A set of top quark spin correlation and polarization observables for the LHC: Standard Model predictions and new physics contributions

We consider top-antitop quark ($t{\bar t}$) production at the Large Hadron Collider (LHC) with subsequent decay into dileptonic and lepton plus jets final states. We present a set of leptonic angular correlations and distributions with which experiments can probe all the independent coefficient functions of the top-spin dependent parts of the $t{\bar t}$ production spin density matrices. We compute these angular correlations and distributions for LHC center-of-mass energies 8, 13, and 14 TeV within the Standard Model at next-to-leading order in the QCD coupling including the mixed QCD-weak corrections and for the transverse top-quark polarization and the $t{\bar t}$ charge asymmetry also the mixed QCD-QED corrections. In addition we analyze and compute the effects of new interactions on these observables in terms of a gauge-invariant effective Lagrangian which contains all operators relevant for hadronic $t{\bar t}$ production up to mass dimension six.


Introduction
All available experimental data on top quarks show that this particle behaves as a bare quark that does not hadronize. This fact implies that observables associated with or induced by top-quark spin effects can be used as reliable tools for investigating the interactions of this quark. As far as hadronic top-antitop quark (tt) production is concerned, the correlation of the t andt spins and polarization of the t andt samples in tt events were first experimentally explored at the Tevatron [1][2][3][4]. Evidence for tt spin correlations was first reported by the D∅ experiment [3] and they were first observed by the ATLAS experiment [5]. The ATLAS and CMS experiments at the LHC (7 and 8 TeV) have performed a number of top spin correlation and polarization measurements, using various sets of observables [5][6][7][8][9][10][11]. These measurements have reached a level of precision that allow for a meaningful interpretation. All measurements of tt spin correlations and top-quark polarization so far are in agreement with respective Standard Model (SM) predictions [12][13][14][15].
The degree of top-quark polarization in a tt sample and the strength of the correlation of the t andt spins depends on the production dynamics and, for a 'given' production dynamics, on the choice of observable, that amounts to a choice of reference axes. (These axes may be interpreted as t andt spin quantization axes.) In order that future measurements at the LHC will be able to extract complete information about the tt production JHEP12(2015)026 dynamics from their data as far as top-spin effects are concerned, it is desirable to devise a set of observables, in some basis, which probe all of the independent coefficient functions of the top-spin dependent parts of the tt production spin density matrices. In this paper we present such a set. The members of this set are constructed such that they have definite properties with respect to discrete symmetries. For definiteness we consider, as far as tt spin correlation effects are concerned, the dileptonic tt decay modes and, as far as polarization effects are concerned, also the lepton plus jets final states. These modes are the most sensitive channels for the investigation of top-spin effects. We compute the distributions and expectation values of these observables at next-to-leading order (NLO) QCD with respect to tt production and decay and take into account also the NLO mixed QCD weak-interaction contributions and for the transverse t andt polarizations also the mixed QCD-QED corrections of order α 2 s α. In addition, we assess the sensitivity of our set of spin correlation and polarization observables to new physics (NP) effects in the framework of the effective field theory approach. We take into account all gauge invariant operators of dimension six which are relevant for hadronic tt production [16][17][18][19][20][21][22]. We compute the contributions of the associated anomalous couplings to our observables. There is already an extensive literature on phenomenological studies, including  and [15, 49-68, 70, 71], on how anomalous couplings, in particular, a non-zero chromo-magnetic and/or chromoelectric dipole moment of the top quark would affect tt production at hadron colliders. In fact, if one considers only the two chromo-moments and puts all other anomalous couplings to zero, it suffices to take only the experimental results for the tt cross section at the Tevatron and the LHC (7 and 8 TeV) and the respective NNLO QCD predictions [72] in order to obtain sensitive bounds on these two anomalous couplings [50,55,68]. The point we want to make with this part of our analysis is that our set of observables, including the tt charge asymmetry 1 is i) large enough to get information on all anomalous couplings of our effective Lagrangian (in fact the number of observables in this set exceeds the number of couplings) and ii) these observables allow to disentangle the various anomalous contributions to a considerable extent. This paper is organized as follows. In the next section we decompose the tt production density matrices of the reactions gg, qq → tt in terms of an orthonormal basis and classify the coefficient functions of these matrices with respect to parity P, CP, naive time reversal T N , and Bose symmetry. The decomposition applies to all parton reactions ij → ttX. In section 3 the effective Lagrangian L NP is introduced that we use for describing non-resonant new physics in hadronic tt production. We use the operator basis employed in [22], albeit with different normalization conventions for the anomalous couplings. Possible new physics contributions to the t → W b decay vertex do not affect our leptonic angular correlations and distributions defined in section 4. We briefly recall the presently available bounds on the anomalous couplings contained in L NP and discuss the domain of applicability of the linear approximation used in section 4. In this section, we consider tt production and decay into dileptonic and lepton plus jets final states at the LHC. For the dileptonic final states, we present a set of leptonic angular correlation variables that are constructed such that each JHEP12(2015)026 member of this set probes a different tt spin correlation coefficient of the production density matrices. In addition, we list a set of single lepton distributions/observables which probe the t andt polarization coefficients of the production density matrices. These observables can be applied both to dileptonic and lepton plus jets final states. We compute, for tt production at the LHC at 8, 13, and 14 TeV, the two-and one-dimensional distributions associated with these observables at NLO QCD (tt production and decay) including the NLO mixed QCD weak-interaction corrections, for the transverse t andt polarizations also the mixed QCD-QED corrections, and, in addition, the contributions induced by L NP which are linear in the anomalous couplings. For completeness we determine also the contributions of L NP to the tt cross section σ tt and the tt charge asymmetry at the LHC, A C . For the observables considered in this paper we estimate the application range of the linear approximation in the anomalous couplings. Our results show that our set of correlation and polarisation observables, including the tt charge asymmetry A C , allows to determine all anomalous couplings of L NP and disentangles these contributions to a considerable extent. We conclude in section 5.

