Search

Inspired by the recent observation of the Higgs boson radiative decay into $Z^0$ by {\tt ATLAS} and {\tt CMS} Collaborations, we investigate the next-to-leading-order (NLO) electroweak correction to this rare decay process in Standard Model (SM). Implementing the on-shell renormalization scheme, we find that the magnitude of the NLO electroweak correction may reach $7\%$ of the leading order (LO) prediction, much more significant than that of the NLO QCD correction, which is merely about $0.3\%$. After incorporating the ${\cal O}(\alpha)$ correction, the predicted partial width from various $\alpha$ schemes tend to converge to each other. Including both NLO electroweak and QCD corrections, the SM prediction for the branching fraction shifts from the LO value of $(1.40-1.71)\times 10^{-3}$ to $(1.55\pm 0.06)\times 10^{-3}$, considerably lower than the measured value ${\cal B}_{\rm exp}[H\to Z^0\gamma]=(3.4\pm 1.1)\times 10^{-3}$. Resolving this alarming discrepancy clearly calls for further theoretical investigations, and, more importantly, experimental efforts from {\tt HL-LHC} and the prospective Higgs factories such as {\tt CEPC} and {\tt FCC-ee}., Comment: 7 pages, 2 figures, 1 table; References added, text improved

Within the framework of nonrelativistic QCD (NRQCD), we calculate the fragmentation function for a charm quark into an $S$-wave fully-charmed tetraquark, denoted as $T_{4c}$. The charm-to-$T_{4c}$ fragmentation function can be expressed as a sum of products of the perturbatively calculable short-distance coefficients and the nonperturbative long-distance matrix elements. The short-distance coefficients are ascertained through the perturbative matching procedure at lowest order in $\alpha_{s}$ and $v$ expansion. Incorporating the celebrated QCD factorization and the charm-to-$T_{4c}$ fragmentation function, we are able to predict the $T_{4c}$ production rate at high transverse momentum $p_T$ in colliders. In phenomenology, we approximate the NRQCD matrix elements with phenomenological four-body wave functions at the origin. After applying appropriate kinematic constraints, both the differential distribution over $p_T$ and the integrated cross sections are predicted at the \texttt{LHC}. The cross section can reach approximately $100$ fb, indicating the potential for a significant number of $T_{4c}$ events to be produced at the \texttt{LHC}. Additionally, we provide predictions for the photoproduction of $T_{4c}$ in electron-proton ($ep$) collisions. However, it is observed that the cross sections for these processes are relatively small at the \texttt{HERA}, \texttt{EIC} as well as \texttt{EicC}. Given these findings, the prospect of detecting these fully heavy tetraquarks at $ep$ colliders appears to be rather challenging., Comment: 17 pages, 2 figures, 5 tables

In this work we investigate the inclusive photoproduction of the $C$-odd $S$-wave fully-charmed tetraquark at electron-ion colliders within the nonrelativistic QCD (NRQCD) factorization framework, at the lowest order in velocity and $\alpha_s$. The value of the NRQCD long-distance matrix element is estimated from two phenomenological potential models. Our studies reveal that the photoproduction of $1^{+-}$ fully-charmed tetraquark may be difficult to observe at HERA and EicC, nevertheless its observation prospect at EIC appears to be bright., Comment: 12 pages, 1 table, 3 figures

