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UNIVERSIT Y OF TIR ana FA CULTY OF NATURAL SCIENCE PHYSICS DEPARTMENT LATTICE QCD

UNIVERSIT Y OF TIR ana FA CULTY OF NATURAL SCIENCE PHYSICS DEPARTMENT LATTICE QCD. Quark- antiquark potential from FermiQCD MSc . Dafina Xha ko. MOTIVATION. Implementation and application of computational techniques in parallel to study the properties of lattice QCD Because:

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UNIVERSIT Y OF TIR ana FA CULTY OF NATURAL SCIENCE PHYSICS DEPARTMENT LATTICE QCD

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  1. UNIVERSITYOF TIRanaFACULTY OF NATURAL SCIENCEPHYSICS DEPARTMENTLATTICE QCD Quark- antiquark potential from FermiQCD MSc. Dafina Xhako

  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 methodsare very expensive. Solution: • Using computational techniques in parallel to gain in time and cost computations.

  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 !)

  4. II. Lattice QCD • Advantages: • Finite lattice, physical quantity can be solved numerically by Monte Carlo methods • Studies properties of QCD that can’t 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 π/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) 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) • -

  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) Gluon part Fermionic part

  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

  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) V0,K (string tension), alpha are coefficients which will be found numerically solving Ax = b system

  8. String tension, setting the scale • To setting the scale of theory we have used the new method from Sommer relation, with r0=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

  9. V FermiQCD • Numerical calculation in LQCD are very expensive Required: • Calculations in computer clusters (we have access on BG – HPC cluster as part of HP-SEE;High-Performance Computing Infrastructure for South East Europe’s Research Communities) • 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

  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

  11. VI. Results of calculations with FermiQCD

  12. 1. Scalability test of FermiQCD • The computation time fall exponentially (for example for lattice volumes 8^4, 16^4)

  13. 2. 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) Thespeedupis 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)

  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.

  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.

  16. 3. Computation time from lattice volume

  17. 4.Effective quark-antiquark potential from planar Wilson loops Step 1. We have write the code in FermiQCD that calculates r x tplanar Wilson loops r=t=1,..6 Step 2. We have made simulation for 100 configurationsstatistically independent, for lattice 8^4, 12^4, 16^4, (lattice volume N4), taking constant physical volume (L4=(aN)4) Step 3. For each simulation we have changed β=6/g, in order to keep constant physical volume, from: (14)

  18. Step 4. We write the script in Matlab/Octave, that calculate: - effective potentials - the coefficients V0,K ,  - statistical errors with Jackknife method - lattice parameter - graph of quark-antiquark from distance between them - extrapolation in continuum limit of the string tension

  19. 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)

  20. 6.2 calculated data 6 ] 2 extrapolation 5.8 5.6 5.4 String tension (K ) [1/fm 5.2 5 4.8 4.6 4.55 4.4 0 0.05 0.1 0.15 0.2 Lattice distance (a) [fm] Extrapolation in continuum limit (a0)

  21. 3 theoretical model 2.5 data in lattice unit 2 4 Lattice 8 1.5 1 Quark-antiquark potential V(r) 0.5 0 -0.5 0 1 2 3 4 5 6 7 8 Distance quark-antiquark (r) Quark-antiquark potential (lattice 8^4)in lattice unit and physical unit r>>, V(r) ~Kr r<<,V(r)~1/r

  22. 2 10000 Theoretical model data in physical unit Data in lattice unit Theoretical model 8000 1.5 4 4 Lattice 12 Lattice 12 6000 1 Quark-antiquark potential V(r) • Quark-antiquark potential V(r) [MeV] [ 4000 0.5 2000 0 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 1 2 3 4 5 6 7 Distance quark-antiquark (r) [fm] Distancequark-antiquark (r) Quark-antiquark potential (lattice 12^4)in lattice unit and physical unit

  23. 1.2 6000 theoretical model data in physical unit 1 Data in lattice unit 5000 theoretical model 4 0.8 Lattice=16 4000 0.6 Quark-antiquark potential V(r) 4 3000 Lattice 16 0.4 Quark-antiquark potential V(r) [MeV] 0.2 2000 0 1000 -0.2 0 2 4 6 8 0 Distance quark-antiquark (r) -1000 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Distance quark-antiquark (r) [fm] Quark-antiquark potential (lattice 16^4)in lattice unit and physical unit

