Dilepton azimuthal correlations in tt̄ production

The dilepton azimuthal correlation, namely the difference φ between the azimuthal angles of the positive and negative charged lepton in the laboratory frame, provides a stringent test of the spin correlation in tt̄ production at the Large Hadron Collider. We introduce a parameterisation of the differential cross section dσ/dφ in terms of a Fourier series and show that the third-order expansion provides a sufficiently accurate approximation. This expansion can be considered as a ‘bridge’ between theory and data, making it very simple to cast predictions in the Standard Model (SM) and beyond, and to report measurements, without the need to provide the numbers for the whole binned distribution. We show its application by giving predictions for the coefficients in the presence of (i) an anomalous top chromomagnetic dipole moment; (ii) an anomalous tbW interaction. The methods presented greatly facilitate the study of this angular distribution, which is of special interest given the 3.2(3.7)σ deviation from the SM next-to-leading order prediction found by the ATLAS Collaboration in Run 2 data.


Introduction
The production of tt pairs at the large hadron collider (LHC) provides a sensitive probe of the properties of the top quark, both in the production and the decay [1][2][3]. Among many observables investigated by the ATLAS and CMS Collaborations, the correlation between the spins of the top quark and anti-quark is particularly subtle and difficult to measure. It is well known that the Standard Model (SM) predicts a sizeable tt spin correlation [4][5][6]. The spins of t andt are not directly measurable but, due to their short lifetime, they can be accessed through the angular distributions of their decay products. For the decay of a top quark t → W + b, W + → ℓ + ν/du, with ℓ = e, µ, τ , the decay products have the angular distribution 1 Γ dΓ dcos θ i = 1 2 (1 + P α i cos θ i ) , with θ i the polar angle between the momentum of the decay product i = ℓ + , ν,d, u, b, W + in the rest frame of the parent top quark, and some reference axisŝ t ; P is the top polarisation along that axis, and α i are constants that, because of angular momentum conservation, must satisfy |α i | ≤ 1. For the charged leptons the SM prediction is α ℓ = 1 at the tree level, with next-to-leading (NLO) corrections at the permille level [7]. Therefore, the correlation between the charged lepton distribution and the top polarisation is (nearly) maximal. Other top quark decay products have smaller spin analysing power, e.g. α d = 0.96, α u = −0.32, α b = −0.39 at NLO. The angular distributions for the decay of a top antiquark are as in (1) with αī = α i but reversing the sign of the cos θ i term.
For the production and subsequent decay of a tt pair, the normalised doubly differential cross section reads 1 σ dσ dcos θ i dcos θ j = 1 4 (1 − Cα i α j cos θ i cos θ j ) , with θ i , θ j the polar angles between the momenta of the decay products i, j, in the rest frame of the parent top (anti-)quark, and some reference axesŝ t andŝt, respectively.
In the above equation we have neglected the small polarisation of t andt, which yields terms linear in cos θ i and cos θ j . The constant C, with |C| ≤ 1, gives the spin correlation between the top quark and anti-quark for the axesŝ t andŝt. By choosing orthonormal reference systems in the t andt rest frames, it can be seen that there are nine independent spin correlation coefficients [8], corresponding to various combinations of axes for t and t. For example, in the so-called 'helicity basis', that is, takingŝ t andŝt in the direction of the respective top (anti-)quark momenta k t , kt in the tt centre-of-mass (CM) frame, the SM prediction at NLO in QCD and electroweak interactions is [8,9] C kk = 0.310 at a CM energy of 7 TeV, C kk = 0.318 at 8 TeV, and C kk = 0.331 at 13 TeV. The nine spin correlation coefficients have been measured by the ATLAS Collaboration at 8 TeV [10]. Previously, the ATLAS and CMS Collaborations measured C kk in the helicity basis at 7 and 8 TeV [11][12][13]. The measurements are consistent with the SM predictions, see Table 1, though the uncertainties are large. These uncertainties partly arise from the need to reconstruct the t andt rest frames, as well as the tt CM frame, from their decay products. In the dilepton channel tt → ℓ + νb ℓ −νb , the reconstruction faces a combinatoric ambiguity due to the two missing neutrinos. In the semileptonic mode tt → qq ′ b ℓ −νb , q = u, c, q ′ = d, s (and the charge conjugate decay) with one neutrino the reconstruction is easier but the discrimination between light quarks, based on tracking variables and jet transverse momentum p T , is quite difficult.
