V ub without shape functions (really!) Zoltan Ligeti, Lawrence Berkeley Lab SLAC, Dec. 7, 2001 Introduction... V ub is very important to overconstrain CKM It doesn t really matter in this program whether you measure a CP violating quantity or not. The length of a side is as good as an angle. (M. Wise, hep-ph/0111167) Semileptonic B X u l ν rate and spectra... Optimal cuts to eliminate b c background Few comments on nonleptonic B X c/s/ rate ( Campagnari) Summary I will not talk about: Exclusive decays ( Postler / Kronfeld) I will not talk about: Lepton endpoint ( Rothstein)
Inclusive semileptonic B decay Operator Product Expansion (OPE): expand decay rates in Λ QCD /m b and α s (m b ) model independent results for sufficiently inclusive observables ( ) { b quark dγ = 1 + 0 + f(λ } 1, λ 2 ) decay m b m 2 +... + α s (...) + αs(. 2..) +... b Interesting quantities computed to order α s, α 2 sβ 0, and 1/m 2 (1/m 3 used to estimate uncertainties) Good news: Total rates known at few (< 5) percent level (duality...) Improvements: better m b from Υ sum rules / moments of B decay spectra / Lattice Upsilon expansion: (Hoang, ZL, Manohar, PRL 82 277, PRD 59 074017) V ub = ( ( ) 3.04 ± 0.06 (pert) ± 0.08 (mb ) 10 3 B(B Xu l ν) 0.001 ) 1/2 1.6 ps τ B Central value: 3.24 (Bigi, Shifman, Uraltsev, ARNPS 47 591); 3.08 (Uraltsev, IJMP A14 4641) Zoltan Ligeti p.1
Inclusive B X u l ν decay and V ub Bad news: In certain restricted regions of phase space the OPE breaks down Stringent cuts required to eliminate b c background... and the troubles begin! Proposals to measure V ub : Lepton spectrum: E l > (m 2 B m2 D )/2m B Hadronic mass spectrum: m X < m D Dilepton mass spectrum: q 2 > (m B m D ) 2 q 2 (GeV 2 ) 25 20 15 10 b c allowed Ee>(m2 B -m 2 D )/2m B q 2 >(m B -m D ) 2 (GeV 2 q2 ) 25 20 15 10 b c allowed mx < m D q 2 >(m B -m D ) 2 ν 5 O theory 5 O =breaks= down 0.5 1 1.5 2 5 10 15 20 25 2 Ee (GeV) m X (GeV 2 ) Zoltan Ligeti p.2
B X u l ν spectra Three qualitatively different regions of phase space: 1) m 2 X E XΛ QCD Λ 2 QCD : the OPE converges, first few terms can be trusted 2) m 2 X E XΛ QCD Λ 2 QCD : infinite set of terms equally important, 2) m 2 X E XΛ QCD Λ 2 QCD : the OPE becomes a twist expansion 3) m X Λ QCD : resonance region cannot compute reliably Both E l > (m 2 B m2 D )/2m B and m X < m D are in (2) since m B Λ QCD m 2 D 0.8 0.6 _1 dγ Γ de l 0.4 (GeV -1 ) 0.2 1 0.8 0.6 _1 dγ _1 dγ Γ dq 2 2 0.04 Γ dm X 0.4 (GeV -2 ) (GeV -2 ) b c b c 0.2 0.02 b c 0.08 0.06 0.5 1 1.5 2 E l (GeV) 2.5 b quark decay to O(α s ) incl. Fermi-motion (model) 1 2 3 4 5 6 2 m X (GeV 2) Experiment happy Theory happy 5 10 15 20 25 q2 (GeV 2 ) Zoltan Ligeti p.3
V ub : q 2 spectrum In large q 2 region, first few terms in OPE can be trusted (Bauer, ZL, Luke, PLB 479 395) Reason: q 2 > (m B m D ) 2 cut implies E X < m D [ m 2 X E XΛ QCD ] Leading and subleading logs of x = m b /(m b q 2 ) were summed (αs n+1 ln n x, x αs n ln n x); results consistent within 1σ (Becher, Neubert, hep-ph/0105217) Unknown corrections are O(Λ QCD /m b ) 3 Weak annihilation dominates (Voloshin, PLB 515 74) Guesstimate: 2 3% of b u semileptonic rate; delta-function at maximal q 2 and maximal E l 0.5 0.4 0.3 0.2 0.1 0 WA 6 8 10 12 14 o q2 (GeV 2 ) ther O(Λ/m b ) 3 Constrain WA by comparing D 0 vs. D cut s SL widths, or V ub from B ± vs. B 0 decay G G relative error as a function of q 2 Zoltan Ligeti p.4
V ub : combine q 2 & m X cuts Can get V ub with theoretical error at 5 10% level, from up to 45% of the events (Bauer, ZL, Luke, PRD 64 113004) Such precision can be achieved even with cuts away from the b c threshold Cuts on (q 2, m 2 X ) included fraction error of V ub of b ul ν rate δm b = 80/30 MeV 6 GeV 2, m D 46% 8%/5% 8 GeV 2, 1.7 GeV 33% 9%/6% (m B m D ) 2, m D 17% 15%/12% Compared to pure q 2 cut: expansion in Λ QCD /m c m b Λ QCD /(m 2 b q2 cut) reduced uncertainties from perturbation series and nonperturbative corrections uncertainty from the b quark light-cone distribution function only turns on slowly Zoltan Ligeti p.5
Uncertainties (1): perturbation series ε ε included fraction: 1.21 G(q 2 cut, m cut ) (a) The O(ɛ) and O(ɛ 2 BLM ) contributions to G(q2 cut, m cut ), normalized to tree level result, for m cut = 1.86 GeV (solid), 1.7 GeV (short dashed), 1.5 GeV (long dashed) (b) Scale variation: the difference between the perturbative corrections to G(qcut, 2 m cut ), normalized to the tree level result, for µ = 4.7 GeV and µ = 1.6 GeV Zoltan Ligeti p.6
Uncertainties (2): b quark mass!"$#%& Fractional effect of ±80 MeV (left) and ±30 MeV (right) uncertainty in m 1S b on G(qcut, 2 m cut ) for m cut = 1.86 GeV (solid line), 1.7 GeV (short dashed line), and 1.5 GeV (long dashed line) Zoltan Ligeti p.7
Uncertainties (3): higher dimension operators (showed this before:) 3 ' ' Λ! "$#&%('*),+ #.- /10(2 Estimate of the uncertainties due to dimension-six terms in the OPE as a function of q 2 cut from weak annihilation (solid) and other operators (dashed) Effect of a model structure function on G(q 2 cut, m cut ) as a function of q 2 cut for m cut = 1.86 GeV (solid), 1.7 GeV (short dashed) and 1.5 GeV (long dashed) f(k + ) = aa Γ(a) (1 x)a 1 e a(1 x) a and x related to Λ and λ 1 Zoltan Ligeti p.8
V ub from inclusive nonleptonic decay? Disclaimer: I had not thought much about this! C. Campagnari, O. Long et al.: Can one predict B(B X c/s/ )? Some issues: Perturbation series contains logs of m W /m b ; probably better to work to fixed order (αs) 2 and get all terms, than to sum any series ( in the literature) Measurement would eliminate b uūd + s s; this is tiny as predicted by the OPE + perturbation theory makes me more and more and more worried! b c cd charmless: B(B ψ(ns)x s ) = few %, X s X d wins λ 2 20 b dg: seems OK assuming no new physics (ask Alex Kagan) b cūd followed by c d: experimental problem... No obvious show-stopper... [doesn t mean that there isn t a non-obvious one] Zoltan Ligeti p.9
Summary Situation for V ub may become similar to present V cb at or below 10% level: Inclusive: neutrino reconstruction seems crucial to obtain q 2 and m X Exclusive: possibly only when unquenched lattice gets there Optimized q 2 and m X cuts can eliminate kinematically allowed region of b c background, while keeping the theory error of V ub close to that of B(B X u l ν) Strategy: (i) reconstruct q 2 and m X ; make cut on m X as large as possible Strategy: (ii) for a given m X cut, reduce q 2 cut to minimize overall uncertainty Would reduce SM allowed range of sin 2β very significantly Zoltan Ligeti p.10
Two slides on exclusive V cb (B. Grinstein and Z.L., hep-ph/0111392)
B D ( ) l ν shapes and V cb (1) Correlation between ρ 2 and V cb F (1) is very large assume F ( ) (1) known How to best use constraints from comparing shapes of B D and B D? ρ 2 2.2 2 1.8 1.6 1.4 1.2 1 B D l ν Belle. DELPHI.. OPAL CLEO. ρ 2 1.2 1 0.8 0.6 0.4 B Dl ν CLEO 1997 CLEO 1999 BELLE 2001 0.8 0.6 0.4. ALEPH 0.2 28 30 32 34 36 38 40 42 44 46 V cb F * (1) 10 3 0.2 0-0.2 ALEPH 1997 2.5 3 3.5 4 4.5 5 V cb F(1) 10 2 Zoltan Ligeti p.11
B D ( ) l ν shapes and V cb (2) Predictions: [QCD SR: χ 2 (1) 0.04, χ 3(1) 0.02, η(1) 0.6, η (1) 0] ρ 2 F ρ 2 F = 0.20 + 0.05 ɛ 0.01 ɛ 2 BLM + 0.08 η(1) + 0.14 η (1) + 1.0 χ 2 (1) 3.0 χ 3(1) 0.02 λ 1 /GeV 2 0.19 ρ 2 A 1 ρ 2 F = 0.12 0.03 ɛ + 0.01 ɛ 2 BLM + 0.06 η(1) 0.14 η (1) Data: 0.75 χ 2 (1) + 3.0 χ 3(1) + 0.01 λ 1 /GeV 2 0.02 Fitted slope parameter CLEO BELLE B D l ν, unitarity constrained fit to ρ 2 A 1 1.67 ± 0.11 ± 0.22 1.35 ± 0.17 ± 0.19 B D l ν, linear fit to ρ 2 F 0.98 ± 0.09 ± 0.07 0.89 ± 0.09 ± 0.05 B Dl ν, unitarity constrained fit to ρ 2 F 1.30 ± 0.27 ± 0.14 1.16 ± 0.25 ± 0.15 B Dl ν, linear fit to ρ 2 F 0.76 ± 0.16 ± 0.08 0.69 ± 0.14 ± 0.09 Ultimately want to fit: ρ 2 A 1, ρ 2 F, [c2 A 1, c 2 F ], R 1, R 2 V cb Zoltan Ligeti p.12