Electroweak Loops as a Probe of New Physics in $t\bar{t}$ Production at the LHC

We calculate the $\mathcal{O}(\alpha)$ weak corrections to top quark pair production at the LHC and include anomalous electroweak interactions from dimension-six operators. The loop calculation and renormalization are consistently done within the Standard Model Effective Field Theory. Sensitivity to the involved operators is exposed through the virtual corrections, which receive enhancement from electroweak Sudakov logarithms. We investigate the prospects of using this feature for probing New Physics at the LHC that so far has only been studied in final states with on-shell sensitivity such as $t\bar{t}+Z$ or $t \to b W$. We find that the large $t\bar{t}$ production rate and the excellent perturbative control allow compensating the loop suppression and yield remarkably strong constraints that are competitive with those from $t\bar{t}+Z$.


Introduction
Run-2 data taking of the Large Hadron Collider (LHC) lasted from early 2015 to mid 2018 and culminated in a data set of about 160 fb −1 integrated luminosity at 13 TeV collision energy. In this period the top quark was produced approximately 240 millions times. This is more than a thousand times more frequent than it was ever produced at its discovery machine -the Tevatron. Soon, around the year 2023, the integrated LHC luminosity is expected to have doubled reaching up to 400 fb −1 . The wealth of this experimental data yields unprecedented precision in hadron collider physics and enables powerful probes of New Physics in the top quark sector, if combined with precise theory predictions.
In this paper, we ask the question whether one can exploit the high precision in the top quark sector to probe New Physics that enters through virtual loop corrections. Historically, this is a very fruitful approach as, for example, the top quark was postulated by considering kaon mixing through loops, decades before its discovery [38,39]. Also the Higgs boson mass was indicated accurately by LEP precision fits arising from electroweak loops [40]. Similarly, many New Physics searches at B-factories harvest loop sensitivity and place incredibly strong limits on deviations from the Standard Model (see e.g., Ref. [41]). Top quark pair production at the LHC might be a similarly powerful probe. Besides resonance searches in the tt mass spectrum and modifications of QCD interactions, the most interesting corner to look for New Physics is the electroweak top quark interactions. Prominent examples are the analyses of tt + Z/W/H [42][43][44][45] final states or the W -helicity fractions in top quark decays [46,47], which yield direct on-shell sensitivity to the couplings between top quarks and the electroweak bosons. Global fits of New Physics contributions in terms of dimensionsix operators relevant for the top quark sector have been carried out in Refs [48][49][50]. The resulting constraints on New Physics are substantial (see e.g. Ref. [44]), but the above analyses have their limitations: 1) The associated production processes tt + X suffer from relatively small cross sections because of coupling suppression, a high production threshold and penalties from branching fractions; 2) Sensitivity in top quark decay dynamics is often diminished by cancellations between the squared matrix element in the numerator and the total width in the denominator. It is therefore essential to explore alternative avenues such as sensitivity to New Physics from electroweak loop corrections. Let us substantiate this reasoning by presenting a first estimate of such prospects. We compare the tree level pp → tt + Z process at O(α 2 s α) with the electroweak correction to the pp → tt process, also at O(α 2 s α). For a final state with a leptonic Z boson decay and semi hadronic top quark decays, one finds for the ttZ process [51] and for the next-to-leading order (NLO) electroweak contribution to tt production 1 [51] α × σ tt × B tt→ ν+jets ≈ 1/128 × 840 pb × 33% ≈ 1800 fb. (1. 2) The striking difference between ttZ and tt production mainly arises because of the factor B Z→ that is missing for the virtual corrections in Eq. (1.2). Hence, the study of New Physics in electroweak corrections to tt seems very promising and appears competitive to similar studies in tt + Z. This rationale is of course very dependent on the respective backgrounds. While the pp → tt + Z → νbbjj + process has relatively small backgrounds, the tt final state is dominantly arising from QCD dynamics which act as irreducible background for our purposes. Fortunately, the QCD production dynamics are known to very high perturbative order (NNLO+NNLL QCD [1,[25][26][27]31]). This should allow for a precise extraction of the electroweak effects which have been found to be comparable or even exceeding the current QCD uncertainties in some phase space regions (in particular for transverse momenta between 500 GeV and 2 TeV, see, e.g., Fig. 4 in Ref. [21]). Building upon this idea, we present a comprehensive investigation in this paper and compare our results with conventional results. We calculate the next-to-leading order weak Figure 1: Electroweak top quark interactions that receive modifications from dimensionsix operators in our work.
corrections and, for the first time, consistently include the effects of dimension-six operators of a Standard Model Effective Field Theory (SMEFT) [56][57][58][59][60][61][62][63] in this loop calculation. A consistent ultraviolet (UV) renormalization within SMEFT guarantees a finite result. Our analysis makes use of differential distributions and allows expanding the cross section up to dimension six and dimension eight from squared dimension-six contributions. We benchmark our results with independent constraints obtained from the pp → tt+Z process.

Dimension-six coupling parameterization
The main aim of this work is the investigation of anomalous top quark electroweak interactions. Hence, we focus on the ttZ and ttW couplings and allow deviations from their SM value within the framework of the Standard Model Effective Field Theory (SMEFT) with contributions from dimension-six operators. In the SM, the ttZ vertex is given by where T 3 , Q, and c w = 1 − s 2 w are the weak isospin, electric charge, and the cosine of the weak mixing angle, respectively. An extension to the SU(2) × U(1) symmetric effective field theory in the Warsaw basis [60] leads to the following shifts of left-handed and right-handed couplings where v is the vacuum expectation value and Λ is the characteristic scale of the Effective Field Theory. The above Wilson coefficients correspond to the operators where τ I are the Pauli matrices. For simplicity, we eliminate one of the two coefficients in d Z L of Eq. (2.2) and set C ϕq1 33 = −C ϕq3 33 . This corresponds to neglecting corrections to the left-handed Zbb vertex that has limited impact on our analysis. As a consequence, shifts in the couplings of ttZ and ttW are correlated, as are their corresponding would-be Goldstone bosons. We also neglect dipole operators and the operator Q ϕW B [60] which contributes to d Z L . They can potentially be studied in ttγ final states. The operator in Eq. (2.4) also modifies the W tb vertex by shifting the SM contributions d W,SM We assume that there is no contribution to an anomalous right-handed W -boson current.
The vertices connecting the would-be Goldstone bosons φ ± and χ 0 to top (and bottom) quarks receive modifications from the operators in Eqs. (2.3-2.5) as well. We find The / p is the incoming boson momentum. In the EFT, there are also four-point vertices Γ φφtt of two would-be Goldstone bosons and two top quarks which, in principle, contribute to our process. However, it turns out that they yield a vanishing contribution after loop integration because their coupling structure only yields tensor rank-1 loop integrals of the kind d D µ ( 2 − m 2 + iε) = 0.

O(α) corrections and UV renormalization in tt production
The tree level amplitudes contributing to qq/gg → tt can be categorized into O(α s ) strong production and O(α) electroweak production. As a result, the pure electroweak qq initiated production cross section is parameterically suppressed by about three orders of magnitude compared to strong production. It is further suppressed by the qq parton luminosities as compared to gluon fusion, which yields more than 80% of the total tt rate at the LHC. Therefore, electroweak production of top quark pairs is not observable at the LHC and does not yield sufficient sensitivity for the study of electroweak couplings. Even tree level interference between strong and electroweak amplitudes at O(αα s ) is too small: It only exists for the parton luminosity suppressed process bb → tt process [64], which yields two permille of the total rate. Fortunately, there is the possibility to access O(α) effects at higher orders in perturbation theory. At next-to-leading order they are the O(αα 2 s ) corrections, which comprise electroweak one-loop corrections to the QCD induced process and the corresponding single real gluon emission contributions to QCD-electroweak mixed amplitudes (see e.g., Refs. [15,19]). Moreover, it is well-known that one-loop diagrams with virtual Z and W bosons are enhanced by logarithms ofŝ M 2 W , which dominate the electroweak correction and enhance sensitivity at high energies. Within the SM these corrections have been known for a long time [14][15][16][17][18][19][20] and feature negative corrections that indeed grow with energy. In fact, close to the production threshold the corrections are approximately −2% and they decrease to −20% at top quark transverse momenta of about 2 TeV [19]. Real emission corrections with Z and W bosons are typically neglected because they lead to a different final state and their unresolved contribution is small [65]. Pure photonic corrections have been calculated in Refs. [66,67] and are found to be constantly small at the level of O(1%) over a wide range of energy.
We extend the existing results in the literature and calculate, for the first time, the weak corrections within a SM Effective Field Theory, accounting for the contributions of the dimension-six operators in Eqs. (2.3-2.5). Hence, we allow for arbitrary deviations from the SM top quark weak interactions that are consistent with the symmetries of the SM. Our calculation is of O(αα 2 s ) and for completeness, we also include the above mentioned suppressed tree level contributions of O(α 2 ) and O(αα s ). We neglect all higher-order contributions that are suppressed even stronger. External fields are renormalized in the on-shell scheme, and there is no coupling renormalization, nor effects of operator mixing at the given order. Our set of input parameters is M Z , M W , m t and G F . For both, the qq and gg initial state, we calculate final state corrections to s-channel diagrams as shown in Fig. (2), using the anomalous couplings in Eqs. (2.2), (2.7) and (2.9-2.12). The gg inital state receives corrections to the t-channel diagrams in Fig 3, as well. Loop amplitudes with would-be Goldstone boson exchange increase the integral tensor rank by up to two due to the additional / p terms in the EFT contributions in Eqs. (2.9-2.12). For example, the first diagram in Fig. 2 has a contribution of the form where D i = ( + k i ) 2 − m 2 i and the k i are linear combinations of the external momenta. This expression contains a rank-four three point integral. However, in this case the tensor rank can be lowered by two ranks using (2.14) We can reduce all other higher-rank tensor integrals arising from the EFT contributions using Eq. (2.14) and arrive at the set of scalar integrals required for the SM calculation. We note, however, that additional UV poles appear since the higher-rank tensor integrals yield additional divergencies. These UV poles are canceled by a proper renormalization within the effective field theory. This is achieved by wave function and mass renormalization which accounts for the extended SMEFT Lagrangian. Following Ref. [68], the renormalization constants for the bare top quark field t 0 = (1 + 1 2 δZ t )t and the bare top quark mass m t,0 = m t + δm t are given by , The UV finite box diagram contributions, shown in Fig. 4, contain additional infrared (IR) poles from soft and collinear gluons in the loop. They are canceled by mixed QCDelectroweak amplitudes with real gluon emission, which are also sensitive to the EFT operators. Hence, we consistently include them in our analysis and confirm a cancellation of all IR poles for arbitrary top quark couplings.

Benchmark process pp → tt + Z
To benchmark our results obtained from pp → tt we also consider the pp → tt + Z process, which yields sensitivity to the same kind of EFT operators in Eqs. (2.3-2.5). In contrast to the virtual corrections in tt production, O(α) dependence arises directly from the onshell coupling of the Z boson to the top quarks at tree level. In the case that Z → the final state is a clean experimental signature with relatively small backgrounds. On the theory side, higher-order predictions such as the NLO corrections to the SM process have been known for many years. In this work we build upon the TOPAZ framework 2 [69], which was used in Refs. [70][71][72] to calculate the NLO QCD corrections to tt + Z production and decay including anomalous couplings between the top quark and the Z boson. These anomalous couplings can be related directly to the EFT coefficients in Eq. (2.2) and allow an immediate comparison with the tt process.

Numerical evaluation
We implement the analytic results of our calculation in Fortran and make it available as an external add-on to the Monte-Carlo simulator MCFM [73]. MCFM allows us to benefit from its easy-accessible, well-tested and established simulation framework. Moreover, the electroweak correction in the SM are already incorporated [74] and allow us to do valuable cross checks. Of course, we fully reproduce the MCFM results when setting C ϕq3 33 = C ϕu 33 = 0 in our calculation. Our add-on can be incorporated into the publicly available version by simply replacing eight files and re-compiling the code 3  We use the parton distribution functions NNPDF30_nnlo_as_0118_qed [75] and its corresponding value of α s . For simplicity, the top quarks are simulated as stable particles in our calculations. We therefore multiply the cross sections with the branching fraction of the lepton+jet final state Br( + jets) = 8/27 ≈ 30% and allow electrons and muons in the final state. This final state allows an unambiguous reconstruction of the top quark momenta and justifies the approximation of stable top quarks. To account for the state-of-the-art QCD predictions for tt production in an approximate way, we match the scale setting of Ref. [21] µ = m T,t /2 and multiply our weak corrections by the NNLO QCD K-factor of 1.67 [76]. This corresponds to the so-called multiplicative approach, which was found to be preferred over an additive approach [21]. The respective QCD scale uncertainty of +4 % −6 % is taken from Ref. [76] and set to ±5 % in our analysis. For the pp → tt + Z process, we multiply with a K-factor of 1.23 and assume an uncertainty of ±15 % [69]. The Z boson is assumed to decay into either electron or muon pairs. For estimating the sensitivity to New Physics in our processes, we need to find the functional dependence of the production rate on the EFT coefficients C ϕq3 33 and C ϕu 33 . Since we also want to exploit the effects from electroweak Sudakov logarithms that grow with energy, we even need to find this dependence for differential distributions. If we truncate all terms beyond O(v 4 Λ 4 ), we can parameterize each bin i of a differential distribution as N SMEFT i = n 0i + n 1i C ϕq3 33 + n 2i C ϕu 33 + n 3i (C ϕq3 33 ) 2 + n 4i (C ϕu 33 ) 2 + n 5i C ϕq3 33 C ϕu 33 , (2.26) with coefficients n j i . These coefficients can be determined by generating six differential distributions for six different values of C ϕq3 33 and C ϕu 33 . Once this fit is obtained, it can be used to predict the differential distribution for any combination of the EFT coefficients and it can even be truncated at O(v 2 Λ 2 ) in order to remove dimension-eight terms from squared dimension-six terms. If not stated otherwise, we keep the squared dimension-six terms in the SMEFT predictions in this work. Using the parameterization in Eq. (2.26) we perform Pearson-χ 2 hypothesis tests on the histogrammed distributions with N SM i = n 0i , where the sum runs over all bins with N i ≥ 5. The null hypothesis is that the observed data follows the expected prediction. In the following, we assume that the observed data is given by the SM prediction and the expected result is from the SMEFT. This allows us to quote the p-value for the case that the SM result is a fluctuation of the SMEFT distribution, which can be translated into a 95 % confidence level (C.L.) exclusion for the given hypothesis. To account for systematic uncertainties ∆ sys. from scale variation we rescale the N H i → f H · N H i with factors This brings the total cross sections of two hypotheses closest while maintaining their shape differences.

Results
In the following, we present the phenomenological results of our calculation for the 13 TeV LHC, assuming an integrated luminosity of 300 fb −1 . For a direct comparison we present our findings for tt and tt + Z side by side. We choose kinematic variables that show strong sensitivity to New Physics for each of the respective processes. In the case of tt + Z production the azimuthal opening angle of the two leptons ∆φ from the Z boson decay was identified as the most sensitive observable in Refs. [70,71]. For tt production we choose the transverse momentum p t T of the top quark because it features the largest electroweak correction for a given energy scale. In Fig. 5 we show their distributions for the SM prediction (blue line), including the statistical uncertainties and the systematic uncertainties from QCD scale variation. For the steeply falling p t T distribution we also show the relative O(α) correction with respect to the leading-order prediction, which emphasizes the electroweak Sudakov enhancement (with negative sign) of corrections growing with energy. At tt × Br( ℓ +jets) Figure 6: Size of the relative modifications due to SMEFT contributions (green, magenta, red, and cyan) with respect to Standard Model expectation (blue), after rescaling to account for systematic uncertainties. The light blue band shows statistical uncertainties.
first glance, the significantly different numbers of events between the two processes stick out: While the tt + Z histogram is saturated with O(100) events per bin, the distribution for tt peaks with more than 10 7 at lower p t T . Evidently this directly impacts the respective statistical uncertainties and it is one of the major impact factors on the New Physics constraints presented below. Systematic uncertainties also differ because tt + Z production is currently known at NLO QCD whereas tt is known at NNLO QCD. In Fig. 5 we also present four selected New Physics predictions for some values of EFT coefficients. The differences in both normalization and shape with respect to the SM prediction are clearly visible and suggest good sensitivity to New Physics.
Before quantifying the exclusion power on the EFT coefficients, we study the origin of this sensitivity in more detail. Therefore, in Fig. 6 we plot the relative SMEFT correction with respect to the NLO SM prediction, after accounting for systematic uncertainties through the rescaling described around Eq. (2.28) The ∆φ distribution (left) of tt + Z features shape variations between −10 % and +15 % depending on the SMEFT parameter point and show almost constant uncertainty of about ±8% in the whole range. Hence, we roughly expect between 1-2 σ C.L. exclusion limits for these points. For the same SMEFT parameter points, the p t T distribution in tt production shows shape changes that are significantly larger than the statistical uncertainties over a wide energy range. Only beyond p t T ≈ 600 GeV, there are less and less events that let statistical uncertainties grow to wash out the sensitivity. This is a first hint that we can expect strong New Physics constraints from the electroweak loop corrections to tt.
Let us now come to our main results. Using the fit in Eq. (2.26) for the kinematic variables discussed above, we scan a wide two-dimensional parameter space for the EFT coefficients C ϕq3 33 and C ϕu 33 and perform a Pearson-χ 2 hypothesis test on the histograms presented in Fig. 6. This allows us to quote exclusion contours for a given set of SMEFT parameter values. In the left plot in Fig. 7 we show two such exclusion contours as a function of the (v/Λ) 2 ·C EFT for tt production at the 13 TeV LHC with 300 fb −1 . The orange contour corresponds to the strict expansion up to dimension-six, whereas the yellow contour contains the squared dimension-six terms of order (v/Λ) 4 . Apart from the lower right edge, the differences are minimal, suggesting sufficient convergence of the EFT expansion in terms of powers of Λ −1 . The constraints appear quite strong as the 2-σ exclusion contour ranges only between ± 0.1 4 . The right plot of Fig. 7 contrasts this exclusion limit with the one from tt + Z production 5 , where we keep dimension-eight contributions in both predictions and apply the same statistical analysis. This direct comparison shows the strikingly stronger limits from tt production. For example, the four selected points in Fig. 7 (right) are lying inside the 2 σ C.L. contour of tt + Z production and, therefore, cannot be excluded using this process. Contrarily, the same points are well outside the corresponding tt contour and can be strongly excluded. It is also remarkable that C ϕu 33 is so much stronger bounded. Its allowed range is more than a factor of five smaller in tt than in tt + Z. This is particularly important because C ϕu 33 only modifies the right-handed ttZ coupling d Z R in Eq. (2.2) and does not affect the tbW vertex. Therefore, it cannot be further constrained from single top quark production or top quark decay analyses. Refs. [48,49] report comparatively loose bounds obtained from studying tZ production which also confirm that electroweak corrections to tt are apparently the strongest probe available at the LHC.
Finally, we compare our projected constraints with the most up-to-date experimental results. Members of the CMS experiment recently published an analysis of tt + Z using 77.5 fb −1 integrated luminosity at 13 TeV [44]. Similar to our work here, they assume the decay Z → with electrons and muons. However, they allow three or four charged leptons in the final state such that also di-leptonic tt decays are included. They interpret their results in terms of the same EFT coefficients that we use here (c − ϕQ= C ϕq1 33 − C ϕq3 33 and c ϕt= C ϕu 33 ), so that a rough comparison is possible. If we take Fig. 7 of Ref. [44] and project the edges of the (upper) 2-dimensional 95 % C.L. contour onto the axes, we find in our notation These numbers are in decent agreement (although somewhat weaker) with our presented results in Fig. 7 (right), keeping in mind that we assumed a higher integrated luminosity and just the lepton plus jet tt decay channel. Yet, we take this agreement as an indication for 4 In appendix A the scaling of the exclusion contour with tighter and tighter cuts on the transverse momentum of the top quark is shown. 5 We note that the exclusion contour for tt+Z production matches very well the one in Ref. [70], although a different input parameter set and a different statistical analysis is used.  Figure 7: Left: Exclusion limits on EFT coefficients including (yellow, solid) and neglecting (orange, dashed) the squared dimension-six terms. Right: Exclusion limits obtained from the p t T distributions for tt production including EW corrections (yellow, solid) and the ∆φ + − distributions for tt + Z production (light green, dashed). Colored dots correspond to particular values of the EFT coefficients.
the robustness of our analysis and our conservative uncertainty treatment. It strengthens our proposal to explore the tt final state in future analyses and exploit its superiority.

Conclusions
In this paper, we investigate the question whether New Physics can be probed through electroweak loop corrections in tt production at the LHC and how it compares to direct on-shell probes such as the pp → tt + Z process. The answer is not at all obvious because of two competing effects: The available cross section for top quark pair production is very large and perturbatively under excellent control, allowing for a very precise analysis. Yet, sensitivity only arises at loop level and through off-shell degrees of freedom. For the tt + Z process the opposite is true, at the respective coupling order.
To tackle this question, we calculate the O(α) weak correction to top quark pair production and include New Physics through a Standard Model Effective Field Theory parameterization. In particular, we allow for anomalous interactions between top quarks and the weak gauge bosons. We study differential distributions to exploit enhanced sensitivity from electroweak Sudakov logarithms. After carefully accounting for the most dominant statistical and systematic uncertainties, we find that the tt process at O(αα 2 s ) is significantly more sensitive to New Physics than the tt + Z process. Hence, virtual loop corrections are prevailing over probes with on-shell degrees of freedom. This opens up a promising new way for electroweak top quark studies in tt production at the LHC.
Our study leaves room for further theoretical investigations and future refinements. For example, a calculation including top quark decays will improve the physics modeling and yield additional sensitivity to a subset of the studied EFT operators here. EFT operators affecting the QCD production dynamics or the top quark Yukawa interaction can also be added to extend the current analysis. On should also keep in mind that a realistic modeling of experimental aspects such as detector effects and efficiencies will, to some extend, weaken the sensitivity. Yet, a first comparison with real experimental data for tt + Z production shows that our projections are solid. Therefore, we are looking forward to the first application of our proposal in an experimental analysis of the pp → tt process using real LHC data.

A Dependence of the exclusion limits on p t T cuts
In Fig. 8 we illustrate the dependence of the exclusion limits from Fig. 7 on cuts on the top quark transverse momentum between p t T < 400 GeV and p t T < 100 GeV. We note that these cuts correspond to upper bounds of the invariant mass of the tt system between m tt 870 GeV and m tt 400 GeV indicating the maximal reach in energy of the respective measurement.