# On a Possible Origin of the Gamma-ray Excess around the Galactic Center
## 2 Modeling the CR Interaction with Molecular Clouds
The flux of CRs \(S(E)=v_{A}f-D\frac{\partial f}{\partial z}\) propagating through the envelope and entering the dense interior of the cloud is derived as (see Equations (1) and (2))  
\[S(E)=\frac{v_{A}f_{IS}(E)}{1-\delta e^{-\eta_{0}(E)}} \tag{5}\]  
where  
\[\eta_{0}(E)=v_{A}\int\limits_{0}^{H}\frac{dz}{D(E,z)}\,, \tag{6}\]  
\(H\) is the size of the envelope, and \(f_{IS}\) is the CR spectrum in the the interstellar medium. Coordinate \(z=0\) corresponds to the boundary between ISM and the envelope, while \(z=H\) corresponds to the conditional boundary between the envelope and the dense interior. The parameter \(\delta<1\) in Equation (5) describes opacity of the cloud and depends on its column density \(N_{H}\) (see [18] for details). The expression for \(\eta_{0}\) can be estimated from a balance of the growth rate between MHD wave excitation and damping rates due to ion-neutral collisions, \(\Gamma=\
u\), which leads to the following equation for \(\eta_{0}\):  
\[1-\delta e^{-\eta_{0}(E)}\propto\delta vpf_{IS}(E)\,. \tag{7}\]  
Since \(\delta<1\) and \(f_{IS}(E)\) decreases with energy quite steeply, there is a certain value of \(E=E_{ex}\), where \(\eta_{0}(E_{ex})=0\). Furthermore, at higher energies the balance can no longer be maintained. This threshold energy, \(E_{ex}\), is a boundary between range of free penetration into the cloud core and the range of self modulated CRs. At sufficiently high energies, \(E>E_{ex}\), the CR flux is not affected by MHD-turbulence and converges to the regime of the free-streaming (since \(\eta_{0}=0\) and \(D\rightarrow\infty\)). Within the range of wave excitation by CRs, \(E<E_{ex}\), the spectrum of MHD fluctuations is provided by excitation-damping balance described by Equation (7). As a result, below \(\sim\)\(E_{ex}\), a universal spectrum of CRs \(f_{p}(E)\propto E^{-1}\) is formed inside the dense regionsof MCs. This spectrum is independent of external parameters outside the cloud. Above \(\sim\)\(E_{ex}\), the spectrum in the core is the same as in the intercloud medium:  
\[f_{p}(E)\propto\left\{ \begin{array}{ll}E^{-1}\,,&\text{if $E\lesssim E_{ex}$}\,,\\ E^{-\gamma_{p}}\,,&\text{if $E\gtrsim E_{ex}$}\,,\end{array} \right. \tag{8}\]  
where \(\gamma_{p}\) is the spectral index of the CRs spectrum in the intercloud medium of the CMZ. Self-excited MHD turbulence may also impact the gamma-ray emission produced by relativistic electrons, since relativistic protons and electrons interact with turbulence in the same way. This is also very relevant for GeV excess interpretations, since the bremsstrahlung and inverse Compton emission from relativistic electrons may comprise a significant part of the total emission intensity in the GC region (see, e.g., [15; 19]). Hence any minor modifications of those emission components may have a big impact on the GC excess amplitude and spectrum. Even if the intensity of bremsstrahlung from electrons is lower as compared to the emission from proton-proton collisions, the energy break in the spectrum of bremsstrahlung emission is expected to be located at higher energies. Therefore visible effect of CRs self-modulation on bremsstrahlung emission should appear at much lower column densities. To illustrate the potential influence of self-generated MHD turbulence on the observed spectra of gamma-ray emission we showed in Figure 1 spectra of proton-proton emission \(\left(\frac{dN}{dE}\right)_{pp}\) and electron bremsstrahlung emission \(\left(\frac{dN}{dE}\right)_{br}\) from the particles inside the cloud. For comparison, we displayed the experimental data points of the GeV excess taken from [PERSON] et al. [15]. We used the following expressions to calculate the gamma-ray emission spectra:  
\[\left(\frac{dN}{dE_{\gamma}}\right)_{i}=\frac{M_{i}c}{4\pi R_{GC}^{2}m_{p}}\int dE \;f_{j}(E)\left(\frac{d\sigma(E,E_{\gamma})}{dE_{\gamma}}\right)_{i}, \tag{9}\]  
where \(i\) stands for either \(pp\) or \(br\) and \(j\) is either \(p\) in the case of proton-proton interactions or \(e\) in the case of electron bremsstrahlung.