ESC NN-Potentials in Momentum Space. I. PS-PS Exchange Potentials

23 Pages·0.3 MB·

by  Th. A. Rijken

ESC NN-Potentials in Momentum Space
I. PS-PS 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 8, 2008)
A momentum space representation is derived for the Nijmegen Extended-Soft-Core (ESC) in-
teractions. The partial wave projection of this representation is carried through, in principle for
Two-Meson-Exchange (TME) in general. Explicit results for the momentum space partial wave
NN-potentials from PS-PS-Exchangeare given.
2 PACSnumbers: 13.75.Cs,12.39.Pn,21.30.+y
0
0
2 I. INTRODUCTION basis for the Nijmegen soft-core models.
In this paper, hereafter referred to as I, a representa-
n
a tionforthe ESC-interactionsinmomentum spaceis pre-
Fornucleon-nucleon[1,2,3,4]andhyperon-nucleon[7]
J sented, which is very elegant and useful for applications
scattering we have shown that the Extended-Soft-Core
8 in momentum space computations.
(ESC)models for baryon-baryoninteractionsgiveanex-
Wesolvebasicallytheproblemoffindingasuitablerep-
1 cellent description of the NN and YN data. So far, the resentationoftheTME-potentials inmomentumspacein
v applications of the ESC-modelhas been in configuration general, and in particularly with gaussian form factors.
8 space. In two papers we give a momentum space repre-
As is apparant from the configuration space representa-
1 sentation of the ESC-potentials. Also, we describe a fit
tion, given in [2, 3, 12], we have to evaluate forms like
0
to the pp- and np-data for 0 T 350 MeV, having
lab
01 χbe2i/nNgdcaotaup=lin1.g15c.onHsetaren,tswaenuds≤ecdut2-0offp≤hmyassicseasl.parameters, 2 ∞dλ F˜(∆2,λ) G˜((∆ k)2,λ) ,
2 π −
0 In synopsis, an account of a modern theoretical basis Z0
h/ forthe soft-coreinteractionshasbeengivenin[1]. Start- which at first sight means an extra numerical integral,
t ing from the so called Standard Model and integrating besides the usual convolutive integration over the ∆-
l- outthe heavyquarksonearrivesataneffective QCDfor momentum. Here, F˜ and G˜ contain the couplings, gaus-
c the u,d,s quarks. The generally acceptedscenariois now sian form factors, and momentum space Yukawa func-
u
n that the QCD-vacuum becomes unstable for momenta tions for the two exchanged mesons. Fortunately, the
: transfers for which Q2 Λ2 1 GeV2 [5], caus- integrations over the λ-parameter can be carried through
v ≤ χSB ≈ analytically, as we shall show in the sequel. The rep-
ing spontaneous chiral-symmetry breaking (χSB). The
i
X vacuumgoesthroughaphase-transition,generatingcon- resentations obtained in this paper contain only two pa-
r stituentquarkmassesvia 0ψ¯ψ 0 =0,andreducingthe rameters, henceforth called t and u, to be integrated over
a h | | i6 numerically. Moreover, the(t,u)-integration region is ef-
gluon coupling α . Viewing the pseudo-scalar mesons π
s
fectivelyoverafinitedomain, thisinviewoftheexponen-
etc. as the Nambu-Goldstone bosons originating from
tials exp( m2t) and exp( m2u) in the integrands. Also,
the spontaneousχSB, makesit naturalto assume quark − 1 − 2
thestronggaussian-likefalloffintheoff-shellregion,typ-
dressingbypseudo-scalarmesons,andalsoothertypesof
ical for the soft-core interactions, is easily realized. To
mesons in general. In this context, baryon-baryoninter-
realize the latter property of the momentum space poten-
actionsaredescribednaturallybymeson-exchangeusing
tials is problematic, when these potentials have to begen-
form factors at the meson-baryonvertices. As a working
erated numerically from configuration space via Fourier
hypothesis, we restrict ourselves for low energy scatter-
transformation.
ingtoatreatmentofthethehadronic-phase. Integrating
The partial wave projection formulas on the LSJ-
out the heavy mesons and baryons using a renormaliza-
partial wave basis are developed for the TME-potentials
tion procedure a la Wilson [6], we restrict ourselves to
mesons with M 1 GeV/c2, arriving at an so-called ef- in momentum space. Like for the OBE-potentials in [8],
≤ we expand the momentum space potentials in the Pauli-
fective field theory as the proper arena to describe low
spinor invariants. The partial wave basis is choosen ac-
energy hadron scattering. This is the general physical
cording to the convention of [9]. Explicit formulas are
workedoutforgeneralpseudoscalar-pseudoscalar(ps-ps)
exchange.
