Probing flavor constrained SMEFT operators through tc production at the Muon collider

: We investigate flavor violating four-Fermi Standard Model Effective Field Theory (SMEFT) operators of dimension-six that can be probed via tc (¯ tc + t ¯ c ) production at the multi-TeV muon collider. We study different FCNC and FCCC processes related to B , B s , K and D decays and mixings, sensitive to these operators and constrain the corresponding couplings. The tensor operator turns out to be most tightly bound. We perform event simulation of the final state signal from tc production together with the SM background to show that operators after flavor constraint can reach the discovery limit at 10 TeV muon collider. We further adopt the optimal observable technique (OOT) to determine the optimal statistical sensitivity of the Wilson coefficients and compare them with the flavor constraints. We use the limits to predict the observational sensitivities of the rare processes like K L → π 0 ℓℓ , D 0 → µµ , t → cℓℓ .


Introduction
Search for physics beyond the Standard Model (BSM) at the Large Hadron Collider (LHC) and preceding experiments has not provided a conclusive result yet despite extensive search and data analysis.This suggests that BSM physics is either weakly coupled, or exists at energy scales considerably removed from the electroweak scale, or possesses signals unaccounted from the standard searches done so far, or all of them.Therefore exploring BSM physics via Standard Model Effective Field Theory (SMEFT) has gained considerable attention in recent times.The construction of effective theory assumes the knowledge of only low scale theory (SM here), so that the Lagrangian involving higher dimensional operators can be written as [1][2][3]: where O i 's are the effective operators constructed out of the Standard Model (SM) fields respecting the SM gauge symmetry, C i 's are the new physics (NP) couplings or Wilson coefficients (WCs), through which the effect of BSM scenarios can be realised.Λ denotes the NP scales integrated out, which has power (n − 4) depending on the mass dimension n of the operator O i under consideration.SMEFT has been studied extensively via operators of dimension-five [1], six [2,3], seven [4,5], and eight [6,7] to estimate NP deviations in a model-independent way.In this study, we will use the dimension-six SMEFT operators that contribute to tc ( tc + tc) production at collider1 after taking the flavor constraints into account.
As we know that the renormalization-group equations (RGE) allow one to compute the running and mixing between the BSM scale down to the electroweak scale and further down to the scale of low-energy precision experiments, the SMEFT can be matched to a low energy EFT.It could be done following a two step matching procedure.The coefficients C i 's generated at the scale Λ will be related to their values at the electroweak scale v ∼ 246 GeV, and these coefficients can further be related to the WCs of a low-energy EFT through RGE running.Therefore, any constraints obtained on the WCs from low energy data will in turn limit C i (Λ)'s at any given BSM scale Λ.The flavor changing neutral current (FCNC) processes are loop suppressed in the SM, and any tree level NP contributions to such processes will have limited parameter spaces allowed by the data.Similarly, the availability of precise data on the flavor changing charged current (FCCC) processes at the low energy could play an essential role to constrain the NP parameter space contributing to such processes at the tree level.Therefore, study of the FCNC and FCCC processes at the low energy will play a crucial role in exploring flavor violating NP effects under the SMEFT framework.
FCNC processes such as b → s(d) transitions are one of the most important probes in this connection, as there is no such interactions within the SM at the tree level 2 .Such flavor observables therefore provide an important bound on the corresponding SMEFT operators, see for example, [11][12][13][14][15][16].In this study, we focus on the observables related to the semi leptonic and leptonic decays of B, B s , K mesons via the processes b → s(d)ℓ + ℓ − and s → dℓ + ℓ − where the data is available only for ℓ = µ or e.In addition, we have considered the available data on B − B and B s − Bs mixing amplitudes to constrain the NP parameter spaces.Among the FCCC processes, we mainly consider the available data on semi leptonic and leptonic decays of B, K, D and D s mesons via b → c(u)ℓ + ν, s → uℓ + ν and c → d(s)ℓ + ν transitions, respectively.
Along with flavor observables, top-quark physics also plays an important role under SMEFT framework as it has O(1) Yukawa coupling, which is crucial to explain the origin of electroweak symmetry breaking (EWSB).Several analyses under SMEFT framework in the topquark sector have been performed in [17][18][19].This motivates us to choose tc(tc) production as an example process to study.
The stringent constraint on the dimension-six effective operators derived from the flavor observables indicate that their production at the LHC is small, while the SM background is huge and mostly irreducible, so that a prediction requires thorough analysis.If we choose the conservative limits, it is also difficult to probe them in a future electron-positron collider with a maximum centre-of-mass (CM) energy of 3 TeV.Therefore we examine the process at high energy muon collider [20].Muons, being fundamental particles, provide the advantage of directing their entire energy towards short-distance scattering rather than having it distributed among the partons.As a result, a 14 TeV muon collider can exhibit effectiveness on par with a 100 TeV proton-proton collider [21].This high-energy capability is particularly advantageous for both the exploration of new heavy particles along with indirect measurements at elevated energy levels.Hence, a multi-TeV moun collider acts as both discovery and precision machine altogether.Limited studies on tc production at lepton colliders have been documented in the existing literature [22][23][24][25].In a somewhat similar analysis [25], the flavor constraints are obtained from rare B decay processes, applicable to the vector operators only, while ours is done considering all possible flavor observables applicable to all of the vector, scalar and tensor type four-Fermi operators.
The paper is arranged in the following way.In Section 2, we discuss the relevant phenomenological framework for our analysis.Then in Section 3, we study the constraints from flavor physics, followed by collider simulation at µ + µ − collider in Section 4. Section 5 elaborates OOT framework and the optimal limits on flavor violating dimension-six effective couplings.In Section 6, we discuss the collider probe of scalar operator contributing to 3 → 2 (hadronic) transition at e + e − machine.Finally, we summarize our conclusion in Section 7.

Phenomenological framework
Our goal of the analysis is to probe the dimension-six flavor violating effective (EFT) operators that contributes to tc ( tc + tc) production at the future muon collider.The Feynman graphs for the tc production are shown in figure 1 ( tc graphs are similar with same contributions and not shown here).There are essentially two different contributions: (i) via Ztc vertex and (ii) four-Fermi µµtc contact interaction.As elaborated later, operators contributing to Ztc affect the process very mildly, while the main contribution arises from the four-Fermi operators.
Feynman diagrams that induce t c production at the muon collider; left: effective four-Fermi (µµtc) couplings, right: effective Ztc couplings.The conjugate processes look alike with exactly same cross-section.
There are three types of four-Fermi operators containing i) four leptons, ii) four quarks and iii) two leptons and two quarks, where the last one serves our purpose.There are seven operators that contribute to the µ + µ − → tc production, as given by [3,24], O where l (e) is the left (right)-handed lepton doublet (singlet), q is the left-handed quark doublet, and u (d) is the up (down)-type right-handed quark singlets.Apart, p, r, i, j indicates flavor indices, τ I are the Pauli matrices, Using Eq. (2.1), the most general four-Fermi effective Lagrangian for µµtc contact interaction can be written as, where vector-like (V st ), scalar-like (S st ) and tensor-like (T st ) couplings can be expressed in terms of the WCs of the seven four-Fermi operators as in Eq. (2.1) as, (2.3) Note that for vector operators we have all the helicity combination of fermions appear for both lepton and quarks, while for the scalar and tensor couplings only RR combination is non zero.In Eq. (2.2), the S LL and T LL also get a matching condition from the hermitian conjugate SMEFT operators, leading to −C (2.4) Therefore, the effective Lagrangian containing Ztc vertex3 is, where, P L(R) = 1−(+)γ 5

2
, a L and a R are the functions of C (1)ij ϕq , C (3)ij ϕq and C ij ϕu as, where, c w = cos θ w , θ w is weak mixing angle.

Flavor constraints and predictions
In this section, we study all possible flavor and top quark related experimental constraints that put bound on the dimension-six effective couplings of our interest as described above.
We then make some observable predictions for some rare processes based on the constraints obtained for the effective operators.

Flavor and top-quark constraints
Note that the operators defined in Eq. (2.1) will contribute to various low energy observables related to the FCNC and FCCC processes.In the following we will discuss different inputs which we have considered in this analysis.

FCNC processes:
Most important constraints will be obtained from the semileptonic and leptonic decays of via b → sℓ + ℓ − decays.The corresponding low energy effective Hamiltonian is given by [42,43] where ℓ stands here for all the three lepton fields: e, µ and τ .Here, e 0 and G F are the U (1) em and Fermi coupling constants, λ bs t = V tb V * ts is the CKM combination.The four-Fermi operators as in Eq. (3.1) are expressed as,  (2.1).Following a tree level matching procedure, one can express the WCs defined in Eq. (2.1) in terms of the WCs of in the low energy effective Hamiltonian given above [44].
• b → dµµ decay: b → dµµ transition is another FCNC process which also plays an important role to constrain NP scenarios [62,63].Like b → sµµ transition, here also O (1) lq , and O qe operators contribute.Consequently, relations between the WCs in two different parametrizations are similar to that in Eq. (3.3), just with a trivial substitution of λ bd t = V * tb V td and appropiate indices on WCs.We utilize the experimentally determined value of ∆C µ 9 = −0.53from the global fits of , and radiative B → X d γ decays data sourced from [63], to relate the dimension-six effective couplings of our choice (Eq.(3.3)).The constraints from this decay channel for these three dimension-six effective couplings are tabulated in Table 1.
• K L → µµ decay: The expression for the branching ratio of K L → µ + µ − decay using the Hamiltonian in Eq. (3.1) can be expressed as [64,65], (3.4)In the above Eq.(3.4), τ L represents the mean life of the K L meson, m µ and m K denote the masses of the muon and kaon respectively, G F stands for the Fermi constant, and β µ and λ sd q are defined as: lq and O qe that contribute to this observables.∆C µ 10 respects Eq. (3.3).Resulting limits on dimension-six effective couplings are written in Table 1.
• B q − Bq mixing (with q = d, s): B 0 or B s mesons exhibit oscillatory behaviour between particle and antiparticle states, a phenomenon arising from the influence of flavor-changing weak interactions.These meson-antimeson mixings are the ∆B = 2 FCNC processes.In presence of NP, the mixing amplitude or the frequency of mesonantimeson oscillations is defined as, with The dominant contribution to M q,SM 12 in the SM arises from the top mediated box diagrams [68][69][70], and the corresponding contribution is given by, We already mentioned that the SMEFT operators in Eq.
The expression of DisB[a,b,c] function in the above equation is written in the appendix.As we have discussed earlier, the contributions to b → sτ + τ − will come from O (1) lq and O qe .The amplitude is calculated using the Hamiltonian described in Eq. (3.1).Therefore, effective couplings related to these operators obey Eq. (3.3).The relevant constraints from this observable are presented in Table 1.

• B 0
s → µµ decay: Following Eq. (3.1), the branching ratio of The branching ratio of this rare decay is 2.69 lq , and O qe contribute to this branching ratio.Using Eq. (3.3), the constraints obtained on the effective couplings for the relevant operators are tabulated in Table 1.

FCCC processes:
• b → qℓν ℓ decay: The most general effective Hamiltonian for b → c(u)ℓν ℓ transitions is written as [82], with the operators where q = u or c.If we match the ones in Eq. (3.17) with those mentioned in Eq. (2.1) for b → cℓν ℓ transition, we find the following relations4 among the couplings [44], (3.19) FCCC processes like b → qℓν have been widely studied on [81,[83][84][85].In this analysis, we borrow the constraints from [81], where the experimental results of B → D(D * )ℓ − νℓ , R(D ( * ) ), B → πℓ − νℓ (ℓ = µ, e) along with lattice inputs are taken into account.We find that from b → cµν µ and b → uµν µ decays, the constraints on GeV −2 respectively.As the uncertainties are very high here, we may infer that the corresponding operator is insensitive to these decays and zero consistent.O (3) lequ is also insensitive for these two decay modes for the same reason.
• P → ℓν ℓ decay: Leptonic decays from pseudoscalar mesons (P ) provide one of cleanest probes to constrain NP.The hadronic matrix elements for these decays with different Lorentz structures are defined as [86], where m P , f P , and m q 1(2) are the mass, decay constant, and mass of constituent quarks of P -meson, resepectively.These inputs for different pseudoscalar mesons are taken from the FLAG review [71].O (3) lequ operators contribute to these decays.Using the effective Hamiltonian in Eq. (3.17), the branching ratio of P → ℓν ℓ is expressed as; (3.21) where, τ P is the lifetime of P meson, m ℓ is the mass of the lepton (ℓ), and V q 1 q 2 is CKM matrix element.C V 1 and C S 2 follow same relations as expressed in Eq. (3.19).Constraints on C V 1 and C S 2 couplings are tabulated in Table 3.Using these observables, we note the constraints on C i /Λ 2 in Table 1 and 2 following Eq.(3.19).
• t → bℓν ℓ decay: Along with the flavor observable described above, semi leptonic decays of top quark is instrumental in constraining NP.The t → bℓν ℓ decay gets contribution from few operators in Eq. (2.1), thereby imposing constraints on the associated dimension six effective couplings.This decay process is governed by W mediated charge current interaction within SM whereas for NP contribution arises from O (3) lequ , and O (3) lequ operators.The total decay width (Γ t ) for this process is given by, where m t is mass of top quark, E b and E ℓ are the energy of b-quark and lepton (ℓ), respectively.SM and NP contribution to the total amplitude are written in the appendix.Constraints on O (3) lq from this decay is tabulated in Table 1.For scalar (C (3) ℓequ /Λ 2 ) and tensor (C ℓequ /Λ 2 ) mediated couplings, orders of the constraints are 10 −3 and 10 −4 respectively.For t → cℓℓ decay (Eq.(3.23)), upper limit of these operstors is 10 −5 .Therefore constraints obtained from t → bℓν on C (1) ℓequ /Λ 2 and C (3) ℓequ /Λ 2 are ruled out from t → cℓℓ decay.As our interest lies mostly in 3 → 2 transitions, WCs of the contributing operators are more constrained from present data involving electron or muon final states compared to tau.However, both electron and muon final states provide similar bounds.Therefore, we do not consider lepton generation indices in the WCs and add indices pertaining to quark sectors only.Utilizing experimental data from various flavor observables and top quark measurements, individual constraints on different WCs are presented in column three of Table 1 and 2. Simultaneous fit is performed to constrain the WCs, incorporating all available flavor violating observables pertaining to specific i → j transitions, see column four of Table 1 and 2. If we further perform a simultaneous fit by considering all the flavor violating contributions, then the constraints on vector, scalar, and tensor couplings turn out to be: These bounds are of similar order for vector and scalar couplings but more stringent for the tensor one, all of which we probe at the muon collider (Section 4).We would like to note however, that the scalar coupling from 3 → 2 transition are less constrained, C (1)32 lequ /Λ 2 ∼ 10 −7 GeV −2 , see in Table 2. Therefore, it can be individually proved with less CM energy (preferably at e + e − colliders) which we discuss in Section 6.However, with more data for 3 → 2 transition in future, C (1)32 lequ /Λ 2 constraints could reach above mentioned simultaneous fit bound.Note that no suitable observable has been found to constrain O lu and O eu operators, resulting an upper bound on them from t → cℓℓ decay, as pointed out in Table 1.We make predictions for t → cℓℓ transition and other flavor observables as we discuss next.

Prediction on different observables
Using the constraints on the EFT operators obtained in Table 1, Table 2 and Table 3, we provide future prediction of following observables, with numerical estimates noted in Table 4.
• B(t → cℓℓ): The branching ratio of t → cℓℓ decay is given by 5 , As of now, there hasn't been any specific experimental searches conducted to observe t → cℓℓ decays.However, an indirect upper limit on t → cℓℓ decays can be obtained by t → Zq searches at ATLAS [87].At 95% C.L., bounds on t → cℓℓ branching ratios are [19], These upper limits of the branching ratio provide upper bounds on the dimensionsix effective vector and scalar operators on the order of 10 −5 whereas for tensor operator the upper bound is 10 −7 .Constraints determined from experimental inputs on dimension-six effective couplings listed in Table 1 and 2 respect this upper bound.
Using the obtained constraints on dimension-six effective couplings, we provide the prediction on this observable in Table 4.
• B(K L → π 0 ℓ l): The branching fraction of K L → π 0 ℓ l can be expressed as [88,89], The details of a s , C ℓ dir , C ℓ int , C ℓ mix and C ℓ γγ are discussed in [88,89].Constraint on C (1) lq /Λ 2 and C (3) lq /Λ 2 couplings provide prediction on this observable.• B(D 0 → µµ): Branching ratio of D 0 → µµ decay is noted as [90], with, (3.28) Prediction on different observables as noted in Table 4 suggests that all the experimental predictions on dimension-six effective couplings are consistent with existing upper bounds.Table 4: Prediction of different observables using more conservative flavor contraints WCs from the simultaneous fit using all possible the flavor violating contributions and existing upper bound.

Collider analysis
In this section, we analyze the possibility of probing the NP at collider in terms of the effective operators as in Eq. (2.1).For that we focus on those most stringent limits of the dimension-six effective operators stemmed from the flavor observables.If the most stringent limits on the NP couplings can be probed for a given √ s then the others can be probed at a smaller √ s.From Section 3.1, we see that the flavor constraint on C (3)32 lequ /Λ 2 , the tensor coupling is one of the most stringent ones which are on the order of ∼ 10 −11 GeV −2 followed by a conservative estimate of C (1)32 lequ /Λ 2 ∼ 10 −9 GeV −2 and C (1)32 lq /Λ 2 ∼ 10 −9 GeV −2 .Therefore, in our subsequent analysis, we will focus on these three WCs mostly.From Eq. (4.2), it is clear that the signal cross-section exhibits a linear growth with the square of the CM energy, conversely, SM background processes are expected to decrease as √ s increases.Therefore, at high √ s, a muon collider should have the capability to detect a such NP scenarios where the CM energy of the machine is expected to go upto 30 TeV.For our analysis, we consider √ s = 10 TeV with an integrated luminosity (L int ) of 1 ab −1 .This choice is within the reach of the future muon collider projections [20].We further note that when we are probing C/Λ 2 ∼ 10 −11 GeV −2 at √ s = 10 TeV, we assume, Λ ≳ 10 TeV, so that the WC C ≳ 10 −3 , to keep the effective theory framework validated.As mentioned before, the constraint on scalar operator C (1)32 lequ /Λ 2 from 3 → 2 transition observable is less stringent, therefore, it can be probed at the electron-positron colliders at much lower CM energy, which we discuss in Section 6.

tc ( tc + tc) production cross-section
The differential tc( tc) production cross-section at µ + µ − collider, governed by the effective four-Fermi contact interaction as shown in figure 1, in terms of vector, scalar and tensor couplings is given by 6 , where . Therefore, total production cross-section is, The variation of the signal cross-section (σ prod ) for the vector, scalar and tensor four Fermi operators used in this model with CM energy ( √ s) is shown in the right side of the figure 4. Here, Λ = 50 TeV and WCs (C = 1) are kept constants.From Eq. (4.1) we see that the total cross-section (σ prod ) is proportional to s/Λ 4 , therefore, larger CM energy yields larger cross-section, provided we are in the effective theory limit, i.e. √ s < Λ.We also note from figure 4, that the production cross-section is largest for the tensor coupling and the lowest for the vector coupling.For vector coupling, the cross-section ∼ 0.1 fb at √ s = 5 TeV and that is down to 0.001 fb at √ s = 1 TeV for C/Λ 2 ∼ 10 −10 GeV −2 , making it a necessity to probe these couplings at √ s = 10 TeV.The variation is shown in one operator scenario, that implies σ prod ∝ C 2 i /Λ 4 .Therefore the variation of total cross-section with C/Λ 2 is symmetric as depicted in the left side of figure 1. Apart from the four-Fermi couplings, Ztc couplings (a Z L and a Z R ) via O (1) ϕq and O ϕu also contribute to tc production by interfering with the four-Fermi vector couplings V ij .The contribution of Ztc couplings can be incorporated to the total cross-section by redefining, handed electron, respectively.On contrary to four-Fermi couplings, the cross-section via Z mediation drops as ∼ 1/s due to the s-channel mediation.In the left side of figure 4, we show the variation of σ prod with varying C/Λ 2 in GeV −2 .From the figure, we again see that the tensor mediated four-Fermi coupling (T RR ) provides the maximum contribution to the tc production and vector mediated production provides the least.In one operator scenario, all the vector operator contribute equally to the total cross-section.If we assume Ztc effective couplings to be of the same order to that of vector coupling, C/Λ 2 ∼ 10 −9 GeV −2 , then the contribution from Ztc is way milder than the four-Fermi operator contribution, see Table 5, where we see that the production cross-section (σ prod ) for vector coupling surpasses that of Ztc coupling by an order of 10 8 in fb at 10 TeV muon collider.Therefore in the following analysis we neglect the contribution of Ztc and study the four-Fermi operators.
We refrain from studying dimension-eight contributions, which in principle could contribute at the same order as our considered dimension-six contributions.

Signal and background processes
In high-energy muon colliders, collision events take place at a CM energy surpassing that achieved by the present LHC (parton level) or potential future electron-positron colliders.Consequently, it is anticipated that jets which are closely clustered together in these collisions, will exhibit collimation, effectively coalescing into a singular, consolidated jet.A notable illustration of this behavior can be observed in the jets stemming from the hadronic decay processes of top quarks or W/Z bosons, where they converge to form a single "top" or "W/Z" jet [95].Since, the signal process is µ − µ + − → tc (by tc production we always mean tc + tc production), for this analysis, we consider a di-jet signal with no leptons.The corresponding SM background processes are
The lepton isolation criteria is set ∆R > 0.5 from another lepton or jet.
The jet radius is taken to be 0.5 with the minimum p T of the jet set as 20 GeV.The jet reconstruction is done using Delphes3.

Cut based Analysis
For the analysis, we use the following sequential cuts for the signal and background processes: • Cut 1: N j = 2 and N l = 0.
Here, M jj is the invariant mass of the di-jet, M h j is the invariant masses of the reconstructed particles of the heavier jet.The major background, in case of our analysis, are the diboson processes.In Cut 1, which is our signal selection cut, we demand only processes with 2 jets.We further remove events with detected leptons.This will remove the detected leptonic and semi-leptonic decay processes for W + W − , ZZ as well as tt.The invariant mass of the di-jet pair is expected to peak near the CM energy of the incoming particles for jets coming from the production level processes.For jets coming from branching or radiation, the invariant mass is expected to peak at a lower value.Imposing Cut 2, we can remove further semileptonic and leptonic processes from di-boson production, where the leptons were not detected.Now, due to the collimation of boosted jets in high energies, as we mention above, we expect multiple jets branching from top or W/Z to appear as a single jet in the detector.The invariant mass of the reconstructed particles of such jets are expected to peak at the mass of their respective sources, i.e. top mass or W/Z mass.In tc signal process, the heavier jet is expected to peak at the top mass, as such Cut 3, will significantly remove W + W − and ZZ backgrounds as evident from the distribution in figure 5.The signal can be further separated from the background by using a charm tagging algorithm on the lighter jet of tc signal process.However, with the current c-tagging efficiency at the LHC [102,103], it looks like a far-fetched possibility.This is primarily due to high miss-tagging efficiency of the b-jets as c-jets.After employing sequential cuts as mentioned above, signal significance S/ √ S + B and efficiency factor are noted in Table 6.We see that the benchmark point with vector couplings have most signal significance, while the tensor one has the least.6: Cut flow and signal significance for signal at the chosen benchmark points and background processes for analysis at muon collider at 10 TeV and 1 ab −1 .Here ϵ s/b denotes the cut efficiency for signal and background following the final cut (Cut 3).

Optimal Observable Technique
The optimal observable technique (OOT) is a statistical tool that enables optimal estimation of NP couplings via minimizing the covariance matrix, which has been used studied for different cases, but mostly without a non-interfering SM background [26][27][28][29].In general, a collider observable (e.g., differential cross-section) contains contribution from both the signal and background.If we wish to write the observable at the production level having the cut efficiencies of signal analysis embedded, then it can be written as, Here g i are the non-linear functions of NP couplings and f i are the function of phasespace co-ordinate ϕ and ϵ s (ϵ b ) is the signal (background) efficiencies in estimating the signal (and background) after using judicious cuts as tabulated in Table 6.In this analysis as our focus on 2 → 2 processes, the phase space co-ordinate will be ϕ = cos θ, where θ is angle between the outgoing particles with respect to the beam axis.However, the choice of ϕ can vary depending on the specific process.We importantly note that the signal of the processes under consideration are essentially di-jet events (as elaborated in the previous section), stemming from the produced particles in the high CM energy of the muon collider, enabling us to use Eq.(5.1), without much problem.However, for processes where the decay products are not collimated, say for example, at smaller CM energy, the usage of Eq. (5.1) is limited.For a generic procedure of inclusion of SM background, see [104].
The essential idea is to determine the NP coefficients g i as precisely as possible.In case of a realistic experimental scenario, the event numbers obey Poisson distribution, we can estimate g i with the application of appropriate weighting functions (w i (ϕ)), There's one particular choice of w i (ϕ) that optimizes the covariance matrix (V ij ) in a sense that statistical uncertainties in g i 's are minimal.In this case, V ij is expressed as: O(ϕ) dϕ. ( Then, the optimal covariance matrix reads with σ T = O(ϕ)dϕ defining the total cross-section and N is total number of events (N = σ T L int ) including the SM background contributions.L int indicates the integrated luminosity of the collider over a period.The χ 2 function, which quantifies the precision of NP couplings, is defined as: where g 0 's are the 'seed values' of NP model inputs.Here we use them as the ones predicted from the flavor constraints.

OOT Sensitivities
Here, we discuss the optimal uncertainties and correlations of different dimension-six effective operators in context of tc( tc) production.As mentioned before, we focus on the probe of most stringent vector, scalar, and tensor mediated couplings that contribute to the tc production at the µ + µ − colliders.The χ 2 variation dimension-six effective operators are shown in the figure 6.At CM energy √ s=10 TeV and luminosity L int = 1 ab −1 , we observe that for all the couplings, optimal uncertainty is narrower than flavor uncertainty.It is intriguing to note that for the tensor-mediated coupling C (3)32 lequ /Λ 2 , the flavor uncertainty is way larger than the optimal collider sensitivity of this coupling at muon collider (see the right most panel in figure 6).This suggests that the pertinent operator is less-consistent in flavor observable, yet in a collider scenario, it can be measured with larger precision.This is because the tensor operator having most stringent flavor bound, has the largest contribution to the tc production compared to scalar and vector operators, as already evidenced by figure 4 and Table 5.  7.
The correlations between the two vector couplings and correlations among scalar and tensor couplings for different background efficiencies (ϵ b = {0.1,0.05}) are shown in figure 7. The benchmark cases with seed values are listed in Table 7.It is clear that the presence of background via ϵ b plays a crucial role in determining the optimal uncertainty of NP couplings, the lesser the contamination, the better the precision.Moreover, the optimal uncertainty of NP couplings depends on the relative NP signal and non-interfering SM background contribution to the final state.Contribution to the final state from two vectors operators is greater than the scalar and tensor operators together by a factor of 25 (Table 6), due to the choice of the seed values of the benchmark points.Therefore, the relative NP signal is larger than the background contribution for the vector operators than the scalar and tensor ones.That is why the vector operator correlation is less affected by the change in background contamination as shown in the figure 7.  7.
In figure 8 we show the variation of 1σ regions for different CM energies and integrated luminosities.In our scenario, the increase of CM energy ( √ s) is more effective than the increase of integrated luminosity.If we increase the CM energy twice, then the signal crosssection is increased by a factor of 4 which in turn suggest four times enhancement in signal events.However, we must remember that when we enhance √ s, we choose Λ > √ s to be consistent with EFT framework, reducing the WC appropriately to keep C/Λ 2 is same ballpark.Larger CM energy also reduces the non-interfering SM backgrounds.However, if we double the luminosity, both the signal and background events will enhance twice, suggesting larger CM energy helps reducing optimal uncertainty for EFT frameworks as considered here.
In figure 9, we show the optimal statistical separation of different cases listed in Table 7 from the 'base model' SM.Here, Case-I and Case-II are followed from flavor constraints noted in Table 1 and    Case-I, Case-II, and Case-III are 28.18σ(24.93σ), 76.97σ (1.18σ), and 87.34σ (23σ) away from SM respectively, so that the distinction of vector operators is comparatively easier than the scalar and tensor operators once we adhere to the benchmark points respecting flavor constraints.Estimated values of vector, scalar, and tensor couplings are noted in Table 8 for different CM energies and luminosities at 5σ separation.To achieve the 5σ separation for tensor operator with current flavor bound, we require 18 ab −1 integrated luminosity for 10 TeV CM energy.

