Combined Description of $\bf{\overline{N}N}$ Scattering and Annihilation With A Hadronic Model

9 Pages·0.17 MB·English

by  V. Mull

Combined Description of NN Scattering and Annihilation With A Hadronic Model
V. Mull1) and K. Holinde1),2)
1) Institut fu¨r Kernphysik(Theorie), Forschungszentrum Ju¨lich GmbH, D-52425 Ju¨lich, Germany
2)Departamento de Fisica Teorica, Facultad de Ciencias Fisicas, Universidad de Valencia, Burjasot (Valencia), Spain
(February 9, 2008)
edly governed by short-distance physics.
A model for the nucleon-antinucleon interaction is pre-
There is generalconsensushoweverthat, for largeand
sented which is based on meson-baryon dynamics. The elas-
medium distances (r > 1fm) the elastic NN interac-
tic part is the G-parity transform of the Bonn NN poten-
tion is well described in terms of meson exchanges and
tial. Annihilation into two mesons is described in terms of
can be reliably obtained from a G-parity transform of
microscopic baryon-exchange processes including all possible
5 combinations of π,η,ρ,ω,a ,f ,a ,f ,a ,f ,K,K∗. The re- suitable NN models. On the other hand, for short dis-
0 0 1 1 2 2 tances, there is at present no reliable theory in this sec-
9 maining annihilation part is taken into account by a phe-
9 nomenological energy- and state independent optical poten- tor; therefore, the common attitude (taken e.g. by the
1 Nijmegen [3] and Paris [4] group) is to content oneself
tial of Gaussian form. Themodel enablesa simultaneous de-
withaphenomenologicalparameterizationofthisregion.
n scription of nucleon-antinucleon scattering and annihilation
Both groups have about 30 parameters at their disposal
a phenomenawith fair quality.
J andobtainimpressivefitstothewealthofexistingexper-
4 13.75.Cs,14.20.Dh,21.30.+y imental data. The hope (expressed by the Paris group)
2 is that the short range NN interaction so determined
provides “valuable hints in the elaboration of a deeper

2 theoretical model” [4].
v I. INTRODUCTION One should realize of course that such a method can
4
only provide constraints but no unique answer. First
1
Quantum Chromodynamics (QCD) is the theory of of all, there are still differences and ambiguities in the
0
1 strong interactions with quarks and gluons representing medium range part of the G-parity transformed poten-
1 thefundamentaldegreesoffreedom. Nevertheless,inthe tials: different NN potentials can be used to start from;
4 low energy regime, an effective theory in terms of collec- there areuncertaintiesdue to missing contributions(like
9 tive, hadronic degrees of freedom is probably the most e.g. correlated ρπ exchange [5]) and vertex form factor
/
h efficient way to quantitatively describe most strong in- effects, which all reach out up to 1.5fm or so. All these
t teraction phenomena. In principle, the formulation and topics should affect the result for the short range piece.
-
l treatment of QCD can be done in terms of either the Moreover,theusualparameterizationinRef.[4]assumes
c
fundamental or the collective variables. It is a matter of a very restricted non-locality structure. Consequently
u
n convenience which set to choose under specific circum- webelievethatareliabletestofamicroscopicmodelcan
: stances. Of course, due to the enormous complexity of only be made by confronting it to the experimental data
v
the theory, this issue is of decisive importance when it directly.
i
X comes to actual calculations. The development of a dynamical model for the short
r Underthisviewpoint,quarkeffectsinlowandmedium range region is undoubtedly a formidable challenge, but
a energy physics have to be defined as phenomena which it has to be met if we want to learn something about
cannot be understood in terms of only a few hadronic the short-range dynamics and not give up from the be-
variables but, on the other hand, have a simple quark- ginning. Furthermore, in order to prove the relevance
gluoninterpretation. Inordertounambigouslyprove(or of quark effects, it is not sufficient to construct a quark-
disprove)the existence of such signals in nuclear physics gluon model which reasonably well describes the empir-
itisessentialto treatasmanyhadronicreactionsaspos- ical situation. In addition, the (possible) breakdown of
sible from a conventionalviewpoint, in terms of baryons theconventionalhadronicpicture,wellestablishedinthe
andmesons. Onlyinthiswayonewillbeabletoreliably outer range part, has to be shown by pointing out spe-
explorethelimitsoftheconventionalframeworkandpos- cific discrepanciesbetweenmodelpredictionsandempir-
siblyestablishdiscrepancieswiththeempiricalsituation, icaldata. Thisispreciselythemotivationforourstudies
which mightthen be identified with explicit quark-gluon of the NN sector, in the conventionalframework, which
effects. we began almost 10 years ago.