tt production density matrices in an orthonormal basis
We consider in this section, for the purpose of setting up our notation, hadronic tt production by 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 andt quarks, respectively. The production density matrices of these reactions are defined by where I ≡ ab = gg, qq and the bar denotes averaging over the spins and colors of I and summing over the colors of t,t. Moreover, α, β are spin labels referring to t andt, respectively. The matrices R I can be decomposed in the spin spaces of t andt as follows: where N c denotes the number of colors. The first (second) factor in the tensor products of the 2 × 2 unit matrix 1 and of the Pauli matrices σ i refers to the t (t) spin space. The functionsB I± i andC I ij can be further decomposed, using an orthonormal basis which we choose as follows. Here and in the followingk denotes the top-quark direction of flight in the tt zero-momentum frame (ZMF). Herep =p 1 denotes the direction of JHEP12(2015)026 one of the incoming partons in this frame. Then the following set forms a right-handed orthonormal basis: r,k,n :r = 1 r p − yk ,n = 1 r p ×k , y =k ·p, r = 1 − y 2 . (2.5) Using rotational invariance we decompose the 3-vectorsB I± and the 3 × 3 matricesC I ij (which have a symmetric and antisymmetric part with 6 and 3 entries, respectively) with respect 2 to the basis (2.5): 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 top-quark scattering angle cos θ * t . The terms in the antisymmetric part of (2.7) may also be represented as follows: Because the initial gg state is Bose symmetric, the matrix R gg must satisfy If CP invariance holds then The implications of (2.9), (2.10) and of parity invariance on the structure functions defined in (2.6), (2.7) are given in table 1. In this table the properties of these functions with respect to a naive T N transformation (reversal of momenta and spins) and CPT N transformation are collected, too, where just for this purpose, the absorptive part of the respective scattering amplitude has been neglected. The entries in table 1 are to be read as follows: for example, for a T -invariant interaction like QCD one has b I± n (y) = −b I± n (y) and thus b I± n = 0 at Born level, whereas absorptive parts generated at 1-loop render this function non-zero. The above decomposition of the spin density matrices R I and classification with respect to discrete symmetries is analogous to the analysis of [73] where a non-orthogonal basis was used.
The spin density matrices of the 2 → 3 reactions that contribute to tt production at NLO in the gauge couplings and are inclusive in tt have the same structure as those discussed above (with coefficient functions which depend on more variables) and can also be classified with respect to discrete symmetries. Table 1 is useful as a guide for the 2 The decomposition of theC I ij is not unique because δij =ninj +rirj +kikj. construction of observables at the level of dileptonic and lepton plus jets final states that project onto contributions to the differential cross section which are even and odd with respect to the above discrete symmetries. These observables apply also to investigations beyond LO, cf. section 4. Strictly speaking, a classification of observables with respect to the discrete symmetries C and CP is not possible in proton proton collisions because the initial pp state is neither an eigenstate of C nor of CP. Yet, for the main tt production modes, gg, qq → tt + X it can be done. The 'contamination' of CP-odd observables by contributions from parity-odd mixed QCD weak contributions to gq(q) → ttq(q) is at the level of a few per mill [14].

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We add a few remarks on the coefficient functions and on the information contained in table 1.
i) The functions A I , c I nn , c I rr , c I kk , and c I rk are generated by P-and CP-conserving interactions, in particular by QCD already at tree level. The functions A I determine the tt cross section, the p t T distribution, etc., while the functions c I ab (a, b = r, k, n) induce P-and CP-even tt spin correlations.
ii) The functions b ± r , b ± k are generated by P-violating interactions and induce longitudinal t andt polarizations (i.e., a polarization in the production plane). Non-zero which are P-, CP-, and CPT N -odd, require CP violation and absorptive parts.

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iii) The coefficient functions b ± n are generated by P-and CP-even absorptive parts in the scattering amplitudes, which is the case at 1-loop in the SM, and already at tree level by P-invariant, but CP-violating (effective) interactions. If interactions of the latter type are present then b + n − b − n = 0. The functions b ± n induce a transverse t andt polarization (i.e., a polarization normal to the production plane), which would differ for t andt if non-standard CP-violating interactions are present. iv) Non-zero coefficient functions c kn , c rn require P-odd but CP-even absorptive contributions to the scattering amplitudes. In the SM they are generated by the absorptive parts of the 1-loop mixed QCD-weak corrections, and they are very small, see section 4.
v) The coefficient functions c r , c k are generated by P-and CP-violating interactions.
As is well known, in the SM CP violating effects in flavor-diagonal reactions like ab → tt generated by the Kobayashi-Maskawa phase are tiny. Observable CP effects in hadronic tt production requires sizeable NP CP violation.
vi) A non-zero coefficient function c n requires P-even but CP-odd absorptive contributions to the scattering amplitudes. In the SM this coefficient function is tiny.
As an aside we recall that in a time-reversal invariant theory, expectation values of T N -odd observables can be generated by the absorptive part whereî,f denote the initial and final state with 3-momenta and spins reversed. The left-hand side of this equation has the generic form of a T N -odd observable.

Effective NP Lagrangian for hadronic top quark pair production
We assume that new physics (NP) effects in hadronic tt production and decay are induced by new heavy particle exchanges, characterized by a mass scale Λ. Then one may construct a local effective Lagrangian L NP in terms of the SM degrees of freedom which respects the SM (gauge) symmetries and describes possible non-resonant new physics interaction structures as long as the moduli of the kinematic invariants of the tt production and decay processes Λ. Recent analyses in the context of tt production include [18][19][20][21][22]. Here, we follow [22], but take also CP-violating effective interactions into account.

tt production
We concentrate on new interactions described by operators of mass-dimension dimO ≤ 6 which manifest themselves in hadronic tt production. We consider only that set of operators which contribute to tt production through interference with the dominant SM amplitudes, i.e., the QCD tree-level amplitudes.

