CHARYBDIS: A Black Hole Event Generator
CHARYBDIS is an event generator which simulates the production and decay of miniature black holes at hadronic colliders as might be possible in certain extra dimension models. It interfaces via the Les Houches accord to general purpose Monte Carlo programs like HERWIG and PYTHIA which then perform the parton evolution and hadronization. The event generator includes the extra-dimensional `grey-body' effects as well as the change in the temperature of the black hole as the decay progresses. Various options for modelling the Planck-scale terminal decay are provided.
Cavendish Laboratory, University of Cambridge, Madingley Road,
Cambridge, CB3 0HE, UK
- P. Richardson
Theory Division, CERN, 1211 Geneva 23, Switzerland
- B.R. Webber
Cavendish Laboratory, University of Cambridge, Madingley Road,
Cambridge, CB3 0HE, UK and
Theory Division, CERN, 1211 Geneva 23, Switzerland
Table of Contents
Models with extra dimensions have become an area of much interest since the
work of Arkani-Hamed, Dimopoulos and Dvali (ADD)  and Randall and
Sundrum (RS) . Like most models of physics beyond the Standard Model
they are seen as a more natural way of explaining the hierarchy problem, that
is, why there are about sixteen orders of magnitude between the electroweak
energy scale and the Planck scale at which gravity becomes large. Such extra
dimension models can also be motivated from string theory.
In extra dimension models, the usual 4-dimensional Planck scale is no longer
considered to be a fundamental scale - instead it is derived from the
fundamental D-dimensional Planck scale which can be as low as current
experimental limits allow (~ 1 TeV).
If the fundamental Planck scale is of order a TeV, gravity is strong at such scales and cannot be ignored as is usually the case in particle physics. The possibility then arises of particle accelerators at TeV-scale energies being able to produce miniature black holes. These would then decay rapidly by Hawking evaporation, giving rise to characteristic high-multiplicity final states.
There has already been much discussion in the literature on this issue, but
little work has been done trying to realistically simulate black holes at
the Large Hadron Collider (LHC). In this work we implement a simple model of
black hole production and decay which can be interfaced to existing Monte Carlo
programs using the Les Houches accord . The major new theoretical input to the generator is the inclusion of the recently calculated `grey-body' factors for black holes in extra dimensions [4, 5, 6]. We also take account of the recoil and change of temperature of the black hole during decay, and provide various models for the termination of the decay process.
2 Black Hole Production and Decay
The details of production and decay of black holes in extra dimension models
are complicated and not particularly well understood. Here we outline the
theory and mention some of the assumptions which are usually made.
In theories with extra dimensions the ~ TeV energy scale is considered
as fundamental - the 4D Planck scale (Mp(4)~ 1018 GeV) is then derived from it. The relationship between the two energy scales is determined by the volume of the extra dimensions. If R is the size of all n extra
dimensions it can be shown, using Gauss' Law, that for r« R then
whereas for r» R
In these expressions Mp is the (4+n)-dimensional Planck mass (throughout this paper the conventions of  are used for Mp). They show that the two energy scales are related (up to volume factors of order unity) by
which allows the sizes of extra dimensions to be calculated for
different values of n . Short scale gravity experiments and particle collider experiments provide limits on the fundamental Planck scale. However for the smaller values of n, the more stringent constraints come from astrophysical and cosmological data, albeit with larger uncertainties. It is widely agreed that both n=1 and n=2 are ruled out by such data. For a comprehensive recent review of these constraints see, for example, .
As the fundamental Planck scale is as low as ~ TeV, it is possible for
tiny black holes to be produced at the LHC when two partons pass within the
horizon radius set by their centre-of-mass energy. The black holes being
considered in this work are in the r« R regime, so an analogous approach
to the usual 4D Schwarzschild calculation  shows the horizon
radius for a non-spinning black hole to be
where MBH is the mass of the black hole.
There has been much discussion in the literature (e.g. [10, 11, 12, 13, 14]) about what the cross section for black hole production
is, but the consensus opinion seems to be that the classical
s~p rh2 is valid (at least for black hole masses
MBH » Mp). It is unclear for exactly what mass this cross section
estimate starts to become unreliable, but for MBH close to the
fundamental Planck scale a theory of quantum gravity would be required to
determine the cross section. The black holes produced may have any gauge and spin quantum numbers so to determine the the p-p or p_-p production cross section it is necessary to sum over all possible quark and gluon pairings. Although the parton-level cross sections grows with black hole mass, the parton distribution functions (pdfs) fall rapidly at high energies and so the cross section also falls off quickly.
Once produced, these miniature black holes are expected to decay
instantaneously on LHC detector time scales (typical lifetimes are
~ 10-26 s). The decay is made up of three major phases:
It has been shown in  that the majority of energy in Hawking
radiation is emitted into modes on the brane (i.e. as Standard Model
particles) but a small amount is also emitted into modes in the bulk
(i.e. as gravitons).
