MIKROTEZ (MSc)

UNIVERSITY OF TIRanaFACULTY OF NATURAL SCIENCEPHYSICS DEPARTMENT

LATTICE QCDQuark- antiquark potential from FermiQCD

MSc. Dafina Xhako

MOTIVATIONImplementation 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.

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 actionThe 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 !)

3II. Lattice QCD

Solution: Wilson (1974) proposed introduction of a non perturbativeapproximation based on discretization of space-time in a hypercubic finitelattice, 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)

III. From QCD to LQCD theoryfields 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 partGluon partIV. Simulations of pure gauge theoryIn 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

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 plattoCalculated 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

String tension, setting the scaleTo 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

V FermiQCDNumerical 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 Europes Research Communities)Parallel calculations

Solution: FermiQCDis 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 TheoryFermiQCD communications are based on MPI, but MPI calls are hidden to the high level algorithms that constitute FermiQCDPrograms are easier to debug because the usage of FermiQCD objects and algorithms does not require explicit use of pointersThe goal of this program is to develop a toolkit for computations and visualizations of Lattice Quantum Chromodynamics (LQCD)

The lower componentsare referred to as MatrixDistributed Processing and they define the language used in FermiQCD.The upper components are the algorithms. The top components represent examples, applications and other tools

VI. Results of calculations with FermiQCD 1. Scalability test of FermiQCD The computation time fall exponentially (for example for lattice volumes 8^4, 16^4)

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

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.

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.

3. Computation time from lattice volume

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 N4), taking constant physical volume (L4=(aN)4)

Step 3. For each simulation we have changed =6/g, in order to keep constant physical volume, from:

(14)

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

N^4W.loopsLattice distance from parameterizationLattice distance calculatedString tensiona2KStatistical error of aStatistical error of a2K

8^45.736000.17070.2221(83)0.3009 (39)9.6796(41)e-053.4427(74)e-0212^45.8536000.12300.1837(57)0.16702(81)3.4302(16)e-051.0552(22)e-0216^4636000.09310.1060(69)0.0596(12)6.1446(90)e-058.0894(26)e-03Results: 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)Extrapolation in continuum limit (a0)

00.050.10.150.24.44.64.855.25.45.65.866.2Lattice distance (a) [fm]String tension (K ) [1/fm2] calculated data extrapolation4.55Quark-antiquark potential (lattice 8^4)in lattice unit and physical unit

012345678-0.500.511.522.53Distance quark-antiquark (r)Quark-antiquark potential V(r) theoretical modeldata in lattice unitLattice 84r, V(r) ~KrQuark-antiquark potential (lattice 12^4)in lattice unit and physical unit0123456700.511.52Distance quark-antiquark (r) Quark-antiquark potential V(r) Theoretical modelData in lattice unitLattice 12400.050.10.150.20.250.30.350200040006000800010000Distance quark-antiquark (r) [fm]Quark-antiquark potential V(r) [MeV] [ data in physical unitTheoretical modelLattice 124Quark-antiquark potential (lattice 16^4)in lattice unit and physical unit02468-0.200.20.40.60.811.2Distance quark-antiquark (r) Quark-antiquark potential V(r) theoretical modelData in lattice unitLattice=16400.050.10.150.20.250.30.35-10000100020003000400050006000Distance quark-antiquark (r) [fm]Quark-antiquark potential V(r) [MeV] data in physical unittheoretical modelLattice 1646. Effective quark-antiquark potential from 3-D Wilson loops

We have write the code in FermiQCD that calculates r1 x r2 x t volumes Wilson loops for 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 constant4) Build - A 4-D lattice mdp_lattice lattice(4,L); - A gauge field gauge_field U(lattice,n); - A random configuration

Loop over N- Monte Carlo steps for (int k=0; k