# Effective axial-vector coupling of gluon as an explanation of the top quark asymmetry

###### Abstract

We explore the possibility that the large forward-backward asymmetry measured by the CDF detector at Tevatron could be due to a universal effective axial-vector coupling of gluon. Using an effective field theory approach we show model independently how such a log-enhanced coupling occurs at 1-loop level. The interference with QCD gluon vector coupling naturally induces the observed positive forward-backward asymmetry that grows with invariant mass and is consistent with the cross section measurements. This scenario does not involve new flavor changing couplings nor operators that interfere with QCD, and, therefore, is not constrained by the LHC searches for 4-quark contact interactions. We predict top quark polarization effects that grow with energy and allow to test this scenario at the LHC. Our proposal offers a viable alternative to new physics scenarios that explain the forward-backward asymmetry anomaly with the interference between QCD and tree level new physics amplitudes.

Introduction. The CDF measurement Aaltonen:2011kc () of large forward-backward (FB) asymmetry for invariant mass GeV came as a surprise. It is unexpectedly large compared to the standard model (SM) next-to-leading order prediction Bowen:2005ap (); Antunano:2007da (); Almeida:2008ug (), it grows with invariant mass since for GeV Aaltonen:2011kc (), and its sign is positive, i.e., opposite to that predicted by the most natural new physics scenarios with axi-gluons axigluon (); Ferrario:2008wm (); Haisch:2011up (). Because of those unusual properties the numerous specific model dependent Rodrigo:2010gm (); wang:2011taa () and model independent Blum:2011up (); Delaunay:2011gv (); AguilarSaavedra:2011vw () new physics (NP) solutions that explain with the interference between QCD and tree level NP amplitudes all suffer from similar problems. They predict large asymmetry also for GeV, large increase of cross section at high energies due to the QCD interference with NP amplitudes, they are strongly constrained Bai:2011ed () by the LHC bounds on four-quark contact interactions Khachatryan:2011as (); ATLAS (), and often involve new flavor changing (FC) couplings to top quark. Therefore it is not surprising that the latter class of models is already now strongly disfavored by the LHC Collaboration:2011dk ().

In this paper we suggest that, if the anomaly with the measured properties persists, it could be induced by anomalously large effective axial-vector coupling of the gluon, that is poorly tested today. We show that NP at the scale TeV can generate large with the following properties: the sign of is automatically positive; the asymmetry grows with due to the gauge invariance of the axial-vector form-factor; the asymmetry is enhanced by large logarithm squared; no dangerous FC couplings are needed to exist; present LHC bounds on four-quark contact interactions do not constrain this scenario; the predicted top quark pair production cross sections at Tevatron and LHC can be consistent with all measurements. Because of the latter, our most interesting prediction for the LHC experiments are related to the top quark polarization effects Cao:2010nw (); Jung:2010yn (); Choudhury:2010cd (); Krohn:2011tw () that should be significantly different from the SM predictions. Therefore this scenario can be fully tested by the LHC using those observables.

Theoretical framework. The most general effective Lagrangian for quark-gluon interactions, compatible with gauge and CP-invariance, is

(1) | |||||

where is the strong coupling constant, is the gluon field, are the color matrices, is the scale of NP, is the invariant momentum-squared carried by the gluon and denotes a generic quark field. At the moment we do not make any assumption on the origin of the form factors . In the most general case the form factors depend also on quark masses that can be neglected for The last term in Eq.(1) is the contribution of the chromomagnetic dipole operator that does not significantly contribute to Blum:2011up ().

Model independently the QCD gauge invariance requires that , thus

(2) |

since no singularities are present in . Equation (2) does not pose any constraint on the form factors and , which could have different magnitudes at arbitrary . Therefore, gauge invariance does not prevent us to have as long as We stress here once again that the QCD gauge invariance is not broken and gluon remains massless because and are induced via the form factors in Eq. (1) that are subject to the condition in Eq.(2). Thus the exist even in the SM, where they are induced by electroweak radiative corrections, but are numerically too small to have significant impact on the observables we consider. However, if the origin of large is due to NP that has currents as in the SM, large and can be generated. This is phenomenologically unacceptable because is strongly constrained by the total cross section that depends quadratically on but only linearly on . Therefore, from now on, we will neglect the contribution of the vectorial form factor in Eq.(1), and consider only NP scenarios that generate with the hierarchy .

