Searching for Extra Z' from Strings and Other Models at the LHC with Leptoproduction
aa r X i v : . [ h e p - ph ] M a y Searching for Extra Z ′ from Strings and OtherModels at the LHC with Leptoproduction Claudio Corian`o a,b , Alon E. Faraggi c and Marco Guzzi aa Dipartimento di Fisica, Universit`a del Salento, andINFN Sezione di Lecce, Via Arnesano 73100 Lecce, Italy b Department of Physics and Institute of Plasma PhysicsUniversity of Crete, 71003 Heraklion, Greece c Department of Mathematical Sciences,University of Liverpool, Liverpool L69 7ZL, United Kingdom
Abstract
Discovery potentials for extra neutral interactions at the Large Hadron Colliderin forthcoming experiments are analyzed using resonant leptoproduction. For thispurpose we use high precision next-to-next-to-leading order (NNLO) determinationsof the QCD background in this channel, at the tail of the Drell-Yan distributions,in the invariant mass region around 0 . < Q < . Z ′ , obtained in left-right symmetric freefermionic heterotic string models and whose existence at low energies is motivatedby its role in suppressing proton decay mediation. We analyze the parametric de-pendence of the predictions and perform comparison with other models based onbottom up approaches, that are constructed by requiring anomaly cancellation andenlarged Higgs structure. We show that the results are not particularly sensitive tothe specific charge assignments. This may render quite difficult the extraction ofsignificant information from the forward-backward asymmetries on the resonance,assuming that these are possible due to a sizeable width. The challenge to discoverextra (non anomalous) Z ′ in this kinematic region remains strongly dependent onthe size of the new gauge coupling. Weakly coupled extra Z ′ will not be easy toidentify even with a very good theoretical determination of the QCD backgroundthrough NNLO. Introduction
The search for neutral currents mediated by extra gauge bosons ( Z ′ ) at the Large HadronCollider will gather considerable attention in the next few years [1]. Additional Abeliangauge interactions arise frequently in many extensions of the Standard Model, like inleft–right symmetric models, in Grand Unified Theories (GUTs) and in string inspiredconstructions [1]. It has also been suggested that the existence of a low scale Z ′ mayaccount for the suppression of proton decay mediating operators in supersymmetric theo-ries and otherwise [2, 3, 4]. Abelian gauge structures may also play a considerable role infixing the structure of the flavor sector, for instance in pinning down the neutrinos massmatrix. Anomaly cancellation conditions, when supported also by an extended Higgsand fermion family structure - for instance by the inclusion of right-handed neutrinos -may allow non-sequential solutions (i.e. charge assignments which are not proportional tothe hypercharge) that are phenomenologically interesting and could be studied by ATLASand CMS. Furthermore, within left–right symmetric models, and their underlying SO (10)embedding, the global baryon minus lepton number ( B − L ) of the Standard Model ispromoted to a local symmetry. Abelian gauge extensions are therefore among the mostwell motivated extensions of the Standard Model. For these reasons, the identificationof the origin of the extra neutral interaction in future collider experiments will be animportant and challenging task. In particular, measurements of the charge asymmetries- both for the rapidity distributions and for the related total cross section - and of theforward-backward asymmetries, may be a way to gather information about the structureof these new neutral currents interactions, although in the models that we have studiedthis looks pretty difficult, given the low statistics.As an extra Z ′ is common in model building, the differences among the various con-structions may remain unresolved, unless additional physical requirements are imposedon these models in order to strengthen the possibility for their unique identification. Inthis work we analyze the potential for the discovery of an extra Z ′ arising in a specificstring construction, which is motivated not only by an anomaly-free structure, as in mostof the bottom–up models considered in the previous literature, but with some additionalrequirements coming from an adequate suppression of proton decay mediation. Bottomup approaches based only on anomaly cancellation are, in this respect, less constrainingcompared to models derived either from a string construction or from theories of grandunification (GUTs) and can only provide a basic framework within which to direct theexperimental searches. At the same time the search for extra neutral interactions has toproceed in some generality and be unbiased, looking for resonances in several complemen-tary channels. In this work we will investigate the relation between more constrained and1ess constrained searches of extra neutral gauge bosons by choosing as a channel lepto-production and proceed with a comparison of some proposals that have been presentedin the recent literature. Our main interest is focused around an extra Z ′ which has beenderived using the free fermionic formulation of string theory in a specific class of left–rightsymmetric string models. The new abelian structure is determined not just as an attemptto satisfy some additional physical requirements, on which we elaborate below, but isnaturally derived from a class of string models which have been extensively studied indetail in the past two decades [6, 7, 8, 9].Our paper is organized as follows: in section 2 we discuss the origin of Z ′ in heterotic–string models. We discuss in some details the origin of the charge assignment under the Z ′ , which is motivated from proton decay considerations and differs from those that havetraditionally been discussed in the literature. Then we move to define the conventions inregard to the charge assignments and the Higgs structure of the models that we consider,which are characterized by a gauge structure which enlarges the gauge group of the Stan-dard Model by one extra U (1). Our numerical analysis of the invariant mass distributionsfor leptoproduction is performed by varying both the coupling of the extra U (1) and themass of the new gauge boson. The dependence on these parameters of the models thatwe discuss are