LHCb Physics prospects
description
Transcript of LHCb Physics prospects
LHCb Physics prospects
Marta CalviUniversità Milano-Bicocca and INFN
On behalf of the LHCb collaboration
Interplay of Collider and Flavour Physics WorkshopCERN,3-4 December 2007
M. Calvi Interplay of Collider and Flavour Physics, 04/12/2007 (2/27
LHCb detector in place
Muon detector
Calorimeters
RICH-2
Magnet
OT
VELO
RICH-1
The cavern is full: construction on schedule. Commissioning ongoing. Will be ready for data-taking at 2008 LHC start-up.
M. Calvi Interplay of Collider and Flavour Physics, 04/12/2007 (3/27
A possible running scenario2008 Calibration and Trigger commissioning Assume to integrate ~ 0.1 fb–1
2009 Start first significant physics data taking: >~ 0.5 fb–1
2010– Stable running. Expect ~ 2 fb–1/ year
LHCb should collect an integrated luminosity 10 fb–1 for year 2014±1
If found to be advantageous for physics, pushaverage luminosity from 21032 to 51032 cm–2s–1
pp interactions/crossing
LH
Cb
n=0
n=1
M. Calvi Interplay of Collider and Flavour Physics, 04/12/2007 (4/27
Physics with “first year” data (0.5 fb–1)
Very interesting Bs results already with first 0.5 fb–1
Examples are:
CP violation in Bs J/s measurement )
Search for Bs decays, extending CDF+D0 limit
s0(AFB=0) in Bd K0* overtaking B-Factories
Before, high statistics channels will be used to calibrate the detector
performance and to demonstrate LHCb physics capabilities
Huge bb production at s=14TeV, in the forward region
230 b in 300 mrad
Corresponding to 1011 bb events in L=0.5 fb-1
bb angular correlation in
pp collisions at s = 14 TeV
(Pythia)
M. Calvi Interplay of Collider and Flavour Physics, 04/12/2007 (5/27
LHCb expected performance
Results obtained from MC simulation (full Detector simulation):
b-decay track resolutions: impact parameter 30 m momentum resolution 0.36%
Particle ID performance with RICH
kaon ID eff. 88%, pion mis-ID 3%
Good K/ separation in 2-100 GeV/c range
Flavour Tagging performance
from combination of several methods:
D2 =45% for Bd
D2 =79 % for Bs depending on channel
Reconstructed B
resolutions: mass res. 14-18 MeV/c2
proper time res. 40 fs
Qvtx
Bs
B–
D
l-K–
K+PV
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The search strategy for New Physics Measurements with New Physics discovery potential
FCNC transitions and rare decays, where standard model contributions are
suppressed enough to allow potential small NP effects to emerge:
• Bs mixing phase (s)
• Very rare leptonic decays: eg. Bs
• Rare semi-leptonic decays: b sℓℓ (eg. Bd K0*BuKee/Bu K)
• Radiative decays: b s (eg. BdK*BsB , …)
• LFV decays (eg. Bs,d e
• CPV in D0 decays and rare D decays….
• Compare gamma from B(s) D(s) K decays and gamma from Bd and Bs KK
• Compare sin(2) from Bd J/KS and sin(2) from Bd KS
• Hadronic penguin b sssdecays ( eg.Bs
Precision measurements of CKM parameters, including angle
determination from tree level decays.
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B Mixing phases with bccs
2sinhcos
2cosh
)sin(sin)(
tt
tmtA
SSf
S
SSfCP
• Very small in SM: s = -2 = -0.037 ± 0.002 rad
• Could be much larger if New Physics contributes to Bs0-Bs
0 transitions
• No CP violation observedyet: s= 0.79 ± 0.56stat+ 0.01
-0.14 syst D0 with 1.1 fb-1
Measure time-dep. asymmetry in decay rates:
Use flavour tagged and untagged events.
Need very good proper time resolution to resolve Bs
0 oscillations.
Non pure CP modes (as BsJ/ ) need angular analysis to disentangle the mixture of CP-even (f=-1, A0, A||) and CP odd (f=+1, A)
1-angle analysis: tr
Increased precision from full 3-angles analysis under study.
total
CP even
CP oddflat background
M. Calvi Interplay of Collider and Flavour Physics, 04/12/2007 (8/27
BsJ/and CP eigenstatesBsJ/()(KK) is the golden channel. Can add pure CP modes, but much lower statistics.
