---

Production of three Vector Bosons

arXiv:hep-ph/9207277v3 16 Nov 1992Production of three Vector Bosonsin e+e−Annihilationas a Test of W±, Z, γ Self-InteractionsC. Grosse-Knetter∗andD.

SchildknechtUniversit¨at BielefeldFakut¨at f¨ur PhysikD-4800 Bielefeld 1GermanyBI-TP 92/30July 1992(Enlarged Version October 1992)AbstractWe study the vector-boson production processes e+e−→W+W−Zand e+e−→W+W−γ which are directly affected by the trilinear andquadrilinear self couplings of the W±, Z and γ. Our analysis is based upona single-parameter effective-Lagrangian model for these self interactionswhich contains the standard model as a special case.

Consequences for thephenomenology at an e+e−collider of 500 GeV (NLC) are discussed, andfits of the free parameter around its standard model value are carried out.∗Partially supported by Deutsche Forschungsgemeinschaft0

1IntroductionThe standard SU(2)L × U(1)Y model of electroweak interactions [1] has beenconfirmed by all experiments up to now.However, this empirical evidence isessentially restricted to vector-boson–fermion interactions. The vector-boson selfinteractions, which are a consequence of the non-Abelian structure of the gaugegroup SU(2)L × U(1)Y (in its minimal realization), and which are essential forthe renormalizability and unitarity of the theory, contribute to reactions realizedat present collider energies only indirectly via vector-boson-loop corrections.Future colliders like LEP II (e+e−at √s = 200 GeV) and NLC (e+e−at√s = 500 GeV) [2] will make direct tests of W±, Z, γ self interactions possibleby the measurement of processes which get tree level contributions from theseself couplings.

The mainly studied process of this type is the two-vector-bosonproduction process e+e−→W+W−(e.g. [3, 4, 5]) as a test of the trilinear self cou-plings.

A full confirmation of the non-Abelian vector-boson sector of the model,however, needs a test of the quadrilinear couplings as well. One class of processeswhich is directly effected by tri- and quadrilinear self couplings, is vector-bosonscattering (V1, V2 →V3, V4 with Vi = W ±, Z, γ) [6, 7], but, unfortunately, it canonly be measured in reactions of the type e+e−→(e+e−, ¯νe−, νe+, ν¯ν) + V1V2 atCM-energies in the TeV region, which cannot be realized experimentally yet.The reactions with direct contributions of tri- and quadrilinear self couplingswhich are of greatest phenomenological interest are the three-gauge-boson pro-duction processes e+e−→W+W−Z and e+e−→W+W−γ [8, 9, 10], which willbe measurable at the expected new linear collider (NLC) with a CM-energy of√s = 500 GeV and a luminosity of 20 fb−1a−1.

The standard model cross sec-tions for these processes were calculated by Barger, Han and Phillips [8] andby Tofighi-Niafiand Gunion [9]. To examine the sensitivity of these cross sec-tions to deviations of the self couplings from their standard model values and tofind experimental bounds on such deviations, we need a more general parameter-dependent model which includes the standard model as a special case.

The mostgeneral procedure would start from an effective Lagrangian containing all vector-boson self interactions which can be constructed in agreement with Lorentz in-variance.Such an analysis was performed for trilinear self couplings and theW+W−-production process in [3] and [4].For three-vector-boson production,however, this seems to be a complicated procedure involving elaborate multi-parameter fits and large individual errors of the fit parameters as a consequenceof limited future statistics.For simplification we base our analysis on the less extended KMSS model [5, 6],which rests upon global SU(2)WI symmetry broken by electromagnetism. Electro-magnetic interactions are assumed to be P- and C- invariant and only dimension-less coupling constants are allowed, thus suppressing dimension-six quadrupoleterms.

