The CKM matrix and CP Violation

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 

• Theory developments: semileptonic 

and nonleptonic decays in SCET 

Semileptonic form factors 

Nonleptonic decays 

Precision test of CKM; search for NP 

Best present α, γ methods are new 

First significant constraints 

Few applications, connections between 

semileptonic and nonleptonic 

• Future / Conclusions 

Z. Ligeti — p. 1

Why is flavor physics and CPV interesting? 

– Sensitive to very high scales 

2 

(s ¯d) 

ɛ K : 

Λ 2 ⇒ Λ NP 

> ∼ 10 4 TeV, B d mixing: 

NP 

(b ¯d) 

2 

Λ 2 NP 

⇒ Λ NP 

> ∼ 10 3 TeV 

– 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 S ψKS ≠ S φKS 

– Enhanced or suppressed CP violation, e.g., sizable S Bs →ψφ or A sγ 

– FCNC’s at unexpected level, e.g., B → l + l − or B s 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 (λ = sin θ C ≃ 0.22) 

⎛ 

⎞ ⎛ 

⎞ 

V ud V us V ub 

1 − 1 2 

⎜ 

⎟ ⎜ 

λ2 λ Aλ 3 (ρ − iη) 

V = ⎝ V cd V cs V cb ⎠ = ⎝ −λ 1 − 1 2 λ2 Aλ 2 ⎟ 

⎠ + O(λ 4 ) 

V td V ts V tb Aλ 3 (1 − ρ − iη) −Aλ 2 1 

• A “language” to compare overconstraining measurements 

V ud V ub 

* 

V Vcb * cd 

(ρ,η) 

α 

V td V* 

tb 

V cd Vcb * 

V ud V ∗ ub + V cd V ∗ 

cb + V td V ∗ 

tb = 0 

Goal: “redundant” measurements sensitive 

to different short distance physics 

(0,0) 

γ 

CPV in SM ∝ Area 

β 

(1,0) 

E.g.: B d mixing and b → dγ given by different 

op’s in H, but both ∝V tb V ∗ 

td 

in SM 

Z. Ligeti — p. 4

Tests of the flavor sector 

• For 35 years, until 1999, the only unambiguous measurement of CPV was ɛ K 

1.5 

excluded area has CL > 0.95 

γ 

∆m d 

η _ 

1 

1 

sin 2β 

0.9 

0.8 

0.5 

∆m s 

& ∆m d 

0.7 

0.6 

0.5 

0.4 

η 

0 

ε K 

|V ub 

/V cb 

| 

γ 

α 

β 

α 

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 

(BABAR Physics Book, 1998) 

ρ _ 

-1 

α 

C K M 

f i t t e r 

LP 2005 

γ 

ε K 

-1.5 

-1 -0.5 0 0.5 1 1.5 2 

ρ 

Z. Ligeti — p. 5

Tests of the flavor sector 

• For 35 years, until 1999, the only unambiguous measurement of CPV was ɛ K 

1.5 


η _ 

1 

0.9 

0.8 

0.7 

0.6 

0.5 

0.4 

0.3 

0.2 

0.1 

0 

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 

(BABAR Physics Book, 1998) 

ρ _ 

η 

1 

0.5 

0 

-0.5 

-1 

ε K 

sin 2β 

|V ub 

/V cb 

| 

α 

γ 

γ 

γ 

α 

-1.5 

-1 -0.5 0 0.5 1 1.5 2 

ρ 

β 

α 

∆m d 

∆m s 

& ∆m d 

• sin 2β = 0.687 ± 0.032, order of magnitude smaller error than first measurements 

C K M 

f i t t e r 

LP 2005 

ε K 

Z. Ligeti — p. 5

What are we after? 

• Flavor and CP violation are excellent probes of New Physics 

– Absence of K L → µµ predicted charm 

– ɛ K predicted 3rd generation 

– ∆m K predicted charm mass 

– ∆m B 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 (ɛ K accommodated with O(1) CKM phase) 

• Hadronic uncertainties preclude precision tests (ɛ ′ K notoriously hard to calculate) 

• K → πνν: Theoretically clean, but rates small ∼ 10 −10 (K ± ), 10 −11 (K L ) 

Observation (3 events): B(K + → π + ν¯ν) = (1.5 +1.3 

−0.9 ) × 10−10 — need more data 

• 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 

y CP = 

Γ(CP even) − Γ(CP odd) 

Γ(CP even) + Γ(CP odd) 

= (0.9 ± 0.4)% 

• At present level of sensitivity, CPV would be the only clean signal of NP 

Can lattice help to understand SM prediction for ∆m D , ∆Γ D ? (SD part for sure) 

Z. Ligeti — p. 7

CP Violation

CPV in decay 

• Simplest, count events; amplitudes with different weak (φ k ) & strong (δ k ) phases 

|A f 

/A f | ≠ 1: A f = 〈f|H|B〉 = ∑ A k e iδ k e iφ k, A f 

= 〈f|H|B〉 = ∑ A k e iδ k e −iφ k 

• Unambiguously established by ɛ ′ K 

≠ 0, last year also in B decays: 

A K − 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 ɛ ′ K and A K − π + 

prove or to rule out that NP contributes 

to either 

Sensitive to NP when SM prediction is model independently small (e.g., A b→sγ ) 

Z. Ligeti — p. 8

CPV in interference between decay and mixing 

• Can get theoretically clean information in some 

cases when B 0 and B 0 decay to same final state B0 

|B L,H 〉 = p|B 0 〉 ± q|B 0 〉 λ fCP = q A fCP 

p A fCP 

q/p 

A 

B 0 

A 

f CP 

Time dependent CP asymmetry: 

a fCP = Γ[B0 (t) → f] − Γ[B 0 (t) → f] 

Γ[B 0 (t) → f] + Γ[B 0 (t) → f] = 2 Im λ f 

1 + |λ f | 

} {{ 2 sin(∆m t) − 1 − |λ f| 2 

1 + |λ 

} 

f | 

} {{ 2 

} 

S f C f (−A f ) 

cos(∆m t) 

• If amplitudes with one weak phase dominate, hadronic physics drops out from λ f , 

and a fCP measures a phase in the Lagrangian theoretically cleanly: 

a fCP = Im λ f sin(∆m t) 

arg λ f = phase difference between decay paths 

Z. Ligeti — p. 9

The cleanest case: B → J/ψ K S 

• Interference between B → ψK 0 (b → c¯cs) and B → B → ψK 0 (¯b → c¯c¯s) 

Penguins with different than tree weak phase are suppressed 

[CKM unitarity: V tb V ∗ 

ts + V cb V ∗ 

cs + V ub V ∗ us = 0] 

A ψKS = V cb V ∗ 

cs } {{ } 

O(λ 2 ) 

T + V ub V ∗ us } {{ } 

O(λ 4 ) 

First term ≫ second term ⇒ theoretically very clean 

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arg λ ψKS = (B-mix = 2β) + (decay = 0) + (K-mix = 0) 

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• World average: sin 2β = 0.687 ± 0.032 — a 5% measurement! 

Z. Ligeti — p. 10

S ψK : a precision game 

Standard model fit without S ψK 

1.5 


1 

∆m d 

∆m s 

& ∆m d 

0.5 

η 

0 

ε K 

|V ub 

/V cb 

| 

γ 

α 

β 

-0.5 

ε K 

-1 

C K M 

f i t t e r 

LP 2005 

-1.5 

-1 -0.5 0 0.5 1 1.5 2 

ρ 

Z. Ligeti — p. 11

S ψK : a precision game 

Standard model fit including S ψK 

First precise test of the CKM picture 

1.5 


Error of S ψK near |V cb | (only |V us | better) 

1 

sin 2β 

∆m d 

∆m s 

& ∆m d 

Without V ub 4 sol’s; ψK ∗ and D 0 K 0 data 

show cos 2β > 0, removing non-SM ray 

η 

0.5 

0 

ε K 

γ 

α 

β 

Approximate CP (in the sense that all 

CPV phases are small) excluded 

|V ub 

/V cb 

| 

-0.5 

sin 2β is only the beginning 

-1 

C K M 

f i t t e r 

LP 2005 

-1.5 

-1 -0.5 0 0.5 1 1.5 2 

ρ 

ε K 

Paradigm change: look for corrections, 

rather than alternatives to CKM 

⇒ 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 (φK S , η ′ K S , etc.) 

