Light and Not So Light Scalar Mesons
aa r X i v : . [ h e p - ph ] M a y Light and Not So Light Scalar Mesons ∗ George Rupp ∗ , Susana Coito Centro de F´ısica das Interac¸c˜oes Fundamentais, Instituto Superior T´ecnico,Technical University of Lisbon, P-1049-001 Lisboa, Portugal andEef van Beveren
Centro de F´ısica Computacional, Departamento de F´ısica, Universidade deCoimbra, P-3004-516 Coimbra, PortugalA multichannel description of the light scalar mesons in the frame-work of the Resonance-Spectrum Expansion is generalised by includingvector-vector and scalar-scalar channels, besides the usual pseudoscalar-pseudoscalar channels. Experimental data for the isoscalar, isodoublet andisovector cases are fitted up to energies well above 1 GeV. The resultingpole positions of the light and intermediate scalar mesons are compared tothe listed resonances. Possible further improvements are discussed.PACS numbers: 14.40.Cs, 14.40.Ev, 11.80.Gw, 11.55.Ds, 13.75.Lb
1. Introduction
The light scalar mesons represent nowadays one of the hottest topicsin hadronic physics. Despite the growing consensus on the existence ofa complete light scalar nonet, comprising the f (600) (alias σ ), K ∗ (800)(alias κ ), a (980) and f (980), which are now all included [1] in the PDGtables, their interpretation and possible dynamical origin in the context ofQCD-inspired methods and models remains controversial. Moreover, theirclassification with respect to the intermediate scalars f (1370), K ∗ (1430), a (1450) and f (1500) [1] is also subject to continued debate. For a brief his-torical discussion of the main theoretical and phenomenological approachesto the light scalars and the corresponding references, see Refs. [2]. ∗ Talk at Workshop “Excited QCD”,
Zakopane, Poland, 8–14 Feb. 2009 (1)
Rupp printed on December 10, 2018
In the present work, the successful multichannel description of the lightscalar mesons in Ref. [3] is further generalised by including, besides theusual pseudoscalar-pseudoscalar (PP) channels, also all vector-vector (VV)[2] and scalar-scalar (SS) channels comprising light mesons. This is crucialfor an extension of the applicability of the approach to energies well above1 GeV, so as to make more reliable predictions for the intermediate scalarresonances as well.
2. Resonance-Spectrum Expansion
We shall study the scalar mesons in the framework of the Resonance-Spectrum-Expansion (RSE) model [4], in which mesons in non-exotic chan-nels scatter via an infinite set of intermediate s -channel q ¯ q states, i.e., akind of Regge propagators [5]. The confinement spectrum for these bare q ¯ q states can, in principle, be chosen freely, but in all successful phenomenolog-ical applications so far we have used a harmonic-oscillator (HO) spectrumwith flavour-indepedent frequency, as in Refs. [6] and [7]. Because of theseparability of the effective meson-meson interaction, the RSE model canbe solved in closed form. The relevant Born and one-loop diagrams aredepicted in Fig. 1, from which it is obvious that one can straightforwardly V = MM MMq ¯ q V Ω V = MM MMq ¯ q q ¯ q Fig. 1. Born (left) and one-loop (right) term of the RSE effective meson-mesoninteraction (see text). sum up the complete Born series. For the meson-meson–quark-antiquarkvertex functions we take a delta shell in coordinate space, which amountsto a spherical Bessel function in momentum space. Such a transition poten-tial represents the breaking of the string between a quark and an antiquarkat a certain distance a , with overall coupling strength λ , in the context ofthe P model. The fully off-energy-shell T -matrix can then be solved as T ( L i ,L j ) ij ( p i , p ′ j ; E ) = − aλ U i j iL i ( p i a ) N X m =1 R im n [11 − Ω R ] − o mj j jL j ( p ′ j a ) U ′ j , with R ij ( E ) = X α ∞ X n =0 g ( α,n ) i g ( α,n ) j E − E ( α ) n , Ω ij = − iaλ µ j k j j jL j ( k j a ) h (1) jL j ( k j a ) δ ij , upp printed on December 10, 2018 and where E ( α ) n is the discrete energy of the n -th recurrence in q ¯ q channel α , g ( α,n ) i is the corresponding coupling to the i -th meson-meson channel, k j and µ j are the (relativistically defined) on-shell momentum and reducedmass of meson-meson channel j , respectively, j jL j ( k j a ) and h (1) jL j ( k j a ) arethe L j -th order spherical Bessel function and spherical Hankel function ofthe first kind, respectively, and U i = √ µ i p i .Spectroscopic applications of the RSE are manifold. In the one-channelformalism, the κ meson was once again predicted, before its experimentalconfirmation, in Ref. [8], 1st paper, after its much earlier prediction inRef. [7]. In the 2nd paper of Ref. [8], the low mass of the D ∗ s (2317) wasshown to be due to its strong coupling to the S -wave DK threshold, anexplanation that is now widely accepted. The 3rd paper of Ref. [8] presenteda similar solution to the whole pattern of masses and widths of the charmedaxial-vector mesons.Multichannel versions of the RSE model have been employed to producea detailed fit of S -wave PP scattering and a complete light scalar nonet [3],with very few parameters (also see below), and to predict the D sJ (2860) [9],shortly before its observation was publicly announced.Finally, the RSE has recently been applied to production processes [10]as well, in the spectator approximation. Most notably, it was shown thatthe RSE results in a complex relation between production and scatteringamplitudes (papers 1–3 in Ref. [10]). Successful applications include theextraction of κ and σ signals from data on 3-body decay processes (4th paperin Ref. [10]), the deduction of the string-breaking radius a from productionprocesses at very different energy scales (5th paper), and even the discoveryof signals hinting at new vector charmonium states in e + e − → Λ c ¯Λ c data(6th paper).
