Motivated by very recent measurement of the radiative decays of $Υ(1S)$ to $χ_{c1}$ at Belle, we use the nonrelativistic QCD factorization theory and calculate the branching fractions of the radiative decays of bottomonium into S-wave and P-wave charmonium, i.e. $Υ(nS) \to η_c(nS)+γ$ and $Υ(nS)\to χ_{cJ}+γ$. We systematically studied the branching fractions of the radiative decays of bottomonium into charmonium. Compared to the previous calculation, we obtained the analytical expression for the decay widths and considered the color-octet contributions. For $Υ(nS) \to η_c(nS)+γ$, the relativistic corrections are also obtained. Through the calculation, the theoretical prediction for $Υ(1S)\toχ_{c1}+γ$ is still smaller than the recent Belle measurement. Further theoretical work and experimental analysis are necessary to understand the $χ_{c1}$ production mechanism in upsilon decays.
In this work we discuss the practical and conceptual issues related to quark-hadron-duality in heavy-heavy systems. Recent measurements in the charmonium region allow a direct test of quark-hadron-duality. We present a formula for non-resonant background production in e^+ e^- \to D{\bar D} and extract the resonance parameters of the ψ(3S)-ψ(6S). The obtained results are used to investigate the upsilon energy range.
The recent measurements by CLEO Collaboration for the inclusive J/Psi and Psi(2S) production in Upsilon(1S) decay and our previous calculation are analyzed. The J/Psi momentum spectrum and the production ratio of Psi(2S) versus J/Psi favour Upsilon to J/Psi(Psi(2S)) + ccbar g as the dominant contribution. We point out that the differences between the experimental data and our previous results are mainly originated from the setting of the parameter charm quark mass. We further suggest the associate strange particle enhancement as a probe for the open charm particles in Upsilon to J/Psi (Psi(2S)…
We comment on the recent observation of chi_b states by the ATLAS Collaboration and contest their peremptory interpretation of the highest of the three observed structures as the chi_b(3P) system. Moreover, we do not agree that from their data the mass barycenter of the chi_b(3P) system can be determined.
In this paper, we study double charmonia production in Upsilon peaks, especially, a S-wave charmonium $η_c$ and a P-wave charmonium $h_c(^1P_1)$, or a S-wave charmonium $J/ψ$ and the $X(3940)$ and $X(4160)$ within the nonrelativistic QCD (NRQCD) approach which is a powerful tool to realize the factorization of double charmonia production in electron-positron annihilation. The $J^{PC}=1^{--}$ state $Υ(nS)$ can provide an ideal laboratory for studying the properties of double-heavy quarkonium and also can separate the perturbative and nonperturbative parts due to the large heavy quark mass compa…
The heavy quark spin symmetry implies that in addition to the recently observed Z(10610) and Z(10650) molecular resonances with $I^G=1^+$ there should exist two or four molecular bottomonium-like states with $I^G=1^-$. Properties of these G-odd states are considered, including their production in the radiative transitions from Upsilon(5S), by applying the same symmetry to the Upsilon(5S) resonance and the transition amplitudes. The considered radiative processes can provide a realistic option for observing the yet hypothetical states.
We present a theoretical prediction for the photon spectrum in radiative Upsilon decay including the effects of resumming the endpoint region, E_γ-> M_Υ/2. Our approach is based on NRQCD and the soft collinear effective theory. We find that our results give much better agreement with data than the leading order NRQCD prediction.
We analyze the end-point region of the photon spectrum in semi-inclusive radiative decays of very heavy quarkonium (m alpha_s^2 >> Lambda_QCD). The S- and P-wave octet shape functions are calculated. When they are included in the analysis of the photon spectrum of the Upsilon (1S) system the agreement with data becomes excellent.
We discuss in detail the photon spectrum of radiative Upsilon 1S decays taking into account a number of results that have recently appeared in the literature. In particular, we show how to consistently combine expressions which are valid in the upper end-point region, where NRQCD factorization breaks down, with those of the central region, where NRQCD factorization holds. An excellent description of data is achieved, but theoretical errors are large.
We improve on a recent determination of alpha_s from Gamma(Upsilon(1S)-->X gamma)/Gamma(Upsilon(1S)-->X) with CLEO data by taking into account color octet contributions and avoiding any model dependence in the extraction. We obtain alpha_s(M_Upsilon(1S))= 0.184+0.015-0.014, which corresponds to alpha_s(M_Z)=0.119+0.006-0.005.
