The Unitarity Triangle Angle γ
An Overview
Tim Gershon
University of Warwick
11th September 2008
CKM2008, Roma, Italy
Content
●
Why measure γ?
–
●
How to measure γ?
–
●
How precisely do we need to measure it?
The theoretically pristine B→DK approach
Where do we stand & what remains to be done?
–
Introduce/provoke talks in the working group
CKM2008 11th September 2008
2
Why Measure γ?
●
Name of the game in flavour physics is to
overconstrain the CKM matrix
measure fundamental parameters
●
constrain new physics effects
Measure the 4 free parameters in various ways
–
CP conserving
{|Vus|, |Vcb|, |Vtd|, |Vub|}
–
CP violating
{εK, φs, β, γ}
–
Tree level
{..., ..., |Vub|, γ}
–
Loop processes
{..., ..., |Vtd|, β}
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... many other possible combinations3
Unitarity Triangle Comparisons
“tree”
“loop”
CP conserving
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CP violating
4
Importance of γ
●
γ plays a unique role in flavour physics
the only CP violating parameter that can be
measured through tree decays
(*)
(*)
●
A benchmark Standard Model reference point
–
●
more-or-less
doubly important after New Physics is observed
How precise is precise enough?
–
–
–
10% Ⓧ
1% ☆
0.1% Ⓧ
CKM2008 11th September 2008
At 3 sigma hardly exclude anything
Seems the right level to test NP
Good luck if you can get the funding ...
5
How To Measure γ
●
Focus on theoretically pristine measurement
–
Interference between
∗
∝ V cb V us
●
●
∗
∝ V ub V cs
colour allowed
final state contains D 0
●
●
colour suppressed
final state contains D 0
Relative magnitude of suppressed amplitude is rB
Relative weak phase is –γ, relative strong phase is δB
CKM2008 11th September 2008
6
One Method, Many Modes
●
●
B→DK with any D decay mode that is
—0
0
accessible to both D and D is sensitive to γ
–
M.Gronau & D.Wyler, PLB 253, 483 (1991)
–
M.Gronau & D.London, PLB 265, 172 (1991)
–
D.Atwood, I.Dunietz and A.Soni, PRL 78, 3257 (1997); PRD 63, 036005 (2001)
Different D decay modes in use
–
–
–
–
–
CP eigenstates
(eg. K+K–, KSπ0)
“GLW”
Doubly-suppressed decays (eg. Kπ)
“ADS”
Singly-suppressed decays (eg. KK*)
“GLS”
Three-body decays
(eg. Ksπ+π–) “GGSZ / Dalitz”
Other possibilities exist ...
CKM2008 11th September 2008
PRD 67, 071301 (2003)
PRD 68, 054018 (2003) & Belle
7
How It Works
●
Consider D→CP eigenstates as an example
1
1r B ei   −  

