Pseudo Nambu-Goldstone Dark Matter: Examples of Vanishing Direct Detection Cross Section

We consider cases where the dark matter-nucleon interaction is naturally suppressed. We explicitly show that by extending the standard model scalar sector by a number of singlets, can lead to a vanishing direct detection cross section, if some softly broken symmetries are imposed in the dark sector. In particular, it is shown that if said symmetries are $SU(2)$ ($SU(N)$) and $U(1) \times S_N$, then the resulting pseudo-Nambu-Goldstone bosons can constitute the dark matter of the Universe, while naturally explaining the missing signal in nuclear recoil experiments.


Introduction
The current status of direct detection experiments reduces the allowed number of dark matter (DM) models with DM particle masses around the electroweak (EW) scale (typically O(GeV) − O(TeV)), as indicated by recent results from the XENON1T collaboration [1]. The main reason for this is the incompatibility of the experimental results with what one would expect from dimensional arguments (i.e. the so-called WIMP miracle [2]), indicating that a DM particle with mass around the EW scale, its interactions should have an EW strength. That is, if the dark matter particle has a mass accessible to direct detection experiments, why its interaction rate with nucleons is not within the detectable range?
There are various ways that the missing direct detection signal can be explained, e.g. via suppressed interaction with the nucleons due to a heavy (integrated-out) mediator [3,4,5,6,7,8], the appearance of "blind spots" [9,10,11] or the smallness of the DM mass [12,13,14,15,16,17,18] which makes the DM particle inaccessible to such experiments. 1 Among the most appealing scenarios, however, direct detection experiments are unable to detect the WIMP due to symmetry arguments [25,26,27,28,29]. In such models there is a symmetry that is responsible for the suppression of the DM-nucleon crosssection, usually through the cancellation of the tree-level DM-nucleon interaction.
In the present work, we focus on the pseudo-Nambu-Goldstone boson (PNGB) DM scenario, which can explain the lack of direct detection signal. The general idea is that Nambu-Goldstone bosons (NGB), which result from a spontaneous breaking of a global symmetry, have derivative couplings with other particles, and so their interactions vanish at zero momentum. On the other hand, a PNGB (a DM cannot be NGB, since it should be massive) is a result of a spontaneously broken approximate global symmetry, which could induce new interactions resulting to a non-vanishing direct detection cross section. However, there are examples of a cancellation that allows the tree-level DM-nucleon interaction to vanish at the zero momentum transfer [27,29], making models featuring such cancellation suitable DM candidates. Furthermore, since such models belong to the family of Higgs portal DM (e.g. [30,31,32,33,34]), this expands the ongoing search for simple models that can explain the DM content of the Universe while respecting experimental constraints.
In our effort to identify PNGB models featuring the aforementioned cancellation, we extent the SM by a scalar field (singlet under the SM gauge symmetry) and doublet under a softly broken SU (2) global symmetry. 2 We also show that the PNGBs in this case remain stable due to the symmetry properties of the interaction terms. Furthermore, we show how these arguments apply to a softly broken SU (N ) global symmetry.
Then, we move to another case, where we add two scalar fields (again singlet under the SM), and we note that the cancellation of PNGB-nucleon interaction occurs assuming a permutation symmetry. However, in contrast to the minimal case [27], the PNGB is not naturally stable unless a dark CPsymmetry is imposed. We also show that this model can be generalized to an arbitrary number (N ) of scalar fields, provided an S N symmetry assumption.
The outline of the paper is the following: in section 2, we discuss the DM content and the natural suppression of the DM-nucleon cross section in the SU (2). At the end of this section, we also show how these results are generalized in the SU (N ) case. In sec.3, we consider the U (1) × S 2 case, and show how the cancellation of the direct detection cross section takes place, which we then generalize to U (1) × S N . Finally, in section 4 we summarize our results, and comment on possible future directions.

The SU (2) case
In this section examine a dark sector with a softly broken SU (2) symmetry, in order to determine if the cancellation takes place. Specifically, the SM is extended by a scalar (Φ) which is a gauge singlet under the SM gauge group, and a doublet under a softly broken SU (2). We show that indeed this model can provide us with naturally stable (multi-component) DM, which exhibits a cancellation of the DM-nucleon interaction. We also show that this holds for SU (N ) and Φ in the fundamental representation. 3 The potential and mass terms The potential is comprised of two parts, the symmetric and the soft breaking ones. The symmetric part (global SU (2) invariant) is while the softly breaking part of the potential can be written as with m 2 Φ 12 = m 2 Φ 21 , m 2 Φ 12 = (m 2 Φ 21 ) * , and m 2 Φ 11,22 ∈ R. Also, note that the potential, Assuming that both H and Φ develop VEVs, where, without loss of generality we have assumed that the lower component of Φ obtains a non-zero VEV. By minimizing the potential, we obtain the following relations The Lagrangian mass terms can be written as where G = (χ, s, φ) T are the PNGBs and S = (h, ρ) T . The mass matrices become with λ Φ given by eq. (2.4). It is also evident that, as expected, M 2 G becomes a zero matrix (i.e. the pseudo-Nambu-Goldstone bosons become massless) in the limit of SU(2) invariance.

