Study of $\chi_{c0}(1P) -->J/\psi \gamma $ and $\chi_{b0}(1P) -->\Upsilon(1S) \gamma $ decays via QCD sum rules

In this study we present the first theoretical calculation of the form factor of the $\chi_{c0}(1P) \rar J/\psi \gamma $ and $\chi_{b0}(1P) \rightarrow \Upsilon(1S) \gamma $ decays in the frame work of QCD sum rules. We also find branching ratio ${\cal B}_r(\chi_{c0}(1P) \rightarrow J/\psi \gamma)=(1.07\times \pm 0.14) \times 10^{(-2)} $ which is in agreement with the experimental data. Furthermore, we estimate the $\Gamma_{tot}(\chi_{b0}(1P)) =5.5\pm 0.5 $ MeV, where experimental bound for full width of $\chi_{b0}$ is $\Gamma_{tot}(\chi_{b0}(1P))<6 $ MeV.


I. INTRODUCTION
Heavy quarkonium states( like b b and cc) and their decay modes offer a laboratory to study the strong interaction in the non-perturbative regime.Charmonium in particular has served as a calibration tool for the corresponding techniques and models [1].Heavy quarkonium states can have many bound states and decay channels used to study and determine different parameters of Standard Model(SM) and QCD from the theoretical perspective.In particular, the calculation of bottomonium masses [2], total widths, coupling constants [3][4][5][6] and branching ratio can serve as benchmarks for the low energy predictions of QCD.In addition, the theoretical calculations on the branching ratio of radiative decays of heavy quarkonium states, are relatively clean with respect to the hadronic or semileptonic decays, and their comparison with experimental data could provide inportant insights into their nature and hyperfine interaction.In this regard, we present the first theoretical study on the branching ratio of inclusive χ c0 → J/ψγ and χ b0 → Υγ decays.Note that in order to calculate the branching ratio we have to get information about the masses and decay constants of the participating particles.It is worth mentioning that the masses can be obtained either by means of the experimental results i.e, the Particle Data Group [21] or by the theoretical methods .The decay constants, on the other hand, can be calculated theoretically via different non-perturbative methods.In this respect, masses and decay constants and spectrums of heavy qurakonium states are calculated in the various approaches( see for instance [7]- [15]).Here, firstly, we calculate the form factor of χ c0 → J/ψγ and χ b0 → Υγ decays in the framework of three-ponit QCD sum rules, secondly we calculate the branching ratio of inclusive χ c0 → J/ψγ and χ b0 → Υγ decays.In section 2, we introduce the QCD sum rules technique for the form factors of inclusive χ c0 → J/ψγ and χ b0 → Υγ decays.Last section is devoted to the numerical analysis and discussion.

II. QCD SUM RULES FOR THE FORM FACTORS
The three-point correlation function associated with the χ c0 (1P ) → J/ψγ and χ b0 (1P ) → Υ(1S)γ vertex is given by where T is the time ordering operator and q is momentum of photon.Each meson and photon field can be described in terms of the quark field operators as follows: We calculate the correlation function Eq. (1) in two different methods.In phenomenological or physical side, it can be evaluated in terms of hadronic parameters such as masses, decay constants and form factors.In theoretical or QCD side, on the other hand, it is calculated in terms of QCD parameters, which are quark and gluon degrees of freedom, by the help of the operator product expansion (OPE) in deep Euclidean region.Equating the structure calculated in two different approaches of the same correlation function, we get a relation between hadronic parameters and QCD degrees of freedom.Finally, we apply double Borel transformation with respect to the momentum of initial and final mesons(p 2 and p ′2 ).This final operation suppresses the contribution of the higher states and continuum.

A. Phenomenological side
We insert the complete sets of appropriate vector meson(|V V |) and scalar meson(|S S|) states (regarding the conservation of the quantum numbers of corresponding interpolating currents) inside correlation functions Eq.(1).Here, vector state is either j/ψ or Υ and scalar state is χ c(b) state.After integrating over the x and y, we get the following result for the correlation function Eq.( 1): where .... contains the contribution of the higher and continuum states with the same quantum numbers .
The matrix elements of the above equation are related to the hadronic parameters as follows: where F (q 2 ) is the form factor of transition and ǫ ′ is the polarization vector associated with the vector meson.Using Eq. ( 4) in Eq. ( 3) and considering the summation over polarization vectors via, the result of the phenomenological or physical side is as follows: We are going to compare the coefficient of g µν structure for further calculation from different approaches of the correlation functions.