∗Present address: Kyushu International University, Fukuoka 805- In paper II [10], the momentum space representation
8512,Japan for the ESC Meson-Pair-Exchange (MPE) potentials is
2
described. Here, we moreover present a fit to the NN- respectivelybyp andp . So,p0 =E (p )= p2+M2
i f a a i i a
daraetathfoerc0o≤ntrTilbaubt≤ion3s50frMomeVO.IBnEt-h,isPESS-PCS--mexocdhealningcel,uadnedd and p0a′ = Ea′(pf) = p2f +Ma2′, and simiplarly for b
MPE. and b’. Because of traqnslation-invariance P = P′ and
The contents of this paper are as follows. In sectionII W = W′ = Ea(pi)+Eb(pi) = Ea′(pf)+Eb′(pf). The
two-particle states we normalize in the following way
wereviewthedefinitionoftheESC-potentialsinthecon-
text of the relativistic two-body equations, the Thomp- p′,p′ p ,p = (2π)32E(p )δ3(p′ p )
son, and Lippmann-Schwinger equation. Here, we ex- h 1 2| 1 2i 1 1− 1 ·
(2π)32E(p )δ3(p′ p ) .(2.4)
ploit the Macke-Klein [11] framework in Field-Theory. × 2 2− 2
For the Lippmann-Schwinger equation we introduce the The relativistic two-body scattering-equation reads
usual potential forms in Pauli spinor space. We include [13, 14, 15]
herethecentral(C),thespin-spin(σ),thetensor(T),the
spin-orbit (SO), the quadratic spin-orbit (Q ), and the ψ(p,P) = ψ0(p,P)+G(p;P)
12
antisymmetric spin-orbit (ASO) potentials. For TME-
d4p′ I(p,p′) ψ(p′,P) , (2.5)
exchange, in the approximations made in [2, 3] only the ×
central, spin-spin, tensor, and spin-orbit potentials oc- Z
where ψ(p,P) is a 4 4-matrix in Dirac-space. The
cur. In section III the special representation for the
contributions to the ke×rnel I(p,p′) come from the two-
ESC-interactions in momentum space is developed and
nucleon irreducible Feynman diagrams. In writing (2.5)
illustrated using some basic examples. In section IV
we have taken out an overall δ4(P′ P)-function and
the general ps-ps exchange is presented. In section V
−
the total four-momentum conservation is implicitly un-
theprojectionontheplanewavespinor-invariantsiscar-
derstood henceforth.
ried through for general ps-ps exchange. The adiabatic
Thetwo-particleGreenfunctionG(p;P)in(2.5)issim-
terms, non-adiabatic, pseudovectorvertex, and off-shell
plytheproductofthefreepropagatorsforthebaryonsof
1/M-corections are given explicitly. In section VI the
line (a) and (b). The nucleon and more general for any
partialwaveanalysisisperformed. FinallyinsectionVII
baryon the Feynman propagators are given by the well
the partial wave contributions from the adiabatic terms
known formula
and the mentioned 1/M-correctionsare given.
Appendix A contains some basic integrals which are
G(s) (p) = d4x 0T(ψ(s) (x)ψ¯(s)(0))0 eip·x
employed in the paper. In Appendix B a dictionary {µ},{ν} h | {µ} {ν} | i
is given containing the results of the convolutive inte- Z
grations for the operators that occur in ps-ps exchange. Πs(p)
= (2.6)
Appendix C contains the LSJ partial wave matrix ele- p2 M2+iδ
−
mentsofseveralimportantoperators. InAppendixDthe
Fourier connection between coordinate and momentum where ψ(s) is the free Rarita-Schwinger field which de-
{µ}
space is illustratedin particularlyfor the ps-ps exchange scribes the nucleon (s = 1), the ∆ -resonance (s = 3),
