Scalar 1-loop Feynman integrals as meromorphic functions in space-time dimension d, II: Special kinematics

Based on the method developed in [K.~H.~Phan and T.~Riemann, Phys.\ Lett.\ B {\bf 791} (2019) 257], detailed analytic results for scalar one-loop two-, three-, four-point integrals in general $d$-dimension are presented in this paper. The calculations are considered all external kinematic configurations and internal mass assignments. Analytic formulas are expressed in terms of generalized hypergeometric series such as Gauss $_2F_1$, Appell $F_1$ and Lauricella $F_S$ functions.Comment: 31 pages, references are added, typos are correcte

Scalar 1-loop Feynman integrals as meromorphic functions in space-time dimension d

The long-standing problem of representing the general massive one-loop Feynman integral as a meromorphic function of the space-time dimension $d$ has been solved for the basis of scalar one- to four-point functions with indices one. In 2003 the solution of difference equations in the space-time dimension allowed to determine the necessary classes of special functions: self-energies need ordinary logarithms and Gauss hypergeometric functions $_2F_1$, vertices need additionally Kamp\'{e} de F\'{e}riet-Appell functions $F_1$, and box integrals also Lauricella-Saran functions $F_S$. In this study, alternative recursive Mellin-Barnes representations are used for the representation of $n$-point functions in terms of $(n-1)$-point functions. The approach enabled the first derivation of explicit solutions for the Feynman integrals at arbitrary kinematics. In this article, we scetch our new representations for the general massive vertex and box Feynman integrals and derive a numerical approach for the necessary Appell functions $F_1$ and Saran functions $F_S$ at arbitrary kinematical arguments.Comment: 9 pages, 3 table

Scalar one-loop vertex integrals as meromorphic functions of space-time dimension d

Representations are derived for the basic scalar one-loop vertex Feynman integrals as meromorphic functions of the space-time dimension $d$ in terms of (generalized) hypergeometric functions $_2F_1$ and $F_1$. Values at asymptotic or exceptional kinematic points as well as expansions around the singular points at $d=4+2n$, $n$ non-negative integers, may be derived from the representations easily. The Feynman integrals studied here may be used as building blocks for the calculation of one-loop and higher-loop scalar and tensor amplitudes. From the recursion relation presented, higher n-point functions may be obtained in a straightforward manner.Comment: 9 pages, talk presented by TR at workshop "Matter To The Deepest", XLI International Conference on Recent Developments in Physics of Fundamental Interactions (MTTD 2017), September 3-8, 2017, Podlesice, Poland, to appear in the proceeding

General ε\varepsilon-representation for scalar one-loop Feynman integrals

A systematic study of the scalar one-loop two-, three-, and four-point Feynman integrals is performed. We consider all cases of mass assignment and external invariants and derive closed expressions in arbitrary space-time dimension in terms of higher transcendental functions. The integrals play a role as building blocks in general higher-loop or multi-leg processes. We also perform numerical checks of the calculations using AMBRE/MB and LoopTools/FF.Comment: 5 pages Latex,Contribution to the Proceedings of QCD 15, Montpellier, July 201

Thermodynamics for Trajectories of a Mass Point

On the basis of information theory, a new formalism of classical non-relativistic mechanics of a mass point is proposed. The particle trajectories of a general dynamical system defined on an (1+n)-dimensional smooth manifold are treated geometrically as dynamical variables. Statistical mechanics of particle trajectories are constructed in a classical manner. Thermodynamic variables are introduced through a partition function based on a canonical ensemble of trajectories. Within this theoretical framework, classical mechanics can be interpreted as an equilibrium state of statistical mechanics. The relationships between classical and quantum mechanics are discussed from this statistical mechanical viewpoint. The maximum entropy principle is shown to provide a unified view of both classical and quantum mechanics.Comment: 22 pages, 1 figur

One-loop formulas for H→ZνlνˉlH\rightarrow Z \nu_l\bar{\nu}_l for l=e,μ,τl = e,\mu, \tau in 't Hooft-Veltman gauge

In this paper, we present analytical results for one-loop contributing to the decay processes $H\rightarrow Z \nu_l\bar{\nu}_l$ (for $l = e, \mu, \tau$). The calculations are performed within the Standard Model framework in 't Hooft-Veltman gauge. One-loop form factors are then written in terms of scalar one-loop functions in the standard notations of {\tt LoopTools}. As a result, one-loop decay rates for the decay channels can be evaluated numerically by using the package. Furthermore, we analyse the signals of $H\rightarrow Z \nu_l\bar{\nu}_l$ via the production processes $e^-e^+ \rightarrow ZH^* \rightarrow Z (H^* \rightarrow Z \nu_l\bar{\nu}_l)$ including the initial beam polarizations at future lepton collider. The Standard Model background such as the processes $e^-e^+ \rightarrow \nu_l\bar{\nu}_l ZZ$ are also examined in this study. In numerical results, we find that one-loop corrections are about $10\%$ contributions to the decay rates. They are sizeable contributions and should be taken into account at future colliders. We show that the signals $H\rightarrow Z\nu_l\bar{\nu}_l$ are clearly visible at center-of-mass energy $\sqrt{s}=250$ GeV and it is hard to probe at higher-energy regions due to the dominant of the background.Comment: 25 page

One-loop contributions for A0→ℓℓˉVA^0 \rightarrow \ell \bar{\ell} V with ℓ≡e,μ\ell \equiv e, \mu and V≡γ,ZV\equiv \gamma, Z in Higgs Extensions of the Standard Model

We present one-loop formulas for the decay of CP-odd Higgs $A^0 \rightarrow \ell \bar{\ell} V$ with $\ell \equiv e, \mu$ and $V\equiv \gamma, Z$ in Higgs Extensions of the Standard Model, considering two higgs doublet model with a complex (and real) scalar, two higgs doublet model as well as triplet higgs model. Analytic results for one-loop amplitudes are expressed in terms of Passarino-Veltman functions following the standard notations of {\tt LoopTools}. As a result, physical results can be generated numerically by using the package. In phenomenological results, the total decay widths and the differential decay rates with respect to the invariant mass of lepton pair are analyzed for two typical models such as two higgs doublet model and triplet higgs model.Comment: 35 page

Tensor Two-loop Self-energy Integrals

Based on the technique for evaluating tensor two-loop self-energy  integrals in parallel and orthogonal spaces \cite{kreimer94}, we present a new UV subtraction procedure for  a subset of integrals which contains overall UV-divergence. The numerical and analytical results of the sun-rise integral are discussed in comparison with other methods