B. Theoretical(QCD) side
Theoretical side consists of perturbative(bare loop see fig. (1) and non-perturbative parts(the contributions of two gluon condensate diagrams fig.(2) ).We calculate it in the deep Euclidean space (p 2 → −∞ and p ′ 2 → −∞).We consider this side as: The perturbative part is a double dispersion integral as follows: where, ρ(s, s ′ , q 2 ) is called spectral density.We aim to evaluate the spectral density with the help of the bare loop diagram in Fig. (1).One of the generic methods to calculate this bare loop integral is the Cutkosky method, where the quark propagators of Feynman integrals are replaced by the Dirac delta functions: Then, using the Cutkosky method we get spectral density as: where λ(a, b, c) = a 2 + b 2 + c 2 − 2ac − 2bc − 2ab and N c = 3 is the color number.Note that, since three δ functions of integrand must vanish simultaneously, the physical regions in the s -s ′ plane must satisfy the following inequality:

Two Gluon Condensates
We consider the two gluon condensate diagrams.Note that, we do not add the heavy quarks condensates, because the heavy quark contributions are suppressed by the inverse of the heavy quark mass, so they can be safely neglected.Now, as a nonperturbative part, we must add contributions coming from the gluon condensates presented in (a), (b), (c), (d), (e) and (f) in Fig. (2).These diagrams are calculated in the Fock-Schwinger fixed-point gauge [16][17][18] where, 2. Two gluon condensate diagram as a radiative corrections for the χc0 → J/ψγ and χ b0 → Υγ decays; the vacuum gluon field is as follows: where k ′ is the gluon momentum and A a µ is the gluon field.In addition, the quark-gluon-quark vertex is used as: We come across the following integrals in calculating the gluon condensate contributions [19,20]: where k is the momentum of the spectator quark m c(b) .These integrals are calculated by shifting to the Euclidean space-time and using the Schwinger representation for the Euclidean propagator: This kind of expression is very easy for the Borel transformation since where M is Borel parameter.We perform integration over the loop momentum and over the two parameters which we use in the exponential representation of propagators [17].As a final operation we apply double Borel transformations to p 2 and p ′2 .We get the Borel transformed form of the integrals in Eq. ( 14) as: where and M 2 1 and M 2 2 are the Borel parameters.The function U 0 (a, b) is as follows: where where the circumflex of Î in the equations is used for the results after the double Borel transformation.As a result of the lengthy calculations we obtain the following expressions for the two gluon condensate: Now, we can compare g µν coefficient of Eq. ( 6) and Eq.(7) .Our result related to the sum rules for the corresponding form factor is as follows: Note that, finally we have to set q 2 = 0 for the real photon.
The continuum thresholds, s 0 and s ′ 0 must not be greater than the energy of the first excited states with the same quantum numbers.In our numerical calculations the following regions for the continuum thresholds in s and s ′ channels are used: (m S + 0.3) 2 ≤ s 0 ≤ (m S + 0.7) 2 and (m V + 0.3) 2 ≤ s ′ 0 ≤ (m V + 0.7) 2 for s and s ′ channels, respectively.Here, m S is the mass of either χ c0 or χ b0 meson and m V is the mass of either j/ψ or Υ meson.Note that, we follow the standard procedure in the QCD sum rules, where the continuum thresholds are suppossed to be independent of Borel mass parameters and q 2 .However, this assumption is not free of uncertainties(see for instance [23]).
The detailed analysis of the form factor shows us that dependence of form factor best fits into the following function: where we a = 0.73 ± 0.01, b = 0.2 ± 0.01 and c = 0.012 ± 0.001 for χ c0 → J/ψγ and a = 0.4812 ± 0.03, b = 0.2 ± 0.001 and c = 0.0084 ± 0.0003 for χ b0 → Υγ decays.Using Q 2 = 0 in Eq. ( 22), we obtain the F (0) = 0.75 ± 0.05 GeV −1 and the F (0) = 0.47 ± 0.03 GeV −1 for χ c0 → J/ψγ and χ b0 → Υγ decays, respectively.The errors in our numerical calculation are the results of both the interval of the working regions for the auxiliary parameters and the uncertainties of the input parameters.
The matix element for the decay of χ c0 → J/ψγ and χ b0 → Υγ is as follows: where p ′ and ǫ ′ are the momentum and polarization of final state vector meson i.e., either j/ψ or Υ mesons, ǫ is the polarization of real photon and p is the momentum of initial scalar meson.Using this matix element, we get: where m S is mass of either χ c0 or χ b0 meson and m V is either mass of j/ψ or Υ meson.The decay width for χ c0 → J/ψγ decays is as: Γ(χ c0 → j/ψγ) = (11.2± 0.3) × 10 (−5) GeV.