physics.txt

H. Aihara1†, T. Barklow2, U. Baur3,4†, J. Busenitz5, S. Errede6, T. A. Fuess7, T. Han8,D. London9†, J. Ohnemus8, R. Szalapski10, C. Wendt11, and D. Zeppenfeld11†1Lawrence Berkeley Laboratory, Berkeley, CA 947202Stanford Linear Accelerator Center, Stanford, CA 94309 3Physics Department, SUNY Buffalo, Buffalo, NY 142604Physics Department, Florida State University, Tallahassee, FL 323065Department of Physics and Astronomy, University of Alabama, Tuscaloosa, AL 35487 6Physics Department, University of Illinois at Urbana – Champaign, Urbana, IL 61801 7Argonne National Laboratory, Argonne, IL 60439 8Physics Department, University of California, Davis, CA 95616 9Physics Department, University of Montreal, Canada H3C 3J710Theory Group, KEK, Tsukuba, Ibaraki 305, Japan11Physics Department, University of Wisconsin, Madison, WI 53706ABSTRACTWe discuss the direct measurement of the trilinear vector boson couplings in presentand future collider experiments. The major goals of such experiments will be theconfirmation of the Standard Model (SM) predictions and the search for signals ofnew physics. We review our current theoretical understanding of anomalous trilineargauge-boson self interactions. If the energy scale of the new physics is∼1 TeV,these low energy anomalous couplings are expected to be no larger thanO(10−2).Constraints from high precision measurements at LEP and low energy charged andneutral current processes are critically reviewed.1. IntroductionOver the last five yearse+e− collision experiments at LEP and at the SLAClinear collider have beautifully confirmed the predictions of the Standard Model(SM). At present experiment and theory agree at the 0.1 – 1% level in the determinationof the vector boson couplings to the various fermions [1], which mayrightly be considered a confirmation of the gauge boson nature of theW and theZ.Nevertheless the most direct consequences of the SU(2)L×U(1)Y gauge symmetry,∗Summary of the Working Subgroup on Anomalous Gauge Boson Interactions of the DPF Long-RangePlanning Study, to be published in “Electroweak Symmetry Breaking and Beyond the Standard Model”,eds. T. Barklow, S. Dawson, H. Haber and J. Siegrist. †Co-convener 1the non-abelian self-couplings of the W, Z, and photon, remain poorly measured todate.A direct measurement of these vector boson couplings is possible in presentand future collider experiments, in particular via pair production processes likee+e− → W+W−, Zγ and qq¯ → W+W−, W γ, Zγ, W Z. The first and major goal ofsuch experiments will be a confirmation of the SM predictions. A precise and directmeasurement of the trilinear and quartic couplings of the electroweak vector bosonsand the demonstration that they agree with the SM would beautifully corroboratespontaneously broken, non-abelian gauge theories as the basic theoretical structuredescribing the fundamental interactions of nature.At the same time, such measurements may be used to probe for new physics.Since the gauge boson self-couplings have not yet been measured with good precision,it is possible in principle that signals for physics beyond the SM will appearin this sector through the discovery of anomalous trilinear (or quartic) gauge-bosonvertices (TGV’s). This possibility immediately raises a number of other questions.What are the expected sizes of such anomalous effects in different models of newphysics? Will the new physics which gives rise to anomalous gauge-boson couplingsmanifest itself in other observables and/or other channels? Are there significantconstraints from low-energy measurements? We address these questions in Section2.For the most part, however, we are interested in the accuracy of various colliderexperiments for a direct measurement of the self-interactions of electroweakvector bosons, so as to evaluate how well the SM predictions can be tested (Section3). For simplicity, we shall restrict ourselves to trilinear vector boson couplings,in particular the WW V , and ZγV , V = γ, Z couplings. Possibilities to test quarticcouplings in collider experiments are discussed in Ref. [2].Analogous to the introduction of arbitrary vector and axial vector couplingsgV and gA for the coupling of gauge bosons to fermions, the measurement of theWW V couplings can be made quantitative by introducing a more general WW Vvertex. For our discussion of experimental sensitivities in Section 3 we shall use aparameterization in terms of the phenomenological effective Lagrangian [3]iLWW Vef f = gWW V hgV1W†µνWµ − W† µWµνVν + κV W†µWνVµν + (1)λVm2WW†ρµWµνVνρ + igV5εµνρσ (∂ρW† µ)Wν − W† µ(∂ρWν)Vσi.Here the overall couplings are defined as gWW γ = e and gWW Z = e cot θW , Wµν =∂µWν −∂νWµ, and Vµν = ∂µVν −∂νVµ. Within the SM, at tree level, the couplings aregiven by gZ1 = gγ1 = κZ = κγ = 1, λZ = λγ = gZ5 = gγ5 = 0. For on-shell photons, gγ1 = 1and gγ5 = 0 are fixed by electromagnetic gauge invariance; gZ1 and gZ5 may, however,differ from their SM values. Deviations are given by the anomalous TGV’s∆gZ1 ≡ (gZ1 − 1) , ∆κγ ≡ (κγ − 1) , ∆κZ ≡ (κZ − 1) , λγ , λZ , gZ5. (2)As we discuss in Section 2, theoretical arguments suggest that these anomalousTGV’s are at most of O(m2W /Λ2), where Λ is the scale of new physics (some are2expected to be considerably smaller). Thus, for Λ ∼ 1 TeV, the anomalous TGV’sare O(10−2), which will make their observation difficult. Conversely, if large anomalousTGV’s are discovered, this implies that the new physics responsible for themis likely to be found directly below the TeV scale.The effective Lagrangian of Eq. (1) parameterizes the most general Lorentzinvariant and CP conserving WW V vertex which can be observed in processes wherethe vector bosons couple to effectively massless fermions. Apart from gV5, all couplingsconserve C and P separately. If CP violating couplings are allowed, threeadditional couplings, gV4, κ˜V and λ˜V , appear in the effective Lagrangian [3] andthey all vanish in the SM, at tree level. For simplicity, these couplings are notconsidered in this report. Terms with higher derivatives are equivalent to a dependenceof the couplings on the vector boson momenta and thus merely lead to aform-factor behaviour of these couplings (see Section 2.3). The C and P conservingterms in LWW γef f correspond to the lowest order terms in a multipole expansion ofthe W−photon interactions, the charge QW , the magnetic dipole moment µW andthe electric quadrupole moment qW of the W+ [4]:QW = egγ1, (3)µW =e2mW(gγ1 + κγ + λγ) , (4)qW = −em2W(κγ − λγ) . (5)Analogous to the general WW V vertex it is possible to parameterize anomalousZγV, V = γ, Z couplings. We shall be interested in constraints from Zγ productionprocesses in Section 3, i.e. we may treat the photon and the Z as being on-shell.As before we are only considering CP-even couplings. Let us denote the Feynmanrule for the Vµ(P) → Zα(q1)γβ(q2) vertex by ieΓαβµZγV (q1, q2, P). The most general suchvertex compatible with Lorentz invariance has been discussed in Ref. [3] and it canbe parameterized in terms of two free parameters, hV3 and hV4,ΓαβµZγV (q1, q2, P) = P2 − m2Vm2ZhhV3εµαβρq2ρ +hV4m2ZPαεµβρσPρq2σi. (6)Within the SM, at tree level, hV3 = hV4 = 0. If CP violating couplings are allowed, twoadditional couplings, hV1 and hV2, appear in the effective Lagrangian [3] which alsovanish in the SM, at tree level. For simplicity, these couplings are not consideredhere. The overall factor P2 − m2Vin Eq. (6) is implied by Bose symmetry for onshellV and/or by gauge invariance for V = γ. These additional factors indicate thatanomalous ZγV couplings can only arise from higher dimensional operators thanthe WW V couplings and hence their effects should be suppressed in any scenario ofnew physics beyond the SM.In Section 3 present measurements from pp¯ and e+e− collider experimentsare summarized. In addition the sensitivity of future Tevatron, LHC, LEP II and3NLC experiments is analysed in detail. Our conclusions are presented in Section 4.2. Theoretical Background2.1 Effective Lagrangians: General ConsiderationsIn this Section we discuss theoretical ideas which lead to anomalous gaugeboson self-interactions, and analyze constraints from low energy and high precisionmeasurements. In the absence of a specific model of new physics, effectiveLagrangian techniques are extremely useful. An effective Lagrangian [5] parameterizes,in a model-independent way, the low-energy effects of the new physics tobe found at higher energies. It is only necessary to specify the particle content andthe symmetries of the low-energy theory. Although effective Lagrangians containan infinite number of terms, they are organized in powers of 1/Λ, where Λ is thescale of new physics. Thus, at energies which are much smaller than Λ, only thefirst few terms of the effective Lagrangian are important.The Fermi theory of the weak interactions is perhaps the best-known exampleof an effective Lagrangian. Within the SM, the charged-current interaction betweentwo fermions is described by the exchange of a W-boson:g28Ψγµ(1 − γ5)Ψ 1q2 − m2WΨγµ(1 − γ5)Ψ , (7)where q2is the momentum transfer (energy scale) of the interaction. We can expandthe W-propagator in powers of q2/m2W :1q2 − m2W= −1m2W"1 +q2m2W+ ...#. (8)The interaction of Eq. (7) can thus be written as the sum of an infinite number ofterms. However, we note that, for energies well below the W mass, only the firstterm is important. This is simply the 4-fermion interaction of the Fermi theory:−GF√2Ψγµ(1 − γ5)Ψ Ψγµ(1 − γ5)Ψ , (9)where GF /√2 = g2/8m2W . In other words, the Fermi theory is the effective theoryproduced when one “integrates out” the heavy degrees of freedom (in this case, theW boson). It is valid at energy scales much less than the scale of heavy physics(q2 ≪ m2W ).Note that, as q2 approaches m2W , one can no longer truncate after the lowestorderterm in q2/m2W . This is evidence that the effective Lagrangian is breakingdown – each of the infinite number of terms becomes equally important as oneis approaching energy scales where the heavy degrees of freedom can be directlyproduced, i.e. they cannot be integrated out. Note also that the truncated effectiveLagrangian (the Fermi theory) violates S-matrix unitarity for q2 > m2W /(g2/4π).Unitarity is restored in the full theory by propagator (form factor) effects and the4scale at which unitarity is apparently violated gives an upper bound for the massesof the heavy degrees of freedom (here the W mass). In a weakly coupled theorylike the SM this upper bound substantially overestimates the masses of the heavydegrees of freedom. Apart from resonance enhancement one needs strong interactiondynamics to obtain cross sections in the full theory which approach the unitaritylimits. As the energy scale is increased, new channels will open up in addition(e.g. WW and W Z production in the case of the Fermi theory). However, the crosssections of these new channels may be too low to be observable, especially if theunderlying dynamics is perturbative in nature. These features, which are easilyunderstood in the context of the SM and the Fermi theory, are general propertiesof all effective Lagrangians.2.2 Power CountingIn order to define an effective Lagrangian, it is necessary to specify the symmetryand the particle content of the low-energy theory. Since all experimentalevidence is consistent with the existence of an SU(2)L × U(1)Y gauge symmetry itis natural to require the effective Lagrangian describing anomalous TGV’s to possessthis invariance. Inspecting Eq. (1), the phenomenological effective LagrangianLWW Vef f describing anomalous WW V couplings appears not to respect this constraint.This impression is wrong, however, since Eq. (1) can be interpreted as the unitarygaugeexpression of an effective Lagrangian in which the SU(2)L × U(1)Y gaugesymmetry is manifest [6]. How this symmetry is realized depends on the particlecontent of the effective Lagrangian. If one includes a Higgs boson, the symmetrycan be realized linearly, otherwise a nonlinear realization of the gauge symmetry isrequired. We will discuss each of these options in turn.