Unique heavy lepton signature at e + e − linear collider with polarized beams
UUnique heavy lepton signature at e + e − linear collider withpolarized beams G. Moortgat-Pick, a,b, ∗ P. Osland, c † A. A. Pankov d, ‡ A. V. Tsytrinov d, § a II. Inst. of Theor. Physics, University of Hamburg,Luruper Chaussee 149, 22761 Hamburg, Germany b DESY, Notkestrasse 85, 22607 Hamburg, Germany c Department of Physics and Technology,University of Bergen, Postboks 7803, N-5020 Bergen, Norway d The Abdus Salam ICTP Affiliated Centre,Technical University of Gomel, 246746 Gomel, Belarus
Abstract
We explore the effects of neutrino and electron mixing with exotic heavy leptons in the process e + e − → W + W − within E models. We examine the possibility of uniquely distinguishing andidentifying such effects of heavy neutral lepton exchange from Z - Z (cid:48) mixing within the same classof models and also from analogous ones due to competitor models with anomalous trilinear gaugecouplings (AGC) that can lead to very similar experimental signatures at the e + e − InternationalLinear Collider (ILC) for √ s = 350, 500 GeV and 1 TeV. Such clear identification of the model ispossible by using a certain double polarization asymmetry. The availability of both beams beingpolarized plays a crucial role in identifying such exotic-lepton admixture. In addition, the sensitivityof the ILC for probing exotic-lepton admixture is substantially enhanced when the polarization ofthe produced W ± bosons is considered. PACS numbers: 12.60.-i, 12.60.Cn, 14.70.Fm, 29.20.Ej ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] a r X i v : . [ h e p - ph ] M a r . INTRODUCTION Detailed examination of the process e + + e − → W + + W − (1)at the ILC is a crucial one for studying the electroweak gauge symmetry, in particular,electroweak symmetry breaking and the structure of the gauge sector in general, and allowsto observe a manifestation of New Physics (NP) that may appear beyond the Standard Model(SM). In the SM, the process (1) is described by the amplitudes mediated by photon and Z boson exchange in the s -channel and by neutrino exchange in the t -channel. This reaction isquite sensitive to both the leptonic vertices and the trilinear couplings to W + W − of the SM Z and of any new heavy neutral boson or a new heavy lepton that can be exchanged in the s -channel or t -channel, respectively. A popular example in this regard, is represented by E models [1–6]. In particular, an effective SU (2) L × U (1) Y × U (1) Y (cid:48) model, which originatesfrom the breaking of the exceptional group E , leads to extra gauge bosons. Indeed, in thebreaking of this group down to the SM symmetry, two additional neutral gauge bosons couldappear and the lightest Z (cid:48) is defined as Z (cid:48) = Z (cid:48) χ cos β + Z (cid:48) ψ sin β (2)and can be parametrized in terms of the hypercharges of the two groups U (1) ψ and U (1) χ which are involved in the breaking of the E group into a low-energy group of rank 6: E → SO (10) × U (1) ψ → SU (5) × U (1) χ × U (1) ψ → SU (3) c × SU (2) L × U (1) Y × U (1) ψ × U (1) χ . (3)For a sufficiently large vacuum expectation value of the Higgs field an effective rank-5 model,which leads to the decomposition (see, for example Ref. [7]) SU (3) c × SU (2) L × U (1) Y × U (1) Y (cid:48) can be deduced from the rank-6 model (see below) so that one of the new gauge bosonsdecouples from low energy phenomenology. The remaining (lighter) new gauge bosons Z (cid:48) is in general a mixture of Z ψ and Z χ and is assumed to lead to measurable effects at thecollider, and an angle β specifies the orientation of the U (1) (cid:48) generator in the E groupspace, where the values β = 0 and β = π/ Z (cid:48) χ and Z (cid:48) ψ bosons, while the value β = − arctan (cid:112) / Z (cid:48) η boson originatingfrom the direct breaking of E to a rank-5 group in superstring inspired models.2nother characteristic of extended models, apart from the Z (cid:48) , is the existence of newmatter, new heavy leptons and quarks. In E models the fermion sector is enlarged, sincethe matter multiplets are in larger representations (the 27 fundamental representation),that contains, in particular, a vector doublet of leptons. From the phenomenological pointof view it is convenient to classify the fermions present in E in terms of their transformationproperties under SU (2). We denote the particles with unconventional isospin assignments(right-handed doublets) as exotic fermions. We here consider two heavy left- and right-handed SU (2) exotic lepton doublets [8, 9] NE − L , NE − R , (4)and one Z (cid:48) boson, with masses larger than M Z and coupling constants that may be differentfrom those of the SM. These leptons are called vector leptons because both the left- andright- handed components transform identically under SU (2). We also assume that the new,“exotic” fermions only mix with the standard ones within the same family (the electron andits neutrino being the ones relevant to process (1)), which assures the absence of tree-levelgeneration-changing neutral currents [10].Current lower limits on M Z (cid:48) obtained from dilepton pair production at the LHC with √ s = 8 TeV and L int ≈
