Binned top quark spin correlation and polarization observables for the LHC at 13.6 TeV.

We consider top-antitop quark ( t ¯ t ) production at the Large Hadron Collider (LHC) with subsequent decays into dileptonic final states. We use and investigate a set of leptonic angular correlations and distributions with which all the independent coefficient functions of the top-spin dependent parts of the t ¯ t production spin density matrices can be experimentally probed. We compute these observables for the LHC center-of-mass energy 13.6 TeV within the Standard Model at next-to-leading order in the QCD coupling including the mixed QCD-weak corrections. We determine also the t ¯ t charge asymmetry where we take in addition also the mixed QCD-QED corrections into account. In addition we analyze and compute possible new physics (NP) effects on these observables in terms of a gauge-invariant effective Lagrangian that contains the operators up to mass dimension six that are relevant for hadronic t ¯ t production. First we compute our observables inclusive in phase space. In order to investigate which region in phase space has, for a specific observable, a high NP sensitivity, we determine our observables also in two-dimensional ( M t ¯ t , cos θ ∗ t ) bins, where M t ¯ t denotes the t ¯ t invariant mass and θ ∗ t is the top-quark scattering angle in the t ¯ t zero-momentum frame.

The exploration of top-quark spin effects in hadronic top-quark pair production has become an established tool for investigating Standard Model (SM) interactions and for searches of new physics (NP).Evidence for t t spin correlations was found first by the D∅ experiment [1] at the Tevatron and they were first observed by the ATLAS experiment [2] at the Large Hadron Collider (LHC).Subsequently, the ATLAS and CMS experiments at the LHC performed at center-of-mass-energies (c.m.) 7 and 8 TeV, and more recently also at 13 TeV, a number of top spin correlation and polarization measurements in the dileptonic and lepton plus jets final states, using various sets of observables [3][4][5][6][7][8][9][10]. Recently, the ATLAS collaboration used top spin correlations to claim quantum entanglement in top quark pairs [11].This was confirmed by CMS [12].On the theory side, SM predictions for spin correlations and polarizations were made at next-to-leading order (NLO) QCD including electroweak interactions for a number of t t spin correlation and top polarization observables in dileptonic and lepton plus jets final states [13][14][15][16].Radiative corrections to off-shell t t production and decay were determined in [17,18].In [19,20] several t t spin correlations were computed at next-to-next-to-leading order (NNLO) QCD for dileptonic final states.
A suggestion was made in [21] to perform a comprehensive study of spin effects in hadronic t t production by measuring all coefficients of the t t production spin density matrices.For this aim the gg, q q → t t spin density matrices were decomposed in a suitable orthonormal basis and a set of spin observables was proposed that project onto the different coefficients of these density matrices.The t t spin correlation and t and t polarization observables were computed at NLO QCD including weak interaction corrections, and possible NP effects on these coefficients and observables were analyzed by using an effective NP Lagrangian.Other recent analyses of using t t spin correlations at the LHC for probing new physics include [22][23][24][25].
The suggestions of [21] were taken up first by the ATLAS experiment in [26] that measured at √ s had = 8 TeV a subset of the proposed spin observables and compared it with SM predictions [21].A more comprehensive analysis was performed by the CMS experiment [10] at √ s had = 13 TeV.Both experiments measured these spin observables for dileptonic final states inclusive in phase space and found agreement with the SM results.Employing the NP computations of [21] the CMS Collaboration used their data also to constrain anomalous NP top-quark couplings, in particular the anomalous chromomagnetic and chromoelectric dipole moments of the top quark [10].Ref. [24] calculated for a CP-even subset of the proposed spin observables in [21] the NP contributions to quadratic order in the anomalous couplings and determined these observables differentially in phase space.
In this paper we extend the analysis of [21] in several ways.We consider t t production and decay into dileptonic final states at the present LHC c.m. energy √ s had = 13.6 TeV.We compute the t t charge asymmetry A C and the set of top spin observables [21] both in the SM at NLO QCD including weak interaction corrections and in the framework of an effective Lagrangian L NP describing new physics effects in t t production [27][28][29][30][31].We analyze two additional t t spin correlations besides those considered in [21] that are useful in pinning down two anomalous couplings of L NP .First, we calculate our observables inclusive in phase space.Then, in order to explore which areas in phase space are particularly sensitive to the various anomalous couplings, we choose a set of two-dimensional (M t t, cos θ * t ) bins, where M t t denotes the t t invariant mass and θ * t is the top-quark scattering angle in the t t zero-momentum frame, and compute our observables bin by bin.
Our paper is organized as follows.In section 2 we briefly recapitulate the description of t t production and decay in the spin-density matrix framework.Section 3 contains the SU (3) c × SU (2) L × U (1) Y invariant effective NP Lagrangian that we use.Our observables are introduced in section 4. In section 5 we present our results for the charge asymmetry A C and the spin observables, both inclusive in phase space and in two-dimensional bins of (M t t, cos θ * t ).The results for the bins are given in detail in appendix A. We conclude in section 6.

