Markovian Monte Carlo program EvolFMC v.2 for solving QCD evolution equations

We present the program EvolFMC v.2 that solves the evolution equations in QCD for the parton momentum distributions by means of the Monte Carlo technique based on the Markovian process. The program solves the DGLAP-type evolution as well as modified-DGLAP ones. In both cases the evolution can be performed in the LO or NLO approximation. The quarks are treated as massless. The overall technical precision of the code has been established at 5×¹⁰−4. This way, for the first time ever, we demonstrate that with the Monte Carlo method one can solve the evolution equations with precision comparable to the other numerical methods.

New version program summary

Program title: EvolFMC v.2Catalogue identifier: AEFN_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEFN_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including binary test data, etc.: 66 456 (7407 lines of C++ code)No. of bytes in distributed program, including test data, etc.: 412 752Distribution format: tar.gzProgramming language: C++Computer: PC, MacOperating system: Linux, Mac OS XRAM: Less than 256 MBClassification: 11.5External routines: ROOT (http://root.cern.ch/drupal/)Nature of problem: Solution of the QCD evolution equations for the parton momentum distributions of the DGLAP- and modified-DGLAP-type in the LO and NLO approximations.Solution method: Monte Carlo simulation of the Markovian process of a multiple emission of partons.Restrictions:

1.

Limited to the case of massless partons.

2.

Implemented in the LO and NLO approximations only.

3.

Weighted events only.

Unusual features: Modified-DGLAP evolutions included up to the NLO level.Additional comments: Technical precision established at 5×¹⁰−4.Running time: For the 10⁶ events at 100 GeV: DGLAP NLO: 27s; C^′-type modified DGLAP NLO: 150s (MacBook Pro with Mac OS X v.10.5.5, 2.4 GHz Intel Core 2 Duo, gcc 4.2.4, single thread).     

Constrained MC for QCD evolution with rapidity ordering and minimum kT

With the imminent start of LHC experiments, development of phenomenological tools, and in particular the Monte Carlo programs and algorithms, becomes urgent. A new algorithm for the generation of a parton shower initiated by the single initial hadron beam is presented. The new algorithm is of the class of the so-called “constrained MC” type algorithm (an alternative to the backward evolution MC algorithm), in which the energy and the type of the parton at the end of the parton shower are constrained (predefined). The complete kinematics configurations with explicitly constructed four momenta are generated and tested. Evolution time is identical with rapidity and minimum transverse momentum is used as an infrared cut-off. All terms of the leading-logarithmic approximation in the DGLAP evolution are properly accounted for. In addition, the essential improvements towards the so-called CCFM/BFKL models are also properly implemented. The resulting parton distributions are cross-checked up to the 10⁻³ precision level with the help of a multitude of comparisons with other MC and non-MC programs. We regard these tests as an important asset to be exploited at the time when the presented MC will enter as a building block in a larger MC program for W/Z production process at LHC.     

Numerical methods for the QCD overlap operator IV: Hybrid Monte Carlo

The computational costs of calculating the matrix sign function of the overlap operator together with fundamental numerical problems related to the discontinuity of the sign function in the kernel eigenvalues are the major obstacle towards simulations with dynamical overlap fermions using the Hybrid Monte Carlo algorithm. In a previous paper of the present series we introduced optimal numerical approximation of the sign function and have developed highly advanced preconditioning and relaxation techniques which speed up the inversion of the overlap operator by nearly an order of magnitude.In this fourth paper of the series we construct an HMC algorithm for overlap fermions. We approximate the matrix sign function using the Zolotarev rational approximation, treating the smallest eigenvalues of the Wilson operator exactly within the fermionic force. Based on this we derive the fermionic force for the overlap operator. We explicitly solve the problem of the Dirac delta-function terms arising through zero crossings of eigenvalues of the Wilson operator. The main advantage of scheme is that its energy violations scale better than O(Δτ²) and thus are comparable with the violations of the standard leapfrog algorithm over the course of a trajectory. We explicitly prove that our algorithm satisfies reversibility and area conservation. We present test results from our algorithm on 4⁴, 6⁴, and 8⁴ lattices.     

Vbfnlo: A parton level Monte Carlo for processes with electroweak bosons

Vbfnlo is a fully flexible parton level Monte Carlo program for the simulation of vector boson fusion, double and triple vector boson production in hadronic collisions at next-to-leading order in the strong coupling constant. Vbfnlo includes Higgs and vector boson decays with full spin correlations and all off-shell effects. In addition, Vbfnlo implements CP-even and CP-odd Higgs boson via gluon fusion, associated with two jets, at the leading-order one-loop level with the full top- and bottom-quark mass dependence in a generic two-Higgs-doublet model.A variety of effects arising from beyond the Standard Model physics are implemented for selected processes. This includes anomalous couplings of Higgs and vector bosons and a Warped Higgsless extra dimension model. The program offers the possibility to generate Les Houches Accord event files for all processes available at leading order.

Program summary

Program title:VbfnloCatalogue identifier: AEDO_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEDO_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: GPL version 2No. of lines in distributed program, including test data, etc.: 339 218No. of bytes in distributed program, including test data, etc.: 2 620 847Distribution format: tar.gzProgramming language: Fortran, parts in C++Computer: AllOperating system: Linux, should also work on other systemsClassification: 11.1, 11.2External routines: Optionally Les Houches Accord PDF Interface library and the GNU Scientific libraryNature of problem: To resolve the large scale dependence inherent in leading order calculations and to quantify the cross section error induced by uncertainties in the determination of parton distribution functions, it is necessary to include NLO corrections. Moreover, whenever stringent cuts are required on decay products and/or identified jets the question arises whether the scale dependence and a k-factor, defined as the ratio of NLO to LO cross section, determined for the inclusive production cross sections are valid for the search region one is interested in.Solution method: The problem is best addressed by implementing the one-loop QCD corrections in a fully flexible NLO parton-level Monte Carlo program, where arbitrary cuts can be specified as well as various scale choices. In addition, any currently available parton distribution function set can be used through the LHAPDF library.Running time: Depending on the process studied. Usually from minutes to hours.     

A numerical algorithm for efficiently obtaining a Feynman parameter representation of one-gluon loop QCD Feynman diagrams for a large number of external gluons

A numerical program is presented which facilitates a computation pertaining to the full set of one-gluon loop diagrams (including ghost loop contributions), with M attached external gluon lines in all possible ways. The feasibility of such a task rests on a suitably defined master formula, which is expressed in terms of a set of Grassmann and a set of Feynman parameters. The program carries out the Grassmann integration and performs the Lorentz trace on the involved functions, expressing the result as a compact sum of parametric integrals. The computation is based on tracing the structure of the final result, thus avoiding all intermediate unnecessary calculations and directly writing the output. Similar terms entering the final result are grouped together. The running time of the program demonstrates its effectiveness, especially for large M.

Program summary

Program title:DILOG2Program identifier:ADXN_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADXN_v1_0Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandProgramming language:FORTRAN 90Computer(s) for which the program has been designed:Personal ComputerOperating system(s) for which the program has been designed: Windows 98, XP, LINUXNumber of processors used:oneNo. of lines in distributed program, including test data, etc.:2000No. of bytes in distributed program, including test data, etc.:16 249Distribution format:tar.gzExternal routines/libraries used:noneCPC Program Library subprograms used:noneNature of problem:The computation of one gluon/ghost loop diagrams in QCD with many external gluon lines is a time consuming task, practically beyond reasonable reach of analytic procedures. We apply recently proposed master formulas towards the computation of such diagrams with an arbitrary number (M) of external gluon lines, achieving a final result which reduces the problem to one involving integrals over the standard set, for given M, of Feynman parameters.Solution method:The structure of the master expressions is analyzed from a numerical computation point of view. Using the properties of Grassmann variables we identify all the different forms of terms that appear in the final result. Each form is called “structure”. We calculate theoretically the number of terms belonging to every “structure”. We carry out the calculation organizing the whole procedure into separate calculations of the terms belonging to every “structure”. Terms which do not contribute to the final result are thereby avoided. The final result, extending to large values of M, is also presented with terms belonging to the same “structure” grouped together.Restrictions:M is coded as a 2-digit integer. Overflow in the dimension of used array is expected to appear for M?20 in a processor that uses 4-bytes integers or for M?34 in a processor with 8-bytes integers.Running time:Depends on M, see enclosed figures.     

