Exclusive B s decays to the charmed mesons D + s (1968,2317) in the standard model
aa r X i v : . [ h e p - ph ] M a y Exclusive B s decays to the charmed mesons D + s (1968 , in the standardmodel Run-Hui Li a,b , Cai-Dian L¨u a and Yu-Ming Wang a a Institute of High Energy Physics and Theoretical Physics Centerfor Science Facilities, P.O. Box 918(4) Beijing 100049, China b School of Physics, Shandong University, Jinan 250100, China
The transition form factors of ¯ B s → D + s (2317) and ¯ B s → D + s (1968) at large recoil regionare investigated in the light cone sum rules approach, where the heavy quark effective theoryis adopted to describe the form factors at small recoil region. With the form factors obtained,we carry out a detailed analysis on both the semileptonic decays ¯ B s → D + s (1968 , l ¯ ν l and nonleptonic decays B s → D + s (1968 , M with M being a light meson or a charmedmeson under the factorization approach. Our results show that the branching fraction of¯ B s → D + s (2317) µ ¯ ν µ is around 2 . × − , which should be detectable with ease at theTevatron and LHC. It is also found that the branching fractions of ¯ B s → D + s (1968) l ¯ ν l are almost one order larger than those of the corresponding B s → D + s (2317) l ¯ ν l decays.The consistency of predictions for B s → D + s (1968 , L ( L denotes a light meson) in thefactorization assumption and k T factorization also supports the success of color transparencymechanism in the color allowed decay modes. Most two-charmed meson decays of B s mesonpossess quite large branching ratios that are accessible in the experiments. These channelsare of great importance to explore the hadronic structure of charmed mesons as well as thenonperturbative dynamics of QCD. PACS numbers: 14.40.Lb, 13.20.He, 11.55.Hx
I. INTRODUCTION
Enthusiasm for the open charm spectroscopy has been renewed since the announcement ofa narrow low mass state D s (2317) with unexpected and intriguing prosperities, observed in the D s π decay mode by BaBar collaboration[1]. The analysis of these charmed resonances can beconsiderably simplified in the limit of infinite heavy quark mass, when the heavy quark acts as astatic color source so that its spin is decoupled from the total angular momentum of the residuallight degrees of freedom. Weak production of charmed mesons in the B s meson decays induced bythe b → c transition serves as an ideal platform to scrutinize the KM mechanism of the standardmodel (SM), explore the dynamics of strong interactions as well as probe the signals of new physics.Moreover, valuable information on the inner structures of the exotic charmed mesons can also beextracted from the rare decays realized via the b → c transition.On the experimental aspect, B s meson will be copiously accumulated at the LHC, which makesthe investigations of the B s ’s static prosperities and its decay characters promising. On the theo-retical side, the heavy quark symmetry can put stringent constraint on the form factors responsiblefor ¯ B s → D + s (1968 , B s meson transitions to the lowest lying charmedmesons, one needs to introduce a universal Isgur-Wise function ξ ( v · v ′ ), whose normalization is ξ ( v · v ′ = 1) = 1 as a consequence of the flavor conserving vector current. However, the heavyquark symmetry could not predict the normalization of the universal form factor τ / responsiblefor the decays of B q meson to the doublet J Ps l = (0 + , + ) / [2], therefore one has to rely on somenonperturbative methods to deal with the ¯ B s → D + s (1968 , B s → D + s (1968 , l ¯ ν l ranging from phenomenological model [3] to QCD sum rules approach [4, 5, 6], PQCD approach[7] and Lattice QCD [8, 9, 10]. It could be found that the available theoretical predictions varyfrom each other, hence the investigation of these modes in the framework that is well rooted in thequantum field theory is in demand.Light cone sum rule(LCSR) offers an systematic way to compute the soft contribution to thetransition form factor almost model-independently[11, 12, 13, 14, 15]. As a marriage of the stan-dard QCD sum rule (QCDSR) technique [16, 17, 18] and the theory of hard exclusive process,LCSR cures the problem of QCDSR applying to the large momentum transfer by performing theoperator product expansion (OPE) in terms of the twists of revelent operators rather than theirdimensions [19]. Therefore, the principal discrepancy between QCDSR and LCSR consists in thatnon-perturbative vacuum condensates representing the long-distance quark and gluon interactionsin the short-distance expansion are substituted by the light cone distribution amplitudes (LCDAs)describing the distribution of longitudinal momentum carried by the valence quarks of hadronicbound system in the expansion of transverse-distance between partons in the infinite momentumframe. Phenomenologically, LCSR has been applied widely to the investigation of the transition ofmesons and baryons in recent years [20, 21, 22, 23, 24, 25, 26, 27, 28].In this work, we will employ the LCSR approach to compute the ¯ B s → D + s (1968 , B s → D sJ M , with M being a light meson or a charmed meson, under the factorization approach.It is expected that we can win the double benefit from such decays: gain better understanding onthe dynamics of strong interactions and clarify the inner structures of D + s (1968 , D + s (1968 , B s → D + s (1968 , B s → D + s (1968 , B s → D + s (1968 , l ¯ ν l and nonlep-tonic decays B s → D + s (1968 , M are calculated in section V. In particular, some remarks onthe factorization of nonleptonic modes are given here. The last section is devoted to the conclusion. II. EFFECTIVE HAMILTONIAN AND LIGHT CONE DISTRIBUTION AMPLITUDESA. Effective Hamiltonian for the b quark decays In this subsection, we would like to collect the effective Hamiltonian for b quark decays afterintegrating out the particles including top quark, W ± and Z bosons above scale µ = O ( m b ). Forthe semileptonic b → cl ¯ ν l transition, the effective Hamiltonian can be written as H eff ( b → cl ¯ ν l ) = G F √ V cb ¯ cγ µ (1 − γ ) b ¯ lγ µ (1 − γ ) ν l . (1)For the nonleptonic transition with ∆ B = 1, the effective Hamiltonian is specified as H eff = G F √ X p = u,c λ ( D ) p (cid:18) C Q p + C Q p + X i =3 ,..., C i Q i + C γ Q γ + C g Q g (cid:19) + h.c. , (2)where the CKM factors are λ ( D ) p ≡ V pb V ∗ pD = V pb V ∗ pd , for b → d transition ; V pb V ∗ ps , for b → s transition . (3)The function Q i are the local four-quark operators: • current-current (tree) operators Q p = (¯ pb ) V − A ( ¯ Dp ) V − A , Q p = (¯ p i b j ) V − A ( ¯ D j p i ) V − A , (4) • QCD penguin operators: Q = ( ¯ Db ) V − A X q (¯ qq ) V − A , Q = ( ¯ D i b j ) V − A X q (¯ q j q i ) V − A ,Q = ( ¯ Db ) V − A X q (¯ qq ) V + A , Q = ( ¯ D i b j ) V − A X q (¯ q j q i ) V + A , (5) • electro-weak penguin operators: Q = ( ¯ Db ) V − A X q e q (¯ qq ) V + A , Q = ( ¯ D i b j ) V − A X q e q (¯ q j q i ) V + A ,Q = ( ¯ Db ) V − A X q e q (¯ qq ) V − A , Q = ( ¯ D i b j ) V − A X q e q (¯ q j q i ) V − A , (6) • electromagnetic and chromomagnetic dipole operators : Q γ = − e π m b ¯ Dσ µν (1 + γ ) F µν b , Q g = − g s π m b ¯ Dσ µν (1 + γ ) G µν b , (7)where i and j are the color indices, (¯ q q ) V ± A = ¯ q γ µ (1 ± γ ) q and the sum runs over all activequark flavors in the effective theory, i.e., q = u, d, s, c, b . The combinations a i of Wilson coefficientsare defined as usual [29]: a = C + C / , a = C + C / , a = C + C / , a = C + C / , a = C + C / ,a = C + C / , a = C + C / , a = C + C / , a = C + C / , a = C + C / . (8) B. Distribution amplitudes of D + s (1968) The distribution amplitudes of pseudoscalar meson D + s can be defined as [30] h D + s ( P ) | ¯ c ( y ) γ µ γ s ( w ) | i = − if D s p µ Z due − i ( P − k ) · y − ik · w φ vD ( u ) − i f D s m D s z µ P · z Z due − i ( P − k ) · y − ik · w g D ( u ) , h D + s ( P ) | ¯ c ( y ) γ s ( w ) | i = − im f D s Z due − i ( P − k ) · y − ik · w φ pD ( u ) , h D + s ( P ) | ¯ c ( y ) σ µν γ s ( w ) | i = i f D s m (1 − m D s m )( P µ z ν − P ν z µ ) Z due − i ( P − k ) · y − ik · w φ σD ( u ) , (9)where z = y − w and u = 1 − k + P + is the longitudinal momentum fraction carried by the charmquark. In the heavy quark limit, the chiral mass can be simplified as m = m D s m c + m s = m D s + O ( ¯Λ) , (10)which indicates that the contribution from the distribution amplitude φ σD ( u ) is suppressed by O ( ¯Λ /m D s ) compared with that from φ vD ( u ) and φ pD ( u ). It can also be observed that the twist-4distribution amplitude g D ( u ) contributes at the power of r with r = m Ds m Bs , therefore it can besafely neglected in the numerical calculations.In the next place, we would like to derive the relations between the distribution amplitudes φ vD ( u ) and φ pD ( u ) in the heavy quark limit with the help of the equation of motion. Following theRef. [30], the nonlocal matrix element with the insertion of pseudotensor current can be rewrittenas h D + s ( P ) | ¯ c ( y ) σ µν γ s ( w ) | i = h D + s ( P ) | ¯ c ( y ) γ µ γ ν γ s ( w ) | i − ig µν h D + s ( P ) | ¯ c ( y ) γ s ( w ) | i . (11)Differentiating both sides of the above equation with respect to w ν for µ = − and to y µ for ν = +,we have Z du ¯ uφ pD ( u ) e − i ( P − k ) · y − ik · w = O ( ¯Λ /m D s ) , (12) Z du [ φ pD ( u ) − φ vD ( u )] e − i ( P − k ) · y − ik · w = O ( ¯Λ /m D s ) , (13)with ¯ u ≡ − u . As shown in Eq. (12), the distribution amplitude φ pD peaks at the region of¯ u ∼ O ( ¯Λ /m D s ). Eq. (13) indicates that the distribution amplitudes φ vD ( u ) and φ pD ( u ) have thesame normalizations Z duφ vD ( u ) = Z duφ pD ( u ) ≡ Z duφ D ( u ) = f D s . (14)In this way, one can express the nonlocal matrix elements relevant to the pseudoscalar D s mesonin the heavy quark limit as h D + s ( P ) | ¯ c ( z ) l s (0) j | i = i Z due iuP · z φ D ( u )[ γ ( P + m D s )] jl , (15)The model of φ D ( u ) adopted in this work is φ D ( u ) = f D s u (1 − u )[1 − C D (1 − u )] , (16)where the shape parameter C D = 0 .
78 is determined to fit the requirement that φ D ( u ) has amaximum at ¯ u = m Ds − m c m Ds . C. Distribution amplitudes of D + s (2317) Following the same philosophy, the distribution amplitudes of scalar charmed meson D ∗ s canbe defined by [31] h D + s (2317)( P ) | ¯ c ( z ) j s (0) l | i = 14 Z due iuP · z {− ( P ) lj Φ D ( u ) + m D s ( I ) lj Φ D ( u ) } , (17)where D ∗ s denotes the D + s (2317) meson and the normalizations of distribution amplitudes are Z du Φ D ( u ) = f D ∗ s , Z du Φ D ( u ) = e f D ∗ s . (18)The decay constants f D ∗ s and e f D ∗ s are given by h | ¯ sγ µ c | D ∗ s ( P ) i = f D ∗ s P µ , h | ¯ sc | D s ( P ) i = m D s e f D ∗ s , (19)where f D ∗ s = ( m c − m s ) e f D ∗ s /m D ∗ s with m c and m s being the current masses of charm quark andstrange quark, respectively.Again, with the help of equation of motion, one can find that the distribution amplitudesΦ D ( u ) and Φ D ( u ) differ at the order of ¯Λ /m D ∗ s ∼ ( m D ∗ s − m c ) /m D ∗ s . Hence, for the leadingpower calculation, it is reasonable to parameterize the distribution amplitudes Φ D ( u ) and Φ D ( u )in the following form Φ D ( u ) = Φ D ( u ) = e f D ∗ s u (1 − u )[1 + a (1 − u )] (20)in the heavy quark limit. e f D ∗ s = (225 ± a = − .
21 is fixed under the condition that the distribution amplitudesΦ D i ( u ) possess the maximum at ¯ u = m D ∗ s − m c m D ∗ s with the charm quark mass m c = 1 . b dependence of the charmed meson distributionamplitudes has been neglected in the above analysis, which will introduce more free parameters. III. LIGHT CONE SUM RULES FOR FORM FACTORSA. Sum rules for ¯ B s → D + s (2317) transition form factors The hadronic matrix element involved in the ¯ B s → D ∗ + s transition can be parameterized as h D ∗ + s ( P ) | ¯ cγ µ γ b | ¯ B s ( P + q ) i = − i [ f + D ∗ s ( q ) P µ + f − D ∗ s ( q ) q µ ] . (21)Following the standard procedure of sum rules, the correlation function for f + D ∗ s ( q ) and f − D ∗ s ( q )is chosen as Π µ ( P, q ) = − Z d xe iq · x h D ∗ + s ( P ) | T { j µ ( x ) , j (0) }| i , (22)where the current j µ ( x ) = ¯ c ( x ) γ µ γ b ( x ) describes the b → c weak transition and j (0) =¯ b (0) iγ s (0) denotes the ¯ B s channel.Inserting the complete set of states between the currents in Eq. (22) with the same quantumnumbers as B s , we can arrive at the hadronic representation of the correlation functionΠ µ ( P, q ) = i h D ∗ + s ( P ) | ¯ c (0) γ µ γ b (0) | ¯ B s ( P + q ) ih B s ( P + q ) | ¯ b (0) iγ s (0) | i m B s − ( P + q ) + X h i h D ∗ + s ( P ) | ¯ c (0) γ µ γ b (0) | ¯ h ( P + q ) ih h ( P + q ) | ¯ b (0) iγ s (0) | i m h − ( P + q ) , (23)where the definition of B s meson decay constant is h B s | ¯ biγ s | i = m B s m b + m s f B s . (24)Combining (21), (24) and (23), we haveΠ µ ( P, q ) = m B s f B s ( m b + m s )[ m B s − ( P + q ) ] [ f + D ∗ s ( q ) P µ + f − D ∗ s ( q ) q µ ]+ Z ∞ s ¯ Bs ds ρ h + ( s, q ) P µ + ρ h − ( s, q ) q µ s − ( P + q ) , (25)where we have expressed the contributions from higher states of the B s channel in the form ofdispersion integral with s B s being the threshold parameter corresponding to the B s channel.On the theoretical side, the correlation function (22) can be also calculated in the perturbativetheory with the help of the OPE technique at the deep Euclidean region P , q = − Q ≪ µ ( P, q ) = Π
QCD+ ( q , ( P + q ) ) P µ + Π QCD − ( q , ( P + q ) ) q µ (26)= Z ∞ ( m b + m s ) ds π ImΠ
QCD+ (q , (P + q) ) s − ( P + q ) P µ + Z ∞ ( m b + m s ) ds π ImΠ
QCD − (q , (P + q) ) s − ( P + q ) q µ . Making use of the quark-hadron duality ρ hi ( s, q ) = 1 π ImΠ
QCD i ( q , ( P + q ) )Θ( s − s h ) , (27)with i = “+ , − ” and performing Borel transformation on both sides of Eq. (27) with respect to( P + q ) , the sum rules for the form factors can be written as f i ( q ) = m b + m s πf B s m B s Z ∞ ( m b + m s ) ds ImΠ
