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Vacuum stability in a two-Higgs model
Physics Letters B 280 (1992) 267-270 North-Holland
PHYSICS LETTERS B
Vacuum stability in a two-Higgs model Jan Freund, Georg Kreyerhoff and Rudolf Rodenberg Physikalisches Institut ILIA, RWTH Aachen, Physikzentrum, W-5100Aachen, FRG Received 25 October 1991; revised manuscript received 30 January 1992
Rising experimental lower bounds on the top quark mass raises the question of vacuum stability. Using the renormalization group improved effective potential we investigate this question and look for bounds on the mass spectrum.
1. Introduction It is well known that large Yukawa couplings offermions tend to destabilize the effective potential o f the scalar sector o f the standard model. This fact has become especially interesting after recent analysis of experimental results seem to favour a top quark mass ranging from 90-180 GeV [ 1 ]. In the framework of the standard model with one Higgs doublet this question has been discussed in refs. [2,3 ], where also some comments on the two-doublet case are given. They obtained lower bounds on the Higgs mass using a renormalization group improvement o f the effective potential formalism in order to sum up leading logarithmic quantum corrections [4]. Here we extend their analysis to obtain a lower b o u n d on the mass of the heavier neutral scalar in the two-Higgs model. In the following we require vacuum stability up to the Planck scale, but our results remain unchanged when the cut-offscale is lowered down to ~ l0 m GeV #1
2. Stability bounds in the two-Higgs model The model we like to discuss here is the simplest extension of the standard model as discussed in refs. [ 3,5 ]. It consists o f two Higgs doublets both with hypercharge Y-- + 1. The Higgs potential reads *~ We do not consider the possibility of our universe being in a metastable vacuum slate here.
+23(0+¢,)(~0~)+2,(¢)t¢)~)(¢+0,) + ½25 [ ( ~ i ~ 2 ) 2 + (02+0,)21 •
( 1)
The Yukawa couplings are chosen such that 02 couples to the up-like quarks, so that instabilities o f the potential are expected in the ~1 = 0 direction. After SSB five physical Higgs bosons survive: two neutral scalars, a pair o f charged scalars and one neutral pseudoscalar. Essentially the neutral scalar sector with the tree-level mass spectrum m ~ = ½[ 2 , v ,~ +22v~ +x/(2,v2-22v2)2+4(23+24+2s)2v~v21
(2)
will be affected by the destabilizing effects o f the Yukawa-couplings (The subscripts h, ~ denote the heavier and the lighter one o f the two scalars, respectively) while the mass o f the charged and pseudoscalar scalars are given by m+------½(24+25)v 2 ,
mp----.--25v2
(3)
with v 2 = v 2 + v22. Stability bounds will be found on mh and it is obvious that an absolute lower bound on mh will be achieved for 2 = 2 3 + 2 4 + 2 5 = 0 if the influence o f a variation o f 2 on the renormalization group flow is not too large. Numerically we found, that 2 = 0 gives an absolute lower bound on mh for tan P=v2/vl ,~ 1. For larger values o f tan fl a variation o f 2 has only a
0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
267
Volume 280, number 3,4
PHYSICS LETTERS B
small influence on the mass spectrum and we found that for tan fl> 4 an absolute lower bound on mh is found, if2 is made as large as possible. With the choice 2 = 0 the neutral scalar components of the Higgs doublets ~2 and form mass eigenstates with masses ~1
m E=max(21 v21,42/)
2)
,
rn~ = min (2, v2, 22v22) .
(4)
On tree level the remaining quartic couplings can now be written as a function of the mass spectrum. We like now to include radiative corrections to these masses which are obtained from the second derivatives of the effective potential #2. The easiest way to do this is to choose renormalization conditions such that the VEV's are not shifted by radiative corrections. For the top and gauge contributions it is possible to rescale the logarithms for every mode independently so that not only the sum of the radiative corrections to the VEV shift vanishes but also every single contribution to the shift. Their corrections to the mass matrix are then given by (see also ref. [ 6 ] ) AM2 = 32n 21
0M 20M],
( -- )FCk 00i
OOj '
(5)
where the sum runs over the eigenmodes of the generalized mass matrix (interaction matrix) of the model, Ck is a spin and colour dependent factor and F=0,1 for bosons or fermions, respectively. E.g. the top contributions, which are the most dominant, to the neutral scalar mass matrix can then be written as AM2=
-
_3
mt4op (0
2n2v2sin2p\O ~).
