Computational methods of elementary particle physics

Transcription

1 Computational methods of elementary particle physics Markus Huber Lecture in SS017 Version: March 7, 017

2 Contents 0.0 Preface Introduction Overview of computational methods in elementary particle physics Overview of this course Basic techniques 7.1 Numerical integration Newton-Cotes formula Gauss quadrature Interpolation Polynomial interpolation Chebyshev interpolation Spline interpolation

3 0.0. PREFACE Contents 0.0 Preface These lecture notes are based on the lecture Computational methods of elementary particle physics held at the University of Graz in the summer term 017. I welcome everybody to send remarks or report any errors to markus.huber@unigraz.at. 3 Version: March 7, 017

4 1 Introduction 1.1 Overview of computational methods in elementary particle physics Computers play a crucial role in elementary particle physics, or in physics in general. As a matter of fact, physics was always pushing the development of faster and better computers. Computers were used from early ballistic calculations for military use to data analysis of events in particle collisions at the LHC. Nowadays they are omnipresent in modeling, data analysis, simulations and analytic calculations. Applications range from one-liners in Mathematica to complex simulations on super computers. Some examples for computer applications in elementary particle physics are the following: Lattice field theory: In field theory a central quantity is the path integral. E.g., for quantum chromodynamics (QCD) it reads: S[A, ψ,ψ] Z = D[A ψψ]e (1.1) The integral is over the elementary fields and S[A, ψ, ψ] is the action of QCD. It can be approximated by putting space-time on a lattice with lattice spacing a and length L, see Fig However, the remaining sum can still not be performed exactly and is calculated using Monte Carlo methods, viz., using a stochastic sampling of configuration space. Since continuum physics corresponds to a = 0 and L =, the corresponding limits have to be performed. In practice, this is done by repeating simulations with different values for a and L and extrapolating to the so-called continuum limit. Lattice simulations account for the most resource intensive calculations in theoretical physics, but their predictive power makes them an extremely useful tool. Monte Carlo event generators: It is not possible to describe collisions of particles fully from first principles because of the complexity of the perturbative part and difficulties in describing the nonperturbative part. However, this is necessary to confront experimental results with theoretical predictions and to separate signals from the background. Event generators produce large numbers of possible outcomes which are then compared to what is actually measured. Data analysis: Experiments as those carried out at the LHC produce immense amounts of data (estimated PB per year for the LHC). Processing and storing them requires large computational resources. In the case of the LHC this is handled via a worldwide network of computing centers, the so-called Worldwide LHC Computing Grid. It handles the distribution of the data and provides distributed computing resources. 4

5 1.1. OVERVIEW OF COMPUTATIONAL METHODS IN ELEMENTARY PARTICLE PHYSICS CHAPTER 1. INTRODUCTION Figure 1.1: Discretizing space-time on a rectangular lattice with lattice spacing a and lattice size L. Functional methods: Functional methods consist of integro-differential, differential or integral equations which, for most quantitative cases, need to be solved numerically. Compared to other methods they are far less resourceintensive. However, the size of the systems of equations solved increases steadily and with that the required computational resources. Depending on the problem, various architectures are used for computations. For some problems desktop PCs are sufficient. However, even then parallelization is advantageous due to the increasing number of cores in modern CPUs. Computer clusters naturally take advantage of parallelized programs. Clusters can have a few nodes, with several CPUs each, up to millions. In some cases it is even useful to run several single-core instances of the same program, e.g., for Monte Carlo methods when the same program is run over and over again to accumulate statistics. Possibilities for parallelization are MPI or OpenMP. In the last decade graphical processing units (GPUs) have become also very important. Programs are written with CUDA or OpenCL. Another possibility is Intel s Xeon Phi coprocessor, which offers a highly-parallel multicore environment similar to a GPU. Various programming languages are used in elementary particles physics. Historically, FORTRAN was always strong and, despite constant rumors to the contrary, it is still widely used. One reason is that many libraries, e.g., LAPACK, used in physics are written in FORTRAN. The fact that FORTRAN was heavily used in physics, or in science in general, is reflected in many physics friendly features, e.g., a complex number type was included early on. Another often used language is C++. For GPU and coprocessor architectures languages like CUDA and OpenCL are required. On the symbolic side Mathematica must be mentioned, which was founded by Stephen Wolfram, a particle physicist, in Special purpose packages can add additional functionalities useful for particle physicists. Special mention deserves the program FORM. It has a narrow focus on use in particle physics, where it is unrivaled in speed when it comes to perform large symbolic calculations. Finally, for (larger) projects additional auxiliary tools exist that can be very useful: make (for the compilation of the code), IDEs (integrated development environments, or at least editors that offer more support than syntax highlighting), version control 5 Version: March 7, 017

