Variation of $\alpha$ from a Dark Matter Force

We consider a long range scalar force that mainly couples to dark matter and unstable Standard Model states, like the muon, with tiny strength. Probing this type of force would present a challenge to observations. We point out that the dependence of the induced background scalar field on dark matter number density can cause the mass of the unstable particles to have spatial and temporal variations. These variations, in turn, leave an imprint on the value of the fine structure constant $\alpha$, through threshold corrections, that could be detected in astronomical and cosmological measurements. Our mechanism can accommodate the mild preference of the Planck data for such a deviation, $(\alpha_{_{\rm CMB}}-\alpha_{\rm present})/\alpha_{\rm present} = (-3.6\pm 3.7)\times 10^{-3}$. In this case, the requisite parameters typically imply that violations of Equivalence Principle may be within reach of future experiments.


INTRODUCTION
Though dark matter (DM) makes up about a quarter of the energy budget in the Universe, its properties remain mostly unknown [1]. In particular it is not known whether DM has any long range interactions other than gravity. If such a "dark" force exists, it could affect the long distance dynamics of DM, potentially providing a better understanding of the observed large scale structure. In any event, given the existing data, such interactions must be quite weak; if they extend over galactic scales, likely they are not allowed to be much stronger than gravity.
Once one accepts that DM may have long range interactions, it is natural to ask what other states are coupled to such a force. If the particles in question are the stable constituents of atoms, the electron and nucleons, the strength of their coupling to the long range force is extremely well constrained by tests of the Equivalence Principle and "fifth force" searches, requiring the strength of those interactions to be sub-gravitational. This situation could limit the effects of the new interactions, though there are potentially interesting scenarios that can arise in this case [2]. However, one could also entertain the possibility that the long range interactions of DM couple more strongly to other more elusive Standard Model (SM) particles, like neutrinos [3] or unstable particles, such as the muon. In the latter case, the absence of these particles on macroscopic scales does not allow very stringent experimental constraints on their new long range interactions. For the same reason, it seems quite challenging to envision how one may uncover a new long distance force between unstable particles and DM.
In this work, we consider the coupling of a long range force, mediated by a light scalar φ to DM and an electrically charged unstable SM fermion f ; for concreteness we will focus on the muon. Scalar long range forces may * email: hooman@bnl.gov † email: pgiardino@bnl.gov be motivated from top down or phenomenological points of view [4][5][6][7][8][9][10][11][12][13]. We show that the the background field φ sourced by the cosmic population of DM can result in variations of the fermion mass m f , in space and time, which leaves its imprint as a threshold effect in the running of fine structure constant α of quantum electrodynamics.

LONG RANGE FORCE
In this section, we describe our mechanism. The basic interactions of interest for our analysis are given by where X, a Dirac fermion, is the DM, φ is a light scalar that mediates the long-range force, and µ is the SM muon. In general, other SM fermions could enter in Eq. (1), but we found the choice of the muon particularly interesting and we will concentrate on it for the rest of this letter. We will assume that the effective dimension-4 operator φμµ is the low energy result of some wellbehaved but un-known UV theory. The relevant mass terms, in vacuo, are given by in an obvious notation.
Let us consider what happens when a sufficiently large density of DM X fermions is present. The equation of motion for φ is then given by (see, for example, Ref. [8]) where n X is the number density of X, . . . denotes an average, and v is the velocity of X. Here, we assume that the population of µ states is negligible. The second equation contains a factor √ 1 − v 2 , sinceXX is Lorentz invariant. We are interested in DM well after its relic density has been set, and hence we can assume v ≈ 0.
If the distribution of DM is static and uniform, and has a characteristic size that is larger than all other distance arXiv:1804.01098v1 According to Eq. (1), the contributions to the mass of µ and X from the scalar force are given by where F = µ, X. An interesting consequence of the modification of the mass of the muon is that, due to threshold corrections, 1 the fine structure constant α changes as well according to The coupling of φ to the muon typically implies tiny couplings between φ and other SM fermions, via radiative corrections. In particular, the diagram in Fig. 1 contributes to the coupling between φ and stable fermions (electron and proton, respectively g e and g p ) even if they are zero in the tree-level Lagrangian 2 . The presence of this long range interaction is observable in tests of the Equivalence Principle, thus providing indirect bounds on the g µ coupling. From Refs. [16,17], we find that |g p | < ∼ 10 −24 and |g e | < ∼ 10 −25 . At the same time, given the 2-loop diagram in Fig. 1 we would expect the coupling to protons to be corresponding to an upper bound |g µ | < ∼ 10 −17 . Notice that this would ensure for the electron coupling |g e | < ∼ 10 −25 , due to an O(m e /m µ ) suppression. Limits on the g X coupling are less strict. It is reasonable to require g X 1 For other works in different contexts see, for example, Refs. [14,15]. 2 We thank W. Marciano for pointing out the potential significance of these diagrams. to be (sub-)gravitational if the range of the force is of galactic scale, in order to avoid conflict with our present understanding of large scale structures. On the other hand, we can relax this requirement if we consider smaller ranges (i.e. heavier mass) for φ.

