15948639
S-bases as a tool to solve reduction problems for Feynman integrals
We suggest a mathematical definition of the notion of master integrals and present a brief review of algorithmic methods to solve reduction problems for Feynman integrals based on integration by parts relations. In particular, we discuss a recently suggested reduction algorithm which uses Grobner bases. New results obtained with its help for a family of three-loop Feynman integrals are outlined.
119320601
The Number of Master Integrals is Finite
For a fixed Feynman graph one can consider Feynman integrals with all possible powers of propagators and try to reduce them, by linear relations, to a finite subset of integrals, the so-called master integrals. Up to now, there are numerous examples of reduction procedures resulting in a finite number of master integrals for various families of Feynman integrals. However, up to now it was just an empirical fact that the reduction procedure results in a finite number of irreducible integrals. It this paper we prove that the number of master integrals is always finite.
55043207
DESIGN OF THE NUCLOTRON BOOSTER IN THE NICA PROJECT
NICA is the new complex being constructed on the JINR aimed to provide collider experiments with ions up to uranium at energy of 3.5x3.5 GeV/u. The NICA layout includes Electron String Ion Source, 6.2 Mev/u linac, 600 MeV/u booster synchrotron, upgraded Nuclotron and ion collider with average luminosity of 10 27 cm -2 s -1 . The main goals of the Booster are the following: accumulation of 4·10 9 Au 32+ ions; acceleration of the heavy ions up to energy required for effective stripping; forming of the required beam emittance with electron cooling system. The present layout makes it possible to place the Booster having 211 m circumference and four fold symmetry lattice inside the yoke of the Synchrophasotron (shut down in 2002). The features of this booster, the requirement to the main synchrotron systems and their parameters are presented in this paper.
2365986
FIESTA 2: Parallelizeable multiloop numerical calculations
Abstract The program FIESTA has been completely rewritten. Now it can be used not only as a tool to evaluate Feynman integrals numerically, but also to expand Feynman integrals automatically in limits of momenta and masses with the use of sector decompositions and Mellin–Barnes representations. Other important improvements to the code are complete parallelization (even to multiple computers), high-precision arithmetics (allowing to calculate integrals which were undoable before), new integrators, Speer sectors as a strategy, the possibility to evaluate more general parametric integrals. Program summary Program title: FIESTA 2 Catalogue identifier: AECP_v2_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AECP_v2_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: GNU GPL version 2 No. of lines in distributed program, including test data, etc.: 39 783 No. of bytes in distributed program, including test data, etc.: 6 154 515 Distribution format: tar.gz Programming language: Wolfram Mathematica 6.0 (or higher) and C Computer: From a desktop PC to a supercomputer Operating system: Unix, Linux, Windows, Mac OS X Has the code been vectorised or parallelized?: Yes, the code has been parallelized for use on multi-kernel computers as well as clusters via Mathlink over the TCP/IP protocol. The program can work successfully with a single processor, however, it is ready to work in a parallel environment and the use of multi-kernel processor and multi-processor computers significantly speeds up the calculation; on clusters the calculation speed can be improved even further. RAM: Depends on the complexity of the problem Classification: 4.4, 4.12, 5, 6.5 Catalogue identifier of previous version: AECP_v1_0 Journal reference of previous version: Comput. Phys. Comm. 180 (2009) 735 External routines: QLink [1], Cuba library [2], MPFR [3] Does the new version supersede the previous version?: Yes Nature of problem: The sector decomposition approach to evaluating Feynman integrals falls apart into the sector decomposition itself, where one has to minimize the number of sectors; the pole resolution and epsilon expansion; and the numerical integration of the resulting expression. Solution method: The sector decomposition is based on a new strategy as well as on classical strategies such as Speer sectors. The sector decomposition, pole resolution and epsilon-expansion are performed in Wolfram Mathematica 6.0 or, preferably, 7.0 (enabling parallelization) [4]. The data is stored on hard disk via a special program, QLink [1]. The expression for integration is passed to the C-part of the code, that parses the string and performs the integration by one of the algorithms in the Cuba library package [2]. This part of the evaluation is perfectly parallelized on multi-kernel computers. Reasons for new version: 1. The first version of FIESTA had problems related to numerical instability, so for some classes of integrals it could not produce a result. 2. The sector decomposition method can be applied not only for integral calculation. Summary of revisions: 1. New integrator library is used. 2. New methods to deal with numerical instability (MPFR library). 3. Parallelization in Mathematica . 4. Parallelization on multiple computers via TCP-IP. 5. New sector decomposition strategy (Speer sectors). 6. Possibility of using FIESTA to for integral expansion. 7. Possibility of using FIESTA to discover poles in d. 8. New negative terms resolution strategies. Restrictions: The complexity of the problem is mostly restricted by CPU time required to perform the evaluation of the integral Running time: Depends on the complexity of the problem References: [1] http://qlink08.sourceforge.net , open source. [2] http://www.feynarts.de/cuba/ , open source. [3] http://www.mpfr.org/ , open source. [4] http://www.wolfram.com/products/mathematica/index.html .
