Probing pseudo-Goldstone dark matter at the LHC

Pseudo-Goldstone dark matter coupled to the Standard Model via the Higgs portal offers an attractive framework for phenomenologically viable pseudo-scalar dark matter. It enjoys natural suppression of the direct detection rate due to the vanishing of the relevant (tree level) Goldstone boson vertex at zero momentum transfer, which makes light WIMP-like dark matter consistent with the strong current bounds. In this work, we explore prospects of detecting pseudo-Goldstone dark matter at the LHC, focusing on the vector boson fusion (VBF) channel with missing energy. We find that, in substantial regions of parameter space, relatively light dark matter ($m_\chi<100$ GeV) can be discovered in the high luminosity run as long as it is produced in decays of the Higgs-like bosons.


I. INTRODUCTION
The dark matter (DM) puzzle remains one of the pressing issues in modern physics.
Various particle physics models have been constructed which fit known properties of dark matter. Among these, the weakly interacting massive particle (WIMP) paradigm remains one of the frontrunners.
Recently, the electroweak-scale dark matter models have come under increasing pressure from direct detection experiments which have so far found null results [1,2]. These constrain the nucleon-dark matter interactions at (effectively) zero momentum transfer. An interesting option to evade such bounds is to employ the property of Goldstone bosons that the relevant vertices vanish at zero momentum transfer ( Fig. 1), while otherwise are unsuppressed [3]. It is important that this statement also applies to massive pseudo-Goldstone bosons, which allows one to use this mechanism to suppress direct detection rates of WIMPlike dark matter. In this case, the nucleon-dark matter interaction arises at one loop level and satisfies the XENON1T bound naturally. This persists for a massive χ as well [3].
In this work, we explore the prospects of probing pseudo-Goldstone DM 1 at the LHC.
A promising channel with a relatively low background is vector boson fusion (VBF) Higgs production followed by its invisible decay into DM pairs [13]. The minimal model contains two Higgs-like scalars whose invisible decays contribute to this missing E T signature. Since pseudo-Goldstone bosons are naturally light, these channels are expected to be allowed kinematically. The relevant constraints that restrict the efficiency of DM production are 1 Pseudo-Goldstone DM in a different context has been considered in [4][5][6][7][8][9][10][11][12].
due to the current bounds on the Higgs invisible decay, heavy Higgs searches and indirect DM detection. We find that despite the strong constraints, light DM can be probed efficiently in the high luminosity run of the LHC.

II. OVERVIEW OF THE MODEL AND EXISTING CONSTRAINTS
Consider a simple extension of the Standard Model with a complex scalar field S carrying a global U(1) charge [3]. The most general renormalizable scalar potential invariant under global U(1) transformations S → e iα S is The U(1) gets broken spontaneously when S acquires a vacuum expectation value (VEV).
This would result in the presence of a massless Goldstone boson. To avoid it, we introduce a soft breaking mass term for S: with the full scalar potential being In this case, a non-zero S VEV generates a pseudo-Goldstone boson with mass µ S .
The parameter µ S 2 can always be made real and positive by phase redefinition. The scalar potential V is therefore invariant under "CP-symmetry" It is easy to show that S develops a real VEV such that this symmetry is unbroken by the vacuum [3]. As a result, ImS couples to other fields in pairs and is therefore stable.
The scalar fields are parametrized as and The stability of the pseudoscalar χ is guaranteed by the "CP-symmetry" and it will play the role of pseudo-Goldstone dark matter in our model.
The potential minimization conditions read Using these relations the parameters µ 2 H and µ 2 S can be eliminated from the scalar potential. The CP-even scalars h and s mix due to the presence of the portal coupling λ HS in the potential and the mass matrix in the (h, s)-basis is It is diagonalized by the orthogonal transformation with where θ is the mixing angle, The eigenvalues of the mass-squared matrix are Here h 1 is identified with the SM-like Higgs boson observed at the LHC. The pseudoscalar mass is given by the soft mass term: The 6 input parameters of the scalar potential are subject to 2 experimental constraints: v = 246 GeV and m h 1 = 125 GeV, leaving 4 free parameters which we choose as m h 2 , θ, v s and m χ .
We note that a number of variations of the Higgs portal model with a complex scalar singlet have been considered in the literature [14][15][16][17][18].

A. Cancellation of the direct DM detection amplitude
The main feature of our dark matter framework is that the direct detection amplitude vanishes at tree level and zero momentum transfer [3]. The leading DM-nucleon scattering process is shown in Fig. 2. Let us examine how the cancellation comes about in the mass eigenstate basis. The relevant interaction terms are given by where f denotes the SM fermions. Thus, the tree-level direct-detection scattering amplitude is given by since the momentum transfer in this process is negligibly small, t 0. Thus, the contributions from h 1 and h 2 exchange cancel each other up to tiny corrections of order t/(100 GeV) 2 .
This cancellation does not require any special relation between m h 1 and m h 2 and occurs for any parameter choice. It is of course a result of the pseudo-Goldstone nature of our dark matter. In terms of the polar coordinates, S = ρe iφ , where φ is identified with dark matter, one finds that the φφρ vertex vanishes for φ on shell and zero momentum of ρ. This statement is specific to the explicit U(1) breaking by a mass term S 2 and does not hold for higher dimensional operators. 2 The cancellation is spoiled by loop corrections which generate U(1) breaking terms of dimension 4, e.g. S 4 . The resulting direct detection cross section is in the ballpark of 10 −49 cm 2 [3], which is significantly below the current bounds. A detailed analysis of these loop corrections has recently been performed in [19,20].
In our framework, the observed DM relic density can have both thermal and non-thermal origin. The DM annihilation cross section does not suffer from the above cancellation since the momentum transfer is large in this case. Thus, the correct relic abundance can be achieved through the usual WIMP annihilation mechanism [3]. In this work however, we consider a more general possibility that the DM production mechanism may be non-thermal, which allows for a wider range of DM masses including m χ as low as 10 GeV.

B. Invisible decay branching ratio
In this study, we aim to probe the invisible decay of the CP-even Higgses h 1 and h 2 into a pair of DM particles χ. When such decays are allowed, the VBF Higgs production with missing energy provides a promising channel for dark matter detection. The χ-χ-h 1,2 couplings are given by leading to the invisible decay widths 2 As detailed in [3], this effective set-up can be obtained from a UV theory in which U(1) is broken at high energies to Z 2 by a VEV of another scalar carrying an even charge. This residual Z 2 forbids odd powers of S in the potential, e.g. the linear term, while higher even-dimension operators are suppressed by a high scale.
The decay h 1 → χχ is quite constrained by the LHC Higgs data, with BR(h 1 → χχ) not exceeding 10% as required by the Higgs signal strength observations [21]. On the other hand, the decay h 2 → χχ can be very efficient. It is important to include both of these DM production channels since either of them can dominate depending on the parameter choice. Eq. 18 shows that the DM couplings to the CP-even scalars grow with m h 1,2 and 1/v s .
Given that sin θ 0.3 as required by the (h 1 ) Higgs coupling measurements [22], one concludes that h 2 couples to DM significantly stronger than h 1 does. For h 2 , the competing decay modes are the sin θ-suppressed h 2 → V V ( * ) and, for larger m h 2 , h 2 → h 1 h 1 and h 2 → tt. The variation of BR(h 2 → χχ) for typical parameter choices is shown in Fig. 3.
We keep m h 2 below 600 GeV to have a substantial h 2 -production cross section and choose m χ = 64 GeV to evade the BR(h 1 → χχ) constraint. We see that for v/v s > 0.1, the invisible decay mode is significant and often dominant.

C. Constraints
The model parameters m h 2 , m χ , v s and θ are constrained by various experiments. Perturbative unitarity considerations exclude small values of v s v, which for fixed scalar masses imply large quartic couplings. The mixing angle θ and dark matter mass m χ are constrained by the LHC Higgs coupling data. In addition, depending on the choice of θ, the heavy CPeven Higgs boson mass m h 2 is subject to the LHC direct search bound. 3 While the direct DM detection constraint is weak and superseded by that from perturbative unitarity in the relevant parameter range, the indirect DM detection constraint from the Fermi satellite is significant for relatively light DM. Finally, when χ is assumed to have been produced thermally, there is a PLANCK constraint on the DM annihilation cross section, which requires a substantial DM-scalar couplings away from the resonance regions.
Below we delineate parameter space consistent with all of the constraints, keeping the spectrum at the electroweak scale and the mixing angle below 0.3. Figure 4 shows the results of our numerical analysis at fixed representative values of sin θ and m h 2 . The grey, purple and orange regions are excluded by the perturbative unitarity constraint λ S < 8π/3 [24], the Higgs invisible decay bound [21] and the gamma-ray observations from dwarf spheroidal galaxies (dSphs) with 6 years and 15 dSphs data by the Fermi-LAT Collaboration [25,26], respectively.

Constraints from unitarity, invisible Higgs decay and dark matter detection experiments
To constrain the invisible h 1 Higgs decay, we use the Higgs signal strength value µ = 1.09 +0.11 −0.10 obtained with 7 + 8 TeV LHC data [21]. In our model, the effective µ is given by where BR inv is the h 1 invisible decay branching ratio. The bound on BR inv is θ-dependent: The direct 13 TeV bound BR inv < 26% [22,27] is weaker, while the 13 TeV constraints on the Higgs signal strength 1.13 +0.09 −0.08 (ATLAS) [27] and 1.17 ± 0.10 (CMS) [22] are only consistent with the SM at 2σ level. It is therefore reasonable to use a conservative bound quoted above.
For the gamma-ray constraint, a Navarro-Frenk-White (NFW) dark matter profile is assumed [28]. The red band shows the allowed parameter range for thermal dark matter, whose abundance lies within a 3σ interval of Ω χ h 2 = 0.1197 ± 0.0022 as reported by the PLANCK Collaboration [29]. One observes the presence of the characteristic dips associated with the resonant DM annihilation through h 1 and h 2 . The DM couplings are allowed to be very small around these dips. The low mass end of the red bands, m χ < m h 1 /2, is excluded by the invisible Higgs decay constraint, while the Fermi bound does not significantly affect the allowed parameter space for thermal DM. The direct DM detection constraint is loose and superseded by that from perturbative unitarity (grey area) [3].
The uncolored regions are allowed by all of the constraints as long as the DM production mechanism is non-thermal. In particular, m χ = O(10) GeV is consistent with the Higgs invisible decay bound for small DM-Higgs couplings.

Constraints from the LHC direct search
Further constraints are imposed by the direct LHC search for Higgs-like states in various channels. We have taken into account the following 13 TeV LHC results: • The h 2 → γγ searches by ATLAS [30] and CMS [31]. The ATLAS search probes the heavy Higgs masses above 200 GeV, whereas the CMS lower bound is 500 GeV.
• Searches for h 2 → W W and h 2 → ZZ in ATLAS [32][33][34] and CMS [35]. The CMS h 2 → ZZ search [35] probes the lowest mass range: m h 2 ≥ 130GeV. The ATLAS h 2 → W W and h 2 → ZZ searches are sensitive to masses above 200 GeV. The combined ATLAS limits from h 2 → W W and h 2 → ZZ searches impose a limit on m h 2 ≥ 300 GeV.
• The ATLAS searches for h 2 → h 1 h 1 in 4b (1804.06174), 2γ 2b [36] and 2W 2γ final states, and the CMS searches in 2γ 2b [37] and 4b [38] final states. The ATLAS searches and the CMS 4b search probe the heavy Higgs masses above 260 GeV and the CMS 2γ 2b search probes those above 250 GeV.
• The ATLAS search for the invisible h 2 decay in the VBF production channel [39]. This probes the heavy Higgs mass range starting from 100 GeV.
We are interested in the low mass range 150-300 GeV as this leads to the strongest signal.
The above searches set a bound on the production cross section of the final state in question.
In the Narrow Width Approximation (NWA), it is given by: The h 2 production cross section is proportional to sin 2 θ, where σ SM pp→h is the production cross section of the SM Higgs boson. The branching ratio for the h 2 decay to a given SM final state is where Γ h→SM is the SM Higgs decay rate to the final state and Γ tot h is the SM Higgs total decay width.
It is clear that σ prod receives two suppression factors: sin 2 θ and the presence of nonstandard decay channels. For the mixing angle values sin θ = 0.1, 0.2 and 0.3, the cross section σ prod is below the limit for the γγ, W W and h 1 h 1 final states. The ZZ searches in both CMS [35] and ATLAS [34] and the combined W W , ZZ search in ATLAS [33] are more sensitive and for sin θ = 0.2, 0.3 impose a non-trivial constraint shown in Fig. 5.
Below the h 2 → h 1 h 1 kinematic threshold, the necessary suppression is provided by the invisible decay channel h 2 → χχ as long as m h 2 > 2m χ . While the sin θ = 0.1 case is It is interesting that at higher m h 2 ∼ 300 GeV, the above constraint disappears due to the additional decay channel h 2 → h 1 h 1 . The corresponding coupling is quite large and remains unsuppressed at large v s making the standard Higgs decay channels much less efficient. Thus, both panels of Fig. 4 are consistent with the h 2 -constraint.

III. COLLIDER ANALYSIS
We aim to probe our DM model via the missing energy signature in the VBF Higgs production channel (Fig. 6). There are of course other options as well. For example, Ref.
[40] has explored gluon fusion with the initial or final state radiation accompanied by missing energy, within a somewhat different Higgs portal DM model. The problematic aspect in this case is the large QCD background. For the VBF mode, the background is lower since the jets produced in this process are forward and easier to tag on. Thus it appears to be a promising channel. We find that the process is efficient only when the DM pair is produced by on-shell h 1 and/or h 2 decays. Thus we will focus on the region m h 2 > 2m χ . In addition, due to the sin 2 θ suppression, the heavy Higgs production cross section is significant only for the electroweak range masses. Although the h 1 production is unsuppressed by the mixing angle, the kinematic reach of h 1 → χχ is smaller than that for the h 2 decay and, furthermore, its branching ratio is already strongly constrained. As we detail below, both h 1 and h 2 can give a dominant contribution to DM pair production, depending on the parameter region.

A. Simulation details
The ATLAS and CMS Collaboration have explored the possibility of detecting invisible decay of the Higgs boson in the VBF channel and have optimised the corresponding cuts [39,41]. The definitions of the signal region and kinematic cuts are very similar in both of the analyses. In this work, we have adopted the CMS analysis of the VBF channel with √ s = 13 TeV data [41]. The VBF signal events are required to produce at least two jets with transverse momentum p T > 80(40) GeV for the leading (subleading) jet and rapidity |η| < 4.7. Further, at least one of the two leading jets must have |η| > 3. Events are also required to have a large transverse missing energy, The dominant SM background contribution to the signal region arises from the W + jets and Z + jets channels, with W and Z decaying leptonically into ν and ν ν , respectively.
The W + jets channel contributes due to a non-zero lepton misidentification probability combined with a large production cross section. The next largest contributions are due to the top production (single and pair) and gauge boson pair production. Contributions from the top production channels are suppressed by the small lepton misidentification probability, whereas the gauge boson pair production channels suffer from smaller cross-sections.
QCD jets and γ + jets are the other two potentially large contributors to the background. is also taken into account. We have summarised the selection criteria for the final state in Table I.

B. Results and discussion
In this section, we discuss the results of our collider simulation for two choices of the heavy CP-even Higgs mass, m h 2 = 150 GeV and 300 GeV. Since the h 2 production is suppressed by small sin 2 θ, we restrict the h 2 mass to the electroweak range.
In order to estimate the statistical significance factor S, we use where S, B and σ B represent the number of signal events, SM background events and the uncertainty in the background measurement. The CMS Collaboration quotes B ± σ B = 1779±96 at 35.9 fb −1 integrated luminosity [41]. To estimate S at high integrated luminosity, we have scaled B accordingly and taken two choices of σ B to reflect our lack of knowledge of how the realistic uncertainty may evolve. The "best case scenario" corresponds to a negligible background uncertainty, σ B √ B, and the second choice is σ B = √ B which effectively reduces the significance by √ 2. We expect that the resulting significance gives an idea of the realistic signal strength and the detection prospects.
Below we consider two representative values of m h 2 and the resulting signals. We fix sinθ to be 0.1 and 0.2 since for larger sinθ ≥ 0.3 a 150 GeV CP-even Higgs is nearly excluded by the LHC data [32,34]. We vary the ratio v/v s within the range [0.01 -5.0], where the upper limit is set by the perturbative unitarity constraint [24]. The DM mass m χ is bounded from above by m h 2 /2 as required by the on-shell decay of h 2 into a DM pair.
In Fig. 7    Let us consider the benchmark points with the same m χ and sinθ as in BP1 and BP2 in order to assess the quantitative changes.
• BP3 (sinθ = 0.2, v/v s = 0.05, m χ 50 GeV Let us close this section with a note. The CMS Collaboration has recently published its projected reach for the invisible decay of the 125 GeV Higgs at high luminosity LHC. The 95% CL upper bound on BR inv assuming SM-like production of the Higgs boson is expected to be 3.8% with 3 ab −1 integrated luminosity at 14 TeV [59]. They have optimized the sensitivity using the cuts E T > 190 GeV and |m jj | > 2500 GeV for this analysis. We have checked that these cuts coupled with the ones in Table I improve the significance factor in our scenario only slightly.

IV. SUMMARY AND CONCLUSIONS
We have studied the dark matter discovery prospects in the Higgs portal framework with pseudo-Goldstone DM. The model is particularly attractive due to its simplicity and elegant cancellation of the direct detection amplitude, which allows for a wide range of DM masses consistent with XENON1T. We have focused on the VBF production of the Higgs-like scalars which decay into DM pairs, thereby producing the "missing E T " signature. Taking into account the current LHC bounds along with the indirect DM detection constraint from Fermi, we find that relatively light, m χ 100 GeV, dark matter can be probed in this channel with the signal significance at L = 3 ab −1 reaching the discovery threshold in certain regions of parameter space.
The model predicts the existence of a heavier Higgs-like boson h 2 with suppressed couplings. This would provide a complementary test of the model, although its detection is hindered by the strong mixing angle suppression and a large invisible decay width. It is noteworthy that h 2 couples to dark matter much stronger than the SM-like Higgs h 1 does, hence its invisible decay can be very efficient even though the invisible decay of h 1 is severely constrained.
In this scenario, dark matter can be light quite naturally since its mass is provided by a symmetry breaking term. The direct detection constraints are very weak, so the lower bound of the order of a few GeV is only set by the B-meson decays. Although we focus on the DM mass range above 10 GeV, essentially all of our results apply to lower masses as well.