Inclusive determinations of V ub and V cb - a theoretical perspective

Transcription

1 Inclusive determinations of V ub and V cb - a theoretical perspective Michael Luke University of Toronto March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 1

2 Global fit, summer 4:.7 η excluded area has CL <.5 ε K m d α m s & m d ε K α CKM f i t t e r ICHEP 24.1 V ub /V cb γ β CKMfitter inputs: ρ sin 2β =.726 ±.37 V ub = (3.9 ±.8 ±.68) 1 3 (~2% uncertainty) March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 2

3 Global fit, summer 4:.7 η α CKM f i t t e r ICHEP 24 - need a better determination of Vub to check for consistency with sin 2β.1 V ub /V cb γ β CKMfitter inputs: ρ sin 2β =.726 ±.37 V ub = (3.9 ±.8 ±.68) 1 3 (~2% uncertainty) March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 3

4 Theorists love inclusive decays... B q e ν e Decay: short distance (calculable) Hadronization: long distance (nonperturbative) - but at leading order, long and short distances are cleanly separated and probability to hadronize is unity p X X dγ d(p.s.) parton model + n ( ) n ΛQCD C n m b Most of the time, details of b quark wavefunction are unimportant - only averaged properties (i.e. k 2 ) matter Γ( B X u l ν l ) = G2 F V ub 2 m 5 b 192π 3 ( α s π 21.3 ( αs π ) 2 + λ 1 9λ 2 2m 2 b Fermi motion k µ Λ QCD ( + O α 2 s, Λ3 QCD m 3 b )) b March 17, the basic theoretical tools are more than a decade old CKM 25 - Workshop on the Unitarity Triangle 4

5 Vcb: PRECISION What progress has been made (a) in the past decade? moment fits to determine nonperturbative matrix elements extensive tests of consistency (limits possible duality violations) data have improved to the level that theory is required to (ΛQCD/ mb) 3 Vub: MODEL INDEPENDENCE moved beyond lepton endpoint to theoretically cleaner cuts (hadronic invariant mass, lepton invariant mass, combined cuts, P+,...) SCET et. al.: unravels scales relevant for cut spectra, generalizes shape function analysis beyond leading order, sums Sudakov logs... theoretical errors now much better understood March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 5

6 What progress has been made (b) since CKM 3? Vcb: Moment fits are better, inconsistencies have gone away with new data, error in Vcb down slightly. Vub: Further development of SCET/subleading theory Perturbative and nonperturbative corrections & uncertainties are better understood. New (possibly large) subleading effects discovered P + cut on spectrum added to list - some useful features March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 6

7 Vcb March 17, 25 7 CKM 25 - Workshop on the Unitarity Triangle

8 Inclusive semileptonic b c decay: V cb B q e ν e Γ(B X c l ν) = G2 F V cb 2 [ 1 192π 3 (.534) ( mυ 2 ) 5 p X X March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 8

9 Inclusive semileptonic b c decay: V cb B q e ν e Γ(B X c l ν) = G2 F V cb 2 [ ( ) Λ1S MeV 192π 3 (.534) ( mυ 2 ) 5 p X X O(Λ QCD /m b ) : ~2% correction March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 9

10 Inclusive semileptonic b c decay: q e V cb B ν e Γ(B X c l ν) = G2 F V cb 2 [ ( ) Λ1S MeV 192π 3 (.534).11 ( Λ1S ( mυ 2 ) 5 p X ) 2 ( ) ( ) λ 1 λ MeV (5 MeV) 2 (5 MeV) 2 X O(Λ QCD /m b ) : O(Λ 2 QCD /m2 b ) : ~2% correction ~5-1% correction March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 1

