The CKM matrix and CP Violation
(in the continuum approximation)
Zoltan Ligeti
Lawrence Berkeley Lab
Lattice 2005, Dublin
July 24–30, 2005
Plan of the talk
•
Introduction
Why you might care
•
CP violation — present status
β: from b → c and b → s modes
α & γ: interesting results last year
Implications for NP in B − B mixing
•
Theory developments: semileptonic
and nonleptonic decays in SCET
Semileptonic form factors
Nonleptonic decays
•
Future / Conclusions
Z. Ligeti — p. 1
Plan of the talk
•
Introduction
Why you might care
•
CP violation — present status
β: from b → c and b → s modes
α & γ: interesting results last year
Implications for NP in B − B mixing
Precision test of CKM; search for NP
Best present α, γ methods are new
First significant constraints
•
Theory developments: semileptonic
and nonleptonic decays in SCET
Semileptonic form factors
Few applications, connections between
Nonleptonic decays
semileptonic and nonleptonic
•
Future / Conclusions
Z. Ligeti — p. 1
Why is flavor physics and CPV interesting?
– Sensitive to very high scales
¯2
(sd)
4
10
TeV,
K : 2 ⇒ ΛNP >
∼
ΛNP
¯2
(bd)
3
Bd mixing: 2 ⇒ ΛNP >
10
TeV
∼
ΛNP
– Almost all extensions of the SM contain new sources of CP and flavor violation
(e.g., 43 new CPV phases in SUSY [must see superpartners to discover it])
– A major constraint for model building
(flavor structure: universality, heavy squarks, squark-quark alignment, ...)
– May help to distinguish between different models
(mechanism of SUSY breaking: gauge-, gravity-, anomaly-mediation, ...)
– The observed baryon asymmetry of the Universe requires CPV beyond the SM
(not necessarily in flavor changing processes in the quark sector)
Z. Ligeti — p. 2
How to test the flavor sector?
• Only Yukawa couplings distinguish between generations; pattern of masses and
mixings inherited from interaction with something unknown (couplings to Higgs)
• Flavor changing processes mediated by O(100) nonrenormalizable operators
⇒ intricate correlations between different decays of s, c, b, t quarks
Deviations from CKM paradigm may result in:
– Subtle (or not so subtle) changes in correlations, e.g., B and K constraints
inconsistent or SψKS 6= SφKS
– Enhanced or suppressed CP violation, e.g., sizable SBs→ψφ or Asγ
– FCNC’s at unexpected level, e.g., B → `+`− or Bs mixing incompatible w/ SM
• Question:
does the SM (i.e., virtual W , Z, and quarks interacting through CKM
matrix in tree and loop diagrams) explain all flavor changing interactions?
Z. Ligeti — p. 3
CKM matrix and unitarity triangle
• Convenient to exhibit hierarchical structure

Vud

V =  Vcd
Vtd
Vus
Vcs
Vts


(λ = sin θC ' 0.22)
1 2
2λ
Vub
1−
λ
 
Vcb  = 
−λ
1 − 12 λ2
Vtb
Aλ3(1 − ρ − iη) −Aλ2
3

Aλ (ρ − iη)

Aλ2
 + O(λ4)
1
• A “language” to compare overconstraining measurements
(ρ,η)
∗
Vud Vub
+ Vcd Vcb∗ + Vtd Vtb∗ = 0
α
Vud Vub*
Vcd Vcb*
Goal: “redundant” measurements sensitive to different short distance physics
Vtd Vtb*
Vcd Vcb*
γ
(0,0)
CPV in SM ∝ Area
β
(1,0)
E.g.: Bd mixing and b → dγ given by different op’s in H, but both ∝VtbVtd∗ in SM
Z. Ligeti — p. 4
Tests of the flavor sector
• For 35 years, until 1999, the only unambiguous measurement of CPV was K
1.5
excluded area has CL > 0.95
∆md
γ
sin 2β
1
∆ms & ∆md
0.9
0.5
0.8
0.7
0.5
α
α
εK
0.6
η
_
η
1
γ
0
β
|Vub/Vcb|
0.4
0.3
0.2
-0.5
0.1
0
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
_
ρ
εK
α
-1
(BABAR Physics Book, 1998)
CKM
γ
fitter
LP 2005
-1.5
-1
-0.5
0
0.5
ρ
Z. Ligeti — p. 5
1
1.5
2
Tests of the flavor sector
• For 35 years, until 1999, the only unambiguous measurement of CPV was K
1.5
excluded area has CL > 0.95
∆md
γ
_
η
1
sin 2β
1
∆ms & ∆md
0.9
0.5
0.8
0.7
η
0.5
α
α
εK
0.6
γ
0
β
|Vub/Vcb|
0.4
0.3
0.2
-0.5
0.1
0
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
_
ρ
εK
α
-1
(BABAR Physics Book, 1998)
CKM
γ
fitter
LP 2005
-1.5
-1
-0.5
0
0.5
1
1.5
2
ρ
• sin 2β = 0.687 ± 0.032, order of magnitude smaller error than first measurements
Z. Ligeti — p. 5
What are we after?
• Flavor and CP violation are excellent probes of New Physics
– Absence of KL → µµ predicted charm
– K predicted 3rd generation
– ∆mK predicted charm mass
– ∆mB predicted heavy top
If there is NP at the TEV scale, it must have a very special flavor / CP structure
• What does the new B factory data tell us?
Z. Ligeti — p. 6
SM tests with K and D mesons
• CPV in K system is at the right level (K accommodated with O(1) CKM phase)
• Hadronic uncertainties preclude precision tests (0K notoriously hard to calculate)
• K → πνν:
Theoretically clean, but rates small ∼ 10−10(K ±), 10−11(KL)
−10
Observation (3 events): B(K + → π +ν ν̄) = (1.5+1.3
— need more data
−0.9 ) × 10
• D system complementary to K, B:
Only meson where mixing is generated by down type quarks (SUSY: up squarks)
CPV & FCNC both GIM and CKM suppressed ⇒ tiny in SM and not yet observed
yCP =
Γ(CP even) − Γ(CP odd)
= (0.9 ± 0.4)%
Γ(CP even) + Γ(CP odd)
• At present level of sensitivity, CPV would be the only clean signal of NP
Can lattice help to understand SM prediction for ∆mD , ∆ΓD ? (SD part for sure)
Z. Ligeti — p. 7
CP Violation
CPV in decay
• Simplest, count events; amplitudes with different weak (φk) & strong (δk) phases
|Af /Af | =
6 1: Af = hf |H|Bi =
P
iδk iφk
Ak e
e
Af = hf |H|Bi =
,
P
Ak eiδk e−iφk
• Unambiguously established by 0K 6= 0, last year also in B decays:
AK − π + ≡
Γ(B → K −π +) − Γ(B → K +π −)
Γ(B →
K −π+)
+ Γ(B →
K +π−)
= −0.115 ± 0.018
– After “K-superweak”, also “B-superweak” excluded: CPV is not only in mixing
– There are large strong phases (also in B → ψK ∗); challenge to some models
• Current theoretical understanding insufficient for both 0K
and AK −π+ to either
prove or to rule out that NP contributes
Sensitive to NP when SM prediction is model independently small (e.g., Ab→sγ )
Z. Ligeti — p. 8
CPV in interference between decay and mixing
• Can get theoretically clean information in some
0
0
cases when B and B decay to same final state
0
0
|BL,H i = p|B i ± q|B i
λfCP
q AfCP
=
p AfCP
A
B0
q/p
B0
fCP
A
Time dependent CP asymmetry:
afCP
Γ[B 0(t) → f ] − Γ[B 0(t) → f ]
2 Im λf
1 − |λf |2
=
=
sin(∆m t) −
cos(∆m t)
2
2
0
0
1
+
|λ
|
1
+
|λ
|
Γ[B (t) → f ] + Γ[B (t) → f ] | {z f }
| {z f }
Sf
Cf (−Af )
• If amplitudes with one weak phase dominate, hadronic physics drops out from λf ,
and afCP measures a phase in the Lagrangian theoretically cleanly:
afCP = Im λf sin(∆m t)
arg λf = phase difference between decay paths
Z. Ligeti — p. 9
The cleanest case: B → J/ψ KS
• Interference between B → ψK 0 (b → cc̄s) and B → B → ψK 0 (b̄ → cc̄s̄)
Penguins with different than tree weak phase are suppressed
∗
∗
[CKM unitarity: VtbVts∗ + VcbVcs
+ VubVus
= 0]
Bd
AψKS =
∗
VcbVcs
| {z }
O(λ2 )
T+
∗
VubVus
| {z }
c
b
c
W
s
d
P
d
O(λ4 )
First term second term ⇒ theoretically very clean
Bd
arg λψKS = (B-mix = 2β) + (decay = 0) + (K-mix = 0)
⇒ aψKS (t) = sin 2β sin(∆m t) with <
∼ 1% accuracy
• World average: sin 2β = 0.687 ± 0.032 — a 5% measurement!
