Lepton Flavour Violating top decays at the LHC
Sacha Davidson ***E-mail address: email@example.com, Michelangelo L. Mangano †††E-mail address:firstname.lastname@example.org, Stéphane Perries ‡‡‡E-mail address: email@example.com and Viola Sordini §§§E-mail address: firstname.lastname@example.org
CERN PH-TH, CH-1211 Genève 23, Switzerland
IPNL, CNRS/IN2P3, 4 rue E. Fermi, 69622 Villeurbanne cedex, France; Université Lyon 1, Villeurbanne; Université de Lyon, F-69622, Lyon, France
We consider lepton flavour violating decays of the top quark, mediated by four-fermion operators. We compile constraints on a complete set of SU(3)U(1)-invariant operators, arising from their loop contributions to rare decays and from HERA’s single top search. The bounds on - flavour change are more restrictive than -; nonetheless the top could decay to a jet with a branching ratio of order . We estimate that the currently available LHC data (20 fb at 8 TeV) could be sensitive to + jet) , and extrapolate that 100 fb at 13 TeV could reach a sensitivity of .
Lepton Flavour Violation(LFV) , meaning local interactions that change the flavour of charged leptons, should occur because neutrinos have mass and mix. This motivates sensitive searches for processes such as  and . However, the model responsible for neutrino masses is unknown, so it is interesting to parametrise LFV with contact interactions, and to look for it everywhere. In this context, the LHC could have the best sensitivity to LFV processes involving a heavy leg, such as the [4, 5, 6], the Higgs [6, 7, 8, 9], or a top . In this paper, we study the LFV top decays , where .
We suppose that these decays are mediated by a four-fermion interaction, and outline in Section 2 the current bounds on LFV branching ratios of the top. The bounds arise from rare decays and HERA’s single top search, and are discussed in more detail in the Appendices. We find that, while these bounds place strong constraints on some specific Lorentz structures for the 4-fermion interactions, they still allow for decays with rates within the LHC reach. In Section 3, we estimate the LHC sensitivity to , with 20 fb of LHC data at 8 TeV. This estimate relies on simulations of the background and signal, and is inspired by the CMS search for . The extrapolation to higher energies and luminosities is discussed in Section 4.
Quark-flavour-changing top decays, such as and , have been studied in the context of explicit models [9, 12, 13] or described by contact interaction parametrisations , and have been searched for at the LHC [11, 15, 16, 17]. Quark-flavour-changing (but lepton-flavour-conserving) three-body decays of the top, , where is a lepton or quark, have also been calculated in explicit models . Top interactions that change quark and lepton flavour, and, in addition, baryon and lepton number, have been explored in  and searched for by CMS . In models with weak-scale neutrinos , there can be lepton number- and flavour-changing decays: , which could appear in the final state of top decays. In the presence of this decay, production could give a final state with three leptons, missing energy and jets, as in the decay we study (see Figure 2). However, a different combination of leptons and jets should reconstruct to the top mass. Finally, Fernandez, Pagliarone, Ramirez-Zavaleta and Toscano  studied almost the same process as us, , but mediated by a (pseudo)-scalar boson. They obtain separately the low-energy bounds on the quark-and lepton-flavour-changing couplings of their boson, and obtain that LFV top branching ratios can be if the boson mass is .
2 Current bounds
We are interested in the decays of a top (or anti-top) to a jet and a pair of oppositely charged leptons of different flavour. In this work, we focus on the processes , where , because the and are easy to identify at the LHC, and flavour violation is the most strictly constrained at low energy. We leave the decays to and for a later analysis.
We suppose that these decays are mediated by four fermion contact interactions. A complete list of the invariant operators that we study is given in Appendix A. We do not impose SU(2) on our operators, because the scale we will probe is not far from the electroweak scale. We refer to these LFV operators as “top operators”. Here, as an example, consider the exchange of a heavy SU(2) singlet leptoquark with couplings and , which (after Fierz rearrangement) generates the dimension-6 contact interaction
Alternatively, we could define the operator coefficient as , in which case because (we take GeV). We will quote low energy bounds on such interactions as limits on the dimensionless s. In the case of our leptoquark example, 1 TeV to satisfy current bounds on second generation leptoquarks from the LHC [22, 23], and thus, for , one expects .
