Why the f 0 (980) is mostly s s

Transcription

1 Why the f 0 (980) i motly Eef van Beveren Departamento de Fíica, Univeridade de Coimbra P-3000 Coimbra, Portugal eef@teor.fi.uc.pt George Rupp Centro de Fíica da Interacçõe Fundamentai Intituto Superior Técnico, Edifício Ciência P-1096 Liboa Codex, Portugal george@ajax.it.utl.pt Michael D. Scadron Phyic Department Univerity of Arizona Tucon, AZ, 8571 USA cadron@phyic.arizona.edu PACS number(): C, 1.39.Pn, Lb hep-ph/ September 6, 000 Abtract We exploit the W -emiion proce to tudy the meaured weak decay D + f 0 (980)π +. We conclude that the calar f 0 (980) meon i motly, which i upported by different model tudie. 1

2 be motly becaue of it nearne to the vector meon φ(100), which i known to be motly. On the other hand, the qq q q cheme[6,7]uggetf 0 (980) and alo a 0 (980) are motly nontrange, by analogy with the vector ω(780) and ρ(770), which are both known to be motly nontrange. We may tudy the prior quetion from the recently meaured weak decay rate [8] Γ(D + f 0(980)π + ) = (.39 ± 1.06) GeV, (1) due to the branching ratio of (1.8 ± 0.8) percent and the lifetime of (0.496 ± 0.01) Given the rate of equation (1), we may expre the decay amplitude magnitude a (p = 73 MeV and m D + = GeV ) 8πΓ M (D + f 0 (980)π + ) = m D + = (1.78 ± 0.40) 10 6 GeV. () p Auming that the f 0 (980) i pure, we may account for the experimental amplitude of equation () via the W + -emiion graph of figure (1). W + c D + f 0 (980) u d π + Figure 1: Contribution of W + emiion to the proce D + f 0 (980)π + Thi predict, uing the value given in [8] for the Fermi coupling contant, G F, the coine of the phae from the CKM weak mixing matrix, c 1, and the pion formfactor, f π, M (D + f 0(980)π + ) W + -emiion = G F c ( ) 1 f π m D m f GeV. (3)

3 at the accuracy of thi paper. For the pion formfactor, f π,wetakethevalue93mev. Not only i equation (3) compatible with the data of equation (), but the analogou I = 3/ W + -emiion proce K + π + π 0 i, too [9]: M (K + π + π 0 ) W + -emiion = G F 1 c 1 f ( π m K ) m π GeV ; (4) the data being [8] (1.834 ± 0.007) 10 8 GeV. By comparing formula () with the prediction of formula (3), we could infer a mall calar mixing of n n and, which i cloe to the theoretical etimate. Since σ f 0 =0,amixing angle, φ,ofabout14 [10] to 0 [11, 1], predict an amplitude ratio M (D + f 0(980)π + ) G F c ( ) 1 f π m D m f 0 co (φ ) 0.9 ± 0., (5) which certainly how good agreement between the experimental decay amplitude ( ) and the W + -emiion reult ( 3). A for earlier interpretation of f 0 (980) being motly, we refer to the 198 and 1986 paper of the preent author, [11] and [13]. In the latter cae the nontrange and trange low-lying calar meon are modelled jointly with the peudocalar and vector including meon which contain c and b quark. The reulting four-parameter model, which fit very well low-energy S- andp -wave meon-meon cattering cro ection a well a the bound tate and reonance poition, predict two complete calar-meon nonet, one below and one above 1 GeV. Alo we note that the motly tructure of the f 0 (980) i treed in the more recent work of Törnqvit [] and of Törnqvit and Roo [3], the latter of which focue on the nontrange σ( ). Let u recall here the DM data [14] meauring the nontrange large σ bump in the ππ ma ditribution for J/ψ ωππ, but only a mall f 0 (980) pimple, which how the mallne of the nontrange content of the f 0 (980) meon. Alo the recent Fermilab E791 collaboration [15] ugget that the f 0 (980) i motly, but that the f 0 (1370) i motly nontrange a hinted by the naive quark model ince D + K + K π + i not een. It hould be noted that the ame i predicted in the unitarized meon model of Ref. [13] (ee alo Ref. [5]), contrary to e.g. Ref. [] and [3], which interpret the f 0 (1370) a motly. Thi dicrepancy between two unitarized model i quite triking and deerve a little more attention. 3

