Table Of Content

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. [1] H.-n.Li and H. L.Yu,Phys. Rev.Lett. 74, 4388 (1995); Phys.Lett. B353, 301 (1995); H.-n.Li, ibid. 348, 597 (1995); H.n. Li and H.L. Yu,Phys.Rev. D53, 2480 (1996). [2] M.Beneke,G.Buchalla,M.Neubert,andC.T.Sachrajda,Phys.Rev.Lett.83,1914(1999); Nucl.Phys.B591,313(2000). [3] C.-D. Lu¨, K. Ukai,and M.-Z. Yang,Phys. Rev.D63, 074009 (2001). [4] X.-Q.Yu,Y.Li and C.-D. Lu¨, Phys.Rev. D71, 074026 (2005). [5] V.M. Braun and I.E. Filyanov, Z. Phys. C44, 157 (1989); Z. Phys.C48, 239 (1990); P.Ball, J. High Energy Physics, 01, 010 (1999). [6] P.Ball, V.M. Braun, Y.Koike and K. Tanaka, Nucl. Phys.B529, 323 (1998). [7] C.-D. Lu¨ and M.-Z. Yang, Eur. Phys.J. C23, 275 (2002). [8] Y.Li, C.-D. Lu¨, Z. J. Xiao, and X.-Q.Yu,Phys. Rev.D 70, 034009 (2004). [9] G. Buchalla, A.J. Buras, and M. E. Lautenbacher, Rev.Mod. Phys.68, 1125 (1996). [10] H.-n.Li, Phys. Rev.D66, 094010(2002); T. Kurimoto, H.-n. Li, and A.I. Sanda,ibid. 65, 014007 (2002). [11] M. Beneke and M. Neubert , Nucl. Phys.B 675, 333 (2003) [12] D.Gross and F. Wilczek, Phys. Rev.Lett. 30, 1343 (1973); Phys. Rev.D.8, 3633 (1973); H.D. Politzer, Phys. Rev.Lett. 30, 1346 (1973); Phys.Rep. 14C, 129 (1974). [13] Y.Li, C.-D. Lu¨ and Z.J. Xiao, J. Phys.G 31, 273(2005). [14] M. Gronau, Phys.Lett. B 233, 479 (1989); R.Aleksan, I.Dunietz,B. Kayserand F. Le Diberder,Nucl. Phys.B 361, 141 (1991).