Notice that and are mediated by different operators. Most of the bounds we quote will apply to , and we will study the LHC sensitivity assuming equal rates for and .
2.1 Decay of the top
In the Standard Model, the top decays almost always to , at a tree-level rate given by:
In the presence of the operator of Eq. (1), the three-body decay rate is:
so the branching ratio, allowing for all the operators listed in the Appendix A, and neglecting fermion masses other than the top (to remove interferences), is
where , and we approximated . This is small (), due to three-body phase space. For the leptoquark example discussed above, for .
The phase space distribution of the and depends on the Lorentz and spinor structure of the contact interaction, and could affect the efficiency of experimental searches for this decay. The squared matrix-elements for the individual contact interactions have the form , where and is the invariant mass-squared of a pair of final state fermions and . Our study will not take this into account, since we found, in some explicit examples, only a small relative effect on the selection efficiency (of the order of 5%).
2.2 Bounds from flavour physics and HERA
Low energy constraints on four fermion operators involving two leptons and two quarks have been estimated and compiled for many operators taken one at a time [24, 25, 26], and carefully studied for selected flavour combinations (see e.g. , or global fits to operators ). However, even in the more recent compilations [25, 26], bounds on LFV operators involving a single top are not quoted. In the Appendices B and C, we estimate bounds on such operators from their possible contributions, inside a loop, to rare , and decays. In Appendix D, we estimate bounds from single top searches at HERA. Here, we summarise the resulting bounds, and list in Tables 1 and 2 the best limits on the coefficients of the various operators. We will find that only the coefficients of some operators are stringently constrained, while others could mediate LFV top decays within the sensitivity of the LHC.
The current upper limit  severely restricts flavour change. For our top operators to contribute, the quark lines must be closed, which requires at least two loops and a CKM factor, see the second diagram of Figure 5. Nonetheless, in the case of scalar or tensor operators involving this diagram can overcontribute to by several orders of magnitude, because the lepton chirality flip is provided by the operator (rather than ), so the diagram is enhanced by a factor . We make order-of-magnitude estimates in Appendix B, and quote the resulting bounds in Table 2.
Exchanging a between the and quark legs of the top operator will generate an operator with down-type external quark legs, see the left diagram of Figure 1. The coefficient of this light quark operator will be suppressed by a loop, CKM factors, and various masses. Numerical values for these suppression factors are given in Table 9 of the Appendix; however, their approximate magnitude is simple to estimate. If the top is singlet (), then the loop is finite; in the case of interactions, this is because mass insertions are required on both internal quark lines to flip chirality. In the case of scalar operators, one internal quark mass for chirality flip is still required, then terms linear in the loop momentum vanish, so the diagram is also proportional to an external quark mass. For scalar operators involving (which require an insertion to connect the to the line), the best limit can arise from exchanging a between the and a charged lepton leg, which generates a charged-current operator as represented in the second diagram of Figure 1. In the Appendix are also given current bounds on the coefficients of the various light quark operators that the top operators can induce. Comparing these bounds to the induced coefficients, gives the limits of Tables 1 and 2 that are labelled with ()s.
HERA collided protons with positrons (or electrons) at a centre of mass energy of 319 GeV, and searched for single tops in the final state. The H1 collaboration had a few events with energetic isolated leptons and missing energy, consistent with followed by leptonic decay of the top [29, 30]. However, this signal was not confirmed by the ZEUS experiment , and neither collaboration had a signal consistent with hadronic top decays. Both collaborations set bounds on ; we follow H1, since they had some events and a weaker bound:
It can be seen from Tables 1 and 2, that rare decays give very weak bounds on some contact interactions of the form and . Such interactions might have contributed a signal at HERA via leptonic decays, so we make some approximate estimates in Appendix D.4, and include the bounds in the tables with a .
The current bounds on LFV branching ratios of the top can be obtained from Tables 1 and 2, and Eq. (4). In these tables, the bound on appears on the first row of each box, where the flavour indices are given in the left column, and the operator label is given in the first row (see Appendix A for operator definitions). In parentheses below the bound, is a clue to where it comes from: HERA means the single top searches at HERA that are discussed in Appendix D, and the ()s mean the bound comes from dressing the top operator with a loop. For instance, the bound 0.0037 in the second row and third column of Table 1, arises by exchanging a between the and legs of the top operator, which gives the operator . This would mediate the unobserved decay , so its coefficient is bounded above, as indicated in Table 6. (The bounds on lighter-quark operators relevant to constraining top operators are given in Tables 6 - 8, and Table 9 gives the loop suppression factors with which the top operators generate the lighter-quark operators). Translated back to the top operator, the upper limit on gives the quoted limit on the top operator coefficient.
In this paper, we are interested in top decays to , the bounds on which are given in the second and third rows of the tables. For many111The exception is the operator involving , whose loop suppression factor would involve two light quark masses. of the operators involving the doublet component of the top (; recall that the last index in the operator label is the top chirality), the rare decay bounds are restrictive, implying that these operators could only induce . Scalar and tensor operators involving , and would overcontribute to . However, there remain operators which are weakly or not constrained, allowing a branching ratio . It is therefore interesting to explore the sensitivity of the LHC to decays.
Finally, it is interesting to consider how large the coefficient of the top operators can be. Some of the upper bounds quoted in Tables 1 and 2 are 1, and should not be interpreted as relevant constraints 222They are given so that in the future, if the experimental bounds improve, the limits can be obtained by simply rescaling the number in the tables. For instance, if the upper bound on decays were to improve by two orders of magnitude, the limit on some s would be divided by 10, and become marginally relevant.. Indeed, the width of the top is given by D0  as GeV (the theoretical decay rate to is 1.3 GeV), which constrains -. Furthermore, phenomenological prejudice and the leptoquark example of Eq. (1), suggest that , because the three-body decay should be mediated by sufficiently heavy ( particles, with sufficiently small couplings to have not yet been detected. We therefore quote in bold face the “relevant” bounds that impose .
3 at the 8 TeV LHC
In this Section, we estimate the sensitivity of current LHC data to the LFV top decays , where . We consider strong production of a pair, because this is the most abundant source of tops at the LHC, followed by the leptonic decay on one top, and the LFV decay of the other. This is illustrated in Figure 2, and gives a final state containing three isolated muons or electrons 333The final states where the decays to are not directly targeted by this search. The fact that such processes, followed by leptonic decays, can pass our selection is taken into account in the signal efficiency, as explained in the following., which has small Standard Model backgrounds.
3.1 Simulation setup
This study is performed for proton-proton collisions at the LHC, with center of mass energy of 8 TeV and an integrated luminosity of 20 , corresponding to the LHC Run1. The details about the signal and background generation are given in Section 3.2. The detector simulation is carried out by Delphes  using a CMS setup parametrization.
Delphes uses particle-flow-like reconstruction. The relative isolation of leptons is calculated from the total of the particles inside a cone of around the lepton direction ( for electrons and for muons), divided by the of the lepton. Jets are clustered using the fastjet package  with the Anti-kt  algorithm with distance parameter . The b-tagging performances are tuned on the typical efficiency and fake rate obtained in CMS.
For this study, no additional interactions in the same or neighbouring bunch crossing (pileup) are simulated.
3.2 Signal and SM backgrounds generation
The signal is generated with PYTHIA 8.205  using tune 4C. Top quarks are pair produced, then one top is forced to decay to charm, , and , with equal probability between and . The decay products are distributed according to the available phase space. 100k events have been generated both for LFV top and anti-top decays.
