Effects of Electroweak Symmetry Breaking on Axion Like Particles as Dark Matter

Axion like particles (ALPs) are generically classified as the pseudo Nambu-Goldstone bosons (pNGB) associated to the spontaneous breaking of a global symmetry, prevalent in many extensions of the Standard Model (SM) $e.g.$ in superstrings theories Svrcek and Witten (2006); Arvanitaki et al. (2010); Demirtas et al. (2020). Though they are analogous to the pNGB of the $U(1)$ Peccei Quinn symmetry, originally introduced to solve the strong CP problem of QCD Peccei and Quinn (1977a, b); Weinberg (1978); Wilczek (1978), ALPs belong to a more general class and hence not necessarily be connected to the solution of the strong CP problem. They are typically very light, neutral pseudo-scalar particles having at most a derivative coupling with the Standard Model at an effective level, suppressed by the scale of spontaneous symmetry breaking ($f_{a}$) of the global symmetry (also referred to as the ALP decay constant), thereby emerging as ideal candidates for explaining the dark matter of the Universe. The mass of an ALP may originate from non-perturbative instanton effect in a strongly interacting hidden sector (analogous to the QCD axion) as well as from explicit symmetry breaking. Correspondingly, contrary to the QCD axion, an ALP does not in general carry any specific relation between its mass and the decay constant. As a result, their mass and decay constant may span over a wide range making them attractive from detection point of view.

Most commonly in the literature, an ALP is considered to be produced non-thermally via the so called $\it{misalignment}$ mechanism Preskill et al. (1983); Abbott and Sikivie (1983); Dine and Fischler (1983); Turner (1983); Arias et al. (2012). The ALP, considered as a classical field in this context, is expected to have an initial non-zero field value and gets frozen there till the Hubble ($\mathcal{H}$) induced friction remains larger that its mass ($m_{a}$). Once $\mathcal{H}\sim m_{a}$, the ALP starts to oscillate at a temperature $T_{{osc}}$ about the minimum of a periodic potential characteristics of a pNGB. Such an oscillatory field mimics as cold dark matter as the associated energy density ($\rho_{a}$) scales as $R^{-3}$, hence behaving like ordinary matter ($R$ is the scale factor of the Universe) provided it remains stable over cosmological time scale. The relic density satisfaction of such ALP provides a standard correlation involving $m_{a}$ and $f_{a}$ which turns out to be well restricted by several cosmological constraints as well from ALP search experiments as summarised in Arias et al. (2012); Cadamuro and Redondo (2012); Ringwald (2012); Workman et al. (2022); Marsh (2018). Considering all such aspects, it turns out that the ALPs being DM should be very light (below keV).

In this work, we propose to include a higher order shift symmetry breaking Higgs portal interaction of ALPs which contributes to its mass ($m_{aE}$) after the electroweak symmetry breaking (EWSB) on top of its existing mass, $m_{a0}$, presumably followed from non-perturbative instanton effect. Inclusion of such additional mass $m_{aE}$ is expected to modify the frequency of ALP oscillation after the EWSB. This would have profound effect on the relic density and hence on the ALP parameter space in $m_{a},f_{a}$ plane, where $m_{a}^{2}={m_{a0}}^{2}+{m_{aE}}^{2}$. We find that depending upon the onset of conventional ALP oscillation (connected with its mass $m_{a0}$ only) before or after the EWSB and associated modification of it after EWSB, the standard correlation between ALP mass and decay constant can be altered significantly leading to opening up of otherwise excluded (restricted) parameter space ($e.g.$ keV-GeV range) for ALPs.

