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Anomalous couplings in the Higgs-strahlung process
18 July 1996
PHYSICS
EZSEVIER
LETTERS I3
Physics Letters B 381 (1996) 243-247
Anomalous couplings in the Higgs-strahlung process W. Kilian, M. Krfimer, P.M. Zerwas Deutsches Elektronen-Synchrotron DESI: D-22603 Hamburg, FRG
Received 28 March 1996 Editor: PV. Landshoff
Abstract The angular distributions
in the Higgs-strahlung
process e’e-
--+ HZ 4
Hff
are uniquely determined
in the Standard
Model. We study how these predictions are modified if non-standard couplings are present in the ZZH vertex, as well as lepton-boson contact term. We restrict ourselves to the set of operators which are singlets under standard SUs x SU2 x UI tinsformations, CP conserving, dimension 6, helicity conserving, and custodial SU2 conserving.
1. Introduction The Higgs-strahlung eie-
process
---f HZ + Hff
[ 1] (1)
together with the WW fusion process, are the most important mechanisms for the production of Higgs bosons in e+e- collisions [ 2,3]. Since the ZZH vertex is uniquely determined in the Standard Model (SM), the production cross section of the Higgs-strahlung process, the angular distribution of the HZ final state as well as the fermion distribution in the Z decays can be predicted if the mass of the Higgs boson is fixed [ 41. Deviations from the pointlike coupling can occur in models with non-pointlike character of the Higgs boson itself or through interactions beyond the SM at high energy scales. We need not specify the underlying theory but instead we will adopt the usual assumption that these effects can globally be parameterized by introducing a set of dimension-6 operators C = CSM +
C
(2)
ZCJi
i
0370-2693/96/$12.00
The coefficients are in general expected to be of the order l/h2, where A denotes the energy scale of the new interactions. However, if the underlying theory is weakly interacting, the ai can be significantly smaller than unity, in particular for loop-induced operators. [It is assumed a priori that the ratio of the available c.m. energy to A is small enough for the expansion in powers of l/A to be meaningful.] If we restrict ourselves to operators [ 51 which are singlets under Xl3 x SU2 x UI transformations of the SM gauge group, CP conserving, and conserving the custodial SU2 symmetry, the following bosonic operators are relevant for the Higgs-strahlung process: Qap = ~l~j~(~+Y)l~
(3)
Qqw = &P+w;,q
(4)
c?cp~= $P+B&
(5)
where the gauge fields W3, B are given by the Z, y fields. This set of operators is particularly interesting because it does not affect, at tree level, observables which do not involve the Higgs particle explicitly. [It is understood that the fields and parameters are (re-)
Copyright 0 1996 Elsevier Science B.V. All rights reserved.
PII SO370-2693(96)00537-O
f~+
tl
f+
f-
// A% II
LH contact terms in the Higgs-strahlung process.
normalized in the Lagrangian f2 in such a way that the particle masses and the electromagnetic coupling retain their physical values. 1 In addition, wc consider the I’ollowing helicityconserving I‘ermionic operators which induce contact terms contributing to P ’ f’ ~ ~- ZH:
o/., = ( cpiiDpp) 0,~ =
( L,.y@i, j
-c l1.c..
(p+~“iD,p)(?,,f’yV,_ )
‘3~ = ( &D,p)
( PRypeR)
+
h.c.
A h.c.
(Ii = A,!!;- (C&Y,)$J+ s;~,cu,R)/!\’
c I’)
(h 1
h, = J,y;, ‘C\c’S,~( -apw + a&j 1 /iI2
(13)
(7)
where SW and cw are the sine and cosine of the weak mixing angle, respectively. In the same way the eeHZ contact interactions t Fig. 1. right diagram) can be defined for left/righthanded electrons and right/left-handed positrons
(8)
[ /l, and em denote the left-handed lcpton doublet and the right-handed singlet. respcctivcly. The vatuum expectation value of the Higgs field is given by (9) = (O,I~/&) with 11 = 146GcV, and the covariant derivative acts on the Higgs doublet as D, = ~9~ - +p-“W’~ + ig’fl,.] Helicity-violating fermionic opeiators do not interfere with the SM amplitude, so that their contribution to the cross section is suppressed by another power of ,2’. The helicity-conserving fermionic operators modify the SM Zee couplings and are therefore constrained by the measurements at LEPI; however, it is possible to improve on the existing limits hy measuring the Higgs-strahlung process at a high-energy c * ep collider since the impact on this process increases with energy [h]. The effective ZZH and the induced yZH interactions (Fig. 1, left diagram) may he written ~%ZII = R.z MZ
CyZN = gzMz
where
yz
=
(
1 i- (10 _ -L~Z/lH 2
$Z,,,.rZ”.H
Mz d&?%$.
