International Journal of Modern Physics C

We report our experience using the formal method VDM⁺⁺ in high energy physics real-time applications. VDM⁺⁺ is an extension of the established formal method VDM to include object-oriented, concurrency and real-time features. VDM⁺⁺ is supported by the VENUS tool-set, comprising an OMT graphical editor, VDM⁺⁺ syntax and type checker, and a C⁺⁺ code generator. Formal specification meta-languages allow formal data modeling, algorithm and system behavior specification at a highly abstract level. Once an abstract design has converged it is refined towards a particular implementation, with formal validation of each step if desired. We have applied VDM⁺⁺ to the design of a fast track pattern recognition algorithm; the design of a global second- level trigger system for LHC experiments; the specification of simulated physics data; and the design of a data router hardware unit for a LHC second-level trigger. We are encouraged by the application of mathematics in an engineering discipline and conclude that formal methods have the potential to make a valuable contribution to the systems development process in high energy physics.

Modern symbolic computing programs address not only the manipulation of mathematical functions, but also process control. Symbolic computing can thus be a powerful tool for developing algorithmic engines that can be fully integrated in the controls environment and facilitates modular design of control systems. This technique has been applied to a special class of accelerator problems, namely the beam steering in transfer lines and accelerator rings in the CERN PS Complex. An optimizer for beam trajectories has first been developed off line then linked on line to process interface routines for prototyping. A beam optics program with a full graphics environment and symbolic functions was used to generate necessary information for the optimizer. An application program has then been written to integrate the system into the controls environment.

A common problem for small experiments is that the design of the data acquisition, the simulation of the experiment and the reconstruction and analysis of real data is broken up into several disjoint development phases. Software development is necessary in all phases, yet continuity and an overall design strategy is lacking. This leads to duplication of software components and interface problems from one phase to the next and prohibits the generation of code which is reusable over the lifetime of many experiments. We address this problem within an object-oriented paradigm.

The common third level trigger and forth level reconstruction farm for the future HERA-B experiment will have to perform full on-line event reconstruction and calibration for an expected input rate of 2000 events/s. More than a hundred powerful RISC processors connected in a network capable of distributing several hundreds of MB/s with low latency are likely to be necessary for this task.

Proper simulation of the real time multi-processor systems is central for an optimal design (hardware and software protocol) of a scalable and flexible parallel data processing architecture. A discrete event, process oriented simulation developed in concurrent μC++ is used as a framework for modelling and evaluating different farm architectures. An object oriented graphic interface to the simulation allows the monitoring of various features and provides an easier way to optimize the system.

In a recent paper¹ a new powerful method to calculate Feynman diagrams was proposed. It consists in setting up a Taylor series expansion in the external momenta squared, a certain conformal mapping and subsequent resummation by means of Padé approximants. I present numerical examples.

We start with a short discussion of the content of a term Renormalisation Group in modern use. By treating the underlying solution property as a reparametrisation symmetry, we relate it with the self-similarity symmetry well-known in mathematical physics and explain the notion of Functional Self-similarity. Then we formulate a program of constructing a regular approach for discovering RG-type symmetries in different problems of mathematical physics. This approach based upon S. Lie group analysis allows one to analyse a wide class of boundary problems for different type of equations. Several examples are mentioned.

We discuss methods and techniques that were recently used in the following high order perturbative calculations: the large quark mass expansion of the decay rates Z⁰→ hadrons and τ→ν+ hadrons and the calculation of various quantities for deep inelastic lepton hadron scattering. Progress on the automatization of some general procedures is discussed.

We evaluate all the primitive divergences contributing to the 7-loop β-function of ɸ⁴ theory, i.e. all 59 diagrams that are free of subdivergences and hence give scheme-independent contributions. Guided by the association of diagrams with knots, we obtain analytical results for 56 diagrams. The remaining three diagrams, associated with the knots 10₁₂₄, 10₁₃₉, and 10₁₅₂, are evaluated numerically, to 10 sf. Only one satellite knot with 11 crossings is encountered and the transcendental number associated with it is found. Thus we achieve an analytical result for the 6-loop contributions, and a numerical result at 7 loops that is accurate to one part in 10¹¹. The series of ‘zig-zag’ counterterms, , previously known for n=3, 4, 5, 6 loops, is evaluated to 10 loops, corresponding to 17 crossings, revealing that the n-loop zig-zag term is , where are the Catalan numbers, familiar in knot theory. The investigations reported here entailed intensive use of REDUCE, to generate O(10⁴) lines of code for multiple precision FORTRAN computations, enabled by Bailey’s MPFUN routines, running for O(10³) CPUhours on DecAlpha machines.

Extending the method successful for one-loop integrals, the computation of two-loop diagrams with general internal masses is discussed. For the two-loop vertex of nonplanar type, as an example, we show a calculation related to vertex.

Complete Feynman diagram automatic computation systems are now coming of age after many years of development. They are made available to the high energy physics community through user-friendly interfaces. Theorists and experimentalists can benefit from these powerful packages for speeding up time consuming calculations and for preparing event generators. The general architecture of these packages is presented and the current development of the one-loop diagrams extension is discussed. A rapid description of the prominent packages and tools is then proposed. Finally, the necessity for defining a standardization scheme is heavily stressed for the benefit of developers and users.

Triggering at the HERA ep collider is challenging because of the high bunch crossing rate and an expected large background. In the H1 experiment, a trigger decision is made in four steps (level 1–4), stepwise decreasing the event rate and allowing for more sophisticated trigger decisions. The time available for L2 is about 20 μs. We have proposed to use an artificial neural network (ANN) for the L2 trigger based on the CNAPS-1064 chip available from Adaptive Solutions, (Oregon, USA). The intrinsic parallelism of the ANN algorithm together with the dedicated hardware offers fast processing of the trigger informations. The trigger system uses up to 10 decision units, each consisting of a Pattern Recognition Module (PRM) and a Data Distribution Board (DDB). A DDB receives the L2 data stream and generates the network inputs used by the algorithms on the PRM. A PRM is a commercial VME board carrying the CNAPS processors.