CONSEQUENCES
We now consider a particular scenario where the range of the force mediated by φ is 100 kpc, so that it spans the Milky Way and the majority of its halo; m φ = 1/100 kpc ∼ 10 −28 eV. We set the mass of DM m X = 1 GeV, for concreteness, and since we require the force mediated by φ to be sub-gravitational this corresponds to imposing |g X | < ∼ 10 −19 ; thus we fix g X = 5 × 10 −20 . Notice that since g X ∼ 10 5 g p the contribution of common matter to the value of φ is negligible and Eq. (3) is valid. We also set the coupling to the muons at g µ = −2 × 10 −18 so that the contribution to its mass is positive, as implied by Eq. (5). Later, we will also consider an interesting case with g µ > 0.
For the DM distribution in the Milky Way we consider the NFW and Burkert profiles [18,19], respectively and where we took R = 20 kpc and r c = 10 kpc. Here, ρ n and ρ b are chosen so that the local density of DM in the solar system (r = 8.5 kpc) is 0.3 GeV/cm 3 . We assume a spherical distribution. We solved Eq. (3) numerically, assuming ∂ r φ| r=0 = φ(∞) = 0 and the above DM profiles, and obtained the value of ∆α/α as a function of distance from the center of the Galaxy. Here, the variation is with respect to the value in vacuum: ∆α ≡ α − α vac . In Fig. 2, we plot our results. We consider particularly interesting the fact that the value of ∆α/α at the center of the Milky Way is O(10) times larger than its value at the outskirts. This hierarchy has a very mild dependence on the DM distribution and on any parameter of our model if one assumes that m φ < ∼ 1/10 kpc −1 . Focusing on the solar system, we find that the mass of the µ lepton receives a contribution due to DM in the Milky Way of ∆m µ /m µ ∼ 10 −5 , that corresponds to a variation of α from its value in vacuo: While ∆m µ corresponds to a deviation of the SM muon Yukawa too small to be accessible at the LHC, ∆α/α is close to the present bounds obtained from the Oklo natural reactor: see for example Ref. [20][21][22] and references therein. The above result can be interpreted in our scenario as a constraint on how much the density of DM of our Galaxy changed in the course of the last 2 billions years, since the activity period of the Oklo reactor. So we can conclude that, in our scenario, variations of order O(1) in the overall mass density of the Milky Way halo are allowed. This is likely much more than the amount of DM accreted through mergers with the satellites of the Milky Way. On the other hand, the above results imply that an O(10) more stringent constraint from Oklo or other similar measurements can be sensitive to ∼ 10% DM accretion by the Milky Way, over time scales of O(10 9 ) years.
Measurements of α in other galaxies are usually less constraining [23,24] and the current bounds are generally of order ∼ 10 −6 for ∆α/α, that would easily accommodate a few orders of magnitude of difference in the density of DM among various galaxies.
Another interesting consequence of this scenario is that the values of the muon mass and α depend on the cosmological era. Since the density of DM is proportional to the cube of the temperature of the Universe, if we go back in time (i.e. at higher temperatures) we expect the mass of the muon and the value of α to change. However, the horizon size, d hor , also depends on the temperature of the Universe and shrinks as we go towards earlier times. Thus, we would eventually reach a point in time where d hor < m −1 φ is the meaningful scale in the calculation of φ. We have, up to O(1) corrections, where n X ∝ T 3 and d hor ∝ 1/H or 2/H if the Universe is either radiation or matter dominated, respectively. Here, H denotes the Hubble scale. In what follows, we will assume the Universe is dominated by matter or radiation when the corresponding energy density dominates by a factor of 10. In between these two regimes, we use a simple linear function to interpolate between 2/H and 1/H. As H grows with T , φ eventually decreases.
In the scenario that we explore here φ reaches its maximum at T ∼ 1 eV at which point ∆m µ ∼ 600 MeV. This large value for the mass of the muon is not problematic by itself, since at those temperatures muons are out of equilibrium and do not play a role in cosmological evolution anymore. Also ∆m µ is large only in a small window around T ∼ 1 eV and ∆m µ /m µ 10 −3 during the Big Bang Nucleosynthesis and earlier epochs. However such a large value of the µ lepton mass affects the fine structure constant and we have, for T ∼ 0.3 − 1 eV This result is particularly interesting if we consider that the Planck experiment [25] found a difference between the value of α at the CMB era with respect to today's measurement of ∆α/α = (−3.6 ± 3.7) × 10 −3 (note that our convention for ∆α differs by a minus sign from that of Ref. [25]). Our benchmark parameters are compatible with this measurement, within 2σ. Alternatively, one could assume the central value of the above Planck result to furnish a mild indication that ∆α/α ∼ −10 3 is preferred. This can be achieved in our scenario by modifying the benchmark parameters of our model. Taking m φ = 1/300 kpc −1 , g µ = 10 −18 and g X = 2 × 10 −21 we obtain m µ ∼ 20 MeV at T = 0.3 eV. In Fig. 3, we plot ∆α/α as a function of temperature for three values of m X = 0.85, 1.0, 1.2 GeV. As one can see, our model can accommodate the central value of the Planck measurement, for m X < ∼ 1 GeV. Whether or not this mild hint will grow in significance, our results point to the possibility of constraining DM long-range interactions through measurements of the variations of physical constants in different eras. Notice also that for a larger g X the muon could become lighter than the electron for a short period before and after CMB, which would allow the electron to decay into a muon and neutrinos! This would have unusual effects on cosmology that we will not further consider in this letter. Here, we add that for the first and second sets of benchmark parameters considered above the DM mass does not vary by more than ∼ 10 −2 and 10 −4 , respectively, which are allowed by the current percent level determinations of the DM energy density [1].
If the central value of the Planck measurement for ∆α/α holds near its current value with improved measurements, the scenario discussed above could typically imply violations of the Equivalence Principle, not far from the current limits. To see this, note that increasing g X by more than an order of magnitude will lead to conflict with the CMB measurements of the DM energy density, as this would change m X more than ∼ 1% for T ∼ 1 eV. Therefore, to stay near the Planck central value we need g µ > ∼ 10 −19 . Then, Eq. (7) implies that g p > ∼ 10 −26 , which is within two oder of magnitudes of the current limits.
Lastly, let us mention that the large positive change in the mass of the muon around CMB era can have another interesting consequence for light thermal relic DM. If the DM thermal relic density is dominantly set through the annihilation into µ + µ − final states 3 , the process could be allowed in early and late cosmology, but become forbidden during the CMB era, thus relaxing the current bounds [26,27] on the thermal relic abundance of light DM.

CONCLUDING REMARKS
In this work, we have examined a possible signal of a long range force, mediated by a light scalar, that couples to DM with order gravitational strength, but could have somewhat larger couplings to unstable SM particles. Given the feebleness of the assumed interactions and the lack of significant populations of the unstable states, this scenario can pose a significant challenge to experimental verification. We show that if the SM particles have electric charge, the scalar potential sourced by DM can modify the threshold effects in the running of the finestructure constant α and lead to its variations in space and time, as a function of DM density. Focusing on the muon for concreteness, we found that for phenomenologically allowed values of parameters existing bounds on variations of α can be satisfied.
In the early Universe, when the density of DM was much larger, we expect sizable deviations in α, however our benchmark parameters are consistent with the current Planck bound from the CMB era. Depending on the sign of the Yukawa couplings to the mediating scalar, one could effect a positive or negative deviation; the latter choice is modestly preferred by the Planck data and can be accommodated by our scenario. We conclude that future improvements in these or other astrophysical data can potentially uncover the effect of the long range scalar force on α. If the Planck hint holds, our mechanism typically predict violation of the Equivalence Principle, not far from present bounds. Our proposal hence provides a handle on an otherwise extremely elusive possible phenomenon, whose discovery would have a revolutionary impact on our understanding of particle physics and cosmology.