Sep 19, 2006 - sJ (2317) and X(3872) considering them as four-quark states in a diquark-antidiquark configuration. The results obtained for their mass...

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Instituto de F´ısica, Universidade de S˜ ao Paulo, C.P. 66318, 05389-970 S˜ ao Paulo, SP, Brazil 2 Instituto de F´ısica, Universidade do Estado do Rio de Janeiro, Rua S˜ ao Francisco Xavier 524, 20550-900 Rio de Janeiro, RJ, Brazil

arXiv:hep-ph/0609184v1 19 Sep 2006

Using the QCD sum rule approach we investigate the possible four-quark structure of the recently + observed mesons DsJ (2317), firstly observed by BaBaR, X(3872), firstly observed by BELLE and ∗0 D0 (2308) observed by BELLE. We use diquark-antidiquark currents and work in full QCD, without relying on 1/mc expansion. Our results indicate that a four-quark structure is acceptable for these mesons. + + The recent observations of the very narrow resonances DsJ (2317) by BaBar [1], DsJ (2460) by CLEO [2], and X(3872) by BELLE [3], all of them with masses below quark model predictions, have stimulated a renewed interest in the spectroscopy of open charm and charmonium states. Due to their narrowness and small masses, these new mesons were considered as good candidates for four-quark states by many authors [4]. The idea of mesons as four-quark states is not new. Indeed, even Gell-Mann in his first work about quarks had mentioned that mesons could be made out of (q q¯), (qq q¯q¯) etc. [5]. The best known example of applying the idea of four-quark states for mesons is for the light scalar mesons (the isoscalars σ(500), f0 (980), the isodublet κ(800) and the isovector a0 (980)) [6, 7]. In a four-quark scenario, the mass degeneracy of f0 (980) and a0 (980) is natural, the mass hierarchy pattern of the nonet is understandable, and it is easy to explain why σ and κ are broader than f0 (980) and a0 (980). In refs. [8, 9] the method of QCD sum rules (QCDSR) [10, 11, 12] was used to study the two-point functions for the + mesons DsJ (2317) and X(3872) considering them as four-quark states in a diquark-antidiquark configuration. The results obtained for their masses are given in Table I.

Table I: Numerical results for the resonance masses resonance DsJ X mass (GeV) 2.32 ± 0.13 3.93 ± 0.15 + Comparing the results in Table I with the resonance masses given by: DsJ (2317) and X(3872), we see that it is possible to reproduce the experimental value of the masses using a four-quark representation for these states. + The study of the three-point functions related to the decay widths DsJ (2317) → Ds+ π 0 and X(3872) → Jψπ + π − , using the diquark-antidiquark configuration for DsJ and X, was done in refs. [13, 14]. The results obtained for their partial decay widths are given in Table II.

Table II: Numerical results for the resonance decay widths + decay DsJ → Ds+ π 0 X → J/ψπ + π − Γ (MeV) (6 ± 2) × 10−3 50 ± 15 exp Γtot (MeV) <5 < 2.3

+ From Table II we see that the partial decay width obtained in ref. [13], supposing that the mesons DsJ (2317) is a four-quark state, is consistent with the experimental upper limit. However, in the case of the meson X(3872), the partial decay width obtained in ref. [14] is much bigger than the experimental upper limit to the total width. In ref. [14] some arguments were presented to reduce the value of this decay width, by imposing that the initial four-quark state needs to have a non-trivial color structure. In this case, its partial decay width can be reduced to Γ(X → J/ψπ + π − )) = (0.7 ± 0.2) MeV. However, that procedure may appear somewhat unjustified and, therefore, more study is needed until one can arrive at a definitive conclusion about the structure of the meson X(3872). + In ref. [8], besides the four-quark state (cq)(¯ sq¯) representing the meson DsJ (2317), it was also studied the con¯ figuration (cq)(¯ ud) associated with a possible scalar meson that we will call D(0+ ) (the 0+ stands for J P ). The mass obtained for this state is: mD(0+ ) = (2.22 ± 0.21) MeV, in a very good agreement with the prediction made in ref. [15] for the D(0+ ) scalar meson: mD(0+ ) = (2.215 ± 0.002) MeV. This value was obtained in ref. [15] by supposing that the meson D(0+ ) is the chiral partner of the meson D, with the same mass difference as the chiral + + pair DsJ (2317) − Ds . The authors of ref. [15] have also evaluated the decay widths DsJ → Ds+ π 0 and D(0+ ) → Dπ ± + + 0 2 + ± 2 obtaining: Γ(DsJ → Ds π ) = 21.5GA keV and Γ(D(0 ) → Dπ ) = 326GA MeV, where they expect GA ∼ 1. Here, we extend the calculation done in refs. [8, 13] to study the vertex associated with the decay D0 (0+ ) → D+ π − .

