Pseudo-Nambu-Goldstone dark matter and two-Higgs-doublet models

TL;DR

This paper proposes a dark matter model where a pseudo-Nambu-Goldstone boson, arising from a complex scalar singlet, interacts weakly with nucleons, making it difficult to detect directly but consistent with relic abundance and Higgs data.

Contribution

It introduces a novel dark matter candidate based on a pseudo-Nambu-Goldstone boson within a two-Higgs-doublet framework, highlighting its suppressed direct detection signals.

Findings

The dark matter candidate evades direct detection due to vanishing scattering amplitude.

The model remains consistent with relic abundance and Higgs measurement constraints.

Indirect detection constraints are also considered in the analysis.

Abstract

We study a dark matter model with one singlet complex scalar and two Higgs doublets. The scalar potential respects a softly broken global symmetry, which makes the imaginary part of the singlet become a pseudo-Nambu-Goldstone boson acting as a dark matter candidate. The pseudo-Nambu-Goldstone nature of the boson leads to the vanishing of its tree-level scattering amplitude off nucleons at zero momentum transfer. Therefore, although the interaction strength could be sufficiently large to yield a viable relic abundance via thermal mechanism, direct detection is incapable of probing this candidate. We further investigate the constraints from Higgs measurements, relic abundance observation, and indirect detection.

Tables

Table 1. Table 1: Coefficients ξ h i f superscript subscript 𝜉 subscript ℎ 𝑖 𝑓 \xi_{h_{i}}^{f} and ξ a f superscript subscript 𝜉 𝑎 𝑓 \xi_{a}^{f} in the four types of Yukawa couplings.

	Type I	Type II	Lepton specific	Flipped
${\xi }_{h_i}^{{\ell }_j}$	$O_{2i}/\sin \beta$	$O_{1i}/\cos \beta$	$O_{1i}/\cos \beta$	$O_{2i}/\sin \beta$
${\xi }_{h_i}^{d_j}$	$O_{2i}/\sin \beta$	$O_{1i}/\cos \beta$	$O_{2i}/\sin \beta$	$O_{1i}/\cos \beta$
${\xi }_{h_i}^{u_j}$	$O_{2i}/\sin \beta$	$O_{2i}/\sin \beta$	$O_{2i}/\sin \beta$	$O_{2i}/\sin \beta$
${\xi }_a^{{\ell }_j}$	$\cot \beta$	$-\tan \beta$	$-\tan \beta$	$\cot \beta$
${\xi }_a^{d_j}$	$\cot \beta$	$-\tan \beta$	$\cot \beta$	$-\tan \beta$
${\xi }_a^{u_j}$	$-\cot \beta$	$-\cot \beta$	$-\cot \beta$	$-\cot \beta$

Equations

$$V_1$$

$$V_2=-m_S^2|S{|}^2+\frac{{\lambda }_S}{2}|S{|}^4+{\kappa }_1|{\Phi }_1{|}^2|S{|}^2+{\kappa }_2|{\Phi }_2{|}^2|S{|}^2.$$

$$V_{\mathrm{soft}}=-\frac{m_S^{\prime 2}}{4}{S}^2+H.c.$$

$$V=V_1+V_2+V_{\mathrm{soft}}.$$

$${\Phi }_1=\left(\begin{array}{c}{\varphi }_1^{+}\\ (v_1+{\rho }_1+i{\eta }_1)/\sqrt{2}\end{array}\right),{\Phi }_2=\left(\begin{array}{c}{\varphi }_2^{+}\\ (v_2+{\rho }_2+i{\eta }_2)/\sqrt{2}\end{array}\right),S=\frac{v_s+s+i\chi }{\sqrt{2}}.$$

$$m_{11}^2$$

$$m_{22}^2$$

$$m_S^2$$

$${\lambda }_{345}\equiv {\lambda }_3+{\lambda }_4+{\lambda }_5.$$

$$|S{|}^2=\frac{1}{2}(v_s^2+s^2+{\chi }^2)+v_{s}s,S^2+(S^{\ast }{)}^2=v_s^2+s^2-{\chi }^2+2v_{s}s,$$

$$m_{\chi }^2=-m_S^2+\frac{1}{2}{m}_S^{\prime 2}+\frac{1}{2}{\lambda }_S{v}_s^2+\frac{1}{2}{\kappa }_1{v}_1^2+\frac{1}{2}{\kappa }_2{v}_2^2=m_S^{\prime 2},$$

$$-{\mathcal{L}}_{\mathrm{mass},\varphi }=\left[m_{12}^2-\frac{1}{2}({\lambda }_4+{\lambda }_5)v_1{v}_2\right]\left(\begin{array}{cc}{\varphi }_1^{-},& {\varphi }_2^{-}\end{array}\right)\left(\begin{array}{cc}{v}_2/v_1& -1\\ -1& v_1/v_2\end{array}\right)\left(\begin{array}{c}{\varphi }_1^{+}\\ {\varphi }_2^{+}\end{array}\right),$$

$$-{\mathcal{L}}_{\mathrm{mass},\eta }=\frac{1}{2}(m_{12}^2-{\lambda }_5{v}_1{v}_2)\left(\begin{array}{cc}{\eta }_1,& {\eta }_2\end{array}\right)\left(\begin{array}{cc}{v}_2/v_1& -1\\ -1& v_1/v_2\end{array}\right)\left(\begin{array}{c}{\eta }_1\\ {\eta }_2\end{array}\right).$$

$$\left(\begin{array}{c}{\varphi }_1^{+}\\ {\varphi }_2^{+}\end{array}\right)=R(\beta )\left(\begin{array}{c}{G}^{+}\\ H^{+}\end{array}\right),\left(\begin{array}{c}{\eta }_1\\ {\eta }_2\end{array}\right)=R(\beta )\left(\begin{array}{c}{G}^0\\ a\end{array}\right),R(\beta )=\left(\begin{array}{cc}\cos \beta & -\sin \beta \\ \sin \beta & \cos \beta \end{array}\right),$$

$$\tan \beta =\frac{v_2}{v_1}.$$

$$m_{H^{+}}^2=\frac{v_1^2+v_2^2}{v_1{v}_2}\left[m_{12}^2-\frac{1}{2}({\lambda }_4+{\lambda }_5)v_1{v}_2\right],m_a^2=\frac{v_1^2+v_2^2}{v_1{v}_2}(m_{12}^2-{\lambda }_5{v}_1{v}_2).$$

$$-{\mathcal{L}}_{\mathrm{mass},\rho s}=\frac{1}{2}\left(\begin{array}{ccc}{\rho }_1,& {\rho }_2,& s\end{array}\right){\mathcal{M}}_{\rho s}^2\left(\begin{array}{c}{\rho }_1\\ {\rho }_2\\ s\end{array}\right),$$

$$({\mathcal{M}}_{\rho s}^2{)}_{11}$$

$$({\mathcal{M}}_{\rho s}^2{)}_{12}$$

$$O^{\mathrm{T}}{\mathcal{M}}_{\rho s}^{2}O=\mathrm{diag}(m_{h_1}^2,m_{h_2}^2,m_{h_3}^2).$$

$$\left(\begin{array}{c}{\rho }_1\\ {\rho }_2\\ s\end{array}\right)=O\left(\begin{array}{c}{h}_1\\ h_2\\ h_3\end{array}\right).$$

$${\mathcal{L}}_{\mathrm{kin}}=(D^{\mu }{\Phi }_1{)}^{†}{D}_{\mu }{\Phi }_1+(D^{\mu }{\Phi }_2{)}^{†}{D}_{\mu }{\Phi }_2,$$

$${\mathcal{L}}_{\mathrm{mass},WZ}=\frac{g^2}{4}(v_1^2+v_2^2)W^{-,\mu }{W}_{\mu }^{+}+\frac{1}{2}\frac{g^2}{4c_{\mathrm{W}}^2}(v_1^2+v_2^2)Z^{\mu }{Z}_{\mu },$$

