. Roberts, arXiv:1002.4666 [nucl-th], • Rev. Mod. Phys. 82 (2010) pp. 2991-3044 Necessary Precondition Essentially nonperturbative Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD • Experiment ↔ Theory comparison leads to an understanding of long-range behaviour of strong running-coupling • However, if one wants to draw reliable conclusions about Q2-dependence of QCD’s running coupling, • Then, approach must veraciously express Q2-dependence of QCD’s running masseS • True for ALL observables • From spectrum … • through elastic & transition form factors … • to PDFs and GPDs … etc.

23. Dichotomy of the pionGoldstone mode and bound-state fπ Eπ(p2) = B(p2) Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD • 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

24. Looking deeply Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD

25. Empirical status of the Pion’svalence-quark distributions Pion Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD • Owing to absence of pion targets, the pion’s valence-quark distribution functions are measured via the Drell-Yan process: π p → μ+μ− X • Three experiments: CERN (1983 & 1985) and FNAL (1989). No more recent experiments because theory couldn’t even explain these! • Problem Conway et al. Phys. Rev. D 39, 92 (1989) Wijesooriyaet al. Phys.Rev. C 72 (2005) 065203 PDF behaviour at large-x inconsistent with pQCD; viz, expt. (1-x)1+ε cf. QCD (1-x)2+γ

26. Models of the Pion’svalence-quark distributions Pion Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD • (1−x)β with β=0 (i.e., a constant – any fraction is equally probable! ) • AdS/QCD models using light-front holography • Nambu–Jona-Lasinio models, when a translationally invariant regularization is used • (1−x)β with β=1 • Nambu–Jona-Lasinio NJL models with a hard cutoff • Duality arguments produced by some theorists • (1−x)β with 0<β<2 • Relativistic constituent-quark models, with power-law depending on the form of model wave function • (1−x)β with 1<β<2 • Instanton-based models, all of which have incorrect large-k2behaviour

27. Models of the Pion’svalence-quark distributions Pion Completely unsatisfactory. Impossible to suggest that there’s even qualitative agreement! Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD • (1−x)β with β=0 (i.e., a constant – any fraction is equally probable! ) • AdS/QCD models using light-front holography • Nambu–Jona-Lasinio models, when a translationally invariant regularization is used • (1−x)β with β=1 • Nambu–Jona-Lasinio NJL models with a hard cutoff • Duality arguments produced by some theorists • (1−x)β with 0<β<2 • Relativistic constituent-quark models, depending on the form of model wave function • (1−x)β with 1<β<2 • Instanton-based models

28. DSE prediction of the Pion’svalence-quark distributions Pion Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD Consider a theory in which quarks scatter via a vector-boson exchange interaction whosek2>>mG2behaviour is(1/k2)β, Then at a resolving scale Q0 uπ(x;Q0) ~ (1-x)2β namely, the large-x behaviour of the quark distribution function is a direct measure of the momentum-dependence of the underlying interaction. In QCD, β=1 and hence QCDuπ(x;Q0) ~ (1-x)2

29. DSE prediciton of the Pion’svalence-quark distributions Pion Completely unambigous! Direct connection between experiment and theory, empowering both as tools of discovery. Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD Consider a theory in which quarks scatter via a vector-boson exchange interaction whosek2>mG2behaviour is(1/k2)β, Then at a resolving scale Q0 uπ(x;Q0) ~ (1-x)2β namely, the large-x behaviour of the quark distribution function is a direct measure of the momentum-dependence of the underlying interaction. In QCD, β=1 and hence QCDuπ(x;Q0) ~ (1-x)2

30. “Model Scale” Essentially nonperturbative domain Pion • Modern DSE studies have exposed a natural value for the model scale; viz., the gluon mass • Q0≈ mG ≈ 0.6GeV ≈ 1/0.33 fm which is the location of the inflexion point in the chiral-limit dressed-quark mass function Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD • At what scale Q0 should the prediction be valid? • Hitherto, PDF analyses within models have used the resolving scale Q0 as a parameter, to be chosen by requiring agreement between the model and low-moments of the PDF that are determined empirically.

31. QCD-based DSE calculation = (1-x)2+γ Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD

32. Hecht, Roberts, Schmidt Phys.Rev. C 63 (2001) 025213 Reanalysis of qvπ(x) • New experiments were proposed … for accelerators that do not yet exist but the situation remained otherwise unchanged • Until the publication of Distribution Functions of the Nucleon and Pion in the Valence Region, Roy J. Holt and Craig D. Roberts,arXiv:1002.4666 [nucl-th], Rev. Mod. Phys. 82 (2010) pp. 2991-3044 Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD • After first DSE computation, the “Conway et al.” data were reanalysed, this time at next-to-leading-order (Wijesooriyaet al. Phys.Rev. C 72 (2005) 065203) • The new analysis produced a much larger exponent than initially obtained; viz., β=1.87, but now it disagreed equally with model results and the DSE prediction • NB. Within pQCD, one can readily understand why adding a higher-order correction leads to a suppression of qvπ(x) at large-x.

