Precision Determination of | V ub | Gil Paz Institute for Advanced - PowerPoint PPT Presentation

Precision Determination of | V ub | Gil Paz Institute for Advanced Study, Princeton

Motivation 1.5 1.5 excluded at CL > 0.95 excluded area has CL > 0.95 φ 3 ∆ m 1 1 d m & m ∆ ∆ sin2 φ s d 1 0.5 0.5 φ φ ε 2 2 K φ φ 0 0 3 η η 1 V /V ub cb -0.5 -0.5 ε K φ 2 -1 -1 CKM sol. w/ cos2 < 0 φ f i t t e r 1 (excl. at CL > 0.95) φ BEAUTY 2006 3 -1.5 -1.5 -1 -1 -0.5 -0.5 0 0 0.5 0.5 1 1 1.5 1.5 2 2 ρ ρ 2 σ “tension” between sin 2 φ 1 and | V ub | : (4 . 10 ± 0 . 09 ± 0 . 39) · 10 − 3 Measured | V ub | = (3 . 59 +0 . 17 − 0 . 18 ) · 10 − 3 Fit | V ub | = Inclusive | V ub | gives the smallest error How is | V ub | determined from ¯ B → X u l ¯ ν decays? CKM 2006: Precision Determination of | Vub | - Gil Paz 2

Kinematics • Hadronic tensor W µν in ( v, n ) basis: (Lange, Neubert, GP [PRD 72, 073006 (2005)]): v = (1 , 0 , 0 , 0) n = (1 , 0 , 0 , 1) [¯ n = 2 v − n = (1 , 0 , 0 , − 1)] • Motivates: n · P = P − = E X + | � n · P = P + = E X − | � P l = M B − 2 E l , ¯ P X | , P X | • Exact triple rate: y = ( P − − P + ) / ( M B − P + ) = G 2 F | V ub | 2 d 3 Γ u � ( P − − P l )( M B − P − + P l − P + ) ˜ ( M B − P + ) W 1 16 π 3 dP + dP − dP l ˜ W 2 � y W 4 + 1 �� W 3 + ˜ ˜ ˜ +( M B − P − )( P − − P + ) + ( P − − P l )( P l − P + ) W 5 2 4 y M 2 π • Simplest phase space: P − ≤ P + ≤ P l ≤ P − ≤ M B • No explicit dependence on m b ! Can predict partial rates instead of fractions (Pedestrian introduction to inclusive | V ub | , chapter 1 of GP hep-ph/0607217) CKM 2006: Precision Determination of | Vub | - Gil Paz 3

Kinematics 5 5 4 4 3 3 ✌ ✌ ☞ ☞ ☛ ☛ ✡ ✡ ✠ ✠ ✟ ✟ ✞ ✞ 2 2 1 1 0 0 0 0 1 1 2 2 3 3 4 4 5 5 ✂ ✂ ✝ ✝ � � ✄ ✄ ✆ ✆ ☎ ☎ ✁ ✁ q 2 = ( M B − P − )( M B − P + ) • P + P − = M 2 X • Experimental cuts ⇒ P + ∼ Λ QCD ∼ 0 . 5 GeV P − ∼ m b ∼ 5 GeV • In order to calculate d 3 Γ we need to know ˜ W i CKM 2006: Precision Determination of | Vub | - Gil Paz 4

Dynamics - OPE region • If we had no charm background... Integrate over P + , P − up to M B , and use HQET based OPE � O 2 � � O 3 � ˜ W i ∼ c 0 � O 0 � + c 2 + c 3 + · · · m 2 m 3 b b • c i calculable in PT: – c 0 known at O ( α s ) (De-Fazio, Neubert ’99) – c 2 known at O ( α 0 s ) (Blok, Koyrakh, Shifman, Vainshtein ; Manohar, Wise ’93) – c 3 known at O ( α 0 s ) (Gremm, Kapustin ’96) – c 4 known at O ( α 0 s ) (Dassinger, Mannel, Turczyk ’06) • � O i � are HQ parameters, taken from experiment: � O 0 � = 1 – B ) 2 − ( M B ) 2 ] / 4 � O 2 � → µ 2 π , µ 2 – G = [( M ∗ � O 3 � → ρ 3 LS , ρ 3 – D • OPE works very well for ¯ B → X c l − ¯ ν ⇒ Error on | V cb | is 2%, know HQ parameters • Similar OPE for total ¯ B → X s γ rate (almost..), which we can’t measure. CKM 2006: Precision Determination of | Vub | - Gil Paz 5

Dynamics - SF Region • Because of the charm background, forced into regions of phase space where HQET based OPE is not valid (”OPE breaks down”) • We do have a systematic 1 /m b expansion, calculated using SCET: 1 � ˜ h k u · j k u ⊗ s k W i ∼ H u · J ⊗ S + u + · · · m b k • H - physics at scale µ h ≥ m b - Calculable in PT J - physics at scale µ i ∼ � m b Λ QCD - Calculable in PT S - physics at scale µ 0 ∼ Λ QCD - Non perturbative function • For ¯ B → X s γ near endpoint: d Γ 1 � h k s · j k s ⊗ s k dE ∼ H s · J ⊗ S + s + · · · m b k CKM 2006: Precision Determination of | Vub | - Gil Paz 6

Dynamics - SF Region • Currently: – H u known at O ( α s ) (Bauer, Manohar ’03; Bosch, Lange, Neubert, GP ’04) – H s known at O ( α s ) (Neubert ’04) – J known at O ( α 2 s ) (Becher, Neubert ’06) u known at O ( α 0 – h k u · j k s ) and s k u classified (K.S.M. Lee, Stewart ’04; Bosch, Neubert, GP ’04; Beneke, Campanario, Mannel, Pecjak ’04; Earlier partial studies) s known at O ( α 0 – Q 7 γ − Q 7 γ contribution: h k s · j k s ) and s k s classified (Loc. cit.) – The rest of s k s are being calculated (S.J. Lee, Neubert, GP in preparation) preliminary results in hep-ph/0609224 • Relation between the two regions: – Moments of SFs related to HQ parameters, e.g.: First moment of S ↔ m b , known at O ( α 2 s ) (Neubert ’04) Second moment of S ↔ µ 2 π , known at O ( α 2 s ) (Loc. cit.) ⇒ Good knowledge of HQ parameters, constrain the SFs – Integrate over large enough regions of phase space, recover OPE result CKM 2006: Precision Determination of | Vub | - Gil Paz 7

BLNP Approach (2005): Principles • BLNP approach (Lange, Neubert, GP [PRD 72, 073006 (2005)]): Use all that we know (2005) about ¯ ν and ¯ B → X u l ¯ B → X s γ : – LO in 1 /m b : H u , H s , J at O ( α s ): � P + W (0) ˜ ω ) , µ i ) ˆ ( P + , y ) = U y ( µ h , µ i ) H u ( y, µ h ) d ˆ ω ym b J ( ym b ( P + − ˆ S (ˆ ω, µ i ) 1 0 – 1 /m b subleading SFs at O ( α 0 s ): ( P + , y ) = U y ( µ h , µ i ) t ( P + ) + (ˆ u ( P + ) − ˆ v ( P + ))(1 − y ) � � W hadr(1) ˜ ( P + − ¯ Λ) ˆ S ( P + ) + 2 ˆ 1 M B − P + y – Known 1 /m b · α s terms from OPE (convoluted with ˆ S ): � P + ( P + , y ) = U y ( µ h , µ i ) C F α s (¯ µ ) ω, µ i ) f ( P + − ˆ ω W kin(1) ˜ ω ˆ d ˆ S (ˆ , y ) 1 ( M B − P + ) 4 π M B − P + 0 b terms from OPE (convoluted with ˆ – Known 1 /m 2 S ): U y ( µ h , µ i ) � 4 λ 1 − 6 λ 2 − λ 1 + 3 λ 2 � W hadr(2) ˜ ˆ ( P + , y ) = S ( P + , µ i ) 1 ( M B − P + ) 2 3 y 2 3 CKM 2006: Precision Determination of | Vub | - Gil Paz 8

BLNP Approach (2005): Principles • Similar expansion can be constructed for ¯ B → X s γ • Absorb the SSF into the LO SF without changing the moment expansion: ω ) + 2(¯ ω ) ˆ ω ) − ˆ Λ − ˆ S (ˆ t (ˆ ω ) + ˆ u (ˆ ω ) − ˆ v (ˆ ω ) ⇒ d Γ s ˆ ω ) ≡ ˆ = · · · ˆ S (ˆ S (ˆ S (ˆ ω, µ i ) m b dE γ • Extract ˆ S from ¯ B → X s γ and use as input for ¯ B → X u l − ¯ ν • Model subleading SFs using moment constraints • Subleading SFs: 3 functions, 9 models each, scan over 9 3 = 729 combinations 1 0.5 0 ☞ � 0.5 ☛ ✞ ✡ ✠ ✟ ✞ � 1 � 1.5 � 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 � ✂ ✄ ✆ ✝ ✁ ☎ • BLNP formalism smoothly and unambiguously interpolates between OPE and SF regions CKM 2006: Precision Determination of | Vub | - Gil Paz 9

BLNP Approach (2005): Experimental Cuts • Lepton Energy endpoint: E l > 2 . 31 GeV Γ (0) Γ kin(1) Γ hadr(1) Γ kin(2) Γ hadr(2) + + + + u u u u u | V ub | 2 ps − 1 = 6 . 810 + 0 . 444 − 3 . 967 + 0 . 042 − 0 . 555 • P + spectrum: P + < ( M 2 D /M B ) ≈ 0 . 66 GeV Γ (0) Γ kin(1) Γ hadr(1) Γ kin(2) Γ hadr(2) + + + + u u u u u | V ub | 2 ps − 1 = 53 . 225 + 4 . 646 − 11 . 862 + 0 . 328 − 0 . 227 • M X spectrum: M X < M D ≈ 1 . 87 GeV Γ (0) Γ kin(1) Γ hadr(1) Γ kin(2) Γ hadr(2) + + + + u u u u u | V ub | 2 ps − 1 = 58 . 541 + 8 . 027 − 9 . 048 + 2 . 100 − 0 . 318 CKM 2006: Precision Determination of | Vub | - Gil Paz 10

BLNP Approach (2005): Results • Error Analysis – LO SF taken from experiment – Perturbative error – SSF error by varying > 700 models – WA: take as fixed % of rate • Experimental implementation: – Belle: E l cut, M X cut, M X & q 2 cut, P + cut & E l cut, M X & q 2 cut, E l cut – BaBar: S max H • HFAG average (ICHEP 2006): | V ub | = (4 . 49 ± 0 . 19 ± 0 . 27) · 10 − 3 with – 4 . 2% HQ error – 3 . 8% Theory error (Perturbative + Subleading SFs) – 1 . 9% WA CKM 2006: Precision Determination of | Vub | - Gil Paz 11

Improved | V ub | • Today! Eliminate WA error – Cut on high q 2 < q 2 max e.g. q 2 max = ( M B − M D ) 2 , combined with M X or P + cut (Lange, Neubert, GP ’05) – Loose efficiency but also the WA error and its uncertainty, Preliminary study gives smaller error with such a cut – Still waiting for experimental implementation! • Today! High precision weight functions – See talk by B.O. Lange (WG 2) – Still waiting for experimental implementation! • Future: – Q 7 γ for ¯ B → X s γ is known at O ( α 2 s ), other ops. are being calculated Once they are known, want ¯ ν at O ( α 2 B → X u l − ¯ s ): ”Only” need H u at O ( α 2 s ) ⇒ full 2 loop inclusive | V ub | – Subleading SFs at order O ( α s ) ⇔ OPE at O ( α s ) – Can we find a way to extract subleading SFs from data? – Complete subleading SF basis for ¯ B → X s γ : CKM 2006: Precision Determination of | Vub | - Gil Paz 12

Complete SSF Basis for ¯ B → X s γ • Q 7 γ − Q 7 γ for ¯ B → X s γ and ¯ B → X u l − ¯ ν SSF: – 1 /m b correction for d Γ – SSF integrate to zero • Recent new result: α s · 1 /m b corrections to Γ( ¯ B → X s γ )! (Lee, Neubert, GP: hep-ph/0609224) • See talk by M. Neubert (WG 2/3/6 joint session) • What is the impact on inclusive | V ub | ? CKM 2006: Precision Determination of | Vub | - Gil Paz 13