Dark matter electromagnetic dipoles: the WIMP expectation

We perform a systematic study of the electric and magnetic dipole moments of dark matter (DM) that are induced at the one-loop level when DM experiences four-fermion interactions with Standard Model (SM) charged fermions. Related to their loop nature these moments can largely depend on the UV completion at the origin of the four-fermion operators. We illustrate this property by considering explicitly two simple ways to generate these operators, from $t$- or $s$-channel tree-level exchange. Fixing the strength of these interactions from the DM relic density constraint, we obtain in particular a magnetic moment that, depending on the interaction considered, lies typically between $10^{-20}$ to $10^{-23}$ ecm or identically vanishes. These non-vanishing values induce, via photon exchange, DM-nucleus scattering cross sections that could be probed by current or near future direct detection experiments.


Introduction
Weakly interacting massive particles (WIMPs) are among the best motivated dark matter (DM) candidates. As is well known, DM particles annihilating into lighter particles with coupling strength of order unity undergo a non-relativistic freeze-out in the primordial thermal bath of the Universe, leaving a relic density of the order of the observed one if the DM mass is roughly around the electroweak scale. This "WIMP miracle" has triggered vast experimental effort in DM searches-see [1][2][3][4][5][6] for reviews. In particular, over the past years, direct and indirect detection experiments have reached the sensitivity necessary to probe this paradigm in many different contexts. Collider experiments also offer possibilities of tests. A number of explicit models have been already excluded, whereas many other ones could be seriously tested in the near future.
In many models WIMPs annihilate into SM fermions via a tor s-channel mediator. If this mediator is sufficiently heavy, it can be integrated out, leading to a local effective interaction. Thus in this case the (tree level) phenomenology of the model reduces to the one that can be obtained from the effective field theory (EFT) for DM annihilation. As is well-known too, in this case one can get a one-to-one relation between the annihilation rate fixed by the relic density constraint and direct, indirect as well as collider signals. 1 In this work we are interested in effective interactions involving charged SM fermions (f ) and DM fermions (χ), of the general form L ⊃ GχOχf O f with O and O any possible operators. From the relic density constraint the dimensional coupling G is typically of the order of 10 −1 G F × (m χ /100 GeV) where G F is the Fermi constant and m χ is the DM mass. If the SM fermion is a light quark and DM lies around the electroweak scale, such values of G have already been ruled out by recent direct detection experiments for operators that lead to spin-independent (SI) cross sections on nuclei. These are in particular the XENON-1T [10], LUX [11] and PandaX-II [12] experiments which have now put upper limits on the SI DM-nucleon cross section down to ∼ 10 −46 cm 2 for m χ ranging from tens to hundreds of GeV. However, WIMPs are not necessarily expected to dominantly couple to light quarks. For other SM fermions (e.g. f = e, µ, τ, c, b, t, ν e , ν µ , ν τ ), direct detection bounds are generally weaker. In addition, various operators may lead to spin-dependent (SD) cross sections, for which the experimental sensitivity is weaker.
An interesting possibility to improve the direct detection sensitivity in such cases stems fom the fact that WIMPs might have electromagnetic dipole moments. In fact, various electromagnetic form factors (electric/magnetic dipoles, anapole, charged radius) of WIMPs have been considered in the literature [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32]. Early studies [17][18][19][20]24] considered them as a solution to resolve the discrepancy between DAMA/CoGeNT signals and null results of other DM searches, though this has been since then well excluded. Collider, γ-ray, and CMB searches for dipole interacting DM have been studied in Ref. [20,21]. More recently, Ref. [28] considered leptophilic DM and showed that its loop-induced electromagnetic dipoles led to restrictive direct detection bounds. 2 In this work, instead of considering that the annihilation induced by the dipole (into SM charged particles via photon exchange) is responsible for the relic density, as in many of these works (e.g. [15,[17][18][19][20]), or instead of assuming a specific model, we will instead start, as in Ref. [16], from the effective four-fermion operators. Once the coefficients of the effective operators are fixed by the relic density, we can compute the dipoles they lead to at the oneloop level (simply from closing the charged fermion line, and attaching an external photon). Actually, since dipoles are loop-level effects, the use of an effective theory to compute them is not necessarily consistent with what we would obtain in UV complete models. If the effective theory holds for arbitrarily high energy scales, the loop integral that leads to the dipoles would be divergent and such divergences cannot be canceled (absorbed) by any counterterms. Unlike for the annihilation process, one thus needs to open the effective interactions. We consider two straightforward ways to generate these effective interactions at tree level, namely sor t-channel exchange. This leads to two general classes of models depending on whether they give a vanishing (as in the s-channel case) or a non-vanishing finite (as in the t-channel case) result. We argue that the results obtained for non-vanishing dipoles are generic, which is illustrated by comparing these results with the ones obtained in a UV complete model.