The QCDSR calculation for the vertex, D0 (0+ )D+ π − , centers around the three-point function given by Z ′ ′ Tµ (p, p , q) = d4 xd4 y eip .x eiq.y h0|T [jD (x)j5µ (y)j0† (0)]|0i, where j0 is the interpolating field for the scalar D0 (0+ ) meson [8]: ud γ5 C d¯Te ) , j0 = ǫabc ǫdec (dTa Cγ5 cb )(¯

(1)

(2)

where a, b, c, ... are colour indices and C is the charge conjugation matrix. In Eq. (1), p = p′ +q and the interpolating fields for the π − and D+ mesons are given by: j5µ = u ¯a γµ γ5 da ,

jD = id¯a γ5 ca .

(3)

The calculation of the phenomenological side proceeds by inserting intermediate states for D, π and D(0+ ), and m2D fD by using the definitions: h0|j5µ |π(q)i = iqµ Fπ , h0|jD |D(p′ )i = m , h0|j0 |D(0+ )(p)i = λ0 . We obtain the following c relation: Tµphen (p, p′ , q) =

λ0 m2D fD Fπ gD(0+ )Dπ /mc (p2 − m2D(0+ ) )(p′ 2 − m2D )(q 2 − m2π )

qµ + continuum contribution ,

(4)

where the coupling constant, gD(0+ )Dπ , is defined by the on-mass-shell matrix element: hDπ|D(0+ )i = gD(0+ )Dπ . The continuum contribution in Eq.(4) contains the contributions of all possible excited states. In the case of the light scalar mesons, considered as diquark-antidiquark states, the study of their vertices functions using the QCD sum rule approach at the pion pole [11, 12, 16], was done in ref.[17]. It was shown that the decay widths determined from the QCD sum rule calculation are consistent with existing experimental data. Here, we follow refs. [13, 17] and work at the pion pole. The main reason for working at the pion pole is that one does not have to deal with the complications associated with the extrapolation of the form factor [18]. The pion pole method consists in neglecting the pion mass in the denominator of Eq. (4) and working at q 2 = 0. In the OPE side one singles out the leading terms in the operator product expansion of Eq.(1) that match the 1/q 2 term. Since we are working at q 2 = 0, 2 2 we take the limit p2 = p′ and we apply a single Borel transformation to p2 , p′ → M 2 . In the phenomenological side, in the structure qµ we get [13]: T

phen

Z ∞ λ0 m2D fD Fπ gD(0+ )Dπ −m2 /M 2 2 −m2D(0+ ) /M 2 −s0 /M 2 D e −e (M ) = ρcc (u) e−u/M du, +A e + 2 2 mc (mD(0+ ) − mD ) u0 2

(5)

where A and ρcc (u) stands for the pole-continuum transitions and pure continuum contributions, with s0 and u0 being the continuum thresholds for D(0+ ) and D respectively. For simplicity, one assumes that the pure continuum contribution to the spectral density, ρcc (u), is given by the result obtained in the OPE side. Therefore, one uses the ansatz: ρcc (u) = ρOP E (u). In Eq.(5), A is a parameter which, together with gD(0+ )Dπ , has to be determined by the sum rule. In the OPE side we single out the leading terms proportional to qµ /q 2 . Transferring the pure continuum contribution to the OPE side, the sum rule for the coupling constant, up to dimension 7, is given by: " # 2 Z u0 1 mc h¯ q qi −m2c /M 2 m2c −m2D(0+ ) /M 2 −s0 /M 2 −m2D /M 2 −u/M 2 +A e = 2h¯ q qi 4 2 C e −e , (6) e − du e u 1− 2 π m2c u 6 with C=

