We compare the phenomenology of two models, the so-called “minimal Z'” and an effective model an SM-like Higgs is realised as a composite state of a new strong interaction, at a multi-TeV linear collider in the hypothesis that the new physics is at a scale beyond the direct reach of the machine

Battaglia, M.

Coradeschi, F.

De Curtis, S.

Dominici, D.

English

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DOI 10.1393/ncc/i2011-11006-9Colloquia: LC10IL NUOVO CIMENTO Vol. 34 C, N. 5 Suppl. 1 Settembre-Ottobre 2011Indirect sensitivity to Z ′s in high-energy e+e− collisions:A “standard” vs. “composite” studyM. Battaglia(1)(2)(3), F. Coradeschi(4)(5)(∗), S. De Curtis(5)and D. Dominici(4)(5)(1) University of California at Santa Cruz - Santa Cruz, CA, USA(2) Lawrence Berkeley National Laboratory - Berkeley, CA, USA(3) CERN - Geneva, Switzerland(4) Dipartimento di Fisica, Universita` di Firenze - Firenze, Italy(5) INFN, Sezione di Firenze - Firenze, Italy(ricevuto il 20 Luglio 2011; pubblicato online il 19 Ottobre 2011)Summary. — We compare the phenomenology of two models, the so-called “min-imal Z′” and an effective model an SM-like Higgs is realised as a composite state ofa new strong interaction, at a multi-TeV linear collider in the hypothesis that thenew physics is at a scale beyond the direct reach of the machine.PACS 13.66.Fg – Gauge and Higgs boson production in e+e− interactions.PACS 14.70.Pw – Other gauge bosons.1. – IntroductionAdditional heavy neutral gauge bosons, or “Z ′s”, appear in many extensions of theStandard Model (SM). Current limits from direct searches at hadron colliders constraintheir mass to be above ∼ 1TeV, with an exact limit which is significantly model-dependent. The LHC will of course push this limit much further; however, even if we getno signal at the LHC, a multi-TeV lepton collider such as ILC or CLIC could still obtainessential information on an eventual Z ′s heavy enough to be out of both its and the LCHreach, by the precision study of electroweak observables, which are in fact sensitive to theeffects of new particles at mass scales well above the collision center-of-mass energy (√s).In this work, we wish to compare the phenomenology of two different models, bothcontaining Z ′s, at an e+e− collider and in the hypothesis that the scale of new physics(NP) is beyond the direct reach of the machine, by analyzing electroweak precision ob-servables in e−e+ → ff¯ processes. The first model is the so-called “minimal Z ′” [1](see also [2]), in which it is assumed that some new physics at a very high energy scale(∗) E-mail: coradeschi@fi.infn.itc© Societa` Italiana di Fisica 139140 M. BATTAGLIA, F. CORADESCHI, S. DE CURTIS and D. DOMINICI(perhaps as high as the grand unification scale) only manifests itself at the TeV scalethrough a single Z ′. In the second model, first studied in [3], the Higgs field (along withseveral other fields, including three Z ′s) is realised as a composite state from a stronginteraction at the TeV scale; for easy of reference, we dub this model Effective CompositeHiggs Model (ECHM). The models depict completely different physical situations andeach one is among the simplest realization of the corresponding scenario, so they arewell-suited to be used as tests of a collider sensitivity.The work is organised as follows. In sect. 2 we discuss the details of the analysis.In sect. 3 we present the models and show results for electroweak observables, thensummarizing our conclusions.2. – Analysis set-upWe will focus on the case of a 3TeV collider with generic beam polarization, andconsider ff¯ production, with f = μ, t. This choice of final states is ideal to emphasizethe difference between the Z ′ model and the ECHM: in the former, the sensitivity tomuons and t quarks is comparable, while in the latter, the sensitivity to the t is muchenhanced.2.1. Observables. – In the SM, e−e+ → ff¯ processes can be fully parametrized interms of four helicity amplitudes, which can be in turn determined by measuring fourobservables: the total production cross-section, σff¯ , the forward-backward asymmetryAFB , the left-right asymmetry, ALR, and the polarized forward-backward asymmetry,ApolFB . These observables still characterize the e−e+ → ff¯ process if Z ′s are the only newneutral state; in fact, in the case of a single Z ′ of known mass (that is, if an high-massZ ′ candidate is observed at the LHC), they can be used to determine the new vectorcouplings to both e and f up to a sign ambiguity [4]. The quality measurement of ALRand ApolFB depends strongly on the degree of beam polarization:(1)ΔpolALRALR=ΔpolApolFBApolFB=ΔPeffPeff, Peff =Pe− − Pe+1 + Pe−Pe+,where Pe−(Pe+) gives the degree of polarization of the e−(e+) beam, with Pe− = −1meaning a fully left polarized and Pe+ = +1 a fully right polarized beam. In the case ofno beam polarization, ALR and ApolLR cannot be directly measured at all, and one has torely only on the first two observables.2.2. Details of the calculation. – Our analysis has been performed with the help of theCalcHEP [5] package. The model files for the ECHM files have been generated from theLagrangian using the FeynRules [6] package in Mathematica [7]; also for the ECHM, thecomputation of the couplings in one of the models has been performed by implementingan external library (in C) to obtain a numerical diagonalization of the mass matrices.While CalcHEP generates matrix elements at tree level, it can be set to incorporatecorrections from Initial State Radiation (ISR) and beamstrahlung, which are known tobe significant at an high-energy linear collider. ISR is implemented via the expression ofref. [8]. For beamstrahlung, we used the preliminary parameters specified for CLIC [9]:horizontal beam size: 45 nm, vertical beam size: 1 nm, bunch length: 44μm, particlesper bunch: 3.7 · 109.“STANDARD” VS. “COMPOSITE” Z′ INDIRECT SENSITIVITY 141Fig. 1. – Cross-section for the e+e− → μ+μ− in the SM at the Born level (solid line), withISR and beamstrahlung with no cuts (dotted line), and with ISR, beamstrahlung and the cutEf,f¯ > 0.8Ebeam (dashed line). The cut selects final state fermions which did not lose muchenergy to radiation, bringing the cross-section within 5–10% of the Born value.We have imposed an acceptance cut on the scattering angle, | cos θ| < 0.9, to ensurethe observability of final fermions in the detector, and also a cut in final-state energy,Ef,f¯ > 0.8Ebeam, to select “pure” high energy events, that is events that have notexperienced large energy loss from ISR and beamstrahlung. As shown in fig. 1, thisensures that the corrected cross-section is within ∼ 10% of the Born one.2.3. Sensitivity scaling law . – In the indirect limit in which the mass scale of NP ismuch greater than the center-of-mass energy of the collider, sensitivity to NP is domi-nated by interference with the SM, which leads to a shift in the observables of the form(2) ΔO ∝ g2X/M2Z′X + O(s/M2Z′X),with O a generic observable and gX , MZ′X generic coupling and mass scale of the Z ′model, respectively. Since the statistical uncertainty scales as√Lint s with integratedluminosity, this leads to the sensitivity scaling law:(3) MZ′X/gX ∝ (Lint s)1/4if the uncertainty is statistic-dominated, which is a safe assumption at least in the caseof asymmetries. As already noted, a potential exception are ALR and ApolFB which canreceive a large contribution to the systematic uncertainty from polarization.3. – ResultsWe present the results of our analysis by giving deviations of the observables listedin sect. 2 with respect to the SM for the two chosen models, in function of the modelparameters. We plot the relative deviation δσ/σSM of the cross-section and the absolutedeviation δAX of the asymmetries (δO = ONP − OSM for every observable O and AXis a generic asymmetry).Before moving on to the actual models, it is useful and instructive to show the devi-ations in function of the Z ′ mass in the so-called Sequential SM (SSM), which includesa Z ′ by replicating the SM couplings to fermions of the Z0 for the new state. The SSMis not a realistic model, but it is useful as a benchmark and serves to illustrate the massscaling of the observable deviations (2), which follows a similar pattern also in every Z ′model. Deviations are shown in fig. 2.142 M. BATTAGLIA, F. CORADESCHI, S. DE CURTIS and D. DOMINICIFig. 2. – Deviations with respect to the SM for e+e− → μ+μ− (left) and to e+e− → tt¯ (right) inthe SSM model at√s = 3TeV. The continuos line represents the relative cross-section deviationδσ/σSM ; the dashed line gives δAFB , the dotted line δALR and the dot-dashed line δApolFB .3.1. Minimal Z ′ model . – “Minimal Z ′” is semi model-independent parametrizationof a light Z ′ and its couplings, first proposed in [1]. The model phenomenology at LHChas been recently studied in [2]. The basic assumption of the description is the presenceof a single new vector boson state with a mass of order TeV plus the minimum amount ofextra non-SM fields needed to make the model renormalisable and anomaly-free, hencethe adjective “minimal”.This approach automatically takes into account effects from the most general possiblekinetic and mass mixings; a Z-Z ′ mixing is automatically induced, and the mixing angleθ′ is defined in terms of the free parameters of the model (see [1] for details). Before themixing, the coupling of the Z ′ to fermions can be written as(4) LZ′int = igZZ ′μf¯γμ(g˜Y Y + g˜BL(B − L))f,where gZ is the standard Z coupling, and Y , B and L are the usual hypercharge, baryonand lepton numbers. The model has thus three free parameters: the Z ′ mass MZ′ andthe couplings g˜Y and g˜BL (in fact, the theory has N−1 additional free parameters, whichfix the couplings of the Z ′ to the N right-handed neutrinos N iR; for general values of g˜Yand g˜BL, at least three heavy neutrinos are needed to cancel anomalies. However, theseare irrelevant at low energy if one chooses—as it is customary—the mass of the N iR tobe greater than MZ′/2). The Z ′-Higgs coupling is equal to gZ g˜Y .Several well-known simple Z ′ models can be reproduced in this framework by fixingthe ratio g˜Y /g˜BL [1,2] (for instance the B−L and the E6−χ models). However, the wholeof the model parameter space is potentially interesting. Since there are no theoreticalreasons to predict a particularly large value for the couplings, we have restricted theanalysis to the region |g˜Y |, |g˜BL| < 1, which is enough to clearly illustrate the modelbehaviour. Also, for the sake of brevity we only show results for a fixed value of theZ ′ mass, MZ′ = 5TeV; for higher mass values, deviations scale in a similar way as theone shown in fig. 2 for the SSM. Our results are shown in figs. 3 and 4. Similarly towhat happens in the benchmark model SSM, both fermion channels display a similarsensitivity; furthermore, deviations are very significant, being > 10% in the majority ofthe parameter space for all observables. By eq. (3), regions well beyond 10TeV could beexplored if precise enough measurements are available.3.2. Effective composite Higgs model . – The second scheme we are going to consider,first proposed in [3], is based on the deconstruction of an extra-dimensional scenario.“STANDARD” VS. “COMPOSITE” Z′ INDIRECT SENSITIVITY 143Fig. 3. – Deviations with respect to the SM for e+e− → μ+μ− in the minimal Z′ model at√s = 3TeV. In the upper row we show |δσ|/σSM and |δAFB |, and in the lower row theabsolute value for the deviations of the polarization-based observables ALR and ApolFB . Darkerregions imply larger deviations, and contour lines correspond to the printed fixed deviationvalues. The middle of the X-axis corresponds to g˜Y = ˜gBL = 0, that is, to the SM.Fig. 4. – Deviations with respect to the SM for e+e− → tt¯ in the minimal Z′ model at √s =3TeV. In the upper row we show |δσ|/σSM and |δAFB |, and in the lower row the absolute valuefor the deviations of the polarization-based observables ALR and ApolFB . Darker regions implylarger deviations; contour lines correspond to fixed deviation values. The middle of the X-axiscorresponds to the SM.144 M. BATTAGLIA, F. CORADESCHI, S. DE CURTIS and D. DOMINICIThe model includes two four-dimensional sectors connected by mass mixing. Thefirst one, the “elementary” sector, is an SU(3)⊗ SU(2)L ⊗U(1)Y SM-like gauge theory,with the usual fermion content (plus right-handed neutrinos to help generate neutrinomasses) but no Higgs and no mass terms. The second one is an effective description ofa composite sector via an SU(3)∗ ⊗ SU(2)L ⊗ SU(2)R ⊗ U(1)X gauge theory, explicitlybroken by the insertion of mass terms and including a scalar (Higgs) bi-doublet and anexpanded fermion sector.The free parameters of the model, which are relevant for the analysis, are the following.– In the gauge sector, two composite gauge couplings, g∗(1,2), which we have forsimplicity chosen to be equal to a common value g∗, and a composite gauge bosonmass scale M∗ (the gauge couplings in the elementary sector have to be fixed inorder to reproduce the SM at low energy).– The fermion sector has many parameters: we have 18 mixing angles, one for eachleft-handed SU(2)L doublet and right-handed singlet in the SM plus three right-handed neutrinos; the Yukawa parameters, described by four complex 3×3 matrices;and the composite fermion mass scale. By contrast with the SM case, Yukawamatrices can be chosen to be “natural”, that is, with entries all of O(1): thehierarchy in mass is created by the different values of the mixing angles. Theseimply that most of the mixing angles are very small, so that they are not relevantto this analysis. For our purposes, it will be sufficient to consider the fermionmass scale m∗ and the Yukawa and mixing parameters relative to the third quarkgeneration, which have to be chosen in order to satisfy the relation:(5)√2mtv sinϕQ L3Y ∗U33 sinϕt R,√2mbv sinϕQL3Y ∗D33 sinϕbR,where mt and mb are the t and b mass, ϕQ L3, ϕt R and ϕbR are the mixing anglesof the (tL, bL) doublet, the tR and the bR respectively, and Y ∗U33 and Y∗D33 areYukawa matrix entries.In the present analysis, since we are interested in studying the modifications fromZ ′ exchanges to standard-4 fermion operators, we have assumed the composite fermionsmass scale, m∗, to be slightly greater than M∗ ∼ m∗ to avoid decay of the Z ′s into heavyfermions; in particular, we have chosen m∗ = 1.5M∗. For the 3rd-generation quarks, wehave assumed, following [3], full tR compositeness, fixing sinϕU R3 = 1. By eq. (5), weare left with four relevant free parameters, which can be chosen to be Y ∗U33, the ratioY ∗D33/Y∗U33, the coupling g∗, and the mass scale M∗. The remaining two mixing anglesare then fixed to be(6) sinϕQL3 =√2mtv Y ∗U33, sinϕbR =mbmtY ∗U33Y ∗D33.The values of Y ∗U33 and Y∗D33/Y∗U33 have to be chosen of order unity, and are subject torelatively strong experimental constraint since they contribute to a modification to thecoupling ZbLbL. We have chosen to fix them to the values Y ∗U33 = 4 and Y∗D33/Y∗U33 =1/2, since their variation has little effect on the phenomenology for the purposes of thisstudy. M∗ has to be  3TeV, and g∗ has to be greater than gSM  0.65 and notso large as to spoil the perturbativity of the model. In fig. 5 we show the constraints“STANDARD” VS. “COMPOSITE” Z′ INDIRECT SENSITIVITY 145Fig. 5. – On the left: allowed region for the ECHM in the (g∗,M∗)-plane for Y ∗U33 = 4 andY ∗D33 = 2, based on a 99% CL fit to the S, T and U parameters and the deviation δgZbb of theZbLbL coupling. On the right: estimate of the degree of perturbativity of the ECHM based onthe rate of the width Γ of the broadest state (which is one of the three Z′s), to its mass M∗.The dashed portion of the curve denotes the region in which Γ/M∗ > 1.on the parameter space from low-energy precision observables as well as an estimate ofthe degree of the perturbativity based on the analysis of the width-to-mass ratio of thebroadest state.The model contains three Z ′ bosons, originating from the SU(2)L⊗SU(2)L⊗U(1)Xcomposite subsector. They are nearly degenerate by construction, all three having massof order M∗. Their couplings to fermions depend strongly on the fermion mixing angle;for the (chiral) fermion f they are generically of order(7) gfZ′  g(g∗gsin2 ϕf − gg∗cos2 ϕf),where g is a generic SM coupling. Since for every fermion except the third-generationquarks the mixing angle is very small, by eq. (7) the corresponding couplings are sup-pressed by factors g/g∗ and are smaller than the SM values; this is also true for bR (seeeq. (6)). The situation is opposite for the tR, which has ϕt R = 1 implying a couplingof order g∗. The left-handed third generation quarks tL and bL are in an intermediateFig. 6. – Deviations with respect to the SM for e+e− → μ+μ− (left) and to e+e− → tt¯ (right)in the ECHM at√s = 3TeV with the mass scale fixed at M∗ = 5TeV and Y ∗U33 = 4, Y∗D33 = 2.The continuos line represents the relative cross-section deviation δσ/σSM ; the dashed line givesδAFB , the dotted line δALR and the dot-dashed line δApolFB . For g∗ > 2, the sensitivity in thetop channel is overwhelmingly greater.146 M. BATTAGLIA, F. CORADESCHI, S. DE CURTIS and D. DOMINICIsituation, with a mixing angle approximately equal to 1/Y ∗U33. However, since their cou-pling depend on two contributions of opposite sign, they are in general not significantlyenhanced with respect to the SM ones. In the end, the best sensitivity is expected inthe top channel. As we show in fig. 6, this is indeed the case, especially as g∗ grows tobe larger than ∼ 2. Again, deviations can be very significant, and the sensitivity can beexpected to extend well beyond 10TeV.3.3. Conclusions. – We have examined the phenomenology of two different physicalscenarios, both including Z ′s, at a linear collider when the mass scale of the new physics isbeyond the direct reach of the collider, by looking at precision observables on e+e− → ff¯processes for f = μ, t. In the first case, the minimal Z ′ model [1], which describesa “typical” Z ′ scenario, we found similar deviations from the SM predictions in bothfermion channels. In the second case, that of the model of ref. [3] which is based onthe presence of a composite sector at the TeV scale, we found that deviations in themuon channel are generally suppressed, while those in the top channel are enhanced.In both cases deviations are sizable and the sensitivity may extend to very high massscales depending on experimental conditions; furthermore the very different observablebehaviour in the two scenarios can let us distinguish them even if no direct signal isobserved.∗ ∗ ∗The authors wish to thank R. Contino for useful discussions regarding the ECHMmodel.REFERENCES[1] Appelquist T., Dobrescu B. A. and Hopper A. R., Phys. Rev. D, 68 (2003) 035012[arXiv:hep-ph/0212073].[2] Salvioni E., Villadoro G. and Zwirner F., JHEP, 0911 (2009) 068 [arXiv:0909.1320[hep-ph]].[3] Contino R., Kramer T., Son M. et al., JHEP, 0705 (2007) 074 [hep-ph/0612180].[4] Riemann S., in 2nd ECFA/DESY Study 1998-2001, pp. 1451-1468.[5] Pukhov A., e-Print: hep-ph/0412191 (2004).[6] Christensen N. D. and Duhr C., Comput. Phys. Commun., 180 (2009) 1614[arXiv:0806.4194 [hep-ph]].[7] Mathematica is a trademark of Wolfram Research Inc.[8] Jadach S. and Ward B. F. L., Comput. Phys. Commun., 56 (1990) 351; Skrzypek M.and Jadach S., Z. Phys. C, 49 (1991) 577.[9] CLIC Collaboration, Preliminary data retrieved from the collaboration public websitehttp://clic-study.org.

Indirect sensitivity to Z′s in high-energy e+e− collisions: standard vs. composite study

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Indirect sensitivity to Z′s in high-energy e+e− collisions: standard vs. composite study

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