Analysis of unstable particle with the maximum entropy method in O(4) $phi^4$ theory on the

a r X i v :h e p -l a t /0210022v 1 15 O c t 2002

UTHEP-465UTCCP-P-131

Analysis of unstable particle with the maximum entropy method

in O (4)φ4theory on the lattice

T.Yamazaki 1and N.Ishizuka 1,2

1Institute

of Physics,University of Tsukuba,Tsukuba,Ibaraki 305-8571,Japan

2Center for Computational Physics,University of Tsukuba,Tsukuba,Ibaraki 305-8577,Japan

(February 7,2008)

Abstract

We explore applications of the maximum entropy method (MEM)to de-termine properties of unstable particles using the four-dimensional O (4)φ4theory as a laboratory.The spectral function of the correlation function of the unstable σparticle is calculated with MEM,and shown to yield reliable results for both the mass of σand the energy of the two-pion state.Calcula-tions are also made for the case in which σis stable.Distinctive di?erences in the volume dependence of the σmass and two-pion energy for the stable and unstable cases are analyzed in terms of perturbation theory.PACS number(s):12.38.Gc,11.15.Ha,11.80.-m,13.25.Jx

Typeset using REVT E X

In contrast to many successes in understanding of stable hadron states by numerical simulations,unstable hadrons and hadronic decay processes are not understood well on the lattice.An important example in QCD isρ→ππdecay.There are a number of di?culties in dealing with such decay processes.One of the di?culties is computer power.In full QCD lattice simulations accessible today,the pion is so heavy that theρmeson cannot decay.A more crucial problem is the di?culty?rst pointed out by Maiani and Testa[1].Because the ρmeson and theππstate in the isospin I=1channel have the same quantum numbers,the correlation function of theρmeson behaves as a multi-exponential function corresponding toππstates with various relative momenta as well as theρmeson.In general,it is very di?cult to decompose the states and extract the energy of each state from such a correlation function.

In this article we explore application of the maximum entropy method(MEM)to extract the mass of unstable particle and the energies of states to which the unstable particle can decay.MEM has already been applied to extract the ground and?rst excited states of the stable mesons in QCD[2,3].The masses of the two states were simultaneously extracted from the peak positions of the spectral function of the meson.This experience leads us to expect MEM to be useful for the multi-exponential correlation function of unstable particles such as theρmeson.

We employ the four-dimensional O(4)φ4theory for the present exploratory study.It is an e?ective theory of QCD,containing the pion andσparticle.Theσcan be made either stable or unstable by choice of parameters.In the non-linear formulation with the constraint φα(x)φα(x)=1,the action is given by

S= x ?κ·φα(x)Dφα(x)?Jφ4(x) ,(1)

whereφα(x)=(πi(x),σ(x))for i=1,2,3,and Dφ(x)= μφ(x+?μ)+φ(x??μ).For mσ>2mπ,theσparticle is unstable and can decay to the I=0two-pion state,similarly to theσmeson in QCD.We examine the e?ciency of MEM for extracting the energies ofσandππstates for both the unstable mσ>2mπand stable mσ<2mπcases.Very recently a similar study for the three-dimensional four-fermion model has been carried out by Allton et al.[4].

Numerical simulations are carried out at(κ,J)=(0.308,0.0012)and(0.30415,0.003), corresponding to mσ/mπ≈3.7and mσ/mπ≈1.8,for several spatial lattice sizes in the range 103?283to investigate the volume dependence of the spectral functions and energies.The temporal lattice size is?xed to T=64.Con?gurations are generated by the multi-cluster algorithm[5].The periodic boundary condition is imposed in all directions.We perform 0.6×106–1.2×106iterations per simulation point(κ,J)and volume.

We calculate the correlation function matrix[5,6]given by C ij(τ)= (O i(τ)?O i(τ+ 1))O j(0) for i,j=σandππ,where O i is the interpolating operator either for theσor the I=0two-pion state with zero momentum,and the subtraction O i(τ)?O i(τ+1)is made for eliminating the vacuum contribution.We apply MEM to the diagonal parts of the correlation function matrix C ii(τ).

In Fig.1we illustrate the correlation functions in theσandππchannels for two volumes in the unstable case.The errors are estimated by the jackknife method with the bin size of8000to16000iterations.For the smaller volume theππcorrelation function exhibits a

change of slope at t≈10,from a larger slope corresponding toσto a smaller one of theππstate.Theσcorrelator decays parallel to the latter,indicating dominance of theππstate for small volumes.For the larger volume,the trend is opposite,theππcorrelation function showing a change of slope from theσparticle toππstate at t≈20,while theππcorrelator is dominated by theππstate.The decrease of“o?-diagonal”contributions with increasing volume in the correlation functions can be understood from a perturbative analysis:the overlaps| 0|σ|ππ |2and| 0|ππ|σ |2are proportional to1/[L3(mσ?Eππ)2]where Eππis the two-pion energy.

The spectral function f(ω)in theσandππchannels is de?ned in terms of the correlation function C ii(τ)(i=σ,ππ)through

L3C ii(τ)= dωf i(ω)K(ω,τ),(2)

where K(ω,τ)=e?τω(1?e?ω)+eω(τ?T)(1?eω).In our MEM analysis the model function m(ω)is chosen as m(ω)=A(ω0)·ω40/ω4,whereω0is a reference point and A(ω0)is the perturbativeσspectral function at a reference pointω0,which is given by

M I(ω0)

A(ω0)=

1?(4 m2α/ω20)θ(ω0?2 mα) forα=σandπ.The constant C in Eqs.(4)and(5)is given by C=3Zπ( m2σ? m2π)2/32π2v2, where v= σ ,and Zα(α=σ,π)is the wave function renormalization factor forσandπ, respectively.We apply the same model function to theσandππcorrelation functions.

We chooseω0=2,and mαand Zαin Eq.(3)are?xed to the values estimated from inverse correlation functions in momentum space.The input parameters for the model function are compiled in Table I.The reconstruction is carried out in the region0≤ω≤3.We choose ?ω=5×10?4around the peaks corresponding to theσandππstates to determine the energies accurately,and?ω=10?2in other regions.The number of data is taken as large as possible0≤τ≤T/2.We also check that the?nal results for the spectral functions do not depend much on the model function.

In Fig.2we show the spectral functions for theσandππcorrelation functions,fσ(ω)and fππ(ω),in the unstable case.Since energy eigenvalues are discrete on a?nite volume,the spectral function is a sum ofδfunctions.We indeed observe sharp peaks for both spectral functions,which can be identi?ed as theππstate with zero momentum at the?rst peak and theσstate at the second peak.The decrease of the peak height for theππstate in fσ(ω) and for theσstate in fππ(ω)agrees with the volume dependence of the correlation functions discussed above.In the?gure for fσ(ω)the position of peak expected for theππstate with momentum p=2π/L is indicated by a downward arrow.The absence of peak in our data implies that the overlap ofσwith this state is very small.

Fig.3shows the spectral functions in the stable case.For fππ(ω)theσcontribution decreases with volume similar to the unstable case.We observe only a single peak in fσ(ω), indicating that| 0|σ|ππ |2is very small in this case.

Theσmass mσand theππstate energy Eππobtained from the peak positions of the spectral functions as functions of spatial lattice size L are shown in Fig.4(unstable case)and 5(stable case)by circles.For larger volumes mσfrom fππ(ω)and Eππfrom fσ(ω)su?er from large errors or are not available.This is because the overlaps| 0|σ|ππ |2and| 0|ππ|σ |2 decrease as the volume increases.

We have seen in Figs.2and3that theσandππcorrelation functions are dominated by theσandππstates with zero momentum,and other states are negligible.In this case we can apply the diagonalization method[6]to the2×2correlation function matrix C(τ)for extraction of the energy eigenvalues of these states.We diagonalize the matrix D(τ,τ0)=C?1/2(τ0)C(τ)C?1/2(τ0)at eachτ,whereτ0is some reference time chosen to be τ0=0in this work.The eigenvalue of D(τ,τ0)is given byλν(τ,τ0)=K(Wν,τ)/K(Wν,τ0) with K(Wν,τ)=e?τWν(1?e?Wν)+e Wν(τ?T)(1?e Wν),where Wνis the energy of the states ν=σ,ππ.

The energies obtained by the diagonalization method are plotted by cross symbols in Figs.4and5.The errors are estimated by the jackknife method with8000to16000elimina-tions.The results for mσand Eππare consistent with those with MEM,but the statistical error with the diagonalization method is much smaller.This is because the diagonalization makes full use of the2×2correlation function matrix while MEM utilizes on the diagonal element.

For the application of the diagonalization method,it is essential that the dominant states in the correlation functions are known.Otherwise we need to consider the correlation function matrix of all possible states with the same quantum numbers,which is a di?cult task.MEM is useful for identifying the dominant states,as exempli?ed with our example.