There are some studies focusing on modification of the PQ axion or ALP potential. For example, in ref. Jeong et al. (2022), the authors examine the effect of tiny (limited by neutron electric dipole moment) explicit Peccei-Quinn (PQ) symmetry breaking on the PQ axion dynamics and its role as DM. In this case, the PQ axion initially starts oscillating about a wrong minimum guided by the explicit PQ breaking term and afterward it oscillates about the true minimum leading to a modification of the conventional PQ parameter space. Ref. Li (2023) recently analyses the effect of introducing a PQ breaking term on axiverse that encompasses the PQ axion, an ALP and a hypothetical mixing between them. In another work, ref. Chatrchyan et al. (2023) discusses the effect of adding a non-periodic potential to ALP and thereby find the possibility of accommodating a large misalignment angles which may change the conventional $(m_{a},f_{a})$ parameter space of ALPs. Another modification of axion or ALP potential is referred as kinetic misalignment (Co et al. (2020); Chang and Cui (2020); Co et al. (2021)). In this scenario, a higher dimensional explicit PQ breaking potential and a large initial field value for the PQ symmetry breaking field lead to a nonzero initial ALP velocity which in turn, triggers a delayed ALP oscillation bringing modifications in the relic abundance as well as in parameter space of ALPs. A more general treatment of the effect of initial conditions of ALP and its effects on the $(m_{a},f_{a})$ parameter space can be found in ref. Eröncel et al. (2022). Ref. Chao et al. (2022) discusses about the impact of ALP mass modification on its relic abundance, within the context of the type-II seesaw mechanism.

Our proposal differs from the existing works in a sense that it relies on the electroweak symmetry breaking phase (most natural and unavoidable phase) for modifying the ALP potential, without changing its minimum though. The new mass term for ALP that originates at EWSB may provide a dominant or sub-dominant contribution to the effective final mass of the ALP ($m_{a}$). Depending on the onset of ALP oscillation before or at EWSB, the standard misalignment mechanism gets modified so as to obtain a new parameter space, in $(m_{a},f_{a})$ plane, carrying significant differences with the standard one which might be interesting from ALP search experiments. It turns out that such a Higgs portal interaction of ALPs allows us to probe for light ALPs. Interactions involving ALP and Standard Model (SM) Higgs have also been exercised in few references Im and Jeong (2019); Dev et al. (2019); Oikonomou (2023), however in different contexts. Specifically, a higher dimensional operator consisting of axion and Higgs are discussed in Oikonomou (2023) which is responsible for a new minimum of ALP inducing an epoch of kination and generation of gravitational waves. Contrary to our proposal, the ALP there neither plays the role of DM nor obtains a shift in its mass during EWSB.

The outline of the work is as follows. In the next section, we discuss the standard ALP scenario and following this, we move for studying the evolution of ALPs in our modified scenario. The observation and constraints are elaborated in sections III and IV respectively. Finally we conclude in section VI.

In this section, we first elaborate on the production of ALPs in the early Universe via misalignment mechanism followed by standard ALP cosmology and the related parameter space. We presume the existence of ALPs prior to the end of primordial inflation as a result of spontaneous breaking of a global $U(1)$ symmetry during inflation. After inflation (followed by a reheating era), the ALP field $a({\bf{x}},t)$ is expected to be spatially homogeneous, hence described by $a(t)$ only, and gets frozen at an initial value $a_{I}$ parametrised by the misalignment angle $\theta_{I}=a_{I}/f_{a}$ (with $\dot{\theta}_{I}=0$), as long as the Hubble remains larger compared to its mass $m_{a0}$. Note that, the ALP being a Nambu Goldstone Boson, the origin of $m_{a0}$ is related to the breaking of shift symmetry which can possibly be connected to the non-perturbative dynamics, (analogous to PQ axion) and independent of temperature. This can be realised if the ALP couples to a hidden $SU(N)$ sector (the global $U(1)$ symmetry is anomalous under such non-abelian group) which may not be thermalised.

