The main objective of high energy physics is to unravel the nature of fundamental particles and their interactions. For this, precise comparisons of theoretical predictions to experimental results are necessary. The next energy frontier, the TeV scale is starting to being probed at the Tevatron collider. In a few years experiments at the CERN Large Hadron Collider (LHC) as well as at the planned future linear colliders will provide lots of new information about the physics at this energy scale. One will be able to test even further the standard model and in particular to probe the yet unknown scalar sector. One will get clues about the symmetry breaking mechanism and eventually find signals for new physics, in the form of supersymmetry, extra dimensions or any other unexpected signal.
Theoretical predictions for physical observables in the framework of the standard model are based on the computation of Feynman diagrams, a perturbative approach for calculations of high energy processes. Although the method is well standardized, the actual computation of processes involving many particles in the final states or higher order calculations of simple processes become rapidly cumbersome and lengthy. The task of computing the large number of processes that could constitute the signals for new physics or the standard model processes that appear as backgrounds to these processes require the use of advanced computational techniques. To this end, a vast research program was undertaken at many institutions around the world for automating as much as possible the computation of high energy processes [l]. The two leading teams in that field are the Japanese GRACE collaboration and the Russian CompHEP collaboration which have both developed independently a software for automatic calculations of Feynman diagrams at tree-level [2, 3].
As one reaches the TeV scale, the use of software packages for computation of high- energy processes become even more relevant. On the one hand, multiparticle processes, particularly those involving Quantum ChromoDynamics (QCD) , will play an important role. For example at the future LHC, the production of four, and even five or six jets will not be marginal. Beside providing a refined probe of the dynamics of color, such QCD processes constitute a background to the search for new particles. Indeed, the search for many of these new particles at hadronic colliders often relies on signatures based on cascade decays. The latter end up with final states involving a large jet multiplicity. On the other hand, the exploitation of precision measurements aimed at TeV hadronic colliders requires the account of higher-order, especially QCD, corrections in standard and beyond standard model predictions. Higher-order electroweak corrections become also relevant especially for the high luminosity electron-positron machines that are currently under discussion. The main physics objectives of the present proposal concern phenomenological studies within the Standard model and its extensions both at colliders (Tevatron, LHC and linear collider) and in astro-particle experiments, in particular :
In order to reach these physics objectives, progress in development of software for automatic calculation will be pursued, new methods and new programs will be created in particular concerning :
To reach our objectives, it is essential to bring together a group of theoreticians, experimentalists as well as specialists of software development for high-energy physics. Indeed, one of the strengths of this collaboration is that it includes the two teams that have developed independently a software package for automatic calculations of Feynman diagrams, CompHEP[3]and GRACE[2]. The two teams have followed a very different approach. CompHEP relies on the trace technique for evaluation of matrix elements and is basically a tree-level program while GRACE is based on the helicity method and aims at a systematic, automated computation of loop corrections [l4]. The complete independence of the two approaches is a desirable feature and this collaboration will pursue in that direction. Indeed one of the difficulties in performing these complex computations is to ensure the correctness of the numerical results obtained. Having completely independent checks of a given calculation certainly helps having confidence in the results.
While the development of specific tools for hadron colliders has already started [15] and event generators for given processes are available [16, 17], the collaboration will strive for a fully automatic approach to computation of processes starting from a given model (Standard model or beyond) all the way to the complete simulation of events in a detector. At the same time, the high precision that can be reached at colliders motivates taking into account higher order effects. GRACE has recently completed a full one-loop electroweak computation [18], and will continue to strive for automated higher order loop level calculations. The CompHEP approach to loop corrections is directed towards the incorporation of large effects in selected processes.
Beside the phenomenological studies of high energy physics at colliders which motivated their development initially, these computing tools start to be applied successfully in a new direction, namely astroparticle physics. In the last year, a French-Russian group (micrOMEGAs collaboration) has developed software based on CompHEP for the calculation of the relic density of dark matter in supersymmetric models [19]. Important constraints on supersymmetric models can be obtained from dark matter searches so these experiments are complementary to the ones performed at colliders searching for signals of supersymmetric extensions of the standard model. The development of this specific software for astroparticle experiments is underway.
The physical issues to be tackled drive the planned developments of software and the progresses towards either goal are intimately interwoven in practice. For sake of synthesis, in this proposal the technical issues which the collaboration intends to address regarding the developments of automated calculations will be presented first. Then, a selection of physically relevant items, whose studies require or benefit from the use of automated calculations, and which the collaboration wishes to study, will be discussed.
1. the one loop correction to the N-leg partonic subprocesses,
2. the tree level N+1-leg partonic subprocesses,
3. an efficient method to merge the real and virtual contributions and cancel the infra-red singularities,
4. a parton shower generator at Next-to-Leading Logarithmic (NLL) accuracy
An extended effort has been devoted to deal with items 2 and 3 already, whereas a substantial progress would still be welcome regarding items 1 and 4. A new general algorithm for the recursive computation of one-loop tensor integrals has been recently developed by the LAPTH team [34] and applied to the calculation of the one-loop, six-leg scalar amplitude [35]. In order to tackle item 1 in the case of external leg with non-zero spins, the application of this algorithm has to be improved in order to circumvent the proliferation of terms coming from the tensor reduction. The Japanese team is addressing item 4 with the development of a NLL parton shower. Bringing together these expertises will allow to complete this program from 1) to 4) and apply it to the study of processes of particular physical interest.
Among the large numbers of relevant QCD processes, the associated production of a boson vector pair plus a jet (P+P→γ+γ+jet, Z+Z+jet, W++W-+jet) is of great importance for the Higgs boson search. Partial results are known already for photon pairs but not for massive vector boson pairs. An even more ambitious project would be the four-jet production at NLO accuracy. Phenomenological counterparts of these calculations are discussed in section 2.1.3.
The GRACE collaboration has started as well to tackle the problem of one loop calculations for 2→4 processes. Although in principle the same formalism that has been used for 2→2 or 2→3 processes can be applied here, the complexity of the problem (thousands of loop diagrams) makes it necessary to improve the method used in order to greatly reduce the size of computations, so that they can be done on a realistic time scale. In the present version part of the loop calculation is done with symbolic manipulation programs such as REDUCE [40] and FORM [41]. The matrix elements thus computed are huge and one quickly reaches the limit of the computer as one tries to go beyond processes with three particles in the final state. The collaboration considers that a completely numerical approach to 3 and 4 points loop integrals will in the end be necessary to complete the program of computing 2→4 processes in the standard model.
Finally, the GRACE collaboration has also developed a program to calculate any one-loop processes in the MSSM, being thus the only program able to achieve this. However, before releasing the program on a wide scale, one needs to ensure the correctness of the results. The package for loop integrals has been extensively tested in the standard model case and the vertices of the supersymmetric models have also been tested in nearly half a million tree-level processes. Nevertheless, the French and Japanese teams consider it necessary to implement the non-linear gauge fixing in the MSSM, just as was done for the standard model.
One such typical case happens when a quantity gets a contribution from a new mechanism appearing at a loop order. In this case, part of the higher order contributions which are known to be large can be accounted for in terms of effective parameters and vertices, thus allowing providing “improved tree-level calculations”. A notable example is the Higgs boson mass in supersymmetric models. Its tree-level contribution is bound to be much below the present direct experimental limit. Yet this mass receives large contributions from SUSY breaking loop effects in the top quark and squark sector, which are known at two- loops already [42]. Another typical example is the QCD corrections to the Higgs width or QCD corrections to Higgs production [43, 44]. Even in a tree-level calculation one then has to device ways to incorporate these higher order effects. In one specific case, an effective potential for the scalar sector of a two-Higgs model has been written [45]. The higher order corrections to the potential can be parameterized with just a handful of parameters thus taking into account higher order effects in tree-level calculations.
One of the main goals of the CompHEP collaboration in the near future is to incorporate the leading one-loop corrections by defining effective vertices for specific interactions. Note that this requires a major rewriting of the symbolic manipulation part of CompHEP. The association of the French experts of higher-order corrections with the Russian experts of automated calculations will be essential to the successful completion of this project.
In the next few years this collaboration will work on the following items:
As concerns the full one-loop electroweak processes, as was already mentioned, the first results of a full one-loop electroweak 2→2 process were obtained by this collaboration. After this success, we can now pursue the study of many other processes relevant for linear colliders. The high precision of the measurements that will be performed at these high-luminosity machines makes it necessary to make higher-order theoretical predictions. Among various processes we mention, e+e→t h, a process that is sensitive to the Higgs-top quark coupling or e+e-→WWγ which is a sensitive probe of the self-couplings of gauge bosons (couplings of vector bosons). The probing of these couplings is relevant for our understanding of the symmetry breaking mechanism.
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-- DenisPerretGallix - 15 Apr 2005
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