Scattering Amplitudes For All Masses and Spins

We introduce a formalism for describing four-dimensional scattering amplitudes for particles of any mass and spin. This naturally extends the familiar spinor-helicity formalism for massless particles to one where these variables carry an extra SU(2) little group index for massive particles, with the amplitudes for spin S particles transforming as symmetric rank 2S tensors. We systematically characterise all possible three particle amplitudes compatible with Poincare symmetry. Unitarity, in the form of consistent factorization, imposes algebraic conditions that can be used to construct all possible four-particle tree amplitudes. This also gives us a convenient basis in which to expand all possible four-particle amplitudes in terms of what can be called "spinning polynomials". Many general results of quantum field theory follow the analysis of four-particle scattering, ranging from the set of all possible consistent theories for massless particles, to spin-statistics, and the Weinberg-Witten theorem. We also find a transparent understanding for why massive particles of sufficiently high spin can not be "elementary". The Higgs and Super-Higgs mechanisms are naturally discovered as an infrared unification of many disparate helicity amplitudes into a smaller number of massive amplitudes, with a simple understanding for why this can't be extended to Higgsing for gravitons. We illustrate a number of applications of the formalism at one-loop, giving few-line computations of the electron (g-2) as well as the beta function and rational terms in QCD. "Off-shell" observables like correlation functions and form-factors can be thought of as scattering amplitudes with external "probe" particles of general mass and spin, so all these objects--amplitudes, form factors and correlators, can be studied from a common on-shell perspective.Comment: 79 page

Duality and Axionic Weak Gravity

The axionic weak gravity conjecture predicts the existence of instantons whose actions are less than their charges in appropriate units. We show that the conjecture is satisfied for the axion-dilaton-gravity system if we assume duality constraints on the higher derivative corrections in addition to positivity bounds which follow from unitarity, analyticity, and locality of UV scattering amplitudes. On the other hand, the conjecture does not follow if we assume the positivity bounds only. This presents an example where derivation of the weak gravity conjecture requires more detailed UV information than the consistency of scattering amplitudes.Comment: 17 pages, 4 figures, version published in PRD: comments added and typos fixed, generalised arguments in section 2.2, results unchange

Topological Field Theory with Haagerup Symmetry

We construct a (1+1)$d$ topological field theory (TFT) whose topological defect lines (TDLs) realize the transparent Haagerup $\mathcal{H}_3$ fusion category. This TFT has six vacua, and each of the three non-invertible simple TDLs hosts three defect operators, giving rise to a total of 15 point-like operators. The TFT data, including the three-point coefficients and lasso diagrams, are determined by solving all the sphere four-point crossing equations and torus one-point modular invariance equations. We further verify that the Cardy states furnish a non-negative integer matrix representation under TDL fusion. While many of the constraints we derive are not limited to the this particular TFT with six vacua, we leave open the construction of TFTs with two or four vacua. Finally, TFTs realizing the Haagerup $\mathcal{H}_1$ and $\mathcal{H}_2$ fusion categories can be obtained by gauging algebra objects. This note makes a modest offering in our pursuit of exotica and the quest for their eventual conformity.Comment: 41+11 pages, 1 figure, 3 tables; v2: corrected statements about the literature, revised Appendix

Deformation and reverse snapping of a circular shallow shell under uniform edge tension

AbstractIn this paper we study the deformation and stability of a shallow shell under uniform edge tension, both theoretically and experimentally. Von Karman’s plate model is adopted to formulate the equations of motion. For a shell with axisymmetrical initial shape, the equilibrium positions can be classified into axisymmetrical and unsymmetrical solutions. While there may exist both stable and unstable axisymmetrical solutions, all the unsymmetrical solutions are unstable. Since the unsymmetrical solutions will not affect the stability of the axisymmetrical solutions, it is concluded that for quasi-static analysis, there is no need to include unsymmetrical assumed modes in the calculation. If the shell is initially in the unstrained configuration, it will only be flattened smoothly when the edge tension is applied. No snap-through buckling is possible in this case. On the other hand, if the shell is initially in the strained position, it will be snapped back to the stable position on the other side of the base plane when the edge tension reaches a critical value. Experiment is conducted on several free brass shells of different initial heights to verify the theoretical predictions. Generally speaking, for the range of initial height H<10 the experimental measurements of the deformation and the reverse snapping load agree well with theoretical predictions

AGE-DIFFERENCES IN THE FREE VERTICAL MOMENTS DURING STEPPING DOWN - PILOT

The aim of this study was to understand age-differences in body control during stepping down by investigating free vertical moments (FVMs). Two older adults and two young adults participated in this study. During each trial, lower extremities kinematics were measured using a 10 camera Vicon system (250Hz) and ground reaction forces were recorded using two Kistler force platforms (1000Hz). FVM was calculated by ground reaction forces using Visual3D software. The results indicated young adults showed adduction-FVM (ADD-FVM) but older adults presented abduction-FVM (ABD-FVM) during double-stance phase. Older adults seemed to exert more ABD-FVM than young adults while in the single support phase. It was concluded that the FVMs seemed to point to different strategies between older adults and young adults

STEPPING CHARACTERISTICS BEFORE STAIR WALKING TRANSITONS IN TAICHI ELDERLY

The purpose of this study was to investigate the difference between TC exerciser and normal elderly in stepping characteristics before stair walking transition. There were 12 TC practitioner elderly and 14 matched controls participated in this stady. Ten Vicon high-speed cameras (250Hz) were utilised to collect kinematic data. Results showed that TC group presented faster CoM velocity during descending and following walk. At the moment of just before transition, TC group showed faster resultant / horizontal CoM velocity, TOE resultant / vertical velocity. TC group also demonstrated greater stride length while contacting the ground. We concluded TC group had better abilities of body control. Faster horizontal CoM velocity and vertical TOE velocity negotiated before transiton in TC group, would be order to transit the unstable situation more efficient

The F-Symbols for Transparent Haagerup-Izumi Categories with G = Z_(2n+1)

The notion of a transparent fusion category is defined. For the Haagerup-Izumi fusion rings with G=Z_(2n+1) (the Z_3 case is the Haagerup H_3 fusion ring), the transparent property reduces the number of independent F-symbols from order O(n6) to O(n^2), rendering the pentagon identity practically solvable. Transparent fusion categories are constructed up to Z_(15), and the explicit F-symbols are compactly presented. The potential construction of categories for new families of fusion rings is discussed

The F-Symbols for Transparent Haagerup-Izumi Categories with G = Z_(2n+1)

The notion of a transparent fusion category is defined. For the Haagerup-Izumi fusion rings with G=Z_(2n+1) (the Z_3 case is the Haagerup H_3 fusion ring), the transparent property reduces the number of independent F-symbols from order O(n6) to O(n^2), rendering the pentagon identity practically solvable. Transparent fusion categories are constructed up to Z_(15), and the explicit F-symbols are compactly presented. The potential construction of categories for new families of fusion rings is discussed