IRB-TH-4/99

July, 1999.

Enhancement of preasymptotic effects in

inclusive beauty decays

B. Guberina∗, B. Melić†, H. Štefančić‡

Theoretical Physics Division, Rudjer Bošković Institute,

P.O.Box 1016, HR-10001 Zagreb, Croatia

Abstract

We extend Voloshin’s recent analysis of charmed and beauty hyperon decays based on SU (3) symmetry

and heavy quark effective theory, by introducing a rather moderate model-dependence, in order to obtain

more predictive power, e.g. the values of lifetimes of the (Λb, Ξb) hyperon triplet and the lifetime of Ωb.

In this way we obtain an improvement of the ratio τ (Λb)/τ (B0d ) ∼ 0.9 and the hierarchy of lifetimes
τ (Λb) ' τ (Ξ0b ) < τ (Ξ−b ) < τ (Ωb) with lifetimes of Ξ−b and Ωb exceeding the lifetime of Λb by 22% and
35%, respectively.

PACS: 14.20.Mr, 13.20.He, 12.39.Hg, 12.39.Jh

Keywords: beauty baryons, lifetimes, inclusive decays, four-quark operators

∗guberina@thphys.irb.hr
†melic@thphys.irb.hr
‡shrvoje@thphys.irb.hr

1



Weak decays of beauty mesons and baryons are believed to be a nice playground where a

variety of phenomena should be well described and understood in the framework of the operator

product expansion (OPE) and heavy quark effective theory (HQET) [1, 2]. The essential underlying

idea in both theories is the expansion in inverse powers of heavy-quark mass – the mass of the

beauty quark, mb ∼ O(5 GeV ), is considered to be heavy compared with the typical hadron scale
of 0.5 − 1 GeV . This is to be compared with the case of charmed mesons and baryons, where the
mass of the charmed quark, mc ∼ 1.3 GeV , is hardly an ideal expansion parameter.

The rate of the beauty-hadron decay is given by

Γ(Hb → f ) =
G2Fm

5
b

192π3
|V |2 1

2MHb
{cf3〈Hb|O3|Hb〉 + cf5

〈Hb|O5|Hb〉
m2b

+
∑

i

c
f
6

〈Hb|Oi6|Hb〉
m3b

+ O(1/m4b) + ...} , (1)

where c
f
j are the Wilson coefficients and

1

mD−3b
〈Hb|OD|Hb〉 (2)

are matrix elements of the D-dimensional operators which appear in the OPE multiplied by the

appropriate power of inverse quark mass. The sum in (1) starts with D = 3, i.e. with O3 = bb

giving

1

2MHb
〈Hb|O3|Hb〉 = 1 + O(1/m2b ) . (3)

Clearly, in the asymptotic limit mb → ∞, one recovers the parton model result – as long as mb
is large enough, one expects all corrections to stay moderate. Furthermore, it is obvious from (3)

that there are no 1/mb corrections – a consequence of the nonexistence of independent operators

of dimension four.

The experimental situation is as follows: the lifetimes of beauty hadrons follow the simple

theoretical mb → ∞ prediction within 5 − 10%:

τ (B+) = τ (B0d ) = τ (B
0
s ) = τ (Λb) , (4)

2



except for the lifetime of Λb, which appears to be by 15 − 25% smaller than predicted in (4). More
precisely [3],

τ (B+)

τ (B0d )
= 1.07 ± 0.03 , (5)

τ (Λb)

τ (B0d )
= 0.81 ± 0.05 . (6)

The lifetimes of b hadrons are

τ (B0d ) = (1.54 ± 0.03) ps ,

τ (B+) = (1.65 ± 0.03) ps ,

τ (Ξb mixture) = (1.39
+0.34
−0.28 ) ps

τ (Λb) = (1.24 ± 0.08) ps . (7)

Theoretical estimates [4] predict the ratio (5) to be 1 + 0.05(fB/200MeV )
2, in accordance

with experiment, but the ratio (6) is predicted to be in the range

τ (Λb)

τ (B0d )
∼ 0.95 − 0.98 , (8)

which seems to be an overestimate.

