1. Michael Luke, University of Toronto FPCP '02, May 18, 2002 1 Determination of |V ub |: Theoretical Issues Michael Luke University of Toronto

2. Michael Luke, University of TorontoFPCP '02, May 18, 20022 Outline 1.Introduction - why we care 2.Approaches: 3.Summary Exclusive: lattice B  º(Ω)`∫ Inclusive: B  X u `∫ + cuts (q 2, m X ) plane (sometimes less is more…) E ` endpoint

3. Michael Luke, University of TorontoFPCP '02, May 18, 20023 - the unitarity triangle provides a simple way to visualize SM relations:  already consistent at ~30% level  want to test at ≤10% level Introduction B s -B s mixing (Tevatron) _ CPV in B 0   K S |V cb |, sin 2 , |V td /V ts |: “easy” (theory and experiment both tractable) |V ub |, ,  : HARD - our ability to test CKM depends on the precision with which these can be measured

4. Michael Luke, University of TorontoFPCP '02, May 18, 20024 World Average ‘02: sin 2  =0.780  0.077: any deviation from SM will require precision measurements!

5. Michael Luke, University of TorontoFPCP '02, May 18, 20025 The problem (of course …): HADRONIC PHYSICS large model dependence Definition: for the purposes of this talk, MODEL DEPENDENT  theoretical uncertainty is NOT parametrically suppressed - theorists argue about O(1) effects MODEL INDEPENDENT  theoretical uncertainty is parametrically suppressed (typically by (  QCD /m b ) n,  s (m b ) n ) - theorists argue about O(1)  (small number) PDG: To believe small discrepancy = new physics, need model independent predictions

6. Michael Luke, University of TorontoFPCP '02, May 18, 20026 Exclusive Decays on lattice: B  º`∫, B  Ω`∫ |p º | small (currently <1 GeV) (lattice spacing errors ~ a | p º | ) - experimental data is best at low q 2 /large p º chiral extrapolation to m q ~0 quenched approximation (no light quark loops) B  º`∫ much easier than B  Ω`∫ ( Ω width vanishes in quenched approximation; chiral extrapolation tricky in unquenched) Need unquenched calculation to be model-independent. (NB unquenched calculations of f B are now being performed) Lattice issues: vanishes for m ` =0 nonperturbative: calculate on lattice

7. Michael Luke, University of TorontoFPCP '02, May 18, 20027  need to measure d  /dq 2 for B  º`∫ at high q 2 /low p 

8. Michael Luke, University of TorontoFPCP '02, May 18, 20028  quenching statistical chiral extrapolation lattice spacing matching misc. (lattice units, …) (A. El-Khadra et. al., PRD64, 014502) - except for HQET matching, all uncertainties can be reduced with (finitely) more computer time (including unquenched calculation) “... there do not appear to be any technical roadblocks to reducing the uncertainties... to a few percent or better, over the course of the present round of experiments.”  V ub ≈15-18% + quenching error

9. Michael Luke, University of TorontoFPCP '02, May 18, 20029 |V ub | from Inclusive Decays { nonperturbative corrections (free quark decay) Good: relation between & |V ub | known to ~5% … and here the troubles begin... (~100  background from charm!)  need to impose stringent cuts to eliminate charm background … Bad: can’t measure it!

10. Michael Luke, University of TorontoFPCP '02, May 18, 200210 B  X u `∫ phase space

11. Michael Luke, University of TorontoFPCP '02, May 18, 200211 perturbative singularity (real+virtual gluons) real gluon emission NO rate at parton level (purely nonperturbative) b  c allowed

12. Michael Luke, University of TorontoFPCP '02, May 18, 200212 PROBLEM:  QCD m B and m D 2 aren’t so different!  kinematic cut and singularity are perilously close …

13. Michael Luke, University of TorontoFPCP '02, May 18, 200213 f(k + ) ~ parton distribution function: nonperturbative! f(k + ) must be modeled moment of f(k + ) are related to matrix elements of local operators (constrains models) k + = k 0 + k 3 (light cone momentum) What smooths out the singularity?

14. Michael Luke, University of TorontoFPCP '02, May 18, 200214 singularity is smeared out by b quark light-cone distribution function f(k + ) rate is sensitive to details of f(k + ) unless (bad for m X <m D !) - introduces model dependence unless we know f(k + )

15. perturbative singularity (real+virtual gluons) real gluon emission NO rate at parton level (purely nonperturbative) b  c allowed

16. Michael Luke, University of TorontoFPCP '02, May 18, 200216  lepton q 2 spectrum is insensitive to Fermi motion (usual OPE holds) - eliminates model dependence! (counterintuitive … LESS inclusive = BETTER behaved)

17. Michael Luke, University of TorontoFPCP '02, May 18, 200217 (V. Barger et. al., PLB251 (1990) 629; A. Falk, Z. Ligeti and M. Wise, PLB406 (1997) 225; I. Bigi, R.D. Dikeman and N. Uraltsev, E.P.J C4 (1998) 453) Pure m X cut: - gets ~80% of B  X u `∫ rate (for ideal cut m X < m D ) BUT - current theoretical predictions are strongly model dependent (cf DELPHI determination has cut m X <1.6 GeV) - f(k + ) can be extracted from photon spectrum in B  X s  error goes from formally O(1) to O(1/m b )... but size of higher twist terms is unknown (30%?) (more on this later) Pure q 2 cut: - insensitive to f(k + )... nonperturbative effects are subleading - theory known to O(1/m b 3,  s 2 ) - leading & subleading renormalization group improvement known BUT - nonperturbative corrections are large (reduced phase space  ~O(1/m c 3 ), not O(1/m b 3 )) (Neubert, JHEP 0007:022 (2000)) - only ~20% of B  X u `∫ rate (C. Bauer, Z. Ligeti and ML, PLB479 (2000) 395) (Czarnecki and Melnikov, PRL88:131801,2002) (Neubert & Becher, hep-ph/0105217)

