– 1–
DALITZ PLOT ANALYSIS FORMALISM
Written January 2006 by D. Asner (Pacific Northwest National
Laboratory)
Introduction:
Weak nonleptonic decays of
D
and
B
mesons
are expected to proceed dominantly through resonant twobody
decays [1]; see Ref. [2] for a review of resonance phenomenology.
The amplitudes are typically calculated with the Dalitzplot
analysis technique [3],
which uses the minimum number of
independent observable quantities. For threebody decays of a
spin0 particle to all pseudoscalar final states,
D
or
B
→
abc
,
the decay rate [4] is
Γ =
1
(2
π
)
3
32
√
s
3
M
2
dm
2
ab
dm
2
bc
,
(1)
where
m
ij
is the invariant mass of particles
i
and
j
. The
coeﬃcient of the amplitude includes all kinematic factors, and
M
2
contains the dynamics. The scatter plot in
m
2
ab
versus
m
2
bc
is the Dalitz plot. If
M
2
is constant, the kinematically allowed
region of the plot will be populated uniformly with events. Any
variation in the population over the Dalitz plot is due to
dynamical rather than kinematical effects. It is straightforward
to extend the formalism beyond threebody final states. For
N
body final states with only spin0 particles, phase space
has dimension 3
N
−
7. Other decays of interest include one
vector particle or a fermion/antifermion pair (
e.g.
,
B
→
D
∗
ππ
,
B
→
Λ
c
pπ
,
B
→
K
) in the final state. For the first case,
phase space has dimension 3
N
−
5, and for the latter two the
dimension is 3
N
−
4.
Formalism:
The amplitude for the process,
R
→
rc, r
→
ab
where
R
is a
D
or
B
,
r
is an intermediate resonance, and
a
,
b
,
c
are pseudoscalars, is given by
M
r
(
J, L, l, m
ab
, m
bc
) =
λ
ab

r
λ
T
r
(
m
ab
)
cr
λ

R
J
(2)
=
Z
(
J, L, l, p, q
)
B
R
L
(

p

)
B
r
L
(

q

)
T
r
(
m
ab
)
.
The sum is over the helicity states
λ
of
r
,
J
is the total angular
momentum of
R
(for
D
and
B
decays, J=0),
L
is the orbital
CITATION: K. Nakamura
et al.
(Particle Data Group), JPG
37
, 075021 (2010) (URL: http://pdg.lbl.gov)
July 30, 2010
14:34
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– 2–
angular momentum between
r
and
c
,
l
is the orbital angular
momentum between
a
and
b
(the spin of
r
),
p
and
q
are the
momenta of
c
and of
a
in the
r
rest frame,
Z
describes the
angular distribution of the finalstate particles,
B
R
L
and
B
r
L
are the barrier factors for the production of
rc
and of
ab
, and
T
r
is the dynamical function describing the resonance
r
. The
amplitude for modeling the Dalitz plot is a phenomenological
object. Differences in the parametrizations of
Z
,
B
L
, and
T
r
,
as well as in the set of resonances
r
, complicate the comparison
of results from different experiments.
Usually the resonances are modeled with a BreitWigner
form, although some more recent analyses use a
K
matrix for
malism [5,6,7] with the
P
vector approximation [8] to describe
the
ππ
Swave.
The
nonresonant
(NR)
contribution
to
D
→
abc
is
parametrized as constant (Swave) with no variation in magni
tude or phase across the Dalitz plot. The available phase space
is much greater for
B
decays, and the nonresonant contribu
tion to
B
→
abc
requires a more sophisticated parametrization.
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 Particle Physics, Angular Momentum, phase space, Babar, Dalitz plot

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