Decay Channel Yield
2fb-1
B/S Sensitivity s
J/(-+) () 8.5k 2.0 0.109
J/(-+) (+-0) 3.0k 3.0 0.142
J/(-+) ’(0) 4.2k <0.42 0.080
c(h-h+h-h+)(K+K-) 3.0k 0.6 0.108
Ds(K+K--) Ds(K+K-+) 4.0k 0.3 0.133
Pure CP modes 0.048
J/() 131k 0.12 0.023
All modes 0.021
Parameter Sensitivity with 2 fb-1
Γs/Γs 0.0092
RT 0.00040
After 10 fb-1 stat(s) = 0.009
> 3 evidence of non-zero s, even if only SM
Sensitivity to other fit parameters (from J/)
In 0.5 fb–1: (s)0.046
from BsJ/()
M. Calvi Interplay of Collider and Flavour Physics, 04/12/2007 (9/27
New Physics in Bs mixing
2006 including ms measurement
With s from
Bs J/LHCb 2fb-1
Ligeti, Paucci,Perez, hep-ph/0604112
M12 1 hs exp(2is) M12SM
New Physics in Bs mixing amplitude M12 parameterized with hs and s:
Additional constraints can come from semileptonic Asymmetry. In SM: ASL
s 10-5.
hs
s
Preliminary results on the LHCb measurement of time dependent charge asymmetry in BsDs
Expect 109 events/ 2fb-1 (ASLs )2x10-3 in 2fb-1
ASLs
M. Calvi Interplay of Collider and Flavour Physics, 04/12/2007 (10/27
Bs
- Highly suppressed in SM: BR(Bs→) =(3.55±0.33)x10-9
- Could be strongly enhanced by SUSY:
BR(Bs→ ) tan6/MH2
- Current Limit Tevatron ~2 fb-1: CDF BR < 4.7 x 10-8 90% CL D0 BR < 7.5 x 10-8 90% CL
- Within Constrained MSSM: current g-2 measurement (which deviates by 3.4 from SM) suggest gaugino mass in the range 450-650GeV at tan50 BR(Bs→ ) in the range 10-8 to10-7
SM prediction
W
W
b
s
t
SM
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Bs LHCb: high efficiency trigger for the signal, but main issue is background rejection.
Exploit good mass resolution and vertexing, and good particle ID.
Largest background is b , b . Specific background dominated by Bc
± →J/
Analysis in a Phase Space with 3 axis: • Geometrical Likelihood (GL) (impact parameters, distance of closest approach between , lifetime, vertex isolation) • Particle-ID Likelihood • Invariant Mass Window around Bs peak
- Sensitive Region: GL > 0.5
- Divide in N bins
- Evaluate expected number of events for signal/background in each bin.
signalbb inc.b μ, b μBc
+ J/Ψμν
(arbitrary normalization)
M. Calvi Interplay of Collider and Flavour Physics, 04/12/2007 (12/27
Integrated luminosity (fb–1)
BR
(x1
0–9)
Uncertainty in background prediction
Expected final CDF+D0 limit
SM prediction
90% CL imit on BR (only bkg is observed)
Integrated luminosity (fb–1)
5 observation
3 evidence
SM predictionBR
(x1
0–9)
Sensitivity to signal(signal+bkg is observed)
10-7
2x10-8
(~0.05 fb-1)
5x10-9
(~ 0.4 fb-1)
J.Ellis et al. arXiv:0709.0098v1 [hep-ph] (2007)
2 fb–1 3 evidence of SM signal
6 fb–1 5 observation of SM signal
0.5 fb–1 exclude BR values down to SM
Bs
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b sℓℓSuppressed loop decay in SM.
NP could contribute at the same levels, could modify BR and angular distributions.
Sensitive to SUSY, gravitation exchange, extra-dimensions.
Inclusive decay difficult to access at hadron collider.
