Subexponential estimations in Shirshov's height theorem (in English)

In 1993 E. I. Zelmanov asked the following question in Dniester Notebook: "Suppose that F_{2, m} is a 2-generated associative ring with the identity x^m=0. Is it true, that the nilpotency degree of F_{2, m} has exponential growth?" We show that the nilpotency degree of l-generated associative algebra with the identity x^d=0 is smaller than Psi(d,d,l), where Psi(n,d,l)=2^{18} l (nd)^{3 log_3 (nd)+13}d^2. We give the definitive answer to E. I. Zelmanov by this result. It is the consequence of one fact, which is based on combinatorics of words. Let l, n and d>n be positive integers. Then all the words over alphabet of cardinality l which length is greater than Psi(n,d,l) are either n-divided or contain d-th power of subword, where a word W is n-divided, if it can be represented in the following form W=W_0 W_1...W_n such that W_1 >' W_2>'...>'W_n. The symbol >' means lexicographical order here. A. I. Shirshov proved that the set of non n-divided words over alphabet of cardinality l has bounded height h over the set Y consisting of all the words of degree <n. Original Shirshov's estimation was just recursive, in 1982 double exponent was obtained by A.G.Kolotov and in 1993 A.Ya.Belov obtained exponential estimation. We show, that h<Phi(n,l), where Phi(n,l) = 2^{87} n^{12 log_3 n + 48} l. Our proof uses Latyshev idea of Dilworth theorem application.Comment: 21 pages, Russian version of the article is located at the link arXiv:1101.4909; Sbornik: Mathematics, 203:4 (2012), 534 -- 55

Measurement of the ratios of branching fractions R(D∗)\mathcal{R}(D^{*}) and R(D0)\mathcal{R}(D^{0})

The ratios of branching fractions $\mathcal{R}(D^{*})\equiv\mathcal{B}(\bar{B}\to D^{*}\tau^{-}\bar{\nu}_{\tau})/\mathcal{B}(\bar{B}\to D^{*}\mu^{-}\bar{\nu}_{\mu})$ and $\mathcal{R}(D^{0})\equiv\mathcal{B}(B^{-}\to D^{0}\tau^{-}\bar{\nu}_{\tau})/\mathcal{B}(B^{-}\to D^{0}\mu^{-}\bar{\nu}_{\mu})$ are measured, assuming isospin symmetry, using a sample of proton-proton collision data corresponding to 3.0 fb${ }^{-1}$ of integrated luminosity recorded by the LHCb experiment during 2011 and 2012. The tau lepton is identified in the decay mode $\tau^{-}\to\mu^{-}\nu_{\tau}\bar{\nu}_{\mu}$. The measured values are $\mathcal{R}(D^{*})=0.281\pm0.018\pm0.024$ and $\mathcal{R}(D^{0})=0.441\pm0.060\pm0.066$, where the first uncertainty is statistical and the second is systematic. The correlation between these measurements is $\rho=-0.43$. Results are consistent with the current average of these quantities and are at a combined 1.9 standard deviations from the predictions based on lepton flavor universality in the Standard Model.Comment: All figures and tables, along with any supplementary material and additional information, are available at https://cern.ch/lhcbproject/Publications/p/LHCb-PAPER-2022-039.html (LHCb public pages

Measurement of prompt D+D^+ and Ds+D^+_{s} production in pPbp\mathrm{Pb} collisions at sNN=5.02 \sqrt {s_{\mathrm{NN}}}=5.02\,TeV

