Revisiting the $B^{0} \to \pi^{0}\pi^{0} $ decays in the perturbative QCD approach

0.17 MB·

by  Yun-Feng Li

Revisitingthe B0 π0π0 decays inthe perturbative QCD approach
→
Yun-Feng Li1,∗ and Xian-Qiao Yu1,†
1School of Physical Scienceand Technology, Southwest University, Chongqing 400715, China
(Dated:July4,2016)
WerecalculatethebranchingratioandCPasymmetryforB¯0(B0)→π0π0decaysinthePerturbativeQCD
approach. Inthisapproach,weconsiderallthepossiblediagramsincludingnon-factorizablecontributionsand
annihilationcontributions, andidentityprincipleisalsotakenintoaccount. Weobtainthebranching ratioof
B0 → π0π0 is about 1.1×10−6. Our result is in agreement with the latest measured branching ratio of
B0→π0π0bytheBelleandHFAGCollaborations.WealsopredictlargedirectCPasymmetryandmixingCP
asymmetryinB0 →π0π0decays,whichcanbetestedbytherunningLHC-bexperiments.
PACSnumbers:13.25.Hw,11.10.Hi,12.38.Bx
6
1
0 The detailed study of B meson decays is a key source of testing the Standard Model(SM), exploring CP violation and in
2 searching of possible new physics beyond the SM. The theoretical studies of B meson decays have been studied widely in
l the literature, oneofthe challengesis thatthemeasuredbranchingratio [1–3] forthe decayofB mesonto neutralpionpairs
u B0 π0π0 is significantly larger than the theoretical predictions obtained in the QCD factorization approach [4–7] or a
J →
perturbativeQCDapproach(PQCD)[8].
1 ThebranchingratioofB0 π0π0hasbeenmeasured,whosedata[9]are
→

] (1.83 0.21 0.13) 10−6;(BABAR),
h (0.9±0 0.1±2 0.10×) 10−6;(Belle), . (1)
p
-  (1.1±7 0.1±3) 10×−6,(HFAG). 
p ± ×
e Inthelastmorethan10years,manytheoreticalteamshavecalculatedthisdecaysindifferentapproach.BenekeandNeubert
h madetheanalysisofB0 π0π0 decaybasedonQCDfactorizationin2003[5]. Recently,QinChang[10],XinLiu[11]and
[ Cong-FengQiao[12]eta→l. recalculatedthisdecaymodelusingdifferentmethod. Thenext-leading-order(NLO)contributions

from the vertex corrections, the quark loops, and the magnetic penguins have also been calculated in the literature [13–16].
2
v By comparing their results, we find the agreement between the theoretical predictions and the experimental data is still not
2 satisfactory,sowerevisitthedecaysofB0 π0π0 inthispaper. WeusethePQCDapproachtocalculatedirectlythisdecays,
→
3 non-factorizablecontributionsandannihilationcontributionandidentityprinciplearealltakenintoaccount.
6 FortheconsideredB¯0 π0π0decays,thecorrespondingweakeffectiveHamiltonianisgivenby [17].
3 →
0 G 10
1. Heff = √F2 Vu∗dVub[C1(µ)O1(µ)+C2(µ)O2(µ)]−Vt∗dVtb[ Ci(µ)Oi(µ)] +H.c., (2)
(cid:26) i=3 (cid:27)
0 X
6 whereCi(µ)areWilsoncoefficientsattherenormalizationscaleµandOi arethelocalfour-quarkoperators
1 (1)current-current(tree)operators
:
v
Xi O1 = (u¯αuα)V−A(d¯βbβ)V−A , O2 = (u¯αbα)V−A(d¯βuβ)V−A ; (3)
r (2)QCDpenguinoperators
a
O = (d¯ b ) (q¯ q ) , O = (d¯ b ) (q¯ q ) ,
3 α α V−A β β V−A 4 α β V−A β α V−A
Xq Xq (4)
O = (d¯ b ) (q¯ q ) , O = (d¯ b ) (q¯ q ) ;
5 α α V−A β β V+A 6 α β V−A β α V+A
q q
X X
(3)electroweakpenguinoperators
3 3
O = (d¯ b ) e (q¯ q ) , O = (d¯ b ) e (q¯ q ) ,
7 α α V−A q β β V+A 8 α β V−A q β α V+A
2 2
3 Xq 3 Xq (5)
O = (d¯ b ) e (q¯ q ) , O = (d¯ b ) e (q¯ q ) .
9 α α V−A q β β V−A 10 α β V−A q β α V−A
2 2
q q
X X
∗Electronicaddress:[email protected]
†Electronicaddress:[email protected]
2
Hereαandβ areSU(3)colorindices. Thenthecalculationofdecayamplitudeistocomputethehadronicmatrixelementsof
thelocaloperators.
InthePQCDapproach,thedecayamplitudecanbewrittenas
Amplitude∼ d4k1d4k2d4k3Tr[C(t)ΦB¯0(k1)Φπ0(k2)Φπ0(k3)H(k1,k2,k3,t)]e−S(t). (6)
Z
In our following calculations, the B0 meson wave function, and the wave functionsof pion mesons and relevant distribution
amplitudesΦA,P,T areofthesameformasthoseadoptedinRefs.[18]. theGegenbauermomentsaπ andotherparametersare
π i
adoptedfromRefs.[18,19]
aπ =0, aπ =0.25, aπ = 0.015,
1 2 4 −
ρ =m /m , η =0.015, ω = 3.0 (7)
π π 0π 3 3
−
withm thechiralmassofthepion.
0π
Fig.1showsthelowestorderdiagramstobecalculatedinthePQCDapproachforB¯0 π0π0decay.Thesumcontributions
→
ofthenon-factorizablediagrams(a)and(b)whichcomefromtheoperatorO are
2
M4 1 1 ∞
= B − 32πC dx dx dx b db b db Φ (x ,b ) [(x 2)ΦA(x )ΦA(x )
Ma 2 √2N F 1 2 3 1 1 2 2 B 1 1 { 2− π 2 π 3
c Z0 Z0
+rπ(1−2x2)ΦTπ(x2)ΦAπ(x3)+rπ(1−2x2)ΦPπ(x2)ΦAπ(x3)]αs(t1a)h1a(x1,x2,x3,b1,b2) (8)
exp[ S (t1) S (t1) S (t1)]C(t1) 2r ΦP(x )ΦA(x )α (t2)
− B a − π a − π a a − π π 2 π 3 s a
h2(x ,x ,x ,b ,b )exp[ S (t2) S (t2) S (t2)]C(t2) ,
a 1 2 3 1 2 − B a − π a − π a a }
where C = 4/3 is the group factor of the SU(3) gauge group and r = M /M , the Sudakov factor S (t)(X =
F c π 0π B X
B¯0,π0,π0) can be found in the appendix of Ref.[20]. The functions h1,2(x ,x ,x ,b ,b ) come from the Fourier transfor-
a 1 2 3 1 2
mationofpropagatorsofvirtualquarkandgluon.Theyaredefinedby
hj(x ,x ,x ,b ,b )=
a 1 2 3 1 2
θ(b b )I (M x (1 x )b )K (M x (1 x )b )
1 2 0 B 1 2 2 0 B 1 2 1
− − −
(cid:26)
p (Kp(M F b ), for F2 >0
0 B a(j) 1 a(j)
+(b b ) , (9)
1 ↔ 2 (cid:27)× π2iH(01)(MB |Fa2(j)|b1), for Fa2(j) <0!
