introduction to dalitz-plot - tifr
TRANSCRIPT
Gagan Mohanty
Introduction to Dalitz-plot
Student Seminar @ TIFR January 27, 2012
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2
Few Diversions
3
Hierarchical expansion of CKM
d s b
u
c
t
d s b
u
c
t
magnitudes phases
(1983)
+ O(¸ 4)
4
tbtstd
cbcscd
ubusud
VVV
VVV
VVV
V
d•b* = 0 B Mesons
A triangle at the heart
tdub VV argarg
(1,0) f3
f2
f1
(,h)
(0,0)
*
cb
td
V
V
*
*
cb
ub
V
V
Goal: check consistency of the CKM paradigm by overconstraining the Unitarity Triangle:
measure three angles and two side lengths
look for possible new physics contributions
5
B Meson: A threefold way
• CP violation in decay: direct – can take place both for
neutral and charged B’s – can have time-dependent and -independent
manifestations – Need two competing diagrams of different
CP-violating and –conserving phases
• CP violation in mixing: indirect – only neutral B’s are possibly
affected – SM predicts very small effects
• violation from mixing/decay interference: – only neutral B’s possibly affected – purely time-dependent effect – arises due to interference between decay
with and without mixing
1f
f
A
A
1q
p
Im 0 =, 1PCP C
6
Enter the Charmless
7
PRO : - Larger BF than two-body decays - Correct way to study interference - Some modes in a well-defined CP eigenstate
CON : - large phase space with low event density; difficult to identify all phase-space structures - mixture of CP-even and -odd final states - more complicated analysis needed
Two vs. Three
Gershon, Hazumi PLB 596, 163
3-body
2-body
BF[10-6]
BF[10-6]
8
Analysis Strategy
Signal extraction (kinematics)
Background fighting:
Continuum (event topology)
Other types of B decays (PID,
charm and charmonia veto)
Dalitz plot technique (three-body decays having reasonable
signal size)
Time-dependent analysis in neutral B meson decays to
determine CP violation parameters at each point of the
phase space
Inclusive
Full (3body)/partial (Q2B)
Time-dependent DP (3body)
uds:cc:bb =
2.1:1.3:1.1
Complexity
From four vectors 12
Conservation laws −4
Meson masses −3
Free rotation −3
Independent variable 2
1
2
3
Usual choice
Invariant mass m12
Invariant mass m23
r ) , ( 3 3 3
p E p m
) 0 , ( M p m
r ) , ( 1 1 1
p E p m
) , ( 2 2 2
p E p m r
3
2
3
22
3
2
21
2
12 2)()( MEmMppppm mmmm
1
2
1
22
1
2
32
2
23 2)()( MEmMppppm mmmm
B
Lorentz invariant phase space:
2
23
2
12dmdmdN
Decay rate
2
23
2
12
2dmdmM
Invariant amplitude
Three-Particle Phase Space
R. Dalitz had applied this technique for the first
time to KL decays
To determine spin and parity of the then known
τ/θ particle in its decay to three pion final state
Q=T1+T2+T3
x=(T2-T1)/√3
y=T3-Q/3
A Dalitz plot is nothing but a scatter plot between any two of the three invariant
mass squared variables, 2
ijm
Decay rate:
2
23
2
12
2dmdmM
If |M|2 is constant, the allowed
region in the Dalitz-plot will be
uniformly populated with events
Any non-uniformity gives
information about dynamics
Now Enter the Dalitz-plot
3
2
3
22
12 2MEmMm
1
2
1
2
2
23
2
ME
m
M
m
1 2
3
1
3
2
2
1
3
1
2
3
Pictorially...
11
12
Isobar Model in Dalitz-plot
B
1
2
3
B
2 1
3
+
Resonance
{13}
K+π+π- toy expt.
Zemach PR 133,
B1201 (1964)
K*0(892)
Resonance mass part
Nonresonant
3
2
4
H12B
34
Angular distribution 1 34 : 2
1 )(cos HsP
for s = 0
for s = 1
for s = 2 2
2
2
1cos3
cos
H
H
uniform
min
2
23max
2
23
2
23min
2
23max
2
23
)()(
2)()(cos
mm
mmmH
Helicity angle of a resonance
KB
KfB
KB
KB
KB
KB
c0
0
0
0*
0*
)980(
)770(
)1430(
)892(
cosθ
H
−1
0
+1 0 ), 980 (
1 ), 770 (
0 ), 1430 (
1 ), 892 (
0
0
0 *
0 *
s f
s
s K
s K
Angular Part (Helicity Angle)
13
14
What do we do in the end?
Extract ck,NR and θk,NR by performing a maximum
likelihood fit
Measure CP violation asymmetries by comparing B
and B amplitudes
Fit fraction is the ratio of the integral of a single decay
amplitude squared to the coherent sum of all
Interference of two states Ma Mb
sin)Im(2cos)Re(2 **2222
bababab
i
a MMMMMMMeMM
2
23
2
12
2dmdmM
Decay rate interference terms
The effect of interference
terms is proportional to the
area of overlap between
resonances in DP
) 1430 ( 0 *
K ) 892 ( 0 *
K
How Interference comes to play?
