Incoherent Fermionic Dark Matter Absorption with Nucleon Fermi Motion

We investigate the incoherent regime of the fermionic dark matter absorption by nuclei using the relativistic Fermi gas model and nuclear form factors. With the momentum transfer being roughly equal to the dark matter mass $m_\chi$, the incoherent regime contributes significantly to the absorption process for $m_\chi \gtrsim 100$\,MeV with a spin-independent operator and for even smaller mass with a spin-dependent one. We also compare the situations for various target nuclei ($^{131}$Xe, $^{72}$Ge, $^{40}$Ar, $^{20}$Ne and $^4$He) that are typically used in the dark matter direct detection. A heavier nucleus actually has the advantage of probing the incoherent scattering of the fermionic absorption dark matter. Observing both the coherent and incoherent contributions would be an important justification of the fermionic dark matter absorption.

We investigate the incoherent regime of the fermionic dark matter absorption by nuclei using the relativistic Fermi gas model and nuclear form factors.With the momentum transfer being roughly equal to the dark matter mass mχ, the incoherent regime contributes significantly to the absorption process for mχ ≳ 100 MeV with a spin-independent operator and for even smaller mass with a spindependent one.We also compare the situations for various target nuclei ( 131 Xe, 72 Ge, 40 Ar, 20 Ne and 4 He) that are typically used in the dark matter direct detection.A heavier nucleus actually has the advantage of probing the incoherent scattering of the fermionic absorption dark matter.Observing both the coherent and incoherent contributions would be an important justification of the fermionic dark matter absorption.

I. INTRODUCTION
More than 80% of the matter in our Universe today is dark matter (DM) [1,2].But its nature still remains unknown.The direct detection [3,4] experiments searching for the DM particle candidates have already explored a substantial region of the parameter space in the 1 GeV ∼ 1 TeV mass range [5,6].For sub-GeV DM, the energy deposit by the DM scattering off a nucleus is typically below the experimental threshold [7,8].This has motivated a new wave of DM model building studies to seek viable mechanisms that could provide a detectable signal.
From this point of view, the DM absorption scenarios where the DM mass is efficiently converted to recoil energies are particularly attractive for probing sub-GeV DM.Both bosonic [9][10][11][12][13][14][15][16][17][18] and fermionic [19][20][21][22][23] DM can be absorbed.The bosonic DM can be fully absorbed in processes such as the dark photoelectric effect a + e → e to ionize the atomic electron and the inverse Primakoff process a+ A Z X → γ + A Z X or the Compton-like scattering a + e → γ + e that converts an axion (a) to a photon.Since both electron and photon are visible, all the axion energy can deposit in detector.
In comparison, the fermionic DM absorption requires a fermion in the final state to conserve the fermion number or equivalently the angular momentum.This does not necessarily mean that only part of the fermion mass can convert to deposit energy.To be more precise, the final-state particles of the charged current processes χ + A Z X → e ± + A Z∓1 X * are all visible in the DM direct detection detector [19,20].On the other hand, the neutral current version χ + A Z X → ν + A Z X has a neutrino (ν) in the final state and hence can deposit only a fraction of the DM mass as target recoil energy [19][20][21][22]24].Nevertheless, the mass to deposit energy conversion rate is still much higher than the classic elastic scattering.Both nuclei [25][26][27] and electron [28][29][30] targets have been searched in direct detection experiments.Concrete models for the neutral current absorption scenario and the current bounds can be found in [19][20][21][22][23].There is still a significant allowed parameter space for the neutral current fermionic DM absorption.
The neutral fermionic absorption DM has a deep root in the history of hunting neutrinos.After Pauli proposed neutrino to explain the missing energy of beta decay [31], verifying its existence became a necessary and urgent issue.Being neutral, it is very difficult for neutrinos to be directly measured.Kan Chang Wang suggested a twobody final state process with K-shell electron capture, e − + A Z X → ν + A Z−1 X, to probe neutrinos [32,33].This is analogous to the neutral current fermionic absorption process with the electron mass converted to the nuclei recoil energy for detection.At that time, electrons and nuclei were known particles while neutrinos were unknown.Since neutrinos are well established now, the same process can be used to probe the unknown fermionic DM particle in the initial state.
The neutral fermionic DM absorption can occur with either electron [21,22] or nucleus [19,20,24] as target.Since the absorption process intrinsically consumes all the DM mass (m χ ) rather than just its kinetic energy as in the conventional elastic scattering, the initial velocity is negligibly small for halo DM.If the target particle is also initially at rest, it then has a fixed recoil energy where m T is the target mass [22].However, the target electron is always bound in an atom.Its binding energy is intrinsically at the keV scale for heavy elements such as xenon.Being comparable with the detection threshold, it is then necessary to take the atomic effects into consideration.The monoenergetic recoil energy would broaden into a peak whose shape is determined by the initial bound and final ionized electron wave functions [21,22].A consistent calculation framework of atomic effects with second quantization can be found in [34].Nevertheless, the initial nucleon motion and binding effect have not been taken into consideration for the nuclei target yet.Since the initial DM particle is approximately at rest with nonrelativistic velocity, the momentum transfer q is actually determined by the neutrino momentum p ν .With tiny mass, the neutrino momentum actually has the same size as its energy.As a first estimation, let us assume the initial nucleon is also at rest and leave the effect of nucleon Fermi motion to be discussed later.Then, the neutrino energy is the DM mass subtracting the nucleon recoil energy, where m N ≫ m χ is the target nucleon mass.Note that the three-momentum transfer and its four-dimensional counterpart have roughly the same value, q 2 ≡ (p χ − p ν ) 2 ≈ −m 2 χ where p χ and p ν are the DM and neutrino momenta, respectively.For m χ ≳ 1/R N ∼ O (10) MeV that corresponds to the typical nuclei size R N ≳ 5 fm, the fermionic DM absorption process can see individual nucleons.For a typical DM direct detection experiment with energy threshold T r = m 2 χ /2(m A + m χ ) ≳ 1 keV [22] for a nuclei target with mass m A , the sensitive mass range is m χ ≳ O(10) MeV.In other words, the individual nucleon can be readily resolved and it is necessary to consider the effect of incoherent scattering with individual nucleons.
In this paper, we would explore the incoherent neutral current fermionic DM absorption with the Fermi motion of individual nucleon inside the target nuclei.In addition to kinematics, the effect on the transition rate can be estimated with Pauli blocking assuming Fermi gas description or form factor.

II. FERMI GAS AND RECOIL ENERGY BROADENING
With individual nucleons being resolved, the first effect to be considered is their Fermi motion inside nuclei and the broadening of the corresponding recoil energy.We describe the incoherent DM absorption using the impulse approximation, i.e., only a single nucleon participates in the process [35].Furthermore, the nucleus is assumed to be a fully degenerate Fermi gas with momentum distribution, where θ(p) is a step function and p F the Fermi momentum.With the prefactor 3/4πp 3 F that is the inverse of the momentum space volume, f (p N )d 3 p N = 1.The values of p F can be obtained from the electron-nucleus scattering experiments [36,37].A formula based on fitting these data can be extracted from the GENIE code [38], where A is the atomic mass.The proton and neutron Fermi momenta p p F and p n F in general differ from each other, especially for heavy nuclei, where p F is the Fermi momentum calculated according to Eq. ( 3) and Z is the charge number.Once measured, the extent of broadening can be used to identify whether the neutral current fermionic DM absorption happens with a proton or neutron target.With a Fermi momentum p F ∼ 250 MeV, the typical nucleon velocity v N ∼ 10 −1 in the nucleus rest frame is much larger than its DM counterpart v χ ∼ 10 −3 .So assuming the DM particles to be at rest in the lab frame is still a good approximation while the initial nucleon needs to be relaxed to have velocity v N .Consequently, the nucleon recoil energy with E N (E N ′ ) being the total energy for the initial (final) nucleon N (N ′ ) and s ≡ m 2 N + 2m χ E N + m 2 χ .Note that the energy boundaries E ± N ′ are independent of the nucleon momentum direction.Inside the allowed energy range above, the event rate has a flat distribution.
It is a good approximation to take For the realistic measurement, the cross section σ times the initial relative velocity v N is the one that directly enters event rate, where |M| 2 is the spin averaged matrix element.The nucleon velocity has been canceled with the relative velocity in the denominator of the cross section that originally comes from the flux factor.Since the finalstate nucleon energy spans a range proportional to |p N | as shown above, the two-body phase space integration dΦ 2 ≡ dT N ′ /8π|p N | would not vanish with |p N |.Intuitively thinking, the final-state phase space of an absorption process is supported by the consumed DM mass m χ , dΦ = 2m χ /m N , rather than the initial nucleon momentum.
As an example, below we assume vector-type operator the experimental analysis [20,[25][26][27].The corresponding spin averaged matrix element is where Λ is the cutoff scale for the operator O V .When replacing the momentum by variables in the lab frame, 25, the nucleon momentum can only induce ≲ 10% correction.So it is a good approximation to take |M| 2 ≈ 4m 2 χ m 2 N /Λ 4 as a constant.Then the integration over E N ′ gives a constant σv N ≈ m 2 χ /4πΛ 4 and hence the same for ⟨σv N ⟩ [22].
Figure 1 illustrates the effect of Fermi motion on the differential cross section ⟨dσv N /dT N ′ ⟩ for a xenon target (A = 131) with p F = 262 MeV.For easy comparison, the differential cross section has been normalized to the unit of (σv N ) 0 ≡ m 2 χ /4πΛ 4 .With this normalization, the curves are independent of the cutoff scale Λ and the small DM mass region is enhanced to make it more explicit.As demonstrated in Eq. ( 6), the span m χ v N of the nucleon recoil energy is the initial nucleon kinetic energy, scales linearly with the DM mass m χ .With vanishing m χ , the recoil energy T N ′ reduces to a fixed value, T N ′ = T N +m χ .For a given T N ′ , the initial nucleon momentum by reversing Eq. ( 6).Then the averaged differential cross section becomes For a given T N ′ , the initial nucleon momentum lim- respectively.This approximation in Eq. ( 9) describes the behaviors in Fig. 1 quite well.With very light DM, the Fermi motion significantly changes the monoenergetic recoil to a wide spectrum with peak around the Fermi energy T F ≡ p 2 F /2m N shown as the vertical dashed line.Heavier DM induces a wider spectrum with a smooth peak and the DM recoil energy can go beyond the Fermi energy by larger extent.
However, the nucleon bound inside nucleus cannot be treated as a free particle.So the effect of Fermi motion on scattering kinematics is not the only consequence.By analogy with the atomic effect for the fermionic DM absorption on electrons inside atom [21,22], the cross section must also be affected.Below we use two approaches, Pauli blocking and form factors, to evaluate this effect.

III. PAULI BLOCKING
Another consequence of the Fermi gas distribution in Eq. ( 2 Figure 2 shows the Pauli blocking factor defined as the ratio of the averaged cross sections with and without the Pauli blocking effect.One may see that the blocking effect is more significant for light DM whose mass determines the energy gain.With larger m χ , it is much easier for a nucleon to gain enough energy to pass the Fermi motion threshold.With larger Fermi momentum, neutrons have larger blocking effect than the proton counterparts.
An intuitive estimation of the Pauli blocking effect can also be obtained geometrically.For a given momentum transfer q, the fraction of nucleons unaffected by Pauli blocking is determined by the overlap between the initial Fermi sphere and the sphere shifted by q [39], The corresponding Pauli blocking factor P RFG for a relativistic Fermi gas (RFG) model is shown as dashed line in Fig. 2. Since |q| ≈ m χ , and a typical Fermi momentum is p F ≈ 262 MeV for 131 Xe, one can expect suppression of DM absorption for m χ ≲ 500 MeV which is roughly the place where P RFG starts to deviate from 1.
