A Tale of Two Anomalies

A recent improved determination of the fine structure constant, $\alpha= 1/137.035999046(27)$, leads to a $\sim 2.4 \sigma$ negative discrepancy between the measured electron anomalous magnetic moment and the Standard Model prediction. That situation is to be compared with the muon anomalous magnetic moment where a positive $\sim 3.7 \sigma$ discrepancy has existed for some time. A single scalar solution to both anomalies is shown to be possible if the two-loop electron Barr-Zee diagrams dominate the scalar one-loop electron anomaly effect and the scalar couplings to the electron and two photons are relatively large. We also briefly discuss the implications of that scenario.

So far, neither the LHC experiments nor direct searches for dark matter have uncovered any signs of a "natural" Higgs sector nor weak scale dark matter states. However, there have been mild deviations from the Standard Model (SM) predictions over the years, some of which have disappeared with time. Of these, a longstanding one is the ∼ 3.7 σ discrepancy between theory and experiment for the muon anomalous magnetic moment g µ − 2 [1,2], which has withstood various theoretical refinements and is being currently remeasured at Fermilab with higher precision. While the final word on g µ − 2 remains to be decided by the new measurements and ongoing theoretical improvements of the SM prediction (see, for example, Refs. [3,4]), the deviation has been a subject of intense phenomenological interest. As new physics at the TeV scale gets more constrained, the parameter space for weak scale models that could explain g µ − 2 starts to close.
Meanwhile, the search for new "dark" or "hidden" states at low mass scales < ∼ 1 GeV has recently been getting increasing attention [5,6], partially spurred by astrophysical considerations related to DM models [7] and perhaps also by the dearth of indications for new high energy phenomena. In fact, g µ − 2 has emerged as an interesting target for dark sector searches, since light states with feeble couplings to the SM can in principle explain the anomaly. An early and motivated possibility was offered by the "dark photon" hypothesis, where a new vector boson that kinetically mixes with the photon [8] could have provided a solution [9]. This ideas and its simple extensions have now been essentially ruled out. However, other light states from a dark sector, for example a scalar that very weakly couples to muons could still furnish a potential solution [10].
A recent precise determination of the fine structure constant, α, has introduced a new twist to this story. An improvement in the measured mass of atomic Cesium used in conjunction with other known mass ratios and the Rydberg constant leads to the new now most precise * email: hooman@bnl.gov † email: marciano@bnl.gov value [11] α −1 (Cs) = 137.035999046(27).
(For a detailed explanation of that prescription and its use in determining the SM prediction for the electron anomalous magnetic moment, a e = (g e − 2)/2, see the articles by G. Gabrielse in Ref. [12].) As a result, comparison of the theoretical prediction of a SM which represents a 2.4 σ discrepancy that is opposite in sign from the long standing muon discrepancy previously mentioned. Note that the current discrepancy in Eqs. (2) and (3) results from an improvement in α −1 from 137.035998995(85) which previously [13] gave ∆a e = −130(77) × 10 −14 and represented a 1.7 σ effect.
The central value has decreased in magnitude, but its significance has increased. The errors from the experimental determinations of a e and α are now the dominant sources of uncertainty. Interestingly, simple dark photon models [9] and their extensions predict a positive deviation. Therefore, the negative ∼ 2.4 σ deviation in g e − 2 cannot be simultaneously explained together with the ∼ 3.7 σ anomaly in g µ − 2 in those models, even if one could circumvent experimental constraints.
In this short letter, we would like to point out that a minimal model based on a single light real scalar φ, can in principle explain the inferred values of both g µ − 2 and g e − 2, in a relatively economical fashion. We will show that a two-loop process Barr-Zee diagram [16,17] can explain g e − 2 while a one-loop contribution could be the primary origin of g µ − 2 [10,18], mediated by the same scalar φ. For a more detailed discussion of these loop processes and their contributions to the electron and muon anomalous magnetic moments see Ref. [19], where the authors discuss the relative contributions of one-and two-loop diagrams, but focus on the case of a pseudoscalar boson. Here, we focus on the effect of a light scalar where the Barr-Zee contribution represents an extension of earlier work in Ref. [10]. Work on the contribution of Barr-Zee type diagrams to g µ − 2 in the context of two Higgs doublet models and supersymmetry can be found in Ref. [20].
Let us consider the following Lagrangian for the real scalar φ of mass m φ where we only include an explicit coupling to a fermion f with strength λ f and have omitted various kinetic terms and fermion masses. In this work, we allow f to correspond to known quarks and leptons, as well as other potential more massive charged fermions. The λ f are constrained by phenomenology, as will be discussed later.
We assume that the coupling to photons, through the field strength tensor F µν , is governed by the the constant κ γ which has mass dimension −1. The sum over couplings to f will induce a contribution to κ γ , but we do not specify all charged states that couple to φ. Later, when we choose numerical values for our parameters we will discuss the range of values that may be expected on general grounds, consistent with various phenomenological constraints. We will start with g µ − 2, assumed to be dominated by the one-loop diagram in Fig.1, which is given by [10,21,22] for lepton of mass m and x ≡ m /m φ . Current experimental constraints, as illustrated in Ref. [23], allow 0.25 GeV < ∼ m φ < ∼ 1 GeV and λ µ ∼ (1 − 3) × 10 −3 , roughly corresponding to a range of parameters that can explain the 3.7σ deviation in g µ − 2, given by [3] which we will approximate as ∆a µ ≈ 3 × 10 −9 . Let us choose, for concreteness, m φ = 250 MeV and λ µ = 10 −3 , which according to Eq. (5) yields ∆a µ ≈ 3 × 10 −9 .
Since we will focus on m φ values significantly larger than m µ = 105.7 MeV in this work, we roughly have ∆a µ ∝ (λ µ /m φ ) 2 not far from our reference parameters in Eq. (7). However, for x 1, one finds ∆a ∝ −x 2 (ln x 2 + 7/6) which deviates from the 1/m 2 φ scaling. Let us now address the mild deviation in Eq. (3). Here, we will concentrate on the "Barr-Zee" diagram in Fig.2, whose leading contribution for a lepton is given by [19,25] ∆a ≈ λ κ γ m 4 π 2 ln(Λ/m φ ), where Λ is an ultraviolet cutoff scale. We will later discuss the form of κ γ obtained from integrating out heavy charged fermions of mass m f in the two-loop Barr-Zee diagram. In that case, one can show ln(Λ/m φ ) = 13/12 + ln(m f /m φ ) [24]. Assuming that the charged states that contribute to κ γ could be of GeV scale masses (e.g. the τ ), we roughly have ln(Λ/m φ ) ∼ 2 − 3. For m φ = 250 MeV, as above, and λ e κ γ ≈ −3×10 −8 GeV −1 , Eq. (8) yields ∆a e ≈ −9× 10 −13 , roughly at the level of the apparent experimental anomaly in Eq. (3). The negative sign can be obtained from the choice of various same sign Yukawa couplings, including those that contribute to κ γ . For concreteness, in the case of the electron, with m e = 0.511 MeV, we will take λ e = 3 × 10 −4 and κ γ = −10 −4 GeV −1 .
Here, we have assumed that the sign of the κ γ is negative, which is a choice corresponding to positive fermion Yukawa couplings to φ, as will be described later.
Note that the chosen value of λ e in Eq. (9) does not follow naive scaling with the lepton mass, λ e /λ µ = m e /m µ , in reference to that of λ µ in Eq. (7). However, since φ is not assumed to control the masses of the leptons, this is not an inconsistent choice and can be easily obtained from a simple effective theory that does not have hierarchic charged lepton interactions. We would also like to mention that for values of ln(Λ/m φ ) larger than those assumed in the preceding discussion, one could choose smaller values of |λ e,µ | due to the enhanced contributions of the Barr-Zee diagrams to both a e and a µ , for λ e κ γ < 0 and λ µ κ γ > 0, respectively. This would presumably originate from the coupling of φ to heavy charged states of mass > ∼ few × 100 GeV (the new charged states cannot be much lighter given the fair agreement of the TeV scale LHC data with the SM predictions).
For the above reference values, we note that the contribution from Eq. (5) is O (10 −14 ), which is negligible compared to that required by the apparent anomaly in Eq. (3). Similarly, the Barr-Zee diagram contribution to a µ from Eq. (8) is ∼ −6×10 −10 , a factor of ∼ 5 too small to and of the wrong sign to account for the anomaly in a µ from Eq. (6). To compensate for this ∼ 20% effect we could change our reference parameters in Eq. (7) very slightly, but the values chosen here suffice to illustrate that a simultaneous resolution of both the current a e and a µ anomalies can be obtained in our framework.
Aspects of phenomenology related to the coupling of φ to muons, including extensions to CP violating couplings, have been discussed before [10,18]. The coupling of light φ to leptons could lead to signals for "bump hunt" searches in scattering processes, as well as rare meson or tau decays involving muons or electrons, if kinematically allowed. Emission of φ from initial or final state electrons will be typically sub-dominant to those involving muons, for our above choice of parameters.
Regardless of the production process for φ, its dominant decay modes play an important role in its phenomenology and experimental implications. Let us first consider the decay of φ into an on-shell fermion pair f f that interact with it through the Yukawa coupling in Eq. (4). The partial width for this decay is given by The decay of φ into photons, assuming the coupling κ γ in Eq. (4), is given by (See, for example, Ref. [23].) The coupling κ γ is generated by the interactions of φ with charged fermions as will be discussed later. However, rather than specify all such charged states, we parametrize the φγγ overall interaction in terms of the effective coupling κ γ . We find that for the above chosen reference values (7) and (9), we get Γ(φ → µ + µ − ) ≈ 2 eV, Γ(φ → e + e − ) ≈ 1 eV, Γ(φ → γγ) ≈ 8 × 10 −4 eV, hence φ decays into muons and electrons with branching fractions of approximately 2/3 and 1/3, respectively. However, as m φ gets larger than 250 MeV, the phase space suppression for the µ + µ − final state becomes less important and the ratio of those branching fractions approaches ∼ 10, for our chosen values of λ µ , λ e , and m φ < ∼ 1 GeV. Let us briefly discuss potential models that could give rise to the types of couplings we have assumed. The coupling of φ to µ and e can be obtained from an effective operator of the form where M is the typical scale of new physics leading to the effective interaction above, L is a lepton doublet of the SM, and R is a right-handed charged lepton. To avoid constraints from flavor changing neutral current data, we generally assume that the structure of these interactions are flavor-diagonal. Also, for typical parameters similar to our reference values assumed before, the underlying interactions generating the above operators are roughly flavor universal, that is c e ∼ c µ . We then have λ ≡ c H /M . Assuming M ∼ 1 TeV, we find that c < ∼ 10 −2 . The scale M in Eq. (12) could be identified with the mass of a vector-like fermion F , with quantum numbers of R . In Ref. [10] a similar setup was assumed, however, here we simply take φ to be a singlet and not responsible for "dark" gauge symmetry breaking. Then, couplings of the form y H HLF R and y φ φF L R can generate the operator in Eq. (12), with c = y H y φ .
The φ coupling to photons κ γ ∼ 10 −4 GeV −1 can be generated from the assumed and implicit couplings of φ to charged fermions. For example, for possible couplings of φ to τ and charm of λ f ∼ few × 10 −2 , we expect to obtain the requisite photon coupling, to obtain the a e discrepancy for λ e = 3 × 10 −4 . One can see this from the estimate (see, for example, Ref. [26]) where Q f is the charge of the fermion, m f ∼ 1 GeV is of the order τ or charm mass, and N f c is the number of colors of fermion f . The above formula for κ γ is obtained in the limit that m f m φ . We note that, with our conventions, for λ f > 0 one obtains κ γ < 0. Other couplings, such as the one to muons, can enhance the above estimate for κ γ , assuming constructive interference from same sign Yukawa couplings.
In summary, we have shown that a simple model, comprising a singlet scalar φ of mass > ∼ 250 MeV and ∼ 10 −3 , 10 −4 couplings to the muon and the electron, respectively, can account for a ∼ 3.7 σ discrepancy in the muon g − 2 and a ∼ 2.4 σ discrepancy in the electron g − 2 of opposite sign. The former anomaly is mediated through a one-loop digram, whereas the latter originates from a two-loop Barr-Zee diagram, using a phe-nomenologically allowed coupling of the scalar to photons. The model could give rise to lepton pair signals in rare meson or tau decays, as well as those in electron and muon scattering processes. A simple effective theory that does not lead to naive scaling of the scalar-lepton couplings with the lepton mass can realize our scenario. The effective theory, in turn, could arise from TeV-scale charged vector-like fermions coupled to the SM Higgs and φ. In that case, the LHC could potentially discover those fermions, which would shed further light on the underlying physics manifested in the possible deviations of the electron and muon g − 2.
Work supported by the US Department of Energy under Grant Contract de-sc0012704.