We numerically investigate the formation of D-dimensional black holes in high-energy particle collision with the impact parameter and evaluate the total cross section of the black hole production. We find that the formation of an apparent horizon occurs when the distance between the colliding particles is less than 1.5 times the effective gravitational radius of each particles. Our numerical result indicates that although both the one-dimensional hoop and the (D-3)-dimensional volume corresponding to the typical scale of the system give a fairly good condition for the horizon formation in the higher-dimensional gravity, the (D-3)-dimensional volume provide a better condition to judge the existence of the horizon. I. INTRODUCTION The brane world scenario is paid much attention in the context of the unified theory of elementary particles. This scenario regards our space as the 3- brane with large extra dimensions on which gauge particles and interactions are confined. I n this scenario, the Planck energy can be at the O(TeV) scale [1]. One of the consequences of lowering the Planck scale is that the properties of a small black hole whose radius is smaller than the size of the extra dimensions are substantially altered; the black hole s is well-described as a D-dimensional black hole centered on the brane, but extending out in to the ( D − 4)- dimensional extra space and the radius of such a black hole is ∼ 1032 times larger than that of the usual black hole with the same mass. Hence it becomes much ea sier to produce black holes using a future planned accelerator such as the CERN Large Ha dron Collider and this possibility has been discussed by many authors [2]. The possibility of producing black holes by the collision of particles can b e estimated by using the hoop conjecture [3], which states that an apparent ho rizon forms when and only when the mass M of the system gets compacted into a region whose circumference C satisfies HD ≡ C 2πrh(M) ≲ 1, (1) where rh(M) is the Schwarzschild horizon radius for the mass M. If we assume that the inequality (1) give the condition for black hole formation in the higher- dimensional gravity, we can evaluate the impact parameter b of the colliding particles which leads to black hole production. The circumference that surrounds two particles at the instant of collision is C ∼ 2b. By setting 2 b/2πrh(2µ) ∼ 1 where 2 µ is the center of mass energy of the system, the maximal impact parameter bmax that leads to the black hole formation becomes bmax ∼ rh(2µ). By introducing a numerical factor F (D) close to unity, the total cross section for black hole production is written as the following form: σb.h.production = F (D)πr 2 h(2µ). (2) Obtaining F (D) is necessary to improve experimental predictions of the black hole produc- tion in collider physics and observations of ultra-high energy cosmic r ays. In our previous paper [4], we investigated the black hole formation in h igh-energy head-on collisions of particles in the D-dimensional gravity and found that HD becomes a parameter 2 geometry of the extra dimensions. The metric with a high-energy po int particle is obtained by infinitely boosting a Schwarzschild black hole with fixed total energ y µ. The result- ing system becomes a massless point particle accompanied by a plane- fronted gravitational shock wave which is the Lorentz-contracted longitudinal gravitat ional field of the particle. Combining two shock waves, we can set up the collision of high-energy two particles moving in ±z direction. This system was originally developed by D’Eath and Payne [7] and recently analyzed in [4, 6]. The metric of this system outside the future light co ne of the colliding shocks is given as ds2 = −dudv + ( H (+) ik H (+) jk + H (−) ik H (−) jk − δij ) dxidxj, H (+) ij = δij + u 2 Θ( u)∇i∇jΦ( x − x+), (4) H (−) ij = δij + v 2 Θ( v)∇i∇jΦ( x − x−), where u = t − z, v = t + z, Θ is the Heviside step function and x ≡ (xi) is the point in flat ( D − 2)-space ( x1, · · · , xD−2) that is transverse to the direction of particle motion. The function Φ( x) depends only on r ≡ |x| = √ xixi and takes the form Φ( x) = −8G4µ log r, for D = 4 (5) Φ( x) = 16πµGD Ω D−3(D − 4) 1 rD−4 , for D > 4 (6) where Ω D−3 is the volume of a unit ( D − 3)-sphere and x± denote the points of two particles in ( D − 2)-dimensional space and we take x± = ( ±b/2, 4 5 6 7 8 9 10 11 D 0.2 0.4 0.6 0.8 1 1.2 1.4 bmax rh(2 )µ FIG. 3: The value of bmax (crosses) as a function of the spacetime dimension D. The dotted line rh(2µ) which comes from the hoop conjecture. Although the ratio bmax/rh(2µ) takes the value around unity, the hoop conjecture does not explain the incre ase of bmax/rh(2µ) with D. The solid line bmax/rh(2µ) = 1 .5rh(µ)/rh(2µ) ∼ 2−1/ (D−3) gives a good fit of bmax. can be written as HD ≡ [ VD−3 Ω D−3rD−3 h (M) ]1/ (D−3) ≲ 1. (18) This condition reduces to the hoop conjecture (1) for D = 4 case. As the characteristic length of the system is b in x1-direction and rh(µ) in other directions, the characteristic (D − 3)-volume becomes VD−3 ∼ b rD−4 h (µ) and HD becomes ∼ [ brD−4 h (µ)/rD−3 h (2µ) ]1/ (D−3) . By setting HD ∼ 1, the maximal impact parameter is given by bmax rh(2µ) ∼ ( rh(2µ) rh(µ) ) D−4 ∝ 2−1/ (D−3). (19) Although this is rough order estimation, the calculation indicates tha t the volume conjecture provides a better condition than the hoop conjecture. To test the validity of the two conjectures further, we consider h ow much energy is trapped by the black hole. This is related to the definition of the mass in the system. In our previous paper [8], we discussed the existence of the gravitatio nal wave energy that does not contribute to the horizon formation in the system with mot ion and found that the hoop conjecture with Hawking’s quasilocal mass provides a good con dition for the horizon formation in four-dimensional gravity. In the present system, we calculate the quantity MA.H. ≡ (D − 2)Ω D−2 16πGD ( AD−2 Ω D−2 ) (D−3)/ (D−2) , (20) 9