Production of Spinning Black Holes at colliders

When the Planck scale is as low as TeV scale, there will be chances to produce Black holes (BH’s) at future colliders. Generally, BH’s produced via pariticle collisions could have non-zero angular momentum. We estimate the production cross section of spinning and non-spinning BH’s for future colliders. Although the production cross section for the rotating BH is much suppressed by angular momentum dependent factor, the total cross section could be ∼ 2 − 3 times enhanced for the case of δ = 4 − 6. PACS number(s): 11.30.Pb, 11.30.Er [email protected] [email protected] Introduction.– Some of the most intriguing phenomena of the TeV-scale gravity [1], [2] is the production of Black holes(BH’s) at the future colliders like Large Hadronic Collider (LHC) of the CERN [3], [4] and at the atmosphere by cosmic rays [5]. The mass of the BH, MBH , produced at LHC could be larger than the D-dimensional gravity scale, MD, which is related with four dimensional Planck scale, MP , as M δ+2 D R δ = M P , where D = 4 + δ and R is the size of extra dimensions. When MD ∼ O(1) TeV, the size of the extra dimensions is much larger than the Schwarzschild radius, RS, of D-dimensional BH of mass MBH ∼ MD: RBH ∼ 1 MD ( MBH MD ) << R. (1) Since the Compton wavelength of the BH(LBH) is smaller than RBH when MD < MBH, the following relation is assumed to be valid for our interests: LBH < RBH << R. (2) In Ref. [3] and [4], the production cross section of BH via particle collisions were estimated by semi-classical arguments. Semi-classical approximation is strictly valid when LBH << RBH and as MBH approaches MD unknown stringy effects might be important. As a first try to estimate, they just considered the case for the non-spinning, non-charged BH production. It is important to note that such tiny black holes could be generated only provided their electric charge is zero, otherwise they would be naked singularities [6]. In the contrary, any object which is produced via fusion process of colliding particles could have angular momentum or spin. Non-spinning case could occur only when the orbital and internal angular momenta are exactly cancelled. We may imagine the case of exactly head-on collision of initial particles for the production of non-spinning BH. The main objective of this letter is re-estimation of the the spinning and non-spinning BH production cross section for the improved estimation. We will point out that even though the spinning BH production cross section is suppressed, the total cross section for the spinning and non-spinning BHs could be much enhanced. That means we may have bigger chance to see the BH at our future lab experiments. 2 Non-spinning BH production.– We first review the production process of non-spinning BHs. Classically speaking, BH can be formed when and only when the energy M is compacted into a region whose circumference in every direction is less than 2πRBH. RBH is Schwarzschild radius of BH which is determined by mass (M), angular momentum (J) and possiblly (electric) charge (Q). From the above classical argument, we deduce the simple picture of BH production at particle collision as following. Consider two partons with the center of mass (CM) energy √ s = MBH moving in opposite directions. When the impact parameter (b) is less than the Schwarzschid radius, the particles effectively inject the critical mass of BH in the radius. Then a BH is formed and almost stationay in the CM frame. From this semi-classical arguments, we could find the geometrical approximation for the cross section for producing a BH of mass MBH as σ(MBH) ≈ πR BH. (3) The Schwarzschild radius for the non-spinning BH is given by [9] RBH(J = 0) = 1 √ πMD ( MBH MD 8Γ( δ+3 2 ) δ + 2 ) 1 δ+1 , (4) so the larger MBH provides the larger RS. From the above expression for the parton level cross sections, we can obtain the total cross section by convoluting parton distribution function (PDF) for the initial partons: dσ(pp → BH+X) dMBH = 2MBH s ∑ a,b ∫ 1 M BH /s fa(xa)fb( M BH xas )σ̂(ab → BH)|s=M2 BH . (5) There are several uncertainties in this semi-classical estimation. Firstly, there could be quantum probability that BH is produced even when b > RS. However, as Voloshin argued in Ref. [7], this probability might be exponentially suppressed by Euclidean BH action( see also Ref. [8]). Secondly, as MBH approaches MP (or MD for D-dimensional BH), BH becomes stringy. Unfortunately we do not have enough information of this unknown stringy correction, yet. In this study, we simply ignore such uncertanties. 3 Spinning BH production. – Now, let us consider the case with the spinning BH. In the Ref. [9], the rotating BH solution in 4+ δ dimensional space-time is obtained. The rotating BH generally has inner and outer horizons. They are described not only by MBH but also by the angular momentum of the BH parametrized by dimensionless paramter a like RBH(J)= 1 √ πMD ( MBH MD 1 1 + a2 8Γ( δ+3 2 ) δ + 2 ) 1 δ+1 = ( 1 1 + a2 ) 1 RS(J = 0), (6) where a ≡ (δ+2)J 2MBHRBH and J is the angular momentum of BH. For the produced BH from the particle collision, the angular momentum of the BH is allowed upto (MBHRBH) at the classical limit. The range of the a parameter is given as 0 ≤ a ≤ (δ + 2) 2 . (7) As we will show below, the upper bound for the angular momentum does not much affect the estimation of total cross section. To estimate the BH production cross section for rotating BH, we assume that the semiclassical reasoning for the non-rotating BH might be still valid for rotating BH. Semi-classical reasoning suggests that, if the impact parameter is less than the size of BH given in Eq.(6), a BH with the mass MBH and angular momentum J forms. The total cross section can be estimated and is of order