aa r X i v : . [ h e p - ph ] M a y The Nature of the X (2175) ∗ S. Coito ∗ , G. Rupp Centro de F´ısica das Interac¸c˜oes Fundamentais, Instituto Superior T´ecnico,Technical University of Lisbon, P-1049-001 Lisboa, Portugal andE. van Beveren
Centro de F´ısica Computacional, Departamento de F´ısica, Universidade deCoimbra, P-3004-516 Coimbra, PortugalWe study the puzzling vector meson X(2175) in a multichannel gener-alisation of the Resonance-Spectrum-Expansion model. Besides the usual P -wave pseudoscalar–pseudoscalar, pseudoscalar–vector, and vector–vectorchannels that couple to mesons with vector quantum numbers, we also in-clude the important S -wave vector–scalar, pseudoscalar–axial-vector andvector–axial-vector channels, including the observed φ (1020) f (980) decaymode. The strong coupling to nearby S -wave channels originates dynami-cally generated poles, two of which come out close to the energy region ofthe X (2175), viz. at (2 . − i . . − i .
20) GeV. Furtherimprovements are proposed.PACS numbers: 14.40.Cs, 11.80.Gw, 11.55.Ds, 13.75.Lb
1. Introduction
The X (2175) was first observed by BABAR [1] in the process e + e − → φ (1020) f (980), and identified as a 1 −− resonance, with M = (2 . ± . ± . ± ±
20) MeV. This state was then con-firmed by BES and denoted Y (2175) [2], from the decay J/ Ψ → ηφf (980),with M = (2 . ± . ± . ± ±
17) MeV. It isnow included in the PDG Particle Listings [3] as the φ (2170).On the theoretical side, the X (2175) has been descibed as a three-mesonresonance in a Faddeev calculation for the φK ¯ K system [4], obtaining a ∗ Talk at Workshop “Excited QCD”,
Zakopane, Poland, 8–14 Feb. 2009 (1)
Coito printed on November 21, 2018 narrow peak around 2150 MeV, about 27 MeV wide. Earlier, a conven-tional Resonance-Chiral-Perturbation-Theory calculation [5] failed to pro-duce such a peak, which led to the former 3-body model. Other approachesinclude QCD-sum-rule calculations for a tetraquark state [6], and a pertur-bative multichannel analysis to distinguish between a strangeonium hybridand a normal 2 D s ¯ s state [7].
2. Resonance Spectrum Expansion and the X (2175) In the present study of the X (2175), we use the Resonance-Spectrum-Expansion (RSE) model [8] to unitarise a normal s ¯ s spectrum. In theRSE approach, non-exotic mesons are described as regular quark-antiquarkstates, but non-perturbatively dressed with meson-meson components. Animportant feature is the inclusion of a complete q ¯ q confinement spectrumin the intermediate state [8, 9], resulting for the multichannel case in aneffective meson-meson potential V ( L i ,L j ) ij ( p i , p ′ j ; E ) = λ j iL i ( p i a ) j jL j ( p ′ j a ) ∞ X n =0 g i ( n ) g j ( n ) E − E n , (1)where λ is an overall coupling constant, a a parameter mimicking the aver-age string-breaking distance, j iL i a spherical Bessel function, p i and p ′ j therelativistically defined relative momenta of initial channel i and final chan-nel j , respectively, g i ( n ) the coupling of channel i to the n -th q ¯ q recurrence,and E n the discrete energy of the latter confinement state. Note that thecouplings g i ( n ), evaluated on a harmonic-oscillator (HO) basis for the P model [10], decrease very rapidly for increasing n , so that practical con-vergence is achieved with the first 20 terms in the infinite sum. Moreover,the separable form of the effective potential (1) allows the solution of the(relativistic) Lippmann-Schwinger equation in closed form.In the present first study, we restrict ourselves to the S channel for the q ¯ q system. Moreover, we assume ideal mixing, so that only s ¯ s states areconsidered. For the confinement mechanism, we take an HO potential. Thischoice is not strictly necessary, as Eq. (1) allows for any confinement spec-trum, but the HO has shown to work fine in practically all phenomenologicalapplications. The flavour-dependent HO spectrum reads E n = m q + m ¯ q + ω (2 n + 3 / ℓ ) . (2)The parameter values ω = 190 MeV and m s = 508 MeV are kept unchangedwith respect to all previous work (see e.g. Ref. [11]). In Table 1, we listsome of the eigenvalues given by Eq. (2). As for the decay sector thatcouples to J P C = 1 −− φ states, we take all pseudoscalar–pseudoscalar (PP), oito printed on November 21, 2018 n s ¯ s Table 1. HO eigenvalues (2) in GeV, for ω = 190 MeV, m s = 508 MeV, ℓ = 0. pseudoscalar–vector (PV), and vector–vector (VV) channels, which are in P -waves, as well as all vector–scalar (VS), pseudoscalar–axial-vector (PA),and vector–axial-vector (VA) channels, being in S -waves. These 15 channelsare listed in Table 2, including the observed [1, 2] φf (980) mode, with therespective orbital angular momenta, spins, and thresholds.Channel Relative L , Total S Threshold KK , . KK ∗ , . ηφ , . η ′ φ , . K ∗ K ∗ , . K ∗ K ∗ , . φf (980) 0 , . K ∗ K ∗ (800) 0 , . ηh (1380) 0 , . η ′ h (1380) 0 , . KK (1270) 0 , . KK (1400) 0 , . K ∗ K (1270) 0 , . K ∗ K (1400) 0 , . φf (1420) 0 , . Table 2. Thresholds in GeV of included meson-meson channels (see Ref. [3]).
