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Signatures of flipped SU(5)
Volume 236, number 3
PHYSICS LETTERS B
22 February 1990
S I G N A T U R E S O F F L I P P E D SU(5) S.A. ABEL H.H. Wills Physics Laboratoo,, Royal Fort, g),ndallAvenue, BristolBS8 1TL, UK and I. W H I T T I N G H A M Department of Physics. James Cook University, Townsville, QLD 4811, Australia Received 2 November 1989
We examine the electric dipole moment of the neutron and flavour changing Z decays in the flipped SU(5 ) grand unification scheme. We find that the new couplings at energy scales above Mcux can lead to neutron electric dipole moments Idl ~ 10-25 ecm and decays of the form Z~bs, sl~ with a branching ratio B ( z ~ , s b ) ~ 10 -6, of which a large proportion (~< ¼) may be CP violating. The first two effects are found to be slightly suppressed in the currently popular no-scale theories, but the CP violation parameter, ~, is relatively theory independent.
1. Introduction
The construction o f four-dimensional fermionic superstring theories [ 1 ] has led to the derivation o f a n u m b e r o f p r o m i s i n g candidates for unification below the Planck scale. O f these, perhaps the most interesting is flipped S U ( 5 ) [2]. This model is unusual in that the Higgs fields do not transform under the adjoint representation. We have the following particle content: - t h r e e generations o f m a t t e r fields denoted by F(10, ~),f(~, - ~3) , and {~C(l, 2); s - a pair o f Higgs fields to break the symmetry clown to SU ( 3 ) × (SU ( 2 ) × U ( 1 ) denoted by H ( 10, ½) and H(10, -~); - a pair o f Higgs to break S U ( 2 ) × U ( 1 ) denoted, h (5, - 1 ) and ~(5, - 1 ), and four gauge singlet fields ~,,, (where m = (0 ..... 3) ). The superpotential of the model is given by W = 2 ~FFh + 2 2~f~f+ 2 3f~ch + )~4HHh + 2sI~It]l+ 26FtIq)+ ).7hh(b+ ).sf)O0,
(1)
where 21 ( h ), -~2(]'l) and 2 3 ( h ) are the mass matri-
ces o f the down and up quarks, and electrons respectively (the generation indices have been suppressed). Its many interesting features, such as a natural d o u b l e t - t r i p l e t mass splitting mechanism for the Higgs fields h a n d / t - a n d a suitable neutrino seesaw mechanism. All the Yukawa couplings o f minimal SU (5) × U ( 1 ) have been derived from renormalizable interactions o f the fermionic string ( a n d are o f order xfi2g), apart from the )-6 coupling which is p r o v i d e d by nonrenormalizable terms in the superpotential, and which gives rise to the see-saw mechanism [3,4]. Subsequent running o f the R G E ' s leads to a low energy superpotential consistent with experiment [5]. The ,).6 coupling has particularly i m p o r t a n t consequences for the low energy phenomenology o f the theory because it carries a flavour index. W h e n the supersymmetric theory becomes softly broken, the R G E ' s give additional terms in the scalar mass matrices, whereas the renormalization o f the Yukawa couplings is unchanged (by virtue o f the no-renormalization theorem [ 6 ] ). This leads to the introduction o f 2 6 flavour changing terms and hence new K o b a y a s h i - M a s k a w a mass mixing matrices. These matrices are unknown, and their off-diagonal ele-
0370-2693/90/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland )
305
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ments are not necessarily (and indeed, unlikely to be) small. The RGE's for the squark masses take the form
22 February 1990
be dominated by linear terms, and to be approximately of the form #H~ .=-#H//2 ( Co q- CI ,~,~-)q q- C2,~ ~)-2 q- C6 [-~6-4~ ] ),
(4)
dm2/dt= (1/8~ 2 ) [3Atl(tH 2 J-A 2)/].1 + ; ~ ( m 2 +m~ +~4~),~1,
dm~/dt= (1/87r 2) [42~(m2v+m 2 +A2)22],
(2) (3)
where we have omitted terms diagonal in the generation indices and neglected the electron masses. Possibly important contributions to flavour changing processes arise in such theories from diagrams where we have a gaugino-fermion-sfermion vertex, due to the fact that the fermions and sfermions are now no longer simultaneously diagonalizable [7 ]; the G I M suppression mechanism no longer operates. For most supersymmetric theories these diagrams still suffer some suppression due to the smallness of the off-diagonal terms in the Kobayashi-Maskawa mixing matrices. However, flipped SU (5) introduces two unknown Kobayashi-Maskawa matrices describing the mixing between the 26 couplings and the up and down quark masses. This has been pointed out for N = 1 minimal supergravity models [ 8 ], and more recently for the string derived no-scale theory [ 9 ], where rare flavour changing processes such as in B decays have been shown to be possibly enhanced. In the minimal supergravity models sofl-SUSY breaking occurs for both the scalars and gauginos and is of the order of the gravitino mass, whereas in the no-scale theory, supersymmetry is softly broken at a scale considerably below the Planck scale by gaugino masses only. This could be achieved by, for example, condensation in the hidden sector [10]. In such models the effects described above are reduced because the G U T scale and the super-unification scale are driven close together dynamically [5] and the scalar masses remain small in this region; they receive only small contributions from the 26 couplings. Choosing regions of parameter space which give a low energy theory consistent with phenomenology (as determined in re['. [ 5 ] ), we have run the RGE's down to the low energy scale with various choices of Yukawa couplings, in order to make a crude estimate of their contributions to the scalar masses. These were found to 306
for the left-handed scalar doublet. In the above equation, C,, are constants; ( o is 0 ( 2 ) ; ('~ and ('2 are O( 1 ). 6"6 is sensitive to our choice of strong coupling constant within the present experimental bounds, but is less than O@o ) for the no-scale theory, and less than O(~ ) for minimal N = 1 supergravity models. Similar equations may be obtained for the right-handed scalar fields, but these mass matrices are dominated by the pieces which are diagonalized in the SKM basis and so give only small contributions to flavour changing decays. This is as expected, since the RGE's can only give off-diagonal contributions in the renorrealization region above ,.I/c;tjT.
2. The electric dipole moment of the neutron
It is well known that in the standard model the electric dipole moment of the neutron is a two loop effect, and takes values o f d < 1 0 37 ecru [ I l l . To date, the best available limits on this are d < 6 × 10-25 e cm [ 12 ] although it should be noted that recent reports [13] claim to have measured d = ( 3 _ + 5 ) × 10 -26 e cm. It has been shown that for the generic supersymmetric theory, in addition to contributions coming from one loop corrections to the imaginary ( CP violating) part of the quark mass, the electric dipole moment of the neutron may receive direct contributions from diagrams such as fig. la [7 ]. Assuming that the m o m e n t u m of the quark in the diagram is small compared with the masses of the scalars and gauginos allows us to expand, and extract the dipole moment relatively easily. For n mass insertions in the gluino diagram, we obtain a contribution of
d=8qg~K,(nq-l)(-l)"(Sni2~" 3(47r)2m S \~/ ,
(5)
where q is the electronic charge of the internal squark, 8m 2 is the off-diagonal piece of the squark masssquared matrix, and m 2 is the diagonal piece. K,, is given by
Volume 236, number 3
PHYSICS LETTERS B
22 February 1990
( S m 2 ) 4 ~ I m ( V2L2,2"~222:~mu2*t2, V:~R)m 7 { C2C2"]
(8)
a
b Fig. I. (a) The conventional diagram for supersymmetric contributions to the neutron electric dipole moment; the blobs represent arbitrary numbers of mass insertions. (b) A contribution involving the wino and charged higgsino; this is expected to be an order of magnitude smaller.
I
( CoX)'+ ' [ ( C o - n ) x - n J
K,, = f dx x~o(n+ 0
1) [ ( C o - 1 ) x + 1 ] , + 2 '
(6)
Any insertions with less Yukawa couplings may be reduced to the product of a matrix and its hermitian conjugate, and is therefore real. For large gaugino masses ( > 100 GeV) the low energy superpotential of flipped SU (5) is such that the two Higgs fields, h and/7, can have vacuum expectation values that differ by a factor of upto ten [ 5 ]. This explains the large mr~rob ratio, and means that the value of 2b may be larger than in the standard model. Because o f this, as a worst case we shall take Ab = 0.1 and the gluino mass a s ml/2= 100 GeV. Substituting accepted values of the couplings and KM matrix elements into (8) gives a worst case contribution to the dipole moment o f d ~ 10 -33 ecru.
(9)
In the case o f minimal flipped S U ( 5 ) the dipole moments are potentially much larger since we have an extra matrix in generation space. Here the leading term has a mass insertion of the form
(
(fire 2) 3 Im( V2c21'2, mu [2626*] V_~R)trts5 \ ~ o 2 and is found to be relatively Co independent and 0 ( 0 . 0 2 ) . Interestingly, in addition to fig. la, we also have contributions from the diagrams in fig. l b, where the charged higgsino couples the wino to the righthanded fields, however we expect this diagram to be an order o f magnitude smaller. For many theories, notably those where supersymmetry is broken by F-terms in the potential, such contributions are greatly reduced. For example, consider a softly-broken minimal SUSY standard model theory. In such a theory we have only two (non-trivial) flavour changing couplings, 21 and 22 (since we may diagonalize the lepton mass matrix) which may be diagonalized by the usual bi-unitary transformations caused by rotating the fields in generation space,
(/~1)diag = VIL/~I ~F~R, (/'[2)cliag = ~2L22 V~R-
(7)
Consider the up-quark/up-squark/gluino diagram. The leading contribution to the dipole moment comes from four mass insertions in fig. I a giving
(10) Not only do we have one less mass insertion in this equation, but also we have three unknown quantities, namely the mixing angles and CP violation parameters of the new KM matrices, and the magnitude of the 2 6 couplings. Taking our worst case value of 2b, the top quark mass to be 100 GeV, and inserting the above and its permutations into ( 5 ) gives a worst case contribution of d~ -6)<
10-25((6A2 sin20 sin d)e cm,
( 11 )
where sin20 sin d is some combination of the unknown KM i.,.ameters and ( A 6 ) 2 is the largest eigenvalue of [262*6]. Because of the large top quark mass and the relatively large value of 2 8 , this diagram dominates over the down-quark/down-squark/gluino diagram. Sipce the value o f A 6 may have to be quite large in ore' "o give satisfactory G U T breaking, and its value at present as derived from fermionic string theories is not well known, it is possible for flipped SU (5) to give neutron electric dipole moments ap307
Volume 236, number 3
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proaching the present day experimental limits in a fairly natural way.
3. Flavour changing decays of the Zo It is known that the new Kobayashi-Maskawa matrices, introduced by the superpotential of flipped SU ( 5 ), can lead to some interesting flavour changing effects such as an enhanced B decay and ~t-e7 [8]. We shall now show that similar considerations can lead to large flavour changing decay modes for the Zo of the form Z~t3s and possibly Z--,tc with a significant CP violation, Such decay modes are of particular interest since they would be directly detectable in forthcoming experiments at LEP. The leading contribution to flavour changing Zo decays comes from the diagram in fig. 2. The analysis for Z ~ b s may be simplified by assuming that the masses of the products are approximately zero compared with the Zo mass so we have concentrated on this process; our arguments apply equally to Z ~ t c modulo the suppression caused by the large top quark mass. Calculating the diagram for a single mass insertion on the left or right squark branch and summing over gluinos gives a branching ratio for this process of g4K2 2 B { z ~ b s , ~ - - 12(4=) 4 \ m~ ] '
(12)
where K is given by K= -
d),dx 0,0
1-x+,,/Co(2y-xy-),2+x)
(13)
~
and the remaining symbols are as in (4) and (5). K is found numerically to be ~ ( - 0 . 3 ) . First we shall consider the contributions from the top quark matrix, )~2. Taking 6m 2 from (4) and substituting it in (12) we find that 2 12 B~z,s~) ~ 7 × 10 -7 (C2~ \ C o o J [U23(/]'I)2U~3
bL
(14)
and similar for Z~bg, where the U,j are the usual Kobayashi-Maskawa matrices and 2t is the top quark Yukawa coupling. Using the accepted values of the U0, and 2 ~ O ( 1 ) we find that, in both the minimal supergravity and no-scale theories, the top quark insertions give contributions to the Z-,bg, sb branching ratios of B(Z~bs,sb) ~ 10 - 9 .
(15)
Such additions to the usual standard model effects are well known in supersymmetric theories [14], however a further enhancement may occur, at least in the minimal N = 1 supergravity theories, from the ,'~6 piece in (4). This gives an additional contribution to the branching ratio of B ( z ~ s b ) ~ 7 × 1 0 - 7 \~oo j [UI3(A6)2U~t3] 2,
(16)
where UIj is the new and unknown KobayashiMaskawa matrix due to/]-6. The above is a possibly dominating contribution to such decays because A 6 is unknown. As an example, taking A6 ~ 2, the above gives B(Z~bs,sb ) ~< 10 -6,
'
22 February 1990
( 17 )
for N = 1 supergravity ( 10- 8 for the no-scale theory ). Obviously, since (15) is proportional to (A6) 4, it is possible that such decays will be much more enhanced than this, perhaps to observable magnitudes, even in a no-scale theory. These decays will show a C P violation directly related to the neutron electric dipole moment calculated earlier, which may be defined by = F(sb) -- F(b~)
( 18 )
F~b) +F~b~) " Fig. 2. Diagram showing the supersymmetric contribution to a flavour changing Z decay.
308
The leading contribution to e comes from diagrams with two mass insertions which have the form
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(8,n 2) 2~ Im( VTL2~;22[),62; ] V,L)m 4 IC2 / C6 |\
\c~/
(19) T h i s is m u c h m o r e significant t h a n the d i p o l e m o m e n t due to the large top q u a r k mass. T a k i n g 2 2 = O ( 1 ) we find
C2~2
~:= 2 2 s i n d c o J
~
sin2d 4 '
(20)
w h e r e d is the u n k n o w n CP v i o l a t i o n p a r a m e t e r . T h i s is the s a m e in b o t h the m i n i m a l s u p e r g r a v i t y and noscale theories, as long as the 2 6 c o n t r i b u t i o n s to flav o u r c h a n g i n g Zo decays are d o m i n a n t , and can lead to an unusually large a m o u n t o f CP v i o l a t i o n .
4. Conclusion W h i l s t it is a c c e p t e d that the g e n e r i c s u p e r s y m m e t tic t h e o r y m a y lead to s o m e interesting p h e n o m e n o logical i m p l i c a t i o n s , in p r a c t i c e it is f o u n d that in m a n y t h e o r i e s these effects are r e d u c e d by such factors as the s m a l l n e s s o f the f l a v o u r c h a n g i n g in the K M matrices. In the p r e s e n t p a p e r we h a v e s h o w n that flipped S U ( 5 ) is able to realize s o m e o f these effects, m a i n l y d u e to the existence o f the n e w and u n k n o w n f l a v o u r matrices, 26/].t6 . T h e m a g n i t u d e and gauge h i e r a r c h y o f this c o u p l i n g (as d e r i v e d f r o m the f o u r - d i m e n s i o n a l f e r m i o n i c string) are not well k n o w n at present. T h i s allows the possibility o f a n e u t r o n electric dipole m o m e n t a p p r o a c h i n g the present day experim e n t a l limit and a f l a v o u r c h a n g i n g d e c a y for the Zo
22 February 1990
with a b r a n c h i n g ratio B ~ 10 . 6 and a possibly large CP v i o l a t i o n p a r a m e t e r ( u p t o ]-). T h e first two effects are suppressed in the no-scale t h e o r y due to the fact that the scalar masses r e m a i n small in the renorm a l i z a t i o n region b e t w e e n the s u p e r u n i f i c a t i o n scale and M o u r , thus m a k i n g it a physically m o r e attractive proposition.
Acknowledgement T h e a u t h o r s w o u l d like to a c k n o w l e d g e the assist a n c e o f N o e l C o t t i n g h a m a n d to t h a n k J o h n Ellis for useful discussions.
References [ 1] H. Kawai et al,, Nucl. Phys. B 288 (1987) l; for a review see G.G. Ross, CERN preprint CERN TH.5109/88. [2] S.M. Barr, Phys. Lett. B 112 (1982) 219; I. Antoniades et al., Phys. Lett. B 194 (1987) 231. [ 3 ] D. Bailin et al., Phys. Len. B 212 (1988) 23. [41 I. Antoniades et al., Phys. Len. B208 (1988) 209. [5] J. Ellis et al., Nucl. Phys. B 311 (1988/89) 1. [6] N.K. Falck, Z. Phys. C 30 (1986) 247. [7] M.J. Duncan, Nucl. Phys. B 221 (1983) 285; M. Scholl, Z. Phys. C 28 (1985) 545: J. Ellis et al., Phys. Len. B 114 ( 1982 ) 231 ; D.V. Nanopoulos and M. Srednicki, Phys. Lett. B 128 (1983) 61. [ 8 ] F. Gabbiano and A. Masiero, Phys. Lett. B 209 ( 1988 ) 289. [9] A.E. Farragi et al., Phys. Lett. B 221 (1989) 337. [ 10] J.P. Derendinger et al., Phys. Lett. B 155 (1985) 65. [ 11 ] R. Shabalin, Sov. J. Phys. 28 (1978) 75. [ 12] I.S. Altarev et al., Phys. Len. B 102 ( 1981 ) 13. [ 13] K. Green el al., RAL preprint (October 1989). [ 14] W. Bernreuther et al., CERN preprint CERN TH.5485/89.
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22 February 1990
S I G N A T U R E S O F F L I P P E D SU(5) S.A. ABEL H.H. Wills Physics Laboratoo,, Royal Fort, g),ndallAvenue, BristolBS8 1TL, UK and I. W H I T T I N G H A M Department of Physics. James Cook University, Townsville, QLD 4811, Australia Received 2 November 1989
We examine the electric dipole moment of the neutron and flavour changing Z decays in the flipped SU(5 ) grand unification scheme. We find that the new couplings at energy scales above Mcux can lead to neutron electric dipole moments Idl ~ 10-25 ecm and decays of the form Z~bs, sl~ with a branching ratio B ( z ~ , s b ) ~ 10 -6, of which a large proportion (~< ¼) may be CP violating. The first two effects are found to be slightly suppressed in the currently popular no-scale theories, but the CP violation parameter, ~, is relatively theory independent.
1. Introduction
The construction o f four-dimensional fermionic superstring theories [ 1 ] has led to the derivation o f a n u m b e r o f p r o m i s i n g candidates for unification below the Planck scale. O f these, perhaps the most interesting is flipped S U ( 5 ) [2]. This model is unusual in that the Higgs fields do not transform under the adjoint representation. We have the following particle content: - t h r e e generations o f m a t t e r fields denoted by F(10, ~),f(~, - ~3) , and {~C(l, 2); s - a pair o f Higgs fields to break the symmetry clown to SU ( 3 ) × (SU ( 2 ) × U ( 1 ) denoted by H ( 10, ½) and H(10, -~); - a pair o f Higgs to break S U ( 2 ) × U ( 1 ) denoted, h (5, - 1 ) and ~(5, - 1 ), and four gauge singlet fields ~,,, (where m = (0 ..... 3) ). The superpotential of the model is given by W = 2 ~FFh + 2 2~f~f+ 2 3f~ch + )~4HHh + 2sI~It]l+ 26FtIq)+ ).7hh(b+ ).sf)O0,
(1)
where 21 ( h ), -~2(]'l) and 2 3 ( h ) are the mass matri-
ces o f the down and up quarks, and electrons respectively (the generation indices have been suppressed). Its many interesting features, such as a natural d o u b l e t - t r i p l e t mass splitting mechanism for the Higgs fields h a n d / t - a n d a suitable neutrino seesaw mechanism. All the Yukawa couplings o f minimal SU (5) × U ( 1 ) have been derived from renormalizable interactions o f the fermionic string ( a n d are o f order xfi2g), apart from the )-6 coupling which is p r o v i d e d by nonrenormalizable terms in the superpotential, and which gives rise to the see-saw mechanism [3,4]. Subsequent running o f the R G E ' s leads to a low energy superpotential consistent with experiment [5]. The ,).6 coupling has particularly i m p o r t a n t consequences for the low energy phenomenology o f the theory because it carries a flavour index. W h e n the supersymmetric theory becomes softly broken, the R G E ' s give additional terms in the scalar mass matrices, whereas the renormalization o f the Yukawa couplings is unchanged (by virtue o f the no-renormalization theorem [ 6 ] ). This leads to the introduction o f 2 6 flavour changing terms and hence new K o b a y a s h i - M a s k a w a mass mixing matrices. These matrices are unknown, and their off-diagonal ele-
0370-2693/90/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland )
305
Volume 236, number 3
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ments are not necessarily (and indeed, unlikely to be) small. The RGE's for the squark masses take the form
22 February 1990
be dominated by linear terms, and to be approximately of the form #H~ .=-#H//2 ( Co q- CI ,~,~-)q q- C2,~ ~)-2 q- C6 [-~6-4~ ] ),
(4)
dm2/dt= (1/8~ 2 ) [3Atl(tH 2 J-A 2)/].1 + ; ~ ( m 2 +m~ +~4~),~1,
dm~/dt= (1/87r 2) [42~(m2v+m 2 +A2)22],
(2) (3)
where we have omitted terms diagonal in the generation indices and neglected the electron masses. Possibly important contributions to flavour changing processes arise in such theories from diagrams where we have a gaugino-fermion-sfermion vertex, due to the fact that the fermions and sfermions are now no longer simultaneously diagonalizable [7 ]; the G I M suppression mechanism no longer operates. For most supersymmetric theories these diagrams still suffer some suppression due to the smallness of the off-diagonal terms in the Kobayashi-Maskawa mixing matrices. However, flipped SU (5) introduces two unknown Kobayashi-Maskawa matrices describing the mixing between the 26 couplings and the up and down quark masses. This has been pointed out for N = 1 minimal supergravity models [ 8 ], and more recently for the string derived no-scale theory [ 9 ], where rare flavour changing processes such as in B decays have been shown to be possibly enhanced. In the minimal supergravity models sofl-SUSY breaking occurs for both the scalars and gauginos and is of the order of the gravitino mass, whereas in the no-scale theory, supersymmetry is softly broken at a scale considerably below the Planck scale by gaugino masses only. This could be achieved by, for example, condensation in the hidden sector [10]. In such models the effects described above are reduced because the G U T scale and the super-unification scale are driven close together dynamically [5] and the scalar masses remain small in this region; they receive only small contributions from the 26 couplings. Choosing regions of parameter space which give a low energy theory consistent with phenomenology (as determined in re['. [ 5 ] ), we have run the RGE's down to the low energy scale with various choices of Yukawa couplings, in order to make a crude estimate of their contributions to the scalar masses. These were found to 306
for the left-handed scalar doublet. In the above equation, C,, are constants; ( o is 0 ( 2 ) ; ('~ and ('2 are O( 1 ). 6"6 is sensitive to our choice of strong coupling constant within the present experimental bounds, but is less than O@o ) for the no-scale theory, and less than O(~ ) for minimal N = 1 supergravity models. Similar equations may be obtained for the right-handed scalar fields, but these mass matrices are dominated by the pieces which are diagonalized in the SKM basis and so give only small contributions to flavour changing decays. This is as expected, since the RGE's can only give off-diagonal contributions in the renorrealization region above ,.I/c;tjT.
2. The electric dipole moment of the neutron
It is well known that in the standard model the electric dipole moment of the neutron is a two loop effect, and takes values o f d < 1 0 37 ecru [ I l l . To date, the best available limits on this are d < 6 × 10-25 e cm [ 12 ] although it should be noted that recent reports [13] claim to have measured d = ( 3 _ + 5 ) × 10 -26 e cm. It has been shown that for the generic supersymmetric theory, in addition to contributions coming from one loop corrections to the imaginary ( CP violating) part of the quark mass, the electric dipole moment of the neutron may receive direct contributions from diagrams such as fig. la [7 ]. Assuming that the m o m e n t u m of the quark in the diagram is small compared with the masses of the scalars and gauginos allows us to expand, and extract the dipole moment relatively easily. For n mass insertions in the gluino diagram, we obtain a contribution of
d=8qg~K,(nq-l)(-l)"(Sni2~" 3(47r)2m S \~/ ,
(5)
where q is the electronic charge of the internal squark, 8m 2 is the off-diagonal piece of the squark masssquared matrix, and m 2 is the diagonal piece. K,, is given by
Volume 236, number 3
PHYSICS LETTERS B
22 February 1990
( S m 2 ) 4 ~ I m ( V2L2,2"~222:~mu2*t2, V:~R)m 7 { C2C2"]
(8)
a
b Fig. I. (a) The conventional diagram for supersymmetric contributions to the neutron electric dipole moment; the blobs represent arbitrary numbers of mass insertions. (b) A contribution involving the wino and charged higgsino; this is expected to be an order of magnitude smaller.
I
( CoX)'+ ' [ ( C o - n ) x - n J
K,, = f dx x~o(n+ 0
1) [ ( C o - 1 ) x + 1 ] , + 2 '
(6)
Any insertions with less Yukawa couplings may be reduced to the product of a matrix and its hermitian conjugate, and is therefore real. For large gaugino masses ( > 100 GeV) the low energy superpotential of flipped SU (5) is such that the two Higgs fields, h and/7, can have vacuum expectation values that differ by a factor of upto ten [ 5 ]. This explains the large mr~rob ratio, and means that the value of 2b may be larger than in the standard model. Because o f this, as a worst case we shall take Ab = 0.1 and the gluino mass a s ml/2= 100 GeV. Substituting accepted values of the couplings and KM matrix elements into (8) gives a worst case contribution to the dipole moment o f d ~ 10 -33 ecru.
(9)
In the case o f minimal flipped S U ( 5 ) the dipole moments are potentially much larger since we have an extra matrix in generation space. Here the leading term has a mass insertion of the form
(
(fire 2) 3 Im( V2c21'2, mu [2626*] V_~R)trts5 \ ~ o 2 and is found to be relatively Co independent and 0 ( 0 . 0 2 ) . Interestingly, in addition to fig. la, we also have contributions from the diagrams in fig. l b, where the charged higgsino couples the wino to the righthanded fields, however we expect this diagram to be an order o f magnitude smaller. For many theories, notably those where supersymmetry is broken by F-terms in the potential, such contributions are greatly reduced. For example, consider a softly-broken minimal SUSY standard model theory. In such a theory we have only two (non-trivial) flavour changing couplings, 21 and 22 (since we may diagonalize the lepton mass matrix) which may be diagonalized by the usual bi-unitary transformations caused by rotating the fields in generation space,
(/~1)diag = VIL/~I ~F~R, (/'[2)cliag = ~2L22 V~R-
(7)
Consider the up-quark/up-squark/gluino diagram. The leading contribution to the dipole moment comes from four mass insertions in fig. I a giving
(10) Not only do we have one less mass insertion in this equation, but also we have three unknown quantities, namely the mixing angles and CP violation parameters of the new KM matrices, and the magnitude of the 2 6 couplings. Taking our worst case value of 2b, the top quark mass to be 100 GeV, and inserting the above and its permutations into ( 5 ) gives a worst case contribution of d~ -6)<
10-25((6A2 sin20 sin d)e cm,
( 11 )
where sin20 sin d is some combination of the unknown KM i.,.ameters and ( A 6 ) 2 is the largest eigenvalue of [262*6]. Because of the large top quark mass and the relatively large value of 2 8 , this diagram dominates over the down-quark/down-squark/gluino diagram. Sipce the value o f A 6 may have to be quite large in ore' "o give satisfactory G U T breaking, and its value at present as derived from fermionic string theories is not well known, it is possible for flipped SU (5) to give neutron electric dipole moments ap307
Volume 236, number 3
PHYSICS LETTERS B
proaching the present day experimental limits in a fairly natural way.
3. Flavour changing decays of the Zo It is known that the new Kobayashi-Maskawa matrices, introduced by the superpotential of flipped SU ( 5 ), can lead to some interesting flavour changing effects such as an enhanced B decay and ~t-e7 [8]. We shall now show that similar considerations can lead to large flavour changing decay modes for the Zo of the form Z~t3s and possibly Z--,tc with a significant CP violation, Such decay modes are of particular interest since they would be directly detectable in forthcoming experiments at LEP. The leading contribution to flavour changing Zo decays comes from the diagram in fig. 2. The analysis for Z ~ b s may be simplified by assuming that the masses of the products are approximately zero compared with the Zo mass so we have concentrated on this process; our arguments apply equally to Z ~ t c modulo the suppression caused by the large top quark mass. Calculating the diagram for a single mass insertion on the left or right squark branch and summing over gluinos gives a branching ratio for this process of g4K2 2 B { z ~ b s , ~ - - 12(4=) 4 \ m~ ] '
(12)
where K is given by K= -
d),dx 0,0
1-x+,,/Co(2y-xy-),2+x)
(13)
~
and the remaining symbols are as in (4) and (5). K is found numerically to be ~ ( - 0 . 3 ) . First we shall consider the contributions from the top quark matrix, )~2. Taking 6m 2 from (4) and substituting it in (12) we find that 2 12 B~z,s~) ~ 7 × 10 -7 (C2~ \ C o o J [U23(/]'I)2U~3
bL
(14)
and similar for Z~bg, where the U,j are the usual Kobayashi-Maskawa matrices and 2t is the top quark Yukawa coupling. Using the accepted values of the U0, and 2 ~ O ( 1 ) we find that, in both the minimal supergravity and no-scale theories, the top quark insertions give contributions to the Z-,bg, sb branching ratios of B(Z~bs,sb) ~ 10 - 9 .
(15)
Such additions to the usual standard model effects are well known in supersymmetric theories [14], however a further enhancement may occur, at least in the minimal N = 1 supergravity theories, from the ,'~6 piece in (4). This gives an additional contribution to the branching ratio of B ( z ~ s b ) ~ 7 × 1 0 - 7 \~oo j [UI3(A6)2U~t3] 2,
(16)
where UIj is the new and unknown KobayashiMaskawa matrix due to/]-6. The above is a possibly dominating contribution to such decays because A 6 is unknown. As an example, taking A6 ~ 2, the above gives B(Z~bs,sb ) ~< 10 -6,
'
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( 17 )
for N = 1 supergravity ( 10- 8 for the no-scale theory ). Obviously, since (15) is proportional to (A6) 4, it is possible that such decays will be much more enhanced than this, perhaps to observable magnitudes, even in a no-scale theory. These decays will show a C P violation directly related to the neutron electric dipole moment calculated earlier, which may be defined by = F(sb) -- F(b~)
( 18 )
F~b) +F~b~) " Fig. 2. Diagram showing the supersymmetric contribution to a flavour changing Z decay.
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The leading contribution to e comes from diagrams with two mass insertions which have the form
Volume 236, number 3
PHYSICS LETTERS B
(8,n 2) 2~ Im( VTL2~;22[),62; ] V,L)m 4 IC2 / C6 |\
\c~/
(19) T h i s is m u c h m o r e significant t h a n the d i p o l e m o m e n t due to the large top q u a r k mass. T a k i n g 2 2 = O ( 1 ) we find
C2~2
~:= 2 2 s i n d c o J
~
sin2d 4 '
(20)
w h e r e d is the u n k n o w n CP v i o l a t i o n p a r a m e t e r . T h i s is the s a m e in b o t h the m i n i m a l s u p e r g r a v i t y and noscale theories, as long as the 2 6 c o n t r i b u t i o n s to flav o u r c h a n g i n g Zo decays are d o m i n a n t , and can lead to an unusually large a m o u n t o f CP v i o l a t i o n .
4. Conclusion W h i l s t it is a c c e p t e d that the g e n e r i c s u p e r s y m m e t tic t h e o r y m a y lead to s o m e interesting p h e n o m e n o logical i m p l i c a t i o n s , in p r a c t i c e it is f o u n d that in m a n y t h e o r i e s these effects are r e d u c e d by such factors as the s m a l l n e s s o f the f l a v o u r c h a n g i n g in the K M matrices. In the p r e s e n t p a p e r we h a v e s h o w n that flipped S U ( 5 ) is able to realize s o m e o f these effects, m a i n l y d u e to the existence o f the n e w and u n k n o w n f l a v o u r matrices, 26/].t6 . T h e m a g n i t u d e and gauge h i e r a r c h y o f this c o u p l i n g (as d e r i v e d f r o m the f o u r - d i m e n s i o n a l f e r m i o n i c string) are not well k n o w n at present. T h i s allows the possibility o f a n e u t r o n electric dipole m o m e n t a p p r o a c h i n g the present day experim e n t a l limit and a f l a v o u r c h a n g i n g d e c a y for the Zo
22 February 1990
with a b r a n c h i n g ratio B ~ 10 . 6 and a possibly large CP v i o l a t i o n p a r a m e t e r ( u p t o ]-). T h e first two effects are suppressed in the no-scale t h e o r y due to the fact that the scalar masses r e m a i n small in the renorm a l i z a t i o n region b e t w e e n the s u p e r u n i f i c a t i o n scale and M o u r , thus m a k i n g it a physically m o r e attractive proposition.
Acknowledgement T h e a u t h o r s w o u l d like to a c k n o w l e d g e the assist a n c e o f N o e l C o t t i n g h a m a n d to t h a n k J o h n Ellis for useful discussions.
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