ESC NN-Potentials in Momentum Space. II. Meson-Pair Exchange Potentials

27 Pages·0.34 MB·

by  Th. A. Rijken

ESC NN-Potentials in Momentum Space
II. Meson-Pair Exchange Potentials
Th.A. Rijken and H. Polinder
Institute for Theoretical Physics Nijmegen, University of Nijmegen, Nijmegen, The Netherlands
J. Nagata∗
Venture Business Laboratory, Hiroshima University, Kagamiyama 2-313, Higashi-Hiroshima, Japan
(Dated: version of: February 4, 2008)
The partial wave projection of the Nijmegen soft-core potential model for Meson-Pair-Exchange
(MPE)forNN-scatteringinmomentumspaceispresented. Here,nucleon-nucleonmomentumspace
MPE-potentialsareNN-interactionswhereeitheroneorbothnucleonscontainsameson-pairvertex.
Dynamically, the meson-pair vertices can be viewed as describing in an effective way (part of) the
effectsofheavy-mesonexchangeandmeson-nucleonresonances. Fromthepointofviewof“duality,”
2 these two kinds of contribution are roughly equivalent. Part of the MPE-vertices can be found in
0 the chiral-invariant phenomenological Lagrangians that have a basis in spontaneous broken chiral
0 symmetry. It is shown that the MPE-interactions are a very important component of the nuclear
2 force, which indeed enables a very succesful description of the low and medium energy NN-data.
Here we present a precise fit to the NN-data with the extended-soft-core (ESC) model containing
n
OBE-,PS-PS-,and MPE-potentials. Anexcellent description oftheNN-datafor T 350 MeV
a Lab
J is presented and discussed. Phase shifts are given and a χ2p.d.p. =1.15 is reached. ≤

8
PACSnumbers: 13.75.Cs,12.39.Pn,21.30.+y

1
v
0 I. INTRODUCTION grams. This in order to distinct these from the planar
2 and crossed-box diagrams, which were given Ref. [3].
0 Thetwotypesoftwo-meson-exchangepotentialsTME,
In the previous paper [1], henceforth refered to as pa-
1 see I, and MPE presented here are part of our program
0 per I, the techniques for the momentumspace treatment
toextendtheNijmegensoft-coreone-boson-exchangepo-
2 of the extended soft-core model, hereafter referred to as
tential [5, 6, 7] to arrive at a new extended soft-core
0 the ESC-model, are described. This implies first the de-
/ velopment of a representation of the ESC-model suit- nucleon-nucleon model, hereafter referred to as the ESC
h
potential [4, 8, 9, 10].
t able for the projection onto the Pauli-spinor rotational
- In the introduction to Ref. [4] a rather complete de-
invariant operators, and secondly the partial wave anal-
l
c ysis. This partial wave analysis is organized along simi- scription is given of the physical background behind the
u MPE-potentials, and we refer the interested reader to
lar lines as used for the soft-core OBE-models [2]. In [1]
n
that reference.
thenucleon-nucleonpartialwavecontributionshavebeen
:
v worked out in detail. These are the analogs of the con- We apply the potentials derivedin this work to fit the
Xi figuration space two-meson-exchange (TME) potentials NN-data. IntheTME-potentialswerestrictourselvesto
givenine.g. [3]. Here,theTME-potentialsaredefinedto the ps-ps exchange. Or, phrased differently, we include
r
a containtheplanar-andcrossed-boxtwo-meson-exchange only the Goldstone-boson sector. This because it gives
potentials. thecompletelong-rangecontribution,OPEP+TPEPand
the inclusion of η etc. is necessary for (i) (approximate)
In this second paper on soft-core two-meson-exchange
chiral symmetry, and (ii) for completeness in the sense
potentials in momentum space, occasiously refered to as
ofSU (3),whichallowsanextensiontohyperon-nucleon
paper II in the following, we derive the same representa- f
and hyperon-hyperon [10].
tion as in paper I, but now for the contributions to the
In fact, this fit has been performed in the configuration
nucleon-nucleonpotentialswheneitheroneorbothnucle-
space version. However,the results were checkednumer-
ons contains a pair vertex, i.e. the MPE-potentials. We
ically in momentum space, using the formulas of papers
givethepartialwavepotentialsinthesimilarrepresenta-
I and II.
tionasusedinpaperI.InRef.[4]theMPE-contributions
This paper is organized as follows. In section II and
totheconfigurationspacenucleon-nucleonpotentials,i.e.
III,wegivetheessentialsoftheprocedurefollowedinde-
when either one or both nucleons contains a pair ver-
riving the new momentum space representation. In sec-
tex, have been derived. The corresponding “seagull”
tion IV the projection of the MPE on the Pauli-spinor
diagrams are refered to as one-pair and two-pair dia-
invariants is worked out for the adiabatic contributions.
