Abstract
We use an effective operator framework to study the contributions to the Tevatron asymmetry from arbitrary vector bosons and scalars, and compare with their effect on the tail at LHC. Our study shows, for example, that models reproducing the asymmetry by exchange of and bosons or colourtriplet scalars lead to a large enhancement in the tail at LHC. This fact can be used to exclude these models as the sole explanation for the asymmetry, using the data already collected by CMS and ATLAS. Our analysis is model independent in the sense that we scan over all possible extra particles contributing to the asymmetry, and allow for general couplings. We also explore a class of Standard Model extensions which can accommodate the Tevatron asymmetry without contributing to the total cross section at first order, so that the enhancement of the tail at Tevatron and LHC is moderate.
Probing the Tevatron asymmetry at LHC
J. A. Aguilar–Saavedra, M. PérezVictoria
[0.2cm] Departamento de Física Teórica y del Cosmos and CAFPE,
Universidad de Granada, E18071 Granada, Spain
1 Introduction
The measurement of the forwardbackward (FB) asymmetry at Tevatron [1, 2, 3] has motivated a plethora of models which attempt to accommodate the experimental values, up to larger than the prediction of the Standard Model (SM). This task is not straightforward because the measured cross section is in good agreement with the SM: any “generic” addition to the production amplitude, large enough to produce the observed FB asymmetry, will easily give rise to too large a departure in the total rate. Many of the proposed models circumvent this problem at the expense of a cancellation between (linear) interference and (quadratic) new physics terms in the total cross section,
(1) 
where is the SM cross section, the one corresponding to the new physics and the interference term. The cancellation requires a new large amplitude which, obviously, should have observable effects elsewhere. The ideal candidate to search for these effects is the Large Hadron Collider (LHC).
New physics in which produces such a large cancellation in the Tevatron cross section will likely produce an observable enhancement in the tail at LHC, even if at this collider top pair production is dominated by gluon fusion. (The tail is also enhanced at Tevatron energies but in the majority of the proposed models the deviations are compatible with present measurements [3].) In some cases, this effect should be visible already with the data collected in 2010. Conversely, if large deviations are not reported, a number of candidates to explain the Tevatron asymmetry will be excluded from the list.
In this paper we make these arguments quantitative for a wide class of SM extensions. We consider general new vector bosons and scalars, classified by their transformation properties under the SM gauge group , and study their possible effects in production. We use effective field theory to consistently (i) integrate out the new heavy states and obtain their contribution to in terms of fourfermion operators; (ii) obtain the cross sections in terms of effective operator coefficients. This allows us to find, for each vector boson and scalar representation, a relation between the possible values of the asymmetry at Tevatron and the excess in the tail at LHC. The discussion of all possible vector boson and scalar representations within the modelindependent effective operator framework allows us to make stronger statements than in other modelindependent studies [4, 5, 6, 7]. At the same time, any model with several vector bosons and scalars can be considered in our framework by simply summing the effective operator coefficients corresponding to the integration of each new particle. Previous studies of LHC signals associated to the FB asymmetry within particular models have been presented in Refs. [8, 9, 10, 11, 13, 12, 14, 15, 16].
After this analysis, we explore an alternative way to produce a large asymmetry with moderate effects in the tail at LHC. The key for this mechanism is the observation that one can also obtain a large without significant changes in the total cross section by introducing new physics which only contributes to the latter at quadratic order. This has been studied before, in the effective formalism, in Ref. [5]. If we write
(2) 
for the forward (F) and backward (B) cross sections, the total rate is maintained at first order provided , which can be achieved, for example, with a new vector boson and a scalar. For models fulfilling this cancellation the size of the new physics contributions required to accommodate the experimental value of are smaller. Therefore, these models provide a better agreement of the tail with Tevatron measurements, and predict a much smaller tail at LHC, which is still potentially observable with forthcoming measurements.
We remark that the use of effective field theory (with the assumption that the new physics is too heavy to be directly produced at LHC) does not limit much the generality of our conclusions. For channel exchange of new vector bosons or scalars, integrating out the new particles gives a good estimate for masses TeV. For channel exchange this approximation is worse, but the cross section enhancement produced by the new particle(s) is always larger than the one from the corresponding fourfermion operator(s), and in this sense our predictions are conservative. On the other hand, if new physics in the tail is not seen at LHC, the new resonances are heavy and the effective operator framework can be safely used. It is also worth pointing out that we include corrections arising from the quadratic terms in new physics in the cross sections. The contributions from the interference of operators with the SM can be neglected, as they are suppressed with respect to the former for the values of parameters that are required to explain the asymmetry.
2 Extra bosons, operators and production
There are ten possible representations [17] for new vector bosons contributing to , while for scalars eight representations contribute. They are collected in Table 1, where the first column indicates the label used to refer to them.
Label  Rep.  Interaction Lagrangian  Sym. 

–  
–  
–  
–  
–  
–  
–  
–  
–  
–  
–  –  –  –  –  
–  –  –  –  –  




–  –  –  –  –  
–  –  –  –  –  –  
–  –  –  –  –  –  
–  –  –  –  –  –  
–  –  –  –  –  
–  –  –  –  –  –  
–  –  –  –  
–  –  –  –  
–  –  –  –  
–  –  –  –  –  –  
–  –  –  –  –  –  
–  –  –  –  –  –  
–  –  –  –  –  –  
–  –  –  –  –  
–  –  –  –  – 
–  
–  –  




–  –  
–  –  –  
–  –  –  
–  –  –  
–  –  –  
–  –  –  
–  –  
–  – 
–  
–  –  –  –  
–  
–  –  –  –  
–  
–  –  –  –  
–  –  
The fourfermion operators contributing to production, including those that do not interfere with the SM QCD amplitude and only appear at quadratic level, have been given in Ref. [18]. (The operators which interfere with the SM were given in Ref. [19].) We collect in Tables 2–5 the values of the corresponding coefficients for all the vector and scalar irreducible representations that can induce these operators. For vector bosons the coefficients have previously been obtained in Ref. [17]. The notation for fourfermion operators is given in appendix A. We stress again that including both interference and quadratic terms in our calculations is not inconsistent, despite the fact that we are not considering dimensioneight operators. When quadratic terms are relevant for production (large couplings) the missing dimensioneight terms are subleading in the classes of SM extensions we consider. A related discussion about the importance of and contributions from dimensionsix operators has been presented in Ref. [20].
We evaluate the FB asymmetry
(3) 
using the SM predictions [21]
(4) 
and the new contributions from fourfermion operators , parameterised in terms of effective operator coefficients and numerical constants. The explicit expressions are collected in appendix A. It is important to point out that positive operator coefficients always increase at first order ( interference with the SM), as it follows from Eqs. (13). We choose, among the recently reported measurements of the FB asymmetry [3],
(5) 
the one for high invariant masses which exhibits the largest deviation () with respect to the SM prediction. The total cross section at LHC is evaluated including fourfermion operators in a similar way. In order to display the effect of new contributions on the tail we evaluate the cross section for invariant masses larger than 1 TeV. We note that our calculation in terms of effective operators gives a larger (smaller) tail than the exact calculation for channel (channel) resonances. In the former case the differences are not dramatic but in the latter our results can be quite conservative, depending on the mass and width of the new resonance. A detailed comparison is presented in appendix B.
The relation between the predictions for the Tevatron asymmetry and the tail at LHC is tested by performing a random scan over the relevant couplings corresponding to each new particle or multiplet. The results for the ten vector boson representations are presented in Fig. 1. (For , , and the regions are onedimensional because there is only one coupling involved.) There are several interesting conclusions which can be drawn from these plots:

For and the allowed regions are inside the corresponding ones for , , respectively. This is expected because the interactions of the former correspond to a particular case of the latter, with only lefthanded couplings.

