Production of $\sigma$ in $\psi(2S)\to \pi^{+}\pi^{-}J/\psi$

Using 14M $\psi(2S)$ events accumulated by BESII at the BEPC, a Covariant Helicity Amplitude Analysis is performed for $\psi(2S)\to\pi^+\pi^-J/\psi, J/\psi\to \mu^+\mu^-$. The $\pi^+\pi^-$ mass spectrum, distinctly different from phase space, suggests $\sigma$ production in this process. Two different theoretical schemes are used in the global fit to the data. The results are consistent with the existence of the $\sigma$. The $\sigma$ pole position is determined to be $(552^{+84}_{-106})-i(232^{+81}_{-72})$ MeV/$c^2$.


Introduction
We report here a study of the process ψ(2S) → π + π − J/ψ, which is the ψ(2S) decay mode with the largest branching fraction [1], using very clean ψ(2S) → π + π − J/ψ (J/ψ → µ + µ − ) events. Early investigations of this decay by Mark I [2] found that the π + π − mass distribution is strongly peaked toward higher mass in contrast to what is expected from phase space. Furthermore, angular distributions favored S-wave production between the ππ system and J/ψ, as well as an S-wave decay of the dipion system.
BESI studied this process with much higher statistics (3.8 million ψ(2S) events) and found that an additional small D-wave component was required in the decay of the dipion system [3]. Also various heavy quarkonium models were fitted, and the parameters for these models determined [3].
Here, we fit the π + π − system from ψ(2S) → π + π − J/ψ decays with the J P C = 0 ++ σ meson. In this decay, the interaction between the ππ system and ψ(2S) or J/ψ is small since these charmonium states are very narrow, so the dipion system is a quasi-isolated system [4].
The σ meson was introduced theoretically in the linear σ model [5], and its existence was first suggested in a one-boson-exchange potential model of nuclear forces [6]. The σ meson is important due to its relation with dynamical chiral symmetry breaking of QCD [7].
There was evidence for a low mass pole in early DM2 [8] and BESI [9] data on J/ψ → ωππ. A huge event concentration in the I = 0 S-wave ππ channel was observed in a pp central production experiment in the region from m ππ = 500 to 600 MeV/c 2 [10]. This peak is too large to be explained as background [11]. Many studies on the possible resonance structure in ππ scattering have appeared in the literature [12]. It was proved that the existence of a light and broad resonance is unavoidable even with nonlinear realization of chiral symmetry [13]. Careful theoretical analyses were made to determine the pole location, which was found to be M − iΓ/2 = (470 ± 30) − i(295 ± 20) MeV/c 2 [14] and Renewed experimental interest arose from E791 data on D + → π + π − π + [16], where it was found that M = 478 +24 −23 ± 17 MeV/c 2 , Γ = 324 +42 −40 ± 21 MeV/c 2 . In the recent partial wave analysis of the decay J/ψ → ωπ + π − [17] by BE-SII, the pole position of σ was determined to be (541 ± 39) − i(252 ± 42) MeV/c 2 . All these experimental results still have large uncertainties. Fig. 1 shows the decay mechanism of ψ(2S) → π + π − J/ψ in the S-matrix formalism. There are three main contributions including an S-wave resonance (σ), a D-wave term (2 + ), and a contact term which is the destructive background required by chiral symmetry [18]. The total amplitude is the sum of these three components. The decay ψ(2S) → π + π − J/ψ can also be described with an effective Lagrangian for the vector pseudo-scalar pseudo-scalar (VPP) vertex, along with the ππ final state interaction (FSI) obtained from ππ scattering data in a Chiral Unitary Approach (ChUA) [19]. In such an approach, the σ resonance is generated dynamically as a pole of the unitarized t-matrix, and the pole position is 469 − i203 MeV/c 2 [20]. A fit to the BESI ψ(2S) → π + π − J/ψ data shows that the ππ FSI plays an important role in this process [19]. A similar result was obtained in Ref. [21] with a comparison of the cases with and without the ππ FSI. We fit our data with both the S-matrix and ChUA schemes.

BESII Experiment
The data sample used for this analysis is taken with the BESII detector at the BEPC storage ring operating at the ψ(2S) resonance. The number of ψ(2S) events is 14.0±0.6 million [22], determined from the number of inclusive hadrons.
mentum resolution is σ p /p = 0.017 1 + p 2 (p in GeV/c), and the dE/dx resolution is ∼ 8%. An array of 48 scintillation counters surrounding the MDC measures the time-of-flight (TOF) of charged tracks with a resolution of ∼ 200 ps for hadrons. Radially outside the TOF system is a 12 radiation length, lead-gas barrel shower counter (BSC). This measures the energies of electrons and photons over ∼ 80% of the total solid angle with an energy resolution of σ E /E = 22%/ √ E (E in GeV). Outside the solenoidal coil, which provides a 0.4 Tesla magnetic field over the tracking volume, is an iron flux return that is instrumented with three double layers of counters (MUID) that identify muons of momentum greater than 0.5 GeV/c.
A GEANT3 based Monte Carlo (MC) simulation program [25] with detailed consideration of detector performance (such as dead electronic channels) is used to simulate the BESII detector. The consistency between data and MC simulation has been carefully checked in many high purity physics channels, and the agreement is quite reasonable [25].

