Single-top production and rare top interactions

The study of the top quark's properties is an important part of the LHC programme. In earlier work, we have studied the rare decay $t \to b \bar b c$, using effective operators to capture the effects of physics beyond the Standard Model. However top decay is primarily sensitive to new physics in the sub-TeV energy regime. If this new physics resides at a higher energy scale, then one needs to turn to single-top production. In this paper, we use the $s$-channel and $t$-channel single-top production measurements to constrain the new physics parameter space associated with such contact interactions. We also study the net top polarization as a means to distinguish between contributions from operators involving different fermion chiralities and Lorentz structures.


I. INTRODUCTION
The top quark has long been believed to be colluding with new physics (NP). However, intense scrutiny of the top quark's properties at the Tevatron and at the LHC has so far not revealed any conclusive departures from the Standard Model (SM). For a few years, the Tevatron experiments reported a large forward-backward asymmetry in top pair-production.
With the accumulation of more statistics and improved calculation of the SM predictions, however, this anomaly disappeared [1]. Nevertheless, the top quark remains a likely suspect -its mass differs from those of other SM fermions by orders of magnitude, so much so that it threatens to push the electroweak vacuum beyond the edge of stability. Moreover, several anomalies in the B-sector [2][3][4][5] continue to fuel speculations that third generation quarks may be the much-sought window to physics beyond the SM.
In earlier work [6], we proposed a study of rare decay modes of the top quark. Since all the top quark measurements to date have been made in channels involving the dominant decay modes of the top, they would, naturally, have missed signs of new physics that only manifests in the rare decay modes. We examined the sensitivity of the LHC to the rare decay t → bb c and found that with 3000 fb −1 of data, the LHC would be able to set statistically significant limits on such decays. However, it is evident that top decay would be most sensitive to new physics effects arising at the energy scale of a few hundred GeV at most. If the new physics contributions originate at higher energy scales, the impact on top decay would be too small to be discernible. In order to probe such interactions further, one must turn to single-top production.
In Ref. [6], NP contributions to the rare decay t → bb c were parametrized in terms of various four-Fermi operators. In this paper, we examine the impact of that same set of operators on single-top production. While the top quark decay in the mode t → bb c has not been observed, single-top production has been measured and these measurements can, in principle, be used to constrain the strength of these interactions. A priori, it may seem that the contribution of such operators to single-top production would be diminished by the parton densities of the heavy quarks in the initial state. While this is true in general, the situation is salvaged somewhat by the fact that the competing SM mode is driven by electroweak interactions and not by strong interactions. A detailed numerical study 2 shows that it is possible to set meaningful limits on the parameters of the interactions using existing LHC data. We also present a futuristic scenario in which very stringent limits may be obtained. This possibility, however, is contingent upon the development of reliable techniques to determine the charge of an outgoing b-quark on an event-by-event basis. We further examine the possibility of distinguishing between the contributions of new physics operators with different Lorentz structures and fermion chiralities using the polarization of the top quark.
Single top production has received significant attention as a direct probe of physics beyond the SM [7,8]. Since the present analysis focusses on the effects of four-quark operators, it is worth mentioning that such operators have been studied quite extensively, particularly in the context of flavor-changing neutral currents involving the top quark [9].
The remainder of this paper is organised as follows. In Section II, we discuss the theoretical generalities related to single top production at the LHC and introduce the effective operators that we use to parametrize new physics contributions, though the detailed analytic expressions are consigned to the Appendix. In Section III, we discuss our numerical analyses and results for s-channel as well as t-channel single-top production at the LHC at center-of-mass energies of 8 TeV and 13 TeV. We conclude in Section IV.

