Analysis of the strong decay $X(5568) \to B_s^0\pi^+$ with QCD sum rules

In this article, we take the $X(5568)$ to be the scalar diquark-antidiquark type tetraquark state, study the hadronic coupling constant $g_{XB_s\pi}$ with the three-point QCD sum rules by carrying out the operator product expansion up to the vacuum condensates of dimension-6 and including both the connected and disconnected Feynman diagrams, then calculate the partial decay width of the strong decay $ X(5568) \to B_s^0 \pi^+$ and obtain the value $\Gamma_X=\left(20.5\pm8.1\right)\,\rm{MeV}$, which is consistent with the experimental data $\Gamma_X = \left(21.9 \pm 6.4 {}^{+5.0}_{-2.5}\right)\,\rm{MeV}$ from the D0 collaboration.

The calculations based on the QCD sum rules indicate that both the scalar-diquark-scalarantidiquark type and axialvector-diquark-axialvector-antidiquark type interpolating currents can give satisfactory mass m X to reproduce the experimental data [2,3,4,8]. In Ref. [9], Agaev, Azizi and Sundu choose the axialvector-diquark-axialvector-antidiquark type interpolating current, calculate the hadronic coupling constant g XBsπ with the light-cone QCD sum rules in conjunction with the soft-π approximation and other approximations, and obtain the partial decay width for the process X(5568) → B 0 s π + . In Ref. [7], Dias et al choose the scalar-diquark-scalar-antidiquark type interpolating current, calculate the hadronic coupling constant g XBsπ with the three-point QCD sum rules in the soft-π limit by taking into account only the connected Feynman diagrams in the leading order approximation, and obtain the partial decay width for the decay X(5568) → B 0 s π + . In previous work [2], we choose the scalar-diquark-scalar-antidiquark type interpolating current to study the mass of the X(5568) with the QCD sum rules. In this article, we extend our previous work to study the hadronic coupling constant g XBsπ with the three-point QCD sum rules by carrying out the operator product expansion up to the vacuum condensates of dimension-6 and including both the connected and disconnected Feynman diagrams, then calculate the partial decay width of the strong decay X(5568) → B 0 s π + . The article is arranged as follows: we derive the QCD sum rule for the hadronic coupling constant g XBsπ in Sect.2; in Sect.3, we present the numerical results and discussions; and Sect.4 is reserved for our conclusion.
2 QCD sum rule for the hadronic coupling constant g XB s π We can study the strong decay X(5568) → B 0 s π + with the three-point correlation function Π(p, q), where the currents interpolate the mesons B s , π and X(5568), respectively, the i, j, k, m, n are color indexes, the C is the charge conjugation matrix. In Ref. [7], the axialvector current is used to interpolate the π meson.
At the hadron side, we insert a complete set of intermediate hadronic states with the same quantum numbers as the current operators J Bs (x), J π (y) and J X (0) into the three-point correlation function Π(p, q) and isolate the ground state contributions to obtain the following result, where p ′ = p + q, the f Bs , f π and λ X are the decay constants of the mesons B s , π and X(5568), respectively, the g XBsπ is the hadronic coupling constant. In the following, we write down the definitions, The two unknown functions ρ Xπ (p 2 , t, p ′2 ) and ρ XBs (t, q 2 , p ′2 ) have complex dependence on the transitions between the ground state X(5568) and the excited states of the π and B s mesons, respectively. We introduce the parameters C Xπ and C XBs to parameterize the net effects, and rewrite the correlation function Π(p, q) into the following form, We set p ′2 = p 2 and take the double Borel transform with respect to the variable P 2 = −p 2 and Q 2 = −q 2 respectively to obtain the QCD sum rule at the left side (LS), In calculations, we neglect the dependencies of the C Xπ and C XBs on the variables p 2 , p ′2 , q 2 therefore the dependencies of the C Xπ and C XBs on the variables M 2 1 and M 2 2 , take the C Xπ and C XBs as free parameters, and choose the suitable values to eliminate the contaminations so as to obtain the stable sum rules with the variations of the Borel parameters [12,13]. Now we carry out the operator product expansion at the large Euclidean space-time region −p 2 → ∞ and −q 2 → ∞, take into account the vacuum condensates up to dimension 6 and neglect the contribution of the three-gluon condensate, as the three-gluon condensate is the vacuum expectation of the operator of the order O(α 3/2 s ). In other words, we calculate the Feynman diagrams shown in Fig.1. For example, the first diagram is calculated in the following ways, The operator product expansion converges for large −p 2 and −q 2 , it is odd to take the limit q 2 → 0. Then we set p ′2 = p 2 , take the quark-hadron duality below the continuum thresholds, and perform the double Borel transform with respect to the variables P 2 = −p 2 and Q 2 = −q 2 respectively to obtain the perturbative term, where the s 0 and u 0 are the continuum threshold parameters for the X(5568) and π, respectively. Other Feynman diagrams are calculated in analogous ways, finally we obtain the QCD sum rules at the right side (RS), The terms qq ss disappear after performing the double Borel transform, the last Feynman diagram in Fig.1 have no contribution.
