Scalar meson in dynamical and partially quenched two-flavor QCD lattice results and chiral

a r X i v :h e p -l a t /0407037v 1 29 J u l 2004IJS-TP-11/04

RBRC-424

BNL-HET-04/8

KANAZAWA-04-11

MIT-CTP-3517

Scalar meson in dynamical and partially quenched

two-?avor QCD:lattice results and chiral loops S.Prelovsek a,b ,C.Dawson c ,T.Izubuchi c,d ,https://www.360docs.net/doc/772523220.html,inos e and A.Soni f a Department of Physics,University of Ljubljana,Jadranska 19,1000Ljubljana,Slovenia b Institute Jozef Stefan,Jamova 39,1000Ljubljana,Slovenia c RIKEN-BNL Research Center,Brookhaven National Laboratory,Upton,NY 11973,USA d Institute of Theoretical Physics,Kanazawa University,Ishikawa 920-1192,Japan e Center for Theoretical Physics,Laboratory for Nuclear Science and Department of Physics,MIT,Cambridge,MA 02139-4307,USA f Physics Department,Brookhaven National Laboratory,Upton,NY 11973,USA ABSTRACT This is an exploratory study of the lightest non-singlet scalar q ˉq state on the lattice with two dynamical quarks.Domain wall fermions are used for both sea and valence quarks on a 163×32lattice with an inverse lattice spacing of 1.7GeV.We extract the scalar meson mass 1.58±0.34GeV from the exponential time-dependence of the dynamical correlators with

m val =m sea and N f =2.Since this statistical error-bar from dynamical correlators is rather large,we analyze also the partially quenched lattice correlators with m val =m sea .They are positive for m val ≥m sea and negative for m val

1Introduction

The interest in the light scalar mesons has been renewed recently[1].The existence of the

scalar mesons above1GeV is well established experimentally and there are enough scalar

states between1GeV and2GeV to represent the scalar qˉq nonet[2].The excess of one observed state in this region has been suggested as an indication for the glueball[3].The

lightest iso-triplet state above1GeV is a0(1450).The only scalar states below1GeV,which are experimentally well established,are iso-triplet a0(980)and iso-singlet f0(980)[2].The

existence of the complete scalar qˉq nonet roughly below1GeV would require another iso-

singlet and two strange iso-doublets.The experimental evidence for the existence of a broad iso-singletσmeson around600MeV is growing[2,4],while the existence of the strange

iso-doubletκreported in[5]is even more controversial at present.This raises a question

whether the lightest scalar qˉq states lie below1GeV or above1GeV.In the latter case,the observed scalar states below1GeV have to be interpreted as exotic states like qqˉqˉq[6],ππ

or KˉK molecules,etc.

In this paper we address the determination of the mass of the lightest scalar qˉq state

with non-singlet?avor structure(referred to as the a0meson[2]),the long-term goal being

to relate this state to the observed resonance a0(1450)or a0(980).We determine the mass of the a0meson using a lattice simulation of dynamical QCD(m val=m sea)and partially

quenched QCD(m val=m sea)with N f=2degenerate sea quarks in both cases.Since our aim is qˉq state composed of the light u and d quarks,we employ Domain Wall Fermion

(DWF)formalism,which has good chiral properties[7].We comment also on the mass of

the sˉu and sˉd scalar mesons.

While DWF formulation ought to be helpful in the long run,at present our numerical

work has serious limitations.We are working with two dynamical?avors,which is not full

QCD.Furthermore we have results only at one lattice spacing on a lattice box that is not very large and also quarks are relatively heavy.For these reasons this is an exploratory work. These issues can of course be improved with more computing resources.

Before we introduce our work,we brie?y review the recent lattice simulations of the light non-singlet scalar states.We quote only the statistical error-bars on masses since the continuum and in?nite-volume extrapolations have not been performed in these simulations:?Fully quenched simulations of qˉq:

The quenched qˉq correlator in the chiral limit was simulated by Bardeen et al.[8, 9]with Wilson fermions.The correlators were found to be negative at small quark masses,which was attributed to the similar mechanism as observed in the present partially quenched study.The e?ects of quenching were modelled using the Quenched Chiral Perturbation Theory and subtracted in order to extract the scalar meson mass m a0=1.326(86)GeV[9].

