BlackMax: A black-hole event generator with rotation, recoil, split branes and brane tension
De-Chang Dai, Glenn Starkman, Dejan Stojkovic, Cigdem Issever, Eram Rizvi, Jeff Tseng
aa r X i v : . [ h e p - ph ] A p r BlackMax: A black-hole event generator with rotation, recoil, split branes and branetension.
De-Chang Dai , Glenn Starkman , Dejan Stojkovic , Cigdem Issever , Eram Rizvi , Jeff Tseng Case Western Reserve University, Cleveland OH 44106-7079, USA Department of Physics, SUNY at Buffalo, Buffalo NY 14260-1500, USA University of Oxford, Oxford, UK and Queen Mary, University of London, London, UK
We present a comprehensive black-hole event generator, BlackMax, which simulates the experi-mental signatures of microscopic and Planckian black-hole production and evolution at the LHC inthe context of brane world models with low-scale quantum gravity. The generator is based on phe-nomenologically realistic models free of serious problems that plague low-scale gravity, thus offeringmore realistic predictions for hadron-hadron colliders. The generator includes all of the black-holegray-body factors known to date and incorporates the effects of black-hole rotation, splitting be-tween the fermions, non-zero brane tension and black-hole recoil due to Hawking radiation (althoughnot all simultaneously).The generator can be interfaced with Herwig and Pythia.The main code can be downloaded from [1].
PACS numbers: 04.50.Gh, 04.70.Dy
I. INTRODUCTION
Models with TeV-scale quantum gravity [2, 3, 4, 5]offer very rich collider phenomenology. Most of themassume the existence of a three-plus-one-dimensionalhypersurface, which is referred as “the brane,” whereStandard-Model particles are confined, while only gravityand possibly other particles that carry no gauge quantumnumbers, such as right handed neutrinos can propagatein the full space, the so called “bulk”. Under certain as-sumptions, this setup allows the fundamental quantum-gravity energy scale, M ∗ , to be close to the electroweakscale. The observed weakness of gravity compared toother forces on the brane ( i.e. in the laboratory) is aconsequence of the large volume of the bulk which di-lutes the strength of gravity.In the context of these models of TeV-scale quantumgravity, probably the most exciting new physics is theproduction of micro-black-holes in near-future accelera-tors like the Large Hadron Collider (LHC) [6]. Accord-ing to the “hoop conjecture” [7], if the impact parame-ter of two colliding particles is less than two times thegravitational radius, r h , corresponding to their center-of-mass energy ( E CM ), a black-hole with a mass of theorder of E CM and horizon radius, r h , will form. Typi-cally, this gravitational radius is approximately E CM /M ∗ Thus, when particles collide at center-of-mass energiesabove M ∗ , the probability of black-hole formation is high.Strictly speaking, there exist no complete calcula-tion (including radiation during the process of forma-tion and back-reaction) which proves that a black-holereally forms. It may happen that a true event horizon and singularity never forms, and that Hawking (or ratherHawking-like) radiation is never quite thermal. In [8] thisquestion was analyzed in detail from a point of view of anasymptotic observer, who is in the context of the LHC themost relevant observer. It was shown that though suchobservers never observe the formation of an event hori-zon even in the full quantum treatment, they do registerpre-Hawking quantum radiation that takes away energyfrom a collapsing system. Pre-Hawking radiation is non-thermal and becomes thermal only in the limit when thehorizon is formed. Since a collapsing system has onlya finite amount of energy, it disappears before the hori-zon is seen to be formed. While these results have im-portant implications for theoretical issues like the infor-mation loss paradox, in a practical sense very little willchange. The characteristic time for gravitational collapsein the context of collisions of particles at the LHC is veryshort. This implies that pre-Hawking radiation will bequickly experimentally indistinguishable from Hawkingradiation calculated for a real black hole. Also, calcula-tions in [8] indicate that the characteristic time in whicha collapsing system losses all of its energy is very similarto a life time of a real black-hole. Thus, one may proceedwith a standard theory of black-holes.Once a black-hole is formed, it is believed to decay viaHawking radiation. This Hawking radiation will consistof two parts: radiation of Standard-Model particles intothe brane and radiation of gravitons and any other bulkmodes into the bulk. The relative probability for theemission of each particle type is given by the gray-bodyfactor for that mode. This gray-body factor depends onthe properties of the particle (charge, spin, mass, momen-tum), of the black-hole (mass, spin, charge) and, in thecontext of TeV-scale quantum gravity, on environmentalproperties – the number of extra dimensions, the locationof the black-hole relative to the brane (or branes), etc. . Inorder to properly describe the experimental signatures ofblack-hole production and decay one must therefore cal-culate the gray-body factors for all of the relevant degreesof freedom.There are several black-hole event generators availablein the literature [9] based on particular, simplified mod-els of low-scale quantum gravity and incorporating lim-ited aspects of micro-black-hole physics. Unfortunately,low-scale gravity is plagued with many phenomenolog-ical challenges like fast proton decay, large n ¯ n oscilla-tions, flavor-changing neutral currents and large mixingbetween leptons [10, 11]. For a realistic understanding ofthe experimental signature of black hole production anddecay, one needs calculations based on phenomenologi-cally viable gravity models, and incorporating all neces-sary aspects of the production and evolution of the black-holes.One low-scale gravity model in which the above men-tioned phenomenological challenges can be addressed isthe split-fermion model [12]. In this model, the Stan-dard Model fields are confined to a “thick brane”, muchthicker than M − ∗ . Within this thick brane, quarks andleptons are stuck on different three-dimensional slices(or on different branes), which are separated by muchmore than M − ∗ . This separation causes an exponen-tial suppression of all direct couplings between quarksand leptons, because of exponentially small overlaps be-tween their wave-functions. The proton decay rate willbe safely suppressed if the spatial separation betweenquarks and leptons is greater by a factor of at least 10than the widths of their wave functions. Since ∆ B = 2processes, like n ¯ n oscillations, are mediated by operatorsof the type uddudd, suppressing them requires a furthersplitting between up-type and down-type quarks. Sincethe experimental limits on ∆ B = 2 operators are muchless stringent than those on ∆ B = 1 operators, the u andd-type quarks need only be separated by a few times thewidth of their wave functions [12].Current black-hole generators assume that the black-holes that are formed are Schwarzschild-like. However,most of the black-holes that would be formed at the LHCwould be highly rotating, due to the non-zero impact pa-rameter of the colliding partons. Due to the existenceof an ergosphere (a region between the infinite redshiftsurface and the event horizon), a rotating black-hole ex-hibits super-radiance: some modes of radiation get am-plified compared to others. The effect of super-radiance[13] is strongly spin-dependent, with emission of higher-spin particles strongly favored. In particular the emissionof gravitons is enhanced over lower-spin Standard-Modelparticles. Since graviton emission appears in detectors asmissing energy, the effects of black-hole rotation cannot be ignored. Similarly, black-holes may be formed withnon-zero gauge charge, or acquire charge during theirdecay. This again may alter the decay properties of theblack-hole and should be included.Another effect neglected in other generators is the re-coil of the black-hole. A small black-hole attached to abrane in a higher-dimensional space emitting quanta intothe bulk could leave the brane as a result of a recoil . Inthis case, visible black-hole radiation would cease. Alter-nately, in a split-brane model, as a black-hole traversesthe thick brane the Standard-Model particles that it isable to emit will change depending on which fermionicbranes are nearby.It is also the case that virtually all the work in this fieldhas been done for the idealized case where the brane ten-sion is negligible. However, one generically expects thebrane tension to be of the order of the fundamental en-ergy scale, being determined by the vacuum energy con-tributions of brane-localized matter fields[14]. As shownin [15], finite brane tension modifies the standard gray-body factors.Finally, it has been suggested [16] that more commonthan the formation and evaporation of black-holes will begravitational scattering of parton pairs into a two-bodyfinal state. We include this possibility.Here we present a comprehensive black-hole event gen-erator, BlackMax, that takes into account practically allof the above mentioned issues , and includes almost allthe necessary gray-body factors . Preliminary studiesshow how the signatures of black-hole production anddecay change when one includes splitting between thefermions, black-hole rotation, positive brane tension andblack-hole recoil. Future papers will explore the implica-tions of these changes in greater detail.In section II and III we discuss the production of black-holes and the gray-body factors respectively. The evap-oration process and final burst of the black-holes is dis-cussed in section IV and V. Sections VI and describe theinput and output of the generator. Section VII showssome characteristic distributions of black-holes for differ-ent extra dimension scenarios. The reference list is exten-sive, reflecting the great interest in the topic [6, 7, 9, 10,11, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 29,30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46,47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62,63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78,79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94,95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107],but by no means complete. Although if the black-hole carries gauge charge it will be pre-vented from leaving the brane. although not necessarily simultaneously Except in the one case of the graviton gray-body factor for arotating black-hole, where the calculation has yet to be achieved.
II. BLACK-HOLE PRODUCTION
We assume that the fundamental quantum-gravity en-ergy scale M ∗ is not too far above the electroweak scale.Consider two particles colliding with a center-of-mass en-ergy E CM . They will also have an angular momentum J in their center-of-mass (CM) frame. By the hoop conjec-ture, if the impact parameter, b , between the two collid-ing particles is smaller than the diameter of the horizonof a ( d + 1)-dimensional black-hole (where d is the totalnumber of space-like dimensions) of mass M = E CM andangular momentum J , b < r h ( d, M, J ) , (1)then a black-hole with r h will form. The cross sectionfor this process is approximately equal to the interactionarea π (2 r h ) .In Boyer-Lindquist coordinates, the metric for a ( d +1)-dimensional rotating black-hole (with angular momen-tum parallel to the ˆ ω in the rest frame of the black-hole)is: ds = (cid:18) − µr − d Σ( r, θ ) (cid:19) dt − sin θ (cid:18) r + a (cid:18) + sin θ µr − d Σ( r, θ ) (cid:19)(cid:19) dφ + 2 a sin θ µr − d Σ( r, θ ) dtdφ − Σ( r, θ )∆ dr − Σ( r, θ ) dθ − r cos θd d − Ω (2)where µ is a parameter related to mass of the black-hole,while Σ = r + a cos θ (3)and ∆ = r + a − µr − d . (4)The mass of the black-hole is M = ( d − A d − πG d µ, (5)and J = 2 M ad − A d − = 2 π d/ Γ( d/
2) (7)is the hyper-surface area of a ( d − G d is defined as G d = π d − M d − ⋆ . (8) FIG. 1: Horizon radius (in GeV − ) of a non-rotating black-hole as a function of mass for 4-10 spatial dimensions. The horizon occurs when ∆ = 0. That is at a radiusgiven implicitly by r ( d ) h = " µ a/r ( d ) h ) d − = r ( d ) s h a/r ( d ) h ) i d − . (9)Here r ( d ) s ≡ µ / ( d − (10)is the Schwarzschild radius of a ( d +1)-dimensional black-hole, i.e. the horizon radius of a non-rotating black-hole.Equation 10 can be rewritten as: r ( d ) s ( E CM , d, M ∗ ) = k ( d ) M − ∗ [ E CM /M ∗ ] / ( d − , (11)where k ( d ) ≡ (cid:20) d − π ( d − / Γ[ d/ d − (cid:21) / ( d − . (12)Figure 1 shows the horizon radius as a function ofblack-hole mass for d from 4 to 10. We see that the hori-zon radius increases with mass; it also increases with d .Figure 2 shows the Hawking temperature of a black-hole T H = d − πr h (13)as a function of the black-hole mass for d from 4 to 10.The Hawking temperature is a measure of the character-istic energies of the particles emitted by the black-hole. T H decreases with increasing mass. However, the be-haviour of T H with changing d is complicated, reflectingthe competing effect of an increasing horizon radius andan increasing d − r ( t ) h = r s B / , (14) FIG. 2: Hawking temperature(in GeV) of a non-rotatingblack-hole as a function of mass for 4-10 spatial dimensions.FIG. 3: Horizon radius (in GeV − ) of a black-hole as a func-tion of mass for different B in d=5 spatial dimensions. with B the deficit-angle parameter which is inverse pro-portional to the tension of the brane.Figure 3 shows the horizon radius as a function ofblack-hole mass for the model with non-zero tensionbrane. As the deficit-angle parameter increases, the sizeof the black-hole increases.Figure 4 shows the Hawking temperature of a black-hole for the model with non-zero tension brane. TheHawking temperature decreases as the deficit angle de-creases.Figure 5 shows the horizon radius as a function ofblack-hole mass for a rotating black-hole. The angularmomentum decreases the size of the horizon and increasesthe Hawking temperature (see figures 5 and 6).If two highly relativistic particles collide with center-of-mass energy E CM , and impact parameter b , then theirangular momentum in the center-of-mass frame beforethe collision is L in = bE CM /
2. Suppose for now that
FIG. 4: Hawking temperature(in GeV) of a black-hole as afunction of mass for different B in d=5 spatial dimensions.FIG. 5: Horizon radius (in GeV − ) of a rotating black-holeas a function of mass for different angular momentum in d=5spatial dimensions. Angular momentum J is in unit of ~ .FIG. 6: Hawking temperature(in GeV) of a rotating black-hole as a function of mass for different angular momentum ind=5 spatial dimensions. Angular momentum J is in unit of ~ . the black-hole that is formed retains all this energy andangular momentum. Then the mass and angular momen-tum of the black-hole will be M in = E CM and J in = L in .A black-hole will form if: b < b max ≡ r ( d ) h ( E CM , b max E CM / . (15)We see that b max is a function of both E CM and thenumber of extra dimensions.We can rewrite condition (15) as b max ( E CM ; d ) = 2 r ( d ) s ( E CM ) h (cid:0) d − (cid:1) i d − . (16)There is one exception to this condition. In the casewhere we are including the effects of the brane tension,the metric (and hence gray-body factors) for a rotatingblack-hole are not known. In this case we consider onlynon-rotating black-holes. Therefore, for branes with ten-sion b tensionmax ( E CM , d ) = 2 r ( d ) s ( E CM ) . (17)Also, for branes with tension only the d = 5 metric isknown.At the LHC, each proton will have E = 7 TeV in theCM frame. Therefore, the total proton-proton center ofmass energy will be √ s = 14 TeV. However, it is not theprotons that collide to make the black-holes, but the par-tons of which the protons are made. If two partons haveenergy vE and uEv , much greater than their respectivemasses, then the parton-parton collision will have s ′ = | p i + p j | = | v ( E, E ) + uv ( E, − E ) | = 4 uE = us . (18)We define a quantity Q ′ Q ′ = E CM = √ s ′ = √ us (19)The center-of-mass energy for the two colliding partonswill be √ us , as will be the 4-momentum transfer Q ′ .The largest impact parameter between the two partonsthat can form a black-hole with this mass will thereforebe b max ( √ us ; d ), as given by equation 16.The total proton-proton cross section for black-holeproduction is therefore σ pp → BH ( s ; d, M ⋆ ) = Z M ⋆ /s du Z u dvv π (cid:2) b max ( √ us ; d ) (cid:3) × X ij f i ( v, Q ′ ) f j ( u/v, Q ′ ) . (20)Here f i ( v, Q ′ ) is the i-th parton distribution function.Loosely this is the expected number of partons of type iand momentum vE to be found in the proton in a colli-sion at momentum transfer Q ′ . In [16] it is argued that strong gravity effects at ener-gies close to the Planck scale will lead to an increase inthe 2 → → πb max in equation (20)with πb max ( √ s ′ > M min ) ≈ πr s P (21)where P = e −
Within BlackMax, the probability of creating a black-hole of center-of-mass energy √ us , in the collision of two FIG. 9: Cross section for production of a non-rotating black-hole as a function of the number of fermion brane-splittingdimensions for d = 10.FIG. 10: Cross section for formation of a black-hole (rotatingor non-rotating) as function of the minimum mass of black-hole, for a zero-tension brane, with no fermion brane-splitting.The vertical lines are the error bars. protons of center-of-mass-energy √ s , is given by P ( Q ′ ) = Z u dvv X ij f i ( v, Q ′ ) f j ( uv , Q ′ ) . (27)According to the theory, there will be some minimummass for a black-hole. We expect M min ∼ M ∗ , but leave M min ≥ M ∗ as a free parameter. Therefore, a black-holewill only form if u > ( M min /Q ) . The type of partonsfrom which a black-hole is formed determines the gaugecharges of the black-hole. Clearly, the probability to cre-ate a black-hole in the collision of any two particularpartons i and j with energies (momenta) vE and uEv , isgiven by P ( vE, uEv , i, j ) = f i ( v, Q ′ ) f j ( u/v, Q ′ ) (28) FIG. 11: Cross section for the two-body final-state scenarioas a function of number of spatial dimensions where M min = M ∗ = 1 TeV, M min = M ∗ = 3 TeV and M min = 5 TeV. The energies and types of the two colliding partons de-termine their momenta and affect their locations withinthe ordinary and extra dimensions. For protons movingin the z -direction, we arbitrarily put one of the partons atthe origin and locate the second parton randomly withina disk in the xy-plane of radius b max ( E CM = √ us ; d ).We must, however, also take into account that the par-tons will be separated in the extra dimensions as well.Each parton type is given a wave function in the ex-tra dimension. For fermions, these wave functions areparametrized by their centers and widths which are in-put parameters (cf. Fig 14). In the split-fermion case, thecenters of these wave functions may be widely separated;but even in the non-split case, the wave functions havenon-zero widths. For gauge bosons, the wave functionsare taken to be constant across the (thick) brane.The output from the generator (described in greaterdetail below) includes the energies, momenta, and typesof partons that yielded black-holes. The locations in timeand space of the black-holes are also output.The formation of the black-hole is a very non-linearand complicated process. We assume that, before set-tling down to a stationary phase, a black-hole loses somefraction of its energy, linear and angular momentum. Weparameterize these losses by three parameters: 1 − f E ,1 − f P and 1 − f L . Thus the black-hole initial state thatwe actually evolve is characterized by E = E in f E ; P z = P z in f P ; (29) J ′ = L in f L ;where E in , P z in and L in are initial energy, momentumand angular momentum of colliding partons, while f E , f P and f L are the fractions of the initial energy, momentum and angular momentum that are retained by the station-ary black-hole. We expect that most of the energy lost inthe non-linear regime is radiated in the form of gravita-tional waves and thus represents missing energy. Yoshinoand Rychkov [46] have calculated the energy losses bynumerical simulation of collisions. Their results will beincorporated in a future upgrade of BlackMax.For a small black-hole, the numerical value for the an-gular momentum is of the order of several ~ . In thatrange of values, angular momentum is quantized. There-fore a black-hole cannot have arbitrary values of angu-lar momentum. We keep the actual angular momentumof the black-hole, J , to be the nearest half-integer, i.e.2 J = (cid:2) J ′ + (cid:3) .The loss of the initial angular momentum in the non-linear regime has as a consequence that the black-holeangular momentum is no longer in the transverse planeof the colliding protons. We therefore introduce a tilt inthe angular-momentum θ ≡ cos − ( J p J ( J + 1) ) . (30)Figure 12 illustrates this geometry.In this version of the generator, we have assumed thatthe angular-momentum quantum numbers of the black-hole were ( J, J m = J ) . We next randomly choose anangle φ , and then reset the angular-momentum axis to( θ, φ ). FIG. 12: Angular momentum tilt geometry. Future versions of the generator may randomize the choice of J m . III. GRAY-BODY FACTORS
Once the black-hole settles down to its stationary con-figuration, it is expected to emit semi-classical Hawk-ing radiation. The emission spectra of different parti-cles from a given black-hole depend in principle on themass, spin and charge of the black-hole, on the “en-vironment” and on the mass and spin of the partic-ular particle. Wherever possible we have made use ofthe correct emission spectrum often phrased in terms ofthe gray-body factor for black-holes in 3+1-dimensionalspace-time. In most cases, these were extant in the lit-erature, but we have calculated the spectra for the split-fermion model ourselves, and reproduced existing spectraindependently. The sources of the gray-body factors aresummarized in Table I. • Non-rotating black-hole on a tensionless brane:
For a non-rotating black-hole, we used previouslyknown gray-body factors for spin 0 , / i.e. gravitons)in the bulk. • Rotating black-hole on a tensionless brane:
For ro-tating black-holes, we used known gray-body fac-tors for spin 0 , / As discussed be-low, this remains a serious shortcoming of cur-rent micro-black-hole phenomenology, since super-radiance might be expected to significantly increasegraviton emission from rotating black-holes, andthus increase the missing energy in a detector. • Non-rotating black-holes on a tensionless branewith fermion brane splitting:
In the split-fermionmodels, gauge fields can propagate through thebulk as well as on the brane, so we have calculatedgray-body factors for spin 0 and 1 fields propagat-ing through the bulk, but only for a non-rotatingblack-hole for the split-fermion model. These areshown in Figures 50-61. • Non-rotating black-holes on a non-zero tensionbrane:
The bulk gray-body factors for a branewith non-zero tension are affected by non-zero ten-sion because of the modified bulk geometry (deficitangle). We have calculated gray-body factors forspin 0 , dimensionality and geometry of the bulk, brane tension, locationof fermionic branes again only for the non-rotating black-hole for abrane with non-zero tension and d = 5. • Two particle final states:
We use the same gray-body factors as a non-rotating black-hole to cal-culate the cross section of two-particle final states(excluding gravitons).In all cases, the relevant emission spectra are loadedinto a data base as described in appendices A.
IV. BLACK-HOLE EVOLUTION
The Hawking radiation spectra are calculated for theblack-hole at rest in the center-of-mass frame of the col-liding partons. The spectra are then transformed to thelaboratory frame as needed. In all cases we have not(yet) taken the charge of the black-hole into accountin calculating the emission spectrum, but have includedphenomenological factors to account for it as explainedbelow.The degrees of freedom of the Standard-Model parti-cles are given in Table II. Using the calculated Hawkingspectrum and the number of degrees of freedom per par-ticle, we determine the expected radiated flux of eachtype of particle as a function of black-hole and environ-mental properties. For each particle type i we assign toit a specific energy, ~ ω i with a probability determined bythat particle’s emission spectrum. (The particle “types”are listed in Table II.)Assume a black-hole with mass M bh emits a masslessparticle with energy ~ ω i . The remaining black-hole willhave energy and momentum like( M bh − ~ ω i , − ~ ω i ) (31)Here we ignore the other dimensions. We use a classicalmodel to simulate the events. The mass of the remainingblack-hole should remain positive. So from equation (31)one gets ~ ω i < M bh / . (32)Combining this with the observation that energy of aparticle is larger than its mass, leads us to require that M i < ~ ω i (33)We next need to determine whether that particle withthat energy is actually emitted within one generatortime-step ∆ t . The time-step itself is an input param-eter (cf. figure 14). We choose a random number N r from the interval [0 , L F i , the total number fluxof particles of type i , and N i , the number of degrees offreedom of that particle type, the particle will be emittedif L F i N i ∆ t > N r . (34) FIG. 13: The emitted fermion intensity (normalized to one)as a function of the distance between the black-hole and thecenter of the gaussian distribution of a fermionic brane. As ablack-hole increases the distance from a fermionic brane dueto recoil the intensity of the emitted fermions of that type fallsdown quickly. The radius of black-hole is set to be 2 M − ∗ . Thewidth of the fermionic brane is M − ∗ . The plot is for the caseof one extra dimension. In the single-brane model, L F i is derived from thepower spectrum of the Hawking radiation. In thesplit-fermion model, we include a suppression factor forfermions. The factor depends on the overlap betweenthe particular fermion brane and the black-hole whenthe black-hole is not located on that fermion’s brane.Fig. 13 shows how the spectrum of emitted particleschanges as the black-hole drifts away from the center ofthe fermion brane. As a black-hole increases its distancefrom a fermion brane due to recoil, the intensity of theemitted fermions of that type declines quickly.If the particle is to be emitted, we choose its angular-momentum quantum numbers ( l, m ) according to: P em ( i, l, m, E ) = L i,l,m ( E ) / X l ′ ,m ′ L i,l ′ ,m ′ ( E ) . (35)Here P em ( i, l, m, E ) is the probability that a type i particle with quantum numbers ( l, m ) will be emitted. L i,l,m ( E ) is the emission spectrum of a particle of type i with quantum numbers ( l, m ). This step is omitted inthe case of non-rotating black-holes since we do not fol-low the angular-momentum evolution of the black-hole.Once the quantum numbers of the emitted particle aredetermined, we calculate the direction of emission ac-cording to the corresponding spheroidal wave function: P em ( θ, φ ) = | Ψ lm ( θ, φ ) | sin θ ∆ θ ∆ φ (36)Here P em ( θ, φ ) is the probability of emission in the( θ, φ ) direction for the angular quantum numbers ( ℓ, m ).Ψ lm ( θ, φ ) is the (properly normalized) spheroidal wave function of the mode with those angular quantum num-bers.Once the energy and angular-momentum quantumnumbers are determined for the i-th particle type, then, ifthat particle type carries SU(3) color we assign the colorrandomly. The color is treated as a three-dimensionalvector ~c i = ( r, b, g ), in which a quark’s color-vector is ei-ther (1 , , , , , , − − A. Electric and Color Charge Suppression
A charged and highly rotating black-hole will tend toshed its charge and angular momentum. Thus, emis-sion of particles with charges of the same sign as thatof the black-hole and angular momentum parallel to theblack-hole’s will be preferred. Emission of particles thatincrease the black-hole’s charge or angular momentumshould be suppressed. The precise calculation of theseeffects has not as yet been accomplished. Therefore, toaccount for these effects we allow optional phenomeno-logical suppression factors for both charge and angularmomentum.The following charge-suppression factors can currentlybe used by setting parameter 19 (cf. section VI) equal to2. F Q = exp( ζ Q Q bh Q em ) (37) F a = exp( ζ c bha c ema ) a = r , b , g . (38) Q bh is the electromagnetic charge of the black-hole, Q em is the charge of the emitted particle; c bha , is the colorvalue for the color a , with a = r , b , g, of the black-hole,and c ema , is the color value for the color a , with a = r , b , g,of the emitted particle. ζ Q and ζ are phenomenologicalsuppression parameters that are set as input parametersof the generator.We estimate ζ Q = O ( α em ) and ζ = O ( α s ), where α em and α s are the values of the electromagnetic and strongcouplings at the Hawking temperature of the black-hole.Note that we currently neglect the possible restorationof the electroweak symmetry in the vicinity of the black-hole when its Hawking temperature is above the elec-troweak scale. Clearly, since α em ≃ − we do notexpect electromagnetic (or more correctly) electroweakcharge suppression to be a significant effect. However,since α s (1 TeV) ≃ .
1, color suppression may well play arole in the evolution of the black-hole.0Once we have determined the type of particle to beemitted by the black-hole, we draw a random number N r between 0 and 1 from a uniform distribution. If N r > F Q then the emission process is allowed to occur,if N r < F Q then the emission process is aborted. We re-peat the same procedure for color suppression factor, F a .Thus, particle emission which decreases the magnitudeof the charge or color of the black-hole is unsuppressed;this suppression prevents the black-hole from acquiring alarge charge/color, and gives preference to particle emis-sion which reduces the charge/color of the black-hole. B. Movement of the Black-Hole duringEvaporation
We choose the direction of the momentum of the emit-ted particle ( ˆ P e ) according to equation (36) in the center-of-mass frame and then transform the energy and mo-mentum to their laboratory frame values ~ ω ′ and ~P ′ e . Theblack-hole properties (energy, momentum, mass, colors,and charge) are then accordingly updated for the nexttime step: E ( t + ∆ t ) = E ( t ) − ~ ω ′ (39) ~P ( t + ∆ t ) = ~P ( t ) − ~P ′ e (40) M ( t + ∆ t ) = q E ( t + ∆ t ) − ~P ( t + ∆ t ) (41) ~c bh ( t + ∆ t ) = ~c bh ( t ) − c i (42) Q bh ( t + ∆ t ) = Q bh ( t ) − Q i (43)Here ~c bh is the color 3-vector of the black-hole and canhave arbitrary integer entries.Due to the recoil from the emitted particle, the black-hole will acquire a velocity ~v and move to a position ~x : ~v ( t ) = ~P ( t ) /E (44) ~x ( t + ∆ t ) = ~x ( t ) + ~v ( t )∆ t (45)Since fermions are constrained to live on the 3+1-dimensional regular brane, the recoil from fermions isnot important. Only the emission of vector fields, scalarfields and gravitons gives a black hole momentum in ex-tradimension. Once a black hole gains momentum inextradimension, it is able to leave the regular brane ifit carries no gauge charge. In the split fermion case, itcan move within the mini-bulk even if it carries gaugecharge. In the case of rotating black holes, because thegray-body factor for gravitons is not yet known, gravi-ton emission is turned off in the generator and the blackholes experience no bulk recoil.Recoil can in principle change the radiation spectrumof the black-hole in two ways. First, the spectrum willnot be perfectly thermal or spherically symmetric in thelaboratory frame, but rather boosted due to the motion of the black-hole. However, as we shall see, the black-hole never becomes highly relativistic, so the recoil doesnot significantly affect the shape of the spectrum.As the lifetime of a small black-hole is relatively short,and its recoil velocity non-relativistic, it does not movefar from its point of creation. However, even a recoil ofthe order of one fundamental length ∼ M − ∗ in the bulkdirection could dislocate the black-hole from the brane .In single-brane models this would result in apparent miss-ing energy for an observer located on the brane ableto detect only Standard Model particles. In the split-fermion model, as the black-hole moves off or on partic-ular fermion branes, the decay channels open to it willchange. C. Rotation
Since two colliding particles always define a singleplane of rotation, rotating black-holes are formed witha single rotational parameter. For two particles collidingalong the z -axis, there should be only one rotation axisperpendicular to the z -axis. However, due to angular-momentum loss both in the formation process, and sub-sequently in the black-hole decay, three things can hap-pen: i) the amount of rotation can change, ii) the rota-tion axis can be altered, and iii) more rotation axes canemerge,because there are more than three spatial dimen-sions. Also, if the colliding particles have a non-zero im-pact parameter in bulk directions the plane of rotationwill not lie entirely in the brane direction. Because solu-tions do not exist for rotating black-holes with more thanone rotation axis, we forbid the emergence of secondaryrotation axes. We do, on the other hand, allow the sin-gle rotation axis of the black-hole to evolve. However, nogray-body factors are known if the single rotation axis ac-quires components in the extra dimensions, therefore welimit the rotation axis to the brane dimensions. Relaxingthese limitations is a subject for future research.We next must determine the rotational axis of theblack-hole. The rotation parameter of a black-hole withangular-momentum quantum numbers ( j, j ) is taken tobe a = JM n + 22 , (46)where J = p j ( j + 1) ~ . The direction of the black-holeangular momentum is taken to be ~J = j ~ ˆ ω + p j ~ ˆ l ⊥ , (47) This is very unlikely because most of the black-holes have gaugecharges. Due to the finite thickness of the single brane or splitting betweenthe quark branes. l ⊥ is a unit vector in the plane perpendicular toˆ ω . We chose the direction of ˆ l ⊥ randomly.When the black-hole emits a particle with angular-momentum quantum numbers ( l, m ), there are severalpossible final states in which the black-hole can end up.We use Clebsch-Gordan coefficients to find the probabil-ity of each state. | j, j> = | j − l | X j ′ ≤ j + l C ( j, j ; l, m, j ′ , j − m ) | l, m> | j ′ , j − m > (48)We use | C ( j, j ; l, m, j ′ , j − m ) | as the probability thatthe new angular-momentum quantum numbers of theblack-hole will be ( j ′ , j − m ). From angular-momentumconservation ~J = ~L + ~J ′ , we can calculate the tilt angleof the black-hole rotation axis as: cosθ = j ( j + 1) + j ′ ( j ′ + 1) − l ( l + 1)2 p j ( j + 1) j ′ ( j ′ + 1) . (49)We randomly choose a direction with the tilt angle θ as a new rotation axis and change quantum numbers to( j ′ , j ′ ).In calculating the gray-body factors, the black hole isalways treated as a fixed unchanging background. Thepower spectrum of emitted particles can be calculatedfrom dEdt = X l,m | A l,m | ωexp (( ω − m Ω) /T H ) ∓ dω π . (50)Here l and m are angular momentum quantum numbers. ω is the energy of the emitted particle. Ω is definied byΩ = a ∗ (1 + a ∗ ) r h . (51)The exponential factor in the denominator of (50)causes the black hole to prefer to emit high angular mo-mentum particles. However, since the TeV black holesare quantum black holes, the gray-body factors shouldreally depend on both the initial and final black-hole pa-rameters. The calculation of the gray-body spectra ona fixed background can cause some problems. In par-ticular, in the current case, the angular momentum ofthe emitted particle (as indeed the energy) may well becomparable to that of the black hole itself. There shouldbe a suppression of particle emission processes in whichthe black hole final state is very different from the initialstate. We therefore introduce a new phenomenologicalsuppression factor, parameter 17, to reduce the probabil-ity of emission events in which the angular momentumof the black hole changes by a large amount.If parameter 17 is equal to 1 (cf. section VI), we donot take into account the suppression of decays whichincrease the angular momentum of the black-hole. If we are using ∆Area suppression (parameter 17 equal to 2)then F L = exp( ζ L ( r bhh ( t + ∆ t ) /r bhh ( t ) − . (52)If we are using J bh suppression (parameter 17 equal to 3)then F L = exp( − ζ L | J bh ( t + ∆ t ) | ) . (53)If we are using ∆ J bh suppression (parameter 17 equal to4) then F L = exp( − ζ L | J bh ( t + ∆ t ) − J bh ( t ) | ) . (54)We might expect ζ L ∼
1, however there is no detailedtheory to support this; as indeed there is no detailedtheory to choose among these three phenomenologicalsuppression factors. It is also worth noting that, while for d = 3 and d = 4 there is a maximum angular momentumthat a black-hole of a given mass can carry, for d ≥ a ≤ R s / d = 3 and a ≤ R s for d = 4.As for the charge and color suppression, we choose arandom number N r between 0 and 1. If N r > F L thenthe particle emission is aborted.The procedure described in this section is then re-peated at each time step with each particle type, andthen successive time steps are taken until the mass ofthe black-hole falls below M ∗ . In practice, the time stepshould be set short enough that in a given time step theprobability that particles of more than one type are emit-ted is small. We set the time step to ∆ t = 10 − GeV − .In two-body final states, one expects no black-hole,and hence no black-hole decay by emission of Hawkingradiation. The generator therefore proceeds directly tothe final burst phase. V. FINAL BURST
In the absence of a self-consistent theory of quantumgravity, the last stage of the evaporation cannot be de-scribed accurately. Once the mass of black-hole becomesclose to the fundamental scale M ∗ , the classical black-hole solution can certainly not be used anymore. Weadopt a scenario in which the final stage of evaporationis a burst of particles which conserves energy, momentumand all of the gauge quantum numbers. For definiteness,we assume the remaining black-hole will decay into thelowest number of Standard-Model particles that conserveall quantum number, momentum and energy.A black-hole with electromagnetic charge Q bh andcolor-vector ~c bh = ( r bh , b bh , g bh ) will be taken to emit N − / down-type quarks (i.e. d,s or b quarks), N / up-type quarks (u,c, or t), N − charged leptons and W2bosons , N gl gluons, and N n non-charge particles (ie. γ , Zan Higgs). We use the following procedure to determines ~N burst ≡ ( N − / , N / , N − , N gl , N n ).Step 1: preliminary solution: • Search all possible solutions with N n = 0. • Choose the minimum number of particles as pre-liminary solution.Step 2: Actual charged/colored emitted particle count: • The preliminary ~N burst ≡ ( N − / , N / , N − , N gl , N n ) having now beendetermined. If the minimum number of solutionis less than 2, we then add N n to keep the totalnumber equal to 2. Later we choose one of themrandomly according to the degrees of freedom ofeach particle. • After obtaining the number of emitted particles,we randomly assign their energies and momenta,subject to the constraint that the total energy andmomentum equal that of the final black-hole state.We currently neglect any bulk components of thefinal black-hole momentum.
VI. INPUT AND OUTPUT
The input parameters for the generator are read fromthe file parameter.txt, see Fig.14:1.
Number of simulations : sets the total numberof black-hole events to be simulated;2.
Center of mass energy of protons : sets thecenter-of-mass energy of the colliding protons inGeV;3.
M ph : sets the fundamental quantum-gravity scale( M ∗ ) in GeV;4. Choose a case : defines the extra dimensionmodel to be simulated:(a) 1: non-rotating black-holes on a tensionlessbrane with possibility of fermion splitting,(b) 2: non-rotating black-holes on a brane withnon-zero positive tension,(c) 3: rotating black-holes on a tensionless branewith d=5,(d) 4: two-particle final-state scenario;5. number of extra dimensions : sets the numberof extra dimensions; this must equal 2 for branewith tension (
Choose a case =2); 6. number of splitting dimensions : setsthe number of extra split-fermion dimensions(
Choose a case =1);7. extradimension size : sets the size of the mini-bulk in units of 1 / TeV (
Choose a case =1);8. tension : sets the deficit-angle parameter B [14, 15]( Choose a case =2);9. choose a pdf file : defines which of the differ-ent CTEQ6 parton-distribution functions (PDF) touse;10.
Minimum mass : sets the minimum mass M min inGeV of the initial black-holes;11. fix time step : If equal to 1, then code uses thenext parameter to determine the time interval be-tween events; if equal to 2 then code tries to opti-mize the time step, keeping the probability of emit-ting a particle in any given time step below 10%.12. time step : defines the time interval ∆ t in GeV − which the generator will use for the black-hole evo-lution;13. Mass loss factor : sets the loss factor 0 for theenergy of the initial black-hole, as defined in equa-tion 29;14. momentum loss factor : defines the loss factor0 ≤ f p ≤ Angular momentum loss factor : sets the lossfactor 0 ≤ f L ≤ Seed : sets the seed for the random-number gener-ator (9 digit positive integer);17.
L suppression : chooses the model for suppress-ing the accumulation of large black-hole angularmomenta during the evolution phase of the black-holes (cf. discussion surrounding equations 52-54); •
1: no suppression; •
2: ∆ Area suppression; • J bh suppression; •
4: ∆ J suppression; This is the distance between fermion branes where only gaugebosons and Higgs field can propagate in split-fermion brane sce-nario. FIG. 14: Parameter.txt is the input file containing the parameters that one can change. The words in parentheses are theparameters that are used in the paper. angular momentum suppression factor : de-fines the phenomenological angular-momentumsuppression factor, ζ L (cf. discussion surroundingequation 52-54);19. charge suppression : turns the suppression of ac-cumulation of large black-hole electromagnetic andcolor charge during the black-hole evolution pro-cess on or off (cf. dicussion surrounding equation37) •
0: charge suppression turned off; •
1: charge suppression turned on;20. charge suppression factor : sets the electro-magnetic charge suppression factor, ζ Q , in 37;21. color suppression factor : sets the colorcharge suppression factor, ζ in 37;21-94 (odd entries:) the widths of fermion wave func-tions (in M − ∗ units); and (even entries:) centersof fermion wave functions (in M − ∗ units) in split-brane models, represented as 9-dimensional vectors(for non-split models, set all entries to 0).When the code terminates, the file output.txt with allthe relevant information (i.e. input parameters, cross sec-tion) is output to the working directory. This file containsalso different segments of information about the gener-ation of black-holes which are labelled at the beginningof each line with an ID word (Parent, Pbh, trace, Pem,Pemc or Elast): • Parent : identifies the partons whose collision re-sulted in the formation of the initial black-hole (seeFig. 16). – column 1: identifies the black-hole; – column 2: PDGID code of the parton; – column 3: energy of the parton; – columns 4-6: brane momenta of the parton. • Pbh : contains the evolution of the charge, color,momentum and energy of the black-holes, and, forrotating black-holes, their angular momentum (cf.Fig. 17). – column 1: identifies the black-hole; – column 2: time at which the black-hole emit-ted a particle; – column 3: PDGID code of a black-hole; – column 4: three times the electromagneticcharge of the black-hole; – columns 5 to 7: color-charge vector compo-nents of the black-hole; – columns 8: energy of the black-hole in the lab-oratory frhame; – columns 9 to 11: brane components ofthe black-hole momentum in the laboratoryframe; – columns 12 to (8+d): bulk components of theblack-hole momentum; – column (9+d): angular momentum of theblack-hole, in the case of rotating black-holes;empty otherwise. • trace : contains the evolution history of the black-holes’ positions (cf. Fig. 18): – column 1: identifies the black-hole; – column 2: the times at which the black-holeemitted a particle; – columns 3 to 5are the brane components of theblack-hole position vector when the black-holeemitted a particle; – columns 6 to (2+d): the bulk components ofthe black-hole position vector, when the black-hole emitted a particle. • Pem : contains a list of the black-holes, with thehistory of their evolution (cf. Fig. 19): – column 1: identifies the black-hole; – column 2: the times at which the black-holeemitted a particle; – column 3: PDGID code of the emitted parti-cle; – column 4: three times the charge of the emit-ted particle; – columns 5 to 7: color-vector components ofthe emitted particle; – columns 8: energy of the emitted particle inthe laboratory frame; – columns 9 to 11: brane components of the mo-mentum of the emitted particle, in the labo-ratory frame; – columns 12 to (8+d): bulk components of themomentum of the emitted particle. • Pemc : contains the same information as Pem, butin the center-of-mass frame of the collision. • Elast : contains the same information as Pem forthe particles emitted in the final decay burst of theblack-hole. Column 12 and onwards are omitted asthese particles have no bulk momentum.5
FIG. 15: Output.txt: There are three parts to this file. The first part is a copy of parameter.txt. The second part includesinformation about the black-hole and the emitted particles. The first column identifies the what type of information each rowis supplying – parent is information about the two incoming partons; Pbh is information on the energy and momenta of theproduced black-holes; trace describes the location of the black-holes; Pem characterizes the emitted particles in the lab frame;Pemc characterizes the emitted particles in the center-of-mass frame; Elast describes the final burst.The third part of the file is the black-hole production cross-section as inferred from the events in this generator run.
VII. RESULTS
We choose the following parameters for the distribu-tions shown in this section and normalize them to anintegrated luminosity of 10 f b − , unless otherwise stated.The values of the parameters are chosen to be the sameas in figure 14, except where a parameter is varied tostudy its effect. • Number of simulations = 10000; • Center of mass energy of protons = 14000 GeV; • M ph = 1000 GeV; • extradimension size = 10 TeV − ; • choose a pdf file =0; • Minimum mass =5000 GeV; • time step =10 − GeV − ; • Mass loss factor =0.0; • momentum loss factor =0.0; • Angular momentum loss factor =0.2; • Seed =123589341; • L suppression =1;6
FIG. 16: Lines in the output file headed by the ID =
Parent contain information about the initial partons which formed theblack-hole.FIG. 17: Lines in the output file headed by the ID =
Pbh contain the energies and momenta of the black-holes for each emissionstep. In case of rotating black-holes, the last column in the line is the angular momentum. • charge suppression =1; • charge suppression factor irrelevant since charge suppression =1; • color suppression factor irrelevant since charge suppression =1; • widths of fermion wave functions = 1 M − ∗ ; • centers of quark wave functions:(10 − / , , , .... )(GeV − ) (i.e. all quarks haveGaussian wave functions centered on a pointdisplaced from the origin by 10 − / − in thefirst splitting direction); • distribution center of leptons:( − − / , , , .... )(GeV − ).In this section, we present some distributions of prop-erties of the initial and evolving black-holes and of theparticles which are emitted by them during the Hawkingradiation and final-burst phases. A. Mass of the Initial Black-Holes
Figures 21, 22 and 23 show the the initial black-holemass distribution for three different extra dimension sce-narios: non-rotating black-holes on a tensionless brane,non-rotating black-holes on a non-zero tension brane andnon-rotating black-holes with split fermion branes respec-tively. Because we chose 5 TeV as the minimum mass ofthe initial black-hole, the distributions have a cut off at5 TeV.
B. Movement of Black-Holes in the Bulk
The generator includes recoil of black-holes due toHawking radiation. The recoil modifies the spectrum dueto the Doppler effect. Even if the effect is small, the highenergy tail of the emitted particle’s energy spectrum islonger than for pure Hawking radiation. Fig. 24 showsthe random motion in the mini-bulk for 10 ,
000 black-holes as consequence of recoil. While most of the black-holes remain on the brane where they were formed, asignificant number of them are capable of drifting all theway to the lepton brane. There are also a few eventswhere the black holes leave the mini-bulk completely.Since Standard-Model charges are confined to the mini-bulk, a black-hole needs to carry zero charge in order tobe able to leave the mini-bulk. Once out of the mini-bulk,a black-hole cannot emit Standard-Model particles any-more. Models with an additional bulk Z symmetry (e.g.Randall Sundrum models) do not allow for a black-holerecoil from the brane [27]. Unfortunately, the number ofblack-holes which escape the mini-bulk is so small thatexperimentally we are unlikely to be able to distinguishbetween models on this basis. C. Initial Black-Hole Charge Distribution
Most of the initial black-holes are created by u and dquarks. Denote by N Q the number of black-holes thathave electromagnetic charge Q . Fig. 25 is a histogramof 3 Q for n s = 0, n s = 4, and n s = 7 in d = 10 space.Since at these parton momenta, there are roughly twiceas many u-quarks in a proton as d-quarks, we expect thatmost of the black-holes have 3 Q = 4. i.e. are made of two7 FIG. 18: Lines in the output file headed by the ID = trace contain the location of the black-hole fo reach emission step.FIG. 19: Lines in the output file headed by the ID =
Pem contain the types of the emitted particles, their energies and momentain the lab frame and the times of their emission.FIG. 20: Lines in the output file headed by the ID =
Elast contain the types, energies and momenta of particles of the finalburst. FIG. 21: Mass distribution of initial black-holes (rotating andnon-rotating) on a tensionless brane for various numbers ofextra dimension.FIG. 22: Mass distribution of initial (non-rotating) black-holes on a non-zero tension brane for B = 1 . B = 0 . B = 0 . u-quarks ( f uu ). One does indeed see a large peak at 3 Q =4 in figure 25. A second peak at 3 Q = 1 corresponds toblack-holes made of one d and one u-quark, or from oneanti-d and one gluon. Since there are only a few gluonsor anti-quarks at these momenta, f ≃ f ud .Similarly, the small peak at 3 Q = − dd and not ¯u-gluon.We expect that f ud ≃ √ f uu f dd , and thus f ≃ p f f − . This relation is roughly satisfied in Fig. 25.In the split-fermion case, since gluons can move in themini-bulk, there is a further suppression of the gluon con-tribution due to the wave-function-overlap suppressionbetween the gluons and fermions. In particular 3 Q = 2and 3 Q = −
1, which are dominated by gluon-quark col-lisions, are suppressed, as can be seen in Fig.25. Fora large number of split dimensions, there are almost no
FIG. 23: Mass distribution of initial (non-rotating) black-holes on a tensionless brane for d = 10 and different numbersof split-fermion branes.FIG. 24: black-hole movement in the mini-bulk due to recoil. X , Y are coordinates in two extra dimensions. The redcircle indicates the width of the quark brane. The blue circleindicates the width of the lepton brane. The black lines areblack-holes traces. The size of the mini-bulk is 10 TeV − ×
10 TeV − . black-holes with non-zero standard model gaugecharges bounce back from the wall of the mini-bulk. black-holes with zero Standard Model gauge charges can leave themini-bulk. gluon-gluon or gluon-quark black-holes. The decline inthe gluon-quark configurations accounts for the simulta-neous rise in the fraction of quark-only configurations( i.e. Q = 4 , , − D. Initial Black-Hole Color Distribution
The colliding partons that form the black-hole carrygauge charges, in particular color and electromagnetic9
FIG. 25: Electromagnetic charge distribution of the initialblack-holes. charge. From the PDFs [32] we see that, at the relevantparton momentum, most of the partons are u and d typequarks – essentially the valence quarks. Contributionsfrom “the sea” – other quarks, antiquarks, gluons andother partons – are subdominant. We therefore estimatethe distribution of the colors of the initial black-holes tobe N i (0) : N i (1) : N i (2) = 4 : 4 : 1 . (55)Here N i ( p ) is the number of black-holes whose i-th color-vector component (i=1 is red, i=2 blue, i=3 green)has the value p . This agrees very well with the graphin Fig. 26. N i ( −
1) and N i ( −
2) refer to black-holescreated from collisions involving gluons or anti-quarks.Their numbers are hard to estimate, but we expect that N i ( − ≪ f i ( − ≪ N i (0 ≤ p ≤ M min significantly lower (at or below 1 TeV), thenblack-hole production by gluon-gluon scattering wouldbe more important, significantly altering the color distri-bution, and making it more sensitive to fermion brane-splitting (which lowers the gluon-gluon contribution). E. Evolution of Black-Hole Color and Chargeduring the Hawking Radiation Phase
Figure 27 shows the color distribution of the black-holes which they accumulate during the evaporationphase. From equation 55, the expected average initialcolor of the black-holes is 2 /
3. Since the colors of emit-ted particles are assigned randomly, we expect the cumu-lative color distribution (CCD) to be symmetric around2 / FIG. 26: Initial color distribution of the created black-holes.The vertical lines are error bar.FIG. 27: Cumulative color distribution for non-rotating black-holes on a tensionless brane with d = 4 and no fermion branesplitting. Histogram with the black squares (open circles) iswith (without) color suppression. The width of the CCD depends on the total number ofparticles emitted by the black-hole during its evaporationphase.As discussed above, we allow for the possibility of sup-pressing particle emission which increases the charge,color or angular momentum of the black-hole excessively(cf. discussion around equations 37, and 52-54). Figure27 shows also the cumulative black-hole color distribu-tion where we suppressed the accumulation of large colorcharges during the evaporation phase. In order to am-plify the effect of color suppression, we have set f = 20instead of the expected f ≃ .
1. We see that the num-ber of black-holes with a color charge larger than 1 isdecreased.0
F. Number of emitted particles
FIG. 28: Number of particles emitted by a non-rotating black-hole on a tensionless brane prior to the “final burst” as func-tion of number of extra dimension. Here n s = 0. The errorbars denote one standard deviation range.FIG. 29: Number of particles emitted by a non-rotating black-hole in the split-fermion model prior to the “final burst” asa function of the number of brane-splitting dimensions, with d = 10. Error bars denote one standard deviation range. Figures 28 through 31 show the number of particlesthat are emitted by a microscopic black-hole during thedecay process before its final burst for a variety of models.In the single tensionless brane model (Fig. 28), thenumber of emitted particles first increases with the num-ber of dimensions for a non-rotating black-hole, but thendecreases. This behaviour is a result of the complicatedinterplay of a number of effects: the horizon size of ablack-hole of a given mass as a function of d , and its FIG. 30: Number of particles emitted by a non-rotating black-hole on a non-zero tension brane prior to the “final burst” asfunction of defizit angle parameter B with d = 5 and n s = 2.Error bars denote one standard deviation range.FIG. 31: Number of particles emitted by a rotating black-hole prior to the “final burst” as function of number of extradimension with n s = 0. Error bars denote one standard devi-ation range. effect on the particle emission spectra; the dependenceof the Hawking temperature on r ( d ) h ; the existence dueto energy-momentum conservation of an upper limit of M bh / M min , the minimum initial mass of a black-hole.In the split fermion model (Fig. 29), the number ofemitted particles decreases with the number of extra di-mensions. This is because, even for a fixed Hawking tem-perature, the average energies of emitted gauge bosonsand scalar fields increase as n s increases.In the model with a finite-tension brane (Fig. 30),1 FIG. 32: Number of particles emitted at the final burst: theaverage number of final burst is about 3.4. the number of particles decreases as the parameter B increases, i.e. with decreasing tension. As the tensionincreases, B gets smaller but the horizon radius of theblack-hole increases. The Hawking temperature thereforedecreases, and, as a consequence, the average energy ofemitted particles falls. More particles will therefore beemitted in the evolution of the black-hole.For rotating black-holes (Fig. 31) (on a tensionless,unsplit brane), the number of emitted particles first in-creases, then decreases, and finally reaches a plateau.This is due to similar reasons as for non-rotating black-holes. Compared to non-rotating black-holes of the samemass, rotation shifts the energy of emitted particles tohigher values because it decreases the horizon radiusand increases the emission of higher angular-momentummodes. This decreases the total number of emitted parti-cles. It also means that the effect of the upper kinematiclimit of M bh / d = 5 and n s = 0. Theaverage number of emitted particles is about 3. Duringthe Hawking radiation phase a black-hole emits about 10particles, so approximately 30% of the emitted particleswill be from the final burst stage. In the case exam-ined in Figure 32, we did not include suppression of largeblack-hole color or electric charge. Thus some black-holesacquire large color and electric charges by the end of theHawking radiation phase. These black-holes then mustdecay into a large number of particles ( >
5) in the finalburst.
G. Energy Distributions of the Emitted Particles
Once formed, the black-holes decay by emission ofHawking radiation, a process which continues until themass of the black-hole falls to the fundamental quantum-gravity scale. At this stage we chose the black-holes toburst into a set of Standard-Model particles as describedin section V. The observable signatures of the decay willdepend on the distributions of energy, momentum andparticle types of the emitted particles.Figures 33 through 36 show the relation between themass of the evolving black-hole and the average energyof emitted particles for different extra dimension models.The error bars denote 1 / √ N times the standard devia-tion of the mean energy. The minimum mass of the initialblack-hole is taken to be M min = 5 TeV.We see from figure 33 that, for a single (i.e. unsplit)brane, when M bh >> M ∗ . a black-hole in d = 10 emitshigher energy particles than a black-hole in lower dimen-sions. For black-hole masses closer to M ∗ , the highestenergy particles are emitted when the dimensionality ofspace is low, i.e. d = 4. This reversal can be understoodfrom figure 2. In the LHC energy range, the curves ofHawking temperature as a function of black-hole massfor different dimensions cross. At high mass, high d ex-hibits the highest Hawking temperature; at low mass,low d exhibits the highest Hawking temperature. It iseasy from this figure to estimate the number of emittedparticles, and to roughly reproduce the results shown infigure 28.The main difference between the curves for different d ,comes from the changing size of the black-hole horizon.For low d , the horizon radius increases more quickly withthe mass than for higher d , as seen in figure 1. The Hawk-ing temperature of the black-hole is inversely propor-tional to r h . So long as the Hawking temperature remainswell below the black-hole mass (here for M bh > ∼ d . By d = 10, the energy of emittedparticles is almost constant from 2 T eV to 5
T eV .The increase of the energy of emitted particles stopsat about 2 TeV and then a decrease begins. The rea-son, as stated above (equation (32)), is that by energy-momentum conservation, a black-hole can only emit par-ticles with less than half of its mass.In the split-fermion model (Fig. 34), the average en-ergy of the emitted particles increases as the number ofdimensions in the mini-bulk increases. The energy shiftcomes from the gauge bosons and scalar fields, which ac-cess the higher-dimensional phase space of the mini-bulk(cf. appendix A: figure 50 through 61).For the brane with non-zero tension (Fig. 35), theradius of the black-hole increases with tension hence theenergy of emitted particles decreases with tension.2
FIG. 33: Average energy of the particles emitted by (non-rotating) black-holes on unsplit branes versus the mass of theblack-hole at the time of emission. Here d = 4, d = 7 and d = 10. Note that the final burst is not included, because itoccurs when the mass of the black-hole is less than 1 TeV.FIG. 34: Average energy of the particles emitted by (non-rotating) black-holes on split branes versus the mass of theblack-hole at the time of emission. Here d = 10 and n s = 0, n s = 3, n s = 7. For a rotating black-hole (Fig. 36), angular momentumdecreases the size of the horizon. Thus since black-holesare typically formed with some initial angular momen-tum, they emit higher energy particles than non-rotatingblack-holes of the same mass. However, the black-holetends to shed its angular momentum rapidly as it emitsparticles. This increases the horizon size, lowers theHawking temperature, and lowers the average energy ofthe emitted particles. The rapid shedding of angular mo-mentum thus leads to a drop in the average emitted par-ticle energy around M min .If one compares rotating with non-rotating black-holes,one finds that the energy of the emitted particles is al-ways larger for the rotating black-holes. By the time FIG. 35: Average energy of the particles emitted by (non-rotating) black-holes on a brane with tension, versus the massof the black-hole at the time of emission. Here d = 5, n s = 2and B = 1, B = 0 . B = 0 . d = 4, d = 7 and d = 10, with atensionless unsplit brane. the mass of the black-hole has dropped well below M min (here to approximately 1 − M ob = s M p + X e P e (56)Here M ob is the observed mass, M p is the true mass ofthe particle, and P e is the particles extra-dimensional3 FIG. 37: Energy distribution of emitted particles in the Hawk-ing radiation step for single-brane non-rotating black-hole.FIG. 38: Energy distribution of emitted particles at the finalburst step for single-brane non-rotating black-hole. momentum. Clearly M ob ≥ M p . A Standard-Model par-ticle, however, cannot leave the Standard Model brane.Its extra-dimensional momentum must therefore be ab-sorbed by the brane or carried away by bulk particles(such as gravitons). We therefore calculate an emittedparticle’s “energy on the brane” according to: E b = vuut M p + X i =1 P i (57)where P i is the regular 3-momentum. We will assumethat the shedding of extra-dimensional momentum israpid, and henceforth we will refer to E b (rather thanthe initial emission energy) as the energy of the emittedparticle.Figure 37 shows the energy distribution of the particlesfrom Hawking radiation in the single-brane model. The cross section for black-hole production increases with d (figure 7). The area under the curves also increase with d .The peaks of the curves are around 200GeV to 400GeV.One can compare, for example, the energy distributionsfor d = 9 and d = 10. A black-hole in high r d tends toemit particles with higher energy, so the curve for d = 10has a longer higher energy tail than for d = 9.Figure 38 shows the energy distribution of the particlesfrom the final burst in the single-brane model. The en-ergy these particles share is much smaller than the energyin the earlier Hawking radiation phase. The peak in theenergy distribution of these particle is around 200GeVto 300GeV. The tails extend just to 1TeV, which is themass at which the black hole is taken to be unstable andundergo its final burst.Figures 39 through 42 show the energy distribution ofemitted particles (including final burst particles) in thevarious models that BlackMax can simulate.Figure 39 shows the energy distribution in the single-brane model. The cross section of black-hole increaseswith d (figure 7). The area under the curves also increaseswith d . The peaks of the curves are around 200 GeV to400 . Again comparing d = 9 and d = 10, a black-holein higher d tends to emit particles with higher energy, sothe curve for d = 10 has a longer high energy tail thanfor d = 9.Figure 40 illustrates the split-fermion model. In thisfigure, we keep the total number of dimensions d fixedbut change the dimensionality n s of the mini-bulk. Thisaffects the spectra of only the gauge boson and scalarfields, as only their propagation is affected by the themini-bulk’s dimensionality. (Gravitons propagate in thefull bulk; other Standard-Model particles propagate onlyon the brane.) As explained above, these spectra willshift to higher energies as the number of splitting dimen-sions is increased. One can see that the curve in n s = 7has the longest high energy tail.Figure 41 illustrates the brane with tension model.The energies of the emitted particles shift to lower en-ergy as B decreases.The energy distribution of emitted particles for rotat-ing black-holes (figure 42) has the same general charac-teristics as the distribution for non-rotating black-holes.Angular momentum causes a black-hole to tend to emithigher energy particles than a non-rotating black-hole.The curves have longer higher energy tails than the non-rotating black-holes.Figure 43 shows the Energy distribution of emittedparticles from two-body final-states scenario. The energyof the emitted particles is about the half of the incomingpartons.In figure 44 we show the energy distribution of differenttypes of particles in the d = 5 single-brane model. Thearea of each curve is dependent on the degree of freedomof each particle and its power spectrum. One can com-pare the ratio of the same type of particles. For example,4 FIG. 39: Energy density distribution of emitted particlesfrom non-rotating black-holes on a tensionless brane withno splitted fermion branes. Shown are the distributions for4 ≤ d ≤
10. The spectra include the final burst particles.FIG. 40: Energy density distribution of emitted particles fromnon-rotating black-holes on a tensionless brane with fermionbrane splitting. Shown are the distributions for d = 10 and0 ≤ n s ≤
7. The spectra include the final burst particles. the area of gluons should be 8 times as large as the area of photons. It is roughly the same as what the figureshows.
H. Pseudorapidity Distributions of the EmittedParticles
Figures 45 to 48 show the pseudorapidity distributionsof the emitted particles for different extra dimension sce-narios.Most of black-holes are made of two u quarks and havecharge 3 /
4. For the shown figures we did not include
FIG. 41: Energy distribution of emitted particles for non-rotating black holes on a non-zero tension brane with B = 1, B = 0 . B = 0 .
6. The spectra include the final burstparticles.FIG. 42: Energy distribution of emitted particles for rotatingblack-holes for 4 ≤ d ≤ charge suppression, because of that the black-holes tendto emit the same number of particles and antiparticlesduring Hawking radiation phase. This can be seen fromthe curves without final burst in figure 45 through figure48. The majority of the black-holes are positive chargedand will tend to emit positive particles in the final burst.That is why there are more positrons than electrons forthe distributions which include the final burst particles.Figure 49 shows the pseudorapidity distribution in thetwo-body final-state scenario. The distribution is muchwider than the equivalent distribution from Hawking ra-diation. In this model we do not consider the angularmomentum of the black hole. We therefore take the de-cay process in the two-body final state scenario to beisotropic in the ordinary spatial directions, just as inother models without angular momentum. In the center5 FIG. 43: Energy distribution of emitted particles for two-body final states.FIG. 44: Energy distribution of each particle type in the d = 5single brane model. of mass frame, particles are therefore emitted in direc-tions uncorrelated with the beam direction. Neverthe-less, because the threshold energy of the two-body final-state model is much lower than that of other models, theintermediate state tends to have a higher velocity downthe beam-pipe. The decay products are therefore emit-ted with larger pseudorapidity. In truth, one may expectthat the intermediate state of the two-body final-statescenario has non-negligible angular momentum. How-ever, since the intermediate state is not a real black hole,it is unclear exactly what role the angular momentum ofthe intermediate state plays.In the final state scenario, the momenta of the twoemitted particles are correlated with the initial partonmomenta, and hence the pseudorapidity distribution ofthe emitted particles reflects that of the initial partons.The ratio of the number of events for pseudorapidity be- FIG. 45: Pseudorapidity distribution of charge leptons andanti leptons for non-rotating black-holes on a tensionlessbrane with and without the final burst particles; d = 5.FIG. 46: Pseudorapidity distribution of quarks and antiquarks for non-rotating black-holes on a tensionless branewith and without the final burst particles; d = 5. tween 0 and 0 . . .
1. This is much higher thanthe asymptotic QCD value of 0.6, as predicted by [16, 26].If the ratio is found not to equal 0 .
6, then this would sug-gest new physics beyond the Standard Model.
I. Emitted Particle Types
Table III shows, for a variety of representative extra-dimension scenarios the fraction of emitted particleswhich are of each possible type – quarks, gluons,(charged) leptons, (weak) gauge bosons, neutrinos, gravi-tons, Higgs bosons and photons. One notable feature isthat the intensity of gravitons relative to other particlesincreases with the number of extra dimensions. Note thatthe absence of gravitons in the case of a rotating black-hole is not physical, but rather reflects our ignorance ofthe correct gray-body factor.6
FIG. 47: Pseudorapidity distribution of charged leptons andanti-leptons for rotating black-holes, on a tensionless brane,with and without the final burst particles. Here d = 5.FIG. 48: Pseudorapidity distribution of quarks and anti-quarks for rotating black-holes, on a tensionless brane, withand without the final burst particles. Here d = 5. VIII. CONCLUSION
Hitherto, black-hole generators for the large-extra-dimension searches at the LHC have made many simpli-fying assumptions regarding the model of both our three-dimensional space and the extra-dimensional space, andsimplifying assumptions regarding the properties of theblack-holes that are produced. In this paper we havediscussed a new generator for black-holes at the LHC,BlackMax, which removes many of these assumptions.With regard to the extra-dimensional model it allows forbrane tension, and brane splitting. With regard to theblack-hole, it allows for black-hole rotation, charge (bothelectro-magnetic and color) and bulk recoil. It also in-troduces the possibility of a two-body final state that isnot a black-hole.Although BlackMax represents a major step forward,there remain important deficiencies that will need to be
FIG. 49: Pseudorapidity distribution for the two-body final-state scenario. The distribution in 2-body final states is muchflatter near η = 0 than emission due to Hawking radiation. addressed in the future. BlackMax continues to insist ona flat geometry for the bulk space, whereas there is con-siderable interest in a warped geometry [3] or in a com-pact hyperbolic geometry [4]. While BlackMax allowsfor black-hole rotation, the absence of either an analyticor a numerical gray-body factor for the graviton in morethan three space dimensions for rotating black-holes isa serious shortcoming that can be expected to materi-ally change the signature of black-hole decay for rotatingblack-holes. Other issues include how to properly ac-count for the likely suppression of decays that cause ablack-hole to acquire very “large” color, charge or angu-lar momentum. (As oppposed to the somewhat contrivedphenomenological approach currently taken.) These arebut a few of the fundamental issues that remain to beclarified.Despite these (and no doubt other) shortcomings, weexpect that BlackMax will allow for a much improved un-derstanding of the signatures of black-holes at the LHC.Work in progress focuses on using BlackMax to explorethe consequences for the ATLAS experiment of more re-alistic black-hole and extra-dimension models.We thank Nicholas Brett for discussions in the earlystages of this paper. We thank Daisuke Ida, Kin-ya Oda,and Seong Chan Park for providing some of the spectraof rotating black-holes. DCD, DS and GDS thank Ox-ford’s Atlas group for its hospitality at various stages ofthis project. DCD, DS and GDS have been supported inpart by a grant from the US DOE; GDS was supportedin part by the John Simon Guggenheim Memorial Foun-dation and by Oxford’s Beecroft Institute for ParticleAstrophysics and Cosmology. [1] ∼ issever/BlackMax/blackmax.html [2] Nima Arkani-Hamed, Savas Dimopoulos, G.R. Dvali,Phys. Rev. D59:086004, 1999. hep-ph/9807344.Ignatios Antoniadis, Nima Arkani-Hamed, Savas Di-mopoulos, G.R. Dvali, Phys. Lett. B436:257-263, 1998.hep-ph/9804398.Nima Arkani-Hamed, Savas Dimopoulos, G.R. Dvali,Phys. Lett. B429:263-272, 1998. hep-ph/9803315.[3] Lisa Randall, Raman Sundrum, Phys.Rev.Lett.83:4690-4693,1999. hep-th/9906064.[4] N. Kaloper, J. March-Russell, G. D. Starkmanand M. Trodden, Phys. Rev. Lett. , 928 (2000)[arXiv:hep-ph/0002001]; G. D. Starkman, D. Stojkovicand M. Trodden, Phys. Rev. Lett. , 231303 (2001)[arXiv:hep-th/0106143]. G. D. Starkman, D. Stojkovicand M. Trodden, Phys. Rev. D , 103511 (2001)[arXiv:hep-th/0012226].[5] I. Antoniadis, Phys. Lett. B , 377 (1990); K. R. Di-enes, E. Dudas and T. Gherghetta, Phys. Lett. B ,55 (1998) [arXiv:hep-ph/9803466].[6] T. Banks, W. Fischler, hep-th/9906038 ; S. Dimopoulos,G. Landsberg, Phys. Rev. Lett. D65 , 024005 (2007) [arXiv:gr-qc/0609024].T. Vachaspati and D. Stojkovic, arXiv:gr-qc/0701096.[9] TRUENOIR: Savas Dimopoulos, Greg L. Landsberg,Phys. Rev. Lett. 87:161602,2001. hep-ph/0106295CHARYBDIS: C.M. Harris, P. Richardson, B.R. Web-ber, JHEP 0308:033, 2003. hep-ph/0307305.Catfish: M. Cavaglia, R. Godang, L. Cremaldi, D. Sum-mers, hep-ph/0609001.[10] D. C. Dai, G. D. Starkman and D. Stojkovic, Phys. Rev.D , 104037 (2006) [arXiv:hep-ph/0605085]; D. Sto-jkovic and G. D. Starkman,D. C. Dai, Phys. Rev. Lett. , 041303 (2006) [arXiv:hep-ph/0505112][11] D. Stojkovic, F. C. Adams and G. D. Starkman, Int. J.Mod. Phys. D , 2293 (2005) [arXiv:gr-qc/0604072];C. Bambi, A. D. Dolgov and K. Freese, Nucl.Phys. B , 91 (2007) [arXiv:hep-ph/0606321];arXiv:hep-ph/0612018.[12] N. Arkani-Hamed, M. Schmaltz, Phys. Rev. D61
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Siopsis, arXiv:0708.3363 [hep-th].[107] D. Kiley, arXiv:0708.1016 [hep-th]. Appendix A: Particle Emission SpectraNowadays gray-body factors can be found in many pa-pers. We collect the relevant papers in table I. Wefollow these papers to calculate energy power spectrain our database except for the split-fermions model (al-though we independently confirm the results as well). Weperform an original calculation of the spectra of gaugebosons and scalar fields in the split-fermions model as9a function of the number of dimensions n s in which thefermion branes are split. These are shown in figures 50to 61.Anyone who has a better spectrum can upgrade ourdatabase. For example, we only calculate the spectra upto the l = 9 mode. FIG. 50: Spectra of scalar fields in d=5 space.FIG. 51: Spectra of scalar fields in d=6 space. FIG. 52: Spectra of scalar fields in d=7 space.FIG. 53: Spectra of scalar fields in d=8 space.FIG. 54: Spectra of scalar fields in d=9 space. FIG. 55: Spectra of scalar fields in d=10 space.FIG. 56: Spectra of gauge bosons in d=5 space.FIG. 57: Spectra of gauge bosons in d=6 space. FIG. 58: Spectra of gauge bosons in d=7 space.FIG. 59: Spectra of gauge bosons in d=8 space.FIG. 60: Spectra of gauge bosons in d=9 space. FIG. 61: Spectra of gauge bosons in d=10 space. TABLE I: Literature sources for particle emission spectraType of particle Type of black hole Brane model ReferencesStandard-Model particles non-rotating unsplit; tensionless [18][19]gravitons non-rotating split/unsplit; tensionless [17]Standard-Model particles non-rotating split/unsplit; with tension [15]gravitons non-rotating split/unsplit; with tension [15]scalars and gauge bosons non-rotating split; tensionless figures 50-61fermions rotating unsplit; tensionless [23][24]gauge bosons rotating unsplit; tensionless [22][24]scalar fields rotating unsplit; tensionless [18] [20][21][24]TABLE II: Degrees of freedom of Standard-Model particles which are emitted from a black hole. For gravitons, the tableshows 1, because the appropriate growth in the number of degrees of freedom is included explicitly in the graviton emissionspectrum. n s is the number of extra dimensions in which vector and scalar fields can propagate.particle type d d / d d Quarks 0 6 0 0Charged leptons 0 2 0 0Neutrinos 0 2 0 0Photons or gluons 0 0 2 + n s Z n s W + and W − n s ) 0Higgs boson 1 0 0 0Graviton 0 0 0 1 TABLE III: The fraction of emitted particles of different types (including final burst particles) in a variety of extra dimensionscenarios. ∗ Note that the absence of gravitons in the case of a rotating black hole is due exclusively to our current ignoranceof the correct gray-body factor.Scenario quarks gluons leptons gauge bosons neutrinos gravitons Higgs bosons photons d = 4 n s = 0 non-rotating black hole 68.21 10.79 9.45 5.72 3.87 2.00e-01 8.99e-01 8.61e-01 d = 5 n s = 0 non-rotating black hole 65.37 13.29 9.04 6.12 3.76 4.60e-01 8.26e-01 1.13 d = 6 n s = 0 non-rotating black hole 63.63 14.51 8.93 6.58 3.51 7.59e-01 7.76e-01 1.30 d = 7 n s = 0 non-rotating black hole 61.25 15.94 8.75 7.17 3.45 1.20 7.89e-01 1.44 d = 8 n s = 0 non-rotating black hole 60.99 15.94 8.56 6.99 3.35 1.93 7.63e-01 1.47 d = 9 n s = 0 non-rotating black hole 59.40 16.26 8.26 7.08 3.26 3.48 7.45e-01 1.51 d = 10 n s = 0 non-rotating black hole 57.56 16.15 7.68 6.82 3.17 6.46 6.97e-01 1.46 d = 10 n s = 1 split fermions model 61.58 19.76 1.64 7.30 3.83e-01 6.95 5.88e-01 1.80 d = 10 n s = 2 split fermions model 61.28 20.40 1.66 7.09 3.49e-01 6.91 4.56e-01 1.86 d = 10 n s = 3 split fermions model 62.10 20.33 1.65 6.76 4.46e-01 6.43 4.15e-01 1.87 d = 10 n s = 4 split fermions model 62.70 19.76 1.72 6.60 4.66e-01 6.62 3.28e-01 1.80 d = 10 n s = 5 split fermions model 63.67 19.34 1.73 6.14 5.17e-01 6.50 2.65e-01 1.83 d = 10 n s = 6 split fermions model 64.96 18.24 1.82 5.85 5.56e-01 6.69 2.35e-01 1.65 d = 10 n s = 7 split fermions model 66.38 17.23 1.83 5.44 5.50e-01 6.68 2.51e-01 1.64 d = 2 n s = 2 B = 1 . d = 2 n s = 2 B = 0 . d = 2 n s = 2 B = 0 . d = 2 n s = 2 B = 0 . d = 2 n s = 2 B = 0 . d = 2 n s = 2 B = 0 . d = 2 n s = 2 B = 0 . d = 4 n s = 0 rotating black hole 64.82 15.41 7.94 6.25 3.50 0.00 ∗ d = 5 n s = 0 rotating black hole 61.38 18.11 7.89 7.06 3.30 0.00 ∗ d = 6 n s = 0 rotating black hole 59.21 20.38 7.27 7.59 3.08 0.00 ∗ d = 7 n s = 0 rotating black hole 57.52 21.82 7.08 8.08 2.87 0.00 ∗ d = 8 n s = 0 rotating black hole 54.41 24.02 6.75 9.09 2.77 0.00 ∗ d = 9 n s = 0 rotating black hole 51.88 24.05 7.98 9.27 3.55 0.00 ∗ d = 10 n s = 0 rotating black hole 52.43 25.67 6.45 9.62 2.52 0.00 ∗ d = 4 n s = 0 two-body final states 58.88 16.89 1.38 6.08 11.14 0.00e+00 1.91 3.71 d = 5 n s = 0 two-body final states 57.71 17.46 1.37 6.22 11.53 0.00e+00 1.91 3.80 d = 6 n s = 0 two-body final states 57.16 17.72 1.39 6.27 11.59 0.00e+00 1.93 3.94 d = 7 n s = 0 two-body final states 57.02 17.91 1.37 6.22 11.59 0.00e+00 1.98 3.91 d = 8 n s = 0 two-body final states 56.58 18.08 1.35 6.31 11.79 0.00e+00 1.92 3.97 d = 9 n s = 0 two-body final states 56.51 18.02 1.41 6.40 11.75 0.00e+00 1.96 3.95 d = 10 n ss