Virtual bilepton effects in polarized Moller scattering
aa r X i v : . [ h e p - ph ] F e b Virtual bilepton effects in polarized Møllerscattering
B. Meirose and A. J. RamalhoInstituto de F´ısicaUniversidade Federal do Rio de JaneiroCaixa Postal 68528, 21945-970 Rio de Janeiro RJ, BrazilOctober 29, 2018
Abstract:
We investigate the indirect effects of heavy vector bileptons being exchangedin polarized Møller scattering, at the next generation of linear colliders. Considering bothlongitudinal and transverse beam polarization, and accounting for initial state radiation,beamstrahlung and beam energy spread, we discuss how angular distributions and asymme-tries can be used to detect clear signals of virtual bileptons, and the possibility of distin-guishing theoretical models that incorporate these exotic particles. We then estimate 95%C. L. bounds on the mass of these vector bileptons and on their couplings to electrons.1
Introduction
Several schemes have been put forward to address important questions left unanswered inthe standard model. There are models in which some of the standard model fundamentalparticles, such as quarks and leptons, are thought to be in fact composite. Another approachis to extend the symmetry by enlarging the local gauge group. In an effort to incorporategravity, more exotic proposals have been made, such as those which depend on the existenceof extra spatial dimensions. Since the standard model has passed severe experimental testsat the e + e − and hadron colliders, showing its validity up to the electroweak scale, oneexpects that it could be seen as an effective theory, valid up to some large mass scale Λ.Most theoretical arguments point to Λ > ∼ T eV . Extended electroweak models predict theexistence of new particles and interactions. An interesting class of exotic particles knownas bileptons [1] is present in several of these models, such as left-right symmetric models,technicolor and theories of grand unification. The bileptons in which we are interestedare vector bosons which couple to standard model leptons and carry two units of leptonnumber. In particular, heavy gauge bileptons may be found in models where the standard SU (2) L × U (1) group is embedded in a larger gauge group. This is the case of some of the331 gauge models [2]. In this paper we propose to look for indirect signals of doubly-chargedvector bileptons in Møller scattering, with polarized beams, and at the new linear colliderenergies. The next generation of e + e − colliders will be designed to operate at center-of-massenergies ranging from the Z mass to a few TeV, with very high luminosities. Electronand positron beam polarizations are expected to be available at these facilities, providingexperimentalists with powerful tools to carry out precision tests of the standard modeland to explore new physical phenomena. Currently, the best lower bound on the vectorbilepton mass is M Y > GeV , a result which was established from muonium-antimuoniumconversion [3]. Another useful lower bound M Y > GeV has been derived [4] from currentexperimental limits on fermion pair production at LEP and lepton-flavor violating chargedlepton decays. While less stringent, this limit does not depend on the assumption that thebilepton coupling is flavor-diagonal.Studies of bilepton effects in Møller scattering have been made before [5], mostly in thecontext of SU(15) grand unified theories. In this paper we work in the framework of the 331models, concentrating on the minimal version, and extend the previous studies by takinginto account (i) important beam effects such as initial-state radiation, beamstrahlung andbeam energy spread; (ii) longitudinal and transverse polarization of the colliding beams,which are expected to be available in the next generation of linear colliders; (iii) Gaussiansmearing of the four-momenta of the final-state leptons, simulating the uncertainties in theenergy measurements in the electromagnetic calorimeters. All these items are significantfor a realistic comparison with experimental data. We analyze several distributions andasymmetries in Møller scattering at the Next Linear Collider (NLC) energies, with thepurpose of searching for indirect signals of gauge bileptons. The predictions of an SU (15)GUT model [6] for these observables are also shown for comparison. From the angulardistribution of final-state electrons we establish bounds on the couplings and mass of thesebileptons at the 95% confidence level. If evidence of the existence of these vector bileptonsis found in Møller scattering, one can also verify the validity of relations between the masses2f the vector bileptons and new neutral gauge bosons, which are connected to the Higgsstructure of the 331 models. In section II we give a brief review of the 331 models [7],outlining only those features that are most relevant to our analysis. Section III describesin detail the numerical simulation of the Møller events and the corresponding analysis ofthe results. 95% confidence level limits are established in section IV. Our conclusions aresummarized in section V. models The 331 models are based on the SU (3) C ⊗ SU (3) L ⊗ U (1) X gauge group and predict newphysics at the TeV energy scale. They arrange the ordinary leptons in SU (3) L antitriplets,two generations of quarks in triplets and a third generation in an antitriplet. Anomalycancelation, essential to a gauge theory, takes place not within each generation but when allthree families of quarks and leptons are taken together. The fact that one quark generationtransforms differently from the other two is essential in these models, since the number oftriplets must equal the number of antitriplets to ensure that the models remain anomaly-free.This implies that the number of generations is divisible by the number of colors. Besidesstandard gauge bosons, the models predict a new neutral gauge boson Z ′ and four vectorbileptons Y ± and Y ±± . These gauge bosons are expected to be considerably heavier thanthe standard gauge bosons. In addition to the ordinary quarks, the class of models we areconsidering here contains three new heavy quarks, with exotic electric charges 5 / − / Y ++ to electrons and positronsis given by the interaction Lagrangian L int = − g l √ Y ++ µ e T Cγ µ γ e − g l √ Y −− µ ¯ eγ µ γ C ¯ e T . (1)For the minimal 331 model, symmetry breaking can be accomplished by three complex SU (3) L triplets and a complex sextet, allowing these nonstandard gauge bosons to acquireplausible masses. In this case, the mass M Z ′ of the neutral vector boson Z ′ and the mass M Y of the doubly-charged bilepton Y ++ are related by M Y M Z ′ = p − θ W )2 cos θ W (2)Alternative Higgs structures are possible in different 331 models, but then the relation aboveno longer holds. One substantial source of beam energy degradation is the initial-state radiation (ISR). Thisis a QED effect, and corresponds to the emission of photons by the incoming electrons and3ositrons. For a considerable fraction of events, this bremsstrahlung emission lowers theeffective center-of-mass energy available for the hard scattering process. To account for ISR,we used the structure function approach discussed in [9]. In order to achieve high luminosi-ties at the new linear colliders, the beams must have very small transverse dimensions. Theparticles in a colliding bunch suffer considerable transverse acceleration due to the collectiveelectromagnetic fields produced by the particles in the opposite colliding bunches, whichgives rise to the emission of synchrotron radiation, the so-called beamstrahlung. The effec-tive energy available for the reaction is then smaller than the nominal value. The averageenergy loss of a positron or electron by beamstrahlung depends on the design parametersof the accelerator. For some designs of a 1
T eV
NLC, for instance, the colliding beamsmay lose about 13% of the nominal energy from beamstrahlung emissions. This loss mayreach 31% at a 3
T eV
CLIC [10]. To obtain the beamstrahlung spectrum, we followed theapproach of ref. [11], which is based on the Yokoya-Chen approximate evolution equationfor beamstrahlung [12]. In this paper, the calculations of the beamstrahlung energy spectrawere carried out starting from the energy-dependent sets of NLC design parameters [13].The spectra corresponding to ISR and beamstrahlung emissions were convoluted, and theresulting distribution was used to compute all the required differential cross sections. Wealso considered a possible beam energy spread, which was taken to be Gaussian distributed,with a width of 1% of the nominal beam energy.In addition to the standard Feynman diagrams for Møller scattering, in 331 models theprocess also proceeds via an s-channel exchange of a doubly-charged bilepton, and a t-channelexchange of a Z ′ boson as well. By ignoring beam effects, neglecting the Z ′ exchanges andconsidering only unpolarized beams, we verified that our numerical calculations agree withthe trace calculation of ref. [5]. Likewise, we cross-checked our calculations with those ofref. [14], which were carried out in the framework of the standard model, but with arbitrarybeam polarization.The differential cross sections were calculated with Monte Carlo techniques, with the sim-ulated events selected according to the following set of cuts : (i) the final-state electronsand positrons were required to be produced within the angular range | cosθ i | < .
