#### Abstract

Based on a recent proposal to build an electron-muon collider, we study two-to-two production processes , that originate from dimension 6 and 8 operators. We compare the sensitivity to those effective couplings obtained at the collider with that of low energy measurements of , , and conversion that have recently been reported in the literature. Whereas for the production of first family fermions, the sensitivity of the collider processes is much weaker; for the second and third family fermions, it is similar or stronger than that of low-energy processes. In the case of , the sensitivity to a dimension 8 contact operator turns out to be the strongest in comparison.

#### 1. Introduction

With the main objective of measuring Higgs boson properties, there has been renewed interest in muon colliders [1–3]. At the low energy level, there have been numerous works concerning from the theoretical and experimental point of view [4–8]. In addition, the recent confirmation of a significant deviation from the Standard Model (SM) value of at Fermilab [9] has also encouraged the proposal of new physics models [10–17] as well as the calculation of higher order QED contributions to the scattering process [18, 19]. The idea of electron-muon collisions has been analyzed in past decades [20–25], and recently, it has been brought up again in a study of an extra boson with generic couplings that could be searched for through a Lepton Flavor Violating (LFV) process like [26]. In a subsequent article, the construction of a high energy collider has been proposed [27]. Besides the clear capability of probing muon LFV effects like the coupling, this machine could even test the Higgs-bottom quark coupling [27]. One important advantage of such a machine is the very low level of SM background processes as they come mostly from vector boson fusion. Other than the elastic there is no two-to-two process to contend with, and this is what motivates our study. We are interested in fermion pair and two-photon production that can come from contact terms and .

LFV operators are already being strongly constrained from low energy measurements like muon decays and transitions [28–30]. We will show the comparison of the potential limits estimated here with these precision measurements. For , we consider the four-fermion dimension-6 operators of the Standard Model Effective Field Theory (SMEFT) as given by the well known *Warsaw* basis [31]. We will also consider some dimension 8 operators that have been constrained in [29]. Given the chiral structure of the SMEFT, it is straightforward to obtain the corresponding amplitudes in the helicity basis. For the two-photon production , we address the contribution from the LFV dipole operator, that is very strongly constrained by , and we consider as well a dimension 8 contact operator that is bound more strictly by our results than by low-energy measurements.

The paper is organized as follows. In Section 2, we analyze the two-body processes induced by four-fermion operators. We explicity provide the corresponding helicity amplitudes and cross sections. Then, we estimate limits on the Wilson coefficients for 13 dimension 6 and 5 dimension 8 four-fermion operators. In Section 3, we consider production originating from two different contributions: a trivalent dimension 6 dipole and a dimension 8 contact operator. We write down the sum of squared amplitudes and the total cross sections. We estimate limits on the coefficients in the same way we did in the previous section, but then, we carry out a detailed Monte Carlo analysis of signal and background processes. In this way, we show that our simplified strategy to obtain limits is realistic. Finally, in Section 4, we summarize our results.

#### 2. Four Fermion Operators

Since we are dealing with massless chiral fermions, the amplitudes are most conveniently written in the helicity basis (see, for example, Ref. [32]). Assuming some degree of polarization in the incoming beams, the general form of the cross section can be divided in four terms:

In this section, we will address four fermion contact operators, and we will obtain sensitivity limits with the following method: first, for each operator, we will assume a total polarization in the cross section. For instance, in that comes from the right-chirality operator, the amplitude gives rise to the cross section term, and we assume , and . Second, we neglect backgrounds and obtain sensitivities in optimal conditions. Third, we will assume an integrated luminosity of order (as in Ref. [27]) and require a minimum cross section of from the operator contribution. Such a cross section yields a significant amount of 40 events. In this way, since we are working with ideal conditions, we will avoid being too optimistic. There are 13 dimension 6 four-fermion operators and 9 of them only contribute to one of the four polarized cross sections. In their case, if there is only partial beam polarization the actual number of events could be reduced by about half or even by one quarter in case of unpolarized beams. On the other hand, we do not expect the presence of background to reduce the sensitivity significantly. In the next section, we will corroborate that this is true for production, where we perform a full signal vs. background analysis.

##### 2.1. Helicity Amplitudes

In this work, we are interested in the process that does not exist in the SM at tree level, but that is generated by dimension six four-fermion operators in the SMEFT. Hereafter, will stand for any of the three charged leptons , , and or any of the quarks except the top quark. The list of four fermion effective operators in Ref. [31] includes arbitrary flavor indices to take into account. However, this does not mean that there are so many possible different helicity amplitudes as there are also Fierz identities that relate them. For instance, the operator gives rise to the same amplitude for any combination 2111, 1211, 1121, and 1112. We choose to work with 2111 that we denote as . For the processes and , the amplitudes between and are clearly the same. In general, we shall take the flavor indices as 2111 for all the operators. This specific choice may exclude some other nonequivalent combinations in some operators, but we have found that this assignment, in the end, covers all the possible helicity amplitudes. The purpose of this preliminary study, rather than being comprehensive, is to get a first glimpse of the potential sensitivity of the collider in two-to-two processes generated by contact operators and compare with the sensitivities of low energy measurements.

The helicity amplitudes are shown in Table 1. We see that operators , , and generate exactly the same helicity amplitudes as they all involve left-chiral spinors. On the other hand, the operator gives rise to two possible combinations of final state chiralities: and . They are not equivalent as shown in Table 1. Notice that many helicity amplitudes are actually equal up to a phase factor related to azimuthal angles. The phase factor is important only in the case of more than one diagram due to interference effects. We also point out that as a preliminary analysis, we will be considering the contribution of each operator separately. Because of the relations [32]:

there are really only three different amplitude structures in this study: , , and . Their squares are proportional to the Mandelstam , , and , respectively. So, they are actually not independent. In fact, there is an identity that is easy to verify [32]: .

Now let us turn our attention to dimension 8 operators. There are multiple structures, but we will pay attention to those specific operators that have been bounded from processes [28, 29]. Moreover, there are dimension 8 operators that coincide with some dimension 6 operator except for an additional term. Obviously, their amplitudes would be equal except for some rescaling factor. The amplitudes of the dimension 8 operators that give rise to chiral structures that do not appear at dimension 6 are shown in Table 2.

##### 2.2. Cross Sections and Limits on Coefficients

As mentioned above, there are only three different types of helicity amplitudes squared. Each of them gives rise to one specific expression for the cross section. For , after dividing by the energy scale and integrating over the phase space we obtain: . For the center-of-mass (CM) collider energy TeV ( GeV), we obtain fb. We will show the corresponding cross sections for each operator in terms of this common . The other amplitudes yield . The amplitudes obtained here grow with the collision energy, but our effective theory calculation is valid for energies below the cut-off scale TeV.

Three benchmark collision energies are proposed in [27]: (1) GeV; (2) GeV; and (3) GeV that correspond to CM energies of TeV, respectively. Since all cross sections are proportional to computing the bound at an energy assuming we know the bound at an energy is straigthforward: we just multiply by the ratio . For instance, , and so we see that the limits at benchmark (3) will be more than twice stronger than benchmark (2). We shall focus on the GeV benchmark in this study. However, in Section 3, we will see that the cross sections for are not proportional to , but one is constant in energy, and the other is proportional to . We will provide limits obtained with the benchmark (3) GeV and for greater energies.

The dimension 6 operators of Table 1 yield the following cross sections: where the term does not appear for the flavor assignment 2111, but the operators that generate would also generate with the assignment 1211.

By requiring that the value of a coefficient be enough to yield the minimum fb of production cross section, we obtain the following lower limits for GeV:

We can compare with the limits from low energy processes , , and conversion in nuclei as recently reported in Tables 6 and 7 of Ref. [28]. We would like to point out that in almost all cases the most stringent bounds in those tables come from conversion in nuclei [33]. The exceptions are first family four-lepton coefficients that are constrained by (Tables 6 [28]) and that is constrained by (Table 7 [28]). Moreover, these bounds are in fact around four orders of magnitude stronger than the ones from and . This brings up one important observation: that if we ignored , the conclusion would be that the collider would yield much stronger constraints than any of the low energy measurements. In Table 3, we show the limits to each operator coefficient (at the scale ) and the ratio for each possible final state. Not surprisingly, we can observe that for the first family fermions the limits are very stringent. However, for the second and third family modes that get their low energy limits via loop contributions, the bounds are weaker and about the same order of magnitude as the potential limits from the production. Strictly speaking, the collider sensitivity for each coefficient is taken at a scale of order 1 TeV, about ten times higher than the electroweak scale. In a more precise analysis, one should take into account the renormalization group dependence on the scale. However, one should bear in mind that such corrections are usually of order a few percent. For instance, the coefficient associated to the operator: . An upper limit from its contribution to decay is reported as at the muon mass scale [28]. The same limit becomes at the scale, which is only a numerical variation. For another example, let us consider the dimension 6 coefficient above. In a series of articles [34–37], we can find the renormalization group evolution of the dimension 6 SMEFT operators listed in the Warsaw basis. The running of in general depends on several other operators, but let us take the term proportional to itself. Then, we find that , that is only a correction. As we are only interested in a preliminary assessment of sensitivities and comparisons with low energy experiment limits, we shall not take corrections of scale dependence into account. Recently, limits on the and coefficients have been published based on the LHC dilepton production that are approximately for first family quarks, for second family, and for [38]. They project that with a hundred times more luminosity the HL-LHC could reduce these limits to one-third of the current value. From what we have found here, the collider would have at least one order of magnitude greater sensitivity than the HL-LHC for these operators.

The dimension 8 operators of Table 2 yield the following cross sections:

A comment on the tensor operators , , and is in order here. , , and the contributions of generate the same amplitude . Except for the additional factor, the cross sections are equal. However, for the helicity mode, the amplitude is . In the CM frame, the functional dependence on the polar angle’s is . If we compare with , we can see that the latter yields a much greater cross section.

As in Table 3 and in Table 4, we show the ratio for the dimension 8 operators. Comparing with the dimension 6 coefficients, there is a suppresing factor of order , and the minimum values have to be much bigger. In contrast, according to Table 5 of Ref. [29], the limits from low energy experiments are still very stringent for dimension 8 couplings.

What we have learned from Table 3 is that for first family fermions the sensitivity of the low energy measurement of conversion in nuclei is two or more orders of magnitude higher. Maybe all the operator coefficients are indeed very suppressed, regardless of potential cancellations. However, for most of the second and third family states, the collider sensitivities are of the same order of magnitude as the low energy limits. The electron-muon collider should be able to provide additional and competitive limits to constrain the set of dimension 6 four fermion LFV operators. The same may not be true for dimension 8 operators, at least for the ones that can be constrained by the low energy experiments.

#### 3. The Process

The amplitude can be generated by the dimension 6 flavor-changing magnetic dipole operator through and channel diagrams where one of the photons is emitted by the effective coupling. Another possibility comes from the dimension 8 operator

We have two chiral versions: and (, ) referring to left-handed and right-handed electron, respectively. In Figure 1, we show the Feynman diagrams associated to each operator.

**(a)**

**(b)**

Both operators give rise to amplitudes that do not depend on angles:

They yield the total cross sections: where the numerical value on the first line is independent of the collision energy, but the numerical value on the second line is taken at TeV.

We see, then, that coefficients and of order and , respectively, would give us a fb value that is our minimum acceptable cross section. Let us notice that the current limit from is of order for the dipole coefficient , well below the potential collider sensitivity. On the other hand, for the dimension 8 operator coefficient , the potential sensitivity from transitions is six orders of magnitude less stringent: [29]. We point out that, in this study, this last coefficient is the one with the highest sensitivity by the collider as compared to the low energy measurements.

Seeing the relatively high sensitivity to the electromagnetic operator, one may wonder what about the effective vertex ? We can use the calculation with the analogous gluon operator

The production cross section is now times greater than , and a coefficient of order would yield the minimum observable production. However, according to [29], the low energy limit is three orders of magnitude more stringent in this case: .

As mentioned above, the amplitudes squared for do not depend on the polar angle. That means that in terms of rapidity, for instance, in the CM frame: where is a constant. The shape of the rapidity distribution in the CM frame is then centered around zero with a width of approximately units. In the lab frame, the center is shifted towards (see next subsection).

##### 3.1. Monte Carlo Analysis of Production and Its SM Background

In the SM, production is given by the two-to-four process , involving 13 Feynman diagrams in unitary gauge. There is also production , involving 24 Feynman diagrams. In Figure 2, we show two representative diagrams for the and resonant production in the SM. Signal and background have very different kinematics, and this makes the separation straightforward. Our goal is to show how a basic set of cuts can reduce the potential background dramatically. We point out here that in the lab frame the rapidities are shifted with respect to those in the CM frame: where is the shift value for . The muon beam goes in the direction of , and so the event products usually appear on the backward hemisphere.

**(a)**

**(b)**

Thus, to study production in collisions, we consider the three processes, where the first one is our signal process as depicted in Figure 1 and (15) and (16) the SM backgrounds shown in Figure 2. We implemented the effective interactions (6) and (7) in our Monte Carlo simulations by means of feynrules 2.0 [39]. We simulated the signal and background processes with madgraph 2.6 [40], with beam energies (100, 3000), (150, 4500), and (200, 6000) GeV. We then have , 1.643, and 2.191 TeV, respectively. In the two-body signal process (14), the CM energies of the final photons are fixed at . This is unlike what happens with the backgrounds (15) and (16), where there is a continuous spectrum for and . Similarly, for the signal process , but for the backgrounds which has a continuous range of values. Furthermore, we observe the final-state photons in the signal process to be very central in the CM frame, with for the vast majority of events, as expected in view of the analytical distribution (12) and as shown in Figure 3. For the process (15), without restrictions on , the photon rapidity distribution in the lab frame is symmetric about , but very broad. If we require in (15) to be large, however, the final-state photons must be very forward or backward. For example, if GeV, the rapidities will have maxima at with as shown in Figure 3, and gets larger for greater values of . For the process (16), the electron rapidity distribution presents essentially the same features but is less forward-backward symmetric, as seen in the figure.

We are, thus, led to consider the following set of phase-space cuts,

The cut is necessary to control infrared divergences in (15) and (16). As discussed above, the cuts in (17) have only small effects on the signal cross section, but they do substantially decrease the cross section for the backgrounds. The effect of the cuts (17) on the cross sections for the processes (14), (15), and (16) is illustrated at (100, 3000) GeV in Table 5. The cross section refers to the anomalous process (14) with only the trivalent vertex and the Wilson coefficients . Similarly, refers to (14) through the vertex and the coefficients . The numerical results agree with (9).

We expect the results for cross sections with cuts in Table 5 to be quite realistic, although detector efficiencies and acceptances have not been allowed for in those results. However, we expect the rapidity acceptance effects to be taken into account by the cut in (17), and we also expect the efficiency for photon identification to be no less than 90%, so that detector effects should be modest. The important exception to this, however, is the background process (16), which in Table 5 seems to represent one-third of , but which must actually be adjusted for the electron-photon misidentification probability. In order to settle this issue, we carried out a detector simulation using Delphes 3.4 [41].

An collider is highly asymmetric, so we assume its detector to have a correspondingly asymmetric design. We obtain a simple but effective asymmetric detector simulation in the lab frame by assuming that in the CM frame, in which the collisions are forward-backward symmetric on average, the detector possesses the same capabilities as the symmetric Muon Collider detector implemented in Delphes 3.4 (which corresponds to the configuration file cards/delphes_card_MuonColliderDet.tcl in the Delphes distribution). We simulate the collisions with madgraph 5 with a loosened version of the cuts (17) in order to adequately populate the phase space,

We run Pythia 6 [42] on these events for QED showering, followed by Delphes 3.4 with Muon Collider configuration. In Figure 3, we display normalized differential cross sections with respect to the CM rapidity , the transverse momentum , and the CM energy for the final-state photons in processes (14) and (15), and the final-state for (16), at three different collision energies. Those differential cross sections correspond to detector-level events generated with the cuts (18) at the parton level. We notice here that the differential cross sections shown in the figure for the signal process (14) correspond to the contact interaction (7) represented by the symbol in Tables 5 and 6. For the kinematic variables considered in Figure 3, however, identical results would have been obtained with the interaction (6). We also point out here the good agreement of the analytical rapidity distribution (12) with the Monte Carlo data in Figure 3(c).

We then apply a preselection cut to the Delphes events. Notice that (19) implicitly includes a cut in absolute rapidity , corresponding to the detector rapidity acceptance range. Finally, we apply the cuts from (17) to the preselected events. The cross sections obtained for the detector-level events are summarized in Table 6. We notice here that the cross section for the process (14) induced by the dim 6 operator (6), as displayed by the diagrams in Figure 1(a), has an energy-independent cross section at the parton level but shows a slight decrease with increasing in Table 6. This is due to the fact that the upper limit on we are using in (17) is fixed. This effect, however, is more than compensated for by the increase in the partonic cross section in the case of the signal process induced by the dim 8 operator (7), as displayed in Figure 1(b), leading to a increasing cross section also at the detector level. The cross section for the process (15) shows a very modest growth with in Table 6, and process (16) actually decreases at the highest energy.

Furthermore, comparing the results in Table 6 at GeV with those in Table 5, we see that detector effects result in an effective efficiency of 84% for the production processes given in the tables by and . We see also that detector efficiencies reduce the cross section for the background process (16) to the same level as the background (15). Completely analogous results are obtained at the two higher energies considered in Table 6.

#### 4. Conclusions

We have obtained individual limits on LFV four fermion operators by looking at the two-to-two production processes they induce at the collider. For operators where is a second or third family fermion, the sensitivity of the collider is of the same order of magnitude as, and for some operators even somewhat stronger than, that of the conversion in nuclei. On the other hand, the collider would have higher sensitivity than the other low energy measurements and even for first family fermions. In the particular case of the Wilson coefficients , for example, the expected sensitivity at an collider would be at least as strong as that of all low-energy measurements, and an order of magnitude larger than that projected for Drell-Yan processes at the HL-LHC. This leads us to expect that, given the large number of independent effective four-fermion operators, the additional information obtained from the collider will certainly be invaluable. In the case of production, the limits from the collider are significantly more stringent than those from the low energy processes for the case of the dimension 8 contact operator (7).

In the SM, the and production involves an additional pair of neutrinos, and this makes the separation of signal and background straightforward. We have made a detailed analysis of background and signal for the case of production, including detector simulation. We observe that with appropiate cuts on the photon energies, the photon pair transverse momentum and the photon rapidities we can dramatically lower the SM background with very little reduction of the signal.

We point out, finally, that our conclusions are based on the assumptions of an integrated luminosity of 1 ab and beam energies (100 GeV, 3 TeV), (150 GeV, 4.5 TeV), (200 GeV, 6 TeV), respectively. Clearly, higher luminosities and/or beam energies would lead to stronger sensitivities to the contact-interaction effective couplings considered here.

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Disclosure

The work has appeared as a preprint on arXiv: https://arxiv.org/abs/2109.02769.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

We are grateful to Georgina Espinoza Gurriz for her assistance with our computer hardware. We acknowledge support from Sistema Nacional de Investigadores de Conacyt, México. We also acknowledge that a preprint has previously been published at arXiv [27].