2011 QCD and High Energy Interactions - Rencontres de Moriond ...

functions h are matrix elements of non-local operators. They cannot be extracted from data,
and must be modeled. How do the strong phases arise for the resolved photon contributions?
It can be shown 11 that the functions h are real by using parity, time reversal, and heavy
quark symmetry. The functions J, on the other hand, are complex since they arise from uncut
propagators and loops.
At the lowest order in αs and ΛQCD/mb, the resolved photon contribution to the CP asym-
metry is
with
A res
Xsγ = π
Im (1 + ɛs)
mb
C1
˜Λ
C7γ
c
17 − Im ɛs
˜Λ u 17 = 2
3 h17(0)
˜Λ c 17 = 2
∞
3
4m2 c/mb
˜Λ ¯ ∞
B dω1
78 = 2
−∞ ω1
dω1
ω1
C1
C7γ
m2 c
f
mb ω1
˜Λ u 17 + Im C8g
4παs
C7γ
˜ Λ ¯ B
78
h17(ω1)
h (1)
78 (ω1, ω1) − h (1)
78 (ω1, 0)
,
, (5)
where f(x) = 2x ln[(1 + √ 1 − 4x)/(1 − √ 1 − 4x)]. The soft functions hij are in light-cone gauge
¯n · A = 0,
h17(ω1, µ) =
h (1)
78 (ω1, ω2, µ) =
dr
2π e−iω1r 〈 ¯ B| ¯ h(0) /¯n iγ ⊥ α ¯nβ gG αβ
s (r¯n) h(0)| ¯ B〉
2MB
dr
2π e−iω1r
du
2π eiω2u 〈 ¯ B| ¯ h(0) T A /¯n h(0) q eq ¯q(r¯n) /¯n T Aq(u¯n)| ¯ B〉
.
2MB
Using the modeling of the soft function as in 11 allows us to estimate the size of resolved photon
contributions. We need to estimate each of the ˜ Λij in (5).
In order to estimate ˜ Λ ¯ B 78 , one can use Fierz transformation and the Vacuum Insertion Ap-
proximation (VIA) to express h (1)
78 as a the square of B meson light-cone amplitudes φB +. This
allows us to write
˜Λ ¯ B
78
VIA
= espec
2f 2 B MB
9
∞
0
φB +(ω1, µ)
dω1
2 ,
where espec denotes the electric charge of the spectator quark in units of e (espec = 2/3 for
B − and −1/3 for ¯ B 0 ). Using 12 to constrain the integral, one finds that in the VIA, ˜ Λ ¯ B 78 ∈
espec[17 MeV, 190 MeV].
Both ˜ Λ u 17 and ˜ Λ c 17 depend on h17. Being a soft function, h17 has support over a hadronic
range. Since in the expression for ˜ Λc 17 , the integral starts at at 4m2c/mb ≈ 1 GeV, we can expect
a small overlap. Indeed one finds that 13 , −9 MeV < ˜ Λc 17 < +11 MeV. For ˜ Λu 17 there is no such
suppression and we have −330 MeV < ˜ Λu 17 < +525 MeV. The range is not symmetric since
the normalization of h17 is 2λ2 ≈ 0.24 GeV2 . This is the same result as one would obtain from
estimating ˜ Λu 17 using naive dimensional analysis.
Including both the direct and resolved contributions and using µ = 2 GeV for the factorization
scale, we find that the total CP asymmetry in the SM is
A SM
C1 ˜Λ u
Xsγ ≈ π
17 −
Im ɛs
˜ Λc 17
+ 40αs
Λc
= 1.15 ×
9π
˜ Λu 17 − ˜ Λc
17
+ 0.71 % .
300 MeV
C7γ
mb
The direct contribution is slightly higher then the 0.5% mention before, since we are using a
slightly lower factorization scale. The conclusion is that the asymmetry is actually dominated
mb
ω1
(6)
(7)