6 1.. OVERVIEW OF THIS COURSE CHAPTER 1. INTRODUCTION S(k ) Γ(p, P ) = K(k, p, P ) S(k + ) Γ(k, P ) Figure 1.: Diagrammatic representation of a meson BSE. systems (git can be considered state-of-the-art, but also subversion or others offer the basic features), debuggers, memory error detection programs and profilers. 1. Overview of this course In this course we will develop the methods necessary to calculate a meson mass in QCD. This will be done by solving the appropriate bound state equation, a Bethe- Salpeter equation (BSE, which, however, could also mean bound state equation), see Fig. 1.. This equation depends on several quantities. Some of them will be taken from models, another one, the quark propagator, will be calculated in this course. On the technical side this will require numeric integration, interpolation and solving an eigenvalue system. The complete code for this project will be developed step by step and should result in a final program. The following tasks need to be implemented which will be done in individual projects: 1. Functions for numeric integration. Functions for interpolation 3. Implementation of the quark propagator 4. Implementation of the BSE 5. Solution of the BSE through an eigenvalue system Programmes suitable for numeric computation should be used, e.g., FORTRAN, C, C++, Python, Julia. 6 Version: March 7, 017

7 Basic techniques.1 Numerical integration A numeric evaluation of an integral is necessary if its solution is not known in closed form or if the integrand itself is only known numerically. Depending on the problem, one can choose from many methods for numerical integration. Alternatively one speaks of (numerical) quadrature, which can refer to one- or multi-dimensional integration, whereas cubature is used for multi-dimensional integration only. In this section we will first look at some simple integration formulas to explain the basic idea and get some feeling for errors. Then we will turn towards integration techniques which are actually used for functional integrals..1.1 Newton-Cotes formula A simple integration method is the Newton-Cotes formula. To calculate an integral I one replaces the integrand f(x) by an approximating function p m (x), typically an m-th order polynomial: I = b a dx f(x) b a dx p m (x). (.1) The approximating function p m (x) is chosen such that it agrees with f(x) at a fixed number of points. For a polynomial this determines the values for the coefficients a i : p m (x) = a 0 + a 1 x a m x m + a m x m. (.) For practical calculations one splits the integration interval [a, b] into smaller intervals with width and defines the points h = x = b a n (.3) x i = a + i h, i = 0,..., n. (.4) The simplest approximation is to use a constant p m (x). This is called the rectangular rule. It is rarely used, because its truncation error is quite large for general functions. We can approximate the function f(x) in the interval x i x x i+1 either by f i = f(x i ) or f i+1 = f(x i+q ). The value of the integral in the interval [x i, x i+1 ] is then h f i and h f i+1, respectively. The total integral can be calculated correspondingly as n I left h f i (.5) i=0 7