In Refs. [13,14], the width of the Z c (4200) is studied with the three-point QCD sum rules by including both the connected and disconnected Feynman diagrams, which is contrary to Ref. [15], where only the connected Feynman diagrams are taken into account to study the width of the Z c (3900). In this article, the contributions come from the connected diagrams can be written as RS c , which is too small to account for the experimental data [1]. Finally, we obtain the QCD sum rule, There appear some energy scale dependence at the hadron side (or LS) of the QCD sum rule according to the factors m u + m d and m b + m s , we can eliminate the energy scale dependence by using the currents J Bs (x) and J π (y), then and the resulting QCD sum rule at the right side also acquires a factor (m b + m s ) (m u + m d ), a equivalent QCD sum rule is obtained, the predicted hadronic coupling constant g XBsπ is not changed. We can also study the strong decay X(5568) → B 0 s π + with the three-point correlation function Π µν (p, q), where the currents interpolate the mesons B s and π, respectively. At the hadron side, we insert a complete set of intermediate hadronic states with the same quantum numbers as the current operators ηs b µ (x) and ηū d ν (y) into the three-point correlation function Π µν (p, q) and isolate the ground state contributions to obtain the following result, where p ′ = p + q, the f Bs1 , f Bs , f a1 and f π are the decay constants of the mesons B s1 (5830), B s , a 1 (1260) and π, respectively, the g XBs1π and g XBs1a1 are the hadronic coupling constants. In the following, we write down the definitions, where the ε µ and ǫ ν are polarization vectors of the axialvector mesons B s1 (5830) and a 1 (1260), respectively. From the values m X = 5567.8 ± 2.9 +0.9 which corresponds to taking the pseudoscalar currents J Bs (x) and J π (y) according to the following identities, The axialvector currents ηs b µ (x) and ηū d ν (y) can also be chosen to study the strong decay X(5568) → B 0 s π + . We also expect to study the strong decay X(5568) → B 0 s π + with the light-cone QCD sum rules using the two-point correlation function Π(p, q), where the π(q)| is an external π state. At the QCD side, we obtain the following result after performing the wick's contraction, where the S kl s (−x) and S ln b (x) are the full s and b quark propagators, respectively. The u andd quarks stay at the same point x = 0, the light-cone distribution amplitudes of the π meson are almost useless, the integrals over the π meson's light-cone distribution amplitudes reduce to overall normalization factors. In the light-cone QCD sum rules, such a situation is possible only in the soft pion limit q → 0, and the light-cone expansion reduces to the short-distance expansion [17]. In Ref. [9], Agaev, Azizi and Sundu take the soft pion limit q → 0, and choose the Cγ µ ⊗ γ µ C type current to interpolate the X(5568), and use the light-cone QCD sum rules to study the strong decay X(5568) → B 0 s π + . The light-cone QCD sum rules are reasonable only in the soft pion approximation.
The unknown parameter is chosen as C XBs = −0.00059 GeV 8 . There appears a platform in the region M 2 1 = (4.5 − 5.5) GeV 2 . Now we take into account the uncertainties of the input parameters and obtain the value of the hadronic coupling constant g XBsπ , which is shown explicitly in Fig.2, Now we obtain the partial decay width, The decays X(5568) → B +K 0 are kinematically forbidden, so the width Γ X can be saturated by the partial decay width Γ X(5568) → B 0 s π + , which is consistent with the experimental value Γ X = 21.9 ± 6.4 +5.0 −2.5 MeV from the D0 collaboration [1]. The present work favors assigning the X(5568) to be the scalar diquark-antidiquark type tetraquark state.

Conclusion
In this article, we take the X(5568) to be the scalar diquark-antidiquark type tetraquark state, study the hadronic coupling constant g XBsπ with the three-point QCD sum rules, then calculate the partial decay width of the strong decay X(5568) → B 0 s π + and obtain the value Γ X = (20.5 ± 8.1) MeV, which is consistent with the experimental data Γ X = 21.9 ± 6.4 +5.0 −2.5 MeV from the D0 collaboration. In calculation, we carry out the operator product expansion up to the vacuum condensates of dimension-6, and take into account both the connected and disconnected Feynman diagrams. The present prediction favors assigning the X(5568) to be the diquark-antidiquark type tetraquark state with J P = 0 + . However, the quantum numbers J P = 1 + cannot be excluded according to decays X(5568) → B * s π + → B 0 s π + γ, where the low-energy photon is not detected.