Z. Ligeti — p. 10
g
b
ψ
c
c
KS
ψ
u,c,t
W
s
d
d
KS
SψK : a precision game
Standard model fit without SψK
1.5
excluded area has CL > 0.95
∆md
1
∆ms & ∆md
0.5
α
η
εK
γ
0
β
|Vub/Vcb|
-0.5
εK
-1
CKM
fitter
LP 2005
-1.5
-1
-0.5
0
0.5
1
1.5
2
ρ
Z. Ligeti — p. 11
SψK : a precision game
Standard model fit including SψK
First precise test of the CKM picture
1.5
Error of SψK near |Vcb| (only |Vus| better)
excluded area has CL > 0.95
∆md
1
Without Vub 4 sol’s; ψK ∗ and D0K 0 data
show cos 2β > 0, removing non-SM ray
sin 2β
∆ms & ∆md
0.5
η
Approximate CP (in the sense that all
CPV phases are small) excluded
α
εK
γ
0
β
|Vub/Vcb|
sin 2β is only the beginning
-0.5
εK
Paradigm change: look for corrections,
rather than alternatives to CKM
-1
CKM
fitter
LP 2005
-1.5
-1
-0.5
0
0.5
ρ
1
1.5
2
⇒ Need detailed tests
⇒ Theoretical cleanliness essential
Z. Ligeti — p. 11
CPV in b → s mediated decays
• Measuring same angle in decays sensitive to different short distance physics may
give best sensitivity to NP (φKS , η 0KS , etc.)
W
Amplitudes with one weak phase expected to dominate:
A=
∗
[P
VcbVcs
| {z } c
O(λ2 )
− Pt + T c ] +
∗
VubVus
[P
| {z } u
Bd − Pt + Tu]
b
SM: expect: SφKS − SψK and CφKS
s
s
d
d
<
∼ 0.05
NP: SφKS 6= SψK possible
NP: Expect different Sf for each b → s mode
NP: Depend on size & phase of SM and NP amplitude
u, c, t
g
O(λ4 )
φ
s
s
Bd b
d
Κ q
q
π s
q
s
q
s
K S
φ
K
S
d
NP could enter SψK mainly in mixing, while SφKS through both mixing and decay
• Interesting to pursue independent of present results — there is room left for NP
Z. Ligeti — p. 12
Status of sin 2βeff measurements
-1
2
-1.4
-1.2
-1
-0.8
LP 2005
PRELIMINARY
LP 2005
0.00 ± 0.23 ± 0.05
-0.14 ± 0.17 ± 0.07
-0.09 ± 0.14
-0.21 ± 0.10 ± 0.02
HFAG
LP 2005
HFAG
H F AG
0.04 ± 0.08 ± 0.06
-0.08 ± 0.07
LP 2005
0.06 ± 0.21
0.06 ± 0.18 ± 0.03
LP 2005
HFAG
0.23 ± 0.23 ± 0.13
-0.11 ± 0.18 ± 0.08
-0.02 ± 0.13
-0.56 +-00..2297 ± 0.03
LP 2005
HFAG
-0.24 ± 0.31 ± 0.15
-0.19 ± 0.39 ± 0.13
HFAG
LP 2005
-0.44 ± 0.23
0.23 ± 0.12 ± 0.07
-0.6
-0.4
0.06 ± 0.11 ± 0.07
0.14 ± 0.10
-0.10 ± 0.25 ± 0.05
LP 2005
π0 KS
K+ K- K0
-1.6
BaBar
Belle
Average
BaBar
Belle
Average
BaBar
Belle
Average
BaBar
Belle
Average
BaBar
Belle
Average
BaBar
Belle
Average
BaBar
Belle
Average
HFAG
η′ K0
φ K0
f0 KS
ω KS
LP 2005
LP 2005
LP 2005
1
KS KS KS
HFAG
LP 2005
LP 2005
LP 2005
HFAG
0
LP 2005
PRELIMINARY
0.69 ± 0.03
0.50 ± 0.25 +-00..0074
0.44 ± 0.27 ± 0.05
0.47 ± 0.19
0.30 ± 0.14 ± 0.02
0.62 ± 0.12 ± 0.04
0.48 ± 0.09
0.95 +-00..2332 ± 0.10
0.47 ± 0.36 ± 0.08
0.75 ± 0.24
0.35 +-00..3303 ± 0.04
0.22 ± 0.47 ± 0.08
0.31 ± 0.26
0.50 +-00..3348 ± 0.02
0.95 ± 0.53 +-00..1125
0.63 ± 0.30
0.41 ± 0.18 ± 0.07 ± 0.11
0.60 ± 0.18 ± 0.04 +-00..1192
0.51 ± 0.14 +-00..1018
0.63 +-00..2382 ± 0.04
0.58 ± 0.36 ± 0.08
0.61 ± 0.23
HFAG HFAG
BaBar
Belle
Average
BaBar
Belle
Average
BaBar
Belle
Average
BaBar
Belle
Average
BaBar
Belle
Average
BaBar
Belle
Average
BaBar
Belle
Average
HFAG HFAG HFAG
KS KS KS
K+ K- K0
ω KS
π0 KS
f0 KS
η′ K0
φ K0
b→ccs World Average
Cf = -Af
H F AG
HFAG
LP 2005
HFAG
LP
2005
sin(2βeff)/sin(2φ1eff)
-0.2
-0.50 ± 0.23 ± 0.06
-0.31 ± 0.17
0
0.2
0.4
0.6
0.8
1
1.2
• Largest hint of deviations from SM: Sη0KS (2σ) and SψK −hSb→si = 0.18±0.06 (3σ)
(Averaging somewhat questionable; although in QCDF the mode-dependent shifts are mostly up)
Z. Ligeti — p. 13
Status of sin 2βeff measurements
fCP
SM allowed range of ∗
|− ηfCP SfCP − sin 2β|
sin 2βeff
Cf
b → cc̄s
ψKS
< 0.01
+0.687 ± 0.032
+0.016 ± 0.046
b → cc̄d
ψπ 0
∼ 0.2
+0.69 ± 0.25
−0.11 ± 0.20
D ∗+D ∗−
∼ 0.2
+0.67 ± 0.25
+0.09 ± 0.12
D+D−
∼ 0.2
+0.29 ± 0.63
+0.11 ± 0.36
φK 0
< 0.05
+0.47 ± 0.19
−0.09 ± 0.14
η0K 0
< 0.05
+0.48 ± 0.09
−0.08 ± 0.07
K +K −KS
∼ 0.15
+0.51 ± 0.17
+0.15 ± 0.09
KS KS KS
∼ 0.15
+0.61 ± 0.23
−0.31 ± 0.17
π 0KS
∼ 0.15
+0.31 ± 0.26
−0.02 ± 0.13
f 0KS
∼ 0.25
+0.75 ± 0.24
+0.06 ± 0.21
ωKS
∼ 0.25
+0.63 ± 0.30
−0.44 ± 0.23
Dominant
process
b → sq̄q
∗
My estimates of reasonable limits (strict bounds worse, model calculations better [Buchalla, Hiller, Nir, Raz; Beneke])
• No significant deviation from SM, still there is a lot to learn from more precise data
In SM, both |SψK − Sη0KS | and |SψK − SφKS | < 0.05 [model estimates O(0.02)]
Z. Ligeti — p. 13
Model building more interesting
• The present Sη0KS
and SφKS central values can be reasonably accommodated
with NP (unlike an O(1) deviation from SψKS two years ago)
sR,L
d
(δ23
)RR
~
bR
_
~
sR
sR,L
~
sR
sR
bR
• Other constraints: B(B → Xsγ) = (3.5±0.3)×10−6
l
mainly constrains LR mass insertions
Now also B(B → Xs`+`−) = (4.5 ± 1.0) × 10−6
agrees with the SM at 20% level
⇒ new constraints on RR & LL mass insertions
d
(δ23
)RR
~
bR
Z
~
sR
bR
• Models must satisfy growing number of constraints simultaneously
Z. Ligeti — p. 14
_
~
sR
