# Testing $\chi_c$ properties at BELLE II

Article (PDF Available)inPhysics Letters B 771(C) · December 2016with 63 Reads
DOI: 10.1016/j.physletb.2017.05.091
Abstract
We have shown that at BELLE II it will be possible to study in detail $\chi_{c_i}-\gamma^*-\gamma$ form factors through measurements of the reaction $e^+e^-\to e^+e^- \chi_{c_i} (\to J/\psi (\to \mu^+\mu^-)\gamma)$. The results were obtained using the newly updated Monte Carlo generator EKHARA.
arXiv:1612.07509v1 [hep-ph] 22 Dec 2016
Testing χcproperties at BELLE II
Henryk Czy˙z, Patrycja Kisza
Institute of Physics, University of Silesia, PL-40007 Katowice, Poland.
Abstract
We have shown that at BELLE II it will be possible to study in detail χciγγ
form factors through measurements of the reaction e+ee+eχci(J/ψ(
µ+µ)γ). The results were obtained using the newly updated Monte Carlo
generator EKHARA.
Keywords: χciproperties, Monte Carlo generators
1. Introduction
Soon the BELLE II experiment [1] will start to operate with unprecedented
luminosity allowing to access information not available before. In this letter
we show that the integrated luminosity of 20-50 ab1will allow BELLE II
collaboration to study in detail the χciγγform factors. These form5
factors are used in the calculations of the electronic widths of the χci, which
were not yet measured. The theoretical predictions available for these widths
[2, 3, 4, 5, 6] depend strongly on the details of the form factors modeling and are
diﬀerent, up to one order of magnitude, despite the good agreement of all the
models with experimental data [7] on the χciJ/ψγ, i = 1,2 and χc2γγ10
partial decay widths.
To give realistic predictions for event selections close to the experimen-
tal ones, we have implemented the model amplitudes in the event generator
EKHARA [8, 9]. The generator can also help to measure cross sections of the
Work supported in part by the Polish National Science Centre, grant number DEC-
2012/07/B/ST2/03867.
Preprint submitted to Physics Letters B December 23, 2016
reactions e+ee+eχciand e+ee+eχci(J/ψ(µ+µ)γ). The15
newly updated code is available from EKHARA web page. The details and al-
gorithms used in the event generation will be described in a separate publication
[10].
The layout of this letter is the following: In Section 2 we describe the model
used in the presented simulations, in Section 3 we give predictions for the event20
rates of the χciproduction cross sections at BELLE II and event rates for the
form factor measurements. Conclusions are presented in Section 4.
2. The model
The model used in this letter is an extension of the model built to describe
χc1and χc2decays to J/ψγ, the χc2decay to γγ and ψdecays to χc1(2) γ25
[6]. The basic assumptions used to construct the amplitudes for χc0decays to
J/ψγ and γγ as well as ψdecay to χc0γare the same as in [6]. We start from
the amplitudes calculated in [2] for γγdecays and assume that the Lorentz
structure, as well as the form factor, are identical also for J/ψγdecays, allowing
for diﬀerent coupling constants. From these assumptions one gets the following30
amplitudes for the decays involving χc0
Aαβ
0γγ (p1, p2)ǫ1
αǫ2
βp2
1=p2
2=0
=c0
γA(p1, p2),
Aαβ
0γJ(p1, p2)ǫ1
αǫ2
βp2
1=0, p2
2=M2
J/ψ
=c0
J/ψ A(p1, p2),
Aαβ
ψ0γ(p1, p2, ǫ)ǫ1
αǫ2
βp2
1=0, p2
2=M2
ψ
=c0
ψA(p1, p2),(1)
where ǫiǫ(pi) are the appropriate polarisation vectors,
A(p1, p2) = 2
6Mχc0(ǫ1ǫ2)(p1p2)(ǫ1p2)(ǫ2p1)M2
χc0+ (p1p2),
c0
γ=16πα
m·(a+f·a0
J
M2
J/ψ
+f·a0
ψ
M2
ψ
)·1
(M2
χc0/4 + m2)2,
2
c0
J/ψ =4·e·a0
J
m·1
(M2
χc0/4 + m2M2
J/ψ /2)2,
c0
ψ=4·e·a0
ψ
m·1
(M2
χc0/4 + m2M2
ψ/2)2.
(2)
The coupling constants can be extracted adding to the set of experimental
data used in [6] the following widths (x=M2
J/ψ /M2
χc0,y=M2
χc0/M2
ψ)
Γ(χc0γγ) = 3
128π|c0
γ|2M5
χc0,
Γ(χc0J/ψγ) = 1
192π|c0
J/ψ |2M5
χc0(3 x)2(1 x)3,
Γ(ψχc0γ) = 1
576π|c0
ψ|2(1 y)3(1 3y)2M5
ψ
y.