In the framework of nonrelativistic QCD (NRQCD) factorization, we compute both the polarized and the unpolarized decay widths for the processes $\eta_b(\chi_{bJ})\to J/\psi J/\psi$, accurate up to next-to-next-to-leading-order (NNLO) in $\alpha_s$. For the first time, we confirm that the NRQCD factorization does hold at NNLO for the process involving triple quarkonia. We find the radiative corrections are considerable. In particular for $\chi_{b2}$, both $\mathcal{O}(\alpha_s)$ and $\mathcal{O}(\alpha_s^2)$ corrections are sizable and negative, and can significantly reduce the leading order prediction. At NNLO, the branching fractions are $8.2\times 10^{-7}$, $6.2\times 10^{-6}$, $7.2\times 10^{-7}$ and $2.7\times 10^{-6}$ for $\eta_b$, $\chi_{b0}$, $\chi_{b1}$ and $\chi_{b2}$ decay, respectively. Our theoretical predictions are consistent with the upper limits measured by the {\tt Belle} Collaboration. Moreover, we investigate the dependence of the theoretical predictions on the ratio of the charm quark mass and the bottom quark mass. By fixing $m_b$ and varying $m_c$ from $1.25$ to $1.9$ GeV, we find the branching fraction can change a factor of $2$, $3$, and $6$ for $\eta_b$, $\chi_{b0}$, and $\chi_{b1}$, respectively. In the phenomenological analysis, with the integrate luminosity $\mathcal{L}=100\,{\rm fb}^{-1}$, we expect about $(5-10)\times 10^3$ $\eta_b(\chi_{bJ})\to J/\psi J/\psi\to \ell \bar{\ell}\ell \bar{\ell}$ events produced at the {\tt LHC}, thus it might be hopeful to search for these processes. On the other hand, there are less than $100$ $\eta_b(\chi_{bJ})\to J/\psi J/\psi$ signal events at the B factory, so it seems the experimental measurements on these channels are quite challenging based on current dataset., Comment: 17 pages, 3 figures, 4 tables; The affiliation and acknowledgement of the first author are updated

In this study, we compute the electromagnetic and hadronic decay widths of the S-wave fully heavy tetraquark $T_{4Q}$ ($Q=c$ or $b$) at lowest order in $\alpha_s$ and $v$, in the framework of nonrelativistic QCD. The short-distance coefficients are determined through the standard procedure of matching. The nonperturbative long-distance matrix elements are related to the phenomenological four-body Schr\"odinger wave functions at the origin, whose values are taken from literature. The branching fractions are predicted to be around $10^{-3}$ and $10^{-6}$ for the $T_{4c}$ hadronic decay and electromagnetic decay, respectively. Combing our results with the $T_{4c}$ production cross sections at the LHC, we also predict the event numbers for various decay channels. With integrated luminosity $\mathcal{L}=100 \,{\rm fb}^{-1}$, it is expected that the event numbers can reach $10^3- 10^4$ for $T_{4c}\to \gamma\gamma$, and $10^5- 10^6$ for $T_{4c}\to {\rm LH}$, at the LHC. The detecting prospect is promising., Comment: 12 pages, 1 figures, 3 tables, errors in Eqs. (11c) and (12c) are corrected, numerical values in Tabs.(II) and (III) (for 0^++ channel) are slightly changed, all the conclusions do not change

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

The $X(6900)$ resonance, originally discovered by the \texttt{LHCb} collaboration and later confirmed by both \texttt{ATLAS} and \texttt{CMS} experiments, has sparked broad interests in the fully-charmed tetraquark states. Relative to the mass spectra and decay properties of fully-heavy tetraquarks, our knowledge on their production mechanism is still rather limited. In this work we investigate the inclusive production of fully-charmed $S$-wave tetraquarks at \texttt{LHC} within the nonrelativistic QCD (NRQCD) factorization framework. The partonic cross sections are computed at lowest order in $\alpha_s$ and velocity, while the long-distance NRQCD matrix elements are estimated from phenomenological potential models. We predict the differential $p_T$ spectra of various fully-charmed $S$-wave tetraquarks at the \texttt{LHC}, and compare with the results predicted from the fragmentation mechanism at large $p_T$ end., Comment: 12 pages, 5 figures, 2 tables. SDCs in .m attachment

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., Comment: 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 ${\mathcal O}(\alpha_s^2)$ 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 $\mathcal{O}(\alpha_s^2)$ corrections on the polarized and unpolarized cross sections, as well as the $J/\psi$ angular distributions, which largely reduce the renormalization scale dependence but increase the ${\mathcal O}(\alpha_s)$ NRQCD predictions to some extent for $\chi_{c0,1}$. With high numerical accuracy, our predictions for $\sigma(J/\psi+\chi_{c1,2})$ through ${\mathcal O}(\alpha_s^2)$ are compatible with the upper limit of the \texttt{Belle} measurement. Although the theoretical 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 NRQCD 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: v3, 15 pages, 4 tables, 4 figures, matching the published version