  24. 6.Effective quark-antiquark potential from 3-D Wilson loops We have write the code in FermiQCD that calculates r1 x r2 x tvolumes Wilson loopsfor r1 =r2 =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

  25. Loop over N- Monte Carlo steps for (int k=0; k<N; k++){ WilsonGaugeAction::heatbath(U,gauge,N); } 6) Make a generic path 7) Loop over all possible paths for mu, nu, t for(int mu=0; mu<4; mu++) for(int nu=0; nu<4; nu++) for(int t=0; t<4; t++){ if (mu!=nu&&nu!=t&&mu!=t) 8) Calculate real part of the trace of time ordered product of link variables. wloop+=real(average_path(U,length,path))/24; 9) Save the Wilson loops in a file .dat 10) Close communications mdp.close_wormholes();

  26. Results: Lattice distance, string tension with their statistical errors for quenched simulation with 8^4, 12^4, 16^4 (3-D loops) Preliminary: Statistical error of a, is also like in case of planar loops, small to justify the difference between a_ calc and a from parameterization (it would be of the range ~ 10-2)

  27. Extrapolation in continuum limit (a0)

  28. 0.2 1.4 theoretical model theoretical model 1.3 data in lattice unit 0.18 data in lattice unit 4 1.2 Lattice 12 4 0.16 Lattice 8 1.1 Quark-antiquark potential V(r) 0.14 1 Quark-antiquark potential V(r) 0.9 0.12 0.8 0.1 0.7 0.08 4 4.5 5 5.5 6 6.5 7 7.5 1 2 3 4 5 6 Distance quark-antiquark (r) Distance quark-antiquark (r) Quark-antiquark potential in lattice unit (lattice 8^4, 12^4)

  29. 0.35 theoretical model 0.3 4 Lattice 16 0.25 Quark-antiquark potential V(r) 0.2 0 2 4 6 8 10 Distance quark-antiquark (r) Quark-antiquark potential in lattice unit (lattice 16^4) data in lattice unit

  30. V. CONCLUSIONS 1. Techniques of parallel calculation are effective in QCD calculations 2.FermiQCD is one of the best programs that we can use in QCD to calculate in parallel. It has e very good scalability. 3. The calculations of quark –antiquarkpotentialin pure gauge theory is used to setting the scale 4. The methods with effective potential from planar Wilson loops is better than from 3-D loops, but we have to check the calculation of statistical errors of lattice spacing.

  31. 6. The quark- antiquark potential is very important to study properties of QCD in low energy regimes, like quark confinement 7. New codes that we have written in FermiQCD are our contribute in this project of LQCD

  32. DAFINA XHAKOTHANK YOU FOR YOUR ATTENTION! 08 October 2012, Tirana

  33. Jeminëkushtet e regjimevetëultaenergjitike Langrazhianii QCD (1) me tenzorin e fushësgluonike (2) kuderivatikovariantkalibrues (3) dheveprimiifushës (4) Vërejme: Langrazhianii QCD-sëpërmbantermakubikë e kuadratikëtë , ndryshenga QED, QCD-ja ka natyrëjolineare. • ështëfushagluonike, • g - ështëkonstantja e çiftimittëfortë, • f - indeksonaromat e kuarkeve (up, down, strange, charm, top, bottom). • Duketmjaftingjashëm me QED përveçtermittëfunditnëekuacionin e dytë.

  34. Potenciali kuark-antikuark prejlaqeve të Wilson-it (Creutz 1974) • Gjendetprej sjelljes laqevetë Wilson-it për kohë të gjata. • Energjia e njëçiftikuark-antikuarkjepetngafunksioniikorrelimittëoperatoritkuark-antikuarknëkohëtëndryshme (8) (9) (10)

  35. Table 1: Computational time for different number of processors used and for different lattice sizes

  36. The code that generate planar Wilson loops #include "fermiqcd.h" // include FermiQCD libraries #include <fstream> int main(intargc, char** argv) { mdp.open_wormholes(argc,argv); // START int L[]={8,8,8,8}; // lattice volume int n=3; // SU(n) gauge group int N=100; // number of gauge configurations mdp_lattice lattice(4,L); // make a 4D lattice gauge_field U(lattice,n); // make a gauge field U coefficients gauge; // set physical parameters gauge["beta"]=5.7; // beta=6/g^2 sets lattice spacing set_hot(U); // make a random gauge configuration int maxsize1=6; int maxsize2=6; int size1; int size2; double wloop; ofstreamfout; fout.open("w8.dat");