q
whereF ’saredefinedby
a(j)
F2 =1 x ,
a(1) − 2
F2 =x . (10)
a(2) 1
Thetotalcontributionforthenon-factorizablediagrams(c)and(d)is
M4 1 1 ∞
= B − 32πC dx dx dx b db b db Φ (x ,b ) [ΦA(x )ΦA(x )(1 x x )
Mc 2 √2N F 1 2 3 2 2 3 3 B 1 3 { π 2 π 3 − 1− 3
c Z0 Z0
+rπΦPπ(x2)ΦAπ(x3)(1−x2)+rπΦTπ(x2)ΦAπ(x3)(1−x2)]αs(t1c)h1c(x1,x2,x3,b2,b3) (11)
exp[ S (t1) S (t1) S (t1)]C(t1)+[ ΦA(x )ΦA(x )(1+x x x ) r ΦP(x )ΦA(x )(1 x )
− B c − π c − π c c − π 2 π 3 3− 1− 2 − π π 2 π 3 − 2
+r ΦT(x )ΦA(x )(1 x )]α (t2)h2(x ,x ,x ,b ,b )exp[ S (t2) S (t2) S (t2)]C(t2) ,
π π 2 π 3 − 2 s c c 1 2 3 2 3 − B c − π c − π c c }
where
hj(x ,x ,x ,b ,b )=
c 1 2 3 2 3
θ(b b )I (M x (1 x )b )K (M x (1 x )b )
2 3 0 B 1 2 3 0 B 1 2 2
− − −
(cid:26)
p (Kp(M F b ), for F2 >0
0 B c(j) 3 c(j)
+(b b ) , (12)
2 ↔ 3 (cid:27)× π2iH(01)(MB |Fc2(j)|b3), for Fc2(j) <0!
q
3
FIG.1.TypicalFeynmandiagramscontributingtotheB¯0 →π0π0decaysinthePQCDapproachatleadingorder.
andF ’saredefinedby
c(j)
F2 =x +x +x x x x x 1,
c(1) 1 2 3− 1 2− 2 3−
F2 =x x x x +x x . (13)
c(2) 1− 3− 1 2 2 3
Thefactorizableannihilationdiagrams(e)and(f)whichcomefromtheoperatorsO ,O ,O ,O ,O ,O ,O ,O ,O in-
1 3 4 5 6 7 8 9 10
volveonlytwolightmesonswavefunctions. M isfor(V A)(V A)and(V A)(V +A)typeoperators,andMp isfor
e − − − e
4
(1+γ )(1 γ )typeoperators:
5 5
−
M4 1 ∞
= B8SπC dx dx b db b db [ ΦA(x )ΦA(x )x 2r2ΦP(x )ΦP(x )(1+x )
Me 2 F 2 3 2 2 3 3{− π 2 π 3 2− π π 2 π 3 2
Z0 Z0
+2rπ2ΦTπ(x2)ΦPπ(x3)(x2−1)]αs(t1e)h1e(x2,x3,b2,b3)exp[−Sπ(t1e)−Sπ(t1e)]C(t1e) (14)
+[ΦA(x )ΦA(x )x +2r2ΦP(x )ΦP(x )(1+x )+2r2ΦP(x )ΦT(x )(1 x )]
π 2 π 3 3 π π 2 π 3 3 π π 2 π 3 − 3
α (t2)h2(x ,x ,b ,b )exp[ S (t2) S (t2)]C(t2) ,
s e e 2 3 2 3 − π e − π e e }
M4 1 ∞
P = B8SπC dx dx b db b db [ r ΦP(x )ΦA(x )x r ΦT(x )ΦA(x )x
Me 2 F 2 3 2 2 3 3{− π π 2 π 3 2− π π 2 π 3 2
Z0 Z0 (15)
2r ΦA(x )ΦP(x )]α (t1)h1(x ,x ,b ,b )exp[ S (t1) S (t1)]C(t1)+[ 2r ΦP(x )ΦA(x )
− π π 2 π 3 s e e 2 3 2 3 − π e − π e e − π π 2 π 3
r ΦA(x )ΦP(x )x r ΦA(x )ΦT(x )x ]α (t2)h2(x ,x ,b ,b )exp[ S (t2) S (t2)]C(t2) ,
− π π 2 π 3 3− π π 2 π 3 3 s e e 2 3 2 3 − π e − π e e }
whereS =2comesfromtherequirementofidentityprincipleand
h1e(x2,x3,b2,b3)= St(x2)K0(MB√x2x3b3)
θ(b2 b3)I0(MB√x2b2)K0(MB√x2b3)+(b2 b3) (16)
×{ − ↔ }
h2e(x2,x3,b2,b3)= St(x3)K0(MB√x2x3b2)
θ(b2 b3)I0(MB√x3b3)K0(MB√x3b2)+(b2 b3) (17)
×{ − ↔ }
The non-factorizableannihilation diagrams (g) and (h) come from the operators O ,O ,O ,O . M is the contribution
4 6 8 10 g
containingtheoperatoroftype(V A)(V A),andMP isthecontributioncontainingtheoperatoroftype(1+γ )(1 γ ).
