D_{sJ}^+(2317)\to D_s^+\pi^0 decay width

We use the QCD sum rules to analize the hadronic decay $D_{sJ}^{+}(2317)\to D_s^+\pi^0$, in the hypotesis that the $D_{sJ}^{+}(2317)$ can be identified as a four-quark state. We use a diquak-antidiquark current and work to the order of $m_s$ in full QCD, without relying on $1/m_c$ expansion. We find that the partial decay width of the hadronic isospin violating mode is proportional to the isovector quark condensate, $\langle0|\bar{d}d-\bar{u}u|0\rangle$. The estimated partial decay width is of the order of 6 keV.


Marina Nielsen
Instituto de Física, Universidade de São Paulo, C.P. 66318, 05389-970 São Paulo, SP, Brazil We use the QCD sum rules to analize the hadronic decay D + sJ (2317) → D + s π 0 , in the hypotesis that the D + sJ (2317) can be identified as a four-quark state. We use a diquak-antidiquark current and work to the order of ms in full QCD, without relying on 1/mc expansion. We find that the partial decay width of the hadronic isospin violating mode is proportional to the isovector quark condensate, 0|dd −ūu|0 . The estimated partial decay width is of the order of 6 keV.
Because of their low masses, these two states are lower than the DK and D * K thresholds. Therefore, their strong decays must proceed through isospin violating effects. There have been some discussions of their decays within the quark model [6,13,14,15,20] and QCD sum rules [16]. In all these studies but [14], the isospin violating effects were considered through the η − π 0 mixing. However, if these mesons are considered as four-quark states, in a QCD sum rule calculation, the isospin violating effects can be introduced through the mass and quark condensate difference between the u and d quarks.
In a recent calculation [23] the scalar-isoscalar meson D + sJ (2317) were considered as a S-wave bound state of a diquark-antidiquark pair. As suggested in ref. [28], the diquark was taken to be a spin zero colour anti-triplet. The corresponding interpolating field is: where a, b, c, ... are colour indices and C is the charge conjugation matrix. In ref. [23], using the QCD sum rule (QCDSR) formalism [29,30,31], it was shown that it is possible to reproduce the experimental mass of the meson D + sJ (2317) using this four-quark state picture. Here, we extend the calculation done in ref. [23] to study the vertex associated with the decay D + sJ (2317) → D + s π 0 . The QCDSR calculation for the vertex, D + sJ (2317)D + s π 0 , centers around the three-point function given by where p = p ′ + q and the interpolating fields for the pion and D s mesons are given by: The fundamental assumption of the QCD sum rule approach is the principle of duality. Specifically, we assume that there is an interval over which the above vertex function may be equivalently described at both, the quark level and at the hadron level. Therefore, the underlying procedure of the QCD sum rule technique is the following: on one hand we calculate the vertex function at the quark level in terms of quark and gluon fields. On the other hand, the vertex function is calculated at the hadronic level introducing hadron characteristics such as masses and coupling constants. At the quark level the complex structure of the QCD vacuum leads us to employ the Wilson's operator product expansion (OPE). The calculation of the phenomenological side proceeds by inserting intermediate states for D s , π 0 and D sJ , and by using the definitions: We obtain the following relation: where the coupling constant g DsJ Dsπ is defined by the on-mass-shell matrix element The continuum contribution in Eq.(5) contains the contributions of all possible excited states. For 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 [11,30,32,33], was done in ref. [34]. In Table I we show the results obtained for the different vertices studied in ref. [34], as well as the experimental values.
From Table I we see that, although not exactly in between the experimental error bars, the hadronic couplings determined from the QCD sum rule calculation are consistent with existing experimental data. The biggest discrepancy is for g f0π + π − and this can be understood since the f 0 → π + π − decay is mediated by gluon exchange and, therefore, probably in this case α s corrections, which were not considered, could play an important role.
Here, we follow ref. [34] and work at the pion pole. The main reason for working at the pion pole is that the matrix element in Eq.(6) defines the coupling constant only at the pion pole. For q 2 = 0 one would have to replace g DsJ Dsπ , in Eq.(6), by the form factor g DsJ Dsπ (q 2 ) and, therefore, one would have to deal with the complications associated with the extrapolation of the form factor [35,36]. The pion pole method consists in neglecting the pion mass in the denominator of Eq. (5) and working at q 2 = 0. In the OPE side one singles out the leading terms in the operator product expansion of Eq.(2) that match the 1/q 2 term. In the phenomenological side, in the structure qµ q 2 we get: In Eq. (7), ρ cont (p 2 , u), gives the continuum contributions, which can be parametrized as ρ cont (p 2 , u) = b(u) s0−p 2 Θ(u − u 0 ) [37,38], with s 0 and u 0 being the continuum thresholds for D sJ and D s respectively. Since we are working at q 2 = 0, we take the limit p 2 = p ′ 2 and we apply the Borel transformation to p 2 → M 2 to get: where stands for the pole-continuum transitions and pure continuum contributions. For simplicity, one assumes that the pure continuum contribution to the spectral density, ρ cc (u), is given by the result obtained in the OPE side. Asymptotic freedom ensures that equivalence for sufficiently large u. Therefore, one uses the ansatz: ρ cc (u) = ρ OP E (u). In Eq.(8), A is a parameter which, together with g DsJ Dsπ , have to be determined by the sum rule.