For new coloursinglet neutral bosons , the linear terms have negative coefficients and decrease , which can only reach the experimental value for large couplings when quadratic terms dominate. Hence, accommodating a large asymmetry automatically implies a large tail. For example, for the enhancement is more than a factor of five. This implies that these possible explanations for can be probed, and eventually excluded, with the luminosity collected in the 2010 run of LHC. The same conclusion applies to the vector bosons and , as they only contribute in initial states and require a huge coupling to produce a large asymmetry.

For colouroctet isosinglet bosons it is possible to have a large and still a moderate tail at LHC. This model provides an example of cancellation of linear terms in the cross section, provided that , i.e. the vector boson couples as an axigluon. However, in order to have positive coefficients in the interference terms the couplings for the third and first generation must have opposite sign. This does not happen for the isotriplet boson which only has lefthanded couplings, and for these SM extensions the predicted tail is large.

Another interesting candidate is a colour triplet , which has positive coefficients in interference terms as well. Its inclusion gives some enhancement to the asymmetry, which can reach the experimental value with the further addition of a scalar. Note that quadratic terms from operators decrease the asymmetry, as it can be derived from Eqs. (LABEL:ec:quad) and is clearly seen in Fig. 1. Hence, this model is interesting only for moderate couplings.

The rest of vector bosons, , and have little interest for the asymmetry because they do not allow for a value appreciably larger than the SM prediction.
Aside from these remarks, we also note that for (i) and ; (ii) and ; (iii) and , the linear terms have opposite sign, which explains the behaviour observed in the plots for these vector bosons.
The results for the eight scalar representations are presented in Fig. 2. Except for the isodoublets , , the regions are onedimensional because there is only one coupling involved. We point out that:

A coloursinglet isodoublet (with the same quantum numbers as the Higgs boson) can give an asymmetry compatible with the experimental value, and still produce a moderate tail at LHC. (Note that quadratic terms involving the operators , decrease the asymmetry.) On the other hand, a colour octet produces an asymmetry smaller than the SM value because the interference terms have opposite sign.

For coloursextets and the interference terms increase the asymmetry because the operator coefficients are positive.
^{2} However, producing an asymmetry requires large couplings and implies a large tail at LHC, which might be already excluded. For the colour triplets , the situation is worse because the interference terms have negative operator coefficients, and even larger couplings are required to produce a large asymmetry. 
For the two scalars , which only contribute in , the deviations in are always very small.
To conclude this survey, we remark again that this correlation between and the tail at LHC applies to SM extensions with a single vector boson or scalar (as many of the ones proposed in the literature) but when more than one particle is present the contributions can add up or cancel, making it easier to fit the experimental data and predict moderate effects in the tail, as we discuss in detail in the next section.
3 A large asymmetry with a small tail
By inspection of Eqs. (13) it is clear that the cancellation of the linear contributions to the cross section takes place provided
(6) 
(see also Ref. [5]). Notice that in the lefthand side of both equations we have and couplings, whereas on the righthand side we have and ones. As we have mentioned, one simple example where both equalities are fulfilled is an axigluon with flavourdiagonal couplings , with the additional requirement that first and third generation couplings have opposite sign, to have positive coefficients. (This model may be excluded by lowenergy measurements, however [25].) But there are many other possibilities which can be constructed combining particles in Table 2, for instance, a colour triplet together with a colour sextet or .
In these SM extensions with vanishing (or very small) contributions to the total cross section at first order, the FB asymmetry is
(7) 
plus smaller corrections from quadratic terms, which depend on the specific operators which yield and .
Remarkably, one can obtain a good fit to both asymmetry measurements in Eqs. (5) with values of , of order unity.
(8) 
including linear and quadratic terms, with a of 1.72. For the best fit is found for , giving similar predictions for the asymmetry.
We have investigated the effects on the tails by implementing fourfermion operators in the generator Protos [26]. We have first checked that this class of SM extensions does not produce a too large tail at Tevatron. Figure 3 shows the invariant mass distributions for the SM and with , fourfermion contributions corresponding to , which yield the asymmetries in Eqs. (8). Above GeV the cross section is enhanced by , which is consistent with data [3].
For LHC the invariant mass distributions are presented in Fig. 4. The cross section above 1 TeV is a factor of above the SM one. This deviation could be visible with the luminosity to be collected in 2011, provided that the systematic uncertainties (jet energy scale, jet energy resolution, jet energy scale, etc.) are low enough. Here it is necessary to mention that for Tevatron a smaller efficiency for channel new physics at high has been recently claimed [27, 28], which could help maintain the cross section at the high bins in agreement with measurements while reproducing the FB asymmetry. For LHC the efficiency decrease at TeV is not significant because of the larger detector coverage up to for charged leptons and for jets (see appendix B).
4 Conclusions
If the asymmetry measured at Tevatron corresponds to new physics, this new physics should also manifest at the tail at LHC. The size of the effect of course depends on the new physics itself which gives rise to the FB asymmetry, and it can serve to discriminate among different explanations. These issues have been investigated here using an effective operator framework and classifying all possible new vector bosons and scalars by their transformation properties under the SM gauge group. Particular models in the literature attempting to explain the observed asymmetry often fall into one of these classes.
For models which reproduce with  and/or channel exchange [29, 30, 10, 11, 12, 27] we have found that the tail above 1 TeV should be enhanced by a factor of five at LHC, at least for bosons heavier than 1 TeV. With pb at the tree level and a luminosity of approximately 35 pb collected in the 2010 run, the SM predicts 6.3 events in the semileptonic decay channel, not including the detector acceptance nor efficiencies. Then, it seems likely that an enhancement by a factor of five in the tail could be excluded just by analysing 2010 data. The same argument applies to channel exchange [31, 32, 33, 27] or a mixture of both [34].
More exotic models explain the observed asymmetry by the exchange of a colourtriplet isosinglet scalar (see for example Refs. [22, 23, 10, 24, 27]) or its coloursextet counterpart [22, 23, 35, 15, 24, 27].
These models could also be tested, and eventually excluded, with data already analysed by CMS and ATLAS in the search for resonances.
On the other hand, models with axigluons [36, 37, 32, 13, 10] or other types of colouroctet bosons [38, 39, 40, 41, 35, 27] can in principle accommodate the measured asymmetry predicting a moderate tail at LHC. (Our discussion obviously does not directly apply to models where the
new physics produces plus other particles in the final state, see for example Ref. [42].)
Besides, we note that Ref. [7] has recently found that SM extensions explaining the FB asymmetry without predicting too large tails must have interference with the SM amplitudes.
Finally, in this paper we have investigated the conditions under which the first order contributions to the total cross section cancel while still producing a FB asymmetry compatible with experimental data. (A previous study at this order has been presented in Ref. [5].) Clearly, in this situation the tails of the invariant mass distribution are much smaller, both at Tevatron and LHC, as it has been shown explictly. A popular example of a model fulfilling these conditions is an axigluon with opposite couplings to the first and third generation, but there are many other possibilities which can be worked out from Table 2. All these SM extensions can accommodate the Tevatron asymmetry and cross section with small couplings, and predict a moderate enhancement of the tail at Tevatron and LHC, which is not in contradiction with experiment. Interestingly, a small excess in the tail with boosted tops has been already observed at Tevatron [43]. In any case, these possible departures will soon be tested with forthcoming LHC data.
Acknowledgements
This work has been partially supported by projects FPA201017915 (MICINN), FQM 101 and FQM 437 (Junta de Andalucía) and CERN/FP/116397/2010 (FCT).
Appendix A Fourfermion operators in production
We use the minimal basis in Ref. [18] for gaugeinvariant fourfermion operators. Fermion fields are ordered according to their spinorial index contraction, and subindices , indicate the pairs with colour indices contracted, if this pairing is different from the one for the spinorial contraction. Our basis consists of the following operators:
(i) operators
(9) 
(ii) operators
(10) 
(iii) operators