λ0 m2D fD Fπ g + . mc (m2D(0+ ) − m2D ) D(0 )Dπ

(7)

In the numerical analysis of the sum rules, the values used for the meson masses, quark masses and condensates are: mD(0+ ) = 2.2 GeV, mD = 1.87 GeV, mc = 1.2 GeV, h¯ q qi = −(0.23)3 GeV3 . For the meson decay constants √ coupling, λ0 , we we use Fπ = 2 93 MeV and fD = 0.20 GeV [19]. We use u0 = 6 GeV2 and for the current meson √ are going to use the result obtained from the two-point function in ref. [8]. Considering 2.6 ≤ s0 ≤ 2.8 GeV we get λ0 = (3.3 ± 0.3) × 10−3 GeV5 . In Fig. 1 we show, through the dots, the right-hand side (RHS) of Eq.(6) as a function of the Borel mass. We use the same Borel window as defined in ref.[8]. To determine gD(0+ )Dπ we fit the QCDSR results with the analytical

7

RHS X LHS (GeV )

1.5e−04

1.0e−04

5.0e−05

0.0e+00

1

1.2

1.4 1.6 2 2 M (GeV )

1.8

2

FIG. 1: Dots: the RHS of Eq.(6), as a function of the Borel mass. The solid line gives the fit of the QCDSR results through the LHS of Eq.(6).

√ expression in the left-hand side (LHS) of Eq.(6). Using s0 = 2.7 GeV we get: C = 1.25 × 10−3 GeV7 and 7 A = 1.47 × 10−3 GeV . Using the definition of C in Eq.(7) and λ0 = 3.3 × 10−3 GeV5 (the value obtained for √ √ s0 = 2.7 GeV) we get gD(0+ )Dπ = 6.94 GeV. Allowing s0 to vary in the interval 2.6 ≤ s0 ≤ 2.8 GeV, the corresponding variation obtained for the coupling constant is 5 GeV ≤ gD(0+ )Dπ ≤ 7.5 GeV. The coupling constant, gD(0+ )Dπ , is related to the partial decay width through the relation: Γ(D0 (0+ ) → D+ π − ) =

q 1 2 g λ(m2D(0+ ) , m2D , m2π ), + 16πm3D(0+ ) D(0 )Dπ

(8)

where λ(a, b, c) = a2 + b2 + c2 − 2ab − 2ac − 2bc. Allowing s0 to vary in the range discussed above we get: Γ(D0 (0+ ) → D+ π − ) = (120 ± 20) MeV.

(9)

In Table III we show the partial decay width obtained in ref. [15], in ref. [13] and here for different decays. From the results in Table III we see that if one uses GA = 0.6, the result presented here and the result in ref. [13] are consistent with the results presented in ref. [15] for both decays. Table III: Numerical results for the resonance partial decay widths from different approaches decay ref. [15] ref. [13] this work + DsJ → Ds+ π 0 21.5 G2A keV (6 ± 2) keV D0 (0+ ) → D+ π − 326 G2A MeV 120 ± 20 MeV

It is important to notice that the BELLE Collaboration [20] has reported the observation of a rather broad scalar meson D0∗0 (2308) in the decay mode D0∗0 (2308) → D+ π − with a total width Γ ∼ 270 MeV. Although both, the mass and the total decay width reported in [20], are bigger than the values found for the meson D(0+ ) studied here, we can not discard the possibility that the BELLE’s resonance can be interpreted as a four-quark state. We have presented a QCD sum rule study of the vertex function associated with the strong decay D0 (0+ ) → D+ π − , where the scalar D(0+ ) meson was considered as diquark-antidiquark state. We get for the partial decay width: Γ(D0 (0+ ) → D+ π − ) = (120 ± 20) MeV. Acknowledgements: This work has been supported by CNPq and FAPESP.

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