Dimensional renormalization: ladders to rainbows

Renormalization factors are most easily extracted by going to the massless limit of the quantum field theory and retaining only a single momentum scale. We derive factors and renormalized Green functions to all orders in perturbation theory for rainbow graphs and vertex (or scattering diagrams) at zero momentum transfer, in the context of dimensional renormalization, and we prove that the correct anomalous dimensions for those processes emerge in the limit D -> 4.Comment: RevTeX, no figure

Lectures on multiloop calculations

I discuss methods of calculation of propagator diagrams (massless, those of Heavy Quark Effective Theory, and massive on-shell diagrams) up to 3 loops. Integration-by-parts recurrence relations are used to reduce them to linear combinations of basis integrals. Non-trivial basis integrals have to be calculated by some other method, e.g., using Gegenbauer polynomial technique. Many of them are expressed via hypergeometric functions; in the massless and HQET cases, their indices tend to integers at $\epsilon\to0$. I discuss the algorithm of their expansion in $\epsilon$, in terms of multiple $\zeta$ values. These lectures were given at Calc-03 school, Dubna, 14--20 June 2003.Comment: 52 pages, 49 figures. Lectures at Calc-03 school, Dubna, 14--20 June 2003. v2: 2 references added, minor typos corrected. v3: methodical improvements, typo in eq. (3.19) corrected, 2 references adde

A Free-Form Lensing Grid Solution for A1689 with New Mutiple Images

Hubble Space Telescope imaging of the galaxy cluster Abell 1689 has revealed an exceptional number of strongly lensed multiply-imaged galaxies, including high-redshift candidates. Previous studies have used this data to obtain the most detailed dark matter reconstructions of any galaxy cluster to date, resolving substructures ~25 kpc across. We examine Abell 1689 (hereafter, A1689) non-parametrically, combining strongly lensed images and weak distortions from wider field Subaru imaging, and we incorporate member galaxies to improve the lens solution. Strongly lensed galaxies are often locally affected by member galaxies, however, these perturbations cannot be recovered in grid based reconstructions because the lensing information is too sparse to resolve member galaxies. By adding luminosity-scaled member galaxy deflections to our smooth grid we can derive meaningful solutions with sufficient accuracy to permit the identification of our own strongly lensed images, so our model becomes self consistent. We identify 11 new multiply lensed system candidates and clarify previously ambiguous cases, in the deepest optical and NIR data to date from Hubble and Subaru. Our improved spatial resolution brings up new features not seen when the weak and strong lensing effects are used separately, including clumps and filamentary dark matter around the main halo. Our treatment means we can obtain an objective mass ratio between the cluster and galaxy components, for examining the extent of tidal stripping of the luminous member galaxies. We find a typical mass-to-light ratios of M/L_B = 21 inside the r<1 arcminute region that drops to M/L_B = 17 inside the r<40 arcsecond region. Our model independence means we can objectively evaluate the competitiveness of stacking cluster lenses for defining the geometric lensing-distance-redshift relation in a model independent way.Comment: 23 pages with 25 figures Replced with MNRAS submitted version. Some figures have been corrected and minor text edit

Young Red Spheroidal Galaxies in the Hubble Deep Fields: Evidence for a Truncated IMF at ~2M_solar and a Constant Space Density to z~2

The optical-IR images of the Northern and Southern Hubble Deep Fields are used to measure the spectral and density evolution of early-type galaxies. The mean optical SED is found to evolve passively towards a mid F-star dominated spectrum by z ~ 2. We demonstrate with realistic simulations that hotter ellipticals would be readily visible if evolution progressed blueward and brightward at z > 2, following a standard IMF. The colour distributions are best fitted by a `red' IMF, deficient above ~2 M_solar and with a spread of formation in the range 1.5 < z_f < 2.5. Traditional age dating is spurious in this context, a distant elliptical can be young but appear red, with an apparent age >3 Gyrs independent of its formation redshift. Regarding density evolution, we demonstrate that the sharp decline in numbers claimed at z > 1 results from a selection bias against distant red galaxies in the optical, where the flux is too weak for morphological classification, but is remedied with relatively modest IR exposures revealing a roughly constant space density to z ~ 2. We point out that the lack of high mass star-formation inferred here and the requirement of metals implicates cooling-flows of pre-enriched gas in the creation of the stellar content of spheroidal galaxies. Deep-field X-ray images will be very helpful to examine this possibility.Comment: 6 pages, 3 figures, submitted to Astrophysical Journal Letters, typographical errors corrected, simulated images with different IMFs illustrated at http://astro.berkeley.edu/~bouwens/ellip.htm

