Extra dimensions at the CERN LHC via mini-black holes: Effective Kerr-Newman brane-world effects

We solve Einstein equations on the brane to derive the exact form of the brane-world-corrected perturbations in Kerr-Newman singularities, using Randall-Sundrum and Arkani-Hamed-Dimopoulos-Dvali (ADD) models. It is a consequence of such models that Kerr-Newman mini-black holes can be produced in LHC. We use this approach to derive a normalized correction for the Schwarzschild Myers-Perry radius of a static $(4+n)$-dimensional mini-black hole, using more realistic approaches arising from Kerr-Newman mini-black hole analysis. Besides, we prove that there are four Kerr-Newman black hole horizons in the brane-world scenario we use, although only the outer horizon is relevant in the physical measurable processes. Parton cross sections in LHC and Hawking temperature are also investigated as functions of Planck mass (in the LHC range 1–10 TeV), mini-black hole mass, and the number of large extra dimensions in brane-world large extra-dimensional scenarios. In this case a more realistic brane-effect-corrected formalism can achieve more precisely the effective extra-dimensional Planck mass and the number of large extra dimensions—in the Arkani-Hamed-Dimopoulos-Dvali model—or the size of the warped extra dimension—in Randall-Sundrum formalism.

Figure 1

Graphic of the brane-effect-corrected normalized inner horizon $K(M,a,Q)\times a$ for different values of $Q^{*}$. For the very short dashed line: $Q^{*}=-0.1$; for the short dashed line: $Q^{*}=-0.5$; for the full gray line: $Q^{*}=-1$; for the full black line: $Q^{*}=-1.5$; and for the long dashed line: $Q^{*}=-2$.

Figure 2

Graphic of the brane-effect-corrected normalized horizon $K(M,a,Q)\times a$ for different values of $Q^{*}$. These are solutions for an intermediate horizon. For the very short dashed line: $Q^{*}=-0.1$; for the short dashed line: $Q^{*}=-0.5$; for the full gray line: $Q^{*}=-1$; for the full black line: $Q^{*}=-1.5$; and for the long dashed line: $Q^{*}=-2$. In this case, the spin of the mini-BH is constrained by the electric and tidal charge. The maximum $a$ obtained for each graphic is: $a=0.32806$ for $Q^{*}=-2$, $a=0.52572$ for $Q^{*}=-1.5$, $a=0.76550$ for $Q^{*}=-1$, $a=0.93842$ for $Q^{*}=-0.5$, and $a=0.99753$ for $Q^{*}=-0.1$ (which practically coincides with the $Q^{*}=-0.5$ case). In these respective values for $a$, the derivative of $K(M,a,Q)$ related to $a$ tends to infinity.

Figure 3

Graphic of the brane-effect-corrected Kerr-Newman outer horizon in the form $c^2{R}_{\mathrm{K}\mathrm{\text{−}}\mathrm{N}}/GM\times a$ for different values of $Q^{*}$. For the very short dashed line: $Q^{*}=-0.1$; for the short dashed line: $Q^{*}=-0.5$; for the full gray line: $Q^{*}=-1$; for the full black line: $Q^{*}=-1.5$; and for the long dashed line: $Q^{*}=-2$. In this case, the spin of the BH is constrained by the electric and tidal charge. The maximum $a$ obtained for each graphic is: $a=0.32791$ for $Q^{*}=-2$, $a=0.52766$ for $Q^{*}=-1.5$, $a=0.76585$ for $Q^{*}=-1$, $a=0.95826$ for $Q^{*}=-0.5$, and $a=0.99866$ for $Q^{*}=-0.1$.

Figure 4

Brane-effect-corrected Kerr-Newman cross section $\sigma$ for the inner horizon (full line) and for the outer horizon (dashed line) for various angular momenta $a$. Significant contributions by the inner horizon just happen when $a\to 1$. Thus, the effective contribution for the total cross section is only given by the outer horizon. This graphic presents an example for the $n=3$ extra-dimensional ADD model, $M_{\mathrm{P}}=1\mathrm{TeV}$, $M_{\mathrm{BH}}=5\mathrm{TeV}$, $Q^{*}=-0.1M_{\mathrm{BH}}$.

Figure 5

Graphics of the brane-effect-corrected Kerr-Newman horizon $R_{\mathrm{K}\mathrm{\text{−}}\mathrm{N}}$ of the produced mini-BH and its associated cross section, for various momenta $a$ versus BH masses, and its associated parton cross section for the Randall-Sundrum brane-world model, where there is one warped extra dimension. Here we present an example for $M_{\mathrm{P}}=4\mathrm{TeV}$, and $Q^{*}=-0.1M_{\mathrm{BH}}$, with the short dashed line for $a=0.1$, the dashed line for $a=0.3$, the long dashed line for $a=0.6$, the gray full line for $a=0.9$, and the full black line for $a=0.997$.

Figure 6

Graphics of the brane-effect-corrected Kerr-Newman radius and its associated parton cross section for $n=2$ ADD model for some BH masses, with $M_{\mathrm{P}}=1\mathrm{TeV}$, and for various angular momenta: short dashed line for $a=0.1$, dashed line for $a=0.3$, long dashed line for $a=0.6$, gray full line for $a=0.9$, and full black line for $a=0.997$.

