# Next-to-leading order QCD corrections to \(W\gamma \) production in association with two jets

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## Abstract

The QCD-induced \(W^\pm \gamma \) production channels in association with two jets are computed at next-to-leading order QCD accuracy. The W bosons decay leptonically and full off-shell and finite width effects as well as spin correlations are taken into account. These processes are important backgrounds to beyond Standard Model physics searches and also relevant to test the nature of the quartic gauge couplings of the Standard Model. The next-to-leading order corrections reduce the scale uncertainty significantly and show a non-trivial phase space dependence. Our code will be publicly available as part of the parton level Monte Carlo program VBFNLO.

## Keywords

Charged Lepton Vector Boson Fusion Real Photon Lead Order Result Real Emission Contribution## 1 Introduction

Di-boson production in association with two jets constitutes an important set of processes at the LHC. They are backgrounds to many Standard Model (SM) searches. For example, W-, Z- and photon-pair production with two accompanying jets are irreducible backgrounds of Higgs production via vector boson fusion. Furthermore, they are sensitive to triple and quartic gauge couplings, thereby providing us with an excellent avenue to understand the electroweak (EW) sector of the SM and possibly to get hints of physics beyond the SM.

There are two mechanisms to produce them, namely, EW-induced channels of order \(\mathcal {O}\!\left( \alpha ^4\right) \) and QCD-induced processes of order \(\mathcal {O}\!\left( \alpha _s^2 \alpha ^2\right) \) for on-shell production at leading order (LO). Additionally, the EW mode is classified into “vector boson fusion” (VBF) mechanism, which involves \(t\) and \(u\) channel exchange, and \(s\) channel contributions corresponding mainly to \(VVV\) production with one \(V\) decaying into two jets.

The VBF production modes include vector boson scattering, \(VV\rightarrow VV\), as a basic topology. For massive gauge boson scattering, the main interest will be to elucidate whether the recently discovered Higgs boson unitarizes this process as predicted in the SM. Processes with a real photon in the final state are also interesting since they are sensitive to triple and quartic gauge couplings and have a higher cross section.

The next-to-leading order (NLO) QCD corrections to the VBF processes have been computed in Refs. [1, 2, 3, 4, 5] for all combinations of massive gauge bosons, including leptonic decays of the gauge bosons as well as all off-shell and finite width effects. A similar calculation with a \(W\) boson and a real photon in the final state has been done in Ref. [6]. For the \(s\) channel contributions, the NLO QCD corrections with leptonic decays were computed in Refs. [7, 8, 9, 10, 11, 12] and are available via the VBFNLO program [13, 14] (see also Refs. [15, 16, 17] for on-shell production and Ref. [18] for NLO EW corrections).

NLO QCD corrections to the QCD-induced processes have been computed for \(W^+W^+jj\) [19, 20, 21, 22], \(W^+W^-jj\) [23, 24], \(W^\pm Zjj\) [25], and \(\gamma \gamma jj\) [26] production. Results for \(\gamma \gamma jjj\) production at NLO QCD have been very recently presented in Ref. [27].

In this paper, we provide first results for the QCD-induced \(W\gamma jj\) production channel. The calculation is based on our previous implementation of NLO QCD corrections to \(WZjj\) production processes [22], where the off-shell photon contribution was included. The interference effects between the QCD and EW induced amplitudes are generally small for most applications [5, 21, 22] and are not considered here. Leptonic decays of the \(W\) boson as well as all off-shell effects are consistently taken into account. This includes also the radiative decay of the \(W\) with a real photon radiated off a charged lepton, which diminishes the sensitivity of the EW-induced \(W\gamma jj\) production mode to anomalous couplings. In this paper, we follow the approach of Ref. [28] (see also references therein) to reduce this contribution by imposing a cut on the transverse mass of the \(W\gamma \) system.

To define the \(W\gamma jj\) signature, since our study is done at the jet cross section level and fragmentation contributions are not taken into account, the real photon has to be isolated from the partons to avoid collinear singularities due to \(q \rightarrow q \gamma \) splittings. While a similar issue with the charged lepton can be resolved by imposing a simple cut on \(R_{l\gamma }=[(y_\gamma - y_l)^2 + (\phi _\gamma - \phi _l)^2]^{1/2}\) (\(y\) and \(\phi \) being the the rapidity and azimuthal angle, respectively) to separate the photon from the charged lepton, it cannot be applied to partons because doing so would also remove events with a soft gluon. These events are needed at NLO (or beyond) to cancel soft divergences in the virtual amplitudes. To solve this problem, we use the smooth cone isolation cut proposed by Frixione [29]. This approach preserves IR safety without the use of fragmentation functions and thereby allows us to focus on the physics of the hard photon.

The QCD-induced \(W\gamma jj\) production process has been implemented within the VBFNLO framework, a parton level Monte Carlo program which allows the definition of general acceptance cuts and distributions.

This paper is organized as follows: In the next section, the major points of our implementation will be provided. In Sect. 3 the setup used for the calculation and the numerical results for inclusive cross sections and various distributions will be given. Conclusions are presented in Sect. 4 and in the appendix results at the amplitude squared level for a random phase-space point are provided.

## 2 Calculational details

At NLO QCD, there are the virtual and the real corrections. We use dimensional regularization [30] to regularize the ultraviolet (UV) and infrared (IR) divergences and use an anticommuting prescription of \(\gamma _5\) [31]. The UV divergences of the virtual amplitude are removed by the renormalization of \(\alpha _s\). Both the virtual and the real corrections are infrared divergent. These divergences are canceled using the Catani–Seymour prescription [32] such that the virtual and real corrections become separately numerically integrable. As mentioned in the introduction, collinear singularities that result from a real photon emitted off a massless quark are eliminated using the *photon isolation cut* proposed by Frixione, which preserves the IR QCD cancelation and eliminates the need of introducing photon fragmentation functions. The real emission contribution includes, allowing for external bottom quarks, \(186\) subprocesses with six particles in the final state.

Our calculation has been carefully checked as follows. The present code is adapted from our previous implementation of the \(WVjj\) (\(V=Z,\gamma ^*\)) production processes [22], which has been crosschecked at the amplitude level by two independent calculations. The adaptation includes removing the \(Z\) contribution, disallowing the decay \(\gamma ^* \rightarrow l^+l^-\) and adding the radiative decay \(W^\pm \rightarrow l^{\pm }{{\bar{\nu }}_{l}} \gamma \). These trivial changes are universal and have been crosschecked. Moreover, the real emission contributions have been crosschecked against Sherpa [42, 43] and agreement at the per mill level was found. A non-trivial 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 and obtained full agreement at the amplitude level. The first implementation uses FeynArts-3.4 [44] and FormCalc-6.2 [45] to obtain the virtual amplitudes. The in-house library LoopInts is used to evaluate the scalar and tensor one-loop integrals.

In the following, we sketch the second implementation, which will be publicly available via the VBFNLO program and is the one used to obtain the numerical results presented in the next section. As customary in all VBFNLO calculations, the spinor-helicity formalism of Ref. [46] is used throughout the code.

The leptonic decays of the EW gauge bosons, which are common for all subprocesses, are calculated once for each phase-space point and stored. In addition we pre-calculate parts of Feynman diagrams that are common to the subprocesses of the real emission and use a caching system to compute Born amplitudes appearing in different dipole terms [32] only once.

For the virtual amplitudes, we use generic building blocks, computed with the in-house program described in Ref. [40], which include groups of loop corrections to Born topologies with a fixed number and a fixed order of external particles, i.e. all self-energy, triangle, box, pentagon, and hexagon corrections to a quark line with four attached gauge bosons are combined into a single routine. The scalar and tensor integrals are computed as described in Ref. [40].

The control of the numerical instabilities is done as customary in our calculations using Ward identities. By replacing a polarization vector with the corresponding momentum, one can build up identities relating \(N\)-point integrals to lower point integrals. This property is transferred to the building blocks as described in Ref. [40], providing an additional check of the correctness on the calculation of the virtual amplitudes. This procedure is possible because we factorize the color and EW couplings from the building blocks and assume the polarization vector of the external gauge bosons as an effective current without using special properties like transversality or on-shellness. These identities are called gauge tests and are checked for every phase space point with a small additional computing cost by using a cache system. If the gauge tests are true by less than \(2\) digits with double precision, the program recalculates the associated building blocks with quadruple precision and the point is discarded if the gauge tests still fail. After this step, the number of discarded points is statistically negligible for a typical calculation with the inclusive cuts specified in the next section. This strategy was also successfully applied in, e.g., Refs. [22, 25, 47, 48, 49]. With this method, we obtain the NLO inclusive cross section with statistical error of \(1\,\%\) in three hours on an Intel \(i5\)-\(3470\) computer with one core and using the compiler Intel-ifort version \(12.1.0\).

To obtain this level of speed, it is important to notice that there are two contributions dominating in two different phase space regions associated with the two decay modes of the \(W\) bosons, namely \(W^\pm \rightarrow l^{\pm }{{\bar{\nu }}_{l}}\) and \(W^\pm \rightarrow l^{\pm }{{\bar{\nu }}_{l}} \gamma \). This means that there are two different positions of the on-shell \(W\) pole in the phase space. For efficient Monte Carlo generation, we divide the phase space into two separate regions to consider these two possibilities and then sum the contributions to get the total result. The regions are generated as double EW boson production as well as \(W\) production with (approximately) on-shell \(W^+\rightarrow \ell ^+\nu _l\gamma \) (or \(W^-\rightarrow \ell ^-\bar{\nu }_l\gamma \)) three-body decay, respectively, and are chosen according to whether \(m(\ell ^+\nu _l\gamma )\) or \(m(\ell ^+\nu _l)\) is closer to \(M_W\).

## 3 Numerical results

The result depends on the factorization and renormalization scales since we only calculate at fixed order in perturbative QCD. Figure 3 shows, both for \(W^+ \gamma jj \) and \(W^- \gamma jj\) production, that the dependence of the cross section on the factorization and renormalization scale, which are set equal for simplicity, is significantly reduced when calculating the NLO QCD corrections. If we vary the two scales separately, a small dependence on \(\mu _F\) is observed, while the \(\mu _R\) dependence is similar to the behavior shown in Fig. 3.

The differential distributions are less sensitive at NLO to the scale variation than at LO and the relative scale uncertainty is equally distributed in the entire \(p_{T,j_1}\) and \(p_{T,\gamma }\) spectrum. The phase space shows a non-trivial dependence with \(K\)-factor s varying, for \(\mu = \mu _0\), from \(1.2\) to \(0.8\) for the \(p_{T}\) distribution of the hardest jet and from \(0.95\) to \(0.8\) for the transverse momenta of the photon in the ranges shown.

In the bottom panels, we observe a similar significant reduction of the scale uncertainties for the \(m_{jj}\) (left) and the \(\Delta y_\text {tags}\) (right) differential distributions, with the \(K\)-factor of the invariant mass distributions varying from about 0.9 to 1.2 at 2.4 TeV and with a fairly constant slope and the \(K\)-factor for the rapidity difference of the two leading tagging jets varying from 0.85 to 1.4 in the range showed.

The \(R_{l\gamma }\) distribution in Fig. 5 shows a sudden increase of the \(K\)-factor starting at \(\pi \), which correlates to the sudden fall of the differential cross section. This discontinuity in the slope can be explained as follows. The \(R\) separation is defined as \(R_{l\gamma } = [(\Delta y_{l\gamma })^2 + (\Delta \phi _{l\gamma })^2]^{1/2}\) where \(\Delta \phi _{l\gamma } \in [0,\pi ]\). For \(0 < R_{l\gamma } < \pi \), the dominant contribution comes from the \(\Delta y_{l\gamma } \approx 0\) region (see the \(\Delta y_{l\gamma }\) distribution in Fig. 5), and the behavior of the \(K\)-factor is given by the one of the \(\Delta \phi _{l\gamma }\) distribution also displayed in Fig. 5, which is rather flat. For \(R_{l\gamma } > \pi \), the rapidity separation must increase and the \(K\)-factor is similar to the one of the \(\Delta y_{l\gamma }\) distribution.

The above results for various differential distributions show that our default scale choice defined in Eq. (5) and the text can make the LO results quite similar to the NLO ones, with the difference being smaller than \(20\,\%\) in most cases. The exceptional cases are the distributions of \(\Delta y_\text {tags}\) (see Fig. 4) and \(\Delta y_{l\gamma }\) (see Fig. 5). Here we observe that the \(K\)-factor increases with large rapidity separations. This indicates that the default scale choice is too large at large rapidity separations, making the LO results too small. We have tried a different scale choice, using Eq. (5) with \(a=1/2\) and \(b=0\), and found that the NLO results, for the distributions shown, agree with the ones obtained with the default scale within \(10\,\%\), while the two scale choices at LO produce differences as large as a factor of 2 for the \(m_{jj}\) and \(\Delta y_\text {tags}\) distributions. We also found that the new scale choice makes the \(K\)-factor s decrease well below one with increasing invariant mass or rapidity separation of the two hardest jets.

## 4 Conclusions

In this paper, we have reported first results for \(W^\pm \gamma jj + X\) production at order \(\mathcal {O}\!\left( \alpha _s^3 \alpha ^3\right) \), including the leptonic decays, full off-shell and finite width effects as well as all spin correlations. The NLO QCD corrections to the total cross section are small but they exhibit non-trivial phase space dependencies, reaching up to \(40\,\%\), and lead to shape changes of the distributions. Hence, they should be taken into account for precise measurements at the LHC.

Our code will be publicly available as part of the VBFNLO program [13, 14], thereby further studies of the QCD corrections with different kinematic cuts can easily be done.

## Notes

### Acknowledgments

We acknowledge the support from the Deutsche Forschungsgemeinschaft via the Sonderforschungsbereich/Transregio SFB/TR-9 Computational Particle Physics. FC is funded by a Marie Curie fellowship (PIEF-GA-2011-298960) and partially by MINECO (FPA2011-23596) and by LHCPhenonet (PITN-GA-2010-264564). MK is supported by the Graduiertenkolleg 1694 “Elementarteilchenphysik bei höchster Energie und höchster Präzision”.

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