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

SLIDE 1

Precision Determination of |Vub|

Gil Paz Institute for Advanced Study, Princeton

SLIDE 2

Motivation

ρ

1
0.5

0.5 1 1.5 2

η

1.5
1
0.5

0.5 1 1.5

2

φ

1

φ

3

φ

ρ

1
0.5

0.5 1 1.5 2

η

1.5
1
0.5

0.5 1 1.5

3

φ

3

φ

2

φ

2

φ

d

m ∆

K

ε

K

ε

d

m ∆ &

s

m ∆

cb

/V

ub

V

1

φ sin2

< 0

1

φ

sol. w/ cos2

(excl. at CL > 0.95)

excluded area has CL > 0.95 excluded at CL > 0.95

BEAUTY 2006

CKM

f i t t e r

2σ “tension” between sin 2φ1 and |Vub|: Measured |Vub| = (4.10 ± 0.09 ± 0.39) · 10−3 Fit |Vub| = (3.59+0.17

−0.18) · 10−3

Inclusive |Vub| gives the smallest error How is |Vub| determined from ¯ B → Xul¯ ν decays?

CKM 2006: Precision Determination of |Vub| - Gil Paz 2

SLIDE 3

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 = 2v − n = (1, 0, 0, −1)]

Motivates:

Pl = MB − 2El, ¯ n · P = P− = EX + | PX|, n · P = P+ = EX − | PX|

Exact triple rate: y = (P− − P+)/(MB − P+)

d3Γu dP+ dP− dPl = G2

F |Vub|2

16π3 (MB − P+)

(P− − Pl)(MB − P− + Pl − P+) ˜

W1 +(MB − P−)(P− − P+) ˜ W2 2 + (P− − Pl)(Pl − P+)

y

4 ˜ W3 + ˜ W4 + 1 y ˜ W5

Simplest phase space:

M2

π

P− ≤ P+ ≤ Pl ≤ P− ≤ MB

No explicit dependence on mb ! Can predict partial rates instead of

fractions (Pedestrian introduction to inclusive |Vub|, chapter 1 of GP hep-ph/0607217)

CKM 2006: Precision Determination of |Vub| - Gil Paz 3

SLIDE 4

Kinematics

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

✁

✂ ✄ ☎ ✆ ✝

✁

✂ ✄ ☎ ✆ ✝ ✞ ✟ ✠ ✡ ☛ ☞ ✌ ✞ ✟ ✠ ✡ ☛ ☞ ✌

P+P− = M2

X

q2 = (MB − P−)(MB − P+)

Experimental cuts ⇒

P+ ∼ ΛQCD ∼ 0.5 GeV P− ∼ mb ∼ 5 GeV

In order to calculate d3Γ we need to know ˜

Wi

CKM 2006: Precision Determination of |Vub| - Gil Paz 4

SLIDE 5

Dynamics - OPE region

If we had no charm background...

Integrate over P+, P− up to MB, and use HQET based OPE ˜ Wi ∼ c0O0 + c2 O2 m2

b

+ c3 O3 m3

b

+ · · ·

ci calculable in PT:

– c0 known at O(αs) (De-Fazio, Neubert ’99) – c2 known at O(α0

s) (Blok, Koyrakh, Shifman, Vainshtein ; Manohar,

Wise ’93) – c3 known at O(α0

s) (Gremm, Kapustin ’96)

– c4 known at O(α0

s) (Dassinger, Mannel, Turczyk ’06)

Oi are HQ parameters, taken from experiment:

– O0 = 1 – O2 → µ2

π, µ2 G = [(M∗ B)2 − (MB)2]/4

– O3 → ρ3

LS, ρ3 D

OPE works very well for ¯

B → Xc l−¯ ν ⇒ Error on |Vcb| is 2%, know HQ parameters

Similar OPE for total ¯

B → Xsγ rate (almost..), which we can’t measure.

CKM 2006: Precision Determination of |Vub| - Gil Paz 5

SLIDE 6

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/mb expansion, calculated using SCET:

˜ Wi ∼ Hu · J ⊗ S + 1 mb

k

hk

u · jk u ⊗ sk u + · · ·

H - physics at scale µh ≥ mb - Calculable in PT

J - physics at scale µi ∼ mbΛQCD - Calculable in PT S - physics at scale µ0 ∼ ΛQCD - Non perturbative function

For ¯

B → Xsγ near endpoint: dΓ dE ∼ Hs · J ⊗ S + 1 mb

k

hk

s · jk s ⊗ sk s + · · ·

CKM 2006: Precision Determination of |Vub| - Gil Paz 6

SLIDE 7

Dynamics - SF Region

Currently:

– Hu known at O(αs) (Bauer, Manohar ’03; Bosch, Lange, Neubert, GP ’04) – Hs known at O(αs) (Neubert ’04) – J known at O(α2

s) (Becher, Neubert ’06)

– hk

u · jk u known at O(α0 s) and sk u classified

(K.S.M. Lee, Stewart ’04; Bosch, Neubert, GP ’04; Beneke, Campanario, Mannel, Pecjak ’04; Earlier partial studies) – Q7γ − Q7γ contribution: hk

s · jk s known at O(α0 s) and sk s classified

(Loc. cit.) – The rest of sk

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 ↔ mb, known at O(α2

s) (Neubert ’04)

Second moment of S ↔ µ2

π, known at O(α2 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

SLIDE 8

BLNP Approach (2005): Principles

BLNP approach (Lange, Neubert, GP [PRD 72, 073006 (2005)]):

Use all that we know (2005) about ¯ B → Xul¯ ν and ¯ B → Xsγ: – LO in 1/mb: Hu, Hs, J at O(αs): ˜ W (0)

1

(P+, y) = Uy(µh, µi)Hu(y, µh)

P+

dˆ ω ymbJ(ymb(P+ − ˆ ω), µi) ˆ S(ˆ ω, µi) – 1/mb subleading SFs at O(α0

s):

˜ W hadr(1)

1

(P+, y) = Uy(µh, µi) MB − P+

(P+ − ¯

Λ) ˆ S(P+) + 2 ˆ t(P+) + (ˆ u(P+) − ˆ v(P+))(1 − y) y

– Known 1/mb · αs terms from OPE (convoluted with ˆ

S): ˜ W kin(1)

1

(P+, y) = Uy(µh, µi) (MB − P+) CF αs(¯ µ) 4π

P+

dˆ ω ˆ S(ˆ ω, µi)f( P+ − ˆ ω MB − P+ , y) – Known 1/m2

b terms from OPE (convoluted with ˆ

S): ˜ W hadr(2)

1

(P+, y) = Uy(µh, µi) (MB − P+)2

4λ1 − 6λ2

3y2 − λ1 + 3λ2 3

ˆ

S(P+, µi)

CKM 2006: Precision Determination of |Vub| - Gil Paz 8

SLIDE 9

BLNP Approach (2005): Principles

Similar expansion can be constructed for ¯

B → Xsγ

Absorb the SSF into the LO SF without changing the moment expansion:

ˆ S(ˆ ω) ≡ ˆ S(ˆ ω) + 2(¯ Λ − ˆ ω) ˆ S(ˆ ω) − ˆ t(ˆ ω) + ˆ u(ˆ ω) − ˆ v(ˆ ω) mb ⇒ dΓs dEγ = · · · ˆ S(ˆ ω, µi)

Extract ˆ

S from ¯ B → Xsγ and use as input for ¯ B → Xu l−¯ ν

Model subleading SFs using moment constraints
Subleading SFs: 3 functions, 9 models each, scan over 93 = 729

combinations

0.2 0.4 0.6 0.8 1 1.2 1.4 2 1.5 1 0.5 0.5 1

✁

✂ ✄ ☎ ✆ ✝ ✞ ✟ ✠ ✡ ✞ ☛ ☞

BLNP formalism smoothly and unambiguously interpolates between OPE

and SF regions

CKM 2006: Precision Determination of |Vub| - Gil Paz 9

SLIDE 10

BLNP Approach (2005): Experimental Cuts

Lepton Energy endpoint: El > 2.31 GeV

Γ(0)

u

+ Γkin(1)

u

+ Γhadr(1)

u

+ Γkin(2)

u

+ Γhadr(2)

u

= 6.810 + 0.444 − 3.967 + 0.042 − 0.555 |Vub|2 ps−1

P+ spectrum: P+ < (M2

D/MB) ≈ 0.66 GeV

Γ(0)

u

+ Γkin(1)

u

+ Γhadr(1)

u

+ Γkin(2)

u

+ Γhadr(2)

u

= 53.225 + 4.646 − 11.862 + 0.328 − 0.227 |Vub|2 ps−1

MX spectrum: MX < MD ≈ 1.87 GeV

Γ(0)

u

+ Γkin(1)

u

+ Γhadr(1)

u

+ Γkin(2)

u

+ Γhadr(2)

u

= 58.541 + 8.027 − 9.048 + 2.100 − 0.318 |Vub|2 ps−1

CKM 2006: Precision Determination of |Vub| - Gil Paz 10

SLIDE 11

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: El cut, MX cut, MX & q2 cut, P+ cut – BaBar: Smax

H

& El cut, MX & q2 cut, El cut

HFAG average (ICHEP 2006): |Vub| = (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

SLIDE 12

Improved |Vub|

Today! Eliminate WA error

– Cut on high q2 < q2

max e.g. q2 max = (MB − MD)2, combined with MX

r 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:

– Q7γ for ¯ B → Xsγ is known at O(α2

s), other ops. are being calculated

Once they are known, want ¯ B → Xu l−¯ ν at O(α2

s):

”Only” need Hu at O(α2

s) ⇒ full 2 loop inclusive |Vub|

– Subleading SFs at order O(αs) ⇔ OPE at O(αs) – Can we find a way to extract subleading SFs from data? – Complete subleading SF basis for ¯ B → Xsγ:

CKM 2006: Precision Determination of |Vub| - Gil Paz 12

SLIDE 13

Complete SSF Basis for ¯ B → Xsγ

Q7γ − Q7γ for ¯

B → Xsγ and ¯ B → Xu l−¯ ν SSF: – 1/mb correction for dΓ – SSF integrate to zero

Recent new result: αs·1/mb corrections to Γ( ¯

B → Xsγ)! (Lee, Neubert, GP: hep-ph/0609224)

See talk by M. Neubert (WG 2/3/6 joint session)
What is the impact on inclusive |Vub|?

CKM 2006: Precision Determination of |Vub| - Gil Paz 13

SLIDE 14

New SSF Impact on |Vub| - Preliminary!

New SSF for ¯

B → Xsγ, e.g.

dω e− i

2 ω¯

n·xf78(ω) =

q

eq

−∞

ds

−∞

dt ¯ B|¯ h(0)...Γih(x−)¯ q(s¯ n)...Γiq(t¯ n)| ¯ B Contribution to dΓs/dEγ: dΓs dEγ ∝ 4παs mb · ˆ f78(MB − 2Eγ) – New SSF do not integrate to zero – Harder to estimate moments – E.g. zeroth moment of f78: non local matrix element of a 4 − q op.

New contributions to the rate Γ( ¯

B → Xsγ) are important They are the only 1/mb correction to the rate

New contributions to dΓs/dEγ have to ”compete” with known SSF not

suppressed by αs (t, u, v) and other αs·1/mb corrections

Maybe more important for cuts with high values of P+
For cuts with lower values of P+ effect probably already included in SSF

error

CKM 2006: Precision Determination of |Vub| - Gil Paz 14

SLIDE 15

Inclusive |Vub|: Comparison of Approaches

BLNP approach is based on more than 12 years of heavy quark

expansion(s) (HQET, SCET). It is the most comprehensive approach

There are other approaches used by HFAG: “BLL”, “DGE”, “LLR”

– |Vub| = (5.02 ± 0.26 ± 0.37) · 10−3 (Bauer, Ligeti, Luke ’01) – |Vub| = (4.46 ± 0.20 ± 0.20) · 10−3 (Andersen, Gardi ’05) – |Vub| = (4.43 ± 0.45 ± 0.29) · 10−3 (Leibovich, Low, Rothstein ’00) – |Vub| = (4.49 ± 0.19 ± 0.27) · 10−3 (Lange, Neubert, GP ’05)

All approaches seem to agree (is it a result of Γu ≈ Γ(0)

u ?)

But central values are not the whole story...
It is time to take a critical look at the error bars!

– BLL: Considering theoretical advances in control over LO SF and SSF, should reevaluate SF(s) sensitivities(s) – DGE: No power corrections are included or estimated! – (LLR measurement would become obsolete with experimental implementation of new weight functions)

Important to resolve considering the 2σ “tension” between sin 2φ1 and

|Vub|

CKM 2006: Precision Determination of |Vub| - Gil Paz 15

SLIDE 16

Inclusive |Vub|: Summary

Impressive improvement in determination of |Vub|

Result of hard experimental and theoretical work

Error on inclusive |Vub|: 18% in PDG 2004 ⇒ 8% in PDG 2006
Improve |Vub| today!

– Cut on high q2 to eliminate WA – Advanced two loop relations between ¯ B → Xu l−¯ ν and ¯ B → Xsγ

New SSF for ¯

B → Xsγ

Need to compare approaches: assumptions, perturbative corrections, non

perturbative corrections

More room for theoretical improvement

CKM 2006: Precision Determination of |Vub| - Gil Paz 16