We then study the implication of non-vanishing dipole for direct detection. We find that the magnitude of the non-vanishing loop-induced dipoles, typically of the order of 10 −20 ecm (or 10 −20 m f /m χ ecm), implies that DM-nucleus scattering via dipole interactions could be 2 Beyond the WIMP regime, there has been growing interest in electromagnetic dipoles of sub-GeV DM due to potential connections with CMB/LSS observations, stellar physics, and the intensity frontier searches-see, e.g., [29][30][31][32].
probed within current and future experimental sensitivities. In particular for operators involving heavy quarks or charged leptons, or when the DM-nucleus cross section is SD at tree level, this might provide the best possibility of probing these interactions and thus possibly the origin of the DM relic density. This stems from the fact that for low nuclear recoil energies the cross section is considerably enhanced by the exchange of a massless (photon) mediator.
The paper is organized as follows. In Sec. 2, we present a complete description of the most general four-fermion interactions of DM fermions with SM fermions, and determine the interaction strength required to produce the observed relic abundance. Given the determined interaction strength, in Sec. 3, we compute the loop-induced electromagnetic dipoles of DM by closing charged fermion loops in the four-fermion interactions, assuming that they are induced by either t-channel or s-channel tree level exchange. There we also compare the results obtained in this way to the ones obtained from considering an explicit UV complete model. In Sec. 4, the resulting magnitude of electromagnetic dipoles is confronted with direct detection limits obtained by investigating the recoil spectra of dipole-interacting DM. We conclude in Sec. 5 and delegate the loop calculation details to the appendix.

Effective interactions of DM
We start with the most general four-fermion interactions of Dirac DM (χ) and SM fermions (f ): where the Γ a matrices (with a = S, P , V , A, T ) span all the 16 possible independent combinations of Dirac matrices: We refer to the above five possible bi-linear products of Dirac spinors as scalar, pseudoscalar, vector, axial-vector, and tensor interactions. In Eq. (2.1), we have normalized the interaction strength by the Fermi constant G F since in the WIMP paradigm, the interactions are typically of this magnitude. Potential deviations are absorbed into the dimensionless constants a and˜ a . Note that in Eq. (2.1) we have inserted an i a factor, which is defined as i S,P,T = i and i V,A = 1, so that the various terms are hermitian, with a and˜ a real numbers-for further discussions see e.g. Refs. [33,34]. For tensor interactions, one could consider adding γ 5 between χ and χ but the operator χσ µν γ 5 χ f σ µν f is actually identical 3 to χσ µν χ f σ µν γ 5 f . Hence Eq. (2.1) provides a complete description of all possible Lorentz-invariant four-fermion interactions. This set of effective operators has also been frequently used for DM searches at colliders-see e.g. Ref. [35]. Note importantly that the S, P and T operators are not SM gauge invariant, but could be generated through electroweak symmetry breaking, see the discussion in Sec. 3.2.
In the SM fermion chiral basis, one can also write Eq. (2.1) as where f L,R ≡ P L,R f , P L,R ≡ (1 ∓ γ 5 )/2, L a and R a are linear combinations of a and a . Given the chiral structure of the SM, and the fact that most results are symmetric under L ↔ R, in this work we will adopt the chiral basis. Note that while a and˜ a in Eq. (2.1) are real and independent of each other, L a and R a in the chiral basis are either complex conjugate of each other ( R S = L * S , R P = L * P , R T = L * T ), or real and independent . Hence the full set of 's in the chiral basis still contains 10 real independent parameters.