The backgrounds for this search, listed in Table 3, are processes that can give rise to three isolated leptons and at least 2 jets in the final state. Most of them are related to the production of real isolated leptons, e.g. from a top pair or vector bosons in the final state. In the table, are also shown the details about the number of generated events and production cross section for 8 TeV and 13 TeV proton-proton collisions. The number of generated events refers to the generation at 8 TeV. The cross section is calculated with the Top++2.0 program to next-to-next-to-leading order in perturbative QCD, including soft-gluon resummation to next-to-next-to-leading-log order (see  and references therein), and assuming a top-quark mass of GeV. When an explicit calculation was not available, the cross sections have been calculated with the MCFM package , version . The kinematic cuts used for the calculation are also shown in the table.
|Process||Events||[pb] (8 TeV)||[pb] (13 TeV)||Source|
|(2l22b)||12M||26.19||86.26||Top++2.0 (NNLO) |
|WW+jets (2l2, )||1M||5.84||11.57||MCFM |
|ZZ+jets (2l2q, )||2.5M||2.71||5.35||MCFM |
|ZZ+jets (2l2, )||1M||0.774||1.53||MCFM |
|ZZ+jets (4l, )||2.5M||0.390||0.738||MCFM |
|WZ+jets (2l2q, )||1M||2.37||4.60||MCFM |
|WZ+jets (3l, )||1M||1.15||2.23||MCFM |
The leading order (LO) matrix element generator, MADGRAPH 5 , with CTEQ6 parton distribution functions, is used to generate top pair production, and associated production of a top pair and a vector boson (, ). MADGRAPH, interfaced with tauola for decays, is used to generate vector-vector production (, and ) and the contribution of weak processes giving rise to final states with one top quark, one quark and a boson (decaying to leptons). For the vector-vector production, we only consider final states with at least two real charged leptons. This means that for the system, the considered final states are 2 charged leptons and 2 neutrinos; for , they are 3 charged leptons and one neutrino or 2 charged leptons and 2 quarks; and for , they are 2 charged leptons and 2 neutrinos, 2 charged leptons and 2 quarks, or 4 charged leptons. In all cases, MADGRAPH accounts for the presence of up to two additional jets at matrix-element level, and the hadronisation is carried out by PYTHIA 8.205. The details about the SM backgrounds simulation and cross sections are shown in Table 3.
3.3 Event selection
The signal for this search is production, followed by the lepton flavor violating decay of one top (which will be denoted as LFV top in the following), and the leptonic decay of the other (standard top, in the following). The number of expected signal events is given by:
where , the integrated luminosity, the selection efficiency on the signal, and pb (see  and references therein).
The considered signature is 3 isolated leptons (with one pair of opposite sign and opposite flavor from the LFV decay), 2 jets (one of which is a -jet), and missing transverse energy. For the event selection, we consider only muons of GeV and , electrons of GeV and , and jets of GeV and . These criteria are comparable to those used in real analyses by the CMS or ATLAS collaborations. A muon is considered isolated if its relative isolation value is less that 0.12, and an electron is considered isolated if its relative isolation value is less that 0.1. We select events containing:
exactly 3 isolated charged leptons (electrons or muons), two of which must be of opposite sign and opposite flavor.
Events are requested to contain at least 2 jets, and
exactly one b-tagged jet, and
the missing transverse energy has to be higher than 20 GeV.
In order to exclude events where two of the isolated leptons come from a real Z boson, we reject events containing any pair of opposite sign isolated muons or electrons with invariant mass between 78 and 102 GeV/c. This cut is particularly helpful in rejecting background arising from associated production with a Z.
The charged lepton that does not belong to the pair of opposite sign and opposite flavor leptons is assigned to the standard top in the event, and assumed to come from the W decay (bachelor lepton). Following a common procedure in reconstruction of semi-leptonic events, the and components of the missing transverse energy are taken as a measurement of the neutrino and , and the longitudinal component of the neutrino momentum is calculated imposing that the invariant mass of the system composed of the bachelor lepton and the neutrino must equal the mass of the W boson. The bachelor lepton and the neutrino 4-momenta are then combined with that of the b-tagged jet, to build a candidate standard top. When more choices of the bachelor lepton are possible (there can be up to 2 possible pairs of opposite sign opposite flavor charged leptons in one event), all are considered and the one giving the best standard top mass is chosen. We reject events in which the invariant mass of the standard top candidate is more than 45 GeV away from the nominal top mass. After the choice of the bachelor lepton, there is only one possible pair of opposite sign and opposite flavor leptons in each event. This is combined with all good (non b-tagged) jets present in the event to build a list of candidates for the LFV top.
Events are requested to have at least one combination giving a LFV top mass within 25 GeV of the nominal value.
The efficiency of the final selection on signal events is
where the uncertainty is statistical only. The signal efficiency is calculated on events where one top decays through , and the other one decays to a quark and a , which subsequently decays to a charged lepton (e, or ) and a neutrino, and is defined as the fraction of such events passing the selection criteria.
The number of expected events, for the signal and for each background category, on 20 fb of proton-proton data at 8 TeV is shown in Table 4, for different subsequent selection requirements.
|Process||no selection||step 1||step 2||step 3||step 4||step 5||step 6|
|Signal||202.57||32.98 0.17||22.66 0.14||9.12 0.09||8.20 0.09||7.50 0.09||3.75 0.06|
|542806||14.78 0.81||10.51 0.69||4.36 0.44||4.36 0.44||3.55 0.40||0.63 0.17|
|WW+jets||116760||0.93 0.33||0.35 0.20|
|ZZ+jets||72900||353.74 0.95||82.50 0.47||3.74 0.10||1.60 0.07||0.25 0.03||0.03 0.01|
|WZ+jets||63360||852.21 4.04||182.96 1.90||8.70 0.42||7.62 0.39||0.74 0.12||0.04 0.03|
|W||4240||9.36 0.24||7.67 0.22||3.59 0.15||3.45 0.14||3.10 0.14||0.27 0.04|
|Z||3840||17.25 0.33||16.44 0.32||7.72 0.22||7.16 0.21||1.85 0.11||0.22 0.04|
|tbZ||282||5.75 0.03||3.59 0.02||1.51 0.01||1.37 0.01||0.13 0.01||0.01 0.01|
3.4 Results and expected limits on the branching ratio
The selection and its efficiency, on signal and background, are discussed in Section 3.3, and summarised in Table 4. Assuming a branching ratio of for the signal, an uncertainty of 2.5% on the luminosity, and 20 fb of data, we would expect signal events, to compare to the expected events from known backgrounds.
In Figure 3, we show the invariant mass of the LFV top candidate (left) and the standard top candidate (right), in events passing all the cuts except those on the masses themselves.
In order to evaluate the sensitivity of this search for , we calculate the expected upper limit that could be set, in the case of absence of the signal. The calculation is based on the number of expected background events surviving the final selection, in 20 fb of 8 TeV LHC data, so the result can be interpreted as the possible upper limit the CMS or ATLAS collaborations (alone) could expect to set with Run1 data, if the signal were not there.
A 95 confidence level (CL) upper limit on the branching fraction of is calculated using the modified frequentist approach (CL method [43, 44]), as it is implemented in the RooStats framework . Based on the number of expected background events, summarised in Table 4, and on Equation (6), the obtained limit is:
Alternative techniques for limit calculations, as implemented in RooStats, have been tried, leading to compatible results. An eventual variation of 100 in the number of expected background events would lead, in the worst case, to an expected limit of @ 95 % CL.
As explained in appendix E, we emulate in our framework the published CMS search for . The reason for this exercise is twofold: on one hand it allows to validate our procedure on simulated samples, by comparing with the CMS background expectations. On the other hand, it provides an estimate of the constraint set on from this previous analysis, which is found to be , on the verge of probing LFV top decays mediated by a four-fermion operator. As proven in the present study, a dedicated analysis would set a limit 50 times stronger (of the order of ), showing that the existing LHC data from Run1 can still be used to obtain interesting constraints on lepton flavor violation.