33. Distribution Functions of the Nucleon and Pion in the Valence Region, Roy J. Holt and Craig D. Roberts, arXiv:1002.4666 [nucl-th], Rev. Mod. Phys. 82 (2010) pp. 2991-3044 Reanalysis of qvπ(x) Aicher, Schäfer, Vogelsang, “Soft-Gluon Resummation and the Valence Parton Distribution Function of the Pion,”Phys. Rev. Lett. 105 (2010) 252003 Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD • This article emphasised and explained the importance of the persistent discrepancy between the DSE result and experiment as a challenge to QCD • It prompted another reanalysis of the data, which accounted for a long-overlooked effect: viz., “soft-gluon resummation,” • Compared to previous analyses, we include next-to-leading-logarithmic threshold resummation effects in the calculation of the Drell-Yan cross section. As a result of these, we find a considerably softer valence distribution at high momentum fractions x than obtained in previous next-to-leading-order analyses, in line with expectations based on perturbative-QCD counting rules or Dyson-Schwinger equations.

34. Trang, Bashir, Roberts & Tandy,“Pion and kaon valence-quark parton distribution functions,” arXiv:1102.2448 [nucl-th],Phys. Rev. C 83, 062201(R) (2011) [5 pages] Current status of qvπ(x) Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD Data as reported byE615 DSE prediction (2001)

35. Trang, Bashir, Roberts & Tandy,“Pion and kaon valence-quark parton distribution functions,” arXiv:1102.2448 [nucl-th],Phys. Rev. C 83, 062201(R) (2011) [5 pages] Current status of qvπ(x) Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD Data after inclusion of soft-gluon resummation DSE prediction and modern representation of the data are indistinguishable on the valence-quark domain Emphasises the value of using a single internally-consistent, well-constrained framework to correlate and unify the description of hadron observables

36. Pion’s Light-Front Distribution Amplitude Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD

37. Expression exact in QCD – no corrections Pion’s valence-quark Distribution Amplitude Expose DCSB in a quantum mechanical wave function Pion’s Bethe-Salpeter wave function • Work now underway with sophisticated rainbow-ladder interaction: Chang, Cloët, Roberts, Schmidt & Tandy Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD Reconstruct φπ(x) from moments: entails Contact interaction (1/k2)ν , ν=0 Straightforward exercise to show ∫01 dxxmφπ(x) = fπ1/(1+m) , hence φπ(x)= fπ Θ(x)Θ(1-x)

38. Pion’s valence-quark Distribution Amplitude Leading pQCDφπ(x)=6 x (1-x) Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD • Using simple parametrisations of solutions to the gap and Bethe-Salpeter equations, rapid and semiquantitatively reliable estimates can be made for φπ(x) • (1/k2)ν=0 • (1/k2)ν =½ • (1/k2)ν =1 • Again, unambiguous and direct mapping between behaviour of interaction and behaviour of distribution amplitude

39. Chang, Cloët, Roberts, Schmidt & Tandy,in progress; Si-xue Qin, Lei Chang, Yu-xin Liu, Craig Roberts and David Wilson, arXiv:1108.0603 [nucl-th], Phys. Rev. C 84 042202(R) (2011) Pion’s valence-quark Distribution Amplitude Leading pQCDφπ(x)=6 x (1-x) • Such behaviour is only obtained with • (1) Running mass in dressed-quark propagators • (2) Pointwise expression of Goldstone’s theorem Eπ(k2) but constant mass quark a2>0 a2<0 Reconstructed from 100 moments Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD Preliminary results: rainbow-ladder QCDanalyses of renormalisation-group-improved (1/k2)ν =1 interaction – humped disfavoured but modest flattening

40. Pion’s valence-quark Distribution Amplitude Leading pQCDφπ(x)=6 x (1-x) • Preliminary results, rainbow-ladder QCD analyses of (1/k2)ν =1 interaction • humped disfavoured but modest flattening Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD • x ≈ 0 & x ≈ 1 correspond to maximum relative momentum within bound-state • expose pQCD physics • x ≈ ½ corresponds to minimum possible relative momentum • behaviour of distribution around midpoint is strongly influence by DCSB

41. Pion’s valence-quark Distribution Amplitude Leading pQCDφπ(x)=6 x (1-x) These computations are the first to offer the possibility of directly exposing DCSB – pointwise – in the light-front frame. • Preliminary results, rainbow-ladder QCD analyses of (1/k2)ν =1 interaction • humped disfavoured but modest flattening Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD • x ≈ 0 & x ≈ 1 correspond to maximum relative momentum within bound-state • expose pQCD physics • x ≈ ½ corresponds to minimum possible relative momentum • behaviour of distribution around midpoint is strongly influence by DCSB