l
sR
New last year: α and γ
∗
[γ = arg(Vub
) , α ≡ π − β − γ]
α measurements in B → ππ, ρρ, and ρπ
γ in B → DK: tree level, independent of NP
[The presently best α and γ measurements were not talked about before 2003]
α from B → ππ
• Until ∼ ’97 the hope was to determine α from:
Γ(B 0(t) → π +π −) − Γ(B 0(t) → π +π −)
Γ(B 0(t)
→
π+π−)
+
Γ(B 0(t)
→
π+π−)
= S sin(∆m t) − C cos(∆m t)
arg λπ+π− = (B-mix = 2β) + (A/A = 2γ + . . .) ⇒ gives sin 2α if P/T were small
[expectation was P/T ∼ O(αs/4π)]
Kπ and ππ rates ⇒ comparable amplitudes in B → ππ with different weak phases
• Isospin
analysis: 6 measurements determine 5 hadronic parameters + weak phase
Bose statistics ⇒ ππ in I = 0, 2
Triangle relations between B +, B 0 (B −, B 0)
decay amplitudes
Z. Ligeti — p. 15
[Gronau, London]
α from B → ππ: Isospin analysis
• Tagged B → π0π0 rates are the hardest input
B(B → π 0π 0) = (1.45 ± 0.29) × 10−6
B → ππ (S/C+– from BABAR)
B → ππ (S/A+– from Belle)
Combined
no C/A00
CK M
1.2
fitter
Moriond 05
1
1 – CL
Γ(B → π 0π 0) − Γ(B → π 0π 0)
= 0.28 ± 0.39
0
0
0
0
Γ(B → π π ) + Γ(B → π π )
Need lot more data to pin down ∆α from
isospin analysis... current bound:
0.8
0.6
0.4
0
◦
|∆α| < 39 (90% CL)
CKM fit
0.2
no α meas. in fit
0
20
40
60
80
α
• Constraint on α weak (measurements 2.3σ apart):
B → π+π−
Sπ + π −
Cπ + π −
BABAR (227m)
−0.30 ± 0.17
−0.09 ± 0.15
BELLE (275m)
−0.67 ± 0.17
−0.56 ± 0.13
average
−0.50 ± 0.12
−0.37 ± 0.11
Z. Ligeti — p. 16
100
(deg)
120
140
160
180
B → ρρ: the best α at present
• Lucky2: longitudinal polarization dominates (CP -even; could be even/odd mixed)
Isospin analysis applies for each L, or in transversity basis for each σ (= 0, k, ⊥)
• Small rate: B(B → ρ0ρ0) < 1.1 × 10−6 (90% CL) ⇒ small penguin pollution
B(B→π 0 π 0 )
B(B→π + π 0 )
= 0.26 ± 0.06 vs.
B(B→ρ0 ρ0 )
B(B→ρ+ ρ0 )
30
1
CK M
(deg)
20
α – αeff
B → ρρ isospin analysis
fitter
ICHEP 2004
0.9
0.8
0.7
10
Isospin bound: |∆α| < 11◦
Sρ+ρ− yields:
α = (96 ± 13)◦
0.6
0
0.5
0.4
-10
0.3
-20
-30
< 0.04 (90% CL)
0.2
exp. upper
limit at CL=0.9
0
0.05
0.1
0.1
0.15
0.2
0 0
BR(B → ρ ρ )
0.25
Ultimately, more complicated than ππ,
I = 1 possible due to finite Γρ, giving
O(Γ2ρ/m2ρ) effects [can be constrained]
0
0.3
-5
x 10
Z. Ligeti — p. 17
[Falk, ZL, Nir, Quinn]
B → ρπ: Dalitz plot analysis
• Two-body B → ρ±π∓:
1
two pentagon relations
from isospin; would need rates and CPV in all
ρ+π −, ρ−π +, ρ0π 0 modes to get α — hard!
Aπ−ρ+ = −0.47+0.13
−0.14
Direct CPV:
Aπ+ρ− = −0.15 ± 0.09
0.75
0.5
Aρπ
+–
0.25
0
-0.25
-0.5
-0.75
3.4σ from 0, challenges some models
Interpretation for α model dependent
-0.75 -0.5 -0.25
0
0.25
0.5
0.75
1
–+
Aρπ
• Last year:
◦
Result: α = (113+27
−17 ± 6)
H F AG
ICHEP 2004
-1
-1
BABAR
1
preliminary
0.8
C.L.
Dalitz plot analysis of the interference regions in B → π +π −π 0
1σ
2σ
3σ
0.6
0.4
0.2
0
0
30
60
90
α
Z. Ligeti — p. 18
120
(deg)
150
180
Combined α measurements
• Sensitivity mainly from Sρ+ρ− and ρπ Dalitz, ππ has small effect
◦
◦
Combined result: α = (99+12
−9 ) — better than indirect fit 92 ± 15 (w/o α and γ)
1 – CL
1.5
CK M
1.2
WA
B → ππ
B → ρπ
B → ρρ
fitter
LP 2005
1
Combined
CKM fit
no α meas. in fit
0.5
sin 2β
0.6
α
0.6
γ
0
β
0.5
0.4
-0.5
0.4
0.2
0
0.8
0.7
0.8
η
1 – CL
0.9
CKM fit
1
1
0.3
0.2
-1
CKM
α from B → ππ, πρ, ρρ
fitter
LP 2005
0
20
40
60
80
α
100
120
140
160
180
-1.5
-1
(deg)
-0.5
0
0.5
ρ
Z. Ligeti — p. 19
1
1.5
0.1
2
0
γ from B ± → DK ±
• Tree level: interfere b → c (B − → D0K −) and b → u (B − → D0K −)
Need D0, D0 → same final state; determine B and D decay amplitudes from data
Many variants depending on D decay: DCP
, DCS/CA [ADS], CS/CS [GLS]
[GLW]
Sensitivity crucially depends on: rB = |A(B − → D0K −)/A(B − → D0K −)| —
• Best measurement now: D
0
0
+ −
, D → KS π π
Both amplitudes Cabibbo allowed; can integrate
over regions in mKπ+ − mKπ− Dalitz plot
◦
+14
γ = 68−15 ± 13 ± 11
[BELLE, 275 m]
γ = (67 ± 28 ± 13 ± 11)◦
[BABAR, 227 m]
σ(γ ) degree
↓
70
BABAR
60
Monte Carlo study
50
40
30
20
10
0
0
0.05
↑
0.1
0.15
BABAR
↑
0.2
BELLE
• Need more data to determine γ more precisely (and settle value of rB )
Z. Ligeti — p. 20
0.25
0.3
rB
Overconstraining the CKM matrix
• SM fit: α, β determine ρ, η nearly as precisely as all data combined
η
0.4
0.3
0.6
fitter
LP 2005
0.5
0.4
0.3
α
α
0.2
0.1
0
-0.4
γ
-0.2
0
β
0.2
0.4
0.6
0.8
1
excluded area has CL > 0.95
0.5
CKM
γ
sin 2β
η
0.6
0.7
excluded area has CL > 0.95
0.7
sin 2β ∆md
0.2
0.1
|Vub/Vcb|
ρ
CKM
∆ms & ∆md
fitter
LP 2005
εK
εK
0
-0.4
γ
-0.2
α
α
γ
0
β
0.2
0.4
0.6
0.8
ρ
• New era: constraints from angles surpass the rest; will scale with statistics
(By the time ∆ms is measured, α may be competitive for |Vtd| side)
• K , ∆md, ∆ms, |Vub|, etc., can be used to overconstrain the SM and test NP
Let’s see how it works...
Z. Ligeti — p. 21
1
The “new” CKM fit
• Include measurements that give meaningful constraints and NOT theory limited
• α from B → ρρ and ρπ Dalitz
• 2β + γ from B → D(∗)±π ∓
• γ from B → DK (with D Dalitz)
• cos 2β from ψK ∗ and ASL (for NP)
1.5
1.5
excluded area has CL > 0.95
excluded area has CL > 0.95
∆md
1
∆md
γ
1
sin 2β
sin 2β
∆ms & ∆md
∆ms & ∆md
0.5
0.5
α
γ
0
β
γ
0
|Vub/Vcb|
β
|Vub/Vcb|
-0.5
-0.5
εK
εK
α
-1
-1
CKM
fitter
LP 2005
-1.5
α
α
εK
η
η
εK
-1
-0.5
CKM
fitter
LP 2005
Fit with “traditional” inputs
0
0.5
1
1.5
2
-1.5
ρ
-1
-0.5
γ
Fit with above included
0
0.5
1
1.5
2
ρ
[+20.0]
−1
∆ms = (17.9+10.5
at 1σ [2σ]
−1.7 [−2.8] ) ps
Z. Ligeti — p. 22
[+8.6]
−1
∆ms = (17.9+3.6
−1.4 [−2.4] ) ps
Constraining NP in mixing: ρ − η view
• NP in mixing amplitude only, 3 × 3 unitarity preserved:
(SM)
⇒ ∆mB = rd2∆mB
, SψK = sin(2β+2θd), Sρρ = sin(2α−2θd), γ(DK) unaffected
Constraints with |Vub|, ∆md, SψK
Add: α, γ, 2β + γ, cos 2β
1 – CL
1.5
1
0.8
0.8
0.5
0.6
α
β
γ
0
0.4
-0.5
0.2
CKM
fitter
Lattice 05
-1
-0.5
-1
CKM
New Physics in B0B0 mixing
0
0.5
1
1.5
fitter
Lattice 05
2
0
-1.5
ρ
β
0.4
First determinations of
(ρ, η) from “effectively”
tree level processes
-0.5
-1
0.6
α
η
γ
0
-1.5
1
SM CKM Fit
1
0.5
η
1 – CL
1.5
SM CKM Fit
1
(SM) 2 2iθd
rd e
M12 = M12
-1
-0.5
0.2
New Physics in B0B0 mixing
0
0.5
1
1.5
2
0
ρ
Only the SM region left even in the presence of NP in mixing
Z. Ligeti — p. 23
[Similar fits also by UTfit]
Constraining NP in mixing: ρ − η view
• NP in mixing amplitude only, 3 × 3 unitarity preserved:
(SM)
⇒ ∆mB = rd2∆mB
, SψK = sin(2β+2θd), Sρρ = sin(2α−2θd), γ(DK) unaffected
Constraints with |Vub|, ∆md, SψK
Add: α, γ, 2β + γ, cos 2β and ASL
1 – CL
1.5
1
0.8
0.8
0.5
0.6
α
β
γ
0
0.4
-0.5
0.2
CKM
fitter
Lattice 05
-1
-0.5
-1
CKM
New Physics in B0B0 mixing
0
0.5
1
1.5
fitter
Lattice 05
2
0
-1.5
ρ
β
0.4
First determinations of
(ρ, η) from “effectively”
tree level processes
-0.5
-1
0.6
α
η
γ
0
-1.5
1
SM CKM Fit
1
0.5
η
1 – CL
1.5
SM CKM Fit
1
(SM) 2 2iθd
rd e
M12 = M12
-1
-0.5
0.2
New Physics in B0B0 mixing
0
0.5
1
1.5
2
0
ρ
• Only the SM region left even in the presence of NP in mixing
Z. Ligeti — p. 23
[Similar fits also by UTfit]
Constraining NP in mixing: rd2 − θd view
• NP in mixing amplitude only, 3 × 3 unitarity preserved:
(SM)
⇒ ∆mB = rd2∆mB
, SψK = sin(2β+2θd), Sρρ = sin(2α−2θd), γ(DK) unaffected
Constraints with |Vub|, ∆md, SψK
Add: α, γ, 2β + γ, cos 2β and ASL
1 – CL
0.8
0
0.4
SM
-100
0.2
0 0
New Physics in B B mixing
-150
0
1
2
3
4
5
6
(deg)
0.6
fitter
Lattice 05
50
0.6
0
-50
0
0.4
SM
-100
0.2
New Physics in B0B0 mixing
-150
7
0.8
100
2θd
(deg)
50
1
CKM
150
fitter
Lattice 05
100
2θd
1 – CL
1
CKM
150
-50
(SM) 2 2iθd
rd e
M12 = M12
0
rd2
1
2
3
4
rd2
5
6
7
• New data restrict rd2 , θd significantly for the first time — still plenty of room left
Z. Ligeti — p. 24
0
NP in mixing: hd − σd view
• Previous fits: |M12/M12SM| can only differ significantly from 1 if arg(M12/M12SM) ∼ 0
(SM) 2 2iθd
rd e
More transparent parameterization: M12 = M12
Modest NP contribution can still have arbitrary phase
1 – CL
CKM
150
fitter
Lattice 05
0.8
(1 + hd e2iσd )
[Agashe, Papucci, Perez, Pirjol, to appear]
180
1
160
0.9
140
0.8
0.7
120
50
0.6
σd
(deg)
100
2θd
1
(SM)
≡ M12
0
-50
0.4
SM
0.2
0 0
New Physics in B B mixing
0
1
2
0.5
80
0.4
60
-100
-150
0.6
100
3
4
rd2
5
6
7
0
0.3
40
0.2
20
0.1
0
0
0.2
0.4
0.6
0.8
1
hd
1.2
1.4
1.6
1.8
2
0
• For |hd| < 0.2, the phase σd is unconstrained; if |hd| < 0.4, σd can take half of (0, π)
Z. Ligeti — p. 25
Intermediate summary
• sin 2β = 0.687 ± 0.032
⇒ good overall consistency of SM, δCKM is probably the dominant source of CPV
⇒ in flavor changing processes
• SψK − Sη0KS = 0.21 ± 0.10 and SψK − hSb→si = 0.18 ± 0.06
⇒ Decreasing deviations from SM (same values with 5σ would still signal NP)
• AK −π+ = −0.12 ± 0.02
⇒ “B-superweak” excluded, sizable strong phases
• Measurements of α =
+12 ◦
99−9
+16 ◦
64−13
and γ =
⇒ Angles start to give tightest constraints
⇒ First serious bounds on NP in B–B mixing; ∼30% contributions still allowed
Z. Ligeti — p. 26
Theoretical developments
Significant steps toward a model independent theory of
certain exclusive decays in the mB ΛQCD limit
Factorization for B → M form factors for q 2 m2B
and certain B → M1M2 nonleptonic decays
Determinations of |Vcb| and |Vub|
• Inclusive and exclusive |Vcb| and |Vub| determinations rely on
heavy quark expansions; theoretically cleanest is |Vcb|incl
G2F |Vcb|2
Γ(B → Xc`ν̄) =
192π 3
"
mΥ
2
ν
5
(0.534)×
2
Λ1S
Λ1S
λ2
λ1
0.22
−0.011
− 0.052
− 0.071
500 MeV
500 MeV
(500 MeV)2
(500 MeV)2
λ1 Λ1S
λ2 Λ1S
ρ1
ρ2
0.006
+ 0.011
− 0.006
+ 0.008
(500 MeV)3
(500 MeV)3
(500 MeV)3
(500 MeV)3
T2
T3
T4
T1
+ 0.002
0.011
− 0.017
− 0.008
(500 MeV)3
(500 MeV)3
(500 MeV)3
(500 MeV)3
#
Λ
2
1S
0.096 − 0.030BLM + 0.015
+ ...
500 MeV
1−
−
+
+
Corrections:
O(Λ/m): ∼ 20%,
O(αs): ∼ 10%,
O(Λ2/m2): ∼ 5%, O(Λ3/m3): ∼ 1 − 2%,
Unknown terms: < few %
Matrix elements determined from fits to many shape variables
• Error of |Vcb|incl ∼ 2%!
New small parameters complicate expansions for |Vub|incl
Z. Ligeti — p. 27
Exclusive b → u decays
• In the hands of LQCD, less constraints from heavy quark symmetry than in b → c
– B → `ν̄: measures fB × |Vub| — need fB from lattice
– B → π`ν̄: useful dispersive bounds on form factors
– Ratios = 1 in heavy quark or chiral symmetry limit (+ study corrections)
• Deviations of “Grinstein-type double ratios” from unity are more suppressed:
fD s
fB
×
— lattice: double ratio = 1 within few %
⇒
fB s
fD
B(B → `ν̄)
B(Ds → `ν̄)
⇒
×
— very clean... after 2010?
B(Bs → `+`−)
B(D → `ν̄)
⇒
f (B→ρ`ν̄)
[Grinstein]
∗
f (D→K `ν̄)
× (D→ρ`ν̄) or q 2 spectra — accessible soon?
f
[ZL, Wise; Grinstein, Pirjol]
f (B→K ∗`+`−)
New CLEO-C D → ρ`ν̄ data still consistent w/ no SU (3) breaking in form factors
Could lattice do more to pin down the corrections?
