Table Of ContentExtended-soft-core Baryon-Baryon Model ESC08
II. Hyperon-Nucleon Interactions
M.M. Nagels
Institute of Mathematics, Astrophysics, and Particle Physics
University of Nijmegen, Nijmegen, The Netherlands
Th.A. Rijken
Institute of Mathematics, Astrophysics, and Particle Physics
University of Nijmegen, Nijmegen, The Netherlands and
Nishina Center for Accelerator-Based Science, Institute for Physical
and Chemical Research (RIKEN), Wako, Saitama, 351-0198, Japan
5
Y. Yamamoto
1
0 Nishina Center for Accelerator-Based Science, Institute for Physical
2 and Chemical Research (RIKEN), Wako, Saitama, 351-0198, Japan
(Dated: version of: January 28, 2015)
n
a
TheYNresultsarepresentedfromanewversionoftheExtended-soft-core(ESC)potentialmodel
J
for Baryon-baryon(BB) scattering. Thepotentials consist of local- and non-local-potentials dueto
7 (i)One-boson-exchanges(OBE),whicharethemembersofnonetsofpseudoscalar-,vector-,scalar-,
2
and axial-vector mesons, (ii) Pomeron and Odderon exchanges, (iii) Two pseudoscalar exchange
(PS-PS),and (iv) Meson-Pair-exchange (MPE). Both the OBE- and Pair-vertices are regulated by
]
h gaussian form factors producing potentials with a soft behavior nearthe origin. The assignment of
t the cut-off masses for the BBM-vertices is dependent on the SU(3)-classification of the exchanged
- mesons for OBE, and a similar scheme for MPE. In addition to these standard ingredients of the
l
c ESC-models also the possible short range repulsion due to the quark Pauli-principle in the BB-
u channelsis included in the analysis, for the first time in a systematic way, in this paper.
n The present version of the ESC-model, called ESC08, describes nucleon-nucleon (NN) and
[
hyperon-nucleon (YN) as well as the S=-2 hyperon-hyperon/nucleon (YY) in a unified way us-
1 ing broken SU(3)-symmetry. Major novel ingredients with respect to the former version ESC04
v are the inclusion of (i) short-range gaussian odderon-potentials corresponding to the odd number
6 gluon-exchange,(ii)exceptionalshortrangerepulsioninspecificYNandYYchannelsduetoPauli-
3 forbidden six-quark cluster (0s)6-configurations. Further new elements are (i) the extension of the
6 JPC =1++ axial-vector meson coupling, (ii) the inclusion of the JPC =1+− axial-vector mesons,
6 and (iii) a completion of the 1/M-corrections for the meson-pair-exchange (MPE) potentials. Like
0 in the ESC04-model, the octet and singlet coupling constants and F/(F +D)-ratio’s of the model
. are conform the predictions of the quark-antiquark pair-creation (QPC) model with dominance of
1
the 3P -mechanism. This not only for the OBE-couplings but also for the MPE-couplings and
0 0
F/(F +D)-ratio’s.
5
BrokenSU(3)-symmetryservestoconnecttheNN,theYN andtheYY channels. Thefittingof
1
: NN dominatesthedeterminationofthecouplingsandthecut-offmasses. Onlyafewparametersare
v stronglyinfluencedbytheYN data,andbytheconstraintsfortheYY-interactionsfollowingfromG-
Xi matrix analyses of hypernuclei. In particular, the meson-baryon coupling constants are calculated
via SU(3) using the coupling constants of the NN-analysis as input. In contrast to ESC04, we
r
do not consider medium strong flavor-symmetry-breaking (FSB) of the coupling constants. The
a
charge-symmetry-breaking (CSB) in the Λp and Λn channels, which is an SU(2) isospin breaking,
is included in theOBE-, TME-, and MPE-potentials.
For the ESC08-model we performed a simultaneous fit to the combined NN and YN scattering
data,suppliedwithconstraintsontheYNandYYinteraction originating from theG-matrixinfor-
mation on hypernuclei. Inaddition totheusualset of35 YN-dataand 3Σ+pcross-sections from a
recentKEK-experimentE289,weadded11elasticandinelasticΛpand3elasticΣ−pcross-sectionsat
higherenergy. Weobtainedwithinthissimultaneousfitχ2/NN =1.081andχ2/YN =1.08.
data data
Inparticular,wewereabletofitthepreciseexperimentaldatumr =0.468 0.010 fortheinelastic
R
Σ−p capture ratio at rest rather well. ±
Besides the results for the fit to the scattering data, which defines the model largely, also the
applicationtohypernuclearsystems,usingtheG-matrixmethod,isratherimportantinestablishing
the ESC-model. Different versions of e.g. the ESC08-model give different results for hypernuclei.
The reported G-matrix calculations are performed for YN (ΛN, ΣN, ΞN) pairs in nuclear matter
and also for some hypernuclei. The obtained well depths (U , U , U ) reveal distinct features of
Λ Σ Ξ
theESC-model.
The inclusion of a quark core Pauli-repulsion can make the Σ-nucleus interaction sufficiently
repulsive,asseemstoberequiredbytheavailableexperimentalevidence. Furthermore,theESC08-
2
model gives small spin-orbit splittings in Λ-hypernuclei,which is also indicated byexperiment.
