Running coupling with minimal length

In models with large additional dimensions, the GUT scale can be lowered to values accessible by future colliders. Because of modification of the loop corrections from particles propagating into the extra dimensions, the logarithmic running of the couplings of the Standard Model is turned into a power law. These loop corrections are divergent and the standard way to achieve finiteness is the introduction of a cutoff. The question remains, whether the results are reliable as they depend on an unphysical parameter. In this paper, we show that this running of the coupling can be calculated within a model including the existence of a minimal length scale. The minimal length acts as a natural regulator and allows us to confirm cutoff computations.

Figure 1

The result for the running of the gauge-couplings for $M_{\mathrm{f}}=100$, 500, 1000, 5000  TeV and fixed $d=3$.

Figure 2

The result for the running of the gauge-couplings for $d=1$, 2, 3, four and fixed $M_{\mathrm{f}}=100\mathrm{T}\mathrm{e}\mathrm{V}$.

Figure 3

$d=2$, 3, 4, 5, $M_{\mathrm{f}}=100\mathrm{T}\mathrm{e}\mathrm{V}$, $M_{\mathrm{S}\mathrm{U}\mathrm{S}\mathrm{Y}}=1\mathrm{T}\mathrm{e}\mathrm{V}$. The dotted lines show the result with a hard cutoff, the solid lines the result from the minimal length.

Figure 4

The values of the fitting parameter ${\chi }_d$. ${\chi }_2=2,1\pm 0.1$, ${\chi }_3=5.2\pm 0.2$, ${\chi }_4=10.1\pm 0.4$, ${\chi }_5=21.7\pm 0.8$, and ${\chi }_6=52\pm 1.9$.

Figure 5

The linear dotted line shows the case of no modification $k=p$.