Direct detection of Light Anapole and Magnetic Dipole DM

We present comparisons of direct detection data for"light WIMPs"with an anapole moment interaction (ADM) and a magnetic dipole moment interaction (MDM), both assuming the Standard Halo Model (SHM) for the dark halo of our galaxy and in a halo-independent manner. In the SHM analysis we find that a combination of the 90% CL LUX and CDMSlite limits or the new 90% CL SuperCDMS limit by itself exclude the parameter space regions allowed by DAMA, CoGeNT and CDMS-II-Si data for both ADM and MDM. In our halo-independent analysis the new LUX bound excludes the same potential signal regions as the previous XENON100 bound. Much of the remaining signal regions is now excluded by SuperCDMS, while the CDMSlite limit is much above them. The situation is of strong tension between the positive and negative search results both for ADM and MDM. We also clarify the confusion in the literature about the ADM scattering cross section.


Introduction
Some of the dark matter (DM) candidates most actively searched-for are weakly interacting massive particles (WIMPs), i.e. particles with weakly interacting cross sections and masses between about 1 GeV/c 2 and 10 TeV/c 2 . Particular attention has been given in recent times to "light WIMPs" with masses around 1-10 GeV/c 2 . In 2004 they were first shown [1] to provide a viable DM interpretation of the DAMA annual modulation [2] compatible with all negative searches at the time, assuming spin-independent isospin-conserving interactions with nuclei and the Standard Halo Model (SHM) for the dark halo of our galaxy. The interest in these candidates intensified after the first DAMA/LIBRA 2008 results [3], and even more in recent years, due to other potential DM hints in the same mass and cross section region.
We have so far mentioned only potential contact interactions. However, an interesting possibility explored recently is that of neutral DM particle candidates which interact with photons through higher electromagnetic moments. For example, these can occur if the DM particle couples at tree level to heavier charged states, and thus to the photon at loop level, as it happens with neutrinos in the Standard Model. Another possibility is that the neutral DM particle is a composite object made of charged particles, as a "dark neutron". In these cases, the interaction at low energy is described by non-renormalizable effective operators suppressed by the scale of new physics, expected to be proportional to the mass of the particles running in the loop, or otherwise, to the compositeness scale of the ultraviolet theory.
In particular, anapole moment DM (ADM) has been studied in the context of direct detection [25,[43][44][45][46] and colliders [48]. Assuming the SHM, Ref. [45] found that ADM could explain the recent CDMS-II-Si excess while still being compatible with the XENON10 and XENON100 null results, for WIMPs with mass m 7 GeV/c 2 . In Ref. [46], the CDMS-II-Si allowed region in the ADM parameter space assuming the SHM was found to be in tension with the LUX bound, although WIMPs with masses m 7 GeV/c 2 were found to be compatible with the LUX null results when a more conservative choice than that adopted by the XENON100 and LUX collaborations was used for the effective luminosity.
Here we consider ADM and MDM as potential explanations of the signal found in direct detection data, both assuming the SHM, and in a halo-independent analysis [47]. The halo-independent analysis was proposed in Refs. [45,[49][50][51][52][53] (see also [54][55][56]) and recently generalized in [47] to be applied to WIMP scattering cross sections with any type of speed dependency. This method consists in mapping the rate measurements and bounds onto v min space, where v min is the minimum WIMP speed necessary to impart to the target nucleus a recoil energy E R . This method allows to factor out a function of v min which gives the dependency of the rate on the DM velocity distribution, and use this as a detector independent variable. Since v min is also a detector-independent quantity, while E R is not (it depends on the target mass), outcomes from different direct detection experiments can be directly compared in v min space.

Data analysis
The data analysis in this paper is the same as in Ref. [53], except for the recent CoGeNT 2014 data [8,9].
The CoGeNT 2014 [8] region and bound in our SHM analysis is obtained using only the unmodulated rate. Adding the modulation data does not change the region significantly. In our halo-independent analysis we use both the rate and modulation amplitude data [8].
