The phase between three-gluon and one-photon amplitudes in psi(2S) and
psi(3770) decays is analyzed.Comment: 5 pages, 4 figures, Talk given at Hadron 03: 10th International
  Conference on Hadron Spectroscopy, Aschaffenburg, Germany, 31 Aug - 6 Sep
  200

Wang, P.

English

arXiv

Ping Wang

Crossref

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The phase between the three gluon and one photon amplitudes in
quarkonium decays
Ping Wang∗
Abstract
The phase between three-gluon and one-photon amplitudes in ψ(2S) and ψ(3770) decays is ana-
lyzed.
1 Motivations
It has been known that in J/ψ decays, the three gluon amplitude a3g and one-photon amplitude aγ
are orthogonal for the decay modes 1+0− (90◦) [1], 1−0− (106 ± 10)◦ [2], 0−0− (89.6 ± 9.9)◦ [3], 1−1−
(138± 37)◦ [4] and NN (89± 15)◦ [5].
J. M. Gérard and J. Weyers [6] augued that this large phase follows from the orthogonality of three-
gluon and one-photon virtual processes. The question arises: is this phase universal for quarkonium
decays? How about ψ(2S), ψ(3770) and Υ(nS) decays?
2 Quarkonium produced in electron-positron colliding experi-
ments
Recently, more ψ(2S) data has been available. Most of the branching ratios are measured in e+e− colliding
experiments. For these experiments, there are three diagrams [7, 8], as shown in Fig. 1, which contribute
to the processes. Although such formulas were written in the early years after J/ψ was discovered, but
e+
e-
c
c
_ hadron
g g g
(a) three-gluon annihilation
e+
e-
c
c
_ hadron
γ*(c c
_
)
(b) one-photon annihilation
e+
e-
hadron
γ*(e+e-)
(c) one-photon continuum
Figure 1: The Feynman diagrams of e+e− → light hadrons at charmonium resonance.
the diagram in Fig. 1(c) is usually neglected. This reflects a big gap between theory and the actual
experiments.
How important is this ampitude? For ψ(2S), at first glance, σBorn = 7887nb; while σc ≈ 14nb.
But for e+e− processes, initial state radiation modifies the Breit-Wigner cross section. With radiative
correction, σr.c. = 4046nb; more important, the e
+e− colliders have finite beam energy resolution, with
∆ at the order of magnitude of MeV; while the width of ψ(2S) is only 300KeV. Here ∆ is the standard
deviation of the guassian function which describes the C.M. energy distribution of the electron-positron.
This reduces the observed cross section by an order of magnitude. For example, with ∆ = 1.3MeV
∗Institute of High Energy Physics, CAS, Beijing 100039, China
1
(parameter of BES/BEPC at the energy of ψ(2S) mass), σobs = 640nb. If ∆ = 2.0MeV (paramters of
DM2/DCI experiment at the same energy), σobs = 442nb.
The contribution from direct one-photon annihilation is most important for pure electromagnetic pro-
cess, like µ+µ−, where the continuum cross section is as large as the resonance itself and the interference
is apparent. This is seen in the µ+µ− cross section curve in the experimental scan of ψ(2S) resonance,
as shown in Fig. 2.
5
6
7
8
9
10
20
3.665 3.67 3.675 3.68 3.685 3.69 3.695 3.7 3.705
Ecm(GeV)
σ(n
b)
ψ(2S) µ+µ-→
Figure 2: µ+µ− curve at ψ(2S) resonance scaned by BES
The observed cross section depends on
experimental details: sm, ∆, etc. [8].
The resonance cross section depends on
the beam energy resolution of the e+e−
collider; on the other hand, the contin-
uum cross section depends on the in-
variant mass cut sm in the selection cri-
teria. This is seen from the treatment
of the radiative correction [9]:
σr.c.(s) =
1− sm
s
∫
0
dxF (x, s)
σ0(s(1− x))
|1 −Π(s(1 − x))|2 .
3 Pure electromagnetic decay
BES reports B(ψ(2S) → ωπ0) = (3.8±1.7±1.1)×10−5. What it means is the cross section of e+e− → ωπ0
at ψ(2S) mass is measured to be (2.4 ± 1.3) × 10−2 nb. About 60% of this cross section is due to
continuum [10]. This gives the form factor Fωπ0(M2ψ(2S))/Fωπ0(0) = (1.6±0.4)×10−2. It agrees well with
the calculation by J.-M. Gérard and G.López Castro [11] which predicts it to be (2πfπ)
2/3s = 1.66×10−2
with fπ the pion decay constant. Similarly π form factor at ψ(2S) is revised [10].
4 ψ(2S) → 1−0− and 0−0− decays
The ψ(2S) → 1−0− decays are due to three-gluon amplitude a3g and one-photon amplitude aγ . With
these two amplitudes, a previous analysis [12] yielded a3g ≈ −aγ , i.e. the phase φ between a3g and aγ is
180◦ and φ = 90◦ is ruled out. Here the SU(3) breaking amplitude ǫ is small compared with a3g. But
these branching ratios so far are all measured by e+e− experiments. So actually we have three diagrams
and three amplitudes. The analysis should be based on Table 1:
modes amplitude B.R.(in 10−4)
ρ+π− a3g + aγ + ac < 0.09
(ρ0π0)
K∗+K− a3g + ǫ+ aγ + ac < 0.15
K∗0K0 a3g + ǫ− 2(aγ + ac) 0.41± 0.12 ± 0.08
ωπ0 3(aγ + ac) 0.38± 0.17 ± 0.11
Table 1: e+e− → ψ(2S) → 1−0− process
In Table 1, a3g interferes with
aγ + ac, destructively for ρπ and
K∗+K−, but constructively for
K∗0K0 (ǫ is a fraction of a3g).
Fitting measured K∗+K− and
ρπ modes with different φ’s are
listed in Table 2.
It shows that a −90◦ phase between a3g and aγ is still consistant with the data within one standard
deviation of the experimental errors [13].
2
φ C =
∣
∣
∣
∣
a3g
aγ
∣
∣
∣
∣
σpre(K
∗+K−)(pb) B0
K∗+K−
(×10−5)1 σpre(ρ0π0)(pb) B0ρ0π0(×10−5)
+76.8◦ 7.0+3.1
−2.2 37
+24
−23 5.0
+3.2
−3.1 64
+43
−41 9.0
+6.1
−6.0
−72.0◦ 5.3+3.1
−2.6 19
+14
−14 3.1
+2.3
−2.3 33
+25
−24 5.5
+4.1
−4.0
−90◦ 4.5+3.1
−2.6 12
+9
−9 2.0
+1.5
−1.5 22
+17
−17 3.7
+2.9
−2.9
180◦ 3.4+3.0
−2.2 4.0
+4.3
−3.2 0.39
+0.42
−0.31 7.8
+8.6
−6.7 1.0
+1.1
−0.8
BES observed < 9.6 < 5.8
Table 2: Calculated results for ψ(2S) → K∗+K− and ρ0π0 with different φ.
The newly measured ψ(2S) → KSKL from BES-II [15], together with previous results on π+π− and
K+K−, is also consistant with a −90◦ phase between a3g and aγ [14]. This is discussed in more detail
by X.H. Mo in this conference.
5 ψ(3770) → ρπ
J.L.Rosner [16] proposed that the ρπ puzzle is due to the the mixing of ψ(2S) and ψ(1D) states, with the
mixing angle θ = 12◦. In this scenario, the missing ρπ decay mode of ψ(2S) shows up instead as decay
mode of ψ(3770), enhanced by the factor 1/sin2θ. He predicts Bψ(3770)→ρπ = (4.1± 1.4)× 10−4 . With
the total cross section of ψ(3770) at Born order to be (11.6±1.8) nb , σBorn
e+e−→ψ(3770)→ρπ = (4.8±1.9) pb .
But one should be reminded that for ψ(3770), the resonance cross section, with radiative correction is
only 8.17nb, while the continuum is 13nb. So to measure it in e+e− experiments, we must know the cross
section e+e− → γ∗ → ρπ. The cross section σe+e−→γ∗→ρπ(s) can be estimated by the electromagnetic
form factor of ωπ0, since from SU(3) symmetry, the coupling of ωπ0 to γ∗ is three times of ρπ [17]. The
ωπ0 form factor measured at ψ(2S) is extrapolated to
√
s = Mψ(3770) by |Fωπ0(s)| = 0.531 GeV/s .
With this, the continuum cross section of ρπ production at ψ(3770) σBorn
e+e−→γ∗→ρπ
= 4.4 pb . Compare
the two cross sections, the problem arises : how do these two interfere with each other?
1
10
− 90 o
 90 o
 0 o
180 o
σ ρ
π (
pb
)
ob
s
(a)
10
-1
1
10
0 1 2 3 4 5 6 7 8 9 10
ρ π
K*+ K − + c.c.
K*o Ko
__
 + c.c.
Bρπ (10
 −4)
σ  
(φ
= 
− 
90
 ) 
(p
b)
ob
s
(b)
Figure 3: (a) e+e− → ρπ cross section as a function of Bψ(3770)→ρπ
for different phases, and (b) e+e− → K∗0K0+c.c., K∗+K−+c.c.,
and ρπ cross sections as functions of Bψ(3770)→ρπ.
If the phase between a3g and aγ
is −90◦, then as in the case of
ψ(2S), the interference between
a3g and ac is destructive in ρπ
and K∗+K− modes, but con-
structive in K∗0K0 mode [18].
The e+e− → ρπ cross sec-
tion at ψ(3770), as a function
of Bψ(3770)→ρπ for different φ’s
are shown in Fig. 3(a); while
the e+e− → ρπ,K∗+K−,K∗0K0
cross sections as functions of
Bψ(3770)→ρπ for φ = −90◦ are
shown in Fig. 3(b). To measure
ψ(3770) → ρπ in e+e− collision,
we must scan the ψ(3770) peak
(as we measure Γee, Γtotal and
Mψ(3770)).
Fig. 4(a) shows the e+e− → ρπ cross section vs C.M. energy for different φ’s. Fig. 4(b) shows the
e+e− → K∗0K0 cross section with φ = −90◦.
MARK-III gives σe+e−→ρπ(
√
s =Mψ(3770)) < 6.3 pb, at 90% C.L. [19]. It favors −90◦.
3
0
2
4
6
8
10
12
14
16
18
3.72 3.73 3.74 3.75 3.76 3.77 3.78 3.79 3.8 3.81 3.82
Ecm (GeV)
σ ρ
π 
(p
b
)
ob
s
− 90 o
 90 o
  0 o
180 o
(a) e+e− → ρπ
0
2.5
5
7.5
10
12.5
15
17.5
20
3.72 3.73 3.74 3.75 3.76 3.77 3.78 3.79 3.8 3.81 3.82
Ecm (GeV)
σ 
   
