Electro-weak physics with forward dynamics at LEP

Introduction
lep1 collider began its operation in the fall 1989. The energies of the electronand positron beams were chosen to be approximately equal to 45 GeV, insuch a way that the available centre of mass energy was centered aroundthe mass of the Z boson. lep1 was terminated in the fall 1995 in order toallow the second phase of the lep experimental program, lep2, at energiesabovetheZ peak. The high-energy reactions mainly studied at lep1 by thefour experiments aleph, delphi, l3 and opal are two-fermion productionprocesses in e+e collision, i.e.e+e ! 
;Z ! f fAbout 17 millions of Z events have been detected at lep. Thanks tothe very large data sample, a remarkable precision has been reached in themeasurements of the Z-boson properties and observables. Three years ofdata collecting at lep1 (1991, 1993 and 1995) were devoted to a precisionscan around the resonance, doing measurements also o-peak in the range88-95 GeV; the rest of data collecting was performed exclusively on the Zpeak. Several technological ingredients contributed to reach the fenomenalperformances of the lep1 program.The installation of precision luminometers also contributed signicantlyto the success of the lep program allowing a luminosity measurements atthe level of per mille or better, which is unavoidable to make optimal use ofthe high statistics collected at lep.In parallel with the technological and experimental progress, a huge eortwas undertaken by the theorists in the calculation of the radiative corrections.The excellent performance of the machine, associated with the above theo-retical eort, allowed to test the ne structure of the Standard Model ofelectro-weak interactions with a never reached level of precision. Moreover,the availability of both accurate measurements and calculations provided theopportunity of putting limitations on models predicting physics beyond theStandard Model.
IntroductionThe luminosity monitoring at lep is performed by studying a well-knownprocess such as the small-angle Bhabha scattering (hereafter sabh). Themain reason, for adopting such a process, is that the sabh scattering in sub-stantially a qed process, dominated by t-channel photon exchange. This inturn implies that its theoretical cross section is dominated byacontributionthat is in principle calculable by means of perturbative qed at arbitrary pre-cision. Moreover, the Z-boson exchange contributions to its cross section,via Z-boson annihilation, Z-boson t-channel exchange and Z-
 interference,is very small; hence a detailed knowledge of the Z-boson properties, liketheprecise value of its mass and decay width, to be determined in large angleprocesses, has a negligible in
uence on the luminosity monitoring. A secondmotivation is given by the fact that sabh scattering cross section is large,and can be rendered much larger than the typical Z-boson annihilation peakcross section provided the detector angular region for sabh scattering eventsis suciently close to the beam pipe.After the long data collecting at the Z pole and a short run at interme-diate energy in the 1995 fall (lep1.5 phase), the second phase of lep, lep2,started operating in 1996. The main reason for the energy upgrade of thelep machine was the precise measurement of the properties of the W boson,through the study of the reactione+e ! W+W ! f1 f2f3 f4The lep2 running was scheduled at several center-of-mass energies, rang-ing from the 161:3 GeV, i.e. 0:5 GeV above the nominal WW productionthreshold, of the rst data collection in 1996 to the present date value of189 GeV, and, as a foreseeing, to the 200 GeV of the last running in 2000.Compared to lep1, the lep2 physics potential will be characterized by astatistical error of the order of 1%, instead of 0:1% of lep1. Nevertheless,also lep2 has to be considered a machine for precision physics, since it willallow to measure the mass of the W -boson with an envisaged precision of 40-50 MeV, and to study in detail the coupling of the W with the other gaugebosons, by directly probing the non-Abelian nature of the theory. Need-less to say, besides the precision measurements of the W -boson, lep2 canalso provide some knowledge on the Higgs boson and on physics beyond theStandard Model.The data, whichwere collected at lep2,allowed a suciently high statis-tics in order to study also reactions involving only one resonant gauge bosonas e+e ! W+e ! f1 f2 e2
Introduction
Figure 0.1: Standard Model cross sections at lep2 (on the left side) [1] andbeyond (on the rightside)[2].e+e ! Z=