20 fb − [11, 12] range in the interval ∼ . − . Z (cid:48) model being tested. Already these masses are too high for a Z (cid:48) to be directlyseen at the ILC. However, even at such high masses, Z (cid:48) exchanges can manifest themselvesindirectly via deviations of cross sections, and in general of the reaction observables, fromthe SM predictions.In this paper, we study the indirect effects induced by heavy lepton exchange in W ± pair production (1) at the ILC, with a center of mass energy √ s = 0 . − L int = 0 . − − . We also present results for a lower energy runat √ s = 350 GeV. For early papers on these effects, see Refs. [13–15]. We allow for effectsdue to extra Z (cid:48) gauge boson exchange. Indirect effects may be quite subtle, both when itcomes to distinguishing an effect from the SM, and also as far as the identification of thesource of an observed deviation is concerned, because a priori different NP scenarios maylead to the same or similar experimental signatures. Clearly, then, the discrimination of oneNP model (in our case the E ) from other possible ones needs an appropriate strategy foranalyzing the data. 3ecently, the problem of distinguishing the Z (cid:48) effects, once observed in process (1), fromthe anomalous gauge couplings, has been studied in [16]. In the AGC models, there is nonew gauge boson exchange, but the W W γ , W W Z couplings are modified with respect tothe SM values, this violates the SM gauge cancellation too and leads to deviations of thecross sections. We consider the CP-conserving set of such couplings, often referred to as κ γ , κ Z , λ γ , λ Z and δ Z [17, 18]. An alternative effective-field-theory approach to these effectswas recently presented [19].In this note, we extend the analysis of Ref. [16], considering the possibility of uniquelyidentifying the effects of heavy neutral lepton exchange from Z - Z (cid:48) mixing within the sameclass of E models. This is relevant, since in this class of models lepton mixing and Z - Z (cid:48) mixing can be simultaneously present. We also distinguish them from analogous ones due tocompetitor models with anomalous trilinear gauge couplings in the process (1) by exploitinga double polarization asymmetry that will unambiguously identify the heavy exotic-leptonmixing effects and is only accessible with the availability of both beams being polarized[21].While the high precision observables determined at LEP severely constrain the elec-troweak sector [22], they leave room for effects at the energies that are discussed here.The paper is organized as follows. In Section II, we briefly review the E models involvingadditional Z (cid:48) bosons and new heavy charged and neutral leptons and emphasize the roleof the heavy neutral lepton and boson mixings in the process (1). Then, in Sect. III wereview the structure of the polarized cross section. In Sect. IV we determine the discoveryreach on the N W e coupling constants, and in Sect. V we determine the identification reach,i.e., down to what coupling strength such a heavy neutral lepton can be distinguished fromother new-physics effects. Then, in Sect. VI we comment on the 350 GeV option, beforeconcluding in Sect. VII. This approach was recently exploited for uniquely identifying the indirect effects of s -channel sneutrinoexchange against other new physics scenarios described by contact-like effective interactions in high-energy e + e − annihilation into lepton pairs [20]. I. LEPTON AND Z − Z (cid:48) MIXINGA. Weak basis