Recent advances in the theoretical description of radiative Upsilon decays are reviewed, including the calculation of next-to-leading order QCD corrections to the photon spectrum.
We review selected recent Belle results in quarkonium physics, that include precision measurement of the eta_b(1S) parameters, evidence for the eta_b(2S); evidence for the psi_2(1D); observation of the psi(4040) and psi(4160) transitions to J/psi eta with anomalously high rates; observation of the Upsilon(5S) transitions to Upsilon(1D)pi+pi- and Upsilon(1S,2S)eta. The low excitations of charmonium and bottomonium are in agreement with the Lattice QCD and effective theories calculations, while high excitations show unexpected properties.
We have calculated the next-to-leading order perturbative QCD corrections to the photon energy spectrum in radiative Upsilon decays. The higher-order corrections significantly modify the shape of the spectrum, in particular at large photon energies, and thereby reduce the discrepancy between experimental data and previous leading-order predictions. The next-to-leading order calculation of the photon energy spectrum allows a more reliable determination of the strong coupling constant from radiative Upsilon decays by restricting the analysis to the region of the energy spectrum that can be descr…
We study possible effects of a light CP-odd Higgs boson on radiative Upsilon decays in the Next-to-Minimal Supersymmetric Standard Model. Recent constraints from CLEO on radiative Upsilon(1S) decays are translated into constraints on the parameter space of CP-odd Higgs boson masses and couplings, and compared to constraints from B physics and the muon anomalous magnetic moment. Possible Higgs - eta_b(nS) mixing effects are discussed, notably in the light of the recent measurement of the eta_b(1S) mass by Babar: The somewhat large Upsilon(1S) - eta_b(1S) hyperfine splitting could easily be expl…
Using a sample of 102 million $Υ(1S)$ events collected with the Belle detector, we report on the first search for charge-parity-even charmonium and charmonium-like states in $Υ(1S)$ radiative decays. No significant $χ_{cJ}$ or $η_c$ signal is observed and 90% C.L. limits on $\BR(Υ(1S)\to γχ_{c0})<6.5 \times 10^{-4}$, $\BR(Υ(1S)\to γχ_{c1})<2.3\times 10^{-5}$, $\BR(Υ(1S)\to γχ_{c2})<7.6 \times 10^{-6}$, and $\BR(Υ(1S)\to γη_c)<5.7\times 10^{-5}$ are obtained. The product branching fraction limits $\BR(Υ(1S)\to γX(3872)) \BR(X(3872)\toπ^+π^-\jpsi)< 1.6 \times 10^{-6}$, $\BR(Υ(1S)\to γX(3872)) \B…
Using a bound-state model of weakly bound gluons for glueballs made of two gluons and a natural generalization of the perturbative QCD formalism for exclusive hadronic processes, we present results for glueball production in radiative J/psi, Upsilon decays into several possible glueball states, including L \not= 0 ones. We perform a detailed phenomenological analysis, presenting results for the more favored experimental candidates and for decay angular distributions.
NRQCD results for Upsilon and Charmonium using both dynamical and quenched configurations are presented. We investigate dependence on the light dynamical quark mass. Preliminary dynamical ($n_f = 2$) Charmonium data are combined with quenched results to extract the strong coupling constant $α_P^{(n_f)}$ for the physical number of light dynamical quarks, $n_f = 3$. Good agreement is found with calculations based on the Upsilon system. We show that a discrepancy in $α_P^{(n_f=0)}$, found between the Upsilon and Charmonium systems in the quenched theory, disappears upon extrapolating to the physi…
In this talk I present the results of two calculations that make use of Non-Relativistic QCD and the newly developed Soft-Collinear Effective Theory. The first process considered is inclusive radiative $Υ$ decay. The second process considered is the leading color-octet contribution to $e^+ e^- \to J/ψ+ X$.
The standard NRQCD power counting breaks down and the OPE gives rise to color-octet shape functions at the upper endpoint of the photon energy spectrum in radiative Upsilon decay. Also in this kinematic regime, large Sudakov logarithms appear in the octet Wilson coefficients, ruining the perturbative expansion. Using SCET, the octet shape functions arise naturally and the Sudakov logarithms can be summed using the renormalization group equations. We derive an expression for the resummed octet energy spectrum.
