Amplitude analysis of $B^+\to J/\psi \phi K^+$ decays

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Abstract

The first full amplitude analysis of $B^+\to J/\psi \phi K^+$ with $J/\psi\to\mu^+\mu^-$, $\phi\to K^+K^-$ decays is performed with a data sample of 3 fb$^{-1}$ of $pp$ collision data collected at $\sqrt{s}=7$ and $8$ TeV with the LHCb detector. The data cannot be described by a model that contains only excited kaon states decaying into $\phi K^+$, and four $J/\psi\phi$ structures are observed, each with significance over $5$ standard deviations. The quantum numbers of these structures are determined with significance of at least $4$ standard deviations. The lightest has mass consistent with, but width much larger than, previous measurements of the claimed $X(4140)$ state. The model includes significant contributions from a number of expected kaon excitations, including the first observation of the $K^{*}(1680)^+\to\phi K^+$ transition.

Figures and captions

 Distribution of $m_{K^+K^-}$ near the $\phi$ peak before the $\phi$ candidate selection. Non-$B^+$ backgrounds have been subtracted using sPlot weights [36] obtained from a fit to the $m_{ { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} K^+K^-K^+}$ distribution. The default $\phi$ selection window is indicated with vertical red lines. The fit (solid blue line) of a Breit--Wigner $\phi$ signal shape plus two-body phase space function (dashed red line), convolved with a Gaussian resolution function, is superimposed. mkk-20[..].pdf [18 KiB] HiDef png [211 KiB] Thumbnail [161 KiB] *.C file Mass of $B^+\rightarrow { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} \phi K^+$ candidates in the data (black points with error bars) together with the results of the fit (blue line) with a double-sided Crystal Ball shape for the $B^+$ signal on top of a quadratic function for the background (red dashed line). The fit is used to determine the background fraction under the peak in the mass range used in the amplitude analysis (indicated with vertical solid red lines). The sidebands used for the background parameterization are indicated with vertical dashed blue lines. dataSW.pdf [21 KiB] HiDef png [188 KiB] Thumbnail [156 KiB] *.C file Definition of the $\theta_{ K ^* }$, $\theta_{ { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} }$, $\theta_{\phi}$, $\Delta{\phi}_{ K ^* , { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} }$ and $\Delta{\phi}_{ K ^* ,\phi }$ angles describing angular correlations in $B^+\rightarrow { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} K^{*+}$, ${ J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} \rightarrow \mu^+\mu^-$, $K^{*+}\rightarrow \phi K^+$, $\phi\rightarrow K^+K^-$ decays (${ J \mskip -3mu/\mskip -2mu\psi \mskip 2mu}$ is denoted as $\psi$ in the figure). Matrix[..].jpg [68 KiB] HiDef png [254 KiB] Thumbnail [76 KiB] *.C file Parameterized efficiency $\epsilon_1(m_{\phi K},\cos\theta_{ K ^* })$ function (top) and its representation in the Dalitz plane $({m_{\phi K }^2},{m_{ { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} \phi }^2})$ (bottom). Function values corresponding to the color encoding are given on the right. The normalization arbitrarily corresponds to unity when averaged over the phase space. effm1cosKs.pdf [23 KiB] HiDef png [278 KiB] Thumbnail [300 KiB] *.C file effm1m2.pdf [38 KiB] HiDef png [436 KiB] Thumbnail [378 KiB] *.C file Parameterized efficiency $\epsilon_2(\cos\theta_{\phi }|m_{\phi K})$, $\epsilon_3(\cos\theta_{ { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} }|m_{\phi K})$, $\epsilon_4(\Delta{\phi}_{ K ^* ,\phi }|m_{\phi K})$, $\epsilon_5(\Delta{\phi}_{ K ^* , { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} }|m_{\phi K})$ functions. Function values corresponding to the color encoding are given on the right. By construction each function integrates to unity at each $m_{\phi K}$ value. The structure in $\epsilon_2(\cos\theta_{\phi }|m_{\phi K})$ present between 1500 and $1600 \mathrm{ Me V}$ is an artifact of removing $B^+\rightarrow { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} K^+K^-K^+$ events in which both $K^+K^-$ combinations pass the $\phi$ mass selection window. effAngles.pdf [52 KiB] HiDef png [628 KiB] Thumbnail [623 KiB] *.C file Parameterized background ${P_{\rm bkg}}_1(m_{\phi K},\cos\theta_{ K ^* })$ function (top) and its representation in the Dalitz plane $({m_{\phi K }^2},{m_{ { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} \phi }^2})$ (bottom). Function