VVector triplets at the LHC (cid:63)
Javier M. Lizana , a and Manuel Pérez-Victoria , b CAFPE and Departamento de Física Teórica y del CosmosUniversidad de Granada, E-18071 Granada, Spain
Abstract.
Several popular extensions of the Standard Model predict extra vector fields that transform as tripletsunder the gauge group SU(2) L . These multiplets contain Z (cid:48) and W (cid:48) bosons, with masses and couplings relatedby gauge invariance. We review some model-independent results about these new vector bosons, with emphasison di-lepton and lepton-plus-missing-energy signals at the LHC. Extra vector bosons are a common feature of all theoriesbeyond the Standard Model (SM) with an extended gaugegroup. The possible extensions of the gauge group havea common feature: they contain the SM group G SM = SU(3) C × SU(2) L × U(1) Y as a subgroup. It is convenientto use this piece of information to systematically organizethe analyses of the new vectors from a model-independentpoint of view [1]. In this phenomenological approach,gauge invariance under G SM plays two crucial, relatedroles: it provides a classification principle and it restrictsthe possible interactions of the new particles, giving rise toa simple and natural parameterization in terms of massesand couplings.Assuming renormalizable interactions to avoid sup-pressions from a higher scale, gauge invariance impliesthat only fifteen di ff erent multiplets of vector bosons canbe singly produced at colliders [1]. Each of these multi-plets has definite quantum numbers under G SM . We canhave, for instance, singlets
B ∈ (1 , , such as the Z (cid:48) bosons associated with an extra U (1) factor, color octets G ∈ (8 , , such as the Kaluza-Klein excitations of glu-ons, etc. Here, we will study another important type ofvector bosons: isospin triplets W ∈ (1 , . These mul-tiplets are formed by a neutral Z (cid:48) boson and a pair of W (cid:48) bosons, of charge ± Experimentally, they are the (cid:63)
Talk given by M. Pérez-Victoria at LHCP 2013, Barcelona, Spain,May 13-18, 2013. a e-mail: [email protected] b e-mail: [email protected] We use the standard notation ( C , I ) Y to denote irreducible represen-tations under G SM , with C and I the dimension of the SU(3) C and SU(2) L representation, respectively, and Y the hypercharge. The simplest gauge extension of the SM that gives rise to vectortriplets is SU(3) C × SU(2) × SU(2) × U(1) Y , spontaneously broken to only vector bosons, together with singlets B , that cangive rise to sizable resonant signals with leptonic finalstates at the LHC [3]. Unlike singlets, they have chargedcomponents that can contribute to lepton-plus-missing-transverse-energy ( (cid:96) + / E T ) events. They can also produceobservable di-jet and di-boson signals.The model-independent study of the collider phe-nomenology of vector triplets was initiated in [3]. Here,we will describe the basic properties of these vectorbosons and review some results about their leptonic sig-natures at the LHC. We will emphasize the possibility ofcombining the data from searches of Z (cid:48) and W (cid:48) bosons inthis context. Let us consider an extension of the SM with a vector field W that belongs to the (1 , representation of G SM . Wemake no assumption about the origin of the new field. Toanalyse the phenomenology of this scenario in a model-independent manner, we consider a general Lagrangianwith gauge-invariant operators of dimension 4, built withthe SM fields and W [1, 4]: L = L SM + L W + L int W , (1)with L SM the SM Lagrangian, L W = −
12 [ D µ W ν ] a [ D µ W ν ] a +
12 [ D µ W ν ] a [ D ν W µ ] a + µ W a µ W µ a , (2) G SM . See [2] for an analysis of a simple model based on this pattern ofsymmetry breaking. We are assuming that there are no extra fermions lighter than thevector bosons. a r X i v : . [ h e p - ph ] J u l HCP 2013 and L int W = g W a µ W µ a φ † φ − g l W µ a ¯ l i γ µ σ a l i − g q W µ a ¯ q i γ µ σ a q i − (cid:32) i g φ W µ a φ † σ a D µ φ + h . c . (cid:33) + g W (cid:15) abc W a µ W b ν W c µν . (3)The derivative D µ is covariant with respect to the SMgauge group; l i and q i denote the left-handed lepton andquark doublets of the i th family, respectively; and W a µν isthe field-strength tensor of the SU(2) L gauge fields. Wehave only written the terms with up to two extra vectorfields, which are the ones relevant for phenomenology, andhave assumed, for simplicity, diagonal and universal cou-plings to the three families of fermions in the interactionbasis. Also for simplicity, we will consider in the follow-ing a real coupling g φ .Electroweak precision tests put limits on the param-eters in this Lagrangian. They were calculated from aglobal fit in [1], and stay mostly unchanged when updatesin the precision observables are taken into account. In Fig-ure 1, we show the limits on the coupling to mass ratios G l , q ,φ W ≡ g l , q ,φ /µ in two di ff erent planes. The flat directionalong G φ W , for vanishing G l W , is due to the fact that thevector triplet preserves custodial symmetry and does notmodify the ρ parameter. However, note that in order tohave leptonic events we need G l W (cid:44)
0. In this case, theplots show that the ratios G φ W and G W have upper bounds.In the second plot, we have also displayed lines of constant | ˜ g/µ | , with ˜ g ≡ g q g l (cid:113) g q + g l . (4)As we explain below, the LHC cross sections dependmostly on this combination of couplings. From the right-hand plot in Fig. 1, we can read the upper bound | ˜ g/µ | (cid:46) .
