1 Introduction - Caltech High Energy Physics - California Institute of ...

274 Semileptonic Decays and Sides of the Unitarity Triangle
constructed from the weak-interaction Lagrangian LW , in terms of which operator (at p2 = m2 Q ) the total
decay rate is given by6 ΓH = 〈HQ|Leff |HQ〉 . (4.33)
Using in Eq. (4.32) the term
Lub = GF Vub
√ (uγµ (1 − γ5) b) ℓµ
(4.34)
2
with ℓµ = ℓγµ (1 − γ5) ν in place of LW , one would find the total inclusive decay rate of B → Xu ℓν. The
effective operator (4.32) is evaluated using short-distance OPE. The leading term in the expansion describes
the perturbative decay rate, while subsequent terms containing operators of higher dimension describe the
nonperturbative contributions. The term of interest for the present discussion is the third one in this
expansion, containing a four-quark operator [74, 75, 76, 77]
L (3)
b→uℓν = −2 G2 F |Vub| 2 m 2 b
3 π
� u
OV −A − O u �
S−P , (4.35)
where the following notation [78] is used for the relevant four-quark operators (normalized at µ = mb):
O q
V −A =(bLγµqL)(qLγµbL) , O q
S−P =(bRqL)(qLbR) ,
T q
V −A =(bLt a γµqL)(qLt a γµbL) , T q
S−P =(bRt a qL)(qLt a bR) . (4.36)
(The operators T , containing the color generators t a , will appear in further discussion.)
The matrix elements of the operators O u over the B mesons can be parameterized in terms of the meson
annihilation constant fB and of dimensionless coefficients B (“bag constants”) as
〈B + |O u V −A|B + 〉 = f 2 B mB
16
(B s 1 + B ns
1 ) , 〈B + |O u S−P |B + 〉 = f 2 B mB
(B
16
s 2 + B ns
2 ) , (4.37)
for the B + meson containing the same light quark (u) as in the operator, and
〈Bd|O u V −A|Bd〉 = f 2 B mB
16
(B s 1 − B ns
1 ) , 〈Bd|O u S−P |Bd〉 = f 2 B mB
(B
16
s 2 − B ns
2 ) , (4.38)
for the Bd meson where the light quark (d) is different from the one in the operator. In the limit of naive
factorization the “bag constants”, both the flavor-singlet (Bs ) and the flavor non-singlet (Bns ) ones are all
equal to one: B s 1 = B ns
1 = B s 2 = B ns
2 = 1, and the matrix elements over the B mesons of the difference of
the operators entering Eq. (4.35) are vanishing. However the expected accuracy of the factorization is only
about 10%, which sets the natural scale for the non-factorizable contributions, i.e., for the deviations from
the naive factorization. (Numerical estimates of non-factorizable terms can be found in [79, 80, 81].) After
averaging the operator in Eq. (4.35) one finds the contribution of the non-factorizable terms to the rates of
the B → Xu ℓν decays in the form
δΓ(B ± → Xu ℓν)= G2F |Vub| 2 f 2 B m3 b δB
12 π
s + δBns ,
2
δΓ(Bd → Xu ℓν)= G2F |Vub| 2 f 2 B m3 b δB
12 π
s − δBns ,
2
(4.39)
where δBs = Bs 2 − Bs 1 and δBns = Bns 2 − Bns 1 . These contributions can be compared with the ‘bare’ total
decay rate Γ0 = G2 F |Vub| 2m5 b /(192π3 ):
δΓ(B ± )
≈
Γ0
16π2 f 2 B
m2 δB
b
s + δBns � �2 s ns
fB δB + δB
≈ 0.03
,
2
0.2 GeV 0.2
δΓ(Bd)
δBs − δBns � �2 s ns
fB δB − δB
≈ 0.03
.
2
0.2 GeV 0.2
(4.40)
Γ0
≈ 16π2 f 2 B
m 2 b
6 The non-relativistic normalization for the heavy quark states is used here: 〈Q|Q † Q|Q〉 =1.
The Discovery Potential of a Super B Factory