Staggered fermion matrix elements using smeared operators

We investigate the use of two kinds of staggered fermion operators, smeared and unsmeared. The smeared operators extend over a $4^4$ hypercube, and tend to have smaller perturbative corrections than the corresponding unsmeared operators. We use these operators to calculate kaon weak matrix elements on quenched ensembles at $\beta=6.0$, 6.2 and 6.4. Extrapolating to the continuum limit, we find $B_K(NDR, 2 GeV)= 0.62\pm 0.02(stat)\pm 0.02(syst)$. The systematic error is dominated by the uncertainty in the matching between lattice and continuum operators due to the truncation of perturbation theory at one-loop. We do not include any estimate of the errors due to quenching or to the use of degenerate $s$ and $d$ quarks. For the $\Delta I = {3/2}$ electromagnetic penguin operators we find $B_7^{(3/2)} = 0.62\pm 0.03\pm 0.06$ and $B_8^{(3/2)} = 0.77\pm 0.04\pm 0.04$. We also use the ratio of unsmeared to smeared operators to make a partially non-perturbative estimate of the renormalization of the quark mass for staggered fermions. We find that tadpole improved perturbation theory works well if the coupling is chosen to be \alpha_\MSbar(q^*=1/a).Comment: 22 pages, 1 figure, uses eps

Quantization of Fayet-Iliopoulos Parameters in Supergravity

In this short note we discuss quantization of the Fayet-Iliopoulos parameter in supergravity theories. We argue that in supergravity, the Fayet-Iliopoulos parameter determines a lift of the group action to a line bundle, and such lifts are quantized. Just as D-terms in rigid N=1 supersymmetry are interpreted in terms of moment maps and symplectic reductions, we argue that in supergravity the quantization of the Fayet-Iliopoulos parameter has a natural understanding in terms of linearizations in geometric invariant theory (GIT) quotients, the algebro-geometric version of symplectic quotients.Comment: 21 pages, utarticle class; v2: typos and tex issue fixe

Non-birational twisted derived equivalences in abelian GLSMs

In this paper we discuss some examples of abelian gauged linear sigma models realizing twisted derived equivalences between non-birational spaces, and realizing geometries in novel fashions. Examples of gauged linear sigma models with non-birational Kahler phases are a relatively new phenomenon. Most of our examples involve gauged linear sigma models for complete intersections of quadric hypersurfaces, though we also discuss some more general cases and their interpretation. We also propose a more general understanding of the relationship between Kahler phases of gauged linear sigma models, namely that they are related by (and realize) Kuznetsov's `homological projective duality.' Along the way, we shall see how `noncommutative spaces' (in Kontsevich's sense) are realized physically in gauged linear sigma models, providing examples of new types of conformal field theories. Throughout, the physical realization of stacks plays a key role in interpreting physical structures appearing in GLSMs, and we find that stacks are implicitly much more common in GLSMs than previously realized.Comment: 54 pages, LaTeX; v2: typo fixe

Unitarity of the infinite-volume three-particle scattering amplitude arising from a finite-volume formalism

In a previous publication, two of us derived a relation between the scattering amplitude of three identical bosons, $\mathcal M_3$, and a real function referred to as the {divergence-free} K matrix and denoted $\mathcal K_{\text{df},3}$. The result arose in the context of a relation between finite-volume energies and $\mathcal K_{\text{df},3}$, derived to all orders in the perturbative expansion of a generic low-energy effective field theory. In this work we set aside the role of the finite volume and focus on the infinite-volume relation between $\mathcal K_{\text{df},3}$ and $\mathcal M_3$. We show that, for any real choice of $\mathcal K_{\text{df},3}$, $\mathcal M_3$ satisfies the three-particle unitarity constraint to all orders. Given that $\mathcal K_{\text{df},3}$ is also free of a class of kinematic divergences, the function may provide a useful tool for parametrizing three-body scattering data. Applications include the phenomenological analysis of experimental data (where the connection to the finite volume is irrelevant) as well as calculations in lattice quantum chromodynamics (where the volume plays a key role).Comment: 19 pages, 4 figures, JLAB-THY-19-2945, CERN-TH-2019-07

On the Equivalence of Three-Particle Scattering Formalisms

In recent years, different on-shell $\mathbf{3}\to\mathbf{3}$ scattering formalisms have been proposed to be applied to both lattice QCD and infinite volume scattering processes. We prove that the formulation in the infinite volume presented by Hansen and Sharpe in Phys.~Rev.~D92, 114509 (2015) and subsequently Brice\~no, Hansen, and Sharpe in Phys.~Rev.~D95, 074510 (2017) can be recovered from the $B$-matrix representation, derived on the basis of $S$-matrix unitarity, presented by Mai {\em et al.} in Eur.~Phys.~J.~A53, 177 (2017) and Jackura {\em et al.} in Eur.~Phys.~J.~C79, 56 (2019). Therefore, both formalisms in the infinite volume are equivalent and the physical content is identical. Additionally, the Faddeev equations are recovered in the non-relativistic limit of both representations.Comment: 13 pages, 5 figure