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Let's first list the operators which involve gluon fields. We do not consider operators that affect only the pure gluon vertices, such as GGG or GGG, where G (G) denotes the (dual) gluon field strength tensor. For a recent bound on the anomalous coupling associated with GGG, see [70]. The strength of GGG is severely constrained by the upper bound on the electric dipole moment of the neutron. For dimO ≤ 6 one then remains with the following three gauge-invariant operators which involve t,t and up to three gluon fields [18,20,22]: Here induce CP-violating interactions, which cannot be expressed in terms of the the 4-quark operators below. One can therefore use a hermitean effective Lagrangian with these two operator combinations and with (3.3) that contains three independent complex coefficients. After spontaneous symmetry breaking, Φ = v/ √ 2, (v ≃ 246 GeV) and restriction to the top-quark gluon sector, one obtains, using operators with definite P and CP properties, real and dimensionless coupling parameters, and the (on-shell) top-quark mass m t for setting the mass scale: 3 and Here t denotes the top-quark Dirac field. In (3.4) the real, dimensionless parametersμ t and d t denote the anomalous chromo-magnetic (CMDM) and chromo-electric (CEDM) dipole moments of the top quark in the convention of [15]. Contrary to our previous analysis in [15] we stay here strictly within the effective field theory approach -i.e., we do not interpret these moments as form factors which may have an absorptive part. 4 The chromo-magnetic

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dipole operator is P-and CP-even, O g (−+) is P-even and CP-odd, while the chromo-electric dipole operator and O g (−−) are both P-and CP-odd. Next, we list the set of gauge-invariant dimO = 6 four-quark operators that yield non-zero tree-level interference terms with the QCD amplitude for qq → tt. We assume universality of the new interactions with respect to the light quarks q = t. Because the respective parton distribution functions suppress tt production by qq annihilation for q = s, c, b as compared to q = u, d, we take into account only the contributions with u, d in the initial state.
One is then left with the following 7 gauge invariant operators [20,22]: Here q L = (u L , u L ) denotes the first generation quark doublet, and σ A denote the Pauli matrices (tr(σ A σ B ) = 2δ AB ). These operators contribute to the effective NP Lagrangian a term of the form Again, the terms which involve b quarks are not of interest to us here. As pointed out in [22] it is advantageous to combine these seven operators such that one gets four isospin-zero operators with definite P and C properties and three isospin-one operators. (In the isospin-one case it is not possible to combine the operators such that they have definite properties with repect to C and P.) In the following q = (u, d) denotes the isospin doublet. One gets where the isosoin-zero part is and The isospin-one contribution can be represented in the form Our isospin-zero and -one couplingsĉ IJ ,ĉ i are related to those of [22] by The relation to the coefficients of the operator basis (3.7)-(3.13) is given in [22]. To summarize, we use The NP contributions to the coefficients of the tt spin density matrices (2.4) induced by interference with the tree-level QCD amplitudes of gg, qq → tt are listed in appendix A. A side remark concerns bb → tt which receives, apart from the order α 2 s QCD term, a SM contribution of order α s α from the interference of the t-channel W -exchangewith possible anomalous W tb couplings -and s-channel gluon exchange diagram. These SM terms, which are small, are taken into account in section 4. The contributions from anomalous W tb couplings are neglected, because i) they are empirically known to be small (see the next subsection) and ii) the respective interference terms are suppressed by the bb parton luminosity.

Top-quark decay
The SM and NP contributions to the parton matrix elements that describe tt production and decay into dileptonic and lepton plus jets final states involve also the top-quark decay density matrices for t → W b → ℓνb, qq ′ b. The top-decay vertex t → W b may also be affected by new physics interactions that can also be parametrized by anomalous couplings (cf., for instance, [20,61]). We use in the following as spin-analyzers of the (anti)top quark only the charged lepton ℓ ± from W ± decay, and we consider below only lepton angular correlations and distributions that are inclusive in the lepton energies. It is known [74][75][76][77] that these observables are not affected by anomalous couplings from top-quark decay if these couplings are small, i.e., if a linear approximation is justified. This is indeed the case in view of the present upper bounds on the moduli of these couplings that can be inferred from the measured W -boson helicity fractions in top-quark decay [78,79]. In other words, the observables which we analyze in the next section are affected only by possible new physics contributions to tt production, which we parametrize by the anomalous couplings discussed above.
We use the results of [80] for the polarized semileptonic and non-leptonic top-quark decays in the SM at NLO QCD.

Experimental bounds and domain of applicability
The size of the anomalous couplings of the NP interactions described by (3.22) are constrained by the experimental data on hadronic tt production which so far show no significant JHEP12(2015)026 sign of new physics. A recent (re)analysis of the contributions of only the chromo-momentŝ µ t andd t to the tt cross section and confrontation with the Tevatron and LHC measurements yields correlated bounds. In terms of our (sign) conventions chosen in (3.4), ref. [50] determined the allowed region (68% CL) in theμ t ,d t plane which may be represented by the bounds −0.03 ≤μ t ≤ 0.02 and |d t | ≤ 0.15. Ref. [55] found a similar result: −0.046 ≤μ t ≤ 0.024 and |d t | ≤ 0.17 (95% CL). Ref. [68] took into account the QCD corrections to the cross section contribution ofμ t and obtained −0.019 ≤μ t ≤ 0.018 (95% CL), puttingd t = 0. The CMS experiment analyzed, for dileptonic tt events at 7 TeV, the distribution of the difference of the leptonic azimuthal angles, and searched for a contribution of a non-zeroμ t . Using the predictions of [15] for interpreting their data, CMS obtained the bound [9] −0.043 ≤μ t ≤ 0.117 (95% CL). Tighter, but indirect bounds ond t andμ t were obtained using the experimental upper bound on the neutron electric dipole moment [42] and data on rare B meson decays [42,81], respectively. (Prospects for correlated bounds on the chromo moments in single top-quark production in the tW mode were recently discussed in [69].) A global fit to tt data using all dimension-six operators that are relevant to hadronic top quark production was recently performed in [70]. The operator set used in this reference contains also the operators contained in (3.22), albeit in a slightly different basis. One may use the correlated bounds of [70] to get order-of-magnitude bounds on the dimensionless couplings of the four-quark operators (3.14), to wit, |ĉ IJ |, |ĉ i | 0.4.
In the next section we will compute the contributions of the NP interactions (3.22) to a number of observables, and we take into account in that section only the NP contributions that are linear in the anomalous couplings. As the anomalous couplings correspond to non-renormalizable effective interactions, their contributions grow in the regime of large energy-momentum transfer to the tt system. In other words, the linear approximation can be applied with confidence only for relatively small anomalous couplings. In order to assess the domain of applicability of this approximation we have considered tt production at the LHC (14 TeV). We have computed, for the tt cross section and for a few spin correlation observables which will be introduced in the next section, the NP contributions to these observables which are linear and quadratic (respectively bilinear) in the anomalous couplings. Following ref. [22] we define the range of applicability of the linear approximation by requiring that the contributions linear in the anomalous couplings are at least twice as large in magnitude as the bilinear ones. Application of this criterion yields |μ t |, |d t | 0.15 and |ĉ IJ | 0.1. (We did not consider the contributions of the anomalous couplingsĉ i because they are subleading, see the next section.) Thus, as far as the chromo-moments are concerned, the empirical bounds discussed above are essentially within the domain of applicability of the linear approximation.

