Download - Dalitz Plot Analysis of D Decays
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Dalitz Plot Analysis of D Decays
Luigi Moroni
INFN-Milano
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Dalitz Analysis of Heavy Flavor Decays
• Infinite power tool!– It provides a “complete observation” of the decay– Everything could be in principle measured
• from the dynamical features of the HF decay mechanism,– Relative importance of non-spector processes
• up to the CP-violating phases, mixing, etc– Just recall from Bo: probably the only clean way to get it
• We already learned a lot on charm
• But, as we know, – strong dynamic effects, if not properly accounted for,
would completely hide or at least confuse the underlying fundamental physics.
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Outline
• In this talk I will address all the key issues of the HF Dalitz analysis– Formalization problems
– Failure of the traditional “isobar” model
– Need for the K-matrix approach
– Implications for the future Dalitz analyses in the B-sector
• Will discuss these issues in the context of the recent D++-+ Dalitz analysis we performed in FOCUS– A lot to be learned
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Formalization Problems
• The problem is to write the propagator for the resonance r– For a well-defined wave with specific isospin and
spin (IJ) characterized by narrow and well-isolated resonances, we know how.
rr3
1
2D r
|1 2
3
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•the propagator is of the simple Breit-Wigner type
1 3 13 2 212
1(cos )
J Jr
D r Jr r r
A F F p p Pm m im
����������������������������
traditionalisobar modelj
j jj
Aia e M
Spin 0
Spin 1
Spin 2 )1cos3()(2
)2(
1
1322
13
13
ppP
ppP
P
J
J
J
21
21
)339(
)1(
1
4422
22
pRpRF
pRF
F
•the decay amplitude is
•the decay matrix element
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when the specific IJ–wave is characterized by large and heavily overlapping resonances (just as the scalars!), the problem is not that simple.
1( )I iK
where K is the matrix for the scattering of particles 1 and 2.
In this case, it can be demonstrated on very general grounds that the propagator may be written in the context of the K-matrix approach as
Indeed, it is very easy to realize that the propagation is no longer dominated by a single resonance but is the result of a complicated interplay among resonances.
i.e., to write down the propagator we need to know the related scattering K-matrix
In contrast
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What is K-matrix?
• It follows from S-matrix and because of S-matrix unitarity it is real
• Viceversa, any real K-matrix would generate an unitary S-matrix
• This is the real advantage of the K-matrix approach:– It drastically simplifies the formalization of any scattering
problem since the unitarity of S is automatically preserved
1/ 2 1/ 22S I i T 1 1K T i 1( )T I iK K
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From Scattering to Production
• Thanks to I.J.R. Aitchison (Nucl. Phys. A189 (1972) 514), the K-matrix approach can be extended to production processes
• In technical language,
– From
– To
• The P-vector describes the coupling at the production with each channel involved in the process– In our case the production is the D decay
1( )T I iK K 1( )F I iK P
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K-Matrix Picture of D++-+
D
P
1
2
3 Multi body
4 =
5 '
K K
1(1 )iK
1( )F I iK P
00++-wave
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Failure of the Isobar Model
• At this point, on the basis of a pretty solid theory, it is very easy to understand when we can employ the traditional Isobar Model and when not.
• It turns out that– for a single pole problem, far away of any threshold, K-
matrix amplitude reduces to the standard BW formula.• The two descriptions are equivalent
– In all the other cases, the BW representation is not any more valid
• The most severe problem is that it does not respect unitarity
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Add BW Add K
The Unitarity circle
Add BW
Add K
An Explicit Example
• Adding BWs ala “Isobar Model”– Breaks the Unitarity
– And heavily modify the phase motion!
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21 0
20 0 0
1 / 2( )
( )(1 )jK A
k kj jA A
g s s s mF I iK f
m s s s s s s
The decay amplitude may be written, in general, as a coherent sumof BW terms for waves with well-isolated resonances plus K-matrix terms for waves with overlapping resonances.
00
1 1
( ) i i
m ni i iBW K
i i i ii i m
A D a e a e F a e F
Can safely say that in general K-matrix formalization is just required by scalars (J=0), whose general form is
KiF
Summarizing
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Where can we get a reliable s-wave scattering parametrization from?
• In other words, we need to know K to proceed.• A global fit to all the available data has been performed!
* p0n,n, ’n, |t|0.2 (GeV/c2)GAMSGAMS
* pn, 0.30|t|1.0 (GeV/c2)GAMSGAMS
* BNLBNL
*p- KKn
CERN-MunichCERN-Munich
::
* Crystal BarrelCrystal Barrel
* Crystal BarrelCrystal Barrel
* Crystal BarrelCrystal Barrel
* Crystal BarrelCrystal Barrel
pp
pp , ,
pp K+K-, KsKs, K+s
np -, KsK-, KsKs-
-p0n, 0|t|1.5 (GeV/c2)E852E852*
At rest, from liquid 2H
At rest, from gaseous
At rest, from liquid
At rest, from liquid
2H
2D2H
“K-matrix analysis of the 00++-wave in the mass region below 1900 MeV’’ V.V Anisovich and A.V.Sarantsev Eur.Phys.J.A16 (2003) 229
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( ) ( ) 200 0
20 0 0
1 2( )
( )(1 )
scatti j scatt A
ij ij scattA A
g g s s s mK s f
m s s s s s s
( )ig is the coupling constant of the bare state to the meson channel
scattijf
0s describe a smooth part of the K-matrix elements
20 0( 2) ( )(1 )A A As s m s s s suppresses the false kinematical singularity
at s = 0 near the threshold
and
is a 5x5 matrix (i,j=1,2,3,4,5)
'
IJijK
K K1= 2= 3=4 4= 5=
A&S
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A&S K-matrix poles, couplings etc.