Within the framework of nonrelativistic QCD (NRQCD) factorization, we calculate the next-to-leading-order (NLO) perturbative corrections to the radiative decay $\Upsilon\to \eta_c(\chi_{cJ})+\gamma$. Both the helicity amplitudes and the helicity decay widths are obtained. It is the first computation for the processes involving both bottomonium and charmonium at two-loop accuracy. By employing the Cheng-Wu theorem, we are able to convert most of complex-valued master integrals (MIs) into real-valued MIs, which makes the numerical integration much efficient. Our results indicate the $\mathcal{O}(\alpha_s)$ corrections are moderate for $\eta_c$ and $\chi_{c2}$ production, and are quite marginal for $\chi_{c0}$ and $\chi_{c1}$ production. It is impressive to note the NLO corrections considerably reduce the renormalization scale dependence in both the decay widths and the branching fractions for $\chi_{cJ}$, and slightly improve that for $\eta_c$. With the NRQCD matrix elements evaluated via the Buchm\"uller-Tye potential model, we find the decay width for $\eta_c$ production is one-order-of-magnitude larger than $\chi_{cJ}$ production, which may provide a good opportunity to search for $\Upsilon\to \eta_c+\gamma$ in experiment. In addition, the decay width for $\chi_{c1}$ production is several times larger than those for $\chi_{c0,2}$. Finally, we find the NLO NRQCD prediction for the branching fraction of $\Upsilon\to \chi_{c1}+\gamma$ is only half of the lower bound of the experimental data measured recently by {\tt Belle}. Moreover, there exists serious contradiction between theory and experiment for $\Upsilon\to \eta_c+\gamma$. The discrepancies between theory and experiment deserve further research efforts., Comment: 24 pages, 2 figures, 6 tables. Significantly improved in writing; Matches version published in JHEP

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 ${}^1S_0^{(1,8)}$ 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., Comment: 11 pages, 2 figures

Inspired by the recent discovery of the $X(6900)$ meson at {\tt LHCb} experiment, we investigate the inclusive production rate of the $C$-odd fully-charmed tetraquarks associated with light hadrons at the $B$ factory within the nonrelativistic QCD (NRQCD) factorization framework. The short-distance coefficient is computed at lowest order in velocity and $\alpha_s$. Employing the diquark-antidiquark model to roughly estimate the long-distance NRQCD matrix elements, we predict the rate for inclusive production of the $1^{+-}$ $T_{4c}$ state and discuss the observation prospects at {\tt Belle 2} experiment., Comment: 11 pages, 3 figures

We apply the newly developed NRQCD factorization formula to analyze the exclusive production of the fully-charmed tetraquark $T_{4c}$ plus a hard photon at $B$ factory. For simplicity, in this work we have concentrated on the associated production of the $0^{++}$ and $2^{++}$ tetraquarks with a photon. With some rough estimates about the size of the unknown nonperturbative NRQCD matrix elements, we predict the production rates of these types of exclusive channels at $\sqrt{s}=10.6$ GeV and also assess their observation prospect at {\tt Belle 2} experiment., Comment: 11 pages, 3 figures

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

The $X(6900)$ resonance, very recently discovered in the double-$J/\psi$ channel at LHCb experiment, has spurred intensive interest in unravelling the nature of the fully charmed tetraquark state. The aim of this paper is to present a model-independent theoretical framework to study the inclusive production of this novel species of exotic hadrons, the resonances composed of four heavy quark (commonly referred to as $T_{4c}$), at large $p_T$ in hadron collision experiments. Appealing to asymptotic freedom and the fact $m_c\gg \Lambda_{\rm QCD}$, we propose that the nonpertubative yet universal gluon-to-$T_{4c}$ fragmentation function, can be decomposed into the product of the perturbatively calculable short-distance coefficient and the long-distance NRQCD matrix elements. We compute the short-distance coefficient at lowest-order in $\alpha_s$ and velocity expansion. Adopting the diquark ansatz to roughly estimate those not-yet-known NRQCD matrix elements, together with the standard QCD factorization theorem, we predict the differential production rates for the $T_{4c/4b}(0^{++})$ and $T_{4c/4b}(2^{++})$ at large $p_T$ in $pp$ collision, which eagerly awaits the confrontation with the future LHC experiments., Comment: 9 pages, 1 table, 3 figures

