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arXiv:1212.3272v2 [hep-ph] 9 Apr 2013

(a) NICPB, R¨avala 10, Tallinn 10143, Estonia (b) INFN sezione di Trieste, via Valerio 2, I-34127 Trieste, Italy (c) Institute of Physics, University of Tartu, Riia 142, 51014 Tartu,Estonia

ABSTRACT

We analyze the impact of effective axial-vector coupling of the gluon on spin polarization observables in tt¯ pair production at the LHC. Working at leading order in QCD, we compute the tt¯ spin-correlation and left-right spin asymmetry coefficients in the helicity basis in the laboratory frame as functions of the new physics scale Λ associated with this coupling. We found that the tt¯ invariant mass dependent asymmetries are more sensitive to the scale Λ than the corresponding inclusive ones, in particular when suitable cuts selecting high tt¯ invariant mass regions are imposed. In the context of this scenario, we show that the LHC has potential either to confirm or to rule out the Tevatron FB top asymmetry anomaly by analyzing the tt¯ spin-correlation and left-right polarization asymmetries. On the other hand, stringent lower bound on the new physics scale Λ can be set in this scenario if no significant deviations from the SM predictions for those observables will be measured.

1

On leave of absence from Dipartimento di Fisica Universit`a di Trieste, Strada Costiera 11, I-34151 Trieste

1

Introduction

Top-quark physics is undoubtedly the best framework where to study polarized processes at the level of fundamental interactions [1]. Discovered at Tevatron in 1995 [2], [3] and copiously produced both at the Tevatron and the Large Hadron Collider (LHC) [4], the top-quark is the heaviest elementary fermion with the measured mass of mt = 172.9 ± 0.6 ± 0.9 GeV and decay width of Γt = 2.0+0.7 −0.6 GeV [5]. Since the top life-time is shorter than the characteristic hadronization time-scale ∼ 1/ΛQCD , with ΛQCD the characteristic QCD energy scale, this guarantees that it will always decay before hadronizing. Indeed, the top decay, which is dominated by the weak decay channel t → W b, is expected to occur before its spin is flipped by strong interactions. This ensures that top spin-polarization at production level will be fully transferred to its decay products. Then, the spin of the top-quark can be accessed by measuring the angular distributions of the final state decay products. The QCD corrections to the tt¯ pair production at hadron colliders can be safely computed at high orders in perturbation theory [6, 7, 8, 9, 10], which allow us to determine the top-quark polarization with high accuracy. An interesting observable that can be measured with high precision at hadron colliders is the spin-correlation in the tt¯ pair production. This observable was analyzed in [11, 12, 13] at the leading order (LO), and is now known at the next to leading order (NLO) [14] in the strong coupling in QCD, while more recently the NLO weak corrections [15] have been included. The standard model (SM) predicts that spins of the top- and antitop-quarks are strongly correlated, which is just a consequence of the partonic tt¯ production mechanisms at hadron colliders. The tree-level partonic processes, contributing to the tt¯ production at the LO in QCD, are the quarkantiquark and gluon-gluon annihilation processes, namely q q¯ → tt¯ and gg → tt¯ respectively [1]. While at Tevatron the first mechanism dominates, the second one is the leading tt¯ production mechanism at the LHC, contributing to almost 90% of the pp → tt¯X total cross section. Therefore, at low tt¯ invariant mass, the top- and antitop-quarks are mainly produced at the LHC experiments in the left-left and right-right helicity configurations, due to the spin-1 nature of gluons [16]. The different production mechanisms and collision energies in the Tevatron and the LHC make the measurements of tt¯ spin correlations in those experiments complementary to each other. This observable is also a very sensitive probe of new physics scenarios that contribute to the partonic tt¯ production mechanisms, whilst keeping the tt¯ production cross section at hadron colliders within experimental and theoretical bounds [12, 17]. Both CDF [18] and D0 [19, 20] Collaborations at Tevatron have performed measurements of the tt¯ spin-correlation which, within experimental errors, are in agreement with the NLO SM predictions. In particular, D0 Collaboration has reported an evidence for the spin-correlation in tt¯ with a significance of 3.1σ [20]. From the LHC, the ATLAS √[21] and CMS [22] Collaborations ¯ have recently analyzed the tt spin correlation by analyzing s = 7 TeV data corresponding to an integrated luminosity of about 2.1 fb−1 and 5 fb−1 , respectively. ATLAS has excluded the hypothesis of zero spin-correlation with a significance of 5.1 σ [21], while the CMS has only reported a 2.9 σ evidence [22]. Within experimental errors both measurements are consistent with the NLO SM predictions [15]. However, despite the good agreement between the SM and data in the top-quark sector, 1

the 3σ excess in the tt¯ charge or forward-backward (FB) asymmetry with respect to the SM predictions [23, 24, 25], observed at Tevatron by the CDF [26] and D0 [27] Collaborations, still needs to be clarified. The intriguing property of this anomalous measurement is that the charge asymmetry increases with the tt¯ invariant mass. At the same time the measured tt¯ production cross section is consistent, within experimental errors, with the SM prediction [6, 8, 10], both at Tevatron [28] and at the LHC [29]. Numerous new physics models have been proposed to explain this excess of events. Most of them predict the existence of new particles that have parity violating interactions with quarks. In particular, models with flavor dependent axigluons [30], flavor-changing Z 0 interactions [31] or W 0 [32] have been suggested. However, in order to reduce the tension with the SM prediction, these new particles should be relatively light. In particular, the new particle masses span from a few hundred GeV in the case of weakly interacting particles up to 1-2 TeV for the strongly interacting ones, such as the axigluons. Some of these models are now strongly constrained by negative searches of new heavy particles, like flavor-changing couplings to top quark [33], and contact terms interactions [34] at the LHC. In Ref.[35] it was shown that the Tevatron anomaly could be explained by introducing a universal effective axial-vector coupling of the gluon with quarks. This effective coupling arises also in the SM, being induced at one-loop by weak radiative corrections [36]. However, it is too small to account for the Tevatron anomaly. Although such an anomalous coupling could have different new physics (NP) origins, its main feature is that it naturally predicts the correct sign for the asymmetry and does not necessarily require new light resonances. As shown in [35], the characteristic new physics scale Λ associated to this coupling should lie in a narrow range Λ ' 1 − 1.3 TeV. This range has been found to correctly reproduce the Tevatron anomaly on top-quark charge asymmetry, while the lower bound on Λ > 1 TeV comes mainly from requiring conservative constraints on the total cross section of top-quark pair production at Tevatron. More recently, in [37], the implications of this scenario has been analyzed for various topquark charge asymmetries at the LHC [38, 39]. In particular, it was shown that the LHC with 7-8 TeV center of mass energy has the potential either to rule out or strongly constrain this scenario [37]. This would require to analyze the cut-dependent charge asymmetries at different invariant masses of the tt¯ system, as a function of tt¯ invariant mass mtt . Large deviations from the SM prediction are indeed expected to appear in regions of mtt close to the Λ scale. On the other hand, when inclusive observables in the kinematic range of mtt are considered, the new physics contribution for a scale Λ > 1 TeV turns out to be smaller than the SM contribution. This picture is consistent with present LHC measurements of top-antitop charge asymmetries [40], which are inclusive in mtt and consistent with the SM prediction [38]. The aim of the present work is to extend the analysis of [35, 37], by computing the effect of this scenario on the tt¯ spin observables that can be measured at the LHC. In particular, we will analyze the spin-correlation and the left-right (LR) polarization asymmetry [41, 42, 43, 44, 45] in the laboratory frame, as a function of the new physics scale Λ. Will show that these observable, when computed on the tt¯ high invariant mass (mtt ) regions, are very sensitive to a scale Λ in the TeV range.

