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tokens: we note that the results for -ray spectra present a similar feature they are too soft as compared to observations. We compute the -ray spectral index in the energy range 0.87GeV at different Galactic coordinates ( l , b ) and, for each position in the sky, we estimated the difference between the spectral slope of simulated and observed -ray spectra (two examples of observed and simulated -ray spectra can be found in Fig. 7 , dashed and solid lines, respectively). For the observed spectra, the slope has been computed in the same energy range, from Selig etal. ( 2015 ). The results are shown in Fig. 2 , right-hand panel. The open histogram shows the distribution of the difference between the simulated and observed slopes. We limit this investigation to the region of the map at small latitudes, 5 b 5. The average value is around 0.08, indicating that the simulated -ray spectra are on average softer than what inferred from observations. To solve the inconsistency between observations and both the simulated CR spectrum at Earth and the -ray spectra, we consider a model with a harder injection spectrum . The shaded histograms in Fig. 2 show that in this case predictions of the CR slope around the location of the Earth (left-hand panel) and of the map of -ray radiation (right-hand panel) are in better agreement with observational constraints. For this reason, from now on we present and discuss in detail only the results obtained with 2.25. Fig. 2 (left-hand panel) also shows that the predicted CR spectrum at the location of the Earth (thick vertical line) is not representative of the average CR spectral slope (peak of the distribution). While in these examples the difference between the local and average spectral slopes is only 0.05, larger differences can be obtained by slightly changing the location of the arm as compared to the Earth. In this sense, the model is not predictive, and there is degeneracy among the simulation parameters. Although a complete study of the parameter space is beyond the scope of this paper, we outline that the proposed model gives a natural explanation for the increasing evidence that the local CR spectrum might not be representative of the average Galactic one (Ackermann etal. 2012 ; Neronov Malyshev 2015 ). Figure 2. Left-hand panel: the distribution of the simulated CR spectral index, estimated in the range 103000GeV, in the plane Z 0. The open histogram shows the results assuming an injection spectral index 2.3, while the shaded histogram is obtained for a harder injection index, 2.25. In both cases, the diffusion coefficient is proportional to E 0.4 . The vertical blue and grey lines depict the observed spectrum on the Earth, respectively. Right-hand panel: the distribution of the difference between the -ray spectral index in the plane of the sky obtained from simulations and from observations. A small latitude region has been selected: 5 b 5. The two different histograms respectively refer to the two different