NLO Effects in EFT Fits to $W^+W^-$ Production at the LHC

We study the impact of anomalous gauge boson and fermion couplings on the production of $W^+W^-$ pairs at potential future LHC upgrades and estimate the sensitivity at $\sqrt{S}=14$ TeV with $3~ab^{-1}$ and $\sqrt{S}=27$ TeV with $15~ab^{-1}$. A general technique for including NLO QCD effects in effective field theory (EFT) fits to kinematic distributions is presented, and numerical results are given for $\sqrt{S}=13$ TeV $W^+W^-$ production. Our method allows fits to anomalous couplings at NLO accuracy in any EFT basis and has been implemented in a publicly available version of the POWHEG-BOX. Analytic expressions for the $K$-factors relevant for $13$ TeV total cross sections are given for the HISZ and Warsaw EFT bases and differential $K$-factors can be obtained using the supplemental material. Our study demonstrates the necessity of including anomalous $Z$- fermion couplings in the extraction of limits on anomalous 3-gauge-boson couplings.


I. INTRODUCTION
With the discovery of the Higgs boson, and subsequent measurements of its properties at the LHC, the general features of the Standard Model (SM) electroweak theory have been confirmed experimentally [1]. Measurements of the Higgs couplings and production rates agree with SM predictions at the 10 − 20% level [2][3][4][5][6] and there is no indication of the existence of any new particle at the TeV scale. Going forward, the task is to make comparisons between theory and data at the few percent level. This requires not only high-luminosity LHC running, but also improved theoretical calculations. Furthermore, Higgs physics cannot be studied in isolation, but must be examined in the context of the entire set of SM interactions.
W + W − pair production is an example of a process whose properties are highly restricted by LEP measurements [7], yet still provides relevant information about the Higgs sector from LHC data. The production of W + W − pairs provides a sensitive test of the electroweak gauge structure, since in the SM there are delicate cancellations between contributions from s−channel γ and Z exchange and t− channel fermion exchange that maintain perturbative unitarity. Deviations from the form of the γW + W − and ZW + W − vertices predicted by the SM spoil the cancellations between contributions that enforce unitarity. These deviations have been studied for decades [8,9], while the importance of non-SM fermion-Z interactions in ensuring unitarity in W + W − pair production has only recently been realized [10][11][12][13].
Beyond the Standard Model (BSM) physics effects in W + W − production can be studied using effective Lagrangian techniques where the new physics is parameterized as an operator expansion in inverse powers of a high scale, Λ, where O  [14,15], but only a small subset of these contribute to W + W − production. At high energies, the longitudinally polarized contributions to W + W − grow with energy faster than the SM contributions in the presence of BSM physics. This implies that the LHC can put strong constraints on the coefficients of the dimension-6 SMEFT operators.
We consistently work with a dimension-6 Lagrangian. At dimension-8, there are many other possible operators, not only modifying the triple-gauge boson interactions but also new 4−point ggW + W − interactions [16]. These operators contribute at tree level at order O(1/Λ 4 ) in the EFT and must be included in a study of the dimension-8 Lagrangian.
However, the study of dimension-8 operators is beyond the scope of this work, so that we do not consider operators modifying the partonic cross section gg → W + W − .
The effects of new physics contributions to W + W − gauge boson pair production can be expected to be of the same order of magnitude as QCD corrections, and so these contributions must be included when extracting limits on new physics. QCD effects in the effective field theory can also change the dependence of the experimental kinematic distributions on the coefficients of Eq. (1). The SM QCD corrections to W + W − pair production are known up to NNLO [17,18], including the effects of a jet veto [19,20], and the electroweak corrections exist at NLO [21][22][23]. The SM and dimension-6 gg initial state contributions are formally NNLO and are not included, although at 14 TeV they increase the cross section by roughly 10% (see e.g. refs. [24,25]). We perform an analysis including QCD corrections [26,27] at NLO in the SMEFT, along with modifications of both the 3-gauge-boson and fermion couplings, extending our previous study [11] by including the leptonic decays of the W 's. The effects of anomalous 3-gauge boson couplings exist in the POWHEG-BOX framework [28,29], and we add the additional contributions from anomalous fermion couplings. This public tool can be found at http://powhegbox.mib.infn.it and can be used to perform fits to anomalous couplings including NLO QCD and showering effects.