Z,Zp”“‘d,,H and Z,AI*“J,,H
his: They may he eliminated in favor of the other operators and the contact terms by applying the equations of‘ motion. The remaining coefficients are given by
+ ~%,,,iYH) (9)
(IO) Additional operators are redundant in this ba-
C’( = -2,q/:
’
(‘RZ
’ *K/ii’
- 78;
(CY/.I
3-
Q1.j
)
/iI’
(15)
(16)
Some consequences of these operators for Higgs production in e +e- collisions have been investigated in the past. Most recently, the effect of novel ZZH vertex operators and P?ZH contact terms on the total cross sections for Higgs production has been studied in Ref. 161. The impact of vertex operators on angular distributions has been analyzed in Refs. [ 7 ] and [ 81. We expand on these analyses by studying the angular distributions for the more general case where both novel vertex operators and contact interactions are present. The analysis of angular distributions in the Higgs-strahlung process ( I ) allows us to discriminate between various novel interactions. In fact, the entire set of parameters CIO.LEI,bl and ~1.. cR can be determined by measuring the polar and azimuthal angular distributions as a function of the beam energy if the rlcctron/positron beams are unpolarized. As expected,
W. K&m et al. /Physics Letters B 381 f 19%) 243-247
245
A = [l - (m~+mz)~/s] x [l- (mH - mz)2/s]. The coefficients a( s)L,R and p(s) LR can ea,dy be determined for the interactions in Eqs. (9) and ( 14) : a(
sfyR =2ao+(S-M~)~CL.R c
p(sfqR= a(s) +2y&Mz
Fig. 2. Polar and azimuthal angles in the Higgs-strahlung process. [The polar angle 0, is defined in the Z rest frame. ]
the energy dependence of the polar angular distribution is sufficient to provide a complete set of measurements if longitudinally polarized electron beams are available ’ .
2. Total cross section and polar angular distribution Denoting the polar the 2 boson and the e+e- beam axis by differential be written for process e’eL,R .+ ZH daL+R = dcose
GiMi
(v, * a,)’
K
x aAsin*B
e
(19)
L,R
al + z(l-+]
(20)
where the boost of the Z boson is given by y = (s + M; - M;)/2Mz,h. The modification of the cross section by the new interaction terms has a simple structure. The coefficient aa just renormalizes the SM cross section. By contrast, the contact interactions grow with s. [The ratio s/h* is assumed to be small enough for the restriction to dimension-6 operators to be meaningful.] The operators 0,~~ Ova affect the coefficient in the cross section which is independent of 8. They damp the falloff of this term, changing the l/s2 to a l/s behavior; however, these contributions remain subleading since they are associated with transversely polarized Z bosons which are suppressed at high energies compared with the longitudinal components. To illustrate the size of the modifications a(s) L*Rand p(s) L*R,we have depicted these functions in Fig. 3 (a) for the special choice a; = 1.
Ati 3. Azimuthal
(1+&R)+6(1+/?L,R)M~/~
distributions
(1 - M;/s)* (17) and the integrated ;
a=
$&-
X
cross section
“z (u, *aa,)‘A1/’
A(1 +LYLJ) + 12(1 +pL*R)M$/S (1 - M2,/s)2
(18)
The Z charges of the electron are defined as usual by a, = -1 and v, = -1 + 4.~2,. s is the c.m. energy squared, and A the two-particle phase space coefficient ’ Since we can restrict ourselves to helicity-conserving couplings, as argued before, additional positron polarization need not be required.