Event Selection
Events with π + π − µ + µ − final states and with the invariant mass m µ + µ − constrained to the J/ψ mass are selected for analysis. Each track, reconstructed using hits in the MDC, must have a good helix fit in order to ensure a correct error matrix in the kinematic fit, and the number of tracks are required to be between 4 and 7.
To select a pair of muons, the muon pair candidates tracks are required to have net charge zero; p µ + > 1.3 GeV/c or p µ − > 1.3 GeV/c or p µ + + p µ − > 2.4 GeV/c; | cos θ µ | < 0.6 to ensure that tracks are in the sensitive region of the MUID; the cosine of the angle between these two tracks in their rest frame cos θ cm µ + µ − < −0.975 to guarantee the collinearity of the tracks; the sum of the MUID hits N hit + + N hit − ≥ 3 to ensure that the tracks are muons; and the invariant mass of two candidate tracks m µ + µ − within 0.35 GeV/c 2 of the J/ψ mass.
For π + π − pair selection, the two candidate tracks are also required to have net charge zero. Each track is required to have momentum p π < 0.5 GeV/c, polar angle | cos θ| < 0.75, and transverse momentum p πxy > 0.1 GeV/c to reject tracks that spiral in the MDC. The dE/dx measurement of each track must be within three standard deviations of the dE/dx expected for the pion hypothesis, and the cosine of the laboratory angle between the candidate tracks must satisfy cos θ ππ < 0.9 to eliminate e + e − pairs from γ conversions. The mass recoiling against the candidate π + π − pair, m recoil π + π − , is shown in Fig. 2a. In order to get well reconstructed signal events and to suppress background, |m recoil π + π − − m J/ψ | < 20 MeV/c 2 , corresponding to three times the mass resolution, is required.
With the above selection criteria, about 40,000 ψ(2S) → π + π − J/ψ → π + π − µ + µ − candidate events are obtained. Fig. 2b shows the π + π − invariant mass distribution for these events, where the dots with error bars are data, and the histogram is Monte Carlo simulation with the PP-GEN generator, which is based on chiral symmetry arguments and partially conserved axial vector currents [26]. It describes the low mass ππ spectrum reasonably well but not the high mass region; the inconsistency between data and Monte Carlo will be considered in the systematic errors. The However, MC simulation indicates that their total contribution is less than 0.1%, which can be neglected. The contamination from continuum production e + e − → π + π − µ + µ − is also very small and neglected in this analysis.

Analysis Method
Two different schemes are used to fit our data. In the first, based on the diagrams in Fig. 1 and taking the VPP vertex as a constant, the total differential cross section which describes ψ(2S) → π + π − J/ψ is where A s represents the amplitude for ψ(2S) → XJ/ψ → π + π − J/ψ with the spin of X being s, Ω represents the solid angle, M is the magnetic quantum number along Z-axis of ψ(2S), λ ψ and λ 2 are the helicities of the J/ψ and 2 + components, respectively, and A contact is the amplitude of the contact term.
In the second, considering the VPP vertex and the S-wave ππ FSI, while neglecting the D-wave FSI, the amplitude is [19] where G is the two-pion loop propagator, V 0S is the S-wave part of V 0 , and t I=0 ππ→ππ is the full S-wave I = 0 ππ → ππ t-matrix, which is the same as those defined in Refs. [19,20]; p 1 and p 2 are the four momenta of the two pions, and p 0 1 and p 0 2 are their energies in the lab frame; g 1 , g 2 , and g 3 are free parameters to be determined by data.
The normalized probability density function used to describe the whole decay process is where x represents a set of quantities which are measured by experiment, and α represents unknown parameters to be determined. The total cross section, σ, can be expressed as where ǫ(Ω) is the detection efficiency which is usually a function of detector performance. The total cross section can be determined by MC integration. Re-weighting a total of N generated events based on simulated ψ(2S) → π + π − J/ψ using a phase space generator, the total cross section is then where N mc (< N ) is the number of MC simulated events after applying the selection criteria.
The maximum likelihood function [27,28] is given by the joint probability density of the selected ψ(2S) → π + π − µ + µ − events, and a set of values, α, is obtained by minimizing the function S, For the amplitudes in the first model, the amplitudes for the cascade two-body decay process can be expanded with helicity amplitudes as: is the π + π − recoil mass spectrum, fitted with a double Gaussian function, and (b) is the π + π − invariant mass spectrum. The histogram in (a) is data, the curve is the fit, the events between arrows are selected; in (b) dots with error bars are data, and the histogram is MC simulation.
where F J λν is the helicity amplitude, which can be found in Ref. [28], D J M,λ−ν (φ, θ, 0) is the Dfunction, and BW X (S ππ , m X , Γ X ) is the Breit-Wigner propagator of X, defined as : . (10) The σ particle, a broad structure in the low π + π − mass region, is not a typical Breit-Wigner resonance. In the first model, four types of Breit-Wigner parameterizations are used to describe it: 2) Width containing a kinematic factor, which was used by the E791 Collaboration [16] Γ 3) P.K.U. ansatz [29], which removes the spurious singularity hidden in Eq. (12) 4) Zou and Bugg's approach [30], where the form includes explicitly into Γ X (s) the Adler zero at s = m 2 π /2.
where the definitions of ρ ππ , ρ 4π , and f (s) are the same as Ref. [17].