Formalism
We consider t t production at NLO QCD including weak interaction corrections and subsequent semileptonic decays of t and t quarks.At LO QCD top-pairs are produced by gg fusion and q q annihilation, and at NLO also gq and g q fusion contribute.We analyze t t production and decay in the so-called factorizable approximation and use the narrow width approximation Γ t /m t → 0. The t and t spin degrees of freedom are fully taken into account.In this approximation the squared matrix element |M I | 2 of the parton reactions I → F (where F denotes here the dileptonic final state from t and t decay, F = b bℓ + ℓ ′− ν ℓ νℓ ′ + X, (ℓ = e, µ, τ )) is of the form where R I denotes the density matrix that describes the production by one of the abovementioned initial parton reactions I of on-shell t t pairs in a specific spin configuration, and ρ and ρ are the density matrices that encode the semileptonic decays of polarized t and t quarks, see below.The trace extends over the t and t spin indices.
We recall the structure of the t t production density matrices for the 2 → 2 reactions where p j , k j and s 1 , s 2 refer to the 4-momenta of the partons and to the spin 4-vectors of the t and t quarks, respectively.The production density matrix R I of each of these reactions is defined by the squared modulus of the respective matrix element, averaged over the spins and colors of I and summed over the colors of t, t.The structure of the R I (I = gg, q q) in the spin spaces of t and t is as follows: The first (second) factor in the tensor products of the 2 × 2 unit matrix 1l and of the Pauli matrices σ i refers to the t ( t) spin space.The prefactors are given to LO QCD by , where N c = 3 denotes the number of colors.The functions BI± i and CI ij can be further decomposed, using an orthonormal basis which we choose as in [21].The top-quark direction of flight in the t t zero-momentum frame (ZMF) is denoted by k, and p = p1 denotes the direction of one of the incoming partons in this frame.A right-handed orthonormal basis is obtained as follows: Using rotational invariance we decompose the 3-vectors BI± and the 3×3 matrices CI ij (which have a symmetric and antisymmetric part with 6 and 3 entries, respectively) with respect to the basis (5): The coefficients b I± v , c I vv ′ are functions of the partonic c.m. energy squared, ŝ, and of y = k • p which is equal to the cosine of the top-quark scattering angle in the c.m. frame of the initial partons.Notice that the terms in the antisymmetric part of ( 7) can be written as follows: Bose symmetry of the initial gg state implies that the matrix R gg must satisfy If CP invariance holds then The conditions ( 9) and (10) imply transformation properties of the coefficient functions b I± v , c I vv ′ defined in ( 6) and (7).These properties are listed, together with the implications of parity invariance, in detail in Table 1 of [21].
We briefly recall the structure of the decay density matrices ρ, ρ that we use.In this paper we concentrate on semileptonic top-quark decays.At NLO QCD we have where ℓ = e, µ, τ .Considering a fully polarized ensemble of top quarks in the top rest frame and integrating over all energy and angular variables in the decay matrix element, except over the angle θ between the polarization vector of the top quark and the direction of flight of the charged lepton ℓ + , one obtains a decay distribution of the form dΓ ℓ /d cos θ = Γ ℓ (1 + κ ℓ cos θ) /2, where Γ ℓ is the partial width of the respective semileptonic decay.From this decay distribution one obtains the respective normalized one-particle inclusive t-decay density matrix The factor κ ℓ is the top-spin analyzing power of the charged lepton.Its value is κ ℓ = 0.999 at NLO QCD [32,33].For semileptonic t decays the respective normalized decay density matrix Eqs. ( 12) and ( 13) will be used in (1) for the computation of the spin observables of section 4.
The proportionality factor in (1) contains, in the narrow width approximation, the branching fractions of semileptonic t and t decay.
Assuming that new physics (NP) effects in hadronic t t production and decay are characterized by a mass scale Λ that is significantly larger than the moduli of the kinematic invariants of the t t production and decay processes, one may describe these (non-resonant) effects by a local effective Lagrangian L NP that involves the SM degrees of freedom and respects the SM symmetries.Respective analyses include [27][28][29][30][31].
The gauge-invariant operators with dimO ≤ 6 relevant for t t production that involve gluon fields are [27,29,31] Here ν is the gluon field strength tensor.Furthermore, T a are the generators of SU(3) c in the fundamental representation, with tr(T a T b ) = δ ab /2.
The sums O gt +O † gt and O gQ +O † gQ are given by linear combinations of four-quark operators as can be shown using the equation of motion for the gluons [31].These linear combinations of four-quark operators are redundant in our case, because they are included in the set of four-quark operators given below that we use.
In the convention used in (17) we used the top-quark mass m t for setting the mass scale.The real and dimensionless coupling parameters μt and dt are, respectively, the chromomagnetic and chromoelectric dipole moment of the top quark.
In [21] we used an effective gluon Lagrangian that is SU (3) c × U (1) em invariant, but not SU (3) c × SU (2) L × U (1) Y invariant, for reasons of a more agnostic, phenomenological analysis.Besides (17) it contains two additional CP-odd operators that contribute to Pand CP-odd spin correlations and to a P-even, CP-odd polarization observable, respectively.These observables were measured in [10] and the respective coupling parameters ĉ(−−) and ĉ(−+) were constrained.In this paper we will not use these operators, but stick to the There are a number of gauge-invariant dimO = 6 four-quark operators that generate nonzero tree-level interference terms with the q q → t t QCD amplitude.Assuming universality of the new interactions with respect to the light quarks q ̸ = t and considering only operators with u, d quarks in the initial state that interfere with the tree-level q q → t t QCD matrix elements, seven gauge-invariant operators remain [29,31].It is useful to combine these seven operators such that one obtains four isospin-zero operators with definite P and C properties and three isospin-one operators [31].The resulting NP effective Lagrangian involving the u, d quarks reads (as above we use m t for setting the mass scale): where the isospin-zero part is and Here and in the following q = (u, d) denotes the isospin doublet.The isospin-one contribution can be represented in the form where In the isospin-one case it is not possible to combine the operators such that they have definite properties with respect to C and P. In summary we use the effective NP Lagrangian that contains the real, dimensionless coupling parameters μt , dt , ĉIJ (I, J = V, A), and ĉ1 , ĉ2 , ĉ3 .The NP contributions to the coefficients of the t t spin density matrices (4) induced by interference with the tree-level QCD amplitudes of gg, q q → t t are listed in the Appendix of [21].In the following we will stick to these dependencies to first order in the anomalous couplings.This is justified a posteriori by the results [10] of the CMS experiment.The experimental constraints on the dimensionless anomalous couplings of (26) signify that they are markedly smaller than one.
A remark on NP contributions to top-quark decay t → bℓν ℓ is in order.The top-quark decay vertex t → W b may be affected by new physics interactions, but the upper bounds on the respective anomalous couplings inferred from measurements of the W −boson helicity fractions [34,35] show that these effects are very small if non-zero.In this paper, we consider t t production and decay in the dileptonic channel, pp → t tX → ℓℓ ′ X.We use as top-spin analyzers the charged lepton from W decay and we analyse only charged-lepton angular observables that are inclusive in the lepton energies.These observables are not affected by anomalous couplings from top-quark decay if a linear approximation is justified, that is, if these couplings are small [36][37][38][39].As just mentioned this is the case.Thus for the observables that we analyse in this paper only contributions to t t production matter as far as NP effects are concerned.