The azimuthal angle & of the fermion f is defined as the angle between the [e-, Z] productionplane and the [Z, f] decay plane (Fig. 2). It corresponds to the azimuthal angle of f in the Z rest frame with respect to the [e- , Z] plane. On general grounds, the & distribution must be a linear function of cos #*, cos 24%) and sin&, sin2&, measuring the helicity components of the decaying spin-l Z state. The coefficients of the sine terms vanish for CP invariant theories. The cos #+ and cos 241, terms correspond to P-odd and Peven combinations of the fermion currents. The general azimuthal distributions are quite involved [ 4,7,8]. We therefore restrict ourselves to the simplified case in which all polar angles are integrated out, i.e., the polar angle 6 of the Z boson in the laboratory frame
246
W. Kilian et al. / Physics Letters B 381 (1996) 243-247
and the polar angle 8, off in the 2 rest frame. In this way we find for the azimuthal q5* distribution:
with f,
‘“,-$$2)
(s)LJ = MZfi x [al+s(l--!$)h]
f2(s)
L,R = 2Mzfi
(22) ‘I’:;:’
x [al+z(l-$)bl]
(23)
The cross section flattens with increasing c.m. energy in the Standard Model, i.e. the coefficients of cos #J* and cos 24, decrease asymptotically proportional to 1/fi and 1/s, respectively. The anomalous contributions modify this behavior: The cos & term receives contributions which increase proportional to 4 with respect to the total cross section, while the cos2qL term receive contributions from the transversal couplings that approaches a constant value asymptotically. The size of the new terms in f 1;: is shown in Fig. 3 (b) as a function of the energy. [The special choice CX~ = 1 we have adopted for illustration, implies ft,2 = ft,.]
4. High-energy
behavior of the coefficients
It is instructive to study the high-energy behavior of the coefficients in the limit @ < s < A2. In this case we obtain the simplified relations: a( S)LSR N F .s .8swcw P(s)L.R N a(s>LJ
CL,R + o(b)
(24)
Fig. 3. Coefficients of the angular distributions as a function of the beam energy. Parameters are described in the text; in particular. a; = 1 has been chosen in the effective Lagrangian equation (2). [The L, R coefficients of the azimuthal distribution coincide for the special choice ai = 1.]
Terms which are proportional to v, = -1 + 4~2, are suppressed by an order of magnitude. If longitudinally polarized electrons are available, the asymptotic value of the coefficients al, bl , CL and CR can be determined by measuring the polar angular distribution without varying the beam energy. The analysis of the azimuthal r,k distribution provides two additional independent measurements of the coefficients al and bl. On the other hand, the set of measurements remains incomplete for fixed energy if only unpolarized electron/positron beams are used at high energies; in this case the coefficients cannot be disentangled completely without varying the beam energy within the preasymptotic region.
+ s (at F ~SWCWbt > + WV,) (25)
References and [I]
s fl (s)L,R N - (al i4swcwbt)
2
fg(~)~,~
= S(QI ~4s~c~bt)
+ WV,)
+O(u,)
J. Ellis, M.K. Gaillard and D.V. Nanopoulos, Nucl. Phys. B
(26)
106 (1976) 292; B.L. Ioffe and V.A. Khoze, Sov. J. Part. Nucl. 9 (1978)
(27)
B.W. Lee. C. Quigg
and H.B. Thacker,
Phys.
50; Rev. D 16
W. Kilian et al. /Physics
(1977)
1519;
J.D. Bjorken, Proc. Summer Institute on Particle Physics, SLAC Report 198 ( 1976). [2 1 M. Carena, PM. Zerwas (conv.) et al., Higgs Physics, in: Proc. of the Workshop Physics with LEP2, Eds. 0. Altarelli, T. SjZistrand, and F. Zwirner, CERN Yellow Report 96-01. ]3] Proceedings of the Workshop e+e- Collisions at 500 GeV: The Physics Potential, Munich-Annecy-Hamburg, Ed. PM. Zerwas, Reports DESY 92-123A.B; 93-123C.
Letters B 381 f 19%) 243-247
241
[4] V. Barger, K. Cheung, A. Djouadi, B. Kniehl. and P.M. Zerwas, Phys. Rev. D 49 (1994) 79. [5] W. Buchmtlller and D. Wyler, Nucl. Phys. B 268 (1986) 621, [6] B. Grzadkowski and J. Wudka, Phys. Lea. B 364 (1995) 49. 171 K. Hagiwara and M.L. Stong, Z. Phys. C 62 ( 1994) 99. [S] G.J. Gounaris, EM. Renard and N.D. Vlachos, Nucl. Phys. B 459 (1996) 51; G.J. Gounaris, J. Laissac, J.E. Paschalis, EM. Renard and N.D. Vlachos, Preprint PM/96-08.
PHYSICS
EZSEVIER
LETTERS I3
Physics Letters B 381 (1996) 243-247
Anomalous couplings in the Higgs-strahlung process W. Kilian, M. Krfimer, P.M. Zerwas Deutsches Elektronen-Synchrotron DESI: D-22603 Hamburg, FRG
Received 28 March 1996 Editor: PV. Landshoff
Abstract The angular distributions
in the Higgs-strahlung
process e’e-
--+ HZ 4
Hff
are uniquely determined
in the Standard
Model. We study how these predictions are modified if non-standard couplings are present in the ZZH vertex, as well as lepton-boson contact term. We restrict ourselves to the set of operators which are singlets under standard SUs x SU2 x UI tinsformations, CP conserving, dimension 6, helicity conserving, and custodial SU2 conserving.
1. Introduction The Higgs-strahlung eie-
process
---f HZ + Hff
[ 1] (1)
together with the WW fusion process, are the most important mechanisms for the production of Higgs bosons in e+e- collisions [ 2,3]. Since the ZZH vertex is uniquely determined in the Standard Model (SM), the production cross section of the Higgs-strahlung process, the angular distribution of the HZ final state as well as the fermion distribution in the Z decays can be predicted if the mass of the Higgs boson is fixed [ 41. Deviations from the pointlike coupling can occur in models with non-pointlike character of the Higgs boson itself or through interactions beyond the SM at high energy scales. We need not specify the underlying theory but instead we will adopt the usual assumption that these effects can globally be parameterized by introducing a set of dimension-6 operators C = CSM +
C
(2)
ZCJi
i
0370-2693/96/$12.00
The coefficients are in general expected to be of the order l/h2, where A denotes the energy scale of the new interactions. However, if the underlying theory is weakly interacting, the ai can be significantly smaller than unity, in particular for loop-induced operators. [It is assumed a priori that the ratio of the available c.m. energy to A is small enough for the expansion in powers of l/A to be meaningful.] If we restrict ourselves to operators [ 51 which are singlets under Xl3 x SU2 x UI transformations of the SM gauge group, CP conserving, and conserving the custodial SU2 symmetry, the following bosonic operators are relevant for the Higgs-strahlung process: Qap = ~l~j~(~+Y)l~
(3)
Qqw = &P+w;,q
(4)
c?cp~= $P+B&
(5)
where the gauge fields W3, B are given by the Z, y fields. This set of operators is particularly interesting because it does not affect, at tree level, observables which do not involve the Higgs particle explicitly. [It is understood that the fields and parameters are (re-)
Copyright 0 1996 Elsevier Science B.V. All rights reserved.
PII SO370-2693(96)00537-O
f~+
tl
f+
f-
// A% II
LH contact terms in the Higgs-strahlung process.
normalized in the Lagrangian f2 in such a way that the particle masses and the electromagnetic coupling retain their physical values. 1 In addition, wc consider the I’ollowing helicityconserving I‘ermionic operators which induce contact terms contributing to P ’ f’ ~ ~- ZH:
o/., = ( cpiiDpp) 0,~ =
( L,.y@i, j
-c l1.c..
(p+~“iD,p)(?,,f’yV,_ )
‘3~ = ( &D,p)
( PRypeR)
+
h.c.
A h.c.
(Ii = A,!!;- (C&Y,)$J+ s;~,cu,R)/!\’
c I’)
(h 1
h, = J,y;, ‘C\c’S,~( -apw + a&j 1 /iI2
(13)
(7)
where SW and cw are the sine and cosine of the weak mixing angle, respectively. In the same way the eeHZ contact interactions t Fig. 1. right diagram) can be defined for left/righthanded electrons and right/left-handed positrons
(8)
[ /l, and em denote the left-handed lcpton doublet and the right-handed singlet. respcctivcly. The vatuum expectation value of the Higgs field is given by (9) = (O,I~/&) with 11 = 146GcV, and the covariant derivative acts on the Higgs doublet as D, = ~9~ - +p-“W’~ + ig’fl,.] Helicity-violating fermionic opeiators do not interfere with the SM amplitude, so that their contribution to the cross section is suppressed by another power of ,2’. The helicity-conserving fermionic operators modify the SM Zee couplings and are therefore constrained by the measurements at LEPI; however, it is possible to improve on the existing limits hy measuring the Higgs-strahlung process at a high-energy c * ep collider since the impact on this process increases with energy [h]. The effective ZZH and the induced yZH interactions (Fig. 1, left diagram) may he written ~%ZII = R.z MZ
CyZN = gzMz
where
yz
=
(
1 i- (10 _ -L~Z/lH 2
$Z,,,.rZ”.H
Mz d&?%$.