In the limit of , it is useful to parametrize the axial-vector form factor as

(3) |

where we absorb the NP coupling and loop factor into the NP scale, Because of the breaking of conformal invariance, induced by renormalization, we expect Raidal:1997hq () to contain also logarithm terms This could give a large log enhancement in the case of . In general, the form factor could also develop an imaginary part for . In perturbation theory, this is related to the absorptive part of the loop diagram generating , when is above the threshold of some specific particles pair production.

and cross section. We first show that the large can be generated consistently with the cross section constraints via the operator

(4) |

The FB asymmetry occurs due to the interference between the gluon mediated s-channel SM amplitude for and the analogous s-channel amplitude induced by two vertices of Eq. (4). First, the induced asymmetry grows with the invariant mass of the system exactly as observed. Second, the sign for the asymmetry comes out to be the right one due to the massless gluon, provided the sign of the form factor is universal. Third, it is expected to give only a subdominant contribution to the production cross section that agrees very well with the SM predictions. In fact, it is predicted Stephenson () that the only observable where the operator in Eq. (4) could play a dominant role is the asymmetry at very high energies, and this is exactly what is happening. The question to address now is how large form factors are needed to explain the observed asymmetry and whether there exist new physics models that can induce it naturally.

We have evaluated the FB asymmetry and total cross section of production at the leading order (LO) in QCD by including the contribution of the axial-vector coupling in Eq.(1). In the total cross section, we have also included the contribution due to the gluon-gluon partonic process . However, since this last process is subleading at Tevatron, giving only a 10% effect of the total cross section, we have retained in the amplitude only the SM contribution.

By using the notation of Eq.(1), the partonic total cross section for , in the limit of , is given by

(5) | |||||

(6) | |||||

where and are the effective axial-vector gluon couplings of the top- and -quarks respectively, is the top-quark velocity in the rest frame, with the center of mass energy, and the corresponding fractions of partons momenta. Here indicate the inclusive cross section integrated in the positive/negative range of respectively, where is the angle between the direction of the outgoing top-quark and the initial quark momentum.

Then, after the convolution of parton cross sections in Eq.(6) with parton distribution functions, the FB asymmetry at the LO is given by

(7) |

where , the cross section, while and indicate the differential integrations in convoluted with the quarks and gluon parton distribution functions respectively. The stands for the total cross section of at the LO, whose SM analytical expression can be found in Beenakker ().

We estimate now how large the form factors we need to explain the large FB asymmetry observed. We consider the most simple case assuming and that is neglecting any possible enhancement from large terms. This is a good approximation in case one is interested in estimating the scale that could provide the asymmetry.

In the numerical integration of Eq.(7) we have used the CTEQ6L1 parton distribution functions (PDF) PDF () and the total cross sections evaluated at the LO in QCD. We set the PDF scale with top-quark mass GeV. We present our results in Fig.1, where we have plotted the new physics contribution to FB asymmetry (continuous lines) and the cross section variation defined as (dashed lines), versus the scale , for several regions of integrations in the invariant mass. The results for the FB asymmetry in Fig.1 are not very sensitive to the choice of the PDF scale. Moreover, we also expect that the inclusion of the next-to-leading order QCD corrections will not change dramatically these predictions, due to the expected factorization property of QCD corrections to cross sections in Eqs.(5)-(6).

We plot the result for the range , where the asymmetry for GeV is larger than SM value and the maximum variation of total cross section is below 20%. In Fig.1, the red curves indicated with [a],[b],[c],[d], correspond to the bins of integrated in the ranges GeV, GeV, GeV, GeV, respectively, while the curves [e] stand for GeV. Notice that results in Fig.1 do not include the SM contribution to that further increases the signal.

The main trend of this scenario is characterized by a FB asymmetry that grows with invariant mass. This is expected from the fact that the form factor is proportional to . We stress that our scenario is in perfect agreement with data for low invariant masses, bin [a], while for the last bin [d] we have at TeV. However, a large FB asymmetry also implies a large contribution to the total LO cross section, due to the constructive interference between the SM amplitude and the one with axial-vector gluon couplings in the s-gluon channel. Still, for the bins [a,b,c] that dominate statistically, the variation of the total cross-section remains small, below the level. The same is true for the observables for GeV, shown in the region [e]. Those results are consistent with the total inclusive asymmetry Jung:2009pi (); Degrande:2010kt () and with cross section measurements since the characteristic uncertainty associated to cross section in the lowest bins can be still of the order of . In the last bin of integration [d], tends to be larger than for TeV. However, we should take into account that the experimental uncertainty of the cross section for the bin [d] is larger than the corresponding ones at lower values of , due to the lack of statistics at high .

