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Conservation laws for neutral currents in spontaneously broken supersymmetric theories
Volume 110B, number 3,4
PHYSICS LETTERS
1 April 1982
CONSERVATION LAWS FOR NEUTRAL CURRENTS IN SPONTANEOUSLY BROKEN SUPERSYMMETRIC THEORIES ~ R. BARBIERI Scuola Normale Superiore, Pisa, Italy and Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, Pisa, Italy and R. GATTO Ddpartement de Physique Thdorique, Universit~ de Genbva, 1211 Geneve 4, Switzerland Received 28 December 1981
We study the diagonalization of the general mass matrix for quarks and their scalar superpartners in models based on a gauge group SU(3) X SU(2) × U(1) X Gwith spontaneous breakdown of supersymmetry. We characterize a class of models where the Cabibbo-Kobayashi-Maskawa unitary matrices occurring in the mass-eigenstate basis are the same in the gaugino as in the vector boson couplings. Unlike the general case, this class of models appears to be safe from the point of view of induced flavour changing neutral interactions.
1. Many physicists share nowadays the point of view that the quadratic divergence in the mass of the physical Higgs particle of the standard electroweak model although absorbable by the renormalization procedure may signal the inadequacy of the model to describe physics beyond 1 0 0 - 1000 GeV. Such inadequacy constitutes a possible theoretical motivation for expecting a supersymmetric world to become manifest at these energies. Supersymmetric models can in fact avoid the quadratic divergence problem. A realm of phenomena where the standard model is particularly successful concerns the observed dramatic suppression of the strangeness-changing neutral currents. A most striking limit comes from the smallness of the K 01 - K 02 mass difference. It is therefore natural to ask whether the same suppression will take place in supersymmetric models of the strong and electroweak interactions. In this paper we will address this question for the general class of models based on a possible gauge group G = SU(3) X SU(2) × U(1) × ~ where supersymmetry, Partially supported by the SwissNational Science Foundation. 0 0 3 1 - 9 1 6 3 / 8 2 / 0 0 0 0 - 0 0 0 0 / $ 0 2 . 7 5 © 1982 North-Holland
as G itself, is spontaneously broken at the tree level by the scalar potential. The need of extending the group is by now an established fact [1] in order to get an accept. able mass spectrum in broken supersymmetry. We shall assume that a consistent model of this type - yet to be found - may in fact be invented. We shall concentrate on the possible effects introduced by the couplings +1 - - g ~ o ; t ~ b X T C - l ~ b + h.c.,
(1)
where the Ca are the left-handed fermion fields associated with the chiral supermultiplets, the ~0a are the corresponding complex scalar fields, the k s are the fermionic supersymmetric partners (gauginos) of the gauge fieldsA~ and finally the ta~b are the generators of G, with coupling constantsga, as acting on the representation of the ff's. We are supposing that the gauge couplings dominate over the Yukawa couplings from the supersymmetric potential. The couplings (1) are in some sense the supersymmetric companions of the standard gauge interactions ¢1 The notation follows that of Ferrara et al. [2]. 211
Volume ll0B, number 3,4 •
O: T
lg~A u ~a tab Tu t~b "
PHYSICS LETTERS (2)
1 April 1982
.t2M (o, Z) = o?F?Fo + ~
gaDs(x) o?tO)o
C~