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

In the paper, we present QCD predictions for $\eta_{c} + \gamma$ production at an electron-position collider up to next-to-next-to-leading order (NNLO) accuracy without renormalization scale ambiguities. The NNLO total cross-section for $e^{+}+e^{-}\to\gamma+\eta_{c}$ using the conventional scale-setting approach has large renormalization scale ambiguities, usually estimated by choosing the renormalization scale to be the $e^+ e^-$ center-of-mass collision energy $\sqrt{s}$. The Principle of Maximum Conformality (PMC) provides a systematic way to eliminate such renormalization scale ambiguities by summing the nonconformal $\beta$ contributions into the QCD coupling $\alpha_s(Q^2)$. The renormalization group equation then sets the value of $\alpha_s$ for the process. The PMC renormalization scale reflects the virtuality of the underlying process, and the resulting predictions satisfy all of the requirements of renormalization group invariance, including renormalization scheme invariance. After applying the PMC, we obtain a scale-and-scheme independent prediction, $\sigma|_{\rm NNLO, PMC}\simeq 41.18$ fb for $\sqrt{s}$=10.6 GeV. The resulting pQCD series matches the series for conformal theory and thus has no divergent renormalon contributions. The large $K$ factor which contributes to this process reinforces the importance of uncalculated NNNLO and higher-order terms. Using the PMC scale-and-scheme independent conformal series and the $\rm Pad\acute{e}$ approximation approach, we predict $\sigma|_{\rm NNNLO, PMC+Pade} \simeq 21.36$ fb, which is consistent with the recent BELLE measurement $\sigma^{\rm obs}$=$16.58^{+10.51}_{-9.93}$ fb at $\sqrt{s} \simeq 10.6$ GeV. This procedure also provides a first estimate of the NNNLO contribution., Comment: 8 pages, 5 figures. Discussions improved and references updated. Revised version to be published in JHEP

In this paper, we study the azimuthal asymmetry in the $J/\psi$ leptoproduction in unpolarized $ep$ collisions. There are two independent azimuthal asymmetry modulations, namely $\mathrm{cos}(\psi)$ and $\mathrm{cos}(2\psi)$, where $\psi$ is the azimuthal angle of the lepton scattering plane with respect to the hadron-interacting plane. We calculate the two modulations as functions of four kinematic variables, and find that they provide a very good laboratory to distinguish several models describing the heavy quarkonium production, including the color-singlet (CS) model, the nonrelativistic QCD (NRQCD) associated with the $^1S_0^{[8]}$ dominance picture, and the NRQCD in which the values of all the three color-octet (CO) long-distance matrix elements are of the same order. In order to make definite conclusions, we restrict our calculation in a specific kinematic region, where the CS and CO mechanisms can be distinguished by scrutinizing the values of the $\mathrm{cos}(\psi)$ modulation, while the $^1S_0^{[8]}$ dominance picture can be tested by measuring the values of the $\mathrm{cos}(2\psi)$ modulation. Calculating their values and carrying out a meticulous statistical analysis, we find that at an integrated luminosity $\mathcal{L}=1000\mathrm{pb}^{-1}$, the statistical uncertainties of the two quantities are small enough to tell the three models apart. When this experiment is implemented at the future $ep$ colliders such as the EIC, crucial information for the $J/\psi$ production mechanism might be discovered.