The RBC Collaboration simulated non-singlet and singlet scalar qˉq states with Domain Wall Fermions[10].The quenching e?ect,which leads to negative correlators at small quark masses,was subtracted as in[8,9].The result is m a0=1.43(10)GeV if only

2

the leading chiral loop(one bubble)is taken into account,and m a0=1.04(7)GeV if next-to leading chiral corrections are included by resummation1.

Mixing of the glueball and qˉq was studied in[3].The quark mass was around m s and no attempt was made to go to the chiral limit.

?Dynamical simulations of qˉq:

The SCALAR Collaboration made an extensive simulation of the singletˉq q state[11] and extracted also the mass of the non-singlet state to be m a0～1.8GeV at mπ/mρ～

0.7.They consider this estimate as an upper bound on the mass since they?tted the

correlators at relatively low times,where contribution of the excited state might be sizable.

UKQCD extracted m a0～1.0(2)GeV from the dynamical and the partially quenched simulation of qˉq[12].Since they simulated only m val≥m sea,they did not observe the striking e?ect of partial quenching discussed below.For this reason they were able to extract the scalar mass from the exponential time-dependence.

MILC[13]simulated qˉq state with three dynamical?avors and saw an indication for the intermediate stateπη,since this state is lighter than a0state at the lightest quark masses.

?Alford and Ja?e reported an indication for the bound singlet and octet qqˉqˉq states below1GeV[6].

All simulations above employed Wilson fermions,except for RBC and MILC simulations, which employed Domain Wall and staggered fermions,respectively.

The only simulation which employed chiral fermions to study light scalar mesons is the quenched simulation of RBC[10].Chiral symmetry is expected to be particularly important for the singlet scalar mesonσ,which is intimately connected with the chiral symmetry breaking.Good understanding of the non-singlet correlator in the chiral limit is the?rst step toward the controlled study of theσmeson.As already mentioned,the present paper presents the dynamical simulation(m sea=m val)of the non-singlet qˉq correlator with Domain Wall fermions.We also simulate partially quenched QCD with m val=m sea.Two degenerate sea quarks have the range of masses corresponding to Mπ～500?700MeV[14].The scalar correlators for m sea=0.02and m sea=0.03at various m val are shown in Fig. 1.The correlators for m val≥m sea are positive and have more or less exponential time-dependence. On the other hand,the correlators for m val

in Fig.3b and has no unknown parameters.We show that the bubble diagram gives a positive contribution for m val ≥m sea and a negative contribution for m val

t ?0.0015?0.001

?0.0005

0.0005

0.001

0.0015

0.002

scalar correlator, m sea =0.02

t

?0.0002?0.0001

00.00010.00020.0003scalar correlator, m sea =0.03Figure 1:The scalar correlators from lattice at (a)m sea =0.02and (b)m sea =0.03for various m val .

The remainder of this paper is organized as follows.The details about the lattice simu-lation are presented in section 2.The dynamical correlators are analyzed in section 3.The resulting error on the scalar mass is rather large,which motivates us to analyze also the partially quenched correlators.The partially quenched artifacts on the scalar correlator are derived within PQChPT in section 4and used to analyze the partially quenched correlators in section 5.Section 6summarizes the conclusions on the mass of a 0meson and brie?y comments on mass of the κresonance,while section 7summarizes the general conclusions.2Numerical simulation

The RBC Collaboration has undertaken a large-scale simulation with N f =2?avors of dynamical Domain Wall quarks with degenerate masses [14].This is an improvement over the quenched simulations and represents an important step toward the simulation of QCD with three dynamical quarks of physical masses.The scalar correlators were calculated on

the dynamical con?gurations with the volume N 3L N T =16332and a single lattice spacing,

so we will not be able to extrapolate the scalar mass to the continuum and to the in?nite volume in the present work.The con?gurations were generated using DBW2gauge action

4

[16]2withβ=0.80and Domain Wall fermion action[7]with M5=1.8and L s=12 (L s is the extent in5th dimension)[14].The separate evolutions were performed for three di?erent bare sea-quark masses m sea=0.02,0.03,0.04,which correspond approximately to

Mπ～500?700MeV.The measurements of the correlators were performed on con?gurations separated by50HMC trajectories.Dynamical Domain Wall fermions have good chiral

properties even at?nite L s with the additive shift in the mass due to the residual chiral symmetry breaking,m res,being approximately0.0014[14],much smaller than either the input sea or valence quark masses.The inverse lattice spacing was determined from the ρ-meson mass and the preliminary result is a?1≈1.7GeV[14].The current uncertainty of the lattice spacing has a small e?ect on the scalar mass;the uncertainty of the scalar meson mass in the present work is dominated by the statistical errors of the scalar correlators. The scalar correlators were measured for the degenerate valence quark masses m1=m2≡m val in the range m val=0.01?0.05.These valence quark masses correspond approximately to Mπ～380?770MeV(Fig.4).We simulated the correlators with p=0,point source and point sink via

1

N3

L x, y 0|ˉq( x,t)Γq( x,t)ˉq( y,0)Γq( y,0)|0 .(1) This method enabled us to calculate also the singlet scalar and singlet pseudoscalar correla-tors and to determine the hairpin insertion m0.

The summary of the scalar and pseudoscalar correlators analyzed in the present paper is

given in the Table1.

3Analysis of dynamical correlators with m val=m sea

The mass m a0and the unrenormalized decay constant f a0can be extracted from the dy-namical scalar correlators using the exponential?t in the conventional way.Indeed,we will verify that the additional contribution from the exchange of two pseudoscalar?elds in PQChPT(Fig.3b)exactly vanishes for m val=m sea,N f=2and m0→∞(12),so the simple exponential?t is well justi?ed.The extracted masses and decay constants are shown

?0.0100.010.020.030.040.05m val =m sea 0.60.70.80.9

1

1.11.2m a 0 = a m a 0p h y ?0.0100.010.020.030.040.05

m val =m sea 0.020.030.040.050.06

0.07

0.08

f a 0 = a f a 0

p h y

Figure 2:The asterisks represent the mass m a 0and decay constant f a 0in the lattice units at the dynamical point m val =m sea ,which are obtained from the exponential ?t of the scalar correlators.The dashed line and the value at m q =0are obtained with the linear ?t.

in Fig.2,while Table 2presents also the time ranges t =t min ?t max and χ2of the ?t 3.The uncorrelated ?ts are used throughout this work and the error-bars are obtained using the jack-knife method.The linear extrapolation to the chiral limit m val =m sea →0gives results in lattice units 4

m a 0=0.93(20),f a 0=0.049(20),(2)

where the jack-knife error-bars are calculated as described in Appendix B of [18].

The resulting errors are rather large,which motivates us to extract the mass also from the partially quenched data with m val =m sea .This forces us to understand the e?ect of partial quenching in the following sections.The use of the Partially Quenched ChPT is crucial for this purpose since it enables us to subtract the signi?cant partially quenched artifacts from the negative lattice correlators in case of m val

4Scalar correlator in partially quenched ChPT

In this section we derive the non-singlet scalar correlator in the Partially Quenched ChPT (PQChPT)within the so-called p -expansion regime.We consider the theory with N val valence quarks q i (which can have di?erent masses m i )and N f degenerate sea quarks q S of mass m sea .The theory incorporates also N val valence ghost-quarks ?q i of mass m i ,which cancel the closed valence-quark loops.PQChPT enables us to study of the partially quenched artifacts,which arise if the valence and the sea quark masses are not equal and if N f =N val .

Our few lowest quark masses are low enough that M 2π/(4πf )2?1,while they are still large enough that M πL ?1and we do not enter ?-regime on our

lattice.

qq qq qq

qq o 128 μ o f ao 128 μ f ao o 2o

??(a)(b)

Figure 3:The contributions to the non-singlet scalar correlator in PQChPT:(a)The exchange of the scalar meson a 0;(b)The bubble diagram is responsible for the unphysical e?ect of partial quenching and represents the exchange of two pseudoscalar ?elds ΦΦ′.The intermediate pseu-doscalar ?elds Φand Φ′can have the ?avor structure Φ,Φ′～ˉq i q j ,ˉq i ?q j ,ˉ?q i q j ,ˉq i q S ,ˉq S q i ,where q i,j are the valence quarks and q S the sea quark.

Non-physical contributions to the scalar correlator in PQChPT arise from the exchange of pseudoscalar ?elds between ˉq q source and ˉq q sink.The leading contribution in the chiral expansion comes from the exchange of two pseudoscalar ?elds and is represented by the so-called bubble diagram in Fig.3b.The two pseudoscalar ?elds Φand Φ′can be mesons Φij ～ˉq i q j and ΦiS ～ˉq i q S with Boson statistics,or mesons Φi ?j ～ˉq i ?q j with fermionic statistics.We do not consider the next-to-leading chiral corrections in PQChPT,which are

suppressed by O (M 2π/(4πf )2)in comparison to the bubble diagram 5.The lattice correlators

can be interpreted as a sum of the a 0-exchange at the tree level in Fig.3a and the bubble diagram in Fig.3b.

For our purpose,we need the strong interactions of pseudoscalar ?elds in PQChPT [15]as well as the kinetic and the mass term for the a 0?eld 6

L =f 2

6(str Φ)2+?μa 0?μa ?0?m 2a 0a 0a ?0(3)

with the physical values f ～95MeV and three-?avor m 0～600MeV ?1000MeV [20].The ?eld U =exp[

√

5

The NLO chiral corrections were taken into account for fully quenched scalar correlator by resummation

[8–10].They are more complicated in partially quenched theory because partially quenched theory implies several intermediate states ΦΦ′in Fig.3b,while quenched theory implies a single intermediate state πη′.6Similar Lagrangian was used for the case of fully quenched ChPT in [8–10].There μ0is denoted by 1

As is standard,we neglect the α(?str Φ)2term in the Lagrangian since αseems to be small

[9,20].We also need the coupling of a 0?eld and the pseudoscalar ?elds to the non-singlet scalar current 7ˉq 2q 1～?μ0f 2(U +U ?)12?

√128μ0f a 0.

The point-point scalar correlator with external momentum p =( 0,E )

C lat (t )≡ x 0|ˉq 2( x ,t )q 1( x ,t )ˉq 1( 0,0)q 2( 0,0)|0

(6)

is computed on the lattice and can be related to the prediction of PQChPT

C P QChP T (t )=F.T.

128μ20f 2a 02m a 0

(e ?m a 0t +e ?m a 0(N T ?t ))The ?rst term is the conventional a 0-exchange,while the second term B (t )=F.T.[B (p )]represents the contribution of the bubble diagram in Fig.3b.The lattice Euclidean momenta p =2sin(p E /2)and the discrete Fourier Transform F.T.on p E are used when we compare PQChPT predictions (7)with the lattice correlators 8.The bubble diagram is calculated from the Lagrangian (3)and the current (5)in the Appendix A,giving

B (p )=4μ20 k N f 1k 2+M 22S (8)?1(k +p )2+M 212k 2+M 2SS N f m 20/3

1(k 2+M 222)2+27Similar current was used for the case of fully quenched ChPT in [8].There μ0is denoted by 1

the pseudoscalar meson masses M andμ0=M2/(4m q)from pion correlators on the same lattices.The hairpin insertion m0～600MeV?1000MeV(normalized for3-?avors)has been determined from theη′correlator in a number of references[20],but the exact value of m0 is not essential for the present work since the extracted scalar mass is almost independent of m0in the wide range m0=[600MeV,∞]as will be demonstrated below.

In order to understand the e?ect of the bubble contribution on the lattice correlators,we derive the asymptotic form of B(t)at large t for a correlator with p=0on a lattice with a→0,aN T→∞and?nite aN L.The asymptotic form for degenerate valence quarks with mass m val is

B(t)t→∞

?→2μ20M2

V S N f

M V V Mη′

2N f m40

M2

V V m20

3

N f m20(M2V V+M2SS)

M2

η′

?M2V V

M V V t ,

where M2η′≡M2SS+1

9These?ndings apply as long as the condition Mη′>M V V is satis?ed,which was assumed in the derivation of(9).

9

B QChP T(p)=?16μ20 k1(k2+M2)2,(10)

B QChP T(t)t→∞

?→?2μ203M2(1+Mt)e?2Mt

N f k1k2+M2N f=2?→0for m0→∞,(12) B ChP T(t)t→∞

?→μ20MM

η′

+(N2f?4)e?2Mt

2(M2aa+M2bb),where M aa is determined

from the pseudoscalar correlator with m val=m a and the sea quark mass of interest. The Mπfrom our lattice correlators are listed in Table3and shown in Fig.4.?We?xμ0=M2val,val/(4m val)for given m val and m sea,where m res is neglected since

it is much smaller than either m val or m sea.Here m val is the input bare mass of the valence quark.M val,val is the mass of the pion with two valence quarks of mass m val at m sea of interest.

2

r0in[8–10].

10

m val 00.05

0.1

0.15

0.2

0.25

M π

2m sea =0.02

Figure 4:M 2π(m val )at m sea =0.02in lattice units.

The bubble diagram vanishes in the case of the dynamical theory (m val =m sea )with N f =2and m 0→∞(12).In this case,the lattice correlator is interpreted solely by the a 0-exchange (7),it has exponential time-dependence and the corresponding m a 0was extracted in Section 3.The bubble diagram is non-zero in general,so it has to be taken into account when the lattice correlators are ?tted by equation (7)in order to extract m a 0and f a 0.The bubble contribution incorporates the physical contributions from the exchange of two pseudoscalars and also the unphysical e?ects of partial quenching when m val =m sea .

We note that our scalar correlators and the quark masses,used to determine μ0=M 2/(4m q ),are not renormalized.We would like to emphasize that this does not prevent us from extracting the physical mass m a 0.This can be seen by rearranging Eqs.(6)and (7):

m 2q x

0|ˉq 2q 1( x ,t )ˉq 1q 2( 0,0)|0 =coef. e ?m a 0t +e ?m a 0(N T ?t ) +F.T M 4πμ20 .(14)The product of quark mass and the scalar current m q ˉq q is invariant under renormalization.The second term on the RHS of (14)depends only on the hadron masses M πand m 0,which are also invariant under renormalization.The scalar mass m a 0can be therefore extracted from the ?rst term on RHS without ambiguity from the renormalization.

5.1Analysis of dynamical correlators taking into account πη′in-

termediate state

The two-?avor dynamical scalar correlator receives a contribution from the exchange of a 0and from the exchange of πη′.The contribution from πη′state vanishes in the limit m 0→∞and the ?t reduces to the standard exponential ?t used in section 3.The contribution of πη′intermediate state at ?nite m 0is given by the bubble contribution (8)with m val =m sea and N f =2.The ?t of the dynamical correlators to PQChPT prediction (7)at various m 0gives m a 0and f a 0in Fig.5.The results are almost independent of m 0and agree with the result

11

of the conventional exponential ?t in section 3.For this reason we will refer to section 3for our dynamical results in the Conclusions.

m val =m sea 0.60.70.80.9

1

1.11.2m a 0 = a m a 0p h y

m val =m sea

0.010.020.03

0.040.05

0.06

0.07

f a 0 = a f a 0p h y Figure 5:The m a 0and f a 0in lattice units obtained from the ?t of the dynamical scalar correlators with the PQChPT prediction (7)for various m 0.The PQChPT prediction for two-?avor dynamical correlators incorporates a 0and πη′exchange.The asterisks represent the ?t for m 0→∞,when the contribution from πη′state vanishes and the ?t reduces to the standard exponential ?t (asterisks are the same as asterisks in Fig.2).The di?erent data points are slightly shifted from m q =0.02,0.03,0.04in horizontal direction for clarity.

