On the Evaluation of Powers and Monomials

Let $y_1 , \cdots ,y_p$ be monomials over the indeterminates $x_1 , \cdots ,x_q$. For every $y = (y_1 , \cdots ,y_p )$ there is some minimum number $L(y)$ of multiplications sufficient to compute $y_1 , \cdots ,y_p$ from $x_1 , \cdots ,x_q$ and the identity 1. Let $L(p,q,N)$ denote the maximum of $L(y)$ over all $y$ for which the exponent of any indeterminate in any monomial is at most $N$. We show that if $p = (N + 1^{o(q)} )$ and $q = (N + 1^{o(p)} )$, then $L(p,q,N) = \min \{ p,q\} \log N + H/\log H + o(H /\log H)$, where $H = pq\log (N + 1)$ and all logarithms have base 2

Continuum Moment Equations on the Lattice

An analysis is given as to why one can not directly evaluate continuum moment equations, i.e., equations involving powers of the position variable times charge, current, or energy/momentum operators, on the lattice. I examine two cases: a three point function evaluation of the nucleon magnetic moment and a four point function (charge overlap) evaluation of the pseudoscalar charge radius.Comment: 9 pages; 1 ps figur

Analytical Result for Dimensionally Regularized Massless On-Shell Planar Triple Box

The dimensionally regularized massless on-shell planar triple box Feynman diagram with powers of propagators equal to one is analytically evaluated for general values of the Mandelstam variables s and t in a Laurent expansion in the parameter \ep=(4-d)/2 of dimensional regularization up to a finite part. An explicit result is expressed in terms of harmonic polylogarithms, with parameters 0 and 1, up to the sixth order. The evaluation is based on the method of Feynman parameters and multiple Mellin-Barnes representation. The same technique can be quite similarly applied to planar triple boxes with any numerators and integer powers of the propagators.Comment: 8 pages, LaTeX with axodraw.st

Analytical Results for Dimensionally Regularized Massless On-shell Double Boxes with Arbitrary Indices and Numerators

We present an algorithm for the analytical evaluation of dimensionally regularized massless on-shell double box Feynman diagrams with arbitrary polynomials in numerators and general integer powers of propagators. Recurrence relations following from integration by parts are solved explicitly and any given double box diagram is expressed as a linear combination of two master double boxes and a family of simpler diagrams. The first master double box corresponds to all powers of the propagators equal to one and no numerators, and the second master double box differs from the first one by the second power of the middle propagator. By use of differential relations, the second master double box is expressed through the first one up to a similar linear combination of simpler double boxes so that the analytical evaluation of the first master double box provides explicit analytical results, in terms of polylogarithms \Li{a}{-t/s}, up to $a=4$, and generalized polylogarithms $S_{a,b}(-t/s)$, with $a=1,2$ and $b=2$, dependent on the Mandelstam variables $s$ and $t$, for an arbitrary diagram under consideration.Comment: LaTeX, 16 pages; misprints in ff. (8), (24), (30) corrected; some explanations adde

Evaluating multiloop Feynman integrals by Mellin-Barnes representation

The status of analytical evaluation of double and triple box diagrams is characterized. The method of Mellin-Barnes representation as a tool to evaluate master integrals in these problems is advocated. New MB representations for massive on-shell double boxes with general powers of propagators are presented.Comment: 5 pages, Talk given at the 7th DESY workshop on Elementary Particle Theory, "Loops and Legs in Quantum Field Theory", April 25-30, 2004, Zinnowitz, Germany, to appear in the proceeding

Electron Self Energy for the K and L Shell at Low Nuclear Charge

A nonperturbative numerical evaluation of the one-photon electron self energy for the K- and L-shell states of hydrogenlike ions with nuclear charge numbers Z=1 to 5 is described. Our calculation for the 1S state has a numerical uncertainty of 0.8 Hz in atomic hydrogen, and for the L-shell states (2S and 2P) the numerical uncertainty is 1.0 Hz. The method of evaluation for the ground state and for the excited states is described in detail. The numerical results are compared to results based on known terms in the expansion of the self energy in powers of (Z alpha).Comment: 21 pages, RevTeX, 5 Tables, 6 figure

Analogs of noninteger powers in general analytic QCD

In contrast to the coupling parameter in the usual perturbative QCD (pQCD), the coupling parameter in the analytic QCD models has cuts only on the negative semiaxis of the Q^2-plane (where q^2 = -Q^2 is the momentum squared), thus reflecting correctly the analytic structure of the spacelike observables. The Minimal Analytic model (MA, named also APT) of Shirkov and Solovtsov removes the nonphysical cut (at positive Q^2) of the usual pQCD coupling and keeps the pQCD cut discontinuity of the coupling at negative Q^2 unchanged. In order to evaluate in MA the physical QCD quantities whose perturbation expansion involves noninteger powers of the pQCD coupling, a specific method of construction of MA analogs of noninteger pQCD powers was developed by Bakulev, Mikhailov and Stefanis (BMS). We present a construction, applicable now in any analytic QCD model, of analytic analogs of noninteger pQCD powers; this method generalizes the BMS approach obtained in the framework of MA. We need to know only the discontinuity function of the analytic coupling (the analog of the pQCD coupling) along its cut in order to obtain the analytic analogs of the noninteger powers of the pQCD coupling, as well as their timelike (Minkowskian) counterparts. As an illustration, we apply the method to the evaluation of the width for the Higgs decay into b+(bar b) pair.Comment: 29 pages, 5 figures; sections II and III extended, appendix B is ne