2
1
A  B −  D2 K −  ∝
1−r B ei   −  

2
1
A  B D 1 K   ∝
1r B ei     

2
1
A  B   D2 K   ∝
1−r B ei     

2
A  B− D 1 K −  ∝
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B

  B −  D1 K −  ∝ 1r 2B2 r B cos B−
B

  B −  D2 K −  ∝ 1r 2B−2 r B cos B−
B

  B   D1 K   ∝ 1r 2B2 r B cos B
B

  B   D2 K   ∝ 1r 2B−2 r B cos B
In practice, measure asymmetries and ratios
where possible to reduce systematics
8
Theoretically Pristine
●
Try to draw a diagram for B–→DK– with a different
weak phase to those on previous slide
–
●
if you succeed, estimate effect on γ extraction
Largest effects due to
charm mixing
–
charm CP violation
PRD 72 031501 (2005)
B mixing (ΔΓ) for neutral B decays
–
⇒ can be controlled
●
}
–
BUT,
–
negligible
PRD 69 113003 (2004), PLB 649 61 (2007)
must obtain hadronic parameters from data
Includes (rB, δB) as well as D decay model
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9
Even more modes ...
●
As well as different D decays, can use
–
–
Different B decays (DK, D*K, DK*)
●
different hadronic factors (rB, δB) for each
●
benefit from D*→Dπ0 & D*→Dγ
●
some care required due to K* width
different hadronic factors (rB, δB) – larger rB
Useful rule-of-thumb: NIMSBHO principle (A.Soffer)
–
●
PLB 557, 198 (2003)
Neutral B decays
●
●
PRD 70, 091503 (2004)
Not Inherently More Sensitive But Helps Overall
Best sensitivity by combining all measurements
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10
Where Do We Stand?
Total data sizes to date: BABAR 465 M BB, Belle ~800 M BB, CDF ~4/fb
only charged B results shown
Huge & impressive effort
● Still much to be done
●
CKM2008 11th September 2008
NEW today!
see V.Tisserand's talk
11
How to Determine D Decay Models
●
To extract γ need to understand D decays
ψ(3770) → DD (CLEOc & BESIII)
–
GLW modes
●
–
need constraints on direct CP violation in D decay
ADS modes
●
rD – can be measured from flavour-tagged D mesons
●
δD – need CP tagged D mesons
multibody modes have additional coherence parameter
NEW tomorrow – see J.Libby's talk
Dalitz modes
● model dependent – need model, including phase variation
● model independent – need c and s parameters
i
i
●
–
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NEW tomorrow – see J.Rademacker's talk
12
Model-Independent Dalitz Measurements
●
Revert to a “counting” analysis by binning the DP
–
–
●
●
PRD 68, 054018 (2003)
EPJC47, 347 (2006)
& arXiv:0801.0840
Model not needed ... BUT
Best to use model to define bins
Measure cos and sin of average D0 – D0 strong phase
difference in each bin
–
ci from KSπ+π– vs. CP tags
–
si (and ci) from KSπ+π– vs. KSπ+π–
}
ψ(3770) → DD
(CLEOc & BESIII)
How to implement this?
–
–
Define common model & binning; expts count events per bin
OR Expts make data available in common format
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13
Alternatives to B→DK
●
B→D(*)π measures sin(2β+γ) (interference between
decays with and without mixing)
PLB 427, 179 (1998)
–
rB very small (~0.02)
●
●
●
small modulation on a large signal ⇒ systematics!
cannot extract rB (only rB2) ⇒ need input, eg. SU(3)
B→D*ρ similar but can get rB from interference
between helicity amplitudes
–
●
PRL 80, 3706 (1998)
but now you need to deal with slow π/π0 related systematics
Bs→DsK similar (measures sin(φs+γ)) but now rB is larger
–
a promising channel for LHCb
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NPB 671, 459 (2003)
14
Definitions of parameters
●
GLW:
ACP± =
  B−  D ± K −  −  B D ± K  
  B  D ± K    B D ± K
−
−
  B  D ± K   B D ± K
−
RCP±=
●
ADS:
A ADS =
R ADS =
●
Dalitz:
●
Notes
–
–
–

−


  B D fav K   B D fav K
−
−

  B−  D ADS K −  −  B   D ADS K  
  B−  D ADS K −    B  D ADS K  
  B  Dfav K    B  Dfav K
−

x  = r B cos B
y  = r B sinB



±2 r B sin B sin 
1r 2B±2 r B cos B cos 
2


=
  B−  D ADS K −    B   D ADS K  
−


=
= 1r B ±2 r B cos B cos 
2 r B r D sin B D sin 
r 2Br 2D 2 r B r D cos B  D  cos 
= r 2Br 2D 2 r B r D cos B D cos 
x − = r B cos B−
y − = r B sin B−
Dalitz parameters (x+,y+), (x–, y–) from B+, B– independently
GLW-Dalitz relation: x ± =  RCP  1∓ ACP −RCP− 1∓A CP−   /4
Additional coherence factors needed for DK* or D→Kππ0
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15
Putting It All Together
●
Evidence for CP violation in both GLW & Dalitz
GLW results can be translated into x+ = –0.082 ± 0.045, x– = 0.103 ± 0.045, rB2 = 0.08 ± 0.07
Combining GLW & Dalitz gives x+ = –0.085 ± 0.026, x– = 0.103 ± 0.027
CKM2008 11th September 2008
x– – x+ = 0.189 ± 0.037 5.1σ from zero
16
Making Constraints on γ
●
Previous discussion suggests we should get a
fairly precise world average for γ
(at least, neglecting model uncertainties)
●
However, extracting γ is non-trivial
–
●
simple trigonometry fails (beware non-Gaussian errors)
Complicated statistical treatment is necessary
–
From Dalitz modes,
●
●
PRD 78 (2008) 034023
BaBar obtain γ = (76 ± 22 ± 5 ± 5)° (from DK−, D*K− & DK*−)
Belle obtain φ3 = (76
CKM2008 11th September 2008
arXiv:0803.3375
+12
−13
± 4 ± 9)° (from DK− & D*K−)
17
Concluding Questions
●
●
Why has neither experiment published a combined
constraint on γ from its B→DK measurements?
What auxiliary measurements should be made?
–
●
How can we solve the problem of model dependence in
Dalitz plot analyses?
–
●
eg. DK* hadronic parameters in DKSπ DP analysis?
Will we reach necessary agreement to enable model
independent analysis?
How much data is needed before statistical issues in γ
extraction become irrelevant?
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18
B→DK
●
●
Following
*
PLB 557 198 (2003)
Suppressed & favoured amplitudes vary across
B→DKπ phase space in both magnitude and phase
A CP± =
ACP± =
±2 r B sin  B sin 
2
R CP± = 1r B ±2 r B cos  B cos  
1r 2B±2 r B cos B  cos 
±2 r s sin s sin 
2
R CP± = 1r s ±2 r s cos  s cos 
1r 2s ±2  r s cos  s cos 
rs =

2
∫K ∣A∣ dPS
∫K ∣A∣2 dPS
∗
∗
e
i s
=
∫K ∣ A∣∣A∣ ei  arg A −arg  A   dPS
2
2
∣
A
∣
dPS
∣
A
∣
dPS
∫
∫
 K
K
∗
∗
∗
–
rs, κ, δs depend on K* selection
–
In DKSπ– do not expect (DKS) or (Dπ–) resonances
CKM2008 11th September 2008
19
0
D→Kππ , D→Kπππ
●
●
●
Following PRD 68 033003 (2003)
Suppressed & favoured amplitudes vary across phase
space in both magnitude and phase
Usual expressions get modified
A ADS =
2 r B r D sin B D sin 
2
2
r B r D 2 r B r D cos  B  D cos
F
A ADS =
R ADS = r 2B r 2D 2 r B r D cos  B D cos
F
2 r B R F r D sin BD sin 
2
r 2B r FD 2 2 r B R F r D cos  B  FD cos
F
rD =

∫PS ∣A∣2 dPS
2
∫PS ∣A∣ dPS
CKM2008 11th September 2008
F 2
F
R ADS = r B r D  2 r B R F r D cos B  D cos  
−i  arg A −arg  A  
∣
A
∣∣
A
∣
e
dPS
∫
−i 
PS
RF e =
2
2
∣
∣
∣
∣
A
dPS
A
dPS
∫
∫
 PS
PS
F
D
Coherence factor
measure from ψ(3770) → DD
20
B→DK, D→πππ
●
●
0
In typical Dalitz analysis, parameters (x+,y+), (x–, y–)
determined independently from B+, B– samples
However, imagine extreme example: D→(XYZ)CP
DP distributions contain no sensitivity to γ
– Rates & asymmetries are sensitivity to γ (as GLW)
Parameter of “CP-specificity”
–
●
x 0 = −∫DP ℜ A A ∗ dDP
–
–
x0 = 0.850 for D→πππ0
BaBar fit for ±=∣z ±−x 0∣ ±=tan
PRL 99 (2007) 251801
CKM2008 11th September 2008
−1

z ±= x±iy ±
ℑ  z±
ℜ z±− x 0

21