aa r X i v : . [ h e p - ph ] N ov B → D ∗∗ – puzzle 1/2 vs 3/2 Benoˆıt Blossier
Laboratoire de Physique Th´eoriqueCNRS/Universit´e Paris-Sud, Bˆat 210, F-91405 Orsay Cedex, FRANCE
Understanding the composition of final states in B → X c lν could help to get afeedback on the persisting disagreement between exclusive and inclusive determinationsof V cb . In particular the series of orbital excitations D ∗∗ and radial excitations ( D ′ , D ∗ ′ )has received a lot of attention; a misinterpretation as a scalar state of the ( D ′ → Dπ )spectrum tail could have induced an experimental overestimate of the broad statescontribution to the total B → X c lν width with respect to theoretical expectations,all of them made however in the infinite mass limit: it is the so-called 1/2 vs 3/2puzzle. We describe first attempts to measure on the lattice form factors of B → D ∗∗ lν at realistic quark masses. Cleaner processes, like hadronic decays B → D ∗∗ π andsemileptonic decays B s → D ∗∗ s lν in the strange sector have recently been examined byphenomenologists, putting new interesting ideas on those issues with, again, the need oflattice inputs. PRESENTED AT the 8th International Workshop on the CKM Unitarity Triangle (CKM2014), Vienna, Austria, September 8-12, 2014 ass (MeV) Width (MeV) j Pl J P D ± ± − S : D ( ∗ ) D ∗± ± ± − − D ∗ ±
50 261 ±
50 0 + D ∗ ± ±
25 384 +107 − ±
12 + + P : D ∗∗ D ± . +3 . − . + D ∗ ± ±
32 + + Table 1: Low-lying spectrum in the D sector; it is convenient to decompose the total orbitalmomentum as J = ⊕ j l , where j l is the orbital momentum of the light degrees of freedom. Understanding the long-distance dynamics of QCD is crucial in the control of the theoretical system-atics on low-energy processes that are investigated at LHCb and, in the next years, at Super Belle,to detect indirect effects of New Physics. It is particularly relevant for processes involving excitedstates, that occur often in experiments. With that respect beauty and charmed mesons represent avery rich sector. An intriguing question concerns the origin of the ∼ σ discrepancy between | V cb | excl and | V cb | incl [1]: expressed differently, it is welcome to know more about the composition of the finalhadronic state X c in the semileptonic decay B → X c lν . We sketch in Table 1 the low-lying spectrumof D mesons. The D states of the j Pl =
12 + doublet are broad while those of the j Pl =
32 + doubletare narrow: indeed, the main decay channels are the non leptonic transitions D ∗∗ → D ( ∗ ) π . Parityconservation implies that the pion has an even angular momentum ℓ with respect to D ( ∗ ) . Orbitalmomentum conservation implies that ℓ = 0 or 2. That’s why D ∗ and D ∗ decay with a pion in the S wave and D ∗ decays with the pion in the D wave. The decay D → D ∗ π occurs with the pionin the S or D waves; however, thanks to Heavy Quark Symmetry, the latter is favored. Therefore,decays of the j Pl =
32 + doublet are suppressed compared to decays of the
12 + doublet. But X c couldbe made of radial excitations as well: the Babar Collaboration claimed to have isolated a bench ofnew D states [2]. Among them, a structure in the D ∗ π distribution is interpreted as D (2550) ≡ D ′ .After a fit, experimentalists obtain m ( D ′ ) = 2539(8) MeV and Γ( D ′ ) = 130(18) MeV. A questionraised about the correctness of this interpretation because, in theory, quark models predict approxi-mately the same D ′ mass (2.58 GeV) but a quite smaller width (70 MeV) [3]. However a well knowncaveat is that excited states properties are very sensitive to the position of the wave functions nodes,themselves depending strongly on the quark model. We collect in Table 2 the branching ratios ofthe B → X c semileptonic decays. We are interested by ∼
