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## 2021 Vol. 45, No. 2

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2021, 45(2): 021001. doi: 10.1088/1674-1137/abccab
Abstract:
Nucleon properties and structure should be modified by short-range correlations (SRC) among nucleons. By analyzing SRC ratio data, we extract the mass of a nucleon in an SRC pair and the expected number of pn-SRC pairs in deuterium, under the assumption that the SRC nucleon mass is universal for different nuclei. The nucleon mass of a two-nucleon SRC pair is \begin{document}$m_{\rm{SRC}}$\end{document}= 852 ± 18 MeV, and the number of pn-SRC pairs in deuterium is \begin{document}$n^{d}_{\rm{SRC}}$\end{document}=0.021 ± 0.005. The mass deficit of the strongly overlapping nucleon can be explained by the trace anomaly contribution to the mass in QCD or alternatively by the vacuum energy in the MIT bag model.
2021, 45(2): 021002. doi: 10.1088/1674-1137/abce14
Abstract:
The Geiger-Nuttall (GN) law of α decay is commonly explained in terms of the quantum tunneling phenomenon. In this study, we show that such an explanation is actually not enough regarding the α particle clustering. Such a conclusion is drawn after exploring the involved coefficients of the GN law based on the conventional description of α decay, namely the formation of an α cluster and its subsequent penetration. The specific roles of the two former processes, in the GN law, manifest themselves via the systematical analysis of the calculated and experimental α decay half-lives versus the decay energies across the \begin{document}$Z=82$\end{document} and \begin{document}$N=126$\end{document} shell closures. The α-cluster preformation probability is then found to behave in a GN-like pattern. This previously ignored point is explicitly demonstrated as the product of an interplay between the mean-field and pairing effect, which in turn reveals the structural influence on the formation of the α cluster in a simple and clear manner. In addition to providing an effective method to evaluate the amount of surface α clustering in heavy nuclei, the present conjecture supports other theoretical treatments of the α preformation probability.
2021, 45(2): 021003. doi: 10.1088/1674-1137/abcfaa
Abstract:
The \begin{document}$X_0(2900)$\end{document}, recently observed by the LHCb Collaboration in the \begin{document}$D^-K^+$\end{document} invariant mass of the \begin{document}$B^+\to D^+D^-K^+$\end{document} process, is the first exotic candidate with four different flavors, beginning a new era for the hadron community. Under the assumption that the \begin{document}$X_0(2900)$\end{document} is a \begin{document}$I(J^P)=0(0^+)$\end{document} \begin{document}$\bar{D}^*K^*$\end{document} hadronic molecule, we extracted the whole heavy-quark symmetry multiplet formed by the \begin{document}$\left(\bar{D},\bar{D}^*\right)$\end{document} doublet and the \begin{document}$K^*$\end{document} meson. For the bound state case, there would be two additional \begin{document}$I(J^P)=0(1^+)$\end{document} hadronic molecules associated with the \begin{document}$\bar{D}K^*$\end{document} and \begin{document}$\bar{D}^*K^*$\end{document} channels, as well as one additional \begin{document}$I(J^P)=0(2^+)$\end{document} \begin{document}$\bar{D}^*K^*$\end{document} molecule. In the light quark limit, they are \begin{document}$36.66~{\rm{MeV}}$\end{document} and \begin{document}$34.22~{\rm{MeV}}$\end{document} below the \begin{document}$\bar{D}K^*$\end{document} and \begin{document}$\bar{D}^*K^*$\end{document} thresholds, respectively, which are unambiguously fixed by the mass position of the \begin{document}$X_0(2900)$\end{document}. For the virtual state case, there would be one additional \begin{document}$I(J^P)=0(1^+)$\end{document} hadronic molecule, strongly coupled to the \begin{document}$\bar{D}K^*$\end{document} channel, and one additional \begin{document}$I(J^P)=0(2^+)$\end{document} \begin{document}$\bar{D}^*K^*$\end{document} molecule. Searching for these heavy quark spin partners will help shed light on the nature of the \begin{document}$X_0(2900)$\end{document}.
Abstract:
We discuss the proper definition for the chiral crossover at finite temperature, based on Goldstone's theorem. Different from the commonly used maximum change in chiral condensate, we propose defining the crossover temperature using the Mott transition of pseudo-Goldstone bosons, which, by definition, guarantees Goldstone's theorem. We analytically and numerically demonstrate this property in the frame of a Pauli-Villars regularized NJL model. In an external magnetic field, we find that the Mott transition temperature shows an inverse magnetic catalysis effect.
2021, 45(2): 022001. doi: 10.1088/1674-1137/abcd8e
Abstract:
Hexaquarks constitute a natural extension of complex quark systems, just as tetra- and pentaquarks do. To this end, the current status of \begin{document}$d^*(2380)$\end{document} in both experiment and theory is reviewed. Recent high-precision measurements in the nucleon-nucleon channel and analyses thereof have established \begin{document}$d^*(2380)$\end{document} as an indisputable resonance in the long-sought dibaryon channel. Important features of this \begin{document}$I(J^P) = 0(3^+)$\end{document} state are its narrow width and deep binding relative to the \begin{document}$\Delta(1232)\Delta(1232)$\end{document} threshold. Its decay branchings favor theoretical calculations predicting a compact hexaquark nature of this state. We review the current status of experimental and theoretical studies on \begin{document}$d^*(2380)$\end{document} as well as new physics aspects it may bring in future. In addition, we review the situation at the \begin{document}$\Delta(1232) N$\end{document} and \begin{document}$N^*(1440)N$\end{document} thresholds, where evidence for a number of resonances of presumably molecular nature has been found – similar to the situation in charmed and beauty sectors. Finally, we briefly discuss the situation of dibaryon searches in the flavored quark sectors.
2021, 45(2): 023001. doi: 10.1088/1674-1137/abcf1f
Abstract:
Precise determination of the \begin{document}$B_c \to \tau\nu_\tau$\end{document} branching ratio provides an advantageous opportunity for understanding the electroweak structure of the Standard Model, measuring the CKM matrix element \begin{document}$|V_{cb}|$\end{document}, and probing new physics models. In this paper, we discuss the potential of measuring the process \begin{document}$B_c \to \tau\nu_\tau$\end{document} with \begin{document}$\tau$\end{document} decaying leptonically at the proposed Circular Electron Positron Collider (CEPC). We conclude that during the Z pole operation, the channel signal can achieve five-\begin{document}$\sigma$\end{document} significance with \begin{document}$\sim 10^9$\end{document} Z decays, and the signal strength accuracies for \begin{document}$B_c \to \tau\nu_\tau$\end{document} can reach around 1% level at the nominal CEPC Z pole statistics of one trillion Z decays, assuming the total \begin{document}$B_c \to \tau \nu_\tau$\end{document} yield is \begin{document}$3.6 \times 10^6$\end{document}. Our theoretical analysis indicates the accuracy could provide a strong constraint on the general effective Hamiltonian for the \begin{document}$b \to c\tau\nu$\end{document} transition. If the total \begin{document}$B_c$\end{document} yield can be determined to \begin{document}${\cal{O}}$\end{document}(1%) level of accuracy in the future, these results also imply \begin{document}$|V_{cb}|$\end{document} could be measured up to \begin{document}${\cal{O}}$\end{document}(1%) level of accuracy.
2021, 45(2): 023002. doi: 10.1088/1674-1137/abcfab
Abstract:
Using a dedicated data sample taken in 2018 on the J/ψ peak, we perform a detailed study of the trigger efficiencies of the BESIII detector. The efficiencies are determined from three representative physics processes, namely Bhabha scattering, dimuon production and generic hadronic events with charged particles. The combined efficiency of all active triggers approaches 100% in most cases, with uncertainties small enough not to affect most physics analyses.
2021, 45(2): 023003. doi: 10.1088/1674-1137/abd16d
Abstract:
A sensitivity study on the measurement of the CKM angle \begin{document}$\gamma$\end{document} from \begin{document}${{B^0_s}}\rightarrow \tilde{D}^{(*)0}\phi$\end{document} decays is conducted using the D-meson reconstructed in the quasi flavour-specific modes \begin{document}$K\pi$\end{document}, \begin{document}$K3\pi$\end{document}, and \begin{document}$K\pi{{\pi^{0}}}$\end{document}, as well as CP-eigenstate modes \begin{document}$KK$\end{document} and \begin{document}$\pi\pi$\end{document}, where the notation \begin{document}$\tilde{D}^0$\end{document} corresponds to a \begin{document}${{D^0}}$\end{document} or \begin{document}${{{\kern 0.2em\overline{\kern -0.2em D}{}}{}^0}}$\end{document} meson. The LHCb experiment is presented as a use case. A statistical uncertainty of approximately \begin{document}$8^{\circ}-19^{\circ}$\end{document} can be achieved with the \begin{document}$pp$\end{document} collision data collected in the LHCb experiment from 2011 to 2018. The sensitivity to \begin{document}$\gamma$\end{document} should be of the order \begin{document}$3^{\circ}-8^{\circ}$\end{document} after accumulating 23 fb-1 of \begin{document}$pp$\end{document} collision data by 2025, whereas it is expected to improve further by 300 fb-1 by the second half of the 2030 decade. The accuracy is dependent on the strong parameters \begin{document}${{r^{(*)}_{B}}}$\end{document} and \begin{document}${{\delta^{(*)}_{B}}}$\end{document}, which together with \begin{document}$\gamma$\end{document}describe the interference between the leading amplitudes of the \begin{document}${{B^0_s}}\rightarrow \tilde{D}^{(*)0}\phi$\end{document} decays.
2021, 45(2): 023004. doi: 10.1088/1674-1137/abd92a
Abstract:
