A. Kryukov

CompHEP 4.4—automatic computations from Lagrangians to events

Abstract We present a new version of the CompHEP program (version 4.4). We describe shortly new issues implemented in this version, namely, simplification of quark flavor combinatorics for the evaluation of hadronic processes, Les Houches Accord-based CompHEP-PYTHIA interface, processing the color configurations of events, implementation of MSSM, symbolical and numerical batch modes, etc. We discuss how the CompHEP program is used for preparing event generators for various physical processes. We mention a few concrete physics examples for CompHEP-based generators prepared for the LHC and Tevatron.

CompHEP-PYTHIA interface: Integrated package for the collision events generation based on exact matrix elements

CompHEP, as a partonic event generator, and PYTHIA, as a generator of final states of detectable objects, are interfaced. Thus, integrated tool is proposed for simulation of (almost) arbitrary collision processes at the level of detectable particles. Exact (multiparti-cle) matrix elements, convolution with structure functions, decays, partons hadronization and (optionally) parton shower evolution are basic stages of calculations. The PEVLIB library of event generators for LHC processes is described. In the widely used generators PYTHIA [1], ISAJET [2] and HERWIG [3] data bases of matrix elements of hard subprocesses are built in. It means that matrix elements are stored as formulas. Furthermore, the matrix element squared |M| 2 is represented by means of some function modelling the behaviour of the integrand to get effective Monte-Carlo integration and events generation. Thus, as one can see, mainly 2 → 2 subprocesses are included in these data bases. However, the generation of events with 3, 4 and more bodies in the final states of hard subprocesses is needed for the Tevatron, LHC and future linear collider physics. One can note, in particular, that for such states there is no possibility to construct simple analytical formulae to match singular behaviour of |M| 2. Multidimensional phase space (4 dimensions in the 3-body case plus 2 dimensions in case of hadron collisons for convilution with parton distributions, 7+2 dimensions in the 4-body case etc) with untrivial regions corresponding to the singularities of the martrix element leads to complicated symbolic structures. Thus, one needs a new approach to the generation of events at the partonic level. Partonic level final states with top quarks, Higgs bosons and intermediate vector bosons, like W tj, ttH, W bb, ttbb and tttt, give practical examples of multidimensional phase space 1CompHEP, as a partonic event generator, and PYTHIA, as a generator of final states of detectable objects, are interfaced. Thus, integrated tool is proposed for simulation of (almost) arbitrary collision processes at the level of detectable particles. Exact (multiparticle) matrix elements, convolution with structure functions, decays, partons hadronization and (optionally) parton shower evolution are basic stages of calculations. The PEVLIB library of event generators for LHC processes is described.

CompHEP 4.5 Status Report

We present a new version of the CompHEP program package, version 4.5. We describe new options and techniques implemented in the version: interfaces to ROOT and HERWIG, parallel calculations, generation of the XML-based header in event files (HepML), full implementation of the Les Houches agreements (LHA I, SUSY LHA, LHA PDF, Les Houches Event format), cascade matching for intermediate scalar resonances, etc.We present a new version of the CompHEP program package, version 4.5. We describe new options and techniques implemented in the version: interfaces to ROOT and HERWIG, parallel calculations, generation of the XML-based header in event files (HepML), full implementation of the Les Houches agreements (LHA I, SUSY LHA, LHA PDF, Les Houches Event format), cascade matching for intermediate scalar resonances, etc.

ATENSOR — REDUCE program for tensor simplification

The paper presents a REDUCE program for the simplification of tensor expressions that are considered as formal indexed objects. The proposed algorithm is based on the consideration of tensor expressions as vectors in some linear space. This linear space is formed by all the elements of the group algebra of the corresponding tensor expression. Such approach permits us to simplify the tensor expressions possessing symmetry properties, summation (dummy) indices and multiterm identities by unify manner. The canonical element for the tensor expression is defined in terms of the basic vectors of this linear space. The main restriction of the algorithm is the dimension of the linear space that is equal to N!, where N is a number of indices of the tensor expression. The program uses REDUCE as user interface.The paper presents a REDUCE program for the simplification of tensor expressions that are considered as formal indexed objects. The proposed algorithm is based on the consideration of tensor expressions as vectors in some linear space. This linear space is formed by all the elements of the group algebra of the corresponding tensor expression. Such approach permits us to simplify the tensor expressions possessing symmetry properties, summation (dummy) indices and multiterm identities by unify manner. The canonical element for the tensor expression is defined in terms of the basic vectors of this linear space. The main restriction of the algorithm is the dimension of the linear space that is equal to N!, where N is a number of indices of the tensor expression. The program uses REDUCE as user interface.