42. Grand Unification Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD

43. Unification of Meson & Baryon Properties Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD • Correlate the properties of meson and baryon ground- and excited-states within a single, symmetry-preserving framework • Symmetry-preserving means: • Poincaré-covariant • Guarantee Ward-Takahashi identities • Express accurately the pattern by which symmetries are broken

44. R.T. Cahill et al., Austral. J. Phys. 42 (1989) 129-145 Faddeev Equation quark exchange ensures Pauli statistics quark diquark composed of strongly-dressed quarks bound by dressed-gluons Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD • Linear, Homogeneous Matrix equation • Yields wave function (Poincaré Covariant FaddeevAmplitude) thatdescribes quark-diquark relative motion within the nucleon • Scalar and Axial-Vector Diquarks . . . • Both have “correct” parity and “right” masses • In Nucleon’s Rest Frame Amplitude has s−, p− & d−wave correlations

45. ContactInteraction Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD Symmetry-preserving treatment of vector×vector contact interaction is useful tool for the study of phenomena characterised by probe momenta less-than the dressed-quark mass, M. Because: For experimental observables determined by probe momenta Q2<M2, contact interaction results are not realistically distinguishable from those produced by the most sophisticated renormalisation-group-improved kernels. Symmetry-preserving regularisation of the contact interaction serves as a useful surrogate, opening domains which analyses using interactions that more closely resemble those of QCD are as yet unable to enter. They’re critical in attempts to use data as tool for charting nature of the quark-quark interaction at long-range; i.e., identifying signals of the running of couplings and masses in QCD.

46. Symmetry-preserving treatment of vector-vector contact-interaction: series of papers establishes strengths & limitations. Contact Interaction Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD arXiv:1204.2553 [nucl-th] Spectrum of hadrons with strangenessChen Chen, L. Chang, C.D. Roberts, Shaolong Wan and D.J. Wilson arXiv:1112.2212 [nucl-th], Phys. Rev. C85 (2012) 025205 [21 pages] Nucleon and Roper electromagnetic elastic and transition form factors, D. J. Wilson, I. C. Cloët, L. Chang and C. D. Roberts arXiv:1102.4376 [nucl-th], Phys. Rev. C 83, 065206 (2011) [12 pages] , π- and ρ-mesons, and their diquark partners, from a contact interaction, H.L.L. Roberts, A. Bashir, L.X. Gutierrez-Guerrero, C.D. Roberts and David J. Wilson  arXiv:1101.4244 [nucl-th], Few Body Syst. 51 (2011) pp. 1-25 Masses of ground and excited-state hadronsH.L.L. Roberts, Lei Chang, Ian C. Cloët and Craig D. Roberts arXiv:1009.0067 [nucl-th], Phys. Rev. C82 (2010) 065202 [10 pages] Abelian anomaly and neutral pion productionHannes L.L. Roberts, C.D. Roberts, A. Bashir, L. X. Gutiérrez-Guerrero & P. C. Tandy arXiv:1002.1968 [nucl-th], Phys. Rev. C 81 (2010) 065202 (5 pages) Pion form factor from a contact interaction L. XiomaraGutiérrez-Guerrero, AdnanBashir, Ian C. Cloët and C. D. Roberts

47. arXiv:1204.2553 [nucl-th], Spectrum of hadrons with strangeness, Chen Chen, L. Chang, C.D. Roberts, Shaolong Wan and D.J. Wilson Spectrum of Hadronswith Strangeness Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD Solve gap equation for u & s-quarks Input ratio ms /mu = 24 is consistent with modern estimates Output ratio Ms /Mu = 1.43 shows dramatic impact of DCSB, even on the s-quark: Ms-ms = 0.36 GeV = M0 … This is typical of all DSE and lattice studies κ = in-hadron condensate rises slowly with mass of hadron

48. arXiv:1204.2553 [nucl-th], Spectrum of hadrons with strangeness, Chen Chen, L. Chang, C.D. Roberts, Shaolong Wan and D.J. Wilson Spectrum of Mesonswith Strangeness Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD Solve Bethe-Salpeter equations for mesons and diquarks

49. arXiv:1204.2553 [nucl-th], Spectrum of hadrons with strangeness, Chen Chen, L. Chang, C.D. Roberts, Shaolong Wan and D.J. Wilson Spectrum of Mesonswith Strangeness Perhaps underestimate radial-ground splitting by 0.2GeV • Computed values for ground-states are greater than the • empirical masses, where they are known. • Typical of DCSB-corrected kernels that omit resonant • contributions; i.e., do not contain effects that may • phenomenologically be associated with a meson cloud. Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD Solve Bethe-Salpeter equations for mesons and diquarks

50. arXiv:1204.2553 [nucl-th], Spectrum of hadrons with strangeness, Chen Chen, L. Chang, C.D. Roberts, Shaolong Wan and D.J. Wilson Spectrum of Diquarkswith Strangeness Craig Roberts: Deconstructing Hadrons in Continuum Strong QCD Solve Bethe-Salpeter equations for mesons and diquarks