$$m_W=\frac{gv}{2},m_Z=\frac{gv}{2c_{\mathrm{W}}},$$

$$\text{Type I:}{\mathcal{L}}_{\mathrm{Y},\mathrm{I}}=-y_{{\ell }_i}{\bar{L}}_{i\mathrm{L}}{\ell }_{i\mathrm{R}}{\Phi }_2-{\tilde{y}}_d^{ij}{\bar{Q}}_{i\mathrm{L}}{d}_{j\mathrm{R}}^{\prime }{\Phi }_2-{\tilde{y}}_u^{ij}{\bar{Q}}_{i\mathrm{L}}{u}_{j\mathrm{R}}^{\prime }{\tilde{\Phi }}_2+H.c.$$

$$\text{Type II:}{\mathcal{L}}_{\mathrm{Y},\mathrm{II}}=-y_{{\ell }_i}{\bar{L}}_{i\mathrm{L}}{\ell }_{i\mathrm{R}}{\Phi }_1-{\tilde{y}}_d^{ij}{\bar{Q}}_{i\mathrm{L}}{d}_{j\mathrm{R}}^{\prime }{\Phi }_1-{\tilde{y}}_u^{ij}{\bar{Q}}_{i\mathrm{L}}{u}_{j\mathrm{R}}^{\prime }{\tilde{\Phi }}_2+H.c.$$

$$\text{Lepton specific:}{\mathcal{L}}_{\mathrm{Y},\mathrm{L}}=-y_{{\ell }_i}{\bar{L}}_{i\mathrm{L}}{\ell }_{i\mathrm{R}}{\Phi }_1-{\tilde{y}}_d^{ij}{\bar{Q}}_{i\mathrm{L}}{d}_{j\mathrm{R}}^{\prime }{\Phi }_2-{\tilde{y}}_u^{ij}{\bar{Q}}_{i\mathrm{L}}{u}_{j\mathrm{R}}^{\prime }{\tilde{\Phi }}_2+H.c.$$

$$\text{Flipped:}{\mathcal{L}}_{\mathrm{Y},\mathrm{F}}=-y_{{\ell }_i}{\bar{L}}_{i\mathrm{L}}{\ell }_{i\mathrm{R}}{\Phi }_2-{\tilde{y}}_d^{ij}{\bar{Q}}_{i\mathrm{L}}{d}_{j\mathrm{R}}^{\prime }{\Phi }_1-{\tilde{y}}_u^{ij}{\bar{Q}}_{i\mathrm{L}}{u}_{j\mathrm{R}}^{\prime }{\tilde{\Phi }}_2+H.c.$$

$${\mathcal{L}}_{\mathrm{Y}}$$

$${\mathcal{L}}_{\mathrm{tri},{\chi }^2}=-\frac{1}{2}({\kappa }_1{v}_1{\rho }_1+{\kappa }_2{v}_2{\rho }_2+{\lambda }_S{v}_{s}s){\chi }^2=\frac{1}{2}\sum _{i=1}^3{g}_{h_i{\chi }^2}{h}_i{\chi }^2,$$

$$g_{h_i{\chi }^2}=-{\kappa }_1{v}_1{O}_{1i}-{\kappa }_2{v}_2{O}_{2i}-{\lambda }_S{v}_s{O}_{3i}.$$

$$i\mathcal{M}=\frac{m_q}{v{s}_{\beta }}\bar{u}(k_2)u(k_1)\left(g_{h_1{\chi }^2}\frac{i}{t-m_{h_1}^2}{O}_{21}+g_{h_2{\chi }^2}\frac{i}{t-m_{h_2}^2}{O}_{22}+g_{h_3{\chi }^2}\frac{i}{t-m_{h_3}^2}{O}_{23}\right),$$

$$i\mathcal{M}\to -i\frac{m_q}{v{s}_{\beta }}\bar{u}(k_2)u(k_1)\left(\begin{array}{ccc}{g}_{h_1{\chi }^2},& g_{h_2{\chi }^2},& g_{h_3{\chi }^2}\end{array}\right)({\mathcal{M}}_h^2{)}^{-1}{O}^{\mathrm{T}}\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right),$$

Full text

1. I Introduction

2. II Model details

3. II.1 Scalar potential

4. II.2 Mass eigenstates

5. II.3 Yukawa couplings

6. II.4 Vanishing of the DM-nucleon scattering amplitude

7. II.5 Alignment limit

8. III Phenomenological constraints

9. III.1 Parameter scan and Higgs measurements

10. III.2 DM relic abundance

11. III.3 Indirect detection

12. IV Conclusions and outlook

13. A Scalar and gauge trilinear couplings

14. B BSM decay widths of the SM-like Higgs boson

I Introduction

Astrophysical and cosmological observations suggest that the majority of matter in the present Universe consists of a nonluminous component called dark matter (DM). In the conventional paradigm, dark matter is a thermal relic remaining from the early Universe, implying that the interaction strength between DM and standard model (SM) particles may be comparable to the strength of weak interactions Bertone:2004pz ; Feng:2010gw ; Young:2016ala . However, null signal results from recent direct detection experiments have put rather stringent constraints on the DM-nucleon scattering cross section Akerib:2016vxi ; Cui:2017nnn ; Aprile:2018dbl . This has become a great challenge to the thermal DM paradigm.

A natural way out is to suppress DM-nucleon scattering in direct detection experiments without suppressing DM annihilation in the early Universe. One possibility is that there are some blind spots with particular parameters leading to the suppression of the DM couplings relevant to direct detection Cheung:2012qy ; Banerjee:2016hsk ; Cai:2017wdu ; Han:2018gej ; Altmannshofer:2019wjb . Additionally, the relevant DM couplings could vanish due to special symmetries Dedes:2014hga ; Tait:2016qbg ; Arcadi:2016kmk ; Cai:2016sjz ; Xiang:2017yfs ; Wang:2017sxx . Moreover, DM-nucleon scattering mediated by pseudoscalars can evade direct detection constraints Ipek:2014gua ; Berlin:2015wwa ; No:2015xqa ; Goncalves:2016iyg ; Haisch:2016gry ; Bauer:2017ota ; Tunney:2017yfp . Furthermore, the DM-nucleon scattering amplitude could be greatly suppressed if the DM particle is a pseudo-Nambu-Goldstone boson (pNGB) protected by an approximate global symmetry Barducci:2016fue ; Gross:2017dan ; Balkin:2018tma ; Huitu:2018gbc ; Alanne:2018zjm ; Kannike:2019wsn ; Karamitros:2019ewv ; Cline:2019okt .

In the last case, tree-level interactions of a pNGB are generally momentum suppressed. As direct detection experiments essentially operate in the zero momentum transfer limit, the amplitude of pNGB dark matter scattering off nucleons vanishes at tree level Gross:2017dan . Loop corrections could break the global symmetry, resulting in nonvanishing scattering. Nevertheless, further investigations have shown that the DM-nucleon cross section at one-loop level is rather small, far away from the capability of current direct detection experiments Azevedo:2018exj ; Ishiwata:2018sdi . Therefore, such a pNGB DM framework seems very appealing for thermal DM.

Previous studies in this framework assumed that the Higgs sector just involves one Higgs doublet as in the SM  Barducci:2016fue ; Gross:2017dan ; Balkin:2018tma ; Huitu:2018gbc ; Alanne:2018zjm ; Kannike:2019wsn ; Karamitros:2019ewv ; Cline:2019okt ; Azevedo:2018exj ; Ishiwata:2018sdi . In this work, we would like to extend the study to two Higgs doublets Branco:2011iw . A Higgs sector with two $\mathrm{SU}(2)_{\mathrm{L}}$ doublets has fairly good motivations. First, two Higgs doublets are typically required for constructing realistic supersymmetric Haber:1984rc and axion Kim:1986ax models. Second, the flexible scalar mass spectrum and additional $CP$ violation sources in two-Higgs-doublet models may be helpful for generating a desired baryon asymmetry of the Universe through the baryogenesis mechanism Turok:1990zg . Finally, two Higgs doublets could provide an available portal to thermal dark matter with attractive phenomenological features Ipek:2014gua ; Ko:2015fxa ; Berlin:2015wwa ; No:2015xqa ; Goncalves:2016iyg ; Bell:2016ekl ; Bauer:2017ota ; Tunney:2017yfp ; Chang:2017gla ; Bell:2017rgi ; Dey:2019lyr ; Altmannshofer:2019wjb .

