tetraquarks - why it is so difficult to model themcfif.ist.utl.pt/~rupp/eef70/talks/bicudo.pdf ·...

IntroductionTetraquarks potentials in Lattice QCD

Problems and solutions in modeling tetraquarksConclusion

Tetraquarkswhy it is so difficult to model them

Pedro Bicudo *work partly done with Marco Cardoso, Nuno Cardoso and Marc Wagner

* CFTP, Instituto Superior Tecnico, Universidade de Lisboa, Portugal

September 1, 2014

Pedro Bicudo Tetraquarks Eef 70, Coimbra 2014 1 / 37



Complementing Eef’s work: systems without annihilation/disconnecteddiagrams, with a four-body tetraquark flux tube.

-10

-5

0

5

10 -10

-5

0

5

10

-2.00e-04

0.00e+00

2.00e-04

4.00e-04

6.00e-04

8.00e-04

1.00e-03

1.20e-03

1.40e-03

1.60e-03

x

y

0.00e+001.57e-043.14e-044.71e-046.29e-047.86e-049.43e-041.10e-031.26e-031.41e-031.57e-03

Colour Fields Computed in SU(3) Lattice QCD for the Static Tetraquark System, N. Cardoso, M. Cardoso, P. Bicudo, Phys.Rev. D84 (2011)

054508




Outline

1 IntroductionTetraquark models since the onset of QCDExperimental tetraquarks candidatesTetraquarks computed in lattice QCD

2 Tetraquarks potentials in Lattice QCDStatic tetraquarks potentials and flux tubesdynamical-antistatic dynamical-antistatic potentials in latticeQCD

3 Problems and solutions in modeling tetraquarksLattice QCD inspired potentialsFlip-flop potentialsNon-orthogonality, and the colour excited statesUnitarity, phase shifts and resonances

4 Conclusion




Tetraquark models since the onset of QCDExperimental tetraquarks candidatesTetraquarks computed in lattice QCD

Outline




4 Conclusion





The discovery of quarks

In the 1950’s, 1960’sand 1970’s the bestparticle physicists,including many NobelPrize winners, workedout the hadron puzzlepiece by piece.

they finally discoveredthe quarks andproduced a new theory,QuantumChromoDy-namics(QCD).

Gell-Mann, M. and Ne’eman, Y. The Eightfold Way, New York: W. A. Benjamin, 1964.





The bag model and tetraquarks

Jaffe’s bag model tetraquarks

Already at the onset of QCD, the bag model predicted many tetraquarks.

Multi-Quark Hadrons. 1. The Phenomenology of (2 Quark 2 anti-Quark) Mesons, Robert L. Jaffe , Phys.Rev. D15 (1977) 267





Recent studies of tetraquark binding in the quark model

Improved flip-flop confining potential

Lattice QCD changed the paradigm for quark confinement.Nevertheless, a flip-flop potential with meson meson and tetraquarklinear confinement, also produces tetraquark binding.

Stability of multiquarks in a simple string model, J. Vijande, A. Valcarce, J.-M. Richard, Phys.Rev. D76 (2007) 114013





Outline




4 Conclusion





the π1 with Jpc = 1−+

Observation of a hybrid, or tetraquark, by COMPASS at CERN

COMPASS at CERN observe an exotic hadron resonance with M= 1660(74) MeV , Γ= 269 (85) MeV, JPC I = 1−+1

Observation of a J**PC = 1-+ exotic resonance in diffractive dissociation of 190-GeV/c pi- into pi- pi- pi+, COMPASS Collaboration, M.

Alekseev et al., Phys.Rev.Lett. 104 (2010) 241803





the charged quarkonia Z+ and Z−

Confirmation of Zc(4430)− byLHCb at CERN: π J/ψ Argandplot

Several charged quarkonia Z qc and Z q

bare observed in different collaborations,after the initial observation by BELLE.

LHCb at CERN confirms the thecrypto-exotic (exotic if we neglect ccannihilation) hadron resonanceZc(4430)− with a resonance mass of4475 MeV and width of 172 MeV.

In a talk at Scadron 70, organized byGeorge, we predicted the correct partialdecay width to π J/ψ with simpleResonant Group Method calculations.

