NUCLEAR
PHYSICS A
Nuclear Physics A.563 (1993) 549-583
North-Holland
The Bethe-Salpeter
relation technique
equation and the dispersion
V.V. Anisovich
Petersburg Nuclear Physics Institute, Gatchina, 188350 St. Petersburg, Russian Federation
D.I. Melikhov
Safety Institute, Russian Academy of Sciences, Moscow, Russian Federation
Nuclear
B.Ch. Metsch,
H.R. Petry
Institut fiir 7’heoretische Kernphysik, Universitiit Bonn, Bonn, Germany
Received
7 April
1993
Abstract
We compare results for two-particle
scattering
amplitudes
and for composite
particle wave
functions and form factors obtained in the framework of a dispersion relation approach with the
corresponding
solutions of the BS equation.
1. Introduction
The development
of a covariant technique
is an important
description
of composite particles and for scattering processes
issue both for the
at low and interme-
diate energies. Relativistic effects are important
even for nuclei, the weakly bound
systems of nucleons. They should play a crucial role in highly excited meson and
baryon states considered
as composite systems of quarks.
Here we discuss the Bethe-Salpeter
CBS) equation [ll, which is widely used for
scattering processes and bound systems (see, for example, recent papers [2-41 and
references therein), and compare it with a treatment
of the same amplitudes based
on a dispersion
relation.
The dispersion-relation
method
has some technical
advantages
which allow one to perform calculations
in a comparatively
simple
form. For example, there are no problems with mass-of-shell
amplitudes.
So, it
seems important
to clarify the common points of these approaches
as well as to
stress the differences.
We show that the dispersion-relation
expression
for the scattering
amplitude
may be obtained as a solution of the BS equation with separable interaction
of the
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550
V. V. Anisovich et al. / Bethe - Salpeter equation
special form. For a composite system interacting with the external field, the form
factor obtained within the standard light-cone dynamics is identical to that of the
dispersion-relation
approach for spinless constituents. For a more complicated
case of the spin constituents these two approaches give the same results up to
some subtraction terms.
The paper is organized as follows. In the next section we consider the BS
equation within the conventional Feynman diagram technique. We discuss the
non-homogeneous and homogeneous equations and the connection between them.
In sect. 3 we consider the same issues in the framework of the dispersion-relation
technique and compare both approaches for the scattering amplitudes. A composite system description in the infinite momentum frame (IMF) is considered in
detail both for the scattering amplitude and the amplitude of the interaction with
an external field (sect. 4). We discuss the dependence of the integral representation for various amplitudes including form factors on the particular choice of the
reference frame. A brief summary is given in sect. 5. Appendices contain the
application of the dispersion technique to fermion-fermion
scattering diagrams.
2. Bethe-Salpeter
equations
The non-homogeneous
A(P;,
p;;
PI,
P2)
in the momentum
representation
BS equation in momentum representation
= v(P;,
Pii
reads
d4k, d4k,
Pl, Pz) +/
i(2Tj4
api,
pi;
klk2)
cS4(k,+k,-P)
x ( m2
_
k;
_
iO)(
m2
_
k2’
_
io)
‘(
k17
k2’
‘1’
‘2)
(2.1)
or in graphical form:
Here the momenta of the constituents obey the momentum conservation
Pl
+P2
=P;
+p;
=p,
law
(2.3)
551
K K Anisouichet al. / Bethe- Salpeterequation
~(p,, pz; k,, k,) is a two-constituent
kl
V( PI, P2;
k,y k2) = k
irreducible kernel.
Pl
a
For example, it can be a kernel induced by a meson-exchange
g2
pcL2
- (k,
In this case
k1
-pJ2
V(p,,
= k2
~2;
I
(2.4)
P2
2
interaction:
Pl
P2
k,, k2) is an infinite sum of two-particle
irreducible graphs
We would like to stress that the amplitude A, determined by the BS equation is
the off-shell amplitude. Even if we set pf =p;‘=p$
=pi2 = m2 in the l.h.s., Eq.
(2.0, the r.h.s. contains the amplitude A(k;, k,; pi, pi) for kf # m2, k; # m2.
Restricting to one-meson exchange in the irreducible kernel I/, but iterating Eq.
(2.1) we come to infinite series of ladder diagrams:
(-j = I + a+m+.*.
Let
these
states
of the
(2.6)
us investigate the intermediate states in these ladder diagrams. First of all,
diagrams have two-particle intermediate states which can appear as real
at c.m. energies 6 > 3m (here s = (Q, +p2j2). it corresponds to the cutting
ladder diagrams across constituent lines:
m-mtE
(2.7)
Such a two-particle state manifests itself as singularity of the scattering amplitude at s = 4m2. However, the amplitude A considered as a function of s has not
only this singularity but also an infinite set of singularities which correspond to the
ladder diagram cuts across meson lines of the type:
(2.8)
K V. Ankouich et al. / Bethe- Salpeterequation
552
The diagrams which appear after this cut procedure
diagrams, e.g., one-meson production diagrams:
are meson production
F+IlYF
(2.9)
Hence, the amplitude A(p;, pi;
complex s-plane (s = (pl +p,)*):
s=4m2
pl,
p2)
has the following cut singularities in the
(2.10)
9
which is related to the rescattering process, Other singularities are related to the
meson production processes with cuts starting at
s = (2m +
np)*;
(2.11)
n = 1, 2, 3,. . . .
The four-point amplitude depends on six variables
2
2
12
P1,P*,P1,P*,
The
I2
s = (PI
+P*)*
= (Pi
t = ( p1
-pi)’
= (P2 -Ph)*.
seventh
variable,
+p$,
u = (p,
-pi>”
(2.12)
= (pi
-p2)*,
is not independent
because Of
the relation
s + t + u =p:
+p;
+p;*
(2.13)
+p;*.
It is possible to decrease the number of variables in Eq. (2.1) if we consider the
amplitude with definite angular momentum. For this purpose we consider Eq. (2.1)
in the c.m.s. of particles 1 and 2 and expand the amplitude A(p;, pi; p,, p2> as
well as the interaction term over the angular momentum states
(L’M’IA(p;,
p;;
PI, P*)ILM)=AL(S;
(L’M’(v(p;,
p;;
PI, p*)ILM)=
I/L@;
P:,
P:,
Pi,
Pi’9 P;*)b%iA4~~
P227 Pi’,
P;*)%Lh.f’~
(2.14)
I/V. Anisovich
et al. / Bethe - Salpeter equation
553
Fig. 1. Ladder diagram of the mass-on-shell
scattering
amplitude
and the inner block which is the
subject of consideration
in Eq. (2.2). (b) Pole diagram which corresponds
to composite particle and the
vertices of the transition “composite
particle + constituents”.
For spinless particles the states I LM) are spherical harmonics Y&0,
get for the amplitude A, the equation
A,(?
cp>.We thus
Pi”, Ph2, Pi, P22)
= qs;
+
x
Pi’,
Pl;‘, Pf, 1);)
d4k
yvL(s;
P;~, P;‘,
/ i(277)
(m’
_
k:
_
iO)(m2
_
k2’
k:,
_
k:)
iO)A,(s’kt’ k22’
p:’ ‘2”).
(2.15)
Here k = k,, k, = P - k and P =pl +pz = (6, 0, 0, 0).
If a bound state of the constituents exist, the scattering partial amplitude has a
pole at s = (pl +p2)2 = M2, where M is the mass of the bound state. This pole
appears both in the on- and off-shell scattering amplitudes. This means that the
infinite sum of diagrams of Fig. la type may be rewritten as a pole term of Fig. lb
plus some regular terms at s = M2.
The left and right blocks in Fig. lb x(pl,
p2; P) and x(p;, ~5; P) satisfy the
homogeneous BS equation
d4k, d4k2
X(P*,
P2i
PI
= /
i(2T)4
VP,,
P2;
kl,
k2)
a4(k, + k, - P)
x (m2 _ kf _ iO)(m2 _ kz _ i()) x(k17
k2’ “’
(2.16)
whose graphical form is
(2.17)
5.54
K V. Anisouich et al. / Bethe - Salpeter equation
n iterations give
(2.18)
The same cutting procedure of the interaction block in the r.h.s. of Eq. (2.18)
shows US that the amplitude x(pi, p2; P) contains all the singularities of the
amplitude A in Eqs. (2.10) and (2.11).
The three-point amplitude x(pi, pz; P> depends on three variables
P2 (or s), P:, ~22,
(2.19)
and again as in the case of the scattering amplitude A the BS equation contains
the mass-of-shell amplitude x(k,, k,; P). x is a solution of the homogeneous
equation, hence normalization condition should be added independently.
The connection between x and A in the pole has a sense of such a normalization condition. It is convenient to normalize x as follows. For P2 --) M2
A( PI,
P2i
Pi, PiI =
x( PI,
P2;
P)x(P;
Pi, PiI
P2-MM2
+ regular terms.
(2.20)
In the formulation of the scattering theory we start from the set of asymptotic
states, containing constituent particles (with mass m) and mesons (with mass CL)
only. We do not include in such a formulation of the scattering theory composite
particles as asymptotic states; we simply cannot know beforehand whether such
bound states exist or not. But if we consider processes of the production of new
constituent-anticonstituent
outgoing pairs which can form bound states, the latter
should be included into the asymptotic state set.
3. Scattering
amplitude
in the dispersion
relation
N/D
method
Now let us consider the scattering amplitude for real particles:
(3.1)
It depends on two variables, s and t. We denote it as A(s, 2). The amplitude,
considered as a function of s, has the singularities given by Eqs. (2.10) and (2.11):
the singularities which are related to s-channel scattering and meson production
processes. We consider the case with a bound state so there is a pole singularity at
s=M2.
V. V. Anisouich et al. / Bethe - Salpeter equation
Fig. 2. The singularities
of partial
amplitudes
in the complex
S-plane
5.55
and related
processes.
The ladder diagrams of the Eq. (2.6) type have t-channel singularities at
t =
(n/_g;
n= 1,2,3...,
(3.2)
These singularities correspond to one-meson exchange, two-meson exchange and
so on.
In the N/D method we deal with partia1 wave amplitudes.
Partial wave amplitudes in the s-channel depend only on s. They have all the
s-channel singularities of A(s, t): right-hand singularities at s = M2, s = 4m2,
s = (2m + p), . . . of Fig. 2.
Left-hand singularities of the partial amplitudes are connected with t-channel
singularities of A(s, t). The S-wave partial amplitude is equal to
where t(z)
spond to
= -
2($ - m2X1 - z>, and z = cos 8. Left-hand
t( z = - 1) = (n&.
singularities
corre-
(3.4)
and are located at s = 4m2 - p2, s = 4m2 - 4~’ and so on.
The dispersion relation N/D method provides the possibility to construct the
relativistic two-particle partial amplitude in the region of low and intermediate
energies.
Let us restrict ourselves to the consideration of the region in the vicinity of
s = 4m2. Here the unitarity condition gives us for the partial wave amplitude (we
consider an S-wave amplitude as an example):
Im A(s) =P(s)IA(.s)]*.
(3.5)
VTV. Anisovich et al. / Bethe - Salpeter equation
556
Here p(s) is the two-particle phase space integrated at fixed s:
p(s) = $/d@,(P;
k,, kz) = $
d@,( P; k,, k2) = (2r)464(
In the N/D
A(s)
s-4m2
T
~
P - k, - k2)
s
(3.6)
’
d3k2
d3k,
(2rr)32k,,,
method the amplitude A(s) is represented
(2a)32k2,
’
as
N(s)
D(s) .
(3.7)
= -
Here N(s) has only left-hand singularities whereas D(s) has only right-hand
singularities. So the N-function is real in the physical region s > 4m2. The
unitarity condition can be rewritten as
Im D(s)
= -p(s)N(s).
(3.8)
The solution of this equation is
m
D(s)=l-1
ds’ ,o(S)N(?)
s_s
(3%
=1-B(s).
4mZ %-
In Eq. (3.9) we neglect the so-called CDD-poles
condition: D(s) + 1 as s -+ m.
Let us introduce the vertex function
[6] and normalize
N(s) by the
(3.10)
G(s) = ,/m.
We assume here that N(s) is positive (the cases with negative N(s) or with
changing-sign N(s) need a special and more cumbersome treatment). Then the
partial wave amplitude A(s) can be expanded in a series
A(s) =G(s)[l
+B(s)
Its graphic interpretation
+B’(s)
+B3(s)
+ . ..]G(s).
(3.11)
is
(3.12)
B(s) is a loop-diagram
B(s) =
0
(3.13)
so the terms in Eq. (3.11) are amplitudes with different numbers of rescatterings.
557
I/ VI Anisouich et al. / Bethe - Safpeter equation
This series is analogous to the series of the ladder diagrams given in Eq. (2.6) for
the Bethe-Salpeter
amplitude.
Loop diagram B(s) plays the central role for the whole dispersion amplitude
and we compare in detail dispersion and Feynman expressions for B(s).
