CP violation at ATLAS in effective field theory

CP violation beyond the Standard Model (SM) is a crucial missing piece for explaining the observed matter-antimatter asymmetry in the Universe. Recently, the ATLAS experiment at the Large Hadron Collider performed an analysis of electroweak Zjj production, thereby excluding the SM locally at 95% confidence level in the measurement of CP -sensitive observables. We take the excess interpretation in terms of anomalous gauge-Higgs interactions at face value and discuss further steps that are required to scrutinize its origin. In particular, we discuss the relevance of multiboson production using adapted angular observables to show how they can be used to directly tension the reported Zjj excess in a more comprehensive analysis. To connect the excess to a concrete UV scenario for which the underlying assumptions of the Zjj analysis are valid, we identify vectorlike leptons as a candidate theory consistent with the observed CP -odd Wilson coefficient hierarchy observed by ATLAS. We perform a complete one-loop matching calculation to motivate further model-specific and correlated new physics searches. In parallel, we provide estimates of the sensitivity reach of the LHC ’ s high luminosity phase for this particular scenario of CP violation in light of electroweak precision and Run-2 Higgs data. These provide strong constraints on the model ’ s CP -even low-energy phenomenology, but also inform the size of the CP -odd SM deformation indirectly via our model hypothesis.


I. INTRODUCTION
The search for non-Standard Model (SM) sources of CP violation is a crucial missing piece in connecting the phenomenological success of the SM so far with its apparent shortcomings related to the observed baryon antibaryon asymmetry [1]. Searches for CP violation in various channels at the Large Hadron Collider (LHC) are therefore a key part of the ongoing experimental program (see, e.g., [2,3] for recent analyses in the context of Higgs physics).
In particular, ATLAS has recently performed a detailed analysis of electroweak Z þ 2j production in Ref. [4], where it also interprets measurements in terms of effective field theory (EFT) deformations of the SM using dimension six CP-violating operators in the Warsaw basis [5], where W, B denote the field strengths of weak SUð2Þ L and hypercharge Uð1Þ Y , H is the Higgs doublet, τ a are the Pauli matrices, and the tilde refers to the dual field strength tensorX μν ¼ ϵ μνδρ X δρ =2 (X ¼ W, B, G). Using the effective Lagrangian ATLAS provides the observed 95% confidence level constraints on the following CP-violating operators [4]: CW TeV 2 Λ 2 ∈ ½−0.11; 0.14; C HWB TeV 2 Λ 2 ∈ ½0.23; 2.34; based on dimension six interference-only contributions arising from matrix elements This leads to asymmetries in P-sensitive distributions, such as the "signed" (according to rapidity) azimuthal angle difference of the tagged jets ΔΦ jj . The benefit of such observables and the study of their asymmetries is that the CP-even deformations do not contribute to the exclusion constraints directly, which also extends to CP-even modifications arising from "squared" dimension six contributions. In Eq. (5), M d6 denotes the amplitude contribution from the operators of Eq. (1); thus, it is a linear function of CW=Λ 2 ; C HWB =Λ 2 (as we are keeping terms up to order 1=Λ 2 ). The constraint on the Wilson coefficient (WC) C HWB in Eq. (4) indicates a tension with the SM while the observed cross section agrees well with the SM expectation with 39.5 fb data [6][7][8]. This prompts us to the following interesting questions.
First, the tension of Eq. (4) seems to rule out the SM at an SM-compatible cross section. Experimental analyses of asymmetries are challenging, and systematics are crucial limiting factors of distribution shape analyses. Nonetheless, the result of Ref. [4] could indeed be the first glimpse of a phenomenologically required and motivated extension of the SM, thus deserving further experimental and theoretical scrutiny.
Second, limiting ourselves to a subset of the dimension six operators that could in principle contribute to physical process can be theoretically problematic, in particular when we wish to interpret the experimental findings in a truly model-independent fashion. While concrete UV scenarios can be expected to exhibit hierarchical Wilson coefficient patterns, it is not a priori clear that limiting oneself to anomalous gauge boson interactions has a broad applicability to UV scenarios.
Addressing these two questions from a theoretical and phenomenological perspective is the purpose of this work. In Sec. II, we motivate additional diboson analyses of the current Oð100Þ fb −1 data set that will allow us to tension or support the results of Eq. (4) straightforwardly. This is particularly relevant as the ATLAS constraints amount to a large, and as it turns out nonperturbative, amount of CP violation associated with a single direction in the EFT parameter space. In Sec. III, we show that the ATLAS assumptions of considering two operators are consistent for models of vectorlike leptons, which cannot only reproduce a hierarchy jC HWB j=Λ 2 > jCWj=Λ 2 as suggested by Ref. [4], but also collapse the analysis-relevant operators to those modifying the gauge boson self-interactions for the considered analyses. Combining both aspects, in Sec. IV, we assess the future of diboson and Z þ 2j analyses from a perturbative perspective and discuss the high-luminosity (HL) sensitivity potential of the LHC in light of the electroweak precision constraints. We conclude in Sec. V.