Z. Ligeti — p. 28
[ZL, Stewart, Wise]
One-page introduction to SCET
• Effective theory for processes involving energetic hadrons, E Λ
[Bauer, Fleming, Luke, Pirjol, Stewart, + . . . ]
Introduce distinct fields for relevant degrees of freedom, power counting in λ
modes
fields
collinear
ξn,p, Aµn,q
qq , Aµs
qus, Aµus
soft
usoft
p2
p = (+, −, ⊥)
2
2
2
SCETI: λ =
p
Λ/E — jets (m∼ΛE)
E(λ , 1, λ)
E λ
E(λ, λ, λ)
E 2λ2
SCETII: λ = Λ/E — hadrons (m∼Λ)
E(λ2, λ2, λ2)
E 2λ4
Match QCD → SCETI → SCETII
• Can decouple ultrasoft gluons from collinear Lagrangian at leading order in λ
ξn,p =
(0)
Yn ξn,p
An,q =
(0)
Yn An,q
Yn†
Rx
Yn = P exp ig −∞ ds n · Aus(ns)
Nonperturbative usoft effects made explicit through factors of Yn in operators
New symmetries: collinear / soft gauge invariance
• Simplified / new (B → Dπ, π`ν̄) proofs of factorization theorems
Z. Ligeti — p. 29
[Bauer, Pirjol, Stewart]
Semileptonic B → π, ρ form factors
• Issues: endpoint singularities, Sudakov effects, etc.
At leading order in Λ/Q, to all orders in αs, form factors
for q 2 m2B written as (Q = E, mb; omit µ-dep’s)
[Beneke & Feldmann; Bauer, Pirjol, Stewart; Becher, Hill, Lange, Neubert]
mB fB fM
F (Q) = Ck (Q) ζk (Q) +
4E 2
p 2~ Q 2
B
p 2 ~ Λ2
M
p 2 ~ Λ2
p 2~ QΛ
Z
dzdxdr+ T (z, Q) J(z, x, r+, Q) φM (x)φB (r+)
(n)
(m)
Matrix elements of distinct d x T J (0) Lξq (x) terms (turn spectator qus → ξ)
R
4
• Symmetries ⇒ nonfactorizable (1st) term obey form factor relations
[Charles et al.]
Symmetries ⇒ 3 B → P and 7 B → V form factors related to 3 universal functions
• Relative size?
SCET: 1st ∼ 2nd
QCDF: 2nd ∼ αs×(1st)
PQCD: 1st ∼ 0
Some relations between semileptonic and nonleptonic decays can be insensitive
to this, while other predictions may be sensitive (e.g., AFB = 0 in B → K ∗`+`−?)
Z. Ligeti — p. 30
|Vub| from B → π`ν̄
• Lattice is under control for large q2 (small |~pπ |),
experiment loses a lot of statistics
dΓ(B 0 → π +`ν̄)
G2F |~
pπ |3
2
2 2
=
|V
|
|f
(q
)|
ub
+
dq 2
24π 3
Best would be to use the q 2-dependent data and
its correlation (both lattice and experiment) to get
|Vub|, reducing role of model-dependent fits
• Dispersion relation and a few points for f+(q2)
give strong constraints on shape
[Boyd, Grinstein, Lebed]
0.8
(1- q 2) f (q2 )
f = f+
0.6
B → ππ using factorization constrains |Vub|f+(0)
0.4
[Bauer et al.]
• Can combine dispersive bounds with lattice and
possibly B → ππ
0.0
[Fukunaga, Onogi; Arnesen et al.]
Z. Ligeti — p. 31
f = f0
0.2
0
5
10
15
20
q2
25
Tension between sin 2β and |Vub|?
• SM fit favors slightly smaller |Vub| than inclusive determination, or larger sin 2β
Inclusive average (error underestimated?)
(HFAG)
|Vub|incl
= (4.38 ± 0.19 ± 0.27) × 10−3
CK M
0.8
1 – CL
Lattice π`ν average
[HPQCD & FNAL from Stewart @ LP’05]
−3
|Vub| = (4.1 ± 0.3+0.7
−0.4 ) × 10
Depends on whether only q 2 > 16 GeV2 is used
1
fitter
LP 2005
CKM fit
(sin2β)
0.6
0.4
Inclusive
average
1σ
0.2
2σ
Light-cone SR [Ball, Zwicky; Braun et al., Colangelo, Khodjamirian]
−3
|Vub| = (3.3 ± 0.3+0.5
−0.4 ) × 10
0
0.3
0.35
0.4
0.45
|Vub|
0.5
-2
x 10
Statistical fluctuations? Problem with inclusive? New physics?
Precise |Vub| crucial to be sensitive to small NP entering sin 2β via mixing
• To sort this out, need precise and model independent fB and B → π form factor
Z. Ligeti — p. 32
Chasing |Vtd/Vts|: B → ργ vs. B → K ∗γ
• Factorization formula:
hV γ|H|Bi =
TiIFV
+
dx dk TiII(x, k) φB (k) φV (x) + . . .
R
[Bosch, Buchalla; Beneke, Feldman, Seidel; Ali, Lunghi, Parkhomenko]
2
1 Vtd −2
B(B → ρ γ)
= ξ (1 + tiny)
B(B 0 → K ∗0γ)
2 Vts 0
0
1 – CL
σ(ξ) = 0.2
1
No weak annihilation in B 0, cleaner than B ±
0.8
CKM fit
0.7
0.6
α
γ
0
β
0.5
[Ball, Zwicky; Becirevic; Mescia]
Conservative? ξ − 1 is model dependent
0.4
-0.5
0.3
0.2
-1
σ(ξ) = 0.2 doubles error estimate
CKM
fitter
LP 2005
Could LQCD help more?
1
0.9
0.5
η
SU (3) breaking: ξ = 1.2 ± 0.1 (CKM ’05)
[New Belle observation + Babar bound]
1.5
-1.5
-1
-0.5
BR(B0 → ρ0γ) / BR(B0 → K*0γ)
0
0.5
1
ρ
• Mild indication that ∆ms might not be right at the current lower limit?
Z. Ligeti — p. 33
1.5
0.1
2
0
B → τ ν might also precede ∆ms
• ∆ms
is not the only way to eliminate the fB
error in ∆md; fB cancels in Γ(B → τ ν)/∆md
Shown are 1 and 2σ contours with
fB = 216 ± 9 ± 21 MeV [HPQCD]
1.5
excluded area has CL > 0.95
If no exp. errors: determine |Vub/Vtd| independent of fB (left with Bd; ellipse for fixed Vcb, Vts)
CKM fit
1
B+ → τ+ν
0.5
• Nailing down fB will remain essential
α
η
If fB is known: get two circles that intersect at
α ∼ 100◦ ⇒ powerful constraints
∆md
γ
0
β
-0.5
Ru / Rt = const
-1
Recall: ∆ms remains important to constrain NP
entering Bs and Bd mixing differently (not just
to determine |Vtd/Vts|)
CKM
fitter
Lattice 2005
-1.5
-1
-0.5
Constraint from B+ → τ+ντ and ∆md
0
0.5
1
1.5
2
ρ
(B → τ ν usually quoted as upper bounds)
• Error of Γ(B → τ ν) will improve incrementally (precise only at a super B factory)
∆ms will be instantly accurate when measured
Z. Ligeti — p. 34
Photon polarization in B → K ∗γ
• SM predicts B(B → Xsγ) correctly to ∼10%; rate does not distinguish b → sγL,R
µν
SM: O7 ∼ s̄ σ Fµν (mbPR +msPL)b , therefore mainly b → sL
Photon must be left-handed to conserve Jz along decay axis
Exclusive B → K ∗γ
Inclusive B → Xsγ
γ
b
γ
s
K*
∗
... quark model (sL implies JzK = −1)
... higher K ∗ Fock states
Assumption: 2-body decay
Does not apply for b → sγg
• Only measurement so far; had been expected to give
Γ[B 0(t) → K ∗γ] − Γ[B 0(t) → K ∗γ]
Γ[B 0(t) → K ∗γ] + Γ[B 0(t) → K ∗γ]
• What is the SM prediction?
B
SK ∗γ = −2 (ms/mb) sin 2β
[Atwood, Gronau, Soni]
= SK ∗γ sin(∆m t) − CK ∗γ cos(∆m t)
What limits the sensitivity to new physics?
Z. Ligeti — p. 35
Right-handed photons in the SM
• Dominant source of “wrong-helicity” photons in the SM is O2
Equal b → sγL, sγR rates at O(αs); calculated to
[Grinstein, Grossman, ZL, Pirjol]
γ
O(αs2β0)
(brem)
Inclusively only rates are calculable: Γ22
/Γ0 ' 0.025
p
Suggests: A(b → sγR)/A(b → sγL) ∼ 0.025/2 = 0.11
g
c
b
O2
s
• Exclusive B → K ∗γ: factorizable part contains an operator that could contribute
at leading order in ΛQCD/mb, but its B → K ∗γ matrix element vanishes
Subleading order: several contributions to B 0 → K 0∗γR, no complete study yet
We estimate:
A(B 0 → K 0∗γR )
C2 ΛQCD
=O
3C7 mb
A(B 0 → K 0∗γL)
∼ 0.1
• Data: SK ∗γ = −0.13±0.32 — both the measurement and the theory can progress
Z. Ligeti — p. 36
Nonleptonic decays
Some motivations
• Two hadrons in the final state are also a headache for us, just like for you
Lot at stake, even if precision is worse
Many observables sensitive to NP — can we disentangle from hadronic physics?