The Jiangmen Underground Neutrino Observatory (JUNO) features a 20 kt multi-purpose underground liquid scintillator sphere as its main detector. Some of JUNO's features make it an excellent location for \begin{document}$^8$\end{document}B solar neutrino measurements, such as its low-energy threshold, high energy resolution compared with water Cherenkov detectors, and much larger target mass compared with previous liquid scintillator detectors. In this paper, we present a comprehensive assessment of JUNO's potential for detecting \begin{document}$^8$\end{document}B solar neutrinos via the neutrino-electron elastic scattering process. A reduced 2 MeV threshold for the recoil electron energy is found to be achievable, assuming that the intrinsic radioactive background \begin{document}$^{238}$\end{document}U and \begin{document}$^{232}$\end{document}Th in the liquid scintillator can be controlled to 10\begin{document}$^{-17}$\end{document} g/g. With ten years of data acquisition, approximately 60,000 signal and 30,000 background events are expected. This large sample will enable an examination of the distortion of the recoil electron spectrum that is dominated by the neutrino flavor transformation in the dense solar matter, which will shed new light on the inconsistency between the measured electron spectra and the predictions of the standard three-flavor neutrino oscillation framework. If \begin{document}$\Delta m^{2}_{21} = 4.8\times10^{-5}\; (7.5\times10^{-5})$\end{document} eV\begin{document}$^{2}$\end{document}, JUNO can provide evidence of neutrino oscillation in the Earth at approximately the 3\begin{document}$\sigma$\end{document} (2\begin{document}$\sigma$\end{document}) level by measuring the non-zero signal rate variation with respect to the solar zenith angle. Moreover, JUNO can simultaneously measure \begin{document}$\Delta m^2_{21}$\end{document} using \begin{document}$^8$\end{document}B solar neutrinos to a precision of 20% or better, depending on the central value, and to sub-percent precision using reactor antineutrinos. A comparison of these two measurements from the same detector will help understand the current mild inconsistency between the value of \begin{document}$\Delta m^2_{21}$\end{document} reported by solar neutrino experiments and the KamLAND experiment.
2021, 45(2): 023101. doi: 10.1088/1674-1137/abca2c
Abstract:
In this study, we obtain the universal function corresponding to the diffractive process and show that the cross section exhibits geometrical scaling. It is observed that diffractive theory according to the color dipole approach at small-x is a convenient framework that reveals the color transparency and saturation phenomena. We also calculate the contribution of heavy quark production in the diffractive cross section at high energy that is determined by the small size dipole configuration. The ratio of the diffractive cross section to the total cross section in electron-proton collision is the other important quantity that is computed in this work.
2021, 45(2): 023102. doi: 10.1088/1674-1137/abcc58
Abstract:
The resonance state of the \begin{document}$\Delta$\end{document} baryon, which exists in four isospin (\begin{document}$I= {3}/{2}$\end{document}) states, has been studied using the hypercentral constituent quark model (hCQM) with a simple linear potential with added first order correction. The calculated data ranges for 1S-5S, 1P-5P, 1D-4D and 1F-2F are given, with possible spin-parity assignments for all the states. The magnetic moments have also been obtained for all four configurations. The \begin{document}$N\pi$\end{document} decay channel width has been calculated for a few states. The linear nature of the data has been verified through Regge trajectories.
2021, 45(2): 023103. doi: 10.1088/1674-1137/abcc5b
Abstract:
We study the scattering of \begin{document}$J/\psi$\end{document} - \begin{document}$J/\psi$\end{document} mesons using quadratic and Cornell potentials in our tetraquark (\begin{document}${{c\bar cc\bar c}}$\end{document}) system. The system’s wavefunction in the restricted gluonic basis, which is written by utilizing the adiabatic approximation and Hamiltonian, is used via a quark potential model. The resonating group technique is used to obtain the integral equations, which are solved to obtain the unknown inter-cluster dependence of the total wavefunction of our tetraquark system. T-Matrix elements are calculated from the solutions, and eventually, the scattering cross sections are obtained using the two potentials. We compare these cross sections and find that the magnitudes of scattering cross sections of quadratic potential are higher than the Cornell potential.
Abstract:
QCD theory predicts the existence of glueballs, but so far all experimental endeavors have failed to identify any such states. To remedy this discrepancy between QCD, which has proven to be a successful theory for strong interactions, and the failure of experimental searches for glueballs, one is tempted to accept the promising interpretation that the glueballs mix with regular \begin{document}$q\bar q$\end{document} states of the same quantum numbers. The lattice estimate of the masses of pure \begin{document}$0^{++}$\end{document} glueballs ranges from 1 to 2 GeV, which is the region of the \begin{document}$f_0$\end{document} family. Thus many authors suggest that the \begin{document}$f_0$\end{document} mesonic series is an ideal place to study possible mixtures of glueballs and \begin{document}$q\bar q$\end{document}. In this paper, following the strategy proposed by Close, Farrar and Li, we try to determine the fraction of glueball components in \begin{document}$f_0$\end{document} mesons using the measured mass spectra and the branching ratios of \begin{document}$J/\psi$\end{document} radiative decays into \begin{document}$f_0$\end{document} mesons. Since the pioneering papers by Close et al., more than 20 years have elapsed and more accurate measurements have been done by several experimental collaborations, so it is time to revisit this interesting topic using new data. We suppose \begin{document}$f_0(500)$\end{document} and \begin{document}$f_0(980)$\end{document} to be pure quark states, while for \begin{document}$f_0(1370)$\end{document}, \begin{document}$f_0(1500)$\end{document} and \begin{document}$f_0(1710)$\end{document}, to fit both the experimental data of \begin{document}$J/\psi$\end{document} radiative decay and their mass spectra, glueball components are needed. Moreover, the mass of the pure \begin{document}$0^{++}$\end{document} glueball is phenomenologically determined.
2021, 45(2): 023105. doi: 10.1088/1674-1137/abcd2c
Abstract:
Measuring the fermion Yukawa coupling constants is important for understanding the origin of the fermion masses and their relationship with spontaneously electroweak symmetry breaking. In contrast, some new physics (NP) models change the Lorentz structure of the Yukawa interactions between standard model (SM) fermions and the SM-like Higgs boson, even in their decoupling limit. Thus, the precise measurement of the fermion Yukawa interactions is a powerful tool of NP searching in the decoupling limit. In this work, we show the possibility of investigating the Lorentz structure of the bottom-quark Yukawa interaction with the 125 GeV SM-like Higgs boson for future \begin{document}$e^+e^-$\end{document} colliders.
2021, 45(2): 023106. doi: 10.1088/1674-1137/abcd2f
Abstract:
Neutrinos stand out among the elementary particles because of their unusually small masses. Various seesaw mechanisms attempt to explain this fact. In this work, applying insights from matrix theory, we are in a position to treat variants of seesaw mechanisms in a general manner. Specifically, using Weyl's inequalities, we discuss and rigorously prove under which conditions the seesaw framework leads to a mass spectrum with exactly three light neutrinos. We find an estimate of the mass of heavy neutrinos to be the mass obtained by neglecting light neutrinos, shifted at most by the maximal strength of the coupling to the light neutrino sector. We provide analytical conditions allowing one to prescribe that precisely two out of five neutrinos are heavy. For higher-dimensional cases the inverse eigenvalue methods are used. In particular, for the CP-invariant scenarios we show that if the neutrino sector has a valid mass matrix after neglecting the light ones, i.e. if the respective mass submatrix is positive definite, then large masses are provided by matrices with large elements accumulated on the diagonal. Finally, the Davis-Kahan theorem is used to show how masses affect the rotation of light neutrino eigenvectors from the standard Euclidean basis. This general observation concerning neutrino mixing, together with results on the mass spectrum properties, opens directions for further neutrino physics studies using matrix analysis.
2021, 45(2): 023107. doi: 10.1088/1674-1137/abcd30
Abstract:
We study the relation between the symmetry group of a Feynman diagram and its reduced diagrams. We then prove that the counterterms in the BPHZ renormalization scheme are consistent with adding counterterms to the interaction Hamiltonian in all cases, including that of Feynman diagrams with symmetry factors.
2021, 45(2): 023108. doi: 10.1088/1674-1137/abcd8c
Abstract:
We consider the positivity bounds on dimension-8 four-electron operators and study two related phenomenological aspects at future lepton colliders. First, if positivity is violated, probing such violations will revolutionize our understanding of the fundamental pillars of quantum field theory and the S-matrix theory. We observe that positivity violation at scales of 1-10 TeV can potentially be probed at future lepton colliders even if one assumes that dimension-6 operators are also present. Second, the positive nature of the dimension-8 parameter space often allows us to either directly infer the existence of UV-scale particles together with their quantum numbers or exclude them up to certain scales in a model-independent way. In particular, dimension-8 positivity plays an important role in the test of the Standard Model. If no deviations from the Standard Model are observed, it allows for simultaneous exclusion limits on all kinds of potential UV-complete models. Unlike the dimension-6 case, these limits apply regardless of the UV model setup and cannot be removed by possible cancellations among various UV contributions. This thus consists of a novel and universal test to confirm the Standard Model. We demonstrate with realistic examples how all the previously mentioned possibilities, including the test of positivity violation, can be achieved. Hence, we provide an important motivation for studying dimension-8 operators more comprehensively.
2021, 45(2): 023109. doi: 10.1088/1674-1137/abcd8f
Abstract:
On a lattice with 2+1-flavor dynamical domain-wall fermions at the physical pion mass, we calculate the decay constants of \begin{document}$D_{s}^{(*)}$\end{document}, \begin{document}$D^{(*)}$\end{document}, and \begin{document}$\phi$\end{document}. The lattice size is \begin{document}$48^3\times96$\end{document}, which corresponds to a spatial extension of \begin{document}$\sim5.5$\end{document} fm, with a lattice spacing of \begin{document}$a\approx 0.114$\end{document} fm. For the valence light, strange, and charm quarks, we use overlap fermions at several mass points close to their physical values. Our results at the physical point are \begin{document}$f_D = 213(5)$\end{document} MeV, \begin{document}$f_{D_s} = 249(7)$\end{document} MeV, \begin{document}$f_{D^*} = 234(6)$\end{document} MeV, \begin{document}$f_{D_s^*} = 274(7)$\end{document} MeV, and \begin{document}$f_\phi = 241(9)$\end{document} MeV. The couplings of \begin{document}$D^*$\end{document} and \begin{document}$D_s^*$\end{document} to the tensor current (\begin{document}$f_V^T$\end{document}) can be derived from ratios \begin{document}$f_{D^*}^T/f_{D^*} = 0.91(4)$\end{document} and \begin{document}$f_{D_s^*}^T/f_{D_s^*} = 0.92(4)$\end{document}, respectively, which are the first lattice quantum chromodynamics (QCD) results. We also obtain ratios \begin{document}$f_{D^*}/f_D = 1.10(3)$\end{document} and \begin{document}$f_{D_s^*}/f_{D_s} = 1.10(4)$\end{document}, which reflect the size of heavy quark symmetry breaking in charmed mesons. Ratios \begin{document}$f_{D_s}/f_{D} = 1.16(3)$\end{document} and \begin{document}$f_{D_s^*}/f_{D^*} = 1.17(3)$\end{document} can be taken as a measure of SU(3) flavor symmetry breaking.
