- Email: [email protected]
Leptonic signals for compositeness at hadron colliders
Nuclear Physics B269 (1986) 421-444 North-Holland, Amsterdam
LEPTONIC SIGNALS FOR COMPOSITENESS
AT H A D R O N COLLIDERS* I. BARS Department of Phystcs, Unwersttyof Southern Cahforma, Los Angeles, CA 90089-0484, USA J F GUNION
Instztute for TheoreticalSezence, UmverstO:of Oregon, Eugene, OR 97403-1274, USA and Department of Phystcs, UC Daws, Daws, CA 95616, USA
M. KWAN Department of Physics, UC Dams, Dams, CA 95616, USA Recewed 16 September 1985
We consider composite models in which quarks and leptons have constttuents in common. Detaded amplitudes for subprocesses of the type quark + antiquark--, lepton + antilepton are constructed using the presumed similarity between QCD and compositeness/pre-color interactions. We demonstrate that sensitivity to composlteness scales, A, as high as 100 TeV to 300 TeV may be achievable at a supercollider with center-of-mass energy, ~/~ = 40 TeV For moderate values of A ( £ 15-20 TeV) cross sections are several orders of magmtude larger than the background Drell-Yan estimate, reflecting the presence of the new underlying strong interaction. Furthermore, some models exhibit a resonant structure m lepton anti-lepton pair mass spectra, near Met;- A, corresponding to heavy preon-anttpreon composite states If A is m the SSC range, compositeness would probably dominate the cross sections of most processes and could readily be explored experimentally. Finally, at S~pS energies, ~ = 540 GeV, observation of lepton-antilepton pair mass spectra with no deviation from the standard-model Drdl-Yan prediction at Mr?= 150 GeV would place limits on A in the range 1.5 to 3 TeV.
1. Introduction T h e l a r g e n u m b e r of o b s e r v e d quarks a n d l e p t o n s a n d their repetitive f a m i l y s t r u c t u r e h a s l e d to speculations that they m a y b e c o m p o s e d o f p r e o n s s t r o n g l y b o u n d b y p r e - c o l o r forces closely a n a l o g o u s to Q C D . T h e p r e - c o l o r c o u p l i n g a p is p r e s u m e d to b e c o m e large at a scale, A ~ 1 TeV, which is m u c h l a r g e r t h a n the scale A QCD - 2 0 0 M e V characteristic o f Q C D . T h e p r e o n lag,r a n # a n will have various * Research supported by the US Department of Energy, under contract no DE-FG03-84ER40168 0550-3213/86/$03.50 © Elsevier Science Publishers B V. (North-Holland Physics Publistung Division)
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chiral symmetries associated with the preon flavors. The primary difference between QCD and pre-color is that these chiral symmetries must remain unbroken [1] in order that the quarks and leptons remain massless [2]. The unbroken symmetry group G F must include SU(3) × SU(2)L × U(1); it is gauged with elementary gauge bosons, and may be used to classify states. In general these consist of: (i) 3 or more massless generations; (ii) massless exotics (absent in most models); and (iii) massive ( M v >=A) states that are approximately degenerate in flavor. The lagrangian of the low-energy effective theory can be written in the form Let r
=
LSM + Lr~R,
(1.1)
where SM and NR refer to the standard model and non-renormalizable portions of L. LsM may include additional generations and exotic composites. In general LNR will depend on both computable coupling constants, ~, and the compositeness scale, A. For large values of A, LNR can be expressed as a series of contact interactions and anomalous magnetic moment interactions. The first 4-Fermi term in the convention of ref. [2] has the form ~2
__
L R-
(1.2)
Lower bounds on A are based on the 4-Fermi interaction of (1.2). The most restrictive lower bounds arise when Lr~R is presumed to have flavor changing interactions [2]. They include (i)
Proton decay
A > h × 101J TeV,
(ii)
Ko - Ko mixing
A > h x 400 TeV,
(iii) D O- D O mixing
A > h x 50 TeV,
(iv)
K + ~ 1r+g~
A ~_h × 30 TeV,
(v)
K L~/~
A>Xx25TeV.
(1.3)
The first three types of processes are easily forbidden by imposing additional symmetries on the underlying theory. The lasttwo, involving the 4-Fermi term ds/~e, are less easily avoided. One possibility [3] is that the mass matrix mixes the leptons and quarks belonging to various families (as defined by Len). For instance the mass matrix could be chosen so that the fighter (u, d) quarks are associated with the heavier 0',, "r) leptons and the heavy (t,b) with the light (re, e); then, instead of (d~#e-) we have (dg#~) and (iv), (v) do not occur. Another possibility [3,4] is that the
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423
preon responsible for quark family repetition is different than the one that repeats the leptons. This would forbid the dangerous terms since leptons would have family symmetries that are different than those of the quarks. In either case, in some models, Cabibbo-Kobayashi-Maskawa mixing of quarks can be arranged so that (iv) and (v) are forbidden either by SU(3) L × SU(3)R symmetries in LNR, or by mixing (u c t) instead of (d s b) since in LNa these produce different results. We shall consider both models in which the bounds (iv), (v) apply, so that A > ~ × 25 TeV, and also models in which these bounds are avoided, so that A >= 1-2 TeV. These two classes of theory will be denoted by (A) and (B) respectively. Within class (B) the two mechanisms of avoiding (iv), (v) could lead to interesting observable differences, we will therefore consider models of both types. The importance of LNR is critically dependent on the values of-both h and A. The 4-Fermi form, however, depends only on }~/A. In the conventions of Eichten, Hinchliffe, Lane and Quigg (EHLQ) X2= 8~r is chosen. With this choice (iv), (v) imply that A must lie above 100 TeV for models of class (A) that allow sd -o #~. We shall see that for some of the detailed models to be discussed the super collider at v~- = 40 TeV can begin to probe such large scales. For models of class (B) we will focus on lower values of A since the only significant constraints are those arising from Bhabha scattering. The analysis of refs. [5] and [6] using the 4-Fermi form (1.2) indicates a lower limit A ___>1-2 TeV. These bounds need to be re-evaluated for the more general models discussed here. It is interesting to note that an upper bound on A can be obtained in many models by considering the cosmological role of stable superheavy particles [7]:
=<_250 TeV.
(1.4)
In class (A) this leaves a narrow range of allowed A values, some portion of which can be probed by a supercollider operating at full luminosity, L = 104°/cm2/year. In this paper we will focus, however, primarily on models of class (B) for which lower values of A could be appropriate. For A < 15 TeV the ¢s-= 40 TeV supercollider is kinematicaUy capable of probing subprocesses at and above the compositeness scale itself. In this domain the simple contact form of interaction, (1.2), is dearly inadequate. At the very least resonances with masses M V, of order A should be present. In addition it is possible that multibody final states (qCl ~ n particles with n > 2) could begin to play a role. We will argue in the next section, however, that two-body final states should remain prominant and perhaps dominant until subprocess energies, ~-, become significantly larger than the compositeness scale, A. Thus we shall concentrate on developing a more reliable description of the 2--* 2 subprocesses, for the range ¢~-- A. In ref. [8] dual model type amplitudes were developed for the 2 ---, 2 subprocesses which display a reasonable resonant structure as well as appropriate high-energy behavior in the region V~-> A. In this reference, however, only quark final states were considered. Typically backgrounds to such
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424
final states are significant and it is thus of interest to consider final states containing leptons. In this paper we will extend the four models of ref. [8] to the processes q1~12 --+/1~2. In addition we will fold these subprocesses with the EHLQ [6] structure functions in order to make actual cross section predictions for the lepton pair mass spectra at V~ = 40 TeV. The corresponding analysis for jet final states is performed in ref. [9]. In sect. 2 we review and extend the 2 ~ 2 amplitudes while sect. 3 turns to cross section predictions and model comparisons.
2. Models for qlqz --' ~'1~2 To explore the effects of LNR for leptonic final states we shall extend the four models constructed in ref. [8] for q1~12--, q3q4, to the qlq2 ~ ~'1~ sector. We first recall the conventions employed there. The model-independent amplitude for the scattering of any four chirally projected fermions is written in the form Mfx~2 _., f3~4 ALLu3LY~tI41L02LY/~U4L-I- ARRu3RYttUlRU2RY~tl)4R ~-
+ BLRU3L'YtLUlLO2RVI~U4R+ BRLu3R'Y~UlRO2L'~ttU,IL -- cLR~3L~CP/)4LU2RY~UlR -- cRLu3Ry~tU4RU2LY#UlL ,
(2.1)
where L, R refer to (1 + y5)/2 chiral projections and spinor normalization is chosen such that the spin sums and averages yield a cross section do 1 "~t (flf2 -~ f3f4) = 16~rS 2 { U 2 [[ALLI 2 + IARRI 2] +S2[IBLRI2+ IB~I 2] +t2[IcLRI2+ IC~12]}
(2.2)
when masses are neglected. Here four terms forbidden by chiral symmetry, (~3LO4L~2RUIL) and three more terms, are dropped. For qlq2 --'/1/2 the color of ql must be equal to that of q2- In computing the ~'1~2 signal at a hadron collider the cross section (2.2) is folded together with color-summed quark structure functions. The net color factor of ~ for Drell-Yan annihilation therefore applies here. In a parity conserving theory one has A LL= A RR, etc. We shall only consider models of this type even though general considerations suggest that some parity violation is required to maintain unbroken chiral symmetries. However for an analysis sensitive to parity violation we would have to construct ALL and A aR separately. This will be avoided here. In ref. [8] four models were developed for the amplitudes ALL... of (2.1). The essential idea in all models is to presume that Regge/dual-model like amplitudes can be used to describe resonant and low-momentum-transfer/high-energy scattering behavior of composite quarks and leptons in close analogy to the phenomenology of
L B a r s et a L /
Compostteness
425
composite hadron scattering in QCD. Typically the amplitudes will have both a dual-model/Veneziano form and, in channels characterized by vacuum quantum numbers at the pre-color constituent level, a pomeron contribution. These two contributions will be taken in the form
l _2a, r(1 - a ( s ) ) r ( 1 - a( t )) =
-- ~gv
.
