Analysis of the 1S and 2S states of Λ Q and Ξ Q with QCD sum rules

In this article, we study the ground states and the first radial excited states of the flavor antitriplet heavy baryon states and with the spin-parity by carrying out operator product expansion up to vacuum condensates of dimension in a consistent way. We observe for the first time that the higher dimensional vacuum condensates play an important role, and obtain very stable QCD sum rules with variations of the Borel parameters for the heavy baryon states. The predicted masses , , and for the first radial excited states , , and , respectively, are in excellent agreement with the experimental data and support assigning , , and to be the first radial excited states of , , and , respectively. The predicted mass for can be confirmed using experimental data in the future.


I. INTRODUCTION
Recently, the CMS collaboration observed a broad excess of events in the region of in the invariant mass spectrum based on a data sample corresponding to an integrated luminosity of up to [1]. If it is fitted with a single Breit-Wigner function, the obtained mass and width are and , respectively. Subsequently, the LH-Cb collaboration observed a new excited baryon state in the invariant mass spectrum with high significance using a data sample corresponding to an integrated luminosity of . The measured mass and natural width are and , respectively, which are consistent with the first radial excitation of the baryon, the resonance [2].
can be assigned to be the state [3,4], or assigned to be the lowest -mode excitation in the family [5].
In 2001, at the charm sector, the CLEO collaboration observed or in the invariant mass spectrum using a data sample recorded by the CLEO detector at CESR [6]. The Belle collaboration determined the isospin of or to be zero using a data sample in the annihilation around , and established it to be a res- onance [7]. can be assigned to be the state [8,9]; however, there are several other possible assignments [10]. In 2006, the Belle collaboration reported the first observation of two charmed strange baryon states that decay into the final state ; the broader one has a mass of and a width of [11]. Subsequently, the BaBar collaboration confirmed or [12]. can be assigned to be the state [8,9]; however, there are several other possible assignments [10].
The mass spectrum of the single heavy baryon states has been studied intensively in various theoretical models . If , and are the first radial excited states of , and , respectively, the mass gaps between the ground states and first radial excited states are less than , which are significantly lower than the amount that is expected by the 3-dimensional harmonic oscillator model. In the QCD sum rules for the single heavy baryon states, if we carry out the operator product expansion up to the vacuum condensates of dimension 6, we have to choose the continuum threshold parameters as or to reproduce the experimental data [26][27][28][29][30][31][32], where the subscript stands for the ground states. The energy gaps and are much µ γ The attractive interaction induced by one-gluon exchange favors forming diquark states or quark-quark-correlations in the color antitriplet [39,40]. The color antitriplet diquark operators have five structures in the Dirac spinor space, where , C, , , and for the scalar, pseudoscalar, vector, axialvector, and tensor diquarks, respectively, and couple potentially to the corresponding scalar, pseudoscalar, vector, axialvector, and tensor diquark states, respectively. The calculations via the QCD sum rules indicate that the favored quark-quark configurations are the scalar and axi- alvector diquark states, while the most favored quarkquark configurations are the scalar diquark states [41]. We usually resort to the light-diquark-heavy-quark model to study the heavy baryon states. In the diquark-quark models, the angular momentum between the two light quarks is denoted as , while the angular momentum between the light diquark and the heavy quark is denoted as . If the two light quarks in the diquark are in relative S-wave or , then the heavy baryon states with the spin-parity and diquark constituents are called -type and -type baryons, respectively [42]. In this article, we study the ground states and first radial excited states of -type heavy baryons with -type interpolating currents.
We can interpolate the corresponding spin-parity flavor antitriplet heavy baryon states with thetype currents and without introducing the relative P-wave explicitly, because multiplying with the currents and changes their parity [43]. We now write the correlation functions, where and .
We insert a complete set of intermediate baryon states with the same quantum numbers as the current operat ors , , and into the correlation functions to obtain the hadronic representation [44][45][46]. After isolating the pole terms of the ground states and the first radial excited states, we obtain the following results: where and are the masses of the ground states and first radial excited states with the parity respectively, and and are the corresponding pole residues defined by , and . We rewrite the correlation functions as according to the Lorentz covariance, and obtain the hadronic spectral densities through the dispersion relation ρ H,0 (s) =lim ϵ→0 ImΠ 0 (s + iϵ) π , where we add the subscript H to denote the hadron side of the correlation functions.
We now carry out the operator product expansion up to the vacuum condensates of dimension 10 in a consistent way, and consider the vacuum condensates, which are quark-gluon operators of the order with . Again, we obtain the corresponding QCD spectral densities through the dispersion relation QCD where we add the subscripts to denote the QCD side of the correlation functions.
Then, we choose the continuum thresholds and to include the ground states and the ground states plus first radial excited states, respectively, and introduce the weight function to suppress the contributions of the higher resonances and continuum states. We take the combination to exclude the contaminations from the heavy baryon states with negative parity, and match the hadron side with the QCD side of the correlation functions. The combinations pick up the heavy baryon states with positive and negative parities, respectively. Finally, we obtain two QCD sum rules: 2M Analysis of the 1S and 2S states of Λ Q and Ξ Q with QCD sum rules Chin. Phys. C 45, 013109 (2021) 013109-3 where is the Borel parameter.
We derive the QCD sum rules in Eq. (10) with regard to , and then eliminate the pole residues and obtain the masses of the ground states and , Hereafter, we will refer to the QCD sum rules in Eq. (10) and Eq. (15) as QCDSR I.
We introduce the notations , , and use the subscripts and to represent the ground states , , and the first radially excited states , , respectively, for simplicity.
where , , we introduce the subscript to denote the QCD representation of the correlation functions below the continuum thresholds . Firstly, let us derive the QCD sum rules in Eq. (16) with respect to to obtaiñ From Eqs. (16)-(17), we can obtain the QCD sum rules i j where the sub-indexes . We can then derive the QCD τ sum rules in Eq. (18) with respect to to obtain , The squared masses satisfy the equation with the indexes and . Finally, we solve the equation in Eq. (20) analytically to obtain two solutions [47][48][49], From the QCD sum rules in Eqs. Hereafter, we will refer to the QCD sum rules in Eq. (18) and Eq. (23) as QCDSR II.

MS
At the QCD side, we take the vacuum condensates to be the standard values , , , , , at the energy scale [44][45][46][47][48][49][50], and take the masses , and from the Particle Data Group [51]. Moreover, we consider the energy-scale dependence of the quark condensates, mixed quark condensates, and masses according to the renormalization group equation, where t = log , and for the flavors , , and , respectively [51,52]. For the charmed baryon states and , we choose the flavor number , while for the bottom baryon states and , we choose the flavor number .
In QCDSR I, we choose the continuum threshold parameters to be rather than or as a constraint to exclude contaminations from the first radial excited states [26][27][28][29][30][31][32], where the subscript denotes the ground states and . Furthermore, we choose the energy scales of the QCD spectral densities in the QCD sum rules for , , and to be the typical energy scales , , , and , respectively, where we subtract from the energy scale for to account for the finite mass of the s-quark. After trial and error, we obtain the Borel parameters , continuum threshold parameters , pole contributions of the ground states, and perturbative contributions, which are shown explicitly in Table 1. From the table, we can see that the pole contributions are approximately 40%-60% or 40%-70%, so the pole dominance is satisfied. The perturbative contributions are larger than 50% except for , although the perturbative contribution is approximately 43%-46% in that case; the contributions of the vacuum condensates of dimension 10 are tiny, and the operator product expansion is well convergent.  Table 1.

Borel parameters and continuum threshold parameters
for the heavy baryon states, where "pole" stands for the pole contributions from the ground states or the ground states plus first radial excited states, and "perturbative" stands for the contributions from the perturbative terms. We now consider all uncertainties of the input parameters, and obtain the values of the masses and pole residues of the ground states of the flavor antitri plet heavy baryon states and , which are shown in Figs. 1-2  continuum threshold parameters are large enough to consider all the ground state contributions but small enough to suppress the first radial excited state contaminations sufficiently. Furthermore, they meet our naive expectations.
In this article, we have neglected the perturbative corrections; if we consider the perturbative corrections, the perturbative terms should be multiplied by a factor , where are coefficients. Although we cannot estimate the uncertainties originating from the corrections with confidence   an important role, and re-determine the Borel windows to extract the heavy baryon masses, as in the case of the heavy mesons, in which the perturbative corrections to the quark condensates are also calculated [54]. Overall, neglecting the perturbative corrections cannot notably impair the predictive ability, because as we obtain the heavy baryon masses from fractions, the perturbative corrections in the numerators and denominators cancel each other out to a certain extent; see Eq. (15).  Fig. 3, we plot the predicted mass of the ground state with variations of the Borel parameter by considering the vacuum condensates up to dimension 6, 8, and 10, respectively, for the continuum threshold parameter . From the figure, we can see that the truncation fails to lead to a flat platform or to reproduce the experimental value of the mass of , whereas the truncations and both lead to very flat platforms and reproduce the experimental value. In fact, the truncations and make little difference, which indicates that the vacuum condensates of dimension 8 (10) play an important (a minimal) role. We should consider the vacuum condensates up to dimension 10 for consistency. If we insist on taking the truncation , we have to choose a much larger continuum threshold parameter , and the predicted mass increases monotonically with the increase of the Borel parameter ; we can reproduce the experimental value of the mass of with a suitable Borel parameter but large uncertainty. In QCDSR II, we can borrow some ideas from the conventional charmonium states. The masses of the ground state, first radial excited state, and second excited state of the charmonium states are , , and , respect- T 2 s 0 ively, from the Particle Data Group [51]. The energy gaps are and , and we can choose the continuum threshold parameters tentatively to avoid contaminations from the second radial excited states. Furthermore, we choose the energy scales of the QCD spectral densities in the QCD sum rules for , , , and to be the typical energy scales , , , and , respectively; again we subtract from the energy scale for to account for the finite mass of the s-quark. After trial and error, we obtain the Borel parameters , continuum threshold parameters , pole contributions, and perturbative contributions, which are shown explicitly in Table 1. From the table, we can see that the pole contributions vary from 40% to 80%, so the pole dominance is satisfied. The perturbative contributions are larger than 70%, so the operator product expansion is well convergent. Again we consider all uncertainties of the input parameters, and obtain the values of the masses and pole residues of the first radial excited states of the flavor antitriplet heavy baryon states, which are also shown in Figs. 1-2 and Table 2. From Table 1 and Figs. 1-2, we can see that rather flat platforms appear in the Borel windows, and the uncertainties originating from the Borel parameters are rather small. The predicted masses , , and are in excellent agreement with the experimental data , , and , respectively [2,51], and support assigning , and to be the first radial excited states of , and , respectively. The prediction can be confronted by experimental data in the future.
If the masses of the ground states, first radial excited states, third radial excited states, etc. of the heavy baryon   , and "Expt" denotes the experimental value.