These assumptions imply a four-parameter Lagrangian; the free parame-ters can be reduced to a single one, which is chosen as the anomalous magnetic1

moment κ of the W± boson if, in addition, one imposes the requirement thatthose terms in the vector-boson scattering amplitudes which grow most stronglywith energy (as s2) are absent (BKS model) [6].It turns out, that this single-parameter model (extended by appropriate cou-plings of the Higgs boson to the vector bosons) can be derived by adding oneextra SU(2)L × U(1)Y invariant dimension-six interaction term to the standardmodel Lagrangian. This extention of the BKS model provides a mechanism whichprotects LEP I observables against (strong) deviations from their standard modelvalues.In Sect.

2, we explain the KMSS model and the reduction to a single parametereffective Lagrangian (BKS model). In Sect.

3 we give an SU(2)L×U(1)Y invariantderivation of the model and discuss its sensitivity to LEP I experiments.InSect. 4, the calculated cross sections are presented and discussed.

We show thedependence of the cross sections on κ. A fit of the free parameter κ on the basisof the expected luminosity and systematic error of future data to be taken at theNLC is performed.

Sect. 5 contains our final conclusions.2The effective LagrangianAs mentioned, our investigations are based on the KMSS model for vector-bosonself interactions [5, 6].

The effective Lagrangian is constructed in the followingway:1. One starts with a SU(2)WI triplet field ⃗Wµ and an (unphysical) photon field˜Aµ.

Kinectic terms and a mass term for the ⃗W field are introduced.2. The most general self interaction of ⃗Wµ in conformity with global SU(2)WIsymmetry and the restriction to dimension-four couplings is constructed.3.

The photon field ˜Aµ is coupled to the ⃗Wµ field by minimal substitution inthe ⃗Wµ kinetic term. This assures electromagnetic gauge invariance andbreaks global SU(2)WI.4.

Mixing between the neutral weak vector boson W3µ and ˜Aµ is added.5. Finally, an anomalous magnetic-moment term for the interaction of theelectromagnetic field with the charged vector bosons is added.The Lagrangian obtained by this procedure is given in [5].

It has four free pa-rameters (of which only two affect the trilinear couplings).The next step is the analysis of the vector-boson scattering processes V1, V2 →V3, V4 with Vi = W ±, Z performed in [6].The requirement of vanishing ofthe most strongly unitarity violating terms of order s2 in the tree amplitudes(which is eqivalent to the demand of vanishing quartic divergences in one-loop2

corrections [11]) leads to three conditions on the four free parameters, so theycan be expressed in terms of a single one, namely the W± anomalous magneticmoment κ. The (higgsless) standard model is obtained by choosing κ = 1.The final Lagrangian LSI for the self interactions is then given byLSI=iehAµ(W −µνW +ν −W +µνW −ν ) + κFµνW +µW −νi+ie κ −sin2 θWsin θW cos θW"Zµ(W −µνW +ν −W +µνW −ν ) + κ cos2 θWκ −sin2 θWZµνW +µW −ν#−e2(AµAµW +ν W −ν −AµAνW +µW −ν)−2e2 κ −sin2 θWsin θW cos θWAµZµW +ν W −ν −12AµZν(W +µW −ν + W −µW +ν)−e2 (κ −sin2 θW)2sin2 θW cos2 θW(ZµZµW +ν W −ν −ZµZνW +µW −ν)+12e21sin2 θWκ2(W −µ W +ν W −µW +ν −W −µ W +ν W +µW −ν)(1)with Fµν = ∂µAν −∂νAµ, W ±µν = ∂µW ±ν −∂νW ±µ , etc., and a single free parameterκ.

Considering small deviations from the SU(2)L × U(1)Y value of κ = 1,κ ≡1 + ∆κ,∆κ ≪1,(2)the last two terms in (1) may be approximated by terms linear in ∆κ, i. e.,−e2cos2 θWsin2 θW1 +2∆κcos2 θW(ZµZµW +ν W −ν −ZµZνW +µW −ν)+12e21sin2 θW(1 + 2∆κ)(W −µ W +ν W −µW +ν −W −µ W +ν W +µW −ν)(3)The Lagrangian (1) is a single-parameter extension of the standard model La-grangian and a reasonable reduction of the multi-parameter Lagrangian of amodel with arbitrary self interactions, since it is constructed using the above-mentioned physical considerations, and it is the most general Lagrangian in thisformalism in which the vector-boson scattering amplitudes do not contain s2terms. It can be used to analyze how changes of vector-boson self couplings interms of a single free parameter influence the cross sections and to carry out aparameter fit.3

3Derivation of the Model from an SU(2)L×U(1)Y-invariant Lagrangian via the Higgs mecha-nism1As mentioned, the interactions (1) of the vector bosons with one another aresuch that tree-level scattering amplitudes do not contain any (tree-)unitarity-violating terms growing as s2 at high energies. Corresponding to the absenceof such terms, there are no Λ4 divergences at the one loop level [11].

For thestandard-model value of κ = 1, the remaining linear high-energy growth of thescattering amplitudes (proportional to s) is compensated by adding the Higgsscalar with sufficiently low mass MH. Vector-boson loops depend logarithmicallyon MH.One may pose the question wether a simple convergence-producing mechanismalso exists in case of the Lagrangian (1) for κ ̸= 1.

In this section we will show,that the quadratic loop divergences can indeed be removed. Addition of suitablenon-standard interactions of the Higgs scalar with the vector bosons allows toembed the self interactions (1) into an SU(2)L × U(1)Y symmetric framework.Even though the added Higgs-scalar–vector-boson interactions involve nonrenor-malizable dimension-six interactions, they will nevertheless provide a sufficientlydecent behaviour of the loop corrections to protect LEP I observables from violentdeviations from standard predictions.The SU(2)L × U(1)Y invariant Lagrangian, which yields the vector bosoninteractions (1), isLSI = LSM + ∆κ gM2WOW Φ ,(4)where ∆κ = κ −1 as in (2), LSM denotes the standard-model Lagrangian, g =e/ sin θW and OW Φ is given byOW Φ = i(DµΦ)†⃗τ · ⃗W µν(DνΦ) = −iv22 tr(Wµν(DµUDνU†))(5)with ⃗Wµν = ∂µ ⃗Wν −∂ν ⃗Wµ −g ⃗Wµ × ⃗Wν, Wµν = ⃗τ2 · ⃗Wµν, U =√2v˜Φ : Φ,˜Φ = iτ2Φ∗and DµU = ∂µU +ig ⃗τ2 · ⃗WµU −ig′U τ32 Bµ.

Explicitly, upon introducing the Z andphoton fields via the usual diagonalization, OW Φ is given by1We thank the referee of this paper whose questions initiated the present section.4

gM2WOW Φ=ieFµνW +µW −ν+ie1sin θW cos θWhZµ(W −µνW +ν −W +µνW −ν ) + cos2 θWZµνW +µW −νi−2e21sin θW cos θWAµZµW +ν W −ν −12AµZν(W +µW −ν + W −µW +ν)−2e21sin2 θW(ZµZµW +ν W −ν −ZµZνW +µW −ν)+e21sin2 θW(W −µ W +ν W −µW +ν −W −µ W +ν W +µW −ν)+Higgs couplings. (6)Substituting this explicit form of OW Φ into (4), one easily verifies that the gauge-boson self interactions contained in (4) are identical to those of the BKS La-grangian (1) with the linear approximation (3) for the quadrilinear interactions.In the present paper we only consider small values of ∆κ, so we can identify theBKS self couplings (1) with those of the SU(2)L × U(1)Y invariant model (4).The effect of boson loops on LEP I observables due to non-standard interac-tions such as OW Φ and other dimension-six terms was investigated in [12, 13].Typically, in distinction from the SU(2)L × U(1)Y spontanously broken standardmodel, for OW Φ one finds a quadratic dependence on the Higgs-boson mass, M2H,and a logarithmic dependence on the neessary cut-off, ln Λ.Anomalous cou-plings of the kind of extra dimension-six terms considered here are not restrictedvery much by LEP I data, as these data are not very sensitive to logarithmiccut-off-dependent loop corrections [13].In summary, the BKS Lagrangian, originally derived from global SU(2)WIweak isospin symmetry broken by electromagnetism and vanishing of the moststrongly (as s2) rising contribution to vector-boson-scattering amplitudes, canbe embedded into an SU(2)L × U(1)Y symmetric theory of the simple and com-pact form (4) with (dimension-six) non-standard Higgs self interactions.

Thisembedding provides an example of how non-standard trilinear and quadrilinearvector-boson-scattering amplitudes can coexist with “standard” empirical LEP Iresults.In the calculations of the present paper the Higgs sector of the theory is disre-garded (although it contributes to three-gauge-boson-production processes) sincewe only want to study the effect of anomalous gauge-boson self interactions. Thisis justified if we assume that MH lies above the energy region to be investigatedin e+e−annihilation at the NLC2 so that the contributions of Higgs exchange tothe cross sections are negligible.2So our cross sections for √s = 2000 GeV should not be taken too seriously, as they wouldeventually be changed by the Higgs effects.5

4Cross-SectionsFigure 1 shows schematically the tree-level Feynman diagrams which contributeto the three-vector-boson production processes e+e−→W+W−Z and e+e−→W+W−γ3. Altogether there are 15 diagrams.

We calculated the cross-sectionsin the same way as Barger, Han and Phillips [8] evaluating the amplitudes byemploying helicity techniques [14] and integrating numerically over the phasespace. All standard-model couplings were derived via the 1-loop relations amongthe masses and coupling constants in the standard model (e.g.

see [15]) from theZ-boson mass MZ, the Fermi coupling GF and the electromagnetic fine structureconstant αem.The latter was set equal to αem = 1/128.8, which is the highenergy value of the running coupling constant. As in [8] we imposed the followingtransverse-momentum and pseudorapidity cuts for the photon produced in thereaction e+e−→W+W−γ:pt,γ > 20 GeV,|ηγ| < 2 .

(7)First we study the total cross-sections for the production of unpolarized vec-tor bosons in the reactions e+e−→W+W−Z and e+e−→W+W−γ. Figure 2shows these cross-sections as a function of the energy for different values of thefree parameter κ around the standard-model value of κ = 1.

We find with κ = 1at √s = 500 GeV the results σtot = 39 fb for e+e−→W+W−Z and σtot = 135 fbfor e+e−→W+W−γ. Beyond the threshold region standard model cross-sectionsdecrease with increasing energy as a consequence of gauge cancellations, i.e., thecancellations of those parts of the amplitudes for the production of longitudi-nally polarized vector bosons which grow as nonegative powers of s. These arecaused by the relations among the self couplings of the vector bosons and thecouplings to fermions in the standard model4.

If the free parameter κ is set κ ̸= 1these relations are violated. Therefore, in those cases there are no complete can-cellations, and σtot is growing with energy, so that unitarity is violated at highenergies and must eventually be restored by “new physics” contributions.

Thatis why the deviations of the cross sections in the general BKS model from thestandard model increase with the energy √s and with ∆κ, which characterizesthe magnitude of the deviations of the self coupling constants from the standardmodel values. Although we find differences of some orders of magnitude from thestandard model at the TeV energy scale, at the NLC energy of √s = 500 GeVthere are just some % differences for our choices of κ from 0.9 to 1.1.

We willcome back to this point later.To illustrate the effect of violation of the gauge cancellations, we show thecross-sections for the production of exclusively transversely and of exclusively3Except for the Higgs boson contribution which we do not consider here as explained above.For Higgs boson effects see [8, 9].4In distinction to vector-boson scattering, in these processes the Higgs boson is not neededfor good high energy behavior.6

longitudinally polarized vector bosons in the reaction e+e−→W+W−Z (Fig.3). Production of tranverse vector bosons yields a large contribution to the totalcross section in the standard model.

However, for κ ̸= 1 the deviations from theSU(2)L × U(1)Y predicts are very small, because the amplitudes of the differentFeynman graphs do not grow with energy and no cancellations are necessary. Incontrast, for the production of longitudinal vector bosons, the standard modelcross section is very small, but the non-cancellation of the leading amplitudesleads to enormous deviations, when κ departs from κ = 1.We turn to the question of how well trilinear and quadrilinear self inter-actions will be measurable in future experiments.We determine the empir-ical limits which can be assigned to κ if the standard model cross sectionswill be confirmed in experiments.First, we perform an analysis at the NLCenergy of √s = 500 GeV.Our investigations are based on the cross-sectionse+e−→W+W−Z, e+e−→W+W−γ (production of unpolarized vector bosons)and e+e−→W+L W−L ZL (production of longitudinal vector bosons).The ex-pected luminosity of NLC is 20 fb−1a−1, so there would be 800 annual eventsof e+e−→W+W−Z.

Following the analysis of [8], 20% of them will be recon-structable from the decay products of the final vector bosons. If we determinethe statistical error to 90% confidence level and assume 2% systematic error wefind a total error of 10% after three years of collecting data.

For the reactione+e−→W+W−γ the statistical error is smaller because of the larger cross-section. By the same reasoning we find a total error of 5% after three years of run-ning.

For the process e+e−→W+L W−L ZL the cross-section of 0.5 fb is extremelysmall and causes a large statistical error. The total error in this case is 70%.As an outlook we perform the same analysis at the energy of √s = 2000 GeV inorder to see, how the precision of the parameter fit increases with energy5.

Sinceno precise parameters of a 2000 GeV machine are available, we assume an exper-imental error of 10% for production of unpolarized vector bosons and again 70%error for the reaction e+e−→W+L W−L ZL. Figure 4 shows the total cross sectionsfor these processes at the two abovementioned fixed energies as a function of κ.We see, how small deviations of κ from its standard model value of κ = 1 affectthe total cross-sections.

From these results, we now can find the interval aroundthe standard model value of κ = 1 to which κ is restricted if the deviations fromthe standard model do not exceed the experimental error. The results of theseparameter fits are given in table 1.

We see that NLC results can restrict κ to aregion of a few % around its standard model value of κ = 1. Accuracy increasesby one order of magnitude if √s is raised from 500 GeV to 2000 GeV.

The pro-duction of exclusively longitudinal vector bosons e+e−→W+L W−L ZL yields limitsin the same order as production of unpolarized gauge bosons, because the effect5We are aware of the fact that these estimates should eventually be refined by taking intoaccount the effects of a Higgs scalar of unknown mass MH ≤2 TeV. Our results have to beconsidered as crude exploratory values.7

of the larger deviations from the standard model is compensated by the largerexperimental error.Process√s = 500 GeV (NLC)√s = 2000 GeVe+e−→W+W−Z0.95 ≤κ ≤1.060.997 ≤κ ≤1.006e+e−→W+W−γ0.98 ≤κ ≤1.090.997 ≤κ ≤1.009e+e−→W+L W−L ZL0.98 ≤κ ≤1.070.999 ≤κ ≤1.003Table 1: Results from the fit of the free parameter κ based on the total cross-sections for e+e−→W+W−Z, e+e−→W+W−γ and e+e−→W+L W−L ZL at√s = 500 GeV and √s = 2000 GeV.5Conclusions• The three vector-boson production processes e+e−→W+W−Z and e+e−→W+W−γ supply the phenomenological easiest way to test the tri- and qua-drilinear self interactions of the electroweak vector bosons directly.• The BKS model reduces the most general form of vector-boson self inter-actions by some physically resonable assumptions to a single parametermodel, which can be used to study the sensitivity of the standard modelcross sections to variations of the self couplings.• The BKS model is not restricted very much by present LEP I data sinceits anomalous vetor-boson self couplings can be obtained from a simpleone-parameter locally SU(2)L × U(1)Y-invariant interaction term.As aconsequence, loop-contributions in this model (extended by appropriateHiggs-couplings) diverge at most logarithmically.• The cross-sections of the three-gauge-boson-production processes are verysensitive to variations of the free parameter. At the NLC energy of √s =500 GeV variations of κ at the per-cent level lead to measurable differencesfrom the standard model.Measurement of the polarisation of the finalvector bosons yields only slighly stricter limits on κ.• Although κ can be determined in e+e−→W+W−, even with stricter pa-rameter limits due to better statistics (see [4] Table 4.2), this does notimply the structure of the quadrilinear couplings.

Indeed, the agreementof the values of κ deduced from e+e−→W+W−and e+e−→W+W−Z,γmeasurements is essential for establishing the full non-Abelian Yang–Millsstructure.8

• In summary: Future e+e−colliders at 500 GeV energy or more will supplygood empirical possibilities to explore the self interactions of the vectorbosons and to test the Yang–Mills structure of the electroweak standardmodel.References[1] S. L. Glashow, Nucl. Phys.

B22 (1961) 579;S. Weinberg, Phys. Rev.

Lett. 19 (1967) 1264;A. Salam, Proc.

8th Nobel Symposium, ed. N. Svartholm (Almquits andWiksells, Stockholm, 1968) p. 367[2] “e+e−Collisions at 500 GeV, the Physics Potential”, ed.

P. W. Zerwas, DESY92-123[3] K. Gaemers and G. Gounaris, Z. Phys. C1 (1979) 259;K. Hagiwara, R. D. Peccei and D. Zeppenfeld, Nucl.

Phys. B282 (1987) 253[4] G. Gounaris, J. L. Kneur, J. Layssac, G. Moultaka, F. M. Renard and D.Schildknecht, in [2] p. 735[5] M. Kuroda, J. Maalampi, K. H. Schwarzer and D. Schildknecht, Nucl.

Phys.B284 (1987) 271[6] C. Bilchak, M. Kuroda and D. Schildknecht, Nucl. Phys.

B299 (1988) 7[7] M. Kuroda, F. M. Renard and D. Schildknecht, Z. Phys. C40 (1988) 575[8] V. Barger, T. Han and R. J. N. Phillips, Phys.

Rev. D39 (1989) 146[9] A. Tofighi-Niaki and J. F. Gunion, Phys.

Rev. D39 (1989) 720[10] F. M. Renard, Z. Phys.

C2 (1979) 17;E. N. Agryresi, O. Karakianitis, C. G. Papadopoulos and W. J. Stirling Phys.Lett. B259 (1991) 195;W. Benakker, F. A. Berends and T.Sack, Nucl.

Phys. B367 (1991) 287;J. Kalinowski, in [2] p. 217;G. B´elanger and F. Boudjema, in [2] p. 783;G. Cvetiˇc, C. Grosse-Knetter and R. K¨ogerler, in [2] p. 775 and Bielefeld-Preprint BI-TP 92/32, submitted to Nucl.

Phys B[11] H. Neufeld, J. D. Stroughair and D. Schildknecht, Phys. Lett.

B198 (1987)563[12] A. de R´ujula, M. B. Gavela, P. Hernandez and E. Mass´o, Nucl. Phys.

B384(1992) 39

[13] K. Hagiwara, S. Ishihara, R. Szalapski and D. Zeppenfeld, Phys. Lett.

B283(1992) 353[14] K. Hagiwara and D. Zeppenfeld, Nucl. Phys.

B274 (1986) 1[15] J. L. Kneur, M. Kuroda and D. Schildknecht, Phys. Lett.

B262 (1991) 93Figure CaptionsFigure 1: Feynman diagrams for three-vector-boson productionFigure 2: Total cross-sections for the reactions (a) e+e−→W+W−Z and (b)e+e−→W+W−γ as a function of the CM-energy for different valuesof κ. The solid lines show the results for the standard-model value ofκ = 1.Figure 3: Cross-sections for the production of (a) exclusively transversely po-larized vector bosons e+e−→W+TW−TZT and (b) exclusively longitu-dinally polarized vector bosons e+e−→W+L W−L ZL as a function of√s for different κ.Figure 4: Total cross-sections for (a) e+e−→W+W−Z, (b) e+e−→W+W−γand (c) e+e−→W+L W−L ZL for fixed √s = 500 GeV and √s =2000 GeV as a function of κ (solid lines).

The dashed lines borderthe regions where the results agree with the standard model valuewithin the estimated experimental error.10

---

출처: arXiv:9207.277 • 원문 보기