At the Collider Detector at Fermilab (CDF), we have designed and implemented a trigger for tau leptons using analog neural network electronics. Tau leptons offer a fertile area of research both for standard model tests and for new physics searches. Because the bulk of tau leptons decay into hadrons, it is challenging to distinguish them from ordinary hadron jets. Neural networks are well suited to this type of difficult classification problem. In this case, software simulations show that an efficiency of 15% with a rejection factor of 100 could be obtained. The input to the network is a 5×5×2 array of calorimeter tower energies surrounding the seed tower of a cluster. If the network’s single output exceeds a tunable threshold, the event is passed to the next stage of the trigger. An existing system based on the Intel ETANN (Electrically Trainable Analog Neural Network) chip was used to implement the tau lepton neural network trigger. The performance of the trigger in current CDF data will be presented.

The training of a Multi-Layer Perceptron (MLP) classifier is considered as a Combinatorial Optimization task and solved using the Reactive Tabu Search (RTS) method. RTS needs only forward passes (no derivatives) and does not require high precision network parameters. TOTEM, a special-purpose VLSI chip, was developed to take advantage of the limited memory and processing requirements of RTS: the final system effects a very close match between hardware and training algorithm. The RTS algorithm and the design of TOTEM are discussed, together with the operational characteristics of the VLSI chip and some preliminary training and generalization tests on triggering tasks.

In the second level triggering for ATLAS “Regions of Interest” (RoIs) are defined in (etha, phi) corresponding to possibly interesting particles. For each RoI physically meaningful parameters are extracted for each subdetector. Based on these parameters a classification of the particle type is made. A feed-forward neural net with 12 input variables, a 6-node intermediate layer, and 4 output nodes has earlier been suggested for this task.

The reported work consists of an implementation of this neural net using a DECPeRLe-1, a Programmable Active Memory (PAM). This is a reconfigurable processor based on Field Programmable Gate Arrays (FPGAs), which has also been used for real-time implementation of feature extraction algorithms for second level triggering.

The implementation is pipelined, runs with a clock of 25 MHz, and uses 0.64 microseconds for one particle classification. Integer arithmetic is used, and the performance is comparable to a floating point version.

Results from a neural-network trigger based on the digital MA16 chip of Siemens are reported. The neural trigger has been applied to data from the WA92 experiment, looking for beauty particles, which have been collected during a run in which a neural trigger module based on Intel’s analog neural chip ETANN operated, as already reported. The MA16 has a precision of 16 bits for input variables, 16 bits for weights, 16 bits for scalar multipliers modifying the transfer function shape, 47 bits for thresholds, 53 bits for internal calculations, 38 bits for the output. Of the latter only 16 bits are used in the MA16 board, as input to the transfer function inplemented on 16-bit addressable EPROM’s. The MA16 board operated at 50 MHz, yielding a response time for a 16 input variable net of 3 μs for a Fisher discriminant (1-layer net) and of 6 μs for a 2-layer net. Results are compared with those previously obtained with the ETANN trigger.

We present the application of a high speed analog neural network chip of the MESA Research Institute in the first level RZ-Trigger of the H1-Experiment at HERA. Using this neural network chip with a settling time of about 50 ns we can reach a first level trigger decision within the HERA bunch-chrossing time of 96 ns.

The new IBM Zero Instruction Set Computer (ZISC) provides a radial basis function neural network. The first generation chip (ZISC036) allows for 64 8-bit inputs, 36 RBF neurons in the middle layer, and up to 16383 possible output categories. Forward processing takes 4μs with a 20MHz clock. Cascading multiple chips increases the number of available RBF’s with no increase in processing time. The chip also executes a learning algorithm. We report on tests of the ZISC with a high energy physics related task.

The selection of samples of events enriched in hadronic decays of the Z into pairs is made difficult by the poor knowledge of jet fragmentation. A Feed-Forward Neural Network can help in maximizing the performance of the classification. This will offer the possibility to test the universality of the Standard Model of the electroweak interaction, and to perform measurements in the sector of nonperturbative QCD.

The efficiency and robustness of neural feed-forward nets in particle searches is studied using the search for the Standard Higgs Boson at LEP-200 as an example. Methods to select the most efficient variables, to define standard cuts, and to recognize significant differences between the training-data sample and a test-data sample are presented. The efficiencies of the neural nets are significantly better than those of standard methods.

The analysis of a particle physics experiment requires the correct selection of the desired multiplicity of charged (pions or kaons) and neutral (photons) particles. Due to fluctuations in hadronic and electromagnetic showers often too many photons are reconstructed by the analysis software. The combination of a feed forward neural network and a genetic algorithm is used to separate photons and fluctuations.

Following an idea derived from techniques used to simulate biological processes I developped a method to optimize selection algorithms in particle physics.

A general method is discussed, the δ-test, which establishes functional dependencies given a table of measurements. The approach is based on calculating conditional probabilities from data densities. Imposing the requirement of continuity of the underlying function the obtained values of the conditional probabilities carry information on the variable dependencies. The power of the method is illustrated on synthetic time-series with different time-lag dependencies and noise levels. For N data points the computational demand is N². Also, the same method is used for estimating nonlinear regression errors and their distributions without performing regression. Comparing the predicted residual errors with those from linear models provides a signal for nonlinearity. The virtue of the method in the context of feedforward neural networks is stressed with respect to preprocessing data and tracking residual errors.