• Linear RealizationWe first consider the linear realization scenario, in which a Higgs doublet fieldΦ is included in the low-energy particle content. This is also called the “decouplingphysics” scenario in the literature because, with the inclusion of a light Higgs boson,the scale of new physics is allowed to be arbitrarily large, even Λ ∼ 1015 GeV wouldbe self-consistent. In addition to the Higgs field the building blocks of the effectiveLagrangian are covariant derivatives of the Higgs field, DµΦ, and the field strengthtensors Wµν and Bµν of the W (SU(2)L) and the B (U(1)Y ) gauge fields:[Dµ, Dν ] = Bˆµν + Wˆµν = ig′2Bµν + i gσa2Waµν . (10)Here, σa, a = 1, . . . , 3 denote the Pauli matrices, and g and g′ are the SU(2)L andU(1)Y gauge coupling constants, respectively. Considering dimension-6 operatorsonly, 11 independent such operators can be constructed [7, 8] of which only 7 arerelevant for our discussion:Lef f =X7i=1fiΛ2 Oi =1Λ2fΦ,1 (DµΦ)†Φ Φ†(DµΦ) + fBW Φ†BˆµνWˆ µνΦ5+ fDW T r([Dµ, Wˆνρ] [Dµ, Wˆ νρ]) − fDBg′22(∂µBνρ)(∂µBνρ)+ fB (DµΦ)†Bˆµν(DνΦ) + fW (DµΦ)†Wˆ µν(DνΦ)+ fWWW T r[WˆµνWˆ νρWˆρµ]. (11)The first four operators, OΦ,1, OBW , ODW , and ODB, affect the gauge bosontwo-point functions at tree level [9] and as a result the coefficients of these fouroperators are severely constrained by present low energy data. The remaining three,OB, OW and OWWW , give rise to non-standard triple gauge boson couplings. Theirpresence in the effective Lagrangian leads to deviations of the WW V couplings fromthe SM, namely [8, 10]∆κγ = (fB + fW )m2W2Λ2, ∆κZ =fW − s2(fB + fW ) m2Z2Λ2, (12)∆gZ1 = fWm2Z2Λ2= ∆κZ +s2c2 ∆κγ , (13)λγ = λZ = λ =3m2W g22Λ2fWWW , (14)with s = sin θW and c = cos θW . Note that all anomalous TGV’s are suppressedby a factor m2W /Λ2 and hence they vanish in the decoupling limit. In fact thisbehaviour is required by unitarity considerations with, typically, |fi| <∼ 32π [11].In general, the coefficients fi are expected to be numbers of order unity. Hence,taking Λ ∼ 1 TeV, one might expect anomalous TGV’s of O(10−2). As pointed outby Einhorn and collaborators [12], the dimension six operators OWWW , OW andOB which lead to anomalous TGV’s cannot be generated at the tree level by anyrenormalizable underlying theory which leads to the effective Lagrangian of Eq. (11).Thus, in this scenario, the expected size of the anomalous TGV’s would be tiny,∼ 1/(16π)2(m2W /Λ2), and only small scales Λ would be accessible experimentally.In the same scenario dimension 8 operators leading to TGV contributions can begenerated at tree level and, thus, they might dominate over the dimension 6 termsconsidered above if Λ is sufficiently small. Since the correlations between differentanomalous WW V couplings exhibited in Eqs. (13) and (14) are due to the truncationof the effective Lagrangian at the dimension six level [8] these relationships wouldnot even be approximately correct in this case.Anomalous ZγV couplings originate only from terms of dimension 8 or higherin the effective Lagrangian and, therefore, are expected to be O(m4Z/Λ4).• Nonlinear RealizationLet us now turn to the scenario in which the SU(2)L ×U(1)Y gauge symmetryis realized non-linearly (“non-decoupling physics”). In this case, one includes onlythe would-be Goldstone bosons (WBGB’s) which give masses to the W- and Zbosons.Since there is no Higgs boson, the low-energy Lagrangian violates unitarityat a scale of roughly 4πv ∼ 3 TeV, so that the new physics must appear at a scaleΛ <∼ 4πv.6A number of nonlinear realizations appear in the literature, all of which aresimilar [13]. For the purpose of illustration we will choose one which conservesthe custodial SU(2)C symmetry of the SM in the limit g′ → 0. Using the matrixΣ ≡ exp(i~ω · ~σ/v), where the ωi are the WBGB’s, we define the SU(2)L × U(1)Ycovariant derivative:DµΣ ≡ ∂µΣ + i2gWaµ σaΣ −i2g′BµΣσ3 . (15)One then constructs terms in the effective Lagrangian using field strengths (Wµν,Bµν ) and covariant derivatives. This effective Lagrangian is known generically asa “chiral Lagrangian,” due to its similarity to low-energy QCD (and chiral perturbationtheory). In the unitary gauge the covariant derivative becomes a linearcombination of gauge bosons. Thus, a gauge-boson field can be constructed by takingthe trace of DµΣ with the appropriate σ matrix, e.g. Zµ ∼ T r[σ3Σ†DµΣ]. In thisway, we can write the Lagrangian of Eq. (1) in terms of SU(2)L × U(1)Y -invariantquantities.Our experience with QCD tells us how to estimate the size of any term in achiral Lagrangian. This estimate is called “naive dimensional analysis” (NDA) [14].It states that a term having b WBGB fields, f (weakly-interacting) fermion fields,d derivatives and w gauge fields has a coefficient whose size iscn(Λ) ∼ v2Λ21vb 1Λ3/2f 1Λd gΛw. (16)Applying NDA to the terms in Eqs. (1) and (2), we see that ∆gV1 and ∆κV areof O(m2W /Λ2). In other words, just as in the linear realization, these terms areeffectively of dimension 6 (in the sense that there is an explicit factor of 1/Λ2). Onthe other hand, we see that the W†ρµWµνVνρ term is effectively of dimension 8, i.e.the coefficient λV is expected to be of order m4W /Λ4. Thus, within the nonlinearrealization scenario, the λV terms are expected to be negligible compared to thoseproportional to ∆gV1 and ∆κV .Within the nonlinear realization scenario, there are two operators whichcontribute to anomalous TGV’s (and not to two-point functions) at lowest order[15, 16]. Writing the heavy mass dependence explicitly, they are:− igv2Λ2L9L T r hWµνDµΣDνΣ†i− ig′v2Λ2L9R T r hBµνDµΣ†DνΣi. (17)These are related to ∆gV1 and ∆κV by:∆gZ1 =e22c2s2v2Λ2L9L,∆κγ =e22s2v2Λ2(L9L + L9R), (18)∆κZ =e22c2s2v2Λ2(L9Lc2 − L9Rs2),7where s2c2 = πα(mZ )/√2GF m2Z. (The gZ5coupling is studied in Ref. [17].) Note thatas far as these three TGV’s are concerned the linear and the nonlinear realization areobtained from each other by identifying L9L = 2fW and L9R = 2fB. In particular thecorrelation between TGV’s as given in Eqs. (13) and (14) holds in both frameworksas long as higher-dimensional operators can be neglected.Anomalous ZγV couplings again originate only from higher order terms inthe effective Lagrangian.2.3 Form FactorsAlthough the anomalous TGV’s ∆gV1, ∆κV , etc. appear as constants inEqs. (2) and (6), they should rather be considered as form factors. Consider the∆κV term, W†µWνVµν . One can write down similar higher-order terms such as1Λ2 W†µWν Vµν, (19)which has the same Feynman rules as W†µWνVµν, except for a multiplicative q2dependence due to the derivatives in . Taking into account all such operators, theoverall coefficient of the Feynman rule is not ∆κV , but rather a form factor∆κ0V1 + aq2Λ2+ bq2Λ2!2+ ... , (20)where a, b, etc. are O(1). Since a constant anomalous TGV would lead to unitarityviolation at high energies [18] such a form factor behaviour is a feature of any modelof anomalous couplings. When studying W+W− production at an e+e− collider atfixed q2 = s this form factor behaviour is of no consequence. Weak boson pairproduction at hadron colliders, however, probes the TGV’s over a large q2 = ˆsrange and is very sensitive to the fall-off of anomalous TGV’s which necessarilyhappens once the threshold of new physics is crossed. Not taking this cutoff intoaccount results in unphysically large cross sections at high energy (which violateunitarity) and thus leads to a substantial overestimate of experimental sensitivities.In our analysis in Section 3 we will assume a simple power law behaviour, e.g.∆κV (q2) = ∆κ0V(1 + q2/Λ2F F )n, (21)and similarly for the other TGV’s. Here ΛF F is the form factor scale which isa function of the scale of new physics, Λ. For WW V couplings we shall use theexponent n = 2, which will be referred to as the ‘dipole form factor’ below. For ZγVcouplings we choose n = 3 (n = 4) for hV3(hV4). Due to the form factor behaviourof the anomalous couplings, the experimental limits extracted from hadron colliderexperiments explicitly depend on ΛF F .The values ∆κ0Vetc. of the form factors at low energy are constrained by partialwave unitarity of the inelastic vector boson pair production amplitude in fermion8antifermion annihilation at arbitrary center of mass energies. Assuming that onlyone anomalous coupling is nonzero at a time, one finds, for ΛF F ≫ mW , mZ [11, 19]|∆gZ01| ≤ nn(n − 1)n−10.84 TeV2Λ2F F, |gZ05| ≤ (2n)n(2n − 1)n−1/23.2 TeVΛF F, (22)|∆κ0γ| ≤ nn(n − 1)n−11.81 TeV2Λ2F F, |∆κ0Z| ≤ nn(n − 1)n−10.83 TeV2Λ2F F, (23)|λ0γ| ≤ nn(n − 1)n−10.96 TeV2Λ2F F, |λ0Z| ≤ nn(n − 1)n−10.52 TeV2Λ2F F, (24)hZ30 ≤(23 n)n(23 n − 1)n−3/20.126 TeV3Λ3F F, |hγ30| ≤ (23 n)n(23 n − 1)n−3/20.151 TeV3Λ3F F, (25)hZ40 ≤(25 n)n(25 n − 1)n−5/22.1 · 10−3 TeV5Λ5F F, |hγ40| ≤ (25 n)n(25 n − 1)n−5/22.5 · 10−3 TeV5Λ5F F.(26)The bounds listed in Eqs. (22) – (26) have been computed with mW = 80 GeV andmZ = 91.1 GeV. In order to satisfy unitarity, n ≥ 1 for ∆gZ1, ∆κV and λV , n ≥ 1/2 forgZ5, n ≥ 3/2 for hV3, and n ≥ 5/2 for hV4. If more than one coupling is varied at a time,cancellations between the TGV’s may occur, and the unitarity limits are weakenedsomewhat. For ΛF F ≫ mW , mZ, the unitarity limits drop like a power of 1/ΛF F withincreasing values of the form factor scale. The experimental limits obtained fromhadron collider experiments must be compared with the bounds derived from Smatrixunitarity. Experiments constrain the WW V and ZγV couplings non-triviallyonly if the experimental limits are more stringent than the unitarity bounds, for agiven value of ΛF F .Strictly speaking the appearance of form factor effects implies that the effectiveLagrangian description in terms of a small set of low-dimensional operatorsbreaks down, i.e. one is probing weak boson pair production at the scale of newphysics. New channels are expected to open up as well. However, the correspondingcross sections might be too small to be observable immediately or the experimentalsignatures might be obscured by backgrounds (compare e.g. W Z productionin the Fermi theory). Thus form factors are a tool to extend the use of effectiveLagrangians to the entire energy range which is accessible at hadron colliders.2.4 Phenomenological Bounds from High Precision ExperimentsIn Section 2.2, we have discussed the reasons why anomalous TGV’s areexpected to be O(m2W /Λ2) at most in an effective Lagrangian approach. However,it is also interesting to ask what is known about anomalous TGV’s from experiment.The errors of present direct measurements, via pair production of electroweakbosons, are still very large (of order 100%, see Section 3). More preciseconstraints might then arise from loop contributions to precisely measured quantitiessuch as (g − 2)µ [12, 20], the b → sγ decay rate [21, 22], B → K(∗)µ+µ− [23], theZ → b¯b [24] rate and oblique corrections (i.e. corrections to the two point functions)to 4-fermion S-matrix elements. Oblique corrections combine information from the9recent LEP/SLD data, neutrino scattering experiments, atomic parity violation, µ-decay, and the W-mass measurement at hadron colliders. These analyses have beenperformed for WW V couplings in the context of linear and nonlinear realizations,and we discuss both of these in turn.• Linear RealizationA complete analysis of low energy constraints on the coefficients of the effectiveLagrangian of Eq. (11) was performed in 1992 [8]. Here we update theseresults by using the comprehensive 1994 analysis of electroweak data by Hagiwaraet al. [25]. With α = 4πe2(0) taken as an input parameter, the neutral- and chargedcurrentdata may be parameterized in terms of three effective form-factors, g¯2Z(q2)and g¯2W (q2) defining the coupling strength of the Z and the W at momentum transferq and the square of the effective weak mixing angle, s¯2(q2). For mt = 174 GeV andαs(m2Z) = 0.12 the LEP and SLD data can be summarized in terms ofg¯2Z(m2Z) = 0.55673 ± 0.00087 , s¯2(m2Z) = 0.23051 ± 0.00042 , ρ = 0.28 , (27)where ρ is the correlation of the two values. In a similar fashion the low-energy dataon neutrino scattering and atomic parity violation determine the same form-factorsat zero momentum transfer:g¯2Z(0) = 0.5462 ± 0.0036 , s¯2(0) = 0.2353 ± 0.0044 , ρ = 0.53 . (28)Finally, the W-mass measurement at hadron colliders together with the input valueof GF can be translated into a measurement of g¯2W (0):g¯2W (0) = 0.4225 ± 0.0017 . (29)These five measurements are closely related to other formulations of theoblique corrections, like the S, T, and U parameters of Peskin and Takeuchi [26]. Thenew feature here is the inclusion of the q2 dependence of the form – factors [8, 25, 27].Indeed, new physics contributions like the operators ODW or ODB do lead to anontrivial q2 dependence of the form-factors, and the more general analysis is neededto constrain these operators. Low energy bounds are obtained by fittingS = SSM(mt, mH ) + ∆S , (30)T = TSM(mt, mH) + ∆T etc. (31)to the data. Here the SM contributions (SSM etc.) introduce a significant dependenceon the values of the Higgs boson and the top quark masses.The four operators ODW , ODB, OBW , and OΦ,1 contribute already at treelevel,∆δρ = α∆T = −v22Λ2fΦ,1 , (32)∆S = −32πs2 m2WΛ2(fDW + fDB) − 4πv2Λ2fBW , (33)10with similar results for the other form-factors. Fitting these to the five data pointsone obtains measurements of the coefficients of the operators in the effective Lagrangian.The central values depend on the top quark and Higgs boson masseswhich we parameterize in terms ofxt =mt − 175 GeV100 GeV , xH =mH100 GeV . (34)Within better than 5% of the 1σ errors, and in the ranges 140 GeV < mt < 220 GeVand 60 GeV < mH < 800 GeV this dependence is given byfDW = −0.35 + 0.012 log xH − 0.14 xt ± 0.62 , (35)fDB = −11 + 0.11 log xH − 0.58 xt ± 11 , (36)fBW = 3.1 + 0.072 log xH + 0.22 xt ± 2.6 , (37)fΦ,1 = 0.23 − 0.031 log xH + 0.36 xt ± 0.17 , (38)assuming Λ = 1 TeV. The correlation matrix C is found to beC =VijpViiVjj=1. -0.323 0.151 -0.2281. -0.979 -0.8061. 0.9051.. (39)Both the 1σ errors and the correlation matrix elements are independent of mHand mt to high precision. Note the strong correlations between the coefficientsof the dimension six operators, in particular between fDB, fBW and fΦ,1. Whilethe contributions of these four operators are already constrained at the tree level,the anomalous WW V couplings only contribute at the 1-loop level to the obliquecorrection parameters. Neglecting all terms which are not logarithmically divergentfor Λ → ∞, the leading effects are given by replacing fDW etc. in Eqs. (32) and (33)by the renormalized quantitiesfrDW = fDW −1192π2fW logΛ2µ2, (40)frDB = fDB −1192π2fB logΛ2µ2, (41)frBW = fBW +α32πs2logΛ2µ2fB203+73c2+m2Hm2W!− fW4 +1c2−m2Hm2W!+ 12g2fWWW !, (42)frΦ,1 = fΦ,1 +3α8πc2logΛ2µ2fBm2Hv2+3m2Wv2(fB + fW )!. (43)Here, µ denotes the unit-of-mass of the dimensional regularization which has beenused to regulate the divergencies which appear in the calculation. The log Λ2µ2 terms11Figure 1: Constraints on a) ∆κγ vs. λ and b) ∆κZ vs. λ at 95% confidence level (CL).All coefficients of the dimension six operators in Eq. (11) are assumed to vanish except fora) fB = fW and fWWW and b) fB = −fW and fWWW . Correlations are shown for threerepresentative Higgs boson masses.in Eqs. (40) – (43) describe mixing of the operators between the new physics scaleΛ and the weak boson mass scale µ = mW .The results of Eqs. (40) – (43) nicely illustrate the problem of deriving lowenergy bounds for the TGV’s. The dominant contributions of the coefficients fB,fW and fWWW are merely renormalizations of those 4 operators which already contributeat tree level. Also, the precision electroweak data are barely sufficient toconstrain all four coefficients frDW , frDB, frBW , and frΦ,1. Hence, indirect bounds onthe TGV’s are only possible if one makes stringent assumptions on the size of these‘tree level’ coefficients. An analogous problem appears when considering 1-loopcontributions of the TGV’s to (g − 2)µ, Z → b¯b or b → sγ and hence data on thoseobservables do not provide model-independent bounds either.Nevertheless, one may proceed and assume that cancellations between treelevel and 1-loop contributions or between any of the coefficients of the dimension 6operators are unnatural. In practice one assumes that all fi vanish at the scale ofnew physics Λ except for the one or two whose effect one wants to analyse. Theresult of such an exercise is shown in Fig. 1. Clearly anomalous TGV’s of O(1)are still allowed by the data. Note that these bounds become more stringent asthe Higgs boson mass increases, pointing to more severe bounds in the nonlinearrealization scenario. If the top quark mass is varied between 150 GeV and 200 GeV,the range allowed for the anomalous couplings increases by up to 50%.12Other processes which at the 1-loop level are sensitive to anomalous gauge bosoncouplings also give constraints of O(1) at best. The current CLEO measurementof the inclusive b → sγ decay rate [22], for example, still allows −2.6 < ∆κγ < −1.2and −0.5 < ∆κγ < 0.4 (for λγ = 0), and −1.7 < λγ < 1.0 (for ∆κγ = 0) at 95% CL.A more stringent assumption on the coefficients of the dimension 6 operatorshas been proposed by De R´ujula et al. [10]. There are no obvious symmetrieswhich distinguish the tree level operators OBW , ODW , ODB, and OΦ,1 from theremaining ones. For a generic model of the underlying dynamics one may henceexpect e.g. |fB + fW | ≈ |fBW | which with fBW /Λ2 = (3.1± 2.6) TeV−2implies |∆κγ| =|fB +fW | m2W /2Λ2 < 0.03 at “95% CL”, a value too small to be observable in W+W−production at LEP II, but still in the interesting range for future linear colliders.Although this naturalness argument is compelling, it is clearly not a proof thatanomalous TGV’s are indeed small.• Nonlinear RealizationThere are several analyses in the literature which discuss the bounds onanomalous TGV’s in the context of the nonlinear realization scenario [16, 28, 29].All conclusions are quite similar. The limits obtained in the nonlinear realizationframework are very similar to those obtained in the linear realization scenario fora large Higgs boson mass. In the following, we will briefly review the results ofRef. [28].As in the linear realization case the effective Lagrangian is nonrenormalizable,and therefore the loop diagrams diverge. Conceptually, this is not a problem– the effective Lagrangian already contains an infinite number of terms, so one canjust add a counterterm to cancel the infinities found in any loop calculation. Inother words, if an anomalous TGV contributes at loop level to an observable, thedivergence of the calculation just renormalizes the coefficient of that observable. Atthe calculational level, however, one has to decide how to deal with the infinities. Inthe past it was common to simply use a cutoff Λ˜ to regulate the divergence. Withthis technique one often obtained a cutoff dependence of the form Λ˜ 2 or even Λ˜4,resulting in extremely stringent constraints on anomalous TGV’s. However, it wasargued in Ref. [6] that the use of cutoffs was incorrect, and often gave misleadingresults. Instead the authors of Ref. [6] advocate the use of dimensional regularization,along with the decoupling-subtraction renormalization scheme. This is theprocedure used in Ref. [28].For the calculational details, we refer the reader to Ref. [28]. Here we onlypresent the results of the global fit. First, consider the case in which only one of theanomalous TGV couplings, ∆gZ1, ∆κV and λV , is nonzero. (The coupling gZ5 wasnot considered in this paper.) The fit gave the following constraints at 1σ:∆gZ1 = −0.033 ± 0.031,∆κγ = 0.056 ± 0.056,∆κZ = −0.0019 ± 0.044, (44)λγ = −0.036 ± 0.034,13λZ = 0.049 ± 0.045.If taken at face value, these limits would imply that most anomalous TGV’s are toosmall to be seen at LEP II or in future Tevatron experiments (see Section 3). TheLHC and NLC, on the other hand, will be able to improve these bounds considerably.However, one should keep in mind that these bounds are rather artificial. Itis very hard to imagine that physics beyond the SM will produce only one anomalousTGV. In general, all such couplings will be produced. In a fit to all five anomalouscouplings simultaneously, the constraints virtually disappear, due to the possibilityof cancellations. At best, one can only conclude that the anomalous TGV couplingsare less than O(1) and even here one must assume that tree level contributions donot cancel the TGV effects.Even so, the bounds of Eq. (44) are interesting. These values represent thesensitivity of the global fit of electroweak data to specific anomalous couplings.Once all of the couplings are allowed to vary simultaneously, no significant boundremains. This obviously implies that, in that part of the allowed region for which theTGV couplings are large, cancellations occur among the contribution of the variousanomalous couplings to low-energy observables. Equation (44) gives an indicationof the level of cancellation required to account for the low-energy data in the eventthat an anomalous TGV at the 10% level is discovered.2.5 SummaryWe have discussed our theoretical understanding of, and the phenomenologicalconstraints from high precision experiments on, the anomalous TGV’s ofEqs. (1) and (2). The phenomenological effective Lagrangian describing anomalouscouplings appears not to respect the SU(2)L × U(1)Y gauge symmetry. However,it can be interpreted as the unitary-gauge expression of an effective Lagrangian inwhich the SU(2)L × U(1)Y symmetry is manifest. How this symmetry is realizeddepends on the particle content of the effective Lagrangian. Regardless of how onerealizes the SU(2)L × U(1)Y gauge symmetry, the anomalous TGV’s ∆gZ1, ∆κV , λVand gZ5 are expected to be at most O(m2W /Λ2), where Λ is the scale of new physics.(In the nonlinear-realization scenario, λV is O(m4W /Λ4)). ZγV couplings are at mostO(m4Z/Λ4). Thus, for Λ ∼ 1 TeV anomalous TGV’s of O(10−2) or less are expected.The discovery of larger anomalous TGV’s at present or future colliders would indicatethat the new physics responsible for them originates below the 1 TeV scale.It is therefore likely, though not certain, that the new physics will first be founddirectly, rather than through (indirect) contributions to anomalous TGV’s.There is indirect evidence from constraints on oblique correction parameters(2-point functions) that anomalous TGV’s are indeed <∼ O(10−2). The limitsobtained from these constraints, however, do depend strongly on other parameters,such as the Higgs boson and top quark mass (in the framework where theelectroweak symmetry is realized linearly; see Fig. 1). They also strongly dependon naturalness arguments which, though compelling, cannot be considered a proofthat large anomalous TGV’s do not exist. Strictly speaking, anomalous TGV’s are14unconstrained by the electroweak precision data since the possibility of large cancellationscannot be excluded. Thus, it is necessary for experiments to search directlyfor evidence of anomalous TGV’s, even though, in light of our current theoreticalunderstanding, such experiments will likely yield null results.3. Measuring WW V and ZγV Couplings in Collider Experiments3.1 General OverviewIn this Section we discuss possibilities to measure the WW V and ZγV couplingsdirectly in collider experiments. To simplify our discussion, we assume thatgγ1 = 1 and gγ5 = gZ5 = 0 in the following. As we have mentioned in the Introduction,electromagnetic gauge invariance requires gγ1 = 1 and gγ5 = 0 for on-shell photons.In contrast to the other couplings, gV5, V = γ, Z, violates charge conjugation andparity. Possibilities to measure gZ5in e+e− collisions are discussed in Ref. [17].At hadron colliders (Tevatron, LHC), di-boson production offers the bestopportunity to probe the WW V and ZγV vertices. The generic set of Feynmandiagrams contributing to di-boson production is shown in Fig. 2. Whereas W+W−production is sensitive to WW γ and WW Z couplings, only the WW γ (WW Z) vertexis tested in W±γ (W±Z) production. ZγV couplings are probed in pp, pp¯ → Zγ. Inorder to reduce the QCD background, one has to require that at least one of the Wand/or Z bosons decays leptonically. In pp, pp¯ → W+W−, tt¯ production representsan additional background. W±γ and W±Z production are of special interest due tothe presence of amplitude zeros [30, 31].Electroweak boson pair production at hadron colliders will be discussed indetail in Section 3.2. We present the general strategy in extracting informationon three vector boson couplings, summarize the current limits on WW V and ZγVcouplings from CDF and DØ, and investigate the prospects of measuring thesecouplings in future Tevatron and LHC experiments. We also discuss possibilities tosearch for the amplitude zeros in W±γ and W±Z production.q1q¯2V1V2q1q¯2V2V1q1q¯2V1V2VFigure 2: Generic Feynman diagrams contributing to di-boson production in hadroniccollisions. V, V1, V2 = W, γ, Z.At LEP, ZγV couplings can be tested in single photon production (e+e− →ννγ ¯ ) and radiative Z decays. Single photon production, in principle, is also sensitiveto the WW γ vertex [32]. WW Z couplings can be probed in the rare decay Z →W f ¯f′[33]. In both cases, however, the sensitivity is not sufficient to competewith the existing limits from CDF and DØ (see below). The constraints on ZγV15couplings from LEP experiments will be discussed in Section 3.3.1. At LEP II,e+e− → W+W− and e+e− → Zγ are the prime reactions to test WW V and ZγVcouplings (see Section 3.3.2). W pair production at a linear e+e− collider with acenter of mass energy of 500 GeV or more [“Next Linear Collider” (NLC)] willbe discussed in Section 3.3.3. Using laser backscattering [34], the NLC can alsobe operated as a eγ or γγ collider, with a center of mass energy of up to ∼ 80%of that available in the e+e− mode, and comparable luminosity. This opens thepossibility of testing the WW V couplings in processes such as eγ → W ν [35, 36],or γγ → W+W− [36, 37] in addition to e+e− → W+W−. ZγV couplings can beinvestigated in Zγ production and in γe → Ze [38]. The NLC could even be operatedas an e−e− collider. Possibilities to probe the three vector boson couplings in e−e−collisions have been explored in Ref. [39]. The limits on anomalous WW V couplingsexpected from reactions accessible in eγ, γγ and e−e− collisions are similar to thosefrom e+e− → W+W−. Alternative e+e− processes, such as e+e− → e+e−W+W−,W+W−V (V = γ, Z), or ννZ ¯ are significantly less sensitive to three gauge bosoncouplings than W pair production. For a summary of these modes see Ref. [40].The sensitivity bounds obtained from e+e− → W+W− are therefore representativefor the limits on anomalous gauge boson couplings which can be achieved at theNLC.The WW V couplings can, in principle, also be tested in single W and Zproduction at HERA [41]. However, in order to achieve limits which are comparableto the current CDF/DØ bounds (see Section 3.2.2), integrated luminosities of theorder 1 fb−1 are needed. Since it is not expected that those can be achieved withinthe next few years, anomalous gauge boson couplings at HERA will not be discussedin this report.3.2 Di-boson Production at Hadron Colliders3.2.1 Theoretical BackgroundFrom the phenomenological effective Lagrangian [see Eqs. (1) and (6)] it isstraightforward to derive cross section formulas for the di-boson production processes,qq¯ → W+W−, Zγ, (45)andqq¯′ → W±γ, W±Z. (46)For our subsequent discussion we find it convenient to briefly discuss the contributionsof anomalous couplings to the helicity amplitudes of the processes listedin (45) and (46). In qq¯′ → W γ, for example, the anomalous contributions ∆MλγλW ,(λγ and λW are the photon and W helicities, respectively) to the helicity amplitudesare given by [42]∆M±0 =e2sin θW√sˆ2mW(∆κγ + λγ)12(1 ∓ cos Θ) , (47)16∆M±± =e2sin θW12sˆm2Wλγ + ∆κγ!√12sin Θ , (48)where Θ denotes the scattering angle of the photon with respect to the quark direction,measured in the W γ rest frame, and √sˆ is the invariant mass of the W-photonsystem. Similar expressions can be derived for the anomalous contributions to theW Z, W+W− and Zγ helicity amplitudes.While the SM contribution to the di-boson amplitudes is bounded from abovefor fixed scattering angle Θ, the anomalous contributions rise without limit as sˆincreases, eventually violating unitarity. This is the reason the anomalous couplingsmust show a form factor behavior at very high energies (see Section 2.3). Anomalousvalues of λV , V = γ, Z, are enhanced by s/mˆ2W in the amplitudes M±± for alldi-boson production processes. Terms containing ∆κV mainly contribute to M±0in W V production and grow only with √s/mˆ W . In qq¯ → W+W−, on the otherhand, the ∆κV term mostly contributes to the (0,0) amplitude and is enhancedby a factor s/mˆ2W [3]. Non-standard values of ∆gZ1 mostly affect the (0,0) [(±, 0)and (0, ±)] amplitude in W Z [W+W−] production, and are proportional to s/mˆ2W[√s/mˆ W ] [3, 43]. The best limits on ∆κV (∆gZ1) are therefore expected from qq¯ →W+W− (qq¯′ → W Z). In Zγ production, terms proportional to hV3(hV4) grow like(√s/mˆ Z)3((√s/mˆ Z)5) [19].For large values of the di-boson invariant mass √sˆ, the non-standard contributionsto the helicity amplitudes would dominate, and would suffice to explaindifferential distributions of the photon and the W/Z decay products. Due to the factthat anomalous couplings only contribute via s-channel W, Z or photon exchange,their effects are concentrated in the region of small vector boson rapidities, and thetransverse momentum distribution of the vector boson should be particularly sensitiveto non-standard WW V and ZγV couplings. This is demonstrated in Fig. 3,where we show the photon pT distribution in pp¯ → W+γ → e+νeγ, and the Z bosontransverse momentum distribution in pp¯ → W+Z → ℓ+1ν1ℓ+2ℓ−2, ℓ1,2 = e, µ, at theTevatron for the SM and various anomalous WW V couplings. A dipole form factor(see Section 2.3) with scale ΛF F = 1 TeV has been assumed. Only one couplingis assumed to deviate from the SM at a time. To simulate detector response, thefollowing cuts have been imposed in Fig. 3:pT (γ) > 10 GeV, |η(γ)| < 1,pT (ℓ) > 20 GeV, |η(ℓ)| < 2.5, ℓ = e, µ, (49)/pT > 20 GeV, ∆R(ℓ, ℓ) > 0.4,mT (ℓγ;/pT) > 90 GeV, ∆R(γ, ℓ) > 0.7.Here, /pTdenotes the missing transverse momentum, η the pseudorapidity, ∆R =[(∆φ)2 + (∆η)2]1/2the separation in the pseudorapidity – azimuthal angle plane, andmT is the cluster transverse mass defined bym2T(ℓγ;/pT) = m(ℓγ)2 + |pT(ℓγ)|21/2+ /pT2− |pT(ℓγ) + p/T|2, (50)17Figure 3: The differential cross section for the transverse momentum of a) the photon inpp¯ → W+γ, and b) of the Z boson in pp¯ → W+Z at the Tevatron in the SM case (solidline) and for various anomalous WW V couplings. The cuts imposed are described in thetext.with m(ℓγ) being the ℓγ invariant mass. The large lepton photon separation andthe mT cut together strongly suppress photon radiation from the final state leptonline (radiative W decays) [42].Information on anomalous WW V and ZγV couplings can be obtained bycomparing the shape of the measured and predicted pT distribution, provided thatthe signal is not overwhelmed by background. If the background is much largerthan the SM prediction, limits on anomalous couplings can still be extracted if aphase space region can be selected where the effects of non-standard three vectorboson couplings dominate.Besides di-boson production, radiative W (Z) decays are also sensitive toWW γ (ZγV ) couplings. However, the parton center of mass energy in these processesis restricted to values around √sˆ = mW (mZ), and the expected limits onanomalous couplings are significantly worse than those obtained from W γ and Zγproduction where much larger values of √sˆ are accessible.3.2.2 Di-boson Production at the Tevatron: Current Results and Future ProspectsBoth, the CDF and DØ Collaboration have searched for W γ [44, 45], Zγ [46,47], W+W− [48, 49], and W Z [48] production in the data samples accumulated inrun 1a. CDF has also searched for W γ and Zγ events in the data of the 1988 – 89run [50]. For a recent summary of electroweak boson pair production results fromCDF and DØ see Ref. [51].18CDF (DØ) extract W γ/Zγ data samples from inclusive e/µ channel W/Zsamples by requiring an isolated photon in a fiducial region of their central (central+ endcap) electromagnetic (EM) calorimeters with ET (γ) ≥ 7 (10) GeV. A minimumlepton − photon angular separation of ∆R(ℓγ) > 0.7 suppresses final-state QEDbremsstrahlung. To reduce the QCD background from W/Z+jets production, excesscalorimeter transverse energy, ET , within a cone of ∆R = 0.4 centered on the photonwas required to be less than 15% (10%) of the photon ET . CDF also required thesum of the transverse momenta of all charged tracks within this cone to be lessthan 2 GeV/c, and also rejected events with a track pointing directly at the EMcluster. Both experiments required transverse/longitudinal EM shower developmentconsistent with a single photon. The selection criteria yield 25 (23) W γ and 8 (6)Zγ candidate events for CDF (DØ).The level of W/Z+jet background, where a jet “fakes” an isolated photon,in each of the W γ/Zγ data samples is determined by use of QCD jet data samplesto obtain a jet misidentification probability Pj→γ(ET ). For the photon selectioncriteria used by CDF, Pj→γ(ET = 9 GeV) ∼ 8 × 10−4, decreasing exponentially toPj→γ(ET = 25 GeV) ∼ 10−4, whereas for the photon selection criteria used by DØ,Pj→γ(ET ) ∼ 4 × 10−4(6 × 10−4) in the central (endcap) calorimeter, and varies onlyslowly with ET . The jet fragmentation probability distribution was then convolutedwith the jet ET spectrum in each of the inclusive W/Z data samples. The Zγbackground in the W γ data arising from non-observation of one of the Z decayleptons is estimated from Monte Carlo simulations. The contributions to W γ and Zγproduction from W/Z decays into τ leptons are also estimated from MC simulationsand found to be small.The SM cross sections for W+W−, W±Z and ZZ production‡ at the Tevatron,including NLO QCD corrections [52], are 9.5 pb, 2.5 pb and 1.4 pb, respectively.Decay modes where one of the weak bosons decays hadronically have significantlylarger branching ratios than all leptonic decays:Br(WW → eνeeνe, µνµµνµ) = 2.4%, Br(WW → eνeµνµ) = 2.4%, (51)Br(W Z → ℓ1ν1ℓ+2ℓ−2) = 1.5%, ℓ1,2 = e, µ, (52)Br(WW → ℓνjj) = 29%, ℓ = e, µ, (53)Br(W Z → ℓνjj) = 15%, Br(W Z → jjℓ+ℓ−) = 4.5%, (54)BR(ZZ → ℓ+ℓ−jj) = 9.4%, Br(ZZ → ℓ+1ℓ−1ℓ+2ℓ−2) = 0.4%. (55)Due to the larger cross section and branching ratio, the ℓνjj final state is completelydominated by W+W− production. W±Z and ZZ production contributeapproximately equally to the ℓ+ℓ−jj final state. All semihadronic channels sufferfrom a large W/Z+ jets background. tt¯ production contributes non-negligibly to thebackground for W+W− production. In contrast, the ℓ1ν1ℓ+2ℓ−2 final state is relativelybackground free.‡ZZ production is, in principle, sensitive to ZZV , V = γ, Z couplings, which vanish in the SM at treelevel [3]. We will not discuss the ZZV couplings accessible in ZZ production in this report.19W+W− and W Z data samples are also extracted from inclusive e/µ W/Z data.CDF has analyzed the WW, W Z → ℓνjj and ZW → ℓ+ℓ−jj (ℓ = e or µ) channels usingstandard W/Z lepton selection cuts, and requiring 60 GeV/c2 < m(jj) < 110 GeV/c2.For leptonic W (Z) events, CDF eliminates W/Z+jets background events by requiringpT (jj) > 130 (100) GeV/c, which also eliminates the SM signal but retainsgood sensitivity for non-zero WW V anomalous couplings. One event passes thecuts in the ℓνjj channel. In the ℓ+ℓ−jj channel no events survive. A clean candidateevent for pp¯ → W+Z → e+νee+e− has also been observed in the CDF dataset [48]. DØ has analyzed the WW → ℓ1ν1ℓ2ν2, ℓ1,2 = e, µ, channels using standardlepton cuts for selection of W pairs. The Z mass region in the ee channel,77 GeV/c2 < m(ee) < 105 GeV/c2, is excluded. To suppress the Z → µ+µ− background,a cut of E/ηT > 30 GeV is imposed, where E/ηTis the projection of the missingET vector onto the bisector of the decay angles of the two muons. To reduce thett¯ background, the total hadronic transverse energy in the event is required to beless than 40 GeV. Backgrounds from Z decay and fake electrons are estimated fromdata and MC simulations. One eeνν and one eµνν event pass all cuts.SM and anomalous coupling predictions for the W γ and Zγ processes areobtained using the event generators of Ref. [42] and [19], and detailed detectorsimulations. MRSD−′structure functions [53] are used for event generation as theybest match the recent W lepton asymmetry measurements from CDF [54]. SM andanomalous coupling predictions for W+W− and W Z production are obtained usingthe event generator of Ref. [55] and MC detector simulations. Presently, a completecalculation of the di-boson transverse momentum distribution, including soft gluonresummation effects, does not exist, except for the ZZ case [56]. Higher order QCDcorrections are therefore approximated in the experimental analysis by a k-factorand by smearing the transverse momentum of the di-boson system according to theexperimentally determined W/Z boson pT spectrum.Direct experimental limits on WW γ and ZγV anomalous couplings for theW γ/Zγ processes are obtained via binned maximum likelihood fits to the ET (γ)distribution. The observed ET (γ) distribution is compared to the sum of expectedsignal plus background(s) prediction, calculating the Poisson likelihood that thissum would fluctuate to the observed number of events in each ET bin, and convolutingwith a Gaussian distribution to take into account systematic uncertaintiesassociated with backgrounds, luminosity normalization, structure function choice,Q2-scale and uncertainties in the shape of the pT (W γ/Zγ) distribution, efficiencies,etc. The 95% CL CDF [44] and DØ [45] limits on anomalous WW γ couplings fromW γ production are shown in Fig. 4a. The bounds on ∆κ0γ and λ0γ extracted byDØ (solid curve) are about 20% better than those obtained by CDF (short dashedcurve). For comparison, we have also included the limits obtained by UA2 [57],and CDF from the 1988-89 data [50]. Due to the smaller center of mass energy(√s = 630 GeV), the correlations between the two couplings at the CERN pp¯ colliderare much more pronounced than at Tevatron energies. The bounds obtainedfrom the 1992-93 data have been obtained using a dipole form factor with scaleΛF F = 1.5 TeV. The CDF limits from the 1988-89 data are for ΛF F = 1 TeV.20Figure 4: Present limits on anomalous WW V couplings from hadron collider experiments.CDF extracts direct experimental limits on WW γ and WW Z anomalous couplingsfrom the ℓνjj and ℓ+ℓ−jj final states via comparison of observed events tothe expected signal within cuts, including systematic uncertainties due to luminositynormalization, jet energy scale and resolution, structure function choice andhigher order QCD corrections, etc. DØ extracts direct experimental limits on WW Vanomalous couplings from the WW → ℓ1ν1ℓ2ν2 mode via comparison of their 95%CL upper limit of σ(WW)expt < 91 pb with σ(WW)pred as a function of anomalouscouplings.The limits obtained from W+W− → ℓ1ν1ℓ2ν2 and WW, W Z → ℓνjj are summarizedand compared to those obtained from W γ production in Fig. 4b. In extractinglimits on non-standard WW V couplings from W pair production, CDF (DØ) assumeda dipole form factor with scale ΛF F = 1.5 TeV (0.9 TeV), ∆κ0γ = ∆κ0Z, λ0γ = λ0Z,and ∆gZ1 = 0. Due to the selection of a phase space region which is particularly sensitiveto WW V couplings and the larger branching ratios for WW, W Z → ℓνjj,the bounds obtained from the semihadronic WW and W Z final states are signifi-cantly stronger than those found from analyzing the WW → ℓ1ν1ℓ2ν2 channel. InSection 3.2.1 we have mentioned that the contributions to the W+W− helicity amplitudesproportional to ∆κV grow like s/mˆ2W whereas the ∆κγ terms in the W γamplitudes are proportional to √s/mˆ W . In contrast, the λV terms always grow likes/mˆ2W . This explains why the limit on ∆κ0Vobtained from the semihadronic WWand W Z final states is significantly better than that found from pp¯ → W γ while thebounds on λ0Vfrom WW, W Z → ℓνjj and W γ production are almost identical.Limits on ∆κ0γ and λ0γextracted from W γ production have the advantage ofbeing independent of assumptions about the WW Z vertex. Similarly, informationon the WW Z couplings, independent from assumptions on the WW γ couplings, canbe obtained from W Z production. From the W Z → jjℓ+ℓ− channel, CDF finds [48]21Figure 5: Comparison of current experimental bounds on WW V couplings and limits obtainedfrom S-matrix unitarity for a dipole form factor.−8.6 < ∆κ0Z < 9.0 for ∆gZ01 = λ0Z = 0 and −1.7 < λ0Z < 1.7 for ∆gZ01 = ∆κ0Z = 0. Theselimits were obtained for a form factor scale ΛF F = 1.5 TeV. The ZZ → jjℓ+ℓ− crosssection was assumed to be given by the SM prediction.In Section 2.3 we have seen that constraints from S-matrix unitarity severelyrestrict the values of the low energy anomalous couplings allowed. For sufficientlysmall values of the form factor scale, the experimental limits on non-standard threevector boson couplings are substantially better than those found from S-matrixunitarity [see Eqs. (22) – (26)]. However, for ΛF F ≫ mW , the unitarity boundsdecrease like 1/ΛnF F , with n = 1, 2 for the WW V couplings, and n = 3, 5 for the ZγVcouplings whereas the experimental limits depend less sensitively on ΛF F [50]. Thisimplies that for sufficiently large form factor scales unitarity bounds eventually willbe stronger than the limits extracted from experimental data. In Fig. 5a we comparethe current experimental limits on WW γ couplings from W γ production with thebounds derived from unitarity for ΛF F = 1.5 TeV. In Fig. 5b a similar comparisonis carried out for WW/W Z → ℓνjj with ΛF F = 1.5 TeV, and WW → ℓ1ν1ℓ2ν2 withΛF F = 0.9 TeV. These values of ΛF F were chosen just large enough that the unitaritybounds would approach the experimental limits. One concludes that the maximumscale which can be probed with the current experimental data on W γ, WW andW Z production is of order 1.5 – 2 TeV.The current CDF [46] and DØ [47] 95% CL limit contours for anomalousZZγ couplings are shown in Fig. 6, together with the constraints from S-matrixunitarity. The limit contours for Zγγ couplings are similar. For completeness, wehave also included the CDF result from the 1988-89 run [50]. The DØ limits onhV30 and hV40 are about 30% more stringent than those obtained by CDF. In order toderive these limits, generalized dipole form factors with ΛF F = 0.5 TeV, and powers22Figure 6: Present limits on anomalous ZZγ couplings from hadron collider experiments,and constraints from S-matrix unitarity.n = 3 (n = 4) for hV3(hV4), are assumed (see Section 2.3). Since the anomalouscontributions to the Zγ helicity amplitudes grow faster with energy than those inW γ production, the experimental limits on hV30 and hV40 depend rather sensitively onthe form factor scale chosen. The maximum form factor scale which can be probedin Zγ production with present experimental data is ΛF F ≈ 500 GeV.Table 1 summarizes the current results on anomalous WW V and ZγV couplingsfrom hadron colliders. With the limited statistics of di-boson events currentlyavailable, deviations from the SM cross section have to be large at least in someregions of phase space in order to lead to an observable effect. The best direct limitson ∆κ0Vare currently obtained from the ℓνjj final state. W γ production results insomewhat better bounds on λ0γthan pp¯ → WW, W Z → ℓνjj. So far, no attempt hasbeen made to combine the limits of CDF and DØ and/or from different channels.During the current data taking period (run 1b) at the Tevatron, one hopesto collect an integrated luminosity of about 100 pb−1 per experiment. For theMain Injector Era, integrated luminosities of the order of 1 fb−1 are envisioned [58].The first run with the Main Injector is currently planned for the period of 1998 –2003. Through further upgrades of the Tevatron accelerator complex, an additionalfactor 10 in luminosity may be gained (TeV*). The substantial increase in integratedluminosity will make it possible to test the WW V and ZγV vertices with muchgreater precision than in current experiments. In Fig. 7 we show the 95% CLlimits on anomalous WW γ and ZZγ couplings expected for CDF from W γ and Zγproduction at the Tevatron (√s = 2 TeV) for 1 fb−1 and 10 fb−1. Here, and in allsubsequent sensitivity plots, we assume that no deviation from the SM prediction isobserved in future experiments. To derive bounds on non-standard WW V couplingsa dipole form factor is assumed. For the ZγV couplings we use form factor powers23Table 1: 95% CL limits on anomalous WW V , V = γ, Z, and ZZγ couplings from CDFand DØ. Only one of the independent couplings is allowed to deviate from the SM at atime. The bounds obtained for Zγγ couplings are very similar to those derived for the ZZγcouplings and are therefore not shown.experiment channel limitCDF pp¯ → W±γ → ℓ±νγ −2.3 < ∆κ0γ < 2.2ℓ = e, µ −0.7 < λ0γ < 0.7DØ pp¯ → W±γ → ℓ±νγ −1.6 < ∆κ0γ < 1.8ℓ = e, µ −0.6 < λ0γ < 0.6CDF pp¯ → W±Z → ℓ+ℓ−jj −8.6 < ∆κ0Z < 9.0ℓ = e, µ −1.7 < λ0Z < 1.7CDF pp¯ → W+W−, W±Z → ℓ±νjj −1.0 < ∆κ0V < 1.1ℓ = e, µ, κγ = κZ, λγ = λZ −0.8 < λ0V < 0.8DØ pp¯ → W+W− → ℓ1ν1ℓ2ν2 −2.6 < ∆κ0V < 2.8ℓ1,2 = e, µ, κγ = κZ, λγ = λZ −2.2 < λ0V < 2.2CDF pp¯ → Zγ → ℓ+ℓ−γ −3.0 < hZ30 < 2.9ℓ = e, µ, ΛF F = 0.5 TeV −0.7 < hZ40 < 0.7DØ pp¯ → Zγ → ℓ+ℓ−γ −1.9 < hZ30 < 1.8ℓ = e, µ, ΛF F = 0.5 TeV −0.5 < hZ40 < 0.5of n = 3 (hV3) and n = 4 (hV4). The curves shown in Fig. 7 are obtained from abinned likelihood fit of the photon ET distribution. In the Zγ case we also show theconstraint from unitarity for ΛF F = 1.5 TeV. The expected experimental limits arecalculated for the same value of ΛF F . The limits on Zγγ couplings are very similarto those found for ZZγ couplings and are therefore not shown. Only W → eνe andZ → e+e− decays are taken into account in our analysis. Electrons are required tohave |η| < 3.6, with at least one electron in the central region of the detector (|η| <1.0). A pseudorapidity cut of |η(γ)| < 2.4 is imposed on photons. The acceptancesare calculated using the following transverse energy and separation cuts:ET (e) > 25 GeV, E/T > 25 GeV, (56)ET (γ) > 10 GeV, ∆R(e, γ) > 0.7. (57)In addition, a cut on the transverse W mass of mWT > 50 GeV and a cluster transversemass cut of mT (eγ; E/T) > 90 GeV were imposed in the W γ case. For Zγ production,we require m(e+e−γ) > 100 GeV and m(e+e−) > 70 GeV. The efficiencies for electron24Figure 7: Projected 95% CL sensitivity limits for a) WW γ couplings from W γ productionand b) ZZγ couplings from Zγ production at the Tevatron for integrated luminosities of1 fb−1 and 10 fb−1.and photon identification were taken from the current CDF analysis, as well asthe probability for a jet to fake a photon, Pj→γ(ET ). The systematic uncertaintyfrom the integrated luminosity, parton densities, and higher order QCD correctionswas assumed to be 5%. From Fig. 7a (7b) one observes that the current limits onanomalous gauge boson couplings can be improved by about a factor 5 – 15 (10 –100) in W γ (Zγ) production in the Main Injector Era. An additional factor 10 inintegrated luminosity leads to roughly a factor 2 improvement in the sensitivitieswhich can be achieved. The maximum form factor scale which can be probed in Zγproduction with 1 fb−1(10 fb−1) is about a factor 2.6 (3) larger than that accessiblewith the current data. The limit contours shown in Fig. 7 can be improved by about20 – 40% if W → µν and Z → µ+µ− decays are included in the analysis.The bounds on ZγV couplings could be further improved by analyzing thereaction pp¯ → Zγ → ννγ ¯ . Here the signal consists of a single high pT photonaccompanied by a large amount of missing transverse energy. Compared to thecharged lepton decay mode of the Z boson, the decay Z → νν¯ offers potentialadvantages. Due to the larger Z → νν¯ branching ratio, the differential cross sectionis about a factor 3 larger than that for qq¯ → e+e−γ and qq¯ → µ+µ−γ combined.Furthermore, final state bremsstrahlung and timelike virtual photon diagrams donot contribute to the ννγ ¯ final state. On the other hand, there are several potentiallyserious background processes which contribute to pp¯ → γp/T, but not to the ℓ+ℓ−γfinal state. The two most important background processes are prompt photonproduction, pp¯ → γj, with the jet rapidity outside the range covered by the detectorand thus “faking” missing transverse momentum, and two jet production whereone of the jets is misidentified as a photon while the other disappears through the25beam hole. A parton level simulation of the γj and jj backgrounds in pp¯ → γp/Tsuggests [19] that those backgrounds can be eliminated by requiring a sufficientlylarge transverse momentum for the photon.To estimate the sensitivity of W+W−, W±Z → ℓνjj and W Z → ℓ+ℓ−jj, ℓ =e, µ, to non-standard WW V couplings in future Tevatron experiments, we requirecharged leptons to have ET > 20 GeV and |η(ℓ)| < 2, and impose a missing transverseenergy cut of 20 GeV. The two leading jets are required to have ET (j) > 30 GeVand 60 GeV < m(jj) < 110 GeV. Events containing an extra jet with ET > 50 GeVare vetoed in order to suppress the top quark background and to reduce the effectof QCD corrections [52, 59]. To suppress the W/Z+ jets background, a cut onthe transverse momentum of the jet pair is imposed, similar to the requirement inthe current CDF analysis. The value of the pT (jj) cut varies with the integratedluminosity assumed:pT (jj) > 150 GeV for ZLdt = 100 pb−1, (58)pT (jj) > 200 GeV for ZLdt = 1 fb−1, (59)pT (jj) > 250 GeV for ZLdt = 10 fb−1. (60)The number of signal events expected is calculated using the event generator ofRef. [55]. The trigger and particle identification efficiencies are assumed to be thesame as in the current CDF data analysis. To estimate the tt¯ and W/Z+ jetsbackground, ISAJET and VECBOS [60] are used. The top quark mass is taken tobe mt = 170 GeV.Confidence levels are obtained by counting events above the pT (jj) cut. Theresulting 95% CL contours at √s = 1.8 TeV for integrated luminosities of 100 pb−1,1 fb−1 and 10 fb−1 are shown in Fig. 8a. To calculate the sensitivity limits in Fig. 8a,we have assumed a form factor scale of ΛF F = 2 TeV and the effective Lagrangianscenario of Section 2.2 where the SU(2)L × U(1)Y symmetry is linearly realized withfB = fW (“HISZ scenario” [8]), which reduces the number of independent WW Vcouplings from five to two. Choosing ∆κγ and λγ as independent parameters, theWW Z couplings are then given by [see Eqs. (12) – (14)]:∆gZ1 =12 cos2 θW∆κγ, (61)∆κZ =12(1 − tan2θW ) ∆κγ, (62)λZ = λγ. (63)The sensitivity limits depend only marginally on the value of ΛF F assumed. Thebounds obtained in this scenario are compared in Table 2 with those derived fordifferent relations between the WW V couplings. The sensitivity limits found in theHISZ scenario are seen to be representative. If the Tevatron center of mass energy26Figure 8: Expected 95% CL sensitivity limits for the WW V couplings in the HISZ scenario[see Eqs. (61) – (63)] a) from pp¯ → WW, W Z → ℓνjj and ℓ+ℓ−jj, and b) from pp¯ →W±Z → ℓ±1ν1ℓ+2ℓ−2at the Tevatron.Table 2: 95% CL limits on anomalous WW V , V = γ, Z from pp¯ → WW, W Z → ℓνjj andℓ+ℓ−jj at √s = 1.8 TeV for RLdt = 1 fb−1 and RLdt = 10 fb−1. Only one coupling at atime is varied, except for the dependencies noted.dependent couplings limit limitRLdt = 1 fb−1RLdt = 10 fb−1Eqs. (61) and (62) −0.31 < ∆κ0γ < 0.41 −0.17 < ∆κ0γ < 0.24λγ = λZ −0.19 < λ0γ < 0.19 −0.10 < λ0γ < 0.11∆κγ = ∆κZ −0.23 < ∆κ0V < 0.29 −0.12 < ∆κ0V < 0.16– −0.35 < ∆gZ01 < 0.43 −0.19 < ∆gZ01 < 0.25– −0.30 < ∆κ0Z < 0.37 −0.16 < ∆κ0Z < 0.20– −0.22 < λ0Z < 0.22 −0.11 < λ0Z < 0.11– −0.56 < λ0γ < 0.56 −0.28 < λ0γ < 0.2927can be increased to 2 TeV the results shown in Fig. 8a and Table 2 improve by afew per cent.For integrated luminosities ≥ 1 fb−1, WW and W Z production with all leptonicdecays can also be used to constrain the WW V vertices. In contrast to thesemihadronic WW, W Z production channels, double leptonic W Z decays are relativelybackground free and thus provide an excellent testing ground for non-standardWW Z couplings. Using a recent calculation of W±Z production which includes NLOQCD corrections [59], sensitivity limits for the pp¯ → W±Z → ℓ±1ν1ℓ+2ℓ−2, ℓ1,2 = e, µ,channel were estimated. No full detector simulation was carried out, however, leptonidentification cuts of pT (ℓ1,2) > 20 GeV and |η(ℓ1,2)| < 2.5, and a missing pTcut of 20 GeV have been imposed to roughly simulate detector response. Particlemomenta are smeared according to the resolution of the CDF detector. The 95%CL limit contours for √s = 1.8 TeV and ΛF F = 1 TeV, obtained from a χ2 fit tothe pT (Z) distribution are displayed in Fig. 8b. Here we have again assumed therelations of Eqs. (61) – (63) for WW γ and WW Z couplings. If the center of massenergy of the Tevatron can be increased to 2 TeV, slightly better limits can be obtained.For RLdt = 1 fb−1, the small number of ℓ±1ν1ℓ+2ℓ−2events severely limits thesensitivity, and the limits obtained from WW, W Z → ℓνjj and ℓ+ℓ−jj are signifi-cantly better than those from double leptonic W Z decays for most of the parameterspace. For 10 fb−1, the non-negligible background starts to negatively influencethe semihadronic channels, and double leptonic and WW, W Z → ℓνjj and ℓ+ℓ−jjfinal states yield comparable results. In contrast to double leptonic W Z decays, theWW → ℓ1ν1ℓ2ν2 final states are plagued by background from tt¯production, and thuswere not studied in detail. The contour limits shown in Figs. 7a and 8 depend onlymarginally on the form factor scale assumed; only the limits on the ZγV couplingsare more sensitive to the value of ΛF F chosen.The expected sensitivity bounds from future Tevatron experiments, varyingonly one of the independent couplings at a time, are summarized in Table 3. Futureexperiments at the Tevatron can measure ∆κγ with a precision of about 0.1 – 0.2.λγ can be determined to better than about 0.1 for RLdt ≥ 1 fb−1. The limits forZγV couplings are of order 10−2 − 10−3.3.2.3 Di-boson Production at the LHCSince terms proportional to the non-standard WW V and ZγV couplings inthe di-boson production amplitudes grow with energy like a power of √s/mˆ W , oneexpects [61] that experiments at the LHC (pp collisions at √s = 14 TeV; L =1.7 · 1034 cm−2s−1[62]) will be able to improve significantly the limits which can beobtained at the Tevatron. To simulate the sensitivity of W γ and Zγ production atthe LHC to non-standard three vector boson couplings, we use the photon, electronand E/Tresolutions of the current ATLAS detector design [63]. Similar results areobtained if CMS [64] specifications are employed. Only W → eνe and Z → e+e−decays are studied. Acceptances are obtained using the following set of cuts:ET (e) > 40 GeV, E/T > 25 GeV, (64)28Table 3: Expected 95% CL limits on anomalous WW V , V = γ, Z, and ZZγ couplingsfrom future Tevatron experiments. Only one of the independent couplings is assumed todeviate from the SM at a time. The limits found for Zγγ couplings are very similar to thoseobtained for hZ3and hZ4.channel limit limitRLdt = 1 fb−1RLdt = 10 fb−1pp¯ → W±γ → e±νγ −0.38 < ∆κ0γ < 0.38 −0.21 < ∆κ0γ < 0.21√s = 2 TeV −0.12 < λ0γ < 0.12 −0.057 < λ0γ < 0.057pp¯ → W+W−, W±Z → ℓ±νjj, ℓ+ℓ−jj −0.31 < ∆κ0γ < 0.41 −0.17 < ∆κ0γ < 0.24ℓ = e, µ, HISZ scenario −0.19 < λ0γ < 0.19 −0.10 < λ0γ < 0.11pp¯ → W±Z → ℓ±1ν1ℓ+2ℓ−2 −0.26 < ∆κ0γ < 0.70 −0.09 < ∆κ0γ < 0.32ℓ1,2 = e, µ, HISZ scenario −0.24 < λ0γ < 0.32 −0.10 < λ0γ < 0.13pp¯ → Zγ → e+e−γ −0.105 < hZ30 < 0.105 −0.044 < hZ30 < 0.044√s = 2 TeV, ΛF F = 1.5 TeV −0.0064 < hZ40 < 0.0064 −0.0025 < hZ40 < 0.0025ET (γ) > 25 GeV, ∆R(e, γ) > 0.7, (65)mWT > 50 GeV, mT (eγ;/eT) > 90 GeV, (66)andm(e+e−) > 70 GeV, m(e+e−γ) > 110 GeV. (67)Since most of the sensitivity to anomalous couplings originates from the high ETtail, the limits which can be obtained change only very little if the ET (γ) (E/T) cutis raised to 50 – 100 GeV (40 – 50 GeV). For the electron and photon identificationefficiencies, the values obtained in the current CDF analysis were used. Thesystematic uncertainty from the integrated luminosity, parton densities, and higherorder QCD corrections was varied between 5% and 10%. NLO QCD correctionsare known to be large at LHC energies, and significantly reduce the sensitivity toanomalous couplings, unless a jet veto is imposed [65]. All jets in W γ and Zγ eventswith a transverse energy larger than 50 GeV were therefore vetoed. This cut alsohelps to reduce to an acceptable level the background from pp → ttγ ¯ → W γ +X and,together with the photon and lepton isolation cuts, the pp → ¯bbγ background [66, 67].The 95% CL limit contours from a binned likelihood fit of the photon ETdistribution for an integrated luminosity of 100 fb−1 are shown in Fig. 9. To obtainthe results shown in this figure, we have assumed a systematic uncertainty of 5%.Almost identical curves are obtained if the systematic uncertainty is increased to10%. In contrast to the sensitivities obtained at Tevatron energies, the limits on29Figure 9: 95% CL sensitivity limits for a) WW γ couplings from W γ production and b)ZZγ couplings from Zγ production at the LHC. Results are displayed for an integratedluminosity of 100 fb−1 and two different form factor scales.WW γ couplings found for pp collisions at √s = 14 TeV depend non-negligibly on theform factor scale. The bounds on ∆κ0γ(λ0γ) are about a factor 3 to 4 (∼ 10) betterthan those possible at the Tevatron with 10 fb−1. The limits on ZγV couplings canbe improved by a factor ∼ 10 (hV30) to ∼ 30 (hV40) for ΛF F = 1.5 TeV. The 95% CLlimit contours for the Zγγ couplings are almost identical to those found for hZ30 andhZ40 and are therefore not shown in Fig. 9b. The limits obtained for ZγV couplingsdepend very strongly on the value of ΛF F assumed. Increasing the form factor scalefrom 1.5 TeV to 3 TeV, the limits improve by a factor 5 to 10. The results shownin Fig. 9 can be improved by about 20 – 40% if W → µν and Z → µ+µ− decays areincluded in the analysis. The limits on anomalous ZγV couplings could be furtherstrengthened if the Z → νν¯ decay can be utilized.Using the NLO calculation of Ref. [59], sensitivity limits for the reaction pp →W±Z → ℓ±1ν1ℓ+2ℓ−2 were estimated by performing a χ2 fit to the pT (Z) distribution.No complete detector simulation was carried out, however, a transverse momentumcut of 25 GeV and a rapidity cut of |η(ℓ1,2)| < 2.5, ℓ1,2 = e, µ, were imposed onthe charged leptons, together with a missing transverse energy cut of 50 GeV. Therelatively large E/Tcut was chosen to reduce backgrounds e.g from event pileupwhich at LHC luminosities may result in a non-negligible amount of “fake” missingtransverse energy [68], and from processes such as pp → Zb¯b → ℓ1ν1ℓ+2ℓ−2 + X. Thelarge E/Tcut has only very little impact on the sensitivity limits which can beachieved. In addition, leptons of the same charge are required to be separated by∆R > 0.4. To reduce the effect of QCD corrections, and the pp → tt¯→ ℓ1ℓ2ℓ2 +X [63]and pp → ttZ¯ [66] backgrounds on the sensitivity limits, jets with pT (j) > 50 GeV and|η(j)| < 2.5 are vetoed. Particle momenta are smeared according to the resolution30Figure 10: 95% CL sensitivity limits from W±Z → ℓ±1ν1ℓ+2ℓ−2at the LHC a) in the HISZscenario and b) if only ∆κZ and λZ are allowed to deviate from the SM.expected for the ATLAS detector [63]. A 50% normalization uncertainty of the SMpT (Z) distribution was taken into account in the derivation of the 95% CL limitcontours, which are shown in Fig. 10 for RLdt = 100 fb−1 and two choices for theform factor scale. In Fig. 10a we show 95% CL limits for the HISZ scenario [seeEqs. (61) – (63)]. Figure 10b displays sensitivity bounds for the case where only∆κZ and λZ are varied.At the LHC, the tt¯ production rate for top quark masses in the range from150 GeV to 200 GeV is about a factor 10 to 30 larger than the pp → W+W−cross section [69]. Unless the top quark background can be reduced very efficiently,one does not expect that W+W− and semihadronic W Z production yield limits onanomalous WW V couplings which can compete with those obtained from pp → W γand double leptonic W Z decays.Table 4 compares the sensitivities which can be achieved in W γ, W Z and Zγproduction at the LHC with 100 fb−1. If the integrated luminosity is reduced by afactor 10, the bounds listed in Table 4 are weakened by about a factor 2. ∆κV andλV in general can be probed to better than 0.1 and 0.01 at the LHC, respectively.The limits which are obtained in the HISZ scenario for ∆κγ from W±Z → ℓ±1ν1ℓ+2ℓ−2are O(10−2) and thus much stronger than those from W γ production. This is dueto the relation between ∆gZ1 and ∆κγ in the HISZ scenario [see Eq. (61)], and thefact that the terms proportional to ∆gZ1in W Z production grow like s/mˆ2W withenergy, whereas the terms proportional to ∆κγ in W γ production only increase like√s/mˆ W at most. Varying the form factor scale from 3 TeV to 10 TeV, the limits onWW V couplings improve by about 30%. For ΛF F < 3 TeV, the bounds deterioraterather quickly; for ΛF F = 1 TeV they are a factor 2 – 5 weaker than those foundfor ΛF F = 3 TeV. As mentioned before, the sensitivities obtained for ZγV couplings31Table 4: Expected 95% CL limits on anomalous WW V , V = γ, Z, and ZZγ couplings fromexperiments at the LHC (pp collisions at √s = 14 TeV; RLdt = 100 fb−1). Only one of theindependent couplings is assumed to deviate from the SM at a time. The limits obtainedfor Zγγ couplings almost coincide with those found for hZ3and hZ4.channel limit limitΛF F = 3 TeV ΛF F = 10 TeVpp → W±γ → e±νγ −0.080 < ∆κ0γ < 0.080 −0.065 < ∆κ0γ < 0.065−0.0057 < λ0γ < 0.0057 −0.0032 < λ0γ < 0.0032pp → W±Z → ℓ±1ν1ℓ+2ℓ−2 −0.0060 < ∆κ0γ < 0.0097 −0.0043 < ∆κ0γ < 0.0086ℓ1,2 = e, µ, HISZ scenario −0.0053 < λ0γ < 0.0067 −0.0043 < λ0γ < 0.0038pp → W±Z → ℓ±1ν1ℓ+2ℓ−2 −0.064 < ∆κ0Z < 0.107 −0.050 < ∆κ0Z < 0.078ℓ1,2 = e, µ, ∆gZ1 = 0 −0.0076 < λ0Z < 0.0075 −0.0043 < λ0Z < 0.0038channel limit limitΛF F = 1.5 TeV ΛF F = 3 TeVpp → Zγ → e+e−γ −0.0051 < hZ30 < 0.0051 −0.0013 < hZ30 < 0.0013−9.2 · 10−5 < hZ40 < 9.2 · 10−5 −6.8 · 10−6 < hZ40 < 6.8 · 10−632depend even more strongly on the form factor scale. A maximum scale of ∼ 10 TeVcan be probed in W γ and W Z production, whereas scales up to 6 TeV are accessiblein Zγ production at the LHC. The limits from W γ and W Z production listed inTable 4 are consistent with those found in Ref. [63].3.2.4 Amplitude Zeros and Rapidity Correlations in W γ and W Z ProductionW γ and W Z production in hadronic collisions are of special interest due tothe presence of amplitude zeros. It is well known that all SM helicity amplitudes ofthe parton-level subprocess q1q¯2 → W±γ vanish for [30]cos θ =Q1 + Q2Q1 − Q2, (68)where θ is the scattering angle of the W-boson with respect to the quark (q1) directionin the W γ rest frame, and Qi (i = 1, 2) are the quark charges in units ofthe proton electric charge e. This zero is a consequence of the factorizability [70] ofthe amplitudes in gauge theories into one factor which contains the gauge couplingdependence and another which contains spin information. Although the factorizationholds for any four-particle Born-level amplitude in which one or more of thefour particles is a gauge-field quantum, the amplitudes for most processes may notnecessarily develop a kinematical zero in the physical region. The amplitude zeroin the W±γ process has been further shown to correspond to the absence of dipoleradiation by colliding particles with the same charge-to-mass ratio [71], a realizationof classical radiation interference.Recently, it was found [31] that the SM amplitude of the process q1q¯2 → W±Zalso exhibits an approximate zero at high energies. The (±, ∓) amplitudes M(±, ∓)vanish forgq1−uˆ+gq2−tˆ= 0, (69)where gqi− is the coupling of the Z boson to left-handed quarks, and uˆ and tˆ areMandelstam variables in the parton center of mass frame. For sˆ ≫ m2Z, the zero inthe (±, ∓) amplitudes is located at cos θ0 = (gq1− + gq2− )/(gq1− − gq2− ), orcos θ0 ≃(+13tan2θw ≃ +0.1 for du¯ → W−Z ,−13tan2θw ≃ −0.1 for u ¯d → W+Z .The existence of the zero in M(±, ∓) at cos θ0 is a direct consequence of the contributingFeynman diagrams and the left-handed coupling of the W-boson to fermions.At high energies, strong cancellations occur, and, besides M(±, ∓), only the(0, 0) amplitude remains non-zero. The combined effect of the zero in M(±, ∓) andthe gauge cancellations at high energies in the remaining helicity amplitudes resultsin an approximate zero for the q1q¯2 → W±Z differential cross section at cos θ ≈ cos θ0.Non-standard WW V couplings in general destroy the amplitude zeros in W γand W Z production. Searching for the amplitude zeros thus provides an additionaltest of the gauge theory nature of the SM.33Unfortunately, the radiation zero in q1q¯2 → W γ → ℓνγ and the approximateamplitude zero in q1q¯2 → W Z → ℓ1ν1ℓ+2ℓ−2are not easy to observe in the cos θ distributionin pp or pp¯ collider experiments. Structure function effects transform thezero in the W γ case into a dip in the cos θ distribution. The approximate zero inW Z production is only slightly affected by structure function effects. Higher orderQCD corrections [72] and finite W width effects [73] tend to fill in the dip. In W γproduction photon radiation from the final state lepton line also diminishes thesignificance of the effect.The main complication in the extraction of the cos θ distribution, however,originates from the finite resolution of the detector and ambiguities in reconstructingthe parton center of mass frame. The ambiguities are associated with the nonobservationof the neutrino arising from W decay. Identifying the missing transversemomentum with the transverse momentum of the neutrino of a given W γ or W Zevent, the unobservable longitudinal neutrino momentum, pL(ν), and thus the partoncenter of mass frame, can be reconstructed by imposing the constraint that theneutrino and charged lepton four momenta combine to form the W rest mass [74].The resulting quadratic equation, in general, has two solutions. In the approximationof a zero W decay width, one of the two solutions coincides with the truepL(ν). On an event to event basis, however, it is impossible to tell which of the twosolutions is the correct one. This ambiguity considerably smears out the dip causedby the amplitude zeros.Instead of trying to reconstruct the parton center of mass frame and measurethe cos θ or the equivalent rapidity distribution in the center of mass frame, onecan study rapidity correlations between the observable final state particles in thelaboratory frame [75]. Knowledge of the neutrino longitudinal momentum is not requiredin determining the rapidity correlations. Event mis-reconstruction problemsoriginating from the two possible solutions for pL(ν) are thus automatically avoided.In 2 → 2 reactions differences of rapidities are invariant under boosts. Onetherefore expects that the rapidity difference distributions dσ/d∆y(V, W), V = γ, Z,where ∆y(V, W) = y(V ) − y(W) and y(W), y(V ) are the rapidities in the laboratoryframe, exhibit a dip signaling the SM amplitude zeros [75]. In W±γ production, thedominant W helicity is λW = ±1 [76], implying that the charged lepton, ℓ = e, µ,from W → ℓν tends to be emitted in the direction of the parent W, and thus reflectsmost of its kinematic properties. As a result, the dip signaling the SM radiationzero should manifest itself in the ∆y(γ, ℓ) = y(γ) − y(ℓ) distribution.The SM ∆y(γ, ℓ) differential cross section for pp¯ → ℓ+/pTγ at the Tevatronis shown in Fig. 11a. To simulate detector response, transverse momentum cutsof pT (γ) > 5 GeV, pT (ℓ) > 20 GeV and /pT > 20 GeV, rapidity cuts of |y(γ)| < 3and |y(ℓ)| < 3.5, a cluster transverse mass cut of mT (ℓγ;/pT) > 90 GeV and a leptonphoton separation cut of ∆R(γ, ℓ) > 0.7 have been imposed. The SM radiationzero is seen to lead to a strong dip in the ∆y(γ, ℓ) distribution at ∆y(γ, ℓ) ≈ −0.3.Next-to-leading QCD corrections do not seriously affect the significance of the dip.However, a sufficient rapidity coverage is essential to observe the radiation zero inthe ∆y(γ, ℓ) distribution [75].34Figure 11: Rapidity difference distributions in the SM at the Tevatron. a) The photonlepton rapidity difference spectrum in pp¯ → ℓ+p/Tγ. b) The y(Z) − y(ℓ+1) and y(ℓ+2) − y(ℓ+1)distributions in pp¯ → W+Z.In contrast to the situation in W γ production, none of the W helicities dominatesin W Z production [76]. The charged lepton originating from the W decay,W → ℓ1ν1, thus only partly reflects the kinematical properties of the parent W boson.As a result, a significant part of the correlation present in the y(Z) − y(W)spectrum [77] is lost, and only a slight dip survives in the y(Z) − y(ℓ1) distribution,which is shown for the W+Z case in Fig. 11b. The dip in the SM y(Z) − y(ℓ1)distribution will thus be more difficult to observe experimentally than that in they(γ) − y(ℓ) distribution in W γ production. Next-to-leading order QCD correctionshave only little impact on the shape of the y(Z) − y(ℓ1) distribution [59]. The cutsused in Fig. 11b are the same as those in Fig. 3a except for the lepton rapidity cutwhich has been replaced by |y(ℓ1,2)| < 2.5.Although the Z boson rapidity, y(Z), can readily be reconstructed from thefour momenta of the lepton pair ℓ+2ℓ−2originating from the Z decay, it would beeasier experimentally to directly study the rapidity correlations between the chargedleptons originating from the Z → ℓ+2ℓ−2 and W → ℓ1ν1 decays. The dotted line inFig. 11b shows the y(ℓ+2) − y(ℓ+1) distribution for W+Z production at the Tevatron.The y(ℓ−2)−y(ℓ+1) spectrum almost coincides with the y(ℓ+2)−y(ℓ+1) distribution. Sincealso none of the Z boson helicities dominates [76] in q1q¯2 → W Z, the rapidities of theleptons from W and Z decays are almost completely uncorrelated, and essentially no35Figure 12: Rapidity difference distributions in the SM at the LHC. a) The photon leptonrapidity difference spectrum in pp → ℓ+p/Tγ. b) The y(Z) − y(ℓ+1) distribution in pp →W+Z.trace of the dip signaling the approximate amplitude zero is left in the y(ℓ+2)− y(ℓ+1)distribution.In pp collisions, the dip signaling the amplitude zeros is shifted to ∆y = 0.Because of the large qg luminosity, the inclusive QCD corrections are very large forW γ and W Z production [59, 65]. At the LHC, they enhance the cross section by afactor 2 – 3. The rapidity difference distributions for W+γ and W+Z production inthe SM for pp collisions at √s = 14 TeV are shown in Fig. 12. Here we have imposedthe following lepton and photon detection cuts:pT (γ) > 100 GeV, |η(γ)| < 2.5, (70)pT (ℓ) > 25 GeV, |η(ℓ)| < 3, (71)/pT > 50 GeV, ∆R(γ, ℓ) > 0.7, (72)together with a ∆R(ℓ, ℓ) > 0.4 requirement on leptons of the same charge in W Zproduction. The inclusive NLO QCD corrections are seen to considerably obscurethe amplitude zeros. The bulk of the corrections at LHC energies originates fromquark gluon fusion and the kinematical region where e.g. the photon or Z bosonis produced at large pT and recoils against a quark, which radiates a soft W bosonwhich is almost collinear to the quark. Events which originate from this phasespace region usually contain a high pT jet. A jet veto therefore helps to reduce36the QCD corrections, as demonstrated by the dotted lines in Fig. 12. Here a jet isdefined as a quark or gluon with pT (j) > 50 GeV and |η(j)| < 3. Nevertheless, theremaining QCD corrections still substantially reduce the visibility of the radiationzero in W γ production at the LHC. In pp → W Z, the difference in significance ofthe dip between the LO and the NLO 0-jet ∆y(Z, ℓ1) distribution is quite small.Given a sufficiently large integrated luminosity, experiments at the Tevatronstudying lepton photon rapidity correlations offer a much better chance to observethe SM radiation zero in W γ production than experiments at the LHC. Searching forthe approximate amplitude zero in W Z production will be difficult at the Tevatronas well as the LHC.Indirectly, the radiation zero can also be observed in the Zγ to W γ crosssection ratio [78]. Many theoretical and experimental uncertainties at least partiallycancel in the cross section ratio. On the other hand, in searching for the effects ofthe SM radiation zero in the Zγ to W γ cross section ratio, one has to assume thatthe SM is valid for Zγ production. Similarly, the ZZ to W Z cross section ratioreflects the approximate amplitude zero in W Z production, whereas the ratio ofW Z to W γ cross sections measures the relative strength of the zeros in W Z and W γproduction [59].3.3 Probing WW V and ZγV Couplings in e+e− Collider Experiments3.3.1 Single Photon Production at LEPIn e+e− collisions at center of mass energies near the Z boson mass, anomalousZγV couplings would affect the production of f¯fγ final states. At LEP energies,the production of single photons is the process which is most sensitive to anomalousZZγ couplings, due to the large branching ratio for Z → νν¯ decays and the absenceof background from final state radiation or final state π0’s misidentified as photons.In order to probe Zγγ couplings one has to study ℓ+ℓ−γ or jjγ final states.The L3 Collaboration has searched for anomalous ZZγ couplings in singlephoton events in the data collected in 1991 – 93 [79]. Non-standard ZZγ couplingsmostly affect the production of energetic single photon events whereas the photonenergy spectrum in the SM process e+e− → ννγ ¯ is peaked at low energies. Therefore,a cluster in the BGO electromagnetic calorimeter with energy greater than half thebeam energy was required. In order to further reduce the SM contribution andto eliminate the background from QED events in which all final state particlesexcept the photon escape undetected down the beampipe or into a detector crack,it was required that the polar angle of the most energetic cluster lies between 20and 160 degrees (excluding the ranges between 34.5 and 44.5, and 135.5 and 145degrees due to gaps between the forward and barrel BGO calorimeters). To suppressthe background from cosmic events, the transverse shape of the BGO cluster wasrequired to be consistent with a photon originating from the interaction point.Apart from the energetic BGO cluster, all other activity in the detector had to beconsistent with noise. In terms of equivalent integrated luminosity at the peak ofthe Z resonance, the data sample corresponds to 50.8 pb−1. One event was selected.The number of events expected in the SM is 1.2.37Figure 13: Present limits on anomalous ZZγ couplings from Z → ννγ ¯ , and from Zγproduction at the Tevatron.Since the level of energetic single photon production is consistent with what isexpected in the SM, upper limits on the ZZγ couplings can be derived. To extractlimits, a modified version of the event generator of Ref. [19] was used. Eventswere generated for various combinations of ZZγ couplings, and passed through thedetector simulation and analysis procedure. Figure 13 shows the 95% CL upperlimits on hZ30 and hZ40 for a form factor scale of ΛF F = 500 GeV. Also shown are thecurrent limits from DØ and CDF. Table 5 summarizes the numerical values, if onlyone of the couplings deviates from the SM at a time. The limits obtained fromZ → ννγ ¯ on hZ30 are significantly better than those found from Zγ production at theTevatron. On the other hand, because of the larger center of mass energy and thestrong increase of the terms proportional to hZ4in the helicity amplitudes, hadroncollider experiments give much better bounds on hZ40 than single photon productionat LEP. LEP and Tevatron experiments thus yield complementary information onZZγ couplings. LEP will discontinue to run on the Z peak in 1996. Final integratedstatistics are expected to increase by perhaps a factor 3 over that used in the currentanalysis. Consequently, the present limits on hZ30 and hZ40 from Z → ννγ ¯ are expectedto improve by not more than about a factor 2 in the future. In contrast to thelimits obtained from hadron collider experiments, the sensitivity bounds derivedfrom Z → ννγ ¯ only marginally depend on the form factor scale.An analysis of ℓ+ℓ−γ final states is in progress.3.3.2 W+W− and Zγ Production at LEP IIW pair production and Zγ production at LEP II (√s = 176 − 190 GeV) offerideal possibilities to probe WW V [3, 80, 81] and ZγV [82] couplings. In contrastto pp, pp¯ → W+W− → ℓνjj, the reaction e+e− → W+W− → ℓνjj is not plagued by38Table 5: 95% CL limits on anomalous ZγV , V = γ, Z, couplings from L3, CDF and DØ.Only one of the independent couplings is allowed to deviate from the SM at a time. Theform factor scale is chosen to be ΛF F = 500 GeV.experiment channel limitL3 e+e− → Z → ννγ ¯ −0.85 < hZ30 < 0.85−2.32 < hZ40 < 2.32CDF pp¯ → Zγ → ℓ+ℓ−γ −3.0 < hZ30 < 2.9ℓ = e, µ −0.7 < hZ40 < 0.7DØ pp¯ → Zγ → ℓ+ℓ−γ −1.9 < hZ30 < 1.8ℓ = e, µ −0.5 < hZ40 < 0.5large backgrounds. Furthermore, the reconstruction of the leptonically decayingW boson is easier than in hadronic collisions, where the longitudinal momentumof the neutrino can be reconstructed only with a twofold ambiguity. At hadroncolliders, limits on non-standard couplings are derived from distributions such as thetransverse momentum distribution of one of the vector bosons which make use of thehigh energy behaviour of the anomalous contributions to the helicity amplitudes.At LEP II, on the other hand, angular distributions are more useful. Differentanomalous couplings contribute to different helicity amplitudes and therefore affectthe angular distributions in a characteristic way (see Ref. [3]).In W+W− production, 5 angles are available from each event. These are theW production angle, ΘW , and the angles of the W± → f¯f′ decay products in the W±rest frames, θ± and φ±. In the extraction of these angles, two problems have to befaced: First, the imperfect detection of W decay products gives rise to uncertaintiesin the reconstructed directions of the W’s and their decay products; second, in thecase of hadronic W decays, the absence of a readily recognizable quark tag impliesthat the W decay angles can only be determined with a two-fold ambiguity fromthe data, resulting in symmetrized angular distributions. Complete information fora W+W− event is only available if it is possible to distinguish the W+ and W−direction.Of the three final states available in W pair production, ℓ1ν1ℓ2ν2, ℓνjj, ℓ =e, µ, and jjjj, we have only studied the ℓνjj channel. The purely leptonic channel isplagued by a small branching ratio (≈ 4.7%) and by reconstruction problems due tothe presence of two neutrinos. In the jjjj final state it is difficult to discriminate theW+ and W− decay products. Due to the resulting ambiguities in ΘW and the W±decay angles, the sensitivity bounds which can be achieved from the 4-jet final stateare a factor 1.5 – 2 weaker than those found from analyzing the ℓνjj state [80]. Inthe ℓνjj channel, on the other hand, the identification of the charged lepton allows39the W+ and W− decays to be distinguished unambiguously.Events in the ℓνjj channel were selected from simulated Monte Carlo data at√s = 176 GeV and √s = 190 GeV using the event generator of Ref. [3]. Initial stateradiation and detector smearing, using the L3 specifications are taken into accountin the simulations. The following cuts were imposed:• Number of calorimetric clusters > 16. This requirement eliminates almost allWW → ℓ1ν1ℓ2ν2, ℓ1,2 = e, µ events. It also helps to suppress the WW → τντ ℓν,ℓ = e, µ and WW → τντ τντ channels where at least one of the τ leptonsdecays hadronically. Furthermore it provides some rejection of e+e− → γγ ande+e− → τ+τ−(γ) events.• A visible energy Evis > 80 GeV. This cut mainly reduces the backgroundfrom e+e− → γγ and e+e− → τ+τ−(γ), removes signal events which are poorlyreconstructed, and further suppresses WW → τντ ℓν, ℓ = e, µ and WW → τντ τντevents where at least one of the τ leptons decays hadronically.• E/T/Evis > 0.1. It reduces the WW → jjjj and Z/γ∗(γ) → jj(γ) backgrounds.Here, “(γ)” denotes a photon from initial state radiation.• The momentum of the most energetic lepton, positively identified as an electronor muon, is pmax > 20 GeV. This cut provides most of the suppression ofthe WW → τντ jj and jjjj, and Z/γ∗ → jj(γ) backgrounds.• 65 GeV < m(ℓν) < 125 GeV. The neutrino momentum was calculated frommomentum balance in the event. This requirement mostly suppresses theWW → τντ jj background.With these cuts, the selection efficiency is about 70%, and the ratio of signal tobackground is approximately 20.Sensitivities to the WW V couplings are calculated for the HISZ scenario [seeEqs. (61) – (63)] from the results of a binned maximum log likelihood fit to eventdistributions, assuming an integrated luminosity of 500 pb−1 which corresponds toseveral years of running. Figure 14a shows the 95% CL limit contours obtained at176 GeV and 190 GeV from a fit to the cos ΘW , cos θℓ, φℓ, cos θj and φj distributions,where the (down type) jet j was chosen at random from the jet pair, i.e. it wasassumed that quarks cannot be tagged. Close to the W pair threshold, the gaugetheory cancellations are not fully operative and the sensitivity to anomalous WW Vis limited. If the LEP II center of mass energy can be increased to 190 GeV, thesensitivity bounds improve by about a factor 1.5. The limits on ∆κ0γ and λ0γfor√s = 176 GeV and 190 GeV in the HISZ scenario are summarized in Table 6 forthe case where only one of the two couplings deviates from the SM at a time.Note that the limits on ∆κ0γ and λ0γ at LEP II are quite strongly correlated,in contrast to those obtained from W γ and WW, W Z production production inhadronic collisions. The much reduced correlations at hadron colliders are due tothe high Tevatron and LHC center of mass energies, and the different high energy40Figure 14: 95% CL sensitivity limits from e+e− → W+W− → ℓνjj at LEP II for anintegrated luminosity of 500 pb−1. a) Limit contours for √s = 176 GeV and 190 GeVfrom fitting all five angular distributions, assuming no quark tagging. b) Contours obtainedassuming that no information about the hadronically decaying W is used (dashed line), usingall five angles assuming no quark tagging (solid line), and contours found for the hypotheticalsituation that all five angles are used and quarks are tagged with 100% efficiency (dottedline).Table 6: Expected 95% CL limits on anomalous WW V , V = γ, Z, couplings from experimentsat LEP II in the HISZ scenario [see Eqs. (61) – (63)] for two center of mass energies.The integrated luminosity assumed is RLdt = 500 pb−1. Only one of the independent couplingsis assumed to deviate from the SM at a time. The limits are obtained from a binnedlog likelihood fit to all five angles, assuming no quark tagging.dependent couplings limit limit√s = 176 GeV √s = 190 GeVEqs. (61) and (62) −0.19 < ∆κ0γ < 0.21 −0.13 < ∆κ0γ < 0.14λγ = λZ −0.18 < λ0γ < 0.19 −0.13 < λ0γ < 0.1441behavior of terms proportional to ∆κγ and λγ in the helicity amplitudes. Figure 14bshows limit contours at √s = 176 GeV for binning events in cos ΘW , cos θℓ, and φℓonly (dashed line), all five angles assuming that quarks cannot be tagged (solidline), and for the hypothetical case where the quarks of the hadronically decayingW boson are always tagged correctly (dotted line). The dotted line thus correspondsto the ultimate theoretical precision with which the anomalous couplings could bedetermined. Whereas the information obtained from the hadronically decaying Wdoes not affect the limits if only one of the two couplings is varied at a time, itreduces the correlations between ∆κ0γ and λ0γ by approximately a factor 1.5. Due tothe relatively low center of mass energy, the limits which can be achieved at LEP IIare very insensitive to the form factor scale and power assumed.The contributions from Z and photon exchange in e+e− → W+W− tend tocancel. Therefore, if the WW γ or WW Z couplings only are allowed to deviate fromthe SM, somewhat more stringent limits are obtained than in the HISZ scenarioused in our simulations.Single photon production [32] at LEP II yields sensitivity limits on the WW Vcouplings which are substantially weaker than those derived from W pair production.The limits estimated from single W production, on the other hand, are comparableto those obtained from e+e− → W+W− [83].ZγV couplings can be probed in Zγ production at LEP II. To illustrate thesensitivities which might be expected, 95% CL limit contours for the ZZγ andZγγ couplings were derived from e+e− → Zγ → ννγ ¯ and e+e− → Zγ → µ+µ−γ,respectively. For both processes a photon energy Eγ > 60 GeV, and | cos θγ| < 0.8 wasrequired. For single photon production, the cut on the photon energy significantlysuppresses [32] the contribution from t-channel W exchange to the ν¯eνeγ final state,which is not included in the calculation used. The muon scattering angle, θµ, ine+e− → µ+µ−γ was required to satisfy | cos θµ| < 0.927 which corresponds to the L3angular coverage for muons at LEP II. Muons are also required to have pT (µ) >10 GeV and to be well isolated from the photon; ∆R(µ, γ) > 0.35. In addition, a cuton the di-muon mass of m(µµ) > 10 GeV is imposed. A simplified model of the L3detector is used to simulate detector effects.Sensitivity bounds are calculated from a fit to the total cross section withincuts. The resulting 95% limit contours for a center of mass energy of 180 GeV,ΛF F = 1 TeV, and an integrated luminosity of 500 pb−1 are shown in Fig. 15. Sincethe LEP II center of mass energy will be significantly above the Zγ threshold, thebounds derived on anomalous ZγV couplings vary only little within the expectedrange of center of mass energies expected (√s = 176 GeV – 190 GeV), in contrastto the situation encountered for W pair production. The limits on hV30 and hV40 aresummarized in Table 7 for the case where only one of the two couplings deviatesfrom the SM at a time. For the ZZγ couplings, we also include the present L3 limitsfrom Z → ννγ ¯ for comparison. Due to the higher LEP II center of mass energythe present limits on ZZγ couplings from Z → ννγ ¯ improve by a factor 1.6 (hZ3) to4.6 (hZ4). The improvement is more pronounced for hZ4, due to the stronger growthwith energy of the terms proportional to hZ4in the helicity amplitudes. The limits42Figure 15: 95% CL sensitivity limits from e+e− → Zγ at LEP II for an integrated luminosityof 500 pb−1. a) Limit contours for ZZγ couplings from single photon production. b)Sensitivity limits for Zγγ couplings from e+e− → µ+µ−γ.Table 7: Expected 95% CL limits on anomalous ZγV V = γ, Z, couplings from experimentsat LEP II for √s = 180 GeV. The integrated luminosity assumed is RLdt = 500 pb−1. Onlyone of the two couplings is assumed to deviate from the SM at a time. For comparison, wehave also included the limits on hZ30 and hZ40 from Z → ννγ ¯ at LEP [79]. The form factorscale chosen is ΛF F = 1 TeV.reactions limitse+e− → Z → ννγ ¯ −0.79 < hZ30 < 0.79 −2.08 < hZ40 < 2.08e+e− → Zγ → ννγ ¯ −0.50 < hZ30 < 0.50 −0.45 < hZ40 < 0.45e+e− → Zγ → µ+µ−γ −0.55 < hγ30 < 0.55 −0.48 < hγ40 < 0.4843on hV30 and hV40 which can be achieved are quite similar at LEP II. The sensitivitybounds on Zγγ couplings are about 10% weaker than those found for ZZγ couplings.However, they are expected to significantly improve, if the angular distributions ofthe final state particles are analyzed instead of the total cross section.3.3.3 W+W− Production at the Next Linear ColliderSince the LEP II center of mass energy is only slightly above the W pairthreshold, the SM gauge cancellations are not fully operative, and the sensitivity toanomalous gauge boson couplings is limited. Much better limits on WW V and ZγVcouplings will be possible at an e+e− collider operating in the several hundred GeVrange or above. Such a machine will presumably be a linear collider. Current designstudies for such a “Next Linear Collider” (NLC) foresee an initial stage with a centerof mass energy of 500 GeV and a luminosity of 8 · 1033 cm−2s−1. In a second stage,the energy is increased to √s = 1.5 TeV, with a luminosity of 1.9 · 1034 cm−2s−1[84].As we have mentioned in Section 3.1, such a linear collider could also beoperated as a eγ, γγ and e−e− collider, and a variety of processes can be used toconstrain the vector boson self-interactions at the NLC. Since the limits obtainedfrom W pair production in the e+e− mode [85] are comparable or better than thoseobtained from other processes, we restrict ourselves to the process e+e− → W+W−in the following.The extraction of limits on the WW V couplings at the NLC [85, 86] followsthe same strategy employed at LEP II. Again, only the ℓνjj final state is analyzed.All five angles are used in the maximum likelihood fits. Two cuts are imposed.First, we require | cos ΘW | < 0.8. This ensures that the event is well within thedetector volume. The second cut forces the W+W− invariant mass to be within afew GeV of the nominal e+e− center of mass energy, and ensures that the W+ andW− invariant masses each are within a few GeV of the W pole mass, mW . In orderto impose the second cut, we reconstruct the mass of the leptonically decaying W(mW1), and the mass of the hadronically decaying W (mW2). mW2is reconstructed byimposing four energy momentum constraints and solving for the momentum vectorof the neutrino from the leptonically decaying W, and mW2. mW1is then given bymW1 =h(Eℓ + Eν)2 − (pℓ + pν)2i1/2. (73)We then requireχ2 < 2, (74)where χ2is defined byχ2 =(mW1 − mW )2Γ2W+(mW2 − mW )2Γ2W. (75)Figure 16 shows the 95% CL contours for ∆κγ and λγ at √s = 500 GeV with80 fb−1, and at √s = 1.5 TeV with 190 fb−1for the HISZ scenario [see Eqs. (61)44Figure 16: The 95% CL limit contours for ∆κγ and λγ from e+e− → W+W− at √s =500 GeV with 80 fb−1(solid line), and at √s = 1.5 TeV with 190 fb−1(dashed line) for theHISZ scenario [see Eqs. (61) – (63)].Table 8: Expected 95% CL limits on anomalous WW V , V = γ, Z, couplings from experimentsat the NLC in the HISZ scenario [see Eqs. (61) – (63)] for two center of mass energiesand integrated luminosities. Only one of the independent couplings is assumed to deviatefrom the SM at a time. The limits are obtained from a log likelihood fit to all five angles,assuming no quark tagging.dependent couplings limit limit√s = 500 GeV √s = 1.5 TeVRLdt = 80 fb−1RLdt = 190 fb−1Eqs. (61) and (62) −0.0024 < ∆κγ < 0.0024 −5.2 · 10−4 < ∆κγ < 5.2 · 10−4λγ = λZ −0.0018 < λγ < 0.0018 −3.8 · 10−4 < λγ < 3.8 · 10−445– (63)]. The limits for the case that only one of the two independent couplingsdeviates from the SM are summarized in Table 8. Depending on the energy andintegrated luminosity of the NLC, the LEP II limits could be improved by two tothree orders of magnitude. No form factor effects are taken into account in thebounds listed. However, due to the fixed center of mass energy, these effects caneasily be incorporated. They result in a simple rescaling of the limits quoted.The sensitivities for ZγV couplings are expected to be of O(10−3) at theNLC [87].4. ConclusionsIn this report, we have discussed the direct measurement of WW V and ZγVcouplings in present and future collider experiments. These couplings are definedthrough a phenomenological effective Lagrangian [see Eqs. (1) and (6)], analogouslyto the general vector and axial vector couplings, gV and gA, for the coupling of gaugebosons to fermions. The major goal of such experiments will be the confirmation ofthe SM predictions. We have also reviewed our current theoretical understandingof anomalous gauge boson self-interactions. If the energy scale of the new physicsresponsible for the non-standard gauge boson couplings is ∼ 1 TeV, these anomalouscouplings are expected to be no larger than O(10−2).Rigorously speaking, the three gauge boson vertices are unconstrained bycurrent electroweak precision experiments. Such experiments only lead to bounds onthe anomalous couplings if one assumes that cancellations between the coefficientsof the effective Lagrangian of the underlying model are unnatural. Even in thiscase, the resulting bounds depend quite strongly on other parameters (mH, mt),and anomalous couplings of O(1) are still allowed by current data (see Section 2.4).Present data from di-boson production at the Tevatron and from single photonproduction at LEP yield bounds typically in the range of 0.5 – 3.0. They aresummarized in Tables 1 and 5. ∆κγ is currently constrained best by the processpp¯ → W+W−, W Z → ℓνjj (CDF), whereas the best bound on λγ originates fromW γ production at the Tevatron (DØ). The most precise limits on the ZγV couplingsresult from e+e− → ννγ ¯ (L3; hZ3) and Zγ production at the Tevatron (DØ;hV4, V = γ, Z). Although the present limits on WW V and ZγV couplings are morethan two orders of magnitude larger than what one expects from theoretical considerationsif new physics exists at the TeV scale, these limits still provide valuableinformation on how well the vector boson self-interaction sector is tested experimentallyat present.Within the next 10 years, the limits on WW V couplings are expected toimprove by more than one order of magnitude by experiments conducted at theTevatron and at LEP II. In Fig. 17 we compare the limits expected from e+e− →W+W− → ℓνjj, pp¯ → W±γ → e±νγ, pp¯ → W±Z → ℓ±1ν1ℓ+2ℓ−2and pp¯ → WW, W Z →ℓνjj, ℓ+ℓ−jj in the HISZ scenario [see Eqs. (61) – (63)] for the envisioned energiesand integrated luminosities. The limits expected from future Tevatron and LEP IIexperiments for ∆κγ are quite similar, whereas the Tevatron enjoys a clear advantagein constraining λγ, if correlations between the two couplings are taken into46Figure 17: Comparison of the expected sensitivities on anomalous WW V couplings inthe HISZ scenario from e+e− → W+W− → ℓνjj at LEP II and various processes at theTevatron.47account. It should be noted, however, that the strategies to extract information onvector boson self-interactions at the two machines are very different. At the Tevatronone exploits the strong increase of the anomalous contributions to the helicityamplitudes with energy to derive limits. At LEP II, on the other hand, informationis extracted from the angular distributions of the final state fermions. Data fromthe Tevatron and LEP II thus yield complementary information on the nature ofthe WW V couplings.Because of the much higher energies accessible at the Tevatron and the steepincrease of the anomalous contributions to the helicity amplitudes with energy,Tevatron experiments will be able to place significantly better limits (of O(10−2 −10−3)) on the ZγV couplings than LEP II (≈ 0.5). The Tevatron limits, however, dodepend non-negligibly on the form factor scale assumed.At the LHC one expects to probe anomalous WW V couplings with a precisionof O(10−1−10−3) (see Table 4) if the form factor scale ΛF F is larger than about 2 TeV.Therefore, it may be possible to probe anomalous WW V couplings at the LHC atthe level where one would hope to see deviations from the SM. The limits on theZγV couplings are very sensitive to the value of ΛF F . For ΛF F ≥ 1.5 TeV, the boundswhich can be achieved are of O(10−3) for hV3, and of O(10−5) for hV4. At the NLC,WW V and ZγV couplings can be tested with a precision of 10−3 or better. Detailsdepend quite sensitively on the center of mass energy and the integrated luminosityof the NLC. If new physics exists at the TeV scale, the NLC has the best chance toobserve deviations from the SM through anomalous WW V couplings.AcknowledgementsWe would like to thank our colleagues in CDF, DØ, and L3 for many valuableand stimulating discussions. We also thank Martin Einhorn for his criticalcomments. This work has been supported in part by the Department of Energyand by the N. S. E. R. C. of Canada and les Fonds F. C. A. R. du Qu´ebec.References1. D. Schaile, CERN-PPE/94-162 (preprint, October 1994), to appear in the Proceedingsof the “27th International Conference on High Energy Physics”, Glasgow,Scotland, July 1994; The LEP Collaborations, CERN-PPE/94-187 (preprint,November 1994).2. F. Cuypers and K. Kolodziej, Phys. Lett. B344, 365 (1995); G. Abu Leiland W. J. 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