In this paper, we consider that the scalar sector involves two $\mathrm{SU}(2)_{\mathrm{L}}$ Higgs doublets as well as a complex scalar $S$, which is a SM gauge singlet. Most terms in the scalar potential obey a global $\mathrm{U}(1)$ symmetry $S\to e^{i\alpha}S$. The exception is a quadratic term that softly breaks this symmetry and gives mass to the imaginary part of $S$, denoted as $\chi$. The real scalar $\chi$ is what we call pNGB dark matter. Its pNGB nature makes its scattering amplitude off nucleons vanish at tree level, evading direct detection constraints. Nonetheless, it is able to obtain an observed DM relic abundance via the thermal production mechanism. We will perform a random scan in the parameter space to investigate reasonable parameter points that satisfy current Higgs measurements at the Large Hadron Collider (LHC), observation of the DM relic abundance, and constraints from indirect detection experiments.

The paper is organized as follows. In Sec. II, we describe the details of the pNGB DM model with two Higgs doublets, including the scalar potential, mass eigenstates, four types of Yukawa couplings, the vanishing of the DM-nucleon scattering amplitude, and the alignment limit. In Sec. III, we perform a random scan in the parameter space and investigate phenomenological constraints from LHC Higgs measurements, relic abundance observation, and indirect detection. Section IV gives the conclusions and outlook. In Appendix A, we write down the scalar and gauge trilinear couplings. Appendix B gives some expressions for decay widths of the SM-like Higgs boson.

II Model details

In this section, we study the model details. As explained above, we assume that the scalar sector involves two Higgs doublets and one SM gauge singlet, and there is a softly broken global $\mathrm{U}(1)$ symmetry leading to pNGB dark matter. The fermion content is assumed to be the same as in the SM. Analogous to generic two-Higgs-doublet models, there are four types of Yukawa couplings that do not induce flavor-changing neutral currents (FCNCs) at tree level. We find that these four types are all applicable to our purpose.

II.1 Scalar potential

The two $\mathrm{SU}(2)_{\mathrm{L}}$ Higgs-doublet fields are denoted as $\Phi_{1}$ and $\Phi_{2}$, both carrying hypercharge $+1/2$. The complex scalar $S$ is a $\mathrm{SU}(2)_{\mathrm{L}}$ singlet and carries no hypercharge. For simplicity, we make two common assumptions for the scalar potential. The first assumption is that $CP$ is conserved in the scalar sector, leading to only real coefficients. The second one is that there is a $Z_{2}$ symmetry $\Phi_{1}\to-\Phi_{1}$ or $\Phi_{2}\to-\Phi_{2}$ forbidding quartic terms that are odd in either $\Phi_{1}$ or $\Phi_{2}$, but such a symmetry can be softly broken by quadratic terms.

Under these assumptions, the general terms in the scalar potential constructed with $\Phi_{1}$ and $\Phi_{2}$ are given by Branco:2011iw

[TABLE]

And we can write down the potential terms that involve $S$ and respect a global $\mathrm{U}(1)$ symmetry $S\to e^{i\alpha}S$,

[TABLE]

In addition, we introduce a quadratic term softly breaking the global $\mathrm{U}(1)$ symmetry,

[TABLE]

Note that even if $m^{\prime 2}_{S}$ is complex, we can always make it real and positive by a phase redefinition of $S$. Then $V_{2}$ and $V_{\mathrm{soft}}$ respect a dark $CP$ symmetry $S\to S^{*}$ Gross:2017dan ; Karamitros:2019ewv . The soft breaking term $V_{\mathrm{soft}}$ can be justified by treating $m^{\prime 2}_{S}$ as a spurion, arising from a more fundamental theory that does not induce other soft breaking terms involving odd powers of $S$ Gross:2017dan ; Huitu:2018gbc .

Now the whole scalar potential is

[TABLE]

In particular regions of the parameter space, $\Phi_{1}$, $\Phi_{2}$, and $S$ develop nonzero vacuum expectation values (VEVs) $v_{1}$, $v_{2}$, and $v_{s}$. They can be expanded as

[TABLE]

By minimizing the potential, we find the following stationary point conditions:

[TABLE]

where

[TABLE]

Note that all terms in $V_{2}$ and $V_{\mathrm{soft}}$ are products of $|S|^{2}$ or of ${S^{2}}+{({S^{*}})^{2}}$. As their expansions are

[TABLE]

the real scalar $\chi$ always appears in pair in the scalar potential. Therefore, $\chi$ cannot decay, becoming a stable DM candidate.

II.2 Mass eigenstates

After the scalar fields obtain their VEVs, the mass squared of $\chi$ is

[TABLE]

where the terms with VEVs are totally canceled by the third stationary point condition (8). If $m^{\prime 2}_{S}=0$, there is no soft breaking term, and $\chi$ is a massless Nambu-Goldstone boson. If $m^{\prime 2}_{S}>0$, $\chi$ would have a physical mass $m_{\chi}=m^{\prime}_{S}$, behaving as a pseudo-Nambu-Goldstone boson. This is exactly what we want.

The mass terms for the charged scalars are derived as

[TABLE]

while those for the $CP$-odd scalars are given by

[TABLE]

The above mass terms can be diagonalized by rotations

[TABLE]

where the rotation angle $\beta$ satisfies

[TABLE]

Now $G^{\pm}$ and $G^{0}$ are massless Nambu-Goldstone bosons eaten by the weak gauge bosons $W^{\pm}$ and $Z$, while $H^{\pm}$ and $a$ are physical states with masses

[TABLE]

The $CP$-even scalars $\rho_{1}$, $\rho_{2}$, and $s$ mix with each other. Their mass terms are

[TABLE]

where the elements of the $3\times 3$ symmetric mass-squared matrix $\mathcal{M}_{\rho s}^{2}$ are given by

[TABLE]

$\mathcal{M}_{\rho s}^{2}$ can be diagonalized by a $3\times 3$ real orthogonal matrix $O$,

[TABLE]

The mass eigenstates $h_{i}$ ($i=1,2,3$) are then related to the interaction eigenstates by

[TABLE]

One of $h_{i}$ should behave like the SM Higgs boson in order to be consistent with observation. Below we adopt a convention with $m_{h_{1}}\leq m_{h_{2}}\leq m_{h_{3}}$.

From the covariant kinetic terms

[TABLE]

we derive the mass terms for the weak gauge bosons,

[TABLE]

where $c_{\mathrm{W}}\equiv\cos\theta_{\mathrm{W}}$ with $\theta_{\mathrm{W}}$ denoting the Weinberg angle, and $g$ is the $\mathrm{SU}(2)_{\mathrm{L}}$ gauge coupling. Defining $v\equiv\sqrt{v_{1}^{2}+v_{2}^{2}}$, the masses of $W$ and $Z$ bosons become

[TABLE]

just as in the SM. From the Fermi constant ${G_{\mathrm{F}}}={g^{2}}/(4\sqrt{2}m_{\mathrm{W}}^{2})$, we obtain $v={(\sqrt{2}{G_{\mathrm{F}}})^{-1/2}}=246.22~{}$\mathrm{G}\mathrm{e}\mathrm{V}$$. Note that $v_{1}$ and $v_{2}$ satisfy $v_{1}=vc_{\beta}$ and $v_{2}=vs_{\beta}$, where we have used the shorthand notations $s_{\beta}\equiv\sin\beta$ and $c_{\beta}\equiv\cos\beta$.

The scalar and gauge trilinear couplings of the scalar mass eigenstates can be found in Appendix A.