Figure 7

Graphics of the brane-effect-corrected Kerr-Newman outer horizon and its associated parton cross section in the $n=4$ ADD model for some mini-BH masses, with $M_{\mathrm{P}}=1\mathrm{TeV}$, and for various angular momenta: short dashed line for $a=0.1$, dashed line for $a=0.3$, long dashed line for $a=0.6$, gray full line for $a=0.9$, and full black line for $a=0.997$.

Figure 8

Graphic of the brane-effect-corrected Kerr-Newman outer horizon for the extra-dimensional $n=6$ ADD model for some BH masses, with $M_{\mathrm{P}}=1\mathrm{TeV}$, and for various angular momenta: short dashed line for $a=0.1$, dashed line for $a=0.3$, long dashed line for $a=0.6$, gray full line for $a=0.9$, and full black line for $a=0.997$.

Figure 9

Cross sections for $n=6$ extra dimensions in ADD model, short dashed line for $a=0.1$, dashed line for $a=0.3$, long dashed line for $a=0.6$, gray full line for $a=0.9$, and full black line for $a=0.997$.

Figure 10

Graphic of parton cross section in picobarn for various values of Planck mass $M_{\mathrm{P}}$ and BH mass $M_{\mathrm{BH}}$ for the Schwarzschild Myers-Perry model. The mini-BH formed by the scattering has $a=0.5$ and is charged ($Q^{*}=-1M_{\mathrm{BH}}$), for the Randall-Sundrum brane-world model.

Figure 11

Graphic of parton cross section in picobarn for various values of Planck mass $M_{\mathrm{P}}$ and BH mass $M_{\mathrm{BH}}$. The mini-BH formed by the scattering has $a=0.5$ and is charged with $Q^{*}=-1M_{\mathrm{BH}}$, for the Randall-Sundrum brane-world model.

Figure 12

Graphics of parton cross section in picobarn for various values of Planck mass $M_{\mathrm{P}}$ and BH mass $M_{\mathrm{BH}}$. The mini-BH formed by the scattering has $a=0.5$ and is charged with $Q^{*}=-1M_{\mathrm{BH}}$, in respectively $n=2$ and $n=6$ extra dimensions in ADD model.

Figure 13

Graphic of parton cross section in picobarn for various values of spinning parameter $a$ and BH mass $M_{\mathrm{BH}}$. The mini-BH formed by the scattering is charged with $Q^{*}=-1M_{\mathrm{BH}}$. In this case we consider the Randall-Sundrum brane-world model.

Figure 14

Graphic of parton cross section in picobarn for various values of $a$ and the BH mass $M_{\mathrm{BH}}$. Note the interesting effect caused by the spinning parameter $a$ in the cross section. Here we consider the $n=2$ extra-dimensional ADD model, the mini-BH is electrically and tidally charged by the bulk with $Q^{*}=-0.1M_{\mathrm{BH}}$.

Figure 15

Graphic of parton cross section in picobarn for various values of $a$ and the BH mass $M_{\mathrm{BH}}$. Note the interesting effect caused by the spinning parameter $a$ in the cross section. In this case, i.e., the $n=6$ extra-dimensional ADD model, the bending is more intense. The mini-BH has $Q^{*}=-0.1M_{\mathrm{BH}}$.

Figure 16

Parton cross sections in picobarn for various values of $Q^{*}$ and $a$, explicitly showing the influence in the scattering if one considers a realistic Kerr-Newman mini-BH. The charge constrains the spin.

Figure 17

Range of Hawking temperature, in TeV, produced in LHC by a charged spinning BH evaporation versus BH mass interval. The range covers some values of charge for a fixed angular momentum: full line: $Q^{*}=-2M$; dotted line: $Q^{*}=-1.5M$; dashed-dotted line : $Q^{*}=-M$; dashed line: $Q^{*}=-0.5M$, all for spinning parameter $a=0.5$. This means the BH charge is an important parameter in the determination of exact BH decay. Here, the Planck mass is 4 TeV and $n=6$ for the Randall-Sundrum brane-world model.

Figure 18

Range of Hawking radiation, in TeV, produced in LHC by a charged spinning BH evaporation versus BH spinning parameter $a$, for a fixed BH mass (1 TeV) for the Randall-Sundrum brane-world model. Here we present an example for $M_{\mathrm{P}}=4\mathrm{TeV}$. For the very short dashed line: $Q^{*}=-0.1$; for the short dashed line: $Q^{*}=-0.5$; for the full gray line: $Q^{*}=-1$; for the full black line: $Q^{*}=-1.5$; and for the long dashed line: $Q^{*}=-2$. In this case, the spinning of the BH is constrained by the electric and tidal charge.

Figure 19

Range of Hawking radiation, in TeV, produced in LHC by a charged spinning BH evaporation versus BH spinning parameter $a$, for a fixed BH mass (1 TeV) for the Randall-Sundrum brane-world model, where there is one extra warped dimension. Here we present an example for Planck mass $M_{\mathrm{P}}=1\mathrm{TeV}$. For the very short dashed line: $Q^{*}=-0.1$; for the short dashed line: $Q^{*}=-0.5$; for the full gray line: $Q^{*}=-1$; for the full black line: $Q^{*}=-1.5$; and for the long dashed line: $Q^{*}=-2$. In this case, the spinning of the BH is constrained by the electric and tidal charge.