.
.
.
.
.
r ( 2 - ,~(s) - , ~ ( t ) )
,
(2.3)
with
a ( s ) = ½ +a's 1 + - - i
,
Mv
1 a ' = 2---~z
(2.4)
and (see refs. [8, 9]) 2~rgp2 1 + e '"P(') F ( c + la'lls I - Itll) P'~= M~ cos(½~rap(s)) F ( c + ½a'(Isl + I t l ) ) '
ap(s) = 1 + a's,
(2.5) (2.6)
respectively. Note that the pomeron form given is appropriate to s-channel pomeron exchange (for large positive t and small negative s the last factor takes the form (½a't)"P(~)); in the case of q1~12""}did2 scattering there is always a change of quantum numbers that forbids vacuum exchange in the t-channel. Thus the pomeron term employed for q1~12---}did2 is a crossed channel remnant of a term that is most naturally associated with q¢'1 "--'qz:l scattering. The coupling constant g~ is estimated, by analogy to QCD, from the coupling of the first resonance in nst at s = Mv2, 1 2 ~gv Bst
"¢'
s - M ~ + iMv/'v '
g2
~-"
8~r.
(2.7)
The parameter g~ is difficult to estimate reliably. We consider values in the range gpZ= 0 to 17. At high M~ >> s, B~t approaches the 4-Fermi contact form s,,<
Bst
)
~r g2 4 2m{"
(2.8) m
If LNR of eq. (1.2) arises from a single term of the form -Bst#lyj,ff,2~t,3yzeg4 then the compositeness scale A of (1.2) is related to Mv of (2.8) by X2
¢r
gvz
-Bst-- 2A~ -- ~" 2---~v 2.
(2.9)
426
L Bars et aL / Compostteness TABLE 1
qq --, e-e + in class (A) (A > 100 TeV) or qcl -"*~'-r÷ in class (B) (A >__1-2 TeV) Model I
u~ dd c~ fi b-b
II
III
IV
ALL
BLR
cLR
ALL
BLR
cLR
ALL
BLR
CLR
ALL
BLR
C LR
A1 At A3 A3 A3 A3
0 0 0 0 0 0
C1 C1 C3 C3 C3 C3
A~ A1 A3 A3 A3 A3
0 B1 0 0 0 0
C3 C3 C3 C3 C3 C3
A4 A4 0 0 0 0
0 0 0 0 0 0
C4 C4 0 0 0 0
0 A5 0 A5 0 A5
0 B2 0 B2 0 B2
0 0 0 0 0 0
F o r ~2 = 8~r (the E H L Q c o n v e n t i o n [6]) a n d u s i n g (2.7) we find
Uv =
(2.1o)
I n actual models, however, several Bst type forms c o n t r i b u t e to a typical 4 - F e r m i i n t e r a c t i o n of the form (1.2) and, i n addition, several different c h i r a l / h e l i c i t y s t r u c t u r e s of the 4 - F e r m i form are present as A --* oo. T h u s if we define A b y (2.10) the limits (1.3(iv)) a n d (1.3(v)) a p p r o p r i a t e to class (A) models require u p w a r d revision d e p e n d i n g u p o n the precise model. However, for the eg final state the m u l t i p l i c i t y of terms is n o t large a n d the b o u n d s o n A are increased b y at most a factor of vr2. W e t u r n n o w to a t a b u l a t i o n of the c o n t r i b u t i o n s to various q1~12 ~ dlZ2 c h a n n e l s of interest. W e use the n o t a t i o n of ref. [8]. W e give i n tables 1 - 5 the r e s o n a n t / R e g g e c o n t r i b u t i o n s for models of class (A). Rules for the p o m e r o n c o n t r i b u t i o n will b e g i v e n shortly. T o facilitate this t a b u l a t i o n we have d e f i n e d the following recurring
TABLE2 qcl --' #-/~+ m both classes (A) and (B) Model I
II
III
IV
ALL
BLR
cLR
ALL
BLR
cLR
ALL
BLR
cLR
ALL
BEY,
cLR
dd c~
A3 A3 A~
0 0 0
(73 ¢3 C~
A3 A3 A3
0 0 0
C3 (73 C3
O 0 A4
0 O 0
0 O C4
0 As 0
0 B2 0
0 O O
fi bb
A1 A3 A3
0 0 0
Ct C3 (73
At As A3
B1 0 0
(73 (73 C3
A4 0 0
0 0 0
C4 0 0
A5 0 As
B2 0 B2
0 0 0
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Compostteness
427
TABLE 3 qF=l--. r - ~ "+ in class (A) (A _> 100 TeV) or q~ --, e - e + m class (B) (A > 1-2 TeV) Model I
U~ dd c~" Sg ti bb
II
III
IV
ALL
BLR
cLR
ALL
BLR
cLR
`4LL
BLR
cLR
ALL
BLR
cLR
A3 ,43 ,43 `43 `41 ,41
0 0 0 0 0 0
C3 C3 C3 C3 (71 C1
A3 ,43 ,43 `43 ,43 ,41
0 0 0 0 0 B1
C3 C3 C3 C3 (73 (72
0 0 0 0 `44 ,44
0 0 0 0 0 0
0 0 0 0 Ca Ca
0 ,45 0 ,45 0 ,45
0 B2 0 B2 0 B2
0 0 0 0 0 0
TABLE 4 q~?:12 -" ~e, v,,~, vfi Model Class (A) A>_100TeV ud ~ v8 c~, - , v~ ffg ~ v~ u d ~ v~ CS ~ v~ {-b ~ v~. ua---, v.r c~ ---, v~r (b ~ v~
Class (B) A>I-2TeV
I
II
III
IV
A LL B LR C LR A LL B LR C LR A LL B LR C LR A LL B LR C LR
ud -o v'~ c~ --* v'~ ffo---* v;r ud ~ v~ Cg ~ v~ t--b --* v~ u~]-, v~ c?, ~ vC" tb "* v~"
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
A2 0 0 0 A2 0 0 0 A2
B1 0 0 0 B1 0 0 0 Bt
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
As AS A5 As A5 As As As AS
B2 B2 B2
B2 B2 B2 B2 B2
B2
0 0 0 0 0 0 0 0 0
TABLE 5 qlq2 -o e~ or #fi Model I
II
III
IV
Class(A)
Class(B)
A LL
B LR
C LR
A LL
B LR
C LR
A LL
B LR
C LR
A LL
B LR
C LR
c~ --, eg u~ --, #~
ci - , eg t~ --, #~
A2
0
(72
0
0
0
A4
0
C4
0
0
0
' sd ~ e )
sb --, e~
A2
0
C2
,42
B1
0
.44
0
C4
0
0
0
L Bars et al. / Compostteness
428
forms: A 1 =--2A"~ +A ~ , A2=
AVV+A '~,
A3=- A vv,
A 4 -- 3A~',, A 5 -= 3A'~,
(2.11a)
B 1 - + (A x~ + A ~ V ) , ___3Az~,
B2-
(2.11b)
C1 - +_(2A v2 + A~z),
C2= + (AVY"+ A~Z), C3 = +A v~, C4 = + 3A~z.
(2.11c)
The different Regge/Veneziano amplitudes A "~, A ~', Az~, A2v, A v2, and A ~z could be characterized by different trajectories and different g2 parameters. However we shall work in the approximation where we take A ,~
=
Aw
=
Ar.~
=
A~V
=
A v~
=
AVZ
=
Bst ,
with only one trajectory form and g~ as specified in (2.3)-(2.7). Some rationale for this approximation is given in ref. [8]. Only the width parameter, F / M ~ , will be allowed to be channel/model dependent in the expression for B,t. For the pomeron contribution the rules are straight-forward. For d~e and e~, bt~ final states there is no pomeron contribution. For ee, /~/~ and ~'q final states A LL = (ALL)dual_mode I +
P,~,
B LR = ( wLR)dual-model, cLR=
(cLR)dual-model-{'- (-')'?ts),
(2.12)
where the 5: sign in C LP" corresponds to the model dependent + sign in (2.11b), (2.11c).
L BarsetaL/ Compostteness
429
N o t e that the amplitudes for models of class (A) (A ~ 100 TeV) and (B) (A > 1-2 TeV), are interchanged by the e ~ ~" interchange due to the assumed mass-matrix mixing (see sect. 1). However, model type IV has, naturally, no sd --, #6 transitions (since lepton and quark family preons are separate, as described in sect. 1). Thus mass matrix mixing is not required to avoid the bounds (1.3(iv), (v)). In fact model IV is invariant under the e --->• interchange as can be seen from the tables so that its predictions under the class (A) assumption are the same as under the class (B) assumption. Finally under the approximation of parity invariance we take A RR ~_ ./I LL , B RL =
B LR,
(2.13)
C R L = C LR .
The standard model contributions to the qT:1 ---, Yd channel are well known. In the notation of E H L Q [6] AL L =
eqe¢ +
s
2coS0w!
ARR= eqe¢ ( s
g
Sz
'
]2RqRe
2 cos 0 w /
Sz
B LR = B RL = 0 ,
cLR= --eqe¢ ( s
g
)2Rq Le
2 co-s 0 w
C R L = - - e2q eL¢ ( gq)2COS0 R ew s
sz
Sz
,
(2.14)
with L = T 3 -
2Q sin 2 0 w ,
s z = s - M 2 + iFzMz, R = - 2Q sin 2 0 w,
(2.15)
where Q is the appropriate charge relative to [e I. The !,6 final state standard model amplitudes are similarly obtained.
430
I Bars et al. / Compostteness
Let us now consider in more detail the remaining parameters at our disposal. First the _+ model-dependent sign of (2.11) and (2.12) is only of significance in cases where interference with the normal Drell-Yan amplitude is of importance. This region covers such a small range of the pair mass, Me?, as we shall see, that we have chosen not to explore this sensitivity and have taken the + sign always. The second parameter is the ratio gI,/gv. 2 2 We shall see that for gv2 = 8~" values of gp2 as high as 17 are probably too large. If the width factor F / M v is small, so that narrow resonances are present in Bst terms, then a small g~ would be required before this resonant structure would be visible above the background of the crossed-channel Pt~ form. The absolute size of g~ could also be considered a parameter although it should remain large to represent strong interactions. It is independent of A2, M~ for regions of subprocess energy, ~ where resonant effects in But are important. However, we shall see that the most likely f ' / M v width values are sufficiently large that the primary dependence is upon the ratio g v2/ M v. 2 Thus we have chosen to maintain (2.7) throughout. Finally, we turn to the width parameter I'/Mv, appearing in the Regge trajectory (2.4). There are several considerations that require attention. At low energies, such that ~, ~"< Mv2, dimensional analysis makes it apparent that the two-body final states we are considering will dominate the final state of the q~l scattering processes. As discussed in the introduction we anticipate that this will continue to be the case until is significantly larger than Mv2 at which point multibody final states, n > 3, will begin to play the dominant role. We argue that multibody final states can emerge only when the preons involved in qcl scattering are forced apart with sufficient energy (v~- >> My) that the pre-color string, whose tension is measured by a' = iM~,X 2 is forced to break more than once. To make an analogy to QCD strings, whose tension is a' ~- (1 GeV2) -1, we consider data from e+e - collisions. The multiplicity of pions in e+e - ~ nrr is (n) - 2-2.5 for v~ < 2-3 GeV, and rising like (n) - 2 + b ln(a's) with a small value of b. Thus, 2-body hadronic final states dominate at lower energies (i.e. (a's) t/2 < 2-3). In other words the QCD strings prefer to make as few splits as possible when not forced to be too long by energetic quarks. This conclusion is also supported by the dominance of O ~ 2¢r over #--, 4¢r decay. Indeed, in the decay of a heavy particle, phase space suppression for massless multibody final states is enormous (see ref. [7]) and can only be overcome by a rapid increase in the multiplicity of diagrams (as in perturbation theory, which is not applicable here) or string breakings that contribute. Evidently these multiple string breakings do not occur in low-energy QCD. By analogy, we argue that preon-preon resonances at M~ (analogous to M 0 in QCD) would prefer to decay predominantly to 2-body final states consisting of quarks and leptons. Also, as for e + e - ~ few pions, we would expect few body final states consisting of quarks and leptons to dominate the compositeness cross sections for the SSC range of energies (when a's is not large).
431
L Bars et a L / Compostteness
TABLE6 Number of 2-bodymodes for a resonancein a gavenpreon channel Model Preon channel
I
II
PP ap~ FF (PF)( PF ) (Pap)(Pap) (apF)(apF)
48 3 16 16 3 1
48 4
III
IV
3
12
16
4
12 12 4 1
Thus we focus only on 2-body final states. In preon models there is a large multiplicity of 2-body final states accessible to a resonance in any given channel. The width of a resonance arising from a single massless two-body mode to which it couples strongly as in (2.7), ~gv 1 2 = 2,t, can be estimated to be Fl-ch~,,~e, = ~Mv.
(2.16)
We now present a tabulation of the number of 2-body quark and lepton decay modes accessible to resonances in the various dual model a m p l i t u d e s - A w, A w, A zv, A ~v, A v~, A ve - appearing in (2.11). The number of modes depends on the preon quantum number's and on the model. The results appear in table 6. The preon notation is established in ref. [8]. In the approximations of ref. [8] the amplitudes referred to above were taken equal even though different preon quantum numbers appeared in the s and t channel for various different contributing diagrams. We can refine the model by dividing up the various contributions to the amplitude combinations in (2.11) according to the preons exchanged in the s and t channels. Given the s-channel preon resonance decay multiplicity, ns, the t-channel decay multiplicity, n t, is easily obtained as n t = 48/n
s ,
(2.17)
where 48 is, of course, the total number of preon degrees of freedom (corresponding to (3 families) × (3 colors + 1 lepton number) × (L or R) × (up or down)). We give only one example of this procedure. Other cases follow a similar pattern. Consider model I contributions to u ~ e - e + for the ALL amplitude. There are three contributing diagrams yielding the A 1 -- 2A v~ + A w structure of (2.11a) appearing in table 1. Three distinct preon channels actually contribute. The corresponding mode
432
L Bars et al. / Compositeness
multiplicities and I ' / M v factors are:
(i) PP:
(ii) ~k~:
r,
r,
My = 6 '
M~
n, = 1 6 ,
F, = ~ Mv 8,
--Ft = 2, M~
nt = 3
Fs = 2 , My
- - - - F_t3 My 8-
n~=48,
nt
n, = 3
1,
'
(iii) FF:
n s = 16,
'
s,
(2.18)
In the following phenomenological survey we will consider both the cases where
F / M v is taken to be the same for all channels and contributions and also the case where the detailed correlated widths of (2.18) and analogous results for other models and channels are used.
3. Phenomenological considerations The formalism for performing the folding of the subprocess cross sections of sect. 2 with the quark and antiquark distributions is well established and will not be reviewed here. We have employed EHLQ N=t = 2 distribution functions for all our calculations. One cautionary note is necessary, however. We have extended cross section plots as a function of lepton pair invariant mass, Me?, to a range Me?< 20 TeV. Conventionally Me? is taken as the Q value at which to evaluate the distribution functions. We also adopt this convention. However, this means that for Me?> 10 TeV the Q value is above the 104 GeV range for which EHLQ claim reliability. However the EHLQ distribution functions continue to behave moderately reasonably over this extended range at the x values appropriate to our calculations. We begin our study of the compositeness models of sect. 2 by presenting predictions at two typical S~pS/tevatron energies ~ = 540 GeV and 900 GeV. Our goal is to see what kind of limits could conceivably be placed on A using optimistic estimates for accumulated luminosity. In figs. 1 and 2, we show predictions for p~ collisions yielding e - e + pairs in model I, class (B); i.e. the ~---, e flip has been performed so that low values of A could be relevant. We have plotted the results for two different cases: gg = 17, A = 6 TeV and g2 = 0, A = 1.5 TeV. It is clear from the graphs that these two cases yield very similar enhancements in the e-e + pair spectrum at Me? masses above the Z. To obtain a total event rate one multiplies the d o / d M d y plotted by Ay = 3 and an appropriate AMe?. The above two cases yield an excess cross section in the range of 0.15 < Me?< 0.20 TeV of order 0.11 pb at x/~ = 540 GeV and 0.2 pb to 0.3 pb at v~--- 900 GeV. Thus, with several thousand inverse nanobarns of accumulated luminosity, limits on A competitive with the
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Bhabha scattering limits can be deduced if the predicted 5 to 10 event excess in this region is not observed. As part of figs. 2a and 2b we also illustrate the minor role of interference effects between standard model, Z + ~,, contributions and the compositeness terms. With a 0.01 TeV resolution there is essentially only one bin where interference is of any significance whatever. This one bin contains too few events to allow any use of angular distribution differences (and the like) which arise due to the interference of the above contributions. We continue next with a study of lepton/anti-lepton pairs at supercoUider energy ¢~- = 40 TeV. We focus first on class (B) models with complete • ~ e mass-matrix mixing. Only for class (B) can the value of A be small in models I-IlL In model IV A can be small for both class (A) and class (B). Indeed its predictions are the same for these two classes as discussed in sect. 2. In the following we shall present do/dMe? spectra where Mr2 is the invariant mass of the lepton and anti-lepton pair. The cross section is integrated over rapidity and other variables subject only to the detector acceptance constraint
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Fig. 3 Comparison of g~ values for composite model I(B) prechctions in qV= 1 ~ e - e +, at F / M v A = 12 TeV
= 0.2,
where Ye, Yt are the rapidities of the lepton and anti-lepton. We have considered pp collisions. The structure functions used are those of EHLQ, (Ns~t = 2). For the moment we consider a single F / M v value for all B,t trajectories. In fig. 3 we begin by illustrating the role of the diffractive/pomeron term Pts of eq. (2.5) for the case of a narrow width parameter, F/Mv = 0.2. We choose A = 12 TeV, model I(B) and consider the e - e + final state. We observe that at g~= 0 resonance structure at Met = M v = 12(¼~r)1/2 TeV-~ 9.86 TeV is readily apparent with a second small enhancement at the first recurrence, Met= v~M V (see ref. [8] for Regge plots of recurrences). As g2 increases the structure is enhanced by interference between Bst and Pts; g2 = 1 maximizes the structure. For high values of g2 the cross section is considerably enhanced but the resonant structure is totally obscured. The parameter g2 cannot be reliably estimated. However we regard the high g2 values as unlikely. Experience with hadron elastic scattering suggests that the first resonances should not be obscured by the pomeron contribution. In addition g2 < 1 can be regarded as yielding conservative compositeness cross section estimates. Also shown in fig. 3 is the Dreil-Yan background. Note the negligible region where interference between Drell-Yan and compositeness terms could be relevant and the very small background to compositeness at large Met. Also note the
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substantial cross section magnitude, fully accessible to detailed Met spectrum measurements at L = 104°/cm 2. Fig. 4 illustrates sensitivity to the F/M v width parameter. Obviously as F/Mv increases above 0.5 the resonance structure is washed out. In fig. 5 we compare model I(B) predictions for qU:1 ~ e - e + at gp2 = 1.0 for two different A values (3 TeV and 12 TeV) and two different width parameters (F/Mv = 0.2 and 3.0). At small F/Mv the resonant structure, of course, moves to lower Me2 values for the lower A value. At large F/M~ the main effect of smaller A is to substantially magnify the cross section. One hope for distinguishing between models I-IV(B) lies in differences in the do/dMt2 spectra. In fig. 6a we again consider the e - e + final state at F/M~ = 0.2, A = 12 TeV. The spectra of models I and II are very similar. A slight decrease in g~ (to -- 0.5) makes the model III spectrum very similar to that of the former two. The spectrum of model IV can be made to take on a shape similar to that for I - I I I by decreasing g~, but only at a lower over all cross section. Fig. 6b repeats these comparisons for the probably unrealistically narrow width parameter I'/M~ = 0.05. N o t e the apparent but probably unobservable second resonance recurrence. Fig. 6c compares models for the t~-/~ + final state at F/M v = 0.05 and A = 12 TeV with similar conclusions.
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In fig. 7a we maintain F/My = 0.2, A = 12 TeV, g~ = 1 and compare predictions of model I(B) for e - e +, # - # + , T-z + and # + e - final states. Note that the e-/~ + final state receives no diffractive Pt~ contribution and is of particular interest for this reason. At A = 12 TeV the cross section is certainly observable. Note however that the ~'-~'+ cross section is the largest of the four (see tables of amplitudes) and that some feed down from ~'+ ~ e+pe~¢, i"- ~ / ~ - ~ p ~ decay to t h e / t - e + channel appears inevitable. We have not computed the magnitude of this background at present. In fig. 7b these same spectra are repeated at F/Mv = 3.0. These same general features also apply in the case of models II-IV(B) with the exception that the e - # + final state is absent for model IV. In figs. 8 and 9 we turn to the incorporation of the more realistic correlated widths discussed in sect. 2. We have taken F/M~ to be diagram and channel dependent according to the predictions appropriate to a particular model. The narrowest widths are predicted for the # - e + channel (present for models I(B), II(B), Ill(B)). In fig. 8a we present d a / d M e / a t A = 12 TeV for models I-III(B). Only in model II(B), where all s-channel F / M v values are 0.5 for the/~-e + case, is any remnant of the resonant structure visible, and even this enhancement is at a marginal cross-section level. In fig. 8b the # - e + and e - e + spectra for model II(B) are compared at A = 12 TeV. In this correlated width case the e - e + average F / M v values are too large to leave more
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than a hint of structure. For the other models, I, II and IV e - e + spectra show no structure for correlated widths; this is illustrated in fig. 8c. In fig 9 we return to model I(B) predictions for the e - e + final state spectra at gp2 = 1.0, with full width correlation included, as a function of A. From these curves it is apparent that for class (B) A values as high as 50 TeV may be probed at V~- = 40 TeV, L = 104°/cm 2. It should be obvious that h = 12 TeV was not randomly chosen for so many of our plots; this value of A is close to being the largest that allows a full exploration of the Met spectrum, out to Met = 15 TeV, for all models and all final states, (at L = 104°/cm2). We now turn briefly to models of class (A) with no family mixing. As discussed in the introduction only A values above = 100 TeV are of interest for models I-III. In this region the contact form of Bst applies. If g2 __ 0 the only new feature of the models considered in this paper relative to earlier calculations is the appearance of Bs, terms in various of the amplitudes A LL, A L R . . . in a manner correlated with the model. The Met spectra are completely smooth and so we choose not to present them. Instead we present (in table 7) integrated cross sections dZ J y=0 =
,~, =A y-~y
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Figure (a): Models Comparison of q ~ - > p - e + . . . .
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M~ (TeV) Fig. 8. Predictions at A = 12 TeV when full correlated widths, as discussed in the text, are included (a) q~l ~ # - e + in models I-III(B). (b) qcl -" # - e + compared to qcl -" e - e + m model II(B). (c) q~l - ' e - e + in models I-IV(B)
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Figure (e) Models Comparison of q~->e-e* ~
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L Bars et al. / Compostteness
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TAat~ 7 Z in pb as a function of A, model and g~
Model
e- e +
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e-# + 0 27 X 10 -3 0.3 × 10 -3 0.6 × 10 -3 none
A = 100 TeV, g~ = 17 x × x x
10 -1 10-1 10-1 10-1
035 0.35 0.35 0 35
× × × x
10 -1 10-1 10- t 10-1
A ~ 300 TeV, g~ = 17 0.66 × 10-3 066 × 10-3 0.66 × 10-3 066 x 10-3
same as g2 = 0
unobservable
at v~- = 40 T e V for an acceptance in y, the ¢2 pair rapidity, of a y = 3.
(3.3)
N o Ye or Ye-- cuts are imposed. The lower limit of Mee>_, 4 TeV is the point at which Drell-Yan b a c k g r o u n d falls below the compositeness signal in the two most extreme cases considered below. The results for ~ as a function of the model are given in two extreme cases gvz = 0 and g2 = 17 at A ~- 100 TeV. At A = 300 TeV only g2 = 17 is considered. A t L = 1 0 4 ° / c m 2 it is clear that A = 100 TeV with gp: = 0 implies only a few events, a b o u t 1 0 - 1 5 combining e - e +, p - g + and e - g + channels. A t g~ = 17, in c o m p a r i s o n , A = 100 TeV yields an e n o r m o u s signal. A value of A = 300 TeV at g~ = 17 again approaches the margin of detectability yielding 13 events in the e - e + a n d # - g ÷ channels. Despite the very marginal event rates the importance of probing a b o v e A = 100 TeV in order to exceed the constraints (1.3(iv)) and (1.3(v)) warrants an intense experimental effort.
4. Conclusion
W e have argued that 2 - b o d y quark and lepton final states (as o p p o s e d to m u l t i b o d y final states) f o r m a substantial part of the cross section of compositeness for SSC energies. We have seen that realistic models o f the qx~H ~ g1~2 amplitudes
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could conceivably yield interesting structure in the lepton/anti-lepton pair mass spectra. The presence or absence of resonance structures is critically dependent on the width parameter, F / M v. However, in any case, if A is in the ~< 15 TeV range the cross sections are expected to be quite large, at SSC energies. In general we believe that the model-dependent F / M v parameter is not small enough for resonance structure to be striking; however, the possibility of a small average E l M v value should not be totally discarded. Even if F / M v is large these realistic qlU:t2~ did2 models generally yield a multiplicity of terms in the various helicity amplitudes of (2.1). This implies an ability to probe significantly larger values of the compositeness scale, A, than previously estimated (see the convention for defining A in the introduction). In the class of models (A), for which the bounds (1.3) require A >/100 TeV, the lepton pair final state is potentially capable of beginning to probe the required A >t 100 TeV region. In the (B) class of models A may be any value (above 2 to 3 TeV). In this case lepton pairs provide a comprehensive survey of compositeness at moderate A values ~<20 TeV while continuing to be sensitive to values of A up to 50 TeV. These results are competitive with those obtained using two jet hadronic final states where compositeness cross sections, but also backgrounds, are larger [9]. High efficiency for lepton final states should, in our opinion, be given a high priority in detector development for the super collider. This work was completed during the Oregon SSC Summer workshop. We thank the organizers for the hospitality and support extended to us. References [1] G 't Hooft, m Recent developments in gauge theories, ed. G. 't Hooft et al. (Plenum, NY 1980) [2] I. Bars, Nucl. Phys. B208 (1982) 77; Proc. Moriond Conf, 1982, Quarks, leptons and supersymmetry, ed. Tranh Than Van, p 541 [3] I Bars, m Proc. 1984 DPF Summer Study for the Design and utilization of the SSC, ed. R. Donaldson and J. Morfin, p. 832 [4] O W Greenberg, R. Mohapatra and S. Nussinov, Maryland preprint (1984) [5] E Eichten, K. Lane and M. Peskin, Phys. Rev. Lett. 45 (1980) 255 [6] E. Eachten, I. Hmchhffe, K. Lane and C Quigg, Rev. Mod. Phys. 56 (1984) 579 [7] I. Bars, M Bowick and K. Freese, Phys. Lett. 138B (1984) 159 [8] I Bars, Proc. of the 1984 DPF Study for the Design and utilization of the SSC, ed R. Donaldson and J Morfin, p 38 [9] I Bars and I Hmchhffe, The effects of quark composlteness at the SSC, LBL-19890 and USC-85/21 (1985)
LEPTONIC SIGNALS FOR COMPOSITENESS
AT H A D R O N COLLIDERS* I. BARS Department of Phystcs, Unwersttyof Southern Cahforma, Los Angeles, CA 90089-0484, USA J F GUNION
Instztute for TheoreticalSezence, UmverstO:of Oregon, Eugene, OR 97403-1274, USA and Department of Phystcs, UC Daws, Daws, CA 95616, USA
M. KWAN Department of Physics, UC Dams, Dams, CA 95616, USA Recewed 16 September 1985
We consider composite models in which quarks and leptons have constttuents in common. Detaded amplitudes for subprocesses of the type quark + antiquark--, lepton + antilepton are constructed using the presumed similarity between QCD and compositeness/pre-color interactions. We demonstrate that sensitivity to composlteness scales, A, as high as 100 TeV to 300 TeV may be achievable at a supercollider with center-of-mass energy, ~/~ = 40 TeV For moderate values of A ( £ 15-20 TeV) cross sections are several orders of magmtude larger than the background Drell-Yan estimate, reflecting the presence of the new underlying strong interaction. Furthermore, some models exhibit a resonant structure m lepton anti-lepton pair mass spectra, near Met;- A, corresponding to heavy preon-anttpreon composite states If A is m the SSC range, compositeness would probably dominate the cross sections of most processes and could readily be explored experimentally. Finally, at S~pS energies, ~ = 540 GeV, observation of lepton-antilepton pair mass spectra with no deviation from the standard-model Drdl-Yan prediction at Mr?= 150 GeV would place limits on A in the range 1.5 to 3 TeV.
1. Introduction T h e l a r g e n u m b e r of o b s e r v e d quarks a n d l e p t o n s a n d their repetitive f a m i l y s t r u c t u r e h a s l e d to speculations that they m a y b e c o m p o s e d o f p r e o n s s t r o n g l y b o u n d b y p r e - c o l o r forces closely a n a l o g o u s to Q C D . T h e p r e - c o l o r c o u p l i n g a p is p r e s u m e d to b e c o m e large at a scale, A ~ 1 TeV, which is m u c h l a r g e r t h a n the scale A QCD - 2 0 0 M e V characteristic o f Q C D . T h e p r e o n lag,r a n # a n will have various * Research supported by the US Department of Energy, under contract no DE-FG03-84ER40168 0550-3213/86/$03.50 © Elsevier Science Publishers B V. (North-Holland Physics Publistung Division)
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chiral symmetries associated with the preon flavors. The primary difference between QCD and pre-color is that these chiral symmetries must remain unbroken [1] in order that the quarks and leptons remain massless [2]. The unbroken symmetry group G F must include SU(3) × SU(2)L × U(1); it is gauged with elementary gauge bosons, and may be used to classify states. In general these consist of: (i) 3 or more massless generations; (ii) massless exotics (absent in most models); and (iii) massive ( M v >=A) states that are approximately degenerate in flavor. The lagrangian of the low-energy effective theory can be written in the form Let r
=
LSM + Lr~R,
(1.1)
where SM and NR refer to the standard model and non-renormalizable portions of L. LsM may include additional generations and exotic composites. In general LNR will depend on both computable coupling constants, ~, and the compositeness scale, A. For large values of A, LNR can be expressed as a series of contact interactions and anomalous magnetic moment interactions. The first 4-Fermi term in the convention of ref. [2] has the form ~2
__
L R-
(1.2)
Lower bounds on A are based on the 4-Fermi interaction of (1.2). The most restrictive lower bounds arise when Lr~R is presumed to have flavor changing interactions [2]. They include (i)
Proton decay
A > h × 101J TeV,
(ii)
Ko - Ko mixing
A > h x 400 TeV,
(iii) D O- D O mixing
A > h x 50 TeV,
(iv)
K + ~ 1r+g~
A ~_h × 30 TeV,
(v)
K L~/~
A>Xx25TeV.
(1.3)
The first three types of processes are easily forbidden by imposing additional symmetries on the underlying theory. The lasttwo, involving the 4-Fermi term ds/~e, are less easily avoided. One possibility [3] is that the mass matrix mixes the leptons and quarks belonging to various families (as defined by Len). For instance the mass matrix could be chosen so that the fighter (u, d) quarks are associated with the heavier 0',, "r) leptons and the heavy (t,b) with the light (re, e); then, instead of (d~#e-) we have (dg#~) and (iv), (v) do not occur. Another possibility [3,4] is that the
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preon responsible for quark family repetition is different than the one that repeats the leptons. This would forbid the dangerous terms since leptons would have family symmetries that are different than those of the quarks. In either case, in some models, Cabibbo-Kobayashi-Maskawa mixing of quarks can be arranged so that (iv) and (v) are forbidden either by SU(3) L × SU(3)R symmetries in LNR, or by mixing (u c t) instead of (d s b) since in LNa these produce different results. We shall consider both models in which the bounds (iv), (v) apply, so that A > ~ × 25 TeV, and also models in which these bounds are avoided, so that A >= 1-2 TeV. These two classes of theory will be denoted by (A) and (B) respectively. Within class (B) the two mechanisms of avoiding (iv), (v) could lead to interesting observable differences, we will therefore consider models of both types. The importance of LNR is critically dependent on the values of-both h and A. The 4-Fermi form, however, depends only on }~/A. In the conventions of Eichten, Hinchliffe, Lane and Quigg (EHLQ) X2= 8~r is chosen. With this choice (iv), (v) imply that A must lie above 100 TeV for models of class (A) that allow sd -o #~. We shall see that for some of the detailed models to be discussed the super collider at v~- = 40 TeV can begin to probe such large scales. For models of class (B) we will focus on lower values of A since the only significant constraints are those arising from Bhabha scattering. The analysis of refs. [5] and [6] using the 4-Fermi form (1.2) indicates a lower limit A ___>1-2 TeV. These bounds need to be re-evaluated for the more general models discussed here. It is interesting to note that an upper bound on A can be obtained in many models by considering the cosmological role of stable superheavy particles [7]:
=<_250 TeV.
(1.4)
In class (A) this leaves a narrow range of allowed A values, some portion of which can be probed by a supercollider operating at full luminosity, L = 104°/cm2/year. In this paper we will focus, however, primarily on models of class (B) for which lower values of A could be appropriate. For A < 15 TeV the ¢s-= 40 TeV supercollider is kinematicaUy capable of probing subprocesses at and above the compositeness scale itself. In this domain the simple contact form of interaction, (1.2), is dearly inadequate. At the very least resonances with masses M V, of order A should be present. In addition it is possible that multibody final states (qCl ~ n particles with n > 2) could begin to play a role. We will argue in the next section, however, that two-body final states should remain prominant and perhaps dominant until subprocess energies, ~-, become significantly larger than the compositeness scale, A. Thus we shall concentrate on developing a more reliable description of the 2--* 2 subprocesses, for the range ¢~-- A. In ref. [8] dual model type amplitudes were developed for the 2 ---, 2 subprocesses which display a reasonable resonant structure as well as appropriate high-energy behavior in the region V~-> A. In this reference, however, only quark final states were considered. Typically backgrounds to such
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final states are significant and it is thus of interest to consider final states containing leptons. In this paper we will extend the four models of ref. [8] to the processes q1~12 --+/1~2. In addition we will fold these subprocesses with the EHLQ [6] structure functions in order to make actual cross section predictions for the lepton pair mass spectra at V~ = 40 TeV. The corresponding analysis for jet final states is performed in ref. [9]. In sect. 2 we review and extend the 2 ~ 2 amplitudes while sect. 3 turns to cross section predictions and model comparisons.
2. Models for qlqz --' ~'1~2 To explore the effects of LNR for leptonic final states we shall extend the four models constructed in ref. [8] for q1~12--, q3q4, to the qlq2 ~ ~'1~ sector. We first recall the conventions employed there. The model-independent amplitude for the scattering of any four chirally projected fermions is written in the form Mfx~2 _., f3~4 ALLu3LY~tI41L02LY/~U4L-I- ARRu3RYttUlRU2RY~tl)4R ~-
+ BLRU3L'YtLUlLO2RVI~U4R+ BRLu3R'Y~UlRO2L'~ttU,IL -- cLR~3L~CP/)4LU2RY~UlR -- cRLu3Ry~tU4RU2LY#UlL ,
(2.1)
where L, R refer to (1 + y5)/2 chiral projections and spinor normalization is chosen such that the spin sums and averages yield a cross section do 1 "~t (flf2 -~ f3f4) = 16~rS 2 { U 2 [[ALLI 2 + IARRI 2] +S2[IBLRI2+ IB~I 2] +t2[IcLRI2+ IC~12]}
(2.2)
when masses are neglected. Here four terms forbidden by chiral symmetry, (~3LO4L~2RUIL) and three more terms, are dropped. For qlq2 --'/1/2 the color of ql must be equal to that of q2- In computing the ~'1~2 signal at a hadron collider the cross section (2.2) is folded together with color-summed quark structure functions. The net color factor of ~ for Drell-Yan annihilation therefore applies here. In a parity conserving theory one has A LL= A RR, etc. We shall only consider models of this type even though general considerations suggest that some parity violation is required to maintain unbroken chiral symmetries. However for an analysis sensitive to parity violation we would have to construct ALL and A aR separately. This will be avoided here. In ref. [8] four models were developed for the amplitudes ALL... of (2.1). The essential idea in all models is to presume that Regge/dual-model like amplitudes can be used to describe resonant and low-momentum-transfer/high-energy scattering behavior of composite quarks and leptons in close analogy to the phenomenology of
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425
composite hadron scattering in QCD. Typically the amplitudes will have both a dual-model/Veneziano form and, in channels characterized by vacuum quantum numbers at the pre-color constituent level, a pomeron contribution. These two contributions will be taken in the form
l _2a, r(1 - a ( s ) ) r ( 1 - a( t )) =
-- ~gv
.
.
.
.
.
.
r ( 2 - ,~(s) - , ~ ( t ) )
,
(2.3)
with
a ( s ) = ½ +a's 1 + - - i
,
Mv
1 a ' = 2---~z
(2.4)
and (see refs. [8, 9]) 2~rgp2 1 + e '"P(') F ( c + la'lls I - Itll) P'~= M~ cos(½~rap(s)) F ( c + ½a'(Isl + I t l ) ) '
ap(s) = 1 + a's,
(2.5) (2.6)
respectively. Note that the pomeron form given is appropriate to s-channel pomeron exchange (for large positive t and small negative s the last factor takes the form (½a't)"P(~)); in the case of q1~12""}did2 scattering there is always a change of quantum numbers that forbids vacuum exchange in the t-channel. Thus the pomeron term employed for q1~12---}did2 is a crossed channel remnant of a term that is most naturally associated with q¢'1 "--'qz:l scattering. The coupling constant g~ is estimated, by analogy to QCD, from the coupling of the first resonance in nst at s = Mv2, 1 2 ~gv Bst
"¢'
s - M ~ + iMv/'v '
g2
~-"
8~r.
(2.7)
The parameter g~ is difficult to estimate reliably. We consider values in the range gpZ= 0 to 17. At high M~ >> s, B~t approaches the 4-Fermi contact form s,,<
Bst
)
~r g2 4 2m{"
(2.8) m
If LNR of eq. (1.2) arises from a single term of the form -Bst#lyj,ff,2~t,3yzeg4 then the compositeness scale A of (1.2) is related to Mv of (2.8) by X2
¢r
gvz
-Bst-- 2A~ -- ~" 2---~v 2.
(2.9)
426
L Bars et aL / Compostteness TABLE 1
qq --, e-e + in class (A) (A > 100 TeV) or qcl -"*~'-r÷ in class (B) (A >__1-2 TeV) Model I
u~ dd c~ fi b-b
II
III
IV
ALL
BLR
cLR
ALL
BLR
cLR
ALL
BLR
CLR
ALL
BLR
C LR
A1 At A3 A3 A3 A3
0 0 0 0 0 0
C1 C1 C3 C3 C3 C3
A~ A1 A3 A3 A3 A3
0 B1 0 0 0 0
C3 C3 C3 C3 C3 C3
A4 A4 0 0 0 0
0 0 0 0 0 0
C4 C4 0 0 0 0
0 A5 0 A5 0 A5
0 B2 0 B2 0 B2
0 0 0 0 0 0
F o r ~2 = 8~r (the E H L Q c o n v e n t i o n [6]) a n d u s i n g (2.7) we find
Uv =
(2.1o)
I n actual models, however, several Bst type forms c o n t r i b u t e to a typical 4 - F e r m i i n t e r a c t i o n of the form (1.2) and, i n addition, several different c h i r a l / h e l i c i t y s t r u c t u r e s of the 4 - F e r m i form are present as A --* oo. T h u s if we define A b y (2.10) the limits (1.3(iv)) a n d (1.3(v)) a p p r o p r i a t e to class (A) models require u p w a r d revision d e p e n d i n g u p o n the precise model. However, for the eg final state the m u l t i p l i c i t y of terms is n o t large a n d the b o u n d s o n A are increased b y at most a factor of vr2. W e t u r n n o w to a t a b u l a t i o n of the c o n t r i b u t i o n s to various q1~12 ~ dlZ2 c h a n n e l s of interest. W e use the n o t a t i o n of ref. [8]. W e give i n tables 1 - 5 the r e s o n a n t / R e g g e c o n t r i b u t i o n s for models of class (A). Rules for the p o m e r o n c o n t r i b u t i o n will b e g i v e n shortly. T o facilitate this t a b u l a t i o n we have d e f i n e d the following recurring
TABLE2 qcl --' #-/~+ m both classes (A) and (B) Model I
II
III
IV
ALL
BLR
cLR
ALL
BLR
cLR
ALL
BLR
cLR
ALL
BEY,
cLR
dd c~
A3 A3 A~
0 0 0
(73 ¢3 C~
A3 A3 A3
0 0 0
C3 (73 C3
O 0 A4
0 O 0
0 O C4
0 As 0
0 B2 0
0 O O
fi bb
A1 A3 A3
0 0 0
Ct C3 (73
At As A3
B1 0 0
(73 (73 C3
A4 0 0
0 0 0
C4 0 0
A5 0 As
B2 0 B2
0 0 0
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427
TABLE 3 qF=l--. r - ~ "+ in class (A) (A _> 100 TeV) or q~ --, e - e + m class (B) (A > 1-2 TeV) Model I
U~ dd c~" Sg ti bb
II
III
IV
ALL
BLR
cLR
ALL
BLR
cLR
`4LL
BLR
cLR
ALL
BLR
cLR
A3 ,43 ,43 `43 `41 ,41
0 0 0 0 0 0
C3 C3 C3 C3 (71 C1
A3 ,43 ,43 `43 ,43 ,41
0 0 0 0 0 B1
C3 C3 C3 C3 (73 (72
0 0 0 0 `44 ,44
0 0 0 0 0 0
0 0 0 0 Ca Ca
0 ,45 0 ,45 0 ,45
0 B2 0 B2 0 B2
0 0 0 0 0 0
TABLE 4 q~?:12 -" ~e, v,,~, vfi Model Class (A) A>_100TeV ud ~ v8 c~, - , v~ ffg ~ v~ u d ~ v~ CS ~ v~ {-b ~ v~. ua---, v.r c~ ---, v~r (b ~ v~
Class (B) A>I-2TeV
I
II
III
IV
A LL B LR C LR A LL B LR C LR A LL B LR C LR A LL B LR C LR
ud -o v'~ c~ --* v'~ ffo---* v;r ud ~ v~ Cg ~ v~ t--b --* v~ u~]-, v~ c?, ~ vC" tb "* v~"
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
A2 0 0 0 A2 0 0 0 A2
B1 0 0 0 B1 0 0 0 Bt
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
As AS A5 As A5 As As As AS
B2 B2 B2
B2 B2 B2 B2 B2
B2
0 0 0 0 0 0 0 0 0
TABLE 5 qlq2 -o e~ or #fi Model I
II
III
IV
Class(A)
Class(B)
A LL
B LR
C LR
A LL
B LR
C LR
A LL
B LR
C LR
A LL
B LR
C LR
c~ --, eg u~ --, #~
ci - , eg t~ --, #~
A2
0
(72
0
0
0
A4
0
C4
0
0
0
' sd ~ e )
sb --, e~
A2
0
C2
,42
B1
0
.44
0
C4
0
0
0
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428
forms: A 1 =--2A"~ +A ~ , A2=
AVV+A '~,
A3=- A vv,
A 4 -- 3A~',, A 5 -= 3A'~,
(2.11a)
B 1 - + (A x~ + A ~ V ) , ___3Az~,
B2-
(2.11b)
C1 - +_(2A v2 + A~z),
C2= + (AVY"+ A~Z), C3 = +A v~, C4 = + 3A~z.
(2.11c)
The different Regge/Veneziano amplitudes A "~, A ~', Az~, A2v, A v2, and A ~z could be characterized by different trajectories and different g2 parameters. However we shall work in the approximation where we take A ,~
=
Aw
=
Ar.~
=
A~V
=
A v~
=
AVZ
=
Bst ,
with only one trajectory form and g~ as specified in (2.3)-(2.7). Some rationale for this approximation is given in ref. [8]. Only the width parameter, F / M ~ , will be allowed to be channel/model dependent in the expression for B,t. For the pomeron contribution the rules are straight-forward. For d~e and e~, bt~ final states there is no pomeron contribution. For ee, /~/~ and ~'q final states A LL = (ALL)dual_mode I +
P,~,
B LR = ( wLR)dual-model, cLR=
(cLR)dual-model-{'- (-')'?ts),
(2.12)
where the 5: sign in C LP" corresponds to the model dependent + sign in (2.11b), (2.11c).
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429
N o t e that the amplitudes for models of class (A) (A ~ 100 TeV) and (B) (A > 1-2 TeV), are interchanged by the e ~ ~" interchange due to the assumed mass-matrix mixing (see sect. 1). However, model type IV has, naturally, no sd --, #6 transitions (since lepton and quark family preons are separate, as described in sect. 1). Thus mass matrix mixing is not required to avoid the bounds (1.3(iv), (v)). In fact model IV is invariant under the e --->• interchange as can be seen from the tables so that its predictions under the class (A) assumption are the same as under the class (B) assumption. Finally under the approximation of parity invariance we take A RR ~_ ./I LL , B RL =
B LR,
(2.13)
C R L = C LR .
The standard model contributions to the qT:1 ---, Yd channel are well known. In the notation of E H L Q [6] AL L =
eqe¢ +
s
2coS0w!
ARR= eqe¢ ( s
g
Sz
'
]2RqRe
2 cos 0 w /
Sz
B LR = B RL = 0 ,
cLR= --eqe¢ ( s
g
)2Rq Le
2 co-s 0 w
C R L = - - e2q eL¢ ( gq)2COS0 R ew s
sz
Sz
,
(2.14)
with L = T 3 -
2Q sin 2 0 w ,
s z = s - M 2 + iFzMz, R = - 2Q sin 2 0 w,
(2.15)
where Q is the appropriate charge relative to [e I. The !,6 final state standard model amplitudes are similarly obtained.
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I Bars et al. / Compostteness
Let us now consider in more detail the remaining parameters at our disposal. First the _+ model-dependent sign of (2.11) and (2.12) is only of significance in cases where interference with the normal Drell-Yan amplitude is of importance. This region covers such a small range of the pair mass, Me?, as we shall see, that we have chosen not to explore this sensitivity and have taken the + sign always. The second parameter is the ratio gI,/gv. 2 2 We shall see that for gv2 = 8~" values of gp2 as high as 17 are probably too large. If the width factor F / M v is small, so that narrow resonances are present in Bst terms, then a small g~ would be required before this resonant structure would be visible above the background of the crossed-channel Pt~ form. The absolute size of g~ could also be considered a parameter although it should remain large to represent strong interactions. It is independent of A2, M~ for regions of subprocess energy, ~ where resonant effects in But are important. However, we shall see that the most likely f ' / M v width values are sufficiently large that the primary dependence is upon the ratio g v2/ M v. 2 Thus we have chosen to maintain (2.7) throughout. Finally, we turn to the width parameter I'/Mv, appearing in the Regge trajectory (2.4). There are several considerations that require attention. At low energies, such that ~, ~"< Mv2, dimensional analysis makes it apparent that the two-body final states we are considering will dominate the final state of the q~l scattering processes. As discussed in the introduction we anticipate that this will continue to be the case until is significantly larger than Mv2 at which point multibody final states, n > 3, will begin to play the dominant role. We argue that multibody final states can emerge only when the preons involved in qcl scattering are forced apart with sufficient energy (v~- >> My) that the pre-color string, whose tension is measured by a' = iM~,X 2 is forced to break more than once. To make an analogy to QCD strings, whose tension is a' ~- (1 GeV2) -1, we consider data from e+e - collisions. The multiplicity of pions in e+e - ~ nrr is (n) - 2-2.5 for v~ < 2-3 GeV, and rising like (n) - 2 + b ln(a's) with a small value of b. Thus, 2-body hadronic final states dominate at lower energies (i.e. (a's) t/2 < 2-3). In other words the QCD strings prefer to make as few splits as possible when not forced to be too long by energetic quarks. This conclusion is also supported by the dominance of O ~ 2¢r over #--, 4¢r decay. Indeed, in the decay of a heavy particle, phase space suppression for massless multibody final states is enormous (see ref. [7]) and can only be overcome by a rapid increase in the multiplicity of diagrams (as in perturbation theory, which is not applicable here) or string breakings that contribute. Evidently these multiple string breakings do not occur in low-energy QCD. By analogy, we argue that preon-preon resonances at M~ (analogous to M 0 in QCD) would prefer to decay predominantly to 2-body final states consisting of quarks and leptons. Also, as for e + e - ~ few pions, we would expect few body final states consisting of quarks and leptons to dominate the compositeness cross sections for the SSC range of energies (when a's is not large).
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TABLE6 Number of 2-bodymodes for a resonancein a gavenpreon channel Model Preon channel
I
II
PP ap~ FF (PF)( PF ) (Pap)(Pap) (apF)(apF)
48 3 16 16 3 1
48 4
III
IV
3
12
16
4
12 12 4 1
Thus we focus only on 2-body final states. In preon models there is a large multiplicity of 2-body final states accessible to a resonance in any given channel. The width of a resonance arising from a single massless two-body mode to which it couples strongly as in (2.7), ~gv 1 2 = 2,t, can be estimated to be Fl-ch~,,~e, = ~Mv.
(2.16)
We now present a tabulation of the number of 2-body quark and lepton decay modes accessible to resonances in the various dual model a m p l i t u d e s - A w, A w, A zv, A ~v, A v~, A ve - appearing in (2.11). The number of modes depends on the preon quantum number's and on the model. The results appear in table 6. The preon notation is established in ref. [8]. In the approximations of ref. [8] the amplitudes referred to above were taken equal even though different preon quantum numbers appeared in the s and t channel for various different contributing diagrams. We can refine the model by dividing up the various contributions to the amplitude combinations in (2.11) according to the preons exchanged in the s and t channels. Given the s-channel preon resonance decay multiplicity, ns, the t-channel decay multiplicity, n t, is easily obtained as n t = 48/n
s ,
(2.17)
where 48 is, of course, the total number of preon degrees of freedom (corresponding to (3 families) × (3 colors + 1 lepton number) × (L or R) × (up or down)). We give only one example of this procedure. Other cases follow a similar pattern. Consider model I contributions to u ~ e - e + for the ALL amplitude. There are three contributing diagrams yielding the A 1 -- 2A v~ + A w structure of (2.11a) appearing in table 1. Three distinct preon channels actually contribute. The corresponding mode
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L Bars et al. / Compositeness
multiplicities and I ' / M v factors are:
(i) PP:
(ii) ~k~:
r,
r,
My = 6 '
M~
n, = 1 6 ,
F, = ~ Mv 8,
--Ft = 2, M~
nt = 3
Fs = 2 , My
- - - - F_t3 My 8-
n~=48,
nt
n, = 3
1,
'
(iii) FF:
n s = 16,
'
s,
(2.18)
In the following phenomenological survey we will consider both the cases where
F / M v is taken to be the same for all channels and contributions and also the case where the detailed correlated widths of (2.18) and analogous results for other models and channels are used.
3. Phenomenological considerations The formalism for performing the folding of the subprocess cross sections of sect. 2 with the quark and antiquark distributions is well established and will not be reviewed here. We have employed EHLQ N=t = 2 distribution functions for all our calculations. One cautionary note is necessary, however. We have extended cross section plots as a function of lepton pair invariant mass, Me?, to a range Me?< 20 TeV. Conventionally Me? is taken as the Q value at which to evaluate the distribution functions. We also adopt this convention. However, this means that for Me?> 10 TeV the Q value is above the 104 GeV range for which EHLQ claim reliability. However the EHLQ distribution functions continue to behave moderately reasonably over this extended range at the x values appropriate to our calculations. We begin our study of the compositeness models of sect. 2 by presenting predictions at two typical S~pS/tevatron energies ~ = 540 GeV and 900 GeV. Our goal is to see what kind of limits could conceivably be placed on A using optimistic estimates for accumulated luminosity. In figs. 1 and 2, we show predictions for p~ collisions yielding e - e + pairs in model I, class (B); i.e. the ~---, e flip has been performed so that low values of A could be relevant. We have plotted the results for two different cases: gg = 17, A = 6 TeV and g2 = 0, A = 1.5 TeV. It is clear from the graphs that these two cases yield very similar enhancements in the e-e + pair spectrum at Me? masses above the Z. To obtain a total event rate one multiplies the d o / d M d y plotted by Ay = 3 and an appropriate AMe?. The above two cases yield an excess cross section in the range of 0.15 < Me?< 0.20 TeV of order 0.11 pb at x/~ = 540 GeV and 0.2 pb to 0.3 pb at v~--- 900 GeV. Thus, with several thousand inverse nanobarns of accumulated luminosity, limits on A competitive with the
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433
q ~ - > e - e + m M o d e l 1B 104 . . . .
I
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Fig 1. Composite and Drell-Yan compared to Drell-Yan only for q~ --* e - e + at ~ = 540 G e V Curves are for model I(B) at gp2 ~ 0, A = 1 5 TeV and at gg = 17, A = 6 TeV.
Bhabha scattering limits can be deduced if the predicted 5 to 10 event excess in this region is not observed. As part of figs. 2a and 2b we also illustrate the minor role of interference effects between standard model, Z + ~,, contributions and the compositeness terms. With a 0.01 TeV resolution there is essentially only one bin where interference is of any significance whatever. This one bin contains too few events to allow any use of angular distribution differences (and the like) which arise due to the interference of the above contributions. We continue next with a study of lepton/anti-lepton pairs at supercoUider energy ¢~- = 40 TeV. We focus first on class (B) models with complete • ~ e mass-matrix mixing. Only for class (B) can the value of A be small in models I-IlL In model IV A can be small for both class (A) and class (B). Indeed its predictions are the same for these two classes as discussed in sect. 2. In the following we shall present do/dMe? spectra where Mr2 is the invariant mass of the lepton and anti-lepton pair. The cross section is integrated over rapidity and other variables subject only to the detector acceptance constraint
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Fig. 3 Comparison of g~ values for composite model I(B) prechctions in qV= 1 ~ e - e +, at F / M v A = 12 TeV
= 0.2,
where Ye, Yt are the rapidities of the lepton and anti-lepton. We have considered pp collisions. The structure functions used are those of EHLQ, (Ns~t = 2). For the moment we consider a single F / M v value for all B,t trajectories. In fig. 3 we begin by illustrating the role of the diffractive/pomeron term Pts of eq. (2.5) for the case of a narrow width parameter, F/Mv = 0.2. We choose A = 12 TeV, model I(B) and consider the e - e + final state. We observe that at g~= 0 resonance structure at Met = M v = 12(¼~r)1/2 TeV-~ 9.86 TeV is readily apparent with a second small enhancement at the first recurrence, Met= v~M V (see ref. [8] for Regge plots of recurrences). As g2 increases the structure is enhanced by interference between Bst and Pts; g2 = 1 maximizes the structure. For high values of g2 the cross section is considerably enhanced but the resonant structure is totally obscured. The parameter g2 cannot be reliably estimated. However we regard the high g2 values as unlikely. Experience with hadron elastic scattering suggests that the first resonances should not be obscured by the pomeron contribution. In addition g2 < 1 can be regarded as yielding conservative compositeness cross section estimates. Also shown in fig. 3 is the Dreil-Yan background. Note the negligible region where interference between Drell-Yan and compositeness terms could be relevant and the very small background to compositeness at large Met. Also note the
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substantial cross section magnitude, fully accessible to detailed Met spectrum measurements at L = 104°/cm 2. Fig. 4 illustrates sensitivity to the F/M v width parameter. Obviously as F/Mv increases above 0.5 the resonance structure is washed out. In fig. 5 we compare model I(B) predictions for qU:1 ~ e - e + at gp2 = 1.0 for two different A values (3 TeV and 12 TeV) and two different width parameters (F/Mv = 0.2 and 3.0). At small F/Mv the resonant structure, of course, moves to lower Me2 values for the lower A value. At large F/M~ the main effect of smaller A is to substantially magnify the cross section. One hope for distinguishing between models I-IV(B) lies in differences in the do/dMt2 spectra. In fig. 6a we again consider the e - e + final state at F/M~ = 0.2, A = 12 TeV. The spectra of models I and II are very similar. A slight decrease in g~ (to -- 0.5) makes the model III spectrum very similar to that of the former two. The spectrum of model IV can be made to take on a shape similar to that for I - I I I by decreasing g~, but only at a lower over all cross section. Fig. 6b repeats these comparisons for the probably unrealistically narrow width parameter I'/M~ = 0.05. N o t e the apparent but probably unobservable second resonance recurrence. Fig. 6c compares models for the t~-/~ + final state at F/M v = 0.05 and A = 12 TeV with similar conclusions.
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In fig. 7a we maintain F/My = 0.2, A = 12 TeV, g~ = 1 and compare predictions of model I(B) for e - e +, # - # + , T-z + and # + e - final states. Note that the e-/~ + final state receives no diffractive Pt~ contribution and is of particular interest for this reason. At A = 12 TeV the cross section is certainly observable. Note however that the ~'-~'+ cross section is the largest of the four (see tables of amplitudes) and that some feed down from ~'+ ~ e+pe~¢, i"- ~ / ~ - ~ p ~ decay to t h e / t - e + channel appears inevitable. We have not computed the magnitude of this background at present. In fig. 7b these same spectra are repeated at F/Mv = 3.0. These same general features also apply in the case of models II-IV(B) with the exception that the e - # + final state is absent for model IV. In figs. 8 and 9 we turn to the incorporation of the more realistic correlated widths discussed in sect. 2. We have taken F/M~ to be diagram and channel dependent according to the predictions appropriate to a particular model. The narrowest widths are predicted for the # - e + channel (present for models I(B), II(B), Ill(B)). In fig. 8a we present d a / d M e / a t A = 12 TeV for models I-III(B). Only in model II(B), where all s-channel F / M v values are 0.5 for the/~-e + case, is any remnant of the resonant structure visible, and even this enhancement is at a marginal cross-section level. In fig. 8b the # - e + and e - e + spectra for model II(B) are compared at A = 12 TeV. In this correlated width case the e - e + average F / M v values are too large to leave more
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Fig. 6 C o m p a r i s o n of predictions for q~l -" e - e + or ~t-~ + for the four different models (all class (B)) at different F / M v values (a) q~ ~ e - e + at F / M ~ = 0.2 (b) qcl ~ e - e + at F / M v ~ 0 05 (c) q~ ~ ~ - ~+ at F / M v = 0.05 All plots are with A = 12 TeV.
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than a hint of structure. For the other models, I, II and IV e - e + spectra show no structure for correlated widths; this is illustrated in fig. 8c. In fig 9 we return to model I(B) predictions for the e - e + final state spectra at gp2 = 1.0, with full width correlation included, as a function of A. From these curves it is apparent that for class (B) A values as high as 50 TeV may be probed at V~- = 40 TeV, L = 104°/cm 2. It should be obvious that h = 12 TeV was not randomly chosen for so many of our plots; this value of A is close to being the largest that allows a full exploration of the Met spectrum, out to Met = 15 TeV, for all models and all final states, (at L = 104°/cm2). We now turn briefly to models of class (A) with no family mixing. As discussed in the introduction only A values above = 100 TeV are of interest for models I-III. In this region the contact form of Bst applies. If g2 __ 0 the only new feature of the models considered in this paper relative to earlier calculations is the appearance of Bs, terms in various of the amplitudes A LL, A L R . . . in a manner correlated with the model. The Met spectra are completely smooth and so we choose not to present them. Instead we present (in table 7) integrated cross sections dZ J y=0 =
,~, =A y-~y
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A yJ4 rev d y--d-'Me?dMe?
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Figure (a): Models Comparison of q ~ - > p - e + . . . .
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Figure (e) Models Comparison of q~->e-e* ~
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L Bars et al. / Compostteness
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TAat~ 7 Z in pb as a function of A, model and g~
Model
e- e +
~-#+
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e-# + 0 27 X 10 -3 0.3 × 10 -3 0.6 × 10 -3 none
A = 100 TeV, g~ = 17 x × x x
10 -1 10-1 10-1 10-1
035 0.35 0.35 0 35
× × × x
10 -1 10-1 10- t 10-1
A ~ 300 TeV, g~ = 17 0.66 × 10-3 066 × 10-3 0.66 × 10-3 066 x 10-3
same as g2 = 0
unobservable
at v~- = 40 T e V for an acceptance in y, the ¢2 pair rapidity, of a y = 3.
(3.3)
N o Ye or Ye-- cuts are imposed. The lower limit of Mee>_, 4 TeV is the point at which Drell-Yan b a c k g r o u n d falls below the compositeness signal in the two most extreme cases considered below. The results for ~ as a function of the model are given in two extreme cases gvz = 0 and g2 = 17 at A ~- 100 TeV. At A = 300 TeV only g2 = 17 is considered. A t L = 1 0 4 ° / c m 2 it is clear that A = 100 TeV with gp: = 0 implies only a few events, a b o u t 1 0 - 1 5 combining e - e +, p - g + and e - g + channels. A t g~ = 17, in c o m p a r i s o n , A = 100 TeV yields an e n o r m o u s signal. A value of A = 300 TeV at g~ = 17 again approaches the margin of detectability yielding 13 events in the e - e + a n d # - g ÷ channels. Despite the very marginal event rates the importance of probing a b o v e A = 100 TeV in order to exceed the constraints (1.3(iv)) and (1.3(v)) warrants an intense experimental effort.
4. Conclusion
W e have argued that 2 - b o d y quark and lepton final states (as o p p o s e d to m u l t i b o d y final states) f o r m a substantial part of the cross section of compositeness for SSC energies. We have seen that realistic models o f the qx~H ~ g1~2 amplitudes
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could conceivably yield interesting structure in the lepton/anti-lepton pair mass spectra. The presence or absence of resonance structures is critically dependent on the width parameter, F / M v. However, in any case, if A is in the ~< 15 TeV range the cross sections are expected to be quite large, at SSC energies. In general we believe that the model-dependent F / M v parameter is not small enough for resonance structure to be striking; however, the possibility of a small average E l M v value should not be totally discarded. Even if F / M v is large these realistic qlU:t2~ did2 models generally yield a multiplicity of terms in the various helicity amplitudes of (2.1). This implies an ability to probe significantly larger values of the compositeness scale, A, than previously estimated (see the convention for defining A in the introduction). In the class of models (A), for which the bounds (1.3) require A >/100 TeV, the lepton pair final state is potentially capable of beginning to probe the required A >t 100 TeV region. In the (B) class of models A may be any value (above 2 to 3 TeV). In this case lepton pairs provide a comprehensive survey of compositeness at moderate A values ~<20 TeV while continuing to be sensitive to values of A up to 50 TeV. These results are competitive with those obtained using two jet hadronic final states where compositeness cross sections, but also backgrounds, are larger [9]. High efficiency for lepton final states should, in our opinion, be given a high priority in detector development for the super collider. This work was completed during the Oregon SSC Summer workshop. We thank the organizers for the hospitality and support extended to us. References [1] G 't Hooft, m Recent developments in gauge theories, ed. G. 't Hooft et al. (Plenum, NY 1980) [2] I. Bars, Nucl. Phys. B208 (1982) 77; Proc. Moriond Conf, 1982, Quarks, leptons and supersymmetry, ed. Tranh Than Van, p 541 [3] I Bars, m Proc. 1984 DPF Summer Study for the Design and utilization of the SSC, ed. R. Donaldson and J. Morfin, p. 832 [4] O W Greenberg, R. Mohapatra and S. Nussinov, Maryland preprint (1984) [5] E Eichten, K. Lane and M. Peskin, Phys. Rev. Lett. 45 (1980) 255 [6] E. Eachten, I. Hmchhffe, K. Lane and C Quigg, Rev. Mod. Phys. 56 (1984) 579 [7] I. Bars, M Bowick and K. Freese, Phys. Lett. 138B (1984) 159 [8] I Bars, Proc. of the 1984 DPF Study for the Design and utilization of the SSC, ed R. Donaldson and J Morfin, p 38 [9] I Bars and I Hmchhffe, The effects of quark composlteness at the SSC, LBL-19890 and USC-85/21 (1985)