A lattice study of light scalar tetraquarks with I=0, 1/2, 1

Sasa Prelovsek Lattice 05 1

A lattice study of light scalar tetraquarks A lattice study of light scalar tetraquarks with I=0, 1/2, 1with I=0, 1/2, 1

Lattice 08, WilliamsburgLattice 08, Williamsburg

Sasa Prelovsek Sasa Prelovsek University of Ljubljana

[email protected]

Lattice data from collaboration with

Bern-Graz-Regensburg Coll. (BGR)

(Daniel Mohler, Christian Lang, Christof Gattringer)


OutlineOutline

motivation challenges present simulation and its results previous lattice simulations


Puzzle of light scalar mesons:Puzzle of light scalar mesons:

2/11 II mm

2/11 II mm

2/11 II mm

[Jaffe, Maiani, t’Hooft, .....]

?

?or qqqqqq

Interpolators used: Interpolators used: diquark anti-diquarkfcfc

qqqq 3,33,3 ][][

2/11 II mm

0++ nonet with inverted spectrum

][][ 55 cTbc

Tbabca dCssCdds

I=1 I=0 I=1/2

0PCJ

I=1

I=1/2


Challenge of the simulation:Challenge of the simulation:to distinguish one-particle (tetraquark) state and scattering states in C(t)

0

0

IC

p

I flavor of source/sink scattering states

0 udud

1/2 udds K

usds K K,

,..2,1,0

2

nL

nk


we distinguish one-particle and scattering states by considering:

En

volume dependence of wn

How to distinguish tetraquark from scattering?How to distinguish tetraquark from scattering?

tEtE ewewtC 1010)(

properties of scattering:

3

32

32

1

1)(,)(:2/1

4

7)(,)(2:0

Lw

LfLdELdEmmEI

LfLdELdEmEI

treeK

tree

property of (one-particle) tetraquark:

)( 0LOw

),(1

),()2(

)( 33

3

tkfL

tkfkd

tCk

Mathur et al.

Kentucky (2006)

k


Lattice simulations of tetraquarksLattice simulations of tetraquarksPrevious studies (briefly reviewed at the end): [BGR Coll, PRD 73 (2006)]

only I=0 channel (Jaffe studied also I=2) all consider a single correlator

Present study:

a whole flavour pattern: I=0, 1/2, 1I=0, 1/2, 1

3x3 correlation matrix3x3 correlation matrix evaluated:

3 different smearings at source and the sink:

spatially symmetric Jacobi smearing on quarks: narrow (n) & wide (w)

]][[

]][[

]][[

wnwn

wwww

nnnn

qqqq

qqqq

qqqq

tEn

tn

nnn

newt

ntvttvtC

)(

,..1,0,)()()()(

anti-diquark diquark

variational method


use quenched approximation:

discard single and double annihilation contractions

There is a good excuse to use these two approximations as we are interested in state with 4 valence quarks: these two

approximations discard

Chirally Improved Fermins [BGR Coll.]

a=0.148 fm, V=163x32, 123x24

ms with physical mass; mu,d correspond to mpi= 340-570 MeV

As in all previous studies we:

SimulationSimulation

.vacqqqqqq


E0 close to shifted 2m

E1,2>2 GeV: to heavy correspond to

f0(980)

meff of ground state not flat

does not have its own eigenvalue

tower of (k)(-k) present

in the ground state eigenvalue

Results for I=0Results for I=0

)](1[ tEEnn eOew n

Lnkkk

2)()(

324

72

Lfm

)](1[ tEEnn eOew n

??)980(,)600( 0f


Results for I=0: ground stateResults for I=0: ground state

if all tree sources behave close to point-like:

then three eigenvalues of 3x3 matrix are:

the whole tower of scattering states comes in a single eigenvalue!


Results for I=0: Results for I=0: ground stateground state

parameters of fit: mpi, w

0)(22 ELdEmm treefitted

we can not exclude weakly coupling lighter state at large t

scattering

particle-one

: weightsspectral

3

0

/1

)(

Lw

LOw

k

),(1

),()2(

)( 33

3

tkfL

tkfkd

tCk


Results for I=1/2 Results for I=1/2 similar conclusions as in I=0 channelsimilar conclusions as in I=0 channel

parameters of fit: mpi, w

LnkkkK

2)()(


Results for I=1 Results for I=1 analysis of ground state is more complicated:

two towers of scattering states KK, pi etass:

conventional fit of mass at large t

)()(

2)()(

kk

LnkkkK

ss


Summary of our results for I=0,1/2,1Summary of our results for I=0,1/2,1

?

ss

ssdudssu

KK

dssudssu

)()(

)()(


Summary of our results for I=0,1/2,1Summary of our results for I=0,1/2,1 excited states:

to heavy to correspond to light tetraquark candidates:

I was not looking for interpretation of these states

(they may be also some excited scattering states) ground state:

effective mass and volume dependence of spectral weights

roughly consistent with tower of scattering states

we find no evidence for light tetraquark at mpi=340-570 MeV

slight fall of meff at large times: we can not completely exclude possibility of very light weakly coupling tetraquark;

requires further study with larger time extent and larger operator basis

there may still exist possibility for tetraquarks at mpi<340 MeV (Kentucky group found I=0 tetraquark only for mpi<300 MeV)


Previous tetraquark simulationsPrevious tetraquark simulations all quenched, all discard annihilation contr. study only I=0 channel (Jaffe studies also exotic I=2 channel)

all consider single correlator

Alford & Jaffe, 2000 interpolator one relatively heavy quark mass different L only ground state exploredconclusion: shift does not completely agree with FULL (!) scattering prediction: possible indication of tetraquark

I=0


Previous tetraquark simulations:Previous tetraquark simulations: Suganuma, Tsumura, Ishii, Okiharu , 2007 0707.3309 [hep-lat]

• diquark antidiquark interpolator

• conventional and hybrid boundary conditions

• only ground state studied

• conclusion: ground state corresponds to scattering

physdu

phys mmm 2,


N. Mathur, K.F. Liu et al. (Kentucky, XQCD Collaboration) [hep-ph/0607110, PRD, 2006] interpolator range of very small quark masses (overlap fermions) two volumes three lowest states explored: sequential Bayes method conclusion: indication for tetraquark around sigma mass for mpi<300 MeV

Previous tetraquark simulations:Previous tetraquark simulations:

)0()0(

?)600(

)1()1(


Concusion and outlookConcusion and outlook I did not find indication for tetraquarks in I=0, ½, 1

channels at mpi>300 MeV. There are still hope to find tetraquark on the lattice !!

Prompts for tetraquark search with dynamical simulation, light quark masses, preferably on various volumes and using variational method !!

the Kentucky group fond indication that

is tetraqurk at mpi<300 MeV

our ground state in I=1 channel seems to be

slightly below the scattering state (at lightest quark mass)

Documents

A lattice study of light scalar tetraquarks with I=0, 1/2, 1