We study the evolution of flavor-singlet, light-cone amplitudes in the soft-collinear effective theory (SCET), and reproduce results previously obtained by a different approach. We apply our calculation to the color-singlet contribution to the photon endpoint in radiative Upsilon decay. In a previous paper, we studied the color-singlet contributions to the endpoint, but neglected operator mixing, arguing that it should be a numerically small effect. Nevertheless the mixing needs to be included in a consistent calculation, and we do just that in this work. We find that the effects of mixing are…
We present a calculation of the two-photon and two-gluon widths for the equal mass quarkonium states $^1S_0$, $^3P_0$ and $^3P_2$ of the charmonium and upsilon systems. The approach taken is based on using the full relativistic $q\bar{q}\to\ga\ga$ amplitude together with a wave function derived from the instantaneous Bethe-Salpeter equation. Momentum space radial wave functions obtained from an earlier fit of the charmonium and upsilon spectra are used to evaluate the necessary integrals.
In this article, we study the radiative decays among the charmonium states with the heavy quark effective theory, and make predictions for the ratios among the radiative decay widths of an special multiplet to another multiplet. The predictions can be confronted with the experimental data in the future and put additional constraints in identifying the $X$, $Y$, $Z$ charmonium-like mesons.
A calculation is presented of the radiative decay of the Upsilon(nS) into a bound state of bottom squarks. Predictions are provided of the branching fraction as a function of the masses of the bottom squark and the gluino. Branching fractions as large as several times 10^{-4} are obtained for supersymmetric particle masses in the range suggested by the analysis of bottom-quark production cross sections. Data are shown that limit the range of allowed masses. Forthcoming high-statistics data from the CLEO Collaboration offer possibilities of discovery or significant new bounds on the existence a…
We analyze the pipi production amplitudes in the excited Upsilon decay processes, Upsilon (2S) to Upsilon (1S)pi^+pi^-, Upsilon (3S) to Upsilon (1S)pi^+pi^- and Upsilon (3S) to Upsilon (2S)pi^+pi^-, and the pipi and KKbar production amplitudes in the charmonium decay processes, psi (2S) to Jpsi pi^+pi^- and Jpsi to phi pi^+pi^-, phi K^+K^-, including the possible effect of light sigma production. The amplitudes are parametrized by the sum of Breit-Wigner amplitudes for the sigma and the other relevant particles and of the direct 2pi-production amplitude, following the VMW method. All the pipi …
In this work, we investigate the decay widths and the line shapes of the open-charm radiative and pionic decays of Y(4274) with the $D_s\bar{D}_{s0}(2317)$ molecular charmonium assignment. Our calculation indicates that the decay widths of $Y(4274)\to D^{+}_{s}D^{*-}_{s}γ$ and $Y(4274)\to D^+_{s}D^-_{s}π^0$ can reach up to 0.05 keV and 0.75 keV, respectively. In addition, the result of the line shape of the photon spectrum of $Y(4274)\to D_s^+ {D}_s^{*-} γ$ shows that there exists a very sharp peak near the large end point of photon energy. The line shape of the pion spectrum of $Y(4274)\to D_…
Bottomonium-like resonances Zb(10610) and Zb'(10650) are good candidates of hadronic molecules composed of BBbar* (or B*Bbar) and B*Bbar*, respectively. Considering Zb (Zb') as heavy meson molecules, we investigate the decays of Zb(')^+ -> Upsilon (nS) pi^+$ in terms of the heavy meson effective theory. We find that the intermediate B (B*) and Bbar (Bbar*) meson loops and the form factors play a significant role to reproduce the experimental values of the decay widths. We also predict the decay widths of Zc^+ -> J/ψpi^+ and psi(2S) pi^+ for a charmonium-like resonance Zc which has been reporte…
We investigate the effects of including the full three-loop QCD correction to the static short distance $1/r$ potential on the spectroscopy and decays in the charmonium and upsilon systems. We use a variational technique with the full three-loop corrected potential to determine a set of unperturbed trial wave functions and treat the relativistic and one-loop corrections as perturbations. The perturbed results are compared to the subset of the charmonium and upsilon spectra using a $χ^2$ test. This approach results in more accurate descriptions of the hyperfine splittings in both the $b\bar{b}$…
We analyze the pi pi production amplitudes in the excited Upsilon decay processes, Upsilon(2S)-->Upsilon(1S)pi+pi-, Upsilon(3S)-->Upsilon(1S)pi+pi- and Upsilon(3S)-->Upsilon(2S)pi+pi-, and the pi pi and KKbar production amplitudes in the charmonium decay processes, psi(2S)-->J psi pi+pi- and J psi-->phi pi+pi-, phi K+K-. The amplitudes are parametrized by the sum of Breit-Wigner amplitudes for the sigma and the other relevant particles and of the direct 2pi-production amplitude, following the VMW method. All the pi pi (and KKbar) invariant mass spectra are reproduced well with the sigma Breit-…
We discuss the theoretical status of inclusive radiative decays of charmonium, with a particular emphasis on the QCD description of the photon spectrum, where progress has occurred in recent years. We also comment on the possible extraction of alpha_s (M_J/psi) and on the possibility to gain important information on the nature of J/psi and psi (2S).
