Meson and tetra-quark mixing

0.43 MB·

by  Ping Wang

EPJ manuscript No.
(will be inserted by the editor)
Meson and tetra-quark mixing
Ping Wang1,2, Stephen R. Cotanch1 and Ignacio J. General3
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1 Department of Physics, North Carolina State University,Raleigh, NC 27695-8202, USA
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0 2 Jefferson Laboratory, 12000 Jefferson Ave.,Newport News, VA 23606, USA
2 3 Bayer School of Natural and EnvironmentalSciences, DuquesneUniversity,Pittsburgh, PA 15282, USA

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Received: date/ Revised version: date
a
J
1 Abstract. Themixingbetweenqq¯mesonandqq¯qq¯tetra-quarkstatesisexaminedwithinaneffectiveQCD
3 Coulomb gauge Hamiltonian model. Mixing matrix elements of the Hamiltonian are computed and then
diagonalized yielding an improved prediction for the low-lying JPC =0±+,1−− isoscalar spectra. Mixing
] effects were found significant for the scalar hadrons but not for the 1−− states, which is consistent with
h
the ideal mixing of vector mesons. A perturbativeassessment of theexact QCD kernelis also reported.
p
-
p PACS. 12.39.Mk Glueball and exotic multi-quark/gluon states – 12.39.Pn Potential models – 12.39.Ki
e Relativistic quark model – 12.40.Yx Hadron mass models and calculations
h
[

1 1 Introduction As detailed in this paper, meson-tetra-quark mixing
v is fundamentally due to qq¯ pair annihilation/formation
0 and entails the strong confining interaction. We include
1 Hadronic structure remains an interesting but challeng- this mechanism using our Coulomb gauge (CG) model,
8 ing problem. This is because quantum chromodynamics which has been successfully applied to meson, glueball,
4 [QCD] permits a variety of hadron formations such as qq¯ hybrid and tetra-quark states [14,15,16,17,18,19,20,21].
.
1 mesons, qq¯qq¯tetra-quarks, qq¯g hybrids and gg glueballs. ThemodelHamiltonianisobtainedfromtheQCDCoulomb
0 Thesestateswill,ingeneral,alsomixviaquarkpairanni- gaugeHamiltonianusingafewsimplifications(seebelow).
8 hilation/formation which further complicates this issue. In this way,the original non-perturbative confining inter-
0 Unfortunately experimental information [1] is predomi- action can be rearranged into a calculable effective po-
:
v nantly limited to masses, widths and spectroscopic quan- tentialbetweencolordensities.We utilize powerfulmany-
i tum numbers with little structure insight. Accordingly, body techniques and relativistic field theory in which the
X
theoretical input is needed and this paper reports a con- non-perturbative vacuum is described as a coherent BCS
r sistent, model study of the dynamic mixing between qq¯ groundstatewithquarkandgluonCooperpairs(conden-
a
meson and qq¯qq¯tetra-quark states. sates).TheresultingmodelretainsthekeyQCDelements
There are abundant mixing analyses in the literature and is thus capable of robust predictions as comprehen-
involving mesons, glueballs and hybrids utilizing several sivelydocumentedinnumerouspublications[14,15,16,17,
differentmethodsincludingperturbationtheory,relativis- 18,19,20,21].
tic Feynman-Schwinger path integrals for Green’s func- Weonlyfocusonthemixingbetweenqq¯andqq¯qq¯con-
tions [2], instantons [3], lattice QCD [4] and effective chi- figurations and, for two reasons, omit qq¯g hybrid and gg
ral approaches [5]. There have also been several meson- glueball states. First, we are interested in low mass spec-
glueballphenomenologicalmixingstudies[6,7,8,9]inwhich tra where energy level mixing arguments imply that ef-
the meson-glueball interaction is modeled, or simply pa- fectsfromglueballsandhybridmesonsshouldnotbelarge
rameterized,andthendiagonalizedtoobtainrelationsbe- sincetheseexotichadronshavesomewhatheaviermasses.
