Precision tests of the $J/\psi$ from full lattice QCD: Mass, leptonic width, and radiative decay rate to ${\eta }_c$

We calculate the $J/\psi$ mass, leptonic width and radiative decay rate to $\gamma {\eta }_c$ from lattice QCD including $u$, $d$ and $s$ quarks in the sea for the first time. We use the highly improved staggered quark formalism and nonperturbatively normalized vector currents for the leptonic and radiative decay rates. Our results are: $M_{J/\psi }-M_{{\eta }_c}=116.5(3.2)\mathrm{MeV}$; $\Gamma (J/\psi \to e^{+}{e}^{-})=5.48(16)\mathrm{keV}$; $\Gamma (J/\psi \to \gamma {\eta }_c)=2.49(19)\mathrm{keV}$. The first two are in good agreement with experiment, with $\Gamma (J/\psi \to e^{+}{e}^{-})$ providing a test of a decay matrix element in QCD, independent of Cabibbo-Kobayashi-Maskawa uncertainties, to 2%. At the same time results for the time moments of the correlation function can be compared to values from the charm contribution to $\sigma (e^{+}{e}^{-}\to \mathrm{\text{hadrons}})$, giving a 1.5% test of QCD. Our results show that an improved experimental error would enable a similarly strong test from $\Gamma (J/\psi \to \gamma {\eta }_c)$.

Figure 1

Our average $J/\psi$ correlator divided by the ground-state exponential [$\mathrm{fn}(M_0,t)$ from Eq. (2)] as a function of lattice time. Lines are drawn to join the points (which include statistical errors) for clarity. The fitted result for the ground-state amplitude $a_0^2$ is given by the blue band. The fit includes 6 normal exponentials and 6 oscillating ones, which are responsible for the oscillating behavior clearly seen in the results.

Figure 2

Results for the charmonium hyperfine splitting plotted as a function of lattice spacing. For the $x$ axis we use $(m_{c}a{)}^2$ to allow the $a$ dependence of our fit function [Eq. (3)] (blue dashed line with gray error band) to be displayed simply. The data points have been corrected for $c$ quark mass mistuning and sea quark mass effects, but the corrections are smaller than the error bars. We do not include on the plot the deliberately mistuned $c$ mass but it is included in the fit to constrain the $c$ mass dependence. The errors shown include (and are dominated by) uncertainties from the determination of the lattice spacing $a$ (from the physical value of the parameter $r_1$) that are correlated between the points. The experimental average is plotted as the black point at the origin, offset slightly from the $y$ axis for clarity.

Figure 3

Results for the charmonium vector decay constant plotted as a function of lattice spacing. For the $x$ axis we use $(m_{c}a{)}^2$ to allow the $a$ dependence of our fit function [Eq. (3)] (blue dashed line with gray error band) to be displayed simply. The data points have been corrected for $c$ quark mass mistuning and sea quark mass effects, but the corrections are smaller than the error bars. We do not include on the plot the deliberately mistuned $c$ mass but it is included in the fit to constrain the $c$ mass dependence. The errors shown include (and are dominated by) uncertainties from the determination of the current renormalization factor $Z$ that are correlated between the points. The experimental average is plotted as the black point at the origin, offset slightly from the $y$ axis for clarity.

Figure 4

Results for the 4th, 6th, 8th and 10th time moments of the charmonium vector correlator shown as blue points and plotted as a function of lattice spacing. The errors shown (the same size or smaller than the points) include (and are dominated by) uncertainties from the determination of the current renormalization factor $Z$ that are correlated between the points. The data points have been corrected for $c$ quark mass mistuning and sea quark mass effects, but the corrections are smaller than the error bars (the value for the deliberately mistuned $c$ mass on set 2 is not shown). The blue dashed line with gray error band displays our continuum and chiral fit. Experimental results determined from $R_{e^{+}{e}^{-}}$ [Eq. (12)] are plotted as the black points at the origin offset slightly from the $y$ axis for clarity.

Figure 5

A schematic diagram of the connected 3-point function in lattice QCD for $J/\psi$ to ${\eta }_c$ radiative decay. The lines all represent $c$ quark propagators in this case. The propagator labeled 1 is the spectator quark; 2 and 3 are the initial and final active quarks, respectively. 0 and $T$ label the position in time of the ${\eta }_c$ and $J/\psi$ operators. The vector current is inserted at time $t$ which takes all values between $0$ and $T$.

Figure 6

A plot of the ratio $C_{3pt}(t,T)/(C_{2pt,{\eta }_c}(t)C_{2pt,J/\psi }(T-t))$ for the three values of $T$ on the fine ensemble, set 4 and for our two different methods. Lines join the points (which have statistical errors on them) for clarity. We only include points in the central region of $t$, i.e. $t\ge 5$ or $t\le T-5$. The pink shaded band shows the ratio of fit parameters $V_{00}^{nn}/a_0{b}_0$ which is the ground-state contribution to this ratio. These come from a fit that included 6 normal exponentials and 5 oscillating ones (which produce the oscillations evident in the figure). The top plot shows the results for the case where ${\gamma }_0$ is included in the $J/\psi$ operator (jpsigamma0 method) and the lower plot shows the results for the case where ${\gamma }_0$ is included in the ${\eta }_c$ operator (etacgamma0 method).

Figure 7

Results for the vector form factor at $q^2=0$ for $J/\psi \to {\eta }_c$ decay plotted as a function of lattice spacing. The filled blue circles are from our preferred jpsigamma0 method; the open blue circles are from the etacgamma0 method. For the $x$ axis we use $(m_{c}a{)}^2$ to allow the $a$ dependence of our fit function to be displayed simply (blue dashed line and gray band). The fit is to results from the jpsigamma0 method. The errors shown include statistical errors and errors from the $Z$ factor. The experimental result extracted from the branching fraction for $J/\psi \to \gamma {\eta }_c$ is plotted as the black point offset slightly from the origin for clarity.