4 meonic tate with ome kind of quark ubtructure irrepective of the precie configuration and not jut a two-meon reonance effect due to trong t-channel exchange and nearby threhold (ee e.g. Ref. [4]), normally conider the calar above 1 GeV a excited tate. However, in the unitarized model of Ref. [5, 13] and [, 3] all calar meon below 1.5 GeV originate in one and only bare calar q q nonet, omewhere in the ma region GeV. Here, bare refer to the model ituation where the coupling to two-meon tate i witched off. Now, when thi coupling i et to the model value which fit the data, the bare calar pectrum get deformed and even extra tate how up, a for example the f 0 ( ) or σ meon, the a 0 (980), and, of coure, the f 0 (980) (ee Ref. [5] for more detail). In the model of Ref. [5, 13], the extra tate contitute a complete light nonet, including a K0 (77) or κ meon, while the pattern of mae of the heavier calar, which are nonethele dominantly radial ground tate, remain largely unaltered. For intance, the motly trange f 0 tay around 1.5 GeV, while the mainly nontrange f 0 ettle cloe to 1.3 GeV, apparently quite compatible with the f 0 (1370). On the other hand, Ref. [, 3] find neither a light κ, northe f 0 (1500), and interpret the f 0 (1370) a motly. We do not wih to enter here into a detailed dicuion on which interpretation i favored by experiment, and refer to Ref. [5] for an analyi of oberved decay mode of the f 0 (1370) and f 0 (1500) that, we believe, upport our view. In any cae, thi i evidently the mot attractive cenario, in which there i a complete doubling of calar tate, but keeping the internal ma pattern of the nonet intact. To conclude the dicuion of the f 0 (1370), we would like to mention a very recent lattice calculation of calar quarkonium mae [16], which i in agreement with our interpretation of the f 0 (1370) and f 0 (1500). In concluion and ummarizing, by tudying the W -emiion proce to decribe the weak decay D + f 0 (980)π +, we are able to reproduce the recently meaured decay rate, provided that the f 0 (980) i aumed to be motly. A mixing angle of about 14, correponding to amalln n admixture, i predicted. Thi interpretation of the f 0 (980) i in agreement with previou work of the author [1, 5, 11, 13], and alo with other model tudie [, 3, 10, 17, 18, 19]. Finally, it i pointed out that a very recent experiment [15] ugget the f 0 (1370) i motly n n, a predicted by the unitarized meon model of Ref. [5, 13]. 4

5 [1] R. Delbourgo, D. Liu and M.D. Scadron, Phy. Lett. B446, 33 (1999). [] Nil A. Törnqvit, Zeit. Phy. C68, 647 (1995). [3] Nil A. Törnqvit and Matt Roo, Phy. Rev. Lett. 76, 1575 (1996). [4] N. Igur and J. Speth, Phy. Rev. Lett. 7, 33 (1996). [5] E. van Beveren and G. Rupp, Eur. Phy. J. C10, 468 (1999). [6] R. Jaffe, Phy. Rev. D15, 67 (1977). [7] D. Black, A. H. Fariborz, and J. Schechter, arxiv:hep-ph/ [8] Particle Data Group, D.E. Groom et al, Eur. Phy. J. C15, 1 (000). [9] M.D. Scadron, Phy. Rev. D9, 1375 (1984); here, vacuum aturation rather than W + emiion wa ued. [10] M. Napuciale, Scalar meon mae and mixing angle in a SU(3) SU(3) linear igma model, arxiv:hep-ph/ [11] M.D. Scadron, Phy. Rev. D6, 39 (198). [1] R. Delbourgo and M.D. Scadron, Int. J. Mod. Phy. A13, 657 (1998). [13] E. van Beveren, T. A. Rijken, K. Metzger, C. Dullemond, G. Rupp, and J. E. Ribeiro, Zeit. Phy. C30, 615 (1986). [14] DM collaboration, J. Augutin et al, Nucl. Phy. B30, 1 (1989). [15] Fermilab E791 collaboration, E.M. Aitala et al, Study of the D + π π + π + decay and meaurement of f 0 mae and width, arxiv:hep-ex/ [16] W. Lee and D. Weingarten, Phy. Rev. D61, (000). [17] S. Ihida, M. Ihida, T. Ihida, K. Takamatu, and T. Turu, Prog. Theor. Phy. 98, 61 (1997). [18] D. Black, A. H. Fariborz, F. Sannino, and J. Schechter, Phy. Rev. D58, (1998). [19] J. A. Oller, E. Oet, and J. R. Peláez, Phy. Rev. D59, (1999). 5