Good prospects for exclusive decays (B Kℓℓ, K*ℓℓ ). Hadronic uncertainty reduced in:
Predicted shift in the zero of the AFB in Bd→K*in ACD model with a single universal extra dimension, a MFV model.(Colangelo et al PhysRevD73,115006(2006))
SM prediction
• Forward-backward asymmetry AFB
• Position of zero crossing of AFB (s0)
• Transversal asymmetries• Ratio of and ee modes
s0= m2 for which AFB=0
M. Calvi Interplay of Collider and Flavour Physics, 04/12/2007 (14/27
Bd K*
Fast MC, 2 fb–1
AF
B(s
)
s = m2 [GeV2]
L= 2fb–1 (s0) = ±0.46 GeV2
L=10 fb–1 (s0) = ±0.27 GeV2 at the level of present theoretical precision
Measure forward-backward Asimmetry as a function of the invariant mass. Determine s0, the m2
for which AFB=0+
–
In SM: BR(BdK*)=(1.22 +0.38-0.32) x10-6 s0= s0(C7,C9)=4.39 +0.38
-0.35 GeV2 Beneke et al hep-ph/0412400
LHCb: 7200 signal events/2fb–1
Bbb/S = 0.2 ± 0.1 (ignoring non-resonant
K events for the time being).
With L= 0.5 fb–1 1800 events (B-Factories projected 2ab-1 yield 450 events.)
M. Calvi Interplay of Collider and Flavour Physics, 04/12/2007 (15/27
Fit to full angular distributions (K-, , ) expressed in
terms of transversity amplitudes (A, A//, A0). Can measure:
BdK*transverse asymmetries
2
0
2
||
2
2
02 )(AAA
AqFL
2
||
2
2
||
2
2)2( )(AA
AAqAT
2 fb–1 10 fb–1
AT(2) 0.42 0.16
FL 0.016 0.007
AFB 0.020 0.008
Stat. precisions in the region s = m2
[1, 6] (GeV/c2)2
where theory calculations are most reliable
Curves from Lunghi & Matias JHEP 0704(2007)058. Points LHCb 2 fb-1
2 fb-1
Asymmetry AT(2) Longitudinal polarization FL
SM NLOMSSM tan=5MSSM tan=5
2 fb-1
Sensitivity with
M. Calvi Interplay of Collider and Flavour Physics, 04/12/2007 (16/27
RK in B+ K+ℓℓ
001.012max
2
2max
2
4
)(
4
)(
ds
dsR
q
m dseKeBd
q
m dsKBd
K
- Large corrections O(10%) possible in models that distinguish between lepton flavours (eg.MSSM at large tan). Constraints to NP also from RK andBR(Bs→µµ) combined.
in SM (Hiller,Krüger PRD69 (2004) 074020)
LHCb 10 fb-1
Bu eeK 9.2 k events
Bu K 19 k events
stat(RK )= 0.043
• Trigger eff 70% on ee channel
under study - not included.
• Similar sensitivity expected for RK* =Bd K*/ Bd eeK*.
• To be compared with 15% error on RK&RK* combined, expected from B-Factories
with 2ab-1.
4m2< mll
2< 6 (GeV/c2)2
M. Calvi Interplay of Collider and Flavour Physics, 04/12/2007 (17/27
Radiative decays
DecayYield
2 fb–1Bbb/S
Bd K* 68k 0.60
Bs 11.5k < 0.55
b (1116) 0.75k < 42
b (1670) 2.5k < 18
Bd K* ACP < 1% in SM, up to 40% in SUSY
Can measure at <% level.
Reference channel for all radiative decays.
Bs No mixing-induced CP asymmetry
in SM, up to 50% in SUSY.
Sensitivity for ACP(t) measurement under study.
b Right-handed component of photon
polarization O(10%) in SM. Can be higher BSM.
Measure photon asymmetry from angular distributions of and hadron in b (p,pK) decays.
3 evidence of right-handed component to 21% with 10 fb–1
M. Calvi Interplay of Collider and Flavour Physics, 04/12/2007 (18/27
sin(2) from Tree and Penguing
B0 KS:– 920 signal events per 2 fb–1, B/S < 1.1 at 90% CL
– After 10 fb–1: stat(sin(2eff)) = 0.10
to be compared with 0.12 expected from B-Factories with 2ab-1
AC
P(t
)
ACP(t)N B 0 J /KS N B0 J /KS N B 0 J /KS N B0 J /KS
2 fb–1 (fast MC, incl. bkg)
Proper time t (ps)
B0 J/KSTime dependent CP asymmetry in B0J/KS is expected to be one of the first CP measurements at LHCb.