PACSnumbers: 13.75.Cs,12.39.Pn,21.30.+y
I. INTRODUCTION interaction [17].
(v) For the diffractive contribution we have next to
This is the second in a series of papers [1–3] , hence- thePomeron-exchange[18]addedtheOdderon-exchange
forth referredto I, II, and III respectively,on the results [19]. Whereas in QCD the Pomeron can be associ-
of the Extended-soft-core (ESC) model for low and in- ated with colorless even number (2,4, ...) of gluon-
termediate energy baryon-baryon interactions using the exchanges, the Odderon is associated with the colorless
ESC08-model. The first results on the BB-channels and odd number (3,5, ...) of gluon-exchanges. At low ener-
applications to hypernuclei were given in the review [4]. gies the Pomeron has JPC =0++, but the Odderon has
With the ESC04-models [5–7], it was shown that a very JPC =1−−.
successful description of the presently available baryon- Secondly, we have opened the possibility to incorporate
baryonscatteringdatacouldbeachievedwithintheESC- possible effects ofa ’structural’orchannel-dependentre-
approach to the nuclear force problem. Also, such a de- pulsion due to Pauli-blocking. This repulsion originates
scription was obtained with meson-baryon coupling pa- from a ’forbidden-state’ in the SU(6)fs Quark-Cluster-
rameters which can be understood rather nicely within Model(QCM)[20,21]. Thisistheanalogofawellknown
the context of the 3P - quark-pair creation mechanism effectinαα-scatteringdiscoveredinthesixties[22]. This
0
[8,9]. Thislattermechanismhasbeenshowntobedom- ’forbidden-state’ is the [51]-irrep and this irrep occurs
inantintheframeworkoflatticeQCD[10]. Thesimulta- with a large weight in the two JP =1/2+-baryon states
neousandunifiedtreatmentoftheNN andYN channels in the SU(3)f-irreps 10 and 8s . These irreps are
in ESC04, using broken SU(3)-flavor, has given already prominent in the Σ+p{(3S}1)- res{pec}tively the ΣN(1S0)-
arathersuccessfulpotentialmodelforthe lowandinter- states. These are precisely the states where according
mediate energy baryon-baryonscattering data. Further- to e.g. the G-matrix calculations the ESC-models pos-
more, the basic ingredients of the model are physically sibly lack some repulsion. This repulsion seems to be
motivated by the quark-model (QM) and QCD. indicated by experiment [23, 24]. The [51]-irrep also oc-
The G-matrix calculations showed that basic features of curs in the other NN-, YN-, and YY-channels, but with
hypernuclear data are also reproduced rather well, im- roughly equal weights, see [20], apart from a few S=-2
provingseveralweakpoints ofthe soft-coreOBE-models channels, e.g. ΞN(I =1,S =0).
[11–13]. However, there remained the problem that the We account for the ’exceptional-repulsion’ in a phe-
meson-exchangemodels seemto be unable togivea pos- nomenological way by enhancing the ”pure” Pomeron-
itive well depth U . A second problem posed the very coupling. So the effective Pomeron-repulsion consists
Σ
small spin-orbit splittings in Λ-hypernuclei [14, 15]. In of the pure Pomeron-exchange contribution augmented
this paper we extend and refine the ESC-model in order with a fraction of Pauli-blocking repulsion, which varies
to provide improvements and answers to these issues. for the different BB-channels. (The other typical quark-
First, we list the new ingredients of the here presented cluster effects like e.g. one-gluon-exchange (OGE) an-
version ESC08c, which are more or less in line with the nex quark-interchange is in ESC-models taken care of
ESC-approach as presented so far. In this category, the by meson exchange.) In this work we try to deter-
followingadditionstotheESC04-modelaremadeforthe mine the strength of this Pauli-blocking effect in BB-
present ESC08-model: channels. The fit to NN determines the sum of both
(i) For the axial-vector mesons with JPC = 1++, the the pure Pomeron-repulsion and the Pauli-blocking re-
A-mesons, next to the γ γ -coupling also the derivative pulsion. The fit to YN determines the fraction of Pauli-
5 µ
γ k -coupling is exploited. blocking in it.
5 µ
(ii) The axial-vector mesons with JPC = 1+−, the B- The ESC08-model realizes a fusion between the soft-
mesons, are included as well. The latter have potentials coremeson-exchangepotentials andQCM-aspectsofthe
of the same type as the pseudo-scalar mesons, but have baryon-baryon interactions and can be called a ’hybrid’
an opposite sign. We notice that now the set of the ex- ESC-model. The soft-core meson-exchange model has
changed quantum numbers for OBE-potentials is identi- beendescribedindetailinpreviouspapers,[5–7]. There-
cal to that for MPE-potentials. fore, we may refer here to those papers for a description
(iii)Forthemeson-exchangewehaveincludedtheBrown- of(a)the physicalbackground,(b)theemployedformal-
Downs-Iddingsanti-symmetricspin-orbitpotentialsfrom ism,(c)thedescriptionofthepotentials,eitherindetails
pseudoscalar-, vector-, scalar-, and axial- meson ex- orinreferencestopaperswherefurtherinformationmay
change [16]. be obtained. In this paper we will derive (i) the new
(iv) We have completed the 1/M-corrections for meson- OBE-potentials employed here for the first time in the
pair-exchange (MPE), in particular for the JPC = 1++- context of the ESC-model, (ii) the Odderon-potentials,
and JPC = 1+−-axial pairs. This also leads to new and (iii) a derivation of the short-range phenomenology
importantcontributionstotheanti-symmetric-spin-orbit connectedtothequarkPauliprinciplewithinthecontext
3
of the SU(3) -formalism as used in the Nijmegen poten- SU(3)-irreps 10 and 8 .
f s
{ } { }
tials. Nexttotheseitems,wewillalsogivethenew1/M- In section VII the simultaneous NN YN YY fitting
⊕ ⊕
correctionsfor the axial-meson-pair-exchangepotentials, procedure is reviewed. In section VIII the results for the
where we restrict ourselves to the spin-spin and tensor coupling constants and F/(F +D)-ratios for OBE and
contributions. The YN symmetric and anti-symmetric MPE are given. They are discussed and compared with
spin-orbit potentials will be described in another paper. the predictions of the QPC-model. Here, also the values
In [5, 6] a detailed description of the basic features of of the BBM-couplings are displayed for pseudo-scalar,
the ESC-models has been givenandmotivated. Manyof vector, scalar, and axial-vectormesons.