2

Regarding the recent ATLAS [21] and CMS [22] measurements of tt¯ spin-correlations, a direct comparison with these results is not possible in the approach of [35, 37], where a low energy parametrization of the effective gluon axial-vector vertex has been adopted. Indeed, the measurements in [21], [22] are inclusive in the mtt invariant mass, while the low energy approximation, used in [35, 37] to parametrize this effective vertex, breaks down for values of mtt > Λ, due to the breaking of perturbative unitarity. However, this is an artefact of the low energy approximation, since the effective gluon axial-vector coupling, being related to an operator of dimension 4, has a momentum dependence which is valid at any energy scale mtt > 2mt . Indeed, this is the case, for instance, of the SM where the gluon axial-vector coupling is generated at one-loop by the electroweak corrections [36]. In order to circumvent this problem, and extend the predictions to the kinematic regions mtt > Λ, we will assume a particular shape of the form factor that would respect unitarity and perturbation theory. In particular, we will assume that this effective coupling tends to a cut-off in the asymptotic limit mtt Λ, while it satisfies the low energy limit required by QCD Ward identities. In this way, a direct comparison with the results in [21], [22] would be possible, although at the price of introducing a new free parameter and a particular shape of the form factor. The purpose of this test is to check that the mtt inclusive top spin correlation observables are mainly dominated by the kinematic regions mtt < Λ ∼ 1 TeV, and therefore they are not very sensitive to cut-off values of order O(1) and to the shape of the form factor. In particular, we will show that in the context of this scenario, values of Λ > 1 TeV are still consistent, within two standard deviations, with the recent ATLAS and CMS recent measurements. These results suggest that a dedicated experimental analysis at the LHC is needed that studies the tt¯ spin correlation dependence on the tt¯ invariant mass mtt in order to either confirm or strongly constrain this scenario. The paper is organized as follows. In Sec. II we review the theoretical framework and provide the analytical expressions for contribution of the effective gluon axial-vector coupling to the polarized q q¯ → tt¯ and gg → tt¯ total cross sections. In Sec. III we study the effects of this scenario on the tt¯ spin-correlation and left-right top-quark asymmetry at the LHC. Finally, in Sec. IV we give our conclusions. In Appendix we report the analytical expressions for the corresponding amplitudes in the helicity basis, and their square moduli given for all possible final spin configurations, for the q q¯ → tt¯ and gg → tt¯ processes.

2 2.1

Polarized processes Theoretical framework

The most general effective vertex Γa µ (q 2 , M ) for a quark-gluon interaction, in momentum space, containing the contribution of lowest dimensional operators, and compatible with gauge-, CP-,

3

and Lorentz-invariance, is [35] n Γa µ (q 2 , M ) = −igs T a γ µ 1 + gV (q 2 , M ) + γ5 gA (q 2 , M ) o + gP (q 2 , M )q µ γ5 + gM (q 2 , M )σ µν q ν ,

(1)

where gS is the strong coupling constant, and T a are the color matrices. In general, the gV,A,P,M form factors depend by a characteristic energy scale M , typically the largest mass scale running in the loops, and by q 2 which is the invariant momentum-squared carried by the gluon. The gV,A,P,M form factors can also depend by the quark flavor. In the following, we will introduce the dependence on the flavor in the form factors when required. All the effective couplings appearing in Eq.(1) arise also in the SM at the one-loop level due to the weak corrections [36]. The corresponding scale M in that case is connected to the electroweak (EW) scale, being induced by the exchange of W and Z weak bosons in the loop. The SM contribution to the parity-violating gA , gP couplings, which is a typical EW correction to the gluon-quark vertex, is expected to be small and cannot explain the Tevatron anomaly [35]. Recently, the NLO weak corrections to the forward-backward and charge asymmetry at Tevatron and LHC has been computed [46] and their effect account for a few percent. Finally, the last term in Eq.(1) is the contribution of the chromomagnetic dipole operator (with gM the corresponding form factor), that may affect the total cross section [47] but does not significantly contribute to the top-quark FB asymmetry [48]. The QCD gauge invariance requires that qµ U¯f (p1 ) Γa µ (q 2 , M ) Uf (p2 ) = 0 ,

(2)

where in the above equation q = p1 − p2 and the external bi-spinors Uf (p1,2 ) associated to the quark flavor f in momentum space are understood to be on-shell. Model independently, this condition implies the following Ward identity 2mQ gA (q 2 , M ) = q 2 gP (q 2 , M ),

(3)

lim gA,V (q 2 , M ) = 0 ,

(4)

thus q 2 →0

since no 1/q 2 singularities are present in gP . Notice that the Ward identity in Eq.(4) is exact and free from any anomaly contribution, since the vector-axial coupling is an effective vertex and the fundamental theory (QCD) is anomaly free. For a more detailed discussion regarding the origin of the form factors gA,P (q 2 , M ) associated to the quark of flavor f , see Refs.[35],[37]. In Ref.[35] we found that the magnitude of gA , necessary to explain the Tevatron AtF B anomaly, is not compatible with the condition gA ∼ gV , since gV is strongly constrained by the measurements on the p¯ p → tt¯ cross section, which are in good agreement with the SM 4

Figure 1: Feynman diagrams (a)-(d) for the q q¯ → tt¯ process, with the contribution of the gluon effective axial-vector coupling to light quarks (gAq ) and top-quark (gAt ).

prediction. Then, following the same approach as in [35], from now on, we will neglect the contribution of the vectorial form factor gV (q 2 , M ) in Eq. (1), and consider only NP scenarios that generate gA with the hierarchy gV gA . In the limit of q 2 M 2 , it is useful to parametrize the axial-vector form factor as q2 (5) gA (q 2 , M ) = 2 F (q 2 , Λ) , Λ where we absorb the NP coupling αN P and loop factor into the NP scale, Λ2 = M 2 /(4παN P ). Because of the breaking of conformal invariance, induced by renormalization, we expect [49] F (q 2 , Λ) to contain also logarithm terms log(q 2 /Λ2 ). This could give a large log enhancement in the case of |q 2 | Λ2 . In general, the form factor F (q 2 , Λ) could also develop an imaginary part for q 2 > 0. In perturbation theory, this is related to the absorptive part of the loop diagram generating gA , when |q 2 | is above the threshold of some specific particles pair production. Below, we will analyze the contribution of the axial-vector gA anomalous coupling, as defined in Eq. (1), to the polarized partonic cross sections for tt¯ pair production at the LHC, related to the processes q q¯ → tt¯ and gg → tt¯. In order to give more general results, we will introduce in the following the dependence of the quark flavor f = q, t in the effective gluon axial-vector coupling gAf , where symbols q and t stand for a generic light quark and top-quark respectively.

