Anomalous Gauge Boson Couplings in the e+e− → ZZ Process J. Alcaraz and M.A. Falag´an ∗ CIEMAT, Avda. Complutense 22, 28040-MADRID, Spain E. S´anchez 0 CERN, 1211 Gen`eve 23, Switzerland 0 0 (February 1, 2008) 2 n a J 2 Abstract 1 3 v We discuss experimental aspects related to the e+e ZZ process and to − 5 → 3 the search for anomalous ZZV couplings (V= Z,γ) at LEP2 and future e+e 4 − 2 1 colliders. We present two possible approaches for a realistic study of the 8 9 reaction and discuss the differences between them. We find that the optimal / h p method to study double Z resonant production and to quantify the presence - p e of anomalous couplings requires the use of a complete four-fermion final-state h : v calculation. i X 12.60.Cn, 13.10.+q, 13.38.Dg r a Typeset using REVTEX partially supported by CICYT Grant: AEN96-1645 ∗ 1 INTRODUCTION Pair production of Z bosons is one of the new physics processes to be studied at LEP2 and at future high energy e+e colliders. Although it is a process with a rather low cross − section (below 1 pb) and experimentally difficult to observe (large and almost irreducible backgrounds), LEP2 gives the first opportunity to perform a measurement and to look for deviations from the Standard Model (SM). In addition, a good understanding of the process is necessary, since it is one of the relevant backgrounds in the search for the Higgs particle. At future e+e colliders, with luminosities of the order of 100 fb 1, several thousands of − − events will provide stringent tests of the SM. The study of triple-gauge boson couplings is one of the key issues at present and future colliders. Anomalous ZγV couplings have been searched for at the Tevatron and at LEP [1]. The first experimental limits on anomalous ZZZ couplings have been provided by the L3 Collaboration [2]. The report is organized as follows. First, the SM amplitude is presented and the effects of possible anomalous ZZV couplings at LEP2 and at future e+e colliders are discussed. − Second, we describe two reweighting approaches developed for the search for anomalous couplings at LEP2. These approaches are compared and their differences are pointed out. Finally, the fitting techniques used to quantify the possible presence of anomalous couplings are briefly presented. STANDARD MODEL AMPLITUDE FOR THE e+e− → ZZ PROCESS The diagrams contributing at first order to the e+e ZZ process in the Standard − → Model are shown in Figure 1. We will assume a collision in the center-of-mass system with total energy √s and neglect the effect of the electron mass. The following notation is used: Electron four-momentum k and helicity σ: • k = (√s, √s eˆ ) 2 2 z 2 σ 1,1 ∈ {− } Positron four-momentum k and helicity σ: • k = (√s, √s eˆ ) 2 − 2 z σ 1,1 ∈ {− } Z four-momentum q and polarization ǫ : • Z1 Z1 q = (E, E2 M2 qˆ); E = √s + MZ21−MZ22 Z1 − Z1 2 2√s q ǫ ǫ (λ ); λ 1,0,1 Z1 ≡ Z1 Z1 Z1 ∈ {− } Z four-momentum q and polarization ǫ : • Z2 Z2 q = (E , E 2 M2 qˆ); E = √s + MZ22−MZ21 Z2 ′ − ′ − Z2 ′ 2 2√s q ǫ ǫ (λ ); λ 1,0,1 Z2 ≡ Z2 Z2 Z2 ∈ {− } where the electron is assumed to collide along the +z axis (eˆ ), and the Z -with mass M - z Z1 goes along the direction given by qˆ = (sinθ cosφ ,sinθ sinφ ,cosθ ). The masses M Z Z Z Z Z Z1 and M are not assumed to be equal to the on-shell mass m because in the following they Z2 Z will be consider as virtual particles decaying into fermions. The matrix element for the e+e ZZ reaction is determined by the same method − → followed in [3]. It reads: S(ǫ ,q ,ǫ ,σ) S(ǫ ,q ,ǫ ,σ) M (σ,σ,λ ,λ ) = (gZe+e−)2√s δ ∗Z2 Z1 ∗Z1 + ∗Z1 Z2 ∗Z2 (1) ZZ Z1 Z2 − σ σ,−σ "−2(kqZ1)+MZ21 −2(kqZ2)+MZ22 # The functions S(ǫ ,q ,ǫ ,σ) are given by: a b b √s q0 q3, q1 +iq2 ǫ0 ǫ3 S(ǫa,qb,ǫb,+) = (cid:18)ǫ1a +iǫ2a, −ǫ0a −ǫ3a (cid:19) q−1 b i−q2,b −qb0 +q3b  ǫb1− ibǫ2  (2)  − b − b − b b  − b − b        q0 +q3, q1 iq2 ǫ1 iǫ2 S(ǫa,qb,ǫb,−) = (cid:18)ǫ0a +ǫ3a, ǫ1a −iǫ2a (cid:19)−q1b+iq2b, √sb −q0 b q3  ǫb0− ǫ3b  (3)  b b − b − b  b − b        3 where the components of the four-vectors are denoted by superscripts. The left/right effec- tive couplings of fermions to neutral gauge bosons are given by: gZf¯f = 2Q sin2θ √2G m2 1/2 (4) + − f W µ Z (cid:16) (cid:17) gZf¯f = gZf¯f +2I √2G m2 1/2 (5) + 3 µ Z − (cid:16) (cid:17) gγf¯f = Q 4πα(M2 ) 1/2 (6) + f γ∗ (cid:16) (cid:17) gγf¯f = gγf¯f (7) + − where Q is the charge of the fermion f in units of the charge of the positron, and the f electromagnetic coupling constant α(M2 ) is evaluated at the scale of the virtual photon γ∗ mass M2 . I is the third component of the weak isospin ( 1/2), sin2θ is the effective γ∗ 3 ± W value of the square of the sine of the Weinberg angle and G is the value of the Fermi µ coupling constant. The effective couplings to the Z absorb the relevant electroweak radiative corrections at the scale of the Z [4]. They are obtained by the substitutions: sin2θ sin2θ (8) W W → e2 √2G m2 (9) 4sin2θ cos2θ → µ Z W W The experimental signature of a e+e ZZ process is a final state with four fermions, − → due to the unstability of the Z particle. A distinctive feature is that the invariant masses ¯ ¯ of the two pairs, ff and f f , peak at the Z mass m . The angular distribution of the decay ′ ′ Z products keeps information on the average polarization of the Z boson. In addition, the Z decay amplitude has to be included for a correct treatment of spin correlations. Assuming that fermion masses are negligible compared with m , this amplitude is given in the rest Z frame of the Z by: M (λ ,λ,λ) = gZff M δ [ǫ (v iλv )] (10) Zif¯f Zi λ Zi λ, λ Zi 1 − 2 − v = (0,cosθ cosφ ,cosθ sinφ , sinθ ) (11) 1 f f f f f − v = (0, sinφ ,cosφ ,0) (12) 2 f f − 4 where pˆ= (sinθ cosφ ,sinθ sinφ ,cosθ ) is the direction of the fermion momentum and f f f f f λ,λ are the helicities of fermion and antifermion, respectively. ANOMALOUS COUPLINGS IN THE e+e− → ZZ PROCESS Anomalous gauge boson couplings lead to interactions of the type shown in Figure 2. The coupling ZZV, with V = Z or γ, does not exist in the Standard Model at tree level. Only two anomalous couplings are possible if the Z bosons in the final state are on-shell, due to Bose-Einstein symmetry. In principle, five more couplings should be considered if at least one of the Z bosons is off-shell. However, as discussed in Appendix A, the new terms must be of higher dimensionality and are suppressed by orders of m Γ /Λ2, where Λ denotes Z Z a scale related to new physics. We will concentrate on the most general expression of the anomalous vertex function at lowest order [3]: s m2 Γαβµ = − V ifV [(q +q )α gµβ +(q +q )β gµα]+ifV ǫαβµρ (q q ) (13) ZZV m2 { 4 Z1 Z2 Z1 Z2 5 Z1 − Z2 ρ } Z Anon-zerovalueoffV leadstoaC-violating,CP-violatingprocess, whiletermsassociated 4 to fV are P-violating, CP-conserving. Using again the formalism followed in [3] we obtain 5 the explicit expressions for the anomalous contributions: Mf4V (σ,σ,λ ,λ ) = i efVgVee s δ ǫ0 (ǫ1 +iσǫ2 )+ǫ0 (ǫ1 +iσǫ2 ) (14) AC Z1 Z2 − 4 σ m2Z σ,−σ Z∗1 Z∗2 Z∗2 Z∗2 Z∗1 Z∗1 h i Mf5V (σ,σ,λ ,λ ) = i efVgVee √s δ (ǫ1αβρ +iσǫ2αβρ) ǫ ǫ (q q ) (15) AC Z1 Z2 − 5 σ m2Z σ,−σ ∗Z1α ∗Z2β Z1ρ − Z2ρ Note that no (s m2 ) factors are present in the final expressions. Compared to the SM − V amplitude all anomalous contributions increase with the center-of-mass energy of the colli- sion. We want to bring the attention to the fact that these anomalous ZZγ couplings are different from those considered in the e+e Z γ anomalous process [3]. Therefore, not − → onlythetwo anomalousZZZcouplings, but allfour anomalousparameters remainessentially unconstrained at present. Anomalous ZZV couplings manifest in three ways: 5 A change in the observed total cross section e+e ZZ. − • → A modification of the angular distribution of the Z. • A change in the average polarization of the Z bosons. • The effect of anomalouscouplings inthee+e ZZ process at Bornlevel is illustrated in − → Figures 3 and 4 for the center-of-mass energies of √s = 190 GeV and √s = 500 GeV, respec- tively. The anomalous distributions are determined for the values fV = 3;i = 4,5;V = Z,γ. i Both CP-violating and P-violating couplings are found to produce a global enhancement in the number of events. This increase is very clear at √s = 500 GeV. There are moderate changes in the angular shape at √s = 190 GeV. At √s = 500 GeV the situation is different. The copiousanomalousproductionat largepolar anglesstartsto compensate thehugepeaks oftheSMdifferential cross section atlowpolarangles. TheSMdivergent behaviour happens in the limit m √s, where the process tends to a t-channel process with production of Z ≪ massless bosons. Anomalous couplings also modify the average polarizations of the Z bosons, as shown in Figures 5, 6 and 7. At √s = 190 GeV the observed change depends on the particular size and type of the anomalous coupling under consideration. ForCP-violating couplings there is always an increase of the production of bosons with different polarizations (longitudinal ver- sus transverse). At √s = 500 GeV all couplings show a similar behaviour: an enhancement of longitudinal-transverse production and a suppression of transverse-transverse production. This is an interesting feature, since the SM process has the opposite behaviour. The fraction ofstates inwhich thetwo bosonsarelongitudinally polarizedis below0.5%at these energies, both for SM and for anomalous production. What is physically observable is a modification ¯ of the angular distributions in the center-of-mass frame of the Z ff decays: in absence of → anomalous couplings, both Z decays will proceed preferentially along the direction of the Z momenta, whereas one of the decays will preferentially occur at 90 if the process is highly ◦ anomalous. 6 Summarizing, at energies close to the threshold of ZZ production the sensitivity to anomalous couplings is weak. For an integrated luminosity of 200 pb 1, tens of events − are expected to be selected. The main anomalous effect is an increase in the cross section, and one expects small improvements from the variations in the angular distributions. At higher energies, with luminosities of the order of 100 fb 1, all anomalous effects contribute − coherently to enhance the sensitivity: a huge increase in the cross section, especially at large polar angles, and a clear correlation between the angular distributions of the Z decay products. INCLUSION OF ANOMALOUS COUPLINGS. REWEIGHTING PROCEDURE In order to take into account anomalous effects with sufficient accuracy, the correct matrix element structure has to be implemented. In many cases, and from the practical point of view, the generation of events for different values of anomalous couplings is not convenient. A more attractive method is to set up a procedure to obtain the Monte Carlo anomalous distributions as a function of a single set of generated events. This is the role of reweighting methods. Let us consider a set of events generated according to the Standard Model differential cross section. New distributions taking into account the anomalous couplings are obtained when every event is reweighted by the factor: 2 (M +M ) M M (cid:12) ZZ AC Z1f¯f Z2f′f¯′(cid:12) WZZ(σ,λ,λ;Ω¯) (cid:12)(cid:12)λZX1,λZ2 (cid:12)(cid:12) (16) ′ ≡ (cid:12)(cid:12) 2 (cid:12)(cid:12) (cid:12) (cid:12) M M M (cid:12) ZZ Z1f¯f Z2f′f¯′(cid:12) (cid:12)(cid:12)λZX1,λZ2 (cid:12)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) The weight WZZ(σ,λ,λ;Ω¯) depend(cid:12)s on the helicities of the(cid:12)initial electron (σ) and of ′ the final fermions (λ,λ). It also depends on the kinematic variables defining the phase space ′ (Ω¯). For convenience we choose the following set: ¯ ¯ The invariant masses of the ff and f f systems: M , M . • ′ ′ Z1 Z2 7 ¯ The polar and azimuthal angles of the ff system : θ , φ . Z Z • The polar and azimuthal angles of the fermion f after a Lorentz boost to the rest frame • ¯ of the ff system: θ , φ . f f The polar and azimuthal angles of the fermion f after a Lorentz boost to the rest ′ • ¯ frame of the f′f′ system: θf′, φf′. The previous result can be extended to take into account other non-resonant diagrams like e+e Zγ f¯ff f¯ and e+e γ γ f¯ff f¯. These diagrams can not be neglected − ∗ ′ ′ − ∗ ∗ ′ ′ → → → → for a correct analysis of double Z resonant production [5]. The final weight is: 2 (M2 ) (M2 ) M M M DZ1 Z1 DZ2 Z2 AC Z1f¯f Z2f′f¯′ WVV(σ,λ,λ′;Ω¯) (cid:12)(cid:12)1+ λZX1,λZ2 (cid:12)(cid:12) (17) ≡ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) VX◦,V◦′ DV◦(MZ21)DV◦′(MZ22)λV◦X,iλV◦′ MV◦V◦′ MV◦f¯f MV◦′f′f¯′(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12) (cid:12) where a sum on all int(cid:12)ermediate V Z,γ is assumed. The propagator facto(cid:12)rs are (cid:12) ◦ ∈ { ∗} (cid:12) DV◦ defined as follows: 1 (q2) = (18) DZ q2 m2 +i Γ q2/m − Z Z Z 1 (q2) = (19) γ D q2 where the imaginary component takes into account the energy dependence of the Z width around the resonance. The expressions for MZγ∗, Mγ∗γ∗, Mγ∗f¯f are obtained by the same method used for M and M . Explicitly, they can be obtained by substituting Z by γ ZZ Zf¯f ∗ where necessary: ǫZ(λZ) ǫγ∗(λγ) (20) → MZ1 → Mγ∗ (21) gZf¯f gγf¯f (22) + → + gZf¯f gγf¯f (23) − → − Weights according to WVV(σ,λ,λ;Ω¯) have been implemented in a FORTRAN program. ′ The approach is well suited for events generated with the PYTHIA e+e Z/γ Z/γ − ∗ ∗ → → 8 ¯ ¯ fff f generator [6]. This implementation will be identified as the “NC08 approach”, because ′ ′ all eight neutral conversion diagrams are considered. Several checks have been done in order to make sure that the calculations are correct. There is also agreement with the results obtained in [7] and in [8]. After reweighting, distributions according to given values of the anomalous couplings fV,fV areobtained. Detectoreffectsarecorrectlytakenintoaccountifeventsarereweighted 4 5 at generator level. INITIAL STATE RADIATION EFFECTS There are several references providing valuable information on the e+e ZZ process. − → A specific SM generator for e+e Z/γ Z/γ (γ) f¯ff f¯(γ) without anomalous couplings − ∗ ∗ ′ ′ → → exists in PYTHIA [6]. The calculation reported in the previous section is well suited for this MC generator, but initial state radiation effects (ISR) need to be taken into account. We assume that the differential cross section can be expressed as follows: dσ(s) dσ(s) ′ = dsR(s,s) (24) ′ ′ d(Phase Space) d(Phase Space) Z ′ where σ(s) is the (undressed) cross section evaluated at a scale s, and σ(s) is the cross ′ ′ section after inclusion of ISR effects. The radiator factor R(s,s) is a “universal” radiator, ′ that is, independent of specific details of the matrix element. With this assumption ISR effects are accounted for by evaluating the matrix element in the center-of-mass system of the four fermions, at the corresponding scale s. ′ In Reference [7] a specific e+e ZZ(γ) f¯ff f¯(γ) generator for anomalous couplings − ′ ′ → → studies is presented. It takes into account ISR effects with the YFSapproach [9]up to (α2) O leading-log. It has some limitations, like the absence of conversion diagrams mediated by virtual photons. The Standard Model cross section (fV = fV = 0) from [7] shows agreement at the 4 5 percent level with the one determined in Reference [5], where it is shown that all significant 9 radiation effects come from “universal” radiator factors. This implies that an approach based on Equation 24 is justified in terms of the required precision. THE COMPLETE e+e− → f¯ff′f¯′ PROCESS Additional non-resonant diagrams are taken into account in SM programs for general four-fermion production, like EXCALIBUR [10]. Under a reasonable set of kinematic cuts, the relative influence of those diagrams can be reduced, but not totally suppressed. This is duetothelowcrosssectionforresonantZZproduction. Typicalexamplesarethoseinvolving chargedcurrents (relevant ine+e ν ν f¯f, e+e uddu, ...) ormultiperipheral effects in − e e − → → e+e e+e f¯f. In addition, the influence of Fermi correlations in final states with identical − − → fermions (e+e f¯ff¯f) is unclear. − → In order to include these effects, the EXCALIBUR program has been extended. All matrix elements from conversion diagrams with two virtual Z particles MEXC(σ,λ,λ;Ω¯) are ZZ ′ modified in the following way: M M M AC Z1f¯f Z2f′f¯′ ∆ZZ(σ,λ,λ;Ω¯) λZX1,λZ2 (25) ′ ≡ M M M ZZ Z1f¯f Z2f′f¯′ λZX1,λZ2 MEXC(σ,λ,λ;Ω¯) MEXC(σ,λ,λ;Ω¯) 1+∆ZZ(σ,λ,λ;Ω¯) (26) ZZ ′ → ZZ ′ ′ (cid:16) (cid:17) where M ,M ,M and M are the same terms defined in the NC08 approach. Based ZZ AC Z1f¯f Z2f′f¯′ on this modification it is straightforward to define an alternative reweighting. It will be identified as the “FULL approach” in the following. More detailed studies are reported in the next section. THE NC08 APPROACH VERSUS A FULL TREATMENT All checks presented inthis section require a precise definition of a“ZZ signal”. Channels involving electron or electronic neutrino pairs in the final state have a non-negligible contri- bution from non-conversion diagrams. Also final states with fermions from the same isospin 10