Searching for vector bileptons in polarized Bhabha scattering
aa r X i v : . [ h e p - ph ] J u l Searching for vector bileptonsin polarized Bhabha scattering
B. Meirose ∗ Physikalisches InstitutAlbert-Ludwigs-Universit¨at FreiburgHermann-Herder-Str. 3 79104 Freiburg i. Br., Germany
A. J. Ramalho † Instituto de F´ısicaUniversidade Federal do Rio de JaneiroCx.Postal 68528, 21945-970 Rio de Janeiro RJ, Brazil (Dated: October 26, 2018)In this paper we analyze the effects of virtual vector bileptons in polarized Bhabha scattering atthe energies of the future linear colliders. In order to make the calculations of the differential crosssections more realistic, important beam effects such as initial state radiation, beamstrahlung, beamenergy and polarization spreads are accounted for. The finite resolution of a typical electromagneticcalorimeter planned for the new linear colliders is also considered in the simulation. The 95%confidence level limits for bilepton masses in 331 models are evaluated.
PACS numbers: 12.60.Cn, 13.88.+e, 14.70.Pw
I. INTRODUCTION:
Although the standard model (SM) explains all currentexperimental data, it is commonly believed that it is notthe final answer. Many questions have been left open,so many theorists believe that there should be some newphysics lurking at the TeV scale. Many models have beenproposed to explain these unanswered questions. Ex-amples are supersymmetry, composite models and grandunification theories. All extended models have in com-mon the prediction of new particles. This paper concernsnonstandard particles known as bileptons [1] - bosonswhich couple to two leptons and which carry two units oflepton number. They are present in several models, suchas technicolor, left-right symmetric models, and in thegrand unification schemes. Heavy gauge bileptons appearin extended gauge models where the electroweak groupis imbedded in a larger group. The so-called 331 models,based on the gauge group SU (3) C ⊗ SU (2) L ⊗ U (1) Y ,fall in this category. In this paper we discuss how tosearch for signals of an off-mass-shell, doubly-chargedvector bilepton in polarized Bhabha scattering, at thenew linear collider energies. Most of the analysis is madein the framework of 331 gauge models [2], but the pre-dictions of an SU (15) model [3] are also presented, andmay be taken as a measure of the model-dependence ofour results. The detection of such indirect effects of avector bilepton would be strong evidence of new physics.Should any signal of a vector bilepton be detected in ahigh energy collider, it would be important to settle thequestion of the correct underlying model. ∗ Electronic address: [email protected] † Electronic address: [email protected]
For realistic comparison with experiment, we considerimportant beam effects such as (i) initial-state radiation,beamstrahlung and beam energy spread; (ii) longitudi-nal and transverse polarization of the colliding beams,which are expected to be available in the next gener-ation of linear colliders; (iii) Gaussian smearing of thefour-momenta of the final-state leptons, simulating theuncertainties in the energy measurements in the electro-magnetic calorimeters. In section II we give a brief re-view of the 331 minimal model [4]. Section III describesin detail the numerical simulation of the Bhabha events.In section IV we analyze several observables such as an-gular distributions and asymmetries, with the purposeto look for indirect vector bileptons signals. In section Vwe establish bounds on the couplings and masses of thesebileptons at the 95% confidence level, from the angulardistribution of the final-state leptons. As we are workingin the context of the minimal 331 version, one can alsoverify the validity of the relations between the masses ofthe vector bileptons and new neutral gauge bosons, whichare connected to the Higgs structure of the 331 models.
II. REVIEW OF THE
MINIMAL MODEL
The fermionic content of the minimal 331 model ar-ranges the ordinary leptons in SU (3) L antitriplets, twogenerations of quarks in triplets and the third genera-tion in an antitriplet. The anomaly cancellation takesplaces only when all three families of quarks and leptonsare taken together. The model also predicts a new neu-tral gauge boson Z ′ and four vector bileptons Y ± and Y ±± , which acquire masses after spontaneous symme-try breaking (SSB): first SU(3) L × U(1) Y breaks downto SU(2) L × U(1) Y . This can be accomplished withonly one SU(3) L scalar triplet; the next stage of SSB,SU(2) L × U(1) Y → U(1) e , requires an additional scalarSU(3) L triplet. In order to provide quarks and leptonswith acceptable masses, however, two additional tripletsand a sextet would be necessary for symmetry breakingwith a minimal Higgs structure. In this minimal versionof the 331 model, the mass M Y of the doubly-chargedbilepton Y ++ and the mass M Z ′ of the neutral vectorboson Z ′ are correlated parameters given by the expres-sion: M Y M Z ′ = q − θ W )2 cos θ W This relation no longer holds for the SU (15) or anyother model with different Higgs structure. Presently,the most useful lower bounds on the vector bileptonmass can be derived from fermion pair production atLEP and lepton-flavor violating charged lepton decays[5], M Y > GeV , and from muonium-antimuoniumconversion [6], M Y > GeV . While the former is lessstringent, it does not depend on the assumption that thebilepton coupling is flavor diagonal. The data collectedby the CDF Collaboration at the Fermilab Tevatron ex-clude 331 Z ′ masses below 920 GeV [7].It has been argued [8] that the minimal 331 modeldescribed above can not be analyzed with perturbationtheory around an energy scale µ , at which the ratio g X /g L = sin θ W ( µ ) / (1 − sin θ W ( µ )), where g X and g L denote the U (1) X and SU (3) L gauge group couplingconstants, develops a Landau-like pole. The scale µ wasfound to be of the order of 4 T eV for the minimal 331model. Solutions [9] have been put forward to circunventthis possible nonperturbative nature of some 331 modelsat the TeV energy scale. In particular, by adding threeoctets of vector leptons to the particle content, it is possi-ble to make sin θ W ( µ ) decrease with the increase of theenergy scale µ , as far as the light states of the standardmodel are concerned, and thus the perturbative regimeis restored for the TeV energy scale. III. SIMULATION
When a charged particle is accelerated to energiesmuch superior than its own rest energy, the probabilityof photon emission is very high. The so called initial-state radiation (ISR) is the emission of photons by theincoming electrons and positrons due this acceleration.ISR is a dominant QED correction in leptonic scatter-ing. Using the structure function approach discussed in[10], ISR effects were taken into account in this simula-tion. We neglect electroweak loop corrections throughoutthe calculations. For the energies planned for the futurelinear colliders, the first-order weak loop-corrections tothe Bhabha angular distribution [11] are at most of theorder of 10% of the corresponding distribution at Bornlevel, for wide-angle scattering. The remaining weak loop-corrections are smaller. We expect that the mag-nitude of weak loop-corrections in the extended modelswe are studying be similar to that of the standard model.Moreover, to some extent these corrections should cancelout in the asymmetry ratios, and therefore their effect onthe observables discussed in the next section should besmall. In the quest for higher luminosities at the linearcolliders, very dense incoming beams, with transverse di-mensions of the order of a few dozen nm, will be needed.The result is a large charge density, leading to very strongelectromagnetic fields. For this reason, the particles ina colliding bunch suffer considerable transverse accelera-tion, which gives rise to the emission of synchrotron radi-ation, the so-called beamstrahlung. The effective energyavailable for the reaction is then smaller than the nominalvalue. Machines with large luminosity per bunch crossingproduce more beamstrahlung. Thus, the average energyloss of a positron or electron by beamstrahlung dependson the design parameters of the accelerator. For somedesigns the beamstrahlung energy loss may even reach30% of the available energy [12]. Using the approach dis-cussed on ref. [13] we obtained the beamstrahlung struc-ture function, starting from an energy-dependent set ofNLC design parameters [14]. Convoluting the ISR andbeamstrahlung emissions spectra we obtained the distri-bution which was used to compute the required differ-ential cross sections. Our simulations also included aGaussian-distributed beam energy spread, with a widthof 1% of the nominal beam energy.As we are working mostly in the framework of the 331minimal model, two new Feynman diagrams have to beadded to the standard diagrams for Bhabha scattering:a s-channel exchange of a Z ′ boson and a u-channel ex-change of a doubly-charged bilepton. As a partial checkof our calculation, we ”switched off” the beam polar-izations, ISR and beamstrahlung effects and the Z ′ ex-changes as well, and verified that we indeed reproducethe results of the trace calculation of ref. [15]. Like-wise, we cross-checked our calculations with those of ref.[16], which were carried out in the framework of the stan-dard model, but with arbitrary beam polarization. Us-ing Monte Carlo techniques we calculated the differentialcross sections with the simulated events selected accord-ing to the following set of cuts : (i) the final-state elec-trons and positrons were required to be produced withinthe angular range | cosθ i | < .
95, where θ i stands for thepolar angle of the final-state lepton with respect to thedirection of the incoming electron beam; (ii) all events inwhich the acollinearity angle ζ of the final-state e + − e − three-momenta did not pass the cut ζ < ◦ were re-jected; (iii) the ratio of the effective center-of-mass en-ergy to the nominal center-of-mass energy for any ac-ceptable event was required to be greater than 0 .
9. Weconsidered a 500 f b − integrated luminosity, which couldbe achieved in several years of machine operation , con-sidering a maximum center-of-mass energy of 1 T eV .Finally, we simulated the finite resolution of the NLCelectromagnetic calorimeters by Gaussian-smearing thefour-momenta of the produced electrons and positrons[17]. The directions of the lepton three-momenta weresmeared in a cone around the corresponding original di-rections, whose half-angle is a Gaussian with half-widthequal to 10 mrad , whereas the energies of the final-stateleptons were distributed as a Gaussian with half-width∆ E of the form ∆ E/E = 10% / √ E ⊕ IV. OBSERVABLES
In our numerical calculations, the longitudinal(transversal) beam polarizations were taken to be P L =90% ( P T = 90%) for electrons and P L = 60%( P T = 60%) for positrons, with an uncertainty given by∆ P L /P L = 0 .
5% (∆ P T /P T = 0 . M Y = 1 . T eV . For acceptablevalues of the bilepton mass, the deviation from the stan-dard model cross section is small over the energy rangeexpected for the next generation of linear colliders. Thiswas also found to be true for the SU (15) model.Next we looked at the angular distributions in the min-imal 331 model at √ s = 500 GeV and √ s = 1 T eV , whichare displayed in Fig.2 with the corresponding curves forthe standard model and SU (15) shown for comparison.All curves indicate that the final-state electrons are emit-ted mostly in small angles (with respect to the incidentelectron beam), but significant differences among the an-gular distributions still occur over the remaining angularrange. These deviations from the SM predictions couldbe helpful to establish the correct underlying electroweakmodel, should any signal of a vector bilepton be detectedin a high energy collider. In a previous work [20], a sim-ilar pattern was found for the sensitivity of Møller scat-tering total cross section and angular distribution to thecoupling of vector bileptons.As the angular distribution for 331 and standard mod-els are strongly asymmetric, a typical non-identical parti-cle final state behavior, the integrated forward-backwardasymmetry is large: A F B = R − . dz dσdz − R . dz dσdz R − . dz dσdz + R . dz dσdz (1) σ ( pb ) √ s SM331
FIG. 1: Total cross section as a function of the center-of-mass energy √ s , for M Y = 1 . T eV (331); solid line (SM)represents the standard model cross section. d σ / d ( c o s θ ) ( G e V ) cos θ SM-5331-5SU15-5331-10SM-10SU15-10
FIG. 2: Angular distribution of the final-state electrons;Curves (SM-5) and (SM-10) show the angular spectra pre-dicted by the standard model at √ s = 500 GeV and √ s =1 T eV respectively, while (331-5) and (331-10) display the cor-responding angular distributions in the minimal 331 model. where z stands for the cosine of the angle θ between theoutgoing and incoming electrons. This is displayed inFig.3, where A F B is plotted for several bilepton massesat an energy √ s = 500 GeV . As expected the 331 A F B value approaches the standard model A F B value as thebilepton mass increases.Also, we analyze the discovery potential of spin asym-metries.Using the polarization-dependent angular distribu-tions, we define the following 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 | ) (2) A F B M Y (GeV)SM331 FIG. 3: Forward-backward asymmetry A F B for several inputmasses M Y , at √ s = 1 T eV , according to the minimal 331model (331). Curve (SM) shows the standard model predic-tion -0.005 0 0.005 0.01 0.015 0.02 0.025-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 A cos θ FIG. 4: Polar angle dependence of spin asymmetry A ( cosθ )at √ s = 1 T eV ; Curve (SM-10) shows the standard modelpredictions for A ( cosθ ), while (331-10) represents the corre-sponding expectations for the minimal 331 model, for a mass M Y = 1 . T eV . Curve (SU15-10) shows the prediction for theSU(15) model. A ( cosθ ) = 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 (0 , dσ ( −| P aL | , | P bL | ) + dσ (0 ,
0) (4) -0.2-0.1 0 0.1 0.2 0.3 0.4 0.5-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 A cos θ SU15-5331-5SM-5
FIG. 5: Polar angle dependence of spin asymmetry A ( cosθ )at √ s = 500 GeV ; Curve (SM-5) shows the standard modelpredictions for A ( cosθ ), while (331-5) represents the corre-sponding expectations for the minimal 331 model, for a mass M Y = 1 . T eV . Curve (SU15-5) shows the prediction for theSU(15) model.
In the definitions of the three asymmetries above, a and b refer to the electron and positron beams respec-tively. Fig.4 shows the angular dependence of asymmetry A for a bileptonic mass M Y = 1 . T eV , at √ s = 1 T eV .We do not show the corresponding 500
GeV curve, sincethe A sensitivity is considerably lower at this energy.Asymmetry A has proved to be quite sensitive to thepresence of vector bileptons. For √ s = 500 GeV this isshown in Fig.5, where the A dependence on cos θ is seento be weaker in the SM, than in the two other modelsendowed with bileptons. The discriminating power of A is even more evident at √ s = 1 T eV , as depicted inFig.6. The resulting errors associated with the calcula-tion of longitudinal asymmetries A and A were small,and hence the error bars are not shown in the figures. Inparticular, using bootstrap resampling techniques [19] weconfirmed that statistical errors are indeed small. Someof the systematic errors, such as the uncertainties on thedegrees of beam polarization, were directly included inthe simulation. As a matter of fact, we expect some ofthese systematic effects to cancel out in the polarizationasymmetries.Asymmetry A concerns the difference between a po-larized distribution and its unpolarized counterpart, andis displayed in Figs.7 and 8 for both 1 T eV and 500
GeV respectively. For both energies its the 331 minimal modelto give the most different predictions compared with theSM. A curious fact that must be noted it’s the change of -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 A cos θ FIG. 6: Polar angle dependence of spin asymmetry A ( cosθ )at √ s = 1 T eV ; Curve (SM-10) shows the standard modelpredictions for A ( cosθ ), while (331-10) represents the corre-sponding expectations for the minimal 331 model, for a mass M Y = 1 . T eV . Curve (SU15-10) shows the prediction for theSU(15) model. -0.4-0.35-0.3-0.25-0.2-0.15-0.1-0.05 0 0.05 0.1 0.15-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 A cos θ SM-10SU15-10331-10
FIG. 7: Polar angle dependence of spin asymmetry A ( cosθ );Curve (SM-10) shows the standard model predictions for A ( cosθ ) at √ s = 1 T eV , while (331-10) represents the corre-sponding expectations for the minimal 331 model, for a mass M Y = 1 . T eV . Curve (SU15-10) shows the prediction for theSU(15) model. behavior that only the 331 model suffers at 1
T eV . At500
GeV the three curves have qualitative similar distri-butions, but at 1
T eV , although this similarity holds forthe SM and the SU (15) model, this is no longer valid forthe 331 model, as one can see in Fig.7.Finally, we analyze the discovery potential of the spinasymmetries A T and A T for Bhabha scattering. Wedefine: -0.3-0.25-0.2-0.15-0.1-0.05 0 0.05 0.1 0.15-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 A cos θ SM-5SU15-5331-5
FIG. 8: Polar angle dependence of spin asymmetry A ( cosθ );Curve (SM-5) shows the standard model predictions for A ( cosθ ) at √ s = 500 GeV , while (331-5) represents the corre-sponding expectations for the minimal 331 model, for a mass M Y = 1 . T eV . Curve (SU15-5) shows the prediction for theSU(15) model. -0.3-0.25-0.2-0.15-0.1-0.05 0 0.05-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 A T cos θ SMSU15331
FIG. 9: Transverse polarization asymmetry A T ( cosθ ) as afunction of cosθ , at √ s = 1 T eV . The line with error bars(SM) corresponds to the standard model prediction, whilethe other curves represent the asymmetry for the minimal331 model and SU (15) model with M Y = 800 GeV . The errorbars for (SU15) and (331) are similar to those of the SM ona bin-to-bin basis. 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 , (5)where the subscript +( − ) indicates that the integrationover the azimuthal angle φ is to be carried out over theregion of phase space where cos φ is positive (negative).Fig.9 shows the result for A T considering M Y =800 GeV and energy of 1
T eV . Although the errors arebigger than the case with longitudinal polarization, the -0.003-0.002-0.001 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 1200 1300 1400 1500 1600 1700 1800 1900 2000 A T M Y (GeV) SU15331 FIG. 10: Transverse polarization asymmetry A T as a func-tion of M Y at 1 T eV . The solid histogram (331) refers tothe minimal 331 model. The resulting curve for the SU (15)model (SU15) is also shown for comparison. observable allows a clear vector bilepton signal for theconsidered extended models. The changes in the shapesof the curves for A T ( cosθ ) for the two nonstandardmodels are small when the bilepton mass is increasedto 1 . T eV .The integrated version of the asymmetry A T , werealso investigated 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 , (6)where the integrations are consistent with the cuts spec-ified in section III. The result is shown in Fig.10, wherethe mass dependence of A T for √ s = 1 T eV for both the SU (15) and the 331 model were plotted. Qualitatively,there are no differences between the curves. Neverthelessthe absolute value of A T is greater for the SU (15) model.For the sake of comparison, the approximate value of A T for the standard model was found to be − . V. BOUNDS
We estimated lower bounds on the bilepton mass by a χ test, comparing the angular distribution dσ/d ( cosθ )of the final-state electrons, modified by the presence of avector bilepton, with the corresponding standard modeldistribution. Here only longitudinal beam polarizationwas considered. We assumed that the experimental datawill be well described by the standard model predictions.We treat g l and M Y as free parameters in an effectivetheory and define our χ estimator as χ ( g l , M Y ) = N b X i =1 (cid:18) N SMi − N i ∆ N SMi (cid:19) (7) M Y ( G e V ) g / e1 TeV500 GeV FIG. 11: 95% C. L. contour plots on the ( g l /e, M Y ) planefor longitudinally polarized Bhabha scattering, at the NLCcenter-of-mass energies √ s = 500 GeV (lower curve) and √ s =1 T eV (upper curve). where N SMi is the number of standard model events de-tected in the i th bin, N i is the number 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 correspondingtotal error, which combines in quadrature the Poisson-distributed statistical error with the systematic error,which for the latter we assumed to be of ǫ = 5% foreach measurement. The angular range | cosθ | < .
95 wasdivided into N b = 20 equal-width bins. For fixed valuesof the coupling g l the bilepton mass M Y was varied asa free parameter to determine the χ distribution. The95% confidence level bound corresponds to an increaseof the χ by 3 .
84 with respect to the minimum χ min ofthe distribution. Fig.11 presents the resulting 95% C. L.contour plots on the ( g l /e, M Y ) plane for the nominalcenter-of-mass energies √ s = 500 GeV and √ s = 1 T eV .The unpolarized case is shown in Fig.12. We also calcu-lated the corresponding 95% C. L. limits on the bileptonmass at these ILC energies, considering only the minimal331 model. The results are displayed in Table I.
VI. CONCLUSIONS