At the upper endpoint of the photon energy spectrum in Upsilon -> X gamma, the standard NRQCD power counting breaks down and the OPE gives rise to color-octet structure functions. Furthermore, in this kinematic regime large Sudakov logarithms appear in the octet Wilson coefficients. The endpoint spectrum can be treated consistently within the framework of a recently developed effective field theory of collinear and soft particles. Here we show that within this approach the octet structure functions arise naturally and that Sudakov logarithms can be summed using the renormalization group equati…
Assuming that $X(3872)$ is a mixture between $2P$ charmonium and $\bar{D}D^{*}$ molecular states with $J^{PC}=1^{++}$, an analysis of $X(3872)$ radiative decays into $J/ψγ$ and $ψ(2S)γ$ is presented. The modification of the radiative branching ratio due to possible constructive or destructive interferences between the meson-loop and the short-distance contact term, which is modeled by a charm quark loop, is shown. The model predictions are shown to be compatible with the experimentally determined ratio of the mentioned branching fractions for a wide range of the $X(3872)$ charmonium content. I…
We apply a phenomenological Lagrangian approach to the radiative decay Y(4260) -> X(3872) + gamma. The Y(4260) and X(3872) resonances are considered as composite states containing both molecular hadronic and charmonium components. Having a leading molecular component in the X(3872) and a sole molecular configuration for the Y(4260) results in a prediction compatible with present data.
We consider the non-relativistic limit of the QCD Hamiltonian in the Coulomb gauge, to describe radiative transitions between conventional charmonium states and from the lowest multiplet of $c\bar{c}$ hybrids to charmonium mesons. The results are compared to potential quark models and lattices calculations.
We analyze the theoretical description of radiative decays in charmonium. We use an elementary emission decay model to build the most general electromagnetic transition operator. We show that accurate results for the widths can be obtained from a simple quark potential model reasonably fitting the spectroscopy if the complete form of the operator is used instead of its standard p/m approximation and the experimental masses are properly implemented in the calculation.
We study exclusive radiative decays of the $Υ$ using soft-collinear effective theory and non-relativistic QCD. In contrast to inclusive radiative decays at the endpoint we find that color-octet contributions are power suppressed in exclusive decays, and can safely be neglected, greatly simplifying the analysis. We determine the complete set of Lorentz structures that can appear in the SCET Wilson coefficients and match onto them using results from a previous calculation. We run these coefficients from the scale $\mups$ to the scale $Λ\sim 1 \textrm{GeV}$, thereby summing large logarithms. Fina…
We argue that the photon spectra in radiative decays of various heavy quarkonium states provide important information on their nature. If two of these states are in the strong coupling regime, we are able to produce a parameter-free model independent formula, which holds at next-to-leading order and includes both direct and fragmentation contributions. When the formula is checked against recent CLEO data it favors Upsilon (2S) and Upsilon (3S) in the strong coupling regime and disfavors Upsilon (1S) in it.