It appears, however, that the ratio τ (Λb)/τ (B
0
d ) is not easy to lower down to the experimental

value, the reason being that the 1/mb expansion converges rapidly. For example, keeping only

operators with D = 3 and D = 5, one obtains 0.98 for the ratio (6). Thus it seems difficult to

accommodate this ratio with the same mass mb entering the decay rates of both Λb and B
0
d. In fact,

strangely enough, putting the physical hadron masses instead of mb would give τ (Λb)/τ (B
0
d ) = 0.73,

up to the O(1/m2b ), in good agreement with experiment. However, this nice Ansatz, proposed in

[5] completely spoils the OPE and contradicts other OPE predictions confirmed by experiments.

Therefore, the only hope to obtain the ratio (6) in the framework of the OPE and HQET

is to look for the possible larger contributions coming from the operators with dimension D = 6

or higher. These operators are known to play an important role in charmed-meson decays, in

3



which, owing to the Pauli interference effect [6, 7, 8], there is a dilation of the lifetime of the D+

meson. In charmed-baryon decays, their role is even more pronounced: they give the dominant

contribution leading to the well-established lifetime hierarchy which was successfully predicted

prior to experiment [9, 10]. Unfortunately, the calculation of the matrix elements of the operators

with dimension D = 6 requires the use of quark models and is, therefore, strongly model dependent.

Recently, Voloshin [11] proposed the way to avoid the use of phenomenological models. He

showed that using SU(3) symmetry and HQET it was possible to relate the measured lifetimes

of charmed hyperons to the differences in semileptonic decay rates, the differences in the Cabibbo

suppressed decay rates of charmed hyperons and the splitting of the total decay rates of b hyperons,

without invoking the quark model results for the matrix elements [12, 13]. He confirmed the

predicted difference in the semileptonic decay rates between the Ξc and Λc by a factor 2 to 3, and

the enhancement of the semileptonic branching ratio for Λ+c coming from the Cabibbo suppressed

decay rate. When applied to beauty decays, Voloshin’s approach leads to a difference of 14% in

the lifetime of Ξ−b with respect to the lifetime of Λb.

In this paper we extend Voloshin’s analysis introducing a rather moderate model dependence,

in order to obtain more predictive power, e.g. the values for the lifetimes of the (Λb, Ξb) hyperon

triplet and the lifetime of Ωb. Basically, we express the decay rates in terms of the nonrelativistic

(NR) wave function at the origin Ψ(0), the value of which we determine using Voloshin’s method.

Our starting point is the expression (1). It is argued [14] that the beauty mass which enters

the expression (1) is a running mass mb(µ). In the limit µ → 0, one obtains the pole mass
which is very often used in calculations. It would be perfectly legitimate to use the pole mass

in pure perturbative theory (with no nonperturbative contribution). However, the use of the

pole mass is very problematic when nonperturbative corrections are calculated, because of the

renormalon singularities resulting in an irreducibile uncertainty of O(ΛQCD/mb) that is larger

than the nonperturbative corrections we are calculating. Shielding mb(µ) against renormalon

ambiguities by choosing µ > 0.5 GeV , one avoids problems with the pole mass. In fact, a natural

choice for the scale µ is mb/5 ∼ 1 GeV , as argued in [15]. Such a relatively low scale makes the

4



MS mass inadequate for treating the decays. A natural definition of the running mass would be

that with the linear dependence on µ:

d m(µ)

d µ
= −cm

αs(µ)

π
+ ... . (9)

The recent value for mb(µ) at µ ∼ 1 GeV and for cm = 16/9 is given by [16, 17]

mb(µ = 1 GeV ) = (4.59 ± 0.08) GeV , (10)

which is slightly lower than the usual values. In this calculation we use mb(µ = 1 GeV ) in the

range 4.6 GeV < mb(1 GeV ) < 4.8 GeV .

Next, we turn to the calculation of the matrix elements of the Oi6 (four-quark) operators.

We follow the approach given by Voloshin [11] based on HQET and flavor SU(3) symmetry. A

suitable parameter to express these matrix elements is the effective decay constant F
ef f
B which is

an analogue of the static decay constant used to evaluate four-quark matrix elements in decays of

heavy baryons [2, 13, 18].

We use the following two differences of decay rates: For ∆1b = Γ(Λb) − Γ(Ξ0b ), we have

∆1b =
G2F m

2
b

4π
| Vcb |2 (c2 − s2)(

√
1 − 4z − (1 − z)2(1 + z))[C5(mb)x + C6(mb)y], (11)

where z = m2c /m
2
b and c and s stand for cos θc and sin θc, respectively (θc is the Cabibbo angle).

For ∆2b = Γ(Ξ
−
b ) − Γ(Λb), we have

∆2b =
G2F m

2
b

4π
| Vcb |2 [l1x + l2y], (12)

where l1 and l2 are the abbreviations for the following expressions:

l1 = (1 − z)2C1(mb) − [c2
√

1 − 4z + s2(1 − z)2(1 + z)]C5(mb), (13)

l2 = (1 − z)2C2(mb) − [c2
√

1 − 4z + s2(1 − z)2(1 + z)]C6(mb). (14)

In the equations displayed above, Ci stand for special combinations of Wilson coefficients described

in [11], while x and y denote combinations of heavy-baryon matrix elements introduced first in the

5



same reference:

x =
〈 1
2
(bΓµb)[(uΓµu) − (sΓµs)]

〉
Ξ
−
b
−Λb

=
〈 1
2
(bΓµb)[(sΓµs) − (dΓµd)]

〉
Λb−Ξ0b

, (15)

y =
〈 1
2
(biΓµbj )[(ujΓµu

i) − (sj Γµsi)]
〉

Ξ
−
b
−Λb

=
〈 1
2
(biΓµbj )[(sj Γµs

i) − (djΓµdi)]
〉

Λb−Ξ0b
, (16)

Similar relations are valid (through HQET and SU(3) symmetry) for the respective members of

the charmed hyperon triplet [11].

The procedure of extraction of the effective parameter F
ef f
B is based on equating expressions

obtained in two approaches. In the first approach, for the matrix elements x and y we use the

SU(3) hypothesis which basically comprises using values of matrix elements extracted from exper-

imental data on charmed baryons for calculations in the beauty-baryon sector. This approach is

based on the assumptions of SU(3) and heavy-quark symmetry. In the second approach, x and

y are calculated using the nonrelativistic quark model, already frequently employed for similar

calculations [2, 13, 18]. Within this model, for x and y we have

x = −y = − | ΨΛb (0) |2 (17)

Equation (17) clearly shows that the valence approximation is used in the calculation of the matrix

elements. The connection between the wave function squared, | ΨΛb (0) |2, and F ef fB is given by
the relation [19, 20]

| ΨΛb (0) |2= T (F ef fB )2, (18)

where

T = 4
M(Σ0b ) − M(Λ0b )
M 2(B∗) − M 2(B) m

∗
u(

1

12
M(B)κ(µ)−

4
9 ). (19)

Here µ ∼ 1 GeV is a typical hadronic scale of hybrid renormalization κ. The decay rate differences
obtained in the first and in the second approach are denoted by ∆ib,SU (3) and ∆

i
b,model, respectively

(i = 1, 2).

The effective parameter F
ef f
B,i is now extracted from the equation

∆ib,SU (3) = ∆
i
b,model. (20)

6



The expressions obtained for FBef f,i, i = 1, 2, are

F
ef f
B,1 =

√√√√ C5(mb)x + C6(mb)y
T (C6(mb) − C5(mb))

, (21)

F
ef f
B,2 =

√
l1x + l2y

T (l2 − l1)
. (22)

The final numerical value is calculated taking into consideration the errors of the expressions

(21) and (22) and combining all numerical values appropriately. For mc = 1.25 GeV and mb =

4.7 GeV , we obtain

F
ef f
B = (0.441 ± 0.026) GeV . (23)

The parameter F
ef f
B shows a slight mass dependence which was incorporated in numerical calcu-

lations.

Next, the numerical value displayed in (23) is used in (18) to obtain the values of the Oi6

matrix elements. As all the matrix elements in the expression (1) for the decay rate are now

available, we can calculate the lifetimes of beauty baryons accordingly. The lifetimes of B mesons

are calculated in a ”standard” way using the B-meson decay constant fB.

Since absolute results for lifetimes are not so reliable owing to ambiguities in quark mass, we

shall express our results mainly in the form of lifetime ratios.

Our results for the ratio rΛB = τ (Λb)/τ (B
0
d) are shown in Fig.1. The Wilson coefficients in (1)

have been calculated at one loop using ΛQCD = 300 MeV . Other relevant numerical parameters

used throughout the paper are [1, 2] µ2π = 0.5 GeV
2, µ2G(Ωb) = 0.156 GeV

2, µ2G(Λb, Ξb) = 0. The

effect of introducing F
ef f
B , which we have calculated, is to bring the ratio rΛB from 0.96 to 0.9

– still two standard deviations from experiment. It is clear from Fig.1 that variation in mb has

almost no effect. Also, the variation of µ2π does change rΛB at the permile level. If the experimental

ratio (6) persists, then there might be the problem in b decays.

An interesting effect in this approach noticed by Voloshin is the enhancement of the lifetime

of Ξ−b compared with the lifetime of Λb. Using the ”standard” B-meson decay constant fB ∼

7



4.60 4.65 4.70 4.75 4.80
mb (GeV)

0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

1.10

τ(
Λ

b)
/τ

(B
d)

mc = 1.25 GeV
mc = 1.35 GeV
mc = 1.15 GeV

Figure 1: The shaded area represents the experimental value of the ratio rΛB within one standard

deviation. The line with diamonds represents the calculated value of rΛB for the ”standard” value

fB = 160 MeV . The values of rΛB using F
ef f
B are calculated for three different values of mass

mc and are represented by lines without symbols. The significant shift from the ”standard” value

result is visible, but the deviation from the experimental band is still substantial.

160 MeV , instead of F
ef f
B , one obtains for the ratio τ (Ξ

−
b )/τ (Λb) ∼ 1.03. Our calculation gives,

Fig.2,

τ (Ξ−b )
τ (Λb)

' 1.22 , (24)

i.e. a relative enhancement of the τ (Ξ−b )/τ (Λb) ratio of the order 20%, which is in fair agreement

with the preliminary experimental results (7) [3]. The main reason for this enhancement is the large

(positive) exchange contribution to Γ(Λb) versus the large negative Pauli interference contribution

in Γ(Ξ−b ).

Such an enhancement is even more pronounced in the lifetime of Ωb, giving the ratio

τ (Ωb)

τ (Λb)
' 1.35 , (25)

i.e. a relative enhancement of the ratio τ (Ωb)/τ (Λb) of the order 30%, Fig.3. This result is a

8



4.60 4.65 4.70 4.75 4.80
mb (GeV)

1.00

1.05

1.10

1.15

1.20

1.25

1.30

τ(
Ξ

b−
)/

τ(
Λ

b)

mc = 1.25 GeV
mc = 1.15 GeV
mc = 1.35 GeV

Figure 2: The line with diamonds represents the value of the ratio τ (Ξ−b )/τ (Λb) for the value

fB = 160 MeV . Values of the same ratio calculated using F
ef f
B are given for three mc masses and

are represented by lines without symbols. The difference of ∼ 20% is clearly visible.

consequence of the even stronger (compared with Ξ−b ) negative Pauli interference contribution

in Γ(Ωb). The results (24) and (25) should both serve as crucial predictions to be checked in

experiment. The calculation gives τ (Ξ0b ) approximately the same as τ (Λb), which together with

(24) and (25) leads to the following lifetime hierarchy:

τ (Λb) ' τ (Ξ0b ) < τ (Ξ−b ) < τ (Ωb) . (26)

As long as the absolute value of the Λb lifetime is concerned, the effect of F
ef f
B is to lower the

theoretical value of τ (Λb) by ∼ 10%, giving

τ (Λb) ∼ 2.0 ps , (27)

which is too high compared with the measured value (1.24 ± 0.08) ps. To obtain better agreement
with experiment, one needs larger mb. If, for example, we use the pole masses m

pole
b = 5.1 GeV ,

mpolec = 1.6 GeV , the result is τ (Λb)
pole = 1.6 ps – still too high vis-à-vis experiment. However,

9



4.60 4.65 4.70 4.75 4.80
mb (GeV)

1.00

1.05

1.10

1.15

1.20

1.25

1.30

1.35

1.40

τ(
Ω

b)
/τ

(Λ
b)

mc = 1.25 GeV
mc = 1.35 GeV
mc = 1.15 GeV

Figure 3: The value of the ratio τ (Ωb)/τ (Λb) obtained using fB = 160 MeV is represented by the

line with diamonds. Values of the same ratio for F
ef f
B , represented by lines without symbols, are

calculated for three different mc masses. Calculations using fB and F
ef f
B differ by ∼ 30%.

playing with a large pole mass is merely an introduction of an additional parameter – a consistent

treatment requires having the running mass mb(µ) in the expansion and its value could hardly

reach more than 4.7 GeV .

Much the same situation appears in the calculation of B-meson lifetimes, which is not affected

by our approach. Typically, one obtains τ (B) ∼ 2 − 2.5 ps for mb = 4.6 GeV and τ (B) ∼
1.75 − 2.2 ps for mb = 4.7 GeV , the range of values for τ s coming from the variation of mc,
1.15 GeV < mc < 1.35 GeV . Comparing with the results for the calculated value of τ (Λb), one

sees that it is easier to have τ (B0d ) near to the experimental value. This may suggest that the

problem with too large a theoretical value of rΛB lies in the theoretical overestimate of τ (Λb).

To conclude, we point out the following. The calculations presented in this paper rely upon

HQET and flavor SU(3) symmetry and are therefore reliable up to violations of these symmetries.

Still, we expect the effects of these violations to be smaller than the main effect of our approach.

10



The procedure applied above significantly increases the contribution of four-quark operators and

numerical results show a significant, albeit still unsufficient shift towards experimental values,

especially in the case of the rΛB ratio. In our approach, to reach the experimental value of rΛB,

would require F
ef f
B to have the value 0.72 GeV , which can be hardly achieved. The discrepancy, still

remaining after increasing the preasymptotic effects coming from four-quark operators, indicates

that there should be other, yet unknown, sources of enhancement of preasymptotic effects and

that these effects should also produce significant contributions. Also, there remains the possibility

of violation of some of the underlying concepts, such as quark-hadron duality, but a consistent

treatment of these problems is still out of the reach of the present theory. In our approach, the

large contributions of the operators of D = 6 also suggest a much wider spread of lifetimes in the

sector of beauty baryons. The extent of this spread is to be verified by future experiments. At the

end, we state that a systematic application of the OPE, HQET and a moderately model-dependent

procedure of enhancement of preasymptotic effects improves the rΛB ratio significantly, although

it cannot resolve the problem completely. We consider this problem along with the problem of

absolute lifetimes of beauty hadrons to be one of the most important issues that heavy-quark

physics should address in the future.

Acknowledgements

This work was supported by the Ministry of Science and Technology of the Republic of Croatia

under the contract Nr. 00980102.

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