Direct CP violation for B ¯ 0 s → K 0 π + π − decay in QCD factorization
aa r X i v : . [ h e p - ph ] N ov Direct
C P violation for ¯ B s → K π + π − decay in QCDfactorization ∗ Gang L¨u
College of Science,Henan University of Technology,Zhengzhou 450001, China
Bao-He Yuan
North China University of Water Resources and Electric Power,zhengzhou 450011, China
Ke-Wei Wei † Institute of High Energy Physics,Chinese Academy of Sciences,Beijing 100049, China (Dated: June 25, 2018)
Abstract
In the framework of QCD factorization, based on the first order of isospin violation, we studydirect CP violation in the decay of ¯ B s → K ρ ( ω ) → K π + π − including the effect of ρ − ω mixing. We find that the CP violating asymmetry is large via ρ − ω mixing mechanism whenthe invariant mass of the π + π − pair is in the vicinity of the ω resonance. For the decay of¯ B s → K ρ ( ω ) → K π + π − , the maximum CP violating asymmetries can reach about 46%. Wealso discuss the possibility to observe the predicted CP violating asymmetries at the LHC. PACS numbers: 11.30.Er, 12.39.-x, 13.20.He, 12.15.Hh ∗ [email protected] † Electronic address: [email protected] . INTRODUCTION CP violating asymmetry is one of the most important areas in the decays of bot-tom hadrons. In the standard model(SM), a non-zero complex phase in the Cabibbo-Kobayashi-Maskawa (CKM) matrix is responsible for CP violating phenomena. In recentyears CP violation in several B decays such as B → J/ψK S and B → K + π − has indeedbeen found in experiments [1, 2]. Due to its much higher statistics, the Large HadronCollider (LHC) will provide a new opportunity to search for more CP violation signals.Direct CP violating asymmetries in b -hadron decays occur through the interference ofat least two amplitudes with the weak phase difference φ and the strong phase difference δ . The weak phase difference is determined by the CKM matrix while the strong phase isusually difficult to control. In order to have a large CP violating asymmetries signal, wehave to apply some phenomenological mechanism to obtain a large δ . It has been shownthat the charge symmetry violating mixing between ρ and ω can be used to obtaina large strong phase difference which is required for large CP violating asymmetries.Furthermore, it has been shown that the measurement of the CP violating asymmetriescan be used to remove the mod( π ) ambiguity in the determination of the CP violatingphase angle α [3–7].Naive factorization approximation has been shown to be the leading order result in theframework of QCD factorization when the radiative QCD corrections of order O ( α s ( m b ))( m b is the b -quark mass) and the O (1 /m b ) corrections in the heavy quark effective theoryare neglected [8]. In naive factorization scheme, the hadronic matrix elements of four-quark operators are assumed to be saturated by vacuum intermediate states. Since thebottom hadrons are very heavy, their hadronic decays are energetic. Hence the quark pairgenerated by one current in the weak Hamiltonian moves very fast away from the weakinteraction point. Therefore, by the time this quark pair hadronizes into a meson, it isalready far away from other quarks and is unlikely to interact with the remaining quarks.This quark pair is factorized out and generates a meson [9, 10]. This approximationcan only estimate the CP violation order neglecting QCD correction. Furthermore, aspointed out in previous studies [5–7], in order to taken into account the nonfactorizablecontributions, an effective parameter, N c , is introduced. The deviation of the value of N c from the color number, 3, measures the nonfactorizable effects in the naive factorizationscheme. Obviously, N c should depend on the hadronization dynamics of different decaychannels. In this scheme, CP violation depends strongly on N c values, which makes theresults uncertainties.In the heavy quark limit, QCD factorization[8] includs nonfactorization strong interac-tion correction, and the decay amplitudes can be calculated at leading power in Λ QCD m b andat next-to-leading order in α s , which can be expressed in terms of form factors and mesonlight-cone distribution amplitudes. One can take into account the nonfactorizable andchirally enhanced hard-scattering spectator and annihilation contributions which appearat order O ( α s ( m b )) and O (1 /m b ), respectively. In this work we adopt the QCD factor-ization scheme including order- α s correction to compute CP violating asymmetry of thedecay ¯ B s → K π + π − via the ρ − ω mixing mechanism. As will be shown later, the CP violating asymmetries in this decay channel could be large and may be observed in theLHC experiments.The remainder of this paper is organized as follows. In Sec. II, we present the form2f the effective Hamiltonian and the general form of QCD factorization. In Sec. III, wegive the formalism for CP violating asymmetries in ¯ B s → K π + π − decay. In Sec. IV, wecalculate the branching ratio for decay process of ¯ B s → K ρ ( ω ) via ρ − ω mixing. Webriefly discuss the input parameters in Sec. V. The numerical results are given in Sec. VI.In Sec. VII we discuss the possibility to observe the predicted CP violating asymmetriesat the LHC. Summary and conclusions are included in Sec. VIII. II. THE EFFECTIVE HAMILTONIAN
