Revisiting Bounds on Primordial Black Hole as Dark Matter with X-ray Background

We designate the photons emitted directly from Hawking evaporation as prompt radiation, encompassing both primary and secondary emissions. Primary photons are defined by Equ.(2.2), with $s=1$. The secondary component emerges from the decay of hadrons originating from the fragmentation of primary quarks and gluons [2], as well as from the decay of gauge bosons. Furthermore, hadrons originating from PBHs also undergo decay or radiation processes. Hence, the direct number density originating from PBHs can be expressed as:

$$\frac{{\rm d}N^{\rm dir}_{\gamma}}{{\rm d}E{\rm d}t}=\frac{{\rm d}N^{\rm prompt}_{\gamma}}{{\rm d}E{\rm d}t}+\frac{{\rm d}N^{\rm FSR}_{\gamma}}{{\rm d}E{\rm d}t}+\frac{{\rm d}N^{\rm dec}_{\gamma}}{{\rm d}E{\rm d}t},$$

(2.3)

where the latter two terms are adopted from ref.[15]. The Final State Radiation (FSR) denotes radiation that remains independent of the astrophysical setting. The combination of FSR and virtual internal bremsstrahlung forms what is known as the full internal bremsstrahlung (IB) radiation. Nevertheless, we found some discrepancies in the studies conducted by refs.[8, 39, 27], while the original formulation of IB radiation was presented by refs.[7, 6]. Given these circumstances, we are inclined to favor the standard particle process outlined in ref.[15], which is detailed as follows:

$$\begin{split}\frac{{\rm d}N^{\rm FSR}_{\gamma}}{{\rm d}E{\rm d}t}+\frac{{\rm d}N^{\rm dec}_{\gamma}}{{\rm d}E{\rm d}t}&=\sum_{i=e^{\pm},\mu^{\pm},\pi^{\pm}}\int{\rm d}E_{i}\frac{{\rm d}N_{i}^{\rm pri}}{{\rm d}E{\rm d}t}\frac{{\rm d}N_{i}^{\rm FSR}}{{\rm d}E}+\sum_{i=e^{\pm},\pi^{0},\pi^{\pm}}\int{\rm d}E_{i}\frac{{\rm d}N_{i}^{\rm pri}}{{\rm d}E{\rm d}t}\frac{{\rm d}N_{i}^{\rm dec}}{{\rm d}E},\end{split}$$

(2.4)

By means of computational analysis, it has been ascertained that the significant contribution to the X-ray band predominantly originates from $e^{-}e^{+}$ pairs generated by FSR, rather than decay. The comprehensive expression for FSR is as follows:

$$\frac{{\rm d}N_{i}^{\rm FSR}}{{\rm d}E_{\gamma}}=\frac{\alpha}{\pi Q_{i}}P_{i\to i\gamma}(x)\left[\log(\frac{1-x}{\mu_{i}^{2}})-1\right],$$

(2.5)

and

$$P_{i\to i\gamma}(x)=\left\{\begin{aligned} &\frac{2(1-x)}{x},&i=\pi^{\pm}\\ &\frac{1+(1-x)^{2}}{x},&i=\mu^{\pm},e^{\pm}\end{aligned}\right.$$

(2.6)

where the fine structure constant $\alpha=1/137.037$, $x=2E_{\gamma}/Q_{i}$, and $\mu_{i}=m_{i}/Q_{i}$, where $Q_{i}=2E_{i}$represents the energy scale for FSR. The splitting function $P_{i\to i\gamma}(x)$ is utilized to distinguish between bosons and fermions. Consequently, Equ.(2.4) has been streamlined to represent the results of $e^{-}e^{+}$ pairs in the following form:

$$\begin{split}\frac{{\rm d}N_{\gamma}^{\rm FSR}}{{\rm d}E{\rm d}t}&=\frac{\alpha}{2\pi}\int{\rm d}E_{e}\frac{{\rm d}N_{e}}{{\rm d}E_{e}{\rm d}t}\left(\frac{2}{E_{e}}+\frac{E_{\gamma}}{E_{e}^{2}}-\frac{2}{E_{e}}\right)\times\left[{\rm ln}\left(\frac{4E_{e}(E_{e}-E_{\gamma})}{m_{e}^{2}}\right)\right].\end{split}$$

(2.7)

The electrons and positrons discharged from PBHs would be initially relativistic [28]. They decelerate through processes involving Compton scattering and ionization losses, with the cross-section increasing until the positrons transition to a non-relativistic state. The $\gamma$-rays spectrum resulting from the In-flight Annihilation (IA) of positrons quantifies the overall flux of annihilating $e^{+}$s particles before they reach a state of stopping or thermalization, as detailed in ref.[6]. This spectrum is expressed as follows:

$$\begin{split}\frac{dN_{\gamma}^{\mathrm{IA}}}{dE_{\gamma}{\rm d}t}&=\frac{\pi\alpha^{2}n_{H}}{m_{e}}\int_{m_{e}}^{\infty}dE_{e^{+}}\frac{dN_{e^{+}}}{dE_{e^{+}}{\rm d}t}\times\int_{E_{\min}}^{E_{e^{+}}}\frac{dE}{dE/dx}\frac{P_{E_{e^{+}}\rightarrow E}}{(E^{2}-m_{e}^{2})}\\ &\times\left(-2-\frac{(E+m_{e})(m_{e}^{2}(E+m_{e})+E_{\gamma}^{2}(E+3m_{e})-E_{\gamma}(E+m_{e})(E+3m_{e}))}{E_{\gamma}^{2}(E-E_{\gamma}+m_{e})^{2}}\right),\end{split}$$

(2.8)

where the symbol $n_{\rm H}$ represents the number density of neutral hydrogen atoms. The positron spectrum emanating from PBHs, denoted as ${\rm d}N_{e^{\pm}}/({\rm d}E_{e^{\pm}}{\rm d}t)$, can be computed by employing Equ.(2.2) with a parameter value of $s=1/2$. The rate of energy loss for positrons due to ionization interactions in the presence of neutral hydrogen, denoted as ${\rm d}E/{\rm d}x$, can be evaluated utilizing the standard Bethe-Bloch formula.

As positrons gradually deplete a substantial portion of their energy and near their rest mass, indicated by $E_{e^{+}}\simeq m_{e}$, the probability of survival for positrons transitioning from energy $E_{e^{+}}$ to rest mass $m_{e}$ can be estimated as $P=P_{E_{e^{+}}\to m_{e}}$. The expression for this probability is provided as:

$$P_{E_{e^{+}}\to E}=\exp\left(-n_{H}\int_{E}^{E_{e^{+}}}{\rm d}E^{\prime}\frac{\sigma_{\rm ann}(E^{\prime})}{|{\rm d}E^{\prime}/{\rm d}x|}\right),$$

(2.9)

where $\sigma_{\rm ann}$ denotes the cross-section for the annihilation of a positron with an electron at rest. The probability $P_{E_{e^{+}}\to E}$ for positrons with energies below a few MeV consistently falls within the range of approximately 0.95 to 1, indicating that only a minor fraction of these positrons undergo annihilation before transitioning to a non-relativistic state. For instance, at injection energies of 10 MeV, the deviation of $P$ from unity is roughly 11% [7]. Additionally, ref.[22] have computed the proportion of positrons forming positronium in flight within various neutral mediums, revealing that in-flight annihilation contributes minimally to X-ray emission. Nonetheless, we include this component for the sake of comprehensiveness.

The interaction between a positron and an electron results in the conversion of rest mass energy into two or more photons, with a total energy of $1022$ keV. In the scenario where a pair of electrons and positrons are at rest, their direct annihilation gives rise to two photons, each possessing an energy of $511$ keV. The ground state of positronium (Ps) exhibits two discernible spin configurations, contingent upon the relative orientations of the electron and positron spins.

In specific terms, one of these configurations, referred to as para-positronium (pPs), materializes $1/4$ of the time, showcasing antiparallel spins and a total spin of $S=0$, identified as ${}^{1}S_{0}$. This state emits two photons moving in opposite directions, each carrying an energy of $511$ keV, akin to the outcome of direct electron-positron annihilation. The second configuration is ortho-positronium (oPs), characterized by parallel spins and a total spin of $S=1$, denoted as ${}^{3}S_{1}$, and occurs $3/4$ of the time. The oPs state necessitates a final state involving more than two photons. The conservation of momentum allocates the total energy of $1022$ keV among three photons, leading to a continuous spectrum of energy levels extending up to $511$ keV [35].

The one-loop Quantum Electrodynamics (QED) correction to the spectrum in oPs decay has been computed numerically in ref.[32, 34], where the Ore-Powell result is provided:

$$\begin{split}\frac{1}{\Gamma_{0}}\frac{{\rm d}\Gamma}{{\rm d}x}&=\frac{2}{\pi^{2}-9}\left[\frac{2-x}{x}+\frac{(1-x)x}{(2-x)^{2}}\right.\left.-\frac{2(1-x)^{2}{\rm log}(1-x)}{(2-x)^{3}}+\frac{2(1-x){\rm log}(1-x)}{x^{2}}\right].\end{split}$$

(2.10)

In order to differentiate it from parallel positronium, we isolate the expression $\frac{{\rm d}N^{\rm oPs}_{\gamma}}{{\rm d}E{\rm d}t}$, which signifies the continuous flux resulting from the three-photon decay of ortho-positronium. Subsequently, we obtain:

$$\frac{dN_{\gamma}^{\mathrm{oPs}}}{dE_{\gamma}{\rm d}t}=\frac{1}{\Gamma_{0}}\frac{{\rm d}\Gamma}{{\rm d}x}\int_{m_{e}}^{\infty}{\rm d}E\frac{{\rm d}N_{e}}{{\rm d}E{\rm d}t},$$

(2.11)

where $x=E/m_{e}$, and ${\rm d}N_{e}/{\rm d}E{\rm d}t$ denotes the positron spectrum originating from PBHs.

The $511$ keV line flux is from direct annihilation with $e^{-}$s and intermediate formation of positronium, and they produce two quasi-monochromatic photons at its mass, which can be written as

$$\frac{dN_{\gamma}^{\mathrm{511}}}{dE_{\gamma}{\rm d}t}=\frac{dN_{e}}{dE_{e}{\rm d}t}\delta(E_{e}-m_{e}),$$

(2.12)

where $\delta$ is the Dirac-delta function.

To account for the prompt $\gamma$-ray emission from Hawking Evaporation and other processes involving electron and positron by-products, it is essential to delineate the fraction of each emission. The probability, symbolized as $P$ as previously indicated in Equ.(2.9), denotes the chance of survival in a non-relativistic state for a positron. Through the utilization of $P$ and the probability of positronium formation $f_{\rm Ps}$, we can determine the fraction linked to the aforementioned contribution.

According to ref.[7], the flux of internal bremsstrahlung $\gamma$-rays is directly related to the rate of positron production, which can be inferred from the intensity of $511$ keV. Drawing on this information and the elucidation in Section 2.3, where $f_{\rm Ps}$ represents the count of $e^{+}$s forming positronium before annihilation, we can derive the following fractions for each emission:

$$\begin{split}f_{\rm IA}&=1-P,\\ f_{\rm oPs}&=3\times\frac{3}{4}Pf_{\rm Ps},\\ f_{\rm IB/FSR}&=\frac{1}{2}\left(1-\frac{3}{4}f_{\rm Ps}\right)^{-1},\\ f_{511}&=2\times\left[P(1-f_{\rm Ps})+\frac{1}{4}Pf_{\rm Ps}\right].\\ \end{split}$$

(2.13)

The emitted photons are multiplied by the number of oPs and $511$ keV, thus yielding $f_{\rm IA}+f_{\rm oPs}/3+f_{511}/2=1$, all originating from electrons or positrons. We introduce the total annihilation flux, which encompasses these three elements:

$$\frac{{\rm d}N^{\rm Ann}_{\gamma}}{{\rm d}E{\rm d}t}=\frac{{\rm d}N^{511}_{\gamma}}{{\rm d}E{\rm d}t}+\frac{{\rm d}N^{\rm IA}_{\gamma}}{{\rm d}E{\rm d}t}+\frac{{\rm d}N^{\rm oPs}_{\gamma}}{{\rm d}E{\rm d}t}.$$

(2.14)

Consequently, the final contribution can be computed in the subsequent manner:

$$\begin{split}\frac{{\rm d}N^{\rm tot}_{\gamma}}{{\rm d}E{\rm d}t}&=\frac{{\rm d}N^{\rm dir}_{\gamma}}{{\rm d}E{\rm d}t}+\frac{{\rm d}N^{\rm Ann}_{\gamma}}{{\rm d}E{\rm d}t}\\ &=f_{\rm prompt}\frac{{\rm d}N^{\rm prompt}_{\gamma}}{{\rm d}E{\rm d}t}+f_{\rm FSR}\frac{{\rm d}N^{\rm FSR}_{\gamma}}{{\rm d}E{\rm d}t}+f_{511}\frac{{\rm d}N^{511}_{\gamma}}{{\rm d}E{\rm d}t}+f_{\rm IA}\frac{{\rm d}N^{\rm IA}_{\gamma}}{{\rm d}E{\rm d}t}+f_{\rm oPs}\frac{{\rm d}N^{\rm oPs}_{\gamma}}{{\rm d}E{\rm d}t}.\end{split}$$

(2.15)

Here, we set $f_{\rm prompt}=1$ for aesthetic reasons. We illustrate an instance of continuous and line spectra, denoted as ${\rm d}N/{\rm d}E{\rm d}t$, for $M_{\rm PBH}=7\times 10^{16}$ g and $f_{\rm PBH}=1$ for PBH in Fig.1.

Firstly, we consider only the PBHs model, providing a conservative estimation. Assuming that the data follows a normal distribution, the $\chi^{2}$ value is calculated as follows:

$$\chi^{2}=\sum\left[\frac{\phi^{\rm{data}}_{\gamma}-\phi^{\rm PBH}_{\gamma}(\theta)}{\sigma^{\rm{data}}}\right]^{2},$$

(4.2)

where $\theta=\{f_{\rm PBH},m_{\rm PBH}\}$ represents the PBH parameters, $\phi_{\gamma}$ denotes the expected integral $\gamma$-flux model from PBHs, and $\phi_{\gamma}^{\rm data}$ represents the experimental measurements with corresponding uncertainties $\sigma_{\rm data}$. The $\chi^{2}$ value is calculated by summing over all energy bins for the discrepancy between the data and the model. We determined the 95% confidence level (C.L.) bound on $f_{\rm PBH}$ for each PBH mass within the specified range. The permissible $f_{\rm PBH}$ values associated with the flux originating from PBHs should not exceed any errors of the measurement data shown in Fig.3, which translates to $\chi^{2}-\chi^{2}_{\rm min}\leq\chi^{2}_{0.05}(N-1)\simeq 3.84$, where $N-1=1$ represents the number of degrees of freedom.

Secondly, we aim to establish a more realistic estimator for deriving the current constraint on PBHs by incorporating the contributions from the DPL fitting term $\phi^{\rm AGN}_{\gamma}$. The $\chi^{2}$ expression is formulated as follows:

$$\chi^{2}=\sum\left[\frac{\phi^{\rm{data}}_{\gamma}-\phi^{\rm PBH}_{\gamma}(\theta)-\phi^{\rm AGN}_{\gamma}}{\sigma^{\rm{data}}}\right]^{2}.$$

(4.3)

It is evident that the model incorporating the DPL background enforces stricter constraints in all scenarios compared to those without it. This suggests that a more profound comprehension of the astrophysical background sources could lead to a substantial enhancement in setting upper limits on the presence of PBHs. The capability to exclude a significant portion of the dataset with the DPL model results in only a small fraction of the flux being attributed to PBHs, thereby directly boosting the constraint ability. However, it is important to note that the DPL form is purely phenomenological, underscoring the necessity to gain a better understanding of the astrophysical background in future analyses.

It appears that by utilizing Equ.(2.13), we can ascertain the value of $P$, considering $f_{\rm Ps}$ as a variable parameter. Previous works [28, 25] have neglected the contribution from IA due to its minor impact, leading to a slight discrepancy in their expression of $f_{\rm Ps}$. Therefore, in this analysis, we set $f_{\rm Ps}$ as either 0 or 1 and leverage the fractions from Equ.(2.13) to weigh each emission source. The constraints on $f_{\rm Ps}$ are depicted in Fig.4, where $f_{\rm Ps}=1$ corresponds to scenarios with no spin PBHs in the presence of a DPL background (solid blue line) and only PBH model (dashed blue line), while $f_{\rm Ps}=0$ is applied to both models (with DPL is solid black line and without DPL is dashed black line).

It is observed that when the environmental temperature exceeds $8000$ K, positrons are more likely to undergo direct annihilation with both free and bound electrons rather than forming Ps. Therefore, the scenario with $f_{\rm Ps}=1$ represents a more realistic case under Milky Way conditions [22]. In Fig.4, the outcomes for $f_{\rm Ps}=0$ exhibit stronger constraints, indicating a potential underestimation of the contribution from the EG sources in the existing literature.

It is worth noting that, as discussed in ref.[28], the error bars associated with the SMM measurement of the isotropic X-ray flux are considered to be unreliable. Consequently, we also present an analysis based solely on SMM data. The remaining dataset can be categorized into three scenarios: utilizing only SMM data, utilizing all data except SMM, and utilizing all data including SMM. The $\chi^{2}$ calculation follows the same formula as Equ.(4.3), with varying selections of data for $\phi_{\gamma}^{\rm data}$ and their corresponding uncertainties $\sigma_{\gamma}^{\rm data}$. The results are compared in Fig.5. It is observed that SMM data in isolation can effectively constrain a significant portion of the considered PBH masses. Consequently, the outcomes from only using SMM data and employing all data with SMM align closely across most of the range. As depicted in Fig.3, SMM data holds particular relevance for higher energies surpassing approximately $200$ keV. However, for more massive PBHs, the spectral characteristics tend to soften, rendering the data without SMM more crucial. These findings underscore the importance of utilizing data within appropriate energy ranges with minimal error bars to yield robust constraints.

In Fig.5, it is evident that for PBHs with masses exceeding $1\times 10^{18}$ g, not only does the energy peak decrease and fall below $100$ keV, but it also significantly weakens. This indicates that high-energy data sources like SMM may not effectively constrain this mass range, thereby posing a challenge in directly achieving this objective using current keV data.

The concept of a rotated PBH potentially leading to the generation of additional particles during Hawking evaporation is intriguing. In principle, it is assumed that PBHs do not possess spins at the time of their formation. Following their formation, PBHs are expected to remain stable, with only a small fraction, roughly one in a million PBHs, being formed with spins $\gtrsim 0.8$ [13]. The existence and implications of rotated PBHs remain a topic of debate, warranting further exploration to better understand their implications. To shed light on this, a comparable analysis is presented within the context of the monochromatic mass function scenario discussed earlier. By adjusting the intrinsic spin towards the maximum value of $a=0.9999$, which corresponds to the maximum spin value in BlackHawk, the results pertaining to rotating PBHs exhibit significant enhancements. This is attributed to the higher number density resulting from the evaporation process in this particular scenario.

It is fascinating to note the enhancements in constraints depicted in Fig.6 when considering the influence of PBH rotation. These results indicate an expansion in the covered parameter space, leading to improved constraints. Notably, even in the most conservative scenarios where there is no spin and no background modeling, we observe better limits, especially in the $10^{17}$ g range. This improvement can be attributed to the comprehensive consideration of all potential emissions in our calculations, resulting in a slight increase in the total flux and subsequently refining the bounds on PBH abundance.