the role of the tetraquark at nonzero temperature francesco giacosa in collaboration with a. heinz,...

The role of the tetraquark at nonzero temperature

Francesco Giacosa

in collaboration with A. Heinz, S. Strüber, D. H. Rischke

ITP, Goethe University, Frankfurt am Main

Hadrons@Fias- 26/6/08

Outline

• Scalar mesons below and above 1 GeV at zero T Light scalar mesons (< 1 GeV) as tetraquark states and tetraquark-quarkonia mixing

• A chiral model with pions, light scalar quark-antiquark and tetraquark states Description of the model at zero T, quark and tetraquark condensates and mixing

• Results at nonzero temperature Order of the phase transition, behavior of the condensates and mixing angle, role interchange

Francesco Giacosa Scalar Quest

http://www.uni-frankfurt.de/index.html


Part I

Spectroscopy in the vacuum

0I)980(

)600(

0

0

f

f

)1710(

)1500(

)1370(

0

0

0

f

f

f2

1I )800(k

1I )980(0a

M < 1 GeV 1 GeV < M < 1.8 GeV

Too many resonances than expected from quark-antiquark states


0PCJ

)1450(0a

)1450(K

Scalar resonances below 1.8 GeV reported by PDG:


0I)980(

)600(

0

0

f

f

ss

dduu

)(2/1

2

1I )800(k

1I )980(0a

M < 1 GeV interpretation


0PCJ

)(1/2 , , dduuuddu

dssdussu , , ,

qq

Assignment has problems!!!


Chiral partner of ?



List of Problems

• Masses: degeneracy of and

• Strong coupling of to

• The scalar quarkonia are p-wave states (L = S = 1), thus expected to be heavier than 1 GeV as tensor and axial-vector mesons

• Some Lattice results find

• Large behavior of light scalar not compatible with quarkonia

)980(0f


)980(0a

)980(0a KK

GeVMdu

5.14.1 from: Prelovsek et al., Phys. Rev. D 70 (2004), Burch et al., Phys. Rev. D 73 (2006)

cN-from: Pelaez, Phys. Rev. Lett. (2004), Pelaez and Rios, hep-ph/0610397



The light scalars are interpeted as tetraquark state

An example of „good diquark” is:

)(:)(:(:0: BRRBcduudfSpinLSpaceqq

A tetraquark is the bound state of two diquarks

Idea of Jaffe (R.L. Jaffe, Phys. Rev. D 15 (1977)) :

Example: )980(0a du s]][u,s,d[- (and not )


0I)980(

)600(

0

0

f

f ],][,[ dudu

2

1I )800(k

1I )980(0a

M < 1 GeV Tetraquark interpretation


0PCJ

]),][,[],][,[(

],,][,[ ,],][,[

sdsdsusu

sdsusdsu

],][,[ ,],][,[

],,][,[ ,],][,[

sudusudu

sddusddu

]),][,[],][,[( sdsdsusu

It is not the chiral partner of !


0I

)1710(

)1500(

)1300(

0

0

0

f

f

f

ss

glueball

dduu

)(2/1

2

1I )1430(0K

1I )1470(0a

M > 1 GeV interpretation


0PCJ

)(1/2 , , dduuuddu

dssdussu , , ,

qq

Mixing among the isoscalars is expected


Chiral partner of !


0PCJ

These are predominantly quarkonia (with glueball-intrusion) (but not only!)

M < 1 GeV 1 GeV < M < 1.8 GeV

0I

)1710(

)1500(

)1370(

0

0

0

f

f

f2

1I

1I )1450(0a

)1450(K

These are predominantlytetraquarks (but not only!)

)980(

)600(

0

0

f

f

Indeed, mixing will occur, thus the scenario changes slightly as:

)(21 dduu

],][,[ duduNot the chiral partner of !

)800(k

)980(0a

Chiral partner of !



Part II

A chiral model with tetraquark


How does this scenario affect finite temperature behavior?

We study this issue in the SU(2) limit within a simple model:

statescalar -extraan is ],][,[2

1)600( resonance The

.pion theofpartner chiral theis )(2

1)1370( resonance The

0

0

duduf

dduuf

Mixing shall play a crucial role:

)(

2

1

],][,[2

1

)cos()sin(

)sin()cos(

)1370(

)600(

00

00

0

0

dduu

dudu

f

f

4545 0

., , :freedom of degrees Five



g coupling with piece Tetraquark

22

22

ML theasjust

2222

)(2

1)(

4

gMFλ

V

A simple chiral model with tetraquark

m)(quarkoniu d)duu(2

1 k)(tetraquar ]d,ud][[u,

2

1 triplet,

It emerges as an SU(2) limit of the SU(3) case

0V

V

0)( :minimum absolute for theSearch

20202

2

20 ...,22

1

M

gf

F

Mg

F

condensatek tetraquar

condensatequark

0

0

g,M,F,,

:parametersunknown 5

(A. Heinz, S. Strüber, F. G. and D. H. Rischke: arXiv:0805.1134 [hep-ph] )

(F.G.,Phys.Rev.D75:054007,2007 )



)(2

1)(

4 2

222222

2

gMFλ

V

...2

2V

:minimum thearound potential theexpand We

22

21

20

02

21

MMg

gM

0

222

220

2 M ,2

3 where

F

M

gM

diagonal.not ismatrix mass theand potential in the

present is 2 A term .orthogonalnot are and fields that theNotice 0 g

)1370(

)600(

0

0

fS

fHOne must therefore diagonalize the model introducing the mass eigenstates

mixing. quarkonium-k tetraquar thedescribes parameter theThus, g



)cos()sin(

)sin()cos(

)1370(

)600(

)2(

00

00

0

0

SOB

fS

fH

That is, the fields H and S, corresponding to the two physical resonances, are introduced in order to diagonalize the potential:

220

21

0

4arctan

MM

g

...2

22

0

02

21

S

HB

Mg

gMBSH t

2

2

20

02

0

0

2

2

S

Ht

M

MB

Mg

gMB

...2

2V 2

0

02

21

Mg

gM

0222

0

222222 4)4( gMMgMMMM HSHS



Part III

Results at nonzero T


We study this model at nonzero T by using the CJT formalismIn the Hartree approximation. (Only double-bubble diagrams are taken into account)

)( :condensate Tetraquark

)( :condensateQuark

0

0

T

T

)( :angle Mixing 0 T

))1370((S )(MM

))600((H )(MM

)(MM

:Masses

0

0

fT

fT

T

SS

HH

0)0(with T

0)0(with T

0)0( with T

Details in: A. Heinz, S. Strüber, F. G. and D. H. Rischke: arXiv:0805.1134 [hep-ph]



MeV 1200M

MeV 400M :esknown valuely approximat 2

MeV 92.4

MeV 139M :esknown valu- well2

,M,F,, :parametersunknown 5

)1370(fS

)600(fH

0

0

0

f

g

)(2

1)(

4 2

222222

2

gMFλ

V

mixing) quarkonium-k(tetraquar gconstant coupling theand

upon them ationsstudy vari shall Weuncertain! are M and M )1370(fS)600(fH 00



n transitiophase

chiral theoforder study the weand M and g vary We(fixed). GeV 4.0 SHM

0(T)

)(

)( H

:) ( 0

quarkquark-antiS

tetraquark

decouplingtetraquarkg order 1GeV 948.0

over cross GeV 948.0

S

S

M

M

0805.1134 [hep-ph]0g



Quark condensate (order parameter) as function of T for different values of g for MS = 1.0 GeV

Increasing of g (mixing):1) Tc decreases2) First order softened3) Cross-over obtained

for g large enough


n transitiophase chiral theoforder study the weand

M and g vary We(fixed). GeV 2.1 HSMSimilar discussion as before

0805.1134 [hep-ph]



We now turn to one specific case:

GeV 3.4gstudies Latticewith agreement in

over-cross have order toin gfix We

MeV 1200 M

MeV 400 M :use We

)1370(fS

)600(fH

0

0

We study for this set of parameters all the temperature-dependent quantitites: masses, mixing angle and condensates.



Finite Temperature behavior of quark and tetraquark condensates:

increase tostarts )(then

TTfor holds )()( T nonzeroAt c2

2

T

TM

gT

2020 :T zero that Remind

M

g

This property depends on the characteristics of the model. However,It does not influence other quantities

0805.1134 [hep-ph]


454/)( :as defined

))(cos())(sin(

))(sin())(cos(

)1370(

)600(

0

0

ss TT

TT

TT

fS

fH

Finite Temperature behavior of masses and angles:

Two ‘critical temperatures’:

MeV 170140 :example In this cs TT

The mixing angle grows with T up to theMaximal value. Then, it changes sign at Ts and becomes negative. (Second change at higher T)

antiquark-quarkmostly is statelighter theTTFor

e)interchang role mixing, (maximal

s

0805.1134 [hep-ph]



Summary and outlook

• Spectroscopy of light scalars at zero T: if the light scalars are not quark-antiquark, how does chiral restoration change?

• Description of a model with pions, scalar quark-antiquark and tetraquark. Mixing at zero T: f0(600) is predominantly tetraquark and f0(1370) pred. quark-antiquark

• Tetraquark-quarkonium mixing implies: (i) decreasing of the critical temperature, (ii) softer first order, and, if the mixing is large enough, cross over. The latter can be obtained also for a mass of the chiral partner above 1 GeV

• The mixing increases with T. At a certain Ts the mixing angle is maximal and a role interchange takes place. Then, chiral restoration takes place in the standard form.

• This was the first step! One shall go further and include more resonances.


Thank you very much




g=2 GeV. Ts>Tc


8.14.1 GM GeV

0

0

I

J PC

lightest predicted glueball

Lattice:

Morningstar (1999)


ssS

ggG

nnN

f

f

f

97.006.026.0

06.089.049.0

26.045.086.0

)1710(

)1500(

)1370(

0

0

0

Result for the mixed states:Obtained upon fit to the known results of PDG


has the largest gluonic amount!!!)1500(f0

F.G. et al, Phys.Rev.D72:094006, 2005 (hep-ph/0509247)

F.G. et al, Phys.Lett.B622:277-285,2005 (hep-ph/0504033)

F.G. et al, Phys.Rev.C71:025202,2005 (hep-ph/0408085 )


)1370(0f19.046.0

500200

)1500(

)1500(

0

0

f

KKf

MeVstatenn

)1500(0f05.024.0

5109

)1500(

)1500(

0

0

f

KKf

MeV

)1710(0f 2.05

10140

)1500(

)1500(

0

0

f

KKf

MeV

Compatible with a dominant:

)( glueballinert

stategg

statess

Francesco Giacosa Scsalar Quest


Strong decays of a tetraquark state:

Subdominant

P

P

[4q]S

P

P

Dominant

[4q]S

}',,,{ K

}',,,{ K

}',,,{ K

}',,,{ K

Previous works and motivations

Original paper:

Jaffe, Phys. Rev. D 15 (1977),

Revival in:

Maiani et al, Phys. Rev. Lett. (2004)

Experimental study:

D. V. Bugg, EPJC47 (2006)

Systematic evaluation of amplitudes: my work

Phys.Rev.D74:014028,2006



I studied the strong decays with a hadronic model

(never see quarks and gluons, only hadrons)

6

2

3

632

632

ˆ

800

0800

800

KK

K

K

PP aa

Nonet of pseduoscalar states:

Nonet of scalar tetraquark states:

]4[

2

)980(]4[)980(

)980(2

)980(]4[

0

000

0

0

00

]4[

qkk

kaqf

a

kaaqf

S

B

B

B

q

]d,ud][[u,2

1]4[

22

])s,ds][[d,]s,us][([u,]4[

q

qf

B

B

The phys. resonances result from mixing



Write the flavor, P, C invariant interaction Lagrangian for the scalar 4q decays:

PPAATrScPAPATrScL

SPP

jjqij

itjqij

qaa

ˆˆˆˆ

nonetscalar -4q nonet, pseud. ˆ

]4[2

]4[1int

]4[

)(with ijkjkiA

]4[ qS

1c}',,,{ K

}',,,{ K]4[ qS

2c}',,,{ K

}',,,{ K

P

P

Dominant

[4q]

S

Sumdominant

P

P

[4q]

S

The trace structure corresponds to the microscopic diagrams:


• Scalar tetraquark and quarkonia states can mix Black et al, Phys. Rev. D 64 (2001), F.G., Phys.Rev.D 75,(2007)

• Extension of the model: ; consider scalar and pseudoscalar quarkonia meson and scalar tetraquark states


)3()3()3( RLV SUSUSU

)(4

1L , 0

2][ SBqq VVTriPS

Spontaneous symmetry breaking takes place, but no need to specify the potential. The pions emerge as Golstone bosons..

22,

2,

20

][min

FF

FFdiagS K

qq

Going further: tetraquark-quarkonia mixing



:obtained is mixing around expandingWhen

.)(22

0

]4[2*]4[1int

jjqij

ijitjqij AATrS

cAAAATrS

cL

TcNTNcNcTNcTNL

fNN SN

S T

shift

q]q[

q][

200

22

00

4

2

with NN :Shift .quarkoniumd)duu(2

1

tetraquark]d,ud][[u,2

1

decay mixing tq-condensate

:dimension oneIn



10%)(sin :Result

][

]4[

)cos()sin(

)sin()cos(

)1450(

)980(

2

0

0

0

0

qqa

qa

a

a

One relates the tetraquark-decay parameters to the mixing strenght by using the decay widths of PDG; then, one can evaluate the mixing:

Result in the isovector sector

The mixing is small !!!



Consider flavor: 3 antisymmetric combinations

d][u, s][u,- ],[ sd

Under SU(3)-flavor the 3 diquarks behave like antiquarks:

q

s

d

u

],[

],[

],[

du

su

sd

jii

i

U )(qq

qU)(q

j

jij

)3(SUU

)3(SUU

jiji

ji

U

)(

)U( ji



We have the correspondences:

and: ]d,u[ ]s,u[- ],[ sd

s d u

d][u, s][u,- ],[ sd

s d u

Compose a diquark and an antidiquark: full 4-q nonet

Example: du s]][u,s,d[- )980(0a



A tetraquark condensate is generated:

22

]4[

2122

]4[

21 )()(2]4[ F

M

cc

M

ccqσ

qσu

qσb

bb

22,

2,

2,,0

FF

FFdiagdiag Ksuu

55

5

GeV 10)42(

]4[],][,[2

1

qdudu bQCD


the role of the tetraquark at nonzero temperature francesco giacosa in collaboration with a. heinz,...

Documents

scalar quarkonia

tetraquark slide

vacuum slide

t light scalar mesons

nonzero t slide

light scalar quarkantiquark

large behavior of light

hepph0610397 slide