QCD Corrections in SMEFT Fits to $WZ$ and $WW$ Production

We investigate the role of anomalous gauge boson and fermion couplings on the production of $WZ$ and $W^+W^-$ pairs at the LHC to NLO QCD in the Standard Model effective field theory, including dimension-6 operators. Our results are implemented in a publicly available version of the POWHEG-BOX. We combine our $WZ$ results in the leptonic final state $e\nu \mu^+\mu^-$ with previous $W^+W^-$ results to demonstrate the numerical effects of NLO QCD corrections on the limits on effective couplings derived from ATLAS and CMS 8 and 13 TeV differential measurements. Our study demonstrates the importance of including NLO QCD SMEFT corrections in the $WZ$ analysis, while the effects on $WW$ production are smaller. We also show that the $\mathcal{O}(1/\Lambda^4)$ contributions dominate the analysis, where $\Lambda$ is the high energy scale associated with the SMEFT.


I. INTRODUCTION
The properties of the Standard Model (SM) have been experimentally verified at the LHC at the O(10 − 20%) level in the Higgs sector [1] and there is no evidence for the existence of any new particles or interactions at the TeV scale yet. High statistics measurements of gauge boson pair production allow for detailed comparisons with Standard Model predictions and can be used to quantify the restrictions on anomalous interactions.
Gauge boson pair production is particularly sensitive to new 3-gauge boson interactions [2] or new fermion-boson interactions [3]. The current task is to make comparisons between theory and data at the few percent level which requires not only high-luminosity LHC running, but also improved theoretical calculations.
The SM rates for both W + W − and W Z production are well known. QCD corrections to W Z production in the Standard Model have been computed to next-to-leading order (NLO) for on-shell production [4,5] and to NNLO for both on-and off-shell production [6,7]. SM electroweak corrections to the W Z process [8][9][10][11][12] are also known at NLO and can have significant effects in the high p T regime. W + W − pair production is also under good theoretical control in the SM: NNLO QCD [13][14][15] and NLO electroweak [8,9,16] corrections are understood and change the distributions and rates significantly.
Gauge boson pair production can be put under the microscope using an effective Lagrangian, where the new physics is parameterized as an operator expansion in inverse powers of a high scale Λ and the assumption is made that there are no light degrees of freedom.
The operators O operators [17,18] when flavor effects are neglected. We compute the amplitudes for W + W − and W Z pair production including the dimension-6 operators, and then consider results when the cross sections are consistently expanded to both 1/Λ 2 and 1/Λ 4 .
The leptonic decay channel of W + W − pair production has been studied at NLO in the SMEFT in a previous work [19]. Here we extend those results to include the leptonic decays from W Z pair production at the LHC in the presence of anomalous 3-gauge boson and anomalous fermion-gauge boson couplings. QCD effects can affect the dependence of the kinematic distributions on the coefficients of Eq. (1). We include anomalous 3gauge boson couplings and anomalous fermion-gauge boson couplings in the POWHEG-BOX to NLO QCD in the SMEFT approach [20][21][22][23] following previous implementations for the SMEFT 3-gauge boson couplings case [24,25]. This public tool can be found at http://powhegbox.mib.infn.it.
Limits on SMEFT coefficients have been obtained in global fits that include gauge boson pair production, Higgs measurements, electroweak precision measurements, and top quark measurements [26][27][28][29][30]. The SMEFT effects are treated at tree level in these fits, while the SM results include all known higher order SM predictions. Fits attempting to use full NLO electroweak SMEFT predictions quickly observe that the plethora of operators makes such fits problematic [31,32]. On the other hand, the inclusion of NLO QCD SMEFT effects is simpler, due to the smaller number of operators involved [19,33,34].
In Section II, we define our notation in terms of anomalous couplings and present some calculational details. Section III contains a sampling of kinematic distributions with benchmark values of the anomalous couplings and Section IV has the results of a numerical fit to W + W − and W Z data. The NLO SMEFT QCD corrections have a numerically significant effect on many of the results. We point out that fits to O(1/Λ 2 ) or to O(1/Λ 4 ) result in quite different limits on the SMEFT coefficients. We conclude in Section V. We also provide fits using W + W − data only in Appendix A, while a discussion about the truncation at order O(1/Λ 2 ) is carried in Appendix B.

A. Effective Gauge and Fermion Interactions
We begin by reviewing the most general CP and Lorentz invariant Lagrangian for anomalous W + W − Z and W + W − γ couplings [2,35], with V = γ, Z, g W W γ = e, g W W Z = g cos θ W , (s W ≡ sin θ W , c W ≡ cos θ W ). The anomalous couplings are defined as g V and gauge invariance implies δg γ 1 = 0. The effective couplings of quarks to gauge fields are 1 [3,19,31,36], Here, g Z = e/(c W s W ) = g/c W and q is an up-or down-flavor quark. The SM quark interactions are: where T q 3 = ± 1 2 and Q q is the electric charge.
This framework leads to 7 unknown parameters, δg Z 1 , δκ Z , λ Z , δg Zu L , δg Zd L , δg Zu R and δg Zd R , contributing to W + W − production. The anomalous right-handed couplings do not contribute to W Z production, hence reducing the number of unknown parameters down to 5. These parameters are O(1/Λ 2 ) in the SMEFT language. The conversion between the effective Lagrangians of Eqs. 2 and 3 and the dimension-6 interactions in the Warsaw basis can be found in many places [3,19,37] and there is a one-to-one mapping between the two approaches 2 .
It is of interest to study the high energy limits of the helicity amplitudes for W + W − and W Z scattering in order to understand generic features of our results. In the high energy limit (s M 2 Z ), only the longitudinal (00) and transverse (±∓) helicity amplitudes 1 We assume no new tensor structures and neglect CKM mixing and all flavor effects. We assume SM gauge couplings to leptons, since these couplings are highly restricted by LEP data. We further neglect possible anomalous right-handed W -quark couplings, since they are suppressed by small Yukawa couplings in an MFV framework and stringently limited by Tevatron and LHC measurements 2 See for example, Tables 4 and 5 of Ref. [19]. remain non-zero in the SM W Z amplitudes (where s is the partonic center of mass energysquared) [38], where θ is the center of mass angle of the W boson with respect to the up quark direction and g Zu L − g Zd L = c 2 W . The radiation zero in the high energy (±, ∓) amplitude at cos θ 0 = (g Zu L + g Zd L )/(g Zu L − g Zd L ) is clearly seen in Eq. 6. The SMEFT contributions to W + Z production that contribute interference effects with the SM in the high energy limit are [28,38], Note that in the high energy limit, s M 2 Z , the dependence on δκ Z is suppressed and that the energy enhanced longitudinal amplitude peaks at θ = π 2 . Only the longitudinal modes have an energy enhanced interference contribution with the SM. The approximate zero of the SM (±∓) amplitude is weakened in the high energy limit where contributions from the anomalous fermion couplings fill in the dip at cos θ 0 .
The complete helicity amplitudes for W + W − production can be found in [2,33]. The energy enhanced amplitudes for q L,R q L,R → W + W − are, Due to the Goldstone boson nature of the longitudinal modes, the amplitudes of Eqs. 7 and 8 satisfy This implies that the high energy limits of W V production (V=W,Z) are only sensitive to 4 combinations of coefficients, and the dependence on other parameters is suppressed by powers of [39][40][41][42]. The amplitudes for W + W − and W Z production can be schematically written as, The cross section is then, In the results of this paper, we implicitly assume that the 1/Λ 4 contributions from the dimension-8 operators can be neglected, so that terms of O(1/Λ 4 ) in the cross section can be kept in a consistent way. This is the case, for example, in strongly interacting models [43,44]. We present an analysis of the truncation at O(1/Λ 2 ) in the Appendix B, to check whether or not the O(1/Λ 4 ) terms are important.

B. Primitive Cross Sections
We want to compute differential and total cross sections for the W Z scattering process at NLO QCD for arbitrary anomalous couplings with kinematic cuts mimicing the experimental analyses. The current calculation uses identical techniques as in Ref. [19]. The decomposition into primitive cross sections works at both lowest order (LO) and NLO and there are 15 primitive cross sections for the W Z process and 35 for the W + W − process at O(Λ −4 ).

C. Calculational Details
We have implemented the process pp → W Z → (l + l − )(l ν ) into the POWHEG-BOX-V2 including anomalous fermion and gauge boson couplings. The existing implementation [24] does not allow for anomalous fermion couplings. Our new implementation allows the user to chose the order of the Λ −2n expansion and to use either the effective Lagrangians described in this work or the Warsaw basis coefficients. Note that we assume different flavor leptonic decays. The results shown in the following sections use CTEQ14qed PDFs and we fix the renormalization/factorization scales at M Z /2.

III. NLO EFFECTS IN W Z DISTRIBUTIONS
We now present distributions for various kinematic variables at LO and NLO with different values of the anomalous couplings using the methods described in the previous section. In addition to the Standard Model, we present results for two benchmark points: Both of these points are near the boundaries of the allowed regions from fits to W + W − and W Z production and serve to illustrate the effects of anomalous couplings on the NLO QCD corrections.
In Fig. 1, we show the distributions in bins of m W Z T and p Z T along with the corresponding ratios of the NLO and LO predictions using the cuts from Ref. [45], where In the right panel we see that at high p T,Z the K factor 3 for the SM becomes very large as a result of real emission effects that arise at NLO, in agreement with Refs. [12,19].
In contrast, the K factor grows only modestly as a function of m W Z T . For the anomalous coupling benchmarks, we see that the K factor can change quite dramatically, particularly in the higher-momentum bins [22]. In the last p Z T bin in particular, the K factor changes from ∼ 7.8 in the SM to roughly 4.5 for our "Gauge" benchmark, and ∼ 9.4 for the "Fermion" point. Similar, but less dramatic, effects are seen in m W Z T as well.
The results of Fig. 1 clearly demonstrate that using the Standard Model K factor in an analysis of anomalous couplings in W Z production is inaccurate at large p Z T . As the high transverse momentum bins provide most of the constraining power for fits to the anomalous couplings, this can drastically change the resulting limits on the anomalous coefficients, as we demonstrate in the following sections.
We next consider the NLO effects on distributions of the angular variables cos θ * W and φ * W . They are the angular variables of the decayed charged lepton in the W rest frame. We use the helicity coordinate system as defined by ATLAS [45], in which the z direction of the W rest frame is the W direction-of-flight as seen in the W Z center-of-mass frame.
The definitions of the x and y axes are given in Ref. [46] and a graphical representation is given in Ref. [11] (with a slight modification for the z direction). Angular variables in the decay products (particularly cos θ * W ) are useful for extracting maximal sensitivity to the gauge boson polarizations [47]. As emphasized in Refs. [40,41,47], SMEFT effects lead to quadratic energy growth at the interference level only in the amplitude for producing two longitudinally polarized gauge bosons, Eq. 7. This behavior was exploited in Ref. [41] to maximize the sensitivity of the p T,V distribution to anomalous couplings.
In Figs. 2 and 3 we show the normalized distributions of cos θ * W and φ * W for the SM and for our two benchmark points at the fiducial level (left) and with an additional cut requiring p T,Z > 400 GeV (right) to enhance the sensitivity of the distributions to At NLO however, a great deal of this dependence is washed out as a result of the high p T bins being more densely populated due to the real emission present at this order [41,48]. While Ref. [41] suggested this could be ameliorated with a jet veto, it was also demonstrated in Ref. [48] that a hard jet in the process is required to maintain access to the interference terms which grow quadratically with energy and are most sensitive to the SMEFT effects.

IV. FITS
The results of Section III demonstrate that including higher-order QCD effects in W Z production in the presence of anomalous gauge and fermion couplings can lead to significantly different predictions than using the LO SMEFT calculation with the Standard Model K-factor. We now consider how these effects change the observed limits on the anomalous couplings based on a fit to experimental data. We consider the results in the case of a fit to only W Z data, and then, as a step towards a global analysis, fit both W + W − and W Z data.
The existing experimental results on W + W − and W Z production at both 8 and 13 TeV are summarized in Table I  To perform the fits, we construct a χ 2 function with the data from the remaining six data sets from Refs. [45,[52][53][54][55], using the distributions indicated in Table I

A. Fits to W Z Data
We first present the fits to the 8 and 13 TeV W Z data from ATLAS and CMS [45,[53][54][55]]. In Fig. 4 we show the 95% C.L. allowed regions from various two parameter fits to the anomalous couplings, in each case fixing the other three couplings to zero. As anticipated in Section III, the constraints using the LO and NLO predictions for the SMEFT contributions are quite different. The constraints on the different combinations of gauge couplings are weaker, in some directions by a factor of two. This is consistent with the behavior of the distributions with our "Gauge" benchmark point in Fig. 1. The effect is somewhat less dramatic in the case of anomalous fermion couplings, but there is still a large difference between the limits at LO and NLO.
In Section II A, we noted that the helicity amplitudes for W Z production had only a sub-leading (in s/M 2 Z ) dependence on δκ Z in the high energy limit. Measurements of W Z production are thus much less sensitive to δκ Z , and we see in Fig. 4 that the limits on δκ Z are indeed an order of magnitude weaker than those on δg Z 1 and λ Z . We also note that there is a near flat direction in the δg Z 1 -δg Zu L plane, and an even more robust flat direction in the δg Zu L -δg Zd L plane, in agreement with the scalings in Eq. 7.

B. Combined Fits to W + W − and W Z Data
In the previous section, it was demonstrated that treating the SMEFT consistently at NLO significantly changes the anomalous coupling constraints using W Z data only. On the other hand, in Ref. [19], it was shown that the NLO effects on W + W + distributions in the presence of anomalous couplings are relatively mild -in other words, using the K-factor derived at the SM is an adequate approximation for setting limits. It is of interest to understand to what extent the significant changes between LO and NLO fits in Fig. 4 remain when including W + W − data. Note that this is only a first step: the anomalous couplings are also constrained by other measurements both in Higgs data, top quark physics and at LEP.
In Fig. 5, we consider the results for various combinations of couplings with the same setup as in Fig. 4, with the other anomalous couplings fixed to zero. The most obvious result is that, even when combined with W + W − data, the effects of treating the SMEFT In each panel we set the three couplings not shown to zero.

13
FIG. 5: As in Fig. 4, but using both W W and W Z data.
at NLO in W Z and W + W − production on the limits are still quite substantial in many directions in parameter space. The first panel is clearly mostly constrained by W Z data.
As discussed in Subsection IV A, δκ Z is much better constrained when W + W − data is included, and since the NLO effects in W + W − production are very small, the limits in the δκ Z − λ Z plane (with all other couplings fixed to zero) are quite similar at LO and NLO. In the δg Z 1 -δκ Z plane, however, there is a flat direction in W + W − production in the high energy limit (Eq. 8), which is broken by the W Z data and the NLO effects are significant.
We have computed the rates up to quadratic order, O(1/Λ 4 ). This has a theoretical complication, however, because in principle dimension-8 operators may contribute at the same order in 1/Λ 4 in the most general EFT framework and these effects are not con- We consider the effects of marginalizing over the operators not shown in each plot. In practice, this is done by minimizing the χ 2 function at each point with respect to the other five couplings. In Fig. 6, we show the limits for the three combinations of anomalous gauge couplings, and compare the effects of profiling over the other five anomalous couplings in black (blue) for LO (NLO) with the results when profiling over only the last gauge coupling in red (green) for LO (NLO). In both cases, the effects of considering the anomalous couplings at NLO weaken the bounds on λ Z . The limits on δg Z 1 in the δg Z 1 -δκ Z plane are also affected, though the effect is more prominent when profiling only over the gauge couplings. We also see the result, anticipated in Refs. [3,33,56], that the limits on the anomalous gauge couplings are generally much weaker when the fermion couplings are allowed to float within their allowed regions. This is only not true in the λ Z direction, as the introduction of λ Z leads to a fundamentally different scaling at high energies for the production of transversely polarized W Z (see Eq. 7).
In Fig. 7, we show the constraints in various planes including anomalous fermion couplings, profiling over all five additional parameters. The NLO effects are again apparent, particularly in removing the remaining correlation between δg Zu L and δg Zd L . Finally, we summarize our results in the form of one parameter limits on each of the anomalous couplings considered in Table II. 15

C. Validity of our Results
The EFT Lagrangian of Eq. 1 is an expansion in powers of (Energy) 2 /Λ 2 and so is only valid for energies less than the scale Λ [44,57]. We consider in Fig. 8  In this case, the fit is degraded by O(10%) on δκ Z and O(30%) for λ Z and δg Z 1 , and the NLO QCD effects remain important. It would be interesting to have experimental results where the overflow bin is explicitly separated, so that the maximum energy of the data points in the last bin is clear. The result that interesting limits can be obtained even disregarding the highest-energy bin has been demonstrated using machine learning for the case of W H production in Ref. [58].

V. CONCLUSIONS
The SMEFT NLO QCD calculation for pp → W Z → (l ν )(l + l − ) has been included in the POWHEG-BOX and the primitive cross sections needed to reproduce our results at 8 and  ysis, as all of the couplings -especially the fermionic ones -will be further constrained, and some flat directions removed, by data from LEP and Higgs measurements. In this Appendix we present constraints on the anomalous gauge and fermion couplings based on only the W + W − data from ATLAS detailed in Table I. In Fig. 9, we show the two dimensional limits setting the other anomalous couplings to zero. It is apparent that the NLO QCD effects do not have a significant impact on fits to the W + W − data alone.