QCD i ( q , s )exp( m B s − sM ) . (28)To the leading order of α s , the correlation function can be calculated by contracting the bottomquark fields in Eq. (22) and inserting the free b quark propagatorΠ µ ( P, q ) = i Z d x Z d k (2 π ) e i ( q − k ) · x m b − k h D ∗ + s ( P ) | ¯ c ( x ) γ µ γ ( k + m b ) iγ s (0) | i . (29)It should be pointed out that the full quark propagator also receives corrections from the back-ground field [33, 34], which can be written as h | T { b i ( x )¯ b j (0) }| i = δ ij Z d k (2 π ) e − ikx i k − m b − ig Z d k (2 π ) e − ikx Z dv [ 12 k + m b ( m b − k ) G µνij ( vx ) σ µν + 1 m b − k vx µ G µν ( vx ) γ ν ] , (30)where the first term is the free-quark propagator and G µνij = G aµν T aij with Tr[ T a T b ] = δ ab . Sub-stituting the second term proportional to the gluon field strength into the correlation function canresult in the distribution amplitudes corresponding to the higher Fock states of D + s (2317) meson.It is expected that such corrections associating with the LCDAs of higher Fock states do not playany significant roles in the sum rules for transition form factors [35], and hence can be safelyneglected.Substituting Eq. (17) into Eq. (29) and performing the integral in the coordinate space, thecorrelation function in the momentum representation at the quark level can be written asΠ µ ( P, q ) = Z u duu (cid:2) ( m b + um D s ) φ D ∗ s ( u ) P µ + m D s φ D ∗ s ( u ) q µ (cid:3) e − ( m b − ¯ uq + u ¯ uP ) / ( uM ) , (31)with u = ( P + q − s ) + q ( P + q − s ) + 4 P ( m b − q )2 P . (32)Combining Eq. (28) and Eq. (31), we can finally derive the sum rules for form factors f ∗ + D s ( q )and f ∗− D s ( q ) as f + D ∗ s ( q ) = m b + m s m B s f B s e m Bs /M Z u duu [ m b Φ D ∗ s ( u ) + um D ∗ s Φ D ∗ s ( u )] e − ( m b − ¯ uq + u ¯ uP ) / ( uM ) ,f − D ∗ s ( q ) = m b + m s m B s f B s e m Bs /M Z u duu m D ∗ s Φ D ∗ s ( u ) e − ( m b − ¯ uq + u ¯ uP ) / ( uM ) . (33) B. Sum rules for ¯ B s → D + s (1968) transition form factors The form factors responsible for the ¯ B s → D + s transition are defined by h D + s ( P ) | ¯ cγ µ b | ¯ B s ( P + q ) i = f + D s ( q ) P µ + f − D s ( q ) q µ . (34)The correlation function associated with the form factors f + D s ( q ) and f − D s ( q ) can be chosen as˜Π µ ( P, q ) = − Z d xe iq · x h D + s ( P ) | T { ˜ j µ ( x ) , j (0) }| i , (35)where the current ˜ j µ ( x ) is given by ˜ j µ = ¯ c ( x ) γ µ b ( x ) . (36)One can write the phenomenological representation of the correlation function at the hadronic levelsimply by repeating the procedure given above as˜Π µ ( P, q ) = im B s f B s ( m b + m s )[ m B s − ( P + q ) ] [ f + D s ( q ) P µ + f − D s ( q ) q µ ]+ Z ∞ s ¯ Bs ds ρ h + ( s, q ) P µ + ρ h − ( s, q ) q µ s − ( P + q ) . (37)On the other hand, the correlation function at the quark level can be calculated in the frameworkof perturbative theory to the leading order of α s as˜Π µ ( P, q ) = Z u duu i [( m b + um D s ) φ D s ( u ) P µ + m D s φ D s ( u ) q µ ] s − ( P + q ) , (38)where u has been defined in Eq. (32). Matching the correlation function obtained in the twodifferent representations and performing the Borel transformation with respect to the variable( P + q ) , the sum rules for the form factor f + D s ( q ) and f − D s ( q ) can be derived as f + D s ( q ) = m b + m s m B s f B s e m Bs /M Z u duu ( m b + um D s ) φ D ( u ) e − ( m b − ¯ uq + u ¯ uP ) / ( uM ) ,f − D s ( q ) = m b + m s m B s f B s e m Bs /M Z u duu m D s φ D ( u ) e − ( m b − ¯ uq + u ¯ uP ) / ( uM ) . (39) IV. NUMERICAL ANALYSIS OF SUM RULES FOR FORM FACTORS
Now we are going to calculate the form factors f iD s ( q ) and f iD s ( q ) numerically. The inputparameters used in this paper [6, 32, 36, 37, 38, 39, 40] are collected as | V ud | = 0 . , | V us | = 0 . , | V ub | = (3 . +0 . − . ) × − , | V cd | = 0 . , | V cs | = 0 . , | V cb | = (43 . +0 . − . ) × − , | V td | = (8 . +0 . − . ) × − , | V ts | = 40 . × − , | V tb | = 0 . ,α = (90 . +3 . − . ) ◦ , β = (21 . +0 . − . ) ◦ , γ = (67 . +4 . − . ) ◦ .m b = (4 . ± . , m c (1GeV) = 1 . , m s (1GeV) = 142MeV ,m B s = 5 . , m D s = 1 . , m D ∗ s = 2 . ,f B s = (151 ± f D s = (273 ± , ˜ f D ∗ s = (225 ± . (40)0As for the decay constant of B s meson, we use the results f B = 130MeV [38] and f B s /f B =1 . ± .