tween primitive and physical masses. Indeed, in our previous model applications (and others
Centralto this workis the mixing betweenmesonand including lattice results) the lightest hybrid and glueball
tetra-quark states, which has not received much atten- massesarepredictedtobeslightlyabove[14,15]andbelow
tion.The fewpublishedstudiesinclude twodiquark,anti- [19]2GeV,respectively.Thisisincontrasttotetra-quark
diquark cluster applications with tetra-quark mixings be- masses which, due to four different color configurations,
tweeneitherhybrid[10]orquarkonium[11]states,ameson- can be much lighter. In particular, we calculated [14,16]
meson coupled-channels scattering calculation [12] and a the lightest tetra-quark to be in the color singlet-singlet
quark model mixing study [13] which obtained an im- state with mass closer to 1 rather than 2 GeV. The other
provedscalarspectrumbyadjustingphenomenologicalpa- reasonfor omitting quark-hybridand quark-glueballmix-
rameters. ing matrix elements is that for our model Hamiltonian
2 Ping Wang, StephenR. Cotanch and Ignacio J. General: Meson and tetra-quark mixing
(see Section 4) the former are perturbative, and thus ex- spuriousretardationcorrections,aidsidentificationofdom-
pectedweak,whilethelatterentirelyvanish(mixingmust inant, low energy potentials and introduces only physical
proceedvia higher orderintermediate states). This would degrees of freedom (no ghosts) [24].
also suggest that glueball widths might not be large, as The bare parton fields have the normal mode expan-
typically expected, perhaps even narrow, consistent with sions (bare quark spinors u,v, helicity, λ= 1, and color
±
a recent theoretical prediction [22]. The issue of mixing vectors ˆǫ )
=1,2,3
involving gluonic states, however, merits a further study C
which we plan to address in a separate communication. Ψ(x)= dk Ψ (k)eikxǫˆ (11)
Thispaperisorganizedintosevensections.Inthenext Z (2π)3 C · C
saencdtiothnenw,eindeSteaciltiothne3Q, pCrDeseCnotualopmerbtugrabuagteivHeaamnailltyosnisiaonf ΨC(k)=uλ(k)bλC(k)+vλ(−k)d†λC(−k) (12)
dk 1
the exact Coulomb kernel. This motivates our Coulomb Aa(x)= [aa(k)+aa ( k)]eikx (13)
gauge model Hamiltonian described in Section 4. Meson Z (2π)3√2k † − ·
and tetra-quark mixing is treated in Section 5 with nu-
dk k
merical results given in Section 6. Finally, key findings Πa(x)=−iZ (2π)3r2[aa(k)−aa†(−k)]eik·x, (14)
and conclusions are summarized in Section 7.
withtheCoulombgaugetransversecondition,k aa(k)=
( 1)µk aa (k) = 0. Here b (k), d (−k) a·nd aa(k)
2 QCD Coulomb gauge Hamiltonian (µ−=0,µ 1−)µarethebarequarkλ,Canti-quλaCrkandgluonFµock
±
operators, the latter satisfying the transverse commuta-
The exact QCD Hamiltonian in the Coulomb gauge [23] tion relations
is (summation over repeated indices is used throughout)
H = H +H +H +H (1) [aaµ(k),abµ†′(k′)]=(2π)3δabδ3(k−k′)Dµµ′(k) , (15)
QCD q g qg C
with
Hq = Z dxΨ†(x)[−iα·∇+βm]Ψ(x) (2) Dµµ′(k)=δµµ′ −(−1)µkµkk2−µ′ . (16)
1
Hg = dx −1Πa(x) Πa(x)+Ba(x) Ba(x) (3)
2Z J ·J ·
(cid:2) (cid:3)
3 Perturbative expansion
H = g dxJa(x) Aa(x) (4)
qg
Z ·
Before addressing meson and tetra-quark mixing, we first
HC = g2 dxdy −1ρa(x)Kab(x,y) ρb(y) , (5) report a perturbative study of the kernel Kab(x,y). Ex-
− 2 Z J J panding in powers of g yields
wheregistheQCDcouplingconstant,Ψ isthequarkfield −1 2 −1 = −2+2g −2 −2
with current quark mass m, Aa =(Aa,Aa) are the gluon M ∇ M ∇ ∇ A∇
fieldssatisfyingthetransversegaugecondi0tion,∇ Aa =0 +3g2∇−2A∇−2A∇−2+... , (17)
(a = 1,2,...8), Πa = −Eatr are the conjugate m·omenta where ab =fabcAc ∇.Notethefirsttermrepresentsthe
and A ·
simple,long-rangedCoulombinteraction.Wehavepertur-
Eatr = −A˙a+g(1−∇−2∇∇·)fabcAb0Ac (6) bnaiatnivecloymcpaolcnuelnattewditthhetehxispekcetranteiolnfovral0u+e+ofqqt¯hsetaHtaesm(iltthoe-
Ea = −A˙a−∇Aa0 +gfabcAb0Ac (7) Faddeev-Popov terms are also included)
1
Ba = Aa+ gfabcAb Ac , (8) E ΨJPC H ΨJPC =g2EC +g4EC +g6EC +... .