(6)
The evaluation of the scalar contributions is more involved. Since they are very small we do not discuss them here and refer the reader to ref. [ 7 ]. The resulting expression have now to be inverted in order to obtain the parameters of the potential as a function of the mass spectrum. Since this function is quite complicated we adopt the following procedure: First use the tree-level formulae (4) to get an approximation to the quartic couplings and use them to evaluate the radiative corrections to the mass spectrum. ~2 These are not exactly the physical masses which are given by the poles of the propagators, but the difference turns out to be small as long as the mass of the particle inside the self-energy loops is not too large.
268
30 April 1992
Subtracting these from the input masses gives the treelevel masses from which the correct quartic couplings are obtained. With these input parameters the RGE improved effective potential in the ~ = 0 direction is evaluated numerically and checked for stability. To discuss our results we begin with the case tan fl~ 1. We found that the results obtained here approximately hold for ½< tan r < 2. We find here two different lower bounds on mh: a trivial bound stating rnh > m~ and a genuine stability bound, where the latter one turns out to be quite independent from rn, for rn~ < 100 GeV as shown in fig. 1. It is however not possible to choose rn~=0 consistently because there is a Linde-Weinberg-like lower bound on this mass (this means that for small m~ the symmetric minimum or a minimum configuration, where only one of the doublets acquires a VEV is preferred). Since we do not want to discuss this kind of vacuum instability we choose ms = 20 GeV in the following. In the interesting parameter range (rntop large) one then has to choose the heavier scalar to correspond to the neutral component of 02. Otherwise we have 22 very small and thus always an instable potential. If the vacuum is stable or not mainly depends on the initial values of 22 and htop. Therefore the stability bounds become nearly independent from tan fl if one plots mh/ sin (radiative corrections) versus
fl=x/~2v+
mtop/
>180
/
fi'160 140 nn,= 120GeV
120 100
/
m, = 90 GeV
80 m, = 60 GeV
60 4O
m~ = 30 OeV
2O i
060
,
r
i
~
I
80
i
i
i
,
r
i
100
i
J
i
I
120
~
i
i
i
I
140
i
i
i
L
[
i
i
,
J
160 180 m~/GeV
Fig. 1. Stability bound on m h for different fixed mass of the lighter scalar, valid for tan fl~ 1. Other input values: m ± = 80 GeV, mp= 0 GeV, tan fl= 2.
Volume 280, number 3,4
PHYSICS LETTERS B
htopV/xf2
sin #= t,3. Thus the only other free parameters left are the masses of the charged and pseudoscalar Higgs fields. In fig. 2 we plotted our results. As can be expected from the renormalization group ~-function [ 8 ] for 7z an increase of m_+ and mp will decrease the lower bound on mh. Further restrictions on the mass-specn3 The top contributions to the radiative corrections also depend only on mtop/sin#, so that a tan #, dependence enters only through the small gaugeand scalar corrections.
30 April 1992
trum might be obtained from the fact that sin fl can not be arbitrarily small in order to avoid Landau singularities of the Yukawa couplings. This yields sin fl> with h m ~ 1.2 or ~ 1.3 if one requires finite Yukawa couplings up to the Planck- or G U T scale, respectively. For tan fl> 4 we found that an absolute lower bound is obtained if ,l is made as large as possible. On tree level 2 is restricted by
x/~rntop/hma,,V
m -m, [2[ < v2sin 2//"
(7)
(a)
m./sin#
160 140 120 -
iiii
oo806040-
m . / s i n ~ 1201C~0 r~L~.~-~~' GeV
80 6~0
.
ii
80
60
m -I-/GeV
This also holds after top and gauge contributions to the radiative corrections are taken into account, while scalar corrections slightly change this bound. For a given mh a maximal 2 is obtained by the choice rnQ= 0. As long as rn~ is only varied in a range where the quartic couplings remain in the perturbative domain, the choice m~ = 0 gives the most restrictive bound. Also the range of 2 has to be constrained, so that for a given set of masses all couplings remain small enough. Of course the stability bounds now depend on the maximal allowed value for the couplings and if this value is chosen too large there is no stability bound at all, since one of the couplings might diverge before an instability of the potential arises. In fig. 3 we calculated upper bounds on the top mass for a given mh
(b)
m./sin#
m~/GeV 220 ~ 200
80 ........ •.... 60 ......... ~ 40 ........
