On the four-quark operator matrix elements for the lifetime of $\Lambda_{b}$

Heavy quark expansion can nicely explain the lifetime of $\Lambda_{b}$. However, there still exist sizable uncertainties from the four-quark operator matrix elements of $\Lambda_{b}$ in $1/m_{b}^{3}$ corrections, which describe the spectator effects. In this work, these four-quark operator matrix elements are investigated using full QCD sum rules for the first time. At the QCD level, contributions from up to dimension-6 four-quark operators are considered. Our method of calculating high-dimensional operator matrix elements is promising to be used to resolve the $\Omega_{c}$ lifetime puzzle.

At present, the standard framework for understanding weakly decaying heavy flavor hadrons is heavy quark expansion (HQE) [6][7][8][9][10][11][12][13].Under this framework, some attempts have been made to resolve the Ω c lifetime puzzle [14][15][16].However, there is still a lack of more reliable calculation based on QCD for the hadronic matrix elements of high-dimensional operators in HQE.HQE describes inclusive weak decays of heavy flavor hadrons.It is a generalization of the operator product expansion (OPE) in 1/m Q , and nonperturbative effects can be systematically studied.The starting point of HQE is the following transition operator where L W is the effective weak Lagrangian governing the decay Q → X f .With the help of the optical theorem the total decay width of a hadron H Q containing a heavy quark Q can be given as where M H is the mass of H Q .The right hand side of Eq. ( 4) is then calculated using OPE for the transition operator T [5, 14] where ξ is the relevant CKM matrix element, T 6 consists of the four-quark operators ( QΓq)(qΓQ) with Γ representing a combination of Dirac and color matrices.
In fact, there was also a conflict between theory and experiment for the lifetime of Λ b as early as in 1996.Taking τ (B 0 ) = (1.519± 0.004) ps in PDG2022 [17] as a benchmark, for τ (Λ b ) = (1.14 ± 0.08) ps in PDG1996 [18], one can find the ratio: Theoretically, the ratio deviated from unity at the level of 20% is considered to be too large.Nowadays we know that the low value of τ (Λ b )/τ (B d ) or the short Λ b lifetime was a purely experimental issue.The world averages in PDG2022 are which are in good agreement with the HQE prediction [5]: However, it can be seen from Eq. ( 8) that, the main uncertainty of the lifetime ratio comes from the 1/m 3 b corrections of the Λ b -matrix elements.The relevant matrix elements can be parameterized in a model-independent way [14]: where (q 1 q 2 ) V −A ≡ q1 γ µ (1 − γ 5 )q 2 and (q 1 q 2 ) S±P ≡ q1 (1 ± γ 5 )q 2 .In the literature, a B is usually The theoretical predictions for L 1 , L 2 and B are listed in Table I, from which, one can see that there are significant differences in the theoretical predictions for L 1 and L 2 .In this work, we intend to clarify this issue.
In Ref. [24], we performed a QCD sum rules analysis on the weak decay form factors of doubly heavy baryons to singly heavy baryons.However, considering there were few theoretical predictions and experimental data available to compare with, we then applied our calculation method to study the semileptonic decay of Λ b → Λ c lν in Ref. [25].It turns out that, our predictions for the form factors and decay widths are consistent with those of heavy quark effective theory (HQET) and Lattice QCD.In this work, similar computing strategies are performed.At the QCD level, contributions from up to dimension-6 four-quark operators are considered to obtain the matrix elements in Eq. ( 9).
The rest of this article is arranged as follows.In Sec.II, the main steps of QCD sum rules are presented, and in Sec.III, numerical results are shown.For consistency, the pole residue of Λ b is also considered.We conclude this article in the last section.

II. QCD SUM RULES ANALYSIS
The following interpolating current is adopted for Λ b : where Q denotes the bottom quark, a, b, c are color indices, and C is the charge conjugation matrix.
Following the standard procedure of QCD sum rules, one can calculate the correlation function at the hadron level and QCD level.At the hadron level, after inserting the complete set of baryon states, one can obtain where λ H = λ Λ b , M = m Λ b are respectively the pole residue and mass of Λ b , the parameters a and b are introduced to parameterize the hadronic matrix element and the ellipsis stands for the contribution from higher excited states.For the forward scattering matrix element, one can show that where ū(q, s)u(q, s) = 2 m Λ b and ū(q, s)γ 5 u(q, s) = 0 have been used.One can see that, only the parameter a in Eq. ( 14) is relevant to the forward scattering matrix element.
It can be seen from Eq. ( 13) that, there are 8 Dirac structures, but only (at most) 2 parameters need to be determined.By considering the contribution of negative-parity baryons [24][25][26], one can update Eq. ( 13) to Here, M +(−) and λ +(−) respectively denote the mass and pole residue of Λ b (1/2 +(−) ), and a +− is the parameter a for the positive-parity final state Λ b (1/2 + ) and the negative-parity initial state Λ b (1/2 − ), and so forth.When arriving at Eq. ( 16) , we have adopted the following conventions: In Eq. ( 17), these iγ 5 are not necessary, but they are convenient.At the QCD level, the correlation function can be written formally as The coefficients A i are then expressed as double dispersion relations where the spectral function ρ A i (s 1 , s 2 , q 2 ) can be calculated using Cutkosky cutting rules, see Fig. 1 . Sum rules are obtained by equating Eq. ( 16) to Eq. ( 18) and then using quark-hadron duality to eliminate the contribution of excited states.Furthermore, by equating the coefficients of the same Dirac structure, one can have 8 equations to solve 8 unknown parameters a ±± and b ±± .
Especially, after performing the Borel transform, one can arrive at where BA i are doubly Borel transformed coefficients and s 0 and T 2 are respectively continuum threshold parameter and Borel parameter.
In this work, contributions from up to dimension-6 four-quark operators are considered at the QCD level.1For the matrix elements in Eq. ( 9), we find that contributions from quark condensate (dimension-3) and quark-gluon condensate (dimension-5) are proportional to the mass of u/d quark, which is taken to be zero in this work.All nonzero, "independent" diagrams are shown in Fig.
2. Here, "independent" is explained as follows.For example, diagram dim-4-2,5 with quark 2 and quark 5 each emitting a gluon is equal to diagram dim-4-1,4, therefore the former is not listed in Fig. 2.
FIG. 2: All nonzero, "independent" diagrams considered in this work.

A. The pole residue
As can be seen in Eq. ( 20), the pole residue of Λ b is an indispensable input.For consistency, in this work, we also perform an analysis on the pole residue of Λ b , whose sum rule is [25] ( from which, one can obtain the mass formula for Λ b Eq. ( 23) can be viewed as a constraint to Eq. ( 22).Following Ref. [27], in this work, we use Eq. ( 23) to determine the continuum threshold parameter s 0 .Specifically, for a set of fixed parameters (renormalization scale, condensate parameters, etc.), the optimal (s 0 , T 2 + ) are obtained through the following procedure: 1.For a trial s 0 , plot the pole residue curve with respect to the Borel parameter T 2 + using Eq. ( 22).Find the minimum point T 2 + on the curve.(See Fig. 3 for some intuitive impressions.) 2. Substitute the set of (s 0 , T 2 + ) obtained in step 1 into Eq.( 23) to calculate the baryon mass, and compare it with the experimental value.If equal (within a small error range), terminate; otherwise, go to step 1.For different input parameters, one can obtain different optimal (s 0 , T 2 + ), which are listed in Table II.
In this work, contributions from up to dimension-6 four-quark operators are also considered for the sum rule in Eq. ( 22).

A. Inputs
Our main inputs in numerical calculation can be found in Table II.We take the MS mass for the bottom quark [17] and neglect the mass of u/d quark.The condensate parameters are taken from Ref. [28].The renormalization scale is taken as µ b = 3 ∼ 6 GeV with m b (m b ) the central value [27], from which, one can estimate the dependence of the calculation results on the renormalization scale.

B. The pole residue and the continuum threshold parameter
Our predictions for the pole residue λ Λ b , together with the continuum threshold parameter s 0 can be found in Table II.A similar investigation has also been performed in our previous work [25], however, in this work, more contributions from higher dimensional operators are considered.
Numerically, the predictions in this work are close to those in Ref. [25].This is essentially because the contributions from higher dimensional operators are small.A comprehensive study of the pole residues of antitriplet heavy baryons can be found in Ref. [29].
TABLE II: Our predictions of the pole residue λ Λ b , L 1,2,3,4 defined in Eq. ( 9), and B ≡ −L 3 /L 1 .For a set of fixed parameters (renormalization scale, condensate parameters, etc.), the continuum threshold parameter s 0 can be determined, and then the quantities we are interested in can also be determined -by requiring them to have as little dependence on the Borel parameter as possible.• As can be seen in Table II that, for a set of fixed parameters (quark mass, renormalization scale, condensate parameters), the continuum threshold parameter s 0 can be determined, and then the quantities we are interested in can also be determined -by requiring them to have as little dependence on the Borel parameter as possible.
• In our opinion, the continuum threshold parameter s 0 is the most important parameter in QCD sum rules.In fact, once s 0 is fixed, the quantity that we are interested in is almost determined by searching for stability region.In the literature, there exist at least two approaches to determine s 0 .One approach is to empirically select √ s 0 approximately 0.5 GeV larger than the ground state mass.Another approach is to determine s 0 through the mass formula.We have adopted the latter approach in this work.The basic logic behind doing so is that the mass formula can be seen as a constraint to the sum rule of two-point correlation function.Of course, this comes at the cost of abandoning the predictive power of hadron mass.

IV. CONCLUSIONS
Heavy quark expansion can nicely explain the lifetime of Λ b .However, there still exist sizable uncertainties in the four-quark operator matrix elements of Λ b in 1/m 3 b corrections, which describe the spectator effects.In this work, these four-quark operator matrix elements are investigated using full QCD sum rules.At the QCD level, contributions from up to dimension-6 four-quark operators are considered.Stable Borel region can be found.We have also considered the uncertainties from various input parameters, and find that the main source of error is scale dependence, which is about 20%.Our results are close to those of the spectroscopy update of Ref. [19] in 2014.Our method of calculating high-dimensional operator matrix elements is promising to be used to resolve the Ω c lifetime puzzle.

3 FIG. 3 :
FIG. 3: The pole residue λ Λ b as a function of the Borel parameter T 2 + .The red, orange, and blue dots respectively correspond to the renormalization scale µ = m b , µ = 6 GeV, and µ = 3 GeV.The minimum points are marked, and they correspond to the experimental mass of Λ b via Eq.(23).

TABLE I :
[5] , L 2 and B predicted by different theoretical methods.This table is copied from Ref.[5].