Observables and results
In this section we consider tt production at the LHC for center-of-mass energies 8, 13, and 14 TeV. First we determine the SM contributions and those of L NP to the tt cross section σ tt and the tt charge asymmetry A C . Then we introduce, for dileptonic and lepton JHEP12(2015)026 plus jets final states, a set of correlation and polarization observables and compute their distributions.
As far as the SM interactions are concerned we compute the four P-even and CP-even correlation observables that will be defined in section 4.2 at NLO QCD (order α 3 s ) [13] including the weak and mixed QCD-weak corrections of order α 2 , α s α, α 2 s α, α s α 2 [14,15,[98][99][100]. We label these SM contributions by the acronym 'NLOW'. The mixed QCD-QED contributions to these spin correlation observables have not yet been calculated. We expect them to be small, of the order of a few percent of the QCD corrections. The combinations of polarization observables considered in section 4.2 are, apart from one combination, Podd and/or CP-odd. These observables receive no QCD or QED contributions. Only one of these polarization observables, the sum of the transverse t andt polarization, receives, at the order of perturbation theory considered here, QCD contributions of order α 3 s , and mixed QCD-weak and mixed QCD-QED contributions of order α 2 s α, which are taken into account. (Contributions of order α 2 α s to the transverse polarization are very small and are not considered here.) The label 'EW' refers to the electroweak contributions. Our computational set-up is described in [14].
In the SM the total σ tt cross section is known to NNLO QCD (order α 4 s ) including soft-gluon resummation at next-to-next-to-leading logarithmic (NNLL) order [72]. The mixed QCD-weak corrections to the cross section (NLOW) were computed in [98][99][100][101][102] and the mixed QCD-QED corrections were determined in [82]. The NLOW corrections to the LHC tt cross section are negative, about ∼ −2% of the LO QCD cross section, while the QCD-QED corrections are positive, about ∼ 1% according to [82]. The LHC charge asymmetry A C defined below is computed at NLO QCD, including the mixed QCD-weak and mixed QCD-QED corrections [86]. These mixed EW corrections are known to be important for A C . The computation of A C to this order of perturbation theory, which is labeled 'NLO+EW', is present state-of-the-art. The Tevatron charge asymmetry was recently computed to NNLO accuracy [89].
Furthermore, we compute the interferences of the NP amplitudes induced by L NP and the tree-level QCD amplitudes, but keep only the terms linear in the anomalous couplings. We employ the following input parameters: for the top-quark mass in the on-shell scheme we use m t = 173.34 GeV. Moreover, we use Γ t = 1.3 GeV, m Z = 91.2 GeV, m W = 80.4 GeV, Γ W = 2.09 GeV, m H = 125 GeV, α(m t ) = 0.008, and we use the CT10 NLO parton distribution functions (PDF) [83]. This PDF set provides also the QCD coupling α s in the MS scheme. For the renormalization and factorization scale, we use µ R = µ F ≡ µ, µ = m t , and estimate scale uncertainties by varying µ between m t /2 and 2m t .