18145511
On the reduction of Feynman integrals to master integrals
The reduction of Feynman integrals to master integrals is an algebraic problem that requires algorithmic approaches at the modern level of calculations. Straightforward applications of the classical Buchberger algorithm to construct Grobner bases seem to be inefficient. An essential modification designed especially for this problem has been w orked out. It has been already applied in two- and three-loop calculations. We are also suggesting to combine our method with the Laporta’s algorithm.
1669343
Applying Gröbner bases to solve reduction problems for Feynman integrals
We describe how Grobner bases can be used to solve the reduction problem for Feynman integrals, i.e. to construct an algorithm that provides the possibility to express a Feynman integral of a given family as a linear combination of some master integrals. Our approach is based on a generalized Buchberger algorithm for constructing Grobner-type bases associated with polynomials of shift operators. We illustrate it through various examples of reduction problems for families of one- and two-loop Feynman integrals. We also solve the reduction problem for a family of integrals contributing to the three-loop static quark potential.
37716395
Neutrino-Modulino Mixing
We suggest an existence of light singlet fermion, S, which interacts with observable matter only via Planck mass suppressed interaction: {approximately}m{sub 3/2}/M{sub P}, where m{sub 3/2} is the supergravity gravitino mass. If the mass of the singlet equals {approximately}m{sup 2}{sub 3/2}/M{sub P}, then {nu}{sub e}{r_arrow}S resonance conversion solves the solar neutrino problem or leads to observable effects. The {nu}S mixing changes supernova neutrino fluxes and has an impact on the primordial nucleosynthesis. The singlet S can originate as the supersymmetric partner of the moduli fields in supergravity or low energy effective theory stemming from superstrings. The {nu}S mixing may be accompanied by observable R-parity breaking effects. {copyright} {ital 1997} {ital The American Physical Society}
15323094
Neutrino physics: Open theoretical questions
We know that neutrino mass and mixing provide a window to physics beyond the Standard Model. Now this window is open, at least partly. And the questions are: what do we see, which kind of new physics, and how far "beyond"? I summarize the present knowledge of neutrino mass and mixing, and then formulate the main open questions. Following the bottom-up approach, properties of the neutrino mass matrix are considered. Then different possible ways to uncover the underlying physics are discussed. Some results along the line of seesaw, GUT and SUSY GUT are reviewed.
18100102
Reconstructing neutrino mass spectrum
Reconstruction of the neutrino mass spectrum and lepton mixing is one of the fundamental problems of particle physics. In this connection we consider two central topics: (i) the origin of large lepton mixing, (ii) possible existence of new (sterile) neutrino states. We discuss also possible relation between large mixing and existence of sterile neutrinos.
54022118
Phenomenology of maximal and near maximal lepton mixing
ueu,0.3. We show that both probabilities and observables depend on e quadratically when effects are due to vacuum oscillations and they depend on e linearly if matter effects dominate. The most important information on n e mixing comes from solar neutrino experiments. We find that the global analysis of solar neutrino data allows maximal mixing with confidence level better than 99% for 10 28 eV 2 D4s ~99.995% CL! We suggest ways to measure e in future experiments. The observable that is most sensitive to e is the rate @NC#/@CC# in combination with the day-night asymmetry in the SNO detector. With theoretical and statistical uncertainties, the expected accuracy after 5 years is De;0.07. We also discuss the effects of maximal and near-maximal n e mixing in atmospheric neutrinos, supernova neutrinos, and neutrinoless double beta decay.
118412914
Describing neutrino oscillations in matter with Magnus expansion
Abstract We present new formalism for description of the neutrino oscillations in matter with varying density. The formalism is based on the Magnus expansion and has a virtue that the unitarity of the S-matrix is maintained in each order of perturbation theory. We show that the Magnus expansion provides better convergence of series: the restoration of unitarity leads to smaller deviations from the exact results especially in the regions of large transition probabilities. Various expansions are obtained depending on a basis of neutrino states and a way one splits the Hamiltonian into the self-commuting and non-commuting parts. In particular, we develop the Magnus expansion for the adiabatic perturbation theory which gives the best approximation. We apply the formalism to the neutrino oscillations in matter of the Earth and show that for the solar oscillation parameters the second order Magnus adiabatic expansion has better than 1% accuracy for all energies and trajectories. For the atmospheric Δ m 2 and small 1–3 mixing the approximation works well ( sin 2 θ 13 = 0.01 ) outside the resonance region 2.7–8 GeV.