Reactions involving antinucleons (for a review see e.g. Clearly, the goals of such a program are completely
the papers by Amsler and Myhrer [1] and Dover et al. differentfromthosetypicallyadvocatedbytheNijmegen
[2]) have always been considered to be the ideal place and Paris groups [3,4]. For us, the main aim is to test,
for finding quark effects since annihilation phenomena withoutanybias,aconventionaldynamicalmodelforthe
from the nucleon-antinucleon (NN) system are suppos- short range part. Thus it would be even counterproduc-
1
tive to introduce sufficient parametersin orderto obtain fails to describe the empirical NN → NN data quanti-
a quantitative fit to NN data since it would inhibit any tatively. Obviously, the state and isospin dependence of
conclusions about the physical relevance of the model. the corresponding annihilation interaction is too strong.
Also, it is obviously essential to treat the short-range (Certainly we could have improved the fit considerably
piece of both the elastic and annihilation interaction in byarbitrarilymodifyingtheinnerpartofboththeelastic
a consistent scheme. andannihilationinteraction. Suchaprocedure,however,
Throughout, we use the G-parity transformed (full) would completely obscure the message and as we hope
Bonn NN potential [6] as elastic NN interaction. This to have made clear, would be against the spirit of our
interactionhas the advantage(essentialfor our purpose) approach.)
thatitisprescribedeverywhere,i.e.for(arbitrarily)short In a second step [9] we have predicted NN → M M
1 2
distances. Thus we are not forced to introduce any ad transition rates going via baryon-exchange with ad-
hocparametersandinthiswaylosethe predictivenessof justable form factor parameters. For this, a DWBA pro-
ourmodelfromthebeginning,whichwouldsurelyreduce cedure has been applied, with A(BOX) and C as initial
(if not destroy completely) the possibilities for a serious stateinteractionandnofinal(meson-meson)interaction.
test of annihilation mechanisms. A reasonable agreement with the experimental situation
Indeed,agooddescriptionofempiricalNN (elasticas could be achieved. However,there is a serious drawback
wellascharge-exchange)scatteringdatacanbe achieved of such a DWBA approach: The annihilation which oc-
with this model by adding a simple phenomenological, curs both in the initial state interaction and in the final
state-andenergyindependentopticalpotentialwithonly transition process, is treated inconsistently.
3parameterstoaccountforannihilation(modelA(BOX) A consistent treatment can best be done in a cou-
[7]). Thus, no arbitrary adjustment of the inner elastic pled channels framework, which yields both NN →NN
part is a priory necessary in order to describe the em- and NN → M M amplitudes at the same time. Liu
1 2
piricaldata;obviously,thisG-paritytransformautomat- and Tabakin [10] demonstrated the need for such an ap-
ically provides the spin, isospin and energy dependence proach in NN physics and were the first to apply this
phenomenologically required. This is an important find- method for explicit mesonic channels, namely ππ and
ing in itself. KK. With about 20 parameters (used to parameter-
Turningthings around,this elasticNN interactionre- ize further effective channels and the short-range elastic
quires essentially no state, isospin or energy dependence part) they obtained a good simultaneous description of
in the imaginary part and thus seems to support an ab- (elastic and charge-exchange) NN scattering as well as
sorptive disk picture as dominant annihilation mecha- NN →ππ,KK annihilation data.
nism. However, this cannot be the end of the story: In Inthispaper,wepresentaconsistentmodeldescribing
order to come to reliable conclusions, it is essential to NN scattering and annihilation into specific channels at
treat both the elastic and annihilation part of the inter- the same time, along the same lines. Compared to Ref.
action consistently in the same microscopic framework. [10] the set of explicitly included meson channels is con-
ItisclearthatduetothecomplexityoftheNN annihi- siderablyenlarged. Namely,apartfromthe pseudoscalar
lationchannelsthis programbecomes quiteinvolvedand mesons π,η,K we consider all possible combinations of
canonly be pursued in steps. In Ref. [7] we have started the lowest mass mesons with 0++,1−−,1++,2++ quan-
by evaluating a selected set of two-meson annihilation tum numbers for both isospin I =0 and I =1. This en-
diagrams,NN →M M →NN,proceedingviabaryon- largedsetofquantumnumbersincludedandthefactthat
1 2
exchange. All combinations of those mesons whose ex- alltwo-mesonchannelsarenowemployedwitharealistic
changesareconsideredintheelasticNN interaction(i.e. strength(in agreementwith experimentalinformationof
π,ρ,ω,σ,δ) have been included (with the same coupling annihilation) turns out to strongly reduce the state de-
constants)aswellasthestrangemesonsK andK∗gener- pendenceoftheannihilationinteractioncomparedtoour
ated by hyperon exchange (with corresponding coupling formermodelCandtodecisivelyimprovethedescription
constants taken from our hyperon-nucleon model [8]). of the data, as will be demonstrated below.
However,sincethesechannelsaccountatmostonlyfor Apartfromfurthertwo-mesonchannelswithcombina-
about 30% of the total annihilation, their contributions tions of mesons not considered so far, there is a part re-
havebeenartificiallyenhancedinRef.[7]inordertopro- mainingwhichcouldbemadeupbyexplicitthree-meson
videthetotalempiricalannihilationrate. (Thishasbeen channels or possible exotic contributions (glue-balls, hy-
achieved by suitably adjusting form factor parameters.) brids,...);Inthemodeltobepresented,thispartistaken
This procedure ofusing only a few annihilationchannels into account by a phenomenological optical potential of
leadsto a verypronouncedstate andisospindependence similar form as used in model A(BOX) of Ref. [7], but
of the annihilation interaction because of strict selection of course with modified parameters since part of the an-
rulesforeachannihilationprocessasaconsequenceofthe nihilation is described microscopically.
conservation of isospin, total angular momentum, parity The physical strength of the annihilation channels is
andG-parity. Thismodel(calledC inRef.[7])represents determinedbyevaluatingallNN →M M crosssections
1 2
therefore the other extreme compared to the state inde- (at rest and in flight) and adjusting the form factor pa-
pendentmodelA(BOX). AsshowninRef.[7],itactually rametersattheannihilationvertices,whichoccurinboth
2
the initial state interaction and the final NN → M1M2 that VNN→M1M2, which occurs both in the NN interac-
transition, to available empirical information. This con- tion (eq. (4)) and in the actual annihilation process (eq.
sistency in the choice of parameters is one major advan- (2)) is now treated in complete consistency.
tage compared to the former calculation [9], in which The remaining contributions to the annihilation part
modelA(BOX)orC hasbeenusedasinitialstateinter- of the NN interaction (involving e.g. the explicit tran-
action. sition into 3 and more mesons) will now be taken into
In the next section, we describe our model for NN account by an additional phenomenological piece (fig. 1
scatteringandannihilation. InsectionIIIwepresentand (c)), for which we use the following parameterization in
discuss the results and compare these with our former coordinate space
models and experiment. Some concluding remarks are
−r2
made in section IV. Vopt = iW e 2r02 . (5)
It is completely independent of energy, spin and isospin,
II. MODEL FOR NN SCATTERING AND with two parameters (W = −1GeV,r = 0.4fm) ad-
0
ANNIHILATION INTO TWO MESONS justed to the NN →NN cross section data.
In our model, the sum over i,j in eq. (4) goes over
Inprinciple,themicroscopictreatmentoftheNN sys- all possible combinations of π,η,ρ,ω,a0,f0,a1,f1,a2,f2
tem is a complicated problem involving couplings be- (via N,∆ exchange) and K,K∗ (via Λ,Σ,Y∗ exchange),
tween various baryonic and mesonic channels and di- cp. fig. 2. In order to obtain the transition interactions
agonal interactions in all channels. In this paper we VNN→M1M2 we start from interaction Lagrangians given
will suppress any diagonalinteraction except in the NN inappendixA.AsintheBonnpotential,thecorrespond-
channel, the reasonbeing that not much is knownabout ing diagrams have been evaluated within time-ordered
theseinteractions,especiallyinthemesonicsector. Also, perturbationtheory. Explicitexpressionsanddetailscan
this approximation is expected not severely to affect be found in [7,9].
