Nuclear Physics B297 (1988) 133 NorthHolland, Amsterdam
EXOTIC (E6) PARTICLES IN e +e  ANNIHII.ATION* F. DEL AGUILAt, E. LAERMANN 2 and P. ZERWAS 3
t Department of Physics, University of Florida, Gainesoille, FL 32611, USA and Departament de Fisica Tebrica~ Universitat Aut6noma de Barcelona, Bellaterra 08193, Barcelona, Spain 2lnstitut f~r Theoretische Physik, Bergische Universitat, Wuppertal, FR Germany 3lnatitut f~r Theoretische Physik, R WTH, Aachen, FR Germany Received 29 May 1987
Fermion mixing is analyzed in (E6) extensions of the standard model. Cross sections and signatures for the production in present and future e+e  colliders are discussed in detail.
1. Introduction
The standard model has been tremendously successful in the past decade, correctly describing all experimental facts down to distances of O(10 16 era) [1]. Many fundamental features, however, do not follow from this theory but are merely incorporated by hand. It is therefore mandatory to explore possible extensions into which the model can be embedded in a natural way. Such a path has been paved by early versions of the superstring theory [2, 3] which suggest E 6 [4] as a possible framework for a phenomenological successful kind of unification. It predicts an interesting spectrum of fermions and gauge particles which could become accessible at the new accelerators now under construction or being planned. Superstring inspired constraints limit the large number of parameters, masses and coupling constants, so that the study may become more tractable. E 6 has been one of the attractive grand unified models. It contains the standard group SUc x SU/ x Utr and it is anomaly free. Moreover, when the fermions are assigned to the 27dimensional fundamental representation, the only chiral fermions are those already observed in the standard model (SM). It is therefore natural that the other fermions acquire large masses for they in general are not protected by symmetries when the gauge symmetry of the SM is restored above 100 GeV. These features may eventually be deduced from superstrings. These seem to require that the gauge symmetry breaking must be along adjoint representations at the unification/compactification scale (Hosotani breaking) and along fundamental representa* Work partially supported by the USSpanish Joint Committee for Scientific and Technological Cooperation under contract no. CCB8~2069.86. 05503213/88/$03.50©Hsevier Science Publishers B.V. (NorthHolland Physics Publishing Division)
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e + e  annihilation
tions below (Higgs breaking). Any low energy signal confirming these features would support the superstring picture. In this paper we discuss the general features for the production of new vectorlike fermions [5] concentrating on e+e  machines and allowing for their production through standard and new gauge bosons [6]. We will begin with a general analysis of the fermion mass matrices. Although the details of the matrices are not fixed without invoking specific models, their general characteristics follow from the chiral (quantum number) structure of the low energy theory that is described by the SM. Using this chiral structure as a zeroth order, the new parameters can be introduced as perturbations. Our discussion in sect. 2 and appendix A follows that of ref. [7], in particular sect. 4 there. Here we are more concrete in the sense that the only new (vectorlike) fermions are those contained in the fundamental representation of E 6. Besides we discuss the interactions with new neutral E 6 gauge bosons. Let us comment on the assumptions characterizing the models we consider. We will concentrate on superstring inspired E 6 models but our results are quite general. (i) The rank of the gauge group at the compactification scale can be 5 or 6. If there is only one extra Z ' at low energies, the heterotic string admits only two possibilities, regardless of whether CalabiYau [3] or orbifold [8] compactifications are considered [9,10]. With a regular embedding of the standard model the extra U 1 in the observable E 8 can be that of the minimal model [11] or B  L [12]. (Due to the SUEI × Utr degeneracy of the fermions in the sextets of the fundamental E 6 representation 27 = (¢~,2) + (15,1), B  L assignments of known fermions are only fixed by the detailed Yukawa coupling structure. We identify B  L with the unbroken symmetry). Both are contained in E 6. (ii) In fundamental E 6 representations the chiral fields are supplemented by new SU2~ x U~r vectorlike fermions which we will analyze in detail. Since the main results depend only on their vectorlike character, generalizations are straightforward. We will assume three complete 27 representations at low energy, giving three families of 15 standard fermions f, plus two associated families of new vectorlike fermions F and a third vectorlike family of higgsinos. The model is assumed to be supersymmetric down to the TeV region to allow for a natural (though technical) solution of the hierarchy problem. The higgsinos and the gauginos are assumed to decouple from the other fermions, as suggested by limits on neutrino masses and on flavourchanging neutral currents (FCNC) in the leptonic sector. Gauginos and higgsinos will therefore not be considered explicitly [13]. In the rank5 case it is argued that the vacuum expectation value (vev) of the SU S singlet coupling to the lepton doublet direction with zero vev is at most very small [14]. Consequently new vectorlike fermions decay through their superpartners [15]. In the rank6 case with or without a large intermediate scale, no general problem with such large vev arises [16], in particular no axion problem. Mixings then allow for the decays of new vectorlike fermions into standard chiral particles. It must be emphasized, however, that some calculations, such as asymmetries, etc., discussed in the following sections
F. del Aguila et al. / e * e  annihilation
are independent of the particular decay mode as long as the final states allow the initial vectorlike fermion to be tagged. (iii) Three different scenarios must be contemplated due to the neutrino mass (problem) [17]: (I) The rank6 model reduces to the standard model at very large energy but (discrete) symmetries protect some vectorlike fermions from acquiring very large masses [18, 19]. Righthanded (RH) neutrinos and extra singlets can acquire large (Majorana) masses M2/Mc through nonrenormalizable terms (M t being the intermediate mass scale and M c the compactification scale at which the nonrenormaliTable terms are generated). In this case new (vectorlike) fermions can be observed but not new gauge bosons [7]. (II) The rank6 model reduces to a rank5 model at a very large scale. Then the neutral singlets can get a large Majorana mass as before but the RH neutrinos (we call RH neutrinos those SU21x U r singlets protected by the extra gauge invariance) must be light for nonrenormalizable masses are negligible and radiative masses small, due to the corresponding loop suppression factor if supersymmetry is effectively broken around 1 TeV. No sizeable mixing with the lefthanded (LH) neutrinos should be allowed to ensure these to be almost massless. In general, further astrophysical constraints on the extra gauge boson have to be considered [20]. Witten pointed out that E 6 singlets with the proper couplings could solve the neutrino problem in superstring inspired models. However, this seems not to be the case for although such couplings can exist they are not large. Also the E 6 singlets tend to get their own masses and to decouple [21]. (III) For the rank6 model at relatively low energies the situation is similar as for the rank5 case [13]. The extra gauge symmetry protects RH neutrinos and extra singlets from getting a mass. They can only become massive at the weak scale but in order to preserve the observed neutrinos almost massless, their mixing with the SU5 singlets must essentially vanish. These singlets can eventually mix with vectorlike neutrals in SU21 doublets, but their masses cannot be large since they are of weak origin and due to their mixing with the massive doublets. Additional constraints on the Yukawa couplings follow from proton decay and FCNC. Models with a large intermediate scale must include 27 + 27 components with SU5 singlets to preserve supersymmetry [22]. Such extensions can easily be incorporated in our framework. Our concern will be the dominant signals in e+e  collisions [23]. Different cases can be distinguished and we separate them in our analysis. (i) The production and subsequent decay of new fermion pairs FF by standard Z exchange is fixed, except by the definite value of the F mass. (i') Depending on this mass, Z decay into a standard and a new fermion Z~ fF can occur, although suppressed by the small mixing angle to be measured in this process. (ii) The production of known fermions ff through a new light Z ' allows us to fix the parameters introduced by the gauge boson mass matrix. (ii') Decays of Z ' into new fermion pairs FF will exhibit the
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mixing parameters of the new fermions among themselves, which could be large. ( i i ' ) Finally, the decay of Z' into a standard and a new fermion fF will allow the measurement of the small fermion mixing parameter suppressing this process. The explicit lagrangian for the fermiongauge boson interactions, conveniently parametrized, is given in sect. 3. It follows from the E 6 gauge couplings, the gauge symmetry breaking (fixing the light gauge bosons) and the fermion mass matrices (fixing the physical fermion eigenstates). The different patterns of gauge symmetry breaking for E 6 superstring inspired models and the corresponding gauge boson parametrization were presented in ref. [12] (see also ref. [24]), and we adopt these schemes in what follows. The cross sections and asymmetries in e+e  annihilation are collected in sect. 4. In sect. 5 we present some relevant quantitative results, while sect. 6 is devoted to the conclusions.
2. Fermion mass malrices In superstring inspired E 6 models the fight fermions must be in a chiral set of fundamental representations, 27, in a vectorlike set of 27 + ~~ pairs or/and in E 6 singlets and gauginos. Here we restrict ourselves to the minimal (realistic) case of three 27's. (The extension to 27 + ~ pairs would be straightforward.) In this case the standard higgsinos must be part of the 27's. In general, higgsinos mix with the corresponding gauginos but not with the other leptons, as argued before. Then the two systems, (higgsinos + gauginos) and new vectorlike leptons, can be studied separately. Here we discuss in detail the vectorlike fermions  for the production of higgsinos and gauginos see refs. [15, 25]. The three 27 families can be divided into the standard fermions
te
!
IL
e~',
u'] u~
d' L d [ '
(2.1)
plus vectorlike copies
LE  '
L  N EJL ' N~ n'~, DLD{.
(2.2)
We use primed symbols for current eigenstates and follow the conventions of ref. [12], where their transformation properties are given. Vector and matrix notation is employed for the three families, but family indices are generally dropped. One
F. del Aguila et al. / e
+
e  annihilation
(E +',  N A ) doublet and the combination of ~tv'e,e'~)L and ( ~PE,E')L which corresponds to the higgsinos decouple, analogously for the Ne'L and n" L associated to the scalars breaking the extra Ut's. The remaining fermions can mix among each other, as long as they carry the same SUc x U1Q charges, making at this point the assignments in (2.1), (2.2) preliminary. To cover all the possibilities for production and decay (into standard fermions) of the new vectorlike states, we allow in general for two relatively light new Z's. In our plots for production, nevertheless, we concentrate only on the next boson and refer to the minimal model [11] and to two models with intermediate scales [12, 26]. This effectively amounts to assuming a large mass for the second new boson in the rank6 case. The decays of the new fermions are'assumed to proceed through small mixing with the standard particles. This describes those scenarios in which the masses of supersymmetric particles are large and/or the relevant Yukawa couplings small. Note, however, that some of our results, like the numerical examples which we give for illustration in sect. 5, are valid in general, since they are independent of the decay mode as long as this allows the new fermion to be tagged. The full analysis including supersymmetric particles will be discussed in a forthcoming publication. The mass matrices we consider can be written as
n~. muku
,
d~, D~,, (2.3)
E[ e~
PkL C ~¢,
N~L~ N,'L neE
Eh, 0 0
0 0
M~* M~*
m~ 0
0 m~
M~ +
M~ +
0
m~
m~
m~T
0
m~T
M1~
M:~
0
m[ T
m~T
M2~T
M3"
PePL
PEL
NEL
NiL
n'L
(2.4)
All entries are 3 x 3 blocks, their elements carrying family indices (with higgsino
F. del Aguila et al. / e + e  annihilation
states corresponding to index 3). (In what follows, we also use the symbol m for individual masses, depending on the context.) ..¢t'u,d,e are Dirac mass matrices; ~krt = C ( ~ L ) x [ L ( R ) = ½(l(q:)ys) ]. The neutrino mass matrix has Dirac and Majorana components. The resulting neutral spectrum will always contain to a very good approximation Dirac neutrinos in heavy doublets
VEL=V'e,EL,
' V~ = NEE.
(2.5)
Their LH parts will be mainly a combination of V~'Land v ~ and their RH parts will primarily correspond to N ~ . The remaining neutrinos will generally be Majorana particles. For the Dirac ones, we will alternate between Majorana and Dirac notations to simplify the analysis. M entries are SU2/ × U ( invariant and produce new vectorlike fermion masses. Since these have not been observed so far, they must be heavier than the known light chiral fermion masses, which are produced by the m entries violating SU2t × U (. The m entries will be treated as small perturbations. The entries and mY3(4) are zero if no vev along the lepton doublet direction { ~ i ) ( ~ ) ) develops. Keeping them nonzero, the diagonalization would be equally easy, and although the detailed form of the mixing parameters would simplify, no apparent simplification for our firstorder analysis follows. We will characterize mass eigenstates omiting the primes but using the same notation as before (although they are mixtures); except for the Dirac neutrinos, for which we will use vE to denote both chiralities (see (2.5)). The zeros in the neutrino matrix ~¢t', are due to the absence of the corresponding tree level contributions 273 in E 6 models. In fact, also the large M1".2.3 entries get zero contributions from E 6 tree level renormalizable terms. We allow, however, for nonrenormalizable and/or radiative terms to give nonzero contributions. In the phenomenological approach we follow, Majorana entries are thus assumed to be as general and as large as possible, since they are not protected by the symmetries of the SM. When only the standard model survives at low energy all M entries can be large (case (I) in the introduction). For intermediate scale models with one extra gauge boson (II), the broken direction is conventionally assigned to n'. As explained above, the M 1,2 ~ entries are then mainly zero, as are the small m entries in the first row and column, after redefining V~.Ein such a way that the combinations with zero M e entries correspond to the standard neutrinos. (m~ must be negligible unless M~ = 0 [26]. m~ could be larger because its contribution to the neutrino mass is suppressed (m'22/M~).) The resulting spectrum will be heavy Majorana singlets and Dirac doublets and almost massless LH neutrinos, with RH neutrinos not necessarily massless if they mix with the heavy doublets (rn~ or m~ nonzero). For rank5 and 6 models without an intermediate scale (III), M 3" is also negligible. In this case, the only heavy neutrals are those in SU21doublets. The SU5 singlets may then get masses by mixing, and they may become an important decay product. In
mr'e2(1)
F. del Aguila et al. / e + e  annihilation
7
the rank5 case M~  0 (~]~[ep) = 0), rod2 'e  m 3~= 0 ( ~ ' ) = 0) and then m~ ~ 0 must be also required. All the former mass matrices can be diagonalized in a systematic expansion in m/M. Setting first m = 0, the chiral fermions remain massless (and perhaps some vectorlike singlets) while the resulting nonzero vectorlike fermion masses are exact to order (re~M) 2. Since SU / × U ( is preserved in this approximation, the standard electroweak currents remain diagonal in the mass eigenstate basis of the vectorlike fermions, whereas the new U 1 currents become non diagonal, measuring the mixing among massive vectorlike fermions (which is of order one). In a second step we diagonalize the submatrices of order m involving only the (until then) massless (chiral) fields. The resulting masses and fermion fields will be of the same order as the observed ones. Up to this point the mixing between new and old fermions vanishes. Finally, in a third step, we introduce the mixing submatrices of order m and diagonalize the full matrices. This procedure leads to mixing between chiral and vectorlike fermions of order m/M. If there are light SU5 singlets, their mixing with L H neutrinos must be forbidden. The states corresponding to higgsinos decouple in the matrices, and they get their mass from mixing with gauginos. They are not treated in this analysis, and the related matrix elements can formally be set to zero in the present context.
3. Mass eigenstates and iagrangians The (E6) neutral lagrangian for fermion pair production in e+e  annihilation can be written as ff'Nc = eJ,,C,~Z¢,
(3.1)
where J , are the four fermionic currents and Za the three (or four) physical gauge bosons (Z 1 = A, Z 2 = Z, Z3,... = new gauge bosons). The numbers Caa (see table 1 for the case of one extra gauge boson) contain the coupling constants and parametrize the gauge symmetry breaking; they are discussed in detail in ref. [12]. In the current eigenstate basis, Ja~ _ .T,,~'ro~,,~,t,,a ~'L ~a I ~"L ,
(3.2)
where the matrices T. a r e T3L , ~3fi Y, ~3~g', Y" in table 2 of ref. [12] and a = 1,2 . . . . . 27 corresponding to the fermions in the 27dimensional fundamental representation of E 6. (Note the slight change in the normalization of the J2,3 currents and C,a, for later convenience.) Diagonali7ing the mass matrices in eqs. (2.3), (2.4) these currents can be written in the mass eigenstate basis. They are given in tables 2a and b for quarks and leptons. These expressions are derived in appendix A. We have used the abbrevia
F. del Agtala et al. / e + e  annihilation TABLE 1
Neutral current couplings Cat for the photon, Co2 for the Z, and C~3 for a new (Et) gauge boson Cll 1
C2t ~ 1
Cw C12 ~   c 3 Sw Cw CI3 =   S 3 Sw
\
cw t
\
C41= 0
C31 = 0
c32 m
cw 1
S3
[   + ~'d q
1
(733 =    C 3

cw ~ X
+ ~ S l2
)
C4 2 z
c,3
C2

 
cw
$3
 
\
q ( si)  cos O,(sinOi), ~ = g2/g3"
tions Cdlt = COS20d,t, Sd.e= sin20d, t, C2~8= COS2fl and s2j8 = sin2fl. A is a unitary 3 x 3 matrix, and ~ are small mixing matrices of order m / M . Mixing parameters proportional to light neutrino masses are so small that they can be neglected in the present context. In what follows we concentrate on the case of one extra gauge boson Z ' besides the photon and the standard Z. The mass mixing angle 03 in the boson sector (see table 1) is experimentally known to be small. Neglecting this angle for the moment, the neutral current lagrangian is given by e
•~NC = eJ~.~A~, +
SwC w
e
J~Z~ +  
'
Cw J ~ , Z ~ , ,
(3.3)
where
Jx + J2, J z = J1  S w2 J E M ,
(3.4) e is the electric charge, Cw(S,,)=cosO,,(sinOw) with 0,, the Weinberg angle. ¢l,2($t,2)=C, OS01,2(sinOi,2) with 01,2 the new mixing angles in the gauge boson sector and X = gz/$3 a ratio of coupling constants whose value is near 1. Only Jx,: enter in JV.M.Z,whereas Jz, is built up from J2.3.4 Jm~ is the usual electromagnetic current and Jz is, for vanishing mixing 17= 0, the standard Z current. The more general case of a nonzero mixing angle 03 is simply recovered by substituting Jz } cos 03 Jz  sin 03 Jz' and Jz, } sin 03 Jz + cos 03 Jz'.
F. del Aguila et al. / e ’ e  annihilation
9
TABLE2a Neutral currents of the quark fields in the mass eigenstate basis Diagonal u currents Ju”,; B,~“~Ypu,; Ja; = BRT,URy,,uR; Diagonal d currents
J:, =  $&_y,d,, J:, w 2 J&t J;=O, Jp, $J$, Nondiagonal d currents Ji”,;  : GYpT dDL + ho.,
Jp, *
J2R, = J$ = 4
The currents are diagonal in the family indices or the mixing matrix is shown explicitly. TABLE2b Neutral currents of the charged and neutrallepton fieIds in the mass eigenstate basis Diagonal e currents
J1”,=  %..v,,% + &Y$&.), J$= :J1”,, Jp, =  ;&,,ER, J$ =  %(*,y,e,  2%qP,), Nondiagonal e currents JI”,==J$=J+O, Jp, = $,q$E,
J& =  &y,s~E, JR=LJR
+ h.c,,
JR= =P
6
IP
2 4
+ h.c. JR=0 4fe
(1, i)(Ru & Y&)are the Majorana components of YR in eq. (2.5) which, apart from small Majorana admixtures, is a Dirac field. The small mixings involving standard neutrinos are suppressed (& = ny2= 0). gP = fi(rl&.4,  i%(6)), gkl) = fi(n;&
 4&
10
F. del A g u i l a et al. / e + e  annihilation
TABLE 3 Charged currents of the quark and lepton fields in the mass eigenstate basis Quark currents J~ = ~L'/~,KdL + ~LY~,Kr/dDL Lepton currents 

4

t+
t+
J/~ = ~eLy~,AeL + ~EytaE  VERY~TIe+eR  Ney~(g~_ L + gRR)E  neY~(gL L + gR R)E
K is the CabibboKobayashiMaskawa matrix, g~R are introduced in table 2b.
As long as the mixing effects are small, the neutral current lagrangian dominates the production of fermion pairs in e +e annihilation. However, for the decay of the new fermions the charged current lagrangian is needed. Assuming that the only light charged gauge bosons are the standard ones, the charged current lagrangian can be written in terms of the current eigenstates as e
Lacc= ~ W ~ + J " ¢2sw
+ h.c.,
(3.5)
where PC
!
(3.6)
and summation over the family degrees of freedom is understood. Using appendix A, the charged currents are denoted in the mass eigenstate basis in table 3 for quarks and leptons. All necessary ingredients are now collected to calculate production cross sections in e +e collisions and decay distributions of the new fermions.
4. P r o d u c t i o n c r o s s
sections and d e c a y s
4.1. e+e  P R O D U C T I O N OF FERMIONS
Charged vectorlike quarks D and charged vectorlike leptons E plus their neutral isospin partners t, E are pairproduced in e+e  annihilation through schannel boson exchange of the photon, Z and Z' (plus eventually additional new Z bosons) e+e 
e+e 
e+e
, DD, y,Z,Z'
, E+E  ,
T,Z,Z'
z,z' ' PEtE .
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11
e * e  annihilation
Additional tchannel Z, Z' or W exchanges (see e.g. ref. [27]) in the lepton case are quadratically suppressed in the eE mixing parameter and can be safely neglected. To a very good approximation, VE can be treated as a Dirac particle in the production process. A small Majorana admixture from the isoscalar neutral states can, however, lead to exciting equalsign dilepton final states e  e  + jets. The total cross section for Dirac fermion pair production in e+e  annihilation qTOt2
o(e + e  ~ F F ) = N¢~fl [(1 + ~fl2)Q, + (1  fl/)Q2] 4~r~t 2
for ~ >> 2m F ,
Nc~sQ1
(4.1)
can be expressed in terms of generalized charges [28]
Q1 = ¼(IQLLI2 + IQRRI2 + IQLRI2 + IQRLI2), (4.2)
Q2 = ½Re(QLLQ~L + QRRQ~R), where Q:FQ~e Q,VQje s s Qij = eFee + swcw22 s  m 2 + i m z F z + c,2 s m2 ' + imz,Fz,,
(4.3)
with i , j = L,R. ef and Qf = i f Swe 2 t are the conventional electroweak charges while the Q'f denote the new U{ charges; for the sake of convenience they are collected for the leading diagonal elements in table 4, as extracted from the currents in table 2. N c = 1 or 3 is the color factor, fl = (1  4 m E / s ) 1/2 is the velocity of F in the final state (not to be confused with the mixing angle introduced in the last section, always appearing in the sine or cosine form). The angular distribution of F is determined by two parameters [29], d o / d c o s 0  1 + a cos20 + 28cos 0. The a parameter fl2Q 1 fl2)Q 2
a = QI + (I 
(4.4)
vanishes at threshold, leading to an isotropic angular distribution, and approaches 1 asymptotically. The 6 parameter measures the forwardbackward asymmetry 8 = AFB(1 + ~a) which is given by BQ3
AFB= (1 + ~fl2)Q 1 + (1  f l 2 ) Q 2
3 Q3
4 QX
for V~ >> 2mF,
(4.5)
12
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TABLE4 Electroweak and new charges (no mixing assumed) for fermions in the (27dimensional) fundamental representation of E 6
er
minimal Q'
/3( = 7"1)
UL
2
! 2
u R
? 3 1
0
~
1
1
3
dL
~
i
_!
3
~
dR
~
o
~
D L
~ ',
0 o
'
eL e~
1 1
1
2 0
1_
,,~R  N~,~
EC
o ~
~ ~
~,.
o
ncL
0
D~
~
intermediate scale models T2 T3 T4 _t 6
1.
0
_2 3 1
_1. 3 1
0
6
3
0
l
~
1
~
~!.
~
~ ~
0
~ 1.
t
t
l
~ 1
t
Z ~
l
i 0
~ ~
 2'. I
~ ,~
o ~
o
~
o
~,
0
5
0
~
Notation: electric charge
Z charge Z ' charge (minimal model) Z ' charge (intermediate scale) Z ' charge (general) ZZ' mass mixing
6 3~
~
ef 2 13  Swe f
Q ' ,   T 3 ( h = l ; c2=1, s 2  0 ) Q'  slclfh
 ~,1)T 2
+ ( h  z c 2 + hs21)( c2T3 + s 2 d r , )
Q , cos 03 Q  sin 03 Q', Q' * sin 03 Q + cos 03 Q'
where
Q3= ¼(IQu.I 2÷ IQ~I 2  IQRLI2  IQLRI2) •
(4.6)
A c o m p a c t r e p r e s e n t a t i o n for the p o l a r i z a t i o n v e c t o r of t h e final state f e r m i o n c a n be found by introducing mixed generalized charges*
Q w . = ½(QLL + Qpa.),
QAL = X2(QLL Q R L ) ,
QVR = ½(QLR + QRR),
QAR = ~(QLR  QR.R),
* We thank J.H. Klthn for pointing out to us this compact form.
(4.7)
E d e l A g u i l a et al. / e + e  annihilation
13
and
2Dv= IQvLI2+ IQvRI2,
2 E v = IQvLI 2  IQval 2,
2DA= IQALI2+ IQ,~I 2,
2EA= IQALI2  IQARI2,
2DvA = QALQ~:L+ QARQ~rR,
2EvA = QALQ~L Q,~Q~,R.
(4.8)
Defining, as usual, the components of the polarization vector along the fli#t direction (PL), perpendicular to this direction but in the production plane (P2), and normal to the production plane (PN), the following representation can be derived [30] (see also ref. [31]), PL
=

2[Re DVAB(1 + cos20) + ( Ev + EArl 2)cos 0 ]/al r,
mF
P±  ~sss4 sin 0(Re DrAft cos 0 + Ev)/JV', mF
(4.9)
PN = ~4fl Sin0 Im EvA/A/" , Vs where the normalization factor M/= (D v + DA)fl2(1 + cos20) + 2Dv( 1 _ f12) + 4ReEvAflCOS 0
(4.10)
coincides with the differential cross section given earlier. Arguments have been presented in the previous sections, suggesting that the isoscalar neutrals N e and ne could be heavy Majorana particles (generically called N in this paragraph). The production cross sections for these particles. e+e~NN Z'
are built up by Z' schannel exchange. Small admixtures from isodoublet neutral states can be safely neglected. The same is true for tchannel W exchange. We find for Majorana fermions 2~ra2 3 ^ o " ~ s ~ Q1,
(4.11)
where
01 = ' (lO l + IO l ), o t N o~.j te ~
j =
cw2
S s
m 2,
+ imz,rz,
(j = L, R).
(4.12)
14
F d e l A g u i l a et al.
/
e +e  annihilation
Unlike the Dirac case, the cross section for Majorana fermion production is strongly suppressed (B 3) near threshold because two identical fermions cannot be produced in a symmetric 3Sx state. The angular distribution is independent of energy, do
1 + cos20
dcos 0
(4.13)
(and of course no forwardbackward asymmetry occurs). The particles come out unpolarized. In addition to pair production, exotic fermions can also be created in a variety of mixed states, comprising light chiral fermions too. Since the production proceeds through mixing, the cross section will in general be small. Apart from the J4 current that couples to Z' only proportionally to s 2, the dominant mixings occur between singlets and doublets, the corresponding mixing parameter being linear in the mass ratios m / M . The prominent reactions to be analyzed are therefore e+e 
,c+E  ' Z,Z'
e+e  z,z,~ ~EN , e+e~;}N,n~, e+e  ~
Z,Z
dD
plus their Cconjugate counterparts. The reaction e+e  ~ e+E  can be mediated by Z and Z' exchange in the s and tchafinel. The corresponding cross section is then given by do(e+e  . e+E ) d cos OE
=
2S
1
1+
cos20 )Q1 + (1 + ¢ )cosOEQ3]
(4.14) where the charges QL3, defined in the same way as in eqs. (4.2), (4.6), are built up by
2 2 Qo = s~cw t s  m2z + i m z r z
Q2 + rn~
tE ,e 7/:=Q~" ( s 2 + c. ~sm2z,+imz,rz ,
s Q2
+m~z,
)
•
(4.15)
F. del Aguila et al. / e + e  annihilation
15
The mixing parameters can be read off the current tables,
n~ = o, IjE=I e* i~1
~t~==
,
(~ +x#j
I s /~;vl ,s~ , ,
_1
1+(_~
As~)c2]~**.
(4.16)
is the (negative square of the) momentum transfer. Note that for s >> m~ tchannel Z exchange gives the dominant contribution in the forward direction. This channel is absent for # and • family members. The reactions e+e , FEN can only proceed through Z,Z' schannel exchange since W exchange is highly suppressed by small mixing angles. For arbitrary masses their cross sections follow from (N  N, or n,) Q2= ~(sm~X1COS0E)
do(e+e~EN) __ __ era2 hl/2 [( 1 + flNfl,cos20N)01 + 4 m NQm ,2 dcos0 N 2s s [ s ] + (,8~ + K)cos 0NQ3[,
(4.17)
with Q1.2.3 derived from ¢
le
TliQ; s 71iQj s Q i j = swc 2 w 2 s  m 2 + i m z F z +   cTw s  m2, + i m z , F z , ,
(4.18)
• N , ( n _e gl(') ) * L
=
..¢,.., = _g~,',
x [  c ~ "g ~"" + ~ ( ¢ ~ g ~ ' " (+)~2ag~'"" + s2B,;'"I]
(4.19) and A~ = [s  (m N + m,)2][s  (m N  m,)2].
16
F. del Aguila et al. / e + e  annihilation
Finally, the production cross section for a pair of nonidentical Majorana particles, e+e  * Nene, is a slight generalization of (4.11), do ~ra2 )kl/2 1 + flNflnCOS2 0 dcos0 = 2T s
4mNm n ) s
(4.20) Q1,
with (~1 given by eq. (4.12) but with Q'~ replaced by the mixing parameter between the two neutral isoscalars 71' N =
(~
c2 + Xs2 ) s 2 ~ 3 ½s2fl.
(4.21)
Because all Majorana currents are purely axial for real mixing parameters, no forwardbackward asymmetry or polarization effects can occur. The cross section for asymmetric quark pair production e+e  * dD has the same form as (4.17) apart from the color factor and the mixing parameters
n~=
 ~',
~  0,
(4.22) Besides, m r~ is to be replaced by m u, whereas m,, which is to be replaced by m e, can be set equal to zero. 4.2. Z'DECAY WIDTHS
To evaluate the e+e  cross sections, the total widths of Z and Z' are needed. Assuming that the new exotic channels are not yet open at the Z, its width is the same as in the standard model. Pair production of old and new fermions sets the scale for the width of the new Z'.
Dirac fermions amz, 1"3/3 2 F(Z'"FF)Nc12c2B[ 2 (QLF +Q~F) 2 .
.
.
.
.2(e:
+
]

ea~)2 , (4.23)
Majorana fermions amz' 3 t N r(z,, N ~ ) = 7:r_~ ~ (Q) oc;,
2
•
(4.24)
17
F. del Aguila et al. / e + e  annihilation
The cross section for the production of mixed final states as e+E , dD or Nen~ is, in general, quadratically suppressed by the mixing parameter, implying low rates. These production mechanisms can presumably be observed only in resonance decays. Using the mixing parameters defined in eqs. (4.16), (4.19) and (4.22), we find r(z
Ff)
'
amz
F 2 + Idl')(1
_
J, 2
OSwC w °tmz'
1. p2~
F ( Z ' =)Ff') = N;v:r_~ (In~.q 2 + l,)a~l~)(1  ~,~,:)~(I + i/~F ),
OCw
(4.25)
where #~) = mF/m~ ), etc. When both fermions in the final state are massive, the expressions must be generaliTed to, e.g.,
r(Z'*N~E)  "mz'X
+ I,,¢1')
2
6c~ × [ 1  ½(/~2 + # 2 , )  ½ ( ~ 2  # 2 , ) 2] + 6Re~l~N°'qkN/.tN#,), (4.26) with X~  [1  (/~N /x,)2][ 1  (#N + #,)2]. Correspondingly for Majorana particles e t m z , X 1/2
F ( Z ' ~ Nene) =
3c~
())'N)2[1  ½(#~ + #~)  ½(#~ #~)2 3#N/~n] " (4.27)
4.3. D E C A Y S O F T H E N E W F E R M I O N S
New fermions can decay into a plethora of channels. In the minimal model without mixing between new and light chiral fermions, their decays are mediated by superpartners, real or virtual depending on their masses, and coupled via Yukawa interactions [15]. More general versions, however, allow for decays mediated by gauge bosons due to mixing of the heavy and light fermions [7]. These channels will be discussed in the present subsection. Apart from presumably small contributions due to the /4 current, the dominant mixing occurs between isosinglets and doublets while mixing effects between singlets and singlets or doublets and doublets are quadratically small. The dominant decay channels are then D ) u + W or d + Z for L the new isosealar quark and E   ~ e  + Z , vE ~ e  + w + and N,v E + Z or E + W for the new leptons, where L / R denotes the chirality of the currents involved. Depending on their masses and mixings the new isodoublets can also
18
F. d e l
A g u i l a et al. /
e + e  annihilation
decay into the new neutral isosinglets, v E ~ N + Z, bosons may be either virtual or real. We shall present light fermions (for a virtual gauge boson) and into (massive) fermion. We first discuss lepton decays from easily deduced.
E  ~ N + W. The gauge the decay widths into three a real gauge boson and a which quark decays can be
4.3.1. (i) E  ~ e  + f t .
This decay proceeds through a righthanded Z current and a mixture of right and lefthanded Z' currents. The decay distributions then follow from a2mE d F = N~~~3 Ojklqjkl 2 d ~ ,
f#e:
(4.28)
where i
x
Pjk =
0 
x
)0 
x (1 x:)(a ! E Qfll,
for j = k for j * k '
m~
c~,f_,E ~ j ~lk
e=+
fork=R,L,
(4.29)
m2
qj* = CwSw22 m23 _ m 2 + i m z r z + Tc~ m23 _ m2z, + i m z ' F z '
(4.30)
The indices 1, 2, 3 correspond to e, f, f respectively; x~ are the particle energies in units of half the E  mass mE; m223= m2(1  xl), cycl., denote the invariant 23 .... mass squared; s2, etc. are the projections of the spin vector s onto the flight direction of particle 2,... ; d~ = dI21 dq02 dx 1 dx 2 is the phase space element with hi(0 ~, ~ ) , i = 1,2, the flight directions of particles 1 and 2. While the Q's are the standard charges defined before, the mixing parameters are given in eqs. (4.16). f = e: Dirac statistics of the two electrons in the final state requires the symmetrization of the propagators in the generalized charges q~k for j = k, i.e. mE/(m232 2 _ m2v + i m v F v ) is to be replaced by the sum me/(m232 2 _ m2v + imvFv) + m 2 / ( r n 2 3  m2v + i m v r v ) , V = Z,Z'. (ii) v L.~ e  + f 2 f 3 . This decay matrix element involves a fighthanded current in the coupling between ~'E and e  and the ordinary charged lefthanded lightfermion current of f2 and f3. The decay width is therefore
t~2m, dr=
Inel2m~
Nc256~ras ~ (m~3 _ m~v) 2 +
(mwrw)2X2(1
 x2)(1  s 2 ) d ~
(4.31)
with definitions analogous to (4.28)(4.30). (iii) D   , d + f t . T h e partial width for the D quark decay is described by the same general formula as for E decays, eq. (4.28). (In the exceptional case D * ddd, however, the interference term between the symmetrized propagators does not carry the colour factor.) Special attention has to be paid to the final state D ~ du~ that is
19
F. del Aguda et al. / e +e  annihilation
generated by a superposition of neutral and charged boson exchange, D* d + u~ and D, u + dfi, respectively. This case is covered by substituting for the corresponding generalized leftleft neutralcurrent charge 2
*
W
IqLLI 2 ~ [qLL[ 2 + [qLWLI2 + 3Re qLLqLL,
(Kt/d)UD qWL=
2S~
m2
m~3 _ m2 + imwF w .
(4.32)
If the masses of the new fermions are sufficiently large, they decay directly into light standardmodel fermions plus gauge bosons.
4.3.2. (i) Neutralcurrent decays of new Dirac fermions into zero mass fermions occur at a rate r(E, eZ)=
°t E2m3(~zz ] 1 + 2m!/rn~ 8S2~wlnRI 1  m2/2(m2
1,
2, 1  m!1'2[am~] I + 2S1m~ ~ (4.33) F ( E  ~ e  Z ' )  ~c~ ([~_E[ 2 + 171~E[2~m3(:m The same expressions are found for D quark decays D , d + Z, Z ' with the obvious changes in the mixing parameters. (ii) Partial widths of charged current decays are given by
r(,E
eW)
o
~/L5_2Ino2 [ _,.'ST 1 xos w
mw
22
2)
1 [ 1 + 2 .mw _ST_2
m2,] 1
m,
(4.34)
and similarly for D ~ u + W. The distributions for polarized particle decays follow from the previous equations by substituting for the last factor
1+
2..v 2,,,,: m2*l+ m~+ese 1  2 my ,
e = + for R / L ,
(4.35)
V = Z , Z ' , W ; F = E, rE; whereas se stands as before for the spinvector projection onto the flight direction of the electron. (iii) Majorana particles might either decay into heavy Dirac fermions N ~ VE + Z t') and N ~ E + W, or through mixing among each other, N ~ n + Z'. The
20
F. del Aguila et aL / e ÷ e  annihilation
corresponding widths are ~x'/~22 m~ (([r/~12 + 1,1~12)[(1/t2)2 + ~ ( 1 + 2 _ 2/~)]
8swcw m~
P"
 12 Re TI~'~I~/~,# 2 } m 3
F(N ~
{(igLl+
EW)= 16s~m~v
' _
+
+

r ( N ~ nZ')
= "X~2 m~ ( rl'~ )2[(1  / ~ ) 2 4c~ m~.
12 Re g~.gR#EI~2Z),
+ bt2z' (1 + tt~  2bt~:.) + 6#n# ~.], (4.36)
where the notation is the same as in earlier parts of this section. In particular, #, stands for the ratio of the mass m~ to the N mass; g in the second equation corresponds to N e, g' must be used for ne, and N(n) in the third equation is the heavier (lighter) of N,,n,. The width for the Z' decay N * ~,EZ' is given by the first equation except for the factor s~, 2 and obvious redefinitions of the mixing parameters. Finally, if N happens to be light the previous processes proceed backwards, VE~ NZ, E  o NW, with the same expressions for the widths but with the appropriate redefinitions of the mass and mixing parameters.
5. Cross sections and signatures We discuss two different issues for the e+e  machines, the interchange of a new Z boson and the production of new fermions. To measure the parameters associated with the new Z' boson, it is convenient to look at standard fermion pair production, in particular the cross section and the forwardbackward asymmetry in e +e , # ÷p. Their dependence on the mixing angles and on h, a ratio of the coupling constants, is illustrated in figs. 1 and 2 for m z,  180 GeV and 500 GeV. The parameters of the models we consider are collected in table 5. Although m z, = 180 GeV is not excluded by production experiments, limits on neutral currents can require, to one standard deviation, larger values in specific models. This is the case, in particular, for model (b) in table 5 [12]. Nevertheless, since we deal with production experiments, we also include this case in the figures.
F. d e / A g u i l a [
IOO]
:
T
w
tR :
w
I
I
I
I
I
t
,
i
]
1
1
et al. i
1
/
e +e 
21
annihilation I
i
i
,
w
.
I
01=0
e"e~

ii;i,,
 
min.
,',
.....
model ( b )
i'.
......
model(c)
'~'
II
model
m=.=500 i
~
i
"
I
i
I
I
1
I
~
I
T
i~
I
I
I
[
model
standard
=:~
(c)
lOgO
A=I
p.* p.
I
i
J
i
i
GeV
I
I
rain. model ~, =1.2
P ;go
IQ
,, o i,
 
i',
.......
o
1
,i~ ~
..........
* T~/~
'
O~=  ' ~ 1 2
. . . . . .
'KI4
I
i2 (b) I
I
60
t 1
I
50
I
I
tO0
I
I
I
)2C
I
140
I
l
Ibl'l
,
,
2~0
i
!
200
I
I
220
E (Ge~V)
I
i
240
i I

:
.=
 . ~ ......
= ,
. ....
(d) i
330
i
!
500
i
i
i
i
[
tO00
2000
E(OeVI
Fig. 1. Ratio R of the cross section o(e+e  ~ / ~ + / ~  ) to the pointlike QED cross section in models with an additional Z' as a function of the CM energy: (a,c) comparison of the models specified in table 5; Co, d) R in the minimal model for ~  1.2 and various mixing angles 0 i.
Our main concern, however, will be the production of new fermions F. The FF couplings to the standard Z are fixed and these decay modes have been considered in ref. [5]. We therefore discuss the striking features of FF production in the e+e continuum and through a new Z'.* 5.1. NEW Q U A R K S
We only consider new vectorlike down quarks in 27's. The production cross section for new D quarks, compared with the standard d quarks, is shown in fig. 3 for D quark masses of 60 and 100 GeV. * While this manuscript was being written, we received a copy of ref. [32], where some aspects of the material presented here are also discussed.
F. del Aguila e t a / . / e + e  annihilation
22 "]
r /
~3 [
it,
A+B e++e~
Ii
=
;
1
~  i
i
,
"rI
,
i
1t 1
   ~
I~'P+"
.... ''
"'
/ a

I
i b,,
I
',
'
/
z
'.
'
.
.
.
.
.
.
;
' ,
0~=0
X=I
 
min
.
model
model
tb}
m o d e l {c)
. . . .
"
,
j
tl
stondord
'
model
]
/ '
3~;
,
i. ,.~l
i
i
i
I
,
l
i
i~
,
l
i
l
i
i
[
t
l
i
i
I
i
i
l
i
I
i
i
+
i
m,. ~ 500
tel
[ ....
' .................
GeV
(
.......
I
1
' !
/
",.~
rain. m o d e l
.+
]
X=1.2
i

B;=+xl2
 
/
. . . . . . .
i
'rtl4
.......
13
]
.........
+ ~I/..
]
I
i
td) QI
i I:O
+10
t
+
1
It~t~
i
i
120
i 140
!
i lEO
i
i
i
1810
!
i
200
i 220
r
i
.
,
. . . .
, ....
~ ....
3100
2410
!
500
EIGeV)
.
~ ~ , . ~
~50
~.!
. . . . . . . . .
11100
1~+00
E(OeV)
Fig. 2. ~  forwardbackward asymmetry, Ave , for the models in fig. 1.
TABLE 5 Values of the parameters discussed in the text (0 x, 02, X, B) fixing the models 01
(c 2 , s 2 )
~
s~8
~ = 1: no 0 t dependence
(a)
1 ~r) 1 {  ~r,
(1,0)
[1,1.2]

minimal model [11]
(b)
1 [  ~r, ~I )
3 S (~/s' V~)
[1,1.2]
0
intermediate scale Higgs breaking [121
(c)
[ _ 1 ,:=, i~r)
(~f~~, ' ~ s)
[1,1.2]
1
intermediate scale Higgs breaking [26]
In all cases #a. t = f/. For ),  1 the popular models referred in the last column are recovered.
i
210100
del Aguila et aL
F.
/
e +e 
annihilation
23
"
R
_t'l
....
;'
i~
I
e'e
~
da
,1~'.',
!
.I
; II
81 = 0
X= 1
i,,L,
 
minmodel
t
4
', . . . . . . . .
li I,',
m o d e l ( b}
...... H
~l tcl
stondord
//i:": •     ~    ~ _ . _ , ;,
' ~.'_~
. . . . . . . . . . . .
! rn~.: 1 8 0 GeV
la)
I
.........
,__
J
i
i
i
i
i
i
ee~ D[) m o : 6 0 OeV
i. I
i
:
I
1
m , = 5 0 0 GeV
L
'~ '~,
,~i
{el i
4
eoe~DD m 0 : lOOGeV
i
.:
..
i
i t
i
~i
',
'! '.1
~', ~ )/ 'l ,I
,i
,:,'! f
tin
'lll
12N
Iq3
_t~. 1hr..
I
(dl
_ _ _ _ ; _ j . _ _ :. ; ~ i ~ , ~ _ l _ _ , _ _ ~ t.1
,, I
 .:..::...::.~
/
:~o
•.

