ZZ production at the LHC: fiducial cross sections and distributions in NNLO QCD

We consider QCD radiative corrections to the production of four charged leptons in the ZZ signal region at the LHC. We report on the complete calculation of the next-to-next-to-leading order (NNLO) corrections to this process in QCD perturbation theory. Numerical results are presented for $\sqrt{s}=8$ TeV, using typical selection cuts applied by the ATLAS and CMS collaborations. The NNLO corrections increase the NLO fiducial cross section by about $15\%$, and they have a relatively small impact on the shape of the considered kinematical distributions. In the case of the $\Delta\Phi$ distribution of the two Z candidates, the NNLO corrections improve the agreement of the theoretical prediction with the CMS data.

The production of Z-boson pairs at the Large Hadron Collider (LHC) provides an important test of the electroweak (EW) sector of the Standard Model (SM) at the TeV scale. Small deviations in the observed rates or in the kinematical distributions could be a signal of new physics, possibly in the form of anomalous couplings. At the same time, ZZ production is an irreducible background for Higgs boson production and new-physics searches. Particularly important are the off-shell effects below the ZZ threshold, relevant for the Higgs signal region in the four-lepton channel. Various measurements of ZZ hadroproduction have been carried out at the Tevatron and the LHC (for some recent results see Refs. [1][2][3][4][5][6][7]).
From the theory side the first NLO predictions for on-shell ZZ production were obtained long ago [8,9]. The leptonic decays of the Z bosons were included, initially neglecting spin correlations in the virtual contributions [10]. The computation of the relevant one-loop helicity amplitudes [11] enabled the first complete NLO calculations [12,13], including spin correlations and off-shell effects. The loop-induced gluon-fusion production channel, which formally contributes only at the nextto-next-to-leading order (NNLO), was computed in Refs. [14,15]. The corresponding leptonic decays were taken into account in Refs. [16][17][18]. NLO predictions for ZZ production including the gluon-induced contribution, the leptonic decays with spin correlations and off-shell effects were presented in Ref. [19]. The NLO QCD corrections to on-shell ZZ + jet production were discussed in Ref. [20,21], and the EW corrections to ZZ production in Ref. [22]. A decisive step forward was carried out in Ref. [23] where the inclusive NNLO cross section for on-shell ZZ production was presented. This calculation was based on the evaluation of the two-loop amplitude for on-shell ZZ production. Later, the two-loop helicity amplitudes for all the vector-boson pair production processes were presented [24,25]. This computation paves the way to the consistent inclusion of the leptonic decays and off-shell effects in the NNLO computation.
In this Letter we carry out this step by considering ZZ production at NNLO including the leptonic decays of the vector bosons together with spin correlations and off-shell effects. Contributions from Zγ * and γ * γ * production as well as from pp → Z/γ * → 4 leptons topologies are also consistently included with all interference terms. Our calculation allows us to apply arbitrary cuts on the final-state leptons and the associated QCD radiation. Here we present selected numerical results for pp → 4 leptons at the LHC in NNLO QCD, using the typical cuts that are applied in the experimental ZZ analyses.
Our calculation is performed with the numerical program Matrix † , which combines the q T subtraction [26] and resummation [27] formalisms with the Munich Monte Carlo framework [28]. Munich provides a fully automated implementation of the Catani-Seymour dipole subtraction method [29,30], an efficient phase-space integration, as well as an interface to the one-loop generator OpenLoops [31] to obtain all required (spin-and colour-correlated) tree-level and one-loop amplitudes. For the numerically stable evaluation of tensor integrals, OpenLoops relies on the Collier library [32], which is based on the Denner-Dittmaier reduction techniques [33,34] and the scalar integrals of [35]. To deal with problematic phase-space points, a rescue system is provided, which employs the quadruple-precision implementation of the OPP method in CutTools [36] and scalar integrals from OneLOop [37]. Our implementation of q T subtraction and resummation for the production of colourless final states is fully general, and it is based on the universality of the hard-collinear coefficients [38] appearing in transverse-momentum resummation. These coef- † Matrix is the abbreviation of "Munich Automates qT subtraction and Resummation to Integrate Cross Sections", by M. Grazzini, S. Kallweit, D. Rathlev, M. Wiesemann. In preparation.
We consider pp collisions with √ s = 8 TeV. As for the EW couplings, we use the so-called G µ scheme, where the input parameters are G F , m W , m Z . More precisely, consistent with the OpenLoops implementation, we use the complex W and Z boson masses to define the EW mixing angle as cos . In particular, we use the values  [47] sets of parton distributions with α S (m Z ) = 0.118, and the α S running is evaluated at each corresponding order (i.e., we use (n + 1)-loop α S at N n LO, with n = 0, 1, 2). We consider N f = 5 massless quark flavours. The central renormalization (µ R ) and factorization (µ F ) scales are set to µ R = µ F = m Z .
We first consider the ATLAS analysis of Ref.
[5] in the three decay channels e + e − e + e − , µ + µ − µ + µ − , and e + e − µ + µ − . The invariant masses of the two reconstructed lepton pairs are required to fullfil the condition 66 GeV ≤ m ll ≤ 116 GeV. In the case of two lepton pairs with the same flavours there is a pairing ambiguity, which is resolved by choosing the pairing that makes the sum of the absolute distances from m Z smaller. The leptons are required to have p T ≥ 7 GeV and rapidity |η| ≤ 2.7. For any lepton pair we require ∆R(l, l ′ ) > 0.2, independently of the flavours and charges of l and l ′ .
The corresponding cross sections are reported in Tab. 1, where the ATLAS results are also shown. The uncertainties on our theoretical predictions are obtained by varying the renormalization and factorization scales in the range 0.5m Z < µ R , µ F < 2m Z with the constraint 0.5 < µ F /µ R < 2. Independently of the leptonic decay channels, the NNLO corrections increase the NLO result by about 15%, similarly to what was found for the inclusive cross section for on-shell ZZ production [23]. This is as expected because the selection cuts are mild and do not significantly alter the impact of radiative corrections. The scale uncertainties are about ±3% at NLO and remain of the same order at NNLO. As noted for the inclusive cross section [23], the NLO scale uncertainty does not cover the NNLO effect. This is not surprizing since the loopinduced gluon-fusion contribution, which provides ∼ 60% of the O (α 2 S ) correction, opens up only at NNLO. The NNLO corrections improve the agreement of the theoretical prediction with the data for the e + e − µ + µ − channel, whereas they deteriorate the agreement in the case of the 4e and 4µ channels. We note, however, that the predicted fiducial cross sections are still consistent with the ATLAS measurements at the 1σ level within the statistics-dominated uncertainties.
Secondly, we consider the CMS analysis of Ref. [7]. The fiducial region is defined as follows: all muons are required to fulfill p µ T > 5 GeV, |η µ | < 2.4, while all electrons are required to fulfill p e T > 7 GeV, |η e | < 2.5. In addition, the leading-and subleading-lepton transverse momenta must satisfy p l,1 T > 20 GeV and p l,2 T > 10 GeV, respectively. In the case of two lepton pairs with the ‡ The Higgs boson contributes less than 1% to the loop-induced gg → ZZ cross section, whereas its effect on the real-virtual contribution is numerically negligible.  CMS does not report the fiducial cross sections corresponding to the above cuts, but only normalized distributions, to which we compare our results. We start with the invariant-mass distribution of the four leptons, which is depicted in Fig. 1. The lower panels show the theory/data comparison, and the NNLO result normalized to the central NLO prediction. We see that the NNLO corrections have a limited impact in the comparison with the data, which still have large uncertainties. The NNLO effects on the normalized distribution are relatively small: they are completely negligible at low invariant masses, and they increase to −5% in the high mass region. This means that the NNLO corrections make the invariant mass distribution slightly softer. We have checked that this effect is due to the gluon-fusion contribution, whose relative effect decreases at high masses, due to the larger values of Bjorken x that are probed. The NLO (NNLO) scale uncertainties range from about ±2% (±1%) at low m ZZ to ±4% (±2%) at high m ZZ . correspond to scale variations as described in the text.
In Fig. 2 we show the analogous results for the leading-lepton p T distribution (left) and the azimuthal separation (∆Φ) of the two Z candidates (right). As in Fig. 1, we see that the NNLO effects on the p T distribution do not change the comparison with the data in a significant way. The NNLO corrections are also relatively small in most of the range of p T considered, except for the low p T region, where they increase significantly. This effect is due to the gluon-fusion contribution, whose relative impact increases as p T decreases. The situation is different for the ∆Φ distribution. Here the NNLO corrections improve the agreement with the data, except for the first bin, where the CMS measurement is an order of magnitude below the theoretical NNLO prediction. The larger impact of NNLO corrections in the ∆Φ distribution can be understood easily by the observation that at LO the reconstructed Z bosons are always back-to-back, i.e., ∆Φ(Z 1 , Z 2 ) = π. As a consequence, the NNLO calculation is effectively NLO in the region 0 ≤ ∆Φ < π. The NNLO corrections amount to about +25% when ∆Φ ∼ < 1.5, and decrease as ∆Φ increases. We note that this effect is entirely due to the NNLO corrections to the qq channel addressed in this paper, since the loop-induced gluon-fusion contribution, which also enters at NNLO, affects the ∆Φ distribution only at ∆Φ = π. The NLO scale uncertainties are about ±11%, while at NNLO the uncertainties are about ±5% at low ∆Φ, and decrease to about ±2% at high ∆Φ.
We have presented the first complete NNLO QCD calculation for the production of four charged leptons in the ZZ signal region at the LHC. We have studied the impact of NNLO corrections on the fiducial cross sections and distributions measured by ATLAS and CMS at the LHC. As for the fiducial cross sections, we found about +15% NNLO corrections w.r.t. the NLO prediction, consistent with what was found for the inclusive cross section for on-shell ZZ production [23]. The impact on the normalized distributions we considered is small compared to the experimental uncertainties, but leads to an improved agreement with the data in the case of the ∆Φ distribution of the two Z candidates. Our calculation was performed with the numerical program Matrix, which is able to carry out fully exclusive NNLO computations for a wide class of processes at hadron colliders. We look forward to further applications of our framework to other important LHC processes.