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An upper limit on the mass of exotic fermions in superstring inspired E(6) models
Volume
196, number
AN UPPER LIMIT IN SUPERSTRING Manuel
PHYSICS
1
LETTERS
ON THE MASS OF EXOTIC INSPIRED E(6) MODELS
B
24 September
1987
FERMIONS
DREES and Xerxes TATA
Physics Department, University of Wisconsin, Madison, WI 53706, USA Received
11 May 1987; revised manuscript
received
I 1 June 1987
We show that in a large class of superstring inspired E( 6) models considered in the literature the requirement of perturbative unification implies that the lightest neutral exotic fermion ( #v’) is lighter than 115 GeV. In many models this fermion may be considerably lighter and may even be the lightest supersymmetric particle. It is argued that such a particle is compatible with all known experiments and with cosmology. Some phenomenological implications are also discussed.
In the last two years a large amount of interest has been focused on the study of the phenomenology of E( 6) supergravity GUTS #’ which emerge as the low energy limit [ 21 of anomaly free superstring theories [ 31. A common feature of these models is the prediction of many exotic particles as required by the multiplet structure of E( 6). Unfortunately most of these particles are expected to have masses well above the W-mass [ 4-61, with the notable exception of some of the exotic neutral fermions: three of these, which are usually referred to as right-handed neutrinos (since they are SU( 2)-singlets), are massless [ 41 in models without any intermediate scale. Furthermore, in addition to the neutral gaugino-higgsino mixed states the model contains at least six exotic neutral fermions. The reason is that in addition to standard quark and lepton superfields, each 27-dimensional generation contains two SU( 2)-doublets H and H and an SU( 2)-singlet N with the right quantum numbers to spontaneously break the electroweak and additional U( 1) gauge symmetry, respectively. There are thus twelve neutral Majorana fermions which can mix in a complicated fashion. It is, however, always possible to find a basis where only one H, one H and one N has a non-vanishing vacuum expectation value (VEV). Following the notation of ref. [ 71 we denote the fields that acquire a VEV by H3, 8, and NJ. One way to insure that (Z,) =0 (Z= H, H, N, i= 1, 2) is to require [ 7,8] that the superpotential couplings of the terms Z,Z3Z3 as well as the soft breaking mass terms of the form m2zI z3Z ,Z: identically vanish; in this case the 12 x 12 neutral fermion mass matrix breaks up into two 6 x 6 matrices. We note, however, that while the vanishing of these couplings and mass terms is sufficient for the VEVs of Z, to vanish, it is not necessary since in the scalar potential the terms linear in Z, from the F-term contributions can be cancelled against the soft supersymmetry breaking A term as well as the off-diagonal scalar mass terms [ 71. In this case, there is no decoupling of the 12 X 12 matrix into two 6 x 6 blocks; similarly, the charged fermion mass matrix does not decouple into two 2 X 2 blocks [ 71. In general, neither of these solutions is invariant under renormalization group evolution #2 so that a priori neither possibility is preferred. In the remainder of this letter we choose to work in the class of models where the decoupling of the exotic fermion mass matrices does occur since this is the possibility that has been extensively studied in the literature [ 4,5,7,8]. In this case, the six neutral and the #’ See ref. [ 1 ] for reviews. ” This is even true for the solution of ref. [7], 1$,= ,1,,,,/2,,,,=0 (i= 1, 2), since that only guarantees the I,,, to vanish at all scales; however, for finite A,,, (i, j. k= I, 2) nonvanishing off-diagonal soft breaking mass terms will be generated even if these are diagonal and equal at the unification scale. Nevertheless the choice [ 71 A,,,=m$,,, = 0 can be technically natural if all A,,, = 0, since in this case the lagrangian is invariant under the transformation Z ,+ -Z,, Z,+Z,, which forbids all dangerous terms.
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65
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two charged fermions contained in the H I , 2- I7"I1,2- and Nj,2-superfields do not mix [7,8] with the gaugino-higgsino sector. Instead, they get their masses from the superpotential
2 fm= Z
i,J= 1
(2a3HJZljN3 +2,3jHiNjI-I3 +23ijI-liNjH3) •
(1)
For models without any intermediate scale, this leads to the mass matrix (in the basis (HI, IZI~,Nl, H2I?]2, N2))
0 2113X M=12101 0
2113X 0 2311/3 2213X ~2123X 0 \2123/~ 2312U
2131/7 2311U 0 2231/7 2321U 0
0 2213X 2231/7 0 2223x 2232/7
2123X 0 2321U 2223x 0 2322U
2132/7~ 23 U/ 0 2232/7/ , 23 /)] 02
(2)
where v = ( H ° ) , /7= (IzI°), x = ( N ) . F r o m this mass matrix it immediately follows that the mass 3//o of the lightest exotic fermion is b o u n d e d according to M 2 ~
(3)
The b o u n d (3) tells us that M0 is at most of order Mw rather than Mz,, the mass of the additional gauge boson Z', which sets the scale for the mass of most o f the exotic particles predicted by E (6) models. Note that this is true even if we consider the full 12 × 12 matrix discussed above ~3. The mass Mo may well be much smaller than indicated by eq. (3). The reason is that x has to be much larger than v and/7 in order to make the Z ' boson sufficiently heavy [9], especially if one takes into account the requirement that the presence o f the massless right-handed neutrinos mentioned above does not spoil successful predictions for nucleosynthesis in the early universe [ 10]. This translates into the b o u n d Mz, ~>300-400 GeV, which gives x > 700-900 GeV. Note furthermore that the determinant of the mass matrix (2) is only quadratic in x, while four eigenstates acquire masses ~ 2x if all the couplings in eq. (1) are o f the same order. This obviously implies [ 7 ] that the two lightest eigenstates get masses o f order M2w/Mz, ! The only way to avoid such small eigenvalues is to impose a hierarchy between the 2iik such that the hierarchy between the VEVs is cancelled. It is also interesting to note that one gets two massless eigenstates if either 23a = 0 or 2i3j = 0 for i, j = 1, 2. This means that all the three terms shown in eq. (1) have to be present. In this paper we attempt to obtain an absolute upper b o u n d on the mass of the lightest neutral fermion eigenstate of the matrix in eq. (2), independent o f the details of the model. The mass matrix (2) has been written in a completely general basis. It is, however, more convenient to work in a basis [ 7 ] where the mass matrix for the charged exotic fermions of the first two generations is diagonal, i.e. to perform rotations in the (H1, H2) space and the (I-~Ii,I'~I2) space such that there are no HzITIIN3 - and HllZI2N3-terms in the new basis. In this basis the mass matrix has the same form as in eq. (2), but with 2123= 2213= 0. In this basis the nonvanishing x-entries 2~13x and 2223Xare the masses o f the charged exotic fermions and thus must exceed 20 GeV so as to satisfy the P E T R A bound [ 11 ] on the mass of new charged particles. Even in this basis, one has to deal with ten independent parameters, which makes a complete analytic treatment o f the problem of finding a relationship between the ten 2ak that maximizes the magnitude of the smallest eigenvalue very difficult. We proceed by first recognizing that the IXijkl at the weak scale are all b o u n d e d from above by a rather small value since they would otherwise diverge [12,6 ] before the unification scale making perturbative unification impossible. We then searched numerically for the combinations of the ten 2ijk that maximize Mo using the M I N U I T program o f the C E R N software library. After imposing a c o m m o n upper bound ~3This followssince the N 1and N2 columns in the 12 × 12 matrix can only have entries proportional to v or ~, since all the relevant terms in the superpotential are required to have the structure N~HflZIk. 66
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2 on all [2ijk[ we found that [Mol is maximal if and only if (up to trivial phase transformations) 2131 =2132 =2311 = --2312 =2231 = --2321 =2232 =2322 -----~2, 2113= --2223 -- 22vg/(x/~ + t72 "x) .
(4a, b)
In this case one even gets f o u r light eigenstates with the same mass, which is given by (5)
Mo = 2 2 v O / . ~ 2 + O2 ,
which, in this case, is also equal to the mass of both the charged exotic fermions. Note that Mo vanishes if ~=---O/v+ov since the sum v2+02 is fixed by the W-mass, M~v = ~g2 1 2 (v 2 + ~2) where g2 is the SU(2) gauge coupling. Of course, the absolute bound on the smallest eigenvalue depends on how large 2 in eq. (4) is allowed to be; we will address this issue shortly. Unfortunately, we have not been able to prove that eq. (5) indeed gives the maximal value for Mo if one requires 12~jk]~<2. We could, however, show that Mo cannot exceed the value given by eq. (15) by more than 25%. One strategy for deriving bounds of this kind is to compute the "expectation value" E of a suitably chosen normalized vector y, i.e. E 2 = IlMyll2; E is then an upper bound for Mo. If one chooses e.g. yl = (0, 0, 0, 0, cos y, sin ~,), where y is determined by the requirement that the x-dependence of My~ vanishes, one finds E 2 ~<22 [sin2y(2v2 + g2) + 2/.;2COS2y] ~<22(2V2 q_ g2) ,
(6)
which gives Mo.<2
2 .
(7)
Another hound can be derived if one considers the vector Y2= (0, COS O¢, sin o~ COSfl, 0, 0, sin o~ sin fl) where c~ and fl are chosen so that the x-dependence of My2 again vanishes; this gives E 2 ~ 2 2 2 v 2 [ s i n 2 0 ~ ( s i n f l + c o s f l ) 2 +cos20~] 4422/.32 ,
(8)
i.e., M0 < 2 2 v .
(9)
Note that the bound (9) approaches the value given in eq. (5) for oo --, ~ ; for co = 1 eq. (7) gives a better bound which is only a factor xl/q~ larger than the value (5). Up to now we have only performed rotations among (H~, H2) and (H~, H2) in order to get rid o f 2t23 and 22~3. By choosing an appropriate rotation in the (N t, N2)-space we can, without loss of generality, eliminate any one entry proportional to v or ~ in the mass matrix (2). We have not yet used this freedom since this makes the matrix look less symmetric. However, since our results cannot depend on the choice of the basis, we can arbitrarily choose the basis where (e.g.) 2132= 0. Again constraining all 12,jkI from above we now find that ]Mo ] is maximal if 2131 =2311 = 2 3 1 2 : 0
--2231 =2321 =2233=2322 ~ 2 ' , 2223=0 -
(lOa, b)
In this situation the eigenvalues again come in pairs that only differ by their sign, and one finds Mo = xf22' v,
(l l)
the other eigenvalues being given by +2~t3x and + ~,/22' ~ (note that this model always predicts [4,6] ~> v). We see once again that Mo vanishes in the limit (o-~ oo. In this basis, we can indeed prove that eq. (1 1 ) gives the maximal possible value of Mo. To do so we merely have to repeat the steps that lead to the bound (6), since the vector yj is constructed such that the only contribution to its expectation value that is proportional to 17has the coefficient 2~32, which vanishes here. Therefore the bound (7) becomes Mo <~x / ~ 2 ' v ,
(12)
which is saturated by eq. (I 1 )! 67
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We now turn to a discussion of the upper bound on ILjkl alluded to earlier. This upper bound emerges from the requirement that all couplings should remain in the perturbative region, i.e. 22k < 4~, for all energies up to the GUT-scale M x - 1016 GeV. In order to find out what this means for the upper bound on the 12ijkl of the weak scale we have computed the renormalization group equations ( R G E ) for these couplings under the assumption that the whole superpotential is given by eq. (1), i.e. we have set all 2ijk with i, j, k = 1, 2, to zero. This may be achieved by imposing a discrete symmetry under Z~--,-Zi, Z3----~Z3 ( Z = H , ISI, N ) . In this case, our choice 233i=23i3 =2,33 ~-0 is invariant under renormalization group scaling. Furthermore, from our experience, we find that the introduction of additional Yukawa couplings tends to accelerate the scaling of 2,jk of eq. (1); as a result, the 12~jk(Mw) I would have to be chosen smaller than before, which only strengthens the bound. On the other hand it is, of course, possible that all other couplings are indeed negligible. Using the general formulae of ref. [ 13 ] we find
d2ij3/dt= (1/167~2)[2ij3(42~j3+4223 +4223
2 -~-2t3z j "]-2i3l 2 -~-23ji 2 -]-23jk 2 -- 3g22 -- ~g~) + 22lk3
+ 2k j3 ( 22.32kt3 + 2,3j2k3j + 2i312k31) + 2il3 (23ji23ti + 23jk23tk) ] ,
( 13 )
where both U(1 )-couplings have been set equal to gb and the indices k, ! have to be chosen such that i . k = l . j = 2 and t = l n Q, d~i3j/dl=(1]16nz)[2i3)(22j3
+2213 + 243o 2 + 24 23kj + 44 2~3j+ 44 2k3j "~ 44 2m + 2 k3l 2 - 3gZ2 - ~ g ~ )
+ 2i31( 223ij23i l -Jr-223kj23kl + 32k3j2k3! ) + 2k3j (2ij3 ~kj3 + 2il32kl3 ) ] •
(14)
The RGE for 430 can be obtained from eq. (? 4) by simply exchanging the first two indices on all 2's. Starting from the initial conditions given in eq. (4) we find the bound 2(Mw) < 0 . 5 ,
(15)
while the boundary conditions (1 O) result in 2,(Mw) < 0 . 7 .
(16)
This illustrates the general trend that the presence of a large number of Yukawa couplings strengthens the bound on their value. The absolute upper bound on Mo can now be obtained by inserting eq. (16) into eq. (11 ). We find Mo
(17)
From the constraint (15) we see that the ansatz (4) results in even smaller values for Mo. Our result (17) has been obtained using the requirement 12ijkI ~<2' for all i, j, k. In principle it is, however, possible that Mo could be increased by reducing some of the couplings even further which would allow the other couplings to be larger than given in eq. (16). In order to investigate this possibility we performed several runs of M I N U I T where the upper bound on all the Yukawa couplings is not the same constant but depends on the actual ratios of the various Yukawa couplings. This procedure is rather time-consuming since the numerical integration of the RGE is now a part of the function to be minimized by MINUIT. Our results indicate, however, that eq. (17) does give the absolute upper bound. We note that the ansatz (10) implies that one charged exotic fermion is massless, since 4223=0 here. This is obviously not a realistic situation. We have studied the effect that a finite mass of the exotic charged fermion might have on 3//0. To this end, we repeated the previous analysis, imposing the additional constraints 12113Xl >/ 12223Xl = m l qmin +,
(18)
where m mm n + is the mass of the lighter charged fermion. Fig. 1 shows the resulting bound on Mo as a function rain for o) = 1 as well as o) = 3. It turns out that the ans~itze (4) and (10), modified according to the requireofrnn+ 68
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120 100
B A
80 M~°X 60 --B
(GeV)
A
~=I
40
2o
q
I00
200 500 mrnin •~+ (GeV)
400
500
Fig. 1. The upper bound on Mo, the mass of the lightest neutral exotic, is shown as a function of m~+", the mass of the lighter charged exotic H-fermion, for two values of ~o _--O/v. The curves labelled A and B have been obtained using the ansatze (4a) and rnin (10a), respectively, with 12tl3xl >~l)~223xl=mR+. The curves labelled A are not extended into the region of small m ~ since they would lie much below the true m a x i m u m which is here given by the curves labelled B.
ment (18), still give the largest values of M0 ~4 The curves for the modified ansatz (4) have been terminated at m ~ n = 0 . 5 v ~ / , ~ 5 + t72 since for smaller values of m nm~n + , Mo decreases here and is thus far below the result obtained from the modified ansatz (10). Note that the bound on the mass of a sequential charged heavy lepton that has recently been set by the UA 1 collaboration [ 14] does not apply f o r / ~ + since the corresponding neutral fermion might be as heavy as 100 GeV, see eq. (17). But even the requirement [11] mn+ >_-20 GeV reduced the bound on Mo from 125 GeV to 115 GeV. Up to now we have only considered models where all matter fields reside within 3 generational 27's of E (6). There might, however, be [ 2] additional light "survivor" superfields out of non-generational 27 + ~ , some of which might carry the gauge quantum numbers of N, H or IZI;in this case the relevant mass matrix would be larger than shown in eq. (2). It is probable that this would allow for larger values of Mo if one again requires I2~;kl <2; but the existence of additional non-negligible terms in the superpotential (1) would strengthen the bound on 2. Since these effects tend to compensate each other we do not expect the final bound to be substantially altered. In this context it might be interesting to note that even a toy model with only one generation of exotic fermions reproduces the bound (17) for large 09, while for o9 _~ 1 the bound on Mo is about 20% smaller than that of the realistic model. Finally, there also exists the possibility [15] of an intermediate scale M I ~ 10 l° GeV. If one of the superpartners of the right-handed neutrinos gets a VEV of that order the right-handed neutrinos can get [ 5] Majorana masses of order 1 TeV; the mass matrix for the H,IZI,N-sector remains unaltered. However, the bound on the Yukawa couplings, and thus on Mo, might increase by 20%. The reason is that one now can only require the couplings to remain in the perturbative regime for energies up to Mx; at higher energies a new, unknown set of RGE replaces the old one, so that no definite statement is possible. If, on the other hand, one of the Nfields gets a VEV of order MI the (3, 3)- and (6, 6)-entries of our mass matrix can become as large as 1 TeV, and no strong bound can be derived. We have shown that the existence of an exotic fermion state ( ~ v c) with a mass smaller than about 115 GeV is a general feature of many superstring inspired models without any intermediate scale. In fact, in many models this state may be considerably lighter than the bound cited above and may even be the lightest supersymmetric particle (LSP) and hence stable. We note here that the existence of such a stable particle may be dangerous for cosmology since its relic mass density could exceed the observed mass density of the universe unless there is an efficient annihilation mechanism for these particles. ~4 If one requires 3.t23=2213=,~132=0, 12uk] <2 otherwise, M I N U I T finds as best solution2231=0,2131 =A311 =--3.312=2321=2232=2322=2, if rn ~'," > 200 GeV; however, from the RGE analysis one finds .~< 0.58, which results in smaller values of Mo than those shown in the figure.
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24 September 1987
supersymmetric relics has already been studied in the context of supergravity models [ 16 ]. that if the LSP is a higgsino, h, with mixings such that the Z°hh coupling is not significantly h-pairs can annihilate via s-channel Z°-exchange, cosmologically acceptable h relic densities if annihilation into bb pairs is kinematically possible ~5, i.e. if rn~ > m b.
We note that the exotic fermions we have discussed here are similar to h except that their couplings to matter are not proportional to matter fermion masses. In general, they are arbitary mixtures o f Hi, IzI~and Ni ( i - - l , 2) and hence have couplings to Z °. Moreover, the lightest o f these is usually heavier than the b-quark, so that such a stable exotic is cosmologically safe. One significant exception to this is if all the 2ijk in eq. (1) are comparable and x~> v, g. In that case, there are two light states which are dominantly mixtures of just N1 and N2 and masses ~M~v/Mz,. These states couple to Z ° only via a small doublet component and hence their annihilation is strongly suppressed. The singlet exotics can have superpotential couplings only to other exotics so that the coupling o f this light state to ordinary matter is also suppressed by Mw/Mz,. Its annihilation via these couplings is, therefore, small. A stable singlet exotic would lead to cosmological problems with its relic mass density [ 16 ]. Within the cosmological context, there is one important difference between the usual supergravity models and the exotic sector of superstring models that seems worth commenting on. This stems from the fact that the exotic sector may contain several light states, (approximately) degenerate in mass, especially for those sets o f 2ijk that lead to the bounds discussed earlier (see eqs. (4) and (10)). This could also occur due to some as yet unknown symmetry of the couplings 2~jk. An example for such a symmetry is the discrete Z3 symmetry considered in ref. [ 17]. In this case one has 2~3t :•321:2321 :•232=•123=2213 =0- One can then see that for each eigenvector (X~, X2, X3, X4, )(5, )(6) with eigenvalue m there is an eigenvector (X~, - X 2 , X3, - X 4 , )(5, - ) ( 6 ) with eigenvalue - m , so that all eigenvalues come in pairs. In this case, annihilation o f two differentMajorana fermions would dominate that of identical Majorana fermions in the non-relativistic limit since it is not Pwave suppressed the way the annihilation o f identical Majorana fermions is [ 16 ]. An alternative way o f viewing this is that if the masses o f the two fermions are exactly degenerate, they can be combined into one Dirac fermion whose annihilation is unsuppressed. In this case it may well be possible for the light exotic to contain large SU (2) singlet components. This s-wave annihilation may also be important in the discussion of the decoupiing of the various particles, since the cross sections at temperatures close to threshold are largest for the case of distinct fermions. We now turn to a brief discussion o f the phenomenology of the light exotics. As we have already noted, if they interact only via gauge interactions and the superpotential of eq. (1), the transformations H , ~ - H i , I Z I ~ - IZli, N ~ - N , ( i = 1, 2) with the i = 3 fields going to themselves are symmetries of the interactions; in this case the lightest state o f the mass matrix (2) or one o f the charged states is absolutely stable. This, of course, is no longer true if we allow all the 2ijk in the superpotential 2ijkH~IZljNk to be non-vanishing or if we introduce couplings of these exotics to matter supermultiplets ~6. In this case, the lightest exotic is absolutely stable only if it is the LSP. Although it is possible to choose all these couplings to vanish, we will not pursue this any further. What if the exotic is indeed the LSP? In this case, the phenomenology would crucially depend on its direct couplings to matter. In the absence of such couplings, all SUSY particles would decay into the lightest SUSY particle o f the non-exotic sector (presumably a combination of a neutral gaugino and higgsino), which could decay via exotic particle loops into the LSP + photon unless the discrete symmetry discussed in the previous paragraph is maintained. This would lead to a degradation o f the missing transverse m o m e n t u m from the case ,5 The annihilation cross section has an S-wave contribution proportional to m 2 so that the cross section into massless fermions is Pwave suppressed in the non-relativistic limit. ~6 Such couplings would, of course, introduce additional terms in the renormalization group equations (13) and (14) and thus max further reduce the bound on the mass of the lightest exotic. We have neglected this in the present analysis. 70
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where the S U S Y particles directly decay into an LSP. I f there are direct couplings o f the LSP to matter, the lightest non-exotic sparticle would decay into the t h r e e - b o d y m o d e LSP + f-f m e d i a t e d by f exchange. I f these couplings are large enough squarks a n d sleptons would decay via the two-body m o d e f ~ f + L S P a n d ~ via ~ q ~ t + L S P so that the p h e n o m e n o l o g y would be similar to that for the case when the LSP is the photino. Since the couplings o f the exotics to m a t t e r are arbitrary, it is possible for bizarre p h e n o m e n o l o g y to result. To illustrate this, we consider the case where the exotics have direct couplings only to the leptons. In this case even strongly p r o d u c e d SUSY particle events would contain multileptons in the final state. Clearly, the resulting p h e n o m e n o l o g y would be very m o d e l d e p e n d e n t a n d the details are b e y o n d the scope o f this letter. In s u m m a r y , we have shown that in all superstring-inspired E ( 6 ) m o d e l s without any i n t e r m e d i a t e scale, i m p o s i n g the r e q u i r e m e n t o f p e r t u r b a t i v e unification implies that the mass scale for the lightest exotic fermion is set by Mw rather than Mz,. In all m o d e l s that have been considered in the literature, this mass is not larger than about 115 G e V a n d is, very often, smaller. This mass b o u n d holds even in models with an i n t e r m e d i a t e scale so long as the VEV o f the N-field a n d not that o f x7c is responsible for the breaking o f the extra U ( I ) at the weak scale. The d e p e n d e n c e o f the b o u n d on the lightest charged exotic mass is s u m m a r i z e d in fig. 1. As discussed, it m a y well be possible for this exotic to be the LSP without conflict with present experiments or with cosmology. In this case the emerging signals for sparticle p r o d u c t i o n would crucially d e p e n d on the details o f superpotential couplings o f the LSP. We are grateful to C. G o e b e l for discussions on properties o f matrices a n d K. Enqvist for discussions on refs. [7,8]. This research was s u p p o r t e d in part by the U n i v e r s i t y o f Wisconsin Research C o m m i t t e e with funds granted by the Wisconsin A l u m n i Research F o u n d a t i o n , a n d in part by the US D e p a r t m e n t o f Energy u n d e r contract DE-AC02-76ER00881.
References [ 1] J. Ellis, CERN preprints CERN-TH.4255 (1985), CERN-TH.4474 (1986). [2] P. Candelas, G.T. Horowitz, A. Strominger and E. Witten, Nucl. Phys. B 258 (1985) 46; E. Witten, Nucl. Phys. B 258 (1985) 75. [ 3 ] M.B. Green and J.H. Schwarz, Phys. Lett. B 149 (1984) 117. [4] J. Ellis, K. Enqvist, D.V. Nanopoulos and F. Zwirner, Nucl. Phys. B 276 (1986) 14. [5] L.E. Ib~ifiezand J. Mas, CERN preprint TH.4426 (1986). [6] M. Drees, University of Wisconsin preprint MAD/PH/328 (1987). [ 7 ] J. Ellis, D.V. Nanopoulos, S.T. Petcov and F. Zwirner, Nucl. Phys. B 283 (1987) 93. [8] B. Campbell, J. Ellis, K. Enqvist, D.V. Nanopoulos, J.S. Hagelin and K.A. Olive, Phys. Lett. B 173 (1986) 270. [9] V. Barger, N. Deshpande and K. Whisnant, Phys. Rev. Lett. 56 (1986) 30; S.M. Barr, Phys. Rev. Lett. 55 (1985) 2778; E. Cohen, J. Ellis, K. Enqvist and D.V. Nanopoulos, Phys. Lett. B 165 (1985) 76; L.S. Durkin and P. Langacker, Phys. Lett. B 166 (1986) 436. [ 10] J. Ellis, K. Enqvist, D.V. Nanopoulos and S. Sarkar, Phys. Lett. B 167 (1986) 452; G. Steigman, L.A. Olive, D.N. Schramm and M.S. Turner, Phys. Lett. B 176 (1986) 33. [ 11 ] M. Davier, Talk presented Intern. Conf. on High energy physics (Berkeley, CA, 1986). [ 12] H.E. Haber and M. Sher, Phys. Rev. D 35 (1987) 2206; M. Drees, University of Wisconsin preprint MAD/PH/313, Phys. Rev. D, to be published. [ 13] N.K. Falck, Z. Phys. C 30 (1986) 247. [ 14] UA1 Collab., C. Albajar et al., Phys. Len. B 185 (1987) 241. [ 15 ] M. Dine, V. Kaplunovsky, M. Mangano, C. Nappi and N. Seiberg, Nucl. Phys. B 259 (1985 ) 519. [ 16] H. Goldberg, Phys. Rev. Lett. 50 (1983) 1419; J. Ellis, J.S. Hagelin, D.V. Nanopoulos, K. Olive and M. Srednicki, Nucl. Phys. B 238 (1984) 453. [ 17 ] M. Drees and X. Tata, University of Wisconsin preprint MAD/PH/344 (1984).
71
196, number
AN UPPER LIMIT IN SUPERSTRING Manuel
PHYSICS
1
LETTERS
ON THE MASS OF EXOTIC INSPIRED E(6) MODELS
B
24 September
1987
FERMIONS
DREES and Xerxes TATA
Physics Department, University of Wisconsin, Madison, WI 53706, USA Received
11 May 1987; revised manuscript
received
I 1 June 1987
We show that in a large class of superstring inspired E( 6) models considered in the literature the requirement of perturbative unification implies that the lightest neutral exotic fermion ( #v’) is lighter than 115 GeV. In many models this fermion may be considerably lighter and may even be the lightest supersymmetric particle. It is argued that such a particle is compatible with all known experiments and with cosmology. Some phenomenological implications are also discussed.
In the last two years a large amount of interest has been focused on the study of the phenomenology of E( 6) supergravity GUTS #’ which emerge as the low energy limit [ 21 of anomaly free superstring theories [ 31. A common feature of these models is the prediction of many exotic particles as required by the multiplet structure of E( 6). Unfortunately most of these particles are expected to have masses well above the W-mass [ 4-61, with the notable exception of some of the exotic neutral fermions: three of these, which are usually referred to as right-handed neutrinos (since they are SU( 2)-singlets), are massless [ 41 in models without any intermediate scale. Furthermore, in addition to the neutral gaugino-higgsino mixed states the model contains at least six exotic neutral fermions. The reason is that in addition to standard quark and lepton superfields, each 27-dimensional generation contains two SU( 2)-doublets H and H and an SU( 2)-singlet N with the right quantum numbers to spontaneously break the electroweak and additional U( 1) gauge symmetry, respectively. There are thus twelve neutral Majorana fermions which can mix in a complicated fashion. It is, however, always possible to find a basis where only one H, one H and one N has a non-vanishing vacuum expectation value (VEV). Following the notation of ref. [ 71 we denote the fields that acquire a VEV by H3, 8, and NJ. One way to insure that (Z,) =0 (Z= H, H, N, i= 1, 2) is to require [ 7,8] that the superpotential couplings of the terms Z,Z3Z3 as well as the soft breaking mass terms of the form m2zI z3Z ,Z: identically vanish; in this case the 12 x 12 neutral fermion mass matrix breaks up into two 6 x 6 matrices. We note, however, that while the vanishing of these couplings and mass terms is sufficient for the VEVs of Z, to vanish, it is not necessary since in the scalar potential the terms linear in Z, from the F-term contributions can be cancelled against the soft supersymmetry breaking A term as well as the off-diagonal scalar mass terms [ 71. In this case, there is no decoupling of the 12 X 12 matrix into two 6 x 6 blocks; similarly, the charged fermion mass matrix does not decouple into two 2 X 2 blocks [ 71. In general, neither of these solutions is invariant under renormalization group evolution #2 so that a priori neither possibility is preferred. In the remainder of this letter we choose to work in the class of models where the decoupling of the exotic fermion mass matrices does occur since this is the possibility that has been extensively studied in the literature [ 4,5,7,8]. In this case, the six neutral and the #’ See ref. [ 1 ] for reviews. ” This is even true for the solution of ref. [7], 1$,= ,1,,,,/2,,,,=0 (i= 1, 2), since that only guarantees the I,,, to vanish at all scales; however, for finite A,,, (i, j. k= I, 2) nonvanishing off-diagonal soft breaking mass terms will be generated even if these are diagonal and equal at the unification scale. Nevertheless the choice [ 71 A,,,=m$,,, = 0 can be technically natural if all A,,, = 0, since in this case the lagrangian is invariant under the transformation Z ,+ -Z,, Z,+Z,, which forbids all dangerous terms.
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two charged fermions contained in the H I , 2- I7"I1,2- and Nj,2-superfields do not mix [7,8] with the gaugino-higgsino sector. Instead, they get their masses from the superpotential
2 fm= Z
i,J= 1
(2a3HJZljN3 +2,3jHiNjI-I3 +23ijI-liNjH3) •
(1)
For models without any intermediate scale, this leads to the mass matrix (in the basis (HI, IZI~,Nl, H2I?]2, N2))
0 2113X M=12101 0
2113X 0 2311/3 2213X ~2123X 0 \2123/~ 2312U
2131/7 2311U 0 2231/7 2321U 0
0 2213X 2231/7 0 2223x 2232/7
2123X 0 2321U 2223x 0 2322U
2132/7~ 23 U/ 0 2232/7/ , 23 /)] 02
(2)
where v = ( H ° ) , /7= (IzI°), x = ( N ) . F r o m this mass matrix it immediately follows that the mass 3//o of the lightest exotic fermion is b o u n d e d according to M 2 ~
(3)
The b o u n d (3) tells us that M0 is at most of order Mw rather than Mz,, the mass of the additional gauge boson Z', which sets the scale for the mass of most o f the exotic particles predicted by E (6) models. Note that this is true even if we consider the full 12 × 12 matrix discussed above ~3. The mass Mo may well be much smaller than indicated by eq. (3). The reason is that x has to be much larger than v and/7 in order to make the Z ' boson sufficiently heavy [9], especially if one takes into account the requirement that the presence o f the massless right-handed neutrinos mentioned above does not spoil successful predictions for nucleosynthesis in the early universe [ 10]. This translates into the b o u n d Mz, ~>300-400 GeV, which gives x > 700-900 GeV. Note furthermore that the determinant of the mass matrix (2) is only quadratic in x, while four eigenstates acquire masses ~ 2x if all the couplings in eq. (1) are o f the same order. This obviously implies [ 7 ] that the two lightest eigenstates get masses o f order M2w/Mz, ! The only way to avoid such small eigenvalues is to impose a hierarchy between the 2iik such that the hierarchy between the VEVs is cancelled. It is also interesting to note that one gets two massless eigenstates if either 23a = 0 or 2i3j = 0 for i, j = 1, 2. This means that all the three terms shown in eq. (1) have to be present. In this paper we attempt to obtain an absolute upper b o u n d on the mass of the lightest neutral fermion eigenstate of the matrix in eq. (2), independent o f the details of the model. The mass matrix (2) has been written in a completely general basis. It is, however, more convenient to work in a basis [ 7 ] where the mass matrix for the charged exotic fermions of the first two generations is diagonal, i.e. to perform rotations in the (H1, H2) space and the (I-~Ii,I'~I2) space such that there are no HzITIIN3 - and HllZI2N3-terms in the new basis. In this basis the mass matrix has the same form as in eq. (2), but with 2123= 2213= 0. In this basis the nonvanishing x-entries 2~13x and 2223Xare the masses o f the charged exotic fermions and thus must exceed 20 GeV so as to satisfy the P E T R A bound [ 11 ] on the mass of new charged particles. Even in this basis, one has to deal with ten independent parameters, which makes a complete analytic treatment o f the problem of finding a relationship between the ten 2ak that maximizes the magnitude of the smallest eigenvalue very difficult. We proceed by first recognizing that the IXijkl at the weak scale are all b o u n d e d from above by a rather small value since they would otherwise diverge [12,6 ] before the unification scale making perturbative unification impossible. We then searched numerically for the combinations of the ten 2ijk that maximize Mo using the M I N U I T program o f the C E R N software library. After imposing a c o m m o n upper bound ~3This followssince the N 1and N2 columns in the 12 × 12 matrix can only have entries proportional to v or ~, since all the relevant terms in the superpotential are required to have the structure N~HflZIk. 66
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2 on all [2ijk[ we found that [Mol is maximal if and only if (up to trivial phase transformations) 2131 =2132 =2311 = --2312 =2231 = --2321 =2232 =2322 -----~2, 2113= --2223 -- 22vg/(x/~ + t72 "x) .
(4a, b)
In this case one even gets f o u r light eigenstates with the same mass, which is given by (5)
Mo = 2 2 v O / . ~ 2 + O2 ,
which, in this case, is also equal to the mass of both the charged exotic fermions. Note that Mo vanishes if ~=---O/v+ov since the sum v2+02 is fixed by the W-mass, M~v = ~g2 1 2 (v 2 + ~2) where g2 is the SU(2) gauge coupling. Of course, the absolute bound on the smallest eigenvalue depends on how large 2 in eq. (4) is allowed to be; we will address this issue shortly. Unfortunately, we have not been able to prove that eq. (5) indeed gives the maximal value for Mo if one requires 12~jk]~<2. We could, however, show that Mo cannot exceed the value given by eq. (15) by more than 25%. One strategy for deriving bounds of this kind is to compute the "expectation value" E of a suitably chosen normalized vector y, i.e. E 2 = IlMyll2; E is then an upper bound for Mo. If one chooses e.g. yl = (0, 0, 0, 0, cos y, sin ~,), where y is determined by the requirement that the x-dependence of My~ vanishes, one finds E 2 ~<22 [sin2y(2v2 + g2) + 2/.;2COS2y] ~<22(2V2 q_ g2) ,
(6)
which gives Mo.<2
2 .
(7)
Another hound can be derived if one considers the vector Y2= (0, COS O¢, sin o~ COSfl, 0, 0, sin o~ sin fl) where c~ and fl are chosen so that the x-dependence of My2 again vanishes; this gives E 2 ~ 2 2 2 v 2 [ s i n 2 0 ~ ( s i n f l + c o s f l ) 2 +cos20~] 4422/.32 ,
(8)
i.e., M0 < 2 2 v .
(9)
Note that the bound (9) approaches the value given in eq. (5) for oo --, ~ ; for co = 1 eq. (7) gives a better bound which is only a factor xl/q~ larger than the value (5). Up to now we have only performed rotations among (H~, H2) and (H~, H2) in order to get rid o f 2t23 and 22~3. By choosing an appropriate rotation in the (N t, N2)-space we can, without loss of generality, eliminate any one entry proportional to v or ~ in the mass matrix (2). We have not yet used this freedom since this makes the matrix look less symmetric. However, since our results cannot depend on the choice of the basis, we can arbitrarily choose the basis where (e.g.) 2132= 0. Again constraining all 12,jkI from above we now find that ]Mo ] is maximal if 2131 =2311 = 2 3 1 2 : 0
--2231 =2321 =2233=2322 ~ 2 ' , 2223=0 -
(lOa, b)
In this situation the eigenvalues again come in pairs that only differ by their sign, and one finds Mo = xf22' v,
(l l)
the other eigenvalues being given by +2~t3x and + ~,/22' ~ (note that this model always predicts [4,6] ~> v). We see once again that Mo vanishes in the limit (o-~ oo. In this basis, we can indeed prove that eq. (1 1 ) gives the maximal possible value of Mo. To do so we merely have to repeat the steps that lead to the bound (6), since the vector yj is constructed such that the only contribution to its expectation value that is proportional to 17has the coefficient 2~32, which vanishes here. Therefore the bound (7) becomes Mo <~x / ~ 2 ' v ,
(12)
which is saturated by eq. (I 1 )! 67
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We now turn to a discussion of the upper bound on ILjkl alluded to earlier. This upper bound emerges from the requirement that all couplings should remain in the perturbative region, i.e. 22k < 4~, for all energies up to the GUT-scale M x - 1016 GeV. In order to find out what this means for the upper bound on the 12ijkl of the weak scale we have computed the renormalization group equations ( R G E ) for these couplings under the assumption that the whole superpotential is given by eq. (1), i.e. we have set all 2ijk with i, j, k = 1, 2, to zero. This may be achieved by imposing a discrete symmetry under Z~--,-Zi, Z3----~Z3 ( Z = H , ISI, N ) . In this case, our choice 233i=23i3 =2,33 ~-0 is invariant under renormalization group scaling. Furthermore, from our experience, we find that the introduction of additional Yukawa couplings tends to accelerate the scaling of 2,jk of eq. (1); as a result, the 12~jk(Mw) I would have to be chosen smaller than before, which only strengthens the bound. On the other hand it is, of course, possible that all other couplings are indeed negligible. Using the general formulae of ref. [ 13 ] we find
d2ij3/dt= (1/167~2)[2ij3(42~j3+4223 +4223
2 -~-2t3z j "]-2i3l 2 -~-23ji 2 -]-23jk 2 -- 3g22 -- ~g~) + 22lk3
+ 2k j3 ( 22.32kt3 + 2,3j2k3j + 2i312k31) + 2il3 (23ji23ti + 23jk23tk) ] ,
( 13 )
where both U(1 )-couplings have been set equal to gb and the indices k, ! have to be chosen such that i . k = l . j = 2 and t = l n Q, d~i3j/dl=(1]16nz)[2i3)(22j3
+2213 + 243o 2 + 24 23kj + 44 2~3j+ 44 2k3j "~ 44 2m + 2 k3l 2 - 3gZ2 - ~ g ~ )
+ 2i31( 223ij23i l -Jr-223kj23kl + 32k3j2k3! ) + 2k3j (2ij3 ~kj3 + 2il32kl3 ) ] •
(14)
The RGE for 430 can be obtained from eq. (? 4) by simply exchanging the first two indices on all 2's. Starting from the initial conditions given in eq. (4) we find the bound 2(Mw) < 0 . 5 ,
(15)
while the boundary conditions (1 O) result in 2,(Mw) < 0 . 7 .
(16)
This illustrates the general trend that the presence of a large number of Yukawa couplings strengthens the bound on their value. The absolute upper bound on Mo can now be obtained by inserting eq. (16) into eq. (11 ). We find Mo
(17)
From the constraint (15) we see that the ansatz (4) results in even smaller values for Mo. Our result (17) has been obtained using the requirement 12ijkI ~<2' for all i, j, k. In principle it is, however, possible that Mo could be increased by reducing some of the couplings even further which would allow the other couplings to be larger than given in eq. (16). In order to investigate this possibility we performed several runs of M I N U I T where the upper bound on all the Yukawa couplings is not the same constant but depends on the actual ratios of the various Yukawa couplings. This procedure is rather time-consuming since the numerical integration of the RGE is now a part of the function to be minimized by MINUIT. Our results indicate, however, that eq. (17) does give the absolute upper bound. We note that the ansatz (10) implies that one charged exotic fermion is massless, since 4223=0 here. This is obviously not a realistic situation. We have studied the effect that a finite mass of the exotic charged fermion might have on 3//0. To this end, we repeated the previous analysis, imposing the additional constraints 12113Xl >/ 12223Xl = m l qmin +,
(18)
where m mm n + is the mass of the lighter charged fermion. Fig. 1 shows the resulting bound on Mo as a function rain for o) = 1 as well as o) = 3. It turns out that the ans~itze (4) and (10), modified according to the requireofrnn+ 68
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120 100
B A
80 M~°X 60 --B
(GeV)
A
~=I
40
2o
q
I00
200 500 mrnin •~+ (GeV)
400
500
Fig. 1. The upper bound on Mo, the mass of the lightest neutral exotic, is shown as a function of m~+", the mass of the lighter charged exotic H-fermion, for two values of ~o _--O/v. The curves labelled A and B have been obtained using the ansatze (4a) and rnin (10a), respectively, with 12tl3xl >~l)~223xl=mR+. The curves labelled A are not extended into the region of small m ~ since they would lie much below the true m a x i m u m which is here given by the curves labelled B.
ment (18), still give the largest values of M0 ~4 The curves for the modified ansatz (4) have been terminated at m ~ n = 0 . 5 v ~ / , ~ 5 + t72 since for smaller values of m nm~n + , Mo decreases here and is thus far below the result obtained from the modified ansatz (10). Note that the bound on the mass of a sequential charged heavy lepton that has recently been set by the UA 1 collaboration [ 14] does not apply f o r / ~ + since the corresponding neutral fermion might be as heavy as 100 GeV, see eq. (17). But even the requirement [11] mn+ >_-20 GeV reduced the bound on Mo from 125 GeV to 115 GeV. Up to now we have only considered models where all matter fields reside within 3 generational 27's of E (6). There might, however, be [ 2] additional light "survivor" superfields out of non-generational 27 + ~ , some of which might carry the gauge quantum numbers of N, H or IZI;in this case the relevant mass matrix would be larger than shown in eq. (2). It is probable that this would allow for larger values of Mo if one again requires I2~;kl <2; but the existence of additional non-negligible terms in the superpotential (1) would strengthen the bound on 2. Since these effects tend to compensate each other we do not expect the final bound to be substantially altered. In this context it might be interesting to note that even a toy model with only one generation of exotic fermions reproduces the bound (17) for large 09, while for o9 _~ 1 the bound on Mo is about 20% smaller than that of the realistic model. Finally, there also exists the possibility [15] of an intermediate scale M I ~ 10 l° GeV. If one of the superpartners of the right-handed neutrinos gets a VEV of that order the right-handed neutrinos can get [ 5] Majorana masses of order 1 TeV; the mass matrix for the H,IZI,N-sector remains unaltered. However, the bound on the Yukawa couplings, and thus on Mo, might increase by 20%. The reason is that one now can only require the couplings to remain in the perturbative regime for energies up to Mx; at higher energies a new, unknown set of RGE replaces the old one, so that no definite statement is possible. If, on the other hand, one of the Nfields gets a VEV of order MI the (3, 3)- and (6, 6)-entries of our mass matrix can become as large as 1 TeV, and no strong bound can be derived. We have shown that the existence of an exotic fermion state ( ~ v c) with a mass smaller than about 115 GeV is a general feature of many superstring inspired models without any intermediate scale. In fact, in many models this state may be considerably lighter than the bound cited above and may even be the lightest supersymmetric particle (LSP) and hence stable. We note here that the existence of such a stable particle may be dangerous for cosmology since its relic mass density could exceed the observed mass density of the universe unless there is an efficient annihilation mechanism for these particles. ~4 If one requires 3.t23=2213=,~132=0, 12uk] <2 otherwise, M I N U I T finds as best solution2231=0,2131 =A311 =--3.312=2321=2232=2322=2, if rn ~'," > 200 GeV; however, from the RGE analysis one finds .~< 0.58, which results in smaller values of Mo than those shown in the figure.
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supersymmetric relics has already been studied in the context of supergravity models [ 16 ]. that if the LSP is a higgsino, h, with mixings such that the Z°hh coupling is not significantly h-pairs can annihilate via s-channel Z°-exchange, cosmologically acceptable h relic densities if annihilation into bb pairs is kinematically possible ~5, i.e. if rn~ > m b.
We note that the exotic fermions we have discussed here are similar to h except that their couplings to matter are not proportional to matter fermion masses. In general, they are arbitary mixtures o f Hi, IzI~and Ni ( i - - l , 2) and hence have couplings to Z °. Moreover, the lightest o f these is usually heavier than the b-quark, so that such a stable exotic is cosmologically safe. One significant exception to this is if all the 2ijk in eq. (1) are comparable and x~> v, g. In that case, there are two light states which are dominantly mixtures of just N1 and N2 and masses ~M~v/Mz,. These states couple to Z ° only via a small doublet component and hence their annihilation is strongly suppressed. The singlet exotics can have superpotential couplings only to other exotics so that the coupling o f this light state to ordinary matter is also suppressed by Mw/Mz,. Its annihilation via these couplings is, therefore, small. A stable singlet exotic would lead to cosmological problems with its relic mass density [ 16 ]. Within the cosmological context, there is one important difference between the usual supergravity models and the exotic sector of superstring models that seems worth commenting on. This stems from the fact that the exotic sector may contain several light states, (approximately) degenerate in mass, especially for those sets o f 2ijk that lead to the bounds discussed earlier (see eqs. (4) and (10)). This could also occur due to some as yet unknown symmetry of the couplings 2~jk. An example for such a symmetry is the discrete Z3 symmetry considered in ref. [ 17]. In this case one has 2~3t :•321:2321 :•232=•123=2213 =0- One can then see that for each eigenvector (X~, X2, X3, X4, )(5, )(6) with eigenvalue m there is an eigenvector (X~, - X 2 , X3, - X 4 , )(5, - ) ( 6 ) with eigenvalue - m , so that all eigenvalues come in pairs. In this case, annihilation o f two differentMajorana fermions would dominate that of identical Majorana fermions in the non-relativistic limit since it is not Pwave suppressed the way the annihilation o f identical Majorana fermions is [ 16 ]. An alternative way o f viewing this is that if the masses o f the two fermions are exactly degenerate, they can be combined into one Dirac fermion whose annihilation is unsuppressed. In this case it may well be possible for the light exotic to contain large SU (2) singlet components. This s-wave annihilation may also be important in the discussion of the decoupiing of the various particles, since the cross sections at temperatures close to threshold are largest for the case of distinct fermions. We now turn to a brief discussion o f the phenomenology of the light exotics. As we have already noted, if they interact only via gauge interactions and the superpotential of eq. (1), the transformations H , ~ - H i , I Z I ~ - IZli, N ~ - N , ( i = 1, 2) with the i = 3 fields going to themselves are symmetries of the interactions; in this case the lightest state o f the mass matrix (2) or one o f the charged states is absolutely stable. This, of course, is no longer true if we allow all the 2ijk in the superpotential 2ijkH~IZljNk to be non-vanishing or if we introduce couplings of these exotics to matter supermultiplets ~6. In this case, the lightest exotic is absolutely stable only if it is the LSP. Although it is possible to choose all these couplings to vanish, we will not pursue this any further. What if the exotic is indeed the LSP? In this case, the phenomenology would crucially depend on its direct couplings to matter. In the absence of such couplings, all SUSY particles would decay into the lightest SUSY particle o f the non-exotic sector (presumably a combination of a neutral gaugino and higgsino), which could decay via exotic particle loops into the LSP + photon unless the discrete symmetry discussed in the previous paragraph is maintained. This would lead to a degradation o f the missing transverse m o m e n t u m from the case ,5 The annihilation cross section has an S-wave contribution proportional to m 2 so that the cross section into massless fermions is Pwave suppressed in the non-relativistic limit. ~6 Such couplings would, of course, introduce additional terms in the renormalization group equations (13) and (14) and thus max further reduce the bound on the mass of the lightest exotic. We have neglected this in the present analysis. 70
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where the S U S Y particles directly decay into an LSP. I f there are direct couplings o f the LSP to matter, the lightest non-exotic sparticle would decay into the t h r e e - b o d y m o d e LSP + f-f m e d i a t e d by f exchange. I f these couplings are large enough squarks a n d sleptons would decay via the two-body m o d e f ~ f + L S P a n d ~ via ~ q ~ t + L S P so that the p h e n o m e n o l o g y would be similar to that for the case when the LSP is the photino. Since the couplings o f the exotics to m a t t e r are arbitrary, it is possible for bizarre p h e n o m e n o l o g y to result. To illustrate this, we consider the case where the exotics have direct couplings only to the leptons. In this case even strongly p r o d u c e d SUSY particle events would contain multileptons in the final state. Clearly, the resulting p h e n o m e n o l o g y would be very m o d e l d e p e n d e n t a n d the details are b e y o n d the scope o f this letter. In s u m m a r y , we have shown that in all superstring-inspired E ( 6 ) m o d e l s without any i n t e r m e d i a t e scale, i m p o s i n g the r e q u i r e m e n t o f p e r t u r b a t i v e unification implies that the mass scale for the lightest exotic fermion is set by Mw rather than Mz,. In all m o d e l s that have been considered in the literature, this mass is not larger than about 115 G e V a n d is, very often, smaller. This mass b o u n d holds even in models with an i n t e r m e d i a t e scale so long as the VEV o f the N-field a n d not that o f x7c is responsible for the breaking o f the extra U ( I ) at the weak scale. The d e p e n d e n c e o f the b o u n d on the lightest charged exotic mass is s u m m a r i z e d in fig. 1. As discussed, it m a y well be possible for this exotic to be the LSP without conflict with present experiments or with cosmology. In this case the emerging signals for sparticle p r o d u c t i o n would crucially d e p e n d on the details o f superpotential couplings o f the LSP. We are grateful to C. G o e b e l for discussions on properties o f matrices a n d K. Enqvist for discussions on refs. [7,8]. This research was s u p p o r t e d in part by the U n i v e r s i t y o f Wisconsin Research C o m m i t t e e with funds granted by the Wisconsin A l u m n i Research F o u n d a t i o n , a n d in part by the US D e p a r t m e n t o f Energy u n d e r contract DE-AC02-76ER00881.
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