Observables
We consider t t production at the LHC for the present center-of-mass energy √ s had = 13.6 TeV.We compute the t t cross section and the t t charge asymmetry A C defined in (27) below in the SM and determine, in addition, the contributions from L NP .Then we focus on the dileptonic t t decay channels and investigate a number of spin correlation and polarization observables.First we perform our computations within the SM and to first order in L NP inclusive in phase space.Because future experimental investigations at the LHC aim at more detailed analyses, we then determine these observables in two-dimensional bins of the t t invariant mass and the cosine of the top-quark scattering angle in the t t zero-momentum frame (ZMF) that will be specified below.We do not apply acceptance cuts because experiments usually unfold their data for comparison with theoretical (top-spin) predictions [10,26].
We use the LHC t t charge asymmetry defined by where ∆|y| = |y t |−|yt| is the difference of the moduli of the t and t rapidities in the laboratory frame.
For the dileptonic final states we consider the well-known polar angle double distributions [13,14] for a choice of reference axes â, b: where and, as above, the unit vectors l+ , l− are the ℓ + and ℓ ′− directions of flight in the t and t rest frames, respectively.The coefficients B 1 , B 2 and C signify the t, t polarizations and t t spin correlations, respectively.As we apply no acceptance cuts on the final states and consider only factorizable radiative corrections the coefficients in ( 29) can be related to the expectation values of the spin observables at the level of the intermediate top quarks.We have [14] C Here S t and St denote the t and t spin operators and σ i are the Pauli matrices.The value of κ ℓ is listed below Eq. ( 12).The coefficients B 1,2 in ( 29) are given by where P , P denote the polarization degrees of the t and t ensembles in t t events with respect to the reference axes â, b: The relative signs in front of the coefficients B 1,2 in the distribution ( 29) are chosen such that in a CP-invariant theory and for the choice â = − b: For the choice of reference axes â and b at the hadron level one cannot use the orthonormal basis at the parton level introduced in section 2, because the incoming quark will be either in the right-moving or in the left-moving proton.As in [21] we choose the following set: We use the unit vector k which is the top quark direction of flight in the t t ZMF.Moreover, we use the direction of one of the proton beams in the laboratory frame, pp , and define unit vectors rp and np as follows: The angle θ * t is the top-quark scattering angle in the t t ZMF.Only in the case of 2 → 2 parton reactions and if the incoming parton 1 is parallel to pp , the unit vectors defined in ( 5) are the same as those in (35).The set (35) defines our choice of reference axes â and b which we list in Table 1.The factors sign(y p ) are required because of the Bose symmetry of the initial gg state.In the following the label (a, b) refers to the choice of reference axes â and b from Table 1.The correlation coefficient C associated with this choice of axes is denoted by The spin correlations and polarizations of ( 29), and sums and differences for different choices of reference axes.The unit vectors associated with the labels k * and r * are defined in (36) and (37).
Correlation sensitive to (31) and is called t t spin correlation for short.The analogous labeling applies to B 1 (a) and B 2 (b) which we refer to as t and t polarization with respect to the chosen axis.Table 2 contains the set of spin correlations and polarizations that we consider.The second column shows to which coefficient of the gg and q q spin density matrices the respective observable is sensitive.The third column indicates the P-and CP-symmetry properties of the observables.1The label "absorptive" means that the respective observable is generated by absorptive parts in the scattering matrix.
Apart from computing the observables of Table 2 in the SM we determine also their sensitivity to the couplings of the effective NP Lagrangian (26).It contains the following dimensionless, real parameters: μt , dt , ĉV V , ĉAA , ĉAV , ĉV A , ĉ1 , ĉ2 , ĉ3 .The appendices A.1 and A.2 of [21] show on which parameters a specific coefficient of the gg and q q spin density matrices depends. 2 This dependence determines the dependence on specific NP parameters of σ, A C and the spin observables B and C. It turns out that in order to significantly increase the sensitivity to some of these parameters, it is useful to introduce, in addition to those of Table 1, another set of reference axes [21] to which we assign the labels k * and r * : Here ∆|y| denotes the difference of the moduli of the t and t rapidities in the laboratory frame as defined below (27).With these vectors, one may consider the spin observables respectively their sums and differences, see Table 2.The sum of B 1,2 (k * ) and of B 1,2 (r * ) is sensitive to the NP contributions from the operator O AV and from a P-odd combination of the operators O 1 i , while the sum of B 1,2 (k) and B 1,2 (r) projects onto the contributions of O V A and O 1 3 .The spin correlations C(k, k * ) and C(r * , k) + C(k, r * ), which were not computed in [21], project onto different regions of phase space than C(k, k) and C(r, k) + C(k, r).While the latter are sensitive to ĉV V , ĉ1 , and μt , the former probe the couplings ĉAA and ĉ2 .The P-and CP-odd correlations C(n, r) − C(r, n) and C(n, k) − C(k, n) are equivalent to CP-odd triple correlations, cf.[21], and they probe the chromoelectric dipole moment of the top quark.
