Weak decays of doubly heavy baryons:"decay constants"

Inspired by the recent observation of the $\Xi_{cc}^{++}$ by LHCb collaboration, we explore the"decay constants"of doubly heavy baryons in the framework of QCD sum rules. With the $\Xi_{cc}, \Xi_{bc}, \Xi_{bb}$, and $\Omega_{cc}, \Omega_{bc}, \Omega_{bb}$ baryons interpolated by three-quark operators, we calculate the correlation functions using the operator product expansion and include the contribution from operators up to dimension six. On the hadron side, we consider both contributions from the lowest-lying states with $J^P=1/2^+$ and from negative parity baryons with $J^P=1/2^-$. We find that the results are stable and the contaminations from negative parity baryons are not severe. These results are ingredients for the QCD study of weak decays and other properties of doubly-heavy baryons.


I. INTRODUCTION
It is widely believed that doubly heavy baryons with two charm and/or bottom quarks exist in reality, but their experimental search has been a while. SELEX collaboration first reported the discovery of Ξ + cc in the Λ + c K − π + final state sixteen years ago [1,2], with the mass measured as m Ξ + cc = (3519 ± 1) MeV [1,2]. However, the SELEX-like Ξ + cc signal is not confirmed by later experiments [3][4][5][6][7]. In 2017, in the Λ + c K − π + π + final state the LHCb collaboration has observed the doubly charmed baryon Ξ ++ cc with the mass [8]: m Ξ ++ cc = (3621.40 ± 0.72 ± 0.27 ± 0.14) MeV. (1) corresponding quark operators. The correlation function of these operators can be handled using the operator product expansion (OPE), where the short-distance coefficients and long-distance quark-gluon interactions are separated. The former are calculable in QCD perturbation theory, whereas the latter can be parameterized in terms of vacuum condensates. The QCD result is then matched, via dispersion relation, onto the observable characteristics of hadronic states. Due to various advantages, the QCDSR has been used to calculate the masses of doubly heavy baryons in Refs. [9,25,[41][42][43][44][45][46]. The main motif of this work is to study "decay constants" using the QCDSR.
The "decay constants" defined by the interpolating current are mandatory inputs for studies of other properties of doubly heavy baryons in QCDSR, for example the heavy-to-light transition form factors.
The rest of the paper is arranged as follows. In Sec. II, we will present the calculation of correlation function in QCD sum rules, including the explicit expressions of the spectral functions.
We include both the contributions from the J P = 1/2 + baryons and the contamination from the J P = 1/2 − baryons. Sec. III is devoted to the numerical results. A summary is presented in the last section.

II. QCD SUM RULES STUDY
A doubly heavy baryon is made of two heavy quarks and one light quark. The quantum numbers and quark contents for the ground states are given in Table I. In this work we will study the J P = 1/2 + baryons which can only weakly decay.

A. QCD Sum rules with only positive parity baryons
The interpolating current for the Ξ QQ and Ω QQ is chosen as where Q = c or Q = b. For the Ξ bc and Ω bc , we choose In the above equations, we have considered the s π h = 1 + baryons only. The QCDSR analysis starts with the two-point correlator: where the interpolating current has been given in the above, and J is defined as A Lorentz structure analysis implies that the two-point correlation function has the form: On the hadronic side, one can insert the complete set of hadronic states into the correlator and then the correlator can be expressed as a dispersion integral over a physical spectral function: where H can be a ground-state doubly heavy baryon and m H denotes its mass. In obtaining the above expression, the polarization summation for spinors has been used: The pole residue λ H is defined as The mass dimension for λ H is 3, while in analogy with the meson case, it is convenient to use the "decay constant" with the definition In the OPE side, we will work at leading order in α s in this work and include the condensate contributions up to dimension six. The full propagator for the heavy quark is given as with where t n = λ n /2 and λ n is the Gell-Mann matrix, and the i, j are the color indices. The full propagator for light quarks is given as With the quark propagators one can express the correlation function in terms of a dispersion relation as: where the spectral density is given by the imaginary part of the correlation function: After equating the two expressions for Π(q 2 ) based on the quark-hadron duality, and making a Borel transformation, we can write the sum rules as The spectral functions ρ 1 and ρ 2 are given as follows: with The integration limits are given by . For the Ω QQ ′ , one needs to replace the condensates correspondingly. The integration lower bound In Ref. [41], the authors obtained a similar expression with our Eq. (19): A few remarks are in order.
• We did not include the mass corrected quark condensate. This might has some impact in the case of Ω cc,bc,bb .
• However the gluon condensate contribution, which is anticipated more important, is missing in Eq. (25).
• It should be noted that in the massless limit, we have the spectral function: Our result is fully consistent with Ref. [47]: • In Ref. [41], the predicted mass m Ξcc = (4.26 ± 0.19) GeV is much larger than the experimental data m exp Ξ ++ cc = 3.621 GeV.

B. QCD Sum rules with both positive and negative parity baryons
In the above analysis, only the 1/2 + baryons are considered. An interpolating current for the negative parity 1/2 − baryon can be defined as where J + is given in Eqs. (2)(3)(4)(5). When the complete set of hadron states is inserted to the correlation function in Eq. (6), both the positive and the negative parity single-particle states can contribute [48,49].
When taking into account the 1/2 − single-particle states, Eq. (9) is rewritten as where λ ± (m ± ) stands for the "decay constant" (mass) of positive or negative parity baryons.
Apparently, the λ + is the "decay constant" λ H we have defined in Eq. (11). The λ − is defined as At the hadronic level, one can take the imaginary part of the correlation function as follows: with Here the ellipses stand for the contributions from higher resonances and the continuum spectra.
Considering the combination √ sρ had 1 + ρ had 2 , and introducing the exponential function exp(−s/M 2 ) to suppress these contributions, one can separate the λ + contributions: where s 0 is the threshold of the continuum states and M 2 is the Borel parameter.
On the OPE side, we compute the correlation function Π(q) to obtain the QCD spectral densities Taking the quark-hadron duality below the continuum threshold s 0 , we arrive at the following QCD sum rule Here ∆ is the threshold parameter, ∆ = (m Q + m Q ′ ) 2 for Ξ QQ ′ , and ∆ = (m Q + m Q ′ + m s ) 2 for In the numerical analysis, the quark masses are used as [50]: m c = 1.35 ± 0.10 GeV, m b = 4.60 ± 0.10 GeV, m s = 0.12 ± 0.01 GeV, while the u and d quarks are taken as massless. Similar values have been taken in Ref. [42].
Baryon masses used in the analysis of decay constants are given in Table II. For the mass of Ξ ++ cc , we adopt the experimental value [8], and we use the isospin symmetry for the Ξ + cc . For other baryons, we use the Lattice QCD results from Ref. [55]. The continuum threshold √ s 0 is used as 0.4 ∼ 0.6 GeV higher than the corresponding baryon mass, where we have assumed that the energy gap between the ground states and the first radial excited states is approximately 0.5 GeV [56].
Complying with the standard procedure of QCD sum rule analysis, the Borel parameter M 2 is varied to find the optimal stability window, in which the perturbative contribution should be larger than the condensate contributions and meanwhile the pole contribution larger than the continuum contribution.
The sum rule in Eq. (19) will be numerically analyzed since it is expected to have a better convergence in contrast with the sum rule in Eq. (20).

A. Masses
Differentiating Eq. (19) or Eq. (35) with respect to −1/M 2 , one can extract the mass of the doubly heavy baryon as or The optimal stability window for M 2 can be determined as follows. The upper bounds of the Borel parameters M 2 can be determined by the requirement that the pole contribution   [41] and Ref. [42] and the Lattice QCD results from Ref. [55] are listed. Our results are consistent with Ref. [42] and Ref. [55] but somewhat different from Ref. [41].
Baryon This work #1 This work #2 Ref. [41] Ref. [42] Ref. [  Here the uncertainties of the relevant parameters, including M 2 , s 0 , the quark masses and the condensates, have been taken into account. It can be seen that our values are consistent with the experimental value when the errors are taken into account. Our results are also consistent with other estimates for instance Ref. [42]. A collection of the results can be found in Table III.

B. Decay constants
Dependence of the "decay constants" λ H on the Borel parameter M 2 is shown in Figs. 3 and 4, where the sum rules in Eq. (19) and Eq. (35) are adopted, respectively. Numerical results for the "decay constants" can be found in Table IV. A few remarks are given in order.
• It is necessary to point out that when including the contributions from the 1/2 − baryons the threshold parameter might be somewhat higher. In this analysis, we have approximately  use the same values.
• Comparing the two sets of results in Table IV, one can see that the negative parity baryons do not provide significant modifications.
• We can see from Table IV that the decay constants of Ω QQ ′ are slightly larger than those of Ξ QQ ′ .

IV. CONCLUSION
In this work we have calculated the "decay constants" for doubly heavy baryons Ξ cc , Ω cc , Ξ bb , Ω bb , Ξ bc and Ω bc using QCD sum rules. In the calculation we have included both the positive and negative parity baryons, and found that the 1/2 − contamination is not severe. The extracted results for the decay constants are ingredients for the study of weak decays and other properties of doubly heavy baryons, including the lifetimes [57][58][59].