To describe the formalism for mixing among exotic and ordinary leptons, we start fromthe leptonic SU (2) × U (1) × U (1) (cid:48) interaction: − L = e (cid:16) ˜ J µ em A µ + ˜ J µZ Z µ + ˜ J µZ (cid:48) Z (cid:48) µ (cid:17) + g √ (cid:16) ˜ J µW W µ + h.c. (cid:17) , (5)where, in the weak-eigenstate basis, and with V = γ, Z, Z (cid:48) , the currents in Eq. (5) can bewritten as: ˜ J µV = (cid:88) a ¯ ε a γ µ Q ε a ε a , ˜ J µW = (cid:88) a ¯ η a γ µ G η a ε a , (6)where the coupling matrices Q ε a and G η a of the neutral and charged currents are defined byEqs. (8) and (11) below. The superscript “0 (cid:48)(cid:48) labels the weak-eigenstate basis. Furthermore,in Eq. (5) we adopt the following notations: e = √ πα em , g = e/s W , s W = sin θ W . InEq. (6), we have introduced, with a = ( L, R ) the left- and right-handed helicities, thecharged and neutral leptons by means of the notation: ε a = e a E a , η a = ν a N a , (7)where e and ν are the ordinary SM electron and neutrino, and E and N are the exoticcharged and neutral heavy leptons, which we assume to be doublets under electroweak SU (2). Furthermore, the neutral current couplings are represented by the matrices Q ε a = Q ε em ,a ; g ε a ; g (cid:48) ε a , with: Q ε em ,a = − − , g ε a = g e a g E a , g (cid:48) ε a = g (cid:48) e a g (cid:48) E a , (8)for the γ , Z and Z (cid:48) , respectively, where ( ε = e , E ) g ε a = ( T ε a − Q ε em ,a s W ) g Z , (9)and T ε a is the third isospin component. Furthermore, g Z = 1 /s W c W , with c W = cos θ W . The needed fermion mixing formalism has been introduced also, e.g., in [15]. Z (cid:48) couplings to fermions in E models, we follow the notation of [15]: g (cid:48) e L = (3 A + B ) g Z (cid:48) , g (cid:48) e R = ( A − B ) g Z (cid:48) ,g (cid:48) E L = ( − A − B ) g Z (cid:48) , g (cid:48) E R = ( − A + 2 B ) g Z (cid:48) , (10)where g Z (cid:48) = 1 /c W , A = cos β/ (2 √ B = √
10 sin β/ G η a = G ν a G N a (11)with G ν L = 1, G ν R = 0, G N a = − T E a . B. Fermion mass basis
We introduce mass eigenstates in the same notation as (7): ε a = e a E a , η a = ν a N a . (12)These states are related to the weak eigenstates (7) by the following transformations: ε a = U ( ψ a ) ε a ; η a = U ( ψ a ) η a , (13)where the unitary mixing matrices U ( ψ a ) and U ( ψ a ) diagonalize, respectively, the chargedand neutral fermion mass matrices. U ( ψ a ) and U ( ψ a ) can be written as: U ( ψ a ) = cos ψ a sin ψ a − sin ψ a cos ψ a ≡ c a s a − s a c a , (14) U ( ψ a ) = cos ψ a sin ψ a − sin ψ a cos ψ a ≡ c a s a − s a c a . (15)Present limits on s a and s a are in general less than 1-2% [9, 23, 24] and m N >
100 GeV[10]. In the fermion-mass-eigenstate basis one can rewrite the interaction Lagrangian (5) as: − L = e (cid:0) J µ em A µ + J µZ Z µ + J µZ (cid:48) Z (cid:48) µ (cid:1) + g √ J µW W µ + h.c.) , (16)where J µV = (cid:88) a ¯ ε a γ µ Q εa ε a , J µW = (cid:88) a ¯ η a γ µ G ηa ε a . (17)6ince the gauge fields of Eq. (16) are the same as those of (5), we must have Q εa = U ( ψ a ) Q ε a U − ( ψ a ) , G ηa = U ( ψ a ) G η a U − ( ψ a ) , (18)and Q εa = Q εem,a , g εa , g (cid:48) εa , with g εa = g ea g eEa g eEa g Ea , g (cid:48) εa = g (cid:48) ea g (cid:48) eEa g (cid:48) eEa g (cid:48) Ea , G ηa = G νa G νEa G Nea G Na . (19)It is clear that the electromagnetic current remains diagonal under the rotation (18), andtherefore is not affected by lepton mixing.In the weak charged currents of Eq. (17) the exotic-lepton mixings modify not only theleft-handed currents but also induce an admixture with the right-handed currents. The off-diagonal term in J µW of Eqs. (17)–(19) induces N W e couplings which allow an additional t -channel exotic-lepton-exchange contribution for the process (1) (see Fig. 1). Parametrizationof the mixing-modified νW e and the mixing-induced N W e couplings are summarized inEqs. (21) and (22), respectively.From (18) and (19) one can obtain expressions for the lepton coupling constants: g ea = g e a c a + g E a s a , g (cid:48) ea = g (cid:48) e a c a + g (cid:48) E a s a ; (20) G νL = c L c L − T E L s L s L , G νR = − T E R s R s R ; (21) G NeL = − s L c L − T E L c L s L , G NeR = − T E R c R s R . (22) C. Z - Z (cid:48) mixing Concerning Z - Z (cid:48) mixing, it can be parametrized as Z Z = cos φ sin φ − sin φ cos φ ZZ (cid:48) , (23)where Z, Z (cid:48) are weak eigenstates, Z , Z are mass eigenstates and φ is the Z - Z (cid:48) mixingangle. Finally, taking Eq. (23) into account, the lepton neutral current couplings to Z and Z are, respectively [15]: g e a = g ea cos φ + g (cid:48) ea sin φ ; g e a = − g ea sin φ + g (cid:48) ea cos φ. (24)Current limits are of the order φ = (2 − × − [10].7 II. POLARIZED CROSS SECTION
In the Born approximation the process (1) is described by the set of five diagrams shownin Fig. 1 and corresponding to mass-eigenstate exchanges (i.e. γ , ν , N , Z and Z ), withcouplings given by Eqs. (20)-(22) and (24). FIG. 1: Feynman diagrams.
