Improved α4\alpha^4 Term of the Electron Anomalous Magnetic Moment

We report a new value of electron $g-2$, or $a_e$, from 891 Feynman diagrams of order $\alpha^4$. The FORTRAN codes of 373 diagrams containing closed electron loops have been verified by at least two independent formulations. For the remaining 518 diagrams, which have no closed lepton loop, verification by a second formulation is not yet attempted because of the enormous amount of additional work required. However, these integrals have structures that allow extensive cross-checking as well as detailed comparison with lower-order diagrams through the renormalization procedure. No algebraic error has been uncovered for them. The numerical evaluation of the entire $\alpha^4$ term by the integration routine VEGAS gives $-1.7283 (35) (\alpha/\pi)^4$, where the uncertainty is obtained by careful examination of error estimates by VEGAS. This leads to $a_e = 1 159 652 175.86 (0.10) (0.26) (8.48) \times 10^{-12}$, where the uncertainties come from the $\alpha^4$ term, the estimated uncertainty of $\alpha^5$ term, and the inverse fine structure constant, $\alpha^{-1} = 137.036 000 3 (10)$, measured by atom interferometry combined with a frequency comb technique, respectively. The inverse fine structure constant $\alpha^{-1} (a_e)$ derived from the theory and the Seattle measurement of $a_e$ is $137.035 998 83 (51)$.Comment: 64 pages and 10 figures. Eq.(16) is corrected. Comments are added after Eq.(40

Everyone Makes Mistakes - Including Feynman

This talk is dedicated to Alberto Sirlin in celebration of his seventieth birthday. I wish to convey my deep appreciation of his many important contributions to particle physics over 40 years and look forward to many more years of productive research.Comment: 16 pages postscript, also available through http://w4.lns.cornell.edu/public/CLN

Eighth-Order Vacuum-Polarization Function Formed by Two Light-by-Light-Scattering Diagrams and its Contribution to the Tenth-Order Electron g-2

We have evaluated the contribution to the anomalous magnetic moment of the electron from six tenth-order Feynman diagrams which contain eighth-order vacuum-polarization function formed by two light-by-light scattering diagrams connected by three photons. The integrals are constructed by two different methods. In the first method the subtractive counter terms are used to deal with ultraviolet (UV) singularities together with the requirement of gauge-invariance. In the second method, the Ward-Takahashi identity is applied to the light-by-light scattering amplitudes to eliminate UV singularities. Numerical evaluation confirms that the two methods are consistent with each other within their numerical uncertainties. Combining the two results statistically and adding small contribution from the muons and/or tau leptons, we obtain $0.000 399 9 (18) (\alpha/\pi)^5$. We also evaluated the contribution to the muon $g-2$ from the same set of diagrams and found $-1.263 44 (14) (\alpha/\pi)^5$.Comment: 27 page

Tenth-order lepton g-2: Contribution from diagrams containing a sixth-order light-by-light-scattering subdiagram internally

This paper reports the result of our evaluation of the tenth-order QED correction to the lepton g-2 from Feynman diagrams which have sixth-order light-by-light-scattering subdiagrams, none of whose vertices couple to the external magnetic field. The gauge-invariant set of these diagrams, called Set II(e), consists of 180 vertex diagrams. In the case of the electron g-2 (a_e), where the light-by-light subdiagram consists of the electron loop, the contribution to a_e is found to be - 1.344 9 (10) (\alpha /\pi)^5. The contribution of the muon loop to a_e is - 0.000 465 (4) (\alpha /\pi)^5. The contribution of the tau-lepton loop is about two orders of magnitudes smaller than that of the muon loop and hence negligible. The sum of all of these contributions to a_e is - 1.345 (1) (\alpha /\pi)^5. We have also evaluated the contribution of Set II(e) to the muon g-2 (a_\mu). The contribution to a_\mu from the electron loop is 3.265 (12) (\alpha /\pi)^5, while the contribution of the tau-lepton loop is -0.038 06 (13) (\alpha /\pi)^5. The total contribution to a_\mu, which is the sum of these two contributions and the mass-independent part of a_e, is 1.882 (13) (\alpha /\pi)^5.Comment: 18 pages, 3 figures, REVTeX4, axodraw.sty used, changed title, corrected uncertainty of a_mu, added a referenc

Improved α4\alpha^4 Term of the Muon Anomalous Magnetic Moment

We have completed the evaluation of all mass-dependent $\alpha^4$ QED contributions to the muon $g-2$, or $a_\mu$, in two or more different formulations. Their numerical values have been greatly improved by an extensive computer calculation. The new value of the dominant $\alpha^4$ term $A_2^{(8)} (m_\mu / m_e)$ is 132.6823 (72), which supersedes the old value 127.50 (41). The new value of the three-mass term $A_3^{(8)} (m_\mu / m_e, m_\mu / m_\tau)$ is 0.0376 (1). The term $A_2^{(8)} (m_\mu / m_\tau)$ is crudely estimated to be about 0.005 and may be ignored for now. The total QED contribution to $a_\mu$ is $116 584 719.58 (0.02)(1.15)(0.85) \times 10^{-11}$, where 0.02 and 1.15 are uncertainties in the $\alpha^4$ and $\alpha^5$ terms and 0.85 is from the uncertainty in $\alpha$ measured by atom interferometry. This raises the Standard Model prediction by $13.9 \times 10^{-11}$, or about 1/5 of the measurement uncertainty of $a_\mu$. It is within the noise of current uncertainty ($\sim 100 \times 10^{-11}$) in the estimated hadronic contributions to $a_\mu$.Comment: Appendix A has been rewritten extensively. It includes the 4th-order calculation for illustration. Version accepted by PR

Isolated effective coherence (iCoh): causal information flow excluding indirect paths

A problem of great interest in real world systems, where multiple time series measurements are available, is the estimation of the intra-system causal relations. For instance, electric cortical signals are used for studying functional connectivity between brain areas, their directionality, the direct or indirect nature of the connections, and the spectral characteristics (e.g. which oscillations are preferentially transmitted). The earliest spectral measure of causality was Akaike's (1968) seminal work on the noise contribution ratio, reflecting direct and indirect connections. Later, a major breakthrough was the partial directed coherence of Baccala and Sameshima (2001) for direct connections. The simple aim of this study consists of two parts: (1) To expose a major problem with the partial directed coherence, where it is shown that it is affected by irrelevant connections to such an extent that it can misrepresent the frequency response, thus defeating the main purpose for which the measure was developed, and (2) To provide a solution to this problem, namely the "isolated effective coherence", which consists of estimating the partial coherence under a multivariate auto-regressive model, followed by setting all irrelevant associations to zero, other than the particular directional association of interest. Simple, realistic, toy examples illustrate the severity of the problem with the partial directed coherence, and the solution achieved by the isolated effective coherence. For the sake of reproducible research, the software code implementing the methods discussed here (using lazarus free-pascal "www.lazarus.freepascal.org"), including the test data as text files, are freely available at: https://sites.google.com/site/pascualmarqui/home/icoh-isolated-effective-coherenceComment: 2014-02-21 pre-print, technical report, KEY Institute for Brain-Mind Research, University of Zurich, et a

Sixth-Order Vacuum-Polarization Contribution to the Lamb Shift of the Muonic Hydrogen

The sixth-order electron-loop vacuum-polarization contribution to the $2P_{1/2} - 2S_{1/2}$ Lamb shift of the muonic hydrogen ($\mu^{-} p^+$ bound state) has been evaluated numerically. Our result is 0.007608(1) meV. This eliminates the largest uncertainty in the theoretical calculation. Combined with the proposed precision measurement of the Lamb shift it will lead to a very precise determination of the proton charge radius.Comment: 4 pages, 5 figures the totoal LS number is change

Numerical and analytical investigation towards performance enhancement of a newly developed rockfall protective cable-net structure

In a previous companion paper, we presented a three-tier modelling of a particular type of rockfall protective cable-net structure (barrier), developed newly in Japan. Therein, we developed a three-dimensional, Finite Element based, nonlinear numerical model having been calibrated/back-calculated and verified with the element- and structure-level physical tests. Moreover, using a very simple, lumped-mass, single-degree-of-freedom, equivalently linear analytical model, a global-displacement-predictive correlation was devised by modifying the basic equation – obtained by combining the principles of conservation of linear momentum and energy – based on the back-analysis of the tests on the numerical model. In this paper, we use the developed models to explore the performance enhancement potential of the structure in terms of (a) the control of global displacement – possibly the major performance criterion for the proposed structure owing to a narrow space available in the targeted site, and (b) the increase in energy dissipation by the existing U-bolt-type Friction-brake Devices – which are identified to have performed weakly when integrated into the structure. A set of parametric investigations have revealed correlations to achieve the first objective in terms of the structure's mass, particularly by manipulating the wire-net's characteristics, and has additionally disclosed the effects of the impacting-block's parameters. Towards achieving the second objective, another set of parametric investigations have led to a proposal of a few innovative improvements in the constitutive behaviour (model) of the studied brake device (dissipator), in addition to an important recommendation of careful handling of the device based on the identified potential flaw