Lifetimes of b-flavoured hadrons

I discuss the heavy quark expansion for the inclusive widths of heavy-light hadrons, which predicts quite well the experimental ratios of Bq meson lifetimes. As for Λb, current determinations of O(m−3 b ) contribution to τ(Λb) do not allow to explain the small measured value of τ(Λb)/τ(Bd). As a final topic, I discuss the implications of the measurement of the Bc lifetime. 1. Lifetimes of heavy-light hadrons Inclusive particle widths describe the decay of the particle into all possible final states with given quantum numbers f . For weakly decaying heavy-light Qq̄ (Qqq) hadrons HQ, the spectator model considers only the heavy quark Q as active in the decay, the light degrees of freedom remaining unaffected. Hence, all the hadrons containing the same heavy quark Q should have the same lifetime; this picture should become accurate in the mQ →∞ limit, when the heavy quark decouples from the light degrees of freedom. However, the measurement of beauty hadron lifetime ratios [1]: τ(B−) τ(Bd) = 1.066±0.02 , τ(Bs) τ(Bd) = 0.99±0.05 , τ(Λb) τ(Bd) = 0.794±0.053 (1) shows that τ(Λb)/τ(Bd) significantly differs from the spectator model prediction. A more refined approach consists in computing inclusive decay widths of HQ hadrons as an expansion in powers of m−1 Q [2]. Invoking the optical theorem, one can write Γ(HQ → Xf ) = 2Im〈HQ|T̂ |HQ〉/2MHQ, with T̂ = i ∫ dxT [Lw(x)Lw(0)] the transition operator describing the heavy quark Q with the same momentum in the initial and final state, and Lw the effective lagrangian governing the decay Q → Xf . An operator product expansion of T̂ in the inverse mass of the heavy quark allows to write: T̂ = ∑ i CiOi, with the local operators Oi ordered by increasing dimension, and the coefficients Ci proportional to increasing powers of m −1 Q . As a result, for a beauty hadron Hb the general expression of the width Γ(Hb → Xf) is: Γ(Hb → Xf) = Γ0 [ c3〈b̄b〉Hb + cf5 mb 〈b̄igsσ ·Gb〉Hb + ∑ i c f(i) 6 mb 〈O6 i 〉Hb +O ( 1 mb )] , (2) with 〈O〉Hb = 〈Hb|O|Hb〉 2MHb , Γ0 = GFm 5 b 192π3 |Vqb| and Vqb the relevant CKM matrix element. Lifetimes of b-flavoured hadrons 2 The first operator in (2) is b̄b, with dimension D = 3; the chromomagnetic operator OG = b̄g2σμνGb, responsible of the heavy quark-spin symmetry breaking, has D = 5; the operators O i have D = 6. In the limit mb →∞, the heavy quark equation of motion allows to write: 〈b̄b〉Hb = 1 + 〈OG〉Hb 2mb − 〈Oπ〉Hb 2mb +O ( 1 mb ) , (3) with Oπ = b̄(i ~ D)b the heavy quark kinetic energy operator. When combined with (3), the first term in (2) reproduces the spectator model result. O(m−1 b ) terms are absent [3, 4] since D = 4 operators are reducible to b̄b by the equation of motion. Finally, the operators OG and Oπ are spectator blind, not sensitive to light flavour. Their matrix elements can be determined from experimental data; as a matter of fact, defining μG(Hb) = 〈OG〉Hb and μπ(Hb) = 〈Oπ〉Hb , one has: μG(B) = 3(M B∗ −M2 B)/4, while μG(Λb) = 0 since the light degrees of freedom in the Λb have zero total angular momentum relative to the heavy quark. Moreover, from the mass formula: MHb = mb + Λ̄ + μπ − μG 2mb +O(m−2 b ), with Λ̄, μπ and μG independent of mb, and from the experimental data, one can infer μπ(Bd) ' μπ(Λb), as confirmed by QCD sum rule estimates [6]. The O(m−3 b ) terms in (2) come from four-quark operators, accounting for the presence of the spectator quark in the decay. Their general expression is [5]: O V −A = (b̄LγμqL)(q̄LγμbL) T q V −A = (b̄Lγμt qL)(q̄Lγμt bL) O S−P = (b̄RqL)(q̄LbR) T q S−P = (b̄Rt qL)(q̄Lt bR) . (4) Their matrix elements over Bq can be parametrized as: 〈O V −A〉Bq = 〈O S−P 〉Bq (mb +mq MBq ) = f 2 Bq MBq 8 , 〈T q V −A〉Bq = 〈T q S−P 〉Bq = 0 , (5) fBq being the Bq decay constant. As for Λb, one can write: 〈Õ V −A〉Λb = f 2 BMB r/48, 〈O V −A〉Λb = −B̃〈Õ V −A〉Λb (6) with Õ V −A = (b̄LγμbL)(q̄LγμqL). In the valence quark approximation B̃ = 1. Actually, with the computed values of the Wilson coefficients in (2), only large values of the parameter r in (6) (namely r ' 3−4) could explain the observed difference between τ(Λb) and τ(Bd). This, however, seems not to be the case. 2. 〈Õ V −A〉Λb from QCD sum rules The parameter r in (6) can be determined using quark models or lattice QCD [7]. HQET QCD sum rules allow to estimate it from the correlator: ΠCD = (1 + 6v )CDΠ(ω, ω′) = i ∫ dxdy eiωv·x−iω v·y〈0|T [JC(x)Õ V −A(0)JD(y)]|0〉 (7) between Λb interpolating fields JC,D (C, D Dirac indices) [8] and the operator Õ q V −A; ω (ω′) is related to the residual momentum of the incoming (outgoing) current p = Lifetimes of b-flavoured hadrons 3 Figure 1. Sum rule for 〈Õ V−A〉Λb as a function of the Borel variable E. mbv μ + k with k = ωv. The projection of the interpolating fields on the Λb state is parametrized by 〈0|JC|Λb(v)〉 = fΛb(ψv)C (with ψv the spinor for a Λb of velocity v). Saturating the correlator Π(ω, ω′) with baryonic states and considering the lowlying double-pole contribution in the variables ω and ω′, one has: Π(ω, ω′) = 〈Õ V −A〉Λb f 2 Λb 2 × 1 (∆Λb − ω)(∆Λb − ω′) + . . . (8) with ∆Λb defined by MΛb = mb + ∆Λb. Besides, for negative values of ω, ω ′, Π can be computed in QCD in terms of a perturbative contribution and of vacuum condensates: Π(ω, ω′) = ∫ dσdσ′ ρΠ(σ, σ ′) (σ − ω)(σ′ − ω′) (9) with possible subtractions omitted [9]. The sum rule consists in equating Π and Π. Moreover, invoking global duality, the contribution of higher resonances and of continuum to Π can be modeled as the QCD term in the region ω ≥ ωc, ω′ ≥ ωc, with ωc an effective threshold. Finally, a double Borel transform to Π QCD and Π in ω, ω′, with Borel parameter E1, E2, removes the subtraction terms in (9), improves factorially the convergence of the OPE and enhances the contribution of the low-lying resonances in Π. Choosing E1 = E2 = 2E, one gets a sum rule the result of which is depicted in figure 1. Considering the variation with E and the threshold ωc, one has an estimate of 〈Õ V −A〉Λb: 〈Õ V −A〉Λb ' (0.4− 1.20)× 10−3 GeV 3 , (10) corresponding to r ' 0.1 − 0.3 [9]. The same calculation gives B̃ ' 1. This result produces τ(Λb)/τ(Bd) ≥ 0.94, at odds with the experimental result. The discrepancy discloses exciting perspectives both from experimental and theoretical sides [10].