The polarized cross section for the process (1) can be written as [15] dσ (cid:0) P − L , P + L (cid:1) d cos θ = 14 (cid:20)(cid:0) P − L (cid:1) (cid:0) − P + L (cid:1) dσ RL d cos θ + (cid:0) − P − L (cid:1) (cid:0) P + L (cid:1) dσ LR d cos θ + (cid:0) P − L (cid:1) (cid:0) P + L (cid:1) dσ RR d cos θ + (cid:0) − P − L (cid:1) (cid:0) − P + L (cid:1) dσ LL d cos θ (cid:21) , (25)where P − L ( P + L ) are degrees of longitudinal polarization of e − ( e + ), θ the scattering angleof the W − with respect to the e − direction. The superscript “RL” refers to a right-handedelectron and a left-handed positron, and similarly for the other terms. The relevant polarizeddifferential cross sections for e − a e + b → W − α W + β contained in Eq. (25) can be expressed as[15, 25] dσ abαβ d cos θ = C k =2 (cid:88) k =0 F abk O k αβ , (26)where C = πα e.m. β W / s , β W = (1 − M W /s ) / the W velocity in the CM frame, and thehelicities of the initial e − e + and final W − W + states are labeled as ab = ( RL, LR, LL, RR )and αβ = ( LL, T T, T L ), respectively. The O k are functions of the kinematical variablesdependent on energy √ s , the scattering angle θ and the W mass, M W , which characterizethe various possibilities for the final W + W − polarizations ( T T, LL, T L + LT or the sumover all W + W − polarization states for unpolarized W ’s).The F k are combinations of lepton and trilinear gauge boson couplings, g W W Z and g W W Z , including lepton and Z - Z (cid:48) mixing as well as propagators of the intermediate states.8or instance, for the LR case one finds F LR = 116 s W (cid:104) ( G νL ) + r N (cid:0) G NeL (cid:1) (cid:105) ,F LR = 2 [1 − g W W Z g e L χ − g W W Z g e L χ ] ,F LR = − s W (cid:104) ( G νL ) + r N (cid:0) G NeL (cid:1) (cid:105) [1 − g W W Z g e L χ − g W W Z g e L χ ] , (27)where the χ j ( j = 1 ,
2) are the Z and Z propagators, i.e. χ j = s/ ( s − M j + iM j Γ j ), r N = t/ ( t − m N ), with t = M W − s/ s cos θ β W /
2, and m N is the neutral heavy lepton mass.Also, in Eq. (27), g W W Z = g W W Z cos φ and g W W Z = − g W W Z sin φ where g W W Z = cot θ W .Note that Eq. (27) is obtained in the approximation where the imaginary parts of the Z and Z boson propagators are neglected, which is fully appropriate far away from the poles.(Accounting for this effect would require the replacements χ j → Re χ j and χ j → | χ j | onthe right-hand side of Eq. (27).)Since the gauge eigenstate Z (cid:48) is neutral under SU (2) L and does not couple to the W + W − pair, the process (1) is sensitive to a Z (cid:48) only in the case of a non-zero Z - Z (cid:48) mixing. Moreover,as one can easily see from the formulae above, the s -channel Z and the t -channel N exchangeamplitudes arise only in the case of non-vanishing mixing angles. In this case, the expressionfor the SM cross section [25] can be obtained from (25) in the limit of vanishing mixing angles.The first term F LR describes the contributions to the cross section caused by neutrino ν and heavy neutral lepton N exchanges in the t -channel while the second one, F LR , isresponsible for s -channel exchange of the photon γ and the gauge bosons Z and Z . Theinterference between s - and t -channel amplitudes is contained in the term F LR . The RL case is simply obtained from Eq. (27) by exchanging L → R .For the LL and RR cases there is only N -exchange contribution, F LL = F RR = 116 s W r N (cid:0) G NeL G NeR (cid:1) . (28)Concerning the O k αβ multiplying the expression in Eq. (28) (see Eq. (26)) their explicitexpressions for polarized and unpolarized final states W + W − can be found in, e.g. [15]. IV. DISCOVERY REACH ON HEAVY LEPTON COUPLINGS
We take “discovery” of new physics to mean exclusion of the Standard Model at a givenconfidence level. In the following, this will be the 95% C.L.9 . No Z - Z (cid:48) mixing Let us start the analysis with a case where there is only lepton mixing and no Z - Z (cid:48) mixing, i.e., φ = 0. Since the mixing angles are bounded by s i at most of order 10 − , wecan expect that retaining only the terms of order s , s and s s in the cross section (25)should be an adequate approximation. To do that we expand the couplings of Eqs. (20)-(22)taking Eq. (9) into account. We find for E models, where T E L = T E R = − / G NeL = s L − s L , G NeR = s R g eL = g e L , g eR = g e R −
12 ( G NeR ) g Z ,G νL = G ν L − ( G NeL ) , G νR = s R s R . (29)From Eqs. (27)-(29) one can see that in the adopted approximation the cross section (25)allows to constrain basically the pair of heavy lepton couplings squared, (( G NeL ) , ( G NeR ) ), itis not possible to constrain s R , which represents mixing in the right-handed neutral-leptonsector.The sensitivity of the polarized differential cross section (25) to the couplings G NeL and G NeR is evaluated numerically by dividing the angular range | cos θ | ≤ .
98 into 10 equal bins,and defining a χ function in terms of the expected number of events N ( i ) in each bin for agiven combination of beam polarizations [16]: χ = (cid:88) { P − L , P + L } bins (cid:88) i (cid:20) N SM+NP ( i ) − N SM ( i ) δN SM ( i ) (cid:21) , (30)where N ( i ) = L int σ i ε W with L int the time-integrated luminosity. Furthermore, σ i = σ ( z i , z i +1 ) = z i +1 (cid:90) z i (cid:18) dσdz (cid:19) dz, (31)where z = cos θ and polarization indices have been suppressed. Also, ε W is the efficiencyfor W + W − reconstruction, for which we take the channel of lepton pairs ( eν + µν ) plus twohadronic jets, giving ε W (cid:39) . N SM ( i ) and N SM+NP ( i ).The uncertainty on the number of events δN SM ( i ) combines both statistical and system-atic errors where the statistical component is determined by δN statSM ( i ) = (cid:112) N SM ( i ). Con-cerning systematic uncertainties, an important source is represented by the uncertainty on10eam polarizations, for which we assume δP − L /P − L = δP + L /P + L = 0 .
