Dynamical NNLO parton distributions

Utilizing recent DIS measurements (\sigma_r, F_{2,3,L}) and data on hadronic dilepton production we determine at NNLO (3-loop) of QCD the dynamical parton distributions of the nucleon generated radiatively from valencelike positive input distributions at an optimally chosen low resolution scale (Q_0^2 < 1 GeV^2). These are compared with `standard' NNLO distributions generated from positive input distributions at some fixed and higher resolution scale (Q_0^2 > 1 GeV^2). Although the NNLO corrections imply in both approaches an improved value of \chi^2, typically \chi^2_{NNLO} \simeq 0.9 \chi^2_{NLO}, present DIS data are still not sufficiently accurate to distinguish between NLO results and the minute NNLO effects of a few percent, despite of the fact that the dynamical NNLO uncertainties are somewhat smaller than the NLO ones and both are, as expected, smaller than those of their `standard' counterparts. The dynamical predictions for F_L(x,Q^2) become perturbatively stable already at Q^2 = 2-3 GeV^2 where precision measurements could even delineate NNLO effects in the very small-x region. This is in contrast to the common `standard' approach but NNLO/NLO differences are here less distinguishable due to the much larger 1\sigma uncertainty bands. Within the dynamical approach we obtain \alpha_s(M_Z^2)=0.1124 \pm 0.0020, whereas the somewhat less constrained `standard' fit gives \alpha_s(M_Z^2)=0.1158 \pm 0.0035.Comment: 44 pages, 15 figures; minor changes, footnote adde

Variable Flavor Number Parton Distributions and Weak Gauge and Higgs Boson Production at Hadron Colliders at NNLO of QCD

Based on our recent NNLO dynamical parton distributions as obtained in the `fixed flavor number scheme', we generate radiatively parton distributions in the `variable flavor number scheme' where also the heavy quark flavors (c,b,t) become massless partons within the nucleon. Only within this latter factorization scheme NNLO calculations are feasible at present, since the required partonic subprocesses are only available in the approximation of massless initial-state partons. The NNLO predictions for gauge boson production are typically larger (by more than 1 sigma) than the NLO ones, and rates at LHC energies can be predicted with an accuracy of about 5%, whereas at Tevatron they are more than 2 sigma above the NLO ones. The NNLO predictions for SM Higgs boson production via the dominant gluon fusion process have a total (pdf and scale) uncertainty of about 10% at LHC which almost doubles at the lower Tevatron energies; they are typically about 20% larger than the ones at NLO but the total uncertainty bands overlap.Comment: 28 pages, 3 tables, 6 figure

First O(αs3)O(\alpha_s^3) heavy flavor contributions to deeply inelastic scattering

In the asymptotic limit $Q^2 \gg m^2$, the heavy flavor Wilson coefficients for deep--inelastic scattering factorize into the massless Wilson coefficients and the universal heavy flavor operator matrix elements resulting from light--cone expansion. In this way, one can calculate all but the power corrections in $(m^2/Q^2)^k, k > 0$. The heavy flavor operator matrix elements are known to ${\sf NLO}$. We present the last 2--loop result missing in the unpolarized case for the renormalization at 3--loops and first 3--loop results for terms proportional to the color factor $T_F^2$ in Mellin--space. In this calculation, the corresponding parts of the ${\sf NNLO}$ anomalous dimensions \cite{LARIN,MVVandim} are obtained as well.Comment: 6 pages, Contribution to the Proceedings of "Loops and Legs in Quantum Field Theory", 2008, Sondershausen, Germany, and DIS 2008, London, U

The 1958 Pekeris-Accad-WEIZAC Ground-Breaking Collaboration that Computed Ground States of Two-Electron Atoms (and its 2010 Redux)

In order to appreciate how well off we mathematicians and scientists are today, with extremely fast hardware and lots and lots of memory, as well as with powerful software, both for numeric and symbolic computation, it may be a good idea to go back to the early days of electronic computers and compare how things went then. We have chosen, as a case study, a problem that was considered a huge challenge at the time. Namely, we looked at C.L. Pekeris's seminal 1958 work on the ground state energies of two-electron atoms. We went through all the computations ab initio with today's software and hardware, with a special emphasis on the symbolic computations which in 1958 had to be made by hand, and which nowadays can be automated and generalized.Comment: 8 pages, 2 photos, final version as it appeared in the journa

The uses of Connes and Kreimer's algebraic formulation of renormalization theory

We show how, modulo the distinction between the antipode and the "twisted" or "renormalized" antipode, Connes and Kreimer's algebraic paradigm trivializes the proofs of equivalence of the (corrected) Dyson-Salam, Bogoliubov-Parasiuk-Hepp and Zimmermann procedures for renormalizing Feynman amplitudes. We discuss the outlook for a parallel simplification of computations in quantum field theory, stemming from the same algebraic approach.Comment: 15 pages, Latex. Minor changes, typos fixed, 2 references adde

The massless higher-loop two-point function

We introduce a new method for computing massless Feynman integrals analytically in parametric form. An analysis of the method yields a criterion for a primitive Feynman graph $G$ to evaluate to multiple zeta values. The criterion depends only on the topology of $G$, and can be checked algorithmically. As a corollary, we reprove the result, due to Bierenbaum and Weinzierl, that the massless 2-loop 2-point function is expressible in terms of multiple zeta values, and generalize this to the 3, 4, and 5-loop cases. We find that the coefficients in the Taylor expansion of planar graphs in this range evaluate to multiple zeta values, but the non-planar graphs with crossing number 1 may evaluate to multiple sums with $6^\mathrm{th}$ roots of unity. Our method fails for the five loop graphs with crossing number 2 obtained by breaking open the bipartite graph $K_{3,4}$ at one edge

O(αs2O(\alpha_s^2) Polarized Heavy Flavor Corrections}to Deep-Inelastic Scattering at Q2≫m2Q^2 \gg m^2

We calculate the quarkonic $O(\alpha_s^2)$ massive operator matrix elements $\Delta A_{Qg}(N), \Delta A_{Qq}^{\rm PS}(N)$ and $\Delta A_{qq,Q}^{\rm NS}(N)$ for the twist--2 operators and the associated heavy flavor Wilson coefficients in polarized deeply inelastic scattering in the region $Q^2 \gg m^2$ to $O(\varepsilon)$ in the case of the inclusive heavy flavor contributions. The evaluation is performed in Mellin space, without applying the integration-by-parts method. The result is given in terms of harmonic sums. This leads to a significant compactification of the operator matrix elements and massive Wilson coefficients in the region $Q^2 \gg m^2$ derived previously in \cite{BUZA2}, which we partly confirm, and also partly correct. The results allow to determine the heavy flavor Wilson coefficients for $g_1(x,Q^2)$ to $O(\alpha_s^2)$ for all but the power suppressed terms $\propto (m^2/Q^2)^k, k \geq 1$. The results in momentum fraction $z$-space are also presented. We also discuss the small $x$ effects in the polarized case. Numerical results are presented. We also compute the gluonic matching coefficients in the two--mass variable flavor number scheme to $O(\varepsilon)$.Comment: 58 pages Latex, 12 Figure

O ( α2s^s_2 ) polarized heavy flavor corrections to deep-inelastic scattering at Q2^2 ≫ m2^2

We calculate the quarkonic O(α$^2_s$) massive operator matrix elements $\Delta$A$_{Qg}$ (N),$\Delta$A$^{PS}_{Qq}$(N) and $\Delta$A$^{NS}_{qq}$,$_Q$(N) for the twist–2 operators and the associated heavy flavor Wilson coefficients in polarized deeply inelastic scattering in the region Q$^2$ ≫ m$^2$ to O(ε) in the case of the inclusive heavy flavor contributions. The evaluation is performed in Mellin space, without applying the integration-by-parts method. The result is given in terms of harmonic sums. This leads to a significant compactification of the operator matrix elements and massive Wilson coefficients in the region Q$^2$ ≫ m$^2$ derived previously in [1], which we partly confirm, and also partly correct. The results allow to determine the heavy flavor Wilson coefficients for g$_1$(x, Q$^2$) to O(α$^2_s$ ) for all but the power suppressed terms ∝ (m$^2$/Q$^2$)$^k$ , k ≥ 1. The results in momentum fraction z-space are also presented. We also discuss the small x effects in the polarized case. Numerical results are presented. We also compute the gluonic matching coefficients in the two–mass variable flavor number scheme to O(ε)

The massless two-loop two-point function

We consider the massless two-loop two-point function with arbitrary powers of the propagators and derive a representation, from which we can obtain the Laurent expansion to any desired order in the dimensional regularization parameter eps. As a side product, we show that in the Laurent expansion of the two-loop integral only rational numbers and multiple zeta values occur. Our method of calculation obtains the two-loop integral as a convolution product of two primitive one-loop integrals. We comment on the generalization of this product structure to higher loop integrals.Comment: 22 pages, revised version, eq. 9, 10 and 53 correcte