Scale-dependent Terms Research Articles

Scale-dependent stratification: a skyline–shoreline scatter plot

Most performance exercises help identify size-dependent and size-independent indicators which separately represent quantity and quality proxies. A good example is the evaluation of a country’s scientific performance relative to its economic standing. Scientific performance is usually evaluated in terms of publications and citations and economic performance is measured in terms of gross domestic product (GDP). In this paper we use exergy, which is a scalar second-order product of publications and citations as a measure of output, with GDP as input. X and GDP are scale-dependent terms. We shall take the ratio X/GDP as the scale-dependent measure of performance relative to economic strength. A scatter plot of X/GDP to GDP serves as a quality–quantity map and from this it is possible to identify what are called the skyline and shoreline boundaries showing the scale-dependent upper and lower bounds of performance. One lesson that emerges is that quality does not necessarily grow with size; there is a noticeable scale-dependent stratification. It is also possible to identify a few economies that stand out noticeably from the rest.

Pitfalls of iterative pole mass calculation in electroweak multiplets

.The radiatively induced mass splitting between components of an electroweak multiplet is typically of order 100 MeV. This is sufficient to endow the charged components with macroscopically observable lifetimes, and ensure an electrically neutral dark matter particle. We show that a commonly used iterative procedure to compute radiatively corrected pole masses can lead to very different mass splittings than a non-iterative calculation at the same loop order. By estimating the uncertainties of the two one-loop results, we show that the iterative procedure is significantly more sensitive to the choice of renormalisation scale and gauge parameter than the non-iterative method. This can cause the lifetime of the charged component to vary by up to 12 orders of magnitude if iteration is employed. We show that individual pole masses exhibit similar scale-dependence regardless of the procedure, but that the leading scale-dependent terms cancel when computing the mass splitting if and only if the non-iterative procedure is employed. We show that this behaviour persists at two-loop order: the precision of the mass splitting improves in the non-iterative approach, but our results suggest that higher-order corrections do not reduce the uncertainty in the iterative calculation enough to resolve the problem at two-loop order. We conclude that the iterative procedure should not be used for computing pole masses in situations where electroweak mass splittings are phenomenologically relevant.

HatHor for single top-quark production: Updated predictions and uncertainty estimates for single top-quark production in hadronic collisions

We present updated predictions for single top-quark production in hadronic collisions. The analysis is based on next-to-leading order QCD calculations. The input parameters are fixed to recent measurements. We compare different PDF sets and investigate the related uncertainties. The impact of uncalculated higher orders is estimated using an independent variation of the renormalisation and factorisation scale. The theoretical predictions are compared with recent measurements from Tevatron and LHC. Furthermore, the cross section measurements are used to estimate the top-quark mass. To perform the analysis we extended the publicly available HatHor program to single top-quark production. We thus provide a unified framework for the fast numerical evaluation of total cross sections for top-quark production, which may be used for example in Standard Model fits. For future extensions towards NNLO accuracy, we include already all scale dependent terms at NNLO. We briefly describe how to use the program and provide all required tools to repeat the aforementioned analysis.

New gravitational scales in cosmological surveys

In the quasistatic regime, generic modifications to gravity can give rise to novel scale-dependence of the gravitational field equations. Crucially, the detectability of the new scale-dependent terms hinges upon the existence of an effective mass scale or length scale at which corrections to General Relativity become relevant. Starting from only a few basic principles, we derive the general form of this scale-dependence. Our method recovers results previously known in the specific case of Horndeski gravity, but also shows that they are valid more generally, beyond the regime of scalar field theories. We forecast the constraints that upcoming experiments will place on the existence of a new fundamental mass scale or length scale in cosmology.

SELF-CONSISTENT THEORY OF ANDERSON LOCALIZATION: GENERAL FORMALISM AND APPLICATIONS

The self-consistent theory of Anderson localization of quantum particles or classical waves in disordered media is reviewed. After presenting the basic concepts of the theory of Anderson localization in the case of electrons in disordered solids, the regimes of weak and strong localization are discussed. Then the scaling theory of the Anderson localization transition is reviewed. The renormalization group theory is introduced and results and consequences are presented. It is shown how scale-dependent terms in the renormalized perturbation theory of the inverse diffusion coefficient lead in a natural way to a self-consistent equation for the diffusion coefficient. The latter accounts quantitatively for the static and dynamic transport properties except for a region near the critical point. Several recent applications and extensions of the self-consistent theory, in particular for classical waves, are discussed.

Structure formation from non-Gaussian initial conditions: Multivariate biasing, statistics, and comparison withN-body simulations

We study structure formation in the presence of primordial non-Gaussianity of the local type with parameters ${f}_{\mathrm{NL}}$ and ${g}_{\mathrm{NL}}$. We show that the distribution of dark-matter halos is naturally described by a multivariate bias scheme where the halo overdensity depends not only on the underlying matter density fluctuation $\ensuremath{\delta}$ but also on the Gaussian part of the primordial gravitational potential $\ensuremath{\varphi}$. This corresponds to a non-local bias scheme in terms of $\ensuremath{\delta}$ only. We derive the coefficients of the bias expansion as a function of the halo mass by applying the peak-background split to common parametrizations for the halo mass function in the non-Gaussian scenario. We then compute the halo power spectrum and halo-matter cross spectrum in the framework of Eulerian perturbation theory up to third order. Comparing our results against $N$-body simulations, we find that our model accurately describes the numerical data for wave numbers $k\ensuremath{\le}0.1--0.3h\text{ }\text{ }{\mathrm{Mpc}}^{\ensuremath{-}1}$ depending on redshift and halo mass. In our multivariate approach, perturbations in the halo counts trace $\ensuremath{\varphi}$ on large scales, and this explains why the halo and matter power spectra show different asymptotic trends for $k\ensuremath{\rightarrow}0$. This strongly scale-dependent bias originates from terms at leading order in our expansion. This is different from what happens using the standard univariate local bias where the scale-dependent terms come from badly behaved higher-order corrections. On the other hand, our biasing scheme reduces to the usual local bias on smaller scales, where $|\ensuremath{\varphi}|$ is typically much smaller than the density perturbations. We finally discuss the halo bispectrum in the context of multivariate biasing and show that, due to its strong scale and shape dependence, it is a powerful tool for the detection of primordial non-Gaussianity from future galaxy surveys.

Measuring the running top-quark mass

We present the first direct determination of the running top-quark mass based on the total cross section of top-quark pair production as measured at the Tevatron. Our theory prediction for the cross section includes various next-to-next-to-leading order QCD contributions, in particular, all logarithmically enhanced terms near threshold, the Coulomb corrections at two loops and all explicitly scale-dependent terms at next-to-next-to-leading order accuracy. The result allows for an exact and independent variation of the renormalization and factorization scales. For Tevatron and LHC we study its dependence on all scales, on the parton luminosity and on the top-quark mass using both the conventional pole mass definition as well as the running mass in the $\overline{\mathrm{MS}}$ scheme. We extract for the top quark an $\overline{\mathrm{MS}}$ mass of $m(\ensuremath{\mu}=m)={160.0}_{\ensuremath{-}3.2}^{+3.3}\text{ }\text{ }\mathrm{GeV}$, which corresponds to a pole mass of ${m}_{t}={168.9}_{\ensuremath{-}3.4}^{+3.5}\text{ }\text{ }\mathrm{GeV}$.

Sudakov resummation and finite order expansions of heavy quark hadroproduction cross sections

We resum Sudakov threshold enhancements in heavy quark hadroproduction for single-heavy quark inclusive and pair-inclusive kinematics. We expand these resummed results and derive analytical finite-order cross sections through next-to-next-to-leading order. This involves the construction of next-to-leading order matching conditions in color space. For the scale dependent terms we derive exact results using renormalization group methods. We study the effects of scale variations, scheme and kinematics choice on the partonic and hadronic cross sections, and provide estimates for top and bottom quark production cross sections.

A scale-dependent equation for solute transport in porous media

The governing equation describing solute transport in porous media is reformulated using standard volume averaging techniques. The alternative formulation is based on a modified definition of the deviation, which allows for variation of macroscopic velocity across the REV. The new equation contains additional scale-dependent terms which are functions of the size of the averaging volume (REV). This result indicates that the scale-dependent nature of the dispersion phenomenon is inherent even at the scale of the REV.