DM relic abundance
The relic abundance of χ via the standard freeze-out mechanism is approximately given by (see e.g. [36]) is the ratio of the freeze-out temperature T f.o to the WIMP mass m χ ; g is the effective number of relativistic degrees of freedom in the thermal bath at freeze-out; and σv is defined as [37,38] σv ≡ n −2 Here subscripts 1, 2, · · · , and 4 denote quantities of the first, second, · · · , and the fourth particles in χ + χ → f + f ; δ 4 is short for δ 4 (p 1 + p 2 − p 3 − p 4 ); and f 1 (f 2 ) is the thermal distribution function of χ (χ). The squared amplitudes |M| 2 has been evaluated and summarized in Tab. 1. For P , V , T interactions, the annihilation amplitudes are of the s-wave type and consequently are nearly constant in the non-relativistic regime. In this case, we can neglect the velocity dependence and reduce Eq.
For S and A interactions, we have |M| 2 ∝ v 2 (p-wave annihilation) and hence the integration is somewhat more complicated. Assuming Maxwell-Boltzmann distributions for f 1 and f 2 , Eq. (2.5) can be reduced to [38] σv ≡ 1 8m 4  Here K 1 and K 2 are K-type Bessel function of orders 1 and 2, s = (p 1 +p 2 ) 2 = 4m 2 χ +m 2 χ v 2 , and σ is the total annihilation cross section [39]: , with θ the angle between p 1 and p 3 . Integrating , we obtain results for σ that are given in Tab. 1. Plugging the results for σ into Eq.
χ , we can integrate Eq. (2.8) analytically by noticing that for any value of p > −1, where Γ is the Euler gamma function. The results for σv are then expanded in T /m and summarized in Tab. 1.
Using the results for σv with T f.o m χ /23 (the typical freeze-out temperature) in Eq. (2.4), we obtain where a denotes benchmark values: S = 0.49, P = 0.13, V = 0.089, A = 0.43, and T = 0.063. Note that a for a = S or A is generally larger than for other cases because the cross section is velocity suppressed, which implies that they would freeze out at higher temperatures for the same coupling strength. Hence to reach the same relic abundance (i.e. same freeze out temperature), the coupling needs to be larger. This generally leads to loop-induced electromagnetic interactions of χ.
Closing the loop in this way one gets amplitudes which take the general form where u 2 , u 1 , ε µ represent the three external lines; the momenta p's and k's have been defined in Fig. 1 with k = p 1 − k 1 = p 2 − k 2 ; m f and Q f are the mass and the electric charge of f ;Γ a ≡ Γ a P L,R ; G X is the coefficient of the effective operator considered. The most general form of the F µ vertex function that respects Lorentz and electromagnetic gauge invariance can be decomposed as a combination of four terms, each one with its own form factor (see e.g. [40,41]): Here q ≡ p 1 −p 2 and in the limit of q 2 → 0, the four form factors F Q (0), F M (0), F E (0), and F A (0) are the electric charge, magnetic dipole, electric dipole and anapole of χ, respectively. For simplicity, we denote In this work, we do not consider the electric charge and anapole of χ because the former remains zero at loop levels if DM is electrically neutral at tree level and the latter causes suppressed signals in DM direct detection. This suppression can be seen from the form of the F A term of Eq. (3.2), which in the low-q 2 regime is proportional to q 2 . This O(q 2 ) coefficient will be canceled by the photon propagator which is proportional to O(q −2 ). Indeed, Ref. [28] has shown that the effect of anapole in direct detection is nearly equivalent to that of contact interactions. Thus, unlike with dipoles, the direct detection does not profit from the several orders of magnitude enhancement related to the 1/q 2 behavior of the amplitude, see below.