2 33 2
tensor potential. etc. (see for example [16]). For the nucleon, the only
case considered in this paper, µ = and for e.g. the
{ } ∅
∆-resonance µ =µ. For the rest of this paper we deal
II. TWO-BODY INTEGRAL EQUATIONS IN only with nuc{leo}ns.
MOMENTUM SPACE
In terms of these one-particle Green-functions the two-
particle Green-function in (2.5) is
A. Relativistic Two-Body Equations
(a)
i Π(sa)(1P +p)
G(p;P) = 2
We consider the nucleon-nucleon reactions (2π)4 "(12P +p)2−Ma2+iδ# ·
N(pa,sa)+N(pb,sb)→N(pa′,sa′)+N(pb′,sb(′2).1) Π(sb)(1P p) (b)
2 − . (2.7)
wanitdhththeefitnoatlalstaantedsrelative four-momenta for the initial ×"(12P −p)2−Mb2+iδ#
Pp==12p(ap+a−pbpb) ,, Pp′′==21p(ap′a+′ −pbp′b,′) , (2.2) iangUathtsoeirnfigorfsntaowlsipnaienco-osfm(b2ap.rl6ey)toeonnsoeetfffi-onmfdoasnsst--hmsahatesltslh-scehaFenlelbysenpmwinarsint-tspetrnaotapess-
which become in the center-of-mass system (cm-system) [17]
for a and b on-mass-shell
Π(s)(p) M Λ(s)(p)
P =(W,0) , p=(0,p) , p′ =(0,p′) . (2.3) = +
p2 M2+iδ E(p) p E(p)+iδ
" 0
− −
In general, the particles are off-mass-shell in the Green-
Λ(s)( p)
functions. In the following of this section, the on-mass- − − , (2.8)
shellmomentaforthe initialandfinalstates aredenoted −p0+E(p) iδ#
−
3
for s = 1,3,.... Here, Λ(s)(p) and Λ(s)(p) are the where u(p,σ) and v(p,σ) are the Dirac spinors for spin-
2 2 + −
on-mass-shell projection operators on the positive- and 1 particles, and E(p)= p2+M2 with M the nucleon
2
negative-energy states. For the nucleon they are mass. Then, in the cm-system, where P = 0 and P =
0
p
W, the Green-function can be written as
+1/2
Λ (p) = u(p,σ) u¯(p,σ) ,
+
⊗
σ=X−1/2
+1/2
Λ (p) = v(p,σ) v¯(p,σ) . (2.9)
−
− ⊗
σ=X−1/2
i M Λ(sa)(p) Λ(sa)( p)
G(p;W) = a + − −
(2π)4 (cid:18)Ea(p)(cid:19)"12W +p0−Ea(p)+iδ − 12W +p0+Ea(p)−iδ#
M Λ(sb)( p) Λ(sb)(p)
b + − − (2.10)
×(cid:18)Eb(p)(cid:19)"12W −p0−Eb(p)+iδ − 12W −p0+Eb(p)−iδ#
Multiplying out (2.10) we write the ensuing terms in B. Three-Dimensional Equation
shorthand notation
Following the same procedures as in [12] we intro-
G(p;W) = G (p;W)+G (p;W)
++ +− duce the three-dimensional wave-function according to
+G−+(p;W)+G−−(p;W) , (2.11) Salpeter [18] by
where G etc. corresponds to the term with ΛsaΛsb
++ + + E (p)E (p) ∞
etc. Introducing the wave-functions (see [18]) φ(p)= a b ψ(p )dp . (2.16)
µ 0
s MaMb Z−∞
ψ (p′)=ΛsaΛsbψ(p′) (r,s=+, ) , (2.12)
rs r s −
Next, we use for the right-inverse of the dp -operation
0
thetwo-bodyequation,(2.5)canbewrittenfore.g. ψ++ the ansatz proposed by Klein and Macke [11],
as R
M M
ψ (p) = ψ0 (p)+G (p;W) ψ(p′ )= a b A (p′)φ(p′) , (2.17)
++ ++ ++ · µ sEa(p′)Eb(p′) W µ
d4p′ I(p,p′) ψ (p′) ,(2.13)
++,rs rs
× where the right-inverse is given by
Z r,s
X
and similar equations for ψ , ψ , and ψ . 1 W (p′)
+− −+ −− A (p′) = −W ,(2.18)
Invoking ‘dynamical pair-suppression’, as discussed in W µ −2πiF(a)(p′,p′)F(b)( p′, p′)
[12], (2.13) reduces to a four-dimensional equation for W 0 W − − 0
ψ++, i.e. with the frequently used notations
ψ (p′) = ψ0 (p′)+G (p′;W) 1
++ ++ ++ F (p,p )=p E(p)+ W +iδ
W 0 0
− 2
d4pI(p′,p)++,++ψ++(p) , (2.14) (p)=E (p)+E (p) . (2.19)
× a b
W
Z
with the Green-function Applying now the p0 integration to (2.14), and perform-
ingthep′-integration,madepossiblebytheansatzabove,
0
i M M
G (p;W) = a b Λ(sa)(p)Λ(sb)( p) one arrives at the Thompson equation [19]
++ (2π)4 E (p)E (p) + + − ·
(cid:20) a b (cid:21)
−1 φ (p′) = φ(0)(p′)+E(+)(p′;W)
1 ++ ++ 2
W +p E (p)+iδ
× (cid:20)2 0− a (cid:21) d3p Kirr(p′,pW)φ (p)(,2.20)
−1 × (2π)3 | ++
1 Z
W p E (p)+iδ . (2.15)
0 b
× 2 − − where the Green function is
(cid:20) (cid:21)
4
and the kernel is given by
1 Λa(p′)Λb ( p′)
E(+)(p′;W)= + + − , (2.21)
2 (2π)3 (W (p′)+iδ)
−W
1 M M M M
Kirr(p′,pW) = a b a b (W (p′))(W (p))
| −(2π)2sEa(p′)Eb(p′)sEa(p)Eb(p) −W −W
+∞ +∞ −1
dp′ dp F(a)(p′,p′)F(b)( p′, p′)
× 0 0 W 0 W − − 0
Z−∞ Z−∞ (cid:20)n o
−1
[I(p′,p′;p ,p)] F(a)(p,p )F(b)( p, p ) . (2.22)
× 0 0 ++,++ W 0 W − − 0