8 .1. NUMERICAL INTEGRATION CHAPTER. BASIC TECHNIQUES or n I right h f i+1 = h i=0 n f i. (.6) These formulas over- or underestimate integrals of functions that increase or decrease monotonically. This behavior can be improved using the average of the two forms instead: n I r h i=0 i=1 f i + f i+1. (.7) As we will see, this is equivalent to the trapezoidal rule. The trapezoidal rule approximates the integrand piecewise by a polynomial of order one: p 1 (x) = a 0 + a 1 x. The area under the curve in the interval [x i, x i+1 ] is then given by the area of a trapezoid: h (f i + f i+1 )/. Summing up all intervals yields n I t h i=0 f i + f i+1 = h (f 0 + f f n + f n ). (.8) The error for a single segment is with this method given by E 1t = b a dx f(x) f(a) + f(b) (b a). (.9) Using a Taylor expansion around x = (b + a)/ with y = x x, we can perform the integral: b a dx f(x) = f(x) = f( x) + yf ( x) + y! f ( x) +..., (.10) h h ) dy (f( x) + yf ( x) + y! f ( x) +... = = h f( x) h3 f ( x) (.11) We use the Taylor expansion also at the endpoints of the interval: f(a) = f( x) h f ( x) + 1 ( ) h f ( x)..., f(b) = f( x) + h f ( x) + 1 ( ) h f ( x) (.1) Plugging all this into Eq. (.9) we obtain ( E 1t = h f( x) + 1 ) ( 4 h3 f ( x) +... h f( x) + 1 ) 8 h3 f ( x) h3 f ( x). (.13) 8 Version: March 7, 017

9 .1. NUMERICAL INTEGRATION CHAPTER. BASIC TECHNIQUES Thus the error is proportional to f ( x) and h 3. When a multisegment trapezoidal rule is used, the individual errors need to be summed: E t 1 ( ) 3 b a n f ( x i ), (.14) 1 n where x i is the midpoint between x i and x i+1. Using an average value for the second derivative, this can be rewritten as E t 1 1 (b a) ( b a n i=0 f = 1 n f ( x i ), (.15) n n=0 ) f = 1 1 (b a)h f. (.16) Thus, the error is O(h ), since b a is fixed. Repeating a similar analysis for the rectangular rules shows that the error is O(h): E r 1 (b a)hf. (.17) Exercise: Derive eq. Eq. (.17). To improve the precision of the results, the straightforward approach is to lower the step size h. However, the usefulness of this approach is limited by round-off errors becoming too large at some points. Another method is to increase the degree of the employed polynomial n. For n = this leads to Simpson s one-third rule, for n = 3 to Simpson s three-eight rule and for n = 4 to Boole s rule. An alternative possibility is to reduce the error by combining the results from calculations with different step sizes (Richardson s extrapolation). For example, consider the errors obtained for two step sizes h 1 and h with the trapezoidal method: E(h 1 ) 1 1 (b a)h 1f, (.18) E(h ) 1 1 (b a)h f. (.19) We assume that f is independent of the step size. The true value for the integral I can then be obtained from each calculation as I I(h 1 ) + c h 1, (.0) I I(h ) + c h, (.1) where I(h) is the result obtained by using the step size h. equations we can calculate the constant c: Plugging this into Eq. (.1) yields Equating these two c = 1 1 (b a)f I(h ) I(h 1 ). (.) h 1 h I I(h ) + I(h ) I(h 1 ) ( ). (.3) h 1 h 1 9 Version: March 7, 017

10 .1. NUMERICAL INTEGRATION CHAPTER. BASIC TECHNIQUES One can show that the error is now O(h 4 ). A particular application is to use h = h 1 /. This process can be used to combine results from calculations with different step sizes. For example, if we have the results I(h 1 ), I(h ) and I(h 3 ), we can combine I(h 1 ) and I(h ) to obtain a result I(h 1, h ) with an error O(h 4 ), but we can also combine I(h ) and I(h 3 ) to obtain a result I(h, h 3 ). Combining the two new results, we can get a result with an error O(h 6 ). Systematically applying this procedure is known as Romberg integration..1. Gauss quadrature The Newton-Cotes formula calculates an integral from values at equidistant points. Gauss quadrature, on the other hand, requires knowledge of the integrand at not equidistant given points. The general formula is c c 1 dx g(x)f(x) n w i f(x i ). (.4) i=1 Gauss quadrature does not only use the freedom to fix the weights w i, but also the positions x i where the function needs to be known. Due to this enlarged freedom, fewer function evaluations are required compared to the Newton-Cotes method. For a given value of n, the formula is exact for polynomials of degree n 1. Note that the function g(x) appears only on the left-hand side. It allows to make the quadrature exact for polynomials up to order n times the function g(x). The values of the positions x i and of the weights w i depend on the choice of g(x). For now we will choose g(x) = 1. This is called Gauss-Legendre quadrature, which is based on Legendre polynomials. The choice of g(x) also determines the values for the boundaries, c 1 and c. For Gauss-Legendre they are set to c 1 = and c = 1. Hence, an integral with different boundaries must be transformed to this interval. In principle any transformation from an interval z [a, b] is possible that transforms this interval to x [, 1]. For example, the linear transformation reads x = z a b, (.5) b a where z is the original variable. The inverse transformation is z = (b a)x + a + b. (.6) When such a transformation of variables is performed, the Jacobian of the transformation must also be included: Thus, a general integral is calculated as dz = b a dx. (.7) b a dzf(z) b a n w i f(z(x i )). (.8) i 10 Version: March 7, 017