11 Inclusive semileptonic b c decay: q e V cb B ν e Γ(B X c l ν) = G2 F V cb 2 192π 3 (.534) ( mυ 2 ) 5 p X [ ( ) ( ) Λ1S 2 ( ) ( ) Λ1S λ 1 λ MeV 5 MeV (5 MeV) 2 (5 MeV) 2 ( ) ( ) ( ) ( ) λ 1 Λ λ 2 Λ ρ 1 ρ (5 MeV) 3 (5 MeV) 3 (5 MeV) 3 (5 MeV) 3 ( ) ( ) ( ) ( ) T 1 T 2 T 3 T (5 MeV) 3 (5 MeV) 3 (5 MeV) 3 (5 MeV) 3 X O(Λ QCD /m b ) : ~2% correction O(Λ 3 QCD /m3 b ) : ~1-2% correction O(Λ 2 QCD /m2 b ) : ~5-1% correction March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 11

12 Inclusive semileptonic b c decay: q e V cb B ν e Γ(B X c l ν) = G2 F V cb 2 192π 3 (.534) ( mυ 2 ) 5 p X [ ( ) ( ) Λ1S 2 ( ) ( ) Λ1S λ 1 λ MeV 5 MeV (5 MeV) 2 (5 MeV) 2 ( ) ( ) ( ) ( ) λ 1 Λ λ 2 Λ ρ 1 ρ (5 MeV) 3 (5 MeV) 3 (5 MeV) 3 (5 MeV) 3 ( ) ( ) ( ) ( ) T 1 T 2 T 3 T (5 MeV) 3 (5 MeV) 3 (5 MeV) 3 (5 MeV) 3 ( ) ].96 ɛ.3 ɛ 2 BLM +.15 ɛ Λ1S MeV X O(Λ QCD /m b ) : ~2% correction O(Λ 3 QCD /m3 b ) : ~1-2% correction O(Λ 2 QCD /m2 b ) : ~5-1% correction Perturbative: ~ 1% This is now a PRECISION field! March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 12

13 Moments of B Decay Spectra: - like rate, moments of spectra can be calculated as a power series in α s (m b ), Λ QCD /m b, and used to determine nonperturbative parameters... this is an old game by now. fits c. 1995: _ " (GeV).6.4 " 1 [GeV 2 ] ! 1 (GeV2) (Falk, ML, Savage)) hadronic invariant mass moments ! [GeV] (Gremm, Kapustin, Ligeti, Wise) lepton energy moments March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 13

14 fits c. 24: (1) Bauer, Ligeti, ML, Manohar and Trott (up to 1/m 3 ) - fit 92 data points (spectral moments with varying lepton energy cuts - many data points strongly correlated) with 7 free parameters hadronic invariant mass moments lepton energy moments March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 14

15 fits c. 24: (2) BABAR (using results of Gambino & Uraltsev) BR M m x m3 x M1 M m2 x m 4 x E cut (GeV) E cut (GeV) hadronic invariant mass moments E cut (GeV) E cut (GeV) lepton energy moments March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 15

16 Both fits have 7 free parameters, work to.. differences are in details: O(Λ QCD /m b ) 3 expand in ΛQCD/mc or not in kinematics (to get mc) +: moves free parameter from O(1) to O(ΛQCD/mb) 3 - : introduces new expansion in ΛQCD/mc Can do fit both ways; essentially no difference in fit results mass definitions - kinetic vs. 1S. Just scheme dependence; no significant difference in fit results slightly different handling of higher orders in ΛQCD/mb fractional hadronic invariant mass moments - results differ (BABAR fits data better; related to point above?) fractional hadronic invariant mass moments intrinsically involve expansion in Λ QCD m b /m 2 - not as clean theoretically c March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 16

17 Hadronic invariant mass moments: From CKM 3: data theory - for most values of the lepton cut, measured s H is significantly higher than predicted April 8, 23 CKM Workshop, IPPP Durham 24 March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 17

18 22 March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 18

19 24 - new data March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 19

20 24 - fit based on new data March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 2