Total tt cross section and charge asymmetry at the LHC
Our results for σ tt are listed in table 2. The uncertainties refer to the scale uncertainties. In the linear approximation, only the P-and CP-even operators contained in L NP contribute; the CMDM operator, whose contribution is largest, the isospin-zero operator O V V , and the P-even part of the isospin-one operator O 1 1 , whose contribution is smallest. The smallness of the isospin-1 compared to the isospin-0 contribution to σ tt , to the charge asymmetry defined below, and to the observables of the next section is due to the fact that a sizeable  fraction of these contributions comes from the kinematic region near threshold. In this region, the contributions of uū → tt and dd → tt are of the same order. The SM result for the cross section listed in table 2 is given here only for reference purposes. Comparing experimental results to predictions one should use the most precise available SM result on σ tt . This is the result of [72] mentioned above. Rather than performing a three-parameter fit to cross section results at the Tevatron and the LHC (7 and 8 TeV) which is left to a future analysis, we make here a crude estimate of the bound onĉ V V by putting the anomalous couplingsμ t andĉ 1 to zero. Using, for 8 TeV, σ exp. = 241.5 ± 8.5 pb [84] and the NNLO+NNLL QCD prediction [72,85] Table 2 contains also our SM and NP prediction for the tt charge asymmetry at the LHC which is defined by where ∆|y| = |y t | − |yt| is the difference of the moduli of the t andt rapidities in the laboratory frame. Our SM predictions, labeled 'NLO + EW', contain apart from the mixed QCD-weak interaction corrections also the mixed QCD-QED corrections of order α 2 s α [86]. We have computed A C by expanding the ratio (4.1), see eq. (4.22) below. The NLOW predictions in table 2 are in accord with the results of [86,87] where different PDF sets were used. 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 andt momenta while those of the initial (anti)quark are kept fixed. Using the recent CMS measurement [88] with their 8 TeV data, A C = 0.001 ± 0.008, and putting the couplingĉ 2 to zero, we obtain the 95 % C.L. bound −0.06 ≤ĉ AA ≤ 0.01; cf. also [88,93]. This result justifies a posteriori the use of the linear approximation. New physics contributions to the tt charge asymmetry A F B at the Tevatron were analyzed in terms of the above effective interactions in [22,[90][91][92]. The latest experimental results on the Tevatron asymmetry A F B are somewhat larger -
but compatible -with the most precise SM results [89]. This impliesĉ AA > 0 andĉ AA is bounded from above. The interaction described by O AA is the effective field theory version of a number of NP models which were invoked to resolve the seeming discrepancy between the experimental results on the Tevatron A F B and the SM predictions. These models include the axigluon model and flavor-changing t-channel Z ′ exchange in uū → tt; see [94] for a recent overview of the NP models discussed in the context of the tt charge asymmetry. The linear approximation for the contributions of the NP interactions to the cross section and the charge asymmetry is justified if |μ t | < |μ t,max | ≃ 0.15 and |ĉ IJ | < |ĉ IJ,max | ≃ 0.1, see section 3.3. We assume in the following that the latter bound holds also for anomalous couplingsĉ i . With these bounds we may make a crude estimate of the maximal difference between (future) experimental measurements of an observable and the SM prediction that will still allow the the use of the linear approximation. In the case of the tt cross section, we estimate this maximal difference by where the coefficients a i are given in table 2. (We take the values of the a i for µ = m t .) The maximal difference δA C is determined in analogous fashion. In table 3 our estimates of δσ tt and δA C are given for the LHC (13 TeV). As these are crude estimates we have refrained from taking the scale uncertainties of the a i into account. For the other LHC center-ofmass energies, the δσ tt and δA C are obtained in analogous fashion using the coefficients of table 2. Table 4 contains our predictions for σ tt and A C for events with tt invariant mass M tt ≤ 500 GeV. The mixed QCD-QED contributions to σ tt (M tt ≤ 500 GeV), which are not included in this table, add ∼ 1% to the cross section values given in this table. The NNLO QCD corrections are not yet available. For A C , the ratio of the isospin-one and isospinzero NP contributions is approximately the same as in the inclusive case listed in table 2. In order to determine constraints on both anomalous couplingsĉ AA andĉ 2 from data on the LHC charge asymmetry alone, one may compute the differential charge asymmetry in narrower bins of M tt , or bins of the rapidity and transverse momentum of the tt system.
In analogy to (4.2) one may estimate the maximal differences δσ tt (M tt ≤ 500 GeV) and δA C (M tt ≤ 500 GeV) that would allow the use of the linear approximation. They are given in table 3.
In the region where, for instance, the tt invariant mass M tt is large, the linear approximation in the NP couplings can no longer be justified just by assuming that the anomalous couplings are small, see section 3.3. While it is not difficult to compute σ tt and A C be-  yond this approximation, the following complication in the comparison with data arises: there is now a proliferation of contributions (being odd in the scattering angle y defined in section 2) which involve, apart fromĉ AA andĉ 2 , also the other anomalous couplings. Information on bothĉ AA andĉ 2 can be obtained either from considering A C at different c.m. energies and the Tevatron asymmetry simultaneously, or from A C and (future) measurements of the inclusive tt spin correlation and polarization observables defined in the next section.

Angular distributions and correlations for dilepton and ℓ + jets events
We analyze tt production at the LHC and subsequent decay into dileptonic final states, and into lepton plus jets final states, where ℓ, ℓ ′ = e, µ, τ.
For the dileptonic events (4.3) we consider four-fold dilepton angular distributions. Without acceptance cuts, besides cuts on the tt invariant mass, these normalized distributions have the form where dΩ = d cos θdφ. The unit vectorsl + ,l − are the ℓ + and ℓ ′− directions of flight in the t andt rest frames, respectively, which are reached from the tt ZMF by rotation-free boosts. The coefficients, i.e., the vectors B ′ 1 , B ′ 2 and the matrix C ′ contain the information from the tt production and the t andt decay density matrices, which may be computed in the SM or in a NP model. As already mentioned in section 3.2, we consider the main SM

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top-decay mode t → W b. We parametrizel + ,l − in terms of spherical coordinates, but we allow for different polar axes, i.e., different reference axesâ andb. We define Integrating over the azimuthal angles we obtain the polar angle double distributions for a choice of reference axesâ,b: The sign in front of C is chosen such that in the absence of acceptance cuts [12,13]: where the ratio on the right-hand side of this formula is the double spin-asymmetry at the level of the t,t intermediate states. (The first (second) arrow refers to the spin state of the t (t) quark with respect to the axisâ (b).) The top-spin analyzing power of the charged lepton from top decay is denoted by κ ℓ . (In the convention used here, κ ℓ + = κ ℓ − = κ ℓ .) At NLO QCD its value is κ ℓ = 0.999 [80]. The polarization degrees of the ensembles of t andt quarks in tt events with respect to the reference axesâ,b are given by where S t , St denotes the t andt spin operator, respectively. They are related to B 1,2 by That is, the relative signs in front of the B 1,2 in (4.8) are chosen such that in a CP-invariant theory and for the choiceâ = −b we have Our choice of reference axesâ andb which we use here is adapted to the orthonormal basis at the parton level introduced in section 2, but is not identical to it. As in section 2, we use the unit vectork which is the top quark direction of flight in the tt ZMF. Moreover, we use the direction of one of the proton beams in the laboratory frame,p p , and define unit vectorsr p andn p as follows: Only in the case of 2 → 2 parton reactions and if the incoming parton 1 is parallel tô p p , the unit vectors defined in (2.5) are the same as those in (4.13). With the set (4.13)