236k signal events / 2 fb–1 B/S=0.6(bb)+7.7(J/)
stat(sin(2)) = 0.020 in 2 fb–1
Compare to 0.019 expected from B-Factories with 2ab-1
After 10 fb–1: (sin(2)) ~ 0.010 Can also push further the search for direct CP violating term cos(mdt)
M. Calvi Interplay of Collider and Flavour Physics, 04/12/2007 (19/27
bsss hadronic penguin decays
Bs
CP violation < 1% in SM due to cancellation of the
mixing and penguin phase
In SM: Bs SM 2 arg(Vts
*Vtb) – arg(VtbVts*/Vtb
*Vts) 0
NP in general affect differently Bs mixing and bs decays
NP≠0
Combining Bs with Bs
J/ measurements can disentangle NP contributions in mixing & decays.
+ NP?
J/
s
W -b
s
ccsBs
LHCb expects 3.1k signal events / 2 fb–1 (BR=1.410–5), B/S<0.8 at 90%CL
From time dependent angular distribution of flavour tagged events:
stat(NP) = 0.05 in 10 fb–1
M. Calvi Interplay of Collider and Flavour Physics, 04/12/2007 (20/27
Different ways to at LHCb
B mode D mode Method
Bs→DsK KK tagged, A(t)
B+→D K+ K+ K3 KK/ counting, ADS+GLW
B+→D*K+ K counting, ADS+GLW
B+→D K+ Ks Dalitz, GGSZ
B+→D K+ KK 4 body Dalitz
B+→D K+ K 4 body Dalitz
B0→D K*0 K + KK + counting, ADS+GLW
B→,KK Tagged, A(t)
tree decays only
M. Calvi Interplay of Collider and Flavour Physics, 04/12/2007 (21/27
from Bs DsK
Two tree decays which interfere via Bs mixing
Can determine +s, hence in a very clean way
LHCb expects 6.2k signal events in 2 fb–1 Bbb/S < 0.18 at 90% CL Bs Ds
–+ background 15±5 % after PID cuts
Fit 4 tagged and 2 untagged time-dependent rates.
With 10 fb–1: Tagged & untagged
Tagged only
s + 4.6 5.7
4.6 5.4
|| 0.027 0.029
(Inputs: =60, strong phase difference =0, amplitude ratio ||=0.37 )
DsK
tagged as Bs
s
s
b
c
u
s
Bs0
Ds
K
K–
s
s
b
u
c
s
Bs0
Ds
M. Calvi Interplay of Collider and Flavour Physics, 04/12/2007 (22/27
Different ways to at LHCb
B mode D mode Method () 2fb-1
Bs→DsK KK tagged, A(t) 10º
B+→D K+ K+ K3+KK/
counting, ADS+GLW 5º - 13º
B+→D*K+ K counting, ADS+GLW Under study
B+→D K+ Ks Dalitz, GGSZ 7-12º
B+→D K+ KK 4 body Dalitz 18º
B+→D K+ K 4 body Dalitz Under study
B0→D K*0 K + KK + counting, ADS+GLW 9º
B→,KK Tagged, A(t) 10º
Combined LHCb sensitivity to with tree decays only (educated guess):
~ 5º with 2 fb–1 ~ 2.5º with 10 fb–1
tree decays only
M. Calvi Interplay of Collider and Flavour Physics, 04/12/2007 (23/27
Charm physics• LHCb will collect a large tagged D*D0
sample (also used for PID calibration). A dedicated D* trigger is foreseen for this purpose.
– Tag D0 or anti-D0 flavour with pion from D*± D0 ±
• Performance studies not as detailed as for B physics.
• Interesting (sensitive to NP) & promising searches/measurements:– Time-dependent D0 mixing with wrong-sign D0K+– decays– Direct CP violation in D0K+K–
• ACP 10–3 in SM, up to 1% (~current limit) with New Physics• Expect stat(ACP) ~ O(10–3) with 2 fb–1
– D0+–
• BR 10–12 in SM, up to 10–6 (~current limit) with New Physics• Expect to reach down to ~510–8 with 2 fb–1
D*-tagged signal yield in 2 fb–1 (from b hadrons only)
D0K right sign 12.4 M
D0K+– wrong sign 46.5 k
D0K+K– 1.6 M
M. Calvi Interplay of Collider and Flavour Physics, 04/12/2007 (24/27
LHCb physics performance summary
• “DC04” full MC simulation datasets (2004-2006) used for extensive studies of LHCb Physics performance reported in more than 30 public notes (CERN-LHCb-2007-xxx) available on LHCb web page
Can easly found from LHCb page of Physics performance: http://lhcb-phys.web.cern.ch/lhcb-phys/DC04_physics_performance/
Eg. Some of the measurements which I have not mentioned:
Bd Bd CERN-LHCb-2007-046 Bd Bs KK CERN-LHCb-2007-059 ms Bs DsCERN-LHCb-2007-017,2007-041 LFV Bd eBs eCERN-LHCb-2007-028• New set of analysis starting with “DC06” full MC simulation datasets (2006-2007)
(close-to-final detector and trigger description). Additional channels under study (eg. Bs b B+ K+ , B+ K+ + Bu D …)
M. Calvi Interplay of Collider and Flavour Physics, 04/12/2007 (25/27
LHCb upgrade?• LHCb is designed to run at average luminosity of 21032 and be able to handle
51032 cm–2s–1.
– Main physics goals expressed in terms of the reach for 10 fb–1 (i.e. 5 nominal years).
• Investigating upgrade of detector to handle higher luminosity: few 1033 cm–2s–1
– Not directly coupled to SLHC machine upgrade since luminosity already available, but may well overlap in time with upgrades of ATLAS and CMS.
• Working group set up to identify the R&D required to make an upgrade of LHCb feasible ( increase trigger efficiency for hadronic modes by a factor two, fast vertex detection, electronics, radiation dose, pile-up, higher occupancy etc.) and to make the physics case.
• Input welcome from theorists on: What is the effective relevance of a statistical increase from 10 to 100 fb-1 on the constraints to Physics BSM which could be derived?
Which are the measurements which we should push to highest possible precision?
Where do we clash against theoretical uncertainties at (or before) 100 fb-1 ?
M. Calvi Interplay of Collider and Flavour Physics, 04/12/2007 (26/27
LHCb beyond 10 fb–1 ?
• CPV in Bs mixing, in particular in b sss penguins
aim for 0.01 (0.002) precision on Bs (Bs J/) CP asymmetry
• angle with theoretically clean methods, e.g. BsDsK, BD(KS)K, B D(hh)K
aim for < 1 precision on angle
• Chiral structure of b s (bsl+l–) using polarization of real (virtual) photon
more detailed and precise analysis of exclusive modes, e.g. AT(2) in B0 K*
Several measurements limited by statistical precision after 10 fb-1:
M. Calvi Interplay of Collider and Flavour Physics, 04/12/2007 (27/27
Conclusions
• LHCb is ready for data taking at 2008 LHC start-up.
• Very interesting results will come already with first 0.1-0.5 fb-1 of data:
-- Bs J/Bs , Bd K0*
but also: Bs Bd DK, Bd,s ,KK,K….
• Actively preparing for analysis of several channels with high potential
for indirect NP discovery and for elucidating its flavour structure.
• LHCb results will provide in particular a strong improvement to the
knowledge of all Bs sector. Welcome all suggestions from theory side for
new interesting channels to explore.
M. Calvi Interplay of Collider and Flavour Physics, 04/12/2007 (28/27
BACK-UP
M. Calvi Interplay of Collider and Flavour Physics, 04/12/2007 (29/27
from B±→ D0K ± (ADS)
colour suppressedus
cub
u
B
0D
Ku
su
cbu
B
K
0Dcolour favoured
• Charged B decay
u
du
scu
0D K
Cabibbo favoureddoubly Cabibbo suppressedu
us
dcu
0D
K
Atwood, Dunietz and Soni, Phys. Rev. Lett. 78, 3257 (1997).
