CERN-TH/96-19
SHEP 96-03

hep-ph/9603202

Spectator Effects in Inclusive Decays

of Beauty Hadrons

M. Neubert and C.T. Sachrajda∗

Theory Division, CERN, CH-1211 Geneva 23, Switzerland

Abstract

We present a model-independent study of spectator effects, which are
responsible for the lifetime differences between beauty hadrons. These
effects can be parametrized in terms of hadronic matrix elements of four
four-quark operators. For B mesons, the coefficients of the non-factorizable
operators turn out to be much larger than those of the factorizable ones,
limiting considerably the usefulness of the vacuum insertion approxima-
tion. Non-factorizable contributions to the lifetime ratio τ(B−)/τ(Bd)
could naturally be of order 10–20%, and not even the sign of these contribu-
tions can be predicted at present. In the case of the Λb baryon, heavy-quark
symmetry is used to reduce the number of independent matrix elements
from four to two. In order to explain the large deviation from unity in
the experimental result for τ(Λb)/τ(Bd), it is necessary that these baryon
matrix elements be much larger than those estimated in quark models.
We have also reexamined the theoretical predictions for the semileptonic
branching ratio of B mesons and charm counting, finding that, given the
present theoretical and experimental uncertainties, there is no significant
discrepancy with experiment.

(Revised Version)

CERN-TH/96-19
September 1996



∗On leave from the Department of Physics, University of Southampton, Southampton SO17 1BJ, UK

1



1 Introduction

In this paper we study “spectator effects” in inclusive decays of beauty hadrons.
These effects involve the participation of the light constituents in the decay and
thus contribute to the differences in the decay widths and lifetimes of different
species of beauty hadrons. Indeed, one of our goals is to understand theoretically
the experimental results for the lifetime ratios [1]:

τ(B−)

τ(Bd)
= 1.02 ± 0.04 ,

τ(Bs)

τ(Bd)
= 1.01 ± 0.07 ,

τ(Λb)

τ(Bd)
= 0.78 ± 0.05 . (1)

Here τ(Bs) refers to the average Bs-meson lifetime. Our study is performed in
the framework of the heavy-quark expansion, in which these ratios are computed
as series in inverse powers of the mass of the b quark [2]–[5] (for recent reviews,
see refs. [6, 7]). The leading term of this expansion corresponds to the decay
of a free b quark. This term is universal, contributing equally to the lifetimes
of all beauty hadrons. Remarkably, the first correction to this result is of order
(ΛQCD/mb)

2 [3, 4]. This leads to the theoretical predictions (see section 2 below)

τ(B−)

τ(Bd)
= 1 + O(1/m3b ) ,

τ(Bs)

τ(Bd)
= (1.00 ± 0.01) + O(1/m3b) ,

τ(Λb)

τ(Bd)
= 0.98 + O(1/m3b ) . (2)

The deviation from this expectation for τ(Λb)/τ(Bd) is striking, and is the prin-
ciple motivation for this study.

Spectator effects, i.e. contributions from decays in which a light constituent
quark also participates in the weak process, have first been considered in refs. [8]–
[10]. For decays of heavy particles, these effects are strongly suppressed due to the
need for the b quark and a light quark in the heavy hadron to be close together (i.e.
from a factor of the “wave-function at the origin”). As the portion of the volume
that the b quark occupies inside the hadron is of order (ΛQCD/mb)

3, spectator
effects appear only at third order in the heavy-quark expansion, and it might
seem safe to neglect them altogether. However, as a result of the difference in the
phase-space for 2 → 2-body reactions as compared to 1 → 3-body decays, these
effects are enhanced by a factor of order 16π2. It is conceivable that they could be
larger than the terms of order (ΛQCD/mb)

2 included in (2). Moreover, spectator

1



effects explicitly differentiate between different species of beauty hadrons. In
order to understand the structure of lifetime differences, it is therefore important
to reconsider the analysis of such effects. The striking experimental result for the
short Λb lifetime gives an additional motivation to such a study.

Previous studies of spectator effects in the decays of beauty hadrons [6] were
performed using the formalism developed in refs. [8]–[10] and made two simpli-
fying assumptions: first, the hadronic matrix elements of these operators were
estimated employing the vacuum insertion approximation [11] for mesons and
quark models [8, 12] for baryons, and secondly the mass of the charm quark was
neglected in the calculation of the coefficients of the four-quark operators in the
heavy-quark expansion. In order to explore the acceptable range of theoretical
predictions, we do not impose factorization or quark-model approximations on
the hadronic matrix elements. Instead, we parametrize them by a set of hadronic
parameters and see how the lifetime ratios depend upon them. One of our main
conclusions is that only a detailed field-theoretic calculation of the relevant ma-
trix elements can lead to reliable predictions. In addition, we derive the exact
expressions for the coefficients as functions of the charm-quark mass.

The semileptonic branching ratio of B mesons has also received considerable
attention. For many years it appeared that the theoretical predictions for this
quantity [13]–[15] lay above the measured value. More recently, the theoretical
predictions have been refined by including exact expressions for the O(αs) cor-
rections [16]. Using the results of these calculations, we present a new analysis
of the semileptonic branching ratio BSL and the average charm multiplicity nc in
B decays. We find that the freedom in the choice of the renormalization scale
(which reflects the ignorance of higher-order perturbative corrections) allows us
to obtain consistent predictions for both quantities simultaneously. We also cal-
culate the spectator contributions to BSL and nc and show that they could change
the semileptonic branching ratio by an amount of order 1%, whereas their effect
on nc is negligible.

In section 2, we discuss the heavy-quark expansion for inclusive decay rates,
and we present our results for the contributions arising from spectator effects.
In section 3, we introduce a set of hadronic parameters defined in terms of the
relevant four-quark operator matrix elements between B-meson and Λb-baryon
states. Heavy-quark symmetry is used to derive some new relations between
the baryonic matrix elements of these operators. In section 4, we then discuss
the phenomenological implications of our results for the understanding of beauty
lifetimes. We also present a critical discussion of previous estimates of spectator
effects based on the factorization approximation. A detailed discussion of the
semileptonic branching ratio of B mesons is presented in section 5. Section 6
contains the conclusions. The renormalization of the operators and parame-
ters describing the spectator effects, and the behaviour of these parameters with
respect to the large-Nc limit, are discussed in appendix A, while appendix B
contains details about the calculation of the semileptonic branching ratio.

2



The reader who is primarily interested in the phenomenological implications
of our analysis can omit sections 2 and 3 and proceed directly to sections 4, 5
and 6, which are written in a self-contained way.

2 Heavy-quark expansion

Inclusive decay rates, which determine the probability of the decay of a particle
into the sum of all possible final states with a given set of quantum numbers
{f}, have two advantages from the theoretical point of view: first, bound-state
effects related to the initial state can be accounted for in a systematic way using
the heavy-quark expansion; secondly, the fact that the final state consists of a
sum over many hadronic channels eliminates bound-state effects related to the
properties of individual hadrons. This second feature is based on the hypothesis
of quark–hadron duality, i.e. the assumption that cross sections and decay rates
are calculable in QCD after a “smearing” procedure has been applied [17]. We
shall not discuss this hypothesis here; however, if after the non-perturbative eval-
uation of the spectator effects discussed in our analysis there remain significant
discrepancies between theory and experiment (for the lifetime ratio τ(Λb)/τ(Bd),
in particular), one may have to seriously question the assumption of duality. A
recent study of inclusive B decays, in which duality violations are invoked to add
non-perturbative contributions of order ΛQCD/mb not present in the heavy-quark
expansion, can be found in ref. [18].