3. Results
As emphasised above, the T -matrix for the effective potential (1) can besolved in closed form. Bound states and resonances correspond to poles of T on the appropriate sheet of the many-sheeted Riemann surface. For the15 channels considered, there are 2 = 32 ,
768 such sheets. However, therelevant poles are in principle those that correspond to relative momentawith negative imaginary parts with respect to open channels, and positive
Coito printed on November 21, 2018 imaginary parts for closed channels. The simplest example of a latter-typepole is a bound state, for which the real part of the momentum is zero.The only two free parameters of the model, viz. λ and a , we fix bydemanding that the mass and width of the φ (1020) be reasonably repro-duced. For a = 5 . GeV − and λ = 3 . GeV − / we get a theoreticalpole position of E theor = (1 . − i . GeV, to be compared withthe PDG [3] value E exp = (1 . − i . GeV, which is more thangood enough for the present simplified investigation. Of course, besides the φ (1020), there are several other resonance poles, which are given in Ta-ble 3, for energies up to about 2.5 GeV . Confinement poles are those that ℜ e ℑ m Type of Pole1 . − . n = 01 . − .
011 confinement, n = 11 . − .
010 confinement, n = 22 . − .
170 continuum2 . − .
020 continuum2 . − .
123 continuum
Table 3. Pole positions, in GeV. end up at the energies of the confinement spectrum in the limit λ → λ →
0, with ℑ m E → −∞ . The latter polesmostly have large imaginary parts, for physical values of λ , but there areexceptions, like the fifth pole in Table 3. Typical cases of pole trajecto-ries are shown in Fig. 1, with the n = 2 confinement pole in the left-handplot and the first continuum pole in the right-hand one. The jump in thetrajectory of the confinement pole is due to a change of Riemann sheet atthe φf (980) threshold. The trajectory of the second continuum pole isdepicted in Fig. 2, left-hand plot, which shows its highly non-linear andnon-perturbative behaviour. As for the main purpose of the present work,we find two poles in the energy region 2.0–2.4 GeV relevant for the X (2175),namely at (2 . − i . GeV and (2 . − i . GeV, being bothcontinuum poles. The 4 S ( n = 3) confinement pole can be easily followedfrom E = 2 .
441 GeV, up to a value of λ ≈ . − / , but one loses itstrack when switching Riemann sheet at the opening of the φf (1420) chan-nel. In any case, all these precise pole positions are not so important forthis first study. Suffice it to say that dynamical poles can be generated inthe energy region pertinent to the X (2175), and possibly with quite small These results are slightly different from the preliminary ones presented at the work-shop, due to a, now corrected, minor error in the computer code. oito printed on November 21, 2018 S state (left); first continuum pole (right). imaginary parts. Preliminary results for a more complete calculation [12],including the D states, indicate a considerable improvement of the polepositions for the X (2175) candidates. Fig. 2. Second continuum pole (left); elastic K ∗ K (1270) cross section (right). Having the exact T -matrix at our disposal, we can easily compute ob-servables, too, like cross sections and phase shifts. Although this is not ofgreat importance here, in view of the rather tentative pole positions above2 GeV, we show in Fig. 2, right-hand plot, the S -wave K ∗ K (1270) crosssection, just as an illustration. The pole at (2 . − i . φf (980) channel. Inclusion of the D states turns out to significantly improve the description, both for the polepositions and for the φf (980) cross section [12], in line with experiment. Coito printed on November 21, 2018
4. Conclusions and outlook
We have shown that coupling a spectrum of confined s ¯ s states to all S -wave and P -wave two-meson channels composed of light mesons allows togenerate dynamical resonances above 2 GeV, besides roughly reproducingthe mass and the width of the φ (1020). This may provide a framework tounderstand the puzzling X (2175) meson, owing to the large and non-linearcoupled-channel effects, especially from the S -wave channels. Inclusion ofthe D s ¯ s states will then account for a more realistic modelling, as con-firmed by preliminary results [12]. Further improvements may be consideredas well, such as deviations from ideal mixing, smearing out resonances inthe final state, and more general transition potentials. Acknowledgements
We thank the organisers for an inspiring and pleasant workshop. We arealso indebted to K. Khemchandani for very useful discussions. This workwas supported in part by the
Funda¸c˜ao para a Ciˆencia e a Tecnologia ofthe
Minist´erio da Ciˆencia, Tecnologia e Ensino Superior of Portugal, undercontract CERN/FP/83502/2008.REFERENCES [1] B. Aubert et al. [BABAR Collaboration], Phys. Rev. D (2006) 091103.[2] M. Ablikim et al. [BES Collaboration], Phys. Rev. Lett. (2008) 102003.[3] C. Amsler et al. [Particle Data Group], Phys. Lett. B (2008) 1.[4] A. Martinez Torres, K. P. Khemchandani, L. S. Geng, M. Napsuciale andE. Oset, Phys. Rev. D (2008) 074031.[5] M. Napsuciale, E. Oset, K. Sasaki and C. A. Vaquera-Araujo, Phys. Rev. D (2007) 074012.[6] Z. G. Wang, Nucl. Phys. A (2007) 106; H. X. Chen, X. Liu, A. Hosakaand S. L. Zhu, Phys. Rev. D (2008) 034012.[7] G. J. Ding and M. L. Yan, Phys. Lett. B (2007) 49.[8] E. van Beveren and G. Rupp, Int. J. Theor. Phys. Group Theor. Nonlin. Opt. (2006) 179.[9] E. van Beveren and G. Rupp, Annals Phys., in press, DOI 10.1016/j.aop.2009.03.013 [arXiv:0809.1149 [hep-ph]].[10] E. van Beveren, Z. Phys. C (1984) 291.[11] E. van Beveren, G. Rupp, T. A. Rijken and C. Dullemond, Phys. Rev. D27