2021, 45(2): 023110. doi: 10.1088/1674-1137/abcd90
Abstract:
We study the potential of the LHCb 13 TeV single W± and Z boson pseudo-data for constraining the parton distribution functions (PDFs) of the proton. As an example, we demonstrate the sensitivity of the LHCb 13 TeV data, collected with integrated luminosities of 5 \begin{document}${\rm{fb}}^{-1}$\end{document} and 300 \begin{document}${\rm{fb}}^{-1}$\end{document}, to reducing the PDF uncertainty bands of the CT14HERA2 PDFs, using the error PDF updating package EPUMP. The sensitivities of various experimental observables are compared. Generally, sizable reductions in PDF uncertainties can be observed in the 300 \begin{document}${\rm{fb}}^{-1}$\end{document} data sample, particularly in the small-x region. The double-differential cross section measurement of Z boson pT and rapidity can greatly reduce the uncertainty bands of u and d quarks in almost the whole x range, as compared to various single observable measurements.
2021, 45(2): 023111. doi: 10.1088/1674-1137/abce10
Abstract:
By applying the Error PDF Updating Method, we analyze the impact of the absolute and normalized single differential cross-sections for top-quark pair production data from the ATLAS and CMS experiments at the Large Hadron Collider, at a center-of-mass energy of \begin{document}$\sqrt{s} = 8$\end{document} TeV, on the CT14HERA2 PDFs. We find that the top quark pair single differential distributions provide minor constraints on the CT14HERA2 gluon PDF when the nominal CT14HERA2 inclusive jet production data are included in the fit. Larger constraints on the gluon distribution are present when the jet data are removed (CT14HERA2mJ) and/or when increased weights are given to the top data in the CT14HERA2 fits. The weighted \begin{document}$t\bar t$\end{document} data provide significant constraints on the CT14HERA2mJ gluon PDF, which are comparable to those obtained from inclusive jet production data. Furthermore, we examine the top quark mass sensitivity of the top-quark pair single differential distributions.
2021, 45(2): 023112. doi: 10.1088/1674-1137/abce11
Abstract:
Inclusive \begin{document}$\Upsilon(1S,2S,3S)$\end{document} photoproduction at the future Circular-Electron-Positron-Collider (CEPC) is studied, using the non-relativistic quantum chromodynamics (NRQCD) factorization formalism. Including the contributions from both direct and resolved photons, we present different distributions for the \begin{document}$\Upsilon(1S,2S,3S)$\end{document} production. Our results suggest that there will be considerable events, implying that well measurements of the \begin{document}$\Upsilon$\end{document} photoproduction can be performed to further study heavy quarkonium physics at electron-positron colliders, in addition to hadron colliders. This supplemental study is very important for clarifying the current situation regarding the heavy quarkonium production mechanism.
2021, 45(2): 023113. doi: 10.1088/1674-1137/abce4f
Abstract:
The next-to minimal supersymmetric standard model (NMSSM) with non-universal Higgs masses, i.e., the semi-constrained NMSSM (scNMSSM), extends the minimal supersymmetric standard model (MSSM) by a singlet superfield and assumes universal conditions, except for the Higgs sector. It can not only maintain the simplicity and grace of the fully constrained MSSM and NMSSM and relieve the tension they have been facing since the discovery of the 125-GeV Higgs boson but also allow for an exotic phenomenon wherein the Higgs decay into a pair of light (\begin{document}$10\sim 60\;{\rm{GeV}}$\end{document}) singlet-dominated (pseudo)scalars (hereafter, in this paper, we use "scalar" for both scalars and pseudoscalars, considering pseudoscalars can also be called CP-odd scalars). This condition can be classified into three scenarios according to the identitiesof the SM-like Higgs and the light scalar: (i) the light scalar is CP-odd, and the SM-like Higgs is \begin{document}$h_2$\end{document}; (ii) the light scalar is CP-odd, and the SM-like Higgs is \begin{document}$h_1$\end{document}; and (iii) the light scalar is CP-even, and the SM-like Higgs is \begin{document}$h_2$\end{document}. In this work, we compare the three scenarios, checking the interesting parameter regions that lead to the scenarios, the mixing levels of the doublets and singlets, the tri-scalar coupling between the SM-like Higgs and a pair of light scalars, the branching ratio of Higgs decay to the light scalars, and sensitivities in the detection of the exotic decay at the HL-LHC and future lepton colliders such as CEPC, FCC-ee, and ILC. Finally, several interesting conclusions are drawn, which are useful for understanding the different delicate mechanisms of the exotic decay and designing colliders in future.
2021, 45(2): 023114. doi: 10.1088/1674-1137/abce50
Abstract:
We study self-conjugate dark matter (DM) particles interacting primarily with Standard Model (SM) leptons in an effective field theoretical framework. We consider SM gauge-invariant effective contact interactions between Majorana fermion, real scalar and real vector DM with leptons by evaluating the Wilson coefficients appropriate for interaction terms up to dimension 8, and obtain constraints on the parameters of the theory from the observed relic density, indirect detection observations and from the DM-electron scattering cross-sections in direct detection experiments. Low energy LEP data has been used to study sensitivity in the pair production of low mass (\begin{document}$\leqslant$\end{document} 80 GeV) DM particles. Pair production of DM particles of mass \begin{document}$\geqslant$\end{document} 50 GeV in association with mono-photons at the proposed ILC has rich potential to probe such effective operators.
2021, 45(2): 023115. doi: 10.1088/1674-1137/abcf20
Abstract:
Various quantum theories of gravity predict the existence of a minimal measurable length. In this paper, we study effects of the minimal length on the motion of a particle in the Rindler space under a harmonic potential. This toy model captures key features of particle dynamics near a black hole horizon and allows us to make three observations. First, we find that chaotic behavior becomes stronger with increases in minimal length effects, leading predominantly to growth in the maximum Lyapunov characteristic exponents, while the KAM curves on Poincaré surfaces of a section tend to disintegrate into chaotic layers. Second, in the presence of the minimal length effects, it can take a finite amount of Rindler time for a particle to cross the Rindler horizon, which implies a shorter scrambling time of black holes. Finally, the model shows that some Lyapunov characteristic exponents can be greater than the surface gravity of the horizon, violating the recently conjectured universal upper bound. In short, our results reveal that quantum gravity effects may make black holes prone to more chaos and faster scrambling.
2021, 45(2): 023116. doi: 10.1088/1674-1137/abcf22
Abstract:
In this work we calculate the mass spectrum of strangeonium up to the \begin{document}$3D$\end{document} multiplet within a nonrelativistic linear potential quark model. Furthermore, using the obtained wave functions, we also evaluate the strong decays of the strangeonium states with the \begin{document}$^3P_0$\end{document} model. Based on our successful explanations of the well established states \begin{document}$\phi(1020)$\end{document}, \begin{document}$\phi(1680)$\end{document}, \begin{document}$h_1(1415)$\end{document}, \begin{document}$f'_2(1525)$\end{document}, and \begin{document}$\phi_3(1850)$\end{document}, we further discuss the possible assignments of strangeonium-like states from experiments by combining our theoretical results with observations. It is found that some resonances, such as \begin{document}$f_2(2010)$\end{document} and \begin{document}$f_2(2150)$\end{document}, listed by the Particle Data Group, and \begin{document}$X(2062)$\end{document} and \begin{document}$X(2500)$\end{document}, newly observed by BESIII, may be interpreted as strangeonium states. The possibility of \begin{document}$\phi(2170)$\end{document} as a candidate for \begin{document}$\phi(3S)$\end{document} or \begin{document}$\phi(2D)$\end{document} cannot be excluded. We expect our results to provide useful references for looking for the missing \begin{document}$s\bar{s}$\end{document} states in future experiments.
2021, 45(2): 023117. doi: 10.1088/1674-1137/abcfac
Abstract:
We propose a new dihedral angle observable for measuring the CP property of the interaction between the top quark and Higgs boson in \begin{document}$t\bar{t}H$\end{document} production at the 14 TeV Large Hadron Collider (LHC). We consider two decay modes of the Higgs boson, \begin{document}$H\to b\bar{b}$\end{document} and \begin{document}$H\to \gamma\gamma$\end{document}, and demonstrate that the dihedral angle distribution is able to distinguish between the CP-even and the CP-odd hypothesis at a 95% confidence level, with an integrated luminosity of \begin{document}$\sim 180~{\rm{fb}}^{-1}$\end{document}.
2021, 45(2): 023118. doi: 10.1088/1674-1137/abcfae
Abstract:
We present a SUSY model with four Higgs doublets of the "private type," in which all fermion types (up, down, and charged leptons) obtain their masses from a different Higgs doublet \begin{document}$H_f\,( f = u_1, d, e)$\end{document}. The conditions for anomaly cancellation imply that the remaining Higgs doublet of the model (\begin{document}$H_{u_2}$\end{document}) must have the same hypercharge as \begin{document}$H_{u_1}$\end{document}, and thus, can only couple to up-type quarks, which opens the possibility to have FCNCs only in this sector. We study the Lagrangian of the model, and in particular, the Higgs potential, to identify the Higgs mass-eigenstates and their interactions; for the Yukawa matrices, we consider the four-texture case. We obtain constraints on the model parameters by using LHC measurements on the properties of the 125 GeV Higgs boson (h), and identify viable regions of the parameter space. Subsequently, these constraints are used to evaluate the prospects for detecting the FCNC decay mode \begin{document}$t \to ch$\end{document} at the future high-luminosity (HL) option for the LHC, which are compared with current limits from LHC-run2. Moreover, we evaluate the FCNC decay of the next heavier Higgs boson \begin{document}$H_2\to tc$\end{document}, which can typically reach \begin{document}$BR(H_2 \rightarrow tc) \approx {\cal{O}}(10^{-4}-10^{-5} )$\end{document}. The search for the signal at HL-LHC is also studied, and it is found that it may be detectable for specific regions of the model parameter space.
2021, 45(2): 023119. doi: 10.1088/1674-1137/abd01c