The rare process of $W$ radiative decay into the $B_c$ meson can serve as an interesting play ground for testing two influential perturbative QCD approaches: non-relativistic quantum chromodynamics (NRQCD) and ligh-cone (LC) factorization. Since the $B_c$ meson is a genuine heavy quarkonium which is composed of two different species of heavy quarks, it is natural to employ NRQCD factorization to tackle this exclusive quarkonium production process; on the other hand, since the mass of the $W$ boson is much greater than that of the $B_c$, the collinear factorization is also a viable approach. To fully disentangle the contributions from several distinct scales $M_W$, $m_Q$, $m_Q v$ ($v$ represents the typical velocity of the $c/\bar{b}$ quarks inside $B_c$), and $\Lambda_{\rm QCD}$, we carry out a joint analysis that combines both light-cone and NRQCD factorization approaches, and compute the order-$\alpha_s$ correction to $W \to B_c+\gamma$. With the aid of the celebrated Efremov-Radyushkin-Brodsky-Lepage (ERBL) evolution equation, we resum the leading collinear logarithms $\alpha_s \ln\! {M_W\over m_Q}$ to all orders in $\alpha_s$. We also explore some phenomenological implications of our predictions., Comment: 16 pages, 3 figures

We study the $\eta_c$ photoproduction in $ep$ collisions in this paper. The short-distance coefficients for $c\bar{c}(^1S_0^{[1]})$, $c\bar{c}(^1S_0^{[8]})$, $c\bar{c}(^3S_1^{[8]})$, and $c\bar{c}(^1P_1^{[8]})$ photoproductions are evaluated at leading order in $\alpha_s$ expansion, where the color-singlet contribution is achieved for the first time. We have carefully analyzed different kinematic distributions of the cross sections and found that the color-singlet contribution is considerably suppressed comparing with the color-octet parts. This feature renders the $\eta_c$ photoproduction process an ideal laboratory to test the color-octet mechanism in nonrelativistic QCD. By taking different sets of long-distance matrix elements, we have observed some apparently distinguishable predictions, which can be utilized to scrutinize the validity of these matrix elements., Comment: 6pages, 2 figures

Within the nonrelativistic QCD (NRQCD) factorization framework, we compute the long-awaited ${\mathcal O}(\alpha_s^2)$ correction for the exclusive double charmonium production process at $B$ factories, {\it i.e.}, $e^+e^-\to J/\psi+\eta_c$ at $\sqrt{s}=10.58$ GeV. For the first time, we confirm that NRQCD factorization does hold at next-to-next-to-leading-order (NNLO) for exclusive double charmonium production. It is found that including the NNLO QCD correction considerably reduces the renormalization scale dependence, and also implies the reasonable perturbative convergence behavior for this process. Our state-of-the-art prediction is consistent with the BaBar measurement within errors., Comment: v2; 5 pages, 2 figures, 2 tables; numerical precision of predictions greatly improved; two additional authors added

The associated production of Higgs boson with a muon pair, $e^+e^-\to \mu^+\mu^- H$, is one of the golden channels to pin down the properties of the Higgs boson in the prospective Higgs factories exemplified by \textsf{CEPC}. The projected accuracy of the corresponding cross section measurement is about per cent level at CEPC. In this work, we investigate both ${\mathcal O}(\alpha)$ weak correction and the ${\mathcal O}(\alpha\alpha_s)$ mixed electroweak-QCD corrections for this channel, appropriately taking into account the effect of finite $Z^0$ width. The $\mu^+\mu^-$ invariant mass spectrum is also predicted. The mixed electroweak-QCD correction turns out to reach 1.5\% of the Born-order result, and thereby must be included in future confrontation with the data. We also observe that, after including higher-order corrections, the simplified prediction for the integrated cross section employing the narrow-width-approximation may deviate from our full result by a few per cents., Comment: v2, 15 pages, 5 figures, 3 tables. Typos corrected, references added, the version to appear in Chinese Physics C