- The balding phase in which the `hair' (asymmetry and moments due to the violent
production process) is lost;
- A Hawking evaporation  phase (a brief spin-down phase during
which angular momentum is shed from a Kerr black hole, then a longer
- A Planck phase at the end of the decay when the mass and / or the Hawking
temperature approach the Planck scale.
In 4D the phase which accounts for the greatest proportion of the mass loss is the Schwarzschild phase . A black hole of a particular mass is
characterized by a Hawking temperature and as the decay progresses the black
hole mass falls and the temperature rises. It is assumed that a
quasi-stationary approach to the decay is valid, that is the black hole has
time to come into equilibrium at each new temperature before the next particle
For an uncharged, non-rotating black hole the decay spectrum is described by the following expression:
where s is the spin of the polarization degree of freedom being considered, l and
m are angular momentum quantum numbers, and G are the so-called `grey-body'
factors. The last term in the denominator is a spin statistics factor which is -1
for bosons and +1 for fermions. The Hawking temperature in (5) is given by
Equation (5) can be used to determine the decay spectrum for
a particular particle e.g. electrons. Since there are two polarizations for spin-1/2 we obtain:
The expression for a particular flavour of quark would be identical but with an additional colour factor. Slightly more care is required with massive gauge bosons since one of their degrees of freedom comes from the Higgs mechanism. This means that, for example,
The grey-body factors modify the spectrum of emitted particles from that of a
perfect thermal black body even in 4D . They quantify the probability of transmission of the particles through the curved space-time outside the horizon, and can be determined from the absorption cross section for the emitted particle species.
For the Schwarzschild phase with w» TH geometric arguments show that Sl,mGµ(w rh)2 in any number of dimensions, which means that at high energies the shape of the spectrum is like that of a black body. However the low energy behaviour of the grey-body factors is spin-dependent and also depends on the number of dimensions.
In 4D it has long been known that for s=0,1/2 and 1 the grey-body factors reduce
the low energy emission rate significantly below the geometrical optics value
[17, 18]. The result is that both the flux and power spectra peak at
higher energies than those for a black body at the same temperature. The spin
dependence of the grey-body factors mean that they are necessary to determine
the relative emissivities of different particle types from a black hole. Until recently (see [4, 5, 6]) these have only been available in the literature for the 4D case [17, 19]).
The dependence both on energy and the number of dimensions means that the grey-body factors must be taken into account in any attempt to determine the number of extra dimensions by studying the energy spectrum of particles emitted from a black hole. When studying black hole decay, other experimental variables may also be sensitive to these grey-body effects.
Finally although the Planck phase cannot be properly understood without a full theory of quantum gravity, it is suggested that in this phase the black hole will decay to a few quanta with Planck-scale energies .
3 Event Generator
3.1 Features of the event generator
There are a number of features of the CHARYBDIS generator which, within the uncertainties of much of the theory, allow reliable simulation of black hole events. Most notable is that unlike other generators (e.g. ) the grey-body effects are fully included. The generator also allows the black hole temperature to vary as the decay progresses and is designed for simulations with either p-p or p_-p.
Due to the difficulty in modelling the balding phase and the lack of a full
theory for quantum gravity to explain the Planck phase of the decay, the
generator only attempts to model the Hawking evaporation phase (expected to
account for the majority of the mass loss). To provide a further simplification only non-spinning black holes are modelled. This is perhaps a less good
approximation but comparison with the 4D situation suggests that most of the
angular momentum will be lost in a relatively short spin-down phase
It is possible that black hole decay does not conserve baryon number, for example by producing three quarks in a colour singlet. However the treatment of processes which do not conserve baryon number in both the QCD evolution and hadronization is complicated and has only been studied for a few specific processes [22, 23, 24, 25]. At the same time the violation of baryon number is extremely difficult to detect experimentally and therefore the effect of including baryon number violation is not expected to be experimentally observable. Therefore CHARYBDIS conserves baryon number in black hole production and decay.
3.2 General description
The black hole event generator developed attempts to model the theory as
outlined in the previous section. There are several related parameters and switches which can be set in the first part of the Les Houches subroutine UPINIT . No other part of the charybdis1000.F code should be modified.
Firstly the properties of the beam particles must be specified. IDBMUP(1) and IDBMUP(2) are their PDG codes (only protons and anti-protons are allowed) and the corresponding energies are EBMUP(1) and EBMUP(2). Note that these settings will over-write any in the main HERWIG or PYTHIA program files.
