Study of singly heavy baryon lifetimes

We study the inclusive decay widths of singly heavy baryons with the improved bag model in which the unwanted center-of-mass motion is removed. Additional insight is gained by comparing the charmed and bottom baryons. We discuss the running of the baryon matrix elements and compare the results with the non-relativistic quark model (NRQM). While the calculated two-quark operator elements are compatible with the literature, those of the four-quark ones deviate largely. In particular, the heavy quark limit holds reasonably well in the bag model for four-quark operator matrix elements but is badly broken in the NRQM. We predict $1-\tau(\Omega_b)/ \tau(\Lambda_b^0) = (8.34\pm2.22)\%$ in accordance with the current experimental value of $(11.5^{+12.2}_{-11.6})\%$ and compatible with $(13.2\pm 4.7)\%$ obtained in the NRQM. We find an excellent agreement between theory and experiment for the lifetimes of bottom baryons. We confirm that $\Omega_c^0$ could live longer than $\Lambda_c^+$ after the dimension-7 four-quark operators are taken into account. We recommend to measure some semileptonic inclusive branching fractions in the forthcoming experiments to discern different approaches. For example, we obtain ${\cal BF} (\Xi_c^+ \to X e^+ \nu_e) = (8.57\pm 0.49)\% $ and ${\cal BF} (\Omega_c^0 \to X e^+ \nu_e) = (1.88\pm 1.69)\% $ in sharp contrast to ${\cal BF} (\Xi_c^+ \to X e^+ \nu_e) = (12.74^{+2.54}_{-2.45})\% $ and ${\cal BF} (\Omega_c^0 \to X e^+ \nu_e) = (7.59^{+2.49}_{-2.24})\% $ found in the NRQM.


I. INTRODUCTION
The study of lifetimes for ground-state singly charmed and bottom baryons, both experimentally and theoretically, are of great interest.On the experimental side, the current world averages of bottom baryon lifetimes given by (in units of ps) [1] τ (Λ 0 b ) = 1.471 ± 0.009, τ (Ξ 0 b ) = 1.480 ± 0.030, −0.17 (1) indicate the lifetime hierarchy where the uncertainty of the Ω − b lifetime is too large to draw a conclusion yet.deviations larger than the 2018 WA value.In 2021 LHCb [7] reported a further new measurement using promptly produced Ω 0 c and Ξ 0 c baryons with the results shown in the same table.Finally, Belle II has disclosed in 2022 the new measurements of τ (Λ + c ) [9] and τ (Ω 0 c ) [10] and achieved the most precise result of the Λ + c lifetime to date.The current WA values of charmed baryon lifetimes in Table I result from the averages of the data from LHCb (2021), PDG (2022) and Belle II (2022).
On the theoretical side, lifetimes of the heavy baryons are commonly analyzed within the framework of heavy quark expansion (HQE), in which the inclusive rate of the heavy baryon B Q is schematically represented by where G F is the Fermi constant and m Q is the heavy quark mass.An analysis of the lifetimes of charmed baryons in HQE up to 1/m 4 c described by dimension-7 fourquark operators was performed in Ref. [11].While the predictions on τ (Λ + c ) and τ (Ξ + c ) were improved after including corrections from dimension-7 operators, one of us (HYC) has introduced some additional parameters such as y and α to the baryonic matrix elements to ensure the validity of HQE in the Ω 0 c sector.Although the value of τ (Ω 0 c ) ∼ 2.3 × 10 −13 s was predicted even before the first LHCb measurement of the Ω 0 c lifetime [12], the use of the above-mentioned ad hoc parameters was not justified.In this work, we shall show that these difficulties can be circumvented.
Through the optical theorem and HQE, the heavy hadron lifetimes are governed by the transition operators which have the following general expression where the subscript of the parenthesis denotes the number of quark operators.A dimension-four operator is absent due the Luke's theorem [13].The inclusive decay widths evaluated by sandwiching the transition operator read as The Wilson coefficients C 3 , C 5 and C ρ depend only on the heavy flavor, while Γ 6,7 are proportional to Õ6,7 , depending also on the light quarks in the heavy baryon.The quantities µ 2 π , µ 2 G and ρ 3 D are of nonperturbative in nature, which will be introduced below.
Owing to the absence of lattice QCD calculations as input, one has to rely on the models to evaluate the baryon matrix elements.Concerning the heavy baryon lifetimes, the non-relativistic quark model (NRQM) turns out to be the favored one in the literature.Most of the baryon matrix elements can be extracted from the mass spectra, providing at least a certain way for the estimation.Nevertheless, the light quark masses are taken to be the constituent ones in O n but current ones in C n , yielding uncontrollable errors.
In this work, we adopt the bag model (BM), where the up and down quarks are treated to be massless.The model did not receive much attention as its estimate of the baryon matrix elements of four-quark operators is much smaller than that of the NRQM.Recently, we found that once the unwanted center-of-mass motion (CMM) of the bag model is removed in the heavy-flavor-conserving decays of heavy baryons, the four-quark operator matrix elements become twice larger [14].In the meantime, there exist several issues for the NRQM estimates of the baryon matrix elements of two-and four-quark operators.In the spirit of HQET, two-quark operator matrix elements such as µ 2 π , µ 2 G and ρ 3 D , and four-quark operator matrix elements such as L q B Q to be defined below are also independent of the heavy quark mass m Q .In the NRQM, L q B Q is related to the heavy baryon wave function modulus squared at the origin; that is, based on the mass formula for hyperfine mass splittings of the bottom baryon and the B meson.An immediate consequence is that the charmed baryon matrix element is much smaller than the bottom baryon one owing to the smallness of This feature is also manifested in the realistic NRQM calculation, see Table V below.However, a large deviation between the values of |ψ Bc cq (0)| 2 and |ψ B b bq (0)| 2 is not consistent with the expectation of the heavy quark limit.This undesired feature can be overcome in the improved bag model which we are going to elaborate on in this work.Therefore, it is worthwhile to revisit the bag model for the study of heavy baryon lifetimes.This paper is organized as follows.In Sec.II, we use the bag model to compute the baryon matrix elements such as those of µ 2 π , µ 2 G and ρ 3 D .Special attention is paid to the running of the four-quark operator matrix elements.In Sec.III, we decompose the transition operator into several parts and briefly discuss the Wilson coefficients therein.Explicit expressions of the contributions to the decay widths of heavy baryons from four-quark operators at dimension-6 and -7 are given.The numerical results are presented and discussed in Sec.IV and we conclude this work in Sec.V. Appendix A is devoted to the evaluation of four-quark operator matrix elements in the bag model.

II. MATRIX ELEMENTS IN THE BAG MODEL
Under the heavy quark effective field theory (HQET), the decomposition of the baryon mass [15] is related to the quantities where d H = 0, 4 for the antitriplet heavy baryon T Q and the sextet one Ω Q , respectively, and it shall be noticed that the HQET parameters Λ, λ 1 and λ 2 are independent of the heavy quark mass.The Q v is the heavy quark field defined as If not stated otherwise, the matrix elements are evaluated at x = 0 and the first-order correction in Eq. ( 10) is treated as uncertainties, leading to Q v = Q when derivatives are not involved.Due to the reparametrization invariance [16], µ 2 π does not get renormalized in the dimensional regularization.On the other hand, C G (m Q , µ) is obtained by matching the full QCD theory with HQET at the energy scale µ, and the ) is generated by the renormalization effects known as the Appelquist-Carazzone decoupling theorem [18].
The mass corrections Λ, λ 1 and λ 2 correspond to the diquark energy, kinetic energy of the heavy quark and chromomagnetic field energy, respectively.For T Q , as a working principle Λ is often taken to be equal among the baryons and λ 2 is extracted from the mass difference of M Σ Q − M Λ Q .Nevertheless, the diquarks in Σ Q and Λ Q are of spin-1 and spin-0 systems, respectively, and there is no reason to assume that their energies are the same.In general, the assumption of a universal Λ would cause errors of order m Q Λ ∆ /3 to λ 2 , where Λ ∆ ≈ 0.2 GeV is the mass difference of spin-1 and spin-0 diquarks.On the other hand, the assumption is acceptable in the Ω Q sector, as the diquark system is of spin-1 for both Ω Q and Ω * Q .
In terms of the creation operators, the baryon wave functions in the BM read where (q, q ′ ) = (u, d), (u, s), (d, s) are the light-quark flavors of Latin (Greek) alphabets stand for the spinor (color) indices, and Ψ describes the spinorspatial distributions of the quarks.In the static bag limit, Ψ is given by where N is a normalization constant, ω ± q = E k q ± m q with E k q = p 2 q + m 2 q , j 0,1 are the spherical Bessel functions, χ ↑ = (1, 0) T , χ ↓ = (0, 1) T and R is the bag radius.
In Eq. ( 12), p q is the magnitude of the 3-momentum of the bag quark q.From the boundary condition, we have [19] tan The heavy quark limit can be obtained by taking the right-hand side of Eq. ( 13) to be zero, leading to tan(p Q R) = 0 and p Q = π R .To obtain the next-order correction, we 13) and obtain Taking the Taylor expansion of the tangent function at π, we find The first-order corrections in charm and beauty baryons are 7% and 2%, respectively, indicating that HQE works well for p Q .

A. Two-quark operator matrix elements
To the leading-order in α s , the gluonic self-coupling does not contribute and hence gluon fields act as eight independent Abelian ones.The gluonic magnetic fields induced by φ qλq are then given by [19] where the definition of M q (r) and µ q (r) can be found in Ref. [19], λ a are the Gellmann matrices sandwiched by the quark color and λ q ∈ {↑, ↓}.The interaction energy between q 1 and q 2 is equivalent to the energy stored in the magnetic field, given by [19] where F S (λ 1 , λ 2 ) depends on the spin configurations.Explicitly, we have where s 1,2 are the spins of q 1 and q 2 , and 4 s 1 • s 2 = −3, 1 for spin-0 and spin-1 configurations, respectively.In the BM, R −1 plays the role of a typical energy scale of the hadron, and the running of α s with respect to R is [20] α s = 0.296 log(1 + (0.281R) −1 ) .
TABLE II: Inputs of the BM with R and m q in units of GeV −1 and GeV, respectively [20].Here m q should not be confused with the pole quark mass.In particular, we have α s = (0.52, 0.55) for R = (4.6,5.0) GeV −1 .It is interesting to see that 1 α s (1 GeV) = 0.495 is close to Eq. ( 19) with R = 5 GeV −1 , which is a typical bag size of the baryon, indicating that the wave functions in the BM are applicable around the hadronic scale µ H = 1 GeV.In this work, we shall allow µ H to range from 0.8 GeV to 1.2 GeV to take into account the hadronic uncertainties.By substituting Eqs. ( 16) and (18) into Eq.( 17), we find Λ ∆ = 0.207 GeV with R = 5 GeV −1 and m q 1 = m q 2 = 0.
The mass correction λ 1 corresponds to the kinetic energy of the heavy quark in the Newtonian limit with a minus sign.Therefore, we have On the other hand, λ 2 in Eq. ( 8) describes the mass correction due to the spin-spin interactions between the heavy quark and the magnetic field.Comparing to Eq. ( 17), we arrive at 2 The bag parameters are taken from Ref. [20] and collected in Table II.The numerical results of λ 1,2 are listed in Table III.It shall be noted that the final results depend weakly on the input of m Q in the BM.Numerically, λ 1,2 vary less than 2% if the pole quark masses given in Eq. ( 56) below are used.In the limit of large m Q , λ 1,2 shall be independent of the heavy quark mass m Q , and therefore their values for B b and B c systems should be the same.A small deviation of order O(1/m 2 Q ) comes from the fact that heavy baryons have different bag radii.As HQET is more reliable in B b , in the numerical evaluation we shall allow the values of B c to vary from their original values to the ones of B b .For instance, we take λ 1 = −(0.418∼ 0.466) GeV 2 for Λ c , causing 10% uncertainties.Additional 6% and 14% uncertainties from 1/m 2 Q are also included for B b and B c , respectively, as described in Eq. ( 15).We note that the calculated λ 1 is consistent with the literature.On the contrary, λ 2 is about 40% larger for T Q , but the deviation will not reflect in the numerical results as λ 2 is always accompanied by d H in Eq. ( 9) and d H = 0 for T Q .

Besides µ 2
π and µ 2 G , there is an additional two-quark operator whose importance has been stressed recently, known as the Darwin operator where we have used v ν = (1, 0, 0, 0) for the baryon at rest and the equation of motion In terms of the matrix elements L q B Q , S q B Q and P q B Q of the fourquark operators to be defined in the next subsection of Eq. ( 29), the Darwin operator 2 Since the right-hand side of Eq. ( 21) is a leading-order result in α s , we have taken on the left-hand side for reasons of consistency.Using integration by part and keeping the leading term in 1/m Q , it is possible to show that Eq. ( 9) can be recasted to the form J • A , where J = g s Q γQ and ∇ × A = B are the current density and vector potential, respectively.It is identical to −g 2 s B • B to the leading order in α s .

TABLE IV:
The two-quark operator matrix elements, where µ 2 G,π and ρ3 D are in units of 10 −1 GeV 2 and 10 −2 GeV 3 , respectively.The parameters of µ 2 G and ρ 3 D are evaluated at the hadronic scale µ H in the BM.Here, the numbers in the parentheses are the uncertainties counting backward in digits, for example, 4.42(24) = 4.42 ± 0.24.
The values of the NRQM are quoted from Refs.[23,24].
takes the form [22] 3 where q sums over the light quarks of B Q .In deriving the above equation, use of 4[D µ , G µν ] = −g s λ a q qλ a γ ν q has been employed.
By using Table V below in advance, the results of the two-quark operator matrix elements in the BM are summarized in Table IV, where the ones in the NRQM [23,24] are also included for comparison.We see that µ 2 π , µ 2 G and ρ 3 D all depend weakly on the heavy quark flavor in the BM, which is a good sign for HQET.The values of µ 2 π are compatible within the range of uncertainties, whereas the central value of µ 2 G for Ω c deviates around 30%.
It is an appropriate place to discuss the renormalization-scale dependence of µ 2 G .We note that the m Q dependence of C(m Q , µ) can be factorized into two parts [17] where the µ dependence is governed by γ(α s ) and β(α s ).In the BM, both parameters and matrix elements are determined at the hadronic scale.To employ the results for the lifetimes, we use derived from the renormalization-scheme independence of C G µ G .The coefficient ) has been determined up to three-loop in Ref. [25] in the MS scheme, given by Although we can renormalize one of them as unity, it cannot be done simultaneously for both in HQET.We note that C G (m Q , m Q ) is missing in the previous study of the singly heavy baryon lifetimes and the formalism of µ 2 G shall be modified as The ratio of the mass splitting in Ω Q is given by where the anomalous dimension of the µ 2 G evolution can be found in Eq. ( 13) of Ref. [25].It is compatible with the experimental value4 R G exp = 0.77, indicating that the coefficient C G (m Q , µ) cannot be neglected.
The Darwin term ρ 3 D in both the BM and NRQM obeys the same hierarchy Ω Q > Ξ Q > Λ Q , where the differences are induced by the strange-quark mass.More interestingly, ρ 3 D shares similar values for T Q and Ω Q , which is anticipated from Eq. ( 22) as only the heavy quark fields get involved 5 .However, it is a non-trivial result in view of Eq. ( 23) as L q T Q and L q Ω Q differ significantly.It is interesting to point out that Refs. [23] and [24] use α s (B c ) = 1 and α s (B b ) = 0.22 for ρ 3 D .However, the huge ratio α s (B c )/α s (B b ) = 4.5 is compensated by the large difference in the four-quark operator matrix elements, resulting in a weak dependence of ρ 3 D on m Q .On the contrary, both α s and the four-quark operator matrix elements in the BM depend slightly on m Q .
The running of ρ 3 D will be discussed in the next section.
Before ending this section, it has to be pointed out that the wave functions given in Eq. ( 12) encounter some inconsistencies as they are localized in the three-dimensional space.According to the Heisenberg uncertainty principle, it cannot be a 3-momentum eigensate, which is referred to as the CMM problem.However, this problem can be neglected in λ 1,2 .As λ 1 is related to p 2 Q , it remains unchanged after removing CMM [14].The readers who are interested in the interplay between mass corrections and matrix elements are referred to Ref. [27].On the other hand, by an explicit calculation, we find that the CMM will in general decrease λ 2 around 10% for a fixed α s .Nonetheless, the effects can be compensated by increasing α s by 10%.As α s is fitted from the mass spectra without removing CMM, it should not be removed in the calculations of λ 2 either.For ρ 3 D , as we shall see shortly, the matrix elements are evaluated by removing the unwanted CMM.However, as α s is likely to be underestimated by 10%, we thus include 10% uncertainties for ρ 3 D in Table IV.

B. Four-quark operator matrix elements
Recall the shorthand notation for the matrix elements of an arbitrary operator We parameterize the matrix elements of the dimension-6 four-quark operators as and where As strong interactions conserve parity, we do not need to consider the parity-violating operators, i.e.Qγ 5 qqQ = 0.
Although not written explicitly, it is understood that I q B Q depend on the energy scale.Our strategy is to first evaluate them at the hadronic scale µ H , where the valence quark approximation of B = 1 is automatically satisfied by the color structure in Eq. (11).
After that, we evolve them to the heavy quark scale µ Q to compute the lifetimes.
On the other hand, for the dimension-7 four-quark operators [28] and P q i (i = 1, • • • , 4) obtained from P q i by interchanging the colors of q and Q, their matrix elements can be expressed in terms of that of dimension-6 ones: 6 and where βi = 1 under the valence quark approximation.In deriving Eq. ( 32), we have applied the approximation 6 The operators P q 1 and P q 2 have the same baryonic matrix elements as the latter is related to the former by hermitian conjugation; that is, P q 2 = (P q 1 ) † [29].
where E q is the energy of the bag quark, taken to be E u,d = M p /3 ≈ 0.32 GeV and  7 We note that similar approximations have also been employed in Ref. [23] by substituting Λ QCD = 0.33 GeV for E q . 8The choice of E u,d = 0.32 GeV was proved to be suitable in describing the exclusive semileptonic decays of heavy baryons [27,30].
As mentioned in the previous section, the wave functions in Eq. ( 12) are problematic as they are localized, inducing a nonzero CMM.To compute the four-quark operator matrix elements, we have to remove the CMM for a consistent procedure.To this end, we have to distribute the wave functions homogeneously over all the space.
Consequently, Eq. ( 12) is modified as Here, (HB) and (SB) stand for the homogeneous bag and static bag approaches, respectively.It is straightforward to show that the wave function is invariant under the space translation which is not respected by Ψ (SB) .A great advantage of Ψ (HB) is that it does not require additional parameter input.However, the calculation becomes much more tedious.
The methodology of the computation is given in Appendix A.
The results of I q B Q in the BM and NRQM [23,24] are summarized in Table V.As stressed in passing, I q B Q depends on the energy scale and the results exhibited in the table are evaluated at µ H , where the valence quark approximation is valid.Besides the 7 Strictly speaking, E q can only be defined when quark energies are not entangled, i.e. interactions between quarks are negligible.In the BM, it is true by constructions in Eq. (12) where quark wave functions are independent to each other.In addition, the normalization condition in Eq. (A1) requires that E u + E d + m c = M Λc which is satisfied with the use of m c in Table II. 8The relation P q 3 ≈ (p Q • p q /m Q ) QΓqqΓQ has been used in Refs.[11,23] with p Q • p q taken to be [11] with m diq being the diquark mass.

TABLE V:
The matrix elements of the four-quark operators in units of 10 −3 GeV 3 with q I = u, d evaluated at the hadronic scale µ H .The results of the NRQM are quoted from Refs.[23] and [24] for charm and beauty baryons, respectively.
a Corresponding to the results in which the CMM is removed from the bag model.
uncertainties from the parameter input, additional 30% uncertainties are put by hand in the NRQM to be conservative as described in Refs.[23,24].Several remarks are in order: • The SU(3) flavor symmetry breaking can be examined by comparing I q I Ξ Q and I s Ξ Q .In the BM and NRQM, the breaking effects are around 10% and 30%, respectively.
• Since the light quark q has to be left-handed due to chiral symmetry, it is the linear combination of S q B Q − P q B Q rather than S q B Q + P q B Q that appears in the lifetimes to the leading-order of four-quark operators.Therefore, it suffices to consider S q B Q − P q B Q in discussing the finite heavy quark mass corrections.From the table, we see that L q B Q and S q B Q − P q B Q in the BM vary less than 10% with respect to the heavy flavor.Accordingly, we shall assign 10% uncertainties to I q B b when computing the lifetimes.
• In the NRQM, L q B Q is related to the heavy baryon wave function modulus squared at the origin, for example, L q GeV 3 for the bottom baryon Λ 0 b , while |ψ Λc cq (0)| 2 = 0.51 × 10 −2 GeV 3 for Λ + c .This is very annoying as |ψ(0)| 2 for hyperons is of the same order of magnitude as the bottom baryons (see Ref. [32] for detail).Thus it is not comfortable to have the charmed baryon wave function at the origin substantially smaller than those in bottom and hyperon systems. 9Fortunately, this is no longer an issue in the BM where L q B Q varies less than 10% from the bottom to the charm sector.
• Both the BM and NRQM agree with each other in I q Bc but differ largely in I q B b .Meanwhile, the HQET and QCD sum rules give smaller values of L q Λ b = −(3.2± 1.6) [33] and −(2.38 ± 0.11 ± 0.34 ± 0.22) [34], respectively, in units of 10 −3 GeV 3 , which are much closer to the values of BM than that of NRQM.
To compute the lifetimes, we have to evolve I q B Q to the energy scale µ Q of the heavy quark where the formalism is derived.Unfortunately, the matrix elements of the four-quark operators diverge to the leading-order of α s .To make sense out of the calculation, one has to regularize them.In turn, subtracting the infinity induces a renormalization-scheme dependence.In HQET, the heavy quark Q is treated as a static one.The renormalization-group evolution of the four-quark operator matrix elements are then given by [35] where κ = α s (µ H )/α s (µ Q ) and the relation 2t a ij t a kl = δ il δ jk − δ ij δ kl /3 has been used in the derivation.We then arrive at whereas Ĩq B Q remains unchanged, leading to B(µ Q ) = 0.64 ± 0.09 and 0.79 ± 0.11 for B b and B c , respectively.
To explore the renormalization-scheme dependence of I q B Q , we consider the full QCD operator matrix elements in the MS scheme.The evolution of I q B Q can be obtained by the fact that the anomalous dimension matrix is diagonalized in the operator basis Therefore, we shall have where C ± (µ) are the Wilson coefficients and we have neglected the mixing with the penguin operators.The equality is derived by the fact that the amplitudes shall not depend on µ.Taking (µ 1 , µ 2 ) = (µ Q , µ H ), we obtain In this renormalization scheme, a straightforward conclusion is that Lq In other words, B(µ) = 1 holds independently of µ for L q B Q in this scheme.Let us return back to Eq. ( 41) and obtain In the leading-logarithmic approximation, the results are We see that the evolution of L q B Q are similar in both schemes, where L q B b and L q Bc are enhanced by 50% and 20%, respectively.Accordingly, we assume the evolution in Eq. ( 42) is also applicable for both S q B Q and P q B Q .
FIG. 1: The leading-order Feynman diagram for C 3 , where the ⊗ represents the insertion of the effective Hamiltonian for Q → q 1 q 2 q 3 .

III. DECAY WIDTHS
The nonperturbative baryonic matrix elements are collected in Tables IV and V and their values at the energy scale µ Q are obtained through Eqs. ( 25), (37) for HQET and ( 42) for full QCD.With these building blocks, we are in a position to compute the lifetimes of heavy baryons.In the beauty baryon decays, (m s /m b ) 2 can be safely neglected, while shall be taken into account in charmed baryon decays.We briefly discuss the expressions of C n , Õ6 and Õ7 with n = 3, 5, ρ from the literature.

A. Two-quark operators
As the experiments are able to probe the semileptonic inclusive decay widths, it is natural to decompose C n into the form where c 1 and c 2 are the leading Wilson coefficients for the effective Hamiltonian with ξ Q being the CKM matrix elements, and c 3-6 are the Wilson coefficients for QCD penguin operators.Note that the byproducts of c i c j with i, j ≥ 3 are discarded in the next-to-the-leading order (NLO).To evaluate C 3 , we take Q → q 1 q 2 q 3 as an illustration, depicted in Fig. 1, where the ⊗ represents the insertion of H ef f .To the leadingorder (LO), C 3 can be obtained by matching Fig. 1 to C 3 QQ.Clearly, the coefficient C 3 depends on the masses of q 1,2,3 appearing in the loop integral.To the NLO with i, j ∈ {1, 2}, the results of K 3 with a single massive q 3 can be found in Ref. [37], and the doubly massive cases with q 1 = q 3 are available in Ref. [38].The expressions with semitauonic decays at the LO are found in Refs.[39,40].The coefficients for the penguin operators are given in Ref. [41].On the other hand, C 5 is available at NLO for the semileptonic and nonleptonic decays with massive and massless final state quarks, respectively [42,43].We use the LO coefficients with massive q 1 and q 3 [44,45].We note that C 5 slightly affects the inclusive decay widths.In particular, µ 2 G = 0 for T Q so their results are not affected by C 5 .On the other hand, the explicit values of K NL D,ij and K SL D are given in Refs.[29,[46][47][48], where we shall only use the LO results.

B. Four-quark operators
The dimension-six operator of Õ6 consists of four types of topologies where the subscripts we, int+, int− and SL stand for W -exchange, Pauli constructive interference, Pauli destructive interference and semileptonic, respectively.Their topological diagrams without QCD corrections are depicted in Fig. 2, where Õint+ and ÕSL share the same topology, i.e.Fig. 2(c).Four-quark operators for W -exchange are decomposed according to their spinor structure Further, the coefficients are disintegrated as Thus far we have taken Õwe and C V −A we as an example and the rest of them can be defined in the same manner.The explicit expressions of K we and K int− to the NLO are given in Refs.[49] and [28] for the nonleptonic and semileptonic decays, respectively, while C int+ is obtained by the Fierz transformation with massless q 1,2,3 .
It should be noticed that the full QCD theory and HQET have different coefficients of K's which are related by Eq. ( 16) in Ref [49].In HQET, one takes the limit of m Q → ∞ and the series of the HQE is truncated to a specific order of 1/m Q .In the BM, we do not take the heavy quark limit to evaluate I q B Q .Therefore, the central values of the computed lifetimes are evaluated with the Wilson coefficients given in the full QCD theory, whereas the deviations from the HQET ones are treated as the uncertainties.
In the numerical evaluation we have taken the NLO and the Cabibbo suppressed contributions into account.As an illustration, we write down the expressions of the Cabibbo-favored decays to LO.We decompose Õ6 into several parts where Γ contains two massive quarks in the loop integral, while the others without tilde have only one massive quark.In the cases of (Q, q 1 , q 2 , q 3 ) ∈ {(c, d, u, s), (b, u, d, c)}, q 1 The dimension-7 operators share the same topological diagrams with the dimension-6 ones, but the NLO correstions are currently still absent.In analogy to Eq. ( 49), we decompose their contributions as [52] Contrary to the previous dimension-6 case, the subscripts int+ and int− here are referred to Figs. 2(b) and 2(c), respectively.We find [11,28] Γ for (Q, q 1 , q 2 , q 3 ) = (b, c, s, c).In Eqs. ( 53) and ( 54), P q 1,2,3,4 are understood as P q 1,2,3,4 B Q instead of dimension-7 operators for simplicity.The discussions are parallel to the dimension-6 ones.By putting x b = 0, we find that ΓB b ,s 7,int− = Γ B b ,s 7,int− and that x Q is absent in Γ B Q ,q 3 7,int− as q 3 does not get involved in the loop integral.
Eqs. ( 53) and ( 54) are derived from the full QCD theory.If one alternatively adopts HQET instead, there will be several extra terms coming from the 1/m Q corrections to the dimension-6 operators [11,49].After including them, the deviations between two theories start at the dimension-8 level.
We next turn to the semileponic inclusive decay widths.Notice that it is possible to have (q 1 , q 2 ) = (ℓ, ν e ) in Fig. 2(c).Therefore, the inclusive semileptonic decay widths also receive corrections from dimension-6 and -7 operators [53].Ignoring the small ratio z = (m µ /m c ) 2 ≈ 1%, the matrix elements are obtained by setting (c 6,int+ and Γ Bc,s 7,int− , given by This completes the investigation in the contributions of four-quark operators.We stress again that the NLO contributions of the dimension-6 operators and Cabibbosuppressed decays have been taken into account in the numerical evaluation.

IV. NUMERICAL RESULTS AND DISCUSSIONS
Since the total inclusive decay widths are proportional to m 5 Q to the leading-order of the 1/m Q expansion, the numerical results are very sensitive to how well the heavy quark mass is under control.Unfortunately, the choice of the heavy quark mass definition is quite arbitrary to the first-order in QCD, and the pole mass definition, with which the formalism is derived, suffers from the divergences due to the infrared renormalon, imposing a minimal ambiguity proportional to Λ QCD [54].
For the heavy quark mass m Q in Eq. ( 6), we shall take the pole mass 10   (m b , m c ) pole = (4.70 ± 0.10, 1.59 ± 0.09) GeV , where the upper and lower bounds correspond to the two-loop and one-loop results.
Here we do not consider the other mass schemes such as MS, kinetic, MSR and 1S which proves to be suitable for the B-meson inclusive decays [55], see Eq. (3.1) and the subsequent discussion in Ref. [55].
schemes as they have to be matched with the pole mass one eventually.The deviations among them are mainly generated by truncating the series of the α s expansion (see Eq. (2.54) in Ref. [23], for instance).As we have allowed a wide range for the pole mass, it will cover the corrections to the order of O(α 2 s ). 11On the other hand, for the quark masses appearing in the loop integral, we fix them with the MS scheme [56] (m c (m c ), m s (2 GeV)) MS = (1.3,0.09) GeV . ( For the ratio m ℓ /m Q in the loop integral we take m Q as the ones in Eq. (56).
To be consistent with the NLO corrections in Γ 6 , we use the LO values of the Wilson coefficients from Tables VII and XIII of Ref. [57] (c 1 , c 2 ) = (1.298,−0.565) , at µ b = 4.4 GeV for bottom quark decays.The calculated results of heavy baryon decay widths are summarized in Table VI, where Γ 3 stands for the two-quark operator contributions with ρ 3 D = 0 and Γ ρ is the decay width contributed from the Darwin operator.The semileptonic inclusive branching fractions are defined by with ℓ = e, µ, τ .Since the NLO corrections to Γ 7 are still absent currently, we shall use the LO value of Γ 7 for τ at the NLO.In the table, the uncertainties arising from m Q , µ H and I q B Q are denoted by the subscripts m, µ and 4, respectively.The central values are evaluated with full QCD operators, and the deviations from HQET ones are treated as uncertainties denoted by the subscript s.Error analyses are not provided for Γ 6,7 in the table for simplicity as their dependence on m Q is rather weak .The errors from µ H and I q B Q will cause around 11% and 10% uncertainties to Γ 6,7 and ρ 3 D , respectively.As ρ 3 D is proportional to α s , the Darwin operator does not contribute at the LO and its effect is negligible compared to other hadronic uncertainties at the NLO.
We see that the differences among the B Q 's are mostly ascribed to the four-quark operators.In particular, Γ 3 is the same for all antitriplet baryons T Q .For B c , the NLO corrections are roughly 50% in both Γ NL 3 and Γ NL 6 .Assuming the α s expansion behaves like a geometric series, there will be roughly 33% deviations in the lifetimes once the α s corrections are included to all orders.On the other hand, the NLO corrections are found to be around 20% in B b .From Table VI we see that all the predicted lifetimes are improved to the NLO except for τ (Ξ + c ).A good sign of the α s expansion is that the uncertainties caused by µ H are systematically reduced once the NLO corrections are included.
It is evident from Table VI that Γ SL 7 contributes destructively to the total semileptonic rate Γ SL , while Γ NL 7 contributes constructively to Γ NL for Λ + c and Ξ 0 c and destructively for Ω 0 c and Ξ + c .Owing to the presence of two strange quarks in the Ω 0 c , Γ SL 7 (Ω 0 c ) and Γ NL 7 (Ω 0 c ) are the largest in magnitude among the charmed baryons.Consequently, the lifetime hierarchy pattern τ The Ω 0 c could live longer than the Λ + c due to the suppression from 1/m c corrections arising from dimension-7 four-quark operators.
The 1/m c expansion in the matrix elements of four-quark operators seems to work reasonably well for the antitriplet charmed baryon T c where Γ 7 is around 40% compared to Γ 6 .However, the 1/m c expansion could be problematic for the Ω c if Γ SL 7 (NLO) is larger than Γ SL 6 (NLO) in magnitude, leading to a smaller or even negative semileptonic rate Γ SL (Ω 0 c ).Also recall that the deviation of the full QCD from HQET is huge for the Ω c .A future study of NLO corrections to Γ 7 and dimension-8 operators will be useful to clarify the issue.Although the computed results for τ (B c ) are compatible with the TABLE VI: Results for the lifetimes of heavy baryons, where the decay widths and lifetimes are in units of 10 −12 (10 −13 ) GeV and 10 −13 s (10 −12 s ), respectively, for B c (B b ).Uncertainties arising from m Q , µ H , I q B Q and the deviation of full QCD from HQET are denoted by the subscripts m, µ, 4 and s, respectively.In the table, NLO in the first column stands for the numerical results to the NLO precision.The NLO values of Γ 7 are taken to be the same as the ones at the LO.[23] obtained in the pole mass scheme for charmed baryons and Ref. [24] in the kinetic mass scheme for bottom baryons, of which the uncertainties are added quadratically.The baryon matrix elements in Refs.[23,24] are evaluated using the NRQM.Experimental results are quoted from Ref. [1] and Table I.The lifetimes are in units of 10 where the predictions of NRQCD are quoted from Ref. [24].We see that while the To compare the BM and NRQM, we collect the calculated BF SL e and τ in Table VII.The numerical values of Λ + c in both models are consistent with experiment.Nevertheless, the results of BF SL e (Ξ + c ) and BF SL e (Ω 0 c ) deviate largely.Notice that if we focus only on the Cabibbo-favored semileptonic decays of c → se + ν e , then we will have under the isospin symmetry, which is well respected by the BM.As for BF SL e (Ω 0 c ), the difference stems from the treatment for dimension-7 operators.If we take E s = 0.33 GeV instead of 0.5 GeV (see the discussions below Eq. ( 34)), our value will turn out to be compatible with Ref. [23].Conversely, if we take Λ QCD = 0.5 GeV for the NRQM, the LO value of Γ SL 7 will be larger than Γ SL 6 in magnitude in Table 24 of Ref. [23], resulting in a much smaller value of BF SL e (Ω 0 c ).
As a final remark, we point out that Ξ Q − Ξ ′ Q mixing is of the order of m s /m Q .It will induce a nonzero µ 2 G in Ξ Q [58] but its effect is beyond the scope of this work.

V. CONCLUSION
We have studied the inclusive decay widths of singly heavy baryons with the final results summarized in Table VI.The nonperturbative baryon matrix elements are estimated using the improved bag model in which the unwanted CMM has been removed.
The running of µ 2 G and I q B Q was discussed under the full QCD theory and HQET.To the leading-order of α s , we found that B = 1 holds irrespective of the energy scale in the MS scheme of the full QCD theory.The results of µ 2 π , µ 2 G and ρ 3 D are in good agreement with the literature but a large discrepancy is found in I q B b .The requirement of I q B b ≈ I q Bc from the heavy quark limit holds nicely in the BM but is badly broken in the NRQM.In particular, L s Ω b obtained in the latter is nearly four times larger than the one in the former.As a result, we have τ ∆ (Ω − b ) = (8.34± 2.22)% in contrast to (13.2 ± 4.7)% obtained in the NRQM, while the current data lead to (11.5 +12.2 −11.6 )%.
We found an excellent agreement between theory and experiment for the lifetimes of bottom baryons even at the dimension-6 level.Effects of dimension-7 operators are rather small.As for charmed baryons, the calculated lifetimes are consistent with the current experiments and we found that the established new hierarchy τ (Ξ + c ) > τ (Ω 0 c ) > τ (Λ + c ) > τ (Ξ 0 c ) is traced back to the destructive contributions from the dimension-7 operators in the Ω 0 c .This confirms the speculation made in Ref. [11], namely, the Ω 0 c , which is naively expected to be shortest-lived in the charmed baryon system owing to the large constructive Pauli interference, could live longer than the Λ + c due to the suppression from 1/m c corrections arising from dimension-7 four-quark operators.
To discriminate between the BM and NRQM approaches, we recommend to measure BF SL e (Ξ + c ) and BF SL e (Ω 0 c ) in the forthcoming experiments as the BM and NRQM differ significantly mainly due to the treatment of dimension-7 operators.A possible sign for the failure of HQE in the Ω 0 c will occur if Γ SL 7 (NLO) is larger than Γ SL 6 (NLO) in magnitude but opposite in sign, leading to a smaller or even negative semileptonic rate Γ SL (Ω 0 c ).Hence, a study of Γ 7 at the NLO and dimension-8 operators is needed to settle down the issue.Foundation of China under Grant No. 12205063.
50 GeV.The spatial derivatives have been omitted as they are proportional to O(1/m Q ) and the last equation follows as the bag quarks are approximately in the energy eigenstates with

FIG. 2 :
FIG. 2: The topological diagrams for the spectator effects: (a) W -exchange, (b) destructive (constructive) Pauli interference and (c) constructive (destructive) Pauli interference at the dimension-6 (dimension-7) level.There is no 4-point vertices between the W bosons and quarks in the diagrams (b) and (c).

10
To the leading-order of α s , m pole b = 4.70 GeV corresponds to m kin b = 4.57 GeV with µ cut = 1 GeV,

11
For example, (m MS c , m kin c , m MSR c ) = (1.28,1.40, 1.36) GeV correspond to m pole c = (1.52,1.67, 1.49) GeV to the first-order correction in α s , where we have used α s (m c ) = 0.38 and µ cut = 1 GeV for the kinetic and MSR mass schemes.For detailed discussions, see Sec. 2.4 of Ref. [23].

17 a 3 ≫Γ 6 >
Corresponding to the results in which the CMM is removed from the bag model.current data, we need to keep in mind the possible shortcomings in the higher-order 1/m c expansion.As for the bottom baryon B b , HQE works nicely, obeying the hierarchy 12 Γ Γ 7 .Nonetheless, m b ≫ Λ QCD is a double blade.Although it ensures the validity of HQE, the numerical results are extremely sensitive to m Q .If we alternatively take τ exp (B) as input to fix m b (see the footnote of Eq. (56)), then the uncertainties denoted by the subscript m can be discarded in Table VI for B b .To acquire the predictive power, we compute the lifetime ratios where the m Q dependence is largely canceled.We define τ ∆ (B b ) ≡ 1 − τ (B b )/τ (Λ b ) and find that τ ∆ (Ξ 0 b ), τ ∆ (Ξ − b ), τ ∆ (Ω − b lifetime pattern τ (Ω − b ) > τ (Ξ − b ) > τ (Ξ 0 b ) > τ (Λ0b ) is respected by theory and partially by experiment, we predict smaller values for τ ∆ (Ξ − b ) and τ ∆ (Ω − b ) compared to the NRQM due to the difference in the ratio I s B b /I u,d B b and the treatment of C G (m Q , µ).

TABLE I :
Evolution of the charmed baryon lifetimes measured in units of fs.
d m s m c m b

TABLE III :
The parameters λ 1 and λ 2 calculated in the bag model in units of GeV 2 .

TABLE VII :
Comparison of our results with Ref.
−13s for B c and 10 −12 s for B b .