Vector leptoquark pair production in e+e− annihilation

The cross section for vector leptoquark pair production in e + e annihilation is calculated for the case of finite anomalous gauge boson couplings �,Z and �,Z. The minimal cross section is found to behave ∝ � 7 , leading to weaker mass bounds in the threshold range than in models studied previously.

Symbolic simplification of tensor expressions using symmetries, dummy indices and identities

The algorithm based on simple geometrical ideas is suggested for simplification O! tensor expressions which takes into account symmetries, dummy indices, and linear identities with many terms. The results of the realization in REDUCE system are adduced.

Program “color” for computing the group-theoretic weight of Feynman diagrams in non-Abelian gauge theories

Abstract Title of program : COLOR Catalogue number : AAXC Program obtainable from : CPC Program Library, Queens University of Belfast, N. Ireland (see application form in this issue) Computer : IBM 360/370 Operating system : OS, VM Programming language used : REDUCE [1] High-speed storage required : depends on the problem; minimum about 400 Kbytes Number of bits in word : 32 Number of lines in combined program and test deck : 200

Construction of rational approximations by means of REDUCE

1. In recent years the rational approximations have been widely used to solve physical and computational problems /1,2/. When a real function f(x) is repeatedly calculated on a ≤ × ≤ b, it is reasonable to replace it by a polynomial or rational approximation on [a,b]. For example, if f(x) is a composite combination of elementary and special functions any of which can be calculated by means of the corresponding standard program, the f(x) values are obtainable by these programs. This method, however, involves unjustified losses in the computing time and often provides a too high accuracy inadequate to the problem in question. In this case it is more convenient to use the corresponding approximation. There exist iteration algorithms which ensure the best (in the sense of absolute or relative error) rational approximations based on P.L. Chebyshev theory /2,3/. Unfortunately, these algorithms are cumbersome and do not guarantee the convergence if the choice of the initial approximation is unsuccessful, see ref./2/. The present paper treats simple algorithms (Padé-Chebyshev approximation /1/ and Paszkowski algorithm /4/) providing approximations similar to the best ones with a relatively moderate computer resources required. In this case the calculation of the approximation coefficients reduces to the solution of a system (generally speaking, ill-conditioned) of linear algebraic equations. The errors of the Padé-Chebyshev approximations and the corresponding best approximations are compared in the paper /5/ where one of the methods of computation of the Padé-Chebyshev approximations is described. 2. The Analytic Computations System Reduce is a rather convenient tool of realization of algorithms of the rational approximation construction. This system saves one the trouble of inventing an effective algorithm of approximated-function computation if this function can be given in an analytic form or if the terms in the Taylor series expansion are known or determined analytically by the differential equation. The possibility of using the rational arithmetic (without round-off errors) is essential because the coefficients of rational approximations are not stable with respect to the perturbations of initial data and to the round-off errors. Specifically, the error is minimized which arises in solving the ill-conditioned systems of linear equations and when converting a power series into a series of the Chebyshev polynomials and vice versa. The ALGOL - like input language and the convenient tools of solving the problems of linear algebra ensure the simplicity and compactness of programs. For example, the program of computation of the Padé-Chebyshev coefficients occupies sixty two cards. 