In Section II, we review the basics of the effective field theory framework for W pair production and discuss the effects of NLO QCD in the SMEFT. The determination of NLO QCD effects in a theory with anomalous couplings has typically been done on a case by case basis, or alternatively by allowing one SMEFT coupling at a time to vary. In Section II B, we present a general method for deriving NLO expressions for the total cross section and for distributions in an SMEFT in terms of a fixed number of sub-amplitudes, which we term "primitive cross sections". These results can be used to obtain either total or differential K-factors in any SMEFT basis. In Section III, we compare projections for the measurements of anomalous 3-gauge-boson couplings at the high-luminosity LHC with those of a future 27 TeV collider and demonstrate the critical importance of including anomalous fermion couplings in the fits (see also ref. [30] for SMEFT projections in a global fit at a 27 TeV hadron collider). We provide numerical results in Section IV for K-factors for the leading lepton p T (p ,lead T ) and m ll distributions for W + W − → e ± µ ∓ νν at 13 TeV as an illustration of our technique, along with analytic results for the total cross section, as functions of arbitrary EFT coefficients.

A. Effective Gauge and Fermion Interactions
Assuming CP conservation, the most general Lorentz invariant 3−gauge boson couplings are [9,31], where V = γ, Z, g W W γ = e and g W W Z = g cos θ W , with θ W being the weak mixing angle, (s W ≡ sin θ W , c W ≡ cos θ W ). The fields in Eq.(2) are the canonically normalized mass eigenstate fields. We define g V 1 = 1 + δg V 1 , κ V = 1 + δκ V and in the SM δg V 1 = δκ V = λ V = 0. Gauge invariance requires δg γ 1 = 0. The effective couplings of quarks to gauge fields can be written as 1 , (assuming no new tensor structures), g Z = e/(c W s W ) = g/c W , Q q is the electric charge of the quarks, and q denotes up-type or down-type quarks. We assume the anomalous fermion couplings, δg Zq L,R , along with the anomalous W -fermion couplings are flavor independent. We also neglect CKM mixing.
The SM quark couplings are: 1 We assume SM gauge couplings to leptons, since these couplings are highly restricted by LEP data.
At high energy scales the dominant contributions to W + W − come from longitudinally polarized W s. Keeping only the terms linear in the anomalous couplings, the amplitudes where r, r , λ, λ label the respective particle helicities, have the high energy limits [9,[11][12][13][31][32][33], where √ s is the partonic sub-energy. From Eq. (1). For future convenience, we consider the mapping to the Warsaw basis [15,34] and the HISZ basis [9]. In the Warsaw basis [15] , the dimension-6 operators relevant for our analysis are, Warsaw Basis  [15] and HISZ [9]. δv is given in Table II. where f can be either a quark or a lepton, Φ stands for the Higgs doublet field with a vacuum expectation value Φ = In the HISZ basis, the fermion couplings are unchanged, while the 3-gauge-boson couplings are, whereŴ µν = i g 2 σ a W a,µν andB µν = i g 2 B µν . Expressions for the anomalous 3-gauge-boson couplings are given in Table I and for the anomalous fermion couplings in Table II. 3

B. Primitive Cross Sections
We want to compute differential and total cross sections for a hadronic scattering process at NLO QCD for arbitrary anomalous couplings. Since these calculations can be numerically intensive, it is desirable not to have to repeat the calculation over and over again for different values of the anomalous couplings. Here we discuss a technique for generating results in terms of a set of primitive cross sections which need to be calculated only once for a given process and set of cuts. The primitive cross sections can be reweighted to allow for rapid scans over the anomalous couplings at NLO order.
Consider an arbitrary differential cross section dσ n ( C) that is calculated to O(Λ −2n ).
It depends on m EFT coefficients C = (C 1 , C 2 , . . . , C m ) and the relevant momenta, p.
We assume C i ∼ O(Λ −2 ). It is important to note that in general dσ 4 is the cross section to order O(Λ −4 ), but is not the amplitude-squared. When both Z− fermion and three gauge boson (3GB) couplings are non-zero, the amplitude-squared contains terms up to Calculating the cross section to O(Λ −2 ), where R i are m-dimensional vectors with R 1 = (1, 0, 0.....0), R 2 = (0, 1, 0....), R m = (0, 0....1), etc. The primitive cross section dσ(n; R i ) is the cross section obtained to arbitrary order O(Λ −2n ) when C i = 1 and all other C j = 0, j = i: The SM cross section with C = 0 is dσ SM . For the process pp → W + W − under con- Evaluating the cross section with 2 non-zero C i coefficients, C i = 1, C j = 1, and C k = 0 if j = i or k = , to arbitrary order O(Λ −2n ) yields the primitive cross sections We can calculate the primitive cross sections, dσ(n; R i ) and dσ(n; M ij ) once and then apply Eq.(11) to get the general result for arbitrary anomalous couplings. Eqs. (9) and The unprimed primitive cross sections are defined according to Eq.(10). To order O(Λ −2n ) the primed primitive cross sections dσ (n, R i ) are defined with the vector R i evaluated relative to the new basis C dσ (n; We need the dσ 's in terms of the already computed dσ's so that we can avoid recalculating the cross sections. Including a complete basis of dimension-6 operators, the input parameters are related by a linear transformation, where α is a 2 × 2 matrix. The dσ matrices are: Now consider the general case of a change of EFT input basis. Assume we have two minimum sets 4 of independent parameters C = (C 1 , C 2 , · · · , C m ) and C = (C 1 , C 2 , · · · , C m ) that can be related linearly, where α −1 is the inverse matrix of α and α −1 ij is its {i, j} th element. Although the two parameter bases C i , C i must give identical results for physical quantities and, 5 the primitive cross sections are not the same since setting C i = 1 is not the same as taking , we define the primitive cross sections with two non-zero C j in the C bases to be where M ij evaluated relative to the C basis. The primitive cross sections dσ(n; R i ), dσ(n; M ij ), and dσ (n; R j ) are defined in Eqs.(10), (12), and (14), respectively.
The cross sections can now be expanded in terms of either set of parameters and primitive cross sections. At O(Λ −2 ), we have the cross sections 6 where σ SM is the SM cross section with C = C = 0. At order O(Λ −4 ), we have Finally, we find the relationships between the primitive cross sections using Eqs. (18,21,22) that can be used to calculate cross sections in terms of an arbitrary EFT basis: or equivalently, The above results are found using which simplifies for i = j We illustrate the procedure and the utility of the results in this section by transforming from the Warsaw to HISZ basis in Sec. IV.

III. FUTURE PROJECTIONS
In this section, we apply the results of the previous sections to projecting allowed of the NLO QCD corrected predictions for the process pp → W + W − → 4 [28,29] 7 , that contains only the 3-gauge-boson anomalous couplings, as originally found in Refs. [26,27] and implemented also in MCFM [38]. In our implementation in the POWHEG-BOX-V2, we also include the anomalous Z-fermion couplings, there is the option to choose the order of the Λ −2n expansion and the results can be generated using the effective interactions of Eqs. (2) and (3) or the Warsaw basis coefficients of Eq.(7). Our projections assume the Λ −4 expansion. Note that our extension works for the case of different-flavor leptonic final states. In the case of same-flavor charged leptons the contribution from ZZ production as well as the interference between W + W − and ZZ contributions should be included.
We apply the basic cuts, where p T is charged lepton transverse momentum, η is charged lepton rapidity, m is the invariant mass of the two charged leptons, and / E T is the missing energy of the event.
Since we do not include detector effects, in our case the missing transverse energy is the 7 Our extension is built upon an updated private version. We thank Giulia Zanderighi for it.
transverse energy of the two final state neutrinos. We work at the parton level and veto jets with where p jet T is the jet transverse momentum and η jet is the jet rapidity. We further assume a 50% efficiency, in line with the experimental results of Ref. [39]. We use CT14QED Inclusive PDFs [40], and take the renormalization/factorization scales equal to M W W 2 .
We assume a systematic uncertainty of δ syst = 16% and find the point where the sys- The p T cut is defined to be the point where, L is the integrated luminosity and N is the number of events passing the cuts. This Retaining δ stat = 16% with the corresponding p T cuts of Eq.(30), we also consider the effect of reducing the systematic error to δ sys = 4%.
We begin by setting the fermion couplings to their SM values. In this case the expansion to O(Λ −4 ) is the full amplitude-squared, including both SM and SMEFT contributions.
The projections at NLO QCD for 14 TeV are shown in Fig. 1 and for 27 TeV in Fig. 2, in black for δ sys = 16% and in red for δ sys = 4%, named "3GB". We see a significant improvement going from 14 TeV to 27 TeV, while the improvement from reducing the systematic error, δ sys = 16% → 4%, is marginal. Compared to Ref. [11], the improvement from 8 TeV to HL-LHC and HE-LHC is important. The coefficient λ Z in particular is highly constrained, |λ Z | < 2 × 10 −3 at the HL-LHC and improved to |λ Z | < 6 × 10 −4 at the HE-LHC. or a Z and a t-channel diagram. The s-channel Z and t-channel diagrams are separately unitarity violating. In the SM, the violation is cancelled once the diagrams are summed and the production rate is unitarized. However, the t-channel and s-channel diagrams have different dependencies on the Z-quark and W -quark couplings. Hence, if there are anomalous quark-gauge boson couplings, the cancellation between the diagrams is spoiled and perturbative unitarity is violated. Although LEP strongly constrains these couplings [42][43][44], these constraints are at the Z-pole. At the higher energies of the LHC, the effects of the anomalous Z-quark and W -quark couplings grow with energy and their effect becomes important. While the perturbative unitarity violation is relevant for the W + W − production, for W ± decays into leptons the leading order is one diagram and the process occurs at the W pole. Hence, there is no cancellation between diagrams to guarantee unitarity conservation and the process occurs at LEP energies. The very strong LEP constraints on W -lepton couplings [7] are relevant, so we set the lepton couplings to their SM values. However we consider the effects of anomalous Z−quark couplings, assuming flavor universality. We display in Fig. 1 and Fig. 2 the constraints we obtain on the 3-gauge-boson couplings while allowing for Z−quarks anomalous couplings ranging over values that are constrained by global fits to LEP limits [42][43][44], We allow the fermion couplings to vary within the 2σ limits given in Eq.  Fig. 1. This is expected as the scan is performed relative to the SM value of the cross section: In order to get a cross section compatible with the SM while allowing at the same time non-zero anomalous fermion couplings, it is necessary to allow for non-zero anomalous triple gauge boson couplings. Comparing 14 TeV to 27 TeV limits we also see that there is no noticeable improvement going from δ sys = 16% down to δ sys = 4%. The 2σ bounds are already saturated, and in particular indicate non-zero values for δκ Z and δg 1 Z at more than 3σ when taking into account the anomalous fermion couplings according to the fit to LEP data as given in Eq. (31). The HE-LHC will thus be able to test the LEP fit and distinguish clearly between SM and non-SM Z−quark couplings, as the curves look quite different. An important implication of our study is that the anomalous fermion couplings have a major result on the allowed regions and cannot be neglected. When comparing our results with Ref. [13], a rough agreement is obtained.
Our limits at the LHC are not exactly the same and the differences can be explained by the different assumptions in the two studies: Ref. [13] does a background+signal study at 13 TeV including also contributions from the W ± Z, while we work at 14 TeV without taking into account the backgrounds and focusing only on W + W − ; Ref. [13] performs a fit on differential bins and profile over the variables not shown in their plots, while we perform a fit on the last bin without profiling.
We have also performed a second scan where the Z−quark couplings are centered around their SM value, using the same 1σ limits as in Eq. (31), The results are shown in Figs  pp → W + W − → µ ± e ∓ νν, where K SM EF T are evaluated in the SMEFT to O(Λ −2n ) and LO and NLO refer to the order in QCD. The notation dσ can represent either differential or total cross sections, where the numerators and denominators must be evaluated with identical cuts. Applying the cuts of Eqs. (26) and (27), the SM total cross sections are, To O(Λ −2 ), the SMEFT K-factor is, where we have normalized, A useful form for expressing the results is, where dσ can be distributions or total cross sections. At 13 TeV with the usual cuts, see Eqs. (26) and (27), the S SM EF T factor for the total cross section is, S where the values of X ij are given in Table III and we define the coefficients, C = (δg Z 1 , λ Z , δκ Z , δg Zu L , δg Zu R , δg Zd L , δg Zd R ). The largest sensitivity of S SM EF T is to the lefthanded Z-quark couplings, as also observed in Ref. [13]. Eq. (39)   The standard cuts given in Eqs. (26) and (27) are applied.
For comparison, we present S SM EF T in the HISZ basis of Eq.(8) using the primitive cross sections discussed above. Using the results of the previous section, C i = Σ j α ij C j , with, and SM EF T factor in a transformed basis is, For i, j = 4 − 7, α ij = δ ij . The primitive cross sections dσ N LO (n, R i ) and dσ N LO (n, M ij ) are defined in Eqs. (10,12) and are evaluated in the original anomalous couplings basis.   (26) and (27) are applied.
For the total cross section in the HISZ basis applying our basic cuts in Eqs. (26) and (27) where the values ofX ij are given in Table IV. The primitive cross sections can also be used to study distributions with arbitrary SMEFT coefficients and we show some sample results for the transverse momentum of the leading charged lepton, p ,lead T , and for the invariant mass distribution of the charged leptons, m . In Fig. 7, we show the distribution of the leading lepton p T in a scenario with only anomalous 3-gauge-boson couplings (LHS) and with only anomalous Z− quark couplings (RHS). The values of the coefficients were chosen to be allowed by experimental limits from W + W − pair production [39,46] and from fits to LEP data [7], and to give similar p T distributions. It is apparent that the anomalous 3-gauge-boson-only and  terms at large p T . SM and SMEFT K-factors are shown in Fig. 9 and it is clear that the SM and SMEFT scale similarly with small variations. Similarly, the K-factors for the anomalous-fermion-only scenario (RHS or Fig. 9) and the anomalous-3GB-only scenario (LHS of Fig. 9) are also similar with small variations. We note that this is for specific choices of the anomalous couplings and for different choices the K-factors have to be checked using the primitive cross sections.
In Fig. 10 are qualitatively similar to theirs for the scenario with only anomalous 3-gauge-boson couplings.

V. CONCLUSIONS
We have extended our previous NLO calculation of the contribution of anomalous couplings to pp → W + W − to include the leptonic decays, pp → W + W − → µ ± e ∓ νν  26) and (27) are applied.