Neutral χ might possess a non-vanishing charge radius defined as dF Q (q 2 )/dq 2 | q 2 →0 . Its effect in direct detection is also suppressed for the same reason. 4 Table 2. Electromagnetic dipoles generated by the loop diagram in Fig. 1. Here the "t" and "s" indices refer to the results obtained considering the corresponding channel in Eq. (3.4). C (t) and C (s) are given in Eq. (3.5).
Since they involve a loop where momenta runs from 0 to infinity, the use of the effective theory to compute these dipoles does not necessarily lead to consistent results. Related to that, two explicit UV complete theories leading to the same operators at low energy does not necessarily lead to the same dipoles. Explicit calculations shows that indeed the calculation of the dipoles from Fig. 1, i.e. with G X a constant, is not consistent, since it leads to loop integral divergent results. Thus one must open the four-fermion interactions. Here we will open them along the two simplest possible ways, from the exchange of a t-channel or s-channel heavy mediator giving momentum-dependent G X functions where m med and y are the mediator mass and coupling, s = (p 1 − p 2 ) 2 and t = (p 1 − k 1 ) 2 . 5 There could be more complex scenarios for the internal structure of the effective vertex, where e.g. G X is generated by a box diagram, for which the current framework does not apply since it typically requires two-loop calculations (which are beyond the scope of this work). Substituting Eq. (3.4) in Eq. (3.1) and performing the loop integration (see Appendix A), we obtain the results in Tab. 2 where C (t) and C (s) in the last column are defined as Here C (t) is finite but C (s) contains a UV divergence, with µ and ε defined by the dimen- 6 The results are obtained assuming , coupling it in pairs to a pair of charged fermions, would lead to suppressed direct detection in a similar way, as it would induce only a charged radius. For a Majorana DM candidate, as is well known, dipole interactions identically vanish, and an anapole leads to suppressed direct detection in a similar way than for a Dirac fermion. 5 Note that when the left diagram is interpreted as DM annihilation, we flip the direction of p2 and obtain s = (p1 + p2) 2 which is the conventional definition of s as the Mandelstam variable. 6 Note however that the tensor operator cannot result from a simple tree level s-channel exchange, but must be induced e.g. from a loop diagram coupling the s-channel mediator to the pair of DM particles and from another loop diagram coupling the s-channel mediator to the pair of SM fermions. This case is thus of the heavy mediator limit: m med m χ,f . It is noteworthy that the loop-induced dipoles for the S, P and T cases are proportional to m f while for the other two cases they are proportional to m χ . This is due to the well-known chirality-flipping nature of S, P , T interactions-see discussions in Ref. [42].
For the t-channel case a magnetic or an electric dipole is always induced (even if never both), depending on the operator considered. In all cases this allows non-suppressed direct detection signals as we will see below. Baring cancellations this implies that any UV complete model generating any one of these operators through a t-channel transition can be efficiently probed via direct detection, see below. This is presumably also the case for models where the effective interactions would be induced at loop level, such as through box diagrams (but we will not explicitly check this statement here). Note that for this t-channel case all the results are obtained finite as it should obviously be. In the s-channel case instead no dipoles at all are obtained for the S, P , V and A cases, as a result of the fact that in this case the loop is a self-energy which cannot give rise to a σ µν . For the tensor case one can get a dipole as the effective operator already contain a σ µν to start with.
In summary, we get two general classes of scenarios, the one leading for simple reasons to vanisihing dipoles and the ones leading to non-vanishing dipoles. For the second class we have where the O( a ) part has been specified in Tab. 2.

A UV complete example
The results obtained above for the dipoles by replacing in Fig. 1 the four fermion interaction by a t-channel propagator have no reasons to give exactly what we would get in a UV complete model leading to these operators through a t-channel heavy mediator exchange. Since DM is neutral and the SM fermions are charged, the t-channel heavy mediator has necessarily a non-vanishing electric charge. Thus in a UV complete model there are necessarily extra diagrams that may modify the dipoles, simply attaching the photon to the heavy mediator rather than to the SM charged fermion. However, for a mediator much heavier than the other particles we do not expect in general that these extra diagrams could induce any large destructive interference for the dipole induced (given in particular the chiral structure of the SM), or even largely change the results. To illustrate this, we consider a simple UV complete model leading to the 4-fermion interactions through t-channel exchange. Consider a charged scalar φ ± that couples to f R and χ: limited interest. Tensor interactions are generated in an easier way from tree level t-channel (through Fierz transformation) or one loop box diagrams. Thus, even if this divergence, that we get only for this s-channel tensor case, means that the result is inconsistent (and that in UV complete models this dipole interaction necessarily never comes without other interactions), we will not elaborate more on this problem. At a very rough level one can expect constraints on this case similar to the ones obtained below for the t-channel T case.