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labelâb transverse n sign(y p )n p −sign(y p )n p r axis r sign(y p )r p −sign(y p )r p helicity kk −k Table 5. Choice of reference axes. The unit vectorsn p ,r p and the variable y p are defined in (4.13).
we define the reference axesâ andb listed in table 5. The factors sign(y p ) are required because of the Bose symmetry of the initial gg state. Instead of extracting the spin correlation coefficients C (for a choice of axes) from the double distributions (4.8) one may analyze the 1-dimensional distributions of the observables ξ = cos θ + cos θ − . (4.14) Labels are suppressed here, i.e., we consider in the following ξ nn , ξ rr , ξ kk , ξ rk , ξ rn , etc. The one-dimensional distribution is obtained in standard fashion by inserting into the integral d cos θ + d cos θ − over the double distribution (4.8). One arrives at which may be used for fits to unfolded experimental results. We recall that C = −9 ξ . The slope of this distribution, B 1 (B 2 ), measures the polarization of t (t) quarks with respect to the chosen reference axis. One may also compute/measure these polarizations using the relation B 1,2 = 3 cos θ ± . If no acceptance cuts are applied, the coefficient B 1 (B 2 ) is the same both for the distribution of ℓ + (ℓ − ) from dilepton and from lepton plus jets events. From now on we use the following notation. The label (a, b) refers to the choice of reference axesâ andb from table 5. The correlation coefficient C(a, b) in (4.8) or (4.16) is associated with this choice of axes, and likewise B 1 (a) and B 2 (b).
In addition to (4.18), we compute also the top and antitop polarization with respect to another set of reference axes:

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where the labels k * and r * refer to the following vectors: r * :â = sign(∆|y|) sign(y p )r p ,b = −sign(∆|y|) sign(y p )r p . (4.21) Here ∆|y| = |y t | − |yt| denotes the difference of the moduli of the t andt rapidities in the laboratory frame. The reason for introducing these additional polarization observables is that they are sensitive to the NP contributions from the operator O AV and from a Podd combination of the operators O 1 i (defined in (3.16) and (3.19), respectively), while B 1,2 (k), B 1,2 (r) project onto the contributions of O V A and O 1 3 . The coefficients C(a, b), B 1 (a), and B 2 (b) (and sums and differences) are listed in table 6. They can be measured with the 2-dimensional distributions (4.8) or with the 1dimensional distributions of ξ ab , respectively with (4.18). Column 2 of this table indicates which coefficient of the ij → ttX production density matrices introduced in section 2 contributes to the respective correlation coefficient. The type of interactions, i.e., the type of contributions in the differential parton cross sections to which a correlation is sensitive, is given in column 3.
Tables 7, 8 contain our results for the inclusive correlation coefficients C(a, b), B 1 (a), B 2 (b) and B 1 (a * ), B 2 (b * ) for the LHC at 8, 13, and 14 TeV, while the coefficients in tables 9, 10 refer to events with tt invariant mass M tt ≤ 500 GeV. Several comments are in order.
1. All normalized differential distributions, respectively the correlation coefficients are ratios. The P-and CP-even coefficients C are, in the SM at NLOW and to first order in the anomalous couplings, schematically of the form where N 0 (δN 1 ) and σ 0 (δσ 1 ) are the contributions at LO QCD (NLOW). A Taylor expansion at NLO in the SM and anomalous couplings yields We use this expansion which is in the spirit of perturbation theory. 5 2. As expected (cf. section 4.1), the tables 7-10 show that for all observables the isospin-1 NP contribution from L NP,q is about one order of magnitude smaller than the isospin-0 contribution. C(r, r) is, in the inclusive case, the smallest of the four P-and CP-even correlations. This can be qualitatively understood with simple treelevel arguments. At the LHC (8TeV) the t andt quarks in the inclusive tt sample JHEP12(2015)026

Within the Standard Model
P-odd, CP-odd, absorptive Table 6. The correlation coefficients of (4.8), respectively of (4.17), (4.18) and (4.19), and sums and differences for different choices of reference axes.
are on average only moderately relativistic. At or very close to threshold the tt system produced at LO QCD by gluon fusion is in a 1 S 0 state. Thus, not too far away from the threshold, the spin configuration of the tt with respect to the r-axeŝ a r andb r (which point into opposite directions) is preferentially ↑↑ or ↓↓, while for relativistic tt the configuration ↑↓ or ↓↑ dominates. This statement applies also to the gg → tt contribution to the helicity correlation C(k, k). The net effect for the inclusive sample is that gluon production contributes positively both to the correlation C(r, r) and C(k, k), but the correlation of the t andt spins with respect to the r-axes is significantly smaller than their correlation with respect to the helicity axesâ =k, b = −k. The t,t produced by qq annihilation make a negative contribution both to C(r, r) and C(k, k), and it turns out that the magnitude of the contribution is in JHEP12(2015)026   Table 7. The correlation coefficients C at NLOW in the SM and the non-zero contributions of the dimensionless anomalous couplings defined in section 3. The central values correspond to the scales µ = m t . The uncertainties refer to µ = m t /2 and µ = 2m t .
both cases approximately the same. This is because in this case, at LO QCD, tt near threshold is in a 3 S 1 state. Thus both with respect to the r-axes and the helicity axes, the spin configuration ↑↓ or ↓↑ dominates, both for non-relativistic and relativistic JHEP12(2015)026 8 TeV (2.9 +1.9 −1.4 ) · 10 −3 0.298 +0.014  Table 8. The correlation coefficients B at NLOW in the SM and the non-zero contributions of the dimensionless anomalous couplings defined in section 3. The central values correspond to the scales µ = m t . The uncertainties refer to µ = m t /2 and µ = 2m t . t,t. As a result, the contributions from gg and qq initial states almost cancel in the case of C(r, r), while in the case of C(k, k) the positive contribution from the gg channel is significantly larger in magnitude than the negative one from qq → tt. This line of arguments suggests that the ratio of C(r, r) and C(k, k) becomes larger if the tt sample is restricted to events not too far away from the tt threshold. This is corroborated by the numbers of table 9. The fact that the correlation C(n, n) JHEP12(2015)026  Table 9. The correlation coefficients C at NLOW in the SM and the non-zero contributions of the dimensionless anomalous couplings defined in section 3 for M tt ≤ 500 GeV.
is rather large in the SM can also be understood from total angular momentum conservation. The measurement of these four correlations would provide sufficient information to simultaneously determine or constrain the CMDMμ t and the anomalous fourquark couplingsĉ V V andĉ 1 .
As an aside we remark that the NLOW and the CMDM (μ t ) contribution to the helicity correlation C(k, k) at 8 TeV and the NLOW result for 14 TeV listed in table 7 agree with those computed in [15] and in [14], respectively, with a different PDF set.  4. The opening angle distribution, σ −1 dσ/d cos ϕ = (1 − D cos ϕ)/2 [13], which is also a useful spin correlation observable, can be obtained from the above diagonal correlation coefficients. Because the reference axes defined in

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This relation can be applied both to the SM and the NP contributions given in the tables below. The correlation coefficients D computed with (4.23) agree with those computed in [15] and [14] for 8 TeV and 14 TeV, respectively. 5. One may also consider the spin correlations C(r * , k), C(k, r * ), and C(k, k * ), where the k * and r * axes are defined in (4.20) and (4.21). These observables are spin JHEP12(2015)026 correlation analogues of the charge asymmetry A C . They receive contributions from the entries of the tt production density matrices that are odd under interchange of the t andt. In the SM these correlations, which receive no contributions from the gg initial state and no contributions, at LO QCD, from qq initial states, are expected to be small. They receive NP contributions proportional to the anomalous couplingŝ c AA andĉ 2 .
6. As long as one considers CP-invariant interactions, C(r, k) = C(k, r). A non-zero difference C(r, k) − C(k, r) requires P-even but CP-odd absorptive contributions. In the SM at NLO there are no contributions of this type. Because our NP couplings are real and we do not take into account radiative corrections to the NP amplitudes, there are also no NP contributions to C(r, k) − C(k, r) in our approximation.
7. The sums C(n, r) + C(r, n) , C(n, k) + C(k, n) (4.24) probe P-odd, CP-even, naive T-odd absorptive (SM) contributions. The coefficients C(n, r) and C(n, k) receive SM contributions from P-odd absorptive parts of the mixed QCD-weak corrections. We have computed the respective 1-loop absorptive contributions to the gg → tt and qq → tt spin density matrices. The resulting effects on (4.24) are very small, cf. tables 7 and 9. Because all anomalous couplings are real in the effective Lagrangian approach and because we do not take into account radiative corrections to the NP amplitudes, there are no NP contributions to (4.24).
8. The SM contributions referred to in the previous item cancel in C(n, r) − C(r, n) , C(n, k) − C(k, n) . (4.25) These differences probe P-and CP-odd (dispersive) NP contributions, that is, nonstandard CP violation. These differences in the correlation coefficients can be related to CP-odd triple correlations. Defining Alternatively one may consider associated CP asymmetries, which are easier to measure. They are related to the expectation values of (4.26) by [15,95]: is less sensitive to the CP-violating anomalous couplings than the left-hand sides of (4.27), but experimentally more robust. The measurement of the two correlations (4.27) or the two asymmetries (4.28) would allow to simultaneously determine or constrain the CEDMd t and the anomalous CP-odd gluon-top-quark couplingĉ (−−) .
Putting the couplingĉ (−−) = 0, the expectation value O CP 1 at 8 TeV computed with the 8 TeV value for C(n, r) − C(r, n) given in table 7 agrees with the result of [15]. 9. As already emphasized above, for pp collisions a classification of observables with respect to CP is, strictly speaking, not possible because |pp is not a CP eigenstate. However, there will be no SM contributions from qq and gg initial states to the CPodd observables listed in the previous item and in (4.30) -assuming no acceptance cuts or cuts which are CP-invariant. Moreover, within our approximation we found no contribution from P-violating but CP-even terms induced by gq production.
10. As to the longitudinal top-polarization coefficients B(k), B(r): the sums receive contributions from the P-odd weak-interaction corrections, that is, from the P-odd terms of the NLOW corrections. However, the SM-induced t andt polarizations are very small to tiny, as the numbers in  [15].) The sums (4.29) receive also contributions from the P-odd NP four-quark operators. Thus, the measurement of the four polarization observables (4.29) would allow to simultaneously determine or constrain four anomalous four-quark couplings, for instanceĉ V A ,ĉ AV ,ĉ 2 , andĉ 3 , assuming that input onĉ 1 is provided from the measurement of the four CP-even spin correlations, see item 3 above. In the differences which are P-and CP-odd and T N -even, the SM contributions to B 1,2 cancel. Nonzero differences require, in addition to CP-violating NP contributions, also absorptive parts. Because our anomalous couplings are real and we do not take into account radiative corrections to the NP contributions, these differences remain zero within our approach. In [15] the chromo moments were interpreted as form factors which may have an absorptive part, and the differences of the longitudinal t andt polarizations were computed in terms of Imd t .

As to the combinations
The sum is generated by P-and CP-even absorptive terms in the differential cross section. In the SM such terms arise from a number of one-loop amplitudes for gg, qq →

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tt which involve parton, weak gauge-boson, Higgs boson, and photon exchange. The QCD contributions given in tables 8 are in accord with the 8 TeV result of [15]. (In this work, the vector perpendicular to the scattering plane was used in un-normalized form and a different PDF set was employed.) In addition we list in tables 8 and 10 also our results for the electroweak contributions, which arise from the absorptive parts of the mixed QCD-weak interaction and QCD-QED contributions to gg, qq → tt.
They turn out to be non-negligible in comparison to the QCD contribution. Thus, for the inclusive tt sample, the transverse polarization of the ensemble of t andt quarks with respect to the reference axis sign(y p )n p is P nt = P nt ≃ 0.5%. In the difference B 1 (n) − B 2 (n) the SM contributions cancel. The difference is sensitive to P-even, but CP-odd dispersive NP contributions, which in the framework of our effective Lagrangian approach are generated at Born level by the operator O g defined in (3.5).
12. Finally we add a remark on the scale dependence of our SM predictions for the correlation and polarization observables. As the numbers given in tables 8 and 10 show, the parity-odd combinations of the B coefficients are very small in the SM but exhibit large scale variations relative to the values for µ = m t . The reason for this is as follows. For the numerators N of the B coefficients that are generated by the P-odd weak interaction contributions, one gets N (µ = 2m t ) > N (µ = m t ) > N (µ = m t /2), due to factorization-scale dependent logarithms in some of the qg induced mixed QCD-weak corrections [100], while for the denominators D of the B coefficients, which are taken into account at lowest order QCD, one has D(µ = 2m t ) < D(µ = m t ) < D(µ = m t /2). This produces the large scale variations in the ratios B = N/D. The P-and CP-even correlation observables given in tables 7 and 9 are generated predominantly by QCD. Here the scale variations are moderate to small and, as is well known, the Taylor expansion (4.22) reduces the scale dependence of the results at NLO QCD with respect to LO. The weak interaction contributions with their rather large scale dependence do not spoil this because they are small compared to the QCD contributions. The correlation C(r, r) is special in that there are cancellations among separate contributions, keeping C(r, r) rather small and more sensitive to scale variations than C(n, n) and C(k, k).
The above spin-correlation and polarization observables probe all entries of the topspin dependent parts of the tt production density matrices. In addition, the various NP contributions are disentangled to a large extent, as the above discussion and tables show. Moreover, a measurement of this set of spin observables -and experimental results on A C , which yields information on the couplingĉ AA -would provide sufficient input for determining all anomalous couplings of the effective Lagrangian (3.22). As mentioned above, the spin correlations C(r * , k), C(k, r * ), and C(k, k * ) are also sensitive toĉ AA .
An important issue is the range of validity of the linear approximation for the NP contributions to the correlation and polarization observables. In analogy to the procedure we applied to the tt cross section and the charge asymmetry A C in section 4.1, -cf. eq. (4.2) and  Table 11. Moduli of maximal differences between future experimental results and SM predictions, for the correlation and polarization observables at the LHC (13 TeV), for which the linear approximation can be applied.
measurements and SM prediction of an observable by adding the moduli of the products of the respective contributions given in tables 7-10 and the maximal values of the anomalous couplings, derived in section 3.3, for which the linear approximation can be applied. The results of these estimates are given, for the LHC (13 TeV) in table 11. The estimates for 8 and 14 TeV can be obtained in completely analogous fashion. The numbers of table 11 show that the range of validity of the linear approximation is quite large. One may compute the above spin observables also for tt events with large tt invariant mass, for instance M tt > 500 GeV. While there is no computational difficulty, one should, as discussed above, in this case abandon the linear approximation in the anomalous couplings. This leads to a proliferation of NP contributions to each of the spin-correlation and polarization coefficients, which will make a comparison with (future) data rather intransparent. One may pursue a pragmatic approach, namely, analyze only selected observables beyond the linear approximation, in particular those which turn out to be measurable with moderate or small experimental uncertainties. Such an analysis is beyond the scope of this paper and will be left to a future investigation.

Summary and conclusions
We have defined a set of correlation and polarization observables with which hadronic tt production dynamics can be probed in detail, that is, all entries of the top-spin dependent parts of the hadronic tt production density matrices, in dileptonic and semileptonic tt events at the LHC. Our observables allow, in particular, to disentangle contributions from interactions which conserve/violate parity and/or CP. We have computed these observables within the Standard Model at NLO QCD including the mixed QCD-(electro)weak

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contributions. In addition we have analyzed possible new physics contributions in terms of an effective Lagrangian that contains all gauge-invariant effective interactions of mass dimension ≤ 6 being relevant for the parton processes at hand. Existing constraints on anomalous couplings which describe the strength of these effective NP interactions justify that only NP terms linear in the anomalous couplings are taken into account in the computation of inclusive (spin) observables for the LHC. Irrespective of these constraints we have estimated the range of validity of the linear approximation for the observables considered in this paper. Some of our polarization and spin correlation observables allow for direct searches of non-standard P-and CP violation in tt events and will provide, once measurements will have been made, direct information on respective interactions which may affect hadronic tt production. The measurement of the complete set of our inclusive spin observables, including A C , would allow to determine, respectively constrain all anomalous couplings contained in L NP . Instead of A C , one may consider the observables C(r * , k), C(k, r * ), and C(k, k * ) which are the spin correlation analogues of the charge asymmetry.
Eventually, it will also be of interest to compute and measure the above tt spin correlations and t andt polarizations in the kinematic regime where the energy-momentum transfer to the tt system is large, for instance for tt events with M tt > 500 GeV. In this regime the contributions of the effective NP interactions grow in general, due to their non-renormalizable nature. In this kinematic regime one should abandon the linear approximation, but then each observable receives a large number of contributions from the anomalous interactions contained in L NP . Such an analysis should be done at some point for selected observables, once experimental feedback on measurement precisions will be available.

A NP contributions to the tt spin density matrices
The tt production spin density matrices to the parton reactions gg, qq → tt(g) and gq(q) → ttq(q) were computed at NLO QCD (dispersive contributions) in [12,13], the contributions of the NLO QCD absorptive parts were determined in [96,97] and the contributions of the mixed QCD-weak corrections (dispersive contributions) in [98][99][100]. The spin-independent term of the mixed QCD-weak corrections to qq, gg → tt, i.e., the respective contributions to the coefficient functions A I of eq. (2.4), were computed also in [101,102] and, without the infrared-divergent box contributions to qq → tt and the corresponding real gluon radiation, in [103], and these results agree with ours. In addition, we have computed the contributions of the absorptive parts of the mixed QCD electroweak corrections. Here we do not list the latter contributions, for the sake of brevity and because they induce only very small effects (see section 4.2). Semileptonic and non-leptonic decays of polarized top-quarks are incorporated at NLO QCD [80].
Here we list the contributions to the tt spin density matrices (2.4) in the decomposition (2.6), (2.7) of the NP interactions described by the effective Lagrangian (3.22). As emphasized above, we take into account only the interferences with the tree-level QCD amplitudes of gg, qq → tt, i.e., only terms linear in the anomalous couplings. We recall that all anomalous couplings are real dimensionless parameters.