With the operator product expansion [11], the effective Hamiltonian in bottom hadrondecays is H eff = G F √ X p = u,c X q = d,s V pb V ∗ pq ( c O p + c O p + X i =3 c i O i + c γ O γ + c g O g ] + H . c . , (1)where c i ( i = 1 , ...., , γ, g ) are the Wilson coefficients, V pb , V pq are the CKM matrixelements. The operators O i have the following form: O p = ¯ pγ µ (1 − γ ) b ¯ qγ µ (1 − γ ) p, O p = ¯ p α γ µ (1 − γ ) b β ¯ q β γ µ (1 − γ ) p α ,O = ¯ qγ µ (1 − γ ) b P q ′ ¯ q ′ γ µ (1 − γ ) q ′ , O = ¯ q α γ µ (1 − γ ) b β P q ′ ¯ q ′ β γ µ (1 − γ ) q ′ α ,O = ¯ qγ µ (1 − γ ) b P q ′ ¯ q ′ γ µ (1 + γ ) q ′ , O = ¯ q α γ µ (1 − γ ) b β P q ′ ¯ q ′ β γ µ (1 + γ ) q ′ α ,O = ¯ qγ µ (1 − γ ) b P q ′ e q ′ ¯ q ′ γ µ (1 + γ ) q ′ , O = ¯ q α γ µ (1 − γ ) b β P q ′ e q ′ ¯ q ′ β γ µ (1 + γ ) q ′ α ,O = ¯ qγ µ (1 − γ ) b P q ′ e q ′ ¯ q ′ γ µ (1 − γ ) q ′ , O = ¯ q α γ µ (1 − γ ) b β P q ′ e q ′ ¯ q ′ β γ µ (1 − γ ) q ′ α ,O γ = − e π m b ¯ sσ µν (1 + γ ) F µν b, O g = − g s π m b ¯ sσ µν (1 + γ ) G µν b, (2)where α and β are color indices, O p and O p are the tree operators, O − O are QCDpenguin operators which are isosinglets, O − O arise from electroweak penguin operatorswhich have both isospin 0 and 1 components. O γ and O g are the electromagneticand chromomagnetic dipole operators. e q ′ are the electric charges of the quarks and q ′ = u, d, s, c, b is implied.The Wilson coefficients can be calculated at a high scale M W and then evolved to scale m b using renormalization group equation. In QCD factorization, We consider weak decay B s → M M ( M , M refer to K and ρ mesons, respectively) in the heavy-quark limit.Up to power corrections of order Λ QCD /m b , the transition matrix element of an operator3 i in the weak effective Hamiltonian is given by[8] h M M |O i | ¯ B i = X j F B → M j ( m ) Z du T Iij ( u ) Φ M ( u )+ ( M ↔ M )+ Z dξdudv T IIi ( ξ, u, v ) Φ B ( ξ ) Φ M ( v ) Φ M ( u )if M and M are both light, (3)Here F B → M , j ( m , ) denotes a B → M , form factor, and Φ X ( u ) is the light-cone distri-bution amplitude for the quark-antiquark Fock state of meson X . T Iij ( u ) and T IIi ( ξ, u, v )are hard-scattering functions, which are perturbatively calculable. The hard-scatteringkernels and light-cone distribution amplitudes (LCDA) depend on a factorization scaleand scheme, which is suppressed in the notation of (3). Finally, m , denote the lightmeson masses.We match the effective weak Hamiltonian onto a transition operator, the matrix ele-ment is given by ( λ ( D ) p = V pb V ∗ pD with D = d or s ) h M ′ M ′ |H eff | ¯ B i = X p = u,c λ ( D ) p h M ′ M ′ |T pA + T pB | ¯ B i . (4)Using the unitarity relation λ ( D ) u + λ ( D ) c + λ ( D ) t = 0 (5)we can get X p = u,c λ ( D ) p T pA = X p = u,c λ ( D ) p " δ pu α ( M M ) A ([¯ q s u ][¯ uD ]) + δ pu α ( M M ) A ([¯ q s D ][¯ uu ]) + λ ( D ) u " ( α u ( M M ) − α c ( M M )) X q A ([¯ q s q ][¯ qD ]) + ( α u , EW ( M M ) − α c , EW ( M M )) X q e q A ([¯ q s q ][¯ qD ]) − λ ( D ) t " α c ( M M ) X q A ([¯ q s D ][¯ qq ]) + α c ( M M ) X q A ([¯ q s q ][¯ qD ]) + α c , EW ( M M ) X q e q A ([¯ q s D ][¯ qq ])+ α c , EW ( M M ) X q e q A ([¯ q s q ][¯ qD ]) (6)where the sums extend over q = u, d, s , and ¯ q s denotes the spectator antiquark. Theoperators A ([¯ q M q M ][¯ q M q M ]) also contain an implicit sum over q s = u, d, s to cover allpossible B -meson initial states.Next we need change the annihilation part4 p = u,c λ ( D ) p T pB = X p = u,c λ ( D ) p × " δ pu b ( M M ) X q ′ B ([¯ uq ′ ][¯ q ′ u ][ ¯ Db ])+ δ pu b ( M M ) X q ′ B ([¯ uq ′ ][¯ q ′ D ][¯ ub ])) − λ ( D ) t " b ( M M ) X q,q ′ B ([¯ qq ′ ][¯ q ′ D ][¯ qb ])+ b ( M M ) X q,q ′ B ([¯ qq ′ ][¯ q ′ q ][ ¯ Db ])+ b , EW ( M M ) X q,q ′ e q B ([¯ qq ′ ][¯ q ′ D ][¯ qb ])+ b , EW ( M M ) X q,q ′ e q B ([¯ qq ′ ][¯ q ′ q ][ ¯ Db ]) (7)where b i , b i, EW and B are given by following. The coefficients of the flavor operators α pi can be expressed in terms of the coefficients a pi defined in [8] as follows: α ( M M ) = a ( M M ) ,α ( M M ) = a ( M M ) ,α p ( M M ) = (cid:26) a p ( M M ) + a p ( M M ) ;if M M = P V ,α p ( M M ) = (cid:26) a p ( M M ) + r M χ a p ( M M ) ;if M M = P V ,α p , EW ( M M ) = (cid:26) a p ( M M ) + a p ( M M ) ;if M M = P V ,α p , EW ( M M ) = (cid:26) a p ( M M ) + r M χ a p ( M M ) ;if M M = P V , (8)For pseudoscalar (P) meson M , the ratios r M χ are defined as r M χ ( µ ) = 2 m M m b ( µ ) ( m q + m s )( µ ) , (9)All quark masses are running masses defined in the MS scheme, and m q denotes theaverage of the up and down quark masses. For vector (V) meson M we have r M χ ( µ ) = 2 m V m b ( µ ) f ⊥ V ( µ ) f V , (10)5here the scale-dependent transverse decay constant f ⊥ V is defined as h V ( p, ε ∗ ) | ¯ qσ µν q ′ | i = f ⊥ V ( p µ ε ∗ ν − p ν ε ∗ µ ) . (11)Note that all the terms proportional to r M χ are formally suppressed by one power ofΛ QCD /m b in the heavy-quark limit.The general form of the coefficients a pi at next-to-leading order in α s is a pi ( M M ) = (cid:18) C i + C i ± N c (cid:19) N i ( M )+ C i ± N c C F α s π (cid:20) V i ( M ) + 4 π N c H i ( M M ) (cid:21) + P pi ( M ) , (12)where N c is the number of colors, the upper (lower) signs apply when i is odd (even). Itis understood that the superscript ‘ p ’ is to be omitted for i = 1 ,
2. The quantities V i ( M )account for one-loop vertex corrections, H i ( M M ) for hard spectator interactions, and P pi ( M M ) for penguin contractions. The N i ( M ) and C F are given by N i ( M ) = ( i = 6 , M = V ,1 ; all other cases. (13) C F = N c − N c . (14)The vertex corrections are given by[8] V i ( M ) = R dx Φ M ( x ) h
12 ln m b µ −
18 + g ( x ) i ( i = 1 − , , , R dx Φ M ( x ) h −
12 ln m b µ + 6 − g (1 − x ) i ( i = 5 , , R dx Φ m ( x ) h − h ( x ) i ( i = 6 , , (15)with g ( x ) = 3 (cid:16) − x − x ln x − iπ (cid:17) + h ( x ) − ln x + 2 ln x − x − (3 + 2 iπ ) ln x − ( x ↔ − x ) i , (16) h ( x ) = 2 Li ( x ) − ln x − (1 + 2 πi ) ln x − ( x ↔ − x ) . (17)The constants −
18, 6, − γ . The light-cone distribution amplitude (LCDA) Φ M is the leading-twist amplitude6f M , whereas Φ m is the twist-3 amplitude. LCDA for pseudoscalar and vector mesonsof twist-2 are Φ P ( x, µ ) = 6 x (1 − x ) " ∞ X n =1 a Pn ( µ ) C / n (2 x − , Φ V k ( x, µ ) = 6 x (1 − x ) " ∞ X n =1 a Vn ( µ ) C / n (2 x − , Φ V ⊥ ( x, µ ) = 6 x (1 − x ) " ∞ X n =1 a ⊥ ,Vn ( µ ) C / n (2 x − , (18)and twist-3 ones Φ p ( x ) = 1 , Φ σ ( x ) = 6 x (1 − x ) , Φ v ( x, µ ) = 3 " x − ∞ X n =1 a ⊥ ,Vn ( µ ) P n +1 (2 x − , (19)where C n ( x ) and P n ( x ) are the Gegenbauer and Legendre polynomials, respectively. a n ( µ )are Gegenbauer moments that depend on the scale µ . Φ V ⊥ ( x, µ ) and Φ V k ( x, µ ) are thetransverse and longitudinal quark distributions of the polarized mesons.At order α s a correction from penguin contractions is present only for i = 4 ,
6. For i = 4 we obtain P p ( M ) = C F α s πN c ( C (cid:20)
43 ln m b µ + 23 − G M ( s p ) (cid:21) + C (cid:20)
83 ln m b µ + 43 − G M (0) − G M (1) (cid:21) + ( C + C ) " n f m b µ − ( n f − G M (0) − G M ( s c ) − G M (1) − C eff8 g Z dx − x Φ M ( x ) ) , (20)where n f = 5 is the number of light quark flavors, and s u = 0, s c = ( m c /m b ) are massratios involved in the evaluation of the penguin diagrams. The function G M ( s ) is given7y G M ( s ) = Z dx G ( s − iǫ, − x ) Φ M ( x ) , (21) G ( s, x ) = − Z du u (1 − u ) ln[ s − u (1 − u ) x ]= 2(12 s + 5 x − x ln s )9 x − √ s − x (2 s + x )3 x / arctan r x s − x . (22)For i = 6, if M is a vector meson, the result for the penguin contribution is P p ( M ) = − C F α s πN c ( C ˆ G M ( s p ) + C h ˆ G M (0) + ˆ G M (1) i + ( C + C ) " ( n f −
2) ˆ G M (0) + ˆ G M ( s c )+ ˆ G M (1) . (23)In analogy with (21), the function ˆ G M ( s ) is defined asˆ G M ( s ) = Z dx G ( s − iǫ, − x ) Φ m ( x ) . (24)Electromagnetic corrections are present for i = 8 ,
10 and correspond to the penguindiagrams. For i = 10 we obtain P p ( M ) = α πN c ( ( C + N c C ) "
43 ln m b µ + 23 − G M ( s p ) − C eff7 γ Z dx − x Φ M ( x ) ) . (25)For i = 8 P p ( M ) = − α πN c ( C + N c C ) ˆ G M ( s p ) , (26)if M is a vector meson.The correction from hard gluon exchange between M and the spectator quark is givenby H i ( M M ) = B M M A M M m B λ B Z dx Z dy " Φ M ( x )Φ M ( y )¯ x ¯ y + r M χ Φ M ( x )Φ m ( y ) x ¯ y , (27)8or i = 1–4,9,10. H i ( M M ) = − B M M A M M m B λ B Z dx Z dy " Φ M ( x )Φ M ( y ) x ¯ y + r M χ Φ M ( x )Φ m ( y )¯ x ¯ y , (28)for i = 5 ,
