A Mathematica based Version of the CKMfitter Package

36 Chapter 5. Probing the Standard Model
Next-to-Leading Order
In NLO, higher corrections in z and 1/mb are taken into account as well as penguin
contributions and QCD corrections. The most recent progress is summarized in
Reference [53] and references therein.
The expansion of Γ12/M12 up to the order of λ 2 u/λ 2 t is given by [53]:
Γ12
M12
= λ2 �
t
−Γ
M12
cc
12 + 2 (Γ uc
12 − Γ cc
12) λu
+ (2Γ
λt
uc
12 − Γ cc
12 − Γ uu
12 ) λ2u λ2 �
t
(5.24)
where λi = V ∗
id Vib, for i = u, c, t. The mixing matrix element M12 is predicted in the
SM as [54]:
λ 2 t
G 2 F
12π 2 mBd ηBB(mb)bB(mb)f 2 Bd m2 W S(xt) . (5.25)
The coefficients Γab 12 are expressed through [53]:
Γ ab
12 = G2 F m2 b
−
24π f 2 Bd MBd
� �
F ab (z) + P ab (z)
�
F ab
S (z) + P ab
�
5
S (z)
3 B′ S(mb)
�
8
3 B(mb)
�
, (5.26)
+ Γ ab
12,1/mb
where the short-distance coefficients F ab
(S) (z) contain the contributions from |∆B| = 1
(z) coefficients contain the contributions from penguin opera-
have been computed in Ref. [53], Γcc 12 is given in Ref. [55]
is derived from Γcc 12 taking the limit z → 0. In addition, they depend on
the Wilson coefficients C1, . . . , C6 and C8, which are also given in Table 5.2. The
contains the corrections of order 1/mb.
operators and the P ab
(S)
tors. The coefficients in Γ uc
12
and Γ uu
12
term Γ ab
12,1/mb
A comparison of the SM predictions of ASL at LO and NLO in QCD is shown in the
Section 5.2. The implementation of the NLO prediction for Γ12/M12 has finished
immediately before the end of this thesis and therefore not all cross checks have been
performed up to this point. As a consequence, the theory prediction for Γ12/M12 in
this thesis is used at LO to produce quantitative results if not stated otherwise.