Figure 8

A comparison of results for the charmonium hyperfine splitting from lattice QCD with experiment. We show only results that include sea quarks and make use of multiple lattice spacings to derive a continuum value. The experimental average [7] is given at the top, followed by the result for HISQ quarks from this paper. The Fermilab clover [55] and twisted mass [10] results follow. Neither of these lower two results include an error for missing ${\eta }_c$ annihilation effects. This error is the dominant error for our calculation. Here we show our error bar excluding this effect as a solid line and the total error including this effect as a dotted line.

Figure 9

A comparison of results for the decay constant of the $J/\psi$ from lattice QCD with experiment. We include only results that include sea quarks and make use of multiple lattice spacings to derive a continuum value. The experimental average [7] is given at the top, followed by the result for HISQ quarks from this paper. The twisted mass [10] results follow.

Figure 10

A comparison of results for the vector form factor $V(0)$ for $J/\psi \to {\eta }_c\gamma$ from lattice QCD with experiment. We include only results that include sea quarks and make use of multiple lattice spacings to derive a continuum value. The experimental result [26] is given at the top, followed by the result for HISQ quarks from this paper. The twisted mass [10] results follow.

Figure 11

Difference in mass in MeV for different meson tastes for the ${\eta }_c$ and $J/\psi$ used here, plotted against the square of the lattice spacing. Red open circles show the mass difference between the local non-Goldstone and Goldstone ${\eta }_c$ mesons. For the vector we have the mass difference between the local $J/\psi$ meson and the 1-link ${\gamma }_i\otimes {\gamma }_i{\gamma }_j$ vector (blue crosses) and the 1-link ${\gamma }_0{\gamma }_i\otimes {\gamma }_0{\gamma }_i{\gamma }_j$ vector (blue open triangles). The local $J/\psi$ meson is the lighter in both cases. Results are for sets 1, 2 and 4 from Tables 2 and 6. Errors are statistical.

Figure 12

The ratio of the 3-point function for the $B_c$ to $B_c$ vector form factor to the $B_c$ 2-point function against the source-current separation $t/a$. The top plot shows results for $m_{b}a=2.0$ on fine lattices, set 4, for various values of $T$. Lines join the points for clarity. The shaded red band gives our fit result for the ground-state matrix element. The lower plot shows results, also on set 4, for two different values of $m_{b}a$. The shaded red and blue bands show the fit results for the ground-state matrix elements.

Figure 13

The speed of light $c^2$ calculated from zero and finite momentum ${\eta }_c$ correlators using HISQ $c$ quarks. The top figure shows $c^2$ as a function of the square of the spatial momentum for coarse lattices, set 2, where $m_{c}a=0.63$ (gray open circles) and for fine lattices, set 4, where $m_{c}a=0.413$ (pink crosses). The lower figure shows the resulting values of $c^2$ as ${\overrightarrow{p}}^2{a}^2\to 0$ as a function of $(m_{c}a{)}^2$. The dashed straight line is drawn to guide the eye.

Figure 14

The rest (static) and kinetic masses for the ${\eta }_c$ meson compared between the HISQ formalism (this paper) and the twisted mass formalism (Fig. 3 from [10]) and plotted against the square of the lattice spacing. The rest mass is given by open circles and the kinetic mass by open triangles, red for twisted mass and blue for HISQ. Errors include the full lattice spacing error on each point.

Figure 15

$R_{J/\psi }-1$ plotted against $a^2$ in ${\mathrm{fm}}^2$ for three different quark formalisms for the $c$ quark: HISQ (this paper, blue open circles), twisted mass ([10], red open squares) and Fermilab clover ([55], green open triangles, showing results on the two finest lattices only). $R_{J/\psi }$ is $M_{J/\psi }/M_{{\eta }_c}$ so $R_{J/\psi }-1=\Delta M_{\mathrm{hyp}}/M_{{\eta }_c}$. For the twisted mass and Fermilab clover results the heaviest and lightest sea quark masses are plotted at each value of the lattice spacing. Only statistical errors are shown. Additional errors from (twice) the lattice spacing error amount to 2% for HISQ and Fermilab clover and 4%–7% for twisted mass. The black cross is the experimental average [7], offset slightly from the origin for clarity.

Figure 16

$f_{J/\psi }$ in GeV is plotted against $a^2$ in ${\mathrm{fm}}^2$ for the HISQ formalism (this paper, blue open circles) and for twisted mass ([10], red open squares). For the twisted mass case the heaviest and lightest sea quark masses are plotted at each value of the lattice spacing. Only statistical errors are shown for the twisted mass results. For the HISQ results we show the raw data from Table 2 with statistical and uncorrelated lattice spacing errors. There is an additional error of 1.3% from correlated lattice spacing and $Z$ factor uncertainties. The black cross is the experimental result from the average leptonic width [7], offset slightly from the origin for clarity.

Figure 17

The vector form factor for $J/\psi \to {\eta }_c$ decay at $q^2=0$, $V(0)$, is plotted against $a^2$ in ${\mathrm{fm}}^2$ for the HISQ formalism (this paper, jpsigamma0 method, blue open circles) and for twisted mass ([10], red open squares). For the twisted mass case the heaviest and lightest sea quark masses are plotted at each value of the lattice spacing. Errors include statistical errors and uncertainties in the $Z$ factors. The black cross is the experimental result from the rate for $J/\psi \to {\eta }_c\gamma$ decay [26], offset slightly from the origin for clarity.