–1–
DALITZ PLOT ANALYSIS FORMALISM
Written January 2006 by D. Asner (Pacifc 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. ±or threebody decays oF a
spin0 particle to all pseudoscalar fnal 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 e²ects. It is straightForward
to extend the Formalism beyond threebody fnal states. ±or
N
body fnal 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 fnal state. ±or the frst 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
)=
X
λ
±
ab

r
λ
²
T
r
(
m
ab
)
±
cr
λ

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

²
p

)
B
r
L
(

²
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