International audienceThe production of prompt $D^+$ and $D^+_{s}$ mesons is studied in proton-lead collisions at a centre-of-mass energy of $\sqrt {s_{\mathrm{NN}}}=5.02\,$TeV. The data sample corresponding to an integrated luminosity of $(1.58\pm0.02)\mathrm{nb}^{-1}$ is collected by the LHCb experiment at the LHC. The differential production cross-sections are measured using $D^+$ and $D^+_{s}$ candidates with transverse momentum in the range of $0<p_{\mathrm{T}} <14\,\mathrm{GeV}/c$ and rapidities in the ranges of $1.5<y^*<4.0$ and $-5.0<y^*<-2.5$ in the nucleon-nucleon centre-of-mass system. For both particles, the nuclear modification factor and the forward-backward production ratio are determined. These results are compared with theoretical models that include initial-state nuclear effects. In addition, measurements of the cross-section ratios between $D^+$, $D^+_{s}$ and $D^0$ mesons are presented, providing a baseline for studying the charm hadronization in lead-lead collisions at LHC energies

Precision measurement of CP\it{CP} violation in the penguin-mediated decay Bs0→ϕϕB_s^{0}\rightarrow\phi\phi

International audienceA flavor-tagged time-dependent angular analysis of the decay $B_s^{0}\rightarrow\phi\phi$ is performed using $pp$ collision data collected by the LHCb experiment at $\sqrt{s}=13$ TeV, the center-of-mass energy of 13 TeV, corresponding to an integrated luminosity of 6 fb^{-1}. The $\it{CP}$-violating phase and direct $\it{CP}$-violation parameter are measured to be $\phi_{s\bar{s}s} = -0.042 \pm 0.075 \pm 0.009$ rad and $|\lambda|=1.004\pm 0.030 \pm 0.009$, respectively, assuming the same values for all polarization states of the $\phi\phi$ system. In these results, the first uncertainties are statistical and the second systematic. These parameters are also determined separately for each polarization state, showing no evidence for polarization dependence. The results are combined with previous LHCb measurements using $pp$ collisions at center-of-mass energies of 7 and 8 TeV, yielding $\phi_{s\bar{s}s} = -0.074 \pm 0.069$ rad and $|\lambda|=1.009 \pm 0.030$. This is the most precise study of time-dependent $\it{CP}$ violation in a penguin-dominated $B$ meson decay. The results are consistent with $\it{CP}$ symmetry and with the Standard Model predictions

Precision measurement of CP\it{CP} violation in the penguin-mediated decay Bs0→ϕϕB_s^{0}\rightarrow\phi\phi

International audienceA flavor-tagged time-dependent angular analysis of the decay $B_s^{0}\rightarrow\phi\phi$ is performed using $pp$ collision data collected by the LHCb experiment at $\sqrt{s}=13$ TeV, the center-of-mass energy of 13 TeV, corresponding to an integrated luminosity of 6 fb^{-1}. The $\it{CP}$-violating phase and direct $\it{CP}$-violation parameter are measured to be $\phi_{s\bar{s}s} = -0.042 \pm 0.075 \pm 0.009$ rad and $|\lambda|=1.004\pm 0.030 \pm 0.009$, respectively, assuming the same values for all polarization states of the $\phi\phi$ system. In these results, the first uncertainties are statistical and the second systematic. These parameters are also determined separately for each polarization state, showing no evidence for polarization dependence. The results are combined with previous LHCb measurements using $pp$ collisions at center-of-mass energies of 7 and 8 TeV, yielding $\phi_{s\bar{s}s} = -0.074 \pm 0.069$ rad and $|\lambda|=1.009 \pm 0.030$. This is the most precise study of time-dependent $\it{CP}$ violation in a penguin-dominated $B$ meson decay. The results are consistent with $\it{CP}$ symmetry and with the Standard Model predictions

Search for Bc+→π+μ+μ−B_c^+\to\pi^+\mu^+\mu^- decays and measurement of the branching fraction ratio B(Bc+→ψ(2S)π+)/B(Bc+→J/ψπ+){\cal B}(B_c^+\to\psi(2S)\pi^+)/{\cal B}(B_c^+\to J/\psi \pi^+)