We analyze the CoGeNT 2014 data [8] in the same way we analyze the older 2011-2012 data [5][6][7] following the procedure specified in Ref. [7], but with a different choice of the residual surface event correction C(E). Using the analytic expression C(E) = 1 − exp(−aE) with E in keVee and a = 1.21 ± 0.11 we find that the errors in the resulting bulk rate are underestimated. Thus we use the data points for C(E) in Fig. 21 of Ref. [7] with their corresponding error bars. As we can see in Fig. 1, the CoGeNT 2014 region using the analytic C(E) form (labeled 'CoGeNT 2014 Analytic C(E)' in Fig. 1a) is much smaller than the region obtained by using the 12 data points of Fig. 21 of [7] (shown in Fig. 1b and labeled 'CoGeNT 2014'). Fig. 1 shows the regions and bounds in the usual WIMP-proton cross section σ p vs WIMP mass m plane assuming the SHM for WIMPs with spin-independent isospin-conserving interactions. For comparison we include the CoGeNT 2014 regions and limit taken from Ref. [9]. These regions (shown in dark blue) are shifted toward larger masses . 90% CL bounds and 68% and 90% CL allowed regions in the spin-independent WIMP proton cross section, σ p , vs WIMP mass plane, assuming the SHM (see section 4). The CRESST-II low mass allowed region, taken from Ref. [10], is only shown at 2σ CL in dark gray. The dark blue 'CoGeNT 2014 Ref. [9]' region and limit are taken from Ref. [9]. The CoGeNT 2011-2012 modulation data yield the blue regions. In Fig. 1a (left panel) the black contour CoGeNT 2014 region corresponds to using the analytic form for C(E) while Fig. 1b (right panel) shows the same region but with C(E) taken from the data points of Fig. 21 of Ref. [7]. and smaller cross sections respect to the regions we obtain, but the two are compatible within the error bars.
Although the C(E) in Fig. 21 of [7] was derived only using the CoGeNT 2011-2012 data, we think it is still reasonable to use it for the cumulative 2014 data. In particular the C(E) value derived from the two log-normal curves in Fig. 1a of [9] for the 0.5 to 0.7 keVee energy interval is C(E) = 0.49, entirely compatible with the C(E) in the same energy interval in Fig. 21 of [7], C(E) = 0.47 ± 0.34.
Notice that in Figs. 1 and 2 we also include the CoGeNT 2011-2012 m-σ regions derived from the annual modulation (only labeled 'CoGeNT' and shown in blue) centered at larger cross sections but still compatible within error bars.
For the halo-independent analysis of the CoGeNT 2014 unmodulated rate data we apply the C(E) in the 12 energy bins of Fig. 21 of Ref. [7] and then combine the resulting bulk rate into only four energy bins, 0.5 to 1.1 keVee, 1.1 to 1.7 keVee, 1.7 to 2.3 keVee and 2.3 to 2.9 keVee. For the modulation data, the C(E) correction is not needed, since it is assumed that the surface events are not annually modulated. With our choice of modulation phase (DAMA's best fit value of 152.4 days from January 1 st ) and modulation period of one year in our fit of the data, the modulation amplitude we find in the first and fourth energy bins are negative, thus we show their modulus in the corresponding figures with thinner solid lines. However, in both bins the modulation amplitude is compatible with zero, at the 0.7σ and 1.8σ confidence level for the first and fourth, respectively. Also in the second and third energy bins the amplitudes are compatible with zero, at the 1.3σ and 0.8σ CL respectively.

Anapole moment dark matter
The possibility that particles could have an anapole moment was first proposed by Ya. Zel'dovich in 1957 [57]. The anapole moment interaction breaks C and P, but preserves CP. It was first measured experimentally in cesium atoms in 1997 [58]. The interaction with photons of a spin- 1 2 Majorana fermion χ due to its anapole moment can be expressed in a Lorentz-invariant form as where g is a dimensionless coupling constant and Λ is the new physics mass scale. In the non-relativistic limit, this contains the interaction of the particle spin with the curl of the magnetic field, H = −(g/Λ 2 ) σ · ∇ × B. Notice thatχγ µ χ vanishes for Majorana fermions.