Stability of PNGBs
It is straightforward to show that the PNGBs in the mass eigenstate basis are stable, by simply writing down the interaction terms. The reason for the stability comes from the fact that SU (2) is explicitly broken by dimension-2 terms, and so the interaction terms are still SU (2) invariant. After SSB, the interaction terms involving the PNGBs become O(3) invariant. Therefore, the diagonalization of their mass matrix, does not induce any non-diagonal three-point interactions, e.g. only ξ 2 i h threepoint interactions (with ξ i the PNGBs in the mass eigenstate basis). This, in turn, means that the Lagrangian, is Z symmetric (i.e. each PNGB carries its own Z 2 parity) that not only forbids decays of the PNGBs, but also PNGB conversions as well. Thus, all pseudo-Nambu-Goldstone bosons are stable, resulting to a three-component DM content.

The pseudo-Nambu-Goldstone-nucleon interaction
Since all PNGBs are stable, we need to calculate three amplitudes for the direct detection cross section. However, due to the O(3) symmetry of the interaction terms, the amplitude for the ξ i -nucleon elastic scattering (ξ i n → ξ i n) is proportional to G i -nucleon elastic scattering amplitude and it is independent of i. In general, the interaction of the three-point terms pertinent to this interaction can be written as But due to the O(3) symmetry, we expect that Figure 1: The Feynman diagram for the elastic scattering between a quark (q) and a PNGB.
From the potential 2.1 and the relations 2.4, we obtain Since we are interested in the zero-momentum transfer limit, the propagator is proportional to the inverse of the mass-matrix M 2 S . Then the direct detection amplitude for all PNGBs (the Feynman diagram is shown in Fig. 1

) becomes
which concludes the proof of the claim that the DM-nucleon cross section vanishes at tree-level and zero momentum transfer. However, this only indicates that the direct detection cross section is "naturally suppressed". In practice, loop corrections need to be included as well, since these effects could allow for a possible direct detection signal [27,37,38].

Generalization to SU (N )
It is straightforward to generalize the above result in the case where Φ is in the fundamental representation of a softly broken SU (N ) global symmetry, since the form of V 0 is the same as in eq. (2.1), with the soft breaking terms being Assuming that the N th component of Φ develops a VEV, one can show that the minimization of the potential requires This results to 2N −1 PNGBs, χ, φ i and s i with i = 1, 2, . . . , N −1, where, similarly to the SU (2) case, all pseudo-Nambu-Goldstone bosons are stable particles. The Lagrangian pertinent to the pseudo-Nambu-Goldstone boson-nucleon interaction takes the familiar form Since the mass matrix M 2 S is independent of N (i.e. it is always given by eq. (2.6)), the amplitude for the process ξ i N → ξ i N at tree-level and zero momentum transfer, vanishes as in the SU (2) case.
One should keep in mind that the cancellation takes place only if Φ is in the fundamental representation of SU (N ). It is not clear if A DD would cancel if another (irreducible) representation of Φ was assumed, as there are additional interactions, corresponding to all the possible contractions of the SU (N ) indices. For example, for N = 2 and Φ in the adjoint representation, there is an interaction term of the form which can potentially change the mixing between the particles in a non-trivial way. Since the number of such interactions increases greatly with the dimension of each representation of SU (N ), it becomes hard to generalize. Thus, we postpone such analysis for the future. 4 3 The U (1) × S 2 case In this section, we examine another case, which we denote as U (1)×S 2 . In this case, the SM is extended by two scalars (S 1,2 ) charged only under a softly broken global U (1). For the desired cancellation to occur, we impose a permutation symmetry on S 1,2 . As we will see, this symmetry provides a sufficient condition for the vanishing of the PNGB-nucleon cross section.