Table 2 contains a number of P-odd observables that require absorptive parts.In the SM they result from absorptive parts of weak-interaction contributions and are very small and we do not compute them.There are also no such contributions from our L NP at tree-level.The observables B 1 (n) − B 2 (n) and C(r, k) − C(k, r) are P-even, but CP-odd.The latter requires in addition absorptive parts.Neither SM nor NP interactions from (26) contribute to these observables.
We close this section with a remark on the opening angle distribution [13,14] σ −1 dσ/d cos φ = (1 − D cos φ)/2 where cos φ = l+ • l− is the scalar product of the two lepton directions determined in their parent t and t rest frames.Measurements by ATLAS and CMS have shown (see, for instance, [10,26]) that this distribution is highly sensitive to t t spin correlations.It can be obtained from the diagonal spin correlation coefficients.Using that the vectors defined in Table 1 form orthonormal sets one gets [21] The opening angle distribution can be determined with this formula from the diagonal correlations that will be computed in the next section.
5 Results for 13.6 TeV We compute the cross section, the charge asymmetry A C , and the expectation values of the above spin observables for pp collisions at the c.m. energy of 13.6 TeV.As already emphasized above we do not apply acceptance cuts on the final states, because experiments compare with theory predictions by correcting their measurements to the parton level and extrapolating to the full phase space (see, e.g., [10,26]).We use the CT18 NLO parton distribution functions [40].This set provides also the NLO QCD coupling α s in the MS scheme.We use the on-shell top-quark mass m t = 172.5 GeV.Moreover, we use GeV, and α(m t ) = 0.008.We perform our computations for three values of the renormalization and factorization scale First we compute our observables inclusively, i.e., by integrating over the complete phase space.Then we determine their values in two-dimensional bins of the t t invariant mass M t t and y p = cos θ * t , where θ * t is the top-quark scattering angle in the t t ZMF.We choose the four M t t intervals 600 GeV < M t t ≤ 800 GeV, 800 GeV < M t t .
For each of the four M t t bins we select four bins in y p = cos θ * t : Our SM computations are performed at NLO QCD including the weak-interaction corrections.We refer to it with the acronym NLOW.In the calculation of the charge asymmetry A C also the mixed QCD-QED corrections of order α 2 s α [41] are taken into account in addition. 3he charge asymmetry and the polarization and spin correlation observables B, C used in this paper are ratios.They are, in the SM at NLOW and to first order in the anomalous couplings schematically of the form where N 0 (N 1 ) and σ 0 (σ 1 ) are the contributions at LO QCD (NLOW) and δN NP and δσ NP denote the first-order anomalous contributions to the numerator of the respective observables and the t t cross section, respectively.We use this schematic notation both for results inclusive in phase space and for bins in M t t and cos θ * t .A priori, it is not clear how to evaluate these ratios where the numerator and denominator consist of truncated perturbation series.A typical Monte Carlo analysis would determine the numerators and denominators of (42) to the attainable order and evaluate the ratio without expanding it.In the spirit of perturbation theory one may expand the ratio.One gets at NLOW and to first order in the anomalous couplings: The difference in the two prescriptions ( 42) and ( 43) for computing X are nominally of higher order in the SM and NP couplings.It may be considered as an additional theory uncertainty.
The results of our inclusive calculations will be given in expanded form.When computing our observables for the two-dimensional bins (40), (41) all the six quantities in the ratio (42) will be separately determined.This allows for evaluation of the ratios in either way.From the binned results listed in Appendix A one can also obtain the inclusive results for these quantities which allows for an unexpanded evaluation of the ratios.Moreover, we display our results for the three scale choices µ = m t /2, m t , 2m t .This allows to correctly account for the correlations of theory uncertainties when different observables of Table 2 are combined, which is advantageous for measurements (cf., e.g.[26]).
Table 3 contains the t t cross section at NLOW for the three scales µ.Theory predictions for the cross section are, as is well known, available at NNLO QCD [45,46], including EW corrections [47].We need the NLOW result for the normalization of our observables.There are three contributions to σ t t from the NP Lagrangian (26).The effect of the chromomagnetic dipole operator is most significant and it is dominated by the contribution to t t production by gg fusion.The contributions from the four-quark operators O V V and O 1 3 are subdominant.Notice that the contributions from O 1 3 to uū → t t and d d → t t have opposite sign and thus tend to cancel. 4 Table 3 contains also the charge asymmetry A C in expanded form (43).We recall that there is no contribution to its numerator at LO QCD.The NP contributions result from the isospin-zero operator O AA and the P-even part of the isospin-one operator O 1 2 .These operators induce contributions to the differential cross section that are odd under interchange of the t and t momenta while those of the initial (anti)quark are kept fixed.
Tables 4 and 5 contain our results in expanded form for the spin correlations and polarizations at NLOW in the SM and the NP contributions from the effective Lagrangian (26).The dominant NP contribution to the first four spin correlations in table 4, which are P-and CP-even, is from the chromo-dipole operator that affects also gg → t t besides q q → t t.In particular, the correlation C(r, r) which is small in the SM appears to have a good sensitivity to μt .
The observables C(k, k * ) and C(r * , k) + C(k, r * ) are the spin correlation analogues of A C .The effect of using the vectors k * and r * is that these correlations project onto different y p Likewise, the use of the vectors r, k and r * , k * in the polarization observables B play analogous roles.The respective variables project onto different y p intervals and are thus sensitive to different (combinations of) four-quark operators, as the results of Table 5 show.
The observable B 1 (n) + B 2 (n) corresponds to the sum of the t and t polarizations normal to the scattering plane and is generated by QCD absorptive parts [48][49][50].The absorptive parts of the electroweak corrections to the t t production matrix elements contribute also, but are not shown here.They are roughly half of the QCD contributions and have the same sign.There are no contributions from the hermitean effective NP Lagrangian to LO QCD.
The charge asymmetry A C and the spin correlation and polarization observables of Tables 4 and 5 provide a set that is large enough to measure, respectively constrain, the couplings of the effective NP Lagrangian (26).
One may expect that the sensitivity to a specific anomalous coupling is not uniform in phase space.Therefore, we compute our observables also more differentially, namely, within the two-dimensional (M t t, y p ) bins specified in (40), (41) in order to investigate which region in phase space provides the highest sensitivity to a specific NP coupling.
One may ask whether any of these observables will depend, within a (M t t, y p ) bin, on additional NP parameters besides those shown in Tables 3 -5.For instance, none of the four y p bins ( 41) is parity-symmetric; thus it could be that the P -even observables have additional NP-parameter dependencies within a bin that cancel in the sum over bins.We checked for all our observables that within the above (M t t, y p ) bins no significant additional parameter dependencies appear; that is to say the numerical dependence on an additional NP parameter is at least 3 orders of magnitude smaller than the significant dependencies displayed in the tables of the appendix and are therefore discarded.
Our results for the binned σ t t, A C , and the spin observables are given in Tables 9 -33 of appendix A. As already mentioned above, we compute for each observable each of the six quantities (if non-zero) in the ratio (42) separately.This allows to compute A C and the spin observables either in unexpanded or expanded form.
An inspection of the bins in Tables 22 -33 of the NP contributions to our observables indicates that in almost all cases the two bins at large t t invariant mass M t t > 800 GeV in the central region, −0.5 ≤ y p ≤ 0.5 seem to have the highest sensitivity to the parameters of the effective NP Lagrangian.In order to eventually obtain a reasonable large dileptonic t t data sample, we suggest here to consider the phase-space region M t t > 600 GeV and −0.5 ≤ y p ≤ 0.5.Assuming that at the LHC at 13.6 TeV an integrated luminosity of 300 fb −1 will eventually be collected, and using the t t cross section given in Table 6 for an estimate, one expects about 5 × 10 5 dileptonic ℓℓ ′ (ℓ, ℓ ′ = e, µ) events in this region.We use the four two-dimensional bins M t t > 600 GeV and −0.5 ≤ y p ≤ 0.5 of Tables 9 -33 and compute, by summing the respective numbers of the four bins, the charge asymmetry and the normalized spin observables in this phase-space region in expanded form (43).The results are given Tables 6 -8.Comparing the coefficients that multiply the contributions of the NP parameters ĉIJ , (I, J = V, A), ĉ1 , ĉ2 , ĉ3 in these tables with the respective numbers in Tables 3 -5 of the inclusive results, one sees that the sensitivity to the couplings of the four-quark operators increases significantly in the high M t t, central region.In particular, the spin correlations C(k, k * ) and C(r * , k) + C(k, r * ) appear to be useful for disentangling the contributions from the operators associated with ĉAA and ĉ2 .Not much is gained in this phase-space region for the sensitivity to the chromo-dipole moments μt and dt compared with the inclusive case.

Summary
We have elaborated on a set of spin correlation and polarization observables, proposed previously by two of the authors of this paper, that allows to probe the hadronic t t production dynamics in detail.These observables project out all entries of the hadronic production spin density matrices.We considered t t production and decays into dileptonic final states at the LHC for the present c.m. energy 13.6 TeV.We computed these observables within the Standard Model at NLO QCD including the mixed QCD weak-interaction contributions.Possible new physics effects were incorporated by using an SU (3) c × SU (2) L × U (1) Y effective Lagrangian with operators that generate tree-level interferences with the LO QCD gg, q q → t t amplitudes.The effect of these NP operators on our observables was taken into account to linear order in the anomalous couplings.This can be justified by the results of the CMS experiment at 13.6 TeV [10] which show that these dimensionless couplings must be markedly smaller than one.We considered also the LHC charge asymmetry A C and two additional spin correlations that turn out to be very useful in disentangling contributions from NP four-quark operators.We emphasize that several our of observables allow for direct searches of non-standard P and CP violation in t t events.
In addition to computing our observables inclusive in phase space, we determined them also in two-dimensional (M t t, cos θ * t ) bins, where M t t denotes the t t invariant mass and θ * t is the top-quark scattering angle in the t t zero-momentum frame.Our analysis shows that the sensitivity to a number of anomalous couplings significantly increases in the high-energy central region.Experimental measurements of these observables were so far made only inclusively, and no deviation from the SM was found.The contributions of anomalous couplings to an observable are, however, not uniform in phase space as our results show.More differential measurements in the future, especially in the high-energy central region, promise to significantly increase our knowledge about top-quark interactions beyond the Standard Model.

A SM and NP values of the binned observables
In the tables of this appendix, we present our results for the cross section σ t t, the charge asymmetry A C , and the spin correlations and polarization observables defined in Section 4 and listed in Table 2 for the two-dimensional bins (40), (41) at √ s had = 13.6 TeV.For the cross section, we list the respective values at LO QCD, the contributions at NLOW, and to first order in the NP couplings.For A C and the spin observables, we list the value N 0 of the respective numerator (cf.( 42)) -if it is significantly different from zero -, the contributions N 1 at NLOW, and those of the NP operators δN N P .These quantities allow to compute A C and the spin observables either in unexpanded or expanded form, cf.eqs.( 42) and ( 43).Tables 9 -21 contain our SM results for the three chosen scales µ, while Tables 22 -33 contain the various NP contributions to each observable.Here the values for the three scales are shown in separate tables.
Table 9: The t t cross section at 13.6 TeV in bins of M t t and y p = cos θ * t for 3 scales µ.For each invariant mass bin the first column displays the range of the y p bin.The numbers in the 2nd, 3rd, and 4th column are the values of σ t t at LO QCD (σ 0 ) for µ = m t /2, m t , and 2m t , respectively.The 5th, 6th, and 7th column contain the NLO QCD plus weak-interaction contributions to σ t t (σ 1 ) for µ = m t /2, m t , and 2m t , respectively.All cross-section numbers are in units of pb.27) at 13.6 TeV in bins of M t t and y p = cos θ * t for 3 scales µ.For each invariant mass bin the first column displays the range of the y p bin.The numbers in the 2nd, 3rd, and 4th column are the values of A C at NLO QCD plus electroweak interactions for µ = m t /2, m t , and 2m t , respectively.All numbers are in units of pb.
( -1.0, -0.5) -2.166 -1.628 -1.259 3.028 1.677 0.934 ( -0.5, 0.0) -2.861 -2.130 -1.625 1.458 0.432 -0.097 ( 0.0, 0.5) -2.861 -2.131 -1.627 1.489 0.447 -0.084 ( 0.5, 1.0) -2.173 -1.630 -1.257 2.985 1.647 0.899 Table 14: The numerator of the spin correlation C(r, k) + C(k, r) = N rk /σ t t in the SM at 13.6 TeV in bins of M t t and y p = cos θ * t for 3 scales µ.For each invariant mass bin the first column displays the range of the y p bin.The numbers in the 2nd, 3rd, and 4th column are the values of N 0,rk at LO QCD for µ = m t /2, m t , and 2m t , respectively.The 5th, 6th, and 7th column contain the NLO QCD plus weak-interaction contributions N 1,rk for µ = m t /2, m t , and 2m t , respectively.All numbers are in units of pb.Table 17: The numerator of the polarization observable B 1 (r) + B 2 (r) = N r /σ t t at 13.6 TeV in bins of M t t and y p = cos θ * t for 3 scales µ.For each invariant mass bin the first column displays the range of the y p bin.The numbers in the 2nd, 3rd, and 4th column are the values N 1,r at NLOW for µ = m t /2, m t , and 2m t , respectively.All numbers are in units of pb.

Table 1 :
(35)ce of reference axes at the hadron level.The unit vectors np , rp and the variable y p are defined in(35).

Table 3 :
(27)SM and NP contributions to the t t cross section and the LHC charge asymmetry(27)at √ s had = 13.6 TeV for three renormalization and factorization scales µ. = 2m t 6.48 × 10 −3 0.357 7.40 × 10 −2 intervals than their un-starred analogues.They are sensitive to the couplings ĉAA and ĉ2 .Therefore, they are useful, together with A C , to obtain information about these couplings from experimental data, once they are available.

Table 4 :
(26)spin correlations C at NLOW in the SM and the non-zero contributions of the NP Lagrangian(26)for the c.m. energy √ s had = 13.6 TeV and three renormalization and factorization scales µ.

Table 5 :
(26)polarizations B at NLOW in the SM and the non-zero contributions of the NP Lagrangian(26)for the c.m. energy √ s had = 13.6 TeV and three renormalization and factorization scales µ.

Table 6 :
As in Table3but for the two-dimensional bin M t t > 600 GeV and −0.5 ≤ y p ≤ 0.5.

Table 7 :
As in Table4but for the two-dimensional bin M t t > 600 GeV and −0.5 ≤ y p ≤ 0.5.

Table 8 :
As in Table5but for the two-dimensional bin M t t > 600 GeV and −0.5 ≤ y p ≤ 0.5.

Table 10 :
The numerator of the LHC charge asymmetry A C in the SM defined in Eq. (

Table 24 :
Continuation of Table23.The NP contributions to the binned numerators of the displayed spin observables at 13.6 TeV for the scale µ = m t /2.All numbers are in units of pb.

Table 25 :
Continuation of Table24.The NP contributions to the binned numerators of the displayed spin observables at 13.6 TeV for the scale µ = m t /2.All numbers are in units of pb.

Table 26 :
22 in table22, but for the scale µ = m t .All numbers are in units of pb.

Table 28 :
24 in table24, but for the scale µ = m t .All numbers are in units of pb.ĉ2 to numerator of C(r * , k) + C(k, r * ).

Table 32 :
24 in table24, but for the scale µ = 2m t .All numbers are in units of pb.