On the derivative of the associated Legendre function of the first kind of integer order with respect to its degree

In our recent works [R. Szmytkowski, J. Phys. A 39 (2006) 15147; corrigendum: 40 (2007) 7819; addendum: 40 (2007) 14887], we have investigated the derivative of the Legendre function of the first kind, $P_{\nu}(z)$, with respect to its degree $\nu$. In the present work, we extend these studies and construct several representations of the derivative of the associated Legendre function of the first kind, $P_{\nu}^{\pm m}(z)$, with respect to the degree $\nu$, for $m\in\mathbb{N}$. At first, we establish several contour-integral representations of $\partial P_{\nu}^{\pm m}(z)/\partial\nu$. They are then used to derive Rodrigues-type formulas for $[\partial P_{\nu}^{\pm m}(z)/\partial\nu]_{\nu=n}$ with $n\in\mathbb{N}$. Next, some closed-form expressions for $[\partial P_{\nu}^{\pm m}(z)/\partial\nu]_{\nu=n}$ are obtained. These results are applied to find several representations, both explicit and of the Rodrigues type, for the associated Legendre function of the second kind of integer degree and order, $Q_{n}^{\pm m}(z)$; the explicit representations are suitable for use for numerical purposes in various regions of the complex $z$-plane. Finally, the derivatives $[\partial^{2}P_{\nu}^{m}(z)/\partial\nu^{2}]_{\nu=n}$, $[\partial Q_{\nu}^{m}(z)/\partial\nu]_{\nu=n}$ and $[\partial Q_{\nu}^{m}(z)/\partial\nu]_{\nu=-n-1}$, all with $m>n$, are evaluated in terms of $[\partial P_{\nu}^{-m}(\pm z)/\partial\nu]_{\nu=n}$.Comment: LateX, 40 pages, 1 figure, extensive referencin

Hopf algebras and Markov chains: Two examples and a theory

The operation of squaring (coproduct followed by product) in a combinatorial Hopf algebra is shown to induce a Markov chain in natural bases. Chains constructed in this way include widely studied methods of card shuffling, a natural "rock-breaking" process, and Markov chains on simplicial complexes. Many of these chains can be explictly diagonalized using the primitive elements of the algebra and the combinatorics of the free Lie algebra. For card shuffling, this gives an explicit description of the eigenvectors. For rock-breaking, an explicit description of the quasi-stationary distribution and sharp rates to absorption follow.Comment: 51 pages, 17 figures. (Typographical errors corrected. Further fixes will only appear on the version on Amy Pang's website, the arXiv version will not be updated.

Observation of Scaling Violations in Scaled Momentum Distributions at HERA

Charged particle production has been measured in deep inelastic scattering (DIS) events over a large range of $x$ and $Q^2$ using the ZEUS detector. The evolution of the scaled momentum, $x_p$, with $Q^2,$ in the range 10 to 1280 $GeV^2$, has been investigated in the current fragmentation region of the Breit frame. The results show clear evidence, in a single experiment, for scaling violations in scaled momenta as a function of $Q^2$.Comment: 21 pages including 4 figures, to be published in Physics Letters B. Two references adde

Search for strong gravity in multijet final states produced in pp collisions at √s=13 TeV using the ATLAS detector at the LHC

A search is conducted for new physics in multijet final states using 3.6 inverse femtobarns of data from proton-proton collisions at √s = 13TeV taken at the CERN Large Hadron Collider with the ATLAS detector. Events are selected containing at least three jets with scalar sum of jet transverse momenta (HT) greater than 1TeV. No excess is seen at large HT and limits are presented on new physics: models which produce final states containing at least three jets and having cross sections larger than 1.6 fb with HT > 5.8 TeV are excluded. Limits are also given in terms of new physics models of strong gravity that hypothesize additional space-time dimensions

Measurements of fiducial cross-sections for t\bart production with one or two additional b-jets in pp collisions at √s =8 TeVusing the ATLAS detector

Fiducial cross-sections for $t\bar{t}$ production with one or two additional $b$-jets are reported, using an integrated luminosity of 20.3 fb$^{-1}$ of proton--proton collisions at a centre-of-mass energy of 8 TeV at the Large Hadron Collider, collected with the ATLAS detector. The cross-section times branching ratio for $t\bar{t}$ events with at least one additional $b$-jet is measured to be 950 $\pm$ 70 (stat.) $^{+240}_{-190}$ (syst.) fb in the lepton-plus-jets channel and 50 $\pm$ 10 (stat.) $^{+15}_{-10}$ (syst.) fb in the $e \mu$ channel. The cross-section times branching ratio for events with at least two additional $b$-jets is measured to be 19.3 $\pm$ 3.5 (stat.) $\pm$ 5.7 (syst.) fb in the dilepton channel ($e \mu$,\,$\mu\mu$, and \,$ee$) using a method based on tight selection criteria, and 13.5 $\pm$ 3.3 (stat.) $\pm$ 3.6 (syst.) fb using a looser selection that allows the background normalisation to be extracted from data. The latter method also measures a value of 1.30 $\pm$ 0.33 (stat.) $\pm$ 0.28 (syst.)\% for the ratio of $t\bar{t}$ production with two additional $b$-jets to $t\bar{t}$ production with any two additional jets. All measurements are in good agreement with recent theory predictions.Comment: 41 pages plus author list + cover page (58 total), 9 Figures, 16 tables, submitted to EPJC, all figures including auxiliary figures are available at https://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/PAPERS/TOPQ-2014-10

Measurement of the branching ratio Γ(Λb⁰ → ψ(2S)Λ0)/Γ(Λb⁰ → J/ψΛ0) with the ATLAS detector

An observation of the $\Lambda_b^0 \rightarrow \psi(2S) \Lambda^0$ decay and a comparison of its branching fraction with that of the $\Lambda_b^0 \rightarrow J/\psi \Lambda^0$ decay has been made with the ATLAS detector in proton--proton collisions at $\sqrt{s}=8\,$TeV at the LHC using an integrated luminosity of $20.6\,$fb$^{-1}$. The $J/\psi$ and $\psi(2S)$ mesons are reconstructed in their decays to a muon pair, while the $\Lambda^0\rightarrow p\pi^-$ decay is exploited for the $\Lambda^0$ baryon reconstruction. The $\Lambda_b^0$ baryons are reconstructed with transverse momentum $p_{\rm T}>10\,$GeV and pseudorapidity $|\eta|<2.1$. The measured branching ratio of the $\Lambda_b^0 \rightarrow \psi(2S) \Lambda^0$ and $\Lambda_b^0 \rightarrow J/\psi \Lambda^0$ decays is $\Gamma(\Lambda_b^0 \rightarrow \psi(2S)\Lambda^0)/\Gamma(\Lambda_b^0 \rightarrow J/\psi\Lambda^0) = 0.501\pm 0.033 ({\rm stat})\pm 0.019({\rm syst})$, lower than the expectation from the covariant quark model.Comment: 12 pages plus author list (28 pages total), 5 figures, 1 table, published on Physics Letters B 751 (2015) 63-80. All figures are available at https://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/PAPERS/BPHY-2013-08

Dijet production in √s = 7 TeV pp collisions with large rapidity gaps at the ATLAS experiment

A 6.8 nb−¹ sample of pp collision data collected under low-luminosity conditions at √s = 7 TeV by the ATLAS detector at the Large Hadron Collider is used to study diffractive dijet production. Events containing at least two jets with pT > 20 GeV are selected and analysed in terms of variables which discriminate between diffractive and non-diffractive processes. Cross sections are measured differentially in ΔηF, the size of the observable forward region of pseudorapidity which is devoid of hadronic activity, and in an estimator, ξ˜, of the fractional momentum loss of the proton assuming single diffractive dissociation (pp → p X). Model comparisons indicate a dominant non-diffractive contribution up to moderately large ηF and small ξ˜, with a diffractive contribution which is significant at the highest ΔηF and the lowest ξ˜. The rapidity-gap survival probability is estimated from comparisons of the data in this latter region with predictions based on diffractive parton distribution functions