I=0 scalar channel

a r X i v :h e p -l a t /0112012v 1 10 D e c 2001

1

I =0scalar channel

SCALAR Collaboration:S.Muroya a ,A.Nakamura b ,C.Nonaka b ,M.Sekiguchi c ,and H.Wada d

a Tokuyama Women’s College,Tokuyama 745-8511,Japan b

IMC,Hiroshima University,Higashi-Hiroshima 739-8521,Japan

c

Faculty of Engineering,Kokushikan University,Tokyo 154-8515,Japan d

Laboratory of Physics,Nihon University,Chiba 274-8501,Japan

Using lattice QCD with dynamical Wilson fermions,we study I =0and J P =0+channel which is constructed

by 12(ˉu u +ˉdd

)|0 ,in order to search for the σmeson.Our preliminary result shows that the connected and disconnected diagrams contribute to the σmeson propagator in the same order.

1.INTRODUCTION

In QCD,the chiral symmetry is spontaneously broken (restored)at the con?nement (decon?ne-ment)phase where sigma meson plays an essen-tial role.We do not doubt this mechanism,and yet no one is sure whether the sigma meson ex-ists or not.It might have large mass and/or wide width,or might be simply π?πcorrelation e?ect.The light σmeson had disappeared from the ta-bles of Particle Data Group (PDG)for over 20years.However,the I =0and J P C =0++me-son,“f 0(400-1200)or σ”,appeared bellow 1GeV mass region in PDG recently[1].This is probably because of recent π-πscattering phase shift re-analyses;Especially,Igi and Hikasa constructed a general model-independent framework,which re-spects the analyticity,unitarity and crossing sym-metry together with chiral symmetry low energy theorem,to describe the ππelastic scattering be-low 1GeV mass region and investigated the ex-istence of σmeson [2].See Ref.[3]for a good review on the situation and physical meaning of the sigma meson.

Now it is very desirable to investigate whether σmeson appears as a pole based on lattice https://www.wendangku.net/doc/ca4427108.html,ing the quenched approximation,Alfold and Ja?e discussed the possibility of the light scalar mesons as ˉq 2q 2states rather than ˉq q [5].Mc-Neile and Michael computed the mixed iso-singlet scalar masses of q ˉq and glueball states in two kind

of situation,i.e,with and without the dynami-cal quark e?ects [6].The σmeson masses which

are obtained with consideration of the dynamical quark e?ects are much lower than the quenched results.The goal of our project is to conclude whether the σmeson exists or not below 1GeV in QCD.

2.σPROPAGATOR

We construct I =0scalar channel by σ|0 .The operator σis given as

σ(x )≡

3 c =1

4 α=1

ˉu c α(x )u c α(x )+ˉd c α(x )d c α(x )2,(1)

where u and d are the u -quark and d -quark Dirac

spinors,respectively.This operator has the same quantum number as the vacuum,I =0and J P =0+.The indices c and αdenote color and Dirac spinor indices,respectively.The σmeson propagator is given by,

G (y,x )=? T rW ?1(x,y )W ?1(y,x )

+2 T rW ?1(y,y )T rW ?1(x,x ) ?2 T rW ?1(y,y ) T rW ?1(x,x ) .

(2)

Here ”Tr”represents summation over color and Dirac spinor indices and W ?1is u ,d quark prop-agator.The third term of Eq.(2), σ(y ) σ(x ) ,corresponds to the subtraction of vacuum contri-

2

10.91

10.91510.9210.92510.9310.93510.9410.9450200

4006008001000

q u a r k l o o p W ^{-1}(x ,x )

No. of Noise

"noise.plot""Fluctuation0"

Figure 1.T rW ?1(x,x )evaluated by the noise method as a function of the number of Z2noise sources.

bution.1The second and the third terms are the same order and the high precision numerical sim-ulations and careful analyses are required.3.NUMERICAL SIMULATIONS

We calculate the σpropagator by using Hy-brid Monte Carlo algorithm.We use Z 2noise method to calculate the disconnected diagrams.The 1000random Z 2numbers are generated.The two-?avors Wilson fermion is simulated on the 83×16lattice.Based on ref.[7],we set β=4.8,κ=0.1846(a =0.197(2)fm,κc =0.19286(14))[7].After the thermalization trajectories,σprop-agators are calculated on a con?guration in every 10trajectories.

3.1.NUMERICAL ACCURACY

Since there are big numerical cancellation in σ(y )σ(x ) ? σ(y ) σ(x ) ,we must be careful to controle the numerical accuracy.

In Fig.1,we show the values of σ(x )for a typical con?guration as a function of the num-ber of Z2noise sources.Dotted lines represent

3 Figure3.The comparison between theσpropa-

gator andρpropagator.

withρpropagator.We can see that theσmeson

mass could be the same order ofρmeson mass,

i.e.,we obtain the suggestion of the existence of

lightσmeson,though the error bar ofσmeson

propagator is very large.

4.CONCLUDING REMARKS

We investigated the property of I=0and

J P C=0++scalar meson(σmeson)whose oper-

ator is1

2(ˉu u+ˉdd).In theσmeson propagator

the contribution of disconnected diagram is the same order of connected diagram;Quenched ap-proximation is not reliable for the investigation of theσmeson.The evaluation of the discon-nected diagram is done by using Z2noise method.

A statistical error ofσpropagator which comes from the disconnected diagram mainly is large in the present stage.As preliminary results,we obtain the following properties ofσpropagator: (1)Both the connected and disconnected parts equally contribute to theσpropagator.(2)σmeson could have mass of the same order of the ρmeson.

It is necessary to generate much more gauge con?gurations and improve the statistical preci-sion of the estimation ofσpropagator.We also plan to improve the source of the sigma propaga-tors.

Furthermore we must investigate the mixing state of theσmeson and glueball if we obtain the result thatσmeson mass is greater than1 GeV region[8,?].

ACKNOWLEDGMENT

We would like to thank T.Kunihiro for the useful discussions and encouragement.This work is supported by Grant-in-Aide for Sci-enti?c Research by Monbu-Kagaku-sho,Japan (No.11440080,No.12554008and No.13135216). This work is performed by SX5at RCNP,Osaka Univ.One of the authors(C.N.)would like to ac-knowledge the?nancial support by the Soryushi Shogakukai.

REFERENCES

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