Abstract:
Using the light-cone QCD sum rules, we evaluate the magnetic moment of the \begin{document}$P_c(4312)$\end{document} pentaquark state by considering both the \begin{document}$\bar D\Sigma_c$\end{document} molecular and diquark-diquark-antiquark state, with quantum numbers \begin{document}$J^P = \dfrac{1}{2}^-$\end{document}. In the calculations, we use the diquark-diquark-antiquark and molecular form of the interpolating currents for the \begin{document}$P_c(4312)$\end{document} pentaquark and the distribution amplitudes of the photon. The numerical results for the magnetic moment obtained using the two different pictures are quite different from each other, which can be used to pin down the underlying structure of \begin{document}$P_c (4312)$\end{document}. Any experimental measurement of the magnetic moment in the near future will provide an understanding of the internal structure of this pentaquark state.
2021, 45(2): 023120. doi: 10.1088/1674-1137/abd084
Abstract:
The strong coupling constants of hadronic multiplets are fundamental parameters which carry information about the strong interactions among participating particles. These parameters can help us construct the hadron-hadron strong potential and gain information about the structure of the involved hadrons. Motivated by the recent observation of the doubly charmed \begin{document}$\Xi_{cc}$\end{document} state by LHCb, we determine the strong coupling constants among the doubly heavy spin-1/2 baryons, \begin{document}$\Xi^{(\prime)}_{QQ^\prime }$\end{document}, \begin{document}$\Omega^{(\prime)}_{QQ^\prime}$\end{document} and light pseudoscalar mesons, \begin{document}$\pi$\end{document}, K, \begin{document}$\eta$\end{document} and \begin{document}$\eta^\prime$\end{document} within the framework of the light cone QCD sum rules. The obtained results may help experimental groups in analysis of the related data at hadron colliders.
2021, 45(2): 023121. doi: 10.1088/1674-1137/abd085
Abstract:
The thermodynamics and overcharging problem in RN-AdS black holes with a cloud of strings and quintessence are investigated by the absorption of scalar particles and fermions in extended phase space. The cosmological constant is treated as the pressure of the black hole. The parameters related to quintessence and the cloud of strings are treated as thermodynamic variables. We find that the first law of thermodynamics is satisfied and the second law of thermodynamics is indefinite. Furthermore, we find that near-extremal and extremal black holes cannot be overcharged.
2021, 45(2): 023122. doi: 10.1088/1674-1137/abd087
Abstract:
The asymptotic symmetries in the Brans-Dicke theory are analyzed using Penrose's conformal completion method, which is independent of the coordinate system used. These symmetries, indeed, include supertranslations and Lorentz transformations for an asymptotically flat spacetime. With the Wald-Zoupas formalism, “conserved charges” and fluxes of the Bondi-Metzner-Sachs algebra are computed. The scalar degree of freedom contributes only to the Lorentz boost charge, even though it plays a role in various fluxes. The flux-balance laws are further applied to constrain the displacement memory, spin memory, and center-of-mass memory effects.
2021, 45(2): 024001. doi: 10.1088/1674-1137/abca1d
Abstract:
The thick-target yield of the 9Be(d, α0)7Li and 9Be(d, α1)7Li* reactions has been first directly measured over deuteron energies from 66 to 94 keV. The obtained S(Ei) of α0 and α1 have similar trends calculated by the thin-target yield, consistent with Yan’s report within the errors. Furthermore, the parametric expression of S(E) was obtained to calculate the theoretical thick target yield, and it roughly agrees with the experimental thick target yield.
2021, 45(2): 024002. doi: 10.1088/1674-1137/abce13
Abstract:
Novel measurements of the neutron energy spectra of the 9Be(d,n)10B reaction with a thick beryllium target are performed using a fast neutron time-of-flight (TOF) spectrometer for the neutron emission angles \begin{document}$\theta=0^\circ$\end{document} and \begin{document}$45^\circ$\end{document}, and the incident deuteron energies are 250 and 300 keV, respectively. The neutron contributions from the 9Be(d,n)10B reaction are distributed relatively independently for the ground state and the first, second, and third excited states of 10B. The branching ratios of the 9Be(d,n)10B reaction for the different excited states of 10B are obtained for the neutron emission angles \begin{document}$\theta=0^\circ$\end{document} and \begin{document}$45^\circ$\end{document}, and the incident deuteron energies are 250 and 300 keV, respectively. The branching ratio of the 9Be(d,n)10B reaction for the third excited state decreases with increase in the incident deuteron energy, and the branching ratios for the ground state and the second excited state increase with increase in the neutron emission angle.
2021, 45(2): 024003. doi: 10.1088/1674-1137/abce52
Abstract:
Heavy ion collisions near the Fermi energy produce a ‘freezout’ region where fragments appear and later decay, emitting mainly neutrons, protons, alpha particles, and gamma rays. These decay products carry information about the decaying nuclei. Fragmentation events may result in high yields of boson particles, especially alpha particles, and may carry important information about the nuclear Bose Einstein condensate (BEC). We study ‘in medium’ four α correlations and link them to the ‘fission’ of 16O in two 8Be in the ground state or 12C*(Hoyle state)+α. Using novel techniques for the correlation functions, we confirm the resonance of 16O at 15.2 MeV excitation energy, and the possibility of a lower resonance, close to 14.72 MeV. The latter resonance is the result of all α particles having 92 keV relative kinetic energies.
2021, 45(2): 024101. doi: 10.1088/1674-1137/abca1b
Abstract:
In the context of the double folding optical model, the strong refractive effect for elastic scattering of 11Li + 12C and 11Li + 28Si systems at incident energies of 29, 50, and 60 MeV/n is studied. Real folded potentials are generated based on a variety of nucleon-nucleon interactions with the suggested density distributions for the halo structure of 11Li nuclei. The rearrangement term (RT) of the extended realistic density dependent CDM3Y6 effective interaction is considered. The imaginary potential was taken in the traditional standard Woods-Saxon form. Satisfactory results for the calculated potentials are obtained, with a slight effect of the RT in CDM3Y6 potential. Successful reproduction with a normalization factor close to one for the observed angular distributions of the elastic scattering differential cross section has been achieved using the derived potentials. The obtained reaction cross-section is studied as a guide by extrapolating our calculations and previous results.
2021, 45(2): 024102. doi: 10.1088/1674-1137/abca2b
Abstract:
In this paper, we investigate the medium modifications of girth distributions for inclusive jets and \begin{document}$Z^0$\end{document} tagged jets with a small radius (\begin{document}$R=0.2$\end{document}) in Pb+Pb collisions with \begin{document}$\sqrt{s}=2.76$\end{document} TeV at the LHC. The partonic spectra in the initial hard scattering of elementary collisions are obtained by an event generator POWHEG+PYTHIA, which matches the next-to-leading order (NLO) matrix elements with parton showering, and the energy loss of a fast parton traversing the hot/dense QCD medium is calculated by Monte Carlo simulation within the higher-twist formalism of jet quenching in heavy-ion collisions. We present the model calculations of event normalized girth distributions for inclusive jets in p+p and Pb+Pb collisions at \begin{document}$\sqrt{s}=2.76$\end{document} TeV, which give good descriptions of ALICE measurements. It is shown that the girth distributions of inclusive jets in Pb+Pb are shifted to lower girth regions relative to those in p+p. Thus, the nuclear modification factor of girth distributions for inclusive jets is larger than unity at small girth regions and smaller than unity at large girth regions. This behavior results from softer fragments inside a jet as well as the fraction alteration of gluon/quark initiated jets in heavy-ion collisions. We further predict the girth distributions for \begin{document}$Z^0$\end{document} boson tagged jets in Pb+Pb collisions at \begin{document}$\sqrt{s}=2.76$\end{document} TeV and demonstrate that the medium modification on girth distributions for \begin{document}$Z^0$\end{document} tagged jets is less pronounced compared to that for inclusive jets because the dominant components of \begin{document}$Z^0$\end{document} tagged jets are quark-initiated jets.
2021, 45(2): 024103. doi: 10.1088/1674-1137/abcc59
Abstract:
We use the two lowest weight states to fit E2 strengths connecting the \begin{document}$0\leftrightarrow 2$\end{document} and \begin{document}$2\leftrightarrow 4$\end{document} transitions in \begin{document}$^{96,98}$\end{document}Mo. Our results confirm that the \begin{document}$2^+$\end{document} and \begin{document}$4^+$\end{document} states are maximally mixed, and that the \begin{document}$0^+$\end{document} states are weakly mixed in both nuclei. An appropriate Hamiltonian to represent the band mixing is found to be exactly solvable, and its eigenstates can be expressed as the basis vectors in the configuration mixing scheme and interacting boson model. The interacting boson model and coexistence mixing configuration under the solvable methods are suitable models for analyzing the band mixing with high accuracy.
2021, 45(2): 024104. doi: 10.1088/1674-1137/abcc5a
Abstract:
It is universally acknowledged that the Generalized Liquid Drop Model (GLDM) has two advantages over other α decay theoretical models: introduction of the quasimolecular shape mechanism and proximity energy. In the past few decades, the original proximity energy has been improved by numerous works. In the present work, the different improvements of proximity energy are examined when they are applied to the GLDM for enhancing the calculation accuracy and prediction ability of α decay half-lives for known and unsynthesized superheavy nuclei. The calculations of α half-lives have systematic improvements in reproducing experimental data after choosing a more suitable proximity energy for application to the GLDM. Encouraged by this, the α decay half-lives of even-even superheavy nuclei with Z=112-122 are predicted by the GLDM with a more suitable proximity energy. The predictions are consistent with calculations by the improved Royer formula and the universal decay law. In addition, the features of the predicted α decay half-lives imply that the next double magic nucleus after 208Pb is 298Fl.
2021, 45(2): 024105. doi: 10.1088/1674-1137/abcc5c
Abstract:
A radial basis function network (RBFN) approach is adopted for the first time to optimize the calculation of \begin{document}$\alpha$\end{document} decay half-life in the generalized liquid drop model (GLDM), while concurrently incorporating the surface diffuseness effect. The calculations presented herein agree closely with the experimental half-lives for 68 superheavy nuclei (SHN), achieving a remarkable reduction of 40% in the root-mean-square (rms) deviations of half-lives. Furthermore, using the RBFN method, the half-lives for four SHN isotopes, 252-288Rf, 272-310Fl, 286-316119, and 292-318120, are predicted using the improved GLDM with the diffuseness correction and the decay energies from WS4 and FRDM as inputs. Therefore, we conclude that the diffuseness effect should be embodied in the proximity energy. Moreover, increased application of neural network methods in nuclear reaction studies is encouraged.
2021, 45(2): 024106. doi: 10.1088/1674-1137/abcd8d
Abstract:
The ALICE Collaboration measured three- and four-pion Bose-Einstein correlations (BECs) for Pb-Pb collisions at the Large Hadron Collider (LHC). It is speculated that the observed significant suppression of multi-pion BECs is owing to a considerable degree of coherent pion emission in these collisions. Here, we study multi-pion BEC functions for granular sources with coherent pion-emission droplets. We find that the intercepts of the multi-pion correlation functions at the relative momenta near zero are sensitive to the number of droplets in the granular source. They decrease with the droplet number. The three-pion correlation functions for evolving granular sources with momentum-dependent partially coherent pion-emission droplets basically agree with the experimental data for Pb-Pb collisions at \begin{document}$\sqrt{s_{NN}}=2.76$\end{document} TeV at the LHC. However, the model results for the four-pion correlation function are inconsistent with the experimental data. Investigations into normalized multi-pion correlation functions of granular sources suggest an interesting enhancement of the normalized four-pion correlation function in the moderate relative-momentum region.
2021, 45(2): 024107. doi: 10.1088/1674-1137/abce12
Abstract:
The temperature dependence of the shell corrections to the energy \begin{document}$\delta E_{\rm{shell}}$\end{document}, entropy \begin{document}$T \delta S_{\rm{shell}}$\end{document}, and free energy \begin{document}$\delta F_{\rm{shell}}$\end{document} is studied by employing the covariant density functional theory for closed-shell nuclei. Taking \begin{document}$^{144}$\end{document}Sm as an example, studies have shown that, unlike the widely-used exponential dependence \begin{document}$\exp(-E^*/E_d)$\end{document}, \begin{document}$\delta E_{\rm{shell}}$\end{document} exhibits a non-monotonous behavior, i.e., first decreasing 20% approaching a temperature of \begin{document}$0.8$\end{document} MeV, and then fading away exponentially. Shell corrections to both free energy \begin{document}$\delta F_{\rm{shell}}$\end{document} and entropy \begin{document}$T \delta S_{\rm{shell}}$\end{document} can be approximated well using the Bohr-Mottelson forms \begin{document}$\tau/\sinh(\tau)$\end{document} and \begin{document}$[\tau \coth(\tau)-1]/\sinh(\tau)$\end{document}, respectively, in which \begin{document}$\tau\propto T$\end{document}. Further studies on the shell corrections in other closed-shell nuclei, \begin{document}$^{100}$\end{document}Sn and \begin{document}$^{208}$\end{document}Pb, are conducted, and the same temperature dependencies are obtained.
2021, 45(2): 024108. doi: 10.1088/1674-1137/abd01e
Abstract:
In the present work, a two-parameter empirical formula is proposed, based on the Geiger-Nuttall law, to study two-proton (\begin{document}$2p$\end{document}) radioactivity. Using this formula, the calculated \begin{document}$2p$\end{document} radioactivity half-lives are in good agreement with the experimental data as well as with calculated results obtained by Goncalves et al. [Phys. Lett. B 774, 14 (2017)] using the effective liquid drop model (ELDM), Sreeja et al. [Eur. Phys. J. A 55, 33 (2019)] using a four-parameter empirical formula, and Cui et al. [Phys. Rev. C 101: 014301 (2020)] using a generalized liquid drop model (GLDM). In addition, this two-parameter empirical formula is extended to predict the half-lives of 22 possible \begin{document}$2p$\end{document} radioactivity candidates with \begin{document}$2p$\end{document} radioactivity released energy \begin{document}$Q_{2p} > 0$\end{document}, obtained from the latest evaluated atomic mass table AME2016. The predicted results are highly consistent with those obtained using other theoretical models such as the ELDM, GLDM and four-parameter empirical formula.
2021, 45(2): 024109. doi: 10.1088/1674-1137/abd083
Abstract:
Exploiting the concept of the dinuclear system, the interaction potential energy of two fragments that are quite close to each other is analyzed. A semi-classical method is used to calculate fission fragment yields using a simplified two-dimensional scission-point model. By considering the tip-to-tip orientation at the scission point of the fission process, we investigate the mass, charge, and kinetic-energy distributions of the fission fragments, for excitation energies in the 0-20 MeV range. Our results show that the fission fragment distributions are reproduced quite well, including the recent experimental results for the isotone chain [D Ramos et al. Phys. Rev. C 97, 054612 (2018)]. Thus, the simplified model is useful for multi-parameter global measurements of fission products.
2021, 45(2): 024110. doi: 10.1088/1674-1137/abd086
Abstract:
The mean total kinetic energy as a function of fission fragments, the <TKE> distribution, is presented for neutron-induced fission of \begin{document}$^{235-239}$\end{document}Np using the scission point model. The calculated results of <TKE> for neutron-induced fission of \begin{document}$^{237,238}$\end{document}Np are compared with the available experimental data to obtain the deformation parameters in the scission point model. The deformation parameters of fission fragments are discussed at the scission point. The calculated results are also compared with the results from other methods and with experimental data. The behavior of the <TKE> distribution is then studied for the neutron-induced fission of actinides. This indicates that the <TKE> values for neutron-induced fission of actinides with odd mass numbers are greater than for those with even mass numbers.
2021, 45(2): 025001. doi: 10.1088/1674-1137/abccae
Abstract:
China Jinping Underground Laboratory (CJPL) is ideal for studying solar, geo-, and supernova neutrinos. A precise measurement of the cosmic-ray background is essential in proceeding with R&D research for these MeV-scale neutrino experiments. Using a 1-ton prototype detector for the Jinping Neutrino Experiment (JNE), we detected 264 high-energy muon events from a 645.2-day dataset from the first phase of CJPL (CJPL-I), reconstructed their directions, and measured the cosmic-ray muon flux to be \begin{document}$(3.53\pm0.22_{\rm{stat.}}\pm0.07_{\rm{sys.}})\times10^{-10}$\end{document} cm\begin{document}$^{-2}$\end{document}s\begin{document}$^{-1}$\end{document}. The observed angular distributions indicate the leakage of cosmic-ray muon background and agree with simulation data accounting for Jinping mountain's terrain. A survey of muon fluxes at different laboratory locations, considering both those situated under mountains and those down mine shafts, indicates that the flux at the former is generally a factor of \begin{document}$(4\pm2)$\end{document} larger than at the latter, with the same vertical overburden. This study provides a convenient back-of-the-envelope estimation for the muon flux of an underground experiment.
2021, 45(2): 025002. doi: 10.1088/1674-1137/abd01b
Abstract:
A sub-array of the Large High Altitude Air Shower Observatory (LHAASO), KM2A is mainly designed to observe a large fraction of the northern sky to hunt for γ-ray sources at energies above 10 TeV. Even though the detector construction is still underway, half of the KM2A array has been operating stably since the end of 2019. In this paper, we present the KM2A data analysis pipeline and the first observation of the Crab Nebula, a standard candle in very high energy γ-ray astronomy. We detect γ-ray signals from the Crab Nebula in both energy ranges of 10\begin{document}$-$\end{document}100 TeV and \begin{document}$>$\end{document}100 TeV with high significance, by analyzing the KM2A data of 136 live days between December 2019 and May 2020. With the observations, we test the detector performance, including angular resolution, pointing accuracy and cosmic-ray background rejection power. The energy spectrum of the Crab Nebula in the energy range 10-250 TeV fits well with a single power-law function dN/dE = (1.13\begin{document}$\pm$\end{document}0.05\begin{document}$_{\rm stat}$\end{document}\begin{document}$\pm$\end{document}0.08\begin{document}$_{\rm sys}$\end{document})\begin{document}$\times$\end{document}10\begin{document}$^{-14}$\end{document}\begin{document}$\cdot$\end{document}(E/20 TeV)\begin{document}$^{-3.09\pm0.06_{\rm stat}\pm0.02_{\rm sys}}$\end{document} cm\begin{document}$^{-2}$\end{document} s\begin{document}$^{-1}$\end{document} TeV\begin{document}$^{-1}$\end{document}. It is consistent with previous measurements by other experiments. This opens a new window of γ-ray astronomy above 0.1 PeV through which new ultrahigh-energy γ-ray phenomena, such as cosmic PeVatrons, might be discovered.
2021, 45(2): 025101. doi: 10.1088/1674-1137/abca1c
Abstract:
In the present work, we used five different versions of the quark-meson coupling (QMC) model to compute astrophysical quantities related to the GW170817 event and the neutron star cooling process. Two of the models are based on the original bag potential structure and three versions consider a harmonic oscillator potential to confine quarks. The bag-like models also incorporate the pasta phase used to describe the inner crust of neutron stars. With a simple method studied in the present work, we show that the pasta phase does not play a significant role. Moreover, the QMC model that satisfies the GW170817 constraints with the lowest slope of the symmetry energy exhibits a cooling profile compatible with observational data.
2021, 45(2): 025102. doi: 10.1088/1674-1137/abccac
Abstract:
Anti-de Sitter (AdS) black holes with lattices are essential for optical conductivity studies in the holographic approach. We investigate the instability of these black holes that can result in the holographic description of charge density waves. In the presence of homogeneous axion fields, we show that the instability of AdS-Reissner-Nordström (AdS-RN) black holes is always suppressed. However, in the presence of Q-lattices, we find that the unstable region becomes the smallest in the vicinity of the critical region for the metal/insulator phase transition. This novel phenomenon is reminiscent of the behavior of the holographic entanglement entropy during quantum phase transitions.
2021, 45(2): 025103. doi: 10.1088/1674-1137/abccaf
Abstract:
It has been shown that the Christodoulou version of the strong cosmic censorship (SCC) conjecture can be violated for a scalar field in a near-extremal Reissner-Nordstrom-de Sitter black hole. In this paper, we investigate the effects of higher derivative corrections to the Einstein-Hilbert action on the validity of SCC, by considering a neutral massless scalar perturbation in \begin{document}$5$\end{document}- and \begin{document}$6$\end{document}-dimensional Einstein-Maxwell-Gauss-Bonnet-de Sitter black holes. Our numerical results show that the higher derivative term plays a different role in the \begin{document}$d = 5$\end{document} case than it does in the \begin{document}$d = 6$\end{document} case. For \begin{document}$d = 5$\end{document}, the SCC violation region increases as the strength of the higher derivative term increases. For \begin{document}$d = 6$\end{document}, the SCC violation region first increases and then decreases as the higher derivative correction becomes stronger, and SCC can always be restored for a black hole with a fixed charge ratio when the higher derivative correction is strong enough. Finally, we find that the \begin{document}$C^{2}$\end{document} version of SCC is respected in the \begin{document}$d = 6$\end{document} case, but can be violated in some near-extremal regimes in the \begin{document}$d = 5$\end{document} case.
2021, 45(2): 025104. doi: 10.1088/1674-1137/abcd2d
Abstract:
We studied the instability of regularized 4D charged Einstein-Gauss-Bonnet de-Sitter black holes under charged scalar perturbations. The unstable modes satisfy the superradiant condition, but not all of the modes satisfying the superradiant condition are unstable. The instability occurs when the cosmological constant is small and the black hole charge is not too large. The Gauss-Bonnet coupling constant further destabilizes black holes when both the black hole charge and the cosmological constant are small and further stabilizes black holes when the black hole charge is large.
2021, 45(2): 025105. doi: 10.1088/1674-1137/abcd2e
Abstract:
We investigate the axion-like particle (ALP)-photon oscillation effect in the high-energy \begin{document}$\gamma$\end{document}-ray spectra of PG 1553+113 and PKS 2155−304 measured by Fermi-LAT and H.E.S.S. The choice of extragalactic background light (EBL) model, which induces the attenuation effect in observed \begin{document}$\gamma$\end{document}-ray spectra, affects the ALP implications. For the ordinary EBL model that prefers a null hypothesis, we set constraints on the ALP-photon coupling constant at 95% C.L. as \begin{document}$g_{a\gamma}\lesssim 5\times 10^{-11} ~{\rm{GeV}}^{-1}$\end{document} for the ALP mass \begin{document}$\sim 10$\end{document} neV. We also consider the CIBER observation of the cosmic infrared radiation, which shows an excess at wavelengths of \begin{document}$\sim 1~\mu$\end{document}m after the substraction of foregrounds. High-energy gamma-rays from extragalactic sources at high redshifts would suffer from a more significant attenuation effect caused by this excess. In this case, we find that the ALP-photon oscillation would improve the fit to the observed spectra of PKS 2155−304 and PG 1553+113 and find a favored parameter region at 95% C.L..
2021, 45(2): 025106. doi: 10.1088/1674-1137/abce51
Abstract:
We obtain an exact slowly rotating Einstein-bumblebee black hole solution by solving the corresponding \begin{document}$rr$\end{document} and \begin{document}$t\phi$\end{document} components of the gravitational field equations for both cases: A) \begin{document}$b_\mu=(0,b(r),0,0)$\end{document} and B) \begin{document}$b_\mu= (0,b(r), \mathfrak{b}(\theta),0)$\end{document}. Then, we check the other gravitational field equations and the bumblebee field motion equations using this solution. We find that for case A, there indeed exists a slowly rotating black hole solution for an arbitrary LV (Lorentz violation) coupling constant \begin{document}$\ell$\end{document}; however, for case B, this slowly rotating solution exists if and only if coupling constant \begin{document}$\ell$\end{document} is as small as or smaller than angular momentum a. Thus far, no full rotating black hole solution has been published; hence, the Newman-Janis algorithm cannot be used to generate a rotating solution in the Einstein-bumblebee theory. This is similar to the Einstein-aether theory, wherein only some slowly rotating black hole solutions exist. To study the effects of this broken Lorentz symmetry, we consider the black hole greybody factor and find that, for angular index \begin{document}$l=0$\end{document}, LV constant \begin{document}$\ell$\end{document} decreases the effective potential and enhances the absorption probability, which is similar to the results for the non-minimal derivative coupling theory.
2021, 45(2): 025107. doi: 10.1088/1674-1137/abce53
Abstract:
We probe the universality of acceleration scale \begin{document}$a_0$\end{document} in Milgrom's modified Newtonian dynamics (MOND) using the recently released rotation curve data from SPARC galaxies. We divide the SPARC data into different subsamples according to the morphological types of galaxies, and fit the rotation curve data of each subsample with the theoretical prediction of MOND. MOND involves an arbitrary interpolation function which connects the Newtonian region and the MOND region. Here we consider five different interpolation functions that are widely discussed in the literature. It is shown that the best-fitting \begin{document}$a_0$\end{document} significantly depends on the interpolation functions. For a specific interpolation function, \begin{document}$a_0$\end{document} also depends on the morphological types of galaxies, implying that \begin{document}$a_0$\end{document} may be not a universal constant. Introducing a dipole correction to \begin{document}$a_0$\end{document} can significantly improve the fits. The dipole directions for four of the five interpolation functions point towards an approximately consistent direction, but \begin{document}$a_0$\end{document} still varies for different interpolation functions.
2021, 45(2): 025108. doi: 10.1088/1674-1137/abcf21
Abstract:
A universal relation between the leading correction to the entropy and extremality was proposed in the work of Goon and Penco. In this paper, we extend this work to massive gravity and investigate thermodynamic extremality relations in a topologically higher-dimensional black hole. A rescaled cosmological constant is added to the action of the massive gravity as a perturbative correction. This correction modifies the extremality bound of the black hole and leads to shifts in the mass, entropy, etc. Regarding the cosmological constant as a variable related to pressure, we obtain the thermodynamic extremality relations between the mass and entropy, pressure, charge, and parameters ci by accurate calculations. Finally, these relations are verified by a triple product identity, which shows that the universal relation exists in black holes.
2021, 45(2): 025109. doi: 10.1088/1674-1137/abcf23
Abstract:
In this paper, the Joule-Thomson expansion of Born-Infeld AdS black holes is studied in the extended phase space, where the cosmological constant is identified with the pressure. The Joule-Thomson coefficient, the inversion curves and the isenthalpic curves are discussed in detail using a 4-dimensional black hole. The critical point of a Born-Infeld black hole is depicted with varying parameter \begin{document}$\beta$\end{document} and the charge Q. In \begin{document}$T-P$\end{document} plane, the inversion temperature curves and isenthalpic curves are obtained with different parameter \begin{document}$\beta$\end{document} and charge Q. We find that the missing negative slope is still conserved in Born-Infeld black holes. We also extend our discussion to arbitrary dimension higher than 4. The critical temperature and the minimum of inversion temperature are compared. The ratio is asymptotically \begin{document}$1/2$\end{document} as Q increases or \begin{document}$\beta\to\infty$\end{document} with \begin{document}$D = 4$\end{document}, and reproduces previous results at higher dimensions.
2021, 45(2): 025110. doi: 10.1088/1674-1137/abd01a
Abstract:
In this study, we implemented the type II seesaw mechanism into the framework of the \begin{document}$U(1)_{\rm B-L}$\end{document} gauge model. To achieve this, we added a scalar triplet, \begin{document}$\Delta$\end{document}, to the canonical particle content of the \begin{document}$U(1)_{\rm B-L}$\end{document} gauge model. By imposing that the \begin{document}$U(1)_{\rm B-L}$\end{document} gauge symmetry be spontaneously broken at TeV scale, we show that the type II seesaw mechanism is realized at an intermediate energy scale, more precisely, at approximately \begin{document}$10^9$\end{document} GeV. To prevent heavy right-handed neutrinos from disturbing the mechanism, we evoke a \begin{document}$Z_2$\end{document} discrete symmetry. Interestingly, as a result, we have standard neutrinos with mass around eV scale and right-handed neutrinos with mass in TeV scale, with the lightest one fulfilling the condition of dark matter. We developed all of these in this study. In addition, we show that the neutral component of \begin{document}$\Delta$\end{document} may perform unproblematic non-minimal inflation with loss of unitarity.
2021, 45(2): 025111. doi: 10.1088/1674-1137/abd01d
Abstract:
We study the linear instability of the charged massless scalar perturbation in regularized 4D charged Einstein-Gauss-Bonnet-AdS black holes by exploring their quasinormal modes. We find that the linear instability is triggered by superradiance. The charged massless scalar perturbation becomes more unstable with increasing Gauss-Bonnet coupling constant or black hole charge. Decreasing the AdS radius, on the other hand, will make the charged massless scalar perturbation more stable. The stable region in parameter space (α, Q, Λ) is given. Moreover, we find that the charged massless scalar perturbation is more unstable for larger scalar charge. The modes of multipoles are more stable than that of the monopole.
2021, 45(2): 025112. doi: 10.1088/1674-1137/abd088
Abstract:
In this study, we examine the laws of thermodynamics and the weak cosmic censorship conjecture in the normal and extended phase spaces of Born-Infeld-anti-de Sitter black holes by considering a charged particle absorption. In the normal phase space, the first and second laws of thermodynamics as well as the weak cosmic censorship are still valid. However, in the extended phase space, the second law of thermodynamics is violated for double-horizon black holes and part of single-horizon black holes. The first law of thermodynamics and the weak cosmic censorship conjecture are still valid for all types of black holes. In addition, we found that the shift of the metric function, which determines the locations of the horizons, takes the same form at the minimum point in both the normal and extended phase spaces, indicating that the weak cosmic censorship conjecture is independent of the thermodynamic phase space.


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# HLSL - Division instead of Multiplication?

This topic is 2509 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

## Recommended Posts

Hi guys,

i have a wild question, i once read that divisions are faster then multiplication. Therefore in some relations

velocity *= delay;

could be transformed into

velocity /= velocity - (velocity / n)  // where n = some power of 2

is this right, or is this old technology? Compilers more intelligent?

Thank you very much

schiho.com

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I have not heard of these cases. At any rate, the HLSL compilers always seems to prefer multiplication when possible and will potentially represent divisions in code as multiplications in assembly. I would imagine that if those cases did exist, either the HLSL compiler or the JIT compiler in the driver would try to handle it.

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i have a wild question, i once read that divisions are faster then multiplication.

I have no idea what your source is, but you need to completely forget this “nugget”. It is exactly the opposite.
Divisions are to be avoided whenever possible as they are up to 25 times slower than multiplies.

Hence the above post which explains that compilers always try to perform a multiplication instead of a division when possible.

L. Spiro

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In assembly division will translate to at least 2 instructions - rcp and mul (and rcp can only operate on scalar values so it may need up to 4 if you're using all register elements). Also - division is scary! You get nasty things like divide by zero to watch out for. So I don't really see how division could be faster.

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Hi guys, thank you for the answer

A: 100/2
B: 100*0.5
What is faster?

I got this from :

OR is CPU and GPU different?

i can agree with most of you, thats why i was asking the question.

thanks

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A is going to be faster on the CPU because the compiler can detect that it's integer division by 2 and convert it into a simple bitshift: 100 >> 1. A compiler might also be able to do the same for B but it might not - because we're into floating point country (where bit shifts don't exist) it might do the full thing.

A GPU will normally operate in floating point internally so B would be expected to be faster.

For a proper comparison you need to pick numbers that won't be susceptible to this kind of coptimization: 100.174 / 2.625 for example. Then you'll see the real difference.

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So,

having this texture lookups:
 //sample neighbour Pixels float4 colorLeft = tex2D(memTex0, biasedTex0.xy-float2(pixel.x, 0)); float4 colorTop = tex2D(memTex0, biasedTex0.xy-float2(0, pixel.y)); float4 colorRight = tex2D(memTex0, biasedTex0.xy+float2(pixel.x, 0)); float4 colorSub = tex2D(memTex0, biasedTex0.xy+float2(0, pixel.y));

And doing calculation by a division:
 buffer2.rgba = (colorLeft + colorTop + colorRight + colorSub) /2 - (currentbuffer2);

should be replaced by a more gpu friendly version ? (division replaced by multiplication)
 buffer2.rgba = (colorLeft + colorTop + colorRight + colorSub)*0.5 - (currentbuffer2);

Is this the bette way of writing hlsl code or will the compiler do that automaticly? I just want to make it a good habit, by choosing the "better way"

thank you very much all

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Checking in PIX, I tested "return color * 0.5f" and "return color / 2.0f". In both cases it translated the code to a multiplication by 0.5, so yes - when you're dividing by a constant value the compiler will automatically detect that and convert it to multiplication for you. If you're dividing by non-constant (the result of another calculation or lookup) the compiler won't be able to make this optimization.

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Thanks mhagain for the testing it with PIX, with non-constant you mean, when the value for the division variable is not hardcoded. I see...thank you a lot, that information was very usefull.

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# How to prove the following results?

I was researching about the Bayes theorem, I would like to prove it in order to understand more I found that the prove uses the following equations:

$$P(A ∩ B) = P(A)P(B|A)$$ $$P(A ∩ B) = P(B)P(A|B)$$

I would like to see if some one could help me to prove that equations using Venn diagrams in order to be more graphic and also providing me the theorems that I would need to prove that result,

Thanks any how, I appreciate the support.

• This is the usual definition of conditional probability. There's no "proof" involved, usually. Are you starting from a different definition? – lulu Aug 29 '16 at 13:10
• Thanks, I see but maybe there would be a empirical fact to state that, what would be the reason to define it in that way? – neo33 Aug 29 '16 at 13:17
• Well, the definition exists for a reason. If you carve up a region, so that the probability of event $A$ is the area of the region we call "$A$" and so on, then the conditional probability is defined by restricting your attention to the region defined by the conditioning event, and scaling up so that the total probability is $1$. The wiki article has some illustrations. – lulu Aug 29 '16 at 13:22
• These come immediately from the definition $$P(A|B)=\tfrac{P(A\cap B)}{P(B)}$$ (and the companion equation obtained by swapping $A$ and $B$). To me, this definition is more intuitive. – MPW Aug 29 '16 at 14:11
• Thanks @MPW this is more clear, but Why I have a question, is there a Venn diagram to see that result? I mean why the people define the conditional probability like that. – neo33 Aug 29 '16 at 14:15

It seems you may be looking for an intuitive justification of the definition $$P(A|B) = P(A \cap B)/P(B).$$ Here are three examples that might help match the definition with intuition. I hope that, somewhere among the three, there is something that persuades you that the definition of conditional probability is intuitively sound.

At an imaginary small college enrollments in schools of Arts, Business, and Science according to gender are as follows:

College     A       B       S         Total
-------------------------------------------
Male       100     500     400        1000
Female     200     400     600        1200
-------------------------------------------
Total      300     900    1000        2200

Suppose a student is chosen at random, given that a Science major was chosen, what is the probability a woman was chosen?

Method 1: Logic. Knowing that a Science major was chosen, we are interested only in the column headed 'S'. Ignore the rest of the table. There are 1000 Science majors of whom 600 are women, so $P(W|S) = 600/1000 = 0.6.$

Method 2: Use definition.
$$P(W|S) = \frac{P(W \cap S)}{P(S)} = \frac{600/2200}{1000/2200} = \frac{600}{1000} = 0.6.$$

The definition gets the same answer as we got by logic. It may look trivial on account of the highly rounded numbers, but this shows that the definition is in accord with real applications.

An urn contains three Red balls and two Green balls. What is the probability of getting two Red balls if two balls are selected at random without replacement?

Method 1. Combinaations. $$P(R_1 \cap R_2) = \frac{{3 \choose 2}{2 \choose 0}}{{5 \choose 2}} = \frac 3 {10} = 0.3.$$

Method 2. The definition of conditional probability implies the 'general multiplication rule': $P(R_1 \cap R_2) = P(R_1)\cdot P(R_2|R_1).$

Obviously, $P(R_1) = 3/5.$ Then on the second draw, it is intuitive to say $P(R_2|R_1) = 2/4$ because the remaining contents of the urn are four balls of which two are red. Then multiplying, we have $$P(R_1 \cap R_2) = \frac{3}{5}\cdot\frac{2}{4} = \frac{6}{20} = 0.3.$$

Again here an intuitive view of a conditional probability $P(R_2|R_1)$ leads to an answer obtained by another method (combinatorics).

A fair die is rolled. We are told that the result is a 'big' number, where $B = \{4,5,6\}.$ Given this conditional information, what is the probability that the result is 'even', where $E=\{2,4,6\}?$ Because we know the result is Big, it seems reasonable to assign probability $1/3$ to each of the outcomes 4, 5, and 6. Of these, two are even. So intuitively, $P(E|B) = 2/3.$

But the definition of conditional probability gives $$P(E|B) = \frac{P(E \cap B)}{P(B)} = \frac{P\{4,6\}}{1/2} = \frac{2/6}{1/2} = 2/3.$$

• Yes that is exactly what I wanted, Thanks a lot for the support I really appreciate the help. – neo33 Aug 30 '16 at 2:39


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# where does nova.conf comes in for nova services?

When I read the source code of nova-scheduler, I don't find the entry where the nova.con comes in. Here is my searching path:

/etc/init.d/nova-scheduler : nothing, just run nova-scheduler(in /usr/bin/nova-scheduler)

/usr/bin/nova-scheduler : run nova.cmd.scheduler main()

/usr/share/pyshared/nova/cmd/scheduler.py:

Well, I guess this code is where nova.conf comes in. And the most probably entry is:

config.parse_args(sys.argv)

if we assign --config-file=/etc/nova/nova.conf option when running nova-scheduler, then this is the entry where nova.conf comes in manually

The reason why I emphasise "manually" is because when we run the service, we don't need to assign such option.(We can see the openstack installation guide of docs.openstack.org with version icehouse, that when executing the nova services, we just only run:service nova-xxxx restart)

Well, this is not a big deal since when creating nova services, all of the options has the default values:(for instance, /usr/share/pyshared/nova/service.py, we can see that there are settings of default options at the beginning of the file). But even if so, then that seems there are no entry of nova.conf for the nova services. Then why do we need it?

Anyone has any idea?

edit retag close merge delete

Sort by » oldest newest most voted

Got the answer: if we run the nova services without assigning --config-file=/etc/nova/nova.conf like:

service nova-scheduler start

Then scheduler.py in /usr/share/pyshared/nova/cmd/ will run with nova.conf.

Actually yes, the entry is config.parse_args(sys.argv) in scheduler.py, lets trace parse_args() in /usr/share/pyshared/nova/config.py:

cfg.CONF(argv[1:],
project='nova',
version=version.version_string(),
default_config_files=default_config_files)

Well if we don't manually assign default_config_file when running nova service, then this value is None. So let's find out the entry of nova.con in cfg.CONF:

/usr/share/pyshared/oslo/config/cfg.py:

CONF = ConfigOpts()

let's see the caller in ConfigOpts(), first note that we have this pre_setup to assign the config_file(s):

prog, default_config_files = self._pre_setup(project,
prog,
version,
usage,
default_config_files)

We can find that the nova.conf enters in:

if default_config_files is None:
default_config_files = find_config_files(project, prog)

If we trace the find_config_file, we can see that it will find the nova.conf in the directories:

~/.${project}/ ~/ /etc/${project}/
/etc/
more


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# Gradient help

#### Juntao

Consider the function z=f(x,y). If you start at the point (4,5) and move toward the point (5,6), the direction derivative is sqrt(2). Starting at (4,5) and moving toward (6,6), the directional derivative is sqrt(5). Find gradient f at (4,5).

Okay, this is probably a simple problem, but I don't know how to start it. Help appreciated.

#### Hurkyl

Staff Emeritus
Science Advisor
Gold Member
Do you know a formula that relates gradients and directional derivatives?

What do you get when you plug what you know into that formula? (represent unknowns with variables)

#### Juntao

Forumula...*scratches head*

Um, I think this is the one, hopefully:
The derivative of f at Po in the direction of u= gradient f dot with u = magnitude grad f * magnitude u * cos theta = magnitude grad f * cos theta.

Or like D_u f= grad f (dot) u = |grad f| |u| cos theta = |grad f| cos theta

Maybe that'll help me out?

#### HallsofIvy

Science Advisor
Homework Helper
D_u f= grad f (dot) u
Yes, that's true as long as u is a unit vector (has length 1).

Another way of saying this is that the directional derivative is
f_x u_x+ f_y u_y where f_x is the partial derivative and u_x is the x component of unit vector u.

Find unit vectors in both the given directions and write out
f_x u_x+ f_y u_y= √(2) and
f_x v_x+ f_y v_y= √(5).

That gives you two equations for f_x and f_y.

#### Hurkyl

Staff Emeritus
Science Advisor
Gold Member
D_u f= grad f (dot) u
That's the one of which I was thinking.

You know two instances of D_u f and u. (don't forget that u is a unit vector!), so grad f is the unknown. You have two equations, so you should be able to solve for the two components of grad f.

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text:
## anonymous 3 years ago Find the solution of gamma and beta function!

1. anonymous

Find the solution of gamma and beta function for $\int\limits_{0}^{\pi/2} \frac{ d \theta }{ \sqrt{\sin \theta} }$

2. anonymous

3. amistre64

i dont have enough experience with beta and gamma

me neither :(

5. ajprincess

I am sorry @gerryliyana I havnt learnt this.:(

6. anonymous

I don't understand your original question, but this is an elliptic integral.

7. abb0t

If the root wasn't there I would have an idea but it's making it a problem with it there.

8. anonymous

@gerryliyana can you state more clearly what you wish to do?

9. abb0t

Isn't it $\frac{ \Gamma (m) }{ 2 \times \Gamma (m) }$

10. abb0t

11. anonymous

ok ok...,for eample i have $\int\limits_{0}^{\pi/2} \frac{ d \theta }{ \sqrt{\sin \theta} }$ based on Beta function; $B(p,q) = 2 \int\limits_{0}^{\pi/2} (\sin \theta)^{2p-1} (\cos \theta)^{2q-1} d \theta$ then, i have $\int\limits_{0}^{\pi/2} \frac{ d \theta }{ \sqrt{\sin \theta} }$ $\int\limits_{0}^{\pi/2} (\sin \theta)^{-1/2} (\cos \theta)^{0} d \theta$ $\int\limits\limits_{0}^{\pi/2} (\sin \theta)^{2 (1/4)-1} (\cos \theta)^{2 (1/2)-1} d \theta$ then i have $B(p,q)=2 \int\limits\limits\limits_{0}^{\pi/2} (\sin \theta)^{2 (1/4)-1} (\cos \theta)^{2 (1/2)-1} d \theta$ $\int\limits\limits\limits_{0}^{\pi/2} (\sin \theta)^{2 (1/4)-1} (\cos \theta)^{2 (1/2)-1} d \theta = \frac{ 1 }{ 2 } B(p,q)$ with p = 1/4 and q = 1/2; then (relation Beta and gamma) $B(p,q) = \frac{ \Gamma (p) \Gamma (q) }{ \Gamma (p+q) }$ $B(p,q) = \frac{ \Gamma (1/4) \Gamma (1/2) }{ \Gamma (1/4+1/2) }$ then i have $\int\limits\limits\limits_{0}^{\pi/2} (\sin \theta)^{2 (1/4)-1} (\cos \theta)^{2 (1/2)-1} d \theta = \frac{ 1 }{ 2 } B(p,q)$ $\int\limits\limits\limits\limits_{0}^{\pi/2} (\sin \theta)^{2 (1/4)-1} (\cos \theta)^{2 (1/2)-1} d \theta = \frac{ 1 }{ 2 } \frac{ \Gamma(1/4)\Gamma (1/2) }{ \Gamma (1/4+1/2 )}$ finally The solution of gamma and Beta function for $$\int\limits_{0}^{\pi/2} \frac{ d \theta }{ \sqrt{\sin \theta} }$$ is $\frac{ 1 }{ 2 } \frac{ \Gamma (1/4) \Gamma (1/2)}{ \Gamma (\frac{ 1 }{ 4 }+\frac{ 1 }{ 2 }) }$

12. anonymous

Oh... okay. You didn't clarify that's what you wanted.

13. anonymous

Ok guys actually, i wanna check my work.., hei how about these "find the solution of gamma and beta function for $\int\limits_{0}^{\pi/2} (\tan^{3}\theta + \tan^{5} \theta) e^{-\tan^{2}\theta} d \theta$

14. anonymous

let $$u=\tan^2 \theta$$ see what happens, have u tried it yet?

15. anonymous

yes., i've tried for this one, but it didn't work, oh my bad :(

16. anonymous

it becomes$\frac{1}{2}\int_{0}^{\infty} u e^{-u} du$right?

17. anonymous

wait a sec..,

18. anonymous

then, $\frac{ 1 }{ 2 } \int\limits_{0}^{\infty} ue^{-u} du = \frac{ 1 }{ 2 } \int\limits_{0}^{\infty} u^{2-1} e^{-u} du$ then p=2

19. anonymous

$= \frac{ 1 }{ 2 } \Gamma(2)$

20. anonymous

and u know that$\Gamma(n+1)=n!$we are done :)

21. anonymous

ah yeah.., thank you so much @mukushla and other woow we are done :)

22. anonymous

very welcome :)


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# Exploring Definite Integral as Limit of Sum

Explore

## Drag the slider for 'n', what do you observe?

Create
In the following applet : Use following commands in the input bar to get the Lower and Upper sum.(Tech Tip:Start typing Lower(or Upper) in the input bar on the left side, the respective command will show up. Select it and enter the respective values. LowerSum[ , , , ] LowerSum[ f, x(A), x(B),n ] UpperSum[ , , , ] UpperSum[ f, x(A), x(B),n ]