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Stability of two-fermion bound states in the explicitly covariant light-front dynamics
SUPPLEMENTS Nuclear
ELSEVJER
Physics
B (Proc.
Suppl.) 108 (2002) 259-263
Stability of two-fermion light-front dynamics
bound states
M. Mangin-Brineta,
and V. A. Karmanovb
J. Carbonell”
aInstitut des Sciences NuclCaires, 53, avenue des Martyrs, 38 026 Grenoble, bLebedev Leninsky
Phys ical Institute, Pr. 53, 119991 Moscow,
in the explicitly
www.elsevier.com/locale/npe
covariant
France
Russia
1. Introduction The system of two bound fermions covers a huge number of interesting problems from atomic, nuclear and subnuclear physics. It is one of the most difficult problems in field theory due to the fact that bound states necessarily involve an infinite number of diagrams. We studied this problem in the framework of the explicitly covariant light-front dynamics [l] (CLFD). In this approach, the state vector is defined on an hyperplane given by the invariant equation w .x = 0 with w2 = 0. The standard light-front, reviewed in [2], is recovered for w = (1, 0, 0, -1). The CLFD equations have been solved exactly for a two fermion system with different boson exchange ladder kernels [3,4]. We have considered separately the usual couplings between two fermions (scalar, pseudo-scalar, pseudo-vector, and vector) and we were interested in states with given angular momentum and parity J” = Of, l*. Each coupling leads to a system of integral equations, which in practice are solved on a finite momentum domain [O; k,,,]. If the solutions necessarily exist when the integration domain is finite - for the kernels are compact, it is not a priori obvious that the equations admit stable solutions when k maz goes to infinity. Particular attention must therefore be paid to the stability of the equations relative to the cutoff k,,,. We develop hereafter an analytical method to study the cutoff dependence of the equations and to determine whether they need to be regularized or not.
The method will here be detailed for a J = O+ state in the Yukawa model but it can be applied to any coupling. Results will be presented for scalar and pseudo-scalar exchange. This latter furthermore exhibits some strange particularities which will be discussed. 2. Scalar
exchange
Let us consider a system of two fermions in a J” = O+ state, bound by a scalar exchange, whose Lagrangian density is given by L = g,G@Q. Its wave function, constructed using all possible spin structures, is determined in the O+ case by two components [6], fi and fi, which depend on the two scalar variables k and 8: a’.
[/cx fi]
ksin0
f2
t
ffYW,I
z is the momentum of one particle in the system of reference where & + & = 0, fi is the spatial part of the normal w_to the light-front plane, 0 is the angle between k and fi, and w is the two component spinor. The appearance of a second component compared to the non relativistic case is due to vector ?L, which induces additional spin structures. fi and f2 satisfy the system of coupled equations : [Ad2 - 4(k2 + m2)] fl(k,b’) =
$/[Knfi(k’$)+Klzfz(k',e')l
0920-5632/02/s - see front matter 0 2002 Elsevier Science B.V. All rights reserved PI1 SO920-5632(02)01340-3
*&k'
M. Mangin-Brinet
260
et al. /Nuclear
Physics B (Proc. Suppl.) 108 (2002) 259-263
Since Kss is repulsive and does not generate collapse, we consider only the first channel. have
[M” - 4(k2 + m2)] f2(k, e> =~J[K21fi(k’,e’)+K2212(k’iBoi~(l~ &k’
M2 is the total mass squared
of the system,
the constituent mass and &k = JG. kernels Kij result from a first integration elementary quantities:
m is
The of more
Allvuw
=
=
s
-
fill -
--cm
J .
