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Treating G (B Æ l n X c ) with the OPE -- Presenting it the “Royal” Way

SLAC 12/’03. Treating G (B Æ l n X c ) with the OPE -- Presenting it the “Royal” Way. Ikaros Bigi Notre Dame du Lac. Wilsonian cut-off scale ~ 1 GeV. G (H Q Æ l n X q ) = G F 2 |KM| 2 m Q 5 ( m ) /192 p 3. HQP € moments constraints systematic uncert.

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Treating G (B Æ l n X c ) with the OPE -- Presenting it the “Royal” Way

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  1. SLAC 12/’03 TreatingG(BÆln Xc) with the OPE--Presenting it the “Royal” Way Ikaros Bigi Notre Dame du Lac Wilsonian cut-off scale ~ 1 GeV G(HQÆln Xq)= GF2|KM|2mQ5( m ) /192p3 • HQP€ moments • constraints • systematicuncert. These results are not classified

  2. First Step: Width in terms of HQP Benson,Mannel, Uraltsev,IB, NuPhy. B665,367 G(BÆln Xc) =F(V(cb),HQP:mQ, mp2,…) ±1-2%,th. limiting factor: perturb. correct. to nonpert. contrib. • Caveat: do not rely on expansion in 1/mch! • do not impose constraint a priori mb-mc= <MB> - <MD>+mp2(1/2mch-1/2mb)+nonlocal op. can check it a posteriori • all order BLM and second order non-BLM contrib. to bb • nonpert. contrib. through order 1/mQ3 • including intrinsic charm (b…c)(c…b)

  3. mb,kin(m) €MS mass mb(mb) Benson,Mannel, Uraltsev,IB, NuPhy. B665,367

  4. pole mass: intrinsic uncertainty ~ LQCD • need short distance mass • MS mass mb(m): m = mb “unnaturally” high scale m << mb - diverges for m Æ 0 • prefer `low-scale running’ mass • ‘kinetic’ mass Voloshin SR dmb,kin(m)/dm= -(16/9)(aS (m)/p) -(16/9)(aS (m)/p)(m/mb) +…

  5. HQ Sum Rules • r2(m) - 1/4 = Sn |t1/2(n) |2 + 2 Sm |t3/2(m) |2 Bj 1990 • 1/2= - 2 Sn |t1/2(n) |2 + Sm |t3/2(m) |2 U 2000 • L(m) = 2 (Snen |t1/2(n) |2 + 2 Smem |t3/2(m) |2) Vo 1992 • m2p(m)/3=Snen2 |t1/2(n) |2 + 2 Smem2|t3/2(m) |2 BiSUVa 1994 • m2G(m)/3=-2Snen2 |t1/2(n) |2 + 2Smem2|t3/2(m) |2 BSU 1997 • r3D(m)/3=Snen3 |t1/2(n) |2 + 2Smem3|t3/2(m) |2ChPir 1994 • -r3LS(m)/3=-Snen3 |t1/2(n) |2 + 2Smem3|t3/2(m) |2BSU 1997 where:t1/2&t3/2denote transition amplitudes for BÆlnD(sq = 1/2 or 3/2)with excitation energyek£m • rigorous definitions, inequalities +experim. constraints

  6. BSUV, Phys.Rev. D56 (1997) 4017 … A. Hoang, hep-ph/0204299 M. Battaglia et al., hep-ph/0304132

  7. Second Step: Determining HQP On the power of the OPE • a host of observables expressed by a universal cast of Heavy Quark Parameters (=expect. values of local operators): mQ,kin, mp2, mG2, rD3, rLS3, … memento: only2local operators in O(1/mQ3 ) ! mQ,kin, mp2, mG2, …€mQ,PS, l1, l2,…: Benson et al., NuPhy. B665,367 CKM Unitarity Triangle WS Proceed., hep-ph/0304132 • energy & mass moments yield HQP to be used universally • caveat: in general not a 1-to-1 correspondence moments €HQP • need linear combinations of moments