Observation of a resonance-like structure in the pi +- psi’ mass distribution inexclusive B to K pi +- psi decays, Belle collaboration, S. Choi et al., Phys. Rev. Lett.

100 (2008) 142001,Observation of the resonant character of the Z(4430)- state, LHCb Collaboration,

Roel Aaij et al., Phys.Rev.Lett. 112 (2014) 222002M. Cardoso and P. Bicudo, Microscopic calculation of the decay of Jaffe-Wilczek

tetraquarks, and the Z(4433), [arXiv:0805.2260]





The difficulty of observing exotic hadrons

Tetraquarks, or other exotics, aredifficult to observe experimentally.Some observations ended up beingunconfirmed.

Clearly they are excited states, withcomplicated flavour or other quantumnumbers, and theorist cannot excatlypredict where to look for them.

Whether their complexity makesthem really hard to observe or theysimply are rare, is open to debate.

The Evidence for a pentaquark signal and kinematic reflections , A.R. Dzierba, D. Krop, M. Swat, S. Teige, A.P. Szczepaniak, Phys.Rev.D69 (2004) 051901

A Partial wave analysis of the pi- pi- pi+ and pi- pi0 pi0 systems and the search for a J**PC = 1-+ meson, A.R. Dzierba, R. Mitchell, E. Scott,

P. Smith, M. Swat, S. Teige, A. Szczepaniak, S.P. Denisov, V. Dorofeev, I. Kachaev ,Phys.Rev. D73 (2006) 072001





Outline




4 Conclusion





Discretizing QCD

To discretize of QCD in a finite space-time lattice,we utilize finite differences.

Different points in the lattice must be connectedby gauge links to preserve gauge invariance.Thus the quarks field are placed on the latticevertices and the gluon fields on the links.

Each link is a SU(3) matrix.

While the quark action is obvious, the guonaction is obtained with the plaquette∑

c

F cµνF c

µν =2βa4 Pµν +O(a) ,

a closed loop in the lattice,

Pµν (r) = 1− 13

Re Tr[Uµ(r)Uν(r + µ)U†µ(r + ν)U†ν(r)

].

http://www.jicfus.jp/en/promotion/pr/mj/guido-cossu/Pedro Bicudo Tetraquarks Eef 70, Coimbra 2014 13 / 37




Ljubljana-Graz tetraquarks

14 meson-meson components and 4 diquark-diquark components

Lattice QCD shows evidence the Z−c has a large tetraquark component.

Evidence for a charged charmonium-like Z+c from QCD, Sasa Prelovsek, C.B. Lang, Luka Leskovec, Daniel Mohler, [arXiv:1405.7623]





Ljubljana-Graz tetraquarks





The difficulty of studying excited tetraquarks in lattice QCD

ρ in ππ phase shiftExcited tetraquarks are resonances,decaying into many channels, 30 forthe last experimental Z−c candidates.

For an absolute evidence, thedifferent partial decay widths shouldbe computed as well in lattice QCD.

With present computers and thephase shift Luscher method, onlyresonances with just one open decaychannel have been studied in latticeQCD, with sufficient detail.

Coupled channel analysis of the rho meson decay in lattice QCD, C.B. Lang, Daniel Mohler, Sasa Prelovsek, Matija Vidmar, Phys.Rev. D84

(2011) 054503




Static tetraquarks potentials and flux tubesdynamical-antistatic dynamical-antistatic potentials in lattice QCD

Outline




4 Conclusion





Tetraquark 4-body potential

Generalized Wilson loop to a tetraquark

A four-body potential is computed in lattice QCD, consistent with the4-body confining potentials used in confining quark models.

Detailed analysis of the tetraquark potential and flip-flop in SU(3) lattice QCD, Fumiko Okiharu, Hideo Suganuma, Toru T. Takahashi,

Phys.Rev. D72 (2005) 014505





Flux Tubes

-10

-5

0

5

10 -10

-5

0

5

10

-2.00e-04

0.00e+00

2.00e-04

4.00e-04

6.00e-04

8.00e-04

1.00e-03

1.20e-03

1.40e-03

1.60e-03

x

y

0.00e+001.57e-043.14e-044.71e-046.29e-047.86e-049.43e-041.10e-031.26e-031.41e-031.57e-03

At T = 0, we find a 4-body tetraquark flux tube.

Colour Fields Computed in SU(3) Lattice QCD for the Static Tetraquark System, N. Cardoso, M. Cardoso, P. Bicudo, Phys.Rev. D84 (2011)

054508





Flux Tubes

-10

-5

0

5

10

x

-10

-5

0

5

10

y

-1.00e-040.00e+001.00e-042.00e-043.00e-044.00e-045.00e-046.00e-047.00e-048.00e-049.00e-041.00e-03

0.00e+009.41e-051.88e-042.82e-043.76e-044.70e-045.64e-046.58e-047.52e-048.46e-049.41e-04

By the way, we also find a 5-body pentaquark flux tube.

Colour Fields of the Static Pentaquark System Computed in SU(3) Lattice QCD, Nuno Cardoso, Pedro Bicudo, Phys.Rev. D87 (2013)

034504





Outline




4 Conclusion





B meson - B meson potential

Screening in the light-light -antistatic-antistatic tetraquark

Mechanism: the light quarks screen the antistatic-antistatic interaction;

however screening only acts at a distance typical of the light quarkwavefunction.

Lattice QCD signal for a bottom-bottom tetraquark, Pedro Bicudo, Marc Wagner, Phys.Rev. D87 (2013) 11, 114511





B meson - B meson potential

With lattice QCD there are attractive channels

In some channels we find attraction, in others we find attraction.

We utilize this potential computed in lattice QCD to solve a Schrodingerequation.

We find binding in the B − B system, but not in D − B or D −D systems.




Lattice QCD inspired potentialsFlip-flop potentialsNon-orthogonality, and the colour excited statesUnitarity, phase shifts and resonances

Outline




4 Conclusion





Only the spin-scalar and static potentials have beencomputed for exotic systems

A possible structure of theS · S potential V3(r).

The spin-tensor (hyperfine, spin-orbit,tensor) potentials have beencomputed in lattice QCD only forstatic mesonic systems.

Already in baryons we are not surewhat potential to utilize.

In exotic systems hybrids, glueballs,tetraquarks, pentaquarks ...) we haveno idea on the spin-tensor potentials.

Approximation: start by ignoringspin-tensor potentials.

Spin-dependent potentials from lattice QCD, Yoshiaki Koma, Miho Koma, Nucl.Phys. B769 (2007) 79-107





Even the spin-scalar potential is not exactly determined yet

Widening of the SU(3) flux tube

datafit: A + B * ln(R)A = 0.1477 ± 0.0035B = 0.0762 ± 0.0090χ2/dof = 0.383

w2 (R

/2)

(fm

2 )

0

0.05

0.1

0.15

0.2

R (fm)0 0.25 0.5 0.75 1 1.25 1.5

Static quark potentials may bedifferent from light/dynamicalpotentials.

There are Coulomb andLogarithmic corrections to theconfining linear, spin-scalarpotential.

The constant shift isundetermined.

The constituent quark mass isgauge dependent.

Approximation: start with linearand Coulomb potentials

Inside the SU(3) quark-antiquark QCD flux tube: screening versus quantum widening, N. Cardoso, M. Cardoso, P. Bicudo, arXiv:1302.3633[hep-lat]

Lattice QCD computation of the colour fields for the static hybrid quark-gluon-antiquark system, and microscopic study of the Casimirscaling, M. Cardoso, N. Cardoso, P. Bicudo, Phys.Rev. D81 (2010) 034504





Fermat points - where the flux tubes match

mesonbaryon

tetraquark

pentaquark

hexaquark

Q1

Q2

Q1

Q1

Q1

Q1

Q2

Q2

Q2

Q2

Q3

Q3

Q3

Q3

Q4

Q4

Q4

Q5

Q5

Q6

FI

FI

FI

FI

FII

FII

FII

FIII

FIII

FIV

V12

FI

Q1

Q2

Qs V34

Q4

FII

The string sections linking the multiquarks join at Fermat points.There is a geometrical method to construct the two Fermat points FI andFII of a tetraquark. Notice in general the quarks are not coplanar.Solution: We develop a numerical sequence converging to the Fermatpoints.

