Generic tests of CP-violation in high-p T multi-lepton signals at the LHC and beyond

We introduce a modification to the standard expression for tree-level CP-violation in scattering processes at the LHC, which is important when the initial state in not self-conjugate. Based on that, we propose a generic and model-independent search strategy for probing tree-level CP-violation in inclusive multi-lepton signals. We then use TeV-scale 4-fermion operators of the form tuℓℓ and tcℓℓ with complex Wilson coefficients as an illustrative example and show that it may generate O (10%) CP asymmetries that should be accessible at the LHC with an integrated luminosity of O (1000) fb − 1 .

The nature of CP-violation (CPV), which is closely related to the flavor structure, is one of the major unresolved problems in particle physics.Indeed, the search for new CP-violating sources, beyond the standard model (SM), may be the key to a deeper understanding of particle physics and the evolution of the universe, since CPV has far-reaching implications for cosmology [1][2][3]; in particular, the strength of CPV effects in the SM is insufficient to explain the observed baryon asymmetry of the universe (BAU), see e.g., [4][5][6].It is, for these reasons, that the search for CPV beyond the SM is a very important component of the on-going effort for unveiling the physics that underlies the SM, even if the latter has already been observed.
In this paper we re-examine the formulation of treelevel CP-violating effects in scattering processes at the LHC, introducing a new term to the "master" CPV expression, which properly identifies the genuine CP violating signal and also takes into account "fake" CPviolating effects that arise when the initial state in not self-conjugate.We then present a generic test of CPV in scattering processes, which is potentially sensitive to a wide variety of underlying new physics (NP) scenarios.We are particularly interested in CPV in the inclusive tri-lepton and four-lepton signals: where ℓ, ℓ ′ = e, µ, τ (preferably ℓ ̸ = ℓ ′ , see below) and X 3 , X3 and X 4 contain in general jets and missing energy.These include the e ± µ + µ − and µ ± e + e − final states for ℓ, ℓ ′ = e, µ and similarly for the pairs ℓ, ℓ ′ = e, τ and ℓ, ℓ ′ = µ, τ , as well as the 3-flavor final state pp → eµτ + X.As an example, we will consider below CPV in the e ± µ + µ − tri-lepton signals, but it should be clear that it is equally important to search for CPV in multi-leptons final states with as many different combination of flavors as possible.
The available momenta of the charged leptons in the final state of these multi-lepton signals allow a straightforward construction of CPV observables in the laboratory frame, as will be shown below.We note, though, that special care is needed for CPV tests at pp colliders, where the initial state is not self-conjugate and the parton distribution functions (PDF's) of the incoming partons may, therefore, have an asymmetric structure.This will be discussed below.
It should be emphasized that a sizeable, say O(≳ 1%) manifestation of CPV in multi-leptons events of the type (1), (2) or (3) will be strong evidence for NP, since the CP-odd CKM-phase of the SM (which is responsible for CPV in the quark sector and has been measured [41]) is expected to yield negligible CP-violating effects in these processes, as it can only arise from EW processes at higher loop orders [42]. 2 Furthermore, new CP-violating effects in leptonic systems may shed light on Leptogenesis, where the BAU is generated from a lepton asymmetry via a decay of a heavy neutral lepton [43,44].
