$Z\gamma$ production in association with two jets at next-to-leading order QCD

Next-to-leading order QCD corrections to the QCD-induced $pp \rightarrow l^+l^- \gamma j j +X$ and $pp \rightarrow \bar{\nu}_l \nu_l \gamma jj+X$ processes are presented. The latter is used to find an optimal cut to reduce the contribution of radiative photon emission off the charged leptons in the first channel. As expected, the scale uncertainties are significantly reduced at NLO and the QCD corrections are phase space dependent and important for precise measurements at the LHC.


I. INTRODUCTION
The production of a prompt photon in association with two charged leptons and two jets at the LHC is an attractive mechanism to study weak boson scattering, namely W + W − → γV with V = Z/γ * . It is also relevant to the study of anomalous gauge boson couplings, which may provide hints of new physics beyond the Standard Model (SM).
At leading order (LO), the process pp → jjγl + l − + X is classified into two mechanisms: the electroweakinduced mechanism of order O α 5 , which is sensitive to W + W − → γV scattering and the QCD-induced channel of order O α 2 s α 3 , which can be considered as a background. The EW contributions can be further classified into t-channel vector-boson fusion contributions known at NLO QCD [1] and other contributions including notably tri-boson production processes with one boson decaying hadronicaly. The NLO QCD corrections to triboson production with leptonic decays were computed in Refs. [2,3] and the hadronic decay modes are available via the VBFNLO program [4]. The interference effects between these contributions are expected to be negligible for most measurements at the LHC [5].
In this paper, we consider the QCD-induced mechanism for the processes pp → l + l − γjj + X, "Z l γjj" (1) pp →ν l ν l γjj + X, "Z ν γjj" (2) and will present the first theoretical prediction at NLO QCD accuracy 1 . Some representative Feynman diagrams at LO are displayed in Fig. 1. Since the dominant contribution comes from the phase space region where the intermediate Z boson is resonant, the above processes are usually referred to as Z l γjj and Z ν γjj production, * francisco.campanario@ific.uv.es † matthias.kerner@kit.edu ‡ duc.le@kit.edu § dieter.zeppenfeld@kit.edu 1 Very recently, in Ref. [6], results for the total cross section level for on-shell Zγjj production have been reported.
accounting for the charged-lepton and neutrino pair production processes, respectively. With this result, all the QCD-induced V V jj production processes are known at NLO QCD [5,[7][8][9][10][11][12][13][14][15][16]. The signature of an isolated photon together with two jets and missing energy is difficult to study in experiment but is, as will be shown later, useful in a Monte Carlo analysis to find (by comparing the two processes) an optimal cut on the invariant mass of the two-charged lepton and photon system to remove the radiative QED contribution (where the photon is emitted off the final charged leptons). This contribution is unwanted because it reduces sensitivity to the weak boson scattering. The focus of this paper is therefore on process (1), however, a comparison of normalized distributions to process (2) will be performed.
We have implemented the QCD-induced processes (1) and (2) within the VBFNLO framework [4], a parton level Monte Carlo program which allows the definition of general acceptance cuts and distributions. As customary in VBFNLO, all off-shell effects, virtual photon contributions and spin-correlation effects are fully taken into account. Our code will be included in the next release of VBFNLO.
In the next section we sketch the calculational setup and in Section III we define our physical observables with a set of cuts and present numerical results for the total cross section as well as various kinematical distributions. Conclusions are presented in Section IV. Values of the virtual amplitudes at a random phase space point are provided in the appendix to facilitate future comparisons with our results.

II. CALCULATIONAL SETUP
The calculational method of the present paper follows closely the one presented in Ref. [16] for the process pp → l + 1 l − 1 l + 2 l − 2 jj + X (called from now on ZZjj for simplicity). As explained there, the gauge invariant class of closed-quark loop diagrams with EW gauge bosons directly attached to the loop are discarded. This contribution is at the few per mille level, hence negligible for all phenomenological purposes. The diagrams with a closed-quark loop and two or three gluons attached to it are however included. We work in the five-flavor scheme and virtual top loops are taken into account. We use the Frixione isolation criteria [17] for the photon and therefore photon fragmentation functions are not included.
Technically, the code for the Z x γjj processes is adapted from the ZZjj code with some modifications. This is possible because we use the effective current approach and the spinor-helicity formalism [18,19] factorizing the leptonic tensor containing the EW information of the system from the QCD amplitdue. For the l + 1 l − 1 l + 2 l − 2 jj case, the generic amplitudes for V 1 V 2 jj with V i = Z/γ * (i = 1, 2) andV jj withV = Z/γ * are first created. Then the leptonic decays are incorporated via effective currents. In this way, all off-shell effects and spin correlation are fully taken into account. This approach also makes it straightforward to obtain the l + l − γjj andν l ν l γjj final states by picking the relevant generic amplitudes and changing the effective currents, namely, only the Zγjj generic amplitude and Z →ν l ν l effective current are needed for the neutrino channel. For the charged-lepton case, we use the V 1 γjj andV ′ jj generic amplitudes withV ′ → l + l − γ effective current. These trivial changes are universal and have been crosschecked. Additionally, the phase-space generator has to be modified for a fast convergence of the Monte-Carlo integration. For this purpose, it is important to notice that, for on-shell photon production, there are two contributions dominating in two different phase space regions associated with the two decay modes of the Z bosons, namely Z → l + l − and Z → l + l − γ. This means that there are two different positions of the on-shell Z pole in the phase space for the process (1). For efficient Monte Carlo generation, we divide the phase space into two separate regions to consider these two possibilities and then sum the two integrals to get the total result. The regions are generated as double EW boson production as well as Z production with (approximately) on-shell Z → l + l − γ three-body decay, respectively, and are chosen according to whether m(l + l − γ) or m(l + l − ) is closer to M Z . The virtual photon contribution, which is far off-shell, does not pose additional problems and is always calculated together with the corresponding Z contribution. Another nontrivial change arises in the virtual amplitudes where we have to calculate a new set of scalar integrals which do not occur in the off-shell photon case. We have again checked this with two independent calculations (as explained in Ref. [16]) and obtained full agreement at the amplitude level. Further details of our calculation and implementation and checks can be found in Ref. [16]. Furthemore, we have crosschecked the LO and real emission contributions without subtraction term against Sherpa [20,21] and agreement at the per mill level was found for integrated cross sections.
With this method, we obtain the NLO inclusive cross section with statistical error of 1% in 4 hours on an Intel i7-3970X computer with one core and using the compiler Intel-ifort version 12.1.0. The distributions shown below are based on multiprocessor runs with a total statistical error of 0.03% at NLO.

III. PHENOMENOLOGICAL RESULTS
For the numerical evaluation of the processes at the LHC operating at 14 TeV center-of-mass energy, we use the MSTW2008 parton distribution function [22] with α LO( NLO) s (M Z ) = 0.13939(0.12018) and the anti-k T cluster algorithm with a cone radius of R = 0.4. We consider jets with transverse momenta p T,j > 20 GeV and rapidity |y j | < 4.5. To simulate experimental detector capabilities, we require hard and central charged leptons with p T,l > 20 GeV and |y l | < 2.5 and photons with p T,γ > 30 GeV and |y γ | < 2.5. We impose minimal sep- To avoid the need of including photon fragmentation functions, we use the photon isolation criterià a la Frixione [17] with a cone radius of δ 0 = 0.7. Events are accepted if For the neutrinos of the "Z ν γjj " channel, we do not apply any cut.
Other input parameters are chosen as M Z = 91.1876 GeV, M W = 80.385 GeV and G F = 1.16637 × 10 −5 GeV −2 . The electromagnetic coupling constant and the weak-mixing angle are calculated via tree level relations. All fermions are taken to be massless, except the top quark with m t = 173.1 GeV. The width of the Z is fixed at Γ Z = 2.508905 GeV. The strong coupling constant is renormalized using the MS scheme. The topquark contribution is decoupled from the running, but is explicitly included in the one-loop amplitude. As a central factorization and renormalization scale, we use the sum of the transverse energy E T = (p 2 T + p 2 ) 1/2 of the two tagging jets and of the reconstructed Zγ system, The first term interpolates between m jj and p T,jets for large and small ∆y jj values, characterizing the dynamics of these processes appropriately.
In the following, we present results for the first generation of leptons. Taking into account both the electron and the muon yields an extra factor of two. Summing over three generations of neutrinos gives a factor of three.
To evaluate the scale uncertainties associated to a fixed order calculation, we plot in Fig. 2 the cross section for the "Z ν γjj " and "Z l γjj " channels varying the central scale in the range µ ∈ (10 −1 , 10)µ 0 simultaneously for the factorization and the renormalization scale, which are set equal for simplicity. At the central scale, we obtain σ LO = 500.82(3) +24% −18% fb and σ NLO = 510.6(1) +2.6% −5.3% fb for the e + e − γjj channel and σ LO = 765.58(3) +26% −19% fb and σ NLO = 840.8(3) +5.3% −7% fb for theν e ν e γjj one. The upper and lower numbers correspond to the scale uncertainties in percentage for variations of a factor 2 around the central scale and the number in brackets is the Monte-Carlo statistical error. At the central scale, we observed very mild K-factors, defined as the ratio of the NLO over LO predictions of the order of 1.01 and 1.1 for the "Z l γjj" and "Z ν γjj" channels, respectively.
Next we investigate the radiative photon emission off the charged leptons in the "Z l γjj " channel (see the middle Feynman diagrams of Fig. 1). These radiative decays present in both the EW and QCD induced processes reduce the sensitivity to anomalous-coupling searches and therefore it is desirable to suppress them. This contribution dominates in the phase-space region where the reconstructed invariant mass of the Zγ system is close to the Z mass. Thus, imposing a cut on M Zγ around the Z mass should remove this contribution. The optimum value of the cut is a priori uncertain. We therefore use the "Z ν γjj " channel, where radiative decays are absent, to determine it.
In the left panel of Fig. 3, as functions of the m Zγ cut, we plot the integrated NLO cross sections for the "Z l γjj" and "Z ν γjj" channels, the latter being multiplied by the ratio of the charge-lepton versus neutrino branching ratios of the Z, Br(Z → e + e − )/Br(Z →ν e ν e ) = 0.506. The LO cross section is also shown for the "Z l γjj" channel. In the bottom panel, the ratios of the modified neutrino cross sections to the LO and NLO electron cross sections are plotted. They do not converge to one in the tails due to the different cuts applied for the charged leptons and the neutrinos. As expected, for the chargedlepton case, one observes that the cross section sharply decreases when the cut value is greater than the Z mass. In the middle panel, the K-factors are plotted. We observe that m cut Zγ = 120 GeV is a good value since the Kfactor exhibits a plateau and the slope of the cross section curves are approximately equal for both processes beyond this value (see bottom panel). This is confirmed in the right panel of Fig. 3, where the normalized differential distributions of the reconstructed rapidity-azimuthal angle separation of the Zγ system are plotted for the two channels. One observes that the cut m Zγ > m Z + Γ Z reduces considerably the effect of the radiative decay in the charged-lepton channel, but some remnant is still clearly visible by comparing to the neutrino channel. Increasing the cut value to 120 GeV makes the NLO distribution of the "Z l γjj" channel very similar to the corresponding "Z ν γjj" one. This is better seen in the bottom panel, where the ratios of the normalized differential distributions between the two channels are plotted. The ratio of the m cut Zγ = 120 GeV curve versus the "Z ν γjj" distribution is rather flat and close to one till R Zγ reaches values of around 3 and then decreases. This difference is probably again due to the different cuts applied between the charged leptons and the neutrinos.
In the following, we impose an additional cut m Zγ > 120 GeV and plot some relevant differential distributions for the two tagging jets and the photon at LO and NLO in the large panels of Fig. 4. The tagging jets are defined as the two jets with highest transverse momenta and are ordered by hardness. The bands show the scale uncertainty in the range µ 0 /2 ≤ µ F = µ R ≤ 2µ 0 . The small panels always show the differential K-factors where the bands represents the scale variations of the NLO result, with respect to σ LO (µ 0 ). In the top row, the differential distributions of the transverse momentum of the two tagging jets (left) and the photon (right) are plotted. The bottom row displays the invariant mass (left) and the rapidity difference (right) of the two tagging jets. As expected, the scale uncertainty decreases considerably at NLO. Note that the rapidity-separation distribution receives large NLO QCD corrections in the region selected for vector boson fusion scattering, ∆y tags > 3. In general, the size of the K-factors range from 0.8 to 1.9, showing that NLO predictions are necessary for accurate measurements. Zγ cut. The middle and lower panels show the differential K-factor plots and the ratios of the normalized electron versus neutrino pair production channels.

IV. CONCLUSIONS
In this article, we have presented first results at NLO in QCD for the pp → l + l − γjj + X and pp →ν l ν l γjj + X processes. With this result, all the QCD-induced V V jj production processes are known at NLO QCD.
By comparing against the neutrino production process, we have been able to efficiently remove the contribution of radiative photon emission off the charged leptons, which diminishes the sensitivity of EW-induced processes to anomalous couplings. As expected, the scale uncertainty is significantly reduced at NLO, which is visible both at the total and differential cross section level. The size of the NLO QCD corrections are phase space dependent ranging from −20% to +90%, and are particularly large in the region where the vector-boson scattering signal is enhanced. NLO corrections are therefore needed for reliable predictions.  In the following, we present results at one phase-space point for the virtual amplitudes to facilitate comparison with future calculations. We chose the same phase-space momentum configuration as Ref. [14], Table I, and give results for the process j 1 j 2 → j 3 j 4 e + e − γ at the squaredamplitude level, averaging over the initial-state helicities and colors.
We include all UV counterterms and all closed-quark loops with gluons attached to it. Diagrams including a closed-quark loop with the Z/γ * directly attached to it are excluded. The top quark is decoupled from the running of α s . However, its contribution is explicitly included in the one-loop amplitudes. Here we use α = α s = 1 for simplicity. With this set up and measur- The bands show the scale variations around the central scale, µ 0 /2 ≤ µ F = µ R ≤ 2µ 0 . In the small panels, the differential K-factors are plotted. The bands reflect the NLO scale variations with respect to σ LO (µ 0 ). The inclusive cuts described in the text are used together with the cut on the invariant mass of the Zγ system, m cut Zγ = 120 GeV, which eliminates the final radiative emission off the charged leptons.
ing energies in GeV, we get at tree level,   For the one-loop integrals, we use the convention with D = 4 − 2ǫ. Additionally, the conventional dimensional regularization method [23] with µ R = M Z is used. With this, the interference amplitudes 2Re(A NLO A * LO ), for the one-loop corrections and the I-operator contribution as defined in Ref. [24], are given in Table II. Switching from the conventional dimensional regular-ization to dimensional reduction method induces a finite shift, which can be calculated noting that the sum |A LO | 2 + 2Re(A NLO A * LO ) should be constant [25]. The shift on the Born amplitude squared comes from the change in the strong coupling constant, see e.g. Ref. [26], Finally, using the rule given in Ref. [24], the shift on the I-operator contribution can be calculated.