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We compute the O(αs2) corrections to the exclusive channels e+e−→J/ψ+χcJ (J=0,1,2) at s=10.58 GeV within the nonrelativistic QCD (NRQCD) factorization framework. The validity of NRQCD factorization at O(αs2) has been confirmed for these double-charmonium exclusive production processes. We analyze the impact of the O(αs2) corrections on the polarized and unpolarized cross sections, as well as the J/ψ angular distributions, which largely reduce the renormalization scale dependence but increase the O(αs) NRQCD predictions to some extent for χc0,1. With high numerical accuracy, our predictions for σ(J/ψ+χc1,2) through O(αs2) are compatible with the upper limit of the Belle measurement. Although the theoretical prediction for σ(J/ψ+χc0) is consistent with both Belle and BaBar measurements within uncertainties, there still exists serious tension between the predicted and the measured profiles for the J/ψ angular distribution. Regarding the bright observation prospect of the e+e−→J/ψ+χc1,2 channels in Belle 2 experiment, it is interesting to compare the future measurements with our NRQCD predictions. The more accurate measurement of e+e−→J/ψ+χ0 at Belle 2 will also help to clarify the long-standing puzzle of J/ψ angular distribution.

The failure of observing the $e^+ e^- \to J/\psi+J/\psi$ events at $B$ factories to date is often attributed to the significant negative order-$\alpha_s$ correction. In this work we compute the ${\cal O}(\alpha^2_s)$ correction to this process for the first time. The magnitude of the next-to-next-to-leading order (NNLO) perturbative correction is substantially negative so that the standard NRQCD prediction would suffer from an unphysical, negative cross section. This dilemma may be traced in the fact that the bulk contribution of the fixed-order radiative corrections stems from the perturbative corrections to the $J/\psi$ decay constant. We thus implement an improved NRQCD factorization framework, by decomposing the amplitude into the photon-fragmentation piece and the non-fragmentation piece. With the measured $J/\psi$ decay constant as input, which amounts to resumming a specific class of radiative and relativistic corrections to all orders, the fragmentation-induced production rate can be predicted accurately and serves a benchmark prediction. The non-fragmentation type of the amplitude is then computed through NNLO in $\alpha_s$ and at lowest order in velocity. Both the ${\cal O}(\alpha_s)$ and ${\cal O}(\alpha^2_s)$ corrections in the interference term become positive and exhibit a decent convergence behavior. Our finest prediction is $\sigma(e^+ e^- \to J/\psi+J/\psi)= 2.13^{+0.30}_{-0.06}$ fb at $\sqrt{s}=10.58$ GeV. With the projected integrated luminosity of 50 ${\rm ab}^{-1}$, the prospect to observe this exclusive process at \texttt{Belle} 2 experiment appears to be bright., Comment: 13 pages, 8 figures, 3 tables

Within the framework of nonrelativistic QCD (NRQCD) factorization formalism, we compute QCD next-to-next-to-leading order (NNLO) corrections to the helicity amplitudes as well as the decay width of $Z\to H+\gamma$, where $H$ can be $\eta_Q (Q=c,b), J/\psi$, or $\Upsilon$. In addition, we resum the next-to-leading logarithms (NLL) of ${m_Z^2}/{m_Q^2}$ to all orders of $\alpha_s$ for the leading-twist helicity amplitude by employing the light-cone factorization approach. It is worth mentioning that we obtain the analytic expressions of the truncated NLL at $\alpha_s^2$. We find that the $\mathcal{O}(\alpha_s)$ corrections are around 10\% for $\eta_c$ and $\Upsilon$ productions, however insignificant for $J/\psi$ and $\eta_b$ productions. The $\mathcal{O}(\alpha_s^2)$ corrections are moderate for charmonium production, while very small for bottomonium production. Moreover, it is found that the NLL resummation can considerably alter the NRQCD prediction, especially for $J/\psi$ production. Combining the NRQCD and light-cone computation, we make phenomenological predictions on the decay widths and branching fractions. In addition, we investigate the dependence of the theoretical results on the heavy quark mass, and find the branching fraction of $Z\to H+\gamma$ monotonically decreases as $m_Q$ increases., Comment: 21 pages, 2 figures, 3 tables. correct typos, add the comparison with LC models, match the published version in PRD. arXiv admin note: text overlap with arXiv:2208.10118

