Table Of ContentEFI 11-XX
arXiv:1101.nnnn
ANGULAR DISTRIBUTIONS IN D∗ DECAYS 1
s
Jonathan L. Rosner 2
Enrico Fermi Institute and Department of Physics
University of Chicago, 5640 S. Ellis Avenue, Chicago, IL 60637
1
1
0 The reaction e+e− → (D∗+D− + c.c.) has been used with great success at
s s
2 the CLEO-c detector, and will be a source of valuable future data from the
n BES-III detector, to obtain information about the properties and decays of
a ∗ ∗
J the Ds and Ds. The angular distributions of the Ds and the final-state
6 particles in D∗ → D γ are needed to model the decays properly, to confirm
s s
the spin-parity assignment JP(D∗) = 1−, and to model the acceptance for
] s
h the rare process D∗ → D e+e−. This note presents some of the necessary
s s
p
expressions.
-
p
e
h PACS numbers: 14.40.Lb, 13.20.Fc, 13.66.Bc, 13.40.Hq
[
1 The reaction e+e− → γ∗ → (D∗+D− +c.c.) has been used with great success at the
s s
v CLEO-c detector, and will be a source of valuable future data fromthe BES-III detector,
7 ∗
to obtain information about the properties and decays of the D and D . The angular
1 s s
3 distributions of the D∗ and the final-state particles in D∗ → D γ are needed to model
s s s
1 the decays properly, to confirm the spin-parity assignment JP(D∗) = 1−, and to model
. s
1 the acceptance for the rare process D∗ → D e+e−. The purpose of the present note is
0 s s
1 to give some of the necessary expressions for angular distributions.
1 We shall assume unpolarized electron-positron collisions, with the e+ beam defining
:
v the +ˆz direction. The density matrix describing the initial virtual photons γ∗ arising
i
X from e+e− collisions is then proportional to the dyadic xˆxˆ+yˆyˆ, where we shall always
r consider linear polarization states.
a
Let us assume that the D∗+ produced opposite a D− (charge-conjugate reactions
s s
are always assumed) lies in the xz plane, making an angle θ with the z axis. If it were
produced along the z axis, its density matrix would be of the form xˆxˆ+yˆyˆ just like
∗ ∗ ∗
that of the virtual photon γ . This may be seen from the form of the γ D D coupling,
s s
∗ ∗
involving the cross product of γ and D polarization vectors dotted into the momentum
s
∗
of the D . (A single power of 3-momentum is required for parity invariance.) This form
s
∗
of coupling may be seen by a short calculation to entail a D production distribution of
s
the form
3
2
W(cosθ) = (1+cos θ) . (1)
8
∗
If the D is produced in the xz plane at an angle θ with respect to the z axis, its density
s
matrix is now of the form xˆ′xˆ′ +yˆyˆ, where xˆ′ ≡ xˆcosθ−ˆzsinθ. Thus we may write (in
1To be submitted to Phys. Rev. D
2rosner@hep.uchicago.edu.
1
the x,y,z basis)
cos2θ 0 −sinθcosθ
∗
ρ(D ) = 0 1 0 . (2)
s
−sinθcosθ 0 sin2θ
We now consider a photon emitted in the decay D∗ → D+γ. A photon emitted along
s s
the z axis can have two polarization states ǫ = xˆ, ǫ = yˆ. We shall perform an Euler
1 2
rotation on the photon, first by an angle β about the y axis and then by an angle α
about the z axis. The photon direction then becomes
pˆ = xˆsinβcosα+yˆsinβsinα+ˆzcosβ , (3)
γ
while the polarization vectors orthogonal to it become
ǫ′ = xˆcosβcosα+yˆcosβsinα−ˆzsinβ , (4)
1
ǫ′ = −xˆsinα +yˆcosα . (5)
2
These polarization vectors are also uniquely specified by the condition that they be
orthogonal to each other and to pˆ , and that ǫ′ have no z component.
γ 2
We now note that the decay D∗ → D γ involves the scalar triple product ǫ(D∗) ·
s s s
ǫ(γ) ×p , so that a photon with momentum p and no polarization component along
γ γ
the z axis is produced with angular distribution
W (θ,α,β) = ǫ′ ·ρ(D∗)·ǫ′ = cos2θcos2αcos2β +cos2βsin2α (6)
2 1 s 1
2 2
+ sin θsin β +2sinθcosθsinβcosβcosα . (7)
The use of ǫ′ rather than ǫ′ comes from the scalar triple product form of the D∗+D+γ
1 2 s s
coupling. A photon with polarization orthogonal to the one specified above is produced
with angular distribution
W (θ,α,β) = ǫ′ ·ρ(D∗)·ǫ′ = cos2α+cos2θsin2α . (8)
1 2 s 2
Thesum over photonpolarizationsthengives aphotonangulardistributionproportional
to W = W +W . Special cases of the above results are
1 2
2
W (θ = 0 or π,α,β) = 1 , W (θ = 0 or π,α,β) = cos β , (9)
1 2
2 2 2 2
W (θ = π/2, π,α,β) = cos α, W (θ = π/2, π,α,β) = cos βsin α+sin β.(10)
1 2
If the final photon undergoes internal conversion, γ → e+e−, with the plane of e+e−
oriented at an angle ψ with respect to the final photon polarization, the dependence
on ψ will be of the form cos2ψ. Hence choosing as a reference polarization direction a
unit vector perpendicular to p with no component in the ˆz direction, the full angular
γ
distribution may be written
2 2
W(θ,α,β,ψ) = W (θ,α,β)sin ψ +W (θ,α,β)cos ψ . (11)
1 2
I thank Anders Ryd for asking the question leading to this investigation. This work
was supported in part by the United States Department of Energy under Grant No. DE
FG02 90ER40560.
2