these were alreadypresentin the Nijmegen soft-core[12] In section IX the YN-results for ESC08c from the com-
and hard-core [25] OBE-models. We refer the reader to bined NN YN YY-fit are discussed. In section X
⊕ ⊕
these references for the description and discussion of the we discuss the fit to the YN scattering data. In sec-
items such as: (broken) SU(3)-flavor, charge-symmetry- tion X, the hypernuclear properties of ESC08 are stud-
breaking (CSB) in YN, meson-mixing in the pseudo- iedthroughtheG-matrixcalculationsforYN (ΛN,ΣN,
scalar-, vector-, scalar- meson SU(3)-nonets, the role of ΞN) and their partial-wave contributions. Here, the im-
the quark-antiquark pair-creation 3P -model for BBM- plications of possible three-body effects for the nuclear
0
and BBMP-couplings. Also, in e.g. [6] one finds a reca- saturation and baryon well-depths are discussed. Also,
pitulation of the goals of our continued investigation of the ΛΛ interactions in ESC08 are demonstrated to be
the baryon-baryonsystems. consistentwiththe observeddataof 6 He. InsectionXI
ΛΛ
we finish by a final discussion, draw some conclusions,
In the soft-core Nijmegen OBE- and ESC-models the
andanoutlook. InAppendixAwedisplaythefullSU(3)
form factors are taken to be of the gaussian-type. In
contents of the MPE-couplings, and in Appendix B for
the (non-relativistic) QM’s a gaussian behavior of the
completeness the JPC = 1+− axial-pair potentials are
form factors for ground-state baryons is most natu-
given. Finally, in Appendix C the antisymmetric spin-
ral. Thetwo-particlebranchpoints,correspondingtoe.g.
orbit potentials are derived explicitly for strange meson-
ππ,πρ, Kρ-etc., are in the ESC-models accounted for
exchange K, K∗, κ, and K .
by the MPE-potentials. Gaussian residue functions are 1
used in regge-pole models for two-particle reactions at
high-energy and low momentum-transfers.
Aspointedoutin[5,6]SU(3)-symmetryandtheQPC- II. SCATTERING FORMALISM, THE
model give strong constraints on the coupling param- LIPPMANN-SCHWINGER EQUATION,
eters. The 3P -model also offers the possibility to in- POTENTIALS
0
troduce a scheme for hypercharge breaking a la Gell-
Mann-Okubo for the BBM-couplings. In order to keep In this paper we treat the nucleon-nucleon (NN) and
some more flexibility in distinguishing the NN- and the hyperon-nucleon (YN) reactions with strangeness S =
YN(S = 1)-channels, such a medium-strong breaking 0, 1. Since in general there are both ’direct’ and ’ex-
− −
was explored in the NSC97 [13] and ESC04 [6]. In the change’potentials, the orderingof the baryonsin the in-
present study we do not apply such a breaking. The re- coming and outgoing states needs special attention. For
sults show that a scheme of SU(3) symmetric couplings keeping this ordering clear, we consider for definiteness
withonlymassbreakingcangiveanexcellentdescription the hyperon-nucleon reactions
of all BB interactions.
The content of this paper is as follows. In section Y(p1,s1)+N(p2,s2)→Y′(p′1,s′1)+N′(p′2,s′2) . (2.1)
II we review very briefly the scattering formalism, the
Like in [12], whose conventions we will follow in this pa-
Lippmann-SchwingerequationfortheT-andV-matrices.
per, we will also refer to Y and Y′ as particles 1 and 3
Similarly,insectionIIItheNN andS = 1YN-channels
− and to N and N′ as particles 2 and 4. The four momen-
ontheisospinandparticlebasis,andtheuseofthemulti-
tum of particle i is p =(E ,k ) where E = k2+M2
channel Schr¨odinger equation is mentioned. The poten- i i i i i i
andM isthe mass/The transitionamplitude matrixM
tials in momentum and configuration space are defined i
p
is related to the S-matrix via
by referring to the description given in [5]. Also SU(3)-
breaking is reviewed briefly. In section IV on the OBE-
f S i = f i i(2π)4δ4(P P ) f M i , (2.2)
potentials, the additions for ESC08 in comparison with f i
h | | i h | i− − h | | i
the ESC04-model are described. Here, we give the new
where P =p +p and P =p′ +p′ represent the total
potentialsinmomentumandconfigurationspace. Insec- i 1 2 f 1 2
fourmomentumfortheinitialstate i andthefinalstate
tionVtheSU(3)structureoftheMPE-potentialsisgiven
| i
f . The latter refer to the two-particle states, which we
and the additions in comparison with the ESC04-model
| i
are listed. The latter are the axial JPC = 1+− (πω)- normalize in the following way
pair potentials, which is the content of Appexdix B. In
section VI the short-range phenomenology is discussed.
We derive the incorporation of the ’exceptional’ Pauli- hp′1,p′2|p1,p2i=(2π)32E(k1)δ3(p′1−p1)·
repulsion, which shows up ’exceptionally’ large in the (2π)32E(k )δ3(p′ p ) . (2.3)
× 2 2− 2
4
We follow section II of [12] in detail. The transfor- Similarly to (2.9) the potentials are expanded as
mation to the non-relativistic normalization of the two-
particle states leads to states with 6
V = V (k2,q2)P . (2.11)
i i
(p′1,s′1;p′2,s′2|p1,s1;p2,s2)=(2π)6δ3(p′1−p1)· Xi=1
×δ3(p′2−p2) δs′1,s1δs′2,s1 . (2.4) The potentials in configuration space are described in
Pauli-spinor space as follows
For these states we define the T-matrix by
V(r) = V (r)+V (r)σ σ +V (r) S +V (r)
(f|T|i)={4M34(E3+E4)}−12hf|M|ii{4M12(E1+E2)(}2−.512), ×CL·S++σVALS1(r·) L2 ·S−T+VQ1(2r) Q1S2L,S(2.12·)
which satisfies the Lippmann-Schwinger equation [12]
whereS =(σ σ )/2,andseee.g. [12]foradefinition
± 1 2
±
of the operators S and Q .
1 12 12
(3,4T 1,2)=(3,4V 1,2)+ d3k
| | | | (2π)3 n·
n Z
X
III. CHANNELS, POTENTIALS, AND SU(3)
2M
(3,4V n ,n ) n1,n2 (n ,n T 1,2) , SYMMETRY
× | | 1 2 p2 k2 +iε 1 2| |
n− n
(2.6) A. Channels and Potentials
and where analogously to Eq. (2.5) the potential V is
On the physical particle basis, there are three charge
defined as
NN-channels:
(f|V|i)={4M34(E3+E4)}−12hf|W|ii{4M12(E1+E2)}−12 . q =+2,+1,0: pp pp , pn pn , nn nn(3..1)
(2.7) → → →
Above, we denoted the initial- and final-state CM-
Similarly, there are four charge YN-channels:
momenta by p and p . Using rotational invariance and
i f
parity conservation we expand the T-matrix, which is a q =+2: Σ+p Σ+p,
4 4-matrixinPauli-spinorspace,into a complete setof →
× q =+1: (Λp,Σ+n,Σ0p) (Λp,Σ+n,Σ0p),
Pauli-spinor invariants ([12, 26]) →
q = 0: (Λn,Σ0n,Σ−p) (Λn,Σ0n,Σ−p),
→
8 q = 1: Σ−n Σ−n. (3.2)
T = T (p2,p2,p .p ) P . (2.8) − →
i f i i f i
i=1 Like in [12, 13], the potentials are calculated on the
X
isospin basis. For S = 0 nucleon-nucleon systems there
Introducing
are two isospin-channels, namely I = 1 and I = 0. For
S = 1 hyperon-nucleon systems there are also two
1
q= 2(pf+pi), k=pf−pi, n=pi×pf =q×k isospin−channels: (i) I = 12 : (ΛN,ΣN → ΛN,ΣN),
(2.9) and (ii) I = 3 : ΣN ΣN.
2 →
with , of course, n =q k, we choose for the operators For the OBE-part of the potentials the treatment of
×
P in spin-space SU(3)fortheBBMinteractionLagrangiansandthecou-
i
plingcoefficientsoftheOBE-graphshasbeengiveninde-
P = 1 , (2.10a) tailinpreviousworkoftheNijmegengroup,e.g. [12]and
1
P = σ σ , (2.10b) [13],FortheTME-andtheMPE-partsthecalculationof
2 1 2
· the coupling coefficients has been exposed in our paper
1
P = (σ k)(σ k) (σ σ )k2 , (2.10c) ontheESC04-model[6]. Therewedescribedthemethod
3 1 2 1 2
· · − 3 ·
ofan automatic computerizedcalculationof these coeffi-
i
P = (σ +σ ) n , (2.10d) cients by exploiting the ’cartesian-octet’-representation.
4 1 2
2 ·
Also in this work we do not solve the Lippmann-
P5 = (σ1 n)(σ2 n) , (2.10e) Schwinger equation, but the multi-channel Schr¨odinger
· ·
i equation in configuration space, completely analogous
P = (σ σ ) n , (2.10f)
6 2 1− 2 · to [12]. The multichannel Schr¨odinger equation for
the configuration-space potential is derived from the
P = (σ q)(σ k)+(σ k)(σ q) (2.10g)
7 1· 2· 1· 2· Lippmann-Schwinger equation through the standard
P = (σ q)(σ k) (σ k)(σ q) .(2.10h) Fourier transform, and the equation for the radial wave
8 1 2 1 2
· · − · ·
function is found to be of the form [12]
Here we follow [12, 26], except that we have chosen here
P to be a purely ‘tensor-force’ operator. u′′ +(p2δ A )u B u′ =0, (3.3)
3 l,j i i,j − i,j l,j − i,j l,j
5
whereA containsthepotential,nonlocalcontributions, SU(3)-symmetry breaking.
i,j
and the centrifugal barrier, while B is only present
i,j
when non-local contributions are included. The solution
in the presence of open and closed channels is given, for IV. OBE-POTENTIALS IN ESC08
example,inRef.[27]. TheinclusionoftheCoulombinter-
action in the configuration-space equation is well known The OBE-potentials in ESC08 are those contained al-
and included in the evaluation of the scattering matrix. ready in ESC04 [5, 6], and some new additional contri-
The momentum space and configuration space poten- butions. The additions to the OBE-potentials w.r.t. the
tials for the ESC-models have been described in paper ESC04-models consist of the following elements: (i) ex-
I [5] for baryon-baryon in general. Here, we will only tension of the baryon-baryon-meson vertex of the axial-
give the new contributions to these potentials, both in vector mesons (JPC = 1++) by adding the derivative
momentum and configuration space. coupling, (ii) inclusion of the axial-vector mesons of the
2nd kind, having JPC = 1+−. In paper I [1] the poten-
tials for non-strange meson exchange have been given.
B. SU(3)-Symmetry and -Breaking, Form Factors Here, we list the additions and the basic potentials for
meson exchange with non-zero strangeness.
The treatment of the mass differences among the
baryonsishandledinthesamewayasforESC04,whichis
A. Additions to the OBE-Potentials in ESC08
exactlythatofotherNijmegen models[12,13,25]. Also,
exchange potentials related to strange meson exchange
K,K∗ etc. , can be found in these references. The interaction Hamiltonian densities for the new
The breaking of SU(3)-symmetry occurs in several couplings are
places. The physical masses of the baryons and mesons
are used. Noticable is the SU(2) SU(3) breaking due a)Axial-vector-mesonexchange( JPC =1++, 1st kind):
to Λ Σ0-mixing [28]. This Λ Σ⊂0-mixing leads also to
a non−-zero coupling of the Λ to−the other I = 1 mesons: =g [ψ¯γ γ ψ]φµ + ifA[ψ¯γ ψ] ∂ φµ . (4.1)
HA A µ 5 A 5 µ A
ρ(760),a (980),a (1270), as well as to the I = 1-pairs.
0 1 M
ForthedetailsoftheseOBE-couplingsseee.g. [13],equa- In ESC04 the g -coupling was included, but not the
A
tions(2.15)-(2.17). LikeinESC04,thecorrespondingso- derivative f -coupling.
A
called CSB-potentials are included in the ESC08-model
b)Axial-vector-mesonexchange(JPC =1+−,2nd kind):
for OBE, TME, and MPE.
The medium-strong SU(3)-symmetry breaking of the
if
BBM-coupling constants is not tried in ESC08. In the = B[ψ¯σ γ ψ] ∂ φµ . (4.2)
HB m µν 5 ν B
ESC04-model this was considered optional, and regu- B
lated by the 3P0-model by a differentiation between the In ESC04 this coupling was not included. Like for
ss¯-quark pair and the creation of a non-strange quark- the axial-vector mesons of the 1st-kind we include a
antiquark pair. Of course, we could contemplate about SU(3)-nonet with members b (1235),h (1170),h (1380).
1 1 1
such an option here, but we did not investigate this op- In the quark-model they are QQ¯(1P )-states.
1
tion.
Thebaryonmassdifferencesintheintermediatestates
The inclusionofthe gaussianformfactorsis discussedin
for TME- and MPE- potentials have been neglected for
previous papers [12] etc. For the approximations made
YN-scattering. This, although possible in principle, be-
in deriving the potentials from the relativistic Born-
comesratherlaboriousandisnotexpectedtochangethe
Approximation we refer also to [12]. Due to these ap-
characteristics of the baryon-baryonpotentials much.
proximations the dependence on q2 is linearized and we
Alsointhiswork,likeESC04-[5–7]andintheNSC97-
write
models [13], the form factors depend on the SU(3) as-
signment of the mesons, In principle, we introduce form
V (k2,q2)=V (k2)+V (k2) q2 , (4.3)
factor masses, i.e. cut-off’s, Λ and Λ for the 8 i ia ib
8 1
{ }
and 1 members of each meson nonet, respectively. where i = 1 8. It turns out that to order q2 only
{ }
In the application to YN and YY, we could allow for −
V =0. The additional OBE-potentials are obtained in
1b
SU(3)-breaking,byusingdifferentcut-offsforthestrange 6
the standard way, see [11, 12]. We write the potential
mesons K, K∗, andκ. However,in the ESC08-modelwe
functions V of (2.11) in the form
i
do not exploit this possible breaking, but assign for the
strangeI =1/2-mesonsthesamecut-offasfortheI =1- V (k2,q2)= Ω(X)(k2) ∆(X)(k2,m2,Λ2) , (4.4)
mesons. Moreover, for the I = 0-mesons we assign the i i ·
X
cut-offs as if there were no meson-mixing. For exam- X
ple weassignΛ for the dominantsingletmesonsη′,ω,ǫ, where m denotes the mass of the meson, Λ the cut-off
1
andΛ forη,φ,S∗,etc. This meansaveryslightformof in the gaussian form factor, and X = S,A,B, and O
8
6
(S= scalar, A= axial-vector, B= axial-vector, and O = Here, is a universal scaling mass, taken to be the
M
diffractive/odderon). For the additions whenX=S,B the proton mass, which we also use in the derivative cou-
propagator is plings above,as well as in the f -coupling of the vector-
V
mesons. Wenotethatthepoleforthederivativecoupling
∆(X)(k2,m2,Λ2)=(1 k2/U2)e−k2/Λ2/(k2+m2) , ofthe axial-vectorexchangeis canceledbecause of a fac-
− (4.5) tor kµ g k k /m2 .
µν − µ ν A
andfortheadditionsinthecasesX=A,Othepropagator
(cid:0) (cid:1)
is
∆(O)(k2,m2,Λ2)= 1 e−k2/4m2O . (4.6)
2
M
B. Meson-exchange with Non-zero Strangeness (∆Y =0)
6
For the non-strange mesons the mass differences at the vertices are neglected, we take at the YYM- and the
NNM-vertex the average hyperon and the average nucleon mass respectively. This implies that we do not include
contributions to the Pauli-invariants P and P . For vector-, and diffractive OBE-exchange we refer the reader
7 8
to Ref. [12], where the contributions to the different Ω(X)’s for baryon-baryon scattering are given in detail. These
i
exchangesleadtotheso-called’exchange-potentials’. FortheinvariantsO ,...,O ,theexpressionsanalogoustothose
1 6
for the non-strange mesons given above apply. This with the amendments that (i) in momentum and configuration
spacethereisacompletesymmetricappearanceofM andM ,(ii)inconfugurationspacethereappearsthebaryon-
Y N
echange operator = operator, and (iii) for the antisymmetric spin-orbit potential . The details
x σ x
P −P P P → P
are given in Appendix C. Therefore, the Ω(X) for these potentials can be obtained from those given in paper I Eqs.
i
(4.9)-(4.13),by replacingboth M and M by (M M )1/2, andadding a(-)-sign. Furthermore,inthe caseof using
Y N Y N
the Proca-formalism [29], we get non-negligible contributions from the second part of the vector-meson propagator
(k k /m2) of the K∗ meson giving
µ ν
VK∗ =V(V) (M3−M1)(M4−M2) V(S), (4.7)
− i i − m2 i
where in V(S) the vector-meson-couplings have to be used, and M and M must be replaced by (M M )1/2. In
i Y N Y N
(4.7) M =M =M and M =M =M . Then the additional terms are
1 4 Y 2 3 N
(M M )(M M )
VKA =V(A) 3− 1 4− 2 V(P), (4.8)
− i i − m2 i
For the axial-vector meson K there is no contribution from the second-term in the propagator.
B
For the mesons with non-zero strangeness, K,K∗,κ,K and K , the mass differences at the vertices are not
A B
neglected, we take into account at the YNM-vertices the differences between the average hyperon and the average
nucleon mass. This implies that we do include contributions to the Pauli-invariants P . There do not occur
8
contributions to P .