Chiral Perturbation Theory for the Quenched Approximation of QCD

[This version is a minor revision of a previously submitted preprint. Only references have been changed.] We describe a technique for constructing the effective chiral theory for quenched QCD. The effective theory which results is a lagrangian one, with a graded symmetry group which mixes Goldstone bosons and fermions, and with a definite (though slightly peculiar) set of Feynman rules. The straightforward application of these rules gives automatic cancellation of diagrams which would arise from virtual quark loops. The techniques are used to calculate chiral logarithms in $f_K/f_\pi$, $m_\pi$, $m_K$, and the ratio of $\langle{\bar s}s\rangle$ to $\langle{\bar u}u\rangle$. The leading finite-volume corrections to these quantities are also computed. Problems for future study are described.Comment: 14 page

A mathematical model for mechanotransduction at the early steps of suture formation

Growth and patterning of craniofacial sutures are subjected to the effects of mechanical stress. Mechanotransduction processes occurring at the margins of the sutures are not precisely understood. Here, we propose a simple theoretical model based on the orientation of collagen fibres within the suture in response to local stress. We demonstrate that fibre alignment generates an instability leading to the emergence of interdigitations. We confirm the appearance of this instability both analytically and numerically. To support our model, we use histology and synchrotron x-ray microtomography and reveal the fine structure of fibres within the sutural mesenchyme and their insertion into the bone. Furthermore, using a mouse model with impaired mechanotransduction, we show that the architecture of sutures is disturbed when forces are not interpreted properly. Finally, by studying the structure of sutures in the mouse, the rat, an actinopterygian (\emph{Polypterus bichir}) and a placoderm (\emph{Compagopiscis croucheri}), we show that bone deposition patterns during dermal bone growth are conserved within jawed vertebrates. In total, these results support the role of mechanical constraints in the growth and patterning of craniofacial sutures, a process that was probably effective at the emergence of gnathostomes, and provide new directions for the understanding of normal and pathological suture fusion

The Kaon BB-parameter with Wilson Fermions

We calculate the kaon $B$-parameter in quenched lattice QCD at $\beta=6.0$ using Wilson fermions at $\kappa=0.154$ and $0.155$. We use two kinds of non-local (``smeared'') sources for quark propagators to calculate the matrix elements between states of definite momentum. The use of smeared sources yields results with much smaller errors than obtained in previous calculations with Wilson fermions. By combining results for $\vec p =(0,0,0)$ and $\vec p =(0,0,1)$, we show that one can carry out the non-perturbative subtraction necessary to remove the dominant lattice artifacts induced by the chiral symmetry breaking term in the Wilson action. Our final results are in good agreement with those obtained using staggered fermions. We also present results for $B$-parameters of the $\Delta I = 3/2$ part of the electromagnetic penguin operators, and preliminary results for \bk\ in the presence of two flavors of dynamical quarks.Comment: 39 pages, including 9 PS figures (LA UR-91-3522

Perturbative matching of staggered four-fermion operators with hypercubic fat links

We calculate the one-loop matching coefficients between continuum and lattice four-fermion operators for lattice operators constructed using staggered fermions and improved by the use of fattened links. In particular, we consider hypercubic fat links and SU(3) projected Fat-7 links, and their mean-field improved versions. We calculate only current-current diagrams, so that our results apply for operators whose flavor structure does not allow ``eye-diagrams''. We present general formulae, based on two independent approaches, and give numerical results for the cases in which the operators have the taste (staggered flavor) of the pseudo-Goldstone pion. We find that the one-loop corrections are reduced down to the 10-20% level, resolving the problem of large perturbative corrections for staggered fermion calculations of matrix elements.Comment: 37 pages, no figure, 20 table

Kaon B parameter from quenched Lattice QCD

We present results of a large-scale simulation for the Kaon B parameter $B_K$ in quenched lattice QCD with the Kogut-Susskind quark action. Calculating $B_K$ at 1% statistical accuracy for seven values of lattice spacing in the range $a\approx 0.24-0.04$ fm on lattices up to $56^3\times 96$, we verify a quadratic $a$ dependence of $B_K$ theoretically predicted. Strong indications are found that, with our level of accuracy, $\alpha_{\bar{MS}}(1/a)^2$ terms arising from our one-loop matching procedure have to be included in the continuum extrapolation. We present $B_K$(NDR, 2 GeV)=0.628(42) as our final value, as obtained by a fit including the $\alpha_{\bar{MS}}(1/a)^2$ term.Comment: 8 pages, Latex(revtex, epsf), 2 epsf figure