Comparing Fig.4with Fig.5we?nd an essential di?erence in the volume dependence of theσmass andππenergy between the unstable and stable cases.In the unstable case,the ππenergy increases and theσmass decreases as the volume increasing,while an opposite trend is seen in the stable case.

The volume dependence of theσmass expected from perturbation theory is given by mσ(L,mσ,g R)=mσ+g R(?mσ(L)??mσ(∞)).(6) Here mσis theσmass for in?nite volume,g R is the renormalized coupling constant,and

?mσ(L)=1

Wα(p)

+AαCα(p) ,(7)

with Aα=2,6forα=π,σ,D=1?3(m2σ?m2π)/m2σ,and

Cα(p)=

m2σ?m2π

m2α+4 3i=1sin2(p i/2)with p i being the spatial momenta.

We?t our results for theσmass obtained by the diagonalization method to Eq.(6), taking mσand g R as?t parameters and setting mπto the value obtained from the pion

correlation function at L=28.The?t curves are plotted in Fig.4and Fig.5for each case, where the data at L=10are excluded from the?tting.The?t parameters andχ2are compiled in Table II.The?ts agrees quite well with the simulation results.We then realize that the di?erent volume dependence between the unstable and stable cases originates from an opposite sign of Cπ(p)in Eq.(7)in the two cases.

In order to understand the volume dependence of theππenergy,we consider the scat-tering length a0.It is related to the energy shift of the two-pion state through L¨u scher’s formula[7]given by

Eππ?2mπ=?4πa0

+c2

a20

96πm2σ 7R2+8 1

REFERENCES

[1]L.Maiani and M.Testa,Phys.Lett.B245585(1990).

[2]M.Asakawa,T.Hatsuda,and Y.Nakahara,Prog.Part.Nucl.Phys.46459(2001).

[3]T.Yamazaki et al.(CP-PACS Collaboration),Phys.Rev.D6*******(2002).

[4]C.R.Allton,J.E.Clowser,S.J.Hands,J.B.Kougut,and C.G.Strouthos,het-lat/0208027.

[5]M.G¨o ckeler,H.A.Kastrup,J.Westphalen,and F.Zimmermann,Nucl.Phys.B425413

(1994).

[6]M.L¨u scher and U.Wol?,Nucl.Phys.B339222(1990).

[7]M.L¨u scher,Commun.Math.Phys.104177(1986);105153(1986).

[8]M.L¨u scher and P.Weisz,Nucl.Phys.B318705(1989).

unstable case

100.1138(2)0.961(1)0.309(2)0.956(1)0.112921(7)

120.1084(2)0.9657(8)0.337(2)0.949(1)0.127042(4)

140.1053(2)0.9667(6)0.351(1)0.944(1)0.13351169(4)

160.1041(3)0.9674(6)0.359(1)0.938(1)0.136818(1)

180.1036(2)0.9676(6)0.362(1)0.935(1)0.138598(1)

240.1024(2)0.9678(8)0.365(1)0.928(1)0.140646(2)

280.1024(2)0.9685(7)0.365(1)0.925(2)0.1410983(6)

stable case

100.1945(2)0.9734(4)0.328(1)0.967(1)0.099507(1)

180.1851(2)0.9754(4)0.3320(7)0.951(1)0.1094956(8)

240.1844(2)0.9735(5)0.3344(6)0.949(1)0.1099262(1)

280.1848(2)0.9755(5)0.3324(7)0.945(1)0.110067(2) TABLE I.The input parameters for the model function of the MEM in the unstable and stable cases.

unstable stable

a0g R a0g R

010

2030

10?3

10?2

10?1

100

10?3

10?2

10?1

100

101

2ππσ

L=10

L=18

FIG.1.Correlation functions for the σand ππin the unstable case.

100

300

500

100

200300

400400

8001200σ spectral function

ππ spectral function

FIG.2.Spectral functions reconstructed from σ(left line)and ππ(right line)correlation func-tions in the unstable case.

ω250

750

1250

100300

500700100

200300400σ spectral function

342

465

ππ spectral function

FIG.3.Spectral functions reconstructed from σ(left line)

and ππ(right line)correlation func-tions in the stable case.

0.14

0.160.180.2

0.220.30.4

0.5

0.6

FIG.4.Energies for σand ππwith the MEM and diagonalization in the unstable case.

0.1

0.20.30.40.50.30.40.50.60.70.8FIG.5.Energies for σand ππwith the MEM and diagonalization in the stable case.