# Two-to-Two Processes at an Electron-Muon Collider

Authors: Antonio O. Bouzas; F. Larios
Journal: Advances in High Energy Physics
 (2022)

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## Abstract

Based on a recent proposal to build an electron-muon collider, we study two-to-two production processese−μ+⟶ff¯, γγ 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 μ⟶eγ, μ⟶ee¯e, and μ⟶e 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 e−μ+⟶γγ, the sensitivity to a dimension 8 contact operator turns out to be the strongest in comparison.


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## Body

## 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 g−2μ 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 g−2μ 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 eμ⟶eμ 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 Z′ boson with generic couplings that could be searched for through a Lepton Flavor Violating (LFV) process like e−μ+⟶e+μ− [26]. In a subsequent article, the construction of a high energy e−μ+ collider has been proposed [27]. Besides the clear capability of probing muon LFV effects like the Heμ 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 e−μ+⟶e−μ+ 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 e−μ+ff¯ and e−μ+γγ.LFV operators are already being strongly constrained from low energy measurements like muon decays andμ−e transitions [28–30]. We will show the comparison of the potential limits estimated here with these precision measurements. For e−μ+⟶ff¯, 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 e−μ+⟶γγ, we address the contribution from the LFV dipole operator, that is very strongly constrained by μ⟶eγ, 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 Section2, we analyze the two-body processes induced by four-fermion eμff 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 γγ 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:
(1)σPe−Pμ+=1+Pe−21+Pμ+2σ+++1−Pe−21−Pμ+2σ−−+1+Pe−21−Pμ+2σ+−+1−Pe−21+Pμ+2σ−+.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, ineR−μR+⟶uRu¯R that comes from the right-chirality Qeu operator, the M+−+− amplitude gives rise to the σ+− cross section term, and we assume Pe−=+1, and Pμ+=−1. Second, we neglect backgrounds and obtain sensitivities in optimal conditions. Third, we will assume an integrated luminosity of order 1ab−1 (as in Ref. [27]) and require a minimum cross section of 0.04fb 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 γγ production, where we perform a full signal vs. background analysis.


### 2.1. Helicity Amplitudes

In this work, we are interested in the processe−p1μ+p2⟶fp3f¯p4 that does not exist in the SM at tree level, but that is generated by dimension six four-fermion operators in the SMEFT. Hereafter, ff¯ will stand for any of the three charged leptons e, μ, and τ or any of the quarks except the top quark. The list of four fermion effective operators in Ref. [31] includes arbitrary p,r,s,t 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 Oll=l¯pγνlrl¯sγνlt gives rise to the same amplitude e−μ+⟶e−e+ for any combination 2111, 1211, 1121, and 1112. We choose to work with 2111 that we denote as Oll2111=l¯μγνlel¯eγνle. For the processes e−μ+⟶μ−μ+ and e−μ+⟶τ−τ+, the amplitudes between Oll2122 and Oll2133 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 e−μ+ 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 Table1. We see that L¯LL¯L operators Qll, Qlq1, and Qlq3 generate exactly the same helicity amplitudes as they all involve left-chiral spinors. On the other hand, the L¯LR¯R operator Qll=l¯μγμlee¯eγμee gives rise to two possible combinations of final state chiralities: eR−eR+ and eR−eL+. 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]:
(2)ij=−ji,ij=−ji,ij=ji∗,ij2=2pi·pj,Table 1
Dimension 6 operators with common flavor indices (2111) and the helicity amplitudes generated. TheCO/Λ2 factor to be included. The R,L indices refer to chiralities. Final state e−e+ also stands for μ−μ+ and τ−τ+. Similarly, dd¯ also stands for ss¯ and bb¯, and uu¯ also stands for cc¯. Top quarks excluded.
(L¯L)(L¯L)e−p1μ+p2⟶fp3f¯p4MOlll¯μγνlel¯eγνleeL−μL+⟶eL−eL+21432Olq1l¯μγνleq¯γνqeL−μL+⟶uLu¯L21432Olq3l¯μγντIleq¯γντIqeL−μL+⟶uLu¯L, dLd¯L∓21432(R¯R)(R¯R)e−p1μ+p2⟶fp3f¯p4MOeee¯μγveee¯eγveeeR−μR+⟶eR−eR+21432Oeue¯μγveeu¯γvueR−μR+⟶uRu¯R21432Oede¯μγveed¯γvdeR−μR+⟶dRd¯R21432(L¯L)(R¯R)e−p1μ+p2⟶fp3f¯p4MOlel¯μγvlee¯eγveeeL−μL+⟶eR−eR+22431Oleluγvlee¯eγveeeR−μL+⟶eR−eL+22134Oluluγvleu¯γvueL−μL+⟶uRu¯R22431Oldluγvled¯γvdeL−μL+⟶dRd¯R22431Oqeq¯γvqe¯uγveeeR−μR+⟶uLu¯L,d¯Ld¯L22431(L¯R)(R¯L)e−p1μ+p2⟶fp3f¯p4MOledql¯μeed¯qeR−μL+⟶dRd¯L1243(L¯R)(L¯R)e−p1μ+p2⟶fp3f¯p4MOlq1l¯μjeeεjkq¯kueR−μL+⟶uRu¯L1234Olequ3l¯μjσμveeεjkq¯kσμvueR−μL+⟶uRu¯L41324+1423there are really only three different amplitude structures in this study:1234, 1324, and 1423. Their squares are proportional to the Mandelstam s2, t2, and u2, respectively. So, they are actually not independent. In fact, there is an identity that is easy to verify [32]: 1234=1432+1324.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 fromμ⟶e processes [28, 29]. Moreover, there are dimension 8 operators that coincide with some dimension 6 operator except for an additional H†H 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.Table 2
Helicity amplitudes generated by dimension 8 operators with common flavor indices2111 associated to four-fermion contact vertices that do not appear with dimension 6 operators. A v2/2CO/Λ4 factor is to be included. As in Table 1, final state e−e+ also stands for μ−μ+ and τ−τ+, dd¯ also stands for ss¯ and bb¯, and uu¯ also stands for cc¯.
(L¯R)(L¯R)e−p1μ+p2⟶fp3f¯p4MOle8l¯μHeel¯eHeeeR−μL+⟶eR−eL+1243OTle8l¯μHσμνeel¯eHσμνeeeR−μL+⟶eR−eL+43124+4123Oleqd18l¯μHeeq¯HdeR−μL+⟶dRd¯L1243Oleqd38l¯μHσμνeeq¯HσμνdeR−μL+⟶dRd¯L43124+4123Ole8l¯μHeel¯eHeeeR−μL+⟶eL−eR+2431OTle8l¯μHσμνeel¯eHσμνeeeR−μL+⟶eL−eR+42134+1423(L¯R)(R¯L)e−p1μ+p2⟶fp3f¯p4MOleuq8l¯μHeeu¯H~†qeR−μL+⟶uRu¯L1243


### 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. For12342=s2, after dividing by the energy scale Λ4 and integrating over the phase space we obtain: σ1234=s/16πΛ4. For the center-of-mass (CM) collider energy s=1.095 TeV (Ee=100,Eμ=3000 GeV), we obtain σ1234s=1.095TeV=36.3 fb. We will show the corresponding cross sections for each operator in terms of this common σ1234. The other amplitudes yield σ1324=σ1423=σ1234/3. The amplitudes obtained here grow with the collision energy, but our effective theory calculation is valid for energies below the cut-off scale Λ=4 TeV.Three benchmark collision energies are proposed in [27]: (1) Ee=20,Eμ=200 GeV; (2) Ee=50,Eμ=1000 GeV; and (3) Ee=100,Eμ=3000 GeV that correspond to CM energies of s=2EeEμ=0.126,0.447,1.095 TeV, respectively. Since all cross sections are proportional to sCO2 computing the bound at an energy s2 assuming we know the bound at an energy s1 is straigthforward: we just multiply by the ratio s1/s2. For instance, 0.447/1.095=0.4, and so we see that the limits at benchmark (3) will be more than twice stronger than benchmark (2). We shall focus on the Ee=100,Eμ=3000 GeV benchmark in this study. However, in Section 3, we will see that the cross sections for e−μ+⟶γγ are not proportional to s, but one is constant in energy, and the other is proportional to s2. We will provide limits obtained with the benchmark (3) Ee=100,Eμ=3000 GeV and for greater energies.The dimension 6 operators of Table1 yield the following cross sections:
(3)σ++σ1234=4Cle2+NcCledq2+NcClequ12+163NcClequ32,σ+−σ1234=43Cee2+43NcCeu2+43NcCed2+43NcCqe2,σ−+σ1234=43Cll2+43Cle2+43NcClq12+43NcClq32+43NcClu2+43NcCld2,where the σ−− term does not appear for the flavor assignment 2111, but the operators that generate σ++ would also generate σ−− with the assignment 1211.By requiring that the value of aCQ coefficient be enough to yield the minimum 0.04 fb of production cross section, we obtain the following lower limits for Ee=100,Eμ=3000 GeV:
(4)Cll,Cee≥2.88×10−2,Cle,Clu,Cld,Cqe≥1.66×10−2,Ceu,Ced,Clq1,Clq3≥1.66×10−2,Clequ3≥0.83×10−2.We can compare with the limits from low energy processesμ⟶eγ, μ⟶ee¯e, and μ⟶e conversion in nuclei μA⟶eA 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 μA⟶eA conversion in nuclei [33]. The exceptions are first family four-lepton coefficients Cll,Cle,Cee that are constrained by μ⟶ee¯e (Tables 6 [28]) and Clequ3eμcc that is constrained by μ⟶eγ (Table 7 [28]). Moreover, these μA⟶eA bounds are in fact around four orders of magnitude stronger than the ones from μ⟶eγ and μ⟶ee¯e. This brings up one important observation: that if we ignored μA⟶eA, the conclusion would be that the eμ 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 mW) and the ratio CQLowenergy/CQCollider 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 e−μ+⟶ff¯ 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 CDL associated to the operator: CDLmμ/mt2e¯RσμνμLFμν. An upper limit from its contribution to μ⟶eγ decay is reported as ∣CDL∣<1.05×10−8 at the muon mass scale [28]. The same limit becomes ∣CDL∣<1.12×10−8 at the mW scale, which is only a 7% numerical variation. For another example, let us consider the dimension 6 coefficient Cel 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 Cel in general depends on several other operators, but let us take the term proportional to Cel itself. Then, we find that Cle1TeV~0.96ClemW, that is only a 4% 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 Clq1 and Clq3 coefficients have been published based on the LHC pp⟶eμ dilepton production that are approximately Clq1,3<0.3 for first family quarks, Clq1,3<2.0 for second family, and Clq1,3<5.0 for bb¯ [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 eμ collider would have at least one order of magnitude greater sensitivity than the HL-LHC for these operators.Table 3
Upper limits to the dimension 6 coefficients from the low energy experiments as reported in Tables 6 and 7 of [28]. They have been multiplied by a factor 4/0.1742 to account for the normalization scale mt used by [28] instead of the Λ=4 TeV scale adopted in this study. In addition, the ratio of limits Low/Collider is shown in parenthesis (Ee=100,Eμ=3000 GeV).
CQCQLow (CLow/CColl)e−e+CQLow (CLow/CColl)μ−μ+CQLow (CLow/CColl)τ−τ+Cll4.16×10–41.45×10–20.98×10−2 (0.34)1.97×10−2 (0.69)Cee4.16×10–41.45×10–20.98×10−2 (0.34)1.97×10−2 (0.69)Cle4.92×10–42.96×10−21.99×10−2 (1.20)1.98×10−2 (1.19)dd¯ss¯bb¯Clq11.51×10–50.91×10–31.60×10−2 (0.96)2.49×10−2 (1.50)Clq32.69×10–41.62×10–26.08×10−3 (0.37)2.49×10−2 (1.50)Cld2.80×10−51.69×10−31.97×10−2 (1.19)2.49×10−2 (1.50)Ced2.86×10−51.72×10−31.97×10−2 (1.19)2.49×10−2 (1.50)Cqe1.52×10−50.92×10−31.58×10−2 (0.95)2.50×10−2 (1.51)Cledq5.34×10−62.79×10−41.11×10−4 (5.80×10−3)3.66×10−3 (0.19)uu¯cc¯−−Clu3.30×10−51.99×10−30.88×10−2 (0.53)Ceu3.19×10−51.92×10−30.89×10−2 (0.54)Clequ15.45×10−62.84×10−40.97×10−3 (5.1×10−2)Clequ35.10×10−56.1×10−35.34×10−6 (6.4×10−4)The dimension 8 operators of Table2 yield the following cross sections:
(5)σ++σ1234eR−eL+=v44Λ4Cle82+NcCleqd182+NcCleuq82+163NcCleqd382+163CTle82,σ++σ1234eL−eR+=v44Λ413Cle82+1123NcCTle82.A comment on the tensor operatorsQlequ3, Qleqd3, and QTle is in order here. Qlequ3, Qleqd3, and the σ++eR−eL+ contributions of QTle generate the same amplitude 1324+1423. Except for the v4/4Λ4 additional factor, the cross sections are equal. However, for the σ+−eL−eR+ helicity mode, the QTle amplitude is 2134+1423. In the CM frame, the functional dependence on the p3 polar angle’s cosθ is 1324+1423~cosθ. If we compare with 2134+1423~3+cosθ/2, we can see that the latter yields a much greater cross section.As in Table3 and in Table 4, we show the ratio CLow/CColl for the dimension 8 operators. Comparing with the dimension 6 coefficients, there is a suppresing v2/2Λ2 factor of order 2×10−3, and the CColl 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.Table 4
The ratio of potential limits from low energy processes and minimum observable values at the collider for the dimension 8 coefficients (Table 5 of [29]; Ee=100,Eμ=3000 GeV).
CQCLow/CCollCle85×10−3/14.0CTle82×10−3/1.3Cleqd183×10−5/8.1Cleqd387×10−4/3.5Cleuq83×10−5/8.1What we have learned from Table3 is that for first family fermions ff¯ the sensitivity of the low energy measurement of μA⟶eA 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 ff¯ 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.


## 2.1. Helicity Amplitudes

In this work, we are interested in the processe−p1μ+p2⟶fp3f¯p4 that does not exist in the SM at tree level, but that is generated by dimension six four-fermion operators in the SMEFT. Hereafter, ff¯ will stand for any of the three charged leptons e, μ, and τ or any of the quarks except the top quark. The list of four fermion effective operators in Ref. [31] includes arbitrary p,r,s,t 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 Oll=l¯pγνlrl¯sγνlt gives rise to the same amplitude e−μ+⟶e−e+ for any combination 2111, 1211, 1121, and 1112. We choose to work with 2111 that we denote as Oll2111=l¯μγνlel¯eγνle. For the processes e−μ+⟶μ−μ+ and e−μ+⟶τ−τ+, the amplitudes between Oll2122 and Oll2133 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 e−μ+ 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 Table1. We see that L¯LL¯L operators Qll, Qlq1, and Qlq3 generate exactly the same helicity amplitudes as they all involve left-chiral spinors. On the other hand, the L¯LR¯R operator Qll=l¯μγμlee¯eγμee gives rise to two possible combinations of final state chiralities: eR−eR+ and eR−eL+. 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]:
(2)ij=−ji,ij=−ji,ij=ji∗,ij2=2pi·pj,Table 1
Dimension 6 operators with common flavor indices (2111) and the helicity amplitudes generated. TheCO/Λ2 factor to be included. The R,L indices refer to chiralities. Final state e−e+ also stands for μ−μ+ and τ−τ+. Similarly, dd¯ also stands for ss¯ and bb¯, and uu¯ also stands for cc¯. Top quarks excluded.
(L¯L)(L¯L)e−p1μ+p2⟶fp3f¯p4MOlll¯μγνlel¯eγνleeL−μL+⟶eL−eL+21432Olq1l¯μγνleq¯γνqeL−μL+⟶uLu¯L21432Olq3l¯μγντIleq¯γντIqeL−μL+⟶uLu¯L, dLd¯L∓21432(R¯R)(R¯R)e−p1μ+p2⟶fp3f¯p4MOeee¯μγveee¯eγveeeR−μR+⟶eR−eR+21432Oeue¯μγveeu¯γvueR−μR+⟶uRu¯R21432Oede¯μγveed¯γvdeR−μR+⟶dRd¯R21432(L¯L)(R¯R)e−p1μ+p2⟶fp3f¯p4MOlel¯μγvlee¯eγveeeL−μL+⟶eR−eR+22431Oleluγvlee¯eγveeeR−μL+⟶eR−eL+22134Oluluγvleu¯γvueL−μL+⟶uRu¯R22431Oldluγvled¯γvdeL−μL+⟶dRd¯R22431Oqeq¯γvqe¯uγveeeR−μR+⟶uLu¯L,d¯Ld¯L22431(L¯R)(R¯L)e−p1μ+p2⟶fp3f¯p4MOledql¯μeed¯qeR−μL+⟶dRd¯L1243(L¯R)(L¯R)e−p1μ+p2⟶fp3f¯p4MOlq1l¯μjeeεjkq¯kueR−μL+⟶uRu¯L1234Olequ3l¯μjσμveeεjkq¯kσμvueR−μL+⟶uRu¯L41324+1423there are really only three different amplitude structures in this study:1234, 1324, and 1423. Their squares are proportional to the Mandelstam s2, t2, and u2, respectively. So, they are actually not independent. In fact, there is an identity that is easy to verify [32]: 1234=1432+1324.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 fromμ⟶e processes [28, 29]. Moreover, there are dimension 8 operators that coincide with some dimension 6 operator except for an additional H†H 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.Table 2
Helicity amplitudes generated by dimension 8 operators with common flavor indices2111 associated to four-fermion contact vertices that do not appear with dimension 6 operators. A v2/2CO/Λ4 factor is to be included. As in Table 1, final state e−e+ also stands for μ−μ+ and τ−τ+, dd¯ also stands for ss¯ and bb¯, and uu¯ also stands for cc¯.
(L¯R)(L¯R)e−p1μ+p2⟶fp3f¯p4MOle8l¯μHeel¯eHeeeR−μL+⟶eR−eL+1243OTle8l¯μHσμνeel¯eHσμνeeeR−μL+⟶eR−eL+43124+4123Oleqd18l¯μHeeq¯HdeR−μL+⟶dRd¯L1243Oleqd38l¯μHσμνeeq¯HσμνdeR−μL+⟶dRd¯L43124+4123Ole8l¯μHeel¯eHeeeR−μL+⟶eL−eR+2431OTle8l¯μHσμνeel¯eHσμνeeeR−μL+⟶eL−eR+42134+1423(L¯R)(R¯L)e−p1μ+p2⟶fp3f¯p4MOleuq8l¯μHeeu¯H~†qeR−μL+⟶uRu¯L1243


## 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. For12342=s2, after dividing by the energy scale Λ4 and integrating over the phase space we obtain: σ1234=s/16πΛ4. For the center-of-mass (CM) collider energy s=1.095 TeV (Ee=100,Eμ=3000 GeV), we obtain σ1234s=1.095TeV=36.3 fb. We will show the corresponding cross sections for each operator in terms of this common σ1234. The other amplitudes yield σ1324=σ1423=σ1234/3. The amplitudes obtained here grow with the collision energy, but our effective theory calculation is valid for energies below the cut-off scale Λ=4 TeV.Three benchmark collision energies are proposed in [27]: (1) Ee=20,Eμ=200 GeV; (2) Ee=50,Eμ=1000 GeV; and (3) Ee=100,Eμ=3000 GeV that correspond to CM energies of s=2EeEμ=0.126,0.447,1.095 TeV, respectively. Since all cross sections are proportional to sCO2 computing the bound at an energy s2 assuming we know the bound at an energy s1 is straigthforward: we just multiply by the ratio s1/s2. For instance, 0.447/1.095=0.4, and so we see that the limits at benchmark (3) will be more than twice stronger than benchmark (2). We shall focus on the Ee=100,Eμ=3000 GeV benchmark in this study. However, in Section 3, we will see that the cross sections for e−μ+⟶γγ are not proportional to s, but one is constant in energy, and the other is proportional to s2. We will provide limits obtained with the benchmark (3) Ee=100,Eμ=3000 GeV and for greater energies.The dimension 6 operators of Table1 yield the following cross sections:
(3)σ++σ1234=4Cle2+NcCledq2+NcClequ12+163NcClequ32,σ+−σ1234=43Cee2+43NcCeu2+43NcCed2+43NcCqe2,σ−+σ1234=43Cll2+43Cle2+43NcClq12+43NcClq32+43NcClu2+43NcCld2,where the σ−− term does not appear for the flavor assignment 2111, but the operators that generate σ++ would also generate σ−− with the assignment 1211.By requiring that the value of aCQ coefficient be enough to yield the minimum 0.04 fb of production cross section, we obtain the following lower limits for Ee=100,Eμ=3000 GeV:
(4)Cll,Cee≥2.88×10−2,Cle,Clu,Cld,Cqe≥1.66×10−2,Ceu,Ced,Clq1,Clq3≥1.66×10−2,Clequ3≥0.83×10−2.We can compare with the limits from low energy processesμ⟶eγ, μ⟶ee¯e, and μ⟶e conversion in nuclei μA⟶eA 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 μA⟶eA conversion in nuclei [33]. The exceptions are first family four-lepton coefficients Cll,Cle,Cee that are constrained by μ⟶ee¯e (Tables 6 [28]) and Clequ3eμcc that is constrained by μ⟶eγ (Table 7 [28]). Moreover, these μA⟶eA bounds are in fact around four orders of magnitude stronger than the ones from μ⟶eγ and μ⟶ee¯e. This brings up one important observation: that if we ignored μA⟶eA, the conclusion would be that the eμ 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 mW) and the ratio CQLowenergy/CQCollider 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 e−μ+⟶ff¯ 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 CDL associated to the operator: CDLmμ/mt2e¯RσμνμLFμν. An upper limit from its contribution to μ⟶eγ decay is reported as ∣CDL∣<1.05×10−8 at the muon mass scale [28]. The same limit becomes ∣CDL∣<1.12×10−8 at the mW scale, which is only a 7% numerical variation. For another example, let us consider the dimension 6 coefficient Cel 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 Cel in general depends on several other operators, but let us take the term proportional to Cel itself. Then, we find that Cle1TeV~0.96ClemW, that is only a 4% 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 Clq1 and Clq3 coefficients have been published based on the LHC pp⟶eμ dilepton production that are approximately Clq1,3<0.3 for first family quarks, Clq1,3<2.0 for second family, and Clq1,3<5.0 for bb¯ [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 eμ collider would have at least one order of magnitude greater sensitivity than the HL-LHC for these operators.Table 3
Upper limits to the dimension 6 coefficients from the low energy experiments as reported in Tables 6 and 7 of [28]. They have been multiplied by a factor 4/0.1742 to account for the normalization scale mt used by [28] instead of the Λ=4 TeV scale adopted in this study. In addition, the ratio of limits Low/Collider is shown in parenthesis (Ee=100,Eμ=3000 GeV).
CQCQLow (CLow/CColl)e−e+CQLow (CLow/CColl)μ−μ+CQLow (CLow/CColl)τ−τ+Cll4.16×10–41.45×10–20.98×10−2 (0.34)1.97×10−2 (0.69)Cee4.16×10–41.45×10–20.98×10−2 (0.34)1.97×10−2 (0.69)Cle4.92×10–42.96×10−21.99×10−2 (1.20)1.98×10−2 (1.19)dd¯ss¯bb¯Clq11.51×10–50.91×10–31.60×10−2 (0.96)2.49×10−2 (1.50)Clq32.69×10–41.62×10–26.08×10−3 (0.37)2.49×10−2 (1.50)Cld2.80×10−51.69×10−31.97×10−2 (1.19)2.49×10−2 (1.50)Ced2.86×10−51.72×10−31.97×10−2 (1.19)2.49×10−2 (1.50)Cqe1.52×10−50.92×10−31.58×10−2 (0.95)2.50×10−2 (1.51)Cledq5.34×10−62.79×10−41.11×10−4 (5.80×10−3)3.66×10−3 (0.19)uu¯cc¯−−Clu3.30×10−51.99×10−30.88×10−2 (0.53)Ceu3.19×10−51.92×10−30.89×10−2 (0.54)Clequ15.45×10−62.84×10−40.97×10−3 (5.1×10−2)Clequ35.10×10−56.1×10−35.34×10−6 (6.4×10−4)The dimension 8 operators of Table2 yield the following cross sections:
(5)σ++σ1234eR−eL+=v44Λ4Cle82+NcCleqd182+NcCleuq82+163NcCleqd382+163CTle82,σ++σ1234eL−eR+=v44Λ413Cle82+1123NcCTle82.A comment on the tensor operatorsQlequ3, Qleqd3, and QTle is in order here. Qlequ3, Qleqd3, and the σ++eR−eL+ contributions of QTle generate the same amplitude 1324+1423. Except for the v4/4Λ4 additional factor, the cross sections are equal. However, for the σ+−eL−eR+ helicity mode, the QTle amplitude is 2134+1423. In the CM frame, the functional dependence on the p3 polar angle’s cosθ is 1324+1423~cosθ. If we compare with 2134+1423~3+cosθ/2, we can see that the latter yields a much greater cross section.As in Table3 and in Table 4, we show the ratio CLow/CColl for the dimension 8 operators. Comparing with the dimension 6 coefficients, there is a suppresing v2/2Λ2 factor of order 2×10−3, and the CColl 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.Table 4
The ratio of potential limits from low energy processes and minimum observable values at the collider for the dimension 8 coefficients (Table 5 of [29]; Ee=100,Eμ=3000 GeV).
CQCLow/CCollCle85×10−3/14.0CTle82×10−3/1.3Cleqd183×10−5/8.1Cleqd387×10−4/3.5Cleuq83×10−5/8.1What we have learned from Table3 is that for first family fermions ff¯ the sensitivity of the low energy measurement of μA⟶eA 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 ff¯ 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. Thee−μ+⟶γγ Process

Thee−μ+⟶γγ amplitude can be generated by the dimension 6 flavor-changing magnetic dipole operator
(6)QeA=l¯μσμνeeHFμν,through t and u channel diagrams where one of the photons is emitted by the effective coupling. Another possibility comes from the dimension 8 operator
(7)QeAA=l¯μeeHFμνFμν.We have two chiral versions:QeAL and QeAR (QeAAL, QeAAR) referring to left-handed and right-handed electron, respectively. In Figure 1, we show the Feynman diagrams associated to each operator.Figure 1
The anomalous two-to-two processe−μ+⟶γγ. Process (a) (t-channel) as given by the trivalent vertex dipole operator QeA. Process (b) as given by the contact term, dimension 8 operator QeAA.
(a)(b)Both operators give rise to amplitudes that do not depend on angles:(8)∑M2=CeAL2+CeAR2e2v2Λ448s+CeAAL2+CeAAR2v2Λ84s3.They yield the total cross sections:(9)σ−−+σ++=CeAL2+CeAR23e2v22πΛ4=CeAL2+CeAR24.32fb,(10)σ−−+σ++=CeAAL2+CeAAR2v28πΛ8s2=CeAAL2+CeAAR20.0206fb,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 s=1.095 TeV.We see, then, that coefficientsCeALR and CeAALR of order 0.1 and 2, respectively, would give us a σ=0.04 fb value that is our minimum acceptable cross section. Let us notice that the current limit from μ⟶eγ is of order 5×10−6 for the dipole coefficient CeALR, well below the potential collider sensitivity. On the other hand, for the dimension 8 operator coefficient CeAALR, the potential sensitivity from μA⟶eA transitions is six orders of magnitude less stringent: CeAALR≤3.2 [29]. We point out that, in this study, this last coefficient is the one with the highest sensitivity by the eμ collider as compared to the low energy measurements.Seeing the relatively high sensitivity to the electromagneticQeAA operator, one may wonder what about the effective vertex eμGG? We can use the QeAA calculation with the analogous gluon operator
(11)QeGG=CeGGΛ4l¯μeeHGaμνGμνa.Thee−μ+⟶gg production cross section is now Nc=8 times greater than e−μ+⟶γγ, and a coefficient of order 0.5 would yield the minimum observable production. However, according to [29], the low energy μA⟶eA limit is three orders of magnitude more stringent in this case: CeGG≤1.6×10−3.As mentioned above, the amplitudes squared fore−μ+⟶γγ do not depend on the polar angle. That means that in terms of rapidity, for instance, y=y3∗ in the CM frame:
(12)dσdy=dσdcθdcθdy=a04exp2y1+exp2y2,where a0 is a constant. The shape of the rapidity distribution in the CM frame is then centered around zero with a width of approximately 2 units. In the lab frame, the center is shifted towards −1.70 (see next subsection).


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

In the SM,γγ production is given by the two-to-four process e−μ+⟶γγνeν¯μ, involving 13 Feynman diagrams in unitary gauge. There is also e−e+ production e−μ+⟶νeν¯μe−e+, involving 24 Feynman diagrams. In Figure 2, we show two representative diagrams for the γγ and resonant e−e+ 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 y are shifted with respect to those y∗ in the CM frame:
(13)y=y∗−y0,withy0=12lnEμEe=1.70,where y0=1.7 is the shift value for Eμ/Ee=30. The muon beam goes in the direction of −k^, and so the event products usually appear on the backward hemisphere.Figure 2
Representative unitary-gauge Feynman diagrams for the SM processes (a)e−μ+⟶γγνeν¯μ and (b) e−μ+⟶e−e+νeν¯μ.
(a)(b)Thus, to studyγγ production in e−μ+ collisions, we consider the three processes,
(14)e−μ+⟶γγ,(15)e−μ+⟶γγν¯μνe,(16)e−μ+⟶e−e+ν¯μνe,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 Ee,Eμ= (100, 3000), (150, 4500), and (200, 6000) GeV. We then have s=1.095, 1.643, and 2.191 TeV, respectively. In the two-body signal process (14), the CM energies of the final photons are fixed at E3∗=E4∗=s/2. This is unlike what happens with the backgrounds (15) and (16), where there is a continuous spectrum for E3∗ and E4∗. Similarly, for the signal process p⟶3T+p⟶4T=0, but for the backgrounds p⟶3T+p⟶4T=ET 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 y3,4∗=y3,4+y0≲2.5 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 E3,4∗, the photon rapidity distribution in the lab frame is symmetric about −y0=−1.70, but very broad. If we require E3,4∗ in (15) to be large, however, the final-state photons must be very forward or backward. For example, if E3,4∗≃250 GeV, the rapidities will have maxima at −y0±Δy with Δy≃2 as shown in Figure 3, and Δy gets larger for greater values of E3,4∗. For the process (16), the electron rapidity distribution presents essentially the same features but is less forward-backward symmetric, as seen in the figure.Figure 3
Normalized differential cross sections for final-state photons for processes (14) and (15) and e± for process (16), with respect to center-of-mass energy (a), transverse momentum (b), and center-of-mass rapidity (c). The beam energies are Ee,Eμ= (100 GeV, 3 TeV) (upper row), (150 GeV, 4.5 TeV) (middle row), and (200 GeV, 6 TeV) (lower row). Blue lines: signal process (14); black line: analytical expression (12); orange lines: background process (15); green lines: background process (16). All differential cross section correspond to detector-level events with cuts (18).We are, thus, led to consider the following set of phase-space cuts,(17)C0:p3T,p4T>1.0GeV,C1:E3∗,E4∗>500.0GeV,C2:pTtot=p⟶3T+p⟶4T<20.0GeV,C3:y3+y0,y4+y0<1.75.The cutC0 is necessary to control infrared divergences in (15) and (16). As discussed above, the cuts C1,2,3 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 Ee,Eμ= (100, 3000) GeV in Table 5. The cross section σγγ3 refers to the anomalous process (14) with only the trivalent eμγ vertex and the Wilson coefficients CeAL=CeAR=1. Similarly, σγγ4 refers to (14) through the eμγγ vertex and the coefficients CeAAL=CeAAR=1. The numerical results agree with (9).Table 5
Cumulative effects of the cuts (17) on the cross sections for (14), (15), and (16) at Ee=100 GeV, Eμ=3 TeV.
PμPeCutsσγγ3 (fb)σγγ4 (fb)σννγγ (fb)σννee (fb)0.00.0C02.160.0103770.2426.60.00.0C0,12.160.01030.00490.0730.00.0C0,1,22.160.01030.00150.00810.00.0C0–32.040.00973.23×10−50.00108+0.4+0.8C0,12.860.01360.00140.081+0.4+0.8C0,1,22.690.01280.000430.0082+0.4+0.8C0–32.690.01289.85×10−60.0038We expect the results for cross sections with cuts in Table5 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 C3 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 σγγ4, 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].Ane−μ+ 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 e−μ+ 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,
(18)C′0=C0,C′1:E3∗,E4∗>250.0GeV,C′2:pTtot=p⟶3T+p⟶4T<40.0GeV,C′3=∅.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 y∗, the transverse momentum pT, and the CM energy E∗ for the final-state photons in processes (14) and (15), and the final-state e± 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 σγγ4 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).Table 6
Cross sections in fb for the signal processes (14) obtained from the operator QeA, see equation (6) and Figure 1(a); (14) obtained from the operator QeAA, see equation (7) and Figure 1(b); and the background processes (15) and (16). All cross sections, at the three energies shown, obtained at the detector-simulation level with cuts C0−3 from (17).
PμPeσγγ3 (fb)σγγ4 (fb)σννγγ (fb)σννee (fb)Ee=100 GeV, Eμ=3 TeV0.00.01.720.0082~10−5~10−5+0.4+0.82.260.011≲10−5≲10−5Ee=150 GeV, Eμ=4.5 TeV0.00.01.610.039~10−4~10−5+0.4+0.82.120.051≲10−4≲10−5Ee=200 GeV, Eμ=6 TeV0.00.01.460.11~10−4≲10−5+0.4+0.81.930.15≲10−4≲10−5We then apply a preselection cut(19)Nγ≥2,Ne±=0,to the Delphes events. Notice that (19) implicitly includes a cut in absolute rapidity ∣y∗∣<2.5, corresponding to the detector rapidity acceptance range. Finally, we apply the cuts C0−3 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 s in Table 6. This is due to the fact that the upper limit on pTtot 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 s in Table 6, and process (16) actually decreases at the highest energy.Furthermore, comparing the results in Table6 at Ee,Eμ=100,3000 GeV with those in Table 5, we see that detector effects result in an effective efficiency of 84% for the γγ production processes given in the tables by σγγ3 and σγγ4. We see also that detector efficiencies reduce the cross section for the e−e+ background process (16) to the same O10−5fb level as the γγ background (15). Completely analogous results are obtained at the two higher energies considered in Table 6.


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

In the SM,γγ production is given by the two-to-four process e−μ+⟶γγνeν¯μ, involving 13 Feynman diagrams in unitary gauge. There is also e−e+ production e−μ+⟶νeν¯μe−e+, involving 24 Feynman diagrams. In Figure 2, we show two representative diagrams for the γγ and resonant e−e+ 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 y are shifted with respect to those y∗ in the CM frame:
(13)y=y∗−y0,withy0=12lnEμEe=1.70,where y0=1.7 is the shift value for Eμ/Ee=30. The muon beam goes in the direction of −k^, and so the event products usually appear on the backward hemisphere.Figure 2
Representative unitary-gauge Feynman diagrams for the SM processes (a)e−μ+⟶γγνeν¯μ and (b) e−μ+⟶e−e+νeν¯μ.
(a)(b)Thus, to studyγγ production in e−μ+ collisions, we consider the three processes,
(14)e−μ+⟶γγ,(15)e−μ+⟶γγν¯μνe,(16)e−μ+⟶e−e+ν¯μνe,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 Ee,Eμ= (100, 3000), (150, 4500), and (200, 6000) GeV. We then have s=1.095, 1.643, and 2.191 TeV, respectively. In the two-body signal process (14), the CM energies of the final photons are fixed at E3∗=E4∗=s/2. This is unlike what happens with the backgrounds (15) and (16), where there is a continuous spectrum for E3∗ and E4∗. Similarly, for the signal process p⟶3T+p⟶4T=0, but for the backgrounds p⟶3T+p⟶4T=ET 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 y3,4∗=y3,4+y0≲2.5 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 E3,4∗, the photon rapidity distribution in the lab frame is symmetric about −y0=−1.70, but very broad. If we require E3,4∗ in (15) to be large, however, the final-state photons must be very forward or backward. For example, if E3,4∗≃250 GeV, the rapidities will have maxima at −y0±Δy with Δy≃2 as shown in Figure 3, and Δy gets larger for greater values of E3,4∗. For the process (16), the electron rapidity distribution presents essentially the same features but is less forward-backward symmetric, as seen in the figure.Figure 3
Normalized differential cross sections for final-state photons for processes (14) and (15) and e± for process (16), with respect to center-of-mass energy (a), transverse momentum (b), and center-of-mass rapidity (c). The beam energies are Ee,Eμ= (100 GeV, 3 TeV) (upper row), (150 GeV, 4.5 TeV) (middle row), and (200 GeV, 6 TeV) (lower row). Blue lines: signal process (14); black line: analytical expression (12); orange lines: background process (15); green lines: background process (16). All differential cross section correspond to detector-level events with cuts (18).We are, thus, led to consider the following set of phase-space cuts,(17)C0:p3T,p4T>1.0GeV,C1:E3∗,E4∗>500.0GeV,C2:pTtot=p⟶3T+p⟶4T<20.0GeV,C3:y3+y0,y4+y0<1.75.The cutC0 is necessary to control infrared divergences in (15) and (16). As discussed above, the cuts C1,2,3 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 Ee,Eμ= (100, 3000) GeV in Table 5. The cross section σγγ3 refers to the anomalous process (14) with only the trivalent eμγ vertex and the Wilson coefficients CeAL=CeAR=1. Similarly, σγγ4 refers to (14) through the eμγγ vertex and the coefficients CeAAL=CeAAR=1. The numerical results agree with (9).Table 5
Cumulative effects of the cuts (17) on the cross sections for (14), (15), and (16) at Ee=100 GeV, Eμ=3 TeV.
PμPeCutsσγγ3 (fb)σγγ4 (fb)σννγγ (fb)σννee (fb)0.00.0C02.160.0103770.2426.60.00.0C0,12.160.01030.00490.0730.00.0C0,1,22.160.01030.00150.00810.00.0C0–32.040.00973.23×10−50.00108+0.4+0.8C0,12.860.01360.00140.081+0.4+0.8C0,1,22.690.01280.000430.0082+0.4+0.8C0–32.690.01289.85×10−60.0038We expect the results for cross sections with cuts in Table5 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 C3 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 σγγ4, 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].Ane−μ+ 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 e−μ+ 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,
(18)C′0=C0,C′1:E3∗,E4∗>250.0GeV,C′2:pTtot=p⟶3T+p⟶4T<40.0GeV,C′3=∅.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 y∗, the transverse momentum pT, and the CM energy E∗ for the final-state photons in processes (14) and (15), and the final-state e± 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 σγγ4 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).Table 6
Cross sections in fb for the signal processes (14) obtained from the operator QeA, see equation (6) and Figure 1(a); (14) obtained from the operator QeAA, see equation (7) and Figure 1(b); and the background processes (15) and (16). All cross sections, at the three energies shown, obtained at the detector-simulation level with cuts C0−3 from (17).
PμPeσγγ3 (fb)σγγ4 (fb)σννγγ (fb)σννee (fb)Ee=100 GeV, Eμ=3 TeV0.00.01.720.0082~10−5~10−5+0.4+0.82.260.011≲10−5≲10−5Ee=150 GeV, Eμ=4.5 TeV0.00.01.610.039~10−4~10−5+0.4+0.82.120.051≲10−4≲10−5Ee=200 GeV, Eμ=6 TeV0.00.01.460.11~10−4≲10−5+0.4+0.81.930.15≲10−4≲10−5We then apply a preselection cut(19)Nγ≥2,Ne±=0,to the Delphes events. Notice that (19) implicitly includes a cut in absolute rapidity ∣y∗∣<2.5, corresponding to the detector rapidity acceptance range. Finally, we apply the cuts C0−3 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 s in Table 6. This is due to the fact that the upper limit on pTtot 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 s in Table 6, and process (16) actually decreases at the highest energy.Furthermore, comparing the results in Table6 at Ee,Eμ=100,3000 GeV with those in Table 5, we see that detector effects result in an effective efficiency of 84% for the γγ production processes given in the tables by σγγ3 and σγγ4. We see also that detector efficiencies reduce the cross section for the e−e+ background process (16) to the same O10−5fb level as the γγ 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 thee−μ+ collider. For operators eμff¯ where f 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 μA⟶eA conversion in nuclei. On the other hand, the eμ collider would have higher sensitivity than the other low energy measurements μ⟶ee¯e and μ⟶eγ even for first family fermions. In the particular case of the Wilson coefficients Clq1,3, for example, the expected sensitivity at an eμ 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 eμ⟶γγ production, the limits from the collider are significantly more stringent than those from the low energy processes for the case of the dimension 8 eμγγ contact operator (7).In the SM, theff¯ and γγ 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 γγ 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−1 and beam energies Ee,Eμ=(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.



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