Structure of quantum chromodynamics, glueballs

and nucleon resonances

H.P. Morsch

(in collaboration with P. Zupranski)

Warsaw, 27.4.2007

Related to the questions: What is the structure of quantum chromodynamics (QCD)?Do glueballs (structures of pure gluons) exist? What is the structure of hadron resonances (radial nucleon N* resonances)

1. What do we know on the structure of QCD?

2. New method to study gluon-gluon systems (not based on postulated QCD Lagrangian!) Basic assumptions Monte-Carlo method to deduce 2-gluon densities 3. Tests of the deduced 2-gluon densities: The QCD gluon propagator (from lattice data) Two-gluon field correlators (from lattice simulations)

4. Two-gluon binding potential and mass of glueballs

5. Structure of the nucleon and N* resonances

6. The Q-dependence of the strong coupling αs(Q)

7. Conclusion

1. What is the structure of QCD?

Three ingredients:1. coupling of gluon fields (due to non-Abelian form)2. qq-coupling by 1gluon exchange3. quark mass term

What is the origin of the quark masses?Dynamical structure, relativistic generation of mass?Binding energies between quarks?Coupling to external Higgs field? Experimental search at LHC.

Strong coupling (constant) αs

αs shows a dependence onthe momentum transfer μ

At large momentum transfersαs is rather small description of QCD by perturbation theorypossible. Experiments at SLAC,LEP,HERA…

How big is αs at small μ?What is the structure of QCDat small momentum transfers?How can we understand theconfinement of quarks and gluons?How can we calculate the properties of QCD at small μ?

Non-perturbative descriptions of QCDNon-perturbative descriptions of QCD

1. Dyson-Schwinger (gap) equation

complicated system of integral equations with not well known quantitites.

2. Lattice QCD approach by solving the QCD Lagrangian on a space-time lattice. Evaluation of results by statistical methods. Problems with lattice spacing, limitation to small momentum transfers, small quark masses, statistical noise, gauge invariance …

3. Method of gluon field correlators (Simonov et al.) many properties derived from gluon field correlators, evaluated by lattice methods.

detailed structure, in particular confinement is not understood!

.);(

,)(

),;()(2)()()( 2

121

vertexgluonquarkdressededrenormalistheispq

propagatorgluondressededrenormalistheiskDwhere

pqqSqpDgZmpiZpS

a

aa

q

bare

2. Our method to study gluon-gluon systems (based on two assumptions about coupling of gluons and of quarks)

Assumption 1: Colour neutral coupling of two gluons gives rise to finite and stable systems of scalar (0+) and tensor (2+)

structure, by L=1 also of vector (1-) form.

From the theory of gluon field correlators, DiGiacomo, Dosch, Shovchenko, Simonov, Phys. Rep. 372, 319 (2002)

Two-gluon field correlator is gauge invariant and non-local, θ(x.x‘) are phase factors. From ass.1 only in the colour neutral coupling the colour transformations of the two gluon fields cancel each other giving rise to stable systems.

Evidence for finite gluon field correlators from lattice QCD simulations (Di Giacomo et al. Nucl. Phys. B 285, 371 (1997).

,)],'()'()',()([)',( ' xxxFxxxFTrxx aa

2-gluon system is massive! What is the origin of mass?

(Cornwall: dynamical mass generation by relativistic effects)

Assumption 2 is equivalent to the conjecture, that the quarks emerge from the decay of the 2-gluon field 2gqq¯+2(qq¯). In this case the elementary q-q interaction v1g(R) is modified by the formed stable 2-gluon density.

Important: Fourier transform of this potential yields a momentum dependent interaction strength (running of the strong coupling αs).Also this form is consistent with αs0 for Q∞ (asymptotic freedom)

Assumption 2 (folding principle): Interaction between two quarks is described by folding the 1-gluon exchange force with a stable 2-gluon density:

')'()'()( 1 drrrRvRV gqq

).()()(4

)(2

QQQQ

RV ss

qq

Deduction of a self-consistent 2-gluon density

by the Monte Carlo method

1. Use of an initial form of ρΦ(r‘) in

')'()'()( 1 drrrRvR gqqV

2. Relativistic Fourier transform to p-space3. Monte Carlo simulation of the decay gg qq¯and 2q2q¯ using V(p1-p2) between quarks with momenta p1, p2 resulting 2-gluon momentum distributions Dqq¯(Q) and D2q2q¯(Q)4. Sum of Dqq¯(Q) and D2q2q¯(Q) retrans-

formed to r-space final 2-gluon density.

Self-consistency condition: Initial and final 2-gluon densitiesshould be the same (also their Fourier transforms).

Two-gluon density in the interaction between quarks must be the same as the colour neutral density formed !

MC-simulation of the decay g+g2gqq¯+2(qq¯) with folding interaction Vqq(R) between the emitted quarks

Details of the folding potential of theeffective q-q interaction

For decay into qq¯ a p-wave density ρpΦ(r)

is needed, which is constrained by <r>=0.

')'()'()( 1 drrrRvR gqqV

using an effective 1-gluon exchange force v1g(R)=-αs/R.

For interaction of the quarks in the limited volume ρΦ(r) the 1-gluon exchange force has to be modified by the size of the 2-gluon density, taken in the simple form

v‘1g(R)=v1g(R)·exp(-aR2).This leads to a finite folding potential Vqq(Q)at Q=0, needed for a self-consistent solution!(in the earlier calculations this effect has not been taken into account. The results on the gluon densities and confinement are the same, but the strong coupling αs is now consistent with other results.)

Deduced 2-gluon density is finite! forms a (quasi) bound state

Folding potential and resulting 2-gluon density

Mass deduced from relativisticFourier transformations: mΦ~0.68GeV

Results of simulation of Q2·ρ(Q) in comparison with the initial values

3. Tests of the deduced two-gluon densities:QCD gluon propagator and two-gluon field correlators

2-gluon field correlatorsDiGiacomo et al. Nucl. Phys. B 483, 371(1997)

Gluon propagatorsBowman et al. Phys. Rev.D 66, 074505 (2002) andD 70, 034509 (2004)

From pole-fits: mΦ~0.64 GeV much lower than lowest glueball massfrom lattice QCD!Vector field (Jπ=1-, L=1) has 14 % of the strength

of the scalar field

(Gluon propagator is most basic 2-point function in Yang-Mills theory.)Gluon propagator and 2g field correlators must be related to the 2-gluon densities.

4. Two-gluon binding potential and eigenstates

Two-gluon system forms a (quasi) bound state Binding potential of two gluonscan be obtained from a 3-dim. reduction of the Bethe-Salpeter eq.

in form of a relativistic Schrödinger equation

2

2

22

)()()()()(2

2rrwithrErrV

dr

d

rdr

dhbari

mmparametermassicrelativistis 4

1

What are the eigenstates in this potential?

Bali et al. Phys.Rev. D 62, 054503(2000)

Resulting binding potentialis consistent with confinementpotential from lattice QCD (Bali et al.) !

Binding potential

The mass of the 2-gluon system can be interpreted as binding

energy of the two gluons!

Eigenstates (glueballs) in the 2g binding (confinement) potential

Absolute binding energies are obtained by fitting the potential by a form Vfit(r)=αs/r+br with the condition αs/r0 for r∞.

Results: Eigenstates (glueballs) exist! Lowest eigenmode at E0=0.68±0.10 GeVRadial excitations at E1=1.69±0.15 GeV E2=2.54±0.17 GeV

E0=m, where m is the mass inserted in the Schrödinger

equation, consistent with the mass required in the relativistic

Fourier transformation!☟

Deduced mass of glueball ground state consistent with σ(550)! Radially excited states consistent with glueball states from lattice QCD.

0++ glueball spectrum in comparison with results from lattice QCD studies

our results

?

mas

s (G

eV

)

1

2

3

0++

Morningstar and Peardon, PRD 60, 034509 (1999)

σ(600)

Is the scalar σ(600) thelowest glueball state?

5. Structure of the nucleon

.

'')'()'(4)(

2

23

datawithconsistentrwith

drrrrrr

N

qN

Nucleon density obtained by foldingtwo-gluon density with density of three quarks

Resulting binding potential becomes more shallow. But attraction betweenemerging quarks increases by a factor 9!

Lowest eigenmode of the nucleon at E0=0.94±0.04 GeV Radial excitations at exp. N* E1=1.42±0.07 GeV P11(1440) E2=1.82±0.12 GeV P11(1710)

Lowest eigenmode has a binding energy consistent with the nucleon mass!

0++ glueballs

?

mas

s (G

eV

)

1

2

3

0++

Nucleon (N*) resonances

σ(600)

Is the scalar σ(600) thelowest glueball state?

Consequence: direct relation of the glueball spectrum to that of radial nucleon resonances(this allows experimental investigations)

1

2

3

1/2+ 3/2+,5/2+. . .

What is the contribution from gluonic and quark excitations?

What can we determine experimentally?

0++ glueball excitations correspond to radial N* resonancesWhat is experimentally known on these excitations?

Comparison with operator sum rules

• Cross section covers maximum monopole strength S1

• Extraction of the nucleon compressibility

KN ~ S1/S-1 ~ 1.3 GeV

First evidence from α-p scattering at SATURNE (Phys. Rev. Lett. 69, 1336 (1992)

Strong L=0 excitation in the region of the lowest P11 at about 1400 MeV

ProjectileΔ excitation

Study of the radial (breathing) mode of the nucleon

22 ||)0( oii

nin YrELS

Study of the breathing mode in p-p scattering at beam momenta 5-30 GeV/c

Contibuting resonancesΔ33 (1232)

D13 (1520), F15 (1680),strong res. at 1400 MeV

Strongest resonance at1400 MeV, width 200 MeV

(breathing mode)

No other resonance seen (high selectivity)

Detailed analysis in terms of a vibration of the valence and multi-gluon densities of the nucleon in Phys. Rev. C 71, 065203 (2005)

Calculated (p,p‘) differential cross section is sensitive to the nucleon transition density

Quantitative description of the p-p data requires a surface peaked transition

density ρtr(r)(consistent with the results from α-p)

Transition density is not consistent with pure valence quark excitation

(deduced from e-p scattering)

Information on the valence quark contribution from the longitudinal e-p amplitude S1/2

C.Smith, NSTAR2004, I.G.Aznauryan, V.D.Burkert, et al., nucl-th/0407021, L.Tiator, Eur.J.Phys.16 (2004)

For the charge transition density

is required 0)( 2drrc

S1/2 amplitude supports breathing mode interpretation !

How do we understanding the observed (p,p‘) transition density ?

1. Excitation of valence quarks deduced from (e,e‘)

2. Strong sea quark contribution due to multi-gluon structure

Conclusion: Multi-gluon contribution of the nucleon breathing mode excitation is ~4 times stronger than the valence quark contribution,

consistent with our model

Study of the strong coupling αs

at small momentum transfer

many different theoretical predictions of αs(Q) for Q0

Strong coupling „constant“ αs(Q) for Q0

Nesterenko and Papavassiliou,PR D 71, 016009 (2005)

Lattice data: Weiß, NP B 47, 71 (1996)Furui and Nakajima, PR D 69, 074505 (2004)and PR D 70, 094504 (2004)

contribution from 2g vector field

Shirkov and Solovtsov,PRL 79, 1209 (1997)

Strong coupling „constant“ αs(Q) for large Q

By adding 2g contributions from smaller and smaller 2-gluon densities corresponding to ss¯, cc¯, bb¯ …αs is better and better described

Above tt¯ further qq¯ system(s) expected (may be in the upper energy range of LHC).

8. Conclusions

1. Our two-gluon field approach gives a consistent and transparent description of the properties of QCD (interactions, confinement, mass, propagators,

glueballs, heavy flavour neutral systems…). All quark masses are compatible with zero no need for coupling to a

scalar Higgs field!

2. Lowest glueball state consistent with scalar meson σ(600).3. Radial nucleon resonances understood as glueballs coupled to 3 quarks.

What is the structure of other nucleon resonances?

4. Q-dependence of the strong interaction described by 1-gluon exchange folded with 2-gluon density. This gives rise to a running of αs(q) consistent

with other results and with asymptotic freedom!Flavour neutral systems heavier than tt¯ should exist.

Big challenge for LHC, GSI, BESIII, CEBAF…

')'()'()( 1 drrrRvR gqqV

Are there eigenstates in the q-q potential?

Only one eigenstate with E0~-0.01 GeV (binding energy appears to correspond

to about 2x the average

„current“ quark mass)

Because of a very low binding energy the glueball states should have a large width! Rough estimate of the width:(Γ≈ 1/E0 ≈ 500 MeV from the systematics of heavy 2-gluon states).

Other flavour neutral and flavoured systems

Generation of other self-consistent 2-gluon densities with radii corresponding to ss¯(b), cc¯(c), and bb¯systems (d).

Self-consistent densities obtained by assuming massless quarks! Mass explained by strong binding of quarks!

Dashed lines obtained by assuming quark masses of 1.3 GeV for (c) and4.5 GeV for (d) (not self-consistent!).

How can flavoured systems be described?

Decay 2gqf1qf1¯+ qf2qf2¯.If the radial size of the 2g-density is the same for decay in qf1 and qf2 2gqf1qf2¯+ qf2qf1¯.

(this may yield a description of pions)

How can we describe baryons?

Decay 4g5(q q¯) (3q qq¯) + (3q¯qq¯) baryon + antibaryon

3. Description of p-p and pion-pion scattering, multi-gluon potential density

and compressibility

Good description of the data obtained in double folding approach, which determines the multi-gluon potential strength.Volume integral of Vpp ~770 MeVfm3

Volume integral of Vππ~130 MeVfm3

From these potentials deduction of potential densities and compressibilities

Multi-gluon potential densities and compressibility

From the multi-gluon potentials we can derive a potential density Vρ(r) for the proton and the pion.

Deduced compressibility is consistentwith that deduced from operator sums

Study of scalar excitation of N* resonancesin the p-α αrec x1x2… reaction at TOF

Experiment was already proposed a long time ago, but the detector for α-particle recoils had to be built.

ΔE-E Si-microstrip detector telescope with 256x256 pixels.

Detector is completed and has been sent to Juelich.Needs installation and cablingfor commissioning during next Experiment at TOF.