Disentangling two- and four-quark state pictures of the charmed scalar mesons

Transcription

1 Physis Letters B 64 5) 17 Disentangling two- and four-quark state pitures of the hared salar esons M.E. Brao a,a.lozea a,r.d.matheus b, F.S. Navarra b,m.nielsen b a Instituto de Físia, Universidade do Estado do Rio de Janeiro, Rua São Franiso Xavier 54, 55-9 Rio de Janeiro, RJ, Brazil b Instituto de Físia, Universidade de São Paulo, C.P , São Paulo, SP, Brazil Reeived 3 May 5; reeived in revised for 4 August 5; aepted 9 August 5 Available online August 5 Editor: M. Cvetič Abstrat We suggest that the reently observed hared salar esons D 38) Belle) and D,+ 45) FOCUS) are onsidered as different resonanes. Using the QCD su rule approah we investigate the possible four-quark struture of these esons and also of the very narrow D sj + 317), firstly observed by BaBar. We use diquark antidiquark urrents and work to the order of s in full QCD, without relying on 1/ expansion. Our results indiate that a four-quark struture is aeptable for the resonanes observed by Belle and BaBar: D 38) and D+ sj 317) respetively, but not for the resonanes observed by FOCUS: D,+ 45). 5 Elsevier B.V. All rights reserved. PACS: Hx; 1.38.Lg; k Reently the first observations of the salar hared esons have been reported. The very narrow D + sj 317) was first disovered in the D+ s π hannel by the BaBar Collaboration [1] and its existene was onfired by CLEO [], Belle [3] and FOCUS [4] Collaborations. Its ass was oonly easured as 317 MeV, whih is approxiately 16 MeV below the predition of the very suessful quark odel for the hared esons [5]. The Belle Collaboration [6] has also reported the observation of a rather E-ail address: nielsen@if.usp.br M. Nielsen). broad salar eson D 38), and the FOCUS Collaboration [7] reported evidene for broad strutures in both neutral and harged final states that, if interpreted as resonanes in the J P = + hannel, would be the D 47) and the D+ 43) esons. While the ass of the salar eson, D 38), observed by Belle Collaboration is also bellow the predition of Ref. [5] approxiately 1 MeV), the asses of the states observed by FOCUS Collaboration are in oplete agreeent with Ref. [5]. Due to its low ass, the struture of the eson D sj + 317) has been extensively debated. It has been interpreted as a s state [8 1], two-eson oleu /$ see front atter 5 Elsevier B.V. All rights reserved. doi:1.116/j.physletb

2 18 M.E. Brao et al. / Physis Letters B 64 5) 17 lar state [13,14], D K ixing [15], four-quark states [16 18] or a ixture between two-eson and fourquark states [19]. The sae analyses would also apply to the eson D 38). In the light setor the idea that the salar esons ould be four-quark bound states is not new [] and, therefore, it is natural to onsider analogous states in the har setor. We propose that the resonanes observed by Belle [6] and FOCUS [7] Collaborations be onsidered as two different resonanes. In this work we use the ethod of QCD su rules QCDSR) [1] to study the two-point funtions of the salar esons, D sj 317), D 38) and D 45) onsidered as four-quark states. The use of the QCD su rules to study the hared salar esons was already done in Refs. [8, 11,1], but in these alulations they were interpreted as two-quark states. In a reent alulation [] soe of us have onsidered that the lowest lying salar esons are S-wave bound states of a diquark antidiquark pair. As suggested in Ref. [3] the diquark was taken to be a spin zero olour antitriplet. We extend this presription to the har setor and, therefore, the orresponding interpolating fields ontaining zero, one and two strange quarks are: j = ɛ ab ɛ de q T a Cγ 5 b )ūd γ 5 C d e T ), j s = ɛ abɛ de [ u T a Cγ 5 b )ūd γ 5 C s T ) ] e + u d, ) j ss = ɛ ab ɛ de s T a Cγ 5 b qd γ 5 C s T ) e, 1) where a,b,,... are olour indies, C is the harge onjugation atrix and q represents the quark u or d aording to the harge of the eson. Sine D sj has one s quark, we hoose the j s urrent to have the sae quantu nubers of D sj, whih is supposed to be an isosalar. However, sine we are working in the SU) liit, the isosalar and isovetor states are ass degenerate and, therefore, this partiular hoie has no relevane here. The QCDSR for the hared salar esons are onstruted fro the two-point orrelation funtion Πq)= i d 4 xe iq.x T [ j S x)j S )]. ) The oupling of the salar eson, S, to the salar urrent, j S, an be paraetrized in ters of the eson deay onstant f S as []: j S S = f S 4 S, therefore, the phenoenologial side of Eq. ) an be written as Π phen q ) = f S 8 S 3) +, S q where the dots denote higher resonane ontributions that will be paraetrized, as usual, through the introdution of the ontinuu threshold paraeter s [4]. In the OPE side we work at leading order and onsider ondensates up to diension six. We deal with the strange quark as a light one and onsider the diagras up to order s. To keep the har quark ass finite, we use the oentu-spae expression for the har quark propagator. We follow Ref. [5] and alulate the light quark part of the orrelation funtion in the oordinate-spae, whih is then Fourier transfored to the oentu spae in D diensions. The resulting light-quark part is obined with the harquark part before it is diensionally regularized at D = 4. We an write the orrelation funtion in the OPE side in ters of a dispersion relation: Π OPE q ) = ds ρs) s q, 4) where the spetral density is given by the iaginary part of the orrelation funtion: ρs) = 1 π I[ Π OPE s) ]. After aking a Borel transfor on both sides, and transferring the ontinuu ontribution to the OPE side, the su rule for the salar eson S an be written as f S 8 S e S /M = where s s/m ρ S s), ρ S s) = ρ pert s) + ρ s s) + ρ qq s) + ρ G s) + ρ ix s) + ρ qq s) + ρ G3 s), with ρ pert s) = 1 1 3π 6 ) 1 3 d s) 4, 5) 6)

3 M.E. Brao et al. / Physis Letters B 64 5) ρ G s) = g G 1 π 6 ρ G3 s) = g3 G 3 1 9π 6 d s)[ 9 + s) 1 1 )] 1 ) + 4, ) 1 3 d 3 s), ) 3 7) 8) whih are oon to all three resonanes and where the lower liit of the integrations is given by = /s. Froj we get: ρ s s) =, ρ qq s) = qq 6 π 4 ) 1 d s), ρ ix s) = qgσ.gq 6 π 4 [ 1 1 ) 1 d s) ρ qq s) = qq 1π d 1 s )], Fro j s we get: ρ s s) =, ρ qq s) = 1 6 π 4 [ qq d s). d 1 s ) 1 s + 9) 1) 11) ) ] + s ss, 1) ρ ix s) = 1 6 π 4 d s)[ s sgσ.gs 6 ) + qgσ.gq s 1 ln1 ) ))], 13) ρ qq s) = qq ss 1π Finally fro j ss we get ρ s s) = s 8 3π 6 ρ qq s) = 1 6 π 4 ρ ix s) = 1 6 π 4 [ ss [ sgσ.gs 1 d s). 1 d ) 3 s) 3, d 1 s ) 1 s d s) s 3 1 s )) 1 1 s qgσ.gq 1 ln1 ) )], ρ qq s) = qq ss 1π d s). 14) 15) ) ] s qq, 16) 17) 18) For the har quark propagator with two and three gluons attahed we use the oentu-spae expressions given in Ref. [6]. In order to get rid of the eson deay onstant and extrat the resonane ass, S, we first take the derivative of Eq. 5) with respet to 1/M and then we divide it by Eq. 5) to get s S = s/m sρ S s) s 19) s/m ρ S s). In the nuerial analysis of the su rules, the values used for the quark asses and ondensates are: s =.13 GeV, = 1. GeV, qq =.3) 3 GeV 3, ss =.8 qq, qgσ.gq = qq with =.8 GeV, g G =.5 GeV 4 and g 3 G 3 =.45 GeV 6. The value for the quark ondensate was obtained using the Gell-Mann Oakes Renner relation, and the ass of the light quarks, u + d =

