High-Performance Computing Infrastructure for South East Europe's Research Communities ”Modeling and Optimization in Science and Technologies” Volume 2, 2014, pp 23-30, ISBN: 978-3-319-01519-4, Springer

Using Parallel Computing to Calculate Static Interquark Potential in LQCD Dafina Xhako1, Rudina Zeqirllari2, Artan Boriçi 2 1

Department of Engineering Science, Faculty of Professional Studies "Aleksander Moisiu" University, Durres, Albania dafinaxhako@yahoo.com

2

Department of Physics, Faculty of Natural Sciences, University of Tirana Tirana, Albania

Abstract. Lattice Quantum Chromodynamics (LQCD) is an algorithmic formulation of QCD, the mathematical model that describes quarks and their interactions. Computations in LQCD are typically very expensive and run on dedicated supercomputers and large computer clusters for many months. In this paper the calculations are performed in one of the clusters for supercomputing of HPSEE (High-Performance Computing Infrastructure for South East Europe’s Research Communities) project, that is located in Bulgaria (BG HPC). We use parallel computing with FermiQCD software, to determine the static quarkantiquark potential. In LQCD the static quark-antiquark potential can be derived from the Wilson loops. The standard method uses rectangular Wilson loops, while we test volume Wilson loops, using simulations with SU(3) gauge field configuration for different values of coupling constant and for different lattice sizes. The calculations are made for 100 statistically independent configurations, of gauge fields of the lattice. Keywords: FermiQCD software, lattice, parallel computing, quark-antiquark potential, string tension, Wilson loops,

1

1

Introduction

Quantum Chromodynamics (QCD) is a theory of strong interaction, a fundamental force that describes the interactions between quarks and gluons. Analytic or perturbative solutions in low-energy QCD are difficult or impossible due to the highly nonlinear nature of the strong force. Wilson (1974) [1] introduced a non perturbative approximation based on discretization of space-time in a hypercubic finite lattice, with N- node per direction separated by a distance a. In LQCD [2], [3], [4], fields representing quarks are defined at lattice sites while the gluon fields are defined on the links connecting neighboring sites. LQCD provides a framework for investigation of non-perturbative phenomena such as confinement. Numerical calculations in lattice using Monte Carlo methods are very expensive because of the spaceâ€“time dimensions (4-dimensional problem). Progress in LQCD requires a combination of improvements in formulation, numerical techniques, and in computer technology. The aim of this paper is the implementation and application of parallel computational techniques to study the properties of QCD in low energy regimes such as quark confinement. The solution is the use of parallel computational techniques to save time and costs in computations.

2

Materials and Methods

2.1

The static quark-antiquark potential

Since the original work by Creutz [5] in the SU(2) gauge theory, many computations of the potential from lattice gauge theories have been performed in SU(3), [6] [7], [8], [9]. Due to the asymptotic freedom of Yang-Mills theory, one expects at short distances a Coulomb-like behavior and the reliability of perturbation theory. At large distances quark confinement shows up and perturbative methods are no longer able to describe the behavior of physical observables; in this context, lattice gauge theories play an important role providing a full non perturbative approach. The quark-antiquark potential can be extracted by large time behavior of the Wilson loops. For any closed path C with length n on the lattice, the Wilson loop is defined as the expectation value of the trace of the product of links around C, (1) The Wilson loops can be written as: (2) The effective potentials are calculated by (3) For a fixed r and for different values of time t, we select the value of effective potential while for long time t it is reached a plateau. Since r =1,..,6 we have 6 values of

effective potential. Finally, for different r we can fit the effective potential according to theoretical model in physical unit: (4) where is a constant, is string tension parameter and is a constant (the coefficient of the Coulomb-like term). Physically, this behavior implies the confinement of color in the flux tubes of the gluonic field. For large distances dominates the linear term and for short distances dominates the Columbic term. In lattice units (non dimensional unit) equation (4) takes the form: (5) where

, , and . For determining the scale of theory we have used the new method from Sommerâ€™s relation, with r0=0.5 fm: (6) where the lattice parameter is: (7) To take physical quantity in continuum we repeat the simulation for different lattice volumes (taking physical length constant ~ L=1.6fm) and extrapolate in continuous limit when aďƒ 0. 2.2

