Craig Roberts Physics Division
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Transcript of Craig Roberts Physics Division
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Imaging Dynamical Chiral Symmetry Breaking
Craig Roberts
Physics Division
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Craig Roberts: Imaging DCSB (38p)
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Collaborators: 2011-Present1. Rocio BERMUDEZ (U Michoácan);2. Chen CHEN (ANL, IIT, USTC);3. Xiomara GUTIERREZ-GUERRERO (U Michoácan);4. Trang NGUYEN (KSU);5. Khépani Raya (U Michoácan);6. Hannes ROBERTS (ANL, FZJ, UBerkeley);7. Chien-Yeah SENG (UW-Mad)8. Kun-lun WANG (PKU);9. J. Javier COBOS-MARTINEZ (U.Sonora);10. Mario PITSCHMANN (ANL & UW-Mad);11. Si-xue QIN (U. Frankfurt am Main);12. Jorge SEGOVIA (ANL);13. David WILSON (ODU);14. Lei CHANG (FZJ); 15. Ian CLOËT (ANL);16. Bruno EL-BENNICH (São Paulo);
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17. Adnan BASHIR (U Michoácan);18. Stan BRODSKY (SLAC);19. Gastão KREIN (São Paulo)20. Roy HOLT (ANL);21. Mikhail IVANOV (Dubna);22. Yu-xin LIU (PKU);23. Michael RAMSEY-MUSOLF (UW-Mad)24. Alfredo RAYA (U Michoácan);25. Sebastian SCHMIDT (IAS-FZJ & JARA);26. Robert SHROCK (Stony Brook);27. Peter TANDY (KSU);28. Tony THOMAS (U.Adelaide)29. Shaolong WAN (USTC)
StudentsPostdocsAsst. Profs.
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Overarching Science Challenges for the coming
decade: 2013-2022 Discover the meaning of confinement Determine its connection with dynamical chiral
symmetry breaking Elucidate their signals in observables
… so experiment and theory together can map the nonperturbative behaviour of the strong interactionIs it possible that two phenomena, so critical in the Standard Model and tied to the dynamical generation of a single mass-scale, can have different origins and fates?
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Craig Roberts: Imaging DCSB (38p)
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Immediate Science Challenges for the coming decade: 2013-
2022 Exploit opportunities provided by new data on hadron elastic and transition form factors– Chart infrared evolution of QCD’s coupling and dressed-masses – Reveal correlations that are key to baryon structure– Expose facts & fallacies in modern descriptions of hadron structure
Precision experimental study of valence region, and theoretical computation of distribution functions and distribution amplitudes– Computation is critical– Without it, no amount of data will reveal anything about the theory
underlying the phenomena of strong interaction physics
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What is QCD?Baryons 2013: 24-28 June 2013
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Very likely a self-contained, nonperturbatively renormalisable and hence well defined Quantum Field TheoryThis is not true of QED – cannot be defined nonperturbatively
No confirmed breakdown over an enormous energy domain: 0 GeV < E < 8000 GeV
Increasingly likely that any extension of the Standard Model will be based on the paradigm established by QCD – Extended Technicolour: electroweak symmetry breaks via a
fermion bilinear operator in a strongly-interacting non-Abelian theory. (Andersen et al. “Discovering Technicolor” Eur.Phys.J.Plus 126 (2011) 81)Higgs sector of the SM becomes an effective description of a more fundamental fermionic theory, similar to the Ginzburg-Landau theory of superconductivity
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Craig Roberts: Imaging DCSB (38p)
(not an effective theory)QCD is a Theory
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What is Confinement?
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Light quarks & Confinement
A unit area placed midway between the quarks and perpendicular to the line connecting them intercepts a constant number of field lines, independent of the distance between the quarks. This leads to a constant force between the quarks – and a large force at that, equal to about 16 metric tons.”
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Craig Roberts: Imaging DCSB (38p)
Folklore … Hall-D Conceptual Design Report(5) “The color field lines between a quark and an anti-quark form flux tubes.
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Light quarks & Confinement
Problem: 16 tonnes of force makes a lot of pions.
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Craig Roberts: Imaging DCSB (38p)
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Light quarks & Confinement
Problem: 16 tonnes of force makes a lot of pions.
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Craig Roberts: Imaging DCSB (38p)
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Light quarks & Confinement In the presence of
light quarks, pair creation seems to occur non-localized and instantaneously
No flux tube in a theory with light-quarks.
Flux-tube is not the correct paradigm for confinement in hadron physics
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Craig Roberts: Imaging DCSB (38p)
G. Bali et al., PoS LAT2005 (2006) 308
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Confinement QFT Paradigm: – Confinement is expressed through a dramatic
change in the analytic structure of propagators for coloured states
– It can almost be read from a plot of the dressed-propagator for a coloured state
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complex-P2 complex-P2
o Real-axis mass-pole splits, moving into pair(s) of complex conjugate singularitieso State described by rapidly damped wave & hence state cannot exist in observable spectrum
Normal particle Confined particle
timelike axis: P2<0
s ≈ 1/Im(m) ≈ 1/2ΛQCD ≈ ½fm
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Dynamical Chiral Symmetry Breaking
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Craig Roberts: Imaging DCSB (38p)
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Dynamical Chiral Symmetry BreakingDCSB is a fact in QCD
– Dynamical, not spontaneous• Add nothing to QCD , no Higgs field, nothing! • Effect achieved purely through the quark+gluon dynamics.
– It’s the most important mass generating mechanism for visible matter in the Universe. • Responsible for ≈98% of the proton’s mass.• Higgs mechanism is (almost) irrelevant to light-quarks.
– Just like gluons and quarks, and for the same reasons, condensates are confined within hadrons. • There are no vacuum condensates.
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Confinement contains condensates, S.J. Brodsky, C.D. Roberts, R. Shrock and P.C. Tandy, arXiv:1202.2376 [nucl-th], Phys. Rev. C85 (2012) 065202
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Craig Roberts: Imaging DCSB (38p)
DCSB
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Mass from nothing!
C.D. Roberts, Prog. Part. Nucl. Phys. 61 (2008) 50M. Bhagwat & P.C. Tandy, AIP Conf.Proc. 842 (2006) 225-227 In QCD, all “constants” of
quantum mechanics are actually strongly momentum dependent: couplings, number density, mass, etc.
So, a quark’s mass depends on its momentum.
Mass function can be calculated and is depicted here.
Continuum- and Lattice-QCD are in agreement: the vast bulk of the light-quark mass comes from a cloud of gluons, dragged along by the quark as it propagates.
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Parton structure of
hadronsBaryons 2013: 24-28 June 2013
Craig Roberts: Imaging DCSB (38p)
Valence quarks
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Parton Structure of Hadrons
Valence-quark structure of hadrons– Definitive of a hadron – it’s how we tell a proton from
a neutron– Expresses charge; flavour; baryon number; and other
Poincaré-invariant macroscopic quantum numbers– Via evolution, determines background at LHC
Sea-quark distributions– Flavour content, asymmetry, intrinsic: yes or no?
Answers are essentially nonperturbative features of QCD
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Craig Roberts: Imaging DCSB (38p)
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Parton Structure of Hadrons Need for calculation is emphasised by Saga of pion’s
valence-quark distribution:o 1989: uv
π ~ (1-x)1 – inferred from LO-Drell-Yan & disagrees with QCD;
o 2001: DSE- QCD predictsuv
π ~ (1-x)2 argues that distribution inferred from data can’t be correct;
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Craig Roberts: Imaging DCSB (38p)
Valence quark distributions in the pion, M.B. Hecht, Craig D. Roberts, S.M. Schmidt, nucl-th/0008049, Phys.Rev. C63 (2001) 025213 .
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Parton Structure of Hadrons Need for calculation is emphasised by Saga of pion’s
valence-quark distribution:o 1989: uv
π ~ (1-x)1 – inferred from LO-Drell-Yan & disagrees with QCD;
o 2001: DSE- QCD predicts uv
π ~ (1-x)2 argues that distribution inferred from data can’t be correct;
o 2010: NLO reanalysis including soft-gluon resummation, inferred distribution agrees with DSE and QCD
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Craig Roberts: Imaging DCSB (38p)
Valence quark distributions in the pion, M.B. Hecht, Craig D. Roberts, S.M. Schmidt, nucl-th/0008049, Phys.Rev. C63 (2001) 025213 .
Soft-gluon resummation and the valence parton distribution function of the pion, M. Aicher, A. Schafer, W. Vogelsang, Phys.Rev.Lett. 105 (2010) 252003, arXiv:1009.2481 [hep-ph]
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Pion’s valence-quark Distribution Amplitude
Same methods can be used to compute φπ(x), projection of the pion’s Poincaré-covariant wave-function onto the light-front
Results have been obtained with rainbow-ladder DSE kernel, simplest symmetry preserving form; and the best DCSB-improved kernel that is currently available.
xα (1-x)α, with α=0.3
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Craig Roberts: Imaging DCSB (38p)
Imaging dynamical chiral symmetry breaking: pion wave function on the light front, Lei Chang, et al., arXiv:1301.0324 [nucl-th], Phys. Rev. Lett. 110 (2013) 132001 (2013) [5 pages].