II.3 Yukawa couplings

Unlike the standard model, Yukawa couplings between the two Higgs doublets and SM fermions generally lead to tree-level FCNCs, which could cause phenomenological problems in flavor physics. This is because diagonalizing the fermion mass matrix cannot make sure that the Yukawa interactions are also diagonalized. Nevertheless, if all fermions with the same quantum numbers just couple to the one same Higgs doublet, the FCNCs will be absent at tree level Glashow:1976nt ; Paschos:1976ay ; Branco:2011iw ; Camargo:2019ukv . This can be achieved by assuming particular $Z_{2}$ symmetries for the Higgs doublets and fermions.

As a result, there are four independent types of Yukawa couplings without tree-level FCNCs, listed as follows.

[TABLE]

Here ${\tilde{\Phi}}_{2}\equiv i\sigma^{2}\Phi_{2}^{*}$, $L_{i\mathrm{L}}\equiv(\nu_{i\mathrm{L}},\ell_{i\mathrm{L}})^{\mathrm{T}}$, and $Q_{i\mathrm{L}}\equiv(u^{\prime}_{i\mathrm{L}},d^{\prime}_{i\mathrm{L}})^{\mathrm{T}}$. The down-type and up-type quark Yukawa matrices $\tilde{y}_{d}^{ij}$ and $\tilde{y}_{u}^{ij}$ can be diagonalized through $({U_{d}})_{ij}^{\dagger}\tilde{y}_{d}^{jk}{({U_{d}})_{kl}}={y_{{d_{i}}}}{\delta_{il}}$ and $({U_{u}})_{ij}^{\dagger}\tilde{y}_{u}^{jk}{({U_{u}})_{kl}}={y_{{u_{i}}}}{\delta_{il}}$. Thus, the interaction eigenstates $u^{\prime}_{i}$ and $d^{\prime}_{i}$ are related to the mass eigenstates $u_{i}$ and $d_{i}$ via ${d^{\prime}_{i}}={({U_{d}})_{ij}}{d_{j}}$ and ${u^{\prime}_{i}}={({U_{u}})_{ij}}{u_{j}}$. The Cabibbo-Kobayashi-Maskawa matrix is defined as ${V_{ij}}\equiv({U_{u}})_{ik}^{\dagger}{({U_{d}})_{kj}}$. As we would not discuss neutrino physics in this work, we assume the lepton sector is the same as in the SM.

After the scalars develop the VEVs, the Yukawa interactions provide mass terms to the fermions. For the mass eigenstates, the four types of Yukawa terms can be expressed in the same form,

[TABLE]

where $P_{\mathrm{L}}$ and $P_{\mathrm{R}}$ are the left- and right-handed projection operators, respectively. The coefficients $\xi_{h_{i}}^{f}$ and $\xi_{a}^{f}$ are listed in Table 1.

II.4 Vanishing of the DM-nucleon scattering amplitude

In this subsection, we verify that the tree-level amplitude of DM scattering off nucleons vanishes at zero momentum transfer. In our case, DM-nucleon scattering is induced by DM-quark scattering. Therefore, we just need to prove that the DM-quark scattering amplitude vanishes in the zero momentum transfer limit.

From the $U(1)$ symmetric potential (2), we obtain the trilinear couplings for the DM candidate $\chi$ as

[TABLE]

where the coupling coefficients for the mass eigenstates are given by

[TABLE]

At tree level, only the $CP$-even Higgs bosons $h_{1}$, $h_{2}$, and $h_{3}$ can mediate $\chi$ scattering off quarks. The Feynman diagram is shown in Fig. 1.

Take the type-I Yukawa couplings as an example. Defining a Lorentz invariant $t\equiv p_{\mu}p^{\mu}$, where $p^{\mu}$ is the 4-momentum of the mediator $h_{i}$, we can write down the DM-quark scattering amplitude as

[TABLE]

where $u({k_{1}})$ and $\bar{u}({k_{2}})$ are the wave functions for the incoming and outgoing quarks, respectively. In the zero momentum transfer limit, $t\to 0$, and the above amplitude can be reexpressed as

[TABLE]

where $(\mathcal{M}_{h}^{2})^{-1}=\mathrm{diag}(m_{{h_{1}}}^{-2},m_{{h_{2}}}^{-2},m_{{h_{3}}}^{-2})$ is the inverse of the diagonalized mass-squared matrix $\mathcal{M}_{h}^{2}\equiv\mathrm{diag}(m_{{h_{1}}}^{2},m_{{h_{2}}}^{2},m_{{h_{3}}}^{2})$. From Eqs. (31) and (20), we have

[TABLE]

Utilizing these equations as well as the orthogonality of $O$, we obtain

[TABLE]

This can be understood as the amplitude expressed in the interaction basis Gross:2017dan .

The inverse of $\mathcal{M}_{\rho s}^{2}$ can be expressed as its adjugate $\mathcal{A}$ divided by its determinant, i.e., ${(\mathcal{M}_{\rho s}^{2})^{-1}}=\mathcal{A}/\det(\mathcal{M}_{\rho s}^{2})$. The relevant elements of $\mathcal{A}$ are

[TABLE]

We then have

[TABLE]

Therefore, we have proven that the tree-level DM-quark amplitude $i\mathcal{M}$ vanishes in the zero momentum transfer limit for the type-I Yukawa couplings. Similarly, we can prove this for the type-II, lepton-specific, and flipped Yukawa couplings.

As the global $\mathrm{U}(1)$ symmetry is softly broken, loop corrections would give a nonvanishing DM-nucleon scattering cross section Gross:2017dan . Nonetheless, we expect that the loop-induced cross section should be typically $\lesssim\mathcal{O}(10^{-50})~{}$\mathrm{c}\mathrm{m}^{2}$$, as suggested by the one-loop evaluation in Ref. Azevedo:2018exj where only one Higgs doublet is considered. Thus, current and near future direct detection experiments should not be able to probe our pNGB DM model.

II.5 Alignment limit

Current LHC Higgs measurements favor a $125~{}$\mathrm{G}\mathrm{e}\mathrm{V}$$ SM-like Higgs boson. If one of the $CP$-even Higgs bosons mimics the SM Higgs boson, the constraints from Higgs measurements can be easily satisfied. For the two Higgs doublets, such a situation can be achieved by requiring that the additional scalars are much heavier than the weak scale so that the lightest $CP$-even Higgs boson reproduces SM-like Higgs signals at the LHC. This is known as the “decoupling” limit Gunion:2002zf . In general, a particular parameter set or relation leading to a $CP$-even Higgs boson mimicking the SM Higgs boson is referred as an “alignment” limit. The decoupling limit is of course an alignment limit, but it is less interesting, as the new particles might be too heavy to be accessed at the LHC.

A more interesting possibility is alignment without decoupling Carena:2013ooa ; Dev:2014yca . In order to find such a possibility, we may rotate the two Higgs doublets $\Phi_{1}$ and $\Phi_{2}$ into the Higgs basis Georgi:1978ri ; Donoghue:1978cj

[TABLE]

and have

[TABLE]

Now $\Phi_{h}$ gains a VEV $v$ and contains a $CP$-even scalar $h$ as well as the Nambu-Goldstone bosons, while $\Phi_{H}$ has zero VEV and contains a $CP$-even scalar $H$ and the physical states $H^{+}$ and $a$. Consequently, the tree-level interactions of the $CP$-even scalar $h$ with weak gauge bosons and SM fermions are totally identical to those of the Higgs boson in the SM. Therefore, the alignment limit means that $h$ does not mix with $H$ and $s$.

In the Higgs basis, the potential terms (1) transform to

[TABLE]

where the new parameters are related to the previous parameters by Bell:2017rgi

[TABLE]

On the other hand, the potential terms (2) transform to

[TABLE]

where the new parameters are given by

[TABLE]

Then the stationary point conditions for the scalar potential are

[TABLE]

As a result, the mass-squared matrix for $CP$-even scalars $(h,H,s)$ is

[TABLE]

In order to prevent $h$-$H$ and $h$-$s$ mixings, the off-diagonal terms $(\mathcal{M}_{hHs}^{2})_{12}$ and $(\mathcal{M}_{hHs}^{2})_{13}$ should be absent, corresponding to

[TABLE]

This is the alignment condition in our model. When this condition is satisfied, the tree-level couplings of $h$ to SM particles are exactly the same as those of the SM Higgs boson.

