Binding of charmonium with two- and three-body nuclei

0.12 MB·English

by  V. B. Belyaev

Binding of charmonium with two- and three-body nuclei
∗
V.B. Belyaev, N.V. Shevchenko
Joint Institute for Nuclear Research, Dubna, 141980, Russia
A. Fix
Institut fu¨r Kernphysik, Johannes Gutenberg-Universita¨t Mainz, D-55099 Mainz, Germany
W. Sandhas
Physikalisches Institut, Universit˝at Bonn, D-53115 Bonn, Germany
(Dated: February 9, 2008)
The energies of the ηcd and ηc3He bound states are calculated on the basis of exact three- and
four-bodyAGSequations. FortheηcN interactionaYukawa-typepotentialhasbeenadopted. The
calculations are done for a certain range of its strength parameter. The results obtained are quite
6
different from calculations based on the folding model.
0
0
PACSnumbers: 21.45.+v,13.75.Gx,13.85.-t,Jz
2

n
a I. INTRODUCTION
J

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The behavior of mesons in a nuclear medium has attracted much attention during the last years [1]. We mention
2
the existence and properties of deeply bound meson-nuclear states [2] as well as the possible restoration of chiral

2 symmetry in the nuclear environment[3], a point closely relatedto modifications of widths andmasses of the mesons
v propagating through nuclear matter [4].
8 It should be emphasized that all these phenomena are usually described in terms of effective degrees of freedom
5
including mesons, nucleons, and isobars. The manifestation of more fundamental quark-gluon degrees of freedom at
0
lowenergiesisveryunlikelybecauseofthestrongmismatchbetweenthe nuclearenergyscale(few tenthofMeV)and
1
the QCD energy scale (few hundreds or thousands of MeV).
0
6 However,aspointedoutin[5]withregardtocharmonium-likemesons,forexampleηc,whichdonotcontainu-and
0 d-quarks,the maincontributiontothe meson-nucleoninteractionisexpectedtooriginatenotfromeffective,butfrom
/ fundamentalQCDdegreesoffreedom,namelyfromthefew-gluonexchange. Inthisway,usingthePomeronexchange
h
model, one can deduce a Yukawa-type potential for the η -nucleon system [5]. Following these ideas, the authors of
t c
- [6] were able to explain the pp-spin correlationat the threshold energy of charm production. Such an analysis allows
l
c one to restrict the uncertainties of the parameters of the charmonium-nucleon potential. Independent theoretical
u
indication on the existence of attraction between charmonium and nucleons has been obtained in [7]. In case of an
n
infinite-mass limit of the heavy quark, these authors have obtained a binding energy of J/ψ in nuclear matter of
:
v the order of 10 MeV. For the (2S) charmonium state this approach predicts an even stronger binding. Despite the
i
X above indications, the data of the charmonium-N interaction are still rather scarce. Bearing this in mind, it appears
reasonable to look for possible formations of the corresponding mesic nuclei, similar to the η-nuclear systems which
r
a wereextensivelydiscussedintheliterature(seee.g.[8]andreferencestherein). Inparticularthestudyofthereactions
′ ′
p¯+A→p¯+p+A →η +A (1)
c
has been planned within the PANDA project [9]. The production of η in electromagnetic processes has already
c
been started in the CLIO collaboration [10]. As explained in [5], the precise knowledge of the binding energy of
meson-nuclear systems is of crucial importance for their experimental observation.
First attempts to calculate the binding of η -mesons with light nuclei have been made in Ref. [5] on the basis of
c
variationalcalculations, and in Ref. [11] where the folding model was used. Both calculations involve a Yukawa form
for the basic η N interaction. Our results for the binding energies of η d and η 3He, presented below, are obtained
c c c
within the AGS theory [12, 13]. Similar to [5, 11] we use the Yukawa type η N potential
c
e−αr
V (r)=−a (2)
ηcN r
∗ Presentaddress: NuclearPhysicsInstitute, 25068 Rˇeˇz,CzechRepublic
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with α = 0.6 GeV. The parameter a is varied between 0.4 and 0.6. Since the two-body potentials act essentially
via their s-wavescontributions, we take into account only s-waveparts of the corresponding η N and NN scattering
c
amplitudes entering the kernel of the AGS equations.
II. FORMALISM
Let us start by giving a brief description of the 3-body η NN formalism. Its main ingredient is a separable
c
representation of the driving η N and NN potentials
c
N
V = |χiihχj|, α∈{η N, NN}. (3)
α α α c
i,j
X
Tooutline the importantaspectsofthis approach,weuse forthe momentanoversimplifiedrepresentationoftheη N
c
potential keeping only the first term in the sum (3), while the actual calculations are performed with a more realistic
representation containing six terms (we will discuss the sensitivity of our results to the number of terms kept in the
separable expansion). For the s-wave triplet state 3S of two nucleons we employ the separable rank-one version of
1
the Paris potential of Ref. [14].
The AGS equations for the η NN system read [8]
c
p2
X (z) = 2Z (z)τ (z− 2 )X (z) (4)
11 21 2 2µ 21
2
p2 p2
X (z) = Z +Z (z)τ (z− 1 )X (z)+Z (z)τ (z− 2 )X (z).
21 21 21 1 2µ 11 23 2 2µ 21
1 2
The functions τ (z), Z , and X (α,β=1,2) can be expressed in terms of the AGS transition operators U and
α αβ αβ βα
form factors |χ i appearing in (3) via
α
Z (z) ≡ (1−δ )hχ |G (z)|χ i (5)
βα αβ β 0 α
X (z) ≡ hχ |G (z)U G (z)|χ i (6)
βα β 0 βα 0 α
where G (z) is the free η NN propagator. As mentioned, for the NN subsystem we use a separable representation
0 c
of the Paris potential
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C
V (k,k′)=−g(k)g(k′), where g(k)= i , (7)
NN k2+β2
i=1 i
X
with parameters C and β listed for the 3S configuration in [14].
i i 1
For the η N interaction we first consider the rank-one representationof the η N potential (3) within the Bateman
c c
method [15]. The corresponding general expression reads
N
V[N](k,k′)= V (k,s )d−1V (s ,k′), (8)
0 0 i ij 0 j
i,j
X
with d =V (s ,s ). Here s is an expansion parameter and
ij 0 i j i
∞
1
V (k,k′)= j (kr)V(r)j (k′r)r2dr (9)
0 2π2 0 0
Z0
the s-wave Fourier transform of the potential V(r). Using expression (2) for the potential V(r) in the integral (9)
and taking N =1 and s =0 in the expansion (8) one can readily obtain
1
V(k,0)V(0,k′) α2
V[1](k,k′)= =−4πa . (10)
0 V(0,0) (k2+α2)(k′2+α2)
IftheBatemanmethodisappliedtoanegativepotential,thenexttermsintheseparableexpansiontendtoincrease
the attraction. Therefore, the neglect of the higher order terms in the sum (8) will result in underestimation of the
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FIG.1: Bindingenergyofηcdsystem,asfunctionofthestrengthaoftheYukawapotential(2). Theresultsobtainedwiththe
Bateman andHilbert-Schmidtrepresentation oftheηcN potentialarepresentedbythedashedandthesolid line,respectively.
absolute value ofthe η d binding energy. An obvious advantageofthe Batemanmethod is that it providesanalytical
c
expressions for all terms in (8).
It is worthwhile to note that the potential (10) predicts an antibound (virtual) state in the η N system lying at
c
2
1 aµα3
ǫ=− −α =−6.1 MeV, (11)
2µ 4π
r !
where µ is η N reduced mass. The corresponding scattering length
c
2
a = ≈0.97 fm (12)
ηcN 4π −α
aµ
is a bit larger than the ηN scattering length predicted by the modern analyses (see e.g. compilation in [16]). Fur-
thermore, there is no absorptive part in the η N potential, which in the ηN case effectively increases the repulsive
c
propertiesofthe interaction. The mostimportantdifference betweenηN andη N is the largermass ofthe η meson,
c c
which should increase the role of attraction in the η few-nucleon dynamics. According to this, one can expect that
c
the η meson should be more tightly bound to the nucleons than the η meson. The energy of the η d bound state,
c c
calculatedwith the potential(10)as a function ofthe strengthparametera, is presentedinFig. 1by the dashedline.
As a next step we employ the Hilbert-Schmidt method for the separable representation of the potential (2). The
corresponding expansion reads
N
′ ′
V (k,k )= λ v (k)v (k ), (13)
ηcN n n n
n=1
X
where the form factors v (k) and strength parameters λ obey the homogeneous integral equation
n n
∞
1 v (k′) k′2dk′
v (k)= V (k,k′) n . (14)
n λn ηcN B− k′2 2π2
Z0 2µ
B = 0 was chosen in the actual calculation.
III. RESULTS
The η d binding energy calculatedfor the potential (13) with N = 6 terms is presentedin Fig. 1. As expected, the
c
last model predicts stronger binding than the leading term in the Bateman expansion (8). Furthermore as one can
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see, in contrast to the results of [5] and [11], we find only one bound state lying in the region −2 MeV ≤ E ≤ −5
b
MeV, when the parameter a is varied between 0.5 and 0.6 GeV.
TABLE I: Dependenceof the ηcd bindingenergy Eb on thenumber of separable terms N in the expansion (13). Parameter a
is equal to 0.55.
N 1 2 3 6
Eb [MeV] -3.429 -3.251 -3.217 -3.196
The results in Table I show the accuracy of the Hilbert-Schmidt method for the η d binding energy. Evidently,
c
already the first two terms in the sum (13) provide quite a satisfactory approximation. Going up to N = 6 results in
only 1.7% increase of |E |.
b
Now we turn to the four-body η 3He system. For the calculation we use the formalism developed in Ref. [17]
c
and Ref. [18] for the interaction of isoscalar-pseudoscalar mesons with three-body nuclei. Since the formal part of
the problem is quite involved we do not present it here and refer the reader to these references. It should only be
recalled that the calculations are based on separable representations of the two- and three-body kernels of the basic
AGS equations so that, as a consequence, the resulting integral equations have the same structure as the three-body
equations (1).
TABLE II:The binding energy of ηc3He system (in MeV).
a 4-Body calculations Variational calculation[5] Folding model [11]
0.4 -1.3 -3.0
0.6 -14.5 -19.0 -0.8
Letuscompareourresultsforthebindingenergyofη 3Hesystemwithvariational[5]andfoldingmodelcalculations
c
[11]. One should emphasize that the variationalmethod used in [5] is not an ab initio many body treatment. Rather
it is a model of meson-nuclear interaction, postulating a Yukawa form for the meson-nuclear potential with a range
depending onthe number ofnucleons inthe nucleus. This model, as wellas the folding model, takesinto accountthe
importanteffectofsmearingthemeson-nucleoninteractionoverthenuclearvolume. However,themultiplescattering
of mesons in the nucleus is not accurately accountedfor. Our results of the microscopicfour-body calculations of the
η 3He binding energy are presented in Fig. 2 and compared with the alternative ones in Table II. The calculations
c
are performed with the value α = 0.6 GeV for the range parameter of the η N potential.
c
In summary,we calculatedthe binding energiesofη d andη 3He using three-and four-bodyequations ofthe AGS
c c
theory. The results exhibit essential differences to the predictions of the folding model as well as to the variational
calculations. The characteristic values of the binding energies are about −3 MeV for the η d system and amount to
c
−14.5 MeV for a = 0.6 in the η 3He case.
c
Acknowledgment
Thisworkwassupportedbythe DeutscheForschungsgemeinschaftandtheRussianFoundationforBasicResearch,
project 436RUS113/761.
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FIG. 2: Dependenceof the ηc3He bindingenergy on theparameter a in (2).
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