Anomalous magnetic moments from asymptotic safety

The measurements of the muon and electron anomalous magnetic moments hint at physics beyond the standard model. We explain why asymptotically safe extensions based on an enlarged scalar sector and Yukawa couplings between leptons and new vector-like fermions explain the data naturally. Models stabilize the Higgs potential, predict the tau anomalous magnetic moment, and feature new particles in the TeV energy range whose signatures at colliders are indicated. With small CP phases, the electron EDM can be as large as the present bound.

The measurements of the muon and electron anomalous magnetic moments hint at physics beyond the standard model. We explain why asymptotically safe extensions based on an enlarged scalar sector and Yukawa couplings between leptons and new vector-like fermions explain the data naturally. Models stabilize the Higgs potential, predict the tau anomalous magnetic moment, and feature new particles in the TeV energy range whose signatures at colliders are indicated. With small CP phases, the electron EDM can be as large as the present bound.
Introduction.-Measurements of the electron and muon anomalous magnetic moments exhibit intriguing discrepancies from standard model (SM) predictions [1][2][3]. Adding uncertainties in quadrature, the deviations ∆a µ ≡ a exp µ − a SM µ = 268(63)(43) · 10 −11 , ∆a e ≡ a exp e − a SM e = −88(28)(23) · 10 −14 (1) amount to 3.5 σ (2.4 σ) for the muon (electron). Recent theory predictions for a µ find up to 4.1 σ [4,5]. There are two stunning features in the data. First, the deviations ∆a µ and ∆a e have opposite sign. Second, their ratio ∆a e /∆a µ = −(3.3 ± 1.6) · 10 −4 is an order of magnitude smaller than the lepton mass ratio m e /m µ and an order of magnitude larger than the square of the mass ratio (m e /m µ ) 2 . Theory explanations of the data (1) with either new light scalars [6][7][8][9], supersymmetry [10][11][12], or bottom-up models [13,14] invariably involve a manifest breaking of lepton flavor universality. In recent years, asymptotic safety has been put forward as a new idea for model building [15,16]. It is based on the discovery [17] that particle theories may very well remain fundamental and predictive in the absence of asymptotic freedom due to interacting high energy fixed points [18][19][20]. For weakly coupled theories, general theorems for asymptotic safety are available [21,22] with templates covering simple [17,23], semi-simple [24], and supersymmetric gauge theories [25]. Yukawa interactions and new scalar fields play a prominent role because they slow-down the growth of asymptotically nonfree gauge couplings, which can enable interacting fixed points [21] including in extensions of the standard model [15,16,26,27].
In this Letter, we show that asymptotically safe extensions of the SM may offer a natural explanation for the data (1). The primary reason for this is that Yukawa interactions, which help generate interacting fixed points, can also contribute to lepton anomalous magnetic moments. We demonstrate this idea in two concrete models by introducing Yukawa couplings between ordinary leptons and new vector-like fermions, and by adding new scalar fields which admit either a flavorful or flavor universal ground state. The stability of SM extensions all the way up to the Planck scale is exemplified using the renormalization group (RG) running of couplings for a wide range of BSM parameters.
New vector-like fermions and scalar matter.-In the spirit of [17], we are interested in SM extensions involving N F flavors of vector-like color-singlet fermions ψ i and N 2 F complex scalar singlets S ij . In their simplest form, the new fermions couple to SM matter only via gauge interactions [15,16]. The new ingredient in this letter are Yukawa couplings between SM and BSM matter. To make contact with SM flavor we set N F = 3. We then consider singlet or doublet models where the new fermions are either SU (2) singlets with hypercharge Y = −1, or SU (2) doublets with Y = − 1 2 . In our conventions, electric charge Q and weak isospin T 3 relate as Q = T 3 + Y . Within these choices, and denoting the SM lepton singlets, doublets and Higgs as E, L and H, respectively, we find three possible Yukawa couplings κ, κ and y with and flavor traces are understood to simplify the subsequent RG analysis. Effects of the Yukawa coupling y have been studied in [15,16,26]. The scalar potential of either model reads where u, v, λ and δ are quartic and portal couplings. We further introduce mass terms for the scalars and vectorlike fermions. The potential (3) admits vacuum configurations V + and V − characterized by Either of these allow for electroweak symmetry breaking. Moreover, in V + , and for suitable mass parameters, the diagonal components of S each acquire the same vacuum expectation value S = 0 and the ground state is flavor universal. In V − a finite vacuum expectation value S = 0 arises only for one flavor direction giving rise to a flavorful vacuum.

arXiv:1910.14062v1 [hep-ph] 30 Oct 2019
Explaining anomalous magnetic moments.-We are now in a position to explain the data (1) in SM extensions with (2) and (3). The relevant leading loop effects due to the couplings κ, κ , and δ are shown in Fig. 1, also using S = S + s. Any lepton flavor = e, µ, τ receives a contribution from BSM scalar-fermion loops with chiral flip on the lepton line induced by the coupling κ (see Fig. 1a). It scales quadratically with the lepton mass, and represents a minimal lepton flavor dependence with f 1 (t) = (2t 3 + 3t 2 − 6t 2 ln t − 6t + 1)/(t − 1) 4 positive for any t, and f 1 (0) = 1. This manifestly positive contribution is the dominant one for a µ . Contributions through Zand W -loops are parametrically suppressed as O(g 2 2 ) and by fermion mixing [28]. Comparing (5) with the muon data for small scalar-to-fermion which is large for TeV-range fermion masses M F . Fixing ∆a µ to the muon data (1) confirms that the corresponding contribution (5) for the electron would come out too small and with the wrong sign ∆a e 6 · 10 −14 (see Fig. 2).
Additionally, chirally enhanced contributions, which are linear in the lepton mass, may arise through a portalmediated scalar mixing where the chiral flip is shifted to a ψ line (Fig. 1b). The key observation is that chiral enhancement naturally explains the electron data (Fig. 2). In practice, this can be realized with either V + or V − . If the ground state is V − , it must point into the electron direction (only S ee = 0) or else (1) cannot be satisfied. Overall, this leads to where m h,s are the Higgs and the BSM scalar mass, and the last term accounts for (5). The loop function f 2 (t) = (3t 2 − 2t 2 ln t − 4t + 1)/(1 − t) 3 is positive for any t and   Fig. 1a (blue band) and Fig. 1b (red band), which, in combination (green band), explain the electron and muon data (cross) simultaneously. The chirally enhanced offset is either flavor universal or points into the electron direction (green arrow). Band widths are indicative of a 20% mass splitting between fermion flavors from leading loops; the hatched region is inaccessible. f 2 (0) = 1. The mixing angle β between the scalar s and the physical Higgs h is fixed via In (6), the term linear in the electron mass provides a unique offset for the electron ∆a e , sketched in Fig. 2.
It dominates parametrically over the quadratic term and can have either sign set by the Yukawas κ, κ and the portal coupling δ.
As an estimate, comparing (6) with the electron data assuming m 2 h /M 2 F 1 and simultaneously fixing (5) to match the muon data, we find |κ sin 2β| (2.9 ± 1.2) · 10 −4 ( M F TeV ) 2 . The full parameter window explaining the  data is indicated in Fig. 3 assuming V − . Corrections from Zand W -exchange, which contribute differently in the singlet and doublet models, are suppressed by small fermion mixing angles and not sizeable enough to be seen in Fig. 3. Also shown are limits on M F (grey) from Drell-Yan processes [27,29,30] and on perturbativity in α κ (red). We observe M F within (0.05 − 2) TeV for α κ within (10 −2 − 1), with κ sin 2β/(4π) deeply perturbative (green) for small portal coupling δ. The dual parameter space (κ κ) where Fig. 1a is replaced by the corresponding Higgs-fermion loops, is ruled out by Z → data [1], which constrains left-handed (righthanded) fermion mixing angles in the singlet (doublet) model to be of O(10 −2 ) or smaller.
If the vacuum is V + , all lepton anomalous magnetic moments receive a chirally enhanced contribution from Fig. 1b, similar to the first term in (6). The offset in Fig. 2 is then slightly tilted and points along the direction of the red band. Due to the smallness of the tilt, results and constraints are similar to those for V − in Fig. 3.
Running of couplings up to the Planck scale.-We now turn to the RG running of couplings and conditions under which models are stable and predictive up to the Planck scale. We normalize couplings to loop factors, where x = g 1 , g 2 , g 3 , y t , y b , y, κ, κ are any of the gauge, top, bottom or BSM Yukawa couplings, and z = λ, u, v, δ are the quartic and portal couplings. Models are matched onto the SM at the scale set by the fermion mass. For the running above M F , we retain all 12 RG beta-functions up to two-loop order in all couplings [31][32][33][34].
The left panel of Fig. 4 shows benchmark trajectories up to the Planck scale M Pl for models starting in the vacuum V − at the scale M F . For some initial conditions α BSM | M F at the low scale, such as those used in Fig. 4, we find that the running is stable up to the Planck scale. We also observe from Fig. 4 that the Higgs potential becomes stable (remains metastable) in the singlet (doublet) model. Higgs stability in the doublet model can be achieved for larger portal and quartic couplings. Some couplings in Fig. 4 run slowly all the way up to the Planck scale. Others show a slow or fast cross-over to near-constant values due to near-zeros of beta functions [35] which arise from a competition between SM and BSM matter. In the absence of quantum gravity, the evolution of couplings ultimately terminates in an interacting UV fixed point corresponding to asymptotic safety (singlet benchmark) with asymptotic freedom prevailing in the weak and strong sectors [15,16,21]. In some cases, trajectories remain safe up to the Planck scale (doublet benchmark) but blow up at transplanckian energies. For other initial conditions we also find unsafe trajectories which terminate in subplanckian Landau poles (see [28] for a detailed study of initial conditions α BSM | M F ).
The right panel of Fig. 4 shows the vacua of singlet and doublet models at the Planck scale in terms of the Yukawa couplings (α κ , α κ )| M F at the matching scale. Integrating the RG between M F and M Pl , we find wide ranges of models whose vacua at the Planck scale are either V + (blue), or a stable V with a metastable Higgs sector (α λ −10 −4 ) such as in the SM [36,37] (yellow). For other parameter ranges we also find V − (green), or unstable BSM potentials (grey), or Landau poles below the Planck scale (light red). Most importantly, the anomalous magnetic moments (1) are matched for couplings in the red-shaded areas. Here, constraints from Higgs signal strength [1] imply an upper bound on α κ corresponding to a lower bound for the scalar mass of about 226 GeV (for M F = 1 TeV). Similar results are found for V + at the low scale (not shown) except that regions with V − in Fig. 4 turn into V + . We conclude that models are stable and Planck-safe for a range of parameters α BSM | M F .
Collider production and decay.-Models predict new scalars and fermions in the TeV energy range. Their phenomenology is characterized by an enlarged flavor sector with a large Yukawa coupling κ and moderate or small couplings κ, δ. We identify collider signatures through production and decay [28]. We denote the fermions in the singlet model by ψ −1 s and the isospin components in the doublet model by ψ 0 d and ψ −1 d ; superscripts show electric charge. The ψ 0 d is lighter than the ψ −1 [38]. All fermion flavors can be pair-produced in pp and machines via s-channel γ or Z exchange, and through W ± exchange at pp-colliders (doublet model only). Lepton colliders allow for pair-production from t-channel S at order κ 2 , which is sizable (see Fig. 3). Single ψ production together with a lepton arises from s-channel Zand W -boson contributions via fermion mixing. S production occurs only via the Higgs portal, or at lepton colliders with t-channel ψ in association with h at order κ κ or in pairs at order (κ ) 2 .
If kinematically allowed, the charged fermions decay as ψ −1 → S and the neutral ones as ψ 0 d → Sν. If these channels are closed, the ψ −1 decay to Higgs plus lepton instead. The decay rate Γ If kinematically allowed, the BSM scalars S undergo tree level decay into ψψ via y, and into ψ via κ . At one-loop arise the decays S → γγ, ZZ, Zγ, and S → W W (doublet model only) from y. Although there is no genuine lepton flavor violation (LFV) as flavor in the S-decay process is conserved, the mixing between the ψ and the SM leptons introduces very distinct LFV-like final states S ij → ± i ∓ j . The LFV-like decays at the order κκ v h /M F or (κ ) 2 (v s v h /2M F ) 2 are the leading ones for negligible y and M S /M F 1. Discussion.-We have shown that extensions of the standard model with new vector-like leptons and singlet scalars (2), (3) explain the muon and electron anomalous magnetic moments (1) simultaneously. Yukawa couplings mixing SM and BSM matter and a Higgs portal coupling are instrumental to generate both minimal (5) and chirally enhanced (6) contributions, which, when taken together, match the data naturally (Fig. 2). Another salient feature is that the Higgs potential remains stable up to the Planck scale, unlike in the SM [36,37]. Further predictions are a strongly and a weakly coupled Yukawa sector, and new matter fields with masses in the TeV range (Fig. 3) which can be tested at colliders.
An intriguing aspect of our models is that they predict the deviation of the tau anomalous magnetic moment from its standard model value solely based on the data (1) and the vacuum, and irrespective of any other specifics. Provided the ground state distinguishes electron flavor we have ∆a τ ≡ a exp τ − a SM τ = (7.5 ± 2.1) · 10 −7 , and ∆a τ = (8.1 ± 2.2) · 10 −7 otherwise. Although the present limit on ∆a τ is four orders of magnitude away [1], it would be very interesting to test this in the future. We also note that with small CP phases, the electric dipole moment of the electron can be as large as the present bound d e < 1.1 · 10 −29 ecm [39]. In settings with flavor universal vacua the bound extends to all lepton electric dipole moments d , which would make an experimental check for the muon and the tau very challenging.
The new ingredients to address the anomalous magnetic moments are key for achieving safe and controlled SM extensions up to the Planck scale (Fig. 4), and extend the ideas for asymptotically safe model building initiated in [15,16]. More work is required to explore the full potential of asymptotic safety for flavor and particle physics.
Acknowledgements.-This work is partly supported by the DFG Research Unit FOR 1873 "Quark Flavour Physics and Effective Field Theories". GH and DL thank the SLAC Theory Group for hospitality during the final stages of this work.