− − g 5 − 5
M4 1 1 ∞
= B 32SπC dx dx dx b db b db Φ (x ,b ) [(x +x )ΦA(x )ΦA(x )
Mg 2 √2N F 1 2 3 1 1 2 2 B 1 1 { 1 3 π 2 π 3
c Z0 Z0
+r2(2+x +x +x )ΦP(x )ΦP(x ) r2ΦP(x )ΦT(x )(x x x )+r2ΦT(x )ΦP(x )(x +x x )
π 1 2 3 π 2 π 3 − π π 2 π 3 2− 1− 3 π π 2 π 3 1 3− 2
−rπ2ΦTπ(x2)ΦTπ(x3)(2−x1−x2−x3)]αs(t1g)h1g(x1,x2,x3,b1,b2)exp[−SB(t1g)−Sπ(t1g)−Sπ(t1g)]C(t1g) (18)
+[ ΦA(x )ΦA(x )x +r2ΦP(x )ΦP(x )(x x x ) r2ΦP(x )ΦT(x )(x x +x )
− π 2 π 3 2 π π 2 π 3 1− 2− 3 − π π 2 π 3 1− 3 2
r2ΦT(x )ΦP(x )(x x +x )+r2ΦT(x )ΦT(x )(x x x )]α (t2)
− π π 2 π 3 1− 3 2 π π 2 π 3 1− 2− 3 s g
h2(x ,x ,x ,b ,b )exp[ S (t2) S (t2) S (t2)]C(t2) ,
g 1 2 3 1 2 − B g − π g − π g g }
M4 1 1 ∞
P = B − 32SπC dx dx dx b db b db Φ (x ,b ) [ ΦA(x )ΦA(x )x
Mg 2 √2N F 1 2 3 1 1 2 2 B 1 1 {− π 2 π 3 2
c Z0 Z0
r2(2+x +x +x )ΦP(x )ΦP(x ) r2ΦP(x )ΦT(x )(x x x )
− π 1 2 3 π 2 π 3 − π π 2 π 3 2− 1− 3
+rπ2ΦTπ(x2)ΦPπ(x3)(x1+x3−x2)+rπ2ΦTπ(x2)ΦTπ(x3)(x1+x2+x3−2)]αs(t1g)h1g(x1,x2,x3,b1,b2) (19)
exp[ S (t1) S (t1) S (t1)]C(t1)+[ ΦA(x )ΦA(x )(x x ) r2ΦP(x )ΦP(x )(x x x )
− B g − π g − π g g − π 2 π 3 1− 3 − π π 2 π 3 1− 2− 3
r2ΦP(x )ΦT(x )(x x +x )+r2ΦT(x )ΦP(x )(x +x x ) r2ΦT(x )ΦT(x )(x +x x )]
− π π 2 π 3 1− 3 2 π π 2 π 3 1 2− 3 − π π 2 π 3 2 3− 1
α (t2)h2(x ,x ,x ,b ,b )exp[ S (t2) S (t2) S (t2)]C(t2) ,
s g g 1 2 3 1 2 − B g − π g − π g g }
where
hj(x ,x ,x ,b ,b )=
g 1 2 3 1 2
θ(b b )I (M √x x b )K (M √x x b )
1 2 0 B 2 3 1 0 B 2 3 2
−
(cid:26)
(K (M F b ), for F2 >0
0 B g(j) 1 g(j)
+(b b ) , (20)
1 ↔ 2 (cid:27)× π2iH(01)(MB |Fg2(j)|b1), for Fg2(j) <0!