In the OPE side we work at leading order and deal with the strange quark as a light one and consider the diagrams up to order m s . To keep the charm quark mass finite, we use the momentum-space expression for the charm quark propagator. We calculate the light quark part of the correlation function in the coordinate-space, which is then Fourier transformed to the momentum space in D dimensions. The resulting light-quark part is combined with the charm-quark part before it is dimensionally regularized at D = 4. Singling out the leading terms proportional to q µ /q 2 , we can write the Borel transform of the correlation function in the OPE side in terms of a dispersion relation: where the spectral density, ρ OP E , is given by the imaginary part of the correlation function. Transferring the pure continuum contribution to the OPE side, the sum rule for the coupling constant, up to dimension 7, is given by: with In Eq. (11), γ measures the isospin symmetry breaking in the quark condensate: and is the source of nonperturbative isospin violation in the OPE of the vertex function. The value of γ has been estimated in a variety of approaches [32,39,40], with results varying over almost one order of magnitude: −1×10 −2 ≤ γ ≤ −2 × 10 −3 . A more recent calculation [41] gives a bigger (in module) value: γ = −2.6 × 10 −2 .
Fixing √ s 0 = 2.7 GeV and varying the quark condensate, the charm quark and the strange quark masses in the intervals: −(0.24) 3 ≤ qq ≤ −(0.22) 3 GeV 3 , 1.1 ≤ m c ≤ 1.3 GeV and 0.11 ≤ m s ≤ 0.15 GeV, we get results for the coupling constant still between the lower and upper limits given above. However, varying the value of γ form −1 × 10 −2 to the more recent value given in [41]: γ = −2.6 × 10 −2 , and keeping the other parameters fixed we get g DsJ Dsπ = 130 MeV. On the other hand, if we use the smallest (in module) value allowed for γ: γ = −2. × 10 −3 , we get g DsJ Dsπ = 10 MeV. Therefore, the biggest source of uncertainty in our calculation is the value of γ. In all cases considered here, the quality of the fit between the LHS and the RHS of Eq.(11) is similar to the one shown in Fig.1.
The coupling constant, g DsJ Dsπ , is related with the partial decay width through the relation: where λ(a, b, c) = a 2 + b 2 + c 2 − 2ab − 2ac − 2bc. Considering γ = −1 × 10 −2 , which was the value found in [40] to be consistent with the neutron-proton mass difference in a QCDSR calculation, and allowing qq , m c , m s and s 0 to vary in the ranges discussed above we get: However, it is important to state that, if the value for γ found in ref. [41] proves to be correct, then the partial decay width could be as large as Γ(D + sJ (2317) → D + s π 0 ) ∼ 40 keV, in agreement with the QCDSR calculation done in ref. [16], where the meson D + sJ (2317) is considered as a ordinary cs state. In Table II we show the partial decay width obtained by different theoretical groups. The first five calculations assume a cs picture for D + sJ (2317), while the last two assume a four-quark picture for it. From the results in Table II we see that we can not get a definitive answer about the structure of the D sJ (2317) meson from its strong decay width, since in both pictures: ordinary cs or four-quark states, different approachs can give results varying from a few keV to a hundred keV.
We have presented a QCD sum rule study of the vertex function associated with the strong decay D + sJ (2317) → D + s π 0 , where the D + sJ (2317) meson was considered as diquark-antidiquark state. We found that the source of isospin violation in our calculation is the parameter γ = 0|dd−ūu|0 0|ūu|0 , which measures the isospin symmetry breaking in the quark condensate. Since, in our approach, the partial decay width is directly proportional to γ 2 , and since there is a large uncertainty in the value of γ, considering γ in the range −2.6 × 10 −2 ≤ γ ≤ −2 × 10 −3 we get the partial decay width in the range 0.2 keV ≤ Γ(D + sJ (2317) → D + s π 0 ) ≤ 40 keV. However, from other QCDSR calculation, we believe that the value of γ is ∼ −1 × 10 −2 , which gives the result shown in Eq. (16).
As a final remark we would like to point out that if, instead of using a isoscalar current, we have used a isovector current for D sJ (2317) (as suggested in ref. [27]), the difference in Eq.(11) would be a factor 2 in the place of γ. In this case the decay would be isospin allowed and the partial width would be ∼ 230 MeV, much bigger than the experimental upper limit to the total width Γ ∼ 5 MeV.