Testing $\chi_c$ properties at BELLE II

The model of the $\chi_{c_i}-\gamma^*-\gamma^*$ and $\chi_{c_i}-J/\psi^*-\gamma^*$ form factors developed in [1] for $\chi_{c_1}$ and $\chi_{c_2}$ is extended to $\chi_{c_0}$ case. The studies performed within this model have shown that at BELLE II it will be possible to study in detail $\chi_{c_i}-\gamma^*-\gamma$ form factors through measurements of the reaction $e^+e^-\to e^+e^- \chi_{c_i} (\to J/\psi (\to \mu^+\mu^-)\gamma)$.The results were obtained using the newly updated Monte Carlo generator EKHARA.


Introduction
Soon the BELLE II experiment [2] will start to operate with unprecedented luminosity allowing to access information not available before. In this letter we show that the integrated luminosity of 20-50 ab −1 will allow BELLE II collaboration to study in detail the χ ci − γ * − γ form factors. These form 5 factors are used in the calculations of the electronic widths of the χ ci , which were not yet measured. The theoretical predictions available for these widths [3,4,5,6,1] depend strongly on the details of the form factors modeling and are different, up to two orders of magnitude, even if of all the models agree with experimental data [7] on the χ ci → J/ψγ, i = 1, 2 and χ c2 → γγ partial 10 decay widths. We advocate here that the experimental studies of the reactions e + e − → e + e − χ ci (→ J/ψ(→ µ + µ − )γ) can differentiate between the proposed ✩ Work supported in part by the Polish National Science Centre, grant number DEC-2012/07/B/ST2/03867.

models.
To give realistic predictions for event selections close to the experimental ones, we have extended the model developed in [1] to cover also χ c0 − γ * − 15 γ * amplitudes and implemented the model amplitudes in the event generator EKHARA [8,9]. The generator can also help in the data analysis of the reactions e + e − → e + e − χ ci and e + e − → e + e − χ ci (→ J/ψ(→ µ + µ − )γ). The newly updated code is available from the EKHARA web page (http://prac.us.edu.pl/∼ekhara/). 20 The layout of this letter is the following: In Section 2 we describe the model used in the presented simulations. In Section 3 we give predictions for the expected number of events of the χ ci production cross sections at BELLE II and the expected number of events for the form factor measurements. The QED non-resonant background is discussed in Section 4. Conclusions are presented 25 in Section 5.

The model
The model used in this letter is an extension of the model [1] built to describe χ c1 and χ c2 decays to J/ψγ, the χ c2 decay to γγ and ψ ′ decays to χ c 1(2) γ. The basic assumptions used to construct the amplitudes for χ c0 decays to J/ψγ 30 and γγ as well as ψ ′ decay to χ c0 γ are the same as in [1]. We start from the χ ci −γ * −γ * amplitudes calculated in [3] and assume that the Lorentz structure, as well as the form factor, are identical also for χ ci − J/ψ * − γ * amplitude. We allow only for different coupling constants. From these assumptions one gets the following amplitudes for the decays χ c0 → γγ, χ c0 → J/ψγ and ψ ′ → χ c0 γ where ǫ i ≡ ǫ(p i ) are the appropriate polarisation vectors, The coupling constants can be extracted from the experimental data adding to the ones used in [1] the following widths where  Table 1: Model parameters and theoretical (th) (this paper, see also [1]), and experimental (exp) [7] values of Γ(χc 0,1,2 → γγ, γJ/ψ) and Γ(ψ ′ → χc 0,1,2 γ).
[1]  The model parameters describing χ c1 and χ c2 are very close to the model pa-50 rameters obtained in [1] and the predictions for the electronic widths (Γ(χ c1 → e + e − ) = 0.37 eV and Γ(χ c2 → e + e − ) = 3.86 eV) did change within the parametric uncertainty of the model, which is about 10%. The a 0 ψ ′ coupling is positive at difference with the negative a ψ ′ . Both signs are the only ones allowed by the fit.

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The χ c1 and χ c2 electronic widths are calculated as loop integrals (see Fig.5 of [1]), thus the χ ci − γ * − γ * , χ ci − J/ψ * − γ * and χ ci − ψ ′ * − γ * form factors are crucial for the theoretical predictions. Table 2 summarises the situation. All the models referenced there give correct predictions for the χ ci → J/ψγ, i = 1, 2 and χ c2 → γγ partial decay widths. Yet, the predictions for the χ c1 and χ c2 60 electronic widths are different up to two orders of magnitude, showing why the experimental studies of the form factors are important. The ψ ′ contribution is taken into account only in [6] and [1]. The contributions taken into account in [6] and [1] are qualitatively the same, yet the differences coming from loop integrals are striking and require 65 further studies. With the couplings obtained from the fit one can predict the rates for the reactions e + e − → e + e − χ ci and e + e − → e + e − χ ci (→ J/ψ(→ µ + µ − )γ). We do consider here only signal processes with the Feynman diagram given in Fig. 1.