Partial wave analysis
The minimization used in the partial wave analysis and to obtain the pole parameters of the σ is based on MINUIT [31]. For the first model, the components considered include amplitudes of σ(0 + ), a D-wave term, and a contact term. The tail of the f 0 (980) has been tried in the fit. However, it has similar behavior to the contact term in this mass region, and therefore it is ascribed to the contact term. All four σ Breit-Wigner parameterizations fit the data well, but have strong destructive interference with the contact term, especially in the low π + π − invariant mass region. The D-wave contribution is only 0.3 to 1%, in agreement with the BESI result [3] based on a different analysis method. Fig. 3 shows the projections of the fit results compared with data for the Breit-Wigner parameterization for the P.K.U. ansatz; other parameterizations give similar results.
The global fits determine the best estimation of the Breit-Wigner parameters for each parameterization. The pole position in the complex energy plane is related to the mass and width of the resonance by The best fit results and the corresponding pole positions for all the parameterizations are listed in Table 1. The statistical error of the resonance mass (width) is determined by a decrease of 1 2 in the log-likelihood from its maximum value with all other parameters fixed to their best solutions. For the second model, where the VPP vertex is represented by an effective Lagrangian and the ππ S-wave FSI is included, the ππ mass spectrum can also be reproduced well. Here the σ requires a much smaller interference between the S-wave FSI and the contact term. In this case, the fit is worse than the fits of the first model; this may due to the fixed pole position of the σ and the neglected D-wave contribution. For the second model, the pole is not measured in the fit, but taken from Ref. [20], which was determined from ππ scattering data.
To check the goodness of fit in our analysis, we construct a variable where N is the number of cells, N DT i and N MC i are the numbers of events in the i'th cell of the Dalitz plot with axes m 2 π + π − and m 2 J/ψπ + for data and MC simulation, respectively. Such a variable should be distributed according to the χ 2 distribution with n = N − K degrees of freedom, where K = 12 is the number of parameters to be determined in our Maximum Likelihood fit. In our case, 15 bins in both m 2 π + π − and m 2 J/ψπ + give 225 cells. To ensure proper χ 2 (n) behavior, cells with less than five events have been merged into adjacent ones. The number of cells becomes N = 208, and the number of degrees of freedom n = 196. From the observed χ 2 value determined using Eq. (16) for each parameterization, the confidence levels (C.L.) are calculated and listed in Table 1.

Systematic Errors
For the first model, the systematic error of the σ pole position arises from the uncertainties of the strength of the 2 + component, the form of the contact term, and the inconsistency between data and of MC simulation. For the 2 + component, we conservatively remove it from the fit, and the difference of the fitted values from the nominal values are taken as systematic errors. Two contact terms, namely, constant amplitude and α 1 + iα 2 ρ, where α 1 and α 2 are two parameters to be fitted, are adopted in the fit; the difference is considered as the systematic error. The MC simulation and data have different mass resolutions in the high mass region of the π + π − system. A modification of π + π − mass resolution is made to improve the fit, and the difference of the fitted pole positions with and without this modification is taken as the systematic error. The systematic error from non-signal backgrounds is neglected.
In order to obtain the m σ and Γσ 2 errors in Eq. (15), we set the denominator in Eq. (10) equal to zero and obtain the pole position and corresponding errors by taking into account the mass and width errors of the Breit-Wigner parameterizations. This is done using a MC sampling method, where the correlation between the mass and width is ignored. Table 2 summarizes the systematic errors from all sources, and Table  1 lists the parameters of pole position and their total errors.  . The right is the π + π − invariant mass fitted by the formula from Ref. [19,20], with no explicit D-wave. Dots with error bars are data, and the histograms are the fit results.
We also fit our data according to the scheme in Ref. [19]. It is found that the ππ S-wave FSI plays a dominant role in ψ(2S) → π + π − J/ψ, while the contribution from the contact term is small. This means that the σ meson has a significant contribution in this process. The σ pole used in this fit, 469 − i203 MeV/c 2 is consistent with the fits to the Breit-Wigner functions. This implies that, although the two theoretical schemes are very different, both of them find the σ meson at similar pole positions.
If the σ meson exists, the pole should occur universally in all ππ system with correct quantum numbers. Our analysis demonstrates that, in ψ(2S) → π + π − J/ψ, even though there is no apparent peak structure, one can still determine the pole location in good agreement with that obtained from J/ψ → ωπ + π − decay [17] by assuming a simple form of the contact term. Hence it provides further evidence for the σ meson.