II. SINGLE TOP PRODUCTION
At a hadron collider, the dominant production mode for top quarks is pp (pp) → tt. Single top production is sub-dominant. Nonetheless, it is important as it provides a cleaner way of measuring the electroweak couplings of the top quark. Within the framework of the SM, single-top production at hadron colliders is classified into 3 production channels as shown in Fig. 1, namely, s-channel, t-channel and W t-channel. At the LHC, t-channel production is the dominant mode, followed by W t associated production. Cross sections for all three channels have been measured at the LHC during the 7-TeV and 8-TeV runs [10][11][12][13][14][15][16]. The 13-TeV run is ongoing and results are already available in some channels [17,18]. These are summarised in Fig. 2. For t-channel single top production some kinematic distributions have also been measured [11,19,20]. All the measurements are, by-and-large, in agreement with the SM predictions, even though certain channels are plagued by large experimental uncertainties. In Ref. [6], the contributions from physics beyond the SM to the rare decay t → bb c were parametrized in terms of the 6-dimensional operators given by L eff , Clearly, these operators can also contribute to single-top production. The W t-channel would remain unaffected by these NP contributions. However, final states identical to those produced in s-channel and t-channel processes can occur via the contact interactions listed above.
B. Contribution to single-top production from L eff The operators listed in Eqs. (1), (2) and (3) can give rise to three possible amplitudes for single top production :bc → tb, bc → tb, bb → tc. In the SM, the first one is an s-channel process, whereas the second and third are t-channel processes. In addition, the three final states get contributions from light-quark initial states in the SM. Some of the key features are as follows: (i) For single-top production due to such operators, the initial states would necessarily consist of bottom and charm quarks. The low densities of these inside the proton tend 5 to suppress the cross section as compared to the SM production rates. This effect is more pronounced in the s-channel than in the t-channel as the SM rate for the latter is CKM supressed unless there is a b quark in the initial state (see Fig. 1).
(ii) The suppression caused by the low initial-state parton densities is compensated to some extent by the growth in the cross section with increase in the parton c.m. energy ( √ŝ ). This, of course, is a generic feature of contact interactions.
(iii) When new physics is parametrized in terms of effective operators, the implicit assumption is that the operators arise due to physics at a very high energy scale (Λ new ) that is be drawn at this stage. In the t-channel, such contributions would manifest, for example, in the form of a harder transverse momentum distribution. However, no such deviations have been found up to p T ∼ 300 GeV [11,19,20].
In our previous work related to t → bb c, the 10 couplings X I AB from the NP effective Lagrangian L eff were found to appear together in six characteristic combinations. We denoted these six combinations byÂ σ i , defininĝ In addition to depending on the above six quantities, various observables were also found to depend on the real and imaginary parts of X V LL and on the combination In the case of single-top production, once again, NP contributions to the various cross sections of interest can be expressed in terms of these same combinations of NP parameters.
Explicit expressions for the matrix elements squared in the different cases may be found in the Appendix. If we restrict our attention to the case in which we sum over the top quark's spin, we find that the 20-dimensional parameter space spanned by the 10 complexvalued parameters X I AB is reduced to a 5-dimensional parameter space composed of ( and Im(X V LL ). 2 If one wishes to consider top quark polarization effects, there is an additional dependence on ( , and Im X T LL X S * LL + X T RR X S * RR (please refer to the Appendix for details). In the present work we shall assume that X V LL and X T LL X S * LL +X T RR X S * RR are both real, thereby reducing the size of the parameter space somewhat. 3 In our numerical work below, we will mostly consider the case in which the polarization of the top quark is ignored (Secs. III A, III B, III C and III E).
A polarization-dependent asymmetry is considered in Sec. III D. Furthermore, throughout the remainder of this paper, we will consider operators with a single Lorentz structure at a time. We will also make certain assumptions about the relative magnitudes of the couplings associated with operators having the same Lorentz structure but different chirality structure.
We close this section with the following comment. The careful reader might have noticed that the effective operators listed in Eqs. (1), (2) and (3) are not SM gauge invariant.
Within a gauge-invariant framework, the presence of new operators, potentially constrained by other observables, could limit the size of the Wilson coefficients corresponding to the 1 The dependence on X V LL comes from SM-NP interference terms. Terms proportional to Im X T LL X S * LL + X T RR X S * RR are associated with triple-product correlations [6]. 2 As we have noted in past work, the situation is somewhat complicated by the fact that the real and imaginary parts of X V LL also appear in the parameterÂ + b . 3 Note that Im(X V LL ) plays an important role in partial rate asymmetries [6]; we neglect such effects here.
operators in our basis. We have examined the gauge-invariant operators involving the second and third quark generations and have found that the observables that they may directly affect are bottom and top pair production. As is the case for single top production within our framework, the new operators lead to effects suppressed by the PDFs of the initial heavy quarks. However, for bottom and top pair production the contributions from the new operators must also compete with the dominant QCD contributions. Therefore we expect these processes to be far less constraining than those considered in the present work.

III. NUMERICAL STUDY
In order to extract limits on theÂ σ i 's (and consequently on X I AB ), we start by implementing L eff alongside the Standard Model in MadGraph5 [22] using FeynRules [23]. This puts us in a position to calculate the tree-level cross section for single-top production in pp collisions, for which we use CTEQ6L parton distributions functions (PDFs) [24], setting both the renormalization and factorization scales to be m t = 173 GeV. In order to approximate higher-order QCD corrections, we estimate K-factors as σ SM N N LO (approx.)/σ SM LO , where σ SM N N LO (approx.) is obtained from the references listed in Table I and σ SM LO is calculated using MadGraph5 in conjunction with MSTW2008LO PDFs [25]. 4 We then compute the tree-level cross sections, including both SM and NP effects, and multiply the results by the corresponding K-factors to obtain estimates of the QCD-corrected values. In the following, "σ SM " denotes the SM cross sections obtained in this manner.

A. t-channel single-top production
At the LHC, t-channel processes yield the dominant contribution to single-top production.
These processes consist of d i u j → td k and d idj → tū k where d i,j,k ∈ {d, s, b} and u j,k ∈ {u, c}.   bc → tb and bb → tc. Figure 3 shows the cross section. As noted above, the operators in L V , L S and L T are considered separately in our analysis. It can be seen that the tensor operators are constrained most tightly, followed by vector and scalar operators. This is expected, given the structure of theÂ σ i s and the fact that large numerical factors accompany X T AB wherever they appear. It is intriguing to note that, except in the case of X T AB , couplings of O(1) are not excluded by the experimental data at 8 TeV. How do we reconcile this with a) our earlier statement that X I AB ∼ O(1) correspond to Λ new in the sub-TeV range, and b) the fact that no exotic physics has been discovered at the LHC so far? We return to this question below, in Sec.III E. Figure 4 shows the analogous analysis for the 13-TeV data. As compared to the 8-TeV data, the 13-TeV data appear to be less constraining. At first glance, one is tempted to attribute this to low statistics given that the 8-TeV measurement is based on 19.7 fb −1 of data while the 13-TeV result is based on 3.2 fb −1 . However, despite the relatively low statistics at 13-TeV, it turns out that the largest component of the uncertainty is due to systematics.
If future analyses can reduce the systematic uncertainty, then tighter constraints can be expected. Presently, for a more effective comparison between the sensitivities to NP couplings horizontal lines depict the CMS measurement [15] along with the total uncertainty. The grey dot-dashed line and the grey shaded region reflect σ SM ± 10%, which is calculated as described in the text. The red dashed, green dot-dashed and blue dotted curves give the cross section in the presence of NP vector, scalar and tensor interactions, respectively. As mentioned in the text, only one Lorentz structure is considered at a time. TeV. In the former case, the black solid and dashed horizontal lines depict the CMS measurement [15] along with the total uncertainty; in the latter case, the black solid and dashed horizontal lines depict the ATLAS measurement [17] along with the total uncertainty. In both cases, the grey band depicts σ SM ± 10%.
and similarly for scalar and tensor operators. In the following, we relax this condition and consider scenarios where X V LR , X V RL and X V RR are smaller than X V LL . As expected, this relaxes the constraint on X V LL (see Fig. 5). In a UV-complete scenario, these operators may not all occur simultaneously and there would exist several possibilities for the relative sizes of the corresponding couplings.
We illustrate the effect using one such possibility. The same exercise can be carried out for scalar and tensor operators; the results are depicted in Figs. 6 and 7.