7, and H i ( M M ) = 0 for i = 6 , λ B is defined by Z dξξ Φ B ( ξ ) ≡ m B λ B (29)with Φ B ( ξ ) is one of the two light-cone distribution amplitudes of the B meson.If M = P , M = V , f refers to decay constant of relevant meson, A M M and B M M are given by A M M = i G F √ − m M ǫ ∗ M · p B F B → M (0) f M , (30) B M M = − G F √ f B s f M f M . (31)where m M and ǫ M are the mass and polarization vector of the vector meson. F B → M isthe form factor for B → M transition.We recall that the term involving r M χ is suppressed by a factor of Λ QCD /m b in heavy-quark power counting. Since the twist-3 distribution amplitude Φ m ( y ) does not vanishat y = 1, the power-suppressed term is divergent. We extract this divergence by defininga parameter X M H through Z dy ¯ y Φ m ( y ) = Φ m (1) Z dy ¯ y + Z dy ¯ y h Φ m ( y ) − Φ m (1) i ≡ Φ m (1) X M H + Z dy [¯ y ] + Φ m ( y ) . (32)The remaining integral is finite (it vanishes for pseudoscalar mesons since Φ p ( y ) = 1),but X M H is an unknown parameter representing a soft-gluon interaction with the spec-tator quark. Since X M H varies within a certain range (specified later) and X MH ∼ ln( m b / Λ QCD )[8], we treat the resulting variation of the coefficients α pi as an uncertainty.We also assume that X M H is universal, i.e., that it does not depend on M and on theindex i of H i ( M M ). For the convolution integrals, one can find the results in Ref. [8].For the annihilation contribution, one can get[8]: b p = C F N c h C A i + C ( A i + A f ) + N c C A f i , (33)9 p , EW = C F N c h C A i + C ( A i + A f ) + N c C A f i . (34)The weak annihilation kernels exhibit endpoint divergences, which we treat in thesame manner as the power corrections to the hard spectator scattering. The divergentsubtractions are interpreted as Z dyy → X M A , Z dy ln yy → −
12 ( X M A ) , (35)and similarly for M with y → ¯ x . The treatment of weak annihilation is model-dependentin the QCD factorization approach. We treat X MA as an unknown complex number of orderln( m b / Λ QCD ) and make the simplifying assumption that this number is independent ofthe identity of the meson M and the weak decay vertex. Here, A i ≈ − A i ≈ πα s (cid:20) (cid:18) X A − π (cid:19) + r M χ r M χ ( X A − X A ) (cid:21) , (36) A i ≈ πα s (cid:20) − r M χ (cid:18) X A − X A − π
3+ 4 (cid:19) + r M χ (cid:18) X A − X A + π (cid:19)(cid:21) , (37) A f ≈ − πα s (cid:20) r M χ (2 X A − − X A ) − r M χ (2 X A − X A ) (cid:21) (38)and A f = A f = 0. Here, M is K meson and M is ρ meson. III. CP VIOLATION IN ¯ B s → K π + π − DECAYA. Formalism
In the vector meson dominance model [12], the photon propagator is dressed by cou-pling to vector mesons. Based on the same mechanism, ρ − ω mixing was proposed [13].The formalism for CP violation in the decay of a bottom hadron, B s , will be reviewed inthe following. The amplitude for B s → K π + π − , A , can be written as A = h π + π − K | H T | ¯ B s i + h π + π − K | H P | ¯ B s i , (39)where H T and H P are the Hamiltonians for the tree and penguin operators, respectively.We define the relative magnitude and phases between these two contributions as follows: A = h π + π − K | H T | ¯ B s i [1 + re iδ e iφ ] , (40)10here δ and φ are strong and weak phase differences, respectively. The weak phasedifference φ arises from the appropriate combination of the CKM matrix elements: φ =arg[( V tb V ∗ ts ) / ( V ub V ∗ us )]. The parameter r is the absolute value of the ratio of tree andpenguin amplitudes, r = (cid:12)(cid:12)(cid:12)(cid:12) h π + π − K | H P | ¯ B s ih π + π − K | H T | ¯ B s i (cid:12)(cid:12)(cid:12)(cid:12) . (41)The amplitude for B s → ¯ K π + π − is¯ A = h π + π − ¯ K | H T | B s i + h π + π − ¯ K | H P | B s i . (42)Then, the CP violating asymmetry, a , can be written as a = | A | − | ¯ A | | A | + | ¯ A | = − r sin δ sin φ r cos δ cos φ + r . (43)We can see explicitly from Eq. (40) that both weak and strong phase differences areneeded to produce CP violation. ρ − ω mixing has the dual advantages that the strongphase difference is large and well known [3, 4]. In this scenario one has h π + π − K | H T | ¯ B s i = g ρ s ρ s ω ˜Π ρω ( t ω + t aω ) + g ρ s ρ ( t ρ + t aρ ) , (44) h π + π − K | H P | ¯ B s i = g ρ s ρ s ω ˜Π ρω ( p ω + p aω ) + g ρ s ρ ( p ρ + p aρ ) , (45)where t V ( V = ρ or ω ) is the tree amplitude and p V is the penguin amplitude for producinga vector meson, V . t aV ( V = ρ or ω ) is the tree annihilation amplitude and p aV is the penguinannihilation amplitude. g ρ is the coupling for ρ → π + π − , ˜Π ρω is the effective ρ − ω mixingamplitude, and s V is from the inverse propagator of the vector meson V, s V = s − m V + im V Γ V , (46)with √ s being the invariant mass of the π + π − pair.The direct ω → π + π − is effectively absorbed into ˜Π ρω , leading to the explicit s de-pendence of ˜Π ρω [14]. Making the expansion ˜Π ρω ( s ) = ˜Π ρω ( m ω ) + ( s − m ω ) ˜Π ′ ρω ( m ω ),the ρ − ω mixing parameters were determined in the fit of Gardner and O’Connell [15]:Re ˜Π ρω (m ω ) = − ±