The extracted m a 0and f a 0are almost insensitive to the presence of the πη′state since the contribution of this state is at least an order of magnitude smaller than the dynamical lattice correlators in the ?tted time-range for m 0≥600MeV.Our dynamical correlators are dominated by the a 0exchange although the mass M π+M η′=M π+(M 2π

+2

024********t ?0.002?0.00100.0010.002s c a l a r c o r r e l a t o r , m s e a =0.02lattice data

024********t PQChPT bubble B(t) for m 0=inf.024********

t

PQChPT bubble B(t) for m 0=800 MeV

Figure 6:The scalar correlator for m sea =0.02and various m val :the lattice data (a)and the bubble contribution B (t )as predicted by PQChPT for m 0→∞(b)and m 0→800MeV (c).The PQChPT prediction (7)in this ?gure represents just B (t )and does not contain the contribution from the a 0-exchange.

?The dynamical lattice correlator with m val =m sea is positive.The bubble contribution describes the exchange of physical πη′and vanishes in the limit m 0→∞(12).?The lattice correlator at m val >m sea is positive and receives a positive and rather small contribution from the bubble,which is less and less important for larger m val .The bubble contribution is positive and relatively small since it falls as a linear combination of +e ?2M V S t and +t ·e ?2M V V t at large t (9),where the pseudoscalar masses M V S and M V V are relatively large for m val ≥0.03and m sea =0.02.

While we do not include it in the extraction of our ?nal results,we have a limited set of partially quenched data for the m sea =0.03evolution,and we have checked that the lattice correlators change sign at m val =m sea also in this case.This can be seen in Fig.1b.

Let us have a closer look at the case of m val =0.01and m sea =0.02in Fig.7,where the e?ect of partial quenching is most striking.The bubble contribution (dot-dashed)is in good quantitative agreement with the data for t ≥8,where the a 0-exchange fades exponentially.The exchange of the a 0scalar meson is dominant at smaller t .The ?t of the data to the PQChPT prediction (7)gives m a 0=0.87(17)and f a 0=0.040(15)at m 0→∞.The PQCHPT prediction with this choice of m a 0and f a 0is given by the solid line in Fig.7and describes the lattice correlator well.Our one-loop analytical formula therefore correctly determines the sign and approximate size of the e?ects when the valence quark mass is lower than the sea quark mass.This gives us con?dence in the veracity of applying this formula to the larger valence quark masses,where the loop e?ects are smaller.

13

t ?0.0015?0.001

?0.0005

0.0005

0.001

0.0015

0.002

scalar correlator (m sea =0.02 , m val =0.01)

Figure 7:The data and PQChPT predictions for the scalar correlator at m sea =0.02,m val =0.01and m 0=∞.

All scalar correlators for m sea =0.02and various m val are ?tted by the PQChPT pre-diction (7),and the resulting m a 0and f a 0are given in Fig.8and Table 4.Figures on the left represent the ?t,which incorporates both the bubble and the a 0-exchange contributions.We ?nd that m a 0and f a 0at m val ≥0.02depend very slightly on the hairpin insertion m 0in the range m 0=[600MeV ,∞].In the case of m val =0.01,the central values of m a 0and f a 0depend signi?cantly on m 0,but they are all consistent for m 0in the range [600MeV ,∞]within the large error-bars 11.The result of the linear extrapolation from m val =0.01?0.05to the chiral limit m val →0is practically independent of whether the m val =0.01data is taken into account due to the large error-bars at m val =0.01with current statistics.The linear extrapolation from m val =0.01?0.05gives

m a 0=0.90(11),f a 0=0.048(11)for m 0→∞m a 0=0.89(9),f a 0=0.044(9)for m 0=800MeV

(15)

m a 0=0.88(9),f a 0=0.043(9)for m 0=600MeV ,which are consistent for m 0=[600MeV ,∞].So the chiral extrapolation m val →0at ?xed m sea =0.02leads to the mass in the lattice units

m a 0=0.89(11),(16)

where the error re?ects the statistical error of the data and the variation of the bubble contribution for m 0in the range m 0=[600MeV ,∞].

00.010.020.030.040.05m val

0.50.6

0.7

0.80.91

1.1

m a 0 = a m a 0p h y fit with a 0?exchange and bubble

00.010.020.030.040.05

m val 0.50.60.70.8

0.9

11.1exponential fit 00.010.020.030.040.05

m val 00.01

0.02

0.030.040.05

0.06

0.07

f a 0 = a f a 0p h y fit with a 0?exchange and bubble

00.010.020.030.040.05m val

00.010.020.03

0.04

0.050.060.07exponential fit Figure 8:The m a 0and f a 0in lattice units obtained from the ?t of the scalar correlators at m sea =0.02with the PQChPT prediction (7).The left ?gures represent the ?t results,when the bubble contribution is taken into account and m 0is varied.The right ?gures represent the conventional exponential ?t e ?m a 0t +e ?m a 0(N T ?t ),which is obtained under the assumption that the bubble contribution vanishes;the correlator with m val =0.01and m sea =0.02is negative and can not be described by e ?m a 0t +e ?m a 0(N T ?t ).The di?erent data points are slightly shifted in horizontal direction for clarity.

The conventional exponential ?t of the scalar correlators for m val ≥m sea =0.02gives m a 0and f a 0in Fig.8on the right.The bubble contribution in Eq.(7)is taken to be zero

15

in this case.The exponential?t obviously does not work for the correlator at m val=0.01,

where the intriguing partially quenched artifact has to be incorporated through the bubble

contribution.However,it gives reasonable m a0and f a0for m val≥m sea:the results from the exponential?t are consistent with the results from the?t to Eq.(7)since the bubble

contribution is zero or relatively small for m val≥m sea.

6Non-singlet scalar meson mass

In this section we collect our main results on the scalar meson mass.

The chiral extrapolation m q→0of the two-?avor dynamical data points m val=m sea=m q gives(Eq.2)

=1.58±0.34GeV,(17) m a0=0.93±0.20or m phy

a0

where only the statistical error is given.The number in GeV is obtained using the preliminary result for the scale a?1≈1.7GeV[14].

We extracted also the scalar meson masses from the partially quenched correlators with m val=m sea.The chiral extrapolation m val→0at?xed m sea=0.02leads to

=1.51±0.19GeV(18) m a0=0.89±0.11or m phy

a0

and we expect that the dependence on the sea quark mass is small.Here the error re?ects

the statistical error of the data and the variation of the bubble contribution for m0in the

range m0=[600MeV,∞](see Eq.16and Fig.8).

The chiral limits of m a0in the dynamical case and in the partially quenched case are

consistent.Note that the error is smaller in the partially quenched case,where the application of the Partially Quenched ChPT was crucial.The mass of the simulated qˉq state is somewhat larger than the mass of a0(980)and it is closer to the mass of a0(1450).We note that our result is consistent with the fully quenched results of Refs.[8–10]within the present accuracy12.

Finally we comment on the scalar iso-doublet mesons sˉu and sˉd,since their relation to the controversial resonanceκis still an open question.We are not able to make a reliable estimate for the mass of the sˉu and sˉd scalar mesons,since we did not simulate non-degenerate valence quarks.We get a rough estimate by extrapolating the mass obtained from the dynamical correlators to1

12The fully quenched results of[8–10]are presented in the Introduction and are consistent with(17,18) if the e?ect of quenching is incorporated at the leading order in the chiral expansion(one bubble).The quenched m a0is somewhat lower if quenching e?ect is incorporated at the next-to-leading order[10].

16

7Conclusions

We presented a lattice study of the lightest scalar qˉq state with non-singlet?avor structure

(a0meson).Good chiral properties of the Domain Wall Fermions are important for the

connected scalar correlator since this is the?rst step toward a controlled investigation of the scalar spectrum,in particular,theσparticle,which is intimately related to the chiral

symmetry breaking.Two degenerate sea-quarks were simulated with masses corresponding to Mπ～500?700MeV.The simulations were done at?xed lattice spacing and one size of the volume.

The value of scalar mass m a0=0.93±0.20in the lattice units was extracted in the conventional way from the dynamical correlators(m val=m sea)and the resulting error is

rather large.The corresponding physical mass m phy

a0=1.58±0.34GeV was obtained using

the preliminary result for the scale a?1≈1.7GeV.

We analyzed also the partially quenched correlators with m val=m sea.They exhibit striking e?ect of partial quenching since they are positive for m val≥m sea and negative for m val

m phy

a0=1.51±0.19GeV from partially quenched correlators,which is consistent with the

mass extracted from our dynamical correlators.

Our current simulation of the qˉq state on the lattice seems to indicate that this state is somewhat heavier than the observed resonance a0(980),and it is closer to the observed resonance a0(1450);however,given the size of our errors this is not conclusive.We must also emphasize that we have only two dynamical?avors,our lattice volume is not very large and also our quark masses are quite heavy.Besides,continuum limit has not been taken as we have data at only one lattice spacing,so the exploratory nature of our study needs to be kept in mind.

We also note that the N f=2theory is likely to have interesting di?erences from QCD (N f=2+1).Recall that the observed resonances a0(980)and a0(1450)decay toηπand KˉK.Bose statistics and isospin conservation restrict a N f=2

o→π+πfor N f=2,though a N f=2

0→η′N f=2+πwould be possible if kinematics allows it.Additional intricacy could be also caused by the presence of a large4-quark component in these channels.Thus the approach to the chiral limit may well exhibit a more involved dependence of the scalar mass on the quark mass than our data(see Fig.2)indicates with relatively heavy quarks.These issues will need to be addressed in future works with more computing resources.

17

Acknowledgments

The dynamical Domain Wall Fermion con?gurations were generated by the RBC Col-laboration and were essential for the lattice results in the present paper.It is a pleasure to thank all the members of the RBC Collaboration,in particular Tom Blum,Yasumichi Aoki, Norman Christ,Bob Mawhinney,Shigemi Ohta and June Noaki.We also thank RIKEN, Brookhaven National Laboratory and U.S.Department of Energy for providing the facilities essential for the completion of this work.All computations were carried out on the QCDSP supercomputers at the RIKEN BNL Research Center and at Columbia University.The re-search of A.S.was supported in part by the USDOE contract number DE-AC02-98CH10886. K.O.was supported in part by D.O.E.grant DFFC02-94ER40818.

18

Appendix A:Calculation of the bubble diagram

In this appendix we derive the result B(p)(Eq.8)for the bubble diagram in Fig.3b.The coupling of the non-singlet scalar current and two pseudoscalar?eldsΦis given by Eq.(5)

ˉq2q1=2μ0(Φ2)12.(19) The scalar correlator receives the following contribution from the exchange of the two pseu-doscalar?elds shown in Fig.3b

B= 0|ˉq2q1ˉq1q2|0 F ig.3b=4μ20 0|Φ1aΦa2Φ2bΦb1|0 ,(20) where indices a and b are summed over all quarks and ghost-quarks of the partially quenched theory:a,b=i,?i,S with i=1,..,N val and S=1,..,N f.The non-zero Wick contractions relevant to the diagram on the Fig.3b are

B=4μ20 Φ1S|ΦS1 ΦS2|Φ2S + Φ1i|Φi1 Φi2|Φ2i + Φ1?i|Φ?i1 Φ?i2|Φ2?i

+ Φ11|Φ22 Φ12|Φ21 + Φ22|Φ11 Φ12|Φ21 .(21)

The propagators for the pseudoscalar?elds follow from the Lagrangian(3).The propagator for the?avor non-diagonal mesons(a=b)in Minkowski space is

Φab|Φba =iδab?a

p2?M2aa?1

(p2?M2aa)(p2?M2bb)

1

N f m20/3 .(23)

The analytical expression(Eq.8)for the bubble diagram in Fig.3is obtained by inserting the propagators(23)and(22)to the expression(21)and by performing the Wick rotation to the Euclidean space.

19

Appendix B:PQChPT correlator for a?nite lattice

The PQChPT prediction for the scalar correlator C P QChP T(t)(7)relevant for a?nite lattice of the volume N3L N T is

C P QChP T(t)=1

N T

t 128μ20f2a022πm4

N3

L N T

N L/2?1

n1,2,3=?N L/2N T/2?1 n4=?N T/2 N f1k2+M22S

?1(k+p)2+M2

12k2+M2SS

N f m20/3

1

(k2+M222)2

+

2

2

2πn i

2

2πn4

2

2πn i

2{2πn4N T})]2.

The notation is given in section4.This correlator is compared with the lattice correlators C lat(t)(6)in section5.

20