values corresponding to the color encoding are given on the right. The normalization arbitrarily corresponds to unity when averaged over the phase space. bkgm1cosKs.pdf [23 KiB] HiDef png [279 KiB] Thumbnail [295 KiB] *.C file bkgm1m2.pdf [38 KiB] HiDef png [430 KiB] Thumbnail [369 KiB] *.C file Parameterized background functions: $P^u_{\rm bkg 2}(\cos\theta_{\phi }|m_{\phi K })$, $P^u_{\rm bkg 3}(\cos\theta_{ { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} }|m_{\phi K })$, $P^u_{\rm bkg 4}(\Delta{\phi}_{ K ^* ,\phi }|m_{\phi K })$, $P^u_{\rm bkg 5}(\Delta{\phi}_{ K ^* , { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} }|m_{\phi K })$. Function values corresponding to the color encoding are given on the right. By construction each function integrates to unity at each $m_{\phi K}$ value. bkgAngles.pdf [56 KiB] HiDef png [770 KiB] Thumbnail [716 KiB] *.C file Background-subtracted and efficiency-corrected data yield in the Dalitz plane of $({m_{\phi K }^2},{m_{ { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} \phi }^2})$. Yield values corresponding to the color encoding are given on the right. PhiK_J[..].pdf [18 KiB] HiDef png [176 KiB] Thumbnail [187 KiB] *.C file Background-subtracted and efficiency-corrected data yield in the Dalitz plane of $({m_{\phi K }^2},{m_{ { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} K }^2})$. Yield values corresponding to the color encoding are given on the right. PhiK_J[..].pdf [18 KiB] HiDef png [178 KiB] Thumbnail [188 KiB] *.C file Background-subtracted and efficiency-corrected data yield in the Dalitz plane of $({m_{ { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} K }^2},{m_{ { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} \phi }^2})$. Yield values corresponding to the color encoding are given on the right. JpsiK_[..].pdf [16 KiB] HiDef png [138 KiB] Thumbnail [139 KiB] *.C file Background-subtracted (histogram) and efficiency-corrected (points) distribution of $m_{\phi K}$. See the text for the explanation of the efficiency normalization. PhiK_o[..].pdf [15 KiB] HiDef png [155 KiB] Thumbnail [138 KiB] *.C file Background-subtracted (histogram) and efficiency-corrected (points) distribution of $m_{ { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} \phi}$. See the text for the explanation of the efficiency normalization. JpsiPh[..].pdf [16 KiB] HiDef png [186 KiB] Thumbnail [168 KiB] *.C file Background-subtracted (histogram) and efficiency-corrected (points) distribution of $m_{ { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} K}$. See the text for the explanation of the efficiency normalization. JpsiK_[..].pdf [15 KiB] HiDef png [156 KiB] Thumbnail [137 KiB] *.C file Kaon excitations predicted by Godfrey--Isgur [53] (horizontal black lines) labeled with their intrinsic quantum numbers: $n{}^{2S+1}L_J$ (see the text). Well established states are shown with narrower solid blue boxes extending to $\pm1\sigma$ in mass and labeled with their PDG names [37]. Unconfirmed states are shown with dashed green boxes. The long horizontal red lines indicate the $\phi K$ mass range probed in $B^+\rightarrow { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} \phi K^+$ decays. kstars[..].pdf [15 KiB] HiDef png [311 KiB] Thumbnail [234 KiB] *.C file Distribution of $m_{ { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} \phi}$ for the data and the fit results with a model containing only $K^{*+}\rightarrow \phi K^+$ contributions. exbKst[..].pdf [20 KiB] HiDef png [214 KiB] Thumbnail [186 KiB] *.C file Distributions of (top left) $\phi K^+$, (top right) ${ J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} K^+$ and (bottom) ${ J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} \phi$ invariant masses for the $B^+\rightarrow { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} \phi K^+$ data (black data points) compared with the results of the default amplitude fit containing $K^{*+}\rightarrow \phi K^+$ and $X\rightarrow { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} \phi$ contributions. The total fit is given by the red points with error bars. Individual fit components are also shown. Displays of $m_{ { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} \phi}$ and of $m_{ { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} K}$ masses in slices of $m_{\phi K}$ are shown in Fig. 20. newbas[..].pdf [30 KiB] HiDef png [485 KiB] Thumbnail [289 KiB] *.C file newbas[..].pdf [30 KiB] HiDef png [434 KiB] Thumbnail [248 KiB] *.C file newbas[..].pdf [56 KiB] HiDef