The next-to-next-to-leading order (NNLO) pQCD prediction for the $\gamma\gamma^* \to \eta_c$ form factor was evaluated in 2015 using nonrelativistic QCD (NRQCD). A strong discrepancy between the NRQCD prediction and the BaBar measurements was observed. Until now there has been no solution for this puzzle. In this paper, we present a NNLO analysis by applying the Principle of Maximum Conformality (PMC) to set the renormalization scale. By carefully dealing with the light-by-light diagrams at the NNLO level, the resulting high precision PMC prediction agrees with the BaBar measurements within errors, and the conventional renormalization scale uncertainty is eliminated. The PMC is consistent with all of the requirements of the renormalization group, including scheme-independence. The application of the PMC thus provides a rigorous solution for the $\gamma\gamma^* \to \eta_c$ form factor puzzle, emphasizing the importance of correct renormalization scale-setting. The results also support the applicability of NRQCD to hard exclusive processes involving charmonium., Comment: 5 pages, 4 figures. References updated. To be published in Phys.Rev.D

We compute the next-to-next-to-leading order (NNLO) QCD corrections to the hadronic decay rates of the pseudoscalar quarkonia, at the lowest order in velocity expansion. The validity of NRQCD factorization for inclusive quarkonium decay process, for the first time, is verified to relative order $\alpha_s^2$. As a byproduct, the renormalization group equation (RGE) of the leading NRQCD 4-fermion operator ${\cal O}_1({}^1S_0)$ is also deduced to this perturbative order. By incorporating this new piece of correction together with available relativistic corrections, we find that there exists severe tension between the state-of-the-art NRQCD predictions and the measured $\eta_c$ hadronic width, and in particular the branching fraction of $\eta_c\to \gamma\gamma$. NRQCD appears to be capable of accounting for $\eta_b$ hadronic decay to a satisfactory degree, and our most refined prediction is ${\rm Br}(\eta_b\to \gamma\gamma) = (4.8\pm 0.7)\times 10^{-5}$., Comment: v3, 6 pages, 3 figures; minor errors corrected, text improved

The associated production of Higgs boson with a hard photon at lepton collider, i.e., $e^+e^-\to H\gamma$, is known to bear a rather small cross section in Standard Model, and can serve as a sensitive probe for the potential new physics signals. Similar to the loop-induced Higgs decay channels $H\to \gamma\gamma, Z\gamma$, the $e^+e^-\to H\gamma$ process also starts at one-loop order provided that the tiny electron mass is neglected. In this work, we calculate the next-to-leading-order (NLO) QCD corrections to this associated $H+\gamma$ production process, which mainly stem from the gluonic dressing to the top quark loop. The QCD corrections are found to be rather modest at lower center-of-mass energy range ($\sqrt{s}<300$ GeV), thus of negligible impact on Higgs factory such as CEPC. Nevertheless, when the energy is boosted to the ILC energy range ($\sqrt{s}\approx 400$ GeV), QCD corrections may enhance the leading-order cross section by $20\%$. In any event, the $e^+e^-\to H\gamma$ process has a maximal production rate $\sigma_{\rm max}\approx 0.08$ fb around $\sqrt{s}= 250$ GeV, thus CEPC turns out to be the best place to look for this rare Higgs production process. In the high energy limit, the effect of NLO QCD corrections become completely negligible, which can be simply attributed to the different asymptotic scaling behaviors of the LO and NLO cross sections, where the former exhibits a milder decrement $\propto 1/s$ , but the latter undergoes a much faster decrease $\propto 1/s^2$., Comment: v4, 11 pages, 6 figures, 2 tables; errors in Appendix are fixed; version accepted for publication at PLB

The prospective Higgs factories, exemplified by ILC, FCC-ee and CEPC, plan to conduct the precision Higgs measurements at the $e^+e^-$ center-of-mass energy around 250 GeV. The cross sections for the dominant Higgs production channel, the Higgsstrahlung process, can be measured to a (sub-) percent accuracy. Merely incorporating the well-known next-to-leading order (NLO) electroweak corrections appears far from sufficient to match the unprecedented experimental precision. In this work, we make an important advancement toward this direction by investigating the mixed electroweak-QCD corrections to $e^+e^-\to HZ$ at next-to-next-to-leading order (NNLO) for both unpolarized and polarized $Z$ boson. The corrections turn out to reach one percent level of the Born-order results, thereby must be incorporated in the future confrontation with the data., Comment: 7 pages, 3 figures, 2 tables; version accepted for publication in PRD RC

