[en] A theoretical framework established recently for the numerical calculation of the eigenvalues of the Schroedinger operator via the Pruefer transformation is extended here to a general class of real central potentials (including long-range ones, such as the Coulomb potential, etc.). Similarly to previous papers, theorems are formulated and proved providing a simple tool for the determination of the eigenvalues, using only the values of the Pruefer ''phase function'' at some distant but finite point. (author)

[en] The matrix generalisation of the Pruefer transformation introduced by Atkinson is applied to a coupled system of radial Schroedinger equations. It is shown that the phase functions corresponding to the matrix case exhibit properties analogous to those of the Pruefer phase function encountered in the scalar case. Rigorous theorems are established which allow a determination of the eigenvalues of the original Schroedinger system with an arbitrary accuracy provided that the asymptotic behaviour of the phase functions is known. The possibility of obtaining the phase functions by means of the integration of an appropriate system of nonlinear first-order differential equations is briefly discussed. (author)

[en] The correlations between event-by-event fluctuations of anisotropic flow harmonic amplitudes have been measured in Pb-Pb collisions at $\sqrt{s}$${}_{NN}$=2.76 TeV with the ALICE detector at the Large Hadron Collider. The results are reported in terms of multiparticle correlation observables dubbed symmetric cumulants. These observables are robust against biases originating from nonflow effects. The centrality dependence of correlations between the higher order harmonics (the quadrangular v ₄ and pentagonal v ₅ flow) and the lower order harmonics (the elliptic v ₂ and triangular v ₃ flow) is presented. The transverse momentum dependences of correlations between v ₃ and v ₂ and between v ₄ and v ₂ are also reported. The results are compared to calculations from viscous hydrodynamics and a multiphase transport (AMPT) model calculations. The comparisons to viscous hydrodynamic models demonstrate that the different order harmonic correlations respond differently to the initial conditions and the temperature dependence of the ratio of shear viscosity to entropy density (η/s). A small average value of η/s is favored independent of the specific choice of initial conditions in the models. The calculations with the AMPT initial conditions yield results closest to the measurements. Correlations among the magnitudes of v ₂, v ₃, and v ₄ show moderate p _T dependence in midcentral collisions. This might be an indication of possible viscous corrections to the equilibrium distribution at hadronic freeze-out, which might help to understand the possible contribution of bulk viscosity in the hadronic phase of the system. Lastly, together with existing measurements of individual flow harmonics, the presented results provide further constraints on the initial conditions and the transport properties of the system produced in heavy-ion collisions.

[en] In ultrarelativistic heavy-ion collision, the event-by-event variation of the elliptic flow v ₂ reflects fluctuations in the shape of the initial state of the system. This allows to select events with the same centrality but different initial geometry. This selection technique, Event Shape Engineering, has been used in the analysis of charge-dependent two- and three-particle correlations in Pb-Pb collisions at $\sqrt{s}$${}_{NN}$ = 2.76 TeV. The two-particle correlator $⟨<!--\; ???\; -->$cos(φ _α-φ _β)$⟩<!--\; ???\; -->$, calculated for different combinations of charges α and β, is almost independent of v ₂ (for a given centrality), while the three-particle correlator $⟨<!--\; ???\; -->$cos(φ _α+φ _β-2ψ ₂)$⟩<!--\; ???\; -->$ scales almost linearly both with the event v ₂ and charge-particle pseudorapidity density. The charge dependence of the three-particle correlator is often interpreted as evidence for the Chiral Magnetic Effect (CME), a parity violating effect of the strong interaction. However, its measured dependence on v ₂ points to a large non-CME contribution to the correlator. Comparing the results with Monte Carlo calculations including a magnetic field due to the spectators, the upper limit of the CME signal contribution to the three-particle correlator in the 10-15% centrality interval is found to be 26-33% at 95% confidence level.