A simpler probe of the tt spin correlation in the dilepton decay mode was pointed out in Ref. [14]: the laboratory frame dilepton azimuthal correlation, namely the difference φ = |φ ℓ + −φ ℓ − | between the azimuthal angles of the two charged leptons, taking theẑ axis in the beam direction. (A predecessor of this correlation was proposed for Z → τ + τ − at the Large Electron Positron Collider, using decay products of the τ leptons [15,16].) The dσ/dφ distribution inherits the top spin correlation, and is presented in Fig. 1, calculated 7 TeV 8 TeV ATLAS 0.315 ± 0.078 [12] 0.296 ± 0.093 [10] CMS 0.08 ± 0.14 [11] 0.276 ± 0.082 [13]   at NLO in QCD interactions for a CM energy of 8 TeV (see the next section for details).
For comparison, we also show the distribution in the absence of spin correlations. This angular distribution allowed to establish the existence of tt spin correlations at the 5.1σ level already with 7 TeV [17]. In order to do so, the experimental distribution was compared to a linear combination of the SM and unpolarised one, i.e. the 7 TeV analogues of the distributions shown in Fig. 1, depending on a parameter f SM , The best-fit value of the parameter f SM was obtained with a likelihood method. The result f SM = 1.30 ± 0.14 (stat) +0.27 −0.22 (sys) allowed to exclude the no correlation hypothesis at the 5.1σ level. Later measurements by the ATLAS and CMS Collaborations have been performed at 7, 8 and 13 TeV [13,[18][19][20] following the same procedure, and the results for f SM are collected in Table 2. In particular, the 13 TeV measurement by the ATLAS Collaboration departs 3.2σ from the NLO prediction (3.7σ without including theory uncertainties), following a trend that was already present in earlier measurements ATLAS CMS 7 TeV 1.19 ± 0.09 (stat) ± 0.18 (sys) [18] -8 TeV 1.20 ± 0.05 (stat) ± 0.13 (sys) [19] 1.14 ± 0.06 (stat) +0.15 −0.17 (sys) [13] 13 TeV 1.250 ± 0.026 (stat) ± 0.063 (sys) [20] - Table 2: Measurements of the best-fit parameter f SM in (3) in the tt dilepton decay mode by the ATLAS and CMS Collaborations.
but is more apparent in this latest one. For the semileptonic tt decay mode a measurement involving the analogous azimuthal angle difference between the charged lepton and the jets has been performed, yielding f SM = 1.12 ± 0.11 (stat) ± 0.22 (sys) [18].
Using 8 TeV data, the CMS Collaboration has placed limits on new physics directly from the shape of the normalised distribution [13], using the SM prediction at NLO and the first-order contribution from an anomalous top chromomagnetic moment, calculated at leading order (LO) [9]. The same has been done in Run 2 at 13 TeV, but using also the normalisation as well as the shape [21]. However, using directly the binned distributions for the theory predictions and to compare with experimental measurements is impractical and difficult to reproduce, for example if one wants to set limits on other types of new physics from experimental data. Instead, it is very convenient to provide the predictions and results in terms of a few numbers, which can then be compared to test the consistency of the SM with data and set limits on new physics. Clearly, the parameter f SM in (3) is not suitable for that, despite its usefulness in establishing the existence of tt spin correlations. First, because the linear combinations g(φ; f SM ) cannot parameterise any normalised dσ/dφ distribution, namely, not all possible distributions can be written in the form (3). In order to make this statement apparent, we generate a distribution with non-SM spin correlation by injecting a top chromomagnetic dipole moment (see section 3 below for details) that yields f SM = 1.15, and compare it in Fig. 2 with the best-fit function g(φ; f SM ) with f SM = 1.15.