We study the electromagnetic (EM) transitions of the $nS$, $nP$ ($n\leq 3$), and $nD$ ($n\leq 2$) charmonium states with a constituent quark model. We obtain a reasonable description of the EM transitions of the well-established charmonium states $J/ψ$, $ψ(2S)$, $χ_{cJ}(1P)$, $h_c(1P)$ and $ψ(3770)$. We find that the M2 transitions give notable corrections to some E1 dominant processes by interfering with the E1 transitions. Our predictions of EM decay properties for the higher charmonium states are also presented and compared with other model predictions. In particular, we discuss the EM deca…
Bottomonium S-wave states were studied using lattice NRQCD. Masses of ground and excited states were calculated using multiexponential fitting to a set of correlation functions constructed using both local and wavefunction-smeared operators. Three-point functions for M1 transitions between vector and pseudoscalar states were computed. Robust signals for transitions involving the first two excited states were obtained. The qualitative features of the transition matrix elements are in agreement with expectations. The calculated values of matrix elements for Y(2S) and Y(3S) decay are considerably…
Coupled-channel effects due to coupling of charmonia to the charmed and anticharmed mesons are of current interest in heavy quarkonium physics. However, the effects have not been unambiguously established. In this paper, a clean method is proposed in order to examine the coupled-channel effects in charmonium transitions. We show that the hindered M1 radiative transitions from the 2P to 1P charmonia are suitable for this purpose. We suggest to measure one or more of the ratios Gamma(h_c'-->chi_{cJ} gamma)/Gamma(chi_{cJ}'-->chi_{cJ} pi^0) and Gamma(chi_{cJ}'-->h_c gamma)/Gamma(chi_{cJ}'-->chi_{c…
We study the photon spectrum in the radiative decays of the charmonium $\frac{1}Γ\frac{dΓ(J/ψ\rightarrowγgg)}{dz}$ as an excellent scene for hunting the exotic mesons (hybrids $q\bar{q}g$ and glueballs $gg$) and an ideal frame for the theoretical and phenomenological investigations of the gluon nature in IR, by subtle estimation of its non-zero effective mass which seems required in Quark-Gluon Constituent Model (QGCM) for identifying the nature of these exotic species.
Using the dynamical diquark model, we calculate the electric-dipole radiative decay widths to $X(3872)$ of the lightest negative-parity exotic candidates, including the four $I=0$, $J^{PC} \! = \! 1^{--}$ ("$Y$") states. The $O$(100--1000 keV) values obtained test the hypothesis of a common substructure shared by all of these states. We also calculate the magnetic-dipole radiative decay width for $Z_c(4020)^0 \! \to \! γX(3872)$, and find it to be rather smaller ($<$~10 keV) than its predicted value in molecular models.
We use QCD sum rules to calculate the width of the radiative decay of the meson X(3872), assumed to be a mixture between charmonium and exotic molecular $[c\bar{q}][q\bar{c}]$ states with $J^{PC}=1^{++}$. We find that in a small range for the values of the mixing angle, $5^0\leqθ\leq13^0$, we get the branching ratio $Γ(X\to J/ψγ)/Γ(X\to J/ψπ^+π^-)=0.19\pm0.13$, which is in agreement, with the experimental value. This result is compatible with the analysis of the mass and decay width of the mode $J/ψ(nπ)$ performed in the same approach.
Assuming the QCD multipole expansion is applicable to hadronic transitions of Upsilon(3S) into lower level bottomonia, we consider the possibility that Upsilon(3S) has a D-wave component. This assumption leads to a natural explanation of the pi-pi spectrum in Upsilon(3S) -> Upsilon(1S) pi-pi. Consequences of this assumption on other hadronic and radiative transitions of Upsilon(3S) are also discussed in the same context.