1
8
1
2
0
"
0
~
~
14o \
~%
m./GeV
80 6 0 " ~ 40
0 m~/s,n# - G-~
120 ~ 0 100 'r~ 20 80
6 -
m,/GeV
Fig. 2. Lower bounds on mJsin ,8 as a function of mtop/Sinfl and m± (a) and mp (b) for m~=0. In (a) we have ?r/p=50 GeV and in (b) m+---90 GeV.
3
tan~
Fig. 3. Upper bound on the top-mass as a function of mh valid for tan #> 4 (for smaller tan tithe bounds are up to l0 GeV lower than indicated here). Other input values: m ± =mp = 100 GeV.
269
Volume 280, number 3,4
PHYSICS LETTERS B
a n d the corresponding m a x i m a l 2. This b o u n d has been calculated for m_+ = mp = 100 GeV. Smaller values for these masses do not lower the b o u n d by m o r e than 10 GeV, while increasing rn± to 150 G e V has effect o f lifting the b o u n d b y up to 20 G e V especially for small values o f mh. The initial values for the quartic couplings were restricted in the range 2 , < 1. Requiring smaller values so that all couplings r e m a i n finite up the G U T - or Planck scale lowers the b o u n d s by an a m o u n t o f 10 GeV, especially for large tan and mh.
3. Conclusion The b o u n d s f o u n d above turn out to be very similar to the b o u n d s found in the one-Higgs model, they are only reduced by the factor sin p. Since this factor can not be arbitrarily small, these b o u n d s are still severely restricting the allowed region in the p a r a m e t e r space. N o t very much, however, can be said on the mass o f the lighter scalar, b u t it is possible to estimate an upper b o u n d on m~ from requiring finite quartic couplings up to a unification scale. F o r the case 2 = 0 these b o u n d s are a p p r o x i m a t e l y the same as in the oneHiggs case (evaluated with vanishing Yukawa couplings), only that they are reduced by a factor cos ft. F r o m the results o f ref. [9] one finds for a cut-off at the Planck scale rn~< 160 GeV × cos ft. This b o u n d is lowered for 2 # 0, so that a mass below or a r o u n d the Z mass seems to be favoured for the lighter scalar. W h a t happens, if particles violating these b o u n d s
270
30 April 1992
are found? F i n d i n g only one relatively light scalar might be an indication that a two- (or m o r e ) - H i g g s m o d e l is realized, either s u p e r s y m m e t r i c or not. It is also possible that the Higgs m e c h a n i s m is realized in a completely different manner, such as in models with composite Higgses, where such stability p r o b l e m s do not exist [ 10 ].
References [ 1] J. Ellis and G.L. Fogli, Phys. Lett. B 232 (1989) 139; P. Langacker, University of Pennsylvania report UPR 0435 T (1990); M. Drees, DESY preprint DESY 91-045 ( 1991 ). [2] N. Cabibbo, L. Maiani, G. Parisi and R. Petronzio, Nucl. Phys. B 158 (1979) 295; M. Lindner, M. Sher and W. Zaglauer, Phys. Left. B 228 (1989) 139. [3] M.J. Duncan, R. Phillipe and M. Sher, Phys. Lett. B 165 (1985) 165; M. Sher, Phys. Rep. 179 (1989) 273. [ 4 ] S. Coleman and E. Weinberg, Phys. Rev. D 7 (1973 ) 1888; J. Illiopoulos, C. Itzykson and A. Martin, Rev. Mod. Phys. 47 (1975) 165; D.J. Amit, Field theory, renormalization group and critical phenomena ( World Scientific, Singapore, 1984). [ 5 ] N.D. Deshpande and E. Ma, Phys. Rev. D 18 ( 1978 ) 2574. [6] R. Rodenberg, Proc. Intern. Seminar Quarks '88 (Tblissi, USSR), eds. A.N. Tavkhelidze et al. (World Scientific, Singapore) p. 316; Nuovo Cimento A 96 (1986) 52; A 99 (1988) 95. [7]G. Kreyerhoff, Aachen internal report (1992), in preparation. [ 8 ] C.T. Hill, C.N. Leung and S. Rao, Nucl. Phys. B 262 (1985) 517. [9] M. Lindner, Z. Phys. C 31 (1986) 295. [ 10] N.V. Krasnikov and R. Rodenberg, Aachen preprint PITHA 91-15, and references therein.