09 [39] determined from QCDSR. The leptonic decay constants of D s (1968) and D s (2317)are borrowed from Ref. [32, 39]. The threshold parameter s can be determined by demanding thesum rule results to be relatively stable in allowed region for Borel mass M , and its value shouldbe around the mass square of the first excited states. As for the heavy-light systems, the standardvalue of the threshold in the X channel would be s X = ( m X + ∆ X ) , where ∆ X is about 0 . . ± .
1) GeV corresponding to s B s = (36 ± M in a complete theory.However, as we truncate the OPE up to leading conformal spin for the distribution amplitudesof D s meson in the leading Fock configuration and keep the perturbative expansion in α s to theleading order, a manifest dependence of the form factors on the Borel parameter M would emerge.Therefore, one should look for a working “window”, where the results only vary mildly with respectto the Borel mass, so that the truncation is acceptable.In the first place, we focus on the form factors at zero momentum transfer. As for the formfactor f + D s (0) associated with B s → D s transition, we require that the contribution from thehigher resonances and continuum states should be less than 30 % in the total sum rules and thevalue of f + D s (0) does not vary drastically within the selected region for the Borel mass. In viewof these considerations, the Borel parameter M should not be too large in order to insure thatthe contributions from the higher states are exponentially damped as can be observed form Eq.(39) and the global quark-hadron duality is satisfactory. On the other hand, the Borel mass couldnot be too small for the validity of OPE near the light-cone for the correlation function in thedeep Euclidean region, since the contributions of higher twist distribution amplitudes amount tothe higher power of 1 /M to the perturbative part. In this way, we indeed find a Borel platform M ∈ [8 , as plotted in Fig. 1. The value of f + D s (0) is 0 . +0 . − . , where we have combined theuncertainties from the variation of Borel mass, the fluctuation of threshold value, the uncertaintiesof quark masses and the errors of decay constants for the involved mesons. Following the sameprocedure, we can further compute the other form factors numerically, whose results have beengrouped in Table I.Now, we can investigate the q dependence of the form factors f iD s ( q ) and f iD ∗ s ( q ). It is knownthat the OPE for the correlation function (22, 34) is valid only at small momentum transfer region0 < q < ( m b − m c ) − QCD ( m b − m c ). As for the case with the large momentum transfer (small1 H GeV L H a L f + H L H GeV L H b L f + H q L FIG. 1: Left figure: The dependence form factor f + D s (0) on the Borel mass M (red solid line), the contribu-tion of higher states in the whole sum rules(magenta short dashing line) and the contributions of the twist-3light cone distribution amplitude’s contribution(blue long dashing line); Right figure: The dependence ofform factor f + D s ( q ) on the momentum transfer q within the whole kinematical region. recoil region), it is expected that HQET works well for the b → c transition. In the framework ofHQET, the matrix elements responsible for ¯ B s → D sx transition can be parameterized as [30] h D ∗ + s ( P ) | ¯ cγ µ γ b | ¯ B s ( P + q ) i = − i p m B s m D ∗ s [ η + D ∗ s ( w )( v + v ′ ) µ + η − D ∗ s ( w )( v − v ′ ) µ ] , h D + s ( P ) | ¯ cγ µ b | ¯ B s ( P + q ) i = √ m B s m D s [ η + D s ( w )( v + v ′ ) µ + η − D s ( w )( v − v ′ ) µ ] , (41)where v = ( P + q ) /m B s and v ′ = P/m D sx are the four-velocity vectors of B s and D sx mesons with D sx being either D s or D ∗ s meson and w = v · v ′ , Combining Eqs. (21), (34) and (41), we have f + i ( q ) = 1 √ m B s m D sx [( m B s + m D sx ) η + i ( w ) − ( m B s − m D sx ) η − i ( w )] ,f − i ( q ) = r m D sx m B s [ η + i ( w ) + η − i ( w )] , (42)with w = ( m B s + m D sx − q ) / m B s m D sx . In the heavy quark limit, the form factors η + D s ( w ) and η − D s ( w ) satisfy the following relations η + D s ( w ) = ξ ( w ) , η − D s ( w ) = 0 , (43)where ξ ( w ) is the Isgur-Wise function[47, 48] with the normalization ξ (1) = 1. Similarly, heavyquark symmetry allows to relate the form factors η + D ∗ s ( w ) and η − D ∗ s ( w ) to a universal function τ / ( w )[49] η + D ∗ s ( w ) + η − D ∗ s ( w ) = − τ / ( w ) , η + D ∗ s ( w ) − η − D ∗ s ( w ) = 2 τ / ( w ) . (44)2An important relation between the B → D ∗∗ form factors at zero recoil region and the slope ρ ofthe B → D ( ∗ ) Isgur-Wise function is ρ = 14 + X n | τ ( n )1 / | + X m | τ ( m )3 / | (45)under the name of the Bjorken sum rule [49]. Here, D ∗∗ denotes the generic L = 1 charmed states,the subscript n , m identify the radial excitations of the states with the same J P . For the B → D ∗∗ transition form factors, the essential difference with the Isgur-Wise function ξ ( y ) is that one cannot invoke heavy quark symmetry arguments to predict the normalization of τ / ( w ) [2].Phenomenologically, one can parameterize the B s → D s (1968 , η ± i ( w ) = η ± i (1) + a ± i ( w −
1) + b ± i ( w − , (46)where the η ± i ( w ) denotes the form factor η ± D s ( w ) and η ± D ∗ s ( w ). The parameters η ± i (1), a ± i and b ± i can be determined under the condition that the form factors derived in the LCSR and HQETapproaches should be connected in the vicinity of region with q ∼ ( m b − m c ) − QCD ( m b − m c ).In this way, we can derive the results of form factors in the whole kinematical region as shown inFig. (1) as an example. The values of all form factors are tabulated in Table I, where the resultsunder the QCDSR approaches are also collected for comparison.As can be observed from Table I, the number of form factor η + D ∗ s ( w ) at the zero-recoil regiondeviates from the zero significantly indicating that the 1 /m c power correction is sizeable for the B → D ∗ s transition. Generally, the expansion of the current¯ c Γ i b = ¯ c v Γ i b v − m c ¯ c v Γ i i D b v + 12 m b ¯ c v Γ i i D b v + ... (47)constitutes the main source of the power corrections. The QCDSR estimation of the form factor f − D ∗ s differs from that obtained in the LCSR approach in sign implying that the power correctionsand radiative corrections of correlation function are in need to reconcile the existing discrepancybetween these two methods. V. SEMILEPTONIC AND NONLEPTONIC DECAYS OF ¯ B s → D s (1968 , With the form factors derived above, we can further investigate the semileptonic and nonleptonicdecays of ¯ B s to D s (1968 , TABLE I: Numbers of f ± i (0) and η ± i ( w ) determined from the LCSR approach, where the uncertainties fromthe Borel mass, threshold value, quark masses and decay constants are combined together. For comparison,the results estimated in the QCDSR are also collected here.this work QCDSR η ± i (1) a ± i b ± i f + D s ( q ) 0 . +0 . − . . ± .
10 [5] η + D s ( w ) 0 . +0 . − . − . +0 . − . . +0 . − . f − D s ( q ) 0 . +0 . − . − . ± .
13 [5] η − D s ( w ) 0 . +0 . − . − . +0 . − . . +0 . − . f + D s ( q ) 0 . +0 . − . . η + D s ( w ) 0 . +0 . − . − . +0 . − . . +0 . − . f − D s ( q ) 0 . +0 . − . . η − D s ( w ) 0 . +0 . − . − . +0 . − . . +0 . − . quark decays was firstly proposed in Ref. [51] as a phenomenological approximation similar tothe vacuum saturation approximation for the four-quark operator matrix elements. Intuitively,factorization might work at least to leading-order approximation, since the partons that eventuallyform the emitted meson escape from the heavy meson remnant as an energetic, low mass, colorsinglet system, therefore independently from the remnant. A. Semileptonic decays of ¯ B s → D + s (1968 , l ¯ ν l With the free quark amplitude given above and the transition form factors derived in the LCSR,we arrive at the differential decay width for ¯ B s → D + s (1968 , l ¯ ν l modes d Γ dq = G F | V cb | π m B ( q − m l ) ( q ) √ λ (cid:20)(cid:0) m l ( λ + 3 q m Dx ) + q λ (cid:1) | f + i ( q ) | +6 q m l ( m B − m Dx − q ) f + i ( q ) f − i ( q ) + 6 q m l | f − i ( q ) | (cid:21) , (48)with λ = ( m B − m Dx − q ) − q m Dx .For convenience, the q dependence of these invariant functions are also plotted in Fig. 2 and3. Integrating Eq. (48), we get the branching fractions of ¯ B s → D + s (1968 , l ¯ ν l as groupedin Table II. It can be observed from this table that the orders of magnitudes for BR ( ¯ B s → D + s (1968 , l ¯ ν l ) obtained in the quark model and sum rule approaches are consistent with eachother. Besides, we can also find that the decay rates for the final state with τ lepton are generally3 − B s → D + s (2317) l ¯ ν l are available, the theoretical predictions presented here can be putto the experimental scrutiny to test the ordinary c ¯ s picture of D s (2317) meson.4 H a L ´ - ´ - ´ - ´ - d G (cid:144) dq H b L ´ - ´ - ´ - ´ - ´ - ´ - d G (cid:144) dq FIG. 2: The q dependence of differential decay width ddq Γ( ¯ B s → D + s l − ¯ ν l ) for the final states with l = e, µ (left figure) and l = τ (right figure). H a L ´ - ´ - ´ - ´ - ´ - ´ - ´ - d G (cid:144) dq H b L ´ - ´ - ´ - ´ - ´ - d G (cid:144) dq FIG. 3: The q dependence of differential decay width ddq Γ( ¯ B s → D + s l − ¯ ν l ) for the final states with l = e, µ (left figure) and l = τ (right figure). B. Nonleptonic decays of ¯ B s → D + s (1968 , M Now, we turn to the calculations of nonleptonic decays ¯ B s → D + s (1968 , M , where M canbe a light meson or a charmed meson. As mentioned above, the factorization assumption will beemployed to decompose the matrix element of four-quark operator h D sx M | Q | ¯ B s i = h M | j | ih D sx | j | ¯ B s i , (49)into the the ¯ B s → D + s (1968 , M .For a light meson M , only the tree-operators in Eq. (2) can contribute to these decay modesinduced by the b → c transition. Then, the decay width for ¯ B s → D + s (1968 , L can be written5 TABLE II: Branching ratios for the semileptonic decays ¯ B s → D + s (1968 , l ¯ ν l with the form factorsestimated in LCSR, where the results calculated in constituent quark model and QCDSR are also displayedfor comparison. ¯ B s → D + s l − ¯ ν l l = e, µ l = τ this work (2 . +1 . − . ) × − (5 . +2 . − . ) × − QCDSR[5] ∼ − ∼ − Constituent Quark Model[3] (4 . − . × − QCDSR in HQET[4] (0 . − . × − ¯ B s → D + s l − ¯ ν l l = e, µ l = τ this work (1 . +0 . − . ) × − (3 . +1 . − . ) × − Constituent Quark Model[3] (2 . − . × − QCDSR [6] (2 . − . × − asΓ( ¯ B s → D + sx L − ) = G F | ~P | πm B s | V cb V ∗ uq a ( µ ) | f L × (cid:26) | f + D sx ( m L ) m L ( ǫ ∗ · P ) | ( L = V ) , | m Bs − m Dsx − m L f + D sx ( m L ) + f − D sx ( m L ) m L | ( L = S, P ) , (50)where L denotes a light meson; V , P and S label the vector, pseudoscalar and scalar mesonsrespectively. The magnitude of the three-momentum for the recoiled charmed meson is | ~P | = [( m B s − ( m D sx + m L ) )( m B s − ( m D sx − m L ) )] / m B s , (51)and the decay constant f L has been collected in Table III. The decay constants for the lightpseudoscalar mesons are taken from the Particle Data Group [36] and the vector meson longitudinaldecay constants are extracted from the data on τ − → ( ρ − , K ∗− ) ν τ . To determine the decayconstants for the scalar meson D ∗ and vector meson D ∗ s , the following relation f D ∗ s f D ∗ ≈ f D ∗ s f D ∗ ≈ f B s f B (52)is assumed in this work. The decay constant of vector meson D ∗ is borrowed from Ref. [39]. Theenergy scale of the Wilson coefficient a ( µ ) is varied from 0 . m b to 1 . m b in the error estimations.Substituting the form factors obtained in the previous sections into Eq. (50), we can get thedecay rates of ¯ B s → D + s (1968 , L as shown in Table IV. From this table, the results evaluatedin the factorization approach are in accord with that predicted in the PQCD approach and theavailable data, which implies that the factorization assumption (FA) works well for these colorallowed modes as expected.6 TABLE III: Decay constants of light and charmed mesons (unit: MeV). f π f K f ρ f K ∗ f D f D ∗ f D ∗ f D ∗ s
131 160 209 ± ± ± ±
10 270 ±
35 312 ± − ) for the nonleptonic decays ¯ B s → D + sx L ( L denotes a light meson)estimated in the FA with the form factors obtained in the LCSR, where we have combined the uncertaintiesfrom the form factors, the scale-dependence and CKM matrix elements. In Ref. [50], the authors alsoemploy the naive factorization but take the transition form factors calculated in the three-point sum rules.Channels this work PQCD[7] Exp.[36] FA [50]¯ B s → D ∗ + s π − . +2 . − . ¯ B s → D ∗ + s K − . +0 . − . ¯ B s → D ∗ + s ρ − +6 − ¯ B s → D ∗ + s K ∗− . +0 . − . ¯ B s → D + s π − +7 − . +10 . . . − . − . − . ± . ± . B s → D + s K − . +0 . − . . +0 . . . − . − . − . . ± . B s → D + s ρ − +17 − . +24 . . . − . − . − . ¯ B s → D + s K ∗− . +1 . − . . +1 . . . − . − . − . . ± . Moreover, it is also helpful to define the following ratios R ≡ BR ( ¯ B s → D ∗ + s π − ) BR ( ¯ B s → D ∗ + s K − ) ≈ BR ( ¯ B s → D + s π − ) BR ( ¯ B s → D + s K − ) ≈ (cid:12)(cid:12)(cid:12)(cid:12) V ud V us f π f K (cid:12)(cid:12)(cid:12)(cid:12) ≈ . ,R ≡ BR ( ¯ B s → D ∗ + s ρ − ) BR ( ¯ B s → D ∗ + s K ∗− ) ≈ BR ( ¯ B s → D + s ρ − ) BR ( ¯ B s → D + s K ∗− ) ≈ (cid:12)(cid:12)(cid:12)(cid:12) V ud V us f ρ f K ∗ (cid:12)(cid:12)(cid:12)(cid:12) ≈ . , (53)which are consistent with those collected in Table IV.As for the two charmed meson decays of B s meson, the decay width in the factorization approachcan be given byΓ( ¯ B s → D + sx X ) = G F | ~P | πm B s | V cb V ∗ cq a ( µ ) − V tb V ∗ tq [ a ( µ ) + a ( µ ) + r q ( a ( µ ) + a ( µ ))] | f X × (cid:26) | f + D sx ( m X ) m X ( ǫ ∗ · P ) | ( X = V ) , | m Bs − m Dsx − m X f + D sx ( m X ) + f − D sx ( m X ) m X | ( X = S, P ) , (54)7with r q = m X ( m b − m c )( m c + m q ) ( ¯ B s → D + s X, X = P ) , m X ( m b − m c )( m c − m q ) ( ¯ B s → D + s X, X = S ) , − m X ( m b − m c )( m c − m q ) ( ¯ B s → D ∗ + s X, X = P ) , − m X ( m b − m c )( m c − m q ) ( ¯ B s → D ∗ + s X, X = S ) , B s → D + sx X, X = V ) , (55)where the quark masses in the above equation are the current quark masses.Combining the Eq. (54) and the form factors listed above, one can easily get the branchingratios of ¯ B s → D + sx X ( X being a charmed meson) as shown in Table V. As can be seen fromthis table, the decay model ¯ B s → D + s D − s possesses a quite large branching ratio of order 10 − ,which should be detected easily at the large colliders such as Tevatron and LHC. Moreover, thetheoretical predictions on the ¯ B s → D + s π − decay can accommodate the experimental data withinthe error bars. As for the ¯ B s → D + s D ∗− s decay, only the upper bound for this mode is available atpresent, which is also respected by our predictions.Subsequently, the ratio of decay rates between the Cabibbo favored and suppressed modes canbe estimated as R ≡ BR ( ¯ B s → D ∗ + s D − s ) BR ( ¯ B s → D ∗ + s D − ) ≈ BR ( ¯ B s → D + s D − s ) BR ( ¯ B s → D + s D − ) ≈ (cid:12)(cid:12)(cid:12)(cid:12) V cs V cd f D − s f D − (cid:12)(cid:12)(cid:12)(cid:12) ,R ≡ BR ( ¯ B s → D ∗ + s D ∗− s ) BR ( ¯ B s → D ∗ + s D ∗− ) ≈ BR ( ¯ B s → D + s D ∗− s ) BR ( ¯ B s → D + s D ∗− ) ≈ (cid:12)(cid:12)(cid:12)(cid:12) V cs V cd f D ∗− s f D ∗− (cid:12)(cid:12)(cid:12)(cid:12) ,R ≡ BR ( ¯ B s → D ∗ + s D ∗− s ) BR ( ¯ B s → D ∗ + s D ∗− ) ≈ BR ( ¯ B s → D + s D ∗− s ) BR ( ¯ B s → D + s D ∗− ) ≈ (cid:12)(cid:12)(cid:12)(cid:12) V cs V cd f D ∗− s f D ∗− (cid:12)(cid:12)(cid:12)(cid:12) , (56)in the naive factorization without the contributions from penguin operators. Such naive estimationsare in good agreement with that presented in Table V, which also indicates that the two charmed-meson decays of B s meson governed by the b → qc ¯ c ( q = s d ) transition are dominated by the treeoperators. VI. DISCUSSION AND CONCLUSION
A detailed analysis of properties about the new charming mesons such as D s (2317) has becomea prominent part of the ongoing and forthcoming experimental programs at various facilities world-wide. The production characters of charmed mesons in the B s decays are especially interesting forhighlighting the understanding of QCD dynamics and enriching our knowledge of flavor physics.8 TABLE V: Branching ratios(unit: 10 − ) for the nonleptonic decays ¯ B s → D + sx X ( X denotes a charmedmeson) estimated in the FA with the form factors obtained in the LCSR, where the uncertainties from theform factors, the scale-dependence and CKM matrix elements have been combined together.Channels This work Exp. [36]¯ B s → D ∗ + s D − s +7 − ¯ B s → D ∗ + s D − . +0 . − . ¯ B s → D ∗ + s D − s . +1 . − . ¯ B s → D ∗ + s D ∗− . +0 . − . ¯ B s → D ∗ + s D ∗− s . +2 . − . ¯ B s → D ∗ + s D ∗− . +0 . − . ¯ B s → D + s D − s +14 − ± B s → D + s D − . +0 . − . ¯ B s → D + s D ∗− s . +2 . − . ¯ B s → D + s D ∗− . +0 . − . ¯ B s → D + s D ∗− s +13 − < B s → D + s D ∗− . +0 . − . More importantly, the theory underlying the description of the decays induced by the b → c tran-sition is mature currently. In view of the large mass of b quark, heavy quark expansion works wellenough to enable a precise determination of the decay amplitude.LCSR approach is employed to compute the ¯ B s → D + s (1968 , η + D s ( w ) responsible for the¯ B s → D + s (1968) at the zero-recoil region transition is numerically small, since this form factor onlyreceive the corrections at order 1 /m b and 1 /m c as indicated by the Luke’s theorem [52]. However,the power correction to the form factor η + D ∗ s ( w ) relevant to the B → D ∗ + s transition is sizeable.Subsequently, we utilize the form factors estimated in the LCSR approach to perform a carefulstudy on the semileptonic decays ¯ B s → D + s (1968 , l ¯ ν l . It has been shown in this work thatthe branching fraction of the semileptonic ¯ B s → D + s (2317) µ ¯ ν µ decay is around 2 . × − , whichshould be detectable with ease at the Tevatron and LHC. The decay rates of semileptonic modesfor the final states with τ lepton are approximately 3 − B s → D + s (1968) l ¯ ν l arealmost one order large than that for the B s → D + s (2317) l ¯ ν l decays.9Nonleptonic decays B s → D + s (1968 , M are also investigated in the framework of factoriza-tion approach in this work. It is found that the theoretical predictions for B s → D + s (1968 , L presented here are in agreement with those obtained in the k T factorization, which supports the suc-cess of color transparence mechanism in the color allowed decay modes. Moreover, ¯ B s → D ∗ + s D − s owns a quite large branching ratio as 1 . × − , which should be accessible experimentally. Moretheoretical results worked out here are expected to be tested by the large colliders in the nearfuture. Acknowledgement
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