odd correlations using jet momenta from events at the Tevatron
Abstract
We discuss odd correlations between jet and lepton momenta in events at the Tevatron that can be used to search for violation. We identify correlations suitable for the lepton plus jets and purely hadronic topquark pair decay channels. As an example of violation we consider the topquark anomalous couplings, including its chromoelectric dipole moment, and we estimate the limits that can be placed at the Tevatron.
pacs:
PACS numbers: 12.15.Ji, 12.15.Mm, 12.60.Cn, 13.20.Eb, 13.20.He, 14.70.PwI Introduction
Looking for new sources of violation remains one of the important goals of high energy colliders. Processes that have been considered before include production and decay of top quark pairsreviews (); Donoghue:1987ax (); ttpairs (); eeanom (); Antipin:2008zx (); Gupta:2009wu (), production and decay of electroweak gauge bosons cpgaugeb (), and production and decay of new particles newpart ().
The Tevatron has now observed hundreds of events and will eventually collect a few thousand. With this in mind, it is interesting to revisit the issue of possible violation in these events. Under optimal conditions, a sample of a few thousand events would have a statistical sensitivity to a violating asymmetry at the few percent level.
In this paper we study odd triple product correlations of the sort first discussed in Ref. Donoghue:1987ax (); tripprods () and find observables suitable for events in which the top quark pairs decay into a lepton plus jets or purely hadronically. Of particular interest are observables that only require reconstruction of two jets (but there is no need to distinguish between and ); one or no hard leptons; and non jets ordered by .
It is well known that violation in the standard model (SM) is too small to induce a signal at an observable level in the processes we consider. We will discuss two violating effective interactions that should serve as benchmarks for the sensitivity of the Tevatron to violation in events.
As in our LHC study Gupta:2009wu (), we parametrize violation using anomalous topquark couplings. The production process is modified relative to the SM by the chromoelectric dipole moment (CEDM) of the topquark via the interaction
(1) 
where is the strong coupling constant and is the usual gluon field strength tensor. The CEDM is induced in principle by any theory that violates , and estimates for its size in several models can be found in Ref. reviews (). Typical estimates presented in Ref. reviews () for the size of suggest that it may be too small to yield observable signals at the Tevatron. Nevertheless, we view the study of this coupling at the Tevatron as a valuable preliminary to future LHC studies.
We also consider violation in the decay vertices and via the anomalous coupling defined by^{1}^{1}1As discussed in the literature, other anomalous couplings will not interfere with the SM and we will not consider them here delAguila:2002nf ().,
(2) 
In Eq. 2 we have explicitly split the phase of into a violating phase and a conserving, unitarity, phase .
The violating anomalous couplings, and , have been recently revisited visavis the upcoming LHC experiments in Ref. Antipin:2008zx () and Ref. Gupta:2009wu (). In Ref. Antipin:2008zx (), general results were derived for the odd correlations induced by these two couplings for both gluon fusion and light annihilation production processes. In the appendix we specialize those general results to the specific processes that are relevant for the Tevatron. In Ref. Gupta:2009wu () a numerical analysis was carried out for LHC concentrating on the dilepton signal, which is not viable at the Tevatron due to the small number of events. The new signals we discuss in this paper pertain to the lepton plus jets and all hadronic decay modes of the topquark pairs and can also be used at LHC.
Ii Observables
In Ref. Antipin:2008zx () all the linearly independent odd correlations induced by anomalous topquark couplings were identified. From these we need to project out the ones that are most suitable for the Tevatron and two considerations come into play. The first one, already discussed in our application to the LHC in Ref. Gupta:2009wu (), is that we want to use only momenta that can be reconstructed experimentally. The second one is that, due to the low statistics at the Tevatron, we will be dealing with at least one hadronic decay of the boson.
We will consider the following correlations ^{2}^{2}2Here we use the LeviCivita tensor contracted with four vectors with the sign convention . We also use to refer to the parton level Mandelstam variables for .:

For the lepton (muon) plus jets process :
(3) 
For the multijet process :
(4)
In Eqs. 3 and 4 we have shown two expressions for each of the correlations. The first one is valid in any frame and in particular can be used in the lab frame. The second one shows the correlation in a particular frame in which it reduces to a simple triple vector product. In these expressions is the sum of the proton and antiproton fourmomenta; is the difference of the proton and antiproton fourmomenta; refers to the or jet momenta; refers to the momenta of a lepton that has been identified as originating from or decay in lepton plus jets events (in our analysis we only consider muons); refer to non jets ordered by (hardness) that reconstruct a ; primes denote the two jets associated with the second in the all hadronic case. In it is not necessary to distinguish and jets. It is only necessary to associate with one of the jets and with the other one when reconstructing the topquark pair event. The hardness of the jet is defined in the usual way, the hardest jet being that with the largest transverse momentum, i.e. , and is the lepton charge (for some of the monoleptonic signals, lepton charge id is needed).
