Study of $B_{s}\to πρ$ decays in the perturbative QCD approach

0.16 MB·English

by  Xian-Qiao Yu

Study of B πρ decays in the perturbative QCD approach
s
→
Xian-Qiao Yu∗, Ying Li†
Institute of High Energy Physics, P.O.Box 918(4), Beijing 100049, China; and
Graduate School of the Chinese Academy of Sciences, Beijing 100049, China
Cai-Dian Lu¨
CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, China; and
Institute of High Energy Physics, P.O.Box 918(4), Beijing 100049, China‡
(Dated: February 2, 2008)
In this note, we calculate thebranching ratio and CP asymmetry parameters of Bs →πρ in the
framework of perturbative QCD approach based on kT factorization. This decay can occur only
via annihilation diagrams in the Standard Model. We find that (a) the charge averaged Br(Bs →
6 π+ρ−+π−ρ+) isabout (9−12)×10−7; Br(Bs→π0ρ0)≃4×10−7;and (b) thereare sizable CP
asymmetriesintheprocesses, whichcan betested inthenear futureLargeHadronCollider beauty
0
experiments(LHC-b) at CERN.
0
2
PACSnumbers: 13.25.Hw,12.38.Bx

n
a
J The B meson rare decays provide a good place for testing the Standard Model (SM), studying CP violation and

1 looking for possible new physics beyond the SM. A lot of theoretical studies have been done in recent years, which
1 are strongly supported by the running B factories in KEK and Stanford Linear Accelerator Center (SLAC). Looking
forwardtothefutureCERNLargeHadronColliderbeautyexperiments(LHC-b),alargenumberofBs andBc mesons
2
can also be produced. So the studies of B meson rare decays are necessary in the next a few years.
v s
In this paper, we study the rare decays B πρ in Perturbative QCD approach (PQCD) [1]. Comparing with
9 s →
QCD factorization approach [2], PQCD approach can make a reliable calculation for pure annihilation diagrams in
6
2 kT factorization. TheendpointsingularityoccurredinQCDfactorizationapproachcanbecuredherebythe Sudakov
1 factor from resummation of double logarithms.
1 In PQCD approach, the decay amplitude can be written as:
5
0
h/ Amplitude∼ d4k1d4k2d4k3 Tr C(t)ΦBs(k1)Φπ(k2)Φρ(k3)H(k1,k2,k3,t) e−S(t). (1)
Z
p (cid:2) (cid:3)
- In our following calculations, the Wilson coefficient C(t), Sudakov factor Si(t)(i=Bs,π,ρ) and the non-perturbative
p
but universal wave function Φ can be found in the Refs. [3, 4, 5, 6, 7]. The hard part H are channel dependent but
e i
fortunately perturbative calculable, which will be shown below.
h
: Likethe Bs π+π− decay[8],the Bs πρ decaysarepureannihilationtype raredecays,whicharedifficulttobe
v → →
calculated in method other than PQCD approach. Fig. 1 shows the lowest order Feynman diagrams to be calculated
i
X in PQCD approach where the big dots denote the quark currents in four quark operators for B π+ρ− according
s
→
to the effective hamiltonian of b quark decay [9]. First for the usual factorizable diagram (a) and (b), the sum of
r
a (V-A)(V-A) current contributions is given by
1 ∞
M [C]=16πC M2 dx dx b db b db [x φA(x )φ (x )+2r r (1+x )φP(x )φs(x )
a F B 2 3 2 2 3 3 3 π 2 ρ 3 π ρ 3 π 2 ρ 3
Z0 Z0
+2r r (x 1)φP(x )φt(x (cid:8))]α (t1)h (x ,x ,b ,b )exp[ S (t1) S (t1)]C(t1)
π ρ 3− π 2 ρ 3 s a a 2 3 2 3 − π a − ρ a a
[x φA(x )φ (x )+2r r (1+x )φP(x )φs(x )+2r r (x 1)φT(x )φs(x )]
− 2 π 2 ρ 3 π ρ 2 π 2 ρ 3 π ρ 2− π 2 ρ 3
α (t2)h (x ,x ,b ,b )exp[ S (t2) S (t2)]C(t2) , (2)
s a a 3 2 3 2 − π a − ρ a a
where r =m /m =m2/[m (m +m )], r =m /m . C =4/3 is the group factor of the SU(3) gauge(cid:9)group.