2.2

Polarized q q¯ → tt¯ process

Let us consider the tree-level scattering q(p1 )¯ q (p2 ) → t(p3 )t¯(p4 ) ,

(6)

where p1−4 are the corresponding particles momenta and q stands for a light quark. The Feynman diagrams (a)-(d) relative to q q¯ → tt¯, including the axial-vector coupling, are shown in Fig. 1. According to Eq. (1), supplemented by the Ward identity in Eq. (4), the Feynman rule ΓaA µ , corresponding to the effective axial-vector gluon couplings to quarks q is mq a µ q a (7) ΓA = igA T γµ γ5 − 2qµ 2 γ5 , q where qµ is the gluon momentum entering the vertex, mq is the quark mass, and T a the color matrix. From now on, to lighten the notation, we will omit the q 2 and any other mass scale dependence in the gAq form factors, unless specified. 5

Below we will give the analytical expressions for the polarized total cross sections for the process q q¯ → tt¯, in the helicity basis and in the q q¯ center of mass frame or zero momentum frame (ZMF). In Appendix we will provide the analytical expressions for the corresponding amplitudes in the helicity basis in the ZMF, and their square moduli given for all possible final spin configurations. The results for the polarized total cross sections are the following 2παS2 βρ 1 + |gAq |2 , 27ˆ s 4παS2 q q¯ σLR (ˆ s) = 1 + |gAq |2 2Re[gAt ](1 − ρ) + β 1 + |gAt |2 (1 − ρ) , 27ˆ s q q¯ q q¯ σRR (ˆ s) = σLL (ˆ s), n o q q¯ q q¯ t t σRL (ˆ s) = σLR (ˆ s) Re[gA ] → −Re[gA ] , q q¯ σLL (ˆ s) =

(8)

√ where we neglect the mass of the initial light quarks, β = 1 − ρ, with ρ = 4m2t /ˆ s, and sˆ = 2 (p1 + p2 ) . The total sum over polarization is in agreement with the unpolarized corresponding result in [35, 37]. As we can see from the results in Eq.(8), the left-right (LR) symmetry, obtained by the simultaneous exchange of left-handed with right-handed top-quark polarizations, is broken at the tree-level by the presence of the axial-vector coupling of the gluon. On the other hand, in pure QCD the top-quark LR symmetry remains exact at any order in perturbation theory due to the parity conservation of strong interactions, while it is broken at one-loop by the effect of weak radiative corrections [41, 42, 43, 44, 45]. On the other hand, the vector-axial coupling of gluon can induce the LR symmetry-breaking on top-quark polarizations at the tree-level. This suggests that any observable based on the LR asymmetry of top-quark polarizations turns out to be a very sensitive probe of this scenario.

2.3

Polarized gg → tt¯ process

The main contribution at the LHC to the top antitop-quark production, is given by the gluongluon fusion process g(p1 )g(p2 ) → t(p3 )t¯(p4 ) .

(9)

The Feynman diagrams (a)-(d) relative to gg → tt¯, including the gluon axial-vector coupling, are shown in Fig. 2

6

Figure 2: Feynman diagrams (a)-(d) for the gg → tt¯ process, with the contribution of the gluon effective axial-vector coupling to the top-quark gAt .

The polarized total cross sections in the helicity basis and in the ZMF are given by ρ 1+β παS2 gg 2 2 16 − 14ρ + 31ρ − (2 + ρ (29 + 2ρ)) log , σLL (ˆ s) = 192 sˆβ β 1−β παS2 1 1+β 2 gg t 2 σLR (ˆ s) = 2 11 (ρ − 4) + 6|gA | (1 − ρ) + (32 + (2 − ρ)ρ) log , 192 sˆβ β 1−β gg gg σRR (ˆ s) = σLL (ˆ s), gg gg σRL (ˆ s) = σLR (ˆ s), (10) where the symbols β and ρ are the same as defined above. We have explicitly checked that the results in Eqs.(8) and (10) are separately gauge invariant for each tt¯ polarizations, including the contribution from the gluon axial-vector coupling. The sum over the tt¯ polarizations reproduces the results for the unpolarized total cross section [37]. In appendix we report the corresponding expressions for the amplitude of gg → tt¯ process in the helicity basis in the ZMF, and their square moduli given for all possible final spin configurations. As we can see from Eq.(10), the gg → tt¯ process turns out to be symmetric under the LR symmetry, even including the effect of the axial-vector coupling. This because of the Cparity of the initial gluon-gluon state. Therefore, the gluon axial-vector contribution to the LR asymmetry purely originates from the quark -antiquark fusion process. Then, the LR polarization asymmetry is very sensitive to the q q¯ production mechanism, as in the case of the FB asymmetry. However, in the SM the FB or charge asymmetry gets the leading contribution from a quantum interference effect in QCD [24, 23, 25, 38], while the LR polarization asymmetry mainly comes from the interference of the tree-level QCD amplitude with the weakly corrected one. Therefore, at the LHC energies, the SM LR polarization asymmetry turns out to be at the level of few permille, while the FB asymmetry can be larger and at the level of few percent. Then, due to the suppressed SM contribution, the LR polarization asymmetry turns out to be a more sensitive probe of the NP scale Λ associated to the gluon axial-vector form factor, with respect to the charge or FB asymmetry. Finally, the corresponding hadronic cross sections pp → tt¯X at LHC for the polarized AB AB processes are obtained by convoluting the polarized partonic cross sections σqq , σgg , in Eqs. (8),(10) respectively (where A, B generically indicate the L,R polarization states of tt¯ ), with

7

the corresponding parton distribution functions (PDF) for quarks and gluons, namely ! Z X AB AB AB dρq σqq (ˆ s) + dρg σgg (ˆ s) , σpp→t t¯X =

(11)

q

where dρq and dρg indicate the differential integrations in dx1 dx2 convoluted with the quarks and gluon PDF, respectively. In the numerical integration of Eq. (11) we have used the CTEQ6L1 parton distribution function (PDF) [50], where we set the PDF scale µ and the strong coupling constant αS (µ) at the same scale µ = mt , with top-quark mass mt = 172 GeV.

3 3.1

Numerical results Spin-correlation

Recently ATLAS [21] and CMS [22] collaborations have reported the measurements of the spin correlations in tt¯ production at the LHC. The degree of correlation A of tt¯ system is defined as the fractional difference between the number of events where the top and antitop quark spin orientations are aligned and those where the top quark spins have opposite alignments, namely A=

N (↑↑) + N (↓↓) − N (↑↓) − N (↓↑) , N (↑↑) + N (↓↓) + N (↑↓) + N (↓↑)

(12)

where the arrows denote the spins of the top and antitop with respect to a chosen quantization axis. In the following we will indicate with Ah the spin correlation A evaluated in the helicity basis and in the ZMF of the tt¯ pair. The ATLAS collaboration has reported the following measurement for A in the helicity basis (Ah ) [21] AATLAS = 0.40+0.09 h −0.08 ,

(13)

corresponding to an integrated luminosity of 2.1fb−1 . Candidate events were selected in the dilepton topology with large missing transverse energy and at least two jets. The hypothesis of zero spin correlation is then excluded at 5.1 standard deviations. On the other hand, the CMS collaboration, by using 5fb−1 of integrated luminosity, has reported the following value for Ah [22] ACMS = 0.24 ± 0.02(stat) ± 0.08(syst) , h

(14)

where systematic and statistical errors are indicated in parenthesis. The above results in Eqs.(13),(14) are inclusive in the available phase space of mtt invariant mass system. √ The corresponding SM prediction for LHC energies S = 8 TeV, at the next-to-leading (NLO) order in QCD is [15] ASM h = 0.31 . 8

(15)

The theoretical uncertainties, after including the NLO QCD corrections, due to the variation of factorization and renormalization scale, including the uncertainties on parton distribution functions (PDF), are small and of the order of 1% [15]. Although, the experimental central values in Eqs.(13) and (14) are quite different, the two measurements are compatible with each other and with the SM prediction within 2 standard deviations. At this point, one may wonder if the above ATLAS and CMS results can provide enough information to constrain the present scenario in the critical range of Λ ∼ 1 − 1.3 TeV, required for explaining the Tevatron top-quark anomaly [35]. Unfortunately, a direct comparison with these results is not possible in the framework of the low energy approximation adopted in Eq.(5) with F (q 2 , Λ) constant, since the ATLAS and CMS measurements in (13) and (14) are inclusive in the mtt invariant mass. Indeed, unitarity and perturbation theory restrict the validity of this approach to the kinematic regions mtt < Λ. In order to extend our predictions to the higher mtt invariant masses mtt > Λ we need to provide a shape for the form factor gA (q 2 ) as a function of q 2 . The price to pay would be in this case the introduction of new free parameters. A simple choice is to assume a particular shape for the gA (q 2 ) function that tends to some fixed cut-off in the regions |q 2 | = m2tt Λ2 , while reproducing the low energy limit of QCD Ward identities in Eq.(4). The purpose of this analysis is to determine the sensitivity of the inclusive top-spin correlation observables to this cut-off, at fixed values of the scale Λ. We will present a detailed discussion on this issue in the last subsection. Now, we will focus on the numerical analysis of the spin-correlation and LR asymmetries, in the low enery limit, that is when we restrict our analysis to the regions mtt < Λ. Following the low-energy approach of Refs.[35],[37], in order to simplify the analysis we will assume a real and universal gluon axial-vector coupling, and reabsorb all the NP effects in the scale Λ defined as follows (q,t)

gA (q 2 ) =

q2 . Λ2

(16)

where we neglected any potential logarithm contribution proportional to q 2 log(q 2 /Λ2 ) and higher powers of q 2 /Λ2 terms. This has the advantage of performing a phenomenological model independent analysis, by introducing only one relevant free parameter. The quark universality of the gluon axial-vector coupling is not only a reduction of the free parameters of the model, but it is actually supported by the explanation of the Tevatron top-quark asymmetry anomaly in terms of this scenario [35]. Therefore, from now on, we will omit from our notations the quark flavor q dependence in the gluon axial-vector coupling. In Figs.3 we present our numerical results for the spin-correlation observable Ah in Eq.(12) (left plot) and √ its corresponding statistical significance S[Ah ] (right plot), evaluated for LHC energies of S = 8 TeV, in the helicity basis, and in the laboratory frame. We show our results for some kinematic ranges of mtt and Λ in the range 1 TeV < Λ < 2 TeV. The kinematic ranges

9

0.4

65

LHC 8 TeV

a

0.3

d [a] 2 mt < mtt < 0.6 TeV

0.2

[c] 0.8 TeV < mtt < 1 TeV [d] 2 mt < mtt < 1 TeV

Ah

S[Ah]

b

0 -0.1

L = 10 fb-1 c

[a] 2 mt < mtt < 0.6 TeV

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[b] 0.6 TeV < mtt < 0.8 TeV

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[c] 0.8 TeV < mtt < 1 TeV [d] 2 mt < mtt < 1 TeV

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LHC 8 TeV

55 50

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0.1

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20 15

-0.3

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-0.5

0 1

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2

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Λ (TeV)

1.4

1.6

1.8

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Λ (TeV)

Figure 3: Top-antitop spin correlation Ah (left plot) and corresponding significance S[Ah ] (right plot) in tt¯ events, in the helicity basis and laboratory frame, for the LHC center of mass energy of 8 TeV and integrated luminosity L=10 fb−1 , as a function of the scale Λ and for some ranges of mtt . Dashed lines in the left plot correspond to the SM predictions at leading order in QCD.

of mtt considered in our analysis are the following [a] [b] [c] [d]

= = = =

2 mt < mtt < 0.6 TeV, 0.6 TeV < mtt < 0.8 TeV, 0.8 TeV < mtt < 1 TeV, 2 mt < mtt < 1 TeV .

(17)

The dashed lines correspond to the SM prediction at the LO in QCD. As we can see from the left plot in 3, the spin-correlation is positive for the [a] and [d] ranges, while it changes sign for the [b] and [c] range. Although we choose the convention Re[gA ] > 0, the spin-correlation Ah and cross sections do not depend on the sign[gA ]. As we can see from the results in Fig.3, the common trend of this scenario is a decrease of Ah with respect to the SM prediction, while the corresponding SM deviations increase by selecting kinematic regions of mtt masses close to the scale Λ. This last property is due to the fact that the axial-vector form factor gA grows quadratically with mtt . On the other hand, the common decrease from the SM prediction can be easily understood by looking at the definition of A in Eq.(12) and at the polarized cross sections in Eqs.(8), (10). The gluon-gluon fusion mechanism dominates at the LHC energies with respect to the quark-antiquark annihilation gg gg combination, since process. In this case, |gA | enters only through the combination σLR + σRL gg gg σLL and σRR do not depend on gA . This results in a positive contribution to the total cross section (cfr. Eqs.(8) and (10)), but a negative one in the numerator of Ah , see Eq.(12), giving rise to a destructive contribution with respect to the SM one. For the [a] and [d] ranges in the left plot of Fig.4, the maximum deviation from the SM value is obtained for Λ = 1 TeV, corresponding to a 10% deviation from the SM prediction, while for 10

0.5

LHC 14 TeV 0.4

100

a

LHC 14 TeV 90

L = 10 fb-1

d 80

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[a] 2 mt < mtt < 0.6 TeV [c] 0.8 TeV < mtt < 1 TeV

S[Ah]

Ah

70

[b] 0.6 TeV < mtt < 0.8 TeV

0.2

[d] 2 mt < mtt < 1 TeV

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[a] 2 mt < mtt < 0.6 TeV

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[c] 0.8 TeV < mtt < 1 TeV [d] 2 mt < mtt < 1 TeV

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Λ (TeV)

Figure 4: Same as in Fig. 3 but for the LHC center of mass energy of 14 TeV and integrated luminosity L=10 fb−1 .