11 .1. NUMERICAL INTEGRATION CHAPTER. BASIC TECHNIQUES The Jacobian can be taken into account directly in the weights. For integrands that stretch over several orders of magnitude a logarithmic transformation is advantageous: x = A + B ln z, z = e x A B. (.9) The coefficients A and B are determined by the conditions x(a) = and x(b) = 1 as A = ln(a b) ln(b/a), B = ln(b/a). (.30) The Jacobian is dz = e x A B B dx. (.31) Let us first consider the two-point formula, viz., the result is exact for polynomials of order 3 and below: dxf(x) = w 1 f(x 1 ) + w f(x ). (.3) From the requirement that it should be exact for f(x) equal to 1, x, x and x 3 we obtain These four conditions leads to dx = = w 1 + w, dx x = 0 = w 1 x 1 + w x, dx x = 3 = w 1 x 1 + w x dx x 3 = 0 = w 1 x w x 3. (.33) w 1 = w = 1, x 1 = x = 1 3. (.34) Note the symmetry x 1 = x. As Simpson s three-eight rule, Eq. (.3) is exact for polynomials up to order three. However, here less evaluations of the function are required ( vs. 4). For higher n the determination of the x i and w i is done using Legendre polynomials, for useful relation see, e.g., [3]. They are given by P 0 (x) = 1, P 1 (x) = x, P n (x) = n 1 x P n (x) n 1 n n P n (x), n =, 3,... (.35) 11 Version: March 7, 017

12 .1. NUMERICAL INTEGRATION CHAPTER. BASIC TECHNIQUES and fulfill the orthogonality relation The weights w i are w i = dxp n (x)p m (x) = δ nm n + 1. (.36) (1 x i )P n(x i ) = (1 x i ) (n + 1) P n+1 (x i ). (.37) The derivative can be calculated from the recurrence relation (1 x )P n(x) = (n + 1)xP n (x) (n + 1)P n+1 (x). (.38) The values for x i correspond to the zeros of the Legendre polynomial of order n: P n (x i ) = 0. There are exactly n zeros. For finding the zeros, the standard Newton method is sufficient, viz., by iterating x i = x i f(x i) f (x i ) (.39) until x i x i < ɛ. ɛ must be chosen sufficiently small, e.g., By using a good initial guess x 0, this method converges in a few steps. For the Legendre polynomials a good initial guess is x j,0 = cos(π (j 0.5)/(n + 0.5)), where j labels the zero. In principle tabulated values for the weights and zeros can be used. However, more flexibility is achieved when they are calculated in the program. The Gauss-Legendre quadrature has an error estimate of E GL n+1 (n!) 4 (n + 1)((n)!) 3 f (n) (x), < ξ < 1. (.40) From this is can be seen that polynomials of order up to n 1 is evaluated exactly, since the derivative f (n) vanishes in that case. Also, convergence is much faster than for Newton-Cotes. In principle it is possible to construct the weights and zeros for a generic function g(x), see []. Some choices lead to well-known orthogonal polynomials. The most useful cases are: g(x) Polynomials Boundaries 1 Legendre [, 1] 1/ 1 x Chebyshev first kind [, 1] 1 x Chebyshev second kind [, 1] e x Laguerre [0, ] e x Hermite [, ] A typical application for Gauss-Chebyshev quadrature is the integration over angles. For example, with u = cos(θ) the following integral, where a factor sin(θ) appears from the integral measure, can be rewritten as π 0 sin(θ) dθ = 1 u du. (.41) 1 Version: March 7, 017

13 .. INTERPOLATION CHAPTER. BASIC TECHNIQUES The generalization to multi-dimensional integrals is straightforward. becomes then a multiple sum. For example: dxdy f(x, y) = n i=1 The sum m wi x w y j f(x i, y i ). (.4) For many applications a fixed order of the quadrature is sufficient. One can easily find out how many points are needed by increasing their number until the result does not change anymore and stay with that number. Alternatively, adaptive integration algorithms exist that rely on subdividing the integral into smaller integrals until a certain convergence criterion is reached. Ideally one can reuse the points already calculated. Gauss-Kronrod quadrature is an example of such an adaptive integration.. Interpolation The dressing function we calculate with functional equations are only known at discrete values. However, in the equations they are typically required for arbitrary arguments. One can try to choose the arguments, where the dressings are calculated, such that they coincide with the arguments used in the equations, but this only works in some cases. Typically we need a way to describe the function also away from the calculated points. This is done via interpolation. In general one can distinguish between two ways to describe a function known at discrete points. One is to try to approximate the general trend of the available data. This is useful when it is known that the data itself has significant error or noise, for example, when it comes from an experimental measurement. Approximating the data in such a generic way is called least-squares regression. Another approach is to make the approximating function pass through every known point. This is done when the data is very precise and known as interpolation. In the following we will only treat this case. Once a method for interpolation is defined, one can determine all the required parameters and use it also outside of the data. This is then known as extrapolation. If no further information about the behavior of the function outside the domain of the available data is known, this is a very delicate procedure and should always be done with caution...1 Polynomial interpolation As we did for integration, one can use a polynomial to describe data. If n + 1 points are known, a polynomial of order n is required to make the polynomial pass through the data: j=1 f(x) = y a 0 + a 1 x a n x n. (.43) A naive implementation requires the following system of equations to be solved for the coefficients a i : f(x i ) = y i = a 0 + a 1 x i a n x n i, i = 0, 1,,..., n. (.44) This system has a unique solution, but typically solving it is not very efficient and problems like an ill-conditioned matrices can appear. 13 Version: March 7, 017

14 .. INTERPOLATION CHAPTER. BASIC TECHNIQUES e -x x Figure.1: Lagrange fit of e x using 101 points between 0 and 100. Clearly some artifacts are seen for high x. If another representation for the polynomial is chosen, the determination of the coefficients is easier. Lagrange interpolation uses this representation: n n f(x) = y (x x j ). (.45) For example, for n =, this yields i=0 b i j=0,j i f(x) = y b 0 (x x 1 )(x x ) + b 1 (x x 0 )(x x ) + b (x x 0 )(x x 1 ). (.46) The coefficients can easily be determined from the f(x i ): f(x 0 ) = y 0 = b 0 (x 0 x 1 )(x 0 x ), f(x 1 ) = y 1 = b 1 (x 1 x 0 )(x 1 x ), f(x ) = y = b (x x 0 )(x x 1 ). (.47) Solving for the b i and plugging these expressions back into Eq. (.46) yields (x x 1 )(x x ) f(x) = y y 0 (x 0 x 1 )(x 0 x ) + y (x x 0 )(x x ) 1 (x 1 x 0 )(x 1 x ) + y (x x 0 )(x x 1 ) (x x 0 )(x x 1 ). For general n the formula is f(x) = y n i=0 y i n j=0,j i (.48) x x j x i x j. (.49) If the function to be described is a polynomial of order n or less, using polynomials is exact. However, for the Lagrange it is difficult to estimate the error in general. Another disadvantage of the Lagrange method is that results of a computation cannot be reused for interpolation at another point. Also, if another point is added to the known data, the complete calculation has to be redone. A method that overcomes the last two problems is using a Newton polynomial. The approximating polynomial is in this case written as f(x) = y c 0 + c 1 (x x 0 ) + c (x x 0 )(x x 1 ) c n (x x 0 )(x x 1 ) (x x n ). (.50) 14 Version: March 7, 017