18. perturbative singularity (real+virtual gluons) real gluon emission NO rate at parton level (purely nonperturbative) b  c allowed Optimized Cuts (C. Bauer, Z. Ligeti and ML, hep-ph/0107074)

19. Michael Luke, University of TorontoFPCP '02, May 18, 200219 Effects of Fermi motion - do not need to know structure function well to have negligible uncertainty on |V ub | (use model to estimate sensitivity to Fermi motion, NOT to get final result!... extract f(k + ) from B  X s  ) Simple model:

20. Michael Luke, University of TorontoFPCP '02, May 18, 200220 Strategy: - combine lepton and hadron invariant mass cuts: larger rate, smaller errors than pure q 2 cut - make cut on m X as large as possible, keeping the background from B  charm under control (depends on detector resolution, modeling of B  (D, D*) `∫ ) - make q 2 cut as low as possible, keeping the contribution from Fermi motion and perturbative uncertainties small

21. Michael Luke, University of TorontoFPCP '02, May 18, 200221 m b - rate is proportional to m b 5 … kinematic cuts also depend on m b, so cut rate is MORE sensitive - need precise value of m b ! (qu: is  80 (  30) MeV error on m b realistic?.. probably not yet) perturbative corrections - known to O(  s 2  0 ) weak annihilation (WA) - a potential problem for ALL inclusive determinations which include large q 2 region (M. Voloshin, hep-ph/0106040) Additional uncertainties for optimized cuts: ~3% (?? guess!) contribution to rate at q 2 =m b 2  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

22. Michael Luke, University of TorontoFPCP '02, May 18, 200222 Representative cuts: (a) q 2 >6 GeV 2, m X <m D 46% of rate (b) q 2 >8 GeV 2, m X <1.7 GeV33% of rate (c) q 2 >11 GeV 2,m X < 1.5 GeV18% of rate UncertaintySize (in V ub )Improvement?  m b  80 MeV:RG improved sum rules, moments of B decay 7%, 8%, 10%spectra, lattice  30 MeV: 3%, 3%, 4%  s 2%, 3%, 7%full two-loop calculation 1/m b 3 3%, 4%, 8%compare B , B 0 (weak annihilation) compare S.L. width of D 0, D S, lattice

23. Michael Luke, University of TorontoFPCP '02, May 18, 200223 E ` endpoint -rate above B  charm endpoint extremely sensitive to Fermi motion (numerically, model dependence is stronger than for  (m X <m D ) ) (lowering the cut (how well is charm background understood?) reduces sensitivity to f(k + ) )

24. Michael Luke, University of TorontoFPCP '02, May 18, 200224 - as with m X spectrum, f(k + ) dependence may be eliminated by relating d  /dE e to photon spectrum in B  X s  (Neubert, PRD49 (1994) 4623) Recent progress: (1) relation between spectra worked out to NLO accuracy (subleading Sudakov logs resummed) (Leibovich, Low, Rothstein, PRD61 (2000) 053006) (2) contribution of operators other than O 7 included (large) (Neubert, hep-ph/0104280) (3) O(1/m b ) (higher twist) corrections relating f(k + ) in B  X s  to B  X u `∫ parametrized (decay Hamiltonians have different Dirac structure!) (Leibovich, Ligeti, Wise, hep-ph/0205148; Bauer, ML and Mannel, hep-ph/0205150)

25. Michael Luke, University of TorontoFPCP '02, May 18, 200225 leading twist terms … sum to f(k + ) subleading twist terms … sum to new distribution functions The effects of subleading “shape functions” are surprisingly large ….  Simple Model: subleading effects are O(20%) (in V ub ) for E cut =2.3 GeV, decrease with E cut (corresponding coefficient in B  X s  is 3)

26. Michael Luke, University of TorontoFPCP '02, May 18, 200226 Additional Caveats: (a)weak annihilation is concentrated at endpoint of E e spectrum ~3% correction to B  X`∫ rate  ~30% correction to rate in endpoint region  ~15% uncertainty in |V ub | (another example of a higher twist effect) (see also Leibovich, Ligeti, Wise, hep-ph/0205148) (b)very restricted phase space - duality problems? CLEO ‘01: exp’t Estimate of theoretical uncertainty (WA not included)

27. Michael Luke, University of TorontoFPCP '02, May 18, 200227 Summary Theory and experiment have now evolved to the level that a model- independent, precision (10% level) determination of |V ub | is possible (!) Exclusive Decays: Lattice B  º`∫ for low |p º | <1 GeV : need unquenched calculation of form factor. Other systematics appear under control... when?? Inclusive Decays: OPE/twist expansion need to design cuts that exclude b  c without introducing large uncertainties Theoretical reliability: (q 2, m X ) cut > q 2 cut > m X cut > E e cut (experimental difficulty is in (roughly) the opposite order...)

28. Michael Luke, University of TorontoFPCP '02, May 18, 200228 Experimental measurements can help beat down the theoretical errors: (a) better determination of m b (moments of B decay distributions) (b) test size of WA (weak annihilation) effects - compare D 0 & D S S.L. widths, extract |V ub | from B  and B 0 separately (c) improve measurement of B  X s  photon spectrum - get f(k + ) - 1/m b corrections?? (d) (most important) measure |V ub | in as many CLEAN ways as possible - different techniques have different sources of uncertainty (c.f. inclusive and exclusive determinations of |V cb |)