[en] In two-particle angular correlation measurements, jets give rise to a near-side peak, formed by particles associated to a higher-p_T trigger particle. Measurements of these correlations as a function of pseudorapidity (Δη) and azimuthal (Δφ) differences are used to extract the centrality and p_T dependence of the shape of the near-side peak in the p_T range 1 < p_T < 8 GeV/c in Pb-Pb and pp collisions at √s_NN = 2.76 TeV. A combined fit of the near-side peak and long-range correlations is applied to the data and the peak shape is quantified by the variance of the distributions. And while the width of the peak in the Δφ direction is almost independent of centrality, a significant broadening in the Δη direction is found from peripheral to central collisions. This feature is prominent for the low-p_T region and vanishes above 4 GeV/c. The widths measured in peripheral collisions are equal to those in pp collisions in the Δφ direction and above 3 GeV/c in the Δη direction. Furthermore, for the 10% most central collisions and 1 < p_T,assoc < 2 GeV/c, 1 < p_T,trig < 3 GeV/c, a departure from a Gaussian shape is found: a depletion develops around the center of the peak. Our results are compared to A Multi-Phase Transport (AMPT) model simulation as well as other theoretical calculations indicating that the broadening and the development of the depletion are connected to the strength of radial and longitudinal flow.

[en] Here, we report on the measurement of freeze-out radii for pairs of identical-charge pions measured in Pb-Pb collisions at √s _NN = 2.76 TeV as a function of collision centrality and the average transverse momentum of the pair k _T. Three-dimensional sizes of the system (femtoscopic radii), as well as direction-averaged one-dimensional radii are extracted. The radii decrease with k _T, following a power-law behavior. This is qualitatively consistent with expectations from a collectively expanding system, produced in hydrodynamic calculations. The radii also scale linearly with < dN _ch/d η > ^1/3. We compare this behavior to world data on femtoscopic radii in heavy-ion collisions. While the dependence is qualitatively similar to results at smaller √s _NN, a decrease in the ratio R _out/R _side is seen, which is in qualitative agreement with a specific prediction from hydrodynamic models: a change from inside-out to outside-in freeze-out configuration. Furthermore, these results provide further evidence for the production of a collective, strongly coupled system in heavy-ion collisions at the CERN Large Hadron Collider.

[en] The polarization of inclusive $\mathrm{J}/\psi$ and $\mathrm{\Upsilon }(1\mathrm{S})$ produced in Pb–Pb collisions at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV at the LHC is measured with the ALICE detector. The study is carried out by reconstructing the quarkonium through its decay to muon pairs in the rapidity region $2.5<y<4$ and measuring the polar and azimuthal angular distributions of the muons. The polarization parameters ${\lambda }_{\theta }$, ${\lambda }_{\varphi }$ and ${\lambda }_{\theta \varphi }$ are measured in the helicity and Collins-Soper reference frames, in the transverse momentum interval $2<p_{\mathrm{T}}<10$ GeV/c and $p_{\mathrm{T}}<15$ GeV/c for the $\mathrm{J}/\psi$ and $\mathrm{\Upsilon }(1\mathrm{S})$, respectively. The polarization parameters for the $\mathrm{J}/\psi$ are found to be compatible with zero, within a maximum of about two standard deviations at low $p_{\mathrm{T}}$, for both reference frames and over the whole $p_{\mathrm{T}}$ range. The values are compared with the corresponding results obtained for pp collisions at $\sqrt{s}=7$ and 8 TeV in a similar kinematic region by the ALICE and LHCb experiments. Although with much larger uncertainties, the polarization parameters for $\mathrm{\Upsilon }(1\mathrm{S})$ production in Pb–Pb collisions are also consistent with zero.