B. s-channel single-top production
Single-top production in the s-channel is due to the processes u idj → td k , with d j,k ∈ {d, s, b} and u i ∈ {u, c}. Of these, the largest contribution comes from ud → tb, since the rest of the processes are CKM suppressed. However, the introduction of L eff enhances the contribution due to cb → tb. This is shown in Fig. 8.
Since the cross section is small (∼ 3.7 pb in the Standard Model), CMS [16] and AT-LAS [13] report the combined cross section due to p p → t X and p p →t X. However,  TeV. The grey dot-dashed line represents σ SM , which is calculated as described in the text. The light-grey and dark-grey shaded regions denote the regions σ SM ± 20% and σ SM ± 10%, respectively. The red dashed, the green dot-dashed and the blue dotted curves give the cross section in the presence of NP vector, scalar and tensor interactions, respectively. As noted above, only one Lorentz structure is considered at a time.

13
in our calculations, we consider only p p → t X. 5 Hence, in this section, the experimental measurements are not indicated on the plots. Instead, the grey band denotes a band of uncertainty around the central value of the corresponding SM cross section. The size of the band is chosen so as to be commensurate with the total uncertainty (statistical + systematic) reported in the actual measurement [13,16]. In Fig. 8, for example, the size of the band is ±40%. While measurements in the s-channel are yet to be reported for 13 TeV, this data set is expected to be several times larger than the 8-TeV data set. Hence we use two bands, of sizes 20% and 10%, to project the limits on X I AB in this case (see Fig. 9). TeV. In the former case, the grey band depicts σ SM ± 40%; in the latter case, the light-grey band depicts σ SM ± 20% and the dark-grey band depicts σ SM ± 10%.
Comparing Fig. 8 with Fig. 3, it is immediately clear that the s-channel process, even when measured with a lower accuracy, yields more stringent bounds than the t-channel process. This is easy to understand. σ SM is much smaller for the s-channel process than for the t-channel process, so even if the relative uncertainty is larger, the absolute deviation allowed is smaller, which leads to tighter constraints on the couplings. A further improvement can be expected to emerge from the s-channel measurements at 13 TeV (see Fig. 9). Here again we consider scenarios where one chiral structure is dominant. The impacts on the corresponding limits are shown in Figs. 10, 11 and 12.

C. Differential Distributions
The ATLAS and CMS collaborations have measured the differential cross section in terms of the transverse momentum (p T ) and rapidity (|y|) of the top quark for t-channel single-top production at 8 TeV [11,19]. CMS has also made a similar measurement at 13 TeV [20]. In Section III A we identified the range of X V AB , X S AB and X T AB that yield a cross section within 10% of the SM prediction. We now confront the predictions from these NP scenarios 6 with the ATLAS data [11]. In doing so, we first compute the χ 2 by comparing the ATLAS 8-TeV data for the unfolded, parton-level distributions (see Figs. 21(a) and 22(a) and Tables 18 and 20 in Ref. [11]) with the distributions computed for the various NP scenarios. This χ 2 is then converted to a p−value by the usual formula where f χ 2 (x; n) is the χ 2 p.d.f. and n is the number of degrees of freedom. The results are plotted in Fig. 13. The p T spectrum proves to have more discriminating power than the |y| spectrum. Using this it is possible to put further constraints on X I AB . For example, if we assume that NP scenarios that yield p < 0.05 are disfavoured, then the resulting restriction on X V AB , X S AB and X T AB is shown in Fig. 15.

D. Top polarization
The polarization of the top quark is an important observable at the LHC and a few measurements of it have already been made [28,29]. The net polarization is usually defined where N ↑ and N ↓ , respectively, denote the number of top quarks with spin aligned along or opposite to a chosen direction. The value of A P depends on the choice of the reference direction. The usual choices for this reference direction are the z-axis or the momentum of the top itself. In the latter case, N ↑ and N ↓ denote the number of top quarks of different helicities. If the top quark is produced in association with another particle (as in the case of single-top production), the momentum of that particle may also be chosen as the reference direction. The utility of A P lies in the fact that it is sensitive to the chiral structure of the coupling at the production vertex. Fortuitously, due to its large mass, the top quark decays before hadronizing. This allows the polarization information (which would otherwise be lost during hadronization) to be gleaned from the angular distribution of the top quark's decay products. Very often, the e or µ coming from the top decay is used for this purpose.
In the preceding sections, we have identified regions of parameter space that are compatible with various single-top production cross section measurements. We now examine the reach of A P as a means to distinguish between the different types of couplings. In particular, we expect A P to deviate from its Standard Model value when there is an increase in the fraction of t R in the ensemble, that is, when there are contributions from X I RL or X I RR . As can be inferred from the expressions in the Appendix, cross section measurements are sensitive to various combinations of sums ofÂ σ i 's. By way of contrast, the spin-dependent contributions to the amplitude squared are dependent on differences ofÂ σ i 's, such asÂ + b −Â − b . For this reason, polarization measurements can yield additional information regarding the types of NP interactions that contribute to the various processes under consideration in this work.
In the following analysis, we choose certain values of X I AB that yield a cross section within a certain "allowed" range and then compute A P according to Eq. (6) above, with N ↑ and N ↓ denoting the number of top quarks of either helicity. 7 Note that we include NP effects in both the numerator and the denominator when calculating A P . Table II lists s-channel single-top polarization asymmetries for various combinations of NP contributions at √ s = 8 TeV. In choosing parameters, we have allowed for an enhancement of ∼20% in the cross section over the SM prediction, noting that the experimental uncertainty is close to 40%. We consider scenarios for which the dominant contribution comes from one operator, 8 keeping the sub-dominant couplings at approximately one-tenth of the dominant one. We find that the deviation in A P can be considerable. Note that the values of A P listed in Table II are based on the calculation of tree-level cross sections, as described at the beginning of Sec. III. K-factors cancel out in the calculation of A P . Estimation of A P based on higher-order calculations is beyond the scope of this work. However, the dominant higher-order corrections would arise from QCD effects and, as such, they are not expected to alter A P significantly.

Dominant Contribution
No Contribution σ (pb) We have not attempted to estimate the accuracy with which the A P can be measured at the LHC. 9 We note, however, that a recent analysis from CMS [29] was able to make a measurement with approximately 38% uncertainty. We also point out that the results from Ref. [29] cannot be used to compare directly with our results because their choice of reference axis is different from ours. Nonetheless, it is clear from Table II that accuracies ∼ 10% or better would be needed in order for A P to be a useful discriminator.
Finally, in Table III, we list the polarization asymmetry for s-channel single-top production at √ s = 13 TeV. This time we allow for ∼10% enhancement in the cross section over the SM prediction. Once again the largest deviations correspond to contributions from X I RL and X I RR , although the net size of the deviation is smaller. This is largely an artefact of the restriction placed on the overall increase in cross section; allowing for a larger variation in the cross section would, in general, allow for greater deviation in A P . It would not, however, guarantee a greater deviation in A P , since the size of A P depends on the chiral structure of the dominant operator.
One can, similarly, obtain the polarization asymmetry for t-channel single-top production. 9 We refrain from making such an attempt, since background rejection depends heavily on the specific algorithm used, which, in turn, often involves boosted decision trees [29] and other sophisticated analysis tools developed and trained by experimentalists to get the best from their respective detectors.

Dominant Contribution
No Contribution σ (pb) If we follow our earlier strategy of choosing benchmark points from well within the allowed band, we would be considering scenarios where ∆σ ∼ 0.05 σ SM . In that case, the deviation in A P is 10% and may not be experimentally discernible. However, as one allows for larger ∆σ, deviations in A P also begin to increase. Tables similar to Tables II and III for the t-channel are not presented here for the sake of brevity.

