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 χci−γ∗−γ

form factors through measurements of the reaction e+e−→e+e−χci(→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 ab−1will allow BELLE II

collaboration to study in detail the χci−γ∗−γform 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 χci→J/ψγ, i = 1,2 and χc2→γγ10

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+e−→e+e−χciand e+e−→e+e−χci(→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 ψ′decays to χc1(2) γ25

[6]. The basic assumptions used to construct the amplitudes for χc0decays to

J/ψγ and γγ as well as ψ′decay to χc0γare the same as in [6]. We start from

the amplitudes calculated in [2] for γ∗γ∗decays and assume that the Lorentz

structure, as well as the form factor, are identical also for J/ψγ∗decays, 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/ψ(p1, p2)ǫ1

αǫ2

βp2

1=0, p2

2=M2

J/ψ

=c0

J/ψ A(p1, p2),

Aαβ

ψ′0γ(p1, p2, ǫ)ǫ1

αǫ2

βp2

1=0, p2

2=M2

ψ′

=c0

ψ′A(p1, p2),(1)

where ǫi≡ǫ(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 + m2−M2

J/ψ /2)2,

c0

ψ′=4·e·a0

ψ‘

√m·1

(M2

χc0/4 + m2−M2

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(χc0→γγ) = 3

128π|c0

γ|2M5

χc0,

Γ(χc0→J/ψγ) = 1

192π|c0

J/ψ |2M5

χc0(3 −x)2(1 −x)3,

Γ(ψ′→χc0γ) = 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(χ→γγ)th 2.24 ·10−3- 5.46 ·10−4

Γ(χ→J/ψγ)th 1.34 ·10−12.82 ·10−13.74 ·10−1

Γ(ψ′→χγ)th 2.96 ·10−22.88 ·10−22.64 ·10−2

Γ(χ→γγ)exp 2.3(2) ·10−3- 5.3(4) ·10−4

Γ(χ→J/ψγ)exp 1.3(1) ·10−12.8(2) ·10−13.7(3) ·10−1

Γ(ψ′→χγ)exp 2.96(11) ·10−22.8(1) ·10−22.7(1) ·10−2

Table 1: Model parameters and theoretical (th) (this paper, see also [6]), and experimental

(exp) [7] values of Γ(χc0,1,2→γγ , γJ/ψ) and Γ(ψ′→χc0,1,2γ).

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(χc1→e+e−) = 0.37 eV and Γ(χc2→e+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+e−→e+e−J/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+e−→e+e−χciand e+e−→e+e−χci(→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+µ−γinvariant

mass is close to the χci mass and µ+µ−invariant mass close to the J/ψ mass

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=2˜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

1−l1¯νl2

2+ǫ¯µ¯νµωl1 ¯µl2 ¯νlν

1−ǫ¯ν¯µνω l1 ¯νl2 ¯µlµ

2,

Bδσ

1=−i

2c1

J/ψ ǫ¯µ¯νσ δ F1

¯µ¯νl2−ǫ¯µ¯ν¯αδ F1

¯µ¯νl¯αlσ,

Aνµωδ

2=−˜c2

γ√2Mχc2gµδlω

1lν

2−gνµ lω

1lδ

2−gνω gµδ (l1·l2) + gνω lµ

1lδ

2,

Bπξσ

2=−c2

J/ψ √2Mχc2F1¯

βπ gσξ l¯

β−F1σπ lξ,

˜c0

γ=16πα

√m(a+fa0

J

M2

J/ψ

+f′a0

ψ′

M2

ψ′

)1

((l1−l2)2/4−m2+iǫ)2,

˜ci

γ=16πα

√m(a+faJ

M2

J/ψ

+f′aψ′

M2

ψ′

)1

((l1−l2)2/4−m2+iǫ)2, i = 1,2,

ci

J/ψ =4·e·aJ

√m·1

(M2

χci/4 + m2−M2

J/ψ /2)2, i = 1,2,

CJ/ψ

σγ =s3ΓJ/ψ→e+e−

α√l2

gσγ

l2−M2

J/ψ +iΓJ/ψ MJ/ψ

, F 1

µν = (ǫ1

µk1ν−ǫ1

νk1µ),

Πχc0=1

q2−M2

χc0+iΓχc0Mχc0

,Πχc1

ωδ =gωδ −qωqδ/M2

χc1

q2−M2

χc1+iΓχc1Mχc1

,

Πχc2

ωδπξ =

1

2(Pωπ Pδξ +Pωξ Pδπ )−1

3(Pωδ Pπξ )

q2−M2

χc2+iΓχc2Mχc2

, Pµν =−gµν +qµqν/M2

χc2.

(5)

The parts of the χci−γ∗−γ∗vertex giving vanishing contribution to the

amplitude due to the contraction with the e−e−γ∗vertices 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 ab−1leads 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 17◦and

150◦the event rates drop to 4.8M, 1.2M and 5.2M. It shows that one has an

access to information about χci−γ∗−γform factors. The expected yields of the70

events for various virtual photon invariant masses (l2

1= (p1−q1)2) are shown

in Fig. 2.

χc2

χc1

χc0

l2

1[GeV 2]

Nev

0−10−20−30−40−50−60

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 17◦and 150◦.

6

In this letter we concentrate on the possible tests of the models of χci−γ∗−γ

form factors using single tag events relaying on identiﬁcation of a simple χcide-

cay mode χci→J/ψ(→µ+µ−)−γ. If one requires identiﬁcation of χciand75

J/ψ through invariant masses of µ+µ−γand µ+µ−ﬁnal states respectively, the

χci−J/ψ∗−γform 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 17◦and

150◦[11].

χc2

χc1

χc0

l2

1[GeV 2]

Nev

0−5−10−15−20−25−30

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 χci−J/ψ∗−γform 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]

0−2−4−6−8−10

l2

1[GeV 2]

0

−2

−4

−6

−8

−10

χc1

Nev

25

20

15

10

5

0

l2

2[GeV 2]

0−2−4−6−8−10

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−

γ∗−γform factors through measurements of the reaction e+e−→e+e−χci(→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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10