The fact that we need a low-energy scale TeV to generate a large contribution to the FB asymmetry of order % for GeV suggests that in the most general case the two scales and , related to the top-quark and light-quark vertex respectively, should be comparable. Indeed, in this case the magnitude of the asymmetry is controlled by the geometric average of the two scales, namely . If we need to be of order of 1 TeV, we cannot push too high, since would be close to the EW scale and the contribution to total cross section would explode. Therefore, results in Fig.1 suggest that the two scales should be comparable, supporting the idea of a universal coupling.

The origin and constraints. Assuming that NP is perturbative, model independently the effective operators Delaunay:2011gv ()

(8) | |||||

(9) |

generate via 1-loop diagrams depicted in Fig. 2. Here and thus both isoscalar and octet operators contribute. Notice that: no is induced due to QCD parity conservation; the 1-loop induced is enhanced by ; the operators , do not induce FC processes; however, there could be different quark flavors in the loop in Fig. 2, extending the operator basis to is straightforward; the operators , do not interfere with the corresponding QCD induced 4-quark processes. The latter point has very important implications for our scenario – the stringent LHC constraints Khachatryan:2011as (); ATLAS () on 4-quark contact interactions do not apply at all. Indeed, those constraints come from the interference between QCD and NP diagrams, and constrain the models that explain with the similar interference very stringently. We stress that our scenario is free from those constraints and NP at 1-2 TeV can induce large as explained above.

Alternatively, large might be generated by new strongly-coupled parity-violating dynamics related to electro-weak symmetry breaking (EWSB) at 1-2 TeV scale. Because this NP is entirely nonperturbative, generating is possible Lane:1996gr () but we are not able to compute it. We are only able to estimate the validity range of the effective coupling parametrization in Eq.(3) that is controlled by , where is expected to be related to as . For TeV, as required by the anomaly, the related scale is 3.5 (4.6) TeV. At this scale a plethora of new resonances should occur at the LHC allowing to test this scenario. Notice that, in the region of large invariant masses , the low-energy ansatz is not valid anymore and the dependence of should be determined by fitting the data.

Observables at LHC and other experiments. First, the natural question to be asked is whether our solution to the Tevatron asymmetry is related to the anomaly observed at pole at LEP. Obviously the answer is no because the initial state at LEP is and gluon does not couple to leptons. In addition, the induced form factors do not affect precision data involving QCD at observable level. Firstly the form factors vanish at low energies according to Eq. (3). Second, at high energies the loop induced is still much smaller than unity and, taking into account relatively large experimental errors in the QCD measurements, does not affect QCD observables.

As we stressed before, the observables that are sensitive to are asymmetries. Because the anomalous axial-vector coupling of gluon grows with , see Eq. (4), the most natural test of our scenario is measuring cross section dependence on However, at the LHC the dominant production process is induced by the s-channel diagram with 3-gluon vertex in addition to the t,u-channel diagrams . The coupling is expected to affect the s-channel diagram since there the gluon coupled to fermions is off-shell us (). This is a peculiar prediction of our scenario which differs from most popular axigluon models where the production mechanism is not affected by the new physics. Moreover, our scenario is testable at the LHC experiments because coupling should induce observable polarization effects of top quarks that also grow with Therefore, studies of top quark polarization at the LHC are sensitive tests of our scenario as well as other models of physics beyond the SM Cao:2010nw (); Jung:2010yn (); Choudhury:2010cd (); Krohn:2011tw (). A dedicated LHC study is needed to discriminate between different sources of asymmetries and polarizations.

Conclusions. Among many model dependent and model independent solutions proposed to explain the measured top quark FB asymmetry, our proposal is the only one that does not involve interference between the QCD and tree level NP contributions mediated by heavy resonances. Instead, we argue that the large is induced by an anomalously large effective axial-vector coupling of the gluon, , described at low energies by the operator in Eq. (4). We have shown that can explain the sign, the magnitude and the behavior of consistently with the cross section measurements. We have shown model independently that logarithmically enhanced can be induced by NP effective operators (8), (9) that do not suffer from flavor constraints and from the LHC constraints on 4-quark contact interactions. While our results are presented in the context of top quark FB asymmetry, our proposal to study anomalous axial-vector coupling of gluon has physics implications beyond that observable. Studying the induced top quark cross sections, distributions and polarization effects at the LHC allows one to test different classes of models beyond the SM.

## Acknowledgement.

We thank the Les Houches 2011 BSM working group members who helped to formulate this study, in particular C. Delaunay and R. Godoble for discussions on top quark asymmetry and polarization, and B. Mele and G. Rodrigo for several discussions. This work was supported by the Estonian Science Foundation under Grants Nos. 8090 and MTT60, and by the Estonian Ministry of Education under the SF0690030s09 project.

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