FermiQCD and BG - HPC cluster

FermiQCD [10], [11] is a library for fast development of parallel applications for Lattice Quantum Field Theories and Lattice Quantum Chromodynamics. It was designed to be easy to use, with a syntax very similar to common mathematical notation, and, at the same time, optimized for PC clusters. All FermiQCD algorithms are parallel, but parallelization is hidden from the high level programmer. FermiQCD components range from low level linear algebra, fitting and statistical functions to high level parallel algorithms specifically designed for lattice quantum field theories such as the Wilson [12], the Asqtad [13] action for KS fermions, the Domain Wall action [14] etc. One of the main differences between FermiQCD and libraries developed by other collaborations is that it follows an object oriented design as opposed to a procedural design. This work makes use of results of the High-Performance Computing Infrastructure for South East Europe's Research Communities (HP-SEE), a project co-funded by the European Commission (under contract number 261499) through the Seventh Framework Programme. HP-SEE deals with multi-disciplinary international scientific communities (computational physics, computational chemistry, life sciences, etc.) stimulating the use of regional HPC infrastructure and its services. 3

The cluster for supercomputing that is located in Bulgaria (BG HPC) has SUSE Linux Enterprise Server 10 and compilers: IBM XL C/C++ Advanced Edition for Blue Gene/P V9.0 and for Blue Gene/P V11.1; GNU Tool chain (gcc, glibc, binutlils, gdb, python). It has 36Intel Xeon X5560 @2.8Ghz; 24 GB per node 576 cores; DDR Infiniband 2.5Îźs 20 Gbps; Small cluster with 4 Nvidia GTX 295 cards, total 1920 GPU cores also connected. Full information is available at [15]. Scalability test of FermiQCD . We first tested the FermiQCD software deployment and allocation of computer time and simulation. We test it for different lattice sizes. The computation time falls exponentially (for example for lattice volumes 16^4) 1800

Computation time [sec]

1600

Lattice 164

1400 1200 1000 800 600 400 200 0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Number of processors

Fig. 1. The computation time from number of processors for lattice 16^4

Speedup and Efficiency test . Let T(n,1) be the run-time of the fastest known sequential algorithm and let T(n,p) be the run-time of the parallel algorithm executed on p processors, where n is the size of the input (lattice volume). The speedup is then defined as

9 8 7

Speedup

6

lattice 84 lattice 124 lattice 164 lattice 204

5 4 3 2

ideal line

1 0 0

2

4

6 8 Number of processors

10

12

14

Fig. 2. Speedup form number of processors for different lattice volumes

The ideal speed up will be S(p) = p, so if we double for example the number of processors, will double the time of execution. Another metric to measure the performance of a parallel algorithm is efficiency, E(p), defined as: 110 lattice 84

Efficiency (%)

100

lattice 124

90

lattice 164

80

lattice 204

70

ideal line

60 50 40 30 0

2

4

6 8 10 Number of processors

12

14

Fig. 3. Efficiency (in percentage) form the number of processors for different lattice volume

5

3

Preliminary Physics Results

3.1

Quark-antiquark potential from planar and volume Wilson loops

We started from the example that estimates a single Wilson loop in FermiQCD and adapted it to calculate 36 rectangular planar Wilson loops for different values of time (t) and direction (r) on a lattice ( r = t = 1,..6), and after that for volume Wilson loops r1 x r2 x t with r1=r2=t=1,..6. We have made simulations for 100 statistically independent configurations, for lattice 8^4, 12^4, 16^4, (lattice volume N^4), taking the physical volume (L4=(aN)^4) as constant. For each simulation we have changed the coupling constant in order to keep constant physical volume, from [16]. Statistical processing of dataâ€™s have not been performed on BG HPC cluster, but in our personal computers, in Matlab. We have written the scripts that calculate effective potentials, the coefficients , K, alpha, statistical errors with Jackknife method, lattice parameter; graph of quark-antiquark from distance between them. Table 1. Lattice distance a, string tension with their statistical errors for quenched simulation with 8^4, 12^4, 16^4 (planar loops)

N^4

Î’

a

from

pa-

a Calculated

String tension

rameterization

Statistical

Statistical

error of a

error of

8^4

5.7

0.1707

0.2221(83)

0.3009 (39)

9.6796e-05

3.4427e-02

12^4

5.85

0.1230

0.1837(57)

0.16702(81)

3.4302e-05

1.0555e-02

16^4

6

0.0931

0.1060(69)

0.0596(12)

6.1446e-05

8.0894e-03

Preliminary result: Statistical error of a, is very small to justify the difference between a_ calculated and a from parameterization (it would be of the range ~ 10 -1). Table 2. Lattice distance a, string tension with their statistical errors for quenched simulation with 8^4, 12^4, 16^4 (volume loops) N^4

Î˛

a

from

pa-

a calculated

String tension

rameterization

8^4

5.7

0.1707

0.1295(23)

0.08469(54)

Statistical

Statistical

error of a

error of

0.9138e-03

0.0503

12^4

5.85

0.1230

0.0990(19)

0.03620(39)

0.4985e-03

0.0339

16^4

6

0.0931

0.0575(99)

0.0217(99)

0.0814e-03

0.0066

Preliminary result: Statistical error of a, is also like in case of planar loops, small to justify the difference between a_ calculated and a from parameterization (it would be of the range ~ 10-2).

Quark-antiquark potential V(r)

3 2.5

theoretical model (fit in range [0.5-8]) data in lattice unit

2

Lattice 84

1.5 1 0.5 0 -0.5 0

1

2

3 4 5 6 Distance quark-antiquark (r)

7

8

Quark-antiquark potential V(r)

Fig. 4. Quark-antiquark potential (lattice 8^4) in lattice unit from planar Wilson loops 1.4 1.3 1.2 1.1

theoretical model data in lattice unit Lattice 84

1 0.9 0.8 0.7 1

2

3 4 Distance quark-antiquark (r)

5

6

Fig. 5. Quark-antiquark potential (lattice 8^4) in lattice unit from volume Wilson loops

7

Conclusions Techniques of parallel computing are effective in lattice QCD calculations. FermiQCD is one of the best programs that we can use in QCD to implement parallel computing. It has e very good scalability. Our first objective was to test FermiQCD program and techniques of parallel computing calculating the static quark-antiquark potential. Our graph results shows that the quark- anti-quark potential confirms a very important properties of QCD in low energy regimes such as quark confinement. The statistical manipulation made in Matlab shows that the method using effective potential from planar Wilson loops (Fig.4) is better than volume Wilson loops (Fig.5), but we have to check the calculation of statistical errors of lattice spacing in our future work. As we can see form Tables 1 and 2 this error is very small to justify the difference between a_ calculated and a from parameterization. In our future work we have to increase the number of configurations, so increasing the statistical data we can take better results. New codes that we have written in FermiQCD are our contribute in this project of LQCD. References 1. Wilson K.G., Phys.rev. D10 (1974) 1445 2. Gupta, R., Introduction to Lattice QCD, Lectures given at the LXVIII Les Houches Summer School "Probing the Standard Model of Particle Interactions", July 28-Sept 5, (1997). 150 pages, arXiv:hep-lat/9807028 3. Gattringer C., Lang B. C., "Quantum Chromodynamics on the Lattice" Springer, ISBN: 3642018491,343 pages (2009) 4. Creutz M. “Quarks, gluons and lattices, Cambridge Univ. Press, Cambridge 1983 5. Creutz M., Phys, Rev, D47 (1993) 661 6. Schilling K., G.S.Bali Int.J.Mod.Phys.C4:1167-1177, (1993), 7. Deldar S., Phys. Rev. D62, 34509 (2000) 8. Bali G., Phys. Rev. D62, 114503 (2000) 9. Parisi G., Petronzio R., and Rapuano F.. Phys.Lett., B128:418, (1983) 10. Di Pierro, M., Matrix Distributed Processing and FermiQCD, Fermilab, Batavia, IL 60510, USA, proceedings of the Workshop on Advanced Computing and Analysis Techniques (Oct 16-20, 2000@ Fermi National Accelerator Laboratory, USA), arXiv:heplat/0011083. 11. Di Pierro, M., FermiQCD: A tool kit for parallel lattice QCD applications, Nucl. Phys. B. Proc. Suppl. 106, pp.1034-1036 (2002), arXiv:hep-lat/0110116. 12. Wilson, K. G., Confinement of Quarks, Phys. rev. D10, pp. 2445–2459 (1974). 13. Bernard, C. et al. MILC Collaboration, Quenched hadron spectroscopy with improved staggered quark action, Phys. Rev. D 58 (1998) 014503, hep -lat/9712010. 14. Kaplan, D. B., A Method for simulating chiral fermions on the lattice, Phys. Lett. B 288 (3), pp. 342-347 (1992) 15. http://www.hp-see.eu 16. Guagnelli, M., Sommer R., Witti, H., Precision computation of a low energy reference scale in quenched lattice QCD. Nucl. Phys. B535, pp. 389–402 (1998), hep-lat/9806005

Using Parallel Computing to Calculate Static Interquark Potential in LQCD

Published on Feb 22, 2014

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