21 Excellent agreement: V cb from exclusive, m b from sum rules (Bauer, Ligeti, ML, Manohar and Trott) V cb (1-3 ) All Moments m b (GeV) Lepton Moments Hadron Moments (BABAR, using results of Gambino and Uraltsev) V cb = (41.4 ±.6 ±.1 τb ) 1 3 V cb = (41.4 ±.4 exp ±.4 HQE ±.6 th ) 1 3 m 1S b = 4.68 ±.3 GeV m b (1 GeV) = (4.56 ±.4) GeV m kin b (1 GeV) = (4.61 ±.5 exp ±.4 HQE ±.2 th ) GeV March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 21

22 Global fits also allow us to make precise predictions of other moments as a cross-check: D 3 D GeV E.7 l 1.5 GeV E1.5 l 1.6 GeV E2.3 l 1.5 GeV E2.9 l dγ de l de l = dγ de l de l dγ de l de l = dγ de l de l {.519 ±.7 (theory).5193 ±.8 (experiment) {.634 ±.8 (theory).636 ±.6 (experiment) (BABAR) (some fractional moments of lepton spectrum are very insensitive to O(1/m 3 ) effects, and so can be predicted very accurately) NB: these were REAL PREdictions (not postdictions) Hadronic physics with < 1% uncertainty! (Bauer and Trott) March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 22

23 Progress PDG (inclusives): V cb = (42 ± 2) 1 3 March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 23

24 Progress PDG (inclusives): 22 (global fits): V cb = (42 ± 2) 1 3 V cb = (4.8 ±.9) 1 3 March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 24

25 Progress PDG (inclusives): 22 (global fits): 24 (global fits): V cb = (42 ± 2) 1 3 V cb = (4.8 ±.9) 1 3 V cb = (41.4 ±.6) 1 3 (.8 ) March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 25

26 Progress PDG (inclusives): 22 (global fits): 24 (global fits): V cb = (42 ± 2) 1 3 V cb = (4.8 ±.9) 1 3 V cb = (41.4 ±.6) 1 3 (.8 ) - looks like we re hitting a wall at 1-2% error - but theory is passing consistency tests with flying colours - we should believe the error more now! - complete O(α 2, O(α s (Λ QCD /m b ) 2 s ) ) corrections can still usefully be done... hard to imagine going to (Λ QCD /m b ) 4 March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 26

27 Vub March 17, CKM 25 - Workshop on the Unitarity Triangle

28 In principle, Vub is as easy as Vcb: V ub = (3.6 ±.8 ±.8) 1 3 ( B(B Xu l ν) ps τ B ) 1/2 5 MeV uncertainty on m b (1S) perturbative uncertainty }{{} combine to a ~5% error (Hoang, Ligeti, Manohar; Uraltsev) - very clean theoretically: greatest uncertainty is b quark mass... nonperturbative effects are small... but this requires cutting out ~1 times larger background from charm March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 28

29 25 2 q 2 15 (GeV 2 ) 1 5 The Classic Method: cut on the endpoint of the charged lepton spectrum Ee (GeV) _1 dγ Γ de l (GeV -1 ) parton model kinematic limit of b c E l (GeV) 2.5 Disadvantages: " only ~1% of rate March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 29

30 25 2 q 2 15 (GeV 2 ) 1 5 The Classic Method: cut on the endpoint of the charged lepton spectrum Ee (GeV) _1 dγ Γ de l (GeV -1 ) parton model kinematic limit of b c including fermi motion (model) E l (GeV) 2.5 Disadvantages: " only ~1% of rate " " " " " sensitivity to fermi motion - local OPE breaks down March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 3

31 25 2 q 2 (GeV 2 ) mx (GeV 2 ) 1 Cutting on the hadronic invariant mass spectrum gives more rate, but has the same problem with fermi motion: (Falk, Ligeti, Wise; Dikeman, Uraltsev).8 parton model including fermi motion (model) kinematic limit of b c _1 dγ 2 Γ dm X (GeV -2 ) m X (GeV 2 ) March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 31

32 25 2 q 2 (GeV 2 ) mx (GeV 2 ) But this doesn t always happen (depends on proximity of cut to perturbative singularities)... the local OPE holds for the leptonic q 2 spectrum: (Bauer, Ligeti, ML) parton model including fermi motion (model) kinematic limit of b c March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 32