Amplitudes with one weak phase expected to dominate: 

A = V cb V ∗ 

cs } {{ } 

O(λ 2 ) 

[P c − P t + T c ] + V ub V 

} {{ us 

∗ [P 

} u − P t + T u ] 

O(λ 4 ) 

B 

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SM: expect: S φKS − S ψK and C φKS 

< ∼ 0.05 

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K 

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NP: S φKS ≠ S ψK possible 

NP: Expect different S f for each b → s mode 

NP: Depend on size & phase of SM and NP amplitude 

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NP could enter S ψK mainly in mixing, while S φKS 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 2β eff measurements 

b→ccs 

η′ K 0 

π 0 K S 

K + K - K 0 

φ K 0 

f 0 

K S 

ω K S 

K S 

K S 

K S 

sin(2β eff )/sin(2φ e 1 ff ) 

HFAG 

HFAG 

HFAG 

LP 2005 

LP 2005 

HFAG 

LP 2005 

LP HFAG 2005 

World Average 0.69 ± 0.03 

BaBar 0.50 ± 0.25 + 0. 

07 

-0. 

04 

Belle 0.44 ± 0.27 ± 0.05 

Average 0.47 ± 0.19 

BaBar 0.30 ± 0.14 ± 0.02 

Belle 0.62 ± 0.12 ± 0.04 

Average 0.48 ± 0.09 

BaBar 0.95 + 0. 

23 

-0. 

32 ± 0.10 

Belle 0.47 ± 0.36 ± 0.08 

Average 0.75 ± 0.24 

BaBar 0.35 + 0. 

30 

-0. 

33 ± 0.04 

Belle 0.22 ± 0.47 ± 0.08 

Average 0.31 ± 0.26 

BaBar 0.50 + 0. 

34 

-0. 

38 ± 0.02 

HFAG 

HFAG 

LP 2005 

HFAG 

-1 0 1 2 

LP 2005 

LP 2005 

LP 2005 

H F A G H F A G 

LP 2005 

PRELIMINARY 

Belle 0.95 ± 0.53 + - 

0. 

12 

0. 

15 

Average 0.63 ± 0.30 

BaBar 0.41 ± 0.18 ± 0.07 ± 0.11 

Belle 0.60 ± 0.18 ± 0.04 + 0. 

19 

-0. 

12 

Average 0.51 ± 0.14 + - 0 0 . . 1 0 1 8 

BaBar 0.63 + 0. 

28 

-0. 

32 ± 0.04 

Belle 0.58 ± 0.36 ± 0.08 

Average 0.61 ± 0.23 

φ K 0 

f 0 

K S 

ω K S 

K S 

K S 

K S 

η′ K 0 

π 0 K S 

K + K - K 0 

HFAG 

C f 

= -A f 

BaBar 0.00 ± 0.23 ± 0.05 

Belle -0.14 ± 0.17 ± 0.07 

HFAG 

LP 2005 

HFAG 

LP 2005 

LP 2005 

Average -0.09 ± 0.14 

BaBar -0.21 ± 0.10 ± 0.02 

HFAG 

LP 2005 

Belle 0.04 ± 0.08 ± 0.06 

Average -0.08 ± 0.07 

BaBar -0.24 ± 0.31 ± 0.15 

HFAG 

Belle 0.23 ± 0.23 ± 0.13 

Average 0.06 ± 0.21 

BaBar 0.06 ± 0.18 ± 0.03 

HFAG 

LP 2005 

Belle -0.11 ± 0.18 ± 0.08 

Average -0.02 ± 0.13 

BaBar -0.56 + - 0 0 . . 2 2 9 7 ± 0.03 

Belle -0.19 ± 0.39 ± 0.13 

Average -0.44 ± 0.23 

BaBar 0.23 ± 0.12 ± 0.07 

LP 2005 

HFAG 

LP 2005 

H F A G H F A G 

LP 2005 

PRELIMINARY 

Belle 0.06 ± 0.11 ± 0.07 

Average 0.14 ± 0.10 

BaBar -0.10 ± 0.25 ± 0.05 

Belle -0.50 ± 0.23 ± 0.06 

Average -0.31 ± 0.17 

-1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 

• Largest hint of deviations from SM: S η ′ K S 

(2σ) and S ψK −〈S b→s 〉 = 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 2β eff measurements 

Dominant 

SM allowed range of ∗ 

process 

f CP |− ηfCP S 

sin 2β eff C f 

fCP − sin 2β| 

b → c¯cs ψK S < 0.01 +0.687 ± 0.032 +0.016 ± 0.046 

b → c¯cd 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 

b → s¯qq φK 0 < 0.05 +0.47 ± 0.19 −0.09 ± 0.14 

η ′ K 0 < 0.05 +0.48 ± 0.09 −0.08 ± 0.07 

K + K − K S ∼ 0.15 +0.51 ± 0.17 +0.15 ± 0.09 

K S K S K S ∼ 0.15 +0.61 ± 0.23 −0.31 ± 0.17 

π 0 K S ∼ 0.15 +0.31 ± 0.26 −0.02 ± 0.13 

f 0 K S ∼ 0.25 +0.75 ± 0.24 +0.06 ± 0.21 

ωK S ∼ 0.25 +0.63 ± 0.30 −0.44 ± 0.23 

∗ 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 |S ψK − S η ′ K S 

| and |S ψK − S φKS | < 0.05 [model estimates O(0.02)] 

Z. Ligeti — p. 13

Model building more interesting 

• The present S η ′ K S 

and S φKS central values can be reasonably accommodated 

with NP (unlike an O(1) deviation from S ψKS two years ago) 

s 

R,L 

b 

R 

d 

(δ 23 

) RR 

~ ~ 

b 

s R 

R 

~ 

s 

R 

_ 

s 

s 

R,L 

R 

• Other constraints: B(B → X s γ) = (3.5±0.3)×10 −6 

mainly constrains LR mass insertions 

Now also B(B → X s l + l − ) = (4.5 ± 1.0) × 10 −6 

agrees with the SM at 20% level 

⇒ new constraints on RR & LL mass insertions 

b 

R 

d 

(δ 23 

) RR 

~ ~ 

b 

s R 

R 

Z 

~ 

s 

R 

l 

_ 

l 

s 

R 

• Models must satisfy growing number of constraints simultaneously 

Z. Ligeti — p. 14

New last year: α and γ 

[γ = arg(V ∗ ub ) , 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 λ π + π− = (B-mix = 2β) + (A/A = 2γ + . . .) ⇒ gives sin 2α if P/T were small 

[expectation was P/T ∼ O(α s /4π)] 

Kπ and ππ rates ⇒ comparable amplitudes in B → ππ with different weak phases 

• Isospin analysis: 6 measurements determine 

5 hadronic parameters + weak phase 

[Gronau, London] 

§ £© 

 

Bose statistics ⇒ ππ in I = 0, 2 

¡¡ 

Triangle relations between B + , B 0 (B − , B 0 ) 

decay amplitudes 

 

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¢¡¤£¦¥ §¨¡© 

Z. Ligeti — p. 15

α from B → ππ: Isospin analysis 

• Tagged B → π 0 π 0 rates are the hardest input 

B(B → π 0 π 0 ) = (1.45 ± 0.29) × 10 −6 

1.2 

1 

C K M 

f i t t e r 

Moriond 05 

B → ππ (S/C +– 

from BABAR) 

