Semicoherent elastic pion-carbon scattering at 25 and 40 GeV/c

In the frame of a systematic study of hadron-nucleus interactions performed at the Serpukhov accelerator (CERN-IHEP 5th experiment), semicoherent elastic scattering on carbon $\text{π}{}^{\mathrm{?12}}\text{C}\text{a}\text{?}\text{π}{}^{\mathrm{?12}}\text{C}{}^1\text{(4.44}\text{MeV}\text{)}$ was investigated with 25 and 40 GeV/c incident pions.The experimental data were obtained with a counter technique, looking at the coincidence between the scattered pion and the 4.44 MeV photon from the J^P = 2⁺ carbon excited state detected by a NaI counter.The value obtained for the integrated cross section at 40 GeV/c in the 0.0032 ? |t| ? 0.27 (GeV/c)² four-momentum transfer range isσ = 1.16 ± 0.11 mb.According to the theoretical models, this t-range at 40 GeV/c covers more than 99% of the angular distribution, so the above value almost corresponds to the total semicoherent elastic cross section.     

Non-commutative Hopf algebra of formal diffeomorphisms

This paper deals with two Hopf algebras which are the non-commutative analogues of two different groups of formal power series. The first group is the set of invertible series with the group law being multiplication of series, while the second is the set of formal diffeomorphisms with the group law being a composition of series. The motivation to introduce these Hopf algebras comes from the study of formal series with non-commutative coefficients. Invertible series with non-commutative coefficients still form a group, and we interpret the corresponding new non-commutative Hopf algebra as an alternative to the natural Hopf algebra given by the co-ordinate ring of the group, which has the advantage of being functorial in the algebra of coefficients. For the formal diffeomorphisms with non-commutative coefficients, this interpretation fails, because in this case the composition is not associative anymore. However, we show that for the dual non-commutative algebra there exists a natural co-associative co-product defining a non-commutative Hopf algebra. Moreover, we give an explicit formula for the antipode, which represents a non-commutative version of the Lagrange inversion formula, and we show that its coefficients are related to planar binary trees. Then we extend these results to the semi-direct co-product of the previous Hopf algebras, and to series in several variables. Finally, we show how the non-commutative Hopf algebras of formal series are related to some renormalization Hopf algebras, which are combinatorial Hopf algebras motivated by the renormalization procedure in quantum field theory, and to the renormalization functor given by the double-tensor algebra on a bi-algebra.