55 V. Vector Boson 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 particles), 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 bosons 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 (photon), 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 fermions 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 hadron 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 fermion'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-quarks, f.i. e+e~ —• ifM, which are not to be considered here. 2. The situation is much better for gauge boson 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.