The cou-
[2k2k’2 + 3k2m2 + 3k12m2 + 4m4
-2kk’Ek&k! COS e COS 8' -kk’(k2
+ k’2 + 2m2) sin 0 sin 8’ cos cp’]
nf2 = -arm(k2 - k’2) (k’ sin 8’ + k sin 0 cos cp’) s - -cr7rm(kr2 - k2) (k sin 0 + k’ sin 8’ cos cp’) n21 s - -cm [(2k2k’2 + 3k2m2 + 3k’2m2 + 4m4 K.22 -
-2kk’EkEkl COSeCOSe')COScp' -
kk’( k2 + k’2 + 2m2) sin 0 sin 0’1
(2)
In practice, the integration region over the momenta is reduced to a finite domain [0, k,,,]. The kinematical term [M2 - 4(k2 + m2)] on 1.h.s. of equation (1) does not generate any singularity and the kernels K, are smooth functions of the 0 variable. Thus, the stability of the solution depends only on the asymptotical behavior of the kernels in the (k, k’) plane. Variables (k, k’) can tend to infinity following different directions: for a fixed value of k, KII decreases as l/k’, and vice versa. As the integration volume contains the factor EL’, this means that the total kernel decreases as 1/k’2, that is like a Yukawa potential. In contrast, Kss does not decrease in any direction of the (k, k’) plane, but tends to a positive constant with respect to k and k’. K22 is thus asymptotically repulsive and does not generate any instability. In the domain where both k, k’ tend to infinity with a fixed ratio $ = y, it is useful to introduce the functions Aij defined by Kij = -2
fi Ai,(e,6y) -& Aij(8, O’, l/y)
ify
x
{2y(1-
cOseCose')
- (1
+~~)sin0sin8’coscp’}
where
“ij d@ ,-J (Q2 + /‘2)m2&k&kl %’
where f~ij depend on the type of coupling. analytical expressions of ~~ij for the scalar pling, read
GD
0
2n
Kij(k, 6’; k’, 0’)
-$J2Tdq+ 1
any We
1 case - COSe’I - cosecose~) 8’ cos ‘p’
D = (1-t r2)(1 +
-
2y sin
e sin
Let us now majorate the function All. For fixed y, the maximum of AlI is achieved at 8 = 0’ and for any 8 = 0’ it reads: AlI (f3 = Of,y) = a’fi. The maximum value of kernel KIT is thus reached for y = 1. The majorated kernel obtained in this way coincides with the non-relativistic potential U(T) = -cr’/r2 in the momentum space with (Y’= cu/(2mn). As is well known [7], for this potential, the binding energy does not depend on cutoff if cy’ < ocr = 1/(4m) what restricts the coupling constant to: Q < n/2. If (Y’ > 1/(4m), the binding energy is cutoff dependent and tends to -co when k,,, + co. A finer majoration of All was done by taking into account its dependence on y [8]. In this way we have found cyCr = r, instead of 7r/2. As the kernel was majorated, the critical coupling constant is expected to be larger than 7r. It can be determined, together with the asymptotical behavior of the wave functions, by considering the limit k + co of equation (1) for fi -4f(k,
z) = dz’K(k, z; yk, z’)f(yk,
$i*ydy[;
z’)
where we have neglected the binding energy, supposing that it is finite, and omitted the indices of fr and KIT. This can also be written 1 1 4f(k,z) = ‘y dz’ dr A(07 6 7) * -1 0
J J
y3’2f(yk,
x
z’) + Y512f(
;, z’)}
{ Looking
for a solution
which behaves
as
(3)
M. Mangin-Brinet et al. /Nuclear Physics B (Proc. Suppl.) 108 (2002) 259-263
we are led for h(z) to the eigenvalue
261
equation
+i h(z) = cr
dz’Hp(z,
2’) h(z’)
s -1
.
with
’ -A(z, dr
Hp(z,z’) =
2.05
z’, y) cash @logy).
M2
2nJ;7
0.6
The relation between the coupling constant (Yand the coefficient /3, determining the power law of the asymptotic wave function, can be found in practice by solving the following eigenvalue equation for a fixed value of ,~3 X&z)
=
+l dz’Q(z, I
j
,_____
2.00
z’) h(z’)
03 0.0 -0.3 -06
(5)
-1
and by taking (Y(P) = l/X@. The relation o(p) obtained in that way is represented in Figure 1. The value p = 0 corresponds to the maximal that is the critical - value of CK CQ = (Y(P = 0) = 3.72, in agreement with the previous analytical estimates. It is independent of the exchanged mass p. 4,...,.,.',..'...',..'...'...'...'...'
Figure 2. Cutoff dependence of the binding energy in the J = O+ state (p = 0.25), in the onechannel problem (fi), for two fixed values of the coupling constant below and above the critical value.
.,, 10'
l.
10" r
._
l
l. lo-'
F
0.
. f&3=0)
l.
0.
l.
lo-'
:j
“;;’
,,,,,,,,,,,,,,,,,,,,,,,,,,, 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.6
0.9
.n*....* lo*
.
.
f,(e=O)
l. . .
l.
0.
I
p=o.eno+/- 0.002 fko.OOo+/- 0.002
.
1
... .
lOA 10.' . .
lOA
1
P
lo-'
1
10
100
k Figure 1. Function a(/?) for LFD Yukawa model with Kii channel only.
Figure 2 shows the two different regimes, when the coupling constant is below ((Y = 3) or above (o = 4) the critical value Q,. As it can be seen
Figure 3. Asymptotical behavior of the J = O+ wave function components fi for B=0.05, cy=1.096, ~=0.25. The slope coefficient are pi = 0.82 and /3z M 0.
M. Mangin-Brinet
262
et al. /Nuclear
Physics B (Proc. Suppl.) 108 (2002) 259-263
in Figure 3, the wave functions accurately follow the power law asymptotical behavior l/lc2+17 with a coefficient p(o) given in Figure 1. It is worth noticing that - at least in the framework of this model - one could measure the coupling constant from the asymptotic behavior of the bound state wave function. A similar study has been done for the J = l+ state, which is shown to be unstable without regularization [8,9]. 3. Pseudo-scalar
90
'.
The sion is Figure k = 0 constant
kernel Kss(Ic, k’, 8,0’, M2), whose expresexplicitly given in [3], is represented in 5 for fixed values of 0,13’. It vanishes for or k’ = 0 and tends towards a positive in all the (k, k’) plane.
Figure
5. Kss kernel in (k, k’) plane.
coupling
The stability of the pseudo-scalar (PS) coupling is analyzed similarly to the scalar one. The same method leads to the conclusion that the equations for the PS coupling are quite surprisingly stable without any regularization. However, the results show a “quasidegeneracy” of the coupling constant, for a One has for wide range of binding energies. instance (see Figure 4) cr = 49.5 for a system with B = 0.001, and o = 48.6 for a system five hundred times more deeply bound (B = 0.5), that is only a 2% difference.
60
from the second channel:
-
8~0.5
---
B=O.OOl
Let us model this kernel by a kind of “potential barrier” in the momentum space (k, k’), displayed in Figure 6, whose advantage is to be analytically solvable.
\
70
:P
i
ci 60
50
40
10
100 Lx
I
10 00
2
Figure 6. Modelling in the (k, k’) plane.
of Ks2 with a simpler
K(k, k’) = $@(k’ Figure 4. Convergence of coupling constants as function of the cutoff Ic,,, for B = 0.001 and B = 0.5. The exchange mass is p = 0.15. This peculiar
behavior
can be shown to come
x [O(k - kl)O(k,
- kl)O(kz
kernel
- k’)
- k) + O(k - kz)O(k,,,
- k)]
+ O(k’ - kz)O(k,,, x O(k:! - k) + $O(k
- kz)@(k,,,
- k)
1
M. Mangin-Brinet
et al. /Nuclear
Physics B (Proc. Suppl.) 108 (2002) 259-263
with O(Z) = 1, 2 > 0 and O(Z) = 0, z <_ 0. This kernel has the same characteristics as K&s since it vanishes when k, k’ + 0, and tends towards a constant when (k, k’) go to infinity with a fixed ratio y = k//k. fi satisfies the Schrijdinger type equation
263
Table 1 Coupling constant as a function of the binding energy for the analytical model of the PS coupling second channel.
(7) =-& / K(k, k’)fdk’)~d3k’
[k2+ ~21fdk)
IE~ = m2 - q. We assume that k1 < kz < km,, et UZ < Ur . The term .skt in the volume element of (6) was replaced by its large momentum behavior, that is by k’. We define l?(k) = [k2 + K2]fi(k). The equation for I’(k), which is analytically solvable, reads:
with
+CC
r(k) =
-$ /_
r(k’)
K(k, k’)m”‘dk’
00
The solution I’(k) is constant kz < k < k,,, : l?(k)
for kl < k < k2 and
=
rlO(k
+
I’zO(k - kz)@(k,,,
The Fi satisfy
- k1)O(k2 - k)
the coupled
- k)
Acknowledgements: The numerical calculations were performed at CGCV (CEA Grenoble) and IDRIS (CNRS). We are grateful to the staff members of these organizations for their constant support.
equations
rl = -aulbr2 (I+ au2b) r2 = -aularl
(I + oura) {
where we have defined
Ui = $lJi
through logarithms in a, b, eq. (8). Besides, the value of IE is much smaller than kl , k2, km,, . This explains the very weak dependence of or(~) V.S. n. We conclude from the above discussion that, even if the PS coupling does not formally need any regularization to insure its stability, calculations without form factors - though analytically understood - lead to results which are hardly interpretable from the physical point of view.
and REFERENCES
(8) Replacing we finally form:
h(k)
= -
I’(k) by its definition in terms of fi(k), get the solution of equation (7) in the
j&y
[W - kl>O(kz- k)
aaul O(k - k2)@(k,,, (1 + cubuz)
where N is a normalisation constant. n the coupling constant is
- k)
1
For a given
CY(~)= (au1 + bu2) + d\/(au, - bu2)2 + 4$ab 2abul (UI and the results are summarized
provided in Table
~2)
by this simple kernel 1. a(~) depends on n
1.
V.A. Karmanov, ZhETF 71 (1976) 399 (transl.: JETP 44 (1976) 210). 2. S.J. Brodsky, H.-C. Pauli and S.S. Pinsky, Phys. Rep., 301 (1998) 299. Thbse Uniuersite’ de Paris 3. M. Mangin-Brinet, (2001). 4. M. Mangin-Brinet, J. Carbonell and V.A. Karmanov, submitted for publication. B. Desplanques, V.A. Kar5. J. Carbonell, manov and J.-F. Mathiot, Phys. Reports 300 (1998) 215. J. Carbonell and V.A. Karmanov, Nucl. Phys. 6. A589 (1995) 713. 7. L.D. Landau, E.M. Lifshits, Quantum mechanics, $35, Pergamon press, 1965. J. Carbonell and V.A. 8. M. Mangin-Brinet, Karmanov, Phys Rev D 64 (2001) 027701. 9. M. Mangin-Brinet, J. Carbonell and V.A. Karmanov, accepted in Phys Rev D (2001).
ELSEVJER
Physics
B (Proc.
Suppl.) 108 (2002) 259-263
Stability of two-fermion light-front dynamics
bound states
M. Mangin-Brineta,
and V. A. Karmanovb
J. Carbonell”
aInstitut des Sciences NuclCaires, 53, avenue des Martyrs, 38 026 Grenoble, bLebedev Leninsky
Phys ical Institute, Pr. 53, 119991 Moscow,
in the explicitly
www.elsevier.com/locale/npe
covariant
France
Russia
1. Introduction The system of two bound fermions covers a huge number of interesting problems from atomic, nuclear and subnuclear physics. It is one of the most difficult problems in field theory due to the fact that bound states necessarily involve an infinite number of diagrams. We studied this problem in the framework of the explicitly covariant light-front dynamics [l] (CLFD). In this approach, the state vector is defined on an hyperplane given by the invariant equation w .x = 0 with w2 = 0. The standard light-front, reviewed in [2], is recovered for w = (1, 0, 0, -1). The CLFD equations have been solved exactly for a two fermion system with different boson exchange ladder kernels [3,4]. We have considered separately the usual couplings between two fermions (scalar, pseudo-scalar, pseudo-vector, and vector) and we were interested in states with given angular momentum and parity J” = Of, l*. Each coupling leads to a system of integral equations, which in practice are solved on a finite momentum domain [O; k,,,]. If the solutions necessarily exist when the integration domain is finite - for the kernels are compact, it is not a priori obvious that the equations admit stable solutions when k maz goes to infinity. Particular attention must therefore be paid to the stability of the equations relative to the cutoff k,,,. We develop hereafter an analytical method to study the cutoff dependence of the equations and to determine whether they need to be regularized or not.
The method will here be detailed for a J = O+ state in the Yukawa model but it can be applied to any coupling. Results will be presented for scalar and pseudo-scalar exchange. This latter furthermore exhibits some strange particularities which will be discussed. 2. Scalar
exchange
Let us consider a system of two fermions in a J” = O+ state, bound by a scalar exchange, whose Lagrangian density is given by L = g,G@Q. Its wave function, constructed using all possible spin structures, is determined in the O+ case by two components [6], fi and fi, which depend on the two scalar variables k and 8: a’.