4.1 Perspectives at 13 TeV and 14 TeV
To estimate the reach of the described search at a center-of-mass energy of 13 TeV, we extrapolate the 8 TeV results, rather than performing a full simulation of signal and background processes at 13 TeV. The increase of the production cross sections for SM processes, from 8 TeV to 13 TeV (see Table 3), is taken into account. The selection requirements and efficiencies are kept the same as for the 8 TeV analysis. For the signal, we have checked on simulated events that the efficiencies at 8 TeV and 13 TeV are consistent within 5.
The sensitivity is estimated by calculating the expected upper limits on , in the absence of signal, for two scenarios: the case of 20 fb and 100 fb proton-proton data collected by the LHC at 13 TeV center-of-mass energy. We also extrapolate the sensitivity to the case of 3000 fb of integrated luminosity at 14 TeV. In this last case, we simply rescale signal and background rates from 13 TeV to 14 TeV, and use the square root of the number of expected background events as an estimate of their uncertainty. The obtained values are summarized in Table 5. The upper limits presented here are derived using statistical uncertainties only, so don’t take into account the possibility for such analyses to become systematically dominated in the future. In order to have an accurate evaluation of the systematics evolution, a deeper study from the LHC experiments would be needed. On the other hand, for an analysis on 13 TeV or 14 TeV data, the selection would have to be re-optimised, possibly leading to an increase in sensitivity.
|8 TeV (20 fb)||13 TeV (20 fb)||13 TeV (100 fb)||14 TeV (3000 fb)|
4.2 Single Top
In addition to mediating LFV top decays, the top operators listed in Appendix A could lead to single top production with an pair, as illustrated in Figure 4. The objects in the final state would be the same as for the process we studied : three leptons, missing energy, and jets, of which one is a . We estimate444The partonic diagrams for are those of , with replaced by par , and by or . So we estimate by comparing to pb . If we neglect spin correlations between the and , then for top operators, can be factorised as where the has coupling to fermions, and the 1/3 is the spin average. Then one has where , and . The three body phase space integral can be written as , where integrates the phase-space of and in the centre-of-mass frame, , and we take the upper limit of the integral as , assuming thhis is a reasonable value for leptons with GeV. Finally, at TeV) TeV, the ratio of to pdfs is of order 6-10, so we estimate that . that at 8 TeV, the cross section for is of similar order to the cross section for , for operators involving a quark and slightly less for a quark.
We neglect this process for two reasons. First, the contact interaction approximation (that the four-momentum in the process ) is more difficult to justify than in top decay, because the energy scale in the process can be . Secondly, finding such events in the backgrounds could be more challenging because the cannot be required to participate in the reconstruction of a top (step 6 of Section 3.3).
We envisage that it makes sense to neglect the LFV single top process in a first search for LFV top decays. This is conservative, because LFV single top production could contribute events that pass our selection.In the absence of a signal, such a search could sufficiently constrain the contact interaction scale , to justify including the single top process in subsequent analyses.
The aim of this paper was to explore the LHC sensitivity to the decay , which is lepton and quark flavour-changing, but baryon and lepton number conserving. We parametrise this decay as occuring via a contact interaction, and list a complete set of invariant dimension-6 operators in Appendix A. We parametrise the coefficient of these interactions, which we refer to as “top operators”, as or equivalently , with
Model-building prejudice (see Section 2.3), suggests that . The top branching ratio is then
These contact interactions are currently constrained from their contribution in loops to rare decays, and from single top searches at HERA. These bounds are discussed in the Appendices, and summarised in section 2. For interactions involving and , rare decay bounds impose for some operators, but others can have .
In Section 3, we evaluate the sensitivity reach of a dedicated search for lepton flavor violation in top decays, at the 8 TeV LHC. The search targets events, where one top decays to an up-type quark ( or ) and a pair of leptons of opposite sign and opposite flavor, and the other one decays to a quark and a , which subsequently decays to a charged lepton and a neutrino. This is illustrated in Figure 2.
