A fully relativistic description of Hypernuclear production in proton- and pion-Nucleus Collisions

Abstract

Exclusive A(p,K^+){_\Lambda}B and A(\pi,K^+){_\Lambda}B^\prime reactions leading to two body final states, have been investigated in a fully covariant model based on an effective Lagrangian picture. The explicit kaon production vertex is described via creation, propagation and decay into relevant channel of N*(1650), N*(1710) and N*(1720) intermediate baryonic resonance states, in the initial collision of the projectile with one of the target nucleons. The bound state wave functions are obtained by solving the Dirac equation with appropriate scalar and vector potentials. The calculated cross sections show strong sensitivity to the final hypernuclear state excited in the reaction. Cross sections of 1 - 2 nb/sr are obtained at peak positions of favored transitions in case of the A(p,K^+){_\Lambda}B reaction on heavier targets.

Full text

Page 1

arXiv:nucl-th/0701005v1 2 Jan 2007
EPJ manuscript No.
(will be inserted by the editor)
A fully relativistic description of Hypernuclear production in
proton- and pion-Nucleus Collisions
R. Shyam1 a
Saha Institute of Nuclear Physics, 1/AF Bidhan Nagar, Kolkata, India
Received: date / Revised version: date
Abstract. Exclusive A(p,K+)ΛB and A(π,K+)ΛB′reactions leading to two body final states, have been
investigated in a fully covariant model based on an effective Lagrangian picture. The explicit kaon produc-
tion vertex is described via creation, propagation and decay into relevant channel of N∗(1650), N∗(1710)
and N∗(1720) intermediate baryonic resonance states, in the initial collision of the projectile with one
of the target nucleons. The bound state wave functions are obtained by solving the Dirac equation with
appropriate scalar and vector potentials. The calculated cross sections show strong sensitivity to the final
hypernuclear state excited in the reaction. Cross sections of 1 - 2 nb/sr are obtained at peak positions of
favored transitions in case of the A(p,K+)ΛB reaction on heavier targets.
PACS. PACS-key 25.40.Ve – PACS-key 13.75.-n,
1 Introduction
Λ hypernuclei have been studied extensively by stopped
as well as in-flight (K−,π−) [1,2] and (π+,K+) reac-
tions [3]. The kinematical properties of the (K−,π−) re-
action allow only a small momentum transfer to the nu-
cleus (at forward angles), thus there is a large probability
of populating Λ-substitutional states in the residual hy-
pernucleus. On the other hand, in the (π+,K+) reaction
the momentum transfer is larger than the nuclear Fermi
momentum, therefore, hypernuclear states with configu-
rations of an outer neutron hole and a Λ hyperon in a
series of orbits covering all bound states can be excited
in this case. The data on the hypernuclear spectroscopy
have been used extensively to extract information about
the hyperon-nucleon interaction within a variety of theo-
retical approaches (see, e.g., [4,5]).
Alternatively, Λ-hypernuclei can also be produced with
high intensity proton beams via the (p,K+)ΛB reaction
where the hypernucleusΛB has the same neutrons and
proton numbers as the target nucleus A. First set of data
for this reaction on deuterium and helium targets, have
already been reported in Ref. [6]. The states of the hyper-
nucleus excited in the (p,K+) reaction may have a differ-
ent type of configuration as compared to those excited in
the (π+,K+) reaction. Thus a comparison of informations
extracted from the study of two reactions is likely to lead
to a better understanding of the hypernuclear structure.
Theoretical studies of the A(p,K+)ΛB reaction are
based on two main approaches; the one-nucleon model
aPresent address:Institut f¨ ur Theoretische Physik, Univer-
sit¨ at Giessen, D-35392 Giessen
(ONM) [Fig. 1(a)] and the two-nucleon model (TNM)
[Fig. 1(b) and 1(c)]. In the ONM the incident proton first
scatters from the target nucleus and emits a (off-shell)
kaon and a Λ hyperon which gets captured into one of
the (target) nuclear orbits. Thus there is only one single
active nucleon (impulse approximation) which carries the
entire momentum transfer to the target nucleus (about 1.0
GeV/c at the outgoing K+angle of 0◦). This makes this
model extremely sensitive to details of the bound state
wave function at very large momenta where its magni-
tude is very small leading to quite low cross sections. In
the ONM calculations of (p,K+) and also of (p,π) reac-
tions the distortion effects in the incident and the outgoing
channels have been found to be quite important [7,8].
In the two-nucleon mechanism (TNM), the kaon pro-
duction proceeds via a collision of the projectile nucleon
with one of its target counterparts. This excites intermedi-
ate baryonic resonant states which decay into a kaon and a
Λ hyperon. The nucleon and the hyperon are captured into
the respective nuclear orbits while the kaon rescatters into
its mass shell. In this picture there are altogether three ac-
tive bound state baryon wave functions taking part in the
reaction process allowing the large momentum transfer to
be shared among three baryons. Consequently, the sensi-
tivity of the model is shifted from high momentum to the
lower momentum parts of the bound state wave functions
where they are relatively better known.
A fully covariant TNM for the A(p,K+)ΛB reaction
has been developed in Refs [9,10] by retaining the field
theoretical structure of the interaction vertices and by
treating the baryons as Dirac particles. The initial inter-
action between the incoming proton and a bound nucleon

Page 2

2R. Shyam: A fully relativistic description of Hypernuclear production in proton- and pion-Nucleus Collisions
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??????????
π
????? ?????
????? ?????
N*
??????
2
??????
p
??????
??????
??????????
??????????
??????????
????????
????????
????????
????????
??????????
??????????
??????????
??????????
??????????
Λ
Λ
K
K
K
Β
Β
Β
Λ
+
+
+
A
p
A
p
A
p
(a)
(b)
(c)
p
q
q
KK
pp
//
i
i
i
i
p
1
π
Fig. 1. Graphical representation of one- and two-nucleon mod-
els of A(p,K+)ΛB reaction. The elliptic shaded area represent
the optical model interactions in the incoming and outgoing
channels.
of the target is described by π, ρ and ω exchange mech-
anisms. This leads to the excitations of N∗(1650)[1
N∗(1710)[1
2
termediate states which decay into kaon and the Λ hy-
peron. We present a brief sketch of this formalism in the
next section.
2
−],
+], and N∗(1720)[3
2
+] baryonic resonance in-
2 Covariant Two-nucleon Model
We provide here only a brief outline of our TNM; de-
tails of this theory are given in Refs. [9,10]. The struc-
ture of the TNM for the (p,K+) reaction is similar to
that of the effective Lagrangian approach for the elemen-
tary pp → pΛK+process as discussed in Ref.[11]. We use
the same effective Lagrangians and vertex parameters to
model the initial interaction between the incoming proton
and a bound nucleon of the target by means of π, ρ and
ω exchange mechanisms. The structure for the resonance-
nucleon-meson vertices were also taken to be the same.
Terms corresponding to the interference between various
amplitudes are retained. After having established the ef-
fective Lagrangians and the coupling constants for various
vertices the amplitudes for the graphs 1b and 1c can be
written by following the well known Feynman rules. These
amplitudes can be evaluated by following the techniques
described in Ref. [10]. It should be noted that the Fig.
1c can be used in a straight forward way to calculate the
amplitudes of the A(π,K+)ΛB′reaction.
The differential cross section for the (p,K+) reaction
is given by
dσ
dΩ=
1
(4π)2
mpmAmB
(Epi+ EA)2
pK
pi
?
mimf
|Tmimf|2,
(1)
where Epiand EAare the total energies of incident proton
and the target nucleus, respectively while mp, mAand mB
10-7
100
10-6
10-5
10-4
10-3
10-2
10-1
100
101
Φ(q) (fm3/2)
|F(q)|
|G(q)|
0
1
23
4
q (fm-1)
10-12
10-10
10-8
10-6
10-4
10-2
ρ(q) (fm3)
0s1/2Λ STATE
5
ΛHe
Fig. 2. (Upper panel) Momentum space spinors [Φ(q)] for 0s1/2
Λ orbit in5
and lower components of the spinor, respectively. (Lower panel)
Momentum distribution [ρ(q) = |F(q)|2+|G(q)|2] for the same
hyperon state calculated with these wave functions.
ΛHe hypernucleus. |F(q)| and |G(q)| are the upper
are the masses of the proton, and the target and residual
nuclei, respectively. The summation is carried out over
initial (mi) and final (mf) spin states. T is the final T
matrix obtained by summing the transition
3 Results and Discussions
The spinors for the final bound hypernuclear state (cor-
responding to momentum pΛ) and for two intermediate
nucleonic states (corresponding to momenta p1 and p2)
are required to perform numerical calculations of various
amplitudes. We assume these states to be of pure-single
particle or single-hole configurations with the core remain-
ing inert. The spinors in the momentum space are ob-
tained by Fourier transformation of the corresponding co-
ordinate space spinors which are the solutions of the Dirac
equation with potential fields consisting of an attractive
scalar part (Vs) and a repulsive vector part (Vv) having
a Woods-Saxon form. The potential parameters are given
in Ref. [10].
In Figs. 2 and 3, we show the lower and upper com-
ponents of the Dirac spinors in momentum space for the
0s1/2hyperon in5
13
ΛC, respectively. In each case, we note that only for mo-
menta < 1.5 fm−1, is the lower component of the spinor
substantially smaller than the upper component. In the
region of momentum transfer pertinent to exclusive kaon
production the lower components of the spinors are not
ΛHe and the 0p3/2and 0s1/2hyperons in

Page 3

R. Shyam: A fully relativistic description of Hypernuclear production in proton- and pion-Nucleus Collisions3
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
101
Φ(q) (fm3/2)
0
1
23
4
q (fm-1)
10-12
10-10
10-8
10-6
10-4
10-2
ρ (fm3)
1
23
4
q (fm-1)
13
ΛC0p3/2Λ STATE
0s1/2Λ STATE
Fig. 3. same as in Fig. 2 for 0p3/2 Λ and 0s1/2 Λ orbits in
hypernucleus13
ΛC.
negligible as compared to the upper component which
demonstrates that a fully relativistic approach is
essential for an accurate description of this reaction. The
spinors calculated in this way provide a good description
of the experimental nucleon momentum distributions for
various nucleon orbits as is shown in Ref. [12]. In the lower
panel of each of Figs. 2 and 3 we show momentum distri-
bution ρ(q)[= F(q)|2+|G(q)|2] of the corresponding Λ hy-
peron. In each case the momentum density of the hyperon
shell, in the momentum region around 0.35 GeV/c, is at
least 2-3 orders of magnitude larger than that around 1.0
GeV/c. In the TNM, therefore cross section will be larger
than the those in the ONM as in the former case the sen-
sitivity of the model gets shifted to lower momenta due
to involvement of three baryon in the momentum sharing
process.
A crucial quantity needed in the calculations of the
kaon production amplitude is the pion self-energy, Π(q),
which takes into account the medium effects on the inter-
mediate pion propagation. Since the energy and momen-
tum carried by such a pion can be quite large (particularly
at higher proton incident energies), a calculation of Π(q)
within a relativistic approach is mandatory. In our study
contributions from the particle-hole (ph) and delta-hole
(∆h) excitations are taken into account. The self-energy
has been renormalized by including the short-range repul-
sion effects by introducing the constant Landau-Migdal
parameter g′which is taken to be the same for ph − ph
and ∆h−ph and ∆h−∆h correlations which is a common
choice.
In Fig. 4, we show the dependence of our calculated
cross sections on pion self-energy. It is interesting to note
that this has a rather large effect. We also note a sur-
prisingly large effect on the short range correlation (ex-
pressed schematically by the Landau- Migdal parameter
g′). Similar results have also been reported in case of the
0
5
10
15
20
25
30
θ (deg)
10-6
10-5
10-4
10-3
10-2
10-1
100
101
dσ/dΩ (nb/sr)
g´ = 0.5
g´ = 0.6
g´ = 0.7
no pion self energy
40Ca (p,K+) 41
ΛCa(0d3/2)
Ep = 2.0 GeV
Fig.
40Ca(p,K+)41
energy of 2.0 GeV. The dotted line shows the results obtained
without including the pion self-energy in the denominator
of the pion propagator while full, dashed and dashed-dotted
lines represent the same calculated with pion self-energy
renormalized with Landau-Migdal parameter of 0.5, 0.6 and
0.7, respectively.
4.
Differential
ΛCa(0d3/2) reaction for the incident proton
cross sectionforthe
(p,π) reactions. However, more definite statements about
the usefulness of (p,K+) reactions in probing the medium
effects on the pion propagation in nuclei must await the
inclusions of distortions in the incident and outgoing chan-
nels.
In Figs. 5 and 6, we show the kaon angular distri-
butions for various final hypernuclear states excited in
12C(p,K+)13
The incident proton energies in the two cases are taken to
be 1.8 GeV and 2.0 GeV, respectively where the angle
integrated cross sections for the two reactions are max-
imum. The calculations are the coherent sum of all the
amplitudes corresponding to the various meson exchange
processes and intermediate resonant states. Clearly, the
cross sections are quite selective about the excited hy-
pernuclear state, being maximum for the state of largest
orbital angular momentum. This is due to the large mo-
mentum transfer involved in this reaction.
It should be noted that in in Fig. 2 the angular distri-
bution has a maximum at angles larger than 0◦. In con-
trast to this, the maximum in Fig 3, occurs at zero degree
and it decreases gradually as the angle increases. This is
due to the fact that in the bound state spinors of13
there are several maxima in the upper and lower compo-
nents of the momentum space bound spinors in the region
of large momentum transfers. Therefore, in the kaon angu-
lar distribution the first maximum may shift to larger an-
gles reflecting the fact that the bound state wave functions
show diffractive structure at higher momentum transfers.
On the other hand, for momentum transfers relevant to
4He(p,K+)5
ΛHe reaction the Dirac spinors are smoothly
varying and are devoid of structures as can be seen in
Fig. 2. In any case, from the purely kinematical arguments
ΛC and4He(p,K+)5
ΛHe reactions, respectively.
ΛC,

Page 4

4R. Shyam: A fully relativistic description of Hypernuclear production in proton- and pion-Nucleus Collisions
0
5
10 15 20 25 30
θK (deg)
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
101
dσ/dΩ (nb/sr)
0p3/2
0s1/2
0p1/2
12C(p,K+)13
ΛC(nlj)
Ep = 1.8 GeV
Fig. 5. Differential cross section for the
action for the incident proton energy of 1.8 GeV for various
bound states of final hypernucleus as indicated in the figure.
12C(p,K+)13
ΛC re-
it is clear that the maximum in the cross section for a
ℓ = 0 transition is expected to occur at smaller angles as
compared that for a ℓ ?= 0 one.
4 Summary and Conclusions
In summary, we have made a study of the A(p,K+)ΛB re-
action on4He,12C, and40Ca targets and of theA(p,K+)ΛB
reaction on the12C target within a fully covariant general
two-nucleon mechanism where in the initial collision of
the projectile with one of the target nucleons, N∗(1710),
N∗(1650), and N∗(1720) baryonic resonances are excited
which subsequently propagate and decay into the rele-
vant channel. Wave functions of baryonic bound states are
obtained by solving the Dirac equation with appropriate
scalar and vector potentials.
In the case of the nucleon projectile, the cross sections
are dominated by the graphs of the type shown in Fig.1b
(target emission graph). The one-pion-exchange processes
make up most of the differential cross section at all angles.
The calculated cross sections are maximum for the hyper-
nuclear state with the largest orbital angular momentum.
For heavier targets, the angular distributions for the fa-
vored transitions peak at angles larger than the 0◦which
in contrast to the results of most of the previous nonrela-
tivistic calculations for this reaction. This reflects directly
the nature of the Dirac spinors for the bound states which
involve several maxima in the region of large momentum
transfer. In case of the light target4He, however, the dif-
ferential cross section still peaks near the zero degree as
in this case the momentum transfers are in the region
where the bound state spinors are smoothly decreasing
with momentum. The energy dependence of the calcu-
lated total production cross section follows closely that
of the pp → pΛK+reaction. We find that the nuclear
0
4
8 1216
θK (deg)
10-4
10-3
10-2
10-1
dσ/dΩ (nb/sr)
4He (p,K+) 5
ΛHe(0s1/2)
Ep = 2.0 GeV
Fig. 6. Differential cross section for the4He(p,K+)5
tion for the incident proton energy of 2.0 GeV for the bound
state of final hypernucleus as indicated in the figure.
ΛHe reac-
medium corrections to the intermediate pion propagator
introduce large effects on the kaon differential cross sec-
tions. There is also the sensitivity of the cross sections
to the short-range correlation parameter g′in the pion
self-energy. Thus, (p,K+) reactions may provide an inter-
esting tool to investigate the medium corrections on the
pion propagation in nuclei.
In case of the pion induced reaction, the hypernuclear
states with stretched spin configurations are preferentially
excited.
The author wishes to thank Horst Lenske and Ulrich
Mosel for several useful discussions and collaboration on
this subject.
References
1. R. E. Chrien and C. B. Dover, Annu. Rev. Nucl. Part. Sci.
39 (1989) 113.
2. H. Band¯ o, T. Motoba and J.ˇZofka, Int. J. Mod. Phys. 5
(1990) 4021.
3. O. Hashimoto, H. Tamura, Prog.Part.Nucl.Phys.57 (2006)
564.
4. E. Hiyama, M. Kamimura, T. Motoba, T. Yamada, and Y.
Yamamoto, Phys. Rev. Lett. 85 (2000) 270
5. C. M. Keil and H. Lenske, Phys. Rev. C 66 (2002) 054307.
6. J. Kingler et al., Nucl. Phys. A634 (1998) 325.
7. B. V. Krippa, Z. Phys. A 351, (1995) 411.
8. E. D. Cooper and H. S. Sherif, Phys. Rev. Lett. 47, 818
(1982); Phys. Rev. C 25, 3024 (1982).
9. R. Shyam, H. Lenske, and U. Mosel, Phys. Rev. C 69 (2004)
065205.
10. R. Shyam, H. Lenske, and U. Mosel, Nuc. Phys. 764 (2006)
313.
11. R. Shyam, Phys. Rev. C 60 (1999) 055213.
12. R. Shyam, W. Cassing and U. Mosel, Nucl. Phys. A586
(1995) 557.