[en] The elliptic and triangular flow coefficients $v_2$ and $v_3$ of prompt ${\mathrm{D}}^{\mathrm{0}}$, ${\mathrm{D}}^{+}$, and ${\mathrm{D}}^{⁎+}$ mesons were measured at midrapidity ($|y|<0.8$) in Pb–Pb collisions at the centre-of-mass energy per nucleon pair of $\sqrt{s_{\mathrm{NN}}}=5.02\text{TeV}$ with the ALICE detector at the LHC. The D mesons were reconstructed via their hadronic decays in the transverse momentum interval $1<p_{\mathrm{T}}<36\text{GeV}/c$ in central (0–10%) and semi-central (30–50%) collisions. Compared to pions, protons, and $\mathrm{J}/\psi$ mesons, the average D-meson $v_{\mathrm{n}}$ harmonics are compatible within uncertainties with a mass hierarchy for $p_{\mathrm{T}}\lesssim 3\text{GeV}/c$, and are similar to those of charged pions for higher $p_{\mathrm{T}}$. The coupling of the charm quark to the light quarks in the underlying medium is further investigated with the application of the event-shape engineering (ESE) technique to the D-meson $v_2$ and $p_{\mathrm{T}}$-differential yields. The D-meson $v_2$ is correlated with average bulk elliptic flow in both central and semi-central collisions. Within the current precision, the ratios of per-event D-meson yields in the ESE-selected and unbiased samples are found to be compatible with unity. All the measurements are found to be reasonably well described by theoretical calculations including the effects of charm-quark transport and the recombination of charm quarks with light quarks in a hydrodynamically expanding medium.

[en] The first measurements of the production of muons and electrons from heavy-flavour hadron decays in Xe–Xe collisions at $\sqrt{s_{\mathrm{NN}}}=5.44$ TeV, using the ALICE detector at the LHC, are reported. The measurement of the nuclear modification factor $R_{\mathrm{AA}}$ is performed as a function of transverse momentum $p_{\mathrm{T}}$ in several centrality classes at forward rapidity ($2.5<y<4$) and midrapidity ($|y|<0.8$) for muons and electrons from heavy-flavour hadron decays, respectively. A suppression by a factor up to about 2.5 compared to the binary-scaled pp reference is observed in central collisions at both central and forward rapidities. The $R_{\mathrm{AA}}$ of muons from heavy-flavour hadron decays is compared to previous measurements in Pb–Pb collisions at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV. When the nuclear modification factors are compared in the centrality classes 0–10% for Xe–Xe collisions and 10–20% for Pb–Pb collisions, which have similar charged-particle multiplicity density, a similar suppression, with $R_{\mathrm{AA}}\sim 0.4$ in the $p_{\mathrm{T}}$ interval $4<p_{\mathrm{T}}<8$ GeV/c, is observed. The comparison of the measured $R_{\mathrm{AA}}$ values in the two collision systems brings new insights on the properties of the quark-gluon plasma by investigating the system-size and geometry dependence of medium-induced parton energy loss. The results of muons and electrons from heavy-flavour hadron decays provide new constraints to model calculations.

[en] Invariant differential yields of deuterons and antideuterons in pp collisions at $\sqrt{s}$ = 0.9, 2.76 and 7 TeV and the yields of tritons, ³He nuclei, and their antinuclei at $\sqrt{s}$ = 7 TeV have been measured with the ALICE detector at the CERN Large Hadron Collider. The measurements cover a wide transverse momentum (p _T) range in the rapidity interval |y| < 0.5, extending both the energy and the p _T reach of previous measurements up to 3 GeV/c for A = 2 and 6 GeV/c for A = 3. The coalescence parameters of (anti)deuterons and H$\stackrel{¯<!--\; ??\; -->}{e}$3 nuclei exhibit an increasing trend with p _T and are found to be compatible with measurements in pA collisions at low p _T and lower energies. The integrated yields decrease by a factor of about 1000 for each increase of the mass number with one (anti)nucleon. Also, the deuteron-to-proton ratio is reported as a function of the average charged particle multiplicity at different center-of-mass energies.