The geometric parton-level cross section, s=p rh2, is used but the parameters MINMSS and MAXMSS allow the mass range for
the black holes produced to be specified. This means that the lower mass limit at
which this expression for the cross section is thought to become valid can be
Three other parameters which must be set before using the event generator are
TOTDIM, MPLNCK and MSSDEF. The total number of dimensions in the model being used is given by TOTDIM (this must be set between 6 and 11). There are a number of different definitions of the Planck mass (set using MPLNCK), but the parameter MSSDEF can be set to three different values to allow easy interchange between the three conventions outlined in Appendix A of . The conversions between these conventions are summarized in Table 1.
It has been suggested that since black hole formation is a non-perturbative
process, the momentum scale for evaluating the pdfs should be the inverse
Schwarzschild radius rather than the black hole mass. The switch
GTSCA should be set to .TRUE. for the first of these options
and .FALSE. for the second. It should be noted that, as confirmed in , the cross sections quoted in reference  were actually calculated with the latter pdf scale. The pdfs to be used are set using the Les Houches parameters PDFGUP and PDFSUF.
Table 1: Definitions of the Planck mass.
As discussed in section 2, the Hawking temperature of the black
hole will increase as the decay progresses so that later emissions will
typically be of higher energy. However to allow comparison with other work
which has ignored this effect, there is a switch TIMVAR which can be
used to set the time variation of the Hawking temperature as on
(.TRUE.) or off (.FALSE.).
The probabilities of emission of different types of particles are set according to the new theoretical results [4, 5, 6]. Heavy particle production is allowed and can be controlled by setting the value of the MSSDEC parameter to 2 for top quark, W and Z, or 3 to include Higgs also (MSSDEC=1 gives only light particles). Heavy particle production spectra may be unreliable for choices of parameters for which the initial Hawking temperature is below the rest mass of the particle being considered.
If GRYBDY is set as .TRUE. the particle types and energies are chosen according to the grey-body modified emission probabilities and spectra. If instead the .FALSE. option is selected, the black-body emission probabilities and spectra are used. The choice of energy is made in the rest frame of the black hole before emission. As overall charge must be conserved, when a charged particle is to be emitted the particle or anti-particle is chosen such that the magnitude of the black hole charge decreases. This reproduces some of the features of the charge-dependent emission spectra in  whilst at the same time making it easier for the event generator to ensure that charge is conserved for the full decay.
Although the Planck phase at the end of decay cannot be well modelled as it is
not well understood, the Monte Carlo event generator must have some way of
terminating the decay. There are two different possibilities for this, each with a range of options for the terminal multiplicity.
If KINCUT=.TRUE. termination occurs when the chosen energy for the emitted particle is ruled out by the kinematics of a two-body decay. At this point an isotropic NBODY decay is performed on the black hole remnant where NBODY can be set between 2 and 5. The NBODY particles are chosen according to the same probabilities used for the rest of the decay. The selection is then accepted if charge and baryon number are conserved, otherwise a new set of particles is picked for the NBODY decay. If this does not succeed in conserving charge and baryon number after NHTRY attempts the whole decay is rejected and a new one generated. If the whole decay process fails for MHTRY attempts then the initial black hole state is rejected and a new one generated.
In the alternative termination of the decay (KINCUT=.FALSE.), particles are emitted according to the energy spectrum until MBH falls below MPLNCK and then an NBODY decay as described above is performed. Any chosen energies which are kinematically forbidden are simply discarded.
In order to perform the parton evolution and hadronization the general purpose event generators require a colour flow to be defined. This colour flow is defined in the large number of colours (Nc) limit in which a quark can be considered as a colour line, an anti-quark as an anti-colour line and a gluon both a colour and anti-colour line. A simple algorithm is used to connect all the lines into a consistent colour flow. This algorithm starts with a colour line (from either a quark or a gluon) and then randomly connects this line with one of the unconnected anti-colour lines (either a gluon or an anti-quark). If the selected partner is a gluon the procedure is repeated to find the partner for its colour line; if it is an anti-quark one of the other unconnected quark colour lines is selected. If the starting particle was a gluon the colour line of the last parton is connected to the anti-colour line of the gluon. Whilst there is no deep physical motivation for this algorithm it at least ensures that all the particles are colour-connected and the showering generator can proceed to evolve and hadronize the event.
After the black hole decay, parton-level information is written into the Les Houches common block HEPEUP to enable a general purpose event generator to fragment all emitted coloured particles into hadron jets, and generate all unstable particle decays (see below).
3.3 Control switches, constants and options
Those parameters discussed in the previous section which are designed to be set by the user are summarized in Table 2.
||PDG codes of beam particles
||Energies of beam particles (GeV)
||PDFLIB codes for pdf author group
||PDFLIB codes for pdf set
||Minimum mass of black holes (GeV)
||Maximum mass of black holes (GeV)
||Planck mass (GeV)
||Convention for MPLNCK (see Table 1)
||Total number of dimensions (4+n)
||Use rh-1as the pdf momentum scale
||rather than the black hole mass
||Allow TH to change with time
||Choice of decay products
||Include grey-body effects
||Use a kinematic cut-off on the decay
||Number of particles in remnant decay
Table 2: List of parameters with brief descriptions, allowed values and default settings.
3.4 Using charybdis1000.F
The generator itself only performs the production and parton-level decay of the black hole. It is interfaced, via the Les Houches accord, to either HERWIG [27, 28] or PYTHIA  to perform the parton shower evolution, hadronization and particle decays. This means that it is also necessary to have a Les Houches accord compliant version of either HERWIG or PYTHIA with both the dummy Les Houches routines (UPINIT and UPEVNT) and the dummy PDFLIB subroutines (PDFSET and STRUCTM) deleted. For HERWIG the first Les Houches compliant version is HERWIG6.500 ; for PYTHIA version 6.220  or above is required.1
The black hole code itself is available as a gzipped tar file at the web address
http://www.ippp.dur.ac.uk/montecarlo/leshouches/generators/charybdis/. The file includes the following code:
The general purpose event generator to be used must be specified in the Makefile (i.e. GENERATOR=HERWIG or GENERATOR=PYTHIA) and also if PDFLIB is to be used (PDFLIB=PDFLIB if required, otherwise PDFLIB= ). The name of the HERWIG or PYTHIA source and the location of the PDFLIB library must also be included.
charybdis1000.F (code for the black hole generator)
- dummy.F (dummy routines needed if not using PDFLIB)
- mainpythia.f (example main program for PYTHIA)
- mainherwig.f (example main program for HERWIG)
- charybdis1000.inc (include file for the black hole generator)
If the code is extracted to be run separately then the following should be taken into account:
charybdis1000.F will produce the HERWIG version by default when compiled, the flag -DPYTHIA should be added if the PYTHIA version is required;
- dummy.F will by default produce the version for use without PDFLIB, the flag -DPDFLIB should be added if PDFLIB is being used.
3.5 List of subroutines
Table 3 contains a list of all the subroutines of the generator along with their functions. Those labelled by HW/PY are
HERWIG / PYTHIA dependent and are pre-processed according to the GENERATOR flag in the Makefile. Many of the utility routines are identical to routines which appear in the HERWIG program.
||Les Houches routines
||Generates a five-body decay
||Generates a four-body decay
||Generates a three-body decay
||Generates a two-body decay
||Hard subprocess and related routines
||Main routine for black hole hard subprocess
||Returns charge of a SM particle
||Returns mass of a SM particle (HW/PY)
||Chooses next particle type if MSSDEC=1
||Chooses next particle type if MSSDEC=2
||Chooses next particle type if MSSDEC=3
||Calculates the pdfs (HW/PY)
||Random number generators
||Randomly rotates a 2-vector
||Random number generator (HW/PY)
||Random number: uniform
||Chooses particle energy from spectrum
||Boost: rest frame to lab, no masses assumed
||Lorentz transformation: rest frame®lab
||Puts mass in 5th component of vector
||Rotation by inverse of matrix R
||Rotation by matrix R
||Square root with sign retention
||Interpolates in a table
Table 3: List of subroutines with brief descriptions.
3.6 Sample plots
Figures 1-3 show the results, at parton level, of neglecting the time variation of the black hole temperature (TIMVAR=.FALSE., dashed line) or the grey-body factors (GRYBDY=.FALSE., dot-dashed line) for initial black hole masses in the range from MINMSS=5000.0 to MAXMSS=5500.0, with the default values for the other parameters. The solid line is for simulations with the default parameter settings (but with the same reduced range of initial black hole masses used in the other two cases).
The effect of time variation is to harden the spectra of all particle species.
However, the effect of the grey-body factors depends on the spin, in this case slightly softening the spectra of scalars and fermions but hardening the spectrum of gauge bosons.
Figure 1: Parton-level energy spectra of Higgs bosons, mH = 115 GeV. Solid: predicted energy spectrum of Higgs bosons from decay of black holes with initial masses 5.0-5.5 TeV. Dashed: neglecting time variation of temperature. Dot-dashed: neglecting grey-body factors.
Figure 2: Parton-level energy spectra of electrons and positrons. As Figure 1
but for electron and positron spectra.
Results of a fuller study of signatures of black hole production and decay at the LHC will be presented elsewhere .
Figure 3: Parton-level energy spectra of photons. As Figure 1
but for photon spectra.
We thank members of the Cambridge SUSY Working Group, members of the ATLAS collaboration, and also P. Kanti and J. March-Russell for helpful discussions. Thanks also to the authors of HERWIG for the code incorporated into this generator, and to T. Sjöstrand for help with running CHARYBDIS with PYTHIA. This work was funded by the U.K. Particle Physics and Astronomy Research Council.
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