Structure of the Meson – Nucleon – N*(1535) Vertices
M. Dillig * ** and C. V. Z. Vasconcellos*
* Instituto di Fisica
Universidade do Rio Grande do Sul, Porto Alegre RS, Brazil
** Institute for Theoretical Physics III
University Erlangen – Nürnberg, Erlangen, Germany
PACS:
Keywords:
Abstract
We investigate the strength of the coupling of
,
,
,
,
and
mesons to the
nucleon – N* S (1535) vertex. Starting from effective meson – baryon Lagrangians we
calculate the influence of open pion, eta and two-pion – nucleon channels, which give
rise to complex off shell corrections (form factors). We find a significant
renormalization of the meson – NN* coupling constants at the
threshold.
Email: mdillig@theorie3.physik.uni-erlangen.de
Email:
Supported in part by the Kernforschungszentrum KFZ, Jülich, Germany
In the last decade, particularly with the advent of modern sychrotrons with a phase
space cooling of the beams (IUCF, CELSIUS, COSY), there has been an increasing
experimental activity on the exclusive production of mesons in proton-proton collisions
(and on light nuclei) near the meson thresholds (1). Characteristic for these reactions is
the very large momentum transfer of 0.5 – 1.5 GeV/c – corresponding to a typical scale
of less than ½ fm in coordinate space - already at threshold. Thus such processes test the
dynamics of meson – exchange currents and baryon resonances and ultimately might
bridge the gap to a more fundamental description in QCD degrees of freedom, i. e.
quarks and gluons. In fact, a variety of specific meson channels, ranging from pion
production up to the
-meson, have been investigated (a recent survey on the meson
production at COSY is given in ref. 2).
In this note we focus on the exclusive production of the eta meson right at it s threshold
at a CMS energy of s = 2.426 GeV, corresponding to a beam momentum of p = 0.768
GeV/c. The – meson production is very interesting from various reasons: among others
it is the isoscalar SU(3) partner of the pion and thus acts as an isospin filter and it s
cross section at excess energies a few MeV above threshold is one order of magnitude
larger than the corresponding cross section for neutral pion production (3) (inspite of the
much larger momentum transfer). This and other unique feature have triggered an
intensive experimental program in the past years (4). In parallel there have been an
increasing number of teoretical models for the production mechanism in the framework
of effective meson – baryon Lagrangians: models with the excitation of specific baryon
resonances (5,..,8), coupled channel models (9) or attempts to relate the production to
the excitation of othere baryonic or more exotic mechanisms (10).
There is concensus in the meson- exchange (MEC) models that threshold eta production
is dominated by the negative parity S – resonance N*(1535) resonance. It has been
found that the exchange of mesons (135), (548), (770),
(783), (550) and
(980)
have a significant influence on the absolute magnitude of the total cross section (5,..8;
presently only data on the total cross section are available). One serious problem of
such caclulations is that the microscopic nature of the coupling constants entering in
these models is not well understood: the relative phases between the various meson
exchanges vary from calculation to calculation (such a problem is typical for most
meson production reactions dominated by intermediate baryon resonances, to mention
only as an example the open strangeness production in pp – pYK processes (11)).
In this note we investigate the influence of open decay channels of the N*(1535) at the
threshold on the meson
NN* vertices. Starting from effective meson – baryon
Lagrangians we caclulate imaginary corrections to the standard real coupling constants
in the dominant one and two loop approximation with
N,
N and
N intermediate
states (Fig. 1). The corresponding fractional N* -- N decay rates are given as (12): N* N 35% – 55%, N 30% – 55%,
N 1% – 10%; the decay rate into other channels is
below 5%. We remark that for the kinematics at eta threshold the coupling to the
channel in the
and
propagator is strictly forbidden (compare Fig. 2 for energy-
momentum flow at the eta threshold).
Fig. 1: Leading loop corrections from the N*(1535) coupling to the open meson –
nucleon channels: one and and two loop vertex corrections (a,b), coupling to the meson
cloud (c).
__________
Fig. 2: Kinematics and notation for the one loop vertex correction at the threshold.
Within standard Feynman rules the calculation of the various diagrams is
straightforward. We examplify the main steps explicitly for the one-loop contri-bution.
With the notation fig. 2 the resulting coupling constants are given as (note p =
With (
=
and
)
(and correspondingly for the
intermediate state). Above
denote the various
meson-baryon-N* and meson-meson vertices (13). With the identity (14)
we reduce the corresponding relativistic
NB Lagrangians to their leading (static)
nonrelativistic limit and evaluate the diagrams in time-ordered perturbation theory,
suppressing antiparticle excitation in the loop. With the substitution
NN to the NN* vertices (6,7,15), we obtain in an obvious notation
L=
with
L
=
L
=
L
=
from the
L
=
Then for the pion loop with an intermediate N we obtain
with E(k ) =
and
(k ) =
.
Next we perform two steps:
-
we isolate the imaginary part of the coupling constant from
with
-
we choose the momentum of the incoming proton and the external meson p
along the z-axis and perform the angular integration analytically.
Combing all factors the final result is given as
with the appreviations A =
,B=
;y =
with y
= y
and from
-y
. In the contributions from the
- isobar intermediate state
exchange the overall structure is preserved (except of the coupling
constants and masses only the spin and isospin factors change).
The contributions from the loop and the
.
triangle are of similar structure and given
as:
-
-
2
loop:
triangle:
All the other
NN* coupling constants are calculated in the same spirit (the final
results are detailed and summarized in ref. 16).
With the imaginary component at hand, we determine the full structure of g** . In a first
step we assume, that the magnitude of the imaginary component is small compared to
the corresponding real part, and we derive the full imaginary part of
each coupling constant as the sum of the various individual contributions (a more
consistent determination of the coupling constants has to start from a selfconsistent
inclusion of the full coupling constant, including the imaginary part at the vertices
NN* in the loop and triangle diagrams itself, resulting in a set of coupled equations for
tlhe coupling constants). In the representation
with the mixing angle
, we fix the absolute value from independent information on
the squared coupling constants, such as from the corresponding N* decay or from
selective scattering processes estimates in the literature.
We collect the various coupling constants as input for our calculation from various
sources in the literature. Concerning the
NN coupling constants we follow the values
of the Bonn group extracted from NN phase shifts (17). The
NN*,
NN* and
NN* strengths are taken from the corresponding partial decay rates of the N* (1535)
(12); for the
,
,
and
coupling we are guided by the recent analysis in refs.
(6,7), based on universality arguments for the NN and
the additional coupling constants we take N
and
N
SU(6) relations (18) from s-wave pi N scattering (19), g
NN* coupling constants. For
from the
and f
width (12) or
from the decay
widths into the channel or related models (20,21); finally we fix g
investigations on the
from the width
squared (except of the
decay (22) or QCD sum rules (23), whereas g
from recent
is taken
( 12). We should remark, that practically all coupling constants
NN coupling), show significant ambiguities up to a factor 2.
A representative set of coupling constants is summarized in Tab. 1.
Table 1: Compliation of coupling constants for our calculation.
We present a few characteristic results of our calculation in Fig. 3 for the complex
angle
. For
NN* the dominant loop correction from fig. 1 is the
triangle diagram,
which dominates the other diagrams by a roughly a factor 3 (Fig. 3(a)). Among the
vertex corrections there is a cancellation between the N and the
intermediate state in
the exchange, resulting from the opposite sign in the isospin factors. From f* = 2 f
the dominates the N contribution; and exchange are of similar importance (the small
NN coupling is compensated by the large g** coupling; note, however, the large
uncertainty in f (17); finally the ontribution is suppressed both by the weak coupling
strength and the large energy denominators.
Fig. 3: (a) Contribution of the various intermediate states (comp. Fig. 1) to the complex
angle of the
NN* coupling constant at the
N threshold; (b) relative phases of the
NN* coupling constants.
Summing all contributions, we find for the various mesons the characteristic pattern for
the complex angle
in the
NN* coupling constants as shown in Fig. 3(b) with a
typical ambiguity up to 50% both in the absolute magnitude and in the mixing angle.
Summarizing, we find a substantial inflluence of the open N, N and
N channels on
the microscopic structure of the meson – nucleon – N*(1535) coupling constants.
Though the calculation of the imaginary component in the N* coupling is well under
control due to the presence of effectively only three open N* decay channels, the results
show still large uncertainties reflecting the poor knowledge of the absolute values of the
various coupling constants in the model. Quantiatively the results will further change
for a selfconsistent evaluation of the couplings (which, however, then requires a model
for details of the full microscopic derivation from the underlying meson-quark coupling
constants, a task, which is much harder to formulate and evaluate selfsonsistently). We
expect that even our preliminary findings have a significant influence on a quantiative
comparison of model calculations with the experiment. We feel that for a detailed
extraction of the nature and the properties nature of the N*(1535) and, more general, for
other baryon resonances in related meson-baryon exchange models for exclusive heavy
meson production as one of the major goals of modern ahdron factories a much more
detailed microscopic understanding of of meson-baryon coupling constants in the
continuum is absolutely necessary.
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