Table Of ContentTetraquark and the flux tube recombination
Marco Cardoso
2 ∗
1 CFTP,InstitutoSuperiorTécnico
0 E-mail: [email protected]
2
Pedro Bicudo
n
a CFTP,InstitutoSuperiorTécnico
J
E-mail: [email protected]
0
3 Nuno Cardoso
CFTP,InstitutoSuperiorTécnico
]
t E-mail: [email protected]
a
l
-
p
Herewestudythestaticpotentialforthetwoquarksandtwoantiquarkssystem. Firstthisisdone
e
h using the tetraquark operator, which has been previously used to calculate the static potential.
[
Thisisfoundtogivegoodresultsintheregionwherethetetraquarkisexpectedtobetheground
1
state,howeverfailingoutsideit.Torepairthis,weresorttoavariationalmethod.Thisletusstudy
v
7 thefirstexcitedstatebesidesthegroundstateofthesystemintwodifferentparticledispositions,
6
onewherethequarksareonthesamesideofarectangleandtheotherwheretheyareatopposite
2
6 sides. Resultsforthefieldcomponentsandforthelagrangiandensityarepresented.
.
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a
InternationalWorkshoponQCDGreen’sFunctions,ConfinementandPhenomenology,
September05-09,2011
TrentoItaly
Speaker.
∗
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
Wilson loop W in a gauge invariant manner as shown in Figs.6(a) and (b),
5Q
respectively.
Tetraquarkandthefluxtuberecombination MarcoCardoso
M
2 L R
3 3
R
2
R
2
L L
2
2
M
R
L 1 L R1
T 1 T 1
M L R
1 4 4
Figure1: Wilsonloopoperatorusedby[2]and[3]tocalculatethestaticpotentialofatetraquarksystem.
Figure 6: (a) The tetraquark Wilson loop W . (b) The pentaquark Wilson
4Q
1. Motivaltoionop W . The contours M, M , R , L (i = 1, 2, j = 3, 4) are line-like and
5Q i j j
R , L (j = 1, 2) are staple-like. The multi-quark Wilson loop physically means
Systems constituted by two quarks and two antiquarks are of extreme importance for strong
j j
interaction physics. Not only because they are a starting point for meson-meson scattering, but
that a gauge-invariant multi-quark state is generated at t = 0 and annihilated
also because of the possible existence of bound-states — tetraquarks, initially predicted by Jaffe
[1]. Thereaarte steve=ral oTbserwveditrehsonqanuceas wrhkosarebceanidnidgatesstpo taettraiqaualrlkys. Tfihxe meodst riencentR3 for 0 < t < T.
candidatesaretheZ+ particlesreportedbytheBellecollaboration. Recently,staticsystemsoftwo
b
quarks and two antiquarks, were studied in the lattice [2, 3] using an operator with the quantum
numbersofthetetTraqhuaerk.tTehetrresaulqtsuindaicraktethWatthielsstaotincpoltoenotiaplofWthissystemancoudldbtehweellpentaquark Wilson loop W are
4Q 5Q
described by the generalized flip-flop potential. This potential, is similar to the ones which were
defined by
first proposed [4, 5] as a device to suppress the non-physical Van der Waals forces, which arise
when potential models based on Casimir Scaling are used, differing by the existence of a third
branchofthepotential—thetetraquarkbranch. 1
˜ ˜ ˜ ˜
W tr(M R M L ),
4Q 1 12 2 12
≡ 3
2. TetraquarkWilsonLoop
1
abc a b c aa ˜ ˜ ˜ bb ˜ ˜ ˜ cc
TheStaticpotentialforthetetrWaquarkhasbeenstudiedǫinthelǫattic′eb′y′[2M]and[3′].(R R R ) ′(L L L ) ′, (4)
5Q 3 12 4 3 12 4
≡ 3!
Forthat,aWilsonLoopoperatorwasconstructed. Itisgivenby
1
where M˜ , M˜ W,4QL=˜ Tar[Mn1dR12RM˜2L12](,i=1,2, j=1,2,3,4()2.1a)re given by
i j3 j
with
˜ ˜ ˜ ˜ µ
M, M , R , L P exp ig dx A (x) SU(3) . (5)
iR1ii2(cid:48) =jεijkεi(cid:48)jj(cid:48)k(cid:48)R≡1jj(cid:48)Rk2k(cid:48) { (2.2) µ } ∈ c
M,M ,R ,L
L1ii2(cid:48) = εijkεi(cid:48)j(cid:48)k(cid:48)L1jj(cid:48)L2kk(cid:48). Z i j j
Thecorrespondingpath˜sarevisuallydes˜cribedonfigure2.