∇× 2 × C ≡h | C| i 2 4 6
arethe non-abelianchromodynamicfields.Thecolorden- Inthis paperallwavefunctionkets, ψ >,haveunitnorm.
sities, ρa(x), and quark currents, Ja, are The leading diagrams corresponding| to g2, g4 and g6 are
shown in Fig. 1. The g2 diagram contributes
ρa(x)=Ψ†(x)TaΨ(x)+fabcAb(x) Πc(x) (9)
· (q,q )
Ja =Ψ†(x)αTaΨ(x), (10) E2C = Z dqdq′F p2 ′ (18)
wtuirtehcsotnasntdaanrtds,SfUab(c3.)TchoeloFradmdaetervic-Peso,pToavd=etλe2arm, ainnadnstt,ruc=- F = Uλ†1(q)Uλ′1(q′)Vλ†′2(q′)Vλ2(q)ΦJλ′1PλC′2†(q′)ΦJλ1PλC2(q) ,
det( ), of the matrix =∇ D with covariantderiJva- withp=q q andq,q theinitial,finalquarkmomenta.
′ ′
tiveMDab = δab∇ gfMabcAc, is· a measure of the gauge Thedressed−spinors, , ,areBCSrotationsofthebare
λ λ
− U V
manifold curvature and the kernel in Eq. (5) is given by spinors and wave function details are given in the follow-
Kab(x,y)= x,a 1 2 1 y,b .TheCoulombgauge ing sections. The above integration is convergent and the
− −
h |M ∇ M | i
Hamiltonian is renormalizable, permits resolution of the numerical value for this Coulomb type interaction energy
Gribov problem, preserves rotational invariance, avoids is EC =25.6 MeV.
2
Ping Wang, StephenR. Cotanch and Ignacio J. General: Meson and tetra-quark mixing 3
ThebottomthreediagramsofFig.1(fromlefttoright)
areproportionalto g6, with respective expectation values
16(1 x2)(1 x2) (q,q )
g2 g4 Zdqdq′dq1dq2p2(p q−)21(p −q )22ωF(q )ω′(q ),
− 1 − 2 1 2
4(1 x2)k2(1 z2) (q,q )
Zdqdq′dq1dq2p2(p q−)4(1p 1q − q )F2ω(q )′ω(q ),
− 1 − 1− 2 1 2
dqdq dq dq (q,q )
′ 1 2F ′
Z p4(p q )2(p q )2(p q q )2ω(q )ω(q )
− 1 − 2 − 1− 2 1 2
k (k 2q ) k q (k 2q ) q /q2
1· 1− 2 − 1· 1 1− 2 · 1 1
g6 g6 g6 k(cid:2) (k 2q ) k q (k 2q ) q /q2 (cid:3),
2· 2− 1 − 2· 2 2− 1 · 2 2
(cid:2) (cid:3)
wherez =kˆ qˆ ,x =pˆ qˆ andk =2p q forj =1,2.
1· 2 j · j j − j
These loop variable integrations are also divergent. Us-
ingthe samerenormalizationprocedureyields the respec-
tivevalues 7.15g6,7.86g6 and0.48g6 MeV. Therefore,the
Fig. 1. Diagrams for thekernel expansion to order g6.
sixth order interaction energy is EC =15.5 MeV and the
6
series takes the form E = (25.6g2 + 13.2g4 + 15.5g6)
C
MeV. Since the coefficients are comparable, g2 must be
less than 1, i.e. α = g2/4π < 0.1, for perturbation the-
s
The g4 diagram reduces to ory to be valid but the strong interaction has α much
s
larger, so the perturbative expansion fails as anticipated.