3. We compute the Padé-Chebyshev approximations by the standard “cross multiplied scheme”/1/. By means of the change of variable x → [(b-a)x+a+b]/2 the approximation on an arbitrary finite range [a,b] is reduced to the approximation on [-1, 1]. We shall, therefore, restrict ourselves to the case when the function f(x) is to be approximated on the [-1, 1] range by the expression of the form R(x) = a<subscrpt>0</subscrpt>+a<subscrpt>1</subscrpt>x+…+a<supscrpt>n</supscrpt>/b<subscrpt>0</subscrpt>+ b<subscrpt>1</subscrpt>x+…+b<subscrpt>m</subscrpt>x<supscrpt>m</supscrpt> (1) where m, n are given non-negative integer numbers, a<subscrpt>0</subscrpt>, a<subscrpt>1</subscrpt>,…, a<subscrpt>n</subscrpt>, b<subscrpt>0</subscrpt>, b<subscrpt>1</subscrpt>,…, b<subscrpt>m</subscrpt> are numerical coefficients to be determined. If the function is specified by a power series ƒ(x)= @@@@ C<subscrpt>k</subscrpt>X<supscrpt>k</supscrpt> (2) the corresponding finite sum (the number of terms is determined by the user on the basis of the accuracy required) is converted, using the well-known economizing procedure, to the polynomial @@@@(x) = @@@@ γ<subscrpt>k</subscrpt> T<subscrpt>k</subscrpt> (3) where T<subscrpt>k</subscrpt> are the k-power Chebyshev polynomials. Solving the system of linear equations 1/2 @@@@ β<subscrpt>j</subscrpt> (γ<subscrpt>i</subscrpt>+j γ<subscrpt>li-jl</subscrpt>) = 0, i=n+1,…, n+m 1/2 @@@@ β<subscrpt>j</subscrpt> (γ<subscrpt>i+j</subscrpt> + γ<subscrpt>li-jl</subscrpt>) = α<subscrpt>i</subscrpt>, i=0,1, …, n (4) we determine the coefficients α<subscrpt>i</subscrpt>, β<subscrpt>i</subscrpt> of the rational approximation R(x) = α<subscrpt>0</subscrpt>+α<subscrpt>1</subscrpt>T<subscrpt>1</subscrpt> + … + α<subscrpt>n</subscrpt>T<subscrpt>n</subscrpt>/β<subscrpt>0</subscrpt> + β<subscrpt>1</subscrpt>T<subscrpt>1</subscrpt> + … + β<subscrpt>m</subscrpt> T<subscrpt>m</subscrpt> (5) As usual /1,4/ @@@@ <italic>dj</italic> denotes that the first term in the sum is to be replaced by d<subscrpt>0</subscrpt>/2. The system (4) is homogeneous and the solution is determined to within the non-zero factor; it is quite natural since the fraction will not change if the numerator and denominator are simultaneously multiplied or divided by a non-zero values. Therefore, the system (4) is completed with the normalization condition, for example β <subscrpt>m</subscrpt>= 1. Finally, the standard transformation reduces (5) to the form (1). The absolute error of the approximation (1) is of the form δ(x) = &PHgr;(x) / @@@@ b<subscrpt>j</subscrpt>x<supscrpt>j</supscrpt> (6) where &PHgr;(x) = @@@@ b<subscrpt>j</subscrpt>x<supscrpt>j</supscrpt> ƒ(x) - @@@@ a<subscrpt>j</subscrpt>x<supscrpt>j</supscrpt> The above-described algorithm is equivalent to the following procedure: the numerator &Pgr;(x) in (6) is expanded in a series of the Chebyshev polynomials and the first m+n+1 terms are equated to zero. The Paszkowski algorithm (4) leads to the rational approximation R(x) of the form (1) such that the first m+n+1 terms in the expansions of f(x) and R(x) coincide (it will be noted that this approximation does not always exist). The program of this algorithm is similar to that of Padé-Chebyshev algorithm. For even functions the approximation may be sought for in the form R(x) = a<subscrpt>0</subscrpt> + a<subscrpt>1</subscrpt>x<supscrpt>2</supscrpt> + … + a<subscrpt>n</subscrpt>(x<supscrpt>2</supscrpt>)<supscrpt>n</supscrpt>/b<subscrpt>0</subscrpt> + b<subscrpt>1</subscrpt>x<supscrpt>2</supscrpt> + … + b<subscrpt>m</subscrpt>(x<supscrpt>2</supscrpt>)<supscrpt>m</supscrpt> (7) For this case the algorithms involved permit a convenient modification. If f(x) is an odd function it is reasonable to find an approximation of the form (7) to the even function f(x)/x and then to multiply the result by x such that the approximation be of the form R(x) = x a<subscrpt>0</subscrpt> + a<subscrpt>1</subscrpt>x<supscrpt>2</supscrpt> + … + a<subscrpt>n</subscrpt>(x<supscrpt>2</supscrpt>)<supscrpt>n</supscrpt>/b<subscrpt>0</subscrpt> + b<subscrpt>1</subscrpt>x<supscrpt>2</supscrpt> + … + b<subscrpt>m</subscrpt>(x<supscrpt>2</supscrpt>)<supscrpt>m</supscrpt> (8) A large relative error at x=0 is thereby avoided. 