Here, R and L are defined by
12 12
This Wilson Loop has the quantum numbers of colour singlet system where the two quarks
formanantitripletandthetwoantiquarksformatriplet.
1 1
˜a a abc a b c bb cc ˜a a abc a b c bb cc
R ′ ǫ ǫ ′ ′ ′R ′R ′, L ′ ǫ ǫ ′ ′ ′L ′L ′. (6)
12 2 1 2 12 1 2
≡ 2 ≡ 2
The multi-quark Wilson loop physically means that a gauge-invariant multi-
quark state is generated at t = 0 and annihilated at t = T with quarks being
spatially fixed in R3 for 0 < t < T.
The multi-quark potential is obtained from the vacuum expectation value of
the multi-quark Wilson loop as
1 1
V = lim ln W , V = lim ln W . (7)
4Q 4Q 5Q 5Q
− T T h i − T T h i
→∞ →∞
4.3 Lattice QCD Result of the Pentaquark Potential
We perform the first study of the pentaquark potential V in lattice QCD
5Q
with (β=6.0, 163 32) for 56 different patterns of QQ-Q¯-QQ type pentaquark
×
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Tetraquarkandthefluxtuberecombination MarcoCardoso
y
(cid:1791)(cid:3365) y (cid:1791)(cid:3365)
α
(cid:1791)(cid:3365) (cid:1791)(cid:3365) (cid:1791)(cid:3365) (cid:1791)(cid:3365)
α
α
(cid:1791)(cid:3365) (cid:1791)(cid:3365) α x
(cid:2200) α
(cid:2779) α (cid:2200)
α α (cid:2779)
(cid:2200) (cid:2200)
(cid:1791) (cid:1791) (cid:2779) (cid:2779)
α α
(cid:2200) α
(cid:1791) (cid:2778) (cid:1791) (cid:1791) (cid:1791)
x
(cid:2200) α(cid:2200)
(cid:2778) (cid:1791) (cid:2778) (cid:1791)
(cid:2200)
Figure2: Herewecanseetheshapeoftheminimalstringwhichlin(cid:2778)ksthefourparticlesinthetetraquark.
Note that when the diquark and the diantiquark are close the five segment structure collapses into a four
segmentone.
Thislatticestudies,indicatethatthestaticpotentialofthetwo-quarkandtwo-antiquarksystem
isageneralizedflip-floppotential:
V =min(V ,V ,V ) (2.3)
FF T M M M M
1 2 3 4
where V and V are the two possible two-meson potentials, given by the sum of two in-
M M M M
1 2 3 4
dependent intra-meson potentials V =V +M , and V is the tetraquark potential which
M M M M T
1 2 1 2
corresponds to the sector where the four particles are confined, linked by a single fundamental
string. Itisgivenby
λ λ
V =C+α∑ i j +σL , (2.4)
T min
2 · 2
i<j
where L in is the minimal distance linking the four particles, which corresponds to the string
m
configurationdisplayedonfig. 2.
For the special case where the four particles form as rectangle as in fig. 2, L is given by
min
L =√3r +r forr > r1 [6,7]. IfweneglecttheCoulombpartofV ,wecouldestimatethat
min 1 2 2 √3 FF
thetetraquarksectorbecomesthegroundstatewhenr (cid:38)√3r .
2 1
3. TetraquarkFields
Thefieldsinthetetraquarksystemcanbecalculatedbyusingthecorrelationofplaquettesand
WilsonLoops. TheSquaredchromoelectricandchromomagneticfields,arecomputedby
WP
E2 = P (cid:104) 0i(cid:105) (3.1)
(cid:104) i (cid:105) (cid:104) 0i(cid:105)− W
(cid:104) (cid:105)
WP
B2 = (cid:104) jk(cid:105) P
(cid:104) i(cid:105) W −(cid:104) jk(cid:105)
(cid:104) (cid:105)
wheretheindexicomplments j andk. P isP =1 1Tr[U (s)U (s+νˆ)U†(s+νˆ)U†(s)].
µν µν −3 µ ν µ ν
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Tetraquarkandthefluxtuberecombination MarcoCardoso
Figure3: Plotofthelagrangiandensityfordifferentvaluesofr withr =14.
1 2
8×10−4
QQ-QQ, r = 6, r = 14
1 2
QQ, r = 14
2
6×10−4
>
2E4×10−4
<
2×10−4
0
−10 −7.5 −5 −2.5 0 2.5 5 7.5 10
x
Figure4: Comparisionbetweenthesquaredchromoelectricfieldinthequark-antiquarkfluxtubecore(in
themeson)andthediquark-antidiquarkfluxtubecore(inthetetraquark).
TheLagrangianandenergydensitiesaregivenby
1
L = E2 B2 (3.2)
(cid:104) (cid:105) 2(cid:104) − (cid:105)
1
H = E2+B2 .
(cid:104) (cid:105) 2(cid:104) (cid:105)
Now, we will present results for the chromo-fields for the tetraquark system for the simple
casewherethefourparticlesformarectangles,asinfig. 2.
Thisresults,andtheoneofthefollowingchapterswereobtainedusingquenchedlatticeQCD
configurations with dimension 243 48 and β =6.2. They where used in GPUs by using a com-
×
binationofCabbibo-Marinari,pseudoheat-bathandover-relaxationalgorithms[8]. APEsmearing
[9]andHypercubicblocking[10]wereusedtoimprovethesignaltonoiseratio.
Theresultsforthelagrangiandensityaregivenonfig. 3forfixedr =14.
2
Aswecanseetheintheleftmostfig. r =0,inwhichcasethesystemthesystemiscollapsed
1
intoameson. Thenwecanseelefttoright,withtheincreaseofr ,theconfiningstringacquiresa
1
shapesimilarwiththeoneonfig. 2forr > r1 ,inagreementwiththeresultobtainedforthestatic
2 √3
potential.
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Tetraquarkandthefluxtuberecombination MarcoCardoso
Figure5: Lagrangiandensityinthefluxtubeforr =r =8andr =8,r =14.
1 2 1 2
In fig. 3 it is compared the squared chromoelectric fields E2 in the flux tube centre of the
(cid:104) (cid:105)
meson and in the centre of the diquark-diantiquark flux tube in the tetraquark. As can be seen
boththeflux-tubeshaveasimilarbehaviour,whichconfirmsthatthestringwhichispresentinthis
systemisafundamentalone,againinagreementwiththeresultsforthestaticpotential.
However,ascanbeseeninfig. 3thisagreementdisappearsforr <√3r —theregionwhere
2 1
thegroundstateofthesystemisatwomesonstateandnotthetetraquarkstate. Ascanbeseenthere,
for r =8 and r =14 (tetraquark region), we obtained the expected ground state. However for
1 2
r =r =8(meson-mesonregion)westillseeastringlinkingthefourparticlesasinthetetraquark
1 2
state, but we should see, to be in agreement with the generalized flip-flop model supported by the
results obtained for the static potential, two strings corresponding to the two mesons. We think
thatthisdisagreementisduetothelowoverlapoftheusedoperatorwiththegroundstate,andalso
to the small temporal extensions used in the Wilson Loop. To obtain the true ground state of the
systeminthisregionandalsothefirstexcitedstateweusedavariationalmethod.
4. VariationalMethodfortheQQQ¯Q¯ system
Now we will improve this method in order to obtain the true ground state of the system and
also to obtain the first excited state. To achieve this, we note that the Wilson Loop operator could
bewrittenasacorrelationofacertainoperatoratdifferenttimesW(t)= Oˆ(t)Oˆ†(0) . Thiscould
(cid:104) (cid:105)
begeneralizedbyconsideringinsteadabaseofoperatorsOˆ. Sothisway,ourwilsonloopbecomes
i
amatrixW = Oˆ Oˆ† . Thiscouldbeusednotonlytoimprovethegroundstateoverlapbutalsoto
ij (cid:104) i j(cid:105)
obtainmoreenergylevelsofthesystem. Thesearegivenbysolvingthegeneralizedeigensystem
W (t) cj(t)=w W (0) cj(t) (4.1)
ij n n ij n
(cid:104) (cid:105) (cid:104) (cid:105)
Thefieldscanthenbecalculatedby(cid:104)P(cid:105)n= (cid:104)WWnnP(cid:105)−(cid:104)P(cid:105)withWn=cinWijcnj.
Wewillconsideragainthecasewhereth(cid:104)ef(cid:105)ourparticlesformarectanglesandalsotwodiffer-
ent alignments of the QQQ¯Q¯ system. A parallel one, where the two quarks are on the same side
oftherectangleandanantiparallelone, wherethequarksareonoppositecornersoftherectangle
(fig. 4).
For both cases we will use a base of two operators, similarly to what was made in [11]. In
the parallel case the operators will be the tetraquark operator Oˆ , the correlation of which is the
4Q
tetraquark Wilson LoopW = Oˆ (t)Oˆ (0) , and a two meson operator. This gives a Wilson
4Q 4Q 4Q
(cid:104) (cid:105)
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Tetraquarkandthefluxtuberecombination MarcoCardoso
Figure6: Left: Parallelalignment. Right: Antiparallelalignment.
Figure7: ElementsoftheWilsonloopmatrixusedfortheparallelalignment.
Figure8: ElementsoftheWilsonloopmatrixusedfortheantiparallelalignment.
loop matrix where the diagonal elements areW and the correlation of two wilson loops, while
4Q
theoff-diagonalelementscorrespondtothetransitionbetweenthetwostates(seefig. 4
Fortheantiparallelalignment,theoperatorsusedwerethetwomeson-mesonoperators,giving
thefourmatrixelementsgivenonfig. 4.
5. Results
Now,wepresenttheresultsobtainedfortheparallelgeometry.
Infig. 5wecanseethecomponentsofthechromoelectricandchromomagneticfieldsforthe
groundstateoftheparallelalignment,withr =6andr =12,whileinfig. 5thecomponentesare
1 2
6
Tetraquarkandthefluxtuberecombination MarcoCardoso
Figure 9: Components of the chromo-electromagnetic field, for the ground state of the parallel geometry
withr =6andr =12.
1 2
presented for the ground state of the antiparallel alignment with r =6 and r =10. The results
1 2
indicatethatwhatwehaveisessencialyatetraquarkinthefirstcaseandatwomesonsysteminthe
second. Besides that we can see features that are common to flux-tubes in general [9] such as the
dominanceofthelongitudinalchromoelectricfield.
In fig. 5 it is shown the evolution of the lagrangian density of the ground state of the parallel
geometry,forfixedr =6. Herewecanseethetransitionbetweenthetwomesonstateinthetop-
1
leftpicture(r =6)tothetetraquarkstateinthebottomrightpicturer =12. Thisisinagreement
2 2
withthepredictionsofthegeneralizedflip-floppotentialcontrarytotheresultsweobtainedusing
only the tetraquark operator W . The same evolution is presented in 5, but for the first excited
4Q
stateofthesameconfigurations. Thisresultsarenotsoreadilyunderstandable.
Tobettercomparethegroundandthefirstexcitedstatewecanlookatfig. 5inthey=0axis.
In fig. 5 we see the lagrangian density for the parallel alignment with varying r and r . For
1 2
r =6 and r =10 we can see the two mesons almost completely separatedly, while for r =r
1 2 1 2
wehaveastatewhenallthefourparticlesarelinked. Weonlyshowpicturesforr r . Thecase
2 1
≥
r <r can be seen by rotating the pictures for r >r . This behaviour is again expected, as we
2 1 2 1
obtaintheseparationintwomesonsofthesystemwhenitcorrespondtotheminimalstringenergy.
Thebehaviourforr =r isalsoexpected, becauseneitherofthetwomesonsystemsispreferred
1 2
overtheother.
Theresultsforthefirstexcitedstatearepresentedinfig. 5. Aswecanseethere,thefirstexcited
stateisdifferentfromatwomesonstate,withallthefourparticlesbeinglinkedtheflux-tube.
Finally we compare the ground state and the first excited state in this geometry in fig. 5 for
y=0, in the left, for r =6 and r =10, in the right for r =r =10. As can be seen, for the
1 2 1 2
first case, the ground state flux-tube in this region is very small, because of the formation of two
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Tetraquarkandthefluxtuberecombination MarcoCardoso
Figure10: Componentsofthechromo-electromagneticfield,forthegroundstateoftheanti-parallelgeom-
etrywithr =6andr =10.
1 2
Figure11: Lagrangiandensityfordifferentvaluesofr andr inthegroundstateoftheparallelgeometry.
1 2
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Tetraquarkandthefluxtuberecombination MarcoCardoso
Figure 12: Lagrangian density for different values of r and r in the first excited state of the parallel
1 2
geometry.
r = 6 and r = 12
1 2
0,0008
n = 0
n = 1
0,0006
0,0004
0,0002
0
-10 -8 -6 -4 -2 0 2 4 6 8 10
Figure13: Comparasionoftheactiondensityinthecenteroffluxtubeintheparallelgeometryforr =6
1
andr =12.
2
mesons.
6. Discussion
Our results for the fields of the two quarks and two antiquarks system agree with the results
obtainedforthestaticpotential,thereforesupportingastringpictureofconfinementwithflux-tube
recombination. Besides this, we were also able to calculate the first excited state of this system.
The low overlap between the tetraquark operator and the two meson ground state may indicate
smallwidthtetraquarkresonances. Wearealsostudyingthefieldsofthepentaquarksandthestatic
potentialsofthissystem.
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Tetraquarkandthefluxtuberecombination MarcoCardoso
Figure 14: Lagrangian density for different values of r and r in the ground state of the anti-parallel
1 2
geometry.
Figure15: Lagrangiandensityfordifferentvaluesofr andr inthefirstexcitedstateoftheanti-parallel
1 2
geometry.
10