15 .. INTERPOLATION CHAPTER. BASIC TECHNIQUES The coefficients c i can be determined recursively starting with a constant interpolation c 0 (x 0 ) = f(x 0 ). Adding the point x 1 one determine the coefficient for linear interpolation: c 1 (x 1, x 0 ) = f(x 1) f(x 0 ) x 1 x 0. (.51) The next point x is used to fix the quadratic coefficient: ( 1 f(x ) f(x 1 ) c (x, x 1, x 0 ) = f(x ) 1) f(x 0 ). (.5) x x 0 x x 1 x 1 x 0 The i-th coefficient can be expressed as: c i (x i, x i,..., x 0 ) = c i(x i, x i,..., x 1 ) c i (x i,..., x 0 ) x i x 0. (.53) The recursive definitions of the c i are known as divided differences and hence this method is also known as Newton s divided-difference interpolation polynomial. The Newton polynomial, Eq. (.50), is in structure similar to a Taylor series. The error can thus be estimated from the next order n + 1: E n Newton c n+1 (x n+1,..., x 0 )(x x 0 ) (x x n ). (.54) Since we already used up all data points, we create a data point (x n+1, y n+1 ) from the polynomial itself from an arbitrarily chosen point x n+1... Chebyshev interpolation One problem of the interpolation methods mentioned up to now is that the error of interpolation is larger at the ends of the domain. To have a uniformly distributed error, one can use orthogonal polynomials as approximating functions. This also belongs to the class of polynomial interpolations. Chebyshev polynomials The recursive definition of Chebyshev polynomials is ( x 1): P 0 (x) = 1, (.55) P 1 (x) = x, (.56) P k (x) = xp k (x) P k (x), k =, 3,... (.57) Chebyshev polynomials can be expressed in terms of cosines: P k (x) = cos(k θ) (.58) with cos θ = x. The equivalence can be seen from P 0 (x) = cos 0 = 1, P 1 (x) = cos(cos x) = x and the trigonometric identity cos((k + 1)θ) + cos((k 1)θ) = cos θ cos(k θ), (.59) 15 Version: March 7, 017

16 .. INTERPOLATION CHAPTER. BASIC TECHNIQUES which yields the recursive relation Eq. (.57). The orthogonality relation of the polynomials is dx P n(x)p m (x) 0 n m = π n = m = 0 1 x π n = m 0 (.60) The k zeros x j can be determined from the zeros of the cosine at ±π/, ±3π/,... in Eq. (.58) as a k j + 1 k θ = π, j = 1,,... k, (.61) ( ) k j + 1 x j = cos π. (.6) k a The indices are put such that the zeros are ordered from (k 1)π/ to π/ which in turn yields the zeros in x from low to high. In principle, one can create a standard polynomial by expressing single polynomials via Chebyshev polynomials. For example: 1 = P 0 (x), (.63) x = P 1 (x), (.64) x = 1 (P 0(x) + P (x)). (.65) However, it is more convenient to make use of the orthogonality properties of the polynomials and write it as f(x) = y d 0 + d i P i (x), (.66) where the coefficients d i can be determined from Eq. (.60): d i = π i=1 dx f(x)p i(x), i = 0, 1,.... (.67) 1 x 16 Version: March 7, 017

17 Bibliography [1] Singiresu S. Rao, Applied numerical methods for engineers and scientists, Prentice Hall, New Jersey, 00. [] William H. Press et al., Numerical recipes - The art of scientific computing, Cambridge University Press, Cambridge, 007. [3] Weisstein, Eric W., Legendre-Gauss Quadrature. From MathWorld A Wolfram Web Resource, Legendre-GaussQuadrature.html 17