We calculate the next-to-next-to-leading-order (NNLO) perturbative corrections to $P$-wave quarkonia annihilation decay to two photons, in the framework of nonrelativistic QCD (NRQCD) factorization. The order-$\alpha_s^2$ short-distance coefficients associated with each helicity amplitude are presented in a semi-analytic form, including the "light-by-light" contributions. With substantial NNLO corrections, we find disquieting discrepancy when confronting our state-of-the-art predictions with the latest \textsf{BESIII} measurements, especially fail to account for the measured $\chi_{c2}\to\gamma\gamma$ width. Incorporating the effects of spin-dependent forces would even exacerbate the situation, since it lifts the degeneracy between the nonperturbative NRQCD matrix elements of $\chi_{c0}$ and $\chi_{c2}$ toward the wrong direction. We also present the order-$\alpha_s^2$ predictions to $\chi_{b0,2}\to\gamma\gamma$, which await the future experimental test., Comment: 6 pages, 2 figures, 1 table; v2, typos fixed, minor changes in text

Unlike the bewildering situation in the $\gamma\gamma^*\to \pi$ form factor, a widespread view is that perturbative QCD can decently account for the recent \textsc{BaBar} measurement of $\gamma\gamma^*\to \eta_c$ transition form factor. The next-to-next-to-leading order (NNLO) perturbative correction to the $\gamma\gamma^*\to \eta_{c,b}$ form factor, is investigated in the NRQCD factorization framework for the first time. As a byproduct, we obtain by far the most precise order-$\alpha_s^2$ NRQCD matching coefficient for the $\eta_{c,b}\to \gamma\gamma$ process. After including the substantial negative order-$\alpha_s^2$ correction, the good agreement between NRQCD prediction and the measured $\gamma\gamma^*\to \eta_c$ form factor is completely ruined over a wide range of momentum transfer squared. This eminent discrepancy casts some doubts on the applicability of NRQCD approach to hard exclusive reactions involving charmonium., Comment: 6 pages, 3 figures and 1 table; adding Eqs.(8) and (9) as well as some references, correcting errors in Table 1, updating Fig.3 to include the "light-by-light" contributions, devoting a paragraph to discuss why our strategy of interpreting the NNLO corrections is justified; Accepted by PRL

Within the framework of nonrelativisitic QCD factorization formalism, we present the next-to-leading-order relativistic corrections to $\Upsilon$ exclusive decay into $\eta_c$ plus $J/\psi$. The double charmonia can be produced through several immediate channels, i.e. $\Upsilon\to g^*g^*g^*\to \eta_c+J/\psi$, $\Upsilon\to g^*g^*\gamma^*\to \eta_c+J/\psi$, and $\Upsilon\to \gamma^*\to \eta_c+J/\psi$. The amplitudes of these three channels are obtained accurate up to ${\cal O}(\alpha_s^3 v^2)$, ${\cal O}(\alpha\alpha_s^2 v^2)$, and ${\cal O}(\alpha\alpha_s^2 v^2)$ respectively, where $v$ indicates the typical heavy quark velocity in bottomonium and/or charmonium rest frame. The decay rates are also presented. We find the next-to-leading-order relativistic corrections to the short distance coefficients as well as the decay rates are both significant and negative, especially for the corrections from bottomonium. More seriously, the decay rates are even brought into negative by including the relativistic corrections, which indicates the poor convergence for the velocity expansion in this kind of processes. Detailed analysis is given in the paper., Comment: 11 pages, 2 figures. v2: added an appendix, corrected some typos

With the recent LHCb data on $\eta_c$ production and based on heavy quark spin symmetry, we obtain the long-distance matrix elements for both $\eta_c$ and $J/\psi$ productions, among which, the color-singlet one for $\eta_c$ is obtained directly by the fit of experiment for the first time. Using our long-distance matrix elements, we can provide good description of the $\eta_c$ and $J/\psi$ hadroproduction measurements. Our predictions on $J/\psi$ polarization are in good agreement with the LHCb data and pass through the two sets of CDF measurements in medium $p_t$ region. Considering all the possible uncertainties carefully, we obtained quite narrow bands of the $J/\psi$ polarization curves.