A fake doublet solution to the muon anomalous magnetic moment

Extensions to the Standard Model that use strictly off-shell degrees of freedom - the fakeons - allow for new measurable interactions at energy scales usually precluded by the constraints that target the on-shell propagation of new particles. Here we employ the interactions between a new fake scalar doublet and the muon to explain the recent Fermilab measurement of its anomalous magnetic moment. Remarkably, unlike in the case of usual particles, the experimental result can be matched for fakeon masses below the electroweak scale without contradicting the stringent precision data and collider bounds on new light degrees of freedom. Our analysis, therefore, demonstrates that the fakeon approach offers unexpected viable possibilities to model new physics naturally at low scales.


Introduction
The longstanding anomaly concerning the magnetic moment of the muon, a µ = (g − 2) µ /2, is generally interpreted as the effect of new degrees of freedom that, coupling to the muon, leave their imprint in this precision observable. At the level of model building, the requirement of perturbative couplings forces the particles responsible for the signal to appear at energies not far above the electroweak scale. Consequently, the proposed (g − 2) µ explanations are often in tension with the null results of complementary collider and precision searches.
To this purpose, we address the (g−2) µ problem in a new framework that overcomes the limitations of conventional model building by relying on new, strictly-virtual, degrees of freedom: the fakeons. Fakeons were originally proposed to solve the problem of ghosts in renormalizable theories of gravity [6] and Lee-Wick theories [7,8,9]. Nevertheless, any new particle can be made a fakeon by adopting the required prescription for its propagator.
Concretely, we consider an extension of the SM that contains a new fake scalar doublet. The latter does not acquire a vacuum expectation value (VEV) and, besides the gauge and Higgs bosons, couples significantly only to muons. Although the scalar sector of the theory matches that of the Fake Inert Doublet Model [10] (fIDM), the resulting phenomenology strongly differs for the presence of a Yukawa coupling that breaks the Z 2 symmetry of the fIDM and singles out the leptons of the second SM generation.
As we show below, the fakeon doublet explains the (g − 2) µ measurement (1) even in a mass range that, for usual particles, is precluded by the measurements of the Zand W -boson decay widths. To demonstrate the case, we check our solution against the constraints that target deviations from lepton universality in the Z and W boson decays. In particular, we study the contributions of the fake doublet to Z → 2µ, Z → 4µ and to the τ and µ lepton decays, showing that these processes do not impose significant constraints on our result.
The present study could be extended to address the anomalies pertaining to the measurement of the muonic proton radius [11] and possible lepton universality violation in the decays of B mesons [12]. It would be also of interest to consider a higher-order kinetic term for the fake doublet, with the purpose of solving also the SM hierarchy problem through the mechanism previously used in Lee-Wick extensions [13,14,15]. In fact, the fakeon prescription can also be used to consistently include fake ghost particles in the theory and prevent their on-shell propagation.

A fake doublet extension of the Standard Model
We consider the following Lagrangian, where µ L is the second-generation left-handed lepton doublet, H is the SM Higgs doublet and Φ = (φ + , φ 0 ) T is a fakeon doublet which transforms as {1, 2, 1/2} under the SU(3) c × SU(2) L × U(1) Y gauge group. We decompose the complex neutral component in its scalar and pseudoscalar parts, Since Φ acquires no VEV, the scalar fakeon masses are where v is the Higgs doublet VEV. We adjust λ 2 so as to ensure that the potential is bounded from below and implicitly set the values of m 2 and λ 3 by specifying m φ ± .
The λ 4 and λ 5 couplings, which regulate the hierarchy of the fakeon masses, are assumed to vanish unless specified otherwise. More details on the scalar sector of the fIDM can be found in [10].
To explain the anomaly, we couple Φ to the muon via a real Yukawa coupling y and assume negligible couplings to the remaining SM fermions. The resulting new interactions suffice to fully capture the fakeon physics of (g − 2) µ .

Fakeons: main phenomenological features
The fakeon propagator relies on a quantization prescription that differs from the Feynman recipe adopted for the SM fields. As a result, fakeons can mediate new interactions in the same way as usual particles do, but cannot be on-shell. Consequently, fakeons do not appear in initial and final states of physical processes, and, in particular, cannot leave any direct imprint in an experimental apparatus after their propagation.
In particular, the fakeon doublet cannot directly contribute to the decay width of the Z boson even for masses below m Z /2: the decay Z → Φ † Φ, allowed by gauge interactions, is forbidden by the fakeon quantization. Therefore, the results of LEP experiments [16] do not affect our solution, contrarily to the case of new ordinary particles. However, constraints on the properties of fakeons can still arise from their virtual contributions at collider and precision experiments. Besides the processes characteristic of fIDM [10], further phenomenological signatures arise from the breaking of the Z 2 symmetry. For instance, the fakeon doublet mediates four-lepton final states decays of the Z boson at the tree level, and contributes to both the di-muon and invisible Z decay widths radiatively. Additionally, loop diagrams that contain fakeons are also modified above every threshold associated with resonant contributions. In the case studied, the most evident impact is on the imaginary parts of the involved amplitude.

The anomalous magnetic moment of the muon
The interactions of the fakeon field in Eq. (8) affect the muon anomalous magnetic moment through the diagrams in Fig. 1, which account for the effect of the neutral and charged fake scalars, respectively. The total correction to the SM result, ∆a Φ µ = ∆a H µ + ∆a A µ + ∆a ± µ ,

THE ANOMALOUS MAGNETIC MOMENT OF THE MUON
The interactions of the fakeon field in Eq. (8) a↵ect the muon anomalous magnetic moment through the diagrams shown in Fig. 1, which account for the e↵ect of the neutral and charged fake scalars, respectively. The total correction a µ = a H µ + a A µ + a ± µ is correspondingly split into three terms that add to the SM result. The contributions of neutral components H and A are computed as where m µ ' 106 MeV is the muon mass and h(x) = arctanh( p x)/ p x for 0 < x < 1, h(x) = arctan( p |x|)/ p |x| for x < 0. The relations above match the expressions presented in the literature for standard neutral scalar and pseudoscalar contributions -see for instance [23] -as kinematics prevent this class of diagrams from developing an imaginary part. In addition, for the charged fakeon contribution we have where r = m 2 µ /m 2 + . This relation does di↵er from the standard charged scalar contribution a ±, std µ on part of the parameter space by an imaginary part given by with ⇥(x) being the Heaviside step function. In fact, for m + < m µ and vanishing neutrino masses, the process generally allows an imaginary part associated with the cuts shown in the second diagram of Fig. 1, which, for a standard scalar, corresponds to the µ ! ⌫ decay. Because fakeons cannot ever be on-shell, the fakeon prescription forces these imaginary parts to vanish, so that a ± µ is necessarily real. For usual scalars, instead, the same cuts yield a finite imaginary contribution that redefines the form factor actually probed by the dedicated experiments [24]. The appearance of such imaginary contributions would furthermore alter the lifetime of the muon in an external magnetic field [25].
In Fig. 2, we show the parameter space where the total fakeon contribution a µ enters the 1 and 2 intervals allowed by Eq. (3), for two benchmark fakeon mass spectra. The orange band corresponds to the degenerate limit m + = m A = m H , whereas the blue region is for solutions that allow a mild splitting of the neutral scalar component m + = m A = 5m H , corresponding to 4 = 5 . O(1) on the analysed range. The remaining shaded areas denote the exclusion bounds due to the experimental searches discussed in the following. As we can see, these do not a↵ect the results obtained for Yukawa couplings y . 0.1, corresponding to charged fakeon masses below ⇠ 10 GeV. In the region m + < m µ , the fakeon solution di↵ers from the one obtained for a standard scalar by the term in Eq. (12). For m ⌧ m µ (corresponding to r 1) the loop functions tend to a is split into three terms. The contributions of neutral components φ H and φ A are where m µ 106 MeV is the muon mass and h(x) = arctanh( √ x)/ √ x for 0 < x < 1, h(x) = arctan( |x|)/ |x| for x < 0. The relations above match the expressions in the literature for standard neutral scalar and pseudoscalar contributions -see e.g. [17,18] -because kinematics prevent this class of diagrams from developing an imaginary part. For the charged fakeon contribution, we have where r = m 2 µ /m 2 φ ± . This relation does differ from the standard charged scalar contribution ∆a ±, std µ on part of the parameter space by an imaginary part given by with Θ(x) being the Heaviside step function. In fact, for m φ ± < m µ and vanishing neutrino masses, the process generally allows an imaginary part associated with the cuts shown in the second diagram of Fig. 1. For a standard scalar field this corresponds to the µ → φν decay. The fakeon prescription forces these imaginary parts to vanish, so that ∆a ± µ is necessarily real. For usual particles, instead, the same cuts yield a finite imaginary contribution that redefines the form factor actually probed Figure 2: Regions of the parameter space (m φ H , y ) where the fake doublet contribution to a µ falls within the 1σ and 3σ bounds on ∆a Fl µ for two representative cases: λ 4 = −0.0003 and λ 5 = 0 (orange) and λ 4 = λ 5 = −0.002 (blue). The remaining shaded regions represent the bounds from the indicated measurements.
by the experiments [19]. Such imaginary contributions would furthermore alter the muon lifetime in an external magnetic field [20].
In Fig. 2, we show the parameter space where the total fakeon contribution ∆a Φ µ enters the 1σ and 3σ intervals allowed by Eq. (3), for two benchmark fakeon mass spectra. The orange band corresponds to a configuration where m φ ± m φ A = m φ H , obtained by setting the quartic couplings as specified in the figure caption. The choice ensures that m φ ± ≥ 3 GeV, as required by electroweak precision tests [10]. The blue region shows the case of a different splitting, m φ H m φ A = m φ ± , obtained for different values of the quartic coupling that result in m φ ± ≥ 10 GeV. The remaining shaded areas denote the exclusion bounds due to the experimental searches discussed below. The results obtained for Yukawa couplings y 0.1, corresponding to (CP-even) neutral fakeon masses below ∼ 10 GeV are mostly constrained only by the mentioned electroweak precision tests, which force a splitting of the charged fakeon component but do not bound the masses of the neutral ones [10]. In both the analyzed cases, the observed values of (g − 2) µ are matched through the dominant contribution of the neutral CP-even fake component.
We remark that our solutions are stable under radiative corrections because the negligible Yukawa couplings assumed for heavier SM fermions, as well as the absence of a VEV for the fake doublet, preclude new large two-loop constant and a µ is well approximated by Focusing on the degenerate limit (orange band), the explanation of the new (g 2) µ measurement requires large values of the Yukawa coupling for fakeon masses above 10 GeV. This is due to a significant cancellation between the contributions of the scalar and pseudoscalar fakeons, which are respectively positive and negative. For the considered non-degenerate case (blue band) the cancellation is weaker and thus allows to match the measured a µ for smaller values of y. The gap close to m + = m µ indicates the presence of a solution dominated by a (negative) charged scalar contribution, which is rejected by the data for a positive overall sign of the fakeon doublet propagator. 1 For lower masses, the positive scalar contribution a H µ dominates resulting in Eq. (13). Before detailing the experimental constraints arising from precision tests, we remark that the presented solution is stable under radiative corrections, because the negligible Yukawa couplings assumed for heavier SM fermions, as well as the absence of a VEV for the fake doublet, preclude new large two-loop Barr-Zee type contributions [26]. We also remark that the additional charged doublet at the muon mass scale changes the running of electromagnetic coupling constant ↵. However, the new e↵ect due to running from ↵(q = m µ ) to ↵(q = M Z ) is unobservably small [27].

COLLIDER AND LEPTON FLAVOR UNIVERSALITY CONSTRAINTS
As we have previously emphasized, the most important experimental fact concerning fakeons is that the decays 1 The fakeon prescription can also be used to consistently include fake ghost particles in the theory by preventing their on-shell propagation. It is possible to turn a fakeon into a fake ghost by simply inverting the signs in front of its propagator and the gauge vertices that involve the fakeon. In our case, a fake ghost doublet would then match the g 2 measurement only in correspondence of the gap for m + ' mµ, through the charged scalar contribution.
Z ! H A and Z ! + , allowed by gauge interactions, cannot occur because fakeons cannot appear as on-shell final states. Therefore, the Z-boson decays [21] can only constrain the fakeon properties through their virtual e↵ects in tree-level or loop processes yielding, in our case, muon final states. In the following we analyze the most important examples of these contributions. The LEP measurements of the Z-boson leptonic decay widths provide stringent constraints on departures from lepton flavor universality. In the model at hand, muon final states receive new loop-level contributions from the diagrams depicted in Fig. 3, which potentially unbalance the yield of the corresponding Z-boson decay channels. The corresponding departure from lepton universality is constrained by [21] for the di-muon final state. Enhancements in the dineutrino final state are, instead, constrained by the Zboson invisible decay width (inv) = 499.0 ± 1.5 MeV.
In order to assess the bound (14), we have calculated the leading order deviations of R µe from the SM value. The fakeon contribution sourced by the diagrams in Fig. 3

is quantified in [4]
where g Z is the neutral current coupling constants, c V = 2 s 2 W 1/2, c A = 1/2 and c V,A are, respectively, the coe cients of the terms proportional to µ and µ 5 in the obtained one-loop amplitude. The kinematics of the diagrams shows that the three internal lines cannot be all simultaneously on-shell. Therefore, the fakeon prescription a↵ects only the imaginary part of the diagram. Because Eq. (15) is sensitive only to the real contribution, we can employ the relation to constrain the properties of the model by requiring that | R µe | < 0.0012. The corresponding bound is indicated by the gray region in Fig. 2, together with the prediction for (g 2) µ . As we can see, the limit is far from excluding our solutions to (g 2) µ . Similarly, the new contribution to the Z-boson invisible width is also not constrained. We remark that for the case of usual scalar fields, these solutions would be precluded by the modifications to the tree-level Z-boson decay width resulting from the direct production of the new scalars.

Collider and lepton flavor universality constraints
As previously emphasized, the decays Z → φ H φ A and Z → φ + φ − , allowed by gauge interactions, cannot occur. Therefore, the Z-boson decays [16] can only constrain the fakeon properties through their virtual effects in tree-level or loop processes yielding, in our case, muon final states. In the following we analyze the most important examples of these contributions.
The LEP measurements of the Z-boson leptonic decay widths provide stringent constraints on departures from lepton flavor universality. In the model at hand, muon final states receive new loop-level contributions from the diagrams depicted in Fig. 3, which potentially unbalance the yield of the corresponding Z-boson decay channels. The corresponding departure from lepton universality is constrained by [16] for the di-muon final state. Enhancements in the di-neutrino final state are, instead, constrained by the Z-boson invisible decay width Γ(inv) = 499.0 ± 1.5 MeV.
In order to assess the bound (13), we have calculated the leading order deviations of R µe from the SM value. The fakeon contribution sourced by the diagrams in Fig. 3 is quantified in [22] where g Z is the neutral current coupling constants, c V = 2 s 2 W − 1/2, c A = −1/2 and c Φ V,A are, respectively, the coefficients of the terms proportional to γ µ and γ µ γ 5 in the obtained one-loop amplitude. The kinematics shows that the three internal lines of the diagrams cannot be all simultaneously on-shell. Therefore, the fakeon prescription affects only the imaginary part of the amplitude. The diagrams of Fig. 4 must be summed, then multiplied by the complex conjugates and integrated over the phase space of the four muons. The result of these operations is a sum of cut diagrams with fakeons circulating in loops, which must be evaluated, as usual, by means of the fakeon prescription. The usual techniques are not immediately applicable, but the impact of the result on our analysis can be estimated by means of simpler arguments. On dimensional grounds, we have contributions of the form whereg is the coupling to the gauge vertex. For y ⇠ 0.1, we obtain a correction to BR(Z ! 4µ) of order 10 6 , which is comparable to the experimental precision. Conservatively, we infer in Fig. 2 by a meshed region that values smaller than y ⇠ 0.1 are compatible with the constraint.
The same process was also recently studied in Ref. [32], which quantified the contribution of the first diagram in Fig. 4 in a toy model where the fake doublet is replaced by a usual scalar. For comparative purposes, we include the result of their analysis in Fig. 2, where it is shown in purple. Due to the specific kinematics of the process, and since the CMS and ATLAS experiments impose a cut on the invariant mass of lepton pairs of m µµ < 4 GeV and m µµ < 5 GeV, respectively, the constraint is peaked in a narrow window around 10 GeV. Regardless of the details pertaining to the treatment of the new di-muon resonance in their simulation, we expect their bound to hold, at least, at the order of magnitude level.
The last precision observables that we analyze probe deviations from lepton flavor universality in the decays of the ⌧ and µ leptons. The former, in particular, provides the most stringent constraints on the solutions to the (g  In our c  the mod  depicted  the limi  the obse  evant lo  prescrip To qu coe cie the two cay of a is [33] L`!  Because Eq. (14) is sensitive only to the real contribution, we can employ the relation to constrain the properties of the model by requiring that |∆R µe | < 0.0012. The corresponding bound is indicated by the gray region in Fig. 2. As we can see, the limit is far from excluding our solutions to (g −2) µ . Similarly, the new contribution to the Z-boson invisible width is also not constraining. We remark that for the case of usual scalar fields, these solutions would be precluded by the modifications to the tree-level Z-boson decay width resulting from the direct production of the new scalars.
The diagrams of Fig. 4 must be summed, then multiplied by the complex conjugates and integrated over the phase space of the four muons. The result is a sum of cut diagrams with fakeons circulating in loops, which must be evaluated with the fakeon prescription. The usual techniques are not immediately applicable, but the impact of the result on our analysis can be estimated by means of simpler arguments. On dimensional grounds, we have contributions of the form whereg is the coupling in the gauge vertex. For y ∼ 0.1, we obtain a correction to BR(Z → 4µ) of order 10 −6 , which is comparable to the experimental precision. Conservatively, we infer that values smaller than y ∼ 0.1 are compatible with the constraint. We highlight this bound in Fig. 2 with a meshed region.
The same process was also recently studied in Ref. [27], which quantified the contribution of the first diagram in Fig. 4 in a toy model where the fake doublet is replaced by a usual scalar. The result is shown in Fig. 2 in pur- The diagrams of Fig. 4 must be summed, then multiplied by the complex conjugates and integrated over the phase space of the four muons. The result of these operations is a sum of cut diagrams with fakeons circulating in loops, which must be evaluated, as usual, by means of the fakeon prescription. The usual techniques are not immediately applicable, but the impact of the result on our analysis can be estimated by means of simpler arguments. On dimensional grounds, we have contributions of the form whereg is the coupling to the gauge vertex. For y ⇠ 0.1, we obtain a correction to BR(Z ! 4µ) of order 10 6 , which is comparable to the experimental precision. Conservatively, we infer in Fig. 2 by a meshed region that values smaller than y ⇠ 0.1 are compatible with the constraint.
The same process was also recently studied in Ref. [32], which quantified the contribution of the first diagram in Fig. 4 in a toy model where the fake doublet is replaced by a usual scalar. For comparative purposes, we include the result of their analysis in Fig. 2, where it is shown in purple. Due to the specific kinematics of the process, and since the CMS and ATLAS experiments impose a cut on the invariant mass of lepton pairs of m µµ < 4 GeV and m µµ < 5 GeV, respectively, the constraint is peaked in a narrow window around 10 GeV. Regardless of the details pertaining to the treatment of the new di-muon resonance in their simulation, we expect their bound to hold, at least, at the order of magnitude level.
The last precision observables that we analyze probe deviations from lepton flavor universality in the decays of the ⌧ and µ leptons. The former, in particular, provides the most stringent constraints on the solutions to the limit of vanishing W squared momentum in which the observables are matched, the computation of the relevant loop contributions is not modified by the fakeon prescription.
To quantify the constraints, we compute the Wilson coe cients of the e↵ective four-fermion interactions for the two processes. The dimension-6 operator for the decay of a charged lepton`= ⌧ (µ) into a lepton`0 = µ(e) is [33] L`!`0 = 4 where G F is the Fermi constant and g S,`0R R and g V,`0L L are the Wilson coe cients due to SM and fakeon contributions. We compare our results to the experiments by using the Michel parameters [34-36]. The most constraining in our case is ⇢, which, for the operators in (17), reduces to For the ⌧ ! µ decay, experiments find [21] ⇢ exp ⌧ !µ = 0.763 ± 0.020, (19) whereas in the case of µ ! e the inferred value of the parameter is [37, 38] The deviation of the ⇢ parameter due to the new physics can be expressed through the quantity where g V,SM LL = 1 and g V,``0 LL is the new contributions due to the presence of the fakeon doublet. The latter is obtained by renormalizing the four-fermion amplitude in the on-shell renormalization scheme used for the SM, and by taking the zeroth order in the expansion of the amplitude in powers of s/m 2 W , where s is the centerof-mass energy squared. In the analysis we focused on the radiative corrections proportional to the new Yukawa couplings, in order to guarantee the that amplitude be finite. We however ignored Yukawa independent corrections that depend on the splitting of the fakeon mass spectrum. For the ⌧ decay, we obtain the bound shown in Fig.2 by requiring | ⇢ ⌧ !µ | < 0.020, while for the µ decay we use | ⇢ µ!e | < 0.00026. The limits we find do not constrain the proposed solution. ple. Due to the specific kinematics of the process, and since the CMS and ATLAS experiments impose a cut on the invariant mass of lepton pairs of m µµ < 4 GeV and m µµ < 5 GeV, respectively, the constraint peaks in a narrow window around 10 GeV. Regardless of the details pertaining to the treatment of the new di-muon resonance in their simulation, we expect the bound to hold, at least, at the order of magnitude level.
The last precision observables that we analyze probe deviations from lepton flavor universality in the decays of the τ and µ leptons. The relevance of these bounds follows from the modifications in the muon sector due to the diagrams in Fig. 5. Since no thresholds are involved in the limit of vanishing W squared momentum in which the observables are matched, the relevant loop contributions are not modified by the fakeon prescription.
To quantify the constraints, we compute the Wilson coefficients of the effective four-fermion interactions for the two processes. The dimension-6 operator for the decay of a charged lepton = τ (µ) into a lepton = µ(e) is [28] where G is the effective Fermi constant and g S, RR and g V, LL are the Wilson coefficients due to SM and fakeon contributions. We compare our results to the experiments by using the Michel parameters [29,30,31]. The most constraining in our case is ρ, which, for the operators in (16), reduces to For the τ → µ decay, experiments find [16] ρ exp τ →µ = 0.763 ± 0.020, whereas for µ → e it is [32,33] ρ exp µ→e = 0.74997 ± 0.00026.
The deviation of the ρ parameter due to the new physics can be expressed through where g V,SM LL = 1 and δg V, LL is the new contribution due to the presence of the fakeon doublet. The latter is obtained by renormalizing the four-fermion amplitude in the on-shell renormalization scheme used for the SM, and taking the zeroth order in the expansion of the amplitude in powers of s/m 2 W , where s is the center-of-mass energy squared. For the τ decay, we obtain the bound shown in Fig.2 by requiring |∆ρ τ →µ | < 0.020, while for the µ decay we use |∆ρ µ→e | < 0.00026. The limits we find do not constrain the proposed solution.

Summary
We have proposed a new solution to the puzzle of the muon anomalous magnetic moment by modelling the physics beyond the SM with purely virtual degrees of freedom: the fakeons. Considering a new fake scalar doublet interacting sizeably only with muons, electroweak gauge bosons and the Higgs field, we have shown that the new (g − 2) µ measurement can be explained in a large part of the parameter space. The predictions, together with the most important bounds arising from complementary collider and precision observables, are collected in Fig. 2.
Unlike for ordinary particles, our results for (g − 2) µ are not significantly impaired by the constraints. Fakeon masses at, or below, the GeV scale are not excluded by the precision measurements of the Z-boson decay width, because the fakeon quantization prescription precludes the production of these particles. The precision tests of lepton universality in Z and W -boson decays, sensitive to deviations at the per-mille level, also fail to exclude our scenario. The collider measurements of Z → 4µ decays test the solution obtained for a degenerate mass spectrum only in a corner of the parameter space with large values of the Yukawa coupling and masses. We find no constraints on the solutions obtained for a mild hierarchy in the fakeon masses.
In conclusion, the analyzed framework allows for new effects well below the electroweak scale without contradicting the available experimental results. Our work motivates further studies of the phenomenology of fake particles in the context of the anomalies in low energy physics observables related to muons. opment Fund CoE program TK133 "The Dark Side of the Universe".