 ",'/!
~~..i 2ao
22e
2fIN
;,41"
5AO
EtGeV}
Fig.
~'
• r,
2ql
E (GeV}
3. T h e r a t i o R f o r l i g h t d q u a r k p r o d u c t i o n
and for the production
of new (E6) D quarks.
In the minimal model D quarks decay into quarks and real or virtual sparticles with a lifetime set by an (unknown) Yukawa coupling constant [15]. In more general schemes these quarks can decay through mixing with standard quarks. Let us first assume that Z' (and possibly Z ' ) is heavy enough to be neglected in D decays. Then the charged to neutral decay ratio is fixed (up to kinematical mass corrections)
£(D, dZ) F ( D * uW)
1 ( m w / 2 ~ _1 2e 2 k m z j 2
(5.1)
using from the tables D * dZ
D  , uW
1
T/d
¢ cw Kn d
(5.2)
24
F. del Ag,uila et aL / e e +
AF8
s
l
annihilation

e°ed
[
d
1.
o.ed~ ".
,'"'""
"'"~
~
,"
',,,~
/ •
[
• s ! t I c t s

I
" " " "
(o) i
i
I__~L_
,
,
,
,
.=
,.
,
J .1.I ,
i
I
I
I
,
.
;
,
I
i
,
i
II
to)
~ , 1
model b)
"
I
I..._L,.~I~._J_~J._..J__
rain.model
 
'
,, I/ '
/
mv=lSOGeV
1
' ,1 t
 7=
mr.= , , [ ....
 . . . . , . .
e*e~DD
ee
ma =GOGeV
m
I , i,! .,
....
. I
•
,
. i . ~ i
i
•
,
• t
•1 ,.I
stondord model
500GeV . . . . . . . . . . . . . . .
i
D15
o = I00
GeV
b 4 2 ,1 2 • =
\
E, I
I
S
0
(b) J
GO
(d) i
i
53
i
1 lOO
t
,
120
i
r
140
~
, 160
i
, 180
i
,
200
,
220
[
,
240
....
i ' .... 300
,.,,,f
i
5o0
,
r
,J
,.,
75r:
EIGeV)
Fig. 4. Forwardbackward asymmetry for d and D quarks. (For D quarks A v e   0 model (b).)
. . . . .
lO00
~ . j
Ibr~o
E[ GeVl
in the
SM and in
and the CabibboKobayashiMaskawa matrix elements K = 13 ×3 Thus the different branching ratios into three fermions are essentially given by the corresponding W, Z* ff branching ratios. The large fraction of D  * de+e/D* ue~ e has been exploited in ref. [33] to develop search strategies for these new states in e+e annihilation. However, also Z' decays could be very important. If the mixing parameters ~d mediating W and Z decays are small, these decay modes could be suppressed relative to Z' decays, because Z' couples to the current ,/4 with a Dd transition matrix element proportional to s2s d that is not necessarily very small. The D quarks decay mainly into three jets, but the semileptonic decays into jet + ev and jet + e+e  are nonnegligible and indeed very interesting. The semileptonic CC decays allow for a direct measurement of the DD forwardbackward asymmetry. This is shown in fig. 4 for the same set of parameters as in the previous figures. Since the new quarks are vectorlike, the asymmetry is zero except for production through the new Z'.
2o0 i'
F. del A g u i l a et al. / e + e  annihilation
25
5.2. NEW LEPTONS
Let us first discuss the isodoublet leptons. The main constraint resulting from their vectorlike character is the mass degeneracy of the neutrals. They form a Dirac neutrino v E almost degenerate with E. The total cross sections for e+e  * E+E and PEtE are shown in fig. 5. The decays of these particles in the lightheavy mixing scheme are characteristic and very exciting. The new charged vectorlike leptons always decay through righthanded neutral currents into a standard charged lepton. In this section we assume that the new neutral isosinglets are not important decay products, for they are too heavy or their mixing small. If Z dominates, E  * e  + Z R
/ [~ e+e , .... jet jet
and the branching ratios are determined by the Z decays. These particles can easily be fully reconstructed in the lepton decay modes. The new neutral heavy lepton v E decays only through (righthanded!) charged currents into d+ W, rE* e+ R
W +
/ ~~ v,e +..... jet jet.
The signatures of both these new vectorlike leptons are characteristically different from those of the charged as well as neutral members of a fourth family. For small enough mixing parameters, the lifetimes of these particles could be long enough to be measurable directly. These clear signatures allow for easy identification and forwardbackward asymmetry measurements. In fig. 6 we plot these asymmetries for both charged and neutral leptons. Due to their vectorlike character the asymmetries are nonvanishing for Z' exchange only. Note that a small Majorana admixture to vE could give rise to spectacular signals like e + e  ~ e  e  W + W + ~ e e +4jets. Nondiagonal Z or Z' decays like Z~eE
÷ [~ e+ee+,...
could lead to exciting events if m e < mz4z,) Even for small mixing parameters ~/~ such a search appears possible given the large number of Z decays expected in SLC and LEP.
del Aguila et a L
F.
26
/
e * e 
IE~O
annihilation
i R
e~e  __E°EmE = 60 OeV
I00
i
, ~
1
i
B =0
X=l
1
...... ,
 .....
model (b) model Jc)
I
]
i~.
i:I
i
i
,
i
IE
L_,?] !3Qn !
f
i e o e  ~ E ° Em E =lOOOeV
;i
l ¢~I ,
 4b
1
stondord
model

:: ,'" " ' m,.SOOGeV I
I ~
l _ .4._.L
.
I
, e*e~vE~
"o!
E
m ~ :100 GeV
m~E =60GeV '
I~

x
t
L
t
, ,.
[
Ib) ~.LbC
i
1 60
i 4
i
i
i00
120
I~C
Ib]
r
l~30
i
230
r
220
200
240
~a00
EO0
E[OeV) Fig.
5.
EIGeV
23C~.
)
Production of charged (E) and neutral (v E) leptons in weak isodoublets.
Finally we comment on the neutral isoscalars N e and no, which we assume to be mainly Majorana [34]. Their production cross section is displayed in fig. 7. In models with a large intermediate scale the order of magnitude for the N masses is 1 TeV, but in the minimal model they are at least one order of magnitude smaller [13]. Then, we include this model in fig. 7 only for comparison. If heavy enough their main decay modes would be (except for the minimal model, where vectorlike fermions are expected to decay supersymmetrically)
N~E+W + L.. e  Z
and
E++W ~ e+ z ,
N ~ ~_~+Z
and
~E+Z
eW
+
~e+W

,
F. d e / A g u i l a
e t aL
/
. e  ~ E " Eb[
mE = 600eV
'/ \\
•
27
e + e  annihilation
,,
E =180GeV
 
'
,
'/~ / \
'
,' " L_:
,I ]

"
"
"
rain model
.....
/'
model(b)
 
 model
~
J__.'._
c)
//standard model
_i,,__ '::
:_ _~=
_~=
. . . .
2
• _ ."
i3
.
(c}
m z. = 1 8 0 G e V
~' F (O) i
" 
i
i 
J.
I
i
. ~
~.
,
J_ ~ 1
I
I . .  L
I
I
mz. = 500 GeV
I
.__
e* e ~
e'e~
V E ~E
r mv~= 60 OeV
vc '~E
m~E = 1 8 0 OeV
/
\
//
/ :
2 "
/ i~
. . . . . .
,I
t
S _ j_ _ . . . . .
.....
.,?
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.........
4
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. . . .
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.
.
.
.
.
.

I/ . . . . . . . . . . . . . . . . . . . .
I l
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"
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......
I
4 !
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i
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l
i
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i
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inp
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l
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i
] 133
•
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I 22,~
I
Idl
24O
I
3~0
5OC
~bq
; F,:r:
15f:3
27~
EIGeV)
E {GeV)
Fig. 6. Forwardbackward asymmetry for E and uE. (AFB = 0 in the SM and in model Co).)
leading to spectacular events when pairproduced in e + e  annihilation e+e  __>e + e  Z Z W  W + , e  e  Z Z W + W
+
, e + e + Z Z W  W  . These final states occur in a ratio 2 : 1 : 1.
6. Condusioas W e have studied the production of new particles, (E6) gauge bosons and vectorlike fermions, and their subsequent decay into standard fermions in e + e  machines. Cross sections and decay rates are given for the different cases. Figures for different
28
E delAguila et al. / e + e  annihilation iO00
R e ' * e  ~ NN
rain model model (b)
' "
....
m N : 2 0 0 C'~V IO0
i 1
ii it 10'
J
,,'
/ 'I
2[~0
23n
',,, ~
/\ I
I x.,i
5110
m=. = 500 GeV
I
i
I lOOfl
2Nl~rl
E (GeV) Fig. 7. Production of (isosinglet) Majorana fermions through Z ' in e+e  annihilation. (o(NN)  0 in the SM and in model (c).) The minimal model is included for comparison.
models are presented for illustration purposes. The main signatures for decays of new vectorlike fermions are also investigated. Although our analysis is rather general, (E6) extended electroweak models inspired by superstrings are emphasized. New (vectorlike) fermions are allowed to mix with standard fermions, and then to decay through gauge interactions. The structure of the mass matrix giving rise to this mixing is analyzed and the gauge interactions mediating the (new) fermion decays are studied in detail. We have followed a phenomenological approach, trying to keep as much generality as possible. In fact many of our results are only based on the general chiral structure of these theories, which is fixed by the standard model. For a suitable choice of parameters our analysis specializes to different scenarios. Models with intermediate scales and the minimal model are particular cases. Constraints from the leptonic sector, in particular from neutrino masses, require in each case different masses (and mixings) for the new neutral isosinglets. All cases are considered but for the rank5 model with characteristic supersymmetric decays for new vectorlike fermions. This case is studied in detail in the literature [15]. We are grateful for discussions with J.H. Kiihn, M. Quir6s, P. Ramond, H. Reno, G.G. Ross, M. RuizAltaba and F. Zwimer. Appendix A In this appendix we describe some details of the diagonalization of the different mass matrices and of the neutral currents in the mass eigenstate basis.
F. del Aguila et al. / e
+
29
e  annihilation
We discuss the quark sector first. The up quarks have the standard content and all the entries in their mass matrix ,/(u, eq. (1.3), go to zero with the standard model breaking. The diagonal matrix is obtained as usual, ~=
U~.U~
(A.1)
+,
where U ~ are 3 x 3 unitary matrices. The down quark mass matrix .,¢t'd is given in eq. (2.3) where smallcase (capital) letters correspond to small (large) A I   ½(0) entries. Its diagonalization proceeds in three steps, (A.2)
[email protected] f U~.~ aU~ + ,
where ULd,R are 6 X 6 unitary matrices that can be factorized into three components each, U = U,~/MUmUM .
(A.3)
UM diagonalizes the large M entries, U,. the resulting upper left small m entries, and Urn~M, which can be obtained as an expansion in m / M , finally determines the small mixing between light and heavy fermions. We find in this way
LMglg~
13×3
0
cosoa 13x3
Ud=[16×61[16×6] _sin0al3x3
o111: o l
13x3
3
sinOa13×3 ] cosOalsx 3 •
U~3×3
(A.4)
(A.5)
In order to avoid flavorchanging neutral currents induced through the heavyfermion sector, we have assumed that the matrix M1d is proportional to M d in eq. (2.3). UR a M has no family structure in this case, and UR a ,~ and Ud m/M (tO first order) are unity. Evaluating
IO'
''6'
m d and M d are diagonal, built up by the masses of the light and heavy down quarks to order ( r e ~ M ) 2, and #d is a matrix order m. Writing the mass eigenstates u = V Uu ',
d   Vdd ' ,
(A.7)
where u'(d') are the current eigenstates, we obtain the mass eigenstate quark currents in table 2a, with ~;j a = #~/Md. Similarly for the charged currents (table 3)
30
F. del Aguila et aL / e + e  annihilation
where K = "L rr ur'JL ra +m is the CabibboKobayashiMaskawa matrix. The lepton sector is more complicated. The charged lepton mass matrix is given in eq. (2.4). Since its structure is the same as that for the down quarks (the matrix is just transposed), the same comments apply. In this way we find
COS0/13× U~=[16×6][16x6]
ud = l M f x~'o
sin 0¢ 13x3 ] COS0I13× 3 '
_sin0e13×3
13×3
13×3
3~
(A.8)
UE3x3
Since we assumed that one combination of the new vectorlike leptons corresponds to higgsinos, they decouple. Thus the indices of the new leptons run over 1, 2 only, when mixing is relevant. With
[m. 0 ] Lm~LM'a'~ t V R M t ' J g m =
#e
Me
and #eir~/Mer=1]:r, + ' i = 1 , 2 , 3 and r = l , 2 , we obtain the currents in the mass eigenstate basis given in table 2b. The neutrino mass matrix, on the other hand, involves many parameters. It can be diagonalized by a unitary matrix
~, = v , j , v .T,
(A.1 l)
which can be factorized into two components,
U~ Urn~mUm, 1
o oA=
0
o
~ ~,
0 0
0 0
0
0
0
~ oo ~, o o 0 0
1 0
0 1
(A.12)
cos0t sin0t
sin0t cos0t
0 0
0 0
0 0
0
0
U~
0
0
0 0
0 0
0 0
cos~ sin~
sin~ cosfl (A.t3)
31
F. del Aguila et al. / e + e  annihilation
diagonalizes the large M entries and
U~/M =
A* 0
0 1
0 0
n~* ~*
n~* ~*
0 n~rA *
0 n~ r
1 n~ r
n5 1
n~* 0
_n TA.
_n r
_n r
0
1
(A.14)
diagonalizes the sum of the resulting diagonal matrix plus the small mixing matrix order m. Each block is 3 x 3. 0e and U~ in (A.13) are the same as in (A.8), (A.9), because they diagonalize the same mass submatrices. For the sake of simplicity we have assumed that the large M~2,a entries are family independent and can be diagonalized by a common angle ft. The subsequent rotation in (A.13) splits the Dirac neutrinos into their degenerate mass eigenstate Majorana components V/~(1, i)(v E ± v~) (see eq. (2.5) in the text). U,~/M in (A.14) is given to first order in m/M. A is the submatrix diagonalizing the light neutrinos, acquiring masses at most of the order m2/M. Therefore A is ineffective. The ~/1,2 mixings are in general very small. As in the case of the charged leptons it is again assumed that the members of the third heavy family decouple (higgsinos). Extending (A.7) to leptons the ensuing mass eigenstate neutral and charged currents can be derived. They are shown in tables 2b and 3b respectively. In diagonalizing J/~ in (2.4) we have assumed that all M entries were large. This is so only for case (I) in the introduction. In case (II) M~2 are negligible, and in case (III) M f are negligible, too. However, the structure of the diagonalizing matrix does not change, except that U~ in (A.13) splits into U2U~ where U2 now contains the B rotations and /.J~ has in their place unit submatrices. The only important difference is that the corresponding eigenvalues are small and that, depending on the ~ values, the new light neutrinos may become important decay products of the heavy leptons. Note that LH neutrinos are always required to mix little with other neutral leptons even if they are light, although then it appears less natural to demand unobservable mixing.
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33