An interesting feature of BESIII data for J/Psi -> gamma-(etaprime-pi-pi)
aa r X i v : . [ h e p - ph ] J a n An interesting feature of BES III data for J/ Ψ → γ ( η ′ ππ ) D V Bugg Queen Mary, University of London, London E1 4NS, UK
Abstract
The η (1835) is confirmed clearly in new BESIII data for J/ Ψ → γ ( η ′ ππ ); the angulardistribution of the photon is consistent with a pseudoscalar. This makes it a candidate foran s ¯ s radial excitation of η ′ and η (1440) (or one or both of η (1405) and η (1475)). However,a conspicuous feature of the BES III data is the absence of evidence for η (1440) → η ′ ππ while it is well known that η (1440) appears in ηππ . Can these facts be reconciled? Thereis in fact a simple explanation. The channel η (1440) → ηππ may be explained by thetwo-step process η (1440) → [ K ∗ K ] L =1 and [ κ ¯ K ] L =0 , followed by K ¯ K → a (980) → ηπ .This process does not produce any significant η ′ π signal because of the Adler zero close tothe η ′ π threshold.PACS numbers: 11.80.Et, ,13.20.Gd, 14.40.Be New BESIII data confirm η (1835) in J/ Ψ → γ ( η ′ ππ ) [1]. Huang and Zhu point out that η (1835)may be explained naturally as an s ¯ s radial excitation of η ′ (958) and η (1440) [2]. However, aconspicuous feature of the BESIII data is the absence of η (1440) → η ′ ππ , although there is asmall and narrow peak attributed to f (1510) → η ′ ππ . The data contrast strongly with theclear η (1440) signal observed in J/ Ψ → γ ( ηππ ) by Mark III [3], DM2 [4] and BES I [5].Figure 1: Triangle graph for η (1440) → ηππ There is a straightforward explanation shown in Fig. 1. The dominant signals attributed to η (1440) are decays to K ∗ (890) ¯ K , κ ¯ K (where κ is the Kπ S-wave), and a weak ηππ channel. The K ¯ K pairs from the first two channels can rescatter through a (980) to ηπ . Triangle graphs forthis process were calculated by Anisovich et al. treating intermediate K ¯ K pairs as real particles[6]. This model fitted Crystal Barrel data on ¯ pp → ηπ + π − π + π − successfully. A feature of thosedata is that the ηπ peak is 30-50 MeV above the K ¯ K threshold, as expected from the phasespace for real intermediate K ¯ K pairs. For the η ′ ππ final state, a (980) → η ′ π is attentuatedstrongly by the Adler zero at s = m η ′ − m π /
2, where m are masses [7]. email: [email protected]
1t will clearly be important to show that the small η ′ ππ peak in BESIII data fitted to f (1510)has J P = 1 + . It looks too narrow and too high in mass to be due to η (1440). It is also obviouslyimportant to look for the η (1835) in J/ Ψ → γ ( K ¯ Kπ ), as a check on the s ¯ s assignment of Huangand Zhu.An important general comment is that fits to η (1405) and η (1475) must include the full s -dependence of opening channels and associated dispersive corrections to resonance amplitudes.Breit-Wigner amplitudes of constant width are seriously misleading. As an example, the fullformula for the production amplitude of η (1475) → [ K ∗ ¯ K ] L =1 is f ∝ kF ( k ) B ( k ) / [ M − s − m ( s ) − ig ρ ( k ) F ( k ) B ( k )] (1) m ( s ) = ( s − M ) π P Z M Γ total ( s ′ ) ds ′ ( s ′ − s )( M − s ′ ) . (2)The ¯ KK ∗ channel opens at a mass of ∼ ρ rises as k , where k isthe momentum of K and K ∗ in the ¯ KK ∗ rest frame; B is the Blatt-Weisskopf centrifugal barrierfactor 1 / (1+ k R ) / and g is a coupling constant. The term m ( s ) is the so-called ‘running mass’and is required to make the amplitude fully analytic; P stands for the principal value integral.The integral includes a subtraction on resonance, making it strongly convergent. The term F ( k )is a form factor. Fits to a large number of data sets with a Gaussian F ( k ) = exp( − αk ) give α = (2 . ± .
25) (GeV/c) − , corresponing to a radius R = 0 .
73 fm for the meson cloud. Thewidth of the K ∗ needs to be folded into the calculation, as does the phase space for productionprocesses such as J/ Ψ → γ ( ¯ KK ∗ ) or ¯ pp → ( ¯ KK ∗ ) σ . s (GeV) m ( s ) ( G e V ) Figure 2: m ( s ) for η (1475) → ¯ KK ∗ .It turns out that the term m ( s ), shown in Fig. (2), varies more rapidly than ( M − s ) nearresonance, because of the k dependence of ρ ( ¯ KK ∗ ). As a result, a fit to the resonance ofconstant width and mass 1476 MeV quoted by the Particle Data Group [8] moves the massdown to 1439 MeV. This raises doubts about the existence of two narrowly spaced J P = 0 − states. The effect on η (1405) is smaller, because the κK phase space varies as k . References [1] M. Ablikim et al., [BESIII Collaboration], arXiv: 1012.3510.[2] J. Huang and S-L. Zhu, Phys. Rev. D 73 (2006) 014023.23] J.Z. Bai et al., Phys. Rev. Lett. 65 (1990) 2507.[4] J.-E. Augustin et al., Phys. Rev. D 46 (1992) 1951.[5] J.Z. Bai et al., Phys. Lett. B 446 (1998) 356.[6] A.V. Anisovich et al.et al.