25% of the total width Γ( B → X c lν ): 1/3of it comes from the channel B → D ∗∗ narrow . Studying the channel B → D ′ lν , assuming it is quitelarge [4] and using the fact that Γ( D ′ → D / π ) ≫ Γ( D ′ → D / π ), one concludes that an excessof B → ( D / π ) lν events could be observed with respect to their B → ( D / π ) lν counterparts. Aquestion is then whether such a potentially large B → D ′ lν width could explain the ”1/2 vs. 3/2”puzzle: [Γ( B → D / lν ) ≃ Γ( B → D / lν )] exp while [Γ( B → D / lν ) ≪ Γ( B → D / lν )] theory [5]. Akinematical factor explains partly this suppression: d Γ B → D / d Γ B → D / = w +1) (cid:16) τ / ( w ) τ / ( w ) (cid:17) . A detailed com-parison between theory and experiment is made in the center panel of Table 2. The main tension1 ( B d → X c lν ) = (10 . ± . B ( B d → [non − D ( ∗ ) ] lν ) = 2 . ± . B ( B d → D ∗∗ narrow lν ) = (0 . ± . B ( B d → D ( ∗ ) πlν ) = (1 . ± . B ( B d → [ Dπ ] broad lν ) = (0 . ± . B ( B d → [ D ∗ π ] broad lν ) = (0 . ± . B d → D ∗∗ eν B exp / B th D ∗ D D ∗ [0, 5] D ∗ ± B d → D ∗∗ π B exp / B th D ∗ ∼ D [0.5, 1] D ∗ no result D ∗ [0.2, 2.6]Table 2: Branching ratio of B → X c lν (left panel); comparison between theory and experiment forthe different B → D ∗∗ lν channels (center panel); comparison between theory and experiment forthe different B → D ∗∗ π channels (right panel).is for B → D ∗ lν . On the experimental side, there are issues about identifying the D ∗ state and thedisagreement in B ( B → D ∗ lν ) between Belle (no events) and BaBar (claim of a signal). On thetheory side, the limitation is that the predictions are made essentially in the infinite mass limit,including lattice QCD calculations of Isgur-Wise functions τ / and τ / . B → D ∗∗ lν and lattice QCD In the Heavy Quark Effective Theory framework, with the trace formalism, the transitions betweentwo heavy-light mesons H j l ,Jv and H j ′ l ,J ′ v ′ are expressed in terms of universal form factors, the Isgur-Wise functions Ξ( w ≡ v · v ′ ), where v is the velocity of the meson. Their number is limitedthanks to Heavy Quark Symmetry: ξ ( w ) parameterizes the elastic transition H − v → H − v ′ and isnormalised at zero recoil: ξ (1) = 1. One has also h H + v ′ | h v ′ γ µ γ h v | H − v i = τ / ( µ, w )( v − v ′ ) µ and h H + v ′ | h v ′ γ µ γ h v | H − v i = √ τ / ( µ, w )[( w + 1) ǫ ∗ µα v α − ǫ ∗ αβ v α v β v ′ µ ]. τ / and τ / are not normalisedat zero recoil; however, any scale dependence vanishes: τ , ( µ, ≡ τ , (1). A quenched latticestudy obtained τ (1) . τ (1), even if the analysis was based on quite short plateaus of the J P = 2 + state effective mass and of τ , (1) data got from ratios of 3-pt and 2-pt correlation functions [6].A similar computation was then led with N f = 2 dynamical quarks, using a set of ETMC gaugeensembles, with acceptable signals for effective masses and τ / , / (1). After a smooth extrapolationto the chiral limit, the authors found again that τ / (1) seems significantly smaller than τ / (1)[7]: lattice results point in the same direction as quark models [8], [9] and Operator ProductionExpansion based sum rules [10], [11]. 2 a M e ff < D ( + ) | A | B > m h =1.5m c m h =3m c Figure 1: Effective mass of D ∗ (left panel) and form factor F A (1) at two b quark masses (rightpanel). b and c quark masses More recently a direct computation in QCD has been tried [12]. The starting point is the definitionof a set of form factors: h D ∗ | A µ | B i = ˜ u + ( p B + p D ) µ + ˜ u − ( p B − p D ) µ , h D ∗ ( ǫ ( λ ) ) | V µ | B i = i ˜ h ǫ µνρσ ǫ ( λ ) ∗ να p αB ( p B + p D ) ρ ( p B − p D ) σ , h D ∗ ( ǫ ( λ ) ) | A µ | B i = ˜ k ǫ ( λ ) µν ∗ p B ν + ǫ ( λ ) ∗ αβ p αB p βB [˜ b + ( p B + p D ) µ + ˜ b − ( p B − p D ) ν ] , with V µ = cγ µ b and A µ = cγ µ γ b . Choosing the kinematical configuration ~p D = ~ ~p B = ( θ, θ, θ )and defining the tensors of polarisation accordingly, it has been shown that the leading form factorsthat contribute to the widths are˜ k = − √ θ F (0) 1 A = − √ θ F (0) 2 A = √ θ F (0) 3 A , ˜ k = 1 θ h F (+2) 1 A + F ( −
2) 1 A i = − θ h F (+2) 2 A + F ( −
2) 2 A i , ˜ u + = − m D ∗ (cid:20) E B − m D ∗ θ ( F A + F A + F A ) − F A (cid:21) , where F ( λ ) µA ≡ h D ∗ ( ǫ ( λ ) ) | A µ | B i and F µA ≡ h D ∗ | A µ | B i . The preliminary study was performed using N f = 2 ETMC ensembles: the charm quark was tuned at the physical point, while several ”light” b quarks were simulated to extrapolate to m b ; cut-off effects were investigated on 2 lattice spacings, athird one will finally be considered. Twisted boundary conditions are required to give a momentumto the B meson in 2-pt and 3-pt correlators. In the twisted-mass formalism it is difficult to isolatethe signal for D ∗ because of the mixing with D state due to a breaking parity cut-off effect: solvinga generalized eigenvalue problem is beneficial. as shown in the left panel of Figure 1. Isolating thesignal for D ∗ is difficult because of the noise, despite averaging over different interpolating fieldsthat belong to the same representation (E or T2) of the O h cubic group. At zero recoil, it seemspossible to isolate the signal for F A but it deteriotates if the b quark mass gets closer to m b , asshown in the right panel of Figure 1. Concerning the decay of D ∗ , it is known that F ( λ ) µA (1) = 0:one needs to inject large momenta, where the data are also noisy.3 B ( s ) → D ( s ) π : a more favorable situation? A comparison between theory and experiment non leptonic B → D decays is made in the right panelof Table 2. Though a (not so conclusive) experimental disagreement in B ( B d → D ∗ π ) between Belleand BaBar, and the fact that theoretical predictions are based on the factorisation approximation,that works well for the so called Class I decays, we globally observe a much better agreementbetween theory and experiment for B d → D ∗ π than for B d → D ∗ lν . B → D ′ lν checked on B → D ′ π It was proposed in [13] to check the hypothesis of a large branching ratio B ( B → D ′ lν ) by studyingnon leptonic decays. By examining the Class I process B → D ′ + π − , one has in the factorisationapproximation B ( B → D ′ + π − ) B ( B → D + π − ) = (cid:18) m B − m D ′ m B − m D (cid:19) (cid:20) λ ( m B , m D ′ , m π ) λ ( m B , m D , m π ) (cid:21) / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f B → D ′ + (0) f B → D + (0) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where λ ( x, y, z ) = [ x − ( y + z ) ][ x − ( y − z ) ] and f B → D ( ′ )+ ( m π ) ∼ f B → D ( ′ )+ (0). With V cb f B → D + (0) =0 . | V cb | incl = 0 . f B → D + (0) = 0 . m D ′ = 2 .
54 GeV, we get B ( B → D ′ + π − ) B ( B → D + π − ) = (1 . ± . × (cid:12)(cid:12) f B → D ′ + (0) (cid:12)(cid:12) . Finally, with B ( B → D + π − ) =0 . B ( B → D ′ + π − ) = (cid:12)(cid:12)(cid:12) f B → D ′ + (0) (cid:12)(cid:12)(cid:12) × (4 . ± . × − . Letting vary the f B → D ′ + (0) form factor in the conservative range [0.1, 0.4], according to the existingtheoretical estimates [4], [15], we conclude that B ( B → D ′ + π − ) th ∼ − : the measurement canbe performed with the B factories samples and at LHCb. Having a look to the Class III process B − → D ′ π − , the factorised amplitude reads: A III fact = − i G F √ V cb V ∗ ud h a f π [ m B − m D ′ ] f B → D ′ ( m π ) + a f D ′ [ m B − m π ] f B → π ( m D ′ ) i . When the corresponding branching ratio is normalised by the Class I counterpart, we find B ( B − → D ′ π − ) B ( B → D ′ + π − ) = τ B − τ B (cid:20) a a × m B − m π m B − m D ′ × f B → π ( m D ′ ) f B → D ′ + (0) f D ′ f D f D f π (cid:21) . The ratio of Wilson coefficients a /a is extracted from B ( B − → D π − ) B ( B → D + π − ) , known experimentally [1], andit remains the computation on the lattice of the ratios of decay constants f D ′ f D and f D f π . CombiningETMC data at different a and m sea in a common fit we get m D ′ s m D s = 1 . , f D ′ s f D s = 0 . ,m D ′ m D = 1 . , f D ′ f D = 0 . . ( D ∗ + s → D ∗ + s π ) = (48 ± B ( D ∗ + s → D + s γ ) = (18 ± B ( D ∗ + s → D + s π + π − ) = (4 . ± . B ( D ∗ + s → D ∗ + s γ ) = (3 . +5 . − . )%Table 3: Branching ratios of non leptonic D ∗ s decays.The experimental result is ( m D ′ /m D ) exp = 1 .
36, 2 σ smaller than our value. For the moment thatdiscrepancy remains unexplained despite several checks described in [13]. With a /a = 0 . τ B /τ B − = 1 . f B → D + (0) = 0 . f B → π ( m D ) = 0 . B ( B − → D ′ π − ) B ( B → D ′ + π − ) = τ B − τ B (cid:20) . f B → D ′ + (0) (cid:21) , B ( B → D ′ + π − ) B ( B → D + π − ) = (1 . ± . × | f B → D ′ + (0) | . Using the experimental value m D ′ m D = 1 .
36, we get B ( B → D ′ + π − ) B ( B → D + π − ) = (1 . ± . × (cid:12)(cid:12) f B → D ′ + (0) (cid:12)(cid:12) : thedependence on m D ′ of that ratio is actually small. Fixing f B → D ′ + (0) = 0 . m D ′ /m D ) exp we have also B ( B → D ′ + π − ) B ( B → D ∗ +2 π − ) = 1 . , B ( B − → D ′ π − ) B ( B − → D ∗ π − ) = 1 . . It means that if f B → D ′ + is large, as claimed by many authors, the measurement of B ( B → D ′ π )should be as feasible as B ( B → D ∗ π ). B s → D ∗∗ s π The situation of the D s spectrum is peculiar: indeed, D ∗ s (2317) and D ∗ s (2460) are below the DK and D ∗ K thresholds. The main consequence is that they are narrow states. Thus it is veryadvantageous to examine them because there is no experimental issue from their broadness. It hasbeen proposed to study hadronic decays B s → D ∗ + s (2317) π − and B s → D ∗ + s (2460) π − [17]. At themoment, only upper limits on B ( D ∗ + s → ... ) are available: B ( D ∗ + s → D + s γ, D ∗ + s → D ∗ + s γγ ) < . B ( D ∗ + s → D + s π ) = (97 ± D ∗ s , that we collect in Table 3. According to [17], at LHCb, onemeasures the cascade B s → D ∗− s π + , D ∗− s → D − s π , D − s → K + K − π − ; the 4-momentum of thenon detected π is extracted from the B s flight direction and the known m B s and m π . Thenarrow peak in the D − s π mass distribution can be observed, depending on the accuracy of trackingcapabilities. Neglecting SU(3) breaking effects, with B ( B s → D + s π − ) = (2 . ± . × − and B ( B s → D ∗− s π + ) = (1 ± . × − , the number of expected events with 1 fb − of integratedluminosity is N ( B s → D ∗− s π + ) = 600 × (1 ± . × B ( D ∗− s → D − s π ) × ǫ π : ∼ . (1)5 eferences [1] J. Beringer et al. [Particle Data Group Collaboration], Phys. Rev. D , 010001 (2012).[2] P. del Amo Sanchez et al. [BABAR Collaboration], Phys. Rev. D (2010) 111101.[3] F. E. Close and E. S. Swanson, Phys. Rev. D (2005) 094004; Z. -F. Sun, J. -S. Yu, X. Liuand T. Matsuki, Phys. Rev. D (2010) 111501.[4] F. U. Bernlochner, Z. Ligeti and S. Turczyk, Phys. Rev. D (2012) 094033.[5] A. Le Yaouanc et al , Phys. Rev. D (1997) 5668; A. K. Leibovich et al , Phys. Rev. D (1998) 308; D. Becirevic et al , Phys. Rev. D , no. 5, 054007 (2013); I. I. Bigi et al , Eur.Phys. J. C , 975 (2007).[6] D. Becirevic, B. Blossier, P. Boucaud, G. Herdoiza, J. P. Leroy, A. Le Yaouanc, V. Morenasand O. Pene, Phys. Lett. B , 298 (2005). [hep-lat/0406031].[7] B. Blossier et al. [European Twisted Mass Collaboration], JHEP , 022 (2009).[arXiv:0903.2298 [hep-lat]].[8] V. Morenas, A. Le Yaouanc, L. Oliver, O. Pene and J. C. Raynal, Phys. Lett. B , 315(1996). [hep-ph/9605206].[9] D. Ebert, R. N. Faustov and V. O. Galkin, Phys. Rev. D , 014016 (2000). [hep-ph/9906415].[10] A. Le Yaouanc, D. Melikhov, V. Morenas, L. Oliver, O. Pene and J. C. Raynal, Phys. Lett. B , 119 (2000). [hep-ph/0003087].[11] N. Uraltsev, [hep-ph/0409125].[12] M. Atoui, B. Blossier, V. Mornas, O. Pne and K. Petrov, [arXiv:1312.2914 [hep-lat]].[13] D. Becirevic et al , Nucl. Phys. B , 313 (2013).[14] B. Aubert et al. [BABAR Collaboration], Phys. Rev. Lett. (2010) 011802.[15] J. Hein et al. [UKQCD Collaboration], Nucl. Phys. Proc. Suppl. (2000) 298; D. Ebert,R. N. Faustov and V. O. Galkin, Phys. Rev. D (2000) 014032; R. N. Faustov andV. O. Galkin, Phys. Rev. D , 034033 (2013); Z. -H. Wang et al , J. Phys. G (2012)085006.[16] G. Duplancic et al , JHEP (2008) 014; P. Ball and R. Zwicky, Phys. Rev. D (2005)014015.[17] D. Becirevic, A. Le Yaouanc, L. Oliver, J. C. Raynal, P. Roudeau and J. Serrano, Phys. Rev.D87