Sensitivity of "flavor-relaxed" C
(1)32 lequ /Λ 2 at the 500 GeV ILC As seen from Table 2, the simultaneous fit on the scalar-mediated WC from 3 → 2 transition observables provides a less stringent bound compared to the other cases or when we combine all the flavor-violating observables together.Specifically, the order of C (1)32 lequ /Λ 2 from 3 → 2 transition observables is ∼ 10 −7 GeV −2 , whereas the best fit value from combining all the flavor-violating observables is of the order ∼ 10 −9 GeV −2 .Therefore, C (1)32 lequ /Λ 2 can be probed at a much lower CM energy preferable at the electron-positron machine such as International Linear Collider (ILC) [105,106].In this segment, we provide analyse probing this operator at the ILC with √ s = 500 GeV.At this CM energy, the outgoing particles are less boosted and hence collimation of decay products, as observed at multi-TeV colliders, is less likely to occur.Hence, an untagged di-jet analysis in such scenario will not be much helpful to segregate the signal processes from the huge two jet background possibilities.This motivates us to revert back to the traditional channels emerging from the heavy particle decay.
For tc production, we choose the signal process t(bW (ℓν))c i.e. di-jet (one b-tagged, other c-tagged) plus one lepton with missing energy ( / E).The relevant non-interfering SM background processes are W (ℓν)W (jj), t(bW (ℓν))t(bW (ℓν)) and W (ℓν)jj.  8 .The invariant mass of the outgoing products turn out to be an important discriminator of the signal and background process, as shown in figure 10.The jets (b and c) in W W and W jj are expected to peak at the W pole which isn't the case for tc and tt.Similarly, the top decay products, bℓ are expected to peak around 100 GeV, taking into account the energy carried away by the invisible neutrino.Finally the c-jet from the signal process results from a contact interaction, in contrary to W W and tt, where it is produced in W decay; thus the invariant mass between c and ℓ/b for the signal is expected to be shifted more towards the CM energy, compared to the backgrounds.This works as an important discriminant in segregating the signal from tt background.The event cross-section after subsequent cutflow is shown in Table 9.After all the kinematical cuts, 5σ significance for this specific final state signal can be achieved at 10 fb −1 luminosity with √ s = 500 GeV at the ILC.   2.

Summary and Conclusion
In this paper, we have explored the SMEFT operators that contribute to tc ( tc + tc) production at colliders.The most significant contribution arises from the quark flavor violating four-Fermi operators.The other operator which provides modification to Ztc vertex has much less contribution at high CM energy and have thus been ignored in this analysis.Such four-Fermi operators contribute significantly to low energy flavor dependent processes and are thus heavily constrained by them.We study all such FCNC and FCCC processes including b → s(d) transitions, P → ℓν, K L → µµ, B − B mixing, top decays etc., to constrain the four-Fermi SMEFT operators.The most stringent bound arises on the tensor operator, having C (3)32 lequ /Λ 2 ∼ 10 −11 GeV −2 , followed by a conservative limit on scalar operator C (1)32 lequ /Λ 2 ∼ 10 −9 GeV −2 , and vector operator C (1)32 lq /Λ 2 ∼ 10 −9 GeV −2 (Although, the order is same, scalar operators is slightly more constrained than vector operator).We also predict observational sensitivities of the processes like K L → π 0 ℓℓ, D 0 → µµ, t → cℓℓ and t → cγ from the obtained limits and show that they are consistent with the existing upper bounds.
Using these constraints, we have examined the future probe of these SMEFT operators at multi-TeV moun collider through tc ( tc + tc) production.The required CM energy is 10 TeV where the production cross-section is of the order of fb, respecting EFT constraint Λ > √ s.Given the high CM energy, the process basically yields di-jet final state (a 'top' jet and a 'charm' jet) with no leptons stemming from the top decay.There is apparently a little chance of tagging them as well, which incorporates several SM backgrounds.There exists very little number of variables at disposal to segregate the signal from background, amongst which invariant di-jet mass and invariant mass of the heavy jet plays an important role to achieve a satisfactory signal significance for vector and scalar operator benchmarks in particular, at high luminosity 1 ab −1 .
Using OOT, we have determined the optimal uncertainties of the vector, scalar and tensor type effective couplings at the benchmarks respecting flavor constraints.We see that at 10 TeV CM energy muon collider with 1 ab −1 integrated luminosity optimal uncertainty of the effective C/Λ 2 is better than the flavor uncertainty.Relative contribution between NP signal and non-interfering SM background play an important role in NP estimation, the less the background, the better the estimation.The dependence of the uncertainty on the CM energy and luminosity has also been compared and the advantages of CM energy to estimate optimal uncertainty pertaining to the EFT limit is discussed.Considering SM as a base model, distinction of different operators from the SM has also been studied.We have observed that at 10 TeV CM energy, 30 fb −1 luminosity is required to segregate (at 5σ level) vector (O  The variation of the dimension-six effective couplings with the renormalization scale (µ) is illustrated in figure 12, with comparisons and validation provided by Wilson [111].
lequ , respectively.The above parametrization will be used for the collider simulation later.The tree-level dimension-six EFT operators that contribute to effective Ztc vertex are, O(1)

3 )
lq , and O qe are the three dimension-six SMEFT operators that contribute to b → sµµ transition.Since we have analyzed the constraints by choosing one operator at a time, therefore, for O qe and O (lq operators, ∆C ℓ 9 = −∆C ℓ 10 ; whereas for O (1)

( 2 . 1 )
contribute to the FCNC processes like b → sµ + µ − and b → dµ + µ − .The four-Fermi operators in Eqs.(3.1) and (3.2), may contribute to B s − Bs and B 0 − B0 mixing amplitudes respectively via the diagram shown in figure 2. In our scenario, the dispersive part of the diagram in figure 2 will contribute to M q 12 .The dominant contribution will arise from a τ mediated loop via the insertion of operators O 9 and O 10 .From the dispersive part of the diagram in figure 2, we obtain, m c is the mass of charm quark, M D and f D are the mass and decay constant of D 0 , respectively.Tensor mediated operator does not contribute to this observable as ℓ + ℓ − |O (3) lequ |D 0 = 0. Constraints on operators written in Eq. (3.27) provide prediction on this observable.

Figure 5 :
Figure 5: Normalized event distribution for invariant di-jet mass (M jj ) (left) and heavy jet mass (M h j ) (right) in case of di-jet final state containing signal and background processes at the muon collider with √ s = 10 TeV and L int = 1 ab −1 .
Here, j corresponds to light jets (u, d, s), c-jets and b-jets collectively.Due to the large uncertainty in c and b tagging, miss-tagging of light jets and c-jets as b-jets and light jets and b-jets as c-jets should be taken into account.The tt background contributes as a result of a missing lepton and miss-tagging of b from top decay as c.The event simulation is done in similar lines with Section 4.2.1.The b and c tagging are done using ILCgen Delphes card based on ILC Snowmass projection [106].The b and c tagging efficiencies are 0.702 and 0.315 respectively

( 1 )
lq ) operators from the SM, whereas for scalar (O (1) lequ ) and tensor (O (3)lequ ) operators we need 40 fb −1 and 18 ab −1 integrated luminosity, respectively, after obeying flavor constraints.However, in a more optimistic case of less stringent scalar operator having C(1)32 lequ /Λ 2 ∼ 10 −7 GeV −2 , 5σ segregation can be achieved at 500 GeV CM energy and 10 fb −1 luminosity at the ILC. the respective 1-loop β functions of the SMEFT operators.The β functions are: → s and b → d transitions and translate them in terms of the WCs of our notation as in Eq.

Table 1 :
[49,63]ints on the different dimension-six effective vector couplings from various flavor and top-quark processes and decay modes.For the details on observables concerning b → sµ + µ − and b → dµ + µ − transitions, see references[49,63]and for the measurements see the text.

Table 2 :
Constraints of scalar and tensor mediated dimension-six effective couplings from various flavor observables, see text for details.

Table 7 :
Different benchmark cases of coupling combinations to study optimal correlations and separability from the SM.

Table 8 :
Values of vector (C Here, γ q = 1 2 y u y † u + y d y † d and γ u = y u y † u .We consider, y u = y d = 0 except for y t .The RGEs for the SM parameters are noted below: