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V. Vector Self-Interactions and Vector Boson Production within the BESS-Model. C Cvctic, C. Grosse-Knelter, II. Kôgerler

Among the models which are presently considered as realistic alternatives to the SM of electroweak interactions the so-called BESS-modelM"'2! constitutes in a certain sense the most radical version. In fact, while most alternatives remain within the framework of Higgs-realized SSB and differ from the SM only either by the underlying gauge group or by the scalar representations (thereby always containing one or more physical Higgs ), the BESS model is characterized by a different mechanism of vector boson mass generation and, consequently, can manage without physical scalars at all. This is effected by realizing the spontaneous symmetry breaking in a nonlinear way. The whole situation can be visualized roughly by taking the Mu —> oo limit of the scalar sector of the SMl3I. The nonlinear nature unavoidably brings an additional local gauge symmetry GR into the gameW which manifests itself (in the most simple case Gn — SU(2)v) by an additional triplet of (heavy) vector V^. These states which are to be interpreted as (resonance) manifestations of the strong self-interactions of the scalar (or, equivalently, longitudinal vector) components, have to be sufficiently heavy in order for the theory to be compatible with low energy experimental data1. They obtain their masses from the same (non-linear) SSB realization as the ordinary vector bosons W11 and Y11 (connected with (5{/(2)t X U(l)y), and thereby also get mixed with the latter ones. Thus the physical content of the theory is described by the following (mass eigenstate-) vector boson fields: W±, V*; AM (), Z°, V?, which are appropriate linear combinations of the original gauge fields W±, V^ and Y,W3, V3, respectively. The exact expressions for vector-boson masses and mixing angles in terms of the parameters of the model can be found in ref. 1. Note that all scalar degrees of freedom (introduced in non-unitarity gauges) get absorbed and therefore do not emerge as physical states. The interaction structure of the new heavy vector bosons is determined by the fact that the unmixed (pure gauge-boson-) fields Vx only interact among themselves (in the well known Yang-Mills-manner) and not with which are 5'(7(2)vr-singlets. Nevertheless, due to mixing, the physical bosons V^, V* get coupled both to fermions and to light bosons already at the tree level. The physical coupling parameters of the BESS-model are the following: g,g',g" (the gauge coupling 2 2 constants connected with SU(2)L, U(1)Y,SU(2)V, respectively) ; / (the overall dimension param- eter which corresponds to the VEV in the linear theory (SM)) and A2 (the "relative weight" of the additional SU(2)V local symmetry). Note that the masses of the additional heavy vector bosons V1x are proportional to A, at least for sufficiently large My, where we have

MVo ~ Mv± ~ Mw(—)A 9 Since the heavy gauge bosons' masses are driven by the coupling constant A, these states do not decouple, i.e. even very massive V-bosons influence low energy physics. This gratifying property of the BESS-model is of considerable importance for phenomenological tests. There is, of course, a price to be paid for avoiding the physical scalar (Higgs) particles. This is renormalizability: the nonlinear realization of SSB, as it stands, leads to a nonrenormalizable theory, which consequently has to be conceived as an effective theory with a bounded region of validity describ- able by a cut-off parameter A. Nevertheless, for energies smaller than A, the theory is perfectly viable even beyond the tree level3. The only (but important) difference to renormalizable theories lies in the

1 In this connection it is also worth emphasizing that the BESS-model reproduces p— IsA tree level since it includes the same custodial SU(2)v as the SM. 2 Since all mixing angles are proportional to g/g", low energy data also demand g" 2> g, g', specifically g/g" < 0.1. 3 The situation is comparable to low energy physics which is effectively (but, in principle, to all loop orders) described by nonrenormalizable chiral Lagrangians. 56

fact that new additional (cut-off dependent) Lagrangian terms emerge at each higher loop levels which cannot completely be absorbed by renormalizing the tree level Lagrangian and, therefore, constitute new observable interactions. If A is not too large (A < 37W), as one would expect from well-known arguments concerning the SB scale of the SM, the 1-loop induced interactions which go proportional to log A are dominant'3). In a series of recent papers!5!'!6]'!7' these 1-loop induced interactions emerging within the BESS model have been calculated, and several interesting (partially unexpected) features have shown up: 1. There are new interactions of vector bosons to fermions (partially right-handed ones); in par- ticular the heavy bosons V±>0 can couple to fermions not only via mixing but also directly due to 1-loop-induced effects4. Unfortunately, however, by chiral symmetry the corresponding fermionic cou- pling constants are proportional to -rd-, where m/ denotes the contributing 's mass!8!. Thus the induced fermionic interactions are strongly flavour dependent and heavily suppressed for all light flavours (m/ < Mw)- Observable (still rather small) effects can only be expected for processes involving top-, f.i. e+e~ —• ifM, which are not to be considered here. 2. The situation is much better for self-interactions where no chiral suppression takes place. The induced self-couplings (still expressed in terms of unmixed fields) are listed in ref. 7, Table 5 (for cubic self-couplings) and Table 7 (for quartic self-couplings). As observed there, the corresponding coupling parameters for the self-interaction of n bosons can be written generically as

1 Boson5 = Mw 2 2 rd where P71(A ) denotes a polynomial in A of degree n, i.e. it is of 3 degree for cubic self-interactions and of 4th degree for quartic ones. This polynomial structure manifests the non-decoupling phenomenon in a cleai way: CnB increases when A (and consequently My) becomes large. More specifically, one observes the following properties of the induced gauge boson self-interactions

- The cubic self-coupling Lagrangians are of pure Yang-Mills type as concerns the Lorentz structure, though they also describe couplings between bosons of different multiplets (induced via 1-loop contributions). This implies in particular that no anomalous magnetic dipole moments, electric quadrupole moments, anapoles etc. are created (besides the SM finite contributions) so that the only structural change relative to the tree level couplings consists in a change of the overall "Yang- Mills"-coupling constant, '..c. a change of the self-covpling of the type Sg of Renard et al.I9I This is sufficient, on the other hand, to lead to a "bad" Vigh energy behaviour of a{e+e~ —> W+W") since no cancellations between the t-channel and the s-channel exchange diagrams occur5. The contributions from induced interactions increase witl> increasing A, so that the corresponding deviations from SM predictions may be measurable even at energies as low as 300 GeV, if My is large enough.

- The situation is more dramatic in the case of f mr-gauge-boson interactions. Here, the induced interactions show genuine deviations from the Yang-Mills structure, which again increase (and even more quickly) with increasing A (or My)-

- Note in this connection that couplings between members of different multiplets (f.i. V —> WW, V —> WV, V —> WWZ) can occur both due to mixing and to induced interactions.

Predictions of the BESS model (including the 1-loop-induced interactions) have been calculated for the two process classes:6

4 Some of these additional interactions are also considered in ref. 1, the corresponding parameters being denoted by b and 6', which, however, are treated as additional free parameters there. 5 Note that it is the inclusion of the 1-loop induced interactions which is responsible for this "bad" asymptotic behaviour. The (tree-level-) mixing effects alone would not affect the cancellation. 6 Calculations of the vector boson fusion processes (f.i. e+e~ —» e+e~ + W+W~) are under progress. 57

{A) W+W-Z, W+W-y, ZZZ, Z

In our calculations we had to take into account the widths of the V-boson, which again have been calcu- lated within the modelé. This is of strong importance since it turns out that the width of both V0 and V+ (which are largely dominated by the two boson modes) increase steeply with My so that V-bosons with mass beyond 2,5 TeV should not be taken seriously since their width exceeds the mass.Let us now

quote the presently available results in detail:

(A) e+e- -* W+W- We have calculated

+ - the total cross section atot(e e- both for unpolarized and polarized final vector bosons - the angular distribution ^ (again for both cases)

- the angular asymmetries AFS and ACE 3^ defined in Ref. 9 for different energies and different values of Mv (i-e. for differentvalues of A2).

The resulting predictions for the energy region up to y/s = 550GeK are plotted in Figs. 1-3 In each case, g/g" has been chosen to be g/g" = 0.10 and the cut-off is A = 5TeV (a choice of A = 2TeK would change the deviations from the SM-values by only few per cents). For comparison, the SM predictions are also given (full lines).

e e yfY 25 Fig. 1: Energy dependence +o- u of a(e w+w- 20 for different values of Mv (dashed-dotted line: A4y = •-£ 15 - 400GeV; dotted line: MV = 700GeV; dashed line: My = 2500GeV; full line: U) 1 'Ti X) - SM prediction) 2 2 "6

150 200 250 300 350 OO 500 550 total CM Energy in GeV

Several remarks are in order: 58

a) The deviations from the SM stem from three effects: the existence of the heavy vector boson Vo the mixing between light and heavy bosons the deviation of the boson self-coupling constants gzww and g^ww from the-fermionic ones due to induced interactions. b) If Mv < 500GeV, the V0 resonance is very narrow and should show up very clearly. In the other case (very heavy V-boson) the V-boson peak becomes very broad and soon (for My > 2TeV) the resonance will be invisible. But then the induced couplings get large and yield sizable deviations from the overall behaviour. c) The relative deviation in the total cross section at JS = 500GeV (if My > 500GeV) can reach from 5 % (for Mv ~ 700) to 15 % (for My ~ 2500GeV) (cf. Fig. 1) and should therefore be observable with a 500 GeV e+e" collider reaching a luminosity of 20 fb l per year where an experimental error smaller than 3 % should be possible. A detailed analysis of the bounds on the model parameters (g",X2) which can be obtained from high precision measurements of W+W--production at a future 500 GeV e+e"-collider will be published elsewhere.

prrr» rrrinn of 3OU bed GCUGS OCSCTS

•SO 3X) ZSQ 300 30 JCO «J 500 =030200 2» JQQ Bl

Fig. 2: Energy dependence of a.)

Wr) c) ff(e+e~ —• Wt Wr) for different

It J ' \ Li LJ ' values of Mv (dotted line: My = 700GeV; daahed line: Mv = 2500GeV; full line: SM prediction)

302O02SOJO0J50JO01803O0 550 tatd CM Eherçy n GeV c) 59

ci) The differences to the SM become more pronounced if the vector boson polarization is ob- served, too. The largest relative deviations (several 100 percent) occur in the LL (longitudinal- longitudinal) channel. However, the absolute values are very small there (Fig. 2). e) The differential cross sections ^ show a similar behaviour. Again, the most drastic effects can be observed in the LL and in the LT + TL channels (Figs. 3). Furthermore the region of small cos•d-values are more conclusive. Unfortunately, both cases are characterized by small absolute values. f) Quantities which are both sensitive to the BESS-model structure and observable to high accuracy are the asymmetries Apg and AcEi although very clear signatures are to be expected only at energies above 500 GeV (1 TeV or larger).

eV - WJT. eV - W. TV,

eV -

ICT W

V -'.O -05 0.0 0.5 QJ90 035 ICQ CCS d Fig. 3: Angular dependence of ^§(e+e~ —• W+W) for y/s = 5QOGeV and for different values of Mv (same choices as in Fig. 2), a) for W^ W? + W^Wj~, b) for

(B) e+e--> three gauge bosons (WWZ, WWf, ZZZ, ZZ1, £77,777) 60

We quote the results of the following quantities:7

- otot(e*c- - W+W-Z,WUV~1, ZZZ) (Fig. 4)

for energies ^/i up to 550 GoV but for a large variety of My-values. Several other distributions fe ^' ^. 5^; 5 = w+izi~/) have also been calculated and will be published elsewhere!11!. In all cases, the parameter choice is the same as before (g/g" = 0.1, A = 5TeV). We again compare with the SM-values (calculated without Higgs or - equivalently - with Mu > ITeV) which have been checked with the SM-calculations of ref. 10. e+e W+W"Z eV

70 i : I nuu I •-..•* t

/ - 60 _- 190 X / \ H— e 180 _ \ \ faC ' /—"\ 50 - i X O ii / \ ^ x T70

40 - - • /// (ft (n 160 -I \\ - -= 30 - o i / 150

20 - - \ / 140 t 10 1"V1 t i I I I . 350 400 450 500 550 250 300 350 400 450 500 550 totd CM energy in GeV totd CM energy in GeV eV - ZZZ Fig. 4: Energy dependence of a)

ted line: MV = 2500GeV)

350 400 450 500 totd GVI energy in GeV : 7 The production of three neutral bosons including at least one photon arc not very sensitive to the BESS model since the corresponding induced couplings are vanishing. Therefore, we don't consider them here We have to make the following remarks: a) The total cross section values (Fig. 4) are small for WWZ (~ 50/6 at y/s = 500GeV) and for WW-y (~ 150/6) and really tiny for ZZZ (~ lfb). But if a future machine (working at v/s = 500GeF) will reach a luminosity of 2 X 1033Cm-2S-1 (i.e. an integrated luminosity of 20/6-1 (per year)) the WWZ and WWj production process should be measurable, even if an appropriate reduction factor (of 0.3 or so [9]) is applied to account for the really reconstratable events. b) Particularly drastic effects are to be expected at y/s = 500GeK, if My is smaller than this energy (Mv = 400GeV, say). They mainly arise from V-resonance contributions both in 3- and 2-body channels (Fig. 4).

ea" — WTTZT eV -

cngtirdsMxiticvioftfieZ E31 = 500Ce/ cngcJtr dstrbtrtibrr of tf» y O00 p

-as a as -LO cos<9 a) c) eV - vnrz

cngjcr dstribution of tie W" Z3= 5Q0G*/ angijcr Au ilxitkyi of the W* •SQCGé/

Fig. S: Angular dependences for \/s = 500GeV: a.) ^ + + + - + + (e+e- _ W W~Z) c) dco'tf (e e~ — W W T), <1) j^ff—(e e~ -*• W W"7) Same choices of Mv as in Fig. 4. T?B denotes the angle of the final B (B = Z, W+, 7) relative to e" direction 62

c) If Mv becomes very large, on the other hand (My > 1.571CV), the (relative) deviations from the SM are again increasing. In fact, they become considerably larger than in the case of two-boson production, which stems from the fact that three (instead of two) vector bosons with longitudinal polarization are produced now. d) The process e+e~ —> ZZZ is of particular interest, since it is apt to test possible direct ZZZZ- (and ZZZV0-) couplings. Such an interaction emerges (as one-loop induced) within the BESS- model, consequently yielding drastic deviations from SM-predictions (see Fig. 4c). Unfortunately, the absolute size of this cross sections is so small (even in the BESS model) that an accurate test of this effect will hardly be obtainable within the nearer future.

e) The angular distributions (for W+W~Z and W+W~f only) (plotted in Fig. 5) are particularly sensitive to the BESS model structure. Even when leaving aside the rather unrealistic case of M\r = 400GeV (which would yield tremendous effects), one obtains deviations from the SM of up to the 50 %. These deviations are peaked in the orthogonal direction (for cos -dz and costi^) and in the forward direction (for cost?^+), respectively. f) The various other distributions (energy-, py-, rapidity-, diboson mass-distribution) will be pub- lished elsewhere!11!. The corresponding deviations are less pronounced here as long as y/s = 500GeV, but increase drastically with increasing CM energy. g) It shouid be mentioned that some of these distributions are very sensitive to the possible existence of a with mass less than 400 GeV. Note that they still can be used to discriminate this case from the BESS model with an equally heavy V since the latter can occur both in the W+W- and in the W±Z channel.

In summary one can state that the specific features of the BESS model (as compared to the SM) should show up in e+e~- vector boson production at ^i = 500GeV if the design luminosity is reached and unless the model parameter g" is not unrealistically large (i.e. if g/g" > 0.03).

References

[1 ] R. Casalbuoni, S. de Curtis, D. Dominici and P.. Gatto, Phys. Lett. B155 (1985) 95; Nucl. Phys. B282(1987)235 [2 ] Tceviews of the BESS model and the calculation of induced interactions can be found in refs. 5-7 [3 ] 1. Appelquist and C. Bernard, Phys. Rev. D22 (1980) 200 [4 ] M. Bando, T. Kugo, S. Uehara, K. Yamawaki and T. Yanagida, Phys. Rev. Lett. 54 (1985) 1215; see also A. P. Balachandran, A. Stern and G. Trahern, Phys. Rev. D19 (1979) 2416 [5 ] G. Cvetic and R. Kôgerler, Nucl. Phys. B328 (1989) 342 [6 ] G. Cvetic and R. Kôgerler, Nucl. Phys. B353 (1991) 462 [7 ] G. Cvetic and R. Kôgerler, Bielefeld preprint BI-TP 90/49 (to appear in Nucl. Phys. B) [8 ] G. Cvetic, R. Kôgerler and J. Trampetic, Phys. Lett. 248B (1990) 128 and to be published [9 ] J. Layssac, G. Moultaka and F.M. Renard, Montpellier-preprint PM/90-42 (1990) [10 ] V. Barger, T. Han and R.J.N. Phillips, Phys. Rev. D39 (1989) 146; A. Tofighi-Niaki and J.F. Gunion, Phys. Rev. D39 (1989) 720

[11 ] G. Cvetic, C. Grosse-Knettcr and R. Kôgerler, to bo published.