III Phenomenological constraints

In this section, we take the type-I Yukawa couplings as an illuminating example to investigate the phenomenological constraints from Higgs measurements, relic abundance observation, and indirect detection.

III.1 Parameter scan and Higgs measurements

There are 12 free parameters in the model, which can be chosen as

[TABLE]

In order to investigate the vast parameter space, we carry out a random scan within the following ranges:

[TABLE]

Then we require the selected parameter points must give positive $m_{h_{1,2,3}}^{2}$, $m_{H^{+}}^{2}$, and $m_{a}^{2}$, ensuring physical scalar masses. Moreover, one of the $CP$-even Higgs bosons $h_{i}$ should have a mass within the $3\sigma$ range of the measured SM-like Higgs boson mass $m_{h}=125.18\pm 0.16~{}$\mathrm{G}\mathrm{e}\mathrm{V}$$ Tanabashi:2018oca . We recognize this scalar as the SM-like Higgs boson and denote it as $h_{\mathrm{SM}}$, and further examine if its properties are consistent with current measurements.

In the $\kappa$ framework Heinemeyer:2013tqa , the couplings of the SM-like Higgs boson to SM particles can be expressed as

[TABLE]

where $g_{hgg}^{{\mathrm{SM}}}$, $g_{h\gamma\gamma}^{{\mathrm{SM}}}$, and $g_{Z\gamma\gamma}^{{\mathrm{SM}}}$ are the loop-induced effective couplings to $gg$, $\gamma\gamma$, and $Z\gamma$, respectively. $\kappa$’s are coupling modifiers, whose values are all equal to 1 in the SM. Equation (72) implies that $\kappa_{W}$ and $\kappa_{Z}$ are equal in our model, and we will use $\kappa_{V}$ representing both of them. Assuming the SM-like Higgs boson is $h_{\mathrm{SM}}=h_{i}$, we have

[TABLE]

The coupling modifiers for fermions can be read off from Table 1. For the type-I Yukawa couplings, all SM fermions have the same coupling modifier, given by

[TABLE]

It is also helpful to define another modifier $\kappa_{H}$ as

[TABLE]

where ${\Gamma_{h}^{{\mathrm{SM}}}}$ is the Higgs total decay width in the SM, ${{\Gamma_{{h_{{\mathrm{SM}}}}}}}$ is the total decay width of the SM-like Higgs boson $h_{\mathrm{SM}}$, and $\Gamma_{h_{\mathrm{SM}}}^{\mathrm{BSM}}$ is the $h_{\mathrm{SM}}$ decay width into final states beyond the SM (BSM). Thus, $\kappa_{H}$ indicates the deviation of the Higgs width decaying into SM final states and is also equal to 1 in the SM. In our model, $\Gamma_{h_{\mathrm{SM}}}^{\mathrm{BSM}}$ can be generally separated into two parts,

[TABLE]

$\Gamma_{{h_{{\mathrm{SM}}}}}^{{\mathrm{inv}}}$ is the $h_{\mathrm{SM}}$ decay width into the invisible final state, i.e., a pair of the DM candidate $\chi$. $\Gamma_{{h_{{\mathrm{SM}}}}}^{{\mathrm{und}}}$ involves decay widths into all kinematically allowed BSM final states that are undetected in current LHC searches. Such final states may include $aa$, ${H^{+}}{H^{-}}$, ${h_{i}}{h_{j}}$, $aZ$, and $H^{\pm}W^{\mp}$. The expressions for these decay widths are listed in Appendix B. Once all the decay widths are evaluated, we can determine the invisible and undetected BSM branching ratios via $\mathrm{BR}_{\mathrm{inv}}=\Gamma_{{h_{{\mathrm{SM}}}}}^{{\mathrm{inv}}}/{{\Gamma_{{h_{{\mathrm{SM}}}}}}}$ and $\mathrm{BR}_{\mathrm{und}}=\Gamma_{{h_{{\mathrm{SM}}}}}^{{\mathrm{und}}}/{{\Gamma_{{h_{{\mathrm{SM}}}}}}}$, respectively.

We utilize a numerical tool Lilith 1.1.4 Bernon:2015hsa to study the constraints from current Higgs measurements. Lilith is able to construct an approximate likelihood based on experimental results of Higgs signal strength measurements. For each selected parameter point in our random scan, we put the corresponding $m_{h_{\mathrm{SM}}}$, $\kappa_{V}$, $\kappa_{f}$, $\mathrm{BR}_{\mathrm{inv}}$, and $\mathrm{BR}_{\mathrm{und}}$ into Lilith. Then Lilith can evaluate $\kappa_{g}$, $\kappa_{\gamma}$, and $\kappa_{Z\gamma}$ involving the loop contributions from SM fermions and gauge bosons whose couplings are modified by $\kappa_{f}$ and $\kappa_{V}$, including next-to-leading-order QCD corrections. Such an evaluation has neglected the loop contributions from the BSM scalars in our model. Nonetheless, these scalars are typically heavy and/or have small couplings. Therefore, their contributions are insignificant for most of the selected parameter points.

We further use Lilith to calculate the likelihood $-2\ln L$ for each parameter point based on Tevatron data Aaltonen:2013ioz , ATLAS run 1 data Aad:2014iia ; Aad:2014eha ; Aad:2014xzb ; ATLAS:2015yda ; Aad:2015gra ; Aad:2015iha ; Aad:2015ona ; Aad:2015gba , CMS run 1 data Chatrchyan:2014tja ; Khachatryan:2014qaa ; Khachatryan:2014jba ; Khachatryan:2015ila ; Khachatryan:2015bnx , ATLAS run 2 data ATLAS:2016bza ; ATLAS:2016ldo ; ATLAS:2016lgh ; ATLAS:2016nke ; ATLAS:2016oum ; ATLAS:2016awy ; ATLAS:2016pkl ; ATLAS:2016gld ; ATLAS:2017syx , and CMS run 2 data CMS:2016nfx ; CMS:2016mmc ; CMS:2016jjx ; CMS:2016ixj ; CMS:2016zbb ; CMS:2017jkd ; CMS:2017lgc . We then transform $-2\ln L$ to a $p$-value and require that the selected parameter points should give $p$-values larger than 0.05. This means that we have rejected the parameter points that are excluded by data at $95\%$ confidence level (C.L.).

Now we can analyze the properties of the remaining parameter points. Figure 2 shows the Lilith $p$-values of the selected parameter points projected in the $\tan\beta$-$\lambda_{1}$ and $\tan\beta$-$\lambda_{2}$ planes. We find that when $\tan\beta\lesssim 0.2$ ($\tan\beta\gtrsim 5$), $\lambda_{1}$ ($\lambda_{2}$) tends to converge on $\lambda_{\mathrm{SM}}=m_{h}^{2}/v^{2}\simeq 0.26$, which is the quartic Higgs coupling in the SM. This is because $\tan\beta\ll 1$ ($\tan\beta\gg 1$) leads to $v_{1}\gg v_{2}$ ($v_{2}\gg v_{1}$) and $\Phi_{1}\simeq\Phi_{h}$ ($\Phi_{2}\simeq\Phi_{h}$), i.e., $\Phi_{1}$ ($\Phi_{2}$) acting as the SM-like Higgs doublet. Since experimental data favor a SM-like Higgs boson, the corresponding quartic coupling would be close to its SM counterpart.