Potential large tree-level CP-asymmetries at the LHC in the tri-and four-lepton production processes (1), ( 2) and (3) can be searched for, using the following triple products (TP) of the lepton momenta (ℓ ̸ = ℓ ′ ): 3,4 which are odd under P and under naive time reversal (T N ): time → -time.Under C and CP they transform as: Thus, to measure a nonzero TP correlation effect for the O CP 's defined in (4), the following T N -odd (and also P -violating) asymmetries can be constructed: where N (O CP > 0) is the number of events for which sign(O CP ) > 0 is measured etc.As will be shown below, a measurement of A T ̸ = 0 and/or ĀT ̸ = 0 may indicate the presence of CPV (CPodd phase(s)), but may also be a signal of some strong or generic CP-even phase, e.g., from final state interactions (FSI) [42,72,73], even if the underlying dynamics that drives the processes under consideration is CPconserving.Therefore, in order to better isolate the pure CPV effect, we use the following observable, sensitive to CPV: A CP may, in fact, also be "contaminated" by CP-even phases when the initial state is not CP-symmetric, as can be the case at the LHC or at pp colliders, in general.To see this, let us consider the underlying (hard) processes of the tri-lepton signals of (1) and (2) (the discussion below applies similarly to the four-lepton signals of (3)): ab → ℓ ′− ℓ + ℓ − + X and āb → ℓ ′+ ℓ − ℓ + + X.We assume for simplicity that there are only 2 interfering amplitudes that contribute to these processes as follows (CPV requires at least two amplitudes with different phases for any given process): where we have factored out the CP-odd phases, ϕ 1,2 , and CP-even phases δ 1,2 .The latter typically arise from FSI at higher loop orders.Also, M i can be complex in general (as in our case below) and the amplitude for the chargeconjugate (CC) channel (ā b → ℓ ′+ ℓ − ℓ + ) is obtained from (9) by changing the sign of the CP-odd phases ϕ i → −ϕ i and replacing M i → M ⋆ i .The corresponding (hard) differential cross-sections can then be schematically written as: and and the 3rd term in (10) arises from ∝ O CP and is where the tree-level CPV resides, i.e., when ∆δ = 0.
We then find for A T and ĀT in ( 6) and ( 7): with , (12) where dΦ is the phase-space volume element, R is the phase-space region of integration and f a , f b are the PDF's of the incoming particles a, b; similarly, for the CC channel, I āb is obtained by replacing As mentioned earlier, we see that A T ̸ = 0 and/or ĀT ̸ = 0 can be observed even in the absence of CPV (i.e., when ∆ϕ = 0), due to the presence of CP-even phases (∆δ ̸ = 0).Also, |A T | ̸ = | ĀT | is possible at the LHC, even with ∆δ = 0, due the different PDF's of the incoming particles in the process and its CC channel, i.e., due to f a , f b ̸ = f ā, fb, when the initial state is not self conjugate.This affects the CP-asymmetry A CP of ( 8), which is given by (using ( 11)): Thus, when the initial state is self-conjuate and I ab = I āb (i.e, the initial state and its CC state have the same PDF's), then the asymmetry appears with the conventional CP-even and CP-odd phase factors, A CP ∝ cos ∆δ sin ∆ϕ; in this case A CP vanishes when the CPodd phase vanishes.The second term in (13), which is ∝ I āb − I ab , deals with the case when the initial state is not self-conjugate and I ab ̸ = I āb, as is the case for the LHC or other future hadron colliders that are being envisioned (see also below).This term is a new correction to the classic expression for tree-level CPV in scattering processes.It is a "fake" CP signal (being ∝ cos ∆ϕ) that can be generated in the presence of a CP-even phase.We note, though, that such a fake CP effect cannot be generated at tree-level in scattering processes at the LHC if there are no resonances involved (for situations involving resonances, see [74]), since then CP-even phases can only arise from FSI at higher loop orders, as opposed to the potentially large tree-level effects in A CP , i.e., the 1st term in (13).It thus follows that, in the absence of resonances, if a large CP asymmetry is measured, say of O(10%), (as shown below), then besides the fact that it will be strong evidence for NP, it will also be a signal of genuine CP-violating tree-level dynamics.
In particular, tree-level exchanges of S 1 and R 2 among the lepton-quark pairs induce the operators in ( 14) and (15), where, in this case, the Wilson coefficients, f i , of the operators in ( 14) and ( 15), satisfy universally for any given set of flavor indices prst in ( 14) and (15), see [13].We will use this relation as a benchmark scenario in the numerical calculations described below.
The scalar and tensor 4-Fermi operators in ( 14) and (15) (or equivalently, tree-level exchanges of the leptoquarks S 1 and R 2 ) generate t tℓ + ℓ − as well as FC tū i ℓ + ℓ − (and the charge-conjugate tu i ℓ + ℓ − ) contact terms, where ℓ = e, µ, τ stands for any one of the SM charged leptons and u i = u, c.The ttℓℓ interaction modifies the process pp → t tℓ + ℓ − , as discussed in detail in [13], and can thus also give rise to tree-level CPV in both the tri-lepton and four-leptons production channels of ( 1)-(3).
In the following, we focus just on the FC tu i ℓℓ 4-Fermi interactions, which can modify (see also [12,31]) and generate CPV in the tri-lepton signals of (1) and (2), via the underlying single-top hard processes u i g → tℓ + ℓ − and the CC channel (see Fig. 1), followed by the t and t decays t → bℓ ′+ ν ℓ and t → bℓ ′− νℓ .
As discussed below, the contribution of the FC tu i ℓℓ effective operators to the tri-lepton signal does not interfere with the SM diagrams, so that the CPV in this case is a pure NP effect; it arises from the imaginary part of the interference between the scalar and the tensor operators, if at least one of the corresponding Wilson coefficients is complex. 6n particular, the numerator of A CP (and of A T and ĀT ) is proportional to the CP-violating part of the crosssection for u i g → tℓ + ℓ − → ℓ ′+ ℓ + ℓ − + X (hereafter we suppress the flavor indices of the operators in ( 14) and ( 15)): and similarly for the CC channel ūi g → tℓ − ℓ + → ℓ ′− ℓ − ℓ + + X by replacing ϵ (p ui , p ℓ ′+ , p ℓ + , p ℓ − ) with ϵ (p ūi , p ℓ ′− , p ℓ − , p ℓ + ), where ϵ (p 1 , p 1 , p 3 , p 4 ) = ϵ αβγδ p α 1 p β 2 p γ 3 p δ 4 and ϵ αβγδ is the Levi-Civita tensor.
In contrast to the numerators, the NP contributions to the denominators of our CP-asymmetries are proportional to the CP-conserving terms , where the dominating term is the pure tensor contribution |f T | 2 .The SM tri-lepton production processes will also contribute to the total number of tri-lepton events which enter the denominators of A CP and T N , TN ; the dominating SM tri-lepton process is pp → W Z + X. 8To assess the feasibility of CP asymmetry measurements in multi-leptons final states at the LHC, we perform a simulation on the tri-lepton signal processes described above, together with the relevant SM background processes, which do not include detector effects other than those modeled by simple threshold and acceptance requirements.Although more elaborated analysis approaches might also be useful, for simplicity, we follow an approach that is completely generic and provides a model-independent test of CPV in multi-lepton final states, which would be designed to be sensitive to any type of underlying CP-violating NP involving charged-leptons.We therefore define the asymmetries for the inclusive multi-lepton signals, with no further event selections on the types or kinematic properties of the other objects in the final state, i.e., X i in (1)-(3).Indeed, in general it is possible to use additional use-ful selections , e.g., in our case a selection of one b-jet (see [13,31,[105][106][107]) will essentially eliminate the dominating pp → W ± Z + X → ℓ ′± ℓ + ℓ − + X SM contribution to the denominators of our asymmetries.Nonetheless, we use only a selection on the minimum invariant mass of the di-leptons involved, m min (ℓ + ℓ − ), which allows us to suppress the SM background without loss of generality.The input for the numerical calculations is further described in Appendix A.
Furthermore, for the NP contribution we study the dependence on the NP scale up to Λ ∼ few TeV; the typical bounds on the natural scale of the operators under investigation, in ( 14) and ( 15), are Λ > ∼ O(1) TeV, see [31].Guided by the relation between the scalar and tensor couplings in ( 16), we set |f S | = 1, |f T | = 0.25 with a maximal CP-odd phase for the tuℓℓ and tcℓℓ operators, so that: Our results are summarized in Fig. 2 and Table I.In Fig. 2 we show the dependence of A CP on m min (ℓ + ℓ − ) and in Table I we give the resulting CP-violating and T N -odd asymmetries for m min (ℓ + ℓ − ) = 400 GeV.The expected inclusive tri-lepton cross-sections for the NP and the dominant SM background, after the event selection criteria have been applied, are given in Appendix B: for m min (ℓ + ℓ − ) = 400 GeV and an integrated luminosity of 1000 fb −1 , we expect an O(100) ℓ ′± ℓ + ℓ − from the SM pp → ZW ± background, whereas the new tuℓℓ and tcℓℓ 4-Fermi operators yield ∼ 10 4 and ∼ 500 ℓ ′± ℓ + ℓ − events, respectively, if Λ ∼ 1 TeV.