5% with the “standard”envisaged values | P − L | = 0 . | P + L | = 0 . W ± pairs which we assume to be δε W /ε W = 0 . W ± pair production in the flux functionapproach [26–30] that assures a good approximation within the expected accuracy of thedata.As a criterion to derive constraints on the coupling constants in the case where no de-viations from the SM were observed within the foreseeable uncertainties on the measurablecross sections, we impose that χ ≤ χ + χ , (32)where χ is a number that specifies the chosen confidence level, and χ is the minimalvalue of the χ function. ! G LNe " ! ! G R N e " ! TL " LTLL UNP
FIG. 2: Discovery reach (95% C.L.) on the heavy neutral lepton couplings ( G NeL ) and ( G NeR ) obtained from differential polarized cross sections with ( P − L = ± . , P + L = ∓ .
6) and different setsof W ± polarizations. Here, √ s = 0 . L int = 0 . − and m N = 0 . From the numerical procedure outlined above, we obtain the allowed regions in ( G NeL ) and ( G NeR ) determined from the differential polarized cross sections with different sets of11olarization (as well as from the unpolarized process (1)) depicted in Fig. 2, where L int =500 fb − has been taken [21].The results of a further potential extension of the present analysis are also shown in Fig. 2where the feasibility of measuring polarized W ± states in the process (1) is assumed. Thisassumption is based on the experience gained at LEP2 on measurements of W polarisation[31]. The method exploited for the measurement of W polarisation is based on the spindensity matrix elements that allow to obtain the differential cross sections for polarised W bosons. Information on spin density matrix elements as functions of the W − productionangle with respect to the electron beam direction was extracted from the decay angles ofthe charged lepton in the W − ( W + ) rest frame. The relevant theoretical framework formeasurement of W ± polarisation was described in [18, 25].In Fig. 2, we consider different cases of polarized W s, with W L and W T referring to longitu-dinally and transversely polarized W s, respectively. As shown in the figure, dσ ( W + L W − L ) /dz is most sensitive to the parameters ( G NeL ) and ( G NeR ) while dσ ( W + T W − T ) /dz has the low-est sensitivity to those parameters. The bounds on heavy lepton couplings obtained from dσ ( W + T W − T ) /dz are not presented here as they are outside of the range shown in Fig. 2. Therole of W polarization is seen to be essential in order to set meaningful finite bounds on the N W e couplings.The obtained bounds are reminiscent of arcs of circles in the ( G NeL ) -( G NeR ) plane. Thisreflects the fact that the deviations in the LR and RL cross sections are approximatelythe same for the right-handed and left-handed couplings (recall that T E L = T E R ) and thusapproximately behave as ( G NeL ) + ( G NeR ) .In this Fig. 2, we considered a fairly low mass, m N = 0 . m N . Thepoint is that the deviation of the cross section induced by the lepton mixing, from the SMprediction can be expressed, e.g., for the LR case, as∆ σ LR ≡ σ NP − σ SM ∝ ( G NeL ) (1 − r N ) , (33)where we have used Eqs. (27) and (29). This structure (1 − r N ) arises from negative inter-ference between a mixing contribution to ν exchange and the N -exchange contribution. Itreflects the decreasing impact of the heavy neutrino exchange contribution to ∆ σ LR , sinceat large values of m N the last term will be small. This leads to a better sensitivity on the12ixing angles with increasing m N . The analogous dependence also holds for ∆ σ RL case. (cid:72) G LNe (cid:76) (cid:45) (cid:72) G R N e (cid:76) (cid:45) m N (cid:61) m N (cid:174)(cid:165) FIG. 3: Same as in Fig. 2 but obtained from the differential polarized cross sections dσ ( W + L W − L ) /dz only, with ( P − L = ± . , P + L = ∓ .
6) and different values of the lepton mass m N = 0 . m N → ∞ . Here, √ s = 0 . L int = 0 . − . B. Including Z - Z (cid:48) mixing Now we turn to the generic case where both lepton mixing and Z - Z (cid:48) mixing occur, so thatthe leptonic coupling constants are as in Eq. (24) and the Z , Z couplings to W ± are as inEq. (27). In this case, in order to evaluate the influence of the Z - Z (cid:48) mixing on the alloweddiscovery region on the heavy lepton coupling plane (( G NeL ) , ( G NeR ) ) one should vary themixing angle φ within its current constraints which depend on the specific Z (cid:48) model [32],namely − . < φ < . ψ model and − . < φ < . χ model.Within a specific Z (cid:48) model and with fixed m N , the χ function basically depends on threeparameters: φ , G NeL and G NeR . In this case, Eq. (32) describes a tree-dimensional surface.Its projection on the (( G NeL ) , ( G NeR ) ) plane demonstrates the interplay between leptonicand Z - Z (cid:48) mixings. Fig. 4 shows, as a typical example, the results of this analysis for the χ -model (left panel) and the ψ -model (right panel), respectively, with fixed m N = 0 . G NeL and G NeR (cid:72) G LNe (cid:76) (cid:45) (cid:72) G R N e (cid:76) (cid:45) Φ(cid:61) Z Χ (cid:253) (cid:72) G LNe (cid:76) (cid:45) (cid:72) G R N e (cid:76) (cid:45) Z Ψ (cid:253) Φ(cid:61) FIG. 4: Discovery reach at 95% CL on the heavy neutral lepton coupling plane (( G NeL ) , ( G NeR ) )at m N = 0 . Z - Z (cid:48) mixing are simultaneouslyallowed for the Z (cid:48) χ model (left panel) and the Z (cid:48) ψ model (right panel), obtained from combinedanalysis of polarized differential cross sections dσ ( W + L W − L ) /dz at different sets of polarization, P − L = ± . , P + L = ∓ .