the [b] and [c] ranges the effect is larger reaching almost a 25% and 100% deviations for the [c] and [d] ranges respectively. For values of Λ = 2 TeV the overall NP effect is strongly reduced and Ah results are much closer to the corresponding SM ones. The numerical values of Ah in Fig.3 for Λ = 1 TeV and Λ = 2 TeV are Ah = (30, −10, −40, 22)% and Ah = (32, −2.1, −21, 26)% respectively. The four series of numbers reported in parenthesis will indicate from now on, if not differently specified, the results corresponding to the mtt integration ranges of [a],[b],[c],[d] respectively. On the right plot of Fig.3, we show the corresponding statistical significance for Ah , that, following the definition of spin-correlation A in (12), is √ (18) S[Ah ] = ∆Ah σ SM+NP L , SM+NP where ∆Ah = |ASM+NP − ASM is the total unpolarized cross section, and L stands for h |, σ h the integrated luminosity, while the SM+NP suffix stands for the full SM and NP contribution. In σ SM+NP we have used the LO QCD cross sections. Notice that the significance in Eq.(18) is a simple theoretical estimation of the true one, since it does not take into account detector efficiencies, acceptance, resolution and systematics. From the results in the right plots of Fig.3, we can see that the corresponding significances for L = 10 fb−1 are quite large. In particular, for Λ = 1 TeV and Λ = 2 TeV we get S[Ah ] = (15, 38, 62, 41) and S[Ah ] = (0.9, 2.3, 3.7, 2.4) respectively. We can see that, for Λ = 2 TeV, the significance is considerably lower, with the maximum effect S[Ah ] ' 4 corresponding to the range [c]. Therefore, we stress that by analyzing the mtt distributions of tt¯ spin correlations at the LHC, the full range up to Λ ∼ 2 TeV can be probed at LHC 8 TeV, even with an integrated luminosity of L = 10 fb−1 . √ In Fig.4 we present the corresponding results of Fig.3, but for LHC energies of S = 14 TeV and integrated luminosity L = 10 fb−1 . By increasing the LHC center of mass energy, we √ see that |Ah | increases by roughly 25-30% with respect to the corresponding values at S = 8 TeV in the regions Ah > 0, while it decreases of roughly the same amount in the

11

regions Ah < 0, for almost all the mtt ranges [a-d], including the SM values. In particular, we get Ah = (39, −0.9, −28, 28)% and Ah = (40, 5.9, −12, 33)% for Λ = 1 and Λ = 2 TeV respectively, where the latter are quite close to the SM values. Due to the larger √ cross sections, the corresponding significances, with respect to the corresponding results at S = 8 TeV , are also increased, roughly by 70% and 30% effects for Λ = 1 TeV and Λ = 2 TeV respectively. In particular, we get S[Ah ] = (25, 63, 104, 73) and S[Ah ] = (1.5, 3.8, 6.1, 4.3) for Λ = 1 TeV and Λ = 2 TeV respectively.

3.2

Left-Right spin asymmetry

Here we consider the LR polarization asymmetry ALR defined as [44] ALR =

N (↑↑) − N (↓↓) + N (↑↓) − N (↓↑) , N (↑↑) + N (↓↓) + N (↑↓) + N (↓↑)

(19)

where the left and right arrows denote the spins of the top and antitop respectively, with respect to a chosen quantization axis. As mentioned in the introduction, the SM contribution to this asymmetry (ASM LR ) is suppressed, being induced by one loop weak radiative corrections to the ¯ QCD q q¯ → tt production. The typical SM value for ASM LR is very small, being of order of 0.5% and 0.04% for the cases of LHC 14 TeV and Tevatron respectively [44]. Therefore, this is a very sensitive probe to any potential parity-violating new physics beyond the SM. The LR polarization asymmetry has been analyzed in [41, 42, 43] for the Tevatron and in [44, 45] for LHC, mainly in the framework of minimal supersymmetric extensions of the SM [44, 41, 43] and more recently in more exotic NP scenarios like axigluons, third-generation enhanced LR models, and supersymmetric models without R-parity [45]. We will see that in our framework, the ALR is at least one order of magnitude larger than the corresponding SM contribution, since it is induced at the tree-level by the effect of the axial-vector coupling of the gluon. For this reason we will neglect the SM contribution to ALR in our analysis. Accordingly, we will use the following formula for the corresponding significance S[ALR ] √ P NP+SM L , (20) S[ALR ] = |AN LR | σ P where in AN LR the leading contribution to the asymmetry is induced by the Re[gA ] terms, which appears in the numerator of the right hand side of Eq.(19), while the denominator clearly includes the NP and SM contributions.

In the left plot of Fig.5 we present our results for the ALR calculated at the LO in QCD and in the helicity basis and laboratory frame, while on the right plot we show the corresponding significance S[ALR ] for L = 10 fb−1 . From these results we can see that the contribution induced by the pure axial-vector coupling to ALR is sizeable. In particular, for Λ = 1 TeV, we get ALR = (4.5, 15, 33, 7.2)%, with a corresponding significance S[ALR ] = (45, 65, 79, 82), while for Λ = 2 TeV the value of ALR lowers considerably, namely ALR = (1.1, 3.4, 7.2, 1.6)% with a corresponding significance S[ALR ] = (11, 14, 15, 18). From these results we can see that, 12

0.4

85

c

L = 10 fb-1

75

2 mt < mtt < 0.6 TeV 0.6 TeV < mtt < 0.8 TeV 0.8 TeV < mtt < 1 TeV 2 mt < mtt < 1 TeV

d

70

[a] 2 mt < mtt < 0.6 TeV

65

S[ALR]

ALR

0.3

[a] [b] [c] [d]

LHC 8 TeV

80

LHC 8 TeV

0.2

60

c

[b] 0.6 TeV < mtt < 0.8 TeV

55

b

[c] 0.8 TeV < mtt < 1 TeV

45

a

40

b

[d] 2 mt < mtt < 1 TeV

50

35 30

0.1

25

d a

20 15

0

10 1

1.2

1.4

1.6

1.8

2

1

Λ (TeV)

1.2

1.4

1.6

1.8

2

Λ (TeV)

Figure 5: Top-antitop left-right polarization asymmetry ALR (left plot) and corresponding significance S[ALR ] (right plot) in tt¯ events, in the helicity basis and laboratory frame, for the LHC center of mass energy of 8 TeV and integrated luminosity L=10 fb−1 , as a function of the scale Λ and for some ranges of mtt .

although ALR is smaller than Ah , its statistical significance is higher than the corresponding one of Ah , mainly due to the fact that in the ALR the SM background is negligible. Therefore, ALR is a more sensitive probe of this scenario than Ah . Notice that the sign of ALR in the right plot of Fig.5 depends on the convention we used for the sign of Re[gA ], namely positive. If we switch this sign, the asymmetry change sign too, being directly proportional to Re[gA ]. Therefore, we stress that a non vanishing measurement of ALR , also determines the sign of Re[gA ] in the framework of this scenario. √ In Fig. 6 we show the corresponding results of ALR for LHC energy of S = 14 TeV. As we can see from the left plot of Fig.6, the trend of ALR by increasing the LHC energy is different with respect to the corresponding Ah behavior, at fixed √ values of Λ. In particular, there is roughly a 45% decrease in ALR , when passing from S = 8 TeV to 14 TeV. This is due to the fact that the total cross section, dominated by the gluon-gluon fusion process, grows faster qq qq than the σLR − σRL contribution, by increasing the center of mass energy. In particular, for the [a-d] mtt ranges we get ALR = (2.8, 8.6, 19, 4.7)% and ALR = (0.7, 1.9, 3.8, 1)% for Λ = 1 TeV and Λ = 2 TeV respectively. √ On the other hand, by comparing the corresponding significances S[ALR ] at S = 8 TeV and 14 TeV in the right plots of Figs.5 and 6 respectively, we see that √ there is an almost 30% increase of S[ALR ] in all integration ranges [a-d], when passing from S = 8 TeV to 14 TeV.

13

0.2

110

LHC 14 TeV LHC 14 TeV

100 [a] [b] [c] [d]

c

2 mt < mtt < 0.6 TeV 0.6 TeV < mtt < 0.8 TeV 0.8 TeV < mtt < 1 TeV 2 mt < mtt < 1 TeV

90 80

S[ALR]

ALR

b

2 mt < mtt < 0.6 TeV 0.6 TeV < mtt < 0.8 TeV 0.8 TeV < mtt < 1 TeV 2 mt < mtt < 1 TeV

60 50

b

[a] [b] [c] [d]

c

70 0.1

L = 10 fb-1

d

a

40

d

30

a

20

0

10 1

1.2

1.4

1.6

1.8

2

1

1.2

1.4

Λ (TeV)

1.6

1.8

2

Λ (TeV)

Figure 6: Same as in Fig. 4 but for LHC center of mass energy of 14 TeV and integrated luminosity L=10 fb−1 .

3.3

Comparison with ATLAS and CMS results

Now we discuss the impact of this scenario on the mtt inclusive measurements of Ah performed √ by ATLAS and CMS collaborations, corresponding to LHC data at S =7 TeV. In particular, we are interested in estimating the maximum effect induced by the axial-vector coupling contribution to these inclusive observables at fixed values of Λ. As mentioned before, this can be done at the cost of introducing a new free parameter in addition to Λ, that should be understood as the upper bound on the gA form factor in the high mtt mass regions mtt Λ. Dimensional analysis and unitarity arguments, suggest that the gA (q 2 ) form factor should not grow with |q 2 | indefinitely and should tend at most to a constant value in the asymptotic limit |q 2 | Λ2 , where in our case this corresponds to q 2 = m2tt Λ2 . In order to implement this parametrization, we replace gA in Eq.(5) by some test function gA (q 2 ) = GF (q 2 ) , that reproduces the low energy limit in Eq.(5), but satisfies the asymptotic condition lim|q2 |→∞ {gA (q 2 )} = g¯A , where g¯A is some constant. By naturalness arguments, we expect g¯A to be at the most of order O(1). For simplifying the analysis, we will restrict to the case in which the constant g¯A is real. Basically, g¯A plays the role here of a new dimensionless free parameter that parametrizes the upper bound of the axial-vector form factor gA , in the kinematic regions mtt Λ. By using some test functions for the gA form factor, satisfying the above criteria, we will show that for Λ > 1 TeV, deviations from the SM results in the inclusive Ah values are very small, at most of order of 10% for asymptotic values of g¯A ≤ 10. As a toy model, we will use the following function to parametrizes the form factor gA (q 2 ) = GF (x), as a function of x = q 2 /Λ2 , namely g¯A xeg¯A e +y , with y = g¯ , (21) GF (x) = g¯A − log 1+y (e A − 1) 14

Gi 1.0

GΘ

0.8 0.6

GF

0.4 0.2 0

2

4

6

8

10

x

Figure 7: The GF (x), evaluated at g¯A = 1, and Gθ (x) versus x = sˆ/Λ2 .

where GF (x) satisfies the required conditions GF (x) = x+O(x2 ) for x 1 and limx→∞ GF (x) = g¯A . In Fig.7, we plot for comparison GF (x), evaluated at g¯A = 1, with the function Gθ (x) defined as Gθ (x) = xθ(1 − x) + g¯A θ(x − 1). In the left plot of Fig.8 we show our prediction for the mtt inclusive spin correlation observable Ah corresponding to Λ = 1 TeV and LHC energy of 7 TeV, as a function of g¯A , for the GF (x) function. The colored bands stand for the 2σ regions for the ATLAS (top region) [21] and CMS (down region) [22] measurements of Ah , while the middle band is the overlap between these two areas. The dashed dot and continuous (red) lines correspond to the SM prediction in Eq.(15) at the NLO in QCD and to the prediction of our scenario for the GF function respectively, suitably rescaled to the SM value at the NLO. In rescaling our predictions we multiplied the results obtained√at the LO in QCD by the SM K factor for the spin correlation defined as S = 7 TeV. K = ANLO /ALO h h at As we can see from these results, the impact of this scenario on the inclusive observable Ah is a decrease of the Ah values with respect to the SM prediction. In the region 4 < g¯A < 10, the Ah approaches to a plateau, namely Ah ∼ 25%. The expected deviations from the SM prediction, for Λ = 1 TeV, are within the 2σ bands of ATLAS and CMS measurements. If we consider larger values of the scale Λ > 1 TeV, the SM deviations are dramatically reduced. The variation (δAh ) of Ah in the range 1 < g¯A < 10 is of order of δAh ∼ 12%, corresponding to Ah = 28% and Ah = 25%, for g¯A = 1 and g¯A = 10 respectively. The range of this variation should be interpreted as the theoretical uncertainty of our scenario on the inclusive observables, at fixed value of Λ = 1 TeV, that becomes smaller by taking larger values of Λ. In the case of the total cross sections, this deviation is even smaller, being of the order of 0.8%, which is a very negligible effect in comparison to the other (QCD and PDF) uncertainties affecting the strong interactions induced cross sections at the LHC. Moreover, the results in Fig.8 are not very sensitive to the choice of the parametrization function. For instance, for g¯A = 1, the difference for Ah evaluated by using GF (x) or Gθ (x) is of order of 8%. In conclusion, we believe that this scenario and in particular the Λ = 1−1.3 TeV region that 15

0.7

0.08

Λ = 1 TeV

LHC 7 TeV

Λ = 1 TeV

2 mt < mtt < 7 TeV

0.6

LHC 7 TeV

2σ ATLAS

0.5

2 mt < mtt < 7 TeV

0.07 0.4

ALR

Ah

2σ CMS ∩ ATLAS SM

0.3

0.06 0.2

2σ CMS 0.1

0

0.05 1

2

3

4

5

− gA

6

7

8

9

10

1

2

3

4

5

− gA

6

7

8

9

10

Figure 8: The mtt inclusive values for Ah (left) and ALR (right) respectively, for the LHC 7 TeV energy and Λ = 1 TeV, versus the gluon axial-vector coupling cut-off g¯A . In left plot, colored (dark blue) bands stand for the ATLAS (top region) and CMS (down region) 2σ regions, while the middle (light blue) band is the overlap between these two areas. The dashed dot and continuous (red) lines correspond to the SM prediction in Eq.(15) at the NLO in QCD and to the prediction of our scenario for the GF function respectively, all multiplied by the NLO rescaling factor.

is required to explain the Tevatron anomaly, is still consistent with the inclusive measurements of Ah reported by the CMS and ATLAS collaborations within 2 standard deviations. This suggests that a dedicated experimental analysis of the mtt distributions of Ah by the CMS and ATLAS collaboration is needed in order to either confirm or ruled out this scenario. On the right plot of Fig.8 we show the corresponding predictions for the mtt inclusive observable ALR . We can see that the general trend is an increase of the ALR values by increasing g¯A . We did not show the SM prediction in the plot since this is about one order of magnitude smaller. We can see that the ALR approaches to a constant value for 4 < g¯A < 10, namely ALR = 7.4%. The variation of ALR in the considered range of g¯A , is of the order of 28%, passing from 5.8% to 7.4%. Therefore, measurements of ALR at the LHC, even if inclusive in mtt , could be crucial for testing this model, although a dedicated analysis of the mtt spectrum would be more effective and less model-dependent in constraining this scenario.

4

Conclusions

We have analyzed the impact of the gluon effective axial-vector coupling on the spin correlations Ah and LR spin asymmetry ALR in top- antitop-quark production at the LHC. We studied these observables at different invariant masses of the tt¯ system and showed that it would be necessary to measure these quantities as function of the tt¯ invariant mass mtt at the LHC. In particular, we found that these observables are very sensitive to the NP scale Λ associated with the effective 16

axial-vector coupling of gluon, in the high tt¯ invariant mass regions close enough to the scale Λ. Moreover, we found that the ALR is the best probe to test this scenario at the LHC since the SM background is negligible. We estimated the potential effect of the gluon effective axial-vector coupling on the mtt inclusive spin correlations measurements obtained by ATLAS and CMS collaboration. We show that this scenario, for a scale Λ ≥ 1 TeV, is still consistent with present measurements within standard deviations. Therefore, a more dedicated analysis of those quantities as a function of mtt is mandatory in order to test this scenario at the LHC. We stress that the 8 TeV LHC has enough sensitivity either to confirm the Tevatron top charge asymmetry anomaly or to rule it out in the context of the considered NP scenario. Acknowledgements We acknowledge useful discussions with W. Bernreuther and Z.-G. Si. E.G. would like to thank the PH-TH division of CERN for its kind hospitality during the preparation of this work. This work was supported by the ESF grants 8090, 8499, 8943, MTT8, MTT59, MTT60, MJD140, JD164, MJD298, by the recurrent financing SF0690030s09 project and by the European Union through the European Regional Development Fund.

A

Matrix elements

Here we give the matrix elements for all possible helicity configurations of the initial and final state particles in partonic processes q(k) q¯(k 0 ) → t(pt ) t¯(pt¯) ,

(22)

g(k) g(k 0 ) → t(pt ) t¯(pt¯) ,

(23)

where k, (k 0 ) and pt , (pt¯) denote 4-momenta of the quark (antiquark) or gluon (gluon) initial and top- (antitop)-quark final states, respectively. The calculations were performed in the zero momentum frame (ZMF), where the z-axis was chosen in the direction of the top and all other momenta are assumed to lie on the xz-plane. In this frame the momenta 4-vectors for the top and antitop are √ √ s s (1, 0, 0, β), pt¯ = (1, 0, 0, −β), (24) pt = 2 2 p where sˆ = (pt¯ + pt )2 and β = 1 − 4m2t /s. We compute the matrix elements for all possible helicity configurations of the initial and final particles. The spinors of helicity eigenstates are constructed by the helicity prescription, where the spin is given in the rest frame of the particle. The state is then boosted in the positive direction of the z-axis and then rotated clockwise in the xz-plane to end up with the chosen 4-momentum of the particle in the ZMF frame.

17

The cross-section is given by dσ i β 2 i ˜i 2 = α c |M | , dΩ 4ˆ s S

(25)

˜ i |2 is a non-normalized color where i ∈ {q q¯, gg}, ci is an overall group theoretic factor and |M averaged squared amplitude for the process. It can be expressed as ˜ i |2 = ρhh0 ρ¯h¯ h¯ 0 Ri 0 ¯ ¯ 0 . |M hh ,hh

(26)

Here R describes the production of on-shell top quark pairs from a given initial state. The matrices ρ, ρ¯ are the density matrices describing the measurement of polarized top and antitop ¯ in Eq. (26) denote the top and antitop quarks in specific final states. The subscripts h and h helicities. In the chosen basis for spin states ρ = (1 + nit σi )/2 and ρ¯ = (1 + nit σ3 σi σ3 )/2, where σi are Pauli matrices. The corresponding covariant spin vectors are st = (γβ nt3 , nt1 , nt2 , γ nt3 ),

st¯ = (γβ nt¯3 , −nt¯1 , nt¯2 , −γ nt¯3 ),

(27)

√

with γ = s/2mt . Helicity eigenstates correspond to ~n = (0, 0, h) for both top and antitop, where h is the sign of helicity. It takes values +1 and −1 denoting right-handed and left-handed fermions, respectively.

A.1

Polarized q q¯ → tt¯ process

The group theoretic factor for this process is cqq¯ =

N2 − 1 1 d(A) = , 4d(F )2 4N 2

(28)

where d(F ) = N and d(A) = N 2 − 1 are the dimensions of the fundamental (F) and adjoint (A) representation, respectively. The momenta of the initial quark and antiquark are √ √ s s 0 (1, − sin(θ)βq , 0, cos(θ)βq ), k = (1, sin(θ)βq , 0, − cos(θ)βq ), (29) k = 2 2 whereqθ the is the angle between the momenta of the initial quark and top in the ZMF and βq = 1 − 4m2q /s. The squared matrix element is given by (25). For the initial q q¯ the production matrix for a given initial state is q q¯ ˜ qq¯¯∗0 ˜ qq¯¯ Rhh M , 0 ,h ¯h ¯ 0 ;hq hq¯ = Mh0 h ;hq hq¯ hh;hq hq¯

(30)

where ˜ qq¯¯ M = δh,h¯ δhq ,hq¯ γq−1 γ −1 cos(θ) hh;hq hq¯ + δh,h¯ δhq ,−hq¯ γ −1 (1 + hq gA βq ) (−hq ) sin(θ) + δh,−h¯ δhq ,hq¯ γq−1 (1 + hgA β) (+h) sin(θ) + δh,−h¯ δhq ,−hq¯ (1 + hq gA βq ) (1 + hgA β) (1 + hq h cos(θ)), 18

(31)

where β =

p √ √ 1 − 4m2t /s, γ = s/2mt , and γq = s/2mq .

After taking the spin sum over initial polarizations, the squared matrix element can be given by: 1 X ˜ qq¯ 2 |M | = C0qq¯ + n1t n1t¯ C1qq¯ + n2t n2t¯ C2qq¯ + n3t n3t¯ C3qq¯ 4 ¯ hq ,hq

q q¯ q q¯ q q¯ , + (n3t − n3t¯ )C03 + (−n1t + n1t¯ )C01 + (n1t n3t¯ + n1t¯ n3t )C13

(32)

where C0qq¯ = + C1qq¯ = C2qq¯ = C3qq¯ = + q q¯ C13 = q q¯ C01 = q q¯ C03 =

1 2(1 + gA2 βq2 )(1 + gA2 β 2 ) + (1 + gA2 βq2 + γq−2 )(1 + gA2 β 2 + γ −2 ) 8 2 2 2 2 βq β (1 + gA ) cos(2θ) + 2βq βgA2 cos(θ) , 1 −2 −2 2 2 −2 2 2 2 − γq γ + βq (1 + gA )(1 − gA β + γ ) sin (θ) , 4 1 −2 2 2 2 2 2 −2 −2 − − γq γ + βq (1 + gA )(1 − gA β − γ ) sin (θ) , 4 1 − 2(1 + gA2 βq2 )(1 + gA2 β 2 ) + (1 + gA2 βq2 + γq−2 )β 2 (1 + gA2 ) 8 2 βq (1 + gA2 )(1 + gA2 β 2 + γ −2 ) cos(2θ) − 2βq βgA2 cos(θ) , 1 γ −1 βq βgA2 + βq (1 + gA2 ) cos(θ) sin(θ) , 2 1 γ −1 βq gA 1 + βq β(1 + gA2 ) cos(θ) sin(θ) , 2 1 2 2 2 2 2 2 gA β + βq (1 + gA β ) cos(θ) + βq β gA − (1 + gA ) sin (θ) . 2