B → ππ (S/A +– 

from Belle) 

Combined no C/A 00 

Γ(B → π 0 π 0 ) − Γ(B → π 0 π 0 ) 

Γ(B → π 0 π 0 ) + Γ(B → π 0 π 0 ) 

= 0.28 ± 0.39 

1 – CL 

0.8 

0.6 

Need lot more data to pin down ∆α from 

isospin analysis... current bound: 

0.4 

0.2 

CKM fit 

no α meas. in fit 

|∆α| < 39 ◦ (90% CL) 

0 

0 20 40 60 80 100 120 140 160 180 

α (deg) 

• Constraint on α weak (measurements 2.3σ apart): 

B → π + π − S π + π − C π + 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

B → ρρ: the best α at present 

• Lucky 2 : longitudinal polarization dominates (CP -even; could be even/odd mixed) 

Isospin analysis applies for each L, or in transversity basis for each σ (= 0, ‖, ⊥) 

• Small rate: B(B → ρ 0 ρ 0 ) < 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→ρ 0 ρ 0 ) 

B(B→ρ + ρ 0 ) 

< 0.04 (90% CL) 

α – α eff 

(deg) 

30 

20 

10 

0 

-10 

-20 

C K M 

f i t t e r 

ICHEP 2004 

exp. upper 

limit at CL=0.9 

B → ρρ isospin analysis 

-30 

0 

0 0.05 0.1 0.15 0.2 0.25 0.3 

x 10 -5 

BR(B → ρ 0 ρ 0 ) 

1 

0.9 

0.8 

0.7 

0.6 

0.5 

0.4 

0.3 

0.2 

0.1 

Isospin bound: 

|∆α| < 11 ◦ 

S ρ + ρ− yields: α = (96 ± 13)◦ 

Ultimately, more complicated than ππ, 

I = 1 possible due to finite Γ ρ , giving 

O(Γ 2 ρ/m 2 ρ) effects [can be constrained] 

[Falk, ZL, Nir, Quinn] 

Z. Ligeti — p. 17

B → ρπ: Dalitz plot analysis 

• Two-body B → ρ ± π ∓ : two pentagon relations 

from isospin; would need rates and CPV in all 

ρ + π − , ρ − π + , ρ 0 π 0 modes to get α — hard! 

{ 

Aπ 

Direct CPV: 

− ρ + = −0.47+0.13 −0.14 

A π + ρ− = −0.15 ± 0.09 

3.4σ from 0, challenges some models 

-1 

Interpretation for α model dependent 

1 

• Last year: Dalitz plot analysis of the interference 

regions in B → π + π − π 0 

0.6 

0.8 

Result: α = (113 +27 

−17 ± 6)◦ 0.4 

C.L. 

A + – 

ρπ 

0.2 

1 

0.75 

0.5 

0.25 

0 

-0.25 

-0.5 

-0.75 

1σ 

2σ 

3σ 

-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 

BABAR 

preliminary 

A – + 

ρπ 

H F A G 

ICHEP 2004 

0 

0 30 60 90 120 150 180 

α 

(deg) 

Z. Ligeti — p. 18

Combined α measurements 

• Sensitivity mainly from S ρ + 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 

1.2 

1 

0.8 

0.6 

0.4 

C K M 

f i t t e r 

LP 2005 

B → ππ 

B → ρπ 

B → ρρ 

WA 

Combined 


η 

1.5 

1 

0.5 

0 

-0.5 

sin 2β 


no α meas. in fit 

γ 

α 

β 

1 – CL 

1 

0.9 

0.8 

0.7 

0.6 

0.5 

0.4 

0.3 

0.2 

0 

0 20 40 60 80 100 120 140 160 180 

α (deg) 

-1 

C K M 

f i t t e r 

LP 2005 

α from B → ππ, πρ, ρρ 

-1.5 

-1 -0.5 0 0.5 1 1.5 2 

ρ 

0.2 

0.1 

0 

Z. Ligeti — p. 19

γ from B ± → DK ± 

• Tree level: interfere b → c (B − → D 0 K − ) and b → u (B − → D 0 K − ) 

Need D 0 , D 0 → same final state; determine B and D decay amplitudes from data 

Many variants depending on D decay: D CP [GLW], DCS/CA [ADS], CS/CS [GLS] 

Sensitivity crucially depends on: r B = |A(B − → D 0 K − )/A(B − → D 0 K − )| 

— ↓ 

• Best measurement now: D 0 , D 0 → K S π + π − 

Both amplitudes Cabibbo allowed; can integrate 

over regions in m Kπ + − m Kπ − Dalitz plot 

γ = ( 68 +14 

−15 ± 13 ± 11) ◦ 

[BELLE, 275 m] 

γ = (67 ± 28 ± 13 ± 11) ◦ [BABAR, 227 m] 

σ(γ) degree 

70 

60 

50 

40 

30 

20 

BABAR 

Monte Carlo study 

10 

↑ ↑ 

0 

0 0.05 0.1 0.15 0.2 0.25 0.3 

r B 

BABAR 

• Need more data to determine γ more precisely (and settle value of r B ) 

BELLE 

Z. Ligeti — p. 20

Overconstraining the CKM matrix 

• SM fit: α, β determine ρ, η nearly as precisely as all data combined 

η 

0.7 

0.6 

0.5 

0.4 

0.3 

0.2 


sin 2β 

γ 

α 

α 

C K M 

f i t t e r 

LP 2005 

η 

0.7 

0.6 

0.5 

0.4 

0.3 

0.2 


sin 2β 

∆m d 

ε K 

γ 

α 

∆m s 

& ∆m d 

ε K 

α 

C K M 

f i t t e r 

LP 2005 

0.1 

γ 

β 

0 

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1 

ρ 

0.1 |V ub 

/V cb 

| 

γ 

β 

0 

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1 

ρ 

• New era: constraints from angles surpass the rest; will scale with statistics 

(By the time ∆m s is measured, α may be competitive for |V td | side) 

• ɛ K , ∆m d , ∆m s , |V ub |, etc., can be used to overconstrain the SM and test NP 

Let’s see how it works... 

Z. Ligeti — p. 21

The “new” CKM fit 

• Include measurements that give meaningful constraints and NOT theory limited 

• α from B → ρρ and ρπ Dalitz • γ from B → DK (with D Dalitz) 

• 2β + γ from B → D (∗)± π ∓ 

• cos 2β from ψK ∗ and A SL (for NP) 

1.5 


1.5 


∆m d 

γ 

∆m d 

1 

sin 2β 

1 

sin 2β 

∆m s 

& ∆m d 

∆m s 

& ∆m d 

0.5 

0.5 

η 

0 

ε K 

|V ub 

/V cb 

| 

γ 

α 

β 

η 

0 

ε K 

|V ub 

/V cb 

| 

γ 

α 

β 

α 

-0.5 

-0.5 

ε K 

α 

ε K 

-1 

-1 

C K M 

f i t t e r 

LP 2005 

-1.5 

-1 -0.5 0 0.5 1 1.5 2 

∆m s = (17.9 +10.5 

−1.7 

Fit with “traditional” inputs 

ρ 

C K M 

f i t t e r 

LP 2005 

γ 

Fit with above included 

-1.5 

-1 -0.5 0 0.5 1 1.5 2 

[+20.0] 

[−2.8] ) ps−1 at 1σ [2σ] ∆m s = (17.9 +3.6 

−1.4 

ρ 

[+8.6] 

[−2.4] ) ps−1 

Z. Ligeti — p. 22

Constraining NP in mixing: ρ − η view 

• NP in mixing amplitude only, 3 × 3 unitarity preserved: M 12 = M (SM) 

12 rd 2 e2iθ d 

⇒ ∆m B = rd 2∆m(SM) 