By analysing tire general mass matrix for the quarks and their scalar superpartners and considering its diagonalization, we shall show that general models with tree level spontaneous breakdown of supersymmetry are safe from the point of view of the induced flavor-changing neutral interactions if the following two conditions are met: (i) All quarks of given charge receive their tree level mass through the coupling of precisely one neutral Higgs boson. (li) All quarks of given charge and helicity have the same value of any generator G with non-vanishing Dterm at the minimum of the scalar potential. These models are characterized by the fact that, unlike the general case, the Cabibbo KobayashiLMaskawa unitary matrices occurring after the introduction of mass eigenstates are the same in the gaugino as in the vector boson couplings (1) and (2), respectively. Flavour changing neutral interactions in broken supersymmetric models are also considered by Dimopoulos and Georgi [1] and by Ellis and Nanopoulos [3] who derive the requirement of almost degenerate masses for the up- and charm-scalar. Our interest here is instead in characterizing a general class of theories for which the experimental suppression of flavour-changing effects is guaranteed. 2. A preliminary problem is the diagonalization of the fermion and scalar mass matrices. To this purpose it is useful to distinguish, among the fields ~p's, all the scalar partners of the quarks (which we call o) from the other components (which we call X). The mass matrix of the cr fields, as well as that of the quarks themselves (called E;), can be studied independently from the other fields. The supersymmetric potential can be written in the general form
f(olX) = V(X) + loiojf(iJ)(x) + oiOjOk ;(ijk) •
(3)
After spontaneous symmetry breaking, some of the Xfields will acquire a nonvanishing vacuum expectation value and eventually give rise, at the minimum, to some non-zero
D~(X) =gax? t(~x)X + ~ . The mass lagrangian for the fields o, Z is
212
(4)
+ i~/z_a [ ~ ; -0f* Of a + h.c.] + I ( 2 T C - 1 F X + h.c.) - - o T ~xa TM
z \ xa 3x*a (5)
in terms of the symmetric matrix
F = (f(iJ)(x)}.
(6)
We specialize this general expression by taking the lefthanded quarks qLa, of given charge and family-index a, together with the charge conjugates of the right-handed ones (qRb) c. Using script letters for the scalar superpartners we will have o:
P) Q~
(7)
.
We assume that the quarks of given charge receive their tree level mass through the coupling of precisely one neutral Higgs meson, as was required [4] to have natural flavour conservation in the standard Yukawa couplings. Under these circumstances, one has F= Q~T
/1
0
'
1 af* X0 3X0* ,
. . . .
~ -/xF, x i}X* ~}X
(8)
where c/~ is the mass matrix in generation space of the quarks q~ and X0 is the vacuum expectation value of the neutral Higgs field coupled to %. The index a in (4) runs over all possible colourless neutral generators of the original gauge group G. In general it includes therefore T3L , Y as well as generators of ~ Let us now consider the case where all quarks of given charge and helicity have the same value of any of these additional generators, as is the case for T3L and Y. This condition is obviously required in order to have flavour conservation in the diagram with the exchange of any one of the gauge bosons associated with these generators whenever
g2/M2 > G F X 10 - 3
(9)
for the corresponding charge and mass. Otherwise it is not necessary ( a family group G b r o k e n at sufficiently high energy is an example). In the case under consideration a simultaneous diagonalization in generation space of both quarks and scalar quarks can be performed. This is because the matrix F can be diagonalized by a unitary transformation
Volume 110B, number 3,4
UTFU = f
PHYSICS LETTERS
d
(10)
acting simultaneously on the fields o and 2;, without affecting the diagonal form of the contribution in (5) o f the D-term. Even though diagonalized in generation space, the scalar mass lagrangian is not yet completely diagonal. For a given flavour, the corresponding mass lagrangian has the form ~ M ( Q , q) = mFqLqR + h.c.
+[Qt
Qc][m;D
m2/~*m*][Q+Dcj Qct]
'
(11)
where
D = ~gaDa(x)
tc~(q),
D e=
~gc~Dc~(X)
ta(qC )
02) are generation independent squared masses. For each quark q, of mass m, there are two scalars with the same SU(3)c X U(1)e m quantum numbers
[°1] E sin'TFe'°lI° 1 Q2
= -sinp
cospJk0
e-iy
Qct
'
1 April 1982
charged "vino" X *3.
-(g/x/2)ql~icodUij)t~( 1 - 3 ' 5 ) dj + h . c . , (16) where cgod, denote the two (~ = 1, 2) scalar partners for each family member (i = 1,2, 3, ...) of the charge 2/3 quarks, and
cli
= cos
Pi ,
c2i = sin Pi "
We might have considered the situation where some of the generators other than those of SU(2) X U(1) contributing to (4) and broken at sufficiently high energy have different eigenvalues on quarks o f the sam.e charge and helicity. In this situation it is no longer true that quarks and scalar quarks are simultaneously diagonalizable in generation space. The unitary transformation which brings the scalars to a diagonal form has typical off-diagonal elements of size/J c~ij/D. For light quarks, taking 100 GeV as a reference supersymmetry-breaking mass scale, crt~ij/(lO0 GeV) is presumably negligible relative to the typical off-diagonal elements in the matrix U F which diagonalizes the fermion mass matrix c-~. For the general gaugino couplings one has therefore, for all SU(3) X SU(2) X U(1) generators,
(13)
--gaX~:~i t~(UF)ij @ + h.c. tgo = I~ml/{~-(D
(17)
(18)
D c)+ [~-(D-DC) 2+l~m1211/2}, Furthermore D ~ (100 GeV) 2 is a reference squared mass difference among scalars of different i, but of the same SU(3) × U(1) numbers.
1
7 = 5arctg(gm) , and masses *2 M 1,2 2 = m2 + ½ 0 + D c) +- [~-(D - D C ) 2 + I/.trn121 1/2 (14) We are finally in the position to write down the gauge coupling (1) in terms of mass eigenstates. In the case under consideration, in all standard gauge couplings (2) corresponding to the generators of SU(3) × SU(2) × U(1), only the charged weak currents have a non-trivial generation mixing matrix
i(g/N/~)W~ fliUij'Y~ 21-(1 -- T5)
dj + h.c.
3. We are in the position of computing the contribution from gaugino exchanges to the effective AS = 2 neutral operator. We consider first the situation where (i) and (ii) are satisfied. In such a case only the charged "vino" interaction (16) is relevant through the diagram of fig. 1. This diagram is readily seen to give rise to the effective lagrangian
(15)
Analogously, the interaction (1) will receive non-trivial angles only in the gauge Yukawa coupling with the
+2 Notice that eq. (14) reduces to the commonly quoted formula M~ m~ =gc~taDa only for ~ = 0. It is true, however, if traces of both sides are taken [2].
#3 Here we use the same notation for the left-handed Weyl spinor h in (1) as for the Dirac spinor h in (16), which has acquired a mass from the coupling (1) after spontaneous symmetry breaking. We have also absorbed the phase factors exp (-+iy) in (13) by redefining the fieldsQlc~i, without affecting the mass terms nor the gauge interactions. Note that we have slightly modified the correct morpheme to substitute the more agreeable "vino" for "wino". 213
Volume ll0B, number 3,4
"'
~ .......
PHYSICS LETTERS
1 April 1982
one obtains
~-
~(z~S=2) eff = _ G 2(cos20csin20c/47r2) m 2
X I(mw/M)4(mc/M)2(aLTuS)(aLYuS). s
Uj~
d
Fig. 1. ~(zXS=2) = _i3~g4 eff
X [" fl4k
k2 C2i C~] J (2rr)4 (k2 - m2) 2 k 2 -- m2 i k 2 -- tn~]
X Uil Ui*2 U]I U;2(gLTudL)(gLT**dL) ,
(19)
where rn L is the mass of the charged "vino". In the limit where the top quark does not couple to the down- and strange-quarks, we get the presumably dominant contribution /2(zxS=2) = - i ~ g 4 cos20c sin20c eff X /-. d4k
k2
(
J ( 2 r r ) 4 (k 2 ~-rn2)2
e21
"42!
k 2 - - mr1 , -
(20)
X (gLY~udL)(gLY~dL) .
Here m21,2 (o~ = 1, 2) are the squared masses of the scalar partners of the up- and charm-quarks, respectively. Notice that (20) is real, as is real the standard contribution coming from up- and charm-quark exchanges. From eqs. (13), (14), (17) one sees that the integrand in (20) goes as m 4 when the charmed quark mass m 2 is small relative to all other dimensioned parameters # 2 D, ID - DC[, rn 2. Putting Fd4k
k2
ai2rr)~-4 (k 2 _ m 2 ) 2
(
C~l k2 - m 2 1
= ( - i/16rr 2) m4/M 6 ,
214
0~2 k2
be
m~ 2]
(21)
(22)
F o r M 2 ~ (50 100 GeV) 2 the zXS = 2 effective lagrangian coming from the exchange of the scalar partners of up- and charm-quarks is about 3 - 4 orders of magnitude smaller than the effective lagrangian coming from the W-exchange diagram and as such completely negligible. On the other hand, the limits to parameters (mixing angles, masses) related to the heavy third generation, coming from their contribution both to the real and imaginary part of the effective lagrangian should be reconsidered. We may ask now what happens if we relax condition (ii), namely we allow for the introduction of "family-like" generators. In general now all gaugino ,~ c~(AS=2) exchanges give a nonvanishing contribution ,~, -Cef f . Two main new things arise in this situation: (A) since the scalar quarks related to up and charm are not anymore almost degenerate in mass, the suppression occurring in (22) as mc/M-+ 0 does not normally take place; (B) there is in general no reason why the effective operator coming from gaugino-exchanges be real relatively to the standard W-exchange diagram. These two facts are in general sufficient to put this last kind of models in serious troubles unlike those of tire first kind which do not seem to present any problem. Clearly the considerations developed here apply as well - mutatis mutandis - to all other standard flayour-changing neutral transitions.
References [11 P. Fayet, Phys. Lett. 84B (1979) 416: S. Dimopoulos and H. Georgi, preprint HUTP-81/A022 (t981). [21 s. Ferrara, L, Girardello and F. Palumbo, Phys. Rev. D20 (1979) 403. [3] J. Ellis and D. Nanopoulos, CERN preprint TH 3216 (1981). 14] S. GIashow and S. Weinberg, Phys. Rev. D15 (1977) 1958.
PHYSICS LETTERS
1 April 1982
CONSERVATION LAWS FOR NEUTRAL CURRENTS IN SPONTANEOUSLY BROKEN SUPERSYMMETRIC THEORIES ~ R. BARBIERI Scuola Normale Superiore, Pisa, Italy and Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, Pisa, Italy and R. GATTO Ddpartement de Physique Thdorique, Universit~ de Genbva, 1211 Geneve 4, Switzerland Received 28 December 1981
We study the diagonalization of the general mass matrix for quarks and their scalar superpartners in models based on a gauge group SU(3) X SU(2) × U(1) X Gwith spontaneous breakdown of supersymmetry. We characterize a class of models where the Cabibbo-Kobayashi-Maskawa unitary matrices occurring in the mass-eigenstate basis are the same in the gaugino as in the vector boson couplings. Unlike the general case, this class of models appears to be safe from the point of view of induced flavour changing neutral interactions.
1. Many physicists share nowadays the point of view that the quadratic divergence in the mass of the physical Higgs particle of the standard electroweak model although absorbable by the renormalization procedure may signal the inadequacy of the model to describe physics beyond 1 0 0 - 1000 GeV. Such inadequacy constitutes a possible theoretical motivation for expecting a supersymmetric world to become manifest at these energies. Supersymmetric models can in fact avoid the quadratic divergence problem. A realm of phenomena where the standard model is particularly successful concerns the observed dramatic suppression of the strangeness-changing neutral currents. A most striking limit comes from the smallness of the K 01 - K 02 mass difference. It is therefore natural to ask whether the same suppression will take place in supersymmetric models of the strong and electroweak interactions. In this paper we will address this question for the general class of models based on a possible gauge group G = SU(3) X SU(2) × U(1) × ~ where supersymmetry, Partially supported by the SwissNational Science Foundation. 0 0 3 1 - 9 1 6 3 / 8 2 / 0 0 0 0 - 0 0 0 0 / $ 0 2 . 7 5 © 1982 North-Holland
as G itself, is spontaneously broken at the tree level by the scalar potential. The need of extending the group is by now an established fact [1] in order to get an accept. able mass spectrum in broken supersymmetry. We shall assume that a consistent model of this type - yet to be found - may in fact be invented. We shall concentrate on the possible effects introduced by the couplings +1 - - g ~ o ; t ~ b X T C - l ~ b + h.c.,
(1)
where the Ca are the left-handed fermion fields associated with the chiral supermultiplets, the ~0a are the corresponding complex scalar fields, the k s are the fermionic supersymmetric partners (gauginos) of the gauge fieldsA~ and finally the ta~b are the generators of G, with coupling constantsga, as acting on the representation of the ff's. We are supposing that the gauge couplings dominate over the Yukawa couplings from the supersymmetric potential. The couplings (1) are in some sense the supersymmetric companions of the standard gauge interactions ¢1 The notation follows that of Ferrara et al. [2]. 211
Volume ll0B, number 3,4 •
O: T
lg~A u ~a tab Tu t~b "
PHYSICS LETTERS (2)
1 April 1982
.t2M (o, Z) = o?F?Fo + ~
gaDs(x) o?tO)o
C~
By analysing tire general mass matrix for the quarks and their scalar superpartners and considering its diagonalization, we shall show that general models with tree level spontaneous breakdown of supersymmetry are safe from the point of view of the induced flavor-changing neutral interactions if the following two conditions are met: (i) All quarks of given charge receive their tree level mass through the coupling of precisely one neutral Higgs boson. (li) All quarks of given charge and helicity have the same value of any generator G with non-vanishing Dterm at the minimum of the scalar potential. These models are characterized by the fact that, unlike the general case, the Cabibbo KobayashiLMaskawa unitary matrices occurring after the introduction of mass eigenstates are the same in the gaugino as in the vector boson couplings (1) and (2), respectively. Flavour changing neutral interactions in broken supersymmetric models are also considered by Dimopoulos and Georgi [1] and by Ellis and Nanopoulos [3] who derive the requirement of almost degenerate masses for the up- and charm-scalar. Our interest here is instead in characterizing a general class of theories for which the experimental suppression of flavour-changing effects is guaranteed. 2. A preliminary problem is the diagonalization of the fermion and scalar mass matrices. To this purpose it is useful to distinguish, among the fields ~p's, all the scalar partners of the quarks (which we call o) from the other components (which we call X). The mass matrix of the cr fields, as well as that of the quarks themselves (called E;), can be studied independently from the other fields. The supersymmetric potential can be written in the general form
f(olX) = V(X) + loiojf(iJ)(x) + oiOjOk ;(ijk) •
(3)
After spontaneous symmetry breaking, some of the Xfields will acquire a nonvanishing vacuum expectation value and eventually give rise, at the minimum, to some non-zero
D~(X) =gax? t(~x)X + ~ . The mass lagrangian for the fields o, Z is
212
(4)
+ i~/z_a [ ~ ; -0f* Of a + h.c.] + I ( 2 T C - 1 F X + h.c.) - - o T ~xa TM
z \ xa 3x*a (5)
in terms of the symmetric matrix
F = (f(iJ)(x)}.
(6)
We specialize this general expression by taking the lefthanded quarks qLa, of given charge and family-index a, together with the charge conjugates of the right-handed ones (qRb) c. Using script letters for the scalar superpartners we will have o:
P) Q~
(7)
.
We assume that the quarks of given charge receive their tree level mass through the coupling of precisely one neutral Higgs meson, as was required [4] to have natural flavour conservation in the standard Yukawa couplings. Under these circumstances, one has F= Q~T
/1
0
'
1 af* X0 3X0* ,
. . . .
~ -/xF, x i}X* ~}X
(8)
where c/~ is the mass matrix in generation space of the quarks q~ and X0 is the vacuum expectation value of the neutral Higgs field coupled to %. The index a in (4) runs over all possible colourless neutral generators of the original gauge group G. In general it includes therefore T3L , Y as well as generators of ~ Let us now consider the case where all quarks of given charge and helicity have the same value of any of these additional generators, as is the case for T3L and Y. This condition is obviously required in order to have flavour conservation in the diagram with the exchange of any one of the gauge bosons associated with these generators whenever
g2/M2 > G F X 10 - 3
(9)
for the corresponding charge and mass. Otherwise it is not necessary ( a family group G b r o k e n at sufficiently high energy is an example). In the case under consideration a simultaneous diagonalization in generation space of both quarks and scalar quarks can be performed. This is because the matrix F can be diagonalized by a unitary transformation
Volume 110B, number 3,4
UTFU = f
PHYSICS LETTERS
d
(10)
acting simultaneously on the fields o and 2;, without affecting the diagonal form of the contribution in (5) o f the D-term. Even though diagonalized in generation space, the scalar mass lagrangian is not yet completely diagonal. For a given flavour, the corresponding mass lagrangian has the form ~ M ( Q , q) = mFqLqR + h.c.
+[Qt
Qc][m;D
m2/~*m*][Q+Dcj Qct]
'
(11)
where
D = ~gaDa(x)
tc~(q),
D e=
~gc~Dc~(X)
ta(qC )
02) are generation independent squared masses. For each quark q, of mass m, there are two scalars with the same SU(3)c X U(1)e m quantum numbers
[°1] E sin'TFe'°lI° 1 Q2
= -sinp
cospJk0
e-iy
Qct
'
1 April 1982
charged "vino" X *3.
-(g/x/2)ql~icodUij)t~( 1 - 3 ' 5 ) dj + h . c . , (16) where cgod, denote the two (~ = 1, 2) scalar partners for each family member (i = 1,2, 3, ...) of the charge 2/3 quarks, and
cli
= cos
Pi ,
c2i = sin Pi "
We might have considered the situation where some of the generators other than those of SU(2) X U(1) contributing to (4) and broken at sufficiently high energy have different eigenvalues on quarks o f the sam.e charge and helicity. In this situation it is no longer true that quarks and scalar quarks are simultaneously diagonalizable in generation space. The unitary transformation which brings the scalars to a diagonal form has typical off-diagonal elements of size/J c~ij/D. For light quarks, taking 100 GeV as a reference supersymmetry-breaking mass scale, crt~ij/(lO0 GeV) is presumably negligible relative to the typical off-diagonal elements in the matrix U F which diagonalizes the fermion mass matrix c-~. For the general gaugino couplings one has therefore, for all SU(3) X SU(2) X U(1) generators,
(13)
--gaX~:~i t~(UF)ij @ + h.c. tgo = I~ml/{~-(D
(17)
(18)
D c)+ [~-(D-DC) 2+l~m1211/2}, Furthermore D ~ (100 GeV) 2 is a reference squared mass difference among scalars of different i, but of the same SU(3) × U(1) numbers.
1
7 = 5arctg(gm) , and masses *2 M 1,2 2 = m2 + ½ 0 + D c) +- [~-(D - D C ) 2 + I/.trn121 1/2 (14) We are finally in the position to write down the gauge coupling (1) in terms of mass eigenstates. In the case under consideration, in all standard gauge couplings (2) corresponding to the generators of SU(3) × SU(2) × U(1), only the charged weak currents have a non-trivial generation mixing matrix
i(g/N/~)W~ fliUij'Y~ 21-(1 -- T5)
dj + h.c.
3. We are in the position of computing the contribution from gaugino exchanges to the effective AS = 2 neutral operator. We consider first the situation where (i) and (ii) are satisfied. In such a case only the charged "vino" interaction (16) is relevant through the diagram of fig. 1. This diagram is readily seen to give rise to the effective lagrangian
(15)
Analogously, the interaction (1) will receive non-trivial angles only in the gauge Yukawa coupling with the
+2 Notice that eq. (14) reduces to the commonly quoted formula M~ m~ =gc~taDa only for ~ = 0. It is true, however, if traces of both sides are taken [2].
#3 Here we use the same notation for the left-handed Weyl spinor h in (1) as for the Dirac spinor h in (16), which has acquired a mass from the coupling (1) after spontaneous symmetry breaking. We have also absorbed the phase factors exp (-+iy) in (13) by redefining the fieldsQlc~i, without affecting the mass terms nor the gauge interactions. Note that we have slightly modified the correct morpheme to substitute the more agreeable "vino" for "wino". 213
Volume ll0B, number 3,4
"'
~ .......
PHYSICS LETTERS
1 April 1982
one obtains
~-
~(z~S=2) eff = _ G 2(cos20csin20c/47r2) m 2
X I(mw/M)4(mc/M)2(aLTuS)(aLYuS). s
Uj~
d
Fig. 1. ~(zXS=2) = _i3~g4 eff
X [" fl4k
k2 C2i C~] J (2rr)4 (k2 - m2) 2 k 2 -- m2 i k 2 -- tn~]
X Uil Ui*2 U]I U;2(gLTudL)(gLT**dL) ,
(19)
where rn L is the mass of the charged "vino". In the limit where the top quark does not couple to the down- and strange-quarks, we get the presumably dominant contribution /2(zxS=2) = - i ~ g 4 cos20c sin20c eff X /-. d4k
k2
(
J ( 2 r r ) 4 (k 2 ~-rn2)2
e21
"42!
k 2 - - mr1 , -
(20)
X (gLY~udL)(gLY~dL) .
Here m21,2 (o~ = 1, 2) are the squared masses of the scalar partners of the up- and charm-quarks, respectively. Notice that (20) is real, as is real the standard contribution coming from up- and charm-quark exchanges. From eqs. (13), (14), (17) one sees that the integrand in (20) goes as m 4 when the charmed quark mass m 2 is small relative to all other dimensioned parameters # 2 D, ID - DC[, rn 2. Putting Fd4k
k2
ai2rr)~-4 (k 2 _ m 2 ) 2
(
C~l k2 - m 2 1
= ( - i/16rr 2) m4/M 6 ,
214
0~2 k2
be
m~ 2]
(21)
(22)
F o r M 2 ~ (50 100 GeV) 2 the zXS = 2 effective lagrangian coming from the exchange of the scalar partners of up- and charm-quarks is about 3 - 4 orders of magnitude smaller than the effective lagrangian coming from the W-exchange diagram and as such completely negligible. On the other hand, the limits to parameters (mixing angles, masses) related to the heavy third generation, coming from their contribution both to the real and imaginary part of the effective lagrangian should be reconsidered. We may ask now what happens if we relax condition (ii), namely we allow for the introduction of "family-like" generators. In general now all gaugino ,~ c~(AS=2) exchanges give a nonvanishing contribution ,~, -Cef f . Two main new things arise in this situation: (A) since the scalar quarks related to up and charm are not anymore almost degenerate in mass, the suppression occurring in (22) as mc/M-+ 0 does not normally take place; (B) there is in general no reason why the effective operator coming from gaugino-exchanges be real relatively to the standard W-exchange diagram. These two facts are in general sufficient to put this last kind of models in serious troubles unlike those of tire first kind which do not seem to present any problem. Clearly the considerations developed here apply as well - mutatis mutandis - to all other standard flayour-changing neutral transitions.
References [11 P. Fayet, Phys. Lett. 84B (1979) 416: S. Dimopoulos and H. Georgi, preprint HUTP-81/A022 (t981). [21 s. Ferrara, L, Girardello and F. Palumbo, Phys. Rev. D20 (1979) 403. [3] J. Ellis and D. Nanopoulos, CERN preprint TH 3216 (1981). 14] S. GIashow and S. Weinberg, Phys. Rev. D15 (1977) 1958.