q
5
andF ’saredefinedby
g(j)
F2 =x +x +x x x x x ,
g(1) 1 2 3− 1 2− 2 3
F2 =x x x x . (21)
g(2) 1 2− 2 3
ThetotaldecayamplitudeofB¯0 π0π0 isthen
→
5 2 1
¯(B¯0 π0π0)=V∗V [C f +C ( + )] V∗V [(2C + C +2C + C + C
A → ud ub 1Me B 2 Ma Mc − td tb 3 3 4 5 3 6 2 7
(22)
1 1 1 1 1 1
+ C + C C ) f +(C C ) Pf +(2C + C ) +(2C + C ) P]
6 8 2 9− 3 10 Me B 6− 2 8 Me B 4 2 10 Mg 6 2 8 Mg
andthedecaywidthisexpressedas
G2M3
Γ(B¯0 π0π0)= F B ¯(B¯0 π0π0)2 (23)
→ 128π |A → |
Thedecayamplitudeof thechargeconjugatechannelforB¯0 π0π0 canbe obtainedbyreplacingV∗ V to V V∗ and
V∗V toV V∗ inEq.(22). ThedecayamplitudeofB¯0 π0π0→inEq.(22)canbeparameterizedas ud ub ud ub
td tb td tb →
¯=V∗ V T V∗V P =V∗ V T[1+zei(−α+δ)], (24)
A ud ub − td tb ud ub
where z = V∗V /V∗ V P/T , and δ = arg(P/T) is the relative strong phase between tree diagrams T and penguin
| td tb ud ub|| |
diagramsP.zandδcanbecalculatedfromPQCD.
Similarly,thedecayamplitudeforB0 π0π0 canbeparameterizedas
→
=V∗V T V∗V P =V∗V T[1+zei(α+δ)]. (25)
A ub ud − tb td ub ud
Thefollowingparametershavebeenusedinournumericalcalculation[1,2,21,22].
Λf=4 =0.25GeV , m =80.41GeV , m =5.280GeV ,
QCD W B
fπ =0.13GeV ,fB =0.19GeV , m0π =1.4GeV , τB0 =1.55 10−12s, (26)
×
V∗V =0.00346, V∗V =0.00885.
| ud ub| | td tb|
1.3
D
6
- 1.2
0
1
@
L
Π0
Π0 1.1
®
0B
Hr
B
1.0
0.9
0 50 100 150
ΑHdegreeL
FIG.2.TheaveragedbranchingratioofB¯0(B0)→π0π0decayasafunctionofCKMangleα.
WeleavetheCabibbo-Kobayashi-Maskawa(CKM)phaseangleαasafreeparametertoexplorethebranchingratioandCP
asymmetry.FromEqs.(24)and(25),wegettheaverageddecaywidthforB¯0(B0) π0π0
→
G2M3 2 ¯2
Γ(B¯0(B0) π0π0)= F B(|A| + |A| )
→ 128π 2 2
G2M3
= F B V∗ V T 2[1+2zcos(α)cos(δ)+z2]. (27)
128π | ud ub |
6
Usingtheaboveparameters,wegetz = 0.52andδ = 106◦ inPQCD.Equation(27)isafunctionofCKMangleα. InFig.2,
weplottheaveragedbranchingratioofthedecayB¯0(B0) π0π0 withrespecttotheparameterα. SincetheCKMangleαis
constrainedasαaround85◦[22]. →
α=(85.4+3.9)◦ (28)
−3.8
WecanarrivefromFig.2
1.08 10−6 <Br(B¯0(B0) π0π0)<1.12 10−6, for80◦ <α<90◦ (29)
× → ×
Thenumberz = V∗V /V∗ V P/T =0.52meansthattheamplitudeofpenguindiagramsisabout0.52timesthatoftree
| td tb ud ub|| |
diagrams,whichshowsthoughthetreecontributiondominatethisdecay,thepenguincontributioncannotbeignored,i. e.,there
arelargecontributionsfrombothtreediagramsandpenguindiagrams.
Intheliterature,therealreadyexistalotofstudiesonB0 π0π0decay.Wegivesomerecentworksdevotedtotheresolution
→
ofthechallenge:
(a)InRef.[10],QinChangandJunfengSunetaldoaglobalfitonthespectatorscatteringandannihilationparametersX (ρ ,φ ),
H H H
Xi(ρi ,φi )andXf(ρf,φf)fortheavailableexperimentaldataforB ππ,πKandKK¯ decaysintheQCDFframework.