300 MeV , Im ˜Π ρω (m ω ) = − ±
300 MeV , and ˜Π ′ ρω ( m ω ) =0 . ± .
04. In practice, the effect of the derivative term is negligible. From Eqs.(40)(41)(44)(45)(46) one has re iδ e iφ = ˜Π ρω ( p ω + p aω ) + s ω ( p ρ + p aρ )˜Π ρω ( t ω + t aω ) + s ω ( t ρ + t aρ ) . (47)11efining p ω + p aω t ρ + t aρ = r ′ e i ( δ q + φ ) , t ω + t aω t ρ + t aρ = αe iδ α , p ρ + p aρ p ω + p aω = βe iδ β , (48)where δ α , δ β , and δ q are strong phases, one finds the following expression from Eq. (47): re iδ = r ′ e iδ q ˜Π ρω + βe iδ β s ω s ω + ˜Π ρω αe iδ α . (49) αe iδ α , βe iδ β , and re iδ will be calculated in the QCD factorization approach later. WithEq. (49), we can obtain r sin δ and r cos δ . In order to get the CP violating asymmetry, a , in Eq. (43), sin φ and cos φ are needed. φ is determined by the CKM matrix elements.In the Wolfenstein parametrization [16], one hassin φ = η p [ ρ (1 − ρ ) − η ] + η , (50)cos φ = ρ (1 − ρ ) − η p [ ρ (1 − ρ ) − η ] + η . (51). B. CP violation via ρ − ω mixing In the following we will study the CP violating asymmetries in the following decay:¯ B s → K ρ ( ω ) → K π + π − . With the Eq. (4)(6)(7)(8), we can calculate the decay am-plitudes in QCD factorization scheme. The expressions for the ¯ B s → K ρ ( ω ) amplitudesare given by √ A ¯ B s → K ρ = A K ρ ( δ pu α − α p + 32 α p ,EW + 12 α p ,EW − β p + 12 β p ,EW ) , (52) √ A ¯ B s → K ω = A K ω ( δ pu α + 2 α p + α p + 12 α p ,EW − α p ,EW + β p − β p ,EW ) , (53)where A K ρ = ( − i G F √ m ρ ε ∗ ρ · p B F B s → K (0) f ρ , (54) A K ω = ( − i G F √ m ω ε ∗ ω · p B F B s → K (0) f ω . (55)12ere F denote B s → K meson form factor. m ρ , m ω are the mass of ρ and ω mesons. ε ∗ ρ , ε ∗ ω correspond to polarizing vectors. f refers to the decay constant. Then we can get √ A ¯ B s → K ρ = A K ρ " δ pu a ,K ρ − a p ,K ρ − γ k χ a p ,K ρ + 32 ( a p ,K ρ + a p ,K ρ ) + 12 ( a ,K ρ + γ k χ a p ,K ρ ) − β p + 12 β p ,EW , (56) √ A ¯ B s → K ω = A K ω " δ pu a ,K ω + 2( a p ,K ω + a p ,K ω )+ a p ,K ω + γ ωχ a p ,K ω + 12 ( a ,K ω + a ,K ω ) −
12 ( a ,K ω + γ ωχ a p ,K ω ) + β p − β p ,EW . (57)where the form of the coefficients a pi at next-to-leading order in α s is given by Eq.(12),which M is K meson and M is ρ meson. β i is the weak annihilation contribution inQCD factorization. γ χ is chirally-enhanced terms which we have denoted above.From Eq. (6)(7)(48), one can get αe iδ α = t ω + t aω t ρ + t aρ = Q Q . (58) Q = A K ω ( δ pu a ,K ω + 2( a u ,K ω − a c ,K ω + a u ,K ω − a c ,K ω ) + ( a u ,K ω − a c ,K ω )+ γ ωχ ( a u ,K ω − a c ,K ω ) + 12 ( a u ,K ω − a c ,K ω + a u ,K ω − a c ,K ω ) − " a u ,K ω − a c ,K ω + γ ωχ ( a u ,K ω − a c ,K ω ) (59)13 = A K ρ ( δ pu a ,K ρ − ( a u ,K ρ − a c ,K ρ ) − γ k χ ( a u ,K ρ − a c ,K ρ ) + 32 ( a u ,K ρ − a c ,K ρ + a u ,K ρ − a c ,K ρ ) + 12 " a u ,K ρ − a c ,K ρ + γ k χ ( a u ,K ρ − a c ,K ρ ) , (60)In a similar way, with the aid of the Fierz identities, we can evaluate the penguinoperator contributions p ρ and p ω . From Eq. (48) we have βe iδ β = p ρ + p aρ p ω + p aω = Q Q , (61)where Q = A K ρ " − a c ,K ρ − γ k x a c ,K ρ + 32 ( a c ,K ρ + a c ,K ρ ) + 12 ( a c ,K ρ + γ k x a c ,K ρ ) − β + 12 β ,EW , (62) Q = A K ω " a c ,K ω + a c ,K ω )+ a c ,K ω + γ ωx a c ,K ω + 12 ( a c ,K ω + a c ,K ω ) −
12 ( a c ,K ω + γ ωx a c ,K ω ) + β − β ,EW . (63)and r ′ e i ( δ q + φ ) = p ω + p aω t ρ + t aρ = Q Q , (64) r ′ e iδ q = Q Q (cid:12)(cid:12)(cid:12)(cid:12) V tb V ∗ td V ub V ∗ ud (cid:12)(cid:12)(cid:12)(cid:12) , (65)where (cid:12)(cid:12)(cid:12)(cid:12) V tb V ∗ td V ub V ∗ ud (cid:12)(cid:12)(cid:12)(cid:12) = p [ ρ (1 − ρ ) − η ] + η (1 − λ / ρ + η ) . (66)14t can be seen that r ′ and δ q depend on both the Wilson coefficients and the CKMmatrix elements, as shown in Eqs. (65). Substituting Eqs. (58) (61) (65) into Eq. (49),we can obtain r , sin δ , and cos δ . Then, in combination with Eqs. (50) and (51) the CP violating asymmetries can be obtained. IV. BRANCHING RATIO OF ¯ B s → K ρ ( ω ) The matrix element for B