We see that the CP-asymmetry increases with the invariant mass cut on the same-flavor di-leptons, m min (ℓ + ℓ − ).This is due to the decrease of the SM contribution with m min (ℓ + ℓ − ) in the denominators of the asymmetries.Also, the asymmetry is larger in the ugfusion case, since the SM background in this case is considerably smaller w.r.t. the signal in this case (see the Appendix and discussion above). 9On the other hand, the asymmetries A T , ĀT and A CP decreases with Λ, as expected.For example, in the tuℓℓ 4-Fermi case, the CP-asymmetry drops from in the ug-fusion case due to the difference between the incoming ug and ūg PDF's, see (11).

TABLE I:
The expected T N -odd and CP asymmetries in tri-lepton events, pp → ℓ ′± ℓ + ℓ − + X, via the ug-fusion and cg-fusion production channels (and the CC ones) at the LHC, for m min (ℓ + ℓ − ) = 400 GeV.Values are given for Λ = 1(2) TeV, Im (f S f ⋆ T ) = 0.25 and the SM background from pp → ZW ± + X, as explained in the text.
To summarize, we have investigated the possible detection of tree-level CPV in scattering processes at the LHC and introduced a modification to the standard formula for such CP-violating effects, which is relevant when the initial state in not self-conjugate.We focused specifically on multi-leptons signals and their sensitivity to new TeV-scale sources of CPV.In particular, we have constructed CP-violating triple-product correlations out of the momenta of the charged leptons in multi-lepton events, which can be used as model-independent tests of tree-level (and therefore large) CPV from any source of underlying CP-violating physics.We have calculated the expected CP-asymmetry in tri-lepton events at the LHC from new TeV-scale FC tuℓℓ and tcℓℓ 4-Fermi interactions, which can be viewed as an EFT parameterization of tree-level TeV-scale leptoquark exchanges in these channels.We showed that an O(10%) CP-asymmetry is naturally expected in this case, if the EFT operators carry a CP-odd phase and the NP scale is of O(T eV ).
The measurement of such O(10%) CP-asymmetry in multi-lepton events is challenging, but if observed, it should stand out as an unambiguous signal of NP that may shed light on the fundamental issue of BAU.We believe that it is quite feasible provided the experimental uncertainties can be kept at the level of O(1%) (see [108]) bearing in mind that such CP-violating effects in the SM are un-observably small in multi-lepton events.Indeed, we estimate the statistical uncertainty in measuring the CP-asymmetry, based on the expected number of tri-lepton events in our NP scenario (see Appendix) to be ∼ 1% − 2% with an integrated luminosity of L ∼ 1000(3000) fb −1 in the tuµµ(tcµµ) NP cases (see Fig. 2).
All event samples (NP signal and SM background) were generated using MadGraph5_aMC@NLO [109] at LO parton-level and with the SMEFTsim model of [110,111] for the EFT framework.The 5-flavor scheme was used to generate all samples, with the NNPDF30_lo_as_0130 PDF set [112] and the default MadGraph5_aMC@NLO LO dynamical scale.
Leptons were required to have transverse momentum of p T > 10 GeV and pseudo-rapidity |η| < 2.5, while for jets we used p T > 20 GeV, |η| < 5.0 and an angular separation of ∆R = 0.4.
In Table II we list the estimated cross-sections for the NP with Λ = 1 TeV (note that the NP cross-section scales as Λ −4 ) and the SM contributions to the inclusive pp → ℓ ′± ℓ + ℓ − + X processes for m min (ℓ + ℓ − ) = 200, 300 and 400 GeV, where m min (ℓ + ℓ − ) is the lower cut on the invariant mass of the same-flavor di-leptons.In particular, the m min (ℓ + ℓ − )-dependent cross-sections are defined as: We note that the simulations were made without parton showering and jet matching, which has no effect on our CP-asymmetry (we confirmed that the calculated CP-asymmetry with and without the extra jet in the trilepton final state is the same within the numerical error).Also, we did not perform any detector simulation which is beyond the scope of this work and is left for a dedicated analysis.Thus, the cross-sections reported in Table II should be viewed as an estimate; a more realistic calculation of the expected total cross-sections for this type of NP and SM background can be found in [31].
TABLE III: The expected T N -odd and CP asymmetries A T , ĀT , A CP and the corresponding axis-dependent asymmetries A i T , Āi T , A i CP (i = x, y, z), for the tri-lepton events pp → ℓ ′± ℓ + ℓ − + X at the LHC with m min (ℓ + ℓ − ) = 400 GeV.Results are given for both the ug-fusion and cg-fusion production channels (and the CC ones).Numbers are presented for Λ = 1 TeV, Im (f S f ⋆ T ) = 0.25 and the dominant SM background from pp → ZW ± + X is included.The cases where an asymmetry is < ∼ 0.5% is marked by an X. A

Āz
T ug-fusion: -5.8% -5.0% -5.6% 3.1% cg-fusion: -4.7% -6.3% X X can be divided into three axis-sensitive triple-products: where i = x, y, z denotes the x, y, z components of the momenta, e.g., p z a and (⃗ p b × ⃗ p c ) z are the z-components of the momenta ⃗ p a and (⃗ p b × ⃗ p c ), respectively.Note that only three out of the four O CP and O x,y,z CP in (C1) and (C2) are independent, since O CP = i=x,y,z O i CP .Furthermore, the axis-sensitive O x,y,z CP transform under P ,C,CP and T N the same as O CP , so that all the discussion and formulae for O CP in the paper applies also to O x,y,z CP .In particular, the axis-dependent CP-asymmetries can be similarly defined as: where A x,y,z T and Āx,y,z T are the axis-dependent T N -odd asymmetries.
In Table III we show a sample of our results for all T N -odd and CP-asymmetries including the axis dependent ones, for the tri-lepton events pp → ℓ ′± ℓ + ℓ − + X at the LHC, which are considered in this paper.The asymmetries are calculated for both the ug-fusion and cg-fusion production channels (and the CC ones), with m min (ℓ + ℓ − ) = 400 GeV , Λ = 1 TeV and Im (f S f ⋆ T ) = 0.25, and the dominant SM background from pp → ZW ± + X is considered.We see that a measurement of the axis-dependent asymmetries can be used to distinguish between the tuℓℓ and the tcℓℓ CP-violating dynam-ics.In particular, in the tuℓℓ case we obtain A z CP → 0 and A x,y CP ∼ 8%, while for the tcℓℓ operator we find A z CP ∼ 5.5% and A x,y CP → 0. Note also that the axisdependent asymmetries may yield a larger effect, e.g., in the cg-fusion case we find that A z CP > A CP .
Appendix D: Differential distributions: signal vs. background In Figs. 4 and 5, we plot the di-muon invariant mass and the triple-product differential distributions, respectively, for an integrated luminosity of L = 1000 fb −1 .We show these distributions for the ug-fusion and the CC ūg-fusion NP cases and the corresponding SM backgrounds, assuming a NP scale of Λ = 1 TeV and/or Λ = 2 TeV.Note that the NP signals scale as Λ −4 and are calculated with our benchmark value for the CPV coupling Im (f S f ⋆ T ) = 0.25.Also, the triple-product distributions in Fig. 5 are calculated with m min (ℓ + ℓ − ) = 300 GeV.

FIG. 2 :
FIG. 2: A CP as a function of m min (ℓ + ℓ − ), for Λ = 1 TeV, Im (f S f ⋆ T ) = 0.25 and including the SM background.The dependence of the asymmetry on Λ is given in Appendix B. The error bars represent the expected statistical uncertainty with an integrated luminosity of 1000(3000) fb −1 for the ug-fusion(cg-fusion) case.