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Jurnal KomputasiJournal of Computational Science

Proceeding Supplement

Lattice QCD Calculation on Hadronic Binding Energy ΛN. Riveli and L.T. Handoko , Phys. J. IPS 1 (2005) 0504

Received: September 7th, 2004; Accepted for publication: September 7th, 2004

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c© 2005 The Indonesian Computational Society ISSN 1829-6424

Jurnal Komputasi Volume 1 (2005) 0504 Proceeding

Lattice QCD Calculation on Hadronic Binding Energy Λ

Nowo Riveli1 and L.T. Handoko1,2

1Department of Physics, University of Indonesia, Depok 16424, Indonesia2Theoretical Physics and Computation Group, Physics Research Center LIPI, Kompleks Puspitek Serpong,

Tangerang 15310, Indonesia

Abstract : We calculate the binding energy of hadron Λ using the lattice QCD. The hadronic binding energy Λ is

defined non-perturbatively trough the Lattice HQET lagrangian. The simulation has been performed in relatively small

lattice volume due to limited computing power. We present the result which can be used to obtain the heavy quark mass

by matching it with the MS renormalisation.

Keyword(s) : Lattice QCD, hadronic binding energy Λ

E-mail : nowo riveli@fisika.ui.ac.id, handoko@lipi.fisika.net

Received: September 7th, 2004; Accepted for publication: September 7th, 2004

1 INTRODUCTION

The basic idea of lattice QCD is to construct the theory for strong interaction in discretized space and time[1, 2, 3]. So it enable to perform any calculations numerically. The main focus is to calculate the quarkmasses which are not calculable directly from the analytic perturbative calculation. This is moreless because aperturbative calculation of quark masses using QCD generates divergences as shown in the effective theory ofQCD known as the heavy quark effective theory (HQET) [4].

Further, it has been proposed to define a term of binding energy, Λ[4], which could be caculated on thelattice space without creating any divergences.

The purpose of this paper is to perform a preliminary calculation for the binding energy using the latticeQCD. The result will be compared to the previous works [5, 6], and the origin of the discrepancies is discussed.

2 DEFINITION OF Λ

Below is definition Λ followed from [4]. To obtain the non-perturbative definition of Λ, we use the HQETlagrangian in lattice formulation,

L =1

1 + aδm

(h(x)D4h(x) + δmh(x)h(x)

). (1)

Also considering a correlation function,

C(t) =∑

~x

〈0|J(~x, t)J†(~0, 0)|0〉 , (2)

which is for sufficiently large time t becomes,

C(t)→ Z2 exp(−Et) , (3)

where Z is constant.

The mass term on the above lagrangian can be choosed in several ways. One of them is to study the propertyof the propagator at large time and at a fixed gauge,

−δm ≡ ln(1 + aδm)

a= limt→∞

1

aln

[Tr (S(~x, T + a))

Tr (S(~x, t))

], (4)

and assuming that the ratio above have a value at the limit of large t. The equation can then be described asa condition where the quark propagator function does not change exponentially at large time.

c© 2005 Indonesian Computational Society 0504-1

N. Riveli, L.T. Handoko Jurnal Komputasi 1 (2005) 0504

Table 1: Simulations Parameter.

simulation volume β number of configurationset A 163 × 32 6.0 50set B 163 × 32 5.8 50

3 DETERMINATION OF RESIDUAL MASS δm

As defined before, to get the value of δm, we should first calculate the quark propagator. One of the possibleform for the propagator is

S(x|0) = δ(x)θ(x4)P(x4|0) , (5)

where P(x4|y4) is path ordered lattice, oftenly called as ”P-line”,

P(x4|y4) =

[x4−y4

a

]∏

n=1

U †(x, x4 − na), x4 > y4 , (6)

where P(x4|y4) = 1 for x4 = y4. According [6], it is required that the definition of improved propagator shouldbe,

PIx(x4|y4 =

[1−

(1

3

)x4−y4+1]Px(x4|y4) . (7)

To reduce statistical error, the poropagator is calculated in a fixed gauge, which we use the Landau gauge inour case, in the form of,

SH(t) =1

3V

∑

x

〈Tr[P(x4 = t|0)

]〉 , (8)

where V is lattice spatial volume.In order to determine the value of δm, further we have to calculate the effective mass defined as,

aδm(t) = − ln

(SH(t+ a)

SH(t)

). (9)

After obtaining values of effective mass at several t, the next step is to fit the result with the one-loopperturbative calculation [4], that is

aδm(t) = aδm+ γ lnt+ a

t. (10)

4 SIMULATIONS PARAMETERS

Throughout the calculation, we perform several configurations called as setA and setB. Each of them with theparameters set written in Tab. 1.