In Appendix A we derive the differential cross section for scattering of an ADM Majorana fermion with mass m and speed v = |v| with a target nucleus at rest. We find where the square of the momentum transfer is q 2 = 2m T E R with m T the nuclear mass, m N is the nucleon mass, µ N and µ T are the DM-nucleon and DM-nucleus reduced mass, respectively, λ N = e/2m N is the nuclear magneton, λ T is the nuclear magnetic moment, J T is the nuclear spin, and we defined the ADM reference cross section where α = e 2 /4π 1/137 is the fine structure constant. In an elastic collision v min is The first term in Eq. (3.2) corresponds to the interaction with the nuclear charge Z and the second term with the nuclear magnetic field. Notice that Eq. (3.2) differs from some of the expressions given for the same cross section in the literature [43,45]. The equation in Refs. [43] and [45] does not make the distinction between the electric and magnetic terms in the cross section and treats instead both as proportional to the nucleus charge squared. Eq. (3.2) agrees with the cross section given in Refs. [44] and [46] except that we do not make the approximation of taking F 2 M,T (q 2 ) equals to one. The form factors we use are the standard longitudinal and transverse nuclear form factors (see e.g. Eq. (1.29) of Ref. [59] or Eqs. (4.32) and (4.33) of Ref. [60]), F 2 L (q 2 ) and F 2 T (q 2 ), but normalized to 1 at q 2 = 0. Thus, using Eqs. (4.44), (4.48) and (4.64) of Ref. [59] we define the electric and magnetic form factors as Here M i and M f are the initial and final projections of the target nuclear angular momentum J T . For small |q|, the monopole component of the charge distribution gives the dominant contribution to the electric form factor F 2 E (q 2 ) (see Eqs. (4.42), (4.52) and (4.53) of Ref. [60]) which we take to be the Helm form factor [61]. The magnetic form factor F M (q 2 ) has contributions from the magnetic moments of the nucleons as well as from the magnetic currents due to the orbital motion of the protons.
For the light WIMPs we consider in the following, the second term in Eq. (3.2) is negligible for all the target nuclei we deal with except sodium. This term is more important for lighter nuclei, such as sodium and silicon, but silicon has a very small magnetic dipole moment. The nuclear magnetic moment of 23 Na is λ Na /λ N = 2.218. We took the magnetic form factor F 2 M (q 2 ) from Fig. 31 of Ref. [60] which is fitted well by the approximate functional , where |q| is in units of fm −1 . One can immediately notice that the velocity dependence of the differential cross section in Eq. (3.2) is different from the dependence of the cross section in the usual case of contact spin-independent interaction, which is dσ T /dE R ∝ 1/v 2 . What is very important for the application of the halo-independent formalism is the presence of two terms with different velocity dependence, which makes it necessary to resort to the generalization of the method, presented in Ref. [47]. There, we studied DM with magnetic dipole moment (MDM) whose cross section also has two terms with different velocity dependence. The MDM reference cross section is defined as with λ χ the DM magnetic dipole moment.

Halo-dependent analysis
For this analysis we use the SHM with a truncated Maxwell-Boltzmann velocity distribution in the galactic reference frame [62], where v is the velocity of the WIMP with respect to Earth and v E is the velocity of Earth with respect to the galaxy, whose average value is v . We use |v | = 232 km/s [62], v 0 = 220 km/s, and v esc = 544 km/s [63] . For the local DM density ρ we use 0.3 GeV/c 2 /cm 3 . In direct DM detection searches, the primary observable is the scattering rate within a detected energy interval E ∈ [E 1 , E 2 ], (see e.g. Ref. [47] for details). Here C T is the mass fraction of the target T , f (v, t) is the DM velocity distribution in the reference frame of Earth, and (E ) is the experimental acceptance. G T (E R , E ) is the detector target-dependent resolution function, expressing the probability that a recoil energy E R is measured as E , and incorporates the mean value  [8,9] and CDMS-II-Si [11] unmodulated rate data. For DAMA, each panel presents three regions, corresponding to three different choices of the sodium quenching factor: from left to right they are Q Na = 0.45, Q Na = 0.30 and the energy dependent Q Na (E R ) from Ref. [64], which is the lowest of the three. The larger values of Q Na yield DAMA regions which overlap more with the CDMS-II-Si and CoGeNT 2014 regions. All the allowed regions overlap in this case for WIMPs of mass 7 to 10 GeV/c 2 for ADM and about 8 to 10 GeV/c 2 for MDM, however the overlapping regions are rejected by several bounds.
In Fig. 2 we include the 90% CL bounds from SIMPLE [19], CDMS-II-Ge low threshold analysis [17], CDMS-II-Si [11], CDMSlite [? ], XENON10 S2-only analysis [14], XENON100 [16], and LUX [13]. For XENON10 and LUX we have multiple lines. For XENON10 the orange solid line is produced by conservatively setting the electron yield Q y to zero below 1.4 keVnr as in Ref. [16], while for the dashed line we do not make this cut [51][52][53]. For LUX the different limits correspond to 0 (dotted line), 1 (double-dotted-dashed line), 3 (dotted-dashed line), 5 (dashed line) and 24 (solid line) observed events (see Ref. [53] for a more complete description of these limits). However, in the range of masses and cross sections presented in Fig. 2 they all overlap except for the 0 observed events bound in the left panel, and also in the right panel except at the bottom of the plot. Fig. 2 shows that, within the SHM both for ADM and MDM, the allowed regions do overlap, as just mentioned, but the regions are entirely rejected by the combined limits of LUX and CDMSlite.

Halo-independent analysis
In this section we proceed as in Ref. [47]. Changing the order of the v and E R integrations in Eq. (4.2) we can write and .
T v 2 /m T is the maximum recoil energy a WIMP of speed v can impart in an elastic collision to a target nucleus initially at rest. We divided the cross section by σ ref , which contains the unknown coupling constants, to factor it intof (v, t), together with ρ. We also factored out from H a 1/v 2 factor in order to write the velocity integral in Eq. (5.1) in the usual formf (v, t)/v of the rate of simple contact interactions. Finally, in the second line we merely changed integration variable from E R to v min using Eq. (3.4).
Introducing the speed distribution we can rewrite Eq. (5.1) as We now define the functionη(v, t) by η(v min , t) goes to zero in the limit of v min going to infinity. Using Eq. (5.6) in Eq. (5.5) we get therefore where in the last equality we integrated by parts and defined the response function The velocity integralη(v min , t) has an annual modulation due to the revolution of Earth around the Sun typically well approximated by the first terms of a harmonic series, where t 0 is the time of maximum signal and ω = 2π/yr. The unmodulated and modulated componentsη 0 andη 1 enter respectively in the definition of the unmodulated and modulated parts of the rate, Once the WIMP mass is fixed, the functionsη 0 (v min ) andη 1 (v min ) are detector-independent quantities that must be common to all (non-directional) direct DM experiments. Therefore we can map the rate measurements and bounds of different experiments into measurements of and bounds onη 0 (v min ) andη 1 (v min ) as functions of v min . Following Ref. [47] we map rate measurementsR i [E 1 ,E 2 ] into measurements ofη i (v min ) (i = 0, 1) as follows. From Eq. (5.7) we get averages ofη (v min ) is proportional to dE max R /dv, which grows linearly with v, times a function that vanishes rapidly for large enough E max R (v) due to the detector resolution function G T (E R , E ) being localized. For this reason, in this case R has a finite width and its integral is finite.
When dσ T /dE R depends on powers of v higher than −2 as is the case of ADM and MDM where a constant term v 0 is present, the integral of the response function H is infinite. In Ref. [47] this problem was circumvented by 'regularizing' the average, i.e. by considering the average of v r minη i (v min ) for some suitable positive power r such that for different experiments. In the last equality we integrated by parts and used the fact that the boundary term vanishes for a good choice of r. In this way one obtains a 'modified response function' v −r min R [E 1 ,E 2 ] (v min ) with finite width. r should be large enough so that the v min ranges of different energy intervals do not overlap too much, but small enough not to give a large weight to the low velocity tail of the response function, which depends on the low energy tail of the resolution function G T (E R , E ) that is never well known. In Fig. 3 we show for ADM some modified response functions v −r min R [E 1 ,E 2 ] (v min ) for various values of r. While the minimum r required to regularize the integral for ADM is r = 2, we see that the modified response function is very wide for r values up to about 10 (the same as for MDM, see Ref. [47]). We use thus r = 10 in the figures (same choice as for MDM in Ref. [47]).
As in Ref. [47], to determine the v min interval corresponding to each detected energy interval [E 1 , E 2 ] for a particular experiment, we choose to use 90% central quantile intervals of the modified response function, i.e. we determine v min,1 and v min,2 such that the area under the function v −r min R [E 1 ,E 2 ] (v min ) to the left of v min,1 and the area to the right of v min,2 are each separately 5% of the total area. This gives the horizontal width of the crosses corresponding to the rate measurements in Figs. 4, 5, 6 and 7. The horizontal placement of the vertical bar in the crosses corresponds to the maximum of the modified response function. The extension of the vertical bar, unless otherwise indicated, shows the 1σ interval around the central value of the measured rate. In the figures, rather than drawing the new averages v r minη i , we prefer to show v −r min v r minη i , so that a comparison can be easily made with the limits onη 0 described below, as well as with the previous literature on the spin-independent halo-independent method.
To compute upper bounds onη 0 , the unmodulated part ofη, from upper limits R lim on the unmodulated rates, we follow the procedure proposed in Ref. [49]. Becauseη 0 (v min ) is by definition a non-decreasing function, the lowest possibleη 0 (v min ) function passing through a point (v 0 ,η 0 ) in v min space is the downward stepη 0 θ(v 0 − v min ). The maximum value ofη 0 allowed by a null experiment at a certain confidence level, denoted byη lim (v 0 ), is determined by the experimental limit on the rate R lim .   to an ADM candidate very close in mass, but looking closely one can see differences in the bounds, due to the difference in cross sections.
In the halo-independent analysis the most constraining limits on ADM and MDM come from LUX, XENON100 and XENON10 while the CDMSlite bound is much above the DMsignal regions.

Results and conclusions
In the SHM analysis of the allowed regions and bounds in the m-σ ref parameter space (Fig. 2 to 7) this limit is much above the DM-signal region. The difference stems from the steepness of the SHMη 0 as a function of v min , which is constrained at low v min by the CDMSlite and other limits. In our halo-independent analysis, although the LUX bound is more constraining than the XENON100 limit, both cover the same range in v min space and are limited to v min 600 km/s for a WIMP mass of 7 GeV/c 2 , 450 km/s for 10 GeV/c 2 and 250 km/s for 20 GeV/c 2 . This is due to the conservative suppression of the response function below 3.0 keVnr assumed in this analysis for both LUX and XENON100 (see Ref. [53] for details). Thus the LUX bound and the previous XENON100 bound exclude mostly the same data for ADM and MDM. In other words, almost all the DAMA, CoGeNT (both the 2011-2012 and 2014 data sets), and CDMS-II-Si energy bins that are not excluded by XENON100 are not excluded by LUX either. The situation remains of strong tension between the positive and negative results, as it was already before the LUX data.
At lower v min values the most stringent bound in our halo-independent analysis comes from the XENON10 S2-only analysis. Even without considering the upper limits, in our halo-independent analysis there are problems in the DM signal regions by themselves: the crosses representing the unmodulated rate measurements of CDMS-II-Si are either overlapped or below the crosses indicating the modulation amplitude data as measured by CoGeNT (2011-2012 as well as 2014 data sets) and DAMA. This indicates strong tension between the CDMS-II-Si data on one side, and DAMA and CoGeNT modulation data on the other (these two seem largely compatible).

A ADM cross section
In general, the differential cross section of the scattering of WIMP off a target nucleus is given by the Golden Rule, where v is the modulus of the relative velocity v of the WIMP with respect to the nucleus, E p , E p and k , k are the final and initial energies of the nucleus and the WIMP respectively, p , p and k ,k are the final and initial momenta of the nucleus and the WIMP, respectively, and an overline denotes an average over the initial and sum over the final spin polarizations.
Here, the transition amplitude T fi at the leading order is defined by where S fi is the scattering matrix element between the initial and final states of the WIMPnucleus system, |k, s, λ and |k, s , λ with the initial (final) spin polarizations s (s ) and λ (λ ) of the WIMP and the nucleus, respectively, andL I (x) is the interaction lagrangian. We use the state normalization which is compatible with Eqs. (A.1) and (A.2), and the conservation of the total momentum. Integrating the ADM interaction lagrangian 1 2 g Λ 2χ γ µ γ 5 χ ∂ νF µν by parts, we can write an equivalent lagrangianL I (x) = −ĵ µ (x)Â µ (x) in terms of the electromagnetic current of the ADM Majorana fermion (notice thatˆdenotes operators unless otherwise stated) With this expression forL I (x), the right-hand side of Eq. (A.2) can be written as whereÂ µ (q) is the four-dimensional Fourier transform ofÂ µ (x), the four-momentum transfer q µ = k µ −k µ with k µ = ( k , k) and k µ = ( k , k ), and |k, s, λ = |k, s |λ with normalization k , s |k, s = (2π) 3 δ s s δ (3) (k − k). Notice that |λ and |λ have an implicit dependence on the nucleus momenta (determined by the ADM momenta, k and k).
The matrix element of the ADM electromagnetic current at the origin between ADM momentum and spin eigenstates can be written as j µ s s (0) = (ρ χ ss (0), j χ ss (0)) = q 2 g µλ − where u ks (u k s ) is the Majorana spinor wave function with momentum k (k ) and polarization s (s ), normalized asū ks u ks = 2mδ s s . In the non-relativistic limit,ū k s γ µ γ 5 u ks = √ 2 k 2 k k+k 2m · s, s , where s = χ † s σχ s with the vector of Pauli matrices σ = (σ 1 , σ 2 , σ 3 ) and the non-relativistic Pauli spinor χ s for the spin polarization s. A simple algebraic calculation shows that the non-relativistic limit of j µ s s (0) is where s T = s − (s ·q)q withq = q/|q| is the component of s transverse to q = k − k .
As a solution to the Heisenberg equation of motion in the Lorenz gauge ∂ µÂ µ (x) = 0, the four-dimensional Fourier transformÂ µ (q) ofÂ µ (x) generated by the nucleus can be written asÂ in terms of the four-dimensional Fourier transformĴ µ (q) of the nucleus electromagnetic current operatorĴ µ (x) (e is the electromagnetic coupling constant). Knowing that |λ and |λ are energy eigenstates with eigenvalues E p and E p , respectively, the matrix element λ |Â µ (q)|λ can be simply written as whereĴ µ (q) is the three-dimensional Fourier transform ofĴ µ (x 0 = 0, x), and q 2 = −|q| 2 in the non-relativistic limit.
in terms of the nuclear current matrix element λ |Ĵ µ (q)|λ = (ρ λ λ (q), J λ λ (q)). Notice that we have replaced s in the ρ λ λ (q) term with s T since for an elastic scattering q · (k + k ) = (k − k ) · (k + k ) = 0. To write the nuclear charge and current density matrix elements ρ λ λ (q) and J λ λ (q) in terms of the quantities defined in de Forest and Walecka [65], we decompose J µ (q) in the reference frame defined as the target nucleus rest frame with the z axis in the direction of the momentum exchange q lab , and the x and y axes in the plane transverse to q lab . Spherical basis vectors e ±1 = (x ± iŷ)/ √ 2 are also introduced. In terms of four-vectors, the de Forest and Walecka frame can be defined by the four mutually orthogonal four-vectors p µ , q µ − (p · q/m 2 T )p µ , e µ +1 and e µ −1 , which in this frame read p µ = (m T , 0), q µ − (p · q)p µ /m 2 T = (0, q lab ), e µ +1 = (0, e +1 ), and e µ −1 = (0, e −1 ). A short algebraic calculation reveals that a four-vector J µ (q) that obeys the conservation law q · J = 0 can be written as where ρ lab = J · p/m T and J lab α = −J · e * α (α = ±1) are the time and spherical components of J µ , respectively, in the de Forest and Walecka frame. Using the fact that p µ − (p · q/q 2 )q µ = (p µ + p µ )/2 and q 2 = −q 2 lab in the non-relativistic limit, we find where p and p are the initial and final momenta of the nucleus in a given frame, and J T,lab λ λ (q) = J lab λ λ,+1 · e +1 + J lab λ λ,−1 · e −1 . Inserting these equations into Eq. (A.11), where V T is the transverse velocity, V T = (k+k )/2m−(p+p )/2m T . Notice that V T ·q = 0 as the name indicates. Using k = k − q, p = p + q, and v = k/m − p/m T , the transverse velocity can also be written as This equation combined with the transversality condition of V T , V T ·q = 0, gives the relation which will be used later.