E. t-channel single-top production -a Futuristic Analysis
We now return to the futuristic analysis that we briefly alluded to in the Introduction. As From the point of view of isolating t-channel subprocesses, bc → tb appears at first glance to be more promising than bb → tc, since bc → tb events could be isolated from other t-channel events by the application of an additional b-tag. In reality, the situation is somewhat complicated.
In order to identify s-channel single-top production and distinguish it from t-channel However, if it were possible to distinguish between b andb quarks, then one would be able to isolate the process bc → tb. To estimate the expected improvement in the limits, one only needs to consider the size of the SM background, which, at 13 TeV, is approximately 135 pb for the usual t-channel production and about 0.014 pb when bc → tb is isolated.
The actual limits are depicted in Fig. 14, which can be compared with Fig. 4 and with the 13-TeV plots in Figs. 5-7. σ SM ± 10% is depicted as a grey band in each of these cases. In Fig. 14(a) and (b), the effect of the interference between the SM and the NP X V LL term is discernible, unlike in the corresponding plots in Sec. III A.
While isolating bc → tb seems to be an exciting possibility, techniques for distinguishing between b-flavored quarks and antiquarks in the final state are still at a nascent stage, although some developments in this direction have been reported [30]. If such techniques can be improved upon sufficiently so as to become reliable even when the statistics are 10 A td final state is approximately 10 4 times more likely to occur than a tb final state in t-channel single-top production. 21 relatively low, then the pp → tb channel will become the primary channel of interest. FIG. 14: Single-top production cross section in the channel pp → tb in the presence of X I AB at 13 TeV. The light-grey band depicts σ SM ± 40%, while the dark-grey band depicts σ SM ± 10%.

IV. SUMMARY AND CONCLUSIONS
Top quark decays are most sensitive to new physics effects at the energy scale of a few hundred GeV. The same effects, if they arise from a higher energy scale, would be more effectively probed in single-top production. In this work, we have focused on NP effects arising from anomalous couplings between the top, bottom and charm quarks. Since we only consider the top's interactions with heavy quarks, it might at first appear that progress would be thwarted by the low densities of heavy quarks inside the proton. Nevertheless, our detailed study shows that it would be possible to place meaningful constraints on the new physics parameters.
We have considered t-channel and s-channel single-top production cross sections to obtain constraints on contact interactions involving t, b and c quarks. Of these two channels, the stronger constraints arise from the s-channel. This is due to the fact that the Standard Model background cross section is smaller for the s-channel. Within a given channel (t or s), the limits are most stringent for tensor operators, followed by vector and scalar operators, respectively. This is essentially due to the additional numerical factors that appear from the Dirac traces for each of these operators.
For t-channel single-top production, data is also available for the p T and |y| differential cross sections. We have examined the compatibility of these measurements with different NP scenarios and found that the p T distribution in particular can be useful in further constraining the NP parameter space.
Apart from the total and differential cross sections, we have also considered the relative contributions to the cross section from top quarks of different helicities. The polarization asymmetry A P , which compares the helicity states of the top quark, can be particularly useful in establishing the presence of operators involving t R , especially since the corresponding Standard Model charged current coupling involves only t L .
Finally, we have considered a futuristic analysis. This analysis rests on one crucial assumption, namely that b quarks in the final state can be distinguished fromb antiquarks on an event-by-event basis. Adopting this assumption, we have obtained limits on the NP fourquark operators that are far more stringent than those obtained from the regular t-channel  and s-channel analyses. The comparison can be seen in Fig. 15. The improvement is indeed startling and perhaps adequate motivation for the pursuit of experimental techniques that will make it possible.
A. s-channel :b(p 1 ) c(p 2 ) −→ t(p 3 , s t )b(p 4 ) − 8m t Im X T LL X S * LL + X T RR X S * RR ǫ (p 1 , p 2 , p 4 , s t ) + 8m t Im X T LL X S * LL + X T RR X S * RR ǫ (p 1 , p 2 , p 4 , s t ) − 8m t Im X T LL X S * LL + X T RR X S * RR ǫ (p 1 , p 2 , p 4 , s t )