Dark Matter, Muon Anomalous Magnetic Moment and the XENON1T Excess

A very economic scenario with just two extra scalar fields beyond the Standard Model is invoked to explain the muon anomalous magnetic moment, the requisite relic abundance of dark matter as well as the Xenon-1T excess through the inelastic down-scattering of the dark scalar.

The observation of an excess in the electronic recoil events at the xenon1t detector [1] has elicited much activity, especially in the context of Dark Matter (DM) . The very structure of the excess demands that not only the DM particle be relatively light, but also that the recoil energy satisfy 1 keV < ∼ E rec. < ∼ 5 keV. To reconcile such a DM with the correct relic abundance and yet survive cosmological constraints emanating from large-scale structure formation, big-bang nucleosynthesis, cosmic microwave background [28], supernovae [29] etc., has been a herculean task. In this Letter, we point out that a relatively simple model can not only satisfy all such constraints but also successfully address another long-standing issue that the Standard Model (SM) faces, namely an explanation of a µ , the anomalous magnetic moment of the muon. Furthermore, it promises exciting signals at currently operating experiments.
Eschewing the more common fermionic DM, we consider the simpler alternative, viz. a complex scalar field φ. The lack of excess events in the first bin at xenon1t [1,3] restricts m φ < ∼ 1 GeV. The dark sector communicates with the SM particles through a light real scalar field ω which also serves to generate a contribution to a µ . There are some advantages to choosing a scalar mediator as opposed to the more popular dark photon. For one, it is the most economic construction in terms of field content. Secondly, ω can both be the mediator as well as potentially engender the mass split required for downscattering. And, finally, having a dark photon generate a substantial a µ would need it to couple to neutrinos as well (at least in the simpler constructions) thereby rendering the heavier component of the DM unstable on cosmological time scales.
Given the field content, the most general scalar potential has many parameters. For the sake of brevity, consider here the relevant part of the same, viz.
Also possible are other cubic and quartic terms, including, possibly a ω 3 one. However, unless their coefficients are large, such terms would not be germane to the issues at hand. While ensuring that φ represent a viable DM demands that its classical value (vev) vanishes identically, we impose an identical (simplifying) condition for ω as well. These conditions and the lightness of the scalars can be easily achieved by suitably adjusting the parameters of the full potential.
The presence of the ∆ 2 term serves to split the two components of φ ≡ (φ 2 + iφ 1 )/ √ 2. For real ∆ 2 (an imaginary component to ∆ 2 does not change anything qualitatively beyond introducing an immaterial mixing), the masses are given by (without loss of generality, ∆ 2 > 0) The xenon1t signal profile requires that while µ φ ∼ O(100 MeV), the splitting is only ∼ 2 keV. The soft trilinear terms in eq.(1) are of great importance as these engender couplings of the form g ij φ i φ j ω While the g ij s play nearly equivalent roles in determining the relic density, g 12 is key to explaining the xenon1t excess. Also note that ω = 0 would generate ∆ 2 .
The messenger ω can have renormalizable interaction terms with only the Higgs field with the H † Hω 2 term constrained by the limits on the invisible decay width of H. Similarly, a ωH † H term would induce a mixing, and is constrained by B and K decays [30][31][32][33][34][35]. However, we do not delve into this and focus, instead, on a leptophilic ω, coupling to fermions through dimension-five operators such as [36] L int (ω/Λ)H ỹ µL2 µ R +ỹ eL1 e R + H.c. , where L 1,2 are the electron and muon doublets and Λ is the cutoff scale, presumably in the multi-TeV range.
(Whileỹ τ could exist as well, it does not largely concern us, and we shall remark on its consequences later.) On symmetry breaking, these lead to effective Yukawa terms We shall, henceforth, parametrize where the scaling factor n s = O(1). Before delving into phenomenological consequences, we must discuss the decays. While φ 1 is absolutely stable, owing to the tiny δ m , φ 2 decays are restricted to (φ 1 + N γ) and (φ 1 + νν) alone. Either of these may occur only at two-loops or higher, and φ 2 is stable on cosmological scales. On the other hand, when allowed, µ and also the constraint from the 4µ final state assuming Br(ω → µ + µ − ) = 1.0. The dotted curve is the projection from BELLE-II experiment [43].
the partial widths of ω into leptonic and scalar channels are given, respectively, by
Owing to its much smaller size, a non-zero y e does not materially affect this conclusion. The situation, though, could change drastically if a y τ were to exist, for it would lead to e + e − → τ + τ − ω → τ + τ − + − at BABAR [44]. In the event of y i ∝ m i , the constraints on y τ could be interpreted in terms of much stronger bounds on y µ and y e . Note, however, that both the BABAR analyses assume Br(ω → µ + µ − ) = 1.0 for m ω > 0.21 GeV or Br(ω → e + e − ) = 1.0 for 0.04 GeV < m ω < 0.21 GeV. In our scenario, whenever it is kinematically allowed to, the ω decays overwhelmingly into a φ i φ j pair (see eq. 6), thereby negating both the aforementioned constraints.
Constraints on y e : We begin by exploring the channels for m ω < 2m φ so as Constraints on the mediator ω coupling to electron. The dotted curves indicate projected sensitivities from HPS [40,49] and Belle-II [36,40,50].
to remove the dependence on the invisible decay modes.
The shape of the disallowed region is largely determined by the energy of the decay electrons and the vertex displacement. For m ω > 2m µ , the small lifetime of ω drastically reduces the sensitivity. Rather, the BABAR search for dark photons via e + e − → γA → γ + − [51] can be used to constrain y e [40]. For y µ y e , only the muonic channel is relevant, and assuming this to be the overwhelmingly dominant mode [36,52] leads to strong limits for m ω ∈ [0.02, 1] GeV (yellow region in Fig. 2). One can similarly reinterpret the BABAR bounds for m ω > 1 GeV in terms of the scalar mediator which would be of the same order as the sub-GeV bounds. However, we refrain from exploring that region as it is of little interest here. The lower energy experiment KLOE [53][54][55][56][57], on the other hand, imposes a comparatively relaxed bound [36,58].
Naturally, all the above constraints are drastically relaxed for m ω > 2m φ . Instead, y e can now be constrained from missing energy/momentum signals. For example, the dark photon search of the NA64 collaboration [59] through nuclei-initiated e − N → e − N A with the A going invisibly, yields constraints. Similarly, the analogous BABAR analysis [60] for dark photons may be used as well. In depicting either in Fig. 2, we have, following refs. [36,40,43], interpreted the constraints rather conservatively, eliminating a slightly larger part of the parameter space than is strictly necessary.
The very structure of eq.(7) ensures that constraints from (g − 2) e [41] are very weak. So are those from fifth force searches [61]. Similarly, the bounds from the cooling of horizontal branch stars or red giants [62] are relevant only for m ω < ∼ 0.1 MeV, while those from SN1978A [52] extend to larger m ω but are weaker. Bounds from nucleosynthesis [52,63] are relevant only for m ω < 1 MeV and are inapplicable in the present context. As Fig. 2 (and Fig.7 of ref. [52]) shows, for m ω > ∼ 0.1 MeV, the (g − 2) µ favoured band of Fig. 1 is unconstrained by considera-tions of y e as long as 1 < ∼ n s < ∼ 10.
Direct Detection via electron recoil: xenon1t excess With the effective Yukawa couplings (4) in place, the triple scalar vertices give rise to three distinct DM initiated processes at a detector, namely φ i D → φ i D (where D is a detector entity, nucleus or electron) and φ 2 D → φ 1 D. The former are elastic in nature with the typical recoil energy for an electron being O(eV) and are, thus, insufficient to explain the E rec. ∼ 2 keV signal at xenon1t . The g 12 term in eq.(1), though, can lead to such events provided the mass-splitting δ m ∼ O( keV).
For an electron recoiling with energy E, the differential cross-section for the atomic ionization induced by DMelectron inelastic scattering is given by [64,65] where a 0 = 1/m e α em and f (v) is the distribution in the DM's velocity v with a Maxwellian form being a very good approximation. The integration limits are given by v min = 2(E − δ m )/m 2 (for E ≥ δ m ) and v max = v ⊕ + v esc where v ⊕ is the Earth's velocity and v esc is the local galactic escape velocity [66]. The form factor |F φ (p)| 2 , as a function of the momentum transfer p, can be approximated to be unity in the case of a heavy mediator. For free electron scattering proceeding through ω-exchange, we have, In evaluating the integral, we use the atomic excitation factor, K(E, p), corresponding to E = 2 keV, from ref. [64]. This choice of E stems from the fact that the xenon1t excess (which has fixed δ m ) is peaked around E ∼ 2 keV. The integration range for p, as determined using momentum conservation, is, for E ≥ δ m , given by The event rate R can be determined using [65] dR Here, N T 4.2 × 10 27 /tonne is the number of Xenon atoms per unit detector mass. Since φ 1,2 are nearly degenerate, the energy density of incident DM particles ρ φ2 ≈ ρ DM /2 ≈ 0.15 GeV/cm 3 [67,68].
At this point we are quite well-equipped to address the xenon1t excess. With (g − 2) µ constraining y µ , a choice for n s (see eq. 5) determines y e . This, in turn, fixes g 12 . The regions of the parameter space that can explain the reported excess within 1σ are depicted in Fig. 3. Note that m ω < ∼ 0.03 GeV is strongly disfavoured by lowenergy data. For a given (m ω , m φ ; y µ ) combination, a larger n s would demand a smaller g 12 so as to maintain the size of the excess, as reflected by the shifting bands. Apart from electrons, the DM will also scatter against the nuclei. However, in the absence of any coupling of ω to the quarks we only have loop-suppressed contributions to the scattering process. This also invalidates the otherwise strong bound set by the cresst collaboration [69] for m φ ∈ [0.3, 1] GeV.

Relic Abundance:
With φ 2 having a lifetime greater than the age of the Universe, the DM comprises equal parts of φ 1,2 . The small δ m ensures that the two decouple chemically well before the heavier one could be annihilated completely or even exponentially suppressed. By virtue of its couplings to φ 1,2 , the ω serves as a portal between the dark and the ordinary sectors.
Post decoupling, the annihilations are crucial in determining the relic abundance. For very light φ i , the only channel available is φ i φ j → e + e − , where the two scalars could either be the same or different. For heavier φ i , the µ + µ − and the ωω modes open up. The last-mentioned, if allowed kinematically, dominates, with propagators (tand u-channel) corresponding to either of φ 1,2 . And, had we included a ω 3 term in the Lagrangian, a further contribution from a s-channel w-exchange would have appeared.
Defining the yield Y φ as the ratio of its number density and entropy-density s(m φ ) of the universe, the relevant Boltzmann equation, in terms of x ≡ m φ /T (T being the temperature) and the Hubble expansion rate H(m φ ), is where R ij ≡ k g ik g kj and all the coupling constants have been factored out of the cross-sections. Since δ m T f , the mass splitting has virtually no effect on the freezeout and we have assumed that Y φ1 = Y φ2 . The factor of 1/2 is occasioned by the ωs being identical particles.
To reduce the number of parameters, we make the simplifying assumption that all three g ij s are numerically very similar, denoting this common value by g ωφφ . With y µ constrained from (g−2) µ , we plot, in Fig. 3, the dependence of the DM relic abundance on g ωφφ as a function of the mediator mass m ω for a given DM mass. The width of the band corresponds to the spread in y µ (see Fig. 1).
For m ω < m φ , the processes φ i φ j → ωω are dominant. With the cross-section having only a mild m ωdependence, so does the requisite g ωφφ . Since y µ plays only a subsidiary role, the band collapses to virtually a single curve. For m ω > m φ , this channel is no more allowed and φ i φ j → µ + µ − dominates. Consequently, g ωφφ must increase with m φ to account for the s-channel suppression. Simultaneously, the allowed spread in y µ becomes relevant. The strong dip around m φ ∼ 2m ω is but a consequence of resonance enhancement.
Understandably, for n s < ∼ 10, the relic abundance has little dependence on it. On the other hand, the parameter space allowed by the xenon1t excess most definitely does. Consequently, it is straightforward to identify the region of parameter space that simultaneously explains all three viz. (g − 2) µ , the xenon1t rate and DM relic abundance.
A larger m φ stipulates a smaller relic number density and, hence, a larger annihilation cross section. The requisite increase in g ωφφ is not as severe as that for maintaining the xenon1t excess (see eq. 8) thereby necessitating a larger n s for ensuring overlap (see Fig. 3). Similarly, m ω m φ would imply a progressively larger g ωφφ . While an overlap with the xenon1t data can still be achieved for n s < 1, a large g ωφφ would result in a large quantum correction to the scalar masses, potentially destabilising the vacuum. Curing this would require the introduction of additional terms in the Lagrangian. For the same reason, m φ < m µ , is disfavoured as annihilation to muons is replaced by that to e − e + ; the smallness of y e translates to a large required g ωφφ .
Processes such as φ i e − → φ j e − maintain kinetic equilibrium and keep the dark sector in thermal contact with the plasma until T kin , when it decouples. For n s ∼ 4, comparing the interaction rate to the expansion (Hubble) rate gives T kin ∼ MeV. After the decoupling, the DM is no longer in kinetic equilibrium with the SM thermal bath and begins to cool more rapidly.
The inter-conversion process φ 2 φ 2 → φ 1 φ 1 , nonetheless, continues to be efficient until the temperature of the dark sector falls below T < T kin . If T < δ m , the fractional abundance of φ 2 would be exponentially suppressed, with N 2 /N 1 ∼ e −δm/T . Using the formalism of refs. [70,71], we find, though, that T > ∼ O(100 keV) and To summarise, we present a very economical model that simultaneously explains the xenon1t excess (through inelastic DM scattering), as well as the anomalous magnetic moment of the muon while producing the requisite dark matter relic density. A single leptophilic scalar ω generates the requisite a µ while serving as a portal between the dark and the visible sectors. The small mass-splitting of O( keV) engendered by a soft term in the scalar potential (or, potentially, by a nonzero ω ) renders the heavier DM component extremely stable on cosmological time scales. While the DM mass is required to be relatively small, viz. O(100 MeV), a sufficient parameter space exists satisfying all constraints, experimental (beam dumps, colliders etc) astrophysical (stellar cooling) and cosmological (BBN, N eff ). The competing constraints render the model eminently testable and, thus, interesting. For example,; in Fig. 2 we have indicated the projected sensitivities for y e from Belle-II [36,40,50] and the Heavy Photon Search (HPS) experiments [40,49]. A similar Belle-II projection for y µ [43] has been indicated in Fig.1. The FASER experiment [72] too can probe such parameters. Clearly, a very large part of the favoured parameter space would be testable in the near future. Also worth studying are the consequences of a nonzero ω , especially in the context of finite-temperature corrections, for this presents intriguing possibilities as far as cosmological history is concerned, whether it be in terms of phase transitions, small late stage inflation etc. We hope to return to such issues at a later date.
DC and SM acknowledge partial support from the SERB, India under research grant CRG/2018/004889. DC also acknowledges the European Union's Horizon 2020 program under Marie Sk lodowska-Curie grant No 690575. DS acknowledges support through the Ramanujan Fellowship grant of DST, India. VS thanks the UGC, India for financial support.