Namely, a Feynman expression B,(s) with a special choice of separable interaction G(4K2 + 4m2) is shown to be equal to the dispersion integral representation
where the four-vector K is defined as
k;-k;
-Pp
P2
2K=k,-kZ-
K2=
k = k,.
$(Pk)“-k’;
(3.14)
The Feynman expression reads
1
BF( P’) = (2~)~i
dk G2(4( Pk)2/P2
(3.15)
/ (mZ-k2-iO)(mZ-(P-k)2-iO)‘
Since a composite system is conveniently
they will be used hereafter:
k_=g(k,-k,);
- 4k2 + 4m2)
handled in the light-cone coordinates,
k+=f$(k,+k&
k2=2k+k_-m;;
m+=m2+k;.
(3.16)
We choose the reference
Pk = P+k_+
frame in which PT = 0. Then
P-k,
(3.17)
and Eq. (3.15) takes the form
dk,
“/(
dk_
2k+k_-m~+iO)(M2-2(P+k_+P_k+)
d2k,
+2k+k_-m$+iO)’
(3.18)
If G = 1 we could have performed the integration
integration contour around the pole
k_=
m%- i0
2k
+
over k-right now, closing the
to obtain the standard
(x = k+/P+):
1
(244i
representation
for a Feynman
loop graph
ds
dx dk;
dx d*k, ( -2ri)
1
_i0
dispersion
= I a(s-W--o)
2x(M2-(m~/x+M2X)+iO)
x
qs-4/w
--m)
167r
= m
I x(1-x)
ds -4s)
I4m2s-(s-M2-iO).
(3.19)
The dispersion integral (3.19) is divergent at s --f 00since we have set G = 1, and
it is G that provides the convergence of B, in Eq. (3.15). Convergence of the
integral (3.19) can be restored by a subtraction procedure. Variable x changes
from 0 to 1, because for x < 0 and x > 1 both poles in k_ are located on the same
side of the integration contour and the integral equals to zero.
For G # 1 some additional steps are required to obtain the dispersion representation, namely, we introduce new variables 5, and [_
P+k_+
P_k,=M~,,
P,k_-
P_k+=hf_.
(3.20)
In terms of these variables Eq. (3.15) takes the form
1
UP21
=
(2T)4i
G*(4([%
x i (5$5”-
m$))
rn+ + iO)(N2
= im2 d{_ n- dk; G2(4(52+
d.$+ d&_ d’k,
- 2M5++
St-
52-
rnc + i0)
113%))
0
x-l(s:-(t1
/
d5+
+m;) +iO)((c+-M)*-
((2+m$) +iO).
(3.21)
The integration over 5, is performed, closing
_-_.- the integration contour in the upper
semi-plane, and two poles .$+= - fi
mt + i0 and 5, = M - /m
+ i0
contribute. The result of integration over 5, is
2rri
1,‘m
(4(5?+77+)
(3.22)
-BP).
V. K Anisouich
Introduction
of the new variable
et al. / Bethe - Salpeter equation
559
s = 4(5! + m$> yields
(3.23)
that is just the dispersion
representation
(3.9).
The separable interaction
G(4K2 + 4m2) has a rather specific form: the vertices
in the c.m. system depend only on the space components
of momenta.
Since we
have demonstrated
that the Feynman diagram calculus with separable interactions
and the dispersion-relation
technique
gives the same expressions
for the loop
diagram, it yields that the BS equation with the separable kernel gives the same
amplitude
as N/D
dispersion
method. Hence, the BS equation
with separable
kernel takes into account
only two-particle
intermediate
states. This is quite
opposite to the case of the Yukawa-type
interaction,
caused by t-channel exchange
of a particle with mass F, g2/(p2 - (k, - /c;)~>, where all intermediate
states
contribute
to the scattering amplitude.
Let us write the unitarity condition
for the scattering amplitude
A, defined as
an infinite sum of loop diagrams
A&
= s)(5 tsp+
Im A(s)
+sm
f-..
(3.24)
=IA(s)12p(.r).
In the 1.h.s. of Eq. (3.25) the energy-off-shell
procedure
of the series (3.24) is performed.
(3.25)
amplitude
emerges
when
the cut
(3.26)
560
V!K Anisouich et a!. / Bethe - Salpeter equation
This amplitude is also represented as an infinite sum of loop diagrams, but
initia1 and final values S and s are different,
A(Q) =
‘f)(s + ;j@(s
t...
(3.27)
It is ener~-off-shell amplitude that has to be considered in the general case and
it contributes to the interaction with the external field, that will be considered in
sect. 4. This amplitude satisfies the equation
(3.28)
Let us stress, that in the dispersion approach we deal with the mass-on-shell
amplitudes, i.e. amplitudes for real constituents, whereas in the BS equation we
had mass-off-shell amplitudes. And the appearance of the energy-off-shell amplitude in the dispersion method is just the charge, that we have to pay for keeping
all the constituents mass-on-shell.
The diagram representation (3.27) yields
45, s) = G(.V
G(s)
1
_B(s)
The physical partial wave amplitude
A(s) =A(s,
(3.29)
.
A(s) is just
(3.30)
s).
The ener~-off-shell
amplitude is in a sense analogous to the Bethe-Salpeter
mass-off-shell amplitude.
Let us consider the partial amplitude near the pole which corresponds to the
bound state. The pole condition is
(3.31)
In the vicinity of this pole
G(M2)
A(s) =G(s)
1
_;+,
G(s) =
JB’O
.-.
1
M2-s
GW2) +
@@iq
(3.32)
K K Anisouich et al. / Bethe - Salpeter equation
Here
we took
homogeneous
G,(s,
into
account
equation
that
1 -B(S)
for the bound-state
- B’(M*Xs
vertex amplitude
-M2>.
G(s), so
of Eq. (3.33) is the factor
M*) =G(s)N-"~(M~).
G,(
A(s, s) =
M2, M*)G,(
The
A4*) reads
(3.33)
(3.34)
The normalization
condition for G,(s, A4*> is the connection
and As, S) in the vicinity of the pole. If we require
then
G,(s,
M2) =G(s)j-;-;G(Z)$G#,M').
G,(s, M2) is the analog of x(p,, p2; P).
The only s-dependence
in the right-side
Gv(s,
= 1 -B(A4’)
561
between
G,(s,
M2)
M2, M2)
+ regular
iv*--s
terms,
(3.35)
N(ZU2) = B’(A4*> and
GAS,M2)=
This amplitude
tion.
4. Interaction
GJs,
G(s)
IIB’O
(3.36)
.
