Nuclear Physics B256 (1985) 218-252 ,e~ North-Holland Publishing Company THE LOW-ENERGY SUPERSYMMETRIC SPECTRUM ACCORDING TO N = 1 SUPERGRAVITY GUTS L.E. I B,~INEZ* CERN, Geneva, Switzerland C. L()PEZ and C. MUIqOZ Departamento de Fisica Tebrwa C-XI, Universidad A utonoma de Madrid. Cantoblanco, 28049 Madrid, Spain Received 1 February 1985 We reexamine the radiative SU(2) × U(1) breaking induced by broken N = 1 supergravity in the light of recent experimental results. Explicit formulae are given for the supersymmetric particle spectrum and the SU(2)x U(1) breaking condition. The experimental results (t-quark mass) constrain the supergravity parameters, particularly the bare higgsino mass Po and the "mixing" Higgs mass term ~. Models with globally supersymmetric "observable" sectors (except for gaugino masses) are excluded in their simplest version. If the identity B = A -- 1 is assumed, further restrictions on the range of the supergravity parameters are found. The particle spectra leading to possible supersymmetric interpretations of the CERN missing PT events are analyzed. We study in detail the following three scenarios: (i) m,~ = 80-140 GeV, M~, = 3 (ieV; (ii) inca = M~, = 40 GeV; (iii) Mg = 40 GeV, m4 >> M~, and give the corresponding spectra of sparticles as a function of the A-parameter. 1. Introduction In the last couple of years, a great deal of effort has been dedicated to the study of grand unified theories (GUTs) coupled to broken N -- 1 supergravity (some review articles are given in [1]). It has been shown that the breaking of N = 1 local supersymmetry may induce radiatively the breaking of the SU(2) x U(1) symmetry for a wide range of parameters [2-5]. In the beginning, it seemed, however, that one could only break the weak symmetry properly for a t-quark mass m t >__ 60 GeV [2]. Soon it was realized [3-5] that one can easily obtain SU(2) × U(1) breaking even with arbitrarily light t-quarks. In particular, two of the authors presented a systematic study [3] of the constraints imposed on the free parameters of the supergravity models (including different rn, values) by requiring the appropriate Weinberg-Salam breaking at the M w scale. Since then the UA1 collaboration has * On leave from l)epartmento de Fisica Teorica, Universidad Autonoma de Madrid, Spain. 218
Transcript
Nuclear Physics B256 (1985) 218-252 ,e~ North-Holland Publishing Company
THE LOW-ENERGY SUPERSYMMETRIC SPECTRUM ACCORDING TO N = 1 SUPERGRAVITY GUTS
L.E. I B,~INEZ*
CERN, Geneva, Switzerland
C. L()PEZ and C. MUIqOZ
Departamento de Fisica Tebrwa C-XI, Universidad A utonoma de Madrid. Cantoblanco, 28049 Madrid, Spain
Received 1 February 1985
We reexamine the radiative SU(2) × U(1) breaking induced by broken N = 1 supergravity in the light of recent experimental results. Explicit formulae are given for the supersymmetric particle spectrum and the SU(2)x U(1) breaking condition. The experimental results (t-quark mass) constrain the supergravity parameters, particularly the bare higgsino mass Po and the "mixing" Higgs mass term ~ . Models with globally supersymmetric "observable" sectors (except for gaugino masses) are excluded in their simplest version. If the identity B = A -- 1 is assumed, further restrictions on the range of the supergravity parameters are found. The particle spectra leading to possible supersymmetric interpretations of the CERN missing PT events are analyzed. We study in detail the following three scenarios: (i) m,~ = 80-140 GeV, M~, = 3 (ieV; (ii) inca = M~, = 40 GeV; (iii) Mg = 40 GeV, m4 >> M~, and give the corresponding spectra of sparticles as a function of the A-parameter.
1. Introduction
In the last couple of years, a great deal of effort has been dedicated to the study of grand unified theories (GUTs) coupled to broken N -- 1 supergravity (some review articles are given in [1]). It has been shown that the breaking of N = 1 local supersymmetry may induce radiatively the breaking of the SU(2) x U(1) symmetry for a wide range of parameters [2-5]. In the beginning, it seemed, however, that one could only break the weak symmetry properly for a t-quark mass m t >__ 60 GeV [2]. Soon it was realized [3-5] that one can easily obtain S U ( 2 )×U ( 1 ) breaking even with arbitrarily light t-quarks. In particular, two of the authors presented a systematic study [3] of the constraints imposed on the free parameters of the supergravity models (including different rn, values) by requiring the appropriate Weinberg-Salam breaking at the M w scale. Since then the UA1 collaboration has
* On leave from l)epartmento de Fisica Teorica, Universidad Autonoma de Madrid, Spain.
218
L.E. Ibahez et al. / Super.'qvmmetrw .~pectrum 219
reported [6] some W-decay events consistent with the process W --* tb with 30 < m t 50 GeV. If m t -- 40 GeV, there is one parameter less in the low-energy supergrav-
ity models and one can find further constraints amongst the supergravity parame- ters. In this paper we study in detail these constraints and present supersymmetric spectra consistent with them. The plan of the paper is as follows. In sect. 2 we fix the notation and conventions for the class of low-energy supergravity models considered in the text. In sect. 3, we give explicit (analytic) formulae for the evolution of the different masses and couplings in the approximation in which only the Yukawa coupling of the t-quark is considered. We present explicitly the dependence of these masses and couplings on the free parameters of the model at the G U T scale: m, M, ~0, A, B. In this section we do not fix still m t - - 40 GeV but let the value of h t free so that it can be fixed appropriately when a more precise value for m t will be measured. In sect. 4 we present numerical formulae for the masses of the s-particles as well as for the Higgs parameters after fixing m t t o the preferred experimental value m t = 40 GeV. We also study the restrictions obtained amongst the different parameters (m, M,/~ o, A, B) in order to be consistent with a stable radiative SU(2) × U(1) breaking. We consider general boundary conditions at the G U T scale as well as the case in which the relationship B = A - 1 holds.
Sect. 5 deals with the low-energy supersymmetric spectrum consistent with the constraints presented in previous sections. We present spectra with squark and gluino masses which could be of interest in trying to give a supersymmetric explanation to the "monoje t" events recently observed at the CERN collider [7]. In this case (and assuming for simplicity B - - A - 1) the possible spectra are parame- trized by a single parameter A. General features of the possible spectra are also discussed. After a short digression in sect. 6 about some non-standard low-energy supergravity scenarios, we present the final comments and conclusions in sect. 7.
2. Low-energy supergravity: some notations
We will closely follow (unless otherwise stated) the notation of ref. [3] and will refer to formulae in that article by (IL. number). Unexplained notations should also be clarified in this paper. We take as our low-energy superpotential
W = e i j h tQ jUn2 , + t.teqHI,H2,, ~c t l =fda 2 w + h.c., (1)
where only the part corresponding to the t-quark Yukawa is shown since, as we will discuss below, it is the only one relevant for the renormalization group equations. The breaking of N = 1 supergravity originates soft terms in the scalar potential:
Ko,t = E m2l~A 2 + mhtAteuOjOa2, + m~Be, jn~,n L + h.c. (2) i k scalars
In the case of "minimal" N = 1 supergravity one has [1], at the Planck scale, B = ( A - 1 ) , At=A and m = g r a v i t i n o mass. In the case of a general K~hler
220 L.E. Ibizhez et a L / Supersymmetric spectrum
manifold, this will not be in general true. Notice that with this notation the relevant trilinear scalar terms in the potential will be
-h t l . tHLO,U + htmA,tqQ_.jHL(_] + h.c. (3)
There will also be gaugino and higgsino mass terms (in Weyl notation): ~ .
EFM = - ~ e u H L H 2 , - M~'~,~ + h.c., ( 4 )
where a is a gauge index. The relevant Higgs scalar potential along the neutral direction is
Notice that in the case of minimal supergravity one has, at the G U T scale,
/ ~ ( 0 ) = ( 1 - A)lxm. (6)
The potential is minimized for [8]
where [3]
v2=v~ + v 2 = 2 (#12- ~ - ( / z ~ + ~ ) cos20) (,g2 + g,2)cos20
cos20-= (~2 _ 1)/(~02 + 1),
, ( 7 )
sin28 2 ~ / ( ~ 2 + 1) 2 2-I~2 = = ( 8 )
and ~ = - v 2 / v 1 = c o t g 8. In principle, ~ may be both positive and negative (implying opposite signs for v 1 and v2) and this may have some relevance in the "minimal" case in which the sign of ~2 3 is related to the sign and size of A. Also M, /~ and A may be both positive and negative, the correct sign on the fermion mass terms must only be imposed on the physical eigenstates. This is eventually possible through a redefinition of the fields.
In order to obtain the SU(2) x U(1) breaking at the right scale, one must impose v 2 = 2 M 2 / g 2 which can be written in terms of ~t 2, /~ and ~ as follows [3]:
( d - - = (9)
Notice that this condition of symmetry breaking does not depend on the sign of o~ (or s in20) and hence applies to both positive and negative ~ .