the main purpose of this work, which is to demonstrate As far as parameters are concerned, part of the cou-
that an increased number of annihilation channels (with pling constants occurred already in the Bonn potential
more meson quantum numbers JPC) treated explicitly [6] and the Juelich hyperon-nucleon model [8] and could
reduces the state dependence of the microscopic annihi- be taken over. Thus they are identical to those used in
lationmodelandbringsinthiswaytheresulttowardsthe theelasticinteraction. Remainingcouplingconstantsare
experiment. The coupled equations for the NN scatter- choseninlinewithempiricalinformation[11],onlythose
ing amplitude TNN→NN and the transition amplitudes withoutanyinformationhadtobefittedtotheNN cross
TNN→M1M2 for the annihilation into two mesons can sections. Furthermore,thevertexfunctionscontainform
factors, parameterized in a monopole type form
then be written as
n
Λ2−M2
TNN→NN = VNN→NN F(p~2) = δ δ . (6)
δ Λ2+p~ 2
+ VNN→NNGNN→NNTNN→NN , (1) δ δ !
TNN→M1M2 = VNN→M1M2 with p~δ (Mδ) being the 3-momentum (mass) of the
+VNN→M1M2GNN→NNTNN→NN . (2) baryon exchanged in VNN→M1M2.
Note that these form factors used in the annihilation
The NN interaction VNN→NN consists of an elastic and diagramshavetobedistinguishedfromthoseusedinthe
an annihilation part, elasticmeson-exchangeprocess(althoughthesameparti-
cles are involved at the vertex) since now the exchanged
VNN→NN = Vel + Vann . (3) baryon is the essential off-shell particle. Therefore the
form factor is needed in a quite different kinematic re-
As stated in the Introduction, we use the G-parity
gion. The parameter Λ should depend on the type of
δ
transform of the (slightly modified, cp. Ref. [7]) full
particles involvedat the vertex. However,in orderto re-
Bonn NN potential [6] for that purpose; corresponding
ducethenumberoffreeparameters,weallowΛtodepend
diagrams are shown in Fig. 1(a). Compared to Ref. [7],
onlyonthe type ofthe exchangedbaryonbut notonthe
V is now split up into two parts
ann produced meson. The values actually used are given in
Table I; they have been fixed in a self-consistency proce-
Vann = VMiMj→NNGMiMjVNN→MiMj + Vopt . duretoreproduceempiricalannihilationdata,seesection
Xij III.
(4) We mention finally that the conservation of parity P
and G-parity G (resp. charge conjugation C) results in
The first part results from a microscopic treatment of thefollowingconditions(herethe primedmagnitudesre-
various two-meson annihilation channels proceeding via fer to the two-meson system, the unprimed ones to the
baryon-exchange (Fig. 1(b)). It is important to realize NN system):
3
G G if M ,M
i j i j The description of the charge exchange polarization
(−1)L+S+I = are G–Eigenstates data(Fig.7)isstillunsatisfactory,especiallyatbackward
((−1)L′+S′+I otherwise angles and higher energies.
(−1)L+1 =P P (−1)L′. (7) At the end of this section we show also the results
i j
for some selected spin observables at two different ener-
The resulting selectionrules are presentedgraphicallyin gies,bothfortheelastic(Fig.8)andthecharge-exchange
Table II. channel (Fig. 9). For the elastic spin-transfer observable
D a few data points have been measured [12], for the
nn
charge-exchange D data have been obtained recently
nn
III. RESULTS AND DISCUSSION [13].
In summary, the increased number of explicit mesonic
Themodelspecifiedinthelastsection(calledmodelD channels included consistently, with a realistic strength,
inthefollowing)providesdefinitepredictionsforboththe has obviously improved the state dependence of our
NN scattering and annihilation amplitude (eqs. (1,2)). baryon exchange annihilation model considerably. On
From these, cross sections and spin observables can be the other hand, the comparison with the phenomeno-
obtained in a straightforward way. Throughout we will logical annihilation model A(BOX) shows that impor-
compare the results of model D with corresponding re- tant physics is still missing. Again, we certainly had
sults of the preceding models A(BOX) and C (Ref. [7]). the possibility to improve the agreement between the
model D results and experimental data e.g. by adding
spin-dependent (spin-orbit, tensor) parts to the optical
A. NN scattering potential (Eq. (5)). For the reasons already discussed in
the Introduction we resisted against this temptation.
Fig. 3 shows the total, integrated elastic and charge
exchange as well as the annihilationcross sections for pp
B. Annihilation into two mesons
scattering. ModelDaswellasA(BOX)ofRef.[7](which
contains absolutely no isospin dependence for the anni-
hilationpart)agreewiththe empiricaldata. Incontrast, In the following section we will look at the results for
the resultoftheeffectivemicroscopicmodelC ofRef.[7] specific annihilation channels.
isbyaboutafactorof2toolargeinthecharge-exchange Mostdatafortheannihilationintotwomesonsareob-
cross section. This is because the isospin dependence of tained from the annihilation at rest in liquid or gaseous
this annihilationmodelistoo strong,beinggeneratedby hydrogen [1,2]. Table III contains all the relative cross
only a few annihilation channels. sections for the 53 annihilation channels included in our
Results for the differential cross sections for elastic consistentannihilationmodelD.Inordertodemonstrate
(pp→pp)scatteringareshowninFig.4. Thedifferential the influence of the initial state interactionwe also show
cross section is essentially flat for low energies, while al- the results (with unchanged transition potential param-
readyformoderateenergiesastrongforwardpeakisseen, eters)whenA(BOX)isusedinsteadofD asinitialstate
whichclearlydemonstratestheimportanceofhigherpar- interaction. The largest contributions are given by the
tialwaves. Forthehighestenergyconsideredhere,amin- combinations of two vector mesons. Vector-pseudoscalar
imum is observedin the cross section because diffractive and vector-axialvectorcombinations also provide sizable
effects become relevant. Throughout, model A(BOX) fractions of the total annihilation. These findings are in-
providesagooddescription,theagreementwiththedata dependentoftheparticularinitialstateinteractionmodel
isstillreasonablefortheconsistentannihilationmodelD used and are determined by the relevant vertex struc-
but fails completely for the effective model C. A simi- tures in the baryon exchange diagrams. Other channels
larsituationis foundfor theelastic polarization(Fig.5). or combinations are of minor importance; however, they
Model D accounts for the basic structures at low ener- tend to weakenthe state dependence of the totalannihi-
gies althoughthere are deficiencies at higher energies. A lation,afeature obviouslyfavoredbythe empiricaldata.
ratherreasonableagreementcanbe achievedwithmodel The channels marked with an asterix can be reached
A(BOX), while model C predicts the wrong sign. for an annihilation at rest only when the width of the
The results of Model D and model A(BOX) for the mesons is taken into account, because the sum of their
differential cross section of the charge-exchange reaction rest masses is larger than twice the nucleon mass.
pp→nn (Fig. 6) almost coincide at higher energies. For The fit of these branching ratios can certainly be im-
low energies, however, differences occur at backwardan- proved by relaxing the condition that the cutoff mass
gles. It has already been noted in the discussion of the Λ in the baryon-exchange diagrams does not depend on
integrated cross section, that the effective annihilation themesonproduced. Forexample,fornucleonexchange,
modelC cannotprovideadescriptionofthis processdue the value ΛN = 1.5GeV is mainly determined from the
to a too strong isospin sensitivity of the annihilation. experimentally well known annihilation channels involv-
ing π,ρ and ω. For spin-2 mesons (a ,f ) the required
2 2
4
dipole-form (n=2) then leads to a relative suppression, in this respect, due to the considerable complexities in-
which could be counterbalanced by a higher value of Λ. troduced by the coupling of various mesonic channels.
As seen from Table III, this would bring the theoretical In order to come to reliable conclusions, a consistent de-
results in better agreement with experiment. scriptionofnotonlyNN scatteringbutalsoannihilation
A great success of the recent experiments done at phenomenainspecificmesonicchannelsisrequired,with
LEAR is the measurement of branching ratios together fullinclusionofinitial- andfinalstateinteractioneffects.
with the determination of the quantum numbers of the Such a program can be best done in a coupled channels
initial NN state [2]. Table IV shows ratios of branching framework. Since it requires an enormous effort it can
ratios for either the same initial NN state into different only be done in steps by increasing the number of ex-
annihilation channels or into the same channel from dif- plicitchannelsand/orincludingmoreandmorediagonal
ferent initial NN states. These ratios express so-called mesonic interactions. In this procedure it is advisable
dynamical selection rules; a famous example is the first to keep the number of free parameters small in order to
one(“πρpuzzle”). Thoseannihilationchannelswhichare avoidfitting the data quantitatively while still missing a
in principle allowed by the fundamental quantum num- lot of important physics in the model.
ber conservation do not occur with equal probability or In this paper, we have presented a conventional ha-
astatisticaldistribution;therateobviouslydependssen- dronic model, in terms of meson and baryon exchange,
sitively on the channels and the involved dynamics. Ra- which enables a simultaneous prediction of NN scatter-
tios like those shown in Table IV are often supposed to ingandannihilationphenomenainthe two-mesonsector
minimize the effects of the initial (and final) state inter- involving the lowest mass JP = 0±,1±,2+ mesons for
actions. However, as clearly seen from the table, there both isospin I = 0 and I = 1. Given that we have
is a strong sensitivity to whether and evenwhich kindof onlyabout10energy-independentfreeparameters(some
initial state interaction is included. It does not drop out open coupling constants, 5 cutoff masses in the baryon-
even if ratios from the same partial waveare considered. exchange diagrams and 2 optical potential parameters)
(Note that the numbers of Table III increase by an or- the results presented are, in our opinion, already quite
der of magnitude if calculated in Born approximation!). encouraging proving at least that also in the NN sector
Thus a consistent description for the transition model, theconventionalhadronicconceptisworthtobepursued
initial state interaction(and probably also final state in- further and is surely a valid alternative to quark-gluon
teraction)isrequiredbeforeonecanseriouslyaddressthe models. Still, remaining discrepancies to empirical data
question about which transition mechanism is preferred. (especiallyinthe spinobservables)areareflectionofthe
Conclusions based on Born approximation appear to be fact that important physics is still missing. Apart from
premature. further mesonic channels, diagonal mesonic interactions
Forthemostimportantannihilationchannelscrosssec- as well as direct couplings between the various mesonic
tion data for the annihilation in flight exist, see Fig. 10, channels have to be included, which represents a chal-
which illustrate the energy dependence of the annihila- lenging task for the future.
tion mechanism. Obviously,model D leads to a satisfac-
tory overalldescription.
For the π+π− and the KK channel some more sensi- APPENDIX A: INTERACTION LAGRANGIANS
tiveobservableshavebeenmeasured,too: thedifferential
cross section and the analyzing power [14]. It has been The following Lagrangians are used in this work for
shown in Ref. [15] that the description of both these ob- the coupling of spin 1 baryons and mesons:
2
servables requires more effort: here the interactions be-
tween the outgoing mesons (which are to some extent LBBS = gBBS ΨαΨβΦj (A1)
knowninthiscase)seemtobeessentialforthereproduc- g
L = BBP Ψ γ5γµ Ψ ∂ Φj (A2)
tion of the experimentally observedfeatures of the data. BBP m α β µ
p
Work is in progress to do a coupled channels calculation
L = g Ψ γµ Ψ Φj
for these annihilation channels including also ππ → ππ, BBV BBV α β µ
ππ →KK, andKK →KK interactions andis this way +fBBV Ψ σµν Ψ (∂ Φj −∂ Φj) (A3)
try to describe also these high quality angle-dependent 4M α β µ ν ν µ
N
data. L = g Ψ γ5γµ Ψ Φj (A4)
BBA BBA α β µ
g
L = BBT iΨ (γµ∂νΨ +γν∂µΨ )
BBT M α β β
IV. CONCLUDING REMARKS N
−i ∂νΨ(cid:8) γµ+∂µΨ γν Ψ Φj (A5)
α α β µν
One of the main topics of current research is to iden- Vertices with(cid:0) a spin1-, a spin3(cid:1)-Bar(cid:9)yon and a meson
tify the relevantdegreesoffreedominlow-andmedium- 2 2
are given by the Lagrangians:
energy strong interaction physics. The short-range part
of the NN interaction represents a particular challenge L = gBDP Ψ Ψµ ∂ Φj (A6)
BDP m α β µ
p
5
g
L = BDV Ψ iγ5γµ Ψν ∂ Φj −∂ Φj (A7) FIG.1. Elastic(a),microscopicannihilation(b),andphe-
BDV mv α β µ ν ν µ nomenological annihilation (c) part of our NN interaction
(cid:0) (cid:1) model.
with the following meaning of the indices:
B : spin 1-baryons
2
D : spin 23-baryons FIG.2. Transition potentialsVNN→MiMj includedexplic-
S : scalar mesons JP =0+ itly in our microscopic annihilation model.