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Pion’s valence-quark Distribution Amplitude
Both kernels agree: marked broadening of φπ(x), which owes to DCSB
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Craig Roberts: Imaging DCSB (38p)
Asymptotic
RL
DB
This may be claimed because PDA is computed at a low renormalisation scale in the chiral limit, whereat the quark mass function owes entirely to DCSB.
Difference between RL and DB results is readily understood: B(p2) is more slowly varying with DB kernel and hence a more balanced result
Imaging dynamical chiral symmetry breaking: pion wave function on the light front, Lei Chang, et al., arXiv:1301.0324 [nucl-th], Phys. Rev. Lett. 110 (2013) 132001 (2013) [5 pages].
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Pion’s valence-quark Distribution Amplitude
Both kernels agree: marked broadening of φπ(x), which owes to DCSB
Baryons 2013: 24-28 June 2013
Craig Roberts: Imaging DCSB (38p)
Asymptotic
RL
DB
This may be claimed because PDA is computed at a low renormalisation scale in the chiral limit, whereat the quark mass function owes entirely to DCSB.
Difference between RL and DB results is readily understood: B(p2) is more slowly varying with DB kernel and hence a more balanced result
These computations are the first to directly expose DCSB – pointwise – on the light-front; i.e., in the infinite momentum frame.
Imaging dynamical chiral symmetry breaking: pion wave function on the light front, Lei Chang, et al., arXiv:1301.0324 [nucl-th], Phys. Rev. Lett. 110 (2013) 132001 (2013) [5 pages].
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Pion’s valence-quark Distribution Amplitude
Baryons 2013: 24-28 June 2013
Craig Roberts: Imaging DCSB (38p)
Established a one-to-one connection between DCSB and the pointwise form of the pion’s wave function.
Dilation measures the rate at which dressed-quark approaches the asymptotic bare-parton limit
Experiments at JLab12 can empirically verify the behaviour of M(p), and hence chart the IR limit of QCD
C.D. Roberts, Prog. Part. Nucl. Phys. 61 (2008) 50
Dilation of pion’s wave function is measurable in
pion’s electromagnetic form factor at JLab12
A-rated: E12-06-10
Imaging dynamical chiral symmetry breaking: pion wave function on the light front, Lei Chang, et al., arXiv:1301.0324 [nucl-th], Phys. Rev. Lett. 110 (2013) 132001 (2013) [5 pages].
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When is asymptotic PDA valid?
Under leading-order evolution, the PDA remains broad to Q2>100 GeV2
Feature signals persistence of the influence of dynamical chiral symmetry breaking.
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Craig Roberts: Imaging DCSB (38p)
Consequently, the asymptotic distribution, φπ
asy(x), is a poor approximation to the pion's PDA at all such scales that are either currently accessible or foreseeable in experiments on pion elastic and transition form factors.
Thus, related expectations based on φπasy(x) should be revised.
asymptotic
4 GeV2
100 GeV2
Pion distribution amplitude from lattice-QCD, I.C. Cloët et al. arXiv:1306.2645 [nucl-th]
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Charged pion elastic form factor
Baryons 2013: 24-28 June 2013
Craig Roberts: Imaging DCSB (38p)
Single interaction kernel, determined fully by just one parameter and preserving the one-loop renormalisation group behaviour of QCD, has unified Fπ(Q2) and φπ(x) (and numerous other quantities)
Prediction of pQCD obtained when the pion valence-quark PDA has the form appropriate to the scale accessible in modern experiments is markedly different from the result obtained using the asymptotic PDA
Pion electromagnetic form factor at spacelike momenta, Lei Chang et al. (in progress)
DSE 2013
pQCD obtained with φπasy(x)
pQCD obtained with φπ(x;2GeV), i.e., the PDA appropriate to the scale of the experiment
15%
Near agreement between the pertinent perturbative QCD prediction and DSE-2013 prediction is striking.
Dominance of hard contributions to the pion form factor for Q2>8GeV2. Normalisation is fixed by a pion wave-function whose dilation with respect to φπ
asy(x) is a definitive signature of DCSB
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Craig Roberts: Imaging DCSB (38p)
Baryon Structure Dynamical chiral symmetry breaking (DCSB)
– has enormous impact on meson properties. Must be included in description
and prediction of baryon properties. DCSB is essentially a quantum field theoretical effect.
In quantum field theory Meson appears as pole in four-point quark-antiquark Green function
→ Bethe-Salpeter Equation Nucleon appears as a pole in a six-point quark Green function
→ Faddeev Equation. Poincaré covariant Faddeev equation sums all possible exchanges and
interactions that can take place between three dressed-quarks Tractable equation is based on the observation that an interaction which
describes colour-singlet mesons also generates nonpointlike quark-quark (diquark) correlations in the colour-antitriplet channel
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R.T. Cahill et al.,Austral. J. Phys. 42 (1989) 129-145
6333 SUc(3):
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Baryon Structure
Remarks Diquark correlations are not inserted by hand
Such correlations are a dynamical consequence of strong-coupling in QCD
The same mechanism that produces an almost massless pion from two dynamically-massive quarks; i.e., DCSB, forces a strong correlation between two quarks in colour-antitriplet channels within a baryon – an indirect consequence of Pauli-Gürsey symmetry
Diquark correlations are not pointlike– Typically, r0+ ~ rπ & r1+ ~ rρ
(actually 10% larger)– They have soft form factors
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Craig Roberts: Imaging DCSB (38p)
SU(2) isospin symmetry of hadrons might emerge from mixing half-integer spin particles with their antiparticles.
Faddeev Equation
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Structure of Hadrons Elastic form factors
– Provide vital information about the structure and composition of the most basic elements of nuclear physics.
– They are a measurable and physical manifestation of the nature of the hadrons' constituents and the dynamics that binds them together.
Accurate form factor data are driving paradigmatic shifts in our pictures of hadrons and their structure; e.g., – role of orbital angular momentum and nonpointlike diquark
correlations– scale at which p-QCD effects become evident– strangeness content– meson-cloud effects– etc.
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Craig Roberts: Imaging DCSB (38p)
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Flavor separation of proton form factors
Very different behavior for u & d quarks Means apparent scaling in proton F2/F1 is purely accidental
Baryons 2013: 24-28 June 2013
Cates, de Jager, Riordan, Wojtsekhowski, PRL 106 (2011) 252003
Q4F2q/k
Q4 F1q
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Diquark correlations!
Poincaré covariant Faddeev equation – Predicts scalar and axial-vector
diquarks Proton's singly-represented d-quark
more likely to be struck in association with 1+ diquark than with 0+
– form factor contributions involving 1+ diquark are softer
Baryons 2013: 24-28 June 2013
Cloët, Eichmann, El-Bennich, Klähn, Roberts, Few Body Syst. 46 (2009) pp.1-36Wilson, Cloët, Chang, Roberts, PRC 85 (2012) 045205
Doubly-represented u-quark is predominantly linked with harder 0+ diquark contributions
Interference produces zero in Dirac form factor of d-quark in proton– Location of the zero depends on the relative probability of finding
1+ & 0+ diquarks in proton– Correlated, e.g., with valence d/u ratio at x=1
d
u
=Q2/M2
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Craig Roberts: Imaging DCSB (38p)
Visible Impacts of DCSB
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Apparently small changes in M(p) within the domain 1<p(GeV)<3have striking effect on the proton’s electric form factor
The possible existence and location of the zero is determined by behaviour of Q2F2
p(Q2) Like the pion’s PDA, Q2F2
p(Q2) measures the rate at which dressed-quarks become parton-like: F2
p=0 for bare quark-partons Therefore, GE
p can’t be zero on the bare-parton domain
I.C. Cloët, C.D. Roberts, A.W. Thomas: Revealing dressed-quarks via the proton's charge distribution, arXiv: 1304.0855 [nucl-th]
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Craig Roberts: Imaging DCSB (38p)
Visible Impacts of DCSB
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Follows that the possible existence and location
of a zero in the ratio of proton elastic form factors
[μpGEp(Q2)/GMp(Q2)] are a direct measure of the nature of the quark-quark interaction in the Standard Model.
I.C. Cloët, C.D. Roberts, A.W. Thomas: Revealing dressed-quarks via the proton's charge distribution, arXiv: 1304.0855 [nucl-th]
Leads to Prediction neutron:protonGEn(Q2) > GEp(Q2) at Q2 > 4GeV2
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Neutron Structure Function at high-x
Valence-quark distributions at x=1– Fixed point under DGLAP evolution– Strong discriminator between theories
Algebraic formula
– P1p,s = contribution to the proton's charge arising from diagrams
with a scalar diquark component in both the initial and final state
– P1p,a = kindred axial-vector diquark contribution
– P1p,m = contribution to the proton's charge arising from diagrams
with a different diquark component in the initial and final state.
Baryons 2013: 24-28 June 2013
Craig Roberts: Imaging DCSB (38p)
I.C. Cloët, C.D. Roberts, et al.arXiv:0812.0416 [nucl-th], Few Body Syst. 46 (2009) 1-36D. J. Wilson, I. C. Cloët, L. Chang and C. D. RobertsarXiv:1112.2212 [nucl-th], Phys. Rev. C85 (2012) 025205 [21 pages]
Measures relative strength of axial-vector/scalar diquarks in proton
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Neutron StructureFunction at high-x
Baryons 2013: 24-28 June 2013
d/u=1/2SU(6) symmetry
pQCD, uncorrelated Ψ
0+ qq only, d/u=0
Deep inelastic scattering – the Nobel-prize winning quark-discovery experiments
Reviews: S. Brodsky et al.
NP B441 (1995) W. Melnitchouk & A.W.Thomas
PL B377 (1996) 11 N. Isgur, PRD 59 (1999) R.J. Holt & C.D. Roberts
RMP (2010)
d/u=0.28DSE: “realistic”
Distribution of neutron’s momentum amongst quarks on the valence-quark domain
DSE: “contact”d/u=0.18
Melnitchouk, Accardi et al. Phys.Rev. D84 (2011) 117501
x>0.9
Melnitchouk, Arrington et al. Phys.Rev.Lett. 108 (2012) 252001
I.C. Cloët, C.D. Roberts, et al.arXiv:0812.0416 [nucl-th], Few Body Syst. 46 (2009) 1-36D. J. Wilson, I. C. Cloët, L. Chang and C. D. RobertsarXiv:1112.2212 [nucl-th], Phys. Rev. C85 (2012) 025205 [21 pages]
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Neutron StructureFunction at high-x
Baryons 2013: 24-28 June 2013
d/u=1/2SU(6) symmetry
pQCD, uncorrelated Ψ
0+ qq only, d/u=0
Deep inelastic scattering – the Nobel-prize winning quark-discovery experiments
Reviews: S. Brodsky et al.
NP B441 (1995) W. Melnitchouk & A.W.Thomas
PL B377 (1996) 11 N. Isgur, PRD 59 (1999) R.J. Holt & C.D. Roberts
RMP (2010)
d/u=0.28DSE: “realistic”
Distribution of neutron’s momentum amongst quarks on the valence-quark domain
DSE: “contact”d/u=0.18
Melnitchouk, Accardi et al. Phys.Rev. D84 (2011) 117501
x>0.9
Melnitchouk, Arrington et al. Phys.Rev.Lett. 108 (2012) 252001
I.C. Cloët, C.D. Roberts, et al.arXiv:0812.0416 [nucl-th], Few Body Syst. 46 (2009) 1-36D. J. Wilson, I. C. Cloët, L. Chang and C. D. RobertsarXiv:1112.2212 [nucl-th], Phys. Rev. C85 (2012) 025205 [21 pages]
NB. d/u|x=1= 0 means there are
no valence d-quarks in the proton!
JLab12 can solve this enigma
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Neutron StructureFunction at high-x
“While it is quite hazardous to extrapolate from our limited xB range all the way to xB = 1, these results appear to disfavor models of the proton with d/u=0 at xB = 1”
Baryons 2013: 24-28 June 2013
Short Range Correlations and the EMC Effect, L.B. Weinstein et al., Phys.Rev.Lett. 106 (2011) 052301, arXiv:1009.5666 [hep-ph]
Figure courtesy of D.W. Higinbotham
Observation: EMC effect measured in electron DIS at 0.35 < xB < 0.7, is linearly related to the Short Range Correlation (SRC) scale factor obtained from electron inclusive scattering at xB > 1.
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EpilogueBaryons 2013: 24-28 June 2013
Craig Roberts: Imaging DCSB (38p)
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Epilogue The Physics of Hadrons is Unique:
– Confronting a fundamental theory in which the elementary degrees-of-freedom are intangible and only composites reach detectors
Confinement in real-world is NOT understood But DCSB is understood, and is crucial to any
understanding of hadron phenomena
They must have a common origin Experimental and theoretical study of the Bound-
state problem in continuum QCD promises to provide many more insights and answers.
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This is not the end
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Table of ContentsI. IntroductionII. Pion valence-quark distributionIII. Pion valence-quark parton distribution amplitudeIV. Charged pion elastic form factorV. Nucleon form factorsVI. Nucleon structure functions at large-xVII. Epilogue
A. DSE cf. Lattice PDAB. When is asymptotic PDA valid?C. GE/GM flavour separationD. Confinement contains condensatesE. Regge Trajectories?
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Lattice comparisonPion’s valence-quark PDA
Employ the generalised-Gegenbauer method described previously (and in Phys. Rev. Lett. 110 (2013) 132001 (2013) [5 pages]).
Baryons 2013: 24-28 June 2013
Craig Roberts: Imaging DCSB (38p)
Lattice-QCD => one nontrivial moment:
<(2x-1)2> = 0.27 ± 0.04 Legend
• Solid = DB (Best) DSE• Dashed = RL DSE• Dotted (black) = 6 x (1-x)• Dot-dashed = midpoint
lattice; and the yellow shading exhibits band allowed by lattice errors
φπ~ xα (1-x)α
α=0.35+0.32 = 0.67- 0.24 = 0.11
DB α=0.31 but 10% a2<0RL α=0.29 and 0% a2
V. Braun et al., PRD 74 (2006) 074501
Pion distribution amplitude from lattice-QCD, I.C. Cloët et al. arXiv:1306.2645 [nucl-th]
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When is asymptotic PDA valid?
φπasy(x) can only be a
good approximation to the pion's PDA when it is accurate to write
uvπ (x) ≈ δ(x)
for the pion's valence-quark distribution function.
This is far from valid at currently accessible scales
Baryons 2013: 24-28 June 2013
Craig Roberts: Imaging DCSB (38p)
Q2=27 GeV2
This is not δ(x)!
Pion distribution amplitude from lattice-QCD, I.C. Cloët et al. arXiv:1306.2645 [nucl-th]
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Flavor separation of proton form factors
Visible Impacts of DCSB
Baryons 2013: 24-28 June 2013
Craig Roberts: Imaging DCSB (38p)
Effect driven primarily by electric form factor of doubly-represented u-quark
u-quark is 4-times more likely than d-quark to be involved in hard interaction
So … GEpu ≈ GEp
Singly-represented d-quark is usually sequestered inside a soft diquark correlation
So, although it also becomes parton-like more quickly as α increases, that is hidden from view
d-quark
u-quark
I.C. Cloët & C.D. Roberts … continuing
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Confinement contains
condensatesBaryons 2013: 24-28 June 2013
Craig Roberts: Imaging DCSB (38p)
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“Orthodox Vacuum” Vacuum = “frothing sea” Hadrons = bubbles in that “sea”,
containing nothing but quarks & gluonsinteracting perturbatively, unless they’re near the bubble’s boundary, whereat they feel they’re trapped!
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u
u
ud
u ud
du
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New Paradigm Vacuum = hadronic fluctuations
but no condensates Hadrons = complex, interacting systems
within which perturbative behaviour is restricted to just 2% of the interior
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Craig Roberts: Imaging DCSB (38p)
u
u
ud
u ud
du
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Regge Trajectories? Martinus Veltmann, “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!
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Craig Roberts: Imaging DCSB (38p)
Phys.Rev. D 62 (2000) 016006 [9 pages]
1993: "for elucidating the quantum structure of electroweak interactions in physics"
Systematics of radial and angular-momentum Regge trajectories of light non-strange qqbar-states“ P. Masjuan, E. Ruiz Arriola, W. Broniowski. arXiv:1305.3493 [hep-ph]
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Hybrid Hadrons & Lattice QCD – Robert Edwards, Baryons13
Heavy pions … so, naturally, constituent-quark like spectra To which potential does it correspond?
Baryons 2013: 24-28 June 2013
arXiv:1104.5152, 1201.2349
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Hybrid meson models – Robert Edwards, Baryons13
With minimal quark content, , gluonic field can in a color singlet or octet
`constituent’ gluonin S-wave
`constituent’ gluonin P-wave
bag model
flux-tube model
arXiv:1104.5152, 1201.2349
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Hybrid baryon models – Robert Edwards, Baryons13
Minimal quark content, , gluonic field can be in color singlet, octet or decuplet
bag model
flux-tube model
Now must take into account permutation symmetry of quarks and gluonic field
arXiv:1104.5152, 1201.2349