International audienceThe first search for nonresonant $B_c^+\to\pi^+\mu^+\mu^-$ decays is reported. The analysis uses proton-proton collision data collected with the LHCb detector between 2011 and 2018, corresponding to an integrated luminosity of 9 fb$^{-1}$. No evidence for an excess of signal events over background is observed and an upper limit is set on the branching fraction ratio ${\cal B}(B_c^+\to\pi^+\mu^+\mu^-)/{\cal B}(B_c^+\to J/\psi \pi^+) < 2.1\times 10^{-4}$ at $90\%$ confidence level. Additionally, an updated measurement of the ratio of the $B_c^+\to\psi(2S)\pi^+$ and $B_c^+\to J/\psi \pi^+$ branching fractions is reported. The ratio ${\cal B}(B_c^+\to\psi(2S)\pi^+)/{\cal B}(B_c^+\to J/\psi \pi^+)$ is measured to be $0.254\pm 0.018 \pm 0.003 \pm 0.005$, where the first uncertainty is statistical, the second systematic, and the third is due to the uncertainties on the branching fractions of the leptonic $J/\psi$ and $\psi(2S)$ decays. This measurement is the most precise to date and is consistent with previous LHCb results

Improved measurement of CPCP violation parameters in Bs0→J/ψK+K−B_s^0\to J/\psi K^+K^- decays in the vicinity of the ϕ(1020)\phi(1020) resonance

The decay-time-dependent $CP$ asymmetry in $B_s^0\to J/\psi(\to \mu^+\mu^-) K^+ K^-$ decays is measured using proton-proton collision data, corresponding to an integrated luminosity of $6 {\rm fb}^{-1}$, collected with the LHCb detector at a center-of-mass energy of 13 TeV. Using a sample of approximately 349 000 $B_s^0$ signal decays with an invariant $K^+ K^-$ mass in the vicinity of the $\phi(1020)$ resonance, the $CP$-violating phase $\phi_s$ is measured, along with the difference in decay widths of the light and heavy mass eigenstates of the $B_s^0$-$\overline{B}_s^0$ system, $\Delta\Gamma_s$, and the difference of the average $B_s^0$ and $B^0$ meson decay widths, $\Gamma_s-\Gamma_d$. The values obtained are $\phi_s = -0.039 \pm 0.022 \pm 0.006$ rad, $\Delta\Gamma_s = 0.0845 \pm 0.0044 \pm 0.0024 ~{\rm ps}^{-1}$ and $\Gamma_s-\Gamma_d = -0.056^{\:+\:0.0013}_{\:-\:0.0015} \pm 0.0014 ~{\rm ps}^{-1}$, where the first uncertainty is statistical and the second systematic. These are the most precise single measurements to date and are consistent with expectations based on the Standard Model and with the previous LHCb analyses of this decay. These results are combined with previous independent LHCb measurements. The phase $\phi_s$ is also measured independently for each polarization state of the $K^+K^-$ system and shows no evidence for polarization dependence.The decay-time-dependent $CP$ asymmetry in $B^0_s\to J/\psi(\to \mu^{+}\mu^{-}) K^{+}K^{-}$ decays is measured using proton-proton collision data, corresponding to an integrated luminosity of 6 $fb^{-1}$, collected with the LHCb detector at a center-of-mass energy of 13 TeV. Using a sample of approximately 349 000 $B^{0}_{s}$ signal decays with an invariant $K^{+}K^{-}$ mass in the vicinity of the $\phi(1020)$ resonance, the $CP$-violating phase $\phi_s$ is measured, along with the difference in decay widths of the light and heavy mass eigenstates of the $B^0_s$-$\bar{B}^0_s$ system, $\Delta\Gamma_s$, and the difference of the average $B^0_s$ and $B^0$ meson decay widths, $\Gamma_s-\Gamma_d$. The values obtained are $\phi_s = \ -0.039 \pm 0.022 \pm 0.006$ rad, $\Delta\Gamma_s = 0.0845 \pm 0.0044 \pm 0.0024$ ps$^{-1}$ and $\Gamma_s-\Gamma_d = -0.0056 ^{+ 0.0013}_{-0.0015} \pm 0.0014$ ps$^{-1}$, where the first uncertainty is statistical and the second systematic. These are the most precise single measurements to date and are consistent with expectations based on the Standard Model and with the previous LHCb analyses of this decay. These results are combined with previous independent LHCb measurements. The phase $\phi_s$ is also measured independently for each polarization state of the $K^{+}K^{-}$ system and shows no evidence for polarization dependence

Measurement of prompt D+D^+ and Ds+D^+_{s} production in pPbp\mathrm{Pb} collisions at sNN=5.02 \sqrt {s_{\mathrm{NN}}}=5.02\,TeV

International audienceThe production of prompt $D^+$ and $D^+_{s}$ mesons is studied in proton-lead collisions at a centre-of-mass energy of $\sqrt {s_{\mathrm{NN}}}=5.02\,$TeV. The data sample corresponding to an integrated luminosity of $(1.58\pm0.02)\mathrm{nb}^{-1}$ is collected by the LHCb experiment at the LHC. The differential production cross-sections are measured using $D^+$ and $D^+_{s}$ candidates with transverse momentum in the range of $0<p_{\mathrm{T}} <14\,\mathrm{GeV}/c$ and rapidities in the ranges of $1.5<y^*<4.0$ and $-5.0<y^*<-2.5$ in the nucleon-nucleon centre-of-mass system. For both particles, the nuclear modification factor and the forward-backward production ratio are determined. These results are compared with theoretical models that include initial-state nuclear effects. In addition, measurements of the cross-section ratios between $D^+$, $D^+_{s}$ and $D^0$ mesons are presented, providing a baseline for studying the charm hadronization in lead-lead collisions at LHC energies

Measurement of prompt D+D^+ and Ds+D^+_{s} production in pPbp\mathrm{Pb} collisions at sNN=5.02 \sqrt {s_{\mathrm{NN}}}=5.02\,TeV

International audienceThe production of prompt $D^+$ and $D^+_{s}$ mesons is studied in proton-lead collisions at a centre-of-mass energy of $\sqrt {s_{\mathrm{NN}}}=5.02\,$TeV. The data sample corresponding to an integrated luminosity of $(1.58\pm0.02)\mathrm{nb}^{-1}$ is collected by the LHCb experiment at the LHC. The differential production cross-sections are measured using $D^+$ and $D^+_{s}$ candidates with transverse momentum in the range of $0<p_{\mathrm{T}} <14\,\mathrm{GeV}/c$ and rapidities in the ranges of $1.5<y^*<4.0$ and $-5.0<y^*<-2.5$ in the nucleon-nucleon centre-of-mass system. For both particles, the nuclear modification factor and the forward-backward production ratio are determined. These results are compared with theoretical models that include initial-state nuclear effects. In addition, measurements of the cross-section ratios between $D^+$, $D^+_{s}$ and $D^0$ mesons are presented, providing a baseline for studying the charm hadronization in lead-lead collisions at LHC energies

Observation of the Bc+→J/ψπ+π0B_c^+ \to J/\psi \pi^+ \pi^0 decay

International audienceThe first observation of the $B_c^+ \to J/\psi \pi^+ \pi^0$ decay is reported with high significance using proton-proton collision data, corresponding to an integrated luminosity of 9fb$^{-1}$, collected with the LHCb detector at centre-of-mass energies of 7, 8, and 13 TeV. The ratio of its branching fraction relative to the $B_c^+ \to J/\psi \pi^+$ channel is measured to be $$\frac{ {\cal{B}}( B_c^+ \to J/\psi \pi^+\pi^0 ) } { {\cal{B}}( B_c^+ \to J/\psi \pi^+ ) } = 2.80 \pm 0.15 \pm 0.11 \pm 0.16 \,,$$ where the first uncertainty is statistical, the second systematic and the third related to imprecise knowledge of the branching fractions for $B^+ \to J/\psi K^{*+}$ and $B^+ \to J/\psi K^+$ decays, which are used to determine the $\pi^0$ detection efficiency. The $\pi^+\pi^0$ mass spectrum is found to be consistent with the dominance of an intermediate $\rho^+$ contribution in accordance with a model based on QCD factorisation