Assuming the scalar boson mass m φ is heavy, by integrating out φ ± , we obtain the effective interaction L eff = −G X χP R f f P L χ, (3.8) where G X = yy * /m 2 φ . One can reformulate it to the form in Eq. (2.1) via Fierz transformation: 7 Eq. (3.9) contains two types (V and A) of effective interactions, with the following 's: 8 According to Tab. 2, by summing up the contributions of the above 's, i.e. from Eq. (3.9) and Tab. 2, we obtain the magnetic dipole in the EFT approach: One can also compute exactly the dipoles directly from the model Lagrangian. As already mentioned above, there are two diagrams that couple χ-χ to the photon, one with the photon coupled to f and the other one with the photon coupled to φ ± . The resulting magnetic dipole from the two diagrams reads (see Appendix A for details on this calculation): Note that a purely left-handed interaction involving a SM fermion doublet and a scalar doublet instead of f R and φ + in Eq. (3.7) gives the same dipoles as the purely right-handed case of Eq. (3.7). By comparing Eq. (3.12) to Eq. (3.11), we see that the magnetic dipole computed in the UV theory in actually 50% higher than the EFT result. Therefore, when using the results of Tab. 2, one should keep in mind that the results may be changed due to new contributions in complete theories. Nevertheless, the EFT driven results of Tab. 2 provide correct estimates of the order of magnitude of the dipoles. In other words, one cannot exclude that specific UV models would give quite different results between both approaches but this explicit example shows that in simple frameworks this is not the case. 7 See, e.g., Ref. [43], page 65. 8 It is not surprising that we get a combination of the SM gauge group invariant V and A operators since the original interactions of Eqs. (3.7) and (3.8) are gauge invariant. Operators of the S, P or T type can be generated in UV complete models, for instance from inducing a gauge invariant dimension 7 operator involving an extra Higgs doublet and electroweak symmetry breaking. In this case the extra interactions involving the Higgs doublet components rather than the Higgs boson vev do not induce an extra contribution to the dipole at same one loop order (but could be relevant for DM annihilation for multi-TeV DM, i.e. for mDM vEW ). Such operators could also be generated if for instance, on top of the interaction of Eq. (3.7), there exists a similar interaction involving a SM left-handed doublet and a scalar doublet rather than fR and φ + , and if the charged component of this scalar doublet mixes with φ + via electroweak symmetry breaking.

Expected magnitude of electromagnetic dipoles
The relic abundance constraint, Ω χ h 2 0.12, requires that the coefficients of the effective operators are typically of the order of a few (or tens of ) percent-see Tab. 1 and Eq. (2.11). By requiring that Ω χ h 2 0.12 is correctly produced, according to Eq. (2.11), we replace L,R a in Tab. 2 with 100 GeV · m −1 χ a and obtain where ecm ≡ e × cm ≈ 15350.3 eV −1 is a commonly used unit for electromagnetic dipoles. The result is almost independent of m χ and m f . It only depends on the electric charge of the SM fermion involved, Q f , which can be 2/3 (for f = u, c, t), −1/3 (for f = d, s, b), or −1 (for f = e, µ, τ ). For S, P , and T interactions, the dipoles depend on m f . Since in this work we require that χχ → f f is responsible for the relic abundance, we concentrate on cases with m χ m f . With this assumption, one can still apply L/R a = 100 GeV · m −1 χ a to the remaining dipoles:

Electromagnetic dipoles in direct detection
In direct detection experiments, the differential event rate of DM-nucleus scattering 9 can be evaluated via (see e.g. [39,44]): Here E r is the nuclear recoil energy; N T is the total number of target nuclei; n χ is the local DM number density; (E r ) is the detection efficiency; dσ dEr is the differential cross section; f ⊕ (v) is the DM velocity distribution in the Earth frame; v min is the minimal velocity to generate a given E r , where m N is the nucleus mass and µ χN ≡ m χ m N /(m χ + m N ) is the DM-nucleus reduced mass. For the local DM density we take n χ = ρ χ /m χ and ρ χ = 0.4 GeV/cm 3 [45].