5 parameters (rB, rD, δB, δD, γ), but only 3 relative decay rates.rD well-measured, but δD poorly constrained by CLEO-c ( expect cosδD~20%)
• D0 and D0 can both decay into K-π+ (or K+π- )
Weak phase difference –Strong phase difference B Amplitude ratio rB ~ 0.08
Strong phase difference DK
Amplitude ratio rD K=0.060±0.003
)cos(2)())((
),cos(2)(1))((
),cos(2)())((
),cos(2)(1))((
22
2
22
2
KDB
KDB
KDBD
KDB
KDB
KDBD
KDB
KDB
KDBD
KDB
KDB
KDBD
rrrrKKB
rrrrKKB
rrrrKKB
rrrrKKB
wrong sign, high sensitivity to
right sign, lower sensitivity to
M. Calvi Interplay of Collider and Flavour Physics, 04/12/2007 (30/27
from B± DK± (ADS+GLW)
ADS+GLW strategy:
Measure the relative rates of B– DK– and B+ DK+ decays with neutral D’s observed in final states K–+ and K+–, and also:
• D0→ K–+–+ and K+–+–: add 3 observables,1 unknown strong phase δK3π, 1 well measured rel. decay rate rD
K3π
• CP eigenstate decays D0→K+K–/ +– : add 1 observable and 0 unknown
Can solve for all unknowns, including the weak phase
Decay 2 fb–1 yield Bbb/S
B–,+ D(K) K–,+ favoured 28k, 28k 0.6
B–,+ D(K) K–,+ favoured 28k, 28k 0.6
B–,+ D(K) K–,+ suppr. 393, 8 2.0, 98
B–,+ D(K) K–,+ suppr. 516, 99 1.5, 8
B–,+ D(hh) K–,+ 4.3k, 3.5k 1.7, 2.1
() = 5–13 with 2 fb–1
Use of B± D* (D0,D) K± under study
depending on D strong phases
Gronau, London, Wyler, PLB. 253, 483 (1991)
( Inputs: =60, rB =0.077, B =130, δKπ = -8.3, δK3π = -60)
M. Calvi Interplay of Collider and Flavour Physics, 04/12/2007 (31/27
(770)
K*(892)
D0
m2(KS+)
m2 (
KS–
)
from B±D0K± (GGSZ) D0 decays into a 3- body CP eigenmode: D0KS
0
Large strong phases between the intermediate resonances allow the extraction of rB, δB, and γ
by studying the D-Dalitz plots from B+ and B– decays
Assume no CP violation in D0 decays
B decay amplitudes:
A(BDK) AD(mKs2,mKs
2)+ rBei(B) AD(mKs
2,mKs2)
A(BDK) AD(mKs2,mKs
2)+ rBei(B) AD(mKs
2,mKs2)
Need to assume a D0 decay model.
Current isobar model used at B factories syst() = 10
Giri, Grossman, Soffer, Zupan, PRD 68, 050418 (2003).
stat() = 7–12 with 2 fb–1
stat() = 4–6 (10 fb–1) syst() = 3–4
Depending on bkg assumptions
At LHCb:
5k signal events in 2 fb–1
B/S = 0.24 (D0±), Bbb/S < 0.7
With more statistics plan to do a model-independent analysis
and control model systematics using CLEO-c data at (3770)
M. Calvi Interplay of Collider and Flavour Physics, 04/12/2007 (32/27
• Treat with same ADS+GLW method as charged case:
– So far used only D decays to K–+, K+–, K+K– and +– final states
• Envisage also GGSZ analysis
Weak phase difference = Magnitude ratio = rB ~ 0.4
D0
d
b
d
s
c
u
B0
K*0
colour-suppressed
d
b
d
s
u
c
B0
D 0
K*0
colour-suppressed
Decay mode (+cc) 2 fb–1 yield Bbb/S
B0 (K+–)D K*0 3400 0.4–2.0
B0 (K–+)D K*0 540 2.2–13
B0 (K+K–)D K*0 470 < 4.1
B0 (–+)D K*0 130 < 14
() = 9 with 2 fb–1
from B0 D0K*0
M. Calvi Interplay of Collider and Flavour Physics, 04/12/2007 (33/27
from B and BsKK
Measure CP asymmetry in each mode:
LHCb 2 fb–1 36k B , Bbb/S ~0.5, Bhh/S =0.07 36k Bs KK, Bbb/S<0.06, Bhh/S =0.07
~ 2x better than current B world average
Adir and Amix depend on mixing phase, angle , and Penguin/Tree amplitude ratio dei
Exploit U-spin symmetry (Fleischer): If assume: d=dKK and =KK 4 measurements and 3 unknowns
can solve for (taking 2 and s from other modes)
ACP
(t ) Adir cos(m t )Amix sin(m t )
cosh t / 2 A sinh t / 2 Assume only 0.8< dKK/d <1.2
and let , KK free
2 fb–1
10 fb–1
stat() = 10+ fake solution
stat() = 5+ decreased fake