e+e ! f1 f2 e+eIt is worth noticing that these processes have avery dierent behaviour,when the centre-of-mass energy of the reaction is increased, if they are com-pared with the two resonant boson processes. Figure 0.1 shows on the leftside the Standard Model cross sections at lep energies, according to ref. [1].The cross section of the double resonant processes can be seen to grow up,after that the boson pair production threshold is passed, and, then, to de-cline, while the single resonant boson processes are monotonically growing.These trends are much more dramatic at higher energies, as the right side ofthe same gure shows for the Linear Collider regime, according to ref. [2]. AtLinear Colliders the single resonant boson cross section become the leadingone, while the double resonant boson ones fade away.This thesis is devoted to the study of electro-weak four-fermion processeswith non-detected particles chie
y at lep energies, but with the intent to pre-pare a subsequent study towards the Linear Collider energies. The processeswhich will be analyzed are1. e+e ! e+el+l, where l = e; , as a correction to the sabh scat-tering. Thus the event denition is given by two electrons seen in the3
Introductionsmall angle electro-magnetic calorimeter and the other two leptons losteither in the beam pipe or at large angle.2. e+e ! eqq0 for a signature of two jets. Thus the electron must bealways lost in the beam pipe, while the quarks may or may not havean angular cut.3. e+e ! e+eqq for a signature of two jets plus an electron. Thus oneof the electrons must be lost in the beam pipe, while the other must beseen at large angle. The quarks may or maynothave an angular cut.The physical motivations which support these studies are reviewed inchapter 1, then chapters 2, 3 and 4 discuss, respectively, each one of theabove processes. Chapter 5 shows conclusions and possible developments.These reactions, although corresponding to dierent physics, are all charac-terized by being four-fermion processes with particles which are scattered atsmall angle, or, in the worst situation, almost at zero angle. In this extremekinematical regime the dynamics is dominated by t-channel photon exchange,which can happen with very small momentum transfer, i.e. with quasi-realphoton exchange.This feature causes many theoretical and numerical problems, whichmustbe solved, in order to calculate the cross section for these processes. The the-oretical problems include the restoration of gauge invariance, because of thepresence of unstable particles, (see section 3.2), the proper treatmentofpho-ton radiation (see sections 3.3 and 3.4), and the dependence of cross sectionson the mass of light-quarks, due to the presence of quasi-on-shell quark prop-agators (see section 4.2). On the other hand, the numerical problems are theneed of building a massive phase space and matrix element (see sections 2.3,3.5 and 4.1), and to deal with very low momentum transfer photons (seesection 3.1).The matrix element, with all the mass terms, is computed by means of theALPHA code [3] for the automatic evaluation of Born scattering amplitudes,based on an original algorithm [3] which avoids the use of Feynman graphtechnique (see section A.2).The numerical results are needed with high precision. A per cent preci-sion is needed for the electro-weak processes, while a per mille accuracy isrequired for the pair correction to sabh scattering, whichis,by far, the moredemanding calculation shown in this thesis, as it can be seen by step-by-stepprocedure kept in sections 2.1, 2.2, 2.4 and 2.5, which, starting from therough soft approximation, arrives at the full calculation.It must be emphasized that, as possible, the numerical results listed in thisthesis are performed only as a rst step within idealized, although realistic,4
Introductionevent selections. Indeed, the numerical simulations are, then, re-run in eventselections (hereafter es) closer to the one considered by the experimentalcollaborations of lep (see sections 2.7 and 4.4). In particular the es adoptedby the opal collaboration are implemented and discussed.Essential list1 of papers and talks:1. [87] \Light pair correction to Bhabha scattering at small angle". Talkgiven byA.Pallavicini at Cortona98 Physics Workshop, Cortona, may27-30, 1998.2. [25] G. Montagna, M. Moretti, O. Nicrosini, A. Pallavicini, F. Pic-cinini \Light Pair Correction to Bhabha Scattering at Small Angle",Nucl. Phys. B, 547:39, 1999.3. [36] \Light pair correction to Bhabha scattering at small angle". Talkgiven byA.Pallavicini at LEP2 Monte Carlo Workshop, CERN, march12-13, 1999.4. [37] \Light pair correction to small-angle Bhabha scattering and thepresent theoretical error". Talk given by A. Pallavicini at XI lepPhysics Workshop, Milano, april 7-9, 1999.5. [39] G. Montagna, M. Moretti, O. Nicrosini, A. Pallavicini, F. Piccinini\Light pair correction to small angle Bhabha scattering in a realisticset-up at lep", Phys. Lett. B, 459:649, 1999.6. [70] \Electro-weak four-fermion processes with a particle lost in thebeam pipe". Talk given by A. Pallavicini at Cortona99 Physics Work-shop, Cortona, june 2-5, 1999.7. [101] \On photon radiation to single-W". Talk given by A. Pallaviciniat LEP2 Monte Carlo Workshop, CERN, october 12-14, 1999.8. [102] G. Montagna, M. Moretti, O. Nicrosini, A. Pallavicini, F. Pic-cinini \Photon radiation in electro-weak four-fermion processes with aparticle lost in the pipe", work in preparation.