Additionally, we project the parameter points in the $m_{h_{\mathrm{SM}}}$-$m_{h_{1}}$ and $m_{h_{2}}$-$m_{h_{3}}$ planes in Figs. 3(a) and 3(b), respectively. In Fig. 3(a), the points with $h_{\mathrm{SM}}=h_{1}$ align along a horizontal line with $m_{h_{1}}\simeq 125~{}$\mathrm{G}\mathrm{e}\mathrm{V}$$, while the remaining points indicate that the SM-like Higgs boson is not the lightest $CP$-even Higgs boson $h_{1}$. On the other hand, two sets of aligned points in Fig. 3(b) correspond to $h_{\mathrm{SM}}=h_{2}$ and $h_{\mathrm{SM}}=h_{3}$.

The projection on the $m_{H^{+}}$-$m_{a}$ plane is presented in Fig. 4. From Eq. (16), we know that the difference between the masses of the charged Higgs boson $H^{+}$ and the $CP$-odd Higgs boson $a$ are due to the $\lambda_{4}$ and $\lambda_{5}$ couplings. If ${m}_{12}^{2}$ is much larger than the $\lambda_{4}$ and $\lambda_{5}$ contributions, the difference would be negligible, as demonstrated in Fig. 4 for $m_{H^{+}},m_{a}\gtrsim 500~{}$\mathrm{G}\mathrm{e}\mathrm{V}$$.

Figure 5(a) shows the projection on the $|\kappa_{V}|$-$|\kappa_{f}|$ plane. We find that the parameter points with $|\kappa_{V}|\simeq|\kappa_{f}|\simeq 1$ have the largest $p$-values, implying that current data still favor that the $125~{}$\mathrm{G}\mathrm{e}\mathrm{V}$$ Higgs boson has SM-like couplings. Nonetheless, $|\kappa_{V}|$ may range from $\sim 0.85$ to $\sim 1$, and $|\kappa_{f}|$ may range from $\sim 0.6$ to $\sim 1.3$. In addition, there are two categories of parameter points approximately aligning along two outstanding lines.

• •

Category 1: One line in Fig. 5(a) corresponds to $|\kappa_{V}|\simeq|\kappa_{f}|$. Actually, the signs of $\kappa_{V}$ and $\kappa_{f}$ are the same for all selected parameter points. This line is thus related to $\kappa_{V}\simeq\kappa_{f}$. The main reason is that if $\tan\beta\gg 1$, we have $s_{\beta}\simeq 1$ and $c_{\beta}\simeq 0$, and Eqs. (59) and (60) become $\kappa_{V}\simeq\kappa_{f}\simeq O_{2i}$, where $\kappa_{V}$ and $\kappa_{f}$ have a nearly total positive correlation. As $|O_{2i}|\leq 1$, in this case both $|\kappa_{V}|$ and $|\kappa_{f}|$ cannot exceed one. Most of the parameter points in this category correspond to the horizontal line with $|O_{2i}|/s_{\beta}\simeq 1$ in the $|O_{1i}|/c_{\beta}$-$|O_{2i}|/s_{\beta}$ plane shown in Fig. 5(b), while the rest give $|O_{2i}|/s_{\beta}<1$.

• •

Category 2: Another line in Fig. 5(a) corresponds to $|\kappa_{V}|\simeq 1$ with varying $|\kappa_{f}|$. This category is related to the vertical line with $|O_{1i}|/c_{\beta}\simeq 1$ in Fig. 5(b). From Eq. (59), we know that $|O_{1i}|\simeq c_{\beta}$ and $|O_{2i}|\simeq s_{\beta}$ could lead to $|\kappa_{V}|\simeq c_{\beta}^{2}+s_{\beta}^{2}=1$. Nonetheless, the second relation $|O_{2i}|\simeq s_{\beta}$ is not important to keep $|\kappa_{V}|\simeq 1$ when $s_{\beta}\ll 1$. Therefore, in the case of $\tan\beta\ll 1$, $|O_{2i}|/s_{\beta}$ could deviate from one, resulting in the vertical line in Fig. 5(b).

There are some scatter points not belonging in the two categories. Most of them correspond to $\tan\beta\sim 1$.

The dominant contributions to $\kappa_{g}$ come from the top and bottom loops, leading to a parametrization of Tanabashi:2018oca

[TABLE]

On the other hand, $\kappa_{\gamma}$ is mainly contributed by the $W$ and top loops, resulting in Tanabashi:2018oca

[TABLE]

In both cases, the interference between the two contributions gives a term with a negative coefficient. In Fig. 6(a), we project the parameter points in the $\kappa_{g}$-$\kappa_{\gamma}$ plane, where the points also align along two lines. One line implies a positive correlation between $\kappa_{g}$ and $\kappa_{\gamma}$, corresponding to category 1. This is because the relation $\kappa_{V}\simeq\kappa_{f}$ gives rise to such a positive correlation via Eqs. (63) and (64). On the other hand, when $|\kappa_{V}|\simeq 1$, $\kappa_{\gamma}$ is negatively correlated to $|\kappa_{f}|$ as all selected parameter points satisfy $\kappa_{V}\kappa_{f}>0$. As $\kappa_{g}$ is positively correlated to $|\kappa_{f}|$, category 2 results in a second line with a negative correlation between $\kappa_{g}$ and $\kappa_{\gamma}$.

$\kappa_{Z\gamma}$ is also dominantly contributed by the $W$ and top loops, given by Tanabashi:2018oca

[TABLE]

The correlations of $\kappa_{Z\gamma}$ to $\kappa_{V}$ and to $\kappa_{f}$ are similar to those of $\kappa_{\gamma}$. In addition, $\kappa_{H}$ can be expressed as Tanabashi:2018oca

[TABLE]

where all the coefficients are positive. Thus, $\kappa_{H}$ is positively correlated to both $|\kappa_{V}|$ and $|\kappa_{f}|$. The projection in the $\kappa_{Z\gamma}$-$\kappa_{H}$ plane are shown in Fig. 6(b). Analogous to Fig. 6(a), category 1 leads to a line indicating a positive correlation between $\kappa_{Z\gamma}$ and $\kappa_{H}$ in Fig. 6(b). Besides, parameter points in category 2 roughly align along a second line with a negative correlation.

In Fig. 7(a), we show the projection in the $m_{\chi}$-$\mathrm{BR}_{\mathrm{inv}}$ plane. When $m_{\chi}>m_{h_{\mathrm{SM}}}/2$, we have $\mathrm{BR}_{\mathrm{inv}}=0$, because the invisible decay $h_{\mathrm{SM}}\to\chi\chi$ is kinematically forbidden. When $m_{\chi}<m_{h_{\mathrm{SM}}}/2$, the invisible branching ratio $\mathrm{BR}_{\mathrm{inv}}$ could be as large as $\sim 25\%$ and still consistent with data at 95% C.L. The projection in the $\Gamma_{h_{\mathrm{SM}}}$-$\mathrm{BR}_{\mathrm{und}}$ plane is presented in Fig. 7(b). We find that the undetected BSM branching ratio $\mathrm{BR}_{\mathrm{und}}$ can be allowed up to $\sim 27\%$, while the total width $\Gamma_{h_{\mathrm{SM}}}$ can range from $\sim 2$ to $\sim 7~{}$\mathrm{M}\mathrm{e}\mathrm{V}$$. There is a line implying a positive correlation between $\Gamma_{h_{\mathrm{SM}}}$ and $\mathrm{BR}_{\mathrm{und}}$. This is reasonable, because opening new decay channels enlarges the total width.

In order to investigate the alignment limit, which corresponds to $\tilde{\lambda}_{6}=\tilde{\kappa}_{1}=0$, the selected parameter points are projected in the $\tan\beta$-$\tilde{\lambda}_{6}$ and $\tan\beta$-$\tilde{\kappa}_{1}$ planes in Figs. 8(a) and 8(b), respectively. We find that most of the selected points satisfy $\tilde{\kappa}_{1}\simeq 0$, showing no particular dependence on $\tan\beta$. On the other hand, $\tilde{\lambda}_{6}$ is typically close to zero for $\tan\beta\gtrsim 20$ and $\tan\beta\lesssim 0.05$. For $0.05\lesssim\tan\beta\lesssim 20$, there is no particular favor in the alignment limit.