A second reason that disfavours f SM as a parameter to report the measurements is that its experimental determination relies on two theory predictions, with their corresponding uncertainties: for the SM and for the hypothetical uncorrelated tt production. It is clearly preferrable to provide the result of experiments via theory-independent measurements, and subsequently compare them to the predictions. We note that, since the nine independent C spin correlation coefficients fully determine the tt spin correlation, the dilepton azimuthal correlation must depend on them. 1 However, the dependence may be 1 Because this distribution in principle depends on all the nine spin correlation coefficients C, and not only the one for the the helicity basis, using a measurement of f SM from the dilepton azimuthal quite complicated, since it involves boosts from the t andt rest frames to the laboratory frame. It is then more convenient to find a direct, fully general parameterisation of the dσ/dφ distribution. This will be our task in section 2, where we show that a Fourier series expansion up to third order suffices to accurately reproduce the actual distributions. Needless to say, the Fourier coefficients are independent of the particular binning used to report the measured distributions, then they make it very easy to compare results from the ATLAS and CMS experiments, as well as to compare these results with theoretical predictions.
Ultimately, one is also interested in how the dilepton azimuthal correlation can constrain (or be a signal of) possible new physics effects. This can easily be accomplished with a theoretical calculation of the dependence of the Fourier coefficients on the coupling(s) of the new physics, as we show in section 3 with the example of anomalous top chromomagnetic moments and in section 4 with the example of an anomalous tbW interaction. With a set of three functions, giving the dependence of the Fourier coefficients on the new physics coupling(s), one can easily reproduce the prediction for the whole distribution including new physics effects. In section 5 we use this framework to address the 13 TeV deviation in detail, in the context of an anomalous top chromomagnetic moment, addressing the interplay between the azimuthal correlation and other observables like the total cross section and spin correlation coefficients.
correlation to determine the latter, as it has often been done in the literature, is conceptually incorrect.
A series expansion of an angular distribution is quite useful as a bridge between theory and data -provided a small subset of coefficients can accurately reproduce the distribution -but also provides a bonus: it allows to investigate subtle effects in experimental data that might manifest in the higher-order coefficients. This type of tests can be performed not only on the dσ/dφ distribution on which we focus here, but on any angular distribution, not only in order to probe the presence of new physics indirect effects, but also to test the modeling of the signal, the unfolding procedure, etc. In our discussion in section 6 we briefly comment on this issue.

Deconstructing the azimuthal correlation
One may extend it to [−π, π] by taking φ = φ ℓ + − φ ℓ − , as some authors do, in which case it would be symmetric around zero in this interval. Therefore, the Fourier expansion of these distributions only contain cosines, The constant term is the overall normalisation. In our case, since φ ∈ [0, π], we have a 0 = 1/π for the normalised distribution.
We calculate the coefficients in the expansion (4) in the SM at NLO in QCD using MadGraph5 [22] with NNPDF 2.3 [23] parton density functions (PDFs), setting dynamic factorisation and renormalisation scales equal to the total transverse mass, Q = i m T i , with the transverse mass defined as m T = m 2 + p 2 T . The scale uncertainty is estimated by using twice and one half of this value. The samples generated have 10 6 events; the number of positive weight events minus the number of negative weight events is around 6.6 × 10 5 . (This would be the typical size of data samples after event selection in the dilepton channel for 50 fb −1 at 13 TeV.) The Monte Carlo statistical uncertainty is estimated by generating two independent samples. Results for the first coefficients, at CM energies of 8 TeV and 13 TeV, are collected in Table 3. For comparison, we also show the coefficients for the SM at LO (using samples of 5 × 10 5 events) and hypothetical unpolarised case. The latter are calculated using MCFM [24] and the uncertainty quoted is from Monte Carlo statistics only.
The 'effective' spin correlation, that is, the slope of the distribution (approximately represented by the best-fit parameter f SM ) mainly depends on the first non-trivial coefficient a 1 . The effect of a 2 is small, and the influence of a 3 and a 4 is marginal. This also  Table 3: Lowest order coefficients in the expansion (4) of the normalised dσ/dφ distribution, for CM energies of 8 TeV (up) and 13 TeV (down).
happens when including new physics contributions of a moderate size in the production or the decay. For a 4 the uncertainty given in Table 3 is dominated by the Monte Carlo statistics. At both energies this coefficient and higher-order ones are very small, therefore the distributions are well approximated by the third-order expansion, and we will do so in the following. In Fig. 3 we compare the actual distribution obtained from the Monte Carlo simulation for the SM at 8 TeV, with the third-order expansion with the coefficients in Table 3, finding very good agreement.
We have explored other possibilities for the parameterisation of the distributions. One obvious candidate would be an expansion in terms of Legendre polynomials, This type of expansion was already used by the CDF Collaboration to investigate the anomalous forward-backward asymmetry observed in tt production at the Tevatron [25].
This is equivalent to the Fourier series (4) we have used, as it can easily be shown using trigonometric identities, but more complicated because the normalisation of the distribu-  Table 3.

Effect of a top chromomagnetic moment
As an example of new physics in tt production that modifies the spin correlation we consider a top chromomagnetic moment. The ttg interaction, including the SM as well as the contribution from gauge-invariant dimension-six operators, can be written as [26] in standard notation, with d V and d A the top chromomagnetic and chromoelectic dipole moments respectively, G a µν the gluon field strength tensor, λ a the Gell-Mann matrices, g s the strong coupling constant and m t the top quark mass. The second term contains both ttg and ttgg interactions and can arise from the dimension-six operator [27] can be constrained from the measurements of inclusive cross sections [28][29][30][31], differential distributions [32][33][34][35][36], and the tt spin correlation [9]. For simplicity we will set d A = 0 and study the influence of a non-zero d V on the dilepton azimuthal correlation. In the SM a chromomagnetic moment d V = 0.007 is generated at the one loop level, mainly arising from QCD corrections to the ttg vertex [37]. We ignore it in our calculations, as these QCD corrections are already included in our NLO calculation for pp → tt, and only consider anomalous contributions to the second term in (7).
The dipole interactions enter at most twice the amplitudes for tt production, therefore the cross section depends quadratically on d V . The dependence of the Fourier coefficients in (4) on d V can be obtained with a simple procedure. One first considers the unnormalised distribution withā 0 = σ/π. Because the functions cos nφ are orthogonal in [0, π], with π 0 cos 2 nφ dφ = π/2, the coefficients can be obtained from a sample of N unweighted events as with j running over the events and φ j the corresponding value of φ. By generating event samples for different values of d V , and extractingā n for each sample, their functional dependenceā n (d V ), which is a fourth-order polynomial too, can be determined. The coefficients of the normalised distribution are .
Our calculations are performed including the SM NLO contribution, the interference between the LO SM and new physics, and the pure new physics contributions at LO. This is the approach taken in Ref. [9], with the difference that we use non-expanded denominators when computing the normalised a n , and keep terms beyond the linear order in d V . There are several arguments [3,38,39] that justify keeping all the terms even if dimension-eight operators are not included. At variance with Ref. [9], we also include a factor K = σ NLO SM /σ LO SM in the LO calculations of the new physics contribution and its interference with the SM, in order to improve the approximation and mimic the effect of calculating higher orders in the new physics contributions too. 2 The new terms in the Lagrangian (7) are implemented in Feynrules [40] and interfaced to MadGraph5 using the universal Feynrules output [41]. At each CM energy seven samples of 5 × 10 5 events are calculated for different values of d V , to determine the quartic dependence of a n with some redundancy and reduce the uncertainty from Monte Carlo statistics. For 8 TeV the fit yields, for the reference factorisation scale Q equal to the total transverse mass,ā Again, d 3 V terms are subdominant and d 4 V terms can safely be neglected. The scale uncertainty is estimated by setting the factorisation and renormalisation scales equal to twice and one half of the dynamic scale Q, and repeating the above procedure. The results for the normalised coefficients a 1 , a 2 and a 3 are presented in Fig. 4, for CM energies of 8 TeV and 13 TeV. The uncertainty bands take into account the scale uncertainty and also the statistical Monte Carlo uncertainty. Furthermore, the bands are symmetrised around the reference predictions by taking the largest deviation with respect to the reference sample, in order to cover a possible bias in the fits (12) and (13) due to the Monte Carlo statistical uncertainty.
Overall, we observe that the scale uncertainty in a 1 , a 2 and a 3 is small. As anticipated, a 1 is the observable governing the effective spin correlation, which for d V small and positive increases up to d V ∼ 0.04, reaching an effective correlation f SM ∼ 1.15, and decreases for larger d V . For negative d V the effective spin correlation decreases from the SM value. Since the 13 TeV ATLAS measurement f SM = 1.250 ± 0.068, is 3.7σ above the SM prediction, one expects tight constraints on negative values of d V . Using this value of f SM as input, together with the dependence of the coefficients we have calculated, we estimate the limit 0.017 ≤ d V ≤ 0.059 at the 95% confidence level (CL). As we have mentioned, with the 8 TeV dataset the CMS Collaboration already has obtained limits on d V from the normalised dσ/dφ distribution, −0.027 ≤ d V ≤ 0.021 at the 95% CL [13]. With 35.9 fb −1 of 13 TeV data the limit using both the normalisation and the shape of the distribution is −0.0018 ≤ d V ≤ 0.012 [21]. For comparison, indirect limits from rare B meson decays imply −0.0012 ≤ d V ≤ 0.0038 at the 95% CL [42]. The latter limits are quite modeldependent, however. A further study of this deviation and its potential explanation in terms of an anomalous top chromomagnetic moment is presented in section 5.

Effect of a tbW anomalous coupling
Although new physics in the top quark decay does not modify the tt spin correlation, it changes the spin analysing power of the the charged lepton α ℓ in (1), as well as the lepton energy distribution in the top quark rest frame, thereby modifying the dilepton azimuthal correlation. New physics in the tbW vertex can affect several observables in the top quark decay, for example the W polarisation fractions [43,44], general W spin observables [45], and the single top production cross sections [46][47][48][49], and are quite constrained by the measurement of those observables in top production and decay. However, there are 'flat directions' in which the constraints are much looser. The tbW interaction including contributions from dimension-six operators can be written as [26] with g the electroweak coupling, M W the W boson mass, and q = p t − p b its momentum; V L equals the Cabibbo-Kobayashi-Maskawa matrix element V tb in the SM, and V R , g L and g R are anomalous couplings, which vanish in the SM at the tree level. One example of a flat direction is a combination of anomalous couplings with M W V R = m t g L . This combination can be generated by the redundant dimension-six operators [27] O 33 An interaction of this type only modifies the diagonal entry in the W spin density matrix corresponding to (0, 0) helicities, therefore the constraints on h L are loose (see Ref. [50] for a detailed discussion). Moreover, a coupling h L in the tbW vertex is equivalent to anomalous couplings V R = m t h L , g L = M W h L in the minimal Lagrangian (14), plus small terms proportional to the b quark mass. Therefore, the insensitivity to the interaction (16) is translated into a flat direction in the (V R , g L ) plane.
We have followed the same procedure outlined in the previous section to calculate the dependence ofā n on the anomalous coupling h L , with one minor difference. A nonzero h L changes the top width Γ t , so that the production × decay cross section does not change in the narrow width approximation. Because we are interested in the normalised distribution, we can for simplicity keep Γ t fixed to its SM value in the calculations, in which case the dependence of the unnormalisedā n on h L is a fourth order polynomial. The difference with respect to the calculation with a varying Γ t is common for allā n , so it cancels when making the ratio to obtain the normalised quantities. We generate seven samples for each CM energy, at the reference factorisation and renormalisation scale Q, and repeat the same for scales Q × 2 and Q/2 to estimate the scale uncertainty. At 8 TeV we find, defining the shorthandĥ L = h L /100, a 0 (pb) = 0.718 − 0.0736ĥ L + 1.
At 13 TeV we find a 0 (pb) = 2.39 − 0.243ĥ L + 6.08ĥ 2 The quadratic and quartic terms are the dominant ones for the range of interest |h L | 0.01. We note that, despite the fact that at leading order α ℓ is not modified by new physics [51][52][53], linear terms in h L appear in the above equations. These are justified by the potential dependence on h L of the lepton energy distribution in the top rest frame, which also affects the dσ/dφ distribution. The predictions for the normalised coefficients a 1 , a 2 and a 3 are presented in Fig. 5.
The 13 TeV ATLAS measurement as well as previous ones indicate an enhanced effective correlation, therefore using the measurements of f SM as input to obtain constraints on h L , which modifies the correlation in the opposite way, is not sensible. Instead, to estimate the potential to set limits on h L -once the source of the present discrepancy is identified -we can use f SM = 1 ± 0.068 as input, centred at the SM prediction and with the uncertainty of the ATLAS 13 TeV measurement. Proceeding this way we obtain a very tight limit, |h L | ≤ 1.1 × 10 −3 . Translated into the couplings in the Lagrangian (14), this amounts to V R ≃ 0.19, g L ≃ 0.088. The only existing limits covering this flat direction have been obtained by the ATLAS Collaboration in Ref. [54] with an analyis of triple differential angular distributions in top quark decays [55]. The point V R = 0.19, g L = 0.088 is well within the 1σ allowed region, which extends up to V R ≃ 0.3, g L ≃ 0.15. This highlights the sensitivity of this distribution to probe anomalous contributions in the top decay, a possibilty that is yet unexplored.

The 13 TeV anomaly on focus
The parameters characterising the dσ/dφ distribution measured by the ATLAS Collaboration can easily be obtained by digitising the plot. At the parton level, in full phase space, they are a 1 = −0.0512 , a 2 = 0.0082 , a 3 = −0.0021 .
and in fiducial phase space (with acceptance cuts) The determination of these coefficients with their correlation and uncertainties requires using the full event dataset, and has to be performed by the ATLAS Collaboration. Still, the values above obtained provide a good starting point to address possible explanations of the anomaly. Here we use the full phase space quantities to compare with put Monte Carlo predictions, noting that, although the corrections from the fiducial to full phase space are sizeable, the discrepancy is already observed in the former, c.f. Ref. [20]. The distributions presented in Fig. 6 correspond to, from steeper to flatter slopes: (a) Uncorrelated tt production, calculated with MCFM using CT14 PDFs [56] and fixed factorisation and renormalisation scales equal to the top mass.
(b) The SM LO prediction, calculated with Madgraph using NN23 PDFs and factorisation and renormalisation scales the total transverse mass.
(c) The SM NLO prediction, also calculated with Madgraph, using the same scales. The shaded band around the line corresponds to the scale uncertainty obtained using one half and twice the total transverse mass. An alternative NLO prediction  (d) The SM NLO prediction above plus a chromomagnetic coupling d V = 0.036, which yields a 1 = −0.0705 (corresponding to f SM ≃ 1.15). As we have mentioned, this is the maximum effective correlation that can be achieved in this context.
(e) The ATLAS result in Ref. [20]. The points correspond to the ATLAS data and their uncertainties.
The continuous lines in Fig. 6 correspond to the third-order Fourier expansion in all cases. As this plot highlights, the difference between data and the various predictions is quite significant. Some comments are in order.
1. The shift in the SM prediction from LO to NLO is much smaller than the difference between the NLO prediction and data, suggesting that the deviation is not due to higher-order corrections unaccounted for. The inclusion of ttj at the partonic level, matched with tt, gives some enhancement of the effective correlation [20], but the Monte Carlo predictions used by the ATLAS Collaboration still deviate from data by 3.2σ.
2. The transverse momentum distributions of the top (anti-)quark measured by the CMS Collaboration also present discrepancies with respect to theory calculations [21] but these do not seem enough to explain the ATLAS deviation. By applying a crude linear reweighting factor of 1.07 − 4.8 × 10 −4 GeV −1 × p t T , obtained from the top p T distribution in fiducial phase space in Ref. [21], the NLO dσ/dφ distribution is mildly modified, from a 1 = −0.0764 to a 1 = −0.0719, still far from the ATLAS measurement.
3. An anomalous top chromomagnetic moment alone seems insufficient to explain the deviation observed. It remains to be seen how large the effective correlation can be by including tt plus jets in the presence of a non-zero d V , higher-order corrections, etc. but reaching the observed distribution seems hard.
With these caveats in mind, let us now discuss other possible effects and constraints on this potential explanation of the anomaly, also as a reference for other SM effects or new physics contributions that can modify the distribution. An anomalous top chromomagnetic moment of the size d V ≃ 0.04, so as to maximise the effective correlation, can be accommodated by the measurements of the total tt cross section, because the predictions are somewhat dependent on the factorisation and renormalisation scales. At 8 TeV our predictions are, for the reference dynamic scale Q = i m T i , twice, and one half of this value, dropping terms of third and fourth order. Using the combination of ATLAS and CMS tt cross section measurements in the eµ dilepton channel σ exp = 241.5 ± 1.4 (stat) ± 5.7 (sys) ±6.2 (lumi) pb [57], we find the loose constraint 0.006 ≤ d V ≤ 0.046 if we require the agreement of any of the predictions in Eqs. (21) with this measurement, within two standard deviations. Next-to-next-to-leading corrections and soft-gluon resummation [58] increase the total cross section by around 8%, relaxing the small tension between the SM predictions (d V = 0) and the measurement, and strengthening the upper bound on d V . At 13 TeV our predictions for the cross section are The naive average of the most precise 13 TeV measurements by the ATLAS [59] and CMS [60] Collaborations is σ exp = 853 ± 24 pb. Requiring 2σ agreement of any of the predictions in Eqs. (22) with this value, we obtain the constraint 0.014 ≤ d V ≤ 0.048. Again, the small tension with the SM predictions (d V = 0) would be relaxed by including higher-order corrections, and the upper limit on d V would be tighter.
A non-zero d V also modifies the spin correlation coefficients C in (2). We restrict ourselves to C kk in the helicity basis, for which the naive average of ATLAS and CMS measurements at 8 TeV (see Table 1) is C kk = 0.284 ± 0.061. Another distribution of interest is the polar angle θ ij between the momenta of the decay products i, j, in the respective rest frame of the parent top (anti-)quark, The spin correlation coefficient D can be written in the basis of nine independent C coefficients [8], as with C rr and C nn the diagonal spin correlation coefficients corresponding to axes orthogonal to the (anti-)top momentum k, within the production plane ( r) and perpendicular to it ( n). The D coefficient has been precisely measured by the CMS Collaboration at 8 TeV, yielding [13] D = −0.204 ± 0.02 (stat) ± 0.024 (sys). The ATLAS Collaboration has not directly measured D from the distribution (23), but it can be obtained from the measurement of the C coefficients [10] by using the relation (24). Ignoring the correlations between experimental uncertainties, we obtain D = −0.229 ± 0.060. The naive average of these two measurements, D = −0.209 ± 0.028, is dominated by the direct determination by the CMS Collaboration.
At 8 TeV our predictions for the reference scale Q are with σ the total cross section in the first of Eqs. (21). At 13 TeV we obtain with σ the total cross section in the first of Eqs. (22). For the d V interval of interest, third and fourth order terms can safely be neglected at both CM energies. The dependence of C kk and D on d V is depicted in Fig. 7, with the uncertainty bands computed using scales Q × 2 and Q/2, as in the previous sections. In the predictions for 8 TeV we include for comparison horizontal bands corresponding to the above obtained averages of experimental measurements, with their 1σ uncertainty. Because the relative contributions of a non-zero d V to C rr and C nn are larger [8] than the contribution to C kk , the variation of D is more pronounced. Therefore, we observe that, although the measurements of C kk are not very restrictive, the measurements of D disfavour values of d V at the few percent level. Still, there is a caveat in the comparison because the measurement [13] of D includes the e + e − and µ + µ − dilepton channels, for which a cut on dilepton masses m ℓℓ around the Z boson mass is applied to reduce the background from Z plus jets. The presence of this cut at the reconstructed level, which affects events with smaller φ, might bias the comparison of the unfolded measurement of D with theory predictions in the presence of new physics. These arguments and caveats are expected to hold for other types of new physics yielding an enhanced spin correlation.

Discussion
The use of a Fourier series to study the behaviour of a function is two centuries old. Still, series expansions have rarely been used in collider phenomenology to parameterise and scrutinise angular distributions. We advocate their use as a bridge between theory and experiments: (i) As a simple method to cast theory predictions for distributions that are otherwise difficult to parameterise. The series expansion allows the experiments to determine the full distribution with any desired binning.
(ii) As a simple output to report experimental measurements (including their correlation when necessary), allowing the theorists for easy reinterpretations by comparing the coefficients predicted by any model with the measured ones.
As an example of (i), we have considered the dilepton azimuthal correlation in tt production and its expansion as a Fourier series, finding that the number of coefficients required to determine the distribution is quite small. We have shown in sections 3 and 4 that theory predictions for dσ/dφ including new physics in the production of tt pairs or in the top quark decay can easily be synthesised in terms of these coefficients. This allows for an easy study and comparison of different predictions with data, as done in section 5. It also allows the experiments to easily investigate potential new physics effects from the measurement of this angular distribution, by using the analytical dependence of the Fourier coefficients on the new physics couplings. The results in section 4 are interesting on their own, as they show the good potential of this distribution to set limits on a flat direction in the parameter space of anomalous tbW couplings.
A third advantage of a series expansion is the potential to pinpoint subtle deviations in the distributions, which might manifest in higher-order coefficients. These deviations might be caused not only by new physics, but also by detector effects. This possibility motivates the use of series expansions in the analysis of other angular distributions, even those that are easily parameterised and for which (i) and (ii) above are not needed. Let us consider for example the well-known angular distribution in top quark decays t → W + b → ℓ + νb, corresponding to the angle θ * ℓ between the momentum charged lepton momentum in the W rest frame and the W boson momentum in the top quark rest frame. The normalised distribution is 1 Γ dΓ dcos θ * = 3 8 (1 + cos θ * ) 2 F + + 3 with F + , F − and F 0 the W helicity fractions [43], satisfying F + + F − + F 0 = 1. The functional form of this distribution is determined by angular momentum conservation, whereas the values of the helicity fractions are given by the interactions, for example F 0 ≃ 0.703, F − ≃ 0.297, F + ≃ 0 in the SM at the tree level. This distribution admits a finite expansion in Legendre polynomials as written in (5) but with x ≡ cos θ * . The non-zero coefficients are However, higher-order coefficients can be generated by detector effects. For example, we have verified that a deficient reconstruction of the W rest frame, arising from a mismeasurement of the missing energy from the neutrino, can generate non-zero b 4 , b 5 , etc. With the high statistics that will be available at the LHC Run 2, it is of the utmost importance to have under very good control the signal modeling, data unfolding, etc. in order to perform precision physics. A series expansion, as proposed in this work, can reveal subtle effects and may become a very useful tool in order to test the robustness of the modeling, especially in case any deviation from the SM is found, as it may be the case with the 13 TeV ATLAS measurement of the azimuthal correlation.
Last, but not least, we have used the proposed framework to investigate in detail this anomaly, within the SM and in the presence of an enhanced top chromomagnetic coupling. We find that the deviation of data from SM predictions is unlikely to be due to missing higher-order corrections, and to explain this deviation in terms of an anomalous coupling is also difficult, though further work in this direction is required.