# On a Possible Origin of the Gamma-ray Excess around the Galactic Center
## 2 Modeling the CR Interaction with Molecular Clouds  
The authors [PERSON] et al. [16] found that the GC excess is correlated with the distribution of MCs. They tested two hypothesis of the GeV-excess: the real excess hypothesis assuming the excess is caused by DM annihilation, and the seeming excess hypothesis assuming that the \"GeV-excess\" is related to a hypothetical depletion of gamma rays from MCs below \(\approx\)2 GeV, as it is directly observed in the Central Molecular Zone (CMZ). The authors [16] did not model in details the origin of the emission depletion below 2 GeV, however stated that it is most likely caused by a magnetic cutoff of cosmic rays in MCs. We have developed the latter idea further. Generally speaking (see, e.g., [15; 16]) the hadronic gamma-ray emission is presented by the two components: \(\pi^{0}\) gamma-ray emission from the diffuse gas around the GC and \(\pi^{0}\) gamma-ray emission from MCs. The investigation by [16] showed that the derived spatial distribution of the emission from MCs correlated nicely with the molecular hydrogen (traced by CO) distribution. Moreover, they found that the gamma-ray emission in the direction on the GC is fully determined by the MC component, where a huge mass of \(5\times 10^{7}M_{\odot}\) of molecular hydrogen is concentrated, while other components have a minor effect there. [PERSON] et al. [16] speculated that this 2 GeV spectral bump is formed due to depletion of the density of CRs with certain energies inside the clouds. [PERSON] et al. [17] and [PERSON] et al. [18] abandoned the phenomenological model of CR propagation and conducted more realistic non-linear simulations of gamma-ray emission from MCs, which naturally produces the \"GeV excess\" through the \"MHD-barrier\", which reflects CRs with energies \(\lesssim\)10 GeV and does not allow them to penetrate into the clouds. We assumed that the apparent GeV excess does not have a real source but rather can be explained through an unaccounted modification of MC spectra at certain energies and overall renormalization of the spectra of all emission components. MC spectra gets depleted below few GeV due to a self-excited MHD turbulence by CRs, when the density of CRs inside the cloud is depleted by magnetic fluctuations around the MCs. A flux of CRs propagating through the diffuse ionized gas can excite self-generated magnetic fluctuations (MHD waves). As for the required renormalization of the emission components, it is not large, taking into account that the overall intensity of the GC excess does not exceed \(\approx\)10% of the total gamma-ray intensity within 10\({}^{\circ}\) from GC (see, e.g., [15]). Such renormalization by \(\approx\)10% could be made relatively easily accounting for the systematic uncertainties related to the gas emissivity. Let us discuss now in detail our model of the modification of MC spectra. The following system of nonlinear kinetic equations describes processes of CR penetration into MCs (see details of the equations in [17]) in one-dimensional approximation. The spectrum of CRs \(f(z,E,t)\) and the spectrum of magnetic fluctuations \(W\) are described by  
\[\frac{\partial f}{\partial t} =\frac{\partial}{\partial z}\left(D\frac{\partial f}{\partial z}-v _{A}f\right)+\frac{\partial}{\partial p}\left(\frac{dp}{dt}f\right) \tag{1}\] \[\frac{\partial W}{\partial t} =\frac{\partial}{\partial k}\left(\frac{W}{T_{\rm nl}}\right)- \frac{\partial}{\partial z}(v_{a}W)+\Gamma W-\
u W\,. \tag{2}\]  
where \(p\) is the particle momentum, the spatial diffusion coefficient  
\[D\simeq\frac{vB^{2}}{6\pi^{2}k^{2}W}\,, \tag{3}\]  
\(\Gamma\) is the frequency of CR collisions with the background gas, \(v_{A}\) is the Alfvenic velocity, the rate of stream instability excited by CR is  
\[\Gamma(k,z)\simeq\pi^{2}\frac{e^{2}v_{A}}{m_{p}c^{2}\Omega}pvD\frac{\partial f }{\partial z}\,, \tag{4}\]  
\((\partial/\partial k)(W/T_{\rm nl}(k))\) describes the nonlinear cascade of magnetic fluctuations and \(\
u\) describes waves damping due to ion-neutral friction.