studied rather carefully in a kinematic region which can be accessed at theLHC. We compare these results with those obtained for a group of 4 different models,introduced in [10], for which we perform a similar analysis using leptoproduction. Fromthis analysis it is quite evident that the search for extra neutral currents at the LHCis a rather difficult enterprise in leptoproduction, unless the coupling of the new gaugeinteraction is quite sizeable. Z ′ Phenomenological string models can be built in the heterotic–string or, using brane con-structions, in the type I string. The advantage of the former is that it produces states inspinorial representations of the gauge group, and hence allows for the SO (10) embeddingof the matter spectrum. The ten dimensional supersymmetric heterotic–string vacua giverise to effective field theories that descend from the E × E or SO (32) gauge groups. Thefirst case gives rise to additional Z ′ s that arise in the SO (10) and E extensions of the Stan-dard Model, and are the cases mostly studied in the literature [1]. A basis for the extra Z ′ arising in these models is formed by the two groups U (1) χ and U (1) ψ via the decomposition E → SO (10) × U (1) ψ and SO (10) → SU (5) × U (1) χ [1]. Additional, flavor non–universal U (1)’s, may arise in heterotic E × E string models from the U (1) currents in the Cartansubalgebra of the four dimensional gauge group, that are external to E . Non–universal2 ′ s typically must be beyond the LHC reach, to avoid conflict with Flavor Changing Neu-tral Currents (FCNC) constraints. Recently [4] a novel Z ′ in quasi–realistic string modelsthat do not descend from the heterotic E × E string has been identified. Under the new U (1) symmetry left–handed components and right–handed components in the 16 spinorial SO (10) representation, of each Standard Model generation, have charge − / / U (1) is family universal and anomaly free. It arises inleft-right symmetric string models [9], in which the SO (10) symmetry is broken directlyat the string level to SU (3) × U (1) B − L × SU (2) L × SU (2) R × U (1) Z ′ × U (1) n × hidden[9]. The U (1) n are flavor dependent U (1)s that are broken near the string scale. TheStandard Model matter states are neutral under the hidden sector gauge group, which inthese string models is typically a rank eight group. It is important to note that the factthat the spectrum is derived from a string vacuum that satisfies the modular invarianceconstraints, establishes that the model is free from gauge and gravitational anomalies.The pattern of U (1) Z ′ charges in the quasi–realistic string models of ref. [9] does not arisein related string models in which the SO (10) symmetry is broken to the SU (5) × U (1)[6], the SO (6) × SO (4) [7], or SU (3) × SU (2) × U (1) [8], subgroups. The reason for thedistinction of the left–right symmetric string models is the boundary condition assignmentto the world–sheet free fermions that generate the SO (10) symmetry in the basis vectorsthat break the SO (10) symmetry to one of its subgroups. The world–sheet fermions thatgenerate the rank eight observable gauge group in the free fermionic models are denotedby { ¯ ψ , ··· , , ¯ η , , } , where ¯ ψ , ··· , generate an SO (10) symmetry, and ¯ η , , produce three U (1) currents . Additional observable gauged U (1) currents may arise at enhanced sym-metry points of the compactified six dimensional lattice. The SO (10) gauge group isbroken to one of its subgroups SU (5) × U (1), SO (6) × SO (4) or SU (3) × SU (2) × U (1) by the assignment of boundary conditions to the set ¯ ψ ··· :1 . b { ¯ ψ ··· ¯ η , , } = {
12 12 12 12 12 12 12 12 } ⇒ SU (5) × U (1) × U (1) , (1)2 . b { ¯ ψ ··· ¯ η , , } = { } ⇒ SO (6) × SO (4) × U (1) . To break the SO (10) symmetry to SU (3) C × SU (2) L × U (1) C × U (1) L both steps, 1and 2, are used, in two separate basis vectors. The breaking pattern SO (10) → SU (3) C × SU (2) L × SU (2) R × U (1) B − L is achieved by the following assignment in two separate basisvectors1 . b { ¯ ψ ··· ¯ η , , } = { } ⇒ SO (6) × SO (4) × U (1) , (2)2 . b { ¯ ψ ··· ¯ η , , } = {
12 12 12 00 12 12 12 } ⇒ SU (3) C × U (1) C × SU (2) L × SU (2) R × U (1) for reviews and the notation used in free fermionic string models see e.g. [5] and references therein. U (1) C = U (1) B − L ; U (1) L = 2 U (1) T R . U (1) symmetries U (1) , , that arise from the three world–sheet fermions ¯ η , , . Inthe free fermionic models, the states of each Standard Model generation fit into the 16representation of SO (10), and are charged with respect to one of the three flavor U (1)symmetries. For the symmetry breaking pattern given in eq. (1) the charge is always+1 / i.e. Q j (16 = { Q, L, U, D, E, N } ) = + 12 (3)whereas for the symmetry breaking pattern in eq. (2) the charges are Q j ( Q L , L L ) = − Q j ( Q R = { U, D } , L R = { E, N } ) = + 12 (4)As a result in the models admitting the symmetry breaking pattern eq. (1) the combina-tion U (1) ζ = U (1) + U (1) + U (1) . (5)is anomalous, whereas in the models admitting the symmetry breaking pattern (2) it isanomaly free. The distinction between the two boundary condition assignments given ineqs. (1) and (2), and the consequent symmetry breaking patterns, is important for thefollowing reason. Whereas the first is obtained from an N = 4 vacuum with E × E or SO (16) × SO (16) gauge symmetry, arising from the { ¯ ψ , ··· , , ¯ η , , ¯ φ , ··· , } world–sheetfermions, which generate the observable and hidden sectors gauge symmetries, the secondcannot be obtained from these N = 4 vacua, but rather from an N = 4 vacuum with SO (16) × E × E gauge symmetry, where we have included here also the symmetry arisingfrom the compactified lattice at the enhanced symmetry point. The important fact fromthe point of view of the Z ′ phenomenology in which we are interested is that the first casegives rise to the type of string inspired Z ′ that arises in models with an underlying E symmetry. Whereas the E may be broken at the string level, rather than in the effectivelow energy field theory, the crucial point is that the charge assignment of the StandardModel states is fixed by the underlying E symmetry. The entire literature on stringinspired Z ′ studies this type of E inspired Z ′ . The second class, however, is novel andhas not been studied in the literature. In this respect it would be interesting to examinehow the symmetry breaking pattern (2) and the corresponding charge assignments (4)can be obtained in heterotic orbifold models in which one starts from a ten dimensionaltheory and compactifies to four dimensions, rather than starting directly with a theoryin four dimensions, as is done in the free fermionic models. This understanding may4ighlight the relevance of ten dimensional backgrounds that have thus far been ignoredin the literature. From the point of view of the Z ′ phenomenology, which is our interesthere, the crucial point will be to resolve between the different Z ′ models and the fermioncharges, which will reveal the relevance of a particular symmetry breaking pattern.The existence of the extra Z ′ at low energies, within reach of the LHC, is motivated byproton longevity, and the suppression of the proton decay mediating operators [2, 3, 4].The important property of this Z ′ is that it forbids dimension four, five and six protondecay mediating operators. The extra U (1) is anomaly free and family universal. It allowsthe fermions Yukawa couplings to the Higgs field and the generation of small neutrinomasses via a seesaw mechanism. String models contain several U (1) symmetries thatsuppress the proton decay mediating operators [3]. However, these are typically non–family universal. They constrain the fermion mass terms and hence must be broken ata high scale. Thus, the existence of a U (1) symmetry that can remain unbroken downto low energies is highly nontrivial. The U (1) symmetry in ref. [9, 4] satisfies all ofthese requirements. Furthermore, as the generation of small neutrino masses in the stringmodels arises from the breaking of the B − L current, the extra U (1) allows lepton numberviolating terms, but forbids the baryon number violating terms. Hence, it predicts that R –parity is violated and its phenomenological implications for SUSY collider searchesdiffer substantially from models in which R –parity is preserved. The charges of theStandard Model states under the Z ′ are displayed in table 9. Also displayed in the tableare the charges under U (1) ζ ′ = U C − U L , which is the Abelian combination of the Cartangenerators of the underlying SO (10) symmery that is orthogonal to the weak hypercharge U (1) Y . The charges under the U (1) combination given in eq. (5) are displayed in table9 as well. These two U (1)’s are broken by the VEV that induces the seesaw mechanism,and the combination U (1) Z ′ = 15 U (1) ζ ′ − U (1) ζ (6)is left unbroken down to low energies in order to suppress the proton decay mediatingoperators. The charges of the Standard Model states under this U (1) Z ′ are displayed intable 9. U (1) Z ′ In this section we fix our conventions and describe the structure of the new neutral sectorthat we are going to analyze numerically in leptoproduction afterwards. The notationsare the same both in the case of the string model and for the other models that we willinvestigate. We show in (9) the field content of the string model obtained within the free5ermionic construction discussed above. Of the 3 extra U (1), we will decouple the twogauge bosons denoted by ζ , ζ ′ and keep only the Z ′ . The assumption of decoupling ofthese extra components are realistic if they are massive enough ( > M Z ′ around 0 . Z ′ interaction is given by X f z f g z ¯ f γ µ f Z ′ µ , (7)where f = e jR , l jL , u jR , d jR , q jL and q jL = ( u jL , d jL ) , l jL = ( ν jL , e jL ). The coefficients z u , z d arethe charges of the right-handed up and down quarks, respectively, while the z q coefficientsare the charges of the left-handed quarks. g z is the Z ′ coupling constant. We can writethe Lagrangean for the Z ′ -lepton-quark interactions as follows L Z ′ = X j g z Z ′ µ h z e jR ¯ e jR γ µ e jR + z l jL ¯ l jL γ µ l jL + z u jR ¯ u jR γ µ u jR + z d jR ¯ d jR γ µ d jR + z q jL ¯ Q jL γ µ Q jL i , (8)with j being the generation index. The low energy spectrum of the model, as discussedabove, is assumed to be the same for the other models that we analyze in parallel. Asshown in (9) the field content of the model is effectively that of the Standard Modelplus 1 additional Higgs doublet. The extra scalars φ , and ζ H , ¯ ζ H and the right handedcomponents N H and ¯ N H are assumed to decouple. In this simplified framework, thestructure of the vertex 6 ield U (1) Y U (1) ζ ′ U (1) ζ U (1) Z ′ Q i
16 12 −
12 35 L i − − −
12 15 U i −
23 12 12 − D i −
32 12 − E i
12 12 − N i
52 12 φ i φ H U − − H D − N H
52 12 N H − ζ H ζ H − − − ig θ W ¯ ψ i γ µ ( g Z,Z ′ V + g Z,Z ′ A γ ) ψV µ , (10)where V µ denotes generically the vector boson. In the Standard Model (SM) v γu = 23 a γu = 0 v γd = − a γd = 0 v Zu = 1 −
83 sin θ W a Zu = − v Zd = − θ W a Zd = 1 . (11)We need to generalize this formalism to the case of the Z ′ .Our starting point is the covariant derivative in a basis where the three electrically-neutral gauge bosons W µ , B µY , B µz areˆ D µ = h ∂ µ − ig (cid:0) W µ T + W µ T + W µ T (cid:1) − i g Y Y B µY − i g z zB µz i (12)and we denote with g, g Y , g z the couplings of SU (2), U (1) Y and U (1) z , with tan θ W = g Y /g . After the diagonalization of the mass matrix we have A µ Z µ Z ′ µ = sin θ W cos θ W θ W − sin θ W ε − ε sin θ W ε sin θ W W µ B Yµ B zµ (13)7here ε is defined as a perturbative parameter ε = δM ZZ ′ M Z ′ − M Z M Z = g θ W ( v H + v H ) (cid:2) O ( ε ) (cid:3) M Z ′ = g z z H v H + z H v H + z φ v φ ) (cid:2) O ( ε ) (cid:3) δM ZZ ′ = − gg z θ W ( z H v H + z H v H ) . (14)Then we define g = e sin θ W g Y = e cos θ W , (15)and we construct the W ± charge eigenstates and the corresponding generators T ± asusual W ± = W ∓ iW √ T ± = T ± iT √ , (16)with the rotation matrix W µ B Yµ B zµ = sin θ W (1+ ε )1+ ε cos θ W ε ε cos θ W ε cos θ W (1+ ε )1+ ε − sin θ W ε ε sin θ W ε ε ε ε A µ Z µ Z ′ µ (17)from the interaction to the mass eigenstates. Substituting these expression in the covariantderivative we obtainˆ D µ = " ∂ µ − iA µ gT sin θ W + g Y cos θ W ˆ Y ! − ig (cid:0) W − µ T − + W + µ T + (cid:1) − iZ µ g cos θ W T − g Y sin θ W ˆ Y g z ε ˆ z ! − iZ ′ µ − g cos θ W T ε + g Y sin θ W ˆ Y ε + g z ˆ z ! (18)where we have neglected all the O ( ε ) terms. Sending g z → ε → Z ′ boson. Hence for thequarks and the leptons we can write an interaction Lagrangean of the type L int = ¯ Q jL N ZL γ µ Q jL