Higgs Decay to Top Quarks at O(\alpha_s^2)

Three-loop corrections to the scalar and pseudo-scalar current correlator are calculated. By applying the large momentum expansion mass terms up to order (m^2/q^2)^4 are evaluated analytically. As an application O(\alpha_s^2) corrections to the decay of a scalar and pseudo-scalar Higgs boson into top quarks are considered. It is shown that for a Higgs mass not far above the $t\bar{t}$ threshold these higher order mass corrections are necessary to get reliable results.Comment: Latex, 20 pages, 14 ps-figures. The complete paper, including figures, is also available via anonymous ftp at ftp://ttpux2.physik.uni-karlsruhe.de/ , or via www at http://www-ttp.physik.uni-karlsruhe.de/cgi-bin/preprints

Three-loop QCD corrections and b-quark decays

We present three-loop (NNNLO) corrections to the heavy-to-heavy quark transitions in the limit of equal initial and final quark masses. In analogy with the previously found NNLO corrections, the bulk of the result is due to the beta_0^2 alpha_s^3 corrections. The remaining genuine three-loop effects for the semileptonic b --> c decays are estimated to increase the decay amplitude by 0.2(2)%. The perturbative series for the heavy-heavy axial current converges very well.Comment: 5 page

Using Weak Lensing Dilution to Improve Measurements of the Luminous and Dark Matter in A1689

The E/SO sequence of a cluster defines a boundary redward of which a reliable weak lensing signal can be obtained from background galaxies, uncontaminated by cluster members. For bluer colors, both background and cluster members are present, reducing the distortion signal by the proportion of unlensed cluster members. In deep Subaru and HST/ACS images of A1689 the tangential distortion of galaxies with bluer colors falls rapidly toward the cluster center relative to the lensing signal of the red background. We use this dilution effect to derive the cluster light profile and luminosity function to large radius, with the advantage that no subtraction of far-field background counts is required. The light profile declines smoothly to the limit of the data, r<2Mpc/h, with a constant slope, dlog(L)/dlog(r)=-1.12+-0.06, unlike the lensing mass profile which steepens continuously with radius, so that M/L peaks at an intermediate radius, ~100kpc/h. A flatter behavior is found for the more physically meaningful ratio of dark-matter to stellar-matter, when accounting for the color-mass relation of cluster members. The cluster luminosity function has a flat slope, alpha=-1.05+-0.07, independent of radius and with no faint upturn to M_i'<-12. We establish that the very bluest objects are negligibly contaminated by the cluster V-i'<0.2, because their distortion profile rises towards the center following the red background, but offset higher by ~20%. This larger amplitude is consistent with the greater estimated depth of the faint blue galaxies, z~=2.0 compared to z~=0.85 for the red background, a purely geometric effect related to cosmological parameters. Finally, we improve upon our earlier mass profile by combining both the red and blue background populations, clearly excluding low concentration CDM profiles.Comment: 17 pages, 21 figures, revised version in response to referee comments,(added some discussion, references), conclusions unchanged. Accepted for publication in Ap

Lattice Green functions in all dimensions

We give a systematic treatment of lattice Green functions (LGF) on the $d$-dimensional diamond, simple cubic, body-centred cubic and face-centred cubic lattices for arbitrary dimensionality $d \ge 2$ for the first three lattices, and for $2 \le d \le 5$ for the hyper-fcc lattice. We show that there is a close connection between the LGF of the $d$-dimensional hypercubic lattice and that of the $(d-1)$-dimensional diamond lattice. We give constant-term formulations of LGFs for all lattices and dimensions. Through a still under-developed connection with Mahler measures, we point out an unexpected connection between the coefficients of the s.c., b.c.c. and diamond LGFs and some Ramanujan-type formulae for $1/\pi.$Comment: 30 page

Heavy-Light Meson Decay Constant from QCD Sum Rules in Three-Loop Approximation

In this paper we compute the decay constant of the pseudo-scalar heavy-light mesons in the heavy quark effective theory framework of QCD sum rules. In our analysis we include the recently evaluated three-loop result of order $\alpha_s^2$ for the heavy-light current correlator. The value of the bottom quark mass, which essentially limits the accuracy of the sum rules for $B$ meson, is extracted from the nonrelativistic sum rules for $\Upsilon$ resonances in the next-to-next-to-leading approximation. We find stability of our result with respect to all types of corrections and the specific form of the sum rule which reduces the uncertainty. Our results $f_B=206\pm 20$ MeV and $f_D=195\pm 20$ MeV for the $B$ and $D$ meson decay constants are in impressive agreement with recent lattice calculations.Comment: minor editorial changes, references added, to appear in PR