II. SCRUTINIZING CH WB WITH DIBOSON PRODUCTION AND CURRENT LHC DATA
Deviations related to the gauge boson self-coupling structure can be scrutinized using abundant diboson production at the LHC. With clear leptonic final states and large production cross sections, these signatures are prime candidates for electroweak precision analyses in the LHC environment with only a minimum of background pollution; see also [9,10]. In particular, radiation zeros observed in Wγ production are extremely sensitive to perturbations of the SM CP-even coupling structures [11][12][13][14][15][16][17]. In this section, we discuss the relevant processes that can be employed to further tension the findings of Eq. (4).

A. Processes
The squared amplitude of Eq. (5) receives interference contributions from dimension six operators that in the special case where they are CP odd, do not change the cross section of a process but appear in CP-sensitive observables. Anomalous weak boson interactions were studied in Ref. [4] through the Zjj channel by the introduction of two CP-violating operators, QW and Q HWB , modifying the differential distribution of the paritysensitive signed azimuthal angle between the two final state jets Δϕ jj ¼ ϕ j 1 − ϕ j 2 , where ϕ j 1 (ϕ j 2 ) is the azimuthal angle of the first (second) jet, as ordered by rapidity. Similar parity-sensitive observables can be constructed for the leptonic final states of the Wγ → lνγ, W þ W − → l þ ν l l −ν l , and WZ → lνl þ l − channels allowing to further constrain the reach of the two Wilson coefficients.

WZ production at the LHC
We study the WZ channel by selecting leptons in the pseudorapidity jηðlÞj < 2.5 and transverse momentum p T > 5 GeV regions. Exactly three leptons are required and at least one same-flavor opposite-charge lepton pair must have an invariant mass within the Z boson mass window m ll ∈ ½60; 120 GeV. In the case of more than one candidate pairs, the one that yields an invariant mass closest to the Z boson is selected. The remaining lepton l 0 is required to have p T ðl 0 Þ > 20 GeV. To obtain a P-sensitive observable, we reconstruct the dilepton pair four-momentum and obtain the rapidity y ll and azimuthal angle ϕ ll . We order the dilepton and third lepton azimuthal angles based on the rapidities of the two reconstructed objects, such that ϕ 1 (ϕ 2 ) is the one with the greatest (smallest) rapidity. The signed azimuthal angle is then constructed as Δϕ The distributions of the signed azimuthal angle for both the SM and the SM-BSM (beyond the Standard Model) interference are normalized to the CMS measured fiducial cross section [25] of the particular phase space region at 13 TeV center of mass energy, 2. WW production at the LHC Turning to the WW channel and following Ref. [26], we produce events decaying to the WW → eν e μν μ final state. The two leptons e and μ are required to satisfy jηðlÞj < 2.5 and p T ðlÞ > 27 GeV with no third lepton in the p T > 10 GeV region. Contributions from the Drell-Yan background are reduced by imposing cuts on the missing energy E T > 20 GeV and on the transverse momentum of the dilepton pair p T ðeμÞ > 30 GeV. The phase space region is constrained further by enforcing the invariant mass condition mðeμÞ > 55 GeV that suppresses the H → WW background. In this channel, the signed azimuthal angle Δϕ ll is then defined directly from the azimuthal angles of the two leptons sorted by rapidity.
The fiducial cross section of WW → eμ þ = E T was measured by ATLAS [26] as σ fid ðpp → WW → lν e μν μ Þ ¼ 379.1 AE 5.0ðstatÞ AE 25.4ðsystÞ AE 8.0ðlumiÞ fb; ð7Þ which is used to normalize the calculated differential distribution of Δϕ ll . The total cross section of the events and the relative statistical and systematic uncertainties are subsequently rescaled to include the final states of all light leptons WW → lνlν.

Wγ production at the LHC
To obtain the cross section of Wγ at 13 TeV, we first use MCFM [27][28][29][30][31] with generation level cuts p T > 10 GeV and jηj < 2.5 for both leptons and photons, requiring the separation ΔRðl; γÞ > 0.4, in order to obtain the cross section at next-to-leading order precision with p T ðγÞ as the renormalization and factorization scale. We have validated these choices against early measurements from ATLAS [9] and CMS [10]. The events are generated as before with MadEvent using the same generation cuts and we rescale the computed MadEvent cross section of the events to the MCFM value, in order to include higher order effects and obtain normalized distributions.
Postgeneration we veto events without at least one lepton (photon) with transverse momentum p T ðlÞ > 35 GeV (p T ðγÞ > 15 GeV and require a separation of ΔRðl; γÞ > 0.7. The azimuthal angles of the photon and the lepton are sorted by rapidity and Δϕ lγ is calculated similarly to the other channels.
We assume that the relative statistical and systematic errors that can be calculated from the measured cross section of Ref. [10], 1 will remain the same for the case of ffiffi ffi s p ¼ 13 TeV and use this in the following statistical analysis.

B. Analysis of CP-sensitive observables
To study the allowed region of the ðCW; C HWB Þ parameter space based on current experimental data at the LHC, we consider the differential distribution where, depending on the process, X ¼ l 0 Z, ll, lγ, and σ HWB and σW are constructed from Q HWB and QW, respectively, and derive from MC integration of Eq. (5). We generate events for each process using the two coupling reference points ðCW; C HWB Þ ¼ ð1; 0Þ and ðCW; C HWB Þ ¼ ð0; 1Þ and can rescale distributions using the linear relation of Eq. (5) to subsequently scan over the space of the two CP-odd Wilson coefficients, performing a χ 2 fit, in order to obtain limits. The χ 2 statistics is defined as where b i SMþd6 ðCW; C HWB Þ is the number of events at a particular luminosity based on the ith bin of the differential distribution Eq. (9) for a set of Wilson coefficients and b i SM is the bin's expected number of events based solely on the SM. The covariance matrix V ij includes the relative statistical and systematic uncertainties 2 from the experimental measurements, obtained from the aforementioned fiducial cross sections Eqs. (6)-(8) for each process and included in V ij as terms of the form ðε 2 rel stat þ ε 2 rel syst Þb i SM b j SM , assuming that both relative and systematic errors are fully correlated. ε rel stat and ε rel stat denote the relative statistical and systematic uncertainties of each process. 1 l for this cross section indicates each type of light lepton (e, μ) and not a sum over them. 2 Luminosity uncertainties are treated as systematics.
We define the confidence intervals with using the χ 2 distribution of k degrees of freedom p k ðxÞ, where k is obtained by subtracting the number of Wilson coefficients from the number of measurements.
We perform a scan based on an integrated luminosity of 139=fb to obtain the 95% confidence level contours shown in Fig. 1. The results are overlapped with the Z þ 2j allowed region from ATLAS [4], as well as the best fit point from experimental data, while the WZ does not constrain the region enough to appear on the plot. To obtain the Z þ 2j contours, we have tuned a covariance matrix on the basis of the information of Ref. [4] to obtain the exclusions reported in their work.
As can be seen the measurement of Z þ 2j is considerably more sensitive to QW than to Q HWB , which results from a combination of accessing t-channel momentum transfers in the weak boson fusion-type selections and the Z boson having a larger overlap with the W 3 field than the photon. The latter is also the reason why Wγ production enhances the sensitivity in the Q HWB direction. We note that electroweak mono-photon production in association with two jets is more challenging due to jet-misidentification and thus does not provide significant sensitivity compared to prompt Wγ production.

C. HL-LHC extrapolation
We repeat the analysis with the same technique but using an integrated luminosity of 3=ab to obtain contours for HL-LHC. Systematic errors could be significantly reduced at HL-LHC; however, for this particular case, we find that the fact that BSM contributions are antisymmetric functions, in contrast to the symmetric SM differential distribution, leads to cancellations of the introduced errors in the χ 2 . 3 Hence, the analysis is predominantly limited only by the statistical fluctuations. The extrapolated contours for 3=ab are shown in Fig. 2.

III. VECTORLIKE LEPTONS AS A MODEL FOR CH WB -TARGETED (DI-)BOSON SIGNALS
Let us return to the examining the excess related to Q HWB from a UV model perspective. To this end, we extend the SM by three heavy vectorlike lepton (VLL) multiplets [32], where the quantum numbers are depicted in SUð3Þ C ⊗ SUð2Þ L ⊗ Uð1Þ Y convention. The most-general gauge-invariant renormalizable Lagrangian involving these heavy VLLs can be written as where m Σ , m η 0 , and m ξ 0 are the masses of Σ, η 0 , and ξ 0 , respectively. P L ðP R Þ are the left (right) chiral projection operator. Y i 's are the complex Yukawa couplings. We will consider m Σ ¼ m η 0 ¼ m ξ 0 ¼ m, i.e., all the VLLs are degenerate in this work. 4 We will see that this class of models provides the appropriate UV backdrop for the Wilson coefficient analysis that we have performed above and on which Ref. [4] relies.

A. Wilson coefficients
We integrate out all three heavy degenerate VLL multiplets; see Eq. (13) leading to the effective Lagrangian, where Q i , C i denote the effective dimension six operators and the Wilson coefficients, respectively. The UV theory in Eq. (13) is suitably matched to the Standard Model effective field theory at the scale m which serves as the cutoff scale of the EFT. Here, the 16π 2 factor signifies that all the effective operators are generated through one loop. And we separate off the loop factor ð16π 2 Þ −1 from the definition of the Wilson coefficients C i , i.e., C i ¼ C i =16π 2 and Λ ¼ m in comparison with Eq. (3). We employ the modified minimal subtraction (MS) renormalization scheme and also set the RG scale at μ ¼ m. Integrating out heavy fermions from UV theories is discussed in Refs. [32,35]. Note that dimension eight CP-violating effects play a subdominant role when perturbative matching is possible in the first place; see [33,36]. We can therefore expect the dimension six deformations to play a dominant role. We present the effective operators in the Warsaw basis [5] and their respective WCs are encapsulated in Table I. We also provide the matching using strongly interacting light Higgs (SILH)-like convention of [32] (see also [37,38]) in Table V of Appendix A for convenience. We compute the most generic results using the complete Lagrangian including CP-conserving and -violating interactions simultaneously. A subset of our generic results (in SILH-like basis) is in well agreement with operators computed in Ref. [32]. The results in WARSAW for this VLL scenario has been computed for the first time in this paper. 5 We find 19 effective operators with nonzero Wilson coefficients (16 CP even þ 3 CP odd). In the renormalizable Lagrangian, the VLLs interact with the SM Higgs doublet and that explain the origin of ten bosonic along with nine fermionic effective operators accompanied by nonzero WCs. These appear due to application of the equation of motion of the SM Higgs doublet on the effective Lagrangian.
Here, we define the following functions to express the WCs in much more compact form [32] in Table I: where i ¼ η, ξ. We further use the additional abbreviations for the same purpose [32], Here, we denote the electron (e)-, up (u)-, and down (d)-types Standard Model Yukawa couplings as Y e SM , Y u SM , and Y d SM , respectively, while we refer to the SM Higgs quartic self-coupling as λ H .
The operators that may affect the couplings of gauge bosons to fermion currents, i.e., the relevant LHC processes are [5,39,40] Q eB ; Q eW ; Q uB ; Q uW ; Q dB ; Hl ; Q Hl ; Q Hq ; Q Hq ; Q Hud ; Q He ; Q Hu ; Q Hd : We have a relevant CP-even operator Q HWB that leads to an additional contribution to oblique corrections [41][42][43][44][45][46][47][48] and in particular the S parameter. In later section, we discuss the impact of all relevant CP-even operators in electroweak precision observables (EWPOs) in detail. At this point, it is worthy to mention that the operators Q HB ; Q HW ; ð20Þ do not modify trilinear gauge interactions as their contributions either vanish due to momentum conservation or can be absorbed into field and coupling redefinitions respecting gauge invariance. By investigating Table I, we also find that our adopted scenario, Eq. (13), predicts CW ¼ 0. Thus, together with our previous observations, we conclude that Q HWB ≠ 0 is the only relevant operator to interpret the results of ATLAS within the vectorlike lepton framework. In passing, we would like to mention that some of the remaining nonzero operators can be probed in Higgs-boson TABLE I. The generated Warsaw basis operators and respective Wilson coefficients after integrating out VLLs using Eq. (13). The CP-odd gauge boson operators are displayed in first three rows. Multiplication with a common factor ð16π 2 m 2 Þ −1 is understood implicitly; see Eq. (14).

Operators
Operator structures Wilson coefficients (C i ) associated final states or (to a lesser extent) through their radiative correction contributions [49,50] (the latter corresponds to a two-loop suppression in the considered vectorlike lepton UV completion). These processes provide additional CP sensitivity, however, at smaller Higgs-boson related production cross sections (see, e.g., the discussion in Ref. [51]) that receive corrections from a range of nonzero Wilson coefficients C HB , C HW . We will not investigate Higgs-CP related effects in this work as neither they contribute to the electroweak precision observables nor impact the discussion of the previous section. 6

B. Constraints from electroweak precision observables and Higgs data
The CP-even Standard Model effective field theory operators contribute to the EWPOs [39,54,55] and to the production and decay of the SM Higgs [56]. We note that all the dimension six operators, generated after integrating out the VLLs, do not leave any impact to these observables; see Table II. We briefly outline the nature of correlations among the relevant effective operators and the EWPOs in Appendix B. Though these observables do not constrain the CP-odd operators directly, we note that within our framework the CP-even and CP-odd operators are related to each other, see Tab. I, through the model parameters, e.g., the Yukawa couplings in Eq. (13). Here, we want to emphasize that instead of considering individual complex Yukawa couplings we prefer to work with a set of Yukawa functions, chosen based on the computed WCs; see Table III. This allows us to avoid unnecessary increase of free parameters in the theory which could have spoiled the quality of the fit without any gain for the earlier choice. Thus, by encapsulating the effects of these observables on CP-even WCs, we can deduce complementary constraints on the CP-odd WCs through the exotic Yukawa couplings in addition to the couplings' phases. We perform a detail χ 2 -statistical analysis 7 using a Mathematica package OptEx [57] to estimate the allowed ranges of the model parameters in the light of the following experimental data: for EWPOs, see Table 2 of Ref. [58] and Higgs data for Run-1 ATLAS and CMS [59,60] and Run-2 ATLAS and CMS [59][60][61][62][63][64][65][66][67][68][69][70]. The statistically estimated parameters which are suitably chosen functions of VLL-Yukawa couplings are depicted in Table III.

IV. INDIRECT VECTORLIKE LEPTONS: FROM RUN-2 TO THE HL-LHC FRONTIER
In Fig. 3, we show a scan over the model parameters when contrasted with the parameter constraints of the ATLAS analysis result of Eq. (4). It can be seen that the large excess in the C HWB 95% constraint that is in tension with the SM favors either low mass scales or very large, potentially nonperturbative couplings. Direct searches for vectorlike leptons have been discussed in [71] and a HL-LHC direct coverage should be possible up to mass scales of 450 GeV which translates into model ImðY i L Y Ã i R Þ ∼ 40, thus probing ReðY i Þ, ImðY i Þ ∼ 6. For such relatively low scales, where the EFT scale is identified with the statistical threshold of a particular analysis, the couplings are still in the strongly coupled, yet perturbative jYj ≲ 4π regime. Such large couplings can lead to potential tension with TABLE II. The CP-even effective operators (in Warsaw basis) after integrating out VLLs: "✓" and "✗" signify that the respective operator is constrained or not, respectively, by the EWPOs and Higgs data. The operators Q ledq , Q   Table I. We assume m ¼ 1 TeV.

VLLs: Yukawa couplings
Fitted values of parameters (@ 68% C.L.) The additional chiral symmetry violation that leads to nonvanishing Wilson coefficients could in principle be traced into a uniform modification of the Higgs two-point function [52] that can in principle be probed at hadron colliders. A related investigation was performed recently by CMS in four top final states [53]. Sensitivity, however, is currently too limited for this effect to play an important role in a global fit. 7 We would like to mention that in our analysis the degree of freedom is 80 and p value is .36. The min-χ 2 is 83.86. other observables that are correlated through our particular model assumption. The constraints outlined in Sec. III B are in fact stronger, in particular for the combination of ImðY ξ L Y Ã ξ R Þ ≲ 40. Returning to the complementary constraints that can be derived from the diboson, and in particular the Wγ analyses, we show the expected sensitivity range to the new physics scenario and its compatibility with the allowed parameter space consistent with the EWPO and Higgs signal strength in Fig. 4. 8 We choose three benchmark FIG. 4. Yukawa values allowed by the 68% and 95% C.L. fits using EWPOs and SM Higgs decays plotted along with the regions allowed from the diboson analysis with 3=ab. In addition, the diboson exclusions obtained for 6=ab (resulting from a ATLAS þ CMS combination) with the same methodology are also included. The 68% and 95% C.L. fits are shown for three benchmark points (see Table IV).  Table I by sampling the Yukawa values and a fixed hypercharge of (SM-like) Y ¼ −1=2 is plotted against the vectorlike lepton mass m. On the right, the relevant combinations of Yukawa values contributing to C HWB are shown. The points for the different exclusions are determined by assessing whether C HWB =m 2 lies within the 95% contours for 139=fb and 3=ab, as well as the allowed range the observed data of the ATLAS experiment [4].
TABLE IV. These are the three benchmark points chosen to analyze the 68% and 95% C.L. allowed parameter space in the Im½Y η L Y Ã η R − Im½Y ξ L Y Ã ξ R plane, using the EWPO and the Higgs data. The shown fit parameters are set to the best-fit values (see Table I) and two other set of points. The corresponding 68% and 95% C.L. regions are shown in Fig. 4 points (see Tab. IV) and show the 65% and 95% C.L. regions in the Im½Y η L Y Ã η R − Im½Y ξ L Y Ã ξ R plane. There it becomes clear that the searches outlined in the beginning of this work will provide important sensitivity to this particular model class in the future in the ImðY ξ L Y Ã ξ R Þ direction, which (for our choice of Y) is relatively unconstrained by Higgs and EWPO data.

V. DISCUSSION AND CONCLUSIONS
The insufficient amount CP violation in the SM to explain the observed matter-antimatter asymmetry is a clear indication of the presence of new physics beyond the SM. Consequently, analyses of CP properties of particle physics interactions are an important part of the current phenomenological program at various energies, reaching up to the current high-energy frontier explored at the LHC. The observation of C HWB -related excess by the ATLAS Collaboration in the recent reference [4] could be the first indication of the presence of such interactions in the gauge boson-Higgs sectors. Taking inspiration from Ref. [4], the focus of this work is twofold which are as follows: (i) We motivate a particular UV model class, namely, that of vectorlike leptons Eq. (13), to provide a minimal and consistent theoretical backdrop to the analysis of Ref. [4]. We perform a complete matching calculation at one-loop order and demonstrate that all relevant CP-odd EFT deformations of the SM amplitude of diboson (and Z þ 2j) are dominantly captured by the Q HWB operator. In parallel, at the given order, we do not induce QW, which impacts the analyses of CPodd observables in (di)boson final states as well, but which is consistent with the SM expectation of CW ¼ 0 given the results of Ref. [4]. The mass scales of the vectorlike lepton scenario that can be directly explored at the LHC [71] constrain the model's parameters to the strong coupling, yet perturbative regime. An analysis of electroweak precision and Run-2 Higgs results indicates that the region of the ATLAS excess could be explained by ImðY η L Y Ã η R Þ ≃ 0 and a significant ImðY ξ L Y Ã ξ R Þ with some tension given the UVmodel's correlation of CP-even and CP-odd couplings for masses that fall into the HL-LHC kinematic coverage. (ii) The excess observed by ATLAS deserves further scrutiny. We show that diboson analyses, and in particular Wγ production will serve as a strong cross check of the excess, in particular because its phenomenology is particularly sensitive to Q HWB -induced deviations. The analysis suggested in Sec. II A will therefore allow the collaborations to directly explain the results of Ref.
[4] as a statistical fluctuation or gather further, strong evidence for a non-SM source of CP violation. Finally, the correlation of different Wilson coefficients as predicted by our matching calculation motivates additional Higgs-based phenomenology probe that can further constrain or solidify the excess through measurements that target, e.g., Q HB , Q HW in a suitable way [51] (see also [72,73]).   [32]). The heavy VLL multiplets are integrated out from the UV complete theory. The WCs are calculated up to one-loop order considering the CPconserving and -violating couplings simultaneously. Note down the additional contributions of CPV couplings to the CP-even operators. Note that we are referring to SILH-like operators as O i , to highlight their difference from the Warsaw-basis operators.