A detailed numerical analysis of the condition (9) on the parameters A, m, M, m t and ~ ( 0 ) was presented in ref. [3]. We considered in that reference a negligible
L E. Ibithez et al. / Supersymmetric spectrum 221
/~-parameter (but did not neglect the bottom Yukawa coupling). Since then the UA1 collaboration has presented [6] evidence for the existence of a t-quark with a mass 30 ~< m t ~< 50 GeV, 40 GeV being apparently the favoured value. We analyze in this paper the further restrictions imposed by setting rrt t --~ 40 GeV on the supergravity parameters. We also include a non-vanishing #-parameter in the equations.
3. Evolution of the couplings and parameters for the case of a "low-mass" t-quark
As pointed out in ref. [3], a low-mass (_< 60 GeV) t-quark necessarily means that at M w one has Io~1 -- 1 and hence I p l l - 1~'21- In this situation, hb/h t -- mb//m t and one can neglect the effect of the h b coupling in the renormalization group equations. This simplifies the formulae quite alot and one can give some analytic expressions for the evolution of couplings and parameters. This we present in this section. Some other formulae were already given in ref. [3]. In appendix A, we show for complete- ness the relevant renormalization group equations [9, 2] in the case of a non-negligi- ble ~t.
Let us start with the t-quark Yukawa coupling. One has for the evolution of Y, = h 2 / ( 4 ~ ' ) 2
The inclusion of a non-vanishing ~ in the equations for the evolution of scalar masses leads to few changes. One has just to do the replacements ~.2 ~ (/t~.2 - t t2) in eqs. (IL.38), (IL.44). Formulae (IL.39) are still valid. Only the masses m S and rn~ of the third generation and /x 2 do depend on Yukawa couplings and it is enough to consider the evolution of one of them (e.g./x 2) since the other masses are related to it:
mS( t ) = l 2 _ 1 2 + 1 2 (13a) i m D 2 m 4 ~ m u ,
m2 (t) = z3~22 _ 2#2_ m 2, (13b)
2 2 2 L.E. Ibithez et aL / Super.~Tmmetrw spectrum
where m~ and m~ are given in (IL.39a, b). The evolution of the (mass) 2 of the other scalars in the theory can be found in (IL.C). One can integrate the evolution equation for ~t~(t) to obtain
t t z 2 ( t ) = l ~ 2 o l ( t ) + M 2 e ( t ) + A m M f ( t ) + m 2 ( h ( t ) - k ( t ) A 2 ) , (14)
where the explicit dependence on the soft parameters M, A, m and ~0 at the G U T scale is shown. The corresponding coefficients are as follows:
I ( t ) =- q 2 ( t ) , h ( t ) = ~ ( 3 / D ( t ) - 1), k ( t ) = 3Y~ ,F /D( t ) 2,
f ( t ) = - 6 Y o H 3 / D ( t ) 2, D ( t ) - (1 +6YoF ) ,
e ( t ) = ~ [ ( G a + Y o G 2 ) / D ( t ) + ( H 2 + 6 Y o H , ) Z / 3 D ( t ) 2 + H ~ ] , (15)
where the functions H2, H3, H4, GI, G 2 and H 8 are given in appendix B and do not depend on Yo but just on gauge coupling constants. Thus eqs. (14) and (15) also show us the explicit dependence on the t-quark Yukawa coupling. We prefer to present here the results corresponding to arbitrary h t since the precise experimental value for rn t is still not available. Moreover, the same formulae could be used in the case of the existence of a heavier fourth generation of fermions. Numerical values for all the functions in eqs. (13)-(15) for the case m t = 40 GeV will be given below. It is convenient to recall here the evolution equation for /~( t ) . It is given in (A.1) and one trivially gets, after integration,
I~( t ) = m 2 + #2ol( t ) + M2g( t ) , (16)
where g( t ) = (3fiz(0)f2(t) + ½fil(0)fl(t)). Obviously one can check that e ( t ) --* g ( t ) when Y0 --* 0. The only other parameter whose evolution equation is of interest to us is ~23. One can integrate its evolution equation to obtain
~z3(t ) = q(t)lx~(O) + r ( t ) # o M + s( t )Arn~t , , , (17)
where q( t ) was already defined and
s ( t ) = 3 q ( t ) Y o F / D ( t ) , r ( t ) = q ( t ) ( 3 Y o H 3 / D ( t ) - HT). (18)
n7, which does not depend on Y0, is given in (B.9). Thus formulae (17) and (18) give us the explicit dependence of ~ ( t ) on the parameters/.t~(0), ~t0, M, m and A the G U T scale as well as on Y0.
The formulae obtained for ~ ( t ) and ~ ( t ) allow us to impose the SU(2)x U(I) breaking condition (9) which can also be rewritten as
1 2 m 2 1 2 6o 2 - tt~ + 2 M z = + 1~2°1 + M2g + 2Mz (19) p,~ + ½M]. r n 2 ( h _ k A 2 ) + l a 2 o l + M 2 e + A m M f + 12M z2 '
1.. E. Ibfhez et al. / Supersvmmetric spectrum 223
implying a constraint amongst the parameters to, m,/t0, M, A (for each given Y0, i.e. top quark mass). The appropriate SU(2) × U(1) breaking depends upon the ratios of the mass parameters at the GUT scale, M/m,/to/m as well as Mz/m, as shown in eq. (19). But, in fact, the dependence on Mz/m is so tiny that for all possible values of m, we can neglect it, the effective resulting parameters being the mass ratios and, of course, A and B. With eq. (19), one can reproduce, for example, figs. 4 and 5 of ref. [3] (except for small effects due to a non-vanishing hb). Notice that if m t -- 40 GeV (implying h t ( 0 ) -- 0 . 0 8 ) / t 2 and/t2 will be very similar and hence, from eq. (19), to = 1. Thus we will approximately have Isin28[ = 1 at the M w scale. From eqs. (14)-(17) one can write
s in20 ( t ) = 2(q/t2(O)+r/t°M+s#°Am) (20) m2(1 + h - kA 2) + 2/t~l+ MZ(g + e) + AmMf'
and hence, imposing [sin20(Mw) [ = 1 will give us a good approximation to the constraints amongst /t~(0), tt 0, M, m and A. Of course, Isin20[ = 1 is only obtained in the h t ~ 0 l imi t . A noteworthy feature of the above equation is that it is invariant under the simultaneous replacement/t0 ~ - / to , M--* - M and A ~ - A implying that the SU(2)× U(I) breaking condition ( Is in20(Mw) I = 1) is invariant under that symmetry.
As was already noticed in ref. [3], a value s i n 2 0 ( M w ) = 1 is only obtainable if at the G U T scale one has 0.6_< s i n 20< 1. One can also see that in order to get s i n 2 0 ( M w ) = - 1 (which is also a possible condition of symmetry breaking) one needs to have, at the GUT scale, - 1 ~<sin20~<-0.6. Thus, in order to have s i n 2 0 ( M w ) = - 1 , one needs to start from a negative /tz(0). All this is especially relevant in the case of "minimal" N = 1 supergravity in which case one can put /t23(0 ) = (1 - A)/tom. Then one has the following possible situations:
(a) /to > O, A < 1 } (b) /to <0 , A > I /t~(O)>O,
(c) / t o > 0 , A > I } (d) /to <0 , A < I / t~ (0 )<0 . (21)
Thus, in cases (a) and (b), one will get SU(2)× U(1) breaking by imposing sin 20 (Mw) = 1 and in cases (c) and (d) it will only be possible if s in20(Mw) = - 1. However, notice that eq. (20) is invariant under the simultaneous replacements (in the minimal case!) s i n 2 0 ( M w ) ~ - s i n 2 0 ( M w) and / t 0 - ~ - / t 0 and hence the SU(2) × U(1) breaking constraint in the cases of going from (a) to (d) and (b) to (c) is the same. Thus we will only study the SUSY breaking conditions for/ t0 > 0.
Before showing numerical formulae for all the above equations, let us first write down some relations for some of the masses of the model. Let us start with the Higgs
224 L.F. Ibi~hez et al. / Supersvmmetric spectrum
sector. There are three neutral higgses and masses [8]
2= d m c
2_( 1 2 m a . b 2 m c
and a charged Higgs with
+ M 2 + V/(m 2 + M 2 ) 2 - 4m~MZzcos228),
(22)
(23)
2 (24) m ~ , = M ~ + m c .
F r o m previous equations, one gets
2 _ m2(1 + h - kA 2) + M2(e + g) + 2#t20 + fmMA. m c - - (25)
In the limit Isin281 = 1 one gets to first order in cos220
22 ( ) mcMz m ~ - 2+Mz~COS220= (Mzcos20)2 12 2 " (26) m~ 1 + M z / m ~
and finally
m a = (27)
Not ice that the Higgs H b mass is very small since for the case at hand, I~l = 1 and cos28 = (to 2 - 1) /( to 2 + 1) ~ 0. Thus with a t-quark mass as low as - 40 GeV there does necessarily exist one very light neutral Higgs [5, 10]. The other two are
2 > 2 m 2, r n ~ > M z z + 2 m 2, and the same is true for the charged quite heavier, m c Higgs, rn~Zl, >__ M2w + 2m 2.
Concern ing the masses of squarks and sleptons, all of them (except for the t-squarks) have simple expressions depending only on m and M. Those relations are given in (IL.C) and numerical formulae will be given below. The masses for the t-squarks have a more complicated expression because the "off -d iagonal" mass terms (/'L -- iR mixing terms) are non-negligible. The t-squark mass matrix is [1]
tR
~_ ( m 2- - m , ( Atm + g/o~) ) IL
[~ - m , ( Atm + g/~0) m -2 |R
(28)
where rn'2t, = rn~ + m t,2 rn2~, = rn~ + m,2 since cos20 = 0. The physical eigenvalues
I,.E. ll~hez et al. /Super, wmmetric spectrum
corresponding to this matrix are
225
m -2 =m2+ ½[(m2+m~+l(m2-m212+4(A,m+./,.~)2m2,]'/2~ (29)
Notice that due to the off-diagonal term one of the eigenvalues may be lighter [11] than the t-quark for A t a n d / o r tt large enough*. We will comment on this possibility later on.
Let us now turn to gauginos and higgsinos. It is well known that the charged winos combine with the charged higgsinos to form a couple of Dirac particles with masses
This simplifies somewhat in the case at hand, in which cosZ20 --- 0 and ]sin20] - 1. As in the case of the s-tops, there is one eigenvalue which may be light if M(#) ~ oo with ~ ( M ) fixed. We will comment on examples below. Finally, there are the neutral gauginos and higgsinos which have a 4 x 4 mass matrix. We thus cannot give a simple expression for the physical eigenvalues but will calculate them numerically when required.
Notice that the situations (a), (d) and (b), (c) discussed in eq. (21) not only have the same numerical SU(2) x U(1) breaking condition but lead to the same "s top" , "chargino" and "neutral ino" spectra. In the case of the t-squark this is because going from (a) to (d) (or (b) to (c)) one changes the sign of both # and ~o and formula (29) remains unchanged. The same happens in the chargino case with ~ and sin 20.
4. Some numerical results
Let us now assume that mr = 40 GeV in the previous equations and see what restrictions we get on the spectrum and on the supergravity parameters. In fact, since the precise value for m t is not yet known we are going to assume instead a fixed hi(0) = 0.08 which gives rise to t-quark masses around 3 7 - 41 GeV corresponding to u2 -- ~-2 I v - This is more simple from the computational point of view. The values of the functions introduced in the previous chapter for h t (0 )=0 .08 and t = 2 I n ( M x / M w) -- 67 are shown in table 1. We get the following numerical formulae.
* Notice also that in this respect the relative sign of At and I~/w is important.
226 L E. Ibahez et al. / Super.9'mmetric spectrum
TABLE 1 N u m e r i c a l v a l u e s o f the coe f f i c i en t s d e f i n e d in the text in the case o f m i n i m a l l o w - e n e r g y c o n t e n t
q = 1.38 / = 1.9 h = 0.9 K = 0.031 f = - 0.15 e = 0.08 g = 0.53 r = - 0.72 s = 0 .046
4.1. S Q U A R K A N D S L E P T O N M A S S E S
For all (t and ,g (except t-squarks), one obtains from eqs. ( IL .CI )
m 2_ = m 2 2 u, + 7.6M - 0.35M~cos20, (31a)
m~d,. = m 2 + 7 .6M 2 + 0 . 4 2 M ~ c o s 2 0 , (31b)
m 2 = m 2 + 7.17M 2 - 0 .15M2cos20 , (31c)
rn 2- - m 2 + 7 .14M 2 + 0.07M~cos 28, dR - - (31d)
m~R = rn 2 + 0.15M 2 + 0 .23M~cos20 , (31e)
m- 2 = m 2 + 0.53M 2 + 0.27Mz2cos28, (31f) et.
2 _ m 2 rn~ - + 0.53M 2 - 0 . 5 M } c o s 2 0 , (31g)
where we remind the reader that cos20 ~ 0 in the case m t = 40 GeV (one numeri- cally finds c o s 2 8 = 0 . 0 - 0 . 1 8 for wide ranges of parameters). To obtain these numbers , we have taken
a¢~ = ~4, M x = 2 . 8 M 1016 GeV, sin20w(Mw) = 0.23, (32)
co r respond ing t o a 3 ( M w ) = ~. One has to add to eqs. (31) the (mass) 2 of the cor respond ing fermionic partner if it is relevant (i.e. m~, m~, m~,). Also, these are the values of the masses at the M w scale, if 2 , mc~.e< Mew one has to let them evolve further till 2 2 2 _ 2 m~.~(Q = mca.~ ) - rnca.~, which corresponds to the physical pole in the propaga tor . On the contrary, if m~.l(Mw)> Mew one has to stop the " runn ing" before it reaches M w. However, this subtlety does not have in general a great numerical influence. Thus, for example, in a case in which 2 m ~ ( M w ) = (40 GeV) 2, the effect o f ignoring the further evolution of m~ leads to an underest imation of the physical met mass by at most - 1 5 % . However, we already have this order of uncer ta in ty by neglecting second-order corrections in the general renormalization g roup equations. In the case of sleptons, this effect is negligible.
L E . Ibahez et aL / Super.~vmmetric spectrum 227
4.2. MASSES OF t-SQUARKS
These are given by eq. (29). Thus we need to know the values of m~, m 2, A t and at the weak scale. One finds for h i (0)= 0.08 the following numerical formulae:
In the cos20---, 0 approximate formula is given by
mg, t~ --" mr2 + m 2 ( 0 . 9 5 -- O.015A2)+7.2M 2
-O.07mMA T [0.94mA + 4.0M + r/1.38~t01,
where r /= 1 ( - 1) for #~ > 0 ( < 0).
A t (Mw) = 0.93A + --M-M4.02, (33) /91
/~(Mw) = 1.38#o, (34)
m~ = (0.93 - 0.02A2)m 2 - O.lmMA + 6.86M 2, (35)
m~ = (0.97 - 0.01A2)m 2 - O.05mMA + 7.45M 2. (36)
limit (and also ( m ~ - m ~ ) / ( m t m A t + ~mt/ to )--, 0), a good
(37)
4.3. HIGGS PARAMETERS AND MASSES
Numerical formulae for the mass parameters in the Higgs potential are given by
With these formulae, one obtains, for the neutral 1t,. Higgs field,
2 m2(1.9 0.03A 2 )+0 .61 M 2 + 3.8/t2o O.15mMA, (41a) m c ~ - -
and the rest of the Higgs masses can be calculated from m 2 using eqs. (23)-(25). In particular, one gets for the lightest Higgs H b the following approximate formula:
m2 = ( ~ M z ) 2 (1 + 0.53y2+ 1.9x2)(1 + 4 . 5 y 2 + 0.3A2 + 1.5yA) 2
As we discussed in the previous section, the charged winos combine with the charged higgsinos to form two Dirac particles. In the approximation cos 20 ---, 0 one gets the eigenvalues
where 3~/= 0 . 8 1 ( M / M w ) , it = 1.38(/t0/Mw) and r /= + 1 ( - 1) for sin20 > 0 ( < 0). This approximation with cos20 ~ 0 is good up to 5% for not too large M or / t 0. Let us remark, however, that the particle spectra shown later on are calculated with the exact formula eq. (30).
The result for the gluino mass is trivial:
a 3 ( t ) M~ = - - M . (43) a G
This gives M~(Mw)= 3M, Mg(10 GeV)= 3.4 M. Concerning the neutral gaugino and higgsino spectra we have to calculate it numerically. However, in some interest- ing limits (M ~ 0) there is an approximate photino eigenstate with mass
a e ( M w ) 8 ae M = O . 5 M , (44) M~,- a e ( M x ) M = ~ a(;
so that typically we will have M g / M ~ - - 6 - 7 . This constraint is important when trying to obtain phenomenological predictions. In this same limit and if a lso/ t is small compared to M w, there is a Majorana higgsino eigenstate with mass
M~o = # = 1.38/t 0 , (45)
and a Dirac zino-higgsino eigenstate with mass - M z. Notice, however, that all these statements are not true for general values of the parameters M and ~t 0.
Let us consider now the condition for SU(2) × U(1) breaking. This is given by eq. (19) after substituting eqs. (38) and (39). We have studied numerically that condition as well as the possible supersymmetric spectra compatible with it. We will show these results later on. In all the wide range of the parameters (#~(0),/t 0, A, m, M) studied one can only get SU(2)× U(1) breaking for 0.98 < [sin2O(Mw)l ~< 1 so that in fact one can approximately use for the SU(2)× U(1) breaking condition
I s in20(Mw) I --- 1, (46)
instead of eq. (19). Notice that this condition is the same as the one imposed in dimensional transmutation schemes so that numerically there will be practically no
LE. Ibi~hez et al. / Super.~vmmetric spectrum
difference between our "tree-level" approach and the one pursued Numer ica l ly condit ion (46) reads
in ref.
229
151.
s i n 2 O ( M w ) - - ± 1 = 2.76#2(0) - 1.44/~0M + 0.09Am#0
which can be rewritten as a scale-independent formula for sin 20 at the G U T scale
s i n 2 8 ( M x ) ± 1 [0.69 - 0.01A 2 + 1.38x 2 ( l + x 2)
+ 0 .22y 2 • 0.03Ax -- 0.05Ay + 0.52xy] , (48)
where x =- # o / m and y ~ M / m . The + ( - ) signs stand for the two possibilities s i n 2 O ( M w ) = +1 .
We plot condi t ion (48) in figs. l a - c in which the lines corresponding to fixed s i n 2 O ( M x ) are drawn in the A - x plane for several values of y ( y = - ±2,0, -~). Not ice that this plot is general and does not assume minimal kinetic terms for the scalars. These figures show us what range of the parameters A, x, y and sin 2O(M x) is al lowed by the conditions of symmetry breaking + stability of the minimum of the scalar potential. There is an allowed domain bounded by forbidden regions* cor responding to s i n 2 8 ( M x ) > 1 (scalar potential unbounded below, leading to M w - Mp) and the conditions [5, 2]
A~m2 <~ 3( m~ + m ~ + ~22 ) , (49)
A 2~< 3(3 + x 2 ) , (50)
which forbid that charge + colour-breaking absolute minima appear. The second bound is just like the first one but taken at the G U T scale (the appearance of these undesired minima must be forbidden at any mass scale). Of course, it could well be that we live on an unstable local (charge + colour conserving) min imum [12] which will eventually decay into one of these undesired minima but we find this possibility contrived.
One draws several interesting conclusions from these plots. (i) The value of s i n 2 0 ( M x ) is forced to be in the range
0.6 .%< [ s in2O(Mx) I ~< 1, (51)
" In fact, the condition Isin20(t)l ~ I must be fulfilled for any t and not only for t = 0. Usually, Isin20(())l ~ 1 implies Isin20(t)l ~ 1 except for values of [sin20(0)p very close to 1 in which case one may obtain values of f sin2O(t)l slightly larger than 1 at intermediate t. Thus the real stability bound from sin28 is slightly more stringent than the one shown in figs. 1-3 although numerically practically identical. We acknowledge discussions on this point with A. Bouquet and J. Kaplan.
230 L.E. Ibi~he: et al. / Super.~Trnmetrw spectrum
which, in terms of the definition of this angle, means
g~(O)= ( 0 . 6 - 1 ) ( m 2 + g ~ ) . (52)
One can check that this is true for arbitrarily large values of M. Restriction (52) was already remarked in ref. [3] and it is a rather puzzling result since it forces two independent parameters of the theory like/t~ and (essentially) m'- to be quite similar in size. This is surprising since /to is just a free parameter in the superpotential whereas m is related to the supersymmetry breaking scale. The bound (51) rests entirely on the assumption of a relatively light ( < 60 GeV) t-quark. In the case of a heavy t-quark /t~(0) and m 2 may be widely different with no problem to get the appropriate SU(2 )x U(1) breaking.
(ii) The bare higgsino mass #o cannot be arbitrarily large for a fixed y = M / m . Thus one gets, for example,
0 4 / t0 ~< 0.8m
0 ~</to < 1.2m
0 -..</to ~< 1.5m
( M / m = 0.5),
( M = 0 ) ,
(M/m = - 0 . 5 ) , (53)
so that the physical higgsino mass in supersymmetric models cannot be made arbitrarily large (e.g. in order to get light charginos). For example, in models with m,~ = rn~---40 GeV (having m = 20 GeV) one will have /to-< 40 GeV, whereas in models with M = 0 one has jus t / to -< too. Another feature shown in fig. 1 is that the larger the ratio M / m , the smal ler / to /m has to be. This is due to t he / toM term of the evolution formula for/t~(t) (eq. (40)). The smaller M is, the faster/t~ increases at low energies. This allows s in20(Mx) to start at M x from smaller values, the fast growing of/ t~ at low energies giving rise to the desired sin28(Mw)-=- 1. Starting at M x from smaller values for sin 28 implies in general the possibility of having larger /to (since s i n 2 0 ( M x ) - 1 / (m2 +/t2o) ). This is the origin of the correlation between the allowed values of / t o and M.
In fig. 1, the line corresponding to the minimal case (i.e. B = A - 1, sin 20 (Mx) = (1 - A ) x / ( 1 + x2)) is also shown. Notice that one always has /to >--0.17m. We will comment below in more detail about the minimal case.
In fig. 1, we have only considered the case in which s in2O(Mw)= + 1 (not - 1 ) and also /to > 0. However, one may easily obtain the figures corresponding to the other cases using the symmetries of eq. (47). Thus, for s in2O(Mw)= - 1 one finds
0 <~ ~to <- 1.5m
0~/t0~< 1.2m
0 ~ / to < 0.8m
( M/m = 0 . 5 ) ,
( M / m -- 0),
( M / m = -
The results for / to < 0 are equal to the ones for / to > 0 but with opposite A and y.
tlttlllllllllllllllll/l~lllllllllTTi7 , ~ , , 1 l J l J 1
0.S x= ~ / m
sin 2 8 (Hx)>l
l I 1
y= H/m= 1/2
l I , r
1.5
Fig. 1. Values of A, P'o, M and sin20(M x ) consistent with radiative SU(2) × U(1) breaking as explained in the text. (a) M/m = - ~, (b) M= 0, (c) M/m = ~. These results are general and do not assume
minimal kinetic terms.
232 L.E. Ibitfiez et al. / Supersymmetric spectrum
Let us now assume the minimal coupling of chiral superfieids to N = 1 supergrav- ity in which we have "canonical" kinetic terms for the scalars in the theory. One has [1,13] in this case B = A - 1 leading to
s i n 2 0 ( M x ) = (1 - A)x/(1 + x2). (54)
We consider this minimal case for two reasons. First, it is the simplest K~hler manifold one can think of. Second it is likely that a proportionality of the type /t2(O)-A/tom may be present in more complicated situations, in particular the dependence o n / t o and m will probably be quite universal. Thus, assuming eq. (54) may give us an idea of the type of constraint, one would get from more general versions of couplings to N = 1 supergravity,
With the above assumption, we are then left with three parameters (A, x and y) related by the symmetry breaking condition. In fig. 2, we plot how A, x and y should be related in this minimal case in order to get the appropriate SU(2) x U(1) breaking. We show the lines corresponding to fixed y = M/m in the A - x plane. The lower lines correspond to the condition s in20(Mw) --- 1 and the upper ones to s i n 2 0 ( M w ) = - 1 . The shaded area below (above) corresponds to the forbidden region s i n 2 0 ( M x ) > 1 ( s i n 2 0 ( M x ) < - 1 ) . These lines correspond to equations + 1 = (1 - A)x/(1 + xZ). Notice also that the lines of fixed y are sometimes cut by a short-barred line. This indicates the stability bounds for not getting charge-colour breaking minima from a large A.
Let us comment on the numerical results for this minimal case starting by the sin 2 0 ( M w ) = 1 case. Several features are noteworthy.
(i) The lines for fixed y have a similar shape as the boundary s i n 2 0 ( M x ) = 1 line. This is simply a reflection of the fact that one always has values of sin 20( M x) not far away from the boundary (0.6 - 1.0).
(ii) Only values A < - 1 are allowed. This is quite a strong restriction since, as shown in fig. 1, A could in principle be any number between = - 3 and + 3.
(iii) The possible values for / to are restricted. Thus one has, for example,
0.15m </t0 < m ( y = 0) ,
0 .17m</ t0 < l . 2 m ( y = - ~ ) ,
0.17m </ t0 < 1.5m ( y = - 1 ) ,
0.22m </ to < 0.5m ( y = ~). (55)
(iv) The values for M are also restricted. Thus, one can check that for y > ¼, the s i n 2 0 ( M w ) < 1 bound is violated. As pointed out in ref. [3] and contrary to statements in ref. [5], the gaugino mass M may be arbitrarily small with no problem at all.
L.E. lbahez et al. / Supersyrnmetric spectrum 233
3 -
2 -
1 -
0 -
-2
-3
-6.
y= 0
- - - - - - y= - I
. . . . . . . . . . . y - -1
y : M/m
Y .."'" sin 2 O (Hx)>1
0.5 1.0 1.5 x= po/m
Fig. 2. Values of A, x = tto/m and y = M / m consistent with radiative SU(2) × U(1) breaking assuming B = A - 1. The shaded regions correspond to Isin20(Mx)l > 1. Results are shown for M / m = O
(continuous line), M / m = - 1 (dashed line) and M/m = + 1 (dotted line).
In the case s i n 2 0 ( M w ) = - 1 , the pa rame te r s are even more restr icted. (i) O n l y va lues 2.6 ~< A _< 3 are a l lowed (for m o d e r a t e y) . (ii) T h e pos s ib l e #o va lues are restr icted. O n e has, for example ,
0.4m_<#0 < 1.2m ( y = ½),
0 .3m _< ~0 < 1 .6m ( y = l ) ,
0 .5m ~</~0 _< 0 .9m ( y = 0 ) , (56)
a n d , for e x a m p l e , the poss ib i l i ty y ~ - ½ is a l ready excluded.
234 I~ E. Ibilhez et al. / Supersvmmetric spectrum
E ' r
i I ! 1 , , . ' ,
6 ,~,L.hL -~ , , , ~ /
5
-2 a)
l 1 1 1 2 3
X= IJ.o/m
-2
-4
_6 ¸
.... sing04 --/~z" b) . -
1 1 1 1 2 3
X= l~o/m
Fig. 3. A l l o w e d values o f the p a r a m e t e r s x = ~ o / m , y = M / m , A in the a s s u m p t i o n B = A - 1. The h a t c h e d a reas a re f o r b i d d e n d u e to the viola t ion of the I s i n 2 0 ( t ) l ~< 1 o r s tabi l i ty c o n s t r a i n t s for (a)
sin 2 0 ( M w ) < O, (b) sin 2 0 ( M w ) > 0.
1_ E. Ibahez et al. / Supersymrnetric spectrum 235
Some of these features are more clearly seen in fig. 3 in which we plot the allowed values of the parameters x = t to /m, y = M / m and A in the x - y plane*. We insist that figs. 2 and 3 assume B = A - 1. Note that for x >_. 1 gauginos cannot be massless. Another important point is that one may have large values o f x, y a n d A , but only by letting them all increase simultaneously. Thus the usual assumption that IA ] _< 3 is only true for small values of x and y.
In all the above cases we took #0 > 0. However the results for #0 < 0 are identical to these, as we stated in sect. 3. All the above restrictions have their reflection on the allowed supersymmetric spectra as we will discuss in detail in the following section.
Before discussing the SUSY spectra, let us make some general comments on the relevance of the above restrictions on general N = 1 supergravity models. A first general statement is that any N = 1 sugra model (with minimal low-energy content) in order to be consistent with a t-quark with m t = 40 GeV needs to verify eq. (51), i.e. #~(0) = (0.6-1)(m 2 + #~) and thus #~(0) cannot be arbitrarily small. This is for example relevant for the recently proposed class of models [14] with an SU(n, 1 ) / S U ( n )× U(1) symmetry in the scalar sector of the theory. These models are characterized by having an "observable sector", which is globally supersymmet- ric (i.e. m = A = B = 0) except for gaugino mass terms. However, B = 0 implies #3(0) = 0 violating condition (50). Starting with a vanishing #3(0) at M~, radiative corrections cannot generate a #3 large enough to get sin 28 = 1 at the M w scale. One needs to complicate appropriately the model in order to obtain the appropriate SU(2) × U(1) breaking. Moreover, if one starts from m = 0 at the G U T scale, the SUSY spectrum will in general be very heavy. This is because experimentally we
2 (20 GeV) 2. Using eq. (31e), one then gets that M >__ 50 GeV, know that m,~ R > implying m s > 150 GeV and m~ >_ 140 GeV.
Another point to remark concerns the parameter A. As we stated above, in the case of minimal kinetic terms one has B = A - 1. Sometimes this equation is assumed although one is not considering minimal kinetic terms. One must be careful in any case because once one assumes B = A - 1 a wide range of A values
- 1 < A _< 3 (including A = 0!) is forbidden if we want to be consistent with m t = 40 GeV.
5. The low-energy supersymmetric spectrum
We discuss in some detail in this section the SUSY spectra consistent with the appropr ia te radiative SU(2) x U(1) breaking and a t-quark with m t = 40 GeV. The
* Notice that figs. 2 and 3 correspond to the approximation [sin 20( M w )1 = 1. In general, relaxing this approximation gives rise to some "bands" instead of the lines drawn for different A. However, these bands are extremely narrow for the regions of m and M of interest. We also remark that in fact in the unphysical rn---, 0o limit, the boundary lines corresponding to I s in20(M w)l < 1 do not close and one may have arbitrarily large A, x and y.
236 LE. lbi~hez et al. / Supersvmmetric spectrum
free parameters in an arbitrary N = 1 supergravity (grand unified) model are
m , M , Izo, A , B , mt . (57)
If we fix m t = 40 GeV and impose SU(2)× U(1) breaking, one still has four free parameters. If one knew, for example, the squark and gluino masses m~ and Mg one would infer from them the values of m and M. We would be left then with two free parameters, for example, A and B, which would parametrize the rest of the supersymmetric spectrum. For the sake of simplicity we are going to further assume that B = A - 1 as in the "canonical" kinetic term case. In fact this assumption has only a certain influence on the Higgs and higgsino sector of the spectrum and it is reasonable to expect that the consideration of more general situations will essentially lead to very similar spectra. For fixed mq and Mg values we will then parametrize the spectra by one single parameter A. On the other hand, A is, as we discussed above, severely restricted by IAI < 3 (for fixed x and y) in the general case (furthermore the region - 1 ~< A ~ 3 is forbidden if B = A - 1) so that not many different spectra are possible within the above hypothesis.
In order to obtain a one-parameter (A) family of spectra, we thus need to fix mq and Mg. We will do that by considering values of these parameters which could be of interest for giving a supersymmetric explanation to the jets + missing PT events recently observed in the p~ collider at CERN [7]. The most outstanding of these events are the "mono-jet" events in which a single (narrow)jet recoils against a large amount of invisible ( - 4 0 GeV) PT" In a SUSY explanation of these events the photino(s) is the natural candidate to carry the missing PT. However, several (essentially three) different SUSY mechanisms have been proposed in order to interpret the data. We will call them the 1~1 (one-squark), 2q (two-squark) and 2g (two-gluino) interpretations. The lgt mechanism [15,16] was in fact proposed [15] well before the monojet events were found. It assumes very light gluinos (say Mg = 3-7 GeV) and heavy squarks (m~ = 80-140 GeV). In this candidate explana- tion, a quark from the proton (antiproton) fuses with a gluino from the "sea" of the antiproton (proton) to form a heavy squark which decays practically at rest. The spectator gluino does not carry PT enough to be detected. The heavy q so created decays ~ 2% of the times into q + 5' giving rise to a jet + missing P'r signature. The dominant decay into q + ~ is difficult to disentangle from the usual QCD back- ground. In the 2~t mechanism [17], one assumes squark masses m~--40 GeV and heavier gluinos (typically slightly heavier than the q) so that the dominant (t decay mode is (t--' q + 5'. The squarks are pair-produced in the collider and then decay into a couple of quarks and a couple of photinos. One would naively expect two-jet + missing PT events from this type of mechanism but the experimental PT cuts and jet criteria make many of these would-be two-jet events look like monojets. Finally in the 2~ mechanism [18] gluinos with m~= 40 GeV are assumed and mq >> rag. Again, the gluinos are pair-produced in the collider and decay into four
L.E. ltmhez et al. / Supers vmmetric spectrum 237
quarks and two photinos. As in the 2C l mechanism the experimental procedure means that much of the time one observes monojets. We discuss the SUSY spectra corresponding to these three scenarios in turn.
5.1. THE 1-SQUARK MECHANISM
In this case [15, 16], one assumes a heavy squark and a very light gluino. We show spectra for various allowed A values in table 2a (m4 -- 80, Mg -- 3 GeV) and table 2b (m,i = 120, M g - - 3 GeV). One can see that in this case the possible A-values are confined to - 2.9 _< A _< - 1 and A --- + 3. We show results for A = - 2.5, - 2, - 1, 2.95 and 3. One has [M[ -- 1 GeV and due to the smallness of M its sign is usually irrelevant. When it is not, we show separately both spectra (if allowed by the SU(2) × U(1) breaking and stability constraints). We see that in both tables 2a and b, the slepton mass is practically equal to the squark mass (it is slightly higher because of the terms proportional to cos28 which are in this case more important than the ones due to a non-vanishing M; in the tables, the q and .~ masses correspond to 1,0 R and eL). There is one lighter (i t) and one heavier ( ih) top squark, most of the time far above present experimental bounds. However, in the m = 80 GeV case and for [A [ = 2.5-3, one may have very light t-squarks as in the example provided for A = - 2 . 5 and A = 2.95. The photino is m~--0 .5 GeV and the rest of the neutralinos are in general much heavier. However, for A = -2 .5 , A = - 2 , there is a neutralino (mostly a higgsino) with mass in the range 20-40 GeV which can be pair produced in Z ° decay. There is also a lighter chargino (~//) with mass in the range 30-65 GeV which can be produced in W and Z ° decays. The heavy chargino ('~/h) lies in the range 90-180 GeV. All the Higgs scalars are very heavy except for the neutral o n e H b with mass in the range 2 -9 GeV. As we remarked in sect. 3, there is always such a light neutral Higgs in any supergravity scheme with m t = 40 GeV. This Higgs could be present in the decay products of the T (through T --, H b + y ) o r
toponium states and couples to fermions ¢~ times more strongly than the standard Higgs. Quite often there is no phase space for H~ appearing in charmonium decays. As a general comment, we should mention that in order to get this scenario with very heavy C t and very light ~ no special teasing of the supergravity parameters is required; it is a scenario which appears naturally for a wide range of values of these parameters.
5.2. THE 2-SQUARK MECHANISM
In this scenario [17], one takes Cl masses m,~ = 40 GeV and heavier gluinos so that the dominant ~ decay mode is ~-- , q + 5,. However, it is not easy to get gluinos which are heavier than squarks in an N = 1 supergravity scheme*. This is easily seen [19] using eqs. (31) and (43), which yield for the average squark mass in terms of
* This has already been remarked on by Ellwanger in ref. [22].
~:.
:-
0'o ~"
• ~_
~
_~'.
~ ~
~
,~
~
~,.
,.,~
'~_
,~
''+~_
~~
~o
~E
~
~Z
'Z~
~~
~
~=~
~<
~ '*
~
~,~
~
~.~
~
,'~
-~
i
,.,,
~
~ ~
~ ~
Yo ~
,..
, ~-~
41
L.E. Ib~hez et al. / Supersymmetric spectrum 239
8 0
6 0
>
e~" / ~ mi" ~ 18 GeV
I ] ' I , I a 20 40 60 80
M~ (GeV)
Fig. 4. rn,~ versus M~ in a general low-energy supergravity model. The shaded area corresponds to Mg > rnq and rot>__ 18 GeV. Only for mq > 35 GeV is this situation possible.
gluino mass
m~ = m 2 + 0.72M~, (58)
so that large M~ implies large m~. One can try to lower the value of m,~ by diminishing m but the experimental constraint m?R > 18 GeV implies from eq. (31)
r n~ = m z + O.O15M~ > (18 GeV) 2, (59)
so that if we set, for example, m,~ = 45 GeV, one obtains from (58) and (59)
Mg < 49 GeV. (60)
The gluino can only be slightly heavier than the squark in a low-energy supergrav- ity model [19]. This is further illustrated in fig. 4 in which we plot rn~ versus Mg for m --- 18 GeV. We observe that the gluino can only be heavier than the squark for m,~ > 35 GeV or so. For higher m (which might be required if the limits on selectron masses are improved) one needs even heavier squarks. The range of supergravity parameters for this scenario to work is extremely restricted, one needs m = 20 GeV, M = 15 GeV and small variations of these parameters spoil the scenario because either m~ < rn~ or m~R < 18 GeV. On the other hand, this possibility implies a light
240 L.E. lbiJfie: et at / Supersvmrnetric spectrum
TABLE 3 Supersymmetric spectra corresponding to the 2~1 scheme
In this case, rn = 20, M = + 15 GeV, leading to m~ -- Mg = 45 GeV. Different signs for M give rise to different spectra although sometimes a given sign is not allowed. The A-values correspond to typical allowed values and sometimes there are two spectra for a given A.
supersymmetr i c spectrum which could be observed in the near future as was remarked by Ellis and Sher [9]. One finds for m,~ = 45 GeV, mg = 45 GeV (rn = 20 GeV, IMI = 15 GeV) the spectra shown in table 3. We show results for A = - 3 , - 1 . 8 , - 1 , 3 and M = + 1 5 GeV which cover the range of al lowed values for B = A - 1. The values A --- - 3 - - 2 are forbidden either due to the stability of the potent ia l or due to having ( / ' ) :~ 0. One of the most prominent features of the spectra is the l ightness of the s lepton masses, m l - - 2 3 - 2 4 GeV which could be tested in e + e - experiments (through, for example , e + e --* ~ y and e ~ e - ---, ~eS') as wel l as in the weak gauge boson decays Z ° ~ . g ~ £ - and W ~ £ ~ at the C E R N col l ider [19]. Concerning the top-squark masses, in this scheme, a general tendency exis ts for obta ining too light t-squarks. As we c o m m e n t e d above, for A = - 3 - - 2 and M = - 1 5 GeV, one gets m ~ < 0 and even for A = 3 and A - - 1 . 8 the obta ined i I mass is experimental ly excluded. Only for A = - 3 and A = - 1 may t-squark masses be safe. W e will c o m m e n t later on about the possible parameters leading to light stops in the three scenarios. The heavy t-squark has a mass mih --- 7 0 - 8 0 GeV. For the photino, one has M , --- 7.5 GeV and concerning the rest of the neutral inos, two of them are usually quite heavy ( 8 5 - 1 0 5 GeV) and corre-
As in
L.E. lb&hez et al. / Superaymmetric spectrum
TABLE 4 tables 2 and 3 but for the 2~, scheme and Mg -- 40. m,~ --- 70 GeV
spond to zeeno-higgsino states. The other neutralino (mostly a higgsino) can be rather light, 7 - 4 0 GeV, and in fact for the A = - 3, M --- + 15 GeV example it is the lightest supersymmetric particle. The lightest chargino is usually quite heavy (68-78 GeV) so that there is not much phase space for the decay W ~ W / + ' ~ nor Z ° ~ "~ + V¢. The heavy chargino weighs 78 - 95 GeV. As in the 1~1 scenario, the Higgs scalars are usually heavy except for the neutral one H b which in this case may be even lighter than in the lq scenario, mH, = 0.3-0.5 GeV. This Higgs could be present in charmonium decays. Let us also mention that there is another neutral scalar (He) which may be relatively light (as light as - 30 GeV) and be produced in toponium decays.
5.3. THE 2-GLUINO MECHANISM
In this case [18], one considers gluino masses M~ = 40 GeV and the missing m o m e n t u m comes from the two photinos appearing in gluino decay. The squarks are assumed to be heavier than the gluino. This scenario, like the 1-squark scenario, may appear with a wide range of supergravity parameters. As shown in fig. 4, it is very easy to get mq >> m~. As an example, we show in table 4 the spectrum for mq = 70 GeV, rng= 40 GeV. The allowed values for A are - 3 _ < A < - 1 and A = 3. The sleptons are a few GeV lighter than the squarks and both stop eigenstates are relatively heavy except for A = 3, M = + 13 in which there may be too light a stop.
242 L.E. lb~hez et al. / Supersvmmetric spectrum
v
,g
80
60
Z,0
20
vl = v 2 ~ H = lt,0
H= 100
H= 0 Limit
~ H = 20
, I ' 1 a I r 1 , I l l , I ' [ 20 Z.O 60 80 100 120 140 160
I,t o (GeV)
Fig. 5. The mass of the lightest chargino versus the ~o parameter for various M-values in a general SUSY theory (in the cos20 = 0 approximation).
The photino weights - 6 . 5 GeV and the qualitative structure of the rest of the neutralino spectrum is similar to that in the 2~1 case, with a relatively light higgsino. The mass of the iightest chargino is = 50 - 68 GeV and thus it can be produced in W-decays (it may be lighter if m~ is increased). The heavier chargino is typically m~h = 90-130 GeV. All higgses are very heavy except for H~, with rot1 b -- 1-4 GeV. This spectrum is a sort of interpolation between those of the 16, and 2q schemes.
The three types of spectra described above give a general flavour of the possibili- ties allowed by the appropriate SU(2) × U(1) breaking from broken N = 1 super- gravity. There are, however, some features which are general and can be discussed without committing oneself to a particular spectrum. For example, it is interesting to know for what ranges of parameters one can expect charginos or stops light enough so that one could produce them in present accelerators. Fig. 5 shows the mass of the iightest chargino versus the ~t 0 parameter for various values of M in the cos20 = 0 approximation. It is clearly seen that relatively light charginos may only be obtained for large M or ~t 0. Thus, for example, if one has m~ --- 100-120 GeV and M~-- 0, one may have ~ o - 1 0 0 GeV leading to a chargino weighing - 3 5 GeV. On the contrary, for m,~ = rng---40 GeV, the lightest chargino should weigh - 6 0 GeV at least. This explains why the lightest chargino is usually lighter in the 1~ I scheme than in the other two, ~0 may be larger in the former case (recall the bounds in eq. (53)).
The case of the lightest t-squark is more complicated since its mass depends on several parameters (A, rn,/,t 0, M). To simplify matters, we assume B =A - 1 and consider the cases of special interest M = 0, + 15 GeV. We show in fig. 6 the mass of
Fig. 6. The mass of the l ightest top squark versus m for ,several (al lowed) A-values. The equal i ty B - A - 1 is a s sumed for simplicity• Resul t s are shown for (a) M = 15, (b) M ~ 0, (c) M = - 15 GeV.
.t " "I..S ll,=.._. -?- i / ~ ~" . I - "~ t " / j -
/ . . . . . . . . . . . . 3"'" o . o ' f "
. . . . . . ~ , ~ A= -1
"'" / " A= -1
) I I l ) ) I I I l ~ I 1 l !
M= -15 GeV . . . . . . . . - - -"
. f f
. . . . . -
1
f A= -1 0 = 1 L l , I i i I L I = 1
20 &.0 60 80 100 120 140 m (5eV)
Fig. 7. M a s s of the l ightest neutral Higgs H b v e r s u s m for several (a l lowed) A-values . The equal i ty B = A - 1 is assumed• Results are s h o w n for (a) M = 0, (b) M = 15 ( ; eV , (c) M = - 15 GeV. Interrupted
curves indicate that beyond that point, charge and colour are broken through (t'/) ¢ 0.
/
A= -1
LI;: Ibahez et al. / Supersyrametric spectrum 245
the lightest s top versus m for several (allowed) A-values. For A = 3 and .4 = - 1, two possible lines (corresponding to two possible/z0) may occur. Notice that the i / m ay be lighter than m t ----- 40 GeV for certain values of A in all the three M = 0, ± 15 GeV cases. For M = + 15 GeV and A = 3, one always gets too light t-squarks unless the sfermions are heavy enough. Only for A --- - 2, - 2.5, may one have m as low as - 30 GeV leading to m~ - 50 GeV, M~ - 45 GeV. For M = 0 and light squarks, one always gets too light t-squarks except for 1.41 < 2. For large enough m, one is safe. Finally, in the case M = - 15 GeV and light squarks, only for A --- - 1, one is safe but one may improve the situation by increasing m. As a general conclusion, the safer way not to have too light t-squarks is by having a large enough m-parameter . Especially dangerous are large 1.41 values. This is obvious from eq. (28).
It is also of some interest to study the possible values of the lightest Higgs field H b mass. As in the i t case, the mass of H b depends onvar ious parameters so that we will assume B = .4 - 1 and take the interesting values M = 0, + 15 GeV. We show in fig. 7 the mass of the H b Higgs versus m for different (allowed) A-values. The general features of the three cases considered ( M = 0, + 15, - 15 GeV) are the same [10]: the smaller m (and I A I) the smaller is mHb. For m = 20 (as in the 2q scheme) this Higgs is a very light m H b - 0 . 5 GeV and should be produced in charmonium decays. Notice, however, that for such a small H b mass, the one-loop corrections to this mass may be relevant.
Let us make a couple of final comments about the spectra shown above. First, in the given numbers, we have assumed ht(0 ) = 0.08 (mt = 37-41 GeV) but the spectra will practical ly remain unchanged in the interval 30 _< m t _< 50 GeV. Second, we have shown the parameters renormalized at the M w scale. As we commented in sect. 4, in the case of Cl or ~ lighter than M w, one needs to further evolve m,~ and mg down to the appropria te scale. However, numerically it makes very little difference.
6. A short digression on some non-standard scenarios
One of the less attractive features of the supergravity class of models considered in the previous sections is the fact we already discussed that one necessarily has # 2 ( 0 ) - m 2 for m t - - - -40 GeV. These two parameters have nothing to do with each other and thus it is rather puzzling that they have to be almost equal in order to get a consistent S U ( 2 ) x U(1) breaking. There are several ways in which one can relax relation (52). One trivial way is to consider the existence of a fourth quark-lepton generat ion. In this case, the Yukawa coupling of this fourth family will be large enough to get ~ (or /x~) negative, leading to S U ( 2 )×U ( 1 ) breaking even if #~(0) << m 2. Another way is to consider models in which the low-energy chiral sector is not minimal but has extra light multiplets which transform like (adjoint + 2(1, 1, + 2) + 2(1, 1, - 2)) under SU(3) x SU(2) X U(1). This type of low-energy matter does appear in some N = 1 supersymmetric G U T s [20]. One can obtain in this case S U ( 2 ) x U(1) breaking with /.t~(0)<< m 2 for m t ~ 4 3 GeV, still compatible with
246 L.E. lbahe z et al. / Super .wmmetrw gpectrum
TABLE 5 S o m e a l l o w e d values of the s u p e r g r a v i t y p a r a m e t e r s in the scena r io d i scussed in sect. 6 wi th a
H i g g s mass roll smal ler t han the c o m m o n sca la r mass m at the G U T scale: this a s sumes ~-o = 0 = ~ ( 0 ) a n d w ~ ~c
expe r imen t . However , one needs to use ext reme supergravi ty parameters : for exam- ple, A = 3, m = 50, M = 216 GeV are consis tent with m t --- 47 GeV but one can get m , - - - 4 3 GeV only for A = 3 , m , M - - - , ~ . Thus in this type of model , all the spar t i c les wou ld be very heavy.
A thi rd in teres t ing possibi l i ty is disposing o f the universality of the scalar masses at the G U T scale and assuming a Higgs mass at the G U T scale m n smal ler than the c o m m o n q, .g masses m. This is not as ar t i f icial as it sounds since the higgses may have coup l ings to other (superheavy) fields which may render m H < m at the G U T scale even if they were equal at the Planck mass [21]. One can easily check that for m H ~: m the formulae (38) and (39) get modi f ied to
# ~ = ( m 2 ( h - k A 2 ) + e M 2 + f m M A ) + ( m ~ - m 2 ) (1 + 3 y ° F ) D ( t )
(61)
t~ = m 2 + g M 2 , (62)
in the ~ --* 0 l imit . In this case, one may break S U ( 2 )× U(1) with a low-mass t -quark and ~ ( 0 ) << m 2. Having m H = ~m at the G U T scale may be sufficient to radia t ive ly b r e a k SU(2) x U(1) with m t = 40 GeV. Some possible values of A, m, rn H, m t and M in this scenar io are shown in table 5. One usual ly needs large values for A and m bu t it is still poss ib le to get relat ively light gaugino masses (e.g. A = 3, m = 150, m n = 37.5, m t = 39, M = 21 GeV). However , the general tendency is to get heavy spect ra . W e hope that the " n o n - m i n i m a l " scenar ios br ief ly discussed here give an idea of wha t one can expect fro/n more sophis t ica ted models.
L.E. lb~hez et al. /Super, wmmetric spectrum 247
7. F inal remarks and c o n c l u s i o n s
We have presented in this paper a detailed study of the constraints imposed on low-energy N = 1 supergravity models by the existence of a low-mass ( m t ~< 60 GeV) t-quark. We give analytical formulae for the scale evolution of masses and couplings in terms of the parameters m, M, #0, A and B (for any h i ) . We find numerical constraints for these parameters when we fix m t ----40 GeV. In particular, we find that #23 = (0.6-1.0)(m 2 +/t20) at the G U T scale. Since, by definition, B = - / t ~ / m / t o, we conclude that models with B ( M x ) = 0 are inconsistent with m t = 40 GeV. This includes the simplest versions of the recently proposed models based on a K~ahler manifold SU(n, 1 ) / S U ( n )× U(1) in which the observable sector is globally super- symmetric (except for gaugino terms) at the G U T scale. One also finds restrictions for the bare higgsino mass parameter/ t0. Depending on the value of gaugino masses at the G U T scale ( M ) one gets bounds on ~to. Thus, for example, one gets (for /t2 > 0)/ t0 < 1.2m (M = 0),/'to < 0.8rn ( M / m = 0.5) and / t o < 1.5m ( M / m = -0 .5) . 3 ~
All these results are general and apply also to models with non-minimal kinetic terms. If one further assumes the relation B =A - 1 , one obtains stronger con- straints. In particular (for the A-convention defined in sect. 2), the range of values - 1 _< A < 2.8 is forbidden. Sometimes in the literature the relationship B = A - 1 is assumed even though the scalar metric is non-minimal. One should be careful in any case since, for example, values like A = 0.0 are inconsistent with radiative SU(2) × U(1) breaking. The restrictions o n / t o and M also get more stringent if one assumes B = A - 1. Thus, for example, one gets (for /t~ > 0) 0.15m _</t o <_ rn ( M / m = 0); 0 . 1 7 m < # o _ < l . 2 m ( M / m = -½ ) ; 0.22m~/t0_<0.5m ( M / m = + ~), etc. Similar bounds (more stringent) are obtained in the/t2 < 0 case. In any case, one always has /to >- 0.15m. Another interesting feature for B = A - 1 is that the gauginos cannot be rnassless for x = / to /m > 1. Also one may have large values for x, y and A but only if we let them all increase simultaneously. Thus the usual stability assumption IAI ~< 3 only applies for relatively small X and Y.
We also discussed the low-energy supersymmetric spectra consistent with the above restrictions. Of the five free parameters / to , M. rn, A and B, one is related to the others through the SU(2) x U(1) breaking condition. We fix M and m in order to obtain squark and gluino masses which could be interesting in trying to give a supersymmetric explanation to the CERN missing PT events. In particular, we consider three cases: (i) lq mechanism with M~ -- 3 GeV, mq = 80-120 GeV; (ii) 2~1 mechanism with rn,~ = M~ = 40 GeV; (iii) 2~ mechanism where M~---40 GeV and m,] >> Mg. Each of the three mechanisms lead to characteristic spectra which are described in detail. By fixing M and m, we are left only with two free parameters A and B. To simplify matters, we show the results in the case B = A - 1 in which a single parameter A parametrizes the possible spectra (for fixed M, rn). We argued, however, that the assumption B = A - 1 only affects essentially the spectra of higgses and higgsinos and that the results should be similar for more general situations. There are some general features of the masses of the sparticles in the three
2 4 8 L.E. Ibi~hez et al. / Super.sTrnmetric spectrum
cases considered. In general, the spectrum in the 2q case is lighter and some SUSY signatures should be soon available [19] in present accelerators (e.g. Z ° ---,£+£-. W - - , , g ~ , e + e - - ' 75'3'). In the other two cases, the sparticles are in general quite heavy. In the lg 1 and 2~ schemes one "ehargino" may be relatively light (~ 30 GeV) and could be produced in W and Z ° decay through W ~ qgll' and Z ° ~ ~/W. In the other (2q) case, these processes have in general little phase space. The photino is practically always the iightest supersymmetric particle but there is also a neutralino (mostly a higgsino) which can be relatively light (even the lightest SUSY particle) and contribute to Z ° and W decay (along with a chargino). The lightest top squark is usually heavier than its spartner in the 1C t and 2~, schemes, but in some restricted ranges of parameters mr, may be very (too) small (even negative!) so that one must be careful and check if one's parameters are safe. This is especially the case in the 2q mechanism where most of the time m -2 is too small. All the Higgs scalars are usually I t
heavier than M w except for the neutral H b which is very light (of the order of a few GeV) especially in the 2q scheme in which it may be as light as - 0.5 GeV and be present in charmonium decay. In the 2q case there is another neutral Higgs which may be as light as - 30 GeV.
Of the three mechanisms proposed to explain the CERN missing p-r events, the 1( t and 2~, schemes can be obtained from an N = 1 supergravity model very easily. In the 2q case the range of supergravity parameters leading to Mg > m,~ is very small but on the other hand the predicted spectra lead to quite dramatic signatures (,g,£ production) which should be seen soon. In the lq and 2~ schemes, one may, on the other hand, have relatively light charginos and neutralinos which could be observed in W and Z ° decays. Moreover, a bump in the multijet cross section [15] should be observed around the squark mass in the case of the lq mechanism. Thus forthcom- ing data may soon decide about the possibilities of these mechanisms. We should remark, however, that the spectra discussed above are just possible (but not the only) supersymmetric spectra. If no trace of supersymmetric particles are found in present accelerators, it could just mean that the range of energies explored is still too small. Let us hope that we will not have to wait for the supercolliders to settle the question.
We acknowledge discussions with A, Bouquet, U. Ellwanger, A. De R/ajula, L. Hall, J. Kaplan, P. Nilles and M. Sher.
Appendix A
The only relevant renormalization group equations with a non-trivial h t depen- dence are
d~ 2 = (21S2 + ~ai - 3 yt)#2 + 3#mYrA ' _ #(3~i2M2 + citMx)" (A.3) d t
dmq2 = ( ~ i 3 M2 + 3a2 M2 + ~al M2) _ Yt(mQ + mb + Ix22 + A2m 2 - #2) d t "
where
d t
(A.4)
? 2 2 _ _ = ( ; ~ a 3 M 2 + ~ a l M 2 ) _ 2 Y t ( m ~ + m Z + # 2 2 + A t m _#2) , (A.5)
d/~ 2 = (36 2 + 6~ - 3 Y , ) t ~ 2 , (A.6) dt
dt = ~ 3 3 m + 362 + ½3-fil - 6 Y t A "
a Mx
(A.7)
(A.8)
dYt = Yt(~63 + 3~ 2 + ~ 6 1 ) - 6 Y t 2, (A.9) d t
2 ai Yt= (47r)2 ' ~i,= ~--~-, t = 2 1 o g ( M x / Q ) , (A.10)
and the convent ions for the couplings are given in sect. 2. Here we have considered a non-negligible # which was not included in ref. [3].
Appendix B
We collect here all the functions H,, F, and Gx. 2 not defined in the text. These only depend on the running couplings a,(t) and not o n h t or any other parameter. We prefer to give here the general expressions for these functions and not just their numerical values in the minimal model because in this way it is easy to see the modif icat ions needed if one considers non-minimal scenarios like, for example,
250 L.E. Ibi~hez et al. / Supersvmmetric spectrum
adding a fourth generation or some extra light higgses. One has
(1 +fl3t)16/3b~(1 +fl2t)3/h2(1 + flit) 13/9h~ . E ( t ) =
F( t ) = fotE(t ' )dt ',
( 3) H~{t)=6(O) 16 1 3 + 13 3 (1 +fl3t) 2 + (1 +f12/) 2 9 (1 + f l i t ) 2 '
fo H2(t) = Hl( t , )d t ,=~(O)( !~h3+ 3h2+ .3 iThl ),
Ha(t ) = f o t d t ' E ( t ' ) n 2 ( t ' ) = t E ( t ) - F ( t ) ,
Ha(t) = F( t )H2( t ) - H3(t) ,
Hs(t ) = 6 ( 0 ) ( - ~f3 + 6f2 - ~ f l ) ,
H6(t ) = fo tH2(t ' )2E(t ' )dt ',
HT(t) = 3fiz(0)h2(t) + 61(O)h,(t),
H d t ) = a ( O ) ( - } f 3 + f 2 - ~f~),
F2(t) = fi(0)(~f3 + ~ f l ) ,
F3(t ) = F(t)F2(t ) - fo'dt' E( t ' )F2( t ' ) ,
F4(t ) = f { /E( t ' )Hs( t ' )d t" ,
a , = F2 - ~14~,
G 2 = 6F 3 - F 4 - 4H2H 4 + 2FH 2 - 2H6,
where the functions f , ( t) and hi(t) have already been defined in (IL):
1 ( 1 ) h i ( t ) _ t f , ( / ) = ~ 1 ( l + f l d ) 2 , ( l + f l d ) ,
(BA)
(B.2)
(B.3)
(B.4)
(B.5)
(B.6)
(B.7)
(B.8)
(B.9)
(B.10)
(B.11)
(B.12)
(B.13)
(B.14)
(BAS)
(B.16)
L.E. IbLthez et al. / Supers vmmetric spectrum 251
and fl, = &(0)b, = ai(O)b,/4~r. Whenever an a, appears in formulae (B.1)-(B.15), it refers to its value at the G U T scale, f i (0)= a(0)/47r corresponds to the G U T coupl ing and the hypercharge normalization is such that a , ( 0 ) = 3 /5a(0) . The numerical values of the functions q, 1, h, k, f , e, g, r and s defined in the text are given in table 1 in the case of the minimal low-energy content of the theory. The numerical values for f, and h i at the M w scale are
./'3 = 803, ] '2= 100, ft = 38,
h 3 = 201, h 2 = 55, h 2 = 27, (B.17)
in the case of minimal content. For the functions F and E relevant for the renormal izat ion of the Yt Yukawa coupling, one has
F = 290, E --- 14. (B.18)
References
[1] H.P. Nilles, Phys. Reports 110 (1984) 1: J. Ellis, CERN preprint TH.3718 (1983); P. Nath, R. Arnowitt and A.H. Chamseddine, Northeastern preprint NUB(2613) (1983): L.E. IbM~ez, Madrid preprint FTUAM 84-7 (1984): H. Haber and G. Kane, Michigan preprint UM-HE 83-17 (1984), Phys. Reports, to appear
[2] L.E. lbk~ez, Nucl. Phys. B218 (1983) 514; L.E. Ib~ff~ez and C. Lopez, Phys. Lett. 126B (1983) 54; L. Alvarez-Gaum~, J. Polchinski and M. Wise, Nu¢l. Phys. B221 (1983) 495: J. Ellis, D.V. Nanopoulos and K. Tamvakis, Phys. Lett. 121B (1983) 123
[3] L.E. Ib~fflez and C. Lope-z, Nucl. Phys. B233 (1984) 511 [4] S. Jones and G .G Ross, Phys. Lett. 155B (1984) 69;
K. Inoue, A. Kakuto and S. Takeshita, Kyushu preprint 83-HE-6 (1983) [5} C. Kounnas, A. Lahanas, D.V. Nanopoulos and M. Quiros, Nucl. Phys. B236 (1984) 438 [6} G. Arnison et al., CERN-EP/84-134 (1984) [7] G. Arnison et al., Phys. Lett. 139B (1984) 115;
P Bagnaia et al., Phys. Lett. 139B (1984) 105 [8] K. Inoue et al., Prog. Theor. Phys. 67 (1982) 1859 {9] K. Inoue et al., Prog. Theor. Phys. 68 (1982) 927; Errata, Kyushu preprint 83-HE-5 (1983)
[10] H.P. Nilles and M. Nushaumer, Phys. Lett. 145B (1984) 73 S. Li and M. Sher, Irvine preprint UCL-TR 84-7 (1984); P. Majumdar and P. Roy, Phys. Rev. D30 (1984) 2432
[11] J. Ellis and S. Rudaz, Phys. Lett. 128B (1983) 248 [12] M. Claudson, L. Hall and I. Hinchliffe, Nucl. Phys. B228 (1983) 501 [13] L. Hall, J. Lykken and S. Wcinberg, Phys. Rcv. D27 (1983) 2359 [14] E. Cremmer et al., Phys. Lett. 133B (1983) 61;
J. Ellis, C. Kounnas and D.V. Nanopoulos, Nucl. Phys. B247 (1984) 373: U. Ellwanger, N. Dragon and M. Schmidt, CERN preprint TH.3794 (1984)
[15] M.J. Herrero, L.E. Ibbiaez, C. Lbpez and F.J. Yndurain, Phys. Lett. 132B (1983) 199: 145B (1984) 430
[16] V. Barger, K. Hagiwara, W. Kcung and J. Woodside, Phys. Rev. Lett. 53 (1984) 641: Madison preprint MAD/PH/197 (1984); A. De Rujula and R. Petronzio, CERN preprint TH.4070 (1984)
252 L E. Ibithez et al. / Super.wmmetric spectrum
[17] J. Ellis and H. Kowalski, DESY Nucl. Phys. B246 (1984) 189 A. Allan, E. Glover and A. Martin, Durham preprint DTRP/84/20 (1984); V. Barger, K. Hagiwara and W. Keung, Madison preprint MADPH/183 (1984)
[18] E. Reya and D.P. Roy, Phys. Lett. 141B (1984) 442; Phys. Rev. Lett. 52 (1984) 881; J. Ellis and H. Kowalski, Phys. Lett. 142B (1984) 441
[19] J. Ellis and M. Sher, Phys. Left. 148B (1984) 309 [20] L.E. IbMaez, Phys. Lett. 126B (1983); 130B (1983) 463;
L.E. IbMaez and G.G. Ross, Phys. Lett. 131B (1983) 335 [21] K. lnoue, A. Kakuto and S. Takeshita, Kyushu preprint 83-HE-6 (1983);
H. Komatsu, Tokyo preprint INS-Rep.-469 (1983); B. (;ato, J. Leon, J. Perez-Mercader and M. Quiros, Madrid preprint IEM.TH 84-2-1 (1984); P. Moxhay and K. Yamamoto, Chapel Hill preprint IFP-215-UNC (1984)
![Introduction to supergravity - arXiv · supersymmetry, but supergravity is introduced as well. The supergravity review [3] is still, 30 years later, a very good introduction. The](https://static.documents.pub/doc/80x56/5ec7a9f876d4fe3f047ef2a9/introduction-to-supergravity-arxiv-supersymmetry-but-supergravity-is-introduced.jpg)
![[Wess Bagger]Supersymmetry and Supergravity](https://static.documents.pub/doc/80x56/55cf8eb6550346703b94d652/wess-baggersupersymmetry-and-supergravity.jpg)