[/cx fi]
ksin0
f2
t
ffYW,I
z is the momentum of one particle in the system of reference where & + & = 0, fi is the spatial part of the normal w_to the light-front plane, 0 is the angle between k and fi, and w is the two component spinor. The appearance of a second component compared to the non relativistic case is due to vector ?L, which induces additional spin structures. fi and f2 satisfy the system of coupled equations : [Ad2 - 4(k2 + m2)] fl(k,b’) =
$/[Knfi(k’$)+Klzfz(k',e')l
0920-5632/02/s - see front matter 0 2002 Elsevier Science B.V. All rights reserved PI1 SO920-5632(02)01340-3
*&k'
M. Mangin-Brinet
260
et al. /Nuclear
Physics B (Proc. Suppl.) 108 (2002) 259-263
Since Kss is repulsive and does not generate collapse, we consider only the first channel. have
[M” - 4(k2 + m2)] f2(k, e> =~J[K21fi(k’,e’)+K2212(k’iBoi~(l~ &k’
M2 is the total mass squared
of the system,
the constituent mass and &k = JG. kernels Kij result from a first integration elementary quantities:
m is
The of more
Allvuw
=
=
s
-
fill -
--cm
J .
The cou-
[2k2k’2 + 3k2m2 + 3k12m2 + 4m4
-2kk’Ek&k! COS e COS 8' -kk’(k2
+ k’2 + 2m2) sin 0 sin 8’ cos cp’]
nf2 = -arm(k2 - k’2) (k’ sin 8’ + k sin 0 cos cp’) s - -cr7rm(kr2 - k2) (k sin 0 + k’ sin 8’ cos cp’) n21 s - -cm [(2k2k’2 + 3k2m2 + 3k’2m2 + 4m4 K.22 -
-2kk’EkEkl COSeCOSe')COScp' -
kk’( k2 + k’2 + 2m2) sin 0 sin 0’1
(2)
In practice, the integration region over the momenta is reduced to a finite domain [0, k,,,]. The kinematical term [M2 - 4(k2 + m2)] on 1.h.s. of equation (1) does not generate any singularity and the kernels K, are smooth functions of the 0 variable. Thus, the stability of the solution depends only on the asymptotical behavior of the kernels in the (k, k’) plane. Variables (k, k’) can tend to infinity following different directions: for a fixed value of k, KII decreases as l/k’, and vice versa. As the integration volume contains the factor EL’, this means that the total kernel decreases as 1/k’2, that is like a Yukawa potential. In contrast, Kss does not decrease in any direction of the (k, k’) plane, but tends to a positive constant with respect to k and k’. K22 is thus asymptotically repulsive and does not generate any instability. In the domain where both k, k’ tend to infinity with a fixed ratio $ = y, it is useful to introduce the functions Aij defined by Kij = -2
fi Ai,(e,6y) -& Aij(8, O’, l/y)
ify
x
{2y(1-
cOseCose')
- (1
+~~)sin0sin8’coscp’}
where
“ij d@ ,-J (Q2 + /‘2)m2&k&kl %’
where f~ij depend on the type of coupling. analytical expressions of ~~ij for the scalar pling, read
GD
0
2n
Kij(k, 6’; k’, 0’)
-$J2Tdq+ 1
any We
1 case - COSe’I - cosecose~) 8’ cos ‘p’
D = (1-t r2)(1 +
-
2y sin
e sin
Let us now majorate the function All. For fixed y, the maximum of AlI is achieved at 8 = 0’ and for any 8 = 0’ it reads: AlI (f3 = Of,y) = a’fi. The maximum value of kernel KIT is thus reached for y = 1. The majorated kernel obtained in this way coincides with the non-relativistic potential U(T) = -cr’/r2 in the momentum space with (Y’= cu/(2mn). As is well known [7], for this potential, the binding energy does not depend on cutoff if cy’ < ocr = 1/(4m) what restricts the coupling constant to: Q < n/2. If (Y’ > 1/(4m), the binding energy is cutoff dependent and tends to -co when k,,, + co. A finer majoration of All was done by taking into account its dependence on y [8]. In this way we have found cyCr = r, instead of 7r/2. As the kernel was majorated, the critical coupling constant is expected to be larger than 7r. It can be determined, together with the asymptotical behavior of the wave functions, by considering the limit k + co of equation (1) for fi -4f(k,
z) = dz’K(k, z; yk, z’)f(yk,
$i*ydy[;
z’)
where we have neglected the binding energy, supposing that it is finite, and omitted the indices of fr and KIT. This can also be written 1 1 4f(k,z) = ‘y dz’ dr A(07 6 7) * -1 0
J J
y3’2f(yk,
x
z’) + Y512f(
;, z’)}
{ Looking
for a solution
which behaves
as
(3)
M. Mangin-Brinet et al. /Nuclear Physics B (Proc. Suppl.) 108 (2002) 259-263
we are led for h(z) to the eigenvalue
261
equation
+i h(z) = cr
dz’Hp(z,
2’) h(z’)
s -1
.
with
’ -A(z, dr
Hp(z,z’) =
2.05
z’, y) cash @logy).
M2
2nJ;7
0.6
The relation between the coupling constant (Yand the coefficient /3, determining the power law of the asymptotic wave function, can be found in practice by solving the following eigenvalue equation for a fixed value of ,~3 X&z)
=
+l dz’Q(z, I
j
,_____
2.00
z’) h(z’)
03 0.0 -0.3 -06
(5)
-1
and by taking (Y(P) = l/X@. The relation o(p) obtained in that way is represented in Figure 1. The value p = 0 corresponds to the maximal that is the critical - value of CK CQ = (Y(P = 0) = 3.72, in agreement with the previous analytical estimates. It is independent of the exchanged mass p. 4,...,.,.',..'...',..'...'...'...'...'
Figure 2. Cutoff dependence of the binding energy in the J = O+ state (p = 0.25), in the onechannel problem (fi), for two fixed values of the coupling constant below and above the critical value.
.,, 10'
l.
10" r
._
l
l. lo-'
F
0.
. f&3=0)
l.
0.
l.
lo-'
:j
“;;’
,,,,,,,,,,,,,,,,,,,,,,,,,,, 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.6
0.9
.n*....* lo*
.
.
f,(e=O)
l. . .
l.
0.
I
p=o.eno+/- 0.002 fko.OOo+/- 0.002
.
1
... .
lOA 10.' . .
lOA
1
P
lo-'
1
10
100
k Figure 1. Function a(/?) for LFD Yukawa model with Kii channel only.
Figure 2 shows the two different regimes, when the coupling constant is below ((Y = 3) or above (o = 4) the critical value Q,. As it can be seen
Figure 3. Asymptotical behavior of the J = O+ wave function components fi for B=0.05, cy=1.096, ~=0.25. The slope coefficient are pi = 0.82 and /3z M 0.
M. Mangin-Brinet
262
et al. /Nuclear
Physics B (Proc. Suppl.) 108 (2002) 259-263
in Figure 3, the wave functions accurately follow the power law asymptotical behavior l/lc2+17 with a coefficient p(o) given in Figure 1. It is worth noticing that - at least in the framework of this model - one could measure the coupling constant from the asymptotic behavior of the bound state wave function. A similar study has been done for the J = l+ state, which is shown to be unstable without regularization [8,9]. 3. Pseudo-scalar
90
'.
The sion is Figure k = 0 constant
kernel Kss(Ic, k’, 8,0’, M2), whose expresexplicitly given in [3], is represented in 5 for fixed values of 0,13’. It vanishes for or k’ = 0 and tends towards a positive in all the (k, k’) plane.
Figure
5. Kss kernel in (k, k’) plane.
coupling
The stability of the pseudo-scalar (PS) coupling is analyzed similarly to the scalar one. The same method leads to the conclusion that the equations for the PS coupling are quite surprisingly stable without any regularization. However, the results show a “quasidegeneracy” of the coupling constant, for a One has for wide range of binding energies. instance (see Figure 4) cr = 49.5 for a system with B = 0.001, and o = 48.6 for a system five hundred times more deeply bound (B = 0.5), that is only a 2% difference.