95, where θ i stands for the polar angle of the final-state lepton with respect to the direction of theincoming electron beam; (ii) all events in which the acollinearity angle ζ of the final-state e + − e − three-momenta did not pass the cut ζ < ◦ were rejected; (iii) the ratio of the ef-fective center-of-mass energy to the nominal center-of-mass energy for any acceptable eventwas required to be greater than 0 . T eV , and several years ofoperation, so as to accumulate an integrated luminosity of 500 f b − . In order to simulatethe finite resolution of the NLC electromagnetic calorimeters, we Gaussian-smeared the four-momenta of the produced electrons and positrons [16]. The energies of the final-state leptonswere distributed as a Gaussian with half-width ∆ E of the form ∆ E/E = 10% / √ E ⊕ mrad .Beam polarization will play a useful role at the new linear colliders [17]. By using polarizedlepton beams, one can effectively reduce backgrounds and increase the sensitivity of spin-4ependent observables to potential new physics. In our calculations we worked with theelectron beam projection operators in the extreme relativistic regime, P ( p a,b ) = lim m →
12 ( p a,b + m )(1 + γ n a,b ) →
12 (1 + P a,bL γ ) p a,b + 12 P a,bT γ ( cosφ a,b n + sinφ a,b n ) p a,b , where n µa,b represent the spin vectors and P a,bL ( P a,bT ) stand for the longitudinal (transverse)polarizations of the incoming electron beams, whose four-momenta for a nominal center-of-mass energy √ s are given by p µa = ( √ s , , , √ s ) and p µb = ( √ s , , , − √ s ). For the numericalcalculations dealing with transverse polarization of the electron beams, the azimuthal anglesof the transverse polarization vectors were taken to be φ a = φ b = 0, and the purely spatialvectors n µ = (0 , , ,
0) and n µ = (0 , , , P aL = − P bL = 0 .
9, with an uncertainty given by ∆ P L /P L = 0 . M Y = 1 . T eV . Only for higher energy values is the total crosssection significantly altered by the exchange of a vector bilepton. The angular distribution dσ/d ( cosθ ) of the final-state electrons is, however, more sensitive to the presence of sucha particle. This is displayed in Fig. 2, where the angular distribution is plotted both for √ s = 500 GeV and √ s = 1 T eV , with the corresponding curves for the standard modelshown for comparison. The Møller scattering angular distribution in the minimal 331 modeldiffers from that of the standard model for most of the angular range, the more so for acenter-of-mass energy √ s = 1 T eV . These deviations from the standard model predictionscan be used to establish bounds on the mass of the gauge bilepton, and its coupling toelectrons. The symmetric shape of the angular distribution suggests that the integratedforward-backward asymmetry A F B should be small. This is indeed the case, as shown inFig. 3, where A F B is plotted for several input values of M Y , at an energy √ s = 1 T eV , andthe one-standard-deviation error bars represent only the statistical errors. Next we analyzethe discovery potential of spin asymmetries.Starting from the polarization-dependent angular distributions, one can compute the follow-ing asymmetries: A ( cosθ ) = dσ ( −| P aL | , −| P bL | ) + dσ ( −| P aL | , | P bL | ) − dσ ( | P aL | , −| P bL | ) − dσ ( | P aL | , | P bL | ) dσ ( −| P aL | , −| P bL | ) + dσ ( −| P aL | , | P bL | ) + dσ ( | P aL | , −| P bL | ) + dσ ( | P aL | , | P bL | ) (3) A ( cosθ ) = dσ ( −| P aL | , −| P bL | ) − dσ ( | P aL | , | P bL | ) dσ ( −| P aL | , −| P bL | ) + dσ ( | P aL | , | P bL | ) (4) A ( cosθ ) = dσ ( −| P aL | , | P bL | ) − dσ (0 , dσ ( −| P aL | , | P bL | ) + dσ (0 ,
0) (5)In the limit | P aL | = | P bL | = 1, A ( cosθ ) and A ( cosθ ) reduce to the familiar parity violatingMøller asymmetries [18] A (1) LR = dσ LL + dσ LR − dσ RL − dσ RR dσ LL + dσ LR + dσ RL + dσ RR (6)5nd A (2) LR = dσ LL − dσ RR dσ LL + dσ RR (7)respectively.The behavior of spin asymmetry A as a function of cosθ is depicted in Fig. 4, for a bileptonmass M Y = 1 . T eV . At √ s = 500 GeV the deviation from the standard model valuesis small. This asymmetry becomes more sensitive to the presence of a vector bilepton atan energy of 1
T eV , and its angular dependence distinguishes the two nonstandard models .Statistical errors for the spin asymmetries discussed in this section are rather small, and mostof the effects of the systematic errors are expected to cancel out in these asymmetries. Fig. 5shows that a similar pattern holds for A , which is represented as a function of cosθ for both500 GeV and 1
T eV . At this latter energy, however, A does not lead to a clear distinctionbetween the minimal 331 model and the SU (15) GUT model. Asymmetry A concerns thedifference between a polarized angular distribution and its unpolarized counterpart, andis displayed in Fig. 6. Unlike A and A , which take values of the order of 5%, A maybecome quite large, reaching a maximum magnitude of about 63% for the minimal 331 modelat √ s = 1 T eV . It can be useful to discriminate a model with vector bileptons from thestandard model, even at a 500
GeV linear collider. At a center-of-mass energy of 1
T eV , thedifference between the prediction of the 331 minimal model for A and the standard valuesis even more striking.We assume that in the next generation of linear colliders, transverse polarization of electronbeams will be available as an extra tool to search for the new physics. This could be achievedby means of spin rotators, which convert longitudinal into transverse polarization. It is notimmediately clear whether the use of transversely polarized electron beams may add anyimportant information, which could not be extracted from longitudinally polarized Møllerscattering. In fact, standard Møller scattering is known not to be particularly sensitive totransverse beam polarization [14], but the presence of an s-channel vector bilepton mightmodify this picture. Effects of transverse beam polarization in the nonstandard Møllerscattering under discussion only materialize if both beams are transversely polarized. Inorder to search for possible advantages of transverse beam polarization for the problem athand, we examined in detail the behavior of the following differential azimuthal asymmetry: A T ( cosθ ) = R (+) dφ d σd ( cosθ ) dφ − R ( − ) dφ d σd ( cosθ ) dφ R (+) dφ d σd ( cosθ ) dφ + R ( − ) dφ d σd ( cosθ ) dφ , (8)where the subscript +( − ) indicates that the integration over the azimuthal angle φ is tobe carried out over the region of phase space where cos φ is positive (negative). As far as A T ( cosθ ) is concerned, and considering an integrated luminosity of 500 f b − , we found thatit would be difficult to separate the signal from the standard model background at center-of-mass energies around 500 GeV , even if M Y is only moderately large. At √ s = 1 T eV ,however, it is feasible to detect the effects of a vector bilepton, as long as its mass is notmuch larger than the center-of-mass energy. This is illustrated in Fig. 7, where A T ( cosθ ) isplotted both for an input mass M Y = 800 GeV and and for M Y = 1 . T eV , along with thestandard model expectation. We also investigated the integrated version of the asymmetry6bove, A T = R (+) d ( cosθ ) dφ d σd ( cosθ ) dφ − R ( − ) d ( cosθ ) dφ d σd ( cosθ ) dφ R (+) d ( cosθ ) dφ d σd ( cosθ ) dφ + R ( − ) d ( cosθ ) dφ d σd ( cosθ ) dφ , (9)where the integrations are consistent with the cuts specified in section III. The mass de-pendence of A T for √ s = 500 GeV and √ s = 1 T eV is presented in Fig. 8. A T is foundto be sizable over a fairly wide range around the bilepton resonance. We checked that theshape of the curve representing the mass dependence of this asymmetry becomes wider as g l increases, while the position of the corresponding minimum remains essentially the same,for a fixed center-of-mass energy. A χ test was applied to estimate discovery limits for a vector bilepton in Møller scatter-ing. Considering only longitudinal beam polarization, we compared the angular distribution dσ/d ( cosθ ) of the final-state electrons, modified by the presence of a vector bilepton, withthe corresponding standard model distribution. Assuming that the experimental data will bewell described by the standard model predictions, we defined a two-parameter χ estimator χ ( g l , M Y ) = N b X i =1 (cid:16) N SMi − N i ∆ N SMi (cid:17) (10)where N SMi is the number of standard model events detected in the i th bin, N i is thenumber of events in the i th bin as predicted by the model with bileptons, and ∆ N SMi = q ( p N SMi ) + ( N SMi ǫ ) the corresponding total error, which combines in quadrature thePoisson-distributed statistical error with the systematic error. For the latter we assumeda conservative value ǫ = 5% for each measurement. The angular range | cosθ | < .
95 wasdivided into N b = 20 equal-width bins. The coupling g l of a vector bilepton to electrons andthe bilepton mass M Y were varied as free parameters to determine the χ distribution. The95% confidence level bound corresponds to an increase of the χ by 5 .
99 with respect to theminimum χ min of the distibution. Fig.9 presents the resulting 95% C. L. contour plots onthe ( g l , M Y ) plane for the nominal center-of-mass energies √ s = 500 GeV and √ s = 1 T eV .The unpolarized case is shown in Fig. 10. Since our differential cross sections contain onlyeven powers of g l , it suffices to use positive values of the coupling in Figs. 9-10. We alsocalculated the corresponding 95% C. L. limits on the bilepton mass at these NLC energies,considering only the minimal 331 model, in which a Z ′ exchange has to be taken into account.The results are displayed in Table I. Future linear colliders will provide an opportunity to look for new particles and their in-teractions. In this paper we discussed how to search for effects of a virtual vector bilepton7able 1: 95% C. L. limits on the bilepton mass in the minimal 331 model, at NLC energiesPolarization √ s =500 GeV √ s =1 TeVunpolarized 1230 GeV 1815 GeVpolarized 2529 GeV 4574 GeVin polarized Møller scattering. Starting from a realistic simulation of this process, we ana-lyzed several polarization-dependent observables that might provide strong evidence of theexistence of vector bileptons, and allow to discriminate models or classes of models whichpredict these particles, should any deviation from the standard expectations be detected.We demonstrated that these polarization-dependent observables are more sensitive to vectorbileptons than their unpolarized counterparts, in a large region of the allowed parameterspace. This sensitivity was found to be stronger at a center-of-mass energy of 1 T eV thanat 500
GeV . The bounds on the masses and couplings of the bileptons, derived from a χ estimator, indicate that it should be possible to probe mass scales of up to several TeV for asignal of a vector bilepton. Longitudinal polarization of the electron beams in Møller scatter-ing has proved useful to improve these bounds on the masses and couplings. Although mostof our calculations were carried out in the context of the minimal 331 model and an SU (15)GUT model, we believe that our overall conclusions could be extended to other models withbileptons. References [1] F. Cuypers and S. Davidson, Eur. Phy. J. C ,503 (1998) [hep-ph/9609487].[2] F. Pisano and V. Pleitez, Phys. Rev. D , 410 (1992); P. H. Frampton, Phys. Rev.Lett. , 2889 (1992).[3] L. Willmann et. al., Phys. Rev. Lett. , 49 (1999).[4] M. B. Tully and G. C. Joshi, Phys. Lett. B , 333 (1993); hep-ph/9905552.[5] Paul H. Frampton and Daniel Ng, Phys. Rev. D , 4240 (1992); Thomas Rizzo, Phys.Rev. D , 910 (1992).[6] P. H. Frampton and B.-H. Lee, Phys. Rev. Lett. , 619 (1990); P. H. Frampton, Int.J. Mod. Phys. A , 2455 (2000).[7] F. Pisano, V. Pleitez and M. D. Tonasse, IFT-UNESP preprint IFT-P.043/97 and ref-erences therein.[8] W. A. Ponce, J. B. Flores and L. A. Sanchez, Int. J. Mod. Phys. A , 643 (2002).[9] M. Skrzypek and S. Jadach, Z. Phys. C , 577 (1991).[10] The CLIC Study Team, ”A 3 T eV e + e − (2001) 1 [hep-ph/0106155].[18] Andrzej Czarnecki and William Marciano, Int. J. Mod. Phys. A13 (1998) 2235[hep-ph/9801394]. 9 σ ( pb ) √ s (GeV) AB F i g u r e : T o t a l c r o sss ec t i o n a s a f un c t i o n o f t h ece n t e r - o f - m a ss e n e r g y √ s , f o r M Y = . T e V ( B ) ; s o li d li n e ( A )r e p r e s e n t s t h e s t a nd a r d m o d e l c r o sss ec t i o n . . - - . - . - . - . . . . . d σ /d(cos θ ) (GeV) c o s θ A B
C D E . - - . - . - . - . . . . . d σ /d(cos θ ) (GeV) c o s θ A B
C D E Figure 2: Angular distribution of the final-state electrons; Curves (A) and (D) show theangular spectra predicted by the standard model at √ s = 500 GeV and √ s = 1 T eV respec-tively, while (B) and (C) display the corresponding angular distributions in the minimal 331model. Curve (E) shows the prediction for the SU(15) model at √ s = 1 T eV .11 . - . . . A FB M Y ( G e V ) A B
Figure 3: Forward-backward asymmetry A F B for several input masses M Y , at √ s = 1 T eV ,according to the minimal 331 model (B). The upper curve (A) shows the expected value forthe standard model 12 . . . . . . . . . . - - . - . - . - . . . . . A c o s θ A B
C D E Figure 4: Polar angle dependence of spin asymmetry A ( cosθ ); Curve (A) shows the pre-diction for the SU(15) model at √ s = 1 T eV . Curves (D) and (B) show the standardmodel predictions for A ( cosθ ) at √ s = 500 GeV and √ s = 1 T eV respectively, while (C)and (E) represent the corresponding expectations for the minimal 331 model, for a mass M Y = 1 . T eV . 13 . . . . . . . . - - . - . - . - . . . . . A c o s θ A B
C D E Figure 5: Polar angle dependence of spin asymmetry A ( cosθ ); Curve (A) shows the pre-diction for the SU(15) model at √ s = 1 T eV . Curves (B) and (D) show the standardmodel predictions for A ( cosθ ) at √ s = 500 GeV and √ s = 1 T eV respectively, while (C)and (E) represent the corresponding expectations for the minimal 331 model, for a mass M Y = 1 . T eV . 14 . - . - . - . - . - . - . - - . - . - . - . . . . . A c o s θ A B
C D E Figure 6: Polar angle dependence of spin asymmetry A ( cosθ ); Curves (B) and (C) showthe standard model predictions for A ( cosθ ) at √ s = 500 GeV and √ s = 1 T eV respectively,while (D) and (A) represent the corresponding expectations for the minimal 331 model, for amass M Y = 1 . T eV . Curve (E) shows the prediction for the SU(15) model at √ s = 1 T eV .15 . - . - . - . - . - . . - - . - . - . - . . . . . A c o s θ A B
C D
Figure 7: Transverse polarization asymmetry A T ( cosθ ) as a function of cosθ , at √ s = 1 T eV .The curve with error bars (D) corresponds to the minimal 331 model with M Y = 800 GeV ,whereas curve (C) corresponds to the SU (15) GUT model prediction for the same mass.Curve (B) represents the behavior of the asymmetry for a 1 . T eV bilepton in the minimal331 model, and the histogram (A) corresponds to the standard model expectation. The errorbars for the upper curves are similar to those of the lower curve on a bin-to-bin basis.16 . - . - . - . - . - . . A M Y ( G e V ) A B C Figure 8: Transverse polarization asymmetry A T as a function of M Y . The solid histogram(A) refers to the minimal 331 model prediction at an energy √ s = 500 GeV and the long-dashed histogram (B) to the SU (15) model at 1 T eV . The resulting asymmetry A T for theminimal 331 model at √ s = 1 T eV is is represented by the dashed histogram (C).17 M Y ( G e V ) g F i g u r e : % C . L . c o n t o u r p l o t s o n t h e ( g l , M Y ) p l a n e f o r l o n g i t ud i n a ll y p o l a r i ze d M ø ll e r s c a tt e r i n g , a tt h e N L C ce n t e r - o f - m a ss e n e r g i e s √ s = G e V ( l o w e r c u r v e ) a nd √ s = T e V ( upp e r c u r v e ) .
500 1000 1500 2000 2500 3000 3500 4000 4500 1 1.5 2 2.5 3 3.5 4 4.5 5 M Y ( G e V ) g F i g u r e : % C . L . c o n t o u r p l o t s o n t h e ( g l , M Y ) p l a n e f o r unp o l a r i ze d M ø ll e r s c a tt e r i n g , a tt h e N L C ce n t e r - o f - m a ss e n e r g i e s √ s = G e V ( l o w e r c u r v e ) a nd √ s = T e V ( upp e r c u r v e ) ..