(3)
The coupling constants and other model parameters were extracted from35
a ﬁt summarised in Table 1. The ﬁt to 8 experimental values with 6 model
a[GeV5/2]m[GeV] aJ[GeV5/2]a0
J[GeV5/2]aψ[GeV5/2]a0
ψ[GeV5/2]
0.0796 1.67 0.129 0.073 -0.078 0.122
widths [MeV] χc0χc1χc2
Γ(χγγ)th 2.24 ·103- 5.46 ·104
Γ(χJ/ψγ)th 1.34 ·1012.82 ·1013.74 ·101
Γ(ψχγ)th 2.96 ·1022.88 ·1022.64 ·102
Γ(χγγ)exp 2.3(2) ·103- 5.3(4) ·104
Γ(χJ/ψγ)exp 1.3(1) ·1012.8(2) ·1013.7(3) ·101
Γ(ψχγ)exp 2.96(11) ·1022.8(1) ·1022.7(1) ·102
Table 1: Model parameters and theoretical (th) (this paper, see also [6]), and experimental
(exp) [7] values of Γ(χc0,1,2γγ , γJ/ψ) and Γ(ψχc0,1,2γ).
parameters gives χ2= 0.94. The model parameters describing χc1and χc2
3
are very close to the model parameters obtained in [6] and the predictions for
electronic widths (Γ(χc1e+e) = 0.37 eV and Γ(χc2e+e) = 3.86 eV)
did change within the parametric uncertainty of the model, which is about 10%.40
The a0
ψcoupling is positive at diﬀerence with negative aψ. Both signs are the
only ones allowed by the ﬁt.
3. The amplitudes and the cross section
e+
p1
e
p2
e+
q1
e
q2
χci
q
J/ψ
l
γ
k1
µ+
q4
µ
q3
p1
p2
χci
q
l1
l2
Figure 1: The Feynman diagram for the amplitude of the reaction e+ee+eJ/ψ(
µ+µ)γ. The notation of four momenta is used in the formulae presented in this letter.
With the couplings obtained from the ﬁt one can predict the rates for the
reactions e+ee+eχciand e+ee+eχci(J/ψ(µ+µ)γ). We do45
consider here only signal processes with the Feynman diagram given in Fig. 1.
The QED background can be suppressed by requiring that the µ+µγinvariant
mass is close to the χci mass and µ+µinvariant mass close to the Jmass
and it will not be considered in this letter. As the χci and J/ψ are almost
on-shell we use a constant χci J/ψ γform factor.50
The relevant amplitudes, with the four momenta denoted in Fig.1, read
M0=e3v(p1)γνv(q1)
l2
1
Aνµ
0
u(q2)γµu(p2)
l2
2
(u(q3)γγv(q4))Πχc0Bσ
0CJ/Ψ
σγ
4
M1=e3v(p1)γνv(q1)
l2
1
Aνµω
1
u(q2)γµu(p2)
l2
2
(u(q3)γγv(q4))Πχc1
ωδ Bδσ
1CJ/Ψ
σγ
M2=e3v(p1)γνv(q1)
l2
1
Aνµωδ
2
u(q2)γµu(p2)
l2
2
(u(q3)γγv(q4))Πχc2
ωδπξ Bπ ξσ
2CJ/Ψ
σγ
(4)
with
Aνµ
0=c0
γ
6Mχc0(gνµ (l1·l2)lµ
1lν
2) (M2
χc0+l1·l2)gνµ l2
1l2
2,
Bσ
0=2c0
J/ψ
6Mχc0
F1σν lν(M2
χc0+k1·l),
Aνµω
1=i˜c1
γǫ¯ν νµω l2 ¯νl2
1lνl2
2+ǫ¯µ¯νµωl1 ¯µl2 ¯νlν
1ǫ¯ν¯µνω l1 ¯νl2 ¯µlµ
2,
Bδσ
1=i
2c1
J/ψ ǫ¯µ¯νσ δ F1
¯µ¯νl2ǫ¯µ¯ν¯αδ F1
¯µ¯νl¯αlσ,
Aνµωδ
2=˜c2
γ2Mχc2gµδlω
1lν
2gνµ lω
1lδ
2gνω gµδ (l1·l2) + gνω lµ
1lδ
2,
Bπξσ
2=c2
J/ψ 2Mχc2F1¯
βπ gσξ l¯
βF1σπ lξ,
˜c0
γ=16πα
m(a+fa0
J
M2
J/ψ
+fa0
ψ
M2
ψ
)1
((l1l2)2/4m2+)2,
˜ci
γ=16πα
m(a+faJ
M2
J/ψ
+faψ
M2
ψ
)1
((l1l2)2/4m2+)2, i = 1,2,
ci
J/ψ =4·e·aJ
m·1
(M2
χci/4 + m2M2
J/ψ /2)2, i = 1,2,
CJ/ψ
σγ =sJ/ψe+e
αl2
gσγ
l2M2
J/ψ +iΓJ/ψ MJ/ψ
, F 1
µν = (ǫ1
µk1νǫ1
νk1µ),
Πχc0=1
q2M2
χc0+iΓχc0Mχc0
,Πχc1
ωδ =gωδ qωqδ/M2
χc1
q2M2
χc1+iΓχc1Mχc1
,
Πχc2
ωδπξ =
1
2(Pωπ Pδξ +Pωξ Pδπ )1
3(Pωδ Pπξ )
q2M2
χc2+iΓχc2Mχc2
, Pµν =gµν +qµqν/M2
χc2.
(5)
The parts of the χciγγvertex giving vanishing contribution to the
amplitude due to the contraction with the eeγvertices are not shown in
the above formulae. The amplitudes were implemented into the event generator55
EKHARA [9, 8]. Two independent codes were built using two diﬀerent meth-
ods of spin summations to cross check the implementation. The phase space
5
generation was also cross checked with an independent computer code. Details
will be given in [10].
The χciproduction cross section (the amplitudes are easy to infer from60
Eq.(4)) integrated over the complete phase space with the integrated luminosity
of BELLE II of 50 ab1leads to event rates of about 140M (χc0), 4.3M (χc1)
and 142M (χc2). These rates allow for detailed studies of many χcidecay modes.
Unfortunately the measurement of electronic width of χc1is out of reach as the
predicted number of events is about 2. For χc2situation is a bit better with65
the expected number of events 284. Further drop is however expected as the
detector does not cover the complete solid angle range.
If one tag an electron or a positron in the angular range between 17and
150the event rates drop to 4.8M, 1.2M and 5.2M. It shows that one has an
events for various virtual photon invariant masses (l2
1= (p1q1)2) are shown
in Fig. 2.
χc2
χc1
χc0
l2
1[GeV 2]
Nev
0102030405060
1e+ 06
100000
10000
1000
100
10
Figure 2: The distributions of expected number of events (Nev ) for χciproduction, when
one observes the positron in the angular range of 17and 150.
6
In this letter we concentrate on the possible tests of the models of χciγγ
form factors using single tag events relaying on identiﬁcation of a simple χcide-
cay mode χciJ/ψ(µ+µ)γ. If one requires identiﬁcation of χciand75
J/ψ through invariant masses of µ+µγand µ+µﬁnal states respectively, the
χciJ/ψγform factors are entering the cross section with ﬁxed invariants,
thus they are almost constant, . All results presented here assume the asymmet-
ric beams of 4 and 7 GeV with half crossing angles of 41.5 mrad. We assume
that the particles (µ+, µ, photon and electron or positron) can be detected80
and their four momenta measured if their polar angles are between 17and
150[11].
χc2
χc1
χc0
l2
1[GeV 2]
Nev
051015202530
10000
1000
100
10
1
0.1
Figure 3: The distributions of expected number of events (Nev) for χciproduction with
subsequent decay to J/ψ(µ+µ)γ. The event selection is described in the text.
The event rates after the applied cuts (950 for χc0, 5610 for χc1and 8849 for
χc2) will allow for testing of the χciJγform factors for the ﬁrst time.
The l2
1invariant mass distribution is shown in Fig. 3. The l2
2invariant mass is,85
as expected, limited to small values with 891 (χc0), 4781 (χc1) and 8083 (χc2)
events with 1GeV 2< l2
2<0. The form factor can be thus extracted with a
7
decent accuracy for one of the invariants close to zero and the second spanning
up to about -25 GeV2. With limited statistics one can even have data for χc1
(473 events) and χc2(303 events) with all the particles observed in the detector90
allowing for an accurate reconstruction of both invariants. The expected event
rates are shown in Fig.4. For χc0one expects only few such events.
Nev
35
30
25
20
15
10
5
0
l2
2[GeV 2]
0246810
l2
1[GeV 2]
0
2
4
6
8
10
χc1
Nev
25
20
15
10
5
0
l2
2[GeV 2]
0246810
l2
1[GeV 2]
0
2
4
6
8
10
χc2
Figure 4: The distributions of expected number of events (Nev) for χc1and for χc2production
with subsequent decay to J/ψ(µ+µ)γwhen both electron and positron are tagged.
The event selection is described in the text.
4. Conclusions
We have shown that at BELLE II it will be possible to study in detail χci
γγform factors through measurements of the reaction e+ee+eχci(95
J/ψ(µ+µ)γ). This should clarify, which of the models giving predictions for
the χc1and χc2electronic widths is correct even without direct measurement of
these widths. In case the electronic widths are measured, it will allow for further
reﬁnements of the models. The expected event rates for the χciproduction show
that detailed studies of the χcibranching ratios will be possible at BELLE II.100
8
The newly updated Monte Carlo generator EKHARA can be of help for the
visibility studies and data analyses.
References
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