III.2 DM relic abundance

The thermal relic abundance of dark matter is essentially determined by the total velocity-averaged annihilation cross section at the freeze-out epoch, which we denote as $\left<\sigma_{\mathrm{ann}}v\right>_{\mathrm{FO}}$. In our model, the DM candidate $\chi$ has the following annihilation channels if kinematically allowed.

• •

Annihilation into a pair of fermions, $\chi\chi\to f\bar{f}$. This channel is mediated by $s$-channel $CP$-even Higgs bosons and suppressed by fermion masses. Thus, $t\bar{t}$ and $b\bar{b}$ are the important final states.

• •

Annihilation into a pair of weak gauge bosons, $\chi\chi\to W^{+}W^{-},ZZ$. This channel is also mediated by $s$-channel $CP$-even Higgs bosons.

• •

Annihilation into a weak gauge boson and a Higgs boson, $\chi\chi\to W^{\pm}H^{\mp},Za$, mediated by $s$-channel $CP$-even Higgs bosons.

• •

Annihilation into a pair of $CP$-even Higgs bosons, $\chi\chi\to h_{i}h_{j}$. This channel can be mediated by $s$-channel $CP$-even Higgs bosons, as well as by $t$- and $u$-channel $\chi$. Additionally, there are contributions from quartic scalar couplings.

• •

Annihilation into a pair of $CP$-odd or charged Higgs bosons, $\chi\chi\to aa,H^{+}H^{-}$. This channel is contributed by the mediation of $s$-channel $CP$-even Higgs bosons and quartic scalar couplings.

Some numerical tools are adopted to calculate the relic abundance. We implement the model with FeynRules 2.3.34 Alloul:2013bka , and import the generated model files to a Monte Carlo generator MadGraph5_aMC@NLO 2.6.5 Alwall:2014hca . Then we utilize a MadGraph plugin MadDM 3 Ambrogi:2018jqj to compute the relic abundance $\Omega_{\chi}h^{2}$ for each parameter point.

The relic abundance predicted by the selected parameter points is shown in Fig. 9, where the color bar denotes the freeze-out annihilation cross section $\left<\sigma_{\mathrm{ann}}v\right>_{\mathrm{FO}}$. We find that the observed value $\Omega h^{2}=0.1186\pm 0.0020$ given by the Planck experiment Ade:2015xua corresponds to $\left<\sigma_{\mathrm{ann}}v\right>_{\mathrm{FO}}\sim\mathcal{O}(10^{-26})~{}$\mathrm{c}\mathrm{m}^{3}\mathrm{/}\mathrm{s}, which is typical for thermal dark matter. Increase in $m_{\chi}$ typically reduces the annihilation cross section and hence increases the relic abundance. Consequently, if the DM candidate is too heavy, say $m_{\chi}\gtrsim 3~{}$\mathrm{T}\mathrm{e}\mathrm{V}, the observed relic abundance could not be achieved.

In Fig. 9, the parameter points predicting $\Omega_{\chi}h^{2}$ over the observed value by $2\sigma$ are denoted with crosses. These points are considered to be excluded by data, because DM overproduction by the thermal mechanism contradicts standard cosmology. On the other hand, if the predicted thermal relic abundance is too low, there could be some nonthermal production Lin:2000qq ; Fujii:2002kr occurring after DM freezes out.

III.3 Indirect detection

In this subsection, we discuss constraints from $\gamma$-ray indirect detection experiments. There are couples of dwarf spheroidal galaxies discovered as satellites of the Milky Way Galaxy. They are considered as the largest substructures of the Galactic dark halo, predicted by the cold DM scenario Springel:2008cc ; Diemand:2008in . As known so far, they are the most DM-dominated systems Strigari:2013iaa . Moreover, $\gamma$-ray emissions from typical astrophysical sources, such as neutral and ionized gases and recent star formation activity, are expected to be rare in such dwarf galaxies Mateo:1998wg ; Gallagher:2003nx ; Grcevich:2009gt . These properties make them perfect targets for searching for $\gamma$-ray emissions from DM annihilation.

The DM velocity dispersion in dwarf galaxies is typically $\sim\mathcal{O}(10^{-5})$ Walker:2009zp , which is smaller than DM velocities at the freeze-out epoch by 4 orders of magnitude. Therefore, if the velocity dependence is significant in DM annihilation, the total velocity-averaged cross section in dwarf galaxies $\left<\sigma_{\mathrm{ann}}v\right>_{\mathrm{dwarf}}$ could be much different from the freeze-out value $\left<\sigma_{\mathrm{ann}}v\right>_{\mathrm{FO}}$.

We further use MadDM to calculate $\left<\sigma_{\mathrm{ann}}v\right>_{\mathrm{dwarf}}$ for each parameter point assuming the average DM velocity is $2\times 10^{-5}$. The ratio of $\left<\sigma_{\mathrm{ann}}v\right>_{\mathrm{dwarf}}$ to $\left<\sigma_{\mathrm{ann}}v\right>_{\mathrm{FO}}$ is demonstrated in Fig. 10(a), where the parameter points excluded by the Planck relic abundance measurement are not shown. Most of the parameter points give $\left<\sigma_{\mathrm{ann}}v\right>_{\mathrm{dwarf}}/\left<\sigma_{\mathrm{ann}}v\right>_{\mathrm{FO}}\sim 1$, indicating that $s$-wave annihilation is dominant. Nonetheless, some points give the ratio away from $\mathcal{O}(1)$, indicating significant dependence on velocity. This is typically due to DM annihilation through the resonances of $CP$-even Higgs bosons, since the resonance effect extremely depends on the difference between the resonance location and the velocity-dependent center-of-mass energy Gondolo:1990dk ; Griest:1990kh .

The vertical dashed line in Fig. 10(a) indicates the location of $m_{\chi}=m_{h_{\mathrm{SM}}}/2$, corresponding to the resonance of the SM-like Higgs boson. We can see that the ratio around this line could range from $\sim 10^{-4}$ to $\sim 30$. On the other hand, locations of the other resonances are not fixed, but their effects are also important.

Figure 10(b) shows the projection of the parameter points in the $m_{\chi}$-$\left<\sigma_{\mathrm{ann}}v\right>_{\mathrm{dwarf}}$ plane, as well as the 95% C.L. upper limits on $\left<\sigma_{\mathrm{ann}}v\right>_{\mathrm{dwarf}}$ given by an analysis of Fermi-LAT and MAGIC $\gamma$-ray observations Ahnen:2016qkx . The analysis combined 6-yr observations of 15 dwarf galaxies from the Fermi-LAT satellite experiment and 158-hr observations of a single dwarf galaxy Segue 1 from the MAGIC Cherenkov telescopes, assuming that the DM distributions in the dwarf galaxies follow the Navarro-Frenk-White profile Navarro:1996gj . The limits were obtained assuming that DM solely annihilates into $b\bar{b}$. However, there are various DM annihilation channels in our model. Fortunately, the $\gamma$-ray spectra yielded from these channels should be similar to the spectrum from the $b\bar{b}$ channel, because they are contributed by similar processes, such as hadronization, hadron decays, and final state radiation. Therefore, we have a good reason to expect that the $b\bar{b}$ limits are approximately applicable to our case.

Some parameter points shown in Fig. 10(b) predict a thermal relic abundance lower than the Planck observed value. In this case, the interpretation of indirect detection constraints depends on the assumption of DM composition. If we assume the DM candidate $\chi$ in our model makes up all dark matter in the Universe, nonthermal production Lin:2000qq ; Fujii:2002kr would be needed to realize the observed abundance. Under such an assumption, the Fermi-MAGIC constraint on the parameter space can be directly read off from Fig. 10(b). We can observe that a large fraction of the parameter points with $m_{\chi}\lesssim 1~{}$\mathrm{T}\mathrm{e}\mathrm{V} are ruled out, while the parameter points with $m_{\chi}\gtrsim 100~{}$\mathrm{G}\mathrm{e}\mathrm{V} and $\Omega_{\chi}h^{2}\sim 0.1$ are not excluded. Additionally, if $m_{\chi}\simeq m_{h_{\mathrm{SM}}}/2$, the resonance effect could both yield a data-allowed relic abundance and lead to a small $\left<\sigma_{\mathrm{ann}}v\right>_{\mathrm{dwarf}}$ evading the indirect detection constraint.

Another reasonable assumption is that the relic abundance of the DM candidate $\chi$ is exactly predicted by the thermal mechanism, and hence it could only constitute a fraction of all dark matter. The fraction is given by the ratio of the predicted value to the Planck observed value, $\xi\equiv\Omega_{\chi}/\Omega$. Thus, the $\chi\chi$ annihilation cross section in dwarf galaxies should be effectively rescaled to $\xi^{2}\left<\sigma_{\mathrm{ann}}v\right>_{\mathrm{dwarf}}$ for comparing with indirect detection constraints. Figure 11 presents the parameter points projected in the $m_{\chi}$-$\xi^{2}\left<\sigma_{\mathrm{ann}}v\right>_{\mathrm{dwarf}}$ plane. Under this assumption, most of the parameter points evade the Fermi-MAGIC constraint.

IV Conclusions and outlook

In this paper, we have studied the pNGB DM framework with two $\mathrm{SU}(2)_{\mathrm{L}}$ Higgs doublets $\Phi_{1}$ and $\Phi_{2}$. The DM candidate $\chi$ is the imaginary part of a complex scalar $S$, which is a SM gauge singlet. Most of the scalar potential terms respect a global $\mathrm{U}(1)$ symmetry $S\to e^{i\alpha}S$, except for a soft breaking term giving mass to $\chi$. As a result, $\chi$ becomes a stable massive pNGB. Mass eigenstates in the scalar sector also include three $CP$-even Higgs boson $h_{i}$, a $CP$-odd Higgs boson $a$, and charged Higgs bosons $H^{\pm}$.

There are four possible types of Yukawa couplings without tree-level FCNCs, just as in usual two-Higgs-doublet models. DM scattering off nucleons is mediated by the $CP$-even Higgs bosons. Because of the pNGB nature of $\chi$, the scattering amplitude vanishes in the limit of zero momentum transfer for all the four Yukawa coupling types. Although loop corrections lead to a small nonvanishing amplitude, current and near future direct detection experiments are incapable of probing such a DM candidate.

Taking the type-I Yukawa couplings as an example, we have performed a random scan in the 12-dimensional parameter space. The selected parameter points are required to provide a SM-like Higgs boson whose properties are consistent with current LHC Higgs measurements. For $\tan\beta\gg 1$ or $\tan\beta\ll 1$, one of the Higgs doublets acts as the SM-like Higgs doublet, i.e., $\Phi_{2}\simeq\Phi_{h}$ or $\Phi_{1}\simeq\Phi_{h}$, and most of the selected parameter points satisfy $\tilde{\lambda}_{6}\simeq 0$ and $\tilde{\kappa}_{1}\simeq 0$, corresponding to the alignment limit. On the other hand, for $\tan\beta\sim 1$ there is no preference to the alignment limit.

We have also calculated the relic abundance and annihilation cross sections predicted by the selected parameter points. For $m_{\chi}\lesssim 3~{}$\mathrm{T}\mathrm{e}\mathrm{V}, it is possible to achieve the observed relic abundance. Because of the resonance effect, the present velocity-averaged annihilation cross section at dwarf galaxies could be rather different from that in the freeze-out epoch. If we assume that $\chi$ makes up all dark matter in the Universe via thermal and nonthermal mechanisms, Fermi-LAT and MAGIC observations of dwarf galaxies have excluded a large fraction of parameter points with $m_{\chi}\lesssim 1~{}$\mathrm{T}\mathrm{e}\mathrm{V}. Nonetheless, for $m_{\chi}\simeq m_{h_{\mathrm{SM}}}/2$ or $100~{}$\mathrm{G}\mathrm{e}\mathrm{V}$\lesssim m_{\chi}\lesssim 3~{}$\mathrm{T}\mathrm{e}\mathrm{V}$$, it is still possible to simultaneously satisfy the constraints from the relic abundance observation and indirect detection. If we assume that $\chi$ only constitutes a fraction of all dark matter when the predicted thermal relic abundance is lower than the observed value, most of the parameter points can evade the Fermi-MAGIC constraint.

Differences among the four types of Yukawa couplings are encoded in the coefficients $\xi_{h_{i}}^{f}$ and $\xi_{a}^{f}$ given by Table 1, which lead to different expressions for $\kappa_{b}$ and $\kappa_{\tau}$. Both $\kappa_{b}$ and $\kappa_{\tau}$ in type II are different from those in type I. Thus, we expect the parameter points in type II favored by Higgs measurements should have distinct behaviors from the result present above. On the other hand, $\kappa_{b}$ in the lepton-specific (flipped) Yukawa couplings is identical to that in type I (type II), but $\kappa_{\tau}$ is different. This could cause minor differences in global fits.

Such a pNGB DM model is strongly related to Higgs physics. The proposed future Higgs factories, such as CEPC CEPCStudyGroup:2018ghi , ILC Baer:2013cma , and FCC-ee Abada:2019zxq , would greatly improve the Higgs measurements. We expect that these measurements could significantly restrict the parameter space in our model. Nevertheless, Higgs measurements are not able to pin down the DM candidate mass $m_{\chi}$, which is solely determined by the soft breaking term that does not affect the rest of the scalar masses. Thus, indirect detection experiments in the future are essentially important for exploring this model.

Acknowledgements.

This work is supported in part by the National Natural Science Foundation of China under Grants No. 11805288, No. 11875327, and No. 11905300, the China Postdoctoral Science Foundation under Grant No. 2018M643282, the Natural Science Foundation of Guangdong Province under Grant No. 2016A030313313, the Fundamental Research Funds for the Central Universities, and the Sun Yat-Sen University Science Foundation.

Appendix A Scalar and gauge trilinear couplings

From the scalar potential (4), we derive the scalar trilinear couplings as

[TABLE]

where $g_{{h_{i}}{\chi^{2}}}$ is already given by Eq. (31), and the other coupling coefficients are given by

[TABLE]

By expanding the Lagrangian (22), we obtain the gauge trilinear couplings for the scalars,

[TABLE]

where $s_{\mathrm{W}}\equiv\sin\theta_{\mathrm{W}}$. The derivative symbol $\overleftrightarrow{\partial^{\mu}}$ is defined as $F\overleftrightarrow{\partial^{\mu}}G=F\partial^{\mu}G-G\partial^{\mu}F$. The coupling coefficients are given by

[TABLE]

Appendix B BSM decay widths of the SM-like Higgs boson

This Appendix gives the decay widths of the SM-like Higgs boson into two-body BSM final states when they are kinematically allowed. Assuming the SM-like Higgs boson is $h_{\mathrm{SM}}=h_{i}$, its invisible decay width at tree level is

[TABLE]

Moreover, its decay widths into $aa$ and $H^{+}H^{-}$ are given by

[TABLE]

Furthermore, the $h_{i}\to aZ$ decay width can be expressed as

[TABLE]

where the $\lambda$ function is defined by

[TABLE]

The decay width of $h_{i}\to H^{+}W^{-}$ is given by

[TABLE]

which is equal to the decay width of $h_{i}\to H^{-}W^{+}$ .

If $h_{\mathrm{SM}}=h_{2}$ or $h_{3}$, it is possible to decay into $h_{1}h_{1}$ and $h_{2}h_{2}$, whose widths can be commonly expressed as

[TABLE]

with ${{\tilde{g}}_{ijj}}={g_{ijj}}+{g_{jij}}+{g_{jji}}$. If $h_{\mathrm{SM}}=h_{3}$, there is another possible decay channel into $h_{1}h_{2}$. The corresponding width is

[TABLE]

where ${{\tilde{g}}_{123}}={g_{123}}+{g_{231}}+{g_{312}}+{g_{213}}+{g_{132}}+{g_{321}}$.