π 0π B π B u d ρ ρ B F c
φ is light cone distribution amplitude, which describes the momentum distribution of the meson wavefunction. The
i
∗ [email protected]
† [email protected]
‡ Mailingaddress
2
ρ
−
u¯ u¯ ρ−
¯b d ¯b d
B B
s s
d¯ d¯
s
u π+ s u π+
(a) (b)
u¯ ρ d u¯ ρ d
− −
¯b ¯b
B B
s s
s s
u π+ d¯ u π+ d¯
(c) (d)
FIG. 1: The lowest orderdiagrams for Bs →π+ρ− decay
function
πi
h (x ,x ,b ,b )=S (x ) H(1)(M √x x b )
a 2 3 2 3 t 3 2 0 B 2 3 2
πi
θ(b b )J (M √x b ) H(1)(M √x b )+(b b ) (3)
× 2− 3 0 B 3 3 2 0 B 3 2 2 ↔ 3
(cid:2) (cid:3)
comes fromthe Fouriertransformationof propagatorsof virtualquarkandgluonin the hardpartcalculations. S (x)
t
resultsfromthethresholdresummationofQCDradiativecorrectionstohardamplitudes[10]. H1(z)=J (z)+iY (z),
0 0 0
J and Y are Bessel functions. The sum of (V-A)(V+A) current contributions of diagrams (a) and (b) is M [C].
0 0 a
−
For the non-factorizable annihilation diagrams (c) and (d), all three meson wave functions are involved. The
(V-A)(V-A) operator’s contribution is:
1 1 ∞
M [C]= − 64πC M2 dx dx dx b db b db φ (x ,b )
c √2N F B 1 2 3 1 1 2 2 B 1 1
c Z0 Z0
[ x φ (x )φA(x )+r r (x x )φt(x )φP(x ) r r (x +x )φs(x )φP(x )
× − 2 ρ 3 π 2 ρ π 3− 2 ρ 3 π 2 − ρ π 2 3 ρ 3 π 2
(cid:8) −rπrρ(x2+x3)φtρ(x3)φTπ(x2)+rπrρ(x3−x2)φsρ(x3)φTπ(x2)]
C(t1)α (t1)h(1)(x ,x ,x ,b ,b )exp[ S (t1) S (t1) S (t1)] [ x φ (x )φA(x )
c s c c 1 2 3 1 2 − B c − π c − ρ c − − 3 ρ 3 π 2
+r r (x x )φt(x )φP(x ) r r (2+x +x )φs(x )φP(x )
ρ π 2− 3 ρ 3 π 2 − ρ π 2 3 ρ 3 π 2
+r r (2 x x )φt(x )φT(x )+r r (x x )φs(x )φT(x )]
π ρ − 2− 3 ρ 3 π 2 π ρ 2− 3 ρ 3 π 2
C(t2)α (t2)h(2)(x ,x ,x ,b ,b )exp[ S (t2) S (t2) S (t2)] . (4)
c s c c 1 2 3 1 2 − B c − π c − ρ c
(cid:9)
3
For the penguin operators, there are also (V A)(V +A) type operators, whose contribution is given as
−
1 1 ∞
MP[C]= 64πC M2 dx dx dx b db b db φ (x ,b )
c √2N F B 1 2 3 1 1 2 2 B 1 1
c Z0 Z0
[ x φ (x )φA(x ) r r (x x )φt(x )φP(x ) r r (x +x )φs(x )φP(x )
× − 3 ρ 3 π 2 − ρ π 3− 2 ρ 3 π 2 − ρ π 2 3 ρ 3 π 2
(cid:8) −rπrρ(x2+x3)φtρ(x3)φTπ(x2)−rπrρ(x3−x2)φsρ(x3)φTπ(x2)]
C(t1)α (t1)h(1)(x ,x ,x ,b ,b )exp[ S (t1) S (t1) S (t1)] [ x φ (x )φA(x )
c s c c 1 2 3 1 2 − B c − π c − ρ c − − 2 ρ 3 π 2
r r (x x )φt(x )φP(x ) r r (2+x +x )φs(x )φP(x )
− ρ π 2− 3 ρ 3 π 2 − ρ π 2 3 ρ 3 π 2
+r r (2 x x )φt(x )φT(x ) r r (x x )φs(x )φT(x )]
π ρ − 2− 3 ρ 3 π 2 − π ρ 2− 3 ρ 3 π 2
C(t2)α (t2)h(2)(x ,x ,x ,b ,b )exp[ S (t2) S (t2) S (t2)] , (5)
c s c c 1 2 3 1 2 − B c − π c − ρ c
where (cid:9)
h(j)(x ,x ,x ,b ,b )=
c 1 2 3 1 2
πi
θ(b b ) H(1)(M √x x b )J (M √x x b )
2− 1 2 0 B 2 3 2 0 B 2 3 1
(cid:26)
K (M F b ), for F2 >0
0 B (j) 1 (j)
+(b b ) , (6)
1 ↔ 2 (cid:27)× π2iH(01)(MB |F(2j)| b1), for F(2j) <0!
q
K is modified Bessel function and F ’s are defined by
0 (j)
F2 =x x x x ; F2 =x +x +x x x x x . (7)
(1) 1 2− 2 3 (2) 1 2 3− 1 2− 2 3
The hardscale t′s in Eqs.(2, 4, 5) are chosenas the largestenergyscale appearingin eachdiagramto kill the large
i
logarithmic corrections:
t1 = max(M √x ,1/b ,1/b ),
a B 3 2 3
t2 = max(M √x ,1/b ,1/b ),
a B 2 2 3
t1 = max(M F2 ,M √x x ,1/b ,1/b ),
c B | (1)| B 2 3 1 2
q
t2 = max(M F2 ,M √x x ,1/b ,1/b ).
c B | (2)| B 2 3 1 2
q
The total decay amplitude is then
1 2 2 1 1 1 1
(B π+ρ−)=f M V∗V (C + C ) V∗V (2C + C 2C C C C + C + C )
A s → B a ub us 1 3 2 − tb ts 3 3 4− 5− 3 6− 2 7− 6 8 2 9 6 10
(cid:20) (cid:21)
1 1
+M V∗V C V∗V (2C + C ) V∗V MP 2C + C , . (8)
c ub us 2− tb ts 4 2 10 − tb ts c 6 2 8
(cid:20) (cid:21) (cid:18) (cid:19)
and the decay width is expressed as
Γ(B π+ρ−)= G2FMB3 (B π+ρ−) 2. (9)
s s
→ 128π A →
(cid:12) (cid:12)
Themostimportantcontributionhereisthefactorizablepen(cid:12)guindiagram,w(cid:12)hichisCKMenhanced. Ifweexchange
the π andρ in Fig. 1,by the same method, we cancompute the B π−ρ+ decay. The expressionsare similar. The
s
→
decay amplitude for B π0ρ0 is
s
→
(B π0ρ0)= (B π+ρ−)+ (B π−ρ+), (10)
s s s
A → A → A →
and the decay width can be written as
Γ(B π0ρ0)= G2FMB3 (B π0ρ0) 2. (11)
s s
→ 512π A →
(cid:12) (cid:12)
(cid:12) (cid:12)
4
Inthefollowing,wefirstgivethebranchingratiosofB πρ. JustasinRef. [8],weleavetheCKMangleγasafree
s
parameter in our numerical calculations. Because there a→re four decay channels: B /B¯ π+ρ−, B /B¯ π−ρ+, it
s s s s
is not possible to distinguish the initial state by detecting the final states. We averageth→e sum of B /B¯ → π+ρ− as
s s
one channel,and B /B¯ π−ρ+ as another, whichis distinguishable by experiments. Using the same par→ametersas
s s
→
refs. [4, 7], we get
Br(B /B¯ π+ρ−)=(5.1+0.8) 10−7,
s s → −0.5 ×
Br(B /B¯ π−ρ+)=(5.4+0.5) 10−7,
s s → −0.8 ×
Br(B /B¯ π0ρ0)=(4.2+0.6) 10−7, (12)
s s → −0.7 ×
where all channels are averagedfor B and B¯ .
s s
In Ref. [11], Beneke etal have estimated the branching ratio of B¯ π+ρ− in the QCD factorization approach.
s
→
Weak annihilation diagrams are power suppressed in the heavy quark limit and, in general, not calculable in QCD
factorization approach. In order to avoid the end-poind singularities, they introduced phenomenological parameters
to replace the divergent integral. With those parameters they estimated that the branching ratio of B¯ π+ρ−
s
is (0.03 0.14) 10−7. In PQCD approach, the annihilation amplitude is calculable. In the rest frame o→f the B
s
meson, t−he d or×d¯quark included in ρ or π has momentum (M /4), and the gluonproducing them has momentum
B
O
q2 = (M2/4). This is a hard gluon, the PQCD can be safely used because of asymptotic freedom of QCD [12]. We
O B
have tested that the bulk of the result comes from the region with α /π <0.2, where a figure was show in Ref. [13].
s
Our predicted result is larger than their estimation, which can be tested by the future experiments.
Using the same definition in Ref. [8], we study the CP violation parameters AdCiPr and aǫ+ǫ′ in the process of
B π0ρ0 decay. We find the direct CP violation parameter Adir(B π0ρ0) is about 2% when γ is near 100◦,
s → CP s →
the small direct CP asymmetry is also a result of small tree level contribution. The mixing CP violation parameter
aǫ+ǫ′(Bs →π0ρ0) is large, whose peak is close to 16% (see Fig. 2).
0.15
0.125
0.1
’
+
a
0.075
0.05
0.025
0
0 25 50 75 100 125 150 175
(degree)
FIG. 2: Mixing CP violation parameter of Bs →π0ρ0 decay as a function of CKM angle γ
The CP violations of B /B¯ π±ρ∓ are very complicated. There are four decay amplitudes, which can be
s s
→
expressed as:
g = π+ρ− H B ; h= π+ρ− H B¯ ;
eff s eff s
h | | i h | | i
g¯= π−ρ+ H B¯ ; h¯ = π−ρ+ H B . (13)
eff s eff s
h | | i h | | i
We introduce four parameters to describe the CP asymmetries in the processes, which are given by [14]
1 1
Cπρ = 2(aǫ′ +aǫ¯′), ∆Cπρ = 2(aǫ′ −aǫ¯′),
1 1
Sπρ = 2(aǫ+ǫ′ +aǫ+ǫ¯′), ∆Sπρ = 2(aǫ+ǫ′ −aǫ+ǫ¯′), (14)
5
where
g 2 h2 2Im(h/g)
aǫ′ = |g|2−+|h|2, aǫ+ǫ′ = −1+ h/g 2 ,
| | | | | |
h¯ 2 g¯2 2Im(g¯/h¯)
aǫ¯′ = |h¯|2−+|g¯|2, aǫ+ǫ¯′ = −1+ g¯/h¯ 2 . (15)
| | | | | |
We calculate the above four CP asymmetry parameters and show the CKM angle γ dependence in Fig. 3. We do
not plot the C in this figure, since its value is near zero. The decay branching ratios depends heavily on the shape
πρ
ofwavefunctions anddecayconstantsetc. Butthe CPasymmetryshouldnotsince the dependence willbe cancelled.
ThedirectCPasymmetrycanbeaffectedbythepowercorrectionsandnext-to-leadingordercontributionseasily. We
investigate the CP asymmetry parameters’sdependence on the hardscale t in Eq. (1), which characterizethe size of
next-to-leading order contribution. The CP asymmetry numbers are shown in Table I with those uncertainties. By
changingthehardscaletfrom0.9tto1.3t,wefindtheCP asymmetriesofB (B¯ ) π±ρ∓ changelittle: forγ =60◦,
s s
→
the uncertainty is less than 1% for C , 4% for ∆C , 3% for S and 7% for ∆S , respectively. The reason is that
πρ πρ πρ πρ
mixing induced CP is dominant here, but the direct CP of B π0ρ0 really changed much, as shown in Table I.
s
→
TABLE I: CP asymmetry parameters using γ =60◦ with uncertainties
Scale Cπρ ∆Cπρ Sπρ ∆Sπρ AdCiPr(Bs →π0ρ0) aǫ+ǫ′(Bs →π0ρ0)
0.9 t 0.1% 73% 10% 21% 0.6% 11%
t 1% 77% 11% 14% 1.6% 13%
1.3 t 1.8% 79% 14% 10% 2.6% 16%
1
0.75
0.5
)
C
0.25
(
S
S 0
-0.25
-0.5
-0.75
0 25 50 75 100 125 150 175
(degree)
FIG. 3: CP violation parameters of Bs0(B¯s) → π±ρ∓ decays: ∆C (solid line), S (dashed line) and ∆S (dotted line) as a
function of CKM angle γ
In conclusion, we study the branching ratio and CP asymmetries of B πρ decays in PQCD approach. We find
s
the branching ratioof B π+ρ−+π−ρ+ is atorder 10−6, which is larger→than QCD factorization’sestimation. We
s
→
also predict CP asymmetries in the process, which may be measured in the future LHC-b experiments.
We thank M.-Z. Yang for useful discussions. This work is partly supported by National Science Foundation of
6
China under Grant No. 90103013,10475085and 10135060.
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