s → P and B s → V (where P and V denote pseudoscalarand vector mesons, respectively) can be decomposed as follows [17]: h P | J µ | B s i = p B s + p P − m B s − m P k k ! µ F ( k )+ m B s − m P k k µ F ( k ) , h V | J µ | B s i = 2 m B s + m V ǫ µνρσ ǫ ∗ ν p ρB s p σV V ( k )+ i ( ǫ ∗ µ ( m B s + m V ) A ( k ) − ǫ ∗ · km B s + m V × ( p B s + p V ) µ A ( k ) − ǫ ∗ · kk m V · k µ A ( k ) ) + i ǫ ∗ · kk m V · k µ A ( k ) , (67)where J µ is the weak current ( J µ = ¯ qγ µ (1 − γ ) b with q = u, d, s ), p B s ( m B s ) , p P ( m P ) , p V ( m V ) are the momenta (masses) of B s , P, V , respectively, k = p B s − p P ( p V ) for B s → P ( V ) transition and ǫ µ is the polarization vector of V . F i ( i = 0 , A i ( i = 0 , , ,
3) in Eq. (67) are the weak form factors which satisfy F (0) = F (0), A (0) = A (0), and A ( k ) = [( m B + m V ) / m V ] A ( k ) − [( m B − m V ) / m V ] A ( k ).With the factorizable decay amplitudes in Eq. (56)(57) we can calculate the decayrate for B s to a pseudoscalar meson ( P ) and a vector meson ( V ) transition by using thefollowing expression [18]:Γ( B s → P V ) = p c πm V | A ( B s → P V ) / ( ǫ · p B s ) | , (68)where p c = q [ m B s − ( m P + m V ) ][ m B s − ( m P − m V ) ]2 m B s is the c.m. momentum of the product particle and A ( B s → P V ) is the decay amplitude.In the QCD factorization approach. Here V T,Pu are the CKM factors, V Tu = | V ub V ∗ uq | , for i = 1 , , (69)15nd V Pu = | V tb V ∗ tq | , for i = 3 , ...., . (70)In our case we take into account the ρ − ω mixing contribution when we calculate thebranching ratios since we are working to the first order of isospin violation. we canexplicitly express the branching ratio for the process ¯ B s → K ρ ( ω ) as the following: BR ( ¯ B s → K ρ ( ω ))= G F p c πm ρ Γ B s | [ V Tu A Tρ ( a , a ) − V Pu A Pρ ( a , ..., a )]+ [ V Tu A Tω ( a , a ) − V Pu A Pω ( a , ..., a )] × ˜Π ρω ( s ρ − m ω ) + im ω Γ ω | , (71)where Γ B s is the total decay width of B s . V. INPUT PARAMETERS
In the numerical calculations, we have several parameters, i.e. N c and the CKM matrixelements in the Wolfenstein parametrization. For the CKM matrix elements, which shouldbe determined from experiments, we use the results of Ref. [2]:¯ ρ = 0 . +0 . − . , ¯ η = 0 . ± . ,λ = 0 . ± . , A = 0 . +0 . − . . (72)In QCD factorization scheme, since power corrections have been considered, N c is onlycolor parameter, hence we use N c = 3. In naive factorization N c includes the nonfatoriz-able effects which are model and process dependent and cannot be theoretically evaluatedaccurately and can be determined by experiment.The running quark masses is taken by the scale µ in B s decay. One has m b ( m b ) = 4 . GeV, m c ( m b ) = 0 . GeV,m u ( m b ) = m d ( m b ) = 0 , m s (2 . GeV ) = 0 . GeV. (73)The values of the scale dependent quantities f ⊥ V ( µ ) and a ⊥ , ( µ ) are given for µ = 1 GeV .The value of Gegenbauer moments are taken from [19]. a ρ = 0 , a ρ = 0 . ± . a ω = 0 , a ω = 0 . ± . a ⊥ ρ = 0 , a ⊥ ρ = 0 . ± . a ⊥ ω = 0 , a ⊥ ω = 0 . ± . a K = 0 . ± . , a K = 0 . ± . f ρ = 216 ± M eV, f ⊥ ρ ( µ ) = 165 ± M eV,f ω = 187 ± M eV, f ⊥ ω ( µ ) = 151 ± M eV, (74)16or B s meson, we use the value[2]: τ = 1 . ps, m B s = 5 . GeV (75)The Wilson coefficients c i can be found in [8]. As discussed in detail in [8], there are largetheoretical uncertainties related to the modeling of power corrections corresponding toweak annihilation effects and the chirally-enhanced power corrections to hard spectatorscattering. So we parameterize these effects in terms of the divergent integrals X H (hardspectator scattering) and X A (weak annihilation). We also model these quantities byusing the parameterization[8] X A = (cid:0) ̺ A e iϕ A (cid:1) ln m B Λ h ; ̺ A ≤ h = 0 . , (76)and similarly for X H . Here ϕ A is an arbitrary strong-interaction phase, which may becaused by soft rescattering. The fitted ̺ A and ϕ A are taken from [20]. For B s → P V decay, ρ P VA ≈ . φ P VA ≈ − ◦ . For the estimate of theoretical