Theory status of four-fermion production at e-e+ colliders
FFrascati Physics Series Vol. XXXX (yyyy)
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Theory status of four-fermion production at e − e + colliders Christian Schwinn ∗ Institute for Theoretical Particle Physics and Cosmology,RWTH Aachen University, D-52056 Aachen, Germany and
Albert-Ludwigs-Universit¨at Freiburg, Physikalisches Institut,D-79104 Freiburg, Germany
Abstract
The status of predictions for four-fermion production at e − e + colliders is re-viewed with an emphasis on the developments after the LEP2 era and an out-look to the challenges posed by the precision program at future colliders. After the discovery of a Higgs boson, the search for physics beyond the StandardModel (SM) of particle physics is one of the main objectives of run 2 of the LHCand of future colliders. In case new particles are not directly accessible at thesecolliders or in non-collider experiments, one can search for indirect evidence fornew physics through precise studies of electroweak (EW) or flavour observables, ∗ Heisenberg Fellow of the German Research Foundation (DFG). a r X i v : . [ h e p - ph ] D ec − e + W − W + ¯ ν ℓ ℓ − ¯ q ′ q e − e + ZZ ℓ + ℓ − ¯ qq e − e + Zγ ℓ + ℓ − ¯ qq (a) W W (b) ZZ (c) Zγ e − ¯ ν e e − γe + W + W + ¯ q ′ q e − e + e − γe + Z ¯ qq (d) W eν (e)
Zee
Figure 1: Classification of signatures in four-fermion production.and the couplings of the gauge bosons and the Higgs boson. Further, accuratemeasurements of input parameters of the SM such as the masses of the W and Z bosons and the top quark are required for the precision-physics program. Herefuture e − e + colliders could play a particularly important role by revisiting theLEP precision measurements at higher statistics, and further measuring top-quark and Higgs-boson properties. Currently linear colliders such as ILC andCLIC as well as circular colliders such as FCC-ee or CEPS are investigated. 1)An important signature at high-energy e − e + colliders is given by four-fermion production processes as shown in Figure 1. They have been exploredat LEP2 2) for centre-of-mass energies √ s = 161 . . W -boson in W -pair production (Fig 1 (a)), andtriple-vector boson couplings in W -pair production, Zγ and single- W produc-tion (Fig 1 (c) and (d), respectively). At future e − e + colliders the precision ofthese measurements could be increased, for instance by up to two magnitudesfor the triple gauge boson couplings. 4) For M W , an accuracy of 3–4 MeV isprojected for an ILC, while 1 MeV may be possible using a threshold scan ofthe W -pair production cross section at a future circular e − e + collider. 4) Four-fermion final states arising from Higgs-boson production with subse-quent decay to b quarks or τ leptons are not considered in this contribution. We W ¯ ν µ µ − ¯ du ee γ/Z WW ¯ ν µ µ − ¯ du ee γ/Z W ¯ d ¯ ν µ µ − uee γ/Z W ¯ d ¯ ν µ µ − u ee γ/Z W ¯ ν µ ¯ duµ − ee Z W ¯ ν µ ¯ duµ − Figure 2: Diagrams contributing at tree-level to the e − e + → u ¯ dµ − ¯ ν µ process.In this contribution, the theoretical challenges and the methods used forfour-fermion production are discussed in Section 2. Recent theoretical resultsare reviewed in Section 3 while an outlook to future developments needed tomeet the requirements of planned colliders is given in Section 4. In the theoretical description of four-fermion production, in general all dia-grams contributing to a given final state must be taken into account for aconsistent, gauge invariant result, resulting in a large number of contributingFeynman diagrams, in particular beyond leading order. These typically in-clude topologies different from the resonant “signal” diagrams of the processesin Figure 1. For instance, as shown in Figure 2, ten tree-diagrams contribute tothe final state u ¯ dµ − ¯ ν µ , where only three diagrams include a resonant W -bosonpair. Similarly, 20 diagrams contribute to the single- W signature u ¯ de − ¯ ν e .The consistent treatment of the W/Z -boson decay-widths poses a furthertheoretical challenge. The
Dyson series allows the resummation of the self-energy Σ V of the vector boson V to all orders into the denominator of the V -boson propagator, ( p − M V +Σ V ( p )). The complex pole µ V of the propagatordefined by µ V − M V + Σ V ( µ V ) = 0 provides a gauge invariant definition of themass M V and width Γ V of the vector bosons, µ V ≡ M V − i M V Γ V . In some cases, gauge invariant subsets of diagrams can be identified. 3)he Dyson summation of the self-energy includes only a subset of higher-order diagrams, but neglects other contributions of the same order. A naiveapplication therefore can lead to inconsistencies such as violations of gaugeinvariance and unitarity, which can can result in dramatically wrong predic-tions, in particular in the case of single- W production at high energies. 5)A simple use of a Breit-Wigner propagator with a fixed width is sufficient inmany leading-order applications, but does not respect electroweak gauge in-variance. In the complex-mass scheme M V → µ V is madein the propagator as well as in the Feynman rules, e.g. in the weak-mixingangle cos θ w = M W /M Z → (cid:112) µ W /µ Z . In this way, algebraic identities amongvertices and propagators required by gauge invariance are satisfied also for afinite width. The fermion-loop scheme , 5) applied in particular to the single- W process at LEP2, 7) uses the fact that diagrams with a closed fermion loopform a gauge invariant subset of diagrams. Finally, the double-pole approxima-tion (DPA) consistently splits the NLO corrections into factorizable correctionsto on-shell vector-boson production and decay, and non-factorizable soft-photoncorrections connecting vector-boson production, propagation and decay. TheDPA has been applied to W - and Z -boson pair production at LEP2. 8)The methods summarized here have been used successfully to describethe LEP2 measurements of four-fermion production with a theoretical accuracybetter than 1% for W -pair production and 2–5% for the other processes. 2 , The high accuracy possible at future e − e + colliders makes it mandatory toimprove the theoretical predictions of four-fermion cross sections beyond thelevel achieved for LEP2. The M W measurement from a threshold scan re-quires a calculation of the W -pair production cross section with a precisionof a few per-mille in the threshold region √ s ∼ M W , where the accuracy ofthe DPA degrades. A complete NLO calculation of charged-current 4-fermionproduction was performed in the complex mass scheme, including loop correc-tions to singly- and non-resonant diagrams. 6) The DPA agrees well with thefull e − e + →
4f calculation for energies 200 GeV (cid:46) √ s (cid:46)
500 GeV while thefull calculation is required near threshold 160 GeV (cid:46) √ s (cid:46)
170 GeV, and for √ s >
500 GeV, where off-shell effects become important. In a further devel-opment, effective-field theory (EFT) methods have been used for a dedicatedalculation of four-fermion production near the W -pair production threshold. 9)This method has allowed to isolate and compute the subset of NNLO correc-tions that is enhanced near threshold due to Coulomb-photon effects. Combin-ing these dominant NNLO corrections, which are of the order of 0 . M W -measurementfrom a threshold scan to ∆ M W (cid:46) √ s (cid:38)
800 GeV, which are particularly relevantfor measurements of triple gauge couplings, higher-order EW corrections areenhanced by Sudakov logarithms. For W -pair production, NNLO correctionsdue to NNLL Sudakov logarithms α log m ( s/M W ) with m = 2 , , √ s = 1 TeV (3 TeV), sothey should be taken into account in the second phase of an ILC or at CLIC.In addition to these precision calculations, the search for indirect signalsof new physics requires a systematic treatment of deviations from the SM inan EFT framework, which has recently been applied to study the sensitivity toanomalous gauge boson couplings in W -pair production. 11) Theoretical methods for higher-order calculations have seen remarkable progressafter the LEP2 era. The theoretical uncertainty on charged-current four-fermion production has been reduced well below the percent level by a full NLOcalculation. 6) The extension of this calculation to the remaining processes ofFig. 1 will be simplified by recent progress on the automation of EW NLOcalculations 12) and may provide sufficient precision for future linear e − e + colliders, if supplemented with dominant NNLO effects in special kinematicregions 9 ,
10) and an improved treatment of initial-state radiation. The preci-sion goals of future circular e − e + colliders may require a full NNLO calculationof EW corrections, where the current state of the art is given by 1 → → e − e + colliderwould stimulate theoretical developments to meet these challenges. eferences
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