Z,Zp”“‘d,,H and Z,AI*“J,,H
his: They may he eliminated in favor of the other operators and the contact terms by applying the equations of‘ motion. The remaining coefficients are given by
+ ~%,,,iYH) (9)
(IO) Additional operators are redundant in this ba-
C’( = -2,q/:
’
(‘RZ
’ *K/ii’
- 78;
(CY/.I
3-
Q1.j
)
/iI’
(15)
(16)
Some consequences of these operators for Higgs production in e +e- collisions have been investigated in the past. Most recently, the effect of novel ZZH vertex operators and P?ZH contact terms on the total cross sections for Higgs production has been studied in Ref. 161. The impact of vertex operators on angular distributions has been analyzed in Refs. [ 7 ] and [ 81. We expand on these analyses by studying the angular distributions for the more general case where both novel vertex operators and contact interactions are present. The analysis of angular distributions in the Higgs-strahlung process ( I ) allows us to discriminate between various novel interactions. In fact, the entire set of parameters CIO.LEI,bl and ~1.. cR can be determined by measuring the polar and azimuthal angular distributions as a function of the beam energy if the rlcctron/positron beams are unpolarized. As expected,
W. K&m et al. /Physics Letters B 381 f 19%) 243-247
245
A = [l - (m~+mz)~/s] x [l- (mH - mz)2/s]. The coefficients a( s)L,R and p(s) LR can ea,dy be determined for the interactions in Eqs. (9) and ( 14) : a(
sfyR =2ao+(S-M~)~CL.R c
p(sfqR= a(s) +2y&Mz
Fig. 2. Polar and azimuthal angles in the Higgs-strahlung process. [The polar angle 0, is defined in the Z rest frame. ]
the energy dependence of the polar angular distribution is sufficient to provide a complete set of measurements if longitudinally polarized electron beams are available ’ .
2. Total cross section and polar angular distribution Denoting the polar the 2 boson and the e+e- beam axis by differential be written for process e’eL,R .+ ZH daL+R = dcose
GiMi
(v, * a,)’
K
x aAsin*B
e
(19)
L,R
al + z(l-+]
(20)
where the boost of the Z boson is given by y = (s + M; - M;)/2Mz,h. The modification of the cross section by the new interaction terms has a simple structure. The coefficient aa just renormalizes the SM cross section. By contrast, the contact interactions grow with s. [The ratio s/h* is assumed to be small enough for the restriction to dimension-6 operators to be meaningful.] The operators 0,~~ Ova affect the coefficient in the cross section which is independent of 8. They damp the falloff of this term, changing the l/s2 to a l/s behavior; however, these contributions remain subleading since they are associated with transversely polarized Z bosons which are suppressed at high energies compared with the longitudinal components. To illustrate the size of the modifications a(s) L*Rand p(s) L*R,we have depicted these functions in Fig. 3 (a) for the special choice a; = 1.
Ati 3. Azimuthal
(1+&R)+6(1+/?L,R)M~/~
distributions
(1 - M;/s)* (17) and the integrated ;
a=
$&-
X
cross section
“z (u, *aa,)‘A1/’
A(1 +LYLJ) + 12(1 +pL*R)M$/S (1 - M2,/s)2
(18)
The Z charges of the electron are defined as usual by a, = -1 and v, = -1 + 4.~2,. s is the c.m. energy squared, and A the two-particle phase space coefficient ’ Since we can restrict ourselves to helicity-conserving couplings, as argued before, additional positron polarization need not be required.
The azimuthal angle & of the fermion f is defined as the angle between the [e-, Z] productionplane and the [Z, f] decay plane (Fig. 2). It corresponds to the azimuthal angle of f in the Z rest frame with respect to the [e- , Z] plane. On general grounds, the & distribution must be a linear function of cos #*, cos 24%) and sin&, sin2&, measuring the helicity components of the decaying spin-l Z state. The coefficients of the sine terms vanish for CP invariant theories. The cos #+ and cos 241, terms correspond to P-odd and Peven combinations of the fermion currents. The general azimuthal distributions are quite involved [ 4,7,8]. We therefore restrict ourselves to the simplified case in which all polar angles are integrated out, i.e., the polar angle 6 of the Z boson in the laboratory frame
246
W. Kilian et al. / Physics Letters B 381 (1996) 243-247
and the polar angle 8, off in the 2 rest frame. In this way we find for the azimuthal q5* distribution:
with f,
‘“,-$$2)
(s)LJ = MZfi x [al+s(l--!$)h]
f2(s)
L,R = 2Mzfi
(22) ‘I’:;:’
x [al+z(l-$)bl]
(23)
The cross section flattens with increasing c.m. energy in the Standard Model, i.e. the coefficients of cos #J* and cos 24, decrease asymptotically proportional to 1/fi and 1/s, respectively. The anomalous contributions modify this behavior: The cos & term receives contributions which increase proportional to 4 with respect to the total cross section, while the cos2qL term receive contributions from the transversal couplings that approaches a constant value asymptotically. The size of the new terms in f 1;: is shown in Fig. 3 (b) as a function of the energy. [The special choice CX~ = 1 we have adopted for illustration, implies ft,2 = ft,.]
4. High-energy
behavior of the coefficients
It is instructive to study the high-energy behavior of the coefficients in the limit @ < s < A2. In this case we obtain the simplified relations: a( S)LSR N F .s .8swcw P(s)L.R N a(s>LJ
CL,R + o(b)
(24)
Fig. 3. Coefficients of the angular distributions as a function of the beam energy. Parameters are described in the text; in particular. a; = 1 has been chosen in the effective Lagrangian equation (2). [The L, R coefficients of the azimuthal distribution coincide for the special choice ai = 1.]
Terms which are proportional to v, = -1 + 4~2, are suppressed by an order of magnitude. If longitudinally polarized electrons are available, the asymptotic value of the coefficients al, bl , CL and CR can be determined by measuring the polar angular distribution without varying the beam energy. The analysis of the azimuthal r,k distribution provides two additional independent measurements of the coefficients al and bl. On the other hand, the set of measurements remains incomplete for fixed energy if only unpolarized electron/positron beams are used at high energies; in this case the coefficients cannot be disentangled completely without varying the beam energy within the preasymptotic region.
+ s (at F ~SWCWbt > + WV,) (25)
References and [I]
s fl (s)L,R N - (al i4swcwbt)
2
fg(~)~,~
= S(QI ~4s~c~bt)
+ WV,)
+O(u,)
J. Ellis, M.K. Gaillard and D.V. Nanopoulos, Nucl. Phys. B
(26)
106 (1976) 292; B.L. Ioffe and V.A. Khoze, Sov. J. Part. Nucl. 9 (1978)
(27)
B.W. Lee. C. Quigg
and H.B. Thacker,
Phys.
50; Rev. D 16
W. Kilian et al. /Physics
(1977)
1519;
J.D. Bjorken, Proc. Summer Institute on Particle Physics, SLAC Report 198 ( 1976). [2 1 M. Carena, PM. Zerwas (conv.) et al., Higgs Physics, in: Proc. of the Workshop Physics with LEP2, Eds. 0. Altarelli, T. SjZistrand, and F. Zwirner, CERN Yellow Report 96-01. ]3] Proceedings of the Workshop e+e- Collisions at 500 GeV: The Physics Potential, Munich-Annecy-Hamburg, Ed. PM. Zerwas, Reports DESY 92-123A.B; 93-123C.
Letters B 381 f 19%) 243-247
241
[4] V. Barger, K. Cheung, A. Djouadi, B. Kniehl. and P.M. Zerwas, Phys. Rev. D 49 (1994) 79. [5] W. Buchmtlller and D. Wyler, Nucl. Phys. B 268 (1986) 621, [6] B. Grzadkowski and J. Wudka, Phys. Lea. B 364 (1995) 49. 171 K. Hagiwara and M.L. Stong, Z. Phys. C 62 ( 1994) 99. [S] G.J. Gounaris, EM. Renard and N.D. Vlachos, Nucl. Phys. B 459 (1996) 51; G.J. Gounaris, J. Laissac, J.E. Paschalis, EM. Renard and N.D. Vlachos, Preprint PM/96-08.