S@M, a mathematica implementation of the spinor-helicity formalism

In this paper we present the package S@M (Spinors@Mathematica) which implements the spinor-helicity formalism in Mathematica. The package allows the use of complex-spinor algebra along with the multi-purpose features of Mathematica. The package defines the spinor objects with their basic properties along with functions to manipulate them. It also offers the possibility of evaluating the spinorial objects numerically at every computational step. The package is therefore well suited to be used in the context of on-shell technology, in particular for the evaluation of scattering amplitudes at tree- and loop-level.

Program summary

Program title: S@MCatalogue identifier: AEBF_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEBF_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 14 404No. of bytes in distributed program, including test data, etc.: 77 536Distribution format: tar.gzProgramming language: MathematicaComputer: All computers running MathematicaOperating system: Any system running MathematicaClassification: 4.4, 5, 11.1Nature of problem: Implementation of the spinor-helicity formalismSolution method: Mathematica implementationRunning time: The notebooks provided with the package take only a few seconds to run.     

Low-lying Wilson Dirac operator eigenvector mixing in dynamical overlap hybrid Monte Carlo

Current dynamical overlap fermion hybrid Monte Carlo simulations encounter large fermionic forces when there is mixing between eigenvectors of the kernel operator with near zero-eigenvalues. This leads to low acceptance rates when there is a large density of near zero eigenvalues. I present a method where these large forces are eliminated and the large action jumps seen when two eigenvalues approach zero are significantly reduced. This significantly increases the stability of the algorithm, and allows the use of larger integration time steps.     

Symmetries and exponential error reduction in Yang-Mills theories on the lattice

The partition function of a quantum field theory with an exact symmetry can be decomposed into a sum of functional integrals each giving the contribution from states with definite symmetry properties. The composition rules of the corresponding transfer matrix elements can be exploited to devise a multi-level Monte Carlo integration scheme for computing correlation functions whose numerical cost, at a fixed precision and at asymptotically large times, increases power-like with the time extent of the lattice. As a result the numerical effort is exponentially reduced with respect to the standard Monte Carlo procedure. We test this strategy in the SU(3) Yang-Mills theory by evaluating the relative contribution to the partition function of the parity odd states.     

Exploring the HMC trajectory-length dependence of autocorrelation times in lattice QCD

We study autocorrelation times of physical observables in lattice QCD as a function of the molecular dynamics trajectory length in the hybrid Monte-Carlo algorithm. In an interval of trajectory lengths where energy and reversibility violations can be kept under control, we find a variation of the integrated autocorrelation times by a factor of about two in the quantities of interest. Trajectories longer than conventionally used are found to be superior both in the Nf=0 and Nf=2 examples considered here. We also provide evidence that they lead to faster thermalization of systems with light quarks.     

Solving Systems of Linear Equations with Relaxed Monte Carlo Method

The problem of solving systems of linear algebraic equations by parallel Monte Carlo numerical methods is considered. A parallel Monte Carlo method with relaxation is presented. This is a report of a research in progress, showing the effectiveness of this algorithm. Theoretical justification of this algorithm and numerical experiments are presented. The algorithms were implemented on a cluster of workstations using MPI.     

BCVEGPY: an event generator for hadronic production of the Bc meson

We have written a Fortran program BCVEGPY, which is an event generator for the hadronic production of the B_c meson through the dominant hard subprocess . To achieve a compact program, we have written the amplitude of the subprocess with the particle helicity technique and made it as symmetric as possible, by decomposing the gluon self couplings and then applying the symmetries. To check the program, various cross sections of the subprocess have been computed numerically and compared with those in the literature. BCVEGPY is written in a PYTHIA-compatible format, thus it is easy to implement in PYTHIA.