4. To estimate the quality of the rational approximation R(x) to the function f(x), the error functions δ(x) = ƒ(x)-R(x) and δ(x)=Δ(x)/ƒ(x) are calculated and the error curves (i.e. the plots of these functions) are built. The absolute error of the approximation coincides with the value of Δ = max \Δ(x)\ and the relative error, with δ = max\δ (x)\. Actually, the error functions are calculated in the finite number of the check points that are uniformly distributed over the range where the function is approximated. To find it out to what extent the approximation R(x) to f(x) differs from the best (in the sense of the absolute and relative error) approximation to f(x) of the same form, one can use the generalized de la Valléé-Poussin theorem /3/. Let us, for example, estimate the similarity of the approximation (1) to the approximation of the same form with the best absolute error. For simplicity, we assume that for the best approximation a<subscrpt>n</subscrpt>≠0, b<subscrpt>m</subscrpt>≠0 (this condition is usually always fulfilled). Then if a given approximation R(x) is close enough to the best one, the function Δ (x) takes on non- zero values λ<subscrpt>1</subscrpt>,-λ<subscrpt>2</subscrpt>,…, (-1)<supscrpt>n+m</supscrpt>λ<subscrpt>n+m+2</subscrpt> with alternating signs at successive points x<subscrpt>1</subscrpt> < x<subscrpt>2</subscrpt> < … < x<subscrpt>n+m+2</subscrpt> in the x range by virtue of the generalized Chebyshev theorem /3/. Assume that λ = min {|λ<subscrpt>1</subscrpt>|,|λ<subscrpt>2</subscrpt>, …, |λ<subscrpt>m+n+2</subscrpt>|} Then, according to the generalized de la Valléé-Poussin theorem /3/, the following inequality is valid: λ ≤ Δ<subscrpt>min</subscrpt> ≤ Δ (9) where Δ <subscrpt>min</subscrpt> is the best absolute error of the approximations. It is clear that Δ coincides with the largest (in the absolute value) extremum of Δ(x) and the least (in the absolute value) extremum of this function (up to a sign) can be used λ. For example, for the Padé-Chebyshev approximation (7) to the function cos(π/4 x) on [-1,1] for m=n=2 the absolute error Δ = =0.68 5 .10<supscrpt>-10</supscrpt> and λ = 0.663 .10<supscrpt>-10</supscrpt>) (2400 check points and the condition of the de la Valléé- Poussin theorem is fulfilled). It is clear that this approximation is rather close to the best one. The errors of the Padé-Chebyshev approximants and the approximants obtained with the help of the Paszkowski algorithm are usually of the same order of magnitude. Consider, as an example, the approximations (1) to the function e<supscrpt>x</supscrpt> on the range [-1,1] for m=n=3. In this case the exponent is replaced with the truncated Taylor series (2) up to x<supscrpt>10</supscrpt>/10! inclusive. For the Padé-Chebyshev approximation Δ = 0.33 .10<supscrpt>-6</supscrpt>, δ = 0.20 . 10<supscrpt>-6</supscrpt>, and Paszkowski algorithm gives Δ = 0.25 .10<supscrpt>-6</supscrpt>, δ = 0.26.10<supscrpt>-6</supscrpt>. The above-described algorithms lead to much lower maximum errors as compared to

Fast algorithm for calculation of Dirac's gamma-matrices traces

This paper describes a fast algorithm for calculation of traces of Diracs gamma-matrices proposed by Kennedy and Cvitanovich. Comparison with standard REDUCE calculations shows that Kennedy-Cvitanovich algorithm is preferable when space-time dimension exceeds 4.

Interactive REDUCE

This paper deals with some distinctive characteristics of the REDUCE-2 computer algebra system operating in interactive mode. A brief description of a new version of the system with extended interactive capabilities is presented.