At very early Universe, after inflation, when $\mathcal{H}\gg m_{a0}$, the $\it{zero~{}mode}$ of the axion field remains overdamped and gets stuck at its initial value $\theta_{I}$ due to Hubble friction. In a radiation dominated era, at some point when

the field rolls toward its potential minimum and starts to oscillate. The onset of oscillation can be characterised by the temperature $T^{0}_{osc}$. Near the minimum of the potential, $V_{\rm{0}}\simeq\frac{1}{2}{m_{a0}}^{2}f_{a}^{2}\theta^{2}$ and the equation of motion (below $T\lesssim T^{0}_{osc}$) reduces to

where the ALP field is considered to be a homogeneous one, $\it{i.e.}$ the spatial variation over Hubble volume, $\nabla^{2}\theta/R^{2}$ vanishes. Using the expression of the Hubble parameter $\mathcal{H}$ in a radiation-dominated universe, $\mathcal{H}(T)=1.66\sqrt{g_{\star}(T)}T^{2}/M_{Pl}$, and the relation in Eq. 3, we can estimate the conventional oscillation temperature $T^{0}_{osc}$ for ALP as

where, $g_{\star}(T)$ is the number of relativistic degrees of freedom ($\it{d.o.f}$) at a temperature $T$ and the value of Planck mass, $M_{Pl}=1.22\times 10^{19}$ GeV is deployed.

At the conventional oscillation temperature $T^{0}_{osc}$, the total energy density of the ALP $\rho_{a}=(\dot{\theta}^{2}f^{2}_{a}+m_{a0}^{2}f_{a}^{2}\theta^{2})/2$ is fully embedded in its potential part only since ALP does not have any initial velocity, $(\dot{\theta}_{I}=0)$ and is given by

For $T\lesssim T^{0}_{osc}$, Eq. 4 implies that field would perform fast oscillations with slowly decreasing amplitude where the average energy density $\langle\rho_{a}\rangle$ scales as $R^{-3}$ and the equation of state $\omega=\langle p_{a}\rangle/\langle\rho_{a}\rangle$ turns out to be zero implying that it behaves like non-relativistic matter Arias et al. (2012); Marsh (2016). Here, $\langle\rangle$ implies averaging over one complete oscillation.

Since the ALP number density in a co-moving volume turns out to be conserved Arias et al. (2012), the present day ALP energy density can be estimated as

where $T_{0}=2.4\times 10^{-4}$ eV is the present temperature and a change of ALP mass at different (later) temperature, if any, is included in the form of $m_{a}(T)$. However as stated above, unlike QCD Axion, ALP masses are generally considered as temperature independent, ${\it{i.e.}}~{}m_{a0}(T^{0}_{osc})=m_{a}(T_{0})=m_{a0}$. In this case, employing Eq. 8 and considering the adiabatic expansion of the Universe $i.e.~{}(R_{osc}/R_{0})^{3}=s(T_{0})/s(T^{0}_{osc})$ where $s$ is the entropy density of the universe given by $s(T)=(2\pi^{2}/45)g_{\star s}T^{3}$, the ALP relic density can be expressed as

where $\rho_{c,0}=1.05\times 10^{-5}h^{2}~{}\textrm{GeV~{}cm}^{-3}$ (present critical energy density) and $g_{\star s}(T_{0})=3.94$ (number of relativistic $\it{d.o.f}$ at present temperature) Bauer and Plehn (2019). Using the expression of Eq. 5 into Eq. 9, the ALP relic density can be estimated as

from which a correlation between the two parameters $m_{a0}$ and $f_{a}$ can easily be obtained. However, once the astrophysical and cosmological bounds are imposed (to be discussed later in section IV), the allowed mass range for the ALPs falls below keV-scale Balázs et al. (2022) only.

In this section, we introduce an explicit higher order shift symmetry breaking term in the ALP potential involving the SM Higgs. As a result, a new contribution toward the mass of ALP originates once the electroweak symmetry breaking takes place. The appearance of such a mass term for ALP at an intermediate phase (during its evolution) in addition to $m_{a0}$ (connected to non-perturbative dynamics) not only enables the effective mass of the ALP ($m_{a}$) and its decay constant ($f_{a}$) to treat as independent parameters, but also modifies the frequency of ALP oscillation at EWSB as we observe below.