4 M.E. Brao et al. / Physis Letters B 64 5) 17 Fig. 1. The D s) ass the lower dashed, solid and dotted lines) and the D 1s) ass the upper dashed, solid and dotted lines), as a funtion of the Borel ass for different values of the ontinuu threshold. Dashed lines: s =.6 GeV; solid lines: s =.7 GeV; dotted lines: s =.8 GeV. Fig.. The D s) ass as a funtion of the Borel ass for different values of the ontinuu threshold. Dashed line: s =.6 GeV; solid line: s =.7 GeV; dotted line: s =.8 GeV. 14 MeV, at the renoralization sale of 1 GeV [7]. Sine the har quark ass introdues a natural sale in the proble, we hose to work at the renoralizationsaleof 1GeV. We all D s), D 1s) and D s) the salar hared esons represented by j, j s and j ss in Eq. 1)), respetively. In Figs. 1 and we show the asses of Fig. 3. The D 1s) ass as a funtion of the Borel ass for different values of the ontinuu threshold and quark ondensate. Solid line: s =.6 GeV and qq 1 GeV) =.4 GeV) 3 ; dotted line: s =.8 GeV and qq 1 GeV) =. GeV) 3 ; dashed line: s =.6 GeV, qq GeV) =.67 GeV) 3 and s GeV) =.1 GeV. these three resonanes as a funtion of the Borel ass for different values of the ontinuu threshold. The Borel window was fixed in suh way that the pole ontribution is always between 8% and % of the total ontribution. Fixing s =.7 GeV and varying the har quark and the strange quark asses in the intervals: GeV and.11 s.15 GeV, we get results for the resonane asses still between the lower and upper lines in Figs. 1 and. A bigger value for the har quark ass akes the results ore stable as a funtion of the Borel ass. One an also vary the value of the quark ondensate. Keeping the ontinuu threshold and the quark asses fixed at s =.7 GeV, = 1. GeV and s =.13 GeV and varying the quark ondensate in the interval: qq =.3 ±.1 GeV) 3, we get a bigger saller) result for the resonane asses using a saller bigger) value of the ondensate. In Fig. 3 we show the ass of the D 1s) state, as a funtion of the Borel ass, for the obination of the values of the ontinuu threshold and quark ondensate that gives the lower and upper liits for the D 1s) ass. In Ref. [8] it was shown that the renoralization sale was an iportant soure of unertainty, in the analysis of the B eson deay onstant. To hek

5 M.E. Brao et al. / Physis Letters B 64 5) 17 1 how the hange of the sale would hange our results we also show, through the dashed line in Fig. 3, the result for the D 1s) resonane ass using the values of the strange quark ass and quark ondensate at the sale GeV: qq GeV) =.67 GeV) 3 and s GeV) =.1 GeV [8]. We see that we get a less stable result for the resonane ass, but it is still opatible with the results at the sale 1 GeV, onsidering the variation in the ontinuu threshold. Therefore, we onlude that it is the variation of the ontinuu threshold that auses the ost signifiant variations in the resonane asses, and it is our ost iportant soure of unertainty. Coparing Figs. 1 and we see that the D 1s) and D s) resonane asses are basially degenerated, while the ass of D s) is around 1 MeV saller than the others. While it is natural to expet that the inlusion of a strange quark would inrease the resonane ass by around the strange quark ass as was the ase when one goes fro D s) to D 1s) ), it is really interesting to observe that this does not happen when one goes fro D 1s) to D s). In ters of the OPE ontributions, we an trae this behavior to the fat that the quark ondensate ter is saller in D s) than in D 1s) due to the hange fro qq to ss ), however the inlusion of the ter proportional to s whih is not present in D 1s) ), opensates this derease. Considering the variations on the quark asses, the quark ondensate and on the ontinuu threshold disussed above, in the Borel window onsidered here our results for the resonane asses are given in Table 1. Coparing the results in Table 1 with the resonane asses given by BaBar, Belle and FOCUS: D sj + 317), D 38) and D,+ 45), we see that we an identify the four-quark states represented by D 1s) and D s) with the BaBar and Belle resonanes, respetively. However, we do not find a four-quark state whose ass is opatible with the FOCUS resonanes, D,+ 45). Therefore, we assoiate the FO- Table 1 Nuerial results for the resonane asses Resonane D s) D 1s) D s) Mass GeV). ±.1.3 ±.18.3 ±. CUS resonanes, D,+ 45), with a salar q state, sine its ass is opletely in agreeent with the preditions of the quark odel in Ref. [5]. Itisalso interesting to point out that a ass of about.4 GeV is also opatible with the QCD su rule alulation for a q salar eson [11]. One an still argue that while a pole approxiation is justified for the very narrow BaBar resonane, this ay not be the ase for the rather broad Belle and FOCUS resonanes. To hek if the width of the resonanes ould odify the pattern observed in the asses of the four-quark states, we have odified the phenoenologial side of the su rule, in Eq. 5), through the introdution of a Breit Wigner-type resonane for Π phen M ) = fs 8 s/m S ρ BW s), π + D ) where ρ BW s) = 1 π with Γs) = Γ s Γs) S s S ) + S Γs), λs, D, π ) λ S, S D, π ) ) 1) s, and λx,y,z) = x + y + z xy xz yz. Of ourse now we annot obtain an expression for the resonane ass as Eq. 19). However, we an still use the resonane ass as a paraeter to opare the opatibility between the right-hand side RHS) and the left-hand side LHS) of the su rule in Eq. ): s π + D ) s/m sρ BW s) s π + D ) s/m ρ BW s) = s s s/m sρ S s) s/m ρ S s). ) In Fig. 4 we show the RHS solid line) and the LHS of Eq. ) for D s), for three different values of the resonane ass, with Γ = 8 MeV and s =.7 GeV. We see that the best agreeent is obtained for S. GeV, whih shows that the inlusion of the width does not hange the value of the ass obtained for the resonane.

6 M.E. Brao et al. / Physis Letters B 64 5) 17 Referenes Fig. 4. The RHS solid line) and the LHS of the su rule in Eq. ) for D s), for different values of the resonane ass. Dashed line: S =.1 GeV; dotted line: S =. GeV; dot-dashed line: S =.3 GeV. We have presented a QCD su rule study of the hared salar esons onsidered as diquark antidiquark states. We found that the asses of the BaBar, D sj + 317), and Belle, D 38), resonanes an be reprodued by the four-quark states q) q s) and s)ū s), respetively. However, the ass of the FOCUS resonane, D,+ 45), whih we believe is not the sae easured by Belle, annot be reprodued in the four-quark state piture onsidered here. Therefore, we interpret it as a noral q state, sine its ass is in oplete agreeent with the preditions of the quark odel in Ref. [5]. We also obtain a ass of. GeV for a four-quark salar state q)ū d) whih was not yet observed, and that should be also rather broad. Aknowledgeents We would like to thank I. Bediaga for fruitful disussions. This work has been supported by CNPq and FAPESP. [1] BaBar Collaboration, B. Auber, et al., Phys. Rev. Lett. 9 3) 41; BaBar Collaboration, B. Auber, et al., Phys. Rev. D 69 4) [] CLEO Collaboration, D. Besson, et al., Phys. Rev. D 68 3) 3. [3] Belle Collaboration, P. Krokovny, et al., Phys. Rev. Lett. 91 3) 6. [4] FOCUS Collaboration, E.W. Vaandering, hep-ex/4644. [5] S. Godfrey, N. Isgur, Phys. Rev. D ) 189; S. Godfrey, R. Kokoshi, Phys. Rev. D ) [6] Belle Collaboration, K. Abe, et al., Phys. Rev. D 69 4) 11. [7] FOCUS Collaboration, J.M. Link, et al., Phys. Lett. B 586 4) 11. [8] Y.-B. Dai, C.-S. Huang, C. Liu, S.-L. Zhu, Phys. Rev. D 68 3) [9] G.S. Bali, Phys. Rev. D 68 3) 7151R). [1] A. Dougall, R.D. Kenway, C.M. Maynard, C. MNeile, Phys. Lett. B 569 3) 41. [11] A. Hayashigaki, K. Terasaki, hep-ph/ [1] S. Narison, Phys. Lett. B 65 5) 319. [13] T. Barnes, F.E. Close, H.J. Lipkin, Phys. Rev. D 68 3) 546. [14] A.P. Szzepaniak, Phys. Lett. B 567 3) 3. [15] E. van Beveren, G. Rupp, Phys. Rev. Lett. 91 3) 13. [16] H.-Y. Cheng, W.-S. Hou, Phys. Lett. B 566 3) 193. [17] K. Terasaki, Phys. Rev. D 68 3) 1151R). [18] L. Maiani, F. Piinini, A.D. Polosa, V. Riquer, Phys. Rev. D 71 5) 148. [19] T. Browder, S. Pakvasa, A.A. Petrov, Phys. Lett. B 578 4) 365. [] R.L. Jaffe, Phys. Rev. D ) 67; R.L. Jaffe, Phys. Rev. D ) 81; R.L. Jaffe, Phys. Rev. D ) [1] M.A. Shifan, A.I. Vainshtein, V.I. Zakharov, Nul. Phys. B ) 385. [] T.V. Brito, F.S. Navarra, M. Nielsen, M.E. Brao, Phys. Lett. B 68 5) 69. [3] R.L. Jaffe, F. Wilzek, Phys. Rev. Lett. 91 3) 33. [4] B.L. Ioffe, Nul. Phys. B ) 317; B.L. Ioffe, Nul. Phys. B ) 591, Erratu. [5] H. Ki, S.H. Lee, Y. Oh, Phys. Lett. B 595 4) 93. [6] L.J. Reinders, H. Rubinstein, S. Yazaky, Phys. Rep ) 1. [7] J. Gasser, H. Leutwyler, Phys. Rep ) 77. [8] M. Jain, B. Lange, Phys. Rev. D 65 ) 563.