Skew Braces of Size p^(2)q I:Abelian Type

This is the first part of a series of two articles.In this paper we enumerate and classify the left braces of size p^(2)q,where p and q are distinct prime numbers,by the classification of regular subgroups of the holomorph of the abelian groups of the same order.We also provide the formulas that define the constructed braces.

EXTENDED FEYNMAN FORMULA FOR HARMONIC OSCILLATOR AND ITS APPLICATION

We report on a new derivation of the extended Feynman propagator for the harmonic oscil-lator at and beyond caustics using the explicit eigenfunction expansion for the propagator.It is shown that the Maslov corrections beyond caustics are directly determined by the parityof the eigenfunctions. The phase correction and related effect of the propagator of a chargedparticle in constant magnetic field and in the presence of Chern-Simon backgraound gaugefield are studied.

Study of the production of Λ_b^0 band ~0 hadrons in pp collisions and first measurement of the Λ_b^0→J/ψpK^- branching fraction

The product of the ∧0/b （-B/0） differential production cross-section and the branching fraction of the decay ∧0/b→ J/ψ pK-（-B/0→ J/ψ-K＊（892）0）is measured as a function of the beauty hadron transverse momentum, PT, and rapidity, y. The kinematic region of the measurements is pT〈20 GeV/c and 2.0 〈g〈4.5.The measurements use a data sample corresponding to an integrated luminosity of 3fb-1 collected by the LHCb detector in pp collisions at centre-of-mass energies √s=7 TeV in 2011 and √s=8 TeV in 2012. Based on previous LHCb results of the fragmentation fraction ratio,f∧0/b/fd,the branching fraction of the decay ∧0/b→ J/ψ pK-is measured to be B（∧0/b→ J/ψ pK-）=（3.17±0.04±0.07±0.34＋0.45/-0.28）×10-4,where the first uncertainty is statistical, the second is systematic, the third is due to the uncertainty on the branching fraction of the decay -B/0 →J/ψ-K＊（892）0,and the fourth is due to the knowledge of f∧0/b/fd.The sum of the asymmetries in the production and decay between ∧0/b and ∧0/bis also measured as a function of PT and y.The previously published branching fraction of ∧0/b→ J/ψ pπ-,relative to that of ∧0/b→ J/ψ pK-,is updated. The branching fractions of ∧0/b→P＋c（→ J/ψp）K-are determined.

Search for the doubly charmed baryon ■

A search for the doubly charmed baryon ■^+cc is performed through its decay to theΛ^+c K^-π^+ final state,using proton-proton collision data collected with the LHCb detector at centre-of-mass energies of 7,8 and 13 TeV.The data correspond to a total integrated luminosity of 9 fb^-1.No significant signal is observed in the mass range from 3.4 to 3.8 GeV/c^2.Upper limits are set at 95%credibility level on the ratio of the ■^+cc production cross-section times the branching fraction to that ofΛ^+c and ■^++cc baryons.The limits are determined as functions of the ■^+cc mass for different lifetime hypotheses,in the rapidity range from 2.0 to 4.5 and the transverse momentum range from 4 to 15 GeV/c.

Search for the rare decay B^(0)→J/ψФ

A search for the rare decay B^(0)→J/ψФis performed using Pp collision data collected with the LHCb dete-ctor at centre-of-mass energies of 7,8 and 13 TeV,corresponding to an integrated luminosity of9 fb.No significant signal of the decay is observed and an upper limitof 1.1x 10^(-7)at 90%confidence level is set on the branching fraction.

Search for the doubly heavy baryons Ω^(0)_(bc) and E^(0)_(bc) decaying to Λ^(+)_(c)π^(-) and E^(+)_(c)π^(-)

The first search for the doubly heavyΩ^(0)_(bc)baryon and a search for the E^(0)_(bc)baryon are performed using pp collision data collected via the LHCb experiment from 2016 to 2018 at a centre-of-mass energy of 13 TeV,corresponding to an integrated luminosity of 5.2 fb^(-1).The baryons are reconstructed via their decays to Λ^(+)_(c)π^(-)and E^(+)_(c)π^(-).No significant excess is fbund for invariant masses between 6700 and 7300 MeV/c^(2),in a rapidity range from 2.0 to 4.5 and a transverse momentum range from 2 to 20 MeV/c.Upper limits are set on the ratio of the Ω^(0)_(bc)and E^(0)_(bc)production cross-section times the branching fraction to Λ^(+)_(c)π^(-)(E^(+)_(c)π^(-))relative to that of the Λ^(0)_(b)(E^(0)_(b))baryon,for different lifetime hypotheses,at 95%confidence level.The upper limits range from 0.5 x 10^(-4)to 2.5 x 10^(-4)for theΩ^(0)_(bc)→Λ^(+)_(c)π^(-)(E^(0)_(bc)→Λ^(+)_(c)π^(-))decay,and from 1.4x 10^(-3)to 6.9 x 10^(-3)for theΩ^(0)_(bc)→E^(+)_(c)π^(-)(E^(0)_(bc)→E^(+)_(c)π^(-))decay,depending on the considered mass and lifetime of theΩ^(0)_(bc)(E^(0)_(bc))baryon.