Axion-like particles, two-Higgs-doublet models, leptoquarks, and the electron and muon g-2

Data from the Muon g-2 experiment and measurements of the fine structure constant suggest that the anomalous magnetic moments of the muon and electron are at odds with standard model expectations. We survey the ability of axion-like-particles, two-Higgs-doublet models and leptoquarks to explain the discrepancies. We find that accounting for other constraints, all scenarios except the Type-I, Type-II and Type-Y two-Higgs-doublet models fit the data well.


Introduction
The high-intensity and high-precision frontiers are ideal for the search for new physics that couples very feebly with the standard model (SM) sector. A long-standing and perhaps best known example that indicates such physics is the 3.7σ anomaly in the anomalous magnetic moment of the muon a µ = (g − 2) µ /2: where a BNL µ = (116592089±63)×10 −11 [1,2] and the SM expectation is a SM µ = (116591810± 43)×10 −11 [3]. A new lattice QCD calculation of the hadronic vacuum polarization suggests that the BNL measurement is compatible with the SM and that no new physics need be incorporated [4]. Until this result is confirmed, we subscribe to the SM value of Ref. [3]. Recently, the Muon g-2 experiment at Fermilab reported the value, a FNAL µ = (116592040 ± 54) × 10 −11 [5], i.e, which is a 3.3σ discrepancy. The combined significance of the anomaly from the Fermilab and BNL measurements is 4.25σ with [5] ∆a µ = a exp µ − a SM µ = (251 ± 59) × 10 −11 . (1.3) Interestingly, new precise measurements of the fine-structure constant α imply a discrepancy in the anomalous magnetic moment of the electron as well. A measurement of α at Laboratoire Kastler Brossel (LKB) with 87 Rb atoms [6] improves the accuracy by a factor of 2.5 compared to the previous best measurement with 137 Cs atoms at Berkeley [7]. The LKB measurement deviates by 5.4σ from the Berkeley result. With these two measurements of α, the SM predictions for the electron anomalous magnetic moments, a LKB e and a B e [8,9] Note the opposite signs of ∆a LKB e and ∆a B e . In this work, we study how well pseudoscalar axion-like particles (ALPs), two-Higgsdoublet models (2HDMs), and leptoquarks (LQs) can provide a common explanation of the anomalies in ∆a µ and ∆a LKB, B e .

Axion-like particles
An ALP, in general, can couple to the photon and leptons via the effective interactions [16], where g aγγ is a dimensionful coupling, and F µν andF µν are the electromagnetic tensor and its dual, respectively. We can take g aγγ to be positive by absorbing a phase into the definition of the field a. Then the sign of y a becomes physical. If Λ is the ultraviolet cut-off of the effective theory, g aγγ = 2 √ 2αc aγγ /Λ with dimensionless coupling c aγγ . The first term in Eq. (2.1) induces the two loop light-by-light (LbL) diagram which is analogous to the SM hadronic contribution from π 0 exchange [11][12][13]. Both terms in Eq.(2.1) contribute to g − 2 via Barr-Zee (BZ) diagrams [14]. By only keeping the leading log, these contributions to a give a ,a = a 1−loop  [16] .
Here, m a is the ALP mass and the one-loop function for a pseudoscalar is [15] f A (r) = ALP masses between 0.1 GeV to 10 GeV are allowed for non-negligible g aγγ [17]. In particular, m a ≤ 0.1 GeV is restricted by beam dump experiments, and LEP data on the decay Z → 3γ constrains m a ≥ 10 GeV via the process e + e − → γ * → aγ → 3γ [18]. Also, Z → 2γ data at LEP provide a constraint if photons from a → 2γ are collimated as a single photon. An upper bound g aγγ O(10 −2 ) GeV −1 is obtained for 1 MeV ≤ m a ≤ 10 GeV [17]. For this coupling, unitarity requires an upper bound, Λ 1 TeV [16]. To obtain parameter values preferred by the data, we separately fit ∆a µ , ∆a LKB e , and ∆a B e , and also fit the combinations, ∆a µ and ∆a LKB e , and ∆a µ and ∆a B e . We do not fit ∆a LKB e and ∆a B e simultaneously. We use the following χ 2 definitions: (2.4) Similar definitions will apply for 2HDMs and leptoquarks. Guided by the constraints mentioned above, we scan the parameter space in two scenarios which have the same number of free parameters: • ALP-1: We fix m a = 0.2, 1 GeV, and vary g aγγ , y aµ , and y ae . The results are shown in Figs. 1 and 2.
• ALP-2: We vary m a , g aγγ and y aµ = y ae . The results are shown in Figs. 3 and 4.
The minimum χ 2 value in all cases is zero indicating that the deviations from the SM predictions can be exactly reproduced. Since the χ 2 distributions are very shallow around the minima, we do do not provide best-fit ALP points.
In the left panel of Fig. 1, the plateau for small Yukawa couplings arises from the LbL contribution. For large negative y aµ , the BZ and 1-loop contributions interfere destructively with the LbL contribution which requires large g aγγ values excluded by beam-dump experiments. For large positive y aµ the BZ and LbL contributions interfere constructively so that the size of g aγγ is reduced to fit the data. However, as y aµ increases, the 1-loop contribution interferes destructively with the BZ and LbL contributions, which causes g aγγ to rise again. A similar reasoning explains the structure of the a LKB e allowed region. A plateau does not appear in the allowed region for ∆a B e because it is negative.

Two-Higgs-doublet models
We now consider two-Higgs-doublet models. In addition to the light Higgs h, the scalar sector is comprised of a heavy Higgs H, pseudoscalar A, and two charge Higgses H ± , which contribute to the electron and muon g − 2 through either 1-loop triangle diagrams or twoloop BZ diagrams. There are five relevant parameters, m A , m H , m H ± , β, and α. The ratio of the vacuum expectation values of the two scalar doublets Φ 1,2 defines tan β ≡ v 2 /v 1 . The mixing between the CP-even neutral components h 1,2 of Φ 1,2 , and the mass eigenstates h, H is given by the angle α [19]: To satisfy the stringent constraints on flavor changing neutral currents, the up-type quarks, down-type quarks and leptons must have Yukawa couplings to Φ 1 or Φ 2 , but not both. This requirement leads to four types of 2HDMs: Type-I (all fermions couple to Φ 2 ), Type-II (only up-type quarks couple to Φ 2 ), Type-X (lepton-specific, in which only leptons couples to Φ 2 ), Type-Y (flipped, in which only down-type quarks couples to Φ 1 ) [15,19]. The rare decay b → sγ requires m H ± 300 GeV for tan β > ∼ 2 for Type-I and Type-X [19] (m H ± 580 GeV for Type-II and Type-Y [20]), which renders their contributions to g − 2 subdominant. Higgs precision measurements from ATLAS and CMS prefer h to be SM-like with cos(β − α) → 0, and H decoupled. After fixing m h = 125.5 GeV, we are left with only two parameters m A and tan β that affect g − 2. The Yukawa interactions in four types of 2HDMs are dictated by tan β: with the normalized Yukawa couplings as listed in Table 1. The contribution to the anomalous magnetic moments is [15]  where   Table 2.
Not surprisingly, the Type-I model does not reproduce the data because it is similar to the SM. Type-Y is similar to Type-I except that the bottom quark contribution is enhanced. However, because of the lightness of the bottom quark, its contribution is not enough for the Type-Y model to explain the data. For Type-X, the large value of tan β enhances the tau-lepton contribution to the two-loop BZ diagram. The contribution from the bottom-quark in the BZ diagram provides a further enhancement in the Type-II model. Note that the Type-II parameters needed are excluded by constraints from B s → µ + µ − and searches for Z → bbA(bb) [15]. Type-X cot β − cot β tan β

Leptoquarks
We consider a scalar leptoquark S 1 ∼ (3, 1, −1/3) and a doublet leptoquark (R 2 ) T = (R 5/3 2 , R 2/3 2 ) ∼ (3, 2, 7/6). Their couplings to quarks and leptons are specified in the "uptype" mass-diagonal basis because the "down-type" basis would violate constraints from µ → eγ [21]. Then, the CKM matrix appears in the couplings with down-type quarks, and the interaction Lagrangian is [22] (4.1) have left-and right-handed couplings to the charge leptons and up-type quarks, and give a large contribution to a under the condition m q m . We neglect the contribution of R 2/3 2 , which only has right-handed couplings to down quarks. In the limit, m q m LQ ,  , and ∆a µ and ∆a B e (lower panels), for m S1 = 10 TeV (black) m S1 = 2 TeV (red). freedom to choose the texture of the Yukawa couplings y L,R ij . According to Eq. (4.2), heavier fermions contribute more to a , so we ignore the u quark and τ lepton. The remaining couplings are y L,R ec, et, µc, µt . However, if both y et and y µt are non-zero, the m t enhancement of µ → eγ becomes incompatible with observation [22]. A non-zero y µc allows the LQ to couple to neutrinos and the s quark, thereby inducing K + → π + νν through CKM mixing. To obey these constraints we set y et = y µc = 0. Finally, we have the Yukawa couplings, Taking the couplings to be real, we scan the parameter space in two scenarios: • S 1 -LQ: We fix m S 1 = 2, 10 TeV, and vary y L ec y R ec and y L µt y R µt . The results are shown in Fig. 9 and Table 3.
• R 2 -LQ: fixing m R 2 = 2, 10 TeV, and varying y L ec y R ec , and y L µt y R µt . The results are shown in Fig. 10 and Table 3.

Summary
In light of the recent measurement of the muon anomalous magnetic moment by the Muon g-2 experiment, we examine three model frameworks as explanations of the (g − 2) e,µ discrepancy with standard model expectations. We considered i) axion-like particles with Table 3. Best-fit points for the scalar and doublet leptoquarks. The minimum χ 2 value in all cases is 0. masses O(1) GeV and couplings to charged leptons and photons, which yields a contribution to the 2-loop light-by-light diagram for (g−2) e,µ . ii) Two-Higgs-doublet models with four Yukawa structures: Type-I, II, X (lepton-specific), and Y (flipped), where the CP-odd scalar with mass O(100) GeV gives the main contribution to (g − 2) e,µ up to 2-loop Barr-Zee diagrams. iii) Scalar leptoquarks, S 1 ∼ (3, 1, −1/3) and R 2 ∼ (3, 2, 7/6), where the Yukawa couplings are assigned as up-type mass-diagonal basis to avoid constraints from µ → eγ. Then the mixed-chiral charm-electron and top-muon Yukawa couplings contribute to (g − 2) e and (g − 2) µ , respectively.
We find that accounting for other constraints, all scenarios except the Type-I, Type-II and Type-Y two-Higgs-doublet models easily accommodate the data.