Z µ + ¯ Q jL N Z ′ L γ µ Q jL Z ′ µ + ¯ u jR N Zu,R γ µ u jR Z µ e-050.00010.0010.010.1799 799.5 800 800.5 801 d σ / d Q [ pb / G e V ] Q [GeV]NNLO FF model, g z = 0.05,tan β =10 NLOLOLO SMNLO SMNNLO SM Figure 1: Plot of the LO, NLO and NNLO cross section for the free fermionic model with M Z ′ = 800 GeV. + ¯ d jR N Zd,R γ µ d jR Z µ + ¯ u jR N Z ′ u,R γ µ u jR Z ′ µ + ¯ d jR N Z ′ d,R γ µ d jR Z ′ µ + ¯ Q jL N γL γ µ Q jL A µ + ¯ u jR N γu,R γ µ u jR A µ + ¯ d jR N γd,R γ µ d jR A µ +¯ l jL N γL γ µ l jL A µ + ¯ e jR N γe,R γ µ e jR A µ +¯ l jL N ZL,lep γ µ l jL Z µ + ¯ l jL N Z ′ L,lep γ µ l jL Z ′ µ +¯ e jR N Ze,R γ µ e jR Z µ + ¯ e jR N Z ′ e,R γ µ e jR Z ′ µ (19)where for the quarks we have N Z,jL = − i g cos θ W T L − g Y sin θ W ˆ Y L g z ε ˆ z L ! N Z ′ ,jL = − i − g cos θ W T L ε + g Y sin θ W ˆ Y L ε + g z ˆ z L ! N Zu,R = − i − g Y sin θ W ˆ Y u,R g z ε ˆ z u,R ! N Zd,R = − i − g Y sin θ W ˆ Y d,R g z ε ˆ z d,R ! , (20)and similar expressions for the leptons. We rewrite the vector and the axial coupling of9he Z and Z ′ bosons to the quarks as − ig c w γ µ g V Z,j = − igc w " c w T L,j − s w ( ˆ Y jL Y jR ε g z g c w ( ˆ z L,j z R,j γ µ − ig c w γ µ γ g AZ,j = − igc w " − c w T L,j − s w ( ˆ Y jR − ˆ Y jL ε g z g c w ( ˆ z R,j − ˆ z L,j γ µ γ − ig c w γ µ g V Z ′ ,j = − igc w " − εc w T L,j + εs w ( ˆ Y jL Y jR g z g c w ( ˆ z L,j z R,j γ µ − ig c w γ µ γ g AZ ′ ,j = − igc w " εc w T L,j + εs w ( ˆ Y jR − ˆ Y jL g z g c w ( ˆ z R,j − ˆ z L,j γ µ γ , (21)where j is an index which represents the quark or the lepton and we have set sin θ W = s w , cos θ W = c w for brevity.The decay rates into leptons for the Z and the Z ′ are universal and are given byΓ( Z → l ¯ l ) = g πc w M Z h ( g Z ,lV ) + ( g Z ,lA ) i = α em s w c w M Z h ( g Z ,lV ) + ( g Z ,lA ) i , Γ( Z → ψ i ¯ ψ i ) = N c α em s w c w M Z h ( g Z ,ψ i V ) + ( g Z ,ψ i A ) i × (cid:20) α s ( M Z ) π + 1 . α s ( M Z ) π − . α s ( M Z ) π (cid:21) , (22)where i = u, d, c, s and Z = Z, Z ′ .For the Z ′ and Z decays into heavy quarks we obtainΓ( Z → b ¯ b ) = N c α em s w c w M Z h ( g Z ,bV ) + ( g Z ,bA ) i × (cid:20) α s ( M Z ) π + 1 . α s ( M Z ) π − . α s ( M Z ) π (cid:21) , Γ( Z → t ¯ t ) = N c α em s w c w M Z s − m t M Z × (cid:20) ( g Z ,tV ) (cid:18) m t M Z (cid:19) + ( g Z ,tA ) (cid:18) − m t M Z (cid:19)(cid:21) × (cid:20) α s ( M Z ) π + 1 . α s ( M Z ) π − . α s ( M Z ) π (cid:21) . (23)The total hadronic widths are defined byΓ Z ≡ Γ( Z → hadrons ) = X i Γ( Z → ψ i ¯ ψ i )10 e-05 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 d σ / d Q [ pb / G e V ] Q [TeV]NNLO FF model, g z = 0.1, tan β = 40 NLOLOLO SMNLO SMNNLO SM Figure 2: Plot of the LO, NLO and NNLO cross section for the free fermionic model with M Z ′ = 800 GeV in the TeVs region.Γ Z ′ ≡ Γ( Z ′ → hadrons ) = X i Γ( Z ′ → ψ i ¯ ψ i ) (24)where we refer to hadrons not containing bottom and top quarks (i.e. i = u, d, c, s ).We also ignore electroweak corrections and all fermion masses with the exception of thetop-quark mass, while we have included the relevant QCD corrections. Similarly to [10]we have considered only tree level decays into fermions, assuming that the decays intoparticles other than the SM fermions are either invisible or are negligible in their branchingratios, then the total decay rate for the Z and Z ′ is given byΓ Z = X i = u,d,c,s Γ( Z → ψ i ¯ ψ i ) + Γ( Z → b ¯ b ) + 3Γ( Z → l ¯ l ) + 3Γ( Z → ν l ¯ ν l )Γ Z ′ = X i = u,d,c,s Γ( Z ′ → ψ i ¯ ψ i ) + Γ( Z ′ → b ¯ b ) + Γ( Z ′ → t ¯ t ) + 3Γ( Z ′ → l ¯ l ) + 3Γ( Z ′ → ν l ¯ ν l ) . (25)We also recall that the point-like cross sections for the photon, the SM Z and thenew Z ′ gauge boson are written as σ γ ( Q ) = 4 πα em Q N c σ Z ( Q , M Z ) = πα em M Z sin θ W cos θ W N c Γ Z → ¯ ll ( Q − M Z ) + M Z Γ Z e-050.00010.0010.010.1799 799.5 800 800.5 801 d σ / d Q [ pb / G e V ] Q [GeV]NNLO free ferm. model, tan β = 40, g z = 0.2 g z = 0.1g z = 0.05 Figure 3: Free fermionic model at the LHC, tan β = 40 σ Z,γ ( Q , M Z ) = πα em − θ W )sin θ W cos θ W ( Q − M Z ) N C Q ( Q − M Z ) + M Z Γ Z , (26)where N C is the number of colours, and σ Z ′ ( Q ) = πα em M Z ′ sin θ W cos θ W N c Γ Z ′ → ¯ ll ( Q − M Z ′ ) + M Z ′ Γ Z ′ σ Z ′ ,γ ( Q ) = πα em N c g Z ′ ,lV g γ,lV sin θ W cos θ W ( Q − M Z ′ ) Q ( Q − M Z ′ ) + M Z ′ Γ Z ′ ,σ Z ′ ,Z ( Q ) = πα em h g Z ′ ,lV g Z,lV + g Z ′ ,lA g Z,lA i sin θ W cos θ W N c ( Q − M Z )( Q − M Z ′ ) + M Z Γ Z M Z ′ Γ Z ′ [( Q − M Z ′ ) + M Z ′ Γ Z ′ ] [( Q − M Z ) + M Z Γ Z ] . (27)The contributions such as Z, γ and similar denote the interference terms. At LO (orleading order) the process proceeds through the q ¯ q annihilation channel and is O (1) inthe strong coupling constant α s . The NLO (or next-to-leading order) corrections involvevirtual corrections with one gluon exchanged in the initial state and real emissions in-volving a single gluon, which is integrated over phase space. These corrections are O ( α s )in the strong coupling. The change induced by moving from LO to NLO amounts to ap-proximately a 20 to 30 % in the numerical value of the cross section that we consider. At12he highest accuracy, we use in our analysis partonic contributions with hard scatteringcomputed at NNLO, or O ( α s ). At this order typical real emissions involve 2 partons in thefinal state - which are integrated over their phase space- and two-loop virtual correctionsat the same perturbative order. The cross section for the invariant mass distributionsfactorizes at a perturbative level in terms of a NNLO (next-to-next-to-leading, or O ( α s ))contribution W V (which takes into account all the initial state emissions of real gluons andall the virtual corrections) and a point-like cross section. The computation of W V can befound in [11] to which we refer for more details. A similar factorization holds also for thetotal cross section if we use the narrow width approximation. At NLO (next-to-leadingorder, or O ( α s )). The colour-averaged inclusive differential cross section for the reaction p + p → l + l + X , is given by dσdQ = τ σ V ( Q , M V ) W V ( τ, Q ) τ = Q S , (28)where all the hadronic initial state information is contained in the hadronic structurefunction which is defined as W V ( τ, Q ) = X i,j Z dx Z dx Z dxδ ( τ − xx x ) P D
Vi,j ( x , x , µ F )∆ i,j ( x, Q , µ F ) , (29)where the quantity P D
Vi,j ( x , x , µ F ) contains all the information about the parton distri-bution functions and their evolution up to the µ F scale, while the functions ∆ i,j ( x, Q , µ F )are the hard scatterings. This factorization formula is universal for invariant mass dis-tributions mediated by s-channel exchanges of neutral or charged currents. The hardscatterings can be expanded in a series in terms of the running coupling constant α s ( µ R )as ∆ i,j ( x, Q , µ F ) = ∞ X n =0 α ns ( µ R )∆ ( n ) i,j ( x, Q , µ F , µ R ) . (30)In principle, factorization and renormalization scales should be kept separate in orderto determine the overall scale dependence of the results. However, as we are going toshow, the high-end of the Drell-Yan distribution is not so sensitive to these higher ordercorrections, at least for the models that we have studied. In our analysis we have decided to compare our results with a series of models introducedin [10]. We refer to this work for more details concerning their general origin. We just13ention that the construction of models with extra Z ′ using a bottom-up approach is, ingeneral, rather straightforward, being based mostly on the principle of cancellation of thegauge cubic U (1) Z ′ and mixed anomalies. One of the most economical ways to proceed isto introduce just one additional SU (2) W Higgs doublet and an extra scalar (weak) singlet,as in [12], and one right-handed neutrino per generation in order to generate reasonableoperators for their Majorana and Dirac masses. However, more general solutions of theanomaly equations are possible by enlarging the fermion spectrum and/or enlarging thescalar sector [13]. In [10] the scalar sector is enlarged with 2 Higgs doublets and one(weak) scalar singlet.Anomalous constructions, instead, require a different approach and several phenomeno-logical analysis have been presented recently [14, 15, 16, 17] that try to identify the sig-nature of these peculiar realizations. In the anomalous models, due to the absence of thenon-resonant behaviour of the s-channel (at least in the double prompt photon produc-tion), the chiral anomaly induces a unitarity growth which should be present in correlatedstudies of other channels [17]. For non anomalous Z ′ the phenomenological predictionsare, as we are going to show, rather similar for all the models - at least in the mass invari-ant distributions in Drell-Yan - and the possibility to identify the underlying interactionrequires a careful study of the forward-backward and/or charge asymmetries [18]. Thisis not going to be an easy task at the LHC, given the size of the cross section at the tailof the invariant mass distribution, the rather narrow widths, and given the presence ofboth theoretical and experimental errors in the parton distributions (pdf’s), unless thegauge coupling is quite sizeable ( O (1)). We refer to [19] for an accurate analysis of theexperimental errors on the pdf’s in the case of the Z peak. It has been shown that theerrors on the pdf’s are comparable with the overall reduction of the cross section as wemove from the NLO to the NNLO.These source of ambiguities, known as experimental errors, unfortunately do not takeinto consideration the theoretical errors due to the implementation of the solution of theDGLAP in the evolution codes, which amount to a theoretical uncertainty [20]. Onceall these sources of indeterminations are combined together, the expected error on theZ peak is likey to be much larger than 3 %. Given the large amount of data that willaccumulate in the first runs (for Q = M Z ), which will soon reduce the statistical errors onthe measurements far below the 0.1 % value, there will be severe issues to be addressedalso from the theoretical side in order to match this far larger experimental accuracy. Thepossibility to use determinations of the pdf’s on the Z peak for further studies of the Z ′ resonances at larger invariant mass values of the lepton pair, have to face several additionalissues, such as the presence of an additional scale, which is Q = M Z ′ , new respect to the Q = M Z scale used as a benchmark for partonometry in the first accelerator runs. We14 e-050.00010.0010.010.1799 799.5 800 800.5 801 d σ / d Q [ pb / G e V ] Q [GeV]NNLO free ferm. model, g z = 0.1, tan β = 10 tan β = 20 tan β = 30 tan β = 40 Figure 4: Free fermionic model at the LHC, g z = 0 . M Z ′ is farlarger than M Z . With these words of caution in mind we proceed with our explorationof the class of models that we have selected, starting from the string model and thenanalizing the bottom-up models mentioned above [10]. These are studied in the limit z H = z H = 0, with the mass of the extra Z ′ generated only by the extra singlet scalar φ .In the string model, as one can see from (9), only the two Higgses H U and H D contributeto the mass of the new gauge boson. The differences between these two types of modelsare, however, not relevant for this analysis, since the mass of the extra gauge boson isessentially a free parameter in both cases.The set of pdf’s that we have used for our analysis is MRST2001 [21], which is givenin parametric form, evolved with CANDIA (see [22]). The models analyzed numericallyare the free fermionic one, “F”, discussed in the previous sections, and the “ B − L ”, “q+u” , “10 + ¯5” and “d-u”, using the notations of [10].Our results are organized in a series of plots on the various resonances and in sometables which are useful in order to pin down the actual numerical value of the variouscross sections at a given invariant mass. 15 .1 M Z ′ = 0 . TeV
We show in Fig. 1 a plot of the Z ′ resonance around a typical value of 800 GeV for the F F model and the SM. The coupling of the extra neutral gauge boson is taken to be 0 .
05, withtan β = 10. We remark that the dependence of the resonance on this second parameter isnegligible. In fact the relevant parameters are the coupling constant g Z and the mass M Z ′ .Notice that the width is very narrow ( ≈ f b − /y after the first 3 years at the LHC(per experiment), we would expect 10 background events versus a signal of approximately30 events. Notice that LO, NLO and NNLO determinations are, essentially, coincidentfor all the practical purposes.In Fig. 2 we show the tail of the distribution for a run with M Z ′ = 800 GeV, wherewe have just modified tan β and we have increased the coupling to g Z = 0 .
1. For Q around 1 . ≈ − fb) the possibility to resolvethese differences experimentally is remote. In Fig. 3 we vary the coupling constants ofthe extra U (1) from a very small value g Z = 0 .
05 up to g Z = 0 .
2. The only variation inthe result is due to the width that increases from 1 to approximately 3-4 GeV’s. Here wehave chosen tan β = 40, and, as shown in Fig. 4 there is essentially no variation on theshape of the resonance due to this variable. In Figs. 5 and 6 we perform a comparativestudy of all the models and the SM background for a resonance mass of 800 GeV. Thereare only minor differences between the 4 bottom-up models and the FF model. The FFmodel shows a resonance curve which sits in the middle of all the determinations but is,for the rest, overlapping with the other curves. The “ B − L ” model, in all the cases,shows a wider width among all, with the “ q + u ” model quite similar to it. The “ d − u ”model has the narrowest width. This feature is particularly obvious from Fig. 7 wherethe result is numerically smoothed out by the increased value of the coupling, which isnow doubled compared to Fig. 6. M Z ′ = 1 . TeV
We illustrate in the next 3 figures our results for the various models for M Z ′ = 1 . g Z = 0 . e-050.00010.0010.010.1799 799.5 800 800.5 801 d σ / d Q [ pb / G e V ] Q [GeV]NNLO free ferm. model, Candia evol.NNLO U(1)
B-L
NNLO U(1) q+u
NNLO U(1)
NNLO U(1) d-u
NNLO SM
Figure 5: Free fermionic model at the LHC, tan β = 40 and g z = 0 . Q values, is to stabilizethe dependence of the perturbative series from the factorization/renormalization scales.In our case we have chosen, for simplicity µ F = µ R = Q , where µ R and µ F are therenormalization and factorization scale, respectively. The separation of this dependencecan be done as in [20], by relating the coupling constants at the two scales ( µ F , µ R ).This separation, in general, needs to be done both in the hard scattering and in theevolution. A zoom of the resonance region is shown in Fig. 9, which shows that thereduction of the signal is by a factor of 10 compared to the case of M Z ′ = 0 . M Z ′ = 2 . e-050.00010.0010.010.1799 799.5 800 800.5 801 d σ / d Q [ pb / G e V ] Q [GeV]NNLO free ferm. model, Candia evol.NNLO U(1)
B-L
NNLO U(1) q+u
NNLO U(1)
NNLO U(1) d-u
NNLO SM
Figure 6: Free fermionic model at the LHC, tan β = 40 and g z = 0 . d σ / d Q [ pb / G e V ] Q [GeV]NNLO free ferm. model, Candia evol.NNLO U(1)
B-L
NNLO U(1) q+u
NNLO U(1)
NNLO U(1) d-u
NNLO SM
Figure 7: Free fermionic model at the LHC, tan β = 40 and g z = 0 . e-050.00010.001 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 d σ / d Q [ pb / G e V ] Q [TeV]NNLO FF model, M Z ' = 1.2 TeV, g z = 0.1, tan β = 40 NLOLOLO SMNLO SMNNLO SM Figure 8: Free fermionic model at the LHC, tan β = 40 and g z = 0 . M Z ′ = 1 . dσ/dQ ( Q = M Z ′ ) (on the peak) as we vary the factorization scale. In Fig.13 the scale µ f has been varied in the interval 1 / M Z ′ < µ f < M Z ′ for a mass M Z ′ = 600GeV. These variations are rather small over all the energy interval that we have analyzedand show consistently the reduction of the scale dependence of the result moving fromLO to NLO and NNLO. The cross section is sizeable in particular above the 4 TeV scale,especially for larger couplings, although the presence of the resonance is not resolved inthis figure given the small width. Finally, in Fig. 14 we plot the total cross section as afunction of the energy for 3 values of the new gauge couplings for M Z ′ =1.2 TeV. Also inthis case the rise of the cross section gets sizeable for larger value of the couplings.19 e-050.00010.0010.011.199 1.1995 1.2 1.2005 1.201 d σ / d Q [ pb / G e V ] Q [TeV]NNLO FF model, M Z ' = 1.2 TeV, g z = 0.1, tan β = 40 NLOLOLO SMNLO SMNNLO SM Figure 9: Free fermionic model at the LHC, tan β = 40 and g z = 0 .
1. Shown are also theSM results through the same perturbative orders.
We have included a set of tables which may be useful for actual experimental searchesand comparisons. In table 2 we show the LO and in table 3 the NLO results for theinvariant mass distributions for the first choice (800 GeV) of the mass of the extra Z ′ inall the models, and the corresponding value also for the SM. In all the cases the proximityamong the various determinations is quite evident, except on the resonance, where thevalues show a wide variability. The pattern at NNLO, shown in table 4 is similar, and thechanges in the cross sections from NLO to NNLO in most of the cases are around 3 % orless. These changes are of the same order of those obtained by a study of the K-factorsin the case of the Z resonance [20]. Also for this kinematical region, as on the Z peak[20], the changes from LO to NLO are around 20-30 %, and cover the bulk of the QCDcorrections. The last several tables describe the relative differences between the results ofthe various models and the SM, normalized to the SM values, at the various perturbativeorders and for 3 values of the coupling constants g Z = 0 . , . .
2. They give anindication of the role played by the changes in the coupling on the behaviour of theseobservables at the tails of the resonance region. In tables 5 and 6 the region that weexplore is between 1 and 1.5 TeV. It is rather clear from these results that for a weakly20 e-051.26 1.28 1.3 1.32 1.34 1.36 1.38 1.4 d σ / d Q [ pb / G e V ] Q [TeV]NNLO FF model, M Z ' = 1.2 TeV, g z = 0.1, tan β = 40 NLOLOLO SMNLO SMNNLO SM Figure 10: Free fermionic model at the LHC, tan β = 40 and g z = 0 . Z ′ ( g Z = 0 .
05) the NLO and NNLO variations respect to the SM result areessentially similar. The differences at NLO between the various models and the NLO SMare a fraction of a percent. Therefore, NNLO QCD corrections will not help in this regionfor such weakly coupled extra Z ′ . The differences are not more sizeable as we increase thenew gauge coupling to 0.1, as shown in 7 and 8. Both at NLO and NNLO the differencebetween the SM background and all the other models is smaller than 1 %. Things are notmuch better for a value of the coupling constant equal to 0.2. The differences between theSM and various models in this region of fast fall-off can be of the order of only 2 %, andjust for one model (“ B − L ”). Given also the small size of these cross sections, which areof the order of 3 × − fb, it is hard to separate the various contributions. Naturally, thesituation will improve considerably if we allow a larger gauge coupling since the differencesbetween signal and background can become, in principle, quite large. We performed a preliminary comparative analysis of the behaviour of several modelscontaining extra neutral currents in anomaly-free constructions and we discussed the21 e-071e-061e-050.00010.0012.499 2.5 2.501 d σ / d Q [ pb / G e V ] Q [TeV]NNLO FF model, M Z ' = 2.5 TeV, g z = 0.1, tan β = 40 NLOLOLO SMNLO SMNNLO SM Figure 11: Free fermionic model and the corresponding SM results at all the three ordersfor M Z ′ = 2 . β also do not play any significant role in these typesof searches. It is reasonable to believe that much of the potentiality for discovering thenew resonance, if found, is its width, and all the models analyzed so far show very similarpatterns, with a gauging of “ B − L ” being the one that has a slightly wider resonantbehaviour. Being the coupling so important in order to identify which model has betterchances to be confirmed or ruled out, it is necessary, especially in bottom-up constructions,to rely on more precise investigations of possible scenarios for the running of the couplings,which are not addressed in approaches of these types. In the case of the free fermionic U (1)that we have analyzed, the possibility to include these models in a more general scenariois natural, since they are naturally produced by a unification scheme, but is left for futurestudies. On the other hand, in these and similar models obtained either in the stringpicture or in Grand Unification, the decoupling of part of the “extra stuff” that would22 e-071e-061e-050.00010.0012.4 2.45 2.5 2.55 2.6 d σ / d Q [ pb / G e V ] Q [TeV]NNLO FF model, M Z ' = 2.5 TeV, g z = 0.1, tan β = 40 g z = 0.4g z = 0.6g z = 0.8g z = 1NNLO SM Figure 12: Free fermionic model and the corresponding SM results at NNLO for M Z ′ = 2 . g z larger than g z = 0 . U (1) ′ s . These assumptions wouldintroduce various alternatives on the choice of the symmetry breaking scales, thresholdenhancements, and so on, which amount, however, to important phenomenological detailswhich strongly affect this search.Since the V-A structure of the couplings exhibits differences with respect to other Z ′ models a measurement of forward-backward asymmetries and/or of charge asymmetriescould be helpful [18], but only if the gauge coupling is sizeable. The discrimination amongthe various models remains a very difficult issue for which NNLO QCD determinations,at least in leptoproduction, though useful, do not seem to be necessary in a first analysis.For those values of the mass of the extra Z ′ that we have considered these correctionscannot be isolated, while the NLO effects remain important.23 d σ / d Q ( Q = M Z ' ) [ pb / G e V ] √ S [TeV] σ Z' at the LHC, LO, 1/2 M Z' < µ f < 2 M Z' NLONNLO (a)
Figure 13: Study of the µ F scale dependence in the total cross section for the U (1) B − L model with M Z ′ = 0 . g z = 0 .
1. Here we have chosen M Z ′ = Q for semplicity. σ [f b ] √ S [TeV]FF model at NLO, g z = 0.1g z = 0.4g z = 0.8 (a) Figure 14: Total cross section for the Free fermionic model at NLO for three differentvalues of g z and for M Z ′ = 1 . µ F = µ R = Q for semplicityand we have integrated the mass invariant distribution on the interval M Z ′ ± Z ′ .24 M Z ′ ( g z ) [GeV] g z M Z ′ = 0 . M Z ′ = 1 . M Z ′ = 2 . .
02 0 .
004 0 .
005 0 . .
05 0 .
024 0 .
036 0 . . .
097 0 .
146 0 . . .
388 0 .
584 1 . . .
875 1 .
314 2 . . .
555 2 .
336 4 . . .
430 3 .
650 7 . . .
500 5 .
256 10 . . .
764 7 .
154 14 . . .
223 9 .
344 19 . . .
876 11 .
82 24 .
611 9 .
723 14 .
60 30 . Table 1: Dependence of the total width on the coupling constant g z for the free fermionicmodel with M Z ′ = 800 GeV, M Z ′ = 1 . M Z ′ = 2 . cknowledgements We thank Simone Morelli for discussions and for various cross-checks in the numericalanalysis. M.G. thanks the Theory Division at the University of Liverpool for hospitalityand the Royal Society for financial support. The work of C.C. was supported (in part)by the European Union through the Marie Curie Research and Training Network “Uni-versenet” (MRTN-CT-2006-035863) and by The Interreg II Crete-Cyprus Program. Hethanks the Theory group at Crete for hospitality. The work of A.E.F. is supported inpart the STFC. 26 σ LO / d Q [pb/GeV], M ′ Z = 800, g z = 0 .
1, tan β = 40, Candia evol. Q [GeV] σ LO ( Q ) FFM σ LO ( Q ) U (1) B − L σ LO ( Q ) U (1) q + u σ LO ( Q ) U (1) σ LO ( Q ) U (1) d − u σ LO ( SM )750 1 . · − . · − . · − . · − . · − . · −
761 1 . · − . · − . · − . · − . · − . · −
773 9 . · − . · − . · − . · − . · − . · −
784 9 . · − . · − . · − . · − . · − . · −
796 9 . · − . · − . · − . · − . · − . · −
800 1 . · − . · − . · − . · − . · − . · −
800 4 . · − . · − . · − . · − . · − . · −
801 1 . · − . · − . · − . · − . · − . · −
839 6 . · − . · − . · − . · − . · − . · −
868 5 . · − . · − . · − . · − . · − . · −
900 4 . · − . · − . · − . · − . · − . · − Table 2: LO invariant mass distributions σ NLO / d Q [pb/GeV], M ′ Z = 800, g z = 0 .
1, tan β = 40, Candia evol. Q [GeV] σ NLO ( Q ) FFM σ NLO ( Q ) U (1) B − L σ NLO ( Q ) U (1) q + u σ NLO ( Q ) U (1) σ NLO ( Q ) U (1) d − u σ NLO ( SM )750 1 . · − . · − . · − . · − . · − . · −
761 1 . · − . · − . · − . · − . · − . · −
773 1 . · − . · − . · − . · − . · − . · −
784 1 . · − . · − . · − . · − . · − . · −
796 1 . · − . · − . · − . · − . · − . · −
800 2 . · − . · − . · − . · − . · − . · −
800 6 . · − . · − . · − . · − . · − . · −
801 2 . · − . · − . · − . · − . · − . · −
839 8 . · − . · − . · − . · − . · − . · −
868 7 . · − . · − . · − . · − . · − . · −
900 5 . · − . · − . · − . · − . · − . · − Table 3: NLO distributions for 750 < Q <
900 GeV d σ NNLO / d Q [pb/GeV], M ′ Z = 800, g z = 0 .
1, tan β = 40, Candia evol. Q [GeV] σ NNLO ( Q ) FFM σ NNLO ( Q ) U (1) B − L σ NNLO ( Q ) U (1) q + u σ NNLO ( Q ) U (1) σ NNLO ( Q ) U (1) d − u σ NNLO ( SM )750 1 . · − . · − . · − . · − . · − . · −
761 1 . · − . · − . · − . · − . · − . · −
773 1 . · − . · − . · − . · − . · − . · −
784 1 . · − . · − . · − . · − . · − . · −
796 1 . · − . · − . · − . · − . · − . · −
800 2 . · − . · − . · − . · − . · − . · −
800 6 . · − . · − . · − . · − . · − . · −
801 2 . · − . · − . · − . · − . · − . · −
839 8 . · − . · − . · − . · − . · − . · −
868 7 . · − . · − . · − . · − . · − . · −
900 6 . · − . · − . · − . · − . · − . · − Table 4: NNLO distributions for 750 < Q <
900 GeV σ SMnlo − σ inlo | /σ SMnlo % , M ′ Z = 800, g z = 0 .
05, tan β = 40, Candia evol. Q [GeV] σ SMnlo ( Q )[pb/GeV] ∆ F F Mnlo % ∆ B − Lnlo % ∆ q + unlo % ∆ nlo % ∆ d − unlo %1000 3 . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − Table 5: Percentage differences at NLO. We define ∆ inlo = | σ SMnlo − σ inlo | /σ SMnlo where i = F F M, B − L, q + u,
10 + ¯5 , d − u . | σ SMnnlo − σ innlo | /σ SMnnlo % , M ′ Z = 800, g z = 0 .
05, tan β = 40, Candia evol. Q [GeV] σ SMnnlo ( Q )[pb/GeV] ∆ F F Mnnlo % ∆ B − Lnnlo % ∆ q + unnlo % ∆ nnlo % ∆ d − unnlo %1000 3 . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − Table 6: Percentage differences at NNLO. We define ∆ innlo = | σ SMnnlo − σ innlo | /σ SMnnlo . σ SMnlo − σ inlo | /σ SMnlo % , M ′ Z = 800, g z = 0 .
1, tan β = 40, Candia evol. Q [GeV] σ SMnlo ( Q )[pb/GeV] ∆ F F Mnlo % ∆ B − Lnlo % ∆ q + unlo % ∆ nlo % ∆ d − unlo %1000 3 . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − Table 7: Percentage differences at NLO for g Z = 0 .
1. Here and in the following we use the same notation of the previous tables. | σ SMnnlo − σ innlo | /σ SMnnlo % , M ′ Z = 800, g z = 0 .
1, tan β = 40, Candia evol. Q [GeV] σ SMnnlo ( Q )[pb/GeV] ∆ F F Mnnlo % ∆ B − Lnnlo % ∆ q + unnlo % ∆ nnlo % ∆ d − unnlo %1000 3 . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − Table 8: Percentage differences at NNLO for g Z = 0 . σ SMnlo − σ inlo | /σ SMnlo % , M ′ Z = 800, g z = 0 .
2, tan β = 40, Candia evol. Q [GeV] σ SMnlo ( Q )[pb/GeV] ∆ F F Mnlo % ∆ B − Lnlo % ∆ q + unlo % ∆ nlo % ∆ d − unlo %1000 3 . · − . · − . · +0 . · +0 . · − . · − . · − . · − . · +0 . · +0 . · − . · − . · − . · − . · +0 . · +0 . · − . · − . · − . · − . · +0 . · +0 . · − . · − . · − . · − . · +0 . · +0 . · − . · − . · − . · − . · +0 . · +0 . · − . · − . · − . · − . · +0 . · +0 . · − . · − . · − . · − . · +0 . · +0 . · − . · − . · − . · − . · +0 . · +0 . · − . · − . · − . · − . · +0 . · +0 . · − . · − . · − . · − . · +0 . · +0 . · − . · − Table 9: Percentage differences at NLO for g Z = 0 . | σ SMnnlo − σ innlo | /σ SMnnlo % , M ′ Z = 800, g z = 0 .
2, tan β = 40, Candia evol. Q [GeV] σ SMnnlo ( Q )[pb/GeV] ∆ F F Mnnlo % ∆ B − Lnnlo % ∆ q + unnlo % ∆ nnlo % ∆ d − unnlo %1000 3 . · − . · − . · +0 . · +0 . · − . · − . · − . · − . · +0 . · +0 . · − . · − . · − . · − . · +0 . · +0 . · − . · − . · − . · − . · +0 . · +0 . · − . · − . · − . · − . · +0 . · +0 . · − . · − . · − . · − . · +0 . · +0 . · − . · − . · − . · − . · +0 . · +0 . · − . · − . · − . · − . · +0 . · +0 . · − . · − . · − . · − . · +0 . · +0 . · − . · − . · − . · − . · +0 . · +0 . · − . · − . · − . · − . · +0 . · +0 . · − . · − Table 10: Percentage differences at NNLO for g Z = 0 . σ nnlo /dQ [pb/GeV] for the FF model with M Z ′ = 2 . β = 40, Candia evol. Q [TeV] g z = 0 . g z = 0 . g z = 0 . g z = 0 . g z = 1 σ SMnnlo ( Q )2 .
400 2 . · − . · − . · − . · − . · − . · − .
423 2 . · − . · − . · − . · − . · − . · − .
446 2 . · − . · − . · − . · − . · − . · − .
469 2 . · − . · − . · − . · − . · − . · − .
492 2 . · − . · − . · − . · − . · − . · − . . · − . · − . · − . · − . · − . · − . . · − . · − . · − . · − . · − . · − . . · − . · − . · − . · − . · − . · − . . · − . · − . · − . · − . · − . · − .
636 1 . · − . · − . · − . · − . · − . · − .
700 1 . · − . · − . · − . · − . · − . · − Table 11: NNLO cross sections for the FF model with a M Z ′ = 2 . g z larger than g z = 0 . eferences [1] For reviews and references therein see e.g. :P. Langacker, arXiv:0801.1345 ;T.G. Rizzo, hep-ph/0610104 ;A. Leike, Phys. Rep. (1999) 143;Yu.Ya.Komachenko and M.Yu.Khlopov,
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