Lattice configurations are downloaded from http://qcd.nersc.gov [8].

5 CALCULATION OF δm

For the first step, we calculate the improved quark propagator as defined in Eqs.(6) ∼ (9). The calculation isconducted with two parameters as written in Tab. 1. Further we obtain the value of residual mass using therelation in Eq. (10). The results are then fitted with Eq. (11). Fig. 1 shows the plotted and fitted results fortwo set of parameters.

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Lattice QCD Calculation on Hadronic... Jurnal Komputasi 1 (2005) 0504

0 5 10 15t/a

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

β = 6.0: δm = 0.36(2)β = 5.8: δm = 0.39(2)

a δm(t)

Figure 1: Plot of effective mass from the improved propagator, for two simulation.

The results for δm are as follows,

aδm = 0.36± 0.02± 0.01 at β = 6.0 , (11)

aδm = 0.39± 0.02± 0.01 at β = 5.8 . (12)

APE Collaboration did not perform the calculation of E for configuration at β = 5.8, so we only use δm atβ = 6.0 (Eq. (13)).

Looking at the similiarity of the incline of two δm plots and impressively close values of δm, we concludethat δm is independent from the parameters used in the simulations. This motivates us to safely take the resultof δm from only one configuration.

6 CALCULATION OF Λ

To get Λ we use,

• δm from eq.(13);

• E from APE collaboration [7], isaE = 0.52± 0.01 , (13)

• a corresponds to β isa−1 = 1.8± 0.2 GeV for β = 6.0 . (14)

So we obtain,Λ = E − δm = 280± 40 MeV pada β = 6.0 . (15)

7 SUMMARY AND DISCUSSION

The results in Tab. 2 are the values of Λ obtained previously by using the same method. Comparing the resultsin Tab. 2, we can make several conclutions,

Table 2: Λ obatined.

author volume configuration δm Λ

ref.[3] 183 × 64 210 0.521± 0.003± 0.010 180± 35 MeV

ref.[2] 243 × 40 500 0.526± 0.003± 0.006 170+30−20 MeV

TA ini 163 × 32 50 0.360± 0.020± 0.010 280± 40 MeV

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N. Riveli, L.T. Handoko Jurnal Komputasi 1 (2005) 0504

• δm obtained in this work has the accuracy up to two decimals, while the other ones mentioned in thispaper have the accuracy up to three decimals. The error mostly comes from the statistical one. Thisshows that the number of configuration and the size of volume used in this paper are not sufficient toconduct more precise calculation.

• Λ differs with two previous results. This is quantities also because of the large statistical error. Actually,combining few configurations will improve the accuracy.

• Using different configurations might cause discrepancies. Different methods and aldorithyms in arisingthe configuration will form gauge configuration with different characteristic, allthough using the sameparameters.

Though the result differs significantly with the previous works, we conclude that the calculation we haveconducted is still acceptable. The reason is the figure shows the same trend with for instance in [6]. Also theerrors could arise due to different configuration we have used.

HKI

REFERENCES

[1] H. J. Rothe, Lattice Gauge Theory, an Introduction, World Scientific Publishing (1997).

[2] G. Munster and M. Walzl, hep-lat/0012005 (2000).

[3] G. P. Lepage, http://www.tech.plym.ac.uk/maths/bus.

[4] G. Martinelli dan C. T. Sachradja, CERN-TH 7517/94, hep-ph/9502352 (1995).

[5] M. Crisafulli, V. Gimenez, G. Martinelli dan C. T. Sachradja, CERN-TH 7521/94, hep-ph/9506210 (1995).

[6] V. Gimenez, G. Martinelli dan C. T. Sachradja, CERN-TH/96-163, hep-ph/9607018 (1996).

[7] The APE Collaboration, C. R. Allton et al., Nucl. Phys. B413 (1994);ibid., Phys. Lett. B326 (1994) 295.

[8] The Gauge Connection, http://qcd.nersc.gov.

• Presented at the Workshop on Computational Science 2K4, Tangerang, Indonesia, 30th August 2004.

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