The DM velocity distribution f ⊕ (v) is often parametrized by a truncated Maxwellian distribution in the frame of the Galaxy and then boosted to the Earth frame. The specific form reads whereṽ = |v + v ⊕ | and |v ⊕ | ≈ 240 km/s is the velocity of the Earth with respect to the Galaxy; v esc ≈ 550 km/s is the escape velocity of the Galaxy; v 0 = 220 km/s is the mean velocity of the Maxwellian distribution. The N f factor normalizes f ⊕ so that f ⊕ (v)d 3 v = 1: The differential cross sections for DM-nucleus scattering via dipoles read [17,21]: Note the 1/E r dependence of these differential cross sections, stemming from the propagator of the (massless) photon. As compared to a standard WIMP case, where the particle exchanged with the nucleon has typically an electroweak scale mass, this will largely boost the number of events in direct detection events, since the recoil energy considered in these experiments is typically of order 5-50 keV (see below). This explains why below the constraints from direct detection through dipoles will be competitive, despite the fact that they involve loop suppressed quantities. In Eqs. (4.5)-(4.6), α = 1/137, Z is the atomic number, F E and F M are two nuclear form factors, and G is a dimensionless quantity depending on the nuclear spin J and the nuclear magnetic dipole d N [17]: where d n = e/(2m p ) is the nuclear magneton, and A is the mass number. For the nuclear form factors, we adopt the following approximate expressions [17]: where r = 1.12A 1/3 fm,r = A 1/3 fm, s = 1 fm, and q = √ 2E r m N . The Θ function takes either the value 1 or the value 0, depending on whether the condition it involves is satisfied. For Xe targets, qr < 4.5 corresponds to E r < 117 keV, which in practice is always satisfied.
In the magnetic dipole cross section (4.5) we have included both SI (∝ F 2 E ) and SD (∝ F 2 M ) parts because they can be equally important. For example, when m χ = m N and v = 1.2v min , the ratio of the two parts at E r = 30 keV is about 1.6. For the electric dipole cross section (4.6) we have neglected a possible SD contribution because it is highly suppressed. The fundamental reason for this is that electric charges of nucleons can be added coherently, unlike magnetic moments of nucleons. As a consequence, the electric dipole cross section is generally much larger than the magnetic one when d M d E . For both Eqs. (4.5) and (4.6), the velocity dependence can be written as follows: where dσ g /dE r and dσ h /dE r are velocity independent. Substituting Eq. (4.10) into Eq. (4.1), we obtain can be computed independently of the cross section and of the kinematics of DM-nucleus scattering. When numerically evaluating the integrals, we take Eq.
where θ is the angle between v ⊕ and v, and integrate θ from 0 to π, v from 0 to v 0 + v esc . The results are presented in Fig. 2. Using Eq. (4.11), we plot in Fig. 3 the differential event rates (dashed curves) for d M = 5 × 10 −20 ecm, d E = 1 × 10 −22 ecm, assuming m χ = 100 GeV and a 10 3 kg liquid Xe target. For comparison, we also present a curve for the following SI contact-interacting cross section:    where µ χn = m χ m n /(m χ + m n ) is the DM-nucleon reduced mass. For the DM-nucleon cross section σ n we have taken the typical value that can be probed by direct detection experiment today, σ n = 10 −46 cm 2 . For the solid curves in Fig. 3, we have included the detection efficiency of XENON-1T, which is taken from Fig. 1 in Ref. [10]. The range of relevant recoil energy in direct detection experiments is relatively narrow, as below ∼ 5 keV and above ∼ 50 keV the efficiency is suppressed (compare dashed and solid lines in Fig. 3).