uncertainties, we shallassign an error of ± . ρ A and ± ◦ to φ A [20].The form factors associated with the weak transitions depend on the inner structureof the hadrons and are hence model dependent. Here we will consider the form factorsobtained in several phenomenological models. For B s decay form factors, we will use theresults (form factors are referred to the ones at q = 0):1). Model 1 [8] F B s → K = 0 . ± . , F B s → K = 0 . +0 . . − . − . , F B s → K = 0 . +0 . − . , F B s → K = 0 . ± . , F B s → K = 0 . , .73 0.74 0.75 0.76 0.77 0.78 0.79 0.80 0.81 0.82-0.5-0.4-0.3-0.2-0.10.00.10.20.30.40.50.60.7 a s (GeV) FIG. 1: Plot of a as a function of √ s corresponding to central parameter values of CKM matrixelements for ¯ B s → K ρ ( ω ) → K π + π − . F B s → K = 0 . ± . ± . . In above Models, the k dependence of the form factors has the following form underthe nearest pole dominance assumption: h ( k ) = h (0)1 − k m h , (77)where h could be F , and m h is the pole mass.It is noted that since the value of k (which is actually the square of the mass of thefactorized light meson) is much smaller than the square of the pole mass which is of order m b , only the values of the form factors at k = 0 are most relevant and hence how theform factors depend on k has little effects (less than 2%). From the above values we seethat the form factor B s → K at q = 0 ranges from 0 .
23 to 0 . VI. NUMERICAL RESULTS AND DISCUSSIONSA. CP violation via ρ − ω mixing in ¯ B s → K π + π − In the numerical calculations, we find the CP violating asymmetry, a , is large whenthe invariant mass of π + π − is in the vicinity of the ω resonance within QCD factorizationscheme.In the respective error ranges, when √ s = 0 . GeV , we get maximum CP violatingasymmetry a = (45 . +16 . . − . − . ) × − (78)18 .73 0.74 0.75 0.76 0.77 0.78 0.79 0.80 0.81 0.82-0.6-0.4-0.20.00.20.40.6 s i n s (GeV) FIG. 2: Plot of sin δ as a function of √ s corresponding to central parameter values of CKMmatrix elements with ρ − ω mixing for ¯ B s → K ρ ( ω ) → K π + π − . r s (GeV) FIG. 3: Plot of r as a function of √ s corresponding to central parameter values of CKM matrixelements with ρ − ω mixing for ¯ B s → K ρ ( ω ) → K π + π − . In QCD factorization, the theoretical errors are large which follows to the uncertaintiesof results. Generally, power corrections beyond the heavy quark limit give the major the-oretical uncertainties. This implies the necessity of introducing 1 /m b power corrections.Unfortunately, there are many possible 1 /m b power suppressed effects and they are gen-erally nonperturbative in nature and hence not calculable by the perturbative method.There are more uncertainties in this scheme. The first error refers to the variation of theCKM parameters. The second error comes from form factors and decay constants. Thethird error corresponds to the Gegenbauer moments. The last error is the wave functionof the B s meson characterized by the parameter λ B , the power corrections due to weakannihilation and hard spectator interactions described by the parameters ρ A,H , φ A,H , re-spectively. Using the central values of above parameters, we first calculate the numericalresults of CP violation and branching ratio, and then add errors according to standarddeviation. In Fig.1, We give the central value of CP violating asymmetry as a function of √ s . From the figure one can see the CP asymmetry parameter is dependent on √ s andchanges rapidly due to ρ − ω mixing when the invariant mass of π + π − is in the vicinity of19he ω resonance. The CP violating asymmetry vary from around −
37% to around 45%.From Eq. (43), one can see that the CP violating asymmetry parameter depends onboth sin δ and r . The plots of sin δ and r as a function of √ s are shown in Fig. 2 and Fig.3. It can be seen that when ρ − ω mixing is taken into account sin δ and r change sharplywhen the invariant mass of π + π − is around 0.782 GeV. From the Fig. 2, one can find ρ − ω mixing make the sin δ value oscillate from − .
56 to 0 .
44 which can not reach thevalue −
1. This result is not in agreement with conclusion from naive factorization whichcan measure the CP violating parameter to remove the mod( π ) phase uncertainty in thedetermination of the CKM angle α arising from the conventional determination throughsin 2 α [7].We have shown that ρ − ω mixing does enhance the direct CP violating asymmetriesand provide a mechanism for large CP violation in QCD factorization scheme. On theother hand, it is important to see whether it is possible to observe these large CP violatingasymmetries in experiments. This depends on the branching ratio for ¯ B s → K ρ ( ω ).We will study this problem in the next section. B. Branching ratios via ρ − ω mixing in ¯ B s → K ρ ( ω ) Including ρ − ω mixing, we calculate the values of branching ratios for ¯ B s → K ρ ( ω ).Base on the reasonable parameters range, we obtain the branching ratio of ¯ B s → K ρ ( ω )is (9 . +2 . . − . − . ) × − which is consistent with the result [20]. In other words, although wecalculate the branching ratio due to ρ − ω mixing in QCD factorization scheme, we findthe contribution of ρ − ω mixing for branching ratio is small and can be neglected. ρ − ω mixing mechanism presents new phase differences and produce extremely small effect forbranching ratio of ¯ B s → K ρ ( ω ). VII. DISCUSSIONS ON POSSIBILITY TO OBSERVE CP VIOLATING ASYM-METRIES AT THE LHC
The LHC is a proton-proton collider currently have started at CERN. With the de-signed center-of-mass energy 14 TeV and luminosity L = 10 cm − s − , the LHC givesaccess to high energy frontier at TeV scale and an opportunity to further improve theconsistency test for the CKM matrix. The production rates for heavy quark flavours willbe large at the LHC, and the b ¯ b production cross section will be of the order 0.5 mb, pro-viding as many as 0 . × bottom events per year [26]. In particular, the LHCb detectoris designed to exploit large number of b -hadrons produced at the LHC in order to makeprecise studies on CP asymmetries and on rare decays in b -hadron systems. The othertwo experiments, ATLAS and CMS, are optimized for discovering new physics and willcomplete most of their B physics program within the first few years [26, 27]. Obviously,the LHC has a great advantage over the current experiments on b -hadrons[28].In the present work, we have predicted possible large CP violating asymmetries indecay channel of ¯ B s → K ρ ( ω ) → K π + π − via the ρ − ω mixing. At the LHC, the b -hadrons are produced from the pp collisions. The possible asymmetry between thenumbers of the b -hadrons, H b , and those of their antiparticles, ¯ H b , has been studied inthe Lund string fragmentation model and the intrinsic heavy quark model [29, 30]. It has20een shown that this asymmetry can only reach values of a few percent. In our followingdiscussions, we will ignore this small asymmetry and give the numbers of H b ¯ H b pairsneeded for observing the CP violating asymmetries we have predicted. These numbersdepend on both the magnitudes of the CP violating asymmetries and the branching ratiosof heavy hadron decays which are model dependent. For one (three) standard deviationsignature, the number of H b ¯ H b pairs we need is [31–33] N H b ¯ H b ∼ BRa (1 − a ) BRa (1 − a ) ! , (79)where BR is the branching ratio for H b → f ρ .For central value of CP asymmetry in Eq. (78), we present the numbers of B s ¯ B s pairs for observing the large CP violating asymmetries at LHC. For the channel¯ B s → K ρ ( ω ) → K π + π − , the numbers of B s ¯ B s pairs are 3 . × (3 . × ) for 1 σ (3 σ ) signature. At the LHC the average B s ¯ B s production is about 10% out of 10 b ¯ b events [26]. From Fig.1, one can see CP violating asymmetries vary sharply at smallenergy range, and reach peak value at √ s = 0 . GeV . Hence, it is very possible toobserve the large CP violating asymmetries in small energy range of ρ ∼ ω resonanceat the peak values of CP violating asymmetries from the LHC experiment. For theexperiments, it is possible to reconstruction π + , π − and K mesons when the invariantmasses of π + π − pairs are in the vicinity of the ω resonance. Therefore, it is very possibleto observe the large CP violating asymmetries in ¯ B s → K ρ ( ω ) → K π + π − at the LHC. VIII. SUMMARY AND CONCLUSIONS
In this paper, we have studied CP violation in ¯ B s → K ρ ( ω ) → K π + π − . It has beenfound that, by including ρ − ω mixing, the CP violating asymmetries can be large whenthe invariant masses of π + π − pairs are in the vicinity of the ω resonance. For the decay¯ B s → K ρ ( ω ) → K π + π − , the maximum CP violation can reach 46%. Furthermore,taking ρ − ω mixing into account, we have calculated the branching ratio of the decay.We have also presented the numbers of B s ¯ B s pairs required for observing the predicted CP violation in experiments at the LHC. We have found the channel is the likely channelin which the large CP violating asymmetries may be observed at LHC. We expect thatour predictions will provide a useful guidance for future investigations and experiments.In our calculations there are some uncertainties. We have worked in the QCD factor-ization which is expected to be a reliable approach in the heavy-quark limit. 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