60
from the second channel:
-
8~0.5
---
B=O.OOl
Let us model this kernel by a kind of “potential barrier” in the momentum space (k, k’), displayed in Figure 6, whose advantage is to be analytically solvable.
\
70
:P
i
ci 60
50
40
10
100 Lx
I
10 00
2
Figure 6. Modelling in the (k, k’) plane.
of Ks2 with a simpler
K(k, k’) = $@(k’ Figure 4. Convergence of coupling constants as function of the cutoff Ic,,, for B = 0.001 and B = 0.5. The exchange mass is p = 0.15. This peculiar
behavior
can be shown to come
x [O(k - kl)O(k,
- kl)O(kz
kernel
- k’)
- k) + O(k - kz)O(k,,,
- k)]
+ O(k’ - kz)O(k,,, x O(k:! - k) + $O(k
- kz)@(k,,,
- k)
1
M. Mangin-Brinet
et al. /Nuclear
Physics B (Proc. Suppl.) 108 (2002) 259-263
with O(Z) = 1, 2 > 0 and O(Z) = 0, z <_ 0. This kernel has the same characteristics as K&s since it vanishes when k, k’ + 0, and tends towards a constant when (k, k’) go to infinity with a fixed ratio y = k//k. fi satisfies the Schrijdinger type equation
263
Table 1 Coupling constant as a function of the binding energy for the analytical model of the PS coupling second channel.
(7) =-& / K(k, k’)fdk’)~d3k’
[k2+ ~21fdk)
IE~ = m2 - q. We assume that k1 < kz < km,, et UZ < Ur . The term .skt in the volume element of (6) was replaced by its large momentum behavior, that is by k’. We define l?(k) = [k2 + K2]fi(k). The equation for I’(k), which is analytically solvable, reads:
with
+CC
r(k) =
-$ /_
r(k’)
K(k, k’)m”‘dk’
00
The solution I’(k) is constant kz < k < k,,, : l?(k)
for kl < k < k2 and
=
rlO(k
+
I’zO(k - kz)@(k,,,
The Fi satisfy
- k1)O(k2 - k)
the coupled
- k)
Acknowledgements: The numerical calculations were performed at CGCV (CEA Grenoble) and IDRIS (CNRS). We are grateful to the staff members of these organizations for their constant support.
equations
rl = -aulbr2 (I+ au2b) r2 = -aularl
(I + oura) {
where we have defined
Ui = $lJi
through logarithms in a, b, eq. (8). Besides, the value of IE is much smaller than kl , k2, km,, . This explains the very weak dependence of or(~) V.S. n. We conclude from the above discussion that, even if the PS coupling does not formally need any regularization to insure its stability, calculations without form factors - though analytically understood - lead to results which are hardly interpretable from the physical point of view.
and REFERENCES
(8) Replacing we finally form:
h(k)
= -
I’(k) by its definition in terms of fi(k), get the solution of equation (7) in the
j&y
[W - kl>O(kz- k)
aaul O(k - k2)@(k,,, (1 + cubuz)
where N is a normalisation constant. n the coupling constant is
- k)
1
For a given
CY(~)= (au1 + bu2) + d\/(au, - bu2)2 + 4$ab 2abul (UI and the results are summarized
provided in Table
~2)
by this simple kernel 1. a(~) depends on n
1.
V.A. Karmanov, ZhETF 71 (1976) 399 (transl.: JETP 44 (1976) 210). 2. S.J. Brodsky, H.-C. Pauli and S.S. Pinsky, Phys. Rep., 301 (1998) 299. Thbse Uniuersite’ de Paris 3. M. Mangin-Brinet, (2001). 4. M. Mangin-Brinet, J. Carbonell and V.A. Karmanov, submitted for publication. B. Desplanques, V.A. Kar5. J. Carbonell, manov and J.-F. Mathiot, Phys. Reports 300 (1998) 215. J. Carbonell and V.A. Karmanov, Nucl. Phys. 6. A589 (1995) 713. 7. L.D. Landau, E.M. Lifshits, Quantum mechanics, $35, Pergamon press, 1965. J. Carbonell and V.A. 8. M. Mangin-Brinet, Karmanov, Phys Rev D 64 (2001) 027701. 9. M. Mangin-Brinet, J. Carbonell and V.A. Karmanov, accepted in Phys Rev D (2001).