33 cut % of rate good bad 25 2 q 2 15 (GeV 2 ) Ee (GeV) E l > m2 B m2 D 2m B ~1% don t need neutrino 25 2 q 2 (GeV 2 ) s H < m 2 D ~8% lots of rate mx (GeV 2 ) 25 2 q 2 (GeV 2 ) q 2 > (m B m D ) 2 ~2% insensitive to f(k + ) mx (GeV 2 ) 25 2 q 2 (GeV 2 ) mx (GeV 2 ) Optimized cut ~45% - insensitive to f(k + ) - lots of rate - can move cuts away from kinematic limits and still get small uncertainties 25 2 q 2 (GeV 2 ) NEW! m X (GeV 2 ) P + > m 2 D /m B ~7% - lots of rate - theoretically simplest relation to b sγ March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 33

34 Theoretical Issues are much the same as in 23: fermi motion - at leading and subleading order (E l, s H, P + cuts) Weak Annihilation (WA) (all) m m 5 b b error in Vub. But restricting phase space increases this - rate is proportional to - 1 MeV error is a ~5% sensitivity - with q 2 cut, scale as ~ m 1 b perturbative corrections - known (in most cases) to (q 2, optimized q 2 s H cuts) O(α 2 s β ) - generally under control. When fermi motion is important, leading and subleading Sudakov logarithms have been resummed. (all) March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 34

35 Theoretical Issues are much the same as in 23: fermi motion - at leading and subleading order (E l, s H, P + cuts) Weak Annihilation (WA) (all) uncertainty in mb is now at 5 MeV level m m 5 b b error in Vub. But restricting phase space increases this - rate is proportional to - 1 MeV error is a ~5% sensitivity - with q 2 cut, scale as ~ m 1 b perturbative corrections - known (in most cases) to (q 2, optimized q 2 s H cuts) O(α 2 s β ) - generally under control. When Fermi motion is important, leading and subleading Sudakov logarithms have been resummed. (all) new insights into all of these March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 35

36 Theoretical Issues: f(k +) shape function f(ω) B b δ(ω i ˆD n)b B }{{} universal distribution function (applicable to all decays) b Options: (i) model Ex: f(k + ) (model) f(k + ) = N(1 x) a e (1+a)x (de Fazio and Neubert.) a, N determined by Λ, λ 1 (gets first two moments right.. but the uncertainty in f(k + ) is not simply given by the uncertainties in Λ, λ 1 ) O(Λ QCD ) k + (GeV) It is very difficult to determine theoretical uncertainties with this approach! March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 36

37 (ii) Better: determine from experiment: the SAME function determines the photon spectrum in B X s γ (at leading order in 1/m) 1 dγ ( Γ B X u l ν l ) = 4 dêl 1 dγ ( Γ dŝ B X u l ν l ) = H 1 Γ s and so can be measured from the photon spectrum in B X s γ: dγ ( B X s γ) = 2f(1 2Êγ) +... dêγ (NB must subtract off contributions of operators other than O 7 ) θ(1 2Êl ω)f(ω) dω ŝ 2 H (3ω 2ŝ H ) ω 4 θ(ω ŝ H )f(ω ˆ ) dω +... Events/1 MeV E* γ [GeV] (BELLE) March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 37 5

38 NB - the smearing approach dγ = dγ parton mb m b +ω f(ω)dω March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 38

39 NB - the smearing approach is not valid beyond tree level... dγ = dγ parton mb m b +ω f(ω)dω some of the radiative corrections which are smeared should properly be included in the renormalization of the shape function this will cancel out in the relations between spectra, but can introduce large spurious radiative corrections in intermediate results March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 39

40 (iii) Best - avoid the shape function altogether, and just relate physical quantities! (leading order shape function cancels out between spectra) ex: mb M dγ u ds H V ub 2 ds H V tb Vts 2 dp γ W sh ( M, P γ ) dγ s dp γ P γ m B 2E γ W sh ( M, P γ ) = θ( M P γ ) + θ(p γ M ) 3 M (2P γ M ) W has an expansion in powers of leading term known α s, Λ QCD /m B, with Pγ 3 +O(α s ) + O(Λ QCD /m B ) - (theoretical) systematic errors accumulate when you include intermediate unphysical quantities like the shape function (i.e. large perturbative corrections cancel out between spectra) - shape function can t fit true spectra, which have resonances - only makes sense when smeared over resonance region March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 4

41 (iii) Best - avoid the shape function altogether, and just relate physical quantities! (leading order shape function cancels out) ex: mb M dγ u ds H V ub 2 ds H V tb Vts 2 dp γ W sh ( M, P γ ) dγ s dp γ P γ m B 2E γ similarly, P dγ u dp + V ub 2 dp + V tb Vts 2 P dp γ W P+ ( P, P γ ) dγ s dp γ (Bosch, Neubert, Lange, Paz) P + m X E X March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 41

42 (iii) Best - avoid the shape function altogether, and just relate physical quantities! (leading order shape function cancels out) ex: mb M dγ u ds H V ub 2 ds H V tb Vts 2 outside region of shape function validity dp γ W sh ( M, P γ ) dγ s dp γ P γ m B 2E γ similarly, P dγ u dp + V ub 2 dp + V tb Vts 2 P dp γ W P+ ( P, P γ ) dγ s dp γ (Bosch, Neubert, Lange, Paz) P + m X E X - P+ cut requires B Xsγ photon spectrum over a smaller region than sh cut - not a big difference in practical terms (W, f(k + ) both suppress large Pγ region) but theoretically cleaner W sh ( M, P γ ) = θ( M P γ ) + θ(p γ M ) 3 M (2P γ M ) P 3 γ March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 42

43 Leading logs - to sum, or not? SCET allows very elegant RGE resummation: (Bauer, Fleming, ML; Bauer, Fleming, Pirjol, Stewart; Bauer, Manohar; Bosch, Neubert, Lange, Paz,... also earlier work by Korchemsky and Sterman, Akhoury and Rothstein, Leibovich, Low and Rothstein) { W NLL P + (, P γ ) = T (a) 1 + C F α s (m b ) H(a) + C [ F α s (µ i ) 4f 2 (a) ln m b( P γ ) 4π 4π at 2 loops: W (α2 s ) P + = C F α 2 s (m b) (4π) 2 µ 2 i a {}}{ leading log next-to-leading log NNLL [ (.83β +3.41) ln 2 m b P γ +(4.67β 19.1) ln ]} 3f 2 (a) + 2f 3 (a) m b P γ (5.19β +c ) (Hoang, Ligeti and ML) O(log 2 ) : O(log) : O(log ) = 1 :.87 : (.86.2c ) not a good expansion! - large Sudakov double logs α n s logm (m b /µ), m = n + 1,..., 2n cancel from W - log m b /µ log 3 is not large enough to justify leading log expansion - more justified to stick to fixed order perturbation theory (cf summing logs of mc/mb in exclusive B D l ν l ) March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 43 ]

44 W: Nonperturbative corrections (Bauer, ML and Mannel; Leibovich, Ligeti and Wise; Burrell, ML and Williamson; Stewart and Lee; Mannel and Tackmann; Bosch, Neubert, Lange, Paz; Beneke, Campanario and Mannel,...) they are there, and we ~understand them (not obvious 5 years ago!) March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 44

45 W: Nonperturbative corrections (Bauer, ML and Mannel; Leibovich, Ligeti and Wise; Burrell, ML and Williamson; Stewart and Lee; Mannel and Tackmann; Bosch, Neubert, Lange, Paz; Beneke, Campanario and Mannel,...) they are there, and we ~understand them (not obvious 5 years ago!) they arise at O(ΛQCD/mb), and require several new subleading shape functions (not just local operators) - so harder to constrain than for Vcb March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 45

46 W: Nonperturbative corrections (Bauer, ML and Mannel; Leibovich, Ligeti and Wise; Burrell, ML and Williamson; Stewart and Lee; Mannel and Tackmann; Bosch, Neubert, Lange, Paz; Beneke, Campanario and Mannel,...) they are there, and we ~understand them (not obvious 5 years ago!) they arise at O(ΛQCD/mb), and require several new subleading shape functions (not just local operators) - so harder to constrain than for Vcb we cannot easily extract subleading shape functions from experiment - forced to model them. Models give expected magnitude of corrections (naively, O(Λ QCD/m) could be 5% or 5%!) Comparison of different cuts indicates which are most sensitive to corrections. March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 46

47 W: Nonperturbative corrections (Bauer, ML and Mannel; Leibovich, Ligeti and Wise; Burrell, ML and Williamson; Stewart and Lee; Mannel and Tackmann; Bosch, Neubert, Lange, Paz; Beneke, Campanario and Mannel,...) they are there, and we ~understand them (not obvious 5 years ago!) they arise at O(ΛQCD/mb), and require several new subleading shape functions (not just local operators) - so harder to constrain than for Vcb we cannot easily extract subleading shape functions from experiment - forced to model them. Models give expected magnitude of corrections (naively, O(Λ QCD/m) could be 5% or 5%!) Comparison of different cuts indicates which are most sensitive to corrections. Corrections are largest for the E l endpoint spectrum (but improve as cuts are loosened), better for sh and P+ March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 47

48 W: Nonperturbative corrections (Bauer, ML and Mannel; Leibovich, Ligeti and Wise; Burrell, ML and Williamson; Stewart and Lee; Mannel and Tackmann; Bosch, Neubert, Lange, Paz; Beneke, Campanario and Mannel,...) they are there, and we ~understand them (not obvious 5 years ago!) they arise at O(ΛQCD/mb), and require several new subleading shape functions (not just local operators) - so harder to constrain than for Vcb we cannot easily extract subleading shape functions from experiment - forced to model them. Models give expected magnitude of corrections (naively, O(Λ QCD/m) could be 5% or 5%!) Comparison of different cuts indicates which are most sensitive to corrections. Corrections are largest for the E l endpoint spectrum (but improve as cuts are loosened), better for sh and P+ Weak annihilation effects can be large - wide variation in estimates of size March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 48

49 (a) spectra 2 Subleading effects (with small WA): Leading order (model) Subleading order (2 models) charm limit (b) integrated spectra March 17, 25 CKM 25 - Workshop on the Unitarity Triangle (Bosch, Neubert, Lange, Paz) 49

50 Theoretical Issues: Weak annihilation... in local OPE b (q 2, optimized q 2 s H cuts) (Bigi & Uraltsev, Voloshin, Leibovich, Ligeti, and Wise) B u soft O ( 16π 2 Λ3 QCD m 3 b factorization violation ).3 ( fb ) ( ) B2 B 1.2 GeV.1 ~3% (?? guess!) contribution to rate at q 2 =m b 2 - an issue for all inclusive determinations - relative size of effect gets worse the more severe the cut - no reliable estimate of size - can test by comparing charged and neutral B s, comparing D and Ds semileptonic widths March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 5

51 Theoretical Issues: Weak annihilation... in nonlocal OPE (E l, s H, P + cuts) phase space enhancement ( factorization violation ) b q q b O 16π 2 Λ2 QCD m 2 B B sub-subleading shape function - enhanced in shape function region to O(ΛQCD/mb) 2 - concentrated in large q 2 region - can easily be >2% shift to integrated rate for El>2.3 GeV (smaller effect for other spectra since more rate included) March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 51

52 Theoretical Issues: Weak annihilation... in nonlocal OPE (E l, s H, P + cuts) b q q March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 52 b O controversial ( 4πα s (µ i ) Λ QCD m b only subleading! Bosch, et. al.; Neubert; Beneke et. al.: colour suppression ε<<1 + no factor of 4 negligible effect (smaller than other 1/m effects) ) ɛ colour suppression - hard to power count... estimates of size vary by almost 2 orders of magnitude! Lee and Stewart: up to 18% of LEADING term for lepton endpoint! (smaller for sh and P+) - would completely mess up shape function expansion

53 cut % of rate good bad 25 2 q 2 15 (GeV 2 ) Ee (GeV) E l > m2 B m2 D 2m B ~1% don t need neutrino - depends on f(k + ) (and subleading corrections) - WA effects largest - reduced phase space - duality issues? 25 2 q 2 (GeV 2 ) s H < m 2 D ~8% lots of rate - depends on f(k + ) (and subleading corrections) - need shape function over large region 2 mx (GeV 2 ) 25 2 q 2 (GeV 2 ) q 2 > (m B m D ) 2 ~2% insensitive to f(k + ) - very sensitive to mb - WA corrections may be substantial - effective expansion parameter is 1/mc 2 mx (GeV 2 ) 25 2 q 2 (GeV 2 ) mx (GeV 2 ) Optimized cut ~45% - insensitive to f(k + ) - lots of rate - can move cuts away from kinematic limits and still get small uncertainties - sensitive to m b (need +/- 6 MeV for 5% error in best case) 25 2 q 2 (GeV 2 ) NEW! m X (GeV 2 ) P + > m 2 D /m B ~7% - lots of rate - theoretically simplest relation to b sγ depends on f(k + ) (and subleading corrections) March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 53

54 Experimental situation: ALEPH 4.12 ±.67 ±.71 L3 5.7 ± 1. ± 1.4 DELPHI 4.7 ±.65 ±.61 OPAL 4. ±.71 ±.71 CLEO (endpoint) 4.69 ±.23 ±.63 BELLE sim. ann. (m X, Q 4.75 ±.46 ±.46 BELLE (endpoint) 4.46 ±.23 ±.61 BABAR (endpoint) 4.4 ±.15 ±.44 BABAR m X 5.22 ±.3 ±.43 2 BABAR (m X, Q ) 5.18 ±.52 ±.42 2 BABAR (E l, Q ) 4.99 ±.34 ±.51 2 BELLE B reco (m X, Q ) 5.54 ±.65 ±.54 Average 4.7 ±.44 HFAG 24 2 χ /dof = 6.7/ 7 (CL = 46.5%) 2 ) V ub [ 1-3 ] quoted uncertainties in any given measurement are approaching the 1% level; theoretical and experimental uncertainties are generally comparable March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 54

55 Bottom line(s): there is no best method - each has its own sources of uncertainty local OPE: b c experience gives us confidence in framework, but we are pushing things to lower momentum scales for Vub - perturbative, nonperturbative effects are more significant nonlocal OPE: reasonable model estimates suggest things are OK, but no experimental test of framework we only believe V cb because of all the checks. Our confidence in Vub will grow if different methods give compatible results. experiments can help beat down theoretical uncertainties improved measurement of B X sγ photon spectrum - lowering cut reduces effects of subleading corrections, as well as sensitivity to details of f(k+) test size of WA (weak annihilation) effects - compare D & DS S.L. widths, extract Vub from B ± and B separately V ub wall is likely to be at the ~5% level via these methods, assuming no inconsistencies March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 55

56 Summary: Theory for V cb from inclusive decays is very mature - many cross-checks, corrections well understood spectral moments are allowing us to test theory, fix nonperturbative corrections at the (ΛQCD/ mb) 3 level uncertainties are ~2% for Vcb, ~5 MeV for mb - values are in excellent agreement with other methods probably hitting the limits of this technique Model-independent determinations of V ub are possible, but require probing restricted regions of phase space - some (but not all!) regions are sensitive to nonperturbative shape function(s) theory of q 2, combined q 2 -mx cut is on the same footing as for b c decays, but at lower momentum transfer much recent progress in theory of shape function region, but not well tested experimentally theoretical uncertainties of ~5% appear feasible March 17, 25 CKM 25 - Workshop on the Unitarity Triangle 56