4 '
0.65100 0.24844 0.52523 0 0.38878 0.36397
1.20720 0.91779 0.55427 0 0.38705 0.29448
1.56122 0.37024 0.23591 0.62605 0.18409 0.18923
1.21257 0.34501 0.39642 0.97644 0.19746 0.00357
1.81746 0.15770 0.179
KKPoles g g g g g
0 11 12 13 14 15
0
15 0.90100 0.00931 0.20689
3.30564 0.26681 0.16583 0.19840 0.32808 0.31193
1.0 0.2
scatt scatt scatt scatt scatt scatt
A A
s f f f f f
s s
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A&S T-matrix poles and couplings
4 '13.1 96.5 80.9 98.6 102.1
116.8 100.2 61.9 140
( , / 2)
(1.019, 0.038) 0.415 0.580 0.1482 0.484 0.401
(1.306, 0.167) 0.406 0.105 0.8912 0.142
KKi i i i i
i i i i
m g g g g g
e e e e e
e e e e
.0 133.0
97.8 97.4 91.1 115.5 152.4
151.5 149.6 123.3 170.6
0.225
(1.470, 0.960) 0.758 0.844 1.681 0.431 0.175
(1.489, 0.058) 0.246 0.134 0.4867 0.100 0
i
i i i i i
i i i i
e
e e e e e
e e e e
133.9
.6 126.7 101.1
.115
(1.749, 0.165) 0.536 0.072 0.160 0.313
i
i i i i i
e
e e e e e
A&S fit does not need a as measured in the isobar fit
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FOCUS D s +
++- analysis
Observe:
•f0(980)
•f2(1270)
•f0(1500) Sideband
Signal
Yield Ds+ = 1475 50
S/N Ds+ = 3.41
PLB 585 (2004) 200
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First fits to charm Dalitz plots in the K-matrix approach!
C.L fit 3 %
sD
Low mass projection High mass projection
+
+20 +
(S - wave)π 87.04 ± 5.60 ± 4.17 0(fixed)
f (1275)π 9.74 4.49 2.63 168.0 18.7 2.5
ρ (1450)π 6.56 ± 3.43 ± 3.31 234.9 ±19.5 ±13.3
decay channel phase (deg)fit fractions (%)
r
j
2iδ 2 2r r 12 13
r 2iδ 2 2j j 12 13j
a e A dm dmf =
a e A dm dm
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Yield DYield D++ = 1527 = 1527 5151
S/N DS/N D++ = 3.64 = 3.64
FOCUS D+ ++- analysis
Sideband Signal
PLB 585 (2004) 200
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2lowm
2highm
D
C.L fit 7.7 %
K-matrix fit results
Low mass projection High mass projection
18 11.7
+
+2
0 +
(S - wave)π 56.00 ± 3.24 ± 2.08 0(fixed)
f (1275)π 11.74 1.90 0.23 -47.5 .7
ρ (770)π 30.82 ± 3.14 ± 2.29 -139.4 ±16.5 ± 9.9
decay channel phase (deg)fit fractions (%)
No new ingredient (resonance) required not present in the scattering!
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With
Without
C.L. ~ 7.5%
Isobar analysis of D+ ++would instead require a new scalar meson:
C.L. ~ 10-6
m = 442.6± 27.0 MeV/c = 340.4 ± 65.5 MeV/c preliminary
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What about -meson then?
• Can conclude that – Do not need anything more than what is already in
the s-wave phase-shift to explain the main feature of D 3 Dalitz plot
• Or, if you prefer,– Any -like object in the D decay should be
consistent with the same -like object measured in the scattering.
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• Just by a simple insertion of KK-1 in the decay amplitude F
• We can view the decay as consisting of an initial production of the five virtual states , KK,’and 4which then scatter via the physical T-matrix into the final state.
• The Q-vector contains the production amplitude of each virtual channel in the decay
1 1 1 1( ) ( )F I iK P I iK KK P TK P TQ
Even more: from P to Q-vector
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Q-vector for Ds
• s-wave dominated by an initial production of and KK-bar states
The two peaks of the ratios correspond to the two dips of the normalizing modulus, while the two peaks due to the K-matrix singularities, visible in the normalization plot, cancel out in the ratios.
The normalizing modulus
Ratio of moduli of Q-vector amplitudes
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Q-vector for D+
• The same!– s-wave dominated by an initial production of
and KK-bar states
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The resulting picture
• The s-wave decay amplitude primarily arises from a ss-bar contribution– Cabibbo favored for Ds
– Cabibbo suppressed together with the competing dd-bar contribution for D+
• The measured fit fractions seems to confirm this picture– s-wave decay fraction, 87% for Ds and only 56% for D+
– The dd-bar contribution in D+ case evidently prefers to couple to a vector state like (770), that alone accounts for about 30% of the decay.
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Conclusions• Dalitz plot analysis is and will be a crucial tool to extract physics
from the HF decays• Nevertheless, to fully exploit this unlimited potential a systematic
revision of the amplitude formalization is required• Thanks to FOCUS, K-matrix approach has been shown to be the
real breakthrough • Its application has been decisive in clearing up a situation which
recently became quite fuzzy and confusing– new “ad hoc” resonances were required to understand data, e.g. (600) and
(900)• Strong dynamics effects in D-decays now seem under control and
fully consistent with those measured by light-quark experiments• The new scenario is very promising for the future measurements of
the CP violating phases in the B sector, where a proper description of the different amplitudes is essential.