Search results for "integral"
showing 10 items of 902 documents
Glueball masses from ratios of path integrals
2011
By generalizing our previous work on the parity symmetry, the partition function of a Yang-Mills theory is decomposed into a sum of path integrals each giving the contribution from multiplets of states with fixed quantum numbers associated to parity, charge conjugation, translations, rotations and central conjugations. Ratios of path integrals and correlation functions can then be computed with a multi-level Monte Carlo integration scheme whose numerical cost, at a fixed statistical precision and at asymptotically large times, increases power-like with the time extent of the lattice. The strategy is implemented for the SU(3) Yang-Mills theory, and a full-fledged computation of the mass and …
Lattice QCD: A Brief Introduction
2014
A general introduction to lattice QCD is given. The reader is assumed to have some basic familiarity with the path integral representation of quantum field theory. Emphasis is placed on showing that the lattice regularization provides a robust conceptual and computational framework within quantum field theory. The goal is to provide a useful overview, with many references pointing to the following chapters and to freely available lecture series for more in-depth treatments of specifics topics.
Reduced hadronic uncertainty in the determination of $V_{ud}$
2018
We analyze the universal radiative correction $\Delta_R^V$ to neutron and superallowed nuclear $\beta$ decay by expressing the hadronic $\gamma W$-box contribution in terms of a dispersion relation, which we identify as an integral over the first Nachtmann moment of the $\gamma W$ interference structure function $F_3^{(0)}$. By connecting the needed input to existing data on neutrino and antineutrino scattering, we obtain an updated value of $\Delta_R^V = 0.02467(22)$, wherein the hadronic uncertainty is reduced. Assuming other Standard Model theoretical calculations and experimental measurements remain unchanged, we obtain an updated value of $|V_{ud}| = 0.97366(15)$, raising tension with …
Examples for Calculating Path Integrals
2001
We now want to compute the kernel K(b, a) for a few simple Lagrangians. We have already found for the one-dimensional case that $$\displaystyle{ K{\bigl (x_{2},t_{2};x_{1},t_{1}\bigr )} =\int _{ x(t_{1})=x_{1}}^{x(t_{2})=x_{2} }[dx(t)]\,\text{e}^{(\mathrm{i}/\hslash )S} }$$ (19.1) with $$\displaystyle{ S =\int _{ t_{1}}^{t_{2} }dt\,L(x,\dot{x};t)\;. }$$ First we consider a free particle, $$\displaystyle{ L = m\dot{x}^{2}/2\;, }$$ (19.2) and represent an arbitrary path in the form, $$\displaystyle{ x(t) =\bar{ x}(t) + y(t)\;. }$$ (19.3) Here, \(\bar{x}(t)\) is the actual classical path, i.e., solution to the Euler–Lagrange equation: $$\displaystyle{ \frac{\partial L} {\partial x}\Big\vert _{…
A Rainich-like approach to the Killing-Yano tensors
2002
The Rainich problem for the Killing-Yano tensors posed by Collinson \cite{col} is solved. In intermediate steps, we first obtain the necessary and sufficient conditions for a 2+2 almost-product structure to determine the principal 2--planes of a skew-symmetric Killing-Yano tensor and then we give the additional conditions on a symmetric Killing tensor for it to be the square of a Killing-Yano tensor.We also analyze a similar problem for the conformal Killing-Yano and the conformal Killing tensors. Our results show that, in both cases, the principal 2--planes define a maxwellian structure. The associated Maxwell fields are obtained and we outline how this approach is of interest in studying …
A diffusion Monte Carlo study of small para-Hydrogen clusters
2007
Abstract An improved Monte Carlo diffusion model is used to calculate the ground state energies and chemical potentials of parahydrogen clusters of three to forty molecules, using two different p-H2-p-H2 interactions. The improvement is due to three-body correlations in the importance sampling, to the time step adjustment and to a better estimation of statistical errors. In contrast to path-integral Monte Carlo results, this method predicts no magic clusters other than that with thirteen molecules.
Monte Carlo calculation of dose rate distributions around the Walstam CDC.K-type137Cs sources
2001
Basic dosimetric data for the Walstam CDC.K-type low dose rate 137Cs sources in water have been calculated using Monte Carlo techniques. These sources, CDC.K1 -K3 and CDC.K4, are widely used in a range of applicators and moulds for the treatment of intracavitary and superficial cancers. Our purpose is to improve existing data about these sources using the Monte Carlo simulation code GEANT3. Absolute dose rate distributions in water have been calculated around these sources and are presented as conventional 2D Cartesian look-up tables. Also the AAPM Task Group 43 formalism for dose calculation has been applied. The calculated dose rate constant for the CDC.K1-K3 source is A = 1.106 +/- 0.001…
On a Class of Feynman Integrals Evaluating to Iterated Integrals of Modular Forms
2019
In this talk we discuss a class of Feynman integrals, which can be expressed to all orders in the dimensional regularisation parameter as iterated integrals of modular forms. We review the mathematical prerequisites related to elliptic curves and modular forms. Feynman integrals, which evaluate to iterated integrals of modular forms go beyond the class of multiple polylogarithms. Nevertheless, we may bring for all examples considered the associated system of differential equations by a non-algebraic transformation to an \(\varepsilon \)-form, which makes a solution in terms of iterated integrals immediate.
Nilpotence of orbits under monodromy and the length of Melnikov functions
2021
Abstract Let F ∈ ℂ [ x , y ] be a polynomial, γ ( z ) ∈ π 1 ( F − 1 ( z ) ) a non-trivial cycle in a generic fiber of F and let ω be a polynomial 1-form, thus defining a polynomial deformation d F + e ω = 0 of the integrable foliation given by F . We study different invariants: the orbit depth k , the nilpotence class n , the derivative length d associated with the couple ( F , γ ) . These invariants bind the length l of the first nonzero Melnikov function of the deformation d F + e ω along γ . We analyze the variation of the aforementioned invariants in a simple but informative example, in which the polynomial F is defined by a product of four lines. We study as well the relation of this b…
Exact solutions for the mutual inductance of circular coils and elliptic coils
2012
An exact solution is presented for the mutual inductance between general noncoaxial thin circular and elliptic coils with parallel axes. The thin coil solution is given as an angular integral of an elliptic integral expression. In addition, for the coaxial case, an exact solution is given for the mutual inductance of a thick circular coil and a thick elliptic coil. The elliptic coil is such that the coil thickness is the same along both elliptic semi-axes. The thick coil solution is given as an integral of an expression involving Bessel and Struve functions. Extensive numerical results for sample geometries are given for both solutions, which are cross checked against each other in the limi…