PHYSICS LETTERS B
Vacuum stability in a two-Higgs model Jan Freund, Georg Kreyerhoff and Rudolf Rodenberg Physikalisches Institut ILIA, RWTH Aachen, Physikzentrum, W-5100Aachen, FRG Received 25 October 1991; revised manuscript received 30 January 1992
Rising experimental lower bounds on the top quark mass raises the question of vacuum stability. Using the renormalization group improved effective potential we investigate this question and look for bounds on the mass spectrum.
1. Introduction It is well known that large Yukawa couplings offermions tend to destabilize the effective potential o f the scalar sector o f the standard model. This fact has become especially interesting after recent analysis of experimental results seem to favour a top quark mass ranging from 90-180 GeV [ 1 ]. In the framework of the standard model with one Higgs doublet this question has been discussed in refs. [2,3 ], where also some comments on the two-doublet case are given. They obtained lower bounds on the Higgs mass using a renormalization group improvement o f the effective potential formalism in order to sum up leading logarithmic quantum corrections [4]. Here we extend their analysis to obtain a lower b o u n d on the mass of the heavier neutral scalar in the two-Higgs model. In the following we require vacuum stability up to the Planck scale, but our results remain unchanged when the cut-offscale is lowered down to ~ l0 m GeV #1
2. Stability bounds in the two-Higgs model The model we like to discuss here is the simplest extension of the standard model as discussed in refs. [ 3,5 ]. It consists o f two Higgs doublets both with hypercharge Y-- + 1. The Higgs potential reads *~ We do not consider the possibility of our universe being in a metastable vacuum slate here.
+23(0+¢,)(~0~)+2,(¢)t¢)~)(¢+0,) + ½25 [ ( ~ i ~ 2 ) 2 + (02+0,)21 •
( 1)
The Yukawa couplings are chosen such that 02 couples to the up-like quarks, so that instabilities o f the potential are expected in the ~1 = 0 direction. After SSB five physical Higgs bosons survive: two neutral scalars, a pair o f charged scalars and one neutral pseudoscalar. Essentially the neutral scalar sector with the tree-level mass spectrum m ~ = ½[ 2 , v ,~ +22v~ +x/(2,v2-22v2)2+4(23+24+2s)2v~v21
(2)
will be affected by the destabilizing effects o f the Yukawa-couplings (The subscripts h, ~ denote the heavier and the lighter one o f the two scalars, respectively) while the mass o f the charged and pseudoscalar scalars are given by m+------½(24+25)v 2 ,
mp----.--25v2
(3)
with v 2 = v 2 + v22. Stability bounds will be found on mh and it is obvious that an absolute lower bound on mh will be achieved for 2 = 2 3 + 2 4 + 2 5 = 0 if the influence o f a variation o f 2 on the renormalization group flow is not too large. Numerically we found, that 2 = 0 gives an absolute lower bound on mh for tan P=v2/vl ,~ 1. For larger values o f tan fl a variation o f 2 has only a
0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
267
Volume 280, number 3,4
PHYSICS LETTERS B
small influence on the mass spectrum and we found that for tan fl> 4 an absolute lower bound on mh is found, if2 is made as large as possible. With the choice 2 = 0 the neutral scalar components of the Higgs doublets ~2 and form mass eigenstates with masses ~1
m E=max(21 v21,42/)
2)
,
rn~ = min (2, v2, 22v22) .
(4)
On tree level the remaining quartic couplings can now be written as a function of the mass spectrum. We like now to include radiative corrections to these masses which are obtained from the second derivatives of the effective potential #2. The easiest way to do this is to choose renormalization conditions such that the VEV's are not shifted by radiative corrections. For the top and gauge contributions it is possible to rescale the logarithms for every mode independently so that not only the sum of the radiative corrections to the VEV shift vanishes but also every single contribution to the shift. Their corrections to the mass matrix are then given by (see also ref. [ 6 ] ) AM2 = 32n 21
0M 20M],
( -- )FCk 00i
OOj '
(5)
where the sum runs over the eigenmodes of the generalized mass matrix (interaction matrix) of the model, Ck is a spin and colour dependent factor and F=0,1 for bosons or fermions, respectively. E.g. the top contributions, which are the most dominant, to the neutral scalar mass matrix can then be written as AM2=
-
_3
mt4op (0
2n2v2sin2p\O ~).
