The scalar pion form factor in two-flavor lattice QCD

We calculate the scalar form factor of the pion using two dynamical flavors of non-perturbatively $\mathcal{O}(a)$-improved Wilson fermions, including both the connected and the disconnected contribution to the relevant correlation functions. We employ the calculation of all-to-all propagators using stochastic sources and a generalized hopping parameter expansion. From the form factor data at vanishing momentum transfer, $Q^2=0$, and two non-vanishing $Q^2$ we obtain an estimate for the scalar radius \left^\pi_{_{\rm S}} of the pion at one value of the lattice spacing and for five different pion masses. Using Chiral Perturbation Theory at next-to-leading order, we find \left^\pi_{_{\rm S}}=0.635\pm0.016 fm$^2$ at the physical pion mass (statistical error only). This is in good agreement with the phenomenological estimate from $\pi\pi$-scattering. The inclusion of the disconnected contribution is essential for achieving this level of agreement.Comment: 15 pages, 10 pdf figures, uses revtex4-1; version to appear in PR

Towards non-linear quadrature formulae

Prompted by an observation about the integral of exponential functions of the form $f(x)=\lambda\mathrm{e}^{\alpha x}$, we investigate the possibility to exactly integrate families of functions generated from a given function by scaling or by affine transformations of the argument using nonlinear generalizations of quadrature formulae. The main result of this paper is that such formulae can be explicitly constructed for a wide class of functions, and have the same accuracy as Newton-Cotes formulae based on the same nodes. We also show how Newton-Cotes formulae emerge as the linear case of our general formalism, and demonstrate the usefulness of the nonlinear formulae in the context of the Pad\'e-Laplace method of exponential analysis.Comment: 14 pages, 3 figures (24 pdf files

Pad\'e and Pad\'e-Laplace Methods for masses and matrix elements

The problem of having to reconstruct the decay rates and corresponding amplitudes of the single-exponential components of a noisy multi-exponential signal is common in many other areas of physics and engineering besides lattice field theory, and it can be helpful to study the methods devised and used for that purpose in those contexts in order to get a better handle on the problem of extracting masses and matrix elements from lattice correlators. Here we consider the use of Pad\'e and Pad\'e-Laplace methods, which have found wide use in laser fluorescence spectroscopy and beyond, emphasizing the importance of using robust Pad\'e approximants to avoid spurious poles. To facilitate the accurate evaluation of the Laplace transform required for the Pad\'e-Laplace method, we also present a novel approach to the numerical quadrature of multi-exponential functions.Comment: 6 pages, 4 PDF figures; poster presented at 39th International Symposium on Lattice Field Theory (Lattice2022), 8-13 August, 2022, Bonn, German

The pion scalar form factor with Nf=2+1N_f=2+1 Wilson fermions

We report preliminary results from an analysis of the pion scalar form factor computed on a set of the $\mathrm{tr}[M]=\mathrm{const}$ CLS gauge ensembles with $N_f=2+1$ Wilson Clover-improved sea quarks. The calculations are carried out for light quarks masses corresponding to $M_\pi\approx 0.130\mathrm{MeV} \ldots 350\mathrm{MeV}$, four values of the lattice spacing $a\approx0.049\mathrm{fm}\ldots0.086\mathrm{fm}$ and a large range of physical volumes. A fine-grained momentum resolution is achieved by allowing for non-vanishing sink momenta and by including two particularly large and fine boxes close to physical quark masses (i.e. $T\times L^3 = 192\times 96^3$, $M_\pi \le 172\mathrm{MeV}$, $a \le 0.064\mathrm{fm}$). The pertinent quark-disconnected contributions have been computed to high precision using a scheme combining 1.) the one-end trick on stochastic volume sources for the computation of differences between two quark flavors with 2.) the hopping parameter expansion and hierarchical probing to evaluate the loops for the heaviest, single quark flavor.Comment: 11 pages, 5 figures, Talk presented at the 40th International Symposium on Lattice Field Theory (LATTICE2023), July 31st - August 4th, 2023, Fermilab, Batavia, Illinois, US

Open Source Software and the “Private-Collective” Innovation Model: Issues for Organization Science

Currently two models of innovation are prevalent in organization science. The "private investment" model assumes returns to the innovator results from private goods and efficient regimes of intellectual property protection. The "collective action" model assumes that under conditions of market failure, innovators collaborate in order to produce a public good. The phenomenon of open source software development shows that users program to solve their own as well as shared technical problems, and freely reveal their innovations without appropriating private returns from selling the software. In this paper we propose that open source software development is an exemplar of a compound model of innovation that contains elements of both the private investment and the collective action models. We describe a new set of research questions this model raises for scholars in organization science. We offer some details regarding the types of data available for open source projects in order to ease access for researchers who are unfamiliar with these, and als

CROSSROADS—Identifying Viable “Need–Solution Pairs”: Problem Solving Without Problem Formulation

Problem-solving research and formal problem-solving practice begin with the assumption that a problem has been identified or formulated for solving. The problem-solving process then involves a search for a satisfactory or optimal solution to that problem. In contrast, we propose that, in informal problem solving, a need and a solution are often discovered together and tested for viability as a “need–solution pair.” For example, one may serendipitously discover a new solution and assess it to be worth adopting although the “problem” it would address had not previously been in mind as an object of search or even awareness. In such a case, problem identification and formulation, if done at all, come only after the discovery of the need–solution pair. We propose the identification of need–solution pairs as an approach to problem solving in which problem formulation is not required. We argue that discovery of viable need–solution pairs without problem formulation may have advantages over problem-initiated problem-solving methods under some conditions. First, it removes the often considerable costs associated with problem formulation. Second, it eliminates the constraints on possible solutions that any problem formulation will inevitably apply

The Shape of Covariantly Smeared Sources in Lattice QCD

Covariantly smeared sources are commonly used in lattice QCD to enhance the projection onto the ground state. Here we investigate the dependence of their shape on the gauge field background and find that the presence of localized concentrations of magnetic field can lead to strong distortions which reduce the smearing radii achievable by iterative smearing prescriptions. In particular, as $a\to 0$, iterative procedures like Jacobi smearing require increasingly large iteration counts in order to reach physically-sized smearing radii $r_{sm}\sim$ 0.5 fm, and the resulting sources are strongly distorted. To bypass this issue, we propose a covariant smearing procedure (``free-form smearing'') that allows us to create arbitrarily shaped sources, including in particular Gaussians of arbitrary radius.Comment: 1+15 pages, 7 figures (24 pdf images

DsD_s physics from fine lattices

We present a preliminary analysis of the charm quark mass and the mass and decay constant $f_{D_s}$ of the $D_s$ meson obtained from dynamical simulations of $N_f = 2$ Wilson QCD on the large and fine lattices simulated by the CLS effort.Comment: 6 pages, 2 figures; talk presented at Lattice 2008, XXVI International Symposium on Lattice Field Theory, July 14-19, 2008, Williamsburg, Virginia, US