M2) enters
all processes
containing
the bound-state
with an external field. Form factor of a composite
In this section
we describe
an external-field
interaction
interac-
system
with
a two-particle
composite
system of spinless constituents.
We compare
the expressions
for a
composite-system
form factor obtained
within the dispersion
method
and the
standard Feynman calculus with separable vertices. We perform calculations
using
light-cone variables (i.e. IMF) as in sect. 3 and discuss the influence
of external
kinematics
choice (i.e. specific IMF) on the form of the integral representation.
In the dispersion
technique
the amplitude
T&P’, P, Q) of the two-particle
system interaction
with an external vector field is described by a series of graphs of
Fig. 3. Dispersion
diagram
for an external
field interaction
with a composite
system.
562
V.V. Anisovich et al. / Bethe- Salpeterequation
Fig. 4. The Feynman
triangle
graph
rE(p’, p, q). Dashed lines are the cuts,
calculation of A,(s’, s, q*).
corresponding
3 type [7], which are obtained by inserting external field into constituent
of the constituent scattering amplitude.
Fig.
T(s’,
qlLc
2,
q
s, q2)+
p
to the
lines
(4.1)
where P2 = s, PI2 = s’, q = P’ -P.
The dispersion relation approach enables one to determine the transverse part
T(s’, s, q2), which is expressed through the energy-off-shell amplitude A(s, S),
introduced in sect. 2 and the double spectral density Av(s’, s, q2) of the triangle
Feynman graph PFF(p’, p, q) with a point-like constituent interaction (Fig. 4)
A(s,
T(s’, S, q2) = /
2) ds’
r(s_-s)
=
G(s)
l-B(s)
G(s)
=
A,(s,
l-B(s)
s’, q2) is the coefficient
ds” A( s”, s’)
A,(& 5’3 q2)
/
Tr(S’-S)
d9 G(S)
dl’ G(.?‘)
?r(S-s)
?r(?-s’)
Pn(s, S’, q2)
G(s’)
1 -B(s’)
(4.2)
’
of the double-cut Feynman graph decomposition
(P’+P-2k),
dk 6(k2--‘)
x6((P-k)2-m2)6((P’-k)2-m2),
(4.3)
-q2(s’
Av(s’,
h(s’,
s, q2) =
+ s - q2)
16A3’2(s’, s,
q2)
e( -ss’q2
- dh(s’,
s, q2) = sf2 + s2 + q4 - 2.7’s - 2sq2 - 2s’q2,
s, q2)),
q2 < 0,
(4.4)
VIiX Anisouich et al. / Bethe - Salpeter equation
and we obtain the dispersion representation
ds’ G(i)
rD(s’,
s, q2) = /
for q2 < 0:
dS’ G( S’)
?T(S--s)
(Y-s’)
The function C in Eq. (4.1) is determined
563
A,@‘,
5, q2).
(4.3
by the Ward identity, that gives [7]
(4.6)
J;(s’, s, q2) determines the form factor of the composite system.
Now let us turn to the Feynman graph with separable vertices as in sect. 3,
corresponding to the three-point function r,. The Feynman expression reads
1
=-
(27f)4i
dk G((Pk)2/s-k2)C((P’k)2,/d-k2)(P’+P-2k),
I ( mZ-k2-iO)(m2-(P-k)2-iO)(m2-(P’-k)’-iO)’
x
(4.7)
Our purpose is to show the transverse part of the expression (4.7) is equal to
(4.9, namely,
rJP’,
P, y) = 2 P(
T(s’,
1 IJ
$q
s, 42) +q,
B(d)
-B(s)
q2
’
(4.8)
where I%‘, S, q21 = r&‘,
s, q2).
Let us first take G = 1, then we come to the conventional graph for a point-Iike
constituent interaction, and we denote all values for this case with the letter F.
2fr(S’,
S, $)
p-
$I
i
+%
US’) -b(S)
cl2
1P
=-
1
(2*)4i
dk (P’i-P-2k),
x
/
t m2 - k2 - iO)(m2 - (P - kj2 - iO)(m’-
(P’ - k)’ - iO> .
(4.9)
564
K V. Anisouich et al. / Bethe - Salpeter equation
Consider the case q2 < 0. We again
the convenient
reference frame
4+=
K(qo
that is possible
P,P_=
p’=
introduce
P,=O;
+ 4) = 0;
p,>
light-cone
variables
and choose
0,
(4.10)
for q* < 0. For this choice
is;
P+q_=
s’+q+
~
2P+
.
$(s’ -s - 42);
P,=
q*=
Pi;
_q+;
(4.11)
The standard
procedure
of the hadron form-factor
calculation
in the light-cone
dynamics is examining the ( + ) component
[81. The ( + ) component
in both sides of
Eq. (4.9) gives
i
(2r)4
dk,
“h
2k+k_-
dk_
d*k,
(P+-
rnt + iO)(s - 2P_k++
x (s’-2PYC++2q,k,+~k_(k+-P+)
Our next step is the integration
k+)/P+
2k_( k+- P+) - rn$ + i0)
-mt+iO)’
over k_. In a complex
k--plane
there
are three
poles
m+- i0
-.
2k,
’
s - 2P_k,2(P+-k,)
rnt + i0
s’ - 2PLk,+
;
2q,k,
2(f’+-k+)
- rnt + i0
.
(4.12)
For k+<O or k+> P, all these singularities
are located on the same side of the
integration
contour and the integral is zero. For 0 < k+< P, one pole (rnc i0)/2k+
is under the contour and the rest two are above it. By closing the contour
565
V.E Anisouichet al. / Bethe- Salpeterequation
in the negative semi-plane
obtain
= &/{dx
and introducing
d2k,}
x(1 -x)((m2
the variable
+ k&+0
-x)
x = k+/F+,
0 <x < 1
-s -iO)
i
X(1-X)
i
=
-1
(m2+@r-4’)
x
-
s'-i0
m2+k+
dS
f T(S-s)
x(1 -x)
T(B’--S’)
m2 + (k,
- qTx)2
=
d:’
dd
7r(S--s)
f3( -s”sq2
Tr(S’-s’)
dx d2k,
(4.13)
x(1-x)
X(1-X)
I
ii
- dA(S’,
16A”“(Y, 5,
s, 42))
q2)
(-q2)(s’+s-q2).
In the general case of Eq. (4.7) a more cumbersome procedure similar to that used
for a loop diagram in sect. 3 gives just the multiplication of the integrand of (4.13)
by G(s)G(s’) and thus precisely the dispersion expression for A,,(s’, s, q2> Eq.
(4.5).
In the case of the bound state with the mass A4 we have for the vector form
factor
ds G,(s)
&(q2)
= /
7+--W)
ds’ G,( s’)
F(S’-W)
A,@‘, s, q2),
(4.14)
where
G(s) =
G(s)
’
\lB’( W)
is the bound state vertex function, introduced in sect. 3. To calculate F,(O) we use
the relation
lim Av(s’,
C+o
s, q2) = T~(s’
-s)p(s),
(4.15)
K V. Anisouich et al. / Bethe - Salpeter equation
566
that yields
F”(O) = /
ds G,2(sMs)
1
ds G2(s)&)
= B’(M2)
?T(s -M2)2
I
=
1
(4.16)
.
?T(s -M2)2
This is just the desired result since F,(O) is a bound state charge in units of the
constituent charge.
We mention that the dispersion representation (4.14) determines Fv(q2) at
q* < 0. To obtain F, at q2 > 0 analytic continuation of (4.14) should be performed, and then anomalous singularities appear in the explicit form.
It may be convenient to use other representations for F,(q2), than we have in
Eq. (4.14). All of them may be obtained from (4.3) for AJs’, s, q2>. Rewrite it
again
1
= K &r
2A,(s’,
x
J
(P’+P-2k),
dk 6(k2-
Let us consider our bound state in the IMF, that corresponds to the introduction
of the light-cone variables. After that we still have the ambiguity connected with
the particular choice of external variables. The convenient choice [81 which was
used above
q+=
0,
P,=O,
,q+=
-q2,
gives (taking p = +)
A,( s’,
m2 + (k, - qTx)2
x(1 -x)
and the conventional
F,(q;)
representation
= ;/$(x1,
x d’k,,
X2
d2k2,
1
k,,,
dx,
(4.17)
!
for the form factor
k&(x,,
dx,
x2 Ikrr + (l
6(1 -x1
-x2)6(k,,
-xl)qT,
+ k,,),
k2T
-x2qT)
(4.18)
561
I/:V Anisouich et al. / Bethe - Salpeter equation
where
rL(% %IklT, kzT)=
4T&
G”(S)
(s_M2)
S=-+-
’
4T
4T
Xl
X2
Of course, the choice (4.10) is not unique. E.g. in ref. 171we have also regarded the
IMF, but the external momenta were chosen in the different way, namely
d-S---q2
4+=
2s
P,=O,
*+,
q$=
We again introduced x+= k+/p+, but the representation
had the form quite different from (4.18),
ds G,(s)
&(q3)
= /-
ds’ G,(s’)
7r(s-M2)
x ! 16ax(l
m2 +
for the form factor
(-q’)(s’+s-q2)
7T(s’-fw)
+‘,
+-
dx d2kT
-X)(2-X)
(4.19)
$(s’-s-q’)2-q2.
s, q2)
.(&)
(kT-qT)2
-
(4.20)
Z-X
One representation can be deduced to another by the appropriate variable change.
This can be trivially shown for a more general case. Let us consider the
invariant integral representation for an arbitrary invariant ampIitude A,
A(q,,... ,qn) =A(qf
,...,
q,”,... ,q,q,
,...)
= ]dk
a(k2,
kq,nkqn),
(4.21)
where dk is the invariant measure on the Lorentz group. Suppose that we have the
integral representation for A in a particular reference frame IQ’) (i.e. some fixed
set of external momenta {q,, . . . , q,} = {qj”, . . . , qi’))) with a specific k, connected
with this reference frame:
A( q$l), . . .,q;“)
= /dk,
*a(k,q(l’),.
. . , klq;“).
(4.22)
568
V. K Anisovich et al. / Bethe - Salpeter equation
Then we turn to the other frame KC2),connected with K(l) by a Lorentz transform
A: {L$‘, . . .) qA2)) = {Aqjl), . . . , AqLl’}. Due to invariance of A we have
A( qp, . . . ) qa)
=
ldk,
. a( k,Aq[l),
. . . , klAq;‘))
= ldk2.a(k2q~‘),...,k2ql;‘)).
(4.23)
Hence, the integrand of our representation has changed its form in the reference
frame KC2’.The old form may be easily restored by the variable change k, = A -II?,.
When hadron-hadron
interactions at high energies are investigated, it may be
convenient or necessary to consider the lab. frame, where the momentum of the
projectile tends to infinity, and the kinematics in this lab. frame is different from
conventionally used (4.10). And the above-mentioned difference between integrands related to representations for different kinematics should be taken into
account. This is important for various processes, e.g. for processes of multiple
scattering.
5. Conclusion
We performed a detailed analysis comparing the Bethe-Salpeter
treatment with
results given by the N/D method of the dispersion-relation
technique. This
method provides a possibility to perform calculations both for scattering amplitudes and for bound systems.
Our results are:
(i) The scattering amplitude obtained in the dispersion-relation technique is
identical to the solution of the BS equation with the separable kernel of the special
form for spinless constituents. For the case of the spin constituents the results of
these two approaches differ by subtraction terms.
(ii) The form-factor dispersion representation
for point-like spinless constituents is equal to the Feynman expression obtained in the framework of the
standard light-cone dynamics. For the spin constituents the results for form
factors, like for scattering amplitudes, differ by subtraction terms. Moreover, for
non-point-like constituents the dispersion expression contains the mass on-shell
form factor of the constituent, whereas the mass off-shell form factor of the
constituent enters the Feynman expression.
The advantages of the dispersion relation method compared to the BS equation
with the kernel of the general form are the possibility to work with mass-on-shell
amplitudes only and to take into account all types of forces (the contribution of the
left-side singularities) on an equal footing independent of their t-channel structure.
Just these advantages enable a good description of pn-scattering phase shift data
as well as deuteron static properties and form factors [71.
V. K Anismich
et al. / &the - Salpeter equation
569
However, up to now the N/D technique has been developed for two-particles
transitions only. Genuine inelastic processes have to be small:.all the many-particle
s-channel singularities are not taken into account in the N/D method. It is just the
point why the N/D method cannot be applied to cases with long-range forces as,
for example, a photon-exchange interaction.
The authors are grateful to V.N. Cribov for helpful discussions. Three of us
(V.V.A., B.Ch.M., H.R.P.) wish to thank the Deutsche Forschungsgemeinschaft for
financial support.
Appendix A
Our discussion was so far based on the assumption of spinless particles. For
completeness we now determine the spin structure for the transitions
NN-,NN,
(A.11
where N is a fermion carrying spin and isospin, i.e. a nucleon or a non-strange
quark. The standard form of matrix elements for these transitions is
(S(P;)ii~~(P*))(~(PI)BAYI(P*)).
(A-2)
Here 6” and 0, are operators which correspond to the state considered. The
momentum notation is shown in Fig. A.1. However, for our considerations, it is
more convenient to choose instead the matrix elements which use the charge-conjugated fields
FCC
-P2) = cw Pd
!Pc( -pi)
= -q
P;)c.
(A.31
Introducing in Eq. (A.2) these fields we have after the Fiertz transformation
W(p,) c, !PJ-p;)
products of two bilinear forms of the following type:
;$f----J;
2
Fig. A.I. Fermion-fermion
scattering
amplitude.
570
I-:J! Anixwich
ei al. / Bethe - Salpeter equation
Here OA and QA denote a related set of operators.
operators which correspond to definite NN states.
To this end consider the bilinear form
(%-P~)QR(P~)
We must construct first the
(A-5)
and introduce the momenta
p=p,
+P2r
Let us concentrate
form
k = +(P,
-Pd.
(A.61
first on the case with isospin I = 1. We will choose QA in the
Ys
for
‘S,
YSG
for
‘DZ,
(A.?
where Kpy = k,k, - $k26,1, and S,$ = SKY- (P,PJ/P*.
Remember that PC and
p have opposite parity hence <?J@> is pseudoscalar and (FCf;;,y,V?>
is scalar.
For P-wave states, QA takes the form
1
for
“PO
~~=~~yaPbkc
for
3P,
kpLrw+ kl,r, - $(kT)S$
for
3P2,
(A-8)
pabc is the totally antisymmetric tensor, and the indices a, b, c run over 0, 1, 2, 3.
The operator
&
produces the bilinear form (@rfi*) a pure S-wave state with spin 1. So the idea of
the construction of the operators Q, is rather simple: states with spins 0 or 1 are
constructed with use ys or r, while angular-momentum states are generated by
the tensors which are constructed from vector k: k,, k,k, and so on. For example,
QA takes the form
(A-9)
for F-states. Here
K &VA= k,k,k,
- $k2(k,6,1, + k,,Sh”,+ k,acl;).
(A.10)
571
V:PCAnisovich et ul. / Bethe - Salpeter equation
k,
Qh
0
ka
OB
Fig. B.l. Fermion-fermion
loop diagram.
The only state under discussion which can mix is ‘P2 (with 3F,); for other states
QA = CIA.
The amplitudes with I = 0 are constructed in the same way: Qa takes the forms
3s,
r
3D,
K wa r”.
F
(A.ll)
These states can mix QA for a P-wave state is
rP 1
(A.12)
Ysk,.
Other states are constructed
in the analogous way.
Appendix B
Fermion-fermion
loop diagram calculations
Here we perform the calculations of the loop diagrams with two fermions in
intermediate state. Let us start from the loop diagram Fig. B.l. The phase space of
fermions in the intermediate state is equal to
d@( NN) = $(2~9~6’( P - k, - k,)
d3k,
d3k,
(2v)42k,,
(2rr)32k,,
W)
’
The matrix element for a real fermion transition reads as
c /d@(NN)(@(p;)oB’U-p;))(%4-WWW,))
spin states
(B-2)
Summing over fermion spins in the intermediate
~!PJk,)~‘(kl)
a
=l;l+m,
state gives
c ?PJ -kz)F:(
u
-k,)
= -f,
+ m,
(B.3)
572
V. V. Anisouich et al. / Bethe - Salpeter equation
so we have for the matrix element of Eq. (B.2):
(%4)Or%(
-&))P,“(K(
bi=/d@(NN)
-P&W(P1))
Tr[O,(&,+m)QA(-i,+m)].
The phase space matrix element pt
loop diagram.
Examples of pi (we will denote
.Zp = O+, l-, 2+ read as follows:
p^(‘S,)=/d@(NN)
(B.4)
is responsible
for the spin structure of the
it also as p(J’>> for the cases Z = 1 and
Tr[y,(fP^+l+m)
XY5(4~+R+m)]
=2,-&
+.
P.5)
r--
Here P* = s and k2 = i(k, - k,)* = m2 - f. The factor 2s appears because of the
trace calculation and (1/167r){(~
is the usual phase space value. Also
$‘(“P,)
= /d@( NN) ElrabcPbkcEIL’n’b’cPb,k,,
= $k2( s + 2m2)~,,b,~~“b’cPbPb~_/d@(
NN)
= -$~(~+2m~)(~~-m~)8~“/d@(NN).
(B.6)
Here we used the result that k,k ” + ik*S,” * because of the phase space
integration. (Recall SW,,= diag(1, -1, - ;, - 1) and “L’= diag(1, 1, 1, 1)). Furthermore
$f;‘( ‘D,) = /d@( NN) K,, Tr[($,
+m)y,(-~2+m)y~]~p’Y’
k,k,k*‘k”’
Here the substitution
used. Define
where operators
-+ &k’(S,“,S L dy’ + 6: “‘~3:Or-t-8; “‘S,,?“‘1 was
I$ for the considered
above states are equal to
The phase-space factors pBA(s) determine
dispersion-relation
These loop diagrams are used for the construction
way as for spinless case.
loop diagrams
of amplitudes in the same
Appendix C
Description of the deuteron in the dispersion-relation technique
Now we turn to the consideration of the two-nucleon composite particle, the
deuteron. The deuteron is a mixture of two neutron-proton
states, S-wave and
D-wave ones, which correspond to two bilinear fermion forms
(~~(-k~)S~~~(k,)~
(qr,( -k2)DFP,,(
S-wave
(C.1)
D-wave.
k,))
Here !PP and WEare fields of the proton and antineutron and operators S, and ZIP
are equal, up to normalization constants, to the operators F, and KPyFP which
were introduced in the previous section:
1
S,=n,(s)p=
D,=n2(s)KILyTy=
-~
dz
I
2k,
2rrl-t~
-y,”1’
-~[(m+~)k,+(ts-m2)r,l].
(C.2)
(C.3)
V. V. Anisovich et al. / Bethe - Salpeter equation
574
Here the normalization
L, = S, or 0,) 1
/t
constants, n1 and n2 are defined by the condition (below
d$$P;k,,k,)Tr[
L,($,+m)~p'(-ri,+m)] =pJs)SI_LIp'
(C-4)
S-wave and D-wave phase space factors are
P,(S)
s-4m2
=& i~
s
5/=
I-
The amplitude pn + pn is a 2
X
(C.5)
2 matrix with elements
(w,(-p,)L,~~(p,))A’,~~(W,(p;)L’“IY,(
-Pa)
-(L P)&'(L'y+.
(C.6)
The same structure has the N-function
A
(S,)Ns,(s)w”)+
q)Nss(s)w)+
N=(D,)N,,(s)w)+
(D,>ND&)W>’
(C.7)
’
According to the general prescription for writing the dispersion relating graphs we
should replace the N-function by the vertexes:
Nss + Gss(s)GSS(s’),
Ns, * Gs&)GSD(s’)
N os + Gos(s)GDS(s’),
Noo --) Goo(s)GDD(s’).
Remind that for energy off-shell transitions
leads to
G,,(s)
s ZS’. The requirement
(C-8)
A,, =A,,
= GDS(s).
(W
In the general case the N-functions are represented
NLL, + C G;lL,( s)G,L”(
n
S)
.
as a sum of the vertexes
(C.10)
Here we discuss in detail the simplest case with II = 1 (Eq. (3.10)) which allows one
to understand the principal point of the consideration. But we conclude this
paragraph giving the sketch of the consideration in the general case.
575
E V. Anisovich et al. / &the - Salpeter equation
Fig. C.l. The amplitude A,,,
of Eq. (C.11).
Now we write down the equation for the amplitudes A,,,. Let us represent
LL,
as a sum of two amplitudes with different states before the last interaction
A
(see Fig. C.l)
A LL'
G L"L'
c
=
aLLlILt
(C.11)
,
L”=S.D
A ss =a
SSS
GSS+a
SDS
(C.12)
GDS .
For amplitudes aLLIf_ with L = S we have the following system of equations:
a SSS = GSS
a SSD = GSD
+ aSDSbDSS*
+aSSSbSSS
+ %ssbsSD
aSDS
= aSDDbDDS
‘SDD
= aSDDbDDD
+ a,Dsbmn
7
+ aSSDbSDD,
(C.13)
+ aSSDbSDD-
It is convenient now to characterize the loop diagram by three indices. The first
and third ones are related to the vertex functions and the second one to the
intermediate state,
biLj
=
Irn
ds’ G'"tr')PJCJl)G,i(s')
.
(C.14)
4mZ ?I-
Eq. (C.13) can be rewritten in a vector form:
a,=q,+a,lj,
g, =
a,
(Gss,
= caSSS2
Gs,,,
aSSDy
0, 0) >
aSDS?
(C.15)
‘SDD>-
Here k is the matrix
bsss
B^= b0 DSS
o
km
0
0
0
bsn
b
0 SDS
km
0
o
bDDS
bDDD
*
(C.16)
576
I/: K Aniwvich et al. / Bethe - Salpeter equation
The analogous equation can be obtained for uLLIL with L, = D:
ud
(aDSS7
aDsD7 aDDSt aDDD)t
=
co,OG,,, G,,)*
i&J =
(C.17)
Eqs. (C.15) and (C.16) can be written in a matrix form
d=$+c?b.
(C.18)
Here
Gsr,t 0,
0
0,
GDS, CD, ’
a^=
as -_
%SS3
aSSD5
USDS)
‘SDD
‘d
aDSS Y
aDSD,
aDDS,
a13DD
I !I
’
(C.19)
The solution of Eq. (C.18) is
-jQ-”
a^=g(l
(C.20)
So for scattering amplitude we get the expression
(C.21)
Here
s^‘=
Gss
0
0
GSD
GDS
0
0
GDD
’
(C.22)
Of course, Eq. (C.21) is a direct consequence of the unitarity condition if we
suppose that the vertices g and g’ have the form given in Eqs. (C.19) and fC.22).
Indeed, let us write the scattering amplitude in the form CC.211 and let the
matrices 8, 2’ and B^ have no singularities in the right and left parts of the
s-plane, respectively. The S-matrix is related to the scattering amplitude by
s”=r+2i&L&lp,
(C.23)
K K Anisooich et al. / Bethe - Salpeter equation
where the matrix fi
571
refers to phase-space volumes of S- and D-states, ps and p,,:
(C.24)
Then from the unitarity condition
amplitude:
(A -A+)
for the .!?-matrix Cs^+ s^ = I) we have for the
= 2&$+&J.
So, since the amplitude A^ is symmetric we can replace the operation
hermitean conjugation by the complex one and rewrite Eq. (C.25) as:
s”j(Z-8)-l - (r-ri)*-‘j C’E2~~(~_~*)-1b~~g(~_fj)-1~1.
(C.25)
of the
(C.26)
Evidently we can omit the matrices 6;’ and g in the right-hand and left-hand sides
of that equation. Then, multiplying the expression by the matrix (I - g>* from the
left-hand side and by the matrix (I - 8) from the right-hand side we have
Im B”=b’$b.
(C.27)
This relation gives us b,,j(s) in the form of Eq. (Cl@, so we obtain the same
result as in the diagram method.
Therefore, the diagram method is a modification of the conventional N/D
method and can be obtained by means of the unitarity condition only. Moreover,
this method can be easily represented by constituent rescattering diagrams which
are very convenient for investigating the interaction of external fields with the
composite system.
The matrix eiements of A^ contain the common factor l/det / 1 - B^1. It is
convenient to separate it. For this end let us introduce the matrix 6:
&=ddetll-81.
(C.28)
Diagonalizing the amplitude a (or ~$1one gets two eigenstates,
(C-29)
578
K V. Anisovich et al. / Bethe - Salpeter equation
by the condition (d, I A I e,) = 0. It gives
The mixing angle 0 is determined
(cos*O - sin%) 6
ADS
+sinOcosO
(3-$)==*.
(C.30)
The diagonal matrix elements are equal to
cos*o
X
2 cos 0 sin 0
-ass
+
sin20
aDS
+
pD
ADD
PS
i
sin*@
X
6
-ass
PS
2 cos 0 sin 0
-
@--&
cos20
aDS
+ --&-%D
s:‘*
ccq31)
I
The amplitude (d, I d I d@‘) has the pole at s = M*, which corresponds to deuteron
bound state. This pole is related to the zero value of det / 1 - 2 1:
(detll-i/),=,z=O.
(C.32)
This pole is absent in the amplitude ( EF I A^I ey’>. It means that the numerator of
this amplitude should be equal to zero at the point s = M*:
(C.33)
It is possible to construct the deuteron vertices either
(I) in a direct way with use of the graph technique or
(II) resolving Eq. (C.20) for the amplitude a^ in the limit s --, M*.
(I) The triangle block in the diagram Fig. C.2 gives us the deuteron
factors.
Fig. C.2a. Triangle
graph for the amplitude
Eqs. (C.34) and (C.37).
form
519
V. K Anisovich et al. / Bethe - Salpeter equation
Fig. C.2a. The double-pole
deuteron
triangle
graph.
The triangle block contains both the transitions
with orbital-momentum
conservation L = L’ and the transitions
with L f L’. To extract the deuteron form factors
from the diagram of Fig. C.2a one should
(i) sum up in the amplitudes
of Fig. C.2a over all right and left loop diagrams
and
(ii) tend the energies of nucleons
in the initial and the final states to the
deuteron mass.
As a result we obtain the double pole diagram of Fig. C.2b with deuteron form
factors as the residue.
The diagram of Fig. C.2a with L = L’ can be obtained from the diagram of Fig.
C.l if one replaces the loop biLj with the triangle block CL;. The removing of the
Fig. Cl
loop biLj is equivalent
to the differentiation
of the scattermg amplitude
with respect to biLj. So the amplitude
of Fig. C.2a type diagrams with L = L’ is
equal to
= - (d,lcSldF)(det(l
(d,Ifldp’)L=Lr
-B^l)-2
x CCL,.,--&(detII-gI)+....
iLj
(C.34)
lLJ
here we write down only the terms contributing
to the double pole as s + M2.
The amplitude
(dy I a”Id”) in Eq. (C.34) is taken at s = M2.
The determinant
det I1 -g I is equal to
-(I-
ks)b
stxAmsb,s,
-
(1 -
~ssDbD&D&DDs
so we have eight terms in the r.h.s. of Eq. (C.34).
The same procedure can be used for the calculation
type with L f L’. However, in this case we should
bm,)bsskd+.m
(C.35)
of the diagrams of Fig. C.2a
use the differentiation
with
K V. Anisouich et al. / Bethe - Salpeter equation
580
respect
to (biLL,bLL,j), see Fig. Cl.
(~oso~sos)~
(&&X,)~
(~sso~sos)~
(&X~oss)~
So the corresponding
Eight combinations
(bnosboso)~
(ksn~soo)~
amplitude
is equal
(k&Jo)7
I1 -&l)-2
a(detll-gI)
c
L.‘j
Eqs. (C.34) and (C.37) determine
form factor of the deuteron
The double discontinuity
+
a( biLL, bLLrj)
iLL’j
C
(C.37)
* *’ ’
the weight of the triangle
[7].
of the triangle
disc, disc,, r;.~L’j (2, s’) =
(C.36)
(~oos~oss).
to
(d,lfld”)L+r=- (d,l&ldP>(det
x
are possible,
diagram
/d@(P,
I?
diagrams
r].LL,j in the
qLLrj is equal to
k,, k;, kz)
spin states
XG’L(s’)(L,)F~(L’,)G~~j(~‘)~T*~q.
(C.38)
Here L, and Lk are the S-wave and D-wave operators,
Fp is the operator of the
electromagnetic
field interaction
with the nucleon,
and E’, 5” are deuteron
polarizations.
The phase-space
integration
d@ is the same as for the case of
= G2 = q2. The spin summaspinless particles with P” = S; PI* = s” and <p’ -P)’
tion gives
disc, disc:, qtLLrj
= jd@(P,
p’; k,, k;, k2) GiL(S-)
+m)F’(kl
xTr[L,(-k,+m)L’,(k;
The photon-nucleon
interaction
can be represented
Fp = rpfl(q2) + (k, + k;)‘f&‘),
where the functions fI and f2 can be expressed
electric and magnetic nucleon form factors
fl
=fh4,
f2
=
24fE
+fivl)/(4m2
+m)]G,~j(S’)S’*S’.
-q2).
(C.39)
in the form:
(C.40)
in the terms on the conventional
(C.41)
581
I/: I/ Anisovich et al. / Bethe - Salpeter equation
The
blocks
deuteron
form-factor
discontinuity
is the superposition
of the set triangle
rJLLrj,
disc, discs,FP(s’,
S’, q2) =
c
SiLLtj
iLL’j
x
(C.42)
disc, disc,, rirL,j.
The coefficients
aiLLTj can be found
calculation
allows us to extract deuteron
in the form
from Eqs. (C.34) and (C.37). The direct
vertices and Eq. (C.42) can be rewritten
discs disc,, FK(s’, ‘Z’, q2)
= _/d@(F,
?;
k,, k,,
k;)
xC(p,,G~“(~))Tr[~,(-R,+rn)L’,(1,+m)F~(l;~+m)]
iL
x 5 (GLtj(“)P,fj)t’*5”*
Let us mention that we have obtained
ing to the initial and final deuteron.
(C.43)
the vertex functions G,PjW)/3,,j
correspondpLj are some new coefficients,
which will be
determined
below.
Now let us turn to the spin structure
of the deuteron
form factor
I
CW) = (~+mJ,,F,(q2) + (4,~,,-q&JF2(q2)
F4q2)
-(~+nJl,4,yj-p-
1
.
The form factors F,,F, and F3 are related
and quadrupole
form factors by
F,=F,+-F
(C.44)
to the conventional
electric,
magnetic
cl2
6M2 Q'
(C.45)
V. V. Anisovich
582
Let us represent
et al. / Bethe - Salpeter equation
the r.h.s. of Eq. (C.43) in the similar form:
disc, discs, P‘(S, S’,
4’)
= [‘*tv
+
(k&T - &6,,)g2(S,
5’7 q2)
(C.46)
The functions gi are calculated using the explicit form of Eq. (C.43). These
functions give us the deuteron form factors Fi:
(C.47)
The form factors Fi give automatically F,(O) = 2.
(II) Now let us turn to another way of finding coefficients piL of Eq. (C.43)
directly from Eq. (C.18). In the limit s + M2 we have det I1 - B^I + 0. It means
that a^ is large and it is possible to neglect g in Eq. (C.18). At s = M2 we have
a(1 -j)
= 0.
(C.48)
This equation breaks into two identical ones for a, and ad Eqs. (C.15), (C.17). Let
us denote the solutions of these equations as a$” and a$‘). As it follows from the
Eq. (C.43) the coefficients pLj are determined by the projections of uL:
up) = Const CpSLiLL[P,
iL
u(do) =
Const Cp,,i,s@.
L
Hence
Pss = Casss y
L&S = CasDsy
The normalization
Pm = %sm
(C.49)
PDD = C+,,-
constant C is determined
from the condition F,(O) = 1.
V. V. Anisouich
et al. / Bethe - Salpeter equation
583
References
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[3]
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