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Quark- antiquark potential from FermiQCD MSc. Dafina Xhako Slide 2 MOTIVATION Implementation and application of computational techniques in parallel to study the properties of lattice QCD Because: In low energy regimes properties of QCD are studied by non-perturbative methods, the common one is lattice QCD. But numerical calculations in lattice using Monte Karlo methods are very expensive. Solution: Using computational techniques in parallel to gain in time and cost computations. Slide 3 I. QCD (Quantum Chromodynamics) Quantum chromodynamics (QCD) is a theory of the strong interaction, a fundamental force that describing the interactions between quarks and gluons. The starting point to study this quantum theory is the partition function in Euclidean space-time (1) Results for physical observables are obtained by calculating expectation values (2) O - is any given combination of operators expressed in terms of time ordered products of gauge and quark fields Z partition function S - Euclidian action The problem: How to calculate these expectation values and to derive from them physical quantity? ( perturbative methods are impossible due to strong coupling in this regime !) Slide 4 II. Lattice QCD Solution: Wilson (1974) proposed introduction of a non perturbative approximation based on discretization of space-time in a hypercubic finite lattice, with N - nodes per direction separated by a distance a (LQCD) - Advantages: Finite lattice, physical quantity can be solved numerically by Monte Carlo methods Studies properties of QCD that cant be seen in high energy regime such as: quark confinement, hadrons spectroscopy etc The discrete space-time lattice acts as a non perturbative regularization scheme. At finite values of the lattice spacing a, which provides an cutoff at /a, there are no infinities. The problems: The action should provide fundamental properties of QCD like gauge invariance etc Finite lattice spacing errors. At finite a, lattice results have discretization errors. Removing these errors: 1) improve the lattice action and operators so that the errors at fixed a are small, 2) repeat the simulations at a number of values of a and extrapolate to a 0. Statistical errors. MC introduce errors ~ 1/sqrt( N ) Slide 5 III. From QCD to LQCD theory fields representing quarks are defined at lattice sites (3) the gluon fields are defined on the links connecting neighboring sites (4) From integral to sum (5) Partial derivative goes as finite difference (6) Full LQCD action, gauge invariant (7) Fermionic part Gluon part Slide 6 IV. Simulations of pure gauge theory In Simulations of pure gauge theory - We take in consideration only gluonic part of action - We have lower computational costs In order to derive physical quantity, we have to construct gauge invariant object in lattice. The only gauge invariant object in simulations of pure gauge theory are Wilson loops. Wilson loops, W(r,t), are trace of time ordered product of link variables along to a close path. The simplest loop is 1x1, which is called plaquette Slide 7 Quark-antiquark potential from LQCD The quark-antiquark potential derive from Wilson loops by calculating effective potential (8) for each r, we select effective potentials when for long time t is reached a platto Calculated quark-antiquark values in lattice are modeled as: or in lattice unit: (9) V 0, K (string tension), alpha are coefficients which will be found numerically solving Ax = b system Slide 8 String tension, setting the scale To setting the scale of theory we have used the new method from Sommer relation, with r 0 =0.5 fm: (10) so the lattice scale parameter is: (11) To take physical quantity in continuum we repeat simulation for different lattice volume (taking physical length constant ~ L=1.6fm ) and extrapolate in continuum limit Slide 9 V FermiQCD Numerical calculation in LQCD are very expensive Required: 1) Calculations in computer clusters (we have access on BG HPC cluster as part of HP-SEE; High-Performance Computing Infrastructure for South East Europes Research Communities) 2) Parallel calculations Solution: FermiQCD is a collection of classes, functions and parallel algorithms for lattice QCD, written in C++. easy to write, read and modify since the FermiQCD syntax resembles the mathematical syntax used in Quantum Field Theory FermiQCD communications are based on MPI, but MPI calls are hidden to the high level algorithms that constitute FermiQCD Programs are easier to debug because the usage of FermiQCD objects and algorithms does not require explicit use of pointers Slide 10 The goal of this program is to develop a toolkit for computations and visualizations of Lattice Quantum Chromodynamics (LQCD) The lower components are referred to as Matrix Distributed Processing and they define the language used in FermiQCD. The upper components are the algorithms. The top components represent examples, applications and other tools Slide 11 VI. Results of calculations with FermiQCD Slide 12 1. Scalability test of FermiQCD The computation time fall exponentially (for example for lattice volumes 8^4, 16^4) Slide 13 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 (12) i.e., the ratio of the sequential execution time to the parallel execution time. Ideally, one would like S(p)=p, which is called perfect speedup, Another metric to measure the performance of a parallel algorithm is efficiency, E(p), defined as: (13) 2. Speedup and Efficiency test Slide 14 Speedup from number of processors 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. Slide 15 Efficiency from number of processors Efficiency is how effectively additional processors are used. The ideal line would be 100%. It isn't uncommon to achieve greater than 100% parallel efficiencies for small numbers of processors. Slide 16 3. Computation time from lattice volume Slide 17 4. Effective quark-antiquark potential from planar Wilson loops Step 1. We have write the code in FermiQCD that calculates r x t planar Wilson loops r=t=1,..6 Step 2. We have made simulation for 100 configurations statistically independent, for lattice 8^4, 12^4, 16^4, (lattice volume N 4 ), taking constant physical volume ( L 4 =(aN) 4 ) Step 3. For each simulation we have changed =6/g, in order to keep constant physical volume, from: (14) Slide 18 Step 4. We write the script in Matlab/Octave, that calculate: - effective potentials - the coefficients V 0, K, - statistical errors with Jackknife method - lattice parameter - graph of quark-antiquark from distance between them - extrapolation in continuum limit of the string tension Slide 19 N^4 W. loops Lattice distance from parameterizat ion Lattice distance calculate d String tension a 2 K Statistical error of a Statistical error of a 2 K 8^45.736000.17070.2221(83) 0.3009 (39) 9.6796(41)e- 05 3.4427(74)e -02 12^45.8536000.12300.1837(57) 0.16702(8 1) 3.4302(16)e- 05 1.0552(22)e -02 16^4636000.09310.1060(69)0.0596(12)6.1446(90)e- 05 8.0894(26)e -03 Results: Lattice distance, string tension with their statistical errors for quenched simulation with 8^4, 12^4, 16^4 ( planar loops ) Preliminary: Statistical error of a, is very small to justify the difference between a_ calc and a from parameterization (it would be of the range ~ 10 -2 ) Slide 20 Extrapolation in continuum limit (a 0) Slide 21 Quark-antiquark potential (lattice 8^4) in lattice unit and physical unit r, V(r) ~Kr Slide 22 Quark-antiquark potential (lattice 12^4) in lattice unit and physical unit Slide 23 Quark-antiquark potential (lattice 16^4) in lattice unit and physical unit Slide 24 6. Effective quark-antiquark potential from 3-D Wilson loops We have write the code in FermiQCD that calculates r 1 x r 2 x t volumes Wilson loops for r 1 =r 2 =t=1,..6 The algorithm of the code follows this steps: 1) include FermiQCD libraries #include "fermiqcd.h" 2) start communication with mdp (matrix distributed process ) mdp.open_wormholes(argc,argv); 3) define this parameters - Lattice volume - Gauge group SU(n) - The number of configurations or MC steps - The coupling constant 4) Build - A 4-D lattice mdp_lattice lattice(4,L); - A gauge field gauge_field U(lattice,n); - A random configuration Slide 25 5) Loop over N- Monte Carlo steps for (int k=0; k