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The QED non-resonant background can be suppressed by requiring that the µ + µ − γ invariant mass is close to the χ ci mass and µ + µ − invariant mass close to the J/ψ mass. Its size is estimated in Section 4. As the χ ci and J/ψ are almost on-shell we use a constant χ ci − J/ψ − γ form factor.
The relevant amplitudes, with the four momenta denoted in Fig.1, read with where ǫ 1 µ is the photon polarisation vector. The parts of the χ ci − γ * − γ * vertex vanishing when contracted with the e − e − γ * vertices are not shown in the above formulae. The A i , i = 0, 1, 2 tensors denote the γ * − γ * − χ * ci vertices, while the B i , i = 0, 1, 2 tensors stand for γ * − J/ψ * − χ * ci vertices. Generally samples. An option of simulating simultaneously of all χ ci production is available in the EKHARA generator to facilitate this simulation. The interferences between amplitudes are not taken into account to speed up the calculations.
For the phase space generation of the e + e − → e + e − χ ci reaction the method used in [9] for the generation of the phase space for reactions e + e − → e + e − P 95 (P = π 0 , η, η ′ ) was adopted. For the simulation of the reaction e + e − → e + e − χ ci (→ J/ψ(→ µ + µ − )γ) the phase space generation is split into two parts: the first part generates the e + , e − and the virtual χ ci four momenta, while the second part generates the µ + , µ − and γ four momenta. In the first part the virtual χ ci invariant mass was generated using the standard change of variables to 100 absorb the Breit-Wigner peak coming from χ ci propagator, while the remaining variables were generated using the same method as in e + e − → e + e − χ ci reaction. In the second part the invariant mass l 2 was generated using the standard change of variables to absorb the Breit-Wigner peak coming from J/ψ propagator, the photon angles were generated flat in the rest frame of the virtual χ ci , 105 while the µ + (µ − ) angles were generated flat in the virtual J/ψ rest frame. In the mode, where all 3 χ ci are generated simultaneously the 3-channel Monte Carlo variance reduction was used. The phase space generation was cross checked with an independent computer code, which uses the representation described in [10], with flat generation of all variables.

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The χ ci production cross section in the reaction e + e − → e + e − χ ci (the amplitudes are easy to infer from Eq.(4)) integrated over the complete phase space with the integrated luminosity of BELLE II of 50 ab −1 leads to the expected number of events of about 140M (χ c0 ), 4.3M (χ c1 ) and 142M (χ c2 ). These rates will allow for detailed studies of many χ ci decay modes. Unfortunately  In this letter we concentrate on the possible tests of the models of χ ci −γ * −γ * form factors using single and double tag events relying on an identification of a simple χ ci decay mode: χ ci → J/ψ(→ µ + µ − )γ. If one requires identification of χ ci and J/ψ through invariant masses of µ + µ − γ and µ + µ − final states respectively, the χ ci − J/ψ * − γ form factors are entering the cross section with 135 fixed invariants, thus they are almost constant. This way the χ ci − γ * − γ form factors can be measured.
In the results presented below we assume the asymmetric beams of 4 and 7 GeV with half crossing angles of 41.5 mrad. We assume also that the particles (µ + , µ − , photon and positron and/or electron) can be detected and their four
For χ c0 production and decay the background is not negligible: 110% for single tag events and 220% for double tag events. It shows that in this case the interference effects between background and signal are important and will have to be studied. Yet, as the signal is small as compared to χ c1 and χ c2 the expected 165 statistical accuracy will also be much worse. For χ c1 and χ c2 the background to signal ratio is much smaller for two reasons: the signals are bigger by one order od magnitude and the decay widths are smaller about one order of magnitude as compared to χ c0 decay width. For single tag events the background to signal ratio is 0.2% for χ c1 and 0.7% for χ c2 , while for double tag events it is 0.1% and 170 1.7% respectively. Thus for χ c1 and χ c2 the interferences between background and signal amplitudes can be neglected and the background can be simulated using the existing Monte Carlo generators. The resonant background, mainly the e + e − → e + e − J/ψ(→ µ + µ − )γ process without χ ci intermediate states involved, should also be studied. None of the existing generators is currently 175 able to generate this process and the main difficulty will be to find an efficient generation algorithm.
It is achieved by event selections, which force the χ ci and J/ψ to be almost on-shell. For these kinematic configurations the QED background is negligible for χ c1 and χ c2 , while for χ c0 it has to be taken into account. The proposed 185 measurements should clarify, which of the models giving predictions for the χ c1 and χ c2 electronic widths is correct, even without direct measurement of these widths. If the electronic widths are measured as well, they will allow for further refinements of the models. The expected number of events for the χ ci production show that detailed studies of the χ ci branching ratios will also be 190 possible at BELLE II. The newly updated Monte Carlo generator EKHARA can be of help for the visibility studies and the data analyses.