25 TeV − .After electroweak symmetry breaking, the neutral andcharged components of the triplet mix with the Z and the W gauge bosons. The resulting neutral mass matrix is M = (cid:32) M Z x cos θ W x cos θ W M W (cid:33) , (5)with M Z = ( g + g (cid:48) ) v / θ W the Weinberg angle, M W = µ + g v and x = gg φ v /
4. Up to terms of order v / M W ,the eigenvalues of this matrix are M Z (cid:39) M Z − x M W cos θ W , M Z (cid:48) (cid:39) M W + x M W cos θ W , (6)and the matrix is diagonalized by a rotation of angle α n with sin α n (cid:39) gg φ θ W v M W . Similarly, the mass matrix in thecharged sector is M = (cid:32) M W xx M W (cid:33) , (7) with M W = g v / W mass. Neglecting againterms of order v / M W , the eigenvalues read M W (cid:39) M W − x M W , M W (cid:48) (cid:39) M W + x M W , (8)and the mixing angle in the 2 × M c is sin α c (cid:39) gg φ W v M W . It is apparent that, as an-nounced before, the mixing with the triplet does not spoilthe custodial-symmetry relation between the mass of the Zand the W. Moreover, the neutral and charged heavy vec-tors stay nearly degenerate: M Z (cid:48) (cid:39) M W (cid:48) + g (cid:48) g φ v M W (cid:48) . (9)On the other hand, the mixing does modify the interac-tions of the mass eigenstates with the fermions, includingthe appearance of a new coupling of the Z (cid:48) to right-handedsinglets. It also induces interactions involving one heavyvector eigenstate and light vectors or the Higgs boson. Allthese e ff ects are suppressed by sin α n , c , which in turn areconstrained by perturbativity of the coupling g φ and by theelectroweak limits above. Note, however, that the decaywidths of the heavy vectors into longitudinal light vectorbosons and Higgs bosons are enhanced by derivative cou-plings, which can lead to a large di-boson branching ratioand an increased total width, even for relatively small mix-ing.In order to simplify the LHC analysis, we will set g φ to zero in the following, so that the mixing vanishes. Thisassumption maximizes the leptonic branching ratios. Thephenomenology of our e ff ective theory is then character-ized by three parameters: M W , g l and g q . (The g W termplays no relevant role when g φ = M Z (cid:48) = M W (cid:48) = M W , while their couplings to fermions in the mass eigenstatebasis are given by L CC W = − √ (cid:16) g q ¯ u Li γ µ V i j d L j + g l ¯ e Li γ µ ν Li (cid:17) W (cid:48) + µ + h . c ., (10) L NC W = − (cid:104) g q (cid:16) ¯ u Li γ µ u Li − ¯ d Li γ µ d Li (cid:17) + g l (cid:16) ¯ ν Li γ µ ν Li − ¯ e Li γ µ e Li (cid:17)(cid:105) Z (cid:48) µ , (11)with V the CKM matrix.We see that the W (cid:48) and Z (cid:48) bosons only interact withthe left-handed fermion chiralities. For this reason, thenotation W (cid:48) L is sometimes used to distinguish the chargedvector in this multiplet from a W (cid:48) R , which couples insteadto the right-handed fermions. Analogously, the Z (cid:48) in thismultiplet may be called a Z (cid:48) L . Within the approximation used to derive the electroweak boundsabove, we can safely interchange µ and M W . A W (cid:48) R vector boson, which appears for instance in Left-Right mod-els, belongs to the complex singlet B ∈ (1 , . See [1, 5] for bounds HCP 2013 -0.4-0.200.20.4 -4 -2 0 2 4 G l W [ T e V − ] G φ W [TeV − ] -0.4-0.200.20.4 -4 -2 0 2 4 G l W [ T e V − ] G q W [TeV − ] Figure 1.
Left: From darker to lighter, confidence regions with ∆ χ ≤ G q W = G φ W = | ˜ g/µ | = . − . For the particular choice g l = g q = g , in the chargedsector we have exactly the sequential W (cid:48) model, com-monly used as a benchmark in the Atlas and CMS analysesof charged vector bosons. On the other hand, the Z (cid:48) is notsequential. It couples to the third component of isospin.The isospin dependent couplings of this neutral vector re-veal that it belongs to a multiplet of dimension higher thanone. For g φ =
0, Drell-Yan is the only relevant productionmechanism of the Z (cid:48) and W (cid:48) bosons in the triplet. Theseheavy bosons can then decay into different final states in-volving quarks or leptons. Here, we concentrate on theleptonic modes, (cid:96) + (cid:96) − and (cid:96) + / E T (with (cid:96) = e , µ ). In thenarrow width approximation, the corresponding cross sec-tions at the LHC can be written as σ ( pp → Z (cid:48) → (cid:96) + (cid:96) − ) = π s (cid:104) c u ω u (cid:16) s , m Z (cid:48) (cid:17) + c d ω d (cid:16) s , m Z (cid:48) (cid:17)(cid:105) , (12) σ ( pp → W (cid:48) → (cid:96) ± ν ) = π s c c ω c (cid:16) s , m W (cid:48) (cid:17) , (13)where the functions ω u , d , c , which contain the informationof parton distribution functions, depend on the colliderenergy and the mass of the heavy bosons, but not on theircouplings. This model-independent parameterization ofthe Z (cid:48) cross section was proposed in [6] (see also [7] foran update for the LHC); the one for the W (cid:48) one is a trivialextension. The phenomenological parameters c u , d , c carrythe information about the Z (cid:48) and W (cid:48) couplings. For vector on its parameters and [4] for a model-independent analysis of its col-lider signatures. Although it does not couple to the SM neutrinos, this W (cid:48) R could give (cid:96) + / E T signals in the presence of very light right-handedneutrinos. triplets, they are given by [3] c u = c d = ˜ g , (14) c c = ˜ g . (15)Therefore, in the narrow width approximation, the crosssections for the leptonic processes mediated by the Z (cid:48) and W (cid:48) bosons in a vector triplet depend on the same two pa-rameters: the e ff ective coupling ˜ g , defined above, and themass M W = M Z (cid:48) = M W (cid:48) . There is thus a complete corre-lation between the (cid:96) + (cid:96) − and (cid:96) + / E T events produced by thetriplets. This simple property would be crucial to identifythis SM extension in case of observation at the LHC. Forthe moment, we have to content ourselves with using theLHC results to put limits on the couplings and masses. InFig. 2, we show the implications of the general bounds on c u and c d obtained by CMS in [8] from a di-lepton analy-sis, for the e ff ective coupling ˜ g of the coupling.While the narrow width approximation captures wellthe basic implications of extra vector triplets, the cross sec-tions can be significantly a ff ected by the shape of the res-onances and the interference with SM amplitudes. Thesee ff ects are specially important for the W (cid:48) , as emphasized in[9], and also for broad Z (cid:48) bosons with masses close to thekinematical reach of the collider [10]. The precise crosssections are thus sensitive to the individual couplings g l and g q , and not only to the combination ˜ g . For instance,when g l and g q have the same sign, the interference of theamplitudes mediated by W and W (cid:48) bosons is negative inthe energy region between their poles, whereas for oppo-site signs, it is positive.A complete general analysis can then be done (for g φ =
0) in terms of three parameters (a mass and two cou-plings). Here, we just present precise bounds in the caseswith g q = g l , for which we also have ˜ g = g l . We use datafrom direct searches of resonances decaying to leptons in HCP 2013 c u c d W ˜ g = g − − − − − − − − Figure 2.
95% C.L. exclusion limits for the cross section of the Z (cid:48) plotted over the c u − c d plane, from [8]. The c u and c d valuesof the triplet as a function of ˜ g are represented by the blue line. ˜ g M W [GeV] % C . L . E W li m it s CombinedLepton + MET (CMS, 7 TeV, 5 fb − )Dilepton (CMS, 7 TeV, 5 fb − ) Figure 3.
95% C.L. exclusion limits for the triplet model with g q = g l = ˜ g and g φ =
0. Regions above the curves are excluded.The bounds obtained using the (cid:96) + (cid:96) − and (cid:96) + / E T channels sep-arately are delimited by the (red) dashed and (blue) dot-dashedlines, respectively. The solid black line represents the limits forthe combination of both channels. Finally, the grey bands repre-sent the systematic uncertainty, corresponding to a ±
10% varia-tion in the signal. We also show in blue the region excluded byelectroweak precision data.
CMS at the LHC ( [8] and [11]). We find limits from boththe (cid:96) + (cid:96) − and the (cid:96) + / E T channels. Moreover, in order totake advantage of the theoretical correlations between theZ (cid:48) and W (cid:48) signals, we also find limits from a combinationof the data in both channels, which involves a commontest-statistic (for more details, see [3]). Figure 3 showsour exclusion limits at the 95% C.L., in the CL s approach,for the individual (cid:96) + (cid:96) − and (cid:96) + / E T channels and for their combination. Note that each curve represents limits on themasses and couplings of both the Z (cid:48) and the W (cid:48) bosons inthe triplet. For comparison, we also display in the sameplot the region excluded by electroweak precision tests.We can notice in this figure several interesting fea-tures. First, comparing the limits from individual chan-nels, we see that the ones from (cid:96) + / E T are stronger, de-spite the fact that this final state requires a transverse-mass,rather than invariant-mass, analysis. The reason is thefactor-4 di ff erence between c u = c d and c c , which comesfrom the couplings in Eqs. (10) and (11). This final statethus gives stronger constraints on the Z (cid:48) inside the tripletthan the ones from the more obvious di-lepton final state.Second, we see in the plot that the combination of neutraland charged channels leads to stronger bounds than theones from the best of the individual channels. In particu-lar, this shows that, even if the combined limits are domi-nated by (cid:96) + / E T , the information from (cid:96) + (cid:96) − data is usefultoo. Finally, we see that in the region of large masses andcouplings, the bounds from electroweak precision data arestronger than the ones from the searches of leptonic reso-nances at the LHC. For example, in the case of the sequen-tial W (cid:48) model, the LHC has only explored so far massesthat had already been excluded by LEP. So, it is no won-der that such a particle has not been discovered in the firstLHC run. Vector triplets appear in many models beyond the SM. Wehave showed how their phenomenology can be studied in amodel-independent fashion, using an e ff ective Lagrangianthat describes their general interactions with the SM par-ticles. The most characteristic signature of a vector tripletat the LHC is a pair of peaks at the same position (andof comparable size) in the invariant-mass and transverse-mass distributions of (cid:96) + (cid:96) − and (cid:96) + / E T events, respectively.The combined analysis of these distributions is useful forlimit setting, discovery and model identification.Searches with other final states are important as well.A vector triplet with small lepton couplings is better seenas a resonance in di-jet events. This channel is sensitive tothe quark couplings, which are poorly constrained by elec-troweak precision data. In this case, however, the Z (cid:48) and W (cid:48) bosons contribute inclusively to the same observable,so this channel alone would not allow to distinguish a vec-tor triplet from many other possibilities. The tt and tb finalstates, on the other hand, receive separate contributionsfrom the Z (cid:48) and W (cid:48) in the triplet, and would be useful toidentify this multiplet. These decay modes are especiallyrelevant when the triplets couple preferentially to the thirdfamily of quarks. Finally, the di ff erent di-boson final statesbecome essential for non-negligible mixing. A combinedanalysis of neutral and charged channels is also possible inthis case. We have used the results of the analysis in [3], which were obtainedwith data from LHC at 7 TeV. However we expect all the qualitativefeatures described in this paragraph to hold for the full LHC-7 and LHC-8 dataset.
HCP 2013
We have only considered here a minimal extension ofthe SM with one vector triplet. In general, all the resultsabove are unchanged by the presence of additional multi-plets of vector bosons. We should mention, however, oneexception. In extensions with a singlet B and a triplet W ,a mixing between the Z (cid:48) bosons in the two multiplets isallowed in the electroweak broken phase [3]. This mix-ing has two e ff ects: it removes the mass degeneracy ofthe triplet and it modifies the couplings of the eigenvec-tors. However, the mixing can only be sizable when thegauge-invariant masses of both multiplets are similar. Asa consequence, one of the neutral eigenstates always staysclose to the W (cid:48) . This scenario can be used to construct aconsistent sequential Z (cid:48) model. Such a consistent, gauge-invariant model necessarily predicts, besides the Z (cid:48) withSM couplings, another Z (cid:48) boson and a W (cid:48) boson, all ofthem with a comparable mass. More details about exten-sions with vector triplets and singlets can be found in [3]. Acknowledgements
MPV thanks the organizers of LHCP 2013 for a pleasantand interesting conference, and both authors thank Jorgede Blas for his collaboration in the study of vector triplets.This work has been supported by the MICINN projectsFPA2006-05294 and FPA2010-17915, and by the Junta deAndalucía projects FQM-101 and FQM-06552.
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