6, at the ILC with √ s = 0 . L int = 1 ab − . The dashed curveslabelled “ φ = 0” refer to the case of no Z - Z (cid:48) mixing. are quite dependent on the Z (cid:48) model and different for these two cases. From the explicitcalculation it turns out that this is due to the different relative signs between the lepton and Z - Z (cid:48) mixing contributions to the deviations of the cross section ∆ σ .Concerning Fig. 4 and the corresponding analysis for the χ and ψ models, we should notethat the bounds on the lepton couplings ( G NeL ) and ( G NeR ) are somewhat looser than inthe case φ = 0 discussed above (roughly, by a factor as large as two), but still numericallycompetitive with the current situation. Also, we can remark that the cross sections forlongitudinal W + W − production provide by themselves the most stringent constraints forthis model.Finally, one should note that although the discovery reach on the lepton couplings ( G NeL ) and ( G NeR ) obtained from polarized differential cross sections is quite dependent on the Z (cid:48) model, this is not the case for the identification reach as the double beam polarizationasymmetry A N double is basically independent of the Z - Z (cid:48) boson mixing.14 . IDENTIFICATION OF HEAVY LEPTON EFFECTS WITH A double By “identification” we shall here mean exclusion of a certain set of competitive mod-els, including the SM, to a certain confidence level. For this purpose, the double beampolarization asymmetry, defined as [20, 33, 34] A double = σ ( P , − P ) + σ ( − P , P ) − σ ( P , P ) − σ ( − P , − P ) σ ( P , − P ) + σ ( − P , P ) + σ ( P , P ) + σ ( − P , − P ) , (34)is very useful. Here P = | P − L | , P = | P + L | , and σ ( ± P , ± P ) denotes the polarized integratedcross section determined within the allowed range of the W − scattering angle (or cos θ ). FromEqs. (25) and (34) one finds for the A double of the process (1) A double = P P ( σ RL + σ LR ) − ( σ RR + σ LL )( σ RL + σ LR ) + ( σ RR + σ LL ) . (35)We note that this asymmetry is only available if both initial beams are polarized. m N (cid:72) TeV (cid:76) A d o ub l e s (cid:61) m N (cid:72) TeV (cid:76) A d o ub l e s (cid:61) FIG. 5: Double beam polarization asymmetry A double for the production of unpolarized W ± as afunction of neutral heavy lepton mass m N for different choices of couplings (cid:113) G NeL G NeR (attached tothe lines) at the ILC with √ s = 0 . √ s = 1 . L int = 1 ab − .The solid horizontal line corresponds to A SMdouble = A Z (cid:48) double = A AGCdouble . The error bands indicate theexpected uncertainty in the SM case at the 1- σ level. It is important to also note that the SM gives rise only to σ LR and σ RL such that thestructure of the integrated cross section has the form σ SM = 14 (cid:2)(cid:0) P − L (cid:1) (cid:0) − P + L (cid:1) σ RL SM + (cid:0) − P − L (cid:1) (cid:0) P + L (cid:1) σ LR SM (cid:3) . (36)15his is also the case for anomalous gauge couplings (AGC) [25], and Z (cid:48) -boson exchange(including Z - Z (cid:48) mixing and Z exchange) [16]. The corresponding expressions for thosecross sections can be obtained from (36) by replacing the specification SM → AGC and Z (cid:48) ,respectively. Accordingly, the double beam polarization asymmetry has a common form forall those cases: A SMdouble = A AGCdouble = A Z (cid:48) double = P P = 0 . , (37)where the numerical value corresponds to the product of the electron and positron degreesof polarization: P = 0 . P = 0 .
