Non-leptonic charmless Bc decays and their search at LHCb
aa r X i v : . [ h e p - ph ] J u l LPT-ORSAY-09-14LAL-09-28
Non-leptonic charmless B c decaysand their search at LHCb S. Descotes-Genon a , J. He b , E. Kou b and P. Robbe ba Laboratoire de Physique Th´eorique, CNRS/Univ. Paris-Sud 11 (UMR 8627)91405 Orsay, France b Laboratoire de l’Acc´el´erateur Lin´eaire, Universit´e Paris-Sud 11, CNRS/IN2P3 (UMR 8607)91405 Orsay, France
Abstract
We discuss the decay of B c mesons into two light mesons ( π, K ( ∗ ) , η ( ′ ) , ρ, ω, φ ). All thesedecay channels come from a single type of diagram, namely tree annihilation. This al-lows us to derive extremely simple SU (3) relations among these processes. The size ofannihilation contributions is an important issue in B physics, and we provide two dif-ferent estimates in the case of non-leptonic charmless B c decays, either a comparisonwith annihilation decays of heavy-light mesons or a perturbative model inspired by QCDfactorisation. We finally discuss a possible search for these channels at LHCb. Introduction
The investigation of the properties of the B c meson started in 1998 when the CDF collab-oration observed 20.4 events containing a B c in the channel B c → J/ψlν [1]. Since then,its mass and width have been measured [2], and bounds on some non-leptonic channelshave been set (
J/ψ with one or three pions, D ∗ + ¯ D . . . ). From the theoretical point ofview, the B c meson shares many features with the better known quarkonia, with the sig-nificant difference that its decays are not mediated through strong interaction but weakinteraction due to its flavour quantum numbers B = − C = ±
1. Theoretical investiga-tions have been carried out on the properties of the B c meson, such as its lifetime, itsdecay constant, some of its form factors [3], based on OPE [4, 5], potential models [6, 7],NRQCD and perturbative methods [8, 9, 10, 11], sum rules [12, 13, 14, 15], or lattice gaugesimulations [16, 17, 18, 19]. The properties of the B c meson will be further scrutinized bythe LHC experiments; the high luminosity of the LHC machine opens the possibility toobserve many B c decay channels beyond the discovery one, in particular at LHCb.1his article is focused on the two-body non-leptonic charmless B c decays. Indeed, thecharmless B c decays with two light mesons ( π, K ( ∗ ) , η ( ′ ) , ρ, ω, φ ) in the final states comefrom a single diagram: the initial b and c quarks annihilate into a charged weak bosonthat decays into a pair formed of a u and a d/s quark, which hadronise into the two lightmesons. This picture is rather different from processes such as B c → J/ψπ for which theinitial c quark behaves as a spectator. The recent high-precision measurements of the B u,d,s and D s decays indicate that such annihilation processes can be significant, contraryto the theoretical expectation of its suppression in the heavy-quark limit. Indeed, fits ofthe data not taking into account annihilation processes are generally of poor quality.But the understanding of these contributions remains limited. The theoretical com-putation of annihilation diagrams is very difficult, so that the annihilation contributionsare often considered as a free parameter in these decays. On the other hand, in many B u,d,s decays, these annihilation contributions come from several different operators (treeand penguin), and they interfere with many different other (non-annihilation) diagrams,making it difficult to obtain an accurate value of annihilation by fitting experimental data.For this reason, the processes such as B d → K + K − , B d → D − s K + , B u → D − s K , whereonly the annihilation diagram contributes, have been intensively worked out while thecurrent experimental measurements are still of limited accuracy.The non-leptonic charmless B c decays which we discuss in this article can have animportant impact on this issue. We have 32 decay channels which come from annihilationonly, as mentioned above. Moreover, these decays involve a single tree operator, whichallows us to derive extremely simple relations among the different decay channels. Finally,when LHCb starts running and observes the pattern of (or at least, provides bounds for)the branching fractions of these decays channels, it will certainly help us to further improveour understanding of the annihilation contributions.The remainder of the article is organized as follows. In section 2, we introduce thedecay processes which we consider in this article. In section 3, we exploit SU (3) flavoursymmetry to derive relations among amplitudes for non-leptonic charmless B c decays. Insection 4, we discuss theoretical issues related to this annihilation diagram and attempt toestimate its size. In section 5, we discuss the prospects of searching non-leptonic charmless B c decays at LHCb and we conclude in section 6. Two appendices are devoted to relatingour work to results from factorisation approaches. Non-leptonic charmless B c decays as pure annihilation pro-cesses The diagram for the non-leptonic charmless B c decays is shown in fig. 1 (the case of singletstates will be discussed below). The initial b and c quarks annihilate into u and d or s quarks, which form two light mesons by hadronising with a pair of qq ( q = u, d, s ) emitted2 cB c ud, sqqW ● xx xx Figure 1: Generic diagram for the non-leptonic charmless B c decaysfrom a gluon. There are 32 decay channels of this kind if we consider only the lightestpseudoscalar and vector mesons. In the case of two outgoing pseudoscalar mesons (PP),there are 4 modes with strangeness one: K + π , K + η, K + η ′ , K π + and 4 modes with strangeness zero: π + π , π + η, π + η ′ , K + ¯ K The same applies for two vectors (VV) up to the obvious changes: K ∗ + ρ , K ∗ + φ, K ∗ + ω, K ∗ ρ + , ρ + ρ , ρ + φ, ρ + ω, K ∗ + ¯ K ∗ In the case of VP decays, one can get two decay modes from one in PP decays, dependingon the pseudoscalar meson which is turned into a vector one, yielding 8 strange decaymodes (∆ S = 1 processes): K ∗ + π , K ∗ + η, K ∗ + η ′ , K ∗ π + , ρ K + , φK + , ωK + , ρ + K and 8 non-strange decay modes (∆ S = 0 processes): ρ + π , ρ + η, ρ + η ′ , K ∗ + ¯ K , ρ π + , φπ + , ωπ + , ¯ K ∗ K + In the next section, we describe these decay channels in terms of a few reduced ampli-tudes using SU (3) flavour symmetry. Similar expressions have been obtained for thecharmless B u,d decays, which have been very useful to disentangle the rather complicateddecay amplitudes of these decays containing many different contributions (tree, penguin,emission, annihilation, etc. . . ) [20, 21, 22]. Comparing to the case of B u,d decays, the SU (3) relations for the B c decays are extremely simple, as it comes from only a singletree-annihilation diagram as mentioned earlier.The theoretical computation of the process shown in Fig. 1 amounts to determiningthe matrix element: h h h |H eff | B c i (1)3here H eff is the effective Hamiltonian which we discuss later on. This matrix elementcontains contributions coming from the qq state i) produced perturbatively from one-gluon exchange linking the dot and one of the crosses in Fig. 1) and ii) produced throughstrong interaction in the non-perturbative regime. Which type of these two contributionsdominates this matrix element is an important issue in the theoretical computation of thehadronic B decays. In many approaches to non-leptonic B decays [23, 24, 25, 26, 27], ithas been pointed out that annihilation diagrams may be sizable, with a large imaginarypart, so that they have an important impact on the phenomenology of CP violationin B decays. Indeed, their contributions seem to be needed to bring agreement betweentheoretical computations and experimental results. There might be a significant differencein the annihilation contributions for B and B c decays since the B c is likely to be consideredas a heavy-heavy system rather than a heavy-light one. We will discuss the theoreticalestimation of the annihilation diagram in more detail in section 4. Relations from SU (3) flavour symmetry In this section, we derive relations among the decay channels relying on the SU (3) flavoursymmetry between u -, d - and s -quarks. Following [28], we first write down the charmless B c decays in terms of the reduced amplitudes using the Wigner-Eckart theorem.Let us first see the possible SU (3) representation of the external states. The initialstate, B + c is a singlet under SU (3), whereas the outgoing state is the product of twomesons, which can be either both in the octet representation or in one octet and onesinglet representations. We have therefore outgoing states which transform as8 × S + 8 A + 10 + 10 ∗ × I (2)where the subscripts allow one to distinguish between the three different octet representa-tions involved. We sandwich the operators induced by the weak interaction Hamiltonianbetween these external states to obtain the amplitudes for the B c decays. The weakHamiltonian for such transitions is given by: H eff = − G F √ (cid:2) V ud V ∗ cb O ∆ S =0 + V us V ∗ cb O ∆ S =1 (cid:3) (3)where the operators are: O ∆ S =0 = uγ µ (1 − γ ) d cγ µ (1 − γ ) b (4) O ∆ S =1 = uγ µ (1 − γ ) s cγ µ (1 − γ ) b (5)These two operators are both SU (3) octets and have the following SU (3) tensor structures: O ∆ S =0 : ( Y, I, I ) = (0 , ,
1) (6) O ∆ S =1 : ( Y, I, I ) = (0 , / , /
2) (7)4here (
Y, I, I ) denotes hypercharge, isospin and isospin projection respectively. Since B c charmless decays involve only operators in an octet representation, one can use theWigner-Eckart theorem to express all the decay amplitudes in terms of three reducedmatrix elements: • a reduced amplitude S = h S ||O || i from the symmetric product of the two incom-ing octet mesons. • a reduced amplitude A = h A ||O || i from the antisymmetric product of the samerepresentations • a reduced amplitude I = h I ||O || i from the product of an octet and a singletmeson.The operator O can be O ∆ S =0 or O ∆ S =1 . Note that the values of the reduced quantities S, A, I are in principle different for the
P P , V P or V V final states. The Wigner-Eckarttheorem requires one to compute the Clebsch-Gordan coefficients describing the projectionof a given 8 × × SU (2) Clebsch coefficientswith the so-called isoscalar coefficients given in ref. [29].Finally, we must consider the different symmetry properties of the out-going states( P and V ) as discussed in ref. [30]. For P P decays, where the wave function of the finalstate is symmetric, only S contributes, apart from the case of final states containing η or η ′ where both S and I are present. For V P decays, the amplitude gets contributionsfrom S and A (and I for final states containing η, η ′ , ω or φ ). For V V decays, thereare three amplitudes corresponding to the three possible polarisations (or equivalentlypartial waves) allowed for the outgoing state. The wave function is symmetric for S and D waves and antisymmetric for P wave, so that the matrix element S contributes to S and D waves, whereas A contributes to P waves. I contributes only to S and D waves ofoutgoing states containing φ, ω mesons.A comment is in order on the mixing of the mesons containing SU (3) singlet states.The η, η ′ , ω, φ mesons are mixtures of the SU (3) octet ( η or ω ) and singlet ( η or ω )flavour states | η ( ω ) i = cos θ p ( v ) | η ( ω ) i + sin θ p ( v ) | η ( ω ) i (8) | η ′ ( φ ) i = − sin θ p ( v ) | η ( ω ) i + cos θ p ( v ) | η ( ω ) i (9)where | η i and | ω i have the flavour composition | u ¯ u + d ¯ d − s ¯ s i / √
6, and | η i and | ω i are | u ¯ u + d ¯ d + s ¯ s i / √
