FCC-he sensitivity estimates on the anomalous electromagnetic dipole moments of the top-quark

In this paper, we study the production of a top-quark in association with a bottom-quark and a electron-neutrino at the Future Circular Collider Hadron Electron (FCC-he) to probe the sensitivity on its magnetic moment $(\hat a_V)$ and its electromagnetic dipole moment $(\hat a_A)$ through the process $e^-p \to e^-\gamma p \to \bar t \nu_e b p$. Assuming a large amount of collisions, as well as of data with cleaner environments, the FCC-he is an excellent option to study new physics, such as the $\hat a_V$ and $\hat a_A$. For our sensitivity study on $\hat a_V$ and $\hat a_A$, we consider center-of-mass energies $\sqrt{s}= 7.07, 10\hspace{0.8mm}TeV$ and luminosities ${\cal L}=50, 100, 300, 500, 1000\hspace{0.8mm}fb^{-1}$. In addition, we apply systematic uncertainties $\delta_{sys}=0\%, 3\%, 5\%$ and we consider unpolarized and polarized electron beam. Our results show that the FCC-he is a very good perspective to probe the $\hat a_V$ and $\hat a_A$ at high-energy and high-luminosity frontier.


I. INTRODUCTION
Electron-proton (e − p) colliders have been and they continue to be considered as ideal machine to probe physics beyond the Standard Model (BSM). These e − p colliders, such as the Future Circular Collider Hadron Electron (FCC-he) [1][2][3][4][5][6]  of matter over antimatter, the neutrino masses, dark matter and dark energy. In summary, e − p + pp deliver high precision of Higgs boson, top-quark and QCD and electroweak physics complementary to e + e − . Furthermore, e − p is an stimulating, realistic option for a next energy frontier collider for particle physics. For a comprehensive study on the physics and detector design concepts we refer the readers to Refs. [1][2][3][4][5][6].
Next the detection of the top-quark, there has been an enormous motivation to investigate the properties and the potential of top-quark in great detail both in production and in decay.
Increasingly sophisticated experimental results of the current colliders are complemented by very precise theoretical predictions in the framework of the SM of particle physics and beyond. Specifically, the anomalous coupling ttγ, which is the subject of this paper, have been studied in detail in hadron colliders and at a future high-luminosity high-energetic linear electron-positron colliders, for a reviews exhaustive see Table I of Ref. [7] and references therein [8][9][10][11][12][13][14][15][16][17][18].
The SM prediction for the Magnetic Moment (MM) and the Electric Dipole Moment (EDM) of the top-quark, that is a t [19] and d t [20][21][22] reads: The a t can be tested in the current and future colliders such as the LHC, CLIC, the Large Hadron-Electron Collider (LHeC) and the FCC-he. In the case of the d t , its value is strongly suppressed as shown in Eq. (1), and is much too hard to be observed. However, it is very attractive for probing new physics BSM. Furthermore, it is considered as a source of CP violation.
With everything already mentioned, the FCC-he potential of e − p collisions at √ s = 7.07 T eV and 10 T eV and high luminosities L = 50 − 1000 f b −1 , offer one of the best opportunities to test and improve our understanding of the top-quark physics. In special their MM and EDM, and as we already mentioned with the latter considered as a source of CP violation. CP violation can explain why there is more matter than antimatter in the universe, which is a topic of great relevance between the scientific community of particles and fields.
The MM and EDM of the top-quark, that isâ V andâ A can be probed in high-energy electron-proton collisions through the neutral current top-quark production, there are mainly two modes, i) Deep Inelastic Scattering (DIS) and ii) Photoproduction. Single top-quark and top-quark pair production is possible by both mechanisms.
In this paper we study in a model-independent way the dipole moments of the top-quark through the process of single top-quark production e − p → e − γp →tν e bp. Fig. 1 shows the schematic diagram for the process e − p → e − γp →tν e bp, while, the Feynman diagrams contributing to the reaction e − γ →tν e b they are shown in Fig. 2.
The rest of the paper is organized as follows. In Section II, we introduce the top-quark effective electromagnetic interactions. In Section III, we sensitivity estimates on top-quark anomalous electromagnetic couplings through e − p → e − γp →tν e bp collisions. Finally, we present our conclusions in Section IV.
In Eq. (2), L ef f is the effective Lagrangian gauge-invariant which contains a series of dimension-six operators built with the SM fields, L SM is the renormalizable SM Lagrangian, Λ is the scale at which new physics expected to be observed, c n are dimensionless coefficients and O (6) n represents the dimension-six gauge-invariant operator. We write the most general effective vertex of ttγ [9, 12, 13, 23, 24] as: this equation includes the SM coupling and contributions from dimension-six effective operators. In addition, g e is the electromagnetic coupling constant, Q t is the top-quark electric charge and the Lorentz-invariant vertex function Γ µ ttγ is given by Here m t is the mass of the top-quark, q is the momentum transfer to the photon and the couplingsâ V andâ A are real and related to the anomalous magnetic moment (a t ) and the electric dipole moment (d t ) of the top-quark, respectively. The relations betweenâ V (â A ) and a t (d t ) are given by The operators contribute to top-quark eletromagnetic anomalous couplings [25][26][27] are whereq L3 is the quark field, σ µν are the Pauli matrices andφ is the Higgs doublet, while W a µν and B µν are the U(1) Y and SU(2) L gauge field strength tensors, respectively. From the parametrization given by Eq. (3), and from the operators of dimension-six given in Eqs. (7) and (8) we obtain the corresponding CP evenâ V and CP oddâ A observables: These observables contain υ = 246 GeV, the breaking scale of the electroweak symmetry and sin θ W (cos θ W ), the sine(cosine) of the weak mixing angle.
The FCC-he will be designed to operate in e − p collision mode, but it can also be operated as a e − γ, γp and γγ collider. A promising mechanism to generate energetic photon beams in a FCC-he is Equivalent Photon Approximation (EPA) [28][29][30]  are emitted by proton is given by [28,31] where x = E γ /E p and Q 2 max is the maximum virtuality of the photon. In our calculations, we use Q 2 max = 2 GeV 2 . The minimum value of the Q 2 min is given by From Eq. (11), the function ϕ is the following where explicitly y, a, b and c are as follows Hence, the total cross-section of the scattering e − p → e − γp →tν e bp can be expressed as where σ(e − γ →tν e b) is the cross-section of the scattering e − γ →tν e b and f γ (x) is the spectrum of equivalent photons which is given in Eq. (11).
It is worth mentioning that, there are different ways to optimize the signal sensitivity e − p → e − γp →tν e bp and reduce the background. This is possible if we apply cut-based optimization, in addition to considering polarized electron beam.
We base our results on the following kinematic acceptance cuts in order to optimize the significance of the signal over all the backgrounds: In Eq. (19), p b T is the transverse momentum of the final state bottom-quark, η b is the pseudorapidity and p νe T is the transverse momentum of the electron-neutrino. The outgoing particles are required to satisfy these isolation cuts.
An essential feature in the design of current and future colliders of high-energy physics, is the implementation of polarized particles beams. Most accelerators have been modified or are being designed with the possibility of using polarized particles sources, such as the FCC-he. The possibility of using polarized electron beams can constitute a strong advantage in searching for new physics [32]. Furthermore, the electron beam polarization may lead to a reduction of the measurement uncertainties, either by increasing the signal cross-section, therefore reducing the statistical uncertainty, or by suppressing important backgrounds. In summary, one another option at the FCC-he is to polarize the incoming beam, which could maximize the physics potential, both in the performance of precision tests and in revealing the properties of the new physics BSM.
The general formula for the total cross-section for an arbitrary degree of longitudinal e − and e + beams polarization is given by [32] σ(P e − , P e + ) = 1 4 where P e − (P e + ) is the polarization degree of the electron (positron) beam, while σ −+ stands for the cross-section for completely left-handed polarized e − beam P e − = −1 and completely right-handed polarized e + beam P e + = 1, and other cross-sections σ −− , σ ++ and σ +− are defined analogously.
The main anomalous electromagnetic couplings affecting top-quark physics that are of interest for our study areâ V andâ A . We have calculated the dependencies of the e − p → e − γp →tν e bp production cross-sections for the FCC-he at 7.07 T eV and 10 T eV on a V andâ A using CalcHEP [31]. Furthermore, for our study we consider unpolarized and polarized electron beam, as well as the basic acceptance cuts given in Eq. (19). Assuming only one anomalous coupling to be non-zero at at time, we obtain the following results for the total cross-section in terms of the dipole moments of the top-quark: i) Total cross-section for √ s = 7.07 T eV and P e − = 0%.
ii) Total cross-section for √ s = 10 T eV and P e − = 0%.
We see that the sensitivities on the total cross-section and on the coefficients ofâ V and a A increase with the centre-of-mass energy, as well as with the polarized electron beam, confirming the expected competitive advantage of the high-energies attainable with the FCC-he.
We first present the total cross-section of the signal e − p → e − γp →tν e bp as a function of theâ V andâ A for the center-of-mass energies of the FCC-he, that is √ s = 7.07 T eV and √ s = 10 T eV , as shown through Figs. 3-6. These results are obtained after applying the kinematic cuts given in Eq. (19) and with unpolarized electron beam P e − = 0%. The results show a clear dependence of the total cross-section of the e − p → e − γp →tν e bp scattering with respect toâ V andâ A , as well as with the center-of-mass energies of the FCC-he.
In the case of the cross-section of the photo-production process e − p → e − γp →tν e bp after application of cuts given by Eq. (19) and with polarized electron beam P e − = −80%, the total cross-section is about 1.8 times larger than that of the photo-production process e − p → e − γp →tν e bp with unpolarized electron beam P e − = 0%, as shown in Figs. 9-12.
Before continuing with our study, it is worth making a discussion about our results obtained in Fgs. 3-6 and 9-12. While the theory predictions forâ V andâ A in Eqs. (5) and (6) as well as the total cross-section that contains the anomalous coupling ttγ have been made in different contexts (see Table I  with respect to the process pp → pγ * γ * p → pttp at LHC, our results show a significant improvement. Furthermore, it is noteworthy that with our process the total cross-sections is a factor O(10 3 ) between pp → pγ * γ * p → pttp and e − p → e − γp →tν e bp, that is, our results project 3 orders of magnitude better than those reported in Ref. [12]. These projections shows that the sensitivity on the anomalous couplingsâ V andâ A can be improved at the FCC-he by a few orders of magnitude in comparison with the projections of the LHC.

III. MODEL-INDEPENDENT SENSITIVITY ESTIMATES ON THEâ V ANDâ A
To determine the sensitivity of the non-standard couplings,â V andâ A , Eqs. (9) and (10) we use the results from Section II, for the process e − p → e − γp →tν e bp at the FCC-he. We consider the center-of-mass energies √ s = 7.07, 10 T eV and luminosities L = 50, 100, 300, 500, 1000 f b −1 with unpolarized and polarized electron beam. Furthermore, we consider the kinematic acceptance cuts given by Eq. (19), take into account the systematic uncertainties δ sys = 0%, 3%, 5% and we follow three different confidence level (C.L.) 68%, 90% and 95% and to make our study more effective we perform a χ 2 test define as: Here σ SM is the cross-section from the SM, while σ BSM ( √ s,â V ,â A ) is the total cross-section which contains contributions from the SM, as well as non-standard contributions which come from the anomalous couplingsâ V andâ A . δ st = 1 √ N SM and δ sys are the statistical and systematic uncertainties. In our study we consider δ sys = 0%, 3%, 5%. The number of events for the process e − p → e − γp →tν e bp is given by The χ 2 (â V ,â A ) analysis due systematic uncertainties is studied for three representative values of δ sys at 0%, 3% and 5%, respectively. And the sensitivity ofâ V andâ A at 95% C.L. is found to be of the order of 10 −1 with √ s = 10 T eV , L = 1000 f b −1 and we consider both cases, that is, unpolarized and polarized electron beam, as shown in Table VI (which includes the acceptance cuts, Eq. (19)). The order of the sensitivity on the anomalous couplingsâ V andâ A for other values of √ s and L varies between 10 −2 − 10 −1 at 68% C.L. and 90% C.L., as shown in Tables I-V. Our study shows that the anomalous ttγ vertex at the FCC-he can be probed to a very good accuracy and is comparable with others existing limits, see Table I, Ref. [7].
For the anomalous magnetic and electric dipole moments, an improvement is reachable in comparison with the constraints obtained from the γe − → tt [11] and pp → pγ * γ * p → pttp [12] searches mentioned previously.

IV. CONCLUSIONS
In this paper, we have study feasibility of measuring the non-standard couplingsâ V andâ A coming from the effective electromagnetic interaction ttγ through the process e − p → e − γp → tν e bp at the FCC-he. Specifically, we assume energies from 7.07 and 10 TeV and integrated luminosities of at least 50, 100, 300, 5000 and 1000 f b −1 . Further our sensitivity study is cut-based, polarized electron beam and sources of systematic uncertainties such as leptons and b-jet identification, as well as in a χ 2 (â V ,â A ) test to extract, enhance and optimize the expected signal cross-section and the sensitivity onâ V andâ A . We find that the total cross-  Table I of Ref. [7]. At this time, the FCC-he is an excellent option for the electron-proton collider. It will be useful for any new physics study.
Fortunately, future of e − p colliders remain promising as it is a natural option like a hybrid between the hadron pp and linear e + e − colliders.
It is worth mentioning that, additional improvements could be achieved on the observables of the top-quark, especially in their electromagnetic properties to the extent that more sophisticated analysis methods are apply. In addition to the improvement in the technology of detection of the current and future high-energy physics colliders.