png [590 KiB] Thumbnail [302 KiB] *.C file Distributions of the fitted decay angles from the $K^{*+}$ decay chain together with the display of the default fit model described in the text. KsAngles.pdf [88 KiB] HiDef png [1 MiB] Thumbnail [653 KiB] *.C file Distributions of the fitted decay angles from the $X$ decay chain together with the display of the default fit model described in the text. XAngles.pdf [87 KiB] HiDef png [1 MiB] Thumbnail [638 KiB] *.C file Distributions of the fitted decay angles from the $Z$ decay chain together with the display of the default fit model described in the text. ZAngles.pdf [87 KiB] HiDef png [1 MiB] Thumbnail [642 KiB] *.C file Distribution of (left) $m_{ { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} \phi}$ and (right) $m_{ { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} K}$ in three slices of $m_{\phi K }$: $<1750 \mathrm{ Me V}$, $1750-1950 \mathrm{ Me V}$, and $>1950 \mathrm{ Me V}$ from the top to the bottom, together with the projections of the default amplitude model. See the legend in Fig. 16 for a description of the components. base_P[..].pdf [111 KiB] HiDef png [1 MiB] Thumbnail [556 KiB] *.C file Mass of $B^+\rightarrow { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} \phi K^+$ candidates in the data with the $p_{\mathrm{ T}} (K)>250 \mathrm{ Me V}$ (default) and $p_{\mathrm{ T}} (K)>500 \mathrm{ Me V}$ selection requirements. KPT500_CMP.pdf [28 KiB] HiDef png [386 KiB] Thumbnail [327 KiB] *.C file Distributions of (top) $\phi K^+$, (middle) ${ J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} K^+$ and (bottom) ${ J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} \phi$ invariant masses for the $B^+\rightarrow { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} \phi K^+$ data after changing the $p_{\mathrm{ T}} (K)>0.25 \mathrm{ Ge V}$ requirement to $p_{\mathrm{ T}} (K)>0.5 \mathrm{ Ge V}$, together with the fit projections. Compare to Fig. 16. KPT500[..].pdf [31 KiB] HiDef png [432 KiB] Thumbnail [266 KiB] *.C file KPT500[..].pdf [35 KiB] HiDef png [385 KiB] Thumbnail [226 KiB] *.C file KPT500[..].pdf [56 KiB] HiDef png [533 KiB] Thumbnail [277 KiB] *.C file Dependence of the real and imaginary parts of the cusp amplitude on the mass in Swanson's model [61]. See the text for a more precise explanation. SwIZPlot.pdf [37 KiB] HiDef png [129 KiB] Thumbnail [86 KiB] *.C file The Argand diagram of the the cusp amplitude in Swanson's model [61]. Motion with the mass is counter-clockwise. The peak amplitude is reached at threshold when the real part is maximal and the imaginary part is zero. The Breit--Wigner amplitude gives circular phase motion, also with counter-clockwise mass evolution, with maximum magnitude when zero is crossed on the real axis. cuspArgand.pdf [24 KiB] HiDef png [176 KiB] Thumbnail [146 KiB] *.C file Distributions of (top left) $\phi K^+$, (top right) ${ J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} K^+$ and (bottom) ${ J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} \phi$ invariant masses for the $B^+\rightarrow { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} \phi K^+$ data (black data points) compared with the results of the amplitude fit containing $K^{*+}\rightarrow \phi K^+$ and $X\rightarrow { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} \phi$ contributions in which $X(4140)$ is represented as a $J^{PC}=1^{++}$ $D_s^+D_s^{*-}$ cusp. The total fit is given by the red points with error bars. Individual fit components are also shown. xonecu[..].pdf [31 KiB] HiDef png [484 KiB] Thumbnail [291 KiB] *.C file xonecu[..].pdf [29 KiB] HiDef png [442 KiB] Thumbnail [255 KiB] *.C file xonecu[..].pdf [57 KiB] HiDef png [601 KiB] Thumbnail [305 KiB] *.C file Distributions of ${ J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} \phi$ invariant mass for the $B^+\rightarrow { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} \phi K^+$ data (black data points) compared with the results of the amplitude fit containing $K^{*+}\rightarrow \phi K^+$ and $X\rightarrow { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} \phi$ contributions in which $X(4140)$ and $X(4274)$ are represented as $J^{PC}=1^{++}$ $D_s^\pm D_s^{*\mp}$ and $0^{-+}$ $D_s^{\pm}D_{s0}^{*}(2317)^{\mp}$ cusps, respectively. The total fit is given by the red points with error bars. Individual fit components are also shown. cusp1h[..].pdf [56 KiB] HiDef png [602 KiB] Thumbnail [306 KiB] *.C file Animated gif made out of all figures. PAPER-2016-019.gif Thumbnail

Tables and captions

 Previous results related to the $X(4140)\rightarrow { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} \phi$ mass peak, first observed in $B^+\rightarrow { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} \phi K^+$ decays. The first (second) significance quoted for Ref. [27] is for the prompt (non-prompt) production components. The statistical and systematic errors are added in quadrature and then used in the weights to calculate the averages, excluding unpublished results (shown in italics). The last column gives a fraction of the total $B^+\rightarrow { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} \phi K^+$ rate attributed to the $X(4140)$ structure. Table_1.pdf [77 KiB] HiDef png [85 KiB] Thumbnail [39 KiB] tex code Previous results related to ${ J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} \phi$ mass structures heavier than the $X(4140)$ peak. The unpublished results are shown in italics. Table_2.pdf [76 KiB] HiDef png [62 KiB] Thumbnail [28 KiB] tex code Results for significances, masses, widths and fit fractions of the components included in the default amplitude model. The first (second) errors are statistical (systematic). Errors on $f_L$ and $f_\perp$ are statistical only. Possible interpretations in terms of kaon excitation levels are given, with notation $n{}^{2S+1}L_J$, together with the masses predicted in the Godfrey-Isgur model [53]. Comparisons with the previously experimentally observed kaon excitations [37] and $X\rightarrow { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} \phi$ structures are also given. Table_3.pdf [67 KiB] HiDef png [187 KiB] Thumbnail [93 KiB] tex code Summary of the systematic uncertainties on the parameters of the $K^{*+}\rightarrow \phi K^+$ states with $J^P=2^-$ and $1^+$. The kaon $p_{\mathrm{ T}}$ cross-check results are given at the bottom. All numbers for masses and widths are in $\mathrm{ Me V}$ and fit fractions in %. Table_4.pdf [57 KiB] HiDef png [196 KiB] Thumbnail [94 KiB] tex code Summary of the systematic uncertainties on the parameters of the $K^{*+}\rightarrow \phi K^+$ states with $J^P=0^-$, $1^-$ and $2^+$. The kaon $p_{\mathrm{ T}}$ cross-check results are given at the bottom. All numbers for masses and widths are in $\mathrm{ Me V}$ and fit fractions in %. Table_5.pdf [56 KiB] HiDef png [239 KiB] Thumbnail [132 KiB] tex code Summary of the systematic uncertainties on the parameters of the $X\rightarrow { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} \phi$ states. The kaon $p_{\mathrm{ T}}$ cross-check results are given at the bottom. All numbers for masses and widths are in $\mathrm{ Me V}$ and fit fractions in %. Table_6.pdf [56 KiB] HiDef png [203 KiB] Thumbnail [107 KiB] tex code Statistical significance of $J^{PC}$ preference for the $X$ states in the default model. The lowest significance value for each state is highlighted. Table_7.pdf [51 KiB] HiDef png [73 KiB] Thumbnail [35 KiB] tex code Significance, in standard deviations, of $J^{PC}$ preference for the $X$ states for dominant systematic variations of the fit model. The label "$L+n$" specifies which $L$ value in Eq. (9) is increased relative to its minimal value and by how much ($n$). The lowest significance value for each state is highlighted. Table_8.pdf [64 KiB] HiDef png [75 KiB] Thumbnail [36 KiB] tex code

Created on 29 May 2020.