3. The determination of the mixing angles θ p,v is an importantphenomenological issue in understanding the nature of these particles. Since we do notaim at a high accuracy in our SU (3) analysis, we will adopt the following values for themixing angles which are not very far from the phenomenological determinations:tan θ p = 12 √ , tan θ v = √ . (10)5hese angles correspond to the ideal mixing for the vector sector: ω = ( u ¯ u + d ¯ d ) / √ φ = s ¯ s (11)and also yield a simple expression of the pseudoscalar mesons: η = ( u ¯ u + d ¯ d − s ¯ s ) / √ η ′ = ( u ¯ u + d ¯ d + 2 s ¯ s ) / √ η, η ′ ) mesons is linked to the U (1) A anomaly (see, e.g.,refs [31, 32]). This value of the pseudoscalar mixing angle θ p = arctan(2 √ − ≃ − . ◦ is close to phenomenological determination, e.g. from the J/ψ radiative decays, θ p ≃− ◦ [33]. Let us stress that for the light mesons, we take the same phase conventionsas in ref. [25], so that some amplitudes have a minus sign with respect to those obtainedfrom ref. [29] ∗ . P P modes
Taking into account the Clebsch-Gordan coefficients together with the issue of octet-singlet mixing, we obtain the following amplitudes for the
P P modes, we haveMode Amplitude Mode Amplitude K + π q S P P π + π K π + q S P P K + ¯ K q S P P K + η − √ S P P + √ I P P π + η √ S P P + √ I P P K + η ′ √ S P P + I P P π + η ′ − q S P P + I P P
Here and in the following tables, these amplitudes must be multiplied by G F / √ V uD V ∗ cb with D = d or s . We notice the relations A ( B + c → K π + ) = √ A ( B + c → K + π ) = ˆ λA ( B + c → K + ¯ K ) (13)with the Cabibbo-suppressing factor ˆ λ = V us /V ud . The above relations are valid in theexact SU (3) limit (for instance, we have S P P = S K + π = S K π + = S K + K ). Obviously,these relations have some interest only if the size of SU (3) breaking remains limited – wewill discuss this issue in Sec. 4. ∗ In detail, ( − ¯ u, ¯ d, ¯ s ) transform as an anti-triplet [30], which means that there is a ( −
1) phase betweenthe conventions of refs. [29] and [25] for the pseudoscalar mesons π − , π , K − , η, η ′ and the vector mesons ρ − , ρ , K ∗− , φ, ω . We have multiplied all the amplitudes by a further ( −
1) factor, so that the differencesbetween our results and those obtained using ref. [29] are limited to a ( −
1) factor for the decay amplitudesfor K π + , K + ¯ K and their vector counterparts. .2 V P modes
For the
V P modes, we have for the strange modesMode Amplitude Mode Amplitude K + ∗ π q S V P + √ A V P ωK + − √ S V P − √ A V P + q I V P ρ K + 12 q S V P − √ A V P φK + 1 √ S V P + √ A V P + √ I V P K ∗ + η − q S V P + √ A V P + I V P ρ + K q S V P − √ A V P K ∗ + η ′ √ S V P − A V P + √ I V P K ∗ π + q S V P + √ A V P and for the non-strange modesMode Amplitude Mode Amplitude ρ + π √ A V P ωπ + 1 √ S V P + q I V P ρ π + − √ A V P φπ + − q S V P + √ I V P ρ + η q S V P + I V P K ∗ + ¯ K q S V P − √ A V P ρ + η ′ − √ S V P + √ I V P ¯ K ∗ K + q S V P + √ A V P providing the simple relations A ( B + c → K ∗ π + ) = √ A ( B + c → K ∗ + π ) = ˆ λA ( B + c → ¯ K ∗ K + ) (14) A ( B + c → ρ + K ) = √ A ( B + c → ρ K + ) = ˆ λA ( B + c → K ∗ + ¯ K ) (15)It should be noted that the amplitude I V P can be significantly different for the pro-cesses involving the vector singlet ( φ, ω ) and the pseudoscalar singlet ( η, η ′ ) since it isknown that the latter should receive a contribution from the anomaly diagram. Thiscould induce a significant breaking of the above relations for channels involving η, η ′ . V V modes
For the
V V modes, we have three different configurations for the outgoing mesons, labeledby their (common) helicity. The left-handedness of weak interactions and the fact thatQCD conserves helicity at high energies suggest that the longitudinal amplitude shoulddominate over the transverse ones (corresponding to helicities equal to ± S , P and D wave amplitudes, and in particular,7he longitudinal amplitude is a linear combination of only S and D waves.Mode S, D
Amplitudes P Amplitude K ∗ + ρ q S V V , √ A V V K ∗ + ω − √ S V V , + √ I V V , − √ A V V K ∗ + φ q S V V , + q I V V , q A V V K ∗ ρ + q S V V , √ A V V ρ + ρ q A V V ρ + ω q S V V , + √ I V V , ρ + φ − √ S V V , + q I V V , K ∗ + ¯ K ∗ q S V V , − √ A V V where the subscript denote the partial wave under consideration ℓ = 0 , ,
2. In particular,we have the interesting relations A ( B + c → K ∗ ρ + ) = √ A ( B + c → K ∗ + ρ ) (16)ˆ λA ( B + c → K ∗ + ¯ K ) = √ − ℓ A ( B + c → K ∗ + ρ ) (17) In the above expressions, the normalization between
S, A and I amplitudes are different:the former is related to the 8 × ×
8. One may relatethese two amplitudes by means of the Zweig rule for the ∆ S = 0 processes involving φ .At the level of quark diagrams, one can see that the B + c → φπ + ( ρ + ) process cannot comefrom the diagram in fig. 1 – since φ is a pure s ¯ s state. The only production process comefrom non-planar diagrams diagram similar to fig. 1, but with a different combination ofquarks into the outgoing mesons: the u ¯ d quarks produced from the W go into π + ( ρ + )whereas the φ meson is made of a s ¯ s -pair produced from vacuum. Such a non-planardiagram is expected to be suppressed, especially for vector mesons such as φ (at leastthree gluons are needed perturbatively, and it is 1 /N c suppressed in the limit of a largenumber of colours).Assuming that the amplitudes for B + c → φπ + and B + c → φρ + vanish, one obtain thefollowing relations: I V P = r S V P I V V , = r S V V , (18)8roviding simpler expressions for the following V P decay amplitudesMode Amplitude Mode Amplitude K ∗ + η √ A V P ρ + η q S V P K ∗ + η ′ √ S V P − A V P ρ + η ′ √ S V P ωK + 12 q S V P − √ A V P ωπ + q S V P φK + q S V P + √ A V P φπ + A ( B + c → ρ + η ) = √ A ( B + c → ρ + η ′ ) (19)Although we assume here that the I V P amplitude is the same for the processes involvingthe vector singlet ( φ, ω ) and the pseudoscalar singlet ( η, η ′ ), as required by SU (3) sym-metry, this assumption is broken for the pseudoscalar singlets due to the anomaly (seenfor instance in the mass difference between η and η ′ ). At the quark level, the detacheddiagram for the pseudoscalar singlet states ( η, η ′ ) cannot be neglected since there is thewell-known anomaly contribution modifying the previous relation: A ( B + c → ρ + η ) −
13 ∆ I V P = √ " A ( B + c → ρ + η ′ ) − √
23 ∆ I V P (20)where ∆ I V P denotes a potentially large 1 /N c -suppressed anomaly contribution.Keeping the same caveat in mind, we can simplify some V V amplitudesMode
S, D
Amplitudes P Amplitude K ∗ + ω q S V V , − √ A V V K ∗ + φ q S V V , q A V V ρ + ω q S V V , ρ + φ Estimating the branching ratios
As mentioned in section 2, a precise estimate of the matrix element for the annihilationdiagram is an important theoretical issue in B physics. Although the domination of theone-gluon exchange diagram has been argued in various theoretical frameworks for B u,d,s decays [25, 23], it has not been investigated whether one-gluon exchange or other (non-perturbative) contributions dominate in B c decays. In this section, we provide branching9atio estimates for the non-leptonic charmless B c decays in two ways, by using experimen-tal data on pure annihilation B decays and by relying on the one-gluon picture `a la QCDfactorisation. The branching ratios can then be readily obtained by the usual formulae Br ( B c → h h ) = q [ M B c − ( m + m ) ][ M B c − ( m − m ) ]Γ tot B c πM B c |h h h |H eff | B c i| (21)and the expression obtained in the section 3 in terms of the reduced amplitudes is relatedto the matrix element through |h h h |H eff | B c i| = G F / √ | V ud ( s ) V ∗ cb | × | R ( h , h ) | (22)where the reduced amplitude R ( h , h ) involves the amplitudes listed in sec. 3, expressedin terms of ( S, A, I ). B d annihilation process There are two pure annihilation processes observed in heavy-light B decays [34, 35, 36]: Br ( B → K + K − ) = (0 . +0 . − . ) × − (23) Br ( B → D − s K + ) = (3 . ± . × − (24)Although the large experimental errors do not allow us to draw any firm conclusion,these data seem to indicate that the annihilation contribution is not negligible. We mayattempt to use these decay channels to very roughly estimate the size of the non-leptoniccharmless B c decays. Since we are interested in charmless final states, let us comparethe B c → K + K and the B → K + K − processes. Assuming naive factorization betweeninitial and final states, the final-state contribution cancels out when taking the ratio ofthe amplitudes. As a result, we find: Br ( B c → K + K ) Br ( B → K + K − ) ≃ (cid:18) V cb V ub (cid:19) | {z } ∼ (cid:18) f B c f B (cid:19) | {z } ∼ τ B c τ B d |{z} ∼ . ξ . (25)The factor ξ represents the difference due to the fact that the B → K + K − process comesfrom a diagram similar to Fig. 1 but the W boson propagates in the t -channel † . In theone-gluon picture, ξ = C /C ≃ B c decay would be more affected relatively, since its Wilson coefficient in the one-gluon picture is smaller. This very naive argument leads to the relation between thesetwo branching ratios: Br ( B c → K + K ) ≃ Br ( B → K + K − ) × . × ξ > ∼ Br ( B → K + K − ) × . † Here we neglect the small penguin diagram contribution to the B → K + K − decay. Br ( B c → K + K ) using this relation depends on the result on Br ( B → K + K − ), which should be improved in the near future. Taking the current central valueof Br ( B → K + K − ), we find a lower limit of Br ( B c → K + K ) at the order of 10 − .Using this result, one can estimate the Wigner-Eckart reduced matrix elements S P P : S P P > ∼ .
085 GeV [( P P ) = ( K K + )] (27)where we used the following CKM central values: V ub = 0 . , V cb = 0 .
041 [37].As mentioned before, there is no good reason to assume that S P P , S P V and S V V should be related. Since we are only looking for order of magnitudes, we will assume as adimensional estimate that | S P P | ≃ √ | S P V | ≃ | S V V | . We emphasise that these relationshave no strong theoretical supports and are just meant as a way to extract order ofmagnitudes for the branching ratios. In addition, we assume the Zweig rule to determinethe singlet contributions I and we neglect the antisymmetric contributions A , as well astransverse V V amplitudes. This provides the following branching ratios of interest, forinstance:[ B d annihil] BR ( B c → φK + ) ≃ O (10 − − − ) , BR ( B c → ¯ K ∗ K + ) ≃ O (10 − ) BR ( B c → ¯ K K + ) ≃ O (10 − ) , BR ( B c → ¯ K ∗ K ∗ + ) ≃ O (10 − ) (28)The suppression of the φK + channel is due to the small CKM factor for the ∆ S = 1processes, Cabibbo-suppressed compared to ∆ S = 0. A second method consists in a model based on one-gluon exchange, in close relation withthe model proposed in QCD factorisation to estimate annihilation contributions for thedecays of heavy-light mesons. In this method, described in more detail in App. A, thematrix element in eq. (1) can be given as: h h h |H eff | B c i = i G F √ V ∗ cb V ud ( s ) N h h b ( h , h ) (29)where N h h = f B c f h f h b ( h , h ) = C F N C C A i ( h h ) . (30)The function A i ( h h ) is estimated as the convolution of the kernel given by one-gluonexchange diagrams and the distribution amplitudes of the initial and final state mesons.While the detailed computation of this function can be found in Appendix A, we wouldlike to emphasize a few differences in B c decays comparing to the B u,d decays that wefound; i) the B c decays are much simpler, since the only operator contributing is O ,and there is only one combination of CKM factors ( V ∗ cb V uD where D = d, s depending onthe strangeness of the outgoing state), ii) the long-distance divergences, which prevents11s from estimating the annihilation contribution in B u,d decays, do not appear in B c annihilation.We can give the numerical results for the following channels: B c → φK + , B c → K ∗ K + , B c → K K + , B c → K ∗ K ∗ + . We start with the function b ( h h ) which turnsout to be quite SU (3) invariant: b ( φK + ) = 1 . , b ( K ∗ K + ) = 1 . , (31) b ( K K + ) = 1 . , b ( K ∗ K ∗ + ) = 1 . b ( V P ) = b ( P V ), and thus A V P = 0. One can see that the SU (3) breakingis rather small as argued in Appendix A, while the difference between P P and
V P ( V V )modes can be as large as +13(+38)%. We next list the normalization factors: N φK + = 0 .
014 GeV , N K ∗ K + = 0 .
014 GeV , (33) N K K + = 0 .
010 GeV , N K ∗ K ∗ + = 0 .
019 GeV (34)For these particular decay channels (no exchange between π and K ), the SU (3) breakingis also small while the difference between P P and
V P ( V V ) modes can be as large as-40(-90)%. Our numerical value for
S, I, A amplitudes for the above processes are: S V P = 0 .
036 GeV , [ V P = φK + , K ∗ K + ] (35) S P P = 0 .
021 GeV , [ P P = K K + ] (36) S V V = 0 .
025 GeV [ V V = K ∗ K ∗ + ] (37)where we have neglected transverse V V amplitudes. We present the relation between N h h and b ( h h ) functions and the Wigner-Eckart reduced matrix elements S, I, A inApp. B. Assuming the SU (3) breaking effect is negligible in b ( h , h ), this relation allowsus to estimate all the other S, I, A amplitudes with the values given in Eq. (32) and theknown values of decay constants for each final state.We obtain finally the following values of branching ratios:[One-gluon] BR ( B c → φK + ) = 5 × − , BR ( B c → ¯ K ∗ K + ) = 9 . × − (38) BR ( B c → ¯ K K + ) = 6 . × − , BR ( B c → ¯ K ∗ K ∗ + ) = 9 . × − (39)The contributions to ρ π + , φπ + and ρ + φ vanish in our approximations, which means thatthese power-suppressed decays must have significantly smaller branching ratios than theabove ones. We do not quote any error bars on these results on purpose: we can easilyestimate the uncertainties coming from our hadronic inputs, but certainly not the sys-tematics coming from the hypothesis underlying our estimate (one-gluon approximation,asymptotic distribution amplitudes, neglect of 1 /m b and 1 /m c suppressed corrections,neglect of soft residual momentum of the heavy quarks in the B c meson).12 .3 Comparison of the two methods Our estimates of the branching ratio for e.g. the B c → K + K in the above two ways arenot consistent. This is clearly because two methods are conceptually different: • The method based on B d annihilation treats the charm quark as massless. It takesinto account some of the non-perturbative long-distance effects expected to occurin B d and B c decays, but treats in a very naive way the relation between matrixelements of the operators O and O . It relies also on extremely naive assumptionsconcerning the respective size of matrix elements for P P , V P and
V V modes. • The method based a perturbative one-gluon exchange treats the charm quark asheavy. It assumes the dominance from a specific set of diagrams computed in aperturbative way, but it provides a consistent framework to perform the estimation.It is well known that both kinds of estimates yield rather different results. This is illus-trated by the fact that the estimate of B d → K + K − in the annihilation models of QCDfactorisation [25] (around 10 − with substantial uncertainties) and perturbative QCD [38]is one order of magnitude below the current experimental average. Therefore, it is notsurprising that our two methods yield branching ratios differing by a similar amount.There are well-known cases where final-state interaction can increase significantly esti-mates based on factorisation, for instance B → Kχ c [39, 40] or D + s → ρ π + [41]. Anobservation of the non-leptonic charmless B c decays will certainly have an important keyto clarify such a controversy as well as further theoretical issues in computational methodsfor the annihilation diagram. Search prospect at LHCb
The LHC pp collider with the center of mass energy of 14 TeV has a large cross sectionfor the bb hadro-production, which can be followed by the production of not only B , B + and B s mesons but also other b hadrons such as, Λ b and B c . The subtraction of theproduction of other known b hadrons [46] bb → ( B d : B u : B s : Λ ′ b s) ≃ (42 . ± .
9% : 42 . ± .
9% : 10 . ± .
9% : 9 . ± . bb → ( B c ) to be less than one %. The LHCb experiment [47], which is dedicatedto B physics analyses with its optimized trigger scheme, allows one to detect the b -decaymodes into hadronic final states.The theoretical estimate of the B c cross section is still under scrutiny. For the dom-inant gg -fusion process, there are two possible mechanisms: gg → b ¯ b followed by thefragmentation, or ¯ b → B c b ¯ c . It is found that the latter dominates in the low- p T regionwhich corresponds to the LHCb coverage [48, 49]. The O ( α s ) computation of the ¯ b → B c b ¯ c σ ( pp → B + c X ) ≃ . − . µb [50] where the error comes from the un-certainty on theoretical inputs such as the choice of the α s scale and the B c distributionfunction. Additional systematics could come from higher-twist and radiative corrections.In the following, we follow the LHCb value for the cross section σ ( B c ) = 0 . µ b but itmust be noted that this value may be affected by a large uncertainty.We can now estimate the expected sensitivity for a specific channel. First, let usdiscuss which channel has the best potential for the detection. The best trigger andreconstruction efficiencies with a large signal over background ratio can be achieved bythe charged K - and/or π -tags (and by avoiding low- p T neutral particles) at LHCb. Sincethe initial B c carries an electric charge, all two-body P P final states contain one neutralparticle. The same remark applies for the
V V channels when one considers the subsequentdecays of the vector particles into pairs of pseudoscalars. In this respect,
P V channelssuch as B + c → φK + , K ∗ K + , K π + , ρ K + , ρ π + , φπ + are the best candidates usingthe vector meson decays, φ → K + K − , K ∗ → K − π + , ρ → π + π − , leading to threecharged tracks. Among these subsequent decays, the small widths of φ and K ∗ make thereconstructing particularly easy comparing to e.g. ρ . On the theoretical side, our Zweigrule argument forbids the B + c → φπ + , whereas the B + c → ρ π + channel comes only fromthe A (asymmetric) amplitude which is also subdominant. Finally, taking into accountthe fact that the ∆ S = 1 channels are Cabibbo suppressed, we draw the conclusion thatthe B + c → K ∗ K + channel might be the best candidate for the detection.Since the selection criteria and trigger efficiencies are different for each channel, de-tailed simulations are necessary in order to estimate the expected sensitivity for differentchannels. For example, such a study has been done for B c → J/ψπ + [50, 51]. Fromthe expected branching ratio Br ( B c → J/ψπ + ) ≃ Br ( B + c → K ∗ K + ) = 10 − yields a few events per year at LHCb. The analysis ofLHCb data will thus allow to set first experimental limits on the non-leptonic charmless B c decays, and give hints on annihilation mechanisms in these decays. Conclusions
In this paper, we have discussed non-leptonic charmless B c decays into two light pseu-doscalar or vector mesons. It turns out that a single tree annihilation diagram is respon-sible for all 32 processes, providing an interesting testing ground for annihilation. Afterdiscussing general aspects of the charmless B c decays, we have shown that the very simplenature of these decays allows us to describe them in terms of a few reduced amplitudesby exploiting SU (3) flavour symmetry to relate various P P , P V and
V V modes.In order to discuss a possible search for charmless non-leptonic B c decays at LHCb,we have proposed two different theoretical estimates of these reduced matrix elements,14ither by comparison with B d annihilation processes or by a perturbative model based onthe exchange of one-gluon. The two models yield a rather wide range of branching ratiopredictions, from 10 − to 10 − . The LHCb experiment has the potential to observe someof the decay channels (such as B c → φK + , K ∗ K + ) if the branching ratio is at the largerside of these estimates.From the theoretical point of view, a better understanding of annihilation diagramsis particularly important. They are often assumed to play a significant role in decaysof heavy-light mesons, but they occur jointly with other kinds of diagrams, making itdifficult to assess precisely their size. Furthermore, for the theoretical estimates of B u,d annihilation diagram in the QCD factorisation, there is an additional uncertainty causedby the infrared divergence occurring in its computation. It is worth mentioning that wefound that such a divergence does not occur in the case of the B c annihilation diagramsuggesting that predictions from models `a la QCD factorisation for the B c decays shouldbe more precise, and thus easier to confirm or reject.On the other hand, it has been discussed that the annihilation diagrams may beenhanced by long-distance effects such as final-state interactions. Although only limitedmodels of such effects have been proposed either for D or for B decays (the former likelymore affected than the latter by such enhancements)[41, 39, 40], the observation of anunexpectedly large branching ratio for the B c annihilation would call for a reassessmentof such long-distance contributions. An observation of charmless non-leptonic B c decaysat LHCb will certainly provide substantial information on these models, in complementwith the observation of other decays such as B d → K + K − or B s → π + π − . Acknowledgments
We would like to thank Marie-H´el`ene Schune for discussion. Work supported in part by EUContract No. MRTN-CT-2006-035482, “FLAVIAnet” and by the ANR contract “DIAM”ANR-07-JCJC-0031. The work of E.K. was supported by the European Commission MarieCurie Incoming International Fellowships under the contract MIF1-CT-2006-027144 andby the ANR (contract ”LFV-CPV-LHC” ANR-NT09-508531).
A Short-distance model for weak annihilation
As highlighted in the introduction, weak annihilation plays a significant role in B u,d,s non-leptonic decays, but it is difficult to estimate it accurately. A model to estimate thiscontribution was provided in the framework of QCD factorisation [42, 25], relying on thefollowing hypothesis : • the diagrams are dominated by the exchange of a single gluon, whose off-shellnessis typically of order O ( √ Λ m b ) 15 hadronisation effects are taken into account through light-cone distribution ampli-tudes (generally taken in their asymptotic form) • soft components are neglected.Being power-suppressed in the heavy-quark limit, the weak-annihilation contributionsto B u,d,s non-leptonic decays cannot be factorised in short- and long-distance effects (ingeneral). Their evaluation within this rough model exhibits endpoint divergences, whichsignals the presence of long- distance contributions not taken into account properly. Thedivergent integrals were regularised on the basis of dimensional analysis, which induces asignificant uncertainty on the estimate of the annihilation contribution.We can follow a similar method to estimate annihilation in the case of the B c decay.Concerning the B c meson, we work in the limit where both b and c quarks are heavy(keeping m c /m b fixed) and we set the momentum of the valence quarks to p µb = m b v µ and p µc = m c v µ , neglecting the soft components of the heavy-quark momenta (and consistentlysetting M B c = m c + m b ). Since we neglect the soft components of p b and p c , the integrationover the B c meson distribution amplitude is trivial and yields f B c . The diagrams tocompute are not very difficult and correspond to a gluon emitted from the b anti-quark orthe c quark from the B c meson and converted into a light quark-anti-quark pair. Followingref. [25], we find in the case where ( M , M ) = ( P, P ) , ( P, V ) , ( V, V ) : A i ( M M ) = πα s Z dx dy (41) ( φ M ( y ) φ M ( x ) (cid:20) y [(¯ x + y ) z b − ¯ xy ] − x [(¯ x + y ) z c − ¯ xy ] (cid:21) + r M r M φ m ( y ) φ m ( x ) (cid:20) − z b )(¯ x + y ) z b − ¯ xy − − z c )(¯ x + y ) z c − ¯ xy (cid:21) If ( M , M ) = ( V, P ), one has to change the sign of the second (twist-4) term above. φ M and φ m are twist-2 and twist-3 two-particle distribution amplitudes of the meson M , and r M is the normalisation of the twist-3 distribution amplitude. In the case of pseudoscalarmesons, we have: r π = 2 m π m b × m q r K = 2 m K m b ( m q + m s ) (42)responsible for the chiral enhancement of twist-4 contributions for pion and kaon outgoingstates. In the case of vector mesons, we have: r V = 2 m V m b f ⊥ V f V (43) z b and z c denote the relative size of the b and c -quark masses: z b = m b m b + m c z c = 1 − z b = m c m b + m c (44)16heir appearance allows one to distinguish the diagram of origin (corresponding to agluon emitted from the b or the c quark line).Eq. (41) is in agreement with the expressions obtained in refs. [25] and [43] in thelimit z b → z c → φ P ( x ) = 6 x (1 − x ) φ V ( x ) = 6 x (1 − x ) φ p ( x ) = 1 φ v ( x ) = 3(2 x −
1) (45)The structure of the singularities in the kernel is due to the propagator of the gluon andseems quite complicated. But if we take as an example Z d ¯ x dy x + y ) z − ¯ xy = Z d ¯ x z − ¯ x log (cid:12)(cid:12)(cid:12)(cid:12) z − ¯ x + ¯ xz ¯ xz (cid:12)(cid:12)(cid:12)(cid:12) (46)The function to be integrated is continuous at x = z and has integrable singularities for x = 0 and x = z/ (1 − z ). The integration can therefore be performed without problem aslong as z is different from 0 , / B c meson,it means that the twist-2 and the twist-4 contributions have no endpoint singularities andyield finite integrals. Therefore, there is no need to introduce models to regularise thedivergent integrals like in the case of heavy-light mesons.The corresponding expressions for the four cases are slightly tedious, but they canbe approximated to a very good accuracy through low-order polynomials in δ , where z b = 0 .
76 + δ and z c = 0 . − δ (corresponding to m b = 4 . m c = 1 . δ = 0): A i ( P P ) = πα s [( − .
83 + 4 . δ + 808 . δ + 2507 δ + 3425 δ ) (47)+ r M r M ( − . − . δ + 73 . δ − . δ + 3575 δ + 16007 δ )] A i ( P V ) = πα s [( − .
83 + 4 . δ + 808 . δ + 2507 δ + 3425 δ ) (48)+ r M r M ( − . − . δ − δ + 58 . δ + 7982 δ + 39778 δ )] A i ( V V ) = πα s [( − .
83 + 4 . δ + 808 . δ + 2507 δ + 3425 δ ) (49)+ r M r M (2 .
44 + 222 . δ + 1565 δ + 3386 δ + 5824 δ − δ − δ )]Within the set of approximations performed here, A i ( P V ) and A i ( P V ) are identical.We use the above formulae to estimate a few branching ratios. We take our inputs forthe vector decay constants and the Wilson coefficient C ( √ m b Λ h ) = − .
288 from ref. [44],and the rest of our inputs from ref. [25]. We take the value of the B c meson decay constant f B c = 395 MeV taking the central value from ref. [3].Let us comment on the SU (3) breaking, which can be included in this QCD computa-tion. We do not have the SU (3)-breaking effects coming from the distribution amplitude:for instance, a small m s correction makes the K and K ( ∗ ) distribution amplitudes slightlyasymmetric. On the other hand, we have the breaking effect in the chiral enhancementparameter r M . The SU (3) breaking (e.g. comparison of r π , r K or r ρ , r K ∗ , r φ ) turns out17o be relatively small. Another SU (3) breaking arises from the following decay constantsin the normalization factor N h h [ ? , 44]: f π = (130 . ± . , f K = (155 . ± . f ρ = (216 ± , f K ∗ = (220 ± , f ω = (187 ± , f φ = (215 ± . The SU (3) breaking in the decay constants for the vector mesons is rather small whilethere is a 24% difference in π and K decay constants. B Identification with results from QCD factorisation
The expressions for all the decay channels considered in Sec. 3 can be recovered fromrefs. [25], [43] and [45] if we identify between the Wigner-Eckart reduced matrix elements
S, I, A and the O reduced coefficients b . In these references, one must take the expres-sions for the decay amplitudes of B u decays into the relevant final state, and pick up the O contribution, which is the only remaining one once the related B c decay is considered.If we perform this identification, we obtain: S P P = r N P P b ( P P ) (51) I P P = r N P P b ( P P ) (52) S V P = r N V P ( b ( P V ) + b ( V P )) (53) A V P = r N V P ( b ( P V ) − b ( V P )) (54) I V P = r N V P ( b ( P V ) + b ( V P )) (55) S V VS,D = r N V V b S,D ( V V ) (56) A V VP = 0 (57) I V VS,D = r N V V b S,D ( V V ) (58)
References [1] F. Abe et al. Observation of the B c meson in p ¯ p collisions at √ s = 1 . Phys.Rev. Lett. , 81:2432–2437, 1998.[2] T. Aaltonen et al. Observation of the Decay B ± c → J/ψπ ± and Measurement of the B ± c Mass.
Phys. Rev. Lett. , 100:182002, 2008.183] N. Brambilla et al. Heavy quarkonium physics. 2004.[4] Ikaros I. Y. Bigi. Inclusive B c Decays As A QCD Lab.
Phys. Lett. , B371:105–110,1996.[5] Martin Beneke and Gerhard Buchalla. The B c Meson Lifetime.
Phys. Rev. , D53:4991–5000, 1996.[6] Pietro Colangelo and Fulvia De Fazio. Using heavy quark spin symmetry in semilep-tonic B c decays. Phys. Rev. , D61:034012, 2000.[7] V. V. Kiselev, A. E. Kovalsky, and A. I. Onishchenko. Heavy quark potential as itstands in QCD.
Phys. Rev. , D64:054009, 2001.[8] Chao-Hsi Chang, Yu-Qi Chen, and Robert J. Oakes. Comparative Study of theHadronic Production of B c Mesons.
Phys. Rev. , D54:4344–4348, 1996.[9] Maurizio Lusignoli, M. Masetti, and S. Petrarca. B c production. Phys. Lett. ,B266:142–146, 1991.[10] Nora Brambilla and Antonio Vairo. The B c mass up to order α s . Phys. Rev. ,D62:094019, 2000.[11] Nora Brambilla, Antonio Pineda, Joan Soto, and Antonio Vairo. Effective field the-ories for heavy quarkonium.
Rev. Mod. Phys. , 77:1423, 2005.[12] V. V. Kiselev. Gold-plated mode of CP-violation in decays of B c meson from QCDsum rules. J. Phys. , G30:1445–1458, 2004.[13] V. V. Kiselev, A. E. Kovalsky, and A. K. Likhoded. Decays and lifetime of B c inQCD sum rules. 2000.[14] V. V. Kiselev, A. E. Kovalsky, and A. K. Likhoded. B c decays and lifetime in QCDsum rules. Nucl. Phys. , B585:353–382, 2000.[15] V. V. Kiselev. Exclusive decays and lifetime of B c meson in QCD sum rules. ((U)).2002.[16] C. T. H. Davies et al. B c Spectroscopy from Lattice QCD.
Phys. Lett. , B382:131–137,1996.[17] I. F. Allison et al. A precise determination of the B c mass from dynamical latticeQCD. Nucl. Phys. Proc. Suppl. , 140:440–442, 2005.[18] H. P. Shanahan, P. Boyle, C. T. H. Davies, and H. Newton. A non-perturbativecalculation of the mass of the B c . Phys. Lett. , B453:289–294, 1999.1919] B. D. Jones and R. M. Woloshyn. Mesonic decay constants in lattice NRQCD.
Phys.Rev. , D60:014502, 1999.[20] Cheng-Wei Chiang, Michael Gronau, and Jonathan L. Rosner. Examination of FlavorSymmetry in
B, B s → Kπ Decays.
Phys. Lett. , B664:169–173, 2008.[21] Cheng-Wei Chiang, Michael Gronau, Jonathan L. Rosner, and Denis A. Suprun.Charmless B → P P decays using flavor SU (3) symmetry. Phys. Rev. , D70:034020,2004.[22] Cheng-Wei Chiang, Michael Gronau, Zumin Luo, Jonathan L. Rosner, and Denis A.Suprun. Charmless B → V P decays using flavor SU (3) symmetry. Phys. Rev. ,D69:034001, 2004.[23] Yong-Yeon Keum, Hsiang-nan Li, and A. I. Sanda. Fat penguins and imaginarypenguins in perturbative QCD.
Phys. Lett. , B504:6–14, 2001.[24] Marco Ciuchini, E. Franco, G. Martinelli, M. Pierini, and L. Silvestrini. CharmingPenguins Strike Back.
Phys. Lett. , B515:33–41, 2001.[25] Martin Beneke and Matthias Neubert. QCD factorization for B → P P and B → P V decays.
Nucl. Phys. , B675:333–415, 2003.[26] Christian W. Bauer, Dan Pirjol, Ira Z. Rothstein, and Iain W. Stewart. B → M (1) M (2): Factorization, charming penguins, strong phases, and polarization. Phys.Rev. , D70:054015, 2004.[27] Junegone Chay, Hsiang-nan Li, and Satoshi Mishima. Possible complex annihilationand B → Kπ direct CP asymmetry. Phys. Rev. , D78:034037, 2008.[28] D. Zeppenfeld. SU (3) Relations for B Meson Decays. Zeit. Phys. , C8:77, 1981.[29] J. J. de Swart. The Octet model and its Clebsch-Gordan coefficients.
Rev. Mod.Phys. , 35:916–939, 1963.[30] Michael Gronau and Jonathan L. Rosner. V td from Hadronic Two-Body B Decays.
Phys. Lett. , B376:205–211, 1996.[31] T. Feldmann, P. Kroll, and B. Stech. Mixing and decay constants of pseudoscalarmesons.
Phys. Rev. , D58:114006, 1998.[32] Thorsten Feldmann. Quark structure of pseudoscalar mesons.
Int. J. Mod. Phys. ,A15:159–207, 2000.[33] J. M. Gerard and E. Kou. η − η ′ masses and mixing: A large N c reappraisal. Phys.Lett. , B616:85–92, 2005. 2034] B. Aubert et al. Improved Measurements of the Branching Fractions for B → π + π − and B → K + π − , and a Search for B → K + K − . Phys. Rev. , D75:012008, 2007.[35] K. Abe et al. Observation of B decays to two kaons. Phys. Rev. Lett. , 98:181804,2007.[36] Michael Morello. Branching fractions and direct CP asymmetries of charmless decaymodes at the Tevatron.
Nucl. Phys. Proc. Suppl. , 170:39–45, 2007.[37] J. Charles et al. CP violation and the CKM matrix: Assessing the impact of the asym-metric B factories (updated results and plots available at: http://ckmfitter.in2p3.fr).
Eur. Phys. J. , C41:1–131, 2005.[38] Chuan-Hung Chen and Hsiang-nan Li. Final state interaction and B → KK decaysin perturbative QCD. Phys. Rev. , D63:014003, 2001.[39] T. N. Pham and Guo-huai Zhu. B → χ c K decay: A model estimation. Phys. Lett. ,B619:313–321, 2005.[40] M. Beneke and L. Vernazza. B → χ cJ K decays revisited. 2008.[41] Svjetlana Fajfer, Anita Prapotnik, Paul Singer, and Jure Zupan. Final state inter-actions in the D + s → ωπ + and D + s → ρ π + decays. Phys. Rev. , D68:094012, 2003.[42] M. Beneke, G. Buchalla, M. Neubert, and Christopher T. Sachrajda. QCD factoriza-tion in B → πK, ππ decays and extraction of Wolfenstein parameters. Nucl. Phys. ,B606:245–321, 2001.[43] Martin Beneke, Johannes Rohrer, and Deshan Yang. Branching fractions, polarisa-tion and asymmetries of B → V V decays.
Nucl. Phys. , B774:64–101, 2007.[44] Patricia Ball, Gareth W. Jones, and Roman Zwicky. B → V γ beyond QCD factori-sation.
Phys. Rev. , D75:054004, 2007.[45] Matthaus Bartsch, Gerhard Buchalla, and Christina Kraus. B → V L V L Decays at Next-to-Leading Order in QCD. 2008[46] C. Amsler et al. [Particle Data Group], Review of particle physics. Phys. Lett. B (2008) 1.[47] A. Augusto Alves et al, LHCb Collaboration. The LHCb detector at the LHC.JINST (2008) S08005.[48] I. P. Gouz, V. V. Kiselev, A. K. Likhoded, V. I. Romanovsky and O. P. Yushchenko.Prospects for the B c studies at LHCb. Phys. Atom. Nucl. (2004) 1559,[Yad. Fiz. (2004) 1581] 2149] K. Kolodziej, A. Leike and R. Ruckl. Production of B(c) mesons in hadronic collisions.Phys. Lett. B , 337 (1995)[50] O. P. Yushchenko. Search for the B + c → J/ψ ( µ + µ − ) π + decay with the LHCbspectrometer. LHCb 2003-113[51] Y. Gao, J. He, Z. Yang. Study of B c lifetime measurement using B ± c → J/ψ ( µ + µ − ) π ±±