3. Light and Intermediate Scalar Mesons S -wave PP scattering In Ref. [3], two of us (E.v.B, G.R.) together with Bugg and Kleefeldapplied the RSE to S -wave PP scattering up to 1.2 GeV, coupling thechannels ππ , K ¯ K , ηη , ηη ′ , η ′ η ′ for I = 0, Kπ , Kη , Kη ′ for I = 1 /
2, and ηπ , K ¯ K , η ′ π for I = 1. Moreover, in the isoscalar case both an n ¯ n and an s ¯ s channel were included, so as to allow dynamical mixing to occur via the K ¯ K channel. The very few parameters, essentially only the overall coupling λ and the transition radius a , were fitted to scattering data from varioussources, for I = 0 and for I = 1 /
2, and to the a (980) line shape, determinedin a previous analysis, for I = 1. Moreover, the parameters λ and a variedless than ±
10% from one case to another. Overall, a good description of
Rupp printed on December 10, 2018 the data was achieved (see Ref. [3] for details). Poles for the light scalarmesons were found at (all in MeV) σ : 530 − i , κ : 745 − i , f (980) : 1007 − i , a (980) : 1021 − i . No pole positions for the intermediate scalars were reported in Ref. [3], asthe fits were only carried out to 1.1 GeV in the isovector case, and to 1.2GeV in the others. Nevertheless, corresponding poles at higher energieswere found, but these were of course quite unreliable.In the following, we shall present preliminary results for fits extended tohigher energies, and with more channels included.
For I = 0, the VV channels that couple to n ¯ n and/or s ¯ s are ρρ , ωω , K ∗ ¯ K ∗ and φφ , for both L = 0 and L = 2, while the SS channels are σσ , f (980) f (980), κκ and a (980) a (980), with L = 0 only. We fit the parame-ters λ and a to sets of S -wave ππ phase shifts compiled by Bugg and Surov-tsev [11], which yield a somewhat larger scattering length than in Ref. [3],viz. 0 . m − π . The results of the fits are shown in Fig. 2, left-hand plot, to-gether with the curve from Ref. [3], where only PP channels were includedand somewhat lower data were used just above the ππ threshold. All threefits are good up to 1 GeV, with only small differences among them. There-above, the VV and SS channels clearly produce very substantial effects,though overshooting between 1.2 and 1.6 GeV. Possible improvements are E CM (GeV) ππ P h a s e ( d e g . ) •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• E CM (GeV) | T η π → η π | ( a ( ) ) Fig. 2. Left: S -wave I = 0 ππ phase shifts; red: PP channels only; blue: also VVchannels; green: also SS channels; data due to Ref. [11]. Right: a (980) line shape;coloured curves: as in left-hand plot; black curve from data analysis by Bugg [11]. upp printed on December 10, 2018 discussed below. With all channels included, we find the first four isoscalarpoles at (all in MeV) σ : 464 − i , f (980) : 987 − i , f (1370) : 1334 − i
185 , f (1500) : 1530 − i . Moreover, there is an extra broad state at (1519 − i f (1370) and the f (1500)are clear improvements with respect to the case with PP and VV channelsonly (see Ref. [2], 1st paper). In particular, the extra pole found here mighthelp to explain the experimental difficulties with the f (1370) and f (1500). a (980) and a (1450) In the isotriplet case, we fit λ and a , as well as the pseudoscalar mixingangle, to the a (980) line shape, just as in Ref. [3], but now with the VV( ρK ∗ , ωK ∗ , φK ∗ ) and SS ( a (980) σ , a (980) f (980), κκ ) channels added.Thus, the quality of the fit is slightly improved, though the differences withthe PP and PP+VV cases are hardly visible in Fig. 2, right-hand plot.The poles we find are (1023 − i
47) MeV (second sheet) for the a (980) and(1420 − i a (1450), which are very reasonable values [1]. K ∗ (800) and K ∗ (1430) Adding the vector ( ρK ∗ , ωK ∗ , φK ∗ ) and scalar ( σκ , f (980) κ , a (980) κ )channels in the isodoublet sector does not allow a stable fit to be obtained.Moreover, the LASS data are known to violate unitarity above 1.3 GeV. Sowe just fit up to 1.3 GeV, with only the PP channels, getting parameters veryclose to the isoscalar case, including a reasonable pseudoscalar mixing angle.See further the conclusions for possible remedies. From the present PP fitwe find the pole postions (722 − i κ and (1400 − i
96) MeVfor the K ∗ (1430), which are again reasonable values [1].
