After a brief review of the electroweak radiative corrections to gauge-boson self-energies, otherwise known as the direct and oblique corrections, a tool for calculation of the oblique parameters is presented. This tool, named OPUCEM, brings together formulas from multiple physics models and provides an error-checking machinery to improve reliability of numerical results. It also sets a novel example for an "open-formula" concept, which is an attempt to improve the reliability and reproducibility of computations in scientific publications by encouraging the authors to open-source their numerical calculation programs. Finally, we demonstrate the use of OPUCEM in two detailed case studies related to the fourth Standard Model family. The first is a generic fourth family study to find relations between the parameters compatible with the EW precision data and the second is the particular study of the Flavor Democracy predictions for both Dirac and Majorana-type neutrinos.
Deep Dive into OPUCEM: A Library with Error Checking Mechanism for Computing Oblique Parameters.
After a brief review of the electroweak radiative corrections to gauge-boson self-energies, otherwise known as the direct and oblique corrections, a tool for calculation of the oblique parameters is presented. This tool, named OPUCEM, brings together formulas from multiple physics models and provides an error-checking machinery to improve reliability of numerical results. It also sets a novel example for an “open-formula” concept, which is an attempt to improve the reliability and reproducibility of computations in scientific publications by encouraging the authors to open-source their numerical calculation programs. Finally, we demonstrate the use of OPUCEM in two detailed case studies related to the fourth Standard Model family. The first is a generic fourth family study to find relations between the parameters compatible with the EW precision data and the second is the particular study of the Flavor Democracy predictions for both Dirac and Majorana-type neutrinos.
The categorization of the electroweak (EW) corrections based on their contribution types dates back to a study of photon propagated four-fermion processes [1]. The corrections to vertices, box diagrams and bremsstrahlung diagrams were all considered as "Direct" whereas the propagator corrections due to vacuum polarization effects were all named as "Oblique" since these participate to the computations in an indirect manner [2].
The EW precision data collected over the last few decades by various particle physics experiments have often been used to constrain many new models of particle interactions. They are particularly useful in checking the allowed parameter space of a given model through its contributions especially to the vacuum polarization corrections to the boson propagators. The main oblique parameters are usually denoted by letters S, T , U and the auxiliaries with letters V , W , Y [3]. As an example, the S parameter estimates the size of the new fermion sector and the T parameter measures the isospin symmetry violation, i.e. the split between the masses of the new up and down-type fermions. The Standard Model is defined by the values S = T = U = 0 for a given top quark and Higgs boson mass.
Together with the detailed review by Peskin and Takeuchi [2], a number of papers were published, calculating the contri-bution of a given model to the oblique parameters. To name a few, the estimation of the number of fermion families and neutral gauge bosons [4], the validity consideration of the Higgsless models [5], and the investigation of the Majorana nature of the neutrinos [6] can be cited. However, a number of such publications suffer from unusual notations with typos in formulas, and errors arising due to utilization of approximations instead of exact calculations (with assumptions such as m H » m Z ) or in some cases from unguarded remarks such as “heavy” for the new fermions.
The goal of this work is two fold: the first is to present a library to compute the oblique parameters S, T and U both with exact one-loop calculations and with some well-defined approximations for a number of models and the second is to scan the available parameter space for the fourth family models. The comparisons between exact and approximate computations, and amongst formulas from different papers provide an error checking machinery which improves the enduser reliability. Implemented in C/C++ languages, we call this library OPUCEM, which stands for Oblique Parameters Using C with Error-checking Machinery. The OPUCEM package consisting of the library and a set of example driver and presentation functions, which are discussed in this manuscript, are publicly available [7].
The next section describes the technical details of the library implementation. Then the following sections are on detailed physics studies demonstrating the use cases of OP-UCEM. Section 3 uses the OPUCEM library to investigate the plausibiliy of a generic fourth Standard Model (SM) family (SM4). Section 4 focuses on the implications of the EW data from the viewpoint of the flavor democracy (FD) hypothesis, which provides a principle theoretical motivation for the potential existence of a fourth SM family. These case studies deal with defining the parameter regions favored by the data and define a set of benchmarking points for the SM4. In both sections, Dirac and Majorana cases of the fourth SM family neutrino are investigated separately. Finally in Section 5, we present our concluding remarks.
The OPUCEM library mainly consists of a header file opucem.h declaring the available function prototypes and opucem.c implementing them. The functions are grouped by the relevant physics cases such as Majorana neutrinos or Higgs bosons. In each case, internal comments are used to document the source code, indicating the reference paper for each of the formula and the nature of the calculations (e.g. exact 1-loop calculations, or approximations valid under certain assumptions such as the new fermions being much heavier than the Zboson). Compilation of the library is straightforward, however a makefile is provided as is customary. The makefile also features additional targets to produce example command-line and graphical-user-interface applications that make use of the library in studying the fermions of a fourth SM family [8,9,10].
One of the primary goals of the implementation is portabil-ity, since we consider portability an auxiliary measure of the reliability of the code. To facilitate this goal, the initial implementation was done in pure C, with the code having no dependencies or requirements beyond a standards-compliant C compiler. Since certain formulas make use of complex numbers, we made use of the complex type found in the C99 standard. However, starting with version 00-00-03, the C99 complex type was dropped in favor of the std::complex template in the C++ standard template library. The main motivation for this change was to provide an easy roadma
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