Electric dipole transitions of $1P$ bottomonia

We compute the electric dipole transitions ${\chi }_{bJ}(1P)\to \gamma \mathrm{\Upsilon }(1S)$, with $J=0$, 1, 2, and $h_b(1P)\to \gamma {\eta }_b(1S)$ in a model-independent way. We use potential nonrelativistic QCD (pNRQCD) at weak coupling with either the Coulomb potential or the complete static potential incorporated in the leading order Hamiltonian. In the last case, the perturbative series shows very mild scale dependence and a good convergence pattern, allowing predictions for all the transition widths. Assuming ${\mathrm{\Lambda }}_{\mathrm{QCD}}\ll m{v}^2$, the precision that we reach is $k_{\gamma }^3/(mv{)}^2\times \mathcal{O}(v^2)$, where $k_{\gamma }$ is the photon energy, $m$ is the mass of the heavy quark and $v$ its relative velocity. Our results are: $\mathrm{\Gamma }({\chi }_{b0}(1P)\to \gamma \mathrm{\Upsilon }(1S))=28_{-2}^{+2}\mathrm{keV}$, $\mathrm{\Gamma }({\chi }_{b1}(1P)\to \gamma \mathrm{{\rm Y}}(1S))=37_{-2}^{+2}\mathrm{keV}$, $\mathrm{\Gamma }({\chi }_{b2}(1P)\to \gamma \mathrm{\Upsilon }(1S))=45_{-3}^{+3}\mathrm{keV}$ and $\mathrm{\Gamma }(h_b(1P)\to \gamma {\eta }_b(1S))=63_{-6}^{+6}\mathrm{keV}$.

Figure 1

Kinematics of the radiative transition $H\to H^{\prime }\gamma$ in the rest frame of the initial-state quarkonium $H$.

Figure 2

For the electric dipole transitions ${\chi }_{bJ}(1P)\to \gamma \mathrm{\Upsilon }(1S)$, with $J=0$ (left panel), $J=1$ (middle panel) and $J=2$ (right panel), we show the leading order decay rate (solid blue curve) and the decay rate obtained including the contributions in Eq. (4) that stem from higher order electromagnetic operators (dashed orange curve).

Figure 3

Matrix elements of the one and two loop corrections to the static potential contributing to the decay width ${\chi }_{bJ}(1P)\to \gamma \mathrm{\Upsilon }(1S)$. In all panels, the solid blue line indicates the LO (no loop corrections) matrix element, the dashed orange line indicates the initial state correction and the dotted green line indicates the final state correction. Left panel: First order correction to the decay width due to the NLO static potential [the term proportional to $a_1(\nu ,r)$]. Middle panel: First order correction to the decay width due to the NNLO static potential [the term proportional to $a_2(\nu ,r)$]. Right panel: Second order correction to the decay width due to the NLO static potential. The additional dot-dashed red line corresponds to a matrix element with a first order correction to both the initial and final states. The matrix elements do not depend on $J$.

Figure 4

For the electric dipole transitions ${\chi }_{bJ}(1P)\to \gamma \mathrm{\Upsilon }(1S)$, with $J=0$ (first column), $J=1$ (second column) and $J=2$ (third column), we show the matrix elements contributing to the decay widths induced by the LO electric dipole operator (3) when the bottomonium wave functions include relativistic corrections due to higher order potentials and kinetic energy in the $1/m$ expansion, see Sec. 2b1. First row: Matrix element at LO, i.e., without relativistic corrections, (solid blue), and at NNLO: Including relativistic corrections due to the $V^{(1)}$ potential (dashed orange and dotted green lines for initial and final state corrections, respectively) and due to the $V_r^{(2)}$ potential (dot-dashed red line for final state correction). Second row: Matrix element at LO (solid blue), and at NNLO: Including relativistic corrections due to the $V_{L^2}^{(2)}$ potential (dashed orange for initial state correction), due to the $V_{LS}^{(2)}$ potential (dotted green for initial state correction), due to the $V_{S^2}^{(2)}$ potential (dot-dashed red for final state correction), and due to the $V_{S_{12}}^{(2)}$ potential (dashed violet and dotted brown for initial and final state corrections, respectively). Third row: Matrix element at LO (solid blue), and at NNLO: Including relativistic corrections due to the $V_{p^2}^{(2)}$ potential (dashed orange and green for initial and final state corrections, respectively), and due to the kinetic energy term $-{\overrightarrow{\nabla }}^4/4m^3$ (dotted red and violet for initial and final state corrections, respectively).

Figure 5

Sums of the matrix elements shown in Figs. 3 and 4, and decay widths at different orders for the E1 transitions ${\chi }_{bJ}(1P)\to \gamma \mathrm{\Upsilon }(1S)$. The three panels in each row refer to the three cases $J=0$, 1, 2, respectively. First row: Sum of the matrix elements at (not up to) LO (solid blue), NLO (dashed orange), NNLO (dotted green) and $\mathrm{NLO}+\mathrm{NNLO}$ (dot-dashed red). Second row: Decay widths at (not up to) LO (solid blue), NLO (dashed orange), NNLO (dotted green) and $\mathrm{NLO}+\mathrm{NNLO}$ (dot-dashed red).

Figure 6

Decay widths of the electric dipole transitions ${\chi }_{bJ}(1P)\to \gamma \mathrm{\Upsilon }(1S)$. The three panels refer to the three cases $J=0$, 1, 2, respectively. In each panel, the dashed blue curve is the LO decay width, the dot-dashed orange one incorporates $\mathrm{LO}+\mathrm{NLO}$ corrections and the solid black curve incorporates $\mathrm{LO}+\mathrm{NLO}+\mathrm{NNLO}$ corrections coming from higher order electromagnetic operators, radiative corrections to the static potential and higher order relativistic corrections to the potential and the kinetic energy. The dotted green curve is similar to the black one but it omits all corrections to the decay width due to radiative corrections of the static potential (one and two loops). We take our central value at $\nu =1.25\mathrm{GeV}$, whereas the gray band indicates the associated uncertainty. The scale setting and the uncertainty estimate are explained in the text.

Figure 7

For the electric dipole transition $h_b(1P)\to \gamma {\eta }_b(1S)$, we show the decay rate at leading order (solid blue curve) and including the contributions in Eq. (5) that stem from higher order electromagnetic operators (dashed orange curve).

Figure 8

For the electric dipole transition $h_b(1P)\to \gamma {\eta }_b(1S)$, we show the matrix elements contributing to the decay width induced by the LO electric dipole operator (3) when the bottomonium wave functions include relativistic corrections due to higher order potentials and kinetic energy in the $1/m$ expansion, see Sec. 2b1. Left panel: Matrix elements at LO (solid blue), and at NNLO: including relativistic corrections due to the $V^{(1)}$ potential (dashed orange and dotted green lines for initial and final state corrections, respectively) and due to the $V_r^{(2)}$ potential (dot-dashed red line for final state correction). Middle Panel: Matrix element at LO (solid blue), and at NNLO including relativistic corrections due to the $V_{L^2}^{(2)}$ potential (dashed orange for initial state correction). Right panel: Matrix element at LO (solid blue), and at NNLO: including relativistic corrections due to the $V_{p^2}^{(2)}$ potential (dashed orange and green for initial and final state corrections, respectively), and due to the kinetic energy term $-{\overrightarrow{\nabla }}^4/4m^3$ (dotted red and violet for initial and final state corrections, respectively).

Figure 9

Sum of the matrix elements shown in Figs. 3 and 8, and decay width at different orders; also final decay width according to Eq. (5), for the E1 transition $h_b(1P)\to \gamma {\eta }_b(1S)$. Left panel: Sum of the matrix elements at (not up to) LO (solid blue), NLO (dashed orange), NNLO (dotted green) and $\mathrm{NLO}+\mathrm{NNLO}$ (dot-dashed red). Middle panel: Decay width at (not up to) LO (solid blue), NLO (dashed orange), NNLO (dotted green) and $\mathrm{NLO}+\mathrm{NNLO}$ (dot-dashed red). Right panel: Description as in Fig. 6.

Figure 10

Leading order decay rate of the electric dipole transition ${\chi }_{b1}\to \gamma \mathrm{\Upsilon }(1S)$. The renormalon subtracted static potential is included at different orders in the Schrödinger equation, which is solved exactly by numerical methods: LO Coulomb-like potential (solid blue), potential up to NLO (dashed orange), potential up to NNLO (dot-dashed green) and potential up to NNNLO (dotted red). The left panel shows the case where the coupling in the static potential is computed at the fixed scale $\nu$ [case ${\nu }_r=\infty$ in Eq. (50)], while the right panel shows the case where the coupling in the static potential is computed at the scale $1/r$ at short distances [case ${\nu }_r=1.0\mathrm{GeV}$ in Eq. (50)].

Figure 11

Decay width, according to Eq. (4), for the transition ${\chi }_{b0}(1P)\to \gamma \mathrm{\Upsilon }(1S)$ in the scheme discussed in the text. The dashed blue curve is the leading order decay rate, the dot-dashed orange curve includes contributions stemming from higher order electromagnetic operators in the pNRQCD Lagrangian and the solid black curve is the final result including both contributions from higher order electromagnetic operators and relativistic corrections to the wave functions of the initial and final states. We take our final value at $\nu =1.25\mathrm{GeV}$ and the gray band indicates the associated uncertainty.

Figure 12

Decay width, according to Eq. (4), for the transition ${\chi }_{b1}(1P)\to \gamma \mathrm{\Upsilon }(1S)$. Description is as in Fig. 11.

Figure 13

Decay width, according to Eq. (4), for the transition ${\chi }_{b2}(1P)\to \gamma \mathrm{\Upsilon }(1S)$. Description is as in Fig. 11.

Figure 14

Decay width, according to Eq. (5), for the transition $h_b(1P)\to \gamma {\eta }_b(1S)$. Description is as in Fig. 11.

Figure 15

Radial wave functions, $R_{n\ell }(r)$, for different values of $n$, $\ell$, and $\nu$ as a function of $r$. $R_{n\ell }(r)$ is the (numerical) solution of the Schrödinger equation for the potential (50).

Figure 16

Radial wave function, $R_{10}(r)$, of the lowest $S$-wave state for $\nu =1.25\mathrm{GeV}$ as a function of $r$. The red solid line shows the (numerical) solution of the Schrödinger equation for the potential (50), while the blue dotted line shows the solution of the Schrödinger equation for the leading order Coulomb potential (11).

Figure 17

As in Fig. 16, but for the radial wave function, $R_{21}(r)$, of the lowest $P$-wave state.