Using the optical theorem, the inclusive decay width of a hadron Hb containing
a b quark can be written as the forward matrix element of the imaginary part of
the transition operator T,

Γ(Hb → X) =
1

mHb
Im〈Hb|T |Hb〉 =

1

2mHb
〈Hb|Γ |Hb〉 , (3)

where T is given by

T = i
∫

d4xT{Leff (x),Leff (0)} . (4)

For the case of semileptonic and non-leptonic decays, the effective weak La-
grangian, renormalized at the scale µ = mb, is

Leff = −
4GF
√

2
Vcb

{
c1(mb)

[
d̄′LγµuL c̄Lγ

µbL + s̄
′
LγµcL c̄Lγ

µbL
]

+ c2(mb)
[
c̄LγµuL d̄

′
Lγ

µbL + c̄LγµcL s̄
′
Lγ

µbL
]

+
∑

`=e,µ,τ

¯̀
Lγµν` c̄Lγ

µbL

}
+ h.c. , (5)

3



where qL =
1
2
(1 −γ5)q denotes a left-handed quark field, d

′ = d cos θc + s sin θc
and s′ = s cos θc − d sin θc are the Cabibbo-rotated down- and strange-quark
fields (sin θc ' 0.2205), and we have neglected b → u transitions. The Wilson
coefficients c1 and c2 take into account the QCD corrections arising from the fact
that the effective Lagrangian is written at a renormalization scale µ = mb rather
than mW . They can be calculated in perturbation theory. The combinations
c± = c1 ±c2 have a multiplicative evolution under change of the renormalization
scale. To leading order, they are given by [19]–[21]

c±(mb) =

(
αs(mW )

αs(mb)

)a±
, a− = −2a+ = −

12

33 − 2nf
. (6)

In the numerical analysis we shall take the values c+(mb) ' 0.86 and c−(mb) =
1/c2+(mb) ' 1.35, corresponding to αs(mZ) = 0.117.

Since the energy release in the decay of a b quark is large, it is possible
to construct an Operator Product Expansion (OPE) for the bilocal transition
operator (4), in which it is expanded as a series of local operators with increasing
dimension, whose coefficients contain inverse powers of the b-quark mass. The
operator with the lowest dimension is b̄b. There is no independent operator with
dimension four, since the only candidate, b̄ i /Db, can be reduced to b̄b by using
the equations of motion [3, 4]. The first new operator is b̄gsσµνG

µνb and has
dimension five. Thus, any inclusive decay rate of a hadron Hb can be written in
the form

Γ(Hb → Xf ) =
G2Fm

5
b

192π3
1

2mHb

{
c
f
3 〈Hb| b̄b |Hb〉 + c

f
5

〈Hb| b̄gsσµνG
µνb |Hb〉

m2b
+ . . .

}
,

(7)
where cfn are calculable coefficient functions (which also contain the relevant CKM
matrix elements) depending on the quantum numbers {f} of the final state. For
semileptonic and non-leptonic decays, the coefficients c

f
3 have been calculated at

one-loop order [22, 23, 16], and the coefficients c
f
5 at tree level [3, 24].

In the next step, the forward matrix elements of the local operators in the
OPE are systematically expanded in inverse powers of the b-quark mass, using
the heavy-quark effective theory (HQET) [25]. One finds [3, 4]

1

2mHb
〈Hb| b̄b |Hb〉 = 1 −

µ2π(Hb) −µ
2
G(Hb)

2m2b
+ O(1/m3b ) ,

1

2mHb
〈Hb| b̄gsσµνG

µνb |Hb〉 = 2µ
2
G(Hb) + O(1/mb) , (8)

where µ2π(Hb) and µ
2
G(Hb) parametrize the matrix elements of the kinetic-energy

and the chromo-magnetic operators, respectively. The purpose of doing this ex-
pansion is that whereas the matrix elements in (7) contain an implicit dependence
on the b-quark mass, the parameters appearing on the right-hand side of (8) are

4



independent of mb (modulo logarithms). These parameters can be determined,
to some extent, from the spectrum of heavy hadron states. Below we shall need
the values

µ2π(Λb) −µ
2
π(B) = −(0.01 ± 0.03) GeV

2 ,

µ2G(B) =
3

4
(m2B∗ −m

2
B) ' 0.36 GeV

2 ,

µ2G(Λb) = 0 . (9)

The difference µ2π(Λb) −µ
2
π(B) can be extracted from the mass formula [6, 7]

(mΛb −mΛc)−(mB −mD) =
[
µ2π(B)−µ

2
π(Λb)

]( 1
2mc
−

1

2mb

)
+ O(1/m2Q) , (10)

where mB =
1
4

(mB + 3mB∗) and mD =
1
4

(mD + 3mD∗) denote the spin-averaged
meson masses. With mΛb = (5625 ± 6) MeV [26], this relation leads to the value
quoted above.

To order 1/m2b in the heavy-quark expansion, the lifetime ratio for two beauty
hadrons is given by

τ(H
(1)
b )

τ(H
(2)
b )

= 1 +
µ2π(H

(1)
b ) −µ

2
π(H

(2)
b )

2m2b
+ cG

µ2G(H
(1)
b ) −µ

2
G(H

(2)
b )

m2b
+ O(1/m3b ) , (11)

where cG ' 1.2 can be obtained using the results of refs. [3, 24]. Using then
the values given in (9), and assuming that in the case of the Bs meson SU(3)-
breaking effects in the values of the matrix elements are of order 20%, we arrive at
the predictions given in (2). Note that in taking a ratio of lifetimes, theoretical
uncertainties related to the values of the b-quark mass (including renormalon
ambiguities) and CKM elements cancel to a large extent. It is for this reason
that we restrict our discussion to the calculation of ratios of lifetimes and decay
rates.

The first two terms in the heavy-quark expansion (7) arise from decays in
which the (light) spectator quarks interact only softly. Additional contributions
of this type appear at the next order through gluonic operators of dimension
six, such as b̄γµ(iDνG

µν)b. Since matrix elements of these operators are blind
to the flavour of the spectator quarks, they can be safely neglected in our anal-
ysis. “Hard” spectator effects manifest themselves first in the matrix elements
of four-quark operators of dimension six. Some examples of the corresponding
contributions to the transition operator T are shown in figure 1. They only
appear in the heavy-quark expansion of non-leptonic decay rates.1 Since these
contributions arise from one-loop rather than two-loop diagrams, they receive a
phase-space enhancement factor of order 16π2 relative to the other terms in the
heavy-quark expansion.

1This is no longer true if b → u transitions or the decays of the Bc meson are considered.

5



�b �i q �q �i b

b b

c

u

d d

b bc

d

u u

Figure 1: Spectator contributions to the transition operator T (left), and the
corresponding operator in the OPE (right). Here Γi denotes some combination
of Dirac and colour matrices.

We have calculated the coefficients of the corresponding four-quark operators
at tree level, including for the first time the dependence on the mass of the
charm quark. This extends the results obtained in ref. [6]. We find that the
corresponding contributions to the non-leptonic widths of mesons and baryons
containing a b quark are given by the matrix elements of the local operator

Γspec =
2G2Fm

2
b

π
|Vcb|

2 (1 −z)2
{(

2c1c2 +
1

Nc
(c21 + c

2
2)
)
OuV−A + 2(c

2
1 + c

2
2) T

u
V−A

}

−
2G2Fm

2
b

3π
|Vcb|

2 (1 −z)2
{(

2c1c2 +
1

Nc
c21 + Nc c

2
2

)[(
1 +

z

2

)
Od
′

V−A − (1 + 2z) O
d′

S−P

]

+ 2c21

[(
1 +

z

2

)
Td
′

V−A − (1 + 2z) T
d′

S−P

]}

−
2G2Fm

2
b

3π
|Vcb|

2
√

1 − 4z

{(
2c1c2 +

1

Nc
c21 + Nc c

2
2

)[
(1 −z) Os

′

V−A − (1 + 2z) O
s′

S−P

]
+ 2c21

[
(1 −z) Ts

′

V−A − (1 + 2z) T
s′

S−P

]}
, (12)

where z = m2c/m
2
b, and Nc=3 is the number of colours. The local four-quark

operators appearing in this expression are defined by

O
q
V−A = b̄LγµqL q̄Lγ

µbL ,

O
q
S−P = b̄R qL q̄L bR ,

T
q
V−A = b̄LγµtaqL q̄Lγ

µtabL ,

T
q
S−P = b̄R taqL q̄L tabR , (13)

where ta = λa/2 are the generators of colour SU(3). For dimensional reasons,
Γspec is proportional to m

2
b rather than m

5
b, in accordance with the fact that

6



spectator effects contribute at third order in the heavy-quark expansion. The
first term in (12) arises from the upper diagram in figure 1, whereas the second
and third terms come from the contributions of the lower diagram with a cū and
cc̄ quark pair in the loop. We note that in the limit z = 0 our results agree with
ref. [6], and with the corresponding expression derived for the lifetimes of charm
hadrons in refs. [8]–[10].

The operators in (13) are renormalized at the scale mb, which will be im-
plicit in our discussion below. This choice has the advantage that logarithms
of the type [αs ln(mb/µhad)]

n, where µhad is a typical hadronic scale, reside en-
tirely in the hadronic matrix elements of the renormalized operators. Using the
renormalization-group equations, the expressions presented in this paper can be
rewritten in terms of operators renormalized at any other scale. However, at
present the scale dependence of the renormalized operators below the scale mb is
known only to leading logarithmic order [27, 28]. It is discussed in detail in ap-
pendix A. Since here we shall treat the matrix elements as unknown parameters,
we can avoid all uncertainties related to the operator evolution by working at the
scale mb.

The hadronic matrix elements of the four-quark operators in (13) contain
the non-perturbative physics of the spectator contributions to inclusive decays of
beauty hadrons. However, the same operators also contribute to the decay of the
b quark, through tadpole diagrams in which the light-quark fields are contracted
in a loop. These “non-spectator” contributions are independent of the flavour of
the light quark q and thus contribute equally to the decay widths of all beauty
hadrons. They are not of interest to our discussion here. In order to isolate the
true spectator effects, we shall implicitly assume a normal ordering of the four-
quark operators, which has the effect of subtracting tadpole-like diagrams. In
practice, this is equivalent to choosing a particular renormalization prescription
for the operators. Alternatively, one may isolate the spectator effects by con-
sidering light-quark flavour non-singlet combinations of the operators; thus, for
example, for matrix elements between Bd states one could take (O

d −Ou) and
(Td −Tu) instead of Od and Td.

3 Parametrization of the matrix elements

In previous analyses [6], [8]–[10], the hadronic matrix elements of the four-quark
operators in (13) have been estimated making simplifying assumptions. Here
we shall avoid such assumptions and, without any loss of generality, express the
relevant matrix elements in terms of a set of hadronic parameters. Clearly, such
an approach has less predictive power; however, it does allow us to find the
range of predictions that can be made in a model-independent way. In view of
the apparent discrepancy between theory and experiment for the Λb lifetime, we
find it worth while to question the model-dependent assumptions made in earlier

7



analyses. Ultimately, the relevant hadronic parameters may be calculated using
some field-theoretic approach such as lattice gauge theory or QCD sum rules. It
has also been suggested that combinations of these parameters may be extracted
from a precise measurement of the lepton spectrum in the endpoint region of
semileptonic B decays, or from a study of spectator effects in charm decays [29].

3.1 Mesonic matrix elements

For matrix elements of the four-quark operators between B-meson states, we
define parameters Bi and εi such that:

1

2mBq
〈Bq|O

q
V−A |Bq〉 ≡

f2BqmBq

8
B1 ,

1

2mBq
〈Bq|O

q
S−P |Bq〉 ≡

f2BqmBq

8
B2 ,

1

2mBq
〈Bq|T

q
V−A |Bq〉 ≡

f2BqmBq

8
ε1 ,

1

2mBq
〈Bq|T

q
S−P |Bq〉 ≡

f2BqmBq

8
ε2 . (14)

This definition is inspired by the vacuum insertion (or factorization) approxi-
mation [11], according to which the matrix elements of four-quark operators are
evaluated by inserting the vacuum inside the current products. This leads to

〈Bq|O
q
V−A |Bq〉 =

(
mb + mq
mB

)2
〈Bq|O

q
S−P |Bq〉 =

f2Bqm
2
Bq

4
,

〈Bq|T
q
V−A |Bq〉 = 〈Bq|T

q
S−P |Bq〉 = 0 , (15)

where fBq is the decay constant of the Bq meson, defined as

〈0 | q̄ γµγ5 b |Bq(p)〉 = ifBq p
µ . (16)

Hence, the factorization approximation corresponds to setting Bi = 1 and εi = 0
at some scale µ (which in general will be different from our adopted choice µ =
mb), where the approximation is believed to be valid. The exact values of the
hadronic parameters are not yet known. However, as discussed in appendix A,
in the large-Nc limit

Bi = O(1) , εi = O(1/Nc) . (17)

An estimate of the parameters εi using QCD sum rules has been obtained by
Chernyak, who finds that ε1 ≈−0.15 and ε2 ≈ 0 [30].

8



In terms of the parameters Bi and εi, the matrix elements of the operator
Γspec in (12) are:

1

2mB
〈B−|Γspec |B

−〉 = Γ0 ηspec (1 −z)
2
{

(2c2+ − c
2
−) B1 + 3(c

2
+ + c

2
−) ε1

}
,

1

2mB
〈Bd|Γspec |Bd〉

= −Γ0 ηspec (1 −z)
2 cos2θc

{
1

3
(2c+ −c−)

2
[(

1 +
z

2

)
B1 − (1 + 2z) B2

]

+
1

2
(c+ + c−)

2
[(

1 +
z

2

)
ε1 − (1 + 2z) ε2

]}

− Γ0 ηspec
√

1 − 4z sin2θc

{
1

3
(2c+ −c−)

2
[
(1 −z) B1 − (1 + 2z) B2

]
+

1

2
(c+ + c−)

2
[
(1 −z) ε1 − (1 + 2z) ε2

]}
, (18)

where c± = c1 ± c2, and

Γ0 =
G2Fm

5
b

192π3
|Vcb|

2 , ηspec = 16π
2 f

2
BmB

m3b
. (19)

The spectator contribution to the width of the Bs meson is obtained from that
of the Bd meson by the replacements sin θc ↔ cos θc and fB,mB → fBs,mBs. Of
course, the values of the parameters Bi and εi for the Bs meson will also differ
from those for the Bd meson due to SU(3)-breaking effects.

Two remarks are in order regarding the result (18). The first concerns the
expected order of magnitude of the spectator contributions to the total decay
width of a B meson. At leading order in the heavy-quark expansion, the total
width of a beauty hadron is Γtot ' 3.7 × Γ0, where the numerical factor arises
from the phase-space contributions of the semileptonic and non-leptonic channels
(we use z = 0.085) [3, 24]. It follows that

Γspec
Γtot

∼
ηspec

4
∼
(

2πfB
mB

)2
∼ 5% . (20)

The second remark concerns the structure of the coefficients in (18). Given
that c+ ' 0.86 and c− ' 1.35, one observes that the coefficients of the colour
singlet–singlet operators are one to two orders of magnitude smaller than those
of the colour octet–octet operators. This implies that at the scale mb even small
deviations from the factorization approximation can have a sizeable impact on
the results.

3.2 Baryonic matrix elements

Next we study the matrix elements of the four-quark operators between Λb-baryon
states. Since the Λb is an iso-singlet, the matrix elements of the operators with

9



q = u or d are the same, and below we drop this label. In the case of baryons,
we find it convenient to use the colour identity

(ta)αβ (ta)γδ =
1

2
δαδ δγβ −

1

2Nc
δαβ δγδ (21)

to rewrite T = −1
6
O + 1

2
Õ and introduce the operators (i,j are colour indices)

ÕV−A = b̄
i
Lγµq

j
L q̄

j
Lγ

µbiL , ÕS−P = b̄
i
R q

j
L q̄

j
L b

i
R . (22)

instead of TV−A and TS−P .
The heavy-quark spin symmetry, i.e. the fact that interactions with the spin

of the heavy quark decouple as the heavy-quark mass tends to infinity, allows us
to derive two relations between the matrix elements of the four-quark operators
between Λb-baryon states. To find these relations, we note that the following
matrix element vanishes in the limit mb →∞:

1

2mΛb
〈Λb| b̄

iγµγ5 b
j q̄kLγ

µqlL |Λb〉 = O(1/mb) . (23)

The physical argument for this is that, because of the spin symmetry for heavy
quarks, the matrix elements for left-handed and right-handed b quarks must be
the same. The above result then follows since b̄γµγ5 b = b̄RγµbR − bLγµbL. A
more formal argument can be given using the covariant tensor formalism of the
HQET [25, 31] to show that the matrix element in (23) is proportional to

〈2Sb · Slight〉 = SΛb (SΛb + 1) −Sb(Sb + 1) −Slight(Slight + 1) = 0 , (24)

since, in the heavy-quark limit, the light degrees of freedom are in a state with
total spin zero.