1Eachentry starts with the bibliographic citation number of the item itself. 5
Chapter 1Four-fermion processes as abackground for two or threebody signatures
In this preliminary chapter the physical motivations for the study of four-fermion processes asabackground for two or three body signatures are pre-sented. In particular three main topics are discussed. In section 1.1 theproblem of luminosity measurements at lep is analyzed with the intent ofstressing the importance of pair contribution to the sabh scattering crosssection. Section 1.2 deals with the features of the single-W process. Boththe theoretical and numerical issues are discussed with a particular emphasison photon radiation. Then, section 1.3 shows the results obtained for thesingle-Z process, which is considered both as a signal on its own, and as abackground for WW physics.1.1 Light pair contribution to small-angle Bha-bha scattering and luminosity measure-mentsThe high-precision determination of the machine luminosity at lep is anessential ingredientofthesuccess of precision tests of the electroweak inter-actions on topoftheZ resonance [4].As is well known, the Bhabha scattering process at small angle (of theorder of a few degrees) is the reference reaction used for luminosity monitoringat lep, owing to its large cross section (dominated by t-channel photonexchange) and its substantial independence of purely electro-weak eects.
Section: 1.1 Chapter: 1Table 1.1: Theoretical error in small-angle Bhabha scattering according toref. [5,6] at typical lep1 and lep2 energies.Type of correction/error lep1 (%) lep2 (%)Missing photonic O(2L) 0:100 0:200Missing photonic O(3L3) 0:015 0:030Vacuum polarization 0:040 0:100Light pairs 0:030 0:050Z-exchange 0:015 0:000Total 0:110 0:250Experimental eorts in the development of ecient, dedicated luminometrydetectors, as well as precision calculations of the sabh scattering cross sectionboth contribute to achieving a measurement of the Z factories luminositywith a total relative error at the 0:1% level [4{6]. On the experimental side,the present total uncertainty is smaller than 0:1% [7,8], close to the 0:05%level [9,10]. As far as the theory contribution to the luminosity measurementis concerned, the estimate of the theoretical errors, according to ref. [5,6],which was used before 1999 summer conferences, is summarized in table 1.1[5,6] for centre-of-mass energies around and above the Z resonance.Some comments on table 1.1 are in order. The components of the theo-retical error refer to the sabh scattering cross section, for any typical eventselection of lep experiments, as computed by the program BHLUMI v4.04[11{13]. The largely dominating source of theoretical error is the missingpart of O(2L) sub-leading photonic corrections, where L = ln(t=m2) isthe collinear logarithm in t-channel scattering. Also the contribution of themissing part of the leading O(3L3) corrections is of photonic nature. Thevacuum polarization entry is the eect of the uncertainty in the hadroniccontribution to the running of QED, when considering the parametrizationand relative error estimate of ref. [14,15]. The next contribution is the un-certainty introduced by the corrections due to the production of light pairs,chie
y e+e ones. The last entry refers to the uncertainty associated to thetreatment of the 
{Z interference. More details about the strategy adoptedto estimate the various sources of theoretical error can be found in ref. [5,6].After the analysis of ref. [5,6], important theoretical developments tookplace. Additional work in the sector of two-loop photonic corrections [16{20]led to the conclusion that the perturbativecontribution from the uncontrolledpart of O(2L) corrections does not exceed the 0:03% level. This conclusion8
Chapter: 1 Section: 1.2has been very recently reinforced by the detailed analysis of ref. [9,10]. Fur-thermore, new determinations [21{24] of QED lower the error on the hadroniccontribution to vacuum polarization in s-channel processes at ps = MZ .This might aect sabh scattering too, although no dedicated analysis forthe low-angle regime exists yet.As a consequence of this progress, it has recently become relevant to re-duce the uncertainty associated to the light-pair contribution. Before thecalculation here presented [25], the theoretical error to sabh scattering, dueto pair production, could be evaluated by approximate means, such as theMonte Carlo (hereafter mc) results based on t-channel approximation [26] orthe analytical calculations in the quasi-collinear approximation [27{32]. Pre-vious leading logarithmic evaluations of the dominantlight-pair contributionto sabh can be found in ref. [33]. Another approach, based on the mccalculation of ref. [25], which includes the exact qed four-fermion matrixelement, two-loop virtual corrections according to ref. [34,35], initial-stateradiation (hereafter isr) in the collinear approximation, and realistic es, isalso available.In this paper the results shown in ref. [25] are reviewed and discussed inchapter 2. The analysis is chie
y presented at lep1 energies (ps = 92 GeV),but numerical results are shown at lep2 energies (ps = 176 GeV) too. Theimpact of the present calculation on the reduction of the theoretical errorfor the lep luminosity measurements is also discussed [36,37]. Moreover,the presence of es's as realistic as possible is carefully addressed, payingparticular attention to the opal es [38], as a signicant case study [39].1.2 Single-W resonant diagrams andW physicsFour-fermions nal states are of special interest for lep2, being entangledwith electro-weak gauge boson production and decay. In this thesis we willstudy the process e+e ! e(e+)()qq0 (1.1)The bulk of the cross section for the process (1.1) is due to two sub-processes, i.e. W -boson pair production and decaye+e ! W+W ! f1 f2f3 f4 (1.2)and the radiation of an almost on shell t-channel photon from the electron(positron) with subsequent production of a W -boson and a neutrino 9
Section: 1.2 Chapter: 1
e+e ! W+e ! f1 f2f3 f4 (1.3)Despite, strictly speaking, both sub-processes (1.2) and (1.3) always occursimultaneously and are indistinguishable, (1.2) dominates if the electron isemitted at large angle whereas (1.3) dominates if the electron is emitted inthe very forward region since in this region the propagator of the t-channelphoton becomes very large.In this thesis we will deal with the process (1.3), i.e. we will restrictthe kinematical range to forwardly emitted electrons. This signal is usuallyreferred to as single-W events since only the two nal jets are detected.The importance of this process has been emphasized since long time ago.In the lep2 energy range it is important in order to study the self inter-action of the gauge bosons, together with the process (1.2), whereas in theLinear Collider energy regime it becomes the most important standard modelprocess. In [40{44] cross sections and distributions are given in the approx-imation of real W -boson production either by [40{42] studying the reactione+e ! eW+ or by [43{47] employing the Weizsacker-Williams [48{50]equivalent-photon approximation for the t-channel photon. In [40{47] it hasbeen pointed out the relevance of this process to the study of trilinear gaugeboson couplings and some assessment of the sensitivityhasbeen given. Thefull four-fermion calculation, including the crucial eect of fermion masses,has been presented in [51] where the lep2 sensitivity to anomalous gaugecouplings has been studied.Measurements of the single-W cross section and the corresponding boundson anomalous gauge couplings have been reported by the lep collaborations[52]. The forecasted nal accuracy is of the order of 1%  2% [53] andtherefore an accurate theoretical prediction for cross section and distributionsis mandatory.The calculation of the cross section for single-W processes poses severalnon trivial theoretical problems. For a realistic account of gauge bosons prop-erties it is mandatory to include the gauge boson width in the propagator.In general this mixes a xed order calculation with an all order resumma-tion of a class of Feynman diagrams and introduces a violation of the Wardidentities of the theory. This issue is of special importance here since, due tothe t-channel photon exchange, even atiny violation of qed Ward identitiesis enhanced by a factor of s=m2e. This is indeed the case if a running widthis used in the calculation. This problem has been extensively studied [54]and several options to address this problem have been explored. By com-paring with the fermion loop scheme, which [54,55] preserves both U(1) andSU(2) Ward identities, it has been shown that the xed width scheme (which10