For m χ = 100 MeV, only about 30% of nucleons will actually contribute.The geometrical picture gives a quite good estimation of the Pauli blocking effect.Nevertheless, the implementation of the Pauli exclusion principle with a fully degenerate Fermi gas model according to Eq. ( 2) would lead to an overestimation of the blocking effect.Naively thinking, the final-state nucleon with momentum below p F should not be completely forbidden since the nucleons inside nuclei are not really fully degenerate.It is necessary to find a different way, namely form factors for the incoherent scattering, to get a better estimation.

IV. FORM FACTORS
The coherent scattering occurs when the momentum transfer is not large enough to resolve the individual nucleons.So it is understandable that the coherent scattering decreases while the incoherent counterpart increases with momentum transfer.To be more specific, the coherent scattering scales as A 2 F 2 (q 2 ) and F 2 (q 2 ) is the coherent nuclear form factor while the incoherent one scales as A[1 − F 2 (q 2 )] 2 [40].The nuclear form factor is normalized such that F (0) = 1 (pointlike nucleus) and F (q 2 ) → 0 for q 2 → ∞ (fully incoherent case).
Figure 3 illustrates the effect of nuclear form factors on the coherent and incoherent scattering cross sections for DM absorption on a 131 Xe target.To make fair comparison, the averaged cross section per nucleon ⟨σv N ⟩/A is shown in the unit of (σv N ) 0 .One may see that the coherent scattering prevails for light DM with m χ ≲ 100 MeV while the incoherent one starts to dominate for heavier DM.Between the two estimations of the incoherent contribution, Pauli blocking with RFG momentum distribution (blue) tends to give smaller values than the form factor approach (green) due to the hard cut at the Fermi momentum as mentioned above.There are various parametrizations of the nuclear form factors.We take the Helm, Klein-Nystrand, and symmetrized Fermi-distribution schemes for comparison.For completeness, we list their detailed parametrizations below.The Helm form factor [41,42], with j 1 (x) being the first order spherical Bessel function contains a scale parameter, r n ≡ c 2 + 7π 2 a 2 /3 − 5s 2 where c ≡ 1.23A 1/3 − 0.60, a = 0.52, and s = 0.9 fm.For comparison, the Klein-Nystrand form factor also has the 3j 1 (|q|r n )/|q|r n factor but the exponential suppression is replaced as [43] F KN (q 2 ) ≡ 3 with a k = 0.7 fm.The symmetrized Fermi-distribution form factor has a more complicated expression [44], with c ≡ 1.23A 1/3 − 0.60 fm and a = 0.52 fm.
As shown in Fig. 3, the cross section of coherent scattering is quickly suppressed by the momentum transfer that increases with the DM mass m χ .For m χ ≲ 100 MeV, the difference among the adopted form factors is not significant.In other words, the three parametrizations work quite well in the coherent regime.With larger momentum transfer, the difference can reach one order of magnitude, especially around the dips.Since when reaching the dip regions the incoherent scattering already starts to dominate, these large differences would not have The coherent (dashed and thin solid) and incoherent (thick solid) cross sections per nucleon as functions of the DM mass mχ for 131 Xe (blue), 72 Ge (orange), 40 Ar (green), 20 Ne (dark red), and 4 He (purple).For the coherent ones, both spin-independent (dashed) and spin-dependent (thin solid) cases have been shown.
a significant impact on the phenomenological studies in this paper.For the incoherent scattering, the difference among form factors is not visible since the corresponding form factor is 1 − F 2 (q 2 ) instead of F 2 (q 2 ).Below we would adopt the Helm form factor for further study.

V. TARGETS AND SPIN DEPENDENCE
With amplification by a factor of A for the coherent scattering, the incoherent contribution starts to dominate only for momentum transfer |q| ≳ 100 MeV.This is especially true for heavy nuclei such as xenon.But it is not necessary for the coherent scattering always to be amplified or even happen.There are two such cases.
One way is using a light nuclei that has smaller nucleon number A. Figure 4 shows the coherent cross section per nucleon for the typical nuclei (including 131 Xe, 72 Ge, 40 Ar, 20 Ne, and 4 He) used for DM direct detection experiments as dashed lines.With smaller nucleon number, the amplification of the coherent scattering compared with its incoherent counterpart would not be that large.The coherent cross section on the left-hand side keeps decreasing with smaller nuclei.For heavier DM mass, the form factor starts to have a dip around m χ ∼ 4.5/r n .Since the radius parameter r n increases with the atomic mass A, a heavier nucleus has its first dip at smaller m χ .Consequently, the dominance of the incoherent scattering appears at lighter DM mass.The heavier nucleus actually has the advantage of seeing the incoherent scattering for m χ ≳ 100 MeV.
Another way is the spin dependence of the effective DM absorption operators.The coherent enhancement A 2 happens for the spin-independent nucleon vertex.Being independent of the nucleon spin, the individual scattering The relative contribution of the incoherent fermionic DM absorption for different nuclear targets including 131 Xe (blue), 72 Ge (orange), 40 Ar (green), 20 Ne (dark red), and 4 He (purple).For the coherent part, the spin-independent vector operator OV with Helm form factor is considered for comparison.
matrix elements add up coherently before being squared to give the cross section.However, the amplification is lost for a spin-dependent nucleon vertex.For comparison with the vector operator used in our earlier discussions, its axial vector counterpart O A = ( N γ µ γ 5 N )(ν L γ µ γ 5 χ L ) would lead to a spin dependent nonrelativistic nucleon vertex.The dashed and dotted lines in Fig. 4 shows the spin-independent O V and spin-dependent O A contributions.Without amplification, the cross section on the left-hand side significantly reduces and its value per nucleon becomes inversely proportional to the atomic mass A. The coherent vs incoherent equality happens at much smaller DM mass now.
It is still true that for a heavier nucleus the incoherent scattering starts to dominate for smaller DM mass.
Figure 5 shows the relative contributions of the incoherent scattering as functions of the DM mass m χ .For comparison, the coherent scattering with spinindependent vector operator O V is shown as solid lines while for the spin-dependent axial vector operator O A the dashed curves significantly move up.One may see that the incoherent contribution increases with the DM mass m χ which roughly corresponds to the momentum transfer.The small wiggle when approaching 1, especially for the 131 Xe target that is shown as blue line, is caused by the dip and peaks in the Helm form factor.For all cases, the incoherent scattering becomes important for m χ ≳ 100 MeV with spin-independent operator.Figure 5 also shows the results with spin-dependent operator as dashed lines.The fraction of incoherent scattering is then significantly enhanced for light DM mass m χ ≲ 100 MeV as expected from Fig. 4.
It is then necessary to study the differential cross section of the incoherent scattering with the Helm form factors as shown in Fig. 6.Comparing it with Fig. 1, the cross section for light DM is significantly suppressed by the incoherent form factor 1 − F 2 (q).Note that the curves shown in both Figs. 1 and 6 are in the unit of (σv N ) 0 ≡ m 2 χ /4πΛ 4 that is already suppressed by the light DM mass.These features are consistent with the quick decreasing of the relative contribution of the incoherent scattering on the left side of Fig. 5.

VI. PROJECTED SENSITIVITIES
Finally, we demonstrate how the experimental sensitivities change if the incoherent channel is taken into account.As concrete examples, we consider the current experiments EXO-200 [29] (exposure 234.1 kg×yr), Majorana Demonstrator [26] (37.5 kg×yr), and PandaX-4T [25] (0.63 t×yr).Figure 7 shows the estimated upper limits on the cross section per nucleon, assuming no more than 10 events occur in the experiment.
As discussed above, for the spin-independent interaction, the coherent and incoherent contributions become comparable at m χ ∼ 150 MeV.Accordingly, for m χ ≳ 150 MeV, the limits obtained with both contributions (red) start to be stronger than those obtained for the coherent-only case (black).
For the spin-dependent interactions, the difference between the both-channels (blue) and coherent-only (green) scenarios is even more dramatic.The incoherent contribution already starts to dominate at m χ ∼ 30 MeV.This is caused by two reasons.First, some isotopes such as 136 Xe with even-even nuclei cannot contribute to the coherent scattering while all isotopes can contribute to the incoherent one.Second, even for those nuclei that can contribute, the coherent channel does not scale with the atomic mass A while its incoherent counterpart scales linearly with A. The difference is especially pronounced for EXO-200 and Majorana which are enriched in even-even isotopes.
into account, the sensitivities to both the spinindependent and spin-dependent interactions improve.The improvement is especially significant for the spindependent case.

VII. DISCUSSION AND CONCLUSION
For the fermionic absorption process, not just the mass conversion to the final-state kinetic energy is efficient but also for the momentum transfer.Being roughly the same size as the absorbed DM mass, the momentum transfer can already enter the incoherent regime for a DM mass m χ ≳ 100 MeV with a spin-independent operator.For comparison, the classical direct detection channel with elastic scattering can only provide a momentum transfer |q| ∼ m χ v χ that is suppressed by the DM velocity v χ ∼ 10 −3 that is just 0.1% of the speed of light.This feature should apply for not just the fermionic absorption process when the DM mass is fully converted but also any exothermic DM scenario.
Our paper is a first attempt to demonstrate and estimate the effects of incoherent scattering.Although a heavier nucleus tends to have larger scattering cross section in the coherent regime with a spin-independent operator, it has the advantage to see the coherent-incoherent equality at a smaller DM mass.For a spin-dependent operator, the coherent scattering cross section per nucleon actually decreases with the nuclei mass number and it is easier for a heavier nuclei target to observe the incoherent scattering.
The Pauli blocking treatment with relativistic Fermi gas model is quite rough but gives a very good illustration.In comparison, the form factor approach gives more features and is a commonly used way in the study of DM direct detection.Future improvement would be a realistic nuclear calculation of the transitions caused by the fermionic DM absorption which would be carried out in our next study.
scale linearly with both the DM mass m χ and the nucleon momentum |p N |.The energy span, 2m χ |p N |/m N , can reach O(10) MeV.Correspondingly, the 3-momentum transfer also broadens around |q| ≈ m χ .

FIG. 1 :
FIG. 1: The differential cross section of the incoherent fermionic DM absorption with vector-type effective operator OV as a function of the final nucleon kinetic energy T N ′ .With the Fermi momentum pF = 262 MeV taken from the xenon nuclei A = 131, its corresponding initial nucleon kinetic energy TN = 36 MeV is shown with the dashed line.
FIG.2:The Pauli blocking factor on the averaged cross section for protons (red) and neutrons (blue) as functions of the DM mass mχ.For comparison, the result with geometrical estimation is shown as dashed line.
) is Pauli blocking.Due to the Pauli exclusion principle, an initial-state nucleon can only transit to the unoccupied final-state levels with |p N ′ | = |p N + q| > p F .Thus, the Pauli blocking effect can be taken into account by introducing a step function θ(|p N ′ | − p F ) to Eq. (7).The kinetic energy T F ≡ p 2 F /2m N corresponding to the Fermi momentum has been shown as the dashed vertical line in Fig. 1.Only those nucleons with kinetic energy T N ′ ≥ T F can actually contribute to the process.
FIG.4:The coherent (dashed and thin solid) and incoherent (thick solid) cross sections per nucleon as functions of the DM mass mχ for 131 Xe (blue), 72 Ge (orange),40 Ar (green),20 Ne (dark red), and 4 He (purple).For the coherent ones, both spin-independent (dashed) and spin-dependent (thin solid) cases have been shown.
FIG.5:The relative contribution of the incoherent fermionic DM absorption for different nuclear targets including 131 Xe (blue), 72 Ge (orange),40 Ar (green),20 Ne (dark red), and 4 He (purple).For the coherent part, the spin-independent vector operator OV with Helm form factor is considered for comparison.
− F 2 H and blue for Pauli blocking with RFG momentum distribution) cross sections per nucleon on 131 Xe target as a function of the DM mass mχ.