B 

, S ψK = sin(2β+2θ d ), S ρρ = sin(2α−2θ d ), γ(DK) unaffected 

Constraints with |V ub |, ∆m d , S ψK 

1.5 

1 – CL 

1 

Add: α, γ, 2β + γ, cos 2β 

1.5 

1 – CL 

1 

1 

SM CKM Fit 

0.8 

1 


0.8 

η 

0.5 

0 

-0.5 

-1 

C K M 

f i t t e r 

Lattice 05 

γ 

α 

-1.5 

-1 -0.5 0 0.5 1 1.5 2 

ρ 

β 

New Physics in B 0 B 0 mixing 

0.6 

0.4 

0.2 

0 

η 

0.5 

0 

-0.5 

-1 

C K M 

f i t t e r 

Lattice 05 

α 

-1.5 

-1 -0.5 0 0.5 1 1.5 2 

Only the SM region left even in the presence of NP in mixing 

γ 

ρ 

β 

First determinations of 

(ρ, η) from “effectively” 

tree level processes 


0.6 

0.4 

0.2 

[Similar fits also by UTfit] 

0 

Z. Ligeti — p. 23

Constraining NP in mixing: ρ − η view 



⇒ ∆m B = rd 2∆m(SM) 

B 



1.5 

1 – CL 

1 

Add: α, γ, 2β + γ, cos 2β and A SL 

1.5 

1 – CL 

1 

1 


0.8 

1 


0.8 

η 

0.5 

0 

-0.5 

-1 

C K M 

f i t t e r 

Lattice 05 

γ 

α 

-1.5 

-1 -0.5 0 0.5 1 1.5 2 

ρ 

β 


0.6 

0.4 

0.2 

0 

η 

0.5 

0 

-0.5 

-1 

C K M 

f i t t e r 

Lattice 05 

α 

-1.5 

-1 -0.5 0 0.5 1 1.5 2 

• Only the SM region left even in the presence of NP in mixing 

γ 

ρ 

β 

First determinations of 

(ρ, η) from “effectively” 

tree level processes 


0.6 

0.4 

0.2 

[Similar fits also by UTfit] 

0 

Z. Ligeti — p. 23

Constraining NP in mixing: r 2 d − θ d view 



⇒ ∆m B = rd 2∆m(SM) 

B 



150 

100 

C K M 

f i t t e r 

Lattice 05 

1 – CL 

1 

0.8 

150 

100 

Add: α, γ, 2β + γ, cos 2β and A SL 

C K M 

f i t t e r 

Lattice 05 

1 – CL 

1 

0.8 

2θ d 

(deg) 

50 

0 

-50 

SM 

0.6 

0.4 

2θ d 

(deg) 

50 

0 

-50 

SM 

0.6 

0.4 

-100 

-150 


0.2 

-100 

-150 

0 

0 1 2 3 4 5 6 7 

0 1 2 3 4 5 6 7 

r d 

2 


• New data restrict r 2 d , θ d significantly for the first time — still plenty of room left 

r d 

2 

0.2 

0 

Z. Ligeti — p. 24

NP in mixing: h d − σ d view 

• Previous fits: |M 12 /M SM 

12 | can only differ significantly from 1 if arg(M 12 /M SM 

12 ) ∼ 0 

More transparent parameterization: M 12 = M (SM) 

12 r 2 d e2iθ d ≡ M (SM) 

12 (1 + h d e 2iσ d) 

Modest NP contribution can still have arbitrary phase 

[Agashe, Papucci, Perez, Pirjol, to appear] 

2θ d 

(deg) 

150 

100 

50 

0 

-50 

-100 

-150 

SM 


0 1 2 3 4 5 6 7 

r d 

2 

C K M 

f i t t e r 

Lattice 05 

1 – CL 

1 

0.8 

0.6 

0.4 

0.2 

0 

σ d 

180 

160 

140 

120 

100 

80 

60 

40 

20 

0 

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 

h d 

1 

0.9 

0.8 

0.7 

0.6 

0.5 

0.4 

0.3 

0.2 

0.1 

0 

• For |h d |< 0.2, the phase σ d is unconstrained; if |h d |< 0.4, σ d can take half of (0, π) 

Z. Ligeti — p. 25

Intermediate summary 

• sin 2β = 0.687 ± 0.032 

⇒ good overall consistency of SM, δ CKM is probably the dominant source of CPV 

⇒ in flavor changing processes 

• S ψK − S η ′ K S 

= 0.21 ± 0.10 and S ψK − 〈S b→s 〉 = 0.18 ± 0.06 

⇒ Decreasing deviations from SM (same values with 5σ would still signal NP) 

• A K − π + = −0.12 ± 0.02 

⇒ “B-superweak” excluded, sizable strong phases 

• Measurements of α = ( 99 +12 ) ◦ ( ) 

−9 and γ = 64 

+16 ◦ 

−13 

⇒ 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 m B ≫ Λ QCD limit 

Factorization for B → M form factors for q 2 ≪ m 2 B 

and certain B → M 1 M 2 nonleptonic decays

Determinations of |V cb | and |V ub | 

• Inclusive and exclusive |V cb | and |V ub | determinations rely on 

heavy quark expansions; theoretically cleanest is |V cb | incl 

Γ(B → X c l¯ν) = G2 F |V cb| 2 

192π 3 

( 

mΥ 

[ ( ) ( Λ1S 

Λ1S 

1 − 0.22 

−0.011 

500 MeV 

500 MeV 

2 

) 5 

(0.534)× 

( ) ( ) 

λ1 Λ 

− 0.006 1S 

λ2 Λ 

(500 MeV) 3 + 0.011 1S 

(500 MeV) 3 

( 

( 

+ 0.011 

) 

T 1 

(500 MeV) 3 + 0.002 

) 2 ) ( 

) 

λ 

− 0.052( 

1 

λ 

(500 MeV) 2 − 0.071 2 

(500 MeV) 2 

( 

) ( 

) 

ρ 

− 0.006 1 

ρ 

(500 MeV) 3 + 0.008 2 

(500 MeV) 3 

) 

T 2 

(500 MeV) 3 − 0.017 

+ 0.096ɛ − 0.030ɛ 2 ( ) Λ1S 

BLM + 0.015ɛ + . . . 

500 MeV 

] 

( 

) 

T 3 

(500 MeV) 3 − 0.008 

( 

) 

T 4 

(500 MeV) 3 

Corrections: O(Λ/m): ∼ 20%, O(Λ 2 /m 2 ): ∼ 5%, O(Λ 3 /m 3 ): ∼ 1 − 2%, 

O(α s ): ∼ 10%, Unknown terms: < few % 

Matrix elements determined from fits to many shape variables 

ν 

• Error of |V cb | incl ∼2%! New small parameters complicate expansions for |V ub | 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 → l¯ν: measures f B × |V ub | — need f B from lattice 

– B → πl¯ν: 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: 

⇒ f B 

⇒ 

⇒ 

f B s 

× f Ds 

f D 

— lattice: double ratio = 1 within few % [Grinstein] 

B(B → l¯ν) 

B(B s → l + l − ) × B(D s → l¯ν) 

B(D → l¯ν) 

— very clean... after 2010? 

f (B→ρl¯ν) 

f (B→K∗ l + l − × f (D→K∗ l¯ν) 

or q 2 spectra — accessible soon? 

) f (D→ρl¯ν) [ZL, Wise; Grinstein, Pirjol] 

New CLEO-C D → ρl¯ν data still consistent w/ no SU(3) breaking in form factors [ZL, Stewart, Wise] 

Could lattice do more to pin down the corrections? 

