Starting from fixed-order perturbation theory (FOPT) we derive expressions for the heavy-flavour components of the deep-inelastic structure functions (F i,H (x, Q 2, mH 2),i = 2, L; H = c, b, t) in the variable-flavour number scheme (VFNS). These expressions are valid in all orders of perturbation theory. This derivation establishes a relation between the parton densities parametrized atn f andn f + 1 light flavours. One of the results is that the heavy quark parton density does not vanish when the factorization scale becomes equal tom H contrary to what is assumed in the literature. Further we observe that in charm electroproduction the exact and asymptotic expressions for the heavy-quark coefficient functions yield identical results forF 2,c(ξ, Q2, mc 2) whenx < 0.01 andQ 2 > 20 (GeV/c)2. From this observation an analysis of the size of the higher order corrections we conclude that in this region the VFNS description and ofF 2,c is better than the one given by FOPT. On the other hand in the charm threshold region i.e.x > 0.01 andQ 2 < 20 (GeV/c)2 it turns out that the reverse is true.

The most important part of the order alpha(s)(2) corrections to the charm component of the charged-current structure functions F-2(x, Q(2)) and F-3(x, Q(2)) has been calculated. This calculation is based on the asymptotic form of the heavy-quark coefficient functions corresponding to the higher order corrections to the W-boson-gluon fusion process. These coefficient functions, which are in principle only valid for Q(2) much greater than m(2), can also be used to estimate the order alpha(s)(2) contributions at lower Q(2) values provided x < 0.1. It turns out that the above corrections are appreciable in the large Q(2) region and they explain the discrepancy found for the structure functions between the fixed-flavour scheme (FFS) and the variable-flavour-number scheme (VFNS). These corrections also hamper the extraction of the strange-quark density from the data obtained for the charged-current and the electromagnetic-current processes. (C) 1997 Elsevier Science B.V.

We examine the charm component F2,c(x,Q2,m2) of the proton structure function F2(x,Q2) in three different schemes and compare the results with the data in the x and Q2 region explored by the HERA experiments. Studied are (1) the three flavour number scheme (TFNS) where the production mechanisms are given by the photon-gluon fusion process and the higher order reactions with three light-flavour parton densities as input (2) the four flavour number scheme (FFNS) where F2,c is expressed in four light flavour densities including one for the charm quark and (3) a variable-flavour number scheme (VFNS) which interpolates between the latter two. Both the VFNS and the TFNS give good descriptions of the experimental data. However one cannot use the FFNS for the description of the data at small Q2.

In this paper we present the analytic form of the heavy flavour coefficient functions for polarized deep inelastic lepton-hadron scattering. The expressions are valid in the kinematical regime Q(2) much greater than m(2) where Q(2) and m(2) stand for the masses squared of the virtual photon and heavy quark respectively, Using these coefficient functions we have computed the next-to-leading order alpha(s) corrections to polarized charm production at HERA collider energies, where bath the electron and proton beams are polarized, We also give an estimate of these corrections at fixed target experiments where the typical Q(2) values are much smaller than at HERA.

In this paper we present the analytic form of the heavy quark coefficient functions for deep inelastic lepton-hadron scattering in the kinematical regime Q2 ⪢ m2. Here Q2 and m2 stand for the masses squared of the virtual photon and heavy quark, respectively. The calculations have been performed up to next-to-leading order in the strong coupling constant αs using operator product expansion techniques. Apart from a check on earlier calculations, which however are only accessible via large computer programs, the asymptotic forms of the coefficient functions are useful for charm production at HERA when the condition Q2 ⪢ mc2 is satisfied. Furthermore, the analytical expressions can also be used when one applies the variable flavour number scheme up to next-to-leading order in αs.

We present analytic formulae for the heavy flavour coefficient functions for polarized deep inelastic lepton-hadron scattering. The expressions are valid in the kinematical regime $Q^2\\gg m^2$ where $Q^2$ and $m^2$ stand for the masses squared of the virtual photon and heavy quark respectively. Using these coefficient functions we have computed the next-to-leading order $\\alpha_s$ corrections to polarized charm production at HERA collider energies, where both the electron and proton beams are polarized. We also give an estimate of these corrections at fixed target experiments where the typical $Q^2$ values are much smaller than at HERA.

Using renormalization group techniques we have derived analytic formulae for the next-to-leading order heavy-quark coefficient functions in deep inelastic lepton hadron scattering. These formulae are only valid in the kinematic regime Q2 ⪢ m2, where Q2 and m2 stand for the masses squared of the virtual photon and heavy quark respectively. Some of the applications of these asymptotic formulae will be discussed.

Motivated by the precision results in the electroweak theory studies of two-loop Feynman diagrams are performed. Specifically this paper gives a contribution to the knowledge of massive two-loop self-energy diagrams in arbitrary and especially four dimensions. This is done in three respects: firstly results in terms of generalized, multivariable hypergeometric functions are presented giving explicit series for small and large momenta. Secondly the imaginary parts of these integrals are expressed as complete elliptic integrals. Finally one-dimensional integral representations with elementary functions are derived. They are very well suited for the numerical evaluations.

Motivated by the results of the electroweak precision experiments, studies of two-loop self-energy Feynman diagrams are performed. An algebraic method for the reduction of all two-loop self-energies to a set of standard scalar integrals is presented. The gauge dependence of the self-energies is discussed and an extension of the pinch technique to the two-loop level is worked out. It is shown to yield a special case of the background-field method which provides a general framework for deriving Green functions with desirable theoretical properties. The massive scalar integrals of self-energy type are expressed in terms of generalized multivariable hypergeometric functions. The imaginary parts of these integrals yield complete elliptic integrals. Finally, one-dimensional integral representations with elementary integrands are derived which are well suited for numerical evaluation.

In this paper the class ofN loop massive scalar self-energy diagrams withN+1 propagators is studied in an arbitrary number of dimensions. As it is known these integrals cannot be expressed in terms of polylogarithms. Here it is shown, however, that they can be described by generalized hypergeometric functions of several variables, namely Laricella functions. These results represent previous small and large momentum expansions in closed form. Numerical comparisons for the finite part in four dimensions with a two-dimensional integral representation show good agreement.

Homoketonization of basketane acetates 6 and 10 on brief treatment with NaOMe in MeOH afforded the seco-basketanones 7 and 11 , respectively, in a stereo-and regiospecific cage opening reaction. As shown by deuterium labeling experiments, both for 6 and 10 , this homoketonization proceeds with retention of configuration. Prolonged basic treatment of 10 led to the exclusive formation of bicyclo[2.2.2]octenyl-acetates 12 . Under identical conditions 6 produced a complex mixture of products. Upon treatment of seco-basketanones 7 and 11 with aq HC1 in MeOH, a rapid regiospecific cationic rearrangement to homobrendanone 14 was observed. This structure was established by X-ray analysis. The effect of a one carbon cage expansion on the base induced cage opening process by extension of the methylene bridge in the homocubane system into an ethylene bridge in the basketane system, is discussed.

The synthesis of 4-substituted basketanes 11 is realized by a regiospecific one carbon-homologation of the readily available 4-substituted homocubanones 10 Subsequent group transformations, starting from basketanone 4-carboxylic acid 11a , leads to an efficient synthesis of basketane 4-acetates 16 .

AbstractKurze Behandlung der Homobasketan‐acetate (I) mit Na‐methoxid‐ Lösung führt zu den seco‐Basketanonen (II), längere Einwirkung der Lösung auf (IIb) und (IIc) zu den Bicyclooctenyl‐acetaten (III).