High $p_T$ correlated tests of lepton universality in lepton(s) + jet(s) processes; an EFT analysis

We suggest a new class of tests for searching for lepton flavor non-universality (LFNU) using ratio observables and based on correlations among the underlying LFNU new physics (NP) effects in several (seemingly independent) di-lepton and single lepton + jet(s) processes. This is demonstrated by studying the effects generated by LFNU 4-Fermi interactions involving 3rd generation quarks. We find that the sensitivity to the scale ($\Lambda$) of the LFNU 4-Fermi operators significantly improves when the correlations among the various di-lepton +jets and single-lepton + jets processes are used, reaching $\Lambda \sim {\cal O}(10)$ TeV at the HL-LHC.

Intriguing hints of lepton-flavor non-universality (LFNU) and therefore of new physics (NP) have appeared in recent years in neutral and charged semileptonic B-decays  (for a recent review see [22]): the R K ( * ) and R D ( * ) anomalies which occur in b → s + − and b → c − ν transitions, respectively.
In this work, we consider testing for LFNU in lepton(s) + jets production at the LHC, by exploiting correlations amongst several LFNU observables. Specifically, we show that an enhanced sensitivity to the scale of the NP can be obtained by combining multiple LFNU tests, based on ratio observables. We demonstrate this for two specific new physics (NP) scenarios, using the so-called SM Effective Field Theory (SMEFT) framework [23][24][25][26], although this approach can be extended to establish a more systematic mapping between the underlying NP dynamics and experimentally realistic observables. The importance of using correlations in the search for NP has recently gained some attention, e.g., in leptoquark searches by combining di-lepton and single lepton production channels [27] and in top-quark systems by using measurements from different top production and decay processes to probe the NP effects [28][29][30][31].
Any evidence of possible LFNU phenomena contradicts the key Standard Model (SM) prediction that the differences in the rates of processes differing only in the flavor of the leptons involved are suppressed by small differences in the Yukawa couplings. In this sense lepton flavor is an accidental (approximate) symmetry of the SM, which may be strongly violated in a variety of well-motivated NP scenarios. Hence, even if the current experimental indications of LFNU have not yet met discovery criteria, providing an accurate probe of these processes, whether confirming such indications or not, will provide a bet-ter understanding of the flavor structure of the physics beyond the SM.
Let us denote generic lepton(s) + jets processes as follows: where m is the number of light jets (j), n is the number of b-jets (j b ) and p is the number of top or anti-top quarks in the final state of the leading-order (LO) hard process; E T denotes missing transverse energy, associated with final state neutrinos. We then define two classes of generic LFU tests at the LHC, involving ratios of the charged dilepton and single-lepton production channels in (1), normalized to the corresponding electron-production channels: where σ mnp and σ mnp are the total cross-sections of the processes (mnp) and (mnp) in (1), respectively. Lepton flavor violation effects of the type pp → i j +· · · (i = j) will not be considered here. For LFNU processes with only neutrinos in the final state, ratios such as (2) are not useful, since the neutrino flavor cannot be detected. In this case a different strategy is needed, which we briefly discuss below. Note that ratio observables such as in (2) provide more reliable probes of NP, since they potentially minimize the effects of theoretical uncertainties involved in the calculation of the corresponding cross-sections.
In the SM (or within NP scenarios which conserve lepton flavor universality) we have T mnp , T mnp → 1, since, as noted above, deviations from unity can only be generated through the non-universal Higgs-lepton Yukawa couplings and through lepton mass dependent polynomials and logarithms from higher order corrections. The former is proportional to the lepton masses and is therefore negligible, while the latter are much smaller than the expected experimental accuracy -as is the case, in particular, for high p T events which is of our interest in arXiv:2005.06457v1 [hep-ph] 13 May 2020 this work. We will include non-universal reconstruction efficiencies for the different leptonic final states in the overall uncertainty of the measurement of T mnp , defined in (2).
As mentioned earlier, we describe the underlying NP responsible for T mnp = 1 and T mnp = 1, using the SMEFT framework, defined by adding to the SM Lagrangian an infinite series of higher-dimensional, gaugeinvariant operators, O (n) i . These operators are constructed using the SM fields and their coefficients are suppressed by inverse powers of the NP scale M [23][24][25][26]: where n is the mass dimension of O (n) i and we assume decoupling and weakly-coupled heavy NP, so that n equals the canonical dimension. The dominating NP effects are then expected to be generated by contributing operators with the lowest dimension (smallest n) that can be generated at tree-level in the underlying theory. The (Wilson) coefficients f i depend on the details of the underlying heavy theory and, therefore, parameterize all possible weakly-interacting and decoupling types of heavy physics.
In what follows we will consider the leading dimension six operators (n = 6) and drop the index n. 1 We also define the "effective scale" Λ = M/ |f | whence where η f = ±1 denotes the sign of f . Thus, for example, Λ = M for "natural" NP with |f | = 1, which we will assume throughout the rest of this work, unless stated otherwise.
In Appendix A we list all the dimension six operators that can potentially violate LFU and that are, therefore, relevant for this study. Here, we will demonstrate our strategy for the two specific SU(2) triplet and singlet 4-Fermi operators (prst are flavor indices): focusing on the case where the heavy underlying NP has a LFNU coupling to 3rd generation quarks and 2nd generation leptons, i.e., on O lq (2233) and O qe (2233); it should be understood, though, that similar effects can be generated in the electron and τ -lepton channels, though, the phenomenology and detection strategies of final states involving the τ -leptons are fundamentally different from those involving the electrons and muons.
We will not consider operators that have a flavor changing quark current involving the 3rd generation quarks, e.g., O lq (3332), which can generate the b → sµ + µ − and b → cτ −ν τ transitions and may, therefore, contribute to R K ( * ) and R D ( * ) , respectively. These operators can generate LFNU collider signals similar to those studied here, see e.g., [33][34][35]. For example, O lq (2232) generates the µ + µ −s b and µ + µ −c t contact terms with the same effective scale, which can contribute to the ratio observables T 010 µµ and T 001 µµ , via sg → bµ + µ − and cg → tµ + µ − , respectively. Furthermore, the operator O (3) lq (3332) generates the contact interactions τ + τ −s b, τ −ν τs t and τ + ν τc b, which can contribute to the T -tests T 010 τ τ , T 110 τ τ as well as T 010 τ , T 110 τ via the hard processes sg → bτ + τ − , gg →sbτ + τ − and cg →bτ −ν τ , gg → cbτ −ν τ , respectively. To study the sensitivity to the flavor non-universal NP we define the following χ 2 -test: where X, Y ∈ (m, n, p) denote the and single channels, respectively, and δT X , δT Y denote the corresponding total experimental plus theoretical 1σ uncertainties, which are assumed to be statistically independent. The experimental uncertainties are assumed to be the dominant ones (see also discussion above).
For the purpose of exacting a bound on Λ we assume that, on average, no NP is observed. We thus generate O(10000) random realizations of the sets of "measured" T -tests, T X,exp and T Y,exp [to be used for the χ 2 -test in (7)], normally distributed with average 1 (i.e., the SM prediction) and standard deviation δT : 2 where N (a, s 2 ) denotes the normal distribution for average a and standard deviation s. The overall uncertainties δT X and δT Y of the data samples are taken as: where δT X,Y stat and δT X,Y sys stand for the statistical and systematic uncertainties expected in the data samples, respectively. The statistical uncertainties are estimated from the expected number of events based on the SM 2 Due to the different detection efficiencies of electrons and muons, we expect the ratios T exp , to deviate from unity even in the absence of NP. This, however, has no effect on our χ 2 -test analysis and will not change our main results. cross-sections: δT X stat = 2/N X (SM ) and δT Y stat = 2/N Y (SM ); for the systematic uncertainties we analyse below 3 different cases: δT X,Y sys = 5%, 10%, 15% for channels involving only light-jets and/or b-jets in the final state and δT X,Y sys = 10%, 20%, 30% for channels with a top-quark in the final state. Without knowing the actual uncertainties of the experiment, the uncertainty scenarios outlined above serve as realistic benchmarks for conveying the main message of this work.
The expected bounds on Λ are then extracted from the randomly distributed range of best fitted values of Λ that minimize the χ 2 -test of Eq. (7) (an example is shown in Appendix B). We use three LHC integrated luminosity scenarios: L = 140, 300, 3000 [fb] −1 , corresponding to the currently accumulated LHC plan, the RUN3 projections and the planned HL-LHC luminosity, respectively. Then, based on the SM cross-sections, we demand at least 100 events for any of the channels X, Y ∈ (m, n, p), i.e., σ SM · L > 100, or else this channel is not included in the χ 2test of Eq. (7). This 100 event criterion is set to ensure that the potential reducible backgrounds (see discussion below) will be sub-leading and, therefore, have a small impact on the overall uncertainty in these measurements.
We demonstrate below our formalism for detecting LFNU based on the ratio observables of (2), using the QCD generated (and therefore dominant) exclusive dimuon + multi-jet and/or top-quarks channels: 3 where the LFNU effects are generated by the operators O  (2) are assumed to be SM-like). We can then define a generic form for the cross-section in (10) with a cut m > m min on the di-muon invariant mass: where σ INT and σ NP are the SM×NP interference and NP 2 terms, respectively. The dominant NP contribution then depends on the di-lepton invariant mass cut and the dilepton channel involved. In particular, the O(Λ −2 ) correction, σ INT , dominates for moderate di-lepton invariant mass cut, for which the SM term is appreciable, whereas the O(Λ −4 ) NP 2 correction, σ NP , is dominant in the high m min -cut regime, where the SM contribution is suppressed. 4 All cross-sections contributing to the LFU T -tests in (2) were calculated exclusively (i.e., separately for each channel without matching) using Mad-Graph5_aMC@NLO [39] at LO parton-level, and a dedicated universal FeynRules output (UFO) model for the EFT framework obtained using FeynRules [40]. In addition, the LO MSTW 2008 parton distribution functions (PDF) set (MSTW2008lo68cl [41]) in the 5 flavor scheme was used with a dynamical scale choice for the central value of the factorization (µ F ) and renormalization (µ R ) scales, corresponding to the sum of the transverse mass in the hard-process. Finally, kinematic cuts (e.g., on the di-lepton invariant mass) were imposed using MadAnalysis5 [42].
Using the four di-lepton + jets channels in (10), we show in Fig. 1 and Table I a (2233), as a function of the di-muon invariant mass cut m min . In particular, the Monte Carlo χ 2 -test analysis of LFNU was repeated for different values of m min , for the three integrated luminosity cases L = 140, 300, 3000 [fb] −1 and the three systematic uncertainty cases, which yield an overall uncertainty of δT ∼ 10%, 15%, 20% for the di-muon multijets production channels (010) µµ , (110) µµ and (020) µµ , i.e., with no top quarks in the final state. The (002) µµ di-muon + top-pair production channel in (10) was not included in the χ 2 -test analysis used to derive the 95% CL bounds listed in Table I Table. This process, i.e., pp → + − + tt, is, however, an important channel that might prove to be a promising direction for the future for disentangling various other types of NP effects, e.g., in leptoquark searches [43,44].
We see that, as expected, the sensitivity to the underlying NP depends on the sign of the Wilson coefficients η f = ±1 and on the overall uncertainty, and it varies    We now consider a complementary analysis where, instead of examining the bounds under the assumption of no NP in the data, we ask what is the discovery potential of a given NP scenario if the NP is assumed to be present in the data. We thus assume that the experimen-tally measured ratios T X,exp are controlled by the NP, so that, in this case, they are normally distributed with a mean value corresponding to the NP expectations T (Λ): where hereΛ is the value of the NP scale injected into the data and tested against the SM prediction. As for the overall uncertainties, δT X , we follow the prescription of (9), where this time the statistical uncertainties are assumed to reflect the NP data, i.e, δT X stat = 2/N X (Λ), where N X (Λ) are the number of events expected for a NP scaleΛ in each of the di-lepton channels X ∈ (mnp). The systematic uncertainties, δT X sys , are kept unchanged, i.e., using the three cases δT X sys = 5%, 10%, 15% for channels involving only light-jets and/or b-jets in the final state. We thus vary Λ in the χ 2 -test of (7) [i.e., with T exp normally distributed around T (Λ) following (12)], from which we generate the distribution of the best fitted NP scale,Λ, for each value ofΛ. 5 This is repeated for different values ofΛ until we find the value that yields a distribution which deviates from the SM prediction at a given CL; we denote this value byΛ(CL). In Table  II we list a sample of our results for the discovery potential of the operators O In particular, we find that a 5σ discovery of the heavy underlying NP that generates these operators can be obtained at the LHC with L = 300 fb −1 , if its scale is in the rangeΛ(5σ) ∼ 2.3 − 2.9 TeV for O At the HL-HLC, the corresponding discovery potential is extended up toΛ(5σ) ∼ 3.7 − 4.6 TeV.
Let us briefly address the potential background for the multi-jets + − production channels (010) µµ , (110) µµ with η f = +1 and O qe with η f = −1, at the LHC with L = 300 fb −1 and m min µµ = 400 GeV and at the HL-LHC with 3000 fb −1 and m min µµ = 700 GeV. Numbers are given for the three different overall uncertainties corresponding to cases 1,2,3 of the systematic uncertainties and the channels that pass the 100 criteria for all cases are (010) µµ , (110) µµ and (020) µµ . See also text. and (020) µµ used in our χ 2 -tests. We note that the irreducible background to these processes such as Z + jets and W + jets production, are included in our calculation since they interfere with our signals. The dominant reducible background for the (010) µµ and (110) µµ channels, pp → µ + µ − + j and pp → µ + µ − + j + j b , are single top + W-boson (tW : pp → tW ) and vector-boson pair production (V V : pp → V V ). On the other hand, for the (020) µµ channel pp → µ + µ − + 2 · j b , the tW and V V background are sub-leading, and the dominant SM reducible background is from top-quark pair production (tt: pp → tt), which is more challenging. However, as opposed to our leading di-lepton signals (010) µµ , (110) µµ and (020) µµ , the reducible background processes, tW, V V and tt, involve large missing energy, which is carried by the neutrinos in the final state. Thus, they can be significantly suppressed with a proper selection cut on the missing transverse energy E T and other acceptance criteria such as lepton isolation criteria and properties of the transverse momenta of the final state particles which can be used, e.g., for a better separation of the tt background from the di-lepton + jets NP signal, see e.g., [45]; see also [46], where the effects of the + −b b contact terms on the (020) µµ channel are considered and a detailed signal to background optimization is performed, which included the tt background.
Finally, we note that an important difference between the two operators O (3) lq and O qe with respect to our χ 2test, is that the former also gives rise to the single-muon + jets and top-quarks production channels (mnp) µ [cf.
(1)] and to neutrino pair-production (with E T signature) in association with jets and top-quarks: (mnp) E T : pp → m · j + n · j b + p · t + E T . In particular, these (mnp) µ and (mnp) E T processes are correlated with the di-lepton + jets production channels (mnp) µµ discussed above. For this case also the best sensitivity to O (3) lq (2233) is expected from the processes generated by QCD interactions, which are depicted in Fig. 2. In particular, note that the QCD-generated single-muon production channels involve a single top-quark in the final state: (001) µ , (101) µ and (011) µ , which affect the ratios T 001 µ , T 101 µ and T 011 µ in (2). Indeed, we find that including these singlemuon + top-quark channels in the χ 2 -test of (7) yields a better sensitivity to the scale of this operator, e.g., for the HL-LHC case, yielding bounds which are ∼ 1 TeV stronger than the ones given in Table I. A detailed analysis which includes the single-lepton channels will be presented in [47]. Furthermore, the dominant (QCD generated) neutrino channels are the processes (010) E T , (110) E T , (020) E T and (002) E T (see Fig. 2), which can be used as well to obtain a better sensitivity to this operator. This requires, however, a different approach (rather than our χ 2 -tests based on ratio observables) for disentangling the NP effects and will, therefore, not be further investigated here. Note, though, that some of these E T + jets and/or topquarks signals are important signals of other well motivated NP scenarios. For example, the processes (020) E T and (002) E T , i.e., pair production of top-quarks and/or b-jets in association with large E T , are also signatures of leptoquark pair-production (see e.g., [44,48]), of pairproduction of the scalar partners of the top or bottom quarks in supersymmetric theories (see e.g., [49]) and may also be useful for dark matter searches (see e.g., [50,51]).
The approach described above improves on the results obtained in previous interesting studies which are based on the analysis of a single process. For example [36] obtains limits of Λ > 1.  (5) and (6), us-ing Drell-Yan di-lepton production qq → + − ; they find a slight improvement for 4-Fermi operators of type O(1133). Note that the current best bound for this last type of operators, i.e., O(1133), was obtained at LEP [52][53][54][55]: Λ > 0.7 − 2.7 TeV. We also note that bounds on O (3) lq derived from the top-quark decays are significantly weaker [56,57] than ours.
To summarize, we have shown that the lepton flavor non-universal ratio observables T mnp and T mnp of (2) can be used to search for new physics using a χ 2 test that is sensitive to the correlations among several lepton + jets and top-quark production channels. We found, for example, that with a realistic assessment of the expected uncertainties involved, a 95%CL bound of Λ > ∼ 3 − 4 TeV can be obtained with the current LHC luminosity, while Λ > ∼ 6 − 7 TeV is expected at the HL-LHC, for the 4-Fermi operators O  (5) and (6), which involve 2nd generation leptons and 3rd generation quarks. These bounds are obtained with a generic di-lepton invariant mass cut for all channels, i.e., without any channel-dependent (specific) optimizations that, we believe, can be further used to better isolate the NP effects and, therefore, to obtain an enhanced sensitivity to its scale. Though the above discussion involves only 3rd generation quarks, our multi-channel correlation LFU tests are expected to yield an improved sensitivity also for lepton flavor non-universality new physics which involves the 1st and 2nd quark generations.