n o (cid:21)
The M/E-factors in (2.22) are due to the difference C. Lippmann-Schwinger Equation
between the relativistic and the non-relativistic normal-
izationofthetwo-particlestates. Inthefollowingwesim- The transformation of (2.24) to the Lippmann-
ply put M/E(p)=1 in the kernel Kirr Eq. (2.22). The Schwinger equation can be effectuated by defining
corrections to this approximation would give (1/M)2-
correctionstothepotentials,whichweneglectinthispa- T(p′,p) = N(p′) M(p′,p;W) N(p) ,
per. Inthesameapproximationthereisnodifferencebe-
V(p′,p) = N(p′) Kirr(p′,p;W) N(p) , (2.25)
tweentheThompson(2.20)andtheLippmann-Schwinger
equation, when the connection between these equations
where the transformation function is
is made using multiplication factors. Henceforth,we will
not distinguish between the two. p2 p2
N(p)= i − . (2.26)
s2MN(E(pi) E(p))
Thecontributionstothetwo-particleirreduciblekernel −
Kirr uptosecondorderinthemeson-exchangearegiven Applicationofthistransformation,yieldstheLippmann-
in detail in [2]. For the definition of the TME-potential Schwinger equation
in the Lippmann-Schwinger equation we shall need the
d3p′′
complete fourth-orderkernelfor the Thompsonequation T(p′,p) = V(p′,p)+
(2π)3
(2.20). In operator notation, we have from (2.20) Z
V(p′,p′′) g(p′′;W)T(p′′,p) (2.27)
×
with the standard Green function
φ++ = φ(+0+) +E2(+) Kirr φ++ g(p;W)= (2π1)3 Λa+(p)Λb+(−p) p2 MpN2+iδ . (2.28)
i −
= φ(0) +E(+) Kirr +Kirr E(+) Kirr +.... φ(0)
++ 2 2 ++ The corrections to the approximation E(+) g(p;W)
(cid:16) (cid:17) are of order 1/M2, which we neglect henc2forth≈.
≡ 1+E2(+) M φ(+0+) (2.23) The transition from Dirac-spinors to Pauli-spinors, is
(cid:16) (cid:17) given in Appendix C of [12], where we write for the
the Lippmann-Schwinger equation in the 4-dimensional
Pauli-spinor space
which implies for the complete kernel M the integral
d3p′′
equation (p′,p) = (p′,p)+
T V (2π)3
Z
(p′,p′′) g˜(p′′;W) (p′′,p) .(2.29)
×V T
The -operator in Pauli spinor-space is defined by
M(p′,pW) = Kirr(p′,pW)+ d3p′′ Kirr(p′,p′′ W) T
| | (2π)3 | · χ(a)†χ(b)† (p′,p)χ(a)χ(b) =u¯ (p,σ )u¯ ( p,σ )
×E2(+)(p′′;W)MZ(p′′,p|W) (2.24) ×σTa(p,σpb′′)Tua(p′′,σa′σ)ua′b(σ−b′p′′,σab′) . a b − (2b.30)
5
and similarly for the -operator. Like in the deriva-
V
p’ -p’ p’ -p’ tionofthe OBE-potentials[20,21]we makeoff-shelland
on-shell the approximation, E(p) = M + p2/2M and
W = 2 p2+M2 = 2M + p2/M , everywhere in the
-p’’ i i
k -p’’’ interaction kernels, which, of course, is fully justified for
2 p
low energies only. In contrast to these kind of approx-
k2 imations, of course the full k2-dependence of the form
factors is kept throughout the derivation of the TME.
k Notice that the Gaussian form factors suppress the high
1 p’’
p’’
k1 momentum transfers strongly. This means that the con-
tributiontothepotentialsfromintermediatestateswhich
p -p p -p are far off-energy-shell can not be very large.
Because of rotational invariance and parity conserva-
( a ) ( b )
tion,theT-matrix,whichisa4 4-matrixinPauli-spinor
×
space,canbeexpandedintothefollowingsetofingeneral
p’ -p’ p’ -p’
8 spinor invariants,see for example [22, 23]. Introducing
[24]
k
2
1
k2 -p’’’ q= (p +p ) , k=p p , n=p p , (2.31)
f i f i i f
2 − ×
p’’ -p’’’
p’’ with, of course, n = q k, we choose for the operators
×
P in spin-space
k k j
1 1
P =1 P =σ σ
1 2 1 2
p -p p -p ·
P =(σ k)(σ k) 1(σ σ )k2 P = i(σ +σ ) n
3 1· 2· − 3 1· 2 4 2 1 2 ·
( c ) ( d )
P =(σ n)(σ n) P = i(σ σ ) n
5 1· 2· 6 2 1− 2 ·
P =(σ q)(σ k)+(σ k)(σ q)
FIG.1: BWtwo-meson-exchangegraphs: (a)planarand(b)– 7 1· 2· 1· 2·
(d) crossed box. The dashed line with momentum k1 refers P8 =(σ1 q)(σ2 k) (σ1 k)(σ2 q)
to thepion and thedashed line with momentum k refers to · · − · ·
2 (2.32)
one of the other (vector, scalar, or pseudoscalar) mesons. To
these we have to add the “mirror” graphs, and the graphs
Here we follow [23], where in contrast to [20], we have
where we interchange thetwo meson lines.
chosen P to be a purely ‘tensor-force’ operator. The
3
expansion in spinor-invariants reads
p’ -p’ p’ -p’
8
(p ,p )= T (p2,p2,p p ) P (p ,p ) . (2.33)
k k T f i j f i f · i j f i
2 2
j=1
X
e
-p’’
Similarly to (2.33) we expand the potentials V. Again
p’’ -p’’ following [23], we neglect the potential forms P and P ,
7 8
p’’ andalsothedependenceofthepotentialsonk q. Then,
·
the expansion (2.33) reads for the potentials as follows
k k
1 1
4
p -p p -p = V˜ (k2,q2) P (k,q) . (2.34)
j j
V
j=1
( a ) ( b ) X
We develop in the following subsections a new repre-
FIG.2: Planar-boxTMOtwo-meson-exchangegraphs. Same sentation of the TME-potentials. Included are the BW-
notation as in Fig. 1. To these we have to add the “mirror” graphs shown in Fig. 1, and the TMO-graphs shown in
graphs, and the graphs where we interchange the two meson Fig.2. Alsotheemployednotationsforthemomentaare
lines. indicated in the figures.
6
III. MOMENTUM SPACE REPRESENTATION TME-POTENTIALS
In this section we give an outline of the procedures in making the partial wave analysis of TME-potentials in
momentum space.
A. TPS Central Potentials
Consider the basic integral
d3k d3k d3∆
V˜(k) = (21π)3 2 δ(k−k1−k2) F˜(k21)G˜(k22)= (2π)3 F˜ ∆2 ·G˜ (k−∆)2 , (3.1)
Z Z Z
(cid:0) (cid:1) (cid:0) (cid:1)
with F˜(k ) and G˜(k ) having the generic soft-core forms
1 2
exp k2/Λ2 exp k2/Λ2
F˜(k2)= − 1 , G˜(k2)= − 2 . (3.2)
k2+m2 k2+m2
(cid:2) 1 (cid:3) (cid:2) 2 (cid:3)
Exploiting the identity
exp −k2/Λ2 =em2/Λ2 ∞ dt exp k2+m2 t (3.3)
k2+m2 Λ2 − Λ2
(cid:2) (cid:3) Z1 (cid:20) (cid:18) (cid:19) (cid:21)
one can write (3.1) as
V˜(k) = em21/Λ21em22/Λ22 ∞ dt ∞ du e−(m21/Λ21)te−(m22/Λ22)u
Λ2 Λ2 ·
Z1 1 Z1 2
d3∆ t u u
exp + ∆2+ 2k ∆ k2 . (3.4)
× (2π)3 − Λ2 Λ2 · − Λ2
Z (cid:20) (cid:18) 1 2(cid:19) 2(cid:21)
(cid:0) (cid:1)
The ∆-integral in (3.4) has been evaluated above, see (D12), and is
tu/Λ2Λ2 t u
(4πa)−3/2 exp 1 2 k2 , with a= + .
−(t/Λ2+u/Λ2) Λ2 Λ2
(cid:20) 1 2 (cid:21) 1 2
Redefining the variables t t/Λ2 and u u/Λ2, we rewrite (3.4) in the form
→ 1 → 2
V˜(k) = (4π)−3/2 em21/Λ21em22/Λ22 ∞dt ∞du exp[−(m21t+m22u)]
(t+u)3/2 ·
Zt0 Zu0
tu
exp k2 (t =1/Λ2, u =1/Λ2) . (3.5)
× − t+u 0 1 0 2
(cid:20) (cid:18) (cid:19) (cid:21)
This form we consider as the basic representation of the soft-core TME-potentials in momentum space.
B. TPS Tensor and Spin-spin Potentials
In this subsection we apply the same technique as used in the previous one to the more complicated case involving
the tensor interaction. For ps-ps in momentum space we have
d3k d3k
V˜(k) = (21π)3 2 δ(k−k1−k2) F˜(k21)G˜(k22)·[σ1·k1×k2][σ2·k1×k2]
Z Z
d3∆
= [σ ∆ k][σ ∆ k]F˜ ∆2 G˜ (k ∆)2 , (3.6)
(2π)3 1· × 2· × · −
Z
(cid:0) (cid:1) (cid:0) (cid:1)
which leads to the basic integral to be evaluated:
d3∆
V˜ (k) = ∆ ∆ F˜ ∆2 G˜ (k ∆)2 . (3.7)
mn (2π)3 m n · −
Z
(cid:0) (cid:1) (cid:0) (cid:1)
7
Repeating the steps taken in the previous subsection, one arrives at the analogue of (3.4):
V˜ (k) = em21/Λ21em22/Λ22 ∞ dt ∞ du e−(m21/Λ21)te−(m22/Λ22)u
mn Λ2 Λ2 ·
Z1 1 Z1 2
d3∆ t u u
∆ ∆ exp + ∆2+ 2k ∆ k2 . (3.8)
× (2π)3 m n − Λ2 Λ2 · − Λ2
Z (cid:20) (cid:18) 1 2(cid:19) 2(cid:21)
(cid:0) (cid:1)
Using the integral
d3∆ 1 1 b2 b2
∆ ∆ exp a∆2+2bk ∆ =(4πa)−3/2 δ + k k exp + k2
(2π)3 m n − · a 2 mn a m n a
Z (cid:20) (cid:21) (cid:20) (cid:21)
(cid:2) (cid:3)
one obtains, again after the redefinitions t t/Λ2 and u u/Λ2, for V˜ the expression
→ 1 → 2 mn
V˜ (k) = (4π)−3/2em21/Λ21em22/Λ22 ∞dt ∞du exp[−(m21t+m22u)]
mn (t+u)5/2 ·
Zt0 Zu0
1 u2 tu
δ + k k exp k2 . (3.9)
mn m n
× 2 (t+u) − t+u
(cid:20) (cid:21) (cid:20) (cid:18) (cid:19) (cid:21)
It is clear that in (3.6) only the δ -term contributes. The result is
mn
1 2
V˜(k) = (σ k) (σ k) (σ σ ) k2 H˜(k2)+ (σ σ ) k2 H˜(k2) (3.10)
1 2 1 2 1 2
− · · − 3 · 3 ·
(cid:20) (cid:21)
with
H˜(k2) = 1(4π)−3/2em21/Λ21em22/Λ22 ∞ dt ∞ du exp[−(m21t+m22u)]
2 (t+u)5/2 ·
Zt0 Zu0
tu
exp k2 . (3.11)
× − t+u
(cid:20) (cid:18) (cid:19) (cid:21)
From equations (3.10)-(3.11) one can read off immediately the projection of (3.6) onto the spinor invariants (2.33).
In Appendix D the same result is derived in an alternative way starting from the configuration space potentials.
IV. GENERAL PS-PS EXCHANGE POTENTIALS
In [2]) the derivation of the ps-ps exchange potentials both in momentum and in configuration space is given. In
this reference the configuration space potentials are worked out fully. The topic of this paper is to do the same for
the momentum space description. In particular, the partial wave analysis is performed leading to a representation
which is very suitable for numerical evaluation.
From [2] it appears that the momentum space TME-potential can be represented in general in the form
2 ∞ N d3∆
V (k) = dλ w (λ) O (k,∆) F(i)(∆2,λ) F(i)((k ∆)2,λ) . (4.1)
αβ π i (2π)3 αβ α β −
Z0 i=1 Z
X
e e e e
Here, stands for the number of different types of λ-functions, and w (λ) are the corresponding weights. The
i i
operators O are for example in the case of ps-ps exchange given by, see [2] Table II,
P αβ
O//(k,∆) = 2(∆ (k ∆))2 2[σ ∆ k][σ ∆ k] ,
e αβ · − − 1· × 2· ×
OX (k,∆) = 2(∆ (k ∆))2+2[σ ∆ k][σ ∆ k] . (4.2)
eαβ · − 1· × 2· ×
In [25] we remind the reader of the precise contents of these operators. As an example for the we give explicitly
e i
the case of πη-exchange, the planar BW- and TMO-graphs,and the BW-crossedgraphs. These can be written as [2]
P
2 ∞ dλ d3∆
V//,X(k) = O//,X(k,∆) F (∆2,m ) F ((k ∆)2,m )
πη ±π λ2 (2π)3 πη π π η − η
Z0 Z h
e −Fπ(∆2, m2π+λ2) Feη((k−∆)2, em2η+λ2) e−eλ2/Λ2πe−λ2/Λ2η . (4.3)
p q i
Here, the (+)-sign refers to the paerallel graphs (//), aend the ( )-sign to the crossed (X) graphs.
−
8
A. Integration over λ-parameter
Since, due to the use of (3.3) and gaussian form factors all integrations over λ become gaussian, they can be
performed analytically. Consider for example, the type of integral appearing in (4.3). We write
2 ∞ dλ d3∆
V (k) = F (∆2,m ) F ((k ∆)2,m )
αβ π λ2 (2π)3 α 1 β − 2
Z0 Z h
e F (∆2, m2+λ2) Fe ((k ∆)2,e m2+λ2) e−λ2/Λ21e−λ2/Λ22 . (4.4)
− α 1 β − 2
q q (cid:21)
Goingthroughthesamestepsasbeetweenequations(3.e3)-(3.5),wegetforthesecondpartbetweenthesquarebrackets,
an extra λ-dependent factor
e−(λ2/Λ21)te−(λ2/Λ22)u e−λ2(t+u) ,
→
where we applied again the same variable redefinition as before, i.e. t t/Λ2 and u u/Λ2. Then, the needed
→ 1 → 2
λ-integral in (4.4) is
2 ∞ dλ 1 e−(t+u)λ2 = 2 √t+u . (4.5)
π λ2 − √π
Z0 h i
Therefore (4.4) becomes, compare with (3.1)-(3.5),
V˜(k) = (2π)−2 em21/Λ21em22/Λ22 ∞dt ∞du exp[−(m21t+m22u)]
(t+u) ·
Zt0 Zu0
tu
exp k2 (t =1/Λ2, u =1/Λ2) . (4.6)
× − t+u 0 1 0 2
(cid:20) (cid:18) (cid:19) (cid:21)
Similarly, all λ-integrations can be performed explicitly.
V. PROJECTION PS-PS EXCHANGE ON SPINOR INVARIANTS
The different contributions, as adiabatic, non-adiabatic, pseudovector-vertex corrections, and off-shell potentials,
are in direct correspondence with their configuration space analogs, given in [2].
A. Adiabatic PS-PS Exchange Potentials
We now have prepared all the necessary ingredients for putting things together and to list the projection of ps-ps-
exchange onto the potentials V . We write
j
V˜(0)(k) = (4π)−3/2 em21/Λ21em22/Λ22 ∞dt ∞du exp[−(m21t+m22u)]
j (t+u)3/2 ·
Zt0 Zu0
tu
exp k2 Ω(0)(k2;t,u) , (5.1)
× − t+u j
(cid:20) (cid:18) (cid:19) (cid:21)
where
2 2
2 f f 15
Ω(0,//)(k2;t,u) = C(I) NNπ NNη +
1 √π NN,// m m · 4
(cid:18) π (cid:19) (cid:18) π (cid:19) (cid:26)
1 t2 8ut+u2 t2u2 √t+u
+ − k2+ k4 ,
2 t+u (t+u)2 · (t+u)2
(cid:18) (cid:19) (cid:18) (cid:19) (cid:27)
2 f 2 f 2 1 √t+u
Ω(0,//)(k2;t,u) = C(I) NNπ NNη k2 ,
2 −√π NN,// m m · 3 · (t+u)
(cid:18) π (cid:19) (cid:18) π (cid:19)
2 f 2 f 2 1√t+u
Ω(0,//)(k2;t,u) = C(I) NNπ NNη , (5.2)
3 √π NN,// m m · 2(t+u)
(cid:18) π (cid:19) (cid:18) π (cid:19)
9
where C (I) denotes the isospin factor.
NN,//
Expressionssimilarto (5.2)holdforthe contributionofthe crossedBW-graphs(X), andaregivenbelow. Note the
difference in the O˜-operators in (4.2) and D (ω ,ω )= D (ω ,ω ).
X 1 2 // 1 2
−
2 2
2 f f 15
Ω(0,X)(k2;t,u) = C(I) NNπ NNη +
1 −√π NN,X m m · 4
(cid:18) π (cid:19) (cid:18) π (cid:19) (cid:26)
1 t2 8ut+u2 t2u2 √t+u
+ − k2+ k4 ,
2 t+u (t+u)2 · (t+u)2
(cid:18) (cid:19) (cid:18) (cid:19) (cid:27)
2 f 2 f 2 1 √t+u
Ω(0,X)(k2;t,u) = C(I) NNπ NNη k2 ,
2 −√π NN,X m m · 3 · (t+u)
(cid:18) π (cid:19) (cid:18) π (cid:19)
2 f 2 f 2 1√t+u
Ω(0,X)(k2;t,u) = C(I) NNπ NNη . (5.3)
3 √π NN,X m m · 2(t+u)
(cid:18) π (cid:19) (cid:18) π (cid:19)
It is convenient to introduce the expansions
Ω(//,X)(k2;t,u)= 2 C(I) (//,X) fNNπ 2 fNNη 2 K Υ(0),1a,1b,1c(t,u) k2 k . (5.4)
j √π NN m m · j,k
(cid:18) π (cid:19) (cid:18) π (cid:19) k=0
X (cid:0) (cid:1)
The functions Υ (t,u) are given in Table I-II.
j,k
B. Non-adiabatic corrections
The denominators D(1)(ω ,ω ) for the non-adiabatic corrections are given in Ref. [2], Table III. In appendix B of
1 2
[12] it is explained how 1/ω4 has to be treated. It follows that
1 ∂ 1 ∂ 1 1
D(1)(ω ,ω ) + (5.5)
// 1 2 ⇒−2 ∂m2 − Λ2 ∂m2 − Λ2 ω2ω2
(cid:18) 1 1 2 2(cid:19) 1 2
and one finds from (3.4) and (5.5) that the contributions to the V˜(1) can be written in the form
j
V˜(1a)(k)=(4π)−3/2 em21/Λ21em22/Λ22 ∞dt ∞du exp[−(m21t+m22u)]
j (t+u)3/2 ·
Zt0 Zu0
1 tu
(t+u) exp k2 Ω(1a)(k2;t,u) . (5.6)
×2 · − t+u j
(cid:20) (cid:18) (cid:19) (cid:21)
The Ω(1a) are given by
j
2 2
f f 1 105
Ω(1a,//)(k2;t,u)=C(I) NNπ NNη +
1 NN,// m m M · − 8
(cid:18) π (cid:19) (cid:18) π (cid:19) (cid:18) (cid:19) (cid:26)
15 t2 5ut+u2 3 t2 5ut+u2 t3u3 1
− k2+ tu − k4+ k6 ,
− 4 t+u 2 (t+u)2 (t+u)3 · (t+u)3
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:27)
f 2 f 2 1 2 5 1 tu 1
Ω(1a,//)(k2;t,u)=C(I) NNπ NNη k2 + k2 ,
2 NN,// m m M ·−3 −4 2t+u · (t+u)2
(cid:18) π (cid:19) (cid:18) π (cid:19) (cid:18) (cid:19) (cid:26) (cid:27)
f 2 f 2 1 5 1 tu 1
Ω(1a,//)(k2;t,u)=C(I) NNπ NNη k2 ,
3 NN,// m m M ·− 4 − 2t+u · (t+u)2
(cid:18) π (cid:19) (cid:18) π (cid:19) (cid:18) (cid:19) (cid:26) (cid:27)
f 2 f 2 1 tu 1
Ω(1a,//)(k2;t,u)=C(I) NNπ NNη 5 2 k2 . (5.7)
4 NN,// m m M ·− − t+u · (t+u)2
(cid:18) π (cid:19) (cid:18) π (cid:19) (cid:18) (cid:19) (cid:26) (cid:27)
10
Again, similar expressions hold for the contribution of the crossed BW-graphs (X), and are given below. See for the
differences in the O˜(1)-operators [2] equations (5.3) and (5.4), and D(1)(ω ,ω )= 2D(1)(ω ,ω ), cfrm. Table III of
X 1 2 − // 1 2
[2].
2 2
f f 1 105
Ω(1a,X)(k2;t,u)= 2C(I) NNπ NNη +
1 − NN,X m m M · − 8
(cid:18) π (cid:19) (cid:18) π (cid:19) (cid:18) (cid:19) (cid:26)
15 t2 5ut+u2 3 t2 5ut+u2 t3u3 1
− k2+ tu − k4+ k6 ,
− 4 t+u 2 (t+u)2 (t+u)3 · (t+u)3
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:27)
2 2
f f 1 2 5 1 tu 1
Ω(1a,X)(k2;t,u)=+2C(I) NNπ NNη k2 + k2 ,
2 NN,X m m M ·−3 −4 2t+u · (t+u)2
(cid:18) π (cid:19) (cid:18) π (cid:19) (cid:18) (cid:19) (cid:26) (cid:27)
2 2
f f 1 5 1 tu 1
Ω(1a<X)(k2;t,u)=+2C(I) NNπ NNη k2 . (5.8)
3 NN,X m m M ·− 4 − 2t+u · (t+u)2
(cid:18) π (cid:19) (cid:18) π (cid:19) (cid:18) (cid:19) (cid:26) (cid:27)
C. Pseudovector-vertex corrections
Here, only the crossed BW-graphs contribute [2]. From the denominators D(1)(ω ,ω ) = 1/ω2ω2 for the
1 2 1 2
pseudovector-vertexcorrections [2] one finds that the contributions to the V˜(1) can be written in the form
j
V˜(1b)(k) = (4π)−3/2 em21/Λ21em22/Λ22 ∞dt ∞du exp[−(m21t+m22u)]
j (t+u)3/2 ·
Zt0 Zu0
tu
exp k2 Ω(1b)(k2;t,u) , (5.9)
× − t+u j
(cid:20) (cid:18) (cid:19) (cid:21)
where
2 2
f f 1 15
Ω(1b,X)(k2;t,u) = C(I) NNπ NNη +
1 NN,X m m M · − 2
(cid:18) π (cid:19) (cid:18) π (cid:19) (cid:18) (cid:19) (cid:26)
5 t2 2tu+u2 t2+u2 1
− k2+tu k4 ,
−2 t+u (t+u)2 · (t+u)2
(cid:18) (cid:19) (cid:18) (cid:19) (cid:27)
f 2 f 2 1 tu 1
Ω(1b,X)(k2;t,u) = C(I) NNπ NNη 1+2 k2 . (5.10)
4 NN,X m m M − t+u · t+u
(cid:18) π (cid:19) (cid:18) π (cid:19) (cid:18) (cid:19)(cid:26) (cid:18) (cid:19) (cid:27)
D. Off-shell corrections in TMO-diagrams
Also in this case the denominators are D(1)(ω ,ω )=1/ω2ω2 [2], and one finds that the contributions to the V˜(1)
1 2 1 2 j
can be written as
V˜(1c)(k) = (4π)−3/2 em21/Λ21em22/Λ22 ∞dt ∞du exp[−(m21t+m22u)]
j (t+u)3/2 ·
Zt0 Zu0
tu
exp k2 Ω(1c)(k2;t,u) . (5.11)
× − t+u j
(cid:20) (cid:18) (cid:19) (cid:21)
The potentials are very similar to the pseudovector-vertexcorrections given above, see [2], section V.C. We have