Program summary

Title of program: BCVEGPYVersion: 1.0 (September, 2003)Catalogue identifier: ADTJProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADTJProgram obtained from: CPC Program Library, Queen's University of Belfast, N. Ireland.Computer: Any computer with FORTRAN 77 compiler. The program has been tested on HP-SC45 Sigma-X workgroups, Linux PCs and Windows PCs with Visual Fortran.Operating systems: UNIX, Linux and Windows.Programming language used: FORTRAN 77.Memory required to execute with typical data: About 2.0 MB.No. of bytes in distributed program, including test data, etc.: 477630No. of lines in distributed program, including test data, etc.: 66461Distribution format: tar gzip fileNature of physical problem: Hadronic production of B_c meson.Method of solution: Improved helicity-approach to the amplitude and symmetries of the amplitude itself have been used to compact the program so as to save cpu as possible as one can. The code with option can generate weighted and un-weighted events. For jet hadronization, an interface to PYTHIA is provided.Restrictions on the complexity of the problem: The hadronic production of the B_c meson in the S-wave states, i.e. pseudo-scalar state () and vector state () are included by the ‘complete calculation’ approach. The hadronic production of B_c meson in P-wave states has not been implemented into the BCVEGPY yet.Typical running time: It depends on which option one chooses to match PYTHIA when generating the B_c events. Typically, if IDWTUP=1, then it takes about 20 hour on a 1.8 GHz Intel P4-processor machine to generate 1000 events; however if IDWTUP=3, to generate 10⁶ events, it takes about 40 minutes only.     

A time saving algorithm for the Monte Carlo method of Metropolis

A time saving algorithm for the Monte Carlo method of Metropolis is presented. The technique is tested with different potential models and number of particles. The coupling of the method with neighbor lists, linked lists, Ewald sum and reaction field techniques is also analyzed. It is shown that the proposed algorithm is particularly suitable for computationally heavy intermolecular potentials.     

Fast recall of state-history in kinetic Monte Carlo simulations utilizing the Zobrist key

We present a novel application of the Zobrist hashing method, known in the computer chess literature, to simulation of diffusional phase transformations in metal alloys. A history of previously visited states can be easily maintained, allowing very fast lookup of energies and transition rates calculated earlier in the simulation. The method has been applied to the simulation of a Fe-1at.%Cu system, with simple potentials and a transition rate for diffusional events approximated from the difference in internal energy between trial states. In this simple model at temperatures of 1073 K we find that 61.2% of the states considered during the simulation have been seen previously, and that this proportion rises to 85.1% at 773 K and even to 99.9% at 373 K. Rapid recall of these states reduces the computational time taken for the same sequence of atom-vacancy exchange moves by a factor of 6.3 at 773 K rising to over 100 at 373 K. We suggest that a similar speedup factor will be found using more sophisticated models of diffusion and that the method can, with small modifications, be applied to a wide range of kinetic Monte Carlo simulations of atomistic diffusion processes.     

Reduction of the self-forces in Monte Carlo simulations of semiconductor devices on unstructured meshes

When using an unstructured mesh for device geometry, the ensemble Monte Carlo simulations of semiconductor devices may be affected by unwanted self-forces resulting from the particle-mesh coupling. We report on the progress in minimisation of the self-forces on arbitrary meshes by showing that they can be greatly reduced on a finite element mesh with proper interpolation functions. The developed methodology is included into a self-consistent finite element 3D Monte Carlo device simulator. Minimising of the self-forces using the proper interpolation functions is tested by simulating the electron transport in a 10 nm gate length, 6.1 nm body thick, double gate metal-oxide-semiconductor field-effect transistor (MOSFET). We demonstrate the reduction in the self-force and illustrate the practical distinction by showing I-V characteristics for the device.     

Monte Carlo simulation of thermodynamic systems with cluster formation under different boundary conditions

Thethermodynamic state function free energy F(T,V,N) as function of the cluster distribution N and the thermodynamic parameters (temperature T and volume V) is calculated in the canonical ensemble. With the help of the Legendre transformation we get all other state functions for different boundary conditions. Monte Carlo simulations with 15000 particles starting from metastable homogeneous initial conditions are made for water and methanol. The Bethe-Weizsäcker ansatz and the Padé approximation are used for the binding energy of clusters. A first-order phase transition is observed under appropriate conditions.     

A Beowulf-class computing cluster for the Monte Carlo production of the LHCb experiment

The computing cluster built at Bologna to provide the LHCb Collaboration with a powerful Monte Carlo production tool is presented. It is a performance oriented Beowulf-class cluster, made of rack mounted commodity components, designed to minimize operational support requirements and to provide full and continuous availability of the computing resources. In this paper we describe the architecture of the cluster, and discuss the technical solutions adopted for each specialized sub-system.