6. Eq. (37) demonstrates that A SMdouble , A AGCdouble and A Z (cid:48) double are indistinguishable for any values of NP parameters, AGC or Z (cid:48) mass and strength of Z - Z (cid:48) mixing, i.e. ∆ A double = A AGCdouble − A SMdouble = A Z (cid:48) double − A SMdouble = 0.On the contrary, the heavy neutral lepton N -exchange in the t -channel will induce non-vanishing contributions to σ LL and σ RR , and thus force A double to a smaller value, ∆ A double = A N double − A SMdouble ∝ − P P r N (cid:0) G NeL G NeR (cid:1) < Z - Z (cid:48) mixing contributions to σ RL and σ LR . A value of A double below P P can provide asignature of heavy neutral lepton N -exchange in the process (1). All those features in the A double behavior are shown in Fig. 5, where we consider unpolarized W s.The identification reach (ID) on the plane of heavy lepton coupling (( G NeL ) , ( G NeR ) ) (at95% C.L.) for various lepton masses m N plotted in Fig. 6 is obtained from conventional χ analysis with A double . In that case the χ function is constructed as χ = (∆ A double /δA double ) where δA double is the expected experimental uncertainty accounting for both statistical andsystematic components. Note that discovery is possible in the green and yellow regions,whereas identification is only possible in the green region. The hyperbola-like limit of theidentification reach is due to the appearance of a product of the squared couplings ( G NeL ) and ( G NeR ) in the deviation from the SM cross section, given by Eq. (28).It should be stressed that the identification reach is independent of the Z (cid:48) model assumed,whereas the discovery reach is not. In fact, in the lower left corner of these figures, we showhow the discovery reach gets modified if we allow for Z - Z (cid:48) mixing within the Z (cid:48) χ model (cf.Fig. 4). 16
10 20 30 40 50 60102030405060 (cid:72) G LNe (cid:76) (cid:45) (cid:72) G R N e (cid:76) (cid:45) IDDIS
Φ(cid:61) (cid:72) G LNe (cid:76) (cid:45) (cid:72) G R N e (cid:76) (cid:45) IDDIS
Φ(cid:61) FIG. 6: Left panel: discovery (DIS) and identification (ID) reaches at 95% CL on the heavyneutral lepton coupling plane (( G NeL ) , ( G NeR ) ), obtained from a combined analysis of polarizeddifferential cross sections dσ ( W + L W − L ) /dz at different sets of polarization, P − L = ± . , P + L = ∓ . m N = 0 . √ s = 0 . L int = 1 ab − . Right panel: similar, with √ s = 1 . m N = 0 . φ = 0” refer to the case of no Z - Z (cid:48) mixing, whereas the outer contour labelled“DIS” refer to the minimum discovery reach in the presence of mixing. VI. DISCOVERY AND IDENTIFICATION REACH AT √ s = 350 GEV
In view of the possibility of a staged ILC construction, we would like to comment on thepossibility of obtaining bounds on heavy neutral leptons at 350 GeV. As illustrated in Fig. 7,polarized beams would already at this low energy allow to place a limit on possible
N W e couplings, in particular at low masses m N . In this figure we explore masses beyond thecorresponding kinematical reach. Even at this rather low energy there is already sensitivityto discover heavy lepton couplings in the range of G ∼ − for low masses and up to G ∼ × − for heavy masses m N and with an assumed integrated luminosity of 500 fb − .It is seen that one can identify heavy-lepton-mixing effects for masses up to m N ∼
400 GeV.Discovery is seen to become more sensitive at higher masses, since the effect is approxi-mately proportional to 1 − r N , whereas for identification the sensitivity is governed by r N ,and thus becomes less efficient at higher masses. For higher beam energy, both sensitivities17 .2 0.4 0.6 0.8 1.0 1.2 1.4151050100 m N (cid:72) TeV (cid:76) G (cid:45) IDDIS 0.35 TeV0.35 TeV 0.5 TeV0.5 TeV 1 TeV1 TeV approximate current limit
FIG. 7: Discovery (DIS) and identification (ID) reach on G ≡ ( G NeL ) = ( G NeR ) . The low-energycase (350 GeV) is compared with the nominal energy cases of 500 GeV and 1 TeV, all at an assumedintegrated luminosity of 500 fb − . The approximate current limit on these couplings is indicatedas a grey band. improve. VII. CONCLUDING REMARKS
In this note we have studied the process e + e − → W + W − and seen how to uniquelyidentify the indirect (propagator and exotic-lepton mixing) effects of a heavy neutral leptonexchange in the t -channel. Discovery of new physics, meaning exclusion of the StandardModel, does not depend on having both initial beams polarized, but the sensitivity is im-proved with beam polarization. Such “discovery” could be due to the existence of a Z (cid:48) ,anomalous gauge couplings, or the effect of a heavy neutral lepton. The potential of theILC to discover heavy lepton effects depends on the possible presence of a Z (cid:48) contribution,and is vastly improved if one is able to determine the polarization of the produced W s.Identification of such new physics effect as being due to a heavy neutral lepton exchange,as opposed to a Z (cid:48) or AGC can be achieved via the determination of a double polarizationasymmetry. This identification of heavy-lepton admixture is independent of the strength ofany Z - Z (cid:48) mixing, as well as the Z (cid:48) model, but requires having both initial beams polarized.18 cknowledgements This research has been partially supported by the Abdus Salam ICTP under the TRILand Associates Scheme and the Belarusian Republican Foundation for Fundamental Re-search. The work of AAP has been partially supported by the Collaborative ResearchCenter SFB676/1-2006 of the DFG at the Department of Physics, University of Hamburg.The work of PO has been supported by the Research Council of Norway. [1] P. Langacker, Rev. Mod. Phys. , 1199-1228 (2009) [arXiv:0801.1345 [hep-ph]].[2] T. G. Rizzo, [hep-ph/0610104].[3] A. Leike and S. Riemann, Z. Phys. C , 341 (1997) [hep-ph/9607306].[4] A. Leike, Phys. Rept. , 143-250 (1999) [hep-ph/9805494].[5] S. Riemann, eConf C , 0303 (2005) [hep-ph/0508136].[6] J. L. Hewett, T. G. Rizzo, Phys. Rept. , 193 (1989).[7] S. Hesselbach, F. Franke and H. Fraas, Eur. Phys. J. C (2002) 149 [hep-ph/0107080].[8] P. H. Frampton, P. Q. Hung and M. Sher, Phys. Rept. (2000) 263 [hep-ph/9903387].[9] P. Langacker and D. London, Phys. Rev. D (1988) 886.[10] J. Beringer et al. [Particle Data Group Collaboration], Phys. Rev. D (2012) 010001.[11] ATLAS Collaboration, “Search for high-mass dilepton resonances with 5 fb − of pp collisionsat √ s = 7 TeV with the ATLAS experiment” , ATLAS-CONF-2012-007; ATLAS-CONF-2012-129;[12] S. Chatrchyan et al. [CMS Collaboration], Phys. Lett. B , 158 (2012) [arXiv:1206.1849[hep-ex]]; S. Chatrchyan et al. [CMS Collaboration], Phys. Lett. B , 3 (2013) [arXiv:1212.6175[hep-ex]]; S. Chatrchyan et al. [CMS Collaboration], EXO-12-061.[13] S. Singh, A. K. Nagawat and N. K. Sharma, Mod. Phys. Lett. A , 1717 (1990).[14] A. K. Nagawat, S. Singh and N. K. Sharma, Phys. Rev. D , 2984 (1990).[15] A. A. Babich, A. A. Pankov and N. Paver, Phys. Lett. B (1993) 351; B (1995) 303.[16] V. V. Andreev, G. Moortgat-Pick, P. Osland, A. A. Pankov and N. Paver, Eur. Phys. J. C (2012) 2147 [arXiv:1205.0866 [hep-ph]].[17] K. Hagiwara, R. D. Peccei, D. Zeppenfeld and K. Hikasa, Nucl. Phys. B , 253 (1987).
18] G. Gounaris, J. L. Kneur, J. Layssac, G. Moultaka, F. M. Renard and D. Schildknecht,Proceedings of the Workshop e + e − Collisions at 500 GeV: the Physics Potential , Ed. P.M.Zerwas (1992), DESY 92-123B, p.735.[19] G. Buchalla, O. Cata, R. Rahn and M. Schlaffer, arXiv:1302.6481 [hep-ph].[20] A. V. Tsytrinov, J. Kalinowski, P. Osland and A. A. Pankov, Phys. Lett. B (2012) 94[arXiv:1207.6234 [hep-ph]].[21] G. Moortgat-Pick, T. Abe, G. Alexander, B. Ananthanarayan, A. A. Babich, V. Bharadwaj,D. Barber, A. Bartl et al. , Phys. Rept. , 131-243 (2008) [hep-ph/0507011].[22] J. Alcaraz et al. [ALEPH and DELPHI and L3 and OPAL and LEP Electroweak WorkingGroup Collaborations], arXiv:0712.0929 [hep-ex].[23] E. Nardi, E. Roulet and D. Tommasini, Phys. Rev. D , 3040 (1992).[24] E. Nardi, E. Roulet and D. Tommasini, Phys. Lett. B , 319 (1994) [hep-ph/9402224].[25] G. Gounaris, J. Layssac, G. Moultaka and F. M. Renard, Int. J. Mod. Phys. A (1993) 3285.[26] W. Beenakker, F. A. Berends and T. Sack, Nucl. Phys. B , 287 (1991).[27] W. Beenakker, K. Kolodziej and T. Sack, Phys. Lett. B , 469 (1991).[28] W. Beenakker and A. Denner, Int. J. Mod. Phys. A , 4837 (1994).[29] A. Denner, S. Dittmaier, M. Roth and L. H. Wieders, Phys. Lett. B , 223 (2005) [Erratum-ibid. B , 667 (2011)] [hep-ph/0502063].[30] A. Denner, S. Dittmaier, M. Roth and L. H. Wieders, Nucl. Phys. B , 247 (2005) [Erratum-ibid. B , 504 (2012)] [hep-ph/0505042].[31] G. Abbiendi et al. , [OPAL collaboration], Phys. Lett. B585 , 223 (2004);P. Achard et al. , [L3 collaboration], Phys. Lett.
B557 , 147 (2003);J. Abdallah et al. , [DELPHI Collaboration], Eur. Phys. J. C , 345 (2008) [arXiv:0801.1235[hep-ex]];J.P. Couchman, A measurement of the triple gauge boson couplings and W boson polarisationin W -pair production at LEP2, Ph.D. thesis, University College London, 2000.[32] J. Erler, arXiv:0909.5309 [hep-ph].[33] T. G. Rizzo, Phys. Rev. D (1999) 113004 [arXiv:hep-ph/9811440].[34] P. Osland, A. A. Pankov and N. Paver, Phys. Rev. D , 015007 (2003) [arXiv:hep-ph/0304123]., 015007 (2003) [arXiv:hep-ph/0304123].