The phenomenologically relevant part of the potential for the ALP or the $\theta$ field, after the spontaneous breaking of the PQ-like global symmetry, then turns out to be

At a temperature sufficiently higher than the electroweak scale, $T>T_{\rm{EW}}\sim$ 150 GeV, the temperature correction to the SM Higgs potential helps the SM Higgs to have a single minimum at origin Kolb and Turner (1990). Hence the Higgs field is expected to settle at origin as a result of which this term does not contribute till the temperature becomes comparable to $T_{\rm{EW}}$ when $H$ gets a vacuum expectation value (vev), $H=(v+h)/\sqrt{2}$ with $v=246$ GeV. After the EWSB, the ALP receives a new contribution to its mass such that its effective mass $m_{a}$ satisfies the relation

where $\theta_{\rm min}=a_{\rm min}/f_{a}$ denotes the newly developed minimum of the ALP potential after EWSB. The value of $\theta_{\rm min}$ can be estimated as a solution to the equation,

obtained by minimising the ALP potential. Clearly, the new potential minimum depends on the choice of $\alpha$ and other parameters ($\Lambda$ and $m_{a0}$). We pursue with $\alpha=\pi$ for the rest of the analysis that results into the effective mass of the ALP given by,

Such a choice of $\alpha=\pi$ is motivated primarily from the fact that the minimum of the ALP potential ($\theta_{\rm min}=0$) remains unaffected due to the presence of this additional contribution to ALP potential via Eq. 12. However, for conventional QCD axion, such explicitly breaking term should be incorporated very carefully as any alteration of the minimum can spoil the resolution of the strong CP problem, the strong-CP violating angle being heavily constrained by neutron electric dipole moment Abel et al. (2020) (widely referred as Axion quality problem Kamionkowski and March-Russell (1992); Holman et al. (1992); Barr and Seckel (1992)). With a general axion like particles, such a solution of strong CP problem is not necessarily be connected to the solution of the strong CP problem. To investigate the impact of different $\alpha$ values on $\theta_{\rm min}$ and hence on relic, we incorporate an analysis in the later part of the following sub-section.

Appearance of such a mass term would affect the evolution of the ALP field, in terms of its change in oscillation frequency, immediately after the EWSB. To analyse it further, we divide the study into two cases depending on whether the ALP starts its oscillation at a temperature $T^{0}_{osc}$ (due to $m_{a0}$) prior to EWSB temperature $T_{\rm{EW}}$ or not, as

Note that the demarcation between these two cases is set by the condition

which translates (assuming a radiation dominated universe) into $m_{a0}>10^{-13}$ GeV (for case [A]) and $m_{a0}\leq 10^{-13}$ GeV (for case [B]) where we use $T_{\rm{EW}}=150$ GeV.

In the previous section, we analyse the relic satisfied parameter space of ALP characterized by its final mass $m_{a}$ and the decay constant $f_{a}$. However, such a parameter space can be further constrained from astrophysical and cosmological limits as well as few laboratory and telescope searches provided one considers an ALP-photon coupling Marsh (2016); Jaeckel and Ringwald (2010) of the form

where $F_{\mu\nu}$ and $\tilde{F}^{\mu\nu}$ are the electromagnetic field strength tensor and its dual respectively. The effective coupling $g_{a\gamma\gamma}$ can be written in terms of the ALP decay constant $f_{a}$ as

Generally, $C_{a\gamma\gamma}$ is expected to be $\mathcal{O}(1)$ and $\alpha$ is the fine-structure constant.

Such an ALP-photon coupling opens up several windows of observation on which a considerable effort is being devoted now-a-days. These ALPs might get produced within the searing plasma of stars via interactions with photons. Such process may subsequently impact the stellar evolution leading to an overall energy loss of a star while escaping. Therefore, the non-observance of any unwanted energy loss in stars sets bounds on the parameter $g_{a\gamma\gamma}$ Turner (1990); Raffelt (2008). A stringent bound on $g_{a\gamma\gamma}<6.6\times 10^{-11}$ GeV^-1 emerges from the study of evolution of the horizontal branch (HB) stars Ayala et al. (2014). Also, the Sun is a likely source of ALPs (solar ALP), which are detectable on Earth in a telescope with a macroscopic magnetic field via reverse Primakoff process, commonly known as the $\it{Helioscope}$ Sikivie (1983). We have used the latest findings with best sensitivity from the CERN Axion Solar Telescope (CAST) which also puts constraints on $g_{a\gamma\gamma}$ similar to those derived from the study of HB stars as, $g_{a\gamma\gamma}<6.6\times 10^{-11}$ GeV^-1 for $m_{a}<0.02$ eV Anastassopoulos et al. (2017). The ALP-photon interaction is also constrained by the measurements of solar neutrino flux as $g_{a\gamma\gamma}<7\times 10^{-10}$ GeV^-1 for $m_{a}\lesssim\mathcal{O}(\rm{keV})$ Gondolo and Raffelt (2009). Other important constraints emerge from the cavity experiments such as Rochester-Brookhaven-Florida and University of Florida (RBF and UF) and Axion Dark Matter Experiment (ADMX) which are sensitive for the ALP mass ranges of $4.5-16.3$ $\mu$eV De Panfilis et al. (1987); Wuensch et al. (1989); Hagmann et al. (1990) and $1.9-3.3~{}\mu$eV Asztalos et al. (2004) respectively. Telescope searches including Visible Multi-Object Spectrograph (VIMOS) and Multi Unit Spectroscopic Explorer (MUSE) further constrain the ALP mass ranges of $4.5-5.5$ eV and $2.7-5.3$ eV respectively Workman et al. (2022); Regis et al. (2021).

Searches for ALP are also actively done by various laboratory experiments. One such experimental approach is the light shining through a wall (LSW) experiment Redondo and Ringwald (2011) where a laser beam is expected to be converted into axion or ALPs after being exposed to a high magnetic field. Subsequently, these converted particles pass through an opaque wall and upon re-converting into photons via a second magnetic field behind the wall, they provide indirect evidence for the presence of ALPs. The current best limit by such LSW experiments are given by OSQAR (Optical Search for QED Vacuum Birefringence, Axions, and Photon Regeneration) as $g_{a\gamma\gamma}<3.5\times 10^{-8}$ GeV^-1 for $m_{a}<0.3$ meV Isleif (2022).

The cosmological constraint of $\Gamma_{a\to\gamma\gamma}^{-1}\geq\tau_{U}$ (with $\tau_{U}$ being the age of the Universe and $\Gamma_{a\to\gamma\gamma}=g_{a\gamma\gamma}^{2}m_{a}^{3}/64\pi$ representing the decay width of ALP) serves as a key condition in guaranteeing the stability of the non-thermally produced cosmic ALPs as a viable dark matter over the universe lifetime Balázs et al. (2022). If $\Gamma_{a\to\gamma\gamma}^{-1}<\tau_{U}$, the extra radiations impart additional limits on $g_{a\gamma\gamma}$ extending to very large ALP masses Masso and Toldra (1995). Additionally, photons from ALP decays can be seen as peaks on top of the known backgrounds in the galactic x-ray spectra and must not surpass the extragalactic background light (EBL) Overduin and Wesson (2004). Also, ALP decaying into photons can lead to the ionization of primordial hydrogen, and the constraint comes from the requirement to prevent this ionization from making a crucial contribution to the optical depth after recombination and hence ensuring the consistency of BBN with observations Masso and Toldra (1995); Cadamuro and Redondo (2012); Depta et al. (2020). Other cosmological constraints comes from the excess photons (when decay occurs during opaque universe) include spectral distortions in the CMB spectrum and increase in $T_{\gamma}$ (photon temperature) relative to $T_{\nu}$ (neutrino temperature), thus modifying the value of $N_{\rm{eff}}$ inferring from WMAP Depta et al. (2020).