A A A A A A u,d →
TheyobtainedlargeB0 π0π0branchingratios1.67+0.33 10−6and2.13+0.43 10−6fordifferentscenarios.
→ −0.30× −0.38×
(b)In Ref. [11], Xin Liu , Hsiang-nan Li and Zhen-Jun Xiao investigate the Glauber-gluoneffect on the B ππ and ρρ
→
decaysbased onthe k factorizationtheorem,they observedsignificantmodificationof B0 π0π0 branchingratio through
T
→
a transverse-momentum-dependent(TMD)wave functionforthe pionwith a weakfalloff in partontransversemomentumk .
T
TheygetthebranchingratiooftheB0 π0π0 0.61 10−6.
→ ×
(c)In Ref. [12], Cong-Feng Qiao et al give a possible solution to the B ππ puzzle using the principle of maximum
→
conformality. They foundthe PQCD prediction is highly sensitive to the choice of the renormalizationscale which enter the
decayamplitude,theyobtainedBr(B π0π0)=(0.98+0.44) 10−6byapplyingtheprincipleofmaximumconformality.
d → −0.31 ×
(d)InRef. [23], Ya-LanZhanget al performeda systematic study forthe B (π+π−,π+π0,π0π0) decaysin the PQCD
→
factorizationapproachwith the inclusion ofall currentlyknownNLO contributionsfromvarioussources. Theygotthe NLO
PQCDpredictionforB0 π0π0 branchingratioBr(B0 π0π0) = [0.23+0.08(ω(b))+0.05(f )+0.04(aπ)] 10−6,itisstill
→ → −0.05 −0.04 B −0.03 2 ×
muchsmallerthanthemeasureddata.
(e)In Ref. [24], Hai-Yang Cheng, Cheng-Wei Chiang and An-Li Kuo used flavor SU(3) symmetry to analyze the data of
charmlessBmesondecaystotwopseudoscalarmesons(PP)andonevectorandonepseudoscalarmesons(VP). Theyfound
thecolor-suppressedtreeamplitudelargerthanpreviouslyknownandhasastrongphaseof 70◦ relativetothecolorfavored
tree amplitude in the PP sector, this large color-suppressed tree amplitude results in the lar−ge B0 π0π0 branching ratios
1.43 0.55 10−6and1.88 0.42 10−6fordifferentscheme. →
Th±ere are×some works on B±0 π×0π0 decay in the framework of PQCD approach before[8, 18, 23]. Ref. [8] is the ear-
→
liest PQCD calculations for B0 π0π0 decay at the leading order, Hsiang-nan Li et al considered partial NLO contribu-
→
tions in Ref. [18]. Based on the work of Ref. [8, 18], Ya-Lan Zhang et al calculated all currentlyknown NLO contributions
from various sources in Ref. [23]. All these calculations got a branching ratio which is much smaller than the experimen-
tal data. In this work, we recalculate the B0 π0π0 decay in the framework of PQCD approach. We find the largest
→
contributions come from the factorizable annihilation diagrams (e) and (f)(see Fig. 1), both tree operator(O ) and penguin
1
operators(O ,O ,O ,O ,O ,O ,O ,O ) contribute to this two diagrams. Our result is much larger than that of previous
3 4 5 6 7 8 9 10
predictions[8,18,23],therearetworeasonsthatmakethedifference.InpreviousPQCDworks[8,18,23],first,thecontributions
ofthefactorizableannihilationdiagrams(e)and(f)comefromtreeoperatorO hadnotbeentakenintoaccount,theauthors
1
only consideredthe non-factorizablediagrams(a) and (b)(small contributions)for operatorO ; second, For O ,O ,O ,O
1 3 4 9 10
operators,previouscalculations[8]showedtheircontributionscancelbetweendiagram(e)and(f), however,werecalculateit
and find their contributionscannotbe canceled between diagram (e) and (f), as shown in Eqs.(14)(15). If we get rid of the
contributionsof and P termsinEq.(22), ourresultisBr(B¯0(B0) π0π0) 0.26 10−6, whichisconsistentwith
Me Me → ∼ ×
previous PQCD predictions[8, 18, 23]. In our recalculations, we consider all the possible diagrams’s contribution, including
non-factorizablecontributionsandannihilationcontributions,andidentityprincipleisalsotakenintoaccount,weobtainbranch-
ingratioofB0 π0π0 around1.1 10−6,whichisstillsmallerthanBABARresult[9],butitisconsistentwiththeBelleand
→ ×
HFAGresults[9]. MoreexperimentalandtheoreticaleffortsshouldbemadetoresolvetheB0 π0π0puzzle.
→
InSM,theCKMphaseangleistheoriginofCPviolation.UsingEqs.(24)and(25),thedirectCPviolatingparameteris
¯2 2 2zsin(α)sin(δ)
dir = |A| −|A| = (30)
ACP ¯2+ 2 1+2zcos(α)cos(δ)+z2
|A| |A|
ItisapproximatelyproportionaltoCKManglesin(α),strongphasesin(δ),andtherelativesizezbetweenpenguincontribution
and tree contribution. We show the direct CP asymmetry dir in Fig. 3. One can see from the figure that the direct CP
asymmetryparameterofB¯0(B0) π0π0 canbeaslargeasAfrCoPm 80%to 78%when80◦ < α < 90◦. ThelargedirectCP
→ − −
asymmetryisalsoaresultoftherearelargecontributionsfrombothtreediagramsandpenguindiagramsinthisdecays.
7
0.0
-0.2
dirCP -0.4
A
-0.6
-0.8
0 50 100 150
ΑHdegreeL
FIG.3.DirectCPviolationparameterofB¯0(B0)→π0π0decayasafunctionofCKMangleα.
For the neutral B0 decays, the B¯0 B0 mixing is very complex. Following notations in the previous literature [25], we
−
definethemixinginducedCPviolationparameteras
2Im(λ )
CP
aǫ+ǫ′ = −1+ λ 2 , (31)
CP
| |
where
V∗V <π0π0 H B¯0 >
λ = tb td | eff| . (32)
CP V V∗ <π0π0 H B0 >
tb td | eff|
Usingequations(24)and(25),wecanderiveas
1+zei(δ−α)
λ =e2iα (33)
CP 1+zei(δ+α)
0.5
aΕ+Ε’ 0.0
-0.5
0 50 100 150
ΑHdegreeL
FIG.4.MixingCPviolationparameterofB¯0(B0)→π0π0decayasafunctionofCKMangleα.
Ifzisaverysmallnumber,i.e.,thepenguindiagramcontributionissuppressedcomparingwiththetreediagramcontribution,
themixinginducedCPasymmetryparameteraǫ+ǫ′ isproportionalto sin2α,whichwillbeagoodplacefortheCKMangle
−
αmeasurement. Howeveraswehavealreadymentioned,z isnotverysmall. WegivethemixingCPasymmetryinFig.4,one
8
canseethataǫ+ǫ′ isnotasimple sin2αbehaviorbecauseoftheso-calledpenguinpollution.Itiscloseto9%whentheangle
near85◦. Atpresent,thereareno−CPasymmetrymeasurementsinexperimentbutthepossiblelargeCPviolationwepredictfor
B¯0(B0) π0π0decaysmightbeobservedintherunningLHC-bexperiments.
Inconc→lusion,werecalculatethebranchingratioandCPasymmetryofthedecaysB¯0(B0) π0π0inPQCDapproach.From
ourcalculation,wefindthebranchingratioofB0 π0π0around1.1 10−6,muchlargerth→anthatofpreviouspredictions[8],
→ ×
andtherearelargeCPviolationintheprocess,whichmaybemeasuredintherunningLHC-bexperiments.Thebranchingratio
wegetisstillsmallerthanBABARresult[9],butitisconsistentwiththelatestBelleandHFAGresults[9].
We wouldto thankDr. Ming-ZhenZhouand Dr. Wen-LongSangfor valuablediscussions. Thisworkis supportedby the
NationalNaturalScienceFoundationofChinaunderGrantNo.11047028andtheFundamentalResearchFundsoftheCentral
Universities,GrantNumberXDJK2012C040.
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