1

Spectator Effects in Inclusive Decays of Beauty Hadrons

UKQCD Collaboration. Massimo Di Pierro a and Chris T. Sachrajda b . Presented by M.D.P.
Department of Physics and Astronomy, University of Southampton, SO17 1BJ, United Kingdom

amdp@hep.phys.soton.ac.uk

bcts@hep.phys.soton.ac.uk

We evaluate the matrix elements of the four-quark operators which contribute to the lifetimes of B-mesons and
the Λb-baryon. We find that the spectator effects are not responsible for the discrepancy between the theoretical
prediction and experimental measurement of the ratio of lifetimes τ(Λb)/τ(B).

1. Introduction

Inclusive decays of heavy hadrons can be stud-
ied in the framework of the heavy quark expan-
sion, in which, for example, lifetimes are com-
puted as series in inverse powers of the mass of
the b-quark [1]. For an arbitrary hadron H

τ−1(H) =
G2Fm

5
b

192π3
|Vcb|

2

2mH

∑
i≥0

cim
−i
b (1)

where

• c0 corresponds to the decay of a free-quark
and is universal.

• c1 is zero because the operators of dimen-
sion four can be eliminated using the equa-
tions of motion.

• c2 can be estimated and is found to be
small.

• c3 contains a contribution proportional to

〈H| bΓq qΓ̃b |H〉 (2)

and is therefore the first term in the expan-
sion to which the interaction between the
heavy and the light quark(s) contribute. Al-
though this is an O(m−3b ) correction, it may
be significant since it contains a phase-space
enhancement.

The aim of our lattice simulation is to compute
c3 for B-mesons and the Λb-baryon, in order to

check whether spectator effects contribute signif-
icantly to the ratios of lifetimes for which the ex-
perimental values are:

τ(B−)

τ(B0)
= 1.06 ± 0.04 (3)

τ(Λb)

τ(B0)
= 0.78 ± 0.04 . (4)

The discrepancy between the experimental
value in eq. (4) and the theoretical prediction
of τ(Λb)/τ(B

0) = 0.98 (based on the Operator
Product Expansion in eq. (1) including terms in
the sum up to those of O(m−2b )) is a major puzzle.
It is therefore particularly important to compute
the O(m−3b ) spectator contributions to this ratio.

The ratios in eqs. (3) and (4) can be expressed
in terms of 6 matrix elements:

τ(B−)

τ(B0)
= a0 + a1ε1 + a2ε2 + a3B1 + a3B2 (5)

τ(Λb)

τ(B0)
= b0 + b1ε1 + b2ε2 + b3L1 + b4L2 (6)

where 1

B1 ≡
8

f2BmB

〈B| bγµLq qγµLb |B〉

2mB
(7)

B2 ≡
8

f2BmB

〈B| bLq qRb |B〉

2mB
(8)

1In terms of the parameters B̃ and r introduced in ref. [2]

r = −6L1

B̃ = −2L2/L1 − 1/3



2

ε1 ≡
8

f2BmB

〈B| bγµLtaq qγµLt
ab |B〉

2mB
(9)

ε2 ≡
8

f2BmB

〈B| bLtaq qRtab |B〉

2mB
(10)

L1 ≡
8

f2BmB

〈Λ| bγµLq qγµLb |Λ〉

2mΛ
(11)

L2 ≡
8

f2BmB

〈Λ| bγµLtaq qγµLt
ab |Λ〉

2mΛ
(12)

and the coefficients ai and bi are given by

value value
a0 +1.00 b0 +0.98
a1 −0.697 b1 −0.173
a2 +0.195 b2 +0.195
a3 +0.020 b3 +0.030
a4 +0.004 b4 −0.252

The values in the table correspond to opera-
tors renormalized in a continuum renormalization
scheme at the scale µ = mB. Their matrix ele-
ments are obtained from those in the lattice reg-
ularization by perturbative matching (see the ap-
pendix).

2. B Decay

The matrix elements B1,B2,ε1,ε2 are com-
puted on a 243×48 lattice at β = 6.2 (correspond-
ing to a lattice spacing a−1 = 2.9(1) GeV) using
the tree-level improved SW action for three values
of κ = 0.14144, 0.14226, 0.14262 and are then ex-
trapolated to the chiral limit (κc = 0.14315) [3].
We find

B1 = 1.06 ± 0.08 (13)

B2 = 1.01 ± 0.06 (14)

ε1 = −0.01 ± 0.03 (15)

ε2 = −0.02 ± 0.02 (16)

which implies that

τ(B−)

τ(B0)
= 1.03 ± 0.02 ± 0.03 (17)

in agreement with the experimental value (3).

3. Λ Decay

The computation the baryonic matrix elements
L1 and L2 is a little more difficult. We have per-

formed an exploratory study in which the light
quark propagators are computed using a stochas-
tic method [4] based on the relation

M−1ij =

∫
[dφ](Mjkφk)

∗φie
−φ∗i (M

+M)ijφj (18)

(rather then using “extended” propagators).
The matrix elements are computed on a 123 ×

24 lattice at β = 5.7 (corresponding to a lattice
spacing a−1 = 1.10(1) GeV) for two values of κ.
We therefore do not attempt an extrapolation to
the chiral limit (κc = 0.14351) but present results
seperately for each value of κ. We find:

L1 =

{
−0.28 ± 0.03 (κ = 0.13843)
−0.20 ± 0.03 (κ = 0.14077)

(19)

L2 =

{
0.13 ± 0.01 (κ = 0.13843)
0.10 ± 0.01 (κ = 0.14077) ,

(20)

which implies that (neglecting the systematic er-
ror due to the chiral extrapolation)

τ(Λb)

τ(B0)
=

{
0.94 ± 0.01 (κ = 0.13843)
0.95 ± 0.01 (κ = 0.14077) .

(21)

These results imply that spectator effects are not
sufficiently large to explain the discrepancy be-
tween the theoretical prediction and experimental
result in eq. (4).

4. Concluding remarks

We find that the matrix elements of the 4-quark
operators (13-16) satisfy the vacuum saturation
hypothesis remarkably well. A similar feature is
true for the ∆B = 2 operators which contribute
to B–B̄ mixing. We do not have a good under-
standing yet of this phenomenon.

The results for the mesonic matrix elements
lead to a prediction for the ratio of the lifetimes
of the charged and neutral mesons which is in
agreement with the experimental result in eq. (3).
Our study of the baryonic matrix elements in-
dicates that spectator effects are not sufficiently
large to explain the experimental ratio of life-
times in eq. (4). This discrepancy between the
theoretical prediction and the experimental mea-
surement remains an important problem to solve.
We do stress, however, that our calculations are



3

Table 1
Lattice perturbative coefficients and operators

pi Pi qi Qi

−4
3

log(λ2a2) + 18.97 bΓqqΓ̃b +1 bΓqbΓ̃q

−1 log(λ2a2) + 2.25 btaΓtaqqΓ̃b + bΓqqtaΓ̃tab +1 btaΓtaqbΓ̃q + bΓqbtaΓ̃taq

−6.89 btaγ0Γγ0taqqΓ̃b + bΓqqtaγ0Γ̃γ0tab +1 btaγ0Γγ0taqbΓ̃q + bΓqbtaγ0Γ̃γ0taq

2 log(λ2a2) − 4.53 btaΓqqΓ̃tab −1 btaΓqbtaΓ̃q

log(λ2a2) − 6.19 bΓtaqqΓ̃tab + btaΓqqtaΓ̃b −1 bΓtaqbtaΓ̃q + btaΓqbΓ̃taq

−6.89 bΓγ0taqqΓ̃γ0tab + btaγ0Γqqtaγ0Γ̃b +1 bΓγ0taqbtaγ0Γ̃q + btaγ0ΓqbΓ̃γ0taq

− log(λ2a2) + 6.12 bΓtaqqγ0Γ̃b −1 bΓtaqbΓ̃taq

−3 log(λ2a2) + 0.96 bΓσµνtaqqtaσνµΓ̃b −1 bΓσµνtaqbΓ̃σµνtaq

−2.43 bΓγµtaqqtaγµΓ̃b +1 bΓγµtaqbΓ̃γµtaq
The analytic expression for pi can be found in [3].

exploratory, and a more precise simulation is nec-
essary, in particular to allow for a reliable extrap-
olation to the chiral limit.

Acknowledgements

It is a pleasure to thank Chris Michael, Hart-
mut Wittig, Jonathan Flynn, Luigi Del Debbio,
Giulia De Divitiis, Vicente Gimenez and Carlotta
Pittori for many helpful discussions. This work
was supported by PPARC grants GR/L29927,
GR/L56329 and GR/I55066.

Appendix: 1-loop lattice perturbative cor-
rections to 4-quark operators

The most difficult component of the evaluation
of the 1-loop perturbative matching between four-
quark lattice operators and those renormalized in
a continuum scheme is the perturbative expan-
sion on lattice. In this appendix we present the
corresponding results for generic operators P0 and
Q0 (whose matrix elements contribute to lifetimes
and B–B̄ mixing respectively):

P0 ≡ bΓq qΓ̃b, (∆B = 0) (22)

Q0 ≡ bΓq bΓ̃q, (∆B = 2) . (23)

Γ⊗Γ̃ represents an arbitrary spinor and color ten-
sor. These operators mix under renormalization

with other 4-quark operators, listed in table 1 2:

P
1 loop
0 = P0 +

αs(a
−1)

4π

∑
i

piPi (24)

Q
1 loop
0 = Q0 +

αs(a
−1)

4π

∑
i

piqiQi . (25)

The Feynman rules correspond to the tree-level
SW-improved action for massless light quarks and
with a small gluon-mass (λ) as the infrared reg-
ulator. The dependence on λ, of course, cancels
when the corresponding continuum calculation is
combined with the lattice one.

REFERENCES

1. I. Bigi, M. Shifman and N. Uralstev,
Ann. Rev. Nucl. Part. Sci. 47 (1997) 591
and references therein.

2. M. Neubert and C. T. Sachrajda,
Nucl. Phys. B483 (1997) 339.

3. M. Di Pierro and C. T. Sachrajda,
hep-lat/9805028 (to be published in
Nucl. Phys. B).

4. C. Michael and J. Peisa, hep-lat/9802015.
5. M. Di Pierro, C. T. Sachrajda and C. Michael,

in preparation.
6. V. Gimenez and J. Reyes, hep-lat/9806023

2These results were also obtained independently in ref. [6].