As can be seen in this figure too, within this range, the dipole lines display, as expected, an extra 1/E r dependence with respect to the contact interaction case. As a result of this relatively narrow range of recoil energy, it is possible to recast the XENON-1T bounds obtained for a contact interaction into bounds holding for the   Fig. 4. The CMB bound is taken from Ref. [21]. The indirect detection bounds, also taken from Ref. [21], are derived from FERMI-LAT constraints on γ-rays from the Galaxy (labelled as indirect-galactic) and its dwarf satellite galaxies (indirect-dwarf). The collider bound is taken from Ref. [20]. massless mediator (∝ 1/E r ) case of interest here. To this end, we apply a spectrumfitting technique previously adopted in Ref. [47], namely using Eq. (4.14) to fit the dipoleinteracting recoil spectra. Specifically, for a given set of d M (d E ) and m χ , one can correspondingly find values of σ n and m χ (usually different from m χ ) that minimizes the integral (dR dipole /dE r − dR contact /dE r ) 2 dE r where dR dipole /dE r and dR contact /dE r are the dipole-and contact-interacting spectra (including the detection efficiency). The minimization is performed under an additional constraint that their total rates are equal. We find that after the minimization, the two spectra are usually very close, with relative differences typically below 20%, which is consistent with the conclusion in Ref. [47]. By mapping d M (d E )-m χ to σ n -m χ and taking the XENON-1T limit from Ref. [10], we obtain the bounds on d M and d E , presented in Fig. 4. For comparison, we also show in Fig. 5 other known bounds on DM electromagnetic dipoles from indirect detection, CMB observations, and collider searches. These bounds in the WIMP regime are known to be much weaker than that from direct detection. Fig. 4 shows that the possibility that the axial operators would be responsible for the observed DM relic density is already excluded by direct detection experiments within the 6.8 GeV < m χ < 1.9 TeV range for charged leptons. Future experiments such as XENON-nT will enlarge this range significantly. For the vector case, although it is beyond the current best limit from XENON-1T, future XENON-nT will be able to probe the range 11.8 GeV < m χ < 205 GeV. The axial case is more constrained than the vector one because it requires a larger coefficient to account for the relic density constraint due to p-wave annihilation, see Eq. (2.11) and Tab. 1. For the S, P and T cases the additional m f /m χ dependence of the dipoles decreases the sensitivity for high values of m χ but boosts it for low values. The sensitivity also splits among generations of fermions. Taking the /s]  Fig. 4 and recast them to bounds on 's according to Tab. 2, and further to bounds on σv according to Tab. 1, assuming a = V /A. For indirect detections, the "indirect-dwarf" bounds are taken from Ref. [48], and "indirect-GC" from [49], assuming that DM annihilates to τ + τ − , µ + µ − or bb. The horizontal line shows the thermal cross section value.
tensor case in the right panel of Fig. 4 as an example, XENON-1T has excluded m χ 189 GeV for f = µ while for f = τ this bound increases to m χ 1.2 TeV. The future experiment XENON-nT will be able to improve the mass bound by roughly a factor of three. The four fermion interactions also induce fluxes of cosmic rays from the annihilation into charged fermion they induce today at tree level in the galactic center and dwarf galaxies. Indirect detection experiments give upper bounds on these fluxes which are generally translated into upper bounds on the annihilation cross section assuming Ω χ h 2 = 0.12 (i.e. not looking at the implications that this annihilation cross section could have on the relic density). In Fig. 6 we show for the V case how these bounds compare with the bounds that can be obtained on the same cross sections from the bound that direct detection set on the dipoles and thus on the coefficient of the four-fermion operators (assuming Ω χ h 2 = 0.12 anyway there too). As Fig. 6 shows, despite that at tree level direct detection experiments are not much sensitive to the four-fermion operators for charged leptons or heavy quarks, when the loop-induced dipoles are taken into account, direct detection offers competitive constraints on such operators in comparison with indirect detection. For the A case (and similarly for the S case) indirect detection constraints are known to be weak as in this case the annihilation is of the p-wave type. But the bounds from dipole induced direct detection remains fully relevant, as given in Fig. 6 too. Explicit UV complete models can be constrained according to the combination of effective operators they lead to. For the model considered in Sec. 3.2, which in a characteristic way gives both V and A interactions with similar weights, we show in Fig. 7 upper bounds on the Yukawa coupling y of Eq. (3.7), using the dipoles obtained in Eq. (3.11) (EFT) and Eq. (3.12) (UV). As previously discussed in Sec. 3.2, the difference between the two approaches shown in Fig. 7 is small. The current XENON-1T limit excludes the mass range 8.1 GeV m χ 94 GeV while future XENON-nT will be able to probe the 6.2 GeV m χ 843 GeV range.
