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Craig Roberts Physics Division

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Imaging Dynamical Chiral Symmetry Breaking

Craig Roberts

Physics Division

Students

Postdocs

Asst. Profs.

- 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)
- Alfredo RAYA (U Michoácan);
- Sebastian SCHMIDT (IAS-FZJ & JARA);
- Robert SHROCK (Stony Brook);
- Peter TANDY (KSU);
- Tony THOMAS (U.Adelaide)
- Shaolong WAN (USTC)

Craig Roberts: Imaging DCSB (38p)

Rocio BERMUDEZ (U Michoácan);

Chen CHEN (ANL, IIT, USTC);

Xiomara GUTIERREZ-GUERRERO (U Michoácan);

Trang NGUYEN (KSU);

Khépani Raya (U Michoácan);

Hannes ROBERTS (ANL, FZJ, UBerkeley);

Chien-Yeah SENG (UW-Mad)

Kun-lun WANG (PKU);

J. JavierCOBOS-MARTINEZ (U.Sonora);

Mario PITSCHMANN (ANL & UW-Mad);

Si-xue QIN(U. Frankfurt am Main);

Jorge SEGOVIA (ANL);

David WILSON (ODU);

Lei CHANG (FZJ);

Ian CLOËT (ANL);

Bruno EL-BENNICH (São Paulo);

Craig Roberts: Imaging DCSB (38p)

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 nonperturbativebehaviour of the strong interaction

Is 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?

Craig Roberts: Imaging DCSB (38p)

- 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

Craig Roberts: Imaging DCSB (38p)

QCD is a Theory

(not an effective theory)

Craig Roberts: Imaging DCSB (38p)

- Very likely a self-contained, nonperturbativelyrenormalisable and hence well defined Quantum Field Theory
This 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

- 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)

Craig Roberts: Imaging DCSB (38p)

- Folklore … Hall-DConceptual Design Report(5)
“The color field lines between a quark and an anti-quark form flux tubes.

Craig Roberts: Imaging DCSB (38p)

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.”

Craig Roberts: Imaging DCSB (38p)

- Problem:
16 tonnes of force

makes a lot of pions.

Craig Roberts: Imaging DCSB (38p)

Problem:

16 tonnes of force

makes a lot of pions.

G. Bali et al., PoS LAT2005 (2006) 308

Craig Roberts: Imaging DCSB (38p)

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

Confined particle

Normal particle

complex-P2

complex-P2

timelike axis: P2<0

s ≈ 1/Im(m) ≈ 1/2ΛQCD≈ ½fm

- Real-axis mass-pole splits, moving into pair(s) of complex conjugate singularities
- State described by rapidly damped wave & hence state cannot exist in observable spectrum

Craig Roberts: Imaging DCSB (38p)

- 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

Craig Roberts: Imaging DCSB (38p)

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

Craig Roberts: Imaging DCSB (38p)

- DCSB 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.

- Dynamical, not spontaneous

C.D. Roberts, Prog. Part. Nucl. Phys. 61 (2008) 50

M. 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

Mass from nothing!

- 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.

Craig Roberts: Imaging DCSB (38p)

Valence quarks

Craig Roberts: Imaging DCSB (38p)

Craig Roberts: Imaging DCSB (38p)

- 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

Valence quark distributions in the pion, M.B. Hecht, Craig D. Roberts, S.M. Schmidt, nucl-th/0008049, Phys.Rev. C63 (2001) 025213 .

Craig Roberts: Imaging DCSB (38p)

- Need for calculation is emphasised by Saga of pion’s valence-quark distribution:
- 1989: uvπ ~ (1-x)1 – inferred from LO-Drell-Yan & disagrees with QCD;
- 2001: DSE- QCD predicts
uvπ ~ (1-x)2

argues that distribution

inferred from data

can’t be correct;

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]

Craig Roberts: Imaging DCSB (38p)

- Need for calculation is emphasised by Saga of pion’s valence-quark distribution:
- 1989: uvπ ~ (1-x)1 – inferred from LO-Drell-Yan & disagrees with QCD;
- 2001: DSE- QCD predicts
uvπ ~ (1-x)2

argues that distribution

inferred from data

can’t be correct;

- 2010: NLO reanalysis including
soft-gluon resummation,

inferred distribution agrees

with DSE and QCD

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].

Craig Roberts: Imaging DCSB (38p)

- Same methods can be used to compute φπ(x), projection of the pion’sPoincaré-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

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].

- 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

Asymptotic

DB

RL

Craig Roberts: Imaging DCSB (38p)

Both kernels agree: marked broadening of φπ(x), which owes to DCSB

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].

These computations are the first to directly expose DCSB – pointwise – on the light-front; i.e., in the infinite momentum frame.

- 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

Asymptotic

DB

RL

Craig Roberts: Imaging DCSB (38p)

Both kernels agree: marked broadening of φπ(x), which owes to DCSB

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

- 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

Craig Roberts: Imaging DCSB (38p)

Pion distribution amplitude from lattice-QCD,

I.C. Cloëtet al. arXiv:1306.2645 [nucl-th]

asymptotic

4 GeV2

100 GeV2

- Consequently, the asymptotic distribution,
- φπasy(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.

Craig Roberts: Imaging DCSB (38p)

Under leading-order evolution, the PDA remains broad to Q2>100 GeV2

Feature signals persistence of the influence of dynamical chiral symmetry breaking.

Pion electromagnetic form factor at spacelikemomenta, Lei Changet al. (in progress)

- 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

DSE 2013

15%

pQCD obtained with φπ(x;2GeV), i.e., the PDA appropriate to the scale of the experiment

pQCD obtained withφπasy(x)

- 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 φπasy(x) is a definitive signature of DCSB

Craig Roberts: Imaging DCSB (38p)

R.T. Cahill et al.,

Austral. J. Phys. 42 (1989) 129-145

SUc(3):

Craig Roberts: Imaging DCSB (38p)

- Dynamical chiral symmetry breaking (DCSB)
– has enormous impact on meson properties.

- Must be included in description
and prediction of baryon properties.

- Must be included in description
- 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.

- Meson appears as pole in four-point quark-antiquark Green function
- 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

Faddeev Equation

SU(2)isospin symmetry of hadrons might emerge from mixing half-integer spin particles with their antiparticles.

Craig Roberts: Imaging DCSB (38p)

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 masslesspion 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

Craig Roberts: Imaging DCSB (38p)

- 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 nonpointlikediquark correlations
- scale at which p-QCD effects become evident
- strangeness content
- meson-cloud effects
- etc.

Q4F2q/k

Cates, de Jager,

Riordan, Wojtsekhowski,

PRL 106 (2011) 252003

Q4 F1q

Craig Roberts: Imaging DCSB (38p)

Very different behavior for u & d quarks

Means apparent scaling in proton F2/F1 is purely accidental

Cloët, Eichmann, El-Bennich, Klähn, Roberts, Few Body Syst. 46 (2009) pp.1-36

Wilson, Cloët, Chang, Roberts,

PRC 85 (2012) 045205

u

d

=Q2/M2

- 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

Craig Roberts: Imaging DCSB (38p)

- 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

- form factor contributions involving

I.C. Cloët, C.D. Roberts, A.W. Thomas: Revealing dressed-quarks via the proton's charge distribution, arXiv: 1304.0855 [nucl-th]

- Apparently small changes in M(p) within the domain 1<p(GeV)<3
- have striking effect on the proton’s electric form factor
- The possible existence and location of the zero is determined by behaviour of Q2F2p(Q2)
- Like the pion’s PDA, Q2F2p(Q2) measures the rate at which dressed-quarks become parton-like:
- F2p=0 for bare quark-partons
- Therefore, GEp can’t be zero on the bare-parton domain

Craig Roberts: Imaging DCSB (38p)

I.C. Cloët, C.D. Roberts, A.W. Thomas: Revealing dressed-quarks via the proton's charge distribution, arXiv: 1304.0855 [nucl-th]

- 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.

Leads to Prediction neutron:proton

GEn(Q2) > GEp(Q2) at Q2 > 4GeV2

Craig Roberts: Imaging DCSB (38p)

I.C. Cloët, C.D. Roberts, et al.

arXiv:0812.0416 [nucl-th], Few Body Syst. 46 (2009) 1-36

D. J. Wilson, I. C. Cloët, L. Chang and C. D. Roberts

arXiv:1112.2212 [nucl-th], Phys. Rev. C85 (2012) 025205 [21 pages]

Measures relative strength of axial-vector/scalar diquarks in proton

Craig Roberts: Imaging DCSB (38p)

- 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.

I.C. Cloët, C.D. Roberts, et al.

arXiv:0812.0416 [nucl-th], Few Body Syst. 46 (2009) 1-36

D. J. Wilson, I. C. Cloët, L. Chang and C. D. Roberts

arXiv:1112.2212 [nucl-th], Phys. Rev. C85 (2012) 025205 [21 pages]

x>0.9

d/u=1/2

SU(6) symmetry

- 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=0.28

DSE: “realistic”

pQCD, uncorrelated Ψ

DSE: “contact”

d/u=0.18

0+qq only, d/u=0

Melnitchouk, Accardiet al.

Phys.Rev. D84 (2011) 117501

Melnitchouk, Arrington et al.

Phys.Rev.Lett. 108 (2012) 252001

Distribution of neutron’s

momentum amongst quarks

on the valence-quark domain

Craig Roberts: Imaging DCSB (38p)

I.C. Cloët, C.D. Roberts, et al.

arXiv:0812.0416 [nucl-th], Few Body Syst. 46 (2009) 1-36

D. J. Wilson, I. C. Cloët, L. Chang and C. D. Roberts

arXiv: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

x>0.9

d/u=1/2

SU(6) symmetry

- 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=0.28

DSE: “realistic”

pQCD, uncorrelated Ψ

DSE: “contact”

d/u=0.18

0+qq only, d/u=0

Melnitchouk, Accardiet al.

Phys.Rev. D84 (2011) 117501

Melnitchouk, Arrington et al.

Phys.Rev.Lett. 108 (2012) 252001

Distribution of neutron’s

momentum amongst quarks

on the valence-quark domain

Craig Roberts: Imaging DCSB (38p)

Short Range Correlations and the EMC Effect,

L.B. Weinstein et al., Phys.Rev.Lett. 106 (2011) 052301, arXiv:1009.5666 [hep-ph]

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.

- “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”

Figure courtesy of

D.W. Higinbotham

Craig Roberts: Imaging DCSB (38p)

Craig Roberts: Imaging DCSB (38p)

Craig Roberts: Imaging DCSB (38p)

- 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.

Craig Roberts: Imaging DCSB (38p)

Craig Roberts: Imaging DCSB (38p)

Introduction

Pion valence-quark distribution

Pion valence-quark parton distribution amplitude

Charged pion elastic form factor

Nucleon form factors

Nucleon structure functions at large-x

Epilogue

DSE cf. Lattice PDA

When is asymptotic PDA valid?

GE/GM flavour separation

Confinement contains condensates

Regge Trajectories?

Pion distribution amplitude from lattice-QCD,

I.C. Cloëtet al. arXiv:1306.2645 [nucl-th]

V. Braun et al., PRD 74 (2006) 074501

- 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

- DBα=0.31 but 10% a2<0
- RL α=0.29 and 0% a2

φπ~ xα (1-x)α

α=0.35

+0.32 = 0.67

- 0.24 = 0.11

Craig Roberts: Imaging DCSB (38p)

Employ the generalised-Gegenbauer method described previously (and in Phys. Rev. Lett. 110 (2013) 132001 (2013) [5 pages]).

Pion distribution amplitude from lattice-QCD,

I.C. Cloëtet al. arXiv:1306.2645 [nucl-th]

Q2=27 GeV2

This is not δ(x)!

Craig Roberts: Imaging DCSB (38p)

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

I.C. Cloët & C.D. Roberts … continuing

- 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

u-quark

d-quark

- 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

Craig Roberts: Imaging DCSB (38p)

Craig Roberts: Imaging DCSB (38p)

u

d

u

u

d

u

u

u

d

Craig Roberts: Imaging DCSB (38p)

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

d

u

u

d

u

u

u

d

Craig Roberts: Imaging DCSB (38p)

Vacuum = hadronic fluctuations

but no condensates

Hadrons = complex, interacting systems

within which perturbativebehaviour is

restricted to just 2% of the interior

1993: "for elucidating the quantum structure of electroweak interactions in physics"

Phys.Rev. D 62 (2000) 016006 [9 pages]

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]

Craig Roberts: Imaging DCSB (38p)

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!

arXiv:1104.5152, 1201.2349

Craig Roberts: Imaging DCSB (38p)

Heavy pions … so, naturally, constituent-quark like spectra

To which potential does it correspond?

arXiv:1104.5152, 1201.2349

With minimal quark content, , gluonic field can in a color singlet or octet

`constituent’ gluon

in S-wave

bag model

`constituent’ gluon

in P-wave

flux-tube model

arXiv:1104.5152, 1201.2349

Minimal quark content, , gluonic field can be in color singlet, octet or decuplet

Now must take into account permutation symmetry of quarks and gluonic field

bag model

flux-tube model