Iterative method to compute the Fermat points and Fermat distances of multiquarks, P. Bicudo, M. Cardoso , Phys.Lett. B674 (2009) 98-102





Outline




4 Conclusion





Two-body potentials and unphysical Van der Waals forces

Since there is no evidence for long distance polarization forces, or Vander Waals forces, in hadron-hadron interactions, the two-bodyconfinement potentials cannot be right, they are wrong!

Van Der Waals Like Forces Between Hadrons Induced By Color Confining Potentials, M.B. Gavela, A. Le Yaouanc, L. Oliver, O. Pene, J.C.Raynal, S. Sood, Phys.Lett.B82 (1979) 431.





Flip-flop potentials have the correct screening

Solution: use a flip-flop potential, where confining flux tubes or stringstake the geometry minimizing the energy of the system.

Quark Confinement And Hadronic Interactions. F. Lenz, et al, Annals Phys.170 (1986) 65.





Outline




4 Conclusion





Fermi Golden Rule cannot be applied to tetraquarks

When the vector basis of the initial state and decay state areorthogonal,

〈i |f 〉 = 0 (1)for instance when the initial state is an excited atom, and the finalstate is an atom in a less excited state plus a photon, 〈e−|e− γ〉 = 0;then it is standard to compute decay widths, either with the FermiGolden Rule, or with Feynman Diagrams,

Ti→f =2π~〈i |H|f 〉2ρ (2)





Meson - meson and tetraquark are not orthogonal

However, when studying the tetraquark decay tomeson-meson channels, all three colour statesare non-orthogonal〈1 1|1 1‘〉 = 1

3

〈1 1|3 3〉 =√

13

〈3 3|1 1‘〉 =√

13 ,

contrary to the decay of 1 meson→ 2 mesons,in orthogonal Fock spaces, here the Fock spaceof on tetraquark and two mesons are notortogonal. This is a orthogonality problem, it maylead to a non-hermitian hamiltonian.

the potential is open in some variables,compact/confined in others.

Solution: use a complete basis.Decays of tetraquark resonances in a two-variable approximation to the triple flip-flop potential Pedro Bicudo, Marco Cardoso, Phys.Rev.

D83 (2011) 094010Pedro Bicudo Tetraquarks Eef 70, Coimbra 2014 33 / 37




Ground state and first excited state of the tetraquark flux tubes

Approximation : Use the second lower potential as the colour excitation,say lowest meson-meson, 1− 1 or 1− 1′, as the excitation of thetetraquark 3 -3, etc .

Variational study of the flux tube recombination in the two quarks and two quarks system in Lattice QCD, M. Cardoso, N. Cardoso, P.

Bicudo, Phys.Rev. D86 (2012) 014503





Outline




4 Conclusion





Study of phase shifts with spherical waves

Our solution shown in Marco Cardoso’s talk this morning consists in :

utilizing a non-orthogonal basis of meson-meson wavefunctions,

we have a flip-flop potentials with 3 sectors: meson-meson,meson-meson and tetraquark potentials,

we also have a flip-flop in the colour excited potential,

in the center of mass we have 9 coordinates, but we fold the 8 confineddirections with integrals in the mesonic wavefunctions,

we are left with coupled channel meson-meson Schrodinger equations,

from the spherical outgoing wavefunctions, we extract phase shifts,

from a simple phase shift analysis we determine the boundstates andresonnances .

Decays of tetraquark resonances in a two-variable approximation to the triple flip-flop potential, P. Bicudo, M. Cardoso, Phys.Rev. D83(2011) 094010.




Summary

Tetraquarks have been suggested by theorists since the onset ofQCD.Tetraquarks have been difficult to observe experimentally, butrecently 1−+π1 (perhaps) and Z− (most likely) have beenconfirmed. Tetraquarks are a main highlight in particle physics.Lattice QCD is crucial for the theoretical confirmation oftetraquarks. The observed tetraquarks are not yet fully computedin Lattice QCD, but lattice QCD progresses constantly and willeventually compute them.Quark models are necessary to understand tetraquarks. Oneencounters several new, interesting, difficult problems intetraquark resonances. Only a perfect model will be able tosimulate tetraquarks quantitavely.

to be continued ...Pedro Bicudo Tetraquarks Eef 70, Coimbra 2014 37 / 37

tetraquarks - why it is so difficult to model themcfif.ist.utl.pt/~rupp/eef70/talks/bicudo.pdf ·...

Documents