15
16
Time-dependent CP asymmetry
Dt = Dz/c
l+
l-
+
-
l+
s-
K+
Ee+=3.1 GeV
Ee-= 9 GeV
KS
Recoil products are used for tagging B: B0 or B0-bar
S C
Time Evolution:
Btag BCP
Tagged as B0 or
B0-bar
DD
DD
D
D
tmtme
tf dd
B
t B
cos1
1sin
1
Im21
42
2
2
/
0
0
056
17
Time-dependent DP
Time-dependent decay rate of B0(B0) → three-body
Determine mixing-induced CP [sine coefficient] and
direct CP [cosine coefficient] at each point in the DP
direct CP
Include detector effects (mistagging and resolution)
18
Dalitz plot analyses of
B+ → π+π-π+ and K+π+π-
19
B+→ π+π+π- Dalitz plot
Phase-space
Coupled BW
ρ(770)
f2(1270)
Single BW
qq BB
PRD 72, 052002 (2005) 232M B pairs
468±35
Fit
20
B+→ π+π+π-: Summary
ρ0(770) is the dominant component
3σ indication for f2(1270) and NR
Little evidence for σ (seen by BES
in the decay J/ψ→ωπ+π -)
No contribution from χc0 not
feasible to measure γ with analysed
dataset
σ
f0(980)
f2(1270)
PLB 597, 39 (2004)
Bediaga et al., PRL 81, 4067 (1998)
PRD 72, 052002 (2005)
21
Time-dependent Dalitz plot
analysis of B0 → KSπ+π-
22
Motivations • Dominantly b → s penguin transition prone to
NP effects
• Provides a test if mixing-induced CP asymmetry
equals to that of tree-level transition b → ccs
• Measure βeff in Q2B modes unambiguously
interference term allows determination of cosine
term (beauty of DP)
• We can determine the relative phase between B0→
K*+(892)π- and B0→ K*-(892)π+ access to CKM
angle γ Deshpande et al., PRL 90, 061802 (2003)
Ciuchini et al., PRD 74, 051301 (2006)
Gronau et al., PRD 75, 014002 (2007)
23
Existing Measurements
Both agree reasonably well
Discrepancy in the nonresonant contribution
Belle also observes structure near 1.3 GeV/c2 in the π+π- spectrum
Time-dependent Q2B
Time-integrated Q2B Time-integrated DP
24
Signal Yield • Simultaneous fit including
– mES, ΔE, NN, Δt and tagged (B0/B0) DP variables
383M B pairs arXiv:0708.2097
Contrib. LP2007
BB
Signal: (2172±70) in total candidate sample of 22525
25
Dalitz plot Content
qq BB
f0(980) ρ0(770)KS Gounaris-Sakurai
f0(980)KS Coupled BW
K*(892)π Single BW
K*0(1430)π LASS shape
fx(1300)KS Single BW
f2(1270)KS ,,
χc0KS ,,
ρ(770)
fx(1300)+f2(1270)
χc0
K*(892)
K*0(1430)
arXiv:0708.2097
Contrib. LP2007
26
Time-dependent CP violation
Charmonium
Charmonium
Time-dependent CP asymmetry measured at each point in
the KSπ+π- Dalitz plot for the first time
f0(980)KS value 2.1σ
above charmonium
ρ0KS consistent with
the world-average
arXiv:0708.2097
Contrib. LP2007
Stat only
Stat + Syst
27
Overall Picture
• Confirmation of the CKM paradigm as the only
source of CP violation in the SM insufficient
to explain the matter-antimatter asymmetry
• Need additional source(s) beyond the realm of the
SM
CKMfitter Group, J. Charles
et al., EPJ C41, 1 (2005)
28
Summary First measurement of the inclusive mode B+ → K+K-π+
DP measurements in the charged Kππ and πππ modes
Evidence of direct CP violation in the ρ0(770)K± decay
βeff measured without any sign ambiguity (thanks to the
time-dependent DP technique)
Measured CP violation parameters agree well with SM
predictions (modulo discrepancy in the loop diagrams)
Look forward to more data (Super flavour factory) to
pin down these holy grails
Because we know already the SM is not the ultimate theory
29
Look another way
Very small in B system
3. CP violation in the decay
(aka "direct" CP-violation)
2 amplitudes, with different weak & strong phases,
to the same final state f
Experimentally seen both in B0 and B± decays
e.g. b c (+) b u
A AA
A A A
1 2
1 2
≠1
2. CPV due to interference between mixing & decay:
( ) q A
mp A
0 iqe
p
2~
1. CP violation in the mixing : * *M iq
p i
D D
12 122
Exp: N(B0B0 ℓ+ℓ+X) ≠ N(B0B0 ℓ-ℓ-X)
The mass eigenstates are not CP eigenstates
≠1 q
p
30
Direct CP asymmetry
i f i f CP
A1 = |A1|
A2 = |A2| ei eif
A1 = |A1|
A2 = |A2| ei eif
(CP-conserving)
f f (CP-violating)
f
+ f
A = A1+ A2 A = A1+ A2 A
Time-integrated direct CP asymmetry requires at least two
amplitudes and 0: