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
Dec 19, 2015
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
Francesco Giacosa Scalar Quest
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
Francesco Giacosa Scalar Quest
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
Francesco Giacosa Scalar Quest
0PCJ
)(1/2 , , dduuuddu
dssdussu , , ,
Assignment has problems!!!
Francesco Giacosa Scalar Quest
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
Francesco Giacosa Scalar Quest
)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
Francesco Giacosa Scalar Quest
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
Francesco Giacosa Scalar Quest
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
Francesco Giacosa Scalar Quest
0PCJ
)(1/2 , , dduuuddu
dssdussu , , ,
Mixing among the isoscalars is expected
Francesco Giacosa Scalar Quest
Chiral partner of !
Francesco Giacosa Scalar Quest
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 !
Francesco Giacosa Scalar Quest
Part II
A chiral model with tetraquark
Francesco Giacosa Scalar Quest
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
Francesco Giacosa Scalar Quest
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 )
Francesco Giacosa Scalar Quest
)(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
Francesco Giacosa Scalar Quest
)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
Francesco Giacosa Scalar Quest
Part III
Results at nonzero T
Francesco Giacosa Scalar Quest
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]
Francesco Giacosa Scalar Quest
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
Francesco Giacosa Scalar Quest
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
Francesco Giacosa Scalar Quest
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
Francesco Giacosa Scalar Quest
n transitiophase chiral theoforder study the weand
M and g vary We(fixed). GeV 2.1 HSMSimilar discussion as before
0805.1134 [hep-ph]
Francesco Giacosa Scalar Quest
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.
Francesco Giacosa Scalar Quest
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]
Francesco Giacosa Scalar Quest
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]
Francesco Giacosa Scalar Quest
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.
Francesco Giacosa Scalar Quest
g=2 GeV. Ts>Tc
Francesco Giacosa Scalar Quest
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
Francesco Giacosa Scalar Quest
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
Francesco Giacosa Scalar 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
Francesco Giacosa Scalar Quest
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
Francesco Giacosa Scalar Quest
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
Francesco Giacosa Scalar Quest
)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
Going further: tetraquark-quarkonia mixing
Francesco Giacosa Scalar Quest
: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
Francesco Giacosa Scalar Quest
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 !!!
Francesco Giacosa Scalar Quest
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
Francesco Giacosa Scalar Quest
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
Francesco Giacosa Scalar Quest
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
Francesco Giacosa Scalar Quest