Z. Ligeti — p. 28

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 p = (+, −, ⊥) p 2 

SCET I : λ = √ Λ/E — jets (m∼ΛE) 

collinear ξ n,p , A µ n,q 

E(λ 2 , 1, λ) E 2 λ 2 

soft q q , A µ s 

E(λ, λ, λ) E 2 λ 2 

SCET II : λ = Λ/E — hadrons (m∼Λ) 

Match QCD → SCET I → SCET II 

• Can decouple ultrasoft gluons from collinear Lagrangian at leading order in λ 

ξ n,p = Y n ξ n,p (0) A n,q = Y n A (0) 

n,q Y n 

† Y n = P exp [ ig ∫ x 

−∞ ds n · A us(ns) ] 

Nonperturbative usoft effects made explicit through factors of Y n in operators 

New symmetries: collinear / soft gauge invariance 

• Simplified / new (B → Dπ, πl¯ν) proofs of factorization theorems 

[Bauer, Pirjol, Stewart] 

Z. Ligeti — p. 29

Semileptonic B → π, ρ form factors 

• Issues: endpoint singularities, Sudakov effects, etc. 

p 2 ~ Q 2 

At leading order in Λ/Q, to all orders in α s , form factors 

for q 2 ≪ m 2 B written as (Q = E, m b; omit µ-dep’s) 

[Beneke & Feldmann; Bauer, Pirjol, Stewart; Becher, Hill, Lange, Neubert] 

F (Q) = C k (Q) ζ k (Q) + m Bf B f M 

4E 2 

∫ 

B 

p 2 

~ Λ2 

p 2 ~ QΛ 

dzdxdr + T (z, Q) J(z, x, r + , Q) φ M (x)φ B (r + ) 

M 

p 2 

~ 

2 

Λ 

Matrix elements of distinct ∫ d 4 x T [ J (n) (0) L (m) 

ξq (x)] terms (turn spectator q us → ξ) 

• 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 ∼ α s ×(1st) PQCD: 1st ∼ 0 

Some relations between semileptonic and nonleptonic decays can be insensitive 

to this, while other predictions may be sensitive (e.g., A FB = 0 in B → K ∗ l + l − ?) 

Z. Ligeti — p. 30

|V ub | from B → πl¯ν 

• Lattice is under control for large q 2 (small |⃗p π |), 

experiment loses a lot of statistics 

dΓ(B 0 → π + l¯ν) 

dq 2 = G2 F |⃗p π| 3 

24π 3 |V ub | 2 |f + (q 2 )| 2 

Best would be to use the q 2 -dependent data and 

its correlation (both lattice and experiment) to get 

|V ub |, reducing role of model-dependent fits 

• Dispersion relation and a few points for f + (q 2 ) 

give strong constraints on shape [Boyd, Grinstein, Lebed] 

0.8 

( 1- q 2 ) f (q2) 

B → ππ using factorization constrains |V ub |f + (0) 

[Bauer et al.] 

• Can combine dispersive bounds with lattice and 

possibly B → ππ 

[Fukunaga, Onogi; Arnesen et al.] 

0.6 

0.4 

0.2 

0.0 

0 

5 

10 

15 

f =f + 

f =f 0 

20 25 

q 2 

Z. Ligeti — p. 31

Tension between sin 2β and |V ub |? 

• SM fit favors slightly smaller |V ub | than inclusive determination, or larger sin 2β 

Inclusive average (error underestimated?) 

|V ub | (HFAG) 

incl 

= (4.38 ± 0.19 ± 0.27) × 10 −3 

Lattice πlν average [HPQCD & FNAL from Stewart @ LP’05] 

|V ub | = (4.1 ± 0.3 +0.7 

−0.4 ) × 10−3 

Depends on whether only q 2 > 16 GeV 2 is used 

1 – CL 

1 

0.8 

0.6 

0.4 

0.2 

1σ 

C K M 

f i t t e r 

LP 2005 


(sin2β) 

Inclusive 

average 

0 

Light-cone SR [Ball, Zwicky; Braun et al., Colangelo, Khodjamirian] 

|V ub | = (3.3 ± 0.3 +0.5 

−0.4 ) × 10−3 

2σ 

0.3 0.35 0.4 

|V ub 

| 

0.45 0.5 

x 10 -2 

Statistical fluctuations? Problem with inclusive? New physics? 

Precise |V ub | crucial to be sensitive to small NP entering sin 2β via mixing 

• To sort this out, need precise and model independent f B and B → π form factor 

Z. Ligeti — p. 32

Chasing |V td /V ts |: B → ργ vs. B → K ∗ γ 

• Factorization formula: 〈V γ|H|B〉 = T I i F V + ∫ dx dk T II 

i (x, k) φ B(k) φ V (x) + . . . 

[Bosch, Buchalla; Beneke, Feldman, Seidel; Ali, Lunghi, Parkhomenko] 

B(B 0 → ρ 0 γ) 

B(B 0 → K ∗0 γ) = 1 2 

∣ V td∣∣∣ 

2 

∣ ξ −2 (1 + tiny) 

V ts 

No weak annihilation in B 0 , cleaner than B ± 

SU(3) breaking: ξ = 1.2 ± 0.1 (CKM ’05) 

[Ball, Zwicky; Becirevic; Mescia] 

Conservative? ξ − 1 is model dependent 

σ(ξ) = 0.2 doubles error estimate 

Could LQCD help more? 

η 

1.5 

1 

0.5 

0 

-0.5 

-1 

[New Belle observation + Babar bound] 


C K M 

f i t t e r 

LP 2005 

γ 

σ(ξ) = 0.2 

α 

-1.5 

-1 -0.5 0 0.5 1 1.5 2 

β 

BR(B 0 → ρ 0 γ) / BR(B 0 → K* 0 γ) 

• Mild indication that ∆m s might not be right at the current lower limit? 

ρ 

1 – CL 

1 

0.9 

0.8 

0.7 

0.6 

0.5 

0.4 

0.3 

0.2 

0.1 

0 

Z. Ligeti — p. 33

B → τ ν might also precede ∆m s 

• ∆m s is not the only way to eliminate the f B 

error in ∆m d ; f B cancels in Γ(B → τν)/∆m d 

If no exp. errors: determine |V ub /V td | independent 

of f B (left with B d ; ellipse for fixed V cb , V ts ) 

If f B is known: get two circles that intersect at 

α ∼ 100 ◦ ⇒ powerful constraints 

Shown are 1 and 2σ contours with 

f B = 216 ± 9 ± 21 MeV [HPQCD] 

η 

1.5 

1 

0.5 

0 


B + → τ + ν 


γ 

α 

β 

∆m d 

• Nailing down f B will remain essential 

-0.5 

R u 

/ R t 

= const 

Recall: ∆m s remains important to constrain NP 

entering B s and B d mixing differently (not just 

to determine |V td /V ts |) 

-1 

C K M 

f i t t e r 

Lattice 2005 

Constraint from B + → τ + ν τ 

and ∆m d 

-1.5 

-1 -0.5 0 0.5 1 1.5 2 

(B → τν usually quoted as upper bounds) 

ρ 

• Error of Γ(B → τν) will improve incrementally (precise only at a super B factory) 

∆m s will be instantly accurate when measured 

Z. Ligeti — p. 34

¢¤£¦¥§£¤¨ 

© 

 

¡ 

Photon polarization in B → K ∗ γ 

• SM predicts B(B → X s γ) correctly to ∼10%; rate does not distinguish b → sγ L,R 

SM: O 7 ∼ ¯s σ µν F µν (m b P R +m s P L )b , therefore mainly b → s L 

Photon must be left-handed to conserve J z along decay axis 

Inclusive B → X s γ 

Exclusive B → K ∗ γ 

γ 

b 

s 

γ 

B K * 

Assumption: 2-body decay 

Does not apply for b → sγg 

... quark model (s L implies J K∗ 

z = −1) 

... higher K ∗ Fock states 

• Only measurement so far; had been expected to give S K ∗ γ = −2 (m s /m b ) sin 2β 

[Atwood, Gronau, Soni] 

Γ[B 0 (t) → K ∗ γ] − Γ[B 0 (t) → K ∗ γ] 

Γ[B 0 (t) → K ∗ γ] + Γ[B 0 (t) → K ∗ γ] = S K ∗ γ sin(∆m t) − C K ∗ γ cos(∆m t) 

• What is the SM prediction? 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 O 2 

Equal b → sγ L , sγ R rates at O(α s ); calculated to O(α 2 sβ 0 ) 

Inclusively only rates are calculable: Γ (brem) 

22 /Γ 0 ≃ 0.025 

Suggests: A(b → sγ R )/A(b → sγ L ) ∼ √ 0.025/2 = 0.11 

[Grinstein, Grossman, ZL, Pirjol] 

γ 

c 

b s 

O 2 

g 

• Exclusive B → K ∗ γ: factorizable part contains an operator that could contribute 

at leading order in Λ QCD /m b , but its B → K ∗ γ matrix element vanishes 

Subleading order: several contributions to B 0 → K 0∗ γ R , no complete study yet 

We estimate: 

A(B 0 → K 0∗ ( ) 

γ R ) 

A(B 0 → K 0∗ γ L ) = O C2 Λ QCD 

3C 7 m b 

∼ 0.1 

• Data: S K ∗ 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 m b ≫ Λ QCD limit, then worry about predictions 

– Need to test accuracy of expansion (even in B → ππ, |⃗p q | ∼ 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 f π is the leading order prediction 

Also holds in large N c , works at 5–10% level, need precise data to test mechanism 

B 0 → D + π − B − → D 0 π − B 0 → D + π − 

B − → D 0 π − B 0 → D 0 π 0 B 0 → D 0 π 0 

SCET: O(1) O(Λ QCD /Q) O(Λ QCD /Q) Q = {E π , m b,c } 

• Predictions: 

Predictions: 

B(B − → D (∗)0 π − ) 

B(B 0 → D (∗)+ π − ) = 1 + O(Λ QCD/Q) , 

B(B 0 → D 0 π 0 ) 

B(B 0 → D ∗0 π 0 ) = 1 + O(Λ QCD/Q) , 

data: ∼ 1.8 ± 0.2 (also for ρ) 

⇒ O(30%) power corrections 

[Beneke, Buchalla, Neubert, Sachrajda; Bauer, Pirjol, Stewart] 

data: ∼ 1.1 ± 0.25 

Unforeseen before SCET 

[Mantry, Pirjol, Stewart] 

Z. Ligeti — p. 38

Color suppressed B → D (∗)0 π 0 decays 

• Single class of power suppressed SCET I 

operators: T { O (0) , L (1) 

ξq , } 

L(1) ξq 

A(D (∗)0 M 0 ) = N M 0 

∫ 


(a) 

b 

d 

dz dx dk + 1 dk+ 2 T (i) (z) J (i) (z, x, k + 1 , k+ 2 ) S(i) (k + 

} {{ 

1 , k+ 2 

} 

) φ M (x) + . . . 

complex − nonpert. strong phase 

c 

u 

d 

d 

( s ) 

(b) 

b 

d 

u 

c 

u ( s ) 

u ( s ) 

Z. Ligeti — p. 39




ξq , } 

L(1) ξq 

A(D (∗)0 M 0 ) = N M 0 

∫ 


(a) 

b 

d 


} {{ 

1 , k+ 2 

} 

) φ M (x) + . . . 


c 

u 

d 

d 

( s ) 

(b) 

b 

d 

u 

c 

u ( s ) 

u ( s ) 

• Not your garden variety factorization formula... S (i) (k + 1 , k+ 2 

) know about n 

〈D 0 (v ′ )|(¯h (c) 

S (0) (k + 1 , k+ 2 ) = v ′ S) n/P L (S † h (b) 

v 

)( ¯dS) k 

+ n/P L (S † u) 

1 

k 

+ | ¯B 0 (v)〉 

2 

√ 

mB m D 

Separates scales, allows to use HQS without E π /m c = O(1) corrections 

(i = 0, 8 above) 

Z. Ligeti — p. 39




ξq , } 

L(1) ξq 

A(D (∗)0 M 0 ) = N M 0 

∫ 


(a) 

b 

d 


} {{ 

1 , k+ 2 

} 

) φ M (x) + . . . 


c 

u 

d 

d 

( s ) 

(b) 

b 

d 

u 

c 

u ( s ) 

u ( s ) 

• Ratios: the △ = 1 relations follow from naive 

factorization and heavy quark symmetry 

A(D*M) 

A(D M) 

D 0 π 0 D 

0 η 

0 0 

D K 

The • = 1 relations do not — a prediction of 

SCET not foreseen by model calculations 

1.0 

π - 

D 

Also predict equal strong phases between 

+ π -D0 D + Ḏ0 Κ 

- 

Κ D + ρ - D 0 ρ - 

0.5 

amplitudes to D (∗) π in I = 1/2 and 3/2 

SCET prediction 

Data: δ(Dπ) = (30 ± 5) ◦ , δ(D ∗ π) = (31 ± 5) ◦ 0.0 

2.0 

1.5 

color allowed 

color suppressed 

* ω + ω 

[Blechman, Mantry, Stewart] 

’ 

D 0 

η 

D 0 ρ 0 

D 0 

ω 

* 

Z. Ligeti — p. 39

Λ b and B s decays 

• CDF measured in 2003: Γ(Λ b → Λ + c π − )/Γ(B 0 → D + π − ) ≈ 2 

Factorization does not follow from large N c , but holds at 

leading order in Λ QCD /Q 

Γ(Λ b → Λ c π − ( 

) 

ζ(w 

Λ 

Γ(B 0 → D (∗)+ π − ) ≃ 1.8 max ) ) 2 

ξ(wmax D(∗) ) 

[Leibovich, ZL, Stewart, Wise] 

Isgur-Wise functions may be expected to be comparable 

Lattice could nail this 

• B s → D s π is pure tree, can help to determine relative size of E vs. C 

[CDF ’03: B(B s → D − s π+ )/B(B 0 → D − π + ) ≃ 1.35 ± 0.43 (using f s /f d = 0.26 ± 0.03)] 

Lattice could help: Factorization relates tree amplitudes, need SU(3) breaking in 

B s → D s l¯ν vs. B → Dl¯ν form factors from exp. or lattice 

Z. Ligeti — p. 40

More complicated: Λ b → Σ c π 

• Recall quantum numbers: 

Σ c = Σ c (2455), Σ ∗ c = Σ c(2520) 

multiplets s l I(J P ) 

Λ c 0 0( 1 2+ ) 

Σ c , Σ ∗ c 1 1( 1 2+ ), 1( 

3 

2 

+ ) 

• Can’t address 

in naive factorization, 

since 

Λ b → Σ c form factor 

vanishes by isospin 

[Leibovich, ZL, Stewart, Wise] 

O(Λ QCD /Q) O(Λ QCD /Q) O(Λ 2 QCD /Q2 ) 

• Prediction: 

Γ(Λ b → Σ ∗ cπ) 

Γ(Λ b → Σ c π) = 2 + O[ Λ QCD /Q , α s (Q) ] = Γ(Λ b → Σ ∗0 

c ρ 0 ) 

Γ(Λ b → Σ 0 cρ 0 ) 

Can avoid π 0 ’s from Λ b → Σ (∗)0 

c 

π 0 → Λ c π − π 0 or Λ b → Σ (∗)+ 

c π − → Λ c π 0 π − 

Z. Ligeti — p. 41

Charmless B → M 1 M 2 decays 

• Limited consensus about implications of the heavy quark limit 

A = A c¯c + N 

[Bauer, Pirjol, Rothstein, Stewart; Chay, Kim; Beneke, Buchalla, Neubert, Sachrajda] 

[ ∫ 

f M2 ζ BM 1 

du T 2ζ (u) φ M2 (u) 

A = A c¯c + N + f M2 

∫ 

dzdu T 2J (u, z) ζ BM 1 

J 

(z) φ M2 (u) + (1 ↔ 2) 

• ζ BM 1 

J 

] B M 

p 2 

~ Λ2 

M’ 

p 2 

~ 

2 

Λ 

p 2 ~ Q 2 

p 2 ~ QΛ 

= ∫ dxdk + J(z, x, k + ) φ M1 (x) φ B (k + ) also appears in B → M 1 form factors 

⇒ Relations to semileptonic decays do not require expansion in α s ( √ 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 

p 2 

~ 

2 

Λ 

• SCET: fit both ζ’s and ζ J ’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 +− = −λ u (T + P u ) − λ c P c − λ t P t = e −iγ T ππ − P ππ 

√ 

2A00 = λ u (−C+P u )+λ c P c +λ t P t = e −iγ C ππ + P ππ 

[many, many authors, sorry!] 