(33) (34) (35)

(36) (37) (38) (39)

The coefficient C0 is proportional to the (final) spin summed result. The quotients Ci /C0 , i ∈ {1, 2, 3} give spin correlations and the quotients Ci /C0 , i ∈ {01, 03} give the spin asymmetry q q¯ for the corresponding quantization axis. Direction ”3” corresponds to helicity. The term C03 is the only source of the spin asymmetry (19) at tree level. The phase space integration is performed over the solid angle. The spin parameters n1t , n2t , n1t , n2t are implicitly dependent on the azimuthal angle, so terms linear in these parameters vanish. Therefore the coefficients C01 and C13 do not contribute to the total cross-section. Only the sum C1 + C2 is relevant after the phase space integration. In conclusion Z 1 q q¯ q q¯ ˜ qq¯|2 dΩ = I0qq¯ + (n1t n1¯ + n2t n2¯ ) I1+2 , (40) + n3t n3t¯ I3qq¯ + (n3t − n3t¯ ) I03 |M t t 4π

19

where 1 (1 + gA2 βq2 + γq−2 /2)(1 + gA2 β 2 + γ −2 /2) , 3 1 = − (1 + gA2 βq2 + γq−2 /2)(1 − gA2 β 2 ) , 3 1 = − (1 + gA2 βq2 + γq−2 /2)(1 + gA2 β 2 − γ −2 /2) , 3 2 = (1 + gA2 βq2 + γq−2 /2)gA β . 3

I0qq¯ = q q¯ I1+2

I3qq¯ q q¯ I03

A.2

(41) (42) (43) (44)

Polarized gg → tt¯ process

The group theoretic overall factor for this process is cgg =

1 d(F )CF2 , = d(A)2 4N

(45)

2

where CF = N2N−1 is the√quadratic Casimir invariant of the fundamental representation. The √ s s 0 gluon momenta are k = 2 (1, − sin(θ), 0, cos(θ)) and k = 2 (1, sin(θ), 0, − cos(θ)), where θ the is the angle between gluon and top momenta in the ZMF. The corresponding spin polarization vectors are ± = √12 (1, ∓ sin(θ), i, ∓ cos(θ)). The production matrix takes a form gg Rhh 0 ,h ¯h ¯ 0 ;λg λ0

g

˜ tu0 ¯ 0 0 M h h ;λg λg =4 ˜g M 0¯0 0

!†

h h ;λg λg

A(A − Cr ) Aβ cos(θ)Cr Aβ cos(θ)Cr Cr

! ˜ tu¯ 0 M hh;λg λg ˜ g¯ 0 , , M

(46)

hh;λg λg

where A = (1 − β 2 cos2 (θ))−1 and Cr =

N2 C2 (G) = , 4CF 2(N 2 − 1)

(47)

is a group theoretic constant, 0 ≤ Cr ≤ 1, with C2 (G) = N being the quadratic Casimir invariant in the adjoint representation. Cr is independent of the normalization of the group generators and for Abelian groups Cr = 0. For abelian gauge theories R is determined entirely ˜ tu , as one would expect. by M ˜ tu and M ˜ g are For gluon spins λg and λ0g the amplitudes M ˜ tu¯ 0 = δh,h¯ δλg ,λ0 γ −1 β sin2 (θ) M hh;λg λg g

˜ g¯ 0 M hh;λg λg

− δh,h¯ δλg ,−λ0g γ −1 (hλg + β) − δh,−h¯ δλg ,λ0g β(λg + h cos(θ)) sin(θ),

(48)

= δh,−h¯ δλg ,−λ0g gA β sin(θ) .

(49)

20

˜ g when top-quarks with opposite The axial coupling appears only in the non-abelian part M helicity are produced from gluons with opposite spin. The effect disappears for low energies and collinear momenta. After taking the spin average over initial polarizations, the squared matrix element can be given in a relatively compact form: 1 X ˜ gg 2 |M | = C0gg + n1t n1t¯ C1gg + n2t n2t¯ C2gg + n3t n3t¯ C3gg 4 λ ,λ0 g

g

gg gg + (−n1t + n1t¯ )C01 , + (n1t n3t¯ + n1t¯ n3t )C13

(50)

where C0gg C1gg C2gg C3gg gg C13 gg C01

= = = = = =

A(A − Cr )[1 − β 4 (1 + sin4 (θ)) + 2β 2 sin2 (θ)] + Cr gA2 β 2 sin2 (θ) , −A(A − Cr )[−γ −4 + (1 − γ −4 ) sin4 (θ)] − Cr gA2 β 2 sin2 (θ) , −A(A − Cr )[γ −4 + β 4 sin4 (θ)] − Cr gA2 β 2 sin2 (θ) , A(A − Cr )[1 − β 4 (1 + sin4 (θ)) − 2β 2 sin2 (θ) cos2 (θ)] − Cr gA2 β 2 sin2 (θ) , A(A − Cr )γ −1 β 2 sin(2θ) sin2 (θ) , Cr AgA γ −1 β 3 sin(2θ) .

(51) (52) (53) (54) (55) (56)

The coefficient C0 is proportional to the (final) spin summed result and the rest are associated with different spin observables. Note that there is no LR-asymmetry for the gg-initial state, because there is no term similar to Eq.(39). Instead, in this process the axial coupling introduces another strong spin asymmetry that is not present in the standard model. It is induced by the gg q q¯ coefficient C01 (and similarly by C01 (38) for the q q¯ initial state). This term is caused by the interference between the axial-vector and the vector couplings, and it is the only term of this kind for the gg initial state. This term could induce azimuthal asymmetries. However, for the symmetric initial state, this effect averages out. In order to observe a physical azimuthal gg asymmetry induced by C01 , initial state polarization is needed. The phase space averaged squared matrix element is given by Z 1 gg ˜ gg |2 dΩ = I0qq¯ + (n1t n1¯ + n2t n2¯ ) I1+2 |M + n3t n3t¯ I3gg , t t 4π

(57)

where I0gg gg I1+2

I3gg

3 1 −2 −2 −4 α 28 + 31γ − (32 + 32γ + 2γ ) + gA2 β 2 , = − 16 β 8 1 α 3 10 + 23γ −2 − γ −2 (32 + γ −2 ) = − − gA2 β 2 , 2 16β β 8 1 3 −2 −4 −2 −4 −6 α = 60 − 25γ + 31γ − (32 + 4γ + 28γ + 2γ ) − gA2 β 2 , 2 16β β 8

(58) (59) (60)

where α = atanh(β) is the rapidity and the substitution Cr = 9/16 corresponding to SU(3) was made. 21

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