Using the Fierz identity

b̄iγµγ5 b
j q̄kLγ

µqlL = −2 b̄
i
R q

l
L q̄

k
L b

j
R − b̄

i
Lγµq

l
L q̄

k
Lγ

µb
j
L , (25)

we then obtain the relations

1

2mΛb
〈Λb|OS−P |Λb〉 = −

1

2

1

2mΛb
〈Λb|OV−A |Λb〉 + O(1/mb) ,

1

2mΛb
〈Λb|ÕS−P |Λb〉 = −

1

2

1

2mΛb
〈Λb|ÕV−A |Λb〉 + O(1/mb) . (26)

The corrections of order 1/mb to these relations contribute at order 1/m
4
b in the

heavy-quark expansion and so are negligible to the order we work in. This leaves
us with two independent matrix elements of the operators OV−A and ÕV−A. The
analogue of the factorization approximation in the case of baryons is the valence-
quark assumption, in which the colour of the quark fields in the operators is

10



identified with the colour of the quarks inside the baryon. Since the colour wave
function for a baryon is totally antisymmetric, the matrix elements of OV−A and
ÕV−A differ in this approximation only by a sign. Hence, we define a parameter
B̃ by

〈Λb|ÕV−A |Λb〉≡−B̃ 〈Λb|OV−A |Λb〉 , (27)

with B̃ = 1 in the valence-quark approximation.
For the baryon matrix element of OV−A itself, our parametrization is guided

by the quark model. We write

1

2mΛb
〈Λb|OV−A |Λb〉≡−

f2BmB

48
r , (28)

where in the quark model r is the ratio of the squares of the wave functions
determining the probability to find a light quark at the location of the b quark
inside the Λb baryon and the B meson, i.e. [9, 10]

r =
|ψΛbbq (0)|

2

|ψ
Bq
bq̄ (0)|

2
. (29)

Guberina et al. have estimated the ratio r for charm decays and find r ' 0.2 in
the bag model, and r ' 0.5 in the non-relativistic quark model [8]. This latter
estimate has been used in more recent work on beauty decays [6]. A similar
result, r ∼ 0.1–0.3, has been obtained by Colangelo and Fazio using QCD sum
rules [32]. Recently, Rosner has criticized the existing quark-model estimates of
r. Assuming that the wave functions of the Λb and Σb baryons are the same,
he argues that the wave-function ratio should be estimated from the ratio of the
spin splittings between Σb and Σ

∗
b baryons and B and B

∗ mesons [33]. This leads
to

r =
4

3

m2Σ∗
b
−m2Σb

m2B∗ −m
2
B

. (30)

If the baryon splitting is taken to be m2Σ∗
b
− m2Σb ' m

2
Σ∗c
− m2Σc = (0.384 ±

0.035) GeV2, this leads to r ' 0.9 ± 0.1. If, on the other hand, one uses the
preliminary result mΣ∗

b
−mΣb = (56±16) MeV reported by the DELPHI Collab-

oration [34], one obtains r ' 1.8 ± 0.5. We conclude that it is conceivable that
r ∼ 1, i.e. larger than previous estimates.

In terms of these parameters, the matrix element of Γspec is given by

1

2mΛb
〈Λb|Γspec |Λb〉 = Γ0 ηspec

r

16

{
4(1 −z)2

[
(c2− − c

2
+) + (c

2
− + c

2
+) B̃

]
−
[
(1 −z)2(1 + z) cos2θc +

√
1 − 4z sin2θc

]
×
[
(c−− c+)(5c+ − c−) + (c− + c+)

2 B̃
]}
. (31)

11



3.3 Numerical results

To illustrate the main features of our results, we calculate the coefficients of
the hadronic parameters Bi, εi, and r and B̃ r in the matrix elements (18) and
(31) in units of Γ0 ηspec. In order to study the dependence on the mass ratio
z = (mc/mb)

2, we first keep the values of the Wilson coefficients in the effective
Lagrangian fixed and vary the mass ratio in the range z = 0.085 ± 0.015. This
leads to the numbers shown in table 1, where the variation with z is indicated
as a change in the last digit(s). Note that for mesons the coefficients of the
parameters Bi are much smaller than those of the parameters εi. It is apparent
that the results are rather stable with respect to the precise value of z. From
now on we shall always use the central value z = 0.085, which is obtained, for
instance, for mc = 1.4 GeV and mb = 4.8 GeV.

Table 1: Coefficients of the hadronic parameters obtained for z = 0.085±0.015.
The values c+ = 0.861 and c− = 1.349 are kept fixed.

Hb B1 B2 ε1 ε2 r B̃ r

B− −0.28(1) — 6.43(21) —
Bd −0.04(0) 0.05(0) −2.12(6) 2.39(2)
Bs −0.03(0) 0.04(0) −1.83(11) 2.32(5)
Λb 0.14(1) 0.26(1)

As another check on the stability of the results we present in table 2 the
coefficients obtained using different “matching” procedures. To this end, we
renormalize the Wilson coefficients c+ and c− of the effective Lagrangian at a
scale µ different from mb. Thus c+ and c− are modified by replacing mb by µ
in (6). This gives us predictions in terms of operators renormalized at µ, which
we then rewrite in terms of those defined at mb by using the evolution equations
given in appendix A. Thus, the meaning of the hadronic parameters is the same
as before, and the differences between the numerical results can be viewed as
an estimate of unknown higher-order perturbative corrections, which we neglect
throughout this paper. The coefficients of the parameters Bi for mesons, as
well as of the parameters r and B̃ r for the Λb baryon, show a significant scale
dependence. To reduce this dependence would require a full next-to-leading order
calculation of radiative corrections, which is beyond the scope of this paper.

4 Phenomenology of beauty lifetimes

We shall now discuss the phenomenological implications of our results for the
calculation of beauty lifetime ratios. The spectator contributions to the decay
widths of Bq mesons and of the Λb baryon are described by a set of hadronic

12



Table 2: Coefficients of the hadronic parameters obtained for the matching
scales µ = mb/2, mb and 2mb, with mb = 4.8 GeV. The value z = 0.085 is kept
fixed.

Hb µ B1 B2 ε1 ε2 r B̃ r

mb/2 −0.62 — 7.26 —
B− mb −0.28 — 6.43 —

2mb +0.02 — 5.90 —
mb/2 −0.04 0.04 −2.32 2.61

Bd mb −0.04 0.05 −2.12 2.39
2mb −0.08 0.09 −1.98 2.24
mb/2 −0.03 0.04 −2.00 2.54

Bs mb −0.03 0.04 −1.83 2.32
2mb −0.07 0.08 −1.72 2.18
mb/2 0.21 0.30

Λb mb 0.14 0.26
2mb 0.09 0.23

parameters: B1,2 and ε1,2 for mesons, and r and B̃ for baryons. The explicit
dependence of the decay rates on these quantities is shown in (18) and (31).
The numerical values of the coefficients multiplying the hadronic parameters are
given in table 2 for three different choices of the matching scale µ. For the
numerical analysis we need the value of the parameter ηspec defined in (19). We
take fB = 200 MeV and mb = 4.8 GeV, so that ηspec ' 0.30, and absorb the
uncertainty in ηspec into the values of the hadronic parameters.

4.1 Lifetime ratio τ(B−)/τ(Bd)

We start by discussing the lifetime ratio of the charged and neutral B mesons.
Because of isospin symmetry, the lifetimes of these states are the same at order
1/m2b in the heavy-quark expansion, and differences arise only from spectator
effects. If we write

τ(B−)

τ(Bd)
= 1 + k1B1 + k2B2 + k3ε1 + k4ε2 , (32)

the coefficients ki take the values shown in table 3. The most striking feature of
this result is the large imbalance between the coefficients of the parameters Bi
and εi, which parametrize the matrix elements of colour singlet–singlet and colour
octet–octet operators, respectively. With εi of order 1/Nc, it is conceivable that
the non-factorizable contributions actually dominate the result. Thus, without a
detailed calculation of the parameters εi no reliable prediction can be obtained.

13



In this conclusion we disagree with the authors of ref. [6], who use factorization
(at a low hadronic scale) to argue that τ(B−)/τ(Bd) must exceed unity by an
amount of order 5%. We will return to this in section 4.4 below.

Table 3: Coefficients ki appearing in (32).

µ k1 k2 k3 k4

mb/2 +0.044 0.003 −0.735 0.201
mb +0.020 0.004 −0.697 0.195
2mb −0.008 0.007 −0.665 0.189

The experimental value of the lifetime ratio given in (1) can be employed to
constrain a certain combination of the parameters:

ε1 =
1

k3

[
(0.02 ± 0.04) −k1B1 −k2B2 −k4ε2

]
. (33)

With the values of the coefficients given in table 3, this relation implies

ε1 ' 0.3 ε2 + δ , (34)

where we expect |δ| < 0.1. This becomes a particularly useful constraint if the
parameters εi turn out to be large. Below, we shall use (33) to eliminate ε1 from
our predictions for spectator effects.

4.2 Lifetime ratio τ(Bs)/τ(Bd)

In the limit where SU(3)-breaking effects are neglected, the spectator contribu-
tions to the decay widths of Bs and Bd mesons are too similar to produce an
observable lifetime difference (see table 2). The corresponding contributions to
the lifetime ratio τ(Bs)/τ(Bd) are of order 10

−3. A precise prediction for SU(3)-
breaking effects is difficult to obtain. Allowing for 30% SU(3)-breaking effects
in the matrix elements of the four-quark operators describing the spectator con-
tributions, we estimate that the resulting contributions to the lifetime ratio are
of order 1–2%. Contributions of order 1% (or less) could also arise from SU(3)-
breaking effects in the matrix elements appearing at order 1/m2b in the expansion
(11). Hence, we agree with ref. [6] that

τ(Bs)

τ(Bd)
= 1 ±O(1%) . (35)

14



4.3 Lifetime ratio τ(Λb)/τ(Bd)

As mentioned in the introduction, the low experimental value of the lifetime ratio
τ(Λb)/τ(Bd) is the primary motivation for our study. We shall now discuss the
structure of spectator contributions to this ratio. It is important that heavy-quark
symmetry allows us to reduce the number of hadronic parameters contributing
to the decay rate of the Λb baryon from four to two, and that these parameters
are almost certainly positive (unless the quark model is completely misleading)
and enter the decay rate with the same sign. Thus, unlike in the meson case, the
structure of the spectator contributions to the width of the Λb baryon is rather
simple, and at least the sign of the effects can be predicted reliably.

For a more detailed discussion, we distinguish between the two cases where
one does or does not allow spectator contributions to enhance the theoretical
prediction for the semileptonic branching ratio, BSL, of B mesons. As we will
discuss below, the theoretical prediction for BSL, which neglects spectator con-
tributions, is slightly larger than the central experimental value. If spectator
effects increased the prediction for BSL further, this discrepancy could become
uncomfortably large.

If we do not allow for an increase in the value of the semileptonic branching
ratio, the explanation of the low value of τ(Λb)/τ(Bd) must reside entirely in a
low value of the Λb lifetime (rather than a large value of the B-meson lifetime).
This can be seen by writing

τ(Λb)

τ(Bd)
= τ(Λb)

(
τ(B−)

τ(Bd)

)1/2
1

[τ(B−) τ(Bd)]
1/2

=
τ(Λb)

BSL

(
τ(B−)

τ(Bd)

)1/2
ΓSL(B) ,

(36)
where BSL is the average semileptonic branching ratio of B mesons, and ΓSL(B)
is the semileptonic width. In the second step we have replaced the geometric
mean [τ(B−) τ(Bd)]

1/2 by the average B-meson lifetime, which because of isospin
symmetry is correct to order 1/m6b in the heavy-quark expansion. Since there
are no spectator contributions to the semileptonic rate ΓSL(B), and since we
do not allow an enhancement of the semileptonic branching ratio, in order to
obtain a small value for τ(Λb)/τ(Bd) we can increase the width of the Λb baryon
and/or decrease (within the experimental errors) the lifetime ratio τ(B−)/τ(Bd).
Allowing for a downward fluctuation of this ratio by two standard deviations,
i.e. τ(B−)/τ(Bd) > 0.94, and using the estimate of 1/m

2
b corrections in (2), we

conclude that

τ(Λb)

τ(Bd)
> 0.97 ×

[
0.98 −

Γspec(Λb)

Γ(Λb)

]
= 0.95 − (d1 + d2B̃) r , (37)

where Γspec(Λb) is the spectator contribution to the width of the Λb baryon. The
values of the coefficients di are given in table 4. If we assume that r and B̃
are of order unity, we find that the spectator contributions yield a reduction of

15



the lifetime of the Λb baryon by a few per cent, and that τ(Λb)/τ(Bd) > 0.9,
in contrast with the experimental result given in (1). If, for example, we try to
push the theoretical prediction by taking the large value B̃ = 1.5 (corresponding
to a violation of the valence-quark approximation by 50%) and choosing a low
matching scale µ = mb/2, we have to require that r > rmin with rmin = 3.1,
2.2 and 1.3 for τ(Λb)/τ(Bd) = 0.78, 0.83 and 0.88 (corresponding to the central
experimental value and the 1σ and 2σ fluctuations). Hence, even if we allow for
an upward fluctuation of the experimental result by two standard deviations, we
need a value of r that is significantly larger than most quark-model predictions
(see the discussion in section 3.2). Clearly, a reliable field-theoretic calculation
of the parameters r and B̃ is of great importance to support or rule out such a
possibility.

Table 4: Coefficients di appearing in (37) and (38).

µ d1 d2 d3 d4
mb/2 0.016 0.023 0.178 −0.201
mb 0.012 0.021 0.173 −0.195
2mb 0.008 0.020 0.167 −0.189

On the other hand, the low experimental value of the semileptonic branching
ratio may find its explanation in a low renormalization scale (see section 5 below),
or it may be caused by the effects of New Physics, such as an enhanced rate for
flavour-changing neutral currents of the type b → sg [35]–[38]. Hence, one may
be misled in using the semileptonic branching ratio as a constraint on the size
of spectator contributions. Then there is the possibility to decrease the value of
τ(Λb)/τ(Bd) by increasing the lifetime of the Bd meson, i.e. in (37) we can allow
for spectator contributions to the width of the Bd meson. From table 2, it follows
that the contributions of the parameters B1 and B2 are very small (of order 10

−3)
and can safely be neglected. Thus, we obtain

τ(Λb)

τ(Bd)
' 0.98 − (d1 + d2B̃) r − (d3ε1 + d4ε2) , (38)

where the coefficients d3 and d4 are also shown in table 4. At first sight, it
seems that with a positive ε1 and a negative ε2 of order 1/Nc one could gain
a contribution of about −0.1, which would take away much of the discrepancy
between theory and experiment. However, the experimental result for the lifetime
ratio τ(B−)/τ(Bd) imposes a useful constraint. Using (33) to eliminate ε1 from
the relation (38), and allowing the parameters Bi to take values between 0 and
2, we find

τ(Λb)

τ(Bd)
' 0.98 ± 0.02 + 0.15ε2 − (d1 + d2B̃) r > 0.88 − (d1 + d2B̃) r . (39)

16



where in the last step we have assumed that |ε2| < 0.5, which we consider to be
a very conservative bound. Even in this extreme case, a significant contribution
must still come from the parameters r and B̃.

In view of the above discussion, the short Λb lifetime remains a potential prob-
lem for the heavy-quark theory. If the current experimental value persists, there
are two possibilities: either some hadronic matrix elements of four-quark opera-
tors are significantly larger than naive expectations based on large-Nc counting
rules and the quark model, or (local) quark–hadron duality, which is assumed
in the calculation of lifetimes, fails in non-leptonic inclusive decays. In the sec-
ond case, the explanation of the puzzle lies beyond the heavy-quark expansion.
Let us, therefore, consider the first possibility and give a numerical example for
some possible scenarios. Assume that µ = mb/2 is an appropriate scale to use
in the evaluation of the Wilson coefficients, and that B̃ = 1.5. Then, to obtain
τ(Λb)/τ(Bd) = 0.8 without enhancing the prediction for the semileptonic branch-
ing ratio requires r ' 3, i.e. several times larger than quark-model estimates. If,
on the other hand, we consider r = 1.5 as the largest conceivable value, we need
ε2 '−0.5, corresponding to a rather large matrix element of the colour-octet op-
erator TS−P. Such a value of ε2 leads to an enhancement of the B-meson lifetime,
and hence to an enhancement of the semileptonic branching ratio of B mesons,
by ∆BSL ' 1%. As we will discuss in section 5, this is still tolerable provided yet
unknown higher-order corrections confirm the use of a low renormalization scale.
Although in both cases some large parameters are needed, we find it important to
note that until reliable field-theoretic calculations of the matrix elements of four-
quark operators become available, a conventional explanation of the Λb-lifetime
puzzle cannot be excluded.

4.4 Relation with the conventional factorization approach

We now discuss in detail the relation of our approach with previous analyses
based on the factorization approximation [6]. This approximation amounts to
setting (see appendix A)

Bi(µhad) =

(
αs(mb)

αs(µhad)

)4/β0
, εi(µhad) = 0 , (40)

at some hadronic scale µhad � mb. Here β0 is the first coefficient of the β
function. The evolution of the operators from mb to µhad is done to leading
logarithmic order [10, 27, 28]. For the purpose of our discussion, we fix µhad
such that αs(µhad) = 0.5. Then the relation between the hadronic parameters
renormalized at the scale mb with those renormalized at µhad, as given in (A.5),
is:

Bi(mb) ' 1.48Bi(µhad) − 0.36εi(µhad) ,

εi(mb) ' −0.08Bi(µhad) + 1.06εi(µhad) . (41)

17



The theoretical prediction for the lifetime ratio τ(B−)/τ(Bd) in terms of the
hadronic parameters renormalized at the scale µhad is obtained by combining (41)
with the numbers given in table 3. We find

τ(B−)

τ(Bd)
' 1 + 0.08B1(µhad) − 0.01B2(µhad) − 0.75ε1(µhad) + 0.20ε2(µhad) . (42)

The factorization approximation Bi(µhad) ' 0.68 and εi(µhad) = 0 leads to
τ(B−)/τ(Bd) ' 1.05, which is close to the value obtained by Bigi et al. [6].
However, it is evident from (42) that even at a low scale the coefficients of the
non-factorizable terms are still much larger than those of the factorizable ones.
Hence, it remains true that non-factorizable corrections are potentially important.
It would be justified to neglect these contributions only if the parameters εi(µhad)
were significantly smaller than 10%. However, to the best of our knowledge there
is currently no compelling argument to support such a strong restriction.

5 Semileptonic branching ratio of B mesons

The semileptonic branching ratio of B mesons has received considerable attention
in the past. It is defined as

BSL =
Γ(B̄ → X eν̄)∑

` Γ(B̄ → X `ν̄) + ΓNL + Γrare
, (43)

where ΓNL and Γrare are the inclusive rates for non-leptonic and rare decays,
respectively. The status of the experimental results on the semileptonic branching
ratio is controversial, as there is a discrepancy between low-energy measurements
performed at the Υ(4s) resonance and high-energy measurements performed at
the Z0 resonance. The average value at low energies is BSL = (10.37 ± 0.30)%

[39], whereas high-energy measurements give B
(b)
SL = (11.11 ± 0.23)% [40]. The

superscript (b) indicates that this value refers not to the B meson, but to a
mixture of b hadrons (approximately 40% B−, 40% B̄d, 12% Bs, and 8% Λb).

Assuming that the corresponding semileptonic width Γ
(b)
SL is close to that of the

B meson,2 we can correct for this and find BSL = (τ(B)/τ(b)) B
(b)
SL = (11.30 ±

0.26)%, where τ(b) = (1.57 ± 0.03) ps is the average lifetime corresponding to
the above mixture of b hadrons [1]. The discrepancy between the low-energy
and high-energy measurements of the semileptonic branching ratio is therefore
larger than 3 standard deviations. If we take the average and inflate the error to
account for this fact, we obtain

BSL = (10.90 ± 0.46)% . (44)

2Theoretically, this is expected to be a very good approximation.

18



In understanding this result, an important aspect is charm counting, i.e. the
measurement of the average number nc of charm hadrons produced per B decay.
Theoretically, this quantity is given by

nc = 1 + B(B̄ → Xcc̄s′) −B(B̄ → no charm) , (45)

where B(B̄ → Xcc̄s′) is the branching ratio for decays into final states containing
two charm quarks, and B(B̄ → no charm) ∼ 0.02 [14, 41, 42] is the Standard
Model branching ratio for charmless decays. Recently, two new measurements of
the average charm content have been performed. The CLEO Collaboration has
presented the value nc = 1.16 ± 0.05 [39, 43], and the ALEPH Collaboration has
reported the result nc = 1.23 ± 0.07 [44]. The average is

nc = 1.18 ± 0.04 . (46)

The naive parton model predicts that BSL ' 15% and nc ' 1.2; however, it
has been known for some time that perturbative corrections could change these
results significantly [14]. With the establishment of the 1/mQ expansion, the
non-perturbative corrections to the parton model could be computed, and their
effect turned out to be very small. This led Bigi et al. to conclude that values
BSL < 12.5% cannot be accommodated by theory, thus giving rise to a puzzle
referred to as the “baffling semileptonic branching ratio” [15]. Recently, Bagan et
al. have completed the calculation of the O(αs) corrections including the effects of
the charm-quark mass [16], finding that they lower the value of BSL significantly.

The analysis of Bagan et al. has been corrected in an erratum [16]. Here we
shall present the results of an independent numerical analysis using the same
theoretical input. As the subject is of considerable importance, we shall explain
our analysis in detail. The semileptonic branching ratio and nc depend on the
pole masses of the heavy quarks, which we allow to vary in the range

mb = (4.8 ± 0.2) GeV , mb −mc = (3.40 ± 0.06) GeV , (47)

corresponding to 0.25 < mc/mb < 0.33. Here mb is the pole mass defined to one-
loop order in perturbation theory. The difference mb −mc is free of renormalon
ambiguities and can be determined from spectroscopy (see, e.g., ref. [7]). Bagan
et al. have also considered the theoretical predictions in a scheme where the quark
masses are renormalized at a scale µ in the ms scheme. We discuss this scheme in
appendix B. At order 1/m2b in the heavy-quark expansion, non-perturbative ef-
fects are described by the single parameter µ2G(B) defined in (9); the dependence
on the parameter µ2π(B) is the same for all inclusive decay rates and cancels out
in BSL and nc. Moreover, the results depend on the scale µ used to renormalize
the coupling constant αs(µ) and the Wilson coefficients c±(µ) entering the non-
leptonic decay rate. They also depend on the value of the QCD scale parameter

19



ΛQCD, which we fix taking αs(mZ) = 0.117 ± 0.004. The corresponding uncer-
tainty is smaller than that due to the variation of the mass parameters and is
added in quadrature. For the two choices µ = mb and µ = mb/2, we obtain

BSL =
{

12.0 ± 1.0%; µ = mb,
10.9 ± 1.0%; µ = mb/2,

nc =
{

1.20 ∓ 0.06; µ = mb,
1.21 ∓ 0.06; µ = mb/2.

(48)

The errors in the two quantities are anticorrelated. Notice that the semileptonic
branching ratio has a much stronger scale dependence than nc. This is illustrated
in figure 2, where we show the two quantities as a function of µ. By choosing a
low renormalization scale, values BSL < 12.5% can easily be accommodated. The
experimental data prefer a scale µ/mb ∼ 0.3–0.5, which is indeed not unnatural.
Using the BLM scale-setting method [45], Luke et al. have estimated that µ '
0.3mb is an appropriate scale in this case [46].

mc/mb = 0.33

0.29

0.25

0.25 0.5 0.75 1 1.25 1.5
µ/mb

8

9

10

11

12

13

14

B
S

L
 (

%
)

mc/mb=0.25

0.29

0.33

0.25 0.5 0.75 1 1.25 1.5
µ/mb

1

1.1

1.2

1.3

1.4
n c

Figure 2: Scale dependence of the theoretical predictions for the semileptonic
branching ratio and nc. The bands show the average experimental values.

The combined theoretical predictions for the semileptonic branching ratio and
charm counting are shown in figure 3. They are compared with the experimental
results obtained at the Υ(4s) and at the Z0 resonance. It was argued that the
combination of a low semileptonic branching ratio and a low value of nc would
constitute a potential problem for the Standard Model [42]. However, with the
new experimental and theoretical numbers, only for the low-energy measurements
a small discrepancy remains between theory and experiment. Note that, using
(45), our results for nc can be used to obtain predictions for the branching ratio
B(B̄ → Xcc̄s′), which is accessible to a direct experimental determination. Our
prediction of (22 ± 6)% for this branching ratio agrees well with the preliminary
result of the CLEO Collaboration: B(B̄ → Xcc̄s′) = (23.9 ± 3.8)% [47].

Having discussed the status of the theoretical predictions obtained to order
1/m2b in the heavy-quark expansion, we now investigate the spectator contribu-
tions to the semileptonic branching ratio and nc. This extends, in the context

20



0.25
0.5

1.0 1.5
0.25

0.29

0.33

µ/mb

mc/mb

LE

HE

8 9 10 11 12 13 14
BSL (%)

1

1.1

1.2

1.3

1.4

n c

Figure 3: Combined theoretical predictions for the semileptonic branching ra-
tio and charm counting as a function of the quark-mass ratio mc/mb and the
renormalization scale µ. The data points show the average experimental values
for BSL and nc obtained in low-energy (LE) and high-energy (HE) measure-
ments.

of the heavy-quark expansion, the phenomenological study presented in ref. [14].
We consider the average of BSL and nc for B

− and Bd mesons,
3 and write for the

spectator contributions to these quantities

∆BSL,spec = b1B1 + b2B2 + b3ε1 + b4ε2 ,

∆nc,spec = n1B1 + n2B2 + n3ε1 + n4ε2 . (49)

The coefficients bi and ni are given in table 5. If, as previously, we eliminate ε1
from these equations using the constraint (33) imposed by the measurement of
τ(B−)/τ(Bd), and we allow that the parameters Bi take values in the range 0 to
2, we obtain

∆BSL,spec ' (−2.1ε2 + 0.2 ± 0.3) % ,

∆nc,spec ' (1.2 ± 0.1) ∆BSL,spec . (50)

For reasonable value of ε2, we expect a contribution to the semileptonic branching
ratio of order 1% or less, and a negligible effect on nc. However, without a detailed
calculation of the hadronic parameters we cannot obtain a quantitative prediction
of the spectator contributions. Nevertheless, we find it interesting that there is
at least a potential to change, and in particular to lower, the value of BSL by
0.5–1%. To achieve such a decrease requires that the hadronic parameter ε2,
which parametrizes the matrix element of the colour octet–octet operator T

q
S−P

3These are approximately the quantities measured experimentally. However, measurements
at LEP receive a contamination from Bs and b-baryon decays.

21



in (13), is positive and of order 0.3–0.5.4 It will be interesting to see if future
calculations of this parameter will confirm or rule out this scenario.

Table 5: Coefficients bi and ni (in %) appearing in (49).

µ b1 b2 b3 b4
mb/2 0.35 −0.02 −2.60 −1.37
mb 0.19 −0.03 −2.56 −1.42
2mb 0.03 −0.05 −2.49 −1.42

µ n1 n2 n3 n4
mb/2 0.50 −0.03 −4.17 −1.54
mb 0.25 −0.03 −3.79 −1.44
2mb 0.03 −0.05 −3.51 −1.36

6 Conclusions

In this paper, we have studied spectator effects in inclusive decays of beauty
hadrons. Although these effects are suppressed by three powers of ΛQCD/mb in
the heavy-quark expansion, they cannot be neglected because of the large phase-
space factor for two-body scattering. The contributions of spectator effects to
inclusive decay rates are given by the hadronic matrix elements of the four local
operators in (13). For mesons, we have expressed these matrix elements in terms
of the hadronic parameters B1,2 and ε1,2 defined in (14). For the Λb baryon, heavy-
quark symmetry reduces the number of independent matrix elements from four
to two, which we parametrize by r and B̃ as defined in (27) and (28). Although
our parametrization is motivated by commonly made simplifications, such as the
vacuum insertion and the valence-quark approximations, we stress that it is intro-
duced without any loss of generality. For a complete understanding of spectator
effects, it will be necessary to evaluate these parameters non-perturbatively, e.g.
in lattice simulations.

We find that in predictions for lifetimes and the semileptonic branching ratio
of B mesons, the coefficients of the colour octet–octet non-factorizable operators
are much larger than those for the colour singlet–singlet factorizable operators.
Thus the contributions from the non-factorizable operators cannot be neglected,
even though their matrix elements are suppressed in the large-Nc limit.

The ratio τ(B−)/τ(Bd) is particularly sensitive to non-factorizable contribu-
tions [see (32) and table 3], making it difficult to predict this quantity with a

4We recall, however, that according to (39) a positive value of ε2 increases the theoretical
prediction for the lifetime ratio τ(Λb)/τ(Bd).

22



precision of better than about 10%. For example, assuming that the magnitudes
of the parameters ε1 and ε2 are smaller than 0.1 or 0.2, we find that the predic-
tions for this lifetime ratio lie in the ranges 0.93–1.11 and 0.84–1.20, respectively.5

However, in our opinion, even if the experimental result had been outside these
ranges, the most likely explanation would have been that the εi parameters are
larger, rather than a failure of the heavy-quark expansion. The experimental
measurement of τ(B−)/τ(Bd) imposes the constraint (33) upon the parameters,
which allows us to eliminate ε1 in other relations. On the other hand, within the
heavy-quark expansion there is only room for a very small deviation of the ratio
τ(Bs)/τ(Bd) from unity due to SU(3)-breaking effects. We estimate these effects
to be of order 1%.

Understanding the low experimental value of the lifetime ratio τ(Λb)/τ(Bd)
remains a potential problem for the heavy-quark theory. If the current exper-
imental value persists, there are two possibilities: either some hadronic matrix
elements of four-quark operators are significantly larger than naive expectations
based on large-Nc counting rules and the quark model, or (local) quark–hadron
duality fails in non-leptonic inclusive decays. In the second case, the explana-
tion of the puzzle lies beyond the heavy-quark expansion. In the first case, it is
most likely that the baryonic parameter r is much larger than most expectations
based on quark-model estimates. It will be interesting to see whether future,
field-theoretic calculations will yield values of r which are sufficiently large. Until
such calculations become available, a conventional explanation of the Λb-lifetime
puzzle cannot be excluded.

Finally, we have performed an analysis of the semileptonic branching ratio
of the B meson (BSL) and of the average number of charmed particles produced
per decay (nc). Our results are summarized in figures 2 and 3. There is a
significant dependence on the predictions for the semileptonic branching ratio
on the renormalization scale µ, which is a manifestation of our ignorance of
higher-order perturbative corrections. The results for nc, on the other hand,
are almost independent of µ. This scale dependence weakens considerably the
anticorrelation in the theoretically allowed values for BSL and nc observed in
ref. [42]. In our view, given the theoretical uncertainties and the disagreement
between the experimental values for the semileptonic branching ratio obtained in
low- and high-energy measurements, there is at present no discrepancy between
theory and experiment for BSL and nc. We have also studied the contributions
of spectator effects for these quantities and find that they are negligible for nc,
whereas they can potentially change the prediction for BSL by up to about 1%.

Note added: After completing this work we became aware of a paper by
I.I. Bigi (preprint UND-HEP-96-BIG01, June 1996 [hep-ph/9606405]), who dis-

5Alternatively, if we assume that the magnitudes of ε1(µhad) and ε2(µhad) renormalized at
a hadronic scale are less than 0.1 and 0.2, then, using (42), we find that the corresponding
predictions lie in the ranges 0.96–1.14 and 0.87–1.23.

23



cusses theoretical predictions for beauty lifetimes, making strong claims concern-
ing the theoretical predictions for the lifetime ratio τ(B−)/τ(Bd). In view of our
discussion in section 4.4, we must disagree with some statements made in this
paper.

Acknowledgements: We thank Patricia Ball, Jon Rosner, Berthold Stech and
Kolia Uraltsev for helpful discussions. C.T.S. acknowledges the Particle Physics
and Astronomy Research Council for its support through the award of a Senior
Fellowship.

24



Appendix A: Renormalization-group evolution

The four-quark operators appearing in the heavy-quark expansion are convention-
ally renormalized at the scale µ = mb. However, one may use the renormalization-
group to rewrite them in terms of operators renormalized at a scale µ 6= mb. The
renormalization-group evolution is determined by the anomalous dimensions of
the four-quark operators in the HQET, where the b quark is treated as static
quark [25]. In the literature, this evolution is sometimes referred to as “hybrid
renormalization” [10, 27, 28].

We find that the operators O
q
V−A and T

q
V−A, and similarly O

q
S−P and T

q
S−P ,

mix under renormalization. At one-loop order, the mixing within each pair (O,T )
is governed by the anomalous dimension matrix

γ̂ =
3αs
2π


 CF 1
−
CF

2Nc

1

2Nc


 , (A.1)

which has eigenvalues 0 and 3Nc. Here Nc is the number of colours, and CF =
(N2c − 1)/2Nc is the eigenvalue of the quadratic Casimir operator in the funda-
mental representation. The operators defined at the scale mb can be rewritten in
terms of those defined at a scale µ 6= mb. In leading logarithmic approximation,
the result is

O(mb) =
[
1 +

2CF
Nc

(κ− 1)
]
O(µ) −

2

Nc
(κ− 1) T (µ) ,

T (mb) =
[
1 +

1

N2c
(κ− 1)

]
T (µ) −

CF

N2c
(κ− 1) O(µ) , (A.2)

where

κ =

(
αs(µ)

αs(mb)

)3Nc/2β0
, (A.3)

and β0 =
11
3
Nc −

2
3
nf is the first coefficient of the β-function (nf = 3 is the

number of light quark flavours).
Given the evolution equations (A.2) for the four-quark operators, it is imme-

diate to derive the corresponding equations for the hadronic parameters defined
in (14), (27) and (28). We obtain:

Bi(mb) =
[
1 +

2CF
Nc

(κ− 1)
]
Bi(µ) −

2

Nc
(κ− 1) εi(µ) ,

εi(mb) =
[
1 +

1

N2c
(κ− 1)

]
εi(µ) −

CF

N2c
(κ− 1) Bi(µ) ,

r(mb) = κr(µ) +
1

Nc
(κ− 1) B̃(µ) r(µ) ,

B̃(mb) r(mb) = B̃(µ) r(µ) . (A.4)

25



Of course, introducing parameters renormalized at a scale µ 6= mb would simply
amount to a reparametrization of the results and as such is not very illuminating.
However, the evolution equations are used in section 3 to study the sensitivity of
our results to unknown higher-order corrections.

As an illustration, we study the evolution from µ = mb down to a typical
hadronic scale µhad � mb, which we choose such that αs(µhad) = 0.5 (corre-
sponding to µhad ∼ 0.67 GeV). We find

Bi(mb) ' 1.48Bi(µhad) − 0.36εi(µhad) ,

εi(mb) ' 1.06εi(µhad) − 0.08Bi(µhad) ,

r(mb) ' [1.54 + 0.18B̃(µhad)] r(µhad) ,

B̃(mb) '
B̃(µhad)

1.54 + 0.18B̃(µhad)
, (A.5)

indicating that renormalization effects can be quite significant. If one assumes
that the matrix elements renormalized at the scale µhad can be estimated using
the vacuum insertion hypothesis for mesons and the valence-quark approximation
for baryons, then

Bi(µhad) '
f2B(µhad)

f2B(mb)
= κ−8/9 ' 0.68 , εi(µhad) ' 0 , B̃(µhad) ' 1 ,

(A.6)
where the factor κ−8/9 in the first equation arises from the anomalous dimension
of the axial current in the HQET [27, 28]. Using (A.4), we then find that the
parameters defined at a renormalization scale mb (as used throughout this paper)
would be B1(mb) = B2(mb) ' 1.01, ε1(mb) = ε2(mb) ' −0.05, r(mb)/r(µhad) '
1.72, and B̃(mb) ' 0.58. Thus, for mesons the violations of the factorization
approximation induced by the evolution from µhad up to mb remain small, i.e. we
find Bi(mb) ' 1 and εi(mb) ' 0. We stress, however, that we do not want to
suggest that the choice of parameters in (A.6) is actually physical.

For mesons, the large-Nc counting rules [48, 49] imply that the parameters εi
are of order 1/Nc, whereas the parameters Bi are of order unity. These results
are respected by the evolution equations (A.4), which in the large-Nc limit take
the form

Bi(mb) = κ∞ Bi(µ) + O(1/Nc) ,

εi(mb) = εi(µ) −
1

2Nc
(κ∞ − 1) Bi(µ) + O(1/N

2
c ) , (A.7)

where κ∞ = [αs(µ)/αs(mb)]
9/22. Under a change of the renormalization scale, the

parameters εi stay of order 1/Nc, whereas the parameters Bi change by a factor
of order unity.

26



Appendix B: BSL and nc in the ms scheme

The semileptonic branching ratio and nc can also be calculated using running
quark masses renormalized in the ms scheme rather than pole masses. To compare
the results in such a scheme to those presented in our work, we have to relate
the ratio of the pole masses to the ratio of the running masses. There is some
freedom in how to do this translation. Since in the expressions for the (partial)
inclusive decay rates radiative corrections are included to order αs(µ) only, it is
consistent to work with the one-loop relation

mc

mb
=
mc(µ)

mb(µ)

(
1 −

2αs(µ)

π
ln
mc(µ)

mb(µ)

)
. (B.1)

We shall refer to this choice as scheme ms1. Alternatively, one may prefer to
resum the leading and next-to-leading logarithms to this relation, which leads to

mc(µ)

mb(µ)
=
mc

mb

(
αs(mc)

αs(mb)

)γ0/2β0 {
1−

αs(mc) −αs(mb)

π

(
γ1β0 −γ0β1

8β20
−

4

3

)}
, (B.2)

where γ0 and γ1 are the one- and two-loop coefficients of the anomalous dimension
of the running quark mass, and β0 and β1 are the coefficients of the β-function.
We shall call this scheme ms2; it has been adopted in the work of Bagan et al. [16].

Our results for these two versions of the ms scheme are:

BSL(ms1) =
{

11.6 ± 0.9%; µ = mb,
10.7 ± 0.9%; µ = mb/2,

nc(ms1) =
{

1.20 ∓ 0.06; µ = mb,
1.20 ∓ 0.06; µ = mb/2,

(B.3)

BSL(ms2) =
{

10.9 ± 0.9%; µ = mb,
10.3 ± 0.9%; µ = mb/2,

nc(ms2) =
{

1.25 ∓ 0.05; µ = mb,
1.24 ∓ 0.06; µ = mb/2,

and the combined predictions for BSL and nc are shown in figure 4. Contrary
to the case of the on-shell scheme, the calculations in the ms scheme become
unstable for low values of the renormalization scale. For this reason, we only
present result for µ ≥ mb/2. The results obtained in the scheme ms1 are close to
those obtained in the on-shell scheme and presented in section 5. In the scheme
ms2, on the other hand, we find lower values for BSL and higher values for nc. We
note that our results for this scheme do not coincide with the numbers presented
in the erratum of ref. [16]; in particular, we do not find the large values of nc
reported there. The numerical differences are mainly due to the fact that Bagan
et al. use lower values for the charm-quark mass (they use mc = 1.33 GeV for

27



the central value of the pole mass rather than 1.4 GeV), and that they multiply
each partial decay rate by the numerical factor

(
mb

mb(µ)

)5
= 1 +

αs(µ)

π

(
20

3
− 5 ln

m2b
µ2

)
+ . . . , (B.4)

which we omit since it cancels trivially in the dimensionless quantities BSL and
nc. Note that, in particular, this factor is responsible for the large apparent scale
dependence of the results presented in ref. [16]. Another difference is that we
include in the calculation an estimate of the contribution from charmless decays,
using Bno charm = (2±1)% [14, 41, 42] and Γ

b→u
SL /Γ

b→c
SL ' 1%. This lowers BSL by

a factor 0.99 and nc by a factor 0.98.

LE

HE

µ/mb
0.5 1.0 1.5

mc/mb

0.33

0.29

0.25

8 9 10 11 12 13 14
BSL (%)

1

1.1

1.2

1.3

1.4

n c

Figure 4: Combined theoretical predictions for the semileptonic branching
ratio and charm counting as a function of the quark-mass ratio mc/mb and the
renormalization scale µ. The solid lines refer to the scheme ms1, the dashed
ones to ms2.

It may be argued that the apparent large scheme dependence of the results
for the semileptonic branching ratio and nc prevent a reliable theoretical predic-
tion. However, as we have shown above the main reason is that the numerical
value of the quark mass ratio mc/mb can be quite different in different schemes
(mc(µ)/mb(µ) ' 0.8 mc/mb in the scheme ms1, and 0.7 mc/mb in ms2). Since
the dependence of BSL and nc on the quark-mass ratio comes simply from phase
space (and is particularly strong for the channel b → cc̄s), we feel that the on-
shell scheme is more adequate for performing the calculation. In other words, we
expect that in the ms scheme one would encounter larger higher-order corrections,
once the calculation is pushed to order α2s and higher.

28



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32