The bounds regarding all these constraints are shown in the Fig. 8 which are taken from the updated online repository AxionLimits O’hare (2020). In Fig. 8, the yellow line acts as the demarkation line below which the ALP may serve as a viable dark matter ($\Gamma_{a\to\gamma\gamma}^{-1}>\tau_{U}\simeq 10^{17}$ sec). On the other hand, the red line corresponds to the ALP dark matter relic satisfaction contour originating from the standard misalignment mechanism (as discussed in section II) representative of the green line displayed in Figs. 2, 3, 6 and 7 with the consideration of $C_{a\gamma\gamma}=1$ (which is followed throughout the analysis). The obtained parameter spaces in our scenario (Figs. 2 and 6) of cases [A] and [B] are further constrained by these bounds and the residual allowed parameter regions in $m_{a}-g_{a\gamma\gamma}$ plane (converted from $m_{a}-f_{a}$) are demonstrated in top and bottom panels of Fig. 8 respectively. Note that as seen from Fig. 8 corresponding to case A, ALP masses ranging from $\mathcal{O}$(keV) to $10^{-13}$ GeV (serves as the lower limit for this case as per our previous discussion) are allowed with larger couplings (by several orders of magnitude) compared to the conventional picture (red line). Similarly, higher ALP-photon couplings are permitted for case B as well, where the upper limit of allowed ALP mass turns out to be $\mathcal{O}(10)$ eV while the lower limit is set at $m_{a}\approx 5\times 10^{-15}$ GeV, which is the minimum mass contribution arising from the dimension-6 operator. It is important to note that for standard misalignment of ALP (without any higher dimensional soft symmetry breaking term), the lower limit of ALP mass can be extrapolated upto very small values ($\sim 10^{-24}$ eV) Marsh (2016).

In the present scenario, where the PQ-like symmetry is assumed to be broken during inflation, the ALP field should experience quantum fluctuations having an amplitude denoted by $\delta a\simeq\mathcal{H}_{inf}/2\pi$ $(or,~{}\delta\theta_{I}\simeq\mathcal{H}_{inf}/2\pi f_{a})$, where $\mathcal{H}_{inf}$ is the Hubble parameter during inflation. These quantum fluctuations give rise to isocurvature perturbation of the cold dark matter Hamann et al. (2009); Beltran et al. (2007); Kawasaki et al. (2018) which is constrained from the measurements of the CMB anisotropies¹¹1These ALP fluctuations however do not play any role in the overall density fluctuations of the universe.. The contribution of ALP to CDM isocurvature perturbation $\mathcal{S}_{CDM}$ can be expressed as

In our scenario, ALP contributes entirely to relic density of CDM $\it{i.e.}$ $\Omega_{CDM}=\Omega_{a}$. The spectrum of CDM isocurvature perturbation in the Fourier space is given by

$k$ is the comoving wavenumber, to be evaluated at the pivot scale $k_{*}$ defined by ${k_{*}}/a_{0}=0.05~{}\textrm{Mpc}^{-1}$. The limit imposed by $Planck$ Aghanim et al. (2020) on CDM isocurvature perturbation with respect to the adiabatic power, $\mathcal{P}_{adi}(k_{*})\approx 2.2\times 10^{-9}$, is expressed as Aghanim et al. (2020)

The ALP density perturbation $\mathcal{S}_{CDM}$ of Eq. 26 can also be recast as

using ${\delta\rho_{a}}/{\rho_{a}}\simeq 2{\delta\theta_{I}}/{\theta_{I}}$ which follows from Eq. 6. Considering the misalignment angle $\theta_{I}$ to be $\mathcal{O}(1)$ and employing the fluctuation of the misalignment angle during inflation, $\delta\theta_{I}\simeq\mathcal{H}_{inf}/2\pi f_{a}$, into Eqs. 29 and 27, we obtain

Following the constraint in Eq. 28 and considering $\theta_{I}=1$ as in the previous sections, we obtain an upper bound on the inflationary Hubble scale $\mathcal{H}_{inf}$ as

Depending on the specific cases in our scenario, we can derive a few more constraints as a consequence of Eq. 31 Arias et al. (2012) as we discuss in the following.

For case A ($T^{0}_{osc}>T_{\rm EW}$), the ALP mass at the onset of oscillation can be constrained by the required condition $3\mathcal{H}_{inf}>3\mathcal{H}(T^{0}_{osc})=m_{a0}$. Using Eq. 31 and setting $\Omega_{a}h^{2}\simeq 0.12$ in Eq. 23, we obtain the following relation

where $g_{\star}(T^{0}_{osc})\simeq 100$ is considered.

On the other hand, for case B ($T^{0}_{osc}\leq T_{\rm EW}$), the constraints can appear in two different ways: ($\it{i}$) when the ALP with effective mass $m_{a}$ starts to oscillate at $T=T_{\rm EW}$ only, the criteria $3\mathcal{H}_{inf}>3\mathcal{H}(T_{\rm EW})$ along with Eq. 31 leads to

($\it{ii}$) Secondly, if the oscillation temperature of the ALP with effective mass $m_{a}$ is itself smaller than $T_{\rm EW}$ (no intermediate change in the ALP mass takes place), we need to utilize the criteria $3\mathcal{H}_{inf}>m_{a}$. Here, similar to case A, using Eq. 31 and setting $\Omega_{a}h^{2}\simeq 0.12$ in Eq. 10 (with $m_{a0}$ replaced by $m_{a}$), we obtain

The correlations obtained in Eq. 32 and constraints on $f_{a}$ as in Eqs. 33-34 are evidently weaker than the other restrictions shown in the previous section and obeyed by the allowed parameter space in the respective cases. The constraint in Eq. 31 is however significant in the context of gravitational wave detection. As inflation can give rise to the generation of gravitational waves through tensor perturbations, the generation of tensor perturbations during inflation is directly correlated with the Hubble parameter Kawasaki et al. (2018) as

which can be translated in our scenario as

which is very small number to be predicted in near future as the current observational constraint on the tensor mode, $r\lesssim 0.1$ , is derived from the Planck measurements of the CMB Ade et al. (2016).

Axion-like particles are well motivated dark matter candidates which are thought to be produced primarily through the misalignment mechanism in the early universe. In this study, we have explored the impact of electroweak symmetry breaking on the evolution of such ALP DM in presence of an explicit shift symmetry breaking dimension-6 Higgs portal interaction of it. We observe that such an operator may significantly contribute to the ALP mass during the EWSB which further initiates a change in the oscillation frequency, thereby deviating from the standard misalignment mechanism in terms of final outcome. We have shown that depending on the standard ALP oscillation temperature (which is determined by the ALP mass originating from non-perturbative dynamics only), the change in the ALP mass across EWSB gives rise to a significant modifications in relic-density allowed parameter space compared to the standard misalignment mechanism.

Our findings can be broadly categorised into two: (a) one in which we obtain an extended parameter space (in $m_{a}-f_{a}$ plane) compared to the standard misalignment mechanism, applicable when the non-perturbative mass of ALP $m_{a0}$ exceeds $10^{-13}$ GeV and (b) secondly, where we obtain a parameter space with a different slope (in $m_{a}-f_{a}$ plane) compared to the standard one which applies when $m_{a0}$ falls below $10^{-13}$ GeV.

Finally, taking into account all the existing constraints from several terrestrial experiments, astrophysical and cosmological bounds on the $m_{a}-g_{a\gamma\gamma}$ plane (translated from $m_{a}-f_{a}$ plane) characterising ALP’s interaction with photons, we have identified a significant residue of newly opened up parameter space (from relic satisfaction point of view) in the sub-keV ALP mass regime to be compatible as non-thermal dark matter. These constraints also play a crucial role in restricting the lower limit on the cut-off scale $\Lambda$ (as evident from Fig. 8), thereby forbidding the other possibility of ALP production via freeze-in from the decay or annihilations of the Higgs, which requires relatively smaller values of $\Lambda~{}(\lesssim\mathcal{O}(10^{9})~{}\textrm{GeV})$. Interestingly, the predicted ALP-photon couplings turn out to be notably larger compared to the case of conventional misalignment, opening up opportunities for exploration through upcoming experiments.