A study of the $p d \to p d \eta$ reaction
A study of the p d → p d η reaction
N.J.Upadhyay,1, ∗ K.P.Khemchandani,1, 2, † B.K.Jain,1, ‡ and N.G.Kelkar3, §
1Department of Physics, University of Mumbai,
Vidyanagari, Mumbai - 400 098, India
2Departamento de F́ısica Teórica and IFIC,
Centro Mixto Universidad de Valencia-CSIC Institutos de Investigación de Paterna,
Aptd. 22085, 46071 Valencia, Spain
3Departamento de Fisica, Universidad de los Andes,
Cra.1E No. 18A-10, Santafe de Bogota, Colombia
(Dated: October 25, 2018)
http://arxiv.org/abs/0704.1530v1
Abstract
A study of the p d → p d η reaction in the energy range where the recent data from Uppsala
are available, is done in the two-step model of η production including the final state interaction.
The η − d final state interaction is incorporated through the solution of the Lippmann Schwinger
equation using an elastic scattering matrix element, Tη d→ η d, which is required to be half off-shell.
It is written in a factorized form, with an off-shell form factor multiplying an on-shell part given
by an effective range expansion up to the fourth power in momentum. The parameters of this
expansion have been taken from an existing recent relativistic Faddeev equation solution for the
ηNN system corresponding to different η −N scattering amplitudes. Calculations have also been
done using few body equations within a finite rank approximation (FRA) to generate Tη d→ η d.
The p − d final state interaction is included in the spirit of the Watson-Migdal prescription by
multiplying the matrix element by the inverse of the Jost function. The η − d interaction is found
to be dominant in the region of small invariant η − d mass, Mηd. The p − d interaction enhances
the cross section in the whole region of Mηd, but is larger for large Mηd. We find nearly isotropic
angular distributions of the proton and the deuteron in the final state. All the above observations
are in agreement with data. The production mechanism for the entire range of the existing data
on the p d → p d η reaction seems to be dominated by the two-step model of η production.
PACS numbers: 25.10.+s, 25.40.Ve, 24.10.Eq
∗Electronic address: neelam@physbu.mu.ac.in
†Electronic address: kanchan@ific.uv.es
‡Electronic address: brajeshk@gmail.com
§Electronic address: nkelkar@uniandes.edu.co
mailto:neelam@physbu.mu.ac.in
mailto:kanchan@ific.uv.es
mailto:brajeshk@gmail.com
mailto:nkelkar@uniandes.edu.co
I. INTRODUCTION
The current great interest in the η-nucleus interaction exists because of the attractive
nature of the η − N interaction in the s-wave [1], and the consequent possibility of the
existence of quasi-bound, virtual or resonant η-nucleus states [2]. The exact nature of
these states, of course, depends upon the precise knowledge of the η −N scattering matrix
at low energies. As the η is a highly unstable meson (lifetime ∼ 10−18s), this precise
information is difficult to obtain directly. It can only be obtained from the eta producing
reactions through the final state interaction. With this motivation, starting with the early
experiments near threshold at Saclay on the p d → 3He η and the p d → p d η reactions,
measurements have been carried out near threshold and beyond at Jülich and Uppsala using
the COSY and Celsius rings respectively. In this series of experiments, the recent data
on the p d → p d η reaction using the Wasa/Promice setup at the Celsius storage ring of
the Svedberg laboratory, Uppsala are thematically complete and cover the excess energy,
Q, (Q =
s − mη − mp − md) ranging from around threshold to 107 MeV. The data
[3] (integrated over other variables) include the invariant mass distribution over the whole
excess energy range for the η− d, η− p and p− d systems and angular distributions for the
proton, deuteron and the eta meson. Like the p d → 3He η reaction, the (inclusive) η − d
invariant mass distribution exhibits a large enhancement near threshold and hence appears
promising to study the η − d interaction. The η − p and p− d invariant mass distributions
do not show any such enhancement. All observed angular distributions are nearly isotropic.
Like in our earlier studies on the p d → 3He η reaction our primary aim in this paper is
to investigate the above mentioned data on the η − d invariant mass distribution to obtain
a better understanding of the η − N as well as the η − d interaction. We speculate, from
our experience on the study of the p d → 3He η reaction [4], that in the region of low η− d
relative energy this set of data will be mainly determined by the η − d interaction, though
the three-body nature of the final state may introduce some uncertainty in this conclusion.
We present a study of the p d → p d η reaction which includes the effect of the final state
interaction. We have investigated two possible diagrams for the production mechanism:
the direct mechanism and the two step process of η production. The direct mechanism
proceeds via an intermediate p n → d η reaction with one of the nucleons in the deuteron
as a spectator. The η meson in the two step model is produced in two steps, namely, p p →
d π+ and π+N → η N , hence involving the participation and sharing of the transferred
momentum by three nucleons. The two step model for η production was first used in [5] and
the data on the p d → 3He η reaction was well explained. The vertices at the two steps have
been described by the corresponding off-shell T -matrices. The T -matrix for π+N → η N is
taken from a coupled channel calculation [1], and that for p p → d π+ is obtained from the
SAID program provided by the authors of Ref. [6].
The final state interaction between the η and the deuteron is explicitly incorporated
through an η − d T -matrix, Tηd. This T -matrix, which is required to be half-off-shell, is
described in two ways. One choice involves taking a “factorized form” which is given by an
off-shell form factor multiplied by an on-shell part given by an effective range expansion up to
the fourth power in momentum. The parameters of this expansion have been taken from an
existing recent relativistic Faddeev equation solution for the ηNN system [7] corresponding
to different η − N scattering amplitudes. The off-shell form factor will be described in the
next sections and is chosen to have a form without any adjustable parameters. The second
prescription involves solving few body equations within the finite rank approximation (FRA)
to obtain Tηd. This approach has been used in literature for the η − d, −3He and −4He
systems [8]. We perform calculations for both the prescriptions using different models of the
elementary coupled channel η-nucleon T -matrix which characterize them.
The interaction between the η meson and the proton in the final state, to a certain
extent is contained implicitly in our calculations. This is due to the fact that we describe
the π+N → η N vertex by a T -matrix, which has been modeled to include the η − N
interaction. This off-shell T -matrix treats the π N , η N and π∆ channels in a coupled
channel formalism [1] and reproduces the experimental data on this reaction very well.
The effect of p−d final state interaction (FSI) is incorporated in the spirit of the Watson-
Migdal FSI prescription [9], in which our model p d → p d η production amplitude is mul-
tiplied by a factor which incorporates the FSI between the proton and the deuteron. This
factor is taken to be the frequently used [10, 11, 12, 13] inverse Jost function, [J(p)]−1, where
p is the relative p− d momentum. The assumption implicit in this approximation that the
mechanism for the primary reaction be short ranged is very well fulfilled in the η-production
reactions. The momentum transfer in these reactions near threshold is around 700 MeV/c.
We include FSI for both doublet (2S1/2) and quadruplet (
4S3/2) p− d states.
The η-nucleon T -matrix, which characterizes our calculations, is not precisely known.
Recent theoretical works on the n p → d η reaction [14] conclude that the data on this
reaction can be reproduced with the strength of the real part of the η-nucleon scattering
length ranging between 0.42 and 0.72 fm. In our earlier work on the p d → 3He η reaction
[4], we found a good agreement with data, with the real part of the scattering length taken
to be around 0.75 fm. This value was also found to be in agreement with the n p → d η
data in a K-matrix calculation of the final state η− d interaction in [15]. The same authors
as in [15], recently performed a fit to a wide variety of data which includes the π N → π N ,
π N → η N , γ N → π N and γ N → η N reactions and gave their best fit value of
the η-nucleon scattering length, aηN to be (0.91, 0.27) fm [16]. The η − d effective range
parameters are given in [7] for aηN up to (1.07 , 0.26) fm. Hence, in the present work we
perform calculations with different models of the η − N interaction, which correspond to
three different values of the η − N scattering length, ranging from aηN = (0.42 , 0.34) fm
to (1.07 , 0.26) fm.
We find that the cross sections calculated using the two-step model and the above in-
puts for the final state interaction reproduce most of the features of the experimental data
reasonably well.
A theoretical effort to understand the Uppsala data [3] was made earlier by Tengblad
et. al. [17]. In [17] the contribution of three different diagrams, namely, the pick-up (a
direct one-step mechanism of η production), the impulse approximation and the two-step
mechanism (here the η meson is produced in two steps via the p p → π+ d and π+N → η N
reactions) to the cross section for the p d → p d η reaction is determined. The authors in
[17] conclude that the impulse approximation is in general negligible as compared to the
other two diagrams, the two-step mechanism is dominant in the near threshold region and
the contribution of the pick-up diagram (referred to as the direct mechanism in the present
work) increases with energy and matches the two-step contribution at an excess energy of
Q = 95 MeV. The latter conclusions regarding the contributions of the two step and pick up
diagrams are in contrast to the findings of the present work as well as to existing literature on
similar kind of reactions. We note here that the authors in [17] do not include the final state
interaction in their calculations in any way. They treat the kinematics and the dependence
of the pion propagator (appearing in the two step model) on the Fermi momenta in an
approximate way. The T -matrices which enter as an input to the two step model are simply
extracted from experimental cross sections and are hence not proper off-shell T -matrices.
As a result of the above approximations, the authors in [17] do not reproduce the observed
enhancement in the η−d invariant mass distribution near threshold, and unlike the observed
isotropic distributions, find anisotropy in their calculated angular distributions.
The contribution from the direct mechanism (or the so-called pick-up diagram of [17]) to
the total cross sections is found to be about four orders of magnitude smaller than the two-
step contribution at threshold in the present work. The one-step contribution does increase
with energy (as also found in [17]), however, even at the highest energy for which data is
available (Tp = 1096 MeV) it remains two orders of magnitude smaller than that due to the
two-step model. This is in contrast to the observations in [17], where the two processes give
comparable contributions at high energies. The difference of orders of magnitude between
the two processes can be understood as a result of the large momentum transfer, q, in
the one-step process. This q, which is very large in the threshold region (∼ 840 MeV/c)
continues to be large even at high energies. For example, it is ∼ 600 MeV/c even at the
highest beam energy of 1096 MeV. This finding of ours is very similar to the previous studies
of the reactions involving high momentum transfer. For example, as mentioned above too,
in [5], for the pd →3 Heη reaction up to 2.5 GeV beam energy, the authors comment that
the one-step cross sections underestimate the data by more than two orders of magnitude.
In yet another calculation [18] of the cross section for the pd →3He X reaction (where
X = η, η′, ω, φ) the two-step model was found to describe the data on these reactions up
to 3 GeV quite well. In [19], in connection with the p d → 3HΛK+ reaction, the authors
claim that for a beam energy of 1 - 3 GeV, the one-step mechanism predicts 2 to 3 orders of
magnitude smaller cross sections as compared to the two-step mechanism. The cross sections
obtained from the one-step model, in Ref. [17] are, however, reported to be only one order
of magnitude less than those due to the two-step model at threshold and comparable to the
two-step ones at high energies.
In the next section, we describe the details of the formalism. In the subsequent sections
we present and discuss the results and finally the conclusions.
II. THE FORMALISM
The differential cross section for the p d → p d η reaction, in the center of mass, can be
written as,
2 (2 π)5 s |~kp|
dΩp′ | ~kp′| dMη d |~kη d| dΩη d
〈 |T |2 〉 (1)
where
s is the total energy in the center of mass and ~kp and ~kp′ are the proton momenta in
the initial and final states respectively. Mη d denotes the invariant mass of the η − d system
and ~kη d and Ωη d denote, in the η − d center of mass, the η momentum and its solid angle,
respectively. Ωp′ represents the solid angle of the outgoing proton. Angular brackets around
|T |2 in Eq. (1) represent the sum over the final and initial spins.
The T -matrix, which includes the interaction between the η and the deuteron is given by
T = 〈ψηd( ~kηd), ~kp′; mp′, md′ | Tpd→pdη | ~kp, ~kd (= − ~kp); mp, md 〉 (2)
where the spin projections for the proton and the deuteron in the initial and final states have
been labeled as mp, md, mp′, and md′ respectively. Tp d→ p d η is the production operator.
The wave function of the interacting η − d in the final state has been represented as
ψηd( ~kηd). In terms of the elastic η − d scattering T -matrix, Tηd, it is written as
〈ψ−ηd | = 〈 ~kηd |+
(2π)3
〈 ~kηd | Tηd | ~q 〉
E(kηd) − E(q) + iǫ
〈 ~q | (3)
The second term here represents the scattered wave. It has two parts originating from
the principal-value and the delta-function part of the propagator in the intermediate state.
Physically they represent the off-shell and the on-shell scattering between the η and the
deuteron. The on-shell part can be shown to be roughly proportional to the η−d momentum
and hence dominant at higher energies. The relative contribution of these terms in our case
would be determined after we substitute the above expression for ψηd( ~kηd) in Eq. (2). We
then get
T = 〈 ~kηd, ~kp′; mp′, md′ | Tpd→pdη | ~kp, ~kd(= −~kp); mp, md 〉 (4)
(2π)3
〈 ~kηd;md′ | Tηd | ~q;m2′ 〉
E(kηd)−E(q) + iǫ
〈 ~q , ~kp′;m2′ , mp′ | Tpd→pdη | ~kp, ~kd; mp, md〉
It can be seen that the Tηd here appears as a half-off-shell T -matrix.
  =  - k
p kd/ 2 - P
FIG. 1: The two step process production mechanism for the p d → p d η reaction.
A. The production mechanism
For evaluating the η production T -matrix, 〈 | Tpd→pdη | 〉, we assume a two-step mechanism
as shown in Fig. 1. In this model, the incident proton produces a pion in the first step
on interacting with one of the nucleons of the target deuteron. In the second step this
pion produces an η meson on interacting with the other nucleon. Both these nucleons
are off-shell and have a momentum distribution given by the deuteron bound state wave
function. To write the production matrix, we resort to certain standard approximations
used in literature [20] (in particular for the triangle diagram appearing in Fig. 1). The
amplitude for the pN → πd process, which in principal is off-shell, is taken at an on-shell
energy. Considering the high proton beam energy, off-shell effects are not expected to be
significant. The π N → η N process is included via an off-shell T -matrix.
The production matrix is written as [4, 5],
〈 | Tpd→ p d η | 〉 =
(2π)3
〈 p n | d 〉 〈 | Tpp→ π+ d | 〉
k2π − m2π + iǫ
〈 | Tπ+ n→ η p | 〉 (5)
where, the squared four momentum of the intermediate pion, k2π = E
π−~k2π, with the energy,
Eπ, calculated at zero fermi momentum and ~kπ = ~kη + ~kp′ − ~kd/2 + ~P . The summation is
over internal spin projections and the matrix element 〈 p n | d 〉 represents the deuteron wave
function in momentum space, which has been written using the Paris parametrization [21].
The factor 3/2 is a result of summing the diagrams with an intermediate π0 and π+.
The integral over the pion momentum in above includes the contribution from the pole
as well as the principal value term. For the pion propagator itself, as we see, we have taken
  =  - k
| = - ( kη +  kd| ) 
-1/2 k
FIG. 2: The direct process production mechanism for the p d → p d η reaction.
the plane wave propagator. This thus excludes any effect in our results due to medium mod-
ification of this propagator due to other nucleons. This aspect may be worth investigating
in future.
The T -matrix for the intermediate p p → π+ d process has been taken from an energy
dependent partial wave analysis of the π+ d → p p reaction from threshold to 500 MeV [6].
The various observables in [6] are given in terms of amplitudes which are parametrized to fit
the existing database. We refer the reader to [6] and the references therein for the relevant
expressions of the helicity and partial wave amplitudes and the notation followed by the
authors in [6].
For the π+ n → η p sub-process, different forms of T -matrices are available. We use
the T -matrix from [1] which treats the π N , η N , and π∆ channels in a coupled channel
formalism. This T -matrix consists of the meson - N∗ vertices and the N∗ propagator as given
below:
Tπ+ n→ η p(k
′ , k ; z) =
gN∗ β
(k′ 2 + β2)
τN∗(z)
gN∗ β
(k2 + β2)
with,
τN∗(z) = ( z − M0 − Σπ(z) − Ση(z) + i ǫ)−1
where Σα(z) (α = π , η) are the self energies from the πN and η N loops. The parameters
of this model are, gN∗ = 0.616, β = 2.36 fm
−1 and M0 = 1608.1 MeV. This T -matrix
reproduces the data on the π+n→ ηp reaction very well.
Although the contribution of the direct mechanism (Fig. 2) is known to be small (owing
to the large momentum transfer involved in the process) [5, 18, 19], for completeness, we
calculate its contribution to the total cross section. The T -matrix for this mechanism can
be written as
〈|Tp d→ p d η|2〉 =
〈|Tpn→ d η(
sη d)|2〉 × | φd(q)|2, (7)
where φd represents the deuteron wave function in the initial state. The spin summed
〈| Tpn→ d η |2〉 is given in terms of the total cross section for the p n → d η reaction by
σT (p n → d η) =
2mpmnmd
| ~pf |
|~pi|
〈| Tpn→ d η|2 〉, (8)
where ~pi and ~pf are the initial and final momenta c.m. system. The momentum transfer ~q,
as shown in Fig. 2, is defined as,
~kp + ~kp′ . (9)
The total cross section, σT , for p n → d η reaction is taken from the experiments [22].
B. Final state interaction
1. η − d interaction
This is incorporated through a half-off-shell η− d T -matrix. We construct this T -matrix
using the following two prescriptions:
1. Factorized form of Tηd
In one ansatz we obtain it by multiplying the on-shell η − d T -matrix by an off-shell
extrapolation factor g(k′ , k). Requiring that this T -matrix goes to its on-shell value
in the case of on-shell momenta, we write
Tη−d(k , E(k0) , k
′) = g(k , k0) Tη−d(E(k0)) g(k
′ , k0), (10)
with g(p , q) → 1 as p → q. For a half-off-shell case, this obviously is the ratio of the
half-off-shell to the on-shell scattering amplitude.
For the on-shell η − d T -matrix we use the effective range expansion of the scattering
amplitude up to the fourth power in momentum,
F (k) =
Rk2 + Sk4 − ik
, (11)
where F is related to T by
Tηd(k, k
′) = − 1
(2π)2µηd
Fηd(k, E(k), k
′). (12)
The effective range expansion parameters (A,R,S) are taken from a recent relativistic
Faddeev equation (RFE) calculation of [7]. This calculation uses the relativistic version
of the Faddeev equations for a three particle mNN system, where m is a meson and it
can be an η, π or a σ meson. These particles interact pairwise, and these interactions
are represented with separable potentials. The parameters of the ηN − πN − σN
potentials are fitted to the S11 resonant amplitude and the π
−p → ηn cross sections.
The η − d effective range parameters obtained from these calculations are listed in [7]
for different sets of the meson-nucleon potentials. Each of these sets gives a specific
value of the η −N scattering length, which is also listed in [7].
Since the half-off-shell extrapolation factor g(k′ , k0) is not known with any certainty,
we choose the following two forms for it.
(i) Following the method in [15] for the final state interaction in the η− d system, we
express the off-shell form factor in terms of the deuteron form factor,
g(k′ , k0) =
d~rj0(rk
′/2)φ2d(r)j0(rk0/2) (13)
where for the deuteron wave function, φd(r), we take the Paris parametrization.
(ii) As a second choice, the form factor is taken to be the ratio of the off-shell η − d
T -matrix to its on-shell value, where both of them are calculated using the three
body equations within FRA. The input to these calculations is the elementary η −N
scattering matrix, the details of which are given in the next Section.
2. Few body equations within the finite rank approximation
The other prescription of η−d FSI involves the use of the half-off-shell η−d T -matrix
obtained by solving few body equations within the finite rank approximation (FRA).
For the details of this formalism and the expression for the η-nucleus T -matrix, we refer
the reader to our earlier works [4]. To mention briefly, the FRA involves restricting
the spectral decomposition of the nuclear Hamiltonian in the intermediate state to
the ground state, neglecting thereby all excited and break-up channels of the nucleus.
This is justified in the η − 4He and possibly in the η − 3He case, but in η−deuteron
collisions, where the break-up energy is just 2.225 MeV, the applicability of the FRA
may be limited. However, it should be noted that a comparative study [23] of the η−d
scattering lengths calculated using the FRA and the exact Alt-Grassberger-Sandhas
(AGS) [24] equations (which include these intermediate excitations) shows that they
are not very different if the real part of the η−N scattering length is restricted up to
about 0.5 fm.
2. p− d interaction
We incorporate the p− d FSI in our calculations by multiplying our model T -matrix by
the inverse Jost function, [J(p)]−1. We include the FSI in both the 1/2 and 3/2 spin states
of p − d and restrict it to the s-wave. Since the p and d are charged we also include the
Coulomb effects. Following standard procedure, we write the Jost function in terms of phase
shifts and use the effective range expansion for the later.
The complete expression for the s-wave inverse Jost function squared is written as,
[Jo (kpd)]
−2 = [Jo (kpd)]
Q + [(1 +
)Jo (kpd)]
D . (14)
Here, to include the effect of the existence of one bound state, namely, the spin 1/2 state
(3He), the doublet Jost function is multiplied by a factor (1 +
), where |EB| is the
separation energy of 3He into p− d. Its value is taken to be 5.48 MeV.
The expressions for spin quadruplet (Q) and doublet (D) [Jo (kpd)]
−2 are given by
[Jo (kpd)]
(k2pd + α
2)2 (bcQ)
3C2o k
1 + cot2 δQ
[Jo (kpd)]
(k2pd + α
2)2 (bcD)
3C2o k
1 + cot2 δD
where,
2 bcµ
and acµ and b
µ are defined as
− 2 γ kpdHγ
bcµ =
where µ stands for either Q or D. The factor C2o in above has its origin in the Coulomb
interaction. The phase shifts δQ ,D are obtained from an effective-range expansion [25, 26],
C2o kpd cot δµ = −
pd − 2 γ kpdHγ (20)
αmred
~ kpd
C2o =
2 π γ
e2 π γ − 1
n(n2 + γ2)
− ln (γ) − 0.57722 (23)
Here mred is the reduced mass in the p− d system, γ the Coulomb parameter and α is the
usual electromagnetic coupling constant. The values of the expansion coefficients aµ, bµ in
Eq. (20) are taken as aQ = 11.88 fm, bQ = 2.63 fm, aD = 2.73 fm, and bD = 2.27 fm. They
have been determined from a fit to the p − d elastic-scattering phase shifts in the relative
p− d momentum range up to around 200 MeV/c [27].
The above expression for the Jost function has the required property that for large p,
J0(p) → 1.
III. RESULTS AND DISCUSSION
Before we discuss the results of the present work, in order to highlight the FSI effects
in the experimental η − d invariant mass distribution we remove the phase space from the
experimental dσ/dMηd and plot in Fig. 3 the |f |2, which is then given by,
|f |2 =
phase space
, (24)
where,
phase space =
12 (2 π)5 s |~kp|
dΩp′ | ~kp′| |~kη d| dΩη d (25)
as a function of the excess energy, Qηd = Mηd − mη − md, where Mηd is the invariant mass
of the η − d system. In this figure we also show the plane wave result (i.e. Tp d→ p d η does
not include any FSI). The cross section, dσ/dMη d in Eq. (24), is evaluated for each Mηd
by performing an integral over the p − d centre of mass momenta, kpd. The range of the
allowed values of kpd at each Mηd is shown by the hashed region. One clearly sees a large
0 10 20 30 40 50 60 70
Qηd ( = Mηd − mη − md ) (MeV)
 = 1032 MeV
Bilger et. al.
Plane wave
FIG. 3: The ratio of experimental differential cross sections [3] to the phase space (Eq. (25)) as a
function of the excess energy, Qηd, along with range of p−d relative momenta, kpd (hashed region),
contributing to |f |2 at each Qηd.
enhancement in the experimental |f |2 near small values of Qη d, which, most likely is due to
the η − d FSI. We also observe a rise at large values of Qηd. Examining the range of p− d
relative momenta which contribute to |f |2 at each Qηd, one can see that this rise occurs at
small values of kpd, indicating thereby the possibility of a large effect of p − d FSI in this
region.
In Fig. 4, we show two sets of the calculated |f |2 along with the experimental results
for a beam energy of 1032 MeV. These results include only η − d FSI. We limit the range
of Qηd up to about 10 MeV, where, this effect is large. In Fig. 4(a) we show results for
the factorized prescription with the off-shell factor generated from the deuteron form factor
and the on-shell part arising from the relativistic Faddeev equation (RFE) calculation of [7].
The results are shown for three different sets of interaction parameters in the RFE. Since
these sets give uniquely different values of the η − N scattering lengths aηN , we identify
them by their corresponding aηN values. For the results presented here, these values are
0.42 + i0.34 fm, 0.75 + i0.27 fm and 1.07 + i0.26 fm. We see that our results reproduce the
enhancement seen in the experimental |f |2 at small values of Qηd. The absolute magnitude
depends upon the choice of the RFE parameters. It increases with aηN , which designate
these parameter sets. The set corresponding to aηN = 1.07 + i0.26 fm, gives results closest
0 2 4 6 8 10 12 14 16
Qηd ( = Mηd − mη − md ) (MeV)
Bilger et. al. 
Plane wave
ηd FSI (aηN = 1.07 + i0.26 fm)
ηd FSI (aηN = 0.75 + i0.27 fm)
ηd FSI (aηN = 0.42 + i0.34 fm)
0 2 4 6 8 10 12 14 16
Qηd ( = Mηd − mη − md ) (MeV)
Bilger et. al. 
Plane wave
ηd FSI (aηN = 0.92 + i0.27 fm)
ηd FSI (aηN = 0.77 + i0.25 fm)
ηd FSI (aηN = 0.4 + i0.3 fm)
FIG. 4: The calculated |f |2 along with the experimental results for a beam energy of 1032 MeV.
(a) The results correspond to the factorized form of Tηd with the off-shell factor generated from the
deuteron form factor. (b) The results correspond to Tηd obtained from few body equations within
the FRA. The data is the same as in Fig. 3.
to the experimental values.
In Fig. 4(b) we show |f |2 calculated using few body equations within the FRA, for η− d
FSI. These results are shown for three different inputs of the η − N T -matrix taken from
[16]. The choice of these T -matrices is such that their scattering length values are close to
those used in Fig. 4(a). Though this model has the limitation of retaining the intermediate
nucleus in its ground state in the η− nucleus elastic scattering, the off-shell re-scattering
effects have been properly included. If we compare Fig. 4(a) and 4(b), the two sets of results
are similar.
In order to check the sensitivity of the results to the off-shell form factor used in the
factorized η − d T -matrix, in Fig. 5(a), we show the |f |2 calculated using two different
off-shell form factors. The on-shell Tηd is obtained from RFE and the off-shell part is either
treated with a deuteron form factor (solid line) or a few body FRA form factor (dash dotted
line) as explained in section II B. The elementary η −N T -matrix parameters required for
the calculation of the FRA form factor are taken from the parametrization of Green and
Wycech [16]. Even though the results (as shown in Fig. 4(a)) corresponding to the aηN =
1.07 + i0.26 fm seem to be the closest to the data, to compare the effect of using different
0 2 4 6 8 10 12 14 16
Qηd ( = Mηd − mη − md ) (MeV)
 = 1032 MeV
Bilger et. al. 
Plane wave
Deuteron form factor
FRA form factor
2 4 6 8 10 12 14
k (fm
 = 51 MeV/c
Deuteron form factor
FRA form factor
FIG. 5: Comparison of the two form factors for aη N = 0.75 + i0.27 fm. (a) Effect of using two
different off-shell extrapolation factors for η − d FSI on |f |2. (b) Two form factors as function of
off-shell momentum (k′).
off-shell form factor, we choose the results corresponding to aη N = 0.75 + i0.27 fm. We
make this choice such that we can compare the two calculations for the inputs corresponding
to a similar η − N scattering length. It should be expected then, that the off-shell form
factors obtained from two different methods should not differ much. This is seen explicitly
in Fig. 5(b) where the two form factors are shown as a function of off-shell momentum (k′)
for an on-shell value, k0 near the low energy peak in the η − d invariant mass distribution
(to be discussed in Fig. 9 later).
Next, we include in our calculations the effect of the p−d FSI. This is done by multiplying
the pd → pdη squared T -matrix (Eq. (4)) used above by the inverse Jost function squared
in Eq. (15) and Eq. (16), and integrating it over the allowed range (as shown in Fig. 3)
of p − d momenta, kpd for each Qη d. We show these results in Fig. 6 for the RFE (with
deuteron form factor) model of η − d FSI, for the parameter set corresponding to aηN =
1.07 + i0.26 fm. We find, that the p− d FSI affects the results in the whole region of Qη d,
while the effect of η − d FSI is confined to small value of Qηd. The large effect of p− d FSI
in the region of small Qηd, however, may not be taken with confidence as the value of kpd
in this region is large (as shown in Fig. 3), where the s-wave effective range expansion (Eq.
(20)) for the calculation of Jost function might not be sufficient. In any case, it appears
0 10 20 30 40 50 60 70
Qηd ( = Mηd − mη − md ) (MeV)
 = 1032 MeV
Bilger et. al. 
Plane wave
ηd FSI
ηd & pd FSI
FIG. 6: The proton-deuteron final state interaction effects on the p d → p d η reaction at the beam
energy of 1032 MeV. The dashed line shows the plane wave results and the dashed dot (solid) line
shows the effect of η − d (η − d & p− d) FSI for aηN = 1.07 + i0.26 fm.
that the effects of both the η− d and the p− d FSI on the η− d invariant mass distribution
are significant. If we disregard the calculated p− d effect for small Qηd, the η− d and p− d
FSI dominate in regions well separated from each other.
Apart from the FSI, another important ingredient of our calculations is the two-step
description of the production vertex. Because of the large momentum transfer, we believe,
as has also been stressed in Ref. [17], that the angular distribution of the outgoing particles
is probably more sensitive to the description of the production vertex. Inclusive angular
distributions have been measured for all the three outgoing particles in the p d → p d η
reaction. In Fig. 7, we show the calculated angular distributions for all the three outgoing
particles along with the measured distributions. We show results without any FSI, with
η− d FSI and with η− d and p− d FSI both included. As each angle has contribution from
a range of Qη d as well as kpd, the calculated results include integration of the cross section
over these variables. We find that the observed nearly isotropic nature of the experimental
angular distributions for the proton, deuteron and eta already gets reproduced by the plane
wave calculations. The effect of both η − d and p − d FSI is large and persists over all
the angles. Their inclusion brings the magnitudes of the proton and deuteron angular
distributions near to experiments. The magnitude of the eta distribution, however, does not
seem to be affected much with the FSI.
Experimental data also exist on the total cross section. In Fig. 8, we compare the total
Bilger et al
Plane wave
ηd FSI
ηd & pd FSI
-0.8 -0.4 0 0.4 0.8
cos θ
 =1032 MeV
FIG. 7: The calculated angular distributions of (a) the deuteron, (b) the proton and (c) the η,
along with the measured cross sections for aηN = 1.07 + i0.26 fm [3].
950 1000 1050 1100
Beam Energy (MeV)
Bilger et al. (PRC 69, 014003 (2004))
Hibou et al. (EPJA 7, 537 (2002))
Two step mechanism
Two step planewave 
The direct mechanism 
0 20 40 60 80 100
Excess Energy (MeV)
FIG. 8: A comparison of the total cross section for the p d → p d η reaction calculated with the
description of the production vertex as a two step mechanism and direct mechanism, along with
the measured cross sections for aη N = 1.07 + i0.26 fm [3, 28].
2430 2440 2450 2460 2470 2480 2490
dη (MeV)
Total contribution 
Off shell contribution
On shell contribution
Plane wave
FIG. 9: Contributions from the off-shell and the on-shell η − d scattering in the final state. The
results are for aη N = 1.07 + i0.26 fm with the inclusion of only the η − d FSI.
cross sections calculated including both the η − d and p − d FSI with the measured cross
sections. The results are shown with the factorized form of η − d FSI with deuteron form
factor for the set corresponding to η −N scattering length equal to 1.07 + i0.26 fm. As we
see, the calculated cross sections are in good agreement with the experimental data.
In Fig. 8 we also give the cross sections calculated for the one-step direct mechanism
(Fig. 2) mentioned in the previous section. Near threshold, these cross sections are about
four orders of magnitude below those obtained from the two-step model and two orders
of magnitude smaller in the high energy range. As mentioned in the Introduction, this
observation is similar to that in other works involving large momentum transfer reactions
[5, 18, 19], and is understandable because the momentum transfer continues to be large
(∼ 600 MeV/c) in the p d → p d η reaction even at an excess energy as large as 100 MeV.
Now we make an observation about the importance of off-shell scattering in treating
η − d FSI near threshold. The scattering part of the η − d wave function (Eq. (3)), gets
contributions from the off-shell as well as the on-shell scattering in the nucleus. To see
quantitatively the relative importance of these two contributions to the cross section for the
p d → p d η reaction, in Fig. 9 we show their contributions separately in the η− d invariant
mass distribution. These results include only the η − d FSI generated from the factorized
prescription using RFE and the deuteron form factor for the η − d T -matrix. We find that
near threshold the off-shell scattering completely dominates the threshold enhancement. At
2430 2440 2450 2460 2470 2480 2490
dη (MeV)
Bilger et. al.
Total contribution 
Plane wave
FIG. 10: A comparison of the calculated results including both the η − d and p − d FSI with the
experimental results. The results are for aηN = 1.07 + i0.26 fm.
higher excess energy, as expected, the on-shell contribution takes over.
Finally we show the nature of agreement of our calculated results with the invariant η−d
mass distribution. In Fig. 10, we compare the calculated results including both the η − d
and p − d FSI with the experimental results. The results are for aηN = 1.07 + i0.26 fm
calculated with the factorized prescription using RFE and the off-shell factor generated from
the deuteron form factor. As we see the overall agreement is reasonably good.
IV. SUMMARY
The invariant η − d mass distribution in the p d → p d η reaction has been studied
by describing the production mechanism in terms of a two step model with a pion being
produced in the intermediate state. The η−d final state interaction (FSI) has been included
in (a) a factorized form involving an on-shell Tηd and two types of off-shell form factors and
(b) by solving few body equations within the FRA. The p−d FSI is included through a Jost
function. The conclusions of this investigation can be summarized as:
1. Experimentally observed large enhancement in the cross section near small η−d excess
energy, Qηd is reproduced by the η − d FSI. The rise in the cross section at large Qηd
(which corresponds to a range of small momenta, kpd) can be accounted for by the
p− d FSI.
2. Quantitative reproduction of the large enhancement requires η− d FSI corresponding
to large values of aηN . In the present calculation it is around 1.07 + i0.26 fm.
3. The calculations successfully reproduce the observed isotropic angular distribution
of the proton and the deuteron in the final state. The total cross sections for the
pd→ pdη reaction are also well reproduced.
4. The off-shell part of the η − d scattering dominates near threshold.
5. The results for two different choices of the off-shell extrapolation factor in the factorized
form of the η − d FSI are similar.
V. ACKNOWLEDGMENTS
The authors wish to thank R. A. Arndt for providing the computer codes for evaluating
the p p → π+ d T -matrix. This work is done under a research grant by the Department of
Science and Technology, Government of India. The authors (NJU, KPK and BKJ) gratefully
acknowledge the same.
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http://arxiv.org/abs/nucl-th/9707044
http://arxiv.org/abs/nucl-th/0506024
http://arxiv.org/abs/nucl-th/9510010
http://arxiv.org/abs/nucl-th/9804050
	Introduction
	The Formalism
	The production mechanism
	Final state interaction
	-d interaction
	p-d interaction
	Results and Discussion
	Summary
	Acknowledgments
	References