In this paper, we evaluate, up to QCD next-to-next-to-next-to-leading order, the $B_c^*$ decay constant, which, within the nonrelativistic QCD (NRQCD) framework, is factorized as the product of the short-distance coefficient (SDC) and the long-distance matrix element. For the first time, the renormalization constant and the anomalous dimension for the NRQCD vector current composed of $c\bar{b}$, which are functions of the charm quark mass $m_c$, the bottom quark mass $m_b$ and the factorization scale $\mu_\Lambda$, are obtained analytically at $\mathcal{O}(\alpha_s^2)$ and $\mathcal{O}(\alpha_s^3)$. The SDC is calculated up to $\mathcal{O}(\alpha_s^3)$ in perturbative expansion. Explicitly, the $\mathcal{O}(\alpha_s^2)$ correction to the SDC is analytically calculated in terms of logarithmic and polylogarithmic functions of $r\equiv m_b/m_c$, and the $\mathcal{O}(\alpha_s^3)$ correction to the SDC is numerically calculated at a series of values of $r$, ranging from $2.1$ to $4.0$. Surprisingly, we find that the nontrivial part of the SDC at $\mathcal{O}(\alpha_s^3)$ can be well estimated by a linear function of $r$. In addition, We find that the $\mathcal{O}(\alpha_s^2)$ and $\mathcal{O}(\alpha_s^3)$ corrections to the decay constant and decay width are considerable and very significant, which indicates a very poor convergence for the perturbative expansion., Comment: 15 pages, 3 figures, 2 tables, a minor typo corrected

In this work, we study the radiative decay of $Z$ boson to a $P$-wave quarkonium $H$ in association with a photon, where $H$ can be $\chi_{QJ}$, $h_Q$ with $Q=c,b$ and $J=0,1,2$. The helicity amplitudes and the unpolarized decay widths are evaluated up to QCD next-to-next-to-leading order (NNLO) within the framework of nonrelativistic QCD (NRQCD). For the first time, we check the NRQCD factorization for $h_Q$ exclusive production at two-loop order. The leading logarithms (LL) of $m_Z^2/m_Q^2$ in the leading-twist short-distance coefficients, which may potentially ruin the perturbative convergence, are resummed to all orders of $\alpha_s$ by employing the light-cone factorization. We find the radiative corrections are considerable for $\chi_{Q2}$ and $h_Q$ productions, while are moderate or even minor for other channels. We also notice that the LL resummation can change the leading-order predictions for decay widths by more than 25\% for $\chi_{c0,2}$ and $h_c$ productions, and by around 50\% for $\chi_{c1}$ production. However, effects of the LL resummation on the next-to-leading-order and NNLO predictions are notably mitigated. Some phenomenological explorations are also performed., Comment: 19 pages, 1 figure, 5 tables, words updated

Within the framework of nonrelativistic QCD (NRQCD) factorization, we compute the three-loop QCD corrections to the decay constant of $B_c$. We reconstruct the analytical expressions for the three-loop renormalization constant and the corresponding anomalous dimension affiliated with the pseudoscalar current composed of two different heavy flavors in NRQCD, which are functions of the ratio between the charm and bottom quark masses. Meanwhile, the short-distance coefficient is obtained with very high numerical accuracy. The three-loop QCD correction turns out to be overwhelmingly large. The phenomenological implication of this new piece of radiative corrections for the $B_c$ leptonic decay is also addressed., 9 pages, 4 figures and 2 tables

Within the nonrelativistic QCD (NRQCD) factorization framework, we compute the order-$\alpha_s^3$ perturbative corrections to the leptonic decay of the $\Upsilon$ and $J/\psi$ with high numerical accuracy, at the lowest order in velocity expansion. We confirm the existing three-loop results in literature. Furthermore, we explicitly consider the complex-valued light-by-light (singlet) contributions. We for the first time also consider the finite charm quark mass effect for $\Upsilon$ leptonic decay, and obtain a new piece of 3-loop contribution to the anomalous dimension related to the composite NRQCD bilinear of vector current that arises from the nonzero charm quark mass. Based on the complete three-loop NRQCD short-distance coefficients, we also present a comprehensive phenomenological analysis for $\Upsilon(J/\psi)$ leptonic width., Comment: 16 pages, 7 figures