(6)
The evaluation of the scalar contributions is more involved. Since they are very small we do not discuss them here and refer the reader to ref. [ 7 ]. The resulting expression have now to be inverted in order to obtain the parameters of the potential as a function of the mass spectrum. Since this function is quite complicated we adopt the following procedure: First use the tree-level formulae (4) to get an approximation to the quartic couplings and use them to evaluate the radiative corrections to the mass spectrum. ~2 These are not exactly the physical masses which are given by the poles of the propagators, but the difference turns out to be small as long as the mass of the particle inside the self-energy loops is not too large.
268
30 April 1992
Subtracting these from the input masses gives the treelevel masses from which the correct quartic couplings are obtained. With these input parameters the RGE improved effective potential in the ~ = 0 direction is evaluated numerically and checked for stability. To discuss our results we begin with the case tan fl~ 1. We found that the results obtained here approximately hold for ½< tan r < 2. We find here two different lower bounds on mh: a trivial bound stating rnh > m~ and a genuine stability bound, where the latter one turns out to be quite independent from rn, for rn~ < 100 GeV as shown in fig. 1. It is however not possible to choose rn~=0 consistently because there is a Linde-Weinberg-like lower bound on this mass (this means that for small m~ the symmetric minimum or a minimum configuration, where only one of the doublets acquires a VEV is preferred). Since we do not want to discuss this kind of vacuum instability we choose ms = 20 GeV in the following. In the interesting parameter range (rntop large) one then has to choose the heavier scalar to correspond to the neutral component of 02. Otherwise we have 22 very small and thus always an instable potential. If the vacuum is stable or not mainly depends on the initial values of 22 and htop. Therefore the stability bounds become nearly independent from tan fl if one plots mh/ sin (radiative corrections) versus
fl=x/~2v+
mtop/
>180
/
fi'160 140 nn,= 120GeV
120 100
/
m, = 90 GeV
80 m, = 60 GeV
60 4O
m~ = 30 OeV
2O i
060
,
r
i
~
I
80
i
i
i
,
r
i
100
i
J
i
I
120
~
i
i
i
I
140
i
i
i
L
[
i
i
,
J
160 180 m~/GeV
Fig. 1. Stability bound on m h for different fixed mass of the lighter scalar, valid for tan fl~ 1. Other input values: m ± = 80 GeV, mp= 0 GeV, tan fl= 2.
Volume 280, number 3,4
PHYSICS LETTERS B
htopV/xf2
sin #= t,3. Thus the only other free parameters left are the masses of the charged and pseudoscalar Higgs fields. In fig. 2 we plotted our results. As can be expected from the renormalization group ~-function [ 8 ] for 7z an increase of m_+ and mp will decrease the lower bound on mh. Further restrictions on the mass-specn3 The top contributions to the radiative corrections also depend only on mtop/sin#, so that a tan #, dependence enters only through the small gaugeand scalar corrections.
30 April 1992
trum might be obtained from the fact that sin fl can not be arbitrarily small in order to avoid Landau singularities of the Yukawa couplings. This yields sin fl> with h m ~ 1.2 or ~ 1.3 if one requires finite Yukawa couplings up to the Planck- or G U T scale, respectively. For tan fl> 4 we found that an absolute lower bound is obtained if ,l is made as large as possible. On tree level 2 is restricted by
x/~rntop/hma,,V
m -m, [2[ < v2sin 2//"
(7)
(a)
m./sin#
160 140 120 -
iiii
oo806040-
m . / s i n ~ 1201C~0 r~L~.~-~~' GeV
80 6~0
.
ii
80
60
m -I-/GeV
This also holds after top and gauge contributions to the radiative corrections are taken into account, while scalar corrections slightly change this bound. For a given mh a maximal 2 is obtained by the choice rnQ= 0. As long as rn~ is only varied in a range where the quartic couplings remain in the perturbative domain, the choice m~ = 0 gives the most restrictive bound. Also the range of 2 has to be constrained, so that for a given set of masses all couplings remain small enough. Of course the stability bounds now depend on the maximal allowed value for the couplings and if this value is chosen too large there is no stability bound at all, since one of the couplings might diverge before an instability of the potential arises. In fig. 3 we calculated upper bounds on the top mass for a given mh
(b)
m./sin#
m~/GeV 220 ~ 200
80 ........ •.... 60 ......... ~ 40 ........