P : pseudoscalar mesons JP =0−
V : vector mesons JP =1−
A : axialvector mesons JP =1+ FIG. 3. Total, elastic, charge-exchange and annihila-
T : tensor mesons JP =2+ tioncrosssectionsforppscattering. Resultsoftheconsistent
modelD aregivenbythesolidlines,dashedlinescorrespond
to the phenomenological model A(BOX) and dashed-dotted
lines result from the effective microscopic model C. The ref-
erences for the data can be found in Ref. [2,16].
FIG. 4. Elastic pp differential cross sections at various
energies. The data are taken from Ref. [17–21]. The same
description of the curvesas in figure3.
FIG. 5. Elastic pp polarizations at some energies. The
data are taken from Ref. [17–21]. The same description of
thecurvesas in figure 3.
FIG.6. Charge-exchange differential cross sections. The
dataaretakenfromRef.[22–24]. Thesamedescriptionofthe
curvesas in figure3.
FIG. 7. Charge-exchange polarizations. The data are
taken from Ref. [22–24]. The same description of the curves
as in figure3.
FIG. 8. Some spin observables in the elastic channel:
Cnn,Dnn and Knn at two different energies. The data are
taken from Ref. [12]. The same description of the curves as
in figure 3.
FIG.9. The spin observables Cnn,Dnn and Knn for the
charge-exchangereaction. Thedataare takenfrom Ref.[13].
The same description of thecurvesas in figure 3.
FIG. 10. Annihilation cross sections “in flight” for the
mostimportantchannels. Thesamedescriptionofthecurves
as in figure 3. The data are calculated from values given in
Ref. [25,26].
6
TABLE I. Coupling constants and cutoff parameters in TABLEII. ConservationofparityandG-parityimposese-
thetransition potential VNN→MiMj. lection rules on the transition from the NN system to the
two meson system: all hashed fields are generally forbidden
by parity conservation. Transitions marked by the letter A
Vertex 4gπ2 f/g Λ n areallowed formeson pair G-parityG′ =(−1)(I+J),whereas
NNπ 0.0778 1500 1 transitions marked byB can occur for G′ =(−1)(I+J+1).
NNη 0.6535 1500 1
NNρ 0.84 6.1 1500 1 meson-state NN-state
NNω 20.0 0 1500 1 L=J L=J L=J±1
NNf 5.723 1500 1 S =0 S =1 S =1
0
NNa 2.6653 1500 1 Sing Trip Coup
0
NNf 10.0 1500 1 PP L′ =J S′ =0 A
1
NNa1 7.0 1500 1 PS L′ =J S′ =0 A B
NNf2 2.0 1500 2 SS L′ =J S′ =0 A
NNa2 4.0 1500 2 VP L′ =J S′ =1 A
N∆π 0.224 1700 1 ...................
L′ =J±1 S′ =1 A B
N∆ρ 20.45 1700 2
NΛK 0.9063 1800 1 VS L′ =J S′ =1 A B
...................
NΛK∗ 2.5217 -5.175 1800 1 L′ =J±1 S′ =1 A
NΣK 0.0313 2000 1 AP L′ =J S′ =1 A B
NNYΣK∗K∗ 00..80430792 2.219 22000000 11 .L.′.=...J..±..1...S.′..=..1. A
NY∗K∗ 3.4077 2000 2 VV .L.′.=...J.......S.′..=..0. A
L′ =J S′ =1 A
...................
L′ =J±1 S′ =1 A B
...................
L′ =J S′ =2 A
...................
L′ =J±1 S′ =2 A B
...................
L′ =J±2 S′ =2 A
AV L′ =J S′ =0 A B
..........................................
L′ =J S′ =1 A B
...................
L′ =J±1 S′ =1 A
...................
L′ =J S′ =2 A B
...................
L′ =J±1 S′ =2 A
...................
L′ =J±2 S′ =2 A B
TP L′ =J S′ =2 A B
...................
L′ =J±1 S′ =2 A
...................
L′ =J±2 S′ =2 A B
TV L′ =J S′ =1 A B
...................
L′ =J±1 S′ =1 A
...................
L′ =J S′ =2 A B
...................
L′ =J±1 S′ =2 A
...................
L′ =J±2 S′ =2 A B
..........................................
L′ =J S′ =3 A B
...................
L′ =J±1 S′ =3 A
...................
L′ =J±2 S′ =3 A B
...................
L′ =J±3 S′ =3 A
7
TABLEIII. Branchingratios“at rest”for53annihilation TABLEIV. Ratios of branching ratios “at rest” for some
channels. The dataare taken from Ref. [1,2]. interestingchannelsasexamplesfordynamicalselectionrules.
The data are taken from Ref. [2,1].
pp→ D A(BOX) EXP.
π+π− 0.54 0.92 0.33±0.017 D A(BOX) Born Exp.
π0π0 0.098 0.33 0.02−0.06
π0η 0.0095 0.01 0.03±0.02 pp (3S1,I=0)→ρ±π∓ 4.67 2.29 3.45 35±16
ππη±η0aηaa0000∓0 0000....00000102332171 0000....00000101774456 00.0.0689±±00..10203 pppppppppp((3(((111P1PSS1S100,02,,,,III,II=====11011)))))→→→→→ρρρfρ±±±2±πππππ0∓∓∓∓ 00..3248 00..1111 00..10956 1.162±.60.23
π0f0 0.067 0.035 pp (3P1,I=1)→f2π0 1.49 1.57 1.23 ≈11
ηf0 0.0090 0.024 pp (3P2,I=1)→f2π0
a+0a−0 0.0017 0.0011 pp (1S0,I=0)→a±2π∓ 0.51 0.93 0.91 3.6−8
a00a00 0.0007 0.0005 pp (3S1,I=1)→a±2π∓
f0a00 0.0002 0.0002 pp (1S0,I=0)→ρ0ρ0 6.30 62.40 1.76 0.1−0.3
f0f0 0.0008 0.0006 pp (1S0,I=0)→ωω
ρ±π∓ 2.32 3.94 3.4±0.2 pp (1S0,I=1)→ρ0ω 13.17 4.23 4.33 0.8
ρωρ00ππη00 000...805597 102...261141 001..65.524±±±000...10145 pppppp(((313SSS101,,,III===100)))→→→ρωω0ωηη 0.47 10.95 0.98 0.55±0.12
ωη 0.20 0.09 0.46±0.14
ρ±a∓ 0.63 0.40
0
ρ0a0 0.24 0.13
0
ρ0f0 0.90 1.82
ωa0 0.17 0.70
0
ωf0 1.28 1.90
a±π∓ 1.03 0.83
1
a0π0 0.22 0.17
1
a0η 0.0078 0.0088
1
f1π0 0.47 0.63
f1η 0.0033 0.0032
ρ+ρ− 4.30 16.8 (<9.5)
ρ0ρ0 1.04 2.20 0.4±0.3
ρ0ω 2.13 3.94 3.9±0.6
ωω 1.07 1.62 1.4±0.6
ρ±a∓ 0.099 0.99
1
ρ0a0 0.028 0.14
1
ωa0 3.76 2.26
1
ρ0f1 0.039 0.029
a±π∓ 0.88 0.74 1.3−2.6
2
a0π0 0.14 0.18
2
f2η 0.0026 0.0032
a0η 0.0025 0.0026
2
f2π0 0.068 0.078 0.41±0.12
a±ρ∓ 0.040 0.028
2
a0ρ0 0.013 0.0052
2
f2ρ0 0.067 0.071 1.57±0.34
f2ω 0.14 0.10 3.05±0.31
K+K− 0.065 0.095 0.1±0.01
K0K0 0.0041 0.024 0.08±0.01
K±K∗∓ 0.050 0.46 0.1±0.016
K0K∗0
/K∗0K0 0.0044 0.025 0.12±0.02
K∗+K∗− 0.055 0.19 0.13±0.05
K∗0K∗0 0.023 0.035 0.26±0.05
Σ 23.77 45.96 30.94±3.91
8
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