In this work, we study the exclusive decay of $\Upsilon$ into $J/\psi$ in association with $\eta_c$ ($\chi_{c0,1,2}$). The decay widths for different helicity configurations are evaluated up to QCD next-to-leading order within the nonrelativistic QCD framework. We find that the QCD corrections notably mitigate the renormalization scale dependence of the decay widths for all the processes. The branching fraction of $\Upsilon\rightarrow J/\psi+\chi_{c1}$ is obtained as $3.73^{+5.10+0.10}_{-2.06-1.19}\times 10^{-6}$, which agrees well with the Belle measurement, i.e., ${\rm Br}(\Upsilon\rightarrow J/\psi+\chi_{c1})=(3.90\pm1.21\pm0.23)\times10^{-6}$. For the other processes, our results of the branching fractions are compatible with the upper limits given by the Belle experiments, except for $\Upsilon(2S)\to J/\psi +\chi_{c1}$, where some tension exists between theory and experiment. Having the polarized decay widths, we study the $J/\psi$ polarization, which turn out to be independent of any nonperturbative parameters. Further, according to our calculation, it is promising to measure all the processes at Super B factory thanks to the high luminosity., Comment: 13 pages, 2 figures, 3 tables, to match the version accepted for publication in PRL

We compute the next-to-next-to-leading-order (NNLO) perturbative corrections to the exclusive channels $e^+e^-\to J/\psi+\chi_{cJ}$ ($J=0,1,2$) at $\sqrt{s}=10.58$ GeV within the nonrelativistic QCD (NRQCD) factorization framework. The validity of NRQCD factorization at ${\cal O}(\alpha_s^2)$ has been confirmed for these double-charmonium exclusive production processes. We analyze the impact of the NNLO perturbative corrections on the polarized and unpolarized cross sections, as well as the $J/\psi$ angular distributions, which largely reduce the renormalization scale dependence but lower the NLO NRQCD predictions to some extent for $\chi_{c0,1}$. With high numerical accuracy, our NNLO predictions for $\sigma(J/\psi+\chi_{c1,2})$ are compatible with the upper limit of the \texttt{Belle} measurement. Although the finest prediction for $\sigma(J/\psi+\chi_{c0})$ is consistent with both \texttt{Belle} and \texttt{BaBar} measurements within uncertainties, there still exists serious tension between the predicted and the measured profiles for the $J/\psi$ angular distribution. Regarding the bright observation prospect of the $e^+e^-\to J/\psi+\chi_{c1,2}$ channels in \texttt{Belle 2} experiment, it is interesting to compare the future measurements with our NNLO predictions. The more accurate measurement of $e^+e^-\to J/\psi+\chi_{0}$ at \texttt{Belle 2} will also help to clarify the long-standing puzzle of $J/\psi$ angular distribution., Comment: 17 pages, 4 tables, 2 figures. b quark loop contributions explicitly included

Within NRQCD factorization framework, in this work we compute, at the lowest order in velocity expansion, the next-to-leading-order (NLO) perturbative corrections to the short-distance coefficients associated with heavy quark fragmentation into the $${}^1S_0^{(1,8)}$$ 1 S 0 ( 1 , 8 ) components of a heavy quarkonium. Starting from the Collins and Soper’s operator definition of the quark fragmentation function, we apply the sector decomposition method to facilitate the numerical manipulation. It is found that the NLO QCD corrections have a significant impact.

Within the framework of nonrelativistic QCD factorization formalism, we compute the helicity amplitude as well as the decay width of ηQ2 (Q=c, b) electromagnetic decay into two photons up to next-to-next-to-leading order in αs expansion. For the first time, we verify the validity of nonrelativistic QCD factorization for the D-wave quarkonium decay at next-to-next-to-leading order. We find that the O(αs) and O(αs2) corrections to the helicity amplitude are negative and moderate, nevertheless both corrections combine to suppress the leading-order prediction for the decay width significantly. By approximating the total decay width of ηQ2 as the sum of those for the hadronic decay and the electric E1 transition, we obtain the branching ratios Br(ηc2→2γ)≈5×10−6 and Br(ηb2→2γ)≈4×10−7. To explore the potential measurement on ηQ2, we further evaluate the production cross section of ηQ2 at LHCb at the lowest order in αs expansion. With the kinematic constraint on the longitudinal rapidity 4.5>y>2 and transverse momentum PT>(2–4)mQ for ηQ2, we find the cross section can reach 2–50 nb for ηc2, and 1–22 pb for ηb2. Considering the integrated luminosity L=10 fb−1 at s=7, 13 TeV, we estimate that there are several hundreds events of pp→ηc2→2γ. Since the background is relatively clean, it is promising to reconstruct ηc2 through its electromagnetic decay. On the other hand, due to the small branching ratio and production cross section, it is quite challenging to detect ηb2→2γ at LHCb.