The relevant signal and SM background processes are simulated for LHC Run1-like conditions: proton-proton collisions at 8 TeV center of mass energy, for an integrated luminosity of about 20 fb. The detector simulation is based on Delphes, with parameters tuned on the CMS detector reconstruction and performances, but does not include pileup. The analysis setup is validated by emulating an existing CMS search for rare top decays to in events, showing reasonable results.
We find that a dedicated search by a single experiment using 20 fb of 8 TeV data could be sensitive to
and extrapolate that a sensitivity of () could be reached with 100 fb at 13 TeV (3000 fb at 14 TeV). From Equation (4), we see that the 100 fb data could impose for operators, and for the and operators. This analysis shows that the existing LHC data from Run1 can still be used to obtain interesting constraints on lepton flavor violation. Although this is understandably not the priority focus in the times of the Run2 startup, let’s not to leave unchecked this possible path to New Physics.
SD thanks P Gambino and Professor Y. Kuno for useful conversations, and the labex OCEVU and the theoretical physics group at Montpellier University-II for hospitality during part of this work. The work of M.L.M. is performed in the framework of the ERC grant 291377 “LHCtheory: Theoretical predictions and analyses of LHC physics: advancing the precision frontier”.
Appendix A Appendix: Operators
Consider the -matrix element mediating , where , and suppose it is induced by local operators with momentum-independent coefficients, such that they can be added to the Standard Model Lagrangian. These operators should respect the gauge symmetry of the Standard Model. However, since the New Physics scale that we explore is not much larger than the electroweak scale , we should include dimension-8 operators constructed from two Higgs fields and four fermions, or two gradients and four fermions. Instead, we choose to work with -invariant, but not -invariant, operators of dimension 6. This is because a dimension-8 -invariant operator, can be a dimension-6 -invariant operator:
where is a quark doublet of th generation, is a lepton singlet, , and SU(2) contraction is in square brackets.
This choice of -non-invariant operators means that the coefficients of and are taken as independent, and in particular, bounds on the first do not apply directly to the second. However, dressing the top operator with a loop will generate the light quark vertex, which gives unavoidable bounds that are estimated in Appendix C. (Treating the as dynamical while the New Physics is a contact interaction is unlikely to be a good approximation, but is the only way we can estimate whether the top operators are in tension with other observables.) In the case of , two or three loops are required to transform the top operators into the dipole operator, so we need an SU(2)-invariant formulation of the top operators. These dimension six and eight operators are given in Appendix A.2.
a.1 invariant “top operators”
We are interested in contact interactions involving two charged leptons of different flavour, a top and a or quark. Such colour-singlet, electric charge-conserving, dimension-6 operators will be referred to as “top operators”. The operators are:
and a redundant list of scalar/tensor operators is:
where , are Dirac spinors, are unequal lepton flavour indices, , and the last chiral superscript of the operators gives the chirality of the top.
The scalar operators can be exchanged for the tensor operators , as shown in Equations (19) and (20). We will use the and operators interchangeably, becuase the are more convenient in top decay, and the tensors at low energy. The tensors are used in the basis of . Notice that operators are defined to have the same chiral projector twice, whereas for they are different. Since the last quark flavour label is fixed to , , and operators appear twice, for both chiralities and (were we using arbitrary quark flavour indices , then ). Finally, the scalar operator contracting quark to lepton spinors is given the name ; there are no such operators of the form , because this is Fierz-equivalent to .
A similar list can be constructed for down-type quarks. These will be relevant, because the top-operators can generate the down-operators at one loop. To reduce index confusion, down-type quarks will have greek flavour indices There are also charge-current (CC) scalar operators
which will be relevant in setting bounds.
These operators are included in the Lagrangian with a dimensionful coefficient
The s have subscripts that identify the operator, and lepton-quark flavour superscripts: would be the coefficient of , and would be the coefficient of (so in particular, the index order is always , even for ).
Our operators can be identified as components of SU(2) invariant dimension 6 operators: