Direct detection of pseudo-Nambu-Goldstone dark matter with light mediator

It has been found that a pseudo-Nambu-Goldstone boson dark matter suppresses the amplitude for elastic scattering with nuclei in non-relativistic limit, and thus can naturally evade the strong constraint of dark matter direct detection experiments. In this paper, we show that non-zero elastic scattering cross section can be induced if the mediator mass is as small as momentum transfer. The predicted recoil energy spectrum can differ from that for usual thermal dark matter. Together with the relevant constraints such as thermal relic abundance, indirect detection and Higgs decays, we investigate the detectability through the current and future dark matter direct detection experiments.


Introduction
Dark matter direct detection experiments impose strong bounds on thermal dark matter. The current strongest bound is given by the XENON1T/PandaX-4T [1,2], and will be further updated by the future XENONnT experiment [3]. The pseudo-Nambu-Goldstone boson (pNGB) dark matter has been proposed as a candidate naturally evading this strong constraint [4]. This is because the amplitude of the elastic scattering with nuclei is suppressed by the small momentum of dark matter through the derivative coupling. A non-zero elastic cross section is induced at loop levels, however it is marginally small compared to future sensitivities [5][6][7]. In addition, a sophisticated global fit of the model with relevant constraints has also been performed [8].
For the pNGB dark matter model, an explicit soft breaking term is required to give a mass for the NGB after the spontaneous breaking of the corresponding global symmetry. It may be natural that the scale of the soft breaking term is much smaller than the scale of the spontaneous symmetry breaking in the sense that the global symmetry is approximate. However this is not necessarily true if a ultra-violet (UV) completion of the pNGB model is considered [9][10][11][12]. In this case, the effective pNGB model derived at low energy can have a soft breaking term larger than the scale of the spontaneous symmetry breaking. As a result, the mass of the particle mediating between dark matter and Standard Model (SM) particles can be much lighter than the dark matter mass.
In this paper, we consider the pNGB dark matter with a light mediator. In this scenario, we will show that non-zero elastic scattering cross section between dark matter and nuclei emerges even though the interactions arise from the derivative coupling. Furthermore, this scenario has a potential to discriminate the pNGB dark matter and the other dark matter candidates such as Weakly Interacting Massive Particles (WIMPs) by comparing the recoil energy spectrum of the event rate at direct detection. We contemplate some relevant constraints such as thermal production of dark matter, indirect detection bounds and Higgs invisible decay. Then, we show that some parameter region has already been excluded by the XENON1T bound, and some other region will be explored by the future XENONnT experiment.

The model
We consider the model extended with a complex singlet scalar S. The model has a global U (1) S symmetry under the transformation S → e iα S, which is broken to the Z 2 parity by introducing the soft breaking term S 2 . The soft breaking term can be derived from a UV completion of the model [9][10][11][12]. The Lagrangian is given by where the scalar potential V is written down as The λ HS term is the Higgs portal coupling giving the interaction between the SM and S. The last term is the soft breaking term giving the mass to the NGB after the spontaneous symmetry breaking. Then, the scalar fields H and S can be parametrized as where v, v s ∈ R are the vacuum expectation values for H and S, respectively. Because of the interests in the low energy dynamics of pNGB and the simplicity of calculation, we use the non-linear representation for S in this paper. These expectation values satisfy the following stationary conditions: The CP-even components h and s mix with each other via the Higgs portal coupling, and the mass matrix is given by This mass matrix is diagonalized by the unitary matrix, and thus the gauge eigenstates h and s can be rewritten by the mass eigenstates h 1 and h 2 as where h 1 is identified as the SM-like Higgs boson with the mass m h 1 = 125 GeV and h 2 is the second Higgs boson whose mass is assumed to be much lighter than m h 1 . The CP-odd component χ is the pNGB with the mass m χ and can be a stable dark matter candidate due to a Z 2 parity associated with the CP symmetry in the dark sector. The dark matter mass is given by the soft breaking parameter m χ . The quartic couplings λ H , λ HS and λ S in the scalar potential can be rewritten in terms of the physical quantities as 9) and the cubic couplings which are relevant to the subsequent sections are given by with the convention (2.14) Note that the interactions between the dark matter and the SM particles come from the kinetic term |∂ µ S| 2 and the soft breaking term − It could be natural from the 't Hooft sense that the soft breaking parameter m χ is much smaller than the scale of the spontaneous symmetry breaking v s because the Z 2 parity is enhanced to the global U (1) S symmetry in m χ → 0 limit. However note that m χ and v s are irrelevant and can be regarded as independent parameters once a UV completion of the model is considered [9][10][11][12]. Therefore, in the following we consider the case of v s , m h 2 m χ . From Eq. (2.4) and m 2 h 2 ≈ λ S v 2 s , it is found that the quadratic mass parameter µ 2 S should be negative and m 2 χ is the trigger of the symmetry breaking in this case. Although the breaking pattern is different, the interactions among the scalars do not change from the ordinal pNGB dark matter model due to v s ∈ R [4]. However, in the light mediator case (m h 2 < m χ ), we will show that non-zero elastic cross section with nuclei is induced and can be tested by the current and future direct detection experiments.
In particular, we focus on the mediator mass range of 20 MeV m h 2 200 MeV. This is because the lighter mass region has already been ruled out by the constraint of big bang nucleosyntheis (BBN) [13] while the heavier mass region is insensitive to dark matter direct detection as will be seen below. In the mass range we focus on, the mixing angle sin θ is roughly constrained in the range of 2 × 10 −5 sin θ 10 −4 [14]. The upper and lower bounds come from the constraints of the meson decays and BBN, respectively. Figure 1: Diagrams for the main dark matter annihilation χχ → h 2 h 2 .

Relic abundance
In the current scenario, the dark matter annihilation channel χχ → h 2 h 2 is dominant in the most of the parameter space. The cross section is calculated from the four diagrams shown in Fig. 1 as where s denotes the Mandelstam variable. In the above, sin θ s are used to simplify the equation. For additional channels χχ → W W, ZZ, f f , the cross sections are calculated as Then, the thermal averaged cross section can be given by [15] σv rel = 1 16m 4 where T is the temperature of the universe and K n (z) (n = 1, 2) denotes the second kind modified Bessel function. The Boltzmann equation is numerically solved by mi-crOMEGAs [16] with the above analytic formulas for the annihilation cross sections, and the relic abundance should accommodate the PLANCK observation Ω χ h 2 ≈ 0.12 [17]. 1 Note that the early kinetic decoupling effect may change the parameter space which can reproduce the observed relic abundance when the dark matter mass is close to the SM-like Higgs resonance [19].

Indirect detection
The main annihilation channel χχ → h 2 h 2 generates cosmic rays in the galaxy via subsequent h 2 decays. In the mass range we focus on (30 MeV m h 2 200 MeV), the main decay channel is h 2 → e + e − , which is constrained by the e + e − observation of AMS-02 [20]. In addition to the e + e − flux, gamma rays are also generated via final state radiation from the produced e + e − and bounded by the observations from dwarf spheroidal galaxies at Fermi-LAT [21]. However this constraint is weaker than the e + e − production [22]. 2 The Cosmic Microwave Background (CMB) may also be distorted by the dark matter annihilation because the produced charged particles and gamma rays ionize the universe after the recombination era. Thus the CMB measurement by PLANCK also sets a bound on the model. We adopt the model independent bounds for multi-step cascade decays derived in the literature [22] (top left panel in Fig. 11 therein). The extracted bounds are shown in Fig. 2. As can be seen, the AMS-02 bound is severe for m χ 100 GeV. For m χ 300 GeV, the CMB bound can be stronger than the AMS-02 bound. However the bound requires σv rel O(10 −25 ) cm 3 /s, which is consistent with thermal relic abundance. Another comment is that the light mediator h 2 does not induce a long range interaction. Thus we do not need to care about the non-perturbative Sommerfeld effects for the dark matter annihilation [23][24][25][26][27]. This is due to the nature of the pNGB dark matter that all the interactions are written by the derivatives couplings.

Higgs decay
Since the mediator h 2 is light enough, the SM-like Higgs boson h 1 decays into h 2 h 2 whose decay width is calculated as where sin θ 1 and m h 2 m h 1 are assumed and m 2 h 2 ≈ λ S v 2 s is used. This decay channel can be regarded as an invisible decay because the lifetime of h 2 is long enough to escape the detector of colliders. In addition to this decay channel, another channel h 1 → χχ is also possible if the dark matter mass is m χ < m h 1 /2 ≈ 62.5 GeV. The decay width is evaluated as The observation of the Higgs signal strength at the LHC is translated into the constraint of the Higgs invisible decay whose branching fraction should satisfy Br inv ≤ 0.11 [28,29].

Direct detection of pNGB dark matter
It is known that the amplitude for the elastic scattering between the pNGB dark matter and a nucleus χA → χA vanishes in non-relativistic limit of dark matter. This argument is based on the premise that the mass of the mediator h 2 is much heavier than the momentum transfer q = √ 2m A E R [4]. However this is not the case in our scenario where the mass of the mediator h 2 can be the same order or smaller than the momentum transfer. Assuming q 2 , m 2 h 2 m 2 h 1 and sin θ 1, the differential cross section for the elastic scattering with a nucleus A is calculated as where t = −q 2 is the Mandelstam variable, v χ is the velocity of dark matter and F (E R ) is the nuclear Helm form factor which is parametrized by [30] F with the spherical Bessel function j 1 (qR), R ≈ √ r 2 − 5s 2 fm, r ≈ 1.2A 1/3 fm, (A is the mass number of the target nucleus) ands ≈ 1 fm. The Helm form factor is normalized as F (0) = 1. The coupling κ A in Eq. (4.1) is given by where A and Z are the mass number and the atomic number of the nucleus, respectively. The coefficients f p q and f n q are the scalar quark form factors of a nucleon, which are chosen as the values in the literature [16]  for the light quarks. The form factors of the heavy quarks can be written in terms of those of the light quarks. Direct detection experiments provide the limits on the elastic scattering cross section with a proton at zero momentum transfer, and the current strongest limit is given by XENON1T/PandaX-4T experiments [1,2]. Note that this limit cannot directly be applied to the current scenario because one cannot simply take zero momentum transfer limit in the differential cross section in Eq. (4.1) due to a strong dependence on the momentum transfer and light mediator mass. However, the experimental limit can be translated into the limit on the total event rate. Then, it is compared with the predicted event rate at a given parameters in the model. 3 The differential event rate is given by where ρ ≈ 0.3 GeV/cm 3 is the local dark matter density, N T is the number of target nucleus, f (v χ ) is the Maxwell-Boltzmann velocity distribution function at the solar and m h 2 = 60 MeV for the pNGB dark matter, and λ hS = 0.006 for the singlet scalar dark matter [33]. The solid (dashed) lines represent the spectra with (without) detector efficiency (XENON1T) [1].
system [32], and v min is the minimum velocity of dark matter at the given recoil energy where µ Aχ = m A m χ /(m A + m χ ) is the reduced mass between dark matter and nucleus. For the experimental limit, assuming no isospin violation [34], one can parametrize the differential cross section between dark matter and a nucleus as where σ exp p is the experimental upper limit on the total elastic cross section with a proton. Using this parametrization, the differential event rate with the experimental limit σ exp p can be rewritten as The left panel of Fig. 3 shows the upper bound on the total event rate R exp obtained from the XENON1T limit on the total elastic scattering cross section σ exp p . The interest , AMS-02 [20] and Higgs decays [28,29], respectively. The red line corresponds to the parameter space which can reproduce the observed relic abundance by PLANCK [17].
of region for the recoil energy is 4.9 keV < E R < 40.9 keV and the exposure 1.0 t × yr is taken into account [1]. The recoil energy spectrum for the pNGB dark matter could be discriminative from the other dark matter candidates thanks to its derivative couplings. The typical energy spectrum for the pNGB dark matter is shown as violet lines in the right panel of Fig. 3. The spectrum for the singlet scalar dark matter (an example of usual WIMPs) is also shown as green lines for comparison [33]. The event rate for the WIMP is enhanced at low energy while that for the pNGB dark matter is suppressed as in the figure. Note that WIMPs may also be able to induce energy spectra similar to the pNGB dark matter if the WIMP mass is heavier and detector efficiency is taken into account because the spectra can have a longer tail at higher energy for heavier WIMPs. 4 However, the shape of the recoil energy spectrum can be complementarily utilized to discriminate the pNGB dark matter and the other candidates combining with the other experiments and observations. Namely, for instance if the dark matter mass is inferred in narrow range by the other experiments or observations, one can use the recoil energy spectrum to discriminate the pNGB dark matter from the usual WIMPs at the end. Fig. 4 shows the parameter space in the (m χ , λ S ) plane where the Higgs mixing is fixed to be sin θ = 10 −4 and the second Higgs mass is m h 2 = 30, 60, 100 and 200 MeV. Fig. 5 shows the same figure with sin θ = 3×10 −5 . The green, orange and violet region are excluded by the limit of XENON1T [1], AMS-02 [20] and Higgs decays [28,29], respectively. The green dashed line represents the future sensitivity of the XENONnT experiment [3]. The red line can reproduce the thermal relic abundance consistent with the PLANCK observation Ω χ h 2 ≈ 0.12 [17]. The constraints of direct detection and Higgs decay tend to be severe for lighter h 2 and larger mixing sin θ as can be seen from Eqs. (3.6), (3.7) and (4.1). On the other hand, the AMS-02 bound does not much depend on m h 2 except for the region close to the Higgs resonance as obvious from Eq. (3.1). As discussed in Ref. [35], the perturbative unitarity constraint imposes the upper bound on the self quartic coupling as λ S < 8π/3, however it is invariably weaker than the constraint from Higgs decay in our scenario. From these plots, we can see that the thermally produced pNGB dark matter with the light mediator can be consistent with all the constraints when the dark matter mass is m χ ∼ 100 GeV for sin θ = 3 × 10 −5 and m h 2 = 60 MeV, and can be tested by the future XENONnT experiment.

Conclusions
In the previous works, the pNGB dark matter has been completely insensitive to dark matter direct detection experiments because the amplitude for the elastic scattering vanishes in non-relativistic limit. In this paper, we have focused on the case that the particle mediating the elastic scattering is light enough, and we have found that a nonzero contribution to the amplitude emerges. Together with the relevant constraints such as the thermal relic abundance, e + e − , gamma ray, CMB and Higgs decays, we have shown the parameter region of this scenario. Some parameter region have already been excluded by the current XENON1T experiment and some other region can be tested by the future XENONnT experiment. The pNGB dark matter can be discriminative from the other dark matter candidates due to complementary study of dark matter direct detection and cosmic ray observations.
In this Appendix, we give analytic expressions for all the annihilation channels for completeness. The annihilation cross sections for χχ → h i h j are given by with the kinematic function λ(x, y, z) = x 2 + y 2 + z 2 − 2xy − 2yz − 2zx and the cubic couplings κ ijk (i, j, k = 1, 2) are given in Eqs. is given by Since the propagator part can be simplified as when √ s, m h 1 m h 2 , Γ h 1 , Γ h 2 , one can verify the above cross sections do not violate the unitarity at s → ∞.