E4C =Z dqdq′dq1p4(21(p−x21q)F)2(ωq(,qq′)) , (19) Wtioentahnedre,fgoureidseedekbyalactatliccuelarbesleulctso,nafidnoinpgt akelrinneealrinptoetreanc--
− 1 1
tial specified in the next section. We note in passing that
where x1 = pˆ·qˆ1. The momentum integration q1 over a subset or class of diagrams may still be amendable to
the loop diverges but replacing the gluon kinetic energy a perturbative treatment. Specifically the chainof bubble
ω(q1) with ω(q1)1+ǫ yields a finite result, A+B/ǫ, for diagrams(i.e.theg4 andfirstg6 diagramsinFig.1)seems
positive ǫ which isolates the divergence.As shown in Fig. tobe converging.Also,laddertype diagrams(thirdg6 di-
2, the integrated value scales as 1/ǫ for small ǫ which is agram)seemmuchsmaller,incontrasttogluondressedor
consistent with dimensional renormalization. Extrapolat- self-energy type diagrams (second g6 diagram), and they
ing the intercept from the linear graph yields the infinite too may be convergent. Hence further perturbative stud-
subtracted renormalized result E4C = A = 13.2 MeV. In ies should be conducted to determine which parts, if any,
this minimal subtraction scheme the coupling g is renor- ofthe exactkernelcanbe treatedas radiative corrections
malized to its physical value by absorbing the infinity. andwhichpartsareresponsibleforconfinementandmust
be included non-perturbatively. This would provide fur-
ther insight into the nature of confinement and also for
improved QCD approximated interactions.

0.05
4 Coulomb gauge model Hamiltonian
0.04 Our model’s starting point is the Coulomb gauge QCD
) Hamiltonian, Eq. (1). In this gauge, the color form of
V
e Gauss’s law, which is essential for confinement, is satis-
G
( 0.03 fied exactly and can be used to eliminate the unphysical
E longitudinal gluon fields. We then make two approxima-
tions:1)replacethe exactCoulombkernelwith acalcula-

ble confiningpotential;2)use the lowestorder,unit value
0.02
for the the Faddeev-Popov determinant. This defines the

CG model Hamiltonian
H = H +HCG+H +HCG (20)
0.01 CG q g qg C
0 3 6 9 12 15 18 1
1 / e HgCG = 2Z dx[Πa(x)·Πa(x)+Ba(x)·Ba(x)](21)

1
Fig. 2. The interaction energy to order g4 versus1/ǫ. HCCG = −2Z dxdyρa(x)Vˆ(|x−y|)ρa(y) . (22)
4 Ping Wang, StephenR. Cotanch and Ignacio J. General: Meson and tetra-quark mixing
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(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)
(anti-quark)Fockoperators,Bλ†1 1 (Dλ†2 2),actingonthe Fig. 3. Diagrams for the meson, tetra-quark mixing term.
Bardeen-Cooper-Schrieffer(BCS)CmodelCvacuum, Ω (see
| i
Refs. [15,17] for full details). For the tetra-quark system,

the quark (anti-quark) cm momenta are q1, q3 (q2, q4) coefficients a,b,c and d are determined by diagonalizing
and the following wave function ansatz is adopted the Hamiltoni a n m atri x , i n whi c h t h e m e s o n -tetra-quark
off-diagonalmixing element is (only HCG contributes)
C
dq dq dq
ΨJPC = 1 2 3 ΦJPC (q ,q ,q ) (23)
| i Z (2π)3(2π)3(2π)3 λ1λ2λ3λ4 1 2 3 M = h q q¯ | HCCG | q q¯ q q¯i, (28)
RC31C42Bλ†1 1(q1)Dλ†2 2(q2)Bλ†3 3(q3)Dλ†4 4(q4)|Ωi . where qq¯ is n n ¯ or s s ¯ , a nd q q¯ q q¯ is n n ¯ n n¯ or nn¯ss¯ .
