QCD for B Physics Hiroyuki Kawamura (RIKEN) KEK Theory meeting “Toward the New Era of Particle Physics ” Dec.12 2007
B Physics Flavor mixing + CP violation via weak interaction NP search by (over)constraining “unitarity triangle” HFAG: LP07
QCD for B Physics • Extraction of |VCKM| and φCP from hadronic weak decays requires a good understanding of QCD effects. ex. (semi-leptonic decay) • QCD effects are even necessary for “direct CP violation” — phase is detected through quantum interference ex. (hadronic decay) + T (tree) P (penguin)
Theoretical tools Light flavor symmetry: isospin, SU(3) symmetry, … ex. isospin analysis for B → ππ etc. MNP Λ/mb expansion ↔ separation of scales (factorization) This talk Mw Heavy quark effective theory (HQET) perturbative QCD factorization for inclusive semi-leptonic decay mb QCD factorization for exclusive hadronic decay μ Soft-Collinear effective theory Λ Lattice simulation Light-cone sum rule (quark hadron duality, spectral function,OPE,…)
HQET Heavy-light meson system • Can be described by an effective field theory which includes only soft modes of “large component “of Q + soft quarks + soft gluon Mw HQET field: QCD mb μ HQET Perturbative matching with full QCD (αs(mb) small) : Λ Wilson coeff. Leading term : SU(2Nf) Spin-Flavor symmetry
Exclusive semi-leptonic decays heavy-heavy form factor Isgur-Wise function At the “zero-recoil limit” :
Experimental result LP07 Extrapolation of the data to ω=1 2007 WA ⇓
Inclusive semi-leptonic decays total rate ↔ OPE (short distance expansion) 1 propagator ⇒
Moments vs. |Vcb| & HQET parameters Buchmuller & Flacher (‘05) Moment → Vcb & HQET parameters ⇓ (kinetic energy)
B→ Xulν differential rate kinematical cuts to avoid Xc background. ex. → outgoing jet has low virtuality (sensitive to soft physics) propagator non-local for b-quark residual momentum: shape function scale of outgoing jet : Factorization Korchemsky, Sterman (’94)
B→ Xulν Shape function: new non-perturbative object — ME of Light-cone non-local operator (similar to parton distribution) Strategies for extracting |Vub| from B → Xulν Lange, Neubert ,Paz (’05) 1. Fit S(ω) from B → Xsγ, use it for B → Xulν 2. Use “shape-function free ” relation between B → Xsγand B → πlν Leibovich,Low,Rothstein (‘99) ex. Lange, Neubert ,Paz (’05) : weight function (calculated at 2-loop) : residual hadronic power corrections
|Vub| from BaBar data ”SF free” analysis by Golubev, Skovpen, Luth : hep-ph/0702072v2 LLR: Leibovich,Low,Rothstein (’99) ↔ 1-loop, without rhc LNP: Lange, Neubert ,Paz (’05) error: exp.(Xulν) +exp. (Xsγ) + th.
Exclusive hadronic decays 4-fermi operators Effective Hamiltonian Naïve factorization from “color transparency” argument, but no μ dependence • QCD factorization holds for decays in which the spectator quark is absorbed into the final heavy meson. Beneke, Buchalla, Neubert, Sachrajda (’00) π T0,8 Фπ B D FB→π
B→Dπ What must be shown? — gluon exchange between (B,D) and π • Soft div. cancel among diagrams • Collinear div. absorbed into universal pion wave function — shown up to 2-loop by BBNS (’00) All-order proof was given using SCET : Bauer et al. (‘01) SCET (Soft-Collinear effective theory) HQET (soft) + collinear modes of quark & gluon + … light-like vector — systematic expansion in — soft mode decouple with col. modes from power counting → Factorization proved at the leading power in operator language
B→ππ, Kπ Key point: B → π form factor asymptotic form — end-point singularity ↔ soft nature of spectator quarks — “hard-collinear scale” 3 formalisms have been developed in recent years (1) QCDF (BBNS:’99 - ) end-point singularity included in the form factor. proof given by SCET. power corrections partly included (parameterized) NLO calculation completed B → M1 M2
B→ππ, Kπ (2) SCET (BPSR ’04-) Bauer, Pirjol, Stewart, Rothstein end-point singularity included in the form factor hard-collinear scale distinguished. → different formula from (1) power correction neglected modest → large number of fitted input (3) PQCD (’01-) Li, Kuem, Sanda, Kurimoto, Mishima, Nagashima, , , kT factorization + Sudakov →end-point suppressed power counting different from (1) & (2). more predictive power NLO calculation has started b:impact parameter
Theory (QCDF) vs. Data Beneke (Beauty06) • Good agreement with B → PP, PV data except “πK puzzle” and large direct CP of π+π- (input set S4: Beneke &Neubert (’03)) BPRS, PQCD are also good. How can different formalism can give similar prediction?
CKM phase from data Beneke (Beauty06) Average: UT fit
B meson LCWF Operator definition light-cone vector: momentum rep. momentum of light quark (at tree level) In SCET soft fields col. fields Hard radiative tail Complicated object which contains soft + “hard” dynamics
Operator relations Kodaira, Tanaka, Qiao, HK (’01) HQ symmetry + EOM → 3-body ops. “Solution” “Wandzura-Wilczek approx.” — “Twist = Dimension - Spin” is not a good quantum number — Higher dim. operators appear in IR region (at large tΛ) “enhanced power correction” Lee,Neubert,Paz (’06) Shape Function
Radiative corrections Lange & Neubert (’03), Braun et al,(03), Li & Liao (’04), Lee & Neubert (’05) 1-loop Cusp singularity → hard radiative tail non-analytic at t=0 UV & IR structure different!
Evolution equation Consistent with Lange & Neubert (’03)
Operator product expansion Separation of UV & IR behavior ⇒ OPE Interpolate B meson LCWF to HQET parameters RG evolution for LCWF 1-loop matching RG evolution for local ops. many higher-dim. ops expansion parameter
Operator product expansion (cont’d) Lee & Neubert PRD72(’05)094028 : Cut-off scheme up to dim-4 ops. → OPE + exponential ansatz “hybrid model” Our calculation (HK & K.Tanaka) MS-bar scheme up to dim-5 ops. +
Calculation 2-point + 3-point functions with non-local operators • Operator identification calculation inx-space • Keep gauge invariance explicitly background method + Fock-Schwinger gauge → decouple from Wilson line
Result dim.3 dim.4 dim.5
: decay constant Matrix elements dim.3 dim.4 dim.5 (covariant tensor formalism) “Chromo-electronic” “Chromo-magnetic”
LCWF from OPE dim.3 dim.4 dim.5 Dim.3&4 terms reproduce the results in cut-off scheme by Lee & Neubert (’05) ↔ Lattice, QCD sum rule Represented by HQET parameter: Evolution & phenomenological studies underway!
Summary • Understanding of QCD effects in B physics has been largely improved in recent years. • B meson wave function for exclusive B decay is quite different from pion wave function. different UV & IR structures • OPE for bilocal operator for B meson LCWF to dim.5 → expressed by 3 HQET parameters Model-independent study of B meson LCWF is underway. Similar analysis for shape function is ongoing.