(K
*0
K
0  
+
 c
.c
.)
 (
pb
)
ob
s
__
__
(b)e+e− → K∗0K0
Figure 4: (a) e+e− → ρπ cross section vs C.M. energy for different phases: φ = −90◦, +90◦, 0◦, and
180◦ respectively. (b) e+e− → K∗0K0 cross section vs C.M. energy with φ = −90◦.
ψ(3770) → 1−0− modes test the universal orthogonal phase between a3g and aγ in quarkonium decays
as well as Rosner’s scenario. A small cross section of e+e− → ρπ at ψ(3770) peak means B(ψ(3770) →
ρπ) ≈ 4 × 10−4. (With radiative correction, the cancellation between a3g and ac cannot be complete.
With a practical cut on the ρπ invariant mass, the cross section is a fraction of 1pb. ) It also implies the
phase of the three gluon amplitude relative to one-photon decay amplitude is around −90◦. These will
be tested by the 20pb−1 of ψ(3770) data by BES-II, or 5pb−1 of ψ(3770) data by CLEO-c.
6 Summary
• The universal orthogonality between a3g and aγ found in various decay modes of J/ψ can be
generalized to ψ(2S) and ψ(3770) decays. A −90◦ phase between a3g and aγ is consistant with the
data on ψ(2S) → 1−0− and 0−0− modes.
• The ψ(3770) → ρπ,K∗+K−,K∗0K0 test the universal −90◦ phase, as well as Rosner’s scenario on
ρπ puzzle. This should be pursued by BES-II and CLEO-c.
• The exisiting Υ(nS) data should be used to test the phase in bottomonium states.
References
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5


The phase between the three gluon and one photon amplitudes in quarkonium decays

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