 

√ 

2A−0 = −λ u (T + C) = e −iγ (T ππ + C ππ ) 

¡¡ 

§ £© 

 

Alternatively, eliminate λ t terms, then e iβ P ππ 

′ 

Diagrammatic language can be justified in SCET at leading order 

• We know: arg(T/C) = O(α s , Λ/m b ), P u is calculable (small), 

¢¡¤£¦¥ §¨¡© 

– P t : “chirally enhanced” power correction in QCDF (treated like others by BPRS) 

– P c : 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 0 π 0 ) is large, so ∆α can be large, but C π 0 π0 is hard to measure 

• Can we use the theory constraint to determine α without C π 0 π 0? 

Z. Ligeti — p. 43

Phenomenology of B → ππ 

• Imposing a constraint on either ɛ ≡ Im(C ππ /T ππ ) or τ ≡ arg[T ππ /(C ππ + T ππ )] 

mixes “tree” and “penguin” amplitudes [expect ɛ, τ = O(α s , Λ/m b )] 

[Bauer, Rothstein, Stewart] 

• CKM fit ⇒ unexpectedly large τ (2σ) 

– large power corrections to T, C? 

– large up penguins? 

– large weak annihilation? 

1 – CL 

1.2 

1 

0.8 

0.6 

0.4 

0.2 

[Höcker, Grossman, ZL, Pirjol] 

isospin analysis 

without C 00 

τ (t) = 0 and w/o C 00 

|τ (t) | < 5 o , 10 o , 20 o and w/o C 00 

τ (t) = 0 and with C 00 


no α in fit 

0 

0 20 40 60 80 100 120 140 160 180 

1 – CL 

1.2 

1 

0.8 

0.6 

0.4 

0.2 

α 

(deg) 

20 o 

10 o 

5 o 

t-convention 

c-convention 

For α ∼ 90 ◦ , 

ɛ ∼ 0.2 ↔ τ ∼ 5 ◦ 

ɛ ∼ 0.4 ↔ τ ∼ 10 ◦ 

For a given τ, theo and exp 

errors highly correlated 

t-convention 

fit without using C 00 

1 

0.8 

0.6 

0.4 

0.2 

0 

1 – CL 

1.2 

t-convention 

c-convention 

May be more applicable to B → ρρ 0 

-60 -40 -20 0 20 40 60 

τ (deg) 

-60 -40 -20 0 20 40 60 

τ (deg) 

Z. Ligeti — p. 44

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 

– e.g., large B → π 0 π 0 rate in SCET accommodated by 〈k −1 

+ 〉 B = ∫ dk + φ B (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 

B → ψK S (β) ∼ 0.2 ◦ 1.6 ◦ 

B → φK S , η (′) K S , ... (β) ∼ 2 ◦ ∼ 10 ◦ 

B → ππ, ρρ, ρπ (α) ∼ 1 ◦ ∼ 15 ◦ 

B → DK (γ) ≪ 1 ◦ ∼ 25 ◦ 

B s → ψφ (β s ) ∼ 0.2 ◦ — 

B s → D s K (γ − 2β s ) ≪ 1 ◦ — 

|V cb | ∼ 1% ∼ 3% 

|V ub | ∼ 5% ∼ 15% 

B → Xl + l − ∼ 5% ∼ 20% 

B → K (∗) ν ¯ν ∼ 5% — 

K + → π + ν ¯ν ∼ 5% ∼ 70% 

K L → π 0 ν ¯ν < 1% — 

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/ψK S 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 B d 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

Additional l Topics

Further interesting CPV modes

B → ρρ vs. ππ isospin analysis 

• Due to Γ ρ ≠ 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 (m i = mass of a pion pair; B = Breit-Wigner) 

A ∼ B(m 1 )B(m 2 ) 1 [ 

f(m1 , m 2 ) ρ + (m 1 )ρ − (m 2 ) + f(m 2 , m 1 ) ρ + (m 2 )ρ − (m 1 ) ] 

2 

= B(m 1 )B(m 2 ) 1 { [f(m1 

, m 2 ) + f(m 2 , m 1 ) ][ ρ + (m 1 )ρ − (m 2 ) + ρ + (m 2 )ρ − (m 1 ) ] 

4 

} {{ } 

I=0,2 

+ [ f(m 1 , m 2 ) − f(m 2 , m 1 ) ][ ρ + (m 1 )ρ − (m 2 ) − ρ + (m 2 )ρ − (m 1 ) ] } 

} {{ } 

I=1 

If Γ ρ vanished, then m 1 = m 2 and I = 1 part is absent 

E.g., no symmetry in factorization: f(m ρ −, m ρ +) ∼ f ρ (m ρ +) F B→ρ (m ρ −) 

• Cannot rule out O(Γ ρ /m ρ ) contributions; no interference ⇒ O(Γ 2 ρ/m 2 ρ) 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 0 〉 = |b d〉, |B 0 〉 = |b d〉 

i d ( |B 0 ) 

(t)〉 

dt |B 0 = 

(t)〉 

( 

M − i 2 Γ )( |B 0 (t)〉 

|B 0 (t)〉 

) 

b 

d 

W 

t 

t 

W 

d 

b 

b 

d 

t 

W 

W 

t 

d 

b 

Mass eigenstates: |B L,H 〉 = p|B 0 〉 ± q|B 0 〉 

• CPV in mixing: Mass eigenstates ≠ CP eigenstates (|q/p| ≠ 1 and 〈B H |B L 〉 ≠ 0) 

Best limit from semileptonic asymmetry (4Re ɛ) 

[NLO: Beneke et al.; Ciuchini et al.] 

A SL = Γ[B0 (t) → l + X] − Γ[B 0 (t) → l − X] 

Γ[B 0 (t) → l + X] + Γ[B 0 (t) → l − X] = 1 − |q/p|4 = (−0.0026 ± 0.0067) 

1 + |q/p| 

4 

⇒ |q/p| = 1.0013 ± 0.0034 

Allowed range ≫ than SM region, but already sensitive to NP 

[dominated by BELLE] 

[Laplace, ZL, Nir, Perez] 

Z. Ligeti — p. ii

B s → ψφ and B s → ψη (′) 

• Analog of B → ψK S in B s decay — determines the phase between B s mixing 

and b → c¯cs decay, β s , as cleanly as sin 2β from ψK S 

β s is a small O(λ 2 ) angle in one of the 

“squashed” unitarity triangles 

1 

0.8 

[update of Laplace, ZL, Nir, Perez] 

C K M 

f i t t e r 

LP 2005 

sin 2β s = 0.0346 +0.0026 

−0.0020 

ψφ is a VV state, so the asymmetry is 

diluted by the CP -odd component 

ψη (′) , however, is pure CP -even 

1 – CL 

0.6 

0.4 

0.2 

1σ 

2σ 

0 

0.025 0.03 0.035 0.04 0.045 

sin(2β s 

) 

• Large asymmetry (sin 2β s > 0.05) would be clear sign of new physics 

Z. Ligeti — p. iii

B s → D ± s K∓ and B 0 → D (∗)± π ∓ 

• Single weak phase in each B s , B s → D ± s K ∓ decay ⇒ the 4 time dependent rates 

determine 2 amplitudes, strong, and weak phase (clean, although |f〉 ≠ |f CP 〉) 

A 

Four amplitudes: B 

1 

s → D s + K − A 

(b → cus) , B 

2 

s → K + Ds 

− 

(b → ucs) 

A 

Four amplitudes: B 

1 

s → Ds − K + A 

(b → cus) , B 

2 

s → K − D s 

+ (b → ucs) 

A D 

+ s K − 

= A ( 

1 Vcb V ∗ ) 

A 

us 

D 

− 

A D 

+ s K − A 2 Vub ∗ V , 

s K + 

= A ( 

2 Vub V ∗ ) 

cs 

cs A D 

− s K + A 1 V ∗ 

cb V us 

Magnitudes and relative strong phase of A 1 and A 2 drop out if four time dependent 

rates are measured ⇒ no hadronic uncertainty: 

( ) V 

∗ 2 ( 

λ D 

+ s K − λ D 

− s K + = tb 

V ts Vcb V ∗ )( 

us Vub V ∗ ) 

cs 

V tb Vts 

∗ Vub ∗ V cs Vcb ∗V 

= e −2i(γ−2β s−β K ) 

us 

• Similarly, B d → D (∗)± π ∓ determines γ + 2β, since λ D + π − λ D − π + = e−2i(γ+2β) 

... ratio of amplitudes O(λ 2 ) ⇒ small asymmetries (and tag side interference) 

Z. Ligeti — p. iv

A near future (& personal) best buy list 

• β: reduce error in B → φK S , η ′ K S , K + K − K S (and D (∗) D (∗) ) modes 

• α: refine ρρ (search for ρ 0 ρ 0 ); ππ (improve C 00 ); ρπ Dalitz 

• γ: pursue all approaches, impressive start 

• β s : is CPV in B s → ψφ small? 

• |V td /V ts |: B s mixing (Tevatron may still have a chance) 

• Rare decays: B → X s γ near theory limited; B → X s l + l − is becoming comparably 

precise 

• |V ub |: reaching < ∼ 10% will be very significant (a Babar/Belle measurement that 

may survive LHCB) 

• Pursue B → lν, search for “null observables”, a CP (b → sγ), etc., for enhancement 

of B (s) → l + l − , etc. 

(apologies if your favorite decay omitted!) 

Z. Ligeti — p. v

More slides removed

£ 

¥ 

£ 

¦ 

¢ 

¤ £ 

¡ 

£ 

 

© 

¨¨ 

 

¨¨ 

© £ 

 

£ 

¡ 

£ 

£ 

¦ 

¢ 

∆m K , ɛ 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 ɛ K 

§ § § § § § 

¤ ¥ 

|ɛ K | ∼ 

∣ Im M 12∣∣∣ 

∣ ∼ 

∆m K 

g 2 Λ 3 QCD 

∣MX 2 ∆m ∣ ⇒ M X ∼ g × 6 · 10 4 TeV 

K 

Similarly, from B 0 − B 0 mixing: M X ∼ g × 3 · 10 2 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 

• (∆m K) SUSY 

(∆m K ) EXP 

( ) 2 ( ) 1 TeV ∆ ˜m 

2 2 ∼ 12 

104 ˜m ˜m 2 Re [ (KL) d 12 (KR) d ] 

12 

KL(R) d : mixing in gluino couplings to left-(right-)handed down quarks and squarks 

Constraint from ɛ K : replace 10 4 Re [ (KL d) 12(KR d ) 12] 

with ∼ 10 6 Im [ (KL d) 12(KR d ) ] 

12 

• Solutions to supersymmetric flavor problems: 

(i) Heavy squarks: ˜m ≫ 1 TeV 

(ii) Universality: ∆m 2˜Q, ˜D 

≪ ˜m 2 

(GMSB) 

(iii) Alignment: |(K d L,R ) 12| ≪ 1 (Horizontal symmetry) 

The CP problems (ɛ (′) 

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 (ɛ K accommodated with O(1) CKM phase) 

Hadronic uncertainties preclude precision tests (ɛ ′ K 

notoriously hard to calculate) 

• K → πνν: Theoretically clean, but rates small B ∼ 10 −10 (K ± ), 10 −11 (K L ) 

⎧ 

⎪⎨ (λ 5 m 2 t) + i(λ 5 m 2 ¡ 

t) t : CKM suppressed 

A ∝ (λ m 2 c) + i(λ 5 m 2 c) c : GIM 

¤¦¥¨§©¥ 

suppressed 

⎪⎩ 

(λ Λ 2 QCD ) u : GIM suppressed 

£ ¦¨© 

¢ 

 

 

 

So far three events observed: B(K + → π + ν¯ν) = (1.47 +1.30 

−0.89 ) × 10−10 

• 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: 

y CP = 

Γ(CP even) − Γ(CP odd) 

Γ(CP even) + Γ(CP odd) 

= (0.9 ± 0.4)% [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 

ρ 0 ρ + 0.95 ± 0.11 0.97 +0.05 

−0.08 

ωρ + 8 +0.12 

−0.15 

ρ 0 K ∗+ 0.96 +0.06 

−0.16 

ρ − K ∗0 0.43 

−0.11 +0.12 0.79 ± 0.09 

φK ∗0 0.45 ± 0.05 0.52 ± 0.05 

φK ∗+ 0.52 ± 0.09 0.46 ± 0.12 

s 

Chiral structure of SM and 

HQ limit claimed to imply 

b 

f L = 1 − O(1/m 2 b ) [Kagan] 

s 

(s b) V-A (s s) V-A 

φK ∗ : penguin dominated — NP reduces f L ? 

s 

d 

Proposed explanations: 

b 

d,s 

c 

.... 

α s(mv) 

α s ( ) 

c 

q µ 

q 

q 

d 

(d b) S-P (s d) S+P 

b 

s 

s 

s 

d 

B 

✲ 

D (∗) 

s 

✒ 

 

❅ 

❘ ❅❅❅ 

D (∗) 

✈ ✲ 

D s 

(∗) 

✻ 

✈ 

✲ 

φ 

K ∗ 

¢¡¢ 

¢¡¢ 

¢¡¢ 

K* T s 

g* T 

b 

φ T 

O(1) O(1/m 2 b 

) × large (model) (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? 

q 

¢¡¢ 

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 ] A CP 

B 0 → π + K − 18.2 ± 0.8 −0.11 ± 0.02 

B − → π 0 K − 12.1 ± 0.8 +0.04 ± 0.04 

B − → π − K 0 24.1 ± 1.3 −0.02 ± 0.03 

B 0 → π 0 K 0 11.5 ± 1.0 +0.00 ± 0.16 

[Fleischer & Mannel, Neubert & Rosner; Lipkin; Buras & Fleischer; Yoshikawa; Gronau & Rosner; Buras et al.; ...] 

R c ≡ 2 B(B+ → π 0 K + ) + B(B − → π 0 K − ) 

B(B + → π + K 0 ) + B(B − → π − K 0 ) 

= 1.00 ± 0.08 

R n ≡ 1 2 

B(B 0 → π − K + ) + B(B 0 → π + K − ) 

B(B 0 → π 0 K 0 ) + B(B 0 → π 0 K 0 ) 

= 0.79 ± 0.08 

R ≡ B(B0 → π − K + ) + B(B 0 → π + K − ) 

B(B + → π + K 0 ) + B(B − → π − K 0 ) 

τ B ± 

τ B 0 

= 0.82 ± 0.06 ⇒ FM bound : γ < 75 ◦ (95% CL) 

R L ≡ 2 ¯Γ(B − → π 0 K − ) + ¯Γ(B 0 → π 0 K 0 ) 

¯Γ(B − → π − K 0 ) + ¯Γ(B 0 → π + K − ) 

= 1.12 ± 0.07 

• Pattern quite different than until 2004: R c 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

The CKM matrix and CP Violation

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