The cancellation mechanism for this model
The Potential In the case of two scalars, each transforming as S i → e −ia S i , the U (1) × S 2 symmetric potential, assuming that all parameters are real numbers (we shall call this assumption dark CP-invariance), is while the S 2 -symmetric soft breaking potential is written as with the total potential given by V = V 0 + V soft . In order to find the minimization conditions, we expand the fields around their VEVs where this particular choice of S 1,2 , ensures that the potential remains symmetric under simultaneous permutations of (s , χ) 1 ↔ (s , χ) 2 . Due to the permutation symmetry, there are only two independent stationary point conditions, which read

Spectrum of the CP-odd scalars
In order to calculate the direct detection amplitude, we first need to identify the PNGB. This can be done by diagonalizing the mass matrix of the CP-odd fields to its eigenvalues. Once the eigenvalues are found, one of them should vanish in the limit where the U (1) is restored, which should correspond to the PNGB. From eq. (3.5), we obtain the mass matrix for the χ's (3.5) from which we find the eigenvalues It is apparent that m 2 ξ 1 vanishes in the limit µ 2 S 1,2 → 0, thus the particle corresponding to this mass can be identified as the would-be Nambu-Goldstone boson of the U (1), i.e. the PNGB of this model. The eigenstates corresponding to these masses are It is worth noting that the PNGB (ξ 1 ) is symmetric under χ 1 ↔ χ 2 . This property of the PNGB, will be proven helpful especially in the N -particle generalization of this model, since it will allow us to calculate the desired direct detection amplitude easily. The imposed dark CP-invariance can potentially keep both of the states ξ 1,2 stable, since there are only interactions involving even numbers of CP-odd particles, e.g. there is no ξ 1 h 2 interaction term while the vertex ξ 2 1 h exists. However, since we are interested in the scenario where the DM particle is a PNGB, we need to impose an extra hierarchy condition, so that ξ 1 will be stable while ξ 2 will be able to decay. This condition is m ξ 1 < m ξ 2 , with their difference (m ξ 2 − m ξ 1 ) at least larger than the mass of the lightest CP-even particle (e.g. m ξ 2 − m ξ 1 > m H ≈ 125GeV if the Higgs boson is the lightest one). This is not too restrictive, and it does not affect the vanishing of the PNGB-nucleon cross section, but it must be pointed out for the sake of completeness.

The direct detection amplitude
The calculation of the quark-ξ 1 scattering amplitude is a relatively straightforward task. We just need to calculate the corresponding Feynman diagram ( fig. 1). In fact, since we are interested in the zero momentum transfer limit, the ingredients that we need in order to show that the direct detection cross section vanishes, are the inverse of the mass matrix of the CP-even scalars and the three-point interaction of a pair of PNGBs with them (i.e. vertices of the form ξ 2 1 h and ξ 2 1 s 1,2 ). The mass terms for the CP-even scalars can be written in a compact form as with Φ = (h, s 1 , s 2 ) T and Observing that only h couples to SM fermions, we only need the following few terms of the inverse of With the interaction term of the Lagrangian terms responsible for the ξ 1 -nucleon elastic scattering being 5 we can show that the the amplitude for the ξ 1 -nucleon elastic scattering vanishes. That is 3.2 Generalization to U (1) × S N As we saw in sec. 3.1, the cancellation mechanism holds when the model consists of two scalars under the assumption that the potential is symmetric under permutations of these scalars. This symmetry fixes the PNGB-s 1,2 interactions and the relevant components of M 2 Φ in such way that A DD vanishes. However, there is no guarantee that this also happens if we add more scalars, since more interaction terms are allowed. In this section, we investigate whether A DD vanishes in a model consisting of an arbitrary number of scalars. We denote this model as U (1) × S N , and it is a direct generalization of U (1) × S 2 with N number of scalars.

The Potential for N Scalars
In the case of N scalar fields, each transforming as S i → e −ia S i (similarly to sec. 3.1), the U (1) × S N symmetric potential, assuming again dark CP-invariance, can be written as where all the sums run over all scalars. This potential has some redundant terms, so we can set some of them to zero: Furthermore, the permutation symmetry, dictates: As previously, we assume soft breaking of U (1). That is, we add the following terms in the potential where, due to the S N symmetry, we have So, from eqs. (3.14), (3.15), and (3.17), the total potential becomes At this point, it becomes clear that the S N symmetry helps keeping the number of new free parameters relatively small. 6 This keeps the model as simple as possible, considering the potential large number of particles.
Similar to the previous, the scalars acquire VEVs where, again, we have assumed that the potential remains symmetric under s, χ i ↔ s, χ j after SSB. From eqs. (3.18) and (3.19), we observe that there are only two independent stationary point conditions, due to the S N symmetry, (similar to 3.1), which are 20) These conditions further reduce the number of new parameters by one, i.e. the maximum number of new parameters introduced is 12 for N ≥ 4 (for N = 2 and 3 these are 9 and 11, respectively).

Spectrum of the CP-odd scalars
As in sec. 3.1, our next step is to find which mass eigenstate corresponds to the PNGB. To do so, we first have to find the mass matrix (M 2 χ ) for the CP-odd scalars. Since the CP-odd and CP-even scalars do not mix (due to the dark CP-invariance), their mass terms are symmetric under permutations of the χ's. As a result, there are only two different entries in the mass matrix for χ's, the diagonal, M 2 χ ii , and the off diagonal, M 2 χ ij , ones. After some algebra, one can show that The eigenvalues of this matrix are The first (m 2 ξ 1 ) corresponds to the particle ξ 1 , which is the PNGB (m 2 ξ 1 → 0 as µ 2 S 1,2 → 0), while the other particles (ξ 2,3,...,N ) are degenerate with mass m ξ 2 = m ξ 3 = · · · = m ξ N . As it turns out (in analogy to sec. 3.1), the PNGB is the S N -symmetric state where the others (not relevant to our discussion) can be found from orthonormality conditions. We also note again that some hierarchy conditions should be imposed in order for the PNGB to be the DM particle.

The Cancellation of the Direct Detection Cross Section
Again the ingredients that we need in order to show that the direct detection cross section vanishes, are the inverse of the mass matrix for the real part of the scalars and the interaction of a pair of pseudo-Nambu-Goldstone particles with them (i.e. ξ 2 1 − h, s i ). As usual the mass terms for the CP-even scalars can be written in a compact form as The interaction term of the Lagrangian which is responsible for the ξ 1 − N elastic scattering is Again, the propagator (i.e. the inverse of the s − h mass matrix) should be multiplied by a column vector ∼ δ 1i (since only h interacts with SM fermions), so the elements of the inverse of M 2 hs relevant to the DM-nucleon interaction are As in sec. 3.1, the Feynman diagram for the elastic PNGB-quark scattering is given fig. 1, with an amplitude proportional to In ref. [27] it was argued that the U (1) case is invariant under S → S † , because there is one phase which can be absorbed by S. This natural symmetry of the model guarantees that the imaginary part of S (the CP-odd scalar) always interact in pairs and as a result it is stable. However, when the scalar sector consists of a larger number of particles, it is not possible to absorb all phases to the scalars, as shown in Table 1. Therefore, in order to guarantee the stability of the DM particle ξ 1 , we have to assume that all parameters are real on top of the S N symmetry.

Conclusion and future direction
Inspired by an Abelian model which introduced a natural mechanism for the vanishing of the direct detection cross section, we have expanded the discussion on the explanation of the smallness of the DM direct detection cross section.
The first case under study (sec. 2) was a softly broken SU (2) global symmetry. In this, we assumed that there is a doublet scalar (singlet under the SM gauge symmetry), which acquires a VEV. We showed that the resulting pseudo-Nambu-Goldstone bosons are all DM candidates, due to a remaining discrete symmetry that keeps them stable. We also showed that the DM-neucleon interaction vanishes. Then, we argued that this case can be generalized in a straightforward fashion to an SU (N ) symmetry, leading to the same result, i.e. vanishing of the DM-neucleon interaction.
Then in sec. 3.2 we examined the U (1)×S N global symmetry, with U (1) being softly broken, where we extended the scalar sector by adding N scalars, charged only under a global U (1). Assuming a dark CP-invariance, we calculated the form of the mass matrices and three-point interactions relevant to the pseudo-Nambu-Goldstone-nucleon interaction, which turned out to vanish.
A parameter space analysis of some simple cases (e.g. U (1) × S 2 or SU (2)), will help us identify potential discovery channels at the LHC and astrophysical observations [39,29]. Also, a calculation of 1-loop corrections will give us with precision the direct detection cross section, which can further be used to probe (or even exclude) the models discussed in this work. In addition, since the cases at hand should be treated as low-energy limits of complete models, an interesting direction would be to determine possible completions. These, can induce (parametrically or energetically suppressed [27,29]) DM-nucleon interactions at the tree-level as well as decays of the PNGBs, allowing for a rich phenomenology, and connection of the DM problem with other open issues in particle physics (e.g. lepton number violation and neutrino masses [40]). Furthermore, there are some cases that we did not consider (i.e. the general irrep of the SU (N ) case), a study of other simple considerations (e.g. SU (2)-triplet) can be insightful, and help us identify similar classes of models. However, since we were only interested in furthering the discussion on the suppression of the DM-nucleon interaction, with a focus on simple realizations, we postpone these for a later project.