– B → ππ, Kπ branching ratios and CP asymmetries (related to α, γ in SM)
– Polarization in charmless B → V V decays
• First derive correct expansion in mb ΛQCD limit, then worry about predictions
– Need to test accuracy of expansion (even in B → ππ, |~
pq | ∼ 1 GeV)
– Sometimes model dependent additional inputs needed
Z. Ligeti — p. 37
B → D (∗)π decays in SCET
• Decays to π±: proven that A ∝ F B→D fπ is the leading order prediction
Also holds in large Nc, works at 5–10% level, need precise data to test mechanism
B 0 → D+π−
B − → D0π−
SCET:
• Predictions:
B 0 → D+π−
B 0 → D0π0
B − → D0π−
B 0 → D0π0
O(1)
O(ΛQCD /Q)
B(B − → D (∗)0π −)
B(B 0 → D (∗)+π −)
O(ΛQCD /Q)
= 1 + O(ΛQCD/Q) ,
Q = {Eπ , mb,c }
data: ∼ 1.8 ± 0.2 (also for ρ)
⇒ O(30%) power corrections
[Beneke, Buchalla, Neubert, Sachrajda; Bauer, Pirjol, Stewart]
0
Predictions:
0
0
B(B → D π )
B(B 0 → D ∗0π 0)
= 1 + O(ΛQCD/Q) ,
data: ∼ 1.1 ± 0.25
Unforeseen before SCET
[Mantry, Pirjol, Stewart]
Z. Ligeti — p. 38
Color suppressed B → D (∗)0π 0 decays
• Single class of power suppressed
SCETI
operators: T
(a)
(∗)0
0
M
M ) = N0
Z
+
+
dz dx dk1 dk2 T
(i)
(z) J
(i)
(b)
u
(1)
(1)
O(0), Lξq , Lξq
[Mantry, Pirjol, Stewart]
A(D
c
b
d (s)
(i)
+
u (s)
u
d
+
u (s)
d
d
+
c
b
+
(z, x, k1 , k2 ) S (k1 , k2 ) φM (x) + . . .
|
{z
}
complex − nonpert. strong phase
Z. Ligeti — p. 39
Color suppressed B → D (∗)0π 0 decays
• Single class of power suppressed
SCETI
operators: T
(a)
(∗)0
0
M
M ) = N0
Z
+
+
dz dx dk1 dk2 T
(i)
(z) J
(i)
(b)
u
(1)
(1)
O(0), Lξq , Lξq
[Mantry, Pirjol, Stewart]
A(D
c
b
d (s)
(i)
+
u (s)
u
d
+
u (s)
d
d
+
c
b
+
(z, x, k1 , k2 ) S (k1 , k2 ) φM (x) + . . .
|
{z
}
complex − nonpert. strong phase
• Not your garden variety factorization formula... S (i)(k1+, k2+) know about n
(c)
¯
hD 0(v 0)|(h̄v0 S) n
/PL(S †h(b)
/PL(S †u)k+ |B̄ 0(v)i
v )(dS)k+ n
(0)
+
+
1
2
S (k1 , k2 ) =
√
mB mD
Separates scales, allows to use HQS without Eπ /mc = O(1) corrections
(i = 0, 8 above)
Z. Ligeti — p. 39
Color suppressed B → D (∗)0π 0 decays
(a)
• Single class of power suppressed
SCETI
operators: T
d (s)
(∗)0
0
M
M ) = N0
+
+
dz dx dk1 dk2 T
(i)
(z) J
(i)
u (s)
u (s)
d
d
[Mantry, Pirjol, Stewart]
A(D
c
b
u
(1)
(1)
O(0), Lξq , Lξq
Z
(b)
c
b
u
d
+
+
(i)
+
+
(z, x, k1 , k2 ) S (k1 , k2 ) φM (x) + . . .
|
{z
}
complex − nonpert. strong phase
• Ratios:
the 4 = 1 relations follow from naive
factorization and heavy quark symmetry
2.0
color allowed
color suppressed
A(D*M)
A(D M)
The • = 1 relations do not — a prediction of
SCET not foreseen by model calculations
◦
∗
◦
Data: δ(Dπ) = (30 ± 5) , δ(D π) = (31 ± 5)
Z. Ligeti — p. 39
π
η
0
DK
D
0
D0η’
1.0
D
D π+
π-
0.5
0
-
D Κ
+
+
D ΚD ρ-
D ρ0
SCET prediction
[Blechman, Mantry, Stewart]
0.0
D 0ω
0
0 0
D
0
Also predict equal strong phases between
amplitudes to D(∗)π in I = 1/2 and 3/2
ω +ω
*
1.5
ρ
0 0
D
*
Λb and Bs decays
• CDF measured in 2003:
−
0
+ −
Γ(Λb → Λ+
π
)/Γ(B
→
D
π )≈2
c
Factorization does not follow from large Nc, but holds at
leading order in ΛQCD/Q
Γ(Λb → Λcπ −)
Γ(B 0 → D (∗)+π −)
' 1.8
Λ
ζ(wmax
)
D (∗) )
ξ(wmax
2
[Leibovich, ZL, Stewart, Wise]
Isgur-Wise functions may be expected to be comparable
Lattice could nail this
• Bs → Dsπ is pure tree, can help to determine relative size of E vs. C
[CDF ’03: B(Bs → Ds−π +)/B(B 0 → D −π +) ' 1.35 ± 0.43 (using fs/fd = 0.26 ± 0.03)]
Lattice could help: Factorization relates tree amplitudes, need SU (3) breaking in
Bs → Ds`ν̄ vs. B → D`ν̄ form factors from exp. or lattice
Z. Ligeti — p. 40
More complicated: Λb → Σcπ
• Recall quantum numbers:
Σc = Σc(2455),
Σ∗c
= Σc(2520)
multiplets
sl
I(J P )
Λc
0
0( 21 )
Σc , Σ∗c
1
1( 21 ), 1( 23 )
+
+
+
• Can’t
[Leibovich, ZL, Stewart, Wise]
address
in naive factorization, since
Λb → Σc form factor
vanishes by isospin
O(ΛQCD /Q)
•
O(ΛQCD /Q)
O(Λ2QCD /Q2 )
0
Γ(Λb → Σ∗0
Γ(Λb → Σ∗c π)
c ρ )
Prediction:
= 2 + O ΛQCD/Q , αs(Q) =
Γ(Λb → Σcπ)
Γ(Λb → Σ0c ρ0)
(∗)0 0
Can avoid π 0’s from Λb → Σc
(∗)+ −
π → Λcπ −π 0 or Λb → Σc
Z. Ligeti — p. 41
π → Λc π 0 π −
Charmless B → M1M2 decays
• Limited consensus about implications of the heavy quark limit
M’
p 2 ~ Λ2
[Bauer, Pirjol, Rothstein, Stewart; Chay, Kim; Beneke, Buchalla, Neubert, Sachrajda]
A = Acc̄ + N fM2 ζ
BM1
Z
A = Acc̄ + N + fM2
•
BM
ζJ 1
p 2~ Q 2
Z
du T2ζ (u) φM2 (u)
BM
dzdu T2J (u, z) ζJ 1 (z) φM2 (u)
+ (1 ↔ 2)
B
p 2 ~ Λ2
M
p 2~ QΛ
p 2 ~ Λ2
R
= dxdk+ J(z, x, k+) φM1 (x) φB (k+) also appears in B → M1 form factors
√
⇒ Relations to semileptonic decays do not require expansion in αs( ΛQ)
• Charm penguins: suppression of long distance part argued, not proven
Lore: “long distance charm loops”, “charming penguins”, “DD rescattering” are
the same (unknown) term; may yield strong phases and other surprises
• SCET: fit both ζ’s and ζJ ’s, calculate T ’s;
QCDF: fit ζ’s, calculate factorizable
(1st) terms perturbatively; PQCD: 1st line dominates and depends on k⊥
Z. Ligeti — p. 42
B → ππ amplitudes
A+− = −λu(T + Pu) − λcPc − λtPt = e−iγ Tππ − Pππ
√
2A00 = λu(−C+Pu)+λcPc +λtPt = e−iγ Cππ + Pππ
√
2A−0 = −λu(T + C) = e−iγ (Tππ + Cππ )
[many, many authors, sorry!]
0
Alternatively, eliminate λt terms, then eiβ Pππ
Diagrammatic language can be justified in SCET at leading order
• We know: arg(T /C) = O(αs, Λ/mb), Pu is calculable (small),
– Pt: “chirally enhanced” power correction in QCDF (treated like others by BPRS)
– Pc: treated as O(1) in SCET (argued to be small by BBNS)
• Isospin analysis: 6 observables determine weak phase + 5 hadronic parameters
B(B → π 0π 0) is large, so ∆α can be large, but Cπ0π0 is hard to measure
• Can we use the theory constraint to determine α without Cπ0π0 ?
Z. Ligeti — p. 43
Phenomenology of B → ππ
• Imposing a constraint on either ≡ Im(Cππ /Tππ ) or τ ≡ arg[Tππ /(Cππ + Tππ )]
mixes “tree” and “penguin” amplitudes [expect , τ = O(αs, Λ/mb)]
[Bauer, Rothstein, Stewart]
[Höcker, Grossman, ZL, Pirjol]
For α ∼ 90◦,
∼ 0.2 ↔ τ ∼ 5◦
∼ 0.4 ↔ τ ∼ 10◦
(t)
τ = 0 and w/o C00
(t)
o
o
o
|τ | < 5 , 10 , 20 and w/o C00
(t)
τ = 0 and with C00
1.2
20o
isospin analysis
without C00
0.8
o
10
0.6
CKM fit
5o
0.2
0
For a given τ , theo and exp
errors highly correlated
no α in fit
0.4
0
20
40
60
80
α
• CKM fit ⇒ unexpectedly large τ (2σ)
May be more applicable to B → ρρ
(deg)
1.2
1 – CL
– large power corrections to T, C?
– large up penguins?
– large weak annihilation?
100 120 140 160 180
t-convention
c-convention
1.2
t-convention
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
-60
-40
-20
0
τ
Z. Ligeti — p. 44
(deg)
t-convention
c-convention
fit without using C00
1 – CL
1 – CL
1
20
40
60
0
-60
-40
-20
0
τ
(deg)
20
40
60
Few comments
• More work & data needed to understand the expansions
Why some predictions work at <
∼ 10% level, while others receive ∼ 30% corrections
Clarify role of charming penguins, chirally enhanced terms, annihilation, etc.
We have the tools to try to address the questions
• Where can lattice help?
– Semileptonic form factors (precision, include ρ and K ∗, larger recoil)
– Light cone distribution functions of heavy and light mesons
– SU (3) breaking in form factors and distribution functions
– Probably more remote: nonleptonic decays, nonlocal matrix elements
R
−1
0 0
– e.g., large B → π π rate in SCET accommodated by hk+ iB = dk+ φB (k+)/k+
Z. Ligeti — p. 45
The future
Theoretical limitations (continuum methods)
• Many interesting decay modes will not be theory limited for a long time
Measurement (in SM)
Theoretical limit
Present error
∼ 0.2◦
1.6◦
B → φKS , η (0)KS , ... (β )
∼ 2◦
∼ 10◦
B → ππ, ρρ, ρπ (α)
∼ 1◦
∼ 15◦
B → DK (γ )
1◦
∼ 25◦
Bs → ψφ (βs)
∼ 0.2◦
—
Bs → DsK (γ − 2βs)
1◦
—
|Vcb|
∼ 1%
∼ 3%
|Vub|
∼ 5%
∼ 15%
B → X`+`−
∼ 5%
∼ 20%
B → K (∗)ν ν̄
∼ 5%
—
K + → π +ν ν̄
∼ 5%
∼ 70%
KL → π 0ν ν̄
< 1%
—
B → ψKS (β )
It would require breakthroughs to go significantly below these theory limits
Z. Ligeti — p. 46
Outlook
• If there are new particles at TeV scale, new flavor physics could show up any time
Belle & Babar data sets continue to double every ∼2 years, will reach ∼1000 fb−1
each in a few years; B → J/ψKS was a well-defined target
• Goal for further flavor physics experiments:
If NP is seen in flavor physics: study it in as many different operators as possible
If NP is not seen in flavor physics: achieve what’s theoretically possible
Even in latter case, powerful constraints on model building in the LHC era
• The program as a whole is a lot more interesting than any single measurement
Z. Ligeti — p. 47
Conclusions
• Much more is known about the flavor sector and CPV than few years ago
CKM phase is probably the dominant source of CPV in flavor changing processes
• Deviations from SM in Bd mixing, b → s and even in b → d decays are constrained
• New era:
new set of measurements are becoming more precise than old ones;
existing data could have shown NP, lot more is needed to achieve theoretical limits
• The point is not just to measure magnitudes and phases of CKM elements (or ρ, η
and α, β, γ), but to probe the flavor sector by overconstraining it in as many ways
as possible (rare decays, correlations)
• Many processes give clean information on short distance physics, and there is
progress toward model independently understanding more observables
Lattice QCD is important; in some cases the only way to make progress
Z. Ligeti — p. 48
Thanks
• To the organizers for the invitation, and for looking after our needs
⇒
• To A. Höcker, H. Lacker, Y. Nir, G. Perez, and I. Stewart for helpful discussions
l
Additional Topics
Further interesting CPV modes
B → ρρ vs. ππ isospin analysis
• Due to Γρ 6= 0, ρρ in I = 1 possible, even for σ = 0
[Falk, ZL, Nir, Quinn]
Can have antisymmetric dependence on both the two ρ mesons’ masses and on
their isospin indices ⇒ I = 1
(mi = mass of a pion pair; B = Breit-Wigner)
1
+
−
+
−
A ∼ B(m1)B(m2) f (m1, m2) ρ (m1)ρ (m2) + f (m2, m1) ρ (m2)ρ (m1)
2
+
1 n
−
+
−
= B(m1)B(m2)
f (m1, m2) + f (m2, m1) ρ (m1)ρ (m2) + ρ (m2)ρ (m1)
|
{z
}
4
I=0,2
+
o
−
+
−
+ f (m1, m2) − f (m2, m1) ρ (m1)ρ (m2) − ρ (m2)ρ (m1)
|
{z
}
I=1
If Γρ vanished, then m1 = m2 and I = 1 part is absent
E.g., no symmetry in factorization: f (mρ− , mρ+ ) ∼ fρ(mρ+ ) F B→ρ(mρ− )
• Cannot rule out O(Γρ/mρ) contributions;
no interference ⇒ O(Γ2ρ/m2ρ) effects
Can ultimately constrain these using data
Z. Ligeti — p. i
CPV in neutral meson mixing
• CPV in mixing and decay: typically sizable hadronic uncertainties
Flavor eigenstates: |B 0i = |b di, |B 0i = |b di
d
i
dt
|B 0(t)i
|B 0(t)i
0
|B (t)i
i
= M− Γ
2
|B 0(t)i
b
t
W
d
d
b
W
t
t
b
d
d
W
t
W
b
Mass eigenstates: |BL,H i = p|B 0i ± q|B 0i
• CPV in mixing: Mass eigenstates 6= CP eigenstates
Best limit from semileptonic asymmetry (4Re )
ASL
(|q/p| =
6 1 and hBH |BLi =
6 0)
[NLO: Beneke et al.; Ciuchini et al.]
Γ[B 0(t) → `+X] − Γ[B 0(t) → `−X]
1 − |q/p|4
=
=
= (−0.0026 ± 0.0067)
1 + |q/p|4
Γ[B 0(t) → `+X] + Γ[B 0(t) → `−X]
⇒ |q/p| = 1.0013 ± 0.0034
[dominated by BELLE]
Allowed range than SM region, but already sensitive to NP
Z. Ligeti — p. ii
[Laplace, ZL, Nir, Perez]
Bs → ψφ and Bs → ψη (0)
• Analog of B
→ ψKS in Bs decay — determines the phase between Bs mixing
and b → cc̄s decay, βs, as cleanly as sin 2β from ψKS
1
βs is a small O(λ2) angle in one of the
“squashed” unitarity triangles
ψη (0), however, is pure CP -even
fitter
LP 2005
0.6
0.4
ψφ is a VV state, so the asymmetry is
diluted by the CP -odd component
CK M
0.8
1 – CL
sin 2βs = 0.0346+0.0026
−0.0020
[update of Laplace, ZL, Nir, Perez]
1σ
0.2
2σ
0
0.025
0.03
0.035
sin(2βs)
• Large asymmetry (sin 2βs > 0.05) would be clear sign of new physics
Z. Ligeti — p. iii
0.04
0.045
Bs → Ds±K ∓ and B 0 → D (∗)±π ∓
• Single weak phase in each Bs, B s → Ds±K ∓ decay ⇒ the 4 time dependent rates
determine 2 amplitudes, strong, and weak phase (clean, although |f i =
6 |fCP i)
A
Four amplitudes: B s →1 Ds+K −
A1
ADs+K −
A
B s →2 K +Ds−
(b → ucs)
A2
(b → cus) ,
Bs → K −Ds+ (b → ucs)
∗
∗
A
−
+
A1 VcbVus
A2 VubVcs
Ds K
=
,
=
∗V
A2 Vub
ADs−K + A1 Vcb∗ Vus
cs
Four amplitudes: Bs →
ADs+K −
(b → cus) ,
Ds−K +
Magnitudes and relative strong phase of A1 and A2 drop out if four time dependent rates are measured ⇒ no hadronic uncertainty:
∗ 2
∗
∗
VtbVts
VcbVus
VubVcs
−2i(γ−2βs −βK )
λDs+K − λDs−K + =
=
e
∗V
VtbVts∗
Vub
Vcb∗ Vus
cs
• Similarly, Bd → D(∗)±π∓ determines γ + 2β, since λD+π− λD−π+ = e−2i(γ+2β)
... ratio of amplitudes O(λ2) ⇒ small asymmetries (and tag side interference)
Z. Ligeti — p. iv
A near future (& personal) best buy list
• β: reduce error in B → φKS , η 0KS , K +K −KS (and D(∗)D(∗)) modes
• α: refine ρρ (search for ρ0ρ0); ππ (improve C00); ρπ Dalitz
• γ: pursue all approaches, impressive start
• βs: is CPV in Bs → ψφ small?
• |Vtd/Vts|: Bs mixing (Tevatron may still have a chance)
• Rare decays: B → Xsγ near theory limited; B → Xs`+`− is becoming comparably precise
• |Vub|: reaching <
∼ 10% will be very significant (a Babar/Belle measurement that
may survive LHCB)
• Pursue B → `ν, search for “null observables”, aCP (b → sγ), etc., for enhancement of B(s) → `+`−, etc.
(apologies if your favorite decay omitted!)
Z. Ligeti — p. v
More slides removed
∆mK , K are built in NP models since 70’s
• If tree-level exchange of a heavy gauge boson was responsible for a significant
fraction of the measured value of K
Im M12 g 2 Λ3QCD ∼
⇒ MX ∼ g × 6 · 104 TeV
|K | ∼ 2 ∆m ∆mK MX
K
Similarly, from B 0 − B 0 mixing: MX ∼ g × 3 · 102 TeV
• New particles at TeV scale can have large contributions in loops [g ∼ O(10−2)]
Pattern of deviations/agreements with SM may distinguish between models
Z. Ligeti — p. vi
K 0 − K 0 mixing and supersymmetry
•
SUSY
(∆mK )
4 1 TeV
∼ 10
m̃
(∆mK )EXP
2 2 2
∆m̃12
m̃2
d
d
Re (KL)12(KR)12
d
KL(R)
: mixing in gluino couplings to left-(right-)handed down quarks and squarks
d
d
4
d
6
d
Constraint from K : replace 10 Re (KL)12(KR)12 with ∼ 10 Im (KL)12(KR)12
• Solutions to supersymmetric flavor problems:
(i) Heavy squarks: m̃ 1 TeV
(ii) Universality: ∆m2Q̃,D̃ m̃2 (GMSB)
d
(iii) Alignment: |(KL,R
)12| 1 (Horizontal symmetry)
(0)
The CP problems (K , EDM’s) are alleviated if relevant CPV phases 1
• With many measurements, we can try to distinguish between models
Z. Ligeti — p. vii
Precision tests with Kaons
• CPV in K system is at the right level (K accommodated with O(1) CKM phase)
Hadronic uncertainties preclude precision tests (0K notoriously hard to calculate)
• K → πνν:
Theoretically clean, but rates small B ∼ 10−10(K ±), 10−11(KL)
 5 2
5
2

 (λ mt ) + i(λ mt ) t : CKM suppressed
A ∝ (λ m2c ) + i(λ5 m2c ) c : GIM suppressed

 (λ Λ2 )
u : GIM suppressed QCD
−10
So far three events observed: B(K + → π +ν ν̄) = (1.47+1.30
−0.89 ) × 10
• Need much higher statistics to make definitive tests
Z. Ligeti — p. viii
The D meson system
• Complementary to K, B: CPV, FCNC both GIM & CKM suppressed ⇒ tiny in SM
– Only meson where mixing is generated by down type quarks (SUSY: up squarks)
– D mixing expected to be small in the SM, since it is DCS and vanishes in the
flavor SU (3) symmetry limit
– Involves only the first two generations: CPV > 10−3 would be unambiguously
new physics
– Only neutral meson where mixing has not been observed; possible hint:
yCP
Γ(CP even) − Γ(CP odd)
= (0.9 ± 0.4)%
=
Γ(CP even) + Γ(CP odd)
[Babar, Belle, Cleo, Focus, E791]
• At the present level of sensitivity, CPV would be the only clean signal of NP
Can lattice help to understand the SM prediction for D − D mixing?
Z. Ligeti — p. ix
Polarization in charmless B → V V
B decay
Longitudinal polarization fraction
BELLE
BABAR
ρ− ρ+
0.98+0.02
−0.03
ρ0 ρ+
0.97+0.05
−0.08
0.95 ± 0.11
8+0.12
−0.15
0.96+0.06
−0.16
+
ωρ
0
ρ K
∗+
ρ− K ∗0
φK ∗0
φK ∗+
+0.12
0.43−0.11
0.79 ± 0.09
0.45 ± 0.05
0.52 ± 0.09
0.52 ± 0.05
0.46 ± 0.12
s
Chiral structure of SM and
HQ limit claimed to imply
fL = 1 − O(1/m2b )
s
b
(s b)V-A (s s)V-A
[Kagan]
s
d
φK ∗: penguin dominated — NP reduces fL?
Proposed explanations:
d
b
d,s
c
....
αs (mv)
c
O(1)
qµ
αs ( )
q
q
s
s
B
-
(d b)S-P (s d)S+P
b
Ds(∗)
s
d
O(1/m2b ) × large
v -
φ
K*T
Ds(∗)
@
6
R
@
D(∗)@@v -
(model)
g*
s
b
T
q
K
φT
∗
(model)
c penguin [Bauer et al.]; penguin annihilation [Kagan]; rescattering [Colangelo et al.]; g fragment. [Hou, Nagashima]
• Can it be made a clean signal of NP?
Z. Ligeti — p. x
B → πK rates and CP asymmetries
Sensitive to interference
between b → s penguin
and b → u tree (and
possible NP)
Decay mode
CP averaged B [×10−6]
ACP
B 0 → π+K −
18.2 ± 0.8
−0.11 ± 0.02
B − → π0K −
12.1 ± 0.8
+0.04 ± 0.04
B − → π−K 0
24.1 ± 1.3
−0.02 ± 0.03
B 0 → π0K 0
11.5 ± 1.0
+0.00 ± 0.16
[Fleischer & Mannel, Neubert & Rosner; Lipkin; Buras & Fleischer; Yoshikawa; Gronau & Rosner; Buras et al.; ...]
+
0 +
−
0 −
Rc ≡ 2
B(B
→ π K ) + B(B
→π K )
B(B + → π + K 0 ) + B(B − → π − K 0 )
= 1.00 ± 0.08
1 B(B 0 → π − K + ) + B(B 0 → π + K − )
= 0.79 ± 0.08
Rn ≡
2 B(B 0 → π 0 K 0 ) + B(B 0 → π 0 K 0 )
B(B 0 → π − K + ) + B(B 0 → π + K − ) τB ±
◦
R≡
= 0.82 ± 0.06 ⇒ FM bound : γ < 75 (95% CL)
B(B + → π + K 0 ) + B(B − → π − K 0 ) τB 0
RL ≡ 2
Γ̄(B − → π 0 K − ) + Γ̄(B 0 → π 0 K 0 )
Γ̄(B − → π − K 0 ) + Γ̄(B 0 → π + K − )
= 1.12 ± 0.07
• Pattern quite different than until 2004: Rc closer to 1, while R further from 1
No strong motivation for NP contribution to EW penguin, will be exciting to sort out
Z. Ligeti — p. xi