Notice that some of the correlations require differentiating between the and jets but others don’t. In addition, requires the reconstruction of the top momenta. This correlation is the one closest to the form that appears in the parton level calculation, and in a perfect reconstruction situation it is identical to . In addition, two of the examples given, and can be used for both processes. Finally we note that there are many other possibilities that we have not listed.
All the correlations listed above are odd, as can be seen most easily in the specific reference frames given. For example, in the lab frame becomes
(5)  
Eq. 5 also clarifies what is meant by : events with a and a decaying to two jets will contribute to the first term in the sum in the first line. Events with a and a decaying to two jets contribute to the second term. The assignment on the second line of Eq. 5 states that if is conserved, the probability for a given jet originating from a quark in a two jet decay to be the hardest one, is equal to the probability for the corresponding jet originating from the antiquark in a two jet decay to be the hardest one. These statements are verified in our numerical simulations both explicitly and by the fact that the asymmetry is induced by violating couplings but vanishes for conserving ones. In an experimental analysis it will be important to implement additional cuts in a way that is blind, typically requiring the same cuts for particles and antiparticles.
Use of the lepton charge in some of the correlations allows us to construct odd and even correlations with the same set of momenta. We exploit this to construct the odd (but even) correlations sensitive to strong phases:
(6) 
The first two have odd analogues in and respectively.
Our observables will be the lab frame distributions for the correlations listed above, as well as their associated integrated counting asymmetries
(7) 
the denominator being just the total number of events in all cases. When our numerical results for the integrated asymmetries are very small we distinguish between very small asymmetries and vanishing asymmetries as described in Ref. Gupta:2009wu ().
Iii Numerical Analysis
Our numerical study in this paper corresponds to the implementation of analytic results presented in Ref. Antipin:2008zx (). The odd correlations for the parton level processes that are relevant at the Tevatron are not explicitly written in Ref. Antipin:2008zx (), so we present them in the Appendix for convenience. The numerical studies are performed with the aid of Madgraph Stelzer:1994ta (); Alwall:2007st (); Alwall:2008pm () following the procedure outlined in Ref. Gupta:2009wu (). For the lepton plus jets channel, we begin with the standard model processes implemented in Madgraph according to the decay chain feature described in Ref. Alwall:2008pm (). This decay chain feature is chosen for consistency with the approximations in the analytical calculation of the violating interference term presented in Ref. Antipin:2008zx (), in which the narrow width approximation is used for the intermediate top quark and boson states. The expressions from Ref. Antipin:2008zx () (Eqs. 1323) are then added to the spin and color averaged matrix element squared for the SM (which Madgraph calculates automatically) and the resulting code is used to generate events. A similar procedure is followed for the purely hadronic decay of with the relevant parton level processes. In this case both ’s decay into a pair of quarks and we only consider the final states without Cabibbo mixing. The code used to generate events is, therefore, missing the terms that are completely due to new physics: those proportional to the anomalous couplings squared. This approximation is justified because those terms do not generate odd correlations. In addition, as long as the conditions that allow us to write the new physics in terms of anomalous couplings remain valid, their contribution to the total crosssection is small.
For event generation we require the top quark and boson intermediate states to be within 15 widths of their mass shell, and two sets of cuts. The first set of cuts includes a minimum transverse momentum for all leptons and jets, a minimal separation between them, and a pseudorapidity acceptance range:
(8) 
with .
For the second set of cuts (in the lepton plus jets channel) we add a missing transverse energy requirement
(9) 
We use SM parameter values as in Madgraph, except for ; and we use the CTEQ6L1 parton distribution functions.
iii.1 Process
We first estimate the counting asymmetries by generating events for each of the four cases: ; ; and . These cases correspond to violation in the production vertex, violation in the decay vertex, strong phases in the decay vertex and the lowest order SM respectively. The relatively large number is chosen to facilitate distinguishing signals from statistical fluctuations. Once we establish a nonzero asymmetry we can cast our result as a function of the anomalous couplings since the asymmetries are linear in them. As mentioned above, we include the new physics only through its interference with the SM. Since these odd terms are also odd, they do not affect the crosssections as they integrate to zero. For this reason the total number of events is the same as in the standard model.