Within the nonrelativistic QCD (NRQCD) factorization framework, we compute the O(αsv2) corrections to the hadronic decay rate of vector quarkonia, exemplified by J/ψ and ϒ. Setting both the renormalization and NRQCD factorization scales to be mQ, we obtain Γ(J/ψ→LH)=0.0716αs3mc2⟨O1(S31)⟩J/ψ[1−1.19αs+(−5.32+3.03αs)⟨v2⟩J/ψ] and Γ(ϒ→LH)=0.0716αs3mb2⟨O1(S31)⟩ϒ[1−1.56αs+(−5.32+4.61αs)⟨v2⟩ϒ]. We confirm the previous calculation of O(αs) corrections on a diagram-by-diagram basis, with the accuracy significantly improved. For J/ψ hadronic decay, we find that the O(αsv2) corrections are moderate and positive, nevertheless unable to counterbalance the huge negative corrections. On the other hand, the effect of O(αsv2) corrections for ϒ(nS) is sensitive to the O(v2) NRQCD matrix elements. With the appropriate choice of the NRQCD matrix elements, our theoretical predictions for the decay rates may be consistent with the experimental data for ϒ(1S,2S)→LH. As a by-product, we also present the theoretical predictions for the branching ratio of J/ψ(ϒ)→3γ accurate up to O(αsv2).

Within the framework of nonrelativistic QCD (NRQCD) factorization formalism, we compute the helicity amplitude as well as the decay width of $\eta_{Q2}$ ($Q=c,b$) electromagnetic decay into two photons up to next-to-next-to-leading order (NNLO) in $\alpha_s$ expansion. For the first time, we verify the validity of NRQCD factorization for the D-wave quarkonium decay at NNLO. We find that the $\mathcal{O}(\alpha_s)$ and $\mathcal{O}(\alpha_s^2)$ corrections to the helicity amplitude are negative and moderate, nevertheless both corrections combine to suppress the leading-order prediction for the decay width significantly. By approximating the total decay width of $\eta_{Q2}$ as the sum of those for the hadronic decay and the electric $E1$ transition, we obtain the branching ratios ${\rm Br}(\eta_{c2}\to 2\gamma)\approx 5\times10^{-6}$ and ${\rm Br}(\eta_{b2}\to 2\gamma)\approx 4\times10^{-7}$. To explore the potential measurement on $\eta_{Q2}$, we further evaluate the production cross section of $\eta_{Q2}$ at LHCb at the lowest order in $\alpha_s$ expansion. With the kinematic constraint on the longitudinal rapidity $4.5>y>2$ and transverse momentum $P_T>(2-4)m_Q$ for $\eta_{Q2}$, we find the cross section can reach $2-50$ nb for $\eta_{c2}$, and $1-22$ pb for $\eta_{b2}$. Considering the integrated luminosity $\mathcal{L}=10\, {\rm fb}^{-1}$ at $\sqrt{s}=7$ TeV and $\sqrt{s}=13$ TeV, we estimate that there are several hundreds events of $pp\to \eta_{c2}\to 2\gamma$. Since the background is relatively clean, it is promising to reconstruct $\eta_{c2}$ through its electromagnetic decay. On the contrary, due to small branching ratio and production cross section, it is quite challenging to detect $\eta_{b2}\to 2\gamma$ at LHCb., Comment: 14 pages, 1 figures, 2 tables

Within the nonrelativistic QCD (NRQCD) factorization framework, we compute the ${\mathcal O}(\alpha_s v^2)$ corrections to the hadronic decay rate of vector quarkonia, exemplified by $J/\psi$ and $\Upsilon$. Setting both the renormalization and NRQCD factorization scales to be $m_Q$, we obtain $\Gamma(J/\psi\to {\rm LH})= 0.0716\frac{\alpha_s^3}{m_c^2} \langle \mathcal{O}_1({}^3S_1)\rangle_{J/\psi} [1-1.19\alpha_s+(-5.32+3.03\alpha_s)\langle v^2\rangle_{J/\psi}]$ and $\Gamma(\Upsilon\to {\rm LH})= 0.0716\frac{\alpha_s^3}{m_b^2}\langle\mathcal{O}_1({}^3S_1)\rangle_{\Upsilon}[1-1.56\alpha_s+(-5.32+4.61\alpha_s)\langle v^2\rangle_{\Upsilon}]$. We confirm the previous calculation of $\mathcal{O}(\alpha_s)$ corrections on a diagram-by-diagram basis, with the accuracy significantly improved. For $J/\psi$ hadronic decay, we find that the ${\mathcal O}(\alpha_sv^2)$ corrections are moderate and positive, nevertheless unable to counterbalance the huge negative corrections. On the other hand, the effect of ${\mathcal O}(\alpha_sv^2)$ corrections for $\Upsilon(nS)$ is sensitive to the $\mathcal{O}(v^2)$ NRQCD matrix elements. With the appropriate choice of the NRQCD matrix elements, our theoretical predictions for the decay rates may be consistent with the experimental data for $\Upsilon(1S,2S)\to {\rm LH}$. As a byproduct, we also present the theoretical predictions for the branching ratio of $J/\psi(\Upsilon)\to 3\gamma$ accurate up to $\mathcal{O}(\alpha_s v^2)$., Comment: 21 pages, 5 figures, 3 tables

Within the nonrelativistic QCD (NRQCD) factorization framework, we have computed the ${\mathcal O}(\alpha_s^2)$ corrections to the exclusive production of $P$-wave spin-triplet charmonia $\chi_{cJ}(J=0,1,2)$ accompanied by a hard photon at $B$ factories. For the first time, we have explicitly verified the validity of NRQCD factorization for exclusive $P$-wave quarkonium production at two-loop order. Unlike the $\chi_{cJ}$ electromagnetic decay processes, the $\mathcal{O}(\alpha_s^2)$ corrections are found to be smaller than the $\mathcal{O}(\alpha_s)$ corrections in all three channels $e^+e^-\to \chi_{c0,1,2}+\gamma$. In particular, the ${\mathcal O}(\alpha_s^2)$ corrections appear moderate for $\chi_{c1}$ case, and insignificant for $\chi_{c0}$. In addition, the next-to-next-to-leading order (NNLO) predictions for the production rates of $\chi_{c0,1}+\gamma$ are insensitive to the renormalization and factorization scales. All of these features may indicate that perturbative expansion in these two channels exhibits a decent convergence behavior. By contrast, both the ${\mathcal O}(\alpha_s)$ and ${\mathcal O}(\alpha_s^2)$ corrections to the $\chi_{c2}+\gamma$ production cross section are sizable, which even reduces the Born order cross section by one order of magnitude after including the NNLO perturbative correction. Taking the values of the long-distance NRQCD matrix elements from nonrelativistic potential model, our prediction to $\chi_{c1}+\gamma$ production rate is consistent with the recent {\tt Belle} measurement. The NNLO predictions for the $\chi_{c0,2}+\gamma$ production rates are much smaller than that for $\chi_{c1}+\gamma$, which seems to naturally explain why the $e^+e^-\to \chi_{c0,2}+\gamma$ channels have escaped experimental detection to date., Comment: typos corrected, some references added

We study the ηc photoproduction in ep collisions in this paper. The short-distance coefficients for cc¯(S0[1]1), cc¯(S0[8]1), cc¯(S1[8]3), and cc¯(P1[8]1) photoproductions via both direct and resolved photons are evaluated at leading order in αs expansion, where the color-singlet contribution is achieved for the first time. The dependence of the cross sections on the renormalization and factorization scales, and the charm quark mass is studied in this paper. As an interesting result, the color-singlet cross section is dominated by the resolved photoproduction, which is almost two orders of magnitude larger than the contributions from the direct photons. The resolved photoproduction also enhances the color-octet cross sections considerably. By including the contributions from direct and resolved photons, we find the color-singlet channel dominates the integrated cross section. Making use of different sets of long-distance matrix elements, we observe some apparently distinguishable predictions, which can be utilized to scrutinize the validity of these matrix elements. Moreover, we carefully analyze different kinematic distributions of the cross sections and find that, in high pt regions, the color-singlet contribution is considerably suppressed comparing with the color-octet parts for most sets of long-distance matrix elements. This feature renders the ηc photoproduction process another laboratory to test the color-octet mechanism in nonrelativistic QCD.