1
8
1
2
0
"
0
~
~
14o \
~%
m./GeV
80 6 0 " ~ 40
0 m~/s,n# - G-~
120 ~ 0 100 'r~ 20 80
6 -
m,/GeV
Fig. 2. Lower bounds on mJsin ,8 as a function of mtop/Sinfl and m± (a) and mp (b) for m~=0. In (a) we have ?r/p=50 GeV and in (b) m+---90 GeV.
3
tan~
Fig. 3. Upper bound on the top-mass as a function of mh valid for tan #> 4 (for smaller tan tithe bounds are up to l0 GeV lower than indicated here). Other input values: m ± =mp = 100 GeV.
269
Volume 280, number 3,4
PHYSICS LETTERS B
a n d the corresponding m a x i m a l 2. This b o u n d has been calculated for m_+ = mp = 100 GeV. Smaller values for these masses do not lower the b o u n d by m o r e than 10 GeV, while increasing rn± to 150 G e V has effect o f lifting the b o u n d b y up to 20 G e V especially for small values o f mh. The initial values for the quartic couplings were restricted in the range 2 , < 1. Requiring smaller values so that all couplings r e m a i n finite up the G U T - or Planck scale lowers the b o u n d s by an a m o u n t o f 10 GeV, especially for large tan and mh.
3. Conclusion The b o u n d s f o u n d above turn out to be very similar to the b o u n d s found in the one-Higgs model, they are only reduced by the factor sin p. Since this factor can not be arbitrarily small, these b o u n d s are still severely restricting the allowed region in the p a r a m e t e r space. N o t very much, however, can be said on the mass o f the lighter scalar, b u t it is possible to estimate an upper b o u n d on m~ from requiring finite quartic couplings up to a unification scale. F o r the case 2 = 0 these b o u n d s are a p p r o x i m a t e l y the same as in the oneHiggs case (evaluated with vanishing Yukawa couplings), only that they are reduced by a factor cos ft. F r o m the results o f ref. [9] one finds for a cut-off at the Planck scale rn~< 160 GeV × cos ft. This b o u n d is lowered for 2 # 0, so that a mass below or a r o u n d the Z mass seems to be favoured for the lighter scalar. W h a t happens, if particles violating these b o u n d s
270
30 April 1992
are found? F i n d i n g only one relatively light scalar might be an indication that a two- (or m o r e ) - H i g g s m o d e l is realized, either s u p e r s y m m e t r i c or not. It is also possible that the Higgs m e c h a n i s m is realized in a completely different manner, such as in models with composite Higgses, where such stability p r o b l e m s do not exist [ 10 ].
References [ 1] J. Ellis and G.L. Fogli, Phys. Lett. B 232 (1989) 139; P. Langacker, University of Pennsylvania report UPR 0435 T (1990); M. Drees, DESY preprint DESY 91-045 ( 1991 ). [2] N. Cabibbo, L. Maiani, G. Parisi and R. Petronzio, Nucl. Phys. B 158 (1979) 295; M. Lindner, M. Sher and W. Zaglauer, Phys. Left. B 228 (1989) 139. [3] M.J. Duncan, R. Phillipe and M. Sher, Phys. Lett. B 165 (1985) 165; M. Sher, Phys. Rep. 179 (1989) 273. [ 4 ] S. Coleman and E. Weinberg, Phys. Rev. D 7 (1973 ) 1888; J. Illiopoulos, C. Itzykson and A. Martin, Rev. Mod. Phys. 47 (1975) 165; D.J. Amit, Field theory, renormalization group and critical phenomena ( World Scientific, Singapore, 1984). [ 5 ] N.D. Deshpande and E. Ma, Phys. Rev. D 18 ( 1978 ) 2574. [6] R. Rodenberg, Proc. Intern. Seminar Quarks '88 (Tblissi, USSR), eds. A.N. Tavkhelidze et al. (World Scientific, Singapore) p. 316; Nuovo Cimento A 96 (1986) 52; A 99 (1988) 95. [7]G. Kreyerhoff, Aachen internal report (1992), in preparation. [ 8 ] C.T. Hill, C.N. Leung and S. Rao, Nucl. Phys. B 262 (1985) 517. [9] M. Lindner, Z. Phys. C 31 (1986) 295. [ 10] N.V. Krasnikov and R. Rodenberg, Aachen preprint PITHA 91-15, and references therein.