  8. |V(cb)/0.042| = 1-0.66[mb- 4.6 GeV] + 0.39(mc-1.15 GeV) + 0.05(mG2-0.35 GeV2) + 0.013(mp2-0.4 GeV2) + 0.09(rD3-0.2 GeV3)- 0.01 (rLS3+0.15 GeV3) energy/had. mass momentsÆHQP M1(El) =G-1ÚdEl El dG/d El Mn(El) =G-1ÚdEl[El- M1(El)]ndG/d El , n > 1 M1(MX) =G-1ÚdMX2 (MX2- MD2) dG/dMX2 Mn(MX) =G-1ÚdMX2 (MX2- <MX2>)ndG/dMX2 , n > 1 `ultimately’ can determine rLS3 [& mG2 in principle] yet for now -- at least as an option -- ‘seed’mG2 & rLS3 mG2 = 0.35 GeV2, rLS3 = - 0.12 GeV 2

  9. |V(cb)/0.042| = 1-0.66[mb- 4.6 GeV] + 0.39(mc-1.15 GeV) + 0.05(mG2-0.35 GeV2) + 0.013(mp2-0.4 GeV2) + 0.09(rD3-0.2 GeV3)- 0.01 (rLS3+0.15 GeV3) |Vcb|=0.0416¥(1±0.017|exp±0.015|G(B)±0.015|HQP) Achille “us” vs. “dmb ~ 2% implyingd|Vcb| > 5%”??? low moments depend on ~ same comb. of HQP! [.=F(mb-0.65mc)] [Gpartµmb2(mb-mc)3] |V(cb)/0.042| = 1- 1.70 [<El>-1.38 GeV]- 0.075 (mc-1.15 GeV) - 0.085(mG2-0.35 GeV2) + 0.07(mp2-0.4 GeV2) - 0.055(rD3-0.2 GeV3)-0.005 (r LS3+0.15 GeV3)

  10. M1-3(El) & M1(MX) & G(BÆln Xc) = F(mb-0.65mc) • facilitates analysis for V(cb) • complicates it for V(ub) -- G(BÆln Xu) = G(mb) -- yet only as a matter of practice, not of principle! M2,3(MX) exhibit different dependence • need higher mass moments

  11. Impact of experimental cuts • Experimental cuts on energyetc. applied for practical reasons • yet they degrade -- even `corrode’ -- `hardness’ Q of transition • `exponential’ contributions exp[-cQ/mhad]missed in usual OPE expressions • quite irrelevant for Q >> mhad • yet relevant for Q ~ mhad! for B Æg Xq: Q = mb - 2 Ecut e.g.: for Ecut~ 2 GeV, Q ~ 1 GeV !

  12. earlier work by C. Bauer • considers verydifferent effects • addresses theor. uncertaint., notbiases! Pilot study (Uraltsev, IB) ~ 1.5 % shift for Ecut=2 GeV ~ 40 % shift for Ecut=2 GeV absolute bias due to experim. cut 2 different ansaetze for distribution function [curves shown for mb=4.6 GeV, mp2 = 0.45 GeV2; bias depends on HQP]

  13. terms ~ O(1/mQ3) irrelevant for this analysis • only 3 dimensional parameters: mb, mp2, Q = mb - 2 Ecut • simplescaling behaviour arises Don Benson more refined studies: at Ecut = 2 GeV mb shifted by ~ 50 MeV mp2 shifted by ~ 0.1 GeV2

  14. hadronic mass moments in B Æl n Xc • encouraging Lessons: • keep the cuts as low as possible • biasin the meas. moments induced by cuts • can be corrected for • not a pretext for inflating theor. uncert. • moments meas. as fction of cuts: important cross check!

  15. Quality control & systematic uncertainties almost alltheoretical uncertainties systematic in nature: • unknown higher order perturb. & nonperturb. contrib. • limitations to quark-had duality Overconstraints most powerful protection against ignorance • Lenin’s dictum: “Trust is good -- control is better!” • measure higher (2nd and 3rd) moments • of different types

  16. M1-3(El) & M1(MX) yield consistent values for mb-0.65mc • M1-3(El) & M1,2[,3](MX) yield consistent values for mb,mc • value of mb thus obtained consistent with value inferred from Y(4S) Æ bb • keep energy cuts as low as possible • yet analyze moments as function of (reasonable) cuts • such overconstraints provide you with measure of “sufficient inclusiveness” case study: study moment analysis as function of P* my guess: inconsistencies emerge when P*Æ 1.6 - 1.7 GeV

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