4. Conclusions and outlook
The preliminary results in this study indicate that a good description ofboth the light and the intermediate scalar mesons is feasible in the RSE, bytaking into account additional sets of coupled channels that should becomerelevant at higher energies. However, some problems persist, like the tooslow rise and subsequent overshooting of the I = 0 ππ phase shift above1 GeV, and the mentioned fitting problems in the isodoublet case. A possiblecause of these difficulties is the assumed sharpness of several thresholdsinvolving broad resonances in their turn, such as σσ , ρρ , σκ , etc., whichmay result in too drastic effects at the opening of these channels. We are Rupp printed on December 10, 2018 now studying ways to account for final-state resonances having non-zerowidths, without destroying unitarity. It might also turn out to be necessaryto consider more general transition potentials.
Acknowledgements
We are grateful to the organisers for a most stimulating workshop. Oneof us (G.R.) thanks J. R. Pel´aez for very useful information on ππ scatter-ing. This work received partial financial support from the Funda¸c˜ao para aCiˆencia e a Tecnologia of the
Minist´erio da Ciˆencia, Tecnologia e EnsinoSuperior of Portugal, under contract CERN/FP/83502/2008.REFERENCES [1] C. Amsler et al. [Particle Data Group], Phys. Lett. B (2008) 1.Note, however, that the K ∗ (800) is still inexplicably omitted from the Sum-mary Table, while its quoted “average” mass of (672 ±
40) MeV makes nosense whatsoever, in view of the recent experimental convergence towards avalue of roughly 800 MeV.[2] S. Coito, G. Rupp and E. van Beveren, invited plenary talk given by G. Ruppat the 19th International Baldin Seminar on High Energy Physics Problems:Relativistic Nuclear Physics and Quantum Chromodynamics (ISHEPP 2008),Dubna, Russia, 29 Sep. – 4 Oct. 2008, arXiv:0812.1527 [hep-ph]; E. van Bev-eren and G. Rupp, Eur. Phys. J. A (2007) 468.[3] E. van Beveren, D. V. Bugg, F. Kleefeld and G. Rupp, Phys. Lett. B (2006) 265.[4] E. van Beveren and G. Rupp, Int. J. Theor. Phys. Group Theory Nonlin. Opt. (2006) 179.[5] E. van Beveren and G. Rupp, Annals Phys., in press, DOI 10.1016/j.aop.2009.03.013 [arXiv:0809.1149 [hep-ph]].[6] E. van Beveren, G. Rupp, T. A. Rijken and C. Dullemond, Phys. Rev. D (1983) 1527.[7] E. van Beveren, T. A. Rijken, K. Metzger, C. Dullemond, G. Rupp andJ. E. Ribeiro, Z. Phys. C (1986) 615.[8] E. van Beveren and G. Rupp, Eur. Phys. J. C (2001) 493; Phys. Rev. Lett. (2003) 012003; Eur. Phys. J. C (2004) 493.[9] E. van Beveren and G. Rupp, Phys. Rev. Lett. (2006) 202001.[10] E. van Beveren and G. Rupp, Annals Phys. (2008) 1215; Europhys. Lett. (2008) 61002. Europhys. Lett. (2008) 51002; J. Phys. G (2007) 1789;arXiv:0712.1771 [hep-ph]; E. van Beveren, X. Liu, R. Coimbra and G. Rupp,Europhys. Lett.85