In section V the same is done for the 1/M-corrections:
thenon-adiabaticandthepseudo-vector-vertexterms. In
∗Present address: Kyushu International University, Fukuoka 805- sectionVIthepartialwaveanalysisisindicated. Thepro-
8512,Japan cedureforthepartialwaveprojectioniscompletelyanal-
2
ogoustothatofpaperI,andcanbetranscribedimmedi- tegrationdictionaryforthe gaussianintegralsthatoccur
ately comparing the invariant contributions Ω (k2;t,u) inMPEbutnotinTME.InAppendix Daderivationfor
j
for MPE to those for TME in I. In section VII the re- the potentials due to the ’derivative scalar pair’ interac-
sultsfromafittotheNN-dataareshownanddiscussed. tion, see the g′ -coupling in A1a is outlined. This for
(ππ)0
Here, phase shifts are givenfor TLab 350MeV and the completeness,sincealthoughwedo notemploythis kind
≤
pair-couplingsare comparedto the values expected from of pair interaction, it occurs occurs often in the current
e.g. chiral lagrangians. literature. InAppendixEthefullSU (3)contentsofour
f
In Appendix A the pair-interaction Hamiltonians are pair interactions is shown.
listed. InAppendixBtheλ-representationsfortheMPE-
denominators are given. In Appendix C we give the in-
II. MOMENTUM SPACE REPRESENTATION MPE-POTENTIALS
Here, we give an outline the essentials of the procedure to derive our new momentum space representation for the
MPE-potentilas. These procedures have been described in I, to which we refer for details. Here, we focuss on the
peculiar features that occur in the application to the MPE-potentials.
The starting point is the basic convolutive integral
d3k d3k
V (k) = 1 2δ(k k k ) F˜ (k2,m )G˜ (k2,m )
M,N (2π)3 − 1− 2 M 1 1 N 2 2
ZZ
d3∆
e = F˜ (∆2,m ) G˜ ((k ∆)2,m ) ,
(2π)3 M 1 N − 2
Z
(2.1)
where F˜ (k2) and G˜ (k2) can be of the form
M N
exp k2/Λ2
M =0: F˜ (k2) = exp k2/Λ2 , M =2: F˜ (k2)= − 1 ,
0 − 1 2 k2+m2
(cid:2) 1 (cid:3)
(cid:2) (cid:3) exp k2/Λ2
N =0: G˜ (k2) = exp k2/Λ2 , N =2: G˜ (k2)= − 2 ,
0 − 2 2 k2+m2
(cid:2) 2 (cid:3)
(cid:2) (cid:3) (2.2)
i.e. M,N =2 is the modified Yukawa type and M,N =0 is the Gaussiantype. Below,we give for the different cases
the momentum space representation, similar to the one that has been developed in paper I:
(i) M =N =2: In paper I using twice the identity
exp −k2/Λ2 =em2/Λ2 ∞dt exp k2+m2 t (2.3)
k2+m2 Λ2 − Λ2
(cid:2) (cid:3) Z1 (cid:20) (cid:18) (cid:19) (cid:21)
the ∆-integral has been carried out. After a redefinition of the variables t t/Λ2 and u u/Λ2 the result in I is
→ 1 → 2
V˜ (k) = (4π)−3/2em21/Λ21em22/Λ22 ∞dt ∞duexp[−(m21t+m22u)]
2,2 (t+u)3/2 ·
Zt0 Zu0
tu
exp k2 (t =1/Λ2,u =1/Λ2) . (2.4)
× − t+u 0 1 0 2
(cid:20) (cid:18) (cid:19) (cid:21)
(ii) M = 2,N = 0: Using the identity (2.3) once, and performing similar steps as in paper I, one easily derives that
for this case
V˜ (k) = (4π)−3/2em21/Λ21em22/Λ22 ∞dt ∞duexp[−(m21t+m22u)]
2,0 (t+u)3/2 ·
Zt0 Zu0
tu
exp k2 δ(u u ) . (2.5)
0
× − t+u · −
(cid:20) (cid:18) (cid:19) (cid:21)
Here,isdefinedδ(u u ) lim δ(u u ),whereu =u ǫ. Thisdefinitionimpliesthatin(2.5)theu-integration
0 ǫ↓0 0,ǫ 0,ǫ 0
− ≡ − −
can simply be performed by the substitution u u in the integrand.
0
→
3
(iii) M =0,N =2: Similarly to the previous case, one has
V˜ (k) = (4π)−3/2em21/Λ21em22/Λ22 ∞dt ∞duexp[−(m21t+m22u)]
0,2 (t+u)3/2 ·
Zt0 Zu0
tu
exp k2 δ(t t ) .
0
× − t+u · −
(cid:20) (cid:18) (cid:19) (cid:21)
(2.6)
Forthe integralsV˜ ofthissection,andsimilarintegralsbelowinthispaper,weintroducethefollowingconvenient
M,N
short-hand notation. We write
∞ ∞ tu
V˜ (k) = dt du w(t,u) v (t,u) exp k2 , (2.7a)
M,N M,N
· − t+u
Zt0 Zu0 (cid:26) (cid:20) (cid:18) (cid:19) (cid:21)(cid:27)
with common weight function w (t,u) defined as
0
w (t,u) (4π)−3/2em21/Λ21em22/Λ22exp[−(m21t+m22u)] . (2.7b)
0 ≡ (t+u)3/2
The form in which these basic integrals appear in MPE depends on two factors:
(i) The denominators D(ω ,ω ). In the next section we will give a catalogue of these.
1 2
(ii) The operators O˜(k ,k ). Also these will be given in the next section.
1 2
III. MESON-PAIR EXCHANGE POTENTIALS 1,2 is given by
g(1)(αβ) = g g g ,
(αβ) NNα NNβ
In [4] the derivation of the pair-exchange potentials
both in momentum and in configuration space is given. g(2)(αβ) = g2 , (3.2)
(αβ)
In this reference the configuration space potentials
with appropriatepowersofm , depending onthe defini-
are worked out fully. The topic of this paper is to π
tion of the Hamiltonians given in [4], section II.
do the same for the momentum space description. In
particular,thepartialwaveanalysisisperformedleading The momentum-dependent operators O(n) are given
αβ,p
to a representation which is very suitable for numerical inTablesIVandVI. Forcompleteness,theseTablesalso
evaluation. contain the isospin factors C(n)(αβ) as derived in Ap-
pendix B of[4]. The momentumoperatorsfor(ππ) and
0
(ππ) both contain a term antisymmetric in k k ,
From[4]andequation(3.1)itfollowsthatthemomen- 1 1 2
↔
whichonly contributes in the nonadiabatic contribution,
tum space MPE-potential can be represented in general
see[4],section4. Intheadiabaticpotential,asexplained
in the form
in [4], they drop out when we integrate over k and k .
1 2
Vα(βn)(k) = C(n)(αβ)g(n)(αβ) d3(k21πd)33k2 detTahileienne[4rg])y, dseecntoiomninIaIt,oirns Dtep(rnm)saoref athlseotdimisceu-osrsdedereind
ZZ processes involvedin one- and two-pairexchange. These
e ×δ(k−k1−k2) F˜0(k21)G˜0(k22) denominators depend on the energies of the exchanged
× O˜α(nβ),p(k1,k2) Dp(n)(ω1,ω2), mdeepseonnds,enic.ee.comωe1sfraonmdvωer2t.icesAwniotthhdeerrisvoautirvsees,oafndω1t,h2e-
p
X
non-adiabatic expansion terms. It appears from [4] that
(3.1)
in general one can write
where the index n distinguishes one-pair (n = 1) and D(n)(ω ,ω ) = c(n) D , (3.3)
two-pair (n = 2) meson-pair exchange, and (αβ) refers {p} 1 2 p1,p2,p3 {p1,p2,p3}
to the particular meson pair that is being exchanged. p1X,p2,p3
The subscript p= ad,na,pv distinguishes respectively where in terms of the integer powers p (i = 1,2,3) the
{ } i
the adiabatic-, the non-adiabatic-, pseudovector vertex-, denominators can be written
and off-shell-contributions. Here, the last three are the
1 1 1
1/M-correctionsto the MPE-potentials. D = . (3.4)
The product of the coupling constants in the cases n = {p1,p2,p3} ω1p1 ω2p2 (ω1+ω2)p3
4
TheenergydenominatorsD(n) arelistedinTablesVand
p
p’ -p’ p’ -p’ VI.
k
2
-p’’
Theevaluationofthemomentumintegrationscannow
k2 k1 readily be performed using the methods given in [4, 11].
There it was shown that the full separation of the k
1
-p’’ and k dependence can be achieved in all cases using
2
k the λ-integral representation, first introduced in [11]. In
1
Appendix B the occurring λ-integrals are listed. From
p -p p -p the listinginBonereadilyseesthatforthe derivationof
the representation similar to that one in Eqs. (2.4)-(2.6)
( a ) ( c ) we need to start out from the generalization of (2.1):
p’ -p’ p’ -p’
k
2 k
2
-p’’
2 ∞ d3∆
k V (k,λ)= dλf (λ)
1 M,N π M,N (2π)3 ·
k Z0 Z
1
eF˜ (∆2, m2+λ2) G˜ ((k ∆)2, m2+λ2()3..5)
p -p p -p × M 1 N − 2
q q
( b ) ( d )
FIG. 1: Time-ordered (a)-(c) one-pair and (d) two-pair dia-
grams. Thedashedlinewithmomentumk referstothepion
1
and the dashed line with momentum k refers to one of the
2
other (vector, scalar, or pseudoscalar) mesons. To these we In paper I it has been shown that all the occurring λ-
have to add the “mirror” diagrams, where for the one-pair integrals can be performed analytically. The result for
diagrams thepair vertex occurs on the othernucleon line.
all cases can be written as
d3∆
V (k) = F˜ (∆2,m ) G˜ ((k ∆)2,m ) D (ω ,ω )
p1,p2,p3 (2π)3 0 1 0 − 2 {p1,p2,p3} 1 2
Z
∞ ∞ tu
e = dt du w (t,u) d (t,u) exp k2 . (3.6)
0 · {p1,p2,p3} − t+u
Zt0 Zu0 (cid:26) (cid:20) (cid:18) (cid:19) (cid:21)(cid:27)
All functions d (t,u) that occur in this work use only representations with M = N = 2, so that no
{p1,p2,p3}
are given in Table VII. As noted in section II we will δ(t t ) or δ(u u ) occurs.
0 0
− −
IV. PROJECTION MPE ON SPINOR INVARIANTS I
ADIABATIC CONTRIBUTIONS
The MPE-contributions from the adiabatic terms, the non-adiabatic- and pseudovector vertex corrections are the
central-, spin-spin-, tensor-, and spin-orbit momentum space analogs of those given in Reference [4] in configuration
space. From(3.1),TablesIV-VI,andTableVitisreadilyverifiedthattheprojectionontothepotentialsV ,similarly
j
5
to the paper I, can be written as
∞ ∞ tu
V(n)(αβ) = dt du w (t,u) exp k2 Ω(n)(k2;t,u) (αβ) . (4.1)
pair 0 − t+u j
Zt0 Zu0 (cid:26) (cid:20) (cid:18) (cid:19) (cid:21) (cid:27)
ThefunctionsΩ(n)eareworkedoutinthesubsectionsbelow. LikeinI,wealsointroduceforconveniencetheexpansion
j
in k2:
K
Ω(ad,na,pv)(k2;t,u)=C(n)(αβ)g(n)(αβ) Υ(ad,na,pv)(t,u) k2 k . (4.2)
j · j,k
k=0
X (cid:0) (cid:1)
Below in this section we give the results for the adiabatic contributions. The coefficients Υad are tabulated in
j,k
Tables VIII-XII.
A. JPC =0++: Adiabatic (ππ) -Exchange potentials
0
The 1-pair and 2-pair contributions are
g f2 3 tu 1
Ω(1)(k2;t,u) = 6 (ππ)0 NNπ d (t,u) + k2 , (4.3a)
1 m m2 · {2,2,0} 2 − t+u · t+u
(cid:18) π (cid:19)(cid:18) π (cid:19) (cid:26) (cid:18) (cid:19) (cid:27)
g 2
Ω(2)(k2;t,u) = 3 (ππ)0 d (t,u) (4.3b)
1 − m2 · {1,1,1}
(cid:18) π (cid:19)
B. JPC =1−−: Adiabatic (ππ) -Exchange potentials
1
(i) 1-pair exchange:
g f2 3 tu 1
Ω(1)(k2;t,u) = 4(τ τ ) (ππ)1 NNπ d (t,u) + k2 , (4.4a)
1 − 1· 2 m2 m2 · 1,1,1 −2 t+u · t+u
(cid:18) π (cid:19)(cid:18) π (cid:19) (cid:26) (cid:18) (cid:19) (cid:27)
g f2 (1+κ ) 1 1
Ω(1)(k2;t,u) = 2(τ τ ) (ππ)1 NNπ 1 d (t,u) + k2 , (4.4b)
2 − 1· 2 m2 m2 M · 2,2,0 · 3 · t+u
(cid:18) π (cid:19)(cid:18) π (cid:19)
g f2 (1+κ ) 1 1
Ω(1)(k2;t,u) = 2(τ τ ) (ππ)1 NNπ 1 d (t,u) , (4.4c)
3 − 1· 2 m2 m2 M · 2,2,0 ·−2 · t+u
(cid:18) π (cid:19)(cid:18) π (cid:19)
g f2 1 1
Ω(1)(k2;t,u) = 2(τ τ ) (ππ)1 NNπ d (t,u) , (4.4d)
4 − 1· 2 m2 m2 M · 2,2,0 · t+u
(cid:18) π (cid:19)(cid:18) π (cid:19)
(ii) 2-pair exchange:
1 g 2
Ω(2)(k2;t,u) = (τ τ ) (ππ)1 [d +d 4d ](t,u) . (4.4e)
1 −2 1· 2 m2 · 1,0,0 0,1,0− 0,0,1
(cid:18) π (cid:19)
C. JPC =1++: Adiabatic (πρ) -Exchange potentials
1
(i) 1-pair exchange:
2 g f g
Ω(1)(k2;t,u) = (τ τ ) (πρ)1 NNπ NNρ d (t,u)
2 M 1· 2 m m · 2,2,0 ·
(cid:18) π (cid:19)(cid:18) π (cid:19)
1 1 u2 1 4 tu 1
+ k2 + (1+κ ) 2 k2 , (4.5a)
ρ
× 2 3 t+u 2 − 3t+u · t+u
(cid:26)(cid:20) (cid:18) (cid:19) (cid:21) (cid:20) (cid:21)(cid:27)
2 g f g
Ω(1)(k2;t,u) = (τ τ ) (πρ)1 NNπ NNρ d (t,u)
3 M 1· 2 m m · 2,2,0 ·
(cid:18) π (cid:19)(cid:18) π (cid:19)
u2 1 2tu 1
+ (1+κ ) , (4.5b)
ρ
× t+u 2 t+u · t+u
(cid:26) (cid:27)
6
(ii) 2-pair exchange:
g 2
Ω(2)(k2;t,u) = (τ τ ) (πρ)1 d (t,u) (4.5c)
2 − 1· 2 m2 · 1,1,1
(cid:18) π (cid:19)
D. JPC =1++: Adiabatic (πσ)-Exchange potentials
(i) 1-pair exchange:
g f g
Ω(1)(k2;t,u) = +(τ τ ) (πσ) NNπ NNσ d (t,u)
2 1· 2 m2 m · 2,2,0 ·
(cid:18) π (cid:19)(cid:18) π (cid:19)
2 tu u2 1
2+ − k2 , (4.6a)
× − 3 t+u · t+u
(cid:20) (cid:18) (cid:19) (cid:21)
g f g
Ω(1)(k2;t,u) = +2(τ τ ) (πσ) NNπ NNσ d (t,u)
3 1· 2 m2 m · 2,2,0 ·
(cid:18) π (cid:19)(cid:18) π (cid:19)
tu u2 1
− , (4.6b)
× t+u · t+u
(cid:18) (cid:19)
(ii) 2-pair exchange:
1 g 2
Ω(2)(k2;t,u) = (τ τ ) (πσ)1 d (t,u)
2 −2 1· 2 m2 · 1,1,1 ·
(cid:18) π (cid:19)
2
2 1 t u
+ − k2 , (4.6c)
×(t+u 3 t+u )
(cid:18) (cid:19)
1 g 2 t u 2
Ω(2)(k2;t,u) = (τ τ ) (πσ)1 d (t,u) − , (4.6d)
3 −2 1· 2 m2 · 1,1,1 t+u
(cid:18) π (cid:19) (cid:18) (cid:19)
E. JPC =1+−: Adiabatic (πω)-Exchange potentials
(i) 1-pair exchange:
g f g
Ω(1)(k2;t,u) = (τ τ ) (πω) NNπ NNω d (t,u)
2 1· 2 m m · 2,2,0 ·
(cid:18) π (cid:19)(cid:18) π (cid:19)
2 tu+u2 1
k2 , (4.7a)
× 3 t+u · t+u
(cid:20) (cid:18) (cid:19) (cid:21)
g f g
Ω(1)(k2;t,u) = +2(τ τ ) (πω) NNπ NNω d (t,u)
3 1· 2 m m · 2,2,0 ·
(cid:18) π (cid:19)(cid:18) π (cid:19)
tu+u2 1
, (4.7b)
× t+u · t+u
(cid:18) (cid:19)
(ii) 2-pair exchange:
1 g 2
Ω(2)(k2;t,u) = (τ τ ) (πω)1
2 −2 1· 2 m2 ·
(cid:18) π (cid:19)
1
d +d k2d (t,u) , (4.7c)
1,0,0 0,1,0 1,1,1
× − 3
(cid:26) (cid:27)
1 g 2
Ω(2)(k2;t,u) = + (τ τ ) (πω)1 d (t,u) . (4.7d)
3 2 1· 2 m2 · 1,1,1
(cid:18) π (cid:19)
7
F. JPC =1++: Adiabatic (πP)-Exchange potentials
ThetreatmentofthePomeronhasbeenexplainedin[3]. ThisimpliestheuseofG /M2 insectionII. Furthermore,
0 N
w.r.t. σ-exchange there is a ( )-sign for P-exchange. Therefore, comparing to (4.6a-4.6d) we obtain the following
−
potentials: e
(i) 1-pair exchange:
g f g 1
Ω(1)(k2;t,u) = (τ τ ) (πP) NNπ NNP d (t,u)
2 − 1· 2 m2 m · M2 · 2,0,0 ·
(cid:18) π (cid:19)(cid:18) π (cid:19) N
2 tu u2 1
2+ − k2 δ(u u ) , (4.8a)
0
× − 3 t+u · t+u · −
(cid:20) (cid:18) (cid:19) (cid:21)
g f g 1
Ω(1)(k2;t,u) = 2(τ τ ) (πP) NNπ NNP d (t,u)
3 − 1· 2 m2 m · M2 · 2,0,0 ·
(cid:18) π (cid:19)(cid:18) π (cid:19) N
tu u2 1
− δ(u u ) . (4.8b)
0
× t+u · t+u · −
(cid:18) (cid:19)
(ii) 2-pair exchange:
1 g 2 1
Ω(2)(k2;t,u) = + (τ τ ) (πP)1 d (t,u)
2 2 1· 2 m · M2 · 1,1,1 ·
(cid:18) π (cid:19) N
2 1 t u 2
+ − k2 δ(u u ) , (4.8c)
0
×(t+u 3(cid:18)t+u(cid:19) )· −
1 g 2 1 t u 2
Ω(2)(k2;t,u) = + (τ τ ) (πP)1 d (t,u) − δ(u u ) . (4.8d)
3 2 1· 2 m2 · M2 · 1,1,1 t+u · − 0
(cid:18) π (cid:19) N (cid:18) (cid:19)
Notice that in (4.8a)-(4.8d) u =1/4m2.
0 P
G. JPC =0++: Adiabatic ’derivative’ (ππ) -Exchange potentials
0
The derivative pair-potentials in coordinate space have been derived in [12] in detail. A summary of this is given
in appendix D. A short derivation of the p-space potentials is also can be found there.
(i) 1-pair exchange:
g′ f 2
Ω(1)(k2;t,u) = 12 (ππ)0 NNπ d (t,u)
1 − m3π !(cid:18) mπ (cid:19) · 2,2,0 ·
15 1t2 8tu+u2 t2u2 1
+ − k2+ k4 , (4.9a)
× 4 2 t+u (t+u)2 · (t+u)2
(cid:20) (cid:21)
(ii) 2-pair exchange:
g′ 2 15 t2 3tu+u2 t2u2 d (t,u)
Ω(2)(k2;t,u) = 6 (ππ)0 + − k2+ k4 1,1,1
1 − m3π ! ·(cid:26)(cid:20) 4 t+u (t+u)2 (cid:21) (t+u)2
1 3 m2t+m2u d (t,u)
+ (m2+m2)+ 1 2 k2+m2m2(t+u) 1,1,1
2 2 1 2 t+u 1 2 t+u
(cid:20) (cid:21)
3 tu d (t,u)
+ k2 0,0,1 . (4.9b)
2 − t+u t+u
(cid:20) (cid:21) (cid:27)
H. JPC =0++: Adiabatic (σσ)-Exchange potentials
8
(i) 1-pair exchange:
g
Ω(1)(k2;t,u) = 2 (σσ) g2 d (t,u) , (4.10a)
1 m · NNσ· 2,2,0
(cid:18) π (cid:19)
(ii) 2-pair exchange:
g 2
Ω(2)(k2;t,u) = (σσ) d (t,u) , (4.10b)
1 − m 1,1,1
(cid:18) π (cid:19)
V. PROJECTION MPE ON SPINOR INVARIANTS II
1/M CORRECTIONS
The non-adiabatic- and pseudovector vertex-corrections have been given in [4], section IV. Similar to Eq. (4.1) we
write these contributions in the form
∞ ∞ tu
V(na,pv)(αβ) = dt du w (t,u) d(n) (t,u) exp k2 Ω(n)(k2;t,u) . (5.1)
pair 0 {p1,p2,p3} − t+u j
Zt0 Zu0 (cid:26) (cid:20) (cid:18) (cid:19) (cid:21) (cid:27)
e
A. Non-adiabatic Corrections
From Eqs. (4.5)-4.8) of [4] one readily obtains the momentum space equivalents using the replacements:
d3k d3k d3k d3k
1 2 ei(k1+k2) 1 2 δ(k k k )
(2π)6 → (2π)3 − 1− 2
Z Z Z Z
Then, by comparison one can easily read off the diverse quantities O˜(na) and D(na)(ω ,ω ) that occur in Eq. (3.1)
αβ,p p 1 2
for the non-adiabatic potentials. The projections onto the Ω(na) in Eq. (5.1) yield
j
(i) (ππ) :
0
g f 2 3 15
Ω(na)(k2;t,u) = (ππ)0 NNπ d (t,u) +
1 − m m M · {na} 4
π (cid:18) π (cid:19) (cid:26)
1 t2 8ut+u2 t2u2 1
+ − k2+ k4 , (5.2a)
2 t+u (t+u)2 · (t+u)2
(cid:18) (cid:19) (cid:18) (cid:19) (cid:27)
g f 2 3 1
Ω(na)(k2;t,u) = (ππ)0 NNπ d (t,u) . (5.2b)
4 − m m M · {na} · t+u
π (cid:18) π (cid:19)
(ii) (ππ) :
1
g f 2 1 15
Ω(na)(k2;t,u) = 2(τ τ ) (ππ)1 NNπ d (t,u) +
1 − 1· 2 m m M · {2,2,0} 4
π (cid:18) π (cid:19) (cid:26)
1 t2 8ut+u2 t2u2 1
+ − k2+ k4 , (5.3a)
2 t+u (t+u)2 · (t+u)2
(cid:18) (cid:19) (cid:18) (cid:19) (cid:27)
g f 2 1 1
Ω(na)(k2;t,u) = 2(τ τ ) (ππ)1 NNπ d (t,u) . (5.3b)
4 − 1· 2 m m M · {2,2,0} · t+u
π (cid:18) π (cid:19)
(iii) (σσ):
g g2 3 tu 1
Ω(na)(k2;t,u) = (σσ) NNσ d (t,u) + k2 . (5.4)
1 m M · {na} −2 t+u · t+u
π (cid:26) (cid:18) (cid:19) (cid:27)
9
(iv) (πσ):
g f g
Ω(na)(k2;t,u) = +(τ τ ) (πσ)1 NNπ NNσ d (t,u)
2 1· 2 m2 m M · {na} ·
π π
5 1t2 13tu+6u2 1tu2(t u) 1
+ − k2+ − k4 , (5.5a)
× 2 6 t+u 3 (t+u)2 · (t+u)2
(cid:26) (cid:27)
g f g
Ω(na)(k2;t,u) = +(τ τ ) (πσ)1 NNπ NNσ d (t,u)
3 1· 2 m2 m M · {na} ·
π π
1t2 7tu+6u2 tu2(t u) 1
− + − k2 . (5.5b)
× 2 t+u (t+u)2 · (t+u)2
(cid:26) (cid:27)
(v) (ππ) (’derivative’):
0
g′ f 2 1
Ω(na)(k2;t,u)= 12 (ππ)0 NNπ
1 − m3π !(cid:18) mπ (cid:19) 2M ·(cid:26)
15 1 t2 8tu+u2 t2u2 d (t,u)
+ + − k2+ k4 1,1,1 ,
4 2 t+u (t+u)2 (t+u)2
(cid:20) (cid:18) (cid:19) (cid:21)
105 15 t2 5tu+u2 3 t2 5tu+u2 t3u3 d (t,u)
+ − k2 tu − k4 k6 na , (5.6a)
− 8 4 t+u − 2 (t+u)2 − (t+u)3 (t+u)3
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:21) (cid:27)
g′ f 2 1 d (t,u) tu d (t,u)
Ω(na)(k2;t,u)= 12 (ππ)0 NNπ 1,1,1 + 5+2 k2 na . (5.6b)
4 − m3π !(cid:18) mπ (cid:19) 2M ·(cid:26) t+u (cid:20)− t+u (cid:21) (t+u)2 (cid:27)
Here, d (t,u) is defined in (B3).
{na}
B. Pseudovector-vertex Corrections
FromEqs.(4.9)-4.11)of[4]likewiseasinthecaseofthenon-adiabaticcorrectionsoneobtainsforthepseudovector-
vertex corrections:
(i) (ππ) :
0
g f 2 3 t2+u2 1
Ω(pv)(k2;t,u) = + (ππ)0 NNπ d (t,u) 3+ k2 , (5.7a)
1 m m M · {1,1,1} t+u · t+u
π (cid:18) π (cid:19) (cid:26) (cid:18) (cid:19) (cid:27)
g f 2 3
Ω(pv)(k2;t,u) = +2 (ππ)0 NNπ d (t,u) . (5.7b)
4 m m M · {1,1,1}
π (cid:18) π (cid:19)
(ii) (ππ) :
1
g f 2 1 3 u2
Ω(pv)(k2;t,u) = +(τ τ ) (ππ)1 NNπ + k2 d (t,u)
1 1· 2 m m M · 2 t+u {2,0,0}
π (cid:18) π (cid:19) (cid:20)(cid:18) (cid:19)
3 t2 1
+ + k2 d (t,u) (5.8a)
2 t+u {0,2,0} · t+u
(cid:18) (cid:19) (cid:21)
g f 2 1 u t
Ω(pv)(k2;t,u) = +2(τ τ ) (ππ)1 NNπ d + d . (5.8b)
4 1· 2 m m M · t+u {2,0,0} t+u {0,2,0}
π (cid:18) π (cid:19) (cid:20) (cid:21)
(iii) (πσ):
g f g 1t2 tu 1
Ω(pv)(k2;t,u) = (τ τ ) (πσ)1 NNπ NNσ d (t,u) 1+ − k2 . (5.9a)
2 − 1· 2 m2 m M · {1,1,1} · 3 t+u t+u
π π (cid:26) (cid:27)
g f g t2 tu 1
Ω(pv)(k2;t,u) = (τ τ ) (πσ)1 NNπ NNσ d (t,u) − . (5.9b)
3 − 1· 2 m2 m M · {1,1,1} · t+u · t+u
π π (cid:18) (cid:19)
10
(iv) (ππ) (’derivative’):
0
Ω(pv)(k2;t,u) = 6 g(′ππ)0 fNNπ 2 1 d (t,u) m2 m2 2
1 − m3π !(cid:18) mπ (cid:19) 2M · 1,1,1 ·(cid:26) 1− 2
(cid:0) (cid:1)
m2(3t2 u2)+m2(3u2 t2) 1
3(m2+m2)+ 1 − 2 − k2
− 1 2 t+u t+u
(cid:20) (cid:21)
t2+2tu+u2 t2+2tu+u2 1
k2+2tu k4 , (5.10a)
− t+u (t+u)2 (t+u)2
(cid:20)(cid:18) (cid:19) (cid:18) (cid:19) (cid:21) (cid:27)
g′ f 2 1
Ω(pv)(k2;t,u) = 24 (ππ)0 NNπ d (t,u)
4 − m3π !(cid:18) mπ (cid:19) 2M · 1,1,1 ·
3 t2+tu+u2 1
m2+m2 + + k2 . (5.10b)
× 1 2 2 t+u t+u
(cid:26) (cid:20) (cid:18) (cid:19) (cid:21) (cid:27)
(cid:0) (cid:1)
The coefficients Υna,pv defined in (4.2) are tabulated in Tables VIII-XII.
j,k
For(πP)-exchange,the 1/M non-adiabaticandpseudo-vectorvertexcorrectionscanbereadofffromthosefor(πσ)
N
and making the same adjustments as given already for the adiabatic contributions. In Table XIII the Ω(pv,ad) for
i
(πP)-pair exchange are given explicitly.
VI. PARTIAL WAVE ANALYSIS
Like the TME-potentials in I, the general form of the MPE-potentials in momentum space is
∞ ∞ tu
V(n)(k) = dt du w(n) (t,u) exp k2 Ω(n)(k2;t,u) , (6.1)
j {p1,p2,p3} − t+u j
Zt0 Zu0 (cid:26) (cid:20) (cid:18) (cid:19) (cid:21) (cid:27)
e
where
w(n) (t,u)=w (t,u) d(n) (t,u) .
{p1,p2,p3} 0 {p1,p2,p3}
Therefore, the partial wave analysis runs along the same lines as described in sections VI and VII of paper I for the
TME-potentials. We refer the reader to this paper for the details.
VII. ESC-MODEL, RESULTS the gaussian cut-off’s Λ. The used α =: F/(F + D)-
ratio’s for the OBE-couplings are: pseudo-scalar mesons
α = 0.355, vectormesons αe = 1.0,αm = 0.275,
The momentum space formulas for the potentials of pv V V
and scalar-mesons α = 0.914, which is computed us-
this paper and paper I have been checked numerically. S
ing the physical S∗ =: f (993) coupling etc.. In Ta-
This is done by solving the Lippmann-Schwinger equa- 0
ble II we show the MPE-coupling constants. The used
tion and comparingthe phase shifts with those obtained
α=:F/(F +D)-ratio’s for the MPE-couplingsare: (πη)
by solving the Schr¨odinger equation using the x-space
etc. and (πω) pairs α( 8 ) = 1.0, (ππ) etc. pairs
equivalent of the potentials. { s} 1
αe ( 8 ) = 0.4,αm( 8 ) = 0.335, (πρ) etc. pairs
After the completion of the p-space formalismwe per- V { }a V { }a 1
α ( 8 )=0.335.
formedaχ2-fitwiththeESC-modeltothe1993Nijmegen A { }a
representationoftheχ2-hypersurfaceoftheNN scatter-
We emphasize that we use the single-energy (s.e.)
ing data below Tlab =350 MeV [15]. phases and χ2-surface [17] only as a means to fit the
This fitting was executed in x-space using the equiva- NN-data. As stressed in [15] the Nijmegen s.e. phases
lent x-space potentials. The reason for this is the much have not much significance. The significant phases are
faster evaluation of the ESC-model in x-space. We ob- the multi-energy (m.e.) ones, see the dashed lines in the
tained a χ2/Ndata = 1.15. The phase shifts are shown figures. One notices that the central value of the s.e.
in Fig.s 2-5. In Table III the results are shown for the phases do not correspond to the m.e. phases in gen-
ten energy bins, where we compare the results from the eral, illustrating that there has been a certain amount
updated partial-wave analysis with the ESC potentials. of noice fitting in the s.e. PW-analysis, see e.g. ǫ and
1
In Table I we show the OBE-coupling constants and 1P at T = 100 MeV. The m.e. PW-analysis reaches
1 lab