  37. for (int k=0; k<N; k++) { // loop over the MCMC WilsonGaugeAction::heatbath(U,gauge,N); // do 10 MCMC steps for (size1=1;size1<=maxsize1;size1++) for (size2=1;size2<=maxsize2;size2++){ int length=2*size1+2*size2; int path[length][2]; // make a generic path for(inti=0; i<size1; i++) { path[i][0]=+1; path[i+size1+size2][0]=-1; } for(inti=size1; i<size1+size2; i++) { path[i][0]=+1; path[i+size1+size2][0]=-1; }

  38. // loop over all possible paths // wloop=0.0; for(int mu=0; mu<4; mu++) for(int nu=mu+1; nu<4; nu++) { // build each path for(inti=0;i<size1;i++) path[i][1]=path[i+size1+size2][1]=mu; for(inti=size1;i<size1+size2;i++) path[i][1]=path[i+size1+size2][1]=nu; wloop+=real(average_path(U,length,path))/6; } // save laqet size1 x size2 fout << size1 << size2 << " " << wloop << endl; } } fout.close(); mdp.close_wormholes(); return 0; }

  39. The code that generate 3-D Wilson loops #include "fermiqcd.h" // include FermiQCD libraries #include <fstream> int main(intargc, char** argv) { mdp.open_wormholes(argc,argv); // START int L[]={8,8,8,8}; // lattice volume int n=3; // SU(n) gauge group int N=10; // number of gauge configurations mdp_lattice lattice(4,L); // make a 4D lattice gauge_field U(lattice,n); // make a gauge field U coefficients gauge; // set physical parameters gauge["beta"]=5.7; // beta=6/g^2 sets lattice spacing set_hot(U); // make a random gauge configuration int maxsize1=6; int maxsize2=6; int maxsize3=6; int size1; int size2; int size3; double wloop; ofstreamfout; fout.open("w_vell8_n.dat");

  40. for (int k=0; k<N; k++) { // loop over the MCMC WilsonGaugeAction::heatbath(U,gauge,N); // do 10 MCMC steps for (size1=1;size1<=maxsize1;size1++) for (size2=1;size2<=maxsize2;size2++) for (size3=1;size3<=maxsize3;size3++){ int length=2*size1+2*size2+2*size3; int path[length][2]; // make a generic path , for(inti=0; i<size1; i++) { path[i][0]=+1; path[i+size1+size2+size3][0]=-1; } for(inti=size1; i<size1+size2; i++) { path[i][0]=+1; path[i+size2+size3+size1][0]=-1; } for(inti=size1+size2; i<size3+size2+size1; i++) { path[i][0]=+1; path[i+size3+size2+size1][0]=-1; }

  41. wloop=0.0; for(int mu=0; mu<4; mu++) for(int nu=0; nu<4; nu++) for(int t=0; t<4; t++){ if (mu!=nu&&nu!=t&&mu!=t) { for(inti=0;i<size1;i++) path[i][1]=path[i+size1+size2+size3][1]=mu; for(inti=size1;i<size1+size2;i++) path[i][1]=path[i+size2+size3+size1][1]=nu; for(inti=size1+size2;i<size1+size2+size3;i++) path[i][1]=path[i+size3+size1+size2][1]=t; wloop+=real(average_path(U,length,path))/24; } } // ruajlaqet size1 x size2 fout << size1 << size2 <<size3<<" "<< wloop << endl; } } fout.close(); mdp.close_wormholes(); return 0; } // loop over all possible paths

  42. 2. Strong Scaling test • In this case the problem size stays fixed but the number of processing elements are increased. In strong scaling, a program is considered to scalelinearlyif the speedup is equal to the number of processing elements used ( N ). • Calculating Strong Scaling Efficiency If the amount of time to complete a work unit with 1 processing element is t1, and the amount of time to complete the same unit of work with N processing elements is tN, the strong scaling efficiency is given as: (13) • In our case we find values for N=1,…,14 and the results

  43. Strong Scaling test

  44. Strong Scaling test

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