We present preliminary calculations towards radiative transitions in charmonium using anisotropic $N_f = 2 + 1$ dynamical ensembles generated by the Hadron Spectrum Collaboration. With the use of newer technologies we aim to investigate transitions between states, including potential exotic charmonium states, lying higher in the spectrum than in previous studies. A crucial ingredient in this work is the use of variationally optimised interpolating operators which allow for a reliable determination of the three-point correlation functions needed. Using these operators, we perform first calculat…
The X(3872) radiative decay is discussed by employing a two-meson multichannel model with the charmonium components, which explains many of the other observed features of X(3872). We have found that the ratio of the branching fractions of the X(3872) radiative decays, $R_γ=\text{Br}(X(3872)\rightarrowψ(2S)γ)$ / Br$(X(3872)\rightarrow J/ψγ)$, is 1.1 $\sim$ 3.4 if one assumes that the decay occurs only from the charmonium components. The invariant mass spectra of $cc^{bar}$($2P$)$\rightarrow J/ψγ$ and $cc^{bar}$($2P$)$\rightarrow ψ(2S) γ$ are also investigated. It is found that an enhancement ap…
The charmonium spectrum is calculated with two nonrelativistic quark models, the linear potential model and the screened potential model. Using the obtained wavefunctions, we evaluate the electromagnetic transitions of charmonium states up to $4S$ multiplet. The higher multipole contributions are included by a multipole expansion of the electromagnetic interactions. Our results are in reasonable agreement with the measurements. As conventional charmonium states, the radiative decay properties of the newly observed charmonium-like states, such as $X(3823)$, $X(3872)$, $X(4140/4274)$, are discus…
We present an evaluation of heavy quarkonium states (bottomium and charmonium) from first principles. We use tree-level QCD (including relativistic corrections) and the full one-loop potential; nonperturbative effects are taken into account at the leading order through the contribution of the gluon condensate <alpha_strong G^2>. We use the values Lambda_QCD= 200 (+80/-60) MeV and <alpha_s G^2> = 0.042 +/- 0.020 GeV^4, but we trade the value of the quark mass by the masses of J/psi, Upsilon as input. We get good agreement in what is essentially a zero parameter evaluation for the masses of the …
Within the framework of nonrelativistic QCD (NRQCD) factorization, we calculate the next-to-leading-order (NLO) perturbative corrections to the radiative decay $Υ\to η_c(χ_{cJ})+γ$. 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 $\mathcal{O}(α_s)$ corrections…
We investigate a new method to probe the helicity of the photon emitted in the b -> s gamma transition. The method relies on the observation of interference effects between two resonance contributions, B -> K*(K gamma) gamma and B -> eta_c(gamma gamma) K or B -> chi_c0(gamma gamma) K to the same final state K gamma gamma. Decays of the type B -> K_res(K gamma) gamma dominate the B -> K gamma gamma yield throughout most of the phase space, and may be accessible at current B meson facilities already.
The exclusive decay of $Υ$ to a vector plus pseudoscalar charmonium is studied in perturbative QCD. The corresponding branching ratios are predicted to be of order $10^{-6}$ for first three $Υ$ resonances, and we expect these decay modes should be discovered in the prospective high-luminosity $e^+e^-$ facilities such as super $B$ experiment. As a manifestation of the short-distance loop contribution, the relative phases among strong, electromagnetic and radiative decay amplitudes can be deduced. It is particularly interesting to find that the relative phase between strong and electromagnetic a…
The angular distributions of the decay products in the successive decays chi_c (chi_b) to J/psi (Upsilon) gamma and J/psi (Upsilon) to l+l- are calculated as a function of the angular momentum composition of the decaying chi meson and of the multipole structure of the photon radiation, using a formalism independent of production mechanisms and polarization frames. The polarizations of the chi states produced in high energy collisions can be derived from the dilepton decay distributions of the daughter J/psi or Upsilon mesons, with a reduced dependence on the details of the photon reconstructio…
We study the potential of radiative decays of the Upsilon(1S) and of the phi mesons to search for a light pseudoscalar Higgs boson, proposed as a possible interpretation of Sigma+ --> p mu+mu- events observed by the HyperCP collaboration at Fermilab. We conclude that the detection of this signal should certainly be possible with the current CLEO Upsilon(1S) data, and is within the reach of KLOE in at least part of the range of couplings suggested by the HyperCP findings.
Using the helicity formalism, we calculate the combined angular distribution function of the two gamma photons ($γ_1$ and $γ_2$) and the electron ($e^-$) in the triple cascade process $\bar{p}p\rightarrow{}^3D_3\rightarrow{}^3P_2+γ_1\rightarrow(ψ+γ_2) +γ_1 \rightarrow (e^- + e^+) +γ_2 +γ_1$, when $\bar{p}$ and $p$ are arbitrarily polarized. We also derive six different partially integrated angular distribution functions which give the angular distributions of one or two particles in the final state. Our results show that by measuring the two-particle angular distribution of $γ_1$ and $γ_2$ and…
We present a study for charmonium radiative transitions: $J/ψ\rightarrowη_cγ$, $χ_{c0}\rightarrow J/Ψγ$ and $h_c\rightarrowη_cγ$ using $N_f=2$ twisted mass lattice QCD gauge configurations. The single-quark vector form factors for $η_c$ and $χ_{c0}$ are also determined. The simulation is performed at a lattice spacing of $a= 0.06666$ fm and the lattice size is $32^3\times 64$. After extrapolation of lattice data at nonzero $Q^2$ to 0, we compare our results with previous quenched lattice results and the available experimental values.