7
(a) PseudoscalarK-meson exchange:
Ω(P) = gPgP k2 , Ω(P) = gPgP 1 . (4.9)
2 − 13 24 12MYMN 3 − 13 24 4MYMN
(cid:16) (cid:17) (cid:16) (cid:17)
7
(b) Vector-meson K∗-exchange:
k2 k2
Ω(V) = gV gV 1 gV fV +fVgV
1a 13 24 − 8M M − 13 24 13 24 4 √M M
(cid:26) (cid:18) Y N(cid:19) M Y N
k4 (cid:0) (cid:1)
+fVfV
13 2416 2M M
M Y N(cid:27)
3 2
Ω(V) = gV gV , Ω(V) = k2Ω(V)
1b 13 24 2M M 2 −3 3
(cid:18) Y N(cid:19)
√M M M M k2
Ω(V) = (gV gV + gV fV +gV fV Y N)+fVfV Y N 1 /(4M M )
3 13 24 13 24 24 13 13 24 2 − M M Y N
(cid:26) M M (cid:18) Y N(cid:19)(cid:27)
(cid:0) (cid:1) √M M 3k2
Ω(V) = 12gV gV +8(gV fV +fVgV ) Y N fVfV /(8M M )
4 − 13 24 13 24 13 24 − 13 24 2 Y N
(cid:26) M M (cid:27)
√M M M M
Ω(V) = gV gV +4(gV fV +fVgV ) Y N +8fVfV Y N /(16M2M2)
5 − 13 24 13 24 13 24 13 24 2 Y N
(cid:26) M M (cid:27)
1
Ω(V) = (gV fV fVgV ) . (4.10)
6 − 13 24− 13 24 √ 2M M
(cid:26) M Y N(cid:27)
(c) Scalar-mesonκ-exchange:
k2 q2 1
Ω(S) = gS gS 1+ , Ω(S) = gS gS
1 − 13 24 8M M − 2M M 4 − 13 242M M
(cid:18) Y N Y N(cid:19) Y N
1
Ω(S) = gS gS , Ω(S) =0. (4.11)
5 13 2416M2M2 6
Y N
(d) Axial-vector K -exchange JPC =1++:
1
2k2 √M M k2 k2
Ω(A) = gAgA 1 + gAfA +fAgA Y N fAfA
2 − 13 24 − 3M M 13 24 13 24 − 13 242 2 6M M
(cid:20) Y N(cid:21) (cid:20)(cid:18) (cid:19) M M (cid:21) Y N
3
Ω(A) = gAgA
2b − 13 24 2M M
(cid:18) Y N(cid:19)
1 √M M k2 1
Ω(A) = gAgA + gAfA +fAgA Y N fAfA
3 − 13 24 4M M 13 24 13 24 − 13 242 2 2M M
(cid:20) Y N(cid:21) (cid:20)(cid:18) (cid:19) M M (cid:21) Y N
Ω(A) = gAgA 1 , Ω(A)′ = gAgA 2 , Ω(A) =0. (4.12)
4 − 13 24 2M M 5 − 13 24 M M 6
(cid:20) Y N(cid:21) (cid:20) Y N(cid:21)
Here, we used the B-field description with α = 1, see paper I, Appendix A. The detailed treatment of the
r
potential proportional to P′, i.e. with Ω(A)′, is given in paper I, Appendix B.
5 5
(e) Axial-vector mesons with JPC =1+−:
(M +M )2 k2 (q2+k2/4 k2
Ω(B) = +fBfB N Y 1 +3
2 13 24 m2 − 4M M 2M M 12M M
B (cid:20)(cid:18) Y N(cid:19) Y N (cid:21)(cid:18) Y N(cid:19)
(M +M )2 1
Ω(B) = +fBfB N Y . (4.13)
3 13 24 m2 4M M
B (cid:20) Y N(cid:21)
(f) Diffractive-exchange (pomeron, K (J =0)):
2
The pomeron carries no strangeness. Therefore, the contribution to the potentials comes from the J=0-part of
K -exchange [18]. The ΩD are the same as for scalar-meson-exchangeEq.(4.11), but with gS gS replaced by
g2DgD, and except for thie zero in the form factor. Since in ESC08-models g =0 ther±e i1s3no24contribution
∓ 13 24 NNa2
to the exchange with non-zero strangeness.
(g) Odderon-exchange: Since the gluons carry no strangeness, there is no contribution to the potentials.
8
As in Ref. [12] in the derivation of the expressions for Ω(X), given above, M and M denote the mean hyperon
i Y N
and nucleon mass, respectively M = (M + M )/2 and M = (M + M )/2, and m denotes the mass of the
Y 1 3 N 2 4
exchanged meson. Moreover, the approximation 1/M2 +1/M2 2/M M , is used, which is rather good since the
N Y ≈ N Y
mass differences between the baryons are not large.
C. One-Boson-Exchange Interactions in Configuration Space I
In configuration space the BB-interactions are described by potentials of the general form
V = V (r)+V (r)σ σ +V (r)S +V (r)L S+V (r)Q
C σ 1 2 T 12 SO Q 12
· ·
(cid:26)
1 1
+V (r) (σ σ ) L ∇2φ(r)+φ(r)∇2 P, (4.14)
ASO 1 2
2 − · − 2 ·
(cid:18) (cid:19)(cid:27)
where
S = 3(σ rˆ)(σ rˆ) (σ σ ), (4.15a)
12 1 2 1 2
· · − ·
1
Q = (σ L)(σ L)+(σ L)(σ L) , (4.15b)
12 1 2 2 1
2 · · · ·
(cid:20) (cid:21)
φ(r) = φ (r)+φ (r)σ σ , (4.15c)
C σ 1 2
·
For the basic functions for the Fourier transforms with gaussian form factors, we refer to Refs. [11, 12]. For the
details of the Fourier transform for the potentials with P′, which occur in the case of the axial-vector mesons with
5
JPC =1++, we refer to paper I, Appendix B.