The results with the set of cuts Eq. 8 are shown in Table 1. After these cuts are applied there remain approximately events, leading to the statistical sensitivity shown in the first column. The results show that all the odd correlations vanish for the two conserving cases (SM and ), and that the even correlations vanish except for the conserving case with unitarity phases, . This establishes numerically that, at least at this level of sensitivity, there is no conserving contamination of the odd signals or vice versa.
3.7  66.9  37.4  100.6  75.8  40.4  3.4  1.8  
3.7  7.2  60.8  8.2  36.7  10.6  0.6  1.9  
3.7  0.8  1.0  0  1.7  1.9  49.3  51.9  
SM  3.7  2.6  0.6  0.4  0.1  0.2  0.4  0.3 
In Table 2 we show the same results with the additional missing requirement of Eq. 9. This additional cut further reduces the number of generated events to about , and slightly decreases the statistical sensitivity. The effect of this cut is minimal on all asymmetries, making it very desirable for reducing background.
3.9  66.4  38.9  102.3  76.5  36.4  3.0  1.4  
3.9  17.2  66.8  18.8  30.8  7.0  0.7  2.3  
3.9  0.4  1.1  1.6  3.1  1.0  44.1  56.8  
SM  3.9  2.5  0.5  0.5  0.2  0.4  0.3  0.6 
Now we summarize our results for the asymmetries in the process with cuts given in Eqs. 8, 9 in terms of the dimensionless anomalous couplings
(10) 
with GeV. We find,
(11) 
In addition to the integrated counting asymmetries one can look for asymmetries in the distributions . In Figure 1 we compare the distributions for induced by (a) and (b) to the SM.
It is instructive to discuss in some detail to understand the role of the hardest jet momenta. The lepton and (non) jet momenta that appear in this correlation act to some extent as the spin analyzers in the and decays. It is well known that the best spin analyzers in the topquark rest frame are the charged lepton momentum (for semileptonic top decay) and the quark momentum (for hadronic decay) Mahlon:1995zn (). Of course, it is not possible to tag the quark jet in experiment, but at the event generator level we can see how things work. To this effect we define , the counting asymmetry corresponding to . These asymmetries, as the original , are interpreted as the sum of processes with from semileptonic decay of and hadronic decay of , and processes with from semileptonic decay of and hadronic decay of . With the cuts in Eqs. 8 and 9 and with , we find and . Interestingly, the asymmetry , is smaller than , which appears in Table 2. To understand what happens, we show in Figure 2 the differential distribution of the numerator of with respect to , the ratio of quark transverse momentum to quark transverse momentum in or decay.
As can be seen in the figure, if one chooses the quark momentum in the lab frame to construct this particular correlation, there is a partial cancellation between the regions with and . This cancellation is removed by choosing instead the hardest jet resulting in the larger . The fact the is larger when using the hardest jet instead of the quark jet appears to be unique to this correlation.
Using our generated events, we estimate that the quark jet has a larger than the quark jet 44% of the time. We can also verify that within statistical errors, the probability of being the hardest jet in is indeed the same as the probability of being in the hardest jet in .
iii.2 Process
In this case we only use the cuts of Eq. 8 as there is no missing energy. The results are shown in Table 3 for about generated events. As expected, the signals and are the same at the parton level and it remains to be seen what dilution there is after hadronization.
3.5  61.2  54.6  61.2  38.8  1.1  
3.5  7.1  7.8  7.1  5.8  1.0  
3.5  1.8  1.5  1.8  0.5  9.6  
SM  3.5  0.7  0.5  0.7  1.0  1.1 
Using our results In Table 3 for the process we find
(12) 
In Figure 3 we compare the distributions for induced by (a) and (b) to the SM.
The results in Eq. 11 and Eq. 12 provide a rough estimate for the sensitivity of the Tevatron to the violating anomalous couplings. The existing Tevatron samples of events with a double tag are of the order of 1000 events tevdata () and this leads to a statistical sensitivity to and of order 1. To account for background, we notice that: a) the experimental cuts to select the events are the same that will be used for a violation study , and b) all the known background processes are conserving. The net effect of the background (apart from possible systematic errors that must be studied by the experiments) is to dilute the asymmetries by a factor . The numerator in does not get additional contributions from the background; but the denominator, which counts the total number of events, does. Similarly, the statistical sensitivity decreases by a corresponding factor . For samples with roughly the same number of background (B) and signal (S) events this amounts to factors of two.
The gluon fusion initiated dilepton channel at TeV leads to Gupta:2009wu (). Comparing this number to those in Eq. 11 and in Eq. 12 we see that the dilepton process at LHC is an order of magnitude more sensitive than the lepton plus jets or all hadronic channels at the Tevatron. This is due to two reasons: first the gluon fusion initiated process is more sensitive to the anomalous couplings. For the case of the Tevatron, only about 15% of topquark pairs are produced via this mechanism. Numerically we have seen that if we restrict the topquark pair sample to that originating from gluon fusion, the asymmetries increase roughly by factors of three. The second reason is that the dilepton channel is more sensitive to these anomalous couplings.
In addition, the statistical sensitivity of the Tevatron with 1000 events is about five times below that of a run at LHC which would produce about 23000 dimuon events after the cuts in Eq. 8 and Eq. 9 are applied. Nevertheless, a study with the available Tevatron data would be extremely valuable in understanding the role of systematic errors in measuring odd asymmetries.
We have performed a series of checks on our numerical analysis as follows. First, we evaluate the asymmetries for a few values of the anomalous couplings to check that they scale linearly. Second, when the estimated asymmetry is small compared to the statistical uncertainty, we repeat the estimate with larger event samples and/or larger values of the anomalous coupling to distinguish between zero asymmetries and numerically small ones. Third, for pair production at the Tevatron the parton process with quarks in the initial state dominates. We have therefore estimated the asymmetries using this parton process only, finding numbers within 10% from the ones obtained when all initial and states are included.
Iv Summary and Conclusion
We have studied the sensitivity of the Tevatron to violating anomalous topquark couplings including its chromoelectric dipole moment . To this effect we have presented a numerical implementation of the results in Ref. Antipin:2008zx () using Madgraph for event generation at the parton level. We have considered processes corresponding to events in the lepton plus jets and all hadronic channels with two btags. In order to generate a statistically clean sample we have performed our numerical simulation for a rather large value of the anomalous couplings (). Using the fact that all the asymmetries are linear in the anomalous couplings we present our final results as equations in terms of these couplings, in Eq. 11, 12.
Numerically, we find a statistical sensitivity to couplings of order one when normalized to the topquark mass: and , Eq. 10. This sensitivity is about two orders of magnitude below what can be accomplished at the LHC with . These results are based on the assumption that there will be one thousand reconstructed events. There could be additional inefficiencies in the reconstruction of our specific observables that must be addressed by a careful experimental study. A few comments are in order: models available in the literature to estimate these anomalous couplings typically yield values too small to be observed at the Tevatron; specific models with new sources of violation may give contributions to the observables we study that cannot be parametrized by the anomalous couplings.
With a long term goal of searching for violation in events at the LHC, it is an important exercise to analyze the available Tevatron data and we urge our experimental colleagues to carry out this study.
Acknowledgements.
This work was supported in part by DOE under contract number DEFG0201ER41155. We thank Sehwook Lee and John Hauptman for useful discussions on the D0 events and David Atwood for useful discussions.Appendix A odd correlations
The spin and color averaged matrix element squared that contains the odd correlations can be easily obtained from the results in Ref. Antipin:2008zx (). For violation in the production process they can be written as
(13) 
in terms of the correlations^{3}^{3}3Notice that these form factors differ from those defined in Ref. Antipin:2008zx () by factors of .
(14) 
In Eq. 13 and in Eq. 14 we have used , the standard parton level Mandelstam variables for production. We have also used the sum and difference of parton momenta
(15) 
Ref. Antipin:2008zx () explicitly gives the result for the case where both s are reconstructed as one jet, in which case the form factors are
(16) 
and , ,
(17) 
Ref. Antipin:2008zx () also indicates how to convert these results into those needed in the case where the s decay leptonically. For the Tevatron we are interested in two additional cases:
In our numerical implementation we rewrite all delta functions as the respective BreitWigner distributions behind them, for example:
(21) 
When violation occurs in the decay vertex, the spin and color averaged matrix element squared containing the odd correlations was written in Ref. Antipin:2008zx () as ^{4}^{4}4Note that there is a typo in Ref. Antipin:2008zx () where and are reversed.
(22) 
The and that appear in Eq. 22 are linear combinations of available momenta and act as spin analyzers for the and respectively. For pairs produced by light annihilation and when both ’s decay leptonically, they are
(23)  
The necessary replacements to obtain the results relevant for us are:
Corresponding changes are needed for the gluon fusion processes to the results in Ref. Antipin:2008zx ().
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