Chapter: 1 Section: 1.3preserves U(1) Ward identities) does provide a satisfactory solution to theproblem (at least at the practical level). In the present paper we will makeuse of the xed width scheme.Another delicate issue is the so called resolved-photon component of thecross section. The quasi-real t-channel photon can split a pair of almostmassless quarks (massless and/or massive) leading to a situation where thepartonic picture of hadrons is inadequate and both perturbative and non-perturbative qcd corrections are large. This issue is raised in [56] wherethe standard approach to this problem is also discussed. To the best of ourknowledge a state-of-the-art calculation of the resolved photon contributionto this process is still missing, however we expect that hadronic uncertaintieswill aect the result at the several per cent level. For single-W like eventsthis does not constitute a severe limitation: once a hard qq invariant masscuts is imposed, the bulk of the signal is kept whereas the resolved photoncontribution becomes almost negligible [57]. We will not discuss this item inthis paper1.Another relevant issue is photon radiation. The goal of the photon radi-ation problem, in line of principle, is to work out the O() complete calcu-lation. Yet, some hints, to implement more correctly the dominant photoncorrection, can be shown without exploiting the full calculation. One of thestandard way of including the leading-log contribution is the structure func-tion formalism [58{65] (hereafter denoted as sf), which can be approachedeither via analytical approximation or via parton-shower numerical methods.This calculation requires, as a starting point, the scale of the process, whichis arbitrary, but a diagrammatic calculation allows to x it. In both theo-retical and experimental analysis presented insofar initial-state radiation hasbeen treated assuming the centre of mass energy as the most appropriatescale for the radiation. Due to the dominance of the t-channel (quasi-real)photon exchange this is not the proper choice in the present case [66].1.3 Single-Z resonant diagrams and di-jet eventsThe process e
 ! eZ=
 has been measured for the rst time [67] in thee+e collision with the opal detector. This is a subprocess of the reactione+e ! e+eZ=
 where a quasi-real photon radiated from one of the beamelectrons scatters o the other electron producing a Z=
. For deniteness,1For new physics searches di-jet events plus missing energy are an importantchanneland the search is carried on with a much softer ( a few GeV) cuts on the di-jet invariantmass. In this case, for a good estimate of the background, the resolved photon issue is animportantone 11
Section: 1.3 Chapter: 1
in the following of this thesis the processe+e ! e+eZ=
 ! e+euu (1.4)is considered and it is named single-Z2. The observable nal state in e+ecollisions, (e)ef f , will consist of the scattered electron e and a fermion pairf f from the Z=
 decay while the other electron (e) remains unobservablein the beam pipe. Thus the momentum transfer of the quasi-real photonbecomes very small.The characteristics of the single-Z signature resemble the ones of thesingle-W process, due to the presence of an electron lost in the beam pipe.On the other hand, the dynamics is controlled by dierent channels, due tothe fact that the Z-boson is not charged and, so, it does not couple withphotons. Among the leading Feynman graphs there are diagrams which giveraise to additional problems both from the numerical and theoretical pointofview, since they areformedby quark propagators, which can become quasi-real, owing to the particular phase-space region being studied. Because ofthis feature the cross section for the single-Z scattering becomes sensitivetothe choice of the quark masses, and so prone to their uncertainties.The experimental collaborations try to avoid these problems by imposingsuitable cuts on the event sample. A discussion of the kinematical cuts isaddressed in the following for a reference idealized, althouh realistic, set-up. Then, as done for the sabh scattering, the numerical results will bespecialized in a more realistic es. For deniteness, in this thesis the opal esis chosen as the reference one [68].The e+e ! e+euu process is important both as a background for WWphysics and as a signal on its own. The dierential distribution for charged-current diagrams may superimpose on the distribution for single-Z processes.Thus the determination of a kinematical discriminantbetween the two pro-cesses is a relevant issue for WW-physics.Recently the opal collaboration has also claimed for some inconsistenciesbetween PYTHIA and Grc4f when used as event generators for the single-Zsignature [69], since these two programs give cross section distribution withrespect to the quark-antiquark invariant mass which disagree at the Z peak,although the experimental error does not allowforadistinction yet [68].A mc code is implemented to cope with the neutral current signature[70]. In particular the aim is to build a cross section integrator and an eventgenerator for the single-Z signature. The program is a natural developmentof the single-W code, so that they are fused in a unique code.2This name convention is not well-established, on the contrary of the single-W case.Thus, it is possible that this process, in the literature, is referred to as the Zee signature.12