(a) Pseudoscalar-mesonK-exchange:
m m2 1
V (r) = gPgP (σ σ ) φ1 +S φ0 . (4.16)
PS 4π 13 244M M 3 1· 2 C 12 T P
(cid:20) Y N (cid:18) (cid:19)(cid:21)
(b) Vector-meson K∗-exchange:
m m2 3
V (r)= gV gV φ0 + φ1 ∇2φ0 +φ0∇2
V 4π 13 24 C 2M M C − 4M M C C
(cid:20)(cid:26) (cid:20) Y N Y N (cid:21)
m2 m(cid:0)4 (cid:1)
+ gV fV +fVgV φ1 +fVfV φ2
13 24 13 24 4 √M M C 13 2416 2M M C
M Y N M Y N (cid:27)
(cid:0) m2 (cid:1) √M M M M m2
+ gV gV + gV fV +gV fV Y N +fVfV Y N φ1 +fVfV φ2 (σ σ )
6M M 13 24 13 13 24 13 13 24 2 C 13 248 2 C 1· 2
Y N (cid:26)(cid:20) M M (cid:21) M (cid:27)
m2 (cid:0) (cid:1)√M M m2
gV gV + gV fV +gV fV Y N φ0 +fVfV φ1 S
−4M M 13 24 13 24 24 13 T 13 248 2 T 12
Y N (cid:26)(cid:20) M (cid:19) M (cid:27)
m2 3 (cid:0) (cid:1) √M M 3 m2
gV gV + gV fV +fVgV Y N φ0 + fVfV φ1 L S
−M M 2 13 24 13 24 13 24 SO 8 13 24 2 SO ·
Y N (cid:26)(cid:20) M (cid:21) M (cid:27)
m4 (cid:0) (cid:1) √M M M M
+ gV gV +4 gV fV +fVgV Y N +8fVfV Y N
16M2M2 13 24 13 24 13 24 13 24 2 ·
Y N (cid:26)(cid:20) M M (cid:21)(cid:27)
3 m2 (cid:0) (cid:1) √M M 1
φ0Q + gV fV fVgV Y N φ0 (σ σ ) L . (4.17)
×(mr)2 T 12 M M 13 24− 13 24 SO · 2 1− 2 · Pσ P
Y N (cid:26) M (cid:27) (cid:21)
(cid:0) (cid:1)
(c) Scalar-meson κ-exchange:
m m2 m2 m4
V (r) = gS gS φ0 φ1 + φ0 L S+
S −4π 13 24 C − 4M M C 2M M SO · 16M2M2 ·
(cid:20) (cid:26)(cid:20) Y N (cid:21) Y N Y N
3 1
φ0Q + ∇2φ0 +φ0∇2 . (4.18)
×(mr)2 T 12 4M M C C P
Y N (cid:27)(cid:21)
(cid:0) (cid:1)
9
(d) Axial-vector K -meson exchange JPC =1++:
1
m 2m2 m2 √M M
V (r)= gAgA φ0 + φ1 + gAfA +fAgA Y Nφ1
A −4π 13 24 C 3M M C 6M M 13 24 13 24 C
(cid:20)(cid:26) (cid:18) Y N (cid:19) Y N M
m4 3 (cid:0) (cid:1)
+fAfA φ2 (σ σ ) ∇2φ0 +φ0∇2 (σ σ )
13 2412M M 2 C 1· 2 − 4M M C C 1· 2
Y NM (cid:27) Y N
m2 √M M(cid:0) (cid:1) m2
gAgA 2 gAfA +fAgA Y N φ0 fAfA φ1 S
−4M M 13 24− 13 24 13 24 T − 13 242 2 T 12
Y N (cid:26)(cid:20) M (cid:21) M (cid:27)
m2 (cid:0) (cid:1)
+ gAgAφ0 L S . (4.19)
2M M 13 24 SO · P
Y N (cid:21)
(e) Axial-vector-mesonexchange JPC =1+−:
m (M +M )2 m2 m2
V (r) = N Y fBfB φ1 + φ2
B −4π m2 13 24 12M M C 4M M C
(cid:20) (cid:26) Y N (cid:18) Y N (cid:19)
m2 m2
∇2φ1 +φ1∇2 + φ0 S . (4.20)
−8M M C C 4M M T 12 P
Y N (cid:20) Y N(cid:21) (cid:27)(cid:21)
(cid:0) (cid:1)
(f)Diffractive-exchange: SinceintheESC08-modelthediffractivepomeronandodderonexchangesareSU(3)singlets
there are no contribution to S =0-exchange potentials.
6
Above, in Eq.’s (4.16-4.20)the exchange operator is defined as
= , (4.21)
x σ
P −P P
where and arethe space-andspin-exchangeoperatorsrespectively. Theextra( )-operatorin(4.17)forthe
x σ σ
P P −P
antisymmetric spin-orbit potential is explained in Appendix C. We note that = , which is well defined for
σ x
−P P P
the coupled singlet-triplet systems.
D. One-Boson-Exchange Interactions in Configuration Space II
Here we give the extra potentials due to the zero’s in the scalar and axial-vector form factors.
a) Again, for X = V,D we refer to the configuration space potentials in Ref. [12]. For X = S we give here the
additional terms w.r.t. those in [12], which are due to the zero in the scalar form factor. They are
m m2 m2 m2
∆V (r)= gS gS φ1 φ2 + φ1 L S
S −4π U2 13 24 C − 4M M C 2M M SO ·
(cid:20) (cid:26)(cid:20) Y N (cid:21) Y N
m4
+ φ1 Q . (4.22)
16M2M2 T 12 P
Y N (cid:27)(cid:21)
b) For the axial-vectormesons, the configuration space potential corresponding to (4.12) is
g2 1
V(1)(r)= A m φ0 (σ σ ) 2φ0 +φ0 2 (σ σ )
A −4π C 1· 2 − 12M M ∇ C C∇ 1· 2
(cid:20) Y N
(cid:0) (cid:1)
3m2 m2
+ φ0 S + φ0 L S . (4.23)
4M M T 12 2M M SO · P
Y N Y N (cid:21)
Theextracontributiontothepotentialscomingfromthezerointheaxial-vectormesonformfactorareobtained
from the expression (4.23) by making substitutions as follows
m2
∆V(1)(r) = V(1) φ0 φ1,φ0 φ1,φ0 φ1 . (4.24)
A A C → C T → T SO → SO · U2
(cid:0) (cid:1)
Note that we do not include the similar ∆V(2)(r) since they involve k4-terms in momentum-space.
A
E. PS-PS-exchange Interactions in Configuration sions for Kirr(BW) and Kirr(TMO) were derived [30],
Space
InFig.’s2and3ofpaperI,theincludedtwo-mesonex-
change graphs are shown schematically. Explicit expres-
10
where also the terminology BW and TMO is explained. the spin-orbit plays no role, therefore we will report on
The TPS-potentials for nucleon-nucleonhave been given the details of the new spin-orbit terms in a separate
in detail in [31]. The generalization to baryon-baryon is paper [32]. However, also new 1/M-corrections for the
similartothatforthe OBE-potentials. So,wesubstitute spin-spin and tensor potentials were obtained for the
M √M M , and include all PS-PSpossibilities with axial-pair interaction of the 2nd kind, i.e. JPC = 1+−.
Y N
→
coupling constants as in the OBE-potentials. As com- These are relevant for the fits presented in this paper,
paredtonucleon-nucleonin[31]wehavehereinaddition and will be given in this section. Below we give the full
the potentials with double K-exchange. The masses are one-pair exchange potential as used at present, because
the physical pseudo-scalar meson masses. For the in- it has not been published before. In the ESC04-models
termediate two-baryon states we take into account the only the leading, i.e. the (1/M)0-terms, were used. For
effects of the different thresholds. We have not included the derivation of the soft-core pair-interactions we refer
uncorrelated PS-vector, PS-scalar, or PS-diffractive ex- the reader to [31]. Below we report on this derivation
change. This because the range of these potentials is for the axial-pair terms of the 2nd kind. The used
similar to that of the vector-,scalar-,and axial-vector- pair-interactionHamiltonian for e.g. the (πω)-pair is
potentials. Moreover, for potentially large potentials,
in particularly those with scalar mesons involved, there =g ψ¯γ σ τψ ∂ν(π φµ)/(m ) , (4.25)
HB (πω) 5 µν · ω πM
willbeverystrongcancellationsbetweentheplanar-and
crossed-box contributions. which gives the BBm1m2-vertex
g
u¯(p′)Γ(2)u(p)=i (πω)1 ( ω ω ) σ ω+σ k ω0 .
F. MPE-exchange Interactions B mπM(cid:20) ± 1± 2 · · (cid:21)
(4.26)
The full SU(3)-structure is given in [6], section IIIA.
In Fig. 4 of paper I the pair graphs are shown. In this
It is assumed that this pair-coupling is dominated by
work we include only the one-pair graphs. The argu-
the SU(3)-octet symmetric coupling, and is given by the
ment for neglecting the two-pair graph is to avoid some
SU(3)-octet symmetric couplings Hamiltonian in terms
’double-counting’. Viewing the pair-vertex as contain-
of SU(2)-isospin invariants and SU(3) isoscalar-factors:
ing heavy-meson exchange means that the contributions
from ρ(750) and ǫ = f (760) to the two-pair graphs is
0
already accounted for by our treatment of the broad ρ
g 1
and ǫ OBE-potential. The MPE-potentials for nucleon- HB8VP = B√8V6P 2 Bµ1 ·ρµ η8+(Bµ1 ·πµ)φ8
nucleon have been given in Ref. [31]. The generaliza- (cid:26)
tion to baryon-baryon is similar to that for the TPS- √3 (cid:2)(cid:0) (cid:1) (cid:3)
+ B (K∗†τK)+h.c.
potentials. For the intermediate two-baryon states we 4 1·
neglect the effects of the different two-baryon thresh- √3(cid:2) (cid:3)
olds. The inclusion of these, although in principle pos- + 4 (K1†τK∗)·π+(K1†τK)·ρ+h.c.
sible, would complicate the computation of the poten- 1 h i
tialsconsiderablyandtheinfluence isnotexpectedtobe −4 (K1†·K∗)η8+(K1†·K)φ8+h.c.
significant. The generalization of the pair-couplings to
1h 1 i
baryon-baryon is described in Ref. [6], section III. Also + H0 ρ π K∗† K+K† K∗ φ η
8 8
2 · − 2 · · −
here in NN, we have in addition to [31] included the (cid:20) (cid:21)(cid:27)
pair-potentials with KK-, KK*-, and Kκ-exchange. The (cid:0) (cid:1) (4.27)
convention for the MPE coupling constants is the same
as in Ref. [31]. Here, B1 ψ¯γ5τσµνψ etc. See for a definition of
∼
the octet-fields η ,φ in terms of the physical mesons
8 8
(cid:2) (cid:3)
[6]. From the pair-interaction Hamiltonian (4.27) one
G. Meson-Pair Potentials, Axial-Pairs (2nd-kind, can easily read off the different meson-pairs that occur
JPC =1+−) from the JPC = 1+−-vertex. In Appendix B we give
the explicit potentials generated by the pair-interaction
Recently we have completed the 1/M,1/M2- (4.27).
corrections to the adiabatic approximation for the
pair-potentials. The main reason is the need for a
carefulevaluationofthe anti-symmetricspin-orbitterms V. SHORT-RANGE PHENOMENOLOGY
for ΛN, in particular for pair-interactions involving
strangeness-exchange like π K,π K∗ etc. From this It is well known that the most extensive study of the
− −
evaluation new contributions emerged, in particular for baryon-baryon interactions using meson-exchange has
the axialpair-interactionsJPC =1++,1+−, leading to a difficultiestoachievesufficientlyrepulsiveshort-rangein-
substantial improvement w.r.t. experimental spin-orbit teractions in two channels. Namely, (i) the Σ+p(I =
splittings [15]. In our fitting procedure for the YN-data 3/2,3S )- and (ii) the ΣN(I = 1/2,1S )-channel. The
1 0