As mentioned in the introduction, Ref. [16] also studied the direct detection signals that could be induced by four-fermion effective operators (involving two charged leptons) via DM-photon interactions induced at the one-loop level. This calculation was done by calculating the loop directly from the EFT and applying an MS prescription. Before concluding let us add a few comments on the several improvements we made here. Beside the fact that, strictly speaking, the divergences obtained at the pure EFT level cannot be removed applying this prescription (since there is no counterterm that could cancel the divergences, i.e. the result must be finite), opening the four-fermion blob as we did above shows that the results can largely depend on the way these effective interactions are generated (as shown by the s-channel example which gives vanishing results). The two-step calculation we did, calculating first the various moments that are induced and subsequently computing what it gives for direct detection (instead of calculating directly the direct detection cross section) is useful as it identifies for each case what kind of electromagnetic interactions is induced (with what it implies for each of these interactions). Phenomeno-logically Fig. 4 also shows that the dramatic improvements of direct detection experiments in the last decade imply that the cases with chirality-flip suppressions (i.e. the S, P and T cases) are also testable (see e.g. the scalar case for f = τ in Fig. 4). As also shown above, electromagnetic interactions induced at the one-loop level are relevant not only for charged leptons but also for the heavy quark case. 10

Conclusion
In the presence of four-fermion effective interactions of dark matter (DM) with Standard Model (SM) fermions, electromagnetic dipoles of DM can easily be generated, due to the loop process illustrated in Fig. 1. This is the case in particular if the operators are generated through the exchange of a t-channel mediator. We study systematically for all possible effective interactions the loop-induced dipoles and find that, if they are not identically vanishing, the electromagnetic dipoles in the WIMP paradigm are typically of the order of 10 −21 (10 −20 ) ecm for vector (axial-vector) interactions, or of 10 −20 m f /m χ ecm for scalar, pseudo-scalar, and tensor interactions, see Eqs. (3.13)-(3.16). Calculations for a UV complete model give very similar results.
Via photon exchange such values imply observable nuclear recoil signals in direct detection experiments. This provides (or will provide) the most stringent constraints for various operators, in particular for axial or scalar operators, as well as for operators involving for instance muons. So far XENON-1T has excluded the loop-induced electromagnetic dipoles for some types of effective interactions in certain mass ranges-see Fig. 4. Future multi-ton liquid xenon experiments with substantially improved sensitivity will be able to probe the dipoles for all types of effective interactions over much broader mass ranges.
If P L in the above traces is replaced by P R , the results are similar except that µνρλ and λµρν flip their signs.
Next, we plug the traces into the loop integral and integrate out k, assuming the mass hierarchy: The loop integral is computed using Package-X [50] and expanded in q 2 , m f , and m χ . Only the leading order term is taken. Note that the integral is free from UV divergences because for k → ∞ the integral behaves like k −5 d 4 k.
After all the cancellations, only the magnetic dipole terms exist. Comparing the last row of Eq. (A.12) to Eq. (3.11), we see that M (i) reproduces the dipole obtained in the EFT approach. When the full theory is taken into account, the additional contribution due to M (ii) is roughly half the size of the previous one, assuming m f m χ . Taking Q φ = −Q f and summing the two diagrams together, we obtain the result in Eq. (3.12).