The inclusive determination of V ub

Transcription

1 The inclusive determination of V ub Paolo Gambino Università di Torino and INFN Torino 1

2 The Unitarity Triangle V =! V * ij jk ik area= measure of CPV V ud V * ub Unitarity determines several triangles in complex plane * * + VcdVcb + VtdVtb V V V V * * ud ub td tb * * VcdVcb VcdVcb = = 0 0 O(λ 3 ) V ub /V cb describes a V td cannot be accessed directly: circle in the (ρ,η) plane we resort to loop transitions FCNC sensitive to new physics R b " V ub V cb 2

3 Only the CKM? ρ = ± η = ± Uncertainties mostly theoretical! Precision flavor physics can progress only together with our understanding of QCD Inclusive B decays are interesting because well measured and accurately predicted 3

4 The present situation UTfit (indirect): V ub = (3.44 ± 0.16)10-3 Exclusive determination: V ub = (3.4 ± 0.4)10-3 Inclusive determinations, HFAG averages: V ub = (4.31 ± 0.17 ±0.35)10-3 (BLNP) V ub = (4.34 ± 0.16 ±0.25)10-3 (DGE) V ub = (3.98 ± 0.15 ±0.30)10-3 (BLNP no bsγ) V ub = (3.98 ± )10-3 NEW 4

5 Exclusive determination Inclusive determination The problem seems to be in the Inclusive determination b 5

6 A set of interdependent measurements b c l ν tree BR~10% V cb b u l ν tree ~10-3 V ub b s γ loop ~ new physics, V ts b d γ loop ~ 10-6 new physics, V td b sl + l - loop ~ new physics Not only BR are relevant: various asymmetries, spectra etc 6

7 What do they have in common? X Simplicity: ew or em currents probe the B dynamics B INCLUSIVE OPE: non-pert physics described by B matrix elemnts of local operators can be extracted by exp suppressed by 1/m b 2 EXCLUSIVE Form factors: in general computed by non pert methods (lattice, sum rules,...) symmetry can provide normalization B factory accuracy challenges our theoretical understanding 7

8 The advantage of being inclusive The decay of a B meson is not the decay of a free b quark! However Λ QCD «m b : inclusive decays via OPE admit systematic expansion in α s and Λ QCD /m b starting with Λ 2 QCD /m b 2. Leading term is parton model. Non-pert corrections are generally small and under control 8

9 A double expansion Differential rate can be related to Im of OPE (HQE): T J ( x) J (0) " c bb + c b D b + c b#! Gb K The leading term is parton model, c i are series in α s New operators have non-vanishing expection values in B and are suppressed by powers of the energy released, E r ~ m b No 1/m b correction! HQE = Heavy Quark Expansion non-pert parameters 1/m b 2 1/m b 3! 3,! 3 D LS These parameters are now known with reasonable accuracy from Inclusive semileptonic B decays 9

10 Leptonic moments 10

11 Hadronic mass moments NEW P.Giordano, PG 11

12 Global fit to V cb, BR sl,hqe Buchmuller & Flacher 06 Based on Gambino & Uraltsev, Benson et al 12

13 Testing parton-hadron duality What is it? For all practical purposes: the OPE. No OPE, no duality Do we expect violations? Yes, problems prevalently arise because OPE must be continued analytically. there are effects that cannot be described by the OPE, like hadronic thresholds. Expected small in semileptonic decays Can we constrain them effectively? in a self-consistent way: just check the OPE predictions. E.g. leptonic vs hadronic moments. Models may also give hints of how it works Caveats? HQE depends on many parameters and we know only a few terms of the double expansion in α s and Λ/m b. 13

14 V ub (not so much) inclusive V ub from total BR(b ulν) almost exactly like incl V cb but we need kinematic cuts to avoid the ~100x larger b clν background: m X < M D E l > (M B2 -M D2 )/2M B q 2 > (M B -M D ) 2... or combined (m X,q 2 ) cuts The cuts destroy convergence of the OPE, supposed to work only away from pert singularities Rate becomes sensitive to local b-quark wave function properties like Fermi motion at leading in 1/m b SHAPE FUNCTION f(k+) 14

15 Luke, CKM workshop

16 Each strategy has pros and cons Luke, CKM workshop

17 Totally inclusive measurements impossible: needs cuts that destroy local OPE, so successful in b c l v. Still non-pert contributions can be resummed into distribution functions (Fermi motion) Different cuts lead to similar V ub. But are present theoretical errors realistic? How much can they be reduced in the future? 17

18 corrections 18 E.Gardi

19 b quark SF emerges from resummed pqcd but needs an IR prescription and power corrections for b B Aglietti et al use analytic coupling E.Gardi 19

20 SF approaches: I. DGE DGE amounts to genuine definition of Fermi motion based on dynamics of on-shell b decay + power suppr.corrs 20

21 Relies on PT to compute the SF! Additional (missing) power corrections can be included, but relation to local OPE needs to be clarified 21

22 Shape function (II) The local OPE breaks down close to boundaries. In the threshold region new dynamics emerges Expanding a shape of width ~Λ in powers of Λ/m b one gets increasingly singular terms d" ' $% E # & m * b ), de # ( 2 + m %' ' E # & m * ' b ),,%'' E b ( 2 + # & m * 1 b 0 ), / ( Describing the shape requires summing most singular terms Into a function of a single variable For m b First few moments of f(k) fixed by local OPE " dk + k n + f (k + ) 22

23 SF approaches: II. Multiscale OPE Neubert et al Many largely unconstrained Subleading SFs ~Δ=m b -2 E cut SF is parameterized 23 perturbative resummation automatic

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25 A new prudent theoretical analysis kinetic scheme. Wilsonian infrared cutoff µ~1 GeV: contribution of soft gluons absorbed into definition of OPE parameters AND distribution function(s) Fermi motion: finite m b SF, includes all available subleading corrections local OPE breaks down at high q 2 : need to model the tail, consistent with positivity, WA naturally emerge. Triple differential distribution including all known pert and nonpert effects, c++ code designed as generator published in PG, P.Giordano, G.Ossola, N.Uraltsev, JHEP10(2007)058 25

26 Perturbative corrections In the kin scheme soft gluon emission is inhibited the spectrum has only collinear singularities no resummation of collinear logs (unnecessary) complete implementation of O(α s 2 β 0 ) corrections -5% in V ub PG,Gardi,Ridolfi 26

27 Fermi motion (I) Leading SF resums leading twist effects, m b universal, q 2 indep Finite m b distribution functions include all 1/m b effects, non-universal no need for subleading SFs This factorization formula perturbatively defines the distribution functions see also Benson, Bigi, Uraltsev for bsγ 27

28 Fermi motion (II) crucial role of moments fit local OPE prediction i=1,2,3 i=1,2,3 The variance gets negative Importance of subleading effects 28

29 The high q 2 tail Higher dimensional operators are not suppressed at high q 2 leading to pathological features. Origin in the non-analytic square root Model I OPE In the integrated rate the 1/m b 3 singularity is removed by the WA operator: needs modelling for q 2 spectrum WA matrix element B WA parameterizes global properties of the tail 29

30 High q 2 (II) Two models of the high q 2 tail q 2 >q 2 * plus a δ(q 2 -m b2 ) for the rest of WA 8.5 < q 2 * <13.5 GeV 2 NB: O(1) mixing between WA and Darwin ops. The isosinglet part of WA can be substantial. Tail analysis is complementary to B + /B d b Σ q u q 30

31 Functional forms About 100 forms considered, large variety. Small uncertainty (1-2%) on V ub 31

32 Theoretical errors Parametric errors generally dominant, in particular m b, 3-4% Perturbative corrections 2-3% Functional form 1-2% Modelling of the q 2 tail and WA depending on cut from 0 to 7%. WA tends to decrease V ub A= M X cut Belle, B= (M x,q 2 ) cut Belle+Babar, C= E l cut Babar Overall theory errors are 5-9%, depending on the cuts. 32

33 Theory errors only This employs the HFAG average for m b etc (bsγ included) Preliminary HFAG Average V ub = (3.98 ± "0.35 ) #10 "3 33

34 Constraining Weak Annihilations An example Preliminary Babar analysis of the q 2 spectrum seems to suggest a small WA contribution and V ub ~

35 V ub determinations with B WA (1GeV)= All determinations consistent with UTfit 35

36 Summary New, relatively simple method available, theory errors investigated in detail high q 2 determination has larger errors, WA tend to decrease V ub possibility to constrain WA using q 2 spectrum + B + /B 0 V ub best determination at present is M X cut, compatible with exclusive and marginally with indirect (fit) determinations. Further improvements will come from an upper cut on q 2. 36

37 Prospects Improvements from knowing the b mass and the OPE pars more precisely lattice, moments, pert calculations, goal 20 MeV for m b from studying all the spectra of b u l v to constrain WA and the SFs from more inclusive measurements to minimize dependence on functional forms AND m b Learning from data a 2% error at super B might be possible 37

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39 Main theoretical desiderata know the b mass and the OPE pars precisely lattice, b c and bsγ moments, pert calculations, goal 20 MeV for m b study all the spectra of b u l v to constrain WA and the SFs, complementary to OPE constraints be as inclusive as possible to minimize dependence on functional forms Present parametric is 3.5% with δm b ~40 MeV, dominates cleanest cuts An almost inclusive measurement is less sensitive to m b. From b c experience, duality violation should be small Therefore, a 2% goal on V ub seems to be realistic. 39

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41 b ulv exclusive There is NO normalization of form f.s from HQ symmetry New first unquenched results lattice errors still ~11-15% Sum rules good at low q 2 lattice at high q 2 : complement each other q 2 extrapolation from theory bounds plus data: FF normalization at 1 point is sufficient Ball-Zwicky, Becher-Hill etc Lattice (distant) goal is 5-6% New strategy using combination of rare B,D decays Grinstein& Pirjol 41

42 V ub from B π l ν full range FF calculation Ball-Zwicky q 2 < 16 HPQCD q 2 > 16 FNAL q 2 > 16 APE q 2 > 16 V ub [10-3 ] Unquenched results probably not yet mature: handle with care 42

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44 The disadvantage of cuts The local OPE breaks down close to boundaries: unreliable unless cut is low enough. In the endpoint region E γ ~ 2 GeV new dynamics: distribution function (Shape Function=SF) and Sudakov resummation. For m b First few moments of f(k) fixed by local OPE OPE certainly valid for E cut ~ GeV. Experiments cut at GeV. What happens in between? Extrapolation to 1.6 GeV is now performed by HFAG using b clv input small 2% error counted in the experimental budget 44

45 Strong dependence on the value of m b known with about MeV uncertainty Experimental situation far from settled! 45

46 Weak annihilation coefficient of Darwin operator adding 1loop corrections b Σ q u µ-dep of WA sets natural scale of q non-factorizable flavor-singlet contributions to B WA see PG,Ossola,Uraltsev BAD: WA 3% in rate but gets enhanced in phase space corners GOOD: WA small but can be experimentally constrained,not only in B + /B 0 46