C C C C C C | i | i | i | i | i | i
There are six off-diagonal matrix elements however two,
The expression for the matrix RC1C2 depends on the spe- ss¯HCG nn¯ a n d ss¯ H C G nn¯n n ¯ , vanis h a nd one (see
cific color scheme selected [14,16C]3.CH4 ere, we focus on the hbelo|wC), |nn¯nin¯ HCGh n|n¯sCs¯ ,|is comiputed very small. The
cthoeloqrq¯sipnagilrest-csoiunpglleettoscchoelmore,si[n(g3l⊗ets¯3,)s1in⊗ce(3it⊗gi¯3v)e1s]1t,hwehloewre- rnemn¯aHinCinGhgnmn¯six s¯ |i n a g Cn dm|asts¯r i H x i Ce lGemnen¯ n s t s ¯ s . a Froe,roh n u nr¯ |m H oCCdGel|nHn¯anmn¯iil,-
h | C | i h | C | i
estmassamongthefourcolorrepresentations.Thisyields tonian,therearetwotypes ofmixingdiagramsillustrated
R1CC131CCλ42 λ=δsC1Cλ2δC3C41.1Tλhλe spsinλpartsofstheλwλaveJfuλnc+tioλn is,, ienxisFtisg.fo3r.qBq¯eac n a n u i sh e ilaotfioc n o l bo re tfwaecet no r d s , i ff neornenz te r s o i n mgliextinqgq¯colnulsy-
h22 1 2| A Ai h22 3 4| B Bi h A B A B| A Bi
a product of Clebsch-Gordan coefficients. Here J is the ters. The expression for the first diagram in Fig. 3 is
total angular momentum, s = s +s , s = s +s ,
A 1 2 B 3 4
1
aarnedzfeorro,sccoanlasrishteandtrownisthatllheorlbowitaesltaenngeurlgayrsmtaotmee(nsetea,SleXc-, M1 = 2Z dq1dq2dq3V(k)Uλ†1(q1)Uλ′1(−q4) (29)
tGioanus6siafnorrapd-wiaalvweapvseefuundcot-isocnaliasruasnedd(vseeceto[1r6]hafodrrodnest)a.ilsA) Uλ†3(q3)Vλ2(q2)ΦJλ1PλC2λ†3λ4(q1,q2,q3)ΦJλ′1PλC4(−2q4),
withq = q k,k=q +q anddressed,BCSspinors
qA2 qB2 qI2 4 − 1− 2 3
f(qA,qB,qI)=e−α2A−α2B−α2I , (24) 1 1+sinφ(q)
= χ (30)
with variational parameters α =α and α determined Uλ √2(cid:18)p1 sinφ(q) σ qˆ(cid:19) λ
A B I − ·
by minimizing the tetra-quark masses p
M = ΨJPC H ΨJPC (25) 1 1 sinφ(q) σ qˆ
JPC h | CG| i λ = − − · χλ . (31)
=M +M +M +M +M , V √2(cid:18) p1+sinφ(q) (cid:19)
self qq q¯q¯ qq¯ annih
p
which were previously calculated [14,16]. The subscripts
The gap angle, φ(q), is the solution to the gap equation
indicate the source of each contribution: the q and q¯self-
that minimizes the energy of the BCS vacuum, i.e., the
energy, the qq, q¯q¯and qq¯scattering, and the qq¯annihila-
vacuum rotated by a Bogoliubov-Valatin transformation
tion, respectively. Finally, the qq¯meson state is
[17].Theeffectiveconfiningpotentialinmomentumspace
dk is V(k). The second diagram in Fig. 3 yields
|ΨJPCi=Z (2π)3ΦJλ1PλC2(k)Bλ†1(k)Dλ†2(−k)|Ωi. (26)
1
M2 = 2Z dq1dq2dq3V(k)Vλ†4(q4)Vλ′4(−q1) (32)
5 Meson and tetra-quark mixing Uλ†3(q3)Vλ2(q2)ΦJλ1PλC2λ†3λ4(q1,q2,q3)ΦJλ1PλC′4(2q1) .
Inthissection,wediscussthemesonandtetra-quarkmix-
ing for the JPC = 0 + and 1 states. As discussed 6 Numerical results
± −−
above, only mixing between flavored (u,d,s) qq¯ mesons
andtetra-quarksisinvestigated.Usingthenotation, qq¯> The twoHamiltonianparametersin ourmodel wereinde-
and qq¯qq¯> for ΨJPC >, the mixed state is given b|y pendently determined while the wavefunctionparameters
| |
were obtained variationally. Because we seek new model
JPC =ann¯ +bss¯ +cnn¯nn¯ +dnn¯ss¯ , (27) masses, the unmixed variational basis states need not be
| i | i | i | i | i
ones producing a minimal, unmixed mass. Hence we can
where nn¯ = 1 (uu¯+dd¯). The state ss¯ss¯ is not included use one of the variational parameters to provide an op-
√2 | i
sinceitsmassismuchhigherthanthemesonmasses.The timal mixing prediction. We have selected α to exploit
I
Ping Wang, StephenR. Cotanch and Ignacio J. General: Meson and tetra-quark mixing 5

0.8 spectrumdescriptionsincethepurenn¯ andss¯stateswere
previously in good agreement [17,18] with observation.
0++ (0x0) Asmentionedabove,thepurelytetra-quarkmatrixel-

0.6 0++ (1x1) ement,hnn¯nn¯|HCG|nn¯ss¯i,wascalculatedtobesmallsince
only annihilation diagrams contribute. Its value was only
- - -
V ) <ss|H|nnss> a few MeV in magnitude for any αI and thus has no ap-
Ge preciable effect in this study.
( 0.4 With the calculated matrix elements and previously
g predicted unmixed meson and tetra-quark masses [14,16,
n
xi 17],thecompleteHamiltonianmatrixwasdiagonalizedto
mi obtaintheexpansioncoefficientsandmassesforthecorre-
0.2 - - - sponding eigenstates. Using αI =0.2, the results for 0++
< n n |H|nnn n >
states are compared in Table 1 to the observed [1] low-
est six 0++ states. Noteworthy, after mixing, the σ me-
- - -
<nn|H|nnss> son mass is shifted from 848 MeV to 776 MeV and the
0.0 strange scalar meson mass also decreases from 1297 MeV
0.0 0.1 0.2 0.3 0.4 0.5 to 1006 MeV, now close to the experimental value of 980
a MeV. Mesonandtetra-quarkmixing clearlyimprovesthe

I
modelpredictions as the massesof the other f states are
0
Fig. 4. The0++ mixingmatrix elementsversusαI.Solidand also in better agreement with data. Figure 6 illustrates
dashed lines are for qq¯spin 0 and 1, respectively. theoverallimproveddescriptionthatmixingprovidesfor
the f spectrum. New structure insight has also been ob-
0
tained from the coefficients, with the predictions that the
this freedom and studied the mixing sensitivity to this
σ/f (600) is predominantly a mixture of nn¯ and nn¯nn¯
0
parameter.
states while the f (980) consists mainly of ss¯ and nn¯ss¯
0
The tetra-quarkparity and chargeparity are givenby
states.
P = ( 1)lA+lB+lI and C = ( 1)lA+sA+lB+sB so for the
lightes−t, unmixed JPC =0 +,−1 states
± −−
Table 2 lists the masses and coefficients for the 0 +
−
0++ lA =lB =lI =0,sA =sB =0 or sA =sB =1, states for αI =0.5.Again, mixing lowers (raises)the pre-
0−+ lA =lB =0,lI =1,sA =sB =1, dictedmassforstatesrelativetounmixedqq¯(qq¯qq¯)states.
All mixed hadron masses are closer to measurement than
1 l =l =0,l =1,s =1 or s =1.
−− A B I A B the unmixed ones, except the most massive state which
presumable could further mix with omitted heavier con-
Forthe1 p-wavestatewechoosel =1sincethisyields
−− I
figurations. Note from the expansion coefficients that fla-
a lower mass than states with l = 1 or l = 1. Note for
A B
the 0++ state, the spin of the two qq¯ clusters are both vor mixing is again weak (i.e. nn¯ with ss¯) and less likely
than meson, tetra-quark mixing having the same flavors.
either 0 or 1 and for each the three mixing matrix ele-
ments versus α are shown in Fig. 4. The mixing term is
I
zerowhenαI iszeroandthenincreaseswithincreasingαI.
Also,mixingwithss¯statesisstrongerthanwithnn¯states. 0.25
In particular, for αI = 0.2, the sA = sB = 0 matrix ele-
mentsare ss¯HCG nn¯ss¯ =365MeV, nn¯ HCG nn¯nn¯ =
166 MeV ahnd| nCn¯ H| CG nin¯ss¯ =45 MehV. | C | i 0.20 - - -
h | C | i <nn|H|nnss>

)
V
Ge 0.15
AgaFinig,uarlel 5msixhionwgsttehremsmiaxriengzevreorsfuosr ααII f=or 00−a+ndstathteesn. ng ( < nn-|H | n n - n n - >
increasewithincreasingα .Incontrasttothe 0++ result, xi 0.10
I mi
nn¯ H nn¯nn¯ nowhasthelargestvalue.Thevalueα =
h | CG| i I - - -
0.5yieldsreasonableηandη masseswithmixingelements <ss|H|nnss>
′
nn¯ HCG nn¯nn¯ =219 MeV, nn¯ HCG nn¯ss¯ =157 MeV 0.05
hand| sCs¯H|CG nn¯iss¯ =138 MehV. | C | i
h | C | i
Forthe1 statesanovelmixingresultwasobtained.
−−
The mixing matrix elements were again 0 for α =0 but, 0.00
I
and very interesting, also essentially 0 for all values of 0.0 0.1 0.2 0.3 0.4 0.5
αI. Our model therefore predicts minimal flavor mixing a
I
for vector mesons which would explain the known ω/φ
ideal mixing. Related, weak mixing also provides a good Fig. 5. The 0−+ mixing matrix elements versus αI.
6 Ping Wang, StephenR. Cotanch and Ignacio J. General: Meson and tetra-quark mixing
7 Summary and conclusions IG(JPC) states, especially glueball and hybrid mesons
with explicit gluonic degrees of freedom. Determining the
WehaveappliedtheestablishedCGmodeltostudyqq¯and level of mixing for these exotic systems will be important
qq¯qq¯mixing for the low-lying 0++, 0 + and 1 spectra. for finally establishing their existence.
− −−
In general, mixing effects are significant and provide an
improvedhadronicdescription.Asimportant,ourfindings
clearly document that mixing is necessary for a complete Acknowledgements. The authors are very appreciative for
understanding of scalar and pseudo-scalar hadrons. the assistanceandadvice fromF. J.Llanes-Estrada.Sup-
The mixed0++ states areasuperpositionofsix states portedinpartbyU.S.DOEgrantsDE-FG02-97ER41048
withcoefficientsobtainedbydiagonalizingtheH Hamil- and DE-FG02-03ER41260.
CG
tonian which decreases the mass for states dominated by
qq¯componentswhile increasingthosepredominantlyhav-
ingtetra-quarkconfigurations.Theresultingf massspec- References
0
trum is in good agreement with observation.
Mixing is notas large for the 0 + spectrum, therefore 1. W.-M. Yao et al, J. Phys.G 33, 1 (2006)
−
the mass shifts are smaller. Again after mixing, predomi- 2. Yu.A.Simonov, Phys. Atom.Nucl. 64, 1876 (2001)
nantlyqq¯qq¯statesincreaseinmasswhiletheqq¯dominated 3. N. Kochelev and D.-P. Min, Phys. Rev. D 72, 097502
masses decrease. All mixed states are closer to measure- (2005)
mentexcepttheheaviestwhichmightbefurthercorrected 4. W.-J. Lee and D. Weingarten, Phys. Rev. D 61, 014015
(2000)
via mixing with omitted higher configurations.It is note-
5. F. Giacosa, Th. Gutsche, V.E. Lyubovitskij and A.
worthy that the CG model provides sufficient flavor mix-
Faessler, Phys.Rev.D 72, 094006 (2005)
ingtoproducereasonablemassesforhistoricallychalleng-
6. L. Burakovsky and P.R. Page, Eur. Phys. J. C 12, 489
ing η,η system.
′ (2000)
Significantly, mixing is calculated to be weak for the
7. L.S. Celenza, B. Huang, H.S. Wang and C.M. Shakin,
1 states. Therefore, minimal flavor mixing for vector
−− Brooklyn College Report No. BCCNT 99/111/283
mesons follows naturally from our model, consistent with
8. F.E. Close and A. Kirk,Phys. Lett. B 483, 345 (2000)
the known ω/φ ideal mixing.
9. D.M. Li, H. Yu and Q.-X. Shen, Commun. Theor. Phys.
Finally, we performed a perturbative investigation of 34, 507 (2000)
the exactQCD Coulombgauge kernel to order g6. As ex- 10. H.NoyaandH.Nakamura,Nucl.Phys.A692,348c(2001)
pected,aseriesexpansioning doesnotconverge,however 11. F. Giacosa, Phys. Rev.D 75, 054007 (2007)
asubsetclassofdiagramsmightbeamendabletoapertur- 12. Evan Beverenand G. Rupp,Phys.Rev.Lett. 93,202001
bative treatment and further study is recommended. Fu- (2004)
tureworkshouldalsoaddressmixingapplicationstoother 13. B. Silvestre-Brac, J. Vijande, F. Fernandez and A. Val-
carce, AIPConf. Proc. 814, 665 (2006)
14. S.R. Cotanch, I.J. General and P. Wang, Eur. Phys. J. A
31, 656 (2007)
15. I.J. General, S.R. Cotanch and F.J. Llanes-Estrada, Eur.
Phys. J. C 51, 347 (2007) [arXiv:hep-ph/0609115]
16. I.J. General, P. Wang, S.R. Cotanch and F.J. Llanes-
Estrada, Phys. Lett.B 653, 216 (2007) [arXiv:0707.1286]
17. F.J.Llanes-EstradaandS.R.Cotanch,Nucl.Phys.A697,
303 (2002)
18. F.J. Llanes-Estrada, S.R. Cotanch, A.P. Szczepaniak and
E.S. Swanson, Phys.Rev.C 70, 035202 (2004)
19. F.J. Llanes-Estrada, P. Bicudo and S.R. Cotanch, Phys.
Rev.Lett. 96, 081601 (2006)
20. F.J. Llanes-Estrada and S.R. Cotanch, Phys. Rev. Lett.
84, 1102 (2000)
21. F.J.Llanes-EstradaandS.R.Cotanch,Phys.Lett.B504,
15 (2001)
22. P. Bicudo, S.R. Cotanch, F.J. Llanes-Estrada and D.G.
Robertson, Eur. Phys. J. C 52, 363 (2007)
23. T.D. Lee, Particle Physics and Introduction to Field The-
ory (Harwood Academic Publishers, New York,1990)
24. D. Zwanziger, Nucl. Phys.B 485, 185 (1997)
Fig. 6. Unmixed and mixed f0 spectrum compared todata.
Ping Wang, StephenR. Cotanch and Ignacio J. General: Meson and tetra-quark mixing 7
Table 1. Mixing coefficients and masses in MeV for 0++ states.
|nn¯ > |ss¯> |nn¯nn¯ >1 |nn¯nn¯ >2 |nn¯ss¯>1 |nn¯ss¯>2
no mixing 848 1297 1282 1418 1582 1718
mixing 776 1006 1329 1440 1676 1918
exp. f0(600) f0(980) f0(1370) f0(1500) f0(1710) f0(2020)
400 - 1200 980±10 1200 - 1500 1507±5 1718±2 1992±16
coeff. a b c1 c2 d1 d2
f0(600) 0.936 -0.075 0.263 0.216 0.030 -0.007
f0(980) 0.057 0.818 -0.022 -0.017 0.549 -0.156
f0(1370) -0.308 0.045 0.922 0.228 0.008 -0.003
f0(1500) -0.139 0.017 -0.282 0.949 0.006 -0.003
f0(1710) -0.031 -0.240 -0.001 -0.002 0.582 0.776
f0(2020) -0.063 -0.514 -0.002 -0.006 0.599 -0.610
Table 2. Mixing coefficients and masses in MeV for 0−+ states.
nn¯ ss¯ |nn¯nn¯ > |nn¯ss¯>
no mixing 610 1002 1252 1552
mixing 531 970 1316 1598
exp. η η′ η(1295) η(1405)
547.51±0.18 957.78±0.14 1294±4 1409.8±2.5
coeff. a b c d
η 0.951 -0.046 0.279 0.126
η′ 0.032 0.973 -0.046 0.223
η(1295) -0.289 0.036 0.953 0.080
η(1405) -0.108 -0.222 -0.105 0.963